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Collective excitations of trapped Fermi or Bose gases
András Csordás
csordas@tristan.elte.hu
Research Group for Statistical Physics of the
Hungarian Academy of Sciences,
Pázmány P. Sétány 1/A, H-1117 Budapest, Hungary
Zoltán Adam
adamzoltan@complex.elte.hu
Department of Physics of Complex Systems, Eötvös University,
Pázmány P. Sétány 1/A, H-1117 Budapest, Hungary
(November 25, 2020)
Abstract
A new method is developed to calculate all excitations
of trapped gases using hydrodynamics
at zero temperature for any equation of state $\mu=\mu(n)$ and for
any trapping potential.
It is shown that a natural scalar product can be defined for the mode
functions, by which the wave operator is hermitian and the mode
functions are orthogonal.
It is also shown that the
Kohn-modes are exact for harmonic trapping in hydrodynamic theory.
Excitations for fermions are calculated in the
BCS-BEC transition region using the equation of
state of the mean-field Leggett-model for isotrop harmonic trap potential.
pacs: 03.75.Kk,03.75.Ss,47.37.+q,05.70.Ce
Several experiments on trapped ultracold gases
probed in the past decade the collective excitations of atomic gases.
Earlier measurements
on bosons Jin97 ; Mewes96 and more recent measurements on fermions
kinast04 ; bartenstein042 near Feshbach resonances can be
explained rather satisfactorily
using hydrodynamics at zero temperature. In his seminal
paper Stringari96 Stringari applied first hydrodynamics for trapped
bosons undergoing Bose-Einstein condensation. His predictions were
confirmed by experiments Jin97 ; Mewes96 . Later,
using the same approach he predicted Stringari04 also the qualitative
behavior of low lying modes
for fermions in the whole crossover region from a BCS type superfluid
Fermi gas to a molecular Bose-Einstein condensation (BEC)
leggett80 ; Engelbrecht97 ; Marini98 .
Now, several recent theoretical papers appeared in the
literature Fuchs03 ; Heiselberg04 ; Hu04 ; Bulgac05 ; Kim04 ; Astrakharchik05
using hydrodynamic theory to better explain
the measurements on the BCS-BEC transition.
In general, no exact
solution to the hydrodynamic equations are known, except when
the equation of state has the polytropic form $\mu(n)\propto n^{\gamma}$
Heiselberg04 .
The hydrodynamic approach leads to a wave
equation for the density oscillations.
In principle, this wave equation can be solved for a single oscillating
mode, if the boundary conditions for the density oscillations are known.
Bulgac et al. Bulgac05 has written the eigenvalue equation for
a single mode in such a way that the two sides were hermitian,
but did not address the question of the function space to which
all the excitations should belong.
Here we use a different approach.
For a general equation of state $\mu=\mu(n)$
it is usually very difficult to prescribe
appropriate boundary conditions at the surface of the gas.
There are a few examples where this problem is circumvented
using some ansatz on the spatial forms of the excitations
Hu04 ; Astrakharchik05 . Here we shall introduce a natural
scalar product, by which
the wave operator itself is hermitian and
the boundary conditions can be treated as in
quantum-mechanical problems: the mode functions are
square integrable functions. The scalar product we shall use automatically
ensures particle conservation. By this way finding excitation frequencies
are relatively easy: the task is to calculate matrix-elements
of the wave operator with the natural scalar product, and then calculate
the eigenvalues of the resulting matrix. We shall demonstrate the whole
procedure for the mean-field model of the BCS-BEC transition for isotropic
trap potential and compare our results with that of the scaling ansatz
approach for the lowest $l=0$ monopole mode.
In hydrodynamic theory for trapped gases at zero temperature
density oscillations are given by the continuity equation
$$\frac{\partial n}{\partial t}+\nabla(n{\bf u})=0,$$
(1)
and the Euler-equation
$$\frac{\partial{\bf u}}{\partial t}+{\bf u}\nabla{\bf u}=-\frac{1}{mn}(\nabla P%
)-\frac{(\nabla V)}{m},$$
(2)
where $\bf u$ is the velocity field, $n$ is the density, $t$ is the time,
$P$ is the pressure, $m$ is the particle mass, and $V$ is the
external trapping potential.
Knowing the
equation of state $\mu=\mu(n)$ at zero temperature in the corresponding
homogeneous system
the equilibrium density in the trapped case can be
determined from the local chemical potential
$$\mu=\mu({\bf r})\equiv\mu(n_{0}({\bf r}))=\mu_{0}-V({\bf r}),$$
(3)
where $\mu_{0}$ is the overall constant chemical potential.
For confining potentials the solution of this equation for
positive $n_{0}(\mathbf{r})$ supplies an equilibrium density,
which has a finite support with a well defined boundary. Typically
$n_{0}$ decreases to zero by approaching the boundary.
Using the thermodynamic identity
$$\frac{\partial\mu}{\partial n}=\frac{1}{n}\left(\frac{\partial P}{\partial n}\right)$$
(4)
valid also at $T=0$ the gradient of Eq. (3)
gives the important vector identity
$$-A_{0}({\bf r})\frac{\nabla n_{0}({\bf r})}{n_{0}({\bf r})}=\nabla V({\bf r}),%
\quad A_{0}({\bf r})\equiv\left(\frac{\partial P}{\partial n}\right)_{n=n_{0}(%
\bf r)}$$
(5)
Eq. (5) plays a central role in the following in simplifying the
linearized hydrodynamics.
In mechanical equilibrium the equilibrium pressure $P_{0}$ must satisfy
$$\nabla P_{0}({\bf r})=-n_{0}({\bf r})\nabla V({\bf r}),$$
(6)
otherwise the right hand side of Eq. (2) will not vanish
for $\mathbf{u}=0$ (this is the local form of the Archimedes-law for
a general external potential).
Close to equilibrium $\bf u$ and
$\delta n({\bf r},t)=n({\bf r},t)-n_{0}({\bf r})$
are small, and $P$ can be expanded
to first order in $\delta n$ as
$$P({\bf r},t)=P_{0}({\bf r})+\left(\frac{\partial P}{\partial n}\right)_{0}%
\delta n({\bf r},t).$$
(7)
An important restriction for $\delta n$ is particle conservation
$\int d^{3}r\,\delta n(\mathbf{r},t)=0$.
Linearizing the continuity equation (1) and the Euler-equation
(2) in $\delta n$ and $\bf u$ using the local form of
the Archimedes-law (6) and the identity (5)
the linearized hydrodynamic equations can be written as
$$\frac{\partial\delta n}{\partial t}+\nabla(n_{0}{\bf u})=0,$$
(8)
$$\frac{\partial{\bf u}}{\partial t}=-\nabla\left[\frac{A_{0}}{n_{0}m}\delta n%
\right].$$
(9)
Let us introduce a new field by
$$\Psi({\bf r},t)=\sqrt{\frac{A_{0}({\bf r})}{n_{0}({\bf r})}}\delta n({\bf r},t)$$
(10)
where $\Psi$ has the same support as $n_{0}$, $A_{0}$ and $\delta n$.
From now on, we allow complex fields $\Psi$
(which are more convenient for problems,
where angular momentum is conserved).
Eliminating $\mathbf{u}$ from Eqs. (8) and (9)
a wave equation
$$\frac{\partial^{2}\Psi}{\partial t^{2}}+{\hat{G}}_{\Psi}\Psi=0,$$
(11)
can be derived for $\Psi$,
where ${\hat{G}}_{\Psi}$ is given by
$$\hat{G}_{\Psi}=-\sqrt{\frac{A_{0}({\bf r})}{n_{0}({\bf r})}}\cdot\nabla\cdot%
\frac{n_{0}({\bf r})}{m}\cdot\nabla\cdot\sqrt{\frac{A_{0}({\bf r})}{n_{0}({\bf
r%
})}}.$$
(12)
The main advantage of the field $\Psi$
is that its wave operator $\hat{G}_{\Psi}$
is manifestly
hermitian hermit with respect to the scalar product
$$\langle\Psi_{1}|\Psi_{2}\rangle=\int_{n_{0}({\bf r})>0}d^{3}r\,\Psi_{1}^{*}({%
\bf r})\Psi_{2}({\bf r})$$
(13)
The scalar product (13) is trivially the correct one for a
homogeneous system with periodic boundary conditions. It was used
for a weakly interacting trapped Bose-gas Csordas99 , where
$\mu(n)\propto n$.
The same idea of finding a proper scalar product
for eigenmodes of a trapped, noninteracting Bose or Fermi gas
at finite temperature using the hydrodynamic approach was applied
in Csordas01 . A single eigenmode
$$\Psi_{i}({\bf r},t)=\sin(\omega_{i}t+\phi_{0})\Psi_{i}({\bf r}),$$
(14)
fulfills the eigenvalue equation
$$\omega^{2}_{i}\Psi_{i}({\bf r})=\hat{G}_{\Psi}\Psi_{i}({\bf r}).$$
(15)
Solutions of (15) are (or for degenerate
eigenvalues can be) orthonormalized with the scalar product
(13)
$$\delta_{ij}=\int_{n_{0}({\bf r})>0}d^{3}r\,\Psi_{i}^{*}({\bf r})\Psi_{j}({\bf r%
}).$$
(16)
$\Psi_{0}(\mathbf{r})=\textrm{Const}\cdot\sqrt{n_{0}(\mathbf{r})/A_{0}(\mathbf{%
r})}$
is always
a solution to (15) with $\omega_{0}=0$. Eq.
(10) implies that $\delta n_{i}=\Psi_{0}\Psi_{i}$, thus
the orthogonality relation (16) shows
that all the modes with $i\neq 0$ are automatically particle conserving,
and the mode $\Psi_{0}$ should be cancelled from the solutions.
Taking a complete orthonormal basis, i.e.,
$\delta_{i,j}=\langle\varphi_{i}|\varphi_{j}\rangle$
the squared excitation frequencies $\omega^{2}$ can be obtained from the
eigenvalues of the matrix
$$G_{i,j}=\langle\varphi_{i}|\hat{G}_{\Psi}|\varphi_{j}\rangle.$$
(17)
The matrix elements in (17)
require the knowledge or the
numerical evaluation of spatial derivatives of the basis functions.
Usually this causes big numerical errors because the high lying
modes are rapidly oscillating functions.
In practice, it is much
better to apply the spatial derivatives to the (spatially varying)
coeffectients of the wave equation, which are usually not oscillating
too much.
The wave operator (12) has the structure
$\hat{G}_{\Psi}=-R\nabla Q\nabla R$. If there exists a similar system
for which the boundary is the same and
the wave operator has also the structure
$\hat{G}^{0}=-R_{0}\nabla Q_{0}\nabla R_{0}$ but with known spectra and
eigenfunctions
$$\hat{G}^{0}\varphi_{i}=\epsilon_{i}^{(0)}\varphi_{i}$$
(18)
then one can eliminate the unwanted spatial derivatives of the basis functions
in the matrix elements if the basis is given by $\varphi_{i}$, $(i=0,1,\ldots)$.
Let us introduce $\alpha$ and $\beta$ by
$$\alpha=\alpha(\mathbf{r})=Q/Q_{0},\quad\beta=\beta(\mathbf{r})=R/R_{0},$$
(19)
then the matrix elements can be written as
$$G_{i,j}=\int d^{3}r\varphi_{i}^{*}(\mathbf{r})\varphi_{j}(\mathbf{r})G_{i,j}(%
\mathbf{r}),$$
(20)
where
$$\displaystyle G_{i,j}(\mathbf{r})=\frac{\epsilon_{i}^{(0)}+\epsilon_{j}^{(0)}}%
{2}\alpha\beta^{2}+R_{0}^{2}Q_{0}\alpha(\nabla\beta)(\nabla\beta)$$
$$\displaystyle+\frac{1}{2}R_{0}^{2}\nabla\left[Q_{0}\beta^{2}(\nabla\alpha)%
\right].$$
(21)
For an isotrop
harmonic trapping potential $V(\mathbf{r})=m\omega_{0}^{2}r^{2}/2$
and for any equation of state $\mu=\mu(n)$ a
whole series of exact solutions of the wave equation can be given.
If $\Psi(\mathbf{r})$ is chosen to be
$$\Psi({\bf r})=\mbox{Const}\sqrt{\frac{n_{0}({\bf r})}{A_{0}({\bf r})}}r^{l}Y_{%
l}^{m}(\vartheta,\phi),\quad l>0$$
(22)
then this mode function fulfills the wave equation with eigenvalue
$\omega^{2}=\omega_{0}^{2}l$. The three $l=1$ modes are the Kohn-modes
(see Ref. Bulgac05 )
for isotropic trapping.
As a specific, nontrivial model let us consider the
mean-field model of Leggett leggett80
for the BCS-BEC transition.
The Leggett model
is fixed in homogeneous systems by the gap equation
$$\sum_{\bf k}\frac{1}{2}\left(\frac{1}{E_{\bf k}}-\frac{1}{\varepsilon_{\bf k}}%
\right)=-\frac{m}{4\pi\hbar^{2}a},$$
(23)
and by the number equation
$$N=\sum_{\bf k}\left(1-\frac{\varepsilon_{\bf k}-\mu}{E_{\bf k}}\right),$$
(24)
where $E_{\bf k}=\sqrt{(\varepsilon_{\bf k}-\mu)^{2}+\Delta^{2}}$,
$\varepsilon_{\bf k}=\hbar^{2}k^{2}/(2m)$,
$\Delta$ is the pairing gap and $a$ is the $s$-wave scattering length.
The equation of state $\mu=\mu(n)$ is implicitly given by the model.
This model captures the essential features of the BCS-BEC transition.
However, recent Monte-Carlo data on the equation of state
show Astrakharchik05 that there are corrections
to the mean-field results of the
Leggett model which should be taken into account for the equation of state,
especially close to unitarity (i.e., around the $a=\infty$ point).
Here we study the above model for simplicity.
In the trapped case we use the $\mu(n)$ function taken from the model
and solve (3) for the density profile keeping
$N=\int d^{3}r\,n(\mathbf{r})$ to be fixed. The equilibrium pressure for any
trap potential $V(\mathbf{r})$ can be calculated from the
local form of the Archimedes law (6). Knowing
the pressure and the density the calculation of $A_{0}(\mathbf{r})$
with help of (5) is straightforward.
The details of the full calculation for the Leggett model will be published
elsewhere Adamunp .
For isotrop harmonic trapping there is a dimensionless coupling
parameter: $\kappa=d/(aN^{1/6})$, where $d=\sqrt{\hbar/m\omega_{0}}$
is the oscillator length. The spectra
depends only on $\kappa$. In three cases the spectra is exactly
known Fuchs03 ; Heiselberg04 ; Stringari04
because the equation of state has a polytropic form:
$\mu\propto n^{\gamma}$. These particular values are $\kappa=-\infty$
(BCS limit), $\kappa=0$ (unitarity limit) and $\kappa=\infty$
(BEC limit). In these cases all the mode functions can be constructed
exactly Fuchs03 , even in the nonisotropic case (The methods
of Refs. Csordas99 ; Csordas00 can be easily
employed to the polytropic equation of state).
We used the $\kappa=-\infty$ mode functions Bruun99 on the
BCS-side and the $\kappa=\infty$ mode functions Heiselberg04
on the BEC-side as basis functions.
Our numerical results for different angular momentum $l$ can
be seen on Fig. 1.
Arrows on both sides show the limiting well-known collective
oscillation frequencies Stringari04 .
In Fig. 2 the behavior of the
lowest $l=0$ quadrupole mode can be seen as a function of $\kappa$. This mode
is the lowest $\kappa$ dependent mode on Fig. 1.
The scaling ansatz
approach Hu04 ; Astrakharchik05 gives quite a good result
for this particular mode. On the scale of Fig. 2
the two curves would be practically indistinguishable.
In order to give a quantitative measure about the quality of the
latter approach
we compare in Fig. 3 our excitation frequencies with
those given by the scaling ansatz.
In the isotrop harmonic trapping potential case $\omega_{sc}^{2}$ is given
Hu04 ; Astrakharchik05 by $\omega_{sc}^{2}/\omega^{2}_{0}=9\langle n\partial\mu/\partial n\rangle/(2%
\langle V\rangle)-1,$
where the average of a quantity like $V$ is taken as
$\langle V\rangle=\int d^{3}r\,V(\mathbf{r})n_{0}(\mathbf{r})$.
From the figure it is clearly seen that scaling ansatz is exact
at $\kappa=0,\pm\infty$, but between these values it is not.
However, the difference in the isotropic case
is so small that is much less than the
experimental resolution.
We have preliminary data Adamunp for the excitation
frequencies for the experimentally relevant
axially symmetric harmonic trapping as well.
Once again the scaling ansatz differs a little for the radial and axial
quadrupole modes for a general intermediate coupling $\kappa$.
For such large anisotropies as in kinast04 ; bartenstein042
however, the deviation $\delta\Omega^{2}$ is much bigger
than in the isotropic case.
Finally, let us turn to the conclusions. We gave a straightforward method
how to solve the hydrodynamic equations in the trapped case
if the equation of state is known. We introduced a natural scalar
product for the transformed wave operator by which the operator is
hermitian. Collective excitations by our method can be found by
simply diagonalizing the matrix of the wave operator on some basis.
The power of our method lies in the fact that we calculate
the whole spectra. We can predict the behavior
of the excitations also for those modes for which no scaling
ansatz is known. The method is not limited to a particular
trap potential (isotropic or not), nor a given mode.
There is no additional approximation, the method calculates
the (numerically) exact modes given by the hydrodynamic
theory.
Here we wanted to present the basic formalism and applied to the simplest
isotrop harmonic case. We compared the numerically
exact excitation frequencies with that of the
scaling ansatz for the monopole mode and showed that the difference
between the two frequencies are extremely small in the whole
region of the BCS-BEC transition for the mean-field Leggett model.
Acknowledgements.
The present work has been partially supported by the Hungarian Research
National Foundation under Grant Nos. OTKA T046129 and T038202.
The authors would like to thank useful discussions with
Prof. P. Szépfalusy and J. Cserti.
References
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(16)
The matrix element can be written as
$\langle\Psi_{1}|\hat{G}_{\Psi}|\Psi_{2}\rangle=\int d^{3}r\,(n_{0}/m)(\nabla%
\Psi_{1}^{*}\sqrt{A_{0}/n_{0}})\cdot(\nabla\Psi_{2}\sqrt{A_{0}/n_{0}})$, provided partial integration is
allowed without a surface term. Usually this is ensured because $n_{0}$
vanishes at the boundary.
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Phys. Rev. A 59, 1477 (1999).
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Semistable Principal Bundles-II
(in positive
characteristics)
V.Balaji111The research of the
first author was partially supported by the DST project no
DST/MS/I-73/97 and A.J.Parameswaran
()
1 Introduction
Let $H$ be a semisimple algebraic group and let $X$ be a smooth
projective curve defined over an algebraically closed field $k$.
One of the important problems in the theory of principal $H$-bundles
on $X$ is the construction of the moduli spaces of semistable
$H$-bundles when the characteristic of $k$ is positive. Over fields of
characteristic $0$ this work was done by A.Ramanathan (cf.[R1]).
For principal $GL(n)$-bundles this is classical, over fields of any
characteristic (cf.[Ses]).
The purpose of this paper is to prove the existence and the
projectivity of the moduli spaces of semistable principal $H$-bundles
on $X$ for fields $k$ of characteristic $p>0$ with precise bounds on
the prime $p$, the restrictions being imposed by the representation
theory of $H$.
It might seem, by the general method of reduction modulo $p$, that the
existence of the moduli space in char.0 implies its existence for
large primes. To the best of our knowledge this is not the case. (cf
Remark 4.10). The representation theoretic considerations
involving heights are essential to the proving of the existence of the
moduli.
The broad strategy of this paper is along the same lines as in the
precursor to this paper ([BS]) where a different approach for
the construction and projectivity of these moduli spaces (in
characteristic zero) was given. However, its implementation involves
several new inputs. The key input for the existence of the
moduli comes from the paper of Ilangovan-Mehta-Parameswaran
([IMP]) which establishes in positive characteristics the links
between the semistability of principal bundles and the concept of a
low height representation. In proving the projectivity of
the moduli space, the key ideas come from a natural interplay of the
recent results of Serre on the representation theory in positive
characteristics ([S2], [S3]), and ideas inspired by the
papers of Ramanan-Ramanathan and Rousseau ([RR], [Rou]).
The principal difficulty is to replace the tensor product theorem
of semistable bundles and unitary representations of
fundamental groups which are central to the characteristic 0
theory. The notions of height and saturated groups provide just the
right replacements.
Let $H$ be a semisimple algebraic group (as coming by reduction from a
Chevalley group scheme defined over $\bf Z$), and fix a faithful
representation $H\hookrightarrow G=SL(n)$ arising as reduction
modulo $p$ of a representation defined over $\bf Z$. Let us denote by
$ht_{H}(G)$ the height of $G$ as an $H$-representation. (cf.
Definition 3.1). We say a representation $H\hookrightarrow G$ is of low height if $char(k)=p>ht_{H}(G)$. Then we
have the following:
Theorem 4.6 Let $H\hookrightarrow G$ be a faithful low
height representation. Then there exists a coarse moduli scheme
$M_{X}(H)$, for semistable principal $H$-bundles. Further, the moduli
space $M_{X}(H)$ is quasi-projective and the canonical morphism
$\overline{\mu}:M_{X}(H)\longrightarrow M_{X}(G)$ is affine.
The proof of the projectivity of the moduli spaces requires more
refined prime bounds. Towards this we introduce a new index which we
term the separable index associated to a $G$-module $W$ (cf.
Definition 5.2). We denote this by $\psi_{G}(W)$ and we say a
$G$-module $W$ is of low separable index if $char(k)=p>\psi_{G}(W)$. We fix throughout, a finite dimensional $G$-module $W$
such that the subgroup $H$ is realised as the isotropy of a closed
orbit, hence giving rise to a closed embedding $G/H\subset W$. We
term these $G$-modules for convenience as affine $(G,H)$-modules
(cf. Def 3.12). Let $A$ be a complete discrete valuation ring
and let $K$ be its quotient field and $k$ its residue field. Then we
have the following theorem:
Theorem 11.1 (Semistable reduction) Let $W$ be a
finite dimensional affine $(G,H)$-module associated to $H$ and $G$
and let $p>\psi_{G}(W)$. Let $H_{K}$ denote the group scheme $H\times{\rm Spec}\,K$, and $P_{K}$ be a semistable $H_{K}$-bundle on $X_{K}$. Then there
exists a finite extension $L/K$, with $B$ as the integral closure of
$A$ in $L$ such that the bundle $P_{K}$, after base change to ${\rm Spec}\,B$,
extends to a semistable $H_{B}$-bundle $P_{B}$ on $X_{B}$.
This in particular implies that the moduli spaces $M_{X}(H)$ are
projective over fields $k$ with $char(k)=p>\psi_{G}(W)$. Together
with Theorem 4.6 we can conclude that the canonical morphism
$\overline{\mu}:M_{X}(H)\longrightarrow M_{X}(G)$ is finite. As a corollary we also
obtain the irreducibility of the moduli spaces when $H$ is semisimple
and simply connected.(Cor 11.10). A large part of this paper is
devoted to proving Theorem 11.1.
The crucial difference between the present approach and the classical
proof of Langton for the properness of the moduli space of semistable
vector bundles can be briefly described as follows. Langton first
extends the family of semistable vector bundles (or equivalently
principal $GL(n)$-bundles) to a $GL(n)$-bundle in the limit although
non-semistable. In other words, the structure group of the limiting
bundle remains $GL(n)$. Then by a sequence of Hecke modifications
the semistable limit is attained without changing the isomorphism
class of the bundle over the generic fibre.
Instead, we extend the family of semistable $H_{K}$-bundles to an
$H_{A}^{\prime}$-bundle with the limiting bundle remaining semistable, but the
structure group scheme $H_{A}^{\prime}$, is non-reductive in the limit. In other
words one loses the reductivity of the structure group scheme. Then,
by using Bruhat-Tits theory (cf §10), we relate the group scheme
$H_{A}^{\prime}$ to the reductive group scheme $H_{A}$ without changing the
isomorphism class of the bundle over the generic fibre as well as the
semistability of the limiting bundle.
We note that the boundedness of semistable principal bundles over
curves in positive characteristics is proved in the preprint
([HN]).222The problem of the construction of the moduli is
being considered independently by V.B.Mehta and S.Subramaniam.
Throughout the paper, we make an effort to specify carefully the
bounds on the characteristic of $k$ that are forced on us. We believe
that our methods can probably be stretched to include more primes and
we indicate at every stage the possible difficulties. The
representation theoretic indices that we have developed here may
possibly be of independent interest.
Before we proceed to describe the contents we pause to remark
that there is some overlap between the present paper and [BS].
The layout of the paper is as follows. In §3 we recall low height
representations and some results from [S2] which we need in later
sections. Here we also define the basic functors for semistable principal
$H$- and $G$-bundles and we prove a technical lemma involving the choice of
a “base point on the curve”which, in some sense gives the motivation for
the rest of the work. In this paper we work with more than one base point
so as to achieve better height bounds.
In §4 we give a simple construction of the moduli space of $H$-bundles
under the right characteristic bounds. The idea of the proof comes from
[BS] and the ingredients involving heights from §3.
The rest of the paper is devoted to proving the semistable reduction
theorem. In §5 some new representation theoretic indices are
introduced and these give the bounds that we need to impose on the
characteristic $p$ in what follows. Here the main point is to give a
criterion for the strong separability of a linear action of a
reductive group. In sections §6 and §7 we construct and study the
flat closure of $H_{K}^{\prime}$ in $G_{A}$ and realise it as isotropy group
schemes along the lines of the classical theorem on
semi-invariants (cf. [B]).
In §8 we prove the key lemmas on the relationship between polystable
bundles and semistable sections inspired mainly by the papers of
Ramanan-Ramanathan ([RR]) and Rousseau ([Rou]). More precisely,
we obtain a notion paralleling that of monodromy subgroup of a
polystable $G$-bundle which is realised as a saturated subgroup of
$G$. This enables us to prove a local constancy for polystable
bundles in char.$p$. In §9 we prove that the family of bundles extends to
a semistable bundle with structure group as a non-reductive group scheme
$H_{A}^{\prime}$ with generic fibre $H_{K}$. In §10 using Bruhat-Tits theory we relate
the non-reductive group scheme $H_{A}^{\prime}$ with the reductive group scheme
$H_{A}$. In §11 we complete the proof of the semistable reduction theorem.
Acknowledgments: We would like to thank the many
people with whom we have had discussions during the course of this
work: S.Kannan, M.S.Narasimhan, M.V.Nori, Gopal Prasad,
M.S.Raghunathan, C.S.Rajan, S.Ramanan, S.Ilangovan, S.Subramaniam, and
V.Uma. We want to especially thank V.B.Mehta and C.S.Seshadri for
their generous help in the paper, from its inception to its
conclusion. Finally we wish to thank the referees for their numerous
suggestions and comments which has led to a considerable improvement
in the exposition.
The first author wishes to thank the School of Mathematics T.I.F.R,
Mumbai and the second author the Chennai Mathemaical Institute and
Institute of Mathematical Sciences, Chennai where much of this work
was carried out. We also wish to thank CAAG for its annual meets where
we got together on this project.
Contents
1.
Introduction
2.
Notations and Conventions
3.
Low height representations
4.
Construction of moduli
5.
Separable index and slice theorem
6.
Towards the flat closure
7.
Affine embedding of $G_{A}/H_{A}^{\prime}$
8.
Semistable bundles, semistable sections and saturated groups
9.
Extension to the flat closure
10.
Potential good reduction
11.
Semistable reduction theorem
2 Notations and Conventions
Throughout this paper, unless otherwise stated, we have the following
notations and assumptions:
(i)
We work over an algebraically closed field $k$ of
characteristic $p>0$.
(ii)
$H$ is a semisimple algebraic group, and $G$, unless
otherwise stated will always stand for the special linear group
$SL(n)$. Their representations are finite dimensional and rational.
(iii)
$A$ is a discrete valuation ring (which could be assumed to be
complete) with residue field $k$, and quotient field $K$.
(iv)
We recall that $\pi:E\longrightarrow X$ is a principal bundle with structure
group $H$, or a principal $H$-bundle for short if $H$ acts on $E$ on the
right and $\pi$ is $H$-invariant and isotrivial, i.e, locally trivial in
the étale topology.
(v)
Let $E$ be a principal $G$-bundle on $X\times T$ where $T$ is
${\rm Spec}\,A$. Let $x\in X$ be a closed point which we fix throughout
and we shall denote by $E_{x,A}$ or $E_{x,T}$ (resp $E_{x,K}$) the
restriction of $E$ to the subscheme $x\times~{}{\rm Spec}\,A$ or $x\times T$ (resp $x\times~{}{\rm Spec}\,K$). Similarly, $l\in T$ will denote the
closed point of $T$ and the restriction of $E$ to $X\times l$ will
be denoted by $E_{l}$.
(vi)
We shall denote $T-l$ by $T^{*}$ throughout this paper.
(vii)
In the case where the structure group is $GL(n)$, when we speak of a
principal $GL(n)$-bundle we identify it often with the associated vector
bundle (and can therefore talk of the degree of the principal
$GL(n)$-bundle).
(viii)
We denote by $E_{K}$ (resp $E_{A}$) the principal bundle $E$ on $X\times{\rm Spec}\,K$ (resp $X\times{\rm Spec}\,A$) when viewed as a principal
$H_{K}$-bundle (resp $H_{A}$-bundle). Here $H_{K}$ and $G_{K}$ (resp $H_{A}$ and
$G_{A}$) are the product group scheme $H\times{\rm Spec}\,K$ and $G\times{\rm Spec}\,K$ (resp $H\times{\rm Spec}\,A$ and $G\times{\rm Spec}\,A$).
(ix)
If $H_{A}$ is an $A$-group scheme, then by $H_{A}(A)$ (resp
$H_{K}(K)$) we mean its $A$ (resp $K$)-valued points. When $H_{A}=H\times{\rm Spec}\,A$, then we simply write $H(A)$ for its $A$-valued
points. We denote the closed fibre of the group scheme by $H_{k}$.
(x)
Let $Y$ be any $G$-scheme and let $E$ be a $G$-principal
bundle. For example $Y$ could be a $G$-module. Then we denote by
$E(Y)$ the associated bundle with fibre type $Y$ which is the
following object: $E(Y)$ = $(E\times Y)/G$ for the twisted action of
$G$ on $E\times Y$ given by $g.(e,y)~{}=~{}(e.g,g^{-1}.y)$.
(xi)
If we have a group scheme $H_{A}$ (resp $H_{K}$) over
${\rm Spec}\,A$ (resp ${\rm Spec}\,K$) an $H_{A}$-module $Y_{A}$ and a
principal $H_{A}$-bundle $E_{A}$, then we shall denote the
associated bundle with fibre type $Y_{A}$ by $E_{A}(Y_{A})$.
(xii)
By a family of $H$ bundles on $X$ parametrised by $T$ we mean a
principal $H$-bundle on $X\times T$, which we also denote by
$\{E_{t}\}_{t\in T}$.
3 Low height representations and some consequences
Let $k$ be an algebraically closed field of characteristic $p>0$. Let
$H$ be a connected reductive algebraic group over $k$. Let $T$ be a
maximal torus of $H$, $X(T):=Hom(T,{\bf G}_{m})$ be the character group of $T$
and $Y(T):=Hom({\bf G}_{m},T)$ be the 1-parameter subgroups of $T$. Let
$R\subset X(T)$ be the root system of $H$ with respect to $T$. Let
${\cal W}$ be the Weyl group of the root system $R$. Let $(~{},~{})$ denote
the ${\cal W}$-invariant inner product on $X(T)\otimes{\bf R}$. For
$\alpha\in R$, the corresponding co-root $\alpha^{\vee}$ is
$2\alpha/(\alpha,\alpha)$. Let $R^{\vee}\subset{X(T)\otimes{\bf R}}$ be the set of all co-roots. Let $B\subset H$ be a Borel subgroup
containing $T$. This choice defines a base $\Delta^{+}$ of $R$ called
the simple roots. Let $\Delta^{-}=-\Delta^{+}$. A root in $R$ is
said to be positive if it is a non-negative linear combination
of simple roots. We take the roots of $B$ to be positive by
convention. Let $\Delta^{\vee}\subset R^{\vee}$ be the basis for
the corresponding dual root system. Then we can define the Bruhat
ordering on ${\cal W}$. The longest element with respect to this
ordering of ${\cal W}$ is denoted by $w_{0}$. A reductive group is
classified by these root-data, namely the character group,
1-parameter subgroups, the root system, co-roots and the ${\cal W}$-invariant pairing.
Let $V$ be a $H$-module, i.e., $V$ is a $k$-vector space together with a
linear representation of $H$ in Aut $(V)$. Then $V$ can be written as
direct sum of eigenspaces for $T$. On each eigenspace $T$ acts by a
character. These are called the weights of the representation. A
weight $\lambda$ is called dominant if
$(\lambda~{},~{}\alpha_{i}^{\vee})\geq 0$ for all simple roots $\alpha_{i}\in\Delta^{+}$. A weight $\lambda$ is said to be $\geq$ another weight $\mu$ if
the difference $\lambda-\mu$ is a non-negative integral linear combination
of simple roots, where the difference is taken with respect to the natural
abelian group structure of $X(T)$. The fundamental weights $\omega_{i}$
are uniquely defined by the criterion
$(\omega_{i}~{},~{}\alpha_{j}^{\vee})=\delta_{ij}$. The element $\rho$ of
$X(T)\otimes{\bf R}$ is defined to be half the sum of positive roots. It
can also seen to be equal to the sum of fundamental weights. The height (cf. [H], Section 10.1) of a root is defined to be the sum of the
coefficients in the expression $\alpha~{}=~{}\Sigma k_{i}\alpha_{i}$. We extend
this notion of height linearly to the weight space and denote this function
by $ht(~{})$. Note that $ht$ is defined for all weights but need not be an
integer even for dominant weights. We extend this notion of height to
representations as follows:
Definition 3.1
$\!\!\!$.
(i)
Given a linear representation $V$ of $H$, we define the
height of the representation $ht_{H}(V)$ (also denoted by $ht(V)$ if
$H$ is understood in the given context) to be the maximum of
$2ht(\lambda)$, where $\lambda$ runs over dominant weights
occurring in $V$.
(ii)
A linear representation $V$ of $H$ is said to be a low
height representation if $ht_{H}(V)<p$, and a weight $\lambda$ is of
low height if $2ht(\lambda)<p$.
Then we have the following theorem (cf. [IMP], [S2])
Theorem 3.2
$\!\!\!$. Let $V$ be a linear representation of $H$ of low height. Then
$V$ is semisimple.
Corollary 3.3
$\!\!\!$. Let $V$ be a low height representation of $H$ and $v\in V$ an
element such that the $H$-orbit of $v$ in $V$ is closed. Then $V$ is a
semisimple representation for the reduced stabiliser $H_{v,red}$ of $v$.
Proposition 3.4
$\!\!\!$. Let $H$ be as above and let $V$ be a low-height
representation of $H$. Then we have the following vanishing of group
cohomology:
$$H^{i}(H,V)=0$$
for all $i\geq 1$.
Proof. We now recall from ([S2] (pp 25,26) )the following
general result on low height modules of connected reductive groups:
Let $V$ be a low height module of $H$. Let $\lambda$ be a dominant
weight which occurs in $V$. Then, if $V(\lambda)=H^{0}(\lambda)$ is
the dual of the Weyl module associated to $\lambda$, by the definition
of height and the low height property of $V$, it follows that
$V(\lambda)$ are also low-height $H$-modules. In particular, it
follows that $V(\lambda)$ are also irreducible and they coincide
with their socle $L(\lambda)$. Therefore, by the semisimplicity of low
height modules, one has $V=~{}\stackrel{{\scriptstyle\lambda}}{{\oplus}}V(\lambda)$.
Therefore by the Vanishing Theorem of Cline-Parshall-Scott-van der
Kallen (cf. [J] pp 237) we have the required cohomology
vanishing since
$$H^{i}(H,V)=~{}\stackrel{{\scriptstyle\lambda}}{{\bigoplus}}H^{i}(H,V(\lambda))=0$$
for all $i\geq 1$. Q.E.D.
3.1 Height and semistability
Let $F$ be a $G$-variety. Then a section $s:X\longrightarrow E(F)$ can be described
as a morphism from $\psi:E\longrightarrow F$ such that $\psi(e.g)=g^{-1}.s(e)$. In
particular, if $H\subset G$ and $F=G/H$ then a section of $E(G/H)$ gives
a reduction of structure group of $E$ to $H$.
We now recall the definitions of semistable, polystable and stable
principal bundles. Note that these definitions make sense for reductive
groups as well.
Definition 3.5
$\!\!\!$. (A. Ramanathan) $E$ is semistable (resp. stable) if for
every parabolic subgroup $P$ of $H$, and for every reduction of structure
group $\sigma_{P}:X\longrightarrow E(H/P)$ to $P$ and for any dominant character
$\chi$ of $P$, the bundle $\sigma_{P}^{*}(L_{\chi}))$ has degree $\leq 0$
(resp.$<0$).(cf.[R1]).
Definition 3.6
$\!\!\!$. A reduction of structure group of $E$ to a
parabolic subgroup $P$ is called admissible if for any character
$\chi$ on $P$ which is trivial on the center of $H$, the line bundle
associated to the reduced $P$-bundle $E_{P}$ has degree zero.
Definition 3.7
$\!\!\!$. An $H$-bundle $E$ is said to be polystable if
it has a reduction of structure group to a Levi $R$ of a parabolic $P$
such that the reduced $R$-bundle $E_{R}$ is stable and the extended $P$
bundle $E_{R}(P)$ is an admissible reduction of structure group for $E$.
Remark 3.8
$\!\!\!$. We note that there is a natural action of the group
$\mbox{{\rm Aut}$\,$}_{G}E$, of automorphisms of the principal $G$-bundle $E$, on $\Gamma(X,E(G/H))$ and the orbits correspond to the $H$-reductions which are
isomorphic as principal $H$-bundles.
Remark 3.9
$\!\!\!$. Let $E_{R}$ be a stable $R$-bundle. Then $E_{R}$ has no
further reduction of structure group to a Levi subgroup $L$ of a parabolic
subgroup in $R$.
Proposition 3.10
$\!\!\!$. Let $E$ be a principal $H$-bundle on $X$. Let $H\hookrightarrow SL(V)$ be a low height faithful representation.
Then the following are equivalent:
(a)
The induced bundle $E(V)=E\times^{H}V$ is semistable.
(b)
The bundle $E$ is semistable as a principal $H$-bundle.
Proof. (b) $\Rightarrow$ (a) follows by ([IMP] Theorem 3.1).
For (a) $\Rightarrow$ (b), we need to proceed as follows. By the Main
Theorem of [S2], any low height representation is actually
semi-simple. Further, if $V=\bigoplus V_{i}$ is the decomposition
into irreducible $H$-modules, then the associated bundle $E(V)$
decomposes as $\bigoplus E(V_{i})$ and the direct sum is of bundles of
degree zero. Therefore it is clear that to prove the converse, we may
as well assume that the representation $\rho$ is an irreducible
representation of $H$ and also of low height. So since we are assuming
that $E$ is non-semistable, there exists a maximal parabolic subgroup
$P\subset H$ and a dominant character $\lambda$ such that the pull
back of $L_{\lambda}$ has degree $deg(L_{\lambda})>0$. Now it is not
very hard to see that there exists a parabolic $P_{1}$ in $SL(V)$ such
that $P=P_{1}\cap H$ (cf [IMP, Lemma 3.5]). Thus we see that,
the reduction of structure group of the vector bundle $E(V)$ to $P_{1}$
is given by a section $\sigma\in\Gamma(E(SL(V)/P_{1}))$ and the line
bundle $L_{\lambda}$ is the restriction of an ample line bundle
$L_{\lambda^{\prime}}$ obtained by a dominant character $\lambda^{\prime}$ of $P_{1}$.
It is clear that $deg(\sigma^{*}(L_{\lambda^{\prime}}))>0$ since
$degL_{\lambda}=deg(\sigma^{*}(L_{\lambda^{\prime}}))$. This implies that
$E(V)$ is also non-semistable, and we are done.
Remark 3.11
$\!\!\!$. This theorem is strict in the sense that given an almost simple group
$H$ and a representation $H\longrightarrow SL(V)$ which is not of low height, there
exists a curve $X$ and a semistable $H$-bundle $E$ on $X$ such that $E(V)$
is not semistable. (The converse works for all but small primes. For more
precise details see [IMP])
3.2 Functorial properness of the evaluation map
The aim of this section is to define the basic functors and prove some
technical lemmas.
Let $G$ be $SL(n)$ and let $H$ be a semisimple algebraic group, $H\subset G$. For our convenience we make the following definition:
Definition 3.12
$\!\!\!$. Define an affine $(G,H)$-module $W$
associated to $(H,G)$ to be a finite dimensional $G$-module, such that
$G/H\stackrel{{\scriptstyle i}}{{\hookrightarrow}}W$ is realised as a closed orbit of
a vector $w\in W$. Observe that since $G/H$ is affine, such a $W$
always exists. We work with this canonical $W$ whenever we refer
to the affine $(G,H)$-module. This is a classical result ( cf. for
example [DM1], p 40 or [B]; also cf. Lemma 7.1
below).
Let
$$F_{G}:(Schemes)\longrightarrow(Sets)$$
be the functor given by
$$F_{G}(T)=\left\{\begin{array}[]{l}\mbox{isomorphism classes of semistable $G$-%
bundles of degree 0}\\
\mbox{on $X$ parametrised by $T$}\\
\end{array}\right.$$
One may similarly define the functor $F_{H}$ (note that since $H$ is
semisimple, for a principal $H$-bundle the associated vector bundles
have degree zero).
Let $x\in X$ be a marked point and let $F_{H,G,x}$ be the
functor
$$F_{H,G,x}(T)=\left\{\begin{array}[]{l}\mbox{isomorphism classes of pairs $(E,%
\sigma_{x})$, $E=\{E_{t}\}_{t\in T}$}\\
\mbox{a family of semistable principal $G$-bundles of
degree 0}\\
\mbox{and $\sigma_{x}:T\longrightarrow E(G/H)_{x}$ a section}\\
\end{array}\right\}$$
(Recall that $E(G/H)_{x}$ denotes the restriction of $E(G/H)$ to $x\times T\approx T$).
Notice that the functor $F_{H}$ is in fact realisable as the following
functor (by Remark 3.8) .
$$F_{H,G}(T)=\left\{\begin{array}[]{l}\mbox{isomorphism classes of
pairs $(E,s)$, $E=\{E_{t}\}_{t\in T}$}\\
\mbox{a family of
semistable $G$-bundles of degree 0 and}\\
\mbox{$s=\{s_{t}\}_{t\in T}$ a section of E(G/H) on $X\times T$}\\
\mbox{or what we may call a
family of sections of $\{E(G/H)_{t}\}_{t\in T}$}.\\
\end{array}\right\}$$
In what follows, we shall identify the functors $F_{H}$ with
$F_{H,G}$. With these definitions we have the following:
Proposition 3.13
$\!\!\!$.
Let $\alpha_{x}$ be the morphism induced by “evaluation of
section” at $x$:
$$\alpha_{x}:F_{H}\longrightarrow F_{H,G,x}.$$
Then $\alpha_{x}$ is a proper morphism of functors.(cf. [DM]).
Proof. Let $T$ be an affine smooth curve and let $l\in T$. Let us
write $T^{*}=T-l$. Then by the valuation criterion for properness, we need
to show the following:
If $E$ is a family of semistable principal $G$-bundles on $X\times T$
together with a section $\sigma_{x}:T\longrightarrow E(G/H)_{x}$; such that for $t\in T^{*}$, we are given a family of $H$-reductions, i.e. a family of
sections $s_{T^{*}}=\{s_{t}\}_{t\in T^{*}}$, where, $s_{t}:X\longrightarrow E(G/H)_{t}$, with the property that, at $x$, $s_{t}(x)=\sigma_{x}(t)$
$\forall~{}t\in T^{*}$; then we need to extend the family $s_{T^{*}}$ to
a section $s_{T}$ of $E(G/H)$ on $X\times T$ such that $s_{l}(x)=\sigma_{x}(l)$ as well.
Let $W$ be an affine $(G,H)$-module associated to $(H,G)$.(see Def
3.12). Thus we get a closed embedding
$$E(G/H)\hookrightarrow E(W)$$
and a family of vector bundles $\{E(W)_{t}\}_{t\in T}$ together with a family of sections $s_{T-l}$ and
evaluations $\{\sigma_{x}(t)\}_{t\in T}$ such that $s_{t}(x)=\sigma_{x}(t)$, $t\neq p$.
For the section $s_{T-l}$, viewed as a section of
$E(W)_{T-l}$ we have two possibilities:
(a)
it extends as a regular section $s_{T}$.
(b)
it has a pole along $X\times l$.
Observe that if (a) holds, then we have
$$s_{T}(X\times(T-l))\subset E(G/H)\subset E(W),$$
since
$E(G/H)$ is closed in $E(W)$, it follows that $s_{T}(X\times l)\subset E(G/H)$. Thus $s_{l}(X)\subset E(G/H)_{l}$.
Further by continuity, $s_{l}(x)=\sigma_{x}(l)$ as well and
this proves the proposition.
If (b) holds the reduction section exists over an open $U_{T}\subset X_{T}$
which contains all the primes of height 1 in $X_{T}$; or equivalently, the
$H$-bundle exists over $U_{T}$. We can now appeal to a theorem of
Colliot-Thélène and Sansuc ([CS] Theorem 6.13 pp 128) which
enables us to extend the principal bundle to $X_{T}$. In other words (b)
cannot occur. Finally observe that the limiting $H$-bundle is semistable
since it arises as a reduction of structure group of a semistable
$G$-bundle and $H\subset G$ is low height. Q.E.D.
Remark 3.14
$\!\!\!$. A different proof involving the semistability of $E(W)$ is given in
[BS]. Here we avoid it so as to improve the prime bounds arising out
of height considerations involved in the construction of the moduli spaces.
4 Construction of the moduli space
The aim of this section is to give a construction of the moduli space
of $H$-bundles. This section is somewhat different from the
corresponding one in[BS] to enable us to provide the best prime
bounds.
Recall that $G=SL(n)$ and $H\subset G$ a semisimple subgroup.
We recall very briefly the Grothendieck Quot scheme used in
the construction of the moduli space of vector bundles
(cf. [Ses]).
Let ${\cal F}$ be a coherent sheaf on $X$ and let ${\cal F}(m)$ be
${\cal F}\otimes{\cal O}_{X}(m)$ (following the usual notations).
Choose an integer $m_{0}=m_{0}(n,d)$ ($n=$ rk, $d=$ deg) such
that for any $m\geq m_{0}$ and any semistable bundle $V$ of
rank $n$ and deg $d$ on $X$ we have
$h^{i}(V(m))=0$ and $V(m)$ is generated by its global
sections.
Let $\chi=h^{0}(V(m))$ and consider the Quot scheme $Q$
consisting of coherent sheaves ${\cal F}$ on $X$ which are
quotients of $k^{\chi}\otimes_{k}{\cal O}_{X}$ with a fixed
Hilbert polynomial $P$. The group ${\cal G}=GL(\chi)$
canonically acts on $Q$ and hence on $X\times Q$ (trivial
action on $X$) and lifts to an action on the universal sheaf
${\cal E}$ on $X\times Q$.
Let $R$ denote the ${\cal G}$-invariant open subset of $Q$
defined by
$$R=\left\{q\in Q\mid\begin{array}[]{l}{\cal E}_{q}={\cal E}\mid_{X\times q}%
\mbox{
is locally free and the canonical map }\\
k^{\chi}\longrightarrow H^{0}({\cal E}_{q})\mbox{ is an isomorphism},\det{\cal
E%
}_{q}\simeq{\cal O}_{X}\end{array}\right\}$$
We denote by $Q^{ss}$ the ${\cal G}$-invariant open subset of $R$
consisting of semistable bundles and let ${\cal E}$ continue to
denote the restriction of ${\cal E}$ to $X\times Q^{ss}$.
Let $q^{\prime\prime}:$ (Sch) $\longrightarrow$ (Sets) be the following functor:
$$q^{\prime\prime}(T)=\left\{(V_{t},s_{t})\mid\begin{array}[]{l}\mbox{$\{V_{t}\}%
$ is a family of semistable principal
$G$-bundles}\\
\mbox{parameterised by $T$ and $s_{t}\in\Gamma(X,V(G/H)_{t})~{}~{}\forall~{}t%
\in T$}\\
\end{array}\right\}.$$
i.e. $q^{\prime\prime}(T)$ consists pairs of rank $n$ vector bundles (or
equivalently principal $G$-bundles) together with a
reduction of structure group to $H$.
By appealing to the general theory of Hilbert schemes, one
can show without much difficulty (cf. [R1, Lemma 3.8.1]) that $q^{\prime\prime}$
is representable by a $Q^{ss}$-scheme, which we denote by $Q^{\prime\prime}$.
Let $W$ be an affine $(G,H)$-module associated to $(H,G)$ (see Def
3.12).
Remark 4.1
$\!\!\!$. Let $E\in Q^{ss}$ and consider $E(W)$ the associated vector
bundle. Then, by the boundedness of the family $E\in Q^{ss}$ it follows
that there exists an $m_{0}$ (independent of $E$) such that if $s$ is
any section of $E(W)$, then ${\#}(zeroes(s))\leq m_{0}$. Fix a subset $J\subset X$ such that ${\#}J>m_{0}$.
The universal sheaf ${\cal E}$ on $X\times Q^{ss}$ is in fact a
vector bundle. Let ${\cal E}_{G}$ denote the associated
principal $G$-bundle, set
$$Q^{\prime}=({\cal E}_{G}/H)_{J}=({\cal E}_{G}/H)_{x_{j}}\times_{Q^{ss}}({\cal E%
}_{G}/H)_{x_{j}^{\prime}}\cdot\cdot\cdot$$
the fibre product being taken over all $j\in J$.
Then in our notation $Q^{\prime}={\cal E}_{G}(G/H)_{J}$ i.e. we take the bundle over
$X\times Q^{ss}$ associated to ${\cal E}_{G}$ with fibre $G/H$ and take its
restriction to $x_{j}\times Q^{ss}\approx Q^{ss}$ and take the product
over $Q^{ss}$. Let $f:Q^{\prime}\longrightarrow Q^{ss}$ be the natural map. Then,
since $H$ is reductive, $f$ is an affine morphism.
Observe that $Q^{\prime}$ parametrises semistable vector bundles
together with “initial values of reductions to $H$ ”.
Define the “evaluation map” of $Q^{ss}$-schemes as follows:
$$\phi_{J}:Q^{\prime\prime}\longrightarrow Q^{\prime}$$
$$(V,s)\longmapsto\{(V,s(x))|x\in J\}.$$
Lemma 4.2
$\!\!\!$. The evaluation map $\phi_{J}:Q^{\prime\prime}\longrightarrow Q^{\prime}$ is proper and injective.
Proof. Let $G/H\hookrightarrow W$ be as in Definition
3.12 and let $(E,s)$ and $(E^{\prime},s^{\prime})\in Q^{\prime\prime}$ such that $\phi_{J}(E,s)=\phi_{J}(E^{\prime},s^{\prime})$ in $Q^{\prime}$. i.e. $(E,s(x))=(E^{\prime},s^{\prime}(x))\forall x\in J$. So we may assume that $E\simeq E^{\prime}$ and that $s$ and $s^{\prime}$
are two different sections of $E(G/H)$ with $s(x)=s^{\prime}(x)\forall x\in J$.
Using $G/H\hookrightarrow W$, we may consider $s$ and $s^{\prime}$ as
sections in $\Gamma(X,E(W))$ and further, as sections of $E(W)$ one
has $s(x)=s^{\prime}(x)\forall x\in J$. By Remark 4.1 this implies $s=s^{\prime}$. This proves that $\phi_{J}$ is injective (since $E(G/H)\hookrightarrow E(W)$ is a closed embedding).
The properness of the map follows easily, the proof being as in
Proposition 3.13. Thus $\phi_{J}$ being proper and injective is
affine. Q.E.D.
Remark 4.3
$\!\!\!$. In [BS] a single base point served the purpose. Here we
employ the standard trick of increasing the number of base points to
achieve injectivity without the semistability of $E(W)$. This enables
us to improve the prime bounds.
Remark 4.4
$\!\!\!$. It is immediate that the ${\cal G}$-action on $Q^{ss}$ lifts to an action
on $Q^{\prime\prime}$.
Recall the commutative diagram
$Q^{\prime\prime}$$Q^{\prime}$$Q^{ss}$$\mu$$f$$\circlearrowright$$\phi_{J}$
By Lemma 4.2, $\phi_{J}$ is a proper injection and hence affine.
One knows that $f$ is affine (with fibres $(G/H)^{\#J}$). Hence $\mu$ is a
${\cal G}$-equivariant affine morphism.
Lemma 4.5
$\!\!\!$. (cum remark) Let $(E,s)$ and $(E^{\prime},s^{\prime})$ be in the same
${\cal G}$-orbit of $Q^{\prime\prime}$. Then we have $E\simeq E^{\prime}$. Identifying $E^{\prime}$ with
$E$, we see that $s$ and $s^{\prime}$ lie in the same orbit of $\mbox{{\rm Aut}$\,$}_{G}E$ on $\Gamma(X,E(G/H))$. Then using Remark 3.8, we see that the reductions $s$
and $s^{\prime}$ give isomorphic $H$-bundles.
Conversely, if $(E,s)$ and $(E^{\prime},s^{\prime})$ such that $E\simeq E^{\prime}$ and the
reductions $s,s^{\prime}$ give isomorphic $H$-bundles, using again Remark 3.8,
we see that $(E,s)$ and $(E^{\prime},s^{\prime})$ lie in the same ${\cal G}$-orbit.
Consider the ${\cal G}$-action on $Q^{\prime\prime}$ with the linearisation
induced by the affine ${\cal G}$-morphism $\mu:Q^{\prime\prime}\longrightarrow Q^{ss}$. It is
seen without much difficulty that, since a good quotient of
$Q^{ss}$ by ${\cal G}$ exists and since $Q^{\prime\prime}\longrightarrow Q^{ss}$ is an affine
${\cal G}$-equivariant map, a good quotient $Q^{\prime\prime}/{\cal G}$ exists
(cf. [R1, Lemma 4.1]).
Moreover by the universal property of categorical quotients,
the canonical morphism
$$\overline{\mu}:Q^{\prime\prime}//{\cal G}\longrightarrow Q^{ss}//{\cal G}$$
is also affine.
Let $M_{X}(H)$ denote the scheme $Q^{\prime\prime}//{\cal G}$. then we have proved the
following theorem.
Theorem 4.6
$\!\!\!$. Let $H\hookrightarrow G$ be a faithful low height
representation. Then there exists a coarse moduli scheme $M_{X}(H)$, for
semistable principal $H$-bundles. Further, the moduli space $M_{X}(H)$
is quasi-projective and the canonical morphism $\overline{\mu}:M_{X}(H)\longrightarrow M_{X}(G)$ is affine.
4.1 Points of the moduli
In this subsection we will recall briefly the description of the
$k$-valued points of the moduli space $M_{X}(H)$. The general functorial
description of $M_{X}(H)$ as a coarse moduli scheme follows by the usual
process.
Proposition 4.7
$\!\!\!$. The “points ” of $M_{X}(H)$ are given by polystable
principal $H$-bundles.
We firstly remark that since the quotient $q:Q^{\prime\prime}\longrightarrow M_{X}(H)$ obtained
above is a good quotient, it follows that each fibre $q^{-1}(E)$ for
$E\in M_{X}(H)$ has a unique closed ${\cal G}$-orbit. Let us denote an orbit
${\cal G}\cdot E$ by $O(E)$. The proposition will follow from the
following:
Lemma 4.8
$\!\!\!$. If $O(E)$ is closed then $E$ is polystable.
Proof. Recall the definition of a polystable bundle Def
3.7 and the definition of admissible reductions Def
3.6. If $E$ has no admissible reduction of structure group to
a parabolic subgroup then it is polystable and there is nothing to
prove.
Suppose then that $E$ has an admissible reduction $E_{P}$, to $P\subset H$. Recall by the general theory of parabolic subgroups that there
exists a 1-PS $\xi:{\bf G}_{m}\longrightarrow H$ such that $P=P(\xi)$. Let
$L(\xi)$ and $U(\xi)$ be its canonical Levi subgroup and unipotent
subgroup respectively. The Levi subgroup will be the centraliser of
this 1-PS $\xi$ and one knows $P(\xi)=L(\xi)\cdot U(\xi)=U(\xi)\cdot L(\xi)$. In particular, if $h\in P$ then $lim\xi(t)\cdot h\cdot\xi(t)^{-1}$ exists. From these considerations one can show that there
is a morphism
$$f:P(\xi)\times{\bf A}^{1}\longrightarrow P(\xi)$$
such that $f(h,0)=m\cdot u$, where $h\in P$ and $h=m\cdot u$,
$m\in L$ and $u\in U$. (see Lemma 3.5.12 [R1])
Consider the $P$-bundle $E_{P}$. Then, using the natural projection $P\longrightarrow L$ where $L=L(\xi)$, we get an $L$-bundle $E_{P}(L)$. Again, using
the inclusion $L\hookrightarrow P\hookrightarrow H$, we get a new
$H$-bundle $E_{P}(L)(H)$. Let us denote this $H$-bundle by $E_{P}(L,H)$.
It follows from the definition of admissible reductions and
polystability that $E_{P}(L,H)$ is polystable.
Further, from the family of maps $f$ defined above, and composing with
the inclusion $P(\xi)\hookrightarrow H$ we obtain a family of
$H$-bundles $E_{P}(f_{t})$ for $t\neq 0$ and all these bundle are
isomorphic to the given bundle $E$. Following ([R1] Prop.3.5 pp
313), one can prove that the bundle $E_{P}(L,H)$ is the limit of
$E_{P}(f_{t})$. It follows that $E_{P}(L,H)$ is in the ${\cal G}$-orbit $O(E)$
because $O(E)$ is closed. Now by Lemma 4.5, $E\simeq E_{P}(L,H)$, implying that $E$ is polystable. Q.E.D.
Remark 4.9
$\!\!\!$. In the above Proposition we have only stated that there is a
surjective map from the set of isomorphism classes of polystable
$H$-bundles to the points of the moduli space. We believe that this
correspondence is a bijection but one possibly needs to discard a few
more primes.
A few remarks are in order regarding the existence and properness of
the moduli spaces of principal bundles for “large” primes.
Remark 4.10
$\!\!\!$. “ A general principle is that if a statement is true
in characteristic zero then it is also true for large
$p$”(cf. ([S2])). One might therefore think that this would imply
the existence and projectivity of the moduli spaces of semistable
$H$-bundles for large primes, since one already knows this in char 0
(cf. for example [R1], [F] or [BS]).
We observe that this principle would indeed hold if one could show that the
subset corresponding to the semistable bundles in a family of $H$-bundles
is open over (Z or large p); for the moduli spaces of $H$-semistable
bundles is realised as a GIT-quotient of a quasi-projective scheme and the
required results would follow by “reduction modulo $p$” for large $p$. To
the best of our knowledge the required “openness result” does not follow
by any general principle.
A key point of this paper is that even for the existence of the moduli
spaces of semistable $H$-bundles as a quasi-projective scheme for large
$p$, one requires height considerations. Moreover we give explicit
bounds for $p$.
Once the moduli space exists as a quasi-projective scheme for large $p$,
its projectivity follows for an unspecified larger $p$. One of the
hard parts of this paper is to give specific representation theoretic bounds
for $p$ for the projectivity of the moduli spaces.
5 Separable index and slice theorem
Let $T$ be a torus and $W$ be a finite dimensional $T$-module.
Further, let ${X}(T)$ be the free abelian group of characters of $T$
and $\cal S$ be the set of distinct characters that occur in $W$.
For every subset $S\subset\cal S$ we have the following map:
$$\nu_{S}:{\bf Z}^{|S|}\longrightarrow{X}(T)$$
given by $e_{s}\longrightarrow\chi_{s}$.
Let $g_{S}$ be the g.c.d of the maximal minors of the map $\nu_{S}$
written under the fixed basis. For any vector $w\in W$, consider the
subset $S_{w}\subset\cal S$, consisting of characters that occur in $w$
with nonzero coefficients. i.e., if $w=\sum a_{\chi}(w)e_{\chi}$, then
$$S_{w}=\{{\chi\in\cal S}|a_{\chi}(w)\neq 0\}$$
Then we have the following:
Lemma 5.1
$\!\!\!$. The characteristic of the field, $p$ does not divide $g_{S_{w}}$ if
and only if the action of $T$ on the vector $w$ is separable.
Proof. Let $T\cdot w$ denote the orbit of $w$ under the $T$
action. Let $T_{w}$ be the stabilis
er. Then $T/T_{w}$ is a torus and the
character group ${X}(T_{w})$ of the stabiliser is the quotient of the
character groups ${X}(T)/{X}(T/T_{w})$. Moreover the image of the dual
map of the quotient map $T\to T/T_{w}$ is canonically identified with
the image of $\nu_{S_{w}}$. Hence the group $T_{w}$ is reduced if and
only if this cokernel, identified with the cokernel of $\nu_{S_{w}}$,
has no $p$-torsion. But this cokernel has $p$-torsion if and only if
the rank of $\nu_{S_{w}}$ drops mod $p$, which in turn happens if and
only if $p$ divides all the maximal minors. Hence the lemma.
Notation
$${{p}}_{T}(W)=\{{\rm largest~{}prime~{}which~{}divides}~{}g_{S}|\forall S%
\subset\cal S\}$$
Definition 5.2
$\!\!\!$. Let $H\longrightarrow SL(W)$ be a finite dimensional
representation of $H$. Define the separable index, $\psi_{H}(W)$
of the representation as follows:
$$\psi_{H}(W)=max\{ht_{H}(W),~{}{{p}}_{T}(W)\}$$
Remark 5.3
$\!\!\!$. When $T$ is a maximal torus of a semisimple group $H$ and the
$T$-module $W$ is actually an $H$-module, then the set of characters
that occur on $W$ can be written down explicitly using Standard
Monomial Theory. From this very explicit form, this separable index is
computable, though it could be tedious or may need a computer. The few
cases where we made some computations indicated that this index is
possibly bounded above by the dimension of $W$. One can easily observe
that the absolute value of each minor of the map $\nu_{S}$ is bounded
above by $l!\cdot h^{l}$, where $l=rank(G)$ and $h=ht_{G}(W)$. Hence the separable index has a weak upper bound given by
$l!\cdot h^{l}$.
Definition 5.4
$\!\!\!$. A representation $H\longrightarrow SL(W)$ is said to be with low
separable index if $p>\psi_{H}(W)$.
Theorem 5.5
$\!\!\!$. If $W$ is a low separable index $H$-module then the
action of $H$ on $W$ is strongly separable i.e., the stabilizer at
any point is absolutely reduced.
Proof. Since the representation is low height, every nilpotent
in the Lie algebra of the $H$ is integrated in $SL(W)$ and hence the
nilpotent part of the Lie algebra of the stabiliser at any $w\in W$
will actually lie in the Lie algebra of the reduced stabiliser. Thus
by the Jordan decomposition of the stabiliser, it is enough to ensure
separability of the action of any maximal torus. Separability index
assures that the given maximal torus $T$ acts separably at all points in
$W$. This implies that every maximal torus acts separably at all points as
all maximal tori are conjugates. Hence the action of $H$ is strongly
separable. Q.E.D.
Remark 5.6
$\!\!\!$. We recall briefly the notions of saturated
subgroups of $GL(V)$. For details cf.pp 524-526 [S3].
We first define a one parameter subgroup defined by an element of
order $p$. Let $V$ be a finite dimensional $k$-vector space, and let
$s\in GL(V)$ be an element such that $s^{p}=1$. One has $s=1+u$
where $u^{p}=0$. If $t\in k$, we can define an element $s^{t}\in GL(V)$ by the truncated binomial formula:
$$s^{t}=1+tu+\frac{t(t-1)u^{2}}{2}+...$$
summed upto $u^{i}$ with $i<p$.
The map $t\longrightarrow s^{t}$ defines a homomorphism of algebraic groups:
$$\phi_{s}:{\bf G}_{a}\longrightarrow GL(V)$$
where ${\bf G}_{a}$ is the additive group. This homomorphism has two characterising properties:
•
$\phi_{s}(1)=s$
•
The map $t\longrightarrow\phi_{s}(t)$ is a polynomial of degree $<p$.
Let $H\subset GL(V)$ be a subgroup. We say that $H$ is saturated if every unipotent element $s\in H$ has the following properties:
•
$s^{p}=1$
•
$s^{t}\in H$ for every $t\in k$
One can see that given any subgroup $H$ there is a smallest saturated
subgroup which contains $H$ called the saturation of $H$.
A property of saturated groups which we need is that if $H$ is
saturated and $H^{0}$ is the connected component of identity of $H$ then
the index $[H:H^{0}]$ is coprime to $p$. (cf.pp 524-526 [S3]).
One can again generalize all these notions for an arbitrary reductive
algebraic group $G$ instead of $GL(V)$. Among the elementary examples
of saturated subgroups are parabolic subgroups, centralizers of any
subgroup, and Levi subgroups (since they can be realised as the
centralizer of a torus). We can isolate a couple of key properties in
the theory of low height representations:
(i)
If $G\longrightarrow GL(V)$ is a
low height representation of $G$ then the isotropy subgroups of closed
orbits in $V$ are saturated.
(ii)
If $S$ is a reductive and saturated subgroup of $G$ and if $G\longrightarrow GL(V)$ is a low height representation of $G$ then $V$ is a low
height module for $S$ as well.(cf. [S2] p.25)
Proposition 5.7
$\!\!\!$. (A version of Luna’s étale slice theorem in
char.$p$) Let $W$ be a low separable index $G$-module. Let $F$ be a
fibre of the good quotient $q:W\longrightarrow W//G$, and let $F^{cl}$ be the
unique closed orbit contained in $F$. Then there exists a $G$-map
$$F\longrightarrow F^{cl}.$$
Proof. Since $\psi_{G}(W)=max\{ht_{G}(W),~{}{{p}}_{T}(W)\}$ the
assumption $p>\psi_{G}(W)$ on the separable index implies the
following:
(i)
Every stabiliser subgroup for the $G$-action on $W$ is reduced,
the action being strongly separable (by Th 5.5).
(ii)
It is saturated, the representation $G\longrightarrow GL(W)$ being
low height.
(iii)
When $w$ is a quasistable point in $W$ or equivalently, the orbit
$G\cdot w$ is closed, then $W$ is a semisimple representation of the
stabiliser $G_{w}$. This is a consequence of the main theorem of
[S3], namely that low height representations are semi-simple.
For more on this (cf. [S2] pp 20-25); it may be kept in mind
that the height of the representation, $ht_{G}(W)$, coincides with
Serre’s index $n_{G}(W)$.
A close examination of Luna’s proof shows that the key point is the
complete reducibility of the tangent space $T_{w}(W)$, of the affine
$G$-module. This is used then to get a splitting of the canonical
injection of the tangent space of the closed $G$-orbit in
$T_{w}(W)$. Once this is achieved the slice can be constructed. The
above proposition is then a corollary to the main slice theorem
applied to a single orbit. (For details cf. [BR] Prop 8.5 p
312). Q.E.D.
6 Towards the flat closure
Fix as in §3.2 a faithful low height representation $H\hookrightarrow G$ defined over $k$ as well as an affine
$(G,H)$-module associated to the pair $(H,G)$.(cf.Def 3.12).
Consider the extension of structure group of the bundle $P_{K}$ via the
induced $K$-inclusion $H_{K}\hookrightarrow G_{K}$ . We denote the
associated $G_{K}$-bundle $P_{K}(G)$ by $E_{K}$.
Then, since $G=SL(n)$, by the projectivity of the moduli space of
semistable vector bundles, there exists a semistable extension
of $P_{K}(G)=E_{K}$ to a $G_{A}$-bundle on $X\times{\rm Spec}\,A$, which we
denote by $E_{A}$. Call the restriction of $E_{A}$ to $X\times l$
(identified with $X$) the limiting bundle of $E_{A}$ and denote
it by $E_{l}$ (as in §1). One has in fact slightly more, which is what
we need.
Lemma 6.1
$\!\!\!$. Let $E_{K}$ denote a family of semistable $G_{K}$-bundles on
$X\times{\rm Spec}\,K$ (or equivalently a family of semistable vector bundles
of rank $n$ and trivial determinant on $X\times T^{*}$). Then (by going to
a finite cover $S$ of $T$ if need be ) the principal bundle $E_{K}$ extends
to $E_{A}$ with the property that the limiting bundle $E_{l}$ is in fact polystable i.e, a direct sum of stable bundles.
Proof. The proof of this Lemma is possibly well known but for
the sake of completeness we give it here. Recall notations as in §4
regarding Quot schemes etc.
Note that the moduli space in question, namely of semistable principal
$G$-bundles, is a GIT quotient $Q^{ss}\longrightarrow M$ by ${\cal G}$, and the family $E_{A}(G)$ is given by a morphism $T\longrightarrow M$. Lift the $K$-valued point, namely, $r_{K}$, given by the family
$E_{K}$, to $Q^{ss}$ and consider the ${\cal G}$-orbit $R_{0}$ of $r_{K}$ in
$Q^{ss}$. Let $\overline{R}_{0}$ be its closure in $Q^{ss}$. Since the
$K$-valued point $r_{K}$ is in fact an $A$-valued point of $M$, the GIT
quotient of $\overline{R}_{0}$ is indeed the curve $T$. Also, observe
that the closure intersects the closed fibre. Consider the morphism
$\psi:\overline{R}_{0}\longrightarrow T$. Since the base is a curve $T$, one has a
multi-section for the morphism $\psi$, and one obtains the curve
$S$. The general fibre has been modified only in the orbit, therefore
the isomorphism class of the bundles remains unchanged. Q.E.D.
Remark 6.2
$\!\!\!$. It is to be noted that the definition of polystability given
here coincides with that in Def 3.7, in the sense that a closed
orbit in the Quot scheme corresponds to a polystable vector bundle.
We observe the following:
•
Note that giving the $H_{K}$-bundle $P_{K}$ is giving a reduction of
structure group of the $G_{K}$-bundle $E_{K}$ which is equivalent to
giving a section $s_{K}$ of $E_{K}(G_{K}/H_{K})$ over $X_{K}$ .
•
We fix a base point $x\in X$ and denote by $x_{A}=x\times{\rm Spec}\,A$, the induced section of the family (which we call the base section):
$$X_{A}\longrightarrow{\rm Spec}\,A$$
•
Let $E_{x,A}$ (resp $E_{x,K}$) be as in §1, the restriction of
$E_{A}$ to $x_{A}$ (resp $x_{K}$). Thus, $s_{K}(x)$ is a section of
$E_{K}(G_{K}/H_{K})_{x}$ which we denote by $E_{x}(G_{K}/H_{K})$.
•
Since $E_{x,A}$ is a principal $G$-bundle on ${\rm Spec}\,A$ and
therefore trivial, it can be identified with the group scheme $G_{A}$
itself. For the rest of the article we fix one such
identification, namely:
$$\xi_{A}:E_{x,A}\longrightarrow G_{A}.$$
•
Since we have fixed $\xi_{A}$ we have a canonical identification
$$E_{x}(G_{K}/H_{K})\simeq G_{K}/H_{K}$$
which therefore carries a natural identity section $e_{K}$ (i.e
the coset $id.H_{K}$). Using this identification we can view $s_{K}(x)$ as
an element in the homogeneous space $G_{K}/H_{K}$.
•
Let $\theta_{K}\in G(K)$ be such that $\theta_{K}^{-1}\cdot s_{K}(x)=e_{K}$. Then we observe that, the isotropy subgroup scheme in $G_{K}$
of the section $s_{K}(x)$ is $\theta_{K}.H_{K}.\theta_{K}^{-1}$.
•
On the other hand one can realise $s_{K}(x)$ as the identity coset of
$\theta_{K}.H_{K}.\theta_{K}^{-1}$ by using the following
identification:
$$G_{K}/\theta_{K}.H_{K}.\theta_{K}^{-1}\stackrel{{\scriptstyle\sim}}{{%
\longrightarrow}}G_{K}/H_{K}.$$
$$g_{K}(\theta_{K}.H_{K}.\theta_{K}^{-1})\longmapsto g_{K}\theta_{K}.H_{K}.$$
Definition 6.3
$\!\!\!$. Let ${H_{K}^{\prime}}$ be the subgroup scheme of $G_{K}$ defined as:
$$H_{K}^{\prime}:=\theta_{K}.H_{K}.\theta_{K}^{-1}.$$
Using $\xi_{A}$ we can have a canonical identification:
$$E_{x}(G_{K}/H_{K}^{\prime})\simeq G_{K}/H_{K}^{\prime}.$$
Then we observe that, using the above identification we get a section
$s_{K}^{\prime}$ of $E_{K}(G_{K}/H_{K}^{\prime})$, with the property that, $s_{K}^{\prime}(x)$ is the
identity section and moreover, since we have conjugated by an
element $\theta_{K}\in G_{A}(K)(=G(K))$, the isomorphism class of the
$H_{K}$-bundle $P_{K}$ given by $s_{K}$ does not change by going to $s_{K}^{\prime}$.
Thus, in conclusion, the $G_{A}$-bundle $E_{A}$ has a reduction to
$H_{K}^{\prime}$ given by a section $s_{K}^{\prime}$ of $E_{K}(G_{K}/H_{K}^{\prime})$, with the
property that, at the given base section $x_{A}=x\times{\rm Spec}\,A$, we
have an equality $s_{K}^{\prime}(x_{A})=e_{K}^{\prime}$, with the identity element
of $G_{K}/H_{K}^{\prime}$ (namely the coset $id.H_{K}^{\prime}$).
Definition 6.4
$\!\!\!$. The flat closure of the reduced group scheme
$H_{K}^{\prime}$ in $G_{A}$ is defined to be the schematic closure of $H_{K}^{\prime}$ in
$G_{A}$ with the reduced scheme structure. Let $H_{A}^{\prime}$ denote the
flat-closure of $H_{K}^{\prime}$ in $G_{A}$.(cf. Lemma 7.1)
We then have a canonical identification via $\xi_{A}$:
$$E_{x}(G_{A}/H_{A}^{\prime})\simeq G_{A}/H_{A}^{\prime}.$$
One can easily check that $H_{A}^{\prime}$ is indeed a subgroup scheme of $G_{A}$
since it contains the identity section of $G_{A}$, and moreover, it is
faithfully flat over $A$. Notice however that $H_{A}^{\prime}$ need not be
a reductive group scheme; that is, the special fibre $H_{k}$ over
the closed point need not be reductive.
Observe further that $s_{K}^{\prime}(x)$ extends in a trivial fashion to a
section $s_{A}^{\prime}(x)$, namely the identity coset section $e_{A}^{\prime}$ of
$E_{x}(G_{A}/H_{A}^{\prime})$ identified with $G_{A}/H_{A}^{\prime}$ .
Remark 6.5
$\!\!\!$. If $H_{A}^{\prime}$ is reductive then the semistable
reduction theorem (Theorem 11.1) follows quite easily. Indeed,
firstly by the rigidity of reductive group schemes over ${\rm Spec}\,A$ (SGA 3, Expose III Cor 2.6 pp 117), by going to a finite cover, we
may assume that $H_{A}^{\prime}=H\times{\rm Spec}\,A$. Secondly, in this case one
can realise $H_{A}^{\prime}$, as the isotropy subgroup scheme for a closed
orbit $w_{A}\in W_{A}$. Then we have a closed $G$-immersion of
$G/H$ in a $G$-module $W$, and one may view $s_{K}$ as a section of
$E_{K}(W_{K})$. Note that $E_{K}(G_{K}/H_{K}^{\prime})\subset E_{K}(W_{K})$.
By choice, along $x_{A}$, the section $s_{K}(x)$ extends regularly to a
section of $E_{A}(G_{A}/H_{A}^{\prime})\subset E_{A}(W_{A})$. Hence by Proposition
3.13, $s_{K}$ extends to a section $s_{A}$ which gives the required
reduction over $X\times{\rm Spec}\,A$.
7 Affine embedding of $G_{A}/H_{A}^{\prime}$
As we have noted, $H_{A}^{\prime}$ need not be reductive and the rest of the proof is
to get around this difficulty. Our first aim is to prove that the
structure group of the bundle $E_{A}(G_{A})$ can be reduced to $H_{A}^{\prime}$ which is
the statement of Proposition 9.1.
We need to prove the following generalisation of a well-known result
(cf. for example [B]).
Lemma 7.1
$\!\!\!$.
There exists a finite dimensional $G_{A}$-module $W_{A}$ such that
$G_{A}/H_{A}^{\prime}~{}\hookrightarrow~{}W_{A}$ is a $G_{A}$-immersion.
Proof. We follow the standard proof. Let $I_{K}$ be the ideal
defining the subgroup scheme $H_{K}^{\prime}$ in $K(G)$ (note that $G_{A}$ (resp
$G_{K}$) is an affine group scheme and we denote by $A(G)$ (resp
$K(G)$) its coordinate ring).
Set $I_{A}=I_{K}\cap A(G)$. Then it is easy to see that since we are over a
discrete valuation ring, $I_{A}$ is in fact the ideal in $A(G)$ defining the
flat closure $H_{A}^{\prime}$. Observe also that $I_{A}$ is a primitive $A$
submodule of $A(G)$, that is, $A(G)/I_{A}$ is torsion free; further, $I_{A}\otimes k=I_{k}$ is the defining ideal in $k(G)$ of $H_{k}^{\prime}$ in $G_{k}$ and
$I_{A}\otimes K$ is $I_{K}$.
Since $A(G)$ and the other modules involved are free over the discrete
valuation ring $A$, a set generates $I_{A}\otimes k=I_{k}$ if and only
if it generates $I_{A}$. Thus we may now choose a finite generating set
$\{f_{i}\}$ of $I_{A}$, such that their images $f_{i,k}$ generate $I_{k}$.
As in the classical proof, one has a finite dimensional $G_{K}$-submodule,
$V_{K}$, containing the $\{f_{i}\}$. Now set $V_{A}=V_{K}\cap A(G)$ and $M=V_{A}\cap I_{A}$. Observe that $I_{A}$, $V_{A}$ and hence $M$ are all
$G_{A}$-submodules of $A(G)$. This can be seen by keeping track of the
co-module operations. Then clearly $V_{A}$ is primitive in $A(G)$ and $M$ is
also primitive in $A(G)$ and in particular, primitive in $V_{A}$.
If we set
$$M_{k}=M\otimes k~{}\mbox{and}~{}V_{k}=V_{A}\otimes k$$
we see that the inclusion $M\hookrightarrow V_{A}$ induces an inclusion
$M_{k}\hookrightarrow V_{k}$ . Observe that
$$f_{i}~{}\in M~{},f_{i,k}~{}\in~{}M_{k}~{}\mbox{and}~{}M~{}\subset~{}I_{A}$$
$$M_{k}~{}\subset~{}I_{k}~{}\mbox{and}~{}M_{k}~{}=~{}V_{k}~{}\cap~{}I_{k}$$
We claim that, for $g\in G_{A}(k)$, one has
$$g\cdot M_{k}\subset M_{k}\iff g\in H_{k}^{\prime}$$
Obviously, if $g\in H_{k}^{\prime}$, then $g\cdot M_{k}\subset M_{k}$,
since $V_{k}$ is $G$-stable and $I_{k}$ is $H_{k}^{\prime}$-stable. Thus, it
suffices to show that
$$f_{i,k}(g)=0~{}\mbox{for all}~{}i$$
that is,
$${f_{i,k}}\mbox{ vanish on }~{}g$$
Since $f_{i,k}\in M_{k}$, it suffices to show that
$$\phi(g)=0~{}\mbox{for}~{}\phi~{}\in~{}M_{k}$$
But $\phi(g)=(g^{-1}\cdot\phi)(id)$, where $g^{-1}\cdot\phi$ is the
action of $G$ on functions on $G$. Now, by hypothesis, $(g^{-1}\cdot\phi)\in M_{k}$. Since $M_{k}\subset I_{k}$, and $id\in H_{k}^{\prime}$ , we
see that $(g^{-1}\cdot\phi)(id)=0$ . This proves the above claim.
Similarly, if we set
$$M_{F}=M\otimes_{A}F~{}\mbox{and}~{}V_{F}=V_{A}\otimes_{A}F$$
where $F$ is any field containing $A$, we see that for $g\in G(F)$
$$g\cdot M_{F}\subset M_{F}\iff g\in H_{A}^{\prime}(F)$$
Let $L$ denote the primitive rank one $A$-submodule $\wedge^{d}M\hookrightarrow\wedge^{d}V=W_{A}$, and [$L$] the $A$-valued point of
${\bf P}(W_{A})$ defined by $L$. Here, ${\bf P}(W_{A})$ is defined by the
functor associated to rank one direct summands of $W_{A}$. Then, the
above discussion means that, we can recover $H_{A}^{\prime}$ as the isotropy
subgroup scheme at [$L$] for the $G_{A}$-action on ${\bf P}(W_{A})$.
Recall that, for any field $F$, the isotropy subgroup of $G_{A}(F)$, at
the point of ${\bf P}(W_{A}(F))$ represented by the base change of $L$ by
$F$, is $H_{A}^{\prime}(F)$ .
Fix a generator $l\in L$ so that $l$ is a primitive element in $W_{A}$
and consider the isotropy subgroup scheme $H_{A}^{\prime\prime}$ at $l$ for the
$G_{A}$-action on $W_{A}$. We claim that, $H_{A}^{\prime\prime}$ coincides with $H_{A}^{\prime}$.
To see this, observe that, $H_{A}^{\prime\prime}$ is the subgroup scheme of $G_{A}$
which leaves the closed subscheme (identified with $Spec(A)$)
determined by $l$ invariant (with the corresponding automorphism on
this subscheme being identity). We see then that, $H_{A}^{\prime\prime}$ is a closed subgroup scheme of $G_{A}$. Further, we see that since
$H_{A}^{\prime\prime}$ is the isotropy subgroup of the vector $l\in L$ and $H_{A}^{\prime}$
that of the line [$L$] we have $H_{A}^{\prime\prime}\hookrightarrow H_{A}^{\prime}$. Since
$H_{K}^{\prime}$ is semi-simple, it has no characters and therefore, the
isotropy subgroup scheme at $(l\otimes K)\in(W_{A}\otimes K)$ is
precisely $H_{K}^{\prime}$. This means that, $H_{K}^{\prime\prime}=H_{K}^{\prime}$. Now, $H_{K}^{\prime}$ is
open (dense) in $H_{A}^{\prime}$ (since $H_{A}^{\prime}$ is the flat closure of $H_{K}^{\prime}$ in
$G_{A}$ ) so that, $H_{K}^{\prime}$ is also dense in $H_{A}^{\prime\prime}$. This implies that,
$H_{A}^{\prime}$ and $H_{A}^{\prime\prime}$ coincide set theoretically. Observe also that
$H_{A}^{\prime}$ is reduced by the definition of flat closure. Thus, it
follows that $H_{A}^{\prime}$ = $H_{A}^{\prime\prime}$. This implies that, $G_{A}/H_{A}^{\prime}\hookrightarrow W_{A}$ is a $G_{A}$-immersion and the above lemma
follows. Q.E.D.
Remark 7.2
$\!\!\!$. Regarding the Lemma 7.1 proved above, we note that
usually the subgroup scheme $H_{A}^{\prime}$ can be realised only as the
isotropy subgroup scheme of a line in a $G_{A}$-module. But here, since
the generic fibre of $H_{A}^{\prime}$ is semisimple, one is able to realise
$H_{A}^{\prime}$ as the isotropy subgroup scheme of a primitive element in a
$G_{A}$-module and the limiting group scheme also as an isotropy
subgroup scheme for an element in a $G_{k}$-module. We note here that
last part of the above proof is seen easily by observing that a
non-trivial character of $H_{A}^{\prime}$ by definition is a non-trivial
character of $H_{K}^{\prime}$ and hence $H_{A}^{\prime\prime}=H_{A}^{\prime}$.
Remark 7.3
$\!\!\!$. We make the following key observations about the
group scheme $H_{A}^{\prime}$. The flat group scheme $H_{A}^{\prime}=Stab(w_{A})$, is the
isotropy subgroup scheme of $G_{A}$ at an $A$-valued point $w_{A}\in W_{A}$, where $W_{A}$ can be realised as $W\otimes A$ (after going to a
finite cover of $A$ if need be) and $W$ is the affine $(G,H)$-module
such that $G/H\subset W$.
Moreover, it is also shown as a part of the proof that the closed
fibre $H_{k}^{\prime}=Stab(w_{k})$, is the isotropy subgroup scheme of $G_{k}$
for a vector $w_{k}\in W$.
Thus if we assume that $p>\psi_{G}(W)$, it follows by Theorem
5.5 that $H_{k}^{\prime}$ is reduced.
8 Semistable bundles, semistable sections and saturated groups
The aim of this section is to prove some general lemmas on polystable
bundles and semistable sections. We assume that $p>\psi_{G}(W)$,
notations as in §5.
Definition 8.1
$\!\!\!$. (following Bogomolov) Let $E$ be a principal
$G$-bundle and let $G\longrightarrow GL(V)$ be a representation of $G$. Let $s$
be a section of the associated bundle $E(V)$. Then we call the section
$s$ stable (resp semistable, unstable) relative to $G$ if at one point
$x\in X$ (and hence at every point on $X$) the value of the section
$s(x)$ is stable (resp semistable, unstable).
(It is easy to see the non-dependence of the definition on the point $x\in X$. Consider the inclusion $k[V]^{G}\hookrightarrow k[V]$ and the
induced morphism $V\longrightarrow V/G$. This induces a morphism $E(V)\longrightarrow E(V/G)$. Observe that $V/G$ is a trivial $G$-module. Thus we have the
following diagram:
$$s:X\longrightarrow E(V)\longrightarrow E(V/G)\simeq X\times V/G$$
Composing with the second projection we get a morphism $X\longrightarrow V/G$
which is constant by the projectivity of $X$. Hence the value of the
section is determined by one point in its $G$-orbit.) (cf. [Rou]
1.10)
Lemma 8.2
$\!\!\!$. Let $E(W)$ be a semistable vector bundle of degree zero and let $R$ be a saturated reductive subgroup of
$GL(W)$. Suppose that $E(W)$ has a reduction of structure group to
$E_{R}$, a stable $R$-bundle, and further suppose that we have a
non-zero section $s:X\longrightarrow E_{R}(W)=E(W)$. Then $s$ is a semistable
section in the sense of Def 8.1.
Proof. Suppose that this is not the case. Then as observed in
the definition, if $s(x)$ unstable for a single $x\in X$ implies
it is unstable for all $x\in X$. In particular for the generic point
$x_{0}\in X$. (cf. [Rou] Prop 1.5)
Since $s$ is a non-zero section of $E_{R}(W)$ and $E_{R}(W)$ is semistable
of degree zero, it is nowhere zero. This section gives a reduction of
structure group of $E_{R}(W)$ to a maximal parabolic subgroup $P_{s}$,
given by the extension:
$$0\longrightarrow{\cal O}_{X}\longrightarrow E_{R}(W)\longrightarrow V\longrightarrow
0$$
for some degree zero vector bundle $V$ and where the first inclusion
is given by the section $s$.
Notice that $SL(W)/P_{s}={\bf P}(W)$. In the language of [RR], the
section $s$ can be thought of as taking values in the cone $W$
and $deg(s)=0$.
We now claim that w.l.o.g we may assume that the representation
$W$ is an irreducible $R$-module.
Since $W$ is a low height $R$-module it is completely reducible, i.e
it can be expressed as
$$W=\bigoplus W^{\alpha}$$
where $W^{\alpha}$ are irreducible $R$-module. Any element $w\in W$ can be
expressed as $w=\oplus w^{\alpha}$ with $w^{\alpha}\in W^{\alpha}$. It is easy to see
that if, $w$ is $R$-unstable and if $\lambda$ is a Kempf 1-PS in $R$
which drives $w$ to 0 then $\lambda$ drives all the $w^{\alpha}$’s to 0 as
well. Further, as bundles
$$E_{R}(W)=\bigoplus E_{R}(W^{\alpha})$$
and since $E_{R}(W)$ is semistable of degree 0 all the $E_{R}(W^{\alpha})$
are semistable of degree 0 being direct summands of $E_{R}(W)$. The
given section also breaks up as $s=\oplus s^{\alpha}$ to give
non-zero (and hence nowhere zero!) sections of $E_{R}(W^{\alpha})$
(since $s(x)=w=\oplus w^{\alpha}$, here of course, not all
$\alpha$’s may be involved!).
Again by Def 8.1, the new sections $s^{\alpha}$ continue to
remain unstable since instability is determined at a point $x\in X$.
This proves the claim.
Once $W$ is irreducible as an $R$-module by Schur Lemma the connected
component $Z^{0}(R)$ of center of $R$ acts as scalars on $W$ and hence
trivially on ${\bf P}(W)$ and as scalars on the ample line
bundle $L$ on it.
Since $m=s(x_{0})$ is unstable we have a Kempf instability flag $P(m)$
and the corresponding 1-PS $\mu$, are also defined over the field
$K(X)$ . This follows by the low separable index assumption, namely $p>\psi_{G}(W)$, which in particular implies $W$ is a low height module
for $G$ and hence for the saturated subgroup $R$ (cf. [RR, Prop
3.13] and [IMP, Theorem 3.1]).
The parabolic subgroup being defined over $K(X)$ gives a reduction of
structure group of $E_{R}$ to a parabolic $P$ of $R$. Let $W=\bigoplus W_{i}$ be the weight space decomposition of $W$ with respect to $\mu$.
Let $m=m_{0}+m_{1}$, with $m_{0}$ of weight $j>0$ and $m_{1}$ the sum of
terms of higher weights. In other words, in the projective space ${\bf P}(W)$ we see that $\mu(t)\cdot m\longrightarrow m_{0}$. It is not too hard to see
that we have an identification of the Kempf parabolic subgroups
associated to the points $m$ and $m_{0}$. i.e $P(m)=P(m_{0})$.(cf. [RR, Proposition 1.9]).
In the generic fibre $E_{R}(W)_{x_{0}}$ we have the projection
$$\bigoplus_{i\geq j}W_{i}\longrightarrow W_{j}$$
which takes $m$ to $m_{0}$. This gives a line sub-bundle $L_{0}$ of degree
zero of $E_{R}(W)$ corresponding to $m_{0}$. It then follows that $m_{0}$ is
in fact semistable for the action of $P/U$, the Levi of $P$, for a
suitable choice of linearisation obtained by twisting the action by a
dominant character $\chi$ of $P$. (This is essentially the content of
[RR, Prop.1.12] and we can apply it since we work in the degree
0.)
The semistability of the point $m_{0}$ with this new linearisation the
forces the degree inequality:
$$deg(L_{0}\otimes L(\chi)^{-1})\leq 0$$
But since $deg(L_{0})=0$, this implies $deg(L(\chi))\geq 0$. This
contradicts the stability of $E_{R}$. Q.E.D.
Remark 8.3
$\!\!\!$. We note that the condition of semistability of the vector bundle
$E_{R}(W)$ is assumed here since [IMP] proves it only for
semisimple groups. But in the situation in which we need (cf. Prop
9.2) this condition automatically holds since we have the
following inclusion
$$R\hookrightarrow G\hookrightarrow GL(W)$$
and therefore $E_{R}(W)=E(W)$ and $E(W)$ is semistable since $W$ is a
low height representation of $G$.
Lemma 8.4
$\!\!\!$. Let $E_{R}$ be a stable $R$-bundle as above and let
$I$ be a saturated reductive (possibly non-connected) subgroup of $R$
such that $E_{R}$ has a reduction of structure group to $I$. Then the
reduced $I$-bundle is also stable.
Proof. We first claim that $I$ is irreducible in
$R$: if not, then by the low height property there exists a parabolic
$P$ and a Levi $L$ in it such that $I\subset L$ and this is
irreducible. This gives a reduction of structure group of $E_{R}$ to $L$
and this again contradicts the stability of $E_{R}$, by Remark
3.9.
Now to prove the Lemma, suppose that the reduced $I$-bundle $E_{I}$ is
not stable. Then, $E_{I}$ has an reduction of structure group
$\sigma$, to a maximal parabolic $P\subset I$. Observe that any
parabolic subgroup of a reductive algebraic group looks like
$P(\lambda)$ for a 1-PS $\lambda:{\bf G}_{m}\longrightarrow I$. Now consider
$P_{R}(\lambda)$ the induced parabolic in $R$. Then, it is clear that
$P_{R}(\lambda)$ gives a reduction of structure group for $E_{R}$.
Notice that $P_{R}(\lambda)$ in $R$ may not be a maximal parabolic, but
there exists a maximal parabolic $Q$ containing it. Now note that
by the irreducibility of $I\subset R$ seen above, $Q\cap I$ is a
proper parabolic in $I$ and contains $P_{I}(\lambda)$. Therefore by the
maximality of $P_{I}$ it follows that $Q\cap I=P_{I}$.
Let $\chi$ be a dominant character of $P(\lambda)$ and let the induced
line bundle be $L_{\chi}$ such that $deg(\sigma^{*}(L_{\chi}))\geq 0$.
Then, since $Q$ is a maximal parabolic a multiple of $\chi$ extends to
a dominant character of $Q$ and the induced line bundle $L_{\chi}$ on
$I/P$ is the restriction of the line bundle from $R/Q$. Therefore, the
degrees of the pull backs to $X$ remain the same. This contradicts the
stability of $E_{R}$. Q.E.D.
Proposition 8.5
$\!\!\!$. Let $E_{R}$ be a stable $R$-bundle and $s$ be a
non-zero section of $E_{R}(W)$ as in Lemma 8.2. Let $s(x)=w$.
Then the $R$-orbit of $w$ is closed and $s$ takes its image in the
closed orbit.
Proof. By Lemma 8.2, since $E_{R}$ is stable, $w\in W^{ss}$. Therefore the section $s$ which can be thought of as a map
$$s:E_{R}\longrightarrow W^{ss}$$
which further takes its values in a fibre $F$ of the GIT quotient:
$$W^{ss}\longrightarrow W^{ss}//R$$
Thus the section $s$ gives the following map:
$$s:E_{R}\longrightarrow F$$
and $F$ contains the vector $w$.
We need to show that the orbit $R\cdot w$ is closed.
We prove this by contradiction.
Suppose then that orbit of $R\cdot w$, is not closed. Let $I$ be the
isotropy at a point $f\in F$ such that $R\cdot f$ is closed. Note that
the identity component $I^{o}$ is reductive and saturated and $I$ is
also reduced.
Then by Proposition 5.7 we have an $I$-invariant “slice”, $S\subset F$ and an $R$-isomorphism
$$\theta:R\times^{I}S\simeq F$$
$$\theta([r,s])=r\cdot s$$
This gives a $R$-equivariant morphism
$$l:F\simeq R\times^{I}S\longrightarrow R/I\simeq F^{cl}.$$
The composition $l\circ s=s_{1}$ of the maps $s$ and $l$ gives a
reduction of structure group, $E_{I}\subset E_{R}$ to the isotropy $I=Stab_{R}(f)$ of a point $f\in F^{cl}$.
By Lemma 8.2 the $I$-bundle $E_{I}$ is stable.
Consider the given section $s$ of $E_{R}(W)$ as obtained via the
reduction of structure group to $I$. This is given as follows:
$$s_{1}:E_{I}\longrightarrow F\hookrightarrow W$$
which is $I$-equivariant. Observe that without loss of generality (by
taking a conjugate of the isotropy $I$) we may assume that $w\in Im(s_{1})$.
(This is easy to see. Indeed, starting with a pair $(I,S)$ namely a
slice and an isotropy subgroup at $f\in S$, the given point $w\in F$
can be expressed as an equivalence class $w=[r,s_{0}]$. Then by
translating the slice $S$ by the element $r\in R$ we get a new slice
$r\cdot S=S^{\prime}$ and a new pair $(I^{\prime},S^{\prime})$ where $I^{\prime}=r\cdot I\cdot r^{-1}$. It is
clear that we have an isomorphism
$$F\simeq R\times^{I}S\simeq R\times^{I^{\prime}}S^{\prime}$$
and under this identification we get a reduction of structure group to
$I^{\prime}$ with the property that the image of the section contains
the given vector $w$.)
Further, by assumption $w\in W-W^{I}$.
Moreover, the $I$-orbit closure of $w$ contains $f\in W^{I}$.
Therefore, if $\overline{w}$ is the image of $w$ in the quotient space
$W/W^{I}$, then clearly $\overline{w}$ is an $I$-unstable vector
in $W/W^{I}$.
Observe also that since $I$ is saturated, by [S2], $W$ is
$I$-cr and hence $W/W^{I}\hookrightarrow W$ obtained as an
$I$-splitting. Note that $E_{I}(W)=E_{R}(W)=E(W)$ is semistable of
degree 0 and since $W/W^{I}$ is an $I$-direct summand of $W$ the
associated bundle $E_{I}(W/W^{I})$ is a direct summand of the degree 0
semistable vector bundle $E_{I}(W)$.
This implies that $E_{I}(W/W^{I})$ is also semistable of degree 0.
Composing the section $s_{1}$ and the $I$-map $W\longrightarrow W/W^{I}$ we have:
$$\overline{s}_{1}:E_{I}\longrightarrow W\longrightarrow W/W^{I}$$
and $\overline{w}\in Im(\overline{s}_{1})$. This gives a non-zero unstable section of $E_{I}(W/W^{I})$ which contradicts the stability
of the bundle $E_{I}$ by Lemma 8.2.
This contradicts the assumption that the orbit $R\cdot w$ is not closed and completes the proof of the Proposition. Q.E.D.
Remark 8.6
$\!\!\!$. The theme in this section fits in with the general theme of
Kempf-Luna in the char.0 case. In char.0 the polystable bundle $E$
comes from an representation of $\pi_{1}(X)\longrightarrow G$. Let $R$ be the Levi
of an admissible parabolic and $E_{R}$ be as in §9. Then $E_{R}$ is
stable. So the representation $\pi_{1}(X)\longrightarrow G$ which factors via $R$
is irreducible . Let $M$ be the Zariski closure of the image. Then the
inclusion $M\hookrightarrow R$ is irreducible in the following
natural sense of [S2] and [S3]: namely, there exists no
parabolic subgroup $P\subset R$ such that $M\hookrightarrow P$.
In this case the proof of Proposition 8.5 now follows easily
by results of Kempf. We need to check that the orbit $R\cdot w$ is
closed. Now $M$ is a reductive subgroup of $R$ which fixes $w$ since
$\pi_{1}(X)$ fixes $w$ (by classical local constancy). If $R\cdot w$ is
not closed then $R$ possesses a non-trivial one-parameter
subgroup and since $M$ fixes $w$ there exists a Kempf parabolic $P$
such that $M\hookrightarrow P\hookrightarrow R$ contradicting
irreducibility of $M\subset R$. (cf. [K, Cor 4.4,4.5])
Remark 8.7
$\!\!\!$. The Proposition 8.5 appears in [RR] but only in
char.0. In [RR] there is an error in the proof of the second
half of their theorem. Here we give a different proof of this and this
works in the situation when the action is separable which in
particular takes care of char.0 as well.
9 Extension to the flat closure
Recall that the section $s_{K}^{\prime}(x)$ extends along the base section
$x_{A}$, to give $s_{A}^{\prime}(x)=w_{A}$. The aim of this section is to prove the
following key theorem.
Theorem 9.1
$\!\!\!$. The section $s_{K}^{\prime}$, extends to a section $s_{A}^{\prime}$ of
$E_{A}(G_{A}/H_{A}^{\prime})$. In other words, the structure group of $E_{A}$ can be
reduced to $H_{A}^{\prime}$; in particular, if $H_{k}^{\prime}$ denotes the closed fibre
of $H_{A}^{\prime}$, then the structure group of $E_{k}$ can be reduced to
$H_{k}^{\prime}$.
9.1 Saturated monodromy groups and Local constancy
Proposition 9.2
$\!\!\!$. Let $E$ be a polystable principal $G$-bundle on
$X$. Let $W$ be a $G$-module of low separable index, $w\in W$ and
$H^{\prime}=Stab(w)$. Let $Y=G/H^{\prime}$ the $G$-subscheme of $W$ defined by the
reduced subgroup $H^{\prime}\subset G$. If $s$ is a section of $E(W)$ such
that for some $x\in X$, the evaluation at $x$, namely $s(x)=w$ is
in $E(Y)_{x}$, then the entire image of $s$ lies in $E(Y)$. In fact we
have a reduction of structure group to a reductive saturated
subgroup $R_{w}$ of $H^{\prime}$ and in particular, the reduced $R_{w}$-bundle is
stable.
Proof. Since the $G$-bundle $E$ is assumed polystable, by Def
3.7, there is an admissible reduction to a parabolic subgroup
$Q\subset G$ and a further reduction of structure group $E_{R}$, to a
Levi subgroup $R\subset Q$ with $E_{R}$ actually stable.
Note that since $R$ is a Levi of a parabolic in $G$, the maximal torus
of $G$ and $R$ are the same.
Further, being a Levi of a parabolic $R$ is a saturated subgroup of
$G$. Since the height of the representation $G\longrightarrow SL(W)$ is low, it
follows that $W$ as an $R$-module is also of low height (cf.
[S2] pp 22).
Thus, we can conclude that $W$ as an $R$-module is also of low
separable index.
Consider the $R$-bundle $E_{R}$ and the $R$-module $W$. We are given a
section $s:X\longrightarrow E(W)=E_{R}(W)$ such that at $x\in X$ $s(x)=w$ is
the given vector in $W$ with $Stab_{G}(w)=H^{\prime}$.
By Proposition 8.5, since $E_{R}$ is stable, the orbit $R\cdot w=F^{cl}$ is a closed orbit. Since the action of $R$ on $W$ is separable
the isotropy, $R_{w}=Stab_{R}(w)$ is reduced, and we have an isomorphism
$R.w\simeq R/R_{w}$. Note further that $R_{w}$ is saturated and
reductive.
As one has observed in the previous proof the section $s$ takes its
values in the fibre $F$ and since $w\in F^{cl}$ we have the
following:
$$s:E_{R}\longrightarrow F^{cl}\simeq R/R_{w}.$$
This gives a reduction of structure group of $E_{R}$ to $R_{w}$. We thus
have the following inclusion of bundles:
$$E_{R_{w}}\hookrightarrow E_{R}\hookrightarrow E$$
Note that $R_{w}=Stab_{R}(w)\subset Stab_{G}(w)=H^{\prime}$. This inclusion
gives the required reduction of structure group of $E$ to $H^{\prime}$ which
indeed comes as an extension of structure group from $E_{R_{w}}$.
Furthermore, $R_{w}$ is saturated and reductive. This complete the proof
of the Proposition. Q.E.D.
9.2 Completion of proof of Theorem 9.1
By Lemma 7.1 ,we have
$$E_{A}(G_{A}/H_{A}^{\prime})\hookrightarrow E_{A}(W_{A}).$$
The given section $s_{K}^{\prime}$ of $E_{K}(G_{K}/H_{K}^{\prime})$ therefore gives a section
$u_{K}$ of $E(W_{K})$. Further, $u_{K}(x)$, the restriction of $u_{K}$ to $x\times T^{*}$, extends to give a section $u_{A}(x)$ of $E_{x}(W_{A})$
(restriction of $E_{A}(W_{A})$ to $x\times T$).Thus, by Proposition
3.13, and by the semistability
of $E_{l}(W_{A})$, the section $u_{K}$ extends to give a section $u_{A}$ of
$E(W_{A})$ over $X\times T$.
Now, to prove the Theorem 9.1 , we need to make sure that:
$$\begin{array}[]{l}\mbox{The image of this extended section $u_{A}$ actually %
lands in
$E_{A}(G_{A}/H_{A}^{\prime})$}\\
\end{array}.$$
(*)( * )
This would then define $s_{A}^{\prime}$.
To prove ($\ast$), it suffices to show that $u_{A}(X\times l)$ lies in
$E_{A}(G_{A}/H_{A}^{\prime})_{l}$ (the restriction of $E_{A}(G_{A}/H_{A}^{\prime})$ to $X\times l$).
Observe that, $u_{A}(x\times l)$ lies in $E_{A}(G_{A}/H_{A}^{\prime})_{l}$ since
$u_{A}(x)=s_{A}^{\prime}(x)=w_{A}$.
Observe further that, if $E_{l}$ denotes the principal $G$-bundle on
$X$, which is the restriction of the $G_{A}$-bundle $E_{A}$ on $X\times T$ to $X\times l$, then $E_{A}(W_{A})_{l}=E_{A}(W_{A})|{X\times l}$, and we
also have
$$\begin{array}[]{ccc}E_{A}(G_{A}/H_{A}^{\prime})_{l}&\stackrel{{\scriptstyle%
\simeq}}{{\longrightarrow}}&E_{l}(G_{k}/H_{k}^{\prime})\\
\Big{\downarrow}&&\Big{\downarrow}\\
E_{A}(W_{A})_{l}&\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}&E_{l}(W)%
\end{array}$$
and the vertical maps are inclusions:
$$E_{A}(G_{A}/H_{A}^{\prime})_{l}\hookrightarrow E_{A}(W_{A})_{l},E_{l}(G_{k}/H_%
{k}^{\prime})\hookrightarrow E_{l}(W)$$
where $E_{l}(W)=E_{l}\times^{H_{k}^{\prime}}W$ with fibre as the $G$-module $W=W_{A}\otimes k$. Note that $G/H_{k}^{\prime}$ is a $G$-subscheme $Y$ of $W$.
Recall that $E_{l}$ is polystable of degree zero. Then, from the
foregoing discussion, the assertion that $u_{A}(X\times l)$ lies in
$E_{A}(G_{A}/H_{A}^{\prime})$, is a consequence of Proposition 9.2 applied to
$E_{l}$. (Note that the group $H_{k}^{\prime}=Stab_{G_{k}}(w_{k})$ satisfies the
hypothesis of Proposition 9.2).
Thus we get a section $s_{A}^{\prime}$ of $E_{A}(G_{A}/H_{A}^{\prime})$ on $X\times T$, which
extends the section $s_{K}^{\prime}$ of $E_{A}(G_{A}/H_{A}^{\prime})$ on $X\times T^{*}$.
This gives a reduction of structure group of the $G_{A}$-bundle $E_{A}$ on
$X\times T$ to the subgroup scheme $H_{A}^{\prime}$ and this extends the given
bundle $E_{K}$ to the subgroup scheme $H_{A}^{\prime}$.
In summary, we have extended the original $H_{K}$-bundle upto
isomorphism to a $H_{A}^{\prime}$-bundle. The extended $H_{A}^{\prime}$-bundle has the
further property that the limiting bundle $E_{l}^{\prime}$ which is an
$H^{\prime}_{k}$-bundle comes with a reduction of structure group to a reductive
and saturated subgroup $R_{w}$ of $H^{\prime}_{k}$. Q.E.D
Remark 9.3
$\!\!\!$. The proof of Theorem 9.1 is not as simple as in the
proof of Proposition 3.13, since
$$E_{A}(G_{A}/H_{A}^{\prime})\hookrightarrow E_{A}(W_{A})$$
is not a closed immersion. The group scheme $H_{A}^{\prime}$ is not
reductive and therefore, we are not given a closed
$G$-embedding of $G_{A}/H_{A}^{\prime}$ in $G_{A}$-module $W_{A}$ (cf. Remark
6.5).
Remark 9.4
$\!\!\!$. The reductive saturated subgroup $R_{w}$ plays the role of
“monodromy” subgroup of the polystable $G$-bundle $E$. (cf.
[BS])
10 Potential good reduction
Recall that by virtue of the separability of the action the group
scheme $H_{A}^{\prime}$ is smooth.
To complete the proof of the Theorem 11.1, we need to extend
the $H_{K}$-bundle to an $H_{A}$-bundle where $H_{A}$ denotes the reductive
group scheme $H\times{\rm Spec}\,A$ over $A$.
Proposition 10.1
$\!\!\!$. There exists a finite extension $L/K$ with the
following property: If $B$ is the integral closure of $A$ in $L$, and
if $H_{B}^{\prime}$ are the pull-back group schemes, then we have a morphism of
$B$-group schemes
$$\chi_{B}:H_{B}^{\prime}~{}\longrightarrow H_{B}$$
which extends the isomorphism $\chi_{L}:H_{L}^{\prime}\cong H_{L}$.
Proof.
Observe first that the lattice $H_{A}^{\prime}(A)$ is a bounded subgroup of $H_{A}(K)$, in the sense of the Bruhat-Tits
theory [BT]. Here, we make the identifications:
$$H_{K}^{\prime}\cong H_{K}~{}\mbox{as K-group schemes}$$
Hence,
$$H_{A}^{\prime}(A)\subset H_{K}^{\prime}(K)\cong H_{K}(K)=H_{A}(K)$$
Then we use the following crucial fact:
$$\left\{\begin{array}[]{l}\mbox{There exists a finite extension $L/K$ and an %
element $g\in H_{A}^{\prime}(L)$ such that}\\
\mbox{$g.H_{A}^{\prime}(A).g^{-1}\hookrightarrow H_{A}(B)$.
}\\
\end{array}\right\}$$
(*)( * )
This assertion is a consequence of the following result from,
([S1] Prop 8, p 546) (cf. also [Gi] Lemma I.1.3.2, or
[La] Lemma 2.4 ).
(Serre) There exists a totally ramified extension $L/K$ having the
following property: For every bounded subgroup $M$ of $H(K)$, there
exists $g\in H(K)$ such that $g.M.g^{-1}$ has good reduction in
$H(L)$ (i.e $h.M.h^{-1}\subset H(B)$, where $B$ is the integral
closure of $A$ in $L$).
For the sake of clarity we gather all the identifications of the
subgroups under consideration:
$$H_{A}^{\prime}(K)=H_{K}^{\prime}(K)~{}\mbox{and}~{}H_{A}^{\prime}(L)=H_{B}^{%
\prime}(L)=H_{L}^{\prime}(L)$$
$$H_{A}^{\prime}(A)\subset H_{B}^{\prime}(B)$$
$$H_{A}(B)=H_{B}(B)$$
Thus, we see that the isomorphism $\chi_{L}:H_{L}^{\prime}\longrightarrow H_{L}$,
given by conjugation by $g$, induces a map $\chi_{L}(B):H_{A}^{\prime}(A)\longrightarrow H_{B}(B)$. The crucial property to note is the
following one:
Given a rational point $\xi_{k}\in H_{k}^{\prime}(k)$, there exists a point
$\xi_{A}\in H_{A}^{\prime}(A)$, and hence in $H_{B}^{\prime}(B)$, which extends $\xi_{k}$,
since $H_{A}^{\prime}$ is smooth over $A$ and $k$ is algebraically closed.
The proposition will follow by the following Lemma. Let $A$, $B$ etc
be as above.
Lemma 10.2
$\!\!\!$. Let $A$ be a complete discrete valuation ring with quotient
field $K$. Let $Z_{A}$ and $Y_{A}$ be $A$-schemes with $Z_{A}$ smooth. Let
$\chi_{L}:Z_{L}\longrightarrow Y_{L}$ be a $L$-morphism such that
$\chi_{L}(B):Z_{A}(A)\longrightarrow Y_{B}(B)$. Then, the $L$-morphism
$\chi_{K}$ extends to a $B$-morphism $\chi_{B}:U_{B}\longrightarrow Y_{B}$,
where $U_{B}$ is an open dense subscheme of $Z_{B}$ which intersects all
the irreducible components of the closed fibre $Z_{k}$.
In particular, if $Z_{A}$ and $Y_{A}$ are smooth and separated group
schemes and if $\chi_{L}$ is a morphism of $L$-group schemes then
there exists an extension $\chi_{B}:Z_{B}\longrightarrow Y_{B}$ as a
morphism of $B$-group schemes.
Proof. Consider the graph of $\chi_{L}$ and denote its schematic
closure in $Z_{B}\times_{B}Y_{B}$ by $\Gamma_{B}$. Let $p:\Gamma_{B}\longrightarrow Z_{B}$ be the first projection. Then $p$ is an
isomorphism on generic fibres. So, it is enough if we prove that $p$
is invertible on an open dense $B$-subscheme $U_{B}$ of $Z_{B}$, which
intersects all the components $C$, of the closed fibre $Z_{k}$.
We claim that, the map $p_{k}:\Gamma_{k}\longrightarrow Z_{k}$ is
surjective onto the subset of $k$-rational points of each components,
and this will imply that $p_{k}$ is surjective since $k$ is
algebraically closed. Note that $Z_{A}$ is assumed to be smooth and
so, the closed fibre is reduced and also $k$ is algebraically
closed. Thus, each $z_{k}\in Z_{k}(k)$ lifts to a point $z\in Z_{A}(A)\subset Z_{B}(B)$, $A$, being a complete discrete valuation ring. Since
$\chi_{L}(B)$ maps $Z_{A}(A)\longrightarrow Y_{B}(B)$, we see that, there exists a $y\in Y_{B}(B)$ such that $(z,y)\in\Gamma_{B}(B)$. Thus, $z_{k}$ lies in
the image of $p_{k}$. This proves the claim.
In particular, by the well-known result of Chevalley on images of
morphisms, the generic points, $\alpha$’s, of all the components $C$
of $Z_{k}$ , lie in the image of $p_{k}$. Let $p_{k}(\xi)=\alpha$. Consider the local rings ${\cal O}_{\Gamma_{B},\xi}$ and
${\cal O}_{Z_{B},\alpha}$. Then by the above claim, the local ring
${\cal O}_{\Gamma_{B},\xi}$ dominates ${\cal O}_{Z_{B},\alpha}$. Since $Z_{B}$ is smooth and hence normal, for every
$\alpha$ the local rings, ${\cal O}_{Z_{B},\alpha}$ are all discrete
valuation ring’s. Further, since $\Gamma_{B}$ is the schematic closure
of $\Gamma_{L}$, it implies that $\Gamma_{B}$ is $B$-flat and $\Gamma_{L}$
is open dense in $\Gamma_{B}$. Moreover, since $p$ is an isomorphism on
generic fibres both local rings have the same quotient
rings. Finally, since ${\cal O}_{Z_{B},\alpha}$ is a discrete
valuation ring, we have an isomorphism of local rings. Therefore
since the schemes are of finite type over $B$, we have open subsets
$V_{i,B}$ and $U_{i,B}$ for each component of $Z_{k}$, which we index
by $i$, such that $p$ induces an isomorphism between $V_{i,B}$ and
$U_{i,B}$. This gives an extension of $\chi$ to open subsets
$U_{i,B}$ for every $i$, with the property that these maps agree on
the generic fibre. Since $Y_{B}$ is separated these extensions glue
together to give an extension $\chi_{B}$ on an open subset, which we
denote by $U_{B}$; this open subset will of course intersect all the
components of the closed fibres of $Z_{k}$.
The second part of the lemma follows immediately, if $Y_{A}$ is affine
(which is our case). More generally, we appeal to the general theorem
of A.Weil on morphisms into group schemes, which says that if a
rational map $\psi_{B}$ is defined in codimension $\leq 1$ and if the
target space is a group scheme then it extends to a global morphism.
(cf. for example [BLR] pp 109). As we have checked above this
holds in our case and this implies that as a morphism of schemes,
$\psi_{L}$ extends to give $\psi_{B}:Z_{B}\longrightarrow Y_{B}$.
Further, by assumption $\chi_{L}$ is already a morphism of $L$-group
schemes and hence it is easy to see that the extension $\chi_{B}$ is
also a morphism of $B$-group schemes. This concludes the proof of the
lemma.
Remark 10.3
$\!\!\!$. Larsen in ([La], (2.7) p 619), concludes from $(\ast)$, in the
$l$-adic case the statement of Proposition 10.1. However, we
give a complete proof.
Remark 10.4
$\!\!\!$. In this section, by the assumption on the
separability index of the affine $(G,H)$-module we were able to conclude
that the flat closure $H_{A}^{\prime}$ is indeed smooth. We observe that since
we are over char.$p$, in general the limiting fibre of the flat
closure $H_{A}^{\prime}$ need not be reduced. This, as one knows is true
in char.0 by virtue of Cartier’s theorem. Indeed more generally, given
a flat group scheme $H_{A}^{\prime}$ with smooth generic fibre $H_{K}^{\prime}$, there is
a construction due to Raynaud of what he calls the Neron-smoothening
of $H_{A}^{\prime}$. This exists as a smooth group scheme $H_{A}^{\prime\prime}$ with generic
fibre $H_{K}^{\prime\prime}\simeq H_{K}^{\prime}$ with the following universal property: given
any smooth $A$-scheme $D_{A}$ and an $A$-morphism $D_{A}\longrightarrow H_{A}^{\prime}$, this
map factors uniquely via an $A$-morphism $D_{A}\longrightarrow H_{A}^{\prime\prime}$. In
particular, $H_{A}^{\prime\prime}(A)=H_{A}^{\prime}(A)$. Thus more generally without any
separability index assumptions, the proof of Proposition 10.1
gives a morphism $H_{B}^{\prime\prime}\longrightarrow H_{B}$. One is unable to make use of this
since the principal bundle $E_{A}^{\prime}$ has structure group $H_{A}^{\prime}$ and there
is no natural reason for its lifting to a principal $H_{A}^{\prime\prime}$-bundle.
(see [BLR]).
11 Semistable reduction theorem
Let H be a semi-simple algebraic group over $k$ an algebraically
closed field of char. p. Let $H\subset G=SL(V)$, be the
representation we have fixed in §1. We retain all the notations of
§7. The aim of this section is to prove the following theorem.
Theorem 11.1
$\!\!\!$. (Semistable reduction) Let $W$ be a finite
dimensional affine $(G,H)$-module associated to $H$ and $G$ and let
$p>\psi_{G}(W)$. Let $H_{K}$ denote the group scheme $H\times{\rm Spec}\,K$, and $P_{K}$ be a semistable $H_{K}$-bundle on $X_{K}$. Then there
exists a finite extension $L/K$, with $B$ as the integral closure of
$A$ in $L$ such that the bundle $P_{K}$, after base change to ${\rm Spec}\,B$,
extends to a semistable $H_{B}$-bundle $P_{B}$ on $X_{B}$.
Proof. First by Proposition 9.1 we have an
$H_{A}^{\prime}$-bundle which extends the $H_{K}$-bundle upto isomorphism. Then,
by Proposition 10.1, by going to the extension $L/K$ we have a
morphism of $B$-group schemes $\chi_{B}:H_{B}^{\prime}\longrightarrow H_{B}$
which is an isomorphism over $L$. Therefore, one can extend the
structure group of the bundle $E_{B}^{\prime}$ to obtain an $H_{B}$-bundle $E_{B}$
which extends the $H_{K}$-bundle $E_{K}$.
We need only prove that the fibre of $E_{B}$ over the closed point is
indeed semistable. This is precisely the content of Proposition
11.3 below. Q.E.D.
Remark 11.2
$\!\!\!$. We remark that this is fairly straightforward in char.0 since it
comes as the extension of structure group of $E_{l}^{\prime}$ by the map $\chi_{k}:H_{k}^{\prime}\longrightarrow H_{k}$. We note that in char.0, $E_{l}^{\prime}$ is the $H_{k}^{\prime}$-bundle
obtained as the reduction of structure group of the polystable vector
bundle $E(V_{A})_{l}$ and so remains semistable by any associated
construction (cf. Proposition 2.6 of [BS]). In our situation
this becomes much more complex and we isolate it in the following
proposition.
Proposition 11.3
$\!\!\!$. The limiting bundle, namely the fibre of $E_{B}$
over the closed point is semistable.
Proof. Recall from Proposition 9.2, that the limiting
bundle of the family $E_{B}^{\prime}$ namely $E_{l}^{\prime}$, had the property that it
had a further reduction of structure group to a reductive and
saturated group $R_{w}$ of $H_{k}^{\prime}$ and hence of $G_{k}=G$. Thus the
representation $R_{w}\longrightarrow G_{k}$ is also low height by ([S2]
pp 25 ). Further, by the low height property, the representation $R_{w}\longrightarrow G=SL(V)$ is completely reducible.
Observe further that since $H_{k}^{\prime}$ is not reductive (cf. Remark
6.5 above), there exists a proper parabolic subgroup $P\subset G_{k}$ such that $H_{k}^{\prime}\subset P$. This follows by the theorem of
Morozov-Borel-Tits (cf. [BoT]). Therefore the subgroup $R_{w}\subset H_{k}^{\prime}\subset P$. Now since $R_{w}\longrightarrow G$ is completely
reducible, $R_{w}\subset P$ implies that $R_{w}\subset L$ for a Levi
subgroup $L\subset P$.
Now $R_{w}$ is a saturated reductive subgroup of $G$. Therefore, since
$p>\psi_{G}(W)$, by Lemma 11.7 (and Remark
11.8) below we see that the modules $LieG_{k}$ and $LieH_{k}^{\prime}$ are low height modules for $R_{w}$ and in particular completely
reducible.
Now $R_{w}$ is a saturated group and the connected component of
identity, $R_{w}^{0}$, is reductive by Proposition 9.2. Since
$R_{w}$ is saturated as a subgroup of $G$ by height considerations, the
modules $LieG_{k}$ and $LieH_{k}^{\prime}$ are low height modules for $R_{w}^{0}$
as well. (cf. Remark 5.6)
Thus by Remark 3.4, we have the following:
$$H^{i}(R_{w}^{0},Lie(H_{k}^{\prime}))=H^{i}(R_{w}^{0},Lie(G_{k}))=0$$
for all $i\geq 1$.
Recall that by Remark 5.6 the saturatedness of $R_{w}$
implies that the index $[R_{w}:R_{w}^{0}]$ is prime to the
characteristic $p$.
Therefore if we denote $R_{w}/R_{w}^{0}$ by $I_{w}$, we see that the order
of $I_{w}$ is prime to $p$. Hence we have the following vanishing of
cohomology:
$$H^{i}(I_{w},Lie(H_{k}^{\prime}))=H^{i}(I_{w},Lie(G_{k}))=0$$
for all $i\geq 1$. (For this classical result cf. [CE]
p. 237.)
Putting together the above results, we can conclude the following:
$$H^{i}(R_{w},Lie(H_{k}^{\prime}))=H^{i}(R_{w},Lie(G_{k}))=0$$
for all $i\geq 1$.
This implies, by the infinitesimal lifting property of ([SGA 3] Exp.III
Cor 2.8) that if we consider the product group scheme $R_{w,B}=R_{w}\times Spec(B)$, then the inclusion
$$i_{k}:R_{w}\hookrightarrow H_{k}^{\prime}\hookrightarrow G_{k}$$
lifts to an inclusion
$$i_{B}:R_{w,B}\hookrightarrow H_{B}^{\prime}\hookrightarrow G_{B}$$
where the generic inclusion is defined upto conjugation by the
inclusion over the residue field.
Denote the above composite by:
$$i_{1,B}:R_{w,B}\hookrightarrow G_{B}$$
By Proposition 10.1, we also have a morphism $\chi_{B}:H_{B}^{\prime}\longrightarrow H_{B}$, which is an isomorphism over the function field $L$. We have the
following diagram:
$R_{w,B}$$H_{B}^{\prime}$$H_{B}$$j_{B}$$\chi_{B}$$i_{B}$
We note that we also have an inclusion $H_{B}\hookrightarrow G_{B}$
coming from the original representation $H\hookrightarrow G$. In other
words we have another morphism
$$j_{1,B}:R_{w,B}\longrightarrow G_{B}$$
Thus, we get the following diagram:
$R_{w,B}$$G_{B}$$G_{B}$$j_{1,B}$$i_{1,B}$
(We remark that there is no vertical arrow to complete the above
diagram!)
Note that over the function field $L$ the maps $j_{1,L}$ and $i_{1,L}$
coincide upto conjugation. Thus by the cohomology vanishing stated
above and the rigidity of maps ([SGA 3] Exp III Cor 2.8), the maps
over the residue fields are also conjugates.
Consider the bundle $E_{l}^{\prime}$ which comes equipped with a reduction to
$R_{w}$ and is semistable as an $R_{w}$-bundle (cf. Prop 9.2).
Since the representations $i_{1,k}:R_{w}\hookrightarrow G_{k}$ and
$j_{1,k}:R_{w}\hookrightarrow G_{k}$ are conjugate it follows that the
associated $G_{k}$-bundles $E_{l}^{\prime}(j_{1,k})$ and $E_{l}^{\prime}(i_{1,k})$ are
isomorphic. Therefore since $E_{l}^{\prime}(i_{1,k})$ is semistable so is
$E_{l}^{\prime}(j_{1,k})$. In particular, since the morphism $j_{1,k}:R_{w}\hookrightarrow G_{k}$ factors via $H_{k}$, the associated $H_{k}$-bundle
$E_{l}^{\prime}(j_{k})$ is semistable. This implies that the induced bundle $E_{B}$
is a family of semistable $H_{B}$-bundles. This completes the proof of
the Theorem 11.1. Q.E.D.
Remark 11.4
$\!\!\!$. Let $H\subset G$, where $G$ is a linear group. In
the notation of §2 let $F_{H}$ and $F_{G}$ stand for the functors
associated to families of semistable bundles of degree zero. (cf.
Proposition 3.13). The inclusion of $H$ in $G$ induces a morphism
of functors $F_{H}\longrightarrow F_{G}$. We remark that the semistable reduction
theorem for principal $H$-bundles need not imply that the
induced morphism $F_{H}\longrightarrow F_{G}$ is a proper morphism of
functors. Indeed, this does not seem to be the case. However, it does
imply that the associated morphism at the level of moduli spaces is
indeed proper (cf. Theorem 4.6).
11.1 Some remarks on low height modules
Lemma 11.5
$\!\!\!$. Let $H\subset G=SL(V)$ be a low height
representation. Let $W$ be a low height $G$-module such that $G/H$ is
embedded as a closed orbit in $W$ (cf. Def 3.12). Suppose
that the subspace $V^{H}\subset V$ of $H$-fixed vector in $V$ is the
zero subspace. Then $W$ contains direct summand different from $V$ and
$V^{*}$. (Note that by the low height assumptions all modules are
completely reducible.)
Proof. For if $W=\oplus V$, then the vector $w\in W$ which
has a closed $G$-orbit and whose isotropy is $H$ projects onto a
vector $v\in V$ fixed by $H$. But by assumption, the subspace $V^{H}=0$. Hence $W$ cannot be a direct sum of copies of $V$. We also
observe that this implies $(V^{*})^{H}=0$ as well and therefore $W$
is not the direct sum of $V^{*}$’s alone.
Lemma 11.6
$\!\!\!$. Let $R\subset G=SL(V)$ be a reductive
saturated subgroup of $G$ that is contained in the Levi of a parabolic
subgroup of $G$. Let $W$ be a low height $G$-module that contains a
component not isomorphic to $V$ and $V^{*}$. Then $Lie(G)$ and $Lie(H)$
are low height $R$-modules, in particular completely reducible.
Proof. Let $n=dim(V)$. Since $W$ contains a component other
than $V$ and $V^{*}$, $ht_{G}(W)\geq 2(n-2)$.
Hence $W$ being a low height $G$-module we have $p>2(n-2)$. Since
$R$ is not irreducible in that $R$ is contained in a certain Levi
subgroup $L\subset P$ of a parabolic subgroup $P\subset G$, it
follows that $ht_{R}(V)<ht_{G}(V)=n-1$.
Hence $ht_{R}(V\otimes V^{*})\leq 2(n-2)<p$. In other words, $V\otimes V^{*}=Lie(G)$ is a low height $R$-module. Note also that
$ht_{R}(Lie(H))\leq ht_{R}(Lie(G))$. Q.E.D.
Lemma 11.7
$\!\!\!$. Let $H\subset G=SL(V)$ be a low height
representation. Let $W$ be a low height $G$-module such that $G/H$ is
embedded as a closed orbit in $W$ (cf. Def 3.12). Let $R\subset G=SL(V)$ be a reductive saturated subgroup of $G$ that is
contained in the Levi of a parabolic subgroup of $G$. Assume that
$V^{H}\subset V^{R}$. Then $Lie(H)$ and $Lie(G)$ are low height
$R$-modules.
Proof. Let $V^{\prime}$ be the subspace complementary to $V^{H}$ in $V$.
Let $n=dim(V)$ and $n^{\prime}=dim(V^{\prime})$. Let $G^{\prime}=SL(V^{\prime})\subset G$.
Then the representation $H\hookrightarrow SL(V)=G$ factors through $H\hookrightarrow SL(V^{\prime})=G^{\prime}$. Moreover $G^{\prime}$ is saturated in $G$ (being the semisimple
part of a Levi subgroup of a parabolic subgroup) and therefore $V$ and
$W$ are low height $G^{\prime}$-modules (by Remark 5.6).
By the choice of $V^{\prime}$, we have $(V^{\prime})^{H}=0$. Since $V^{H}\subset V^{R}$ we see that $R\subset G^{\prime}$. Therefore the $G^{\prime}$-orbit gives a
closed embedding of $G^{\prime}/H$ in $W$. It follows by Lemma
11.5 that $W$ contains summands other than $V^{\prime}$ and
${V^{\prime}}^{*}$.
Hence by Lemma 11.6 $Lie(G^{\prime})$ and $Lie(H)$ are low
height $R$-modules. Now the result follows because $ht_{H}(V)=ht_{H}(V^{\prime})$ and hence $ht_{R}(V)=ht_{R}(V^{\prime})$. This works for the duals as
well, i.e $ht_{R}(V^{*})=ht_{R}({V^{\prime}}^{*})$. By additivity of heights we see
that
$$ht_{R}(V\otimes V^{*})=ht_{R}(V^{\prime}\otimes{V^{\prime}}^{*})<p$$
since $Lie(G^{\prime})$ is a low height $R$-module. Q.E.D.
(cf. [S2] p. 27 for some of the computations made here)
Remark 11.8
$\!\!\!$. We note that the subgroup $R_{w}$ to which we
apply Lemma 11.7 satisfies the condition of the Lemma,
especially the condition that $V^{H}\subset V^{R_{w}}$. This follows
since $H_{A}^{\prime}$ is the flat closure of $H_{K}^{\prime}$ in $G_{A}$ and since
$R_{w}\subset H_{k}^{\prime}$. In fact, for the purposes of Prop 11.3
or the semistable reduction theorem one could have worked with $G^{\prime}=SL(V^{\prime})$ instead of $G$. In that case it is clear that the flat closure
of $H_{K}^{\prime}$ is actually realised in $G^{\prime}_{A}$ itself.
11.2 Irreducibility of the moduli space
We first remark that the semistable reduction theorem Theorem
11.1 holds in fact in a slightly more general setting as well.
Corollary 11.9
$\!\!\!$. Let $\cal X\longrightarrow S$ be a smooth family of curves
parametrised by $S=SpecA$ where $A$ is a complete discrete
valuation ring with char.$K$ = 0 and residue characteristic $p$.
Suppose further that $p>\psi(W)$ as in Theorem 11.1. Let
$H_{S}$ be a reductive group scheme obtained from a split Chevalley
group scheme $H_{\bf Z}$. Suppose further that we are given a family
of semistable principal $H_{K}$-bundles $E_{K}$ over ${\cal X}_{K}$. Then,
there exists a finite cover $S^{\prime}\longrightarrow S$ such that the family after
pull-back to $S^{\prime}$ extends to a semistable family $E_{S^{\prime}}$.
Proof. The proof of Theorem 11.1 goes through with some
minor modifications.
We then have
Corollary 11.10
$\!\!\!$. Let $H$ be simply connected. Then for
$p>\psi_{G}(W)$ the moduli spaces $M(H)$ of principal $H$-bundles is
irreducible.
Proof. The proof of this is now standard once Cor 11.9
is given and one knows the fact over fields of char 0. The argument
very briefly runs as follows: The first point is to observe that given
the prime bounds, namely $p>\psi_{G}(W)$, the moduli scheme can be
constructed as in §4 over $S={\bf Z}-\{p\leq\psi_{G}(W)\}$. Call
this scheme $M(H)_{S}$. Then Cor 11.9 implies that $M(H)_{S}$ is
projective and further, GIT (cf. [Ses1]) implies that the
canonical map $M(H)\longrightarrow M(H)_{S}\otimes k$ is a bijection on $k$-valued
points. Further, since $M(H)_{S}$ is projective and connected over the
generic fibre (by char 0 theory), Zariski’s connectedness theorem
implies that the closed fibre $M(H)_{S}\otimes k$ is also connected and
hence so is $M(H)$. Now observe that the quot scheme $Q^{\prime\prime}$ constructed
in §4 is easily seen to be smooth by some standard deformation
theory. Hence $M(H)$ is normal and connected and therefore irreducible.
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Chennai Mathematical Institute
92, G.N.Chetty Road, Chennai-600017
INDIA
E-mail:
balaji@cmi.ac.in
[2mm]
A.J.Parameswaran
School of Mathematics,
Tata Institute of Fundamental Research.
Homi Bhabha Road,
Mumbai-40005
INDIA
E-mail:
param@math.tifr.res.in |
Multi-Resonator-Assisted Multi-Qubit Resetting in a Network
Xian-Peng Zhang${}^{1,2}$
Li-Tuo Shen${}^{1}$
Zhang-Qi Yin${}^{3}$
Luyan Sun${}^{3}$
Huai-Zhi Wu${}^{1}$
Zhen-Biao Yang${}^{1}$
zbyang@fzu.edu.cn
1.Department of Physics, Fuzhou University, Fuzhou, 350116, P. R. China
2.Department of Physics, National Tsing Hua University, Hsinchu, 300, Taiwan
3.Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, P. R. China
Abstract
We propose a quantum bath engineering method for the initialization of arbitrary number of flux-tunable transmon qubits with a multi-resonator circuit quantum electrodynamics (QED) architecture. Through the application of the microwave drives, we can prepare any number of qubits distributed among the network into arbitrary initial states (on the Bloch sphere). Taking into account the practically experimental parameters, we verify that the initialization process could be achieved in $1\mu s$ with the fidelity in excess of 99%. Moreover, due to the special structure of the circuit network, the initialization efficiency is independent on the number of (ideal, provided) qubits, as only a definite number of bosonic modes are involved in despite of the increasing number of the qubits to be initialized.
The turn of this century has witnessed so many advances in theoretical and experimental frontiers of quantum information processing, including quantum computation CUP2010 ; CUP2007 , simulation LPR2008-2-527 ; NP2012-8-264 ; NP2012-8-267 as well as communication Nature2008-453-1023 ; Science2007-316-1316 , with an outlook that many intellectual breakthroughs and technical innovations are still to be made in the future RMP2013-85-623 ; RMP2013-85-553 ; Science2013-339-1169 . The physical realization of quantum computing generally requires the ability to initialize qubit to arbitrary superposition of computational basis states FP2000-48-771 . Determinately and quickly initializing qubit into any well-defined state provides a convenient avenue for an error-corrected information processor S2011-332-1059 ; N2012-482-382 ; N2008-455-1085 , as well as for quantum memories PRL2010-105-140501 ; PRL2010-105-140502 ; PRB2010-82-024413 .
The traditional route to generate coherence and entanglement relies on minimizing coupling to a dissipative bath. Alternatively, demanding control over the coupling in engineer is replaced by a relatively open environment that allows dissipation to actually assist the generation of coherence and entanglement PRL1996-77-4728 ; PRA2008-78-042307 ; NJP2012-14-055005 . Cavity- or resonator-assisted cooling, which dissipates the kinetic energy in a dissipative environment created by cavity or resonator photon loss, has become eye-catching in artificial atoms APL2010-96-203110 ; PRL2012-109-183602 ; PRL2013-110-120501 ; PRA2015-91-013825 , genuine atoms NL2004-428-50 ; PRL2009-103-103001 , spins PRA2010-82-041804 ; PRL2014-112-050501 , and mechanical objects NL2006-444-71 ; NL2006-444-67 ; NL2006-443-193 . A paradigmatic example is quantum state engineering of a single qubit in a quantum bath, where dissipation may be engineered to relax the system towards any arbitrary states (corresponding to points on the Bloch sphere) PRL2012-109-183602 . Recently, it was demonstrated that cavity-assisted cooling of ensemble spin and artificial atom systems can be implemented on a timescale about microseconds PRL2012-109-183602 ; PRL2014-112-050501 . Resonator-assisted quantum bath engineering may be used to prepare the superconducting flux qubit into any orbital state of the Bloch sphere with a controllable phase factor PRA2015-91-013825 . However, these gratifying schemes all focus on a single qubit. In quantum computation, the operation of logic gates requires direct or indirect coupling between qubits. During the process, one of the most critical challenges is how to effectively initialize designated qubits while with the other qubits unaffected FP2000-48-771 , especially in such an always-connected circuit network Science2013-339-1169 .
We propose here an effective quantum bath engineering method for initializing arbitrary multiple flux-tunable transmon qubits with a multi-resonator circuit quantum electrodynamics (QED) architecture. In this work, independent, rapid and precise control over the internal state of the flux-tunable transmon qubit, allows us to prepare arbitrary qubits into any well-defined states for random initial states. The Markovian master equation describing the hybrid quantum circuits has been solved in the presence of the corresponding (say, to qubits) adjustable microwave drives. Our calculations verified by Monte-Carlo simulation indicate that arbitrary number of qubits are able to be simultaneously prepared into redefined ground states for random initial states with short polarization time in the 0.2-0.8 $\mu$s range for the experimentally feasible sample parameters, which is significantly shorter than the intrinsic energy relaxation time for the superconducting flux qubit in the 6-20 $\mu$s range arXiv:1407.1346v1 ; PRL2014-113-123601 . Our numerical calculations certify that such a method is tolerable of considerable fluctuation of system’s parameters.
The computational basis states of our circuits are realized using the two lowest energy levels, $\left|0\right\rangle$ and $\left|1\right\rangle$, of the flux-tunable transmon qubit PRL2011-107-240501 . As shown in FIG. 1(a), a transmon-type qubit capacitively coupled to two of its adjacent resonators constructs the scalable structural block, N such blocks capacitively coupled to each other form the multi-resonator multi-qubit circuit QED network. Qubits interact through two strongly coupled resonators, served as a Purcell filter, to suppress the off-resonant interactions PRL2008-101-080502 . This multi-resonator multi-qubit circuit QED architecture is feasible for the current experiment technology PRL2015-114-080501 . In the presence of $N$ independently adjustable microwave drives, acting one-to-one on $N$ qubits, oscillation between energy levels $\left|0\right\rangle$ and $\left|1\right\rangle$ of each qubit with regolabile frequency $\omega_{n}$ is induced near resonance. We can write the Hamiltonian for the hybrid quantum circuit within the standard rotating wave approximation (RWA), as well as in the rotating frame $R_{1}=\sum_{m=1}^{2N+2}\varpi_{L}a_{m}^{+}a_{m}+\sum_{n=1}^{N}\varpi_{L}\sigma%
_{z}^{n}/2$, ($\hbar=1$ is assumed)
$$\displaystyle H_{1}$$
$$\displaystyle=$$
$$\displaystyle\sum_{m=1}^{2N+2}\delta\omega a_{m}^{+}a_{m}+\sum_{n=1}^{N+1}\nu a%
_{2n-1}a_{2n}^{+}+\nu a^{+}_{2n-1}a_{2n}$$
(1)
$$\displaystyle+$$
$$\displaystyle\sum_{n=1}^{N}\mathtt{Re}(\Omega_{n})\sigma_{x}^{n}+\mathtt{Im}(%
\Omega_{n})\sigma_{y}^{n}+\delta\varpi_{n}\sigma_{z}^{n}/2$$
$$\displaystyle+$$
$$\displaystyle\sum_{n=1}^{N}g(a_{2n}^{+}+a_{2n+1}^{+})\sigma_{-}^{n}+g(a_{2n}+a%
_{2n+1})\sigma_{+}^{n}$$
with $\delta\omega=\omega_{c}-\varpi_{L}$ and $\delta\varpi_{n}=\omega_{n}-\varpi_{L}$, where $a_{m}$ ($a_{m}^{+}$) is the annihilation (creation) operator of the $m$th resonator with the frequency $\omega_{c}$, $\sigma_{+}^{n}$ ($\sigma_{-}^{n}$) and $\sigma_{j}^{n}$ are the raising (lowering) and $j$-direction ($j=x,y,z$) Pauli operators of the $n$th qubit with the frequency $\varpi_{L}$, $\Omega_{n}$ is the Rabi frequency of the drive acting on the $n$th qubit, $v$ is the filter-filter coupling and $g$ is the qubit-filter interaction.
Our design concept is illustrated in FIG. 1(b)-(c). Firstly, we rotate the Pauli operators of each qubit, as depicted in FIG. 1(b), which makes the low-eigenvalue eigenstate of the rotated Pauli operator to be the target state of the $n$th qubit. Secondly, the photon-loss-assisted driving could stabilize each qubit to its redefined ground state, i.e., the required state, as depicted in FIG. 1(c). For the $n$th qubit, any specified point on the Bloch sphere $\left|-\right\rangle_{n}=\cos(\frac{\theta_{n}}{2})\left|0\right\rangle+e^{i%
\phi_{n}}\sin(\frac{\theta_{n}}{2})\left|1\right\rangle$ with $\theta_{n}\in[0,\pi]$ and $\phi_{n}\in[0,2\pi)$, must be the eigenstate of the Pauli operator after rotation $\sigma_{\mathbf{z}}^{n}=-\sin\theta_{n}\cos\phi_{n}\sigma_{x}+\sin\theta_{n}%
\sin\phi_{n}\sigma_{y}+\cos\theta_{n}\sigma_{z}$ with eigenvalue $-1,$ i.e., ground state. This rotation is able to be realized by the around-$z$-axis rotation with angle $\phi_{n}$ followed by a around-$y$-axis rotation with angle $\theta_{n}$ (As depicted in FIG. 1(b), where the bold subscripts $\mathbf{x},\mathbf{y},\mathbf{z}$ indicate the space basics after rotation). By weakly coupling each qubit to adjacent resonators with capacitance and by introducing a microwave drive that is near resonance with the qubit transition, such a engineered low-temperature quantum bath will drive each flux-tunable transmon qubit to its ground state $\left|-\right\rangle_{n}$.
In analogy to resonator-assisted quantum bath engineering PRA2015-91-013825 , we here introduce a rotating transformation $\mathbf{R}^{n}$ of Pauli operators for each qubit to investigate the polarization efficiency of the arbitrary direction
$$\left[\begin{array}[]{c}\sigma_{\mathbf{x}}^{n}\\
\sigma_{\mathbf{y}}^{n}\\
\sigma_{\mathbf{z}}^{n}\end{array}\right]\texttt{=}\left[\begin{array}[]{ccc}%
\cos\theta_{n}\cos\phi_{n}&\texttt{-}\cos\theta_{n}\sin\phi_{n}&\sin\theta_{n}%
\\
\sin\phi_{n}&\cos\phi_{n}&0\\
\texttt{-}\sin\theta_{n}\cos\phi_{n}&\sin\theta_{n}\sin\phi_{n}&\cos\theta_{n}%
\end{array}\right]\left[\begin{array}[]{c}\sigma_{x}^{n}\\
\sigma_{y}^{n}\\
\sigma_{z}^{n}\end{array}\right],$$
(2)
where the rotation angles $\theta_{n}$ and $\varphi_{n}$ are determined by the Rabi frequency and the detuning of the $n$th drive field with
$$-\frac{\mathtt{Re}(\Omega_{n})}{\sin\theta_{n}\cos\phi_{n}}=\frac{\mathtt{Im}(%
\Omega_{n})}{\sin\theta_{n}\sin\phi_{n}}=\frac{\delta\varpi_{n}}{2\cos\theta_{%
n}}.$$
(3)
We can define this ratio as an effective Rabi frequency
$$\bar{\Omega}\equiv[|\Omega_{n}|^{2}+|\delta\varpi_{n}|^{2}/4]^{1/2},$$
(4)
where we have removed the $n$-dependence of that by adjusting $\Omega_{n}$ and $\delta\varpi_{n}$ for simplicity.
In the rotating frame of $R_{2}=\sum_{n=1}^{N}\bar{\Omega}\sigma_{\mathbf{z}}^{n}+\sum_{m=1}^{2N+2}%
\delta\omega a_{m}^{+}a_{m}+\sum_{n=1}^{N+1}\nu(a_{2n-1}a_{2n}^{+}+a^{+}_{2n-1%
}a_{2n})$, the Hamiltonian generates six modes M$(m,n)(m=\pm 1,n=0,\pm 1)$ with frequencies $\omega_{mn}=\delta\omega+mv+2n\bar{\Omega}$. We here make a brief summation of the functions of these six modes. There is no preference in the $\sigma_{\mathbf{z}}^{n}$ direction for the dynamics of modes M$(m,0)$ at the thermal equilibrium, while those of modes M$(m,\pm 1)$ would drive the qubit to the $\left\langle\sigma_{\mathbf{z}}^{n}\right\rangle=\pm 1$ states, repectively PRL2014-112-050501 ; PRA2015-91-013825 . Therefore, modes M$(m,-1)$ must dominate our polarization process. Here we prefer the mode M$(-1,-1)$ SM . We may set $\Delta=\delta\omega-v-2\bar{\Omega}$ to be close to zero, choose the strong enough filter-filter coupling, and make the effective Rabi frequency satisfy $2\bar{\Omega},2v\gg\Delta$, so that other high-frequency modes M$(m,n)$ separate well from mode M$(-1,-1)$. In the interaction frame of $R_{2}$, the RWA reduces the Hamiltonian into $H_{I}(t)=\sum_{n=1}^{N}H_{n}(t)$ with
$$H_{n}=\sum_{m=2n-1}^{2n+2}\frac{g}{4}(\cos\theta_{n}+1)e^{i(\Delta t+\phi_{n})%
}a_{m}^{+}\sigma_{-}^{n(\mathbf{z})}+H.c..$$
(5)
The RWA is valid in the parameter regime where $2\bar{\Omega},2v$ are large compared to the time scale of interest, which can be dictated by the dissipation rate of the resonator, $\kappa$, and the qubit-filter coupling $g$, i.e., $2\bar{\Omega},2\nu,\gg\kappa,g$.
We now move on to study the polarization efficiency of each qubit dominated by the mode M$(-1,-1)$, i.e., $\Delta\rightarrow 0$, in the Markovian limit. The Markovian process requires that the future process of system depends solely on its present state, which may be validated when the resonator dissipation rate is much larger than the coupling strength between the qubit and resonator in the lowest excitation manifold, i.e., bad resonator condition $(\kappa\gg g)$. The reduced dynamics of flux-tunable transmon qubits in the interaction frame of the dissipator is given to the second order by the TCL master equation (See Appendix B):
$$\dot{\varrho}(t)=-\int_{0}^{t}d\tau e^{-\kappa\tau/2}\mathtt{tr}_{c}[[H_{I}(t)%
,[H_{I}(t-\tau),\varrho(t)\otimes\rho_{eq}]]].$$
(6)
If the condition in Eq. (3) is satisfied, the $nth$ qubit with the target state characterized by the specified point on the Bloch sphere–$(\theta_{n},\phi_{n})$ can be effectively polarized, when the spectrums of the microwave resonator, the filter-fliter coupling and the effective Rabi frequency match, i.e., $\Delta\rightarrow 0$. Then the effective polarization rate of each qubit becomes (See Appendix B):
$$\displaystyle\Gamma_{n}=\frac{(1+\cos\theta_{n})^{2}}{1+4(\Delta/\kappa)^{2}}%
\frac{g^{2}}{\kappa}.$$
(7)
Define $\vec{P}_{\sigma_{\mathbf{z}}^{n}}(t)=[P_{-1}(t),P_{1}(t)]^{T}$, where $P_{\sigma_{\mathbf{z}}^{n}}(t)=\left\langle\sigma_{\mathbf{z}}^{n}\right|%
\varrho(t)\left|\sigma_{\mathbf{z}}^{n}\right\rangle(\sigma_{\mathbf{z}}^{n}=%
\pm 1)$ is the expectation value of the projection operator $\left|\sigma_{\mathbf{z}}^{n}\right\rangle\left\langle\sigma_{\mathbf{z}}^{n}\right|$ at an arbitrary time $t$, the Lindblad master equation will reduce to a rate equation for the state populations of each qubit:
$$\frac{d}{dt}\vec{P}_{\sigma_{\mathbf{z}}^{n}}(t)=\Gamma_{n}\mathbf{M}\vec{P}_{%
\sigma_{\mathbf{z}}^{n}}(t),$$
(8)
with
$$\mathbf{M}=\left[\begin{array}[]{cc}-\bar{n}&\bar{n}+1\\
\bar{n}&-(\bar{n}+1)\end{array}\right].$$
(9)
where $\bar{n}=\mathtt{tr}_{c}[a_{m}^{+}a_{m}\rho_{eq}]$ characterizes the temperature of the bath. The expectation value of the photon number operator at equilibrium is related to the temperature, $T_{c}$, of the bath by
$$\bar{n}=\frac{1}{e^{\omega_{c}/k_{B}T_{c}}-1}$$
(10)
where $k_{B}$ is the Boltzmann constant. The equilibrium state of the flux-tunable transmon qubit satisfies $\partial_{t}\vec{P}_{\sigma_{\mathbf{z}}^{n}}(t)=0$ and the expectation value of the operator $\sigma_{\mathbf{z}}^{n}$ for the equilibrium state is
$$\left\langle\sigma_{\mathbf{z}}^{n}\right\rangle_{eq}=\frac{e^{-\omega_{c}/k_{%
B}T_{c}}-1}{e^{-\omega_{c}/k_{B}T_{c}}+1}.$$
(11)
In the ideal case where all resonators are all cooled to their ground states, i.e., vacuum state ($T_{c}\rightarrow 0$), the final expectation
value is approximately $\left\langle\sigma_{\mathbf{z}}^{n}\right\rangle_{eq}\simeq-1$.
The separability of rate equation (8) for each qubit makes it easy for us to simulate the gratifying results. For the $nth$ qubit initially taken to be maximally mixed in the basis, i.e., $P_{\sigma^{n}_{\mathbf{z}}}(0)=1/2$ $(\sigma^{n}_{\mathbf{z}}=\pm 1)$, the evolutions of the simulated expectation values $\left\langle\sigma^{n}_{\textbf{z}}\right\rangle$ for different temperatures $T_{c}=0.0,0.3,0.4,0.5$K are shown in FIG. 2 with equilibrium expectation values $-1.000,-0.995,-0.979,-0.948$, respectively. The evolution of the simulated expectation value $\left\langle\sigma^{m}_{\mathbf{z}}\right\rangle$ for temperature $T_{c}\leq 0.3$K (within which the thermodynamics effect may not be observed arXiv:1407.1346v1 ; PRL2014-113-123601 ) may be fitted to an exponential function to derive an effective polarization time constant, $T_{n}$. A fit to a model given by $\left\langle\sigma_{\mathbf{z}}^{n}\right\rangle=\mathtt{exp}(-t/T_{n})-1$ yields the parameter
$$T_{n}\simeq\frac{1}{\Gamma_{n}}=\frac{1+4(\Delta/\kappa)^{2}}{(1+\cos\theta_{n%
})^{2}}\frac{\kappa}{g^{2}}.$$
(12)
We find that the resetting efficiency is independent on the number of qubits. Inset of FIG. 2 depicts the effective dissipation rate versus the dimensionless parameters $\Delta/\kappa$ and $\theta_{n}$ in the units of $g^{2}/\kappa$. Apparently, the polarization time increases rapidly, when the Stokes photons are off-resonant with the resonator and the condition $\Delta\gg\kappa$ is satisfied. And the most efficient polarization happens in $z$ direction with the effective dissipation rate $\Gamma_{n}=4g^{2}/\kappa$ ($\theta_{n}=0,\Delta=0$).
Here we consider the experimentally feasible parameter regimes $v/2\pi=100$ MHz, $\bar{\Omega}/2\pi=100$ MHz, $\omega_{L}/2\pi=5.7$ GHz and $\omega_{c}/2\pi=6$ GHz. If we want to initialize the $n$th qubit into the state $\left|-\right\rangle_{n}=\cos(\frac{\theta_{n}}{2})\left|0\right\rangle+e^{i%
\phi_{n}}\sin(\frac{\theta_{n}}{2})\left|1\right\rangle$, the qubit frequency should be set to be $\omega_{n}=\omega_{L}+2\bar{\Omega}\cos\theta_{n}$, and the microwave drive acting on this qubit should be adjusted with $\left[\mathtt{Re}(\Omega_{n}),\mathtt{Im}(\Omega_{n}),\delta\varpi_{n}/2\right%
]=\left[-\bar{\Omega}\sin\theta_{n}\cos\phi_{n},\bar{\Omega}\sin\theta_{n}\sin%
\phi_{n},\bar{\Omega}\cos\theta_{n}\right]$. In such a case, the polarization time, $T_{n}$, of each qubit is of the range of $0.2-0.8\mu$s in the parameter regime $(g,\kappa)/2\pi=(2,20)$ MHz, which is significantly shorter than the intrinsic energy relaxation time for the superconducting flux qubit in the 6-20$\mu$s range arXiv:1407.1346v1 ; PRL2014-113-123601 .
Assume that the hybrid quantum circuit has three qubits ($N=3$) and should be initialized into $\left[\left\langle\sigma_{x}^{1}\right\rangle,\left\langle\sigma_{y}^{2}\right%
\rangle,\left\langle\sigma_{z}^{3}\right\rangle\right]=[-1,-1,-1]$ state. For the initial state $\left[\left\langle\sigma_{y}^{1}\right\rangle,\left\langle\sigma_{z}^{2}\right%
\rangle,\left\langle\sigma_{x}^{3}\right\rangle\right]=[1,1,1]$, a Monte Carlo method is used to simulate the Lindblad master equation of the interaction Hamiltonian $H_{1}$, based on Quantum Toolbox in Python CPC2012-183-1760 ; CPC2013-184-1234 . FIG. 3(a) describes the evolutions of the expectation values $\left\langle\sigma^{n}_{\mathbf{z}}\right\rangle$ of the qubits in the parameter regime $(g,\kappa)/2\pi=(2,20)$ MHz. Apparently, it is consistent with the inset of FIG. 3(a), which shows the evolutions of the expectation values $\left\langle\sigma^{n}_{\mathbf{z}}\right\rangle$ of the qubits for the rate equation (8). As shown in FIG. 3(a), all qubits are able to reach a quite high reliability with the final expectation values $\left[\left\langle\sigma_{x}^{1}\right\rangle,\left\langle\sigma_{y}^{2}\right%
\rangle,\left\langle\sigma_{z}^{3}\right\rangle\right]\approx[-0.9998,-0.9997,%
-1.0000]$. We can obviously observe that the speed of the simulated expectation value $\left\langle\sigma^{3}_{z}\right\rangle$ (approaching $-1$) is four times as fast as those of the simulated expectation values $\left\langle\sigma^{1}_{x}\right\rangle$ and $\left\langle\sigma^{2}_{y}\right\rangle$, which validates the analytical derivation of the effective $\theta_{n}$-dependent dissipation rate (7) with $\Gamma_{3}=4\Gamma_{1}=4\Gamma_{2}$.
The influences of the fluctuations of parameters $\Omega_{n}$ and $\delta\varpi_{n}$ have been previously studied and are normally small enough to be neglected PRA2015-91-013825 . Simulation results show that the considerable fluctuations of parameters $\kappa$ and $g$ are also allowed. The polarization efficiency can be improved when we make an optimization of the parameters $(g,\kappa)/2\pi=(15,10)$ MHz, as shown in FIG. 3(b), where all transmon-type qubits are almost completely driven into their target states after 0.32$\mu$s polarization process with $\left[\left\langle\sigma_{x}^{1}\right\rangle,\left\langle\sigma_{y}^{2}\right%
\rangle,\left\langle\sigma_{z}^{3}\right\rangle\right]\approx\left[-0.998,-0.9%
98,-1.000\right]$. This significant improvement over previous works opens the way to multi-resonator mutli-qubit network quantum protocols across a range of quantum algorithm, photonic memory, and nascent quantum simulation.
Having included the qubit dissipation, spontaneous emission and dephasing will make a disturbance on the equilibrium state. The hybrid quantum system and its dissipative environment can be described by the Lindblad master equation
$$\displaystyle\frac{d}{dt}\rho(t)$$
$$\displaystyle=$$
$$\displaystyle-i[H_{1},\rho])+\sum_{m=1}^{2N+2}\kappa D[a_{m}]\rho(t)$$
$$\displaystyle+$$
$$\displaystyle\sum^{N}_{n=1}\frac{1}{T_{\theta}}D[\sigma^{n(z)}_{-}]\rho(t)+%
\frac{1}{2T_{\phi}}D[\sigma^{n(z)}_{z}]\rho(t)$$
where $1/T_{\theta}$ is spontaneous emission rate, $1/T_{\phi}$ is phase relaxation rate and $D[A]\rho=A\rho A^{+}-(1/2)\{A^{+}A,\rho\}$. The evolutions of the expectation values $\left\langle\sigma^{n}_{\mathbf{j}}\right\rangle$, with the experimentally available parameters $(T_{\theta},T_{\phi})=(20,10)\mu$s PRL2014-113-123601 , are simulated for the master equation in Eq. (Multi-Resonator-Assisted Multi-Qubit Resetting in a Network). We note that a quite high reliability can be obtained with the final expectation values $\left[\left\langle\sigma_{x}^{1}\right\rangle,\left\langle\sigma_{y}^{2}\right%
\rangle,\left\langle\sigma_{z}^{3}\right\rangle\right]\approx\left[-0.995,-0.9%
95,-1.000\right]$, as shown in FIG. 3(c). Consequently, we conclude that the scheme is
in principle feasible with the presently experimental sample parameters.
In conclusion, we have demonstrated a quantum bath engineering method for the initialization of arbitrary number of flux-tunable transmon qubits with a multi-resonator
circuit QED architecture. Precise, rapid and independent control over the internal states of transmon-type qubits, allows us to achieve flexible resetting for any designated qubits in an alway-connected circuit network.
This work is supported by the National Natural Science Foundation
of China under Grants No. 11405031, No. 11305037, and No. 11374054, the Natural Science Foundation
of Fujian Province under Grant No. 2014J05005, and the fund from Fuzhou University. Z.-Q.Y. is supported by the National Natural Science Foundation of China under Grants No. 61435007 and No. 11474177. L.-Y.S. is supported by the National Natural Science Foundation of China under
Grant No. 11474177 and the 1000 Youth Fellowship program in China.
Appendix: Derivation of Markovian Master Equation and Analysis of
Approximations
.1 System Hamiltonian
We here give a derivation of the interaction Hamiltonian with the whole system. Assuming the microwave drive acting on each qubit to be nearly resonant with the corresponding qubit frequency $\omega_{n}$ $(n=1\sim N)$, the composite Hamiltonian of the multi-resonator multi-qubit network for the superconducting circuit is described by $H=H_{0}+H_{r}+H_{d}+H_{h}$ with
$$\displaystyle H_{0}$$
$$\displaystyle=$$
$$\displaystyle\sum^{2N+2}_{m=1}\omega_{c}a_{m}^{+}a_{m}+\sum^{N}_{n=1}\frac{%
\omega_{n}}{2}\sigma^{n}_{z},$$
(.1.1)
$$\displaystyle H_{r}$$
$$\displaystyle=$$
$$\displaystyle\sum^{N}_{n=1}ga^{\texttt{ }}_{2n}\sigma^{n}_{x}+ga^{\texttt{ }}_%
{2n+1}\sigma^{n}_{x}+H.c.,$$
(.1.2)
$$\displaystyle H_{d}$$
$$\displaystyle=$$
$$\displaystyle\sum^{N}_{n=1}\Omega_{n}\sigma^{n}_{-}e^{i\omega_{L}t}+\tilde{%
\Omega}\sigma^{n}_{-}e^{-i\omega_{L}t}+H.c.,$$
(.1.3)
$$\displaystyle H_{h}$$
$$\displaystyle=$$
$$\displaystyle\sum_{n=1}^{N+1}\nu a_{2n-1}a_{2n}^{+}+\nu a^{+}_{2n-1}a_{2n},$$
(.1.4)
where $a_{m}$ ($a_{m}^{+}$) is the annihilation (creation) operator of the $mth$ resonator with the frequency $\omega_{c}$, $\Omega_{n}$ and $\tilde{\Omega}_{n}$ are the Rabi and the counter-rotating Rabi frequencies of the drive acting on the $nth$ qubit, $\sigma_{+}$ ($\sigma_{-}$) and $\sigma_{j}$ ($j=x,y,z$) are the raising (lowering) and $j$-direction ($j=x,y,z$) Pauli operators of the $n$th qubit with the frequency $\omega_{L}$, $v$ is the filter-filter coupling and $g$ is the qubit-filter coupling. $\hbar=1$ is assumed.
After moving into a rotating frame defined by $R_{1}=\sum_{m=1}^{2N+2}\varpi_{L}a_{m}^{+}a_{m}+\sum_{n=1}^{N}\varpi_{L}\sigma%
_{z}^{n}/2$, the system Hamiltonian is transformed to
$$\displaystyle H_{1}$$
$$\displaystyle=$$
$$\displaystyle\sum_{m=1}^{2N+2}\delta\omega a_{m}^{+}a_{m}+\sum_{n=1}^{N+1}\nu a%
_{2n-1}a_{2n}^{+}+\nu a^{+}_{2n-1}a_{2n}$$
$$\displaystyle+$$
$$\displaystyle\sum_{n=1}^{N}\mathtt{Re}(\Omega_{n})\sigma_{x}^{n}+\mathtt{Im}(%
\Omega_{n})\sigma_{y}^{n}+\delta\varpi_{n}\sigma_{z}^{n}/2$$
$$\displaystyle+$$
$$\displaystyle\sum_{n=1}^{N}g(a_{2n}^{+}+a_{2n+1}^{+})\sigma_{-}^{n}+g(a_{2n}+a%
_{2n+1})\sigma_{+}^{n}$$
with $\delta\omega=\omega_{c}-\omega_{L}$ and $\delta\varpi_{n}=\omega_{n}-\omega_{L}$. Here we have made the standard rotating wave approximation (RWA) to remove any time-dependent terms in the Hamiltonian. This RWA is valid when the resonant frequencies of the resonators, the drive, and the qubit transition, $\omega_{c},\omega_{L}$ and $\omega_{n}$, are larger than the inverse time scale of interest. In our case this time scale will be dictated by the dissipation rate of the cavity, $\kappa$, and the Hamiltonian frequencies $g$, $\mu$, $\Omega_{n}$ and $\tilde{\Omega}_{n}$. Hence we require $\omega_{c},\omega_{L},\omega_{n}\gg g,\kappa,\nu,\Omega_{n},\tilde{\Omega}_{n}$.
In analogy to resonator-assisted quantum bath engineering PRA2015-91-013825 , we introduce a rotating transformation $\mathbf{R}^{n}$ of Pauli operators of each qubit to investigate the polarization efficiency of the arbitrary direction
$$\left[\begin{array}[]{c}\sigma_{\mathbf{x}}^{n}\\
\sigma_{\mathbf{y}}^{n}\\
\sigma_{\mathbf{z}}^{n}\end{array}\right]\texttt{=}\left[\begin{array}[]{ccc}%
\cos\theta_{n}\cos\phi_{n}&\texttt{-}\cos\theta_{n}\sin\phi_{n}&\sin\theta_{n}%
\\
\sin\phi_{n}&\cos\phi_{n}&0\\
\texttt{-}\sin\theta_{n}\cos\phi_{n}&\sin\theta_{n}\sin\phi_{n}&\cos\theta_{n}%
\end{array}\right]\left[\begin{array}[]{c}\sigma_{x}^{n}\\
\sigma_{y}^{n}\\
\sigma_{z}^{n}\end{array}\right]$$
(.1.6)
with
$$-\frac{\mathtt{Re}(\Omega_{n})}{\sin\theta_{n}\cos\phi_{n}}=\frac{\mathtt{Im}(%
\Omega_{n})}{\sin\theta_{n}\sin\phi_{n}}=\frac{\delta\varpi_{n}}{2\cos\theta_{%
n}}.$$
(.1.7)
We can define this ratio as an effective Rabi frequency for each qubit
$$\bar{\Omega}\equiv[|\Omega_{n}|^{2}+|\delta\varpi|^{2}/4]^{1/2},$$
(.1.8)
where we have removed the $n$-dependence of that by adjusting $\Omega_{n}$ and $\delta\varpi_{n}$ for simplicity, which is an easy operation in experiment. If we now move into the interaction frame of
$R_{2}=H_{h}+\sum_{n=1}^{N}\bar{\Omega}\sigma_{\mathbf{z}}^{n}+\sum_{m=1}^{2N}%
\delta\omega a_{m}^{+}a_{m}$, the total Hamiltonian reduces to $H_{2}(t)=\sum_{n}H_{n}(t)$ with
$$\displaystyle H_{n}(t)$$
$$\displaystyle=$$
$$\displaystyle ge^{itR_{2}}\left(a_{2n}^{+}+a_{2n+1}^{+}\right)\sigma_{-}^{n}e^%
{-itR_{2}}+H.c.$$
(.1.9)
$$\displaystyle=$$
$$\displaystyle\left[e^{it\bar{\Omega}\sigma_{\mathbf{z}}^{n}}(\Theta_{\mathbf{x%
}}^{n}\sigma_{\mathbf{x}}^{n}+\Theta_{\mathbf{y}}^{n}\sigma_{\mathbf{y}}^{n}+%
\Theta_{\mathbf{z}}^{n}\sigma_{\mathbf{z}}^{n})e^{-it\bar{\Omega}\sigma_{%
\mathbf{z}}^{n}}\right]$$
$$\displaystyle\times$$
$$\displaystyle g\left[e^{itR_{2}}(a_{2n}^{+}+a_{2n+1}^{+})e^{-itR_{2}}\right]+H%
.c.,$$
where
$$\Theta_{\mathbf{x}}^{n}=\frac{1}{2}\cos\theta_{n}e^{i\phi_{n}},\Theta_{\mathbf%
{y}}^{n}=-\frac{i}{2}e^{i\phi_{n}},\Theta_{\mathbf{z}}^{n}=\frac{1}{2}\sin%
\theta_{n}e^{i\phi_{n}}.$$
(.1.10)
Using the Baker-Campbell-Hausdorf expansion, we obtain
$$e^{it\bar{\Omega}\sigma_{\mathbf{z}}^{n}}\sigma_{\mathbf{x}}^{n}e^{-it\bar{%
\Omega}\sigma_{\mathbf{z}}^{n}}=e^{2it\bar{\Omega}}\sigma_{+}^{n(\mathbf{z})}+%
e^{-2it\bar{\Omega}}\sigma_{-}^{n(\mathbf{z})},$$
(.1.11)
$$e^{it\bar{\Omega}\sigma_{\mathbf{z}}^{n}}\sigma_{\mathbf{y}}^{n}e^{-it\bar{%
\Omega}\sigma_{\mathbf{z}}^{n}}=i(e^{-2it\bar{\Omega}}\sigma_{-}^{n(\mathbf{z}%
)}-e^{2it\bar{\Omega}}\sigma_{+}^{n(\mathbf{z})}),$$
(.1.12)
$$\displaystyle e^{itR_{2}}a_{2n}^{+}e^{-itR_{2}}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}\left[e^{i\left(\delta\omega+v\right)t}a_{2n}^{+}+e^{i%
\left(\delta\omega-v\right)t}a_{2n}^{+}\right.$$
$$\displaystyle+$$
$$\displaystyle\left.e^{i\left(\delta\omega+v\right)t}a_{2n+1}^{+}-e^{i\left(%
\delta\omega-v\right)t}a_{2n+1}^{+}\right],$$
$$\displaystyle e^{itR_{2}}a_{2n+1}^{+}e^{-itR_{2}}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}\left[e^{i\left(\delta\omega+v\right)t}a_{2n}^{+}-e^{i%
\left(\delta\omega-v\right)t}a_{2n}^{+}\right.$$
$$\displaystyle+$$
$$\displaystyle\left.e^{i\left(\delta\omega+v\right)t}a_{2n+1}^{+}+e^{i\left(%
\delta\omega-v\right)t}a_{2n+1}^{+}\right],$$
where $\sigma_{\pm}^{n(\mathbf{z})}=(\sigma_{\mathbf{x}}^{n}\pm i\sigma_{\mathbf{y}}^%
{n})/2$ are the ladder operators in the $\mathbf{z}$-basis. Here we emphasize the special structure of the multi-resonator multi-qubit circuit QED architecture. As shown in Eq.s (.1,.1), RWA corresponding to resonators only generates two modes ($\delta\omega\pm v$), which greatly reduces the complexity of analytical derivation of the scheme, and efficiently improve the polarization. Let us make comparisons with other three kinds of network structures. The most general multi-cavity and multi-qubit system is the one at which each qubit is coupled to one cavity, which is directly coupled to its neighboring ones. In this case, the second RWA will generate total $3N$ modes, the resetting efficiency will be greatly reduced, because only small proportion of modes are matched. In comparison with the case where qubits interact through $n(n>2)$ resonators, the second RWA will generate total $3n$ modes. The resetting processes is also influenced by the same reason. The simpler network, where qubits interact through only one resonator is experimentally undesirable due to the off-resonant interaction.
Substituting Eqs. (.1.11-.1) into Hamiltonian (.1.9), we can obtain $H_{n}(t)=\sum_{l=\pm 1}\sum_{k=0,\pm 1}H^{n}_{lk}(t)$ with
$$H^{n}_{lk}(t)=\left\{\begin{array}[]{ll}\sum_{m}A_{mn}^{l}g\Theta_{-}^{n}e^{i%
\omega_{lk}t}a_{m}^{+}\sigma_{-}^{n(\textbf{z})}+H.c.,&\hbox{k=$-$1;}\\
\sum_{m}A_{mn}^{l}g\Theta_{\textbf{z}}^{n}e^{i\omega_{lk}t}a_{m}^{+}\sigma_{z}%
^{n(\textbf{z})}+H.c.,&\hbox{k=0;}\\
\sum_{m}A_{mn}^{l}g\Theta_{+}^{n}e^{i\omega_{lk}t}a_{m}^{+}\sigma_{+}^{n(%
\textbf{z})}+H.c.,&\hbox{k=+1.}\end{array}\right.$$
(.1.15)
where the rotating frame of $R_{2}$ makes the Hamiltonian generate six modes M$(l,k)(l=\pm 1,k=0,\pm 1)$ with frequencies $\omega_{lk}=\delta\omega+lv+2k\bar{\Omega}$, coefficients $A_{mn}^{l}$ are given in FIG. 4(a)-(b) for the case $N=3$, and ($\theta_{n},\phi_{n}$)-dependent coefficients are given by $\Theta_{\pm}^{n}=e^{i\phi_{n}}(\cos\theta_{n}\mp 1)/2$ for the $n$th qubit.
There is no preference in the $\sigma_{\mathbf{z}}^{n}$ direction for the dynamics of modes M$(\pm,0)$ at the thermal equilibrium, while those of modes M$(l,\pm 1)$ will drive the qubit to the $\left\langle\sigma_{\mathbf{z}}^{n}\right\rangle=\pm 1$ states, repectively PRL2014-112-050501 ; PRA2015-91-013825 . Therefore, resonator-assisted cooling requires that we should remove the dynamics of M$(l,+1)$ with appropriate RWA. The ($\theta_{n},\phi_{n}$)-dependent coefficients $\Theta_{+}^{n}$ indicate that this RWA has more warrant for an around-$z$-axis polarization, where coefficients $\Theta_{+}^{n}$ approaches zero and the dynamics of M$(l,+1)$ disappears.
We may set $\Delta_{l}=\delta\omega+lv-2\bar{\Omega}$ to be close to zero, and choose the appropriate filter-filter coupling and the effective Rabi frequency, so that other high-frequency modes M$(l^{\prime},n)(l^{\prime}\neq l$ or $n\neq-1)$ well separate from mode M$(l,-1)$. It happens when $2\bar{\Omega}$ and $v$ are big enough, i.e., $2\bar{\Omega},v\gg\Delta_{l}$. In the interaction frame of $R_{2}$, RWA reduces the Hamiltonian into $H^{l}_{I}=\sum_{n=1}^{N}\sum_{m=1}^{2N+2}H_{mn}^{l}(t)$ with
$$H_{mn}^{l}(t)=\Theta_{mn}^{l}a_{m}^{+}\sigma_{-}^{ln}(t)+\Theta_{mn}^{l\ast}a_%
{m}\sigma_{+}^{ln}(t),$$
(.1.16)
where
$$\displaystyle\Theta_{mn}^{\pm}=gA_{mn}^{l}\Theta_{-}^{n},\sigma_{\pm}^{ln}(t)=%
\sigma_{\pm}^{n(\textbf{z})}e^{\mp i\Delta_{l}t}.$$
(.1.17)
This RWA is valid in the parameter regime where $v,2\bar{\Omega}$ are large compared to the time scale of interest, which can be dictated by the dissipation rate of the resonator, $\kappa$, and the qubit-filter coupling $g$, i.e., $2\bar{\Omega},\nu,\gg\kappa,g$.
.2 Derivation of Markovian master equation
In order to model the ground state cooling of the multi-resonator multi-qubit network for the superconducting circuit, we use an open quantum system description of the resonators and qubits. The dynamics of the joint system can be modeled by the TCL master equation formalism OUP2002 , which allows us to derive an effective dissipator acting on each qubit. The evolution of the hybrid quantum system is described by the Lindblad master equation
$$\frac{d}{dt}\rho(t)=L[H_{I}^{l}(t)]\rho(t)+D_{c}\rho(t),$$
(.2.1)
where index $l=\pm 1$ corresponds to system dominated by mode M$(l,-1)$, $L$ is the superoperator $L[H_{I}^{l}(t)]\rho(t)=-i[H_{I}^{l}(t),\rho]$ describing the evolution under the interaction Hamiltonian $H_{I}^{l}$, and $D_{c}$ is a dissipator describing the quality factor of the filters phenomenologically as a photon amplitude damping channel SV1974
$$D_{c}=\sum_{m=1}^{2N+2}\frac{\kappa}{2}((1+\bar{n}_{m})D[a_{m}]+\bar{n}_{m}D[a%
_{m}^{+}]),$$
(.2.2)
where $D[A]\rho=2A\rho A^{+}-\{A^{+}A,\rho\}$, $\bar{n}_{m}=\mathtt{tr}_{c}[a_{m}^{+}a_{m}\rho_{eq}]$ characterizes the temperature of the bath. The expectation value of the photon number operator at equilibrium is related to the temperature, $T_{c}$, of the bath by
$$\bar{n}_{m}=\frac{1}{e^{\omega_{c}/k_{B}T_{c}}-1}$$
(.2.3)
where $k_{B}$ is the Boltzmann constant.
We now move into the interaction frame defined by the dissipator $D_{c}$. The interaction superoperators in this frame are given by $\tilde{S}(t)=e^{-tD_{c}}S(t)e^{tD_{c}}$, while we have $\tilde{\rho}(t)=e^{-tD_{c}}\rho(t)$ for the density operator. Then the master equation in the dissipator interaction frame (.2.2) becomes
$$\displaystyle\frac{d}{dt}\tilde{\rho}(t)=\tilde{L}[H^{l}_{I}(t)]\tilde{\rho}(t).$$
(.2.4)
We define a projection operator $\hat{P}$ onto the relevant degrees of freedom for the reduced system
$$\hat{P}\rho(t)=\varrho(t)\otimes\rho_{eq},$$
(.2.5)
where $\varrho(t)=\mathtt{tr}_{c}[\rho(t)]$ is the reduced state of $N$ qubits and $\rho_{eq}$ is the equilibrium state of the cavity under the dissipation $D_{c}$ ($D_{c}\rho_{eq}=0$). In the case of weak coupling, the second order TCL master equation is given by OUP2002
$$\frac{d}{dt}\hat{P}\tilde{\rho}(t)=\int_{0}^{t}d\tau\hat{P}\tilde{L}[H^{l}_{I}%
(t)]\tilde{L}[H^{l}_{I}(t-\tau)]\hat{P}\tilde{\rho}(t).$$
(.2.6)
We now explicitly consider the interaction frame of the dissipator. To do this we use the definition of the adjoint channel $D^{+}_{c}$, which satisfies tr${}_{c}[D_{c}^{+}[A]B]=$tr${}_{c}[AD_{c}[B]]$ for all operators A, B on the resonator system. The adjoint channel has the following useful properties:
$$\displaystyle D_{c}^{+}[\mathbf{I}]$$
$$\displaystyle=$$
$$\displaystyle\mathbf{0},e^{tD_{c}^{+}}[\mathbf{I}]=\mathbf{1},$$
(.2.7)
$$\displaystyle D_{c}^{+}[a_{m}]$$
$$\displaystyle=$$
$$\displaystyle-\frac{\kappa}{2}a_{m},e^{tD_{c}^{+}}[a_{m}]=e^{-\kappa t/2}a_{m}$$
(.2.8)
$$\displaystyle D_{c}^{+}[a_{m}^{+}]$$
$$\displaystyle=$$
$$\displaystyle-\frac{\kappa}{2}a_{m}^{+},e^{tD_{c}^{+}}[a_{m}^{+}]=e^{-\kappa t%
/2}a_{m}^{+}.$$
(.2.9)
Hence we have
$$\displaystyle\hat{P}\tilde{\rho}(t)$$
$$\displaystyle=$$
$$\displaystyle\mathtt{tr}_{c}[e^{-tD_{c}}\rho(t)]\otimes\rho_{eq}$$
(.2.10)
$$\displaystyle=$$
$$\displaystyle\mathtt{tr}_{c}[e^{-tD^{+}_{c}}[I]\rho(t)]\otimes\rho_{eq}$$
$$\displaystyle=$$
$$\displaystyle\hat{P}\rho(t).$$
In addition, with $D_{c}P\rho(t)=\mathtt{tr}_{c}[\rho(t)]\otimes D_{c}\rho_{eq}=0$, the reduced dynamics of the composite system in the interaction frame of the dissipator (.2.2) is given to the 2nd order by the TCL master equation PRL2014-112-050501
$$\displaystyle\dot{\varrho}(t)$$
$$\displaystyle=$$
$$\displaystyle\int_{0}^{t}d\tau\mathtt{tr}_{c}[L[H^{l}_{I}(t)]e^{\tau D_{c}}L[H%
^{l}_{I}(t\texttt{-}\tau)]\varrho(t)\otimes\rho_{eq}]$$
$$\displaystyle=$$
$$\displaystyle\int_{0}^{t}d\tau\mathtt{tr}_{c}[e^{\tau D^{+}_{c}}(L[H^{l}_{I}(t%
)])L[H^{l}_{I}(t\texttt{-}\tau)]\varrho(t)\otimes\rho_{eq}]$$
$$\displaystyle=$$
$$\displaystyle\int_{0}^{t}d\tau e^{-\kappa\tau/2}\mathtt{tr}_{c}[L[H^{l}_{I}(t)%
]L[H^{l}_{I}(t\texttt{-}\tau)]\varrho(t)\otimes\rho_{eq}]$$
$$\displaystyle=$$
$$\displaystyle\texttt{-}\int_{0}^{t}d\tau e^{-\kappa\tau/2}\mathtt{tr}_{c}[[H^{%
l}_{I}(t),[H^{l}_{I}(t\texttt{-}\tau),\varrho(t)\otimes\rho_{eq}]]].$$
Starting with the 2nd order TCL master equation (.2), we now expand this in terms of the component Hamiltonian $H_{\imath n}^{l}(t)$.
