pinecone/core-2020-05-10-deduplication huggingface dataset:
Fueter’s theorem for monogenic functions in biaxial symmetric domains

Fueter's theorem discloses a remarkable connection existing between holomorphic functions and monogenic functions when m is odd. It states that is monogenic if is holomorphic and is a homogeneous monogenic polynomial in . Eelbode et al. (AIP Conf Proc 1479:340-343, 2012) proved that this statement is still valid if the monogenicity condition on is dropped. To obtain this result, the authors used representation theory methods but their result also follows from a direct calculus we established in our paper Pea Pea and Sommen (J Math Anal Appl 365:29-35, 2010). In this paper we generalize the result from Eelbode et al. (2012) to the case of monogenic functions in biaxially symmetric domains. In order to achieve this goal we first generalize Pea Pea and Sommen (2010) to the biaxial case and then derive the main result from that
On functional equations characterizing derivations: methods and examples

Functional equations satisfied by additive functions have a special interest not only in the theory of functional equations, but also in the theory of (commutative) algebra because the fundamental notions such as derivations and automorphisms are additive functions satisfying some further functional equations as well. It is an important question that how these morphisms can be characterized among additive mappings in general.\ud
The paper contains some multivariate characterizations of higher order derivations. The univariate characterizations are given as consequences by the diagonalization of the multivariate formulas. The method allows us to refine the process of computing the solutions of univariate functional equations of the form\ud
∑k=1nxpkfk(xqk)=0,\ud
where pk and qk (k=1,…,n) are given nonnegative integers and the unknown functions f1,…,fn:R→R are supposed to be additive on the ring R. It is illustrated by some explicit examples too.\ud
As another application of the multivariate setting we use spectral analysis and spectral synthesis in the space of the additive solutions to prove that such a functional equation characterizes derivations of higher order. The results are uniformly based on the investigation of the multivariate version of the functional equations
Weaving K-frames in Hilbert Spaces

Gavruta introduced $K$-frames for Hilbert spaces to study atomic systems with
respect to a bounded linear operator. There are many differences between
K-frames and standard frames, so we study weaving properties of K-frames. Two
frames $\{\phi_{i}\}_{i \in I}$ and $\{\psi_{i}\}_{i \in I}$ for a separable
Hilbert space $\mathcal{H}$ are woven if there are positive constants $A \leq
B$ such that for every subset $\sigma \subset I$, the family $\{\phi_{i}\}_{i
\in \sigma} \cup \{\psi_{i}\}_{i \in \sigma^{c}}$ is a frame for $\mathcal{H}$
with frame bounds $A, B$. In this paper, we present necessary and sufficient
conditions for weaving $K$-frames in Hilbert spaces. It is shown that woven
$K$-frames and weakly woven $K$-frames are equivalent. Finally, sufficient
conditions for Paley-Wiener type perturbation of weaving $K$-frames are given
$C^\infty$-convergence of conformal mappings on triangular lattices

Two triangle meshes are conformally equivalent if for any pair of incident
triangles the absolute values of the corresponding cross-ratios of the four
vertices agree. Such a pair can be considered as preimage and image of a
discrete conformal map. In this article we study discrete conformal maps which
are defined on parts of a triangular lattice $T$ with strictly acute angles.
That is, $T$ is an infinite triangulation of the plane with congruent strictly
acute triangles. A smooth conformal map $f$ can be approximated on a compact
subset by such discrete conformal maps $f^\varepsilon$, defined on a part of
$\varepsilon T$ for $\varepsilon>0$ small enough, see [U. B\"ucking,
Approximation of conformal mappings using conformally equivalent triangular
lattices, in "Advances in Discrete Differential Geometry" (A.I. Bobenko ed.),
Springer (2016), 133--149]. We improve this result and show that the
convergence is in fact in $C^\infty$. Furthermore, we describe how the
cross-ratios of the four vertices for pairs of incident triangles are related
to the Schwarzian derivative of $f$.Comment: 17 pages, 3 figure
On the computational complexity of the rooted subtree prune and regraft distance.