Defining
$$F^{l}_{\vec{n}\vec{\imath}}(t,s)=\mathtt{tr}_{c}[[H_{\imath n}^{l}(t),H_{%
\imath^{\prime}n^{\prime}}^{l}(s)]\varrho(t)\otimes\rho_{eq}]$$
(.2.12)
with $\vec{n}=(n,n^{\prime})$ and $\vec{\imath}=(\imath,\imath^{\prime})$, we have
$$\frac{d}{dt}\varrho(t)=-\sum_{\vec{n},\vec{\imath}}\int_{0}^{t}d\tau e^{-%
\kappa\tau/2}F^{l}_{\vec{n}\vec{\imath}}(t,t-\tau).$$
(.2.13)
Using the properties of the cavity equilibrium state
$$\displaystyle\mathtt{tr}_{c}[a_{\imath}a_{\imath^{\prime}}^{+}\rho_{eq}]$$
$$\displaystyle=$$
$$\displaystyle(\bar{n}_{\imath}+1)\delta_{\imath,\imath^{\prime}},$$
(.2.14)
$$\displaystyle\mathtt{tr}_{c}[a_{\imath}^{+}a_{\imath^{\prime}}\rho_{eq}]$$
$$\displaystyle=$$
$$\displaystyle\bar{n}_{\imath}\delta_{\imath,\imath^{\prime}},$$
(.2.15)
$$\displaystyle\mathtt{tr}_{c}[a_{\imath}^{+}a_{\imath^{\prime}}^{+}\rho_{eq}]$$
$$\displaystyle=$$
$$\displaystyle 0,$$
(.2.16)
$$\displaystyle\mathtt{tr}_{c}[a_{\imath}a_{\imath^{\prime}}\rho_{eq}]$$
$$\displaystyle=$$
$$\displaystyle 0,$$
(.2.17)
we obtain
$$\displaystyle F^{l}_{\vec{n}\vec{\imath}}(t,s)$$
$$\displaystyle=$$
$$\displaystyle\mathtt{tr}_{c}[\Theta_{\imath n}^{l}a_{\imath}^{+}\sigma_{-}^{ln%
}(t),[\Theta_{\imath^{\prime}n^{\prime}}^{l\ast}a_{\imath^{\prime}}\sigma_{+}^%
{ln^{\prime}}(s),\varrho(t)\otimes\rho_{eq}]]+\mathtt{tr}_{c}[\Theta_{\imath n%
}^{l\ast}a_{\imath}\sigma_{+}^{ln}(t),[\Theta_{\imath^{\prime}n^{\prime}}^{l}a%
_{\imath^{\prime}}^{+}\sigma_{-}^{ln^{\prime}}(s),\varrho(t)\otimes\rho_{eq}]]$$
(.2.18)
$$\displaystyle=$$
$$\displaystyle\Theta_{\imath n}^{l}\Theta_{\imath^{\prime}n^{\prime}}^{l\ast}%
\left\{\mathtt{tr}_{c}[a_{\imath}^{+}a_{\imath^{\prime}}\rho_{eq}][\sigma_{-}^%
{ln}(t)\sigma_{+}^{ln^{\prime}}(s)\varrho-\sigma_{-}^{ln}(t)\varrho\sigma_{+}^%
{ln^{\prime}}(s)]+\mathtt{tr}_{c}[a_{\imath^{\prime}}a_{\imath}^{+}\rho_{eq}][%
\varrho\sigma_{+}^{ln^{\prime}}(s)\sigma_{-}^{ln}(t)-\sigma_{+}^{ln^{\prime}}(%
s)\varrho\sigma_{-}^{ln}(t)]\right\}$$
$$\displaystyle+$$
$$\displaystyle\Theta_{\imath n}^{l\ast}\Theta_{\imath^{\prime}n^{\prime}}^{l}%
\left\{\mathtt{tr}_{c}[a_{\imath^{\prime}}^{+}a_{\imath}\rho_{eq}][\varrho%
\sigma_{-}^{ln^{\prime}}(s)\sigma_{+}^{ln}(t)-\sigma_{-}^{ln^{\prime}}(s)%
\varrho\sigma_{+}^{ln}(t)]+\mathtt{tr}_{c}[a_{\imath}a_{\imath^{\prime}}^{+}%
\rho_{eq}][\sigma_{+}^{ln}(t)\sigma_{-}^{ln^{\prime}}(s)\varrho-\sigma_{+}^{ln%
}(t)\varrho\sigma_{-}^{ln^{\prime}}(s)]\right\}$$
$$\displaystyle=$$
$$\displaystyle\delta_{\imath,\imath^{\prime}}\left\{\Theta_{\imath n}^{l}\Theta%
_{\imath n^{\prime}}^{l\ast}\bar{n}_{\imath}[\sigma_{-}^{ln}(t)\sigma_{+}^{ln^%
{\prime}}(s)\varrho-\sigma_{-}^{ln}(t)\varrho\sigma_{+}^{ln^{\prime}}(s)]+%
\Theta_{\imath n}^{l\ast}\Theta_{\imath n^{\prime}}^{l}\bar{n}_{\imath}[%
\varrho\sigma_{-}^{ln^{\prime}}(s)\sigma_{+}^{ln}(t)-\sigma_{-}^{ln^{\prime}}(%
s)\varrho\sigma_{+}^{ln}(t)]\right.$$
$$\displaystyle+$$
$$\displaystyle\left.\Theta_{\imath n}^{l}\Theta_{\imath n^{\prime}}^{l\ast}(%
\bar{n}_{\imath}+1)[\varrho\sigma_{+}^{ln^{\prime}}(s)\sigma_{-}^{ln}(t)-%
\sigma_{+}^{ln^{\prime}}(s)\varrho\sigma_{-}^{ln}(t)]\texttt{+}\Theta_{\imath n%
}^{l\ast}\Theta_{\imath n^{\prime}}^{l}(\bar{n}_{\imath}+1)[\sigma_{+}^{ln}(t)%
\sigma_{-}^{ln^{\prime}}(s)\varrho-\sigma_{+}^{ln}(t)\varrho\sigma_{-}^{ln^{%
\prime}}(s)]\right\}.$$
Now to calculate the dissipator for these terms, we take the upper limit of the integral to infinity $\int_{0}^{t}d\tau\rightarrow\int_{0}^{\infty}d\tau$, and define the superoperator generators
$$\displaystyle G^{l}_{\vec{n}\vec{\imath}}(t)\varrho(t)$$
$$\displaystyle=$$
$$\displaystyle-\int_{0}^{\infty}d\tau e^{-\kappa\tau/2}F_{\vec{n}\vec{\imath}}^%
{l}(t,t-\tau).$$
(.2.19)
Hence the reduced system master equation is given by
$$\frac{d}{dt}\varrho(t)=\sum_{\vec{n},\vec{\imath}}G^{l}_{\vec{n}\vec{\imath}}(%
t)\varrho(t).$$
(.2.20)
Using the following formula of integration
$$\int_{0}^{\infty}d\tau e^{-\kappa\tau/2}e^{\pm i\tau\Delta_{l}}=\frac{2}{%
\kappa\mp i2\Delta_{l}}=\eta_{l}\pm i\lambda_{l},$$
(.2.21)
with
$$\eta_{l}=\frac{2\kappa}{\kappa^{2}+4\Delta_{l}^{2}},\lambda_{l}=\frac{4\Delta_%
{l}}{\kappa^{2}+4\Delta_{l}^{2}},$$
(.2.22)
we have
$$\displaystyle G^{l}_{\vec{n}\vec{\imath}}(t)\varrho(t)$$
$$\displaystyle=$$
$$\displaystyle-\int_{0}^{\infty}d\tau e^{-\kappa\tau/2}F_{\vec{n}\vec{\imath}}^%
{l}(t,t-\tau)$$
(.2.23)
$$\displaystyle=$$
$$\displaystyle-\delta_{\imath,\imath^{\prime}}\int_{0}^{\infty}d\tau e^{-\kappa%
\tau/2}\left\{e^{i\Delta_{l}\tau}\Theta^{l}_{\imath n}\Theta^{l*}_{\imath n^{%
\prime}}\bar{n}_{\imath}[\sigma^{ln}_{-}\sigma_{+}^{ln^{\prime}}\varrho-\sigma%
^{ln}_{-}\varrho\sigma^{ln^{\prime}}_{+}]+e^{-i\Delta_{l}\tau}\Theta^{l*}_{%
\imath n}\Theta^{l}_{\imath n^{\prime}}\bar{n}_{\imath}[\varrho\sigma^{ln^{%
\prime}}_{-}\sigma^{ln}_{+}-\sigma^{ln^{\prime}}_{-}\varrho\sigma^{ln}_{+}]\right.$$
$$\displaystyle+$$
$$\displaystyle\left.e^{i\Delta_{l}\tau}\Theta^{l}_{\imath n}\Theta^{l*}_{\imath
n%
^{\prime}}(\bar{n}_{\imath}+1)[\varrho\sigma^{ln^{\prime}}_{+}\sigma^{ln}_{-}-%
\sigma^{ln^{\prime}}_{+}\varrho\sigma^{ln}_{-}]+e^{-i\Delta_{l}\tau}\Theta^{l*%
}_{\imath n}\Theta^{l}_{\imath n^{\prime}}(\bar{n}_{\imath}+1)[\sigma^{ln}_{+}%
\sigma_{-}^{ln^{\prime}}\varrho-\sigma^{ln}_{+}\varrho\sigma^{ln^{\prime}}_{-}%
]\right\},$$
$$\displaystyle=$$
$$\displaystyle-\delta_{\imath,\imath^{\prime}}\left\{(\eta_{l}+i\lambda_{l})%
\Theta^{l}_{\imath n}\Theta^{l*}_{\imath n^{\prime}}\bar{n}_{\imath}[\sigma^{%
ln}_{-}\sigma_{+}^{ln^{\prime}}\varrho-\sigma^{ln}_{-}\varrho\sigma^{ln^{%
\prime}}_{+}]+(\eta_{l}-i\lambda_{l})\Theta^{l*}_{\imath n}\Theta^{l}_{\imath n%
^{\prime}}\bar{n}_{\imath}[\varrho\sigma^{ln^{\prime}}_{-}\sigma^{ln}_{+}-%
\sigma^{ln^{\prime}}_{-}\varrho\sigma^{ln}_{+}]\right.$$
$$\displaystyle+$$
$$\displaystyle\left.(\eta_{l}+i\lambda_{l})\Theta^{l}_{\imath n}\Theta^{l*}_{%
\imath n^{\prime}}(\bar{n}_{\imath}+1)[\varrho\sigma^{ln^{\prime}}_{+}\sigma^{%
ln}_{-}-\sigma^{ln^{\prime}}_{+}\varrho\sigma^{ln}_{-}]+(\eta_{l}-i\lambda_{l}%
)\Theta^{l*}_{\imath n}\Theta^{l}_{\imath n^{\prime}}(\bar{n}_{\imath}+1)[%
\sigma^{ln}_{+}\sigma_{-}^{ln^{\prime}}\varrho-\sigma^{ln}_{+}\varrho\sigma^{%
ln^{\prime}}_{-}]\right\},$$
$$\displaystyle=$$
$$\displaystyle-\delta_{\imath,\imath^{\prime}}\left\{\bar{n}_{\imath}\eta_{l}[%
\Theta^{l}_{\imath n}\Theta^{l*}_{\imath n^{\prime}}(\sigma^{ln}_{-}\sigma_{+}%
^{ln^{\prime}}\varrho-\sigma^{ln}_{-}\varrho\sigma^{ln^{\prime}}_{+})+\Theta^{%
l*}_{\imath n}\Theta^{l}_{\imath n^{\prime}}(\varrho\sigma^{ln^{\prime}}_{-}%
\sigma^{ln}_{+}-\sigma^{ln^{\prime}}_{-}\varrho\sigma^{ln}_{+})]\right.$$
$$\displaystyle+$$
$$\displaystyle i\bar{n}_{\imath}\lambda_{l}[\Theta^{l}_{\imath n}\Theta^{l*}_{%
\imath n^{\prime}}(\sigma^{ln}_{-}\sigma_{+}^{ln^{\prime}}\varrho-\sigma^{ln}_%
{-}\varrho\sigma^{ln^{\prime}}_{+})-\Theta^{l*}_{\imath n}\Theta^{l}_{\imath n%
^{\prime}}(\varrho\sigma^{ln^{\prime}}_{-}\sigma^{ln}_{+}-\sigma^{ln^{\prime}}%
_{-}\varrho\sigma^{ln}_{+})]$$
$$\displaystyle+$$
$$\displaystyle(\bar{n}_{\imath}+1)\eta_{l}[\Theta^{l}_{\imath n}\Theta^{l*}_{%
\imath n^{\prime}}(\varrho\sigma^{ln^{\prime}}_{+}\sigma^{ln}_{-}-\sigma^{ln^{%
\prime}}_{+}\varrho\sigma^{ln}_{-})+\Theta^{l*}_{\imath n}\Theta^{l}_{\imath n%
^{\prime}}(\sigma^{ln}_{+}\sigma_{-}^{ln^{\prime}}\varrho-\sigma^{ln}_{+}%
\varrho\sigma^{ln^{\prime}}_{-})]$$
$$\displaystyle+$$
$$\displaystyle\left.i(\bar{n}_{\imath}+1)\lambda_{l}[\Theta^{l}_{\imath n}%
\Theta^{l*}_{\imath n^{\prime}}(\varrho\sigma^{ln^{\prime}}_{+}\sigma^{ln}_{-}%
-\sigma^{ln^{\prime}}_{+}\varrho\sigma^{ln}_{-})-\Theta^{l*}_{\imath n}\Theta^%
{l}_{\imath n^{\prime}}(\sigma^{ln}_{+}\sigma_{-}^{ln^{\prime}}\varrho-\sigma^%
{ln}_{+}\varrho\sigma^{ln^{\prime}}_{-})]\right\}.$$
In view of the identity of $N$ separated qubits, we only discuss the evolution of one qubit, the state of that is diagonal in the coupled angular momentum basis $\varrho^{n}(t)=\sum_{\imath=\pm 1}P_{\imath}(t)\varrho^{n}_{\imath}$. Here we finally consider the diagonal matrix elements $P_{\sigma^{n}_{\mathbf{z}}}(t)=\left\langle\sigma^{n}_{\mathbf{z}}\right|%
\varrho^{n}(t)\left|\sigma^{n}_{\mathbf{z}}\right\rangle(\sigma^{n}_{\mathbf{z%
}}=\pm 1)$ of the reduced density operator $\varrho^{n}(t)$, which corresponds to the expectation value of the projection operator $\varrho_{\sigma^{n}_{\mathbf{z}}}=\left|\sigma^{n}_{\mathbf{z}}\right\rangle%
\left\langle\sigma^{n}_{\mathbf{z}}\right|$ at an arbitrary time $t$. For the $ath$ qubit, we obtain
$$\displaystyle\frac{d}{dt}P_{\sigma^{a}_{\textbf{z}}}(t)$$
$$\displaystyle=$$
$$\displaystyle\sum_{\vec{n},\vec{\imath}}\texttt{tr}_{a}[G^{l}_{\vec{n}\vec{%
\imath}}(t)\varrho(t)\varrho_{\sigma^{a}_{\textbf{z}}}]$$
(.2.24)
$$\displaystyle=$$
$$\displaystyle-\sum_{nn^{\prime}}\sum_{\imath}\left\{\bar{n}_{\imath}\eta_{l}%
\texttt{tr}_{a}[\Theta^{l}_{\imath n}\Theta^{l*}_{\imath n^{\prime}}(\sigma^{n%
}_{-}\sigma_{+}^{n^{\prime}}\varrho\varrho_{\sigma^{a}_{\textbf{z}}}-\sigma^{n%
}_{-}\varrho\sigma^{n^{\prime}}_{+}\varrho_{\sigma^{a}_{\textbf{z}}})+\Theta^{%
l*}_{\imath n}\Theta^{l}_{\imath n^{\prime}}(\varrho\sigma^{n^{\prime}}_{-}%
\sigma^{n}_{+}\varrho_{\sigma^{a}_{\textbf{z}}}-\sigma^{n^{\prime}}_{-}\varrho%
\sigma^{n}_{+}\varrho_{\sigma^{a}_{\textbf{z}}})]\right.$$
$$\displaystyle+$$
$$\displaystyle\left.i\bar{n}_{\imath}\lambda_{l}\texttt{tr}_{a}[\Theta^{l}_{%
\imath n}\Theta^{l*}_{\imath n^{\prime}}(\sigma^{n}_{-}\sigma_{+}^{n^{\prime}}%
\varrho\varrho_{\sigma^{a}_{\textbf{z}}}-\sigma^{n}_{-}\varrho\sigma^{n^{%
\prime}}_{+}\varrho_{\sigma^{a}_{\textbf{z}}})-\Theta^{l*}_{\imath n}\Theta^{l%
}_{\imath n^{\prime}}(\varrho\sigma^{n^{\prime}}_{-}\sigma^{n}_{+}\varrho_{%
\sigma^{a}_{\textbf{z}}}-\sigma^{n^{\prime}}_{-}\varrho\sigma^{n}_{+}\varrho_{%
\sigma^{a}_{\textbf{z}}})]\right.$$
$$\displaystyle+$$
$$\displaystyle\left.(\bar{n}_{\imath}+1)\eta_{l}\texttt{tr}_{a}[\Theta^{l}_{%
\imath n}\Theta^{l*}_{\imath n^{\prime}}(\varrho\sigma^{n^{\prime}}_{+}\sigma^%
{n}_{-}\varrho_{\sigma^{a}_{\textbf{z}}}-\sigma^{n^{\prime}}_{+}\varrho\sigma^%
{n}_{-}\varrho_{\sigma^{a}_{\textbf{z}}})+\Theta^{l*}_{\imath n}\Theta^{l}_{%
\imath n^{\prime}}(\sigma^{n}_{+}\sigma_{-}^{n^{\prime}}\varrho\varrho_{\sigma%
^{a}_{\textbf{z}}}-\sigma^{n}_{+}\varrho\sigma^{n^{\prime}}_{-}\varrho_{\sigma%
^{a}_{\textbf{z}}})]\right.$$
$$\displaystyle+$$
$$\displaystyle\left.i(\bar{n}_{\imath}+1)\lambda_{l}\texttt{tr}_{a}[\Theta^{l}_%
{\imath n}\Theta^{l*}_{\imath n^{\prime}}(\varrho\sigma^{n^{\prime}}_{+}\sigma%
^{n}_{-}\varrho_{\sigma^{a}_{\textbf{z}}}-\sigma^{n^{\prime}}_{+}\varrho\sigma%
^{n}_{-}\varrho_{\sigma^{a}_{\textbf{z}}})-\Theta^{l*}_{\imath n}\Theta^{l}_{%
\imath n^{\prime}}(\sigma^{n}_{+}\sigma_{-}^{n^{\prime}}\varrho\varrho_{\sigma%
^{a}_{\textbf{z}}}-\sigma^{n}_{+}\varrho\sigma^{n^{\prime}}_{-}\varrho_{\sigma%
^{a}_{\textbf{z}}})]\right\}$$
$$\displaystyle=$$
$$\displaystyle-\sum_{nn^{\prime}}\sum_{\imath}\left\{\bar{n}_{\imath}\eta_{l}%
\texttt{tr}_{a}[\Theta^{l}_{\imath n}\Theta^{l*}_{\imath n^{\prime}}(\varrho%
\varrho_{\sigma^{a}_{\textbf{z}}}\sigma^{n}_{-}\sigma_{+}^{n^{\prime}}-\varrho%
\sigma^{n^{\prime}}_{+}\varrho_{\sigma^{a}_{\textbf{z}}}\sigma^{n}_{-})+\Theta%
^{l*}_{\imath n}\Theta^{l}_{\imath n^{\prime}}(\varrho\sigma^{n^{\prime}}_{-}%
\sigma^{n}_{+}\varrho_{\sigma^{a}_{\textbf{z}}}-\varrho\sigma^{n}_{+}\varrho_{%
\sigma^{a}_{\textbf{z}}}\sigma^{n^{\prime}}_{-})]\right.$$
$$\displaystyle+$$
$$\displaystyle\left.i\bar{n}_{\imath}\lambda_{l}\texttt{tr}_{a}[\Theta^{l}_{%
\imath n}\Theta^{l*}_{\imath n^{\prime}}(\varrho\varrho_{\sigma^{a}_{\textbf{z%
}}}\sigma^{n}_{-}\sigma_{+}^{n^{\prime}}-\varrho\sigma^{n^{\prime}}_{+}\varrho%
_{\sigma^{a}_{\textbf{z}}}\sigma^{n}_{-})-\Theta^{l*}_{\imath n}\Theta^{l}_{%
\imath n^{\prime}}(\varrho\sigma^{n^{\prime}}_{-}\sigma^{n}_{+}\varrho_{\sigma%
^{a}_{\textbf{z}}}-\varrho\sigma^{n}_{+}\varrho_{\sigma^{a}_{\textbf{z}}}%
\sigma^{n^{\prime}}_{-})]\right.$$
$$\displaystyle+$$
$$\displaystyle\left.(\bar{n}_{\imath}+1)\eta_{l}\texttt{tr}_{a}[\Theta^{l}_{%
\imath n}\Theta^{l*}_{\imath n^{\prime}}(\varrho\sigma^{n^{\prime}}_{+}\sigma^%
{n}_{-}\varrho_{\sigma^{a}_{\textbf{z}}}-\varrho\sigma^{n}_{-}\varrho_{\sigma^%
{a}_{\textbf{z}}}\sigma^{n^{\prime}}_{+})+\Theta^{l*}_{\imath n}\Theta^{l}_{%
\imath n^{\prime}}(\varrho\varrho_{\sigma^{a}_{\textbf{z}}}\sigma^{n}_{+}%
\sigma_{-}^{n^{\prime}}-\varrho\sigma^{n^{\prime}}_{-}\varrho_{\sigma^{a}_{%
\textbf{z}}}\sigma^{n}_{+})]\right.$$
$$\displaystyle+$$
$$\displaystyle\left.i(\bar{n}_{\imath}+1)\lambda_{l}\texttt{tr}_{a}[\Theta^{l}_%
{\imath n}\Theta^{l*}_{\imath n^{\prime}}(\varrho\sigma^{n^{\prime}}_{+}\sigma%
^{n}_{-}\varrho_{\sigma^{a}_{\textbf{z}}}-\varrho\sigma^{n}_{-}\varrho_{\sigma%
^{a}_{\textbf{z}}}\sigma^{n^{\prime}}_{+})-\Theta^{l*}_{\imath n}\Theta^{l}_{%
\imath n^{\prime}}(\varrho\varrho_{\sigma^{a}_{\textbf{z}}}\sigma^{n}_{+}%
\sigma_{-}^{n^{\prime}}-\varrho\sigma^{n^{\prime}}_{-}\varrho_{\sigma^{a}_{%
\textbf{z}}}\sigma^{n}_{+})]\right\}$$
$$\displaystyle=$$
$$\displaystyle-\sum_{\imath}\left|\Theta^{l}_{\imath a}\right|^{2}\left\{\bar{n%
}_{\imath}\eta_{l}\texttt{tr}_{a}[(\varrho\varrho_{\sigma^{a}_{\textbf{z}}}%
\sigma^{a}_{-}\sigma_{+}^{a}-\varrho\sigma^{a}_{+}\varrho_{\sigma^{a}_{\textbf%
{z}}}\sigma^{a}_{-})+(\varrho\sigma^{a}_{-}\sigma^{a}_{+}\varrho_{\sigma^{a}_{%
\textbf{z}}}-\varrho\sigma^{a}_{+}\varrho_{\sigma^{a}_{\textbf{z}}}\sigma^{a}_%
{-})]\right.$$
$$\displaystyle+$$
$$\displaystyle\left.i\bar{n}_{\imath}\lambda_{l}\texttt{tr}_{a}[(\varrho\varrho%
_{\sigma^{a}_{\textbf{z}}}\sigma^{a}_{-}\sigma_{+}^{a}-\varrho\sigma^{a}_{+}%
\varrho_{\sigma^{a}_{\textbf{z}}}\sigma^{a}_{-})-(\varrho\sigma^{a}_{-}\sigma^%
{a}_{+}\varrho_{\sigma^{a}_{\textbf{z}}}-\varrho\sigma^{a}_{+}\varrho_{\sigma^%
{a}_{\textbf{z}}}\sigma^{a}_{-})]\right.$$
$$\displaystyle+$$
$$\displaystyle\left.(\bar{n}_{\imath}+1)\eta_{l}\texttt{tr}_{a}[(\varrho\sigma^%
{a}_{+}\sigma^{a}_{-}\varrho_{\sigma^{a}_{\textbf{z}}}-\varrho\sigma^{a}_{-}%
\varrho_{\sigma^{a}_{\textbf{z}}}\sigma^{a}_{+})+(\varrho\varrho_{\sigma^{a}_{%
\textbf{z}}}\sigma^{a}_{+}\sigma_{-}^{a}-\varrho\sigma^{a}_{-}\varrho_{\sigma^%
{a}_{\textbf{z}}}\sigma^{a}_{+})]\right.$$
$$\displaystyle+$$
$$\displaystyle\left.i(\bar{n}_{\imath}+1)\lambda_{l}\texttt{tr}_{a}[(\varrho%
\sigma^{a}_{+}\sigma^{a}_{-}\varrho_{\sigma^{a}_{\textbf{z}}}-\varrho\sigma^{a%
}_{-}\varrho_{\sigma^{a}_{\textbf{z}}}\sigma^{a}_{+})-(\varrho\varrho_{\sigma^%
{a}_{\textbf{z}}}\sigma^{a}_{+}\sigma_{-}^{a}-\varrho\sigma^{a}_{-}\varrho_{%
\sigma^{a}_{\textbf{z}}}\sigma^{a}_{+})]\right\}$$
$$\displaystyle=$$
$$\displaystyle-\sum_{\imath}\eta_{l}\left|\Theta^{l}_{\imath a}\right|^{2}\left%
\{2\bar{n}_{\imath}[P_{\sigma^{a}_{\textbf{z}}}(t)\Lambda^{+}_{\sigma^{a}_{%
\textbf{z}}}-P_{\sigma^{a}_{\textbf{z}}-1}(t)\Lambda^{-}_{\sigma^{a}_{\textbf{%
z}}}]\right.$$
$$\displaystyle+$$
$$\displaystyle\left.2(\bar{n}_{\imath}+1)[P_{\sigma^{a}_{\textbf{z}}}(t)\Lambda%
^{-}_{\sigma^{a}_{\textbf{z}}}-P_{\sigma^{a}_{\textbf{z}}+1}(t)\Lambda^{+}_{%
\sigma^{a}_{\textbf{z}}}]\right\},$$
with $\Lambda^{a}_{\beta 1}=\delta_{a,-\beta}$.
With the definition of $\vec{P}_{\sigma^{n}_{\mathbf{z}}}(t)=(P_{-1}(t),P_{1}(t))^{T}$, the master equation (.2.24) reduces to a rate equation for the state populations in terms of atom components:
$$\displaystyle\frac{d}{dt}\vec{P}_{\sigma^{n}_{\mathbf{z}}}(t)$$
$$\displaystyle=$$
$$\displaystyle\Gamma^{l}_{n}\mathbf{M}\vec{P}_{\sigma^{n}_{\mathbf{z}}}(t),$$
(.2.25)
with
$$\displaystyle\mathbf{M}$$
$$\displaystyle=$$
$$\displaystyle\left[\begin{array}[]{cc}-\bar{n}&\bar{n}+1\\
\bar{n}&-(\bar{n}+1)\end{array}\right],$$
(.2.26)
$$\displaystyle\Gamma^{l}_{n}$$
$$\displaystyle=$$
$$\displaystyle\sum_{\imath}2\eta_{l}\left|\Theta^{l}_{\imath n}\right|^{2}.$$
(.2.27)
Substituting (.1.17) and (.2.22) into (.2.27), we obtain the effective dissipation rate of each qubit for different dominated modes M$(l,-1)$
$$\displaystyle\Gamma^{l}_{n}$$
$$\displaystyle=$$
$$\displaystyle\frac{(1+\cos\theta_{n})^{2}}{1+4(\Delta/\kappa)^{2}}\frac{g^{2}}%
{\kappa}.$$
(.2.28)
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Linguistically Regularized LSTMs for Sentiment Classification
Qiao Qian, Minlie Huang, Xiaoyan Zhu
State Key Lab. of Intelligent Technology and Systems, National Lab. for Information Science
and Technology
Dept. of Computer Science and Technology, Tsinghua University, Beijing 100084, PR China
qianqiaodecember29@126.com, aihuang@tsinghua.edu.cn, zxy-dcs@tsinghua.edu.cn
Abstract
Sentiment understanding has been a long-term goal of AI in the past decades. This paper deals with sentence-level sentiment classification. Though a variety of neural network models have been proposed very recently, however, previous models either depend on expensive phrase-level annotation, whose performance drops substantially when trained with only sentence-level annotation; or do not fully employ linguistic resources (e.g., sentiment lexicons, negation words, intensity words), thus not being able to produce linguistically coherent representations. In this paper, we propose simple models trained with sentence-level annotation, but also attempt to generating linguistically coherent representations by employing regularizers that model the linguistic role of sentiment lexicons, negation words, and intensity words. Results show that our models are effective to capture the sentiment shifting effect of sentiment, negation, and intensity words, while still obtain competitive results without sacrificing the models’ simplicity.
Linguistically Regularized LSTMs for Sentiment Classification
Qiao Qian, Minlie Huang, Xiaoyan Zhu
State Key Lab. of Intelligent Technology and Systems, National Lab. for Information Science
and Technology
Dept. of Computer Science and Technology, Tsinghua University, Beijing 100084, PR China
qianqiaodecember29@126.com, aihuang@tsinghua.edu.cn, zxy-dcs@tsinghua.edu.cn
Copyright © 2017,
Association for the Advancement of Artificial Intelligence (www.aaai.org).
All rights reserved.
Introduction
Understanding sentiment has always been one of the goals of AI in the decades. As a small step toward sentiment understanding, sentiment classification aims to classify sentiment to sentiment classes such as positive or negative, or more fine-grained classes such as very positive, positive, neutral, etc.
There has been a variety of approaches for this purpose such as lexicon-based classification (?), and early machine learning based methods (?; ?), and recently neural network models such as convolutional neural network (CNN) (?; ?),
recursive autoencoders (?; ?), Long Short-Term Memory (LSTM) (?; ?; ?; ?), and many more.
In spite of the great success of these neural models,
there are some defects in previous studies. First, tree-structured models such as recursive autoencoders and Tree-LSTM (?; ?), depend on parsing tree structures and expensive phrase-level annotation, whose performance drops substantially when only trained with sentence-level annotation. Second, sequence models such as CNN and recurrent network are not easy to produce competitive results as reported in the literature (?). Third, linguistic knowledge has not been fully employed in neural models, though (?) shows that part-of-speech tags can be quite effective for sentence-level classification.
The goal of this research is to developing simple sequence models but also attempts to fully employing linguistic resources to benefit sentiment classification. Firstly, we attempts to develop simple models that do not depend on parsing trees and avoid phrase-level annotation which is too expensive in real-world applications. Secondly, in order to obtain competitive performance, simple models can benefit from linguistic resources. Three types of resources can be addressed: sentiment lexicon, negation words, and intensity words. Sentiment lexicon offers the prior polarity of a word which can be useful in determining the sentiment polarity of longer texts such as phrases and sentences. Negation words are typical sentiment shifters (?), which constantly shift sentiment expression. Intensify words such as very, and extremely change the valence degree of sentiment, which is important for fine-grained sentiment classification.
In order to model the linguistic role of sentiment, negation, and intensity words, and thus to generate linguistically coherent representations, our central idea is to regularize the difference between the predicted sentiment distribution of the current position, and that of the previous or next positions. For instance, if the current position is a negation word not, the current predicted distribution should be close to the transformed distribution of the next predicted distribution parameterized by the negation transformation matrix.
To summarize, our contributions lie in three folds:
•
We propose several regularizers to model the linguistic role of sentiment, negation, and intensity words in sentiment classification. Experiments show that the regularizers are quite effective.
•
Due to the complexity of sentiment shifting effect of negation and intensify words, we design word-specific transformation matrices to respect the role of each word.
•
Unlike previous models depend on parsing structures and expensive phrase-level annotation, our models are simple and efficient but also obtain competitive performance.
Related Work
Neural Networks for Sentiment Classification
Recently, there are many neural networks proposed for sentiment classification.
The most pioneering (perhaps) model may be the recursive autoencoder neural network which builds the representation of a sentence from subphrases recursively (?; ?; ?; ?). Such recursive models usually depend on a tree structure of input text, and in order to obtain competitive results, usually require heavy annotation on each subphrase.
Sequence models do not depend on a particular tree structure, for instance,
convolutional neural network (CNN) is another type of widely used models for sentiment classification (?; ?). As used similarly in image processing, CNN defines convolution operations on a sequence of a text.
Long short-term memory models are also common for learning sentence-level representation due to its capability of modeling the prefix or suffix context (?). LSTM can be commonly applied to sequential data but also tree-structured parsing trees (?; ?). A complete search for optimal structures of recurrent networks and LSTMs can be seen in (?).
Applying Linguistic Knowledge for Sentiment Classification
Linguistic knowledge and sentiment resources are very helpful for sentiment analysis such as sentiment lexicons, negation words (not, never, neither, etc.), and intensity words (very, extremely, etc.).
Sentiment lexicon, such as Hu and Liu Lexicon (?) and MPQA lexicon (?),
is widely used for sentiment classification (?).
Negation words play a critical role in modifying sentiment of textual expressions. Some early negation models are designed to reverse the sign of sentiment value of the modified text (?).
Since each individual negation word can affect sentiment words in different ways, the shifting hypothesis, is proposed, assuming that negators change the sentiment values by a constant amount (?).
The modified word is also an important factor on the negation effect, for instance, negation words turn positive to negative and turn negative to not negative. (?) incorporate negation words as feature into neural network.
(?) incorporate negation words and other linguistic knowledge into a SVM classifier for composing opposing polarities.
The intensity words can change the valence degree (i.e., sentiment intensity) of the content word.
Sentiment intensity of a phrase indicates the strength of associated sentiment, which is quite important for fine-grained sentiment classification or rating.
(?) directly reverse the polarity of modified words or change the sentiment strength by a fixed value.
(?) predict the valence value for content words using a linear regression model.
(?) introduce a kernel function to combine semantic information for predicting sentiment score.
In the SemEval-2016 task 7 subtask A, (?) propose a learning-to-rank model with a pair-wise strategy to predict sentiment intensity scores.
Long Short-term Memory Network
Long Short-Term Memory (LSTM)
To deal with the notorious issues of gradient explosion and vanishing in recurrent neural network (?), Long Short-term Memory network is proposed by incorporating an additional memory cell $c_{t}\in R^{d}$ at each time step. The hidden states $h_{t}$ and memory cell $c_{t}$ is a function of their previous $c_{t-1}$ and $h_{t-1}$ and input vector $x_{t}$, formally defined as follows:
$$c_{t},h_{t}=g^{(LSTM)}(c_{t-1},h_{t-1},x_{t})$$
(1)
The hidden state $h_{t}\in R^{d}$ denotes the representation of position $t$ while also encoding the preceding context of the position. For more details about LSTM, we refer readers to (?).
Bidirectional LSTM
In LSTM, the representation of each position ($h_{t}$) only encodes the prefix context in a forward direction while the backward context is not respected. Bidirectional LSTM (?) exploited two parallel passes (forward and backward) and concatenated representations of the two LSTMs as the representation of each position. The forward and backward LSTMs are respectively formulated as follows:
$$\overrightarrow{c}_{t},\overrightarrow{h}_{t}=g^{(LSTM)}(\overrightarrow{c}_{t%
-1},\overrightarrow{h}_{t-1},x_{t})$$
(2)
$$\overleftarrow{c}_{t},\overleftarrow{h}_{t}=g^{(LSTM)}(\overleftarrow{c}_{t+1}%
,\overleftarrow{h}_{t+1},x_{t})$$
(3)
where $g^{(LSTM)}$ is the same as that in Eq 1. Particularly, parameters in the two LSTMs are shared.
The representation of the entire sentence is $[\overrightarrow{h}_{n},\overleftarrow{h}_{1}]$, where $n$ is the length of the sentence.
At each position $t$, the joint representation $h_{t}=[\overrightarrow{h}_{t},\overleftarrow{h}_{t}]$, which is the concatenation of hidden states of the forward LSTM and backward LSTM.
In this way, the forward and backward contexts can be considered simultaneously.
Linguistically Regularized LSTM
The central idea of the paper is to output linguistically coherent predictions in sentiment classification by regularizing the outputs at adjacent positions of a sentence. For example, in sentence ‘‘this movie is interesting”, the predicted sentiment distributions at ‘‘this*111The asterisk denotes the current position.”, “this movie*”, and “this movie is*” should almost be the same, while the predicted sentiment distribution at “this movie is very interesting*” should be quite different from the preceeding positions since a sentiment word (“interesting”) is seen.
More formally, the predicted sentiment distribution ($p_{t}$, based on $h_{t}$, see Eq. 4) at position $t$ should be linguistically regularized with respect to that of the preceding ($t-1$) or following ($t+1$) positions. We propose a generic regularizer and three special regularizers based on the following linguistic observations:
•
Non-Sentiment Regularizer: if the two adjacent positions are all non-opinion words, the sentiment distributions of the two positions should be close to each other.
•
Sentiment Regularizer: if the word is a sentiment word found in a lexicon, the sentiment distribution of the current position should be significantly different from that of the next or previous positions.
•
Negation Regularizer: Negation words such as “not” and “never” are critical sentiment shifter (?): usually shifts sentiment polarity from the positive side to the negative one, but sometimes highly depends on the negation word and the words they modify. The negation regularizer models this linguistic phenomena.
•
Intensity Regularizer:
Intensity words such as “very” and “extremely” change the valence degree of a sentiment expression: for instance, from positive to very positive. Modeling this effect is quite important for fine-grained sentiment classification, and the intensity regularizer is designed to formulate this effect.
In order to enforce the model to produce coherent predictions, we propose a new loss function as follows to incorporate these regularizers:
$$E(\theta)=-\sum_{i}y^{i}\log{p^{i}}+\alpha\sum_{i}\sum_{t}L^{i}_{t}+\beta||%
\theta||^{2}$$
(4)
where $y^{i}$ is the gold distribution, $p^{i}$ is the predicted distribution output from a softmax layer taking the sentence representation as input, $L^{i}_{t}$ is one of the above regularizers or combination of these regularizers, $\alpha$ is the weight for the regularization term, and $i,t$ is the index of sentence and position respectively.
Non-Sentiment Regularizer (NSR)
This regularizer constrains that the sentiment distributions of adjacent positions should not vary much if the additional input word $x_{t}$ is not a sentiment word, formally as follows:
$$L_{t}^{(NSR)}=max(0,D_{KL}(p_{t},p_{t-1})-M)$$
(5)
where $M$ is a hyperparameter for margin, $p_{t}$ is the predicted distribution at position $t$ whose representation is $h_{t}$, and $D_{KL}(p,q)$ is a symmetric KL divergence defined as follows:
$$D_{KL}=\frac{1}{2}\sum_{l=1}^{C}{p(l)\log q(l)+q(l)\log p(l)}$$
where $p,q$ are distributions over sentiment labels.
Sentiment Regularizer (SR)
The sentiment regularizer constrains that the sentiment distributions of adjacent positions should drift accordingly if the input word is a sentiment word.
Let’s revisit the example “this movie is interesting” again. At position $t=4$ we see a positive word “interesting” so the predicted distribution at this position would be more positive than that at position $t=3$. This is the issue of sentiment drift.
In order to address the sentiment drift issue, we propose a polarity shifting distribution $s_{c}\in R^{C}$ for each sentiment class defined in a lexicon. For instance, a sentiment lexicon may have class labels like strong positive, weakly positive, weakly negative, and strong negative, and for each class, there is a shifting distribution which will be learned by the model. The sentiment regularizer states that if the current word is a sentiment word, the sentiment distribution drift should be observed in comparison to the previous position, as formulated as follows:
$$p_{t-1}^{(SR)}=p_{t-1}+s_{c(x_{t})}$$
(6)
$$L_{t}^{(SR)}=max(0,D_{KL}(p_{t},p_{t-1}^{(SR)})-M)$$
(7)
where $p_{t-1}^{(SR)}$ is the drifted sentiment distribution after considering the shifting sentiment distribution corresponding to the word at position $t$, $c(x_{t})$ is the prior sentiment class of word $x_{t}$, and $s_{c}\in\theta$ is a parameter to be optimized but could also be set fixed with prior knowledge. Note that in this way all words of the same sentiment class share the same drifting distribution, but in a refined setting,
we can learn a shifting distribution for each sentiment word if large-scale datasets are available.
Negation Regularizer (NR)
The negation regularizer approaches how negation words shift the sentiment distribution of its modifiers.
When the input word $x_{t}$ is a negation word, the sentiment distribution should be shifted accordingly. However, the negation role is more complex than that by sentiment words, for example, the word “not” in “not good” and “not bad” have different roles in polarity shifting. The former changes the polarity to negative, while the latter changes to neutral instead of positive.
To respect such complex negation effects, we propose a transformation matrix $T_{m}\in R^{C\times C}$ for each negation word $m$, and the matrix will be learned by the model. The regularizer assumes that if the current position is a negation word, the sentiment distribution of the current position should be close to that of the next or previous position with the transformation.
$$p_{t-1}^{(NR)}=softmax(T_{x_{j}}\times p_{t-1})$$
(8)
$$p_{t+1}^{(NR)}=softmax(T_{x_{j}}\times p_{t+1})$$
(9)
$$L_{t}^{(NR)}=min\left\{\begin{aligned} \displaystyle max(0,D_{KL}(p_{t},p_{t-1%
}^{(NR)})-M)\\
\displaystyle max(0,D_{KL}(p_{t},p_{t+1}^{(NR)})-M)\end{aligned}\right.$$
(10)
where $p_{t-1}^{(NR)}$ and $p_{t+1}^{(NR)}$ is the sentiment distuibution after transformation, $T_{x_{j}}\in\theta$ is the transformation matrix for a negation word $x_{j}$, a parameter to be learned during training. In total, we train $m$ transformation matrixs for $m$ negation words.
Intensity Regularizer (IR)
The intensify regularizer models how intensity words influence the sentiment valence of a phrase or a sentence.
Intensifier can change the valence degree of the content word.
Sentiment intensity of a phrase indicates the strength of associated sentiment, which is quite important for fine-grained sentiment classification or rating.
The formulation of the intensity effect is quite the same as that in the negation regularizer, but with different parameters of course. For each intensity word, there is a transform matrix to favor the different roles of various intensifiers on sentiment shift. For brevity, we will not repeat the formulas here.
Applying Linguistic Regularizers to Bidirectional LSTM
To make our model simple and elegant in a mathematical form, we do not consider the modification scope of negation and intensity word, which is a quite challenging problem in the NLP community.
However, we can alleviate the problem by leveraging bidirectional LSTM.
For a single LSTM, we employ a backward LSTM from the end to the beginning of a sentence. This is because, at most times, the modified words of negation and intensity words are usually at the right side of the modifiers.
But sometimes, the modified words are at the left side of negation and intensity words. To better address this issue, we employ bidirectional LSTM and let the model determine which side should be chosen.
More formally, in Bi-LSTM, we compute a transformed sentiment distribution on $\overrightarrow{p}_{t-1}$ of the forward LSTM and also that on $\overleftarrow{p}_{t+1}$ of the backward LSTM, and compute the minimum distance of the distribution of the current position to the two distributions. This could be formulated as follows:
$$\overrightarrow{p}_{t-1}^{(R)}=softmax(T_{x_{j}}\times\overrightarrow{p}_{t-1})$$
(11)
$$\overleftarrow{p}_{t+1}^{(R)}=softmax(T_{x_{j}}\times\overleftarrow{p}_{t+1})$$
(12)
$$L_{t}^{(R)}=min\left\{\begin{aligned} \displaystyle max(0,D_{KL}(%
\overrightarrow{p}_{t},\overrightarrow{p}_{t-1}^{(R)})-M)\\
\displaystyle max(0,D_{KL}(\overleftarrow{p}_{t},\overleftarrow{p}_{t+1}^{(R)}%
)-M)\end{aligned}\right.$$
(13)
where $\overrightarrow{p}_{t-1}^{(R)}$ and $\overleftarrow{p}_{t+1}^{(R)}$ are the sentiment distributions transformed from the previous state $\overrightarrow{p}_{t-1}$ and next state $\overleftarrow{p}_{t+1}$ respectively.
Note that $R\in\{NR,IR\}$ indicating the formulation works for both negation and intensity regularizers.
Due to the same consideration, we redefine $L_{t}^{(NSR)}$ and $L_{t}^{(SR)}$ with bidirectional LSTM similarly. The formulation is the same and omitted for brevity.
Discussion
Unlike previous studies on negation and intensity words, which modulate the linguistic effect of these words by some predefined rules, our models address these factors with mathematical operations, parameterized with shifting distribution vectors and transformation matrices.
In the sentiment regularizer, the sentiment shifting effect is parameterized with a class-specific distribution (but could also be word-specific if with more data). In the negation and intensity regularizers, the effect is parameterized with word-specific transformation matrices, meaning that different words have different parameters. Since the mechanism of how negation and intensity words shift sentiment expression is quite complex and highly dependent on individual words, we believe such mathematical operation will be more suitable for addressing complex linguistic roles of these words.
This is a major advantage of our approach over other methods.
Experiment
Dataset and Sentiment Lexicon
Two datasets are used for evaluating the proposed models: Movie Review (MR) (?) which has two classes as negative, positive and Stanford Sentiment Treebank (SST) (?) which has five classes.
For details, we refer readers to the two papers. SST has provided phrase-level annotation on all inner nodes, but we only use the sentence-level annotation since one of our goals is to avoid expensive phrase-level annotation.
The sentiment lexicon contains two parts. The first part comes from MPQA (?), which contains $5,153$ sentiment words, each with polarity rating. The second part consists of the leaf nodes of the SST dataset (i.e., all sentiment words) and there are $6,886$ polar words except neural ones. We combine the two parts and ignore those words that have conflicting sentiment labels, and produce a lexicon of $9,750$ words with 4 sentiment labels. For negation and intensity words, we collect them manually since the number is small, some of which can be seen in Table 2.
Regarding the parameters and initialization details of the models, please refer to the “Supplemental Material” due to the length limit.
Overall Comparison
We include several lines of baselines in the evaluation. The baselines are listed as follows:
•
RNN/RNTN Recursive Neural Network over parsing trees, proposed by (?) and Recursive Tensor Neural Network (?) employs tensors to model correlations between different dimensions of child nodes’ vectors.
•
LSTM/Bi-LSTM Long-short Term Memory (?) and the bidirectional variant as introduced previously.
•
Tree-LSTM Tree-Structured Long Short-Term Memory (?) introduces memory cells and gates into tree-structured neural network.
•
CNN Convolutional Neural Network (?) generates sentence representation by convolution and pooling operations.
Firstly, we evaluate our model on the MR dataset and the results are shown in Table 3. As can be seen, we can make the following statements:
•
Both LR-LSTM and LR-Bi-LSTM outperforms their counterparts substantially (81.5% vs. 77.4% and 82.1% vs. 79.3%, resp.), demonstrating the effectiveness of the linguistic regularizers.
•
LR-LSTM and LR-Bi-LSTM perform slightly better than Tree-LSTM but Tree-LSTM leverages a constituency tree structure while our model is a simple sequence model. As future work, we will apply such regularizers to tree-structured models.
•
On this dataset, our model is comparable to CNN.
For fine-grained sentiment classification, we evaluate our model on the SST dataset which has five sentiment classes { very negative, negative, neutral, positive, very positive} so that we can evaluate the sentiment shifting effect of intensity words. The experiment result is shown in Table 3.
We have the following observations:
•
Similarly, linguistically regularized LSTM and Bi-LSTM are better than their counterparts. It’s worth noting that LR-Bi-LSTM (trained with just sentence-level annotation) is even comparable to Bi-LSTM trained with phrase-level annotation. That means, LR-Bi-LSTM can avoid the heavy phrase-level annotation but still obtain competitive results.
•
Our models are comparable to Tree-LSTM but our models are not dependent on a parsing tree and more simple, and hence more efficient. Further, for Tree-LSTM, the model is heavily dependent on phrase-level annotation, otherwise the performance drops substantially (from 51% to 48.1%).
•
On this dataset, our model is apparently better than CNN.
The Effect of Different Regularizers
In order to reveal the effect of each individual regularizer, we conduct ablation experiments. Each time, we remove a regularizer and observe how the performance varies.
First of all, we conduct this experiment on the entire datasets, and then we experiment with sub-datasets that only contain negation words or intensity words.
The experiment results are shown in Table 4 where we can see that the non-sentiment regularizer and sentiment regularizer play a key role222Kindly note that almost all sentences contain sentiment words, see Tab. 1., and the negation regularizer and intensity regularizer are effective but less important than the previous two regularizers. This may be due to the fact that only 14% of sentences contains negation words in the test datasets, and 23% contains intensity words, and thus we further evaluate the models on two subsets, as shown in Table 5.
The experiments on the subsets show that: 1) With linguistic regularizers, LR-Bi-LSTM outperforms Bi-LSTM remarkably on these subsets;
2) When the negation regularizer is removed from the model, the performance drops significantly on both MR and SST subsets;
3) Similar observations can be made regarding the intensity regularizer.
The Effect of the Negation Regularizer
To further reveal the linguistic role of negation words, we compare the predicted sentiment distributions of a phrase pair with or without a negation word. The experimental results performed on MR are shown in Fig. 1. Each dot denotes a phrase pair (for example, $<$interesting, not interesting$>$), where the x-axis denotes the positive score333 The score is obtained from the predicted distribution, where 1 means positive and 0 means negative. of the phrase without negators (e.g., interesting), and the y-axis indicates the positive score for the phrase with negators (e.g., not interesting).
The curves in the figures show this function: $[1-y,y]=softmax(T_{nw}*[1-x,x])$ where $[1-r,r]$ is a sentiment distribution on $[negative,positive]$, $x$ is the positive score of the phrase without negators (x-axis) and $y$ that of the phrase with negators (y-axis), and $T_{nw}$ is the transformation matrix for the negation word $nw$, see Eq. 8.
We can observe the following statements:
•
All the dots are distributed around the curve, and there is no dot at the up-right and bottom-left blocks, indicating the regularizer plays in a role in sentiment shifting of negators.
•
The dots at the up-left and bottom-right respectively indicates the negation effects: changing negative to positive and positive to negative.Typical phrases include never seems hopelessly (up-left), no good scenes (bottom-right), not interesting (bottom-right), etc. There are also some positive/negative phrases shifting to neutral sentiment such as not so good, and not too bad.
•
The dots located at the center indicate that neutral phrases maintain neutral sentiment with negators. Typical phrases include not at home, not here, where negators typically modify non-sentiment words.
The Effect of the Intensity Regularizer
To further reveal the linguistic role of intensity words, we perform experiments on the SST dataset, as illustrated in Figure 2. We show the confusion matrix that shows how the sentiment shifts after being modified by intensifiers. The number 20 in the first matrix, for instance, means that there are 20 phrases have a sentiment class of negative (-) but shifting to very negative (- -) after being modified by an intensity word “very”.
As can be seen from the results, for “most”,
there are 21/21/13/12 phrases whose sentiment is shifted from negative to very negative (eg. most irresponsible picture), positive to very positive (eg. most famous author), neutral to negative (eg. most plain), and neutral to positive (eg. most closely), respectively. Similar observations can be found with word “very”.
There are also many phrases maintain their sentiment. No surprisingly, for very positive/negative phrases, phrases modified by intensifiers still maintain the strong sentiment.
For the left phrases, they fall into three categories: first, words modified by intensifiers are non-sentiment words, such as most of us, most part; second, intensifiers are not strong enough to shift sentiment, such as most complex (from negative to negative), most traditional (from positive to positive); third, our models fail to shift sentiment with intensifiers such as most vital, most resonant film.
Conclusion and Future Work
We present linguistically regularized LSTMs for sentence-level sentiment classification. The proposed models address the sentient shifting effect of sentiment, negation, and intensity words to produce linguistically coherent representations. Furthermore, our models are sequence LSTMs which do not depend on a parsing tree-structure and do not require expensive phrase-level annotation to obtain competitve results. Results show that our models are able to address the linguistic role of sentiment, negation, and intensity words.
In order to maintain the simplicity of the proposed models, we do not fully consider the modification scope of negation and intensity words, though we partially address this issue by applying a minimization operartor (see Eq. 10, Eq. 13) and bi-directional LSTM. As future work, we plan to apply the linguistic regularizers to tree-LSTM to address the scope issue since the parsing tree is easier to indicate the modification scope explicitly.
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Gauge-Invariant Formalism with a Dirac-mode Expansion
for Confinement and Chiral Symmetry Breaking
Shinya Gongyo
gongyo@ruby.scphys.kyoto-u.ac.jp
Department of Physics, Graduate School of Science,
Kyoto University,
Kitashirakawa-oiwake, Sakyo, Kyoto 606-8502, Japan
Takumi Iritani
iritani@ruby.scphys.kyoto-u.ac.jp
Department of Physics, Graduate School of Science,
Kyoto University,
Kitashirakawa-oiwake, Sakyo, Kyoto 606-8502, Japan
Hideo Suganuma
suganuma@ruby.scphys.kyoto-u.ac.jp
Department of Physics, Graduate School of Science,
Kyoto University,
Kitashirakawa-oiwake, Sakyo, Kyoto 606-8502, Japan
(December 8, 2020)
Abstract
Using the eigen-mode of the QCD Dirac operator
${\ooalign{/\hfil\cr$D$}}=\gamma^{\mu}D^{\mu}$, we develop a manifestly
gauge-covariant expansion and projection
of the QCD operators such as the Wilson loop and the Polyakov loop.
With this method, we perform a direct analysis of
the correlation between confinement and chiral symmetry breaking
in lattice QCD Monte Carlo calculation on $6^{4}$ at $\beta$=5.6.
Even after removing the low-lying Dirac modes,
which are responsible to chiral symmetry breaking,
we find that the Wilson loop obeys the area law,
and the slope parameter corresponding to the string tension or
the confinement force is almost unchanged.
We find also that the Polyakov loop remains to be almost zero
even without the low-lying Dirac modes,
which indicates the $Z_{3}$-unbroken confinement phase.
These results indicate that one-to-one correspondence does not
hold for between confinement and chiral symmetry breaking
in QCD.
pacs: 12.38.Aw, 12.38.Gc, 14.70.Dj
I Introduction
Nowadays, quantum chromodynamics (QCD) has been established
as the fundamental gauge theory of the strong interaction.
However, nonperturbative properties of low-energy QCD
such as color confinement and chiral symmetry breaking NJL61
are not yet well understood, which gives
one of the most difficult problems in theoretical physics.
The nonperturbative QCD has been studied
in lattice QCD W74 ; KS75 ; C7980 ; R05 ; Cr11
and various analytical frameworks
N74 ; tH81 ; BC80 ; C82 ; S94 ; SST95 .
In particular, it is rather interesting and important
to examine the correlation between confinement
and chiral symmetry breaking
SST95 ; M95 ; W95 ; HFGHO08 ; BBGH08 ; SWL08 ; C11 ; CPB11 ,
since the direct relation is not yet shown between them in QCD.
The strong correlation between them has been suggested
by the almost simultaneous phase transitions
of deconfinement and chiral restoration in lattice QCD
both at finite temperature R05 ; K02
and in a small-volume box R05 .
The close relation between confinement and chiral symmetry breaking
has been also suggested in terms of the monopole
degrees of freedom SST95 ; M95 ; W95 .
Here, the monopole topologically appears in QCD
by taking the maximally Abelian (MA) gauge
KSW87 ; SNW94 ; STSM95 ; KMS95 ; AS99 .
For example, by removing the monopoles in the MA gauge,
confinement and chiral symmetry breaking are
simultaneously lost in lattice QCD M95 ; W95 .
(The instantons also disappear without monopoles STSM95 .)
This indicates an important role of the monopole
to both confinement and chiral symmetry breaking,
and these two nonperturbative QCD phenomena seem
to be related via the monopole.
However, as a possibility, removing the monopoles
may be “too fatal” for most nonperturbative properties.
If this is the case, nonperturbative QCD
phenomena are simultaneously lost by their cut.
In fact, there remains an important question:
if only the relevant ingredient of
chiral symmetry breaking is carefully removed,
how will be confinement in QCD?
In this paper, considering this question,
we perform a direct investigation between
color confinement and chiral symmetry breaking in lattice QCD,
using the Dirac-mode expansion
in a gauge-invariant manner SGIY11 .
The organization of this paper is as follows.
In Sec.II, we introduce the gauge-invariant formalism
with the Dirac-mode expansion.
In Sec.III, we present the operator formalism in lattice QCD.
In Sec.IV, we formulate the Dirac-mode expansion and projection.
In Sec.V, we show the lattice results on the
analysis of confinement in terms of the Dirac modes in QCD.
Section VI is devoted to summary and discussions.
II Gauge-Invariant Formalism with Dirac-mode Expansion
We newly develop a manifestly gauge-covariant expansion
of the QCD operator such as the Wilson loop,
using the eigen-mode of the QCD Dirac operator
${\ooalign{/\hfil\cr$D$}}=\gamma^{\mu}D^{\mu}$, and investigate
the relation between confinement and chiral symmetry breaking.
II.1 Gauge Covariant Expansion in QCD
instead of Fourier Expansion
In the previous studies YS0809 ; YS10 ,
we investigated the relevant gluon-momentum region
for confinement in lattice QCD,
and found that the string tension $\sigma$,
i.e., the confining force, is almost unchanged
even after removing the high-momentum gluon component above 1.5GeV
in the Landau gauge.
In fact, the confinement property originates
from the low-momentum gluon component below 1.5GeV,
which is the upper limit to contribute to $\sigma$.
The previous study on the relevant gluonic modes
was based on the Fourier expansion, i.e.,
the eigen-mode expansion of the momentum operator $p^{\mu}$.
Because of the commutable nature of $[p^{\mu},p^{\nu}]=0$,
all the momentum $p^{\mu}$ can be simultaneously diagonalized,
which is one of the strong merits of the Fourier expansion.
Also it keeps Lorentz covariance.
However, the Fourier expansion does not
keep gauge invariance in gauge theories.
Therefore, for the use of the Fourier expansion in QCD,
one has to select a suitable gauge such as
the Landau gauge YS0809 ; YS10 ; IS0911 ,
where the gauge-field fluctuation is strongly suppressed
in Euclidean QCD.
As a next challenge, we consider a gauge-invariant method,
using a gauge-covariant expansion in QCD instead of the
Fourier expansion.
In fact, we consider a generalization of the Fourier expansion or
an alternative expansion with keeping the gauge symmetry.
A straight generalization is to use the covariant derivative operator
$D^{\mu}$ instead of the derivative operator $\partial^{\mu}$.
However, due to the non-commutable nature of $[D^{\mu},D^{\nu}]\neq 0$,
one cannot diagonalize all the covariant derivative
$D^{\mu}$ ($\mu=1,2,3,4$) simultaneously,
but only one of them can be diagonalized.
For example, the eigen-mode expansion of
$D_{4}$ keeps gauge covariance and is rather interesting,
but this type of expansion inevitably breaks the Lorentz covariance.
Then, we consider the eigen-mode expansion of
the Dirac operator ${\ooalign{/\hfil\cr$D$}}=\gamma^{\mu}D^{\mu}$
or $D^{2}=D^{\mu}D^{\mu}$ BI05 , since such an expansion keeps both
gauge symmetry and Lorentz covariance.
In particular, the Dirac-mode expansion is rather interesting because
the Dirac operator
/
$D$
directly connects with
chiral symmetry breaking via the Banks-Casher relation BC80
and its zero modes are related to the topological charge
via the Atiyah-Singer index theorem AS68 .
Here, we mainly consider the manifestly gauge-invariant new method
using the Dirac-mode expansion.
Thus, the Dirac-mode expansion has some important merits:
•
The Dirac-mode expansion method manifestly keeps both
gauge and Lorentz invariance.
•
Each QCD phenomenon can be directly investigated in terms of
chiral symmetry breaking.
II.2 Eigen-mode of Dirac Operator in Lattice QCD
Now, we consider
the Dirac operator and its eigen-modes
in lattice QCD formalism with spacing $a$ in the Euclidean metric.
On the lattice, each site is labeled by
$x=(x_{1},x_{2},x_{3},x_{4})$ with $x_{\mu}$ being an integer.
In lattice QCD, the gauge field is described by
the link-variable $U_{\mu}(x)=e^{iagA_{\mu}(x)}\in{\rm SU}(N_{c})$,
where $g$ is the QCD gauge coupling and
$A_{\mu}(x)\in{\rm su}(N_{c})$ corresponds to the gluon field.
In lattice QCD,
the Dirac operator
${\ooalign{/\hfil\cr$D$}}=\gamma_{\mu}D_{\mu}$ is expressed with $U_{\mu}(x)$ as
$$\displaystyle{\ooalign{/\hfil\cr$D$}}_{x,y}=\frac{1}{2a}\sum_{\mu=1}^{4}\gamma%
_{\mu}\left[U_{\mu}(x)\delta_{x+\hat{\mu},y}-U_{-\mu}(x)\delta_{x-\hat{\mu},y}%
\right],$$
(1)
where the convenient notation
$U_{-\mu}(x)\equiv U^{\dagger}_{\mu}(x-\hat{\mu})$ is used.
Here, $\hat{\mu}$ denotes the unit vector
on the lattice in $\mu$-direction R05 .
In this paper, we adopt the hermite definition of
the $\gamma$-matrix, $\gamma_{\mu}^{\dagger}=\gamma_{\mu}$.
Thus,
/
$D$
is anti-hermite and
satisfies
$$\displaystyle{\ooalign{/\hfil\cr$D$}}_{y,x}^{\dagger}=-{\ooalign{/\hfil\cr$D$}%
}_{x,y}.$$
(2)
The normalized eigen-state $|n\rangle$
of the Dirac operator
/
$D$
is introduced as
$$\displaystyle{\ooalign{/\hfil\cr$D$}}|n\rangle=i\lambda_{n}|n\rangle$$
(3)
with $\lambda_{n}\in{\bf R}$.
Because of $\{\gamma_{5},{\ooalign{/\hfil\cr$D$}}\}=0$, the state
$\gamma_{5}|n\rangle$ is also an eigen-state of
/
$D$
with the
eigenvalue $-i\lambda_{n}$.
The Dirac eigenfunction
$$\displaystyle\psi_{n}(x)\equiv\langle x|n\rangle$$
(4)
obeys ${\ooalign{/\hfil\cr$D$}}\psi_{n}(x)=i\lambda_{n}\psi_{n}(x)$,
and its explicit form of the eigenvalue equation in lattice QCD is
$$\displaystyle\frac{1}{2a}$$
$$\displaystyle\sum_{\mu=1}^{4}\gamma_{\mu}[U_{\mu}(x)\psi_{n}(x+\hat{\mu})-U_{-%
\mu}(x)\psi_{n}(x-\hat{\mu})]$$
(5)
$$\displaystyle=$$
$$\displaystyle i\lambda_{n}\psi_{n}(x).$$
The Dirac eigenfunction $\psi_{n}(x)$ can be
numerically obtained in lattice QCD, besides a phase factor.
According to
$U_{\mu}(x)\rightarrow V(x)U_{\mu}(x)V^{\dagger}(x+\hat{\mu})$,
the gauge transformation of $\psi_{n}(x)$ is found to be
$$\displaystyle\psi_{n}(x)\rightarrow V(x)\psi_{n}(x),$$
(6)
which is the same as that of the quark field.
To be strict, for the Dirac eigenfunction,
there appears an irrelevant $n$-dependent global phase factor
$e^{i\varphi_{n}[V]}$,
according to the arbitrariness of the definition of $\psi_{n}(x)$.
It is notable that the quark condensate
$\langle\bar{q}q\rangle$, the order parameter of
chiral symmetry breaking, is given by
the zero-eigenvalue density $\rho(0)$
of the Dirac operator,
via the Banks-Casher relation BC80 ,
$$\displaystyle\langle\bar{q}q\rangle=-\lim_{m\to 0}\lim_{V\to\infty}\pi\rho(0).$$
(7)
Here, the spectral density of the Dirac operator is defined by
$$\displaystyle\rho(\lambda)\equiv\frac{1}{V}\sum_{n}\langle\delta(\lambda-%
\lambda_{n})\rangle,$$
(8)
with the four-dimensional volume $V$.
Also, the zero-mode number asymmetry of
the Dirac operator
/
$D$
is equal to
the topological charge (the instanton number)
$Q\equiv\frac{g^{2}}{16\pi^{2}}\int d^{4}x\ {\rm Tr}\ (G_{\mu\nu}\tilde{G}_{\mu%
\nu})$,
which is known as the Atiyah-Singer index theorem,
Index(
/
$D$
)=$Q$ AS68 .
In calculating the eigenvalue of the Dirac operator
/
$D$
,
we use the Kogut-Susskind (KS) formalism KS75 ; R05 ,
which is often used to remove the redundant doublers of lattice fermions.
Here, the use of the KS formalism is just the practical reason
to reduce the calculation of the Dirac eigenvalues.
In fact, the result of the Dirac-mode projection,
which will be shown in Sec.IV, is unchanged,
when the Dirac operator is directly diagonalized.
In the KS method, using
$T(x)\equiv\gamma_{1}^{x_{1}}\gamma_{2}^{x_{2}}\gamma_{3}^{x_{3}}\gamma_{4}^{x_%
{4}}$
with $\gamma_{\mu}^{-k}\equiv(\gamma_{\mu}^{-1})^{k}$ ($k=1,2,...$),
all the gamma matrices $\gamma_{\mu}$ are diagonalized as
$T^{\dagger}(x)\gamma_{\mu}T(x\pm\hat{\mu})=\eta_{\mu}(x)1$
with the staggered phase $\eta_{\mu}(x)$ defined by
$$\displaystyle\eta_{1}(x)\equiv 1,\quad\eta_{\mu}(x)=(-1)^{x_{1}+\cdots+x_{\mu-%
1}}\ (\mu\geq 2).$$
(9)
For $\chi_{n}(x)\equiv T^{\dagger}(x)\psi_{n}(x)$,
the Dirac eigenvalue equation has no spinor index,
and the spinor degrees of freedom can be dropped off,
which reduces the lattice-fermion species from 16 to 4 R05 .
In the KS method,
the Dirac operator $\gamma_{\mu}D_{\mu}$ is
replaced by the KS Dirac operator $\eta_{\mu}D_{\mu}$,
$$\displaystyle(\eta_{\mu}D_{\mu})_{x,y}=\frac{1}{2a}\sum_{\mu=1}^{4}\eta_{\mu}(%
x)[U_{\mu}(x)\delta_{x+\hat{\mu},y}-U_{-\mu}(x)\delta_{x-\hat{\mu},y}],$$
(10)
and the spinless eigenfunction $\chi_{n}(x)$ satisfies
$$\displaystyle\frac{1}{2a}\sum_{\mu=1}^{4}\eta_{\mu}(x)[U_{\mu}(x)\chi_{n}(x+%
\hat{\mu})-U_{-\mu}(x)\chi_{n}(x-\hat{\mu})]$$
$$\displaystyle=i\lambda_{n}\chi_{n}(x).$$
(11)
In the KS formalism,
the chiral partner $\gamma_{5}\psi_{n}(x)$
reduces into $\eta_{5}(x)\chi_{n}(x)=(-1)^{x_{1}+x_{2}+x_{3}+x_{4}}\chi_{n}(x)$,
which is an eigenfunction of $\eta_{\mu}D_{\mu}$
with the eigenvalue $-i\lambda_{n}$.
Using the KS formalism KS75 ; R05 ,
the Dirac-mode number $L^{4}\times N_{c}\times$ 4
is reduced to be $L^{4}\times N_{c}$ on the $L^{4}$ lattice.
The actual number of the independent Dirac eigenvalue
$\lambda_{n}$ is about $L^{4}\times N_{c}/2$,
due to the chiral property of the Dirac operator, i.e.,
pairwise appearance of $\pm\lambda_{n}$.
III Operator Formalism in Lattice QCD
To keep the gauge symmetry, careful treatments are necessary,
since naive approximations may break the gauge symmetry.
Here, we take the “operator formalism” SGIY11 ,
as explained below.
We define the link-variable operator $\hat{U}_{\pm\mu}$
by the matrix element of
$$\displaystyle\langle x|\hat{U}_{\pm\mu}|y\rangle=U_{\pm\mu}(x)\delta_{x\pm\hat%
{\mu},y}.$$
(12)
Note that $\hat{U}_{\mu}$ and $\hat{U}_{-\mu}$ are
Hermitian conjugate as the operator in the Hilbert space
in the sense that
$$\displaystyle\langle y|\hat{U}_{\mu}^{\dagger}|x\rangle$$
$$\displaystyle=$$
$$\displaystyle U_{\mu}^{\dagger}(y)\delta_{y+\hat{\mu},x}=U_{\mu}^{\dagger}(x-%
\hat{\mu})\delta_{x-\hat{\mu},y}$$
(13)
$$\displaystyle=$$
$$\displaystyle U_{-\mu}(x)\delta_{x-\hat{\mu},y}=\langle x|\hat{U}_{-\mu}|y\rangle.$$
In the operator formalism,
Eq.(5) for the Dirac eigen-state
is simply expressed as
$$\displaystyle\frac{1}{2a}\sum_{\mu=1}^{4}\gamma_{\mu}(\hat{U}_{\mu}-\hat{U}_{-%
\mu})|n\rangle=i\lambda_{n}|n\rangle.$$
(14)
In the KS method, where the spinor index is dropped off,
one identifies $\chi_{n}(x)=\langle x|n\rangle$,
and then Eq.(11)
for the KS Dirac eigen-state is expressed as
$$\displaystyle\frac{1}{2a}\sum_{\mu=1}^{4}\hat{\eta}_{\mu}(\hat{U}_{\mu}-\hat{U%
}_{-\mu})|n\rangle=i\lambda_{n}|n\rangle,$$
(15)
where
$\hat{\eta}_{\mu}$
is defined by
$\langle x|\hat{\eta}_{\mu}|y\rangle=\eta_{\mu}(x)\delta_{x,y}.$
Owing to $\eta_{\mu}(x\pm\hat{\mu})=\eta_{\mu}(x)$,
one finds $\hat{\eta}_{\mu}\hat{U}_{\pm\mu}=\hat{U}_{\pm\mu}\hat{\eta}_{\mu}$,
so that there is no ordering uncertainty in
the KS Dirac operator in Eq.(15).
In the KS method, the chiral partner $\gamma_{5}|n\rangle$
corresponds to $\hat{\eta}_{5}|n\rangle$, where
$\hat{\eta}_{5}$ is defined by the matrix element
$\langle x|\hat{\eta}_{5}|y\rangle=\eta_{5}(x)\delta_{x,y}=(-1)^{x_{1}+x_{2}+x_%
{3}+x_{4}}\delta_{x,y}$.
Due to $\eta_{5}(x\pm\hat{\mu})=-\eta_{5}(x)$, we note
$\hat{\eta}_{5}\hat{U}_{\pm\mu}=-\hat{U}_{\pm\mu}\hat{\eta}_{5}$.
In the following, we mainly use
the ordinary Dirac operator $\gamma_{\mu}D_{\mu}$
and the spinor eigenfunction $\psi_{n}(x)=\langle x|n\rangle$.
When the KS method is applied, one only has to
use the identification of $\chi_{n}(x)=\langle x|n\rangle$
in the following arguments.
The final results are the same between both calculations
based on $\gamma_{\mu}D_{\mu}$ and $\eta_{\mu}D_{\mu}$.
The Wilson-loop operator $\hat{W}$ is defined as the product of
$\hat{U}_{\mu}$ along a rectangular loop,
$$\displaystyle\hat{W}\equiv\prod_{k=1}^{N}\hat{U}_{\mu_{k}}=\hat{U}_{\mu_{1}}%
\hat{U}_{\mu_{2}}\cdots\hat{U}_{\mu_{N}}.$$
(16)
For arbitrary loops, one finds $\sum_{k=1}^{N}\hat{\mu}_{k}=0$.
We note that the functional trace of the Wilson-loop operator
$\hat{W}$ is proportional to the ordinary vacuum expectation value
$\langle W\rangle$ of the Wilson loop:
$$\displaystyle{\rm Tr}\ \hat{W}$$
$$\displaystyle=$$
$$\displaystyle{\rm tr}\sum_{x}\langle x|\hat{W}|x\rangle={\rm tr}\sum_{x}%
\langle x|\hat{U}_{\mu_{1}}\hat{U}_{\mu_{2}}\cdots\hat{U}_{\mu_{N}}|x\rangle$$
(17)
$$\displaystyle=$$
$$\displaystyle{\rm tr}\sum_{x_{1},x_{2},\cdots,x_{N}}\langle x_{1}|\hat{U}_{\mu%
_{1}}|x_{2}\rangle\langle x_{2}|\hat{U}_{\mu_{2}}|x_{3}\rangle\langle x_{3}|%
\hat{U}_{\mu_{3}}|x_{4}\rangle\cdots\langle x_{N}|\hat{U}_{\mu_{N}}|x_{1}\rangle$$
$$\displaystyle=$$
$$\displaystyle{\rm tr}\sum_{x}\langle x|\hat{U}_{\mu_{1}}|x+\hat{\mu}_{1}%
\rangle\langle x+\hat{\mu}_{1}|\hat{U}_{\mu_{2}}|x+\sum_{k=1}^{2}\hat{\mu}_{k}%
\rangle\cdots\langle x+\sum_{k=1}^{N-1}\hat{\mu}_{k}|\hat{U}_{\mu_{N}}|x\rangle$$
$$\displaystyle=$$
$$\displaystyle\sum_{x}{\rm tr}\{U_{\mu_{1}}(x)U_{\mu_{2}}(x+\hat{\mu}_{1})U_{%
\mu_{3}}(x+\sum_{k=1}^{2}\hat{\mu}_{k})\cdots U_{\mu_{N}}(x+\sum_{k=1}^{N-1}%
\hat{\mu}_{k})\}$$
$$\displaystyle=$$
$$\displaystyle\langle W\rangle\cdot{\rm Tr}\ 1.$$
Here, “Tr” denotes the functional trace,
and “tr” the trace over SU(3) color index.
The Dirac-mode matrix element of the link-variable operator
$\hat{U}_{\mu}$ can be expressed with $\psi_{n}(x)$:
$$\displaystyle\langle m|\hat{U}|n\rangle$$
$$\displaystyle=$$
$$\displaystyle\sum_{x}\langle m|x\rangle\langle x|\hat{U}_{\mu}|x+\hat{\mu}%
\rangle\langle x+\hat{\mu}|n\rangle$$
(18)
$$\displaystyle=$$
$$\displaystyle\sum_{x}\psi_{m}^{\dagger}(x)U_{\mu}(x)\psi_{n}(x+\hat{\mu}).$$
Although the total number of the matrix element is very huge,
the matrix element is calculable and gauge invariant,
apart from an irrelevant phase factor.
Using the gauge transformation (6), we find
the gauge transformation of the matrix element as
$$\displaystyle\langle m|\hat{U}_{\mu}|n\rangle$$
$$\displaystyle=$$
$$\displaystyle\sum_{x}\psi^{\dagger}_{m}(x)U_{\mu}(x)\psi_{n}(x+\hat{\mu})$$
(19)
$$\displaystyle\rightarrow$$
$$\displaystyle\sum_{x}\psi^{\dagger}_{m}(x)V^{\dagger}(x)\cdot V(x)U_{\mu}(x)V^%
{\dagger}(x+\hat{\mu})\cdot V(x+\hat{\mu})\psi_{n}(x+\hat{\mu})$$
$$\displaystyle=$$
$$\displaystyle\sum_{x}\psi_{m}^{\dagger}(x)U_{\mu}(x)\psi_{n}(x+\hat{\mu})=%
\langle m|\hat{U}_{\mu}|n\rangle.$$
To be strict, there appears an $n$-dependent global phase factor,
corresponding to the arbitrariness of the phase in the basis
$|n\rangle$. However, this phase factor cancels as
$e^{-i\varphi_{n}}e^{i\varphi_{n}}=1$
between $|n\rangle$ and $\langle n|$, and does not appear
for QCD physical quantities including
the Wilson loop and the Polyakov loop.
In the practical lattice-QCD calculation,
we adopt the KS formalism to reduce the computational complexity,
as mentioned in Sec II-B.
In the KS method, instead of $\psi_{n}(x)$,
we use the spinless eigenfunction $\chi_{n}(x)$ of
the KS Dirac operator $\eta_{\mu}D_{\mu}$,
with the identification of $\chi_{n}(x)=\langle x|n\rangle$,
and the KS-reduced matrix element of $\hat{U}_{\mu}$ is expressed as
$$\displaystyle\langle m|\hat{U}|n\rangle$$
$$\displaystyle=$$
$$\displaystyle\sum_{x}\langle m|x\rangle\langle x|\hat{U}_{\mu}|x+\hat{\mu}%
\rangle\langle x+\hat{\mu}|n\rangle$$
(20)
$$\displaystyle=$$
$$\displaystyle\sum_{x}\chi_{m}^{\dagger}(x)U_{\mu}(x)\chi_{n}(x+\hat{\mu}).$$
In the arguments in the next section,
the same results are obtained between the calculations
based on the original Dirac operator $\gamma_{\mu}D_{\mu}$
and the KS Dirac operator $\eta_{\mu}D_{\mu}$.
IV Dirac-mode Expansion and Projection
IV.1 General Definition of Dirac-mode Expansion and Projection
From the completeness of the Dirac-mode basis,
$\sum_{n}|n\rangle\langle n|=1$, we get
$$\displaystyle\hat{O}=\sum_{m}\sum_{n}|m\rangle\langle m|\hat{O}|n\rangle%
\langle n|$$
(21)
for arbitrary operators.
Based on this relation, the Dirac-mode expansion and projection
can be defined SGIY11 .
We define the projection operator $\hat{P}$
which restricts the Dirac-mode space,
$$\displaystyle\hat{P}\equiv\sum_{n\in A}|n\rangle\langle n|,$$
(22)
where $A$ denotes arbitrary set of Dirac modes.
In $\hat{P}$, the arbitrary phase cancels
between $|n\rangle$ and $\langle n|$.
One finds $\hat{P}^{2}=\hat{P}$ and $\hat{P}^{\dagger}=\hat{P}$.
The typical projections are
IR-cut and UV-cut of the Dirac modes:
$$\displaystyle\hat{P}_{\rm\ IR}\equiv\sum_{|\lambda_{n}|\geq\Lambda_{\rm IR}}|n%
\rangle\langle n|,\quad\hat{P}_{\rm\ UV}\equiv\sum_{|\lambda_{n}|\leq\Lambda_{%
\rm UV}}|n\rangle\langle n|.$$
(23)
Using the projection operator $\hat{P}$, we define
the Dirac-mode projected link-variable operator,
$$\displaystyle\hat{U}^{P}_{\mu}\equiv\hat{P}\hat{U}_{\mu}\hat{P}=\sum_{m\in A}%
\sum_{n\in A}|m\rangle\langle m|\hat{U}_{\mu}|n\rangle\langle n|.$$
(24)
During this projection, there appears some non-locality in general,
but it would not be important for the argument of
large-distance properties such as confinement.
Each lattice QCD configuration is characterized by the
set of the link-variable $\{U_{\mu}(s)\}$, or equivalently,
the link-variable operator $\{\hat{U}_{\mu}\}$, and then
the Dirac-mode projection is described by the
replacement of $\{\hat{U}_{\mu}\}$ by $\{\hat{U}^{P}_{\mu}\}$.
In fact, the Dirac-mode projection
of QCD physical quantities $\langle O[U_{\mu}(s)]\rangle$ or
${\rm Tr}\hat{O}[\hat{U}_{\mu}]$ can be defined
by the replacement of
$$\displaystyle{\rm Tr}\hat{O}[\hat{U}_{\mu}]\rightarrow{\rm Tr}\hat{O}[\hat{U}^%
{P}_{\mu}].$$
(25)
Also in full QCD, after the integration over the quark degrees of freedom,
all the QCD physical quantities can be written
by $\langle O[U_{\mu}(s)]\rangle$ or ${\rm Tr}\hat{O}[\hat{U}_{\mu}]$,
so that the Dirac-mode projection can be applied in the same way.
IV.2 Dirac-mode expansion and projection of the Wilson loop
In this subsection, we consider the Dirac-mode
expansion and projection of the Wilson-loop
$\langle W(R,T)\rangle\propto{\rm Tr}\hat{W}(R,T)$
corresponding to the $R\times T$ rectangular loop.
For the ordinary Wilson loop $\langle W(R,T)\rangle$,
its area law indicates the confinement phase of
the QCD vacuum and the linear arising potential between static
quark and antiquark in the infrared region R05 .
From the Wilson-loop operator
$\hat{W}\equiv\prod_{k=1}^{N}\hat{U}_{\mu_{k}},$
we get the Dirac-mode expansion of the Wilson loop as
$$\displaystyle{\rm Tr}\hat{W}={\rm Tr}\prod_{k=1}^{N}\hat{U}_{\mu_{k}}={\rm Tr}%
(\hat{U}_{\mu_{1}}\hat{U}_{\mu_{2}}\cdots\hat{U}_{\mu_{N}})={\rm tr}\sum_{n_{1%
},n_{2},\cdots,n_{N}}\langle n_{1}|\hat{U}_{\mu_{1}}|n_{2}\rangle\langle n_{2}%
|\hat{U}_{\mu_{2}}|n_{3}\rangle\cdots\langle n_{N}|\hat{U}_{\mu_{N}}|n_{1}\rangle.$$
(26)
Based on this expression, we investigate the role of
specific Dirac modes to the area law of the Wilson loop.
In fact, if some Dirac modes are essential to reproduce
the area law of the Wilson loop or the confinement property,
the removal of the coupling to these modes leads to
a significant change on the area law.
In this way, we try to answer the question of
“Are there any relevant Dirac modes responsible
to the area law of the Wilson loop?”
To this end, we define the Dirac-mode projected
Wilson-loop operator,
$$\displaystyle\hat{W}^{P}$$
$$\displaystyle\equiv$$
$$\displaystyle\prod_{k=1}^{N}\hat{U}^{P}_{\mu_{k}}=\hat{U}^{P}_{\mu_{1}}\hat{U}%
^{P}_{\mu_{2}}\cdots\hat{U}^{P}_{\mu_{N}}=\hat{P}\hat{U}_{\mu_{1}}\hat{P}\hat{%
U}_{\mu_{2}}\hat{P}\cdots\hat{P}\hat{U}_{\mu_{N}}\hat{P}$$
(27)
$$\displaystyle=$$
$$\displaystyle\sum_{n_{1},n_{2},\cdots,n_{N+1}\in A}|n_{1}\rangle\langle n_{1}|%
\hat{U}_{\mu_{1}}|n_{2}\rangle\langle n_{2}|\hat{U}_{\mu_{2}}|n_{3}\rangle%
\cdots\langle n_{N}|\hat{U}_{\mu_{N}}|n_{N+1}\rangle\langle n_{N+1}|.$$
Then, we obtain the functional trace of
the Dirac-mode projected Wilson-loop operator,
$$\displaystyle{\rm Tr}\ \hat{W}^{P}$$
$$\displaystyle=$$
$$\displaystyle{\rm Tr}\ \prod_{k=1}^{N}\hat{U}^{P}_{\mu_{k}}={\rm Tr}\ \hat{U}^%
{P}_{\mu_{1}}\hat{U}^{P}_{\mu_{2}}\cdots\hat{U}^{P}_{\mu_{N}}={\rm Tr}\ \hat{P%
}\hat{U}_{\mu_{1}}\hat{P}\hat{U}_{\mu_{2}}\hat{P}\cdots\hat{P}\hat{U}_{\mu_{N}%
}\hat{P}$$
(28)
$$\displaystyle=$$
$$\displaystyle{\rm tr}\sum_{n_{1},n_{2},\cdots,n_{N}\in A}\langle n_{1}|\hat{U}%
_{\mu_{1}}|n_{2}\rangle\langle n_{2}|\hat{U}_{\mu_{2}}|n_{3}\rangle\cdots%
\langle n_{N}|\hat{U}_{\mu_{N}}|n_{1}\rangle,$$
which is manifestly gauge invariant.
Here, the arbitrary phase factor
cancels between $|n_{k}\rangle$ and $\langle n_{k}|$.
Its gauge invariance is also numerically checked
in the lattice QCD Monte Carlo calculation.
The original Wilson-loop operator $\hat{W}(R,T)$ couples to
all the Dirac modes, and
${\rm Tr}\ \hat{W}(R,T)$ obeys the area law,
$$\displaystyle{\rm Tr}\ \hat{W}(R,T)\propto\langle W(R,T)\rangle\propto e^{-%
\sigma RT},$$
(29)
for large $R$ and $T$. Here, the slope parameter $\sigma$
corresponds to the string tension or the confinement force.
For the restriction of the Dirac-mode space to be $A$,
we investigate the Dirac-mode projected Wilson-loop operator
$\hat{W}^{P}(R,T)$, which couples to the restricted Dirac modes.
If the removed Dirac modes are essential for the confinement property or
the area-law behavior of the Wilson loop, a large change is expected
on the behavior of ${\rm Tr}\ \hat{W}^{P}(R,T)$.
If not, no significant change is expected on the behavior
of ${\rm Tr}\ \hat{W}^{P}(R,T)$.
In fact, one can investigate the role of the removed Dirac modes
to confinement by checking the area-law behavior of
${\rm Tr}\ \hat{W}^{P}(R,T)$ and the slope parameter $\sigma^{P}$,
which is formally written as
$$\displaystyle\sigma^{P}\equiv-\lim_{R,T\rightarrow\infty}\frac{1}{RT}{\rm ln}%
\{{\rm Tr}\ \hat{W}^{P}(R,T)\}.$$
(30)
IV.3 Corresponding Dirac-mode projected inter-quark potential
For the estimation of
the slope parameter $\sigma^{P}$ from ${\rm Tr}\ \hat{W}^{P}(R,T)$,
we define the corresponding Dirac-mode projected inter-quark potential,
$$\displaystyle V^{P}(R)\equiv-\lim_{T\to\infty}\frac{1}{T}{\rm ln}\{{\rm Tr}\ %
\hat{W}^{P}(R,T)\},$$
(31)
which is also manifestly gauge-invariant.
To be strict, due to the non-locality
appearing in the Dirac-mode projection,
$V^{P}(R)$ does not have a definite meaning of
the static potential.
However, it is still useful to obtain $\sigma^{P}$
in Eq.(30) from ${\rm Tr}\hat{W}^{P}(R,T)$.
In fact, $\sigma^{P}$ is obtained from the infrared slope of $V^{P}(R)$.
Note also that, in the unprojected case of $\hat{P}=1$,
the ordinary inter-quark potential is obtained
apart from an irrelevant constant,
$$\displaystyle V(R)$$
$$\displaystyle=$$
$$\displaystyle-\lim_{T\to\infty}\frac{1}{T}{\rm ln}\{{\rm Tr}\ \hat{W}(R,T)\}$$
(32)
$$\displaystyle=$$
$$\displaystyle-\lim_{T\to\infty}\frac{1}{T}{\rm ln}\langle W(R,T)\rangle+{\rm
irrelevant%
}\ {\rm const.},$$
because of ${\rm Tr}\ \hat{W}=\langle W\rangle\cdot{\rm Tr}\ 1$, as was derived in Eq.(17).
V Analysis of Confinement in terms of Dirac Modes in QCD
We consider various projection space $A$ in the Dirac-mode space,
e.g., IR-cut or UV-cut of Dirac modes.
With this Dirac-mode expansion and projection formalism,
we calculate the Dirac-mode projected Wilson loop ${\rm Tr}W^{P}(R,T)$
in a gauge-invariant manner.
In particular, using IR-cut of the Dirac modes,
we directly investigate the relation between chiral symmetry breaking
and confinement as the area-law behavior of the Wilson loop,
since the low-lying Dirac modes are
responsible to chiral symmetry breaking.
As a technical difficulty of this formalism, we have to deal with
huge dimensional matrices and their products.
Actually, the total matrix dimension of
$\langle m|\hat{U}_{\mu}|n\rangle$ is (Dirac-mode number)${}^{2}$.
On the $L^{4}$ lattice, the Dirac-mode number is
$L^{4}\times N_{c}\times$ 4, which can be reduced to be $L^{4}\times N_{c}$,
using the KS formalism KS75 ; R05 , as mentioned in Sec.II-B.
The actual number of the independent Dirac eigenvalue
$\lambda_{n}$ is about $L^{4}\times N_{c}/2$,
due to the chiral property of
/
$D$
, i.e.,
pairwise appearance of $\pm\lambda_{n}$.
Even for the projected operators, where the Dirac-mode space is
restricted, the matrix is generally still huge.
In addition, we have to deal with the product of
the huge matrices $\langle m|\hat{U}_{\mu}|n\rangle$
in calculating the Wilson loop.
Then, at present, we use a small-size lattice
in the numerical calculation.
In this paper, we perform the SU(3) lattice QCD Monte Carlo calculation
with the standard plaquette action
at $\beta=5.6$ on $6^{4}$ at the quenched level,
using the pseudo-heat-bath algorithm.
The ordinary periodic boundary condition is used for the link-variable.
The gauge configurations are taken
every 500 sweeps after 10,000 sweeps thermalization,
and 20 gauge configurations are used for each analysis.
At $\beta=5.6$, the lattice spacing $a$ is estimated as
$a\simeq 0.25{\rm fm}$, i.e., $a^{-1}\simeq 0.8{\rm GeV}$,
which leads to the string tension $\sigma\simeq 0.89{\rm GeV/fm}$
in the inter-quark potential.
(This estimate is done also on a larger volume lattice.)
Then, the total volume is $V=(6a)^{4}\simeq(1.5{\rm fm})^{4}$,
and the momentum cutoff is $\pi/a\simeq 2.5{\rm GeV}$.
On the $6^{4}$ lattice, the Dirac-mode number is
$6^{4}\times 3\times 4=15,552$, which is reduced
to be $6^{4}\times 3=3,888$ using the KS formalism.
In fact, the KS Dirac operator $\eta_{\mu}D_{\mu}$ and
the KS-reduced matrix element $\langle m|\hat{U}_{\mu}|n\rangle$
are expressed by $3,888\times 3,888$ matrix.
Considering the pairwise appearance of $\lambda_{n}$ and $-\lambda_{n}$,
the actual number of the independent Dirac eigenvalue $\lambda_{n}$
is reduced to be around $6^{4}\times 3/2=1,944$.
To diagonalize the KS Dirac operator $\eta_{\mu}D_{\mu}$,
we use LAPACK LAPACK .
For the statistical error on the lattice data,
we adopt the jackknife error estimate R05 .
We show in Fig.1(a) the spectral density $\rho(\lambda)$ of
the QCD Dirac operator
/
$D$
.
The chiral property of
/
$D$
leads to
$\rho(-\lambda)=\rho(\lambda)$.
Figure 1(b) is the IR-cut Dirac spectral density
$$\displaystyle\rho_{\rm IR}(\lambda)\equiv\rho(\lambda)\theta(|\lambda|-\Lambda%
_{\rm IR})$$
(33)
with the IR-cutoff $\Lambda_{\rm IR}=0.5a^{-1}\simeq 0.4{\rm GeV}$.
Note that, using the eigenvalue $\lambda_{n}$,
the quark condensate
$\langle\bar{q}q\rangle$ is obtained as
$$\displaystyle\langle\bar{q}q\rangle$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{V}{\rm Tr}\frac{1}{{\ooalign{/\hfil\cr$D$}}+m}=-\frac{1%
}{V}\sum_{n}\frac{1}{i\lambda_{n}+m}$$
(34)
$$\displaystyle=$$
$$\displaystyle-\frac{1}{V}\left(\sum_{\lambda_{n}>0}\frac{2m}{\lambda_{n}^{2}+m%
^{2}}+\frac{\nu}{m}\right),$$
where $\nu$ is the total number of the zero mode of
/
$D$
.
Here, the non-zero eigenvalues appear as pairwise,
which makes $\langle\bar{q}q\rangle$ real.
(In lattice QCD, one has to take account of the doubler contribution,
which can be regarded as flavor at the quenched level.)
Then, in the presence of the IR cut $\Lambda_{\rm IR}$
for the Dirac eigen-mode, the quark condensate is obtained as
$$\displaystyle\langle\bar{q}q\rangle_{\Lambda_{\rm IR}}=-\frac{1}{V}\sum_{%
\lambda_{n}\geq\Lambda_{\rm IR}}\frac{2m}{\lambda_{n}^{2}+m^{2}}.$$
(35)
We show in Fig.2 the lattice QCD result of
the quark condensate $\langle\bar{q}q\rangle_{\Lambda_{\rm IR}}$
as the function of the current quark mass $m$
in the presence of IR cut $\Lambda_{\rm IR}$.
By removing the low-lying Dirac modes,
the chiral condensate $\langle\bar{q}q\rangle$
is largely reduced, reflecting the Banks-Casher relation.
Actually, directly from lattice QCD calculation,
we find a large reduction of the chiral condensate
in the presence of the IR cut
$\Lambda_{\rm IR}=0.5a^{-1}\simeq 0.4{\rm GeV}$,
$$\displaystyle\frac{\langle\bar{q}q\rangle_{\Lambda_{\rm IR}}}{\langle\bar{q}q%
\rangle}\simeq 0.02,$$
(36)
around the physical region of
$m\simeq 0.006a^{-1}\simeq 5{\rm MeV}$ PDG , as shown Fig.2.
Now, let us consider the removal of
the coupling to the low-lying Dirac modes
from the Wilson loop $\langle W(R,T)\rangle$.
Figure 3 shows the Dirac-mode projected
Wilson loop $\langle W^{P}(R,T)\rangle\equiv{\rm Tr}\hat{W}^{P}(R,T)$
after removing low-lying Dirac modes,
which is obtained in lattice QCD with the IR-cut of
$\rho_{\rm IR}(\lambda)\equiv\rho(\lambda)\theta(|\lambda|-\Lambda_{\rm IR})$
with the IR-cutoff $\Lambda_{\rm IR}=0.5a^{-1}$.
Even after removing the coupling to the low-lying Dirac modes,
which are responsible to chiral symmetry breaking,
the Dirac-mode projected Wilson loop is found to
obey the area law as
$$\displaystyle\langle W^{P}(R,T)\rangle\propto e^{-\sigma^{P}RT},$$
(37)
and the slope parameter $\sigma^{P}$
corresponding to the string tension
or the confinement force is almost unchanged as
$$\displaystyle\sigma^{P}\simeq\sigma.$$
(38)
In fact, the confinement property seems to be kept
in the absence of the low-lying Dirac modes or
the essence of chiral symmetry breaking SGIY11 .
This result indicates that one-to-one correspondence does not
hold for confinement and chiral symmetry breaking in QCD.
Next, to estimate the slope parameter $\sigma^{P}$,
we consider the potential $V^{P}(R)$ obtained
from $\langle W^{P}(R,T)\rangle$.
Figure 4 shows
the “effective mass” of the inter-quark potential
$V_{\rm eff}(R,T)\equiv{\rm ln}[\langle W^{P}(R,T)\rangle/\langle W^{P}(R,T+1)\rangle]$
after removing the low-lying Dirac modes,
plotted against $T$ at each $R$.
One finds the “plateau” or the stability of
the effective mass $V_{\rm eff}(R,T)$ against $T$,
which means the dominance of the ground-state component.
Similarly in the standard procedure
to obtain potentials in lattice QCD R05 ; TS02 ,
we determine the inter-quark potential $V^{P}(R)$
by the exponential fit of the Wilson loop
$$\displaystyle\langle W^{P}(R,T)\rangle=Ce^{-V^{P}(R)T}$$
(39)
for $T=1,2,3$, which corresponds to
the plateau region of $T=1,2$ in $V_{\rm eff}(R,T)$.
Figure 5 shows the Dirac-mode projected
inter-quark potential $V^{P}(R)$
after removing low-lying Dirac modes below
the IR-cutoff $\Lambda_{\rm IR}=0.5a^{-1}$.
No significant change is observed
on the inter-quark potential besides an irrelevant constant,
that is, the slope parameter $\sigma^{P}$ is almost unchanged,
even after removing the low-lying Dirac modes.
On the potential argument,
we comment on the non-locality stemming from the Dirac-mode projection,
which makes the link-variable extended and makes the potential meaning vague.
This non-locality appears hyper-cubic symmetrically in the four-dimensional
space-time, and its effect would be maximal for the IR-cut case.
As a whole, such a non-locality makes the potential flat,
due to the spatial averaging. (As an extreme example, the “potential”
between wall-like sources is completely flat.)
However, our obtained potential is almost the same as the original
confining one, in spite of the possible flattening effect by the non-locality.
Therefore, regardless of the non-locality, the confinement is kept
after cutting off the low-lying Dirac modes.
(Since no flattening effect is observed in this projection, the non-locality
effect would not be significant, at least for the argument of confinement.)
As another way to clarify the confinement on the periodic lattice,
we also investigate the Polyakov loop $\langle L_{P}\rangle\equiv\langle{\rm tr}\prod_{t=1}^{L}U_{4}(\vec{x},t)%
\rangle/3$
and the center $Z_{3}$-symmetry R05 in terms of the Dirac-mode projection.
The Polyakov loop $\langle L_{P}\rangle$, which is usually used
at finite temperature, can be also applied to our temporally periodic system
on the link-variable, and it physically relates to
the quark single-particle energy and the $Z_{3}$-symmetry R05 .
Note that the non-locality effect is less significant for
the Polyakov loop $\langle L_{P}\rangle$ or the quark single-particle energy.
Now, we calculate the Polyakov loop with cutting off
of the low-lying Dirac modes,
$$\displaystyle\langle L_{P}\rangle_{\rm IR}\equiv\frac{1}{3}\frac{1}{V}\langle{%
\rm Tr}(\prod_{k=1}^{L}\hat{U}_{4}^{P})\rangle=\frac{1}{3}\frac{1}{V}\langle{%
\rm Tr}\{(\hat{U}_{4}^{P})^{L}\}\rangle,$$
(40)
and its scatter plot,
using the same lattice ($6^{4},\beta=5.6$) and
the same IR cutoff $\Lambda_{\rm IR}=0.5a^{-1}$.
In the use of the full Dirac modes, i.e., $\hat{P}=1$,
$\langle L_{P}\rangle_{\rm IR}$ coincides with $\langle L_{P}\rangle$.
We show in Fig.6 the scatter plot of the Polyakov loop
$\langle L_{P}\rangle_{\rm IR}$ after cutting off the low-lying Dirac modes
below $\Lambda_{\rm IR}=0.5a^{-1}$.
We find that the IR-cut Polyakov loop $\langle L_{P}\rangle_{\rm IR}$
remains to be almost zero, i.e., $\langle L_{P}\rangle_{\rm IR}\simeq 0$,
which corresponds to the $Z_{3}$-unbroken phase.
In fact, even after removing the low-lying Dirac modes,
which are responsible to chiral symmetry breaking,
the single-quark energy is extremely large and
the system is in the $Z_{3}$-unbroken confinement phase.
We also investigate the UV-cut of Dirac modes in lattice QCD,
using the UV-cut Dirac spectral density
$\rho_{\rm UV}(\lambda)\equiv\rho(\lambda)\theta(\Lambda_{\rm UV}-|\lambda|)$
with the UV-cutoff $\Lambda_{\rm UV}=2a^{-1}\simeq 1.6{\rm GeV}$.
In this case, unlike the IR cut,
the chiral condensate is almost unchanged,
and chiral symmetry breaking is almost kept.
We show in Fig.7 the UV-cut Wilson loop and
the corresponding inter-quark potential,
after removing the UV Dirac modes.
We find that the area-law behavior of the Wilson loop
and the slope parameter $\sigma^{P}$
are almost unchanged by the UV-cut of the Dirac modes.
This result seems consistent with the pioneering lattice study
of Synatschke-Wipf-Langfeld SWL08 :
they found that the confinement potential is almost reproduced
only with low-lying Dirac modes,
using the spectral sum of the Polyakov loop BBGH08 ; G06BGH07 .
Furthermore, we examine “intermediate(IM)-cut”, where
a certain part of $\Lambda_{1}<|\lambda_{n}|<\Lambda_{2}$
of Dirac modes is removed.
Unfortunately, when the wide region of Dirac modes is removed,
the statistical error becomes quite large
for the Dirac-mode projected Wilson loop.
Here, we remove the IM Dirac modes of
$0.5-0.8[a^{-1}]$, $0.8-1.0[a^{-1}]$,
and $1.0-1.2[a^{-1}]$, respectively,
and investigate the corresponding IM-cut Wilson loop and
the corresponding inter-quark potential
in each case, as shown in Fig.8.
For each case, the area-law behavior of the Wilson loop
and the slope parameter $\sigma^{P}$
are found to be almost unchanged by the IM-cut of the Dirac modes.
Thus, from the above lattice QCD results,
we conclude that there is no specific region of
the Dirac modes responsible to the confinement in QCD,
unlike the chiral symmetry breaking.
Instead, we conjecture that the “seed” of confinement is
distributed not only in low-lying Dirac modes but also
in a wider region of the Dirac-mode space.
VI Summary and Discussions
We have developed a manifestly gauge-covariant expansion and projection
using the eigen-mode of the QCD Dirac operator
${\ooalign{/\hfil\cr$D$}}=\gamma^{\mu}D^{\mu}$.
With this method, we have performed a direct investigation
of correspondence between confinement and chiral symmetry breaking
in SU(3) lattice QCD on the $6^{4}$ periodic lattice
at $\beta$=5.6 at the quenched level.
We have found that
the Wilson loop remains to obey the area law,
and the slope parameter corresponding to the string tension or
the confinement force is almost unchanged,
even after removing the low-lying Dirac modes,
which are responsible to chiral symmetry breaking.
We have also found that the Polyakov loop remains to be almost zero
even without the low-lying Dirac modes,
which indicates the $Z_{3}$-unbroken confinement phase.
These results indicate that one-to-one correspondence does not
hold for between confinement and chiral symmetry breaking in QCD.
As a caution, we have used a coarse and small lattice,
because of the technical difficulty to diagonalize the full Dirac operator.
In particular, the box size of our lattice volume is about 1.5fm.
In fact, to be strict, this region we survey is the intermediate distance,
of which confining behavior is rather important for the quark-hadron physics.
To obtain more definite conclusion, especially
on the asymptotic confining behavior of the potential,
it is desired to perform larger-volume lattice QCD calculations
and to cut various wider region of the Dirac modes,
although it is technically quite difficult.
Our strategy is to investigate the relation
between the nonperturbative properties of QCD,
by extracting or removing
the essence of chiral symmetry breaking.
This is similar to the demonstration of Abelian/monopole dominance
tH81 ; SST95 ; M95 ; W95 ; KSW87 ; SNW94 ; STSM95 ; KMS95 ; AS99 ; EI82
or center/vortex dominance HFGHO08 ; DFGO97 ; KT98
for nonperturbative properties.
However, while the previous scenario has been done
in a specific gauge, our new method is manifestly gauge-invariant.
In this analysis, we have carefully amputated only the
“essence of chiral symmetry breaking”
by cutting off the low-lying Dirac modes.
Then, we have artificially realized the
“confined but chiral restored situation” in QCD.
Recently, Lang and Schrock studied the hadron spectra
after the cut of the low-lying Dirac modes LS11 .
Since the quark propagator is directly expressed
with the Dirac operator
/
$D$
,
the Dirac-mode projection is straightforward,
and complicated projection procedure is not necessary in such studies.
In their study, although the confinement was not checked,
the appearance of hadronic spectra seems to suggest
the existence of the confinement force,
even after cutting the low-lying Dirac modes.
Next, we comment on the possible relation among
confinement, chiral symmetry breaking, and monopoles in QCD.
There is a close relation
between confinement and chiral symmetry breaking
through the monopoles in the MA gauge SST95 ; M95 ; W95 .
The monopole would be essential degrees of freedom for
most nonperturbative QCD: confinement SNW94 ,
chiral symmetry breaking M95 ; W95 , and instantons STSM95 .
In fact, removing the monopole would be “too fatal”
for the nonperturbative properties, so that
nonperturbative QCD phenomena are simultaneously lost by their cut.
On the approximate coincidence of
the critical temperatures of deconfinement and chiral restoration,
a large change of monopoles may lead to both phase transitions M95 ,
since the global connection of the monopole current
seems to be largely changed around the QCD phase transition KMS95 .
As for the recent finite-temperature QCD analysis,
a lattice QCD group has reported
a certain difference between the “critical temperatures” of
deconfinement and chiral restoration,
which are determined by the susceptibility peak of
the Polyakov loop and chiral condensate, respectively AFKS06 .
This may be also an indirect evidence of
“confinement $\neq$ chiral symmetry breaking” in QCD.
Next, we briefly discuss the role of low-lying Dirac modes
in the viewpoint of instantons in QCD.
The Dirac zero-mode associated with an instanton
is localized around it S94 .
However, the localized objects are hard to contribute to
the large-distance phenomenon of confinement,
although such low-lying Dirac modes contribute to
chiral symmetry breaking.
Recall that instantons contribute to chiral symmetry breaking,
but do not directly lead to confinement S94 .
Then, as a thought experiment,
if only instantons can be carefully removed from the QCD vacuum,
confinement properties would be almost unchanged,
but the chiral condensate is largely reduced,
and accordingly some low-lying Dirac modes disappear.
Thus, in this case, confinement is almost unchanged,
in spite of the large reduction of low-lying Dirac modes.
If the relation between confinement and chiral symmetry breaking
is not one-to-one in QCD, richer phase structure is expected in QCD.
For example, the phase transition point can be different
between deconfinement and chiral restoration
in the presence of strong electro-magnetic fields,
because of their nontrivial effect on chiral symmetry ST9193 .
It is also interesting to investigate the similar analysis
at finite temperatures in lattice QCD.
The full QCD calculation in this direction is also an interesting subject.
Acknowledgements
The authors thank Prof. K. Langfeld and Dr. T. Misumi for useful comments.
H.S. is supported in part by the Grant for Scientific Research
[(C) No. 23540306, Priority Areas “New Hadrons” (E01:21105006)]
and T.I. is supported by Grant-in-Aid for JSPS Fellows (No.23-752),
from the Ministry of Education, Culture, Science and Technology
(MEXT) of Japan.
This work is supported by the Global COE Program,
“The Next Generation of Physics, Spun from Universality and Emergence”.
The lattice QCD calculations are done on NEC SX-8R at Osaka University.
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Probing Lepton Flavor Violating decays in MSSM with Non-Holomorphic Soft Terms
Utpal Chattopadhyay,
b
Debottam Das,
a
Samadrita Mukherjee
tpuc@iacs.res.in
debottam@iopb.res.in
tpsm9@iacs.res.in
School of Physical Sciences, Indian Association for the Cultivation of Science,
2A & B Raja S.C. Mullick Road, Jadavpur,
Kolkata 700 032, IndiaInstitute of Physics, Bhubaneswar, Odisha 751005, and HBNI, Mumbai, India
Abstract
The Minimal Supersymmetric Standard Model (MSSM)
can be extended to include non-holomorphic trilinear
soft supersymmetry (SUSY) breaking interactions that may have distinct signatures. We consider non-vanishing off-diagonal
entries of the coupling matrices associated with holomorphic (of MSSM)
and non-holomorphic trilinear terms corresponding to sleptons with elements $A^{l}_{ij}$ and $A^{\prime l}_{ij}$. We first improve the MSSM charge breaking minima
condition of the vacuum to include the off-diagonal entries $A^{l}_{ij}$ (with $i\neq j$).
We further extend this analysis for non-holomorphic trilinear interactions.
No other sources of lepton flavor violation
like that from charged slepton matrices are considered. We constrain the interaction terms
via the experimental limits of processes like charged
leptons decaying with lepton flavor violation (LFV) and Higgs boson decaying to charged leptons with
LFV. Apart from the leptonic decays we compute all the
three neutral LFV Higgs boson decays of MSSM. We find that an analysis with non-vanishing $A^{\prime}_{e\mu}$ involving
the first two generations of sleptons receives the dominant constraint
from $\mu\to e\gamma$. On the other hand, $A^{\prime}_{e\tau}$ or
$A^{\prime}_{\mu\tau}$ can be constrained from the CMS 13 TeV analysis giving limits to the
respective Yukawa couplings via considering
SM Higgs boson decaying into $e\tau$ or $\mu\tau$ final states. Contributions from
$A^{l}_{ij}$ is too little to have any significance compared to the large effect from $A^{\prime l}_{ij}$.
Keywords:Supersymmetry, Non-holomorphic soft terms, Lepton flavor violation
IP-BBSR-2019-8
1 Introduction
The Higgs data is gradually
drifting towards the Standard Model (SM) expectations Khachatryan:2016vau ever since the
first observation of a new resonance at the Large Hadron Collider (LHC)
Aad:2012tfa ; Chatrchyan:2012xdj in 2012. Still, SM is far away to be a complete description
of particle physics in view of many theoretical aspects and a few experimental data.
Indeed, the unknown new physics (NP) has always been the driving force
in studying particle physics for many decades to explain e.g.,
the existence of dark matter, neutrino masses, or matter-anti matter asymmetries. Most of these NP models
offer new particles with new interactions that could possibly be tested at the LHC. Apart from this direct test, one can hope
to find the NP signatures via indirect search experiments involving flavor physics, e.g., through the
dedicated experiments that search for quark
or charged lepton flavor violations (cLFV) like
$b\rightarrow s\gamma$ or $\mu\rightarrow e\gamma$. Among the flavor violating observables,
the cLFV processes are of particular interest. The reason is that in the context of Standard Model (SM) or in the
minimal extension of the SM that includes the Yukawa interactions in the neutral lepton sector, the decay rates involving
cLFV processes are strongly suppressed (e.g., $BR(\mu\rightarrow e\gamma)\sim 10^{-55}$)petkov .
This can be attributed to the tinyness of the neutrino masses which
is the only source of cLFV processes. However, any extension of SM, mainly
in the leptonic sector may offer new particles or new interactions with the
SM leptons. This can potentially change the cLFV decay
rates drastically Cheng:1976uq ; Bilenky:1977du (for a review see for instance Raidal:2008jk ).
The Minimal Supersymmetric Standard Model (MSSM), the supersymmetric extension
of SM with the two Higgs doublets in general may have a large number of
flavor violating couplings through soft SUSY breaking interactions SUSYreviews1 ; SUSYbook1 ; SUSYbook2 ; SUSYreviews2 ; Gabbiani:1996hi .
The lepton sector is particularly important in this context.
Interestingly, the flavor violating soft SUSY breaking parameters may also be generated radiatively.
In fact, guided by the origin of cLFV processes, one may broadly classify a few MSSM and beyond the MSSM scenarios as follows:
•
In the extensions of MSSM, one may connect the origin of the cLFV to the masses for
neutrinos, the existence of which have been strongly established
through neutrino oscillation experiments
deSalas:2017kay ; Esteban:2016qun .
One of the most attractive possibilities would be
to consider a seesaw
mechanism seesaw:I ; seesaw:II ; seesaw:III that can also be generalized
in the framework of SUSY models (i.e. the SUSY seesaw) susy-seesaw .
In the simplest example i.e, type-I SUSY seesaw, the right handed neutrino Yukawa couplings
that generate neutrino masses, may also radiatively induce the SUSY soft-breaking left handed slepton
mass matrices ($M_{{\tilde{L}}_{ij}}^{2}$),
leading to flavor violations at
low energies Borzumati:1986qx ; hisano . This can substantially influence
lepton flavor violating decays of the types
($l_{j}\rightarrow l_{i}\gamma)$) or three-body lepton decays
$l_{j}\rightarrow 3l_{i}$ through photon, $Z$ or Higgs penguins and the flavor
violating decays of the Higgs scalars
(see e.g., hisano ; Hamzaoui:1998nu ; DiazCruz:1999xe ; Isidori:2001fv ; Babu:2002et ; masieroall ; Paradisi:2005tk ; Brignole:2003iv ; Brignole:2004ah ; Han:2000jz ; Arganda:2004bz ; Arganda:2005ji ; Gomez:2015ila ; Gomez:2017dhl ; Aloni:2015wvn ; Arhrib:2012ax ; Arana-Catania:2013xma ; Abada:2014kba ; Hammad:2016bng ).
Other important probes are semileptonic $\tau$-decays, $\mu-e$ conversion of nucleus etc. The flavor changing processes involving
Higgs scalars potentially become large for large $\tan\beta$
Babu:2002et ; Brignole:2003iv .
However here
the typical mass scales
of the extra particles (such as right handed neutrinos) are in
general very high, often close to the gauge coupling unification scale. An attractive alternative is the inverse seesaw
scenario, where the presence
of comparatively light right-handed neutrinos and sneutrinos can enhance
the flavor violating decays
Deppisch ; mypaper1 ; mypaper2 ; Arganda:2014dta ; Arganda:2015naa ; Arganda:2015uca . In addition to generating $M_{{\tilde{L}}_{ij}}^{2}$ radiatively,
the right handed neutrino extended models
may as well be embedded in grand unified theory (GUT) framework
Barbieri:1994pv ; Barbieri:1995tw ; Hall:1985dx ; Dev:2009aw .
•
Though neutrino mass models, particularly in presence
of new states imply cLFV, the later does not necessarily imply neutrino mass
generation.
The simplest example is the R-parity
conserving MSSM. Here
direct sources of flavor violation are in the off-diagonal soft terms of the
slepton mass matrices and
trilinear coupling matrices (see e.g., SUSYreviews2 )(specifically through
$M_{\tilde{e}}^{2},A_{f}$). One may probe the non-zero off-diagonal elements of all the soft SUSY breaking terms
($A_{f},M_{\tilde{L}}^{2},M_{\tilde{e}}^{2}$) which may
induce cLFV processes through loops mediated by sleptons-neutralinos and/or sneutrinos-charginos.
This can also be realized in a High scale SUSY breaking
model, e.g., in a supergravity or superstring inspired scenarios, where
non-universal soft terms can be realized
in the high-scale effective Lagrangian (see, for example Ref:Brignole:1997dp and references therein)
apart from running via Renormalization Group (RG) evolution that itself may generate
flavor violation SUSYreviews2 . 111We also note in passing that
intricacies related to the large number of soft breaking parameters in the
cLFV computations and also the inter-generation mixings in the general MSSM can be evaded completely in a High scale SUSY
model
where SUSY breaking is communicated in a flavor blind manner SUSYreviews2 . Popular examples are mSUGRA, anomaly-mediated supersymmetry
breaking (AMSB) or gauge-mediated supersymmetry breaking (GMSB).
Although, in general, cLFV processes through radiatively induced $M_{\tilde{L}}^{2}$ may look more appealing,
in some cases it may be somewhat restrictive in obtaining any significant amount of flavor violating branching ratios.
On the contrary, being free from the
constraints of the masses of neutrinos, soft SUSY breaking parameters can lead to reasonably large decay rates
which may be interesting and could be testable in near future.
In this analysis, we would go one step further. We would limit ourselves within the MSSM field content
augmented by most general soft SUSY breaking terms
without considering their high scale origin.
In its generic form MSSM includes only holomorphic trilinear soft SUSY breaking
terms SUSYreviews1 ; SUSYbook1 ; SUSYbook2 ; SUSYreviews2 . However,
in a most general framework, it has been shown
that certain non-holomorphic supersymmetry breaking terms may qualify as soft terms when there is no gauge singlet field present
Jack:nh ; Martin:nh ; Jack:nh1 ; Haber:wh ; Hetherington:2001bk . Such a
consideration can be phenomenologically interesting.
For example, if MSSM soft SUSY breaking
sector is extended to include $A_{f}^{\prime}\phi^{2}\phi^{*}$ type of interactions,
one may find that an SM like CP even Higgs boson with mass $\sim$125 GeV can be achieved with relatively
lighter squarks with the help of the specific $A^{\prime}_{t}$
Chattopadhyay:2016ivr , the relevant coupling from non-holomorphic trilinear interaction. Similarly, the non-holomorphic (NH)
terms may also be helpful
to fulfill constraints
from rare B-decays (viz. $Br(B\rightarrow X_{s}\gamma)$, $Br(B_{s}\rightarrow\mu^{+}\mu^{-})$ etc) both in a phenomenological MSSM (pMSSM) like scenario Chattopadhyay:2016ivr
or in some high scale model like constrained MSSM (CMSSM) Un:2014afa ; Ross:2016pml ; Ross:2017kjc or minimal Gauge Mediated Supersymmetry Breaking
(mGMSB)Chattopadhyay:2017qvh .
Another interesting feature is that a small NH trilinear coupling (namely $A^{\prime}_{\mu}$) may be capable to attune the inflexible constraints of $(g-2)_{\mu}$
Chattopadhyay:2016ivr . Focusing only on the leptonic sector,
the playground associated with the NH soft
terms may not be completely free, rather there can be strong constraints appearing
from
different lepton flavor violating decays via their off-diagonal entries. This is of-course similar to holomorphic trilinear
interactions of MSSM, apart from an enhancement by $\tan\beta$ with $A^{\prime}_{i}$ associated with down type of quark and lepton.
We will consider the slepton mass squared matrix to be diagonal, and consider that the only source of cLFV
to be the holomormorphic and non-holomorphic trilinear coupling matrices, namely $A_{f}~{}\&~{}A_{f}^{\prime}$. For the sake of
explicit understanding we will scan either $A_{f}$ or $A_{f}^{\prime}$ at a time
to find the allowance of associated off-diagonal elements
under the present and future experimental sensitivities of
different cLFV observables. In order to perform this analysis, a more important checkpoint is avoidance of any
dangerous charge and color breaking global minima (CCB). It is known that a large trilinear coupling (diagonal) in general,
lead to unphysical or metastable CCB minima.
For lepton flavor it is only the charge breaking (CB) of vacuum that is of concern, but it requires the
involvement of the off-diagonal entries of the trilinear couplings.
In this analyses we will necessarily improve the charge breaking constraint for a general
trilinear coupling matrix having non-vanishing entries in its diagonal and
off-diagonal elements.
So, we should survey if there is any violation in
charge breaking minima condition and always ensure that the electroweak symmetry breaking minimum i.e., the global minimum is a charge conserving one.
The rest of the work is ordered as follows. In Section 2 we discuss the theoretical framework that includes the slepton mass matrices in presence of the
NH terms along with the cLFV observables which we would consider in the analysis.
Section 3 covers the examination of analytical structure of charge breaking minima in the context of our model.
Numerical results are presented in Section 4.
Here we take the inter-generational mixings in the trilinear couplings of the slepton fields for both holomorphic
and non-holomorphic couplings. The free rise of these couplings are limited via charge breaking minima and
also from the non-observation of any signals at the LHC apart from the same experiments that search for cLFV processes. Here
we also consider the future sensitivity of the experiments.
In closing we conclude in Section 5.
2 Theoretical Framework
We focus on the general lepton flavor mixing through left and right
slepton mixing in the MSSM with R-parity conserved. We will further include contributions
from non-standard soft supersymmetry breaking interactions.
The superpotential is given by,
$$\begin{split}\displaystyle W_{MSSM}&\displaystyle=\bar{U}{\bf y_{u}}Q\cdot H_{%
u}-\bar{D}{\bf y_{d}}Q\cdot H_{d}-\bar{E}{\bf y_{e}}L\cdot H_{d}+\mu H_{u}%
\cdot H_{d}.\end{split}$$
(1)
where $y$’s are the Yukawa matrices in flavor space.
The MSSM soft terms read SUSYbook1 ; SUSYbook2 ; SUSYreviews2 ,
$$\begin{split}\displaystyle-\mathcal{L}_{soft}^{MSSM}&\displaystyle=\frac{1}{2}%
(M_{3}\tilde{g}\tilde{g}+M_{2}\tilde{W}\tilde{W}+M_{1}\tilde{B}\tilde{B}+c.c)%
\\
&\displaystyle\quad+({\tilde{u}^{*}}_{iR}{\bf A_{u}}_{ij}\tilde{q}_{jL}\cdot H%
_{u}+{\tilde{d}^{*}}_{iR}{\bf A_{d}}_{ij}\tilde{q}_{jL}\cdot H_{d}+{\tilde{e}^%
{*}}_{iR}{\bf A_{l}}_{ij}\tilde{\ell}_{jL}\cdot H_{d}+h.c.)\\
&\displaystyle\quad+\tilde{q}^{\dagger}_{iL}{\bf M^{2}_{\tilde{q}}}_{ij}\tilde%
{q}_{jL}+\tilde{\ell}^{\dagger}_{iL}{\bf M^{2}_{\tilde{L}}}_{ij}\tilde{\ell}_{%
jL}+{\tilde{\tilde{u}}^{*}}_{iR}{\bf M^{2}_{\tilde{u}}}_{ij}{\tilde{u}^{*%
\dagger}}_{jR}+{\tilde{d}^{*}}_{iR}{\bf M^{2}_{\tilde{d}}}_{ij}{\tilde{d}^{*%
\dagger}}_{jR}\\
&\displaystyle\quad+{\tilde{e}^{*}}_{iR}{\bf M^{2}_{\tilde{e}}}_{ij}{\tilde{e}%
^{*\dagger}}_{jR}+m_{H_{u}}^{2}H_{u}^{*}H_{u}+m_{H_{d}}^{2}H_{d}^{*}H_{d}+(B_{%
\mu}H_{u}.H_{d}+c.c).\end{split}$$
(2)
MSSM can be extended to include a set of possible additional
non-holomorphic trilinear soft SUSY breaking terms as given below Jack:nh ; Jack:nh1 ; Martin:nh ,
$$\begin{split}\displaystyle-\mathcal{L^{\prime}}_{soft}^{NH}&\displaystyle%
\supset\tilde{u}^{*}_{iR}{\bf A_{u}^{\prime}}_{ij}\tilde{q}_{jL}\cdot H_{d}^{*%
}+\tilde{d}^{*}_{iR}{\bf A_{d}^{\prime}}_{ij}\tilde{q}_{jL}\cdot H_{u}^{*}+%
\tilde{e}^{*}_{iR}{\bf A_{e}^{\prime}}_{ij}\tilde{\ell}_{jL}\cdot H_{u}^{*}+h.%
c.~{}.\\
\end{split}$$
(3)
Here $i,j=1,2,3$ denote indices in the fermion (f) family space.
In contrast to eq.2 here the sfermions ${{\tilde{u}}^{*}}_{iR}$
and ${\tilde{q}}_{iL}$
couple with $H_{d}^{*}$ instead of $H_{u}$.
The trilinear coupling matrices are typically scaled and
characterized by the quark/charged lepton Yukawa couplings.
The effect of above non-standard or non-holomorphic terms (Eq:3) are reflected in the mass matrices
and mixing angles of physical sparticles. Since we are concerned with general flavor mixing in the slepton sector,
the general form of $6\times 6$
slepton mass squared matrix is written in the electroweak basis ($\tilde{e}_{L},\tilde{\mu}_{L},\tilde{\tau}_{L},\tilde{e}_{R},\tilde{\mu}_{R},%
\tilde{\tau}_{R}$)
in terms of left and right handed blocks as given below.
$$\displaystyle M_{\tilde{l}}=$$
$$\displaystyle\left(\begin{matrix}M^{2}_{\tilde{l}LL}&M^{2}_{\tilde{l}LR}\\
M^{2}_{\tilde{l}RL}&M^{2}_{\tilde{l}RR}\\
\end{matrix}\right).$$
(4)
In the above, each block is a $3\times 3$ matrix where one has,
$$\displaystyle M^{2}_{{\tilde{l}LL}_{ij}}$$
$$\displaystyle=M^{2}_{\tilde{L}_{ij}}+(M_{Z}^{2}(-\frac{1}{2}+\sin^{2}\theta_{W%
})\cos 2\beta+m_{\ell_{i}}^{2})\delta_{ij},$$
(5)
$$\displaystyle M^{2}_{{\tilde{l}RR}_{ij}}$$
$$\displaystyle=M^{2}_{\tilde{e}_{ij}}+(-M_{Z}^{2}\sin^{2}\theta_{W}\cos 2\beta+%
m_{\ell_{i}}^{2})\delta_{ij}.$$
(6)
Here $\beta$ is defined via $\tan\beta=\frac{v_{u}}{v_{d}}$, the ratio of Higgs vacuum expectation values.
$\theta_{W}$ and $M_{Z}$ refer to the Weinberg angle and the Z-boson mass respectively
whereas $m_{\ell_{i}}$ refers to lepton masses respectively.
The non-holomorphic trilinear couplings modify slepton left-right mixings.
For MSSM with non-holomorphic soft
terms one has the following 222Flavor mixing through trilinear couplings may be generated radiatively in presence
of right handed neutrinos (see e.g.,hisano ; Gomez:2015ila )..
$$\displaystyle M^{2}_{\tilde{l}LR}=$$
$$\displaystyle\left(\begin{matrix}(A_{e}-(\mu+A_{e}^{\prime})\tan\beta)&A_{e\mu%
}-A_{e\mu}^{\prime}\tan\beta&A_{e\tau}-A_{e\tau}^{\prime}\tan\beta\\
A_{\mu e}-A_{\mu e}^{\prime}\tan\beta&(A_{\mu}-(\mu+A_{\mu}^{\prime})\tan\beta%
)&A_{\mu\tau}-A_{\mu\tau}^{\prime}\tan\beta\\
A_{\tau e}-A_{\tau e}^{\prime}\tan\beta&A_{\tau\mu}-A_{\tau\mu}^{\prime}\tan%
\beta&(A_{\tau}-(\mu+A_{\tau}^{\prime})\tan\beta)\\
\end{matrix}\right)$$
(7)
$$M^{2}_{\tilde{l}RL}=(M^{2}_{\tilde{l}LR})^{\dagger}.$$
(8)
With only three sneutrino eigenstates, ${\tilde{\nu}}_{L}$ with $\nu=\nu_{e},\nu_{\mu},\nu_{\tau}$ in MSSM, the sneutrino mass matrix corresponds
to a $3\times 3$ matrix.
We note that the non diagonality in flavor comes exclusively from the soft SUSY-breaking parameters.
The main non-vanishing sources for $i\neq j$ are: the masses $M_{\tilde{L}\,ij}$ for the slepton $SU(2)$ doublets
$(\tilde{\nu}_{Li}\,\,\,\tilde{l}_{Li})$, the masses $M_{\tilde{e}\,ij}$ for the slepton $SU(2)$ singlets $(\tilde{l}_{Ri})$, and the trilinear couplings ${A}_{ij}$.
Our analysis however would only explore the effects of non-diagonal holomorphic or
non-holomorphic trilinear couplings that induce mixing in the slepton mass square
matrices ($M^{2}_{\tilde{\ell}LR}$). Regarding sneutrinos, we may write down
a corresponding $3\times 3$ mass matrix, with respect to the
$(\tilde{\nu}_{eL},\tilde{\nu}_{\mu L},\tilde{\nu}_{\tau L})$ electroweak interaction basis
in the sneutrino sector, and we have
$${\mathcal{M}}_{\tilde{\nu}}^{2}=\left(\begin{array}[]{c}M^{2}_{\tilde{\nu}\,LL%
}\end{array}\right),$$
(9)
where
$$M_{\tilde{\nu}\,LL\,ij}^{2}=M_{\tilde{L}\,ij}^{2}+\left(\frac{1}{2}M_{Z}^{2}%
\cos 2\beta\right)\delta_{ij}\,.$$
(10)
In the above, due to $SU(2)_{L}$ gauge invariance, the same soft mass $M_{\tilde{L}\,ij}$
occurs in both the slepton and sneutrino $LL$ mass matrices.
As is known, within the MSSM, LFV decays
get no tree-level contribution, just like other FCNC decays. They obtain leading
order contributions at loop level via mediation of sleptons (sneutrino)-neutralinos (charginos).
Here the source of lepton flavor violation can be from any one (or all) entries-
$M^{2}_{\tilde{l}LL},M^{2}_{\tilde{l}LR},M^{2}_{\tilde{l}RR}$ of Eq.4. But, we will focus on studying
the impacts of the non-holomorphic trilinear couplings on the cLFV
observables in the NH-MSSM (which would henceforth be called NHSSM), specially
in comparison to their holomorphic counterparts. Thus
the only source of lepton flavor violation which we would consider here
is associated with the
left-right slepton mixing. This means that sneutrino-chargino loops will hardly carry any importance in our analysis.
$l_{j}\rightarrow l_{i}\gamma$ :
Supersymmetric contributions to lepton flavor violating decays $l_{j}\rightarrow l_{i}\gamma$
can be sizable and potentially quite large compared to the same for various other BSM physics models.
The slepton-neutralino and sneutrino-chargino loops mostly
contribute to the amplitude of $l_{j}\rightarrow l_{i}\gamma$,
through charged particles appearing in the
loops. The general amplitude can be written as hisano ,
$$\displaystyle i\mathcal{M}=ie\epsilon^{\mu*}\overline{u}_{i}(p-q)\left[q^{2}%
\gamma_{\mu}(A_{1}^{L}P_{L}+A_{1}^{R}P_{R})+m_{l_{j}}i\sigma_{\mu\nu}q^{\nu}(A%
_{2}^{L}P_{L}+A_{2}^{R}P_{R})\right]u_{j}(p)\ ,$$
(11)
where $\epsilon^{*}$ is the photon polarization vector and $q$ being its momentum.
If the photon is
on-shell, the first part of the off-shell amplitude vanishes. Thus, we only need to focus on $A_{2}^{L}$ & $A_{2}^{R}$.
The coefficients $A_{2}^{L,R}$ that consist of chargino and neutralino contributions are as given below,
$$\displaystyle A_{2}^{L,R}=A_{2}^{(\tilde{\chi}^{0})L,R}+A_{2}^{(\tilde{\chi}^{%
\pm})L,R}.$$
(12)
For the case of our current interest, only
the NH trilinear couplings may change the slepton mass matrices. So, the flavor violating effects would directly
enter from the slepton mass matrix elements into the elements of the diagonalizing matrices.
The $\tilde{\chi}^{\pm}-\tilde{\nu}$ loops are hardly of any importance here because NH couplings do not affect the
sneutrino mass matrix.
So $A^{L}_{2}(\tilde{\chi}^{0})$ which corresponds to the contribution from real photon emission, is given by hisano ,
$$\displaystyle A_{2}^{(\tilde{\chi}^{0})L}$$
$$\displaystyle=\frac{1}{32\pi^{2}}\sum_{A=1}^{4}\sum_{X=1}^{6}\frac{1}{M_{%
\tilde{l}_{X}}^{2}}\left[N_{iAX}^{L}N_{jAX}^{L*}\frac{1}{12}F_{1}^{N}(x_{AX})+%
N_{iAX}^{L}N_{jAX}^{R*}\frac{m_{\tilde{\chi}^{0}_{A}}}{3m_{l_{j}}}F_{2}^{N}(x_%
{AX})\right],$$
(13)
where, $x_{AX}=\frac{m_{\tilde{\chi}^{0}_{A}}^{2}}{M^{2}_{\tilde{l_{X}}}}$. One obtains $A^{R}$
by simply interchanging $L\leftrightarrow R$.
The loop functions denoted by $F_{1}$, $F_{2}$ and
the couplings $N_{AX}^{L,R}$ can be read from hisano ; Rosiek:1995kg .
$$\displaystyle F_{1}^{N}(x)$$
$$\displaystyle=\frac{2}{(1-x)^{4}}\left[1-6x+3x^{2}-6x^{2}\log x\right],$$
$$\displaystyle\&~{}~{}F_{2}^{N}(x)$$
$$\displaystyle=\frac{3}{(1-x)^{3}}\left[1-x^{2}+2x\log x\right],$$
(14)
and
$$\displaystyle N_{iAX}^{L}$$
$$\displaystyle=-\sqrt{2}g_{1}\left(Z_{L}^{i+3,X}\right)^{*}Z_{N}^{1A}+Y_{l_{i}}%
\left(Z_{L}^{i,X}\right)^{*}Z_{N}^{3A},$$
$$\displaystyle N_{iAX}^{R}$$
$$\displaystyle=\frac{(Z_{L}^{i,X})^{*}}{\sqrt{2}}\left(g_{1}\left(Z_{N}^{1A}%
\right)^{*}+g_{2}\left(Z_{N}^{2A}\right)^{*}\right)+Y_{l_{i}}\left(Z_{N}^{3A}%
\right)^{*}\left(Z_{L}^{i+3,X}\right)^{*}.$$
(15)
Clearly, the couplings $N_{AX}^{L}$ involve
$Z_{N}$ & $Z_{L}$ the neutralino and slepton mixing matrices respectively
that transform them from the electroweak basis to the
mass basis. $Z_{L}$ is the
$6\times 6$ slepton mixing matrix that allows for flavor changes in the loop, leading to
the flavor violation. One may also evaluate the process using the mass
insertions in the slepton mixing matrix (through Eq: 7) which
depend on the diagonal and non-diagonal entries of the NH trilinear coupling matrix
$A_{\ell}^{(^{\prime})}$. The associated Feynmann diagram are shown in fig. 1. For example,
in case of $\mu\to e\gamma$, in the slepton mixing
would be induced by $A_{e\mu}-A^{\prime}_{e\mu}\tan\beta$. This indicates a typical domination
of $A^{\prime}_{e\mu}$ unless $A_{e\mu}$ is too large or there is much cancellation.
Finally, the decay rate is given by,
$$\displaystyle\Gamma(l_{j}\rightarrow l_{i}\gamma)=\frac{e^{2}}{16\pi}m^{5}_{l_%
{j}}\left(\left|A^{L}_{2}\right|^{2}+\left|A^{R}_{2}\right|^{2}\right).$$
(16)
$l_{j}\rightarrow 3l_{i}$ :
In the Standard Model (SM), $l_{j}\rightarrow 3l_{i}$ has a vanishingly small branching fraction, e.g. $Br(\tau\rightarrow 3\mu)<10^{-14}$ Pham:1998fq ,
while various models of beyond the SM may predict this particular process
to be of the order of $10^{-10}-10^{-8}$.
The current experimental limit of the same BR
is of the order of few times
$10^{-8}$ Lees:2010ez ; Hayasaka:2010np ; Aaij:2014azz ; Amhis:2014hma which has much better sensitivity compared to the 3-body decays of $\mu$.
The main experimental obstacle to improve
the sensitivity with $\tau$ leptons is the fact that $\tau$ is not produced in large numbers.
The amplitude for $l_{j}\rightarrow 3l_{i}$ comprises of contributions from $\gamma,Z,\phi(=h,H,A)$ penguin diagrams
and the box diagram as
shown in fig. 2
with slepton (sneutrino)-neutralino (chargino) appearing inside the loop. Detailed expressions for
the diagrams may be found in
hisano ; Arganda:2005ji ; Babu:2002et . The effects of NH off-diagonal elements toward LFV appear via slepton mass matrices.
This induces an effective vertex $\phi,\gamma,Z-l_{i}-\bar{l}_{j}$ which in turn leads to
processes like $l_{j}\rightarrow 3l_{i}$. All the penguin processes would further be
boosted in case of non-holomorphic couplings via additional $\tan\beta$ factor
(see Eq: 7)). The usual dominance of $\gamma$-penguins in the cLFV
processes in generic MSSM holds good except in the large $\tan\beta$ domain,
where the Higgs penguin diagrams are expected to be relevant Babu:2002et since they scale as
$\tan^{6}\beta$
Babu:2002et ; Dedes:2002rh . On the other hand,
Z and box contributions are hardly significant at low and moderate $\tan\beta$.
$\phi(h,H,A)\rightarrow l_{i}\bar{l}_{j}$ :
Flavor changing Higgs decays can play significant roles for investigating lepton flavor violation.
The same Higgs mediated penguin diagrams, induced by $\phi-\tilde{l}_{i}-\bar{\tilde{l}}_{j}$ vertex, may effectively contribute in
$\phi-l_{i}-\bar{l}_{j}$ vertex through loops leading to Higgs flavor violating decays.
The effective Lagrangian representing the interaction between neutral Higgs boson and charged leptons is given by Dedes:2002rh ; Babu:2002et ,
$$\displaystyle-\mathcal{L}_{eff}=\bar{e}^{i}_{R}y_{e_{ii}}\big{[}\delta_{ij}H^{%
0}_{d}+(\epsilon_{1}\delta_{ij}+\epsilon_{2ij}(A_{ij}-A^{\prime}_{ij}\tan\beta%
))H_{u}^{0*}\big{]}l_{L}^{j}+h.c.$$
(17)
The first term of the above equation denotes the Yukawa interaction whereas $\epsilon_{1}$ encodes the corrections to the charged lepton Yukawa couplings from flavor
conserving loops Babu:2002et .
The last term in
Eq.17 corresponds to the source of flavor violation through the insertion of $(A_{ij}-A^{\prime}_{ij}\tan\beta)$ in the slepton arms inside the loops.
$\epsilon_{2}$ arises out of loop functions involving neutralino and slepton masses owing to various cLFV processes. The effective Lagrangian in Eq:17 esentially generates all the off-diagonal Yukawa couplings radiatively if the respective
holomorphic (non-holomorphic) trilinear couplings $A_{ij}~{}(A^{\prime}_{ij})$ are non zero. This in turn produces flavor violating decays of
Higgs scalars or lepton 3-body decays induced by the Higgs penguins. Among the Higgs mediated diagrams, typically
dominant contributions come from the CP-odd Higgs exchange $A$ for large $\tan\beta$. This may be understood from the effective Lagrangian describing
couplings of the physical Higgs bosons to the leptons, which can be derived from Eq. (17)Dedes:2002rh ; Babu:2002et .
$$\displaystyle-{\cal L}^{\text{eff}}_{i\neq j}=(2G_{F}^{2})^{1/4}\,\frac{m_{E_{%
i}}\kappa^{E}_{ij}}{\cos^{2}\beta}\left(\bar{e}^{i}_{R}\,l^{j}_{L}\right)\left%
[\cos(\alpha-\beta)h+\sin(\alpha-\beta)H-iA\right]+\text{h.c.}.$$
(18)
Here,
$\alpha$ is the CP-even Higgs mixing angle and $\tan\beta=v_{u}/v_{d}$, and
$$\displaystyle\kappa^{E}_{ij}$$
$$\displaystyle=$$
$$\displaystyle\frac{\epsilon_{2}(A_{ij}-A^{\prime}_{ij}\tan\beta)}{\left[1+%
\left(\epsilon_{1}+\epsilon_{2}(A_{ii}-A^{\prime}_{ii}\tan\beta)\right)\tan%
\beta\right]^{2}}\ .$$
(19)
Since the cLFV branching ratios are proportional to $({\kappa^{E}_{ij}})^{2}$, from
the above equation it is clear that the non-holomorphic trilinear couplings via $\tan\beta$
enhancement may
have greater importance towards Higgs mediated processes. In fact,
all the Higgs mediated flavor violating observables may receive large boost,
while assuming no cancellation in eq. 19 or if holomorphic $A_{ij}$
is negligible compared to $A^{\prime}_{ij}$. In case of flavor violating
Higgs decays the branching fraction $\phi_{k}\to\mu\tau$ where Higgs bosons
$h,H,A$ are denoted as $\phi_{k}$ for $k=1,2,3$ can be related to the flavor
conserving decay $\phi_{k}\to\tau\tau$ as followsBrignole:2003iv .
$$\displaystyle{\text{Br}(\phi_{k}\to\mu\tau)}=\tan^{2}\beta~{}(|\kappa_{\tau\mu%
}^{E}|^{2})~{}C_{\Phi}~{}{\text{Br}(\phi_{k}\to\tau\tau)}\,,$$
(20)
where we used $1/\cos^{2}\beta\simeq\tan^{2}\beta$.
The coefficients $C_{\Phi}$ are given by,
$$\displaystyle C_{h}=\left[\frac{\cos(\beta-\alpha)}{\sin\alpha}\right]^{2},~{}%
~{}~{}~{}C_{H}=\left[\frac{\sin(\beta-\alpha)}{\cos\alpha}\right]^{2},~{}~{}~{%
}~{}C_{A}=1.$$
(21)
3 Analysis of Charge Breaking Minima
Absence of any flavor changing neutral current (FCNC) significantly constrains the off-diagonal elements in the mass and trilinear
coupling matrices.
However, the Charge and Color Breaking(CCB) constraints are more robust than the
corresponding FCNC data Casas:1996de . In a multi-scalar theory, the existence of several vacua and choice of the desired electroweak symmetry breaking
put strong constraints on the allowed parameter space. In this context, we first put the effort to analyze the charge
breaking bounds for two generations of sleptons associated with the
$(\tilde{\mu}-\tilde{\tau})$ sector. Here trilinear couplings can accommodate
the off-diagonal entries of both the holomorphic and
non-holomorphic soft SUSY breaking terms. Then we will generalize it for all the three generations of sleptons.
Three basic components of tree level scalar potential $V_{0}$ are the F-term, D-term and the soft breaking terms, $V_{0}=V_{F}+V_{D}+V_{soft}$.
The constituents of $V_{0}$ are given as follows,
$$\displaystyle V_{F}$$
$$\displaystyle=\sum_{a}\Big{|}\frac{\partial W}{\partial\phi_{a}}\Big{|}^{2},$$
(22)
here the superpotential W is given by Eq. 1. The D-term part gives additional quartic terms for scalar potential
associated with gauge couplings $g_{a}$.
$$\displaystyle V_{D}$$
$$\displaystyle=\frac{1}{2}\sum_{a}g_{a}^{2}~{}(\sum_{a}\phi_{a}^{\dagger}T^{a}%
\phi_{a})^{2}$$
(23)
The Holomorphic and non-holomorphic soft terms in V (and hence in $-\mathcal{L}$) can be written as,
$$\displaystyle V_{soft}$$
$$\displaystyle=\sum M^{2}_{\phi_{a}}|\phi_{a}|^{2}+({\tilde{u}^{*}}_{iR}{\bf A_%
{u}}_{ij}\tilde{q}_{jL}\cdot H_{u}+{\tilde{d}^{*}}_{iR}{\bf A_{d}}_{ij}\tilde{%
q}_{jL}\cdot H_{d}+{\tilde{e}^{*}}_{iR}{\bf A_{e}}_{ij}\tilde{\ell}_{jL}\cdot H%
_{d}+h.c.),$$
(24)
$$\displaystyle V_{soft}^{NH}$$
$$\displaystyle=\tilde{u}^{*}_{iR}{\bf A_{u}^{\prime}}_{ij}\tilde{q}_{jL}\cdot H%
_{d}^{*}+\tilde{d}^{*}_{iR}{\bf A_{d}^{\prime}}_{ij}\tilde{q}_{jL}\cdot H_{u}^%
{*}+\tilde{e}^{*}_{iR}{\bf A_{e}^{\prime}}_{ij}\tilde{\ell}_{jL}\cdot H_{u}^{*%
}+h.c.$$
(25)
In the above, $\phi_{a}$ runs over all the scalar components of chiral superfields.
The full MSSM scalar potential may indeed have several minima where squarks
or sleptons may additionally acquire non-zero vevs which may in turn lead to charge and/or color breaking vacua. Since the violation of charge and/or
color quantum number is yet to be observed, it is understood that the universe at present
is at a ground state which is Standard Model like (SML) Chattopadhyay:2014gfa , with only neutral
components of the Higgs scalars acquiring vevs.
A priori, it indicates that those parts of the multi-dimensional parameter space
corresponding to MSSM scalar potential that allow a deeper charge and color breaking
(CCB) minima Frere1983 ; Casas:1995pd ; Gunion1988 ; Dress1985 ; Komatsu1988 ; Langacker:1994bc ; Strumia:1996pr ; Chattopadhyay:2014gfa should be excluded.
The dangerous directions could be
associated with unacceptably large trilinear couplings, in particular $A_{t}$, $A_{b}$, the ones associated with top and bottom Yukawa couplings.
Thus, one may have a CCB vacuum that is deeper than the desired EWSB vacuum.
Analyses of CCB constraints in MSSM may be seen in Refs Hollik:2016dcm ; Casas:1995pd ; Casas:1996de ; Chattopadhyay:2014gfa ; Chowdhury:2013dka
a related study on non-holomorphic soft terms was made in Ref.Beuria:2017gtf .
Here one should note that the rate of tunneling from SML false vacuum to such CCB true vacuum
is roughly proportional to $e^{-a/y^{2}}$, where
“a” is a constant of suitable dimension that can be
determined via field theoretic calculations and “y”
is the Yukawa coupling. The tunneling rate is enhanced for large Yukawa couplings
Kusenko:1996jn ; Kusenko:1996xt ; Kusenko:1996vp ; Kusenko:1995jv ; Brandenberger:1984cz ; LeMouel:2001ym , thus leading to large effects from the third generation of sfermions.
But it is not always true that, trilinear terms of third generations of squarks/sleptons
are the most important for charge and color breaking minima. With the
variation of non-holomorphic
soft terms, there can be significant changes in all of the corresponding Yukawa couplings through loops Chattopadhyay:2018tqv
and other two generations of quarks can have notable effect in charge and color breaking condition Beuria:2017gtf .
Here we will study the analytic expressions for only charge breaking minima for slepton soft masses and
slepton trilinear couplings (both holomorphic and non-holomorphic) considering all three generations.
We will particularly generalize our studies to include first the effect of non-vanishing, non-diagonal trilinear soft terms.
Furthermore, we will consider only the case of absolute stability of the vacuum, i.e. without trying to analyze any tunneling effect.
3.1 Charge breaking with flavor violation in MSSM
In this subsection we first analyze the effect of non-vanishing off-diagonal entries $A^{(^{\prime})}_{ij}$ on charge breaking in MSSM.
The relevant terms in $V_{F}$, $V_{D}$ and $V_{soft}$ of MSSM related to the slepton sector are as given below,
$$\displaystyle V_{F}$$
$$\displaystyle=|\mu^{*}H_{u}^{-}-\tilde{\nu}^{*}_{iL}y_{ij}\tilde{e}_{jR}|^{2}+%
|\mu^{*}H_{u}^{0}-\tilde{e}^{*}_{iL}y_{ij}\tilde{e}_{jR}|^{2}+\sum_{l}|y_{ij}H%
_{d}^{0}\tilde{l}_{jL}|^{2}+y_{ij}y^{\star}_{ij^{\prime}}\tilde{e}_{jR}^{*}%
\tilde{e}_{j^{\prime}R}(|H_{d}^{0}|^{2}+H_{d}^{+}H_{d}^{-}),$$
$$\displaystyle V_{D}$$
$$\displaystyle=V_{D_{Y}}+V_{\vec{D}}$$
$$\displaystyle=\frac{g^{2}_{1}}{8}(|H_{u}^{0}|^{2}-|H_{d}^{0}|^{2}-|\tilde{l}_{%
iL}|^{2}+2|\tilde{e}_{iR}|^{2})^{2}+\frac{g^{2}_{2}}{8}(|H_{u}^{0}|^{2}-|H_{d}%
^{0}|^{2}+|\tilde{l}_{iL}|^{2})^{2},$$
$$\displaystyle V_{soft}^{\tilde{l}}$$
$$\displaystyle=\tilde{l}_{iL}^{\dagger}(M^{2}_{\tilde{L}})_{ij}\tilde{l}_{jL}+%
\tilde{e}_{iR}^{*}(M^{2}_{\tilde{e}})_{ij}\tilde{e}_{jR}^{\dagger}+[\tilde{e}_%
{iR}^{*}{\bf A_{lij}}\tilde{l}_{jL}\cdot H_{d}+h.c]~{},$$
$$\displaystyle V_{soft}^{H}$$
$$\displaystyle=m^{2}_{H_{d}}|H_{d}^{0}|^{2}+m^{2}_{H_{u}}|H_{u}^{0}|^{2}-2Re(B%
\mu H_{d}^{0}H_{u}^{0}).$$
Below, we collect the terms, originating from $V_{F}$, $V_{D}$, $V^{\tilde{l}}_{soft}$ and $V_{soft}^{H}$ appearing in the diagonal and off-diagonal elements
for the 2nd and 3rd generations of sleptons (viz. smuon and stau).
$$\displaystyle V_{\mu}^{diag}$$
$$\displaystyle=\tilde{\mu}_{L}^{*}(M^{2}_{{\tilde{L}}_{22}}+|y_{\mu}H_{d}^{0}|^%
{2})\tilde{\mu}_{L}+\tilde{\mu}_{R}^{*}(M^{2}_{{\tilde{e}}_{22}}+|y_{\mu}H_{d}%
^{0}|^{2})\tilde{\mu}_{R}+[\tilde{\mu}_{L}^{*}(A_{\mu}h_{d}-\mu^{*}y_{\mu}h_{u%
})\tilde{\mu}_{R}+h.c]+|y_{\mu}|^{2}|\tilde{\mu}_{L}|^{2}|\tilde{\mu}_{R}|^{2}%
~{},$$
$$\displaystyle V_{\tau}^{diag}$$
$$\displaystyle=\tilde{\tau}_{L}^{*}(M^{2}_{{\tilde{L}}_{33}}+|y_{\tau}H_{d}^{0}%
|^{2})\tilde{\tau}_{L}+\tilde{\tau}_{R}^{*}(M^{2}_{{\tilde{e}}_{33}}+|y_{\tau}%
H_{d}^{0}|^{2})\tilde{\tau}_{R}+[\tilde{\tau}_{L}^{*}(A_{\tau}H_{d}^{0}-\mu^{*%
}y_{\tau}H_{u}^{0})\tilde{\tau}_{R}+h.c]+|y_{\tau}|^{2}|\tilde{\tau}_{L}|^{2}|%
\tilde{\tau}_{R}|^{2}~{},$$
$$\displaystyle V_{\mu\tau}$$
$$\displaystyle=\tilde{\mu}_{L}^{*}M^{2}_{{\tilde{L}}_{23}}\tilde{\tau}_{L}+%
\tilde{\mu}_{R}^{*}[M^{2}_{{\tilde{e}}_{23}}+|{\bf y_{\mu\tau}}H_{d}^{0}|^{2}]%
\tilde{\tau}_{R}+[\tilde{\mu}_{L}^{*}(A_{\mu\tau}H_{d}^{0}-\mu^{*}y_{\mu\tau}H%
_{u}^{0})\tilde{\tau}_{R}+h.c]+|y_{\mu\tau}|^{2}|\tilde{\mu}_{L}|^{2}|\tilde{%
\tau}_{R}|^{2}+\tilde{\tau}_{L}^{*}M^{2}_{{\tilde{e}}_{32}}\tilde{\mu}_{L}$$
$$\displaystyle+\tilde{\tau}_{R}^{*}[M^{2}_{\tilde{e}_{32}}+|{\bf y_{\tau\mu}}H_%
{d}^{0}|^{2}]\tilde{\mu}_{R}+[\tilde{\tau}_{L}^{*}(A_{\tau\mu}H_{d}^{0}-\mu^{*%
}y_{\tau\mu}H_{u}^{0})\tilde{\mu}_{R}+h.c]+|y_{\tau\mu}|^{2}|\tilde{\tau}_{L}|%
^{2}|\tilde{\mu}_{R}|^{2}~{},$$
$$\displaystyle V_{H}$$
$$\displaystyle=(m^{2}_{H_{u}}+|\mu|^{2})|H_{u}^{0}|^{2}+(m^{2}_{H_{d}}+|\mu|^{2%
})|H_{d}^{0}|^{2}~{},$$
$$\displaystyle V_{D}$$
$$\displaystyle=\frac{g_{1}^{2}}{8}(|H_{u}^{0}|^{2}-|H_{d}^{0}|^{2}-|\tilde{\mu}%
_{L}|^{2}-|\tilde{\tau}_{L}|^{2}+2|\tilde{\mu}_{R}|^{2}+2|\tilde{\tau}_{R}|^{2%
})^{2}+\frac{g_{2}^{2}}{8}(|H_{u}^{0}|^{2}-|H_{d}^{0}|^{2}+|\tilde{\mu}_{L}|^{%
2}+|\tilde{\tau}_{L}|^{2})^{2}.$$
In the first place we consider non-vanishing vevs for the neutral
components of the two Higgs
scalars and the stau and smuon fields. The latter are responsible for the
generation of charge breaking minima. Allowing both $H_{u}^{0}$ and $H_{d}^{0}$ to fluctuate in the positive and negative directions, we choose to constrain the slepton
fields with a particular scalar field value $\phi$.
In this specific direction one has,
$$\displaystyle|\tilde{\tau_{L}}|=|\tilde{\tau_{R}}|=\alpha\phi~{},$$
$$\displaystyle|\tilde{\mu_{L}}|=|\tilde{\mu_{R}}|=\beta\phi~{},$$
$$\displaystyle H_{d}^{0}=\phi~{},$$
(26)
$$\displaystyle H_{u}^{0}=\eta\phi.$$
with $\eta$ being any real number and $\alpha$, $\beta$ to be real and positive. The total tree-level scalar potential involving Higgs, smuon and stau fields,
assuming $\mu$ to be real and $y_{ij}$ or $A_{ij}$ referring to real symmetric matrices, reduces to,
$$V_{\tilde{l},H}=A\phi^{2}+B\phi^{3}+C\phi^{4},$$
(27)
where,
$$\displaystyle A$$
$$\displaystyle=\alpha^{2}(M_{\tilde{L}_{33}}^{2}+M_{\tilde{e}_{33}}^{2})+\beta^%
{2}(M_{\tilde{L}_{22}}^{2}+M_{\tilde{e}_{22}}^{2})+2\alpha\beta(M_{\tilde{L}_{%
23}}^{2}+M_{\tilde{e}_{23}}^{2})+m^{2}_{H_{d}}+\eta^{2}m^{2}_{H_{u}}+(1+\eta^{%
2})|\mu|^{2}-2B\mu\eta~{},$$
$$\displaystyle B$$
$$\displaystyle=2\alpha^{2}(A_{\tau}-\mu y_{\tau}\eta)+2\beta^{2}(A_{\mu}-\mu y_%
{\mu}\eta)+4\alpha\beta(A_{\mu\tau}-\mu y_{\mu\tau}\eta)~{},$$
$$\displaystyle C$$
$$\displaystyle=\frac{g_{1}^{2}+g_{2}^{2}}{8}(\eta^{2}-1+\beta^{2}+\alpha^{2})^{%
2}+(2+\alpha^{2})\alpha^{2}y_{\tau}^{2}+(2+\beta^{2})\beta^{2}y_{\mu}^{2}+2%
\alpha^{2}\beta^{2}y_{\mu\tau}^{2}.$$
We require that the minima at $\langle\phi\rangle=0$ should be deeper than a minima with $\langle\phi\rangle\neq 0$ and this is possible when
$B^{2}(\alpha,\beta,\eta)<4A(\alpha,\beta,\eta)~{}C(\alpha,\beta,\eta)$.
Here we consider
a scenario with 6 vevs corresponding to L and R components of smuon and stau fields apart from the neutral Higgs fields corresponding to eq. 3.1.
We want to have the most stringent condition that would avoid the charge breaking minima.
Thus, in the D-flat direction, which explicitly demands that all the $g^{2}_{i}$ terms in the tree level scalar potential to be absent, we choose,
$$\displaystyle\alpha$$
$$\displaystyle=\frac{1}{\sqrt{2}},~{}\beta=\frac{1}{\sqrt{2}},~{}\eta=0,$$
so that, $(\eta^{2}-1+\alpha^{2}+\beta^{2})=0$.
Thus we obtain,
$$\displaystyle A$$
$$\displaystyle=\frac{1}{2}(M_{\tilde{L}_{33}}^{2}+M_{\tilde{e}_{33}}^{2})+\frac%
{1}{2}(M_{\tilde{L}_{22}}^{2}+M_{\tilde{e}_{22}}^{2})+(M_{\tilde{L}_{23}}^{2}+%
M_{\tilde{e}_{23}}^{2})+m^{2}_{H_{d}}+|\mu|^{2}~{},$$
$$\displaystyle B$$
$$\displaystyle=A_{\tau}+A_{\mu}+2A_{\mu\tau}~{},$$
$$\displaystyle C$$
$$\displaystyle=\frac{5}{4}(y_{\tau}^{2}+y_{\mu}^{2}+\frac{2}{5}y_{\mu\tau}^{2}).$$
With $B^{2}(\alpha,\beta,\eta)<4A(\alpha,\beta,\eta)C(\alpha,\beta,\eta)$ one obtains the following that would avoid a charge breaking minima
$$\Big{(}A_{\tau}+A_{\mu}+2A_{\mu\tau}\Big{)}^{2}<5(y_{\tau}^{2}+y_{\mu}^{2}+%
\frac{2}{5}y_{\mu\tau}^{2})\times\Big{[}\frac{1}{2}(M_{\tilde{L}_{33}}^{2}+M_{%
\tilde{e}_{33}}^{2})+\frac{1}{2}(M_{\tilde{L}_{22}}^{2}+M_{\tilde{e}_{22}}^{2}%
)+(M_{\tilde{L}_{23}}^{2}+M_{\tilde{e}_{23}}^{2})+m^{2}_{H_{d}}+|\mu|^{2}\Big{%
]}.$$
(28)
Including all the three generations of leptons, Eq. 28 generalizes into the following.
$$\Big{(}\sum_{e,\mu,\tau}A_{i}+2\sum_{i\neq j}A_{ij}\Big{)}^{2}<5\big{(}\sum_{e%
,\mu,\tau}y_{i}^{2}+\frac{2}{5}\sum_{i\neq j}y_{ij}^{2}\big{)}\times\Big{[}%
\frac{1}{2}\sum_{e,\mu,\tau}(M^{2}_{\tilde{L}_{ii}}+M^{2}_{\tilde{e}_{ii}})+%
\sum_{i\neq j}(M^{2}_{\tilde{L}_{ij}}+M^{2}_{\tilde{e}_{ij}})+m^{2}_{H_{d}}+|%
\mu|^{2}\Big{]}.$$
(29)
3.2 Charge breaking condition in NHSSM
With Non-Holomorphic term in $V_{soft}$ involving only the appropriate trilinear NH couplings we will have following extra terms,
$$\displaystyle V^{\tilde{l}}_{NH}=-[\tilde{\mu}_{L}^{*}(A^{\prime}_{\mu}H_{u}^{%
0})\tilde{\mu}_{R}+\tilde{\tau}_{L}^{*}(A^{\prime}_{\tau}H_{u}^{0})\tilde{\tau%
}_{R}+\tilde{\mu}_{L}^{*}(A^{\prime}_{\mu\tau}H_{u}^{0})\tilde{\tau}_{R}+%
\tilde{\tau}_{L}^{*}(A^{\prime}_{\tau\mu}H_{u}^{0})\tilde{\mu}_{R}+h.c].$$
Considering the direction mentioned in 3.1, we find the following for NHSSM.
$$\displaystyle V_{\tilde{l},H}$$
$$\displaystyle=\{\alpha^{2}(M_{L_{33}}^{2}+M_{e_{33}}^{2})+\beta^{2}(M_{L_{22}}%
^{2}+M_{e_{22}}^{2})+2\alpha\beta(M_{L_{23}}^{2}+M_{e_{23}}^{2})+m^{2}_{H_{d}}%
+\eta^{2}m^{2}_{H_{u}}+(1+\eta^{2})|\mu|^{2}-2B\mu\eta\}\phi^{2}$$
(30)
$$\displaystyle+\{2\alpha^{2}(A_{\tau}-A^{\prime}_{\tau}\eta-\mu y_{\tau}\eta)+2%
\beta^{2}(A_{\mu}-A^{\prime}_{\mu}\eta-\mu y_{\mu}\eta)+4\alpha\beta(A_{\mu%
\tau}-A^{\prime}_{\mu\tau}\eta-\mu y_{\mu\tau}\eta)\}\phi^{3}$$
$$\displaystyle+\{\frac{g_{1}^{2}+g_{2}^{2}}{8}(\eta^{2}-1+\beta^{2}+\alpha^{2})%
^{2}+(2+\alpha^{2})\alpha^{2}y_{\tau}^{2}+(2+\beta^{2})\beta^{2}y_{\mu}^{2}+2%
\alpha^{2}\beta^{2}y_{\mu\tau}^{2}\}\phi^{4}.$$
Earlier, we obtained the most stringent bound along D-flat direction which requires $\eta=0$.
But, the same choice is insufficient to provide any bound on the NH trilinear couplings.
This is simply because $\eta$ gets multiplied with the NH trilinear parameters in Eq. 30.
Instead we assume $\alpha=\frac{1}{\sqrt{2}},~{}\beta=\frac{1}{\sqrt{2}},~{}\eta=1$, which leads to a stringent bound on the
$A_{\mu}^{\prime}$ and $A^{\prime}_{\tau}$ for avoiding a deeper charge breaking minima in NHSSM. The bound can be read as shown below:
$$\begin{split}\displaystyle\Big{[}A_{\tau}-(\mu y_{\tau}+A^{\prime}_{\tau})+A_{%
\mu}-(\mu y_{\mu}+A^{\prime}_{\mu})+2\{A_{\mu\tau}-(\mu y_{\mu\tau}+A^{\prime}%
_{\mu\tau})\}\Big{]}^{2}<4.\frac{1}{4}(\frac{g^{2}_{1}+g^{2}_{2}}{2}+5y_{\tau}%
^{2}+5y_{\mu}^{2}+2y_{\mu\tau}^{2})\\
\displaystyle\times\Big{[}\frac{1}{2}(M_{\tilde{L}_{33}}^{2}+M_{\tilde{e}_{33}%
}^{2})+\frac{1}{2}(M_{\tilde{L}_{22}}^{2}+M_{\tilde{e}_{22}}^{2})+(M_{\tilde{L%
}_{23}}^{2}+M_{\tilde{e}_{23}}^{2})+m^{2}_{H_{u}}+m^{2}_{H_{d}}+2|\mu|^{2}-2B%
\mu\Big{]}.\end{split}$$
(31)
Again dealing with all three generations of sleptons Eq 31 becomes,
$$\begin{split}\displaystyle\Big{(}\sum_{e,\mu,\tau}\{A_{i}-(A_{i}^{\prime}+\mu y%
_{i})\}+2\sum_{i\neq j}\{A_{ij}-(A^{\prime}_{ij}+\mu y_{ij})\}\Big{)}^{2}<\Big%
{(}\frac{g^{2}_{1}+g^{2}_{2}}{2}+5\sum_{e,\mu,\tau}y_{i}^{2}+2\sum_{i\neq j}y_%
{ij}^{2}\Big{)}\times\Big{[}\frac{1}{2}\sum_{e,\mu,\tau}(M^{2}_{\tilde{L}_{ii}%
}\\
\displaystyle+M^{2}_{\tilde{e}_{ii}})+\sum_{i\neq j}(M^{2}_{\tilde{L}_{ij}}+M^%
{2}_{\tilde{e}_{ij}})+m^{2}_{H_{u}}+m^{2}_{H_{d}}+2|\mu|^{2}-2B\mu\Big{]}.\end%
{split}$$
(32)
Eq.32 represents the most general condition to avoid a charge breaking minima considering all kinds
of soft breaking terms for three generations of fermions. Stringent constraints
on individual non-holomorphic diagonal and non-diagonal
trilinear couplings may be derived from 32. However, rather
than following the bounds on the individual couplings,
hereafter we will use eq. 29 and 32 in all our results to constrain the trilinear parameters.
4 Status of different LFV decays
Here we would summarize the experimental efforts and the degree of current and future sensitivities of several cLFV processes.
In the radiative decay of $l_{j}\rightarrow l_{i}\gamma$, the experiment
leading to the most
stringent constraint is MEG TheMEG:2016wtm , which is currently operational at
the Paul Scherrer Institute in
Switzerland. This searches for the radiative process $\mu\rightarrow e\gamma$.
The MEG collaboration proclaimed a new limit on the rate for this process based on the analysis of a data set
with $3.6\times 10^{14}$ stopped muons. The non-observation of the cLFV process leads to
$Br(\mu\rightarrow e\gamma)<4.2\times 10^{-13}$ TheMEG:2016wtm ,
which is four times more stringent
than the earlier one, obtained by the same collaboration.
Moreover, the MEG collaboration has announced plans for
future upgrades leading to a sensitivity of about $6\times 10^{-14}$ after 3 years of data
acquisition Baldini:2013ke .
The most interesting results in the near future are
expected in $\mu\rightarrow 3e$ and $\mu-e$ conversion in nuclei.
The Mu3e experiment Blondel:2013ia ; Perrevoort:2016nuv is designed to search for charged lepton flavor violation
in the process $\mu\rightarrow 3e$ with
a branching ratio sensitivity of $10^{-16}$.
The present limit on the $\mu\rightarrow 3e$ has been set by the SINDRUM experiment Bellgardt:1987du .
As no signal was observed, branching fractions larger than $1.0\times 10^{-12}$ were excluded
at 90% confidence limit (CL). For the upcoming Mu3e experiment, in phase I, a branching
fraction of $5.2\times 10^{-15}$ can be measured or excluded at 90% CL Perrevoort:2018ttp .
In the recent times, the most actively studied cLFV processes are
the rare $\tau$ decays. $\tau$-pairs are abundantly produced at
the $B$ factories e.g., in the BELLE Hayasaka:2007vc & BABAR Aubert:2009ag collaborations.
There are significant improvement on most of the cLFV modes of the
$\tau$ decays, though any of them has not been discovered yet.
The LHCb collaboration also announced the first ever
bounds on $\tau\rightarrow 3\mu$ in a hadron collider Aaij:2013fia . The current experimental upper limits on the LFV
radiative decays TheMEG:2016wtm ; Aubert:2009ag ; Hayasaka:2007vc are collected in the table 1 with references.
Apart from the leptonic decays with LFV there are bounds from LFV Higgs decays.
The first direct search of LFV Higgs decays were performed by CMS and ATLAS Collaborations Khachatryan:2015kon ; Aad:2015gha .
A slight excess of signal events with a significance of $2.4\sigma$ was observed by CMS at 8 TeV data
but, that early peak by CMS is not supported at 13 TeV anymore, finding
$Br(h\rightarrow\mu\tau)<1.20\%$ with 2.3 $fb^{-1}$ data LFVCMS13TeV . Subsequently CMS confirmed the disappearance of that excess Sirunyan:2017xzt ; Aad:2019ugc .
Additionally, at 13 TeV with integrated luminosity of $35.9fb^{-1}$, no significant excess over
the Standard Model expectation is observed. The observed (expected) upper limits
on the lepton flavor violating branching fractions of the Higgs boson are
$Br(h\rightarrow\mu\tau)<0.28$% (0.37%) and
$Br(h\rightarrow e\tau)<0.47$% (0.34%) at 95% confidence level. These results are used to derive upper limits on the
off-diagonal $\mu\tau$ and $e\tau$ Yukawa couplings Aad:2019ugc . These limits on the lepton flavor violating branching
fractions of the Higgs boson and on the associated Yukawa couplings are the most stringent to date (see table 2). Similarly, the null search results of
$Br(\tau\rightarrow e/\mu+\gamma)$ and $Br(\tau\rightarrow 3e/\mu)$ pdg translate into bounds on corresponding
objects like $\sqrt{Y_{ij}^{2}+Y_{ji}^{2}}$ Harnik:2012pb .
Limits on LFV Higgs decay processes closely follow the search results of $h\rightarrow\mu\mu/\tau\tau$ channels. Evidence for the 125 GeV Higgs boson
decaying to a pair of $\tau$ (or $\mu$) leptons are presented in Refs. Aad:2012an ; Aad:2012cfr ; Chatrchyan:2014nva ; Aad:2015vsa .
Furthermore, dedicated searches are conducted for additional neutral Higgs bosons decaying to the $\tau^{+}\tau^{-}$ final state in proton-proton
collisions at the 13 TeV LHC Sirunyan:2017khh ; Aaboud:2017sjh ; Sirunyan:2018zut which lead exclusion plots in the $m_{A}-\tan\beta$ plane and also give limits on
$\sigma(gg\rightarrow\phi)\times Br(\phi\rightarrow\tau^{+}\tau^{-})$ with ($\phi=h,H,A$). Since the cLFV processes of heavier Higgs bosons are
proportional to $Br(\phi\rightarrow\tau^{+}\tau^{-})$ Brignole:2003iv these limits are extremely important for any analysis of $Br(\phi\rightarrow\mu\tau)$.
Similar results for $\tau^{+}\tau^{-}$ finals states at $\sqrt{s}=8$ TeV
are available in Khachatryan:2014wca ; Aad:2014vgg .
Recently, some model independent analyses of heavier Higgs boson decaying into $\mu\tau$ channel have been performed Arganda:2019gnv and it is shown that,
at $\sqrt{s}=14$ TeV with $\mathcal{L}=300$ $fb^{-1}$ the sensitivities to the experimental probes increase with heavier Higgs boson
masses. Lepton flavor violating decays of Higgs have also been searched for in the
first-second and first-third generations of leptons, i.e $e\mu$ and $e\tau$ Khachatryan:2016rke ; Aad:2016blu channels in the LHC at $\sqrt{s}=8$ TeV.
These classes of LFV processes are also being studied in LHC through the decays of
neutral heavy Higgs like bosons for different supersymmetric and non-supersymmetric models
Aad:2015pfa ; Aaboud:2018jff ; Aaij:2018mea .
5 Results
We divide our LFV decay analyses into three parts namely,
$l_{i}\rightarrow l_{j}\gamma$, $l_{i}\rightarrow 3l_{j}$ and
$\phi(=h,H,A)\rightarrow l_{i}\bar{l}_{j}$. As mentioned before, we try to explore the LFV effects
of the relevant trilinear coupling matrices related to the holomorphic and
non-holomorphic trilinear interactions.
Simply, out of their
association with the type of interactions we will label the
couplings themselves as “holomorphic” (like $A_{ij}$ or $A_{ii}\equiv A_{i}$)
or “non-holomorphic” (like $A^{\prime}_{ij}$ or $A^{\prime}_{ii}\equiv A^{\prime}_{i}$).
Furthermore, as pointed out earlier, for simplicity,
our analysis probes either (i) a non-vanishing set of
[$A_{ij},A_{i}$] while having vanishing values for the set [$A^{\prime}_{ij},A^{\prime}_{i}$] or (ii) vice-versa.
We will particularly identify the region of parameter space in relation to
a given constraint from an LFV decay where non-holomorphic couplings may
have prominent roles.
Considering the fact that only the trilinear couplings associated with sleptons are
of importance in this analysis, we will either consider the matrices $A_{f}=0$ and $A^{\prime}_{f}\neq 0$
and vice-versa, where $f\equiv e,\mu,\tau$.
Combining the holomorphic and non-holomorphic couplings, we will often use
$A_{f}^{(\prime)}$ to mean either $A_{f}$ or $A^{\prime}_{f}$ depending on the context.
We vary both the diagonal and off-diagonal components of
${\bf{A^{(\prime)}}_{e}}_{ij}$ over broad range of values
(see table 3)
subject to the condition $|\delta^{(\prime)}_{ij}|<1$ with $i\neq j$, where
$\delta^{(\prime)}_{ij}=\frac{{{A^{(\prime)}}_{e}}_{ij}}{{{A^{(\prime)}}_{e}}_{%
ii}}$. Here, we remind that all the off-diagonal entries of the slepton mass matrices
are considered to be zero leading to flavor violation possible only from the trilinear coupling sources.
We will impose the charge breaking
constraint and select the
possible diagonal and off-diagonal
trilinear coupling values that would obey Eq.32.
We must emphasize here that in an analysis that scans
the diagonal and off-diagonal elements of the trilinear coupling matrices,
such as the present one, we do not expect severe restrictions on the off-diagonal
entries $A^{(\prime)}_{ij}$ simply because of the possible cancellation of
terms in the left hand side of Eq.32. On the other hand,
a choice of fixed signs of diagonal and off-diagonal entries of
tri-linear coupling matrix elements would show prominent effects of
charge breaking.
Apart from the above, we impose the experimental
bounds of different cLFV observables and LHC direct search results for
$\phi\rightarrow l_{i}\bar{l}_{j}$.
The values/ranges of relevant soft parameters used in this analysis
are listed in table 3.
We compute the SUSY mass spectra and related branching fractions from SARAH (v4.10.0)Staub:2013tta ; Staub:2015kfa generated
FORTRAN codes that are subsequently used in SPheno (v4.0.3) Porod:2011nf .
In regard to a few relevant SM parameters, we use
$m_{t}^{pole}=173.5$ GeV, $m_{b}^{\overline{MS}}=4.18$ GeV and $m_{\tau}=1.77$ GeV Olive:2016xmw and we use the SUSY mass scale as
$M_{\rm SUSY}=\sqrt{m_{\tilde{t}_{1}}m_{\tilde{t}_{2}}}$.
We further impose the following limits for Higgs mass $m_{h}$333We consider a $\pm$3 GeV theoretical uncertainty in computing lighter Higgs massloopcorrection that arises
from the uncertainties from radiative corrections up to
three loops, top quark pole mass, renormalization scheme and scale dependencies
etc., branching ratios like $Br(b\rightarrow s\gamma)$ and $Br(B_{s}\rightarrow\mu^{+}\mu^{-})$ at $2\sigma$ level, lighter chargino mass bound
from LEP, along with an LHC limit for the direct lighter top squark searchesOlive:2016xmw ; Aaboud:2017ayj .
$$\displaystyle 122.1~{}{\rm GeV}\leqslant m_{h}\leqslant 128.1~{}{\rm GeV},$$
$$\displaystyle 2.99\times 10^{-4}\leqslant Br(b\rightarrow s\gamma)\leqslant 3.%
87\times 10^{-4},$$
$$\displaystyle 1.5\times 10^{-9}\leqslant Br(B_{s}\rightarrow\mu^{+}\mu^{-})%
\leqslant 4.3\times 10^{-9},$$
$$\displaystyle m_{\tilde{\chi}_{1}}^{\pm}\geq 104~{}{\rm GeV},m_{\tilde{t}_{1}}%
\geq 1000~{}{\rm GeV}.$$
•
$\mathbf{l_{j}\rightarrow l_{i}\gamma}$
We show the results of computation of $Br(l_{j}\rightarrow l_{i}\gamma)$ in fig. 4. Here, the cyan and blue colored zones correspond to
the scanning selected for the relevant holomorphic and non-holomorphic
trilinear coupling matrix elements $A_{ij}$ and $A^{\prime}_{ij}$
respectively. Only the parameter points that satisfy the
charge breaking vacuum constraint of
Eq. 32 are shown where
the domain of variations of
$A_{ij}$ and $A^{\prime}_{ij}$ are mentioned in table 3.
Apart from $A_{ij}^{(\prime)}$, the
mass parameters varied are $M_{1}$, the mass of bino and the diagonal soft masses
of the left and right handed sleptons namely, $M_{\tilde{L}_{ii}}$ and $M_{\tilde{e}_{ii}}$ whose ranges are as given in table 3.
As we mentioned earlier, in this analysis
we do not consider any non-vanishing off-diagonal entry for the scalar mass matrices.
The off-diagonal trilinear couplings $A^{\prime}_{e\mu}$ has stronger influence on $Br(\mu\rightarrow e\gamma)$
compared to $A_{e\mu}$. The resulting influence superseed the present upper bound of $Br(\mu\rightarrow e\gamma)$
(see fig.4). The excluded regions that are unavailable via experimental limits are shown as gray bands in the top.
Additionally, the black horizontal lines in fig.4 to fig.4 display the upcoming sensitivities
for the respective channels of $l_{j}\rightarrow l_{i}\gamma$ (see table 1).
We remind that the effect of a non-vanishing $A^{\prime}_{e\mu}$ is associated
with an enhancement
by $\tan\beta$ that in general pushes up the branching ratio
$Br(\mu\rightarrow e\gamma)$ over 2 to 3 orders of magnitude compared to the same
arising out of $A_{e\mu}$. This is also true for the other decay modes namely
$\tau\rightarrow e\gamma$ and $\tau\rightarrow\mu\gamma$ in relation to the corresponding trilinear couplings.
However,
only the present $Br(\mu\rightarrow e\gamma)$ bound
is strong enough to exclude a significant amount of NHSSM
parameter space
depending on the values of $A^{\prime}_{e\mu}$ and the mass parameters
that would simultaneously satisfy the charge breaking constraint.
•
$\mathbf{l_{j}\rightarrow 3l_{i}}$
We will now try to investigate the influence of non-vanishing trilinear couplings $A_{ij}$ and $A^{\prime}_{ij}$ on $Br{(l_{j}\rightarrow 3l_{i})}$ in fig. 5.
This is in spite of the fact that any influence is yet too far to be tested in experiments such as that for $Br(\tau\rightarrow 3e)$ and $Br(\tau\rightarrow 3\mu)$.
Similar to fig. 4, the cyan and blue colored regions refer to parameter
points with given $A_{ij}$ and $A^{\prime}_{ij}$ values respectively that would avoid the charge breaking minima.
The present bounds of $Br(\tau\rightarrow 3e)$ and $Br(\tau\rightarrow 3\mu)$ are of the order of $10^{-8}$
by BELLE collaborationHayasaka:2010np . These are expected to reach $10^{-9}$ in Super B Aushev:2010bq . Clearly as seen in the figures both the limits are significantly
larger compared to the level of contributions under discussion. Regarding $Br(\mu\rightarrow 3e)$, a non-vanishing $A^{\prime}_{e\mu}$ can push it up to
$10^{-12}$ but $Br(\mu\rightarrow e\gamma)$ is more effective a constraint to limit $A^{\prime}_{e\mu}$.
In figure 5 we see the stretch of $Br(\tau\rightarrow 3e/\mu)$ with holomorphic and non-holomorphic trilinear couplings respectively. The color
coding is the same as in the figure 4. The cyan and blue colored points are shown after avoiding the charge breaking minima.
The current sensitivity of these two channels is of the order of $10^{-8}$
by BELLE collaboration Hayasaka:2010np which is expected to reach $10^{-9}$ in Super B Aushev:2010bq . In the following,
after satisfying all the respective bounds as mentioned in Eq:LABEL:mh_bsg and in table 1 and the radiative decays $Br(\tau\rightarrow e\gamma)$,
$Br(\tau\rightarrow\mu\gamma)$ in particular, we find
both $Br(\tau\rightarrow 3e)$ and $Br(\tau\rightarrow 3\mu)$ can reach up to $\sim 10^{-11}$. This is again two orders of magnitude smaller than
the future proposed sensitivity. The maximum reaches are $10^{-10}$ and $10^{-11}$ for $Br(\tau\rightarrow 3e)$ and $Br(\tau\rightarrow 3\mu)$ respectively,
while for $Br(\mu\rightarrow 3e)$ may become $10^{-12}$ with non-holomorphic $A^{\prime}_{e\mu}$.
$Br(\mu\rightarrow e\gamma)$ limits the free increase of $Br(\mu\rightarrow 3e)$ in particular.
We like to comment that virtual Higgs exchange can also induce $\tau$ decaying to $\mu$ along with psedoscalar meson like $\tau\rightarrow\mu(\pi/\eta/\eta^{\prime})$
though the later decay fractions lie
much below the future sensitivity presented in table 1.
With CP conservation in Higgs sector, only the exchange of CP-odd Higgs is expected to be present dominantly because of its enhanced couplings to the
down-type quarks. These interactions and relations to the other LFV processes can be found in Brignole:2004ah ; Sher:2002ew .
•
$\mathbf{\phi(h,H,A)\rightarrow l_{i}\bar{l}_{j}}$:
We focus now on flavor violating Higgs decays that lead to
two oppositely charged leptons where a Higgs boson can be
the SM-like Higgs boson ($h$), CP-even
heavier Higgs ($H$) or CP-odd Higgs ($A$).
We however note that the current experimental level to probe light higgs
decay branching ratios as seen in table 2 is way too
large compared to the branching ratio values shown below. This is unlike
the decays of the heavier Higgs where non-holomorphic parameters may give
large contributions.
$\bullet$ $\mathbf{h\rightarrow l_{i}\bar{l}_{j}}$:
We discuss the
light higgs decay here for completeness before moving to the discussion
of heavy Higgs leptonic decays with flavor violations.
fig.6 shows the
scatter plots of the branching ratios $Br(h\rightarrow e\mu)$, $Br(h\rightarrow e\tau)$ and $Br(h\rightarrow\mu\tau)$ when $A^{(^{\prime})}_{e\mu}$, $A^{(^{\prime})}_{e\tau}$
and $A^{(^{\prime})}_{\mu\tau}$ respectively are varied. All the points satisfy the
mass bounds and flavor constraints of Eq.LABEL:mh_bsg and
table 1 apart from avoiding any charge breaking
minima (Eqs.29,32).
The color convention is same as before, i.e.
the points in blue and cyan are for varying non-holomorphic and holomorphic
trilinear couplings respectively. The spread in the colored points
is the consequence of the random scanning of soft masses
as mentioned in table 3.
One notes that, irrespective of the source of flavor violation
the branching ratio
$Br(\phi\rightarrow l_{i}\bar{l}_{j})$ itself is proportional to
$\tan^{2}\beta$ Brignole:2003iv . An appropriate
NH coupling additionally multiplies the result with $\tan^{2}\beta$
potentially leading to an enhancement by a factor of $\sim 10^{3}$ when
compared to corresponding the MSSM scenario for $\tan\beta\ ^{>}\hbox to 0.0pt{${}_{\sim}$}\ 30$.
Our results for the holomorphic case closely
agree with the results of Arganda:2015uca with LR type flavor
violation in the slepton sector of MSSM.
$Br(h\rightarrow e\tau)$ is
in the ballpark of $\sim 10^{-10}~{}(\sim 10^{-13})$ for
non(-holomorphic) trilinear parameters.
We note that as seen in fig.4(a),
stringency due to $Br(\mu\rightarrow e\gamma)$
causes the extent of allowed
variation $A_{e\mu}^{(^{\prime})}$ to be much smaller than the other trilinear couplings
corresponding to those with $e$-$\tau$ or $\mu$-$\tau$. Consequently,
$Br(h\rightarrow e\mu)$ of fig.6 is smaller by 3 to 4 orders of magnitude with respect to the branching ratios of
fig.6 and fig.6.
We will now briefly study the level of dependence of the associated
sparticle masses
on the above LFV $h$-decays in fig.7.
The relevant loops contain neutralinos and sleptons.
The left and the right panels show $Br(h\rightarrow\mu\tau)$ when
soft masses of bino ($M_{1}$) and the lighter stau ($m_{\tilde{\tau}_{1}}$)
respectively are varied. All the input parameters and ranges are as given in table 3. The branching ratio profile of Fig.7b shows
that the radiative corrections associated with the decay expectedly
fall with $m_{\tilde{\tau}_{1}}$ while the same of
fig.7a hardly shows any correlation with $M_{1}$.
We note that for NHSSM, the regions of larger $Br(h\rightarrow\mu\tau)$
that are typically associated with large $A^{\prime}_{\mu\tau}$ may
lead to tachyonic staus due to large L-R slepton mixing. This is consistent
with what we find to be a small discarded zone
with $m_{\tilde{\tau}_{1}}\sim$ near 1 TeV that could otherwise
have a large branching ratio.
$\bullet$ $\mathbf{H/A\rightarrow l_{i}\bar{l}_{j}}$:
We now turn to explore the dependence on $A_{ij}^{\prime}$’s when
heavier Higgs bosons decay into
leptons with LFV namely, $H/A\rightarrow l_{i}\bar{l}_{j}$.
figure 8(a), figure 8(b) and figure 8(c) refer to the scatter plots of the associated
branching ratios for the decay of H or A bosons
into $e\mu$, $e\tau$ and $\mu\tau$
respectively where $A_{ij}^{\prime}$ are varied.
The figures have the same color convention as
before. Here, all the points satisfy the mass bounds and
flavor constraints of Eq. LABEL:mh_bsg and table 1.
As before, the figures show only the points that satisfy the charge breaking
minima constraints (Eqs. 28, 32) for varying
holomorphic or non-holomorphic trilinear coupling parameters.
The branching ratios are expected to be large because
the couplings of H/A to down-type fermions grow with $\tan\beta$.
As we discussed earlier, the LFV branching fraction
$Br(\phi_{k}\rightarrow e\tau)$ or $Br(\phi_{k}\rightarrow\mu\tau)$
can be cast in
terms of flavor conserving di-tau branching ratio
$Br(\phi_{k}\rightarrow\tau\tau)$
following Eq. 20 where $\phi_{k}$ refers to $h,H,A$ for $k=1,2,3$
respectively.
$Br(\phi_{k}\rightarrow\tau\tau)$
in general depends on slepton and
neutralino massesBrignole:2003iv with hardly any dependence on
trilinear couplings at the lowest order.
We must use the LHC data here, particularly for the
heavier Higgs bosons decaying into the di-tau channelsAaboud:2017sjh ; Aaboud:2016cre . The constraint in the
$[m_{A},\tan\beta]$ plane is rather stringent in the large $\tan\beta$ and small $m_{A}$ region. With our choice of $m_{A}$, that is rather high,
the flavor conserving branching ratio satisfies the LHC
limitAaboud:2017sjh . Apart from this, our computed results of
$\sigma(gg\phi_{k})Br(\phi_{k}\rightarrow\tau\tau)$ for all $k$
fall in the allowed
zones of the LHC limitAaboud:2016cre . The flavor violating decay rates
of $H/A$ that are computed by using Eq. 20
become large because of large $C_{H}$ or $C_{A}$ (Eq.21) when $h$ is chosen
to be SM like in its couplings. The decay rates may potentially
get amplified by a factor of $\sim 10^{3}$ (via $\propto\tan^{2}\beta$)
in presence of non-vanishing non-holomorphic trilinear couplings. Thus,
unlike the case of LFV $h$-decay of figure 6,
the LFV branching ratios of $H/A$ as shown in
figures 8(b) and 8(c)
may scale as high as $10^{-4}$. This may be of significance in relation
to a future high energy
collider.
•
Direct constraints on $Y_{ij}$ from LFV Higgs decay limits of LHC:
With identical Yuakwa couplings that MSSM inherits from SM, we will now use the constraints on relevant off-diagonal Yukawa
couplings arising out of
the null limits of SM-like higgs decays like
$h\rightarrow\mu\tau$, $h\rightarrow e\tau$, and $h\rightarrow e\mu$ as given by the LHC data.
The CMS collaboration performed direct exploration of
$h\rightarrow\mu\tau$, followed by the hunt
for $h\rightarrow e\tau/e\mu$ decays with 8 TeV
corresponding to an integrated luminosity
of 19.7 $fb^{-1}$Khachatryan:2015kon ; Khachatryan:2016rke .
The hadronic, electronic and muonic decay channels for the
$\tau$-leptons were also explored for the above mentioned LFV processes
with 13 TeV LHC dataSirunyan:2017xzt ; Aad:2019ugc .
The null results can effectively put upper limits on the off-diagonal
namely $\mu\tau$ and $e\tau$ Yukawa couplings. We relate this to the
outcome of having non-vanishing Yukawa couplings arising out of radiative
corrections due to the trilinear soft terms of both holomorphic and
non-holomorphic origins. The current LHC
boundsSirunyan:2017xzt ; Aad:2019ugc are $\sqrt{(Y_{\mu\tau}^{2}+Y_{\tau\mu}^{2})}<1.50\times 10^{-3}$ and
$\sqrt{(Y_{e\tau}^{2}+Y_{\tau e}^{2})}<2.26\times 10^{-3}$. We note that in
general for an LFV scenario there can be two independent
Yukawa couplings $Y_{ij}$ and $Y_{ji}$Calibbi:2017uvl . For example, this arises from different possible Yukawa couplings with higgs
for $e_{L}$ with $\mu_{R}$ and $\mu_{L}$ with $e_{R}$ superfields. However,
we consider them to be identical in this analysis for simplicity and
this is consistent with our assumption of a single
trilinear soft parameter $A_{ij}$ which is same as $A_{ji}$.
For simplicity and better understanding, we fix the masses of the soft parameters and present $\sqrt{Y_{ij}^{2}+Y_{ji}^{2}}$ with $A^{(^{\prime})}_{ij}$ in figure 9.
We set $M_{1}=400$ GeV while diagonal soft slepton soft masses
($M_{\tilde{L}}$ & $M_{\tilde{e}}$) are fixed at specific values 1, 2, 3 and 5 TeV (2, 3 and 5 TeV in case of NH couplings). In the left panel we show
the plots for the holomorphic off-diagonal trilinear terms and in the right panel we show the similar terms for the non-holomorphic ones. Clearly, larger radiative corrections are induced
in the case of non-vanishing $A^{{}^{\prime}}_{ij}$, particularly, when sleptons are light.
For smaller soft masses of sleptons $M_{\tilde{L}}$ & $M_{\tilde{e}}\simeq 1$ TeV,
with our choice of high $\tan\beta$, non holomorphic trilinear couplings may even generate unacceptable tachyonic states of sleptons.
The black horizontal lines in each plot relates to the upper bound on respective
$\sqrt{Y_{ij}^{2}+Y_{ji}^{2}}$.
As we will see next that for the first two generations, the black line
corresponds to the upper limit of $Br(\mu\rightarrow e\gamma)$, and for the other two generations they refer to the null results of $Br(h\rightarrow l_{i}\bar{l}_{j})$ searched by the LHC experiment.
With the understanding on how off-diagonal Yukawa couplings can be directly influenced by trilinear parameters, we now present figure 10
that shows the derived bounds on $l_{i}\rightarrow 3l_{j}$, $l_{i}\rightarrow l_{j}\gamma$ and $\sqrt{(Y_{ij}^{2}+Y_{ji}^{2})}$ in the
$(|Y_{ij}|-|Y_{ji}|)$ plane. The observed CMS limits on $\sqrt{(Y_{ij}^{2}+Y_{ji}^{2})}$ for
$\sqrt{s}=13$ TeV which are derived from the CMS direct searches of $Br(h\rightarrow\mu\tau)$ and
$Br(h\rightarrow e\tau)$ Sirunyan:2017xzt are shown as black solid curves in figure 10(b) and (c). These indeed constitute
the most stringent limits concerning the 2nd and 3rd generations of sleptons.
The Higgs boson going to $\mu\tau$ channel gives
$\sqrt{(Y_{\mu\tau}^{2}+Y_{\tau\mu}^{2})}<1.50\times 10^{-3}$ and the same for $e\tau$ channel leads to
$\sqrt{(Y_{e\tau}^{2}+Y_{\tau e}^{2})}<2.26\times 10^{-3}$ at 95% confidence level.
These limits constitute a significant improvement in the $\mu\tau$ channel over the previously obtained limits by CMS
and ATLAS using 8 TeV proton-proton collision data corresponding to an integrated luminosity of about $20~{}fb^{-1}$
Khachatryan:2015kon ; Khachatryan:2016rke shown by the green curves above the black curves. For the $e\tau$ mode, in the
8 TeV analysis, the allowed value of $\sqrt{(Y_{e\tau}^{2}+Y_{\tau e}^{2})}$ was less than $2.4\times 10^{-3}$, so the 13 TeV limit is seen to be
almost overlapped with the 8 TeV one in the middle plot of figure 10. We observe that some of the blue points originated from the
non-holomorphic trilinear couplings exceed the current CMS limits but almost all the cyan points from the holomorphic couplings are safe here. Needless to mention again, all the data points shown here
respect charge breaking minima condition.
For the first two generations, most dominant constraint comes from the absence of $\mu\rightarrow e\gamma$,
which is shown by the solid black curve in $|Y_{e\mu}|-|Y_{\mu e}|$ parameter space.
For completeness, one may find that, the CMS $\sqrt{s}=8$ TeV data puts
95% confidence level constraints on Yukawa couplings derived from $Br(h\rightarrow e\mu)<0.035\%$ which yields
$\sqrt{(Y_{e\mu}^{2}+Y_{\mu e}^{2})}$ to be less than $5.4\times 10^{-4}$ Khachatryan:2016rke but, absence of $\mu\rightarrow e\gamma$ regulates it even more,
implying a limit of $\sqrt{(Y_{e\mu}^{2}+Y_{\mu e}^{2})}<3.6\times 10^{-6}$ Harnik:2012pb .
Such a tiny Yukawa coupling, with $\cos(\beta-\alpha)\sim 10^{-3}/10^{-4}$ leads to $Br(h\rightarrow e\mu)\sim\mathcal{(}O)(10^{-13})$ or so.
444Here we should state that for non-decoupling Higgs, $m_{A}\gtrsim m_{h}$, $\cos(\beta-\alpha)$ can be large which may enhance
$Br(h\rightarrow e\mu/e\tau/\mu\tau)$. For maximum $Br(h\rightarrow l_{i}\bar{l}_{j})$ one may see the reference Aloni:2015wvn ..
One may further combine the results of figures 9 and 10 to conclusively draw upper limits on $A^{\prime}_{ij}$’s obeying the observations related to relevant LFV processes and charge breaking minima bounds. This in turn would determine maximum allowed branching ratios for our concerned cLFV process
which could be tested in the near future. For completeness
we summarize our results in the table 4
which shows the allowed values of $A^{l}_{ij}$ and $A^{\prime l}_{ij}$ in general
and the resulting maximum values of different LFV decay branching ratios. Our results can be summarized as follows:
(i) For first two generations of lepton, rise off-diagonal holomorphic and NH
trilinear couplings i.e., $A^{(\prime)}_{e\mu}$ are visibly
restricted by
the upper bound of $\mu\rightarrow e\gamma$.
(ii) The other two combinations of trilinear coupling parameters, namely, $e\tau$ and $\mu\tau$ are
regulated by the CMS 13 TeV results.
(iii) Finally the derived allowed ranges for
$A_{e\mu}$ and $A^{\prime}_{e\mu}$ are much more restricted compared to $A^{(\prime)}_{e\tau}$ or $A^{(\prime)}_{\mu\tau}$.
6 Conclusion
The Minimal Supersymmetric Standard Model when extended with the most general soft SUSY breaking trilinear terms
may lead to interesting phenomenologies. These additional terms that are non-holomorphic
in nature were analyzed in several studies
in the past as well as in the recent years. We focus on introducing flavor violating lepton decays and Higgs
decaying to charged leptons involving flavor violation due to non-vanishing non-diagonal entries of the
trilinear coupling matrices both of standard and non-standard types. In this analysis, we first upgrade the existing
analytical result involving trilinear couplings for avoiding the appearance of charge breaking minima of vacuum
in MSSM.
In other words, we extend the traditional analytical result for charge breaking in MSSM by including
non-diagonal entries of the soft-breaking trilinear coupling matrices ($A^{l}_{ij}$). We extend the analysis further by
involving non-holomorphic trilinear coupling matrices ($A^{\prime l}_{ij}$). By considering vevs for
appropriate sleptons and non-vanishing values of $A^{\prime l}_{ij}$ we are able to delineate regions of
parameter space that are associated with appearance of charge breaking minima of the vacuum.
On the contrary, we also find plenty of possibilities of evading the charge breaking conditions
even with reasonably large values of $A^{\prime l}_{ij}$ due to cancellation of terms in the analytical
result. We studied the effects of considering non-vanishing off-diagonal trilinear terms of both types,
one by one, on cLFV processes like $l_{j}\to l_{i}\gamma$ or $l_{j}\to 3l_{i}$ and all the variants of Higgs ($h,H,A$)
decays into $l_{i}{\bar{l}}_{j}$ involving flavor violation.
For simplicity, we do not consider any flavor violation effect from
the slepton mass matrices. In this phenomenological work, we include
(i) the present and future experimental sensitivities of cLFV observables, and (ii) the 8TeV and 13TeV CMS
results that search SM Higgs boson decays into flavor violating modes, namely $e\tau$ or $\mu\tau$.
We find that
NH couplings namely $A^{\prime l}_{ij}$ are
better suited in achieving
larger rates for all flavor violating decay observables that can potentially be tested in the near future. In particular, $\mu\to e\gamma$ would be more favourable
to test $A^{\prime l}_{ij}$ involving first two generation sleptons. On the other hand,
MSSM Higgs decays (specially that of the heavier Higgs bosons) into LFV modes may strongly be influenced by
$A^{\prime}_{e\tau}$ or $A^{\prime}_{\mu\tau}$. For most of these
observables the standard trilinear couplings $A_{ij}$ turn out to be inadequate to
produce any significant contribution in relation to the present or future experimental measurements. This indeed emphasizes the usefulness
of including the non-holomorphic trilinear terms for such analyses.
Acknowledgements
SM would like to thank Abhishek Dey for many helpful discussions. The computations was supported in part by the SAMKHYA: The High Performance Computing
Facility provided by Institute of Physics, Bhubaneswar.
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Fixed point theorems for $\alpha$–contractive mappings of Meir–Keeler type and applications
Maher Berzig
maher.berzig@gmail.com
Mircea-Dan Rus
rus.mircea@math.utcluj.ro
Department of Mathematics, Tunis College of Sciences and Techniques, 5 Avenue Taha Hussein, Tunis, Tunisia
Department of Mathematics, Technical University of Cluj-Napoca, Str. Memorandumului 28, 400114 Cluj-Napoca, Romania
Abstract
In this paper, we introduce the notion of $\alpha$–contractive mapping of Meir–Keeler type in complete metric spaces and prove new theorems which assure the existence, uniqueness and iterative approximation of the fixed point for this type of contraction. The presented theorems extend, generalize and improve several existing results in literature. To validate our results, we establish the existence and uniqueness of solution to a class of third order two point boundary value problems.
keywords:
Fixed point, $\alpha$–contractive mapping of Meir–Keeler type, coupled fixed point, cyclic contraction, ordered metric space, two point boundary value problem
MSC: [2010] 47H10, 34B15
††journal: …
1 Introduction
In [7], Meir and Keeler introduced a new contraction condition for
self-maps in metric spaces and generalized the well known Banach contraction
principle as follows.
Theorem 1.1 ([7]).
Let $(X,d)$ be a complete metric space and
$T:X\rightarrow X$. Assume that for every $\varepsilon>0$, there exists
$\delta(\varepsilon)>0$ such that:
$$x,y\in X:\varepsilon\leq d(x,y)<\varepsilon+\delta(\varepsilon)\Rightarrow d(%
Tx,Ty)<\varepsilon.$$
Then $T$ has a unique fixed point $x^{\ast}\in X$ and $T^{n}x\rightarrow x^{\ast}$ (as $n\rightarrow\infty$) for every $x\in X$, where $T^{n}$ denotes
the $n$-th order iterate of $T$.
In another direction, Ran and Reurings [10] extended Banach’s
contraction principle to the setting of ordered metric spaces and obtained
some interesting applications to matrix equations. Later on, the results of
Ran and Reurings were extended and generalized by many authors (e.g.,
[1, 2, 3, 4, 6, 8, 9, 12, 13, 11]
and the references therein). In particular, Harjani et al. [5]
unified these two directions by studying the fixed points of Meir–Keeler type
contractions in ordered metric spaces.
Very recently, Samet et al. [14] took a new approach to the
generalization of Banach’s contraction principle and introduced the concept of
$\alpha-\psi$–contractive type mappings, while establishing various fixed
point theorems for such mappings in the setting of complete metric spaces. In
particular, this new approach contains many of the generalizations considered
in
[1, 2, 3, 4, 6, 8, 9, 12, 13, 11, 5, 10]
as special cases.
In this context, the aim of this paper is to unify the concepts of
Meir–Keeler contraction [7] and $\alpha-\psi$–contractive type
mapping [14] and establish some new fixed point theorems in
complete metric spaces for such mappings. Several consequences of our results
are presented in Section 3. We validate our results with an
application to the study of the existence and uniqueness of solutions for a
class of third order two point boundary value problems.
2 Main results
2.1 Preliminaries
Throughout this paper, let $\mathbb{N}$ denote the set of all non-negative
integers, $\mathbb{Z}$ the set of all integers and $\mathbb{R}$ the set of all
real numbers. We start by introducing the concept of $\alpha$–contractive
mapping of Meir–Keeler type. Subsequently, we prove some lemmas useful later.
In what follows, let $(X,d)$ be a metric space, $T:X\rightarrow X$ and
$\alpha:X\times X\rightarrow[0,+\infty)$, if not stated otherwise.
Definition 2.1.
We say that $T$ is an $\alpha$–contractive mapping of Meir–Keeler type (with
respect to $d$) if for all $\varepsilon>0$, there exists $\delta(\varepsilon)>0$ such that
$$x,y\in X:\varepsilon\leq d(x,y)<\varepsilon+\delta(\varepsilon)\Rightarrow%
\alpha(x,y)d(Tx,Ty)<\varepsilon.$$
(1)
Lemma 2.1.
If $T$ is an $\alpha$–contractive mapping of Meir–Keeler
type, then
$$\alpha(x,y)d(Tx,Ty)<d(x,y)\quad\text{for all }x,y\in X\text{ with }x\neq y.$$
Proof.
Fix $x,y\in X$ with $x\neq y$ and let $\varepsilon:=d(x,y)>0$. Then, by
(1), $\alpha(x,y)d(Tx,Ty)<\varepsilon=d(x,y)$, which concludes the proof.
∎
Definition 2.2 ([14]).
We say that $T$ is $\alpha$–admissible if
$$x,y\in X:\alpha(x,y)\geq 1\Rightarrow\alpha(Tx,Ty)\geq 1.$$
Example 2.1.
Let $X=\mathbb{R}$. Define $\alpha:X\times X\rightarrow[0,+\infty)$ by
$$\alpha(x,y)=\left\{\begin{array}[c]{ll}\mathrm{e}^{x-y}&\text{if}\quad x\geq y%
,\\
0&\text{if}\quad x<y.\end{array}\right.$$
(2)
Then
$$\alpha(x,y)\geq 1\Leftrightarrow x\geq y\text{\quad}(x,y\in X),$$
hence a mapping $T:X\rightarrow X$ is $\alpha$–admissible iff it is nondecreasing.
Lemma 2.2.
Assume that $T$ is $\alpha$–admissible and $\alpha$–contractive of Meir–Keeler type. Let $x,y\in X$ such that $\alpha(x,y)\geq 1$. Then
$$\alpha(T^{n}x,T^{n}y)\geq 1\quad\text{for all }n\in\mathbb{N}\text{,}$$
(3)
the sequence $\left\{d(T^{n}x,T^{n}y)\right\}$ is nonincreasing, and
$$d(T^{n}x,T^{n}y)\rightarrow 0\quad\text{(as }n\rightarrow\infty\text{).}$$
Proof.
Since $T$ is $\alpha$–admissible and $\alpha(x,y)\geq 1$, then (3)
follows simply by induction on $n$.
Next, let $n\in\mathbb{N}$. If $T^{n}x\neq T^{n}y$, then, by (3) and
Lemma 2.1, it follows that
$$d(T^{n+1}x,T^{n+1}y)\leq\alpha(T^{n}x,T^{n}y)d(T^{n+1}x,T^{n+1}y)=\alpha(T^{n}%
x,T^{n}y)d(T(T^{n}x),T(T^{n}y))<d(T^{n}x,T^{n}y).$$
Else, if $T^{n}x=T^{n}y$, then $d(T^{n+1}x,T^{n+1}y)=d(T^{n}x,T^{n}y)$.
Concluding, $\left\{d(T^{n}x,T^{n}y)\right\}$ is nonincreasing, hence
convergent to some $\varepsilon\geq 0$.
Assume that $\varepsilon>0$, and let $p\in\mathbb{N}$ such that $\varepsilon\leq d(T^{p}x,T^{p}y)<\varepsilon+\delta(\varepsilon)$. Then $\alpha(T^{p}x,T^{p}y)d(T(T^{p}x),T(T^{p}y))<\varepsilon$, and further, by
(3), we get $d(T^{p+1}x,T^{p+1}y)<\varepsilon$, which is clearly not
possible, hence our assumption on $\varepsilon$ is wrong. Concluding, we have
necessarily $\varepsilon=0$.
∎
Definition 2.3.
We say that a sequence $\{x_{n}\}$ in $X$ is $(T,\alpha)$–orbital if
$x_{n}=T^{n}x_{0}$ and $\alpha(x_{n},x_{n+1})\geq 1$ for all $n\in\mathbb{N}$.
Definition 2.4.
We say that $T$ is $\alpha$–orbitally continuous if for every $(T,\alpha)$–orbital sequence $\{x_{n}\}$ in $X$ such that $x_{n}\rightarrow x\in X$ as
$n\rightarrow+\infty$, there exists a subsequence $\{x_{n(k)}\}$ of
$\{x_{n}\}$ such that $Tx_{n(k)}\rightarrow Tx$ as $k\rightarrow+\infty$.
Remark 2.1.
Clearly, if $T$ is continuous, then $T$ is $\alpha$–orbitally continuous (for
any $\alpha$).
Definition 2.5.
We say that $(X,d)$ is $(T,\alpha)$–regular if for every $(T,\alpha)$–orbital sequence $\{x_{n}\}$ in $X$ such that $x_{n}\rightarrow x\in X$ as
$n\rightarrow+\infty$, there exists a subsequence $\{x_{n(k)}\}$ of
$\{x_{n}\}$ such that $\alpha(x_{n(k)},x)\geq 1$ for all $k$.
Definition 2.6.
We say that $(X,d)$ is $\alpha$–regular if for every sequence $\{x_{n}\}$ in
$X$ such that $x_{n}\rightarrow x\in X$ as $n\rightarrow+\infty$ and
$\alpha(x_{n},x_{n+1})\geq 1$ for all $n$, there exists a subsequence
$\{x_{n(k)}\}$ of $\{x_{n}\}$ such that $\alpha(x_{n(k)},x)\geq 1$ for all $k$.
Remark 2.2.
Clearly, if $(X,d)$ is $\alpha$–regular, then it is also $(T,\alpha)$–regular (for any $T$).
Example 2.2.
Let $d$ be the usual (Euclidian) distance on $\mathbb{R}$, and $\alpha:\mathbb{R}\times\mathbb{R}\rightarrow[0,+\infty)$ given by
(2). Then $(\mathbb{R},d)$ is $\alpha$–regular.
Definition 2.7.
Let $N\in\mathbb{N}$. We say that $\alpha$ is $N$–transitive (on $X$) if
$$x_{0},x_{1},\dots,x_{N+1}\in X:\alpha(x_{i},x_{i+1})\geq 1\text{ for all }i\in%
\{0,1,\ldots,N\}\Longrightarrow\alpha(x_{0},x_{N+1})\geq 1.$$
In particular, we say that $\alpha$ is transitive if it is $1$–transitive,
i.e.,
$$x,y,z\in X:\alpha(x,y)\geq 1,~{}\alpha(y,z)\geq 1\Longrightarrow\alpha(x,z)%
\geq 1.$$
The following remarks are immediate consequences of the previous definition.
Remark 2.3.
Any function $\alpha:X\times X\rightarrow[0,+\infty)$ is $0$-transitive.
Remark 2.4.
If $\alpha$ is $N$ transitive, then it is $kN$–transitive for all
$k\in\mathbb{N}$.
Remark 2.5.
If $\alpha$ is transitive, then it is $N$–transitive for all $N\in\mathbb{N}$.
Example 2.3.
Let $X=\mathbb{R}$. Then $\alpha$ defined by (2) is transitive.
Example 2.4.
Let $N\in\mathbb{N}\setminus\{0\}$ and $\left\{A_{1},\ldots,A_{N}\right\}$ a family of nonempty sets. Let $X=\bigcup_{i=1}^{N}A_{i}$ and $R=\bigcup_{i=1}^{N}\left(A_{i}\times A_{i+1}\right)$ (with
$A_{N+1}:=A_{1}$). Define $\alpha:X\times X\rightarrow[0,+\infty)$ by
$$\alpha(x,y)=\left\{\begin{array}[c]{ll}1,&\text{if }(x,y)\in R\\
0,&\text{otherwise.}\end{array}\right.$$
Then $\alpha$ is $N$–transitive, but not necessarily transitive (see, also,
Corollary 3.7).
Definition 2.8.
Let $x,y\in X$. A vector $\zeta=(z_{0},z_{1},\ldots,z_{n})\in X^{n+1}$ is
called an $\alpha$–chain (of order $n$) from $x$ to $y$ if $z_{0}=x$,
$z_{n}=y$ and, for every $i\in\{1,2,\ldots,n\}$,
$$\alpha(z_{i-1},z_{i})\geq 1\text{ or }\alpha(z_{i},z_{i-1})\geq 1\text{.}$$
Definition 2.9.
We say that $X$ is $\alpha$–connected if for every $x,y\in X$ with $x\neq y$,
there exists an $\alpha$–chain from $x$ to $y$.
2.2 Existence and uniqueness of fixed points
Now, we are ready to present and prove the first main result of the paper.
Theorem 2.1.
Let $(X,d)$ be a complete metric space, $\alpha:X\times X\rightarrow[0,+\infty)$ a $N$–transitive mapping (for some
$N\in\mathbb{N}\setminus\{0\}$) and $T:X\rightarrow X$ an $\alpha$–contractive mapping of Meir–Keeler type satisfying the following conditions:
(A1)
$T$ is $\alpha$–admissible;
(A2)
there exists $x_{0}\in X$ such that $\alpha(x_{0},Tx_{0})\geq 1$;
(A3)
$T$ is $\alpha$–orbitally continuous.
Then $T$ has a fixed point, that is, there exists $x^{\ast}\in X$ such that
$Tx^{\ast}=x^{\ast}$.
Proof.
Define the sequence $\{x_{n}\}$ in $X$ by $x_{n+1}=Tx_{n}$ for all
$n\in\mathbb{N}$; equivalently, $x_{n}=T^{n}x_{0}$. Since $\alpha(x_{0},Tx_{0})\geq 1$, then by Lemma 2.2 we get
$$\alpha(x_{n},x_{n+1})\geq 1\quad\text{for all }n\in\mathbb{N}$$
(4)
and
$$d(x_{n},x_{n+1})\rightarrow 0\quad\text{as}\quad n\rightarrow+\infty.$$
(5)
Fix $\varepsilon>0$. Without any loss of generality, we may assume that
$\delta(\varepsilon)\leq\varepsilon$. Using (5), there exists $k$
such that
$$d(x_{n},x_{n+1})<\frac{\delta(\varepsilon)}{N}\quad\text{for all }n\geq k\text%
{.}$$
(6)
We introduce the set $Y\subset X$ defined by
$$Y:=\left\{x\in X:\text{ there exists }q(x)\in\{0,1,\ldots,N-1\}~{}~{}\text{%
such
that}~{}~{}d(x_{k+q(x)},x)<\varepsilon+\delta(\varepsilon)\text{ and }\alpha(x%
_{k+q(x)},x)\geq 1\right\}.$$
Fix $x\in Y$. Our first claim is that
$$T^{N}x\in Y\text{ and }q\left(T^{N}x\right)=q(x).$$
(7)
For short, let $q:=q(x)$.
First, we prove that
$$d(x_{k+q},T^{N}x)<\varepsilon+\delta(\varepsilon).$$
(8)
Using the triangle inequality and (6), we obtain
$$d(x_{k+q},T^{N}x)\leq\sum_{i=0}^{N-1}d(x_{k+q+i},x_{k+q+i+1})+d(x_{k+q+N},T^{N%
}x)<\delta(\varepsilon)+d(T^{N}x_{k+q},T^{N}x),$$
while $\alpha(x_{k+q},x)\geq 1$ leads to
$$d(T^{N}x_{k+q},T^{N}x)\leq d(Tx_{k+q},Tx)\leq d(x_{k+q},x)$$
by Lemma 2.2; hence, we conclude that
$$d(x_{k+q},T^{N}x)<d(Tx_{k+q},Tx)+\delta(\varepsilon)\leq d(x_{k+q},x)+\delta(%
\varepsilon).$$
(9)
Clearly, if $d(x_{k+q},x)<\varepsilon$, then (9) leads to
(8), so it is enough to consider the case when $\varepsilon\leq d(x_{k+q},x)$. Then $x\in Y$ leads to $\varepsilon\leq d(x_{k+q},x)<\varepsilon+\delta(\varepsilon)$. Using next that $T$ is an $\alpha$–contractive mapping of Meir–Keeler type, we obtain that $\alpha(x_{k+q},x)d(Tx_{k+q},Tx)<\varepsilon$, and since $\alpha(x_{k+q},x)\geq 1$, we
arrive to
$$d(Tx_{k+q},Tx)<\varepsilon;$$
(10)
hence (8) follows again by (9) and (10).
Next, we prove that
$$\alpha(x_{k+q},T^{N}x)\geq 1.$$
(11)
Indeed,
$$\alpha(x_{k+q+i},x_{k+q+i+1})\geq 1\text{\quad for all }i\in\{0,1,\ldots,N-1\}$$
(12)
by (4). Also, $\alpha(x_{k+q},x)\geq 1$ leads by Lemma 2.2
to
$$\alpha(x_{k+q+N},T^{N}x)\geq 1.$$
(13)
Now, using (12), (13) and the $N$–transitivity of $\alpha$,
we finally get (11).
Concluding, our first claim (7) is proven.
Our second claim is
$$x_{k+i+1}\in Y\text{ and }q(x_{k+i+1})=i\quad\text{for all }i\in\{0,1,\ldots,N%
-1\}\text{.}$$
(14)
Indeed, $d(x_{k+i},x_{k+i+1})<\frac{\delta(\varepsilon)}{N}<\varepsilon+\delta(\varepsilon)$ by (6), while $\alpha(x_{k+i},x_{k+i+1})\geq 1$
by (4), which proves (14).
Now, by (7) and (14), we can easily conclude that
$$x_{n}\in Y\text{ and }q(x_{n})=(n-k-1)\operatorname{mod}N\quad\text{for all
}n\geq k+1.$$
(15)
Finally, let $m,n\geq k+1$ and assume that $q(x_{n})\leq q(x_{m})$ without any
loss of generality. Then, by the triangle inequality, (6) and
(15), it follows that
$$\displaystyle d(x_{n},x_{m})$$
$$\displaystyle\leq d(x_{n},x_{k+q(x_{n})})+\sum_{i=q(x_{n})}^{q(x_{m})-1}d\left%
(x_{k+i},x_{k+i+1}\right)+d(x_{k+q(x_{m})},x_{m})$$
$$\displaystyle<2(\varepsilon+\delta(\varepsilon))+(q(x_{m})-q(x_{n}))\frac{%
\delta(\varepsilon)}{N}\leq 2(\varepsilon+\delta(\varepsilon))+\delta(%
\varepsilon)\leq 5\varepsilon.$$
Concluding, $\{x_{n}\}$ is a Cauchy sequence in the complete metric space
$(X,d)$, hence convergent to some $x^{\ast}\in X$. Moreover, $\{x_{n}\}$ is a
$(T,\alpha)$–orbital sequence by (4), hence, by (A3), there exists
a subsequence $\{x_{n(k)}\}$ of $\{x_{n}\}$ such that $Tx_{n(k)}\rightarrow Tx^{\ast}$ as $k\rightarrow+\infty$. But $Tx_{n(k)}=x_{n(k)+1}\rightarrow x^{\ast}$ as $k\rightarrow+\infty$, hence $Tx^{\ast}=x^{\ast}$ by the
uniqueness of the limit, which concludes the proof.
∎
In the next theorem, we replace the continuity of the mapping $T$ by a
regularity condition over the metric space $(X,d)$.
Theorem 2.2.
In the conditions of Theorem 2.1, if (A3) is replaced with:
(A4)
$(X,d)$ is $(T,\alpha)$–regular,
then the conclusion of Theorem 2.1 holds.
Proof.
Following the proof of Theorem 2.1, we only have to prove that
$x^{\ast}$ is a fixed point of $T$. Since $\{x_{n}\}$ is a $(T,\alpha)$–orbital sequence, then, by (A4), there exists a subsequence $\{x_{n(k)}\}$
of $\{x_{n}\}$ such that
$$\alpha(x_{n(k)},x^{\ast})\geq 1\quad\text{for all}\quad k\in\mathbb{N}.$$
Next, using Lemma 2.1, we get
$$d(Tx_{n(k)},Tx^{\ast})\leq\alpha(x_{n(k)},x^{\ast})d(Tx_{n(k)},Tx^{\ast})\leq d%
(x_{n(k)},x^{\ast})\quad\text{for all}\quad k\in\mathbb{N}$$
(with equality when $x_{n(k)}=x^{\ast}$). As $x_{n(k)}\rightarrow x^{\ast}$,
we obtain that $x_{n(k)+1}=Tx_{n(k)}\rightarrow Tx^{\ast}$. As $\{x_{n(k)+1}\}$ is a subsequence of $\{x_{n}\}$ and $x_{n}\rightarrow x^{\ast}$ we have
$x_{n(k)+1}\rightarrow x^{\ast}$. Now, the uniqueness of the limit gives us
$Tx^{\ast}=x^{\ast}$ and the proof is complete.
∎
To assure the uniqueness of the fixed point, we will consider the following
additional assumption.
(A5)
$X$ is $\alpha$–connected.
This is the purpose of the next theorem.
Theorem 2.3.
If adding (A5) to the hypotheses of Theorem 2.1 (or Theorem
2.2), then $x^{\ast}$ is the unique fixed point of $T$ and
$T^{n}(x)\rightarrow x^{\ast}$ (as $n\rightarrow\infty$) for every $x\in X$.
Proof.
Let $x\in X\setminus\{x^{\ast}\}$. By (A5), there exists $(x^{\ast}=z_{0},z_{1},\ldots,z_{n}=x)$ an $\alpha$–chain from $x^{\ast}$ to $x$.
Since
$$\alpha(z_{i-1},z_{i})\geq 1\text{ or }\alpha(z_{i},z_{i-1})\geq 1\quad\text{%
for
all }i\in\{1,2,\ldots,n\},$$
it follows by Lemma 2.2 and the symmetry of $d$, that
$$d(T^{n}(z_{i-1}),T^{n}(z_{i}))\rightarrow 0\text{ (as }n\rightarrow+\infty%
\text{)\quad for all }i\in\{1,2,\ldots,n\}\text{.}$$
(16)
Now, since $z_{0}=x^{\ast}$ is a fixed point of $T$, it follows that
$T^{n}(z_{0})=x^{\ast}$ for all $n$, which finally leads to
$$T^{n}z_{i}\rightarrow x^{\ast}\text{ (as }n\rightarrow+\infty\text{)\quad for
all }i\in\{1,2,\ldots,n\},$$
using (16); hence, $T^{n}x\rightarrow x^{\ast}$ (as $n\rightarrow+\infty$). In particular, if $x$ is another fixed point of $T$, it follows
that $x=x^{\ast}$ which is a contradiction, and the proof is concluded.
∎
3 Some corollaries
In this section, we will derive some corollaries from our previous theorems.
3.1 Coupled fixed point theorems for bivariate $\alpha$–contractive
mappings of Meir–Keeler type on complete metric spaces
The theorems obtained in the previous section allow us to derive some coupled
fixed point results in complete metric spaces. First, let us recall the
following definitions.
Definition 3.1 ([4]).
Let $X$ be a nonempty set and $F:X\times X\rightarrow X$
be a given mapping. A pair $(x,y)\in X\times X$ is called a coupled fixed
point of $F$ if $F(x,y)=x$ and $F(y,x)=y$.
Also, $x\in X$ is called a fixed point of $F$ if $(x,x)$ is a coupled fixed
point, i.e., $F(x,x)=x$.
Definition 3.2 ([11]).
Let $X$ be a nonempty set, and $F,G:X\times X\rightarrow X$.
The symmetric composition (or, the $s$-composition for short) of $A$
and $B$ is defined by
$$G\ast F:X\times X\rightarrow X,\quad(G\ast F)(x,y)=G(F(x,y),F(y,x))\quad(x,y%
\in X).$$
Remark 3.1 ([11]).
The $s$-composition is an associative law. Also, the
projection mapping
$$P_{X}:X\times X\rightarrow X,\quad P(x,y)=x\quad(x,y\in X)$$
is the identity element with respect to the $s$-composition (i.e., $F\ast P_{X}=P_{X}\ast F=F$ for all $F:X\times X\rightarrow X$). Consequently, for
any $F:X\times X\rightarrow X$ one can define the functional powers (i.e., the
iterates) of $F$ with respect to the $s$-composition by
$$F^{n+1}=F\ast F^{n}=F^{n}\ast F\quad(n\in\mathbb{N}),\quad F^{0}=P_{X}\text{.}$$
We have the following result.
Corollary 3.1.
Let $(X,d)$ be a complete metric space, $\alpha:(X\times X)\times(X\times X)\rightarrow[0,+\infty)$ a $N$–transitive mapping on
$X\times X$ for some $N\in\mathbb{N}\setminus\{0\}$, and $F:X\times X\rightarrow X$ such that for every $\varepsilon>0$ there exists
$\delta(\varepsilon)>0$ for which:
$$(x,y),(u,v)\in X\times X:\varepsilon\leq\frac{d(x,u)+d(y,v)}{2}<\varepsilon+%
\delta(\varepsilon)\Rightarrow\alpha((x,y),(u,v))d(F(x,y),F(u,v))<\varepsilon.$$
(17)
Suppose that
(B1)
for all $(x,y),(u,v)\in X\times X$,
$$\alpha((x,y),(u,v))\geq 1\Longrightarrow\alpha((F(x,y),F(y,x)),(F(u,v),F(v,u))%
)\geq 1;$$
(B2)
there exists $(x_{0},y_{0})\in X\times X$ such that
$$\alpha\left((x_{0},y_{0}),(F(x_{0},y_{0}),F(y_{0},x_{0}))\right)\geq 1\quad%
\text{and}\quad\alpha\left((F(y_{0},x_{0}),F(x_{0},y_{0})),(y_{0},x_{0})\right%
)\geq 1;$$
(B3)
$F$ is continuous.
Then $F$ has a coupled fixed point, that is, there exists $(x^{\ast},y^{\ast})\in X\times X$ such that $x^{\ast}=F(x^{\ast},y^{\ast})$ and $y^{\ast}=F(y^{\ast},x^{\ast})$.
Proof.
Consider
$$D\left((x,y),(u,v)\right):=\frac{1}{2}\left(d(x,u)+d(y,v)\right)\quad\text{for%
all }(x,y),(u,v)\in X\times X.$$
Then, clearly, $(X\times X,D)$ is a complete metric space. Also, let
$T:X\times X\rightarrow X\times X$ be defined by
$$T(x,y)=(F(x,y),F(y,x))\quad\text{for all }(x,y)\in X\times X$$
and $\beta:(X\times X)\times(X\times X)\rightarrow[0,+\infty)$ be given
by
$$\beta((x,y),(u,v))=\min\left\{\alpha((x,y),(u,v)),\alpha((v,u),(y,x))\right\}%
\quad\text{for all }(x,y),(u,v)\in X\times X.$$
(18)
First, we prove that $\beta$ is $N$-transitive. Let $(x_{i},y_{i})\in X\times X$ ($i\in\{0,1,\ldots,N+1\}$) such that $\beta\left((x_{i},y_{i}),(x_{i+1},y_{i+1})\right)\geq 1$ for all $i\in\{0,1,\ldots,N\}$. By the
definition of $\beta$, it follows that
$$\alpha\left((x_{i},y_{i}),(x_{i+1},y_{i+1})\right)\geq 1\text{ and }\alpha%
\left((y_{i+1},x_{i+1}),(y_{i},x_{i})\right)\geq 1\text{ \quad for
all }i\in\{0,1,\ldots,N\},$$
hence, by the $N$–transitivity of $\alpha$, we have that
$$\alpha\left((x_{0},y_{0}),(x_{N+1},y_{N+1})\right)\geq 1\text{ and }\alpha%
\left((y_{N+1},x_{N+1}),(y_{0},x_{0})\right)\geq 1\text{,}$$
which concludes our argument.
We claim next that $T$ is a $\beta$–contractive mapping of Meir–Keeler type
(with respect to $D$). Indeed, let $\varepsilon>0$ and let $\delta(\varepsilon)>0$ for which (17) is satisfied. If $(x,y),(u,v)\in X\times X$ are such that $\varepsilon\leq D\left((x,y),(u,v)\right)<\varepsilon+\delta(\varepsilon)$, then also $\varepsilon\leq D\left((v,u),(y,x)\right)<\varepsilon+\delta(\varepsilon)$ by the definition of
$D$, hence
$$\begin{array}[c]{c}\alpha((x,y),(u,v))d(F(x,y),F(u,v))<\varepsilon\\
\alpha((v,u),(y,x))d(F(v,u),F(y,x))<\varepsilon\end{array}$$
by (17). These two inequalities lead straight to
$$\beta((x,y),(u,v))D\left(T(x,y),T(u,v)\right)<\varepsilon,$$
which proves our claim.
Next, it is easy to check that $T$ is $\beta$–admissible by (B1). Moreover,
(B2) ensures that $\beta((x_{0},y_{0}),T(x_{0},y_{0}))\geq 1$, while (B3)
ensures that $T$ is continuous, hence $\beta$–orbitally continuous.
Concluding, all the hypotheses of Theorem 2.1 applied to the metric
space $(X\times X,D)$, the mapping $T$ and the function $\beta$ are satisfied,
hence $T$ has a fixed point $(x^{\ast},y^{\ast})\in X\times X$, meaning that
$(x^{\ast},y^{\ast})$ is a coupled fixed point of $F$. The proof is now complete.
∎
Corollary 3.2.
In the conditions of Corollary 3.1, if (B3) is replaced with:
(B4)
for every sequence $\{(x_{n},y_{n})\}$ in $X\times X$
such that $x_{n}\rightarrow x\in X$, $y_{n}\rightarrow y\in X$ as
$n\rightarrow+\infty$, and
$$\alpha((x_{n},y_{n}),(x_{n+1},y_{n+1}))\geq 1,\quad\alpha((y_{n+1},x_{n+1}),(y%
_{n},x_{n}))\geq 1\quad\text{for all }n\in\mathbb{N},$$
there exists a subsequence $\{(x_{n(k)},y_{n(k)})\}$ such that
$$\alpha((x_{n(k)},y_{n(k)}),(x,y))\geq 1,\quad\alpha((y,x),(y_{n(k)},x_{n(k)}))%
\geq 1\quad\text{for all }k\in\mathbb{N};$$
then the conclusion of Corollary 3.1 holds.
Proof.
Using the notations in the proof of Corollary 3.1, it easily follows
by (B4) that $(X\times X,D)$ is $\beta$–regular, hence $(T,\beta)$–regular.
By following the proof of Corollary 3.1, the conclusion follows by
Theorem 2.2 applied to the metric space $(X\times X,D)$, the mapping
$T$ and the function $\beta$.
∎
For the uniqueness of the coupled fixed point, we consider the following assumption.
(B5)
$X\times X$ is $\beta$–connected, where $\beta$ is
defined by (18).
Corollary 3.3.
If adding condition (B5) to the hypotheses of Corollary
3.1 (or Corollary 3.2) then $x^{\ast}=y^{\ast}$, $(x^{\ast},x^{\ast})$ is the unique coupled fixed point of $F$ and $x^{\ast}$ is the
unique fixed point of $F$. Moreover, $F^{n}(x,y)\rightarrow x^{\ast}$ as
$n\rightarrow\infty$ for all $x,y\in X$.
Proof.
We use the notations in the proof of Corollary 3.1. Then, by Theorem
2.3, it follows that $(x^{\ast},y^{\ast})$ is the unique fixed point of
$T$, hence the unique coupled fixed point of $F$. Since $(y^{\ast},x^{\ast})$
is also a coupled fixed point of $F$, then $(x^{\ast},y^{\ast})=(y^{\ast},x^{\ast})$, hence $x^{\ast}=y^{\ast}$, meaning also that $x^{\ast}$ is the
unique fixed point of $F$. Since $T^{n}(x,y)=\left(F^{n}(x,y),F^{n}(y,x)\right)$ for all $n\in\mathbb{N}$ and $x,y\in X$, the proof is complete.
∎
We conclude this subsection with a particular form of the above corollaries,
when $\alpha$ is represented as:
$$\alpha\left((x,y),(u,v)\right)=\min\left\{\alpha_{0}(x,u),\alpha_{0}(v,y)%
\right\}\quad\left((x,y),(u,v)\in X\times X\right)\text{,}$$
(19)
where $\alpha_{0}:X\times X\rightarrow[0,+\infty)$. Note that, in this
case, $\beta=\alpha$. We subsume the conclusions of Corollaries 3.1,
3.2 and 3.3 in one single result, as follows:
Corollary 3.4.
Let $(X,d)$ be a complete metric space, $\alpha_{0}:X\times X\rightarrow[0,+\infty)$ a $N$–transitive mapping on $X\times X$ for
some $N\in\mathbb{N}\setminus\{0\}$, and $F:X\times X\rightarrow X$ such that
for every $\varepsilon>0$ there exists $\delta(\varepsilon)>0$ for which:
$$(x,y),(u,v)\in X\times X:\varepsilon\leq\frac{d(x,u)+d(y,v)}{2}<\varepsilon+%
\delta(\varepsilon)\Rightarrow\min\left\{\alpha_{0}(x,u),\alpha_{0}(v,y)\right%
\}d(F(x,y),F(u,v))<\varepsilon.$$
Suppose that
(C1)
for all $(x,y),(u,v)\in X\times X$,
$$\alpha_{0}(x,u)\geq 1,~{}\alpha_{0}(v,y)\geq 1\Longrightarrow\alpha_{0}\left(F%
(x,y),F(u,v)\right)\geq 1;$$
(C2)
there exists $(x_{0},y_{0})\in X\times X$ such that
$$\alpha_{0}\left(x_{0},F(x_{0},y_{0})\right)\geq 1,\quad\alpha_{0}\left(F(y_{0}%
,x_{0}),y_{0}\right)\geq 1.$$
If either
(C3)
$F$ is continuous,
or
(C4)
for every sequence $\{(x_{n},y_{n})\}$ in $X\times X$
such that $x_{n}\rightarrow x\in X$, $y_{n}\rightarrow y\in X$ as
$n\rightarrow+\infty$, and
$$\alpha_{0}(x_{n},x_{n+1})\geq 1,\quad\alpha_{0}(y_{n+1},y_{n})\geq 1\quad\text%
{for all }n\in\mathbb{N},$$
there exists a subsequence $\{(x_{n(k)},y_{n(k)})\}$ such that
$$\alpha_{0}\left(x_{n(k)},x\right)\geq 1,\quad\alpha_{0}(y,y_{n(k)})\geq 1\quad%
\text{for all}\ k\in\mathbb{N};$$
then $F$ has a coupled fixed point, that is, there exists $(x^{\ast},y^{\ast})\in X\times X$ such that $x^{\ast}=F(x^{\ast},y^{\ast})$ and
$y^{\ast}=F(y^{\ast},x^{\ast})$.
Additionally, if
(C5)
$X$ is $\alpha_{0}$–connected,
then $x^{\ast}=y^{\ast}$, $(x^{\ast},x^{\ast})$ is the unique
coupled fixed point of $F$, $x^{\ast}$ is the unique fixed point of $F$ and
$F^{n}(x,y)\rightarrow x^{\ast}$ as $n\rightarrow\infty$ for all $x,y\in X$.
Proof.
It checks easily that the hypotheses of Corollaries 3.1, 3.2
and 3.3 are satisfied, with $\alpha$ defined by (19).
∎
3.2 Fixed point theorems for ${\mathcal{R}}$–contractive mappings of
Meir–Keeler type on a metric space endowed with a $N$–transitive binary
relation
The notions and results in Section 2 easily translate to the setting
of metric spaces endowed with a $N$–transitive binary relation.
In what follows, let $(X,d)$ be a metric space, ${\mathcal{R}}$ be a binary
relation over $X$ and $T:X\rightarrow X$. We first start with some terminology
that is symmetrical to that in Section 2.
Definition 3.3.
We say that $T$ is a ${\mathcal{R}}$–contractive mapping of Meir–Keeler type
(with respect to $d$) if for all $\varepsilon>0$, there exists $\delta(\varepsilon)>0$ such that
$$x,y\in X:x{\mathcal{R}}y,~{}\varepsilon\leq d(x,y)<\varepsilon+\delta(%
\varepsilon)\Rightarrow d(Tx,Ty)<\varepsilon.$$
Definition 3.4.
We say that $T$ is ${\mathcal{R}}$–preserving if
$$x,y\in X:x{\mathcal{R}}y\Rightarrow Tx{\mathcal{R}}Ty.$$
Definition 3.5.
We say that a sequence $\{x_{n}\}$ in $X$ is $(T,{\mathcal{R}})$–orbital if
$x_{n}=T^{n}x_{0}$ and $x_{n}{\mathcal{R}}x_{n+1}$ for all $n\in\mathbb{N}$.
Definition 3.6.
We say that $T$ is ${\mathcal{R}}$–orbitally continuous if for every
$(T,{\mathcal{R}})$–orbital sequence $\{x_{n}\}$ in $X$ such that
$x_{n}\rightarrow x\in X$ as $n\rightarrow+\infty$, there exists a subsequence
$\{x_{n(k)}\}$ of $\{x_{n}\}$ such that $Tx_{n(k)}\rightarrow Tx$ as
$k\rightarrow+\infty$.
Remark 3.2.
Clearly, if $T$ is continuous, then $T$ is ${\mathcal{R}}$–orbitally
continuous (for any ${\mathcal{R}}$).
Definition 3.7.
We say that $(X,d)$ is $(T,{\mathcal{R}})$–regular if for every
$(T,{\mathcal{R}})$–orbital sequence $\{x_{n}\}$ in $X$ such that
$x_{n}\rightarrow x\in X$ as $n\rightarrow+\infty$, there exists a subsequence
$\{x_{n(k)}\}$ of $\{x_{n}\}$ such that $x_{n(k)}{\mathcal{R}}x$ for all $k$.
Definition 3.8.
We say that $(X,d)$ is ${\mathcal{R}}$–regular if for every sequence
$\{x_{n}\}$ in $X$ such that $x_{n}\rightarrow x\in X$ as $n\rightarrow+\infty$ and $x_{n}{\mathcal{R}}x_{n+1}$ for all $n$, there exists a
subsequence $\{x_{n(k)}\}$ of $\{x_{n}\}$ such that $x_{n(k)}{\mathcal{R}}x$
for all $k$.
Remark 3.3.
Clearly, if $(X,d)$ is ${\mathcal{R}}$–regular, then it is also
$(T,{\mathcal{R}})$–regular (for any $T$).
Definition 3.9.
Let $N\in\mathbb{N}$. We say that ${\mathcal{R}}$ is $N$–transitive (on $X$)
if
$$x_{0},x_{1},x_{2},\dots,x_{N},x_{N+1}\in X:x_{i}{\mathcal{R}}x_{i+1}\text{ for
all }i\in\{0,1,\ldots,N\}\Longrightarrow x_{0}{\mathcal{R}}x_{N+1}.$$
In particular, for $N=1$ we recover the usual transitivity property.
Definition 3.10.
Let $x,y\in X$. A vector $\zeta=(z_{0},z_{1},\ldots,z_{n})\in X^{n+1}$ is
called a ${\mathcal{R}}$–chain (of order $n$) from $x$ to $y$ if $z_{0}=x$,
$z_{n}=y$ and
$$z_{i-1}{\mathcal{R}}z_{i}\text{ or }z_{i}{\mathcal{R}}z_{i-1}\text{\quad for
every }i\in\{1,2,\ldots,n\}.$$
Definition 3.11.
We say that $X$ is ${\mathcal{R}}$–connected if for every $x,y\in X$ with
$x\neq y$, there exists a ${\mathcal{R}}$–chain from $x$ to $y$.
The main results in Section 2 translate to the setting of metric
spaces endowed with an arbitrary binary relation as follows.
Corollary 3.5.
Let $(X,d)$ be a complete metric space, ${\mathcal{R}}$ a
$N$–transitive binary relation over $X$ (for some $N\in\mathbb{N}\setminus\{0\}$) and $T:X\rightarrow X$ a ${\mathcal{R}}$–contractive mapping
of Meir–Keeler type. Assume that:
(D1)
$T$ is ${\mathcal{R}}$-preserving;
(D2)
there exists $x_{0}\in X$ such that $x_{0}{\mathcal{R}}Tx_{0}$.
If either
(D3)
$T$ is continuous,
or
(D4)
$(X,d)$ is $(T,{\mathcal{R}})$–regular,
then $T$ has a fixed point $x^{\ast}\in X$. Additionally, if
(D5)
$X$ is ${\mathcal{R}}$–connected,
then $x^{\ast}$ is the unique fixed point of $T$ and $T^{n}(x)\rightarrow x^{\ast}$ (as $n\rightarrow\infty$) for every $x\in X$.
Proof.
Define the mapping $\alpha:X\times X\rightarrow[0,+\infty)$ by
$$\alpha(x,y)=\left\{\begin{array}[c]{ll}1,&\text{if }x\mathcal{R}y\\
0,&\text{otherwise.}\end{array}\right.$$
The conclusions then follows directly from Theorems 2.1, 2.2 and
2.3.
∎
The following result is a consequence of Corollary 3.4 for bivariate
${\mathcal{R}}$–contractive mappings of Meir–Keeler type.
Corollary 3.6.
Let $(X,d)$ be a complete metric space, ${\mathcal{R}}$ a
$N$–transitive binary relation over $X$ (for some $N\in\mathbb{N}\setminus\{0\}$), and $F:X\times X\rightarrow X$ such that for every
$\varepsilon>0$ there exists $\delta(\varepsilon)>0$ for which:
$$x,y,u,v\in X:x{\mathcal{R}}y,~{}v{\mathcal{R}}u,~{}\varepsilon\leq\frac{d(x,u)%
+d(y,v)}{2}<\varepsilon+\delta(\varepsilon)\Rightarrow d(F(x,y),F(u,v))<\varepsilon.$$
Suppose that
(E1)
for all $x,y,u,v\in X$,
$$x{\mathcal{R}}y,~{}v{\mathcal{R}}u\Longrightarrow F(x,y){\mathcal{R}}F(u,v);$$
(E2)
there exists $(x_{0},y_{0})\in X\times X$ such that
$$x_{0}{\mathcal{R}}F(x_{0},y_{0}),\quad F(y_{0},x_{0}){\mathcal{R}}y_{0}.$$
If either
(E3)
$F$ is continuous,
or
(E4)
for every sequence $\{(x_{n},y_{n})\}$ in $X\times X$
such that $x_{n}\rightarrow x\in X$, $y_{n}\rightarrow y\in X$ as
$n\rightarrow+\infty$, and $x_{n}{\mathcal{R}}x_{n+1},~{}y_{n+1}{\mathcal{R}}y_{n}$ for all $n\in\mathbb{N}$, there exists a subsequence $\{(x_{n(k)},y_{n(k)})\}$ such that $x_{n(k)}{\mathcal{R}}x,~{}y{\mathcal{R}}y_{n(k)}$ for
all $k\in\mathbb{N}$,
then $F$ has a coupled fixed point $(x^{\ast},y^{\ast})\in X\times X$. Additionally, if
(E5)
$X$ is ${\mathcal{R}}$–connected,
then $x^{\ast}=y^{\ast}$, $(x^{\ast},x^{\ast})$ is the unique coupled fixed
point of $F$, $x^{\ast}$ is the unique fixed point of $F$ and $F^{n}(x,y)\rightarrow x^{\ast}$ as $n\rightarrow\infty$ for all $x,y\in X$.
Proof.
Define the mapping $\alpha_{0}:X\times X\rightarrow[0,+\infty)$ by
$$\alpha_{0}(x,y)=\left\{\begin{array}[c]{ll}1,&\text{if }x\mathcal{R}y\\
0,&\text{otherwise.}\end{array}\right.$$
The conclusions then follows directly from Corollary 3.4.
∎
3.3 Fixed point results for cyclic contractive mappings of
Meir–Keeler type
In this section, we obtain some fixed point results for cyclic $\alpha$–contractions of Meir–Keeler type. We start by recalling the result
obtained by Kirk, Srinivasan and Veeramani in [6] for cyclic
contractive mappings.
Theorem 3.1 ([6]).
Let $(X,d)$ be a complete metric space, $\left\{A_{1},A_{2},\ldots,A_{N}\right\}$ a family of nonempty and closed subsets of $X$
and $T:X\rightarrow X$. Suppose that the following conditions hold:
(F1)
$T(A_{i})\subseteq A_{i+1}$ for all $i\in\{1,2\dots,N\}$
(where $A_{N+1}=A_{1}$);
(F2)
there exists $k\in(0,1)$ such that
$$d(Tx,Ty)\leq kd(x,y)\text{\quad for all }x\in A_{i},y\in A_{i+1},i\in\{1,2%
\dots,N\}\text{.}$$
Then $\bigcap_{i=1}^{N}A_{i}$ is non-empty and $T$ has a unique fixed point in
$\bigcap_{i=1}^{N}A_{i}$.
The aim of our next result is to weaken the contraction condition (F2) by
considering the following condition of Meir–Keeler type:
(F3)
for every $\varepsilon>0$, there exists $\delta(\varepsilon)>0$
such that
$$x\in A_{i},y\in A_{i+1},i\in\{1,2,\ldots,N\}:\varepsilon\leq d(x,y)<%
\varepsilon+\delta(\varepsilon))\Rightarrow d(Tx,Ty)<\varepsilon.$$
Corollary 3.7.
Let $(X,d)$ be a complete metric space, $\left\{A_{1},A_{2},\ldots,A_{N}\right\}$ a family of nonempty and closed subsets of $X$
and $T:X\rightarrow X$. Suppose that (F1) and (F3) hold.
Then $\bigcap_{i=1}^{N}A_{i}$ is non-empty and $T$ has a fixed point $x^{\ast}\in\bigcap_{i=1}^{N}A_{i}$. Moreover, $x^{\ast}$ is the unique fixed point of
$T$ in $\bigcup_{i=1}^{N}A_{i}$ and $T^{n}(x)\rightarrow x^{\ast}$ for all
$x\in\bigcup_{i=1}^{N}A_{i}$.
Proof.
Let $Y:=\bigcup_{i=1}^{N}A_{i}$. Then $Y$ is a closed part of $X$; hence,
$(Y,d)$ is a complete metric space. Moreover, the restriction $\left.T\right|_{Y}$ of $T$ to $Y$ is a self-map of $Y$, by (F1); for
convenience, we write $T$ instead of $\left.T\right|_{Y}$.
Define the mapping $\alpha:Y\times Y\rightarrow[0,+\infty)$ by
$$\alpha(x,y)=\left\{\begin{array}[c]{ll}1,&\text{if }(x,y)\in R:=\bigcup_{i=1}^%
{N}\left(A_{i}\times A_{i+1}\right)\\
0,&\text{otherwise.}\end{array}\right.$$
We check that the conditions in Theorem 2.2 are satisfied for the
complete metric space $(Y,d)$, the mappings $\alpha$ and $T$.
First, define $A_{i+kN}:=A_{i}$ for all $i\in\{1,2,\ldots,N\}$ and
$k\in\mathbb{Z}$. Then (F1) extends to
$$T(A_{i})\subseteq A_{i+1}\text{\quad for all }i\in\mathbb{Z}\text{.}$$
We check that $\alpha$ is $N$–transitive (see also Example 2.4).
Indeed, let $x_{0},x_{1},\dots,x_{N+1}\in Y$ such that $\alpha(x_{k},x_{k+1})\geq 1$ (i.e., $(x_{k},x_{k+1})\in R$) for all $k\in\{0,1,\ldots,N\}$.
This means that there exists $i\in\{1,\ldots,N\}$ such that
$$x_{0}\in A_{i},~{}x_{1}\in A_{i+1},\ldots,x_{k}\in A_{i+k},\ldots,x_{N+1}\in A%
_{i+N+1}=A_{i+1}\text{,}$$
hence $(x_{0},x_{N+1})\in A_{i}\times A_{i+1}\subseteq R$, which finally leads
to $\alpha(x_{0},x_{N+1})\geq 1$.
Clearly, $T$ is $\alpha$–contractive of Meir–Keeler type, by (F3).
We claim next that $T$ is $\alpha$–admissible, i.e., (A1) is satisfied.
Indeed, let $x,y\in Y$ such that $\alpha(x,y)\geq 1$; hence, there exists
$i\in\{1,2\dots,N\}$ such that $x\in A_{i},y\in A_{i+1}$. Then, by (F1),
$\left(Tx,Ty\right)\in\left(A_{i+1},A_{i+2}\right)\subseteq R$, hence
$\alpha\left(Tx,Ty\right)\geq 1$.
Now, let $x_{0}\in A_{1}$ arbitrary. Then $Tx_{0}\in A_{2}$, hence
$\alpha(x_{0},Tx_{0})\geq 1$ which concludes (A2).
Next, we prove (A4), by showing that $(Y,d)$ is $\alpha$–regular, so let
$\{x_{n}\}$ be a sequence in $Y$ such that
$$x_{n}\rightarrow x\in Y\text{ as }n\rightarrow\infty\quad\text{and\quad}\alpha%
(x_{n},x_{n+1})\geq 1\text{ for all }n\in\mathbb{N}\text{.}$$
It follows that there exist $i,j\in\{1,\ldots,N\}$ such that
$$x_{n}\in A_{i+n}\text{ for all }n\in\mathbb{N}\text{\quad and\quad}x\in A_{j},$$
hence
$$x_{(j-i-1+N)+kN}\in A_{j-1+(k+1)N}=A_{j-1}\quad\text{for all }k\in\mathbb{N};$$
By letting
$$n(k):=(j-i-1+N)+kN\quad\text{for all }k\in\mathbb{N}\text{,}$$
note that $j-i-1+N\geq 0$, and we conclude that the subsequence $\left\{x_{n(k)}\right\}$ satisfies
$$(x_{n(k)},x)\in A_{j-1}\times A_{j}\subseteq R\quad\text{for all }k\in\mathbb{N}$$
hence $\alpha(x_{n(k)},x)\geq 1$ for all $k$, which proves our claim.
Now, all the conditions in Theorem 2.2 (for $(Y,d)$, $\alpha$ and $T$)
are satisfied, hence there exists a fixed point $x^{\ast}\in Y$ of $T$.
Clearly, $x^{\ast}\in\bigcap_{i=1}^{N}A_{i}$, since
$$x^{\ast}\in A_{k}\text{ for some }k\in\{1,2,\ldots,N\}$$
and
$$x^{\ast}\in A_{i}\Rightarrow x^{\ast}=Tx^{\ast}\in A_{i+1}\text{ for all
}i\text{.}$$
Moreover, it is straightforward to check that $Y$ is $\alpha$–connected,
i.e., (A5) is satisfied. Indeed, if $x,y\in Y$ ($x\neq y$) with $x\in A_{i}$,
$y\in A_{j}$ ($i,j\in\{1,2,\ldots,N\}$), then let $z_{0}:=x$, $z_{k}\in A_{k+i}$ arbitrary for every $k\in\{1,2,\ldots,N+j-i-1\}$ and $z_{N+j-i}:=y$.
Note that $N+j-i\geq 1$. Then $(z_{k-1},z_{k})\in R$ (i.e., $\alpha(z_{k-1},z_{k})\geq 1$) for every $k\in\{1,2,\ldots,N+j-i\}$, hence
$(z_{0},z_{1},\ldots,z_{N+j-i})$ is a $\alpha$-chain from $x$ to $y$.
Now, the rest of the conclusion follows by Theorem 2.3.
∎
4 Some consequences in ordered metric spaces
Clearly, the initial result of Meir and Keeler (Theorem 1.1) follows as
a particular case of our Theorems 2.2 and 2.3, by simply
choosing $\alpha(x,y)=1$ for all $x,y\in X$. In what follows, we will also
show that several fixed point and coupled fixed point results in ordered
metric spaces can be easily deduced (and improved) from our theorems.
4.1 Fixed point results in ordered metric spaces
Let $X$ be a nonempty set. Recall that a binary relation $\preceq$ over $X$ is
called a partial order if it is reflexive, transitive and anti-symmetric. If
$\preceq$ is a partial order over $X$, then $x,y\in X$ are called comparable
(subject to $\preceq$) if $x\preceq y$ or $y\preceq x$. Also, $X$ is called
$\preceq$–connected if for every $x,y\in X$, there exist $z_{0},z_{1},\ldots,z_{n}\in X$ such that $z_{0}=x$, $z_{n}=y$ and $z_{i-1},z_{i}$ are
comparable for every $i\in\{1,2,\ldots,n\}$.
In [5], Harjani et al. obtained several fixed point results in
partially ordered sets for mappings satisfying some contraction condition of
Meir–Keeler type. The main results in [5] for the case of
nondecreasing mappings can be summarized as follows.
Theorem 4.1 ([5]).
Let $(X,d)$ be a complete metric space,
$\preceq$ a partial order over $X$ and $T:X\rightarrow X$ such that for all
$\varepsilon>0$ there exists $\delta(\varepsilon)>0$ for which:
$$x,y\in X:x\preceq y,~{}\varepsilon\leq d(x,y)<\varepsilon+\delta(\varepsilon)%
\Rightarrow d(Tx,Ty)<\varepsilon\text{.}$$
Assume that:
(G1)
$T$ is nondecreasing (subject to $\preceq$);
(G2)
there exists $x_{0}\in X$ such that $x_{0}\preceq Tx_{0}$.
If either
(G3)
$T$ is continuous,
or
(G4)
for every nondecreasing sequence $\{x_{n}\}$ in $X$ such
that $x_{n}\rightarrow x\in X$, there exists a subsequence $\{x_{n(k)}\}$ of
$\{x_{n}\}$ such that $x_{n(k)}\preceq x$ for all $k\in\mathbb{N}$,
then $T$ has a fixed point. In addition, if
(G5)
for every $x,y\in X$, there exists $z\in X$ which is
comparable to $x$ and $y$,
then the fixed point of $T$ is unique.
As it can be easily seen, this result follows straight from Corollary
3.5, with $\mathcal{R}$ being the partial order $\preceq$. Moreover,
(G5) can be replaced by the weaker assumption:
(G5a)
$X$ is $\preceq$–connected.
Also, if $x^{\ast}$ is the unique fixed point of $T$, then $T^{n}(x)\rightarrow x^{\ast}$ (as $n\rightarrow\infty$) for every $x\in X$. This
follows by Corollary 3.5 and its an extension of the conclusion in
Theorem 4.1.
4.2 Coupled fixed point results in ordered metric spaces
In [13], Samet studied the coupled fixed points of mixed strict
monotone mappings that satisfied a contraction condition of Meir–Keeler type,
thereby extending the previous work of Bhaskar and Lakshmikantham
[4]. In what follows we present an extension of the results of
Samet [13]; in this direction, we do not require that the mixed
monotone property be strict and we also weaken other assumptions. We also
improve the conclusion.
First, recall the following definition:
Definition 4.1 ([4]).
Let $(X,\preceq)$ be a partially ordered set. A mapping
$F:X\times X\rightarrow X$ is said to have the mixed monotone property if
$$x_{1},x_{2},y_{1},y_{2}\in X:x_{1}\preceq x_{2},~{}y_{1}\succeq y_{2}%
\Longrightarrow F(x_{1},y_{1})\preceq F(x_{2},y_{2}).$$
Our extension of the main results in [13] follows straight from
Corollary 3.6, with $\mathcal{R}$ being the partial order $\preceq$,
and can be stated as follows.
Theorem 4.2.
Let $(X,d)$ be a complete metric space, $\preceq$ a partial order
over $X$ and $F:X\times X\rightarrow X$ such that for every $\varepsilon>0$
there exists $\delta(\varepsilon)>0$ for which:
$$x,y,u,v\in X:x\preceq u,~{}y\succeq v,~{}\varepsilon\leq\frac{1}{2}[d(x,u)+d(y%
,v)]<\varepsilon+\delta(\varepsilon)\Rightarrow d(F(x,y),F(u,v))<\varepsilon.$$
Suppose that:
(H1)
$F$ has the mixed monotone property;
(H2)
there exist $x_{0},y_{0}\in X$ such that $x_{0}\preceq F(x_{0},y_{0})$ and $y_{0}\succeq F(y_{0},x_{0})$.
If either
(H3)
$F$ is continuous,
or
(H4)
$(X,d,\preceq)$ has the following property: if
$\{x_{n}\}$ is a nondecreasing (respectively, nonincreasing) sequence in $X$
such that $x_{n}\rightarrow x$, then $x_{n}\preceq x$ (respectively,
$x_{n}\succeq x$) for all $n$,
then $F$ has a coupled fixed point $(x^{\ast},y^{\ast})\in X\times X$. In addition, if
(H5)
$X$ is $\preceq$–connected,
then $x^{\ast}=y^{\ast}$, $(x^{\ast},x^{\ast})$ is the unique
coupled fixed point of $F$, $x^{\ast}$ is the unique fixed point of $F$ and
$F^{n}(x,y)\rightarrow x^{\ast}$ as $n\rightarrow\infty$ for all $x,y\in X$.
5 Application to a third order two point boundary value problem
We study the existence and uniqueness of solution to the third order
differential equation
$$x^{\prime\prime\prime}(t)+f(t,x(t))=0,\quad t\in(0,1),$$
(20)
where $f\in C([0,1]\times\mathbb{R},\mathbb{R})$, with the boundary value
conditions
$$x(0)=x(1)=x^{\prime\prime}(0)=0.$$
(21)
This problem is equivalent to finding a solution $x\in C([0,1],\mathbb{R})$ to
the integral equation
$$x(t)=\int_{0}^{1}G(t,s)f(s,x(s))\,\mathrm{d}s,\ $$
where
$$G(t,s)=\left\{\begin{array}[c]{ll}\frac{1}{2}(1-t)(t-s^{2}),&0\leq s\leq t\leq
1%
,\\
\frac{1}{2}t(1-s)^{2},&0\leq t\leq s\leq 1.\end{array}\right.$$
Clearly, $G(t,s)\geq 0$ for all $t,s\in[0,1]$. Also, we can verify easily
that
$$\int_{0}^{1}G(t,s)\,\mathrm{d}s=\frac{t-t^{3}}{6}\leq\frac{\sqrt{3}}{27}\text{%
\quad for all }t\in[0,1].$$
(22)
Let $\Phi$ be the set of all nondecreasing functions $\varphi:[0,+\infty)\rightarrow[0,+\infty)$ such that for all $\varepsilon>0$ there exists
$\delta(\varepsilon)>0$ with
$$\varepsilon\leq t<\varepsilon+\delta(\varepsilon)\Longrightarrow\varphi(t)<\varepsilon.$$
Let $\xi:\mathbb{R}^{2}\rightarrow\mathbb{R}$ and $\varphi\in\Phi$. We
consider the following assumptions:
(J1)
there exists $N\in\mathbb{N}\setminus\{0\}$ such that
$$a_{0},a_{1},\dots,a_{N+1}\in[0,1]:\xi(a_{i},a_{i+1})\geq 0\text{ for all
}i\in\{0,1,\ldots,N\}\Longrightarrow\xi(a_{0},a_{N+1})\geq 0.$$
(J2)
for every $a,b\in\mathbb{R}$:
$$\xi(a,b)\geq 0\Longrightarrow\left|f(t,a)-f(t,b)\right|\leq 9\sqrt{3}\varphi(%
\left|a-b\right|)\quad\text{for all }t\in[0,1].$$
(J3)
for every $x,y\in C\left([0,1]\right)$:
$$\inf_{t\in[0,1]}\xi(x(t),y(t))\geq 0\Longrightarrow\inf_{t\in[0,1]}\xi\left(%
\int_{0}^{1}G(t,s)f(s,x(s))\,\mathrm{d}s,\int_{0}^{1}G(t,s)f(s,y(s))\,\mathrm{%
d}s\right)\geq 0.$$
(J4)
there exists $x_{0}\in C\left([0,1]\right)$ such
that
$$\inf_{t\in[0,1]}\xi\left(x_{0}(t),\int_{0}^{1}G(t,s)f(s,x_{0}(s))\,\mathrm{d}s%
\right)\geq 0$$
(J5)
for every $x,y\in C\left([0,1]\right)$, there exist
$z_{0},z_{1},\ldots,z_{n}\in C\left([0,1]\right)$ such that $z_{0}=x$,
$z_{n}=y$ and, for every $i\in\{1,2,\ldots,n\}$:
$$\inf_{t\in[0,1]}\xi(z_{i-1}(t),z_{i}(t))\geq 0\quad\text{or\quad}\inf_{t\in[0,%
1]}\xi(z_{i}(t),z_{i-1}(t))\geq 0.$$
Theorem 5.1.
Let $f:[0,1]\times\mathbb{R}\rightarrow\mathbb{R}$ be continuous
and assume that there exist $\xi:\mathbb{R}^{2}\rightarrow\mathbb{R}$ and
$\varphi\in\Phi$ such that (J1)–(J4) are satisfied. Then the equation
(20) with the boundary conditions (21) has solution. In
addition, if (J5) is satisfied, then the solution is unique.
Proof.
Let $X:=C\left([0,1]\right)$ be endowed with the metric
$$d(u,v)=\max_{t\in[0,1]}|u(t)-v(t)|,\quad u,v\in X.$$
It is well known that $(X,d)$ is a complete metric space. Define the mapping
$T:X\rightarrow X$ by
$$(Tx)(t)=\int_{0}^{1}G(t,s)f(s,x(s))\,\mathrm{d}s\quad(x\in X,t\in[0,1]).$$
The problem reduces to the fixed point problem for $T$.
Let $\alpha:X\times X\rightarrow[0,\infty)$ be defined by
$$\alpha(x,y)=\left\{\begin{array}[c]{ll}1,&\text{if }\xi(x(t),y(t))\geq 0\text{%
for all }t\in[0,1],\\
0,&\text{otherwise.}\end{array}\right.$$
It is easy to observe that $\alpha$ is $N$–transitive by (J1), $T$ is
$\alpha$–admissible by (J3) and $\alpha(x_{0},Tx_{0})\geq 1$ by (J4). Also, it
follows in a standard fashion that $T$ is continuous, hence we omit this proof.
Now, using (J2), (22) and the fact that $\varphi$ is nondecreasing,
it follows that for all $x,y\in X$ with $\alpha(x,y)\geq 1$:
$$\left|(Tx)(t)-(Ty)(t)\right|\leq\int_{0}^{1}G(t,s)\left|f(s,x(s))-f(s,y(s))%
\right|\,\mathrm{d}s\leq 9\sqrt{3}\left(\int_{0}^{1}G(t,s)\,\mathrm{d}s\right)%
\varphi(d(x,y))\leq\varphi(d(x,y))\text{,}$$
hence
$$d\left(Tx,Ty\right)\leq\varphi(d(x,y))\text{\quad for all }x,y\in X\text{
with }\alpha(x,y)\geq 1\text{.}$$
This clearly leads to
$$\alpha(x,y)d\left(Tx,Ty\right)\leq\varphi(d(x,y))\text{\quad for all
}x,y\in X\text{.}$$
(23)
Now, let $\varepsilon>0$. Since $\varphi\in\Phi$, there exists $\delta(\varepsilon)>0$ such that
$$\varepsilon\leq a<\varepsilon+\delta(\varepsilon)\Longrightarrow\varphi(a)<\varepsilon.$$
(24)
Let $x,y\in X$ with $\varepsilon\leq d(x,y)<\varepsilon+\delta(\varepsilon)$.
Then, by (23) and (24), it follows that
$$\alpha(x,y)d\left(Tx,Ty\right)\leq\varphi(d(x,y))<\varepsilon;$$
hence, we conclude that $T$ is $\alpha$–contractive mapping of Meir-Keeler
type.
Now, we can apply Theorem 2.1 and obtain the existence of a fixed point
of $T$, hence the existence of a solution to (20)–(21). In
addition, (J5) ensures that $X$ is $\alpha$–connected and the uniqueness of
the solution follows by Theorem 2.3. The proof is now complete.
∎
Corollary 5.1.
Let $f:[0,1]\times\mathbb{R}\rightarrow\mathbb{R}$ be continuous and assume
there exists $\varphi\in\Phi$ such that the following conditions are satisfied:
(K1)
$0\leq f(t,b)-f(t,a)\leq 9\sqrt{3}\varphi(b-a)$ for all
$t\in[0,1]$ and $a,b\in\mathbb{R}$ with $a\leq b$.
(K2)
there exists $x_{0}\in C\left([0,1]\right)$ such that
for all $t\in[0,1]$, we have
$$x_{0}(t)\leq\int_{0}^{1}G(t,s)f(s,x_{0}(s))\,\mathrm{d}s.$$
Then (20)-(21) has a unique solution.
Proof.
Consider the mapping $\xi:\mathbb{R}^{2}\rightarrow\mathbb{R}$ be defined by
$\xi(a,b)=b-a$ ($a,b\in\mathbb{R}$). Then the result follows straight from
Theorem 5.1. Indeed, $\xi$ clearly satisfies (J1), while (J2) and (J3)
follow by (K1). Condition (K2) ensures (J4), while (J5) follows easily, by
noting that for every $x,y\in C\left([0,1]\right)$, the function
$$z:[0,1]\rightarrow\mathbb{R}\text{,\quad}z(t)=\max\{x(t),y(t)\}\text{ }(t\in[0%
,1])$$
satisfies
$$z\in C\left([0,1]\right),~{}\inf_{t\in[0,1]}\xi(x(t),z(t))\geq 0,~{}\inf_{t\in%
[0,1]}\xi(y(t),z(t))\geq 0\text{.}$$
∎
Remark 5.1.
Condition (K2) can be replaced by
(K2a)
there exists $x_{0}\in C\left([0,1]\right)$ such
that for all $t\in[0,1]$, we have
$$x_{0}(t)\geq\int_{0}^{1}G(t,s)f(s,x_{0}(s))\,\mathrm{d}s,$$
while all the other conditions and conclusions remain unchanged. In this case,
the proof follows similarly, by letting $\xi:\mathbb{R}^{2}\rightarrow\mathbb{R}$ be defined by $\xi(a,b)=a-b$ ($a,b\in{\mathbb{R}}$).
Acknowledgement
The second author is grateful for the financial support provided by the Sectoral
Operational Programme Human Resources Development 2007-2013 of the Romanian
Ministry of Labor, Family and Social Protection through the Financial
Agreement POSDRU/89/1.5/S/62557.
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The VIRMOS deep imaging survey: I. overview and survey strategy
O. Le Fèvre
1Laboratoire d’Astrophysique de Marseille, UMR 6110, Traverse
du Siphon-Les trois Lucs, 13012 Marseille, France
email: olivier.lefevre@oamp.fr
1
Y. Mellier
2Institut d’Astrophysique de Paris, UMR 7095, 98 bis Bvd Arago, 75014 Paris, France
23Observatoire de Paris, LERMA, UMR 8112, 61 Av. de l’Observatoire, 75014 Paris, France
3
H.J. McCracken
1Laboratoire d’Astrophysique de Marseille, UMR 6110, Traverse
du Siphon-Les trois Lucs, 13012 Marseille, France
email: olivier.lefevre@oamp.fr
14Osservatorio Astronomico di Bologna, via Ranzani 1, 40127 Bologna, Italy
4
S. Foucaud
1Laboratoire d’Astrophysique de Marseille, UMR 6110, Traverse
du Siphon-Les trois Lucs, 13012 Marseille, France
email: olivier.lefevre@oamp.fr
1
S. Gwyn
1Laboratoire d’Astrophysique de Marseille, UMR 6110, Traverse
du Siphon-Les trois Lucs, 13012 Marseille, France
email: olivier.lefevre@oamp.fr
1
M. Radovich
2Institut d’Astrophysique de Paris, UMR 7095, 98 bis Bvd Arago, 75014 Paris, France
25Osservatorio di Capodimonte, via Moiariello 16, 80131 Napoli, Italy
5
M. Dantel-Fort
3Observatoire de Paris, LERMA, UMR 8112, 61 Av. de l’Observatoire, 75014 Paris, France
3
E. Bertin
C. Moreau
2Institut d’Astrophysique de Paris, UMR 7095, 98 bis Bvd Arago, 75014 Paris, France
23Observatoire de Paris, LERMA, UMR 8112, 61 Av. de l’Observatoire, 75014 Paris, France
31Laboratoire d’Astrophysique de Marseille, UMR 6110, Traverse
du Siphon-Les trois Lucs, 13012 Marseille, France
email: olivier.lefevre@oamp.fr
1
J.-C. Cuillandre
6Canada-France Telescope Corporation, 65-1238 Mamalahoa Hwy, Kamuela, Hawaii 96743, USA
6
M. Pierre
7Service d’Astrophysique, CE Saclay, L’Orme des Meurisiers, 91191 Gif sur Yevette Cedex , France
7
V. Le Brun
1Laboratoire d’Astrophysique de Marseille, UMR 6110, Traverse
du Siphon-Les trois Lucs, 13012 Marseille, France
email: olivier.lefevre@oamp.fr
1
A. Mazure
1Laboratoire d’Astrophysique de Marseille, UMR 6110, Traverse
du Siphon-Les trois Lucs, 13012 Marseille, France
email: olivier.lefevre@oamp.fr
1
L. Tresse${}^{,}$
The data presented in this paper has been
obtained with the Canada-France-Hawaii Telescope, operated by the
National Research Council of Canada, the Centre National de la
Recherche Scientifique of France, and the University of Hawaii 1Laboratoire d’Astrophysique de Marseille, UMR 6110, Traverse
du Siphon-Les trois Lucs, 13012 Marseille, France
email: olivier.lefevre@oamp.fr
1
(Received May XX, 2003; accepted …, 2003)
Key Words.:
Cosmology: observations – Galaxies: evolution –
Cosmology: gravitational lensing – Cosmology: large scale structure
of universe
††offprints: O. Le Fèvre
This paper presents the CFH12K-VIRMOS survey:
a deep B, V, R and I imaging survey in four fields
totalling more than 17 deg${}^{2}$, conducted with the
$30\times 40$ arcmin${}^{2}$ field CFH-12K camera.
The survey is intended to be a multi-purpose survey used for a variety of
science goals, including surveys of very high redshift galaxies
and weak lensing studies.
Four high galactic latitude fields, each $2\times 2$ deg${}^{2}$, have been
selected along the celestial equator:
0226-04, 1003+01, 1400+05, and 2217+00.
The 16 deg${}^{2}$ of the ”wide” survey are
covered with exposure times of 2h, 1.5h, 1h, 1h ,
while the $1.3\times 1$ deg${}^{2}$
area of the ”deep” survey at the center of the 0226-04 field is covered with
exposure times of 7h, 4.5h, 3h, 3h, in B,V,R and I respectively.
An additional area $\sim 2$deg${}^{2}$ has been imaged in the
0226-04 field corresponding to the area surveyed by the
XMM-LSS program (pierre03, ).
The data is pipeline processed at the Terapix facility at the Institut
d’Astrophysique de Paris to produce large mosaic images.
The catalogs produced contain the positions, shape, total and
aperture magnitudes for the 2.175 million objects measured in the 4 areas
The depth measured as a $3\sigma$ measurement in a 3 arc-second aperture
is $I_{AB}=24.8$ in the “Wide” areas, and $I_{AB}=25.3$
in the deep area. Careful quality control has been applied
on the data to ensure internal consistency and assess the
photometric and astrometric accuracy as described in
joint papers (mccracken03, ).
These catalogs are used to select targets for the VIRMOS-VLT Deep Survey,
a large spectroscopic survey of the distant universe (Le Fèvre et al., 2003).
First results from the CFH12K-VIRMOS survey have been published on
weak lensing (e.g. van Waerbeke & Mellier 2003).
Catalogs and images are available through the VIRMOS
database environment under Oracle (http://www.oamp.fr/virmos).
They will be open for
general use on July 1st, 2003.
1 Introduction
Deep imaging over large areas is required for many
fields of astronomy, to survey large numbers of objects
or to search for rare, low projected density objects.
It is a key tool to our understanding
of the universe, from solar system studies to the most distant galaxies.
The measurement of positions, magnitudes, colors, and shape
are key observable to all astronomical investigations.
From astrometric and photometric measurements, a census of
the number and positions of various classes of objects
can be conducted as a function of magnitude, color, or shape.
Progress in detector area have allowed to design
and build large CCD mosaics covering a significant fraction
of the field available on wide field telescopes
groom00 . These mosaics have enabled
survey work on large areas which had only been possible
previously with photographic plates at the focus of
Schmidt and prime foci of major telescopes. Probing
scales larger than 0.5 degree in one single exposure
with the sensitivity of CCDs
has become possible at a few facilities
in the past few years (cuillandre00, ; boulade00, ; miyazaki02, ; kuijken02, ),
enabling a wealth of new deep survey
initiatives (postman98, ; nonino99, ; mccracken01, ; mcmahon01, ; wilson03, )
This paper is the first of a serie that describes the
deep survey we have undertaken starting when the CFH12K camera became
available at CFHT prime focus in 1999. The main survey covers a total of
16 deg${}^{2}$
in 4 areas, each $2\times 2$deg${}^{2}$, which have been imaged in B,V,R
and I bands, at a depth equivalent to $I_{AB}=24.8$
for a $3\sigma$ detection in a 3 arc-second circular
aperture for all the area, with a deeper area imaged to
$I_{AB}=25.3$.
We focus here on the survey goals and strategy,
and the observations performed, together with
an overview of the pipeline
processing and the content of the
photometric catalogs.
Other papers in this series (mccracken03, ); (Gwyn et al., 2003 in
preparation)
describe in details the procedures
followed to build the astrometry and photometry of the large image mosaics,
and the quality control applied. The VIRMOS survey has been
used already for cosmic shear studies and 8 refereed papers have been
published so far from this data set by the virmos-descart
project (vanwearmel03, ).
2 Survey Goals
The survey has been designed to
address a broad range of astrophysical questions in one single observing
strategy. Several main science drivers have been identified:
•
study the evolution of galaxies from
redshifts $\sim 5$,
•
study the evolution of large scale structures
over 100 $h^{-1}$ Mpc
from redshifts $\sim 5$,
•
measure weak lensing signature of large scale structures,
•
measure the properties of galaxy biasing using together the
dark matter (from weak lensing) and the galaxy distribution,
•
identify new high redshift clusters of galaxies,
•
identify faint AGN and study their evolution,
•
identify new Kuiper belt objects to
give new insights into the formation of the solar system,
•
identify very faint halo white dwarfs,
•
study high redshift Lyman-break galaxies with $3.5<z<4.2$.
using the multi-color data set of the optical and U-band survey
The data is also being used with time series
to search for high redshift supernovae.
This imaging survey is being used to select the targets
of the VLT-VIRMOS deep redshift survey of more than 100000 galaxies with
$0<z<5+$ (lefevre03, ), to
obtain large catalogues of galaxy shapes for cosmic shear studies
and as the optical
counterpart to the XMM medium deep survey
(http://vela.astro.ulg.ac.be/themes/spatial/xmm/LSS/)
being carried out
with XMM pierre03 . The
deep VIRMOS field is also completed by a U-band follow up
done at ESO, with the WFI camera (Radovich et al 2003), as well as
in radio wavelength with the VLA bondi03 . A
tiny area of the deep field has also been observed in
K-band, with SOFI at the ESO/NTT (Iovino et al 2003).
3 Survey Strategy
3.1 Field Selection
Survey fields have been selected with the following criteria:
•
along the celestial equator
to allow for visibility from northern and southern hemispheres
observatories,
•
galactic latitude higher
than $l=45$ deg.,
•
low cirrus absorption as seen from the DIRBE maps available when the
survey started in 1998,
•
visibility of any 2 fields at any time of the year to fill observing
nights.
The four fields selected are listed in Table 1 and their position
in the dust sky map shown in figure 1.
3.2 Survey depth
The survey has been designed to reach a limiting magnitude $I_{AB}=24.5$
(at $5\sigma$ in a 3 arcsec diameter aperture) in
all of the area surveyed, with a smaller 1.3 $deg^{2}$ area in the 0226-04
field observed to $I_{AB}=25$. The depth of this imaging survey is then at least
one magnitude deeper than the limiting magnitude(s) of the VIRMOS-VLT
Deep Survey (VVDS), set to be $I_{AB}=22.5$ for the wide areas and
$I_{AB}=24$ for the deep area (Le Fèvre et al., 2003), and ensure that
the imaging survey will not introduce any bias in galaxy samples
selected for the spectroscopic survey.
The actual measured completeness limits are presented in section 6.2
3.3 The CFH12K camera
Observations have been carried out with the CFHT-12K camera build by
CFHT and the University of Hawaii (cuillandre00, ).
The CFH12K camera
(http://www.cfht.hawaii.edu/Instruments/Imaging/CFH12K/)
is a particularly powerful system.
It is based on
a mosaic of 12 3-edge buttable MIT/Lincoln-Labs. CCID20 thinned backside illuminated CCDs,
each $2048\times 4096$ pixels for a total array size
of $12288\times 8192$ pixels (18.4 cm $\times$ 12.3 cm in physical size).
The pixel size (15 microns) corresponds to
a scale of 0.205 arcsec/pixel at CFHT prime focus, well adapted
to the mean seeing at the CFHT prime focus ($\approx 0.8"$). It covers a field of
view $42\times 28$ arcmin${}^{2}$. The camera is composed of two different
types of CCDs. Three are high resisitivity chips and nine are
epitaxial silicon devices. The high resistivity CCDs have better
efficiency than the epitaxial beyond 500 nm, but are
less efficient in the blue (see figure 2). This difference
is corrected during the pre-processing step that rescales each CCD
with respect to a reference device of the camera.
None of the CCDs has a readout noise higher than 6 electrons
$rms$ nor shows dark current signal over 1 hour time scale, which is largely
sufficient for deep imaging surveys with sky background
noise dominated images. The readout time
is about one minute, which permits to easily split long exposures
into short ones
without increasing significantly the overheads. In addition to
facilitate the construction of master flat fields and fringe
patterns and to simplify the cosmic ray rejection process,
the image splitting also permits a better control of the camera focus and
to reject bad seeing exposures. The autonomy of the
LN2 reservoir is much longer than a full single night observation
process and we never detected any temperature drift during the
45 nights dedicated to the VIRMOS survey.
BVRI broad band filters are available on CFH12K.
VRI are rather standard Mould filters, but B is special
(see figure 2). They have been
designed
in order to cover the whole optical spectral range in a continuous manner.
The photometric informations have therefore no gaps from one band to
the other allowing to probe the whole redshift range from $z=0$ to
$z=5$ continuously. This advantage is however obtained at the expense
of photometric redshift accuracy for high-$z$ galaxies.
3.4 Observing strategy
Each $2\times 2$ deg${}^{2}$ field has been divided in a series of CFH12K
pointings covering the full area, with a slight overlap of $\sim 30$
arcsec between each pointing.
A classical shift-and-add observing strategy
has been adopted, each pointing being observed between
5 and 12 times depending on the filters and depth.
Because during our first 2 observing
runs 2 of the 12 CCDs were of poor quality, the
grid of pointings has had to be adjusted to the available pixels for
these runs. We started operating
with $35\times 28$ arcmin${}^{2}$ fields, switching to the
full $42\times 28$ arcmin${}^{2}$
field when new CCDs were installed in the first observing
period of 2000. The field geometry
is therefore non rectangular in some of the bandpasses.
5by several arcseconds (typically $10-20$ arcsec) between each exposure.
We aimed to bring first the I-band coverage to $2\times 2$deg${}^{2}$
in each of the fields
to ensure that reference catalogs would be available for the magnitude selected
VIRMOS-VLT Deep redshift Survey. We have then completed the coverage
in other bands in a uniform way, proportional to the actual number of
clear nights.
During the acquisition of a sequence of
exposures, the aperture flux of control stars was monitored for changes in
photometric conditions. The exposure times were extended in case a loss of
flux was detected during sub-exposures.
4 Observations
4.1 Observed pointings
The individual pointings observed are listed in
http://www.oamp.fr/virmos,
with the date of observations, the FWHM
measured on point sources images on the detectors,
and the corresponding CFHT archive files.
The final CFH12K mosaics resulting from the combination
of these individual pointings are listed in
Table.2 for each of the 4 fields and 4 bands,
together with the resulting exposure times and limiting
magnitudes ($5\sigma$ in a 3 arcsec circular aperture).
4.2 calibrations
Standard photometric fields from Landolt (1992) have been acquired
each night survey observations have been obtained.
The fields SA92, SA101 and SA110,
have been observed several times over the period 1999-2002 in
order to check the reliability of the photometric calibration
and the stability of the CFH12K performances.
The detailed process of photometric calibration, relative
from exposure to exposure within a set of exposures of a pointing
in each bandpass, as well as the absolute flux calibration
from the observed standards is described in details in
(mccracken03, ).
4.3 Image quality
The image quality of each exposure acquired for the survey is presented in
Figure
3 for each of the bands. The median seeing is FWHM=0.75 arcsec
in the I band but increases to 0.88 arcsec in V and R, and 0.97 arcsec in B.
Moreover, the V and R seeing distributions are significantly broader than
the I and B data sets.
This trend reflects that atmospheric dispersion is negligible in
I-band and increases towards the B-band. However, it mostly
results from our observing strategy since we preferentially did
I-band observations during good seeing periods in order to
get high quality I-band selected catalogues for the VMOS spectroscopic
sample as well as for the cosmic shear studies.
5 Data Processing
5.1 Pre-processing
All data are pre-processed using the FLIPS software package (Cuillandre
et al., 2002). FLIPS (http://www.cfht.hawaii.edu/jcc/Flips/flips.html)
is composed of C-language programs and Cshell commands that
automatically generate master bias, dark and flat images, build
a model of the fringe pattern and subtract
the master bias, dark files and fringe pattern as well as
the overscan to raw CFH12K images. Each pixel and
each image is then rescaled to account for intrinsic efficiency. A
binary mask image provided by CFHT at each observing run
identifies all bad and hot pixels over each CCD of the
camera. They are taken into account during the master file
generation process. The final data products
are pre-calibrated images that includes the description of the
pre-processing history in the FITS header and the
value of the CCD to CCD rescaling coefficient, according to the
different gains of each CCD output and
the quantum efficiency difference between
epitaxial and high resisitivty CCDs. At this stage, all
CCDs are therefore in the same ADU scaling units and
arbitrarily normalised to the CCD#04 efficiency.
The FLIPS package has been installed at the Terapix center
described below in order to handle the pre-processing
locally.
5.2 The Terapix facility
The Terapix facility (http://terapix.iap.fr)
has been set up as a French national center
to handle the processing of large imaging cameras. The facility
includes the hardware and software environment
required to process data on a Terabyte scale. Three COMPACQ
XP1000 workstations with up to 2 GB of RAM memory
and connected to 4 Terabytes of hard discs
installed in a secured raid-5 configuration have been
devoted to the VIRMOS image processing (mellier02, ). Over the past four years, Terapix
has processed more than 10,000 CFH12K images for the VIRMOS survey, totalling
about 2.5 TB of input data.
A database environment allows to streamline
data access and storage.
5.3 Pipeline processing
Pipeline processing allows efficient data processing with
excellent quality control. The stable behavior
of the CFH-12K camera allows for very accurate bias, flat field, and
fringing correction.
The pipeline software has been developped within the Terapix environment.
The main steps are described in (mccracken03, ) as well as
in the Terapix progress reports (mellier02, ). The software tools
developped for the object detection, image astrometry, photometry,
pixel weighting and flaging, image resampling and stacking,
object classification and
catalogue construction
are described at http://terapix.iap.fr/soft/
and can be downloaded freely from this site. The
http://terapix.iap.fr/soft/releases.html also provides
a documentation for all software tools.
The astrometric projection of all images into a common system
used to build the large mosaic images has been developed to ensure
an overall absolute calibration to the USNO reference catalogue
(monet98, ) accurate to
better than 0.3 arcsecond, while the relative position
accuracy within the catalogs is better than 0.1 arcsec r.m.s.
The photometric calibration ensures that all individual images
are calibrated on the same reference. Photometric zero point uncertainties
are better than 0.1 magnitudes across the final mosaics. A detailled description and
application of the pipeline, with a comprehensive
description of quality assessments are described in (mccracken03, ).
Similar procedures applied to the wide VIRMOS
survey and the deep U-band survey have been used by Gwyn et al
(2003, in preparation) and Radovich et al (2003), respectively.
6 Results: survey images and catalogs
6.1 Field coverage
The final multi-wavelength coverage of the four survey fields is shown
in Figure 4 to Figure 7.
6.2 Catalog content
The catalogs are built using the SExtractor (bertinarnou96, ). Particular care has been taken to ensure uniform source
detection, using a matched xi-squared image (see
Fig. 8) produced from the
individual band images, or a local threshold algorithm. The
difference between these two methods has been demonstrated to
be marginal, with no effect in the magnitude range to the
completness limit (mccracken03, ).
Because of the large area covered, bright stars are present in the
mosaics, creating increased background around them. In addition, there
are areas in the images not useable for accurate photometry. These
can be areas for which the signal to noise is too low
like e.g. areas of overlap between individual CCDs, areas affected
by internal reflexions or ghosts from bright stars around the field,
or even areas affected by transient effects like satellite trails.
Masks are therefore computed for each
mosaic.
The catalogs contain the following information for each object extracted from
the SExtractor V.2.3 used with the generic configuration files given in
the Appendix and which were used as templates for all optical data from the
CFH12K part of the survey:
•
Identifier
•
Positions:
–
X-IMAGE (pixel) , Y-IMAGE (pixel), and
–
ALPHA_J2000 (decimal degrees) and DELTA_J2000 (decimal degrees)
•
Flux and Magnitudes in the AB photometric system, defined as
-2.5log(flux in ADU)+ZP (mag. zero point), including the associated errors:
–
the total flux and
magnitude FLUX_AUTO (ADU), MAG_AUTO,
–
the isophotal flux and magnitude,
FLUX_ISO (ADU), MAG_ISO,
–
the corrected isophotal flux and magnitude
FLUX_ISOCOR (ADU), MAG_ISOCOR,
–
the best of mag_auto and mag_isocor,
FLUX_BEST (ADU), MAG_BEST,
–
the aperture flux and magnitude
FLUX_APER(2) (ADU), MAG_APER(2),
Colors can then be derived directly using the aperture magnitudes
computed within the same aperture for all bands.
•
Shapes:
–
image size,
KRON_RADIUS (pixel) and ISOAREA_IMAGE (pixel${}^{2}$),
–
major axis A_IMAGE (pixel) , A_WORLD (decimal degrees),
–
minor axis B_IMAGE (pixel), B_WORLD (decimal degrees) ,
–
position angle of main axes (all angles are measured counter-clockwise from the east.
theta denotes the position angle between the major axis, ”a”,
and the naxis1 (east-west) image axis),
THETA_IMAGE (decimal degrees), THETA_J2000 (decimal degrees),
–
the values of the values of the weighted second moment matrix coefficients,
X2_IMAGE (pixel${}^{2}$), Y2_IMAGE (pixel${}^{2}$),
XY_IMAGE (pixel${}^{2}$),
–
X2_WORLD (decimal degees${}^{2}$),
Y2_WORLD (decimal degees${}^{2}$), XY_WORLD
(decimal degees${}^{2}$),
–
peak surface brightness MU_MAX (mag/arcsec${}^{2}$),
–
detection threshold above the background MU_THRESHOLD (mag/arcsec${}^{2}$).
•
Flags: several flags (FLAGS) are stored for each object.
they indicate:
–
saturated pixels,
–
pixel located inside a masked area, as defined above.
–
Star-Galaxy classification:
A star / galaxy classifier defined from the half-light radius
vs. magnitude diagram (see (mccracken03, )). Stars are selected along
the vertical branch at fixed radius corresponding to seeing disk size.
This method initally described in (falhman94, ) turns out to be
more reliable than the initial CLASS_STAR of SExtractor,
The photometry and astrometry quality have been assessed through many
quality checks.
The positional accuracy has been checked
against the USNO catalog. The photometry accuracy has been
first assessed through
the Elixir queue observing program part of the queue observations
at CFHT, which monitors the photometric zero points accross nights.
The magnitudes of objects common to adjacent CCDs have been compared
and showed differences in the range 0.05–0.1 magnitudes.
The comparison of colors for the bright stellar objects in the fields
with the observed and predicted locus of stars has shown
color errors less than 0.1 magnitudes. Counts of galaxies
are in full agreement with the litterature. Finally, the angular
correlation function shows similar shape and amplitude
as from other imaging surveys. All these quality checks
are extensively described in (mccracken03, ).
They ensure that the data are free of
systematics before science analysis can be conducted.
In the I band, a total of 2.175 million objects have been
detected. Among these, 1.15 million objects have $17.5\leq I_{AB}\leq 24$,
in the range of the VIRMOS-VLT Deep Survey.
An example of a multi-color set is given in Figure 9.
6.3 Database access
All catalogs and mosaic images are stored in an interactive database
implemented under the Oracle-8 environment. Specific
database development have been conducted to allow
for easy database query and data retrieval.
The catalogs are now in direct access to the CFHT-VIRMOS
consortium. They will be released for general use
for CFHT users starting July 1st, 2003 (http://www.oamp.fr/virmos).
To access the database, first a password should be requested
sending an email to virmos.database@oamp.fr.
The database
allows to query the photometric catalog and to retrieve
image sections. Any parameter listed in the photometric catalog
can be used to define a user selection, e.g. magnitudes, colors,
and/or positions. A user catalog is then produced with entries
as selected by the user from the list of catalog parameters.
7 Conclusions
A deep imaging survey has been conducted in 4 high
galacitic latitude areas with the CFH-12K camera.
A total area of more than $17$deg${}^{2}$ has been imaged
in I band, and significant coverage of
the survey area has been performed also in B, V and R.
Pipeline processing has been conducted at the Terapix
facility at IAP to produce large mosaic images
calibrated in astrometry and photometry.
Photometric catalogs containing positions, magnitudes,
colors, shapes of more than 2 million objects
have been produced for more than 17deg${}^{2}$
and quality control has been applied
as described in joint papers. These catalogs and
mosaic images will
be released for general use starting July 1st, 2003,
at http://www.oamp.fr/virmos.
While many scientific programs are under way, the survey
has already been used for cosmic shear. The virmos-descart cfht survey
(http://www.terapix.iap.fr/Descart/cfhtsurvey.html) primarily focusses
on the I-band
sample to measure galaxy shape with high accuracy and probe the dark
matter properties and cosmological parameters. Because these cosmic
shear results provide also an estimate of systematics on galaxy shapes
and can be cross-checked with independent cosmic shear results
from other teams, they also provide quality assessments on the
VIRMOS data set. In addition to
the detection of
cosmic shear signal
(vanwaerbeke00, ), constraints on $\Omega_{m}$,
$\sigma_{8}$ (vanwaerbeke02, ), the biasing properties of the
dark matter (hoekstra02, ; pen03a, ) as
well is its non-Gaussian properties
(bernardeau02, ; pen03b, ) and the 3-dimension
power spectrum of the dark matter
(pen03a, ) have already been provided
using the VIRMOS survey for cosmic shear.
The VIMOS spectroscopic survey is now started at the ESO/VLT
and more than 20,000 spectra have been collected
(lefevre03, ). The joint
U-band, optical, near infrared and spectroscopic informations
are now under process to probe the star formation history,
the clustering history of galaxies up to redshift $z\approx$ 5
and the properties of the light and dark matter relations.
Acknowledgements.
We thank the CFHT time allocation committee for continuous
support of this long term program, the CFHT staff for
the execution of the observations performed in queue
scheduling mode, the Terapix staff for its
continuous help during the VIRMOS image
processing period, the CNRS-INSU, CEA/DAPNIA and
the Programme National de Cosmologie for support of the Terapix
facility and IAP and OAMP for funding of this program.
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A1: SExtractor configuration files
Tables 3 and 4 list the parameters
used for all
$\chi^{2}$-images and all wide field analysis of VIRMOS optical
obervations with the CFH12K. |
Selective multi-photon excitations in the Kerr nonlinear resonator
A R Shahinyan ${}^{1}$, A R Tamazyan${}^{1,2}$ and G Yu Kryuchkyan${}^{1,2,3}$
${}^{1}$ Institute for Physical Researches,
National Academy of Sciences,
Ashtarak-2, 0203, Ashtarak, Armenia
${}^{2}$ Physics Department, Yerevan State University, Alex Manoogian 1, 0025,
Yerevan, Armenia
${}^{3}$ Centre of Quantum Technologies and New Materials, Yerevan State University, Alex Manoogian 1, 0025,
Yerevan, Armenia
anna_shahinyan@ysu.am
a.tamazyan@ysu.am
kryuchkyan@ysu.am
Abstract
We study the Kerr nonlinear resonator (KNR) driven by a continuous wave field in
quantum regimes where the strong Kerr interactions give rise to selective
resonant excitations of oscillatory mode. We use an exact quantum theory of the KNR
without any limitation on quantum noise and without any quantum
state truncation procedure. We focus on understanding of various regimes of
selective multi-photon resonant excitations of mode depending on the detuning,
amplitude of the driving field and the strength of nonlinearity. The analysis
is provided on basis of the photon number distributions and photon-number
correlation function.
pacs: 03.65.Yz, 42.50.-p, 42.50.Ar
††: J. Phys. B: At. Mol. Opt. Phys.
, ,
February 2015
Keywords: selective excitations, Kerr nonlinear resonator, anharmonic oscillator, multiphoton states
1 Introduction
A harmonic oscillator is the simplest system that can be described exactly in
the framework of quantum mechanics. The quantum dynamics of the oscillator is
naturally described by the Fock states, which have a definite number of energy
quanta. However, oscillatory states with equidistant energetic levels can
not be selectively excited. Indeed, excitations usually lead to production of
thermal states or coherent states involving the sum of many oscillatory Fock
states. In this area, the preparation of Fock oscillatory states can be
effectively realized in an anharmonic oscillator, particularly in the Kerr nonlinear resonator (KNR) including direct photon-photon interactions. In this
paper, we analyze selective excitations of photon-number states in the KNR by using an exact quantum
treatment of the system.
The Kerr nonlinearity leading to photon-photon interactions is a well known and widely
used phenomenon in nonlinear optics and quantum optics. It has been particularly
used to generate both squeezed light [1] and ultra-fast pulses
[2]. The Kerr nonlinear resonator could be a cavity containing a
Kerr medium that is driven by the pump field. Another example of the Kerr
nonlinear resonator provides a microwave LC-circuit that involves a Josephson
junction [3] as well as a stripline resonator with a Josephson
junction [4]. The Kerr-interaction in this area has been used for
development of Josephson parametric amplifiers [5] and for the
generation of squeezed states of microwave fields [6].
A strong Kerr nonlinearity makes
frequencies of transitions between adjacent oscillatory energy levels
different, i.e. in an anharmonic oscillator nonlinearity breaks the
equidistance of oscillatory energy levels. Thus, in the case of a strong
nonlinearity the oscillatory energy levels are well resolved and spectroscopic
selective excitations of transitions between Fock states are possible. The
selective excitations can be realized for the Kerr nonlinear resonator by tuning
the frequency of the driving field to the sum of the frequency of the
resonator and the Kerr nonlinear shift of oscillatory energy levels.
An important parameter responsible for the KNR is the
ratio $\chi/\gamma$ between the parameter of the Kerr-type nonlinearity and damping of the
oscillatory mode or the photon decay rate. As a rule, the
efficiency of quantum nonlinear effects requires a high nonlinearity with
respect to dissipation. On the other hand decoherence and dissipation destroy
quantum effects. Thus, the production of quantum states is limited by the
coherence lifetime of the system. In the majority of investigations the ratio
$\chi/\gamma$ has been small. Nevertheless, resonators with a Josephson junction
can be used to reach strong nonlinear regimes with respect to the photon
decay rate. Particularly, the regimes of weak
$\chi/\gamma<<1$, strong $\chi/\gamma>1$ and very strong $\chi/\gamma>>1$ nonlinearities have been demonstrated for a Josephson junction embedded in a transmission-line resonator by adjusting
the parameters of the circuit [7]-[9].
The Kerr nonlinear dissipative resonator is usually studied in the
framework of the master equation in the steady-state limit. In this way, the
master-equation solutions are usually obtained analytically as well as numerically by
using the method of Fock state truncation. This approximation allows to consider a
Kerr-type nonlinear resonator for the case of sufficiently weak driving external fields. In this
approach, the nonlinear quantum scissor effects were discussed earlier
[10](see also the review paper [11]).
The photon-number effects [12], the multi- photon blockade and the
generation of Fock states in the KNR have also been studied numerically in [13]-[16].
In this paper, we analyze excitations of photon states in the KNR driven by a
continuous wave field for strong and very strong Kerr nonlinearities. The
novelty is that an exact quantum solution of the Fokker-Planck equation in
complex $P$ representation [17]-[19] is used. This approach gives a
full quantum description of the Kerr dissipative resonator without any state
truncation procedure. By this approach, we analyse both the excitations of
oscillatory states at the level of few quanta for the case of a strong
nonlinearity, and the multi-photon selective resonant excitations of the Kerr-type
nonlinear resonator for a very strong nonlinearity. The last case is realized by
tuning the frequency of the driving field for wide ranges of the parameters, i.e.
the nonlinearity and the amplitude of the driving field. We clarify the selective
excitations of the mode by considering photon-number effects on the base of mean
photon numbers, probability distributions of photons and the second-order
correlation functions of photons.
The paper is arranged as follows. In Section 2 we describe the KNR by using the potential solution of the Fokker-Planck equation in the
complex P-representation. In Section 3 we analyze multi-photon
excitations in the KNR on the basis of the photon-number state populations and the second-order photon correlation function. This analysis includes the study of non-resonant cascaded excitations as well as selective resonant excitations of the KNR depending on the detuning, the amplitude of the driving field and the strength of the Kerr nonlinearity. We summarize our results in Section 4.
2 Exact quantum treatment of the Kerr nonlinear resonator: short description
The Hamiltonian of the Kerr nonlinear resonator driven by a monochromatic field in the rotating-wave approximation reads as:
$$H_{0}=\hbar\Delta a^{\dagger}a+\hbar\chi a^{\dagger 2}a^{2}+\hbar(\Omega a^{%
\dagger}+\Omega^{*}a),$$
(1)
where $a^{\dagger}$, $a$ are the oscillatory creation and annihilation operators respectively,
$\chi$ is the nonlinearity strength, and $\Delta=\omega_{0}-\omega$ is the
detuning between the mean frequency of the driving field and the frequency of
the oscillator.
The Hamiltonian (1) describing an anharmonic oscillator has been proposed for a wide range of physical systems,
including optical fibers, nano-mechanical oscillators, various Josephson junction based devices, quantum dots, quantum scissors, etc.
The full Hamiltonian of the system reads as $H=H_{0}+H_{\rm{loss}}$, where the term
$H_{\rm{loss}}=a\Gamma^{+}+a^{+}\Gamma$
is responsible for the linear losses of oscillatory modes, due to
couplings with heat reservoir operators giving rise to the
damping rate $\gamma$.
We trace out the reservoir degrees of freedom in the Born-Markov limit assuming that the system and the environment are
uncorrelated at the initial time t = 0. This procedure leads to the master
equation for the reduced density matrix of the oscillatory mode in the following form:
$$\frac{\rmd\rho}{\rmd t}=-\frac{\rmi}{\hbar}[H_{0},\rho]+\sum_{i=1,2}L_{i}\rho L%
_{i}^{+}-\frac{1}{2}L_{i}^{+}L_{i}\rho-\frac{1}{2}\rho L_{i}^{+}L_{i}.$$
(2)
Here $L_{1}=\sqrt{(N+1)\gamma}a$ and $L_{2}=\sqrt{N\gamma}a^{+}$ are the
Lindblad operators, $\gamma$ is the dissipation rate and $N$ denotes the mean
number of quanta of the heat bath.
To study the pure quantum effects we focus on the cases of very low reservoir
temperatures with $N$ approximately equal to zero.
The master equation is then transformed into a Fokker-Planck equation. In this
simplest case analytical results for the Kerr nonlinear dissipative resonator
in a steady state have been obtained in terms of the solution of the
Fokker-Planck equation for the quasi-probability distribution function
$P(\alpha,\alpha^{*})$ in the complex P-representation [20]. This
approach is based on the method of potential equations that leads to the analytic
solution for the quasi-probability distribution function $P(\alpha,\alpha^{*})$ within the framework of the exact nonlinear treatment of quantum
fluctuations. On the whole, the various moments of mode operators as well as the photon distribution function were obtained [17], [18], [19]. For instance, the photon number probability distribution function
$P_{n}=\langle n|\rho|n\rangle$ can be expressed in terms of the complex
P-representation as follows:
$$P_{n}=\frac{1}{n}\int\int_{C}\rmd\alpha\rmd\alpha^{*}\exp(-\alpha\alpha^{*})%
\alpha^{n}{\alpha^{*}}^{n}P(\alpha,\alpha^{*}),$$
(3)
where $C$ is the appropriate integration contour for each of
the variables $\alpha$ and $\alpha^{*}$, in individual complex planes.
The details of calculations involving contour integrations have been presented in [17], [18], [19]. The results for the mean photon number and the photon number probability distribution function read as:
$$\displaystyle P_{n}=\frac{|\varepsilon|^{2n}\Gamma(\lambda)\Gamma(\lambda^{*})%
}{n!_{0}\rm{F}_{2}(\lambda,\lambda^{*},2|\varepsilon|^{2})}\sum_{k=0}^{\infty}%
\frac{|\varepsilon|^{2k}}{k!\Gamma(k+n+\lambda)\Gamma(k+n+\lambda^{*})},$$
(4)
$$\displaystyle\langle n\rangle=\langle a^{\dagger}a\rangle=\frac{\Omega^{2}}{(%
\Delta+\chi)^{2}+(\gamma/2)^{2}}\frac{{{}_{0}\rm{F}_{2}}(\lambda+1,\lambda^{*}%
+1,2|\varepsilon|^{2})}{{{}_{0}\rm{F}_{2}}(\lambda,\lambda^{*},2|\varepsilon|^%
{2})},$$
(5)
where $\varepsilon=\Omega/\chi$, $\lambda=(\gamma+\rmi\Delta)/\rmi\chi$, and ${{}_{0}\rm{F}_{2}}$ is the hypergeometric function:
$${}_{0}\rm{F}_{2}(a,b,z)=\sum_{k=0}^{\infty}\frac{z^{k}\Gamma(a)\Gamma(b)}{k!%
\Gamma(k+a)\Gamma(k+b)}.$$
(6)
In this approach, the second-order correlation function of photon-numbers for zero-delay time
$$g^{(2)}(0)=\frac{<a^{\dagger}a^{\dagger}aa>}{<a^{\dagger}a>^{2}}$$
(7)
is calculated in the following form:
$$g^{(2)}(0)=\frac{|\varepsilon|^{4}\Gamma(\lambda)\Gamma({\lambda^{*}})_{0}\rm{%
F}_{2}(2+\lambda,2+\lambda^{*},2|\varepsilon|^{2})}{\Gamma(2+\lambda)\Gamma(2+%
\lambda^{*})_{0}\rm{F}_{2}(\lambda,\lambda^{*},2|\varepsilon|^{2})}\frac{1}{%
\langle n\rangle^{2}}.$$
(8)
This approach has been applied to a driven anharmonic oscillator to obtain an analytic solution for the Wigner function and to investigate the
quantum-classical correspondence [19], (see, also [17], [18]). Below we analyse
other operational regimes of the KNR considering excitations of
the system by an external coherent field that is resonant to the frequencies of
transitions between oscillatory states.
3 Multi-photon excitations in the Kerr dissipative resonator
In the absence of any driving field, the states of the nonlinear oscillator
are the photon-number states $|n\rangle$, which are spaced in energy $E_{n}=E_{0}+\hbar\omega_{0}n+\hbar\chi n(n-1)$ with $n=0,1,...$. The
levels form an anharmonic ladder with
a nonlinearity parameter that is given by $E_{21}-E_{10}=2\hbar\chi$.
The energy spectrum can be probed with the response of
the system to the driving field when the driving frequency
is tuned. In this case, the spectral lines of the system are resolved if the nonlinear shifts of oscillatory energy levels are larger than the state line-widths, i.e. $\chi/\gamma>1$.
Here we also require the nonlinearity coefficient to be real, since the photon blockade is expected to work only if the intracavity medium is dispersive, rather than absorptive.
According to the formula $E_{n}=E_{0}+\hbar\omega_{0}n+\hbar\chi n(n-1)$,
the resonant frequencies $n\hbar\omega_{n}=E_{n0}$ of the multiphoton
transitions $|0\rangle\rightarrow|n\rangle$ are derived in the following form
$\omega_{n}=\omega_{0}+\chi(n-1)$. Thus, in order to excite a state with one photon
($E_{10}=\hbar\omega_{0}$) the driving frequency is fixed to the resonance
frequency $\omega_{1}=\omega_{0}$. For the two-photon transition
$|0\rangle\rightarrow|2\rangle$, $E_{20}=2\hbar\omega_{0}+2\chi$ and the
resonance frequency is $\omega_{2}=\omega_{0}+\chi$, while for the three-photon
transition $|0\rangle\rightarrow|3\rangle$, $E_{30}=3\hbar\omega_{0}+6\chi$, the resonant frequency is $\omega_{3}=\omega_{0}+2\chi$. Hence, the values of the
detuning at the resonant frequencies read as $\Delta_{n}=\omega_{0}-\omega_{n}=-\chi(n-1)$. This analysis is correct if the driving field amplitude is weak. Otherwise, the spectrum of mode excitations contains frequencies corresponding to oscillatory Raman processes under the external drive, moreover, the resonance frequencies are shifted due to the Stark effects as well. Indeed, in the second-order approximation of the perturbation theory on the interaction of the anharmonic oscillator with a monochromatic field the shift of oscillatory energy $E_{n}$ in transitions through the states $|n-1\rangle$ and $|n+1\rangle$ is calculated as $\Delta E_{n}=\hbar\Omega^{2}\left(\frac{n}{\omega+\chi(2n-1)}-\frac{(n+1)}{%
\omega+\chi(2n+1)}\right)$.
As the calculations based on (4), (5), (8) show, the results strongly depend on all the parameters: the nonlinearity, the detuning and the amplitude of the driving field. Below, we focus on comparative analysis of the regimes of strong and very strong nonlinearities in order to demonstrate selective excitations in the KNR due to formation of well-resolved multi-photon resonance spectral lines. In this way, we concentrate on particular cases with the following values of nonlinearity: $\chi/\gamma=2$ and $\chi/\gamma=20$. Our goal is to estimate the interplay of the following parameters: the ratio $\chi/\gamma$, the detuning $\Delta/\gamma$, the amplitude of the driving field $\Omega/\gamma$,
and the excitations in the Kerr resonator for the both regimes of strong and very strong nonlinearities.
3.1 Non-resonant excitations of oscillatory states
In this subsection, we consider the excitations of the KNR on the basis of (4) and (5) assuming the quantum regime, but not formation of well-resolved energy levels of the KNR. At first, two-dimensional plots of the quantum mechanical mean photon number are depicted in Figure 1,
in dependence on values of the detuning $\Delta/\gamma$ and the driving amplitude $\Omega/\gamma$.
Comparing Figure 1(a) and Figure 1(b) we conclude that the maximum values of the mean photon number in the resonator are essentially small for the large value of the ratio $\chi/\gamma=20$ in comparison with the case of $\chi/\gamma=2$. This means that high nonlinearity with
respect to dissipation leads to formation of a strong quantum operational regime.
Next, we analyse the photon number distribution in dependence of the detuning,
for the case of the strong nonlinearity, $\chi/\gamma=2$. The results of
calculations based on (4) are depicted in Figure 2(a) as
two-dimensional plots of the Fock state populations. These plots show the formation
of Poissonian-type distributions from the thermal distribution, if the
frequency of the driving field is tuned around the frequency of the KNR. Examples of
photon-number distribution functions are plotted in Figure 2(b) for
different values of the detuning.
3.2 Selective resonant excitations of oscillatory states
In this subsection we discuss the case of very strong nonlinearities for which energy levels of the KNR are well resolved and hence selective excitations of the oscillatory mode take place. In this way, the populations
$P_{0},P_{1},P_{2},P_{3}$ of the photon-number states $|n\rangle$ calculated in (5) are demonstrated in Figure 3 for the case of the very strong nonlinearity $\chi/\gamma=20$, in dependence of the dimensionless detuning $\Delta/\gamma$. As we see, the selective resonance excitations of the Fock states $|1\rangle$, $|2\rangle$, $|3\rangle$ are evident for both cases of the driving field: $\Omega/\gamma=5$ and $\Omega/\gamma=20$. Indeed, the populations $P_{1}$ depicted in Figure 3(b) display the maximum 0.5 at $\Delta=0$, that corresponds to
the resonance one-photon transition $|0\rangle\rightarrow|1\rangle$ at the frequency $\omega=\omega_{0}$.
The other maximum $P_{1}=0.38$, for $\Omega/\gamma=5$, corresponds to the
population of $|1\rangle$ state through the Raman process with the energy
conservation $E_{0}+2\omega_{2}=\omega_{k}+E_{1}$. This process
involves the two-photon excitation of the Fock state $|2\rangle$, (in the transition
$|0\rangle\rightarrow|2\rangle$ at the frequency of the pump field $\omega_{2}=\omega_{0}+\chi$), and the decay $|2\rangle\rightarrow|1\rangle$ at the frequency $\omega_{k}=\omega_{0}+2\chi$.
As we see in Figure 3(c), for a comparatively weak driving field, $\Omega/\gamma=5$, the population $P_{2}$ reaches only the value 0.2 due to the two-photon resonance excitation $|0\rangle\rightarrow|2\rangle$ at the frequency $\omega_{2}=\omega_{0}+\chi$. In the range of strong excitations the role of multi-photon processes is increasing. We observe that these populations display peaked structures and the maximum values are realized
for the definite parameters of detuning in accordance with resonant frequencies. Indeed, for the case of the strong field, $\Omega/\gamma=20$, the Raman processes of the population of $|2\rangle$ state become effective. Particularly, the peaks at
$\Delta/\gamma=-40$ and $\Delta/\gamma=-60$ correspond to the Raman processes through the states $|3\rangle$ at the
frequency $\omega_{3}=\omega_{0}+2\chi$ and $|4\rangle$ at $\omega_{4}=\omega_{0}+3\chi$, respectively.
As we see in Figure 3(d), for weak excitations $\Omega/\gamma=5$ (1), the population $P_{3}$ is negligible for all detunings, while for the case of strong excitations $\Omega/\gamma=20$ (2), the selective population takes place.
In general, from the analytically obtained results and numerical analysis we can conclude
that selectively low-energetic states $|1\rangle$, $|2\rangle$ and
$|3\rangle$ can be effectively excited for the very strong nonlinearities.
Nevertheless, we strongly demonstrate that the values of Fock state populations
are limited in continuous-wave (cw) steady-state regime of the KNR. Indeed, in Figure 3(b) the
population $P_{1}$ is limited by the value 0.5 and the populations $P_{2}$,
$P_{3}$ are less than $P_{1}$. This effect of selective excitations can be
interpreted as a blockade of two or more photons in the KNR by production of a
single-photon Fock state. Thus, observation of these limits on the populations
restricts the possibility to observe the photon blockade effects in an over
transient steady-state regime.
We demonstrate this fact also in Figure 4
by showing the dependence of the populations on the parameters of nonlinearity at the fixed detuning $\Delta/\gamma=-20$ and the external field amplitude $\Omega/\gamma=20$. This regime involves a two-photon resonance excitation of the KNR for $\chi/\gamma=20$. We observe that effective excitations of the oscillatory mode take place for the definite ranges of nonlinearity, particularly $P_{1}$ is limited by 0.5, while the populations $P_{2}$ and $P_{3}$ are limited by the values 0.35 and 0.3 correspondingly. Analogously, the populations versus the amplitude of the driving field depicted in Figure 5 demonstrate the limited behaviour in steady-state regimes as well. However the behavior of the populations of two-photon and three-photon number states
depicted in Figure 5(b) differs from $P_{0}$ and $P_{1}$ (Figure 5(a)). The complete analysis of the populations $P_{0}$, $P_{1}$, $P_{2}$, $P_{3}$
for the case of selective excitations of the KNR, $\Omega/\gamma=20$, in dependence on the detuning $\Delta/\gamma$ and the amplitude of the driving field $\Omega/\gamma$ are presented in Figure 6.
In the end of this section, we turn to the investigation of photon statistics on the base of the normalized second-order correlation function (7) for zero delay time. The correlation function describes the ratio of the number of photon
pairs emitted simultaneously to the product of the number of photons emitted independently
from one another and, on the other hand, is expressed through the variance
of the photon number fluctuations as $\langle(\Delta n)^{2}\rangle=\langle n\rangle+\langle n\rangle^{2}(g^{(2)}-1)$. Thus, the condition $g^{(2)}<1$ corresponds to the sub-Poissonian statistics, $\langle(\Delta n)^{2}\rangle<\langle n\rangle$ . The condition $g^{2}(0)=1$ corresponds to the Poissonian statistics and the condition $g^{2}(0)>1$ to the super-Poissonian statistics correspondingly. The results for $g^{2}(0)$ and the mean photon numbers are depicted in Figure 7 for three values of nonlinearities $\chi/\gamma=2$, $\chi/\gamma=10$, $\chi/\gamma=20$. As we see, for the regime of non-resonant excitations, ($\chi/\gamma=2$), the correlation function approximately equals to the unity for a large range of the detuning. This result is in accordance with the results depicted in Figure 2 and reflects the Poissonian statistics of the KNR mode. Note, the correlation function displays a peak around the value of detuning $\Delta/\gamma=-35$ in the critical or the threshold
region of the photons production. The analogous result on critical growth of
the correlation functions in the region of the threshold of generation has been
demonstrated for the non-degenerate parametric oscillator [22]. For the
regimes of very strong nonlinearities $g^{(2)}<1$, i.e. the mode displays
sub-Poissonian nonclassical statistics for the large range of the detuning from
$\Delta/\gamma=-40$ to $\Delta/\gamma=40$ due to selective excitations of the
KNR.
4 Conclusion
In conclusion, we have demonstrated spectroscopic multi-photon resonant
selective excitations of Fock states in the KNR driven by a continuous wave
field for strong Kerr nonlinearities. The selective excitation of the
resonator mode is realized in the cases of well resolved energy spectra of
anharmonic oscillator by tuning the frequency of the driving field for wide
ranges of the following parameters: the nonlinearity, the detuning and the
amplitude of the driving field. The results have been obtained on the basis of
the exact quantum solution of the Fokker-Planck equation in the complex $P$
representation for the KNR. This approach allowed us to calculate the mean
photon numbers as well as to analyse photon statistics of the resonator mode
on the basis of both the probability distributions of photons and the
second-order photon correlation functions. Particularly, we have demonstrated
the limited behaviour of photon number-state populations in steady-state
regimes due to dissipation and decoherence effects. We observed that the
population of the one-photon state is limited by the value 0.5 while the
population of the two-photon number state is limited by 0.35 for all ranges of
the parameters. This conclusion restricts the possibilities of realization of
the photon blockade due to cw excitations in over transient steady-state
regimes.
We acknowledge support from the Armenian State Committee of Science, the
Project No.13-1C031.
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On Monge-Ampére equations with homogenous right hand side
Panagiota Daskalopoulos${}^{*}$
Department of Mathematics, Columbia University, New York
pdaskalo@math.columbia.edu
and
Ovidiu Savin
Department of Mathematics, Columbia University, New York
savin@math.columbia.edu
Abstract.
We study the regularity and behavior at the origin of solutions to the two-dimensional degenerate Monge-Ampére
equation $\det D^{2}u=|x|^{\alpha}$, with $\alpha>-2$. We show that
when $\alpha>0$ solutions admit only two
possible behaviors near the origin, radial and non-radial, which
in turn implies $C^{2,\delta}$ regularity. We also show that the
radial behavior is unstable. For $\alpha<0$ we prove that
solutions admit only the radial behavior near the origin.
$*:$ Partially supported
by NSF grant 0102252
1. Introduction
We consider the degenerate two dimensional Monge-Ampére equation
(1.1)
$${}\det D^{2}u=|x|^{\alpha},\qquad x\in B_{1}$$
on the unit disc $B_{1}=\{\,|x|\leq 1\}$ of $\mathbb{R}^{2}$ and in the range of exponents $\alpha>-2$. Our goal is to investigate the behavior of solutions $u$ near the origin,
where the equation becomes degenerate.
The study of (1.1) is motivated by the Weyl problem with nonnegative curvature,
posed in 1916 by Weyl himself: Given a Riemannian metric $g$ on the 2-sphere
${\mathbb{S}}^{2}$ whose Gauss curvature is everywhere positive, does there
exist a global $C^{2}$ isometric embedding $X:({\mathbb{S}}^{2},g)\to(\mathbb{R}^{3},ds^{2})$, where $ds^{2}$ is the standard flat metric on $\mathbb{R}^{3}$?
H. Lewy [10] solved the problem under the assumption that the metric $g$
is analytic.
The solution to the Weyl problem, under the regularity assumption that $g$ has continuous
fourth order derivatives, was given in 1953 by L. Nirenberg [12].
P. Guan and Y.Y. Li [6] considered the question: If the Gauss curvature of the metric $g$ is nonnegative instead
of strictly positive and $g$ is smooth, is it still possible to
have a smooth isometric embedding ?
It was shown in [6] that for any $C^{4}$-Riemannian metric $g$
on ${\mathbb{S}}^{2}$ with nonnegative Gaussian curvature, there is always a $C^{1,1}$
global isometric embedding into $(\mathbb{R}^{3},ds^{2})$.
Examples show that for some analytic metrics with positive Gauss
curvature on ${\mathbb{S}}^{2}$ except at one point, there exists only a
$C^{2,1}$ but not a $C^{3}$ global isometric embedding into $(\mathbb{R}^{3},ds^{2})$. Note that the phenomenon is global, since C.S. Lin
[11] has shown that for any smooth 2-dimensional Riemannian
metric with nonnegative Gauss curvature there exists a smooth local isometric embedding into $(\mathbb{R}^{3},ds^{2})$.
This leads to the following question, which was posed in [6]: Under what conditions on a smooth metric $g$ on
${\mathbb{S}}^{2}$ with nonnegative Gauss curvature, there is a
$C^{2,\alpha}$ global isometric embedding into $(\mathbb{R}^{3},ds^{2})$,
for some $\alpha>0$, or even a $C^{2,1}$ ?
The problem can be reduced to a partial differential equation of
Monge-Ampére type that becomes degenerate at the points where
the Gauss curvature vanishes. It is well known that in general one
may have solutions to degenerate Monge-Ampére equations which
are at most $C^{1,1}$.
One may consider a smooth Riemannian metric $g$ on ${\mathbb{S}}^{2}$ with
nonnegative Gauss curvature, which has only one non-degenerate
zero. In this case, if we represent the $C^{1,1}$ embedding as a
graph, answering the above question amounts to studying the
regularity at the origin of the degenerate Monge-Ampére
equation
(1.2)
$$\det D^{2}u=f,\qquad\mbox{on}\,\,B_{1}$$
in the case where the forcing term $f$ vanishes quadratically at
$x=0$. More precisely, it suffices to assume that $f(x)=|x|^{2}g(x)$,
where $g$ is a positive Lipschitz function.
This leads to equation (1.1) when $\alpha=2$.
In addition to the results mentioned above, degenerate equations of the form (1.2) on $\mathbb{R}^{2}$ were previously considered by P. Guan in [5] in the case where $f\in C^{\infty}(B_{1})$ and
(1.3)
$$A^{-1}\,(x_{1}^{2l}+B\,x_{2}^{2m})\leq f(x_{1},x_{2})\leq A\,(x_{1}^{2l}+B\,x_%
{2}^{2m})$$
for some constants $A>0,B\geq 0$ and positive integers $l\leq m$. The $C^{\infty}$ regularity of the solution $u$ of (1.2)
was shown in [5], under the additional condition that
$u_{x_{2}x_{2}}\geq C_{0}>0$. It was conjectured in [5] that
the same result must be true under the weaker condition that
$\Delta u\geq C_{0}>0$. This was recently shown by P. Guan and I.
Sawyer in [8].
Equation (1.1) has also an interpretation in the language of
optimal transportation with quadratic cost $c(x,y)=|x-y|^{2}$. In
this setting the problem consists in transporting the density
$|x|^{\alpha}\,dx$ from a domain $\Omega_{x}$ into the uniform density
$dy$ in the domain $\Omega_{y}$ in such a way that we minimize the total
“transport cost”, namely
$$\int_{\Omega_{x}}|y(x)-x|^{2}|x|^{\alpha}dx.$$
Then, by a theorem of Y. Brenier [1], the optimal map $x\mapsto y(x)$ is given by the gradient of a solution of the
Monge-Ampére equation (1.1). The behavior of these
solutions at the origin gives information on the geometry of the
optimal map near the singularity of the measure $|x|^{\alpha}\,dx$.
We will next state the results of this paper. We assume that $u$ is a solution of
equation (1.1). Then, $u$ is $C^{\infty}$-smooth away from the origin. The
following results describe the regularity of $u$ at the origin.
We begin with the case when $\alpha>0$.
Theorem 1.1.
If $\alpha>0$, then $u\in C^{2,\delta}$ for a small $\delta$ depending
on $\alpha$.
Theorem 1.1 is a consequence of Theorem 1.2 which shows
that there are exactly two types of behaviors near the origin.
Theorem 1.2.
If $\alpha>0$, and
(1.4)
$${}u(0)=0,\quad\nabla u(0)=0$$
then, there exist positive constants $c(\alpha)$, $C(\alpha)$ depending
on $\alpha$ such that either $u$ has the radial behavior
(1.5)
$${}c(\alpha)|x|^{2+\frac{\alpha}{2}}\leq u(x)\leq C(\alpha)|x|^{2+\frac{\alpha}%
{2}}$$
or, in an appropriate system of coordinates, the non-radial behavior
(1.6)
$${}u(x)=\frac{a}{(\alpha+2)(\alpha+1)}|x_{1}|^{2+\alpha}+\frac{1}{2a}x_{2}^{2}+%
O\left((|x_{1}|^{2+\alpha}+x_{2}^{2})^{1+\delta}\right)$$
for some $a>0$.
The non-radial behavior (1.6) was first shown by P. Guan
in [5], under the condition that $u_{x_{2}x_{2}}\geq C_{0}>0$
near the origin, and was recently generalized in [8] to
only assume that $\Delta u\geq C_{0}>0$.
The next result states that the radial behavior is unstable.
Theorem 1.3.
Suppose $\alpha>0$, let $u_{0}$ be the radial solution to
$(\ref{eq})$,
$$u_{0}(x)=c_{\alpha}|x|^{2+\frac{\alpha}{2}}$$
and consider the Dirichlet problem
$$\det D^{2}u=|x|^{\alpha},\quad\quad u=u_{0}-\varepsilon\cos(2\theta)\mbox{ on
$\partial B_{1}$}.$$
Then $u-u(0)$ has the nonradial behavior
$(\ref{nonrad})$ for small $\varepsilon$.
Subsequences of blow up solutions satisfying
(1.5) converge to homogenous solutions, as shown next.
Theorem 1.4.
Under the assumptions of Theorem 1.2, if $u$ satisfies
$(\ref{rad})$, then for any sequence of $r_{k}\to 0$ the blow up
solutions
$$r_{k}^{-2-\frac{\alpha}{2}}u(r_{k}x)$$
have a subsequence that converges uniformly on compact sets to a
homogenous solution of $(\ref{eq})$.
In the case $-2<\alpha<0$ solutions have only the radial
behavior. Actually, we prove a stronger result by showing that $u$
converges to the radial solution $u_{0}$ in the following sense.
Theorem 1.5.
If $-2<\alpha<0$ and $(\ref{00})$ holds, then
$$\lim_{x\to 0}\frac{u(x)}{u_{0}(x)}=1.$$
Our results are based on the following argument: assume that a
section of $u$, say $\{u<1\}$, is “much longer” in the $x_{1}$
direction compared to the $x_{2}$ direction. If $v$ is an affine
rescaling of $u$ so that $\{v<1\}$ is comparable to a ball, then
$v$ is an approximate solution of
$$\det D^{2}v(x)\approx c|x_{1}|^{\alpha}.$$
Hence, the geometry of small sections of solutions of this new
equation provides information on the behavior of the small
sections of $u$. For example, if the sections of $v$ are “much
longer” in the $x_{1}$ direction (case $\alpha>0$) then the
corresponding sections of $u$ degenerate more and more in this
direction, producing the non-radial behavior (1.6). If
the sections of $v$ are longer in the $x_{2}$ direction (case
$\alpha<0$) then the sections of $u$ tend to become round and we
end up with a radial behavior near the origin.
We close the introduction with the following remarks.
Remark 1.6.
From the proofs one can see that the theorems above, with the
exception of the instability result, are still valid for the
equation with more general right hand side
$$\det D^{2}u=|x|^{\alpha}g(x)$$
with $g\in C^{\delta}(B_{1})$, $g>0$.
Remark 1.7.
i. We will show in the proof of Theorem 1.1 that solutions of (1.1), with $\alpha>0$, which satisfy the radial behavior (1.5) at the origin are of class $C^{2,\frac{\alpha}{2}}$.
ii. Theorems 1.1, 1.2 and the results of Guan in [5]
and Guan and Sawyer in [8] imply that solutions of (1.1),
with $\alpha$ a positive integer, which satisfy the non-radial behavior (1.6) at the origin are $C^{\infty}$-smooth.
Remark 1.8.
Equations of the form
(1.7)
$$\det D^{2}w=|\nabla w|^{\beta},\qquad\beta=-\alpha$$
for which the set $\{\nabla w=0\}$ is compactly included in the domain of
definition,
can be reduced to (1.1) by defining $u$ to be the Legendre transform
of $w$. Hence, Theorem 1.5 establishes the
sharp regularity of solutions $w$ of equation (1.7) when $0<\beta<2.$
The paper is organized as follows. In Section 2 we introduce
tools and notation to be used later in the paper. In Section 3 we prove
Theorem 1.2. In Section 4 we establish the radial behavior of solutions
when $-2<\alpha<0$, showing Theorem 1.5. In Section 5 we investigate
homogenous solutions and give the proof of Theorem 1.4. In Section 6 we prove Theorem 1.3. Finally, in
Section 7 we show that Theorem 1.2 implies Theorem 1.1.
Acknowledgment: We are grateful to P. Guan and Y.Y. Li for introducing us to this problem and for many useful discussions.
2. Preliminaries
In this section we investigate the geometry of the sections of
$u$, namely the sets
$$S_{t,{x_{0}}}^{u}:=\{u(x)<u(x_{0})+\nabla u(x_{0})\cdot(x-x_{0})+t\}.$$
We omit the indices $u$ and $x_{0}$ whenever there is no possibility
of confusion. We recall some facts about such sections.
John’s lemma (c.f. Theorem 1.8.2 in [9])
states that any bounded convex set $\Omega\subset\mathbb{R}^{n}$ is balanced with
respect to its center of mass. That is, if $\Omega$ has center of
mass at the origin, there exists an ellipsoid $E$ (with center of
mass $0$) such that
$$E\subset\Omega\subset k(n)E$$
for a constant $k(n)$ depending only on the dimension $n$.
Sections $S_{t,{x_{0}}}^{u}$ of solutions to Monge-Ampére equations with
doubling measure $\mu$ on the right hand side also satisfy a balanced
property with respect to $x_{0}$.
We recall the following definition.
Definition 2.1 (Doubling measure).
The measure $\mu$ is doubling with respect to ellipsoids
in $\Omega$ if there exists a constant $c>0$ such that for any
point $x_{0}\in\Omega$ and any ellipsoid $x_{0}+E\subset\Omega$
(2.1)
$${}\mu(x_{0}+E)\geq c\mu\left((x_{0}+2E)\cap\Omega\right).$$
The following theorem, due to L. Caffarelli [2] holds.
Theorem 2.2 (Caffarelli).
Let $u:\Omega\to\mathbb{R}$ be a (Alexandrov) solution of
$$\det D^{2}u=\mu$$
with $\mu$ a doubling measure. Then, for each $S_{t,{x_{0}}}\subset\Omega$
there exists a unimodular matrix $A_{t}$ such that
(2.2)
$${}k_{0}^{-1}A_{t}B_{r}\subset S_{t,{x_{0}}}-x_{0}\subset k_{0}A_{t}B_{r}$$
with
$$r=t\,(\mu(S_{t,{x_{0}}}))^{-1/n},\qquad\det A_{t}=1.$$
for a constant $k_{0}(c,n)>0$.
The ellipsoid $E=A_{t}B_{r}$ remains invariant if we replace $A_{t}$ with $A_{t}\,O$
with $O$ orthogonal, thus we may assume that $A$ is triangular. If (2.2) is
satisfied we write
$$S_{t}\sim A_{t}$$
and say that the eccentricity of $S_{t}$ is proportional to $|A_{t}|$.
The measure that appears in (1.1), namely
$$\mu:=|x|^{\alpha}\,dx$$
is clearly doubling with respect to ellipsoids for $\alpha>0$. We will
see in Section 4 that this property is still true for $-1<\alpha<0$ but fails for $-2<\alpha\leq-1$.
Next we discuss the case when the right hand side in the Monge-Ampére
equation depends only on one variable, i.e
(2.3)
$$\det D^{2}u=h(x_{1}).$$
We will show in Section 3 that such equations are
satisfied by blow up limits of solutions to $\det D^{2}u=|x|^{\alpha}$ at the origin, when $\alpha>0$.
These equations remain invariant under affine transformations. Also, by
taking derivatives along the $x_{2}$ direction one obtains the Pogorelov type estimate
$$u_{22}\leq C$$
in the interior of the sections of $u$.
Assume that $u$ satisfies equation (2.3) in $B_{1}\subset\mathbb{R}^{n}$, in any dimension $n\geq 2$ and perform the following partial
Legendre transformation:
(2.4)
$$y_{1}=x_{1},\quad y_{i}=u_{i}(x)\quad i\geq 2,\qquad u^{*}(y)=x^{\prime}\cdot%
\nabla_{x^{\prime}}u-u(x)$$
with $x^{\prime}=(x_{2},...,x_{n})$.
The function $u^{*}$ is obtained by taking the Legendre transform of
$u$ on each slice $x_{1}=const.$ We claim that $u^{*}$ (which is convex
in $y^{\prime}$ and concave in $y_{1}$) satisfies
(2.5)
$$u^{*}_{11}+h(y_{1})\,\det D^{2}_{y^{\prime}}u^{*}=0.$$
To see this we first notice that by the change of variable
$$v(x_{1},x^{\prime})\rightarrow u(x_{1},x^{\prime}+x_{1}\xi^{\prime})$$
$v$ satisfies the same equation as $u$ and
$$v^{*}(y)=u^{*}(y)-y_{1}\,\xi^{\prime}\cdot y^{\prime}.$$
Thus we may assume that $D^{2}u$ is diagonal at $x$. Now it is easy
to check that
$$u^{*}_{1}=-u_{1},\quad\nabla_{y^{\prime}}u^{*}=x^{\prime}$$
and
$$u^{*}_{11}=-u_{11},\quad D^{2}_{y^{\prime}}u^{*}=[D^{2}_{x^{\prime}}u]^{-1}.$$
Hence $u^{*}$ satisfies (2.5).
Remark 2.3.
The following hold:
$i$.
The partial Legendre transform of $u^{*}$ is $u$, i.e. $(u^{*})^{*}=u.$
$ii$.
The inequality $|u-v|\leq\varepsilon$ implies that $|u^{*}-v^{*}|\leq\varepsilon$ on their common domain of definition.
$iii$.
In dimension $n=2$, the partial Legendre transform of the function $p(x_{1},x_{2})=a\,|x_{1}|^{2+\alpha}+b\,x_{1}x_{2}+d\,x_{2}^{2}$
is given by
(2.6)
$$p^{*}(y_{1},y_{2})=(a\,|x_{1}|^{2+\alpha}+b\,x_{1}x_{2}+d\,x_{1}^{2})^{*}=-a\,%
y_{1}^{2+\alpha}+\frac{1}{4d}(y_{2}-b\,y_{1})^{2}.$$
Notice that $p$ is a solution of the equation $\det D^{2}u=c\,|x_{1}|^{\alpha}$, for an appropriate constant $c$, and $p^{*}$ is a
solution of the equation $w_{11}+c\,|y_{1}|^{\alpha}\,w_{22}=0$.
We will restrict from now on our discussion to dimension $n=2$ and the special case where $h(x_{1})=|x_{1}|^{\alpha}$.
Lemma 2.4.
Assume that for some $\alpha>0$, $w$ solves the equation
$$Lw:=w_{11}+|y_{1}|^{\alpha}\,w_{22}=0\qquad\mbox{in $B_{1}\subset\mathbb{R}^{2%
}$}$$
with $|w|\leq 1$.
Then in $B_{1/2}$, $w$ satisfies
$$\begin{split}\displaystyle w(y)&\displaystyle=a_{0}+a_{1}\cdot y+a_{2}\,y_{1}%
\,y_{2}+\\
&\displaystyle+a_{3}\left(\frac{1}{2}\,y_{2}^{2}-\frac{1}{(\alpha+2)(\alpha+1)%
}\,|y_{1}|^{2+\alpha}\right)+O((y_{2}^{2}+|y_{1}|^{2+\alpha})^{1+\delta})\end{split}$$
with $|a_{i}|$ and $O(\cdot)$ bounded by a universal constant and
$\delta=\delta(\alpha)>0$.
Proof.
First we prove that $w_{2}$ is bounded in the interior.
Since $Lw_{2}=0$, the same argument applied inductively would imply that the derivatives of $w$ with respect to $y_{2}$ of any order are bounded in the interior.
To establish the bound on $w_{2}$, we show that
(2.7)
$${}L(Cw^{2}+\varphi^{2}w_{2}^{2})\geq 0$$
for a smooth cutoff function $\varphi$, to be made precise later. Indeed, a direct computation shows that
$$L(w^{2})=2\,(w_{1}^{2}+|y_{1}|^{\alpha}w_{2}^{2})$$
and
$$\begin{split}\displaystyle L(\varphi^{2}&\displaystyle w_{2}^{2})=L(\varphi^{2%
})w_{2}^{2}+\varphi^{2}L(w_{2}^{2})+2(\varphi^{2})_{1}(w_{2}^{2})_{1}+2|y_{1}|%
^{\alpha}(\varphi^{2})_{2}(w_{2}^{2})_{2}\\
&\displaystyle=L(\varphi^{2})\,w_{2}^{2}+2\varphi^{2}\,(w_{21}^{2}+|y_{1}|^{%
\alpha}w_{22}^{2})+8(\varphi_{1}w_{2})(\varphi w_{21})+8|y_{1}|^{\alpha}(%
\varphi_{2}w_{2})(\varphi w_{22})\end{split}$$
hence
$$\begin{split}\displaystyle L(Cw^{2}+\varphi^{2}w_{2}^{2})\geq 2C&\displaystyle%
|y_{1}|^{\alpha}w_{2}^{2}+2\varphi^{2}\,(w_{21}^{2}+|y_{1}|^{\alpha}w_{22}^{2}%
)\\
&\displaystyle+L(\varphi^{2})w_{2}^{2}+8(\varphi_{1}w_{2})(\varphi w_{21})+8|y%
_{1}|^{\alpha}(\varphi_{2}w_{2})(\varphi w_{22}).\end{split}$$
By choosing the cutoff function $\varphi$ such that $\varphi_{1}=0$
for $|y_{1}|\leq 1/4$, then
$$L(\varphi^{2})\geq-C_{1}|y_{1}|^{\alpha},\quad|\varphi_{1}w_{2}|\leq C_{1}|y_{%
1}|^{\alpha/2}|w_{2}|$$
and we obtain (2.7) if $C$ is large. Therefore
$w_{2}$ is bounded in the interior by the maximum principle.
The equation $w_{11}+|y_{1}|^{\alpha}\,w_{22}=0$ and the bound $|w_{22}|\leq C$ imply
the bound
$$|w_{11}|\leq C\,|y_{1}|^{\alpha}.$$
Thus $w_{1}$ is bounded. The same estimates as above show that
$w_{12}$, $w_{122}$ are bounded as well. By Taylor’s formula, namely
$$f(t)=f(0)+f^{\prime}(0)\,t+\int_{0}^{t}(t-s)\,f^{\prime\prime}(s)\,ds$$
and the equation $Lw=0$, we conclude that
$$w(y_{1},0)=w(0)+w_{1}(0)\,y_{1}-\frac{w_{22}(0)}{(\alpha+2)(\alpha+1)}\,y_{1}^%
{2+\alpha}+O(|y_{1}|^{3+\alpha}),$$
$$w(y_{1},y_{2})=w(y_{1},0)+w_{2}(y_{1},0)\,y_{2}+\frac{w_{22}(0)}{2}\,y_{2}^{2}%
+O(|y_{2}|^{3}+|y_{1}y_{2}^{2}|),$$
and
$$w_{2}(y_{1},0)=w_{2}(0)+w_{12}(0)\,y_{1}+O(|y_{1}|^{2+\alpha})$$
from which the lemma follows.
∎
Notation: By universal constants we understand positive constants that may also depend on the exponent $\alpha$. Also, when there is no
possibility of confusion we use the letters $c$, $C$
for various universal constants that change from line to line.
3. Proof of Theorem 1.2
Throughout this section we assume that $\alpha>0$, that $u$ satisfies
$$u(0)=0,\quad\nabla u(0)=0$$
and we simply write $S_{t}$ for the section $S^{u}_{t,0}$.
Let
$$\Gamma:=\{\,|x_{1}|^{2+\alpha}+x_{2}^{2}<1\,\}$$
be the $1$ section of $|x_{1}|^{2+\alpha}+x_{2}^{2}$ at $0$.
If a set
$\Omega$ satisfies
$$(1-\theta)\Gamma\subset\Omega\subset(1+\theta)\Gamma$$
we write
$$\Omega\in\Gamma\pm\theta.$$
The following approximation lemma constitutes the basic step in the proof of
Theorem 1.2.
Lemma 3.1.
Assume that $u$ in the section $S_{1}$ satisfies
(3.1)
$$\det D^{2}u=c\,f(x),\qquad|f(x)-|x_{1}|^{\alpha}|\leq\varepsilon$$
and
(3.2)
$$S_{1}\in\Gamma\pm\theta$$
with $\varepsilon\leq\varepsilon_{0}$ and $\varepsilon^{1/8}\leq\theta$, $\theta<1$ small. Then, for some small universal $t_{0}$, we have
$$S_{t_{0}}\in A\,D_{t_{0}}(\Gamma\pm\theta\,t_{0}^{\delta})$$
where
$$A:=\begin{pmatrix}a_{11}&0\\
a_{21}&a_{22}\end{pmatrix},\quad D_{t_{0}}:=\begin{pmatrix}t_{0}^{\frac{1}{2+%
\alpha}}&0\\
0&t_{0}^{\frac{1}{2}}\end{pmatrix}$$
and
$$|A-I|\leq C\theta,\qquad\mbox{$C$ universal.}$$
Moreover, the constant $c$ in
(3.1) satisfies
(3.3)
$${}|c-2(1+\alpha)(2+\alpha)|\leq C\theta.$$
Proof.
We consider the solution
(3.4)
$${}v:=\frac{c^{1/2}}{[2(1+\alpha)(2+\alpha)]^{1/2}}(|x_{1}|^{2+\alpha}+x_{2}^{2})$$
of the equation
$$\det D^{2}v=c\,|x_{1}|^{\alpha}$$
and compute that
(3.5)
$${}\det D^{2}(v+\sqrt{c\varepsilon}\,|x|^{2})>c\,(|x_{1}|^{\alpha}+\varepsilon)%
\geq\det D^{2}u$$
and
(3.6)
$${}\det D^{2}(u+\sqrt{c\varepsilon}\,|x|^{2})>c\,(f(x)+\varepsilon)\geq\det D^{%
2}v$$
because $|f(x)-|x_{1}|^{\alpha}|\leq\varepsilon$, by assumption.
We first notice that the assumption (3.2) implies that the constant
$c$ in equation (3.1) is bounded from above by a universal constant, if $\varepsilon_{0}$ is small. This can be easily seen from equation (3.6) which, with the aid of the
maximum principle, implies that $u+\sqrt{c\varepsilon}\,|x|^{2}\geq v$, on $\{u=1\}$
(notice that both $v$ and $w=u+\sqrt{c\varepsilon}\,|x|^{2}$ satisfy $v(0)=w(0)=0$
and $\nabla w(0)=\nabla v(0)=0$). Since $\{u=1\}\in\Gamma\pm\theta$,
this readily gives a bound on $c$, if we assume that $\theta$ is small.
We will next show that
(3.7)
$${}\{v<1\}\in\Gamma\pm 2\theta$$
which implies the bound (3.3).
Indeed, if
$$\{v<1\}\subset(1-2\theta)\,\Gamma$$
then $v>u+\tilde{c}\,\theta\,|x|^{2}$ on $\{u=1\}$, for a universal $\tilde{c}$,
thus
$$v>u+\sqrt{c\varepsilon}\,|x|^{2},\qquad\mbox{on}\,\,\{u=1\}$$
since, by the assumptions of the lemma, $\sqrt{\varepsilon}<\varepsilon^{1/8}\leq\theta$ and
$\varepsilon\leq\varepsilon_{0}$, with $\varepsilon_{0}$ sufficiently small.
We conclude
from the maximum principle (see (3.5)), that $v>u+\sqrt{c\varepsilon}\,|x|^{2}$ in $S_{1}$. This is a contradiction, since $u(0)=v(0)=0$.
If
$$(1+2\theta)S_{1}\subset\{v<1\}$$
then similarly we obtain $v+\sqrt{c\varepsilon}|x|^{2}<u$ in $S_{1}$,
a contradiction.
Let $w$ be the solution of the problem
$$\det D^{2}w=c\,x_{1}^{2},\quad\mbox{in}\,\,S_{1},\qquad w=u\quad\mbox{on $%
\partial S_{1}$}.$$
By the maximum principle
$$w+\sqrt{c\varepsilon}\,(|x|^{2}-2)\leq u\leq w-\sqrt{c\varepsilon}\,(|x|^{2}-2)$$
thus
$$|w-u|\leq C\sqrt{\varepsilon}.$$
Also from (3.7) we obtain
$$|w-v|\leq C\theta.$$
Hence, by Remark 2.3, the corresponding partial Legendre
transforms defined in Section 2 satisfy in $B_{1/2}$
(3.8)
$${}|w^{*}-v^{*}|\leq C\theta$$
(3.9)
$${}|w^{*}-u^{*}|\leq C\sqrt{\varepsilon},\quad u^{*}(0)=0,\quad\nabla u^{*}(0)=0$$
and $w^{*}$ and $v^{*}$ solve the same linear equation
$$w^{*}_{11}+c\,|y_{1}|^{\alpha}w^{*}_{22}=0.$$
Using Lemma 2.4 for the difference $w^{*}-v^{*}$ together with
(2.6), (3.4), (3.3) and (3.8), yields to
(3.10)
$$\begin{split}\displaystyle w^{*}=-&\displaystyle|y_{1}|^{2+\alpha}+\frac{1}{4}%
y_{2}^{2}+a+b_{1}y_{1}+b_{2}y_{2}\\
&\displaystyle+\theta\,\left(c\,y_{1}y_{2}+d_{1}|y_{1}|^{2+\alpha}+d_{2}y_{2}^%
{2}+O((|y_{1}|^{2+\alpha}+y_{2}^{2})^{1+\delta}\,)\right)\end{split}$$
with the
coefficients $a,b_{i},c,d_{i}$ bounded by a universal constant.
From (3.9) we find that
$$w^{*}(0,y_{2})\geq-C\sqrt{\varepsilon}\quad\mbox{and}\quad w^{*}(y_{1},0)\leq C%
\sqrt{\varepsilon}$$
since, from the convexity in $y_{2}$ and concavity in $y_{1}$ of
$u^{*}$,
$$u^{*}(0,y_{2})\geq 0\quad\mbox{and}\quad u^{*}(y_{1},0)\leq 0.$$
This and (3.10) imply the bounds
$$|a|\leq C\varepsilon^{1/2},\quad|b_{1}|\leq C\varepsilon^{1/4},\quad|b_{2}|%
\leq C\varepsilon^{1/4}.$$
Thus, if $|y_{1}|^{2+\alpha}+y_{2}^{2}\leq 10\,t_{0}$, then
$$w^{*}=-(1-d_{1}\theta)\,|y_{1}|^{2+\alpha}+\left(\frac{1}{4}+d_{2}\theta\right%
)y_{2}^{2}+c\theta\,y_{1}y_{2}+O(\varepsilon^{1/4}+\theta\,t_{0}^{1+\delta}).$$
Hence, by performing the
partial Legendre transform on $w^{*}$ (using that $(w^{*})^{*}=w$ and (2.6)), we obtain
(3.11)
$${}\left|w-[e_{1}|x_{1}|^{2+\alpha}+e_{2}\,(x_{2}+e_{3}\,x_{1})^{2}]\right|\leq
C%
(\varepsilon^{1/4}+\theta t_{0}^{1+\delta})$$
for
$$|x_{1}|^{2+\alpha}+4e_{2}^{2}\,(x_{2}+e_{3}x_{1})^{2}\leq 10\,t_{0}$$
with $|e_{1}-1|,|e_{2}-1|,|e_{3}|$ bounded by $C\theta$.
We next observe that if $p(x)=e_{1}|x_{1}|^{2+\alpha}+e_{2}\,(x_{2}+e_{3}\,x_{1})^{2}$,
then the function
$$\tilde{p}(y):=\frac{1}{t_{0}}p(Fy)$$
with $F$ given by
$$F^{-1}:=\begin{pmatrix}t_{0}^{-\frac{1}{2+\alpha}}&0\\
0&t_{0}^{-\frac{1}{2}}\end{pmatrix}\begin{pmatrix}e_{1}^{\frac{1}{2+\alpha}}&0%
\\
e_{3}e_{2}^{\frac{1}{2}}&e_{2}^{\frac{1}{2}}\end{pmatrix}=D_{t_{0}}^{-1}A^{-1}$$
satisfies
$$\tilde{p}(y)=|y_{1}|^{2+\alpha}+y_{2}^{2}.$$
Hence, denoting by
$$\tilde{w}(y)=\frac{1}{t_{0}}w(Fy)$$
we conclude from (3.11) that
$$|\tilde{w}(y)-(|y_{1}|^{2+\alpha}+y_{2}^{2})|\leq C(\varepsilon^{1/4}t_{0}^{-1%
}+\theta t_{0}^{\delta}),\qquad\mbox{for $|y_{1}|^{2+\alpha}+y_{2}^{2}\leq 2$}.$$
Since $|\tilde{w}-\tilde{u}|\leq C\varepsilon^{1/2}t_{0}^{-1}$ (because $|w-u|\leq C\varepsilon^{1/2}$) we find
for $\varepsilon<\min(\theta^{8},\varepsilon_{0})$, with $\varepsilon_{0}$ small, that
$$\{\tilde{u}<1\}\in\Gamma\pm\gamma$$
with
$$\gamma=C(\varepsilon^{1/4}\,t_{0}^{-1}+\theta\,t_{0}^{\delta})\leq\theta\,t_{0%
}^{\delta^{\prime}}.$$
The proof is now completed since
$S_{t_{0}}=F\{\tilde{u}<1\}=A\,D_{t_{0}}\{\tilde{u}<1\}.$
∎
The proof given above also shows the following Lemma.
Lemma 3.2.
Assume that $u$ satisfies
$$\det D^{2}u=c\,f(x),\qquad\mbox{on}\,\,S_{1}$$
and
$$B_{1/k_{0}}\subset S_{1}\subset B_{k_{0}}.$$
Then, given $\theta_{0}$, there exist $\varepsilon_{1}(\theta_{0},k_{0})$ and
$t_{1}(\theta_{0},k_{0})$ small such that if
$$|f(x)-|x_{1}|^{\alpha}|\leq\varepsilon_{1}$$
then
$$S_{t_{1}}\in A_{0}D_{t_{1}}(\Gamma\pm\theta_{0})$$
with
(3.12)
$${}A_{0}:=\begin{pmatrix}a_{0,11}&0\\
a_{0,21}&a_{0,22}\end{pmatrix}$$
and
$$c(k_{0})\leq a_{0,ii}\leq C(k_{0}),\quad|a_{0,12}|\leq C(k_{0})$$
for some universal constants $c(k_{0})$, $C(k_{0})$.
The proof of Theorem 1.2 readily follows from the next
proposition which shows that if the section $S_{\lambda}$ has large
eccentricity, for some $\lambda$, then $u$ enjoys the nonradial
behavior (1.6) at the origin.
Proposition 3.3.
Assume that $u$ solves the equation
$$\det D^{2}u=|x|^{\alpha},\qquad\mbox{on}\,\,S_{1}$$
and that $S_{1}$ has large eccentricity, i.e.
$$FB_{1/k_{0}}\subset S_{1}\subset FB_{k_{0}},\quad F:=c\begin{pmatrix}b&0\\
0&1/b\end{pmatrix}$$
with $b\geq C_{0}$. Then, there exists a $z$-system of coordinates
such that
(3.13)
$${}u(z)=\frac{a}{(\alpha+2)(\alpha+1)}|z_{1}|^{2+\alpha}+\frac{1}{2a}z_{2}^{2}+%
O\left((|z_{1}|^{2+\alpha}+z_{2}^{2})^{1+\delta}\right)$$
for some $a>0$.
Proof.
The proof will be based on an inductive argument, where at each step will
use Lemma 3.1.
Denote by
$$v_{1}(x):=u(Fx),$$
and compute that $v_{1}$ satisfies the equation
$$\det D^{2}v_{1}(x)=(\det F)^{2}|Fx|^{\alpha}=c^{4+\alpha}b^{\alpha}|(x_{1},b^{%
-2}x_{2})|^{\alpha}.$$
Also,
$$\{v_{1}<1\}=F^{-1}S_{1}.$$
If $b$ is large, then $v_{1}$ satisfies hypothesis of the Lemma
3.2. Hence, for some fixed $\theta_{0}$ we obtain
$$S_{t_{1}}=F\{v_{1}<t_{1}\}\in FA_{0}D_{t_{1}}(\Gamma\pm\theta_{0})$$
with $A_{0}$ satisfying (3.12).
We assume by induction that for $t=t_{1}t_{0}^{k}$ we have
$$S_{t}\in FA_{k}D_{t}(\Gamma\pm\theta_{0}t_{0}^{(k-1)\delta})$$
with
$$A_{k}:=\begin{pmatrix}a_{k,11}&0\\
a_{k,21}&a_{k,22}\end{pmatrix}$$
and
(3.14)
$${}\quad c/2\leq a_{k,ii}\leq 2C,\quad|a_{k,21}|\leq 2C.$$
We will show that
$$S_{t_{0}t}\in FA_{k+1}D_{t_{0}t}(\Gamma\pm\theta_{0}t_{0}^{k\delta})$$
where
$$A_{k+1}=A_{k}\,E_{k}$$
and
$$E_{k}:=\begin{pmatrix}e_{k,11}&0\\
e_{k,21}&e_{k,22}\end{pmatrix}$$
with
(3.15)
$${}|e_{k,ii}-1|\leq C\theta_{0}t_{0}^{(k-1)\delta},\quad|e_{k,21}|\,t^{-\frac{%
\alpha}{2(2+\alpha)}}\leq C\theta_{0}t_{0}^{(k-1)\delta}.$$
Notice that condition (3.15) implies the bound
(3.16)
$${}|A_{k+1}-A_{k}|\leq C\theta_{0}t_{0}^{(k-1)\delta}.$$
To prove this inductive step, we observe that the function
$$v_{t}(x):=t^{-1}u(FA_{k}D_{t}x)$$
satisfies in $\{v_{t}<1\}$ the equation
$$\det D^{2}v_{t}=c_{t}\,|\tilde{x}|^{\alpha}$$
with
$$|\tilde{x}|^{\alpha}=\left|\left(a_{k,11}t^{\frac{1}{2+\alpha}}x_{1},b^{-2}(a_%
{k,21}t^{\frac{1}{2+\alpha}}x_{1}+a_{k,22}t^{\frac{1}{2}}x_{2})\right)\right|^%
{\alpha}=c_{t}^{\prime}\,f_{t}(x)$$
and
$$|f_{t}-|x_{1}|^{\alpha}|\leq b^{-2}t^{\frac{\alpha}{2(2+\alpha)}}.$$
Also
$$\{v_{t}<1\}\in\Gamma\pm\theta_{0}t_{0}^{(k-1)\delta}$$
since $S_{t}\in FA_{k}D_{t}(\Gamma\pm\theta_{0}t_{0}^{(k-1)\delta})$ by the inductive assumption.
Hence, if $\delta=\delta(\alpha)$ is chosen small, then $v_{t}$ satisfies the
assumptions of Lemma 3.1, yielding to
$$\{v_{t}<t_{0}\}\in\tilde{A}D_{t_{0}}(\Gamma\pm\theta_{0}t_{0}^{k\delta})$$
with
(3.17)
$${}|\tilde{A}-I|\leq C\theta_{0}t_{0}^{(k-1)\delta}.$$
Thus
$$S_{t_{0}t}\in FA_{k}D_{t}\tilde{A}D_{t_{0}}(\Gamma\pm\theta_{0}t_{0}^{k\delta}).$$
Defining $E_{k}$ such that
$$D_{t}\tilde{A}=E_{k}D_{t},$$
we see from (3.17) that $E_{k}$ satisfies (3.15). We
conclude the proof of
the induction step by first choosing $\theta_{0}$ small so
that (3.12) and (3.16) imply that (3.14) is always
satisfied.
Denote by
$$A^{*}:=\lim_{k\to\infty}A_{k}.$$
We will prove next that
(3.18)
$${}S_{t}\in FA^{*}D_{t}(\Gamma\pm C^{\prime}t^{\delta}).$$
As before, let $t=t_{1}t_{0}^{k}$. Notice that
$$A^{*}=A_{k}E_{k}^{*},\quad E_{k}^{*}:=\Pi_{i=k}^{\infty}E_{i}$$
and it is straightforward to
check from (3.15) that
(3.19)
$${}|e^{*}_{k,ii}-1|\leq C_{1}t^{\delta},\quad|e^{*}_{k,12}|\,t^{-\frac{\alpha}{%
2(2+\alpha)}}\leq C_{1}t^{\delta}.$$
We have
$$A_{k}D_{t}=A^{*}(E_{k}^{*})^{-1}D_{t}=A^{*}D_{t}\tilde{E}$$
with
$$|\tilde{e}_{k,ii}-1|\leq C_{2}t^{\delta},\quad|\tilde{e}_{k,12}|\leq C_{2}t^{%
\delta}.$$
Now (3.18) follows since
$$\tilde{E}(\Gamma\pm C_{2}t^{\delta})\subset\Gamma\pm C^{\prime}t^{\delta}.$$
Finally, from (3.18) we see that
$$u(FA^{*}x)=|x_{1}|^{2+\alpha}+x_{2}^{2}+O((|x_{1}|^{2+\alpha}+x_{2}^{2})^{1+%
\delta})$$
which implies that in a $z$-system of coordinates
$$u(z)=\beta_{1}|z_{1}|^{2+\alpha}+\beta_{2}\,z_{2}^{2}+O((|z_{1}|^{2+\alpha}+z_%
{2}^{2})^{1+\delta}).$$
The rescaled functions
$$r^{-1}u(r^{\frac{1}{2+\alpha}}z_{1},r^{\frac{1}{2}}z_{2})$$
converge, as $r\to 0$, to
$$\tilde{u}(z):=\beta_{1}|z_{1}|^{2+\alpha}+\beta_{2}z_{2}^{2}.$$
Moreover, this function solves the limiting equation
$$\det D^{2}\tilde{u}=|z_{1}|^{\alpha}.$$
Hence
$$2(2+\alpha)(1+\alpha)\beta_{1}\beta_{2}=1$$
which implies (3.13).
∎
4. Negative powers
In this section we consider the equation
(4.1)
$${}\det D^{2}u=|x|^{\alpha},\qquad\mbox{in}\ \Omega\subset\mathbb{R}^{2}$$
in the negative range of exponents $-2<\alpha<0$.
We will assume, throughout the section, that $0\in\Omega$ and
$$u(0)=0,\,\,\nabla u(0)=0.$$
Our goal is to prove the following proposition, which shows that solutions of equation (4.1) admit only the radial behavior near the origin. This is in contrast
with the case $0<\alpha<\infty$, where both the radial behavior and
the non-radial behavior (3.13) occur (see Proposition 3.3).
Proposition 4.1.
There exist positive constants $c$, $C$ (depending on $u$) such that
$$c\,|x|^{2+\alpha/2}\leq u(x)\leq C\,|x|^{2+\alpha/2}$$
near the origin.
We distinguish two cases depending on whether or not the measure
$|x|^{\alpha}dx$ is doubling with respect to all ellipsoids (see
the discussion in Section 2).
i. The case $-1<\alpha<0:$ In this case the measure
$$\mu:=|x|^{\alpha}dx$$
is doubling with respect to ellipsoids. Indeed, it suffices to show that there exists $c>0$ such that for any ellipsoid $E$, we have
(4.2)
$${}\mu(x_{0}+E)\geq c\,\mu(x_{0}+2E).$$
Since $-1<\alpha$, the density
$$(x_{1}^{2}+x_{2}^{2})^{\alpha/2}$$
is doubling on each line $x_{2}=const.$ with the doubling constant
independent of $x_{2}$. This implies that the density $\mu=|x|^{\alpha}\,dx$ is
doubling with respect to any line in the plane. From this and the
fact that $x_{0}+2E$ can be covered with translates of $x_{0}+E/2$
over a finite number of directions we obtain (4.2).
From Theorem 2.2, there exists a matrix $A_{t}$ such that
$S_{t}\sim A_{t}$, i.e
(4.3)
$${}k_{0}^{-1}A_{t}B_{r}\subset S_{t}\subset k_{0}A_{t}B_{r},$$
with
$$r=t\,(\mu(S_{t}))^{-1/2},\quad\quad\det A_{t}=1.$$
In this case Proposition 4.1 follows from the lemma below.
Lemma 4.2.
There exist universal constants $C>0$ large and $\delta>0$,
such that if $S_{t}\sim A_{t}$ with $|A_{t}|>C$, then
(4.4)
$$S_{\delta t}\sim A_{\delta t},\qquad\mbox{ with\, $|A_{\delta t}|\leq|A_{t}|/2%
$}.$$
In particular, $|A_{t}|\leq C|A_{t_{0}}|$, if $t\leq t_{0}$.
Proof.
We will use a compactness argument.
Assume, by contradiction, that the conclusion of the lemma is not
true. Then we can find a sequence of solutions $u_{k}$ of
(4.1) with sections $S^{u_{k}}_{t_{k}}$ at $0$ such that
$S_{t_{k}}^{u_{k}}\sim A_{t_{k}}^{u_{k}}$ with $|A^{u_{k}}_{t_{k}}|\to\infty$ and (4.4) does not hold for any $\delta>0$.