The graph-theoretic operation of rooted subtree prune and regraft is increasingly being used as a tool for understanding and modelling reticulation events in evolutionary biology. In this paper, we show that computing the rooted subtree prune and regraft distance between two rooted binary phylogenetic trees on the same label set is NP-hard. This resolves a longstanding open problem. Furthermore, we show that this distance is xed parameter tractable when parameterised by the distance between the two trees
The laplacian of a graph as a density matrix: a basic combinatorial
  approach to separability of mixed states

We study entanglement properties of mixed density matrices obtained from
combinatorial Laplacians. This is done by introducing the notion of the density
matrix of a graph. We characterize the graphs with pure density matrices and
show that the density matrix of a graph can be always written as a uniform
mixture of pure density matrices of graphs. We consider the von Neumann entropy
of these matrices and we characterize the graphs for which the minimum and
maximum values are attained. We then discuss the problem of separability by
pointing out that separability of density matrices of graphs does not always
depend on the labelling of the vertices. We consider graphs with a tensor
product structure and simple cases for which combinatorial properties are
linked to the entanglement of the state. We calculate the concurrence of all
graph on four vertices representing entangled states. It turns out that for
some of these graphs the value of the concurrence is exactly fractional.Comment: 20 pages, 11 figure
Random and exhaustive generation of permutations and cycles

In 1986 S. Sattolo introduced a simple algorithm for uniform random
generation of cyclic permutations on a fixed number of symbols. This algorithm
is very similar to the standard method for generating a random permutation, but
is less well known.
  We consider both methods in a unified way, and discuss their relation with
exhaustive generation methods. We analyse several random variables associated
with the algorithms and find their grand probability generating functions,
which gives easy access to moments and limit laws.Comment: 9 page
Direct constructions of hyperplanes of dual polar spaces arising from embeddings

Let e be one of the following full projective embeddings of a finite dual polar space Delta of rank n >= 2: (i) The Grassmann-embedding of the symplectic dual polar space Delta congruent to DW(2n 1,q); (ii) the Grassmann-embedding of the Hermitian dual polar space Delta congruent to DH(2n-1, q(2)); (iii) the spin-embedding of the orthogonal dual polar space Delta congruent to DQ(2n, q); (iv) the spin-embedding of the orthogonal dual polar space Delta congruent to DQ(-)(2n+ 1, q). Let H-e denote the set of all hyperplanes of Delta arising from the embedding e. We give a method for constructing the hyperplanes of H-e without implementing the embedding e and discuss (possible) applications of the given construction
The Tchebyshev transforms of the first and second kind

We give an in-depth study of the Tchebyshev transforms of the first and
second kind of a poset, recently discovered by Hetyei. The Tchebyshev transform
(of the first kind) preserves desirable combinatorial properties, including
Eulerianess (due to Hetyei) and EL-shellability. It is also a linear
transformation on flag vectors. When restricted to Eulerian posets, it
corresponds to the Billera, Ehrenborg and Readdy omega map of oriented
matroids. One consequence is that nonnegativity of the cd-index is maintained.
  The Tchebyshev transform of the second kind is a Hopf algebra endomorphism on
the space of quasisymmetric functions QSym. It coincides with Stembridge's peak
enumerator for Eulerian posets, but differs for general posets. The complete
spectrum is determined, generalizing work of Billera, Hsiao and van
Willigenburg.
  The type B quasisymmetric function of a poset is introduced. Like Ehrenborg's
classical quasisymmetric function of a poset, this map is a comodule morphism
with respect to the quasisymmetric functions QSym.
  Similarities among the omega map, Ehrenborg's r-signed Birkhoff transform,
and the Tchebyshev transforms motivate a general study of chain maps. One such
occurrence, the chain map of the second kind, is a Hopf algebra endomorphism on
the quasisymmetric functions QSym and is an instance of Aguiar, Bergeron and
Sottile's result on the terminal object in the category of combinatorial Hopf
algebras. In contrast, the chain map of the first kind is both an algebra map
and a comodule endomorphism on the type B quasisymmetric functions BQSym.Comment: 33 page
On multiplicity-free skew characters and the Schubert Calculus