Without loss of generality we may assume that
(4.5)
$${}A^{u_{k}}_{t_{k}}:=\begin{pmatrix}a_{k}&0\\
0&a_{k}^{-1}\end{pmatrix},\quad a_{k}\to\infty.$$
We renormalize the functions $u_{k}$ as
(4.6)
$$v_{k}(x):=\frac{1}{t_{k}}u_{k}(r_{k}A^{u_{k}}_{t_{k}}x)$$
so that
$$\det D^{2}v_{k}=c_{k}\,|A^{u_{k}}_{t_{k}}x|^{\alpha}=c_{k}^{\prime}\,|x_{1}^{2%
}+a_{k}^{-4}x_{2}^{2}|^{\alpha/2}$$
and
$$k_{0}^{-1}B_{1}\subset S^{v_{k}}_{1}\subset k_{0}B_{1}.$$
Since, $S_{t_{k}}^{u_{k}}\sim A_{t_{k}}^{u_{k}}$, i.e in particular $r_{k}=t_{k}\,(\mu(S_{t_{k}}^{u_{k}}))^{-1/2}$, the Monge Ampére measure $\det D^{2}v_{k}\,dx$
satisfies
$$\det D^{2}v_{k}(S^{v_{k}}_{1})=r_{k}^{2}t_{k}^{-2}\mu(S^{u_{k}}_{t_{k}})=1.$$
Hence, as $k\to\infty$ we can find a subsequence of the $v_{k}$’s that
converge uniformly to a function $v$ that satisfies
(4.7)
$${}\det D^{2}v=c\,|x_{1}|^{\alpha}dx$$
and
$$k_{0}^{-1}B_{1}\subset S^{v}_{1}\subset k_{0}B_{1},\quad\det D^{2}v_{k}(S^{v}_%
{1})=1.$$
Obviously, the constant $c$ in (4.7) is bounded from above and
below by universal constants. Since the right hand side of
(4.7) does not depend on $x_{2}$ and $v$ is constant on
$\partial S^{v}_{1}$, Pogorelov’s interior estimate holds and we
obtain the bound
$$v_{22}<C_{1},\qquad\mbox{in $(2k_{0})^{-1}B_{1}$}.$$
This implies that the section $S^{v}_{\delta}$ contains a segment
of size $\delta^{1/2}$ in the $x_{2}$ direction, namely
(4.8)
$${}\{x_{1}=0,\,|x_{2}|\leq(\delta/C_{1})^{1/2}\}\subset S^{v}_{\delta}.$$
From Theorem 2.2 there exists
(4.9)
$${}A_{\delta}=\begin{pmatrix}a&0\\
b&a^{-1}\end{pmatrix},\quad 0<a<C(\delta),\quad|b|\leq C(\delta)$$
with
(4.10)
$${}k_{0}^{-1}A_{\delta}B_{r}\subset S^{v}_{\delta}\subset k_{0}A_{\delta}B_{r}$$
and
(4.11)
$${}r=\delta\,[\det D^{2}v(S^{v}_{\delta})]^{-1/2}.$$
From (4.8) and (4.10) we have
$$\frac{r}{a}\geq c_{1}\,\delta^{1/2}$$
while from (4.9), (4.10) and (4.11) we get
$$\delta^{2}=r^{2}\,\det D^{2}v(S_{\delta}^{v})\geq c_{2}\,r^{2}\,\frac{r}{a}(ar%
)^{1+\alpha}.$$
From the last two inequalities we obtain
(4.12)
$${}a\leq C_{2}\delta^{\frac{-\alpha}{4(2+\alpha)}}\leq 1/4\quad\mbox{for
$\delta$ small universal}.$$
Since the $v_{k}$’s converge uniformly to $v$, their $\delta$
sections also converge uniformly, thus
$$S^{v_{k}}_{\delta}\sim A_{\delta},\qquad\mbox{for $k$ large}$$
and hence
$$S^{u_{k}}_{\delta t_{k}}\sim A^{u_{k}}_{t_{k}}A_{\delta}.$$
From (4.5), (4.9), (4.12) we conclude
$$|A^{u_{k}}_{t_{k}}A_{\delta}|\leq|A^{u_{k}}_{t_{k}}|/3\quad\mbox{for $k$ large,}$$
which implies that the function $u_{k}$ satisfies (4.4) , a contradiction.
∎
ii. The case $-2<\alpha\leq-1$: In this case the measure $\mu$ is not doubling with
respect to any convex set but it is still doubling with respect to
convex sets that have the origin as the center of mass.
We proceed as in the first case but replacing the sections $S_{t}$
with the sections $T_{t}$ that have $0$ as the center of mass. The
existence of these sections follows from the following lemma due
to L. Caffarelli, Lemma 2 in [3].
Lemma 4.3 (Centered sections).
Let $u:\mathbb{R}^{n}\to\mathbb{R}\cup\{\infty\}$ be a globally defined
convex function (we set $u=\infty$ outside $\Omega$). Also, assume
$u$ is bounded in a neighborhood of $0$ and the graph of $u$ does
not contain an entire line.
Then, for each $t>0$, there exists a “$t-$ section”
$T_{t}$ centered at $0$,
that is there exists $p_{t}$ such that the convex set
$$T_{t}:=\{\,u(x)<u(0)+p_{t}\cdot x+t\,\}$$
is bounded and has $0$ as center of mass.
Using the lemma above one can obtain Theorem 2.2
(similarly as in [2]), with $S_{t}$ is
replaced by $T_{t}$: for every $T_{t}\subset\Omega$ as above, there exists a unitary matrix $A_{t}$, such that
(4.13)
$${}k_{0}^{-1}A_{t}B_{r}\subset T_{t}\subset k_{0}A_{t}B_{r}$$
with
$$r=t\,(\mu(T_{t}))^{-1/2}.$$
If $(\ref{3.10})$ is satisfied we write $T_{t}\sim A_{t}.$
We will next show the analogue of Lemma 4.2 for this case.
Lemma 4.4.
There exist universal constants $C>0$ large and $\delta>0$,
such that if $T_{t}\sim A_{t}$ with $|A_{t}|>C$, then $T_{\delta t}\subset T_{t}$ and
(4.14)
$$T_{\delta t}\sim A_{\delta t},\qquad\mbox{ with $|A_{\delta t}|\leq|A_{t}|/2$}.$$
Proof.
We argue similarly as in the proof of Lemma 4.2. We assume by
contradiction that the conclusion does not hold for a sequence of
functions $u_{k}$. Proceeding as in the
proof of lemma 4.2, we work with the renormalizations $v_{k}$ of
$u_{k}$ defined by (4.6) which satisfy
$$\det D^{2}v_{k}=c_{k}^{\prime}\,|x_{1}^{2}+a_{k}^{-4}x_{2}^{2}|^{\alpha/2}=:%
\mu_{k}$$
and
$$k_{0}^{-1}B_{1}\subset T^{v_{k}}_{1}\subset k_{0}B_{1},\quad\det D^{2}v_{k}(T^%
{v_{k}}_{1})=1.$$
As $k\to\infty$, we can find a subsequence of the $v_{k}$’s which
converges uniformly to a function $v$. Since $a_{k}\to\infty$ and $-2<\alpha\leq-1$, the corresponding measures $\mu_{k}$, when
restricted to a line $x_{2}=const.$, converge weakly to the measure
$c\,|x_{2}|^{1+\alpha}\delta_{\{x_{1}=0\}}$. This implies that the
measures $\mu_{k}$ converge weakly to $c\,|x_{2}|^{1+\alpha}d\mathcal{H}^{1}_{\{x_{1}=0\}},$ where $d\mathcal{H}^{1}$
is the 1 dimensional Hausdorff measure.
Hence, the limit function $v$ satisfies
(4.15)
$${}\det D^{2}v=c\,|x_{2}|^{1+\alpha}\,d\mathcal{H}^{1}_{\{x_{1}=0\}}$$
$$k_{0}^{-1}B_{1}\subset T^{v}_{1}\subset k_{0}B_{1},\qquad\det D^{2}v_{k}(T^{v}%
_{1})=1.$$
Clearly $c$ is bounded from above and below by universal constants.
We notice that the measure $d\mathcal{H}^{1}_{\{x_{1}=0\}}$ is doubling with
respect to any convex set with the center of mass on the line $\{x_{1}=0\}$.
Using the same methods as in the case of classical Monge-Ampére equation
one can show that the graph of $v$ contains no line segments when
restricted to
$\{x_{1}=0\}$ (see the Lemma 4.5 below). From this and the fact that $v$ is the
convex envelope of its restriction on $\partial T_{1}^{v}$ and $\{x_{1}=0\}$
(see (4.15)) we conclude that there exist two supporting
planes with slopes $\beta e_{2}\pm\gamma e_{1}$ to the graph of $v$ at $0$. Moreover,
it follows from the compactness of the equation (4.15) that $\gamma$ can be
chosen universal, and the sections $T^{v}_{\delta}$ satisfy
$$T^{v}_{\delta}\subset(2k_{0})^{-1}B_{1}$$
when $\delta\leq\delta_{0}$, a universal constant. We have
(4.16)
$${}T^{v}_{\delta}\subset\{|x_{1}|\leq c(\gamma)\delta\}.$$
Let $A_{\delta}$ be of the form (4.9) with
$${}k_{0}^{-1}A_{\delta}B_{r}\subset T^{v}_{\delta}\subset k_{0}A_{\delta}B_{r}$$
and
(4.17)
$${}r=\delta\,[\det D^{2}v(T^{v}_{\delta})]^{-1/2}\sim\delta(r/a)^{1+\alpha/2}.$$
On the other hand (4.16) implies
$$a\,r\leq c\delta$$
which together with (4.17) yields
$$a\leq c\,\delta^{\frac{2+\alpha}{6+\alpha}}\leq 1/4$$
for $\delta$ small enough. Now the contradiction follows as in
Lemma 4.2.
∎
Lemma 4.5.
If $v$ satisfies $(\ref{3.11})$, then
$$v_{0}(t):=v(0,t)$$
is strictly convex.
Proof.
Assume that the conclusion does not hold. Then,
after subtracting a linear function, we can assume that
$$v\geq 0\qquad\mbox{in $T^{v}_{1}$}$$
and
$$\quad v_{0}(t)=0\quad\mbox{for $t\leq 0$},\quad\quad v_{0}(t)>0\quad\mbox{for %
$t>0$}.$$
Let
$$l_{\varepsilon}:=\varepsilon t+a_{\varepsilon}$$
be such that
(4.18)
$${}0\in\{v_{0}<l_{\varepsilon}\}=(b_{\varepsilon},c_{\varepsilon})\rightarrow 0%
,\qquad\frac{c_{\varepsilon}}{|b_{\varepsilon}|}\to 0\quad\mbox{as $%
\varepsilon\to 0$}.$$
We consider the linear function $p_{\varepsilon}$ in $\mathbb{R}^{2}$ such that $\{u<p_{\varepsilon}\}$ has center of mass on $\{x_{1}=0\}$ and $p_{\varepsilon}=l_{\varepsilon}$ on
$\{x_{1}=0\}$.
We claim that for $\varepsilon$ small, $\{u<p_{\varepsilon}\}$ is compactly included in
$T_{1}^{v}$. Otherwise, the graph of $v$ would contain a segment passing
through $0$, hence $v=0$ in an open set which intersects the line $\{x_{1}=0\}$ and we contradict (4.15).
Since $d\mathcal{H}^{1}_{\{x_{1}=0\}}$ is doubling with
respect to the center of mass of $\{u<p_{\varepsilon}\}$, we conclude that this
set is also balanced around $0$ which contradicts (4.18).
∎
We are now in the position to exhibit the final steps of the proof
of Proposition 4.1 in the case $-2<\alpha\leq-1$.
Proof of Proposition 4.1:
We choose $t_{0}$ small, such that
$$T_{t_{0}}\subset\Omega.$$
The existence of $t_{0}$ follows from the fact that the graph of $u$
cannot contain any line segments.
From Lemma 4.4 we conclude that there exists a large
constant $K>0$ depending on the eccentricity of $T_{t_{0}}$ such
that
$$T_{t}\sim A_{t},\qquad\mbox{with $|A_{t}|\leq K$
for all $t\leq\delta t_{0}$}.$$
Claim: There exists $\gamma$ depending on $K$ such that $S_{\gamma t}\subset T_{t}.$
To show this, first observe that by rescaling we can
assume that $t=1$. We use the compactness of the problem for
fixed $K$. If there exist a sequence $\gamma_{k}\to 0$ and
functions $u_{k}$ for which the conclusion does not hold then, the
graph of the limiting function $u_{\infty}$ (of a subsequence of
$\{u_{k}\}$) contains a line segment. This is a contradiction since
$u_{\infty}$ solves the Monge-Ampére equation (4.1), which
proves the claim.
If $t=1$, then from simple geometrical considerations and the claim
above we obtain
$$\gamma k_{0}^{-1}K^{-1}B_{1}\subset S_{\gamma}\subset k_{0}KB_{1}.$$
By rescaling, we find that $S_{t}$ has bounded
eccentricity for $t$ small, and the proposition is proved.
∎
5. Homogenous solutions and blowup limits
We will consider in this section homogenous solutions of the equation
$$\det D^{2}w(x)=|x|^{\alpha}\qquad\mbox{in
$\mathbb{R}^{2}$}$$
for $\alpha>-2$, namely solutions of the form
$$w(x)=r^{2+\alpha/2}g(\theta):=r^{\beta}g,\qquad\beta=2+\alpha/2.$$
In the polar system of coordinates
$$D^{2}w(x)=r^{\beta-2}\begin{pmatrix}\beta(\beta-1)g&(\beta-1)g^{\prime}\\
(\beta-1)g^{\prime}&g^{\prime\prime}+\beta g\end{pmatrix}.$$
Thus, the function $g$ satisfies the following ODE
(5.1)
$${}\beta g(g^{\prime\prime}+\beta g)-(\beta-1)(g^{\prime})^{2}=1/(\beta-1).$$
We consider $g$ as the new variable in a maximal interval $[a,b]$ where
$g$ is increasing, and define $h$ on $[g(a),g(b)]$ as
$$g^{\prime}=\sqrt{2h(g)}.$$
We have
$$g^{\prime\prime}=h^{\prime}(g)$$
thus $h$ satisfies
$$\beta t\,(h^{\prime}(t)+\beta t)-2(\beta-1)\,h(t)=1/(\beta-1).$$
Solving for $h$ we obtain
(5.2)
$${}2\,h_{c}(t)=c\,t^{2(1-\frac{1}{\beta})}-\beta^{2}t^{2}-\frac{1}{(\beta-1)^{2}}$$
for some $c$ positive.
The function $g$ on $[a,b]$ is the inverse of
$$a+\int_{g(a)}^{\xi}\frac{1}{\sqrt{2h_{c}(t)}}dt$$
and the length of the interval $[a,b]$ is given by
(5.3)
$${}b-a=\int_{\{h_{c}>0\}}\frac{1}{\sqrt{2h_{c}(t)}}dt:=I_{c}.$$
Solutions of (5.1) are periodic, of period $2(b-a)$, thus
a global solution $g$ on the circle exists if and only if $I_{c}$ equals
$\pi/k$, for some integer $k$. Next we investigate the existence of such
solutions.
First we notice that for any quadratic polynomial $f(s)=-l^{2}\,s^{2}+d_{1}\,s+d_{2}$ of opening $-2\,l^{2}$, we have
(5.4)
$${}\int_{\{f>0\}}\frac{1}{\sqrt{f(s)}}ds=\frac{\pi}{l}.$$
Therefore if $\phi(s)$ denotes any convex function which intersects the parabola
$l^{2}s^{2}$ at two points, and we set $f(s)=-l^{2}s^{2}+d_{1}s+d_{2}$, with
$d_{1}s+d_{2}$ denoting the
line through the intersection points between $\phi(s)$ and $l^{2}s^{2}$, then
$$\int_{\{\phi(s)-l^{2}s^{2}>0\}}\frac{1}{\sqrt{\phi(s)-l^{2}s^{2}}}\,ds\geq\int%
_{\{f>0\}}\frac{1}{\sqrt{f(s)}}\,ds=\frac{\pi}{l}.$$
If $\phi(s)$ is concave we obtain the opposite inequality.
Applying the above to $h_{c}(s)$, we find that depending on the convexity of the first term in (5.2), we
obtain that the integral $I_{c}$ in (5.3) is less (or greater) than $\pi/\beta$
for $\beta<2$ (or $\beta>2$), i.e.,
(5.5)
$$I_{c}<\frac{\pi}{\beta},\quad\mbox{if}\,\,\beta<2\qquad\mbox{and}\qquad I_{c}>%
\frac{\pi}{\beta},\quad\mbox{if}\,\,\beta>2.$$
On the other hand, by performing the change of variable
$$t=s^{\frac{\beta}{2}}$$
in the integral (5.3) we obtain the integral (5.4) with
$$f(s):=c_{1}s-4s^{2}-c_{2}s^{2-\beta}$$
for some positive constants $c_{1}$, $c_{2}$ depending on $c$. Hence,
depending on
the convexity of the last term of $f$, the integral $I_{c}$ is greater (or less) than $\pi/2$
for $\beta<2$ (or $\beta>2$), i.e.,
(5.6)
$$I_{c}>\frac{\pi}{2},\quad\mbox{if}\,\,\beta<2\qquad\mbox{and}\qquad I_{c}<%
\frac{\pi}{2}\quad\mbox{if}\,\,\beta>2.$$
Let $-2<\alpha<0$, or equivalently $1<\beta<2$. It follows from (5.5) and (5.6) that $\pi/2<I_{c}<\pi/\beta$, hence $I_{c}=\pi/k$, for
an integral $k$ only when $k=1$. This readily implies that the only homogeneous solution
in this case is the radial one.
Assume next that $\alpha>0$. We will show next that in this case, depending on the value
of $\beta$, more homogeneous solutions may exist.
To this end, denote by $c_{0}=c_{0}(\alpha)$ the value of $c$ for which the two
functions
$$f_{1}(t)=c\,t^{2(1-\frac{1}{\beta})}\quad\mbox{and}\quad f_{2}(t)=\beta^{2}t^{%
2}+\frac{1}{(\beta-1)^{2}}$$
become tangent. When $c<c_{0}$, then the set were $h_{c}(t)>0$
is empty. As $c\to c_{0}^{+}$ the set $\{t:\,h_{c}(t)>0\}$ approaches the point
$t_{0}$ at which the two functions $f_{1}(t)$ and $f_{2}(t)$ become tangent when $c=c_{0}$. Since $f^{\prime}_{1}(t_{0})=f^{\prime}_{2}(t_{0})$ when $c=c_{0}$, the point $t_{0}$ satisfies
$$2c\,(1-\frac{1}{\beta})\,t_{0}^{1-\frac{2}{\beta}}=2\beta^{2}t_{0}$$
which implies that
$$c\,(1-\frac{1}{\beta})\,t_{0}^{-\frac{2}{\beta}}=\beta^{2}.$$
As $c\to c_{0}^{+}$, $f_{1}(t)$ behaves as its Taylor quadratic
polynomial, namely
$$f_{1}(t)\approx f(t_{0})+f^{\prime}(t_{0})\,t^{2}+\frac{f^{\prime\prime}(t_{0}%
)}{2}\,t^{2}$$
and
$$\frac{f^{\prime\prime}(t_{0})}{2}=c\,(1-\frac{1}{\beta})\,(1-\frac{2}{\beta})%
\,t_{0}^{-\frac{2}{\beta}}=\beta^{2}(1-\frac{2}{\beta}).$$
We conclude that, as $c\to c_{0}^{+}$, $(h_{c})^{+}$ behaves as a quadratic polynomial
of opening $-4\beta$, and thus $I_{c}$ converges to
$\pi/\sqrt{2\beta}$. Hence, $(\frac{\pi}{\sqrt{2\beta}},\frac{\pi}{2})\subset\{I_{c},\,\,c>c_{0}\}$ and also $\{I_{c},\,\,c>c_{0}\}\subset(\frac{\pi}{\beta},\frac{\pi}{2})$, by (5.5), (5.6).
Summarizing the discussion above yields:
Proposition 5.1.
Homogenous solutions to $(\ref{eq})$ are periodic on the unit
circle.
i. If $-2<\alpha<0$, then the only homogenous solution is the radial one.
ii. If $\alpha>0$, then there
exists a homogenous solution of principal period $2\pi/k$ if and only if
$$\frac{\pi}{k}\in\{I_{c},\,\,c>c_{0}(\alpha)\}.$$
In addition,
$$(\frac{\pi}{\sqrt{2\beta}},\frac{\pi}{2})\subset\{I_{c},\,\,c>c_{0}\}\subset(%
\frac{\pi}{\beta},\frac{\pi}{2})$$
with $\beta=2+\alpha/2$.
Using the proposition above, we will now prove Theorem 1.4. We begin
with two useful remarks.
Remark 5.2.
From (5.2) we see that any point in the positive quadrant can
be written as $(t,\sqrt{2h_{c}})$ for a suitable $c$. Hence, given
any point $x_{0}\in\partial B_{1}$ and any positive
symmetric unimodular matrix
$A$, there exists a homogenous solution $w$ in a neighborhood of
$x_{0}$ such that $D^{2}w(x_{0})=A$.
Remark 5.3.
Equation (5.2) gives
$$[(h^{\prime}+\beta t)+\beta(\beta-1)t]\,t^{\frac{2}{\beta}-1}=c(1-\frac{1}{%
\beta})$$
hence
$$\Delta w\,[r^{2}w_{rr}]^{\frac{2}{\beta}-1}$$
is constant for any local homogenous solution $w$. This
quantity will play a crucial role in the proof of Theorem 1.4.
Definition 5.4.
For any solution $u$ of equation (1.1), we define
$$J_{u}(x):=(\Delta u)(r^{2}u_{rr})^{\gamma},\qquad\gamma:=\frac{2}{\beta}-1.$$
Remark 5.5.
The quantity $J_{u}(x)$ remains invariant under the homogenous scaling
$$v(x)=r^{-\beta}u(rx),\quad J_{v}(x)=J_{u}(rx).$$
We denote by $J_{0}$ the constant obtained when we evaluate $J$ on the
radial solution $u_{0}$ of (1.1).
Proposition 5.6.
The function
$$|J_{u}-J_{0}|$$
cannot have an interior maximum in $\Omega\setminus\{0\}$ unless it is
constant.
Proof.
We compute the linearized operator $u^{ij}M_{ij}$ for
$$M=\log J_{u}=\log(\Delta u)+\gamma\log(x_{i}x_{j}u_{ij})$$
at a point
$x\in\Omega\setminus\{0\}$ where $J_{u}(x)\neq J_{0}$.
By choosing an appropriate system of coordinates and by rescaling,
we can assume that $|x|=1$ and $D^{2}u$ is diagonal. By differentiating the
equation (1.1) twice we obtain
(5.7)
$${}u^{ii}u_{kii}=\alpha x_{k}$$
and
$${}u^{ii}u_{klii}=u^{ii}u^{jj}u_{kij}u_{lij}+\alpha(\delta^{l}_{k}-2x_{k}x_{l}).$$
Since the linearized equation of each second derivative of $u$ depends
on $D^{3}u$, $D^{2}u$ and $x$ we see that
(5.8)
$$u^{ij}M_{ij}=H(D^{3}u,D^{2}u,x)$$
where $H$ is a quadratic polynomial in $D^{3}u$ for fixed $D^{2}u>0$ and
$x$.
Let $w$ denote the (local) homogenous solution for which
$D^{2}u(x)=D^{2}w(x)$.
Since $M_{w}=\log J_{w}$ is constant, we have
$$H(D^{3}w,D^{2}w,\cdot)=0$$
in a neighborhood of $x$.
Claim. We have
$$\|D^{3}u(x)-D^{3}w(x)\|\leq C|\nabla M|$$
with the constant $C$ depending on $D^{2}u$ and $x$.
Proof of Claim. From (5.7) and the following equalities
$$M_{k}=\frac{u_{iik}}{\Delta u}+\gamma\frac{x_{i}x_{j}u_{ijk}+2x_{i}u_{ik}}{x_{%
i}x_{j}u_{ij}}$$
we obtain the following system for the third
derivatives of $u$,
$$\begin{pmatrix}\par 1&0&1&0\\
\par 0&1&0&1\\
\par b_{1}&d_{1}&b_{2}&0\\
\par 0&b_{1}&d_{2}&b_{2}\end{pmatrix}\begin{pmatrix}\frac{u_{111}}{u_{11}}\\
\par\frac{u_{112}}{u_{11}}\\
\par\frac{u_{221}}{u_{22}}\\
\par\frac{u_{222}}{u_{22}}\end{pmatrix}=\begin{pmatrix}\alpha x_{1}\\
\par\alpha x_{2}\\
\par M_{1}-2\gamma\frac{x_{1}u_{11}}{u_{rr}}\\
\par M_{2}-2\gamma\frac{x_{2}u_{22}}{u_{rr}}\par\end{pmatrix}$$
and
$$b_{i}=\frac{u_{ii}}{\Delta u}+\gamma\frac{x_{i}^{2}u_{ii}}{u_{rr}},\quad\quad d%
_{i}=2\gamma u_{ii}\frac{x_{1}x_{2}}{u_{rr}}.$$
The third order derivatives of $w$ solve the same system but with no dependence
on $M$ in the right hand
side vector (since the corresponding $M$ for $w$ is constant).
It is enough to show that the determinant of the third order derivatives coefficient matrix above is positive. This determinant is equal to
$$d_{1}d_{2}+(b_{1}-b_{2})^{2}=4\gamma^{2}(\frac{x_{1}x_{2}}{u_{rr}})^{2}+(b_{1}%
-b_{2})^{2}$$
and can vanish only if one of the coordinates, say $x_{2}=0$, and $b_{1}=b_{2}$,
i.e.
$$u^{2}_{11}=\frac{1-\gamma}{1+\gamma}=\beta-1.$$
This implies that $J(x)=J_{0}$ which is a contradiction. Thus, the
determinant is positive and the claim is proved.
Since $H$ depends quadratically on $D^{3}u$ and $D^{2}u=D^{2}w$ at $x$,
the claim above implies that
$$\begin{split}\displaystyle|H(D^{3}u,D^{2}u,x)|&\displaystyle=|H(D^{3}u,D^{2}u,%
x)-H(D^{3}w,D^{2}u,x)|\\
&\displaystyle\leq C(x,D^{2}u)(|\nabla M|+|\nabla M|^{2}).\end{split}$$
Hence, (5.8) implies that on the set where $J(x)\neq J_{0}$ there exists a smooth function
$C(x)$ depending on $u$ such that
$$|u^{ij}M_{ij}|\leq C(x)(|\nabla M|+|\nabla M|^{2}).$$
From the strong maximum principle, we conclude that $M$ cannot
have a local maximum or minimum in this set unless it is constant.
With this the Proposition is proved.
∎
Theorem 1.4 will follow from the proposition below.
Proposition 5.7.
Suppose that $u$ is a solution $u$ of (1.1), with $\alpha>-2$, which satisfies
(5.9)
$${}c|x|^{\beta}\leq u(x)\leq C|x|^{\beta},\qquad\beta=2+\alpha/2.$$
Then the limit
$$J_{u}(0):=\lim_{x\to 0}J_{u}(x)$$
exists. Moreover, if for a sequence of $r_{k}\to 0$ the blow up solutions
$$v_{r_{k}}:=r_{k}^{-\beta}u(r_{k}x)$$
converge uniformly on compact sets to the solution $w$, then $w$ is homogenous of degree $\beta$ with $J_{w}=J_{u}(0)$.
Proof.
From (5.9) we find that as $x\to 0$, $J_{u}(x)$ is
bounded away from
$0$ and $\infty$ by constants depending on $c$, $C$.
We will first show that $\lim_{x\to 0}J_{u}(x)=J(0)$ exists.
We may assume, without loss of
generality, that
$$\limsup_{x\to 0}J_{u}(x):=k>J_{0}.$$
Let $x_{i}$ be a sequence of points for which $\limsup$ is achieved. The
blow up solutions $v_{r_{i}}$, $r_{i}=|x_{i}|$, have a subsequence which converges uniformly
on compact sets of $\mathbb{R}^{2}$ to a solution $v$. Moreover, there
exists a point $y$ on the unit circle for which
$$J_{v}(y)=k\geq\limsup_{x\to 0}J_{v},$$
hence, by Proposition 5.6, $J_{v}$ is constant.
This argument also shows that if
$$J_{u}(z)\leq k-\varepsilon\quad\mbox{then $J_{u}(x)\leq k-\delta(\varepsilon)$%
on the circle $|x|=|z|$}.$$
Thus, if there exists a sequence of points $y_{j}\to 0$ with
$$\lim_{y_{j}\to 0}J_{u}(y_{j})<k$$
then $J_{u}$ would have an interior maximum in the annulus $\{x:\,|y_{j}|\leq|x|\leq|y_{j^{\prime}}|\}$ that contains one of the points $x_{i}$ given above, a contradiction.
This shows that $\lim_{x\to 0}J_{u}(x)$ exists.
It remains to prove that if $J_{v}$ is constant, then $v$ is
homogenous. It suffices to show that $D^{2}v$ is homogenous of degree
$\beta-2$, or more precisely that for each second derivative $v_{ij}$, we have
(5.10)
$$x\cdot\nabla v_{ij}=(\beta-2)\,v_{ij}.$$
To this end, for a fixed point $x$, we
consider the homogenous solution $w$ with $D^{2}w(x)=D^{2}v(x)$. Since
$$\nabla J_{v}(x)=\nabla J_{w}(x)=0$$
the third derivatives of $v$ and $w$
solve the same system. We have seen in the proof of Proposition 5.6 that this system is solvable provided $J_{v}\neq J_{0}$.
Thus $D^{3}v(x)=D^{3}w(x)$ if $J_{v}\neq J_{0}$. Since (5.10)
is obviously true for $w$, this implies that the equality holds for $u$ as well.
If $J_{v}=J_{0}$ we denote by $\Gamma$ the set where $D^{2}u(x)$ does
not coincide with the hessian of the radial solution. From the
proof of Proposition 5.6 we still obtain $D^{3}v(x)=D^{3}w(x)$
if $x\in\Gamma$, and by continuity (5.10)
holds for $x\in\bar{\Gamma}$. If $x$ is in the open set $\bar{\Gamma}^{c}$, then $D^{2}v$ coincides with $D^{2}u_{0}$ and (5.10) is
again satisfied. This finishes the proof of the proposition.
∎
Proof of Theorem 1.4. The proof of the theorem readily follows from Propositions 4.1, 5.1 and 5.7. ∎
6. Proof of Theorem 1.3
We consider the Dirichlet problem
(6.1)
$$\begin{cases}\det D^{2}u=|x|^{\alpha}&\text{in $B_{1}$}\\
u=u_{0}-\varepsilon\cos(2\theta)&\text{on $\partial B_{1}$}\end{cases}$$
in the range of exponents $\alpha>0$. Here
$$u_{0}(x)=c_{\alpha}\,|x|^{\beta},\qquad\beta=2+\alpha/2$$
denotes the radial solution of the equation, i.e, $\det D^{2}u_{0}=|x|^{\alpha}$.
We write the solution as
(6.2)
$$u=u_{0}-\varepsilon v.$$
Heuristically, is $\epsilon$ is small $v$ satisfies the linearized equation
at $u_{0}$, namely
$$(D^{2}u_{0})^{-1}:D^{2}v=0,$$
where we use the notation $A:B=\sum_{ij}a_{ij}\,b_{ij}$ for
the Frobenius inner product between two $n\times n$ matrices $A$
and $B$.
At any point $x_{0}\in B_{1}$, we denote by $\nu$ and $\tau$ the unit
normal (radial) and unit tangential direction, respectively, to
the circle $|x|=|x_{0}|$ at $x_{0}$. In $(\nu,\tau)$ coordinates,
$$D^{2}u_{0}=c_{\alpha}r^{\beta-2}\begin{pmatrix}\beta(\beta-1)&0\\
0&\beta\end{pmatrix}$$
hence, $v$ satisfies
the equation
$$v_{\nu\nu}+(\beta-1)v_{\tau\tau}=0.$$
Solving this equation with boundary data $v=\cos(2\theta)$ we obtain
the solution
$$v=r^{\rho}\cos(2\theta)$$
with
$$\rho(\rho-1)+(\beta-1)(\rho-4)=0.$$
Solving the quadratic equation with respect to $\rho$
gives
$$\rho=\frac{2-\beta\pm\sqrt{\beta^{2}+12\,\beta-12}}{2}.$$
Since $\beta:=2+\alpha/2>2$ the only acceptable solution is
$$\rho=\frac{2-\beta+\sqrt{\beta^{2}+12\,\beta-12}}{2}$$
and it satisfies
(6.3)
$$2<\rho<\beta$$
which suggests that close to the origin the perturbation term
$\varepsilon v$ dominates $u_{0}$.
We wish to show that the solution $u$ of the Dirichlet problem
(6.1) admits at the origin the non-radial behavior
(1.6), if $\varepsilon\leq\varepsilon_{0}$, with $\varepsilon_{0}$ sufficiently
small. We will argue by contradiction. Assume,
that $u$ has the radial behavior
$$c_{0}\,|x|^{\beta}\leq u(x)\leq C_{0}\,|x|^{\beta}$$
with $c_{0}$, $C_{0}$ universal constants. By rescaling, we deduce that
$$c\,I\leq|x|^{2-\beta}D^{2}u(x)\leq C\,I$$
with $I$ denoting the identity matrix.
The function $v$ which is defined by (6.2) satisfies
$$|v|\leq 1,\qquad v=\cos(2\theta)\mbox{ on $\partial B_{1}$}$$
and solves the equation
$$a^{ij}v_{ij}=0$$
with
$$A=(a^{ij})=\int_{0}^{1}(tD^{2}u_{0}+(1-t)D^{2}u)^{-1}dt=\int_{0}^{1}(D^{2}u_{0%
}+\varepsilon(t-1)D^{2}v)^{-1}dt.$$
Hence
(6.4)
$$c\,I\leq r^{\beta-2}A\leq C\,I.$$
The solution $u$ has bounded third order derivatives in $B_{1}\setminus B_{1/2}$, thus
$$|D^{2}v(x)|\leq C\,\|v\|_{L^{\infty}}\leq C\qquad\mbox{
in $B_{1}\setminus B_{1/2}$}.$$
By rescaling we obtain the bound
$$|D^{2}v(x)|\leq C|x|^{-2}.$$
From this we find that
$$r^{\beta-2}|A-D^{2}u_{0}^{-1}|\leq C\varepsilon r^{-\beta}$$
hence, $v$ satisfies the Dirichlet problem
(6.5)
$$\begin{cases}f^{ij}v_{ij}=0&\text{in $B_{1}$}\\
v=\cos(2\theta)&\text{on $\partial B_{1}$}\end{cases}$$
with
$$F:=c\,r^{\beta-2}\,A$$
hence, by (6.4),
$$c\,I\leq F\leq C\,I\quad\mbox{and}\quad|F-F_{0}|\leq C\varepsilon r^{-\beta}$$
with
$$F_{0}:=\nu\otimes\nu+(\beta-1)\,\tau\otimes\tau.$$
(As before, we denote by $\nu$ and $\tau$ the unit normal (radial) and unit tangential directions, to the circle $|x|=|x_{0}|$ at each point $x_{0}\in B_{1}$).
Also,
$$|v|\leq 1,\quad\mbox{ on $B_{1}$}.$$
From the definitions of $A$ and $F$ we also obtain
(6.6)
$${}\|\nabla(F-F_{0})\|\leq C(r_{0})\,\varepsilon\qquad\mbox{ for $|x|\geq r_{0}%
$.}$$
Set
$$w:=r^{\rho}\cos(2\theta).$$
Then, $w$ satisfies the equation
$$F_{0}:D^{2}w=0$$
thus, we have
$$|f^{ij}\,w_{ij}|\leq Cr^{\rho-2}\min\{\varepsilon r^{-\beta},1\}.$$
Applying the Aleksandrov maximum principle on $v-w$ (see Theorem 9.1 in
[4]), we find that
$$|v-w|\leq C\,\varepsilon^{\delta}$$
and therefore (see (6.6))
(6.7)
$${}|D^{2}v-D^{2}w|\leq C^{\prime}(r_{0})\,\varepsilon^{\delta},\qquad\mbox{ for%
$|x|\geq r_{0}$}.$$
We next compute
$$M_{u}(x):=\log(\Delta u)+\gamma\log(r^{2}u_{rr}),\qquad\gamma:=\frac{2}{\beta}-1$$
in terms of $M_{u_{0}}$, for $|x|\geq r_{0}$, with $r_{0}$ small, fixed.
We recall that $M_{u_{0}}$ is constant in $x$. Since $u=u_{0}-\varepsilon\,v$,
we find that
$$M_{u}(x)=M_{u_{0}}-\varepsilon\left(\frac{\Delta v}{\Delta u_{0}}+\gamma\,%
\frac{v_{rr}}{u_{0,rr}}\right)-\frac{\varepsilon^{2}}{2}\left((\frac{\Delta v}%
{\Delta u_{0}})^{2}+\gamma\,(\frac{v_{rr}}{u_{0,rr}})^{2}\right)+O(\varepsilon%
^{3}).$$
Because
$$\det D^{2}u=\det D^{2}u_{0}$$
the function $v$ satisfies the equation
$$-u_{0,rr}\,v_{\tau\tau}-u_{0,\tau\tau}\,v_{rr}+\varepsilon\det D^{2}v=0$$
or equivalently (since $u_{0}(r)=c_{\alpha}\,r^{\beta}$)
$$v_{rr}+(\beta-1)\,v_{\tau\tau}=\varepsilon\,\frac{r^{2-\beta}}{c_{\alpha}\,%
\beta}\,\det D^{2}v.$$
The last equality implies that
$$\frac{\Delta v}{\Delta u_{0}}+\gamma\frac{v_{rr}}{u_{0,rr}}=\varepsilon\frac{r%
^{2(2-\beta)}}{c_{\alpha}^{2}\,\beta^{3}\,(\beta-1)}\det D^{2}v,$$
and also that
$$\begin{split}\displaystyle(\frac{\Delta v}{\Delta u_{0}})^{2}+\gamma\,(\frac{v%
_{rr}}{u_{0,rr}})^{2}&\displaystyle=(1+\frac{1}{\gamma})\,(\frac{\Delta v}{%
\Delta u_{0}})^{2}+O(\varepsilon)\\
&\displaystyle=-\frac{2r^{2(2-\beta)}}{c_{\alpha}^{2}\,\beta^{2}\,(\beta-2)}(%
\Delta v)^{2}+O(\varepsilon).\end{split}$$
From (6.7) and the above we conclude that
$$M_{u}(x)=M_{u_{0}}+\varepsilon^{2}r^{2(2-\beta)}\,[a_{1}\,(-\det D^{2}w)+a_{2}%
\,(\Delta w)^{2}]+O(\varepsilon^{2+\delta})$$
for $|x|\geq r_{0}$, with $O(\varepsilon^{2+\delta})$ depending on $r_{0}$.
The constants $a_{1}$ and $a_{2}$ are given by
$$a_{1}=\frac{1}{c_{\alpha}^{2}\,\beta^{3}\,(\beta-1)}\quad\mbox{and}\quad a_{2}%
=\frac{2}{c_{\alpha}^{2}\,\beta^{2}\,(\beta-2)}.$$
We recall that $w(r,\theta)=r^{\rho}\,\cos(2\theta)$.
Then, a direct computation shows that each term in the square
brackets above is positive. Thus the $\varepsilon^{2}$ term is positive and
homogeneous of degree $2\,(\rho-\beta)$, with $\rho<\beta$ (as
shown in (6.3)). We conclude from Proposition 5.6
that
$$\lim_{x\to 0}M_{u}(x)>M_{u_{0}}.$$
Hence, from Proposition 5.7, the blowup limit
of $u$ at the origin cannot be $u_{0}$. On the other hand, from the
symmetry of the boundary data for $u$ we conclude that the
function $v-v(0)$ has exactly two disconnected components where it
is positive (or negative). Thus the blowup limit at the origin for
$u$ has period $\pi$ on the unit circle which contradicts
Proposition 5.1.
∎
7. Proof of Theorem 1.1
In this final section we will present the last steps of the proof of Theorem 1.1.
We distinguish the two different cases of behavior at the origin,
(1.5) and (1.6).
Case 1: Radial Behavior. We will show that solutions
of (1.1) with the radial behavior
(1.5) are $C^{2,\frac{\alpha}{2}}$.
We begin by observing that solutions of (1.1) satisfy, in $B_{1}\setminus B_{1/2}$, the estimate
(7.1)
$$\|D^{2}u\|_{C^{0,1}(B_{1}\setminus B_{1/2})}\leq C(\alpha)$$
provided that
(7.2)
$$c(\alpha)\,|x|^{2+\frac{\alpha}{2}}\leq u(x)\leq C(\alpha)\,|x|^{2+\frac{%
\alpha}{2}}.$$
For any $r>0$, the rescaled functions
(7.3)
$${}u^{r}(x):=r^{-2-\frac{\alpha}{2}}\,u(rx)$$
solve the equation (1.1). Since $u$ has the radial behavior (1.5) at the origin,
each function $u^{r}$ satisfies (7.2).
Hence, applying (7.1) to $u^{r}$, we obtain for
$x,y\in B_{1}\setminus B_{1/2}$ the estimates
$$|D^{2}u(rx)-D^{2}u(ry)|\leq r^{\frac{\alpha}{2}}|x-y|,\qquad|D^{2}u(rx)|\leq C%
\,r^{\frac{\alpha}{2}}.$$
The above estimates, readily imply that $u\in C^{2,\frac{\alpha}{2}}$.
Case 2: Non-radial Behavior. In the rest of the
section we will show that solutions of (1.1) which satisfy
the nonradial behavior (1.6) are also of class
$C^{2,\delta}$, for some $\delta>0$. The idea is simple: we
approximate $u$ with quadratic polynomials in the $x_{2}$ direction.
However, the proof is quite technical.
In order to simplify the constants, we assume that $u$ solves the equation
(7.4)
$${}\det D^{2}u=2(2+\alpha)(1+\alpha)\,|x|^{\alpha}$$
instead of (1.1) and (after rescaling) that
(7.5)
$${}u(x)=|x_{1}|^{2+\alpha}+x_{2}^{2}+O\left((|x_{1}|^{2+\alpha}+x_{2}^{2})^{1+%
\delta}\right),\qquad\mbox{as}\,|x|\to 0.$$
From now on, we will denote points in $\mathbb{R}^{2}$ with capital letters
$$X=(x_{1},x_{2}).$$
The Hölder continuity of the second order derivatives of $u$ follows easily
from the following proposition.
Proposition 7.1.
Let $\lambda>0$ be small and
$$Y\in\Omega_{\lambda}:=\{\lambda\leq|x_{1}|^{2+\alpha}+x_{2}^{2}\leq 2\lambda\}.$$
Then, there exist $C$, $\mu$ universal constants such that in
$B:=B(Y,\lambda^{1+\alpha})$, we have
$$\|D^{2}u\|_{C^{\mu}(B)}\leq C\quad\mbox{and}\quad\|D^{2}u-D^{2}u(0)\|_{L^{%
\infty}(B)}\leq\lambda^{\mu}.$$
We will show that in the sections
$$S_{{X_{0}},t}:=\{\,X:\,u(X)<u(X_{0})+\nabla u(X_{0})\cdot(X-X_{0})+t\}.$$
of $u$ at the point
$$X_{0}=(0,x_{0}),\quad|x_{0}|\leq 2\lambda^{1/2}$$
we can approximate $u$ by quadratic polynomials of opening
$2$ on vertical segments. We begin by making the following definition.
Definition 7.2.
We say that
$$u\in Q(e,\varepsilon,\Omega)$$
if for any vertical segment $l\subset\Omega$ of length less than $e$,
there exists a quadratic polynomial $P_{x_{1},l}(x_{2})$ of opening $2$, namely
$$P_{x_{1},l}(x_{2})=x_{2}^{2}+p(x_{1},l)\,x_{2}+r(x_{1},l)$$
such that
$$\left|u(x_{1},x_{2})-P_{x_{1},l}(x_{2})\right|\leq\varepsilon e^{2}\quad\mbox{%
on $l$}.$$
Notice that for $c<1$ we have
$$Q(e,\varepsilon,\Omega)\subset Q(ce,c^{-2}\varepsilon,\Omega).$$
The plan of the proof is as follows: We prove Proposition 7.1 for points $Y\in S_{X_{0},t}$, with $t\leq\lambda$. We first show that $u$ belongs to some appropriate
$Q$ classes and distinguish two cases; one when $t\geq\lambda^{\frac{\alpha}{2}+1-\delta_{1}}$ for some fixed
$\delta_{1}>0$, and the other when $t=\lambda^{\frac{\alpha}{2}+1-\delta_{1}}$. In the first case we use the same method as in Lemma
3.1 and approximate the right hand side $|f(X)|^{\alpha/2}$
of the rescaled Monge-Ampére equation with $|x_{1}|^{\alpha}$ (see
Lemma 7.3). In the second case we approximate $f(X)$ with a
more general polynomial $x_{1}^{2}+px_{1}+q$ and obtain a better
approximation ($Q$ class) for $u$ (Lemma 7.4).
The Hölder estimates for points $Y\in S_{X_{0},t}$, $|x_{0}|\geq\lambda^{1/2}$ are obtained in appropriate sections $S_{Y,\sigma}$
in which all the values of $|x|$ are comparable. In these sections
the Monge-Ampére equation is nondegenerate and the classical
estimates apply. To obtain the appropriate section $S_{Y,\sigma}$ we
distinguish two cases, depending on the distance from $Y$ to the
$x_{2}$ axis. If $|y_{1}|\geq\lambda^{1/2}$, then we take $\sigma$ so
that $S_{Y,\sigma}$ is at distance greater than $|y_{1}|/2$ from the
$x_{2}$ axis (Lemma 7.5). If $|y_{1}|\leq\lambda^{1/2}$, then we
take $\sigma=\lambda^{\frac{2+\alpha}{2}}$ and $S_{Y,\sigma}$ is
close enough to the $x_{2}$ axis so that all its points are at
distance comparable to $\lambda^{1/2}$ from the origin (Lemma
7.6).
In what follows we will denote by $A_{t}$, $D_{t}$ the matrices
$$A_{t}=\begin{pmatrix}a_{11}&0\\
a_{21}&a_{22}\end{pmatrix},\qquad D_{t}=\begin{pmatrix}t^{\frac{1}{2+\alpha}}&%
0\\
0&t^{\frac{1}{2}}\end{pmatrix}.$$
Lemma 7.3.
Let $X_{0}=(0,x_{0})$ with $|x_{0}|\leq 2\lambda^{1/2}$, $0<\lambda<1$.
Then, for any $\delta_{1}>0$ and
(7.6)
$${}\lambda^{\frac{\alpha}{2}+1-\delta_{1}}\leq t\leq\lambda$$
there exists a small $\delta_{2}>0$, depending on $\delta_{1}$, such that
(7.7)
$${}S_{X_{0},t}-X_{0}\in A_{t}D_{t}(\Gamma\pm t^{\delta_{2}})$$
with
(7.8)
$$|A_{t}-I|\leq t^{\delta_{2}}.$$
Moreover,
$$u\in Q(t^{\frac{1}{2}},\lambda^{\delta_{2}},S_{X_{0},t}).$$
Proof.
We begin by observing that if $t=\lambda$, then the
conclusion of
the lemma follows from the expansion (7.5) with matrix
$A_{t}=I$. We will show by induction, using at each step the approximation lemma 3.1,
that (7.7) and (7.8) hold for every $t=\lambda\,t_{0}^{k}$, $k\in\mathbb{N}$, which satisfies (7.6).
Assume that (7.7) and (7.8) hold for some $t=\lambda\,t_{0}^{k}$ satisfying (7.6), with
$A_{t}$ bounded and $a_{t,11}$ bounded from below.
Consider the rescaling
(7.9)
$${}v(X):=\frac{1}{t}\,(u(X_{0}+A_{t}D_{t}X)-u(X_{0})-\nabla u(X_{0})\,(A_{t}D_{%
t}X)\,).$$
Since $u$ satisfies (7.4), the function $v$ satisfies the equation
(7.10)
$$\det D^{2}v=2(2+\alpha)(1+\alpha)a_{11}^{2}\,a_{22}^{2}\,t^{-\frac{\alpha}{2+%
\alpha}}\,|X_{0}+A_{t}D_{t}X|^{\alpha}.$$
Since
(7.11)
$$|X_{0}+A_{t}D_{t}X|^{2}=(t^{\frac{1}{2+\alpha}}\,a_{11}x_{1})^{2}+(t^{\frac{1}%
{2+\alpha}}a_{12}x_{1}+t^{\frac{1}{2}}a_{22}x_{2}+x_{0})^{2}$$
and $|x_{0}|\leq 2\,\lambda^{1/2}$, we conclude from the above that $v$ satisfies
(7.12)
$${}\det D^{2}v=c\,|f(X)|^{\frac{\alpha}{2}},\qquad S^{v}_{0,1}\in\Gamma\pm t^{%
\delta_{2}}$$
with
$$|f(X)-x_{1}^{2}|\leq C\,\left(\lambda^{\frac{1}{2}}t^{-\frac{1}{2+\alpha}}+t^{%
\frac{\alpha}{2(2+\alpha)}}\right)\leq t^{\frac{\delta_{1}}{2(2+\alpha)}}.$$
Notice that the last inequality holds if (7.6) is satisfied.
Lemma 3.1 with $\varepsilon=t^{\delta^{\prime}}$,
$\delta^{\prime}(\delta_{1},\alpha)>0$ small, yields
$$S^{u}_{X_{0},t_{0}t}-X_{0}\in A_{t_{0}t}D_{t_{0}t}(\Gamma\pm(t_{0}t)^{\delta_{%
2}})$$
with
$$A_{t_{0}t}=A_{t}E_{t},\quad|E_{t}-I|\leq C\,t^{\delta_{2}}.$$
Thus, (7.7) and (7.8) hold for $t^{\prime}=t\,t_{0}$.
If $t^{\prime}\leq\lambda^{\frac{\alpha}{2}+1-\delta_{1}}$ we stop, otherwise we continue the induction.
From (7.12) we find that
(7.13)
$${}\left|v-(|x_{1}|^{2+\alpha}+x_{2}^{2})\right|\leq Ct^{\delta_{2}}\qquad\mbox%
{in
$S^{v}_{0,1}$}$$
which together with (7.9) and (7.8), yields to
$$u\in Q(t^{\frac{1}{2}},C\,\lambda^{\delta_{2}},S^{u}_{X_{0},t}).$$
The lemma is proved by replacing $\delta_{2}$ with $\delta_{2}/2$.
∎
We will next examine closer the borderline case $t=\lambda^{\frac{\alpha}{2}+1-\delta_{1}}$
and show the better approximation (7.15) of $u$ by quadratic polynomials in the
$x_{2}$ variable. We begin by observing that the conclusion of the previous lemma implies
that
$$S_{X_{0},t}-X_{0}\in A_{t}D_{t}(\Gamma\pm\lambda^{\delta_{2}}),\qquad|A_{t}-I|%
\leq\lambda^{\delta_{2}}$$
for all $\lambda^{\frac{\alpha}{2}+1-\delta_{1}}\leq t\leq\lambda$.
Lemma 7.4.
Assume that for $t=\lambda^{\frac{\alpha}{2}+1-\delta_{1}}$ and
$\delta_{2}\ll\delta_{1}$, we have
(7.14)
$${}S_{X_{0},t}-X_{0}\in A_{t}D_{t}(\Gamma\pm\lambda^{\delta_{2}}),\quad\quad|A_%
{t}-I|\leq\lambda^{\delta_{2}}.$$
Then if $\delta_{1}$ is small, universal, we have
(7.15)
$$u\in Q(e,C\,\lambda^{\delta_{2}},S_{X_{0},\frac{t}{2}}),\quad\quad\mbox{for %
all $e$ with $\lambda^{\frac{2+\alpha}{4}}\leq e\leq t^{1/2}$}.$$
Proof.
Let $v$ be the re-scaling defined in (7.9). It follows from (7.10),
(7.11) and (7.14) that $v$ satisfies
$$\det D^{2}v=c\,f(X)^{\frac{\alpha}{2}},\quad\quad S^{v}_{0,1}\in\Gamma\pm%
\lambda^{\delta_{2}}$$
with
$$|f(X)-x_{1}^{2}-p\,x_{1}-q|\leq t^{\frac{\alpha}{2(2+\alpha)}},\quad\quad|p|,|%
q|\leq\lambda^{\frac{\delta_{1}}{2+\alpha}}$$
thus
$$\left|f(X)^{\frac{\alpha}{2}}-(x_{1}^{2}+px_{1}+q)^{\frac{\alpha}{2}}\right|%
\leq\varepsilon:=t^{\delta_{0}(\alpha)},\quad\quad\delta_{0}(\alpha)=\frac{%
\alpha\min\{\alpha,2\}}{4(2+\alpha)}.$$
Similarly as in the proof of Lemma 3.1 we define
the function $w$ as the solution to
$$\det D^{2}w=c\,(x_{1}^{2}+px_{1}+q)^{\frac{\alpha}{2}},\quad\mbox{$w=1$ on
$\partial S^{v}_{0,1}$}$$
and obtain (see (7.13)) that
$$|v-w|\leq C\varepsilon^{\frac{1}{2}}=Ct^{\frac{\delta_{0}(\alpha)}{2}}$$
and
$$\left|w-(|x_{1}|^{2+\alpha}+x_{2}^{2})\right|\leq C\lambda^{\delta_{2}}.$$
By considering the partial Legendre transform $w^{*}$, one can deduce
from the last inequality, the bounds on $|p|$, $|q|$ and Lemma 2.4 that
$$|w_{22}-2|\leq C\lambda^{\delta_{2}}\quad\mbox{in $S^{v}_{0,1/2}$}.$$
This implies that
$$w\in Q(e,C\lambda^{\delta_{2}},S^{v}_{0,1/2}),\quad\quad\mbox{for any
$e$}$$
hence
$$v\in Q(e,C\lambda^{\delta_{2}},S^{v}_{0,1/2}),\quad\quad\mbox{for $e\geq t^{%
\frac{\delta_{0}(\alpha)}{8}}$}.$$
Then, similarly as at the end of the proof of the previous lemma, we obtain that
$$u\in Q(t^{1/2}e,C\lambda^{\delta_{2}},S^{u}_{0,t/2}),\quad\quad\mbox{for $e%
\geq t^{\frac{\delta_{0}(\alpha)}{8}}$}$$
from which the lemma follows, since
$$t^{\frac{1}{2}}t^{\frac{\delta_{0}(\alpha)}{8}}\leq\lambda^{\frac{2+\alpha}{4}}$$
for
$\delta_{1}$ small, universal (depending only on $\alpha$).
∎
The next lemma proves Proposition 7.1 for a point $Y\in S_{X_{0},\lambda}$ at distance greater than $\lambda^{1/2}$ from the $x_{2}$
axis, assuming the conclusions of lemmas 7.3 and 7.4.
Lemma 7.5.
Assume that for $\lambda^{\frac{\alpha}{2}+1-\delta_{1}}\leq t\leq\lambda$, we have
(7.16)
$${}S_{X_{0},t}-X_{0}\in A_{t}D_{t}(\Gamma\pm\lambda^{\delta_{2}}),\quad\quad|A_%
{t}-I|\leq\lambda^{\delta_{2}}$$
and
$$u\in Q(e,C\,\lambda^{\delta_{2}},S_{X_{0},\frac{t}{2}})\quad\quad\mbox{for %
some $e$,}\quad\lambda^{\frac{2+\alpha}{4}}\leq e\leq t^{\frac{1}{2}}.$$
If
$$Y=(y_{1},y_{2})\in S_{X_{0},\frac{t}{3}},\quad\quad 1\leq|y_{1}|e^{-\frac{2}{2%
+\alpha}}\leq 2$$
then $D^{2}u$ is Hölder continuous in the ball $B:=B(Y,\lambda^{1+\alpha})$, and for
some constant $0<\beta<1$, it satisfies
(7.17)
$$\|D^{2}u\|_{C^{0,\beta}(B)}\leq C\quad\quad\mbox{and}\qquad|D^{2}u(Y)-D^{2}u(0%
)|\leq C\,\lambda^{\beta}.$$
Proof.
Consider the section $S^{u}_{Y,ce^{2}}$ for a small constant $c$.
By Theorem 2.2 there exists a matrix
$$F:=\begin{pmatrix}a&0\\
d&b\end{pmatrix},\quad\quad a,b>0$$
such that
(7.18)
$$FB_{1/{C_{0}}}\subset S^{u}_{Y,ce^{2}}-Y\subset FB_{1},\qquad\mbox{$C_{0}(%
\alpha)>0$ universal}.$$
Using the assumptions of the lemma and (7.18) we will derive bounds
on the coefficients of the matrix $F$.
Clearly,
$$\nu:=\frac{c^{1/2}e}{b}$$
satisfies the bound
(7.19)
$${}\frac{1}{2C_{0}}\leq\frac{c^{1/2}e}{b}\leq 2.$$
Since $e\leq t^{\frac{1}{2}}$, the corresponding section for the
rescaling $v$ (see (7.9), (7.13)) satisfies
$$S^{v}_{\tilde{Y},\frac{ce^{2}}{t}}\subset\{|x_{1}|^{2+\alpha}+x_{2}^{2}\leq 3/4\}$$
or more precisely
$$S^{v}_{\tilde{Y},\frac{ce^{2}}{t}}-\tilde{Y}\subset\left((e^{2}/t)^{\frac{1}{2%
+\alpha}}+\lambda^{\frac{\delta_{2}}{2(2+\alpha)}}\right)B_{1}$$
thus,
$$D_{t}^{-1}A_{t}^{-1}FB_{1/C_{0}}\subset\left(e^{\frac{2}{2+\alpha}}t^{-\frac{1%
}{2+\alpha}}+\lambda^{\delta_{3}}\right)B_{1}.$$
The last inclusion implies the estimate
(7.20)
$${}|d|\leq 2C_{0}\,(e^{\frac{2}{2+\alpha}}t^{\frac{\alpha}{2(2+\alpha)}}+%
\lambda^{\delta_{3}}t^{\frac{1}{2}}+a\lambda^{\delta_{3}})\leq 4C_{0}\,(e^{%
\frac{2}{2+\alpha}}+a)\,\lambda^{\delta_{3}}.$$
The rescaling
$$w(x):=\frac{1}{b^{2}}u(Y+Fx)$$
satisfies
(7.21)
$${}\det D^{2}w=\frac{a^{2}}{b^{2}}f(x)^{\frac{\alpha}{2}},\qquad B_{1/C_{0}}%
\subset S^{w}_{0,\nu^{2}}\subset B_{1}$$
with
(7.22)
$${}f(x):=(y_{1}+ax_{1})^{2}+(y_{2}+dx_{1}+bx_{2})^{2}$$
and
(7.23)
$${}|w-P^{\prime}_{x_{1}}(x_{2})|\leq\lambda^{\delta_{3}},\qquad\mbox{in $S^{w}_%
{0,\nu^{2}}$}.$$
We claim that if $c$ is chosen small, universal, then
(7.24)
$${}2a\leq e^{\frac{2}{2+\alpha}}\leq|y_{1}|.$$
Indeed, otherwise from (7.21), we deduce that
$$\det D^{2}w\geq a^{2+\alpha}b^{-2}\left(x_{1}+\frac{y_{1}}{a}\right)^{\alpha}$$
with
$$(2a)^{2+\alpha}b^{-2}\geq e^{2}b^{-2}\geq c^{-1}\nu^{2}$$
and for small $c$ we contradict $B_{1/C_{0}}\subset S^{w}_{0,\nu^{2}}$, since
$\nu$ is bounded.
From (7.19), (7.20), (7.24) and $|y_{2}|\leq 4\lambda^{1/2}$ we obtain that ${f(x)}/{y_{1}^{2}}$ is bounded away
from $0$ and $\infty$ by universal constants, and also its
derivatives are bounded by universal constants. From (7.21)
we find that
$$c_{1}\leq\frac{a^{2}|y_{1}|^{\alpha}}{b^{2}}\leq C_{1}$$
which implies that $a^{2+\alpha}$, $|y_{1}|^{2+\alpha}$, $b^{2}$, and $e^{2}$ are all
comparable. Moreover, using also (7.23), we have
(7.25)
$${}\|D^{2}w\|_{C^{0,1}}\leq C,\quad|w_{22}-2|\leq\lambda^{\delta_{4}}\quad\mbox%
{in
$S_{0,\nu^{2}}/2$}.$$
Hence
(7.26)
$${}|w_{22}(x)-w_{22}(y)|\leq C\,\lambda^{{\delta_{4}}/2}|x-y|^{1/2}\quad\quad%
\mbox{
for $x,y\in S_{0,\nu^{2}}/2$}.$$
Also, we have
$$D^{2}u(Y+Fx)=b^{2}(F^{-1})^{T}D^{2}w(x)\,F^{-1}$$
with
$$b\,F^{-1}=\begin{pmatrix}b/a&0\\
-d/a&1\end{pmatrix}=\begin{pmatrix}0&0\\
0&1\end{pmatrix}+O(\lambda^{\delta_{3}})$$
which together with (7.25) implies the second part of the
conclusion (7.17).
Finally, since
$$|Fx|\geq\frac{b|x|}{2}\geq\lambda^{1+\alpha}|x|$$
we obtain from (7.25) and (7.26) the estimate
$$|D^{2}u(Y+Fx)-D^{2}u(Y+Fy)|\leq C\lambda^{{\delta_{4}}/2}|x-y|^{1/2}\leq C\,|%
Fx-Fy|^{\beta}.$$
This finishes the proof of the lemma.
∎
The next lemma proves Hölder continuity when $Y$ is
$\lambda^{1/2}$ close to the $x_{2}$ axis.
Lemma 7.6.
Assume that $(\ref{2.6})$ holds for
$t=\lambda^{\frac{\alpha}{2}+1-\delta_{1}}$,
$$u\in Q(e,\lambda^{\delta_{2}},S_{X_{0},\frac{t}{2}})\quad\quad\mbox{for}\quad e%
=\lambda^{\frac{2+\alpha}{4}}$$
and
$$|x_{0}|\geq\lambda^{\frac{1}{2}}/2,\quad Y\in S_{X_{0},\frac{t}{3}},\quad|y_{1%
}|\leq e^{\frac{2}{2+\alpha}}.$$
Then, the conclusion of Lemma 7.5 still holds.
Proof.
The proof is very similar to that of Lemma 7.5. The
only difference is that now the second term of $f$ in (7.22)
dominates the sum.
Indeed, since $\lambda^{1/2}\geq|y_{1}|$ and $|y_{2}|\geq\lambda^{1/2}/4$, the function
${f(x)}/{y_{2}^{2}}$
is bounded away from $0$ and $\infty$ by
universal constants, and also its derivatives are bounded by universal
constants. Hence, $a^{2+\alpha}$, $y_{2}^{2+\alpha}$, $b^{2}$ and $e^{2}$ are
all comparable and the rest of the proof is the same.
∎
Proof of Proposition 7.1.
For $Y\in\Omega_{\lambda}$ we consider the section $S^{u}_{Y,\sigma}$ that
becomes tangent to the $x_{2}$ axis at $X_{0}=(0,x_{0})$. Since $|x|^{\alpha}dx$ is doubling,
there exists $C_{1}$ universal such that
$$Y\in S^{u}_{X_{0},t/3},\quad|x_{0}|\leq 2\lambda^{1/2},\quad t:=C_{1}\sigma%
\leq C_{2}\lambda.$$
We distinguish the following three cases:
i.
If $t\geq t_{0}:=\lambda^{\alpha/2+1-\delta_{1}}$, then the proposition follows from Lemmas
7.3 and 7.5 with
$$e=|y_{1}|^{\frac{2+\alpha}{2}}\geq c_{1}t^{1/2}.$$
ii.
If $t\leq t_{0}$ and $|y_{1}|\geq\lambda^{1/2}$, then we
apply Lemmas 7.4 and 7.5 for $S_{X_{0},t_{0}}$ with $e$
defined as above.
iii.
If $t\leq t_{0}$ and $|y_{1}|\leq\lambda^{1/2}$,
then we apply Lemma 7.4 and Lemma 7.6. We remark that the
hypothesis $|x_{0}|\geq\lambda^{1/2}/2$ is satisfied because $Y\in\Omega_{\lambda}$.
∎
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Zürich 61 (1916), 40-72. |
Apéry-Type Series with Summation Indices of Mixed Parities and Colored Multiple Zeta Values, I
Ce Xu${}^{a,}$ and Jianqiang Zhao${}^{b,}$
a. School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, PRC
b. Department of Mathematics, The Bishop’s School, La Jolla, CA 92037, USA
Email: cexu2020@ahnu.edu.cn, corresponding author, ORCID 0000-0002-0059-7420.Email: zhaoj@ihes.fr, ORCID 0000-0003-1407-4230.
Abstract. In this paper, we shall study Aéry-type series in which the central binomial coefficient appears as part of the summand. Let $b_{n}=4^{n}/\binom{2n}{n}$. Let $s_{1},\dots,s_{d}$ be positive integers with $s_{1}\geq 2$. We consider
the series
$$\displaystyle\displaystyle\sum_{n_{1}>\cdots>n_{d}>0}\displaystyle\frac{b_{n_{1}}}{n_{1}^{s_{1}}\cdots n_{d}^{s_{d}}}$$
and the variants with some or all indices $n_{j}$ replaced by $2n_{j}\pm 1$ and
some or all “$>$” replaced by “$\geq$”, provided the series are defined.
We can also replace $b_{n_{1}}$ by its square in the above series when $s_{1}\geq 3$. The main result is that all such series
are $\mathbb{Q}$-linear combinations of the real and/or the imaginary parts of colored multiple zeta values of level 4, i.e., multiple polylogarithms evaluated at 4th roots of unity.
Keywords: Apéry-type series, colored multiple zeta values, mixed parities, iterated integrals.
AMS Subject Classifications (2020): 11M32, 11B65, 11B37, 44A05, 33B30.
1 Introduction
In his celebrated proof of irrationality of ${\zeta}(2)$ and ${\zeta}(3)$ in 1979, Apéry used crucially the following two identities
$${\zeta}(2)=3\displaystyle\sum_{n\geq 1}\displaystyle\frac{1}{n^{2}{\binom{2n}{n}}}\quad\text{and}\quad{\zeta}(3)=\displaystyle\frac{5}{2}\displaystyle\sum_{n\geq 1}\displaystyle\frac{(-1)^{n-1}}{n^{3}{\binom{2n}{n}}}.$$
(1.1)
Motivated by Apéry’s proof, Leshchiner [14] generalized these to higher weight Riemann zeta values
and some other analogs. However, no irrationality proof has been found so far for other Riemann zeta values at odd positive
integers greater than 4, although a lot of progress has been made (see e.g., [13, 17, 22]. In particular, in 2020, Lai and Yu [13] proved that for any small $\varepsilon>0$, the number of irrationals among the following odd zeta values: $\zeta(3),\zeta(5),\zeta(7),\ldots,\zeta(s)$, is at least $(c_{0}-\varepsilon)\sqrt{s/\log(s)}$, provided $s$ is a sufficiently large odd integer with respect to $\varepsilon$, with constant $c_{0}=1.192507\ldots$.
On the other hand, series generalizing those on the right-hand side of (1.1),
including odd-indexed variations (see Remark 4.2) have appeared in the calculations
of the ${\varepsilon}$-expansions of the Feynman diagrams in recent years (see, e.g., [8, 9, 10]).
In the meantime, the frequent and sometimes unexpected appearance of colored multiple zeta values (see (1.2))
in quite a few different branches of mathematics and physics has attracted the attention of many mathematicians and physicists alike.
These numbers, as vast generalizations of Riemann zeta values, are all conjectured to be not only irrational
but also transcendental. One naturally wonders if the multiple sums, which we call Apéry-type series,
that generalize those in (1.1) can be related to these numbers.
In a series of papers, we will answer some of these questions. As part I of this series, this paper concentrates on
Apéry-type series such as those defined by (4.43) and (4.53) in which
the central binomial coefficients appear only on the denominators.
We will show that a large class of these series can be expressed as $\mathbb{Q}$-linear combinations of the real and/or the imaginary parts
of the colored multiple zeta values of level 4, i.e., multiple polylogarithms evaluated at 4th roots of unity, see Thm. 4.1. Some related results may be found in [4, 11, 18, 19] and references therein.
1.1 Notation.
Let $\mathbb{N}$ be the set of positive integers and $\mathbb{N}_{0}:=\mathbb{N}\cup\{0\}$.
A finite sequence ${\boldsymbol{\sl{s}}}:=(s_{1},\ldots,s_{d})\in\mathbb{N}^{d}$ is called a composition. We define the weight and the depth of ${\boldsymbol{\sl{s}}}$ by
$$|{\boldsymbol{\sl{s}}}|:=s_{1}+\cdots+s_{d},\quad\text{and}\quad\operatorname*{dp}({\boldsymbol{\sl{s}}}):=d,$$
respectively. For any $N$th roots of unity $z_{1},\dotsc,z_{d}$ the colored multiple zeta values (CMZVs) of level $N$
are defined by
$$\operatorname{Li}_{{\boldsymbol{\sl{s}}}}({\boldsymbol{\sl{z}}}):=\displaystyle\sum_{n_{1}>\cdots>n_{d}>0}\displaystyle\frac{z_{1}^{n_{1}}\dots z_{d}^{n_{d}}}{n_{1}^{s_{1}}\dots n_{d}^{s_{d}}},$$
(1.2)
which converge if $(s_{1},z_{1})\neq(1,1)$ (see [16] and [21, Ch. 15]), in which case we call $({\boldsymbol{\sl{s}}};{\boldsymbol{\sl{z}}})$ admissible. The multiple zeta values are CMZVs of level 1, namely, $\zeta({\boldsymbol{\sl{s}}}):=\operatorname{Li}_{{\boldsymbol{\sl{s}}}}(1_{d})$ where $1_{d}$ is the string of 1’s with $d$ repetitions. Moreover, CMZVs can be expressed using Chen’s iterated integrals
$$\operatorname{Li}_{{\boldsymbol{\sl{s}}}}({\boldsymbol{\sl{z}}})=\displaystyle\int_{0}^{1}{\texttt{a}}^{s_{1}-1}{\texttt{x}}_{\xi_{1}}\cdots{\texttt{a}}^{s_{d}-1}{\texttt{x}}_{\xi_{d}},$$
(1.3)
where $\xi_{j}:=\prod_{i=1}^{j}z_{i}^{-1}$, ${\texttt{a}}:=dt/t$ and ${\texttt{x}}_{\xi}:=dt/(\xi-t)$
for any $N$th roots of unity $\xi$, see [21, Sec. 2.1] for a brief summary of this theory. The theory of iterated integrals was developed first by K.T. Chen in the 1960’s. It has played important roles in the study of algebraic topology and algebraic geometry in the past half century. Its simplest form is
$$\displaystyle\displaystyle\int_{0}^{1}f_{1}(t)dtf_{2}(t)dt\cdots f_{p}(t)dt=$$
$$\displaystyle\,\displaystyle\int_{0}^{1}f_{1}(t)dt\circ f_{2}(t)dt\circ\cdots\circ f_{p}(t)dt$$
$$\displaystyle:=$$
$$\displaystyle\,\displaystyle\int\limits_{1>t_{1}>\cdots>t_{p}>0}f_{1}(t_{1})f_{2}(t_{2})\cdots f_{p}(t_{p})dt_{1}dt_{2}\cdots dt_{p}.$$
One can extend these to iterated integrals over any piecewise smooth path on the complex plane via pull-backs.
We refer the interested reader to Chen’s original work [6, 7] for more details.
1.2 Akhilesh’s result.
In [2, 3] Akhilesh discovered some very important and surprising connections between
MZVs and the following Apéry-type series (which he calls multiple Apéry-like sums and which are normalized slightly differently here)
$${\sigma}({\boldsymbol{\sl{s}}};x):=\displaystyle\sum_{n_{1}>n_{2}>\cdots>n_{d}>0}{\binom{2n_{1}}{n_{1}}}^{-1}\displaystyle\frac{(2x)^{2n_{1}}}{(2n_{1})^{s_{1}}\cdots(2n_{d})^{s_{d}}}.$$
(1.4)
His ingenious idea is to study the $n$-tails (and more generally, double tails) of such series.
We reformulate one of his key results as follows to make it more transparent. Set
$$\displaystyle g_{s}(t)=\left\{\begin{array}[]{ll}\tan t\,dt,&\hbox{if $s=1$;}\\
dt\circ(\cot t\,dt)^{s-2}\circ dt,&\hbox{if $s\geq 2$,}\end{array}\right.$$
(1.7)
and their non-trigonometric counterpart
$$G_{s}(t)=\left\{\begin{array}[]{ll}{\omega}_{2}&\hbox{if $s=1$;}\\
{\omega}_{1}{\omega}_{0}^{s-2}{\omega}_{1}&\hbox{if $s\geq 2$,}\end{array}\right.$$
(1.8)
where ${\omega}$’s are defined by (2.10). Further, we set $\binom{0}{0}=1$,
$$\displaystyle b_{n}(x)=4^{n}{{\binom{2n}{n}}}^{-1}x^{2n}\quad\text{and}\quad b_{n}=b_{n}(1)=4^{n}{{\binom{2n}{n}}}^{-1}\quad\forall n\geq 0.$$
Theorem 1.1.
([3, Thm. 4])
For all $n\in\mathbb{N}_{0}$, ${\boldsymbol{\sl{s}}}=(s_{1},\dots,s_{d})\in\mathbb{N}^{d}$ we have
$$\displaystyle{\sigma}({\boldsymbol{\sl{s}}};\sin y)_{n}:=$$
$$\displaystyle\,\displaystyle\sum_{n_{1}>\cdots>n_{d}>n}\displaystyle\frac{b_{n_{1}}(\sin y)}{(2n_{1})^{s_{1}}\cdots(2n_{d})^{s_{d}}}=\displaystyle\frac{d}{dy}\displaystyle\int_{0}^{y}g_{s_{1}}\circ\cdots\circ g_{s_{d}}\circ b_{n}(\sin t)\,dt,$$
where $y\in(-\pi/2,\pi/2)$ if $s_{1}=1$ and $y\in[-\pi/2,\pi/2]$ if $s_{1}\geq 2$. Using non-trigonometric 1-forms, we have
for all $x\in(-1,1)$
$$\displaystyle{\sigma}({\boldsymbol{\sl{s}}};x)_{n}:=$$
$$\displaystyle\,\displaystyle\sum_{n_{1}>\cdots>n_{d}>n}\displaystyle\frac{b_{n_{1}}(x)}{(2n_{1})^{s_{1}}\cdots(2n_{d})^{s_{d}}}=\sqrt{1-x^{2}}\displaystyle\frac{d}{dx}\displaystyle\int_{0}^{x}G_{s_{1}}\circ\cdots\circ G_{s_{d}}\circ b_{n}(t){\omega}_{1}.$$
We will again call the sum $|{\boldsymbol{\sl{s}}}|:=s_{1}+\dots+s_{d}$ the weight of the series ${\sigma}({\boldsymbol{\sl{s}}};\sin y)$ and $d$ the depth.
1.3 Main results.
In Thm 2.3 and Thm 3.1 we will show that the tails of the series
$$\displaystyle{\tau}^{\star}({\boldsymbol{\sl{s}}};x)_{n}:=$$
$$\displaystyle\,\displaystyle\sum_{n_{1}\geq\cdots\geq n_{d}\geq n}{\binom{2n_{1}}{n_{1}}}^{-1}\displaystyle\frac{(2x)^{2n_{1}}}{(2n_{1}+1)^{s_{1}}\cdots(2n_{d}+1)^{s_{d}}},$$
$$\displaystyle\chi({\boldsymbol{\sl{s}}};x)_{n}:=$$
$$\displaystyle\,\displaystyle\sum_{n_{1}>\cdots>n_{d}>n}{\binom{2n_{1}}{n_{1}}}^{-1}\displaystyle\frac{(2x)^{2n_{1}}}{(2n_{1}-1)^{s_{1}}\cdots(2n_{d}-1)^{s_{d}}}$$
can be written as iterated integrals similar to the one in Thm 1.1. As corollaries, we show that the series
${\sigma}({\boldsymbol{\sl{s}}};1)_{0}$, ${\tau}^{\star}({\boldsymbol{\sl{s}}};1)_{0}$, $\chi({\boldsymbol{\sl{s}}};1)_{0}$ and more general similar series defined
by (4.43) and (4.53) with summation indices of any parity pattern
can be expressed as $\mathbb{Q}$-linear combinations of the real and/or the imaginary parts of CMZVs of level 4.
We also consider other Apéry-type series similar to the above by using the
square of central binomial coefficients such as
$$\begin{split}{\sigma}^{(2)}({\boldsymbol{\sl{s}}}):=&\,\displaystyle\sum_{n_{1}>\cdots>n_{d}>0}{\binom{2n_{1}}{n_{1}}}^{-2}\displaystyle\frac{16^{n_{1}}}{(2n_{1})^{s_{1}}\cdots(2n_{d})^{s_{d}}},\\
{\tau}^{\star,(2)}({\boldsymbol{\sl{s}}}):=&\,\displaystyle\sum_{n_{1}\geq\cdots\geq n_{d}\geq 0}{\binom{2n_{1}}{n_{1}}}^{-2}\displaystyle\frac{16^{n_{1}}}{(2n_{1}+1)^{s_{1}}\cdots(2n_{d}+1)^{s_{d}}}\end{split}$$
(1.9)
and show that they also lie in the $\mathbb{Q}$-vector space of CMZVs of level 4.
2 First variant with odd summation indices
In this section, we consider a variation of the Apéry-type series studied in [3] by restricting
the summation indices to odd numbers only and replacing strict inequalities among them by
non-strict ones. Concerning this we have the next well-known result. Define
$$\displaystyle{\omega}_{0}:=$$
$$\displaystyle\,\displaystyle\frac{dt}{t},\quad$$
$$\displaystyle\,{\omega}_{1}:=$$
$$\displaystyle\,\displaystyle\frac{dt}{\sqrt{1-t^{2}}},\quad$$
$$\displaystyle\,{\omega}_{2}:=$$
$$\displaystyle\,\displaystyle\frac{t\,dt}{1-t^{2}},$$
(2.10)
$$\displaystyle{\omega}_{3}:=$$
$$\displaystyle\,\displaystyle\frac{dt}{t\sqrt{1-t^{2}}},\quad$$
$$\displaystyle\,{\omega}_{5}:=$$
$$\displaystyle\,\displaystyle\frac{t\,dt}{\sqrt{1-t^{2}}},\quad$$
$$\displaystyle\,{\omega}_{8}:=$$
$$\displaystyle\,\displaystyle\frac{dt}{1-t^{2}}.$$
(2.11)
Lemma 2.1.
For all $d\in\mathbb{N}$ and ${\boldsymbol{\sl{s}}}=(s_{1},\dots,s_{d})\in\mathbb{N}^{d}$, we have
$$\displaystyle\operatorname{ti}_{\boldsymbol{\sl{s}}}(x)_{n}:=\displaystyle\sum_{n_{1}>\cdots>n_{d}\geq n}\displaystyle\frac{x^{2n_{1}+1}}{(2n_{1}+1)^{s_{1}}\cdots(2n_{d}+1)^{s_{d}}}=\displaystyle\int_{0}^{x}{\omega}_{0}^{s_{1}-1}{\omega}_{2}\cdots{\omega}_{0}^{s_{d-1}-1}{\omega}_{2}{\omega}_{0}^{s_{d}-1}(t^{2n}{\omega}_{8}).$$
Proof.
This follows easily by direct computation.
∎
For all $n\in\mathbb{N}_{0}$ and ${\boldsymbol{\sl{s}}}=(s_{1},\dots,s_{d})\in\mathbb{N}^{d}$ we define
$$\displaystyle{\tau}({\boldsymbol{\sl{s}}};x)_{n}:=$$
$$\displaystyle\,\displaystyle\sum_{n_{1}>\cdots>n_{d}\geq n}{\binom{2n_{1}}{n_{1}}}^{-1}\displaystyle\frac{(2x)^{2n_{1}}}{(2n_{1}+1)^{s_{1}}\cdots(2n_{d}+1)^{s_{d}}},$$
$$\displaystyle{\tau}^{\star}({\boldsymbol{\sl{s}}};x)_{n}:=$$
$$\displaystyle\,\displaystyle\sum_{n_{1}\geq\cdots\geq n_{d}\geq n}{\binom{2n_{1}}{n_{1}}}^{-1}\displaystyle\frac{(2x)^{2n_{1}}}{(2n_{1}+1)^{s_{1}}\cdots(2n_{d}+1)^{s_{d}}}.$$
Theorem 2.2.
For all $d\in\mathbb{N}$ and ${\boldsymbol{\sl{s}}}=(s_{1},\dots,s_{d})\in\mathbb{N}^{d}$, the tail
$$\displaystyle{\tau}({\boldsymbol{\sl{s}}};1/2)_{n}=$$
$$\displaystyle\,\displaystyle\int_{0}^{1/2}\displaystyle\frac{4t}{\sqrt{1-4t^{2}}}{\omega}_{0}^{s_{1}-1}{\omega}_{2}\cdots{\omega}_{0}^{s_{d-1}-1}{\omega}_{2}{\omega}_{0}^{s_{d}-1}(t^{2n}{\omega}_{8}).$$
Proof.
Note that
$$\displaystyle\displaystyle\int_{0}^{1/2}\displaystyle\frac{4t}{\sqrt{1-4t^{2}}}t^{2n}dt=$$
$$\displaystyle\,\displaystyle\int_{0}^{1/4}\displaystyle\frac{2t^{n}}{\sqrt{1-4t}}dt=\displaystyle\frac{\Gamma(n+1)^{2}}{\Gamma(2n+2)}={{\binom{2n}{n}}}^{-1}\displaystyle\frac{1}{2n+1}.$$
Thus
$$\displaystyle\,\displaystyle\sum_{n_{1}>\cdots>n_{d}\geq n}{\binom{2n_{1}}{n_{1}}}^{-1}\displaystyle\frac{1}{(2n_{1}+1)^{s_{1}}(2n_{2}+1)^{s_{2}}\cdots(2n_{d}+1)^{s_{d}}}$$
$$\displaystyle=$$
$$\displaystyle\,\displaystyle\int_{0}^{1/2}\displaystyle\frac{4t}{\sqrt{1-4t^{2}}}\displaystyle\sum_{n_{1}>\cdots>n_{d}\geq n}\displaystyle\frac{t^{2n_{1}}dt}{(2n_{1}+1)^{s_{1}-1}(2n_{2}+1)^{s_{2}}\cdots(2n_{d}+1)^{s_{d}}}$$
$$\displaystyle=$$
$$\displaystyle\,\displaystyle\int_{0}^{1/2}\displaystyle\frac{4t}{\sqrt{1-4t^{2}}}\bigg{(}\displaystyle\frac{d}{dt}\operatorname{ti}_{\boldsymbol{\sl{s}}}(t)_{n}\bigg{)}dt$$
$$\displaystyle=$$
$$\displaystyle\,\displaystyle\int_{0}^{1/2}\displaystyle\frac{4t}{\sqrt{1-4t^{2}}}{\omega}_{0}^{s_{1}-1}{\omega}_{2}\cdots{\omega}_{0}^{s_{d-1}-1}{\omega}_{2}{\omega}_{0}^{s_{d}-1}(t^{2n}{\omega}_{8}),$$
as desired.
∎
It turns out that the star version ${\tau}^{\star}$ behaves better. To study this,
we will extend Chen’s iterated integrals by combining 1-forms and functions as follows.
For any $r\in\mathbb{N}$, 1-forms $f_{1}(t)\,dt,\dots,f_{r+1}(t)\,dt$ and functions $F_{1}(t),\dots,F_{r}(t)$, we define
recursively
$$\displaystyle\,\displaystyle\int_{0}^{1}\big{(}f_{1}(t)\,dt+F_{1}(t)\big{)}\circ\cdots\circ\big{(}f_{r}(t)\,dt+F_{r}(t)\big{)}\circ f_{r+1}(t)\,dt$$
$$\displaystyle:=$$
$$\displaystyle\,\displaystyle\int_{0}^{1}\big{(}f_{1}(t)\,dt+F_{1}(t)\big{)}\circ\cdots\circ\big{(}f_{r-1}(t)\,dt+F_{r-1}(t)\big{)}\circ f_{r}(t)\,dt\circ f_{r+1}(t)\,dt$$
$$\displaystyle+$$
$$\displaystyle\,\displaystyle\int_{0}^{1}\big{(}f_{1}(t)\,dt+F_{1}(t)\big{)}\circ\cdots\circ\big{(}f_{r-1}(t)\,dt+F_{r-1}(t)\big{)}\circ\big{(}F_{r}(t)f_{r+1}(t)\big{)}\,dt.$$
We now set the 1-forms
$$\displaystyle h_{s}(t)=\left\{\begin{array}[]{ll}2\csc 2t\,dt,&\hbox{if $s=1$;}\\
\csc t\,dt\circ(\cot t\,dt)^{s-2}\circ\csc t\,dt,&\hbox{if $s\geq 2$,}\end{array}\right.$$
(2.14)
and their non-trigonometric counter part
$$\displaystyle H_{s}(t)=\left\{\begin{array}[]{ll}{\omega}_{20}:={\omega}_{0}+{\omega}_{2},&\hbox{if $s=1$;}\\
{\omega}_{3}{\omega}_{0}^{s-2}{\omega}_{3},&\hbox{if $s\geq 2$.}\end{array}\right.$$
(2.17)
Theorem 2.3.
For all $n\in\mathbb{N}_{0}$, ${\boldsymbol{\sl{s}}}=(s_{1},\dots,s_{d})\in\mathbb{N}^{d}$ we have the tail
$$\displaystyle{\tau}^{\star}({\boldsymbol{\sl{s}}};\sin y)_{n}=$$
$$\displaystyle\,\displaystyle\sum_{n_{1}\geq\cdots\geq n_{d}\geq n}\displaystyle\frac{b_{n_{1}}(\sin y)}{(2n_{1}+1)^{s_{1}}\cdots(2n_{d}+1)^{s_{d}}}=\displaystyle\frac{d}{dy}\displaystyle\int_{0}^{y}h_{s_{1}}\circ\cdots h_{s_{d}}\circ b_{n}(\sin t)\,dt.$$
(2.18)
Hence
$$\displaystyle{\tau}^{\star}({\boldsymbol{\sl{s}}};x)_{n}=$$
$$\displaystyle\,\sqrt{1-x^{2}}\displaystyle\frac{d}{dx}\displaystyle\int_{0}^{x}H_{s_{1}}\circ\cdots\circ H_{s_{d}}\circ b_{n}(t){\omega}_{1}.$$
Proof.
When $d=s_{1}=1$, the right-hand side of (2.18) is equal to
$$\displaystyle\displaystyle\frac{d}{dy}\displaystyle\int_{0}^{y}2\csc 2t\,dtb_{n}(\sin t)\,dt=\sec y\csc y\displaystyle\int_{0}^{y}b_{n}(\sin t)\,dt.$$
(2.19)
Observe that
$$\displaystyle{\binom{2k+2}{k+1}}^{-1}\displaystyle\frac{1}{k+1}=\displaystyle\frac{(k+1)!^{2}}{(2k+2)(2k+1)(2k)!}\displaystyle\frac{1}{k+1}=\displaystyle\frac{1}{2}{\binom{2k}{k}}^{-1}\displaystyle\frac{1}{2k+1}.$$
By Thm. 1.1 we get
$$\displaystyle\displaystyle\sum_{n_{1}\geq n}{\binom{2n_{1}+2}{n_{1}+1}}^{-1}\displaystyle\frac{(4\sin^{2}y)^{n_{1}+1}}{n_{1}+1}=2\tan y\displaystyle\int_{0}^{y}b_{n}(\sin t)\,dt.$$
Hence
$$\displaystyle\displaystyle\sum_{n_{1}\geq n}\displaystyle\frac{b_{n_{1}}\sin^{2n_{1}+1}y}{2n_{1}+1}=\sec y\displaystyle\int_{0}^{y}b_{n}(\sin t)\,dt,$$
(2.20)
which is exactly (2.19). Thus the case $d=s_{1}=1$ of the theorem is proved.
Now, repeatedly multiplying (2.20) by $\cot y$ and integrating $s-1$ times, we get
$$\displaystyle\displaystyle\sum_{n_{1}\geq n}\displaystyle\frac{b_{n_{1}}\sin^{2n_{1}+1}y}{(2n_{1}+1)^{s}}=\displaystyle\int_{0}^{y}(\cot t\,dt)^{s-2}\,(\csc t\,dt)\,(b_{n}(\sin t)\,dt).$$
(2.21)
Replacing $n$ by $n_{2}$, multiplying by $1/(2n_{2}+1)$ and taking the sum $\displaystyle\sum_{n_{1}\geq n_{2}\geq n}$,
we get
$$\displaystyle\displaystyle\sum_{n_{1}\geq n_{2}\geq n}\displaystyle\frac{b_{n_{1}}\sin^{2n_{1}+1}y}{(2n_{1}+1)^{s}(2n_{2}+1)}=$$
$$\displaystyle\,\displaystyle\int_{0}^{y}(\cot t\,dt)^{s-2}(\csc t\,dt)\displaystyle\sum_{n_{2}\geq n}\displaystyle\frac{b_{n_{2}}(\sin t)}{2n_{2}+1}\,dt$$
$$\displaystyle=$$
$$\displaystyle\,2\displaystyle\int_{0}^{y}(\cot t\,dt)^{s-2}\,(\csc t\,dt)\,(\csc 2tdt)\,(b_{n}(\sin t)\,dt).$$
Multiplying by $1/(2n_{2}+1)^{s_{2}}$ for $s_{2}\geq 2$ we get
$$\displaystyle\,\displaystyle\sum_{n_{1}\geq n_{2}\geq n}\displaystyle\frac{b_{n_{1}}\sin^{2n_{1}+1}y}{(2n_{1}+1)^{s_{1}}(2n_{2}+1)^{s_{2}}}$$
$$\displaystyle=$$
$$\displaystyle\,\displaystyle\int_{0}^{y}(\cot t\,dt)^{s-2}(\csc t\,dt)\displaystyle\sum_{n_{2}\geq n}\displaystyle\frac{b_{n_{2}}(\sin t)}{(2n_{2}+1)^{s_{2}}}\,dt$$
$$\displaystyle=$$
$$\displaystyle\,\displaystyle\int_{0}^{y}(\cot t\,dt)^{s_{1}-2}(\csc t\,dt)(\csc t\,dt)(\cot t\,dt)^{s_{2}-2}(\csc t\,dt)\,(b_{n}(\sin t)\,dt).$$
The theorem follows from doing these repeatedly and can be easily proved by induction. We leave the details to the interested reader.
∎
Corollary 2.4.
For all admissible ${\boldsymbol{\sl{s}}}=(s_{1},\ldots,s_{d})\in\mathbb{N}^{d}$ with $s_{1}\geq 2$, we have
$$\displaystyle t^{\star}({\boldsymbol{\sl{s}}})_{n}:=\displaystyle\sum_{n_{1}\geq\cdots\geq n_{d}\geq n}\displaystyle\frac{1}{(2n_{1}+1)^{s_{1}}\cdots(2n_{d}+1)^{s_{d}}}=\displaystyle\frac{2}{\pi}\displaystyle\int_{0}^{\pi/2}h_{s_{1}}\circ\cdots\circ h_{s_{d}}\circ b_{n}(\sin t)\,dt.$$
In particular,
$$\displaystyle t^{\star}({\boldsymbol{\sl{s}}}):=\displaystyle\sum_{n_{1}\geq\cdots\geq n_{d}\geq 0}\displaystyle\frac{1}{(2n_{1}+1)^{s_{1}}\cdots(2n_{d}+1)^{s_{d}}}=\displaystyle\frac{2}{\pi}\displaystyle\int_{0}^{\pi/2}h_{s_{1}}\circ\cdots\circ h_{s_{d}}\circ\,dt.$$
Proof.
Integrating (2.18) over $(0,\pi/2)$ and noticing the fact that
$$\displaystyle\displaystyle\int_{0}^{\pi/2}(\sin t)^{2n}dt=\displaystyle\frac{\pi}{2b_{n}},$$
we obtain the corollary immediately.
∎
Example 2.5.
Let ${\boldsymbol{\sl{s}}}=(2_{d})$ be the string of 2’s with $d$ repetitions. Then we see that
$$\displaystyle t^{\star}(2_{d})=\displaystyle\frac{2}{\pi}\displaystyle\int_{0}^{\pi/2}\left(\displaystyle\frac{dt}{\sin t}\right)^{2d}dt=\displaystyle\frac{2}{\pi}i\Big{(}\operatorname{Li}_{2d+1}(-i)-\operatorname{Li}_{2d+1}(i)\Big{)}=\displaystyle\frac{4}{\pi}{\beta}(2d+1),$$
where $\beta$ is the Dirichlet beta function
$$\displaystyle{\beta}(s)=\displaystyle\sum_{k\geq 0}\displaystyle\frac{(-1)^{k}}{(2k+1)^{s}}.$$
(2.22)
Moreover, we have
$$\displaystyle t^{\star}(2_{a},3,2_{b})=\displaystyle\frac{2}{\pi}\displaystyle\int_{0}^{\pi/2}(\csc tdt)^{2a+1}\circ\cot tdt\circ(\csc tdt)^{2b+1}\circ dt,$$
where $a,b\in\mathbb{N}_{0}$.
Remark 2.6.
In [15], T. Murakami first proved the analog of Zagier’s 2-3-2 formula of MZVs for multiple $t$-values $t(2_{a},3,2_{b})$. Thereafter, several other proofs have appeared in the literature, see for example [12].
Proposition 2.7.
For every positive integer $d$, we have
$$\displaystyle{\tau}^{\star}(1_{d};\sin y)=$$
$$\displaystyle\,2\csc 2y\operatorname{{{Im}}}\operatorname{Li}_{d}(i\tan y),$$
$$\displaystyle{\tau}^{\star}(2_{d};\sin y)=$$
$$\displaystyle\,2\csc y\operatorname{{{Im}}}\operatorname{Li}_{2d}\big{(}i\tan(y/2)\big{)}.$$
Proof.
By Thm. 2.3 we obtain
$$\displaystyle{\tau}^{\star}(1_{d};\sin y)=$$
$$\displaystyle\,2\csc 2y\displaystyle\int_{0}^{y}\left(\displaystyle\frac{2dt}{\sin 2t}\right)^{d-1}dt=\displaystyle\frac{2\csc 2y}{(d-1)!}\displaystyle\int_{0}^{y}\left(\displaystyle\int_{t}^{y}\displaystyle\frac{2dx}{\sin 2x}\right)^{d-1}dt$$
$$\displaystyle=$$
$$\displaystyle\,\displaystyle\frac{2\csc 2y}{(d-1)!}\displaystyle\int_{0}^{y}\log^{d-1}\left|\displaystyle\frac{\csc 2t+\cot 2t}{\csc 2y+\cot 2y}\right|dt$$
$$\displaystyle=$$
$$\displaystyle\,\displaystyle\frac{2\csc 2y}{(d-1)!}\displaystyle\int_{0}^{y}\log^{d-1}\big{|}\tan y\cot t\big{|}\,dt.$$
(2.23)
By routine differentiation and induction it can be proved easily that
$$\displaystyle\displaystyle\frac{1}{(d-1)!}\displaystyle\int_{0}^{y}\log^{d-1}\big{|}\tan y\cot t\big{|}\,dt=\operatorname{{{Im}}}\operatorname{Li}_{d}(i\tan y)=\displaystyle\frac{i}{2}\Big{(}\operatorname{Li}_{d}(-i\tan y)-\operatorname{Li}_{d}(i\tan y)\Big{)}$$
for all $d\geq 1$. To see that both sides $\to 0$ as $y\to 0$ we can use (2.23).
Similarly,
$$\displaystyle{\tau}^{\star}(2_{d};\sin y)=$$
$$\displaystyle\,\csc y\displaystyle\int_{0}^{y}\left(\displaystyle\frac{dt}{\sin t}\right)^{2d-1}dt=\displaystyle\frac{\csc y}{(2d-1)!}\displaystyle\int_{0}^{y}\left(\displaystyle\int_{t}^{y}\displaystyle\frac{dx}{\sin x}\right)^{2d-1}dt$$
$$\displaystyle=$$
$$\displaystyle\,\displaystyle\frac{\csc y}{(2d-1)!}\displaystyle\int_{0}^{y}\log^{2d-1}\big{|}\tan y/2\cot t/2\big{|}\,dt=2\csc y\operatorname{{{Im}}}\operatorname{Li}_{2d}\big{(}i\tan(y/2)\big{)}.$$
This completes the proof of the proposition.
∎
Example 2.8.
Specializing at $y=\pi/4$ and $\pi/2$ in the two identities of Prop. 2.7 respectively, we see that
$$\displaystyle\displaystyle\sum_{n_{1}\geq\cdots\geq n_{d}\geq 0}{\binom{2n_{1}}{n_{1}}}^{-1}\displaystyle\frac{2^{n_{1}}}{(2n_{1}+1)\cdots(2n_{d}+1)}=$$
$$\displaystyle\,2\operatorname{{{Im}}}\operatorname{Li}_{d}(i)=2{\beta}(d),$$
$$\displaystyle\displaystyle\sum_{n_{1}\geq\cdots\geq n_{d}\geq 0}{\binom{2n_{1}}{n_{1}}}^{-1}\displaystyle\frac{4^{n_{1}}}{(2n_{1}+1)^{2}\cdots(2n_{d}+1)^{2}}=$$
$$\displaystyle\,2\operatorname{{{Im}}}\operatorname{Li}_{2d}(i)=2{\beta}(2d),$$
where ${\beta}$ is the Dirichlet beta function defined by (2.22).
3 Second variant with odd summation indices
In this section, we consider another variation of the Apéry-type series (1.4) by restricting
the summation indices to odd numbers only and keeping the strict inequalities among them. These
series do not behave as well as the first variant studied in the last section but are still of interest.
Define the 1-forms
$$\displaystyle{\kappa}_{s}(t)=\left\{\begin{array}[]{ll}\sin t\,dt\,\csc t\,dt+\tan t\,dt,&\hbox{if $s=1$;}\\
\sin t\,dt(\cot t\,dt+1)(\cot t\,dt)^{s-2}\csc t\,dt,&\hbox{if $s\geq 2$,}\end{array}\right.$$
(3.26)
and their non-trigonometric counterpart
$$\displaystyle K_{s}(t)=\left\{\begin{array}[]{ll}{\omega}_{5}{\omega}_{3}+{\omega}_{2},&\hbox{if $s=1$;}\\
{\omega}_{5}({\omega}_{0}+1){\omega}_{0}^{s-2}{\omega}_{3},&\hbox{if $s\geq 2$.}\end{array}\right.$$
(3.29)
Theorem 3.1.
For all $n\in\mathbb{N}_{0}$ and ${\boldsymbol{\sl{s}}}=(s_{1},\dots,s_{d})\in\mathbb{N}^{d}$ the tail
$$\displaystyle\chi({\boldsymbol{\sl{s}}};\sin y)_{n}:=\displaystyle\sum_{n_{1}>\cdots>n_{d}>n}\displaystyle\frac{b_{n_{1}}(\sin y)}{(2n_{1}-1)^{s_{1}}\cdots(2n_{d}-1)^{s_{d}}}=\displaystyle\frac{d}{dy}\displaystyle\int_{0}^{y}{\kappa}_{s_{1}}\circ\cdots\circ{\kappa}_{s_{d}}\circ b_{n}(\sin t)\,dt.$$
In the above $y\in[-\pi/2,\pi/2]$ if $s_{1}>1$ and $y\in(-\pi/2,\pi/2)$ if $s_{1}=1$. Using
non-trigonometric 1-forms, for $x\in(-1,1)$ we have
$$\displaystyle\chi({\boldsymbol{\sl{s}}};x)_{n}:=\displaystyle\sum_{n_{1}>\cdots>n_{d}>n}\displaystyle\frac{b_{n_{1}}(x)}{(2n_{1}-1)^{s_{1}}\cdots(2n_{d}-1)^{s_{d}}}=\sqrt{1-x^{2}}\displaystyle\frac{d}{dx}\displaystyle\int_{0}^{x}K_{s_{1}}\circ\cdots\circ K_{s_{d}}\circ b_{n}(t){\omega}_{1}.$$
Proof.
With $s=2$ the identity (2.21) yields that
$$\displaystyle\displaystyle\sum_{n_{1}>n}{\binom{2n_{1}-2}{n_{1}-1}}^{-1}\displaystyle\frac{4^{n_{1}-1}\sin^{2n_{1}-1}y}{(2n_{1}-1)^{2}}=\displaystyle\int_{0}^{y}(\csc t\,dt)b_{n}(\sin t)\,dt.$$
Noting that
$${\binom{2n_{1}-2}{n_{1}-1}}^{-1}={\binom{2n_{1}}{n_{1}}}^{-1}\displaystyle\frac{2(2n_{1}-1)}{n_{1}},$$
we have
$$\displaystyle\displaystyle\sum_{n_{1}>n}\displaystyle\frac{b_{n_{1}}(\sin y)}{n_{1}(2n_{1}-1)}=2\sin y\displaystyle\int_{0}^{y}(\csc t\,dt)b_{n}(\sin t)\,dt.$$
Differentiating, we obtain
$$\begin{split}\displaystyle\sum_{n_{1}>n}\displaystyle\frac{b_{n_{1}}\sin^{2n_{1}-1}y\cos y}{2n_{1}-1}=&\,\cos y\displaystyle\int_{0}^{y}(\csc t\,dt)b_{n}(\sin t)\,dt+\displaystyle\int_{0}^{y}b_{n}(\sin t)\,dt.\end{split}$$
(3.30)
Multiplying (3.30) by $\tan y$ we get
$$\displaystyle\displaystyle\sum_{n_{1}>n}\displaystyle\frac{b_{n_{1}}(\sin y)}{2n_{1}-1}=$$
$$\displaystyle\,\sin y\displaystyle\int_{0}^{y}(\csc t\,dt+\sec y)b_{n}(\sin t)\,dt.$$
(3.31)
Dividing (3.30) by $\sin y$ and integrating
$$\displaystyle\displaystyle\sum_{n_{1}>n}\displaystyle\frac{b_{n_{1}}\sin^{2n_{1}-1}y}{(2n_{1}-1)^{2}}=$$
$$\displaystyle\,\displaystyle\int_{0}^{y}(\cot t\,dt)(\csc t\,dt)b_{n}(\sin t)\,dt+\displaystyle\int_{0}^{y}(\csc t\,dt)b_{n}(\sin t)\,dt.$$
Repeatedly multiplying by $\cot y$ and integrating, we see that for all $s\geq 2$
$$\displaystyle\displaystyle\sum_{n_{1}>n}\displaystyle\frac{b_{n_{1}}\sin^{2n_{1}-1}y}{(2n_{1}-1)^{s}}=$$
$$\displaystyle\,\displaystyle\int_{0}^{y}(\cot t\,dt)^{s-1}(\csc t\,dt)b_{n}(\sin t)\,dt+\displaystyle\int_{0}^{y}(\cot t\,dt)^{s-2}(\csc t\,dt)b_{n}(\sin t)\,dt.$$
Hence if $s\geq 2$ then
$$\displaystyle\displaystyle\sum_{n_{1}>n}\displaystyle\frac{b_{n_{1}}(\sin y)}{(2n_{1}-1)^{s}}=$$
$$\displaystyle\,\sin y\displaystyle\int_{0}^{y}(\cot t\,dt+1)(\cot t\,dt)^{s-2}(\csc t\,dt)b_{n}(\sin t)\,dt.$$
(3.32)
The theorem now follows from repeatedly applying (3.31) or (3.32) at each depth.
This concludes the proof of the theorem.
∎
4 Variant with summation indices of mixed parities
By combining Thm. 1.1, Thm. 2.3, and Thm. 3.1 we obtain
the following result easily.
Theorem 4.1.
Suppose $d\in\mathbb{N}$ and ${\boldsymbol{\sl{s}}}=(s_{1},\dots,s_{d})\in\mathbb{N}^{d}$. Let $y\in(-\pi/2,\pi/2)$ if $s_{1}=1$ and $y\in[-\pi/2,\pi/2]$ if $s_{1}\geq 2$.
Set ${\lambda}_{2n,s}(t)=g_{s}(t)$, ${\lambda}_{2n+1,s}(t)=h_{s}(t)$ and ${\lambda}_{2n-1,s}(t)={\kappa}_{s}(t)$ which are defined by (1.7), (2.14) and (3.26), respectively.
Then for any $l_{1}(n),\dots,l_{d}(n)=2n,2n\pm 1$ we have the tails
$$\displaystyle\displaystyle\sum_{n_{1}\ \underset{1}{\succ}\ \dots\underset{d-1}{\succ}n_{d}\ \underset{d}{\succ}\ n}\displaystyle\frac{b_{n_{1}}(\sin y)}{l_{1}(n_{1})^{s_{1}}\cdots l_{d}(n_{d})^{s_{d}}}=\displaystyle\frac{d}{dy}\displaystyle\int_{0}^{y}{\lambda}_{l_{1},s_{1}}\circ\cdots{\lambda}_{l_{d},s_{d}}\circ b_{n}(\sin t)\,dt,$$
(4.33)
where “$\underset{j}{\succ}$” is “$\geq$” if $l_{j}(n)=2n+1$ and is “$>$” otherwise.
Using non-trigonometric 1-forms, we get for all $x\in(-1,1)$
$$\displaystyle\displaystyle\sum_{n_{1}\ \underset{1}{\succ}\ \dots\underset{d-1}{\succ}n_{d}\ \underset{d}{\succ}\ n}\displaystyle\frac{b_{n_{1}}(x)}{l_{1}(n_{1})^{s_{1}}\cdots l_{d}(n_{d})^{s_{d}}}=\sqrt{1-x^{2}}\displaystyle\frac{d}{dx}\displaystyle\int_{0}^{x}{\Lambda}_{l_{1},s_{1}}\circ\cdots{\Lambda}_{l_{d},s_{d}}\circ b_{n}(t){\omega}_{1},$$
(4.34)
where ${\Lambda}_{2n,s}(t)=G_{s}(t)$, ${\Lambda}_{2n+1,s}(t)=H_{s}(t)$ and ${\Lambda}_{2n-1,s}(t)=K_{s}(t)$ which are defined
by (1.8), (2.17) and (3.29), respectively.
Proof.
Define
$$\displaystyle f_{1}(t):=1,\quad f_{2}(t):=\displaystyle\frac{t}{\sqrt{1-t^{2}}},\quad f_{3}(t)=\displaystyle\frac{1}{t},\quad f_{20}(t):=\displaystyle\frac{1}{t\sqrt{1-t^{2}}},\quad f_{5}(t)=t.$$
By Thm. 1.1, Thm. 2.3, and Thm. 3.1
$$\displaystyle\displaystyle\sum_{n_{1}>n}\displaystyle\frac{b_{n_{1}}(x)}{2n_{1}}=$$
$$\displaystyle\,f_{2}(x)\displaystyle\int_{0}^{x}b_{n}(t){\omega}_{1},$$
(4.35)
$$\displaystyle\displaystyle\sum_{n_{1}>n}\displaystyle\frac{b_{n_{1}}(x)}{(2n_{1})^{s}}=$$
$$\displaystyle\,f_{1}(x)\displaystyle\int_{0}^{x}{\omega}_{0}^{s-2}{\omega}_{1}\,b_{n}(t){\omega}_{1}\quad\forall s\geq 2,$$
(4.36)
$$\displaystyle\displaystyle\sum_{n_{1}\geq n}\displaystyle\frac{b_{n_{1}}(x)}{2n_{1}+1}=$$
$$\displaystyle\,f_{20}(x)\displaystyle\int_{0}^{x}b_{n}(t){\omega}_{1},$$
(4.37)
$$\displaystyle\displaystyle\sum_{n_{1}\geq n}\displaystyle\frac{b_{n_{1}}(x)}{(2n_{1}+1)^{s}}=$$
$$\displaystyle\,f_{3}(x)\displaystyle\int_{0}^{x}{\omega}_{0}^{s-2}{\omega}_{3}\,b_{n}(t){\omega}_{1}\quad\forall s\geq 2,$$
(4.38)
$$\displaystyle\displaystyle\sum_{n_{1}>n}\displaystyle\frac{b_{n_{1}}(x)}{2n_{1}-1}=$$
$$\displaystyle\,f_{5}(x)\displaystyle\int_{0}^{x}{\omega}_{3}\,b_{n}(t){\omega}_{1}+f_{2}(x)\displaystyle\int_{0}^{x}b_{n}(t){\omega}_{1},$$
(4.39)
$$\displaystyle\displaystyle\sum_{n_{1}>n}\displaystyle\frac{b_{n_{1}}(x)}{(2n_{1}-1)^{s}}=$$
$$\displaystyle\,f_{5}(x)\displaystyle\int_{0}^{x}({\omega}_{0}+1){\omega}_{0}^{s-2}{\omega}_{3}\,b_{n}(t){\omega}_{1}\quad\forall s\geq 2.$$
(4.40)
For convenience, we call the right-hand side of (4.35) and (4.36)
(resp. (4.37) and (4.38), resp. (4.39) and (4.40))
a ${\sigma}$-block (resp. ${\tau}^{\star}$-block, resp. $\chi$-block). In (4.34),
each $s_{j}$ corresponds to (a variation of) such a block. Observing that $f_{j}(t){\omega}_{1}={\omega}_{j}$ we find that
after starting with a block in (4.35)-(4.40),
all the middle blocks should be modified as follows: (i) change $f_{j}(x)$ to ${\omega}_{j}$, (ii) remove the
integral sign, and (iii) drop the 1-form $b_{n}(t){\omega}_{1}$. Repeating this until the end block,
for which only operations (i) and (ii) are required.
This concludes the constructive proof of the theorem.
∎
Remark 4.2.
We note that Apéry-type series with indices of mixed parity already appeared implicitly
in [9, (1.1)]. Indeed, using their notation one need to consider, for e.g., the following series:
$$\displaystyle\displaystyle\sum_{j=1}^{\infty}\displaystyle\frac{1}{\binom{2j}{j}}\displaystyle\frac{u^{j}}{j^{c}}S_{a}(2j-1)=$$
$$\displaystyle\,2^{c}\displaystyle\sum_{j=1}^{\infty}\displaystyle\frac{1}{\binom{2j}{j}}\displaystyle\frac{u^{j}}{(2j)^{c}}\left(\displaystyle\sum_{k=0}^{j-1}\displaystyle\frac{1}{(2k+1)^{a}}+\displaystyle\sum_{k=1}^{j-1}\displaystyle\frac{1}{(2k)^{a}}\right)$$
$$\displaystyle=$$
$$\displaystyle\,2^{c}\displaystyle\sum_{j>k\geq 0}\displaystyle\frac{1}{\binom{2j}{j}}\displaystyle\frac{u^{j}}{(2j)^{c}(2k+1)^{a}}+2^{c}\displaystyle\sum_{j>k>0}\displaystyle\frac{1}{\binom{2j}{j}}\displaystyle\frac{u^{j}}{(2j)^{c}(2k)^{a}}.$$
Example 4.3.
By composing (4.36) and (4.37) we see that
$$\displaystyle\displaystyle\sum_{n_{1}\geq n_{2}>0}\displaystyle\frac{b_{n_{1}}(x)}{(2n_{1}+1)(2n_{2})^{2}}=\displaystyle\frac{1}{x\sqrt{1-x^{2}}}\displaystyle\int_{0}^{x}{\omega}_{1}^{3}.$$
Taking $x=1/2,\sqrt{3}/2$ we see immediately that
$$\displaystyle\displaystyle\sum_{n_{1}\geq n_{2}>0}\binom{2n_{1}}{n_{1}}^{-1}\displaystyle\frac{1}{(2n_{1}+1)(2n_{2})^{2}}=\displaystyle\frac{4}{\sqrt{3}}\displaystyle\frac{(\sin^{-1}(1/2))^{3}}{3!}=\displaystyle\frac{\pi^{3}}{4\cdot 81\sqrt{3}},$$
$$\displaystyle\displaystyle\sum_{n_{1}\geq n_{2}>0}\binom{2n_{1}}{n_{1}}^{-1}\displaystyle\frac{3^{n}}{(2n_{1}+1)(2n_{2})^{2}}=\displaystyle\frac{4}{\sqrt{3}}\displaystyle\frac{(\sin^{-1}(\sqrt{3}/2))^{3}}{3!}=\displaystyle\frac{2\pi^{3}}{81\sqrt{3}}.$$
These are consistent with the first two identities at the beginning of [1]. Many other evaluations in
the loc. cit. can be verified using similar ideas by repeatedly applying (4.35)–(4.40).
Taking $x=\sqrt{2}/2$ we also get
$$\displaystyle\displaystyle\sum_{n_{1}\geq n_{2}>0}\binom{2n_{1}}{n_{1}}^{-1}\displaystyle\frac{2^{n}}{(2n_{1}+1)(2n_{2})^{2}}=\displaystyle\frac{4}{\sqrt{3}}\displaystyle\frac{(\sin^{-1}(\sqrt{2}/2))^{3}}{3!}=\displaystyle\frac{\pi^{3}}{96\sqrt{3}}.$$
By specializing at $y=\pi/2$ (or taking limit as $x\to 1^{-}$) we obtain the following theorem,
which helps answer two questions at the end of [20] affirmatively in Cor. 5.1.
Theorem 4.4.
Suppose $d\in\mathbb{N}$, ${\boldsymbol{\sl{s}}}=(s_{1},\dots,s_{d})\in\mathbb{N}^{d}$ and $s_{1}\geq 2$.
Set $\delta(l)=0$ if $l(n)=2n$ and $\delta(l)=1$ if $l(n)=2n\pm 1$.
(a)
Suppose $l_{1}(n),\dots,l_{d}(n)=2n,2n+1$.
Then we have
$$\displaystyle\displaystyle\sum_{n_{1}\ \underset{1}{\succ}\ \cdots\ \underset{d-1}{\succ}n_{d}\ \underset{d}{\succ}\ 0}\displaystyle\frac{b_{n_{1}}}{l_{1}(n_{1})^{s_{1}}\cdots l_{d}(n_{d})^{s_{d}}}\in i^{\delta(l_{1})}\mathsf{CMZV}_{|{\boldsymbol{\sl{s}}}|}^{4},$$
where “$\underset{j}{\succ}$” is “$\geq$” if $l_{j}(n)=2n+1$ and is “$>$” otherwise.
(b)
Suppose $l_{1}(n),\dots,l_{d}(n)=2n,2n\pm 1$. If for all $l_{j}(n)=2n-1$ ($j\geq 2$) we have $l_{j-1}(n)\neq 2n$, then
we have
$$\displaystyle\displaystyle\sum_{n_{1}\ \underset{1}{\succ}\ \cdots\ \underset{d-1}{\succ}n_{d}\ \underset{d}{\succ}\ 0}\displaystyle\frac{b_{n_{1}}}{l_{1}(n_{1})^{s_{1}}\cdots l_{d}(n_{d})^{s_{d}}}\in i^{\delta(l_{1})}\Big{(}\mathsf{CMZV}_{|{\boldsymbol{\sl{s}}}|}^{4}+\nu(l_{1})\mathsf{CMZV}_{|{\boldsymbol{\sl{s}}}|+1}^{4}\Big{)},$$
(4.41)
where $\nu(l)=1$ if $l(n)=2n-1$ and $\nu(l)=0$ otherwise. In particular, if $l_{j}(n)\neq 2n$ for all $j$ then
(4.41) holds.
(c)
More generally, $l_{1}(n),\dots,l_{d}(n)=2n,2n\pm 1$ then we have
$$\displaystyle\displaystyle\sum_{n_{1}\ \underset{1}{\succ}\ \cdots\ \underset{d-1}{\succ}n_{d}\ \underset{d}{\succ}\ 0}\displaystyle\frac{b_{n_{1}}}{l_{1}(n_{1})^{s_{1}}\cdots l_{d}(n_{d})^{s_{d}}}\in\mathsf{CMZV}_{\leq|{\boldsymbol{\sl{s}}}|+\nu(l_{1})}^{4}\otimes\mathbb{Q}[i].$$
(4.42)
(d)
Moreover, the claim in (4.42) still holds if one changes any of the strict inequalities
$n_{j}>n_{j+1}$ to $n_{j}\geq n_{j+1}$ and vice versa, provided the series is defined.
Here we set $n_{d+1}=0$. In particular, if $l_{1}(n),\dots,l_{d}(n)=2n,2n\pm 1$ then
$$\displaystyle\displaystyle\sum_{n_{1}\succ n_{2}\succ\cdots\ \succ n_{d}\succ\ 0}\displaystyle\frac{b_{n_{1}}}{l_{1}(n_{1})^{s_{1}}\cdots l_{d}(n_{d})^{s_{d}}}\in\mathsf{CMZV}_{\leq|{\boldsymbol{\sl{s}}}|+\nu(l_{1})}^{4}\otimes\mathbb{Q}[i],$$
(4.43)
where “$\succ$” can be either “$\geq$” or “$>$”, provided the series is defined.
Remark 4.5.
Let $q=\displaystyle\max\{j:l_{j}(n)\neq 2n+1\}$. The series is defined if and only if “$\underset{q}{\succ}$” is “$>$”.
Proof.
Put ${\texttt{x}}_{\xi}=dt/(\xi-t)$ for any $\xi\in\mathbb{C}$ and ${\texttt{d}}_{\xi,\xi^{\prime}}={\texttt{x}}_{\xi}-{\texttt{x}}_{\xi^{\prime}}$.
First, we observe that under the change of variables
$$\displaystyle t\to\sin^{-1}t\quad\text{then}\quad t\to\displaystyle\frac{1-t^{2}}{1+t^{2}},$$
(4.44)
we have
$$\displaystyle\cot t\,dt\to$$
$$\displaystyle\,{\omega}_{0}={\texttt{a}}:=\displaystyle\frac{dt}{t}\to{\texttt{y}},\quad$$
$$\displaystyle\csc t\,dt\to$$
$$\displaystyle\,{\omega}_{3}:=\displaystyle\frac{dt}{t\sqrt{1-t^{2}}}\to{\texttt{d}}_{-1,1},$$
(4.45)
$$\displaystyle dt\to$$
$$\displaystyle\,{\omega}_{1}:=\displaystyle\frac{dt}{\sqrt{1-t^{2}}}\to i{\texttt{d}}_{-i,i},\quad$$
$$\displaystyle\sec t\csc t\,dt\to$$
$$\displaystyle\,{\omega}_{20}:=\displaystyle\frac{dt}{t(1-t^{2})}\to{\texttt{y}}+{\texttt{z}},$$
(4.46)
$$\displaystyle\tan t\,dt\to$$
$$\displaystyle\,{\omega}_{2}:=\displaystyle\frac{t\,dt}{1-t^{2}}\to{\texttt{z}},\quad$$
$$\displaystyle\sec t\,dt\to$$
$$\displaystyle\,{\omega}_{8}:=\displaystyle\frac{dt}{1-t^{2}}\to-{\texttt{a}},$$
(4.47)
where ${\texttt{y}}={\texttt{x}}_{-i}+{\texttt{x}}_{i}-{\texttt{x}}_{-1}-{\texttt{x}}_{1}$ and ${\texttt{z}}=-{\texttt{a}}-{\texttt{x}}_{-i}-{\texttt{x}}_{i}$.
Furthermore, we notice
$$\displaystyle\sin t\,dt\to{\omega}_{5}$$
$$\displaystyle\,=\displaystyle\frac{t\,dt}{\sqrt{1-t^{2}}}\to\displaystyle\frac{dt}{(i-t)^{2}}+\displaystyle\frac{dt}{(i+t)^{2}}.$$
(4.48)
Let
$$\displaystyle{\mathsf{O}}:=\mathbb{Q}\left\langle{\omega}_{j}:0\leq j\leq 3\text{ or }j=5\right\rangle.$$
By repeatedly using the six cases (4.35)–(4.40) we see that every sum in (4.42)
can be expressed as $\mathbb{Q}$-linear combinations of the following form
$$\displaystyle\displaystyle\int_{0}^{1}{\alpha}_{1}\dots\alpha_{m}$$
(4.49)
with $m\leq|{\boldsymbol{\sl{s}}}|$ and ${\alpha}_{j}\in{\mathsf{O}}$.
(a) In this case ${\omega}_{5}$ never appears and the weight in (4.35)–(4.38)
is always the same as the number of 1-forms appearing on the right-hand side so that there is
no weight drop. Thus we only need to consider the number ${\omega}_{1}$’s appearing in (4.34).
From (4.45)–(4.47) we see that
only ${\omega}_{1}$ produces $i$ after the change of variables $t\to(1-t^{2})/(1+t^{2})$.
Among the four cases (4.35)–(4.38) only iteration (4.36) affects the number of ${\omega}_{1}$’s in (4.42), by adding two ${\omega}_{1}$’s. Then ending block $b_{n}(t){\omega}_{1}={\omega}_{1}$ when $n=0$. So we
need to consider the starting 1-form inside the iterated integral of (4.34),
which is chopped off after taking the derivative $d/dx$ and then multiplied by $\sqrt{1-x^{2}}$.
This 1-form is ${\omega}_{1}$ if and only if it is a ${\sigma}$-block with $s_{1}\geq 2$.
Hence, after taking $d/dx$ we find that on the right-hand side of (4.34)
the total number of ${\omega}_{1}$’s is even for a starting ${\sigma}$-block (i.e., $l_{1}(n)=2n$) and the number is odd
for a starting ${\tau}^{\star}$-block (i.e., $l_{1}(n)=2n+1$). The claim of (a) is thus proved.
(b)
We first claim that we may reduce this case to the case where $l_{j}(n)=2n-1$ appears only when $j=1$.
We will prove this by induction on the depth. We have nothing to do when the depth is 1. In general,
we change the index $n_{j}\to n_{j}+1$ for all $j\geq 2$ such that $l_{j}(n)=2n-1$.
Then we need to consider the following three possible blocks in front of $j$-th block.
Setting $k=n_{j-1},m=n_{j},r=s_{j-1},s=s_{j}$, we see that
$$\displaystyle\displaystyle\sum_{k>m}\displaystyle\frac{1}{(2k-1)^{r}(2m-1)^{s}}\Longrightarrow\displaystyle\sum_{k>m}\displaystyle\frac{1}{(2k+1)^{r}(2m+1)^{s}}=\displaystyle\sum_{k\geq m}\displaystyle\frac{1}{(2k+1)^{r}(2m+1)^{s}}-\displaystyle\frac{1}{(2k+1)^{r+s}},$$
$$\displaystyle\displaystyle\sum_{k\geq m}\displaystyle\frac{1}{(2k+1)^{r}(2m-1)^{s}}\Longrightarrow\displaystyle\sum_{k\geq m+1}\displaystyle\frac{1}{(2k+1)^{r}(2m+1)^{s}}$$
$$\displaystyle\hskip 142.26378pt=\displaystyle\sum_{k\geq m}\displaystyle\frac{1}{(2k+1)^{r}(2m+1)^{s}}-\displaystyle\frac{1}{(2k+1)^{r+s}}.$$
For the possible ${\tau}^{\star}$-block after the $j$-th block, setting $k=n_{j+1},m=n_{j},r=s_{j+1},s=s_{j}$ we have
$$\displaystyle\displaystyle\sum_{k>m}\displaystyle\frac{1}{(2m-1)^{s}(2k+1)^{r}}\Longrightarrow\displaystyle\sum_{k+1>m}\displaystyle\frac{1}{(2m+1)^{s}(2k+1)^{r}}=\displaystyle\sum_{k\geq m}\displaystyle\frac{1}{(2m+1)^{s}(2k+1)^{r}}$$
which looks in good shape. We also need to consider the possible ${\sigma}$-blocks after the $j$-th block:
(setting $k=n_{j+1},m=n_{j},r=s_{j+1},s=s_{j}$)
$$\displaystyle\displaystyle\sum_{k>m}\displaystyle\frac{1}{(2k-1)^{s}(2m)^{r}}\Longrightarrow\displaystyle\sum_{k+1>m}\displaystyle\frac{1}{(2k+1)^{s}(2m)^{r}}=\displaystyle\sum_{k\geq m}\displaystyle\frac{1}{(2k+1)^{s}(2m)^{r}},$$
which looks in good shape, too.
To summarize, the above shows that if no ${\sigma}$-$\chi$-block chain appears then we see that
no weight drops can occur in the decomposed sums.
Furthermore, from the above, we can assume the $\chi$-block appears only as the first block, if it ever does. By
the explicit iterated integral expressions of the ${\sigma}$- and ${\tau}^{\star}$-blocks (4.35)–(4.38),
we see that if $\chi$-block does not appear then all the CMZVs involved are of the same weight. If a $\chi$-block
appears at the beginning then (4.40) shows that the weight can increase by one for some CMZVs
and the counting of ${\omega}_{1}$ is the same as the case with a starting ${\tau}^{\star}$-block.
This completes the proof of (b).
(c)
As the proof of (b), we first claim that we may reduce the general case
to the case where $l_{j}(n)=2n-1$ appears, if it ever does, then $j=1$.
We will prove this by induction on the depth. We have nothing to do when the depth is 1. In general,
we change the index $n_{j}\to n_{j}+1$ for all $j\geq 2$ such that $l_{j}(n)=2n-1$.
Then we need to consider the following possible block ${\sigma}$ in front of $j$-th block since
the other two possibilities have been already handled by case (b):
(setting $k=n_{j-1},m=n_{j},r=s_{j-1},s=s_{j}$)
$$\displaystyle\displaystyle\sum_{k>m}\displaystyle\frac{1}{(2k)^{r}(2m-1)^{s}}\Longrightarrow\displaystyle\sum_{k>m+1}\displaystyle\frac{1}{(2k)^{r}(2m+1)^{s}}=\displaystyle\sum_{k>m}\displaystyle\frac{1}{(2k)^{r}(2m+1)^{s}}-\displaystyle\frac{1}{(2k)^{r}(2k-1)^{s}}.$$
Thus by partial fractions we may decompose the second term above as pure powers of either $2k$ or $2k-1$,
thus reducing the depth by 1. We point out that this is also the reason why
the weight may drop due to the partial fractions when ${\sigma}$-$\chi$-block chain appears.
Thus we will assume the $\chi$-block appears only as the first block. Then this case is proved again
by the explicit formula (4.40).
(d) Observe that for any fixed $n$,
$$\displaystyle\displaystyle\sum_{m\geq n}\displaystyle\frac{1}{m^{s}(2n\pm 1)^{t}}=$$
$$\displaystyle\,\displaystyle\frac{1}{n^{s}(2n\pm 1)^{t}}+\displaystyle\sum_{m>n}\displaystyle\frac{1}{m^{s}(2n\pm 1)^{t}},$$
$$\displaystyle\displaystyle\sum_{m>n}\displaystyle\frac{1}{(2m+1)^{t}n^{s}}=$$
$$\displaystyle\,-\displaystyle\frac{1}{n^{s}(2n+1)^{t}}+\displaystyle\sum_{m\geq n}\displaystyle\frac{1}{(2m+1)^{t}n^{s}},$$
$$\displaystyle\displaystyle\sum_{m>n}\displaystyle\frac{1}{(2m+1)^{t}(2n-1)^{s}}=$$
$$\displaystyle\,-\displaystyle\frac{1}{(2n+1)^{t}(2n-1)^{s}}+\displaystyle\sum_{m\geq n}\displaystyle\frac{1}{(2m+1)^{t}(2n-1)^{s}},$$
$$\displaystyle\displaystyle\sum_{m\geq n}\displaystyle\frac{1}{(2m-1)^{t}n^{s}}=$$
$$\displaystyle\,-\displaystyle\frac{1}{(2n-1)^{t}n^{s}}+\displaystyle\sum_{m>n}\displaystyle\frac{1}{(2m+1)^{t}n^{s}},$$
$$\displaystyle\displaystyle\sum_{m\geq n}\displaystyle\frac{1}{(2m-1)^{t}(2n+1)^{s}}=$$
$$\displaystyle\,\displaystyle\frac{1}{(2n-1)^{t}(2n+1)^{s}}+\displaystyle\sum_{m>n}\displaystyle\frac{1}{(2m+1)^{t}(2n+1)^{s}}.$$
By partial fraction decomposition we can reduce all the first terms on the right-hand side of the above to a
$\mathbb{Q}$-linear combination of single powers of either $1/n$ or $1/(2n\pm 1)$. The only complication
is that when $j=1$ non-admissible terms may appear, which requires us to use the shuffle regularization.
This is similar to the proof of [20, Thm. 9.6] using
[20, Lemma. 9.4 and Lemma 9.5]. We leave the details to the interested reader
who may also refer to Example 5.10 (especially the computation for $S_{2}$ on page 5.10)
for the detailed steps to carry this out explicitly.
∎
Corollary 4.6.
Let $d\in\mathbb{N}$ and ${\boldsymbol{\sl{s}}}=(s_{1},\dots,s_{d})\in\mathbb{N}^{d}$. Let ${\boldsymbol{\sl{l}}}=(l_{1},\dots,l_{d})$ where $l_{j}(n)=2n$ or $2n\pm 1$ for all $j=1,\dots,d$.
Let “$\succ$” denote either “$\geq$” or “$>$”. Then for every real algebraic $x$ such that $0<x\leq 1$ the value
$$\displaystyle\Xi({\boldsymbol{\sl{s}}},{\boldsymbol{\sl{l}}};x):=\displaystyle\sum_{n_{1}\succ n_{2}\succ\cdots\ \succ n_{d}\succ\ 0}\displaystyle\frac{b_{n_{1}}(x)}{l_{1}(n_{1})^{s_{1}}\cdots l_{d}(n_{d})^{s_{d}}},$$
if it exists, can be expressed as a $\mathbb{Q}[i,x,\sqrt{1-x^{2}}]$-linear combination of the multiple polylogarithms
evaluated at algebraic points.
Proof.
By the proof of Thm. 4.4, up to factors of $x$ and $\sqrt{1-x^{2}}$ in front
(which can be seen more easily from (4.35)–(4.40)), $\Xi({\boldsymbol{\sl{s}}},{\boldsymbol{\sl{l}}};x)$ can be expressed as
a $\mathbb{Q}$-linear combinations of
$$\displaystyle\displaystyle\int_{0}^{x}\Big{[}{\omega}_{j}:j=0,1,2,3\Big{]}_{\ell}$$
(4.50)
where $\Big{[}{\omega}_{j}:j=0,1,2,3\Big{]}_{|{\boldsymbol{\sl{s}}}|}$ is an iteration of 1-forms of length $\ell\leq|{\boldsymbol{\sl{s}}}|+1$.
Here the 1-form ${\omega}_{5}$ is not needed since the proof of Thm. 4.4 shows that
if $l_{j}(n)=2n-1$ then we may assume $j=1$ (i.e., $\chi$-block only appears at the beginning).
Therefore, after applying the change of variables $t\to\displaystyle\frac{1-t^{2}}{1+t^{2}}$,
by (4.45)–(4.47) we see that (4.50)
is transformed to a $\mathbb{Q}[i]$-linear combination of iterated integrals of the form
$$\displaystyle\displaystyle\int_{{\lambda}(x)}^{1}{\alpha}_{1}\dots\alpha_{\ell},.$$
where ${\lambda}(x)=\sqrt{\displaystyle\frac{1-x}{1+x}},\ {\alpha}_{j}\in\{{\texttt{x}}_{0},{\texttt{x}}_{\mu}:\mu^{8}=1\}.$
Note ${\lambda}(x)\to x$ under the change of variables $t\to\displaystyle\frac{1-t^{2}}{1+t^{2}}$. If $x\neq 1$, to convert this to multiple polylogs we
generally need to use the regularization process. Thus, for an arbitrarily small ${\varepsilon}>0$ we write
$$\displaystyle\displaystyle\int_{{\lambda}(x)}^{1}{\alpha}_{1}\dots\alpha_{\ell}=$$
$$\displaystyle\,\displaystyle\sum_{j=0}^{\ell}\displaystyle\int_{{\lambda}(x)}^{\varepsilon}{\alpha}_{1}{\alpha}_{2}\dots\alpha_{j}\displaystyle\int_{\varepsilon}^{1}{\alpha}_{j+1}\dots\alpha_{\ell}$$
$$\displaystyle=$$
$$\displaystyle\,\displaystyle\sum_{j=0}^{\ell}(-1)^{j}\displaystyle\int_{\varepsilon}^{{\lambda}(x)}{\alpha}_{j}\dots\alpha_{2}{\alpha}_{1}\displaystyle\int_{\varepsilon}^{1}{\alpha}_{j+1}\dots\alpha_{\ell}$$
$$\displaystyle=$$
$$\displaystyle\,\displaystyle\sum_{j=0}^{\ell}(-1)^{j}\displaystyle\int_{{\varepsilon}/{\lambda}(x)}^{1}{\alpha}^{\prime}_{j}\dots\alpha^{\prime}_{2}{\alpha}^{\prime}_{1}\displaystyle\int_{\varepsilon}^{1}{\alpha}_{j+1}\dots\alpha_{\ell}$$
where ${\alpha}^{\prime}_{k}={\texttt{x}}_{\xi/{\lambda}(x)}$ if ${\alpha}_{k}={\texttt{x}}_{\xi}$ where $\xi=0$ or $\xi^{8}=1$.
Note that $\xi/{\lambda}(x)$ is still algebraic.
By the usual regularization procedure we see that the last expression can be written as a polynomial $P(\log({\varepsilon}))$
plus $O({\varepsilon}\log^{\ell}({\varepsilon}))$, such that all the coefficients of $P$ are $\mathbb{Q}$-linear combination of the multiple polylogarithms
evaluated at algebraic points. Here we have used the fact that $\log({\lambda}(x))=\tfrac{1}{2}\big{(}\operatorname{Li}_{1}(-x)-\operatorname{Li}_{1}(x)\big{)}$.
Finally, taking ${\varepsilon}\to 0$ yields the corollary at once.
∎
Theorem 4.7.
Keep notation as in Thm. 4.4. Assume $s_{1}\geq 3$.
(a)
Let $l_{1}(n),\dots,l_{d}(n)=2n,2n\pm 1$. If $l_{1}(n)\neq 2n-1$ and
for all $l_{j}(n)=2n-1$ ($j\geq 2$) we have $l_{j-1}(n)\neq 2n$, then
$$\displaystyle\displaystyle\sum_{n_{1}\ \underset{1}{\succ}\ \cdots\ \underset{d-1}{\succ}n_{d}\ \underset{d}{\succ}\ 0}\displaystyle\frac{b_{n_{1}}^{2}}{l_{1}(n_{1})^{s_{1}}\cdots l_{d}(n_{d})^{s_{d}}}\in\mathsf{CMZV}_{|{\boldsymbol{\sl{s}}}|}^{4}.$$
(4.51)
In particular, if $l_{j}(n)\neq 2n$ for all $j$ then (4.51) holds. If $l_{1}(n)=2n-1$ and
for all $l_{j}(n)=2n-1$ ($j\geq 2$) we have $l_{j-1}(n)\neq 2n$, then
$$\displaystyle\displaystyle\sum_{n_{1}\ \underset{1}{\succ}\ \cdots\ \underset{d-1}{\succ}n_{d}\ \underset{d}{\succ}\ 0}\displaystyle\frac{b_{n_{1}}^{2}}{l_{1}(n_{1})^{s_{1}}\cdots l_{d}(n_{d})^{s_{d}}}\in\mathsf{CMZV}_{|{\boldsymbol{\sl{s}}}|}^{4}+\mathsf{CMZV}_{|{\boldsymbol{\sl{s}}}|+1}^{4}+\mathsf{CMZV}_{|{\boldsymbol{\sl{s}}}|+2}^{4}.$$
(b)
More generally, if $l_{1}(n),\dots,l_{d}(n)=2n,2n\pm 1$ then we have
$$\displaystyle\displaystyle\sum_{n_{1}\ \underset{1}{\succ}\ \cdots\ \underset{d-1}{\succ}n_{d}\ \underset{d}{\succ}\ 0}\displaystyle\frac{b_{n_{1}}^{2}}{l_{1}(n_{1})^{s_{1}}\cdots l_{d}(n_{d})^{s_{d}}}\in\mathsf{CMZV}_{\leq|{\boldsymbol{\sl{s}}}|+2\nu(l_{1})}^{4}\otimes\mathbb{Q}[i].$$
(4.52)
(c)
Moreover, the claim in (b) still holds if one changes any of the strict inequalities
$n_{j}>n_{j+1}$ to $n_{j}\geq n_{j+1}$ in (4.52) and vice versa,
provided the series is defined. In particular,
$$\displaystyle\displaystyle\sum_{n_{1}\ \succ\ \cdots\ \succ\ n_{d}\succ\ 0}\displaystyle\frac{b_{n_{1}}^{2}}{l_{1}(n_{1})^{s_{1}}\cdots l_{d}(n_{d})^{s_{d}}}\in\mathsf{CMZV}_{\leq|{\boldsymbol{\sl{s}}}|+2\nu(l_{1})}^{4}\otimes\mathbb{Q}[i],$$
(4.53)
where “$\succ$” can be either “$\geq$” or “$>$”, provided the series is defined.
Proof.
The key observation is that
$$\displaystyle\displaystyle\int_{0}^{1}\displaystyle\frac{x^{2n+1}}{\sqrt{1-x^{2}}}\,dx=\displaystyle\int_{0}^{\pi/2}\sin^{2n+1}t\,dt=B\Big{(}n+1,\displaystyle\frac{1}{2}\Big{)}=\displaystyle\frac{b_{n}}{2n+1}.$$
(a) When $l_{1}(n)=2n+1$ by (4.34) we see that the sum
$$\displaystyle\displaystyle\sum_{n_{1}\ \underset{1}{\succ}\ \dots\underset{d-1}{\succ}n_{d}\ \underset{d}{\succ}\ 0}\displaystyle\frac{b_{n_{1}}(x)/\sqrt{1-x^{2}}}{l_{1}(n_{1})^{s_{1}}\cdots l_{d}(n_{d})^{s_{d}}}=\displaystyle\frac{1}{x\sqrt{1-x^{2}}}\displaystyle\int_{0}^{x}{\omega}_{0}^{s_{1}-2}{\omega}_{3}\circ{\Lambda}_{l_{2},s_{2}}(t)\circ\cdots{\Lambda}_{l_{d},s_{d}}(t)\circ{\omega}_{1}$$
when $s_{1}\geq 2$. Thus multiplying by $x$ on both sides and integrating over $(0,1)$ we get
$$\displaystyle\displaystyle\sum_{n_{1}\ \underset{1}{\succ}\ \dots\underset{d-1}{\succ}n_{d}\ \underset{d}{\succ}\ 0}\displaystyle\frac{b_{n_{1}}^{2}}{l_{1}(n_{1})^{s_{1}+1}\cdots l_{d}(n_{d})^{s_{d}}}=$$
$$\displaystyle\,\displaystyle\int_{0}^{1}{\omega}_{1}{\omega}_{0}^{s_{1}-2}{\omega}_{3}\circ{\Lambda}_{l_{2},s_{2}}(t)\circ\cdots{\Lambda}_{l_{d},s_{d}}(t)\circ{\omega}_{1}.$$
The claim follows immediately since there are even number of ${\omega}_{1}$’s in this case.
(b) If $l_{1}(n)=2n$ then we see that
$$\displaystyle\displaystyle\sum_{n_{1}\ \underset{1}{\succ}\ \dots\underset{d-1}{\succ}n_{d}\ \underset{d}{\succ}\ 0}\displaystyle\frac{b_{n_{1}}(x)}{l_{1}(n_{1})^{s_{1}}\cdots l_{d}(n_{d})^{s_{d}}}=$$
$$\displaystyle\,\displaystyle\int_{0}^{1}{\omega}_{0}^{s-2}{\omega}_{1}\circ{\Lambda}_{l_{2},s_{2}}(t)\circ\cdots{\Lambda}_{l_{d},s_{d}}(t)\circ{\omega}_{1}$$
when $s_{1}\geq 2$. Then we can divide by $x\sqrt{1-x^{2}}$ and integrate over $(0,1)$ to get
$$\displaystyle\displaystyle\sum_{n_{1}\ \underset{1}{\succ}\ \dots\underset{d-1}{\succ}n_{d}\ \underset{d}{\succ}\ 0}\displaystyle\frac{b_{n_{1}}^{2}}{l_{1}(n_{1})^{s_{1}+1}\cdots l_{d}(n_{d})^{s_{d}}}=$$
$$\displaystyle\,\displaystyle\sum_{n_{1}\ \underset{1}{\succ}\ \dots\underset{d-1}{\succ}n_{d}\ \underset{d}{\succ}\ 0}\displaystyle\frac{b_{n_{1}}b_{n_{1}-1}}{(2n_{1}-1)l_{1}(n_{1})^{s_{1}}\cdots l_{d}(n_{d})^{s_{d}}}$$
$$\displaystyle=$$
$$\displaystyle\,\displaystyle\int_{0}^{1}{\omega}_{3}{\omega}_{0}^{s_{1}-2}{\omega}_{1}\circ{\Lambda}_{l_{2},s_{2}}(t)\circ\cdots{\Lambda}_{l_{d},s_{d}}(t)\circ{\omega}_{1}$$
since
$$\displaystyle\frac{b_{n_{1}-1}}{2n_{1}-1}=\displaystyle\frac{b_{n_{1}}}{2n_{1}}.$$
(4.54)
The theorem holds as well in this case as the number of ${\omega}_{1}$’s is still even.
(c) If $l_{1}(n)=2n-1$ then since $s_{1}\geq 3$ by (4.34) we have
$$\displaystyle\displaystyle\sum_{n_{1}\ \underset{1}{\succ}\ \dots\underset{d-1}{\succ}n_{d}\ \underset{d}{\succ}\ n}\displaystyle\frac{b_{n_{1}}(x)}{l_{1}(n_{1})^{s_{1}}\cdots l_{d}(n_{d})^{s_{d}}}=x\displaystyle\int_{0}^{x}({\omega}_{0}+1){\omega}_{0}^{s-2}{\omega}_{3}\circ{\Lambda}_{l_{2},s_{2}}(t)\circ\cdots{\Lambda}_{l_{d},s_{d}}(t)\circ b_{n}(t){\omega}_{1}.$$
We first differentiate this to get
$$\displaystyle\displaystyle\sum_{n_{1}\ \underset{1}{\succ}\ \dots\underset{d-1}{\succ}n_{d}\ \underset{d}{\succ}\ n}\displaystyle\frac{2n_{1}b_{n_{1}}x^{2n_{1}-1}}{l_{1}(n_{1})^{s_{1}}\cdots l_{d}(n_{d})^{s_{d}}}=$$
$$\displaystyle\,\displaystyle\int_{0}^{x}({\omega}_{0}+1)^{2}{\omega}_{0}^{s-3}{\omega}_{3}\circ{\Lambda}_{l_{2},s_{2}}(t)\circ\cdots{\Lambda}_{l_{d},s_{d}}(t)\circ b_{n}(t){\omega}_{1}.$$
As in the $l_{1}(n)=2n$ case, we can divide by $\sqrt{1-x^{2}}$ and integrate over $(0,1)$ to get
$$\displaystyle\displaystyle\sum_{n_{1}\ \underset{1}{\succ}\ \dots\underset{d-1}{\succ}n_{d}\ \underset{d}{\succ}\ n}\displaystyle\frac{b_{n_{1}}^{2}}{l_{1}(n_{1})^{s_{1}}\cdots l_{d}(n_{d})^{s_{d}}}=\displaystyle\int_{0}^{1}{\omega}_{1}({\omega}_{0}+1)^{2}{\omega}_{0}^{s_{1}-3}{\omega}_{3}\circ{\Lambda}_{l_{2},s_{2}}(t)\circ\cdots{\Lambda}_{l_{d},s_{d}}(t)\circ b_{n}(t){\omega}_{1}$$
by using (4.54) again. This completes the proof of the theorem.
∎
5 A corollary and some examples
In this last section, we will apply our main theorems to compute a few typical Apéry type series to illustrate the power of our method. We can also see how the regularization process is needed in some of the examples.
First, we can answer affirmatively a few questions we posted at the end of [20].
For ${\boldsymbol{\sl{k}}}\in\mathbb{N}^{d}$ and ${\boldsymbol{\sl{l}}}\in\mathbb{N}^{e}$ we define
$$\displaystyle\zeta_{n}({\boldsymbol{\sl{k}}}):=$$
$$\displaystyle\,\displaystyle\sum_{n\geq m_{1}>\dots>m_{d}>0}\displaystyle\frac{1}{m_{1}^{k_{1}}\cdots m_{d}^{k_{d}}},$$
$$\displaystyle t_{n}({\boldsymbol{\sl{l}}}):=$$
$$\displaystyle\,\displaystyle\sum_{n\geq r_{1}>\dots>r_{e}>0}\displaystyle\frac{1}{(2r_{1}-1)^{l_{1}}\cdots(2r_{e}-1)^{l_{e}}}.$$
Corollary 5.1.
For all $m\in\mathbb{N}$, $p\in\mathbb{N}_{\geq 2}$, $q\in\mathbb{N}_{\geq 3}$,
and all compositions of positive integers ${\boldsymbol{\sl{k}}}$ and ${\boldsymbol{\sl{l}}}$ (including the cases ${\boldsymbol{\sl{k}}}=\emptyset$ or ${\boldsymbol{\sl{l}}}=\emptyset$),
we have
$$\displaystyle{\rm(a)}$$
$$\displaystyle\ \ \displaystyle\sum_{n=1}^{\infty}b_{n}\displaystyle\frac{\zeta_{n}({\boldsymbol{\sl{k}}})t_{n}({\boldsymbol{\sl{l}}})}{n^{p}}\in\mathsf{CMZV}_{|{\boldsymbol{\sl{k}}}|+|{\boldsymbol{\sl{l}}}|+p}^{4},$$
$$\displaystyle\quad{\rm(b)}$$
$$\displaystyle\ \ \displaystyle\sum_{n=1}^{\infty}b_{n}^{2}\displaystyle\frac{\zeta_{n}({\boldsymbol{\sl{k}}})t_{n}({\boldsymbol{\sl{l}}})}{n^{q}}\in\mathsf{CMZV}^{4}_{|{\boldsymbol{\sl{k}}}|+|{\boldsymbol{\sl{l}}}|+q},$$
$$\displaystyle{\rm(c)}$$
$$\displaystyle\ \ \displaystyle\sum_{n=0}^{\infty}b_{n}\displaystyle\frac{\zeta_{n}({\boldsymbol{\sl{k}}})t_{n}({\boldsymbol{\sl{l}}})}{(2n+1)^{p}}\in i\mathsf{CMZV}^{4}_{|{\boldsymbol{\sl{k}}}|+|{\boldsymbol{\sl{l}}}|+p},$$
$$\displaystyle\quad{\rm(d)}$$
$$\displaystyle\ \ \displaystyle\sum_{n=0}^{\infty}b_{n}^{2}\displaystyle\frac{\zeta_{n}({\boldsymbol{\sl{k}}})t_{n}({\boldsymbol{\sl{l}}})}{(2n+1)^{q}}\in\mathsf{CMZV}^{4}_{|{\boldsymbol{\sl{k}}}|+|{\boldsymbol{\sl{l}}}|+q}.$$
Proof.
Write
$$\displaystyle\zeta_{n}({\boldsymbol{\sl{k}}})=\displaystyle\sum_{n\geq m_{1}>\dots>m_{d}>0}\displaystyle\frac{1}{m_{1}^{k_{1}}\cdots m_{d}^{k_{d}}},\quad t_{n}({\boldsymbol{\sl{l}}})=\displaystyle\sum_{n>r_{1}>\dots>r_{e}\geq 0}\displaystyle\frac{1}{(2r_{1}+1)^{l_{1}}\cdots(2r_{e}+1)^{l_{e}}}.$$
We only need to note
the following facts: (i) for any summation index $m$ for $\zeta_{n}({\boldsymbol{\sl{k}}})$ and summation index $r$ for $t_{n}({\boldsymbol{\sl{l}}})$ there
are only two possibilities: $m>r$ or $r\geq m$; (ii) we can re-write
$$\displaystyle\displaystyle\sum_{n>r_{1}}\displaystyle\frac{1}{(2n+1)^{q}(2r_{1}+1)^{l_{1}}}=\displaystyle\sum_{n\geq r_{1}}\displaystyle\frac{1}{(2n+1)^{q}(2r_{1}+1)^{l_{1}}}-\displaystyle\frac{1}{(2n+1)^{q+l_{1}}}$$
and obtain similar identities when $n$ and $r_{1}$ are replaced by $r_{j}$ and $r_{j+1}$. Therefore, we see that
(a) and (c) are special cases of Thm. 4.4(a).
(b) and (d) are special cases of Thm. 4.7(a).
∎
In the following we will compute a series of examples using our main theorems.
Example 5.2.
When depth $d=1$, by (4.38) we see that for all $x\in[-1,1]$
$$\displaystyle\displaystyle\sum_{n\geq 0}\displaystyle\frac{b_{n}(x)}{(2n+1)^{m+2}}=$$
$$\displaystyle\,\displaystyle\frac{1}{x}\displaystyle\int_{0}^{x}{\omega}_{0}^{m}{\omega}_{3}{\omega}_{1}$$
(5.55)
for all $m\geq 0$. Applying $t\to\displaystyle\frac{1-t^{2}}{1+t^{2}}$ to (5.55) we have
$$\displaystyle\displaystyle\sum_{n\geq 0}\displaystyle\frac{b_{n}(x)}{(2n+1)^{m+2}}=$$
$$\displaystyle\,\displaystyle\frac{i(-1)^{m}}{x}\displaystyle\int_{{\lambda}(x)}^{1}({\texttt{x}}_{i}-{\texttt{x}}_{-i})({\texttt{x}}_{1}-{\texttt{x}}_{-1}){\texttt{y}}^{m},$$
where ${\lambda}(x)=\sqrt{\displaystyle\frac{1-x}{1+x}}$ and ${\texttt{y}}={\texttt{x}}_{-i}+{\texttt{x}}_{i}-{\texttt{x}}_{-1}-{\texttt{x}}_{1}$ as defined in (4.45).
Taking $x=1$ and applying Au’s Mathematica package [4] we get
$$\displaystyle\displaystyle\sum_{n\geq 0}\displaystyle\frac{b_{n}}{(2n+1)^{2}}=$$
$$\displaystyle\,2\,\operatorname{{{Im}}}(\operatorname{Li}_{1,1}(i,-i)+\operatorname{Li}_{1,1}(-i,-i))=2G\approx 1.83193119,$$
(5.56)
$$\displaystyle\displaystyle\sum_{n\geq 0}\displaystyle\frac{b_{n}}{(2n+1)^{3}}=$$
$$\displaystyle\,2\,\operatorname{{{Im}}}\Big{(}\operatorname{Li}_{1_{3}}(-i,i,i)+\operatorname{Li}_{1_{3}}(-i,i,-i)-\operatorname{Li}_{1_{3}}(-i,i,-1)-\operatorname{Li}_{1_{3}}(-i,i,1)$$
$$\displaystyle-$$
$$\displaystyle\,\operatorname{Li}_{1_{3}}(-i,-i,-i)-\operatorname{Li}_{1_{3}}(-i,-i,i)+\operatorname{Li}_{1_{3}}(-i,-i,-1)+\operatorname{Li}_{1_{3}}(-i,-i,1)\Big{)}$$
$$\displaystyle=$$
$$\displaystyle-\displaystyle\frac{\pi^{3}}{32}-\displaystyle\frac{1}{8}\pi\log^{2}2+4{\rm Im}\operatorname{Li}_{3}\left(\displaystyle\frac{1+i}{2}\right)\approx 1.122690025,$$
where $G={\beta}(2)$ is Catalan’s constant. This sum appears in [5, Example 2.12], too.
Example 5.3.
As an application of Cor. 4.6, we now compute
$$\displaystyle\displaystyle\sum_{n\geq 0}\displaystyle\frac{1}{{\binom{2n}{n}}(2n+1)^{2}}=\displaystyle\sum_{n\geq 0}\displaystyle\frac{b_{n}(1/2)}{(2n+1)^{2}}.$$
From the previous example we see that
$$\displaystyle\,\displaystyle\sum_{n\geq 0}\displaystyle\frac{1}{{\binom{2n}{n}}(2n+1)^{2}}==\displaystyle\int_{\tfrac{1}{\sqrt{3}}}^{1}({\texttt{x}}_{i}-{\texttt{x}}_{-i})({\texttt{x}}_{1}-{\texttt{x}}_{-1})$$
$$\displaystyle=$$
$$\displaystyle\,2i\left(\displaystyle\int_{\tfrac{1}{\sqrt{3}}}^{0}({\texttt{x}}_{i}-{\texttt{x}}_{-i})({\texttt{x}}_{1}-{\texttt{x}}_{-1})+\displaystyle\int_{0}^{1}({\texttt{x}}_{i}-{\texttt{x}}_{-i})\displaystyle\int_{1/\sqrt{3}}^{0}({\texttt{x}}_{1}-{\texttt{x}}_{-1})+\displaystyle\int_{0}^{1}({\texttt{x}}_{i}-{\texttt{x}}_{-i})({\texttt{x}}_{1}-{\texttt{x}}_{-1})\right)$$
$$\displaystyle=$$
$$\displaystyle\,2i\left(\displaystyle\int_{0}^{\tfrac{1}{\sqrt{3}}}({\texttt{x}}_{1}-{\texttt{x}}_{-1})({\texttt{x}}_{i}-{\texttt{x}}_{-i})-\displaystyle\int_{0}^{1}({\texttt{x}}_{i}-{\texttt{x}}_{-i})\displaystyle\int_{0}^{\tfrac{1}{\sqrt{3}}}({\texttt{x}}_{1}-{\texttt{x}}_{-1})\right)+4G\quad(\text{by \eqref{eq-Catalan}})$$
$$\displaystyle=$$
$$\displaystyle\,4\operatorname{{{Im}}}\Big{(}\operatorname{Li}_{1,1}\Big{(}\displaystyle\frac{-1}{\sqrt{3}},i\Big{)}-\operatorname{Li}_{1,1}\Big{(}\displaystyle\frac{1}{\sqrt{3}},-i\Big{)}\Big{)}-\pi\log(2+\sqrt{3})+4G\approx 1.063459833.$$
Example 5.4.
As an easy example of Thm. 4.4(a),
by (4.36) and (4.37) we have
$$\displaystyle\displaystyle\sum_{n_{1}>n_{2}\geq 0}\displaystyle\frac{b_{n_{1}}}{n_{1}^{2}(2n_{2}+1)}=$$
$$\displaystyle\,4\displaystyle\int_{0}^{\pi/2}dt\circ(\csc t\sec tdt)\circ dt$$
$$\displaystyle=$$
$$\displaystyle\,4\displaystyle\int_{0}^{1}{\omega}_{1}\circ{\omega}_{20}\circ{\omega}_{1}\qquad(\text{by $t\to\sin^{-1}t$})$$
$$\displaystyle=$$
$$\displaystyle\,-4\displaystyle\int_{0}^{1}({\texttt{x}}_{-i}-{\texttt{x}}_{i})\circ({\texttt{x}}_{0}+{\texttt{x}}_{-1}+{\texttt{x}}_{1})\circ({\texttt{x}}_{-i}-{\texttt{x}}_{i})=7\zeta(3)$$
by the change of variables $t\to(1-t^{2})/(1+t^{2})$ then using Au’s package [4].
Example 5.5.
For a pure $\chi$-sum, by (4.40) we have
$$\displaystyle\displaystyle\sum_{n>0}\displaystyle\frac{b_{n}}{(2n-1)^{2}}=$$
$$\displaystyle\,\displaystyle\int_{0}^{1}({\omega}_{0}+1){\omega}_{3}{\omega}_{1}=i\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}{\texttt{d}}_{-1,1}(1-{\texttt{y}})$$
$$\displaystyle=$$
$$\displaystyle\,2G-\displaystyle\frac{1}{32}\pi^{3}+4\operatorname{{{Im}}}\operatorname{Li}_{3}\Big{(}\displaystyle\frac{1+i}{2}\Big{)}-\displaystyle\frac{1}{8}\pi\log^{2}2\approx 2.954621213,$$
where we see the weight can increase by one as predicted by
Thm. 4.4(b). Similarly,
$$\displaystyle\displaystyle\sum_{n>0}\displaystyle\frac{b_{n}}{(2n-1)^{3}}=$$
$$\displaystyle\,\displaystyle\int_{0}^{1}({\omega}_{0}+1){\omega}_{0}{\omega}_{3}{\omega}_{1}=i\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}{\texttt{d}}_{-1,1}{\texttt{y}}({\texttt{y}}-1)$$
$$\displaystyle=$$
$$\displaystyle\,-4{\beta}(4)+\displaystyle\frac{1}{96}\bigg{(}2\operatorname{{{Im}}}\operatorname{Li}_{4}\Big{(}\displaystyle\frac{1+i}{2}\Big{)}+4\pi\log^{3}2+3\pi^{3}\log 2$$
$$\displaystyle\,\hskip 85.35826pt-12\pi\log^{2}2-3\pi^{3}-\operatorname{{{Im}}}\operatorname{Li}_{3}\Big{(}\displaystyle\frac{1+i}{2}\Big{)}\bigg{)}\approx 2.1543060048.$$
Example 5.6.
For a sum of mixed parities as examples of
Thm. 4.4(b),
by (4.37) and (4.40) we have
$$\displaystyle\displaystyle\sum_{n_{1}>n_{2}\geq 0}\displaystyle\frac{b_{n_{1}}}{(2n_{1}-1)^{2}(2n_{2}+1)}=$$
$$\displaystyle\,\displaystyle\int_{0}^{1}({\omega}_{0}+1){\omega}_{3}{\omega}_{20}{\omega}_{1}=i\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}({\texttt{a}}-{\texttt{x}}_{-1}-{\texttt{x}}_{1}){\texttt{d}}_{-1,1}({\texttt{y}}-1)$$
$$\displaystyle=$$
$$\displaystyle\,14{\beta}(4)-16\operatorname{{{Im}}}\operatorname{Li}_{4}\Big{(}\displaystyle\frac{1+i}{2}\Big{)}-\displaystyle\frac{1}{12}\pi\log^{3}2-\displaystyle\frac{3}{16}\pi^{3}\log 2$$
$$\displaystyle\,\hskip 56.9055pt+\displaystyle\frac{1}{8}\pi\log^{2}2+\displaystyle\frac{5}{32}\pi^{3}-4\operatorname{{{Im}}}\operatorname{Li}_{3}\Big{(}\displaystyle\frac{1+i}{2}\Big{)}\approx 3.937040753.$$
So we see the weight can increase by one with a starting $\chi$-block.
Example 5.7.
For a sum of mixed parities without ${\sigma}$-block but with a starting ${\tau}^{\star}$-block,
by (4.38) and (4.39) we have
$$\displaystyle\,\displaystyle\sum_{n_{1}\geq n_{2}>0}\displaystyle\frac{b_{n_{1}}}{(2n_{1}+1)^{2}(2n_{2}-1)}=\displaystyle\int_{0}^{1}{\omega}_{3}({\omega}_{5}{\omega}_{3}+{\omega}_{2}){\omega}_{1}$$
$$\displaystyle=$$
$$\displaystyle\,\displaystyle\int_{0}^{1}{\omega}_{3}{\omega}_{20}{\omega}_{1}-\displaystyle\int_{0}^{1}{\omega}_{0}{\omega}_{3}{\omega}_{1}\qquad(\text{since }{\omega}_{5}=-d\sqrt{1-t^{2}},\ \sqrt{1-t^{2}}{\omega}_{3}={\omega}_{0})$$
$$\displaystyle=$$
$$\displaystyle\,-i\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}({\texttt{y}}+{\texttt{z}}){\texttt{d}}_{-1,1}+i\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}{\texttt{d}}_{-1,1}{\texttt{y}}$$
$$\displaystyle=$$
$$\displaystyle\,\displaystyle\frac{3}{16}\pi^{3}-8\operatorname{{{Im}}}\operatorname{Li}_{3}\Big{(}\displaystyle\frac{1+i}{2}\Big{)}+\displaystyle\frac{1}{4}\pi\log^{2}2\approx 1.630404535576,$$
where ${\texttt{y}}+{\texttt{z}}=-{\texttt{a}}-{\texttt{x}}_{-1}-{\texttt{x}}_{1}$. Thus the weight is unchanged as predicted by
Thm. 4.4(b).
Example 5.8.
For a sum of mixed parities with a starting ${\sigma}$-block,
followed by a ${\tau}$-block then a $\chi$-block, by (4.36), (4.37) and (4.40) we get
$$\displaystyle\,\displaystyle\sum_{n_{1}>n_{2}\geq n_{3}>0}\displaystyle\frac{b_{n_{1}}}{(2n_{1})^{2}(2n_{2}+1)(2n_{3}-1)^{2}}=\displaystyle\int_{0}^{1}{\omega}_{1}{\omega}_{20}{\omega}_{5}({\omega}_{1}+1){\omega}_{3}{\omega}_{1}$$
$$\displaystyle=$$
$$\displaystyle\,\displaystyle\int_{0}^{1}{\omega}_{1}\big{(}{\omega}_{20}{\omega}_{3}-{\omega}_{3}{\omega}_{0}\big{)}{\omega}_{3}{\omega}_{1}\qquad(\text{since }{\omega}_{5}=-d\sqrt{1-t^{2}},\ \sqrt{1-t^{2}}{\omega}_{3}={\omega}_{0})$$
$$\displaystyle=$$
$$\displaystyle\,\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}{\texttt{d}}_{-1,1}{\texttt{d}}_{-1,1}({\texttt{y}}+{\texttt{z}}){\texttt{d}}_{-i,i}-\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}{\texttt{d}}_{-1,1}{\texttt{y}}{\texttt{d}}_{-1,1}{\texttt{d}}_{-i,i}$$
$$\displaystyle=$$
$$\displaystyle\,\displaystyle\frac{G}{4}\Big{(}\displaystyle\frac{\pi^{3}}{4}-32\operatorname{{{Im}}}\operatorname{Li}_{3}\Big{(}\displaystyle\frac{1+i}{2}\Big{)}+\pi\log^{2}2\Big{)}-\displaystyle\frac{15}{2}\bigg{(}\operatorname{Li}_{5}\Big{(}\displaystyle\frac{1}{2}\Big{)}+\log 2\operatorname{Li}_{4}\Big{(}\displaystyle\frac{1}{2}\Big{)}\bigg{)}-\displaystyle\frac{1}{4}\log^{5}2$$
$$\displaystyle\,+6\pi{\beta}(4)+24\operatorname{{{Re}}}\operatorname{Li}_{3,1,1}(1,1,I)+\displaystyle\frac{1}{384}\bigg{(}80\pi^{2}\log^{3}2-15\pi^{4}\log 2-87\pi^{2}\zeta(3)-2250\zeta(5)\bigg{)}$$
$$\displaystyle\,\approx 0.98658158829.$$
So weight is unchanged in every step as predicted by
Thm. 4.4(b).
Example 5.9.
For a sum of mixed parities with ${\sigma}$-block followed by a $\chi$-block,
by (4.36) and (4.39) we obtain
$$\displaystyle\,\displaystyle\sum_{n_{1}>n_{2}>0}\displaystyle\frac{b_{n_{1}}}{(2n_{1})^{2}(2n_{2}-1)}$$
$$\displaystyle=$$
$$\displaystyle\,\displaystyle\int_{0}^{\pi/2}dt\,dt\Big{(}\sin t\,dt\csc t\,dt+\tan t\,dt\Big{)}\,dt$$
$$\displaystyle=$$
$$\displaystyle\,\displaystyle\int_{0}^{\pi/2}dt\,dt\Big{(}d(-\cos t)\,\csc t\,dt+\tan t\,dt\Big{)}\,dt$$
$$\displaystyle=$$
$$\displaystyle\,\displaystyle\int_{0}^{\pi/2}dt\,\cos t\,dt\csc t\,dt+dt\Big{(}\cot t\,dt+\tan t\,dt\Big{)}\,dt$$
$$\displaystyle=$$
$$\displaystyle\,\displaystyle\int_{0}^{\pi/2}dt\,(\sin t-1)\csc t\,dt+dt\,\csc t\sec t\,dt\,dt$$
$$\displaystyle=$$
$$\displaystyle\,\displaystyle\int_{0}^{\pi/2}dt\,dt-\csc t\,dt+dt\,\csc t\sec t\,dt\,dt$$
$$\displaystyle=$$
$$\displaystyle\,\displaystyle\int_{0}^{1}{\omega}_{1}{\omega}_{1}-{\omega}_{3}{\omega}_{1}+{\omega}_{1}{\omega}_{20}{\omega}_{1}$$
$$\displaystyle=$$
$$\displaystyle\,-i\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}{\texttt{d}}_{-1,1}+\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}({\texttt{y}}+tz){\texttt{d}}_{-i,i}-{\texttt{d}}_{-i,i}^{2}$$
$$\displaystyle=$$
$$\displaystyle\,\displaystyle\frac{1}{8}\pi^{2}-2G+\displaystyle\frac{7}{4}\zeta(3)\approx 1.5053689423.$$
Note that not only the weight is a mix of 2 and 3, but this is a mix of both real
and imaginary parts of some CMZVs of level 4. The main complication is brought in by the 1-form $\sin t\,dt$
(corresponding to ${\omega}_{5}$)
appearing in the $\chi$-block, which moves to the front after integration by parts if the block in front is a ${\sigma}$-block
but disappears if the block in front is a ${\tau}^{\star}$-block.
Example 5.10.
We apply the idea of proof of Thm. 4.1 to the sum
$$S:=\displaystyle\sum_{n_{1}\geq n_{2}\geq n_{3}\geq 1}\displaystyle\frac{b_{n_{1}}}{n_{1}^{2}(2n_{2}+1)(2n_{3}-1)}.$$
First, we break the sum in two sub-sums $S=S_{1}+S_{2}$ where
$$\displaystyle S_{1}=$$
$$\displaystyle\,\displaystyle\sum_{n_{1}>n_{2}\geq n_{3}\geq 1}\displaystyle\frac{b_{n_{1}}}{n_{1}^{2}(2n_{2}+1)(2n_{3}-1)},$$
$$\displaystyle S_{2}=$$
$$\displaystyle\,\displaystyle\sum_{n_{1}\geq n_{3}\geq 1}\displaystyle\frac{b_{n_{1}}}{n_{1}^{2}(2n_{1}+1)(2n_{3}-1)}.$$
Then by (4.36), (4.37) and (4.39) we see that
$$\displaystyle S_{1}=$$
$$\displaystyle\,4\displaystyle\int_{0}^{1}{\omega}_{1}\displaystyle\sum_{n_{2}\geq n_{3}>0}\displaystyle\frac{b_{n_{2}}(t)}{(2n_{2}+1)(2n_{3}-1)}{\omega}_{1}$$
$$\displaystyle=$$
$$\displaystyle\,4\displaystyle\int_{0}^{1}{\omega}_{1}{\omega}_{20}\displaystyle\sum_{n_{3}>0}\displaystyle\frac{b_{n_{3}}(t)}{2n_{3}-1}{\omega}_{1}$$
$$\displaystyle=$$
$$\displaystyle\,4\displaystyle\int_{0}^{1}{\omega}_{1}({\omega}_{0}+{\omega}_{2}){\omega}_{5}{\omega}_{3}{\omega}_{1}+4\displaystyle\int_{0}^{1}{\omega}_{1}({\omega}_{0}+{\omega}_{2}){\omega}_{2}{\omega}_{1}.$$
By the change of variables $t\to\displaystyle\frac{1-t^{2}}{1+t^{2}}$ we obtain
$$\displaystyle S_{1}=$$
$$\displaystyle\,-4A-4\bar{A}-4\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}({\texttt{a}}+{\texttt{x}}_{-i}+{\texttt{x}}_{i})({\texttt{a}}+{\texttt{x}}_{-1}+{\texttt{x}}_{1}){\texttt{d}}_{-i,i},$$
where $\bar{A}$ is the complex conjugation of
$$\displaystyle A=\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}{\texttt{d}}_{-1,1}\displaystyle\frac{dt}{(i-t)^{2}}({\texttt{a}}+{\texttt{x}}_{-1}+{\texttt{x}}_{1}){\texttt{d}}_{-i,i}.$$
Integration by parts yields
$$\displaystyle A=$$
$$\displaystyle\,\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}\displaystyle\frac{{\texttt{x}}_{-1}-{\texttt{x}}_{1}}{i-t}({\texttt{a}}+{\texttt{x}}_{-1}+{\texttt{x}}_{1}){\texttt{d}}_{-i,i}-\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}{\texttt{d}}_{-1,1}\displaystyle\frac{{\texttt{a}}+{\texttt{x}}_{-1}+{\texttt{x}}_{1}}{i-t}{\texttt{d}}_{-i,i}.$$
Explicitly, for all fourth roots of unity $\xi\neq i$
$$\displaystyle\displaystyle\frac{{\texttt{a}}}{i-t}=i({\texttt{x}}_{i}-{\texttt{a}}),\ \displaystyle\frac{{\texttt{x}}_{\xi}}{i-t}=\displaystyle\frac{{\texttt{d}}_{i,\xi}}{\xi-i}.$$
(5.57)
We obtain
$$\displaystyle A=$$
$$\displaystyle\,\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}\left(\displaystyle\frac{1}{-1-i}{\texttt{d}}_{i,-1}-\displaystyle\frac{1}{1-i}{\texttt{d}}_{i,1}\right)({\texttt{a}}+{\texttt{x}}_{-1}+{\texttt{x}}_{1}){\texttt{d}}_{-i,i}$$
$$\displaystyle-$$
$$\displaystyle\,\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}{\texttt{d}}_{-1,1}\left(-i({\texttt{x}}_{i}+{\texttt{a}})-\displaystyle\frac{1}{1+i}{\texttt{d}}_{i,-1}+\displaystyle\frac{1}{1-i}{\texttt{d}}_{i,1}\right){\texttt{d}}_{-i,i}$$
$$\displaystyle=$$
$$\displaystyle\,\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}\left(-{\texttt{x}}_{i}+\displaystyle\frac{1-i}{2}{\texttt{x}}_{-1}+\displaystyle\frac{1+i}{2}{\texttt{x}}_{1}\right)({\texttt{a}}+{\texttt{x}}_{-1}+{\texttt{x}}_{1}){\texttt{d}}_{-i,i}$$
$$\displaystyle-$$
$$\displaystyle\,\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}{\texttt{d}}_{-1,1}\left(-i{\texttt{a}}+\displaystyle\frac{1-i}{2}{\texttt{x}}_{-1}-\displaystyle\frac{1+i}{2}{\texttt{x}}_{1}\right){\texttt{d}}_{-i,i}.$$
Therefore using Au’s package [4] we find that
$$\displaystyle S_{1}=-4\displaystyle\int_{0}^{1}\Big{(}{\texttt{d}}_{-i,i}({\texttt{a}}+{\texttt{x}}_{-1}+{\texttt{x}}_{1})^{2}{\texttt{d}}_{-i,i}-{\texttt{d}}_{-i,i}{\texttt{d}}_{-1,1}^{2}{\texttt{d}}_{-i,i}\Big{)}=8G^{2}\approx 6.71194375752575.$$
Now we turn to $S_{2}$. Set
$$\displaystyle S_{2}(x)=$$
$$\displaystyle\,\displaystyle\sum_{n_{1}\geq n_{3}\geq 1}\displaystyle\frac{b_{n_{1}}(x)}{n_{1}^{2}(2n_{1}+1)(2n_{3}-1)}.$$
By partial fraction
$$\displaystyle S_{2}(x)=$$
$$\displaystyle\,\displaystyle\sum_{m\geq n>0}\left(\displaystyle\frac{4b_{m}(x)}{(2m+1)(2n-1)}-\displaystyle\frac{2b_{m}(x)}{m(2n-1)}+\displaystyle\frac{b_{m}(x)}{m^{2}(2n-1)}\right)$$
$$\displaystyle=$$
$$\displaystyle\,\displaystyle\sum_{m\geq n>0}\displaystyle\frac{4b_{m}(x)}{(2m+1)(2n-1)}-\displaystyle\sum_{m>n>0}\left(\displaystyle\frac{2b_{m}(x)}{m(2n-1)}-\displaystyle\frac{b_{m}(x)}{m^{2}(2n-1)}\right)-\displaystyle\sum_{n>0}\left(\displaystyle\frac{2b_{n}(x)}{n(2n-1)}-\displaystyle\frac{b_{n}(x)}{n^{2}(2n-1)}\right)$$
$$\displaystyle=$$
$$\displaystyle\,4\displaystyle\sum_{n>0}\left(f_{20}(x)\displaystyle\int_{0}^{x}\displaystyle\frac{b_{n}(t)}{2n-1}{\omega}_{1}-f_{2}(x)\displaystyle\int_{0}^{x}\displaystyle\frac{b_{n}(t)}{2n-1}{\omega}_{1}+\displaystyle\int_{0}^{x}{\omega}_{1}\displaystyle\frac{b_{n}(t)}{2n-1}{\omega}_{1}-\displaystyle\frac{b_{n}(x)}{4n^{2}}\right)$$
$$\displaystyle=$$
$$\displaystyle\,4\big{(}f_{20}(x)-f_{2}(x)\big{)}\displaystyle\int_{0}^{x}({\omega}_{5}{\omega}_{3}{\omega}_{1}+{\omega}_{2}{\omega}_{1})+4\displaystyle\int_{0}^{x}{\omega}_{1}({\omega}_{5}{\omega}_{3}{\omega}_{1}+{\omega}_{2}{\omega}_{1})-4\displaystyle\int_{0}^{x}{\omega}_{1}{\omega}_{1}$$
by (4.35)–(4.39). Note that
$$\displaystyle\displaystyle\lim_{x\to 1^{-}}\displaystyle\int_{0}^{1}$$
$$\displaystyle\,{\omega}_{5}{\omega}_{3}{\omega}_{1}=-i\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}{\texttt{d}}_{-1,1}\bigg{(}\displaystyle\frac{dt}{(i-t)^{2}}+\displaystyle\frac{dt}{(i+t)^{2}}\bigg{)}=2\,\operatorname{{{Re}}}\left(-i\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}\displaystyle\frac{{\texttt{d}}_{-1,1}}{i-t}\right)$$
$$\displaystyle=$$
$$\displaystyle\,2\,\operatorname{{{Im}}}\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}\left(\displaystyle\frac{1}{-1-i}{\texttt{d}}_{i,-1}-\displaystyle\frac{1}{1-i}{\texttt{d}}_{i,1}\right)=\operatorname{{{Im}}}\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}\left(-2{\texttt{x}}_{i}+(1-i){\texttt{x}}_{-1}+(1+i){\texttt{x}}_{1}\right),$$
which is a finite value in $i\mathsf{CMZV}_{2}^{4}$.
Further, setting ${\lambda}(x)=\sqrt{\displaystyle\frac{1-x}{1+x}}$ we get
$$\displaystyle\displaystyle\int_{0}^{x}{\omega}_{2}{\omega}_{1}=$$
$$\displaystyle\,-i\displaystyle\int_{{\lambda}(x)}^{1}{\texttt{d}}_{-i,i}({\texttt{a}}+{\texttt{x}}_{-i}+{\texttt{x}}_{i}).$$
We only need to take care of
$$\displaystyle\displaystyle\int_{{\lambda}(x)}^{1}{\texttt{x}}_{-i}{\texttt{a}}=$$
$$\displaystyle\,\displaystyle\int_{{\lambda}(x)}^{1}{\texttt{x}}_{-i}\displaystyle\int_{{\lambda}(x)}^{1}{\texttt{a}}-\displaystyle\int_{{\lambda}(x)}^{1}{\texttt{a}}{\texttt{x}}_{-i}=-\log\bigg{(}\displaystyle\frac{i+{\lambda}(x)}{i+1}\bigg{)}\log{\lambda}(x)-\displaystyle\int_{{\lambda}(x)}^{1}{\texttt{a}}{\texttt{x}}_{-i}$$
of which the last term $\to-\operatorname{Li}_{2}(i)$ as $x\to 1^{-}$. Hence
$$\displaystyle\displaystyle\lim_{x\to 1^{-}}(f_{20}(x)-f_{2}(x))\displaystyle\int_{{\lambda}(x)}^{1}{\texttt{x}}_{-i}{\texttt{a}}=$$
$$\displaystyle\,-\log\bigg{(}\displaystyle\frac{i}{i+1}\bigg{)}\displaystyle\lim_{x\to 1^{-}}\sqrt{1-x^{2}}\log\sqrt{\displaystyle\frac{1-x}{1+x}}$$
$$\displaystyle=$$
$$\displaystyle\,-\log\bigg{(}\displaystyle\frac{i}{i+1}\bigg{)}\displaystyle\frac{\sqrt{2}}{2}\displaystyle\lim_{{\varepsilon}\to 0^{+}}\sqrt{{\varepsilon}}\log({\varepsilon}/2)=0.$$
Thus
$$\displaystyle S_{2}=$$
$$\displaystyle\,\displaystyle\lim_{x\to 1^{-}}S_{2}(x)=4\displaystyle\int_{0}^{1}{\omega}_{1}({\omega}_{5}{\omega}_{3}{\omega}_{1}+{\omega}_{2}{\omega}_{1})-4\displaystyle\int_{0}^{1}{\omega}_{1}{\omega}_{1}$$
$$\displaystyle=$$
$$\displaystyle\,4\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}\bigg{(}{\texttt{z}}-{\texttt{d}}_{-1,1}\Big{(}\displaystyle\frac{dt}{(i-t)^{2}}+\displaystyle\frac{dt}{(i+t)^{2}}\Big{)}\bigg{)}{\texttt{d}}_{-i,i}-2\left(\displaystyle\int_{0}^{1}{\omega}_{1}\right)^{2}$$
$$\displaystyle=$$
$$\displaystyle\,-4\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}({\texttt{a}}+{\texttt{x}}_{-i}+{\texttt{x}}_{i}){\texttt{d}}_{-i,i}-2B-2\bar{B}-\displaystyle\frac{\pi^{2}}{2},$$
where
$$\displaystyle B=$$
$$\displaystyle\,2\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}\displaystyle\frac{{\texttt{d}}_{-1,1}}{i-t}{\texttt{d}}_{-i,i}-2\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}{\texttt{d}}_{-1,1}\displaystyle\frac{{\texttt{d}}_{-i,i}}{i-t}$$
$$\displaystyle=$$
$$\displaystyle\,2\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}\left(\displaystyle\frac{1}{-1-i}{\texttt{d}}_{i,-1}-\displaystyle\frac{1}{1-i}{\texttt{d}}_{i,1}\right){\texttt{d}}_{-i,i}-\,2\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}{\texttt{d}}_{-1,1}\left(\displaystyle\frac{1}{-2i}({\texttt{x}}_{i}-{\texttt{x}}_{-i})-\displaystyle\frac{1}{i-t}-i\right)$$
$$\displaystyle=$$
$$\displaystyle\,\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}\big{(}{\texttt{x}}_{-1}+{\texttt{x}}_{1}-i{\texttt{d}}_{-1,1}-2{\texttt{x}}_{i}\big{)}{\texttt{d}}_{-i,i}-i\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}{\texttt{d}}_{-1,1}{\texttt{d}}_{-i,i}+2\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}\displaystyle\frac{{\texttt{d}}_{-1,1}}{i-t}+2i\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}{\texttt{d}}_{-1,1}$$
$$\displaystyle=$$
$$\displaystyle\,\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}\big{(}{\texttt{x}}_{-1}+{\texttt{x}}_{1}-i{\texttt{d}}_{-1,1}-2{\texttt{x}}_{i}\big{)}({\texttt{d}}_{-i,i}+1)-i\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}{\texttt{d}}_{-1,1}{\texttt{d}}_{-i,i}+2i\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}{\texttt{d}}_{-1,1}.$$
Hence
$$\displaystyle B+\bar{B}=2\left(\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}\big{(}{\texttt{x}}_{-1}+{\texttt{x}}_{1}-{\texttt{x}}_{-i}-{\texttt{x}}_{i}\big{)}{\texttt{d}}_{-i,i}+\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}{\texttt{d}}_{-i,i}+i\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}{\texttt{d}}_{-1,1}\right).$$
We can see that $S_{2}=7\zeta(3)-8G\approx 1.0866735685$ by Au’s package [4] and therefore
$$\displaystyle\sum_{n_{1}\geq n_{2}\geq n_{3}\geq 1}\displaystyle\frac{b(n_{1})}{n_{1}^{2}(2n_{2}+1)(2n_{3}-1)}=7\zeta(3)+8G^{2}-8G\approx 7.79861732643.$$
In the next three examples, we consider some Apéry-type series which involve the square of the central binomial coefficients.
Example 5.11.
Since $\chi$-block does not appear in this example we see that the weight of the CMZVs
is the same as the weight of the series, as predicted by Thm. 4.7(a):
$$\displaystyle\displaystyle\sum_{n_{1}\geq n_{2}>0}\displaystyle\frac{b_{n_{1}}^{2}}{(2n_{1}+1)^{4}(2n_{2})}=$$
$$\displaystyle\,\displaystyle\int_{0}^{1}{\omega}_{1}{\omega}_{0}{\omega}_{3}{\omega}_{2}{\omega}_{1}=\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}{\texttt{z}}{\texttt{d}}_{-1,1}{\texttt{y}}{\texttt{d}}_{-i,i}=W_{5}\approx 0.04433915814,$$
where
$$\displaystyle W_{5}=$$
$$\displaystyle\,5\log 2\operatorname{Li}_{4}\Big{(}\displaystyle\frac{1}{2}\Big{)}+21\operatorname{Li}_{5}\Big{(}\displaystyle\frac{1}{2}\Big{)}+\pi\bigg{(}16\operatorname{{{Im}}}\operatorname{Li}_{4}\Big{(}\displaystyle\frac{1+i}{2}\Big{)}-17{\beta}(4)+8\operatorname{{{Im}}}\operatorname{Li}_{3}\Big{(}\displaystyle\frac{1+i}{2}\Big{)}\log 2\bigg{)}$$
$$\displaystyle\,+\displaystyle\frac{379}{2880}\pi^{4}\log 2+\displaystyle\frac{1}{30}\log^{5}2-16\operatorname{{{Re}}}\operatorname{Li}_{3,1,1}(1,1,i)-\displaystyle\frac{1}{192}\pi^{2}\Big{(}16\log^{3}2-29\zeta(3)\Big{)}-\displaystyle\frac{27}{4}\zeta(5).$$
Example 5.12.
The sum next has weight 4 but due to the ${\sigma}$-$\chi$-block chain we need
to use CMZVs of weight of both 3 and 4 to express it:
$$\displaystyle\displaystyle\sum_{n_{1}>n_{2}>0}$$
$$\displaystyle\,\displaystyle\frac{b_{n_{1}}^{2}}{(2n_{1})^{3}(2n_{2}-1)}=\displaystyle\int_{0}^{1}{\omega}_{3}{\omega}_{1}({\omega}_{5}{\omega}_{3}+{\omega}_{2}){\omega}_{1}=\displaystyle\int_{0}^{1}{\omega}_{3}({\omega}_{1}{\omega}_{0}-dt{\omega}_{3}){\omega}_{1}+{\omega}_{3}{\omega}_{1}{\omega}_{2}{\omega}_{1}$$
$$\displaystyle\,\ \hskip 56.9055pt\qquad(\text{since }{\omega}_{5}=-d\sqrt{1-t^{2}},\ \sqrt{1-t^{2}}{\omega}_{3}={\omega}_{0},\ \sqrt{1-t^{2}}{\omega}_{1}=dt)$$
$$\displaystyle=$$
$$\displaystyle\,\displaystyle\int_{0}^{1}({\omega}_{3}{\omega}_{1}-{\omega}_{1}{\omega}_{3}){\omega}_{1}+{\omega}_{3}{\omega}_{1}{\omega}_{0}{\omega}_{1}+{\omega}_{3}{\omega}_{1}{\omega}_{2}{\omega}_{1}\qquad(\text{since }t{\omega}_{3}={\omega}_{1})$$
$$\displaystyle=$$
$$\displaystyle\,\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}({\texttt{d}}_{-i,i}{\texttt{d}}_{-1,1}-{\texttt{d}}_{-1,1}{\texttt{d}}_{-i,i})-{\texttt{d}}_{-i,i}({\texttt{y}}+{\texttt{z}}){\texttt{d}}_{-i,i}{\texttt{d}}_{-1,1}$$
$$\displaystyle=$$
$$\displaystyle\,2G^{2}-G\pi\log 2+\displaystyle\frac{1}{64}\pi\bigg{(}3\pi^{3}-128\operatorname{{{Im}}}\operatorname{Li}_{3}\Big{(}\displaystyle\frac{1+i}{2}\Big{)}+4\pi\log^{2}2\bigg{)}+2G\pi-\displaystyle\frac{21}{4}\zeta(3)$$
$$\displaystyle\approx$$
$$\displaystyle\,0.40829155182.$$
Example 5.13.
The series in this example does not have a ${\sigma}$-$\chi$-block chain so that there is no weight drop. But the first block is a
$\chi$-block so the weight can increase by two as predicted by Thm. 4.7(b):
$$\displaystyle\displaystyle\sum_{n_{1}>n_{2}>0}\displaystyle\frac{b_{n_{1}}^{2}}{(2n_{1}-1)^{3}(2n_{2})}=$$
$$\displaystyle\,\displaystyle\int_{0}^{1}{\omega}_{1}({\omega}_{0}+1)^{2}{\omega}_{3}{\omega}_{2}{\omega}_{1}$$
$$\displaystyle=$$
$$\displaystyle\,-\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}{\texttt{z}}{\texttt{d}}_{-1,1}({\texttt{y}}-1)^{2}{\texttt{d}}_{-i,i}=W_{6}+2W_{5}+W_{4}\approx 0.38530528471,$$
where $W_{5}$ is defined in Example 5.11 and
$$\displaystyle W_{4}=$$
$$\displaystyle\,-2G^{2}-\displaystyle\frac{49}{720}\pi^{4}+2\pi\operatorname{{{Im}}}\operatorname{Li}_{3}\Big{(}\displaystyle\frac{1+i}{2}\Big{)}-\displaystyle\frac{11}{48}\pi^{2}\log^{2}2+\displaystyle\frac{1}{6}\log^{4}2+G\pi\log^{2}2+4\operatorname{Li}_{4}\Big{(}\displaystyle\frac{1}{2}\Big{)},$$
$$\displaystyle W_{6}=$$
$$\displaystyle\,68\operatorname{Li}_{6}\Big{(}\displaystyle\frac{1}{2}\Big{)}-\displaystyle\frac{7655}{27648}\pi^{6}+\displaystyle\frac{61}{2}\pi\operatorname{{{Im}}}\operatorname{Li}_{4,1}(i,1)-\displaystyle\frac{41}{2}\pi\operatorname{{{Im}}}\operatorname{Li}_{4,1}(i,-1)+96\pi\operatorname{{{Im}}}\operatorname{Li}_{5}\Big{(}\displaystyle\frac{1+i}{2}\Big{)}$$
$$\displaystyle\,-19\pi{\beta}(4)\log 2+32\pi\operatorname{{{Im}}}\operatorname{Li}_{4}\Big{(}\displaystyle\frac{1+i}{2}\Big{)}\log 2-\displaystyle\frac{181}{2880}\pi^{4}\log^{2}2-\displaystyle\frac{1}{96}\pi^{2}\log^{4}2+\displaystyle\frac{1}{90}\log^{6}2$$
$$\displaystyle\,-\displaystyle\frac{169}{4}\zeta(\bar{5},1)-\displaystyle\frac{5}{12}\pi^{2}\operatorname{Li}_{4}\Big{(}\displaystyle\frac{1}{2}\Big{)}+10\log 2\operatorname{Li}_{5}\Big{(}\displaystyle\frac{1}{2}\Big{)}-24G{\beta}(4)-6\pi^{2}\log 2\zeta(3)$$
$$\displaystyle\,-24\operatorname{{{Re}}}\operatorname{Li}_{4,2}(-1,i)+64\operatorname{{{Re}}}\operatorname{Li}_{3,1,1,1}(1,1,1,i)+\displaystyle\frac{8195}{128}\zeta(3)^{2}+\displaystyle\frac{2821}{32}\log 2\zeta(5).$$
Acknowledgement. Ce Xu is supported by the National Natural Science Foundation of China [Grant No. 12101008], the Natural Science Foundation of Anhui Province [Grant No. 2108085QA01] and the University Natural Science Research Project of Anhui Province [Grant No. KJ2020A0057]. Jianqiang Zhao is supported by the Jacobs Prize from The Bishop’s School.
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Artificial Interference Aided Physical Layer Security in Cache-enabled Heterogeneous Networks
Wu Zhao, Zhiyong Chen, Kuikui Li, and Bin Xia
Cooperative Medianet Innovation Center, Shanghai Jiao Tong University, Shanghai, P. R. China
Email: {zhaowu,zhiyongchen,kuikuili,bxia}@sjtu.edu.cn
Abstract
Caching popular contents is a promising way to offload the mobile data traffic in wireless networks, but so far the potential advantage of caching in improving physical layer security (PLS) is rarely considered. In this paper, we contribute to the design and theoretical understanding of exploiting the caching ability of users to improve the PLS in a wireless
heterogeneous network (HetNet). In such network, the base station (BS) ensures the secrecy of communication by utilizing some of the available power to transmit a pre-cached file, such that only the eavesdropper’s channel is degraded. Accordingly, the node locations of BSs, users and eavesdroppers are first modeled as mutually independent poisson point processes (PPPs) and the corresponding file access protocol is developed. We then derive analytical expressions of two metrics, average secrecy rate and secrecy coverage probability, for the proposed system. Numerical results are provided to show the significant security advantages of the proposed network and to characterize the impact of network resource on the secrecy metrics.
I Introduction
With the thriving development of mobile paying and internet of things, the privacy and security of wireless communication networks have become one of the most important issues. However, the broadcast nature of wireless channel leads to severe security vulnerabilities such as eavesdropping and jamming [1]. To overcome these shortages, physical layer security (PLS) has emerged as a promising technology to complement and augment the security of wireless networks.
In [2], Wyner shows that when the eavesdropping channel is degraded than the main legitimating channel, the secrecy of communication can be perfectly guaranteed at a non-zero rate. And first, characterizes the maximal achievable secrecy rate as ‘secrecy capacity’ of the discrete wiretap channel.
Further, various efficient approaches are proposed to improve the secrecy capacity, e.g., artificial noise adding [3], and relay cooperating[4].
By exploiting multi-input single-output techniques, [3] proposes an artificial noise assisted beamforming scheme, which imposes the artificial noise into the null space of the legitimating channel to degrade the eavesdropping channel.
One source-destination pair with multiple relays intercepted by multiple eavesdroppers (ERs) is considered in [4]. By determining the relay weights, the authors maximize the achievable secrecy rate under different cooperating schemes.
With the popularization of and explosion of small communication equipments, the topology of the wireless network is becoming densely and randomly, which intensifies the concern for secure transmission.
Based on poisson point process (PPP) [5], [6, 7, 8, 9] propose various schemes to improve physical layer security in such wireless heterogeneous network (HetNet).
In [6], the authors consider two transmission strategies based on sectoring and beamforming with artificial noise aided and investigate the secrecy capacity of both schemes.
By exchanging the location information between BSs, [7] analysis the effect of node locations on the achievable secrecy rate.
In [8], the authors develop a tractable framework to analysis the average secrecy rate in a three-tier sensor network consisting of sensors, access points and sinks.
[9] confound ERs with jamming signal from friendly jammers and artificial noise from full-duplex user. By selecting the jammer selection threshold to maximize secrecy probability.
Recently, caching popular contents at base station (BSs) and users has been introduced as a promising technique to address the mobile data tsunami in wireless networks [10, 11].
The authors in [12] propose centralized and decentralized caching algorithms to guarantee secret transmission rate by coded multicast delivery.
Further, [13] utilizes the cached files of users as side information to cancel received interference from. However, the potential of caching in improving physical layer security is rarely considered until recently [14]. The authors study the secure cooperative transmission among multiple cache-enabled BSs with the shared video data. Under the secrecy rate constraint, the total transmit power is minimized by jointly optimizing caching and transmission policies.
In this paper, we propose a heuristic scheme to enhance physical layer security in a cache-enabled HetNet by exploiting the caching ability of users. Instead of sending Gaussian noise as [3, 6], the BS transmits the target message combined with an artificial interference which is a file pre-cached at user. Since the cached file is known perfectly by the user, this part of interference can be erased as [13] while [3] and [6] need the orthogonal space of legitimating channel to isolate noise. Meanwhile, ERs are confused by this part of artificial interference due to absence of this file. Specifically, by using stochastic geometry, we model the node locations (BSs, users and ERs) of the three-tier HetNet as mutually independent PPPs. The file access protocol is then proposed based on whether the file is cached or not and whether the user has cache ability or not. Accordingly, we derive analytical expressions of average secrecy rate and secrecy coverage probability for the proposed system in different transmission schemes. Numerical results show that proposed scheme can achieve promising performance in the both ER-dense and ER-sparse scenarios.
II System Model
II-A Network Structure
As shown in Fig. 1, we consider a general wireless cache-enabled HetNet consisting three tiers of BSs, users and ERs, where the locations of BSs, users and ERs in each tier are spatially distributed based on independent PPPs, denoted as $\Phi\!_{b}$, $\Phi_{u}$ and $\Phi\!_{e}$ with density $\lambda_{b}$, $\lambda_{u}$ and $\lambda_{e}$, respectively. All nodes operate in single-antenna and we consider the downlink transmission, where time is divided into discrete slots with equal duration and we study one slot of the system. Large-scale fading and small-scale fading are both considered. We use $d^{-\beta}$ to denote large scale fading along the distance $d$, where $2\leqslant\beta\leqslant 4$ is the path-loss exponent. For the small-scale fading, we consider the Rayleigh fading channel $h$, i.e., $\mid\!h\!\mid^{2}\sim\exp(1)$.
Consider a file library consisting of $N$ files denoted by $\mathcal{F}\!\triangleq\!\{f_{1},f_{2},\ldots,f_{N}\}$, and all the files are assumed to have equal length $L$. Each user randomly requests a file $f_{i}$ with probability $p_{i}$ and $\sum_{i=1}^{N}{p_{i}}=1$. Without loss of generality (w.l.o.g), we assume $p_{1}\geq p_{2}\geq\cdots\geq p_{N}$. Here, we also consider a file can only be stored entirely rather than partially. Non-colluding ERs intercept information by passive listening signal from BS.
We assume only $\alpha$ part of users have cache ability, where $0\leq\alpha\leq 1$. The cache-enabled users also follow a thinning PPP with density $\alpha\lambda_{u}$. The cache-enabled users have same caching size with $(M\times L)bits$, where $M<N$ and cache the same $M$-most popular files out of $\mathcal{F}$ in this paper, which are marked as set $\mathcal{M}\triangleq\{f_{1},f_{2},\ldots,f_{M}\}$. To the aim of tractability, we assume that BSs can access all the files in $\mathcal{F}$ by directly connect to the core-network and neglect the extra cost for BS to fetch files. The set $\mathcal{M}$ will be broadcasted to all users by BSs at off-peak time then pre-stored at cache-enabled users. ERs considered in this paper have no cache ability.
II-B File Access Protocol
Let $\mathcal{Q}$ be the total amount of request from users in $\Phi_{u}$ at one slot. As indicated in Fig. 1, the file access protocol can be described as follows:
(a)
Self-offloading: When a cache-enabled user happens to request a file in $\mathcal{M}$, the request will be satisfied and offloaded immediately from the user’s local storage, termed as “Self-offloading”. By denoting the cache hit probability of the request fall in $\mathcal{M}$ as $\delta=\sum_{i=1}^{M}p_{i}$, the amount of this request is $\mathcal{Q}_{SO}={\alpha\delta\mathcal{Q}}$.
(b)
Secure-transmission: When a cache-enabled user requests $f_{i}$ ($i>M$) from the complementary set $\mathcal{F}\!\setminus\!\mathcal{M}$ which is denoted as $\mathcal{C}\triangleq\{f_{M+1},f_{M+2},\ldots,f_{N}\}$, the target file $f_{i}$ can be provided by the nearest BS. In order to improve the transmission security, BS can combine the target file $f_{i}$ with a cached file $f_{m}$, i.e., $f_{m}\in\mathcal{M}$.111W.l.o.g, we use $f_{1}$ as $f_{m}$ in this paper which is noticed to all cache-enabled users. Therefore, the transmission signal is $t_{i}\!=\!\sqrt{\theta\!P}x_{i}\!+\!\sqrt{(1\!-\!\theta)P}x_{m}$, where $P$ is the transmission power of BS, $x_{i}$ and $x_{m}$ are the signal of $f_{i}$ and $f_{m}$ with $E(|x_{i}|^{2})=E(|x_{m}|^{2})=1$ respectively, and $\theta\in(0,1]$ is the ratio of power allocation.
Since the pre-cached signal $x_{m}$ is known perfectly by the cache-enabled user, the user can cancel the extra interference $x_{m}$. However, $x_{m}$ is unknown for the ERs and thus can be viewed as a interference. We term this transmission as “Secure-transmission”. And the amount of this request is $\mathcal{Q}_{ST}=\alpha(1-\delta)\mathcal{Q}$
(c)
Normal-transmission: When a user does not have
cache ability and requests $f_{i}$ from $\mathcal{F}$, $f_{i}$ will be transmitted by its nearest BS, termed as “Normal-transmission”. The transmission signal is $t_{i}=\sqrt{P}x_{i}$. Moreover, according to which subset of $f_{i}$ belongs to, the request can be divided into two types: $f_{i}\!\in\!\mathcal{M}$ and $f_{i}\!\in\!\mathcal{C}$. Therefore the amount of these two types request are $\mathcal{Q}_{NT_{\mathcal{M}}}\!=\!{(1\!-\!\alpha)\delta\mathcal{Q}}$, $\mathcal{Q}_{NT_{\mathcal{C}}}\!=\!{(1-\!\alpha)(1-\delta)\mathcal{Q}}$, respectively.
In this paper, we assume all the BSs work in the full loaded state due to $\lambda_{u}\gg\lambda_{b}$ and each BS randomly serve one of user requests with equal probability. Therefore, the locations of BS in different states $\{(b),(c)\}$ are distributed as thinning PPPs $\Phi\!_{b_{1}},\Phi\!_{b_{2}},\Phi\!_{b_{3}}$ with density $\lambda_{b_{1}}\!=\!\frac{\mathcal{Q}_{ST}}{\mathcal{Q}_{ST}+\mathcal{Q}_{NT}}%
\lambda_{b}\!=\!\frac{\alpha(1-\delta)}{1-\alpha\delta}\lambda_{b}$,
$\lambda_{b_{2}}\!=\!\frac{\mathcal{Q}_{NT_{\mathcal{M}}}}{\mathcal{Q}_{ST}+%
\mathcal{Q}_{NT}}\lambda_{b}\!=\!\frac{(1-\alpha)\delta}{1-\alpha\delta}%
\lambda_{b}$ and $\lambda_{b_{3}}\!=\!\frac{\mathcal{Q}_{NT_{\mathcal{C}}}}{\mathcal{Q}_{ST}+%
\mathcal{Q}_{NT}}\lambda_{b}\!=\!\frac{(1-\alpha)(1-\delta)}{1-\alpha\delta}%
\lambda_{b}$ respectively.
III Analysis of Transmission Protocol
According to Slivnyak’s theorem [15], a typical user $u_{0}$ locating at the origin of the Euclidean area does not change the distribution of PPP, no matter with or without caching ability. We also consider that the link between $u_{0}$ and its serving BS $b_{0}$ can be eavesdropped by all ERs in the network.
III-A Normal Transmission
A typical user with no cache ability denoted as $un_{0}$ requests $f_{i}$ from $\mathcal{F}$. The nearest BS $b_{0}$ serves this request within the normal-transmission. Since the interference signals transmitted by other BSs from $\Phi_{b_{1}},\Phi_{b_{2}},\Phi_{b_{3}}$ cannot be cancelled without cached files, the interference $un_{0}$ suffering is equivalent coming from $\{\Phi_{b}\backslash b_{0}\}$ with power $P$.
Therefore the received signal of $un_{0}$ is
$$y\!_{u\!n_{0}}\!\!=\!\!\sqrt{\!P}d_{u\!n_{0},b_{0}}^{-\frac{\beta}{2}}\!h_{u\!%
n_{0},b_{0}}x_{i}+\!\!\!\!\!\!\sum_{k\in\{\!\Phi\!_{b}\!\backslash b_{0}\!\}}%
\!\!\!\!\sqrt{\!P}d_{u\!n_{0},b_{k}}^{-\frac{\beta}{2}}\!h_{u\!n_{0},b_{k}}x_{%
k^{\prime}}\!+n_{0},$$
(1)
where $d_{u\!n_{0},b_{0}}$ denotes the distance between $u\!n_{0}$ and $b_{0}$, $h_{u\!n_{0},b_{0}}$ ($h_{u\!n_{0},b_{k}}$) represents the Rayleigh fading channel between $u\!n_{0}$ and $b_{0}$ ($b_{k}$), $x_{i}$ ($x_{k^{\prime}}$222Note that $x_{k^{\prime}}$ include secure transmission and normal transmission from BSs in $\{\Phi_{b}\backslash b_{0}\}$.) is the transmission signal of $b_{0}$ ($b_{k}$), and $n_{0}\sim\mathcal{CN}(0,\sigma^{2})$ denotes the additive white Gaussian noise (AWGN). W.l.o.g, the variance of AWGN noise $n_{i}$ is $\sigma^{2}$ for $i=0,1,2,3$ in this paper.
Therefore the received signal-to-interference-plus-noise ratio (SINR) at $u\!n_{0}$ is
$$S\!I\!N\!R_{u\!n_{0}}=\frac{Pd_{u\!n_{0},b_{0}}^{-\beta}\!\!\mid\!\!h_{u\!n_{0%
},b_{0}}\!\!\mid^{2}}{\sum\limits_{k\in\{\Phi\!_{b}\!\backslash b_{0}\}}\!\!Pd%
_{u\!n_{0},b_{k}}^{-\beta}\!\!\mid\!\!h_{u\!n_{0},b_{k}}\!\!\mid^{2}\!\!+%
\sigma^{2}}.$$
(2)
For the ER of $un_{0}$, the received signal at an arbitrary ER $e_{j}\in\Phi_{e}$ is similarly given by:
$$y_{e\!_{j}n}\!=\!\sqrt{\!P}d_{e_{j},b_{0}}^{-\frac{\beta}{2}}h_{e_{j}\!,b_{0}}%
x_{i}+\!\!\!\!\sum_{k\in\{\Phi\!_{b}\!\backslash b_{0}\}}\!\!\!\!\sqrt{P}d_{e_%
{j},b_{k}}^{-\frac{\beta}{2}}h_{e_{j},b_{k}}x_{k^{\prime}}\!+n_{1}.$$
(3)
Because $x_{i}$ is eavesdropped signal for $e_{j}$, the SINR of $e_{j}$ can be written as
$$S\!I\!N\!R_{e_{j}n}=\frac{Pd_{e_{j},b_{0}}^{-\beta}\!\!\mid\!\!h_{e_{j},b_{0}}%
\!\!\mid^{2}}{\sum\limits_{k\in\{\Phi\!_{b}\!\backslash b_{0}\}}\!\!Pd_{e_{j},%
b_{k}}^{-\beta}\!\!\mid\!\!h_{e_{j},b_{k}}\!\!\mid^{2}\!+\sigma^{2}}.$$
(4)
III-B Secure Transmission
A typical user with cache ability denoted as $uc_{0}$ requests $f_{i}$ from $\mathcal{C}$. The nearest BS $b_{0}$ will serve this request within the secure-transmission. The received signal at $uc_{0}$ is given by:
$$\displaystyle y\!_{u\!c_{0}}\!=\!\sqrt{\theta\!P}d_{u\!c_{0},b_{0}}^{-\frac{%
\beta}{2}}h_{u\!c_{0}\!,b_{0}}x_{i}+\!\sqrt{(1-\theta)P}d_{u\!c_{0},b_{0}}^{-%
\frac{\beta}{2}}h_{u\!c_{0},b_{0}}x_{m}$$
$$\displaystyle\!{+}\!\!\sum_{j\in\{\!\Phi\!_{b_{1}}\!\!\backslash\!b_{0}\!\}}\!%
\!\!\left\{\!\!\sqrt{\theta\!P}d_{u\!c_{0},b_{j}}^{-\frac{\beta}{2}}\!h_{u\!c_%
{0},b_{j}}x_{j}\!+\!\sqrt{\!(1\!-\!\theta)\!P}d_{u\!c_{0},b_{j}}^{-\frac{\beta%
}{2}}h_{u\!c_{0},b_{j}}x_{m}\!\right\}$$
$$\displaystyle\!{+}\!\!\sum_{k\in\Phi\!_{b_{2}}}\!\!\!\!\sqrt{\!P}d_{u\!c_{0},b%
_{k}}^{-\frac{\beta}{2}}h_{u\!c_{0},b_{k}}x_{k}\!+\!\!\!\sum_{l\in\Phi\!_{b_{3%
}}}\!\!\sqrt{\!P}d_{u\!c_{0},b_{l}}^{-\frac{\beta}{2}}h_{u\!c_{0},b_{l}}x_{l}%
\!{}+n_{2}.$$
(5)
As described in Section II-B, the pre-cached signal $x_{m}$ is known perfectly at $uc_{0}$. And assume that the perfect channel state information (CSI) is fully available at cache-enabled users. Therefore, the $(1\!-\!\theta)$ part of interference from $\Phi_{b_{1}}$ and fully interference from $\Phi_{b_{2}}$ can be cancelled [13]. The SINR of $uc_{0}$ is
$$S\!I\!N\!R_{u\!c_{0}}\!=\!\frac{\theta\!Pd_{u\!c_{0},b_{0}}^{-\beta}\!\!\mid\!%
\!h_{u\!c_{0},b_{0}}\!\!\mid^{2}}{\!\theta\!P\!\!\!\underbrace{\!\!\sum\limits%
_{j\in\{\!\Phi\!_{b_{1}}\!\!\backslash b_{0}\!\}}\!\!\!\!d_{u\!c_{0},b_{j}}^{-%
\beta}\!\!\mid\!\!h_{u\!c_{0},b_{j}}\!\!\!\mid^{2}}_{I_{\Phi\!_{b_{1}}}}{\!+}P%
\underbrace{\!\!\sum\limits_{l\in\Phi\!_{b_{3}}}\!\!d_{u\!c_{0},b_{l}}^{-\beta%
}\!\!\mid\!\!h_{u\!c_{0},b_{l}}\!\!\!\mid^{2}\!}_{I_{\Phi\!_{b_{3}}}}+\sigma^{%
2}}.$$
(6)
For the ER of $uc_{0}$, the received signal $y_{e_{j}c}$ is given by
$$\displaystyle y_{e\!_{j}c}$$
$$\displaystyle\!=\!\sqrt{\!\theta\!P}d_{e\!_{j}\!,b_{0}}^{-\frac{\beta}{2}}h_{e%
\!_{j}\!,b_{0}}x_{i}\!+\!\sqrt{\!(1\!-\!\theta)\!P}d_{e\!_{j}\!,b_{0}}^{-\frac%
{\beta}{2}}h_{e\!_{j}\!,b_{0}}x_{m}$$
$$\displaystyle+\!\!\!\!\sum_{k\in\{\Phi\!_{b}\!\backslash b_{0}\!\}}\!\!\!\!\!%
\!\sqrt{\!P}d_{e\!_{j}\!,b_{k}}^{-\frac{\beta}{2}}h_{e\!_{j}\!,b_{k}}x_{k^{%
\prime}}\!+n_{3},$$
(7)
Thus the SINR of $e_{j}$ can be calculated as
$$S\!I\!N\!R_{e\!_{j}c}\!=\!\frac{\theta\!P\!\mid\!\!h_{e\!_{j},b_{0}}\!\!\mid^{%
2}\!\!d_{e\!_{j},b_{0}}^{-\beta}}{P\!\!\underbrace{\!\!\sum\limits_{k\in\{\Phi%
\!_{b}\backslash b_{0}\}}\!\!\!\!\!d_{e\!_{j},b_{k}}^{-\beta}\!\!\mid\!\!h_{e%
\!_{j},b_{k}}\!\!\mid^{2}\!}_{I_{\Phi\!_{b}}}+(1\!-\!\theta)P\!d_{e\!_{j},b_{0%
}}^{-\beta}\!\!\mid\!\!h_{e\!_{j},b_{0}}\!\!\mid^{2}\!+\sigma^{2}\!}.$$
(8)
Remark 1.
We can observe from $(\ref{SINR_EC})$ that the expression has the form of $\frac{\theta\!X}{C+(1-\theta)X}$, where $X\!=\!P\!\mid\!h_{e_{j},b_{0}}\!\!\mid^{2}\!d_{e_{j},b_{0}}^{-\beta}$ which is a function of variables $h_{e_{j},b_{0}}$ and $d_{e_{j},b_{0}}$, while $C\!=\!P\!I_{\Phi\!_{b}}\!+\!\sigma^{2}$ is not relevant. Therefore we have $S\!I\!N\!R_{e\!_{j}c}\leq\frac{\theta}{1-\theta}\triangleq\gamma_{th_{0}}$.
IV Security Metrics Analysis
In this section, the secrecy performance of two transmission protocols are compared in terms of average secrecy rate and secrecy coverage probability.
IV-A Average Secrecy Rate
Consider a link between the user $u_{0}$ and serving BS $b_{0}$ being intercepted by $E\!R\in\Phi_{e}$. We focus on the most detrimental ER which has the highest receive SINR from $b_{0}$.
The instantaneous secrecy rate $\mathcal{C}$ is thus given as
$$\mathcal{C}\triangleq\lceil C_{u}-C_{e}\rceil^{\dagger},$$
(9)
where$\lceil x\rceil^{\dagger}\!=\max\{x,0\}$. $C_{u}$ and $C_{e}$ are, respectively, the instantaneous capacity of the user’s ($u_{0}$) channel and the most detrimental ER’s channel, which can be expressed uniformly as $C_{i}\!=\!\log_{2}(1\!+\!\gamma_{i})$, $i=u,e$. Here, $\gamma_{e}$ is the instantaneous received SINR of the most detrimental ER, which is given by
$$\gamma_{e}\!=\!\max\limits_{e_{j}\in\Phi_{e}}\{S\!I\!N\!R_{e\!_{j}}\}.$$
(10)
The average secrecy rate is defined as
$$\overline{\mathcal{C}}\triangleq\!\!\int_{0}^{\infty}\!\!\!\!\!\int_{0}^{%
\infty}\!\!\lceil C_{u}\!-C_{e}\rceil^{\dagger}\,d\gamma_{u}\!\,d\gamma_{e},$$
(11)
and can be rewritten as [8]
$$\overline{\mathcal{C}}\!=\!\frac{1}{\ln 2}\!\int_{0}^{\infty}\!\!\!\big{[}1\!-%
\!F_{\gamma_{u}}(\gamma_{th})\big{]}\frac{F_{\gamma_{e}}\!(\!\gamma_{th}\!)}{1%
+\gamma_{th}}\,d\gamma_{th},$$
(12)
where $F_{\gamma_{u}}$ and ($F_{\gamma_{e}}$) are the cumulative probability functions (CDFs) of $\gamma_{u}$ and ($\gamma_{e}$), respectively. Therefore, the $\overline{\mathcal{C}}$ of two transmission protocols are given as follow.
IV-A1 Secure Transmission
Lemma 1.
Let $\gamma_{uc}$ be the SINR of the typical user with cache ability, the CDF of $\gamma_{uc}$ can be calculated as
$$\displaystyle F\!_{\gamma\!_{u\!c}}\!(\!\gamma_{th}\!)\!=\!1\!-\!2\pi\lambda_{%
b}\!\!\!\int_{0}^{\infty}\!\!\!\!\!\!x\!\exp\!\Big{\{}$$
$$\displaystyle\!\!-\!\pi x^{2}\!\big{[}\!\mathcal{Z}(\gamma_{th})\lambda_{b_{1}%
}\!\!+\!\mathcal{Z}(\frac{\gamma_{th}}{\theta})\lambda_{b_{3}}\!+\!\lambda_{b}%
\big{]}$$
$$\displaystyle\!\!-\!\frac{\sigma^{2}}{\theta\!P}\gamma_{th}x^{\beta}\Big{\}}\!%
\,dx,$$
(13)
where $\mathcal{Z}(\!\gamma_{th}\!)\!=\!\frac{2\gamma_{th}}{\beta-2}{}_{2}F_{1}[1,1\!%
-\!\frac{2}{\beta};2\!-\!\frac{2}{\beta};-\gamma_{th}]$, ${}_{2}F_{1}[\cdot]$ is the Gauss hypergeometric function.
Proof.
Please refer to Appendix A.
∎
Lemma 2.
Let $\gamma_{ec}$ be the SINR of the most detrimental ER of the typical cache-enabled user, the CDF of $\gamma_{ec}$ is written as
$$F_{\gamma_{ec}}(\gamma_{th})=\\
\begin{cases}\widetilde{F}_{\gamma_{ec}}\!(\gamma_{th})&0\leq\gamma_{th}\leq%
\gamma_{th_{0}}\\
\qquad 1&\qquad\text{else},\\
\end{cases}$$
(14)
where $\widetilde{F}_{\gamma_{ec}}\!(\gamma_{th})$ is
$$\displaystyle\exp\!\!\Bigg{\{}\!\!\!-\!2\pi\lambda_{e}\!\!\!\int_{0}^{\infty}%
\!\!\!\!\!x\!\exp\!\bigg{\{}\!\!\!-\!\pi\lambda_{b}\Gamma(1\!+\!\frac{2}{\beta%
})\Gamma(1\!-\!\frac{2}{\beta})\!\!\left[\!\frac{\gamma_{th}x^{\beta}}{\theta%
\!-\!(1\!\!-\!\theta)\gamma_{th}}\!\right]\!\!^{\frac{2}{\beta}}$$
$$\displaystyle \!-\!{\frac{\sigma^{2}}{P}}\!\Big{[}%
\frac{\gamma_{th}x^{\beta}}{[\theta-\!(1\!-\theta)\gamma_{th}]}\Big{]}\!\bigg{%
\}}\!\,dx\!\Bigg{\}}.$$
(15)
and $\Gamma[\cdot]$ is the Gamma function.
Proof.
Please refer to Appendix B.
∎
Theorem 1.
In the interference-limited scenario, the average secrecy rate for the secure transmission is given by
$$\displaystyle\overline{\mathcal{C}}\!_{ST}$$
$$\displaystyle\!=\!\frac{1}{\ln 2}\!\int_{0}^{\gamma_{t\!h\!_{0}}}\!\!\frac{%
\exp\!\!\left\{\!-\frac{\lambda_{e}}{\lambda_{b}}\Big{/}\Gamma(1\!\!+\!\!\frac%
{2}{\beta})\Gamma(1\!\!-\!\!\frac{2}{\beta})[\frac{\gamma_{t\!h}}{\theta-(1-%
\theta)\gamma_{t\!h}}]\!^{\frac{2}{\beta}}\!\!\right\}}{(1+\gamma_{t\!h})[%
\mathcal{Z}(\gamma_{t\!h})\frac{\lambda_{b_{1}}}{\lambda_{b}}\!+\!\mathcal{Z}(%
\frac{\gamma_{t\!h}}{\theta})\frac{\lambda_{b_{3}}}{\lambda_{b}}\!+\!1]}\,d%
\gamma_{t\!h}$$
$$\displaystyle+\frac{1}{\ln 2}\!\int_{\gamma_{t\!h\!_{0}}}^{\infty}\!\!\frac{\,%
d\gamma_{t\!h}}{(1\!+\!\gamma_{t\!h})[\mathcal{Z}\!(\!\gamma_{t\!h}\!)\!\frac{%
\lambda_{b_{1}}}{\lambda_{b}}\!+\!\mathcal{Z}\!(\!\frac{\gamma_{t\!h}}{\theta}%
\!)\!\frac{\lambda_{b_{3}}}{\lambda_{b}}\!+\!1]}.$$
(16)
Proof.
By substituting $(\ref{CDF_UC})$ and $(\ref{CDF_EC})$ into $(\ref{Capacity1})$, it is easy to obtain this theorem.
∎
IV-A2 Normal Transmission
Lemma 3.
Let $\gamma_{un}$ as the SINR of the typical user without cache ability. Similar to $(\ref{CDF_UC})$, the CDF of $\gamma_{un}$ is
$$\displaystyle F\!_{\gamma\!_{u\!n}}\!(\!\gamma_{th}\!)\!=\!1\!\!-\!2\pi\lambda%
_{b}\!\!\!\int_{0}^{\infty}\!\!\!\!\!\!x\!\exp\!\!\left[\!-\pi\!\lambda_{b}x^{%
2}\!(\mathcal{Z}\!(\gamma_{th})\!+\!\!1\!)\!-\!\frac{\sigma^{2}}{P}\gamma_{th}%
x^{\beta}\!\right]\!\!\,dx.$$
(17)
Lemma 4.
Let $\gamma_{en}$ as the SINR of the most detrimental ER of the typical user without cache ability. Similar to $(\ref{CDF_EC})$, it is easy to obtain the CDF of $\gamma_{en}$, which is given by
$$\displaystyle F\!_{\gamma_{e\!n}}\!(\!\gamma_{th}\!)\!=\!\exp\!\!\Bigg{\{}\!\!%
\!-\!2\pi\lambda_{e}\!\!\!\int_{0}^{\infty}\!\!\!\!\!x\!\exp\!\bigg{\{}\!\!\!-%
\!\pi\lambda_{b}\Gamma(\!1\!+\!\frac{2}{\beta}\!)\Gamma(\!1\!-\!\frac{2}{\beta%
}\!)\!\!\left[\!\gamma_{th}x^{\beta}\right]\!^{\frac{2}{\beta}}$$
$$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\ \ -\!{\frac{\sigma^{2}}{P}}%
\gamma_{th}x^{\beta}\bigg{\}}\!\!\,dx\Bigg{\}}.$$
(18)
Theorem 2.
In the interference-limited scenario, the average secrecy rate for the normal transmission is derived as
$$\displaystyle\overline{\mathcal{C}}\!_{N\!T}$$
$$\displaystyle{=}\frac{1}{\ln 2}\!\int_{0}^{\infty}\!\frac{\exp\!\left\{\!-%
\frac{\lambda_{e}}{\lambda_{b}}\Big{/}\Gamma(1\!\!+\!\!\frac{2}{\beta})\Gamma(%
1\!\!-\!\!\frac{2}{\beta})\gamma_{t\!h}\!^{\frac{2}{\beta}}\!\right\}}{(1+%
\gamma_{t\!h})[\mathcal{Z}(\gamma_{t\!h})\!+\!1]}\,d\gamma_{t\!h}.$$
(19)
Proof.
By substituting $(\ref{CDF_UN})$ and $(\ref{CDF_EN})$ into $(\ref{Capacity1})$, we obtain this theorem.
∎
By comparing $(\ref{C_UC})$ and $(\ref{C_UN})$, we can find that $\overline{\mathcal{C}}_{ST}$ and $\overline{\mathcal{C}}_{NT}$ are both dependent on $\lambda_{e}/\lambda_{b}$, while $\overline{\mathcal{C}}_{ST}$ is also dependent on the power allocation ratio $\theta$ and the ratio of BS in three different states $\lambda_{b_{i}}/\lambda_{b}$, $i=1,3$. Note that $\lambda_{b_{i}}/\lambda_{b}$, $i=1,3$, are related to the cache-user ratio $\alpha$ and the cache hit ratio $\delta$. Numerical results will be given in Section V to show the effects of these parameters.
IV-B Secrecy Coverage Probability
Let $R_{s}$ be a given secrecy rate threshold. The delivery is securely successful when the instantaneous secrecy rate $\mathcal{C}$ is larger than the threshold $R_{s}$. Thus, the secrecy coverage probability can be expressed as
$$\displaystyle\mathcal{P}$$
$$\displaystyle\triangleq\!P_{r}(\mathcal{C}>\!R_{s})\!=\!P_{r}[\log_{2}(1\!\!+%
\!\gamma_{u})\!-\!\log_{2}(1\!+\!\gamma_{e})\!\!>\!\!R_{s}]$$
$$\displaystyle=\mathbb{E}_{\gamma_{u},\gamma_{e}}\!\left\{\textbf{1}_{[\gamma_{%
u}>(1+\gamma_{e})2^{R\!_{s}}-1]}\right\}$$
$$\displaystyle=\int_{0}^{\infty}\!\!\!\!\int_{(1+\gamma_{e}\!)2^{R\!_{s}}\!-1}^%
{\infty}f_{\gamma_{u}}\!(\gamma_{1})f_{\gamma_{e}}\!(\gamma_{2})\,d\gamma_{1}%
\!\,d\gamma_{2}$$
$$\displaystyle=\int_{0}^{\infty}\!\!\!\!f_{\gamma_{e}}\!(\gamma_{th})\!\big{\{}%
\!1\!-\!F_{\gamma_{u}}\![2^{R_{s}}\!(1\!+\!\gamma_{th}\!)\!-\!1]\big{\}}\!\,d%
\gamma_{th},$$
(20)
where $f_{\gamma_{u}}\!(\gamma_{u})$ and $f_{\gamma_{e}}\!(\gamma_{e})$ are the probability distribution functions (PDFs) of $\gamma_{u}$ and $\gamma_{e}$, respectively.
Theorem 3.
In the interference-limited scenario with the secure transmission, the secrecy coverage probability is
$$\displaystyle\mathcal{P}\!_{S\!T}$$
$$\displaystyle\!=\!\!\int_{0}^{\gamma_{t\!h\!_{0}}}\!\!\!\bigg{\{}\frac{\exp\!%
\!\big{[}\!-\frac{\lambda_{e}}{\lambda_{b}}\Big{/}\Gamma(1\!\!+\!\!\frac{2}{%
\beta})\Gamma(1\!\!-\!\!\frac{2}{\beta})[\frac{\gamma_{t\!h}}{\theta-(1-\theta%
)\gamma_{t\!h}}]\!^{\frac{2}{\beta}}\!\big{]}}{\mathcal{G}(\!{\gamma_{th}}\!)%
\frac{\lambda_{b_{1}}}{\lambda_{b}}\!+\!\mathcal{G}(\!\frac{\gamma_{th}}{%
\theta}\!)\frac{\lambda_{b_{3}}}{\lambda_{b}}\!+\!1}$$
$$\displaystyle\quad\ \ \frac{2\lambda_{e}\theta\gamma_{th}^{-\frac{\beta+2}{%
\beta}}}{\beta\lambda_{b}\Gamma(1+\frac{2}{\beta})\Gamma(1-\frac{2}{\beta})[%
\frac{\gamma_{t\!h}}{\theta-(1-\theta)\gamma_{t\!h}}]^{\frac{\beta-2}{\beta}}}%
\bigg{\}}\,d\gamma_{th},$$
(21)
where $\mathcal{G}(\!\gamma_{t\!h}\!)$ is given as
$$\mathcal{G}(\!\gamma_{t\!h}\!)\!=\!{[(1\!+\!\gamma_{th}\!)2^{R_{s}}\!\!-\!1]}^%
{\frac{2}{\beta}}\!\!\!\int_{{[(1\!+\!\gamma_{th}\!)2^{R_{s}}\!\!-\!1]}^{\frac%
{2}{\beta}}}^{\infty}\!\frac{1}{1\!+\!x^{\frac{\beta}{2}}}dx.$$
(22)
Proof.
By differentiating $(\ref{CDF_EC})$ to get $f_{\gamma_{e}}\!(\gamma_{e})$, then substituting $f_{\gamma_{e}}\!(\gamma_{e})$ and $(\ref{CDF_UC})$ into $(\ref{p_coverage})$, we obtain this theorem.
∎
Theorem 4.
In the interference-limited scenario with the normal transmission, the secrecy coverage probability is
$$\displaystyle\mathcal{P}\!_{N\!T}$$
$$\displaystyle\!=\!\!\int_{0}^{\infty}\!\!\!\bigg{\{}\!\frac{\exp\!\big{[}\!-\!%
\frac{\lambda_{e}}{\lambda_{b}}\Big{/}\Gamma(1\!\!+\!\!\frac{2}{\beta})\Gamma(%
1\!\!-\!\!\frac{2}{\beta})\gamma_{t\!h}\!^{\frac{2}{\beta}}\!\big{]}}{\mathcal%
{G}({\gamma_{th}})+1}$$
$$\displaystyle\qquad\quad\!\frac{2\lambda_{e}\gamma_{th}^{-\frac{\beta+2}{\beta%
}}}{\beta\lambda_{b}\Gamma(\!1+\frac{2}{\beta}\!)\Gamma(1-\frac{2}{\beta})}%
\bigg{\}}\,d\gamma_{th}$$
(23)
Proof.
By differentiating $(\ref{CDF_EN})$ to get $f_{\gamma_{e}}\!(\gamma_{e})$, then substituting $f_{\gamma_{e}}\!(\gamma_{e})$ and $(\ref{CDF_UN})$ into $(\ref{p_coverage})$, we get this theorem.
∎
V Numerical Results
In this section, numerical results are provided to evaluate the performance of the proposed transmission schemes. The BSs, ERs and users are distributed based on PPPs with density $\{\lambda_{b},\lambda_{e},\lambda_{u}\}=\{1,5,100\}/km^{2}$ in the simulation. We consider the transmission power $P=30~{}dBm$ and the noise power $\sigma^{2}=-174~{}dBm$. We consider the path loss exponent $\beta=4$, the total number of files $N=100$, the cache size $M=5$, the power allocation ratio $\theta=0.5$, and the cache user ratio $\alpha=0.5$. In the simulation, the file popularity distribution is modeled as Zipf distribution, i.e., the requested probability of the $i$-th ranked file is given by $p_{i}=\frac{1/{i^{\eta}}}{\sum_{j=1}^{N}{1/j^{\eta}}}$ where $\eta\geq 0$ characterizes the skew of the popularity distribution. We use $\eta=0.8$ in the simulation. These parameters will not change unless specified otherwise.
In Fig.2, the average secrecy rate of the secure transmission $\overline{\mathcal{C}}_{ST}$ versus the power allocation ratio $\theta$ is illustrated. It can be seen that there exists an optimal $\theta^{*}$ to achieve the maximal $\overline{\mathcal{C}}\!_{ST}$ for a given $\alpha$, and different $\alpha$ has different $\theta^{*}$. As presented in (1), $\overline{\mathcal{C}}\!_{ST}$ cannot be expressed in a closed form. As such, we cannot derive $\theta^{*}$ in theory. We can observe that the average secrecy rate $\overline{\mathcal{C}}\!_{ST}$ first increase with $\theta$ when $\theta\in(0,\theta^{*})$, then decreases with $\theta$ when $\theta\in(\theta^{*},1)$. This interesting phenomenon can be well explained from $(\ref{SINR_UC})$ and $(\ref{SINR_EC})$. The increase of $\theta$ improves the SINRs of both user and ER, but the increment at user is dominant in $(0,\theta^{*})$. When $\theta$ is getting larger, the $\overline{\mathcal{C}}\!_{ST}$ will be compromised due to the growing effects of eavesdropping. We can also obtain that the secure transmission can achieve better optimal $\overline{\mathcal{C}}$ in larger $\alpha$ scenario, because more secure transmissions occurs in the network.
In Fig.3, the average secrecy rate $\overline{\mathcal{C}}$ versus the density ratio of $\frac{\lambda_{e}}{\lambda_{b}}$ with different cache size $M$ is illustrated. Note from (19) that $\overline{\mathcal{C}}_{NT}$ is only depend on $\frac{\lambda_{e}}{\lambda_{b}}$ which is considered as a baseline. We can see that with the increase of $\frac{\lambda_{e}}{\lambda_{b}}$, the $\overline{\mathcal{C}}$ is decreased for both with and without cache-enabled transmission schemes, which indicates that more ERs cause more serious eavesdropping. It should be highlighted that, even with $\frac{\lambda_{e}}{\lambda_{b}}\!=\!10$, the $\overline{\mathcal{C}}_{ST}$ is still above $1.5~{}bits/\!s/\!Hz$ which only reduced $25\%$ from above $2~{}bits/\!s/\!Hz$ when $\frac{\lambda_{e}}{\lambda_{b}}=0.1$, while $\overline{\mathcal{C}}\!_{N\!T}$ reduces to $0.3~{}bits/\!s/\!Hz$ from $2~{}bits/\!s/\!Hz$, i.e., reduced by $85\%$. In addition, we can also observe that the $\overline{\mathcal{C}}\!_{S\!T}$ improves with increasing cache size $M$, due to more interference signal can be cancelled with larger ratio of $\Phi_{b_{1}}$ and $\Phi_{b_{2}}$.
In Fig.4, the secrecy coverage probability $\mathcal{P}$ for various $\frac{\lambda_{e}}{\lambda_{b}}$ is presented. We can observe that $\mathcal{P}_{ST}$ is much higher than $\mathcal{P}_{NT}$ for both $\frac{\lambda_{e}}{\lambda_{b}}=5$ and $0.5$, which indicates the promising effect of secure transmission in both ER-dense scenario and ER-sparse scenario. Similar as Fig.3, we can also see that more ERs cause more serious eavesdropping leading lower secrecy coverage probability. The simulation results are presented along with the theoretical ones in Fig.3 and Fig. 4. We can see from the figures that the theoretical results are in excellent agreement with the simulation results.
VI Conclusion
In this paper, we reveal that the caching ability of users can be used to improve the transmission security for physical layer security in the wireless cache-enabled HetNet. The corresponding secure transmission scheme is developed, where the transmitter combines the message signal with the pre-cached file. This scheme can introduce extra interference at ER, but this interference can be cancelled at the cache-enabled users. Based on stochastic geometry, we derive the expression of average secrecy rate and secrecy coverage probability for the secure transmission and the normal transmission. Finally, we show that the secure transmission achieves a significant security gain than the normal transmission.
-A Proof of Lemma 1
By replacing the distance $d_{u_{0},b_{0}}$ between the typical user and its nearest BS with $x$, the PDF of $x$ is $f_{X}(x)\!=\!2\!\pi\lambda_{b}x\exp\{-\pi\lambda_{b}x^{2}\}$[5]. Then the CDF of $\gamma_{uc}$ is derived as
$$\displaystyle F_{\gamma_{uc}}(\gamma_{th})\!\triangleq\!\!P_{r}[S\!I\!N\!R_{uc%
}\leq\gamma_{th}]\!=\!\mathbb{E}_{x}[S\!I\!N\!R_{uc}\leq\gamma_{th}|d_{u_{0},b%
_{0}}\!=\!x]$$
$$\displaystyle=\int_{0}^{+\infty}\!\!\!P_{r}\!\Big{[}\frac{\theta\!P\!\mid\!h_{%
u_{0},b_{0}}\!\mid^{2}\!x^{-\beta}}{\theta\!P\!I_{\Phi\!_{b_{1}}}\!\!+P\!I_{%
\Phi\!_{b_{3}}}\!\!+\!\sigma^{2}}\leq\gamma_{th}\Big{]}\!f_{X}(x)\,dx$$
$$\displaystyle=\int_{0}^{+\infty}\!\!\!\!P_{r}\Big{[}\!\!\mid\!h_{u_{0},b_{0}}%
\!\mid^{2}\leq\!\gamma_{th}x^{\beta}(I_{\Phi\!_{b_{1}}}\!\!+\!\frac{I_{\Phi\!_%
{b_{3}}}}{\theta}\!+\!\frac{\sigma^{2}}{\theta\!P})\Big{]}\!f_{X}(x)\,dx$$
$$\displaystyle\overset{(a)}{=}1\!{-}\!\!\int_{0}^{\infty}\!\!\!\!\mathcal{L}_{I%
_{\Phi\!_{b_{1}}}}\!\!(\gamma_{th}x^{\beta})\mathcal{L}_{I_{\Phi\!_{b_{3}}}}\!%
\!(\frac{\gamma_{th}x^{\beta}}{\theta})e^{-\frac{\gamma_{th}x^{\beta}\!\sigma^%
{2}}{\theta\!P}}\!f\!_{X}(x)\,dx,$$
(24)
where Step (a) follows from $\mid\!h_{u_{0},b_{0}}\!\mid^{2}\sim\exp(1)$. Under the condition of $d_{u_{0},b_{0}}\!=\!x$, the remaining interferences resulted from $\Phi\!_{b_{1}}$ and $\Phi\!_{b_{3}}$ are spatially located at the outside of the circle centered at $u_{0}$ with radius $x$ denoted as $\mathcal{C}_{(u_{0},x)}$. Therefore the Laplace transform $\mathcal{L}_{I_{\Phi\!_{b_{1}}}}$ is derived as
$$\displaystyle\mathcal{L}_{I_{\Phi\!_{b_{1}}}}\![\gamma_{th}x^{\beta}]=\mathbb{%
E}_{I_{\Phi\!_{b_{1}}}}\!\!\Big{[}\exp({-\gamma_{th}x^{\beta}\!\!\!\!\!\sum%
\limits_{j\in\{\Phi\!_{b\!_{1}}\!\!\backslash b_{0}\}}\!\!\!\!\!d_{u_{0},b_{j}%
}^{-\beta}\!\!\mid\!\!h_{u_{0},b_{j}}\!\!\mid^{2}})\Big{]}$$
$$\displaystyle=\mathbb{E}_{I_{\Phi\!_{b_{1}}}}\!\!\Big{\{}\!\prod_{j\in\{\Phi\!%
_{b_{1}}\!\!\backslash b_{0}\!\}}\!\!\!\!\!\!\big{[}\!\exp\!\big{(}\!-\!\gamma%
_{th}x^{\beta}d_{u_{0},b_{j}}^{-\beta}\!\!\mid\!h\!_{u_{0},b_{j}}\!\!\mid^{2})%
\big{]}\!\Big{\}}$$
$$\displaystyle\overset{(a)}{=}\!\exp\!\bigg{\{}\!\!-\!\lambda_{b_{1}}\!\!\!\int%
_{R^{2}\backslash\mathcal{C}_{(u_{0},x)}}\!\!\!\!\Big{[}1\!-\!\mathbb{E}_{\mid
h%
_{u_{0},b_{j}}\mid^{2}}(e^{-\gamma_{th}x^{\beta}r^{-\beta}})\Big{]}\!\,dr\!%
\bigg{\}}$$
$$\displaystyle\overset{(b)}{=}\exp\!\Big{[}\!-\!2\pi\lambda_{b_{1}}\!\!\int_{x}%
^{\infty}\!\!\frac{v}{1+(\gamma_{th}x^{\beta})^{-1}v^{\beta}}\,dv\Big{]}$$
$$\displaystyle\overset{(c)}{=}\exp\!\big{[}\!-\!\pi\lambda_{b_{1}}x^{2}\gamma_{%
th}^{\frac{2}{\beta}}\int_{\gamma_{th}^{-\frac{2}{\beta}}}^{\infty}\frac{\,dy}%
{1+y^{\frac{\beta}{2}}}\big{]}$$
$$\displaystyle=\exp\!\big{[}\!-\!\pi\lambda_{b_{1}}x^{2}\mathcal{Z}(\gamma_{th}%
)\big{]},$$
(25)
where step (a) follows from the probability generating functional (PGFL) of PPP, step (b) is obtained by converting the cartesian coordinates into polar coordinates, step (c) is obtained by replacing $(\gamma_{th}x^{\beta})^{-\frac{2}{\beta}}v^{2}$ with $y$.
Similarly, the Laplace transform of the $\mathcal{L}_{I_{\Phi\!_{b_{3}}}}$ is
$$\displaystyle\mathcal{L}_{I_{\Phi\!_{b_{3}}}}\!\big{[}\frac{\gamma_{th}x^{%
\beta}}{\theta}\big{]}$$
$$\displaystyle=\mathbb{E}_{I_{\Phi\!_{b_{3}}}}\!\!\Big{[}\exp({-\frac{\gamma_{%
th}x^{\beta}}{\theta}\!\!\sum\limits_{l\in\Phi\!_{b\!_{3}}\!\!}\!\!d_{u_{0},b_%
{l}}^{-\beta}\!\!\mid\!\!h_{u_{0},b_{l}}\!\!\mid^{2}})\Big{]}$$
$$\displaystyle=\exp\!\big{[}\!-\!\pi\lambda_{b_{3}}x^{2}\mathcal{Z}(\frac{%
\gamma_{th}}{\theta})\big{]}.$$
(26)
By substituting $(\ref{L_I_B1})$, $(\ref{L_I_B3})$ into $(\ref{F_UC})$, we can obtain Lemma 1 and the proof is completed.
-B Proof of Lemma 2
Let us replace the distance $d_{e_{i},b_{0}}$ with $x$. When $\gamma_{ec}\leq\gamma_{th_{0}}$, by substituting $(\ref{SINR_EC})$ into $(\ref{max_SINR_EC})$, the CDF of $\gamma_{ec}$ is
$$\displaystyle F\!_{\gamma_{ec}}\!(\gamma_{th})\triangleq P_{r}\big{\{}\max%
\limits_{e_{i}\in\Phi_{e}}[S\!I\!N\!R_{e_{i}}\leq\gamma_{th}]\big{\}}$$
$$\displaystyle=P_{r}\!\bigg{\{}\!\max\limits_{e_{i}\in\Phi_{e}}\!\Big{[}\frac{%
\theta P\!\mid\!\!h_{e_{i},b_{0}}\!\!\mid^{2}\!d_{e_{i},b_{0}}^{-\beta}}{P\!I_%
{\Phi\!_{b}}\!+\sigma^{2}\!+\!(1\!-\theta)P\!\mid\!h_{e_{i},b_{0}}\!\!\mid^{2}%
\!d_{e_{i},b_{0}}^{-\beta}}\leq\gamma_{th}\Big{]}\bigg{\}}$$
$$\displaystyle=\mathbb{E}_{\Phi\!_{e}}\!\bigg{[}\!\prod_{i\in\Phi\!_{e}}\!\!\!P%
_{r}\!\Big{(}\frac{\theta P\!\mid\!h_{e_{i},b_{0}}\!\!\mid^{2}\!d_{e_{i},b_{0}%
}^{-\beta}}{P\!I_{\Phi\!_{b}}\!+\sigma^{2}\!+\!(1\!-\!\theta)P\!\!\mid\!\!h_{e%
_{i},b_{0}}\!\!\mid^{2}\!d_{e_{i},b_{0}}^{-\beta}}\leq\gamma_{th}\Big{)}\!%
\bigg{]}$$
$$\displaystyle\overset{(a)}{=}\!\exp{\!\bigg{\{}\!\!\!-\!\lambda_{e}\!\!\!\int_%
{R^{2}}\!\!\!\!\!1\!\!-\!P_{r}\!\Big{[}\!\!\!\mid\!h_{e_{i},b_{0}}\!\!\mid^{2}%
\leq\!\frac{(\!P\!I_{\Phi\!_{b}}\!\!+\!\sigma^{2})\gamma_{th}r^{\beta}}{[%
\theta-\!(1\!-\theta)\gamma_{th}]P}\!\Big{]}\!\,dr\!\!\bigg{\}}}$$
$$\displaystyle\overset{(b)}{=}\exp{\bigg{\{}\!\!-\!2\pi\lambda_{e}\!\!\!\int_{0%
}^{\infty}\!\!\!\!x\mathcal{L}_{I_{\Phi\!_{b}}}\!\!\Big{[}\frac{\gamma_{th}x^{%
\ \beta}}{[\theta-\!(1\!-\theta)\gamma_{th}]}\Big{]}\!\mathrm{e^{\!-\!\frac{%
\sigma^{2}\gamma_{th}\!x^{\beta}}{[\theta-(1-\theta)\gamma_{th}]\!P}}}\!\,dx\!%
\bigg{\}}}$$
(27)
where step (a) follows from the PGFL of PPP, step (b) is obtained by converting the cartesian coordinates into polar coordinates.
Note that the interference comes from $\{\Phi\!_{b}\backslash b_{0}\}$, which is the reduced Palm distribution of PPP $\Phi\!_{b}$. According to Slivnyak-Mecke theorem [16], the reduced Palm distribution of PPP is equivalent of its original distribution, i.e.$\{\Phi\!_{b}\backslash b_{0}\}=\Phi\!_{b}$ as illustrated in [9]. Denoting $S=\frac{\gamma_{th}x^{\beta}}{[\theta-(1-\theta)\gamma_{th}]}$, the Laplace transform of interference $I_{\Phi_{b}}(S)$ is derived as
$$\displaystyle\mathcal{L}_{I_{\Phi\!_{b}}}\!(S)\!=\mathbb{E}_{I_{\Phi\!_{b}}}%
\big{[}\mathbf{e}^{-SI_{\Phi\!_{b}}}\big{]}$$
$$\displaystyle=\mathbb{E}_{I_{\Phi\!_{b}}}\!\Big{[}\exp{\!\big{(}\!-\!S\!\!\!\!%
\sum_{j\in\{\Phi\!_{b}\backslash b_{0}\}}\!\!\!\!\mid\!\!h_{e_{i},b_{j}}\!\!%
\mid^{2}\!\!d_{e_{i},b_{j}}^{-\beta}\big{)}}\Big{]}$$
$$\displaystyle\overset{(a)}{\!=\!}\!\exp{\!\left\{\!\!-\!\lambda_{b}\!\!\int_{R%
^{2}}\!\!\!\!1\!\!-\!\mathbb{E}_{\mid h_{e_{i},b_{j}}\!\mid^{2}}\!\!\left[\exp%
{\!\big{(}\!\!-\!S\!\mid\!\!h_{e_{i},b_{j}}\!\!\mid^{2}\!\!d_{e_{i},b_{j}}^{-%
\beta}\big{)}}\!\right]\!\!\,d(d_{e_{i},b_{j}})\!\right\}}$$
$$\displaystyle\overset{(b)}{=}\exp\!{\Big{\{}\!\!-\!2\pi\lambda_{b}\!\!\int_{0}%
^{\infty}\!\!\frac{v}{1+\frac{v^{\beta}}{S}}\,dv\Big{\}}}$$
$$\displaystyle=\exp\!\Big{[}\!-\!\pi\lambda_{b}\Gamma(1\!+\!\frac{2}{\beta})%
\Gamma(1\!-\!\frac{2}{\beta})S^{\frac{2}{\beta}}\Big{]},$$
(28)
where step (a) follows from the PGFL of PPP,
where step (b) is obtained by converting cartesian coordinates into polar coordinates.
By substituting $(\ref{L_B})$ into $(\ref{F_EC})$, we can get $\widetilde{F}_{\gamma_{ec}}\!(\gamma_{th})\text{as}~{}(\ref{CDF_EC_2})$. When $\gamma_{ec}$ is larger than $\gamma_{th_{0}}$, it is clearly to note that $F_{\gamma_{ec}}(\gamma_{th})=1$. Then the proof is completed.
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Sons, 2013. |
An imputation-based approach for parameter estimation in the
presence of ambiguous censoring with application in industrial
supply chain
[
[
Indiana University–Purdue University
Department of Mathematical Sciences
Indiana University–Purdue University
Indianapolis, Indiana 46202-3216
USA
(#1; 5 2008; 3 2010)
Abstract
This paper describes a novel approach based on “proportional
imputation” when identical units produced in a batch have random but
independent installation and failure times. The current problem is
motivated by a real life industrial production–delivery supply chain
where identical units are shipped after production to a third party
warehouse and then sold at a future date for possible installation.
Due to practical limitations, at any given time point, the exact
installation as well as the failure times are known for only those
units which have failed within that time frame after the
installation. Hence, in-house reliability engineers are presented
with a very limited, as well as partial, data to estimate different
model parameters related to installation and failure distributions.
In reality, other units in the batch are generally not utilized due
to lack of proper statistical methodology, leading to gross
misspecification. In this paper we have introduced a likelihood
based parametric and computationally efficient solution to overcome
this problem.
,
imputation,
maximum likelihood estimation,
proportional sampling,
reliability,
doi: 10.1214/10-AOAS348
††volume: #1††issue: 4
4
2010
{aug}
a]Samiran Ghoshlabel=e1]samiran@math.iupui.edu
Censoring
.
1 Introduction: Background of the problem.
After the production process, consumer goods are often distributed
through multi-step channels, giving rise to the term
“production–delivery” supply chain. An exception to this practice
is “just-in-time” manufacturing where a product is assembled and
shipped directly only upon the request of a customer, which is quite
popular in the personal computer industry. However, for most
consumer products, items produced by a company are not shipped
directly to the final customer. The traditional route for any large
scale industrial operation is to ship the manufactured products to a
warehouse. The warehouses are often maintained by third party
retailer/shops, from where the products are sold and installed at a
future date to the final customer. Due to geographic as well as
company–retailer relationship, once the batch is shipped, it is
often unknown to the producing company whether a specific unit is
working or is still not installed, until and unless the unit stops
working and the final customer claims a warranty at a future date.
At that point in time the data on the failed unit becomes
“complete” in a sense that we know exactly its installation as well
as failure time. For all other units it is not known (hence
“partial” information only) whether they are working or are not at
all installed. The above setup is quite common in practice in many
industrial supply chains, giving rise to a situation where in-house
engineers face a dilemma regarding the optimal usage of available
information. The untimely failure of a unit is always costly
to the producer from the warranty perspective [Abernethy (1996)]. Also,
after infant mortality, reliability assessment and future lifetime
prediction at an early stage of the product lifespan is advantageous
for appropriate customer satisfaction issues.
Reliability estimation requires knowledge of the population at risk
and the reliability of each unit of the population. The major
objective is always to acquire timely information of interest on
failure modes. However, in the presence of both “complete” and
“partial” information, current practice is to estimate relevant
reliability information by using those units which have completed
their life cycle (i.e., “complete” portion only), while not
utilizing the “partial” information [Abernethy (1996); Kececioglu (1993)]. The
primary reason for this is the absence of any established
methodology for dealing with the current situation. This clearly
makes the inferential procedure suboptimal. In this article we
adopt a proportional imputation based approach to yield a practical
solution to the situation described above. The thrust of this paper
is the estimation of the unknown parameters under the assumption
that we know the actual parametric distribution of installation as
well as failure time. The more general problem of unknown
distributional form for either installation or failure time (or
both) is not considered here and is left for future work.
The rest of the article is organized as follows. In the first three
sections we present notation and a theoretical justification of the
proposed methodology. Section 5 presents the algorithm for
proportional imputation. The connection between the exact likelihood
based approach and our proposed algorithm is described in Section 6.
Section 7 describes the simulation performance of our algorithm. We
also include the analysis of industrial furnace data in Section 8.
We conclude the article with some discussion.
2 Notation and mathematical setting.
The problem of interest is motivated from a large industrial company
producing residential furnace components. The units are produced and
shipped within the continental USA via multiple channels. However,
the general description of the problem and our solution is neither
dependent on a specific company nor confined to a specific
commodity. Rather, our proposed solution will have a broader
application since the setup is common to many production delivery
supply chains. Consider a setup in which $N$ identical units are
produced in a batch, which are then shipped to a warehouse. These
units will be installed only after being purchased by the customer
at some future date. We assume there exists no substantial time lag
between purchase and actual installation of unit/units. Purchase and
installation will be considered as the event of interest, and the
time in which this transpires will be referred to as the
“installation time.” Consider a fixed end of study time $T_{0}$. The
general data description at hand is rather simple. For a particular
unit we either know both the installation and failure times or know
nothing at all. In fact, for many units at time $T_{0}$, their current
status will be unknown due to the fact that they have not yet failed
either due to noninstallation or are still in working condition.
Let $X$ $(\sim F_{X}(\cdot))$ and $T$ $(\sim F_{T}(\cdot))$ denote the
continuous random variables corresponding to installation time and
failure time and which are assumed to be independent of each other.
In this paper we assume that $F_{X}(\cdot)$ and $F_{T}(\cdot)$ are completely
specified but with unknown parameters. We denote the random set
$\Omega=\{i\in\{1,2,\ldots,N\}\dvtx X_{i}+T_{i}\leq T_{0}\}$ to be the set
of indices of the completely observed units. Let $C$ denote the
cardinality of $\Omega\dvtx C=|\Omega|=\sum_{i=1}^{N}I\{X_{i}+T_{i}\leq T_{0}\}$. Following standard results in survival/reliability analysis,
the complete likelihood for the above setup is
$$\displaystyle \qquad L(F_{X},F_{T})$$
$$\displaystyle=$$
$$\displaystyle\prod_{i=1}^{N}[f_{X,T}(x_{i},t_{i})I\{x_{i}+t_{i}\leq T_{0}\}]^{%
\tau_{i}}[P\{X+T>T_{0}\}]^{1-\tau_{i}}$$
$$\displaystyle\propto$$
$$\displaystyle\biggl{\{}\prod_{i\in\Omega}f_{X,T}(x_{i},t_{i})\biggr{\}}\biggl{%
\{}S_{X}(T_{0})+\int_{0}^{T_{0}}S_{T}(T_{0}-x)\,dF_{X}(x)\biggr{\}}^{N-C},$$
where $\tau_{i}$ is an indicator of whether the $i$th unit is observed
or not for $i=1,2,\ldots,N$. The above likelihood is difficult to
maximize numerically except for the very restrictive case when $X$
and $T$ are independent and identically distributed (i.i.d.)
according to an exponential distribution. For the other popular
reliability distributions (e.g., Weibull, Gamma), the above
likelihood is difficult to maximize due to excessive flatness,
especially when $C\ll N$. In the furnace data described in Section
8
and also in other simulation studies, the $\frac{C}{N}$ ratio is on
average $40\%$ or below. With only this much data the above
likelihood essentially becomes very flat and brute force
optimization often produces unstable estimates with large variances.
For more details on this see the simulation studies in Section 7.
Next we provide a proportional imputation scheme that has close
connection with the above likelihood, yet it employs a search
strategy parallel to Monte-Carlo-based approaches which is
computationally faster and produces stable estimates.
2.1 Standard practice and an alternative formulation.
For notational simplicity and without loss of generality, we assume
that the first $C$ units are observed or, in other words, we have
complete information for $\{x_{i},t_{i}\}_{i=1}^{C}$. Notably, the
manufacturer knows nothing about a unit under two circumstances.
First, if $X>T_{0}$, that is, the unit is not being installed until time
$T_{0}$ and denoted as event $B$. Second, $X<T_{0}$ but $T>T_{0}-X$, that is,
the unit is installed but still in operation and denoted as event
$D$. Since exact likelihood is difficult to use, traditional
practice is of two forms [Abernethy (1996); Kececioglu (1993)]. The most simplistic
approach is to think that only $C$ units are produced. Since we will
have complete information for all of them, we may use standard
theory to estimate model parameters corresponding to $X$ and $T$
under specific distributional choices. The other practice is to
think that we have $C$ units not from the full distribution but
rather from the truncated distribution of both $X$ and $T$ (i.e.,
observed if $X<T_{0}$ and $T<T_{0}$). Then under some specific
distributional assumptions (popular choices are Exponential, Weibull,
etc.) the MLE or rank egression based approaches are used for
parameter estimation [Wang (2004); Johnson (1964); Michael and Schucany (1986)]. Both of these
approaches will produces erroneous estimates for the setup
considered. The situation will be much simpler if it is also known
for a specific “noninformative” unit whether it is under the event
$B$ or $D$. This knowledge, if available, will enable us to render
the case as Type-1 right censoring at $T_{0}$ either on $X$ (under
$B$) or on $T$ (under $D$) and then follow the usual theory of
estimation with censored data [Meeker and Escobar (1998); Klein and Moeschberger (2005)]. Unfortunately,
practical considerations suggest that even this information will not
be available under most producer–retailer setups resulting in
“ambiguous” censoring. This is unavoidable unless the producer
company has an agreement with the retailer to get in-time unit
specific sales information. This involves monetary implications and
often short-term cost cutting actions get higher priority.
In this article we took an alternative route to impute the
installation time ($X$) for those units under $D$, that is, installed
but not failed. Note that if we know or can successfully impute the
installation time and assume that the unit is still working, this
essentially means the failure time is being censored. This enables
us to use standard methodology to estimate the model parameters [see
Meeker and Escobar (1998); Klein and Moeschberger (2005)]. However, the crucial question is not only how
to impute the unobserved installation time, but also how many units
are needed to be imputed. Next we present the theory of an
interesting computational approach to achieve this task based on a
proportional sampling imputation scheme.
3 How many to sample and where to sample from?
In the
parametric setup we generally assume some distributional form for
$X$ and $T$, Weibull and Exponential being the most popular choice
to reliability engineers [Abernethy (1996)]. Our present methodology is
general in the sense that it does not depend on any specific
distributional choice for both $X$ and $T$. Note that for $C$
complete units we have samples from three conditional distributions,
namely:
[]
1.
$x|X+T\leq T_{0}$;
2.
$t|X+T\leq T_{0}$;
3.
$x+t|X+T\leq T_{0}$.
It is not difficult to formalize an estimation procedure if we have
samples from $\{x|X\leq T_{0}\}$. However, the identity
$$f_{X}(x|X+T\leq T_{0})=\frac{f_{X}(x|X\leq T_{0})F_{T}(T_{0}-x)F_{X}(T_{0})}{F%
_{T+X}(T_{0})}$$
implies
$$\displaystyle f_{X}(x|X\leq T_{0})$$
$$\displaystyle=$$
$$\displaystyle\frac{f_{X}(x|X+T\leq T_{0})F_{T+X}(T_{0})}{F_{T}(T_{0}-x)F_{X}(T%
_{0})}$$
$$\displaystyle\propto$$
$$\displaystyle f_{X}(x|X+T\leq T_{0})F_{T}^{-1}(T_{0}-x)$$
$$\displaystyle\propto$$
$$\displaystyle f_{X}(x|X+T\leq T_{0})\{1-S_{T}(T_{0}-x)\}^{-1}.$$
{remark*}
Note that the number of samples (if available) from
$\{x|X\leq T_{0}\}$ will be larger than that from $\{x|X+T\leq T_{0}\}$. Hence, we have the identity, $\#\mbox{ samples }\{x|X\leq T_{0}\}-\#\mbox{ samples }\{x|X+T\leq T_{0}\}=\#%
\mbox{ samples }\{x|X\leq T_{0}\cap T>T_{0}-X\}$. We will try to impute this difference (or
unobserved installations) via proportional sampling.
The above calculation shows why the assumption that the samples are
from right truncated and independent distributions is not valid.
Even though $X$ and $T$ are assumed to be independent, the very
nature of the “installation-failure” setup will make them
intrinsically dependent. Hence, it will be wrong to carry out
separate estimation of the parameters of the distributions of $X$
and $T$ under the truncation assumption, as in reality we do not
have samples from $\{x|X\leq T_{0}\}$ and $\{t|T\leq T_{0}\}$. Next we
have exploited this mutual dependence of $X$ and $T$ via a sampling
and imputation based approach.
3.1 Proportional imputation scheme.
To estimate the number of imputations necessary, let us denote the
random variable $V=\sum_{j=1}^{N}V_{j}$, where
$$V_{j}=\cases{1,&\quad$\mbox{if $j${th} unit is installed on or before }T_{0},$%
\cr 0,&\quad$\mbox{otherwise.}$}$$
Hence, $P[V_{j}=1]=P[X\leq T_{0}]=F_{X}(T_{0})$ and $V_{j}\sim\operatorname{Bernoulli}(F_{X}(T_{0}))$.Under the assumption that units are identical and
independent, $V\sim\operatorname{Binomial}(N,\break F_{X}(T_{0}))$. Hence, $E[V]=NF_{X}(T_{0})$ and since $C$ units are already observed, we need to impute
for $NF_{X}(T_{0})-C$ units. Of course, $NF_{X}(T_{0})-C$ need not be an
integer and so we round it up to produce a sensible estimate. We use
$[\cdot]$ notation to denote this rounding procedure. All these make
sense provided we know the parameters in $F_{X}(\cdot)$, but, in fact, the
main purpose of this paper is to estimate those parameters. However,
for the time being let us assume that some crude estimates of these
parameters are available. We will describe exactly how to get such
accurate estimates in Section 5.
Without loss of generality, we assume $C$ units are ordered in the
sense that $x_{i}<x_{i+1}$ for $i=1,\dots,C-1$. The observed
installations are depicted in Figure 1.
These installations produce a natural $C+1$ partitioning of the
study interval, that is, $[0,T_{0}]$. Due to the continuous
distributional choice for $X$, we consider the case with no ties.
However, we remark that the case with ties can be handled with minor
modifications. The probability of a unit being installed in the
interval $[x_{k},x_{k+1}]$ is given by
$P[x_{k}<X<x_{k+1}]=F_{X}(x_{k+1})-F_{X}(x_{k})$. An installed unit will
remain unobserved if it does not fail by $T_{0}$. So the conditional
probability of remaining unobserved is given by
$$P[T>T_{0}-X|x_{k}<X<x_{k+1}]=\frac{\int_{x_{k}}^{x_{k+1}}S_{T}(T_{0}-x)f_{X}(x%
)\,dx}{F_{X}(x_{k+1})-F_{X}(x_{k})}.$$
(3)
Next we present a theorem for the above conditional probability if
the interval $[x_{k},x_{k+1}]$ becomes narrower, that is, $x_{k+1}\downarrow x_{k}$.
Theorem 3.1.
$\lim_{x_{k+1}\downarrow x_{k}}\frac{\int_{x_{k}}^{x_{k+1}}S_{T}(T_{0}-x)f_{X}(%
x)\,dx}{F_{X}(x_{k+1})-F_{X}(x_{k})}=S_{T}(T_{0}-x_{k})$, provided $f_{X}{(x_{k+1})}\neq 0$.
{pf}
This follows by application of l’Hospital’s rule.
{remark*} This indicates that if $x_{k+1}\downarrow x_{k}$, then
the probability of survival (i.e., remaining unobserved) for a unit
installed exactly at $x_{k}$ will be $S_{T}(T_{0}-x_{k})$.
Now using equation (3), the joint probability of a unit being
installed in $[x_{k},x_{k+1}]$ and then remaining unobserved is
$$\qquad P[(x_{k}<X<x_{k+1})\cap(T>T_{0}-X)]=\int_{x_{k}}^{x_{k+1}}S_{T}(T_{0}-x%
)f_{X}(x)\,dx.$$
(4)
Due to the nonincreasing property of the survival function, it is
easy to see that
$$\displaystyle S_{T}(T_{0}-x_{k})\int_{x_{k}}^{x_{k+1}}f_{X}(x)\,dx$$
$$\displaystyle\leq$$
$$\displaystyle\int_{x_{k}}^{x_{k+1}}S_{T}(T_{0}-x)f_{X}(x)\,dx$$
$$\displaystyle\leq$$
$$\displaystyle S_{T}(T_{0}-x_{k+1})\int_{x_{k}}^{x_{k+1}}f_{X}(x)\,dx.$$
We would like to use the above inequality to approximate equation
(4) via
$$\displaystyle I_{k+1}$$
$$\displaystyle=$$
$$\displaystyle P[(x_{k}<X<x_{k+1})\cap(T>T_{0}-X)]$$
$$\displaystyle\simeq$$
$$\displaystyle\frac{S_{T}(T_{0}-x_{k})+S_{T}(T_{0}-x_{k+1})}{2}[F_{X}(x_{k+1})-%
F_{X}(x_{k})].$$
{remark*}
Note if $T_{0}\downarrow$ but $[x_{k},x_{k+1}]$ remains
fixed with $x_{k+1}\leq T_{0}$, then $I_{k+1}\uparrow$ due to the
monotone decreasing property of the survival function. Conversely,
if $T_{0}\uparrow$, then $I_{k+1}\downarrow$. The approximation for
$I_{k+1}$ given in equation (3.1) works very well provided the
observed installation times are not very sparse over $[0,T_{0}]$.
Next, we present a theorem characterizing unobserved installation
times over different regions.
Theorem 3.2.
Let $x_{k}\in(x_{k-1},x_{k+1})$. Then $P[T>T_{0}-X|x_{k-1}<X<x_{k}]\leq P[T>T_{0}-X|x_{k}<X<x_{k+1}]$.
The proof is provided in the Appendix. Theorem 3.2 implies that
the probability of remaining
unobserved increases as the installation time gets closer to the end
of study time $T_{0}$. Equation (3.1) characterizes the probability
of a single unit being installed in $[x_{k},x_{k+1}]$ but remains
unobserved until $T_{0}$. Note that we have $C+1$ such intervals in
$[0,T_{0}]$. Hence, the expected number of unobserved installations in
$[x_{k},x_{k+1}]$ is
$$\alpha_{k+1}=\frac{\{NF_{X}(T_{0})-C\}I_{k+1}}{\sum_{j=0}^{C}I_{j+1}},$$
with the identity
$\sum_{k=0}^{C}\alpha_{k+1}=NF_{X}(T_{0})-C$.
Lemma 3.1.
$\sum_{k=0}^{C}I_{k+1}=\sum_{k=1}^{C}\frac{F_{X}(x_{k})}{2}[S_{T}(T_{0}-x_{j-1}%
)-S_{T}(T_{0}-x_{j+1})]+F_{X}(T_{0})\frac{1+S_{T}(T_{0}-x_{c})}{2}$, where $x_{0}=0$ and $x_{C+1}=T_{0}$.
{pf}
Note that $I_{k+1}=\frac{S_{T}(T_{0}-x_{k})+S_{T}(T_{0}-x_{k+1})}{2}[F_{X}(x_{k+1})-F_{X}%
(x_{k})]$. Hence,
$$\displaystyle\sum_{k=0}^{C}I_{k+1}$$
$$\displaystyle=$$
$$\displaystyle\sum_{k=0}^{C}\frac{S_{T}(T_{0}-x_{k})+S_{T}(T_{0}-x_{k+1})}{2}[F%
_{X}(x_{k+1})-F_{X}(x_{k})]$$
$$\displaystyle=$$
$$\displaystyle[F_{X}(x_{1})-F_{X}(x_{0})]\frac{S_{T}(T_{0}-x_{0})+S_{T}(T_{0}-x%
_{1})}{2}$$
$$\displaystyle{}+[F_{X}(x_{2})-F_{X}(x_{1})]\frac{S_{T}(T_{0}-x_{1})+S_{T}(T_{0%
}-x_{2})}{2}$$
$$\displaystyle {}\vdots$$
$$\displaystyle{}+[F_{X}(x_{C+1})-F_{X}(x_{C})]\frac{S_{T}(T_{0}-x_{C})+S_{T}(T_%
{0}-x_{C+1})}{2}.$$
After cancelling successive terms and setting $S_{T}(0)=1$, we
complete the proof.
Note that even if the distributional forms for $X$ and $T$ are
known, $\alpha_{k+1}$ will still not be available if we do not know
the parameters of $F_{X}(\cdot)$ and $F_{T}(\cdot)$. In Section 5 we
will propose a general iterative approach for estimating these
parameters which in turn will yield the estimate
$\widehat{\alpha}_{k+1}$ for $k=0,\ldots,C$. In practice, we use
$[\widehat{\alpha}_{k+1}]$ for obvious reasons. We would like to put
forward a sampling based approach to impute these unobserved
installation times in Section 5. We denote the random set
$\Gamma=\{i\in\{1,2,\ldots,N\}\dvtx(X_{i}\leq T_{0})\cap(X_{i}+T_{i}>T_{0})\}$ with $|\Gamma|=\sum_{k=0}^{C}[\widehat{\alpha}_{k+1}]$ being the
number of imputed samples of $X$. In this situation, by combining the
observed and imputed samples we have the case of type-1 right
censoring for the installation time $X$. The likelihood for $X$ is
then given by
$$L_{X}=\biggl{\{}\prod_{i\in\Omega\cup\Gamma}f_{X}(x_{i})\biggr{\}}S_{X}(T_{0})%
^{N-C-|\Gamma|},$$
(7)
which we need to maximize with respect to the parameters to obtain
the ML estimates.
4 Characterization of failure time.
So far our effort was to characterize the expected number of
unobserved installation times in different partitions of $[0,T_{0}]$.
Once this is known, we want to impute these installation times in an
iterative fashion (see Section 5). For the time being, if we
assume the imputed samples represent the actual unobserved
installation times, it presents the case of random right censoring
for $T$.
This is explained in Figure 2. The left-hand diagram in
Figure 2 represents the possible scenarios with both
installation and failure times. In the right-hand diagram of Figure 2
we plot the time to failure for each unit, taking
installation time as the starting point. For the imputed
installation time (i.e., unobserved due to the fact that the unit is
still working) what we really get is $T_{0}-X$ or the random
censoring time. Hence, the observed variable is $T^{\ast}=\min\{T,T_{0}-X\}$. Note that $X$ and $T$ are assumed to be independent and so
are $T$ and $T_{0}-X$. Let $\delta$ indicate whether $T^{\ast}$ is
censored ($\delta=0$) or it is a real failure ($\delta=1$). For the
current situation we have $C$ real failures and $[NF_{X}(T_{0})-C]$
censored times, while $[N(1-F_{X}(T_{0}))]$ units do not contribute to
the estimation process as they provide no information related to
failure. The data from $n=[NF_{X}(T_{0})]$ units consists of the pair
$(t_{i}^{\ast},\delta_{i})$. Since we are interested in inference about
the parameters of $F_{T}(\cdot)$, the likelihood function for the same is
given by
$$L_{T}=\prod_{i=1}^{n}[f_{T}(t_{i}^{\ast})]^{\delta_{i}}[S_{T}(t_{i}^{\ast})]^{%
1-\delta_{i}}.$$
(8)
5 Iterative algorithm.
All our earlier calculations are solely for the purpose of parameter
estimation in the distributions of $X$ and $T$. The key quantity of
the whole discussion is ${\alpha_{k+1}}$ (see Section 3.1), which
represents the number of unobserved installation times in
$[x_{k},x_{k+1}]$. However, the estimation of ${\alpha_{k+1}}$
requires knowledge of the parameters in the distributions of $X$ and
$T$. We have assumed so far that the distributions of $X$ and $T$
are known; however, the parameters are actually unknown. Hence, an
iterative procedure is proposed.
Begin procedure
[Step 1.]
Step 0.
Find initial parameter estimates of $F_{X}(\cdot)$ and
$F_{T}(\cdot)$ assuming that they are coming from a truncated distribution
($<T_{0}$) for which we have complete knowledge (e.g., Weibull,
Exponential, etc.).
Step 1.
Using the current value of the distribution parameters, find
$\widehat{\alpha}_{k+1}$ for $k=0,\ldots,C$. Note that it is quite
possible to have $\widehat{\alpha}_{k+1}$ not as an integer, say,
$\widehat{\alpha}_{k+1}=\operatorname{int}(\widehat{\alpha}_{k+1})+%
\operatorname{frac}(\widehat{\alpha}_{k+1})=U_{k+1}+V_{k+1}$.
Step 2.
Draw $U_{k+1}$ samples from the interval $[x_{k},x_{k+1}]$ of the distribution $F_{X}(\cdot)$ using current values of the
distribution parameters.
Step 3.
First, draw a sample from a $\operatorname{Bernoulli}(V_{k+1})$. If it is
equal to one, draw another sample as in step 2, otherwise skip to
the next step. Hence, the total number of imputed samples is either
$U_{k+1}$ or $U_{k+1}+1$.
Step 4.
Re-estimate the parameters of $X$ using both imputed
and observed ($C$) samples via MLE under right censoring using
equation (7).
Step 5.
Re-estimate the parameters of $T$ by using both
observed ($C$) and censored samples via equation (8). The
random censoring value for any imputed sample is $T_{0}-X_{\mathrm{imputed}}$.
Step 6.
Return to step 1 until an acceptable
convergence tolerance level is reached on the parameter estimates.
End procedure
Note that the conventional approach stops at “Step 0” without any
further iteration, so we are simply using that as the initial guess.
Details for obtaining the MLE for some of the truncated
distributions (e.g., Exponential and Weibull) are described in the
Appendix. Though this algorithm assumes that the parametric
form of $X$ and $T$ are known, it does not depend upon any specific
distributional choice. Under the assumption that the specific
distributional choices of $F_{X}(\cdot)$ and $F_{T}(\cdot)$ are correct, the
speed of convergence depends upon the actual observed sample size
($C$) and end of study time ($T_{0}$). If $C$ is too small, it will
require many imputations (as $[NF_{X}(T_{0})-C]$ is big). Similarly, if
$T_{0}$ is too small thus representing an early study termination, it
will force $C$ to be quite small. Both of these cases represent very
little available information. This generally results in large
sampling variance with high fluctuations in the iterations resulting
in nonconvergence.
6 Connection with the exact likelihood.
Note that our main goal is to estimate parameters in the
distribution of $X$ and $T$ and typically a likelihood is a function
of those parameters. As noted earlier in Section 3, though $X$ and
$T$ are assumed to be independent, the nature of ambiguous
censoring make their joint distribution dependent, where the
functional component related to respective parameters are
nonseparable. As a consequence, maximum likelihood estimation
requires joint maximization for all parameters over the exact
likelihood function given in equation (2), which is
computationally prohibitive. Thus, a major point in this article is
the separation of the $X$ and $T$ distributions via equations
(7) and (8). A pertinent question is the theoretical
justification of the above in light of the exact likelihood. Note
that $P\{X+T>T_{0}\}=S_{X}(T_{0})+\int_{0}^{T_{0}}S_{T}(T_{0}-x)\,dF_{X}(x)$. In
case there is an oracle which supplies us information about the
$N-C$ unobserved units, that is, whether $\{X>T_{0}\}$ or $\{X\leq T_{0}\}\cap\{T>T_{0}-X\}$, the above expression simplifies
considerably. Suppose that out of those $N-C$ units we know that
$|\Gamma|$ ($\simeq[NF_{X}(T_{0})-C]$) units are installed (with
reported installation times) but have not yet failed by $T_{0}$; then
for those units, $P\{X+T>T_{0}\}=f_{X}(x)S_{T}(T_{0}-x)$. For the remaining
$N-|\Omega|-|\Gamma|$ ($\simeq[N(1-F_{X}(T_{0}))]$) no information is
available, as they are not installed. Hence, we get type-1 right
censoring on $X$ at $T_{0}$, implying $P\{X+T>T_{0}\}=S_{X}(T_{0})$. The
likelihood contribution from the imputed and unobserved units is
$\{\prod_{j\in\Gamma}f_{X}(x_{j})S_{T}(T_{0}-x_{j})\}S_{X}(T_{0})^{N-|\Omega|-|%
\Gamma|}$. Under the above
setup, the complete likelihood for all observed and imputed samples
becomes
$$\displaystyle L(F_{X},F_{T})$$
$$\displaystyle\quad\propto\biggl{\{}\prod_{i\in\Omega}f_{X}(x_{i})f_{T}(t_{i})%
\biggr{\}}\biggl{\{}\prod_{j\in\Gamma}f_{X}(x_{j})S_{T}(T_{0}-x_{j})\biggr{\}}%
S_{X}(T_{0})^{N-|\Omega|-|\Gamma|}$$
(9)
$$\displaystyle\quad\propto\biggl{\{}S_{X}(T_{0})^{N-|\Omega|-|\Gamma|}\prod_{i%
\in\Omega\cup\Gamma}f_{X}(x_{i})\biggr{\}}\biggl{\{}\prod_{i\in\Omega}f_{T}(t_%
{i})\prod_{j\in\Gamma}S_{T}(T_{0}-x_{j})\biggr{\}}.$$
This is what corresponds to equations (7) and (8).
7 Simulation studies.
Next we present some simulation studies with different choices of
reliability distributions to demonstrate the efficacy of the
proposed approach. In particular, we consider exponential and
Weibull distributions for both $X$ and $T$ with different values of
$T_{0}$. To explain the convergence criteria let us assume $\mu$ is a
parameter (in either $X$ or $T$) that needs to be estimated. We stop
the iteration when $|\frac{\mu_{i+p}-\mu_{i}}{\mu_{i+p}}|<\varepsilon$, where $i$ denotes the iteration number, $p$ is a
prespecified positive integer constant and $\varepsilon$ is a
prespecified small value chosen by the end user. For
multi-parameter cases this needs to be satisfied for every
parameter. Alternatively, in the spirit of the Monte-Carlo-based
approach, we may run a fixed but large number of iterations and
discard the first few iterations as nonstabilized (or “burn-in”)
values and keep all the remaining to report the estimated empirical
mean and standard deviation. We took the second approach as we found
that convergence is very fast even for $\varepsilon=0.0005$, except for
the situation when $\frac{C}{N}<20\%$. In every situation we also
report the exact stopping time if we choose to use the first
stopping criterion (i.e., stop if $|\frac{\mu_{i+p}-\mu_{i}}{\mu_{i+p}}|<\varepsilon$). We also report the exact runtime
in every simulation using R code on a Windows-XP-based machine until
convergence. We hope this should give the reader a comprehensive
idea about the run time efficacy of our approach. The computer code
used for the simulation is available as a supplementary material
[Ghosh (2009)].
Table 1 represents the simulation results for different
choices of distributions for $X$ and $T$. We choose $N=200$ for all
experiments. We run the iteration $1000$ times for each model, of
which we discard the first $100$ as burn-in values. The reported
parameter estimates and standard deviations are based on the
remaining $900$ iterations. We also report the convergence iteration
number, which, for the multi-parameter case, represents the maximum
of all iterations taken by individual parameters to satisfy
$|\frac{\mu_{i+p}-\mu_{i}}{\mu_{i+p}}|<\varepsilon$. As we can see
from Table 1, convergence is achieved quickly. For
parameter estimation we used the maximum likelihood approach which
is described briefly in the Appendix section. Again for other
nontrivial distributions with complicated MLE, the method of
moments or rank regression based approaches [Johnson (1964)] could be
used. In each model, following standard practice, we obtain the
initial parameter estimates for the distribution of $X$ and $T$
using the right truncated distribution. These initial estimates are
way off in all cases, which explains why standard practice is
unsatisfactory in this nontrivial situation. We summarize our
simulation result in Table 1. The first two rows in Table
1 are of special interest since we assumed
$X,T\stackrel{{\scriptstyle\mathrm{i.i.d.}}}{{\sim}}\operatorname{Exp}(\lambda=\delta)$. As shown in
Appendix B.5, the exact likelihood given in equation (2) can be solved
numerically in this case. For $T_{0}=6$ the exact likelihood based MLE
yields $\widehat{\lambda}=0.22$ with asymptotic standard deviation
$\widehat{\sigma}_{\lambda}=0.026$. For $T_{0}=5$, we get
$\widehat{\lambda}=0.18$ with asymptotic standard deviation
$\widehat{\sigma}_{\lambda}=0.028$. In both of these cases our
simulation result is very close to the true value
($\lambda=\delta=0.2$) even though we did not use the information
that $\lambda=\delta$ in our proposed algorithm. In Figure
4 we present pictorially the result for these two cases.
This supports the viability of our algorithm. Next we explore
non-i.i.d. cases. Figure 3 presents the case for $X\sim\operatorname{Exp}(\lambda)$ and $T\sim\operatorname{Exp}(\delta)$ with two different observation
times ($T_{0}=4,6$). In the first case, we choose the true model
parameters in such a way that about $50\%$ of the cases are observed
(i.e., $C>100$). Figure 3(a) and (b) present the case when
$\lambda=0.5$ and $\delta=0.2$. We observe $108$ and $66$ units for
$T_{0}=6$ and $4$, respectively. As expected, the case with more units
produces better estimates. Nevertheless, we point out that for
$T_{0}=4$, even though we observe only about $33\%$ of the units, the
final parameter estimates are still noticeably close to the true
parameter values. Similar observations could be made for the other
choice of parameter values in Figure 3(c) and (d). To
elucidate the problem when using the exact maximum likelihood based
approach, we have also plotted the log-likelihood surface (obtained
via equation (2) and numerical integration) in Figure
5 for the case $\lambda=0.5$ and $\delta=0.2$. Figure
5(a) represents the case when we have complete
observations for all units ($C=N$). However, as $T_{0}$ shrinks, $C$
goes down, and, as a result, the likelihood surface becomes very flat.
Hence, searching for the MLE becomes computationally challenging and
often leads to large variance. We have noted this problem earlier in
Section 2. Figure 5(d) presents the log-likelihood surface
obtained via equation (6) when imputation is in use. This
representative plot is obtained for a specific iteration when $97$
units are imputed while running the algorithm described in Section
5. The flatness of the resulting log-likelihood surfaces in Figure
5(c) and (d) is an indicator of computational difficulties
in finding the MLE for each case. Next, in Figure 6 we
describe the iteration result when $X\sim\operatorname{Exp}(\lambda)$ and $T\sim\operatorname{Weibull}(\beta,\theta)$. In Figure 7 we describe the
iteration result when $X\sim\operatorname{Weibull}(\beta,\theta)$ and $T\sim\operatorname{Exp}(\lambda)$. In all cases the final estimates are quite close to
the true model parameters. Though not reported here, we obtain
similar results with the gamma distribution. For details of the
sampling from a truncated gamma distribution, please refer to
Damien and Walker (2001). We have confined our simulation exploration only to
commonly used reliability distributions; however, we are hopeful
that the algorithm presented here will also work for other
distributions with nonnegative support.
8 Motivating application.
The data set that we will analyze using the current procedure came
from an industrial house producing residential furnace components
during one week in May $2001$. We consider a batch with $N=400$
units. The data consist of $C=133$ pairs of points as observed units
(i.e., $\{x_{i},t_{i}\}_{i=1}^{133}$), which have failed within the
observation time of seven years from the date of manufacturing.
Figure 8(a) shows a violin plot for installation and
failure times. The violin plot is a combination of a box plot and a
kernel density plot. There is no specific information available
about the remaining units. We are assuming that there exists no unit
which has failed but was not reported. In practice, this could have
happened for many other reasons. In the present context the
reliability engineers believe that it is appropriate to model
installation time ($X$) using an exponential distribution, while
failure time ($T$) is modeled according to a Weibull distribution [Jager and Bertsche (2004); Zhu (2007)].
It should be noted that seasonality plays an important role
in selling, installation and duty cycles (how rigorously the unit is
being used) of the product. However, since in the present case we
consider only a single batch, we assume that these effects will be
similar for every unit in the batch. When comparing the units
produced under different batches (and possibly produced at different
times of the year), additional care is required as the independence
assumption between $X$ and $T$ becomes questionable. This is due to
the fact that some installation times are associated with severe
duty cycles and more reliability problems.
Before running the algorithm we divide the installation times as
well as failure times by their corresponding standard deviation
estimated from $133$ samples. This rescaling is done for numerical
stabilization only, which results in faster convergence of the
algorithm. Rescaled random variables have straightforward
relationships with the original variables, without any drastic
change to the distributional form. We run the algorithm for $1000$
iterations, however, convergence (with $p=5$, $\varepsilon=0.0005$) was
achieved much earlier. We discard the first $100$ iterations as
burn-in and report the estimates on the basis of the remaining $900$
iterations in Table 2. For model comparison purposes we
have also investigated separately the case where $T$ is assumed to
follow the exponential distribution, without altering the
distribution of $X$. In each case we obtain the initial parameter
estimates using the right truncated distributions. Figure 8
represents the case for the Exponential–Weibull model combination.
Though the Exponential–Exponential model parameter is different from
the previous choice (see Table 2), the density plot of
the two distributions of $T$ are quite similar as depicted in Figure
9(b). We have also compared the predictive performance of
different models in Figure 9(c), including the usual
practice of truncated distributions without any imputation. We
estimated the expected number of failures to be observed for
different observation times over an interval of six months. This
expected failure number is then compared with the observed failure
number for the current data set. This required repeated
re-estimation of model parameters at different time points. As can
be seen, the truncated models have a huge overestimation problem
throughout the study period. This again justifies our earlier
criticism of current practice. Imputed models produce stable
estimates and do much better even at the very early stage of product
lifetime with only limited data. The Exponential–Weibull model
choice does a little better than the Exponential–Exponential model.
However, they are very much comparable as expected from Figure
9(b). It is desirable to estimate the expected failure
number accurately for two main reasons. First, by accurately
estimating warranty claims, an estimate of required financial
reserves can be performed. This has immense implications in terms of
future financial resource management. Second, it is desired to
continuously improve the quality of consumer products, especially at
the very high quality levels enjoyed by many consumer products
today. All these aspects necessarily depend upon the accurate and
efficient estimation of the reliability parameters (in $X$ and $T$).
The method described in this paper provides a first step in this
direction.
9 Concluding remarks.
Unlike electronic commodities, item specific tracking is not a
feasible solution for many large scale industrial operations. Hence,
the availability of both “complete” and “partial” information is
quite common. In addition, except for very rare occasions, there are
hardly any situations where all units in a batch start working at
the same time. Unavailability of the installation time in a timely
fashion is a major challenge to reliability engineers. Because of
confidentiality issues we can not reveal any company specific
information. However, we would like to mention that the above problem
exists in different industrial sectors, and there is no clear
solution thus far. In this paper we have proposed a computational
approach to solve the problem with the optimal usage of partial and
complete information. From a reliability engineer’s perspective,
this current approach is simple, fast and also has straightforward
interpretability.
The primary focus of any reliability analysis is the failure time.
However, the waiting time for the installation is also very important
in the sense that it provides valuable market specific information
from the sales perspective, including seasonality and periodic sales
patterns. In our approach we have targeted simultaneous estimation
for both installation and failure time parameters in a combined
fashion. To the best of our knowledge, this is the first attempt to
do so. Finally, we would like to point out some of the assumptions
that we have made in this paper, a violation of which will require
more research. First, we have assumed that installation time and
failure time are independent. This may be questionable in some
situations as discussed in Section 8. Second, there is no aging
effect for the units installed at different time points. Finally, we
made the assumption that the distributional form of both
installation and failure times is known. While for most of the
legacy industrial products, in-house experts have a good idea about
this from historical knowledge, it is of theoretical interest to see
the effect of convergence and the quality of parameter estimates
under incorrect parametric model specification. One way to avoid
this is to choose a larger class of models. From the reliability
perspective there is considerable effort to generalize Weibull and
other popular reliability distributions [see Bali (2003) and
Shao (2004)]. However, the resultant estimation procedure will be
more involved. Another possibility is a nonparametric extension;
however, the resulting procedure will be much more complex. In an
ongoing work we are also exploring the exact probabilistic and
inferential procedure based on equation (2).
Appendix A Proof of Theorem 3.2.
We can use the inequality (3.1) to argue that the following
holds:
$$\displaystyle S_{T}(T_{0}-x_{k})$$
$$\displaystyle\leq$$
$$\displaystyle P[T>T_{0}-X|x_{k}<X<x_{k+1}]\leq S_{T}(T_{0}-x_{k+1}),$$
$$\displaystyle S_{T}(T_{0}-x_{k-1})$$
$$\displaystyle\leq$$
$$\displaystyle P[T>T_{0}-X|x_{k-1}<X<x_{k}]\leq S_{T}(T_{0}-x_{k}).$$
Combining both of these yields the proof.
Appendix B Maximum likelihood estimation.
We concentrate here on Exponential and Weibull distribution as used
in the simulation, though other distributions with positive support,
such as gamma and log-normal, can also be considered. Most of the
results are published elsewhere and referenced as required.
B.1 Truncated exponential.
Let $X\sim\operatorname{Exp}(\lambda)$ with $0\leq X\leq T_{0}$. The p.d.f. is given
by
$$f(x|\lambda,T_{0})=\frac{\lambda\exp(-x\lambda)}{1-\exp{(-T_{0}\lambda)}}.$$
If we have $n$ observations, then differentiating the log-likelihood
equation with respect to $\lambda$ and equating it to zero yields
$$\frac{1}{\lambda}-\frac{T_{0}\exp{(-T_{0}\lambda)}}{1-\exp{(-T_{0}\lambda)}}-%
\overline{x}=0.$$
The above equation needs to be solved numerically to get the MLE of
$\lambda$.
B.2 Randomly right censored exponential.
Let $T\sim\operatorname{Exp}(\lambda)$ and we observe $T^{\ast}=\min\{T,C_{r}\}$,
where in the current context $C_{r}=T_{0}-X$ and $X$ is another random
variable denoting installation time. Let us denote our samples as
$\{t_{i}^{\ast},\delta_{i}\}_{i=1}^{n}$, where $\delta_{i}=1$ means the
sample is an actual observation and $0$ means it is censored. If we
have $\sum_{i=1}^{n}\delta_{i}=C$ true observations, then the
log-likelihood is given by
$$L(\lambda)=c\log\lambda-\lambda\sum_{i=1}^{C}t_{i}-\lambda\sum_{j=1}^{n-C}(T_{%
0}-x_{j}),$$
which upon equating to $0$ yields
$\widehat{\lambda}=\frac{C}{\sum_{i=1}^{C}t_{i}+\sum_{j=1}^{n-C}(T_{0}-x_{j})}$.
B.3 Truncated Weibull.
The MLE calculation for the truncated Weibull distribution is
somewhat involved and may not always exist. Some explicit
mathematical formulations with the required regularity conditions are
described in Mittal and Dahiya (1989). We briefly mention only the final
result here that has been used in this paper. Suppose $X\sim\operatorname{Weibull}(\beta,\theta)$, but with $0\leq X\leq T_{0}$. Let us denote
by $Y=\frac{X}{T_{0}}$. Unfortunately, the MLE for $\beta$ is not
available in closed form and needs to be solved numerically using
the equation
$$\frac{\sum_{i=1}^{n}y_{i}^{\beta}}{n}-\frac{\sum_{i=1}^{n}y_{i}^{\beta}\log y_%
{i}}{{n}/{\beta}+\sum_{i=1}^{n}\log y_{i}}+\biggl{[}\exp\biggl{\{}\frac{{n}/{%
\beta}+\sum_{i=1}^{n}\log y_{i}}{\sum_{i=1}^{n}y_{i}^{\beta}\log y_{i}}\biggr{%
\}}-1\biggr{]}^{-1}=0.$$
Once we know $\widehat{\beta}$, the MLE of $\theta$ is
$$\widehat{\theta}=T_{0}\biggl{(}\frac{\sum_{i=1}^{n}y_{i}^{\widehat{\beta}}\log
y%
_{i}}{{n}/{\widehat{\beta}}+\sum_{i=1}^{n}\log y_{i}}\biggr{)}^{{1}/{\widehat{%
\beta}}}.$$
B.4 Randomly right censored Weibull.
Suppose $T\sim\operatorname{Weibull}(\beta,\theta)$. Similar to the randomly right
censored exponential case $T^{\ast}=\min\{T,C_{r}\}$, where in the
current context $C_{r}=T_{0}-X$. We denote our data set as $\{t_{i}^{\ast},\delta_{i}\}_{i=1}^{n}$ and $\sum_{i=1}^{n}\delta_{i}=C$. The MLE is given
explicitly in Shao (2004) and Lemon (1975), which again needs to be solved
numerically for $\beta$ using the equation
$$\frac{1}{{\beta}}+\frac{\sum_{i=1}^{n}\delta_{i}\operatorname{log}t_{i}^{\ast}%
}{C}-\frac{\sum_{i=1}^{n}(t_{i}^{\ast})^{{\beta}}\log t_{i}^{\ast}}{\sum_{i=1}%
^{n}(t_{i}^{\ast})^{{\beta}}}=0.$$
Once we know $\widehat{\beta}$, the MLE of $\theta$ is
$$\widehat{\theta}=\biggl{(}\frac{\sum_{i=1}^{n}(t_{i}^{\ast})^{\widehat{\beta}}%
}{C}\biggr{)}^{{1}/{\widehat{\beta}}}.$$
B.5 Derivation of the exact MLE for i.i.d. exponential case.
We assume $X,T\stackrel{{\scriptstyle\mathrm{i.i.d.}}}{{\sim}}\operatorname{Exp}(\lambda)$. The complete
likelihood is given by
$$L(\lambda)\propto(\lambda)^{2C}e^{-\lambda\sum_{i=1}^{C}(x_{i}+t_{i})}[e^{-%
\lambda T_{0}}+T_{0}\lambda e^{-\lambda T_{0}}]^{N-C}.$$
Now differentiating the log-likelihood equation with respect to
$\lambda$ and equating it to zero yields
$$\frac{2C}{\lambda}+\frac{T_{0}(N-C)}{1+\lambda T_{0}}-\sum_{i=1}^{C}(x_{i}+t_{%
i})-(N-C)T_{0}=0.$$
The equation needs to be solved numerically for $\lambda$ to obtain
MLE.
Acknowledgments
Special thanks to Dr. Eric Adams for proposing the problem and for
his many valuable comments. I would also like to thank an anonymous
referee and the Associate Editor, whose comments provided additional
insights and have greatly improved the scope and presentation of the
paper.
{supplement}\stitle
Furnace Data Set and R Code for Furnace Data as well as
Simulation for all Models Considered in the Paper
\slink[doi]10.1214/10-AOAS348SUPP \slink[url]http://lib.stat.cmu.edu/aoas/348/supplement.zip
\sdatatype.zip
\sdescription
R code is used for the simulation as well as real data analysis.
Supplementary material has five files:
1. Furnace data in MS Excel format (data.xls).
2. Code for analyzing furnace data (code_furn.doc).
3. Code for the Exponential–Exponential model
(new_code_Exp(2).doc).
4. Code for the Exponential–Weibull model
(new_code_ExpWeb.doc).
5. Code for the Weibull–Exponential model
(new_code_WebExp.doc).
For the simulation examples data sets are generated on the fly at the
beginning of the code. No special R package is required to run the
codes. All the codes are commented for the ease of understanding.
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667–678.
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alloy in the long lifetime regime. Ph.D. thesis, Univ. Michigan. |
Dissipative Dynamics in a Quantum Register
P. Zanardii ${}^{1,2}$
${}^{1}$ ISI Foundation, Villa Gualino
Viale Settimio Severo 65 -101133 Torino, Italy
${}^{2}$ Unità INFM, Politecnico di Torino,
Corso Duca degli Abruzzi 24, I-10129 Torino, Italy
Abstract
A model for a quantum register dissipatively coupled with a bosonic thermal bath
is studied. The register consists of $N$ qubits (i.e.
spin $\frac{1}{2}$ degrees of freedom), the bath is described
by $N_{b}$ bosonic modes. The register-bath coupling is chosen
in such a way that the total number of excitations is conserved.
The Hilbert space splits allowing the study of the dynamics separately
in each sector. Assuming that the coupling with the bath
is the same for all qubits, the
excitation sectors have a further decomposition
according the irreducible representations of the $su(2)$
spin algebra.
The stability against environment-generated noise of
the information encoded in a quantum
state of the register
depends on its $su(2)$ symmetry content.
At zero temperature we find that states belonging to the vacuum symmetry sector
have for long time vanishing fidelity, whereas
each lowest
spin vector is decoupled from the bath
and therefore is decoherence free.
Numerical results are shown in the one-excitation space in the case
qubit-dependent bath-system coupling.
pacs: 71.10.Ad , 05.30.Fk
I introduction
The unavoidable interaction that each real-world system has with its environment
is one of the major limitations to pratical realization of a quantum computer [1].
Indeed the outstanding potential capabilities of such a device rely heavily on the possibility
of maintaining the quantum coherence in the system, and one of the typical effects
of the coupling with the environment is to destroy phase relations between
quantum states appearing in a linear superposition.
This latter phenomenon is referred to as decoherence [2]: it can take place
also when there is no system-environment energy exchange at all.
To overcome this difficulty, in the last few years there has been a growing interest
in the so-called error correction schemes [3], in which, by means of suitable
encondings and measurement protocols, one is able to disentangle the system
from the environment in order to recover uncorrupted information.
Another possible approach, pointed out in [4], is to make use of symmetry
to protect the information stored in a quantum state against environment-induced noise.
In this case one, rather than to design states that can be easily corrected,
looks for states that cannot easily be disturbed.
The simplest system for which dynamical symmetry provides us these ”safe” states
is a collection of $N$ qubits coupled all in the same way with the environment.
Such a system can be thought of as quantum register $S,$ analogous
to those characteristic of classical computation.
In this paper we study a model Hamiltonian describing the exchange of elementary
quanta between the register and the environment, modelled by a bosonic bath $B$.
The coupling with the bath is realized in term of the off-diagonal
generators of a $su(2)$ dynamical algebra [5].
The marginal dynamics of $S$ is dissipative (i.e. the register energy is not conserved).
The global Hilbert space
decomposes in (dynamically) invariant sectors, characterized by their $su(2)$ symmetry content
as well as their number of excitations, as it will be specified later.
It should be emphasized that the focus
of this paper is on the role played by dynamical-algebraic
structures in providing collective states of the register
intrinsecally stable against environment-induced decoherence.
The adopted physical model is, in a sense, generic and it is not
aimed to describe a specific physical implementation of a Quantum Computer
(as done instead, for example in [6]) but a broad class of open quantum systems
that could eventually turn out to be relevant for quantum data processing
applications.
In sect. II, after recalling the fundamentals of open quantum systems,
the model is introduced and its general feature briefly discussed.
In sect. III the associated Hilbert space structure is analyzed.
In sect. IV the one-excitation subspace is studied,
analytical as well as
numerical results are presented for the fidelity and entropy.
Sect. V contains some preliminary numerical results in the case
of a qubit depending coupling with the bath.
Section VI contains a number of conclusive remarks and perspectives
II the model
Before introducing our model we begin by
briefly recalling a few basic facts about open quantum systems.
Let ${\cal H}_{s},\,{\cal H}_{b}$ denote respectively the system and the environment
Hilbert
spaces. We assume ${\cal H}_{b}$ to be much larger than ${\cal H}_{s}.$
The total Hilbert space is given by the tensor product ${\cal H}={\cal H}_{s}\otimes{\cal H}_{b}.$
A state over ${\cal H}_{\alpha}$ ($\alpha=s,\,b$) is a hermitean non negative operator $\rho_{\alpha}$
of $\mbox{End}({\cal H}_{\alpha})$ with
$\mbox{tr}^{\alpha}(\rho_{\alpha})=1.$
The manifold of the state over ${\cal H}_{\alpha}$ will be denoted by ${\cal S}_{\alpha}.$
The elements of ${\cal S}_{\alpha}$ that are also projectors
($\rho^{2}=\rho$) provide the pure states. The set ${\cal S}_{\alpha}^{P}$ of pure states
generates ${\cal S}_{\alpha}$
as its convex hull, furthermore it is in a one-to-one correspondence with ${\cal H}_{\alpha}.$
According to quantum mechanics, time evolution of the overall (closed) system is unitary, therefore
if the initial state has the separable form $\rho(0)=\rho_{s}\otimes\rho_{b},\,(\rho_{\alpha}\in{\cal S}_{\alpha})$
then for any $t\geq 0,$
the marginal (Liouvillian) evolution on ${\cal S}_{s}$ (open) is given by
$${\cal E}^{\rho_{b}}_{t}\colon{\cal S}_{s}\rightarrow{\cal S}_{s}\colon\rho_{s}%
\rightarrow\mbox{tr}^{b}\,(U_{t}\,\rho(0)\,U^{\dagger}_{t}),$$
(1)
where $\mbox{tr}^{b}$ denotes the partial trace over ${\cal H}_{b}.$
The superoperators $\{{\cal E}^{\rho_{b}}_{t}\}_{t\geq 0}$
are trace-preserving completely positive maps [7], that are the most general
description of the evolution of an open quantum system.
${\cal S}_{\alpha}^{P}$ is not invariant under the action of $\{{\cal E}^{\rho_{b}}_{t}\}_{t\geq 0},$
typically an initial pure state of the subsystem becomes mixed in a very short time scale
depending on the strength of the interaction.
This state can either eventually get pure again or not, but in any case an irreversible
loss of the information stored in the initial preparation has occurred.
It is important to notice that this mechanism is active
even when there is no energy-exchange between the subsystems (i.e. the subystem Hamiltonians
are constants of motion) at finite as well as at zero temperature.
When an energy-exchange occurs we call the resulting dynamics dissipative.
We introduce now the model.
The Hamiltonian is given by $H=H_{s}+H_{b}+H_{I}$ where
$$\displaystyle H_{s}$$
$$\displaystyle=$$
$$\displaystyle\sum_{i=1}^{N}\epsilon_{i}\,\sigma_{i}^{z},$$
(2)
$$\displaystyle H_{b}$$
$$\displaystyle=$$
$$\displaystyle\sum_{k=1}^{N_{b}}\omega_{k}\,(b_{k}^{\dagger}b_{k}+1/2),$$
$$\displaystyle H_{I}$$
$$\displaystyle=$$
$$\displaystyle\sum_{i=1}^{N}\sum_{k=1}^{N_{b}}(g_{ki}b_{k}^{\dagger}\sigma^{-}_%
{i}+\mbox{h.c.}),$$
here the $\sigma_{i}^{\alpha}$’s are the spin $1/2$ Pauli operators ($i$ is the qubit index),
and the $b_{k}$’s bosonic operators.
$H_{s}$ ($H_{b}$) is the Hamiltonian of the register (bath), $H_{I}$ the register-bath interaction.
This model is closely related to the one
known in the literature as the Dicke maser model [8].
The latter, for generic
$N,\,N_{b},$ is not solvable and has a non-trivial ground-state phase diagram.
In order to shed some light on the physics of this system we write down the equation of the motion
for the Heisenberg operators $O(t)\equiv U(t)^{\dagger}\,O\,U(t).$
To simplify the expressions, it turns useful to perform the (unitary) transformation
$\sigma_{j}^{\pm}\mapsto\sigma_{j}^{\pm}\exp(\pm\,i\epsilon_{i}\,t),\,b_{k}%
\mapsto b_{k}\exp(-i\,\omega_{k}\,t)$,
whereby the Heisenberg equations then read
$$\displaystyle i\,\frac{\partial\sigma_{i}^{+}}{\partial t}$$
$$\displaystyle=$$
$$\displaystyle 2\,\sum_{k}g_{ki}(t)b_{k}^{\dagger}\,\sigma_{i}^{z},$$
(3)
$$\displaystyle i\,\frac{\partial\sigma_{i}^{z}}{\partial t}$$
$$\displaystyle=$$
$$\displaystyle-\sum_{k}(g_{ki}(t)b_{k}^{\dagger}\,\sigma^{-}_{i}-\mbox{h.c.}),$$
$$\displaystyle i\,\frac{\partial b_{k}}{\partial t}$$
$$\displaystyle=$$
$$\displaystyle-\sum_{j}g_{kj}(t)\sigma_{j}^{-}.$$
where $g_{ki}(t)\equiv g_{ki}\exp[i\,(\omega_{k}-\epsilon_{i})\,t]$.
By a formal integration of the field equation and the substitution of the result into the
spin equations one obtains
$$\displaystyle i\,\frac{\partial\sigma_{i}^{+}}{\partial t}$$
$$\displaystyle=$$
$$\displaystyle\phi_{i}^{+}(t)-2\,i\sum_{j}\int_{0}^{t}d\tau K_{ij}(t,\tau)%
\sigma_{j}^{+}(\tau)\,\sigma_{i}^{z}(t),$$
(4)
$$\displaystyle i\,\frac{\partial\sigma_{i}^{z}}{\partial t}$$
$$\displaystyle=$$
$$\displaystyle-\phi_{i}^{z}(t)-\sum_{j}\int_{0}^{t}d\tau K_{ij}(t,\tau)\sigma_{%
j}^{+}(\tau)\sigma_{i}^{-}(t)-\mbox{h.c}.$$
Here
$$\displaystyle K_{ij}(t,t^{\prime})$$
$$\displaystyle=$$
$$\displaystyle\sum_{k}g_{ki}(t)\,g_{kj}^{*}(t^{\prime})$$
(5)
$$\displaystyle\phi_{i}^{+}(t)$$
$$\displaystyle=$$
$$\displaystyle 2\,\sum_{k}g_{ki}(t)\,b_{k}^{\dagger}(0)\sigma_{i}^{z}(t),$$
$$\displaystyle\phi_{i}^{z}(t)$$
$$\displaystyle=$$
$$\displaystyle\sum_{k}g_{ki}(t)\,b_{k}^{\dagger}(0)\sigma_{i}^{-}(t).$$
This coupled system of non-linear integro-differential equations describes
the dynamics of the spin subsystem in closed form.
By means of the intermediation of the bath bosons each spin gets interacting
with all the others it via a sort of time-retarded Heisenberg
coupling.
The information about the bath (dynamics as well as
preparation) is contained in the kernels $K_{ij}$,
and in the operators $\{\phi_{i}^{\alpha}\}.$
If $\epsilon_{j}=\epsilon,\,(j=1,\ldots,N)$ and the bath-spin coupling is the same for all the spins,
one has
$$K_{ij}(t,t^{\prime})=\sum_{k}|g_{k}|^{2}\exp[-i(\epsilon-\omega_{k})\,(t-t^{%
\prime})].$$
(6)
Under rather general assumptions this kernel is strongly peaked at $t=t^{\prime}$,
If one has $K_{ij}(t,t^{\prime})\sim\delta(t-t^{\prime})$ then
(II) become a system of coupled non-linear differential equations.
Despite this strong simplification, also in this case the solution, due to non-linearity, remains difficult
and one has to resort to numerical techniques.
An alternative approach based on symmetry considerations,
will be introduced in the next section.
III Hilbert Space Structure
The Hilbert space is given by the tensor product ${\cal H}={\cal H}_{s}^{\otimes\,N}\otimes{\cal H}_{b}^{\otimes\,N_{b}}$, being ${\cal H}_{s}$ (${\cal H}_{b}$) the two (infinite) dimensional single spin (boson)
space.
The coupling of the spin system with the bosonic bath is described
by the hamiltonian
$H_{I}$ is such that the raising (lowering)
of one spin state is associated to the destruction (creation) of one boson.
From this follows that the system admits the constant of motion
$$\displaystyle{\cal I}=\sum_{i=1}^{N}\sigma^{z}_{i}+\sum_{k=1}^{N_{b}}n_{k}+N/2.$$
(7)
The eigenvalues of ${\cal I}$ give the number of elementary (spin as well as bosonic)
excitations over the reference state
$|0\rangle\equiv|0\rangle_{s}\otimes|0\rangle_{b}$.
The latter is a lowest weight vector for the spin as well
as for the boson algebra:
$\sigma^{-}_{\alpha}\;|0\rangle=b_{k}\;|0\rangle=0,\;\forall\alpha,k.$
Its energy is set equal to zero.
The Hilbert space splits into invariant eigen-spaces of $\cal I$,
${\cal H}=\oplus_{I}{\cal H}_{I};$ an elementary combinatorial argument shows that
the dimension
of the $I$-excitations space ${\cal H}_{I},\;(I\in{\bf{N}})$ is given by
$$\displaystyle d_{I}=\sum_{l=0}^{{{min}}(N,I)}\pmatrix{N\cr l}\pmatrix{I-l+N_{b%
}-1\cr N_{b}-1}.$$
(8)
If $N=N_{b}=1,$ one has $d_{0}=1,\,d_{I}=2,\,(I\geq 1),$
the model reduces to the exactly solvable Jaynes-Cummings model of quantum
optics [9].
It is worth noticing that the general spin-boson model considered in the literature
on quantum dissipation, [usually addressed in the framework of the Feynmann-Vernon
influence functional (see [10] for a review)], due to the presence
of terms $b_{k}\,\sigma^{-}_{i},\,b^{\dagger}_{k}\,\sigma^{+}_{i},$ (neglected here in view of the
rotating wave approximation)
does not conserve ${\cal I},$
spoiling the associated dynamical decomposition of the Hilbert space,
on which our subsequent numerical analysis relies.
Nevertheless, since in this papers we are interested
only in the role played by collective effects in stabilizing a quantum state,
this restrictions
does not result in any severe loss of generality.
A basis for ${\cal H}_{I}$ is given by $|\psi^{(n)}_{\alpha,k}\rangle=|\alpha\rangle\otimes|k\rangle,$
where
$$|\alpha\rangle\equiv\prod_{j=1}^{n}\sigma_{\alpha_{j}}^{+}\,|0\rangle_{s},%
\qquad|k\rangle\equiv\prod_{j=1}^{I-n}b_{k_{j}}^{\dagger}\,|0\rangle_{b}.$$
(9)
Where
$n=1,\ldots,{min}(I,N),\;\alpha\in{\cal C}(N,n),\;k\in{\cal C}^{\prime}(N_{b},N%
-n),$
${\cal C}(n,k)$ (${\cal C}^{\prime}(n,k)$) denoting the set of the combinations without
(with) repetitions of $n$ objects $k$ by $k$.
Following the general scheme of [4]
we specialize hereafter the model assuming the parameters $\{\epsilon_{i}\},\;\{g_{ki}\},\;(i=1,\ldots,N),$
independent of the qubit replica index $i.$
The first assumptions follows simply from the fact that qubits are replicas of the same system.
The independence of the coupling constants on the qubit index is analogous to the so-called Dicke limit
of quantum optics [8]; it holds - for example - when the typical bath wave-lengths coupled with the register
are much greater than the distances between the qubits.
In this latter case the environment is no able to probe the internal structure of the register:
as long as the dynamics is concerned it has an effective point-like topology.
The common value $\epsilon$ of the qubit ”magnetic” fields $\{\epsilon_{i}\}$ will be chosen as the unit
of the energy scale; analogously the
”sound velocity” of the boson will be set equal to one, so that
their dispersion relation reads $\omega_{k}=k,\;(k=2\,\pi\,n/N_{b},\;n=1,\ldots,N_{b}).$
The Hamiltonian can then be written as
$$H=\epsilon\,S^{z}+B\,S^{+}+S^{-}\,B^{\dagger}+H_{b},$$
(10)
being $S^{\alpha}=\sum_{j=1}^{N}S_{j}^{\alpha},\,(\alpha=z,\pm)$ global
spin operators, spanning a Lie Algebra $su(2),$ and $B=\sum_{k}g_{k}b_{k}$.
The fact that in (10) the spins appear only
trough the $S^{\alpha}$’s means the all the qubits are treated symmetrically:
the dynamics allows only for coherent excitations of the computational (spin) degrees of freedom.
This is, from the algebraic point of view, a very strong constraint:
the dynamics gets invariant under the action
of the symmetryc group ${\cal S}_{N}$ of the qubit index permutations.
This provides
us one more constant of motion.
Indeed from (10) immediately follows that the total spin $S^{2}$ is conserved and
${\cal H}_{I}$ splits according the $su(2)$-irrep.
The multiplicity of each irrep associated with the total spin quantum number S is given by
$$n(S,N)=\frac{N!\,(2\,S+1)}{(N/2+S+1)!\,(N/2-S)!}.$$
(11)
One finds the following decomposition in invariant subspaces
$$\displaystyle{\cal H}_{I}$$
$$\displaystyle=$$
$$\displaystyle\bigoplus_{S=S_{m}(N,I)}^{N/2}\bigoplus_{r=1}^{n(S,N)}{\cal H}_{I%
}(S,r),$$
(12)
$$\displaystyle{\cal H}_{I}(S,r)$$
$$\displaystyle=$$
$$\displaystyle\bigoplus_{S^{z}=-S}^{min\,(I-N/2,\,S)}|I,\,S,\,r,\,S^{z}\rangle%
\otimes{\cal H}_{b}(N_{b}(I,S^{z})),$$
where $S_{m}(N,I)=\,max\,(N/2-I,s),$ ( $s=0$ for $N$ even, and $s=1/2$ otherwise), ${\cal H}_{b}(N)$ denotes
the eigenspace, in ${\cal H}_{b}$, of $N_{b}=\sum_{k}n_{k}$ corresponding to the eigenvalue $N,$
$N_{b}(I,S^{z})=I-N/2-S^{z},$ $|I,\,S,\,r,\,S^{z}\rangle$ is a simultaneous eigenvector
of ${\cal I},\,S^{2},\,S^{z}$ associated respectively to the eigenvalues $I,\,S\,(S+1),\,S^{z}.$
The reference state $|0\rangle,$ belongs to the subspace $\oplus_{I}{\cal H}_{I}(N/2),$ with maximal total spin eigenvalue $S=N/2:$ this subspace will be denoted
by ${\cal H}^{sym},$ and referred to as the symmetric subspace.
If ${\cal H}^{sym}_{I}\equiv{\cal H}^{sym}\cap{\cal H}_{I},$ one has
$$\displaystyle{\cal H}^{sym}_{I}$$
$$\displaystyle=$$
$$\displaystyle\mbox{span}\{(S^{+})^{n}|\psi^{(I-n)}_{0,k}\rangle\,|\,n=0,\ldots%
,I\}$$
(13)
$$\displaystyle\mbox{dim}({\cal H}_{I}^{sym})$$
$$\displaystyle=$$
$$\displaystyle\sum_{l=0}^{{{min}}(N,I)}\pmatrix{I-l+N_{b}-1\cr N_{b}-1}.$$
The orthogonal complement of ${\cal H}^{sym}_{I}$ will be denoted by ${\cal H}^{A}_{I}.$
It is the direct sum of all the sectors with non-maximal $S^{2}$-eigenvalue.
Before ending this section we notice that an additional term of the form
$H^{\prime}=S^{z}\sum_{k}w_{k}(b_{k}+b_{k}^{\dagger}),$ would destroy the $u(1)$ symmetry
generated by ${\cal I},$ but not the $su(2)$ structure.
Such a term, considered in [4], does not correspond to an energy-exchange
but is a source of pure decoherence.
Since our analysis relies on the invariant decomposition
${\cal H}=\oplus_{I}{\cal H}_{I}$
this term has been omitted here.
IV $1$-Excitation Space
Due to the field-theoretic nature of our model,
the dimensionality formula (8) clearly shows that for
increasing excitation number $I$ the problem of diagonalizing $H$ becomes
rapidly intractable.
In particular, a finite-temperature analysis (arbitrary number of excitations)
is very difficult.
Nevertheless one of the interesting features of quantum noise is to be active
also a $T=0,$ thanks to vacuum fluctuations.
This latter issue can be addressed by exact numerical means, without
an artificious truncation of the bosonic space,
by noticing that
the one-excitation space ${\cal H}^{(1)},$ has dimension $d_{1}=N+N_{b},$
that is only a linear function of the total number of degrees of freedom.
The basis $\{|\psi^{(1)}_{\alpha,k}\rangle\}$ is given by
$|\alpha\rangle\equiv\sigma^{+}_{\alpha}\;|0\rangle,(\alpha=1,\ldots,N)$ and
$|k\rangle\equiv b^{\dagger}_{k}\;|0\rangle,(k=1,\ldots,N_{b}).$
Equation (III) in this case reads
$${\cal H}_{1}={\cal H}_{1}(N/2,1)\bigoplus_{r=1}^{N-1}{\cal H}_{1}(N/2-1,r),$$
(14)
The symmetric space ($S=N/2$) is $N_{b}+1$-dimensional and it is spanned by the vector
$|\psi^{sym}\rangle\equiv N^{-1/2}S^{+}\,|0\rangle,$ and by the whole set $\{|k\rangle\}.$
The subspace ${\cal H}^{A}_{1}$ corresponds to $S=N/2-1.$
An orthonormal basis of ${\cal H}^{A}_{1}$ is given by
$$|\phi_{k}\rangle\equiv S_{k}^{+}\,|0\rangle,\qquad S_{k}^{+}\equiv N^{-1/2}%
\sum_{j=1}^{N}e^{i\,k\,j}\sigma_{j}^{+},$$
(15)
where,
$k=2\,n\pi/N,\,n=1,\ldots,N-1.$
Since in ${\cal H}^{A}_{1}$ the bosonic vacuum factorizes, this subspace, when necessary,
will be identified with its projection over ${\cal H}_{s}$.
Now we observe that the vectors $\{|\phi_{k}\rangle\},$ are
annihilated by $S^{-},$ as they have minimal $S^{z}$-projection, but also by the $\{b_{k}\}$,
as they have empty boson sector. From this follows that
$H_{I}\,|\phi_{k}\rangle=0,(\forall k)$ therefore ${\cal H}^{A}_{1}$ is decoupled from the bath;
it is an energy eigenspace with eigenvalue $E=\epsilon.$
In terms of evolution superoperators, if $\rho$ is a state over ${\cal H}^{A}_{1},$
we have ithe fixed-point relations
${\cal E}_{t}^{0}(\rho)=\rho,\,(t\geq 0),$
where ${\cal E}^{0}_{t}$ denotes the superoperator associated with the bath-vacuum
density matrix $|0\rangle_{b}\langle 0|_{b}.$
The states over ${\cal H}^{A}_{1}$ are unaffected by the decoherence induced by coupling
with the bath vacuum and can therefore to encode information in a safe way.
The space ${\cal H}^{A}_{1}$ is noiseless only at zero temperature;
for finite temperature the $|\phi_{k}\rangle$’s get mixed with all the vectors
belonging to the same $su(2)$-irrep, making the induced dynamics non unitary.
It is important to notice that, for $N\neq 2,$ this states are not the
noiseless ones introduced in [4],
as the latter are associated with spin singlets (i.e. are annihilated by $S^{-}$ and $S^{+}$)
and are decoherence-free at any temperature,
whereas the $|\phi_{k}\rangle$’s belong to $N-1$-dimensional $su(2)$ multiplets.
The spectrum in the symmetric subspace can be obtained by resorting to
exact numerical diagonalization of $H$, that provides the
eigenvectors and eigenvalues $\{|\phi_{i}\rangle,E_{i}\}_{i=1}^{d_{1}}.$
On the other hand the spectrum
in ${\cal H}^{sym}_{1}$ is given by
the $N_{b}+1$ zeros of the expression [11]
$$P_{N,N_{b}}(E)=E-\epsilon-N\,\sum_{k=1}^{N_{b}}\frac{|g_{k}|^{2}}{E-\omega_{k}},$$
(16)
that corresponds to the analogous single spin problem with rescaled coupling
$g_{k}\mapsto\sqrt{N}\,g_{k}.$
This follows from the symmetry constraint that makes $|\psi^{sym}\rangle$
the only state coupled with the bosonic modes.
Let $|\psi_{0}\rangle=\sum_{i}c_{i}^{0}\,|\phi_{i}\rangle,\;(c_{i}^{0}=\langle\phi_%
{i}|\psi_{0}\rangle)$
be the initial state; at $t>0$ we can write, in terms of the chosen basis
$$\displaystyle|\psi(t)\rangle$$
$$\displaystyle\equiv$$
$$\displaystyle e^{-i\,H\,t}|\psi_{0}\rangle=\sum_{\gamma=1}^{d_{1}}C_{\gamma}(t%
)\,|\gamma\rangle\in{\cal H},$$
(17)
$$\displaystyle C_{\gamma}(t)$$
$$\displaystyle=$$
$$\displaystyle\sum_{i=1}^{d_{1}}c^{0}_{i}\,c^{i}_{\gamma}e^{-i\,E_{i}\,t},%
\qquad(c^{i}_{\gamma}\equiv\langle\gamma|\phi_{i}\rangle).$$
The marginal density matrix is given by $\rho_{s}(t)=\mbox{tr}^{b}|\psi(t)\rangle\langle\psi(t)|$.
By using the relations
$$\displaystyle\mbox{tr}^{b}|\alpha\rangle\langle\alpha^{\prime}|$$
$$\displaystyle=$$
$$\displaystyle\sigma^{+}_{\alpha}|0\rangle_{s}\langle 0|_{s}\sigma^{-}_{\alpha^%
{\prime}},\;\mbox{tr}^{b}|k\rangle\langle k^{\prime}|=\delta_{kk^{\prime}}|0%
\rangle_{s}\langle 0|_{s},$$
(18)
$$\displaystyle\mbox{tr}^{b}|\alpha\rangle\langle k|$$
$$\displaystyle=$$
$$\displaystyle\mbox{tr}^{b}|k\rangle\langle\alpha|=0,$$
one obtains
$$\displaystyle\rho_{s}(t)$$
$$\displaystyle=$$
$$\displaystyle\sum_{\alpha\alpha^{\prime}=1}^{N}C_{\alpha}(t)\,\bar{C}_{\alpha^%
{\prime}}(t)|\alpha\rangle\langle\alpha^{\prime}|$$
$$\displaystyle+$$
$$\displaystyle|0\rangle_{s}\langle 0|_{s}\sum_{k=1}^{N_{b}}|C_{k}(t)|^{2}.$$
The first (last) $N$ ($N_{b}$) terms describe a sector with a reversed (excited) spin (boson).
The marginal density matrix can be readily diagonalized, simply by observing
that it can be written in the form $\rho_{s}(t)=P_{1}(t)\,|\psi_{s}(t)\rangle\langle\psi_{s}(t)|+P_{0}(t)|0\rangle%
_{s}\langle 0|_{s},$ where
$$|\psi_{s}(t)\rangle=\frac{1}{\sqrt{P_{1}(t)}}\sum_{\alpha=1}^{N}C_{\alpha}(t)%
\,|\alpha\rangle\in{\cal H}_{s}^{\otimes N},$$
(20)
and
$P_{0}(t)=1-P_{1}(t)=\sum_{k=1}^{N_{b}}|C_{k}(t)|^{2}.$
The von Neumann entropy of $S$ is therefore given by
$S_{s}(t)=-\sum_{i=0}^{1}P_{i}(t)\log_{2}P_{i}(t).$
A completely symmetric expression ($\alpha\leftrightarrow k,\,P_{1}\leftrightarrow P_{0}$) is obtained
for the bath marginal density matrix $\rho_{b}=\mbox{tr}^{s}(\rho),$
from which follows that $S_{b}=S_{s},$ furthermore
we have $S(S|B)=S(B|S)=-S_{s}$ for the conditional entropies, and $S(B:S)=-2\,S_{s}$ for the mutual entropy;
this is a consequence of the purity of the overall
system-bath state.
In order to study the corruption of the information stored in a pure quantum state it appears useful
to study the following quantity, called (input-output) fidelity [12]
$$F(t)=\langle\psi_{0}^{s}|\,\rho_{s}(t)\,|\psi_{0}^{s}\rangle.$$
(21)
The fidelity measures the overlap between the initial state $|\psi^{s}_{0}\rangle$
and the evolved one.
In the following we shall be interested in the evaluation of (21)
with initial data in ${\cal H}_{1}$ of the form $|\psi_{0}\rangle=|\psi_{0}^{s}\rangle\otimes|0\rangle_{b},$
($\rho_{s}(0)=|\psi_{0}^{s}\rangle\langle\psi_{0}^{s}|$). For such initial preparation, if
$|\tilde{\psi}_{s}(t)\rangle=\sqrt{P_{1}(t)}\,|\psi_{s}(t)\rangle,$
one can write
$$\displaystyle F(t)$$
$$\displaystyle=$$
$$\displaystyle\langle\tilde{\psi}_{s}(t)|\,\rho_{s}(0)\,|\tilde{\psi}_{s}(t)\rangle$$
$$\displaystyle=$$
$$\displaystyle|\langle\psi_{0}^{s}|\tilde{\psi}_{s}(t)\rangle|^{2}=|\sum_{%
\alpha=1}^{N}C_{\alpha}(t)\,\bar{C}_{\alpha}(0)|^{2}\equiv|D(t)|^{2}.$$
The ”decoherence” function $D(t)$ is related to the decay of the off-diagonal elements
of $\rho_{s}(t)$; indeed if, at $t=0$, we prepare the system in the state
$2^{-1/2}\,(|0\rangle_{s}+|\psi_{0}^{s}\rangle)\otimes|0\rangle_{b},$
where $S^{z}\,|\psi_{0}^{s}\rangle=(1-N/2)\,|\psi_{0}^{s}\rangle,$
it is immediate to check that
$\langle\psi_{0}^{s}|\rho_{s}(t)|0\rangle_{s}=2^{-1}\,D(t).$
The mechanism responsible for the energy exchange induces also
a dephasing between the zero and one excited spin states: dissipation is associated with decoherence.
In figure (1) is shown the behaviour of $F(t),$ for different values of the coupling with the bath,
with initial data $|\psi_{0}^{s}\rangle=|\psi^{sym}\rangle.$
In this figure and in the subsequent ones $\epsilon^{-1}$ is chosen as the time unit.
These results are obtained by exact diagonalization of $H$ in ${\cal H}_{1},$ that provides
the dynamical functions $\{C_{\gamma}(t)\}$.
For strong bath-system couplings $F(t)$ develops oscillatory structures, due to the back and forth
exchange of energy between the system and the bath.
We report the simulations for weak couplings, since it is the case physically relevant.
Furthermore, since we are essentialy interested in the role played in the large times dynamics,
by the symmetry structure of the initial state, and not in a detailed description
of the bath-system coupling, we have choosen $g_{k}=g_{0},\,(\forall k).$
From the point of view of the energy-information loss
the latter choice is the worst case in that each qubit is coupled equally well with all the bath modes,
which should not, of course, be the case in real systems.
The fidelity in this range of coupling parameters and for intermediate times, vanishes in exponential way
$F(t)\simeq\exp(-t/\tau).$
The relaxation time $\tau,$ which turns out to be inversely proportional to $N\sum_{k}|g_{k}|^{2},$
is the time scale over which the dissipative process takes place.
The real and immaginary parts of $D(t)$ have an exponential damping modulated by oscillations
over a time scale $\epsilon^{-1}.$
For very small times a naive perturbation up to the second order in $H_{I}$
shows that indeed $F(t)\stackrel{{\scriptstyle t\rightarrow 0+}}{{\simeq}}1-t^{2}/2\,\,N\,\Delta,$
where $\Delta=\sum_{k}|g_{k}|^{2}.$
Of course this process is nothing but the relaxation of the excited spin,
whose energy is transferred to the environment; for sufficiently large times one finds
$$U_{t}\,|\psi^{sym}\rangle\otimes|0\rangle_{b}=|0\rangle_{b}\otimes|\psi_{b}\rangle,$$
(23)
where $|\psi_{b}\rangle=\sum_{k}c_{k}\,|k\rangle,$ is a superposition of all one boson states.
Some words of caution are now in order.
The model under consideration is nothing but a multi-mode generalization
of the Jaynes-Cummings model with many atoms. In analogy with the latter,
for long time scale, $t>t_{C},$ it exhibits a complex pattern of collapses and
revivals [13]. Furhermore since in each excitation space we have only a
finite number of degrees of freeedom the phenomenon of the Poincaré recurrences
is also present for $t>t_{R}.$
In the following we will show results for $t\ll t_{C},\,t_{R},$
in other terms we assume that, thanks
to the great number of bosonic modes and the weak coupling,
the physically relevant time-scales are
much smaller than the ones at which this more complex behaviour appear.
The energy exchange of the register with the bath can then be considered
irreversible.
Suppose now that $|\psi_{0}^{s}\rangle=c_{s}\,|\psi^{sym}\rangle+c_{a}\,|\psi^{a}\rangle,$ where
$|\psi^{a}\rangle\in{\cal H}_{1}^{A},$ is a normalized vector, and $|c_{a}|^{2}+|c_{s}|^{2}=1.$
Also, for the sake of concreteness, and without any loss of generality, let us consider the case $N=2.$
The vectors $|\psi^{sym}\rangle,\,|\psi^{a}\rangle,$ are given respectively by the two Bell-basis states
$|\psi^{sym}\rangle=\frac{1}{\sqrt{2}}(|\uparrow\downarrow\rangle+|\downarrow%
\uparrow\rangle),$
$|\psi^{a}\rangle=\frac{1}{\sqrt{2}}(|\uparrow\downarrow\rangle-|\downarrow%
\uparrow\rangle).$
The initial marginal density matrix of $S$ is therefore given by
$$\displaystyle\rho_{s}(0)$$
$$\displaystyle=$$
$$\displaystyle|c_{s}|^{2}\,|\psi^{sym}\rangle\langle\psi^{sym}|+|c_{a}|^{2}\,|%
\psi^{a}\rangle\langle\psi^{a}|$$
$$\displaystyle+$$
$$\displaystyle c_{s}\,\bar{c}_{a}\,|\psi^{sym}\rangle\langle\psi^{a}|+\bar{c}_{%
s}\,c_{a}\,|\psi^{\alpha}\rangle\langle\psi^{sym}|,$$
By using our previous result for the symmetric initial state, and the fact that $|\psi^{a}\rangle$
is an energy eigenstate, it follows easily from (IV), for $t$ large enough, that
$$\displaystyle{\cal E}^{0}_{t}(|\psi^{a}\rangle\langle\psi^{sym}|)$$
$$\displaystyle=$$
$$\displaystyle{\cal E}^{0}_{t}(|\psi^{sym}\rangle\langle\psi^{a}|)=0,$$
(25)
$$\displaystyle{\cal E}^{0}_{t}(|\psi^{a}\rangle\langle\psi^{a}|)$$
$$\displaystyle=$$
$$\displaystyle|\psi^{a}\rangle\langle\psi^{a}|,\;{\cal E}^{0}_{t}(|\psi^{sym}%
\rangle\langle\psi^{sym}|)=|0\rangle_{s}\langle 0|_{s}.$$
Therefore
the large times density matrix is given by
$$\rho_{s}\simeq|c_{a}|^{2}|\psi^{a}\rangle\langle\psi^{a}|+|c_{s}|^{2}|0\rangle%
_{s}\langle 0|_{s},$$
(26)
from which straightforwardly follows for the fidelity the behaviour
$$F{\simeq}|c_{a}|^{4}=(1-|c_{s}|^{2})^{2}=(1-|\langle\psi^{sym}|\psi_{0}^{s}%
\rangle|^{2})^{2}.$$
(27)
In other terms:
the final state depends on the initial preparation symmetry content;
a complete corruption of the initial information
is obtained only if the initial state belongs to the vacuum $S^{2}$-sector ${\cal H}_{1}(N/2)$,
so that the smaller is the projection over it, the closer to one is the fidelity.
The extreme case is $|\psi_{0}^{s}\rangle\in{\cal H}_{1}^{A},$
$F(t)=1,\,\forall t$ in which there is no relaxation at all.
In the intermediate situations the spin system remains partially entangled with the
environment and its state never gets pure.
This situation is illustrated in figures (2), and
(3) where fidelity and entropy are shown as functions of time
in the case of $|\psi_{0}^{s}\rangle=M^{-1/2}\sum_{\alpha=1}^{M}|\alpha\rangle,$ for $M=1,\,2,\,3.$
In this case it is trivial to check, by using equations (26), and (27),
that $F\simeq(1-M/N)^{2},$ and $S_{s}\simeq(M/N-1)\log_{2}(1-M/N)-M/N\log_{2}(M/N).$
Notice that if $|c_{s}|^{2}=1,$ one has the complete de-excitation of the spin system, therefore
the initial state $|\psi^{sym}\rangle$ is maximally entangled
and the final state $|0\rangle_{s},$
with zero mutual entanglement of the qubits.
The system undergoes energy as well as information loss.
On the other hand
if $|c_{a}|^{2}=1,$ the final, and initial, state $|\psi_{a}\rangle$
is maximally entangled: energy and information are conserved.
In the temporal range in which the decay of fidelity is exponential,
the following relation between relaxation times holds: $\tau(c_{s})\,|c_{s}|^{2}=\tau(1).$
V Replica dependent coupling
In this section we present some numerical results in the case in which the system-bath coupling depends on the
qubit replica.
If the coupling functions $\{g_{k}(i)\}$ and/or the qubit energies $\{\epsilon_{i}\}$ depend on
the qubit replica, the total spin operator
$S^{2}$ is no longer a constant of the motion.
This situation is expected to be more realistic than the one previously assumed
in that the latter amounts to have a bath with an infinite (i.e. very large) coherence length.
In this case the decomposition (III) is not invariant: the
dynamics results in a non-trivial mixing of the $su(2)$-irreducible sectors ${\cal H}_{I}(S,r).$
In particular one has $S^{-}{\cal H}_{1}^{A}\neq 0,$ therefore the vectors $|\psi^{a}\rangle$ can decay.
In other words the loss of the symmetry constraint allows the dissipation-decoherence
induced by the bath to invade the whole Hilbert space.
We choose
$g_{k}(i)=g_{0}\,\cos(k\,i/\xi),$
where $\xi$ is a parameter related to the bath coherence length
(so that for $\xi=\infty$ we recover the results of the previous
sections).
In figure (4) are reported the plots of $F(t)$ with initial condition in ${\cal H}_{1}^{A}$
for different $\xi$’s.
Figure (5) shows the behaviour of $F(t)$ for small times with $\xi=1.$
Notably one observes that the initial condition $|\psi_{0}^{s}\rangle\in{\cal H}_{1}^{A},$
exhibits a faster fidelity decay with respect to $|\psi_{0}^{s}\rangle\in{\cal H}_{1}^{sym},$
for short times $t<t_{c}$.
For longer times, with obvious meaning of the notation, $F_{A}(t)>F_{sym}(t).$
The numerical simulations show in any case that $\bar{F}_{A}>\bar{F}_{sym},$ the bar denoting temporal
average.
VI conclusions
In this paper we have presented the study of a physical model for a
quantum register coupled with the environment.
Information is encoded in the quantum state of the register.
The register consists of $N$ non-interacting replicas of a two-level system (i.e. a $N$-qubits register).
The register environment is described by a bosonic bath consisting of $N_{b}$ modes,
with $N_{b}\gg N.$
Its coupling with the register is realized by the exchange of elementary quanta
of energy. The resulting dynamics of the register is dissipative;
in the weak coupling regime, energy and information are irreversibly lost into the bath.
Even though the model is non-trivial, exact
analytical as well as numerical results can be obtained thanks to the existence
of a constant of the motion (excitation number) that leads to a decomposition of the total Hilbert
space in dynamically independent sectors.
Assuming that the environment couples in the same way with all the register qubits
one has a further splitting of the sectors according to the irreducible representations of the
spin $su(2)$ algebra. Each $su(2)$ lowest vector is decoupled from the bath vacuum fluctuations
and therefore, at zero temperature, is decoherence free.
The smallest subspace in which one can have non-trivial physics is the one-excitation
sector ${\cal H}_{1}.$
The dimension of ${\cal H}_{1}$ scales linearly in the total number of degrees of freedom,
therefore a thorough analysis of the dynamics in ${\cal H}_{1},$ can be performed
by means of exact numerical diagonalization of the model Hamiltonian.
The temporal dependence of quantity of interest, such as fidelity and entropy have been studied.
The asymptotic behaviour depends on the symmetry content of the initial state.
Smaller is the projection of the initial state over the vacuum $su(2)$-sector ${\cal H}^{sym},$
greater is the fidelity.
In particular a complete energy-information loss
occurs only when the state
belongs to ${\cal H}^{sym}.$
Some numerical results for bath-system coupling dependent on the qubit are also presented.
In this more realistic situation the $su(2)$-structure is unstable:
dissipation and decoherence affects the whole Hilbert space and then
safe encondings no longer exists.
Nevertheless our results
shows that, on long time scales, the average fidelity of the previously noiseless states
is still greater of the one of the other states.
This suggest that the symmetry-based protection of quantum state suggested in [4]
can be valuable in the general case.
This last issue, along with the necessary
finite temperature generalizations, worth further investigations.
Acknowledgements.
Stimulating discussions with M. Rasetti and R. Zecchina
are gratefully aknowledged.
The author also thanks
C. Calandra and G. Santoro for providing him access to the CICAIA of the Modena University,
and Elsag-Bailey for financial support
References
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A. Ekert and R. Josza, Revs. Mod. Phys. 68, 733, (1996)
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44, 236 (1981) |
No-Cloning Theorem on Quantum Logics
Takayuki Miyadera$\ {}^{*}$
and Hideki Imai$\ {}^{*,\dagger}$
$\ {}^{*}$
Research Center for Information Security (RCIS),
National Institute of Advanced Industrial
Science and Technology (AIST).
Daibiru building 1003,
Sotokanda, Chiyoda-ku, Tokyo, 101-0021, Japan.
(e-mail: miyadera-takayuki@aist.go.jp)
$\ {}^{\dagger}$
Graduate School of Science and Engineering,
Chuo University.
1-13-27 Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan .
Abstract
This paper discusses the
no-cloning theorem in a logico-algebraic approach.
In this approach, an orthoalgebra is considered as
a general structure for
propositions in a physical theory.
We proved that an orthoalgebra admits cloning operation
if and only if
it is a Boolean algebra.
That is, only classical theory
admits the cloning of states.
If unsharp propositions are to be included in the theory,
then a notion of effect algebra is considered.
We proved that an atomic Archimedean
effect algebra admitting cloning operation is
a Boolean algebra.
This paper also presents a partial result indicating a relation
between cloning on effect algebras and hidden variables.
I Introduction
In 1982, Wootters and Zurek WZ and Dieks Dieks elucidated the
no-cloning theorem: unknown quantum states cannot be cloned.
The no-go theorem that prohibits the universal cloning of quantum states
often plays a central role in quantum
information Nielsen .
For instance,
in quantum cryptography BB84 ,
applying this theorem, legitimate users could detect
an eavesdropper who pilfered the information.
Recently, Barnum, Barrett, Leifer and Wilce Barnum
reported a generalization of the theorem
in general probability theory (or the convex approach),
the framework of which is sufficiently broad to
treat both classical theory and quantum theory as
its examples.
The structure of state space is considered
a fundamental object for this framework.
For instance, a classical theory
can be characterized by its convex state space
being a simplex.
Barnum, Barrett, Leifer and Wilce
elucidated that if the state of a system can be cloned,
then the state space is a simplex.
This paper discusses an approach to address
this problem
in the context of quantum logics.
This approach was originated by Birkhoff and von Neumann vN
in 1936.
Here,
the structure of propositions is
the most fundamental object.
A theory is determined by specifying
an algebra consisting of the propositions.
A classical theory is
identified with a Boolean algebra.
Birkhoff and von Neumann studied properties
that are
satisfied by projection operators
in Hilbert space.
The proposition system of quantum theory,
in contrast to classical theory,
does not satisfy the distributive law and thus
is not identified with a Boolean algebra.
Nonetheless, it
satisfies the weaker axioms for an orthomodular lattice.
Since then, numerous studies have been
conducted to justify the common Hilbert space formalism
of quantum mechanics (see TheBook and
references therein).
A typical study in this
direction
starts with a very general algebra and then
certain reasonable conditions are imposed on it.
In this approach, orthoalgebra is considered
as a general structure;
the Boolean algebra and the orthomodular
lattice are examples for the same.
If unsharp propositions are to be included in the theory,
then a notion of effect algebra is considered FoBe1 .
The objective of this paper is to assess the
cloning process on the above-mentioned algebras and the
conditions required for them to satisfy the
no-cloning theorem.
This paper is organized as follows:
Section II presents
a brief review of orthoalgebra.
Section III provides our main result
that can be regarded as a no-cloning theorem on the orthoalgebras.
Here we prove that if an orthoalgebra admits cloning it is definitely
a Boolean algebra. Hence, only classical theory admits
cloning. This result agrees with the result obtained
by Barnum, Barrett, Leifer and Wilce in general probability theory.
Section IV presents
partial results of extension of the earlier result to effect algebras.
II Orthoalgebras
An orthoalgebra, consisting of
sharp propositions, is a generalized structure of
the Boolean algebra
and the orthomodular lattice,
which play an important role in
the investigation of quantum logic.
Its definition is stated as follows:
TheBook
Definition 1
Let us consider $(P,0,1,\oplus)$ consisting of a set
$P$ which contains two special elements $0$ and $1$ and
a partially
defined binary operation $\oplus$.
If the quadruple
satisfies the following conditions for
all $p,q,r\in P$,
then $(P,0,1,\oplus)$ is called an orthoalgebra.
(i)
If $p\oplus q$ is defined (denoted as $p\perp q$), then
$q\oplus p$ is also defined and $p\oplus q=q\oplus p$ holds.
(ii)
If $q\perp r$ and $p\perp(q\oplus r)$ hold,
then $p\perp q$ and $(p\oplus q)\perp r$, and
$p\oplus(q\oplus r)=(p\oplus q)\oplus r$ hold.
(iii)
For every $p\in P$, there exists a unique $q\in P$
such that $p\perp q$ and $p\oplus q=1$ hold.
We represent such uniquely determined $q$ as $p^{\prime}$.
(iv)
If $p\perp p$, then $p=0$.
Example 1
A simple example is the set of projection operators
in Hilbert space. Let ${\cal H}$ be a Hilbert space and
$P({\cal H})$ be the set of all the projection operators on it.
In $P({\cal H})$, a partially defined binary operation
is introduced by
$p\oplus q=p+q$ (summation of operators), for $p,q\in P({\cal H})$
with $pq=0$.
$0$ and $1$ are a null operator and an identity operator on
${\cal H}$, respectively.
Hence from the above definition,
it can be derived that $(P({\cal H}),0,1,\oplus)$ becomes an orthoalgebra.
A partial order in
$(P,0,1,\oplus)$ can be introduced.
Definition 2
If there exists an element $r\in P$ such that
$p\perp r$ and $q=p\oplus r$ hold,
we denote as $p\leq q$ (or equivalently $q\geq p$).
It can be confirmed that $\leq$ forms
a partial order and satisfies
$0\leq p\leq 1$
for every element $p\in P$.
That is, $(P,0,1,\leq)$ forms a bounded poset.
We denote
the least upper and the greatest lower bounds of
$\{p,q\}$
by $p\vee q$ and
$p\wedge q$ (unique), respectively,
if they exist.
If $P$ is an
orthoalgebra, it can be proved
that $p\wedge p^{\prime}=0$ for any element $p\in P$.
If $p\perp q$ and $p\vee q$ exists,
it coincides with
$p\oplus q$.
An orthomodular poset may be defined as an
orthoalgebra $P$ such that the coherence law is
satisfied. That is, for any mutually orthogonal
$p,q$ and $r\in P$, $(p\oplus q)\oplus r$ is defined.
Two elements $p,q\in P$ are said to be compatible
if there exist mutually orthogonal elements $x,y$ and $z$ satisfying
$p=x\oplus z$ and $q=y\oplus z$.
If an orthomodular poset $P$ satisfies
the compatibility condition,
that is, every pair $p,q\in P$ is
compatible, then $P$ becomes a Boolean algebra.
Having defined an orthoalgebra
$P$ which is a set of propositions,
we can now introduce states and dynamics on it.
Definition 3
Let $P$ be an orthoalgebra. A state on $P$ is a map $\mu:P\to{\bf R}$
such that,
for any $p,q\in P$ with $p\perp q$,
$\mu(p\oplus q)=\mu(p)+\mu(q)$ holds, and
$\mu(p)\geq 0$ for any $p$ and $\mu(1)=1$ are satisfied.
A nonnegative value $\mu(p)$ is interpreted as the
probability to obtain ‘Yes’ when a measurement
of $p$ for the state $\mu$ is made.
We assume a sufficient number of states on orthoalgebras,
although there exist orthoalgebras with none; this assumption guarantees the existence of a tensor product of
orthoalgebras that will be defined later FoBe .
The dynamics (or physical process), which is represented as
a morphism between orthoalgebras, is discussed
in the Heisenberg picture.
Definition 4
Let $P_{1}$ and $P_{2}$ be orthoalgebras. A map $\phi:P_{1}\to P_{2}$ is called a morphism if
it satisfies the following conditions:
(i)
For any $p,q\in P_{1}$ with $p\perp q$,
$\phi(p)\perp\phi(q)$ and $\phi(p\oplus q)=\phi(p)\oplus\phi(q)$ hold.
(ii)
$\phi(1)=1$ holds.
In the Schrödinger picture, a state $\mu$ on $P_{2}$ is mapped
onto a state $\mu\circ\phi$ on $P_{1}$.
We often have to consider a composite system of
two (or more) systems. An advantage
of treating orthoalgebra is that a tensor product
is naturally defined in its category.
To define the tensor product we introduce the
notion of bimorphism.
Definition 5
Let $P,Q$ and $L$ be orthoalgebras.
A map $\beta:P\times Q\to L$ is called a bimorphism
if it satisfies the following conditions:
(i)
For $a,b\in P$ with $a\perp b$ and
$q\in Q$, $\beta(a\oplus b,q)=\beta(a,q)\oplus\beta(b,q)$ holds.
(ii)
For $c,d\in Q$ with $c\perp d$ and $p\in P$,
$\beta(p,c\oplus d)=\beta(p,c)\oplus\beta(p,d)$ holds.
(iii)
$\beta(1,1)=1$ holds.
Definition 6
Let $P$ and $Q$ be orthoalgebras. $(T,\tau)$ consisting
of an orthoalgebra $T$ and a bimorphism $\tau:P\times Q\to T$
is called a tensor product of $P$ and $Q$ if the following
conditions are satisfied:
(i)
If $L$ is an orthoalgebra and $\beta:P\times Q\to L$
is a bimorphism, there exists a morphism $\alpha:T\to L$
such that $\beta=\alpha\circ\tau$.
(ii)
Every element of $T$ is a finite orthogonal sum
of elements of the form $\tau(p,q)$ with $p\in P$ and $q\in Q$.
We represent
$T$ and $\tau(p,q)$ as $P\otimes Q$ and $p\otimes q$, respectively.
If there exists at least one bimorphism $\beta:P\times Q\to L$
the above defined tensor product exists.
The existence of the bimorphism is often confirmed by the
existence of
a sufficiently rich set of states FoBe ,
which is assumed
throughout the paper.
III Cloning on orthoalgebras
This section describes a cloning process on orthoalgebras.
Cloning is an operation that produces a pair of copies for
an arbitrary given state.
Classical theory realizes this operation easily.
A simple model is illustrated as follows:
Consider a classical system having a
discrete finite sample space $\Omega_{N}=\{1,2,\ldots,N\}$.
Every state can be described by a probability
distribution $(p_{n})_{n\in\Omega_{N}}$ on it.
There exists an observable that distinguishes
all the pure states. A composite system with
a doubled sample space $\Omega_{N}\times\Omega_{N}$ is considered.
Measurement of the observable that perfectly distinguishes the
pure states followed by the preparation of a pair of the identified state
clones an arbitrary state.
That is, a state $(p_{n})$ is mapped to
a state $(p_{nm})_{(n,m)\in\Omega_{N}\times\Omega_{N}}$
on the composite system that is defined by
$p_{nm}=p_{n}\delta_{nm}$, whose
marginal state on
each system coincides with the original state $(p_{n})$.
The following steps describe the characterization of cloning
operation in the algebraic setting:
Consider an orthoalgebra $P$ that
has a separating state space.
That is, $p=q$ follows if $p,q\in P$ satisfy $\omega(p)=\omega(q)$ for
all the states $\omega$.
If a state $\omega$ on $P$ is cloned,
the cloned state $\omega^{\prime}$ on $P\otimes P$ must satisfy
$\omega^{\prime}(q\otimes 1)=\omega^{\prime}(1\otimes q)=\omega(q)$.
We described here that states are mapped to
other states as time evolves; in the dual picture, however,
observables are mapped backward with respect to time.
That is, a map $\phi:P\otimes P\to P$ describes
the time evolution. It should satisfy
$$\displaystyle\omega\circ\phi(q\otimes 1)=\omega\circ\phi(1\otimes q)=\omega(q)$$
for any $q\in P$. If
$P$ has a separating state space, this condition implies that for every $q\in P$,
$$\displaystyle\phi(q\otimes 1)=\phi(1\otimes q)=q$$
holds.
We apply this relation as a defining property of
a cloning map.
Definition 7
Let $P$ be an orthoalgebra.
A morphism $\phi:P\otimes P\to P$ is called a cloning map
if the following conditions are satisfied:
(i)
For any $p\in P$, $\phi(p\otimes 1)=p$ holds.
(ii)
For any $p\in P$, $\phi(1\otimes p)=p$ holds.
If there exists a cloning map $\phi:P\otimes P\to P$,
then $P$ is said to satisfy a cloning property.
The following is our main theorem:
Theorem 1
Let $P$ be an orthoalgebra.
$P$ has a cloning property if and only if
$P$ is a Boolean algebra.
Because a Boolean algebra can be considered as
a set of sharp propositions in a classical system
according to Stone’s representation theorem,
this theorem essentially claims that
if cloning operation can be performed in
a system, the system is classical.
To prove this theorem, we consider
a lemma with respect to the cloning map.
Lemma 1
Let $P$ be an orthoalgebra with a cloning property
and $\phi:P\otimes P\to P$ be a
cloning map. The following statements are satisfied:
(i)
For $p,q\in P$, $\phi(p\otimes q)=0$ if and only
if $p\perp q$.
(ii)
For any $p\in P$, $\phi(p\otimes p)=p$ holds.
Proof:
Let us begin with the ‘if’ part of (i).
Assume that $p,q\in P$ satisfy $p\perp q$.
Since $1=q\oplus q^{\prime}$ holds, we obtain
$p\otimes 1=p\otimes(q\oplus q^{\prime})=p\otimes q\oplus p\otimes q^{\prime}$.
Since $\phi$ is a morphism, we obtain
$$\displaystyle p$$
$$\displaystyle=$$
$$\displaystyle\phi(p\otimes 1)$$
$$\displaystyle=$$
$$\displaystyle\phi(p\otimes q\oplus p\otimes q^{\prime})$$
$$\displaystyle=$$
$$\displaystyle\phi(p\otimes q)\oplus\phi(p\otimes q^{\prime}).$$
Similarly, with $1=p\oplus p^{\prime}$, we obtain
$$\displaystyle q$$
$$\displaystyle=$$
$$\displaystyle\phi(1\otimes q)$$
$$\displaystyle=$$
$$\displaystyle\phi(p\otimes q)\oplus\phi(p^{\prime}\otimes q).$$
That is, we have $p,q\geq\phi(p\otimes q)$.
In an orthoalgebra, $p\perp q$ implies
$p\wedge q=0$.
Thus we obtain $\phi(p\otimes q)=0$.
Conversely, we assume $\phi(p\otimes q)=0$
for some $p,q\in P$.
We then obtain
$p=\phi(p\otimes q^{\prime})$ and $q=\phi(p^{\prime}\otimes q)$.
Since $(p\otimes q^{\prime}\oplus p\otimes q)\oplus(p^{\prime}\otimes q\oplus p^{%
\prime}\otimes q^{\prime})$ is defined
(equals $1$),
$p\otimes q^{\prime}\perp p^{\prime}\otimes q$ follows.
Therefore, due to the morphism quality of $\phi$,
we obtain $p\perp q$.
Proof of (ii).
Since $p\perp p^{\prime}$ and $1=p\oplus p^{\prime}$ hold,
$p=\phi(1\otimes p)=\phi(p\otimes p)\oplus\phi(p^{\prime}\otimes p)$
follows. Applying (i), $\phi(p^{\prime}\otimes p)=0$ and
$\phi(p\otimes p)=p$ hold.
$\blacksquare$
Applying this lemma, the following two lemmas are proved
as given below:
Lemma 2
Let $P$ be an orthoalgebra with a cloning property.
Then $P$ satisfies coherence law. That is,
if $x,y,z\in P$ are mutually orthogonal with each other,
$(x\oplus y)\oplus z$ is defined.
Proof:
As $x\oplus y$ is defined and $(x\oplus y)\oplus(x\oplus y)^{\prime}=1$ holds,
we have
$$\displaystyle z=\phi(1\otimes z)=\phi((x\oplus y)\otimes z)\oplus\phi((x\oplus
y%
)^{\prime}\otimes z).$$
Decomposing the first term in the right-hand side,
we obtain
$$\displaystyle z=\phi(x\otimes z)\oplus\phi(y\otimes z)\oplus\phi((x\oplus y)^{%
\prime}\otimes z)=\phi((x\oplus y)^{\prime}\otimes z),$$
where Lemma 1 (i) is used.
On the other hand, $x\oplus y=\phi((x\oplus y)\otimes 1)$
holds and $((x\oplus y)\otimes 1)\oplus((x\oplus y)^{\prime}\otimes z)$
is defined. Thus, due to the
morphism quality of $\phi$, $(x\oplus y)\oplus z$ is defined.
$\blacksquare$
Lemma 3
Let $P$ be an orthoalgebra with a cloning property.
Then any two elements are compatible.
That is,
for any $p,q\in P$, there exist
mutually orthogonal elements $a,b,r\in P$
such that $p=r\oplus a$ and $q=r\oplus b$ hold.
In addition, this decomposition
is
unique.
Proof:
Substituting $r:=\phi(p\otimes q)$, we obtain
$$\displaystyle p$$
$$\displaystyle=$$
$$\displaystyle\phi(p\otimes 1)=r\oplus\phi(p\otimes q^{\prime}),$$
$$\displaystyle q$$
$$\displaystyle=$$
$$\displaystyle\phi(1\otimes q)=r\oplus\phi(p^{\prime}\otimes q).$$
Substituting $a=\phi(p\otimes q^{\prime})$ and $b=\phi(p^{\prime}\otimes q)$,
we obtain $p=r\oplus a$ and $q=r\oplus b$.
Since $p\otimes q^{\prime}\perp p^{\prime}\otimes q$ holds,
$a\perp b$ follows.
If
$p=r\oplus a$ and $q=r\oplus b$
are decompositions
with mutually orthogonal $r,a,b$,
we obtain
$$\displaystyle\phi(p\otimes q)=\phi((r\oplus a)\otimes(r\oplus b))=\phi(r%
\otimes r)=r,$$
where Lemma 1 is used.
Thus the decomposition is unique.
$\blacksquare$
Proof of Theorem 1
The above two lemmas prove that
every orthoalgebra with the cloning property satisfies
the coherence law and the compatibility condition and is a
Boolean algebra.
Conversely, let $P$ be a Boolean algebra.
We define $\phi:P\otimes P\to P$ by
$\phi(p\otimes q)=p\wedge q$ and its natural extension
to their orthogonal sums.
Its well-definedness can be proved as follows.
Let $\{p_{m},q_{m},r_{n},s_{n}\}\subset P$ be a finite family
satisfying
$\bigoplus_{m}p_{m}\otimes q_{m}=\bigoplus r_{n}\otimes s_{n}$.
Since $P$ is Boolean, there exist finite mutually orthogonal
nonvanishing elements $\{x_{i}\}_{i=1}^{N}\subset P$ and
subsets $S_{p_{m}},S_{q_{m}},S_{r_{n}},S_{s_{n}}\subset\{1,2,\ldots,N\}$, such that the following are satisfied:
$$\displaystyle p_{m}$$
$$\displaystyle=$$
$$\displaystyle\bigoplus_{i\in S_{p_{m}}}x_{i}$$
$$\displaystyle q_{m}$$
$$\displaystyle=$$
$$\displaystyle\bigoplus_{i\in S_{q_{m}}}x_{i}$$
$$\displaystyle r_{n}$$
$$\displaystyle=$$
$$\displaystyle\bigoplus_{i\in S_{r_{n}}}x_{i}$$
$$\displaystyle s_{n}$$
$$\displaystyle=$$
$$\displaystyle\bigoplus_{i\in S_{s_{n}}}x_{i}.$$
The condition $\bigoplus_{m}p_{m}\otimes q_{m}=\bigoplus r_{n}\otimes s_{n}$
indicates the followings:
(i)
$(S_{p_{m}}\times S_{q_{m}})\cap(S_{p_{m^{\prime}}}\times S_{q_{m^{\prime}}})=\emptyset$ for $m\neq m^{\prime}$ and
$(S_{r_{n}}\times S_{s_{n}})\cap(S_{r_{n^{\prime}}}\times S_{s_{n^{\prime}}})=\emptyset$ for $n\neq n^{\prime}$ hold.
(ii)
$\bigcup_{m}(S_{p_{m}}\times S_{q_{m}})=\bigcup_{n}(S_{r_{n}}\times S_{s_{n}})$
holds. We denote $F:=\bigcup_{m}(S_{p_{m}}\times S_{q_{m}})$.
Applying
$p_{m}\wedge q_{m}=\oplus_{(i,i)\in S_{p_{m}}\times S_{q_{m}}}x_{i}$,
$\bigoplus_{m}p_{m}\wedge q_{m}=\oplus_{(i,i)\in F}x_{i}$ holds, which equals
$\bigoplus_{n}r_{n}\wedge s_{n}$.
Thus $\phi$ is well-defined.
It can be noted that this $\phi$ satisfies the conditions of
the cloning map.
$\blacksquare$
If an orthoalgebra $P$ admits an orthoalgebra $L$ and
a bimorphism $\beta:P\times P\to L$ satisfying the following conditions,
then $P$ is a Boolean algebra.
(i)
There exists a morphism $\phi:L\to P$.
(ii)
For every $p\in P$, $\phi\circ\beta(p,1)=\phi\circ\beta(1,p)=p$
holds.
In fact, according to the definition of $P\otimes P$,
there exists a morphism $\alpha:P\otimes P\to L$
satisfying $\beta(p,1)=\alpha(p\otimes 1)=p$ and
$\beta(1,p)=\alpha(1\otimes p)=p$ for every $p$, and
Theorem 1 can be applied.
It may be worth noting that
some important examples such as $P=P({\cal H})$ and
$L=P({\cal H}\otimes{\cal H})$ for a Hilbert space ${\cal H}$
can be treated in this manner.
IV Cloning on effect algebras
In the previous section, we
proved that if an orthoalgebra
has a cloning property, then it is a Boolean
algebra.
This section considers a possible extension of
this result to
effect algebras.
Let us begin with the definition of an effect algebra.
Definition 8
Let us consider $(L,0,1,\oplus)$ consisting of a set
$L$ which contains two special elements $0$ and $1$ and
a partially
defined binary operation $\oplus$.
If the quadruple
satisfies the following conditions for
all $p,q,r\in L$,
then $(L,0,1,\oplus)$ is called an effect algebra:
(i)
If $p\oplus q$ is defined (denoted by $p\perp q$), then
$q\oplus p$ is also defined and $p\oplus q=q\oplus p$.
(ii)
If $q\perp r$ and $p\perp(q\oplus r)$ hold,
then $p\perp q$ and $(p\oplus q)\perp r$, and
$p\oplus(q\oplus r)=(p\oplus q)\oplus r$ hold.
(iii)
For every $p\in L$, there exists a unique $q\in L$
such that $p\perp q$ and $p\oplus q=1$ hold.
We represent such unique $q$ as $p^{\prime}$.
(iv’)
If $p\perp 1$, then $p=0$.
It may be noted that the condition (iv) in Definition
1 is stronger than
(iv’). That is, every orthoalgebra is an effect algebra;
the converse is not true.
A partial order $\leq$ is introduced as in the orthoalgebra.
While every element of an orthoalgebra is sharp, that is,
$p\wedge p^{\prime}=0$ holds,
elements of an effect algebra may not be sharp.
If all the elements of an effect algebra
are sharp, then this algebra turns out to be an orthoalgebra.
Example 2
Let us consider a quantum system described by a Hilbert space
${\cal H}$. $E({\cal H})$ is defined as a set
of all the positive operators $x$ on ${\cal H}$ that satisfy
$x\leq{\bf 1}$, where ${\bf 1}$ is an identity operator.
$x\oplus y=x+y$ (summation as operators)
is defined if $x+y\leq{\bf 1}$ holds.
$0$ and $1$ are the null operator and identity operator, respectively.
The quadruple $(E({\cal H}),0,1,\oplus)$
becomes an effect algebra.
The notions of state,
dynamics and tensor product
are defined by simply replacing orthoalgebra
by effect algebra in the previous definitions.
Cloning condition is defined as in the orthoalgebra.
Definition 9
Let $L$ be an effect algebra.
$\phi:L\otimes L\to L$ is called a cloning map if
and only if it satisfies $\phi(p\otimes 1)=\phi(1\otimes p)=p$
for any $p\in L$.
An effect algebra for which there exists a cloning map is
said to satisfy a cloning property.
The property stated in Lemma 1,
in contrast to that in orthoalgebras wherein it played
a crucial role,
does not hold in effect algebras.
Certain partial results on effect algebras are
elucidated as follows.
The first result is related to atomic effect
algebras.
Definition 10
Let $L$ be an effect algebra.
A non-zero element $a\in L$ is called an atom if and only if
$[0,a]:=\{x|0\leq x\leq a\}=\{0,a\}$ holds.
Definition 11
Let $L$ be an effect algebra.
If for every non-zero $p\in L$ there exists an atom $a\in L$ satisfying
$a\leq p$, then $L$ is called an atomic algebra.
That is, an atomic algebra has ‘smallest’ units in it.
On the other hand, the ‘boundedness’ of the algebra
is imposed by another condition.
Definition 12
Let $L$ be an effect algebra.