In this paper we classify the multiplicity-free skew characters of the
symmetric group. Furthermore we show that the Schubert calculus is equivalent
to that of skew characters in the following sense: If we decompose the product
of two Schubert classes we get the same as if we decompose a skew character and
replace the irreducible characters by Schubert classes of the `inverse'
partitions (Theorem 4.2).Comment: 14 pages, to appear in Annals. Comb. minor changes from v1 to v2 as
  suggested by the referees, Example 3.4 inserted so numeration changed in
  section
Dense near octagons with four points on each line, III

This is the third paper dealing with the classification of the dense near octagons of order (3, t). Using the partial classification of the valuations of the possible hexes obtained in [12], we are able to show that almost all such near octagons admit a big hex. Combining this with the results in [11], where we classified the dense near octagons of order (3, t) with a big hex, we get an incomplete classification for the dense near octagons of order (3, t): There are 28 known examples and a few open cases. For each open case, we have a rather detailed description of the structure of the near octagons involved
Unsigned state models for the Jones polynomial

It is well a known and fundamental result that the Jones polynomial can be
expressed as Potts and vertex partition functions of signed plane graphs. Here
we consider constructions of the Jones polynomial as state models of unsigned
graphs and show that the Jones polynomial of any link can be expressed as a
vertex model of an unsigned embedded graph.
  In the process of deriving this result, we show that for every diagram of a
link in the 3-sphere there exists a diagram of an alternating link in a
thickened surface (and an alternating virtual link) with the same Kauffman
bracket. We also recover two recent results in the literature relating the
Jones and Bollobas-Riordan polynomials and show they arise from two different
interpretations of the same embedded graph.Comment: Minor corrections. To appear in Annals of Combinatoric
Words and Polynomial Invariants of Finite Groups in Non-Commutative Variables

Let V be a complex vector space with basis {x 1, x 2, . . . , x n } and G be a finite subgroup of GL(V). The tensor algebra T(V) over the complex is isomorphic to the polynomials in the non-commutative variables x 1, x 2, . . . , x n with complex coefficients. We want to give a combinatorial interpretation for the decomposition of T(V) into simple G-modules. In particular, we want to study the graded space of invariants in T(V) with respect to the action of G. We give a general method for decomposing the space T(V) into simple modules in terms of words in a Cayley graph of the group G. To apply the method to a particular group, we require a homomorphism from a subalgebra of the group algebra into the character algebra. In the case of G as the symmetric group, we give an example of this homomorphism from the descent algebra. When G is the dihedral group, we have a realization of the character algebra as a subalgebra of the group algebra. In those two cases, we have an interpretation for the graded dimensions and the number of free generators of the algebras of invariants in terms of those words
A second look at the toric h-polynomial of a cubical complex

We provide an explicit formula for the toric $h$-contribution of each cubical
shelling component, and a new combinatorial model to prove Clara Chan's result
on the non-negativity of these contributions. Our model allows for a variant of
the Gessel-Shapiro result on the $g$-polynomial of the cubical lattice, this
variant may be shown by simple inclusion-exclusion. We establish an isomorphism
between our model and Chan's model and provide a reinterpretation in terms of
noncrossing partitions. By discovering another variant of the Gessel-Shapiro
result in the work of Denise and Simion, we find evidence that the toric
$h$-polynomials of cubes are related to the Morgan-Voyce polynomials via
Viennot's combinatorial theory of orthogonal polynomials.Comment: Minor correction
Mask formulas for cograssmannian Kazhdan-Lusztig polynomials

We give two contructions of sets of masks on cograssmannian permutations that
can be used in Deodhar's formula for Kazhdan-Lusztig basis elements of the
Iwahori-Hecke algebra. The constructions are respectively based on a formula of
Lascoux-Schutzenberger and its geometric interpretation by Zelevinsky. The
first construction relies on a basis of the Hecke algebra constructed from
principal lower order ideals in Bruhat order and a translation of this basis
into sets of masks. The second construction relies on an interpretation of
masks as cells of the Bott-Samelson resolution. These constructions give
distinct answers to a question of Deodhar.Comment: 43 page
Competitively tight graphs