For every element $x\in L$, we define its isotropic index $\iota(x)$
as the maximal nonnegative integer $n$ such that $na:=a\oplus a\oplus\cdots\oplus a$ ($n$ times) is defined.
If $\iota(x)$ is finite for every $x\neq 0$, then
$L$ is called an Archimedean algebra.
The following is the first result on effect algebras.
Theorem 2
If an atomic Archimedean effect algebra $L$ has a cloning
property, $L$ is a Boolean algebra.
Proof:
First let us prove that every atomic element is a sharp element.
Suppose $a$ is an atom.
As $[0,a]=\{0,a\}$, if $a\leq a^{\prime}$ holds then $a\wedge a^{\prime}=a$,
otherwise $a\wedge a^{\prime}=0$.
Assume that $a\leq a^{\prime}$ holds ($a\perp a$ follows),
then $\phi(a\otimes a^{\prime})\geq\phi(a\otimes a)$ holds.
Since $a=\phi(a\otimes 1)\geq\phi(a\otimes a)$ holds,
$\phi(a\otimes a)$ is $a$ or $0$.
If $\phi(a\otimes a)=a$ holds, then it implies
$\phi(a\otimes a^{\prime})\geq a$.
Then the cloning property leads to
$$\displaystyle a=\phi(a\otimes 1)=\phi(a\otimes a^{\prime})\oplus\phi(a\otimes a%
)\geq 2a.$$
Therefore implying $a=0$ which leads to a contradiction.
Thus $\phi(a\otimes a)=0$ should hold
and $\phi(a\otimes a^{\prime})=a$ follows.
Since we assumed $a\perp a$,
$2a=a\oplus a$
is defined and $\phi(2a\otimes a^{\prime})=2\phi(a\otimes a^{\prime})=2a$ holds.
As $a^{\prime}\geq\phi(2a\otimes a^{\prime})$ holds, it means
that $a^{\prime}\geq 2a$ should hold;
thus $3a\in L$ is defined.
Repeating the same arguments,
we obtain $Na\in L$ for arbitrarily large $N$.
As $L$ is Archimedean, $a=0$ follows. This
contradicts the nonvanishing characteristics of $a$.
Thus it can be concluded that $a\wedge a^{\prime}=0$ for every
atom $a$.
Consider an arbitrary element $p\in L$.
Assume a non-zero element $x\in L$
satisfying $x\leq p,p^{\prime}$.
As $L$ is atomic, there exists an atom $a\leq x$;
thus $a\leq p,p^{\prime}$ holds,
implying $a^{\prime}\geq p,p^{\prime}$
and
$a\leq p\leq a^{\prime}$.
It indicates $a\wedge a^{\prime}=a$.
We, however, proved that it does not hold
for an atom $a$; therefore, this leads to a contradiction.
Hence it can be concluded
that $x=0$
and $p\wedge p^{\prime}=0$.
Thus all the elements in $L$ are sharp,
which means that $L$ is an orthoalgebra.
According to Theorem 1,
an orthoalgebra with cloning property is a
Boolean algebra.
$\blacksquare$
In non-atomic effect algebras,
the above theorem does not hold.
The following is a simple example.
$\Omega_{N}:=\{1,2,\ldots,N\}$
is a finite discrete set consisting of $N$ points.
Let $L_{N}:=[0,1]^{\Omega_{N}}=\{f|\Omega_{N}\to[0,1]\}$ be the set of
all functions from $\Omega_{N}$ to $[0,1]$.
On $L_{N}$, a partial binary operation $\oplus$
can be defined by $(f\oplus g)(x)=f(x)+g(x)$ for all $x\in\Omega_{N}$ if
$f(y)+g(y)\in[0,1]$ for all $y\in\Omega_{N}$ holds.
Both $0$ and $1$ are defined in a natural manner.
It can be noted that $L_{N}$ becomes
an effect algebra although it is not an orthoalgebra.
In addition, it can be proved that
$L_{N}\otimes L_{N}$ is isomorphic to
$L_{N^{2}}:=\{f|\Omega_{N}\times\Omega_{N}\to[0,1]\}$.
A cloning map can be defined by
$$\displaystyle\phi(f)(x):=f(x,x)$$
for $f\in L_{N^{2}}$.
It may be worth noting that the sharp elements of this
effect algebra forms a Boolean algebra.
Based on the above example, it may be expected that
the cloning property on effect algebras is related to
the classicality.
We now explain the definition of hidden variable
introduced by Pulmannová Pulmannova .
It uses the following MV-algebra Chang :
Definition 13
Let $M$ be a set with two special elements $0$ and $1$.
If on $M$ a binary operation $+$ and
a unary operation ${}^{\prime}$ are defined and satisfy the following conditions
for all $a,b,c\in M$, then
$(M,+,\ ^{\prime},0,1)$ is called an MV-algebra.
(i)
$a+b=b+a$
(ii)
$(a+b)+c=a+(b+c)$
(iii)
$a+a^{\prime}=1$
(iv)
$a+0=a$
(v)
$(a^{\prime})^{\prime}=a$
(vi)
$0^{\prime}=1$
(vii)
$a+1=1$
(viii)
$(a^{\prime}+b)^{\prime}+b=(a+b^{\prime})^{\prime}+a$.
If we define a partial binary operation $\oplus$ by
$a\oplus b:=a+b$ only for $a,b\in M$ with $a\leq b^{\prime}$,
then $(M,0,1,\oplus)$ becomes an effect algebra.
The states on an MV-algebra are defined by the
states on its corresponding
effect algebra.
Definition 14
(Pulmannova )
Let $L$ be an effect algebra. We consider that $L$ admits hidden
variables if and only if there exists an MV-algebra, $M$ satisfying
the following conditions.
(i)
There exists a morphism
from $L$ to $M$, that is, a map
$h:L\to M$ such that
for all $p,q\in L$ with $p\perp q$, $h(p\oplus q)=h(p)+h(q)$ holds.
(ii)
For every state $\omega$ on $L$, there exists a
state $\overline{\omega}$ on $M$ such that
$\overline{\omega}\circ h(q)=\omega(q)$ for every $q\in L$ holds.
The following theorem is a partial result on
non-atomic effect algebras.
Theorem 3
Let $L$ be an effect algebra with a cloning property.
Suppose there exists a family $\{p_{1},p_{2},\ldots,p_{N}\}\subset L$ such that
$[0,p_{n}]$ is a linearly ordered ideal for every $p_{n}$ and
$1=\oplus_{n=1}^{N}p_{n}$ holds.
$L$ admits a hidden variable.
Proof:
We define a binary operation $+$ on
$[0,p_{n}]$ by $x+y=x\oplus y$ for $x,y\in[0,p_{n}]$ with $x\perp y$
and $x+y=p_{n}$ otherwise.
A unary operation ${}^{\prime}$ is defined by $x^{\prime}:=p_{n}\ominus x$ for
$x\in[0,p_{n}]$. (That is, $x^{\prime}$ is a unique element that satisfies
$x\oplus x^{\prime}=p_{n}$.)
Then it can be noted that $([0,p_{n}],+,^{\prime},0,p_{n})$ becomes an
MV-algebra.
Let us consider their Cartesian product, $\Pi_{n}[0,p_{n}]$,
which becomes again an MV-algebra by defining the summation
and the unary operation
‘pointwise’ and $1:=(p_{n})_{n=1,\ldots,N}$. That is,
the summation is defined as $(x_{n})_{n=1,\ldots,N}+(y_{n})_{n=1,\ldots,N}:=(x_{n}+y_{n})_{n=1,\ldots,N}$, and the unary operation
is defined as $(x_{n})^{\prime}_{n=1,\ldots,N}:=(x_{n}^{\prime})_{n=1,\ldots,N}$. We define a map $h:L\to M$
by $h(x)=(\phi(p_{n}\otimes x))_{n=1,\ldots,N}$.
$h(x)=h(y)$ means $\phi(p_{n}\otimes x)=\phi(p_{n}\otimes y)$ for
all $n$. As $x=\phi(1\otimes x)=\oplus_{n=1}^{N}\phi(p_{n}\otimes x)$
holds, it implies $x=y$.
In addition, for any $(x_{n})_{n=1,\ldots,N}$
satisfying $x_{n}\in[0,p_{n}]$ for each $n$,
$x:=\oplus_{n=1}^{N}x_{n}$ is defined and it satisfies
$\phi(p_{n}\otimes x)=\phi(p_{n}\otimes x_{n})=x_{n}$ for each $n$
since $p_{n}$ is a
sharp element (Lemma 1.9.6 in TheBook ) and $p_{n}\wedge x_{m}=0$
holds for $n\neq m$.
Thus $h$ is a bijection.
If a pair $x,y\in L$ satisfies $x\perp y$,
$x\oplus y=\oplus_{n=1}^{N}\phi(p_{n}\otimes(x\oplus y))$
holds. Therefore $h(x\oplus y)=(\phi(p_{n}\otimes(x\oplus y))_{n=1,\ldots,N}=(\phi(p_{n}\otimes x%
)+\phi(p_{n}\otimes x))_{n=1,\ldots,N}=h(x)+h(y)$ follows.
That is, $h$ satisfies the condition (i).
Conversely, if $h(x)\leq h(y)^{\prime}$ holds,
it means $\phi(p_{n}\otimes x)\oplus\phi(p_{n}\otimes y)\leq p_{n}$ for each $n$.
It entails $\left(\oplus_{n=1}^{N}\phi(p_{n}\otimes x)\right)\oplus\left(\oplus_{n=1}^{N}%
\phi(p_{n}\otimes y)\right)\leq 1$.
That is, $x\leq y^{\prime}$ holds.
Thus we proved that $x\leq y^{\prime}$ if and only if
$h(x)\leq h(y)^{\prime}$.
Let $\omega$ be a state on $L$.
A state on $M$,
$\overline{\omega}$, is defined by
$\overline{\omega}(\{x_{n}\})=\omega(\oplus_{n=1}^{N}x_{n})$.
This $\overline{\omega}$ satisfies
the condition (ii).
$\blacksquare$
V Summary
This paper considers the no-cloning theorem on
orthoalgebras and effect algebras. We proved that
an orthoalgebra admits cloning operation if and only if it is
a Boolean algebra. That is,
cloning operation can be performed only on classical systems.
In addition, we
proved that an atomic Archimedean effect algebra
with a cloning property is a Boolean algebra.
We also
obtained a partial result
that indicates a connection between
the cloning property and hidden variables.
Although we conjecture that effect algebra with
the cloning property admits
a hidden variable,
we have not succeeded in proving it.
Acknowledgements
The authors thank an anonymous referee for fruitful comments.
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Does a radio jet drive the massive multi-phase outflow in the ultra-luminous infrared galaxy IRAS 10565+2448?
Renzhi Su${}^{1,2,3}$,
Elizabeth K. Mahony${}^{3}$,
Minfeng Gu${}^{1}$,
Elaine M. Sadler${}^{3,4,5}$,
S. J. Curran${}^{6}$,
James R. Allison${}^{5,7}$,
Hyein Yoon${}^{4,5}$,
J. N. H. S. Aditya${}^{4,5}$,
Yogesh Chandola${}^{8}$,
Yongjun Chen${}^{11,12}$
Vanessa A. Moss${}^{3}$,
Zhongzu Wu${}^{9}$,
Xi Shao${}^{1,2}$,
Xiang Liu${}^{10}$,
Marcin Glowacki${}^{13}$,
Matthew T. Whiting${}^{3}$,
Simon Weng${}^{4,5}$
${}^{1}$Key Laboratory for Research in Galaxies and Cosmology, Shanghai Astronomical Observatory, Chinese Academy of Sciences,
80 Nandan
Road, Shanghai 200030, China
${}^{2}$University of Chinese Academy of Sciences, 19A Yuquan Road, Beijing 100049, China
${}^{3}$ATNF, CSIRO Space and Astronomy, PO Box 76, Epping, NSW 1710, Australia
${}^{4}$Sydney Institute for Astronomy, School of Physics A28, University of Sydney, NSW 2006, Australia
${}^{5}$ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D)
${}^{6}$School of Chemical and Physical Sciences, Victoria University of Wellington, PO Box 600, Wellington 6140, New Zealand
${}^{7}$First Light Fusion Ltd., Unit 9/10 Oxford Industrial Park, Mead Road, Yarnton, Kidlington OX5 1QU, UK
${}^{8}$Purple Mountain Observatory, Chinese Academy of Sciences, No.10 Yuanhua Road, Qixia District, Nanjing 210023, China
${}^{9}$College of Physics, Guizhou University, 550025 Guiyang, PR China
${}^{10}$Xinjiang Astronomical Observatory, Chinese Academy of Sciences, 150 Science 1-Street, Urumqi 830011, China
${}^{11}$Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, China
${}^{12}$Key Laboratory for Radio Astronomy, Chinese Academy of Sciences, Nanjing 210008, China
${}^{13}$International Centre for Radio Astronomy Research, Curtin University, Bentley, WA 6102, Australia
E-mail: surz@shao.ac.cn
(Accepted XXX. Received YYY; in original form ZZZ)
Abstract
We present new upgraded Giant Metrewave Radio Telescope (uGMRT) H i 21-cm observations of the ultra-luminous infrared galaxy IRAS 10565+2448, previously reported to show blueshifted, broad, and shallow H i absorption indicating an outflow. Our higher spatial resolution observations have localised this blueshifted outflow, which is $\sim$ 1.36 kpc southwest of the radio centre and has a blueshifted velocity of $\sim 148\,\rm km\,s^{-1}$ and a full width at half maximum (FWHM) of $\sim 581\,\rm km\,s^{-1}$. The spatial extent and kinematic properties of the H i outflow are consistent with the previously detected cold molecular outflows in IRAS 10565+2448, suggesting that they likely have the same driving mechanism and are tracing the same outflow. By combining the multi-phase gas observations, we estimate a total outflowing mass rate of at least $140\,\rm M_{\odot}\,yr^{-1}$ and a total energy loss rate of at least $8.9\times 10^{42}\,\rm erg\,s^{-1}$, where the contribution from the ionised outflow is negligible, emphasising the importance of including both cold neutral and molecular gas when quantifying the impact of outflows. We present evidence of the presence of a radio jet and argue that this may play a role in driving the observed outflows. The modest radio luminosity $L_{\rm 1.4GHz}$ $\sim 1.3\times 10^{23}\,{\rm W\,Hz^{-1}}$ of the jet in IRAS 10565+2448 implies that the jet contribution to driving outflows should not be ignored in low radio luminosity AGN.
keywords:
galaxies: active – galaxies: ISM – radio lines: ISM – ISM: jets and outflows
††pubyear: 2022††pagerange: Does a radio jet drive the massive multi-phase outflow in the ultra-luminous infrared galaxy IRAS 10565+2448?–Does a radio jet drive the massive multi-phase outflow in the ultra-luminous infrared galaxy IRAS 10565+2448?
1 Introduction
Neutral atomic hydrogen, H i, plays a significant role in galaxy formation and evolution (e.g. Hopkins &
Beacom, 2006; Driver
et al., 2018; Morganti et al., 2013). With the hyperfine transition structure, H i can be traced either in emission or in absorption against a background radio continuum source. H i emission has been detected in thousands of nearby galaxies (e.g. Lang
et al., 2003; Meyer
et al., 2004; Winkel et al., 2016; Catinella
et al., 2018), while H i absorption has only been detected towards about 200 bright radio sources, see Morganti &
Oosterloo (2018). One advantage of observing H i in absorption is that the strength of the H i absorption line has no dependence on redshift and hence can trace the H i in the distant Universe where H i emission is too weak to be detected with existing telescopes. Furthermore, H i absorption experiments can be conducted with Very Long Baseline Interferometry (VLBI), making it a powerful tool in studying AGN accretion and feedback (e.g. Maccagni et al., 2014; Morganti et al., 2013; Schulz et al., 2021).
Outflows are a key element in galaxy evolution, especially the AGN-driven outflows which can drive the co-evolution between supermassive black holes (SMBH) and their host galaxies (e.g. Ferrarese &
Merritt, 2000; Di
Matteo et al., 2005; Bower et al., 2006; Croton
et al., 2006; Kormendy &
Ho, 2013). The interstellar medium (ISM) in galaxies is multi-phase, including ionised, neutral, and molecular gas. Previous studies of multi-phase outflows have greatly advanced our understanding of the impact of outflows (e.g. Veilleux et al., 2005; Fabian, 2012; Cicone
et al., 2014; Harrison et al., 2014; Mahony et al., 2013; Mahony
et al., 2016; Heckman &
Thompson, 2017; Veilleux et al., 2020; Speranza
et al., 2021).
H i absorption can reveal the neutral counterpart of outflows. A famous example is 4C 12.50 in which a broad and blueshifted H i outflow was found (Morganti et al., 2005). Later VLBI observations spatially resolved the H i absorption and showed that the outflow is driven by a jet (Morganti et al., 2013). To date, due to the high sensitivity required, only a handful of H i outflows have been found in bright radio sources(e.g. Morganti et al., 2005; Mahony et al., 2013; Aditya &
Kanekar, 2018; Aditya, 2019), see also a compilation by Morganti &
Oosterloo (2018).
IRAS 10565+2448 is a $z$=0.0431 ultra-luminous infrared galaxy (ULIRG) with an infrared luminosity of $1.2\times 10^{12}\rm L_{\odot}$ and a star formation rate of $131.8\,\rm M_{\odot}$ yr${}^{-1}$ (Rupke &
Veilleux, 2013; U et al., 2012). It is a merging system involving three galaxies with highly disturbed stellar morphology; see the $gri$ bands composite image from Sloan Digital Sky Survey (SDSS) in Fig. 1 on which the three galaxies have been marked as A, B, and C.
An Arecibo observation with a velocity resolution of 8.3 km s${}^{-1}$ and a spatial resolution of 3.3 arcmin revealed both H i emission and absorption towards the source (Mirabel &
Sanders, 1988). An intriguing H i spectral property of IRAS 10565+2448 is a broad, shallow, and blueshifted H i absorption wing indicating an outflow. However, due to the large Arecibo beam the H i absorption profile is diluted by H i emission, which precludes obtaining a clean H i absorption profile needed to estimate the impact of the outflow. In order to disentangle the H i emission and absorption in this system, we carried out new observations with the upgraded Giant Metrewave Radio Telescope (uGMRT). The much higher spatial resolution of this interferometer should help us obtain a clean H i absorption line and possibly localise the H i outflow. The presence of outflows in IRAS 10565+2448 is also supported by studies of other bands (e.g. Rupke
et al., 2005b; Rupke &
Veilleux, 2013; Cicone
et al., 2014; Fluetsch
et al., 2019). Throughout this paper, we use the $\Lambda$CDM cosmology with $H_{0}=70\,$km s${}^{-1}\,$Mpc${}^{-1}$, $\Omega_{m}$=0.3, and $\Omega_{\Lambda}$=0.7.
2 Observations and data reduction
The uGMRT observations were carried out on 31 August 2021 with a bandwidth of 50 MHz, divided into 4096 channels and centred on 1360 MHz. This setting gives a velocity coverage of -6000 $\sim$ 5000 km s${}^{-1}$ with respect to the systemic velocity (Fluetsch
et al., 2019) of IRAS 10565+2448 and a velocity resolution of $\sim$ 2.7 km s${}^{-1}$. Each scan of IRAS 10565+2448 had a length of about 45 minutes and was bracketed by 5-minute scans of the phase calibrator 1111+199. The total on-source time was 6.75 hours. 3C 147 and 3C 286 were observed at the beginning, middle, and end of the entire experiment for bandpass and amplitude calibration.
The uGMRT data was calibrated using the Astronomical Image Processing System (AIPS; Greisen, 2003). We first inspected the data set and removed any bad data arising from non-working antennas, bad baselines, time-based issues, or radio frequency interference (RFI). We then calibrated the gains and bandpass and applied the solutions. After repeating the flagging and calibrating processes for a few rounds, we masked the channels occupied by the H i line and binned the rest for continuum imaging. We imaged the radio continuum using Difmap (Shepherd, 1997). We cleaned the dirty image gradually. After a few rounds of phase-only and amplitude-only self calibrations, we did both amplitude and phase self calibrations with decreasing time intervals to 0.5 minutes for the solutions. We then applied the solutions obtained from self calibration to the spectral line data, which was later binned to a velocity resolution $\sim$ 21.5 km $\rm s^{-1}$ (every eight channels are binned). Finally, we subtracted the continuum with a linear fit applied to line-free channels and generated the spectral cube. Two different spectral-line cubes were made, with uniform and natural weighting. They have an rms noise of 0.70 mJy beam${}^{-1}$ channel${}^{-1}$ and 0.41 mJy beam${}^{-1}$ channel${}^{-1}$, and a resolution of 2.23 arcsec $\times$ 1.80 arcsec and 2.84 arcsec $\times$ 2.35, respectively.
3 Results
3.1 Continuum image
Using uniform weighting, the continuum image of IRAS 10565+2448 we obtained has a resolution of 2.25 arcsec $\times$ 1.68 arcsec and an rms noise $\sim$ 27 $\mu$Jy beam${}^{-1}$. Galaxy B is not detected at this noise level in our observation, and the radio image of our target (galaxy A) is presented in Fig. 2. The radio image is resolved with a peak flux density of 39.3 mJy beam${}^{-1}$ and a total flux density of 49.6 mJy.
3.2 H i lines
3.2.1 H i emission
Previous Arecibo observations detected both H i emission and absorption towards IRAS 10565+2448 (Mirabel &
Sanders, 1988). However, due to the low spatial resolution provided by a single dish, the H i emission is mixed with H i absorption and galaxies A, B and C were not able to be spatially separated. In our uGMRT observations, those galaxies were well spatially separated. Unlike the Arecibo observation, we did not detect the H i emission in galaxy A, B or C. This is not surprising, as our cubes have a noise level similar to the faint H i emission detected in the Arecibo observation and the uGMRT is not sensitive to the diffuse H i emission due to the missing short spacings. In an integrated intensity map spanning velocities from -500 km s${}^{-1}$ to -250 km s${}^{-1}$ in the uniform-weighting cube, and a similar map spanning velocities from 250 km s${}^{-1}$ to 500 km s${}^{-1}$ in the natural-weighting cube, we found some pixels showing positive emission at the $\sim$2$\sigma$ value, see the middle and right plots in Fig. 5. However, by visually inspecting the mean wide-velocity spectra spanning $\sim$ -800 – 1000 km s${}^{-1}$, extracted from the square 14 and 17 surrounding those pixels, we concluded that these positive values arose from noise or from H i emission that was too faint to be verified in our observation. The mean spectrum from square 17 is presented in Fig. 6. We suggest that more sensitive observations may help to identify the H i emission shown on Arecibo spectrum (Mirabel &
Sanders, 1988).
3.2.2 H i absorption
We have successfully recovered the H i absorption seen in the Arecibo observation, which has two prominent features. One is narrow and deep, while the another is broad and shallow (Mirabel &
Sanders, 1988). To give an overview of the spectra against background radio emission, we have distributed nine red squares to cover the radio emission, see Fig. 2, over which we then extracted the mean spectra from uniform-weighting cube which are presented in Fig. 3.
We only found H i absorption from the central square 5 and 6, see Fig. 3. What is intriguing is that we spatially resolved the two H i absorption features, as the narrow and deep feature comes from square 5 and the broad and shallow feature comes from square 6, although our resolution can not widely separate them.
To better illustrate the positions of the two H i gas, we made integrated intensity maps. For the narrow and deep feature, we used a velocity range from -250 to 250 km s${}^{-1}$ as can be seen in the mean spectrum from square 5. We found that the H i gas comes from the central part, see the left plot in Fig. 5. For the broad and shallow feature which extends from $\sim$ -500 km s${}^{-1}$ to $\sim$ 375 km s${}^{-1}$, we used velocity spanning from -500 to -250 km s${}^{-1}$. The integral intensity map is shown in the middle plot in Fig. 5, with the position of the blueshifted, broad, and shallow component marked with the red square 15.
Similarly, we distributed four squares to extract spectra from the natural-weighting cube that has lower spectral noise, see the positions of the four green squares in Fig. 2. These spectra are shown in Fig. 7. We found a possible redshifted, broad, and shallow H i absorption line with velocity up to $\sim 500\,\rm{km\,s^{-1}}$ in the spectrum from square 10. We then made an integral intensity map using velocities from 250 to 500 km s${}^{-1}$ and the natural-weighting cube, which is presented in the right plot in Fig. 5. The location of the possible redshifted and broad H i absorption is marked with the red square 16. Again, we found a few pixels showing positive intensity marked with the red square 17, see the section 3.2.1.
To give a further view of both the blueshifted and possible redshifted, broad, and shallow H i absorption spectra, we also extracted the mean spectra over squares 15 and 16 from both uniform-weighting and natural-weighting cubes. We note that the positions of the outflows are somewhat uncertain due to the low signal-to-noise ratio. The extracted spectra are shown in Fig. 8. The possible redshifted, broad, and shallow H i absorption feature seen in the spectrum extracted over square 16 from the natural-weighting cube is more evident than that in the spectrum extracted over square 10. Further we smoothed the spectrum from the square 10 and presented the smoothed spectrum in Fig. 4. The redshifted absorption has a peak signal-to-noise ratio of $\sim$ 3 in the smoothed spectrum. To further test the significance of the redshifted absorption in the smoothed spectrum, we calculated the reduced $\chi^{2}$ and found two-component fitting gives a reduced $\chi^{2}$ of 1.16 while one-component fitting, however, gives a 10% smaller value. This slight difference in reduced $\chi^{2}$ indicates that the redshifted absorption is not significant if it is real. We thus state that the detection of a redshifted, broad, and shallow H i absorption is tentative and leave it for future more sensitive observations.
To parametrize the H i absorption lines we detected, we used the Bayesian method developed by Allison
et al. (2012) which adopts a multi-modal nested sampling, a Monte Carlo sampling algorithm, to find and fit any potential absorption line. The Bayes factor $B$ (Kass &
Raftery, 1995) is given to quantify the significance of the absorption lines found. $\ln(B)$ > 1 means a Gaussian spectral line model is preferred over a model with no line, see Allison
et al. (2012) for the details of this method. By applying this method, we obtained the parameters for the spectra from square 5 and 6 which are listed in Tab. 1. We found both spectra are best fitted with a single Gaussian profile via a Bayesian approach as described in Allison
et al. (2015). Although the signal-to-noise ratio of the spectrum from square 6 is low, the whole line has a significance of 6.9$\sigma$. We can also see the smoothed spectrum from square 6 in Fig. 4 or the spectrum of ‘15U’ in Fig. 8, both showing more significant absorption and confirming the detection.
4 Discussion
4.1 The gas traced by the deep and narrow H i absorption
The deep and narrow H i absorption, spanning from $\sim-250\,\rm{km\,s^{-1}}$ to $\sim 250\,\rm{km\,s^{-1}}$, was found only towards the square 5 that has an angular size of 1.32 arcsec $\times$ 1.32 arcsec corresponding to a linear size of 1.12 kpc $\times$ 1.12 kpc. Therefore the gas traced seems to come from a compact region. If we are tracing the neutral gas from the large galactic disc, we should also see the absorption from the surrounding squares, which is not the case here. For example, the peak optical depth of the absorption is 0.14, see Tab. 1, which would let us see the absorption at 2$\sigma$ level from square 6 that has a mean flux density of 9.84 mJy beam${}^{-1}$. However, despite the broad absorption from square 6, there is no other absorption. Also, the FWHM of the deep and narrow absorption is about 200 $\rm km\,s^{-1}$ which is generally much larger than those tracing large galactic disks (e.g. Gallimore et al., 1999; Serra
et al., 2012; Curran et al., 2016; Maccagni et al., 2017). Therefore, the best interpretation is that we are tracing the gas from the circumnuclear disc or torus. The calculated H i column density is $5.13_{-0.36}^{+0.36}\times 10^{19}T_{\rm s}$ where $T_{\rm s}$ is the spin temperature, and we assume that the background radio emission from square 5 is fully covered by the H i gas, i.e. the covering factor is unity.
4.2 The gas traced by the shallow H i absorption
Broad and shallow H i absorption lines are often interpreted as jet-driven outflows (e.g. Morganti et al., 2005; Mahony et al., 2013). In our observations, we successfully recovered the blueshifted, broad and shallow H i absorption, confirming the outflow. Besides, we further detected a possible redshifted, broad and shallow H i absorption line.
In previous studies, redshifted H i absorption is often interpreted as inflow (e.g. Morganti
et al., 2009; Araya et al., 2010; Struve &
Conway, 2012; Maccagni et al., 2014). However, the FWHM of redshifted gas is generally narrow, around 10–20 $\rm km\,s^{-1}$, and much smaller than the FWHM, $310_{-49}^{+49}$ $\rm km\,s^{-1}$, measured from the smoothed redshifted H i line in Fig. 4.
Thus it appears the possible redshifted, broad, and shallow line traces a redshifted outflow, as in principle we could detect the redshifted outflow through absorption provided the H i gas is in front of background radio continuum.
4.3 Comparison with outflows detected in IRAS 10565+2448 in other bands
Studies of the outflows in IRAS 10565+2448 have been carried out at multiple wavelengths. Optical observations of $\rm H\alpha$ emission with the Integral Field Unit (IFU) on the Gemini Multi-Object Spectrograph Rupke &
Veilleux (2013) revealed widespread ionised outflows in IRAS 10565+2448. However, the $\sim 1$ kpc scale ionised outflow extending from the nucleus to southeast is different from those in other regions, reflected by the higher FWHM, see Figure 13 in Rupke &
Veilleux (2013). This indicates that there may be various kinds of outflows driven by different mechanisms. Besides, there is a larger, $\sim$ 2–3 kpc, and more diffuse ionised outflow extending to the southwest, see Figure 17 in Rupke &
Veilleux (2013). The warm ionised outflows in IRAS 10565+2448 have a mean central velocity of -133 km s${}^{-1}$ and mean maximum velocity of -535 km s${}^{-1}$ (Rupke &
Veilleux, 2013). The mass outflow rate of ionised gas was estimated as 1.4 $\rm M_{\odot}\,yr^{-1}$, corresponding to an energy loss rate of $8.5\times 10^{40}\,\rm erg\,s^{-1}$ (Rupke &
Veilleux, 2013).
Na i D absorption, which is often used as a tracer of neutral hydrogen, was also detected in IRAS 10565+2448 with a column density of $3.0\times 10^{21}$ atoms cm${}^{-2}$ in long-slit spectroscopic observations (Rupke
et al., 2005a, b), which is about half of that detected in the our blueshifted H i outflow if adopting $T_{\rm s}=100$ K. Follow up IFU observations spatially resolved the Na i D absorption and found widespread blueshifted Na i D absorption indicating outflow in IRAS 10565+2448 (Rupke &
Veilleux, 2013). Like the ionised outflow, the Na i D outflow has a similar spatial distribution with a diffuse component extending to the southwest, see Figure 17 in Rupke &
Veilleux (2013). Na i D outflows in IRAS 10565+2448 have a mean central velocity of -218 km s${}^{-1}$ and mean maximum velocity of -468 km s${}^{-1}$ (Rupke &
Veilleux, 2013). The mass outflow rate of neutral gas estimated from Na i D absorption is 64.6 $\rm M_{\odot}\,yr^{-1}$, corresponding to an energy loss rate of $2.6\times 10^{42}\,\rm erg\,s^{-1}$ (Rupke &
Veilleux, 2013).
More importantly, IRAM PdBI observations detected both blueshifted and redshifted cold molecular CO outflows, revealed as broad emission wings spanning velocities -300 km s${}^{-1}$ $\sim$ -600 km s${}^{-1}$ and 300 km s${}^{-1}$ $\sim$ 600 km s${}^{-1}$ (Cicone
et al., 2014). The image of the CO outflows shows that the blueshifted outflow extends from the nucleus to the southwest, whereas the redshifted outflow extends from the nucleus to the northeast with an estimated radius of $\sim$ 1.1 kpc (Cicone
et al., 2014). Assuming a spherical geometry and adopting a CO-to-$\rm H_{\rm 2}$ conversion factor of 0.8 $M_{\odot}/(\rm K\,km\,s^{-1}\,pc^{2})$ that is commonly used for the molecular ISM of ULIRGs (Bolatto
et al., 2013), a $\rm H_{\rm 2}$ gas mass outflow rate of 300 $\rm M_{\odot}\,yr^{-1}$ was obtained. Fluetsch
et al. (2019) studied the cold molecular outflows in several nearby galaxies, including IRAS 10565+2448. By assuming the outflow is expelled shell-like, the $\rm H_{\rm 2}$ gas mass outflow rate was estimated as 100 $\rm M_{\odot}\,yr^{-1}$, corresponding to an energy loss rate of $6.3\times 10^{42}\,\rm erg\,s^{-1}$. Please note that the $\rm H_{2}$ gas considered here only comes from wings spanning velocity -300 km s${}^{-1}$ $\sim$ -600 km s${}^{-1}$ and 300 km s${}^{-1}$ $\sim$ 600 km s${}^{-1}$. By using the data from Rupke &
Veilleux (2013) and Cicone
et al. (2014), Fluetsch
et al. (2021) studied this source again. Interestingly, blueshifted OH 199 $\mu$m absorption was detected in IRAS 10565+2448 (Veilleux
et al., 2013), giving a mass outflow rate of 248 $\rm M_{\odot}\,yr^{-1}$ (González-Alfonso et al., 2017).
Our observations have detected the blueshifted and possible redshifted neutral hydrogen outflows through absorption and support the model where the outflows are expelled shell-like, as the broad wing, from -500 km s${}^{-1}$ to -250 km s${}^{-1}$, absorption strength from square 6 is larger than that from square 5 where the radio emission is brightest and so should have stronger absorption if the H i column density is uniform across squares 5 and 6, see also the positions of the outflows in Fig. 5. From our observation, the blueshifted and possible redshifted H i outflows are quite symmetric with respect to the nucleus. To calculate the H i mass outflow rate, we followed the equation (Heckman, 2002):
$$\dot{M}=30\frac{\Omega}{4\pi}\frac{r_{*}}{1\,\mathrm{kpc}}\frac{N_{\mathrm{HI}}}{10^{21}\mathrm{cm}^{-2}}\frac{v}{300\,\mathrm{km\,s}^{-1}}\rm M_{\odot}\,{\rm yr}^{-1}$$
(1)
where the $\Omega$ is the solid angle of outflow, $r_{*}$ is the outflow radius, $N_{\mathrm{HI}}$ is the H i column density, and $v$ is the outflow velocity.
Since the detection of redshifted outflow is not significant, we temporarily exclude the redshfited outflow from our calculation of outflow properties.
We use the spectrum from square 6 as the blueshifted outflow spectrum, and the distance from the radio nucleus to the centre of square 6 as the outflow radius. We chose not to use the spectrum from square 15 because it seems slightly influenced by the narrow and deep absorption from square 5, see Fig. 8, whereas the spectrum from square 6 looks unpolluted. Actually, square 6 is almost overlapped with square 15 and so using the spectrum and position information from either square 6 or 15 would not make a big difference.
Using the spectrum from square 6, we found that the blueshifted outflow has a velocity up to $\sim$ -530 km s${}^{-1}$ with the central velocity at $\sim$ -148 km s${}^{-1}$. The corresponding H i column density is $\sim 7.83\times 10^{19}T_{\rm s}$ atoms cm${}^{-2}$ and the projected distance between the radio nucleus and the centre of square 6 is 1.36 kpc, which was used as the outflow radius. By assuming the blueshifted outflow has a solid angle of $\pi$, we derive a mass outflow rate of $\sim 0.39T_{\rm s}$ $\rm M_{\odot}\,yr^{-1}$ for the blueshifted outflow.
We can calculate the energy loss rate by using the equation (Holt
et al., 2006):
$$\dot{E}=6.34\times 10^{35}\frac{\dot{M}}{2}\left(v^{2}+\frac{{\rm FWHM}^{2}}{1.85}\right){\rm erg}\,{\rm s}^{-1},$$
(2)
where FWHM is the full width at half maximum, which gives a energy loss rate of $\sim 2.6\times 10^{40}T_{\rm s}$ $\rm erg\,s^{-1}$ for the blueshifted H i outflow in IRAS 10565+2448.
The larger, $\sim$ 2–3 kpc, more diffuse, and warm ionised outflow detected through optical $\rm H\alpha$ emission and diffuse neutral outflow extending to southwest detected through Na i D absorption, the cold molecular outflows detected through CO emission, and the neutral hydrogen outflow detected through H i absorption all are aligned from northeast to southwest, indicating they may be multi-phase gas counterparts of the same outflow.
To more precisely understand the impact of outflows in IRAS 10565+2448, we need to combine the results from these observations. For the ionised outflow, we directly quote the mass outflow rate and energy loss rate from $\rm H\alpha$ gas from Rupke &
Veilleux (2013), which is negligible compared to the molecular and neutral outflows. For the neutral hydrogen outflow, we use the results of our own observations. We obtained a mass outflow rate of $\sim 0.39T_{\rm s}$ $\rm M_{\odot}\,yr^{-1}$ corresponding to an energy loss rate of $\sim 2.6\times 10^{40}T_{\rm s}$ $\rm erg\,s^{-1}$. Lastly, we use the calculations from CO emission in Fluetsch
et al. (2019) for the molecular $\rm H_{2}$ outflow. The reason of why we don’t use the OH 199 $\mu$m absorption from Veilleux
et al. (2013) is that absorption can only reveal foreground gas. We don’t use the CO analysis in Cicone
et al. (2014) because spherical geometry was assumed for the outflow whereas our uGMRT observations support expelled shell geometry that used by Fluetsch
et al. (2019). A mass outflow rate of $100\,\rm M_{\odot}\,yr^{-1}$ and an energy loss rate of $\sim 6.3\times 10^{42}\,\rm erg\,s^{-1}$ were given to the $\rm H_{2}$ gas outflow in Fluetsch
et al. (2019). The properties of these outflows are summarised in Tab. 2.
The H i gas spin temperature is a key factor in estimating the impact of the outflow. Previously known measurements of H i spin temperature imply that 100 K should be taken as a lower limit (e.g. Reeves et al., 2015; Reeves
et al., 2016; Kanekar
et al., 2014; Murray et al., 2018; Allison, 2021). If we assume an H i spin temperature of 100 K in IRAS 10565+2448, then the total mass outflow rate is $\sim 140\,\rm M_{\odot}\,yr^{-1}$, corresponding to a total energy loss rate $\sim 8.9\times 10^{42}\,\rm erg\,s^{-1}$.
Please note that $\sim 140\,\rm M_{\odot}\,yr^{-1}$ and $\sim 8.9\times 10^{42}\,\rm erg\,s^{-1}$ should be taken as lower limits due to the reasons below: 1), the low velocity part, $|v|\leq 300\,\rm km\,s^{-1}$, of CO emission were not taken into consideration; 2), the possible redshifted H i outflow has been excluded and the H i spin temperature we used is 100 K which is at the lower end of the range of observed $T_{\rm s}$, especially given the presence of an ionised outflow counterpart; 3), we have not corrected the projection effect because our observed quantities including radial velocity and FWHM are projected value.
4.4 What drives the outflow in IRAS 10565+2448?
Both starbursts and AGN can drive outflows. Although the star formation rate in IRAS 10565+2448 is as high as 131.8 $\rm M_{\odot}$ year${}^{-1}$ (U et al., 2012), our analysis suggests that the radio jet may play a role in driving outflows.
4.4.1 Starburst-driven outflow?
In star-forming galaxies, the kinetic power released by supernovae (SNe) is capable of driving outflows. Rupke &
Veilleux (2013) concluded that a starburst is the dominant source of driving outflows in IRAS 10565+2448 because there is no sign of AGN from optical and X-ray bands (Veilleux et al., 1995; Yuan
et al., 2010; Teng &
Veilleux, 2010) or the AGN is weak whose bolometric luminosity only occupies 17 percent of the total bolometric luminosity of the galaxy based on the polycyclic aromatic hydrocarbon (PAH) strength and mid-infrared (MIR) spectral shape (Veilleux
et al., 2009). However, there is no specific calculation in Rupke &
Veilleux (2013) to test whether a starburst can drive the observed outflows.
Here, we take a cautious approach. Following Veilleux et al. (2005) we estimated the mass outflow rate driven by SNe as $\dot{M}=0.26({\rm SFR}/{\rm M_{\odot}\,yr^{-1})}$ ${\rm M_{\odot}\,yr^{-1}}$, which is $\sim 34.3\,{\rm M_{\odot}\,yr^{-1}}$ – much lower than the value of $\sim 140\,\rm M_{\odot}\,yr^{-1}$ that we have observed in IRAS 10565+2448. Since the mass flow rate derived from the models only considers the mass flux from the hot wind produced by the starburst, the actual outflow rate could be several times higher if (as expected) this wind entrains mass from the host galaxy. If sufficient additional mass is entrained, then the outflow rate could plausibly reach the observed value of $\sim 140\,\rm M_{\odot}\,yr^{-1}$.
From the kinetic power view, the kinetic power associated with the outflows is at least $8.9\times 10^{42}\,\rm erg\,s^{-1}$ whereas SNe can release, $P_{\rm K,SF}=7.0\times 10^{41}\,(SFR/{\rm M_{\odot}\,yr^{-1}})$ ${\rm erg\,s^{-1}}$, $\sim 6.7\times 10^{43}\,\rm erg\,s^{-1}$. If supernovae are the main driver for the outflows, then the thermalization efficiency, the fraction of SNe kinetic power converted to outflow kinetic power, needs to be at least $\sim 0.13$.
IRAS 10565+2448 is classified as a ‘composite’ galaxy based on the $\rm[OIII]5007/H\beta\,vs\,[NII]6583/H\alpha$ diagram and an ‘HII’ galaxy based on $\rm[OIII]5007/H\beta\,vs\,([SII]6716+6731)/H\alpha$ and $\rm[OIII]5007/H\beta\,vs\,[OI]6300/H\alpha$ diagrams (Yuan
et al., 2010). Using the spectral classification scheme presented in Veilleux &
Osterbrock (1987), IRAS 10565+2448 is classified as ‘HII’ as well (Yuan
et al., 2010).
Thus, IRAS 10565+2448 seems most likely to be an ‘HII’ galaxy. (Fluetsch
et al., 2019) observed a large sample of ‘HII’ galaxies and derived a typical thermalization efficiency of about 1–2 percent. In the galaxy M82 the thermalization efficiency could be as high as 100% (Strickland &
Heckman, 2009), but this is an extreme case.
We note that the star formation rate in M82 is only about $\sim 10\rm M_{\odot}\,yr^{-1}$(de Grijs, 2001), which is much lower than that in IRAS 10565+2448.
Although we can’t completely exclude that the thermalization efficiency in IRAS 10565+2448 is as high as 0.13, this would be unusually high in comparison to the star-forming systems studied by Fluetsch
et al. (2019).
Besides the contribution from SNe, the radiation pressure from young starbursts can also produce outflows (e.g. Thompson et al., 2005, 2015). The momentum rate of the outflows is at least $\dot{M}V=3.3\times 10^{35}\,\rm g\,cm\,s^{-2}$, while the radiation momentum rate is $\sim 1.1\times 10^{35}\,\rm g\,cm\,s^{-2}$ estimated from the bolometric luminosity of the host galaxy (Cicone
et al., 2014). We can see that the outflow momentum rate is much higher than the radiation momentum rate, whereas from observations the ratio between outflow momentum rate and radiation momentum rate ranges from 0.1 to 0.5 (Fluetsch
et al., 2019). More evidence can be seen from the H i outflow FWHM which is about 581 km s${}^{-1}$ and much larger than the neutral outflow FWHM, $\sim$275 km s${}^{-1}$, seen from large starbursts (Veilleux et al., 2005).
As stated in Section 4.3, there may be various outflows driven by different mechanisms. Based on the arguments above, we think it is unlikely that the starburst alone could drive the observed outflows.
4.4.2 Evidence for an AGN in IRAS 10565+2448.
Previous optical and X-ray observations found little evidence for the existence of an AGN in IRAS 10565+2448 (Veilleux et al., 1995; Yuan
et al., 2010; Teng &
Veilleux, 2010). However, some radiatively inefficient AGN have little or no observational signature in the optical and X-ray bands Heckman &
Best (2014). The presence of a jet in IRAS 10565+2448 is confirmed by observations from the European VLBI Network (EVN). We downloaded the pipeline-calibrated visibilities at 5 GHz and 8.4 GHz of IRAS 10565+2448 and produced the images in Difmap, which are showed in Fig. 9. We argue that the detected radio emission supports the existence of a jet based on the points below:
1.
The brightness temperature. Using the 8.4 GHz data and following the Equation 10 in Shen
et al. (1997), we get a brightness temperature of $3.3\times 10^{10}$ K that is much higher than those, $<10^{5}$ K, of starbursts (Condon
et al., 1991). Actually, radio emission from normal starbursts are generally resolved out in VLBI observations.
2.
The morphology. In some extreme starburst galaxies, like Mrk 273 (Carilli &
Taylor, 2000; Bondi et al., 2005) and Arp 220 (Smith et al., 1998; Lonsdale et al., 2006; Batejat et al., 2011, 2012; Varenius
et al., 2019), there is indeed radio emission from starbursts detected by VLBI. However, the VLBI scale radio morphologies of those starbursts are very different from AGN jet morphologies. The radio morphologies of those extreme starbursts are a clumpy of compact radio sources, see the Figure 2 in Carilli &
Taylor (2000) and Figure 1 in Bondi et al. (2005) for Mrk 273 and Figure 1 in Smith et al. (1998), Figure 1 in Lonsdale et al. (2006), Figure 1 in Batejat et al. (2011), and Figure 1 in Batejat et al. (2012) for Arp 220. In IRAS 10565+2448, the EVN detected radio emission is however not one of clumpy compact radio sources.
3.
The radio luminosity. From Varenius
et al. (2019) we know that in Arp 220 the average flux density of starburst radio emission at 5 GHz is 0.255 mJy, giving a luminosity of $1.9\times 10^{20}\,\rm W\,Hz^{-1}$. From Bondi et al. (2005), we know that in Mrk 273 the brightest radio emission of starbursts has a peak flux density of 0.74 mJy beam${}^{-1}$ at 5 GHz, giving a maximum luminosity of $2.4\times 10^{21}\,\rm W\,Hz^{-1}$. In IRAS 10565+2448, EVN observations give a flux density of 24.6 mJy at 5 GHz, corresponding to a luminosity of $1.0\times 10^{23}\,\rm W\,Hz^{-1}$ which is much higher than those from starbursts.
On this basis, we argue that the radio emission detected by the EVN observations comes from a jet.
4.4.3 Energy- or momentum-conserving wind-driven or radiation-driven outflow?
AGN-induced outflows are a major feedback mechanism in galaxies, and can be classified as energy-conserving mode, momentum-conserving mode, or radiation-pressure mode, depending on the physical environments surrounding the SMBH (e.g. Silk &
Rees, 1998; King, 2003, 2010; King &
Pounds, 2015; Faucher-Giguère & Quataert, 2012; Zubovas &
King, 2012; Fabian, 2012; Costa
et al., 2014).
In an energy-conserving outflow, the thermal energy of the AGN wind radiates away very slowly and so ‘keeps conserved’ while the wind expels the surrounding interstellar medium (ISM). Assuming a SMBH accreting at the Eddington limit, a high gas covering fraction $\sim$1 and a 100 per cent thermal-to-kinetic conversion efficiency, the kinetic power of an energy-conserving outflow is about 5 per cent of the AGN bolometric luminosity (e.g. King, 2010; Zubovas &
King, 2012; Costa
et al., 2014).
In our target, the kinetic power of the outflows is at least $8.9\times 10^{42}\,\rm erg\,s^{-1}$ and the AGN has a radiation power of $6.5\times 10^{44}\,\rm erg\,s^{-1}$ if the AGN contributes 17 percent of the total bolometric luminosity of the galaxy (Veilleux
et al., 2009; Cicone
et al., 2014). Thus the ratio between outflow kinetic power and AGN radiation power is $\geq 0.014$. Although the lowest ratio is consistent with what expected from the ‘energy conserving’ scenario, we argue that the outflow is not, or mainly, driven by the ‘energy conserving’ wind based on the three points below:
1.
As stated in the Section 4.3, the mass outflow rate and outflow kinetic power we obtained are lower limits. If we assume the H i gas spin temperature is 1000 K that is not high as there is already ionised outflow counterpart, then we would have total mass outflow rate of $\sim\,490\,\rm M_{\odot}\,yr^{-1}$ and outflow kinetic power of $\sim 3.2\times 10^{43}\,\rm erg\,s^{-1}$. This kinetic power is just 4.9 percent of the AGN radiation power. It seems the AGN could drive the observed outflow. However, this is too optimistic because in reality the thermal-to-kinetic conversion efficiency and the gas covering fraction assumed above are upper limits and significant lower couplings are expected in some simulations (e.g. Bourne
et al., 2014; Roos et al., 2015), implying that much less than 5 percent of the AGN bolometric luminosity can be converted into the kinetic power of an outflow. Further, if we add the low velocity part of molecular outflow and the possible redshifted H i outflow and further correct the projection effect, we would obtain a higher outflow mass rate and kinetic power.
2.
A more direct piece of evidence comes from the morphology of the ionised outflows. As stated in the Section 4.3, the 1 kpc scale ionised outflow extending to southeast has higher FWHM indicated by $\rm H\alpha$ emission (Rupke &
Veilleux, 2013). The outflow driven by an ‘energy conserving’ wind generally have a large opening angle. Thus if the 1 kpc scale ionised outflow is driven by an AGN wind, we should see the gas surrounding the nucleus have similar FWHM as the FWHM of the 1 kpc scale ionised outflow, which is not the case for the optical observations (Rupke &
Veilleux, 2013). Similarly, we only detected the H i outflow in square 6, which also does not support the large-opening-angle outflow.
3.
Lastly, the velocity FWHM of the H i outflow is large $\sim 581\,\rm km\,s^{-1}$ (noting that this is just the projected quantity to line of sight), and is very similar to those produced by jet-gas interactions as noted in previous studies (e.g. Morganti et al., 2005; Mahony et al., 2013).
In contrast to an ‘energy conserving’ outflow, if the thermal energy of the AGN wind radiates away very quickly during its expansion, then the energy is not conserved but the momentum is still conserved. This corresponds to a ‘momentum-conserved’ outflow, which is less efficient than the ‘energy conserving’ outflow and the outflow kinetic power is expected to be about 0.1 per cent of the AGN radiation power (e.g. King &
Pounds, 2015). Thus the outflows in IRAS 10565+2448 are not driven by AGN momentum. One property of an AGN momentum-driven outflow is that the outflow momentum rate is close to the AGN radiation momentum rate. In IRAS 10565+2488, the momentum rate of the outflows is at least $\dot{M}V=3.3\times 10^{35}\,\rm g\,cm\,s^{-2}$ which is around 14.1 times the AGN momentum rate $L_{\rm AGN}/c$, where $c$ is the speed of light.
The AGN radiation can directly interact with dust clouds in the ISM and thus drive outflows (e.g. Fabian, 2012; Ishibashi &
Fabian, 2015; Ishibashi et al., 2017, 2018). In this scenario, the outflow kinetic power is expected to be about 1 per cent of the AGN radiation power and the outflow momentum rate is expected to be up to 5 times $L_{\rm AGN}/c$. Based on the observed properties of the outflows, the radiation-pressure mode does not appear to explain them.
Therefore, while we could not exclude the existence of AGN-wind (radiation)-driven outflows, the AGN wind (radiation) should make only a minor contribution to driving the outflows.
4.4.4 Jet-driven outflow?
The final mechanism that can drive massive gas outflows is the radio jet. During the growth of a radio jet it can clear and press the gas on its propagation way and thus drive outflows, and this model is supported by the high spatial resolution observational results that the outflows are found to be at the hot spots of the jet (e.g. Morganti et al., 2013; Mahony et al., 2013; Murthy et al., 2022). From hydrodynamical simulations, a radio jet is capable of accelerating clouds to high velocities if the ratio of jet power to Eddington luminosity $\eta=P_{\rm jet}/L_{\rm Edd}$ is above $10^{-4}$ (e.g. Wagner &
Bicknell, 2011; Wagner
et al., 2012; Mukherjee et al., 2018).
For our target, the integral flux density observed in our uGMRT image is 49.6 mJy at 1.36 GHz which corresponds to a radio luminosity of $\sim 2.1\times 10^{23}\,{\rm W\,Hz^{-1}}$ at 1.40 GHz in the rest frame. However, the lower-resolution NRAO VLA Sky Survey (NVSS) gives a total 1.4 GHz flux density of 57.6 mJy (Condon et al., 1998), implying a slightly higher 1.4 GHz radio luminosity of $2.5\times 10^{23}\,{\rm W\,Hz^{-1}}$. We note some radio emission may come from star formation. If we use the star formation rate in our target and the equation 3 in Sullivan et al. (2001), we estimated the maximum 1.4 GHz radio luminosity contribution from star formation is $1.2\times 10^{23}\,{\rm W\,Hz^{-1}}$. Thus, we still have $1.3\times 10^{23}\,{\rm W\,Hz^{-1}}$ from a jet, corresponding to a jet power $5.3\times 10^{42}\,\rm erg\,s^{-1}$ using the Equation 2 in Best et al. (2006). The estimated SMBH mass of $\sim 2.0\times 10^{7}\,{\rm M_{\odot}}$ implies an Eddington luminosity of $\sim 2.6\times 10^{45}\,\rm erg\,s^{-1}$ (Dasyra
et al., 2006). Then the ratio of jet power to Eddington luminosity is $\sim 2.0\times 10^{-3}$ which is larger than $10^{-4}$.
Therefore, the radio jet in IRAS 10565+2448 can drive the observed outflows. More evidence in support of a jet-driven outflow comes from the morphology of the radio emission and the spatial distribution of the H i and molecular outflows. A Very Large Array (VLA) continuum image at 8.4 GHz with a resolution of 0.25 arcsec (Condon
et al., 1991) shows that the radio emission is elongated from northeast to southwest. This spatial alignment between the H i and molecular outflows and the radio emission supports the jet-driven scenario. Additionally, the velocity width of the outflows is very broad which is a common feature of jet-gas interactions such as that observed in e.g. 3C 293 (Mahony
et al., 2016), 4C 12.50 (Morganti et al., 2013), and 3C 236 (Schulz et al., 2018).
Based on the discussion above, we argue that the radio jet is capable of driving the observed outflows in addition to the outflows driven by the starburst.
4.5 Implications
It is generally accepted that outflows, especially AGN-driven outflows, play a significant role in galaxy evolution. Quantifying the basic parameters regarding mass outflow rate, outflow mass, kinetic power, radius, spatial distribution, and momentum rate of outflows are crucial for understanding their impact and driving mechanism. However, this is often difficult as outflows contain multi-phase gas which requires multi-band observations. In IRAS 10565+2448, the optical warm ionised outflow has a mass outflow rate of $\sim 1.4\,{\rm M_{\odot}\,yr^{-1}}$ which is much smaller than that of the cold molecular outflow and neutral hydrogen outflow. Without the latter two detections, the impact of the outflow in IRAS 10565+2448 would be highly underestimated and the outflow driving mechanism would remain highly uncertain. Previous observations have also shown that the dominant mass in outflows is often in neutral or molecular (e.g. Tadhunter et al., 2014; Morganti &
Oosterloo, 2018), which highlights the necessity of multi-band observations of outflows.
High-resolution observations can help us directly image the outflows and thus determine the outflows radius. Given the outflow mass, if the outflowing gas is uniformly distributed within the outflow cone, the mass outflow rate is given by $\dot{M}=3MV/R$, where $M$ is the outflow mass, $V$ the outflow velocity, and $R$ the outflow radius. Instead, if the outflowing gas is distributed in a thin shocked shell, the mass outflow rate would be reduced by a factor of three.
In our target, we did not find a broad absorption wing towards nucleus or at least the broad absorption strength from square 5 is not as strong as that from square 6, which supports the shell-like outflow scenario. Another key element in outflows is the radius, the distance between nuclei and outflows. In previous low spatial resolution observations from which no accurate radius can be derived, people have to assume a value, e.g. 1 kpc. However, we can directly measure it and thus accurately measure the mass outflow rate if the outflows are spatially resolved. Therefore, high spatial resolution observations are crucial in quantifying the impact of outflows.
Jets play a significant role in AGN feedback. While there is no doubt that bright jets can drive massive outflows and have powerful impact on host galaxies (e.g. Morganti &
Oosterloo, 2018), the influence of less powerful (‘modest’) jets on host galaxies has not been well recognised. A recent notable study of B2 0258+35 showed that the modest jet of $L_{\rm 1.4GHz}\sim 2.1\times 10^{23}\,{\rm W\,Hz^{-1}}$ has carried $\sim 75\%$ gas in its central region (Murthy et al., 2022), showing that the contribution from jets cannot be ignored even in the case of a modest jet like the one in B2 0258+35. Our target also has a jet with 1.4 GHz radio luminosity of $\sim 1.3\times 10^{23}\,{\rm W\,Hz^{-1}}$ which reinforces this point. To make it clear whether jets contribute substantially to AGN-driven outflows, and whether jet-driven outflows are more powerful than AGN-wind (radiation)-driven outflows, we suggest that a larger AGN sample with clear jet evidence and a comparison AGN sample without jets are needed to enable a statistical study.
5 Summary
We have presented uGMRT H i spectral-line observations of the ULIRG IRAS 10565+2448, which was reported to show blueshifted, broad, and shallow H i absorption indicating outflow and faint H i emission in previous Arecibo observations (Mirabel &
Sanders, 1988). The much higher spatial resolution of the uGMRT allowed us to separate the H i absorption from emission and to localise the H i outflow gas.
We did not detect the H i emission see by Arecibo, since our observations have lower sensitivity and also much higher spatial resolution, which is likely to have resolved out the faint H i emission.
We successfully recovered the blueshifted, broad, and shallow H i absorption with rest-frame velocity up to $\sim-530$ km s${}^{-1}$ which is slightly different from the maximum shifted velocity, $\sim-600$ km s${}^{-1}$, in the Arecibo spectrum. This difference may result from our lower sensitivity.
Besides the blueshifted H i outflow, we made a possible detection of a redshifted H i outflow in the opposite direction. By combining our H i data with data on the warm ionised and cold molecular outflows (e.g. Rupke &
Veilleux, 2013; Cicone
et al., 2014), the outflows in IRAS 10565+2448 have a total mass outflow rate of at least 140 $\rm M_{\odot}\,yr^{-1}$ and a total energy loss rate of at least $8.9\times 10^{42}\,\rm erg\,s^{-1}$. By analysing these data, we conclude that the radio jet may play a role in driving the observed outflows.
We stress the importance of multi-band observations to probe multi-phase gas and the necessity of high spatial resolution observations to image outflows, which can help us to accurately measure the properties of outflows and thus understand their driving mechanism and impact on the host galaxies. The modest jet radio luminosity, $L_{\rm 1.4GHz}$ $\sim 1.3\times 10^{23}\,{\rm W\,Hz^{-1}}$, of IRAS 10565+2448 emphasises that we cannot ignore the contribution from radio jets in driving outflows even when the radio luminosity is modest, see also the case of B2 0258+35 (Murthy et al., 2022). Finally, as one of few sources with spatially resolved optical, atomic, and molecular outflow detections, IRAS 10565+2448 can provide new constraints on understanding jet-gas interaction and inform future simulations of feedback processed in galaxy evolution.
Acknowledgements
We thank the anonymous referee for the comments that improve this paper.
We thank the staff of the GMRT who have made these observations possible. The GMRT is run by the National Centre for Radio Astrophysics of the Tata Institute of Fundamental Research.
RZS thanks CSIRO for their hospitality during his stay in Australia and acknowledges support from a joint SKA PhD scholarship.
RZS and MFG are supported by the National Science Foundation of China (grant 11873073), Shanghai Pilot Program for Basic Research–Chinese Academy of Science, Shanghai Branch (JCYJ-SHFY-2021-013), and the science research grants from the China Manned Space Project with NO. CMSCSST-2021-A06. This work is supported by the Original Innovation Program of the Chinese Academy of Sciences (E085021002).
Y.Chandola acknowledges the support from the NSFC under grant No. 12050410259, and Center for Astronomical Mega-Science, Chinese Academy of Sciences, for the FAST distinguished young researcher fellowship (19-FAST-02), and MOST for the grant no. QNJ2021061003L.
Parts of this research were supported by the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013.
MG (Curtin) was partially supported by the Australian Government through the Australian Research Council’s Discovery Projects funding scheme (DP210102103).
6 DATA AVAILABILITY
The EVN data are available through its archive. Other data are available on reasonable request to the corresponding author.
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The band structure of the whole spectrum of an N-body cold system containing atoms with arbitrary integer spin and dominated by singlet pairing force
Y Z He${}^{1}$, Y M Liu${}^{2}$, Z B Li${}^{1}$, C G Bao${}^{1,}$111Corresponding author: C G Bao, stsbcg@mail.sysu.edu.cn
${}^{1}$School of Physics, Sun Yat-Sen University, Guangzhou, 510275, P. R. China
${}^{2}$Department of physics, Shaoguan University, Shaoguan, 510205, P. R. China
Abstract
The spectra of $N$-boson systems with arbitrary nonzero spin $\mathfrak{f}$ have been studied.
Firstly, only the singlet pairing interaction is considered, a set of eigenstates together with the eigenenergies are analytically obtained.
The completeness of this set is proved.
The analytical expression allows us to see clearly the spin structures of various states different in $N$ and/or $\mathfrak{f}$, and to find out the similarity and relationship lying among them.
Secondly, the effect of other interactions is evaluated via exact numerical calculations on the systems with a smaller $N$.
Some features and notable phenomena that might emerge in high-$\mathfrak{f}$ systems, say, the ground band might have extremely high level density, have been discussed.
††: Phys. Scr.
Keywords: spinor Bose-Einstein condensates, singlet pairing interaction, spin structures
1 Introduction
The study of many-body cold systems is a hot topic in recent years.
What we pay attention to and think about are the following points:
•
After the application of optical trapping, the spin degrees of freedom are liberated [1, 2, 3, 4, 5].
Furthermore, due to the progress in techniques, very low temperature (say, $T\leq 10^{-11}K$) can be achieved [6].
At such a temperature the spatial degrees of freedom can be completely frozen, and the systems dominated by spin degrees of freedom can be experimentally created.
It turns out that the spin dependent forces are very weak (nearly two order weaker than the central force).
Thus, an important feature of these systems is the extreme high sensitivity.
•
Due to the progress in techniques, the precise manipulation of a few atoms can be realized [7, 8].
Thus, instead of a very large particle number $N$, the study of cold systems with a smaller $N$ (say, $N\leq 10$) is meaningful because they might be controlled more precisely.
•
The role of [0]-pairs, namely, the particles are two-by-two coupled to spin zero, in fermion systems is well known (say, in superconductivity and in nuclear theory) [9, 10].
For spin-$\mathfrak{f}$ boson systems with $\mathfrak{f}\leq 2$, it has been proved that the number of the [0]-pairs is a good quantum number due to the conservation of seniority [5, 11, 12], and they appear as a basic constituent in spin structures.
The number of them is an important index for classifying the states.
Nonetheless, when $\mathfrak{f}\geq 3$, seniority is in general not conserved (except in some special cases).
Therefore, the role of the [0]-pairs in $\mathfrak{f}\geq 3$ systems remains to be further clarified.
While numerous literatures have been dedicated to the spin-1 and spin-2 systems, the study on $\mathfrak{f}\geq 3$ systems is relatively scarce [13, 14, 15, 16, 17].
It turns out that the spin degrees of freedom would increase greatly with $\mathfrak{f}$.
Let $N=10$ for an example.
If $\mathfrak{f}=1$, 2, 3, and 4, the number of spin states with magnetization $M=0$ is 6, 55, 338, and 1514, respectively.
When $N$ becomes larger, the number is terribly large.
Say, when $N=1000$ and $\mathfrak{f}=3$, this number would be $7.7735\times 10^{11}$.
Due to the great increase of the dimension of the spin space, rich physics would be involved in the high-$\mathfrak{f}$ systems.
Thus, the study of them is worthy.
This paper is dedicated to the study of $N$-boson spin-$\mathfrak{f}$ bound cold systems with arbitrary $N$ and $\mathfrak{f}$ (nonzero).
The study aims beyond the ground state but covers the whole spectrum.
The purpose is to clarify the role of singlet pairing in high-$\mathfrak{f}$ systems, and to find out common characters among these systems.
2 The complete spectra of high-$\mathfrak{f}$ systems under singlet pairing interaction
We assume that the temperature is so low that all the spatial degrees of freedom are frozen, and all particles fall into the same spatial state $\phi$ which is most favorable to binding.
Due to the freezing, the spin-orbit coupling can be neglected.
Then the high-$\mathfrak{f}$ system is completely governed by the spin dependent Hamiltonian
$$\displaystyle H_{\mathrm{spin}}=\sum_{i<j}V_{ij},\ \ \ V_{ij}=\sum_{\lambda}g_{\lambda}P_{\lambda}^{i,j},$$
(1)
where $g_{\lambda}$ is the weighted strength of the $\lambda$-channel ($\lambda=0,2,\cdots,2\mathfrak{f}$), and the factor $\int\phi^{4}(r)\mathrm{d}\bi{r}$ has been included.
$P_{\lambda}^{i,j}$ is the projector to the $\lambda$-channel.
Let the Hamiltonian be dominated by singlet pairing interaction, namely, $g_{0}$ is much more negative than the other strengths.
In this case the effect of the latter is smaller, thus they can be approximately given as being equal to each other.
Since the spin structures will not be changed when the set $\{g_{\lambda}\}$ are shifted as a whole and/or when a new unit is adopted, the case with $g_{0}<g_{2}=\cdots=g_{2\mathfrak{f}}$ is equivalent to the case with $g_{0}=-1$ and $g_{2}=\cdots=g_{2\mathfrak{f}}=0$.
This simplified Hamiltonian is denoted as $H_{[0]}$.
We are first going to find out all the eigenenergies and eigenstates of $H_{[0]}$ analytically.
Then, when $H_{\mathrm{spin}}$ deviates from $H_{[0]}$, the effect of the deviation is evaluated.
Let $\{\Phi_{N_{1},S,l}\}$ be a set of normalized, symmetrized, and orthogonal spin states of a subsystem with $N_{1}$ particles.
$S$ is the total spin and $l$ is just an index to specify further the states.
Let $\chi$ denote the spin state of a particle, and $(\chi\chi)_{0}$ denote a singlet pair ([0]-pair).
Let $N=N_{1}+2J$ and $\mathfrak{P}_{N}$ denote a summation over the $N!$ permutations of particles.
Then, for the product state $\mathfrak{P}_{N}\Phi_{N_{1},S,l}(\chi\chi)_{0}^{J}$, we divide the $N!$ permutations into four parts as
$$\displaystyle\mathfrak{P}_{N}=\mathfrak{P}_{N,a}+\mathfrak{P}_{N,b}+\mathfrak{P}_{N,c}+\mathfrak{P}_{N,d},$$
(2)
where $\mathfrak{P}_{N,x}$ includes those permutations that the particles $i$ and $j$ are both in $\Phi_{N_{1},S,l}$ (if $x=a$), $i$ in $\Phi_{N_{1},S,l}$ and $j$ in $(\chi\chi)_{0}^{J}$ (if $x=b$), $j$ in $\Phi_{N_{1},S,l}$ and $i$ in $(\chi\chi)_{0}^{J}$ (if $x=c$), and both $i$ and $j$ are in $(\chi\chi)_{0}^{J}$ (if $x=d$).
For the case $x=b$, formally we can extract the $i$-th particle from $\Phi_{N_{1},S,l}$ by using the fractional parentage coefficients $\beta_{S^{\prime}}$ as $\Phi_{N_{1},S,l}=\sum_{S^{\prime}}\beta_{S^{\prime}}(\chi(i)\phi_{S^{\prime}})_{S}$.
The details of $\beta_{S^{\prime}}$ and $\phi_{S^{\prime}}$ are irrelevant in the follows.
Thus we have
$$\displaystyle\mathfrak{P}_{N,b}[\Phi_{N_{1},S,l}(\chi\chi)_{0}^{J}]=\mathfrak{P}_{N-2}^{(i,j)}2N_{1}J[\sum_{S^{\prime}}\beta_{S^{\prime}}(\chi(i)\phi_{S^{\prime}})_{S}(\chi(j)\chi)_{0}(\chi\chi)_{0}^{J-1}],$$
(3)
where $\mathfrak{P}_{N-2}^{(i,j)}$ denotes a summation over all $(N-2)!$ permutations (particles $i$ and $j$ are excluded).
Making use of the following specific 9-$j$ symbol
$$\displaystyle\left\{\begin{array}[]{ccc}\mathfrak{f}&S^{\prime}&S\\
\mathfrak{f}&\mathfrak{f}&0\\
0&S&S\end{array}\right\}=\frac{(-1)^{\mathfrak{f}+S^{\prime}+S}}{(2\mathfrak{f}+1)(2S+1)},$$
(7)
We have
$$\displaystyle(\chi(i)\phi_{S^{\prime}})_{S}(\chi(j)\chi)_{0}=\frac{1}{2\mathfrak{f}+1}(\chi(i)\chi(j))_{0}(\chi\phi_{S^{\prime}})_{S}+Z.$$
(8)
In $Z$ the particles $i$ and $j$ are not coupled to zero.
When $H_{\mathrm{spin}}=H_{[0]}$, $Z$ has no contribution to energy and therefore can be neglected.
After the neglect,
$$\displaystyle\mathfrak{P}_{N,b}[\Phi_{N_{1},S,l}(\chi\chi)_{0}^{J}]=\frac{2N_{1}J}{2\mathfrak{f}+1}\mathfrak{P}_{N-2}^{(i,j)}[\Phi_{N_{1},S,l}(\chi(i)\chi(j))_{0}(\chi\chi)_{0}^{J-1}],$$
(9)
and
$$\displaystyle H_{[0]}\mathfrak{P}_{N,b}[\Phi_{N_{1},S,l}(\chi\chi)_{0}^{J}]=\frac{1}{2}\sum_{i\neq j}V_{ij}\mathfrak{P}_{N,b}[\Phi_{N_{1},S,l}(\chi\chi)_{0}^{J}]=-\frac{N_{1}J}{2\mathfrak{f}+1}\mathfrak{P}_{N}[\Phi_{N_{1},S,l}(\chi\chi)_{0}^{J}].$$
(10)
The above formula remains unchanged when $\mathfrak{P}_{N,b}\rightarrow\mathfrak{P}_{N,c}$.
Similarly, we have
$$\displaystyle H_{[0]}\mathfrak{P}_{N,d}[\Phi_{N_{1},S,l}(\chi\chi)_{0}^{J}]=-\frac{(2J+2\mathfrak{f}-1)J}{2\mathfrak{f}+1}\mathfrak{P}_{N}[\Phi_{N_{1},S,l}(\chi\chi)_{0}^{J}].$$
(11)
Let the set $\{\Phi_{N_{1},S,l}\}$ include all those states of the $N_{1}$-body subsystem full in seniority (i.e., all the $N_{1}$ particles are unpaired).
Accordingly, $\Phi_{{}_{N_{1},S,l}}$ has no contribution on energy when $H_{\mathrm{spin}}=H_{[0]}$.
Let $\tilde{\mathfrak{P}}_{N}$ be the operator for normalization and symmetrization, and let
$$\displaystyle\Psi_{J,S,l}^{[N]}\equiv\tilde{\mathfrak{P}}_{N}[\Phi_{N_{1},S,l}(\chi\chi)_{0}^{J}].$$
(12)
From the above formulae we arrive at
$$\displaystyle H_{[0]}\Psi_{J,S,l}^{[N]}=-\frac{(2N-2J+2\mathfrak{f}-1)J}{2\mathfrak{f}+1}\Psi_{J,S,l}^{[N]}\equiv E_{J}^{[N]}\Psi_{J,S,l}^{[N]}.$$
(13)
which is just the Schrödinger equation.
Thus, a series of eigenenergies and eigenstates of $H_{[0]}$ have been analytically obtained.
The completeness of this set is given below.
Each state is a product of a core (filled with $N_{1}$ unpaired particles) together with $J$ [0]-pairs.
In particular, when $N$ is fixed, $E_{J}^{[N]}$ depends only on $J$, thus a group of states with the same $J$ but different in $S$ and $l$ are degenerate.
When $N=2K$ or $2K+1$, $J$ is ranged from 0 to $K$.
Accordingly, the whole spectrum is divided in to $K+1$ bands specified by $J$, the width of each band is zero.
Obviously, when $J$ is larger, more [0]-pairs are contained, therefore the energy $E_{J}^{[N]}$ is lower.
The lowest band has $J=K$ and contains only one state, namely, the ground state (g.s.) $\Psi_{\mathrm{gs}}^{[N]}=\tilde{\mathfrak{P}}_{N}[\chi(\chi\chi)_{0}^{J}]$ or $\tilde{\mathfrak{P}}_{N}[(\chi\chi)_{0}^{J}]$.
The g.s. energy is $E_{\mathrm{gs}}^{[N]}=-[2(K+\mathfrak{f)}\pm 1]K/(2\mathfrak{f}+1)$, where $+(-)$ is for odd (even) $N$.
Let $L\equiv K-J$.
While $J$ denotes the number of [0]-pairs, $L$ is related to the seniority ($L=N_{1}/2$ or $(N_{1}-1)/2$ if $N$ is even or odd).
It denotes also the bands in the order of uprising energy, say, $L=0$ ($K$) is for the bottom (top) band.
The excitation energy
$$\displaystyle E_{L}^{[N]ex}\equiv E_{J}^{[N]}-E_{\mathrm{gs}}^{[N]}=\frac{[2(L+\mathfrak{f)}\pm 1]L}{2\mathfrak{f}+1},$$
(14)
where $+(-)$ is for odd (even) $N$.
As examples, the spectra of $E_{L}^{[N]\mathrm{ex}}$ with $\mathfrak{f}=3$ and $N=5\rightarrow 9$ are shown in figure 1.
The spectra with $\mathfrak{f}=1\rightarrow 5$ and $N\geq 10$ (even) are shown in figure 2.
It turns out that $E_{L}^{[N]\mathrm{ex}}$ does not depend on $N$, but $L$ and $(-1)^{N}$, this is clearly shown in figure 1.
Say, the two spectra with $N=7$ and 9 are similar, except that the latter contains one more band at the top.
We found
$$\displaystyle E_{L+1}^{[N]\mathrm{ex}}-E_{L}^{[N]\mathrm{ex}}=\frac{4L+2\mathfrak{f}+\mu}{2\mathfrak{f}+1},$$
(15)
where $\mu=2-(-1)^{N}$.
Thus, the separation between the higher bands is larger.
Recall that the excitation $L\rightarrow L+1$ is realized by breaking a [0]-pair.
Thus, the solidity of the [0]-pair does not depend on how many other [0]-pairs are surrounding, but depends on how many unpaired particles are surrounding, i.e., on the seniority of the state.
This is shown in figure 1 and figure 2.
The solidity depends also on $\mathfrak{f}$ as shown in (15).
The [0]-pair with a larger $\mathfrak{f}$ would be easier to be broken as shown in figure 2, where the level would be lower when $\mathfrak{f}$ increases.
It is clear from the Schrödinger equation (13) that, if some [0]-pairs are added into (removed from) an eigenstate so that $\Psi_{J,S,l}^{[N]}\rightarrow\Psi_{J^{\prime},S,l^{\prime}}^{[N^{\prime}]}$, where $N^{\prime}-2J^{\prime}=N-2J\equiv N_{1}$.
Then, these two states will both belong to the $L=N_{1}/2$ (if $N$ is even) band and have exactly the same core $\Phi_{N_{1},S,l}$.
They are different only in the number of [0]-pairs.
Consequently, these states different in $N$ are related to each other.
This relationship could be called brotherhood.
Note that $M$, the $Z$-component of $S$, is a good quantum number.
We assume $M=0$ (the cases with $M\neq 0$ are similar when an external field is absent).
When $N$ is fixed, the total number of spin states with $M=0$ denoted as $\mathcal{N}_{N}$ is known (say, for $\mathfrak{f}=3$, $\mathcal{N}_{N}=8,18,32,\cdots$ when $N=3,4,5,\cdots$).
From the brotherhood, we know that, when a [0]-pair is added to every spin state of a $(N-2)$-body system, all the spin states of the $N$-body system can be recovered except those in the top-band.
Therefore, the number of states contained in the top-band is
$$\displaystyle\mathfrak{N}_{K,(-1)^{N}}=\mathcal{N}_{N}-\mathcal{N}_{N-2}.$$
(16)
In general, for the $L$ band,
$$\displaystyle\mathfrak{N}_{L,(-1)^{N}}=\mathcal{N}_{2L}-\mathcal{N}_{2L-2}.$$
(17)
For the bottom band with $L=0$, $\mathfrak{N}_{0,(-1)^{N}}=1$.
Summing up the above numbers, we found
$$\displaystyle\sum_{L=0}^{K}\mathfrak{N}_{L,(-1)^{N}}=\mathcal{N}_{N}.$$
(18)
Equation (18) confirms that the set of eigenstates $\{\Psi_{J,S,l}^{[N]}\}$ is complete.
The numbers $\mathfrak{N}_{L,(-1)^{N}}$ of various states are given in figure 1 and figure 2.
It increases rapidly with $N_{1}$ (i.e., with $L$) as shown in figure 1, and with $\mathfrak{f}$ as shown in figure 2.
3 Evaluation of the effect caused by a deviation of the Hamiltonian from singlet pairing
When $H_{\mathrm{spin}}$ deviates from $H_{[0]}$, it is denoted as $H_{\mathrm{devi}}$.
In the follows we present results from exact numerical calculation to evaluate the effect of $H_{\mathrm{devi}}$.
Examples with $\mathfrak{f}=3$ and $N=8$ are given in figure 3, in which $g_{0}$ remains to be most negative.
Four cases are shown:
(i) $g_{2}=g_{4}\leq g_{6}$, in this case a shift of the strengths as a whole is further performed so that $g_{2}=g_{4}=0$ (figure 3a).
(ii) $g_{2}=g_{4}=0\geq g_{6}$ (figure 3b).
(iii) $g_{2}<g_{4}$ (figure 3c).
(iv) $g_{2}>g_{4}$ (figure 3d).
Let $P_{\psi}^{0}$ be the probability of a pair of atoms coupled to zero.
Since the key point is to check the conservation of seniority, in these figures $P_{\psi}^{0}$ of all the 151 eigenstates with $M=0$ are plotted.
If a group of states have the same seniority, their $P_{\psi}^{0}$ would be exactly the same.
Otherwise, the distribution of $P_{\psi}^{0}$ is diffused.
In figure 3a and figure 3b $g_{2}=g_{4}=0$ is given, while $g_{6}$ is given at 3 values.
In figure 3a we found that the black points (associated with $g_{6}=0$) are exactly distributed on five horizontal lines implying the conservation of seniority and, accordingly, the clear band structure.
The upmost line containing only one point is for the g.s. with seniority zero (all the eight particles are in [0]-pairs), while the lowest line is for the top-band with seniority being 8 (full in seniority).
When $g_{6}>0$ (figure 3a), we found that the points previously belonging to a horizontal line diffuse strongly along horizontal direction but weakly along vertical direction.
In particular, each diffused point has a partner point at the horizontal line, the spin states of this pair are very close to each other (see below).
It implies that all the eigenstates of $H_{[0]}$ have not been seriously disturbed by $H_{\mathrm{devi}}$, and therefore seniority remains nearly conserved.
However, their energies may be seriously affected, the order of the states (i.e., the index $i$) may therefore be changed.
E.g., when $g_{6}=0.3$, the lowest three states of the $L=2$ band ($i=5$, 6, and 7) have $S=3$, 0, and 8.
However, when $g_{6}=0.6$, the indexes of these three states become $i=5$, 9, and 6, respectively.
Due to the diffusion along horizontal direction, the band widths will become broader, and the crossover of bands may occur.
E.g., when $g_{6}=0.6$, the top state of the $L=2$ band has $i=33$, and its energy level goes up deeply into the $L=3$ band.
This state has $S=12$ and is found to be dominated by the component containing two [0]-pairs and four unpaired particles.
Since the four spins coupled to $S=12$ are aligned, this state has a smaller $P_{\psi}^{0}$ and is subjected to a stronger repulsion from $g_{6}$.
This explains why this state is higher.
Similarly, when $g_{6}=0.6$, the highest state of the $L=3$ band has $i=134$, it goes up deeply into the $L=4$ band.
This state has $S=18$ and is dominated by one [0]-pairs and six unpaired particles.
Since the six spins coupled to $S=18$ are all aligned, this state has also a smaller $P_{\psi}^{0}$ and also subjected to a stronger repulsion from $g_{6}$.
Whereas we found that the lowest state of the top-band has $i=41$ and $S=0$.
Therefore, the repulsion from $g_{6}$ can be reduced.
Thus, although the seniority is nearly conserved, the energies are strongly affected by $H_{\mathrm{devi}}$.
To evaluate the deviation quantitatively, let the eigenstates of $H_{[0]}$ be denoted as $\Phi_{i^{\prime}}$, while those of $H_{\mathrm{devi}}$ be $\Psi_{i}$.
For the case given in figure 3a with $g_{6}=0.3$, among the 151 eigenstates, 81 of them can find a partner so that $\langle\Phi_{i^{\prime}}|\Psi_{i}\rangle>0.99$ and other 61 of them have $0.95<\langle\Phi_{i^{\prime}}|\Psi_{i}\rangle<0.99$.
These data confirm that the deviation in the eigenstates as a whole is slight.
In particular, the deviation in high-lying states is even much smaller.
Say, $\langle\Phi_{i^{\prime}}|\Psi_{1}\rangle=0.950$, $\langle\Phi_{i^{\prime}}|\Psi_{21}\rangle=0.993$, and $\langle\Phi_{i^{\prime}}|\Psi_{150}\rangle=1.000$.
When $g_{6}$ increases from $0.3$ to $0.6$, the above value $81\rightarrow 22$ and $61\rightarrow 60$, it implies a larger deviation.
In figure 3b the vertical diffusion of points is also slight, thus the near conservation of seniority also holds, in particular for higher states.
However, the widths of the bands become very broad.
When $g_{6}=-0.6$, the energy of the bottom state of every band is close to the g.s..
For examples, the bottom state of the top-band with $L=4$ has $i=1$ and $S=24$.
Thus, this state is fully polarized and it becomes the g.s..
Accordingly, the width of the top-band covers all the spectrum.
When both $g_{6}$ and $g_{0}$ are negative, there is a competition between alignment and pairing.
When alignment exceeds pairing, as in the above case, a change of phase of the g.s. would occur.
Similarly, the blue point for the bottom state of the $L=3$ band has $i=2$ and $S=18$.
It is composed of 6 aligned particles together with a [0]-pair.
While the blue point for the bottom state of the $L=2$ band has $i=4$ and $S=12$ containing four aligned particles and two [0]-pairs.
Thus, when $g_{6}$ is sufficiently negative, all the unpaired particles are aligned in the bottom states of every band.
Alternatively, the attraction from $g_{6}$ would be minimized in $S=0$ states.
Therefore, among the five bands, four top states have $S=0$ (the appearance of five $S=0$ states is prohibited because, for $N=8$ and $\mathfrak{f}=3$, the multiplicity of $S=0$ state is 4).
For $\mathfrak{f}=3$, it has been pointed out that there is a critical value $g_{6}=(18g_{4}-11g_{2})/7\equiv g_{\mathrm{crit}}$ (disregarding the value of $g_{0}$) at which the seniority is strictly conserved [11].
Figure 3c and figure 3d are for the case in the neighborhood of $g_{\mathrm{crit}}$.
In figure 3c, $g_{2}>g_{4}$, and $g_{\mathrm{crit}}$ becomes negative.
Accordingly, due to the attraction from $g_{6}$, this figure is similar to figure 3b.
Whereas in figure 3d, $g_{2}<g_{4}$, and $g_{\mathrm{crit}}$ becomes positive.
Accordingly, due to the repulsion from $g_{6}$, this figure is more or less similar to figure 3a.
Note that, for figure 3a, figure 3b, and when $g_{6}=0$, all the states belonging to the same band are degenerate.
However, when $g_{6}=g_{\mathrm{crit}}$, the states belonging to the same band are not degenerate.
The band is further divided into a few pieces.
Only the states in a piece are degenerate [11].
One more example with $\mathfrak{f}=4$ is given in figure 4.
For $N=8$, there are totally 526 $M=0$ eigenstates.
$P_{\psi}^{0}$ of the lowest 100 states are plotted.
For higher states with $i>100$, $P_{\psi}^{0}$ of them are $<0.09$.
When $i>250$, most $P_{\psi}^{0}$ are close to zero.
Figure 4 demonstrates that, with $H_{\mathrm{devi}}$, a larger $\mathfrak{f}$ does not further spoil the near conservation of seniority.
The corresponding band structure remains clear.
4 Final remarks
For $H_{[0]}$, we have solved the $N$-boson problem with arbitrary spin $\mathfrak{f}>0$.
The complete spectra for systems different in $N$ and $\mathfrak{f}$ have been obtained analytically.
The completeness of the spectra has been proved.
It is found that the band structure based on the conservation of seniority holds also when $\mathfrak{f}\geq 3$.
Comparison has been made among the spectra different in $N$ and/or $\mathfrak{f}$.
Similarity and relationship among their eigenstates have been demonstrated.
In particular, for $H_{\mathrm{devi}}$, the effect of the deviation has been studied via strict calculations on $P_{\psi}^{0}$.
The following points are mentioned.
•
There is brotherhood among the eigenstates different in $N$.
Therefore, the knowledge from few-body systems (which can be obtained rigorously) can be used to evaluate the low-lying states of many-body systems.
•
The case with a positive $g_{0}$ is noticeable.
When all the strengths change their signs ($g_{0}\rightarrow+1$), all the above discussions remain valid unless that the whole spectrum is reversed.
In this case, the ground band might have an extremely high level density and well protected by a very large energy gap $E_{\mathrm{gap}}=(2N+2\mathfrak{f}-3)/(2\mathfrak{f}+1)$.
Let $N=1000$ as an example.
When $\mathfrak{f}=2$, the number of $M=0$ states contained in the ground band is $1.672\times 10^{5}$.
However, when $\mathfrak{f}=3$, this number is $7.716\times 10^{9}$.
The great difference demonstrates that the increase of $\mathfrak{f}$ could lead to extremely high level density.
If the strengths could be tuned so that $g_{2}$ is close to $g_{4}<g_{0}$ (for $\mathfrak{f}=2$) or $g_{2}$ and $g_{4}$ are both close to $g_{6}<g_{0}$ (for $\mathfrak{f}=3$), the level density of the ground band could be extremely large.
•
For $\mathfrak{f}=3$, there are two zones in the parameter space lying along the line $g_{2}=g_{4}=g_{6}$ and the line $g_{6}=(18g_{4}-11g_{2})/7$ in which the conservation of seniority holds roughly.
In these zones the whole spectra can be divided into bands.
The states in a band have their $P_{\psi}^{0}$ close to each other.
For $\mathfrak{f}\geq 4$, it is believed that there would also be zones in which the band structure holds.
Nonetheless, the case with $\mathfrak{f}\geq 4$ remains to be further clarified.
This work is supported by the National Natural Science Foundation of China under Grants No.11874432, 11372122, 10874122, and by the Key Area Research and Development Program of Guangdong Province under Grant No. 2019B030330001.
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[17]
He L and Yi S
2009 Phys. Rev.A 80 033618 |
On contractible edges in convex decompositions
Ferran Hurtado and Eduardo Rivera-Campo
(Date:: )
Abstract.
Let $\Pi$ be a convex decomposition of a set $P$ of $n\geq 3$ points in general position in the plane. If $\Pi$ consists of more than one polygon, then either $\Pi$ contains a deletable edge or $\Pi$ contains a contractible edge.
Ferran Hurtado worked at Universitat Politècnica de Catalunya until his death in 2014
Eduardo Rivera-Campo, Universidad Autónoma Metropolitana-Iztapalapa, erc@xanum.uam.mx
Partially supported by Conacyt, México
1. Introduction
Let $P$ be a set of $n\geq 3$ points in general position in the plane. A convex decomposition of $P$ is a set $\Pi$ of convex polygons with vertices in $P$ and pairwise disjoint interiors such that their union is the convex hull $CH(P)$ of $P$ and that no point in $P$ lies in the interior of any polygon in $\Pi$. A geometric graph with vertex set $P$ is a graph $G$, drawn in the plane in such a way that every edge is a straight line segment with ends in $P$.
Let $\Pi$ be a convex decomposition of $P$. We denote by $G(\Pi)$ the skeleton graph of $\Pi$, that is the plane geometric graph with vertex set $P$ in which the edges are the sides of all polygons in $\Pi$. An edge $e$ of $\Pi$ is an interior edge if $e$ is not an edge of the boundary of $CH(P)$.
An interior edge $e$ of $\Pi$ is deletable if the geometric graph $G(\Pi)-e$, obtained from $G(\Pi)$ by deleting the edge $e$, is the skeleton graph of a convex decomposition of $P$. Neumann-Lara et al [6] proved that if a convex decomposition $\Pi$ of a set $P$ of $n$ points consists of more that $\frac{3n-2k}{2}$ polygons, where $k$ is the number of vertices of $CH(P)$, then $\Pi$ has at least one deletable edge.
An interior edge $e=uv$ of $\Pi$ is contractible from $u$ to $v$ if the geometric graph
$G(\Pi)/\vec{uv}=(G(\Pi)-\{x_{1}u,x_{2}u,\ldots,x_{m}u,uv\})+\{x_{1}v,x_{2}v,%
\ldots x_{m}v\}$ is a skeleton graph of a convex decomposition of $P\setminus\{u\}$, where $x_{1},x_{2},\ldots,x_{m}$ are the remaining vertices of $G(\Pi)$ which are adjacent to $u$.
A simple convex deformation of $\Pi$ is a convex decomposition $\Pi^{\prime}$ obtained from $\Pi$ by moving a single point $x$ along a straight line segment, together with all the edges incident with $x$, in such a way that at each stage we have a convex decomposition of the corresponding set of points. Deformations of plane graphs have been studied by several authors, both theoretically and algorithmically, see for instance [3, 4, 7] and [1, 2, 5], respectively.
Let $P_{1}$ and $P_{2}$ be sets of $n\geq 3$ points in general position in the plane. A convex decomposition $\Pi_{1}$ of $P_{1}$ and a convex decomposition $\Pi_{2}$ of $P_{2}$ are isomorphic if there is an isomorphism of $G(\Pi_{1})$ onto $G(\Pi_{2})$, as abstract plane graphs, such that the boundaries of $CH(P_{1})$ and $CH(P_{2})$ correspond to each other with the same orientation.
Thomassen [7] proved that if $\Pi_{1}$ and $\Pi_{2}$ are isomorphic convex decompositions, then $\Pi_{2}$ can be obtained from $\Pi_{1}$ by a finite sequence of simple convex deformations. As a tool, Thomassen proved that if $\Pi$ is a convex decomposition with at least two polygons, then there is an isomorphic convex decomposition $\Pi^{\prime}$ that can be obtained from $\Pi$ by a finite number of simple convex deformations that preserve the boundary and such that $\Pi^{\prime}$ contains either a deletable edge or a contractible edge. In this note we prove that every convex decomposition $\Pi$ with at least two polygons contains an edge which is deletable or contractible. Furthermore, if $P$ contains at least one interior point, then $\Pi$ contains a contractible edge.
2. Preliminary results
Let $\Pi$ be a convex decomposition of $P$ containing no deletable edges.
For every interior edge $e$ of $G\left(\Pi\right)$, the graph $G\left(\Pi\right)-e$ has an internal face $Q_{e}$ which is not convex and at least one end of $e$ is a reflex vertex of $Q_{e}$.
We define an abstract directed graph $\overrightarrow{G\left(\Pi\right)}$
with vertex set $P$ in which $\overrightarrow{uv}$ $\in$ $A\left(\overrightarrow{G\left(\Pi\right)}\right)$ if and only if $u$ is a
reflex vertex of $Q_{uv}$. Notice that for each interior edge $uv$ of $G\left(\Pi\right)$, the directed graph $\overrightarrow{G\left(\Pi\right)}$ contains at least one of the arcs $\overrightarrow{uv}$ and $\overrightarrow{vu}$ (see Fig. 1).
Remark 1.
(1)
The outdegree of every vertex $u$ of $\overrightarrow{G\left(\Pi\right)}$ is at most $3$.
(2)
The outdegree of every vertex $u$ in the boundary of $CH\left(P\right)$
is $0$.
(3)
An interior vertex $u$ of $\Pi$ has outdegree 3 in $\overrightarrow{G\left(\Pi\right)}$ if and only if $u$ has degree 3 in $G\left(\Pi\right)$.
(4)
If $\overrightarrow{uv},\overrightarrow{uw}$ $\in$ $A\left(\overrightarrow{G\left(\Pi\right)}\right)$, then $uv$ and $uw$ lie in a
common face of $G\left(\Pi\right)$.
For two points $\alpha$ and $\beta$ in the plane, we denote by $r\left(\alpha\beta\right)$ the ray, with origin $\alpha$, that contains the
segment $\alpha\beta$.
Lemma 2.
An edge $uv$ of $\Pi$ is not contractible from $u$ to $v$ if and only if
there are edges $yx$ and $xu$, lying in a common face of $G\left(\Pi\right)$ that contains vertex $u$, such that the ray $r\left(yx\right)$
meets the edge $uv$ at point $u_{t}$, with $u\neq u_{t}\neq v$, and that the
triangular region defined by $x$, $u_{t}$ and $u$ contains no point of $P$
in its interior.
Proof.
It is easy to see that the existence of such edges $yx$ and $xu$ implies that the edge $uv$ cannot be contracted from $u$ to $v$; we proceed to prove the sufficiency part of the lemma. Let $uv$ be an interior edge of $\Pi$ with $u$ not lying in the boundary of
$CH\left(\Pi\right)$ and let $x_{1},x_{2},\ldots,x_{m}$ be the remaining
vertices of $G\left(\Pi\right)$ which are adjacent to $u$. We contract
the edge $uv$ in a continuous way as follows: Slide the point $u$ along the
ray $r\left(uv\right)$, together with the edges $x_{1}u,x_{2}u,\ldots,x_{m}u$ (see Fig. 2).
If $uv$ is not contractible from $u$ to $v$, then either the transformed
graph $T\left(G\left(\Pi\right)\right)$ becomes non planar or one of
its faces becomes non convex. This implies that we must reach a point $u_{t}=$ $u+t\left(v-u\right)$, with $0<t<1$, such that there are two edges $yx_{i}$ and $x_{i}u_{t}$ lying in a common face, which become collinear in $T\left(G\left(\Pi\right)\right)$ (see Fig. 3).
Notice that two or more different pairs of edges
$yx_{i}$, $x_{i}u_{t}$ and $y^{\prime}x_{j}$, $x_{j}u_{t}$ may become collinear simultaneusly;
in such a case we may choose any of those pairs and proceed with the proof.
The triangular region defined by $x_{i}$, $u_{t}$ and $u$ is the region swept by
the edge $x_{i}u_{s}$, $0\leq s\leq t$ and therefore it contains no point of $P$ in its interior.
The lemma follows since the edges $yx_{i}$ and $x_{i}u$ lie in a common face of $G\left(\Pi\right)$ and the ray $r\left(yx_{i}\right)$ meets the edge $uv$
at the point $u_{t}$.
∎
Let $N$ denote the set of arcs $\overrightarrow{uv}$ of $\overrightarrow{G\left(\Pi\right)}$ such that the edge $uv$ is not contractible from $u$
to $v$ in $\Pi$. For each $\overrightarrow{uv}\in N$ let $y=y_{uv}$, $x=x_{uv}$ and $u_{t}$ be as in Lemma 2. Since the edges $y_{uv}x_{uv}$ and $x_{uv}u$ lie in a common face of $G\left(\Pi\right)$ and the triangular
region, defined by $x_{uv}$, $u_{t}$ and $u$, contains no point of $P$ in its
interior, the geometric graph $G\left(\Pi\right)-x_{uv}u$ contains a face $Q_{x_{uv}u}$ in which $x_{uv}$ is a reflex vertex and therefore $\overrightarrow{x_{uv}u}\in A\left(\overrightarrow{G\left(\Pi\right)}\right)$. This defines a
function $f:N\longrightarrow A\left(\overrightarrow{G\left(\Pi\right)}\right)$ given by $f\left(\overrightarrow{uv}\right)=\overrightarrow{x_{uv}u}$.
Notice that the arcs $f\left(\overrightarrow{uv}\right)$ and $\overrightarrow{uv}$ form a directed path in $\overrightarrow{G\left(\Pi\right)}$ with length 2 and middle vertex $u$. This implies that if $f\left(\overrightarrow{u_{1}v_{1}}\right)=f\left(\overrightarrow{u_{2}v_{2}}\right)$, then $u_{1}=u_{2}$. Moreover, if $uv_{1},uv_{2}$ and $uv_{3}$ are distinct
arcs such that $f\left(\overrightarrow{uv_{1}}\right)=f\left(\overrightarrow{uv_{2}}\right)=f%
\left(\overrightarrow{uv_{3}}\right)=\overrightarrow{xu}$, then $u$ is adjacent in $G\left(\Pi\right)$ to $v_{1},v_{2},v_{3}$ and to $x$, which is not possible by Remark 1, since $u$ has
outdegree 3 in $\overrightarrow{G\left(\Pi\right)}$. It follows that
there are no three arcs in $N$ with the same image under the function $f$
and therefore $\left|\operatorname{Im}\left(f\right)\right|=\left|N\right|-\left|U\right|$,
where $U$ is the set of points $u$ of $P$ for which
there is a pair of arcs $\overrightarrow{uv},\overrightarrow{uw}\in N$ such
that $f\left(\overrightarrow{uv}\right)=f\left(\overrightarrow{uw}\right)$.
Lemma 3.
Let $\Pi$ be a convex decomposition of $P$ with no deletable edges. If $U\neq\emptyset$, then there
is a function $g:U\rightarrow A\left(\overrightarrow{G\left(\Pi\right)}\right)$ such that for each $u\in U$, $g\left(u\right)$ is not in the
image of the function $f$.
Proof.
Let $u\in U$ and let $v,w$ and $x=x_{uv}=x_{uw}$ be points in $P$ such that $f\left(\overrightarrow{uv}\right)=f\left(\overrightarrow{uw}\right)=%
\overrightarrow{xu}$. If $u$ has degree larger than 3 in $G\left(\Pi\right)$, let $z\notin\left\{v,w,x\right\}$ be such that $uz$ is an edge of $G\left(\Pi\right)$. By Remark 1, the outdegree of $u$ in $\overrightarrow{G\left(\Pi\right)}$ is at most 2, therefore $\overrightarrow{uz}$ is not an
arc of $\overrightarrow{G\left(\Pi\right)}$. It follows that $\overrightarrow{zu}$ must be an arc of $\overrightarrow{G\left(\Pi\right)}$.
In this case $g\left(u\right)=\overrightarrow{zu}$ $\notin\operatorname{Im}\left(f\right)$ since $z\neq x$ and $\overrightarrow{xu}$ is the unique arc in $Im(f)$ that ends at $u$.
If $u$ has degree 3 in $G\left(\Pi\right)$, then $u$ has outdegree 3 in $\overrightarrow{G\left(\Pi\right)}$, by Remark 1 and therefore $\overrightarrow{ux}$ is an arc $\overrightarrow{G\left(\Pi\right)}$. We
claim that in this case $g\left(u\right)=\overrightarrow{ux}$ $\notin\operatorname{Im}\left(f\right)$. Let $l_{ux}$ denote the line containing the edge
$ux$, and let $y_{uv}$ and $y_{uw}$ be points in $P$ and such that the rays $r\left(y_{uv}x\right)$ and $r\left(y_{uw}x\right)$ intersect the edges $uv$ and $uw$, respectively.
Without loss of generality we assume that $l_{ux}$ is a vertical line such
that $v$ and $y_{uw}$ lie to the left of $l_{ux}$ and $w$ and $y_{uv}$ lie
to the right of $l_{ux}$ (see Fig. 4). Clearly the angles $\measuredangle y_{uv}xu$ and $\measuredangle y_{uw}xu$ are smaller than $\pi$, it is easy
to see that $\measuredangle y_{uw}xy_{uv}$ is also smaller than $\pi$.
Therefore if $xz$ is an edge of $\Pi$ with $z\notin\left\{u,y_{uv},y_{uw}\right\}$, then $\overrightarrow{xz}$ is not an arc of $\overrightarrow{G\left(\Pi\right)}$. This implies that if $\overrightarrow{ux}\in\operatorname{Im}\left(f\right)$, then $\overrightarrow{ux}=f\left(\overrightarrow{xy_{uv}}\right)$ or $\overrightarrow{ux}=f\left(\overrightarrow{xy_{uw}}\right)$ since $f\left(\overrightarrow{a}\right)$
and $\overrightarrow{a}$ form a directed path of length 2 for each arc $\overrightarrow{a}\in N$.
Suppose $\overrightarrow{ux}=f\left(\overrightarrow{xy_{uv}}\right)$. By
the definition of $f$, there is an edge $y_{xy_{uv}}u$ such that the ray $r\left(y_{xy_{uv}}u\right)$ intersects the edge $xy_{uv}$. Since $v$ and $w$ are the only vertices different from $x$ which are adjacent to $u$ in $G\left(\Pi\right)$, one of them must be the vertex $y_{xy_{uv}}$. Since
both edges $uw$ and $xy_{uv}$ lie in the right halfplane defined by $l_{ux}$
then $r\left(wu\right)$ cannot intersect the edge $xy_{uv}$ and therefore $y_{xy_{uv}}\neq w$. Finally, since $r\left(y_{uv}x\right)$ intersects the
edge $uv$, $r\left(vu\right)$ cannot intersect the edge $xy_{uv}$.
Therefore $\overrightarrow{ux}\neq f\left(\overrightarrow{xy_{uv}}\right)$; analogously $\overrightarrow{ux}\neq f\left(\overrightarrow{xy_{uw}}\right)$.
∎
3. Main results
In this section we prove our main results.
Theorem 4.
Let $P$ be a set of points in general position in the plane. If $\Pi$ is a
convex decomposition of $P$ consisting of more than one polygon, then either
$\Pi$ contains a deletable edge or $\Pi$ contains a contractible edge.
Proof.
Assume the result is false and $\Pi$ contains no deletable edges and no
contractible edges. Define the directed graph $\overrightarrow{G\left(\Pi\right)}$ as in the previous section, notice that $A\left(\overrightarrow{G\left(\Pi\right)}\right)\neq\emptyset$ since $\Pi$ contains at least
two polygons. Since $\Pi$ contains no contractible edges, $N=A\left(\overrightarrow{G\left(\Pi\right)}\right)$.
Let $B=B\left(\overrightarrow{G\left(\Pi\right)}\right)$ be the set of
arcs of $\overrightarrow{G\left(\Pi\right)}$ of the form $\overrightarrow{uw}$, with $w$ in the boundary of $CH\left(P\right)$, and let $\overrightarrow{uw}\in B$. By Remark 1, $w$ has outdegree 0 in $\overrightarrow{G\left(\Pi\right)}$ which implies $\overrightarrow{uw}$ $\notin$ $\operatorname{Im}\left(f\right)$.
If $U=\emptyset$, then $\operatorname{Im}\left(f\right)\subset A\left(\overrightarrow{G\left(\Pi\right%
)}\right)\backslash B$, therefore $\left|N\right|=\left|\operatorname{Im}\left(f\right)\right|\leq\left|A\left(%
\overrightarrow{G\left(\Pi\right)}\right)\backslash B\right|\leq\left|A\left(%
\overrightarrow{G\left(\Pi\right)}\right)\right|-3$, which is not possible since $\Pi$ contains no deletable edges and $\left|B\right|\geq 3$.
And if $U\neq\emptyset$, by Lemma 3 no arc in $\operatorname{Im}\left(g\right)$ lies in $\operatorname{Im}\left(f\right)$, therefore $\operatorname{Im}\left(f\right)\subset A\left(\overrightarrow{G\left(\Pi\right%
)}\right)\backslash\left(\operatorname{Im}\left(g\right)\cup B\right)$. In this case $\left|\operatorname{Im}\left(f\right)\right|\leq\left|A\left(\overrightarrow{G%
\left(\Pi\right)}\right)\right|-\left|\operatorname{Im}\left(g\right)\right|-%
\left|B\right|$, since $g\left(u\right)\notin B$. This is a contradiction since $A\left(\overrightarrow{G\left(\Pi\right)}\right)=N$, $\left|\operatorname{Im}\left(g\right)\right|=\left|U\right|$, $\left|B\right|\geq 3$ and $\left|\operatorname{Im}\left(f\right)\right|=\left|N\right|-\left|U\right|$.
∎
Corollary 5.
Let $\Pi$ be a convex decomposition of a set of points $P$ in general
position in the plane. If $P$ contains at least one interior point, then $\Pi$ contains at least one contractible edge.
Proof.
Let $\Pi^{\prime}$ be a convex decomposition of $P$ obtained from $\Pi$
by removing deletable edges, one at a time, until no such edges remain, and
let $\overrightarrow{G\left(\Pi^{\prime}\right)}$ be the corresponding
directed abstract graph. Since $P$ contains an interior point, $\Pi^{\prime}$ contains at least one interior edge.
By Theorem 4, there is an arc $\overrightarrow{uv}\in A\left(\overrightarrow{G\left(\Pi^{\prime}\right)}\right)$ such that $uv$ is
contractible from $u$ to $v$ in $\Pi^{\prime}$. If $uv$ is not
contractible in $\Pi$, then by Lemma 1 there are edges $yx$ and $xu$ lying
in a common face of $G\left(\Pi\right)$ such that the ray $r\left(yx\right)$ meets the edge $uv$ at an interior point $u_{t}$ and that the
triangular region $yu_{t}u$ contains no point of $P$ in its interior. This
implies that the geometric graph $G\left(\Pi\right)-xu$ contains a face $Q_{x}$ in which $x$ is a reflex vertex and therefore $xu$ is not deletable in $\Pi$ and $\overrightarrow{xu}$ is an arc of $\overrightarrow{G\left(\Pi\right)}$.
Let $R$ be the face of $G\left(\Pi\right)$ which contains both edges $yx$
and $xu$. Since $\Pi^{\prime}$ is obtained from $\Pi$ by deleting edges
but no points, then there is a face $R^{\prime}$ of $G\left(\Pi^{\prime}\right)$ which contains the edge $xu$ and the region bounded by $R$, let $y^{\prime}\in P$ be such that $y^{\prime}x$ is an edge of $R^{\prime}$.
Notice that $y^{\prime}\neq y$ otherwise $uv$ could not be a contractible
edge of $\Pi^{\prime}$ because the ray $r\left(yx\right)$ meets the edge
$uv$ at the point $u_{t}$ (Fig. 5, left). Nevertheless, since the face $R^{\prime}$ contains the edge $xu$ and the region bounded by $R$, the ray $r\left(y^{\prime}x\right)$ also meets the edge $uv$ at an interior point $u_{t^{\prime}}$ (Fig. 5, right) which again is a contradiction.∎
Corollary 6.
Let $\Pi$ be a convex decomposition of a set of points $P$ in general
position in the plane and $Q$ be the set of points in the boundary of $CH\left(P\right)$. There is a sequence $P=P_{0},P_{1},\ldots,P_{m}=Q$ of
subsets of $P$, and a sequence $\Pi_{0},\Pi_{1},\ldots,\Pi_{m}$ of
convex decompositions of $P_{0},P_{1},\ldots,P_{m}$, respectively, such
that $\Pi_{0}=\Pi$, $\Pi_{m}$ consists of the boundary of $CH\left(P\right)$ and for $i=0,1,\ldots,k$, $\Pi_{i+1}$ is obtained from $\Pi_{i}$ by contracting an edge and for $i=k+1,k+2,\ldots,m-1$, $\Pi_{i+1}$ is
obtained from $\Pi_{i}$ by deleting an edge.
Proof.
By Corollary 5 , if $P_{i}$ contains interior points, then $\Pi_{i}$ has a
contractible edge. If $P_{i}$ contains no interior points, then each interior
edge of $\Pi_{i}$ is a deletable edge.
∎
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Simple Permutations Mix Even Better
Alex Brodsky
Department of Computer Science
University of Toronto
abrodsky@cs.toronto.edu
Shlomo Hoory111
Research is supported in part by an NSERC grant and a PIMS postdoctoral
fellowship.
Department of Computer Science
University of British Columbia
shlomoh@cs.ubc.ca
Abstract
We study the random composition of a small family of $O(n^{3})$
simple permutations on $\{0,1\}^{n}$.
Specifically we ask how many randomly selected simple permutations need be
composed to yield a permutation that is close to $k$-wise independent.
We improve on the results of Gowers [12] and
Hoory et al. [13] and show that up to a polylogarithmic factor,
$n^{2}k^{2}$ compositions of random permutations from this family suffice.
In addition, our results give an explicit construction of a degree
$O(n^{3})$ Cayley graph of the alternating group of $2^{n}$ objects with
a spectral gap $\Omega(2^{-n}/n^{2})$, which is a substantial improvement over
previous constructions.
Keywords: Mixing-time, k-wise independent permutations, cryptography,
multicommodity flow, reversible computation.
A naturally occurring question in cryptography is how well the composition
of simple permutations drawn from a simple distribution resembles a random
permutation.
Although such constructions are a common source of security for
block ciphers like DES and AES,
their mathematical justification (or lack thereof) is troubling.
This motivated the investigation of Hoory et al. [13] who considered
the notion of almost
$k$-wise independence. Namely, that the distribution obtained when
applying a permutation from a given distribution to any $k$ distinct
elements is almost indistinguishable from the distribution obtained when
applying a truly random permutation.
Therefore, the question is
how close is the composition of $T$ random simple permutations
to $k$-wise independent?
Another motivation is a fundamental open problem in the theory of expanding
graphs.
222A solution to this problem was announced recently by
Kassabov [15].
Namely, the problem of constructing a constant degree expanding
Cayley graph of the symmetric group.
A possible relaxation of this problem is to ask whether one can find a small
set of simple permutations such that its action on $k$ points yields an
expanding graph.
It turns out that these two problems reduce to bounding the mixing time and
the spectral gap of the random walk on the same graph.
This walk, $P$, is defined on the state space
of $k$-tuples of distinct elements from the $n$-dimensional binary cube.
In each step it randomly selects a simple permutation and
applies it to each of the $k$ elements at its current position.
The mixing time, $\tau({\epsilon})$, is the number of steps needed to
come ${\epsilon}$-close to the uniform distribution (in total variation distance),
and the spectral gap, $\mbox{gap}(P)$, is the difference between
the two largest eigenvalues of $P$’s transition matrix.
Following the construction of DES, and previous work by Gowers [12]
and Hoory et al [13],
we consider the class of width $2$ simple permutation, denoted ${\Sigma}$.
The action of such a permutation on an element of the $n$-dimensional
binary cube is to XOR a single coordinate with a Boolean function of $2$
other coordinates; there are $16n(n-1)(n-2)$ such
permutations.
These problems were first considered by Gowers [12] who gave an
$\tilde{O}(n^{3}k(n^{2}+k)(n^{3}+k))$
333Notation $\tilde{O}$ suppresses a polylogarithmic factor
in $n$ and $k$.
bound on the mixing time, by lower bounding the
spectral gap $1/\mbox{gap}(P)=\tilde{O}(n^{2}(n^{2}+k)(n^{3}+k))$.
Subsequently, Hoory et al. [13] improved the bound on the mixing
time to $\tilde{O}(n^{3}k^{3})$ by proving that $1/\mbox{gap}(P)=\tilde{O}(n^{2}k^{2})$.
Both results were achieved using the canonical paths technique,
and neither result applies for $k>2^{n/2}$.
Using the comparison technique, in conjunction with the theory of reversible
computation, we give better bounds for all values of $k$
up to the largest conceivable value, $k=2^{n}-2$.
Theorem 1.
$\tau({\epsilon})=\tilde{O}(n^{2}k^{2}\cdot\log(1/{\epsilon}))$,
as long as $k\leq 2^{n/50}$.
Theorem 2.
$1/\mbox{gap}(P)=O(n^{2}k)$
for all $k\leq 2^{n}-2$.
Using the well known connection between the mixing time and the
spectral gap Theorem 2 implies:
Corollary 3.
$\tau({\epsilon})=O(n^{2}k\cdot(nk+\log(1/{\epsilon})))$
for all $k\leq 2^{n}-2$.
The proofs of both Theorems are based on the comparison technique for
Markov chains [8].
To prove Theorem 2 we compare the random walk $P$
either to a Glauber dynamics Markov chain or to the random walk on
the alternating group using $3$-cycles.
To prove Theorem 1 we observe that
after a short preamble the random walk $P$ is almost surely in
a “generic” state.
Consequently, it suffices to bound the mixing time of a Markov chain
restricted to “generic” states.
To this end we again employ the comparison technique,
but with a better comparison constant.
In all cases we construct the multicommodity flows required
by the comparison technique using ideas from the
theory of reversible computation.
It follows from [13, 17] that these results
apply also in the more general setting of adaptive adversaries
(see references for a definition).
Corollary 4.
Let $T$ be the minimal number of random
compositions of independent and uniformly distributed permutations from
${\Sigma}$ needed to generate a permutation which is
${\epsilon}$-close to $k$-wise independent against an adaptive adversary.
Then $T=\tilde{O}(n^{2}k^{2}\cdot\log(1/{\epsilon}))$ for $k\leq 2^{n/50}$,
and $T=O(n^{2}k\cdot(nk+\log(1/{\epsilon})))$ for $k\leq 2^{n}-2$.
1 Preliminaries
Let $f$ be a random permutation on some base set $X$.
Denote by $X^{(k)}$ the set of all $k$-tuples of distinct elements from $X$.
We say that $f$ is ${\epsilon}$-close to $k$-wise independent if for every
$(x_{1},\ldots,x_{k})\in X^{(k)}$ the distribution of
$(f(x_{1}),\ldots,f(x_{k}))$
is ${\epsilon}$-close to the uniform distribution on $X^{(k)}$.
We measure the distance between two
probability distributions $p,q$ by the total variation distance, defined by
$$\displaystyle d(p,q)=\frac{1}{2}||p-q||_{1}=\frac{1}{2}\sum_{\omega}|p(\omega)%
-q(\omega)|=\max_{A}\sum_{\omega\in A}p(\omega)-q(\omega).$$
Assume a group $H$ is acting on a set $X$ and let $S$ be a subset of $H$
closed under inversion. Then the Schreier graph
$G=\mbox{sc}(S,X)$ is defined
by $V(G)=X$ and $E(G)=\{(x,xs):x\in X,\,s\in S\}$.
For a sequence $\omega=(s_{1},\ldots,s_{\ell})\in S^{l}$ we denote
$x\omega=xs_{1}\cdots s_{\ell}$, and we sometimes refer by $x\omega$ to
the walk $x,xs_{1},\ldots,xs_{1}\cdots s_{\ell}$.
The random walk $X_{0},X_{1},\ldots$
associated with a $d$-regular graph $G$ is defined by the
transition matrix $P_{vu}=\Pr[X_{i+1}=u|X_{i}=v]$ which is $1/d$ if
$(v,u)\in E(G)$ and zero otherwise.
The uniform distribution $\pi$ is stationary for this Markov chain.
If $G$ is connected and not bipartite, we know that given any initial
distribution of $X_{0}$, the distribution of $X_{t}$ tends to the uniform
distribution. The mixing time of $G$ is
$\tau({\epsilon})=\max_{v\in V(G)}\min\{t:d(P^{(t)}(v,\cdot),\pi)<{\epsilon}\}$,
where $P^{(t)}(v,.)$ is the probability distribution of $X_{t}$ given that
$X_{0}=v$.
It is not hard to prove (see [1, Lemma 20]) that
$$\displaystyle\tau(2^{-\ell-1})\leq\ell\cdot\tau(1/4).$$
(1)
Let $1=\beta_{0}\geq\beta_{1}\geq\cdots\geq\beta_{|V(G)|}$ be the eigenvalues
of the transition matrix $P$.
We say that this random walk is lazy if for some constant $\delta>0$ we have
$P_{vv}\geq\delta$ for all $v\in V(G)$.
We denote the spectral gap $1-\beta_{1}$ of the Markov chain $P$ by $\mbox{gap}(P)$.
Two fundamental results relating the spectral gap of a Markov chain to
its mixing time are the following:
Theorem 5.
([10, Proposition 3])
If the random walk on $G$ is lazy then
$\tau({\epsilon})=O\left(\log(|V(G)|/{\epsilon})\,/\,\mbox{gap}(P)\right).$
Theorem 6.
( [19, Proposition 1.ii]
or [1, Chapter 4])
For any time reversible Markov chain $P$ and ${\epsilon}>0$,
$\mbox{gap}(P)=\Omega(\log(1/2{\epsilon})\,/\,\tau({\epsilon})).$
2 Composing simple permutations
Another building block that we use are results on reversible computation
that enables us to compose simple permutations to construct permutations
that are easier to work with. A classical result of Coppersmith and
Grossman [6] is that for $n>3$ the set of width $2$ simple
permutations generates exactly the alternating group $A_{n}$. Thus, all
compositions must be even permutations.
Formally, we define the set of width $w$ simple permutations$,{\Sigma}_{w}$,
as the set of permutations
$f_{i,J,h}$ where $i\in[n]$, $J=\{j_{1},\ldots,j_{w}\}$ is a size $w$ ordered
subset of $[n]\setminus\{i\}$, and $h$ is a Boolean function on ${\{0,1\}}^{w}$.
The permutation $f_{i,J,h}$ maps
$(x_{1},\ldots,x_{n})\in{\{0,1\}}^{n}$ to
$(x_{1},\ldots,x_{i-1},x_{i}\oplus h(x_{j_{1}},\ldots,x_{j_{w}}),x_{i+1},\ldots%
,x_{n})$.
We are primarily interested in width $2$ simple permutations,
and denote ${\Sigma}={\Sigma}_{2}$.
Theorem 7.
(Barenco et al. [3])
The permutation that flips the $n$-th bit of input $x$ if and only if the first
$w$ bits of $x$ are $1$ can be implemented as a composition
of $O(w)$ permutations from ${\Sigma}$, as long as $w\leq n-2$.
Theorem 8.
(Brodsky [4])
for any distinct $x,y,z\in{\{0,1\}}^{n}$, one can compose $O(n)$
permutations from ${\Sigma}$ to obtain the $3$-cycle $(xyz)$.
A length $\ell$ implementation of the permutation $\sigma$ is a sequence
of permutations $\sigma_{1},\ldots,\sigma_{\ell}$ from ${\Sigma}$ whose composition
is $\sigma$.
Theorem 8 gives a length $O(n)$ implementation
for $3$-cycles.
We would like to use this implementation to construct a multicommodity flow
with low load on all edges. However, Theorem 8
does not guarantee this. We solve this problem by randomizing the
implementation, enabling us to prove a stronger theorem.
A length $\ell$ randomized implementation of the permutation
$\sigma$ is a sequence of random permutations
$\sigma_{1},\ldots,\sigma_{\ell}$ from ${\Sigma}$ whose composition is $\sigma$.
In Theorem 9 we give a randomized implementation
for 3-cycles, such that applying any prefix
$\sigma_{1}\cdots\sigma_{\ell^{\prime}}$ of the randomized implementation of a
uniformly random 3-cycle $(xyz)$ to $x$ yields a string that looks
random. Namely, its min-entropy $H_{\infty}(\cdot)$ is high, which
is the minimum amount of information revealed when exposing the value
of a random variable $X$, that is
$H_{\infty}(X)=\min_{\chi}(-\log_{2}(\Pr[X=\chi]))$.
Theorem 9.
Let $x,y,z\in{\{0,1\}}^{n}$ be uniformly distributed and distinct. Then there
is a length $L=O(n)$ randomized implementation $\sigma_{1}\cdots\sigma_{L}$
of the 3-cycle $(xyz)$ such that for all $\ell\in[L]$
the min-entropy of
$(x\sigma_{1}\cdots\sigma_{\ell-1},\sigma_{\ell})$ (which is a random variable on
${\{0,1\}}^{n}\times{\Sigma}$) is at least $\log_{2}(2^{n}\cdot n^{3})-O(1)$.
Note, this implies that the min-entropy of the marginals is big, i.e.,
$H_{\infty}(x\sigma_{1}\cdots\sigma_{\ell-1})\geq n-O(1)$
and $H_{\infty}(\sigma_{\ell})\geq\log_{2}(n^{3})-O(1)$.
3 Proof of Theorem 2
In order to prove that the composition of random permutations from ${\Sigma}$
approaches $k$-wise independence quickly we construct the Schreier graph
$G_{k,n}=\mbox{sc}({\Sigma},X^{(k)})$, where $X^{(k)}$ is the set of
$k$-tuple with $k$ distinct elements from the base set $X={\{0,1\}}^{n}$.
It is convenient to think of $X^{(k)}$ as the set of $k$ by $n$ binary
matrices with distinct rows.
A simple permutation acts on $X^{(k)}$ by acting on each of the rows.
Then $P$ is the transition matrix of the random walk on $G_{k,n}$.
We prove that the random walk on this graph mixes rapidly.
To prove that $1/\mbox{gap}(P)=O(n^{2}k)$,
we first observe that $\mbox{gap}(P)$ is monotone nonincreasing in $k$.
This follows from the fact that the graph $G_{k+1,n}$ is a lift
of $G_{k,n}$ and therefore inherits the spectrum of $G_{k,n}$.
To see this, observe that any eigenfunction of $G_{k,n}$, can be lifted
to an eigenfunction on $G_{k+1,n}$, where the value of the latter on
some $k+1$ by $n$ matrix is the value of the former on the matrix obtained by
deleting the last row. The eigenvalues of these two eigenfunctions is the
same.
In light of this observation, it is sufficient to prove the following two
lemmas:
Lemma 10.
$1/\mbox{gap}(P)=O(n^{2}\cdot 2^{n})$ for $k=2^{n}-2$.
Lemma 11.
$1/\mbox{gap}(P)=O(n^{2}k)$ for $k\leq 2^{n}/3$.
We obtain the lower bound on the spectral gap of $P$ using the comparison
technique [8]. This technique enables one to lower bound
$\mbox{gap}(P)$ by $\mbox{gap}(\tilde{P})/A$, where $\tilde{P}$ is some other Markov chain,
and $A$ is the comparison constant.
In our case, all chains are walks on regular graphs.
An upper bound on $A$ is obtained by constructing
a multicommodity flow on the underlying graph of $P$.
The flow flows a unit between
all pairs of endpoints of edges of $\tilde{P}$
such that the flow through each edge of $P$ is small.
To prove Lemmas 10 and 11,
we compare $P$ to two different Markov chains.
We start with the first Lemma.
Proof.
(of Lemma 10)
For $k=2^{n}-2$, the state space of $P$ comprises
all even permutations of ${\{0,1\}}^{n}$.
Let $\tilde{P}$ be a Markov chain on this state space, where
in each step
we pick three distinct elements of the cube $x,y,z\in{\{0,1\}}^{n}$
and perform the permutation $(xyz)$.
It follows from a result of Friedman [11],
that $1/\mbox{gap}(\tilde{P})=\Theta(2^{n})$.
Therefore, it is sufficient to prove that the comparison constant of $P$ to
$\tilde{P}$ is $O(n^{2})$.
444
Alternately, one can define a transition of $\tilde{P}$ as performing two random
transpositions (not necessarily disjoint) and use a result of Diaconis and
Shahshahani [9] that $1/\mbox{gap}(\tilde{P})=\Theta(2^{n})$.
To bound the comparison constant $A$, we need to construct a multicommodity
flow $f$ in $G_{k,n}$ that flows a unit between every two matrices $M,M^{\prime}$
such that $\tilde{P}(M,M^{\prime})>0$.
Since the chains $P$ and $\tilde{P}$ correspond to random walks on regular graphs
with degrees $d=\Theta(n^{3})$ and
$\tilde{d}=\Theta(2^{3n})$ respectively,
the formula given in [8, Theorem 2.3] reduces to:
$$\displaystyle A=(d/\tilde{d})\cdot\max_{(N,N^{\prime})\in E(G_{k,n})}\left\{%
\sum_{\gamma:\>(N,N^{\prime})\in\gamma}f(\gamma)\cdot|\gamma|\right\}.$$
(2)
Let $M,M^{\prime}$ be two matrices such that $\tilde{P}(M,M^{\prime})>0$. Then $M^{\prime}$ can be
obtained by applying some $3$-cycle $(xyz)$ to $M$. Recall that the
randomized implementation given by Theorem 9
induces a probability distribution on the length $L$ sequences of
permutations from $\Sigma$ whose composition is $(xyz)$. Such a
distribution naturally translates to a distribution on length $L$
paths from $M$ to $M^{\prime}$. We obtain a unit flow from $M$ to $M^{\prime}$ by
flowing through each such path $\gamma$ an amount equal to its
probability. We claim that the multicommodity flow obtained by
repeating this process for all $M,M^{\prime}$ pairs satisfying $\tilde{P}(M,M^{\prime})>0$
yields a small comparison constant.
Since $|\gamma|\cdot(d/\tilde{d})=\Theta(n\cdot|\Sigma|/2^{3n})$
for all paths $\gamma$ with non-zero flow, the problem of bounding the
sum in (2) reduces to bounding the total flow
through a given edge $e\in E(G_{k,n})$. Let
$\gamma=(M_{0},\ldots,M_{L})$ be a path from $M_{0}$ to $M_{L}$,
where $M_{L}$ is obtained from $M_{0}$ by applying the 3-cycle $(xyz)$.
Assume further that $\gamma$ goes
through the edge $e$ at the $\ell$-th step, and that $x$ is
the $r$-th row of $M$. For any of the $\Theta(2^{4n}\cdot n)$ possible
assignments to $x,y,z,\ell,r$, we can determine the distribution
of the $r$-th row of the matrices $M_{0},\ldots,M_{L}$. In particular, the
probability that $(M_{\ell-1},M_{L})$ is equal to $e$ is bounded by the
probability that they coincide in their $r$-th row. By
Theorem 9, in average over all assignments to
$x,y,z,\ell,r$, this probability is $O(1/2^{n}|\Sigma|)$.
Putting it all together yields that, up to a constant factor, the
comparison constant $A$ is bounded $(n\cdot|\Sigma|/2^{3n})\cdot(2^{4n}\cdot n)\cdot(1/2^{n}|\Sigma|)=n^{2}$, as claimed.
∎
Proof.
(of Lemma 11)
Let $\tilde{P}$ be the a Markov chain on the same state space as $P$,
which is the $k$ by $n$ binary matrices with distinct rows.
If the current state of $\tilde{P}$ is the matrix $M$,
then the next state is determined by picking
a row $r\in\{1,\ldots,k\}$ and setting it to a random new value that
is distinct from all other $k-1$ rows.
The process $\tilde{P}$ is the Markov chain of coloring the clique on
$k$ vertices with $2^{n}$ colors defined in [14, section 4.1].
Proposition 4.5 therein bounds its mixing time by
$\tilde{\tau}({\epsilon})=O(k\log(k/{\epsilon}))$ as long as $k\leq 2^{n}/3$.
Setting ${\epsilon}=1/4k$ in Theorem 6 implies that
$\mbox{gap}(\tilde{P})=\Omega(1/k)$.
Therefore, as in the proof of Lemma 10,
it is sufficient to prove that the comparison constant of $P$ to $\tilde{P}$
is $O(n^{2})$.
Given matrices $M,M^{\prime}$ such that $\tilde{P}(M,M^{\prime})>0$, we know that $M^{\prime}$ is obtained
from $M$ by changing the value of some row $r$ from $x$ to $y$.
To construct paths from $M$ to $M^{\prime}$, we note that $M^{\prime}$ can be obtained by
applying the 3-cycle $(xyz)$ to $M$
for any $z\in{\{0,1\}}^{n}$ that is distinct from all rows of $M,M^{\prime}$.
We choose $z$ at random from the $2^{n}-(k+1)$ allowed values.
As in the proof of Lemma 10, the randomized implementation
of $(xyz)$, given by Theorem 9, defines
a distribution on paths from $M$ to $M^{\prime}$ and therefore a multicommodity
flow. We turn to bound the comparison constant, given by (2).
As before, $|\gamma|\cdot(d/\tilde{d})=\Theta(n\cdot|\Sigma|/k2^{n})$
for all $\gamma$ with non-zero flow, and it suffices to bound the flow through
some edge $e\in E(G_{k,n})$. We enumerate over the choices of the
position $\ell$, row $r$ and distinct $x,y$, which make a total of
$\Theta(nk2^{2n})$ possible values.
Again we apply Theorem 9 to argue that in average,
the probability of agreement with $e$ is bounded by $O(1/|\Sigma|2^{n})$.
555
One should note that $z$ is uniformly distributed only over
$2^{n}-(k+1)>2^{n-1}$ values. However, this is equivalent to conditioning a
uniform $z$ on an event with probability at least half and therefore
(by Lemma 19) can only increase the probability of agreement
with $e$ by a factor of two.
Therefore, up to a constant factor,
$A=(n\cdot|\Sigma|/k2^{n})\cdot(nk2^{2n})\cdot(1/|\Sigma|2^{n})=n^{2}$,
as claimed.
∎
4 Proof of Theorem 1
In light of inequality (1), it is sufficient to
prove that $\tau(1/4)=\tilde{O}(n^{2}k^{2})$.
The outline of the proof is the following.
We start by introducing the notion of a generic matrix,
and as suggested by the name, most matrices are generic.
The proof then proceeds by arguing that after a short random walk
almost surely all matrices encountered are generic.
Therefore, it is sufficient to bound the mixing time of a walk that is
restricted to generic matrices.
For such a walk, we can compare the chain to a chain defined only on
generic matrices and achieve a much smaller comparison constant.
This yields the desired bound, $\tilde{O}(n^{2}k^{2})$.
Let $w=10\cdot(\log k+\log n)$. By assumption, we have $w\leq n/4$
for a sufficiently large $n$, and we set $p=\lceil n/2w\rceil$.
Let $C_{1},\ldots,C_{p},C$ be a partition of $[n]$ such that
$|C_{i}|=w$ for $i=1,\ldots,p$ and $|C|=n-pw$.
Consequently, $n/4\leq n/2-w<|C|\leq n/2$.
We say that a $k$ by $n$ matrix is generic, if for all $j\in[p]$,
its restriction to $C_{j}$ has distinct rows.
It is not difficult to check that a uniformly distributed matrix $M$ is
almost surely generic.
In fact, it is sufficient that the rows of $M$ are $2^{-w}$-close
to $2$-wise independent, since then the probability that
$M$ is not generic is bounded by $p$ times the probability that the restriction
of $M$ to $C_{j}$ doesn’t have distinct rows.
This yields the bound
$p\cdot\binom{k}{2}\cdot(2\cdot 2^{-w})=o(1/n^{3}k^{3})$ and
implies the following lemma:
Lemma 12.
If the rows of a random $k$ by $n$ matrix $M$ are $2^{-w}$-close to $2$-wise
independent, then $M$ is generic with probability $1-o(1/n^{3}k^{3})$.
It follows from a result of Chung and Graham about the mixing time
of the “Aldous Cube” [5], that the number of steps needed
to come close to $2$-wise independence, which is the same as the mixing time of
$G_{2,n}$, is $O(n\log n)$.
This is stated in the following lemma (whose proof is deferred to
Section 6).
Lemma 13.
For all $w\geq 1$ the ${\epsilon}$ mixing time of the Schreier graph
$\mbox{sc}({\Sigma}_{w},X^{(2)})$ is $O(n\log n\log(1/{\epsilon}))$.
Therefore, the matrix obtained after
$T_{1}=O(n\log n\cdot w)=O(n\log n\cdot(\log k+\log n))$
steps is $2^{-w}$-close to $2$-wise independent,
and by Lemma 12 it is
generic with probability $1-o(1/n^{3}k^{3})$.
This implies that if we proceed by $T_{2}=O(n^{3}k^{3})$ steps,
then all $T_{2}$ matrices encountered are generic
with probability $1-o(T_{2}/n^{3}k^{3})>1-{\epsilon}_{1}$, for any fixed ${\epsilon}_{1}>0$ and
sufficiently large $n$.
We introduce a new Markov chain $P^{\prime}$.
The state space of $P^{\prime}$ consists of all generic $k$ by $n$ matrices.
If the chain is currently at the matrix $M$, then the next state is determined
as follows. We pick a uniformly distributed simple permutation $\sigma\in{\Sigma}$.
If $M\sigma$ is generic, we move to $M\sigma$. Otherwise, we remain at $M$.
Let $\tau^{\prime}({\epsilon})$ denote the ${\epsilon}$-mixing time of $P^{\prime}$,
and require that $T_{2}\geq\tau^{\prime}({\epsilon}_{2})$ for some fixed ${\epsilon}_{2}>0$.
We claim that as long as $2{\epsilon}_{1}+{\epsilon}_{2}<1/4$ the mixing time of $P$ can be
bounded by $\tau(1/4)\leq T_{1}+T_{2}$.
To see this,
let $M$ be some $k$ by $n$ matrix with distinct rows, and consider following
two matrices.
The first matrix $M^{\prime}$ obtained when starting at $M$ and walking $T_{1}+T_{2}$
steps using $P$.
The second matrix $M^{\prime\prime}$ is defined as follows.
Let $\hat{M}$ be the matrix obtained
when starting at $M$ and performing $T_{1}$ steps of $P$.
If $\hat{M}$ is not generic, we set $M^{\prime\prime}=\hat{M}$.
Otherwise, $M^{\prime\prime}$ is the matrix reached by the length $T_{2}$ walk using $P^{\prime}$
that starts at $\hat{M}$.
We claim that $d(M^{\prime},M^{\prime\prime})\leq{\epsilon}_{1}$ and that $M^{\prime\prime}$ is
$({\epsilon}_{1}+{\epsilon}_{2})$-close to the uniform distribution over $k$ by $n$ matrices
with distinct rows
666Note that by our assumptions, the distance between the uniform
distribution over matrices with distinct rows and generic matrices is $o(1)$.
Proving those claims will imply that
$$\displaystyle\tau(1/4)\leq\tau^{\prime}({\epsilon}_{2})+O(n\log n\cdot(\log k+%
\log n)),$$
(3)
as long as $\tau^{\prime}({\epsilon}_{2})=O(n^{3}k^{3})$.
We start by checking that indeed $d(M^{\prime},M^{\prime\prime})\leq{\epsilon}_{1}$. It is convenient to
think of the two length $T_{1}+T_{2}$ walks from $M$ to $M^{\prime}$ and $M^{\prime\prime}$ as
defined over the same probability space ${\Sigma}^{T_{1}+T_{2}}$ which is the choice
of a simple permutation in each of the $T_{1}+T_{2}$ steps. Denote the
the $P$-walk by $(M_{0}=M,M_{1},\ldots,M_{T_{1}+T_{2}}=M^{\prime})$.
Then, if all the matrices $M_{T_{1}},\ldots,M_{T_{1}+T_{2}}$ are generic,
it coincides with the walk leading to $M^{\prime\prime}$, and in particular we have
$M^{\prime}=M^{\prime\prime}$. By the previous arguments, this event happens at least
with probability $1-{\epsilon}_{1}$, implying that $d(M^{\prime},M^{\prime\prime})\leq{\epsilon}_{1}$.
The proof that $M^{\prime\prime}$ is $({\epsilon}_{1}+{\epsilon}_{2})$-close to uniform is more delicate.
We know that the matrix $\hat{M}$ is generic with probability at least
$1-{\epsilon}_{1}$.
Also, since $T_{2}\geq\tau^{\prime}({\epsilon}_{2})$, we know that conditioned on $\hat{M}$
being generic, $M^{\prime\prime}$ is ${\epsilon}_{2}$-close to the uniform distribution.
Therefore $M^{\prime\prime}$ is $({\epsilon}_{1}+{\epsilon}_{2})$-close to the uniform distribution over
matrices with distinct rows. This argument can be easily formalized using
Lemma 18 of Section 6.
We are left with the proof of the following lemma.
Lemma 14.
$\tau^{\prime}(1/4)=\tilde{O}(n^{2}k^{2})$.
To bound the mixing time of the Markov chain $P^{\prime}$, we apply the comparison
technique [8]. We compare $P^{\prime}$ to the Markov chain $\tilde{P}$ defined
on the same state space, the $k$ by $n$ generic matrices.
Given that $\tilde{P}$ is at a matrix $M$, we determine the next state as follows.
With probability half we pick a random column $c\in C$ and row $r\in[k]$
and flip the corresponding bit with probability half.
Otherwise, we pick at random an index $i\in[p]$, a row $r\in[k]$ and
a string $\alpha\in{\{0,1\}}^{w}$ that is distinct from all other $k-1$ rows in
the restriction of $M$ to the columns $C_{i}$. We set the bits at row $r$
and columns $C_{i}$ to $\alpha$.
Consequently, the following two lemmas,
imply Lemma 14. Note that we need not worry about the
smallest eigenvalue of $P^{\prime}$ since a random permutation from ${\Sigma}$ is the
identity with probability $1/16$.
Lemma 15.
$\mbox{gap}(\tilde{P})=\Omega(1/nk)$.
Lemma 16.
The comparison constant $A$ of $\tilde{P}$ to $P^{\prime}$ satisfies $A=\tilde{O}(1)$.
Proof.
(of Lemma 15)
Consider two Markov chains $\tilde{P}_{1}$ and $\tilde{P}_{2}$:
1.
The state space of $\tilde{P}_{1}$ are the $k$ by $w$ binary matrices with distinct
rows. At each step one chooses a random row and sets it to a random new
value distinct from all other $k-1$ rows.
This chain is exactly the coloring chain of a clique on $k$ vertices
with $2^{w}$ colors of [14, Proposition 4.5],
and as in the proof of Lemma 11,
it satisfies $\mbox{gap}(\tilde{P}_{1})=\Omega(1/k)$.
2.
$\tilde{P}_{2}$ is the random walk on the $(n-wp)\cdot k$ dimensional binary cube,
where in each step with probability half,
one flips a random coordinate. Therefore, $\mbox{gap}(\tilde{P}_{2})=\Omega(1/nk)$.
One can think of the chain $\tilde{P}$ as the product of $p$ copies of $\tilde{P}_{1}$ and
one copy of $\tilde{P}_{2}$. Indeed the state space of $\tilde{P}$ is the direct product of
the $p+1$ state spaces. Moreover, a step of $\tilde{P}$ performs a move of $\tilde{P}_{2}$
with probability $1/2$ and otherwise performs the move in a randomly selected
copy of $\tilde{P}_{1}$.
It is straight forward to check that the spectral gap of $\tilde{P}$ is
$\min(\mbox{gap}(\tilde{P}_{1})/p,\mbox{gap}(\tilde{P}_{2}))/2$, implying the desired bound.
∎
Proof.
(of Lemma 16)
Let $G^{\prime}$ be the underlying graph of $P^{\prime}$.
The vertices of $G^{\prime}$ are the generic $k$ by $n$ matrices,
and $(N,N^{\prime})$ is an edge of $G^{\prime}$ if $P^{\prime}(N,N^{\prime})>0$.
To bound the comparison constant $A$, we need to construct a multicommodity
flow $f$ in $G^{\prime}$ that flows a unit between every two matrices $M,M^{\prime}$
such that $\tilde{P}(M,M^{\prime})>0$.
The chains $P^{\prime}$ and $\tilde{P}$ correspond to random walks on regular graphs
with degrees $d^{\prime}=\Theta(n^{3})$, $\tilde{d}=\Theta(kn2^{w}/w)$ respectively,
and as before the comparison constant $A$ is defined by (2).
To build a path $\gamma$ from $M$ to $M^{\prime}$ we need to distinguish two types
of $\tilde{P}$ transitions.
Type (i) flips the bit at row $r$ and column $c\in C$.
Type (ii) changes the bits at row $r$ and columns
$C_{i}$ from $\alpha$ to $\alpha^{\prime}$.
We start by constructing the type (i) paths.
Let $j\in[p]$ be a random index, and let $\beta\in{\{0,1\}}^{w}$ be the
restriction of the $r$-th row of $M$ to $C_{j}$.
Also let $S$ be a random sequence of $w-1$ distinct elements from
$C\setminus\{c\}$.
The unit flow from $M$ to $M^{\prime}$ is along paths $\gamma=\gamma_{M,M^{\prime}}^{S,j}$.
Each such path is defined by composing simple permutations from ${\Sigma}$
to achieve the permutation that acts on $x\in{\{0,1\}}^{n}$ by flipping coordinate
$c$ if the restriction of $x$ to $C_{j}$ is $\beta$.
Clearly such a permutation maps $M$ to $M^{\prime}$.
We follow the method of Barenco et al. [3] to build
an AND gate with $w$ inputs.
This gate inverts its output bit (the coordinate $c$) if its $w$ inputs (the
coordinates $C_{j}$) have some fixed value $\beta$.
The coordinates in the set $S$ are used as “scratch”.
Let $C_{j}=\{j_{1},\ldots,j_{w}\}$, $S=\{s_{1},\ldots,s_{w-1}\}$ and
$\beta=(b_{1},\ldots,b_{w})$.
Let $\sigma_{1}$ be the simple permutation that flips coordinate
$s_{1}$ of $x\in{\{0,1\}}^{n}$ if $x_{j_{1}}$ is equal to $b_{1}$,
and let $\sigma_{\ell}$ for $2\leq\ell\leq w-1$ be the simple permutation
that flips coordinate $s_{\ell}$ if $x_{s_{\ell-1}}$ is one and $x_{j_{\ell}}$
is equal to $b_{\ell}$. Also, we denote by $\tau_{c}$ the simple permutation that
flips $x_{c}$ if $x_{s_{w-1}}$ is one and $x_{j_{w}}$ is equal to $b_{w}$.
We claim that the following permutation
flips coordinate $c$ of $x\in{\{0,1\}}^{n}$ if the restriction of $x$ to $C_{j}$ is
equal to $\beta$:
$$\displaystyle\sigma=(\tau_{c}\sigma_{w-1}\cdots\sigma_{2}\sigma_{1}\sigma_{2}%
\cdots\sigma_{w-1})^{2}$$
To see this, one checks by induction that
$\sigma_{\ell}\cdots\sigma_{1}\cdots\sigma_{\ell}$ flips coordinate $s_{\ell}$ if
$x_{j_{1}},\ldots,x_{j_{\ell}}$ is equal to $b_{1},\ldots,b_{\ell}$.
For the type (ii) paths, we need to change the bits at row $r$ and
columns $C_{i}$ from $\alpha$ to $\alpha^{\prime}$.
The problem is that if we change $\alpha$ to $\alpha^{\prime}$ bit by bit,
as suggested by the construction of type (i) paths,
we might violate row distinctness.
To solve this problem, we start our path by applying a length
$L=O(w\log w\cdot(1+2\log k))$ sequence $\phi$ of simple permutations with
indices restricted to $C_{i}$.
Let $\hat{M}=M\phi$ and $\hat{M^{\prime}}=M^{\prime}\phi$, and let $C_{i}^{\prime}$ and $C_{i}^{\prime\prime}$ be the first
and last $\lfloor(w-1)/2\rfloor$ columns of $C_{i}$.
We say that $\phi$ is valid if for both the restriction of $\hat{M}$ to $C_{i}^{\prime\prime}$ and
for the restriction of $\hat{M^{\prime}}$ to $C_{i}^{\prime}$, have distinct rows.
By Lemma 13 we know for a random $\phi$, both $\hat{M}$
and $\hat{M^{\prime}}$ are $1/8k^{2}$-close to $2$-wise independence.
Therefore, a random $\phi$ is not valid with probability bounded by
$k^{2}\cdot(2^{-w/2+1}+1/8k^{2})\leq 1/4$.
If $\phi$ is valid we define a path $\gamma=\gamma_{M,M^{\prime}}^{S,j,\phi}$
from $M$ to $M^{\prime}$, where $j\in[p]\setminus\{i\}$ and $S$ is
a length $w-1$ sequence of elements from $C$.
The path is prefixed by $\phi$ to get from $M$ to $\hat{M}$
and is suffixed by $\phi^{-1}$ to get from $\hat{M^{\prime}}$ to $M^{\prime}$.
Let $\hat{\alpha}$ and $\hat{\alpha}^{\prime}$ be the restriction of the $r$-th row
of $\hat{M}$ and $\hat{M^{\prime}}$ to $C_{i}$ respectively,
and let $\beta$ be the restriction of the $r$-th row of $M$ to $C_{j}$.
Then the middle path connecting $\hat{M}$ to $\hat{M^{\prime}}$ is defined as follows:
$$\displaystyle\sigma=[(\prod_{\{c\in C_{i}^{\prime}\,:\,\hat{\alpha}_{c}\neq%
\hat{\alpha}^{\prime}_{c}\}}\tau_{c})\cdot\sigma_{w-1}\cdots\sigma_{2}\sigma_{%
1}\sigma_{2}\cdots\sigma_{w-1}]^{2}\cdot[(\prod_{\{c\in C_{i}\setminus C_{i}^{%
\prime}\,:\,\hat{\alpha}_{c}\neq\hat{\alpha}^{\prime}_{c}\}}\tau_{c})\cdot%
\sigma_{w-1}\cdots\sigma_{2}\sigma_{1}\sigma_{2}\cdots\sigma_{w-1}]^{2},$$
where $\tau_{c}$ and $\sigma_{\ell}$ are as defined for the type (i) sequences.
Therefore it is guaranteed that the matrices encountered
along the first and second half of the sequence agree with
$\hat{M}$ on the columns $C_{i}^{\prime\prime}$ and with
$\hat{M^{\prime}}$ on the columns $C_{i}^{\prime}$ respectively.
Since $\phi$ is valid, this implies that we never attempt to move to
a non-generic matrix throughout the entire path.
We define the unit flow from $M$ to $M^{\prime}$ by splitting the flow uniformly
between all valid paths $\gamma$ designated by $S,j,\phi$.
There are two points that need special attention in the constructed type (i)
and type (ii) paths.
The first point is that all indices of the simple permutations used in $\phi$
are in $C_{i}$. This is unacceptable for us, as it induces an undue load on
a small subset of ${\Sigma}$. To solve this problem we replace
each simple permutation used in $\phi$ by a constant length sequence that
avoids that problem.
For example, the permutation that flips coordinate $i_{1}$ if $i_{2}$ and $i_{3}$
are $1$, denoted $\chi_{i_{1},i_{2},i_{3}}$, is replaced by the sequence
$(\chi_{s_{2},s_{1},i_{3}}\chi_{s_{1},i_{2}},\chi_{s_{2},s_{1},i_{3}},\chi_{i_{%
1},s_{2}})^{2}$
where permutation $\chi_{i_{1},i_{2}}$ XORs coordinate $i_{1}$ with $i_{2}$.
The second point is that some of the simple permutations used
($\sigma_{1}$ and some of the permutations in $\phi$)
do not use three indices. However, in the definition of ${\Sigma}$,
we have three indices at our disposal even if we don’t use all three.
We use this to guarantee that all simple permutations used have one index in
$C_{j}$ and two from $S$ or $c$ for type (i) paths or $C_{i}$ for type (ii) paths.
To complete the proof, we have to bound the comparison constant $A$
given by (2).
We have $d^{\prime}/\tilde{d}=\theta(n^{2}w/k2^{w})$ and $|\gamma|=O(L)$.
Also, $f(\gamma)$ is $\Theta(w/n(m)_{w-1})$ for type (i) paths
and $\Theta(w/|{\Sigma}^{(w)}|^{L}n(m)_{w-1})$ for type (ii) paths,
where we denote $m=|C|$,
$(m)_{q}=m(m-1)(m-2)\cdots(m-q+1)$,
and ${\Sigma}^{(w)}$ as the width $2$ simple permutations restricted to the
$w$-dimensional cube.
Therefore, we only have to bound the maximal number of $\gamma_{M,M^{\prime}}^{S,j}$
and $\gamma_{M,M^{\prime}}^{S,j,\phi}$ paths through an edge $(N,N^{\prime})$.
We start with type (i) paths.
The first step is to extract as much information as possible about a path
$\gamma$ through $(N,N^{\prime})$ by considering the simple permutation $s$ associated
with $(N,N^{\prime})$.
Note first that $s$ determines $j$.
Moreover, since only one of $\sigma_{1},\ldots,\sigma_{w-1}$ and
$\tau_{c}$ can be equal to $s$, any path $\gamma$ using $s$,
must use it in one of $O(1)$ possible positions.
Since a permutation $\sigma_{\ell}$ for $\ell\in[w-1]$ or $\tau_{c}$ determines
two indices of $S,c$
there are only $\Theta((m)_{w-2})$ choices for $S,c$ that are
consistent with $s$.
The last thing still needed to reconstruct $\gamma$ is the string
$\beta\in{\{0,1\}}^{w}$. Since the columns $C_{j}$ are not modified throughout the
entire sequence, $\beta$ must be the restriction of some row of $N$ to $C_{j}$,
limiting $\beta$ to one of $k$ possible values.
Therefore, the total number of type (i) paths through $(N,N^{\prime})$
is $O(k\cdot(m)_{w-2})$,
and the contribution of the type (i) sequences to $A$ is:
$$\displaystyle A_{(i)}=O(\overset{d^{\prime}/\tilde{d}}{\overbrace{(n^{2}w/k2^{%
w})}}\cdot\overset{f(\gamma)\cdot|\gamma|}{\overbrace{(Lw/(m)_{w})}}\cdot%
\overset{\mbox{choices for }j,S,c,\beta\mbox{ and position}}{\overbrace{(k%
\cdot(m)_{w-2})}})=O(Lw^{2}/2^{w})=o(1).$$
For type (ii) paths we distinguish the cases where $(N,N^{\prime})$ is
in the first middle or last sections of a path
$\gamma_{M,M^{\prime}}^{S,j,\phi}$.
Consider the first section (and similarly the last).
We enumerate over possible positions $\ell\in[L]$.
Then we know two indices of the
sequence $S$ and one of the $3L$ indices in $C_{i}$ that where used by $\phi$.
Therefore, we have $L\cdot(m)_{w-3}\cdot|{\Sigma}^{(w)}|^{L}/w$ possible values for
$S,i,\phi$ and the position.
This enables us to determine $M$ and $\hat{M}$. We still have to determine the
row $r$, the two strings $\alpha,\alpha^{\prime}$ and the index $j$
which have $O(kn2^{w}/w)$ possibilities.
Therefore the contribution of the first and last sections of type (ii)
paths is:
$$\displaystyle A_{(ii.\mbox{first,last})}$$
$$\displaystyle=$$
$$\displaystyle O(\overset{d^{\prime}/\tilde{d}}{\overbrace{(n^{2}w/k2^{w})}}%
\cdot\overset{f(\gamma)\cdot|\gamma|}{\overbrace{(Lw/|{\Sigma}^{(w)}|^{L}(m)_{%
w})}}\cdot\overset{\mbox{choices for }j,S,i,\phi,\alpha,\alpha^{\prime},\beta%
\mbox{ and position}}{\overbrace{(k2^{w}L\cdot(m)_{w-2}\cdot|{\Sigma}^{(w)}|^{%
L}/w^{2})}})$$
$$\displaystyle=$$
$$\displaystyle O(L^{2})=O(w^{2}\log^{2}w\cdot(1+\log k)^{2}).$$
For the middle section of type (ii) paths, as for the type (i) argument,
given $(N,N^{\prime})$ we first determine the position up to $O(1)$ possible choices.
Then we determine the index $i$ or $j$ and two indices from $S$,
then we have $O((m)_{w-2}\cdot|{\Sigma}^{(w)}|^{L}/w)$ possibilities for
$i,j,S,\phi$.
Also we have $k2^{w}$ choices for the row and the strings $\beta,\alpha$ and
$\alpha^{\prime}$.
Therefore,
$$\displaystyle A_{(ii.\mbox{middle})}=O(\overset{d^{\prime}/\tilde{d}}{%
\overbrace{(n^{2}w/k2^{w})}}\cdot\overset{f(\gamma)\cdot|\gamma|}{\overbrace{(%
Lw/|{\Sigma}^{(w)}|^{L}(m)_{w})}}\cdot\overset{\mbox{choices for }j,S,i,\phi,%
\alpha,\alpha^{\prime},\beta\mbox{ and position}}{\overbrace{k2^{w}\cdot(m)_{w%
-2}\cdot|{\Sigma}^{(w)}|^{L}/w}})=O(Lw).$$
∎
5 Proof of Theorem 9
First, we describe the randomized implementation of a $3$-cycle
$(xyz)$ using the simple permutations in $\Sigma$. Second, we show
that this randomized implementation satisfies the statement of the
theorem. The randomness is introduced into the implementation of
$(xyz)$ by using a permutation $\phi\in S_{n}$ and two vectors $v_{4},v_{5}$.
Let $\phi$ be some permutation of the $n$ coordinates.
If $\omega=\sigma_{1}\cdots\sigma_{L}$ implements $(x\phi,y\phi,z\phi)$,
then $\omega^{\phi}$ is an implementation of $(xyz)$,
where $\omega^{\phi}=\phi\omega\phi^{-1}$ is the conjugation of
$\omega$ with $\phi$, i.e. the conjugation each of the permutations
$\sigma_{i}$ used in $\omega$. Note that the set $\Sigma$ of simple
permutations is closed under conjugation by permutations from $S_{n}$, because
this just relabels the indices.
For a vector $v\in{\{0,1\}}^{n}$, we denote the first $n-2$ bits of $v$ by $v^{\prime}\in{\{0,1\}}^{n-2}$ and the last two bits of $v$ by $v^{\prime\prime}\in{\{0,1\}}^{2}$, i.e.,
$v=v^{\prime}v^{\prime\prime}$. We call the last two bits the control bits.
For convenience, the notation
$v^{\prime}00$, $v^{\prime}01$, $v^{\prime}10$, and $v^{\prime}11$ denotes bit vectors comprising
the first $n-2$ bits of $v$ and the control bits $00$, $01$, $10$, and
$11$, respectively. Let $(v)_{j}$ denote the $j$-th bit of a vector $v$.
Finally, let $v_{1}=x\phi$, $v_{2}=y\phi$ and $v_{3}=z\phi$.
If $v^{\prime}_{1}$ is equal to $v^{\prime}_{2}$ or to $v^{\prime}_{3}$ then we say that $\phi$ is
invalid.
This can only occur if $x$, $y$, or $z$ are less than Hamming distance
$3$ apart and $\phi$ maps all indices on which $x$
and $y$ (or $z$) differ to the control indices.
For the rest of the description we assume that $\phi$ is valid.
Let $v_{4},v_{5}\in{\{0,1\}}^{n}$ be two additional vectors satisfying the validity
requirement of being at least Hamming distance $3$ from each other and from
the former three vectors.
Observe that $(v_{1},v_{2},v_{3})=\psi_{1}\psi_{2}$ where
$\psi_{1}=(v_{1},v_{2})(v_{4},v_{5})$ and $\psi_{2}=(v_{1},v_{3})(v_{4},v_{5})$.
Therefore it suffices to implement the two double transpositions
$\psi_{1}$ and $\psi_{2}$. These are implemented in an identical manner.
Each implementation is divided into 15 blocks: a core block, which
implements the permutation ${\rho_{\mathit{core}}}=(v^{\prime}_{5}00,v^{\prime}_{5}01)(v^{\prime}_{5}10,v^%
{\prime}_{5}11)$,
and seven block pairs conjugating it.
The first four of these blocks, called $\pi$-blocks ensure that the
control bits of each of the four vectors are distinct. Specifically, $v^{\prime}_{i}v^{\prime\prime}_{i}$ is mapped to $v^{\prime}_{i}c_{i}$, where $c_{1}=00$, $c_{2}=c_{3}=01$, $c_{4}=10$ and $c_{5}=11$. If $v^{\prime\prime}_{i}=c_{i}$ then the corresponding block,
labeled $\pi_{i}$ performs a nop. Otherwise, block $\pi_{i}$ performs the
permutation $(v^{\prime}_{i}v^{\prime\prime}_{i},v^{\prime}_{i}c_{i})(v^{\prime}_{i}a_{i},v%
^{\prime}_{i}b_{i})$ where
$\{a_{i},b_{i}\}={\{0,1\}}^{2}\backslash\{v^{\prime\prime}_{i},c_{i}\}$.
The remaining three blocks, called $\tau$-blocks, map $v^{\prime}_{1}$, $v^{\prime}_{2}$ (or
$v^{\prime}_{3}$), and $v^{\prime}_{4}$ to $v^{\prime}_{5}$, using the control bits to distinguish
between the four vectors. Block $\tau_{i}$ performs the permutation
$\tau_{i}=\prod_{v^{\prime}\in{\{0,1\}}^{n-2}}(v^{\prime}c_{i},u^{\prime}c_{i})$, where $u^{\prime}=v^{\prime}\oplus v^{\prime}_{i}\oplus v^{\prime}_{5}$. Since it can easily be checked that
$\tau_{i}=\tau_{i}^{-1}$, that $\pi_{i}=\pi_{i}^{-1}$, and that
$$\pi_{1}\pi_{2}\pi_{4}\pi_{5}\tau_{1}\tau_{2}\tau_{4}{\rho_{\mathit{core}}}\tau%
_{4}\tau_{2}\tau_{1}\pi_{5}\pi_{4}\pi_{2}\pi_{1}=\psi_{1}\mathrm{\ \ \ and\ \ %
\ }\pi_{1}\pi_{3}\pi_{4}\pi_{5}\tau_{1}\tau_{3}\tau_{4}{\rho_{\mathit{core}}}%
\tau_{4}\tau_{3}\tau_{1}\pi_{5}\pi_{4}\pi_{3}\pi_{1}=\psi_{2},$$
we need only describe the implementation of each of these blocks.
Each of the blocks is implemented using $O(n)$ simple permutations. Each
$\tau$-block is implemented by concatenating $n-2$ simple permutations,
where for $j=1\cdots n-2$, the $j$-th simple permutation is the
identity if $(v^{\prime}_{i})_{j}=(v^{\prime}_{5})_{j}$, and otherwise flips the $j$-th bit of
vector $v$ if $v^{\prime\prime}=c_{i}$.
The implementation of the ${\rho_{\mathit{core}}}$ and $\pi$ blocks is more involved.
Permutation ${\rho_{\mathit{core}}}$ flips bit $(v^{\prime\prime})_{2}$ if and only if $v^{\prime}=v^{\prime}_{5}$.
Barenco et al [3] showed how such permutations can be
implemented using $O(n)$ simple permutations, comprising four sub-blocks:
$\rho_{\mathit{top}}\rho_{\mathit{bot}}\rho_{\mathit{top}}\rho_{\mathit{bot}}$ where permutation $\rho_{\mathit{top}}$ flips
bit $(v^{\prime\prime})_{1}$ if the first $\lceil(n-2)/2\rceil$ bits of $v^{\prime}$ match
the first $\lceil(n-2)/2\rceil$ bits of $v^{\prime}_{5}$, and where permutation
$\rho_{\mathit{bot}}$ flips bit $(v^{\prime\prime})_{2}$ if the latter $\lfloor(n-2)/2\rfloor$
bits of $v^{\prime}$ match the latter $\lfloor(n-2)/2\rfloor$ bits of $v^{\prime}_{5}$
and $(v^{\prime\prime})_{1}=1$. Each sub-block uses the remaining $\lceil(n-2)/2\rceil$ bits as “scratch”, returning them to their original state by
the end of the sub-block. For details about the construction of the
two sub-blocks see [3] or Lemma 16.
Each block $\pi_{i}$ is implemented in a similar manner using two
permutations that are nearly identical to the implementation of ${\rho_{\mathit{core}}}$.
The first (second) permutation performs the identity if $(v^{\prime\prime})_{1}=(c_{i})_{1}$ (respectively, $(v^{\prime\prime})_{2}=(c_{i})_{2}$) and otherwise flips bit
$(v^{\prime\prime})_{1}$ (respectively, $(v^{\prime\prime})_{2}$) if $v^{\prime}=v^{\prime}_{i}$.
The length of the implementations of $\psi_{1}$ and $\psi_{2}$ is $O(n)$,
since each of the seven blocks can be implemented using $O(n)$ simple
permutations from $\Sigma$. The randomize implementation of $(xyz)$ is
obtained by uniformly choosing at random a valid permutation $\phi$ and
the two valid random vectors $v_{4},v_{5}$.
We now prove that this randomized implementation satisfies the statement of the
theorem.
Let $\Omega=\{x,y,z,v_{4},v_{5},\phi\}$
be the probability space obtained by uniformly
choosing three distinct vectors $x$, $y$, and $z$, and then uniformly
choosing a corresponding implementation, which is fixed by $v_{4}$, $v_{5}$,
and $\phi$. Each point $\omega=(x,y,z,v_{4},v_{5},\phi)\in\Omega$
corresponds to an implementation $\sigma_{1}\cdots\sigma_{L}$ of the
3-cycle $(xyz)$.
The size of $\Omega$ is $\Theta(2^{5n}n!)$, and
although not uniform, the probability
of each point in $\Omega$ is $O(1/2^{5n}n!)$. Thus, our problem
of upper-bounding
$\Pr[x\sigma_{1}\sigma_{2}\cdots\sigma_{\ell-1}=\tilde{x},\,\sigma_{\ell}=%
\tilde{\sigma}]$
reduces to a counting problem.
For all implementations $\omega\in\Omega$, the indices of the $\ell$-th
permutation $\sigma_{\ell}$ depend only on its position, $\ell$, and $\phi$.
Moreover, as we change $\phi$ the indices of the $\ell$-th permutation of the
implementation of $(x,y,z,v_{4},v_{5},\phi)$ agree with the indices of some
fixed permutation $\tilde{\sigma}$ only on a subset of $S_{n}$ that is of
size $O(n!/n^{3})$ and depends only on $\ell$ and $\tilde{\sigma}$.
To establish the theorem we need to prove that for any given $\phi$ the
number of choices of $x$, $y$, $z$, $v_{4}$, and $v_{5}$, such that
$x\sigma_{1}\sigma_{2}\cdots\sigma_{\ell-1}=\tilde{x}$, is $O(2^{4n})$, implying
the number of points in $\Omega$ that agree with $\tilde{x}$
and $\tilde{\sigma}$ is $O(2^{4n}n!/n^{3})=O(|\Omega|/2^{n}n^{3})$. This is
accomplished by the following lemma:
Lemma 17.
Let $\phi\in S_{n}$ be fixed.
Then the set of all $x,y,z,v_{4},v_{5}$ such that implementation
corresponding to $(x,y,z,v_{4},v_{5},\phi)$ satisfies the equality
$x\sigma_{1}\sigma_{2}\cdots\sigma_{\ell-1}=\tilde{x}$
is of size $O(2^{4n})$.
Proof.
(of Lemma 17)
Let $v_{1}=x\phi$, $v_{2}=y\phi$, $v_{3}=z\phi$, and $\tilde{v}=\tilde{x}\phi$.
Let $\Omega_{\tilde{v},\ell}$ be the set of tuples $(v_{1},\ldots,v_{5})$ for
which $x\sigma_{1}\sigma_{2}\cdots\sigma_{\ell-1}=\tilde{x}$ is satisfied.
Note that this set is independent of $\phi$.
Then the claim is that $|\Omega_{\tilde{v},\ell}|=O(2^{4n})$.
The proof is via case analysis with respect to position $\ell$.
Without loss of generality we assume that the position is in the first
half of the implementation, that which realizes permutation $(v_{1},v_{2})(v_{4},v_{5})$, otherwise, swapping $v_{2}$ and $v_{3}$ allows the same argument
to be reused for the latter half of the implementation. Furthermore,
due to symmetry, we assume that the position of $\ell$ is in or to the
left of block ${\rho_{\mathit{core}}}$. There are four main cases: either $\ell$ is on
a boundary between two blocks, $\ell$ is in block $\tau_{i}$, $\ell$ is in the
block ${\rho_{\mathit{core}}}$, or $\ell$ is in block $\pi_{i}$.
In the first case, the position, $\ell$, is on a block boundary.
Since each $\pi$-block only toggles bits $(v^{\prime\prime})_{1}$ and $(v^{\prime\prime})_{2}$,
if position $\ell$ is adjacent to a $\pi$-block, then $\tilde{v}^{\prime}=v^{\prime}_{1}$.
Thus, all but two bits of $v_{1}$ are fixed by $\tilde{v}$. If position,
$\ell$, is on a boundary but is not adjacent to a $\pi$-block, then it
must occur after block $\tau_{1}$. Since block $\tau_{1}$ maps $v^{\prime}_{1}00$
to $v^{\prime}_{5}00$, and none of the remaining blocks, $\tau_{i}$ or ${\rho_{\mathit{core}}}$,
change the $v^{\prime}$ component to any other value, we have $\tilde{v}^{\prime}=v^{\prime}_{5}$.
Thus, all but two bits of $v_{5}$ are fixed by $\tilde{v}$, implying that
$|\Omega_{\tilde{v},\ell}|=O(2^{4n})$.
In the second case, the position, $\ell$, is inside block $\tau_{i}$. If $i\not=1$, then none of the simple permutations in block $\tau_{i}$ flips a
bit. Therefore, the value of $\tilde{v}^{\prime}=v^{\prime}_{5}$; thus fixing all but two bits
of $v_{5}$, as before. If $i=1$, then at position $\ell$, we know exactly
how many of the $n-2$ simple permutations have already been performed.
Let $j$ be this number. Hence we know that $\tilde{v}^{\prime}=(v^{\prime}_{5})_{1},\ldots,(v^{\prime}_{5})_{j},(v^{\prime%
}_{1})_{j+1},\ldots,(v^{\prime}_{1})_{n-2}$. Therefore, $j$ bits of
$v_{5}$ and $n-2-j$ bits of $v_{1}$ are therefore fixed by $\tilde{v}$,
implying that $|\Omega_{\tilde{v},\ell}|=O(2^{4n})$ as well.
In the third case, the position, $\ell$, is inside block ${\rho_{\mathit{core}}}$. In this
case we must look at the sub-blocks of the block ${\rho_{\mathit{core}}}$. If the position
occurs on a sub-block boundary, and since each of the sub-blocks simply
toggles the bits $(v^{\prime\prime})_{1}$ and $(v^{\prime\prime})_{2}$, the remaining bits of $v^{\prime}_{5}$
are fixed by $\tilde{v}^{\prime}$. If the position $\ell$ is inside a sub-block, then
things are only slightly more complicated. Assume that position $\ell$
is in a $\rho_{\mathit{top}}$ sub-block (similar arguments hold for $\rho_{\mathit{bot}}$).
Then, $\rho_{\mathit{top}}$ toggles bit $(v^{\prime\prime})_{1}$ if the first half of $v^{\prime}$
matches the first half of $v^{\prime}_{5}$. The bits being matched are never
modified and the other half of the bits of $v^{\prime}$ are used as “scratch”.
We know that the first half of $v^{\prime}$ and $v^{\prime}_{5}$ coincide throughout the
block $\rho_{\mathit{top}}$, and therefore $\tilde{v}^{\prime}$ determines this half of $v^{\prime}_{5}$.
The operations on the “scratch” half depends only on the fixed half and
the position, and therefore can be reversed, reducing the problem to the
position occurring at the beginning of $\rho_{\mathit{top}}$. Thus, $\tilde{v}$
fixes all but two of the bits of $v_{5}$.
In the last case, the position, $\ell$, is inside a $\pi$-block. Block
$\pi_{i}$ comprises two blocks that are similar to ${\rho_{\mathit{core}}}$. Each of the
two blocks is either the identity or toggles $(v^{\prime\prime})_{1}$ or $(v^{\prime\prime})_{2}$ if
$v^{\prime}=v^{\prime}_{i}$. If $i=1$ then the two blocks in Block $\pi_{i}$ behave
in the same manner as block ${\rho_{\mathit{core}}}$, except that $\tilde{v}$ fixes all
but two of the bits of $v_{1}$ rather than $v_{5}$. If $i\not=1$, then,
for the most part, the argument remains the same. We need only consider
what happens if the position, $\ell$, is in one of the eight sub-blocks.
As mentioned before, half of the bits of $v^{\prime}$ are not modified by the
sub-block, while the other half are used as “scratch”. Again, without
loss of generality, we assume that the sub-block does not modify the
first half of $v^{\prime}$. As before, $\tilde{v}^{\prime}$ fixes the first half of $v^{\prime}_{1}$.
We enumerate on all choices for the first half of $v^{\prime}_{i}$. This enables
us to reverse the operations of the sub-block on the “scratch”, fixing
the second half of $v^{\prime}_{1}$—as in the third case. This implies that
$|\Omega_{\tilde{v},\ell}|=O(2^{4n})$, and completes the proof.
∎
6 Odds and Ends
Proof.
(of Lemma 13)
We have to prove that for all $w\geq 1$ the mixing time of
$G_{2,n}^{(w)}=\mbox{sc}({\Sigma}_{w},X^{(2)})$ is $O(n\log n)$.
Given a $2$ by $n$ matrix with rows $s,t$,
we change basis to $s,u$ with $u=s\oplus t$.
Let $i\in[n]$ be a random coordinate, and consider the action of a width
$w$ permutations XORing the $i$-th bit with a random function $h$ on $w$
distinct coordinates from $[n]\setminus\{i\}$.
We claim that its action on $s,u$ is the same as XORing the $i$-th bit of
$s$ and $u$ with two independent random bits
$\alpha_{s}$ and $\alpha_{u}$ respectively.
The bits $\alpha_{s},\alpha_{u}$ are one with probability
$1/2$ and $p_{\ell}=1-\prod_{j=1}^{w}(1-\frac{\ell}{n-j})$ respectively,
where $\ell$ is the number of ones in $u$ not counting the $i$-th bit.
To see that this is indeed the resulting walk we observe the fact that
if $s$ and $t$ differ on one of the input bits of the random function $h$,
then the value of the $i$-th coordinate of $s$ and of $t$ change
independently with probability half. Otherwise they change simultaneously
with probability $1/2$.
The $u$-component of this walk is a variant of the Aldous cube, and by
the comment at the end of [5] it follows that this walk
mixes in $O(n\log n)$ time. We are left to show that in this time the
walk on both components mixes. The way to see it is to notice that in
$O(n\log n)$ time the event $A$ where the indices $i$ assume all
possible values in $1,2,\ldots,n$ (coupon collector) happens with high
probability. Now since the bits $\alpha_{s}$ are independent of
$\alpha_{u}$, we get that even when we condition over the walk on the
$u$ component, the $s$ component achieves uniform distribution
conditioned on $A$, which ends the proof.
∎
Lemma 18.
Let $A$ be an event such that $\Pr[A]\geq 1-{\epsilon}$,
and let $Z$ be a random variable over a domain $\Omega$ such
that $d(Z|A,\mbox{uniform})\leq{\epsilon}$.
Then $d(Z,\mbox{uniform})\leq 2{\epsilon}$.
Proof.
$$d(Z,\mbox{uniform})=\max_{S\subseteq\Omega}\Pr[Z\in S]-\frac{|S|}{|\Omega|}%
\leq\max_{S\subseteq\Omega}\Pr[Z\in S|A]+\Pr[\overline{A}]-\frac{|S|}{|\Omega|%
}\leq{\epsilon}+d(Z|A,\mbox{uniform})\leq 2{\epsilon}.$$
∎
Lemma 19.
Let $X$ be a random variable and $A$ an event.
Then $\Pr[X|A]\leq\Pr[X]/\Pr[A]$.
(Follows from the definition of conditional probability.)
7 Some concluding remarks
Let us review what we currently know about the spectral gap of the Markov chain
$P=P_{\Sigma}^{(k,n)}$.
By Theorem 2, $\mbox{gap}(P)\leq\Omega(1/n^{2}k)$.
On the other hand, $\mbox{gap}(P)$ is nonincreasing in $k$ by the
lifting argument from Section 3.
Since for $k=1$, $P$ is the standard random walk on the cube,
we have that $\mbox{gap}(P)\geq 1/n$.
In general, a generating set $S$ for which the spectral gap is large becomes
more difficult as $k$ increases, until the largest conceivable $k$, which is
$2^{n}-2$. In this case, this is the random walk on the Cayley graph of the
alternating group $A_{N}$ for $N=2^{n}$ with the generating set $S$.
It is open whether one can find a constant size set
for which $A_{N}$ is an expander, [16, Problem 10.3.4].
777The problem of finding a constant size expanding set for
$A_{N}$ or $S_{N}$ is equivalent.
On the other hand, by Alon and Roichman [2], a random set
of permutations of size $O(N\cdot\log N)$ will almost surely have a constant
spectral gap.
Although smaller expanding sets for $A_{N}$ are not known to exist,
the general belief is that such sets exist;
Rozenman, Shalev, and Wigderson assume the existence of an
$N^{1/30}$ expanding set for $A_{N}$, [18, section 1.4].
Our results suggest that width $2$ permutations may be used to
construct an $O(\log^{3}N)$ expanding set for $A_{N}$.
However, several obstacles stand in the way of achieving this goal.
The first one is to prove that for width $2$ permutations the spectral gap
does not deteriorate with $k$, as we believe, and is $\Omega(1/n)$ for all $k$.
The second problem is to achieve a constant gap.
To this end, one has to overcome the inherent and obvious weakness of the
width $2$ simple permutations.
Namely, that their action depends only on two
coordinates and changes only one.
This leads to poor expansion because there is only a small chance that the
action will flip a specific bit
or increase the distance between two similar vectors.
One approach to avoiding this problem is to replace the standard set
of generators of the cube $e_{1},\ldots,e_{n}$ with some expanding set
of size $O(n)$. Such an expanding set for the cube can readily be
constructed from the generating matrix of a good code [7],
and could then be used to define
an $O(n^{3})$ expanding set of permutations.
References
[1]
D. Aldous and J. A. Fill.
Reversible markov chains and random walks on graphs.
http://stat-www.berkeley.edu/users/aldous/RWG/book.html.
[2]
N. Alon and Y. Roichman.
Random Cayley graphs and expanders.
Random Structures Algorithms, 5(2):271–284, 1994.
[3]
A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor,
T. Sleator, J. A. Smolin, and H. Weinfurter.
Elementary gates for quantum computation.
Phys. Rev. A, 52(5):3457–3467, 1995.
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OGLE-2013-BLG-0578L: MICROLENSING BINARY COMPOSED OF A BROWN DWARF AND AN M DWARF
H. Park${}^{1}$,
A. Udalski${}^{2,5}$,
C. Han${}^{1,6}$,
AND
R. Poleski${}^{2,3}$,
J. Skowron${}^{2}$,
S. Kozłowski${}^{2}$,
Ł. Wyrzykowski${}^{2,4}$,
M. K. Szymański${}^{2}$,
P. Pietrukowicz${}^{2}$,
G. Pietrzyński${}^{2}$,
I. Soszyński${}^{2}$,
K. Ulaczyk${}^{2}$
(The OGLE Collaboration)
${}^{1}$Department of Physics, Institute for Astrophysics, Chungbuk National University, Cheongju 371-763, Korea
${}^{2}$Warsaw University Observatory, Al. Ujazdowskie 4, 00-478 Warszawa, Poland
${}^{3}$Department of Astronomy, Ohio State University, 140 West 18th Avenue, Columbus, OH 43210, USA
${}^{4}$Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK
Abstract
Determining physical parameters of binary microlenses is hampered by the lack of information
about the angular Einstein radius due to the difficulty of resolving caustic crossings. In
this paper, we present the analysis of the binary microlensing event OGLE-2013-BLG-0578, for
which the caustic exit was precisely predicted in advance from real-time analysis, enabling
to densely resolve the caustic crossing and to measure the Einstein radius. From the mass
measurement of the lens system based on the Einstein radius combined with the additional
information about the lens parallax, we identify that the lens is a binary that is composed
of a late-type M-dwarf primary and a substellar brown-dwarf companion. The event demonstrates
the capability of current real-time microlensing modeling and the usefulness of microlensing
in detecting and characterizing faint or dark objects in the Galaxy.
Subject headings:binaries: general, brown dwarfs – gravitational lensing: micro
55footnotetext:
The OGLE Collaboration66footnotetext:
Corresponding author
1. INTRODUCTION
It is known that low-mass stars comprise a significant fraction of stars in the Solar neighborhood
and the Galaxy as a whole. The Galaxy may be teeming with even smaller mass brown dwarfs. Therefore,
studying the abundance and properties of low-mass stars and brown dwarfs is of fundamental importance.
There have been surveys searching for very low-mass (VLM) objects (Reid et al., 2008; Aberasturi et al., 2014), but
these surveys are limited to the immediate solar neighborhood. As a result, the sample of VLM objects
is small despite their intrinsic numerosity and thus our understanding about VLM objects is poor.
Microlensing surveys detect objects through their gravitational fields rather than their radiation
and thus microlensing can provide a powerful probe of VLM objects. However, the weakness of microlensing
is that it is difficult to determine the lens mass for general microlensing events. This difficulty
arises due to the fact that the time scale of an event, which is the only observable related to the
physical lens parameters, results from the combination of the lens mass, distance and the relative
lens-source transverse speed. As a result, it is difficult to identify and characterize VLM objects
although a significant fraction of lensing events are believed to be produced by these objects.
However, it is possible to uniquely determine the physical lens parameters and thus identify VLM
objects for a subset of lensing events produced by lenses composed of two masses. For unique
determinations of the physical lens parameters, it is required to simultaneously measure the
angular Einstein radius $\theta_{\rm E}$ and the microlens parallax $\pi_{\rm E}$ that are related
to the lens mass $M$ and distance to the lens $D_{\rm L}$ by
$$M_{\rm tot}={\theta_{\rm E}\over\kappa\pi_{\rm E}};\qquad D_{\rm L}={{\rm AU}%
\over\pi_{\rm E}\theta_{\rm E}+\pi_{\rm S}},$$
(1)
respectively (Gould, 2000). Here $\kappa=4G/(c^{2}{\rm AU})$, $\pi_{\rm S}={\rm AU}/D_{\rm S}$
is the parallax of the source star, and $D_{\rm S}$ is the distance to the source. The angular
Einstein radius is estimated by analyzing deviations in lensing light curves caused by the finite
size of the lensed source stars (Gould, 1994; Nemiroff & Wickramasinghe, 1994; Witt & Mao, 1994). For a single-lens events,
finite-source effects can be detected only for a very small fraction of extremely high-magnification
events where the lens-source separation at the peak magnification is equivalent to the source size.
On the other hand, light curves of binary-lens events usually result from caustic crossings during
which finite-source effects become important and thus the chance to detect finite-source effects
and measuring the Einstein radius is high. Furthermore, binary-lens events tend to have longer
time scales than single-lens events and this also contributes to the higher chance to measure the
lens parallax. In fact, most known VLM lensing objects were identified through the channel of
binary-lens events (Hwang et al., 2010; Shin et al., 2012; Choi et al., 2013; Han et al., 2013; Park et al., 2013; Jung et al., 2015).
Despite the usefulness of binary-lens events, the chance to identify VLM objects has been low.
One important reason for the low chance is that caustic crossings last for very short periods of
time. The duration of a caustic crossing is
$$t_{\rm cc}={2\rho\over\sin\phi}t_{\rm E},$$
(2)
where $t_{\rm E}$ is the Einstein time scale of the event, $\phi$ is the entrance angle of the
source star with respect to the caustic line, and $\rho=\theta_{\ast}/\theta_{\rm E}$ is the
angular source radius $\theta_{\ast}$ normalized to the angular Einstein radius $\theta_{\rm E}$.
Considering that the Einstein time scale is $\sim({\cal O})10$ days and the Einstein radius is
$\sim({\cal O})$ milli-arcsec
for typical Galactic microlensing events, the duration of a caustic crossing is $\sim({\cal O})$ hours for
Galactic bulge source stars with angular stellar radii of $\sim({\cal O})$ 1–10 $\mu$-arcsec. Therefore,
it is difficult to densely resolve caustic crossings from surveys that are being carried out with
over hourly observational cadences.
Another reason for the low chance of resolving caustic crossings is the difficulty in predicting
their occurrence. Caustics produced by a binary lens form a single or multiple sets of closed curves
and thus caustic crossings always come in pairs. Although it is difficult to predict the first
crossing (caustic entrance) based on the fraction of the light curve before the caustic entrance,
the second crossing (caustic exit) is guaranteed after the caustic entrance. To resolve the short-lasting
caustic exit, it is required to precisely predict the time of the caustic crossing so that observation
can be focused to resolve caustic crossings. This requires vigilant modeling of a lensing event conducted
with the progress of the event followed by intensive follow-up observation.
In this paper, we report the discovery of a VLM binary that was detected from the caustic-crossing
binary-lens microlensing event OGLE-2013-BLG-0578. The caustic exit of the event was precisely predicted
by real-time modeling, enabling dense resolution and complete coverage of the caustic crossing. Combined
with the Einstein radius measured from the caustic-crossing part of the light curve and the lens
parallax measured from the long-term deviation induced by the Earth’s parallactic motion, we uniquely
measure the lens mass and identify that the lens is a VLM binary composed of a low-mass star and
a brown dwarf.
2. Observation
The microlensing event OGLE-2012-BLG-0578 occurred on a star located toward the Galactic Bulge direction.
The equatorial coordinates of the lensed star are $(\alpha,\delta)_{\rm J2000}=(17^{\rm h}59^{\rm m}59^{\rm s}\hskip-2.0pt.85,-29%
^{\circ}44^{\prime}06^{\prime\prime}\hskip-2.0pt.9)$, that correspond to the Galactic coordinates
$(l,b)=(0^{\circ}\hskip-2.0pt.90,-3^{\circ}\hskip-2.0pt.10)$. The event was first noticed on April 22,
2013 from survey observations conducted by the Optical Gravitational Lensing Experiment
(OGLE: Udalski, 2003) using the 1.3m Warsaw telescope at the Las Campanas Observatory in Chile.
Images were taken using primarily in I-band filter and some V-band images were also taken
to constrain the lensed star (source).
In Figure 1, we present the light curve of the event. The light curve shows two distinctive
spikes that are characteristic features of a caustic-crossing binary-lens event. The caustic crossings
occurred at ${\rm HJD}^{\prime}={\rm HJD}-2450000\sim 6426.0$ and $6461.8$. Although the event was first noticed
before the caustic crossings, the caustic entrance was missed. With the progress of the event, it became
clear that the event was produced by a binary lens from the characteristic “U”-shape trough between the
caustic crossings.
Although the first caustic crossing was missed, the second crossing was densely resolved. Resolving the
crossing became possible with the prediction of the caustic crossing from vigilant modeling of the light
curve followed by intensive follow-up observation. It is known that reliable prediction of the second
caustic crossing is difficult based on the light curve before the minimum between the two caustic crossings
(Jaroszyński & Mao, 2001). With the emergence of the correct model after passing the caustic trough, we focused on
the prediction of the exact caustic-crossing time. The first alert of the caustic exit was issued on June 17, 2013,
1.3 days before the actual caustic exit. On the next day, the second alert was issued to predict more
refined time of the caustic exit. In response to the alert, the OGLE experiment, which is usually operated
in survey mode, entered into “following-up” mode observation by increasing the observation cadence.
Thanks to the intensive follow-up observation, the caustic exit was completely and densely covered. See the
upper panel of Figure 1.
In order to securely measure the baseline magnitude and detect possible higher-order effects, observation
was continued after the caustic exit to the end of the Bulge season. From these observations, we obtain
2284 and 27 images taken in I and V bands, respectively. Photometry of the event was done by
using the customized pipeline (Udalski, 2003) that is based on the Difference Imaging Analysis method
(Alard & Lupton, 1998; Woźniak, 2000). We note that the I-band data are used for light curve analysis, while
the V-band data are used for the investigation of the source type.
It is known that photometric errors estimated by an automatic pipeline are often underestimated and thus
errors should be readjusted. We readjust error bars by
$$e^{\prime}=k(e^{2}+e_{\rm min}^{2})^{1/2}.$$
(3)
Here $e_{\rm min}$ is a term used to make the cumulative distribution function of $\chi^{2}$ as a function
of lensing magnification becomes linear. This process is needed to ensure that the dispersion of data points
is consistent with error bars of the source brightness. The other term $k$ is a scaling factor used to make
$\chi^{2}$ per degree of freedom (dof) becomes unity.
3. Analysis
We analyze the event by searching for the set of lensing parameters (lensing solution) that best describe
the observed light curve. Basic description of a binary-lens event requires seven standard lensing parameters.
Three of these parameters describe the source-lens approach including the time of the closest approach of the
source to a reference position of the lens, $t_{0}$, the lens-source separation at $t_{0}$ in units of the
Einstein radius, $u_{0}$, and the time required for the source to cross the Einstein radius, $t_{\rm E}$ (Einstein
time scale). In our analysis, we set the center of the mass of the binary lens as a reference position in the
lens plane. Another two parameters describe the binary lens including the projected binary separation in units
of the Einstein radius, $s$ (normalized separation), and the mass ratio between the lens components, $q$. Due
to the asymmetry of the gravitational field around the binary lens, it is needed to define the angle between
the source trajectory and the binary axis, $\alpha$ (source trajectory angle). The last parameter is the
normalized source radius $\rho$, which is needed to describe the caustic-crossing parts of the light curve
that are affected by finite-source effects.
It is often needed to consider higher-order effects in order to precisely describe lensing light curves and
this requires to include additional lensing parameters. For long time-scale events, such effects are caused
by the positional change of an observer induced by the orbital motion of the Earth around the Sun
(“parallax effect”: Gould, 1992) and/or the change of the binary separation and orientation caused by the
orbital motion of the lens (“lens orbital effect”: Dominik, 1998; Albrow et al., 2000). The analyzed event lasted
throughout the whole Bulge season and thus these effects can be important. The parallax effect is described
by two parameters, $\pi_{\rm E,\it N}$ and $\pi_{\rm E,\it E}$, that are the two components of the lens
parallax vector ${\mbox{\boldmath$\pi$}}_{\rm E}$ projected onto the sky along the north and east equatorial coordinates,
respectively. The direction of the lens parallax vector corresponds to the relative lens-source proper motion
and its magnitude corresponds to the relative lens-source parallax $\pi_{\rm rel}={\rm AU}(D_{\rm L}^{-1}-D_{\rm S}^{-1})$ scaled to the Einstein radius of the lens, i.e.,
$$\pi_{\rm E}={\pi_{\rm rel}\over\theta_{\rm E}}.$$
(4)
To the first-order approximation, the lens orbital effect is described by two parameters, $ds/dt$ and $d\alpha/dt$,
which are the change rates of the normalized binary separation and the source trajectory angle, respectively.
To model caustic-crossing parts of the light curve, it is needed to compute magnifications affected by
finite-source effects. To compute finite-source magnifications, we use the numerical method of the inverse
ray-shooting technique (Kayser et al., 1986; Schneider & Weiss, 1986) in the immediate neighboring region around caustics and
the semi-analytic hexadecapole approximation (Pejcha & Heyrovský, 2009; Gould, 2008) in the outer region surrounding caustics.
We consider the effects of the surface brightness variation of the source star. The surface brightness is
modeled as
$$S_{\lambda}\propto 1-\Gamma_{\lambda}\left(1-{3\over 2}\cos\psi\right),$$
(5)
where $\Gamma_{\lambda}$ is the linear limb-darkening coefficient, $\lambda$ is the passband, and $\psi$ is
the angle between the line of sight toward the source star and the normal to the source surface. We adopt
the limb-darkening coefficients $(\Gamma_{V},\Gamma_{I})=(0.62,0.45)$ from Claret (2000) based on the
source type. The source type is determined based on its de-reddened color and brightness. See Section 4
for details about how the source type is determined.
Searching for the best-fit solution of the lensing parameters is carried out based on the combination of
grid-search and downhill approaches. We set $(s,q,\alpha)$ as grid parameters because lensing magnifications
can vary dramatically with small changes of these parameters. On the other hand, magnifications vary smoothly
with changes of the remaining parameters, and thus we search for these parameters by using a downhill approach.
We use the Markov Chain Monte Carlo (MCMC) method for the downhill approach. Searching for solutions throughout
the grid-parameter spaces is important because it enables one to check the possible existence of degenerate
solutions where different combinations of the lensing parameters result in similar light curves.
In the initial search for solutions, we conduct modeling of the light curve based on the 7 basic binary-lensing
parameters (“standard model”). From this, it is found that the event was produced by a binary with a projected
separation very close to the Einstein radius, i.e. $s\sim 1.0$. Caustics for such a resonant binary form a single
big closed curve with 6 cusps. The two spikes of the light curve were produced by the source trajectory passing
diagonally through the caustic. See Figure 2 where we present the source trajectory with respect to
the caustic777
We note that the source trajectory is curved due to the combination of
the lens-orbital and parallax effects. We also note that the caustic
varies in time due to the positional change of the binary-lens components
caused by the orbital motion.. The estimated mass ratio between the binary components is $q\sim 0.2-0.3$.
Although the standard model provides a fit that matches the overall pattern of the observed light curve, it is
found that there exists some residual in the region around ${\rm HJD}^{\prime}\sim 6410$. See the bottom panel of
Figure 1. We, therefore, check whether higher-order effects improve fit. We find that separate
consideration of the parallax effect (“parallax model”) and the lens-orbital effect (“orbital model”)
improves fit by $\Delta\chi^{2}=579$ and $344$, respectively. When both effects are simultaneously considered
(“orbit+parallax model”), the fit improves by $\Delta\chi^{2}=616$, implying that both effects are important.
In Figure 3, we present the contours of $\Delta\chi^{2}$ in the space of the higher-order lensing
parameters. Contours marked in different colors represent the regions with $\Delta\chi^{2}<1$ (red),
$4$ (yellow), $9$ (green), $16$ (sky blue), $25$ (blue), and $36$ (purple). It shows that the higher-order
effects are clearly detected. Considering the time gap between the caustic crossings that is approximately
a month and the long duration of the event that lasted throughout the whole Bulge season, the importance of
the higher-order effects is somewhat expected.
It is known that lensing events with higher-order effects are subject to the degeneracy caused by the mirror
symmetry of the source trajectory with respect to the binary axis (Skowron et al., 2011). This so-called “ecliptic
degeneracy” is important for Galactic Bulge events that occur near the ecliptic plane. The pair of the solutions
resulting from this degeneracy have almost identical parameters except $(u_{0},\alpha,\pi_{\rm E,\it N},d\alpha/dt)\rightarrow-(u_{0},\alpha,\pi_{{\rm
E%
},N},d\alpha/dt)$. It is found that $u_{0}>0$ is marginally preferred
over the $u_{0}<0$ solution by $\Delta\chi^{2}=25.0$, which corresponds to formally $\sim 5\sigma$ level difference.
However, this level of $\Delta\chi^{2}$ can often occur due to systematics in data and thus one cannot
completely rule out the $u_{0}<0$ solution. In Table 1, we present the best-fit parameters of both
$u_{0}>0$ and $u_{0}<0$ solutions. We note that the uncertainties of the lensing parameters are estimated based
on the distributions of the parameters obtained from the MCMC chain of the solution. We also present the best-fit
model light curve ($u_{0}>0$ solution) and the corresponding source trajectory with respect to the lens and
caustic in Figures 1 and 2, respectively.
4. Physical Parameters
By detecting both finite-source and parallax effects, one can measure the angular Einstein radius and the lens
parallax, which are the two quantities needed to determine the mass and distance to the lens. The lens parallax
is estimated by $\pi_{\rm E}=(\pi_{\rm E,\it N}^{2}+\pi_{\rm E,\it E}^{2})^{1/2}$ from the parallax parameters
determined from light-curve modeling.
In order to estimate the angular Einstein radius, it is needed to convert the measured normalized source radius
$\rho$ into $\theta_{\rm E}$ by using the angular radius of the source star, i.e. $\theta_{\rm E}=\theta_{\ast}/\rho$.
The angular source radius is estimated based on the de-reddened color $(V-I)_{0}$ and brightness $I_{0}$ of the
source star which are calibrated by using the centroid of the Bulge giant clump on the color-magnitude diagram
as a reference (Yoo et al., 2004). By adopting the color and brightness of the clump centroid $(V-I,I)_{0,{\rm c}}=(1.06,14.45)$ (Bensby et al., 2011; Nataf et al., 2013), we estimate that $(V-I,I)_{0}=(0.72,17.68)$ for the source star,
implying that the source star is a G-type subgiant. Figure 4 shows the locations of the source and
the centroid of giant clump in the color-magnitude diagram of stars around the source star. We then translate
$V-I$ into $V-K$ using the color-color relation of Bessell & Brett (1988) and obtain the angular source radius by
using the relation between $V-K$ and $\theta_{\ast}$ of Kervella et al. (2004). The determined angular source radius
is $\theta_{\ast}=0.93\pm 0.06$ $\mu$as. Then, the Einstein radius is estimated as $\theta_{\rm E}=0.97\pm 0.07$
mas for the best-fit solution ($u_{0}>0$ model). We note that $u_{0}<0$ model results in a similar Einstein
radius due to the similarity in the measured values of $\rho$. Combined with the measured Einstein time scale
$t_{\rm E}$, the relative lens-source proper motion is estimated as $\mu=\theta_{\rm E}/t_{\rm E}=4.90\pm 0.35$
mas ${\rm yr}^{-1}$.
With the measured lens parallax and Einstein radius, we determine the mass and the distance to the lens using
Equation (1). In Table 2, we list the physical parameters of the lens system corresponding
to the $u_{0}>0$ and $u_{0}<0$ solutions. We note that the estimated parameters from the two solutions are
similar. According to the estimated mass, the lens system is composed of a substellar brown-dwarf companion
and a late-type M-dwarf primary. The distance to the lens is $D_{\rm L}<1.2$ kpc and thus the lens is located
in the Galactic disk. The projected separation between the lens components is $r_{\perp}=sD_{\rm L}\theta_{\rm E}$
is slightly greater than 1 AU. In order to check the validity of the obtained lensing solution, we compute
the projected kinetic to potential energy ratio $\rm(KE/PE)_{\perp}$ by
$$\left({\rm KE\over\rm PE}\right)_{\perp}={(r_{\perp}/{\rm AU})^{2}\over 8{\pi}%
^{2}(M/M_{\odot})}\left[\left({1\over s}{ds\over dt}\right)^{2}+\left({d\alpha%
\over dt}\right)^{2}\right]$$
(6)
(Dong et al., 2009). The estimated ratio is $\rm(KE/PE)_{\perp}<1$ for both $u_{0}>0$ and $u_{0}<0$ solutions and thus
meets the condition of a bound system.
5. Conclusion
We presented the analysis of a caustic-crossing binary-lens microlensing event OGLE-2013-BLG-0578 that led to
the discovery of a binary system composed of a substellar brown-dwarf companion and a late-type M-dwarf primary.
Identification of the lens became possible due to the prediction of the caustic crossing from vigilant real-time
modeling and resolution of the caustic from prompt follow-up observation. The event demonstrates the capability
of current real-time modeling and the usefulness of microlensing in detecting and characterizing
faint or dark objects in the Galaxy.
Work by C.H. was supported by Creative Research Initiative
Program (2009-0081561) of National Research Foundation of
Korea. We acknowledge funding from the European Research
Council under the European Community’s Seventh Framework
Programme (FP7/2007-2013)/ERC grant agreement No. 246678
to A.U. for the OGLE project.
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Kirchberg-Wassermann exactness vs exactness: reduction to the unimodular totally disconnected case
Chris Cave
Department of Mathematical Sciences, University of Copenhagen,
Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark
chris.cave@math.ku.dk
and
Joachim Zacharias
School of Mathematics and Statistics, University of Glasgow,
University Place, Glasgow G12 8QQ, Scotland
joachim.zacharias@glasgow.ac.uk
Abstract.
We show that in order to prove that every second countable locally compact groups with exact reduced group C*-algebra is exact in the dynamical sense (i.e. KW-exact) it suffices to show this for totally disconnected groups.
The first author is supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92).
1. Introduction
There are two natural notions of exactness for locally compact groups which to our knowledge were first mentioned by Kirchberg in [8]. A weak one, called here $C^{*}$-exactness which says that the reduced group algebra is an exact $C^{*}$-algebra, and a strong one called KW-exactness, which asserts that given any exact sequence of dynamical systems over the group the corresponding sequence of reduced crossed products is exact. The stronger exactness property can thus be regarded as a dynamical form of exactness. Here KW stands for Kirchberg and Wassermann who introduced and studied these notions in [9] and [10]. Since the crossed product by trivial actions is just the tensor product by the reduced group algebra it is evident that KW-exactness implies $C^{*}$-exactness. As announced in [8] and later proved in [9], the two concepts are equivalent for discrete groups but whether the same equivalence holds true in the case of general second countable locally compact groups has been an open problem ever since.
Note that there are numerous other concepts related to exactness such as amenability at infinity or the non-existence of non-compact ghost operators, which have been studied and put forward in the past decades ([1, 12, 14]). There has been considerable recent progress in the understanding of these conditions showing that they are equivalent to KW-exactness ([3]). In view of those developments the question of equivalence of KW-exactness and $C^{*}$-exactness appears more pressing than ever.
In this note we do not answer this question but reduce the problem to the case when the group is unimodular and totally disconnected. Thus if $C^{*}$-exact but non KW-exact groups exist then there must also exist totally disconnected unimodular such groups. This had already been suspected by experts (see the introduction of [1]).
Acknowledgements
We would like to thank Kang Li and Sven Raum for multiple stimulating discussions.
2. Preliminaries
2.1. $C^{*}$-exactness and KW-exactness
As is well-known a $C^{*}$-algebra $A$ is exact if for any exact sequence
$$0\to I\to E\to Q\to 0$$
of $C^{*}$-algebras the sequence of minimal tensor products
$$0\to A\otimes I\to A\otimes E\to A\otimes Q\to 0$$
is exact. Kirchberg and partly Wassermann proved that this property is equivalent to nuclear embeddability and passes to subalgebras and quotients (c.f. [4, Chapter 10]). Exactness of the second sequence can only fail in the middle. That is the kernel of the map onto $A\otimes Q$ is strictly larger than $A\otimes I$. It is easy to check directly from the definition that a minimal tensor product $A\otimes B$ of two $C^{*}$-algebras is exact iff $A$ and $B$ are exact ([4, Proposition 10.2.7]).
Definition 2.1.
Let $G$ be a locally compact group. Then $G$ is said to be $C^{*}$-exact if $C^{*}_{r}(G)$ is an exact $C^{*}$-algebra.
If $A$ is a $C^{*}$-algebra and $G$ is a locally compact group acting on $A$ by $\alpha\colon G\to\mathrm{Aut}(A)$ then the action $\alpha$ is called continuous if for all $a\in A$, the map $g\mapsto\alpha_{g}(a)$ is norm continuous.
Definition 2.2.
Let $G$ be a locally compact group. $G$ is said to be KW-exact (KW for Kirchberg and Wassermann) if for all $C^{*}$-algebras $A$ and all continuous actions $\alpha\colon G\to\mathrm{Aut}(A)$ and for all closed two-sided ideals $I\unlhd A$ such that $\alpha_{g}(I)=I$ for all $g\in G$, the sequence
$$0\to I\rtimes_{\alpha,r}G\to A\rtimes_{\alpha,r}G\to A/I\rtimes_{\alpha,r}G\to
0$$
is exact.
By recent results in [12, 3] it is now known that it suffices to check exactness of only one such sequence, that is, $G$ is KW-exact iff
$$0\to C_{0}(G)\rtimes_{L,r}G\to C_{b}^{lu}(G)\rtimes_{L,r}G\to(C_{b}^{lu}(G)/C_%
{0}(G))\rtimes_{L,r}G\to 0$$
is exact, where $L$ is the left translation action on the $C^{*}$-algebra of bounded left uniformly continuous functions $C_{b}^{lu}(G)$ on $G$.
As already mentioned, since $A\otimes C^{*}_{r}(G)\cong A\rtimes_{\tau,r}G$ where $\tau\colon G\to\mathrm{Aut}(A)$ is the trivial action, we have:
Proposition 2.3.
If $G$ is KW-exact then it is $C^{*}$-exact.
KW-exactness satisfies the following permanence properties.
Proposition 2.4.
Let $G$ be a locally compact group.
(1)
If $G$ is amenable then $G$ is KW-exact [10, Proposition 6.1].
(2)
If $G$ is connected then $G$ is KW-exact [10, Theorem 6.8].
(3)
Let $N\trianglelefteq G$ be a closed normal subgroup. If $N$ and $G/N$ are KW-exact then $G$ is KW-exact [10, Theorem 5.1].
Given a subgroup $H$ of a locally compact group $G$, elements in $H$ and $C_{r}^{*}(H)$ only act as multipliers on $C_{r}^{*}(G)$. However if $H$ is open in $G$ then it is easy to see that $C_{r}^{*}(H)\subset C_{r}^{*}(G)$. Since exactness passes to subalgebras we get.
Proposition 2.5.
If $G$ is a locally compact $C^{*}$-exact group and $H\leq G$ is an open subgroup then $C^{*}_{r}(H)\hookrightarrow C^{*}_{r}(G)$ is an injective $*$-homomorphism and so $C^{*}_{r}(H)$ is also exact.
2.2. Structure of locally compact groups
The following proposition follows from the closure properties of the class of amenable locally compact groups. We indicate the proof for the reader’s convenience.
Proposition 2.6 ([15, Proposition 4.1.12]).
Every locally compact group $G$ has a unique maximal amenable closed normal subgroup.
Proof.
Since unions of directed systems of amenable subgroups of $G$ are amenable one only needs to show that given two closed normal amenable subgroups $H_{1}$ and $H_{2}$ the closed subgroup $H$ generated by them is amenable. Now the semidirect product $H_{1}\rtimes H_{2}$ is amenable and $H$ is the closure of the continuous image of $H_{1}\rtimes H_{2}$. This implies that $H$ is also amenable.
∎
Definition 2.7.
Let $G$ be a locally compact group. Then the amenable radical, denoted by $\mathrm{Rad}(G)$ is the unique maximal amenable closed normal subgroup of $G$.
We have the following characterisation of totally disconnected locally compact groups which is a classical result by van Danzig.
Theorem 2.8 ([13]).
Let $G$ be a locally compact group. Then $G$ is totally disconnected if and only if it admits a neighbourhood basis of the identity consisting of compact open subgroups.
We use the following structure theorem of locally compact groups which is deduced from a solution to Hilbert’s fifth problem [11, Theorem 4.6]. Recall that a subgroup $H\leq G$ is characteristic if it is preserved under every automorphism in $\mathrm{Aut}(G)$.
Theorem 2.9 ([5, Theorem 3.3.3], [6, Theorem 23]).
Let $G$ be any locally compact group. The quotient group $G/\mathrm{Rad}(G)$ has a finite index open characteristic subgroup which splits as a direct product $S\times D$ where $S$ is a connected semi-simple Lie group and $D$ is totally disconnected.
3. Reduction to the unimodular totally disconnected locally compact second countable case
The aim of this section is to prove the following theorem.
Theorem 3.1.
If KW-exactness and $C^{*}$-exactness are equivalent for all unimodular totally disconnected second countable groups then they are equivalent for all locally compact second countable groups.
3.1. Induced representations and weak containment
3.1.1. Induced representations
Let $G$ be a locally compact group and $H\leq G$ a closed subgroup. For a Borel measure $\nu$ on $G/H$ and $g\in G$, denote $\nu_{g}$ to be the measure defined as $\nu_{g}(E)=\nu(gE)$ for all Borel sets $E\subset G/H$. A regular Borel measure $\nu$ is quasi-invariant if $\nu_{g}\sim\nu$ for all $g\in G$ where $\sim$ denotes mutual absolute continuity of measures.
Let $\mu_{H}$ be a Haar measure on $H$ and define a mapping $T_{H}\colon C_{c}(G)\to C_{c}(G/H)$ where
$$T_{H}(f)(xH)=\int_{H}f(xh)\,d\mu_{H}(h).$$
This map is surjective [2, Lemma B.1.2].
Lemma 3.2 ([2, Lemma B.1.3]).
Let $\rho\colon G\to\mathbb{R}^{>0}$ be a continuous function on $G$. Then the following are equivalent
(1)
for all $g\in G$ and $h\in H$ one has
$$\rho(gh)=\frac{\Delta_{H}(h)}{\Delta_{G}(h)}\rho(g);$$
(2)
The functional $\lambda_{\rho}\colon C_{c}(G/H)\to\mathbb{C}$ defined by
$$\lambda_{\rho}\circ T_{H}(f)=\int_{G}f(g)\rho(g)\,d\mu_{G}$$
is well-defined and positive.
If the above conditions hold then the associated regular Borel measure $\mu_{\rho}$ to $\lambda_{\rho}$ under the Riesz representation is quasi-invariant with Radon–Nikodym derivative
$$\frac{d(\mu_{\rho})_{y}}{d\mu_{\rho}}(xH)=\frac{\rho(yx)}{\rho(x)}\quad\forall
x%
,y\in G.$$
Such a function $\rho\colon G\to\mathbb{R}^{>0}$ is called a rho-function for the pair $(G,H)$. For every pair $(G,H)$ there always exists a rho-function for $(G,H)$.
Indeed if $f\in C_{c}(G)_{+}$ then
$$\rho_{f}(x)=\int_{H}\Delta_{G}(h)\Delta_{H}(h)^{-1}f(xh)d\mu_{H}(h)$$
is a continuous rho-function.
Thus there always exists a quasi-invariant regular Borel measure on $G/H$. In fact every quasi-invariant regular Borel measure is associated to a rho-function for $(G,H)$ [2, Theorem B.1.4]. When $H$ is a closed normal subgroup then one takes $\rho=1$ and the associated quasi-invariant regular Borel measure is the usual Haar measure on $G/H$.
Let $\pi\colon H\to\mathcal{U}(\mathcal{H})$ be a unitary representation and fix a quasi-invariant measure $\mu$ on $G/H$. Define a new Hilbert space
$$\mathcal{H}(\pi)=\Set{\xi\colon G\to\mathcal{H}}{\xi(xh)=\pi(h^{-1})\xi(x)%
\mbox{ and }\int_{G/H}\|\xi(x)\|^{2}\,d\mu(x)<\infty}$$
with inner product given by
$$\Braket{\xi,\eta}=\int_{G/H}\Braket{\xi(x),\eta(x)}\,d\mu(x)$$
The representation of $G$ induced from $\pi$ or simply the induced representation is the representation $\mathrm{ind}_{H}^{G}\pi\colon G\to\mathcal{U}(\mathcal{H}(\pi))$ given by
$$\mathrm{ind}_{H}^{G}\pi(x)\xi(y)=\left(\frac{d\mu_{x^{-1}}}{d\mu}(yH)\right)^{%
1/2}\xi(x^{-1}y)\quad\forall\xi\in\mathcal{H}(\pi)\ \forall x,y\in G.$$
Given another quasi-invariant measure on $G/H$ the same construction gives a unitarily equivalent representation so we can call $\mathrm{ind}_{H}^{G}\pi$ the induced representation of $\pi$ to $G$ without trepidation [2, Proposition E.1.5].
When $H=\set{e}$ then $\mathrm{ind}_{H}^{G}(1_{H})=\lambda_{G}$. More generally, the representation $\mathrm{ind}_{H}^{G}(1_{H})$ is called the quasi-regular representation of $G/H$. If $H$ is normal in $G$ then $\mathrm{ind}_{H}^{G}(1_{H})$ is unitarily equivalent to $\lambda_{G/H}\circ q$ where $q\colon G\to G/H$ is the natural surjection and $\lambda_{G/H}$ is the left regular representation of $G/H$ [7, Proposition 2.38].
3.1.2. Weak containment
The following definition for weak containment of representations is not standard; however, it is sufficient for our applications.
Definition 3.3 ([2, Theorem F.4.4]).
Let $\pi$ and $\rho$ be unitary representations of $G$. Then $\pi$ is weakly contained in $\rho$, denoted by $\pi\prec\rho$, if $\|\pi(f)\|\leq\|\rho(f)\|$ for all $f\in L^{1}(G)$.
We have the following properties of weak containment and induced representations.
Proposition 3.4.
Let $G$ be a locally compact group and $H\leq G$ a closed subgroup. Then
(1)
$\mathrm{ind}_{H}^{G}(\lambda_{H})$ is unitarily equivalent to $\lambda_{G}$ [7, Corollary 2.52];
(2)
$G$ is amenable if and only if $1_{G}\prec\lambda_{G}$ [2, Theorem G.3.2];
(3)
if $\sigma$ and $\rho$ are unitary representations on $H$ and $\sigma\prec\rho$ then $\mathrm{ind}_{H}^{G}(\sigma)\prec\mathrm{ind}_{H}^{G}(\rho)$ [2, Theorem F.3.5].
We believe the following is well known but we provide a proof as we could not find a reference.
Lemma 3.5.
Let $G$ be a locally compact second countable group and suppose $H\leq G$ is a closed normal amenable subgroup. If $G$ is $C^{*}$-exact then $G/H$ is $C^{*}$-exact.
Proof.
As $H$ is amenable it follows that $1_{H}\prec\lambda_{H}$ and so $\mathrm{ind}_{H}^{G}(1_{H})\prec\lambda_{G}$. However $\mathrm{ind}_{H}^{G}(1_{H})$ is unitarily equivalent to $\lambda_{G/H}\circ q$ where $q\colon G\to G/H$ is the quotient map and $\lambda_{G/H}$ is the left regular representation of $G/H$. Hence $\lambda_{G/H}\circ q\prec\lambda_{G}$ and so it remains to show that the natural map $T_{H}$ defined by
$$T_{H}\colon C_{c}(G)\to C_{c}(G/H),\quad T_{H}(f)(gH)=\int_{H}f(gh)\,d\mu_{H}(h)$$
extends to a surjective $*$-homomorphism from $C^{*}_{r}(G)\to C^{*}_{r}(G/H)$. So let $\lambda_{G/H}\circ q\colon C_{c}(G)\to\mathcal{B}(L^{2}(G/H))$ be the natural extension of $\lambda_{G/H}\circ q$. That is
$$\lambda_{G/H}\circ q(f)=\int_{G}f(g)\lambda_{G/H}(gH)\,dg$$
for all $f\in C_{c}(G)$. We will show that $\lambda_{G/H}\circ q(f)=\lambda_{G/H}(T_{H}(f))$ for all $f\in C_{c}(G)$. Then it will follow that
$$\|\lambda_{G/H}(T_{H}(f))\|=\|\lambda_{G/H}\circ q(f)\|\leq\|\lambda_{G}(f)\|$$
as $\lambda_{G/H}\circ q\prec\lambda_{G}$ and so $T_{H}$ extends to a surjective $*$-homomorphism from $C^{*}_{r}(G)\to C^{*}_{r}(G/H)$. So for all $f\in C_{c}(G)$ and $\xi\in C_{c}(G/H)$ and $yH\in G/H$ we have
$$\displaystyle\lambda_{G/H}\circ q(f)\xi(yH)=\int_{G}f(g)\xi(g^{-1}yH)\,dg$$
$$\displaystyle=\int_{G/H}\int_{H}f(xh)\xi(h^{-1}x^{-1}yH)\,dhdx$$
$$\displaystyle=\int_{G/H}\int_{H}f(xh)\xi(x^{-1}yH)\,dhdx$$
where the second equality follows from Weil’s integration formula [7, Corollary 1.21] and the final equality follows from normality of $H$. Now
$$\displaystyle\lambda_{G/H}(T_{H}(f))\xi(yH)$$
$$\displaystyle=\int_{G/H}T_{H}f(xH)\lambda_{G/H}(xH)\xi(yH)\,d(xH)$$
$$\displaystyle=\int_{G/H}\int_{H}f(xh)\xi(x^{-1}yH)\,dhdx.$$
Hence $\lambda_{G/H}\circ q(f)=\lambda_{G/H}(T_{H}(f))$ for all $f\in C_{c}(G)$ and so $T_{H}$ extends to a surjection. As $C^{*}_{r}(G)$ is exact it follows that $C^{*}_{r}(G/H)$ is also exact as exactness passes to quotients.
∎
3.2. Reduction to totally disconnected case
Lemma 3.6.
If KW-exactness and $C^{*}$-exactness are equivalent in the class of unimodular totally disconnected locally compact second countable groups then they are equivalent in the class of totally disconnected locally compact second countable groups
Proof.
Let $G$ be a totally disconnected locally compact second countable group and suppose $G$ is $C^{*}$-exact. Let $G_{0}=\mathrm{ker}(\Delta)$. In particular $G_{0}$ is a closed normal unimodular subgroup of $G$. As $G$ is totally disconnected, there exists a compact open subgroup $K\leq G$. Compact groups are unimodular it follows that $\Delta|_{K}=1$. Hence $K\leq G_{0}$ and so $\mu_{G}(G_{0})\geq\mu_{G}(K)>0$ and so $G_{0}$ is open. Thus as $G$ is $C^{*}$-exact it follows that $G_{0}$ is $C^{*}$-exact.
By assumption this implies that $G_{0}$ is KW-exact and as $G/G_{0}$ is abelian it follows that $G$ is also KW-exact.
∎
We are now ready to prove the main result of this section.
Proof of Theorem 3.1.
Let $G$ be a $C^{*}$-exact locally compact second countable group. Let $\mathrm{Rad}(G)$ be the amenable radical of $G$. Then by Lemma 3.5 it follows that $G/\mathrm{Rad}(G)$ is $C^{*}$-exact. By Theorem 2.9 there exists an open normal finite index subgroup $N\leq G/\mathrm{Rad}(G)$ such that $N\cong S\times D$ where $S$ is a connected semisimple Lie group and $D$ is totally disconnected. We have the tensor decomposition where $C^{*}_{r}(N)\cong C^{*}_{r}(S)\otimes C^{*}_{r}(D)$. As $C^{*}_{r}(N)$ is exact if follows that $C^{*}_{r}(D)$ is also exact [4, Proposition 10.2.7].
By assumption and by Lemma 3.6 this implies that $D$ is KW-exact. As connected locally compact groups are KW-exact [10, Theorem 6.8] and KW-exactness is preserved under extensions [10, Theorem 5.1] by closed normal subgroups it follows that $N$ is KW-exact. We know $N$ is open so in particular it is closed in $G/\mathrm{Rad}(G)$. Further $N$ is cocompact in $G/\mathrm{Rad}(G)$ so $G/\mathrm{Rad}(G)$ is KW-exact. As $\mathrm{Rad}(G)$ is a closed normal and amenable subgroup of $G$ and KW-exactness is preserved under extensions it follows that $G$ is KW-exact.
∎
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Real difference Galois theory.
Thomas Dreyfus
Université Paul Sabatier - Institut de Mathématiques de Toulouse,
118 route de Narbonne, 31062 Toulouse
tdreyfus@math.univ-toulouse.fr
(Date:: November 24, 2020)
Abstract.
In this paper, we develop a difference Galois theory in the setting of real fields. After proving the existence and uniqueness of the real Picard-Vessiot extension, we define the real difference Galois group and prove a Galois correspondence.
2010 Mathematics Subject Classification: 12D15,39A05
Work supported by the labex CIMI
Contents
1 Reminders of difference algebra
2 Existence and uniqueness of Picard-Vessiot extensions over real fields
3 Real difference Galois group
Introduction
Let us consider an equation of the form:
(1)
$$\phi Y=AY,$$
where $A$ is an invertible matrix having coefficients in a convenient field $\mathbf{k}$***In all the paper, all fields are of characteristic zero. and $\phi$ is an automorphism of $\mathbf{k}$. A typical example is $\mathbf{k}:=\mathbb{C}(x)$ and $\phi y(x):=y(x+1)$. The aim of the difference Galois theory is to study (1) in an algebraic point of view. See [vdPS97] for details on this theory. See also [BB62, Fra63, HS08, Mor09, MU09]. The classical framework for difference Galois theory is to assume that $C$, the subfield of $\mathbf{k}$ of elements invariant under $\phi$, is algebraically closed. The goal of the present paper is to present a descent result. We explain what happens if we take instead a smaller field $\mathbf{k}$ that satisfies some convenient assumptions, including that $\mathbf{k}$ is a real field and $C$ is real closed, see $\lx@sectionsign\ref{sec2}$ for the definitions.
Assume that $C$ is algebraically closed and let us make a brief summary of the difference Galois theory. An important object attached to (1) is the Picard-Vessiot extension. Roughly speaking, a Picard-Vessiot extension is a ring extension of $\mathbf{k}$ containing a basis of solutions of (1). The Picard-Vessiot extension always exists, but the uniqueness is proved in [vdPS97] only in the case where $C$ is algebraically closed. To the Picard-Vessiot extension, we attach a group, the difference Galois group, that measures the algebraic relations between solutions belonging to the Picard-Vessiot extension. This group may be seen as a linear algebraic subgroup of invertible matrices in coefficients in $C$. We also have a Galois correspondence. Note that several definitions of the difference Galois group have been made and the comparison between different Galois groups can be found in [CHS08].
From now, we drop the assumption that $C$ is algebraically closed, and we make convenient assumptions including that $\mathbf{k}$ is a real field and $C$ is real closed. Our approach will follow [CHS13, CHvdP15], which prove similar results in the framework of differential Galois theory. Let us present a rough statement of our main result, Theorem 7. We prove that is this setting, a real Picard-Vessiot exists, i.e., a Picard-Vessiot extension that is additionally a real ring. Then, we also show a uniqueness result: given $R_{1}$ and $R_{2}$ two real Picard-Vessiot extensions, if $R_{1}\otimes_{\mathbf{k}}R_{2}$ has field of constants equal to $C$, then $R_{1}$ and $R_{2}$ are isomorphic over $\mathbf{k}$. We define a real difference Galois group, which may be seen as a linear algebraic subgroup of invertible matrices in coefficients in $C[i]$, and that is defined over $C$. See Proposition 12. This allows us to prove a Galois correspondence, see Theorem 13. See also [CH15, Dyc05] for similar results in the framework of differential Galois theory.
The paper is presented as follows. In $\lx@sectionsign\ref{sec1}$, we make some reminders of difference algebra. In $\lx@sectionsign\ref{sec2}$, we state and prove our main result, Theorem 7, about the existence and uniqueness of real Picard-Vessiot extensions. In $\lx@sectionsign\ref{sec3}$, we define the real difference Galois group, and prove a Galois correspondence.
1. Reminders of difference algebra
For more details on what follows, we refer to [Coh65]. A difference ring $(R,\phi)$ is a ring $R$ together with a ring automorphism $\phi:R\rightarrow R$. An ideal of $R$ stabilized by $\phi$ is called a difference ideal of $(R,\phi)$. A simple difference ring $(R,\phi)$ is a difference ring with only difference ideals $(0)$ and $R$. If $R$ is a field then $(R,\phi)$ is called a difference field.
Let $(R,\phi)$ be a difference ring and $m\in\mathbb{N}^{*}$. The difference ring $R\{X_{1},\dots,X_{m}\}_{\phi}$ of difference polynomials in $m$ indeterminacies over $R$ is the usual polynomial ring in the infinite set of variables
$$\{\phi^{\nu}(X_{j})\}^{\nu\in\mathbb{Z}}_{j\leq m},\leavevmode\nobreak\ $$
and with automorphism extending the one on $R$ defined by:
$$\phi\left(\phi^{\nu}(X_{j})\right)=\phi^{\nu+1}X_{j}.$$
The ring of constants $R^{\phi}$ of the difference ring $(R,\phi)$ is defined by
$$R^{\phi}:=\{f\in R\ |\ \phi(f)=f\}.$$
If $R^{\phi}$ is a field, the ring of constants we will be called field of constants.
A difference ring morphism from the difference ring $(R,\phi)$ to the difference ring $(\widetilde{R},\widetilde{\phi})$ is a ring morphism $\varphi:R\rightarrow\widetilde{R}$ such that
$\varphi\circ\phi=\widetilde{\phi}\circ\varphi$.
A difference ring $(\widetilde{R},\widetilde{\phi})$ is a difference ring extension of a difference ring $(R,\phi)$
if $\widetilde{R}$ is a ring extension of $R$ and $\widetilde{\phi}_{|R}=\phi$;
in this case, we will often denote $\widetilde{\phi}$ by $\phi$. Two difference ring extensions $(\widetilde{R}_{1},\widetilde{\phi}_{1})$ and $(\widetilde{R}_{2},\widetilde{\phi}_{2})$ of a difference ring $(R,\phi)$ are isomorphic over $(R,\phi)$ if there exists a difference ring isomorphism $\varphi$ from $(\widetilde{R}_{1},\widetilde{\phi}_{1})$ to $(\widetilde{R}_{2},\widetilde{\phi}_{2})$ such that
$\varphi_{|R}=\operatorname{Id}$.
Let $(R,\phi)$ be a difference ring. Until the end of the paper, $R[i]$ will be the ring ${R[i]:=R[X]/(X^{2}+1)}$. When $R\neq R[i]$, we equip $R[i]$ with a structure of difference ring with $\phi(i)=i$.
2. Existence and uniqueness of Picard-Vessiot extensions over real fields
Let $(\mathbf{k},\phi)$ be a difference field of characteristic zero. Consider a linear difference system
(2)
$$\phi Y=AY\hbox{ with }A\in\mathrm{GL}_{n}(\mathbf{k}),$$
where $\mathrm{GL}_{n}$ denotes the group of invertible $n\times n$ square matrices in coefficients in $\mathbf{k}$.
Definition 1.
A Picard-Vessiot extension for (2) over $(\mathbf{k},\phi)$ is a difference ring extension $(R,\phi)$ of $(\mathbf{k},\phi)$ such that
(1)
there exists $U\in\mathrm{GL}_{n}(R)$ such that $\phi(U)=AU$ (such a $U$ is called a fundamental matrix of solutions of (2));
(2)
$R$ is generated, as a $\mathbf{k}$-algebra, by the entries of $U$ and $\det(U)^{-1}$;
(3)
$(R,\phi)$ is a simple difference ring.
We may always construct a Picard-Vessiot extension as follows. Take an indeterminate $n\times n$ square matrix $X:=X_{j,k}$ and consider $\mathbf{k}\{X,\det(X)^{-1}\}_{\phi}$ which is equipped with a structure of difference ring with $\phi X=AX$. Then, for any $I$, maximal difference ideal of $\mathbf{k}\{X,\det(X)^{-1}\}_{\phi}$, the ring $\mathbf{k}\{X,\det(X)^{-1}\}_{\phi}/I$ is a simple difference ring and therefore, is a Picard-Vessiot extension.
According to [vdPS97, $\lx@sectionsign$1.1], when the field of constants $C:=\mathbf{k}^{\phi}$ is algebraically closed, we have also the uniqueness of the Picard-Vessiot extension, up to a difference ring isomorphism. Furthermore, in this case we have $C=R^{\phi}$ and, see [vdPS97, Corollary 1.16], there exist an idempotent $e\in R$, and $t\in\mathbb{N}^{*}$, such that $\phi^{t}(e)=e$, $R=\displaystyle\bigoplus_{j=0}^{t-1}\phi^{j}(e)R$, and for all $0\leq j\leq t-1$, $\phi^{j}(e)R$ is an integral domain.
In [CHS08], it is defined the notion of weak Picard-Vessiot extension we will need in the next section.
Definition 2.
A weak Picard-Vessiot extension for (2) over $(\mathbf{k},\phi)$ is a difference ring extension $(R,\phi)$ of $(\mathbf{k},\phi)$ such that
(1)
there exists $U\in\mathrm{GL}_{n}(R)$ such that $\phi(U)=AU$;
(2)
$R$ is generated, as a $\mathbf{k}$-algebra, by the entries of $U$ and $\det(U)^{-1}$;
(3)
$R^{\phi}=\mathbf{k}^{\phi}=C$.
From what precede, when the field of constants is algebraically closed, a Picard-Vessiot extension is a weak Picard-Vessiot extension. Note that the converse is not true as shows [vdPS97, Example 1.25].
We say that a field $\mathbf{k}$ is real when $0$ is not a sum of squares in $\mathbf{k}\setminus\{0\}$. We say that a field $\mathbf{k}$ is real closed when $\mathbf{k}$ does not admit an algebraic extension that is real. In particular, $\mathbf{k}$ is real closed if and only if $\mathbf{k}[i]$ is algebraically closed and satisfies $\mathbf{k}[i]\neq\mathbf{k}$.
Example 3.
The field $\mathbb{R}((x))$ of formal Laurent series with real coefficients is real. The field $\mathbb{Q}(x)$ is real. The field of real numbers is real closed.
From now we assume that $\mathbf{k}$ is a real field and its field of constants $C:=\mathbf{k}^{\phi}$ is real closed with the transcendence degree $C|C\cap\overline{\mathbb{Q}}$ infinite.
Remind that we have seen that we have the existence of $(R,\phi)$, Picard-Vessiot extension for (2) over $(\mathbf{k},\phi)$.
Lemma 4.
Let $(R,\phi)$, be a Picard-Vessiot extension for (2) over $(\mathbf{k},\phi)$. Then, $(R[i],\phi)$, is a Picard-Vessiot extension for (2) over $(\mathbf{k}[i],\phi)$.
Proof.
If $R=R[i]$ there is nothing to prove. Assume that $R\neq R[i]$. Let $(0)\neq I$ be a difference ideal of $(R[i],\phi)$. Note that $I\cap R$ is a difference ideal of $(R,\phi)$. We claim that $I\cap R\neq(0)$. To the contrary, assume that $I\cap R=(0)$. Let $a,b\in R$ with $0\neq a+ib\in I$. Then, $\phi(a)+i\phi(b)\in I$ and for all $c\in R$, $ac+ibc\in I$. Let $J$ be the smallest difference ideal of $R$ that contains $a$. From what precede, we may deduce that for all $a_{1}\in J$, there exists $b_{1}\in R$ such that $a_{1}+ib_{1}\in I$. Since $(R,\phi)$ is a simple difference ring, we have two possibilities: $J=(0)$ and $J=R$. We are going to treat separately the two cases. Assume that $J=(0)$. Then $a=0$ and $ib\in I$. But $ib\times(-i)=b\in I\cap R\setminus\{0\}$ which proves our claim when $J=(0)$. Assume that $J=R$. Then, there exists $b_{1}\in R$ such that $1+ib_{1}\in I$. But $(1+ib_{1})(1-ib_{1})=1+b_{1}^{2}\in I\cap R$. Since $R\neq R[i]$ we find that $1+b_{1}^{2}\neq 0$ which proves our claim when $J=R$.
Since $I\cap R\neq(0)$ and $(R,\phi)$ is a simple difference ring, $I\cap R=R$. We now remark that $I$ is stable by multiplication by $\mathbf{k}[i]$, which shows that $I=R[i]$. This proves the lemma.
∎
Proposition 5.
Let $(R,\phi)$, be a Picard-Vessiot extension for (2) over $(\mathbf{k},\phi)$. Then, there exist an idempotent $e\in R$, and $t\in\mathbb{N}^{*}$, such that $\phi^{t}(e)=e$, $R=\displaystyle\bigoplus_{j=0}^{t-1}\phi^{j}(e)R$, and for all $0\leq j\leq t-1$, $\phi^{j}(e)R$ is an integral domain.
Proof.
Due to Lemma 4, $(R[i],\phi)$, is a Picard-Vessiot extension for (2) over $(\mathbf{k}[i],\phi)$. The field of constants of $\mathbf{k}[i]$ is $C[i]$, which is algebraically closed. From [vdPS97, Corollary 1.16], we obtain that there exist $a,b\in R$, with $a+ib$ is idempotent, $t\in\mathbb{N}^{*}$, such that $\phi^{t}(a+ib)=a+ib$,
(3)
$$R[i]=\displaystyle\bigoplus_{j=0}^{t-1}\phi^{j}(a+ib)R[i],$$
and for all $0\leq j\leq t-1$, $\phi^{j}(a+ib)R[i]$ is an integral domain. If $R=R[i]$, $a+ib\in R$ and the proof is complete. Assume that $R\neq R[i]$. Let $e:=a^{2}+b^{2}$. A straightforward computation shows that $a-ib$ is idempotent. Since $e=(a+ib)(a-ib)$ is the product of two idempotent elements it is also idempotent. Using $\phi^{t}(a-ib)=a-ib$, we find $\phi^{t}(e)=e$.
Let us prove that for all $0\leq j\leq t-1$, $\phi^{j}(a-ib)R[i]$ is an integral domain. Let $0\leq j\leq t-1$, $c+id\in R[i]$ with $c,d\in R$, such that $\phi^{j}(a-ib)(c+id)=0$. It follows that $\phi^{j}(a+ib)(c-id)=0$ and therefore, $c-id=0=c+id$ since for all $0\leq j\leq t-1$, $\phi^{j}(a+ib)R[i]$ is an integral domain. We have proved that for all $0\leq j\leq t-1$, $\phi^{j}(a-ib)R[i]$ is an integral domain. Let us prove that for all $0\leq j\leq t-1$, $\phi^{j}(e)R[i]$ is an integral domain. Let $0\leq j\leq t-1$, $c\in R[i]$, such that $c\phi^{j}(e)=c\phi^{j}(a+ib)\phi^{j}(a-ib)=0$. We use successively the fact that $\phi^{j}(a+ib)R[i]$ and $\phi^{j}(a-ib)R[i]$ are integral domains to deduce that $c=0$, which shows that $\phi^{j}(e)R[i]$ is an integral domain. Therefore, for all $0\leq j\leq t-1$, $\phi^{j}(e)R$ is an integral domain.
We claim that $\{\phi^{j}(e),0\leq j\leq t-1\}$ are linearly independent over $R[i]$. Let us consider ${c_{0},\dots,c_{t-1}\in R[i]}$ such that $\displaystyle\sum_{j=0}^{t-1}c_{j}\phi^{j}(e)=0$.
We have $\displaystyle\sum_{j=0}^{t-1}c_{j}\phi^{j}(a-ib)\phi^{j}(a+ib)=0$. We use (3) to deduce that for all $0\leq j\leq t-1$, $c_{j}\phi^{j}(a-ib)=0$. We remind that for all $0\leq j\leq t-1$, $\phi^{j}(a-ib)R[i]$ is an integral domain. This shows that for all $0\leq j\leq t-1$, $c_{j}=0$. This proves our claim.
Using (3), to prove the lemma, it is now sufficient to prove the equality
(4)
$$\displaystyle\bigoplus_{j=0}^{t-1}\phi^{j}(a+ib)R[i]=\displaystyle\bigoplus_{j%
=0}^{t-1}\phi^{j}(e)R[i].$$
The inclusion $\displaystyle\bigoplus_{j=0}^{t-1}\phi^{j}(e)R[i]\subset\displaystyle\bigoplus%
_{j=0}^{t-1}\phi^{j}(a+ib)R[i]$ is a direct consequence of the fact that $e=(a-ib)(a+ib)\in(a+ib)R[i]$. Let us prove the other inclusion. Let ${\alpha\in\displaystyle\bigoplus_{j=0}^{t-1}\phi^{j}(a+ib)R[i]}$, and define $f:=\displaystyle\prod_{j=0}^{t-1}\phi^{j}(e)$ which is invariant under $\phi$. Therefore, $fR[i]$ is a difference ideal of $R[i]$. We use $e=(a+ib)(a-ib)$ and the fact that $(a+ib)R[i]$ is an integral domain to obtain that $f\neq 0$ and $fR[i]\neq(0)$. Since $R[i]$ is a simple difference ring, the difference ideal $fR[i]$ equals to $R[i]$. Consequently, there exists $\beta\in R[i]$ such that $f\beta=\alpha$. We again use (3) to find that ${\alpha=f\displaystyle\sum_{j=0}^{t-1}c_{j}\phi^{j}(a+ib)}$ for some $c_{j}\in R[i]$. Since $e=(a-ib)(a+ib)$, we may define for all $0\leq j\leq t-1$, $d_{j}:=f/\phi^{j}(a-ib)\in R[i]$. A straightforward computation shows that $\alpha=\displaystyle\sum_{j=0}^{t-1}c_{j}d_{j}\phi^{j}(e)$, which implies $\alpha\in\displaystyle\bigoplus_{j=0}^{t-1}\phi^{j}(e)R[i]$. We have proved $\displaystyle\bigoplus_{j=0}^{t-1}\phi^{j}(a+ib)R[i]\subset\displaystyle%
\bigoplus_{j=0}^{t-1}\phi^{j}(e)R[i]$. If we combine with the other inclusion, we obtain (4). This completes the proof.
∎
Let $R$ be a difference ring that is the direct sum of integral domains $R:=\displaystyle\bigoplus_{j=0}^{t-1}R_{j}$. We define $K$, the total ring of fractions on $R$, by $K:=\displaystyle\bigoplus_{j=0}^{t-1}K_{j}$, where for all $0\leq j\leq t-1$, $K_{j}$ is the fraction field of $R_{j}$.
We say that $R$ is a real ring if for all $0\leq j\leq t-1$, $K_{j}$ is a real field.
The notion of Picard-Vessiot extension is not well suited in the real case. Following [CHvdP15], let us define:
Definition 6.
A real Picard-Vessiot extension for (2) over $(\mathbf{k},\phi)$ is a difference ring extension $(R,\phi)$ of $(\mathbf{k},\phi)$ such that
(1)
$(R,\phi)$ is a Picard-Vessiot extension for (2) over $(\mathbf{k},\phi)$;
(2)
$(R,\phi)$ is a real difference ring.
The goal of the section is to prove:
Theorem 7.
Let us consider the equation (2) which has coefficients in $(\mathbf{k},\phi)$.
(1)
There exists a real Picard-Vessiot extension for (2) over $(\mathbf{k},\phi)$.
(2)
Let $(R,\phi)$ be a real Picard-Vessiot extension for (2) over $(\mathbf{k},\phi)$. Then, $(R,\phi)$ is a weak Picard-Vessiot extension for (2) over $(\mathbf{k},\phi)$, i.e., the ring of constants of $R$ is $C$.
(3)
Let $(R_{1},\phi_{1})$ and $(R_{2},\phi_{2})$ be two real Picard-Vessiot extensions for (2) over $(\mathbf{k},\phi)$. Let us equip the ring $R_{1}\otimes_{\mathbf{k}}R_{2}$ with a structure of difference ring as follows: $\phi(r_{1}\otimes_{\mathbf{k}}r_{2})=\phi_{1}(r_{1})\otimes_{\mathbf{k}}\phi_{%
2}(r_{2})$ for $r_{j}\in R_{j}$ and assume that $\left(R_{1}\otimes_{\mathbf{k}}R_{2}\right)^{\phi}=C$. Then, $(R_{1},\phi_{1})$ is isomorphic to $(R_{2},\phi_{1})$ over $(\mathbf{k},\phi)$.
Before proving the theorem, we are going to state and prove a lemma which is inspired by a lemma of [Sei58].
Lemma 8.
Consider a difference field $(\mathbf{K},\phi)$ of characteristic zero that is finitely generated over $\mathbb{Q}$ by the elements $u_{1},\dots,u_{m}$, i.e., $\mathbf{K}:=\mathbb{Q}\langle u_{1},\dots,u_{m}\rangle_{\phi}$, where $\mathbb{Q}\langle u_{1},\dots,u_{m}\rangle_{\phi}$ is the fraction field of $\mathbb{Q}\{u_{1},\dots,u_{m}\}_{\phi}$. Let $h:\mathbf{K}\rightarrow\mathbb{C}$ be an injective morphism of field†††Let us prove that such morphism exists. The elements of $\mathbf{K}$, satisfy a list of algebraic equations, which have a solution in an extension of $\mathbb{C}$. Then, the equations have a solution in the algebraic closed field $\mathbb{C}$. In other words, we have the existence of an embedding of $\mathbf{K}$ into $\mathbb{C}$.. For every $1\leq j\leq m$, $k\in\mathbb{Z}$, let us write $c_{j,k}:=h(\phi^{k}(u_{j}))\in\mathbb{C}$. Then, the assignment $u_{j}\mapsto\widetilde{u}_{j}:=(c_{j,k})_{k\in\mathbb{Z}}$ defines an injective difference field
morphism between $(\mathbf{K},\phi)$ and $(\mathbb{C}^{\mathbb{Z}},\phi_{s})$, where $\phi_{s}$ denotes the shift.
Proof of Lemma 8.
Let $P\in\mathbb{Q}\{X_{1},\dots,X_{m}\}_{\phi}$. We have the following equality $P(\widetilde{u}_{1},\dots,\widetilde{u}_{m})=P((c_{1,k})_{k\in\mathbb{Z}},%
\dots,(c_{m,k})_{k\in\mathbb{Z}}).$ Therefore, $P(u_{1},\dots,u_{m})=0$ if and only if $P(\widetilde{u}_{1},\dots,\widetilde{u}_{m})=0$.
∎
Proof of Theorem 7.
(1)
Let us prove the existence of a real Picard-Vessiot extension. We have seen how to construct $(R,\phi)$, Picard-Vessiot extension for (2) over $(\mathbf{k},\phi)$. Let ${U\in\mathrm{GL}_{n}(R)}$ be a fundamental solution.
As we can see in Proposition 5, $R$ is a direct sum of integral domains and we may define $K$, the total ring of fractions of $R$. The ring $K$ is a direct sum of fields $K:=\displaystyle\bigoplus_{j=0}^{t-1}K_{j}$ satisfying $\phi(K_{j})=K_{j+1}$, $K_{t}:=K_{0}$. Therefore, for all $0\leq j\leq t-1$, $(K_{j},\phi^{t})$ is a difference field. Let $(\mathcal{K},\phi)$ be the difference ring generated over $\mathbb{Q}$ by the entries of $U$ and the elements involved in the algebraic difference relations in $K$. In particular, the entries of the matrix $A$ of (2) belong to $\mathcal{K}$. As we can see from Lemma 8, for all $0\leq j\leq t-1$, there exists $\widetilde{h}_{j}$, an embedding of $(\mathcal{K}\cap K_{j},\phi^{t})$ into $(\mathbb{C}^{\mathbb{Z}},\phi_{s})$. If $t>1$, without loss of generalities, we may assume that for all $0\leq j\leq t-2$, (resp. for $j=t-1$), for all $u\in\mathcal{K}\cap K_{j}$, $\widetilde{h}_{j}(u)=\widetilde{h}_{j+1}(\phi(u))$ (resp. $\phi_{s}\left(\widetilde{h}_{0}(u)\right)=\widetilde{h}_{t-1}(\phi(u))$).
We may define $\widetilde{h}$, an embedding of $(\mathcal{K},\phi)$ into $(\mathbb{C}^{\mathbb{Z}},\phi_{s})$ as follows. Let $k=\sum_{j=0}^{t-1}k_{j}$ with $k\in\mathcal{K}$, $k_{j}\in K_{j}$ and define $\widetilde{h}(k)\in\mathbb{C}^{\mathbb{Z}}$ as the sentence which term number $c+dj$, with $0\leq c\leq t-1$, $d\in\mathbb{Z}$, equals to the term number $d$ of $\widetilde{h}_{c}(k_{c})$. From now, we are going to identify the embeddings of $(\mathcal{K},\phi)$ into $(\mathbb{C}^{\mathbb{Z}},\phi_{s})$ with the induced map between $\mathcal{K}$ and $\mathbb{C}$.
Let $\widetilde{U}\in\mathrm{GL}_{n}(\mathbb{C})$ be the image of $U$ under $\widetilde{h}$. Let $\widetilde{C}_{0}\in\mathrm{GL}_{n}(\mathbb{C})$ be such that $\widetilde{V}:=\widetilde{U}\widetilde{C}_{0}\in\mathrm{GL}_{n}(\mathbb{R})$ and such that
–
the entries of $\widetilde{C}_{0}$ are algebraically independent to the image of the elements of $\mathcal{K}$ under the image of $\widetilde{h}$,
–
the entries of $\widetilde{V}$ are algebraically independent to the real part and the imaginary part of the image of the elements of $\mathcal{K}\cap\mathbf{k}$ under $\widetilde{h}$.
Since the transcendence degree $C|C\cap\overline{\mathbb{Q}}$ is infinite, there exists $C_{0}\in\mathrm{GL}_{n}(C)$ such that the embedding $\widetilde{h}$ extends to an embedding of the difference ring generated over $\mathbb{Q}$ by $\mathcal{K}$ and the entries of $C_{0}$ and such that the image of $C_{0}$ under this embedding is $\widetilde{C}_{0}$. We still call $\widetilde{h}$ this embedding. Take as a fundamental solution $V:=UC_{0}$. Let $R_{1}$ be the difference ring generated as a $\mathbf{k}$-algebra, by the entries of $V$ and $\det(V)^{-1}$. We use the fact that the entries of $\widetilde{V}\in\mathrm{GL}_{n}(\mathbb{R})$ are algebraically independent to the real part and the imaginary part of the image under $\widetilde{h}$ of the elements of $\mathcal{K}\cap\mathbf{k}$ and $\mathbf{k}$ is a real field to deduce that $R_{1}$ is a real ring. This implies that $R_{1}[i]\neq R_{1}$. We claim that $R_{1}$ is a simple difference ring. To the contrary, assume that there exists $I$, a difference ideal of $R_{1}$ different from $(0)$ and $R_{1}$. It follows that $I(R_{1}[i])$ is different from $(0)$ and $R_{1}[i]$. With Lemma 4 we find that $(R[i],\phi)$ is a Picard-Vessiot extension for (2) over $(\mathbf{k}[i],\phi)$. By construction, $(R[i],\phi)=(R_{1}[i],\phi)$ and
therefore, $(R_{1}[i],\phi)$ is a simple difference ring. We find a contradiction, since $IR_{1}[i]$ is different from $(0)$ and $R_{1}[i]$. We have proved our claim, that is that $R_{1}$ is a simple difference ring. We additionally use the fact that $R_{1}$ is a real ring to prove that $(R_{1},\phi)$ is a real Picard-Vessiot extension for (2) over $(\mathbf{k},\phi)$.
(2) With Lemma 4 we find that $(R[i],\phi)$ is a Picard-Vessiot extension for (2) over $(\mathbf{k}[i],\phi)$.
Remind that by assumption, $C[i]$ is algebraically closed. As we can deduce from [vdPS97, Lemma 1.8], $R[i]^{\phi}=C[i]$. It follows that $R^{\phi}\subset C[i]$. By assumption, $R$ is a real ring. This implies that $i\notin R$. Therefore, $C=\mathbf{k}^{\phi}\subset R^{\phi}$. Hence, the field of constants of $R$ is $C$.
(3)
Let us prove the uniqueness adapting the proof of [vdPS97, Proposition 1.9]. Let $(R_{1},\phi_{1})$ and $(R_{2},\phi_{2})$ be real Picard-Vessiot extensions for (2) over $(\mathbf{k},\phi)$ and assume that $\left(R_{1}\otimes_{\mathbf{k}}R_{2}\right)^{\phi}=C$.
The canonical maps $R_{1}\rightarrow R_{1}\otimes_{\mathbf{k}}R_{2}$ and $R_{2}\rightarrow R_{1}\otimes_{\mathbf{k}}R_{2}$ are injective since their kernel are difference ideal and $R_{1},R_{2}$ are simple difference rings. Then, the image of the two maps are generated over $\mathbf{k}$ by two fundamental solutions in $R_{1}\otimes_{\mathbf{k}}R_{2}$ which field of constants is $C$.
Consequently the two fundamental solutions $U$ and $V$ of $R_{1}$ and $R_{2}$ in $R_{1}\otimes_{\mathbf{k}}R_{2}$ equal up to the multiplication of a matrix $C_{0}$ in $\mathrm{GL}_{n}(C)$. This concludes the proof.
∎
Remark 9.
In Theorem 7, (3), we may replace the assumption $\left(R_{1}\otimes_{\mathbf{k}}R_{2}\right)^{\phi}=C$ by the assumption that $R_{1}\otimes_{\mathbf{k}}R_{2}$ is a real ring.
Indeed, assume that $R_{1}\otimes_{\mathbf{k}}R_{2}$ is a real ring and let us prove that $\left(R_{1}\otimes_{\mathbf{k}}R_{2}\right)^{\phi}=C$. Take $I$, any maximal difference ideal of $(R_{1}\otimes_{\mathbf{k}}R_{2})[i]$ and consider $R_{3}:=(R_{1}\otimes_{\mathbf{k}}R_{2})[i]/I$. It is straightforward to check that $R_{3}^{\phi}$ contains the ring of constants of $\left(R_{1}\otimes_{\mathbf{k}}R_{2}\right)[i]$. Remind that by assumption, $C[i]$ is algebraically closed.
As we can deduce from [vdPS97, Lemma 1.8], $R_{3}^{\phi}=C[i]$. It follows that $\left(R_{1}\otimes_{\mathbf{k}}R_{2}\right)^{\phi}\subset C[i]$. But $i\notin R_{1}\otimes_{\mathbf{k}}R_{2}$ because $R_{1}\otimes_{\mathbf{k}}R_{2}$ is a real ring. Since $C=\mathbf{k}^{\phi}\subset\left(R_{1}\otimes_{\mathbf{k}}R_{2}\right)^{\phi}$, we find $\left(R_{1}\otimes_{\mathbf{k}}R_{2}\right)^{\phi}=C$.
As shows the following example, who is inspired by [CHS13], the assumption for the uniqueness in Theorem 7, is not superfluous.
Example 10.
Let $\phi:=f(z)\mapsto f(2z)$ and consider ${\phi Y=\sqrt{2}Y}$ which has coefficients in $\mathbb{R}(x)$. Let us consider the following fundamental solutions $(\sqrt{x})$ and $(i\sqrt{x})$. Consider the corresponding difference ring extensions ${R_{1}|\mathbb{R}(x):=\mathbb{R}\left[\sqrt{x},\sqrt{x}^{-1}\right]|\mathbb{R}%
(x)}$ and $R_{2}|\mathbb{R}(x):=\mathbb{R}\left[i\sqrt{x},(i\sqrt{x})^{-1}\right]$. Let us prove that $(R_{1},\phi)$ is a simple difference ring. The proof for $(R_{2},\phi)$ is similar. Let $I\neq(0)$ be a difference ideal of $R_{1}$ and let $P\in\mathbb{R}[X]$ with minimal degree such that $P(\sqrt{x})\in I$. Let $k\in\mathbb{N}$ be the degree of $P$. Assume that $k\neq 0$. We have $\phi(P(\sqrt{x}))=P(\sqrt{2}\sqrt{x})\in I$, which shows that $\phi(P(\sqrt{x}))-\sqrt{2}^{k}P(\sqrt{x})=Q(\sqrt{x})\in I$ where $Q\in\mathbb{R}[X]$ has degree less than $k$. This is in contradiction with the minimality of $k$, and shows that $k=0$. This implies that $I=R_{1}$, which proves that $(R_{1},\phi)$ is a simple difference ring.
Since $R_{1}$ and $R_{2}$ are real rings, $R_{1}|\mathbb{R}(x)$ and $R_{2}|\mathbb{R}(x)$ are two real Picard-Vessiot extensions for $\phi Y=\sqrt{2}Y$ over $(\mathbb{R}(x),\phi)$.
Note that there are no difference ring isomorphism between $(R_{1},\phi)$ and $(R_{2},\phi)$ over $\mathbb{R}(x)$ because $X^{2}=x$ has a solution in $R_{1}$ and no solutions in $R_{2}$. This is not in contradiction with Theorem 7 since $(R_{1}\otimes_{\mathbb{R}(x)}R_{2})^{\phi}\neq\mathbb{R}$, because
$$\left(\sqrt{x}\otimes_{\mathbb{R}(x)}\frac{1}{i\sqrt{x}}\right)\in(R_{1}%
\otimes_{\mathbb{R}(x)}R_{2})^{\phi}\setminus\mathbb{R}.$$
3. Real difference Galois group
In this section, we still consider (2). Let $(R,\phi)$ be a real Picard-Vessiot extension for (2) over $(\mathbf{k},\phi)$ with fundamental solution $U\in\mathrm{GL}_{n}(R)$. Consider the difference ring $(R[i],\phi)$, which is different from $(R,\phi)$, since $R$ is a real ring. Inspiriting from [CHS13], let us make the definition.
Definition 11.
We define $G$, the real difference Galois group of (2), as the group of injective difference ring morphism between $R$ and $R[i]$ letting $\mathbf{k}$ invariant.
Due to Theorem 7, (2), we have an injective group morphism
$$\begin{array}[]{cccc}\rho_{U}:&G&\longrightarrow&\mathrm{GL}_{n}(C[i])\\
&\varphi&\longmapsto&U^{-1}\varphi(U),\end{array}$$
which depends on the choice of the fundamental solution $U$ in $R$. Another choice of a fundamental solution in $R$ will give a representation that is conjugated to the first one.
Remind, see Proposition 5, that there exist an idempotent $e\in R$, and $t\in\mathbb{N}^{*}$, such that $\phi^{t}(e)=e$, $R=\displaystyle\bigoplus_{j=0}^{t-1}\phi^{j}(e)R$, and for all $0\leq j\leq t-1$, $\phi^{j}(e)R$ is an integral domain. Due to Lemma 4, $(R[i],\phi)$ is a Picard-Vessiot extension for (2) over $(\mathbf{k}[i],\phi)$. Furthermore, $R[i]=\displaystyle\bigoplus_{j=0}^{t-1}\phi^{j}(e)R[i]$ and the total ring of fractions of $R[i]$ equals $K[i]$, where $K$ is the total ring of fractions of $R$. Then, we call $G_{K[i]}$, the difference Galois group of (2), the group of difference ring automorphism of $K[i]$ letting $\mathbf{k}[i]$ invariant. See [vdPS97] for more details. The difference Galois group of (2) may also be seen as a subgroup of $\mathrm{GL}_{n}(C[i])$. Furthermore, its image in $\mathrm{GL}_{n}(C[i])$ is a linear algebraic subgroup of $\mathrm{GL}_{n}(C[i])$. We have the following result in the real case.
Proposition 12.
Let $(R,\phi)$ be a real Picard-Vessiot extension for (2) over $(\mathbf{k},\phi)$ with fundamental solution $U\in\mathrm{GL}_{n}(R)$. Let $G$, be the real difference Galois group of (2) and $G_{K[i]}$, be the difference Galois group of (2). We have the following equality
$$\hbox{Im }\rho_{U}=\left\{U^{-1}\varphi(U),\varphi\in G\right\}=\left\{U^{-1}%
\varphi(U),\varphi\in G_{K[i]}\right\}.$$
Furthermore, $\hbox{Im }\rho_{U}$ is a linear algebraic subgroup of $\mathrm{GL}_{n}(C[i])$ defined over $C$. We will identify $G$ with a linear algebraic subgroup of $\mathrm{GL}_{n}(C[i])$ defined over $C$ for a chosen fundamental solution.
Proof.
Let us define $G_{R[i]}$ as the group of difference ring automorphism of $R[i]$ letting $\mathbf{k}[i]$ invariant. We claim that we have the equalities:
$$\left\{U^{-1}\varphi(U),\varphi\in G\right\}=\left\{U^{-1}\varphi(U),\varphi%
\in G_{K[i]}\right\}=\left\{U^{-1}\varphi(U),\varphi\in G_{R[i]}\right\}.$$
Let us prove the equality $\left\{U^{-1}\varphi(U),\varphi\in G\right\}=\left\{U^{-1}\varphi(U),\varphi%
\in G_{K[i]}\right\}$. Remind that $U\in\mathrm{GL}_{n}(R)$. Since an element of $G_{K[i]}$ induces an element of $G$, we obtain the inclusion $\left\{U^{-1}\varphi(U),\varphi\in G_{K[i]}\right\}\subset\left\{U^{-1}\varphi%
(U),\varphi\in G\right\}$. Let $\varphi\in G$. We may extend $\varphi$ as an element $\varphi_{K[i]}\in G_{K[i]}$ by putting $\varphi_{K[i]}(i)=i$ and for all $0\leq j\leq t-1$, $a,b\in\phi^{j}(e)R[i]$, $\varphi_{K[i]}(\frac{a}{b})=\frac{\varphi_{K[i]}(a)}{\varphi_{K[i]}(b)}$. Since $U\in\mathrm{GL}_{n}(R)$, we find that $U^{-1}\varphi(U)=U^{-1}\varphi_{K[i]}(U)$. Therefore, we obtain the other inclusion $\left\{U^{-1}\varphi(U),\varphi\in G\right\}\subset\left\{U^{-1}\varphi(U),%
\varphi\in G_{K[i]}\right\}$ and the equality $\left\{U^{-1}\varphi(U),\varphi\in G\right\}=\left\{U^{-1}\varphi(U),\varphi%
\in G_{K[i]}\right\}$.
With a similar reasoning, we prove the equality $\left\{U^{-1}\varphi(U),\varphi\in G\right\}=\left\{U^{-1}\varphi(U),\varphi%
\in G_{R[i]}\right\}$. This concludes the proof of the claim.
We define $G_{R}$, as the group of difference ring automorphism of $R$ letting $\mathbf{k}$ invariant. Due to Theorem 7, (2), $(R,\phi)$ is a weak Picard-Vessiot extension for (2) over $(\mathbf{k},\phi)$. Applying [CHS08, Proposition 2.2], we find that $\left\{U^{-1}\varphi(U),\varphi\in G_{R}\right\}$ is a linear algebraic subgroup of $\mathrm{GL}_{n}(C)$.
Then, we may use [CHS08, Corollary 2.5], to find that the latter group, viewed as a linear algebraic subgroup of $\mathrm{GL}_{n}(C[i])$, equals to $\left\{U^{-1}\varphi(U),\varphi\in G_{R[i]}\right\}$. We conclude the proof using the claim.
∎
We finish this section by giving the Galois correspondence. See [vdPS97, Theorem 1.29] for the equivalent statement in the case where $C$ is algebraically closed.
Theorem 13.
Let $(R,\phi)$ be a real Picard-Vessiot extension for (2) over $(\mathbf{k},\phi)$ with total ring of fractions $K$, $\mathcal{F}$ be the set of difference rings $\mathbf{k}\subset F\subset K$, and such that every non zero divisor is a unit of $F$. Let $G$, be the real difference Galois group of (2), $\mathcal{G}$ be the set of linear algebraic subgroups of $G$.
(1)
For any $F\in\mathcal{F}$, the group $G(K/F)$ of elements of $G$ letting $F$ invariant belongs to $\mathcal{G}$.
(2)
For any $H\in\mathcal{G}$, the ring $K^{H}:=\{k\in K|\forall\sigma\in H,\sigma(k)=k\}$ belongs to $\mathcal{F}$.
(3)
Let $\alpha:\mathcal{F}\rightarrow\mathcal{G}$ and $\beta:\mathcal{G}\rightarrow\mathcal{F}$ denote the maps $F\mapsto G(K/F)$ and $H\mapsto K^{H}$. Then, $\alpha$ and $\beta$ are each other’s inverses.
Remark 14.
If we replace $G$ by $G_{R}$, see the proof of proposition 12, which is a more natural candidate for the definition of the real difference Galois group, we lose the Galois correspondence. Take for example $\phi_{1}Y(x):=Y(x+1)=\exp(1)Y(x)$, which has solution $\exp(x)$. A real Picard-Vessiot extension for $Y(x+1)=\exp(1)Y(x)$ over $(\mathbb{R},\phi_{1})$ is $(\mathbb{R}[\exp(x),\exp(-x)],\phi_{1})$. Let $K:=\mathbb{R}(\exp(x))$ be the total ring of fractions. We have $G\simeq\mathbb{C}^{*}$ and $G_{R}\simeq\mathbb{R}^{*}$. Note that $G_{R}\subset\mathrm{GL}_{1}(\mathbb{R})$, viewed as a linear algebraic subgroup of $\mathrm{GL}_{1}(\mathbb{C})$, equals to $G$. On the other hand, we have no bijection with the linear algebraic subgroups of $G_{R}$, which are $\{1\},\mathbb{Z}/2\mathbb{Z}$, $\mathbb{R}^{*}$, and the difference subfields of $K$, which are $\mathbb{R}(\exp(kx))$, $k\in\mathbb{N}$.
Proof of Theorem 13.
Let $\mathcal{F}_{[i]}$ be the set of difference rings $\mathbf{k}[i]\subset F\subset K[i]$, such that every non zero divisor is a unit of $F$. Let $\mathcal{G}_{[i]}$ be the set of linear algebraic subgroups of $G_{K[i]}$. Remind that the field of constants of $\mathbf{k}[i]$ is algebraically closed. In virtue of the Galois correspondence in difference Galois theory, see [vdPS97, Theorem 1.29], we find that
(a) For any $F\in\mathcal{F}_{[i]}$, the group $G_{K[i]}(K[i]/F)$ of elements of $G_{K[i]}$ letting $F$ invariant belongs to $\mathcal{G}_{[i]}$.
(b) For any $H\in\mathcal{G}_{[i]}$, the ring $K[i]^{H}$ belongs to $\mathcal{F}_{[i]}$.
(c) Let $\alpha_{[i]}:\mathcal{F}_{[i]}\rightarrow\mathcal{G}_{[i]}$ and $\beta_{[i]}:\mathcal{G}_{[i]}\rightarrow\mathcal{F}_{[i]}$ denote the maps $F_{[i]}\mapsto G_{K[i]}(K[i]/F)$ and $H\mapsto K[i]^{H}$. Then, $\alpha_{[i]}$ and $\beta_{[i]}$ are each other’s inverses.
We use Proposition 5 to find that we have a bijection $\gamma:\mathcal{F}\rightarrow\mathcal{F}_{[i]}$ given by $\gamma(F):=F[i]$. The inverse is $\gamma^{-1}(F)=F\cap K$. Now remark that since the fundamental solution has coefficients in $R$, for all $F\in\mathcal{F}$, $G(K/F)=G_{K[i]}(K[i]/\gamma(F))$. If we combine this fact with (a) and Proposition 12, we find (1).
Proposition 12 tells us that we may identify the groups in $\mathcal{G}$ with the corresponding groups in $\mathcal{G}_{[i]}$. To prove the point (2), we remark that for all $H\in\mathcal{G}$, $K[i]^{H}=K^{H}[i]$. Combined with (b), this shows the point (2) since $K^{H}=\gamma^{-1}(K[i]^{H})\in\mathcal{F}$ .
The point (3) follows from (c) and the fact that for all $F\in\mathcal{F}$ (resp. $H\in\mathcal{G}$) $G(K/F)=G_{K[i]}(K[i]/\gamma(F))$ (resp. $K[i]^{H}=\gamma(K^{H})$).
∎
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On the real representations of the Poincare group
Leonardo Pedro
Centro de Fisica Teorica de Particulas, CFTP,
Departamento de Fisica, Instituto Superior Tecnico, Universidade
Tecnica de Lisboa, Avenida Rovisco Pais nr. 1, 1049-001 Lisboa, Portugal
(January 17, 2021)
Abstract
We study the real representations of the Poincare group
and its relation with the complex representations.
The classical electromagnetic field —
from which the Poincare group was originally defined — is a
real representation of the Poincare group.
We review the map from the complex to the real
irreducible representations—finite-dimensional or unitary—of a Lie
group on a Hilbert space.
We show that all the
finite-dimensional real representations of the restricted
Lorentz group are also representations of the full Lorentz group,
in contrast with many complex representations.
We study the unitary irreducible representations of the
Poincare group with discrete spin and show that for each pair of
complex representations with positive/negative energy, there is one real
representation; we show that there are unitary transformations,
defining linear and angular momenta spaces which are common for the
real and complex representations.
1 Introduction
1.1 Motivation
Henri Poincaré defined the Poincare
group as the set of transformations
that leave invariant the Maxwell equations for the classical
electromagnetic field. The classical electromagnetic field is a real
representation of the Poincare group.
The complex representations of the Poincare group were systematically
studiedwigner ; mackey ; ohnuki ; poincare ; weinberg ; knapp
and used in the definition of
quantum fieldsfeynmanrules ; stringfields .
These studies were very important in the evolution of the
role of symmetry in the Quantum Theorysymmetry ,
which is based on complex Hilbert spacesyang .
We could not find in the literature a systematic study on the real
representations of the Poincare group — even though representation
theoryrealalgebras ; spinorsrealhilbert and Quantum Theory
realqft ; realqftII ; realqftIII ; quantumstatistics ; hestenes_old ; realQM
on real Hilbert spaces were investigated before — as it seems to be
a common assumption that all fields of all modern theories must be
quantum fields and therefore, somehow, every consistent representation
must be complex. However, due to the existence of a map between real
and complex representations, the motivation for this study is
independent of the validity of such assumption.
The reasons motivating this study are:
1) The real representations of the Poincare group play a main role in
the classical electromagnetism and general
relativityclassicalfields . It is reasonable to think that the
real representations of the Poincare group will still play an
important role in the most modern theories based on the classical
electromagnetism and general relativity. As an example, the
self-adjoint quantum fields — such as the Higgs boson, Majorana
fermion or quantum electromagnetic field — transform as real
representations under the action of the Poincare
groupstringfields .
2) The parity — included in the full Poincare group — and
charge-parity transformations are not symmetries of the Electroweak
interactionsbrancocp . It is not clear why the charge-parity is
an apparent symmetry of the Strong interactionsstrongcp or how
to explain the matter-antimatter asymmetryimbalance through the
charge-parity violation. We will show that that all the
finite-dimensional real representations of the restricted Lorentz
group are also representations of the parity; and that there are
linear and angular momenta spaces which are common for the real and
complex representations of the Poincare group, therefore independent
of the charge and matter-antimatter properties. These results may be
useful in future studies of the parity and charge-parity violations.
1.2 On the map from the complex to the real irreducible
representations of a group
Many representations of a group— such as the finite-dimensional
representations of semisimple Lie groupsHall or the unitary
representations of separable locally compact
groupslocallycompact — are direct sums (or integrals) of
irreducible representations.
The study of irreducible representations on complex Hilbert
spaces is in general easier than on real Hilbert spaces, because the
field of complex numbers is the algebraic closure — where any
polynomial equation has a root — of the field of real numbers. Given
a real Hilbert space, we can always obtain a complex Hilbert space
through complexification — extension of the scalar multiplication to
include multiplication by complex numbers.
Yet, given an irreducible representation on a real
Hilbert space V, the representation on the complex Hilbert space
resulting from the complexification of V may be reducible, because
there is a 2-dimensional real representation of the field of complex
numbers. Therefore, the complex representations are not a
generalization of the real representations, in the same way that the
complex numbers are a generalization of the real numbers.
There is a well studied map, one-to-one or two-to-one and surjective
up to equivalence, from the complex to the real linear
finite-dimensional irreducible representations of a real Lie
algebrarealalgebras ; realirrep .
In Section 2, we review a similar map from the complex
to the real irreducible representations—finite-dimensional or
unitary—of a Lie group on a Hilbert space. This section follows
closelyrealalgebras , with the difference that we will also use
the Schur’s lemma for unitary representations on a complex Hilbert
spaceschur .
Related studies can be found in the references
compactlie ; spinorsrealhilbert ; realqft ; realqftII ; realqftIII .
1.3 Finite-dimensional representations of the Lorentz group
The Poincare group, also called inhomogeneous Lorentz group, is the
semi-direct product of the translations and Lorentz Lie
groupsHall . Whether or not the Lorentz and Poincare groups
include the parity and time reversal transformations depends on the
context and authors. To be clear, we use the prefixes full/restricted
when including/excluding parity and time reversal transformations. The
Pin(3,1)/SL(2,C) groups are double covers of the full/restricted
Lorentz group. The semi-direct product of the translations with the
Pin(3,1)/SL(2,C) groups is called IPin(3,1)/ISL(2,C) Lie group — the
letter (I) stands for inhomogeneous.
A projective representation of the Poincare group on a complex/real
Hilbert space is an homomorphism, defined up to a complex phase/sign,
from the group to the automorphisms of the Hilbert space. Since the
IPin(3,1) group is a double cover of the full Poincare group, their
projective representations are the samepin .
All finite-dimensional projective representations of a simply
connected group, such as SL(2,C), are well defined
representationsweinberg .
Both SL(2,C) and Pin(3,1) are semi-simple Lie groups, and so all its
finite-dimensional representations are direct sums of irreducible
representationsHall . Therefore, the study of the
finite-dimensional projective representations of the restricted
Lorentz group reduces to the study of the finite-dimensional
irreducible representations of SL(2,C).
The Dirac spinor is an element of a 4 dimensional complex vector
space, while the Majorana spinor is an element of a 4 dimensional real
vector spacetodorov ; irreducible ; pal ; dreiner . The complex
finite-dimensional irreducible representations of SL(2,C) can be
written as linear combinations of tensor products of Dirac spinors.
In Section 3 we will review the Pin(3,1) and
SL(2,C) semi-simple Lie groups and its relation with the Majorana,
Dirac and Pauli matrices. We will obtain all the real
finite-dimensional irreducible representations of SL(2,C) as linear
combinations of tensor products of Majorana spinors, using the map
from Section 2.
Then we will check that all these real representations are also
projective representations of the full Lorentz group, in contrast with
the complex representations which are not all projective
representations of the full Lorentz group.
1.4 Unitary representations of the Poincare group
According to Wigner’s theorem, the most general transformations,
leaving invariant the modulus of the internal product of a
Hilbert space, are: unitary or anti-unitary operators, defined up to a
complex phase, for a complex Hilbert; unitary, defined
up to a signal, for a real Hilbertwignertheorem . This motivates
the study of the (anti-)unitary projective representations of the full
Poincare group.
All (anti-)unitary projective representations of ISL(2,C) are, up to
isomorphisms, well defined unitary representations, because ISL(2,C)
is simply connectedweinberg . Both ISL(2,C) and IPin(3,1) are
separable locally compact groups and so all its (anti-)unitary
projective representations are direct integrals of irreducible
representationslocallycompact . Therefore, the study of the
(anti-)unitary projective representations of the restricted Poincare
group reduces to the study of the unitary irreducible representations
of ISL(2,C).
The spinor fields, space-time dependent spinors, are solutions of the
free Dirac equationDirac . The real/complex Bargmann-Wigner
fieldsBW ; allspins , space-time dependent linear combinations of
tensor products of Majorana/Dirac spinors, are solutions of the free
Dirac equation in each tensor index. The complex unitary irreducible
projective representations of the Poincare group with discrete spin
can be written as complex Bargmann-Wigner fields.
In Section 4, we will obtain all the real
unitary irreducible projective representations of the Poincare
group, with discrete spin, as real Bargmann-Wigner fields, using the
map from Section 2. For each pair of complex representations with
positive/negative energy, there is one real representation.
We will define the Majorana-Fourier and Majorana-Hankel unitary
transforms of the real or complex Bargmann-Wigner fields.
Then we relate the Majorana transforms to the linear and angular
momenta of a representation of the Poincare group.
The free Dirac equation is diagonal in the Newton-Wigner
representationnewton , related to the Dirac representation
through a Foldy-Wouthuysen transformationrevfoldy ; foldy of
Dirac spinor fields. The Majorana-Fourier transform, when applied on
Dirac spinor fields, is related with the Newton-Wigner representation
and the Foldy-Wouthuysen transformation. In the context of Clifford
Algebras, there are studies on the geometric square roots of -1
hestenes_old ; squareroot and on the
generalizations of the Fourier transformclifford , with
applications to image processingimage .
2 On the map from the complex to the real irreducible
representations of a group
2.1 Representations on real and complex Hilbert spaces
Remark 2.1.
Let $H_{n}$, with $n\in\{1,2\}$, be two Hilbert spaces with internal
products $<,>:H_{n}\times H_{n}\to\mathbb{F}$,($\mathbb{F}=\mathbb{R},\mathbb{C}$).
A linear operator $U:H_{1}\to H_{2}$ is unitary iff:
1) it is surjective;
2) for all $x\in H_{1}$, $<U(x),U(x)>=<x,x>$.
Remark 2.2.
Given two real Hilbert spaces $H_{1}$, $H_{2}$ and an unitary operator
$U:H_{1}\to H_{2}$, the inverse operator $U^{-1}:H_{2}\to H_{1}$ is defined by:
$$\displaystyle<x,U^{-1}y>=<Ux,y>,\ x\in H_{1},y\in H_{2}$$
Definition 2.3.
A representation $(M_{G},V)$ of a Lie group $G$representations on
a real or complex Hilbert space $V$ is defined by:
1) the representation space $V$, which is an Hilbert space;
2) the representation group homomorphism $M:G\to B(V)$
from the group elements to the bounded automorphisms with a bounded
inverse, such that the map $M^{\prime}:G\times V\to V$ defined by
$M^{\prime}(g,v)\equiv M(g)v$ is continuous.
Definition 2.4.
Let $V_{n}$, with $n\in\{1,2\}$, be two Hilbert spaces.
The representations $(M_{n,G},V_{n})$ of a group $G$ on the
Hilbert spaces $V_{n}$ are equivalent iff there is a linear bijection
$\alpha:V_{1}\to V_{2}$ such that for all $g\in G$, $\alpha\circ M_{1,G}(g)=M_{2,G}(g)\circ\alpha$.
Definition 2.5.
Consider a representation $(M_{G},V)$.
An isomorphism of $(M_{G},V)$ is a bijective operator $S:V\to V$
commuting with $M_{G}(g)$, for all $g\in G$.
Definition 2.6.
Consider a representation $(M_{G},V)$.
An isometry of $(M_{G},V)$ is a unitary isomorphism of $(M_{G},V)$.
Definition 2.7.
Let $W$ be a linear subspace of $V$.
$(M_{G},W)$ is a (topological) subrepresentation of $(M_{G},V)$
iff $W$ is closed and invariant under the group action, that is, for all $w\in W$:
$(M(g)w)\in W$, for all $g\in G$.
Definition 2.8.
A representation $(M_{G},V)$ is (topologically) irreducible iff their only
sub-representations are the non-proper or trivial sub-representations:
$(M_{G},V)$ and $(M_{G},\{0\})$, where $\{0\}$ is
the null space. An irreducible representation is called irrep.
Definition 2.9.
Consider a representation $(M_{G},V)$ on a
complex Hilbert space.
A C-conjugation operator of $(M_{G},V)$ is an anti-unitary
involution of $V$ commuting with $M_{G}(g)$,
for all $g\in G$.
Definition 2.10.
Consider a representation $(M_{G},W)$ on a
real Hilbert space.
A R-imaginary operator of $(M_{G},W)$, $J$, is an isometry of
$(M_{G},W)$ verifying $J^{2}=-1$.
Definition 2.11.
Consider an irreducible representation $(M_{G},V)$ on a
complex Hilbert space.
The representation is C-real iff there is a C-conjugation
operator. The subset of
C-real irreducible representations is $R_{G}(\mathbb{C})$.
The representation is C-pseudoreal iff
there is no C-conjugation operator but there is an
anti-isometry of $(M_{G},V)$. The subset of
C-pseudoreal irreducible representations is $P_{G}(\mathbb{C})$.
The representation is C-complex iff
there is there is no anti-isometry of $(M_{G},V)$.
The subset of C-complex irreducible representations is
$C_{G}(\mathbb{C})$.
Definition 2.12.
Consider a representation $(M_{G},W)$ on a
real Hilbert space.
The representation $(M_{G},W^{c})$ is the complexification of the
representation $(M_{G},W)$, defined as
$W^{c}\equiv\mathbb{C}\otimes W$, with the multiplication by scalars
such that $a(bw)\equiv(ab)w$
for $a,b\in\mathbb{C}$ and $w\in W$.
The internal product of $W^{c}$ is defined as:
$$\displaystyle<v_{r}+iv_{i},u_{r}+iu_{i}>_{c}\equiv<v_{r},u_{r}>+<v_{i},u_{i}>+%
i<v_{r},u_{i}>-i<v_{i},u_{r}>$$
for $u_{r},u_{i},v_{r},v_{i}\in W$ and $<v_{r},u_{r}>$ is the internal product of $W$.
Definition 2.13.
Consider a representation $(M_{G},V)$ on a
complex Hilbert space.
The representation $(M_{G},V^{r})$ is the realification of the
representation $(M_{G},V)$, defined as
$V^{r}\equiv V$ is a real Hilbert space
with the multiplication by scalars restricted to reals
such that $a(v)\equiv(a+i0)v$
for $a\in\mathbb{R}$ and $v\in V$.
The internal product of $V^{r}$ is defined as
$$\displaystyle<v,u>_{r}\equiv\frac{<v,u>+<u,v>}{2}$$
for $u,v\in V$ and $<v,u>$ is the internal product of $V$.
Proposition 2.14.
Let $H_{n}$, with $n\in\{1,2\}$, be two complex Hilbert spaces and
$H^{r}_{n}$ its complexification.
The following two statements are equivalent:
1) The operator $U:H_{1}\to H_{2}$ is unitary;
2) The operator $U^{r}:H_{1}^{r}\to H_{2}^{r}$ is unitary, where
$U^{r}(h)\equiv U(h)$, for $h\in H_{1}$.
Proof.
Since $<h,h>=<h,h>_{r}$ and
$U^{r}(h)=U(h)$, for $h\in H_{1}$, we get the result.
∎
2.2 The map from the complex to the real representations
Definition 2.15.
Consider the representation $(M_{G},W)$ on a
real Hilbert space and let $(M_{G},W^{c})$ be its complexification.
$(M_{G},W)$ is R-real iff $(M_{G},W^{c})$ is C-real irreducible.
The set of R-real irreducible representations is $R_{G}(\mathbb{R})$.
$(M_{G},W)$ is R-pseudoreal iff $(M_{G},V)$ is C-pseudoreal irreducible,
with $W^{c}=V\oplus\bar{V}$.
The set of R-pseudoreal irreducible representations is
$P_{G}(\mathbb{R})$.
$(M_{G},W)$ is R-complex iff $(M_{G},V)$ is C-complex irreducible, with
$W^{c}=V\oplus\bar{V}$.
The set of R-complex irreducible representations is
$C_{G}(\mathbb{R})$.
Proposition 2.16.
Any irreducible real representation is R-real or R-pseudoreal or
R-complex.
Proof.
Consider an irreducible representation $(M_{G},W)$
on a real Hilbert space.
There is a C-conjugation operator of $(M_{G},W^{c})$,
$\theta$, defined by $\theta(u+iv)\equiv(u-iv)$ for $u,v\in W$,
verifying $(W^{c})_{\theta}=W$.
Let $(M_{G},X^{c})$ be a proper non-trivial subrepresentation of
$(M_{G},W^{c})$. Then $\theta$ is a C-conjugation operator of the
subrepresentations $(M_{G},Y^{c})$ and $(M_{G},Z^{c})$, where
$Y^{c}\equiv\{u+\theta v:u,v\in X^{c}\}$ and
$Z^{c}\equiv\{u:u,\theta u\in X^{c}\}$. Therefore,
$Y^{c}=\{u+iv:u,v\in Y\}$ and $Z^{c}=\{u+iv:u,v\in Z\}$,
where $Y\equiv\{\frac{1+\theta}{2}u:u\in Y^{c}\}$ and
$Z\equiv\{\frac{1+\theta}{2}u:u\in Z^{c}\}$, are invariant closed subspaces of
$W$.
If $Y=\{0\}$ then $Z=\{0\}$ and $Y^{c}=X^{c}=\{0\}$, in contradiction
with $X^{c}$ being non-trivial.
If $Z=W$ then $Y=W$ and $Z^{c}=X^{c}=W^{c}$, in contradiction with $X^{c}$
being proper.
Therefore $Z=\{0\}$ and $Y=W$, which implies
$Z^{c}=\{0\}$ and $Y^{c}=W^{c}$.
So, $(M_{G},W)$ is equivalent to $(M_{G},(X^{c})^{r})$,
due to the existence of the bijective linear
map $\alpha:(X^{c})^{r}\to W$, $\alpha(u)=u+\theta u$,
$\alpha^{-1}(u+\theta u)=u$, for $u\in(X^{c})^{r}$.
Suppose that there is a C-conjugation operator of $(M_{G},X^{c})$,
$\theta^{\prime}$. Then $(M_{G},W_{\pm})$ is a proper non-trivial
subrepresentation of $(M_{G},W)$, where
$W_{\pm}\equiv\{\frac{1\pm\theta^{\prime}}{2}w:w\in W\}$, in contradiction with
$(M_{G},W)$ being irreducible.
∎
Proposition 2.17.
Any real representation which is R-real or R-pseudoreal or
R-complex is irreducible.
Proof.
Consider an irreducible representation
on a complex Hilbert space $(M_{G},V)$.
There is a R-imaginary operator $J$ of the representation
$(M_{G},V^{r})$, defined by $J(u)\equiv iu$, for $u\in V^{r}$.
Let $(M_{G},X^{r})$ be a proper non-trivial subrepresentation of
$(M_{G},V^{r})$. Then $J$ is an R-imaginary operator of
$(M_{G},Y^{r})$ and $(M_{G}^{r},Z^{r})$,
where $Y^{r}\equiv\{u+Jv:u,v\in X^{r}\}$ and
$Z^{r}\equiv\{u:u,Ju\in X^{r}\}$.
Then $(M_{G},Y)$ and $(M_{G},Z)$ are
subrepresentations of $(M_{G},V)$,
where the complex Hilbert spaces
$Y\equiv Y^{r}$ and $Z\equiv Z^{r}$ have the scalar
multiplication such that $(a+ib)(y)=ay+bJy$,
for $a,b\in\mathbb{R}$ and $y\in Y$ or $y\in Z$.
If $Y=\{0\}$, then $Z=X^{r}=\{0\}$ which is in contradiction with $X^{r}$
being non-trivial.
If $Z=V$, then $Y=V$ and $X^{r}=V^{r}$ which is in contradiction with
$X^{r}$ being non-trivial.
So $Z=\{0\}$ and $Y=V$, which implies that $V=(X^{r})^{c}$.
Then there is a C-conjugation operator of $(M_{G},V)$,
$\theta$, defined by $\theta(u+iv)\equiv u-iv$, for $u,v\in X^{r}$.
We have $X^{r}=V_{\theta}$.
Suppose there is a R-imaginary operator of $(M_{G},V_{\theta})$,
$J^{\prime}$. Then $(M_{G},V_{\pm})$, where
$V_{\pm}\equiv\{\frac{1\pm iJ^{\prime}}{2}v:v\in V\}$,
are proper non-trivial subrepresentations of $(M_{G},V)$,
in contradiction with $(M_{G},V)$ being irreducible.
Therefore, if $(M_{G},V)$ is C-real, then $(M_{G},V_{\theta})$ is
R-real irreducible.
If $(M_{G},V)$ is C-pseudoreal or C-complex, then
$(M_{G},V_{\theta}^{r})$ is R-pseudoreal or R-complex, irreducible.
∎
Definition 2.18.
Consider a representation $(M_{G},W)$ on a
real Hilbert space.
A R-imaginary operator of $(M_{G},W)$, $J$, is an isometry of $(M_{G},W)$
verifying $J^{2}=-1$.
2.3 Finite-dimensional representations
Lemma 2.19 (Schur’s lemma for finite-dimensional representationsschur ).
Consider an irreducible finite-dimensional representation $(M_{G},V)$ of a Lie
group $G$ on a complex Hilbert space $V$. If the representation
$(M_{G},V)$ is irreducible then any isomorphism $S$ of $(M_{G},V)$ is a
scalar.
Lemma 2.20.
Consider an irreducible finite-dimensional representation $(M_{G},V)$ on
a complex Hilbert space. An anti-isomorphism of $(M_{G},V)$, if it
exists, is unique up to a scalar isometry of $(M_{G},V)$.
Proof.
Let $\theta_{1}$,$\theta_{2}$ be two anti-isomorphisms of $(M_{G},V)$. The
product $(\theta_{2}\theta_{1})$ is an isomorphism of $(M_{G},V)$;
since $(M_{G},V)$ is irreducible,
$(\theta_{2}\theta_{1})=re^{i\phi}$; with $\phi,r\in\mathbb{R}$, $r>0$.
We have $1=(\theta_{2})^{2}=r^{2}e^{i\phi}\theta_{1}e^{i\phi}\theta_{1}=r^{2}$.
Therefore $\theta_{2}=\alpha\theta_{1}\alpha^{-1}$; where
$\alpha\equiv e^{i\frac{\phi}{2}}$ is a scalar isometry of $(M_{G},V)$.
∎
Proposition 2.21.
Two R-real irreducible finite-dimensional representations are
isomorphic iff their complexifications are isomorphic.
Proof.
Let $(M_{G},V)$ and $(N_{G},W)$ be C-real irreducible representations,
with $\theta_{M}$ and $\theta_{N}$ the respective C-conjugation operators.
If there is an isomorphism $\alpha:V\to W$ such that
$\alpha M_{G}(g)=N_{G}(g)\alpha$ for all $g\in G$, then
$\vartheta\equiv\alpha\theta_{M}\alpha^{-1}$ is an anti-isomorphism of
$(N_{G},W)$. Since it is unique up to a phase, then
$\theta_{N}=e^{i\phi}\vartheta$. Therefore $e^{i\frac{\phi}{2}}\alpha$ is an
isomorphism between $(M_{G},V_{\theta_{M}})$ and $(N_{G},W_{\theta_{N}})$,
where $V_{\theta_{M}}\equiv\{(1+\theta_{M})v:v\in V\}$.
∎
Proposition 2.22.
Two C-complex or C-pseudoreal irreducible finite-dimensional
representations are isomorphic or anti-isomorphic iff their
realifications are isomorphic.
Proof.
Let $(M_{G},V)$ and $(N_{G},W)$ be R-complex or R-pseudoreal irreducible
representations, with $J_{M}$ and $J_{N}$ the respective R-imaginary
operators. If there is an isomorphism $\alpha:V\to W$ such that
$\alpha M_{G}(g)=N_{G}(g)\alpha$ for all $g\in G$, then
$K\equiv\alpha J_{M}\alpha^{-1}$ is a R-imaginary operator of
$(N_{G},W)$.
When considering $(N_{G},W_{J_{N}})$ and $(M_{G},V_{J_{M}})$, where
$W_{J_{N}}\equiv\{(1-iJ_{N})w:w\in W\}$, we get that
$(1-J_{N}K)(1-KJ_{N})=c$ as an operator of $W_{J_{N}}$.
If $c=0$ then $K=-J_{N}$ and $\alpha$ defines an anti-isomorphism
between $(M_{G},V_{J_{M}})$ and $(N_{G},W_{J_{N}})$.
If $c\neq 0$ then $(1-J_{N}K)\alpha$ is an isomorphism between
$(M_{G},V_{J_{M}})$ and $(N_{G},W_{J_{N}})$.
∎
Definition 2.23.
A finite-dimensional representation is completely reducible iff it can
be expressed as a direct sum of irreducible representations.
Remark 2.24 (Weyl theorem).
All finite-dimensional representations of a semi-simple Lie group
(such as SL(2,C)) are completely reducible.
2.4 Unitary representations
Lemma 2.25 (Schur’s lemma for unitary representationsschur ).
Consider an irreducible unitary representation $(M_{G},V)$ of a Lie
group $G$ on a complex Hilbert space $V$. If the representation
$(M_{G},V)$ is irreducible then any normal operator $N$ of $(M_{G},V)$ is a
scalar.
Lemma 2.26.
Consider an irreducible unitary representation $(M_{G},V)$ on a complex
Hilbert space. An anti-isometry of $(M_{G},V)$, if it exists, is unique
up to a scalar isometry of $(M_{G},V)$.
Proof.
Let $\theta_{1}$,$\theta_{2}$ be two anti-isometries of $(M_{G},V)$. The
product $(\theta_{2}\theta_{1})$ is an isometry of $(M_{G},V)$;
since $(M_{G},V)$ is irreducible,
$(\theta_{2}\theta_{1})=e^{i\phi}$; with $\phi\in\mathbb{R}$.
Therefore $\theta_{2}=\alpha\theta_{1}\alpha^{-1}$; where
$\alpha\equiv e^{i\frac{\phi}{2}}$ is a scalar isometry of $(M_{G},V)$.
∎
Proposition 2.27.
Two R-real irreducible unitary representations are
isometric iff their complexifications are isometric.
Proof.
Let $(M_{G},V)$ and $(N_{G},W)$ be C-real irreducible representations,
with $\theta_{M}$ and $\theta_{N}$ the respective C-conjugation operators.
If there is an isometry $\alpha:V\to W$ such that
$\alpha M_{G}(g)=N_{G}(g)\alpha$ for all $g\in G$, then
$\vartheta\equiv\alpha\theta_{M}\alpha^{-1}$ is an anti-isometry of
$(N_{G},W)$. Since it is unique up to a phase, then
$\theta_{N}=e^{i\phi}\vartheta$. Therefore $e^{i\frac{\phi}{2}}\alpha$ is an
isometry between $(M_{G},V_{\theta})$ and $(N_{G},W_{\theta})$, where
$V_{\theta_{M}}\equiv\{(1+\theta_{M})v:v\in V\}$.
∎
Proposition 2.28.
Two C-complex or C-pseudoreal irreducible unitary
representations are isomorphic or anti-isomorphic iff their
realifications are isomorphic.
Proof.
Let $(M_{G},V)$ and $(N_{G},W)$ be R-complex or R-pseudoreal irreducible
representations, with $J_{M}$ and $J_{N}$ the respective R-imaginary
operators. If there is an isometry $\alpha:V\to W$ such that
$\alpha M_{G}(g)=N_{G}(g)\alpha$ for all $g\in G$, then
$K\equiv\alpha J_{M}\alpha^{-1}$ is a R-imaginary operator of
$(N_{G},W)$.
When considering $(N_{G},W_{J_{N}})$ and $(M_{G},V_{J_{M}})$, where
$W_{J_{N}}\equiv\{(1-iJ_{N})w:w\in W\}$, we get that
$(1-J_{N}K)(1-KJ_{N})=r$ as an operator of $W_{J_{N}}$, where $r$ is a
non-negative null real scalar. If $c=0$ then $K=-J_{N}$ and $\alpha$ defines an
anti-isometry between $(M_{G},V_{J_{M}})$ and $(N_{G},W_{J_{N}})$.
If $c\neq 0$ then $(1-J_{N}K)\alpha c^{-\frac{1}{2}}$ is an isometry between
$(M_{G},V_{J_{M}})$ and $(N_{G},W_{J_{N}})$.
∎
Definition 2.29.
A unitary representation is completely reducible iff it can be expressed as a
direct integral of irreducible representations.
Remark 2.30.
All unitary representations of a separable locally compact group
(such as the Poincare group) are completely reducible.
3 Finite-dimensional representations of the Lorentz group
3.1 Majorana, Dirac and Pauli Matrices and Spinors
Definition 3.31.
$\mathbb{F}^{m\times n}$ is the vector space of $m\times n$ matrices whose
entries are elements of the field $\mathbb{F}$.
In the next remark we state the Pauli’s fundamental theorem of gamma
matrices. The proof can be found in the referencediracmatrices .
Remark 3.32 (Pauli’s fundamental theorem).
Let $A^{\mu}$, $B^{\mu}$, $\mu\in\{0,1,2,3\}$, be two sets of
$4\times 4$ complex matrices verifying:
$$\displaystyle A^{\mu}A^{\nu}+A^{\nu}A^{\mu}$$
$$\displaystyle=-2\eta^{\mu\nu}$$
(1)
$$\displaystyle B^{\mu}B^{\nu}+B^{\nu}B^{\mu}$$
$$\displaystyle=-2\eta^{\mu\nu}$$
(2)
Where $\eta^{\mu\nu}\equiv diag(+1,-1,-1-1)$ is the Minkowski metric.
1) There is an invertible complex matrix $S$ such that
$B^{\mu}=SA^{\mu}S^{-1}$, for all $\mu\in\{0,1,2,3\}$.
$S$ is unique up to a non-null scalar.
2) If $A^{\mu}$ and $B^{\mu}$ are all unitary, then $S$ is unitary.
Proposition 3.33.
Let $\alpha^{\mu}$, $\beta^{\mu}$, $\mu\in\{0,1,2,3\}$, be two sets of
$4\times 4$ real matrices verifying:
$$\displaystyle\alpha^{\mu}\alpha^{\nu}+\alpha^{\nu}\alpha^{\mu}$$
$$\displaystyle=-2\eta^{\mu\nu}$$
(3)
$$\displaystyle\beta^{\mu}\beta^{\nu}+\beta^{\nu}\beta^{\mu}$$
$$\displaystyle=-2\eta^{\mu\nu}$$
(4)
Then there is a real matrix $S$, with $|detS|=1$, such that
$\beta^{\mu}=S\alpha^{\mu}S^{-1}$, for all $\mu\in\{0,1,2,3\}$. $S$ is unique up to a signal.
Proof.
From remark 3.32, we know that there is an
invertible matrix $T^{\prime}$, unique up to a non-null scalar, such that $\beta^{\mu}=T^{\prime}\alpha^{\mu}T^{{}^{\prime}-1}$.
Then $T\equiv T^{\prime}/|det(T^{\prime})|$ has $|detT|=1$ and it is unique up to a
complex phase.
Conjugating the previous equation, we get $\beta^{\mu}=T^{*}\alpha^{\mu}T^{*-1}$.
Then $T^{*}=e^{i2\theta}T$ for some real number $\theta$.
Therefore $S\equiv e^{i\theta}T$ is a real matrix,
with $|detS|=1$, unique up to a signal.
∎
Definition 3.34.
The Majorana matrices, $i\gamma^{\mu}$, $\mu\in\{0,1,2,3\}$, are $4\times 4$ complex unitary matrices verifying:
$$\displaystyle(i\gamma^{\mu})(i\gamma^{\nu})+(i\gamma^{\nu})(i\gamma^{\mu})$$
$$\displaystyle=-2\eta^{\mu\nu}$$
(5)
The Dirac matrices are $\gamma^{\mu}\equiv-i(i\gamma^{\mu})$.
In the Majorana bases, the Majorana matrices are $4\times 4$ real
orthogonal matrices. An example of the Majorana matrices in a
particular Majorana basis is:
$$\displaystyle\begin{array}[]{llllll}i\gamma^{1}=&\left[\begin{smallmatrix}+1&0%
&0&0\\
0&-1&0&0\\
0&0&-1&0\\
0&0&0&+1\end{smallmatrix}\right]&i\gamma^{2}=&\left[\begin{smallmatrix}0&0&+1&%
0\\
0&0&0&+1\\
+1&0&0&0\\
0&+1&0&0\end{smallmatrix}\right]&i\gamma^{3}=\left[\begin{smallmatrix}0&+1&0&0%
\\
+1&0&0&0\\
0&0&0&-1\\
0&0&-1&0\end{smallmatrix}\right]\\
\\
i\gamma^{0}=&\left[\begin{smallmatrix}0&0&+1&0\\
0&0&0&+1\\
-1&0&0&0\\
0&-1&0&0\end{smallmatrix}\right]&i\gamma^{5}=&\left[\begin{smallmatrix}0&-1&0&%
0\\
+1&0&0&0\\
0&0&0&+1\\
0&0&-1&0\end{smallmatrix}\right]&=-\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}%
\end{array}$$
(6)
In reference realgamma it is proved that the set of five
anti-commuting $4\times 4$ real matrices is unique up to
isomorphisms. So, for instance, with $4\times 4$ real matrices it is not possible
to obtain the euclidean signature for the metric.
Definition 3.35.
The Dirac spinor is a $4\times 1$ complex column matrix, $\mathbb{C}^{4\times 1}$.
The space of Dirac spinors is a 4 dimensional complex vector space.
Lemma 3.36.
The charge conjugation operator $\Theta$, is an anti-linear involution
commuting with the Majorana matrices $i\gamma^{\mu}$.
It is unique up to a complex phase.
Proof.
In the Majorana bases, the complex conjugation is a charge conjugation
operator. Let $\Theta$ and $\Theta^{\prime}$ be two charge conjugation operators
operators. Then, $\Theta\Theta^{\prime}$ is a complex invertible matrix commuting with
$i\gamma^{\mu}$, therefore, from Pauli’s fundamental theorem,
$\Theta\Theta^{\prime}=c$, where $c$ is a non-null complex scalar.
Therefore $\Theta^{\prime}=c^{*}\Theta$ and from $\Theta^{\prime}\Theta^{\prime}=1$, we get that
$c^{*}c=1$.
∎
Definition 3.37.
Let $\Theta$ be a charge conjugation operator.
The set of Majorana spinors, $Pinor$, is the set of Dirac spinors
verifying the Majorana condition (defined up to a complex phase):
$$\displaystyle Pinor\equiv\{u\in\mathbb{C}^{4\times 1}:\Theta u=u\}$$
(7)
The set of Majorana spinors is a 4 dimensional real vector space.
Note that the linear combinations of
Majorana spinors with complex scalars do not verify the Majorana
condition.
There are 16 linear independent products of Majorana matrices. These
form a basis of the real vector space of endomorphisms of Majorana spinors,
$End(Pinor)$. In the Majorana bases, $End(Pinor)$ is the vector space of
$4\times 4$ real matrices.
Definition 3.38.
The Pauli matrices $\sigma^{k},\ k\in\{1,2,3\}$ are $2\times 2$
hermitian, unitary, anti-commuting, complex matrices.
The Pauli spinor is a $2\times 1$ complex column matrix. The space of
Pauli spinors is denoted by $Pauli$.
The space of Pauli spinors, $Pauli$, is a 2 dimensional complex vector
space and a 4 dimensional real vector space. The realification of
the space of Pauli spinors is isomorphic to the space of Majorana
spinors.
3.2 On the Lorentz, SL(2,C) and Pin(3,1) groups
Remark 3.39.
The Lorentz group, $O(1,3)\equiv\{\lambda\in\mathbb{R}^{4\times 4}:\lambda^{T}\eta\lambda=\eta\}$, is the set of
real matrices that leave the metric, $\eta=diag(1,-1,-1,-1)$,
invariant.
The proper orthochronous Lorentz subgroup is defined by
$SO^{+}(1,3)\equiv\{\lambda\in O(1,3):det(\lambda)=1,\lambda^{0}_{\ 0}>0\}$.
It is a normal subgroup.
The discrete Lorentz subgroup of parity and time-reversal is
$\Delta\equiv\{1,\eta,-\eta,-1\}$.
The Lorentz group is the semi-direct product of the previous
subgroups, $O(1,3)=\Delta\ltimes SO^{+}(1,3)$.
Definition 3.40.
The set $Maj$ is the 4 dimensional real space of the linear
combinations of the Majorana matrices, $i\gamma^{\mu}$:
$$\displaystyle Maj\equiv\{a_{\mu}i\gamma^{\mu}:a_{\mu}\in\mathbb{R},\ \mu\in\{0%
,1,2,3\}\}$$
(8)
Definition 3.41.
$Pin(3,1)$ pin is the group of endomorphisms of Majorana
spinors that leave the space $Maj$ invariant, that is:
$$\displaystyle Pin(3,1)\equiv\Big{\{}S\in End(Pinor):\ |detS|=1,\ S^{-1}(i%
\gamma^{\mu})S\in Maj,\ \mu\in\{0,1,2,3\}\Big{\}}$$
(9)
Proposition 3.42.
The map $\Lambda:Pin(3,1)\to O(1,3)$ defined by:
$$\displaystyle(\Lambda(S))^{\mu}_{\ \nu}i\gamma^{\nu}\equiv S^{-1}(i\gamma^{\mu%
})S$$
(10)
is two-to-one and surjective. It defines a group homomorphism.
Proof.
1) Let $S\in Pin(3,1)$. Since the Majorana matrices are a basis of the
real vector space $Maj$, there is an unique real matrix $\Lambda(S)$ such that:
$$\displaystyle(\Lambda(S))^{\mu}_{\ \nu}i\gamma^{\nu}=S^{-1}(i\gamma^{\mu})S$$
(11)
Therefore, $\Lambda$ is a map with domain $Pin(3,1)$. Now we can check
that $\Lambda(S)\in O(1,3)$:
$$\displaystyle(\Lambda(S))^{\mu}_{\ \alpha}\eta^{\alpha\beta}(\Lambda(S))^{\nu}%
_{\ \beta}=-\frac{1}{2}(\Lambda(S))^{\mu}_{\ \alpha}\{i\gamma^{\alpha},i\gamma%
^{\beta}\}(\Lambda(S))^{\nu}_{\ \beta}=$$
(12)
$$\displaystyle=-\frac{1}{2}S\{i\gamma^{\mu},i\gamma^{\nu}\}S^{-1}=S\eta^{\mu\nu%
}S^{-1}=\eta^{\mu\nu}$$
(13)
We have proved that $\Lambda$ is a map from $Pin(3,1)$ to $O(1,3)$.
2) Since any $\lambda\in O(1,3)$ conserve the metric $\eta$, the matrices
$\alpha^{\mu}\equiv\lambda^{\mu}_{\ \nu}i\gamma^{\nu}$ verify:
$$\displaystyle\{\alpha^{\mu},\alpha^{\nu}\}=-2\lambda^{\mu}_{\ \alpha}\eta^{%
\alpha\beta}\lambda^{\nu}_{\ \beta}=-2\eta^{\mu\nu}$$
(14)
In a basis where the Majorana matrices are real, from Proposition
3.33 there is a real invertible matrix $S_{\lambda}$,
with $|detS_{\Lambda}|=1$, such that $\lambda^{\mu}_{\ \nu}i\gamma^{\nu}=S^{-1}_{\lambda}(i\gamma^{\mu})S_{\lambda}$.
The matrix $S_{\Lambda}$ is unique up to a sign. So, $\pm S_{\lambda}\in Pin(3,1)$ and we proved that the map
$\Lambda:Pin(3,1)\to O(1,3)$ is two-to-one and surjective.
3) The map defines a group homomorphism because:
$$\displaystyle\Lambda^{\mu}_{\ \nu}(S_{1})\Lambda^{\nu}_{\ \rho}(S_{2})i\gamma^%
{\rho}=\Lambda^{\mu}_{\ \nu}S_{2}^{-1}i\gamma^{\nu}S_{2}$$
(15)
$$\displaystyle=S_{2}^{-1}S_{1}^{-1}i\gamma^{\mu}S_{1}S_{2}=\Lambda^{\mu}_{\ %
\rho}(S_{1}S_{2})i\gamma^{\rho}$$
(16)
∎
Remark 3.43.
The group $SL(2,\mathbb{C})=\{e^{\theta^{j}i\sigma^{j}+b^{j}\sigma^{j}}:\theta^{j},b^{j}%
\in\mathbb{R},\ j\in\{1,2,3\}\}$ is simply connected.
Its projective representations are equivalent to its ordinary representationsweinberg .
There is a two-to-one, surjective map $\Upsilon:SL(2,\mathbb{C})\to SO^{+}(1,3)$, defined by:
$$\displaystyle\Upsilon^{\mu}_{\ \nu}(T)\sigma^{\nu}\equiv T^{\dagger}\sigma^{%
\mu}T$$
(17)
Where $T\in SL(2,\mathbb{C})$, $\sigma^{0}=1$ and $\sigma^{j}$, $j\in\{1,2,3\}$ are the Pauli matrices.
Lemma 3.44.
Consider that $\{M_{+},M_{-},i\gamma^{5}M_{+},i\gamma^{5}M_{-}\}$ and $\{P_{+},P_{-},iP_{+},iP_{-}\}$ are orthonormal basis of
the 4 dimensional real vector spaces $Pinor$ and $Pauli$, respectively, verifying:
$$\displaystyle\gamma^{0}\gamma^{3}M_{\pm}=\pm M_{\pm}$$
$$\displaystyle,\ \sigma^{3}P_{\pm}=\pm P_{\pm}$$
(18)
The isomorphism
$\Sigma:Pauli\to Pinor$ is defined by:
$$\displaystyle\Sigma(P_{+})=M_{+},$$
$$\displaystyle\ \Sigma(iP_{+})=i\gamma^{5}M_{+}$$
(19)
$$\displaystyle\Sigma(P_{-})=M_{-},$$
$$\displaystyle\ \Sigma(iP_{-})=i\gamma^{5}M_{-}$$
(20)
The group $Spin^{+}(3,1)\equiv\{\Sigma\circ A\circ\Sigma^{-1}:A\in SL(2,\mathbb{C})\}$ is a subgroup of $Pin(1,3)$.
For all $S\in Spin^{+}(1,3)$, $\Lambda(S)=\Upsilon(\Sigma^{-1}\circ S\circ\Sigma)$.
Proof.
From remark 3.43, $Spin^{+}(3,1)=\{e^{\theta^{j}i\gamma^{5}\gamma^{0}\gamma^{j}+b^{j}\gamma^{0}%
\gamma^{j}}:\theta^{j},b^{j}\in\mathbb{R},\ j\in\{1,2,3\}\}$.
Then, for all $T\in SL(2,C)$:
$$\displaystyle-i\gamma^{0}\Sigma\circ T^{\dagger}\circ\Sigma^{-1}i\gamma^{0}$$
$$\displaystyle=\Sigma\circ T^{-1}\circ\Sigma^{-1}$$
(21)
Now, the map $\Upsilon:SL(2,\mathbb{C})\to SO^{+}(1,3)$ is given by:
$$\displaystyle\Upsilon^{\mu}_{\ \nu}(T)i\gamma^{\nu}=(\Sigma\circ T^{-1}\circ%
\Sigma^{-1})i\gamma^{\mu}(\Sigma\circ T\circ\Sigma^{-1})$$
(22)
Then, all $S\in Spin^{+}(3,1)$ leaves the space $Maj$ invariant:
$$\displaystyle S^{-1}i\gamma^{\mu}S=\Upsilon^{\mu}_{\ \nu}(\Sigma^{-1}\circ S%
\circ\Sigma)i\gamma^{\nu}\in Maj$$
(23)
Since all the products of Majorana matrices, except the identity, are
traceless, then $det(S)=1$. So,
$Spin^{+}(3,1)$ is a subgroup of $Pin(1,3)$ and $\Lambda(S)=\Upsilon(\Sigma^{-1}\circ S\circ\Sigma)$.
∎
Definition 3.45.
The discrete Pin subgroup $\Omega\subset Pin(3,1)$ is:
$$\displaystyle\Omega\equiv\{\pm 1,\pm i\gamma^{0},\pm\gamma^{0}\gamma^{5},\pm i%
\gamma^{5}\}$$
(24)
The previous lemma and the fact that $\Lambda$ is continuous,
implies that $Spin^{+}(1,3)$ is a double cover of $SO^{+}(3,1)$.
We can check that for all $\omega\in\Omega$, $\Lambda(\pm\omega)\in\Delta$.
That is, the discrete Pin subgroup is the double cover of the
discrete Lorentz subgroup. Therefore, $Pin(3,1)=\Omega\ltimes Spin^{+}(1,3)$
Since there is a two-to-one continuous surjective group homomorphism,
$Pin(3,1)$ is a double cover of $O(1,3)$, $Spin^{+}(3,1)$
is a double cover of $SO^{+}(1,3)$ and $Spin^{+}(1,3)\cap SU(4)$ is a
double cover of $SO(3)$. We can check that $Spin^{+}(1,3)\cap SU(4)$ is
equivalent to $SU(2)$.
3.3 Finite-dimensional representations of SL(2,C)
Remark 3.46.
Since SL(2,C) is a semisimple Lie group, all its finite-dimensional
(real or complex) representations are direct sums of irreducible
representations.
Remark 3.47.
The finite-dimensional complex irreducible representations of SL(2,C)
are labeled by $(m,n)$, where $2m,2n$ are natural numbers.
Up to equivalence, the representation space $V_{(m,n)}$ is the tensor
product of the complex vector spaces $V_{m}^{+}$ and $V_{n}^{-}$, where $V_{m}^{\pm}$ is a
symmetric tensor with $2m$ Dirac spinor indexes, such that
$\gamma^{5}_{\ k}v=\pm v$, where $v\in V_{m}^{\pm}$ and $\gamma^{5}_{\ k}$
is the Dirac matrix $\gamma^{5}$ acting on the $k$-th index of $v$.
The group homomorphism consists in applying the same matrix of
$Spin^{+}(1,3)$, correspondent to the $SL(2,C)$ group element we are
representing, to each index of $v$.
$V_{(0,0)}$ is equivalent to $\mathbb{C}$ and the image of the group
homomorphism is the identity.
These are also projective representations of the time reversal transformation,
but, for $m\neq n$, not of the parity transformation, that is,
under the parity transformation, $(V^{+}_{m}\otimes V^{-}_{n})\to(V^{-}_{m}\otimes V^{+}_{n})$ and under the time
reversal transformation $(V^{+}_{m}\otimes V^{-}_{n})\to(V^{+}_{m}\otimes V^{-}_{n})$.
Lemma 3.48.
The finite-dimensional real irreducible representations of SL(2,C)
are labeled by $(m,n)$, where $2m,2n$ are natural numbers and $m\geq n$.
Up to equivalence, the representation space $W_{(m,n)}$ is defined
for $m\neq n$ as:
$$\displaystyle W_{(m,n)}$$
$$\displaystyle\equiv\{\frac{1+(i\gamma^{5})_{1}\otimes(i\gamma^{5})_{1}}{2}w:w%
\in W_{m}\otimes W_{n}\}$$
$$\displaystyle W_{(m,m)}$$
$$\displaystyle\equiv\{\frac{1+(i\gamma^{5})_{1}\otimes(i\gamma^{5})_{1}}{2}w:w%
\in(W_{m})^{2}\}$$
where $W_{m}$ is a
symmetric tensor with $m$ Majorana spinor indexes, such that
$(i\gamma^{5})_{1}(i\gamma^{5})_{k}w=-w$, where $w\in W_{m}$; $(i\gamma^{5})_{\ k}$
is the Majorana matrix $i\gamma^{5}$ acting on the $k$-th index of $w$;
$(W_{m})^{2}$ is the space of the linear combinations of the
symmetrized tensor products $(u\otimes v+v\otimes u)$, for $u,v\in W_{m}$.
The group homomorphism consists in applying the same matrix of
$Spin^{+}(1,3)$, correspondent to the $SL(2,C)$ group element we are
representing, to each index of the tensor. In the $(0,0)$ case,
$W_{(0,0)}$ is equivalent to $\mathbb{R}$ and the
image of the group homomorphism is the identity.
These are also projective representations of the full Lorentz group,
that is, under the parity or time reversal transformations,
$(W_{m,n}\to W_{m,n})$.
Proof.
For $m\neq n$ the complex irreducible representations of SL(2,C) are
C-complex. The complexification of $W_{(m,n)}$ verifies
$W_{(m,n)}^{c}=(V^{+}_{m}\otimes V^{-}_{n})\oplus(V^{-}_{m}\otimes V^{+}_{n})$.
For $m=n$ the complex irreducible representations of SL(2,C) are C-real.
In a Majorana basis, the C-conjugation operator of $V_{(m,m)}$,
$\theta$, is defined as $\theta(u\otimes v)\equiv v^{*}\otimes u^{*}$,
where $u\in V^{+}_{m}$ and $v\in V^{-}_{m}$.
We can check that there is a bijection $\alpha:W_{(m,m)}\to(V_{(m,m)})_{\theta}$,
defined by $\alpha(w)\equiv\frac{1-i(i\gamma^{5})_{1}\otimes 1}{2}w$;
$\alpha^{-1}(v)\equiv v+v^{*}$, for $w\in W_{(m,m)}$, $v\in(V_{(m,m)})_{\theta}$.
Using the map from chapter 1,
we can check that the representations
$W_{(m,n)}$, with $m\geq n$, are the unique
finite-dimensional real irreducible representations of SL(2,C), up to
isomorphisms.
We can check that $W_{(m,n)}^{c}$ is equivalent to $W_{(n,m)}^{c}$,
therefore, invariant under the parity or time reversal transformations.
∎
As examples of real irreducible representations of $SL(2,C)$ we have
for $(1/2,0)$ the Majorana spinor, for $(1/2,1/2)$ the linear
combinations of the matrices $\{1,\gamma^{0}\vec{\gamma}\}$, for $(1,0)$ the linear combinations of the
matrices $\{i\vec{\gamma},\vec{\gamma}\gamma^{5}\}$. The group homomorphism is defined as $M(S)(u)\equiv Su$ and
$M(S)(A)\equiv SAS^{\dagger}$, for $S\in Spin^{+}(1,3)$,
$u\in Pinor$, $A\in\{1,\vec{\gamma}\gamma^{0}\}$ or $A\in\{i\vec{\gamma},\vec{\gamma}\gamma^{5}\}$.
We can check that the domain
of $M$ can be extended to $Pin(1,3)$, leaving the considered vector
spaces invariant. For $m=n$, we can define the “pseudo-representation”
$W_{(m,m)}^{\prime}\equiv\{((i\gamma^{5})_{1}\otimes 1)w:w\in W_{(m,m)}\}$
which is equivalent to $W_{(m,m)}$ as an $SL(2,C)$ representation, but
under parity transforms with the opposite sign.
As an example, the “pseudo-representation” $(1/2,1/2)$ is defined as
the linear combinations of the matrices $\{i\gamma^{5},i\gamma^{5}\vec{\gamma}\gamma^{0}\}$.
4 Unitary representations of the Poincare group
4.1 Bargmann-Wigner fields
Definition 4.49.
Consider that $\{M_{+},M_{-},i\gamma^{0}M_{+},i\gamma^{0}M_{-}\}$ and $\{P_{+},P_{-},iP_{+},iP_{-}\}$ are orthonormal basis of
the 4 dimensional real vector spaces $Pinor$ and $Pauli$, respectively, verifying:
$$\displaystyle\gamma^{3}\gamma^{5}M_{\pm}=\pm M_{\pm}$$
$$\displaystyle,\ \sigma^{3}P_{\pm}=\pm P_{\pm}$$
Let $H$ be a real Hilbert space.
For all $h\in H$, the bijective linear map
$\Theta_{H}:Pauli\otimes_{\mathbb{R}}H\to Pinor\otimes_{\mathbb{R}}H$ is defined by:
$$\displaystyle\Theta_{H}(h\otimes_{\mathbb{R}}P_{+})=h\otimes_{\mathbb{R}}M_{+},$$
$$\displaystyle\ \Theta_{H}(h\otimes_{\mathbb{R}}iP_{+})=h\otimes_{\mathbb{R}}i%
\gamma^{0}M_{+}$$
$$\displaystyle\Theta_{H}(h\otimes_{\mathbb{R}}P_{-})=h\otimes_{\mathbb{R}}M_{-},$$
$$\displaystyle\ \Theta_{H}(h\otimes_{\mathbb{R}}iP_{-})=h\otimes_{\mathbb{R}}i%
\gamma^{0}M_{-}$$
Definition 4.50.
Let $H_{n}$, with $n\in\{1,2\}$, be two real Hilbert spaces
and $U:Pauli\otimes_{\mathbb{R}}H_{1}\to Pauli\otimes_{\mathbb{R}}H_{2}$ be an operator.
The operator $U^{\Theta}:Pinor\otimes_{\mathbb{R}}H_{1}\to Pinor\otimes_{\mathbb{R}}H_{2}$ is defined as
$U^{\Theta}\equiv\Theta_{H_{2}}\circ U\circ\Theta^{-1}_{H_{1}}$.
The space of Majorana spinors is isomorphic to
the realification of the space of Pauli spinors.
Definition 4.51.
The real Hilbert space
$Pinor(\mathbb{X})\equiv Pinor\otimes L^{2}(\mathbb{X})$
is the space of square integrable
functions with domain $\mathbb{X}$ and image in $Pinor$.
Definition 4.52.
The complex Hilbert space
$Pauli(\mathbb{X})\equiv Pauli\otimes L^{2}(\mathbb{X})$
is the space of square integrable functions with domain $\mathbb{X}$
and image in $Pauli$.
Remark 4.53.
The Fourier Transform
$\mathcal{F}_{P}:Pauli(\mathbb{R}^{3})\to Pauli(\mathbb{R}^{3})$ is an unitary operator defined by:
$$\displaystyle\mathcal{F}_{P}\{\psi\}(\vec{p})\equiv\int d^{n}\vec{x}\frac{e^{-%
i\vec{p}\cdot\vec{x}}}{\sqrt{(2\pi)^{n}}}\psi(\vec{x}),\ \psi\in Pauli(\mathbb%
{R}^{3})$$
Where the domain of the integral is $\mathbb{R}^{3}$.
Remark 4.54.
The inverse Fourier transform verifies:
$$\displaystyle-\vec{\partial}^{2}\ \mathcal{F}_{P}^{-1}\{\psi\}(\vec{x})$$
$$\displaystyle=(\mathcal{F}_{P}^{-1}\circ R)\{\psi\}(\vec{x})$$
$$\displaystyle i\vec{\partial}_{k}\ \mathcal{F}_{P}^{-1}\{\psi\}(\vec{x})$$
$$\displaystyle=(\mathcal{F}_{P}^{-1}\circ R_{k}^{\prime})\{\psi\}(\vec{x})$$
Where $\psi\in Pauli(\mathbb{R}^{3})$ and
$R,R_{k}^{\prime}:Pauli(\mathbb{R}^{3})\to Pauli(\mathbb{R}^{3})$, with $k\in\{1,2,3\}$, are
linear maps defined by:
$$\displaystyle R\{\psi\}(\vec{p})$$
$$\displaystyle\equiv(\vec{p})^{2}\psi(\vec{p})$$
$$\displaystyle R_{k}^{\prime}\{\psi\}(\vec{p})$$
$$\displaystyle\equiv\vec{p}_{k}\ \psi(\vec{p})$$
Definition 4.55.
Let $\vec{x}\in\mathbb{R}^{3}$. The spherical coordinates
parametrization is:
$$\displaystyle\vec{x}=r(\sin(\theta)\sin(\varphi)\vec{e_{1}}+\sin(\theta)\sin(%
\varphi)\vec{e_{2}}+\cos(\theta)\vec{e}_{3})$$
where $\{\vec{e}_{1},\vec{e}_{2},\vec{e}_{3}\}$ is a fixed orthonormal basis of
$\mathbb{R}^{3}$ and $r\in[0,+\infty[$, $\theta\in[0,\pi]$, $\varphi\in[-\pi,\pi]$.
Definition 4.56.
Let
$$\displaystyle\mathbb{S}^{3}\equiv\{(p,l,\mu):p\in\mathbb{R}_{\geq 0};l,\mu\in%
\mathbb{Z};l\geq 0;-l\leq\mu\leq l\}$$
The Hilbert space $L^{2}(\mathbb{S}^{3})$ is the real Hilbert space of real
Lebesgue square integrable functions of $\mathbb{S}^{3}$. The internal product is:
$$\displaystyle<f,g>=\sum_{l=0}^{+\infty}\sum_{\mu=-l}^{l-1}\int_{0}^{+\infty}%
dpf(p,l,\mu)g(p,l,\mu),\ f,g\in L^{2}(\mathbb{S}^{3})$$
Definition 4.57.
The Spherical transform
$\mathcal{H}_{P}:Pauli(\mathbb{R}^{3})\to Pauli(\mathbb{S}^{3})$
is an operator defined by:
$$\displaystyle\mathcal{H}_{P}\{\psi\}(p,l,\mu)\equiv\int r^{2}drd(\cos\theta)d%
\varphi\frac{2p}{\sqrt{2\pi}}j_{l}(pr)Y_{l\mu}(\theta,\varphi)\psi(r,\theta,%
\varphi),\ \psi\in Pauli(\mathbb{R}^{3})$$
The domain of the integral is $\mathbb{R}^{3}$. The spherical
Bessel function of the first kind $j_{l}$ bessel ,
the spherical harmonics $Y_{l\mu}$harmonics and the associated Legendre
functions of the first kind $P_{l\mu}$ are:
$$\displaystyle j_{l}(r)\equiv$$
$$\displaystyle r^{l}\Big{(}-\frac{1}{r}\frac{d}{dr}\Big{)}^{l}\frac{\sin r}{r}$$
$$\displaystyle Y_{l\mu}(\theta,\varphi)\equiv$$
$$\displaystyle\sqrt{\frac{2l+1}{4\pi}\frac{(l-m)!}{(l+m)!}}P_{l}^{\mu}(\cos%
\theta)e^{i\mu\varphi}$$
$$\displaystyle P_{l}^{\mu}(\xi)\equiv$$
$$\displaystyle\frac{(-1)^{\mu}}{2^{l}l!}(1-\xi^{2})^{\mu/2}\frac{\mathrm{d}^{l+%
\mu}}{\mathrm{d}\xi^{l+\mu}}(\xi^{2}-1)^{l}$$
Remark 4.58.
Due to the properties of spherical harmonics and Bessel functions, the
Spherical transform is an unitary operator. The inverse
Spherical transform verifies:
$$\displaystyle-\vec{\partial}^{2}\ \mathcal{H}_{P}^{-1}\{\psi\}(\vec{x})$$
$$\displaystyle=(\mathcal{H}_{P}^{-1}\circ R)\{\psi\}(\vec{x})$$
$$\displaystyle(-x^{1}i\partial_{2}+x^{2}i\partial_{1})\ \mathcal{H}_{P}^{-1}\{%
\psi\}(\vec{x})$$
$$\displaystyle=(\mathcal{H}_{P}^{-1}\circ R^{\prime})\{\psi\}(\vec{x})$$
Where $\psi\in Pauli(\mathbb{S}^{3})$ and $R,R^{\prime}:Pauli(\mathbb{S}^{3})\to Pauli(\mathbb{S}^{3})$ are
linear maps defined by:
$$\displaystyle R\{\psi\}(p,l,\mu)$$
$$\displaystyle\equiv p^{2}\psi(p,l,\mu)$$
$$\displaystyle R^{\prime}\{\psi\}(p,l,\mu)$$
$$\displaystyle\equiv\mu\ \psi(p,l,\mu)$$
Definition 4.59.
The real vector space $Pinor_{j}$, with $2j$ a positive integer, is the
space of linear combinations of the tensor products of $2j$
Majorana spinors, symmetric on the spinor indexes. The real vector
space $Pinor_{0}$ is the space of linear combinations of the tensor
products of $2$ Majorana spinors, anti-symmetric on the spinor
indexes.
Definition 4.60.
The real Hilbert space
$Pinor_{j}(\mathbb{X})\equiv Pinor_{j}\otimes L^{2}(\mathbb{X})$ is the
space of square integrable functions with domain $\mathbb{X}$ and
image in $Pinor_{j}$.
Definition 4.61.
The Hilbert space $Pinor_{j,n}$, with $(j-\nu)$ an
integer and $-j\leq n\leq j$ is defined as:
$$\displaystyle Pinor_{j,n}\equiv\{\Psi\in Pinor_{j}:\sum_{k=1}^{k=2j}(\gamma^{0%
})_{1}\Big{(}\gamma^{0}\gamma^{3}\gamma^{5}\Big{)}_{k}\Psi=2n\Psi\}$$
Where $\Big{(}\gamma^{3}\gamma^{5}\Big{)}_{k}$ is the matrix $\gamma^{3}\gamma^{5}$
acting on the Majorana index $k$. Given
Definition 4.62.
The Spherical transform
$\mathcal{H}_{P}^{\prime}:Pinor_{j}(\mathbb{R}^{3})\to Pinor_{j}(\mathbb{S}^{3})$
is an operator defined by:
$$\displaystyle\mathcal{H}_{P}^{\prime}\{\psi\}(p,l,J,\nu)\equiv\sum_{\mu=-l}^{l%
}\sum_{n=-j}^{j}<l\mu jn|J\nu>\Big{(}\mathcal{H}_{P}^{\Theta}\Big{)}_{1}\{\psi%
\}(p,l,\mu,n),\ \psi\in Pinor_{j}(\mathbb{R}^{3})$$
$<l\mu jn|J\nu>$ are the Clebsh-Gordon coefficients and
$\psi(p,l,\mu,n)\in Pinor_{j,n}$ such that
$\psi(p,l,\mu)=\sum_{n=-j}^{j}\psi(p,l,\mu,n)$. $(j-n)$, $(J-\nu)$ and
$(J-j)$ are integers, with $-J\leq\nu\leq J$ and $|j-l|\leq J\leq j+l$.
$\Big{(}\mathcal{H}_{P}^{\Theta}\Big{)}_{1}$ is the realification of the
transform $\mathcal{H}_{P}$, with the imaginary number replaced by the
matrix $i\gamma^{0}$ acting on the first Majorana index of $\psi$.
Proposition 4.63.
Consider a unitary operator $U:Pinor_{j}(X)\to Pinor_{j}(\mathbb{R}^{3})$,
defined by
$U\{\Psi\}(\vec{x})\equiv\int_{\mathbb{X}}dXU(\vec{x},X)\Psi(X)$ and such that
$H^{2}\circ U=U\circ E^{2}$, where
$$\displaystyle iH\{\Psi\}(\vec{x})\equiv\Big{(}\gamma^{0}\vec{\not{\partial}}+i%
\gamma^{0}m\Big{)}_{k}\Psi(\vec{x})$$
the Majorana matrices act on some Majorana index $k$;
$E^{2}\{\Phi\}(X)\equiv E^{2}(X)\Phi(X)$ with $E(X)\geq m\geq 0$ a real
number.
Then the operator $U^{\prime}:Pinor(X)\to Pinor(\mathbb{R}^{3})$ is unitary,
where $U^{\prime}$ is defined by:
$$\displaystyle U^{\prime}\{\Psi\}(\vec{x})\equiv\int_{\mathbb{X}}dX\frac{E(X)+H%
(\vec{x})\gamma^{0}}{\sqrt{E(X)+m}\sqrt{2E(X)}}U(\vec{x},X)\Psi(X)$$
Proof.
We have that
$$\displaystyle<U^{\prime}\{\Psi\},U^{\prime}\{\Psi\}>=\int d^{3}\vec{x}dXdY$$
$$\displaystyle\Psi^{\dagger}(Y)U^{\dagger}(\vec{x},X)\frac{E(Y)+\gamma^{0}H(%
\vec{x})}{\sqrt{E(Y)+m}\sqrt{2E(Y)}}\frac{E(X)+H(\vec{x})\gamma^{0}}{\sqrt{E(X%
)+m}\sqrt{2E(X)}}U(\vec{x},X)\Psi(X)$$
From the symmetry of
$H^{2}(\vec{x})$, in the integral we can set $E^{2}(X)=E^{2}(Y)$
and hence $E(X)=E(Y)$. Since we have:
$$\displaystyle\frac{E(X)+\gamma^{0}H(\vec{x})}{\sqrt{E(X)+m}\sqrt{2E(X)}}\frac{%
E(X)+H(\vec{x})\gamma^{0}}{\sqrt{E(X)+m}\sqrt{2E(X)}}=1$$
And $U$ is unitary, we get $<U^{\prime}\{\Psi\},U^{\prime}\{\Psi\}>=<\Psi,\Psi>$.
We also have that
$$\displaystyle<U^{-1}\{\Psi\},U^{-1}\{\Psi\}>=\int dXd^{3}\vec{x}d^{3}\vec{y}$$
$$\displaystyle\Psi^{\dagger}(\vec{y})\frac{E(X)+H(\vec{y})\gamma^{0}}{\sqrt{E(X%
)+m}\sqrt{2E(X)}}U(\vec{y},X)U^{\dagger}(\vec{x},X)\frac{E(X)+\gamma^{0}H(\vec%
{x})}{\sqrt{E(X)+m}\sqrt{2E(X)}}\Psi(\vec{x})$$
Since $U$ is unitary, we get
$<U^{\prime-1}\{\Psi\},U^{\prime-1}\{\Psi\}>=<\Psi,\Psi>$.
Therefore, $U^{\prime}$ is unitary.
∎
Definition 4.64.
The Fourier-Majorana transform
$\mathcal{F}_{M}:Pinor_{j}(\mathbb{R}^{3})\to Pinor_{j}(\mathbb{R}^{3})$ is an
unitary operator defined by:
$$\displaystyle\mathcal{F}_{M}\{\Psi\}(\vec{p})\equiv\int d^{3}\vec{x}\Big{(}%
\frac{e^{-i\gamma^{0}\vec{p}\cdot\vec{x}}}{\sqrt{(2\pi)^{3}}}\Big{)}_{1}\prod_%
{k=1}^{2j}\Big{(}\frac{E_{p}+H(\vec{x})\gamma^{0}}{\sqrt{E_{p}+m}\sqrt{2E_{p}}%
}\Big{)}_{k}\Psi(\vec{x}),\ \Psi\in Pinor_{j}(\mathbb{R}^{3})$$
The matrices with the index $k$ apply on
the corresponding spinor index of $\Psi$.
Definition 4.65.
The Hankel-Majorana transform
$\mathcal{H}_{M}:Pinor_{j}(\mathbb{R}^{3})\to Pinor_{j}(\mathbb{S}^{3})$ is an
unitary operator defined by:
$$\displaystyle\mathcal{H}_{M}\{\Psi\}(p,l,J,\nu)\equiv\sum_{\mu=-l}^{l}\sum_{n=%
-j}^{j}<l\mu jn|J\nu>\int d^{3}\vec{x}$$
$$\displaystyle\Big{(}\frac{2p}{\sqrt{2\pi}}j_{l}(pr)Y_{l\mu}(\theta,\varphi)%
\Big{)}_{1}\prod_{k=1}^{2j}\Big{(}\frac{E_{p}+H(\vec{x})\gamma^{0}}{\sqrt{E_{p%
}+m}\sqrt{2E_{p}}}\Big{)}_{k}\Psi(\vec{x},n)$$
The matrices with the index $k$ apply on
the corresponding spinor index of
$\Psi\in Pinor_{j}(\mathbb{R}^{3})$.
$<l\mu jn|J\nu>$ are the Clebsh-Gordon coefficients and
$\Psi(\vec{x},n)\in Pinor_{j,n}$ such that
$\Psi(\vec{x})=\sum_{n=-j}^{j}\Psi(\vec{x},n)$.
The inverse Fourier-Majorana transform verifies:
$$\displaystyle(iH(\vec{x}))_{k}\ \mathcal{F}_{M}^{-1}\{\psi\}(\vec{x})$$
$$\displaystyle=(\mathcal{F}_{M}^{-1}\circ R)\{\psi\}(\vec{x})$$
$$\displaystyle\vec{\partial}_{l}\ \mathcal{F}_{M}^{-1}\{\psi\}(\vec{x})$$
$$\displaystyle=(\mathcal{F}_{M}^{-1}\circ R^{\prime})\{\psi\}(\vec{x})$$
Where $\psi\in Pinor_{j}(\mathbb{R}^{3})$ and
$R,R^{\prime}:Pinor_{j}(\mathbb{R}^{3})\to Pinor_{j}(\mathbb{R}^{3})$ are linear maps
defined by:
$$\displaystyle R\{\psi\}(\vec{p})$$
$$\displaystyle\equiv(i\gamma^{0})_{k}E_{p}\psi(\vec{p})$$
$$\displaystyle R^{\prime}\{\psi\}(\vec{p})$$
$$\displaystyle\equiv(i\gamma^{0})_{1}\vec{p}_{l}\ \psi(\vec{p})$$
The inverse Hankel-Majorana transform verifies:
$$\displaystyle(iH(\vec{x}))_{k}\ \mathcal{H}_{M}^{-1}\{\psi\}(\vec{x})$$
$$\displaystyle=(\mathcal{H}_{M}^{-1}\circ R)\{\psi\}(\vec{x})$$
$$\displaystyle(-x^{1}\partial_{2}+x^{2}\partial_{1}+\sum_{k=1}^{2j}(i\gamma^{0}%
\gamma^{3}\gamma^{5})_{k})\ \mathcal{H}_{M}^{-1}\{\psi\}(\vec{x})$$
$$\displaystyle=(\mathcal{H}_{M}^{-1}\circ R^{\prime})\{\psi\}(\vec{x})$$
Where $\psi\in Pinor_{j}(\mathbb{S}^{3})$ and $R,R^{\prime}:Pinor_{j}(\mathbb{S}^{3})\to Pinor_{j}(\mathbb{S}^{3})$ are
linear maps defined by:
$$\displaystyle R\{\psi\}(p,l,J,\nu)$$
$$\displaystyle\equiv(i\gamma^{0})_{k}E_{p}\psi(p,l,J,\nu)$$
$$\displaystyle R^{\prime}\{\psi\}(p,l,J,\nu)$$
$$\displaystyle\equiv(i\gamma^{0})_{1}\nu\ \psi(p,l,J,\nu)$$
Definition 4.66.
The space of (real) Bargmann-Wigner fields $BW_{j}(\mathbb{R}^{3})$ is defined as:
$$\displaystyle BW_{j}\equiv\{\Psi\in Pinor_{j}:\Big{(}iH(\vec{x})\Big{)}_{k}%
\Psi=\Big{(}iH(\vec{x})\Big{)}_{1}\Psi;1\leq k\leq 2j\}$$
Definition 4.67.
The complex Hilbert space
$Dirac_{j}(\mathbb{X})\equiv Pinor_{j}(\mathbb{X})\otimes\mathbb{C}$
is the complexification of $Pinor_{j}(\mathbb{X})$.
The space of complex Bargmann-Wigner fields is the complexification of
the space of real Bargmann-Wigner fields.
4.2 On the Poincare, ISL(2,C) and IPin(3,1) groups
Definition 4.68.
The $IPin(3,1)$ group is defined as the semi-direct product
$Pin(3,1)\ltimes\mathbb{R}^{4}$, with the group’s product defined as
$(A,a)(B,b)=(AB,a+\Lambda(A)b)$, for $A,B\in Pin(3,1)$ and
$a,b\in\mathbb{R}^{4}$ and $\Lambda(A)$ is the Lorentz transformation
corresponding to $A$.
The $ISL(2,C)$ group is isomorphic to the subgroup of $IPin(3,1)$,
obtained when $Pin(3,1)$ is restricted to $Spin^{+}(1,3)$. The full/restricted
Poincare group is the representation of the $IPin(3,1)/ISL(2,C)$ group on
Lorentz vectors, defined as $\{(\Lambda(A),a):A\in Pin(3,1),a\in\mathbb{R}^{4}\}$.
Consider a Majorana spinor field $\Psi\in Pinor(\mathbb{R}^{3})$.
Let the Dirac Hamiltonian, $H$, be defined in the configuration space by:
$$\displaystyle iH\{\Psi\}(\vec{x})\equiv(\gamma^{0}\vec{\gamma}\cdot\vec{%
\partial}+i\gamma^{0}m)\Psi(\vec{x}),\ m\geq 0$$
In the Majorana-momentum space:
$$\displaystyle iH\{\Psi\}(\vec{p})\equiv i\gamma^{0}E_{p}\Psi(\vec{p})$$
The free Dirac equation is verified by:
$$\displaystyle(\partial_{0}+iH)e^{-iHx^{0}}\{\Psi\}=0$$
Definition 4.69.
Given a Majorana spinor field
$\Psi\in Pinor(\mathbb{R}^{3})$, we define $\Psi(x)\equiv e^{-iHx^{0}}\{\Psi\}(\vec{x})$.
The Majorana spinor field projective representation of the Poincare
group is defined, up to a sign, as:
$$\displaystyle P(\Lambda_{S},b)\{\Psi\}(x)\equiv\pm S\Psi(\Lambda^{-1}_{S}x+b)$$
Where $\Lambda_{S}\in O(1,3)$, $S\in Pin(3,1)$ is such that
$\Lambda^{\mu}_{S\ \nu}\gamma^{\nu}=S\gamma^{\mu}S^{-1}$ and $b\in\mathbb{R}^{4}$.
4.3 Finite mass case
Remark 4.70.
The complex irreducible projective representations of the Poincare
group with finite mass split into positive and negative energy
representations,
which are complex conjugate of each other. They are labeled by one number $j$, with $2j$
being a natural number.
The positive energy representation spaces $V_{j}$ are, up to isomorphisms, written
as:
$$\displaystyle\Psi_{j}(x)\equiv\int\frac{d^{3}\vec{p}}{(2\pi)^{3}}\prod_{k=1}^{%
2j}\Big{(}\frac{\not{p}\gamma^{0}+m}{\sqrt{E_{p}+m}\sqrt{2E_{p}}}\Big{)}_{k}e^%
{-ip\cdot x}\Psi_{j}(\vec{p})$$
where $p^{0}=E_{p}$ and $\Psi_{j}(\vec{p})$ is a symmetric tensor product of Dirac spinor
fields defined on the 3-momentum space, verifying
$(\gamma^{0})_{k}\Psi_{j}(\vec{p})=\Psi_{j}(\vec{p})$.
The matrices with the index $k$ apply in
the corresponding spinor index of $\Psi_{j}$
The representation space $V_{0}$ is, up to isomorphisms, written in a
Majorana basis as:
$$\displaystyle\Psi_{0}(x)\equiv\int\frac{d^{3}\vec{p}}{(2\pi)^{3}}\frac{\not{p}%
\gamma^{0}+m}{\sqrt{E_{p}+m}\sqrt{2E_{p}}}e^{-ip\cdot x}(1+\gamma^{0})(i\gamma%
^{5})\Psi_{0}(\vec{p})\frac{\not{p}\gamma^{0}+m}{\sqrt{E_{p}+m}\sqrt{2E_{p}}}$$
where $p^{0}=E_{p}$ and $\Psi_{0}(\vec{p})$ is a scalar defined on the 3-momentum space.
The representation map consists in applying the spin one-half
representation map to every spinor index.
Proposition 4.71.
The real irreducible projective representations of the Poincare
group with finite mass are labeled by one number $j$, with $2j$
being a natural number.
The representation spaces $W_{j}$ with $j>0$ are, up to isomorphisms, written
as:
$$\displaystyle\Psi_{j}(x)\equiv\int\frac{d^{3}\vec{p}}{(2\pi)^{3}}\prod_{k=1}^{%
2j}\Big{(}\frac{\not{p}\gamma^{0}+m}{\sqrt{E_{p}+m}\sqrt{2E_{p}}}\Big{)}_{k}%
\Big{(}e^{-i\gamma^{0}p\cdot x}\Big{)}_{1}\Psi_{j}(\vec{p})$$
where $p^{0}=E_{p}$ and $\Psi_{j}(\vec{p})$ is a symmetric tensor product of Majorana spinor
fields defined on the 3-momentum space, verifying
$(i\gamma^{0})_{k}\Psi_{j}(\vec{p})=(i\gamma^{0})_{1}\Psi_{j}(\vec{p})$.
The matrices with the index $k$ apply on
the corresponding spinor index of $\Psi_{j}$.
The representation space $W_{0}$ is, up to isomorphisms, written
in a Majorana basis as:
$$\displaystyle\Psi_{0}(x)\equiv\int\frac{d^{3}\vec{p}}{(2\pi)^{3}}\frac{\not{p}%
\gamma^{0}+m}{\sqrt{E_{p}+m}\sqrt{2E_{p}}}e^{-i\gamma^{0}p\cdot x}i\gamma^{5}%
\Psi_{0}(\vec{p})\frac{\not{p}\gamma^{0}+m}{\sqrt{E_{p}+m}\sqrt{2E_{p}}}$$
where $p^{0}=E_{p}$ and $\Psi_{0}(\vec{p})$ is a scalar defined on the 3-momentum space.
The representation map consists in applying the spin one-half
representation map to every spinor index.
4.4 Null mass case
When the mass goes to zero, the representation spaces that we had are
no longer irreducible, since the helicity becomes invariant under
Lorentz transformations. This is independent of whether the
representation is real or complex. The subspaces $V_{j}^{\pm}$ or $W_{j}^{\pm}$, where for
all $\Psi_{j}\in V_{j}^{\pm}$ or $\Psi_{j}\in W_{j}^{\pm}$ ,
$(\vec{\not{p}}\gamma^{5})_{k}\Psi_{j}(\vec{p})=\pm\Psi_{j}(\vec{p})$
for all the indexes $k$, are the irreducible representations for null
mass and discrete helicity. These are not invariant under parity.
There are also massless representations with continuous spin, which
will not be studied.
5 Conclusion
The complex representations are not a generalization of the real
representations, in the same way that the complex numbers are a
generalization of the real numbers. There is a map, one-to-one or
two-to-one and surjective up to equivalence, from the complex to
the real irreducible representations of a Lie group on a Hilbert
space.
All the real finite-dimensional projective representations of the
restricted Lorentz group are also projective representations of the
full Lorentz group, in contrast with the complex representations which
are not all projective representations of the full Lorentz group.
We obtained all the real
unitary irreducible projective representations of the Poincare
group, with discrete spin, as real Bargmann-Wigner fields.
For each pair of complex representations with positive/negative
energy, there is one real representation.
The Majorana-Fourier and Majorana-Hankel unitary
transforms of the real or complex Bargmann-Wigner fields are related
to the linear and angular momenta of a representation of the Poincare
group, that is, there are linear and angular momenta spaces which are
common for the real and complex representations of the Poincare group,
therefore independent of the charge and matter-antimatter properties.
This allows us to define, with a common formalism, momenta spaces for
the classical electromagnetic field or for the quantum Dirac field,
for instance.
Acknowledgements
Leonardo Pedro’s work was supported by FCT under contract
SFRH/BD/70688/2010.
Leonardo thanks R. M. Fonseca and J. Mourao for the important
discussions, suggestions and corrections.
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Classifying the Dynamics of Architected Materials by Groupoid Methods
Bram Mesland
Mathematisch Instituut
Universiteit Leiden
Niels Bohrweg 1
2333CA Leiden, Netherlands
b.mesland@math.leidenuniv.nl
and
Emil Prodan
Department of Physics and
Department of Mathematical Sciences
Yeshiva University
New York, NY 10016, USA
prodan@yu.edu
Abstract.
We consider synthetic materials consisting of self-coupled identical resonators carrying classical internal degrees of freedom. The architecture of such material is specified by the positions and orientations of the resonators. Our goal is to calculate the smallest $C^{\ast}$-algebra that covers the dynamical matrices associated to a fixed architecture and adjustable internal structures. We give the answer in terms of a groupoid $C^{\ast}$-algebra that can be canonically associated to a uniformly discrete subset of the group of isometries of the Euclidean space. Our result implies that the isomorphism classes of these $C^{\ast}$-algebras split these architected materials into classes containing materials that are identical from the dynamical point of view.
This work was supported by the U.S. National Science Foundation through the grants DMR-1823800 and CMMI-2131760 and by U.S. Army Research Office through contract W911NF-23-1-0127.
1. Introduction and main statements
The science of architected materials is a branch of the science of synthetic materials that can be defined as the art of achieving full control over the classical degrees of freedom of the material and of their couplings. With such control, an architected material can be “programmed” to perform specific functions. The attribute “architected” mainly refers to the spatial arrangements of the building blocks of the material and, typically, these building blocks have adjustable internal structures to facilitate various degrees of programming. In an era dominated by information technology, the main applications of such materials are related to sensing, information storage and information processing. All these applications rely on specific dynamical features supported by a given architecture and a central problem in the field is to determine which new architectures bring genuine novelty. More specifically, given two architectures and the means to modify the internal structure of the resonators, one will like to know if there are dynamical effects that can be produced with one architecture but not with the other. This problem is equivalent to classifying the dynamical matrices that can be generated under various architectures. Its solution can be found in this work for a broad class of architected materials in the quadratic regime (as defined in section 2.4).
We build on two seminal papers of Bellissard [5] and Kellendonk [32] that pointed out that atomic systems have canonical underlying $C^{\ast}$-algebras where the dynamics of electrons can be formalized, analyzed and classified. This was an important development in condensed matter physics because it unleashed many of the techniques specific to operator algebras, such as $K$-theory and noncommutative geometry [25], thus bringing much needed analytical tools to the difficult field of aperiodic solids [7, 8]. For example, the observed quantized plateaus in the integer quantum Hall effect and the so called bulk boundary-correspondence were rigorously explained using such tools [6, 33] (see also [18, 19, 40], for more recent developments engaging groupoid frameworks). These ideas penetrated also the world of architected materials, where the dynamics is carried by classical degrees of freedom (see e.g. [1, 2, 42, 21, 22, 47, 20]). The operator algebraic framework has been instrumental in these applications for understanding global aspects of the dynamics, establishing connections between various architectures and for predicting novel effects. In the present work, we focus on architected materials that are assembled from copies of a fixed part, called here the seed resonator. As opposed to the atomic systems, where the nuclei are featureless,111At the energy scales operated in condensed matter physics. these resonators have shapes and orientations and the Bellissard-Kellendonk formalism, which engages only the group of pure translations, is no longer optimal for this context. The present work was motivated by the need of a more general formalism, where the orientations of the resonators are treated more effectively. As we shall see, the architected materials dealt with here can be formalized as point patterns inside the full group ${\rm Iso}({\mathbb{E}}^{d})$ of Euclidean isometries. Besides being a non-abelian group, ${\rm Iso}({\mathbb{E}}^{d})$ contains rotations and reflections that move points over large distances, hence, formalizing the mentioned materials presents conceptual as well as technical challenges.
The mathematical aspects of patterns in non-abelian groups have been steadily attracted an interest from the mathematics community in the past few years [11, 12, 13, 14, 15, 16, 4]. In fact, during the completion of our work, Enstad, Raum and van Velthoven posted the works [27, 28] on frame theory for patterns in locally compact second countable (lcsc) groups. These works not only cleared for us all the technical challenges, but they also supplied the natural mathematical framework we were looking for. Using this input together with empirical facts, for materials consisting of discrete architectures of self-coupling identical resonators, such as mechanical resonators or acoustic and photonic cavities, here are the statements we have to communicate:
1)
The architecture of such a material is specified by a uniformly separated closed subset ${\mathcal{L}}$ of ${\rm Iso}({\mathbb{E}}^{d})$ (see Def. 3.5).
2)
Every such closed set has a canonically associated lcsc étale groupoid ${\mathcal{G}}_{\mathcal{L}}$ and a separable $C^{\ast}$-algebra $C^{\ast}_{r}({\mathcal{G}}_{\mathcal{L}})$ [27].
3)
The dynamical matrix generating the dynamics of the material in the quadratic regime defines a linear operator over the Hilbert space $\ell^{2}({\mathcal{L}},{\mathbb{C}}^{N})$, where $N$ is the number of internal degrees of freedom carried by the seed resonator.
4)
If the physics involved in the coupling of the resonators is Galilean invariant, then any dynamical matrix can be generated from a left regular representation of ${\mathbb{K}}\otimes C^{\ast}_{r}({\mathcal{G}}_{\mathcal{L}})$, where ${\mathbb{K}}$ is the algebra of compact operators.
5)
By changing the internal structure of the seed resonator, the dynamical matrices sample the entire self-adjoint section of ${\mathbb{K}}\otimes C^{\ast}_{r}({\mathcal{G}}_{\mathcal{L}})$. Thus, ${\mathbb{K}}\otimes C^{\ast}_{r}({\mathcal{G}}_{\mathcal{L}})$ is the smallest $C^{\ast}$-algebra that can account for the dynamics of the material. It can be canonically associated to an architected material by both mathematical and physical means.
6)
The isomorphism classes of the canonically associated $C^{\ast}$-algebras sort the architected materials into classes of materials that are identical from dynamics point of view.
Corollary 1.1.
The research of this kind of architected materials essentially reduces to classifying the mentioned étale groupoids with respect to the equivalence relation introduced by Muhly, Renault, Williams [41].
Furthermore, many dynamical effects in materials science are brought to life by deformations of materials, such as when one seeks topological pumping [21] or topological spectral flows [38]. Many of these effects are predicted by means of topology and the biggest challenge for the field, in this respect, is the identification of the topological space in which the deformations of the materials occur. The stabilization of the groupoid $C^{\ast}$-algebra mentioned above gives the solution to this challenge if the deformations engage only the internal structures of the resonators or, more generally, if the deformations keep the material inside a fixed isomorphism class. If this is the case, then specialized tools developed for operator algebras can be employed to detect the dynamical effects that can be generated with an architected material (see section 5), generally with great effectiveness and precision.
The paper is organized as it follows. Section 2 introduces the physical systems and specifies the assumptions made. They are by no means restrictive and they all can be straightforwardly incorporated in the designs of materials. For example, we argue that, to maintain control over the dynamics, the materials should have finite coupling range and display certain continuity against deformations of their architectures. The section also explains how the configurations of these physical systems are connected to patterns inside ${\rm Iso}({\mathbb{E}}^{d})$ and it derives the specific form of the dynamical matrices under the assumption of Galilean invariance, which represents the point of departure for the operator theoretic framework. The first part of section 3 reproduces from [27] the relevant facts about the topological space of closed subsets of lcsc groups and specific classes of patterns. It also introduces the related concepts of hull and transveral of a pattern. The second part of section 3 is devoted to examples, which, among other things, enable us to explain what has been gained by upgrading from the group of translations to the full group of Euclidean isometries. Section 4 introduces the canonical groupoid associated to a closed subset of a lcsc group and reproduces from [27] a fundamental result stating that this groupoid is étale if and only if the closed subset is uniformly separated. The rest of the section is devoted to the associated groupoid algebra and its relation to the space of dynamical matrices discussed in section 2. It is in this part where the physical meaning of the groupoid $C^{\ast}$-algebra is established. The last section discusses possible research directions opened up by the formalism introduced by our work. Since we want to keep the message of this paper entirely focused on the synergy between the mathematical framework and the physics of architected materials, these research directions are only briefly mentioned and actual results along those lines will be presented elsewhere.
Lastly, we point out that there are several works that address point symmetries of tiled spaces (see e.g. [45, 50] and references therein). There is, however, almost no overlap between these works and our, because the rotations considered in the former are emerging symmetries stemming from particular arrangements of featureless tiles (i.e. without internal structure). In our work, the rotations play an active role in the sense that they specify the internal structures the resonators of an ensemble, relative to that of a seed resonator. Patterns of resonators derived from actions of generic discrete groups on ${\mathbb{R}}^{d}$ were considered by one of the authors in [43]. This work also has little overlap with the present work because [43] deals only with point-patterns of ${\mathbb{R}}^{d}$ and the $C^{\ast}$-algebras of the dynamical matrices are crossed products rather than groupoid algebras. Furthermore, all these mentioned works are more about exploring specific interesting classes of materials and no attempts at a general classification are made.
2. Empirical facts
This section supplies a precise description of the physical assumptions that make possible the theoretical predictions presented in the following sections. Along the way, this section lays out our recommendations for experimental physicists and engineers on how to build architected materials that fall under the umbrella of our framework. Regarding the exposition, our plan is to describe in details a rather specific class of physical systems, those of coupled mechanical resonators, and then supply a list other physical systems with similar attributes.
2.1. A laboratory setting
A mechanical resonator is a confined mechanical system carrying a finite set $\bm{q}=\{q_{1},\ldots,q_{N}\}$ of degrees of freedom and displaying a quadratic Lagrangean
$$L_{0}(\dot{\bm{q}},\bm{q})=\tfrac{1}{2}\sum_{x\in{\mathcal{L}}}\left[\dot{\bm{q}}_{x}\cdot\widehat{M}_{0}\cdot\dot{\bm{q}}_{x}^{T}-\bm{q}_{x}\cdot\widehat{V}_{0}\cdot\bm{q}_{x}^{T}\right],\quad\widehat{M}_{0},\widehat{V}_{0}\in M_{N}({\mathbb{C}}).$$
(2.1)
Throughout, $M_{N}({\mathbb{C}})$ denotes the algebra of $N\times N$ matrices with complex entries. The degrees of freedom will always be observed and quantified using equipment that is rigidly attached to the resonator or rather to the frame of the resonator. The equipment can be a sensing device or a simple video camera. Abstractly, this is conveyed by a co-moving frame and by a local research assistant that carries the observations and the measurements in such a rigidly attached frame.
We will be concerned with finite and infinite clusters of identical resonators in the physical space ${\mathbb{E}}^{d}$, $d=1,2,3$. Thus, apart from their locations and orientations, all resonators are identical to a seed resonator. In our framework, the identical resonators carry with them the same equipment, which is attached in exactly the same fashion on the frame of the resonator. As a consequence, the numerical values $\{q_{1},\ldots,q_{N}\}$ recorded by the local research assistants are not affected by translations, rotations or reflections of the resonators.
Remark 2.1.
The above details are quite central to our approach (see Remark 2.5) and should not be taken lightly. For example, what we are proposing here is quite different from the case of atomic systems, where the atomic orbitals are always quantified in a fixed laboratory frame, common to all atoms.
$\Diamond$
The resonators come with their own force fields and a pair of resonators self-couple without any human intervention (see Examples 2.6 and 2.7). In abstract terms, this means that the Lagrangean $L$ of a pair or a cluster of resonators is entirely determined by the positions and orientations of the individual resonators. If ${\mathcal{L}}$ encodes the geometric configuration of a cluster of $r$ identical resonators, referred to as the architecture of the cluster, then the entire physics is encoded in the relation
$${\mathcal{L}}\mapsto L_{\mathcal{L}}(\dot{\bm{q}}_{1},\ldots,\dot{\bm{q}}_{r},\bm{q}_{1},\ldots,\bm{q}_{r}),\quad\dot{\bm{q}}_{i},\ \bm{q}_{i}\in{\mathbb{R}}^{N}.$$
(2.2)
Remark 2.2.
The relation (2.2) can be actually mapped experimentally (see e.g. [1]) and this is highly recommended, especially if an experimental platform is used repeatedly.
$\Diamond$
Remark 2.3.
Eq. (2.2) assumes that degrees of freedom belong to the specific manifold ${\mathbb{R}}^{N}$. This is sufficient for our purposes, because we are interested only in the regime of small oscillations, where we can use a local chart of the configuration manifold of the seed resonator, to map its degrees of freedom into ${\mathbb{R}}^{N}$.
$\Diamond$
2.2. Formalizing the space of architectures
Pictorially, the physical system we just described can be represented as in Fig. 2.1. We are going to exploit the fact that the resonators are identical, in which case the only distinction between two resonators in a cluster is reflected in the positioning of their intrinsic frames. If the laboratory frame is fixed once and for all, then the intrinsic frames of the resonators can be generated from the laboratory frame by applying a rotation followed by a translation. This simple line of reasoning reveals that the architecture ${\mathcal{L}}$ of a cluster of resonators is completely specified by a subset of the topological group ${\rm Iso}({\mathbb{E}}^{d})$ of isometries. We will denote this subset by the same symbol ${\mathcal{L}}$. As indicated in Fig. 2.1, we include in our analysis proper as well as improper space transformations.
Among other important things, this development enables us to label the resonators in a more natural way by elements of ${\rm Iso}({\mathbb{E}}^{d})$. Specifically, we will write the degrees of freedom as $\{\bm{q}_{x}\}_{x\in{\mathcal{L}}}$. The right action of the group on itself will be indicated by a dot. Concretely,
$${\rm Iso}({\mathbb{E}}^{d})\ni g\mapsto g\cdot x:=xg^{-1},\quad\forall\ x\in{\rm Iso}({\mathbb{E}}^{d}).$$
(2.3)
This action extends from points to subsets of ${\rm Iso}({\mathbb{E}}^{d})$. Therefore, it makes sense to consider the action $g\cdot{\mathcal{L}}$ of the group on the architecture. The action of an isometry $g\in{\rm Iso}({\mathbb{E}}^{d})$ on $p\in{\mathbb{E}}^{d}$ will be indicated by the same symbol $g\cdot p$.
Remark 2.4.
The right action distinguishes itself from the left action of the group due to the following property: The configurations of the resonators labeled by $x$ and $gx$ drift apart from each other if $x$ wonders away from the neutral element and $g$ contains a rotation. This, however, is not the case if the right action of the group is used instead, as it can be easily seen by the rule of multiplication in ${\rm Iso}({\mathbb{E}}^{d})$ shown in Appendix 6.$\Diamond$
2.3. Galilean invariance
We now imagine an experimenter engaged in the process of mapping (2.2) and suppose the experimenter examines an architecture ${\mathcal{L}}^{\prime}$ that is related to another architecture ${\mathcal{L}}$ by ${\mathcal{L}}^{\prime}=g\cdot{\mathcal{L}}$, $g\in{\rm Iso}({\mathbb{E}}^{d})$. If the physics involved in the coupling of resonators is Galilean invariant, then the experimentally mapped Lagrangeans will enter the following relation
$$L_{{\mathcal{L}}^{\prime}}\left(\{\dot{\bm{q}}_{g\cdot x}\}_{x\in{\mathcal{L}}},\{\bm{q}_{g\cdot x}\}_{x\in{\mathcal{L}}}\right)=L_{\mathcal{L}}\left(\{\dot{\bm{q}}_{g\cdot x}\}_{x\in{\mathcal{L}}},\{\bm{q}_{g\cdot x}\}_{x\in{\mathcal{L}}}\right).$$
(2.4)
Note that the labeling of the resonators by group elements is quite instrumental here. Indeed, one should be aware that a linear indexing (i.e. some arbitrary ordering) of the resonators cannot be made uniformly for all architectures ${\mathcal{L}}$, because certain deformations of ${\mathcal{L}}$ can switch two resonators. In such cases, it will become ambiguous which ordering to use. Quite the opposite, the indexing by the group elements fits naturally with the deformations of the architectures and this is why the Galilean invariance can be formulated as cleanly as in Eq. (2.4).
2.4. Quadratic regime
It is convenient to reference the Lagrangeans from the individual quadratic contributions of the uncoupled resonators. Hence, we write
$$L_{\mathcal{L}}=\tfrac{1}{2}\sum_{x\in{\mathcal{L}}}\left[\dot{\bm{q}}_{x}\cdot\widehat{M}_{0}\cdot\dot{\bm{q}}_{x}^{T}-\bm{q}_{x}\cdot\widehat{V}_{0}\cdot\bm{q}_{x}^{T}\right]-V_{\mathcal{L}}\big{(}\{\bm{q}_{x}\}_{x\in{\mathcal{L}}}\big{)},$$
(2.5)
where $V_{\mathcal{L}}$ plays the role of the full coupling potential. In the regime of weak coupling, which we can be sure it occurs at least when the resonators are spread apart, the full potential $\sum_{x\in{\mathcal{L}}}\frac{1}{2}\bm{q}_{x}\cdot\widehat{V}_{0}\cdot\bm{q}_{x}^{T}-V_{\mathcal{L}}\left(\{\bm{q}_{x}\}_{x\in{\mathcal{L}}}\right)$ has a unique minimum occurring at $\{\bar{\bm{q}}_{x}({\mathcal{L}})\}_{x\in{\mathcal{L}}}$ in the vicinity of $\{\bm{q}_{x}=0\}_{x\in{\mathcal{L}}}$. If we restrict the experimental observations to the regime of small oscillations around such minima, the Lagrangeans can be approximated by quadratic expressions
$$L_{\mathcal{L}}=\tfrac{1}{2}\sum_{x\in{\mathcal{L}}}\dot{\bm{\xi}}_{x}\cdot\widehat{M}_{0}\cdot\dot{\bm{\xi}}_{x}^{T}-\tfrac{1}{2}\sum_{x,x^{\prime}\in{\mathcal{L}}}\,{\bm{\xi}}_{x}\cdot\widehat{W}_{x,x^{\prime}}({\mathcal{L}})\cdot{\bm{\xi}}_{x^{\prime}}^{T},$$
(2.6)
where $\bm{\xi}_{x}=\bm{q}_{x}-\bar{\bm{q}}_{x}$ and
$$\left(\widehat{W}_{x,x^{\prime}}({\mathcal{L}})\right)_{\alpha\beta}=\delta_{x,x^{\prime}}\left(\widehat{V}_{0}\right)_{\alpha,\beta}+\left.\frac{\partial^{2}V_{\mathcal{L}}}{\partial(\bm{q}_{x})_{\alpha}\partial(\bm{q}_{x}^{\prime})_{\beta}}\right|_{\bar{\bm{q}}({\mathcal{L}})},$$
(2.7)
for $x,x^{\prime}\in{\mathcal{L}}$. We will refer to these objects as the coupling matrices.
A direct consequence of Eq. (2.4) is the following equivariance property of the coupling matrices:
$$\widehat{W}_{g\cdot x,g\cdot x^{\prime}}(g\cdot{\mathcal{L}})=\widehat{W}_{x,x^{\prime}}({\mathcal{L}}),\quad g\in{\rm Iso}({\mathbb{E}}^{d}),$$
(2.8)
which holds whenever the physics involved in the coupling of the resonators is Galilean invariant. For example, this invariance will be broken in the presence of external fields or for couplings involving external frames and fixings.
Remark 2.5.
Our choice of observing and quantifying the degrees of freedom in a co-moving frame is prominently manifested in Eq. (2.8). Indeed, if instead the choice was to observe them from the fixed laboratory frame, then we had to specify how the degrees of freedom transform under the Euclidean transformations and the right side of Eq. (2.8) would contain a conjugation by such transformation.
$\Diamond$
Example 2.6.