The competition graph of a digraph $D$ is a (simple undirected) graph which
has the same vertex set as $D$ and has an edge between two distinct vertices
$x$ and $y$ if and only if there exists a vertex $v$ in $D$ such that $(x,v)$
and $(y,v)$ are arcs of $D$. For any graph $G$, $G$ together with sufficiently
many isolated vertices is the competition graph of some acyclic digraph. The
competition number $k(G)$ of a graph $G$ is the smallest number of such
isolated vertices. Computing the competition number of a graph is an NP-hard
problem in general and has been one of the important research problems in the
study of competition graphs. Opsut [1982] showed that the competition number of
a graph $G$ is related to the edge clique cover number $\theta_E(G)$ of the
graph $G$ via $\theta_E(G)-|V(G)|+2 \leq k(G) \leq \theta_E(G)$. We first show
that for any positive integer $m$ satisfying $2 \leq m \leq |V(G)|$, there
exists a graph $G$ with $k(G)=\theta_E(G)-|V(G)|+m$ and characterize a graph
$G$ satisfying $k(G)=\theta_E(G)$. We then focus on what we call
\emph{competitively tight graphs} $G$ which satisfy the lower bound, i.e.,
$k(G)=\theta_E(G)-|V(G)|+2$. We completely characterize the competitively tight
graphs having at most two triangles. In addition, we provide a new upper bound
for the competition number of a graph from which we derive a sufficient
condition and a necessary condition for a graph to be competitively tight.Comment: 10 pages, 2 figure
Variations on a result of Bressoud

The well-known Rogers-Ramanujan identities have been a rich source of mathematical study over the last fifty years. In particular, Gordon’s generalization in the early 1960s led to additional work by Andrews and Bressoud in subsequent years. Unfortunately, these results lacked a certain amount of uniformity in terms of combinatorial interpretation. In this work, we provide a single combinatorial interpretation of the series sides of\ud
these generating function results by using the concept of cluster parities. This unifies the aforementioned results of Andrews and Bressoud and also allows for a strikingly broader family of q–series results to be obtained. We close the paper by proving congruences for a “degenerate case” of Bressoud’s theorem
The Lecture Hall Parallelepiped

The s-lecture hall polytopes P [subscript s] are a class of integer polytopes defined by Savage and Schuster which are closely related to the lecture hall partitions of Eriksson and Bousquet-Mélou. We define a half-open parallelopiped Par  [subscript s] associated with P  [subscript s] and give a simple description of its integer points. We use this description to recover earlier results of Savage et al. on the δ-vector (or h*-vector) and to obtain the connections to s-ascents and s-descents, as well as some generalizations of these results.National Science Foundation (U.S.) (Grant DMS-1068625
Equality of P-partition generating functions

To every labeled poset (P,\omega), one can associate a quasisymmetric
generating function for its (P,\omega)-partitions. We ask: when do two labeled
posets have the same generating function? Since the special case corresponding
to skew Schur function equality is still open, a complete classification of
equality among (P,\omega) generating functions is likely too much to expect.
Instead, we determine necessary conditions and separate sufficient conditions
for two labeled posets to have equal generating functions. We conclude with a
classification of all equalities for labeled posets with small numbers of
linear extensions.Comment: 24 pages, 19 figures. Incorporates minor changes suggested by the
  referees. To appear in Annals of Combinatoric
On Seymour's Decomposition Theorem

Let $\mathcal M$ be a class of matroids closed under minors and isomorphism.
Let $N$ be a matroid in $\mathcal M$ with an exact $k$-separation $(A, B)$. We
say $N$ is a $k$-decomposer for $\mathcal M$ having $(A, B)$ as an inducer, if
every matroid $M\in \mathcal M$ having $N$ as a minor has a $k$-separation $(X,
Y)$ such that, $A\subseteq X$ and $B\subseteq Y$. Seymour [3, 9.1] proved that
a matroid $N$ is a $k$-decomposer for an excluded-minor class, if certain
conditions are met for all 3-connected matroids $M$ in the class, where
$|E(M)-E(N)|\le 2$. We reinterpret Seymour's Theorem in terms of the
connectivity function and give a check-list that is easier to implement because
case-checking is reduced
0-Hecke algebra action on the Stanley-Reisner ring of the Boolean
  algebra

We define an action of the 0-Hecke algebra of type A on the Stanley-Reisner
ring of the Boolean algebra. By studying this action we obtain a family of
multivariate noncommutative symmetric functions, which specialize to the
noncommutative Hall-Littlewood symmetric functions and their (q,t)-analogues
introduced by Bergeron and Zabrocki, and to a more general family of
noncommutative symmetric functions having parameters associated with paths in
binary trees introduced recently by Lascoux, Novelli, and Thibon. We also
obtain multivariate quasisymmetric function identities, which specialize to
results of Garsia and Gessel on generating functions of multivariate
distributions of permutation statistics.Comment: Added connections with a family of noncommutative symmetric functions
  introduced recently by Lascoux, Novelli, and Thibo
Asymptotics for the number of spanning trees in circulant graphs and
  degenerating d-dimensional discrete tori

In this paper we obtain precise asymptotics for certain families of graphs,
namely circulant graphs and degenerating discrete tori. The asymptotics contain
interesting constants from number theory among which some can be interpreted as
corresponding values for continuous limiting objects. We answer one question
formulated in a paper from Atajan, Yong and Inaba in [1] and formulate a
conjecture in relation to the paper from Zhang, Yong and Golin [21]. A crucial
ingredient in the proof is to use the matrix tree theorem and express the
combinatorial laplacian determinant in terms of Bessel functions. A
non-standard Poisson summation formula and limiting properties of theta
functions are then used to evaluate the asymptotics.Comment: 27pages, 3 figure
The combinatorics of automorphisms and opposition in generalised polygons

We investigate the combinatorial interplay between automorphisms and opposition in (primarily finite) generalised polygons. We provide restrictions on the fixed element structures of automorphisms of a generalised polygon mapping no chamber to an opposite chamber. Furthermore, we give a complete classification of automorphisms of finite generalised polygons which map at least one point and at least one line to an opposite, but map no chamber to an opposite chamber. Finally, we show that no automorphism of a finite thick generalised polygon maps all chambers to opposite chambers, except possibly in the case of generalised quadrangles with coprime parameters
Refined Enumeration of Noncrossing Chains and Hook Formulas

International audienceIn the combinatorics of finite finite Coxeter groups, there is a simple formula giving the number of maximal chains of noncrossing partitions. It is a reinterpre-tation of a result by Deligne which is due to Chapoton, and the goal of this article is to refine the formula. First, we prove a one-parameter generalization, by the considering enumeration of noncrossing chains where we put a weight on some relations. Second, we consider an equivalence relation on noncrossing chains coming from the natural action of the group on set partitions, and we show that each equivalence class has a simple generating function. Using this we recover Postnikov's hook length formula in type A and obtain a variant in type B
Canonical rings of Q-divisors on P^1

The canonical ring $S_D = \bigoplus_{d \geq 0} H^0(X, \lfloor dD \rfloor)$ of
a divisor D on a curve X is a natural object of study; when D is a Q-divisor,
it has connections to projective embeddings of stacky curves and rings of
modular forms. We study the generators and relations of S_D for the simplest
curve X = P^1. When D contains at most two points, we give a complete
description of S_D; for general D, we give bounds on the generators and
relations. We also show that the generators (for at most five points) and a
Groebner basis of relations between them (for at most four points) depend only
on the coefficients in the divisor D, not its points or the characteristic of
the ground field; we conjecture that the minimal system of relations varies in
a similar way. Although stated in terms of algebraic geometry, our results are
proved by translating to the combinatorics of lattice points in simplices and
cones.Comment: 19 pages, 3 figure
The Pieri rule for dual immaculate quasi-symmetric functions

The immaculate basis of the non-commutative symmetric functions was recently
introduced by the first and third author to lift certain structures in the
symmetric functions to the dual Hopf algebras of the non-commutative and
quasi-symmetric functions. It was shown that immaculate basis satisfies a
positive, multiplicity free right Pieri rule. It was conjectured that the left
Pieri rule may contain signs but that it would be multiplicity free. Similarly,
it was also conjectured that the dual quasi-symmetric basis would also satisfy
a signed multiplicity free Pieri rule. We prove these two conjectures here.Comment: 14 pages, corrections from v1 and v