The experimental platform introduced in [1] uses a seed resonator that consists of a rigid body fitted with permanent magnets. If the seed resonator aligned with the laboratory frame, hence the one labeled by the neutral element $e$, produces a magnetic field $\bm{B}_{e}(p)$ at the point $p\in{\mathbb{E}}^{3}$, then the resonator labeled by $x=(t_{x},\bm{r}_{x})\in{\rm Iso}({\mathbb{E}}^{3})$ (see Apendix 6 for notation) produces at the same point $p$ a magnetic field
$$\bm{B}_{x}(p)=\gamma(\bm{r}_{x})\cdot\bm{B}_{e}(x^{-1}\cdot p),$$
(2.9)
where $\gamma$ represents the action of rotations on pseudo-vectors (or 2-forms). Now, let ${\mathcal{L}}$ be a finite point set in ${\rm Iso}({\mathbb{E}}^{3})$ specifying a stable equilibrium configuration of an ensemble of such resonators. If $\tilde{x}$ is an out of equilibrium configuration of the resonator labeled by $x\in{\mathcal{L}}$, then we encode its internal degrees of freedom in the element $\bm{q}_{x}:=x^{-1}\tilde{x}\in{\rm Iso}({\mathbb{E}}^{3})$.222We can view $\bm{q}_{x}$ as an element of ${\mathbb{R}}^{6}$ by using a local chart of the Lie group ${\rm Iso}({\mathbb{E}}^{d})$. Note that, in the actual experiments, $\bm{q}_{x}$ are constraint on sub-manifolds of ${\rm Iso}({\mathbb{E}}^{3})$. In the experimental conditions of [1], the potential energy of the mechanical system comes from the magnetic field’s energy, written below in specific units,333The energy of an object carrying a saturated magnetization $\bm{M}$ in an external magnetic field $\bm{B}_{\rm ext}$ is $-\int_{{\mathbb{E}}^{3}}{\rm}dp\,\langle\bm{M}(p),\bm{B}_{\rm ext}(p)\rangle$. Since the self-iteraction does not vary with the configuration of the object, $\bm{B}_{\rm ext}$ can promoted to the net magnetic field $\bm{B}_{\rm net}$. This together with the fact that $\int_{{\mathbb{E}}^{3}}{\rm d}p\,\langle\bm{B}_{\rm net}(p)-4\pi\bm{M}(p),\bm{B}_{\rm net}(p)\rangle=0$, valid in the absence of free currents, gives Eq. (2.10) up to a multiplicative factor. The latter is absorbed in our chosen units for energy.
$$E=-\int_{{\mathbb{E}}^{3}}{\rm d}p\,\langle\bm{B}_{\rm net}(p),\bm{B}_{\rm net}(p)\rangle,$$
(2.10)
where $\langle\cdot,\cdot\rangle$ is the standard scalar product of ${\mathbb{R}}^{3}$. The net magnetic field corresponding to a dynamical configuration (i.e. out of equilibrium) is a linear superposition
$$\bm{B}_{\rm net}(p)=\sum_{x\in{\mathcal{L}}}\bm{B}_{x\bm{q}_{x}}(p)=\sum_{x\in{\mathcal{L}}}\gamma(\bm{r}_{x\bm{q}_{x}})\cdot\bm{B}_{e}\big{(}\bm{q}_{x}^{-1}x^{-1}\cdot p\big{)}$$
(2.11)
and the mechanical potential energy takes the form
$$V_{\mathcal{L}}=-\int_{{\mathbb{E}}^{3}}{\rm d}p\,\sum_{x,y\in{\mathcal{L}}}\big{\langle}\gamma(\bm{r}_{x\bm{q}_{x}})\cdot\bm{B}_{e}(\bm{q}_{x}^{-1}x^{-1}\cdot p),\gamma(\bm{r}_{y\bm{q}_{y}})\cdot\bm{B}_{e}(\bm{q}_{y}^{-1}y^{-1}\cdot p)\big{\rangle}.$$
(2.12)
Using the invariance of the Lebesgue measure and of the scalar product against the actions of ${\rm Iso}({\mathbb{E}}^{3})$, one can manually check the equivariance of the potential
$$V_{g\cdot{\mathcal{L}}}\left(\{{\bm{q}}_{g\cdot x}\}_{x\in{\mathcal{L}}}\right)=V_{\mathcal{L}}\left(\{\bm{q}_{g\cdot x}\}_{x\in{\mathcal{L}}}\right).$$
(2.13)
In fact, using those simple facts, we can re-write the potential as
$$V_{\mathcal{L}}=-\int_{{\mathbb{E}}^{3}}{\rm d}p\,\sum_{x,y\in{\mathcal{L}}}\big{\langle}\gamma(\bm{r}_{\bm{q}_{x}})\cdot\bm{B}_{e}(\bm{q}_{x}^{-1}\cdot p),\gamma(\bm{r}_{x^{-1}y\bm{q}_{y}})\cdot\bm{B}_{e}(\bm{q}_{y}^{-1}y^{-1}x\cdot p)\big{\rangle},$$
(2.14)
hence making it manifestly equivariant.
$\Diamond$
Example 2.7.
The case of a seed resonator fitted with permanently electrically polarized components can be analyzed in a similar manner. Additional examples are discussed in section 2.8.$\Diamond$
2.5. Dynamical matrix
From now on, we restrict the discussion to the quadratic regimes. Consider generalized driving forces
$$\bm{F}_{x}(t)={\rm Re}\left[(f_{x}^{1},\ldots,f_{x}^{N})e^{\imath\omega t}\right]={\rm Re}[\bm{f}_{x}e^{\imath\omega t}],\quad x\in{\mathcal{L}},$$
(2.15)
applied on each of the resonators. To describe the response of the cluster more effectively, we encode the driving forces in the vector $|\bm{f}\rangle=\sum_{x\in{\mathcal{L}}}\bm{f}_{x}\otimes|x\rangle$ from the Hilbert space ${\mathbb{C}}^{N}\otimes\ell^{2}({\mathcal{L}})$. Then
the system responds as
$$\bm{q}_{x}(t)=\bar{\bm{q}}_{x}+\widehat{M}_{0}^{-\frac{1}{2}}\cdot{\rm Re}\left[\bm{\xi}_{x}e^{\imath\omega t}\right],$$
(2.16)
where $\bm{\xi}_{x}$ are the coefficients of the vector $|\bm{\xi}\rangle=\sum_{x\in{\mathcal{L}}}\bm{\xi}_{x}\otimes|x\rangle$ from ${\mathbb{C}}^{N}\otimes\ell^{2}({\mathcal{L}})$ satisfying the equation
$$(D_{\mathcal{L}}-\omega^{2}\,I)|\bm{\xi}\rangle=\widehat{M}_{0}^{-\frac{1}{2}}\otimes I\,|\bm{f}\rangle,\quad D_{\mathcal{L}}=\sum_{x,x^{\prime}\in{\mathcal{L}}}w_{x,x^{\prime}}({\mathcal{L}})\otimes|x\rangle\langle x^{\prime}|,$$
(2.17)
with
$$w_{x,x^{\prime}}({\mathcal{L}})=\widehat{M}_{0}^{-\frac{1}{2}}\,\widehat{W}_{x,x^{\prime}}({\mathcal{L}})\,\widehat{M}_{0}^{-\frac{1}{2}}\in M_{N}({\mathbb{C}}).$$
(2.18)
Since arbitrary time-dependent driving forces can be decomposed in Fourier components, the conclusion is that the response of the cluster to an external excitation reduces to the study of the spectral properties of the dynamical matrix $D_{\mathcal{L}}$, viewed as a linear operator over the Hilbert space ${\mathbb{C}}^{N}\otimes\ell^{2}({\mathcal{L}})$.
2.6. Controlling dynamics
In the applications of architected materials, one needs control over the coupling matrices, that is, a researcher should be able to measure them with reasonable precision and be able to modify them according to the specifications of a given application. Here, we identify a class of dynamical matrices that have a positive chance to be controlled in a laboratory. Of course, these issues are relevant only for infinite architectures.
A coupling matrix $w_{x,x^{\prime}}({\mathcal{L}})$ is said to have finite coupling range if it vanishes whenever $x^{\prime}\cdot x$ is outside a fixed compact vicinity of the neutral element of ${\rm Iso}({\mathbb{E}}^{d})$. We claim that coupling matrices that have finite range and depend continuously on ${\mathcal{L}}$ can be measured in a laboratory. Indeed, the equivariance relation (2.8) assures us that it is enough to focus the observations only on $x,x^{\prime}$ in a compact vicinity of the neutral element and, due to the assumed continuity, a good approximation of the map ${\mathcal{L}}\to w_{x,x^{\prime}}({\mathcal{L}})$ can be derived by interpolating a finite sampling of ${\mathcal{L}}$’s. This assures us that, even for an infinite architecture, we still have a chance to measure with enough precision every single coupling matrix of the system.
We can go one step beyond the class of systems described above. Indeed, any dynamical matrix that can be approximated by such dynamical matrices with finite coupling ranges can be also controlled in a laboratory. Of course, when mentioning approximations one needs to specify a norm on the dynamical matrices. The correct norm to consider here is the operator norm on ${\mathbb{C}}^{N}\otimes\ell^{2}({\mathcal{L}})$, because convergence of dynamical matrices in this norm implies convergence of their resonant spectra. This gives us control over the spectral properties.
2.7. A look ahead
We encoded the dynamics of the clusters of resonators in the dynamical matrix $D_{\mathcal{L}}$. In the presence of Galilean invariance, the equivariance relation (2.8) implies
$$w_{g\cdot x,g\cdot x^{\prime}}(g\cdot{\mathcal{L}})=w_{x,x^{\prime}}({\mathcal{L}}),\quad g\in{\rm Iso}({\mathbb{E}}^{d}),$$
(2.19)
which can be used to reduce the expression of the dynamical matrix
$$D_{{\mathcal{L}}}=\sum_{x,x^{\prime}\in{\mathcal{L}}}w_{e,x\cdot x^{\prime}}(x\cdot{\mathcal{L}})\otimes|x\rangle\langle x^{\prime}|.$$
(2.20)
Among other things, this shows that the entire dynamics is encoded in $M_{N}({\mathbb{C}})$-valued map $w_{e,g}({\mathcal{L}})$, defined on tuples $(g,{\mathcal{L}})$ with $g\in{\mathcal{L}}$. This together with the discussion in subsection 2.6 is the point of departure for the operator algebraic framework. Indeed, as we shall see in section 4, any dynamical matrix as in Eq. (2.20) can be generated as the left-regular representation of an element from a stable, separable $C^{\ast}$-algebra canonically associated to ${\mathcal{L}}$. This is the stabilization of the groupoid $C^{\ast}$-algebra introduced in [27], which generalizes Bellissard-Kellendonk construction from the context of the abelian group ${\mathbb{R}}^{d}$ to that of general locally compact groups. To formulate this $C^{\ast}$-algbera, we first need to give sense to the continuity of the map (2.2) and of the coupling matrices, that is, to specify natural topologies on the domains of these maps. In fact, the key to the operator theoretic formalism rests on the leap from the physical space to the topological space where the architectures ${\mathcal{L}}$ live.
As already implied above, the $C^{\ast}$-algebra we just mentioned is invariant to modifications of the internal structure of the seed resonator. Furthermore, by changing this internal structure, one can modify the coupling matrices and explore the entire $C^{\ast}$-algebra. Thus, it is possible to identify this $C^{\ast}$-algebra with the smallest $C^{\ast}$-algebra that covers the dynamics of a lattice ${\mathcal{L}}$ of identical resonators with adjustable internal structure. This effort is justified because the representation, structure and K theories of this $C^{\ast}$-algebra filter out all dynamical effects achievable with such mechanical systems (see it at work in e.g. [1]). Furthermore, upon deformations of the architecture ${\mathcal{L}}$, one can find many other mechanical systems that have isomorphic $C^{\ast}$-algebras and, as such, are equivalent from the dynamical point of view. The outstanding conclusion is that the world of architected materials can be divided in well-defined classes and, if we want to discover new dynamical effects in architected materials, the task is reduced to exploring different Morita equivalence classes of $C^{\ast}$-algebras.
Remark 2.8.
At the first sight, there might be an objection to our statement that we can actually achieve all available coupling matrices, given the fact that our exposition was carried in the weak coupling regime. Note, however, that the coupling matrices can be also manipulated via the inertia matrix $\widehat{M}_{0}$, which is not constrained in any way by the weak coupling assumption. In particular, large values of the coupling matrices (2.18) can be generated by reducing the inertia of the resonators.
$\Diamond$
2.8. Other physical systems covered by analysis
Mode-coupling theory is an effective theory applying to weakly coupled discrete resonators. It works in specific windows of frequencies, where only a finite number of resonant modes are active for the un-coupled resonators. Once these modes are identified, the theory assumes that the collective resonant modes can be approximated by linear combinations of the local resonant modes. In other words, the full Hilbert space of collective resonant modes is projected onto the subspace spanned by such linear combinations. This automatically leads to discrete dynamical matrices as in Eq. (2.17). If the coupling of the resonators is mediated by a homogeneous medium, then the equivariance relation (2.19) holds, hence, the dynamics of these systems is covered by our analysis. Concrete examples include coupled acoustic cavities [22], patterning of surfaces [48], coupled plate resonators [38], coupled electromagnetic cavities or photonic resonators [52].
It is important, however, to stress that our analysis and predictions are not bound only to those systems. Discrete resonators can be coupled via man-designed bridges and other synthetic couplings. Our predictions still apply as long as these couplings are designed from the geometric information encoded in ${\mathcal{L}}$ and are insensitive to the rigid displacement and rotation of the coupled pairs. Thus, the task of an experimentalist will be to engineer the pairwise coupling matrices $w_{x,x^{\prime}}({\mathcal{L}})$, by natural or synthetic means, such that they depend continuously on $x$ and $x^{\prime}$, as well as on ${\mathcal{L}}$, and, in addition, satisfy the equivariance relation (2.19).
Given the discussion in the previous paragraph, we will go as far as saying that interesting laboratory models can be constructed from lattices of general locally compact groups (see [37] for more details on such efforts). If one is interested in discovering new dynamical features in a systematic and controlled fashion, this is certainly a path worth exploring. The mathematical formalism of the next sections is developed for this very general context.
3. The space of patterns and associated concepts
The work of Enstad and Raum in [27] introduces a purely topological route to the $C^{\ast}$-algebra canonically associated to a pattern (i.e. closed subset) in a locally compact group. Their approach is extremely general since it employs the standard Fell topology of the space of closed subsets and it relies solely on topological means to define uniformly discrete patterns. The alternative will be to endow the group with a metric and to generalize the induced Hausdorff metric on the space of compact subsets to a metric on the space of closed subsets, as it has been successfully done in the context of the abelian group ${\mathbb{R}}^{d}$ [32, 30, 36]. However, it has been already recognized that those constructions lead exactly to the Fell topology [9], which can be seen from [10][Prop. E.1.3] and [30][pg. 16]. For these reasons, the formalism put forward in [27] feels very natural and fits well with all the details of the applications we have in mind. Therefore, we decided to adopt it here and abandon our earlier efforts on alternative approaches. As an upshot, most of the technical results needed along the way have been already supplied in [27], which means we can focus on physical interpretations and examples.
3.1. Topologizing the space of patterns
For us, a pattern in a topological space is just a closed subset of that space. As we have seen in section 2, the physical systems like the one illustrated in Fig. 2.1 are associated to discrete subsets of the lcsc group ${\rm Iso}({\mathbb{E}}^{d})$. Of course, the main interest is in unbounded subsets and this is the main source of difficulties. Thus, our first task is to describe the space of closed subsets of a lcsc group. Most of what we say is reproduced from [29].
Definition 3.1 ([29]).
Let $X$ be any fixed topological space and let ${\mathcal{C}}(X)$ be the family of all closed subsets, hence including the void set $\emptyset$. For each compact subset $K$ of $X$ and each finite family ${\mathcal{F}}$ of nonvoid open subsets of $X$, let $U(K;{\mathcal{F}})$ be the subset of ${\mathcal{C}}(X)$ consisting of all $Y$ such that: (i) $Y\cap K=\emptyset$, and (ii) $Y\cap F\neq\emptyset$ for each $F\in{\mathcal{F}}$. The Fell topology on ${\mathcal{C}}(X)$ is defined by the subbasis consisting of such $U(K;{\mathcal{F}})$.
Proposition 3.2 ([29]).
If $X$ is locally compact, then the space ${\mathcal{C}}(X)$ of patterns in $X$ is automatically a compact Hausdorff space.
If $X$ is lcsc, then ${\mathcal{C}}(X)$ is second-countable too and its topology can be described by convergence of sequences. The convergence of sequences for the Fell topology can be described in the following concrete terms:
Proposition 3.3 ([10]).
Let $C_{n},C\in{\mathcal{C}}(X)$ for each $n\in{\mathbb{N}}$. Then $C_{n}$ converges to $C$ in the Fell topology if and only if both of the following statements hold: (i) Whenever $x\in C$ then there exist $x_{n}\in C_{N}$ such that $x_{n}\to x$, and (ii) Whenever $(n_{k})_{k\in{\mathbb{N}}}$ is a subsequence of ${\mathbb{N}}$ and $x_{n_{k}}\in C_{n_{k}}$ converges to $x\in X$, then $x\in C$.
As we already mentioned in section 2, one of the tasks for us is to address the continuity of the map (2.2). The practical criterion from Proposition 3.3 reflects exactly what one will intuitively think of a continuous deformation of the pattern ${\mathcal{L}}$, namely, a continuous displacement of its points. Thus, Fell topology on the domain of the map (2.2) gives the answer to our first challenge.
3.2. Classes of discrete patterns
From here on, we specialize the discussion to the case when $X$ is a lcsc topological group $G$. In such instances, $G$ plays both the passive role of hosting a pattern and the active role of space transformations. For the former, we will denote the elements of $G$ by $x,y,\ldots$, and for the latter we will use $g,g^{\prime},\ldots$. We continue to denote its neutral element by $e$. The right action of the group on itself can be elevated to a continuous $G$-action on ${\mathcal{C}}(G)$:
$$g\cdot C=\{xg^{-1},\ x\in C\},\quad g\in G,\quad C\in{\mathcal{C}}(G).$$
(3.1)
The following it is claimed in [27] to be standard terminology for point sets in frame theory, though we were not able to identify the source of it.
Definition 3.4 ([27]).
For ${\mathcal{L}}\in{\mathcal{C}}(G)$ and $S\subseteq G$, one says that ${\mathcal{L}}$ is
1)
$S$-separated if $|{\mathcal{L}}\cap g\cdot S|\leq 1$ for all $g\in G$;
2)
$S$-dense if $|{\mathcal{L}}\cap g\cdot S|\geq 1$ for all $g\in G$.
Definition 3.5 ([27]).
A subset ${\mathcal{L}}\subset G$ is called uniformly separated if there exists a non-empty open set $U\subseteq G$ such that ${\mathcal{L}}$ is $U$-separated. The set ${\mathcal{L}}$ is called uniformly dense if there exists a compact subset $K\subseteq G$ such that ${\mathcal{L}}$ is $K$-dense. If ${\mathcal{L}}$ is both separated and relatively dense, the it is called a Delone set.
From our discussion in section 2, it is evident that, in such applications, we are dealing with uniformly separated subsets of ${\rm Iso}({\mathbb{E}}^{d})$. For example, the resonators in Fig. 2.1 cannot be overlapped and also the regime of small couplings was invoked in section 2. Thus, in these applications, the origin of the seed resonator can be surrounded by an imaginary ball $B$ of the Euclidean space and these rigidly attached balls should not overlap when copies of the seed resonator are incorporated in the designs of the metamaterials. This imaginary ball then leads to the open set $U=B\times O(d)$ in ${\rm Iso}({\mathbb{E}}^{d})$ and to a $U$-separated architecture ${\mathcal{L}}$. This kind of designs, however, do not exploit to the fullest what ${\rm Iso}({\mathbb{E}}^{d})$ has to offer. In principle, any open set $U\subset{\rm Iso}({\mathbb{E}}^{d})$ can be used to shape the resonators and this gives us hopes there is plenty to be explored and discovered with these physical systems.
A Delone pattern is usually associated to the bulk of a material. However, a typical application of the operator theoretic methods is to the so called bulk-defect correspondence principle [44] and a metamaterial with a defect is rather associated to uniformly separated patterns. Hence our focus from now on is on such patterns. The following statement gives a first characterization of these families:
Proposition 3.6 ([27]).
For any non-empty open set $U\subseteq G$, the set of $U$-separated sets is closed in ${\mathcal{C}}(G)$.
Note that, among other things, the above statement assures us that the limit of a sequence of $U$-separated closed subsets is also $U$-separated.
3.3. The hull and transversal of a pattern
The concepts introduced here make sense for generic patterns in a lcsc group $G$. Therefore, we will keep the presentation at that level of generality and then follow with specific examples relevant for the physical systems discussed in section 2.
Definition 3.7.
The closure of the orbit of ${\mathcal{L}}\in{\mathcal{C}}(G)$ under the natural $G$-action is called the hull of ${\mathcal{L}}$:
$$\Omega_{{\mathcal{L}}}=\overline{\{g\cdot{\mathcal{L}},\ g\in G\}}.$$
(3.2)
Since ${\mathcal{C}}(G)$ is compact, $\Omega_{\mathcal{L}}$ is compact for any ${\mathcal{L}}\in{\mathcal{C}}(G)$. Furthermore, since the property of a closed subset of being $U$-separated is unaffected by translations, any member of $\Omega_{\mathcal{L}}$ is $U$-separated if ${\mathcal{L}}$ is $U$-separated. A fine point noted in [27] is that, if a pattern is not uniformly dense, then $\Omega_{\mathcal{L}}$ contains the empty set $\emptyset$ as an invariant point. Thus, the proper space for analysis is:
Definition 3.8 ([27]).
The punctured hull of ${\mathcal{L}}$ is defined as:
$$\Omega^{\times}_{{\mathcal{L}}}=\Omega_{\mathcal{L}}\setminus\emptyset.$$
(3.3)
$\Omega^{\times}_{{\mathcal{L}}}$ is a locally compact space. Most that will be said from now on builds on the dynamical system $(G,\Omega^{\times}_{\mathcal{L}})$. An essential role will be undertaken by its standard Poincaré transversal:
Definition 3.9.
The standard transversal of $(G,\Omega^{\times}_{\mathcal{L}})$ is
$$\Xi_{\mathcal{L}}=\{{\mathcal{L}}^{\prime}\in\Omega_{\mathcal{L}},\ e\in{\mathcal{L}}^{\prime}\}\subset\Omega_{\mathcal{L}}.$$
(3.4)
The standard transversal is closed in $\Omega_{\mathcal{L}}$ and has no intersection with $\emptyset$, hence it is actually a compact subspace of $\Omega^{\times}_{\mathcal{L}}$. The applications of the operator theoretic approach to materials science rests quite heavily on the ability to calculate $\Xi_{\mathcal{L}}$. In the physics literature, this space is called the phason space of the pattern.
Remark 3.10.
For the patterns described in section 2, a practical way to think about the transversal is as follows. Using rigid motions, we can place a resonator of the pattern at the origin, with its intrinsic frame aligned with that of the laboratory. We take a picture of the resulted pattern with a camera located above the origin of the laboratory frame and we place that picture in $\Xi_{{\mathcal{L}}}$. We repeat the process until each resonator gets to sit at the origin once and then we take the topological closure of the discrete set of patterns so obtained. The criterion spelled in Proposition 3.3 is instrumental for this last step.
$\Diamond$
3.4. Examples
The success of the $C^{\ast}$-algebraic framework, as applied to the context discussed in section 2, is very much conditioned by ones ability to map out the standard transversals. While these applied aspects will be scrutinized in our future works, here we illustrate several instances where the standard transversal can be mapped without much effort. We use these examples to highlight the differences emerging when the patterns are seen as subsets of ${\rm Iso}({\mathbb{E}}^{d})$ or of ${\mathbb{R}}^{d}$.
Example 3.11.
The pattern of resonators in Fig. 3.1 was generated by acting with the space-group p2 on an initial triangle slightly displaced from the origin. This displacement is needed in order for the centers of the resonators to generate a uniformly separated point set. According to Remark 3.10, we should move each resonator to the origin of the laboratory frame and align the two. This is done in Figs. 3.1(b,c) for two randomly picked resonators. As one can see, the resulted patterns look the same in the two situations, and evidently will look the same if we sample other resonators. As such, the standard transversal consists of a single point, the pattern that we see either in panel (b) or (c).
$\Diamond$
Example 3.12.
The pattern from Fig. 3.2 was generated as in Example 3.11, but using the wallpaper group p4. The conclusion is the same, namely the transversal of the pattern consists of a single point. These two findings are not singular. In fact, any pattern generated by acting with a wallpaper group on a seed shape will have a singleton tranversal. Examples of architected materials designed with such algorithms can be found in [38].
$\Diamond$
Remark 3.13.
As abstract patterns, the motifs from Figs. 3.1 and 3.1 can be certainly seen as closed subsets of the group ${\mathbb{R}}^{2}$ of pure translations. In this case, their standard transversals, as defined in Def. 3.9, are homeomorphic to two, respectively, four copies of the seeding shape. If, instead, we follow Kellendonk’s recipe for constructing a transversal and take the origin of the local frames as the puctures of the tiles, we find transversals that consists of two, respectively, four points. $\Diamond$
Example 3.14.
Fig. 3.3 displays a pattern of resonators generated by repeated application of $(t,\bm{r})\in{\rm Iso}({\mathbb{E}}^{2})$ on a seed triangle, where $t$ is a translation and $\bm{r}$ is a proper rotation by an angle $\theta$ that is incommensurate with $2\pi$ (see appendix 6 for notation and other details). Fig. 3.3 also shows rigid shifts of the pattern resulted from aligning different resonators with the laboratory’s frame. Each of those patterns represent a point of the transversal $\Xi_{{\mathcal{L}}}$. As one can see, the axis of the pattern rotates and also the local environment of the aligned resonator changes as well as a result of these actions. Still, there is a bijection between the orbit of ${\mathcal{L}}$ generated by these alignments and the points $n\theta$ of the circle $2\pi{\mathbb{R}}/{\mathbb{Z}}$. The latter densely fill the circle and it is not difficult to show (see [34] for such an exercise) that the transversal, which is the closure of the orbit, is homeomorphic with the full circle.
$\Diamond$
Remark 3.15.
Example 3.14 can be viewed as a quasi 1-dimensional physical system without a defect, but note that its natural description and computation of the transversal still involves the Euclidean group ${\rm Iso}({\mathbb{E}}^{2})$. As a point set of ${\rm Iso}({\mathbb{E}}^{2})$, this pattern is uniformly separated but not a Delone set. This also shows why the class of uniformly separated patterns is actually the natural one to consider for this kind of applications.
$\Diamond$
Remark 3.16.
We can use Example 3.14 to showcase some practical advantages brought in by the leap to the full Euclidean group. The quasi-periodic patterns of point resonators are quite popular with the materials scientists. In the existing applications, some already mentioned in section 1, the patterns were generated by pure translations of the seed resonators, following certain algorithms (see e.g. [1] or [22]). In such cases, the physical systems are treated as point sets in ${\mathbb{R}}^{d}$ and the standard transversal can be shaped as circles or tori. Many dynamical effects can be, theoretically, brought to life by driving the phason along topologically nontrivial paths inside such transversals. This, unfortunately, is a very difficult task in practice, especially if only translations are allowed. For the pattern from Example 3.14, however, a driving of the phason can be accomplished by synchronously rotating the resonator while holding their centers fixed, which is a much simpler procedure. This is not an isolated example since upgrading from the group of pure rotations to the full group of Euclidean isometries brings in more opportunities that can be exploited in specific applications (see [39] for specific examples).
$\Diamond$
Remark 3.17.
It will be difficult here to apply Kellendonk’s prescription, which requires tiling of the space.$\Diamond$
Example 3.18.
The pattern in Fig. 3.4 was generated using again the discrete subgroup p2, but each isometry from the p2 group was applied on a different initial shape. These initial shapes were generated by applying $(t,\bm{r})\in{\rm Iso}({\mathbb{E}}^{2})$ on a triangle aligned with the laboratory frame, where $(t,\bm{r})$ were sampled randomly and uniformly from the set $(t_{0},\bm{r}_{0})\cdot B_{\epsilon}\times[-\epsilon,\epsilon]$, where $B_{\epsilon}$ is the $\epsilon$-ball of ${\mathbb{E}}^{2}$ centered at the origin. Here, we used the parametrization (6.9) of ${\rm Iso}({\mathbb{E}}^{2})$. These random fluctuations can be a result of inherent experimental errors. In his case, the transversal is the product space $\left(B_{\epsilon}\times[-\epsilon,\epsilon]\right)^{\times\rm p2}$ equipped with the product topology. This space comes with a continuous action of p2.
$\Diamond$
Example 3.19.
Additional examples can be found in [4].
$\Diamond$
4. Dynamics by groupoid methods
This section describes the groupoid canonically associated to a pattern. We pay equal attention to the algebraic structure and the topology of this groupoid. A Haar system is generally needed to build a $C^{\ast}$-algebra over such groupoid. An important aspect of the present context is that the groupoids associated to uniformly separated patterns are etalé and such groupoids come with an intrinsic Haar system, hence with an intrinsic $C^{\ast}$-algebra. The main statement of the section is that this $C^{\ast}$-algebra generates the dynamical matrices of the resonant physical systems discussed in section 2, through its standard left regular representation.
4.1. Topological groupoid associated to a pattern
To keep the exposition as streamlined as possible, we assume a readership familiar with basic notions related to groupoids. Valuable sources of information on the subject are [46, 49, 51].
Any topological dynamical system, hence $\left({\rm Iso}({\mathbb{E}}^{d}),\Omega^{\times}_{{\mathcal{L}}}\right)$ too, has a canonically associated transformation groupoid. A more general yet simpler structure is that of generalized equivalence and its associated topological groupoid:
Definition 4.1 ([51], p. 5).
Let $G$ be a group and $X$ a set. Then $X\times G\times X$ can be given the structure of a groupoid by adopting the inversion function
$$(x,g,y)^{-1}=(y,g^{-1},x),$$
(4.1)
the range and source maps
$$\mathfrak{r}(x,g,y)=(x,e,x),\quad\mathfrak{s}(x,g,y)=(y,e,y),$$
(4.2)
and by declaring that two triples $(x,g,y)$ and $(w,h,z)$ are composible if $y=w$, in which case the product is
$$(x,g,y)\cdot(y,h,z):=(x,gh,z).$$
(4.3)
If $X$ is a topological space and $G$ is a topological group, then $X\times G\times X$ becomes a topological groupoid if endowed with the product topology.
Using this simple groupoid structure, we can effortlessly describe both the algebraic structure and topology of a transformation groupoid:
Definition 4.2 ([51] p.6).
Let $G$ be lcsc and $X$ be a lcsc topological $G$-space. Then the transformation groupoid of $(X,G)$ is the lcsc topological sub-groupoid of the groupoid introduced in Definition 4.1 consisting of the triples $(x,g,y)$ with $x=g\cdot y$. The topology is that induced from $X\times G\times X$.
From now on, we will fix a pattern ${\mathcal{L}}_{0}\in{\mathcal{C}}(G)$ in a lcsc group $G$. For the start, ${\mathcal{L}}_{0}$ can be any closed subset. When applied to $\left(G,\Omega^{\times}_{{\mathcal{L}}_{0}}\right)$, the above definition leads to the lcsc groupoid $\widetilde{\mathcal{G}}_{{\mathcal{L}}_{0}}$ with source $\tilde{\mathfrak{s}}$ and range $\tilde{\mathfrak{r}}$ maps
$$\tilde{\mathfrak{s}}\left(g\cdot{\mathcal{L}},g,{\mathcal{L}}\right)=({\mathcal{L}},e,{\mathcal{L}}),\quad\tilde{\mathfrak{r}}\left(g\cdot{\mathcal{L}},g,{\mathcal{L}}\right)=\left(g\cdot{\mathcal{L}},e,g\cdot{\mathcal{L}}\right).$$
(4.4)
Its space of units $\widetilde{\mathcal{G}}_{{\mathcal{L}}_{0}}^{0}:=\tilde{\mathfrak{s}}(\widetilde{\mathcal{G}}_{{\mathcal{L}}_{0}})=\tilde{\mathfrak{r}}(\widetilde{\mathcal{G}}_{{\mathcal{L}}_{0}})$ coincides with the punctured hull $\Omega^{\times}_{{\mathcal{L}}_{0}}$.
Definition 4.3.
The canonical topological groupoid ${\mathcal{G}}_{{\mathcal{L}}_{0}}$ associated to the closed pattern ${\mathcal{L}}_{0}\in{\mathcal{C}}(G)$ is the restriction of $\widetilde{\mathcal{G}}_{{\mathcal{L}}_{0}}$ to the transversal $\Xi_{{\mathcal{L}}_{0}}\subset\widetilde{\mathcal{G}}_{{\mathcal{L}}_{0}}^{0}$ of Definition 3.9:
$${\mathcal{G}}_{{\mathcal{L}}_{0}}:=\tilde{\mathfrak{s}}^{-1}(\Xi_{{\mathcal{L}}_{0}})\cap\tilde{\mathfrak{r}}^{-1}(\Xi_{{\mathcal{L}}_{0}}).$$
(4.5)
Its space of units ${\mathcal{G}}_{{\mathcal{L}}}^{0}$ coincides with $\Xi_{{\mathcal{L}}_{0}}$, hence it is a compact space. Among other things, this assures us that its groupoid $C^{\ast}$-algebra has a unit. The canonical groupoid of a pattern can be characterized more concretely in a form that brings it more closely to the ones seen in the physics applications:
Proposition 4.4.
The topological groupoid canonically associated to a pattern ${\mathcal{L}}_{0}\in{\mathcal{C}}(G)$ consists of
1.
The set of tuples
$${\mathcal{G}}_{{\mathcal{L}}_{0}}=\left\{(g,{\mathcal{L}}),\ {\mathcal{L}}\in\Xi_{{\mathcal{L}}_{0}},\ g\in{\mathcal{L}}\right\}\subset G\times\Xi_{{\mathcal{L}}_{0}},$$
(4.6)
equipped with the inversion map
$$(g,{\mathcal{L}})^{-1}=(g^{-1},g\cdot{\mathcal{L}})=g\cdot(e,{\mathcal{L}})$$
(4.7)
and lcsc topology inherited from $G\times\Xi_{{\mathcal{L}}_{0}}$.
2.
The subset of ${\mathcal{G}}_{{\mathcal{L}}_{0}}\times{\mathcal{G}}_{{\mathcal{L}}_{0}}$ of composable elements
$${\mathcal{G}}_{{\mathcal{L}}_{0}}^{2}=\left\{\left((g^{\prime},{\mathcal{L}}^{\prime}),(g,{\mathcal{L}})\right)\in{\mathcal{G}}_{{\mathcal{L}}_{0}}\times{\mathcal{G}}_{{\mathcal{L}}_{0}},\ {\mathcal{L}}^{\prime}=g\cdot{\mathcal{L}}\right\}$$
(4.8)
equipped with the composition
$$(g^{\prime},{\mathcal{L}}^{\prime})\cdot(g,{\mathcal{L}})=(g^{\prime}g,{\mathcal{L}}).$$
(4.9)
The source and range maps of ${\mathcal{G}}_{{\mathcal{L}}_{0}}$ are
$$\mathfrak{s}(g,{\mathcal{L}})=(e,{\mathcal{L}}),\quad\mathfrak{r}(g,{\mathcal{L}})=\left(e,g\cdot{\mathcal{L}}\right).$$
(4.10)
Proof.
Since
$$\tilde{\mathfrak{s}}^{-1}({\mathcal{L}},e,{\mathcal{L}})=\left\{(g\cdot{\mathcal{L}},g,{\mathcal{L}}),\ g\in G\right\}$$
(4.11)
and
$$\quad\tilde{\mathfrak{r}}^{-1}({\mathcal{L}},e,{\mathcal{L}})=\left\{({\mathcal{L}},g^{-1},g\cdot{\mathcal{L}}),g\in G\right\},$$
(4.12)
$(g\cdot{\mathcal{L}},g,{\mathcal{L}})$ belongs to the intersection (4.5) if and only if ${\mathcal{L}}$ and $g\cdot{\mathcal{L}}$ both belong to $\Xi_{{\mathcal{L}}_{0}}$. This automatically constrains $g$ to be on the lattice ${\mathcal{L}}$ and, vice versa, if $g\in{\mathcal{L}}$, then $(g\cdot{\mathcal{L}},g,{\mathcal{L}})$ belongs to the intersection (4.5). Thus ${\mathcal{G}}_{{\mathcal{L}}_{0}}$ can be presented as the sub-groupoid of $\widetilde{\mathcal{G}}_{{\mathcal{L}}_{0}}$ consisting of the triples
$$(g\cdot{\mathcal{L}},g,{\mathcal{L}})\in\widetilde{\mathcal{G}}_{{\mathcal{L}}_{0}},\quad{\mathcal{L}}\in\Xi_{{\mathcal{L}}_{0}}\subset\Omega^{\times}_{{\mathcal{L}}_{0}},\quad g\in{\mathcal{L}},$$
(4.13)
endowed with the topology inherited from $\widetilde{G}_{{\mathcal{L}}_{0}}$. Lastly, we can drop the redundant first entry of the triples.
∎
Example 4.5.
In the case when $\Xi_{{\mathcal{L}}_{0}}$ is generated with a wallpaper group, as in Examples 3.11 and 3.12, the transversal consists of single point and ${\mathcal{G}}_{{\mathcal{L}}_{0}}$ reduces to the wallpaper group itself.
$\Diamond$
Example 4.6.
For the setting from Example 3.18, ${\mathcal{G}}_{{\mathcal{L}}_{0}}$ is isomorphic with the transformation groupoid associated to $\left({\rm p2},\left(B_{\epsilon}\times[-\epsilon,\epsilon]\right)^{\times\rm p2}\right)$.
$\Diamond$
In [41], the authors introduced an equivalence relation on groupoids that ensures that their associated $C^{\ast}$-algebras are Morita equivalent. In this context, they introduced the notion of abstract transversal:
Definition 4.7 ([41]).
Let ${\mathcal{G}}$ be a lcsc groupoid with source and range maps $\mathfrak{s}$ and $\mathfrak{r}$, respectively. An abstract transversal for ${\mathcal{G}}$ is a closed subset $\Xi$ of its space of units ${\mathcal{G}}^{0}$ such that $\Xi\cap\mathfrak{r}\left(\mathfrak{s}^{-1}(x)\right)\neq\emptyset$ for any $x\in{\mathcal{G}}^{0}$ and $\mathfrak{s}$ and $\mathfrak{r}$ are both open maps when restricted to $\mathfrak{r}^{-1}(\Xi)$.
Remark 4.8.
The notion of group action on topological spaces can be generalized to actions of groupoids [41] (see also [51][Def. 2.1]). For the transformation groupoid associated to a dynamical system $(G,\Omega)$, the groupoid action on its space of units is determined by the action of $G$ on $\Omega$. Note that $\mathfrak{r}\left(\mathfrak{s}^{-1}(x)\right)$ appearing in the above definition is just the orbit of $x$ under the $G$-action. Thus, an abstract transversal intersects every $G$-orbit. The other conditions ensure that the notion of transversal intersection is compatible with the topological groupoid structure. Thus, $\Xi$ is the analog of a Poincaré section.
$\Diamond$
Proposition 4.9 ([41]).
If $\Xi$ is an abstract transversal for ${\mathcal{G}}$, then ${\mathcal{G}}$ and ${\mathcal{G}}|_{\Xi}:=\mathfrak{s}^{-1}(\Xi)\cap\mathfrak{r}^{-1}(\Xi)$ are equivalent groupoids. In particular, their respective groupoid $C^{\ast}$-algebras are Morita equivalent.
In our context, the following statement assures us that $\widetilde{\mathcal{G}}_{{\mathcal{L}}_{0}}$ and ${\mathcal{G}}_{{\mathcal{L}}_{0}}$ are equivalent groupoids.
Proposition 4.10 ([27]).
Let ${\mathcal{L}}_{0}$ be a uniformly separated pattern in $G$. Then the standard transversal $\Xi_{{\mathcal{L}}_{0}}$ from Definition 3.9 is an abstract transversal for $\widetilde{\mathcal{G}}_{{\mathcal{L}}_{0}}$.
Remark 4.11.
Both groupoids $\widetilde{\mathcal{G}}_{{\mathcal{L}}_{0}}$ and ${\mathcal{G}}_{{\mathcal{L}}_{0}}$ are equally useful in physical applications [5, 32]. For example, most of the time, the empirical input, such as probability measures on the spaces of units, is readily available for $\widetilde{\mathcal{G}}_{{\mathcal{L}}_{0}}$, but, as explained below, the analysis is simpler on ${\mathcal{G}}_{{\mathcal{L}}_{0}}$.$\Diamond$
Definition 4.12 ([49], Def. 8.4.1).
A topological groupoid ${\mathcal{G}}$ is called étale if the range map is a local homeomorphism.
Remark 4.13.
The étale groupoids can be thought as generalizations of discrete groups. Their fibers $\mathfrak{r}^{-1}(x)$ and $\mathfrak{s}^{-1}(x)$ are discrete topological spaces for any $x\in{\mathcal{G}}^{0}$ and any étale groupoid admits a canonical Haar system supplied by the system of counting measures on each fiber [46, Def. 2.6, Lemma 2.7]. $\Diamond$
The following remarkable result from [27] characterizes the uniformly separated patterns in a lcsc group via their canonical groupoid $C^{\ast}$-algebras:
Proposition 4.14 ([27]).
Consider a lcsc group $G$ and ${\mathcal{L}}_{0}\in{\mathcal{C}}(G)$. Then the groupoid ${\mathcal{G}}_{{\mathcal{L}}_{0}}$ is étale if and only if ${\mathcal{L}}_{0}$ is uniformly separated.
The important conclusion is that any of the physical system described in section 2 has an underlying lcsc étale groupoid.
4.2. The groupoid $C^{\ast}$-algebra of a uniforlmy separated pattern
The linear space $C_{c}\left({\mathcal{G}}_{{\mathcal{L}}_{0}},M_{N}({\mathbb{C}})\right)$ of compactly supported $M_{N}({\mathbb{C}})$-valued continuous functions on ${\mathcal{G}}_{{\mathcal{L}}_{0}}$ can be endowed with the associative multiplication
$$(f_{1}\ast f_{2})(g,{\mathcal{L}})=\sum_{g^{\prime}\in{\mathcal{L}}}f_{1}\left(g^{\prime}\cdot(g,{\mathcal{L}})\right)\cdot f_{2}(g^{\prime},{\mathcal{L}})$$
(4.14)
and $\ast$-operation
$$f^{\ast}(g,{\mathcal{L}})=f\left((g,{\mathcal{L}})^{-1}\right)^{\dagger}=f(g^{-1},g\cdot{\mathcal{L}})^{\dagger}.$$
(4.15)
The resulting $\ast$-algebra is completed to a $C^{\ast}$-algebra by using the natural right-module structure of $C_{c}\left({\mathcal{G}}_{{\mathcal{L}}_{0}},M_{N}({\mathbb{C}})\right)$ over the $C^{\ast}$-algebra $C\left(\Xi_{{\mathcal{L}}_{0}},M_{N}({\mathbb{C}})\right)$. The latter can be promoted to a $C^{\ast}$-Hilbert module structure via the $C\left(\Xi_{{\mathcal{L}}_{0}},M_{N}({\mathbb{C}})\right)$-valued inner product
$$\langle f_{1}|f_{2}\rangle({\mathcal{L}}):=\rho(f_{1}^{\ast}\ast f_{2})({\mathcal{L}})=\sum_{g\in{\mathcal{L}}}f_{1}(g,{\mathcal{L}})^{\dagger}\cdot f_{2}(g,{\mathcal{L}}),$$
(4.16)
where $\rho$ is the restriction map
$$\rho:C_{c}\left({\mathcal{G}}_{{\mathcal{L}}_{0}},M_{N}({\mathbb{C}})\right)\rightarrow C\left(\Xi_{{\mathcal{L}}_{0}},M_{N}({\mathbb{C}})\right),\ \ (\rho f)({\mathcal{L}})=f(e,{\mathcal{L}}).$$
(4.17)
If $E_{{\mathcal{L}}_{0}}$ is the Hilbert $C^{\ast}$-module completion of $C_{c}\left({\mathcal{G}}_{{\mathcal{L}}_{0}},M_{N}({\mathbb{C}})\right)$ in this inner product, then the convolution from the left supplies a left action of the $\ast$-algebra $C_{c}\left({\mathcal{G}}_{{\mathcal{L}}_{0}},M_{N}({\mathbb{C}})\right)$ on $E_{{\mathcal{L}}_{0}}$ by bounded adjointable endomorphisms, extending the action of $C_{c}\left({\mathcal{G}}_{{\mathcal{L}}_{0}},M_{N}({\mathbb{C}})\right)$ on itself.
Definition 4.15 ([35]).
The reduced groupoid $C^{\ast}$-algebra of ${\mathcal{G}}_{{\mathcal{L}}_{0}}$ over the $C^{\ast}$-algebra $M_{N}({\mathbb{C}})$ is the completion of the core algebra $C_{c}\left({\mathcal{G}}_{{\mathcal{L}}_{0}},M_{N}({\mathbb{C}})\right)$ in the norm inherited from the embedding
$$C_{c}\left({\mathcal{G}}_{{\mathcal{L}}_{0}},M_{N}({\mathbb{C}})\right)\rightarrowtail{\rm End}^{\ast}(E_{{\mathcal{L}}_{0}})$$
and is denoted $C^{\ast}_{r}\left({\mathcal{G}}_{{\mathcal{L}}_{0}},M_{N}({\mathbb{C}})\right)$.
Remark 4.16.
If $N=1$, we simplify the notation to $C^{\ast}_{r}({\mathcal{G}}_{{\mathcal{L}}_{0}})$ and refer to this algebra as the groupoid $C^{\ast}$-algebra of ${\mathcal{G}}_{{\mathcal{L}}_{0}}$. There is a standard isomorphism $C^{\ast}_{r}\left({\mathcal{G}}_{{\mathcal{L}}_{0}},M_{N}({\mathbb{C}})\right)\simeq M_{N}({\mathbb{C}})\otimes C^{\ast}_{r}({\mathcal{G}}_{{\mathcal{L}}_{0}})$, so those more general $C^{\ast}$-algebras appearing in Definition 4.15 are just matrix amplifications of the groupoid $C^{\ast}$-algebra of ${\mathcal{G}}_{{\mathcal{L}}_{0}}$. We recall that such matrix amplifications have no impact on the K-theory. The isomorphism we just mentioned will come handy in section 4.4, where we make the connection with the models discussed in section 2.
$\Diamond$
Remark 4.17.
For a second countable, Hausdorff, étale groupoid ${\mathcal{G}}$, the reduced $C^{\ast}$-algebra is separable. In turn, this implies that its $K$-groups are countable [17] and this is of fundamental importance for the sought applications because it enables a sensible enumeration of the topological dynamical states supported by a class of metamaterials. $\Diamond$
Remark 4.18.
For non-amenable groupoids, their reduced and full $C^{\ast}$-algebras are distinct. For the concrete applications discussed in section 2, it is the reduced $C^{\ast}$-algebra that is relevant.$\Diamond$
Example 4.19.
Given the discussion in Example 4.5, it is immediate to see that, when the pattern is generated using a wallpaper group $G$ as in Examples 3.11 and 3.12, there is an isomorphism $C^{*}_{r}(\mathcal{G}_{\mathcal{L}_{0}},M_{N}(\mathbb{C}))\simeq M_{N}({\mathbb{C}})\otimes C^{\ast}_{r}(G)$, where the latter is the reduced group $C^{\ast}$-algebra of $G$.
$\Diamond$
Remark 4.20.
If the patterns from Examples 3.11 and 3.12 are seen as closed subsets of ${\mathbb{R}}^{2}$, as discussed in Remark 3.13, then the associated groupoid algebra will be $M_{\alpha N}({\mathbb{C}})\otimes C^{\ast}_{r}({\mathbb{Z}}^{2})$, where $\alpha=2$ and 4, respectively. These $C^{\ast}$-algebras are strictly larger than the group $C^{\ast}$-algebras of the wallpaper groups that generated the patterns in the first place.$\Diamond$
Example 4.21.
The reduced groupoid $C^{\ast}$-algebra corresponding to the tranformation groupoid of a $G$-space coincides with the corresponding reduced crossed product $C^{\ast}$-algebra [51, Example 1.51]. Therefore, for the setting discussed in Example 3.18, the canonical groupoid $C^{\ast}$-algebra reduces to the crossed product algebra associated to the dynamical system $\left({\rm p2},\left(B_{\epsilon}\times[-\epsilon,\epsilon]\right)^{\times\rm p2}\right)$.
$\Diamond$
Example 4.22.
For the pattern from Example 3.14, the groupoid algebra reduces to the tensor product of $M_{N}({\mathbb{C}})$ and the crossed product $C({\mathbb{S}}^{1})\rtimes{\mathbb{Z}}$, where the action of ${\mathbb{Z}}$ is by rotation by $\theta$.
$\Diamond$
4.3. The left-regular representations
The restriction map $\rho$ introduced above extends by continuity to a positive map between $C^{\ast}$-algebras. By composing $\rho$ with the evaluation maps
$$j_{{\mathcal{L}}}:C\left(\Xi_{{\mathcal{L}}_{0}},M_{N}({\mathbb{C}})\right)\to M_{N}({\mathbb{C}}),\ \ j_{\mathcal{L}}(\chi)=\chi({\mathcal{L}})$$
(4.18)
for some ${\mathcal{L}}\in\Xi_{{\mathcal{L}}_{0}}$, one obtains a family of $M_{N}({\mathbb{C}})$-valued positive maps. By further composing with the trace state on $M_{N}({\mathbb{C}})$, one obtains a family of states
$$\rho_{\mathcal{L}}={\rm Tr}\circ j_{\mathcal{L}}\circ\rho$$
(4.19)
on $C^{\ast}_{r}\left({\mathcal{G}}_{{\mathcal{L}}_{0}},M_{N}({\mathbb{C}})\right)$, indexed by the transversal $\Xi_{{\mathcal{L}}_{0}}$. The GNS-representations corresponding to these states supply the left-regular representations of the $C^{\ast}$ algebra $C^{\ast}_{r}\left({\mathcal{G}}_{{\mathcal{L}}_{0}},M_{N}({\mathbb{C}})\right)$.
4.4. Sorting dynamics with groupoid algebras
We are now in the position to make the connection with the dynamical matrices (2.20) analyzed in section 2. We recall that our stated goal is to determine the smallest $C^{\ast}$-algebra which generates all dynamical matrices associated to a given aperiodic architecture of identical resonators with adjustable internal structure.
We specialize the discussion to the particular case when the lcsc group $G$ is ${\rm Iso}({\mathbb{E}}^{d})$ and ${\mathcal{L}}_{0}$ is one of its uniformly separated patterns, representing a specific configuration of identical resonators.
The Hilbert space corresponding to GNS-representation induced by $\rho_{\mathcal{L}}$ is
$${\mathcal{H}}_{\mathcal{L}}:=\ell^{2}\left(\mathfrak{s}^{-1}({\mathcal{L}}),{\mathbb{C}}^{N}\right)=\ell^{2}\left({\mathcal{L}},{\mathbb{C}}^{N}\right),$$
(4.20)
the space of ${\mathbb{C}}^{N}$-valued square summable sequences over ${\mathcal{L}}$, and the representation acts explicitly as
$$[\pi_{\mathcal{L}}(f)\varphi](g^{\prime})=\sum_{g\in{\mathcal{L}}}f\left(g\cdot(g^{\prime},{\mathcal{L}})\right)\cdot\varphi(g),\quad g^{\prime}\in{\mathcal{L}},\ \varphi\in\ell^{2}\left({\mathcal{L}},{\mathbb{C}}^{N}\right),$$
(4.21)
which follows directly from Eq. (4.14). In particular,444Here, $\alpha\otimes|g^{\prime}\rangle$ represents the map $\varphi(g)=\alpha\,\delta_{g,g^{\prime}}$.
$$\pi_{\mathcal{L}}(f)(\alpha\otimes|g^{\prime}\rangle)=\sum_{g\in{\mathcal{L}}}f\left(g\cdot(g^{\prime},{\mathcal{L}})\right)\cdot\alpha\otimes|g\rangle,\quad\alpha\in{\mathbb{C}}^{N}.$$
(4.22)
Below, we reproduce the expression (2.20) of a generic dynamical matrix
$$D_{{\mathcal{L}}}=\sum_{x,x^{\prime}\in{\mathcal{L}}}w_{e,x\cdot x^{\prime}}(x\cdot{\mathcal{L}})\otimes|x\rangle\langle x^{\prime}|$$
(4.23)
and, obviously,
$$D_{{\mathcal{L}}}(\alpha\otimes|x^{\prime}\rangle)=\sum_{x\in{\mathcal{L}}}w_{e,x\cdot x^{\prime}}(x\cdot{\mathcal{L}})\cdot\alpha\otimes|x\rangle.$$
(4.24)
We now recall the discussion from subsection 2.6 and, by comparing Eqs. (4.21) and (4.23), we see that a dynamical matrix with finite coupling range can be generated as the left-regular representation of an element $f$ from the core sub-algebra, if we take $f(g,{\mathcal{L}}):=w_{e,g}({\mathcal{L}})$ for any $(g,{\mathcal{L}})\in{\mathcal{G}}_{{\mathcal{L}}_{0}}$. Since these sets are dense in both the space of acceptable dynamical matrices and the groupoid $C^{\ast}$-algebra, we can conclude that $C^{\ast}_{r}\left({\mathcal{G}}_{\mathcal{L}},M_{N}({\mathbb{C}})\right)$ generate all dynamical matrices that can be controlled in a laboratory, for a fixed number of internal degrees of freedom.
Various dynamical effects, such as topological pumping, topological phase transitions, etc., require deformations of the internal structures of the resonators. During such deformations, the degrees of freedom carried by the seed resonator may be continuously turned on or off. Hence, we need to define a $C^{\ast}$-algebra that covers such situations and, furthermore, gives a precise sense to what it means to “continuously” turn on and off a degree of freedom. For this, we recall the isomorphism from Remark 4.16 and note the following canonical embedding
$$C^{\ast}_{r}({\mathcal{G}}_{{\mathcal{L}}_{0}},M_{N}({\mathbb{C}}))\simeq M_{N}({\mathbb{C}})\otimes C^{\ast}_{r}({\mathcal{G}}_{{\mathcal{L}}_{0}})\hookrightarrow{\mathbb{K}}\otimes C^{\ast}_{r}({\mathcal{G}}_{{\mathcal{L}}_{0}}),$$
(4.25)
where ${\mathbb{K}}$ is the algebra of compact operators over $\ell^{2}({\mathcal{L}})$. Thus, ${\mathbb{K}}\otimes C^{\ast}_{r}({\mathcal{G}}_{{\mathcal{L}}_{0}})$ is a natural solution for the problem we just described. Furthermore, as argued in subsection 2.7, this is the smallest $C^{\ast}$-algebra among the possible solutions, hence, it is the algebra we were actually looking for. At this point, we have justified the statements 1) through 5) in our introductory section 1 and statement 6) is a direct consequence of those statements.
5. Outlook
This section indicates several directions in materials science and research that can be efficiently investigated with the formalism developed by our work.
Firstly, if the dynamical matrix $D_{{\mathcal{L}}_{0}}$ displays an isolated band $\Delta$ of resonant spectrum, then we can define the spectral projection $P_{\Delta}$ onto $\Delta$, usually called the band projection and observe the time evolution of a state $\psi$ excited from $P_{\Delta}{\mathcal{H}}_{{\mathcal{L}}_{0}}$. Since this time evolution cannot escape this subspace, $P_{\Delta}\mathcal{H}_{{\mathcal{L}}_{0}}$ defines an invariant for dynamics. As it is well known, these invariants are accounted for by the $K_{0}$-theory of $C^{\ast}_{r}({\mathcal{G}}_{{\mathcal{L}}_{0}})$, and here is where the $C^{\ast}$-algebraic framework can make fine contributions to our understanding of the dynamics of these physical system [5, 7]. Our formalism opens such lines of research for contexts that were not explored yet.
Secondly, we already mentioned that the bulk-defect correspondence principle can be formulated very generally and precisely using groupoids, to a point where a full classification of materials defects can be attempted [18, 44]. There is already plenty of evidence that possible point symmetries of defects can highly enhance the bulk-defect correspondences [23]. For example, the point-defects of the patterns examined in Examples 3.11 and 3.12, can still display a symmetry relative to a point group. If this is the case, then our formalism leads to a defect $C^{\ast}$-algebra that is stably isomorphic to the group $C^{\ast}$-algebra of this point group. This means that the bulk-defect correspondence lands in the representation ring of this point group, thus possibly leading to interesting topological defect states. In contradistinction, in a formalism that only engages the group of pure translations, the defect $C^{\ast}$-algebra is always isomorphic to ${\mathbb{K}}$, hence no such refined predictions are possible.
Thirdly, the mathematical framework can be applied to patterns of resonators generated with other lcsc groups, but one has to be careful on the physics side. For example, fractal patterns engage the scaling transformation of the Euclidean space. Thus, an interesting direction will be to upgrade from ${\rm Iso}({\mathbb{E}}^{d})$ to the sub-group ${\rm Sim}({\mathbb{E}}^{d})$ of the affine transformations generated by the isometries and dilation. In this case, however, the equivariance (2.19) is no longer ensured by natural laws and, instead, it needs to be enforced by finely tuning the internal structure of the resonators, which, nevertheless, is within the reach of the present manufacturing capabilities.
Lastly, the cyclic cohomology of the groupoid $C^{\ast}$-algebra and its pairing with the K-theory [24] supplies the venue for us to discover the topological responses to external stimuli for entirely new classes of materials, e.g. by following [18, 19] as models of analysis.
6. Appendix: The Euclidean spaces and their groups of isometries
At the conceptual level, the resonators live in the group of isometries of their ambient physical spaces. As such, we collect in this appendix some known facts about the algebraic and topological characteristics of these groups. The information will come handy when we discuss particular examples, though, it will play little role in the main formal developments.
6.1. Generic facts
The $d$-dimensional Euclidean space ${\mathbb{E}}^{d}$ consists of the set ${\mathbb{R}}^{d}$ equipped with the metric
$${\rm d}(x,y):=\left[\sum\nolimits_{j=1}^{d}(x_{j}-y_{j})^{2}\right]^{\frac{1}{2}}.$$
(6.1)
We will consistently represent a point of ${\mathbb{R}}^{d}$ as a column vector with $d$ entries and use “$\cdot$” to indicate matrix multiplications. Furthermore, we let ${\rm Homeo}({\mathbb{E}}^{d})$ denote the group of homeomorphims of ${\mathbb{E}}^{d}$.
Proposition 6.1 ([3]).
The set ${\rm Homeo}({\mathbb{E}}^{d})$ equipped with the compact-open topology inherited from ${\mathbb{E}}^{d}$ is a topological group, i.e. multiplication and inversion maps are both continuous in this topology.
Remark 6.2.
Certain fine aspects about the above statement are worth recalling. If a topological Hausdorff space is compact, then its group of homeomorphisms is automatically a topological group when equipped with the compact-open topology. This is not always the case for non-compact space. It is true in this particular case because the compact-open topology on ${\rm Homeo}({\mathbb{E}}^{d})$ coincides with the one inherited from the one-point compactification $\alpha{\mathbb{E}}^{d}$ of the Euclidean space [26].
$\Diamond$
Definition 6.3.
The Euclidean group ${\rm Iso}({\mathbb{E}}^{d})$ is defined as the topological sub-group of ${\rm Homeo}({\mathbb{E}}^{d})$ containing the set of isometric homeomorphisms,
$$\operatorname{\mathrm{Iso}}({\mathbb{E}}^{d}):=\left\{f\in{\rm Homeo}\left({\mathbb{E}}^{2}\right)~{}|~{}{\rm d}(f(x),f(y))={\rm d}(x,y)~{}\forall\ x,y\in{\mathbb{E}}^{d}\right\}.$$
(6.2)
By definition, the topology of ${\rm Iso}({\mathbb{E}}^{d})$ is that inherited from ${\rm Homeo}({\mathbb{E}}^{d})$.
6.2. Structure and topology of the Euclidean groups
The topological group ${\mathbb{R}}^{d}$ acts by translations on ${\mathbb{E}}^{d}$ and is a normal sub-group of ${\rm Iso}({\mathbb{E}}^{d})$. The topological group $O(d)$ acts by rotations on ${\mathbb{E}}^{d}$ and is also a sub-group of ${\rm Iso}({\mathbb{E}}^{d})$, though not a normal one. It consists of the set of orthogonal matrices
$$O(d)=\{\bm{r}\in M_{d}({\mathbb{R}}),\ \bm{r}\cdot\bm{r}^{T}=\bm{r}^{T}\cdot\bm{r}=I_{d}\}$$
(6.3)
and the topology inherited from ${\rm Homeo}({\mathbb{E}}^{d})$ coincides with the one induced by the metric
$${\rm d}_{O}(\bm{r},\bm{r}^{\prime})=\|\bm{r}-\bm{r}^{\prime}\|_{\rm HS},$$
(6.4)
where $\|\cdot\|_{\rm HS}$ is the Hilbert-Schmidt norm. The Euclidean group can be presented as the semi-direct product
$${\rm Iso}({\mathbb{E}}^{d})={\mathbb{R}}^{d}\rtimes_{\beta}O(d),\quad\beta_{\bm{r}}(v)=\bm{r}\cdot v.$$
(6.5)
Thus, an element of ${\rm Iso}({\mathbb{E}}^{d})$ can be uniquely presented as a tuple $(v,\bm{r})$, with $v\in{\mathbb{R}}^{d}$ and $\bm{r}\in O(d)$, and composition and inversion rules
$$(v_{1},\bm{r}_{1})(v_{2},\bm{r}_{2})=(v_{1}+\bm{r}_{1}\cdot v_{2},\bm{r}_{1}\cdot\bm{r}_{2}),\quad(v,\bm{r})^{-1}=(-\bm{r}^{-1}\cdot v,\bm{r}^{-1}).$$
(6.6)
We will reserve the symbol $\bm{1}$ for the neutral element $(0,I_{d})$ of ${\rm Iso}({\mathbb{E}}^{d})$. As a topological space, ${\rm Iso}({\mathbb{E}}^{d})={\mathbb{R}}^{d}\times O(d)$, with the group appearing on the right reduced to a topological space.
Regarding the structure of $O(d)$, any orthogonal matrix has determinant ${\rm det}(\bm{r})=\pm 1$ and, since the determinant is a continuous map, this implies that $O(d)$ has at least two connected components, the pre-images of $\pm 1$ through the determinant map, which turn out to be the only connected components of $O(d)$ [31]. Furthermore, since the determinant is a topological group morphism, the connected component of the neutral element $I_{d}$ is a normal sub-group of $O(N)$, called the special orthogonal group and denoted by $SO(N)$. This leads to the following split exact sequence
$$\{e\}\to SO(d)\to O(d)\rightarrow O(d)/SO(d)\simeq{\mathbb{Z}}_{2}\to\{e\}.$$
(6.7)
6.2.1. Dimension $d=2$
In this case, the generator of ${\mathbb{Z}}_{2}$ can be embedded in $O(2)$ via the matrix $\bm{p}=\begin{pmatrix}0&1\\
1&0\end{pmatrix}$, which has a nontrivial action by conjugation. Thus, $O(2)$ is the internal semi-direct product $O(2)=SO(2)\rtimes\left\langle\bm{p},I_{2\times 2}\right\rangle$ and, as a topological space,
$$O(2)\simeq{\mathbb{S}}^{1}\times\{-1,+1\}.$$
(6.8)
Hence, as a topological space,
$${\rm Iso}({\mathbb{E}}^{2})\simeq{\mathbb{R}}^{2}\times{\mathbb{S}}^{1}\times\{-1,+1\}.$$
(6.9)
6.2.2. Dimension $d=3$
In this case, the exact sequence (6.7) leads to a direct product $O(3)=SO(3)\times\{I_{d\times d},-I_{d\times d}\}$ and to a topological space, $O(3)\simeq{\mathbb{R}}{\mathbb{P}}^{3}\times\{-1,+1\}$. Hence, as a topological space,
$${\rm Iso}({\mathbb{E}}^{3})\simeq{\mathbb{R}}^{3}\times{\mathbb{R}}{\mathbb{P}}^{3}\times\{-1,+1\}.$$
(6.10)
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Convergence Rates for Stochastic Approximation on a Boundary
Kody Law
University of Manchester
Neil Walton
Durham University
Shangda Yang
University of Manchester
Abstract
We analyze the behavior of projected stochastic gradient descent focusing on the case where the optimum is on the boundary of the constraint set and the gradient does not vanish at the optimum. Here iterates may in expectation make progress against the objective at each step. When this and an appropriate moment condition on noise holds, we prove that the convergence rate to the optimum of the constrained stochastic gradient descent will be different and typically be faster than the unconstrained stochastic gradient descent algorithm. Our results argue that the concentration around the optimum is exponentially distributed rather than normally distributed, which typically determines the limiting convergence in the unconstrained case.
The methods that we develop rely on a geometric ergodicity proof. This extends a result on Markov chains by Hajek [14] to the area of stochastic approximation algorithms.
As examples, we show how the results apply to linear programming and tabular reinforcement learning.
1 Introduction.
We analyze the behavior of stochastic gradient descent focusing on the case where the optimum is on the boundary of the constraint set and the gradient does not vanish at the optimum.
For a projected stochastic gradient descent algorithm, when the expected value of the objective decreases on each iteration, we will show that a constrained stochastic gradient descent algorithm has a different rate of convergence that would be anticipated by standard results for stochastic gradient descent.
General convergence rates are known for (projected) stochastic gradient descent with a convex objective function. Here to obtain an $\epsilon$-approximation of the optimum, we require $O(\epsilon^{-2})$ iterations [26, 27, 6]. Such results can be improved to require $O(\epsilon^{-1})$ iterations when the objective function is strongly convex [26, 27, 6]; however, even in this case, the distance to the optimum will still be of order $O(\epsilon^{-2})$.
Moreover, there are commonly applied functions such as linear or piecewise linear objectives which are not strongly convex.
Thus standard results would suggest $O(\epsilon^{-2})$ iterations are required to obtain an $\epsilon$-approximation of the optimizer.
In this paper, we argue that a stochastic approximation procedure can achieve the faster rate of $O(\epsilon^{-1})$.
In our main result, we consider a constrained stochastic gradient descent algorithm minimizing $f(\bm{x})$ with iterates $\{\bm{x}_{t}\}_{t=0}^{\infty}$ belong to a constraint set $\mathcal{X}$ and with a learning rate of the form $1/t^{\gamma}$ with $\gamma\in[0,1]$. If the optimum is on the boundary then we can expect the algorithm to make progress against its objective in proportion to the learning rate. This is Condition C1. Further we assume sub-exponential noise. This is Condition C2. When Condition C1 and C2 hold then we prove that
$$\mathbb{E}[f(\bm{x}_{t})]-\min_{x\in\mathcal{X}}f(x)=O\left(\frac{1}{t^{\gamma}}\right)\,.$$
This result is more formally stated in Theorem 1 and, specifically, Corollary 1.
An exponential concentration bounds is also given. Taking $\gamma=1$, we see that an $\epsilon$-approximation of the optimum requires $O(\epsilon^{-1})$ iterations.
The conditions required for Theorem 1 are generic. Thus the results are not intended to be applicable to any one particular stochastic gradient descent algorithm, nor do we place specific design restrictions on the algorithm.
What the result emphasizes is that a different convergence rate may hold for a broad class of stochastic gradient descent algorithms when there are constraints. This differs and may well be faster than the well established convergence rates of stochastic gradient descent in the unconstrained case.
The high-level intuition for this behavior in a stochastic gradient algorithm is as follows. Consider a projected stochastic gradient descent algorithm with a small but fixed learning rate (cf. Section 3 for a formal definition).
When the optimum is in the interior of the constraint set, the progress of the algorithm will slow as the iterates approaches the minimizer in a manner that is roughly proportional to the distance to the optimum.
In this regime, the process behaves in a manner that is similar to an Ornstein-Uhlenbeck (OU) process [23].
An OU process is known to have a normal distribution as its limiting stationary distributions.
This stationary distribution determines convergence to the optimum [8].
If we consider the same iterates but instead these are now projected to belong to a polytope then, assuming that the optimum is on the boundary, the resulting process behaves in a manner that is approximated by a reflected Brownian motion [19, 20].
When the gradient is non-zero on the boundary, it is well known that a reflected Brownian motion with negative drift has an exponential distribution as its stationary distribution [15].
The normal and exponential distributions found have
a very different concentration in probability.
Thus we can anticipate a different convergence behavior.
In particular, for the one-dimensional case, simple calculations using Kingman’s bound [16] indicate that the distance from the optimum should be within $\epsilon$ of the optimum in $O(\epsilon^{-1})$ steps for a projected stochastic gradient descent algorithm.
The argument of Kingman also indicates that an exponential Lyapunov function might be more appropriate than the standard expansion and interpolation methods used for the convergence analysis of stochastic gradient descent.
The aim of this paper is to rigorously confirm this for stochastic approximation algorithms.
The proof method that we develop uses a martingale argument first described by Hajek [14] for Markov chains.
The Lyapunov function bounds of [14] and also [3]
are commonly employed to give exponential tail bounds for a stationary Markov processes. These are often applied in the analysis of queueing networks.
Here we adapt this analysis and apply it to stochastic gradient descent. To the best of our knowledge this is the first paper to apply this approach to gain concentration bounds in the context of stochastic approximation.
Above, we discuss our results for projected stochastic gradient descent; however, the results hold more broadly.
In particular, the main result (Theorem 1) requires a condition on the progress of algorithm and constraints on the noise induced by the gradient descent steps. Thus the results in Theorem 1 are not specific to any one algorithm. Instead they give general conditions for exponential concentration of measure to hold for a stochastic approximation procedure. For this reason, any constrained stochastic approximation procedure may enjoy faster convergence rates than those achieved by the corresponding unconstrained algorithm.
However, it is also important to identify scenarios when these assumptions are known to hold. In Theorem 2, we prove a convergence result for projected stochastic gradient descent when applied to a linear program. Here we show that when the set of constraints are known but the costs in the objective are unknown then the assumptions of Theorem 1 are satisfied and we can expect a faster convergence rate.
Finally as an application, we apply our results to tabular reinforcement learning.
We consider a discounted Markov Decision Process (MDP) where the state transition function is known but the reward function is unknown.
Algorithms such as Q-learning are well known to require $O(\epsilon^{-2})$ iterations to reach an $\epsilon$-approximation of the optimal value function.
It is well known that a MDP can be characterized as a linear program: either in a primal formulation that calculates the value function; or in a dual formulation that gives the occupancy measure of the MDP.
We apply projected stochastic gradient descent to the dual linear programming formulation of the MDP. This can be thought of as a relatively simple policy gradient algorithm. Here we prove a gap dependent bound which requires $O(\epsilon^{-1})$ to find an $\epsilon$-approximation of the optimal occupancy measure.
This policy gradient algorithm is faster than the rate of convergence that would be achieved for instance by $Q$-learning. Certainly, bespoke algorithms exist in recent reinforcement learning literature [33, 35]; however, it is striking that with a relatively simple change –to optimize over the policy’s occupancy measure– a standard projected gradient descent yields a faster algorithm.
Other other applications and extensions of our results would be possible. We have not considered different forms of averaging gradients which would likely yield a faster convergence rate. We assume a somewhat canonical choice of variance in gradient steps. (See Condition C2.) It is possible to investigate the dependence on the moment generating function of the gradient. This may be applicable, for instance, in online reinforcement learning.
Further, there may be different formulations of the exponential martingale argument given in Section 4. For instance, it is possible to prove similar results using the Martingale approach of Kingman [17].
The remainder of the paper is organized as follows. In Section 2, we review relevant literature. In Section 3, we introduce the basic model, notation and algorithms that we consider.
In Section 4, we present our main result: Theorem 1. Theorem 1 gives an exponential concentration bound for any stochastic approximation procedure satisfying two conditions, C1 and C2. We also provide a simpler formulation of the main result, namely, Corollary 1.
In Section 5 and specifically in Theorem 2, we apply our results to linear programs.
In Section 6, we then apply Theorem 2 to solve a tabular reinforcement learning problem.
The proof of Theorem 1 is contained in Section 7. Further results from Section 5 and Section 6 are proved in Section 8.
2 Literature Review.
We now review relevant literature. We review results on the rate of convergence of stochastic gradient descent.
We review the martingale approach that we use to establish exponential tail bound. We discuss its typical usage for establishing exponential ergodicity in Markov chains. We discuss the changes required to adapt these arguments to time-dependent stochastic approximation.
We review results on constrained stochastic approximation.
We apply our results to linear programs with unknown costs. So we briefly review probabilistic concentration results in this setting.
We apply our results to reinforcement learning. We review the linear programming approach to dynamic programming and discuss related results in reinforcement learning.
Lyapunov Bounds.
A key ingredient of our analysis is an exponential tail bound acheived using a geometric Lyanpunov function argument.
We note that such arguments are commonly employed in order to establish the exponential ergodicity of a Markov chain. For instance, see [25, Chapter 15]. Specifically, Hajek [14] provides a proof that converts a linear drift condition into an exponential Martingale that can then be used to gain fast convergence rates for ergodic Markov chains. Extending this argument to stochastic approximation is one of the key contributions of this paper.
The argument of Hajek can be thought of as an extension of the Martingale bound of Kingman for the G/G/1 queue [17]. Indeed it is possible to derive a different version of the proof of the given in this paper using the argument of Kingman; however, the approach of Hajek [14] is cleaner and so we do not peruse this Kingman proof here [17]. There are alternative arguments for achieving an exponential tail bound using a Lyapuov function with linear drift [3]. However, these arguments can apply to time-inhomogeneous Markov chains, and so are not applicable in our context.
As discussed in the introduction, many of these bounds have typically been applied in the context of queueing networks. This because many queueing processes can be viewed as random walks chains constrained to belong to a polytope region. Here it is broadly known that a linear negative drift with constraints leads to an exponential distribution bounds [15].
There are results connecting the limit behavior of stochastic approximation procedures with constrained random walks and diffusions [19].
Nonetheless, the concentration results proven here are, to the best of our knowledge, new in the context of stochastic approximation.
Stochastic Approximation and Stochastic Gradient Descent.
Due to its applicability in machine learning, there is now a vast literature on stochastic gradient descent [5].
The argument used originally by Robbins-Monro [30] as well as more recent approaches used in online convex optimization [37] tend to focus on interpolating the (Euclidean) distance to the optimum. This contrasts the exponential Lyapunov approach discussed above.
As discussed in the introduction, the rate of convergence of a stochastic gradient descent procedure on a convex objective (either with or without projection) is well known to be within $O({1}/{\sqrt{t}})$ of the optimum after $t$-iterations the algorithm. If it is further assumed that the objective function is strongly convex then this rate of convergence can be improved to be $O(1/t)$ [27, 26, 6]. These results are standard and hold for projected stochastic gradient descent.
The convergence of stochastic approximation algorithms on constrained regions is given in detail in the text of Kushner and Clark [18].
See also [19].
The geometric approach taken in this paper is influenced by the early work on projected stochastic gradient descent [12]. See also [1, Chapter 3].
Convergence rates for constrained stochastic approximation algorithms are considered by Buche and Kushner [7]. Here the authors note that analysis typically applied to analyze unconstrained stochastic approximation does not readily apply to the constrained case, and a diffusion analysis is conducted with results depending on which constraints of the optimization are active.
Large deviations analysis is a further method used to better understand boundary behavior in stochastic approximation with constraints [10]. These results [10, 7] concern deviations around a rate of convergence of the order $O(1/\sqrt{t})$, whereas we are interested in establishing conditions for a faster $O(1/{t})$ rate of convergence.
Linear Programs and Markov Decision Processes.
In this article, we apply our results to linear programs and then to solve Markov decision processes which in both cases have unknown random costs. This gives two examples where it is possible to apply the results of this paper.
The study of stochastic linear programs is common within operations research. We refer to Prekopa [29], Birge and Louveaux [4] and Shapiro et al. [32] as standard textbooks that cover both theory as well as variety of linear and non-linear programming examples where stochastic perturbations in their coefficients can occur.
We provide concentration results for our stochastic gradient descent algorithm at the optimum.
The work of [11] focuses on concentration of probability for linear programs with random objectives.
A Markov decision process can be characterized as a linear program [24]. Schweitzer and Seidmann [31] provides a modern formulation of a Markov decision process as a linear program. This is an extended approach and popularized as a method for approximate dynamic program [9].
Such linear programs can be characterized in either primal or dual form. The dual form can be more tractable for specific problem classes for example in scheduling [28]. In our case, this dual form is most convenient.
As an example of our results, we consider tabular reinforcement learning. Here we consider a finite state finite action Markov decision process where costs are unknown and must be estimated by sampling. In our case, we assume that the dynamics of the system are known. There are multiple tabular reinforcement learning algorithms, the most well known being Q-learning [36]. Q-learning requires $O(\epsilon^{-2})$ samples for an $\epsilon$ approximation [22]. General minimax lower-bounds require $O(\epsilon^{-2})$ samples for a $\epsilon$ approximation of the optimal policy [13]. Minimax results consider the best performance on worse case problem. However, problem specific convergence rates can improve these bounds by allowing dependence on gap between the optimal policy and a sub-optimal policy. (This is similar to bandit algorithms where regret bounds are typically $O(\log T)$ but the minimax regret is typical $O(\sqrt{T})$ [21].)
For a recent analysis and bound for gap-dependent bound for tabular RL see [33].
3 Model and Notation
This section acts as a brief recap of a number of well-known optimization formulations.
We define the projected stochastic gradient descent as an algorithm to solve these problems. We also determine conditions that we place on learning rates for such algorithms.
We define linear programs and Markov decision processes as special cases of our optimization formulation which we will later analyze as example applications.
Basic Notation. We apply the convention that $\mathbb{Z}_{+}=\{n:n=0,1,2,...\}$ and $\mathbb{R}_{+}=\{x:x\geq 0\}$. We apply the convention that (implied) multiplication has precedence over division, i.e. $2a/3b=(2\times a)/(3\times b)$. We define $a\vee b=\max\{a,b\}$ and $a\wedge b=\min\{a,b\}$.
Optimization Notation. We let $\mathcal{X}$ denote a closed bounded convex subset of $\mathbb{R}^{d}$. For a function $f:\mathcal{X}\rightarrow\mathbb{R}$, we consider the minimization
$$\min_{\bm{x}\in\mathcal{X}}\,f(\bm{x})\,.$$
(1)
We let $\mathcal{X}^{\star}$ be the set of minimizers of the above optimization. We let $\Pi_{\mathcal{X}}(\bm{x})$ denote the projection of $x$ onto the set $\mathcal{X}$. That is $\Pi_{\mathcal{X}}(\bm{x}):=\operatorname*{argmin}_{\bm{y}\in\mathcal{X}}||\bm{x}-\bm{y}||^{2}$, where $||\cdot||$ denotes the Euclidean norm. We let $d(\bm{x},\mathcal{X})$ denote the distance from the point to its projection. That is $d(\bm{x},\mathcal{X}):=\min_{\bm{y}\in\mathcal{X}}||\bm{x}-\bm{y}||$. We let $F$ be the gap between the maximum and minimum of $f(x)$ on $\mathcal{X}$, that is
$$F:=\max_{x\in\mathcal{X}}f(x)-\min_{x\in\mathcal{X}}f(x)\,.$$
(2)
Stochastic Gradient Descent with Constraints. We consider stochastic iterative procedures for solving the optimization (1). A standard algorithm for this problem is projected stochastic gradient descent.
Projected stochastic gradient descent is a randomized procedure for optimizing the objective (1) when random estimates of the gradient of $f(\bm{x})$ are available. Specifically, the sequence $\{\bm{x}_{t}\}_{t=0}^{\infty}$ obeys the iteration
$$\bm{x}_{t+1}=\Pi_{\mathcal{X}}(\bm{x}_{t}-\alpha_{t}\bm{c}_{t})\,,$$
(PSGD)
where, for each $t\in\mathbb{Z}_{+}$, $\alpha_{t}$ is a positive real number and $\bm{c}_{t}$ is a random variable such that $\mathbb{E}[\bm{c}_{t}|\bm{c}_{1},\bm{x}_{1},...,\bm{c}_{t-1},\bm{x}_{t}]=\nabla f(\bm{x}_{t})$. We let $\mathcal{F}_{t}=(\bm{c}_{1},\bm{x}_{1},...,\bm{c}_{t-1},\bm{x}_{t})$. (More formally we let $\mathcal{F}_{t}$ be the $\sigma$-field generated by $(\bm{c}_{1},\bm{x}_{1},...,\bm{c}_{t-1},\bm{x}_{t})$ and $\{\mathcal{F}_{t}\}_{t\geq 0}$ is its associated filtration.)
When $f(\bm{x})$ is convex and the sequence $\alpha_{t}$ obeys the condition
$$\sum_{t=0}^{\infty}\alpha_{t}=\infty,\quad\text{and }\qquad\alpha_{t}\xrightarrow[t\rightarrow\infty]{}0\,,$$
(3)
then $d(\bm{x}_{t},\mathcal{X}^{\star})\rightarrow 0$ as $t\rightarrow\infty$
[19]. Sequences typically considered take the form $\alpha_{t}=a/(b+t)^{\gamma}$ for $a,b>0$ and $\gamma\in[0,1]$.
Instead of condition (3) in this paper, we will assume that $\{\alpha_{t}\}_{t\geq 0}$ is a deterministic non-increasing sequence such that
$$\sum_{t=0}^{\infty}\alpha_{t}=\infty,\qquad\liminf_{t\rightarrow\infty}\,\frac{\alpha_{2t}}{\alpha_{t}}>0\qquad\text{and }\qquad\lim_{t\rightarrow\infty}\frac{\alpha_{t}-\alpha_{t+1}}{\alpha_{t}}=0\,.$$
(4)
We do not necessarily assume that $\alpha_{t}\rightarrow 0$ as we later wish to consider the case of small but constant step sizes.
The above condition is satisfied by any sequence of the form $\alpha_{t}=a/(b+t)^{\gamma}$ for $a,b>0$ and $\gamma\in[0,1]$.
(Note that for any sequence satisfying (3) it holds that $\liminf_{t\rightarrow\infty}(\alpha_{t}-\alpha_{t+1})/\alpha_{t}=0$.)
The results in the paper are not specific to Projected stochastic gradient descent. In general,
our results consider a random sequence $\{\bm{x}_{t}\}_{t=0}^{\infty}$ adapted to a filtration $\{\mathcal{F}_{t}\}_{t=0}^{\infty}$ with $\bm{x}_{t}\in\mathcal{X}$ for each $t\in\mathbb{Z}_{+}$. The distance between successive terms is determined by the sequence $\{\alpha_{t}\}_{t=0}^{\infty}$. Specifically, we assume $\mathbb{E}[||\bm{x}_{t+1}-\bm{x}_{t}||]=O(\alpha_{t})$. We refer to any such algorithm as a constrained stochastic gradient descent algorithm. Projected stochastic gradient descent is one specific example of a constrained stochastic gradient descent algorithm.
Linear Programming Notation. We will be interested in the special case where $\mathcal{X}$ is a polytope. That is
$\mathcal{X}=\{\bm{x}\in\mathbb{R}^{d}:H\bm{x}\leq\bm{b}\}$
for a matrix $H\in\mathbb{R}^{p\times d}$ and a vector $\bm{b}\in\mathbb{R}^{p}$. We assume that the objective $f$ is linear. That is $f(\bm{x})=\bar{\bm{c}}^{\top}\bm{x}$ for a constant vector $\bar{\bm{c}}\in\mathbb{R}^{d}$ . Thus the above optimization is a linear program:
$$\displaystyle\textrm{minimize}\;\;\bar{\bm{c}}^{\top}\bm{x}\;\;\ \textrm{subject to}\;\;H\bm{x}\leq\bm{b}\;\;\textrm{over}\;\;\bm{x}\in\mathbb{R}^{d}\,.$$
We let $\mathcal{E}$ be the set of extreme points of the set $\mathcal{X}$.
The set $\mathcal{E}$ is finite for any polytope and the optimum of a linear program is alway achieved at an extreme point [2].
We let $\mathcal{E}^{\star}$ be the set of extreme points in $\mathcal{X}^{\star}$.
Markov Decision Process Notation.
A Markov Decision Process (MDP) is a stochastic process $(S_{t}:t\in\mathbb{Z}_{+})$ defined on a set of states $\mathcal{S}$. The states visited are determined by actions $(A_{t}:t\in\mathbb{Z}_{+})$ defined on the set of actions $\mathcal{A}$. Specifically, the transitions to the next state $S_{t+1}$ is determined by the current state $S_{t}$, the action $A_{t}$ and an independent identically distributed random variable $U_{t}$. That is there is a function $F:\mathcal{S}\times\mathcal{A}\times[0,1]\rightarrow S$ such that
$$S_{t+1}=F(S_{t},A_{t};U_{t})\,.$$
We let
$P(s^{\prime}|s,a)=\mathbb{P}(S_{t+1}=s^{\prime}|S_{t}=s,A_{t}=a)\,.$
We assume that the set of states, $\mathcal{S}$, and actions, $\mathcal{A}$, are both finite.
For each state and action pair $(S_{t},A_{t})$, there is a real-valued cost
$$C_{t}=c(S_{t},A_{t}).$$
Costs can be positive or negative.
We assume that the costs are bounded by $c_{\max}<\infty$ that is $|c(s,a)|\leq c_{\max}$ for all $s\in\mathcal{S}$ and $a\in\mathcal{A}$. A policy is a function, $\pi:\mathcal{S}\rightarrow\mathcal{A}$, that selects an action for each state. Specifically under policy $\pi$,
$$A_{t}=\pi(S_{t}).$$
Note that under a policy $\pi$ the process $(S_{t}:t\in\mathbb{Z}_{+})$ is a Markov chain. We let $\mathbb{E}_{s_{0}}^{\pi}[\cdot]$ be the expectation of the Markov chain $S$ under policy $\pi$ conditional on the initial state $S_{0}=s_{0}$.
We let $\mathcal{P}$ denote the set of policies, which is the set of functions from $\mathcal{X}$ to $\mathcal{A}$.
A discounted MDP has the objective to minimize the discounted costs of a MDP:
$$\displaystyle V(s_{0}):=\textrm{minimize}\quad\mathbb{E}_{s_{0}}^{\pi}\bigg{[}\sum_{t=0}^{\infty}\beta^{t}C_{t}\bigg{]}\quad\textrm{over}\quad\pi\in\mathcal{P},$$
where the discount factor $\beta$ is such that $\beta\in(0,1)$.
4 An Exponential Lyapunov Bound
When the slope of the objective function does not go to zero as we approach the boundary, we can anticipate that a stochastic gradient descent algorithm will expect to always make progress towards minimizing its objective proportional to the learning rate $\alpha_{t}$, except perhaps at the optimum or at points within a small region of the optimum.
More precisely, we can expect a constrained stochastic gradient descent sequence $\{\bm{x}_{t}\}_{t=0}^{\infty}$ to satisfy the condition
$$\mathbb{E}[f(\bm{x}_{t+1})-f(\bm{x}_{t})|\mathcal{F}_{t}]\leq-2\alpha_{t}\delta$$
(C1)
whenever $f(\bm{x}_{t})-f(\bm{x}^{\star})\geq\alpha_{t}B$ for some $\delta>0$ and some $B>0$. We also place some bounds on the size of the change in $f(\bm{x}_{t})$. Specifically, we assume that there exists a constant $\eta>0$ and a random variable $Y$ such that
$$\big{[}|f(\bm{x}_{t+1})-f(\bm{x}_{t})|\big{|}\mathcal{F}_{t}\big{]}\leq\alpha_{t}Y\quad\text{and}\quad\mathbb{E}[e^{\alpha_{0}\eta Y}]<\infty\,.$$
(C2)
Condition (C1) need not hold for all projected gradient descent algorithms and problem instances. Nonetheless, the condition is reasonable as one might expect a stochastic gradient descent algorithm to make progress against its objective when away from a point with zero gradient.
The Condition (C2) is a reasonably mild assumption. For example, in projected stochastic gradient descent, this amounts to assuming that the magnitude of gradient estimates $||\bm{c}_{t}||$ have a moment generating function, and thus a sub-exponential tail. (See Lemma 7 for a verification of this claim.)
Shortly we will estaiblish the conditions (C1)
and (C2) when applying projected stochastic gradient descent to linear program. However, for now we leave (C1) and (C2) as general conditions that can be satisfied by a constrained stochastic gradient descent algorithm.
Below we present our main convergence result.
In essence, the result establishes that once the sum $\sum_{s=1}^{t}\alpha_{s}$ is sufficiently large the error of a constrained stochastic gradient descent is of the order of $\alpha_{t}$.
Theorem 1.
When Condition (C1) and (C2) are satisfied by a constrained stochastic gradient descent algorithm,
there exist positive constants $E,G,H$ such that for any $n$ with $t/2^{n}>1$
$$\displaystyle\mathbb{P}(f(\bm{x}_{t+1})-f(\bm{x}^{\star})\geq z)\leq e^{-\frac{\delta G^{n}}{2E\alpha_{t}}(z-\alpha_{t}B-F-\alpha_{0}B+\sum_{s=\lfloor t/2^{n}\rfloor}^{t}\alpha_{s}\frac{\delta}{2})}+H{e^{-\frac{\delta G^{n}}{2E\alpha_{t}}(z-\alpha_{t}B)}}\,.$$
(5)
Further,
for any $t$ such that
$\sum_{s=\lfloor t/2^{n}\rfloor}^{t}\alpha_{s}\delta\geq 2(F+\alpha_{0}B)$
there exists a constant $C$ such that
$$\mathbb{E}[f(\bm{x}_{t+1})-f(\bm{x}^{\star})]\leq C\alpha_{t}\,.$$
(6)
We can interpret the above result as follows.
(When reading the above result, it is best to consider the parameter $n=1$. Though for faster decaying learning rates, e.g. $\alpha_{t}=1/t$, a different choice of $n$ is required to obtain the best bound.)
In (5), we see that the probability that $f(\bm{x}_{t})-f(\bm{x}^{\star})$ is above $z$ can be split into two terms. The first term accounts for the time it takes for the initial condition to be forgotten and for the process $f(\bm{x}_{t})$ to approach the value $f(\bm{x}^{\star})$.
Notice if step sizes $\alpha_{t}$ are kept constant then this term decays exponentially fast.
Following this, the second term, accounts for excursions in the process away from $f(\bm{x}^{\star})$. These excursions exhibit a behavior that is analogous to a G/G/1 queue [17] and has a tail behavior that is exponentially distributed. Notice this is different from typical behavior of an Ornstein-Uhlenbeck process (which we would typically associate with constant step-size stochastic gradient descent) which is know to have a normally distributed stationary distribution.
From (6), we see that once the dependence on initial state of the system has been accounted for then the process $f(\bm{x}_{t})$ will be within a factor of $\alpha_{t}$ of the optimum. In particular, for step sizes of the form $\alpha_{t}=1/t^{\gamma}$, we have that
$$\mathbb{E}[f(\bm{x}_{t})-f(\bm{x}^{\star})]=O\left(\frac{1}{t^{\gamma}}\right)\,.$$
This final observation is confirmed in the following corollary.
Corollary 1.
For learning rates of the form
$$\displaystyle\alpha_{t}=\frac{a}{(b+t)^{\gamma}}$$
with $a,b>0$ and $\gamma\in[0,1]$, if Conditions (C1) and (C2) are satisfied by a stochastic gradient descent algorithm, then
$$\displaystyle\mathbb{P}(f(\bm{x}_{t+1})-f(\bm{x}^{\star})\geq z)\leq I{e^{-t^{\gamma}{J}z}}\,$$
(7)
and
$$\displaystyle\mathbb{E}[f(\bm{x}_{t+1})-f(\bm{x}^{\star})]\leq\frac{K}{t^{\gamma}}\,$$
(8)
for time independent constants $I$, $J$ and $K$.
It is worth remarking that Theorem 1 and Corollary 1 hold for any algorithm for which the generic conditions (C1) and (C2) hold. Thus the results are not intended to applicable to any one particular stochastic gradient descent algorithm, nor do we place specific design restrictions on the algorithm.
Moreover, it is not necessary to verify these conditions in order to implement any algorithm. The result emphasizes that a convergence rate may hold for a broad class of stochastic gradient descent algorithms when there are constraints. This convergence may well be faster than that anticipated by the well established convergence in the unconstrained case.
5 Linear Programming Proofs
In the proof of Theorem 1 and Corollary 1, we did not specify the stochastic approximation procedure used nor did we explore settings where the key Conditions C1 and C2 hold. The aim of this section is to provide a concrete example where we prove that Conditions C1 and C2 hold and thus Theorem 1 and Corollary 1 apply. Here we consider a linear program where the cost function that we wish to minimize must be sampled and where the constraints of the optimization are deterministic. We consider the projected stochastic gradient descent algorithm where the costs sampled are batched. When the batch size is sufficiently large and assuming the sample would cost a reasonably light tailed then it is possible to verify the Conditions C1 and C2. Thus in this case, we prove that projected stochastic gradient descent has a faster convergence rate than (unconstrained) stochastic gradient descent.
As discussed in Section 3, we are interested in solving a linear program of the form
$$\displaystyle\textrm{minimize}\;\;\bar{\bm{c}}^{\top}\bm{x}\;\;\ \textrm{subject to}\;\;H\bm{x}\leq\bm{b}\;\;\textrm{over}\;\;\bm{x}\in\mathbb{R}^{d}\,,$$
(9)
where $\mathcal{X}=\{\bm{x}\in\mathbb{R}^{d}:H\bm{x}\leq\bm{b}\}$
is a bounded polytope.
We suppose that the constraint set $\mathcal{X}$ is deterministic and known, however, the cost vector $\bar{\bm{c}}$ is unknown but can be sampled.
Specifically, $t\in\mathbb{Z}_{+}$, we let $\bm{c}^{i}_{t}$, $i=1,...,B$ be independent sub-Gaussian random vectors in $\mathbb{R}^{d}$, with mean $\bar{\bm{c}}$ that is
$$\displaystyle\mathbb{E}[\bm{c}^{i}_{t}|\mathcal{F}_{t}]=\bar{\bar{\bm{c}}}$$
for $i=1,...,B$ and $t\in\mathbb{Z}_{+}$, and there exists a constant $\lambda>0$ such that
$$\displaystyle\mathbb{E}\big{[}e^{\bm{\eta}^{\top}\bm{c}^{i}_{t}}|\mathcal{F}_{t-1}\big{]}\leq\ e^{\bm{\eta}^{\top}\bar{\bar{\bm{c}}}+\lambda||\bm{\eta}||^{2}}\ ,\qquad\forall\bm{\eta}\in\mathbb{R}^{d}\,.$$
(10)
We think of the set of costs $(\bm{c}^{i}_{t}:i=1,...,B)$ as a batch which we then average in order to make a stochastic gradient descent step. That is, we let
$$\displaystyle\bm{c}_{t}=\frac{1}{B}\sum_{i=1}^{B}\bm{c}_{t}^{i}\,,$$
and then apply projected stochastic gradient descent
$$\displaystyle\bm{x}_{t+1}=\Pi_{\mathcal{X}}\big{(}\bm{x}_{t}-\alpha_{t}\bm{c}_{t}\big{)}\,.$$
(11)
We can prove that the conditions of Theorem 1 are satisfied for linear programs. From this we obtain the following results.
Theorem 2.
For the projected stochastic gradient descent (11) applied to a linear program (9) with
learning rates of the form
$$\displaystyle\alpha_{t}=\frac{a}{(b+t)^{\gamma}}$$
for $a,b>0$ and $\gamma\in[0,1]$, if the batch size $B$ is sufficiently large
then
$$\displaystyle\mathbb{P}(\bar{\bm{c}}^{\top}\bm{x}_{t+1}-\bar{\bm{c}}^{\top}\bm{x}^{\star}\geq z)\leq I{e^{-t^{\gamma}{J}z}}\,$$
(12)
and
$$\displaystyle\mathbb{E}[\bar{\bm{c}}^{\top}\bm{x}_{t+1}-\bar{\bm{c}}^{\top}\bm{x}^{\star}]\leq\frac{K}{t^{\gamma}}\,$$
(13)
where $I$, $J$ and $K$ are time independent constants.
To prove Theorem 2, we must show that the premise of Theorem 1 is satisfied. Specifically, we must prove that projected stochastic gradient descent makes constant progress against its objective.
The key ingredient that we require in order to prove is a lemma, which shows that the angle between a sub-optimal point and the closest optimal point is always bounded away from zero.
Indeed, although we prove this results for polytope regions it is reasonable to expect that the angle between a suboptimal point and the optimum might be bounded below for a wide variety of optimization problems, beyond linear programs.
6 Markov Decision Process Proofs
There are a large number of linear programs for specific problems where the above analysis applied.
However, we now focus in this section on the example of solving discounted Markov decision processes using the results from the last section.
Here we use a linear programming approach to prove the convergence of a simple policy gradient algorithm.
Here we considered a Markov decision process where the dynamics of the system are known but the costs accrude are unknown.
Here we look to optimize a Markov decision process through sampling these costs.
We define a Markov decision process in Section 3.
It is well known that a Markov decision process can be formulated as a linear program, where the primal form of this linear program solves for the optimal value function and the dual form finds the optimal occupancy measure. (See the appendix, Section A.1, for a brief discussion on this.)
In this linear programming formula, this dual problem takes the form:
Minimize
$$\displaystyle\sum_{s\in\mathcal{S}}\sum_{a\in\mathcal{A}}\bar{c}(s,a)x(s,a)$$
(Dual)
subject to
$$\displaystyle\sum_{a\in\mathcal{A}}x(s^{\prime},a)=\xi(s^{\prime})+\beta\sum_{s\in\mathcal{S}}\sum_{a\in\mathcal{A}}x(s,a)P(s^{\prime}|s,a),\qquad\forall s^{\prime}\in\mathcal{S}$$
over
$$\displaystyle(x(s,a):s\in\mathcal{S},a\in\mathcal{A})\in\mathbb{R}_{+}^{\mathcal{S}\times\mathcal{A}}.$$
Here $(\xi(s):s\in\mathcal{S})$ is a positive vector.
We assume that the dynamics as given by $(P(s^{\prime}|s,a):a\in\mathcal{A},\,s,s^{\prime}\in\mathcal{S})$ are known but costs are unknown and must be sampled, then above we have a linear program with an unknown objective and known constraints. For this reason we can apply the analysis developed in the last section.
Here we assume that we are able to sample costs $\bar{\bm{c}}=(\bar{c}(s,a):s\in\mathcal{S},a\in\mathcal{A})$ where the states and actions as distributed according to some predetermined probability distribution $\bm{\pi}=(\pi(s,a):a\in\mathcal{A},s\in\mathcal{S})$. There are a number of ways of sampling the cost vector $\bm{c}_{t}$ for each $t$. The most straight-forward is as follows.
For each $t$, the cost $\bm{c}_{t}=(c_{t}(s,a):s\in\mathcal{S},\,a\in\mathcal{A})$ is assumed as follows. We take IID samples $(s^{i}_{t},a^{i}_{t})$, $i=1,...,B$ with distribution $\bm{\pi}=(\pi(s,a):s\in\mathcal{S},a\in\mathcal{A})$ where $\pi(s,a)>0$ for all $s\in\mathcal{S}$ and $a\in\mathcal{A}$. We then define
$$\displaystyle c_{t}(s,a)=\frac{1}{B}\sum_{i=1}^{B}\frac{\bar{c}(s_{t}^{i},a_{t}^{i})}{\pi(s_{t}^{i},a_{t}^{i})}\mathbb{I}[(s_{t}^{i},a_{t}^{i})=(s,a)]\,.$$
We can think of this method of observing costs as x experience reply. In experience reply a policy is followed and states and actions are observed according to some distribution $\pi$. As samples come from a Markov process, these samples are stored in memory and resampled independently at random. The costs observed would then be approximated by the above dynamics.
We consider the projected gradient descent
$$\displaystyle\bm{x}_{t+1}=\Pi_{\mathcal{X}}\big{(}\bm{x}_{t}-\alpha_{t}\bm{c}_{t}\big{)}\,.$$
(14)
Here the projection above is on to the constraint set of the dual problem:
$$\mathcal{X}=\Big{\{}\bm{x}\in\mathbb{R}_{+}^{\mathcal{S}\times\mathcal{A}}:\sum_{a\in\mathcal{A}}x(s^{\prime},a)=\xi(s^{\prime})+\beta\sum_{s\in\mathcal{S}}\sum_{a\in\mathcal{A}}x(s,a)P(s^{\prime}|s,a),\;\forall s^{\prime}\in\mathcal{S}\Big{\}}\,.$$
Given the description above the following result is a direct consequence of Corollary 1.
Theorem 3.
For the projected stochastic gradient descent (14) applied to a Markov Decision Process (Dual) with
learning rates of the form
$$\displaystyle\alpha_{t}=\frac{a}{(b+t)^{\gamma}}$$
for $a,b>0$ and $\gamma\in[0,1]$, there exists a $B_{0}$ such that for all batch sizes $B\geq B_{0}$
the following bounds hold:
$$\displaystyle\mathbb{P}(\bar{\bm{c}}^{\top}\bm{x}_{t+1}-\bar{\bm{c}}^{\top}\bm{x}^{\star}\geq z)\leq I{e^{-t^{\gamma}{J}z}}\,$$
(15)
and
$$\displaystyle\mathbb{E}[\bar{\bm{c}}^{\top}\bm{x}_{t+1}-\bar{\bm{c}}^{\top}\bm{x}^{\star}]\leq\frac{K}{t^{\gamma}}\,$$
(16)
for time independent constants $I$, $J$ and $K$.
The fastest convergence rate that can be obtained above is when $\gamma=1$. We note that this rate of convergence is faster than would typically be observed when applying a tabular reinforcement learning algorithm such as $Q$-learning. The rate of convergence of $Q$-learning is known to be of the order of $t^{-1/2}$. We note however that the constant $K$ given in the above bound depends on the structure of the polytope $\mathcal{X}$. Thus our bound (13) is an instance dependent bound rather than a minimax bound which would typically be applied to gain a bound across all problem instances.
Although we will focus on the above sampling, we now breifly discuss other methods where costs can be sampled.
The above sampling mimics the behaviour where there is a policy that visits each state and action with distribution $\pi(s,a)$ and then by storing in memory the states and actions visited by the policy we can independently draw from the list of state and actions visited. This method of achieving a batch of (approximately) independent samples is know as experience replay.
Another approach would apply to episodic environments. Here we assumed that the MDP terminates in a finite expected time and the MDP can then be restarted from some fixed state $s_{0}\in\mathcal{S}$. Under this assumption, the costs reached under a fixed policy over successive episodes are independent.
We would consider $B$ episodes and apply the average cost achieved per episode.
It is also possible that a sequence of state action cost triples is sampled directly from a fixed policy. Further, a particular choice is where the costs are sampled directly from the current policy choice as indicated by $\bm{x}$.
7 Proof of Theorem 1 and Corollary 1
In this section, we prove our main result, Theorem 1, as well as Corollary 1, which is a simplified version of Theorem 1.
The following is a brief outline of the proofs of Theorem 1 and Corollary 1. The proof of Theorem 1 uses Lemma 1, Lemma 2 and Proposition 1, which are stated below.
Lemma 1, although not critical to our analysis, simplifies the drift condition C1 by eliminating some additional terms and boundary effects.
Lemma 2, on-the-other-hand, is a key component of our proof. It converts the linear drift condition C1 into an exponential bound which we then iteratively expand. The lemma extends Theorem 2.3 from Hajek [14]
by allowing for adaptive time dependent step sizes.
Proposition 1 applies standard moment generating function inequalities to the results found in Lemma 2.
After Proposition 1 is proven the proof of Theorem 1 is essentially complete. We simply need to translate results back to the notation of our original formulation.
Corollary 1 follows after carefully bounding some of the series stated in Theorem 1. For the learning rates considered, cleaner bounds are obtained in Corollary 1. (Three other generic, technical lemmas with standard proofs are required. These are proven in Section A.2 of the appendix.)
We now proceed with the steps outlined above. We let
$$L_{t}:=f(\bm{x}_{t})-f(\bm{x}^{\star})-\alpha_{t}B$$
(17)
where $\alpha_{t}$ satisfies (4).
First we simplify the above conditions C1 and C2 to give the Lyapunov conditions (18) and (19) stated below.
Lemma 1.
Given Conditions C1 and C2 hold, there exists a constant $t_{0}$ such that the sequence of random variables $(L_{t}:t\geq t_{0})$ satisfies
$$\mathbb{E}\big{[}L_{t+1}-L_{t}\big{|}\mathcal{F}_{t}\big{]}\mathbb{I}[L_{t}\geq 0]<-\alpha_{t}\delta,$$
(18)
and
$$[|L_{t+1}-L_{t}||\mathcal{F}_{t}]\leq\alpha_{t}Z\quad\text{where}\quad D:=\mathbb{E}[e^{\alpha_{0}\eta Z}]<\infty\,.$$
(19)
Proof.
Applying the definition of $L_{t}$ and the drift condition C1 gives
$$\displaystyle\mathbb{E}[L_{t+1}-L_{t}|\mathcal{F}_{t}]=$$
$$\displaystyle\,\mathbb{E}[f(\bm{x}_{t+1})-f(\bm{x}_{t})|\mathcal{F}_{t}]+(\alpha_{t}-\alpha_{t+1})B$$
$$\displaystyle\leq$$
$$\displaystyle-2\alpha_{t}\delta+(\alpha_{t}-\alpha_{t+1})B$$
$$\displaystyle\leq$$
$$\displaystyle-2\alpha_{t}\delta\left[1-(\alpha_{t}-\alpha_{t+1})B/\alpha_{t}\delta\right]$$
Since $(\alpha_{t}-\alpha_{t+1})/\alpha_{t}\rightarrow 0$, there exists a constant $t_{0}$ such that $(\alpha_{t}-\alpha_{t+1})/\alpha_{t}<\delta/2B$ for all $t\geq t_{0}$.
This gives the first drift condition (18).
For the second condition, for $t\geq t_{0}$ with $t_{0}$ as just defined:
$$\displaystyle\big{[}|L_{t+1}-L_{t}|\big{|}\mathcal{F}_{t}\big{]}\leq$$
$$\displaystyle\,\left[|f(\bm{x}_{t+1})-f(\bm{x}_{t})|\big{|}\mathcal{F}_{t}\right]+|\alpha_{t+1}-\alpha_{t}|B$$
$$\displaystyle\leq$$
$$\displaystyle\,\alpha_{t}Y+\alpha_{t}\frac{(\alpha_{t}-\alpha_{t+1})}{\alpha_{t}}B$$
$$\displaystyle\leq$$
$$\displaystyle\,\alpha_{t}(Y+\delta/2)\,.$$
Taking $Z=Y+\delta/2$, it is clear that condition (19) holds for $Z$ as an immediate consequence of the boundedness condition on $Y$ in C2.
∎
Now given (19) holds, we will convert the drift condition (18) into a exponential bound and then iterate to give the bound below.
Lemma 2.
$$\mathbb{E}[e^{\eta L_{t+1}}]\leq\mathbb{E}[e^{\eta L_{t_{1}}}]\prod_{k=t_{1}}^{t}\rho_{t}+D\sum_{\tau=t_{1}+1}^{t+1}\prod_{k=\tau}^{t}\rho_{k}\,,$$
for $t\geq t_{1}\geq t_{0}$ where $\rho_{t}=e^{-\alpha_{t}\eta\delta+\alpha_{t}^{2}\eta^{2}E}$, and $E=\mathbb{E}\left[({e^{\eta\alpha_{0}Z}-1-\alpha_{0}\eta Z})/{\eta^{2}}\right]<\infty$.
Proof.
Let $Z_{t}=(L_{t+1}-L_{t})/\alpha_{t}$. From (19), we have $[|Z_{t}||\mathcal{F}_{t}]\leq Z$ where $\mathbb{E}[e^{\eta\alpha_{0}Z}]<\infty$. From (18), we have $\mathbb{E}[Z_{t}|\mathcal{F}_{t}]\leq-\delta$ on the event $\{L_{t}\geq 0\}$. Thus, on the event $\{L_{t}\geq 0\}$ the following holds:
$$\displaystyle\mathbb{E}[e^{\eta(L_{t+1}-L_{t})}|\mathcal{F}_{t}]=\mathbb{E}[e^{\alpha_{t}\eta Z_{t}}|\mathcal{F}_{t}]$$
$$\displaystyle=1+\alpha_{t}\eta\mathbb{E}[Z_{t}|\mathcal{F}_{t}]+\alpha_{t}^{2}\eta^{2}\mathbb{E}\left[\frac{e^{\alpha_{t}\eta Z_{t}}-1-\alpha_{t}\eta Z_{t}}{\alpha_{t}^{2}\eta^{2}}\Big{|}\mathcal{F}_{t}\right]$$
$$\displaystyle=1+\alpha_{t}\eta\mathbb{E}[Z_{t}|\mathcal{F}_{t}]+\alpha_{t}^{2}\eta^{2}\sum_{k=2}^{\infty}\frac{1}{k!}\mathbb{E}[Z_{t}^{k}|\mathcal{F}_{t}]\eta^{k-2}\alpha_{t}^{k-2}$$
$$\displaystyle\leq 1-\alpha_{t}\eta\delta+\alpha_{t}^{2}\eta^{2}\sum_{k=2}^{\infty}\frac{1}{k!}\mathbb{E}[Z^{k}]\eta^{k-2}\alpha_{0}^{k-2}$$
$$\displaystyle=1-\alpha_{t}\eta\delta+\alpha_{t}^{2}\eta^{2}\mathbb{E}\left[\frac{e^{\eta\alpha_{0}Z}-1-\alpha_{0}\eta Z}{\eta^{2}}\right]$$
(20)
$$\displaystyle=1-\alpha_{t}\eta\delta+\alpha_{t}^{2}\eta^{2}E$$
$$\displaystyle\leq e^{-\alpha_{t}\eta\delta+\alpha_{t}^{2}\eta^{2}E}=:\rho_{t}\,.$$
(21)
In the first equality above, we apply a Taylor expansion. In the first inequality we apply (18) and (19) above, and also recall that $\alpha_{t}$ is decreasing. In the final inequality, we applied the standard bound $1+x\leq e^{x}$. We note that $\rho_{t}$ as define above satisfies $\rho_{t}<1$ whenever $\alpha_{t}<{\delta}/{\eta E}$.
We note that $E$ is finite since by assumption $\mathbb{E}[e^{\eta\alpha_{0}Z}]<\infty$. Also from the expansion given in (20), it is clear that $E$ is positive.
The bound (21) holds on the event $\{L_{t}\geq 0\}$.
Now notice
$$\displaystyle\mathbb{E}[e^{\eta L_{t+1}}|\mathcal{F}_{t}]$$
$$\displaystyle=\mathbb{E}[e^{\eta(L_{t+1}-L_{t})}|\mathcal{F}_{t}]e^{\eta L_{t}}\mathbb{I}[L_{t}\geq 0]+\mathbb{E}[e^{\eta(L_{t+1}-L_{t})}|\mathcal{F}_{t}]e^{\eta L_{t}}\mathbb{I}[L_{t}<0]$$
$$\displaystyle\leq\rho_{t}e^{\eta L_{t}}\mathbb{I}[L_{t}\geq 0]+\mathbb{E}[e^{\alpha_{0}Z}]e^{\eta L_{t}}\mathbb{I}[L_{t}<0]$$
$$\displaystyle\leq\rho_{t}e^{\eta L_{t}}\mathbb{I}[L_{t}\geq 0]+D\mathbb{I}[L_{t}<0]$$
$$\displaystyle\leq\,\rho_{t}e^{\eta L_{t}}+D\,.$$
The first inequality applies the above bound (21) and the second inequality applies the boundedness condition (19). Taking expectations above gives
$$\mathbb{E}[e^{\eta L_{t+1}}]\leq\rho_{t}\mathbb{E}[e^{\eta L_{t}}]+D.$$
By induction, we have
$$\mathbb{E}[e^{\eta L_{t+1}}]\leq\mathbb{E}[e^{\eta L_{t_{1}}}]\prod_{k=t_{1}}^{t}\rho_{t}+D\sum_{\tau=t_{1}+1}^{t+1}\prod_{k=\tau}^{t}\rho_{k}\,,$$
as required.
∎
The following is a technical lemma proven in the appendix that we will require for Proposition 1 below.
Lemma 3.
If $\alpha_{t}$, $t\in\mathbb{Z}_{+}$, is a decreasing positive sequence, then
$$\min_{s=t_{1},...,t}\left\{\frac{\sum_{k=s}^{t}\alpha_{k}}{\sum_{k=s}^{t}\alpha_{k}^{2}}\right\}=\frac{\sum_{k=t_{1}}^{t}\alpha_{k}}{\sum_{k=t_{1}}^{t}\alpha_{k}^{2}}$$
(22)
Moreover, if $\alpha_{t}$, $t\in\mathbb{Z}_{+}$ satisfies the learning rate condition (4) then
$$\min_{s=\lfloor t/2^{n}\rfloor,...,t}\left\{\frac{\sum_{k=s}^{t}\alpha_{k}}{\sum_{k=s}^{t}\alpha_{k}^{2}}\right\}\geq\frac{G^{n}}{\alpha_{t}}$$
(23)
for some positive constant $G$ and for $n\in\mathbb{N}$ such that $t/2^{n}>1$.
With the moment generating function bound in Lemma 2 and the above lemma, we can bound the tail probabilities and expectation of $L_{t}$.
Proposition 1.
For any sequence satisfying (4),
there exists a constant $H$ such that
$$\displaystyle\mathbb{P}(L_{t+1}\geq z)\leq$$
$$\displaystyle\ e^{-\frac{\delta G^{n}}{2E\alpha_{t}}(z-F-\alpha_{0}B+\sum_{s=\lfloor t/2^{n}\rfloor}^{t}\alpha_{s}\frac{\delta}{2})}+H{e^{-\frac{\delta G^{n}}{2E\alpha_{t}}z}}\ $$
for $z\geq 0$. Further,
for $t$ is such that $\sum_{s=1}^{t}\alpha_{s}\frac{\delta}{2}\geq F+\alpha_{0}B$, then
$$\mathbb{E}[L_{t+1}]\leq\frac{2\left(1+H\right)E}{G^{n}\delta}\alpha_{t}\,.$$
Proof.
We apply Lemma 2 with $t_{1}=\lfloor t/2^{n}\rfloor$ which gives
$$\displaystyle\mathbb{P}(L_{t+1}\geq z)$$
$$\displaystyle\leq e^{-\eta z}\mathbb{E}[e^{\eta L_{t}}]$$
$$\displaystyle\leq e^{-\eta z}\mathbb{E}[e^{\eta L_{t_{1}}}]\prod_{k=t_{1}}^{t}\rho_{t}+e^{-\eta z}D\sum_{\tau=t_{1}+1}^{t+1}\prod_{k=\tau}^{t}\rho_{k}$$
$$\displaystyle=e^{-\eta z}\mathbb{E}[e^{\eta L_{t_{1}}}]e^{\sum_{k=t_{1}}^{t}-\alpha_{k}\eta\delta+\alpha_{k}^{2}\eta^{2}E}+e^{-\eta z}D\sum_{\tau=t_{1}+1}^{t+1}e^{\sum_{k=\tau}^{t}-\alpha_{t}\eta\delta+\alpha_{t}^{2}\eta^{2}E}\ .$$
(24)
We let $\eta$ be such that $0\leq\eta\leq\min_{t_{1}\leq s\leq t}\left\{\frac{\sum_{k=s}^{t}\alpha_{k}\delta}{2\sum_{k=s}^{t}\alpha_{k}^{2}E}\right\}$.
Notice, for such $\eta$, it holds that
$$\sum_{k=\tau}^{t}-\alpha_{k}\eta\delta+\alpha_{k}^{2}\eta^{2}E\leq-\frac{1}{2}\sum_{k=\tau}^{t}\alpha_{k}\eta\delta,\qquad\forall\tau=t_{1},...,t\ .$$
Applying this to (24) gives
$$\displaystyle\mathbb{P}(L_{t+1}\geq z)$$
$$\displaystyle\leq e^{-\eta z}\mathbb{E}[e^{\eta L_{t_{1}}}]e^{\sum_{k=t_{1}}^{t}-\alpha_{k}\eta\frac{\delta}{2}}+e^{-\eta z}D\sum_{\tau=t_{1}+1}^{t+1}e^{\sum_{k=\tau}^{t}-\alpha_{k}\eta\frac{\delta}{2}}$$
$$\displaystyle\leq e^{-\eta z}\mathbb{E}[e^{\eta L_{t_{1}}}]e^{\sum_{k=t_{1}}^{t}-\alpha_{k}\eta\frac{\delta}{2}}+e^{-\eta z}D\sum_{\tau=t_{1}+1}^{t+1}e^{-(t-\tau)\alpha_{k}\eta\frac{\delta}{2}}$$
$$\displaystyle\leq e^{-\eta z}\mathbb{E}[e^{\eta L_{t_{1}}}]e^{\sum_{k=t_{1}}^{t}-\alpha_{k}\eta\frac{\delta}{2}}+e^{-\eta z}D\frac{e^{\alpha_{t}\eta\frac{\delta}{2}}}{1-e^{-\alpha_{t}\eta\frac{\delta}{2}}}$$
(25)
In the 2nd inequality above we note that $\alpha_{k}\geq\alpha_{t}$ for all $k\leq t$. In the 3rd inequality, we note that the summation over $\tau$ are terms from a geometric series, so we upper bound this by the appropriate infinite sum.
By Lemma 3, we see that
$$\min_{t_{1}\leq s\leq t}\left\{\frac{\sum_{k=s}^{t}\alpha_{k}\delta}{2\sum_{k=s}^{t}\alpha_{k}^{2}E}\right\}=\frac{\delta}{2E}\frac{\sum_{k=t_{1}}^{t}\alpha_{k}}{\sum_{k=t_{1}}^{t}\alpha_{k}^{2}}\geq\frac{\delta G^{n}}{2\alpha_{t}E}$$
for a constant $G>0$. In particular, we take $\eta=\delta G^{n}/2E\alpha_{t}$.
Thus, the bound (25) becomes
$$\mathbb{P}(L_{t+1}\geq z)\leq\mathbb{E}\Big{[}e^{\frac{\delta G^{n}L_{t_{1}}}{2E\alpha_{t}}}\Big{]}e^{-\frac{\delta G^{n}}{2E\alpha_{t}}(z+\sum_{s=t_{1}}^{t}\alpha_{s}\frac{\delta}{2})}+e^{-\frac{\delta G^{n}}{2E\alpha_{t}}z}D\frac{e^{\frac{\delta^{2}G^{n}}{4E}}}{1-e^{-\frac{\delta^{2}G^{n}}{4E}}}\ .$$
Noting that $L_{t_{1}}\leq\max_{x\in\mathcal{X}}f(x)-\min_{x\in\mathcal{X}}f(x)+\alpha_{0}B=F+\alpha_{0}B$, by the definition of $F$ in (2). We simplify the above expression as follows
$$\displaystyle\mathbb{P}(L_{t+1}\geq z)\leq$$
$$\displaystyle\,e^{\frac{\delta G^{n}}{2E\alpha_{t}}[F+\alpha_{0}B-z-\sum_{s=\lfloor t/2^{n}\rfloor}^{t}\alpha_{s}\frac{\delta}{2}]}+e^{-\frac{\delta G^{n}}{2E\alpha_{t}}z}D\frac{e^{\frac{\delta^{2}G^{n}}{4E}}}{1-e^{-\frac{\delta^{2}G^{n}}{4E}}}$$
$$\displaystyle\leq$$
$$\displaystyle\ e^{-\frac{\delta G^{n}}{2E\alpha_{t}}(z+\sum_{s=\lfloor t/2^{n}\rfloor}^{t}\alpha_{s}\frac{\delta}{2}-F-\alpha_{0}B)}+H{e^{-\frac{\delta G^{n}}{2E\alpha_{t}}z}}\ .$$
(26)
Above we define $H:=D{e^{\frac{\delta^{2}G^{n}}{4E}}}/{(1-e^{-\frac{\delta^{2}G^{n}}{4E}})}$.
Notice if $t$ is such that $F+\alpha_{0}B-\sum_{s=\lfloor t/2^{n}\rfloor}^{t}\alpha_{s}\frac{\delta}{2}\leq 0$, then the above inequality (26) can be bounded by
$$\mathbb{P}(L_{t+1}\geq z)\leq\left(1+H\right)e^{-\frac{\delta G^{n}}{2E\alpha_{t}}z}.$$
Thus
$$\displaystyle\mathbb{E}[L_{t+1}]$$
$$\displaystyle\leq\mathbb{E}[L_{t+1}\vee 0]=\int_{0}^{\infty}\mathbb{P}(L_{t+1}\geq z)dz\leq(1+H)\int_{0}^{\infty}e^{-\frac{\delta G^{n}}{2E\alpha_{t}}z}\,dz$$
$$\displaystyle=2\left(1+H\right)\frac{\alpha_{t}E}{2\delta G^{n}}\ ,$$
as required.
∎
With Proposition 1 in place we can translate our result back into our original notation and prove Theorem 1.
Proof of Theorem 1.
From Proposition 1
$$\displaystyle\mathbb{P}(L_{t+1}\geq z^{\prime})\leq$$
$$\displaystyle\ e^{-\frac{\delta G^{n}}{2E\alpha_{t}}(z^{\prime}-F-\alpha_{0}B+\sum_{s=\lfloor t/2^{n}\rfloor}^{t}\alpha_{s}\frac{\delta}{2})}+H{e^{-\frac{\delta G^{n}}{2E\alpha_{t}}z^{\prime}}}\ $$
for $z^{\prime}\geq 0$ where $f(\bm{x}_{t})=L_{t}+\alpha_{t}B+f(\bm{x}^{\star})$. Taking $z^{\prime}=z-\alpha_{t}B$, gives
$$\displaystyle\mathbb{P}(f(\bm{x}_{t})-f(\bm{x}^{\star})\geq z)$$
$$\displaystyle=\mathbb{P}(L_{t}\geq z-\alpha_{t}B)$$
$$\displaystyle\leq e^{-\frac{\delta G^{n}}{2E\alpha_{t}}(z-\alpha_{t}B-F-\alpha_{0}B+\sum_{s=\lfloor t/2^{n}\rfloor}^{t}\alpha_{s}\frac{\delta}{2})}+H{e^{-\frac{\delta G^{n}}{2E\alpha_{t}}(z-\alpha_{t}B)}}\,,$$
which gives (5) as required.
Also by Proposition 1, we have
$$\displaystyle\mathbb{E}[f(\bm{x}_{t+1})-f(\bm{x}^{\star})]$$
$$\displaystyle=\mathbb{E}[L_{t+1}]+\alpha_{t+1}B$$
$$\displaystyle\leq 2\left(1+H\right)\frac{\alpha_{t}E}{2\delta G^{n}}+\alpha_{t+1}B$$
$$\displaystyle=\left[2\left(1+H\right)\frac{E}{2\delta G^{n}}+B\right]\alpha_{t}.$$
Thus taking $C=\left[2\left(1+H\right){E}/{2\delta G^{n}}+B\right]$, the required bound (6) holds.
∎
We now prove Corollary 1 as a consequence of Theorem 1. This involves carefully bounding the terms in Theorem 1.
Proof of Corollary 1.
First we consider the case $\gamma\in[0,1)$, after we will consider the case where $\gamma=1$.
When $\gamma\in[0,1)$ we take $n=1$ in Theorem 1. Notice that
$$\displaystyle\sum_{s=\lfloor(u-b)/2^{n}\rfloor}^{u-b}\alpha_{s}=$$
$$\displaystyle\sum_{s=\lfloor(u-b)/2^{n}\rfloor}^{u-b}\frac{a}{(b+s)^{\gamma}}$$
$$\displaystyle\geq\,$$
$$\displaystyle\int_{u/2-b}^{u-b}\frac{a}{(b+s)^{\gamma}}ds=\frac{a}{1-\gamma}\left[1-\frac{1}{2^{1-\gamma}}\right]u^{1-\gamma}\,.$$
(27)
When $t=u-b$ is such that
$$t\geq b+\frac{\alpha_{0}B+F}{\frac{a}{1-\gamma}\left[1-\frac{1}{2^{1-\gamma}}\right]}=:t_{0}\,,$$
then applied to the above bound, we have that
$$\sum^{t}_{s=\lfloor t/2^{n}\rfloor}\alpha_{s}\frac{\delta}{2}\geq\alpha_{0}B+F\,,$$
Thus applying bound (5) from Theorem 1 for $t\geq t_{0}$, with $t_{0}$ as define above and $n=1$, we see that
$$\displaystyle\mathbb{P}(f(\bm{x}_{t+1})-f(\bm{x}^{\star})\geq z)\leq(1+H){e^{-\frac{\delta G}{2E\alpha_{t}}(z-\alpha_{t}B)}}\,.$$
Integrating the above bound then gives
$$\displaystyle\mathbb{E}[f(\bm{x}_{t+1})-f(\bm{x}^{\star})]\leq\left[\frac{2c(1+H)}{\delta G}+B\right]\alpha_{t}\,.$$
Now suppose that $\gamma=1$ then the analogous bound to (27) is
$$\displaystyle\sum_{s=\lfloor(u-b)/2^{n}\rfloor}^{u-b}\alpha_{s}=$$
$$\displaystyle\sum_{s=\lfloor(u-b)/2^{n}\rfloor}^{u-b}\frac{a}{(b+t)}$$
$$\displaystyle\geq\,$$
$$\displaystyle\int_{u/2^{n}-b}^{u-b}\frac{a}{(b+t)}ds={a}\left[\log(b+t)\right]^{u-b}_{u/2^{n}-b}=an\log 2\,.$$
We see that if we chose $n$ such that
$$n=\left\lceil\frac{(\alpha_{0}B+F)}{a\log 2}\right\rceil$$
then
$$\sum^{t}_{s=\lfloor t/2^{n}\rfloor}\alpha_{s}\frac{\delta}{2}\geq\alpha_{0}B+F\,,$$
holds. Thus is we choose $n$ as above and $t_{0}=1$ then, again applying (5) from Theorem 1, we have the bound
$$\displaystyle\mathbb{P}(f(\bm{x}_{t+1})-f(\bm{x}^{\star})\geq z)\leq(1+H){e^{-\frac{\delta G^{n}}{2E\alpha_{t}}(z-\alpha_{t}B)}}\,.$$
Thus we see that the bound (12) holds with $I:=H+1$ and $J:={\delta G^{n}}/{2E}$.
Integrating the above bound then gives
$$\displaystyle\mathbb{E}[f(\bm{x}_{t+1})-f(\bm{x}^{\star})]\leq\left[\frac{2E(1+H)}{\delta G^{n}}+B\right]\alpha_{t}\,.$$
Thus we see that (13) holds with $K:=B+{2E(1+H)}/{\delta G^{n}}$.
∎
8 Proof of Theorem 2 and Theorem 3
Theorem 2 and Theorem 3 show how Theorem 1 and Corollary 1 can be applied in the case of linear programming and dynamic programming.
The proof of Theorem 2 is an application of Theorem 1. As such, we are required to show that the conditions of Theorem 1 as satisfied. Specifically we must prove that Conditions C1 and C2 hold. The moment generating function bound in Condition C2 follows in a reasonably straightforward manner when random costs have a sub-Gaussian tail behavior. However, Condition C1 requires a more careful analysis. Here we give a geometric argument which proves that there is a non-zero angle between any sub-optimal point in $\mathcal{X}$ and the nearest optimal point in $\mathcal{X}^{\star}$. This is Lemma 5. (Note that Lemma 5 holds for linear programs, though may hold for many other convex optimization problems.) We then provide an perturbation analysis of this result in Corollary 2. We then apply a series of probabilistic bounds to this in Proposition 2. The results from Proposition 2 can then be collected together to verify that Conditions C1 and C2 hold and thus we prove Theorem 2.
The proof of Theorem 3 relies of the linear programming formulation of a Markov decision process and thus Theorem 3 follows immediately from Theorem 2.
The following result bounds the angle between optimal and sub-optimal points for a polytope. (Indeed other constrain sets $\mathcal{X}$ would satisfy this result and thus would potentially be amenable to our analysis.)
Lemma 4.
If $\mathcal{X}$ is a bounded polytope and
$$\mathcal{X}^{\star}=\text{argmin}_{\bm{x}\in\mathcal{X}}\bar{\bm{c}}^{\top}\bm{x}.$$
Then there exists a constant $K>0$ such that
$${\frac{\bar{\bm{c}}^{\top}(\bm{x}-\bm{x}^{\star})}{||\bar{\bm{c}}||||\bm{x}-\bm{x}^{\star}||}}\geq K,$$
for $\bm{x}^{\star}$ the projection of $\bm{x}$ onto $\mathcal{X}^{\star}$.
Proof.
We assume without loss of generality that $\bar{\bm{c}}^{\top}\bm{x}^{\star}=0$ and $||\bar{\bm{c}}||=1$. Let $\mathcal{E}$ be the extreme points of $\mathcal{X}$. Let $\mathcal{E}^{\star}$ be the extreme points in $\mathcal{X}^{\star}$. Then let $\mathcal{E}^{\prime}:=\mathcal{E}\setminus\mathcal{E}^{\star}$ and $\mathcal{X}^{\prime}$ is the convex closure of $\mathcal{E}^{\prime}$. Let $a:=\min_{\bm{x}\in\mathcal{X}^{\prime}}\bar{\bm{c}}^{\top}\bm{x}$ and $D:=\max_{\bm{x}^{\star}\in\mathcal{X}^{\star},\bm{x}^{\prime}\in\mathcal{X}^{\prime}}||\bm{x}^{\star}-\bm{x}^{\prime}||.$
We will show we can take $K:={a}/{D}$.
For all $\bm{x}\in\mathcal{X}\setminus\mathcal{X}^{\star}$, $\bm{x}$ must be a convex combination of a point in $\mathcal{X}^{\star}$ and a point in $\mathcal{X}^{\prime}.$ Specially,
$$\bm{x}=(1-p)\bm{x}_{0}+p\bm{x}_{1},$$
(28)
for $\bm{x}_{0}\in\mathcal{X}^{\star}$ and $\bm{x}_{1}\in\mathcal{X}^{\prime}$ and $p\in(0,1]$. Then, as required,
$$\frac{\bar{\bm{c}}^{\top}\bm{x}}{||\bm{x}-\bm{x}^{\star}||}\geq\frac{\bar{\bm{c}}^{\top}(\bm{x}-\bm{x}_{0})}{||\bm{x}-\bm{x}_{0}||}=\frac{\bar{\bm{c}}^{\top}(\bm{x}_{1}-\bm{x}_{0})}{||\bm{x}_{1}-\bm{x}_{0}||}\geq\frac{a}{D}=K>0\,.$$
The first inequality above uses the fact that $\bm{x}^{\star}$ is closest to $\bm{x}$. The equality applies (28). Then finally, we apply the definitions of $a$, $D$ and $K$.
∎
The following lemma is a deterministic version of the result that we want. Here we provide a geometric argument which shows that if we apply a projected gradient descent algorithm with the deterministic cost vector $\bar{\bm{c}}$ then we always make progress that is proportional to step size against the objective.
Lemma 5.
If $\bm{x}\in\mathcal{X}$ is such that
$$\bar{\bm{c}}^{\top}\bm{x}-\bar{\bm{c}}^{\top}{\bm{x}}^{\star}\geq 2\alpha||\bar{\bm{c}}||^{2},$$
(29)
then $\hat{\bm{y}}=\Pi_{\mathcal{X}}(\bm{x}-\alpha\bar{\bm{c}})$ is such that
$${\bar{\bm{c}}}^{\top}\hat{\bm{y}}-\bar{\bm{c}}^{\top}\bm{x}\leq-\alpha\big{(}1-\sqrt{1-K^{2}}\big{)}||\bar{\bm{c}}||^{2}.$$
(30)
Proof.
The proof follows by a geometric argument. To help with this we refer the reader to Figure 2.
Now, let
$$\bm{y}^{\prime}=\bm{x}+\alpha\bar{\bm{c}}.$$
(31)
Let $\bm{x}^{\star}$ be the projection of $\bm{x}$ onto $\mathcal{X}^{\star}=\text{argmin}_{\bm{x}\in\mathcal{X}}\bar{\bm{c}}^{\top}\bm{x}$. By Lemma 4,
$$\frac{\bar{\bm{c}}^{\top}(\bm{x}-\bm{x}^{\star})}{||\bm{x}-\bm{x}^{\star}||}\geq K.$$
Let $\bm{y}^{\prime\prime}$ be the projection of $\bm{y}^{\prime}$ onto the line segment between $\bm{x}$ and $\bm{x}^{\star}$. By convexity, $\bm{y}^{\prime\prime}\in\mathcal{X}$. Also, by (29), $\bm{y}^{\prime\prime}\neq\bm{x}^{\star}$. To see this notice that $||\bm{y}^{\prime}-\bm{x}||=\alpha||\bar{\bm{c}}||$ by definition, where as
$$||\bm{y}^{\prime}-\bm{x}^{\star}||\geq\frac{\bar{\bm{c}}}{\|\bar{\bm{c}}\|}(\bm{y}^{\prime}-\bm{x}^{\star})\geq 2\alpha\|\bar{\bm{c}}\|$$
Thus since $\bm{y}^{\prime}$ is closer to $\bm{x}\in\mathcal{X}$ than $\bm{x}^{\star}$. So the projection, $\bm{y}^{\prime\prime}$, cannot equal $\bm{x}^{\star}$.
Further, by Lemma 4
$$\frac{\bar{\bm{c}}^{\top}(\bm{x}-\bm{x}^{\star})}{||\bar{\bm{c}}||||\bm{x}-\bm{x}^{\star}||}>K.$$
(32)
Note that because $\bm{y}^{\prime\prime}$ is the projection of $\bm{y}^{\prime}$ onto the line between $\bm{x}$ and $\bm{x}^{\star}$ we have that
$$||\bm{x}-\bm{y}^{\prime\prime}||=(\bm{x}-\bm{y}^{\prime})^{\top}\frac{(\bm{x}-\bm{x}^{\star})}{||\bm{x}-\bm{x}^{\star}||}.$$
(33)
(Note that the left and right are both $||\bm{x}-\bm{y}^{\prime}||\cos\theta$ where $\theta$ is the angle between $\bm{x}^{\star}$, $\bm{x}$ and $\bm{y}^{\prime}$. See Figure 5.)
Combining (31), (32), and (33) gives
$$\frac{||\bm{x}-\bm{y}^{\prime\prime}||}{||\bm{x}-\bm{y}^{\prime}||}=\frac{(\bm{x}-\bm{y}^{\prime})^{\top}}{||\bm{x}-\bm{y}^{\prime}||}\frac{(\bm{x}-\bm{x}^{\star})}{||\bm{x}-\bm{x}^{\star}||}=\frac{\bar{\bm{c}}^{\top}}{||\bar{\bm{c}}||}\frac{(\bm{x}-\bm{x}^{\star})}{||\bm{x}-\bm{x}^{\star}||}>K.$$
Also by Pythagoras’ Theorem,
$$||\bm{x}-\bm{y}^{\prime}||^{2}=||\bm{y}^{\prime}-\bm{y}^{\prime\prime}||^{2}+||\bm{y}^{\prime\prime}-\bm{x}||^{2}.$$
Thus,
$$\displaystyle||\bm{y}^{\prime}-\bm{y}^{\prime\prime}||^{2}$$
$$\displaystyle=||\bm{x}-\bm{y}^{\prime}||^{2}-||\bm{y}^{\prime\prime}-\bm{x}||^{2}$$
$$\displaystyle\leq||\bm{x}-\bm{y}^{\prime}||^{2}-K^{2}||\bm{x}-\bm{y}^{\prime}||^{2}$$
$$\displaystyle=||\bm{x}-\bm{y}^{\prime}||^{2}(1-K^{2})$$
$$\displaystyle=\alpha^{2}||\bar{\bm{c}}||^{2}(1-K^{2}).$$
(34)
Since $||\bm{y}^{\prime}-\hat{\bm{y}}||\leq||\bm{y}^{\prime}-\bm{y}^{\prime\prime}||$, we have
$$||\bm{y}^{\prime}-\hat{\bm{y}}||\leq\alpha||\bar{\bm{c}}||\sqrt{1-K^{2}}.$$
Since $\bar{\bm{c}}^{\top}(\bm{y}^{\prime}-\bm{x})\leq-\alpha||\bar{\bm{c}}||^{2}$, it must be that
$$\displaystyle\bar{\bm{c}}^{\top}(\hat{\bm{y}}-\bm{x})$$
$$\displaystyle=\bar{\bm{c}}^{\top}(\hat{\bm{y}}-{\bm{y}}^{\prime})+\bar{\bm{c}}^{\top}({\bm{y}}^{\prime}-\bm{x})$$
$$\displaystyle\leq||\bar{\bm{c}}||\,||\hat{\bm{y}}-{\bm{y}}^{\prime}||-\alpha||\bar{\bm{c}}||^{2}$$
$$\displaystyle\leq\alpha||\bar{\bm{c}}||^{2}\sqrt{1-K^{2}}-\alpha||\bar{\bm{c}}||^{2}=-\alpha(1-\sqrt{1-K^{2}})||\bar{\bm{c}}||^{2}.$$
Thus we see that (30) holds.
∎
Lemma 5 shows that projected gradient descent will make constant progress on a linear program when the objective function is known. Of course there is some room for error in this estimate. The corollary below gives such a bound.
Corollary 2.
If $\bm{x}\in\mathcal{X}$ is such that
$$\bar{\bm{c}}^{\top}\bm{x}-\bar{\bm{c}}^{\top}\bm{x}^{\star}\geq 2\alpha||\bar{\bm{c}}||^{2},$$
and $\bm{x}^{\prime}\in\mathbb{R}^{d}$ is such that
$$||\bm{x}^{\prime}-\bm{x}+\alpha\bar{\bm{c}}||\leq\alpha\delta||\bar{\bm{c}}||\qquad\text{for }\quad\delta\leq 1-\sqrt{1-K^{2}}\,,$$
(35)
then $\hat{\bm{x}}=\Pi_{\mathcal{X}}(\bm{x}^{\prime})$ is such that
$${\bar{\bm{c}}}^{\top}\hat{\bm{x}}-\bar{\bm{c}}^{\top}\bm{x}\leq-\alpha p||\bar{\bm{c}}||^{2},$$
(36)
for some constant $p>0$.
Proof.
The argument here follows from the proof of Lemma 5 and we reuse the notation given there. Notice that
$$\displaystyle\bar{\bm{c}}^{\top}(\hat{\bm{x}}-\bm{x})=\,$$
$$\displaystyle\bar{\bm{c}}^{\top}(\hat{\bm{x}}-\hat{\bm{y}})\,+\,\bar{\bm{c}}^{\top}(\hat{\bm{y}}-{\bm{y}})$$
$$\displaystyle\leq\,$$
$$\displaystyle||\bar{\bm{c}}||\cdot||\hat{\bm{x}}-\hat{\bm{y}}||-\alpha(1-\sqrt{1-K^{2}})||\bar{\bm{c}}||^{2}$$
$$\displaystyle\leq\,$$
$$\displaystyle||\bar{\bm{c}}||\cdot||{\bm{x}}^{\prime}-{\bm{y}}^{\prime}||-\alpha(1-\sqrt{1-K^{2}})||\bar{\bm{c}}||^{2}$$
$$\displaystyle\leq\,$$
$$\displaystyle\delta||\bar{\bm{c}}||^{2}-\alpha(1-\sqrt{1-K^{2}})||\bar{\bm{c}}||^{2}\,.$$
The first inequality above applies the Cauchy-Schwartz inequality and Lemma 5. The second follows since projections reduce distances and we apply (35) recalling that $y^{\prime}:=x-\alpha\bar{c}$. The final inequality follows from assumptions on $\delta$ in (35).
∎
The following is a fairly standard bound on sub-Gaussian random variables. Similar bounds can be found in texts such as Vershynin [34]. A proof of Lemma 6 can be found in Appendix B.
Lemma 6.
Given that the costs $\bar{\bm{c}}^{i}_{t}$ are sub-Gaussian,
it holds that
$$\displaystyle\mathbb{P}\left(\Big{\|}\frac{1}{B}\sum_{i=1}^{B}\bm{c}_{t}^{i}-\bar{\bm{c}}\Big{\|}\geq\delta||\bar{\bm{c}}||\right)\leq 3^{d-1}e^{-\frac{\delta^{2}}{16}\frac{||\bar{\bm{c}}||^{2}}{\lambda}B}$$
(37)
for all $\delta>0$
The following proposition applies Lemma 6 to Corollary 2. So that we have a bound on the expected drift of our stochastic gradient descent algorithm.
Proposition 2.
There exist a batch size $B$ such that
$$\mathbb{E}[\bar{\bm{c}}{{}^{\top}}\bm{x}_{t+1}-\bar{\bm{c}}{{}^{\top}}\bm{x}_{t}|\mathcal{F}_{t}]\leq-\alpha p^{\prime}||\bar{\bm{c}}||\quad\text{on the event}\quad\{\bar{\bm{c}}^{\top}\bm{x}_{t}-\bar{\bm{c}}^{\top}\bm{x}^{\star}\geq 2\alpha||\bar{\bm{c}}||^{2}\}$$
for some $p^{\prime}>0.$
Proof.
Let $\bm{x}^{\prime}_{t}=\bm{x}_{t}-\alpha_{t}\bm{c}_{t}$.
By Corollary 2, if
$$\alpha_{t}||\bm{c}_{t}-\bar{\bm{c}}||=||\bm{x}^{\prime}_{t}-\bm{x}_{t}+\alpha_{t}\bar{\bm{c}}||\leq\alpha_{t}\delta||\bar{\bm{c}}||$$
(38)
then
$$\bar{\bm{c}}^{\top}{\bm{x}}_{t+1}-\bar{\bm{c}}^{\top}\bm{x}_{t}\leq-\alpha_{t}p||\bar{\bm{c}}||^{2}\,.$$
Thus when (38) holds progress against the linear programming objective is guarenteed. We now seek to bound the error when (38) does not hold.
Applying the Cauchy-Schwarz Inequality,
$$\displaystyle\bar{\bm{c}}^{\top}\bm{x}_{t+1}-\bar{\bm{c}}^{\top}\bm{x}_{t}$$
$$\displaystyle\leq||\bar{\bm{c}}||||{\bm{x}}_{t+1}-\bm{x}_{t}||$$
$$\displaystyle\leq\alpha_{t}||\bar{\bm{c}}||\cdot||\bm{c}_{t}||.$$
From this upper-bound, observe that the errors are determined by the fluctuations in $\bm{c}_{t}$. Further, notice that the left-hand side of (38) equals $\alpha_{t}||\bm{c}_{t}-\bar{\bm{c}}||$.
Given these two observations, we analyze the term
$$\displaystyle\mathbb{E}\Big{[}||\bm{c}_{t}||\cdot\mathbb{I}\big{[}||\bm{c}_{t}-\bar{\bm{c}}||\geq\delta||\bar{\bm{c}}||\big{]}\Big{]}\,.$$
This term bounds the error in $\bar{\bm{c}}^{\top}\bm{x}_{t+1}-\bar{\bm{c}}^{\top}\bm{x}_{t}$ when (38) does not hold.
Notice that by the Triangle Inequality
$$\displaystyle\mathbb{E}\Big{[}||\bm{c}_{t}||\cdot\mathbb{I}\big{[}||\bm{c}_{t}-\bar{\bm{c}}||\geq\delta||\bar{\bm{c}}||\big{]}\Big{]}$$
$$\displaystyle\leq||\bar{\bm{c}}||\cdot\mathbb{P}\big{(}||\bm{c}_{t}-\bar{\bm{c}}||\geq\delta||\bar{\bm{c}}||\big{)}$$
(39)
$$\displaystyle\quad+\mathbb{E}\Big{[}||\bm{c}_{t}-\bar{\bm{c}}||\cdot\mathbb{I}\big{[}||\bm{c}_{t}-\bar{\bm{c}}||\geq\delta||\bar{\bm{c}}||\big{]}\,\Big{]}\,.$$
(40)
We now bound the two terms (39) and (40).
We can bound (39) using Lemma 6
$$\mathbb{P}\big{(}||\bm{c}_{t}-\bar{\bm{c}}||\geq\delta||\bar{\bm{c}}||\big{)}\leq 3^{d-1}e^{-\frac{\delta^{2}}{16}\frac{||\bar{\bm{c}}||^{2}}{\lambda}B}\,.$$
(41)
We can bound (40) with Lemma 6 also. Specifically,
$$\displaystyle\mathbb{E}\Big{[}||\bm{c}_{t}-\bar{\bm{c}}||\cdot\mathbb{I}\big{[}||\bm{c}_{t}-\bar{\bm{c}}||\geq\delta||\bar{\bm{c}}||\big{]}\Big{]}$$
$$\displaystyle=\delta^{-1}\int_{\delta||\bar{\bm{c}}||}^{\infty}x\mathbb{P}\big{(}||\bm{c}_{t}-\bar{\bm{c}}||\geq x\big{)}\,dx$$
$$\displaystyle\leq\delta^{-1}\int_{\delta||\bar{\bm{c}}||}^{\infty}x3^{d-1}e^{-\frac{B}{16}\frac{x^{2}}{\lambda}}\,dx$$
$$\displaystyle=\frac{16\cdot 3^{d-1}\lambda}{B\delta}e^{-\frac{\delta^{2}}{16}\frac{||\bar{\bm{c}}||^{2}}{\lambda}B}\,.$$
(42)
Applying (41) & (42) to (39) & (40) above gives that
$$\mathbb{E}\Big{[}||\bm{c}_{t}||\cdot\mathbb{I}[||\bm{c}_{t}-\bar{\bm{c}}||\geq\delta||\bar{\bm{c}}||]\Big{]}\leq 3^{d-1}\left[1+\frac{16\cdot 3^{d-1}\lambda}{B\delta}\right]e^{-\frac{\delta^{2}}{16}\frac{||\bar{\bm{c}}||^{2}}{\lambda}B}\,.$$
We can the combine the above inequalities
$$\displaystyle\,\mathbb{E}[\bar{\bm{c}}^{\top}\bm{x}_{t+1}-\bar{\bm{c}}^{\top}\bm{x}_{t}|\mathcal{F}_{t}]$$
$$\displaystyle=\mathbb{E}\Big{[}(\bar{\bm{c}}^{\top}\bm{x}_{t+1}-\bar{\bm{c}}^{\top}\bm{x}_{t})\mathbb{I}[||\bm{c}_{t}-\bar{\bm{c}}||\leq\delta||\bar{\bm{c}}||]\Big{|}\mathcal{F}_{t}\Big{]}+\mathbb{E}\Big{[}(\bar{\bm{c}}^{\top}\bm{x}_{t+1}-\bar{\bm{c}}^{\top}\bm{x}_{t})\mathbb{I}[||\bm{c}_{t}-\bar{\bm{c}}||\geq\delta||\bar{\bm{c}}||]\Big{|}\mathcal{F}_{t}\Big{]}$$
$$\displaystyle\leq-\alpha_{t}p||\bar{\bm{c}}||\cdot\mathbb{P}\big{(}||\bm{c}_{t}-\bar{\bm{c}}||\leq\delta||\bar{\bm{c}}||\big{)}+\alpha_{t}||\bar{\bm{c}}||\cdot\mathbb{E}\Big{[}||\bm{c}_{t}||\mathbb{I}[||\bm{c}_{t}-\bar{\bm{c}}||\geq\delta||\bar{\bm{c}}||]\Big{|}\mathcal{F}_{t}\Big{]}$$
$$\displaystyle\leq-\alpha_{t}p||\bar{\bm{c}}||\left(1-3^{d-1}e^{-\frac{\delta^{2}}{16}\frac{||\bar{\bm{c}}||^{2}}{\lambda}B}\right)+\alpha_{t}||\bar{\bm{c}}||3^{d-1}\left[1+\frac{16\cdot 3^{d-1}\lambda}{B\delta}\right]e^{-\frac{\delta^{2}}{16}\frac{||\bar{\bm{c}}||^{2}}{\lambda}B}$$
$$\displaystyle=-\alpha_{t}p||\bar{\bm{c}}||+\alpha_{t}||\bar{\bm{c}}||e^{-\frac{\delta^{2}}{16}\frac{||\bar{\bm{c}}||^{2}}{\lambda}B}\left[\,2+\frac{16\cdot 3^{d-1}\lambda}{B\delta}\,\right]$$
$$\displaystyle=-\alpha_{t}p^{\prime}||\bar{\bm{c}}||$$
where
$$\displaystyle p^{\prime}=p-e^{-\frac{\delta^{2}}{16}\frac{||\bar{\bm{c}}||^{2}}{\lambda}B}\left[\,2+\frac{16\cdot 3^{d-1}\lambda}{B\delta}\,\right]\,.$$
Since $p$ is positive and independent of $B$ it should be clear that
for $B$ sufficiently large we can choose $p^{\prime}>0$.
∎
The proof of Theorem 2 is now immediate.
Proof of Theorem 2.
By Proposition 2, it holds that
$$\mathbb{E}[\bar{\bm{c}}^{\top}\bm{x}_{t+1}-\bar{\bm{c}}^{\top}\bm{x}_{t}|\mathcal{F}_{t}]\leq-2\alpha_{t}p^{\prime}||c||$$
(43)
whenever $\bar{\bm{c}}^{\top}\bm{x}_{t}-\bar{\bm{c}}^{\top}\bm{x}^{\star}\geq 2\alpha||\bar{\bm{c}}||^{2}$. In otherwords, Condition C1 of Theorem 1 is satisfied with $\delta=2||\bar{\bm{c}}||$ and $B=p^{\prime}||\bar{\bm{c}}||$.
Further we show that the moment generating function condition on increments, C2, is satisfied. In particular notice
$$\big{[}|\bar{\bm{c}}^{\top}\bm{x}_{t+1}-\bar{\bm{c}}^{\top}\bm{x}_{t}|\big{|}\mathcal{F}_{t}\big{]}=\alpha_{t}\bar{\bm{c}}^{\top}\bm{c}_{t}\,.$$
Thus in the context of condition C1 we can take $Y=\bar{\bm{c}}^{\top}\bm{c}_{t}$. Notice by the sub-Gaussian assumption on costs (10), for $\bm{c}_{t}=\sum_{i=1}^{B}\bm{c}^{i}_{t}/B$, we have that
$$\mathbb{E}[e^{Y}]=\mathbb{E}[e^{\sum_{i=1}^{B}\bar{\bm{c}}^{\top}\bm{c}^{i}_{t}/B}]\leq[e^{\bar{\bm{c}}^{\top}\bar{\bm{c}}/B+\lambda||\bar{\bm{c}}||^{2}/B^{2}}]^{B}=e^{||\bar{\bm{c}}||^{2}+\lambda||\bar{\bm{c}}||^{2}/B}<\infty\,.$$
Thus we see that the moment generating function condition C2 is also satisfied. Since Condition C1 and C2 hold Theorem applies (as well as corollary). Thus theorem holds.
∎
Appendix A Proofs of Additional Results
A.1 Linear Programming Formulation of MDPs.
In this section, we briefly recall the linear programming formulation of a Markov decision process. The optimum of a discounted Markov decision process (cf. Section 3) is the unique solution to the Bellman equation:
$$V(s)=\min_{a\in\mathcal{A}}\Big{\{}c(s,a)+\beta\sum_{s^{\prime}\in\mathcal{S}}V(s^{\prime})P(s^{\prime}|s,a)\,\Big{\}}.$$
The value function $\bm{V}=(V(s):s\in\mathcal{S})$ is (component-wise) the largest vector such that the following inequality holds:
$$V(s)\leq c(s,a)+\beta\sum_{s^{\prime}\in\mathcal{S}}V(s^{\prime})P(s^{\prime}|s,a)\qquad s\in\mathcal{S},a\in\mathcal{A}.$$
In other words, the optimal value function $\bm{V}$ can be found by solving the following linear program:
Maximize
$$\displaystyle\sum_{s\in\mathcal{S}}\xi(s)V(s)$$
subject to
$$\displaystyle V(s)\leq c(s,a)+\beta\sum_{s^{\prime}\in\mathcal{S}}V(s^{\prime})P(s^{\prime}|s,a),\;\;\forall s\in\mathcal{S},a\in\mathcal{A}$$
over
$$\displaystyle(V(s):s\in\mathcal{S})\in\mathbb{R}^{\mathcal{S}}$$
where $\bm{\xi}=(\xi(s):s\in\mathcal{S})$ is a positive vector. (Here setting $\xi(s)=1$ for all $s\in\mathcal{S}$ would be a standard choice.)
The dual of the above linear program is
Minimize
$$\displaystyle\sum_{s\in\mathcal{S}}c(s,a)x(s,a)$$
(Dual)
subject to
$$\displaystyle\sum_{a\in\mathcal{A}}x(s^{\prime},a)=\xi(s^{\prime})+\beta\sum_{s\in\mathcal{S}}\sum_{a\in\mathcal{A}}x(s,a)P(s^{\prime}|s,a),\qquad\forall s^{\prime}\in\mathcal{S}$$
over
$$\displaystyle(x(s,a):s\in\mathcal{S},a\in\mathcal{A})\in\mathbb{R}_{+}^{\mathcal{S}\times\mathcal{A}}.$$
Here the variables $x(s,a)$ are sometimes referred to as occupancy measures. For a given variable $\bm{x}$ satisfying the constraints of the dual, we can determine a policy
$$\pi(a|s)=\frac{x(s,a)}{\sum_{a^{\prime}\in\mathcal{A}}x(s,a^{\prime})}\,.$$
A.2 Proof of Additional Lemma’s from Section 4.
Lemma 3.
If $\alpha_{t}$, $t\in\mathbb{Z}_{+}$, is a decreasing positive sequence, then
$$\min_{s=t_{1},...,t}\left\{\frac{\sum_{k=s}^{t}\alpha_{k}}{\sum_{k=s}^{t}\alpha_{k}^{2}}\right\}=\frac{\sum_{k=t_{1}}^{t}\alpha_{k}}{\sum_{k=t_{1}}^{t}\alpha_{k}^{2}}$$
(44)
Moreover, if $\alpha_{t}$, $t\in\mathbb{Z}_{+}$ satisfies the learning rate condition (4) then for $n\in\mathbb{N}$ such that $t/2^{n}>1$
$$\min_{s=\lfloor t/2^{n}\rfloor,...,t}\left\{\frac{\sum_{k=s}^{t}\alpha_{k}}{\sum_{k=s}^{t}\alpha_{k}^{2}}\right\}\geq\frac{G^{n}}{\alpha_{t}}$$
(45)
for some positive constant $G$.
Proof.
It is straight-forward to show that for $a,a^{\prime},A,A^{\prime}>0$
$$\frac{a}{a^{\prime}}\leq\frac{A}{A^{\prime}}\quad\text{if and only if}\quad\frac{a+A}{a^{\prime}+A^{\prime}}\leq\frac{A}{A^{\prime}}\,.$$
(46)
[Note that both expression above are equivalent to $AA^{\prime}+aA^{\prime}\leq AA^{\prime}+a^{\prime}A$.]
Take positive numbers $a_{s},a_{s}^{\prime},\ s=1,...,t$. If
$$\frac{a_{s}}{a_{s}^{\prime}}\leq\frac{a_{k}}{a_{k}^{\prime}}$$
(47)
for $k=s-1,...,t$, then
$$\sum_{k=s-1}^{t}a_{k}^{\prime}a_{s}\leq\sum_{k=s-1}^{t}a_{k}a_{s}^{\prime}.$$
(48)
Thus,
$$\frac{a_{s}}{a_{s}^{\prime}}\leq\frac{\sum_{k=s-1}^{t}a_{k}}{\sum_{k=s-1}^{t}a_{k}^{\prime}}.$$
Thus applying (46) with $A=\sum_{k=s-1}^{t}a_{k}$ and $A^{\prime}=\sum_{k=s-1}^{t}a_{k}^{\prime}$ gives
$$\frac{\sum_{k=s}^{t}a_{k}}{\sum_{k=s}^{t}a_{k}^{\prime}}\leq\frac{\sum_{k=s-1}^{t}a_{k}}{\sum_{k=s-1}^{t}a_{k}^{\prime}}.$$
(49)
Finally, taking $a_{k}=\alpha_{k}$ and $a_{k}^{\prime}=\alpha_{k}^{2}$, we see that (48) holds since $\alpha_{t}$ is decreasing. Thus, from (49), we see that the result (44) holds.
If the condition (4) holds then $\liminf_{t\rightarrow\infty}\alpha_{2t}/\alpha_{t}>0$ implies
$$\frac{\alpha_{t}}{\alpha_{\lfloor t/2\rfloor}}>\sqrt{G}$$
(50)
for some $G>0$. Since the sequence is decreasing and (50) holds, we have that
$$\frac{\sum_{k=\lfloor t/2^{n}\rfloor}^{t}\alpha_{k}}{\sum_{k=\lfloor t/2^{n}\rfloor}^{t}\alpha_{k}^{2}}\geq\frac{(t-\lfloor t/2^{n}\rfloor)\alpha_{t}}{(t-\lfloor t/2^{n}\rfloor)\alpha_{\lfloor t/2^{n}\rfloor}^{2}}=\frac{\alpha^{2}_{t}}{\alpha^{2}_{\lfloor t/2^{n}\rfloor}}\frac{1}{\alpha_{t}}=\frac{\alpha^{2}_{t}}{\alpha^{2}_{\lfloor t/2\rfloor}}\times...\times\frac{\alpha^{2}_{\lfloor t/2^{n-1}\rfloor}}{\alpha^{2}_{\lfloor t/2^{n}\rfloor}}\frac{1}{\alpha_{t}}\geq\frac{G^{n}}{\alpha_{t}}\,.$$
Applying this to (44) with $t_{1}=\lfloor t/2^{n}\rfloor$ gives
$$\min_{s=\lfloor t/2^{n}\rfloor,...,t}\left\{\frac{\sum_{k=s}^{t}\alpha_{k}}{\sum_{k=s}^{t}\alpha_{k}^{2}}\right\}\geq\frac{G^{n}}{\alpha_{t}}\,.$$
Thus (45) holds as required.
∎
Lemma 7.
For projected stochastic gradient descent [cf. (PSGD)], if for all $t\in\mathbb{Z}_{+}$ there exists $a,b>0$ such that
$$\displaystyle\mathbb{P}(||\bm{c}_{t}||\geq z|\mathcal{F}_{t})\leq ae^{-bz}$$
(51)
and $\nabla f(\bm{x})$ is continuous, then (C2) holds, that is
$$\displaystyle\Big{[}|f(\bm{x}_{t+1})-f(\bm{x}_{t})|\big{|}\mathcal{F}_{t}\Big{]}\leq\alpha_{t}Y\,,\quad\text{with}\quad\mathbb{E}[e^{\eta Y}]<\infty$$
(52)
for some $\eta>0$.
Proof.
Notice by Taylor’s Theorem, for some $\bm{y}_{t}$ belonging to the line segment between $\bm{x}_{t}$ and $\bm{x}_{t+1}$, it holds that
$$\displaystyle|f(\bm{x}_{t+1})-f(\bm{x}_{t})|=$$
$$\displaystyle|(\bm{x}_{t+1}-\bm{x}_{t})\cdot\nabla f(\bm{y}_{t})|$$
$$\displaystyle\leq\,$$
$$\displaystyle||\bm{x}_{t+1}-\bm{x}_{t}||\cdot||\nabla f(\bm{y}_{t})||$$
$$\displaystyle\leq\,$$
$$\displaystyle\alpha_{t}||\bm{c}_{t}||\max_{\bm{x}\in\mathcal{X}}||\nabla f(\bm{x})||\,.$$
Thus we can see that the moment generating function condition (52) holds if we can prove that
$$\mathbb{E}[e^{\eta||\bm{c}_{t}||}|\mathcal{F}_{t}]<\infty$$
(53)
for some $\eta>0$.
The condition (53) is a consequence of (51). Specifically,
$$\displaystyle\mathbb{E}[e^{\eta||\bm{c}_{t}||}|\mathcal{F}_{t}]=$$
$$\displaystyle\int_{0}^{\infty}\mathbb{P}(e^{\eta||\bm{c}_{t}||}>y\,|\mathcal{F}_{t})dy$$
$$\displaystyle\leq\,$$
$$\displaystyle 1+\int_{1}^{\infty}\mathbb{P}\Big{(}||\bm{c}_{t}||\geq\frac{1}{\eta}\log y\Big{|}\mathcal{F}_{t}\Big{)}dy$$
$$\displaystyle\leq\,$$
$$\displaystyle 1+\int_{1}^{\infty}ay^{-b/\eta}dy<\infty\qquad\text{ for }0<\eta<b\,.$$
(54)
Thus from (54) we see that (53) holds and thus the required condition (52) holds.
∎
Appendix B Additional Lemmas from Section 5
Lemma 6.
Given that the costs $\bm{c}^{i}_{t}$ are sub-Gaussian,
it holds that
$$\displaystyle\mathbb{P}\left(\Big{\|}\frac{1}{B}\sum_{i=1}^{B}\bm{c}_{t}^{i}-\bm{c}\Big{\|}\geq\delta||\bm{c}||\right)\leq 3^{d-1}e^{-\frac{\delta^{2}}{16}\frac{||\bm{c}||^{2}}{\lambda}B}$$
(55)
for all $\delta>0$
Proof.
The proof follows a reasonably standard argument for random vectors. (For instance, see [34, Chapter 4].)
Letting $\bm{c}_{t}=\sum_{i=1}^{B}\bm{c}^{i}_{t}/B$, first notice that
$$\displaystyle\left\|\bm{c}_{t}-\bm{c}\right\|=\sup_{\bm{\eta}:||\bm{\eta}||=1}\bm{\eta}^{\top}(\bm{c}_{t}-\bm{c})\,.$$
(56)
We let $\mathcal{N}$ be the $\frac{1}{2}$-net of the sphere $S_{d-1}:=\{\bm{\eta}\in\mathbb{R}^{d}:\|\bm{\eta}\|=1\}$. (Recall that $\mathcal{N}$ is an $\epsilon$-net is a subset of $S$ such that all points in $S$ are within $\epsilon$ of a point in $\mathcal{N}$.) The following bound on the cardinality of $\mathcal{N}$ is known
$$\displaystyle|\mathcal{N}|\leq 3^{d}\,,$$
(57)
from [34, Corollary 4.2.13].
Further it is straight-forward to show that
$$\displaystyle\|\bm{c}_{t}-\bm{c}\|\leq 2\sup_{\eta\in\mathcal{N}}\bm{\eta}^{\top}(\bm{c}_{t}-\bm{c})\,.$$
(58)
To see why (58) holds: let $\bm{\eta}_{\star}$ achieve the maximum in (56) and let $\bm{\eta}_{0}$ be the point closest to $\bm{\eta}_{\star}$ in $\mathcal{N}$. Then
$$\displaystyle\left\|\bm{c}_{t}-\bm{c}\right\|=\bm{\eta}^{\top}_{\star}(\bm{c}_{t}-\bm{c})=$$
$$\displaystyle(\bm{\eta}_{\star}-\bm{\eta}_{0})^{\top}(\bm{c}_{t}-\bm{c})+\bm{\eta}_{0}^{\top}(\bm{c}_{t}-\bm{c})$$
$$\displaystyle\leq$$
$$\displaystyle\left\|\bm{\eta}_{\star}-\bm{\eta}_{0}\right\|\left\|\bm{c}_{t}-\bm{c}\right\|+\sup_{\eta\in\mathcal{N}}\bm{\eta}^{\top}(\bm{c}_{t}-\bm{c})$$
$$\displaystyle\leq$$
$$\displaystyle\frac{1}{2}\left\|\bm{c}_{t}-\bm{c}\right\|+\sup_{\eta\in\mathcal{N}}\bm{\eta}^{\top}(\bm{c}_{t}-\bm{c})$$
(59)
In the first inequality above, we apply the Cauchy-Schwartz Inequality; then we apply the definition of $\eta_{0}$; rearranging (59) gives (58), as required.
We now achieve the bound (55) as follows: for $\theta=\delta||\bm{c}||/4\lambda$ ,
$$\displaystyle\mathbb{P}\left(\Big{\|}\frac{1}{B}\sum_{i=1}^{B}\bm{c}_{t}^{i}-\bm{c}\Big{\|}\geq\delta||c||\right)=\ $$
$$\displaystyle\mathbb{P}\Big{(}\sup_{\bm{\eta}\in S_{d-1}}\bm{\eta}^{\top}(\bm{c}_{t}-\bm{c})\geq\delta||\bm{c}||\Big{)}$$
(60)
$$\displaystyle\leq\ $$
$$\displaystyle\mathbb{P}\left(2\sup_{\bm{\eta}\in\mathcal{N}}\bm{\eta}^{\top}(\bm{c}_{t}-\bm{c})\geq\delta||\bm{c}||\right)$$
(61)
$$\displaystyle\leq\ $$
$$\displaystyle|\mathcal{N}|\sup_{\bm{\eta}\in\mathcal{N}}\mathbb{P}\left(\bm{\eta}^{\top}(\bm{c}_{t}-\bm{c})\geq\frac{\delta||\bm{c}||}{2}\right)$$
(62)
$$\displaystyle\leq\ $$
$$\displaystyle 3^{d-1}\sup_{\bm{\eta}\in\mathcal{N}}\mathbb{P}\left(e^{B\theta\bm{\eta}^{\top}(\bm{c}_{t}-\bm{c})}\geq e^{B\theta\frac{\delta||\bm{c}||}{2}}\right)$$
(63)
$$\displaystyle\leq\ $$
$$\displaystyle 3^{d-1}\sup_{\bm{\eta}\in\mathcal{N}}e^{-B\theta\frac{\delta||\bm{c}||}{2}}\mathbb{E}\left[e^{B\theta\bm{\eta}^{\top}(\bm{c}_{t}-\bm{c})}\right]$$
$$\displaystyle=\ $$
$$\displaystyle 3^{d-1}\sup_{\bm{\eta}\in\mathcal{N}}e^{-B\theta\frac{\delta||\bm{c}||}{2}}\mathbb{E}\left[e^{\theta\bm{\eta}^{\top}(\bm{c}^{i}_{t}-\bm{c})}\right]^{B}$$
$$\displaystyle\leq\,$$
$$\displaystyle 3^{d-1}e^{-B\theta\frac{\delta||\bm{c}||}{2}}e^{\theta^{2}\lambda B}$$
$$\displaystyle=\ $$
$$\displaystyle 3^{d-1}e^{-B\frac{\delta^{2}||\bm{c}||^{2}}{16\lambda}}\,.$$
(64)
The equality (60) follows from (56) and the definition of $\bm{c}_{t}$.
The inequality (61) applies (58). The inequality (62) is a union bound. Inequality (63) applies the bound on the cardinality of $|\mathcal{N}|$ from (57). The inequalities from (63) to (64) applies the sub-Gaussian assumption on costs along with standard moment generating function bounds for sums of independent random variables.
∎
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Free wreath products with amalgamation
Amaury Freslon
amaury.freslon@math.u-psud.fr
Laboratoire de Mathématique d’Orsay, CNRS, Université Paris-Saclay, 91405 Orsay, France.
Abstract.
We define and study a notion of free wreath product with amalgamation for compact quantum groups. These objects were already introduced in the case of duals of discrete groups under the name “free wreath products of pairs” in a previous work of ours. We give several equivalent descriptions and use them to establish properties like residual finiteness, the Haagerup property or a smash product decomposition.
Key words and phrases: Compact quantum groups, representation theory, noncrossing partitions
2010 Mathematics Subject Classification: 20G42, 05E10
1. Introduction
Compact quantum groups were introduced by S.L. Woronowicz in [36] and [38] as a generalisation of compact groups, and in particular of compact Lie groups, in the spirit of non-commutative geometry. The study of these objects has continued ever since and revealed unsuspected connections to other fields like combinatorics [2], free probability [22] or graph theory [25]. One essential aspect of these advances is that they were all related to the search for new examples of compact quantum groups.
Another direction of research on this topic is a vast classification program for a subclass of compact quantum groups that can be described combinatorially, the easy [2] (or more generally partition [16]) quantum groups. The study of various types of invariants and constructions for the purpose of their classification led to the discovery of new families of examples of compact quantum groups and to new insight into previously known ones. The present work is an instance of this phenomenon. We will be interested in compact quantum groups which were discovered in the course of the classification of non-crossing partition quantum groups on two self-adjoint colours in [18], and which turned out to be generalizations of objects introduced by J. Bichon in [3] in relation to quantum automorphism groups of graphs.
The free wreath product with amalgamation is a generalization of the free wreath product where instead of considering a single compact quantum group $\mathbb{G}$ and an integer $N$, we consider a dual inclusion of quantum groups (see Section 3 for details). The definition came out of a combinatorial description of the representation theory of these objects when $\mathbb{G}$ is the dual of a discrete group, but free wreath products are know to enjoy several equivalent descriptions, each of them suited to the study of some specific properties. In the present work, we will give similar descriptions and deduce from them several structure results on free wreath products of pairs without always resorting to the partition picture.
Let us now describe more precisely the contents of the paper. After some preliminaries, we will give in Section 3 an abstract definition of free wreath products with amalgamation (Definition 3.4) together with elementary examples. We will then focus on the case of discrete groups to give another characterization in terms of universal $*$-algebras (Proposition 3.8) and use it to deduce several results concerning monoidal equivalence and approximation properties. We will conclude that first part with a specific result concerning discrete abelian groups, which in turn sheds light on the classical counterpart of free wreath products with amalgamation, which we will simply call wreath product with amalgamation.
In Section 4, we make the connection with the original definition involving partitions. This enables us to prove a topological generation result in Theorem 4.4, which can be applied to the question of residual finiteness. We also provide a generalization of a result of F. Lemeux and P. Tarrago [24] in Proposition 4.6 concerning monoidal equivalence, from which results on approximation properties can be deduced.
Eventually, in Section 5 we show that, in the discrete case, when $\Lambda$ is a normal subgroup of $\Gamma$, the free wreath product of the pair fits into an exact sequence of Hopf $*$-algebra. We then give applications to cohomological dimension and determine when the exact sequence splits to yield a smash product decomposition.
Acknowledgement
The author whishes to thank Julien Bichon for reading and commenting a preliminary version of this text, as well as for bibliographical assistance. He was supported by the ANR grants “Noncommutative analysis on groups and quantum groups” (ANR-19-CE40-0002) and “Operator algebras and dynamics on groups” (ANR-19-CE40-0008).
2. Preliminaries
In this preliminary section we will recall some basic facts about compact quantum groups and their combinatorics, mainly to fix terminology and notations. We refer the reader to [31] and [27] for detailed treatments of the theory. It is known since the work of M. Dijkhuizen and T. Koornwinder [13] that compact quantum groups can be treated algebraically through the following notion of a CQG-algebra.
Definition 2.1.
A CQG-algebra is a Hopf $*$-algebra which is spanned by the coefficients of its finite-dimensional unitary corepresentations.
If $G$ is a compact group, then its algebra of regular functions $\mathcal{O}(G)$ is a CQG-algebra. Based on that example, and in an attempt to retain the intuition coming from the classical setting, we will denote a general CQG-algebra by $\mathcal{O}(\mathbb{G})$ and say that it corresponds to the compact quantum group $\mathbb{G}$. If $\Gamma$ is a discrete group and $\mathbb{C}[\Gamma]$ denotes its group algebra, it is easy to endow it with a Hopf $*$-algebra structure with coproduct given by $\Delta(g)=g\otimes g$ for all $g\in\Gamma$. Since this turns each $g\in\Gamma\subset\mathbb{C}[\Gamma]$ into a one-dimensional co-representation, it yields a CQG-algebra. The resulting compact quantum group is called the dual of $\Gamma$ and is denoted by $\widehat{\Gamma}$.
The main feature of compact quantum groups is their nice representation theory, which is just another point of view on the corepresentation theory of the corresponding CQG-algebra. Let us give a definition to make this clear.
Definition 2.2.
An $n$-dimensional representation of a compact quantum group $\mathbb{G}$ is an element $v\in M_{n}(\mathcal{O}(\mathbb{G}))$ which is invertible and such that for all $1\leqslant i,j\leqslant n$,
$$\Delta(v_{ij})=\sum_{k=1}^{n}v_{ik}\otimes v_{kj}.$$
It is said to be unitary if it is unitary as an element of $M_{n}(\mathcal{O}(\mathbb{G}))$.
Given two representations $v$ and $w$, one can form their direct sum by considering a block diagonal matrix with blocks $v$ and $w$ respectively, and their tensor product by considering the matrix with coefficients
$$(v\otimes w)_{(i,k),(j,\ell)}=v_{ij}w_{k\ell}.$$
In this setting, an intertwiner between two representations $v$ and $w$ of dimension respectively $n$ and $m$ will be a linear map $T:\mathbb{C}^{n}\to\mathbb{C}^{m}$ such that $Tv=wT$ (we are here identifying $M_{n}(\mathbb{C})$ with $M_{n}(\mathbb{C}.1_{\mathcal{O}(\mathbb{G})})\subset M_{n}(\mathcal{O}(\mathbb{G}))$). The set of all intertwiners between $v$ and $w$ will be denoted by $\displaystyle\operatorname{Mor}_{\mathbb{G}}(v,w)$.
If $T$ is injective, then $v$ is said to be a subrepresentation of $w$, and if $w$ admits no subrepresentation apart from itself, then it is said to be irreducible. One of the fundamental results in the representation theory of compact quantum groups is due to S.L. Woronowicz in [38] and can be summarized as follows :
Theorem 2.3 (Woronowicz).
Any finite-dimensional representation of a compact quantum group splits as a direct sum of irreducible ones, and any irreducible representation is equivalent to a unitary one.
We will need to consider subgroups in this quantum setting, and this can be done in two ways. First, if $G$ is a compact group and $H$ is a closed subgroup, then the restriction of functions yields a Hopf algebra $*$-homomorphism $\pi:\mathcal{O}(G)\to\mathcal{O}(H)$, leading to the following definition :
Definition 2.4.
Let $\mathbb{G}$ and $\mathbb{H}$ be compact quantum groups. We say that $\mathbb{H}$ is a quantum subgroup of $\mathbb{G}$ if there exists a surjective Hopf algebra $*$-homomorphism $\pi:\mathcal{O}(\mathbb{G})\to\mathcal{O}(\mathbb{H})$. We then write $\mathbb{H}<\mathbb{G}$.
Consider now the dual of a discrete group $\Gamma$ and let $A\subset\mathcal{O}(\widehat{\Gamma})=\mathbb{C}[\Gamma]$ be a Hopf $*$-subalgebra. Then, there exists a subgroup $\Lambda$ of $\Gamma$ such that $A=\mathbb{C}[\Lambda]$. Abstracting this yields
Definition 2.5.
Let $\mathbb{G}$ and $\mathbb{H}$ be compact quantum groups. If $\mathcal{O}(\mathbb{H})$ is a Hopf $*$-subalgebra of $\mathcal{O}(\mathbb{G})$, then $\mathbb{H}$ is said to be a dual quantum subgroup of $\mathbb{G}$.
Remark 2.6.
With a theory of discrete quantum groups at hand together with the corresponding generalization of Pontryagin duality (which we will not introduce because we do not need it later on), the previous definition becomes very natural since it corresponds to the discrete dual $\widehat{\mathbb{H}}$ of $\mathbb{H}$ being a quantum subgroup of the discrete dual $\widehat{\mathbb{G}}$ of $\mathbb{G}$. One could therefore write this as $\widehat{\mathbb{H}}<\widehat{\mathbb{G}}$ but we will avoid that notation in the sequel.
Part of the present work relies on the combinatorial approach to compact quantum groups initally developped by T. Banica and R. Speicher in [2]. We will use here the coloured version of that theory introduced in [16], which we very briefly recall. The basic objects are partitions of finite sets, which we represent by drawing the elements of the set on two parallel rows and connecting them by lines if they belong to the same component of the partition. Such a graphical interpretation enables to concatenate partitions horizontally and vertically (if the number of points matches) and rotate points from one line to another.
A category of partitions is a collection of partitions $\mathcal{C}(k,\ell)$ on $k+\ell$ points for all integers $k,\ell$ which is globally stable under the above operations and contains the identity partition $|$. It is moreover said to be coloured if to each point is attached an element of a colour set $\mathcal{A}$, which is simply a set endowed with an involution $x\mapsto x^{-1}$. When a point of a partition is rotated from one row to another, its colour is changed into its image under the involution. Moreover, vertical concatenation of coloured partitions is only allowed when the colours of the points which are merged match. We also have a coloured version of the identity partition for each $x\in\mathcal{A}$, called the $x$-identity partition. To each coloured partition $p$ we associate the word $w$ formed by the colours of the upper row (from left to right) and the word $w^{\prime}$ formed by the colours of its lower row (again from left to right). We then write $p\in\mathcal{C}(w,w^{\prime})$.
The framework of easy quantum groups generalizes to this setting. In particular, to any category of coloured partitions $\mathcal{C}$ and any integer $N$ is associated a compact quantum group $\mathbb{G}_{N}(\mathcal{C})$. Its CQG-algebra is generated by the coefficients of unitary representations $(u^{x})_{x\in\mathcal{A}}$ indexed by the colours and any irreducible representation is a subrepresentation of a tensor product
$$u^{\otimes w}=u^{w_{1}}\otimes\cdots\otimes u^{w_{n}},$$
where $w=w_{1}\cdots w_{n}$ is a word over $\mathcal{A}$. Moreover, the intertwiner spaces are completely determined by the category of partitions, in the sense that
$$\operatorname{Mor}_{\mathbb{G}_{N}(\mathcal{C})}\left(u^{\otimes w},u^{\otimes w^{\prime}}\right)=\operatorname{span}\left\{T_{p}\mid p\in\mathcal{C}(w,w^{\prime})\right\}.$$
Here, $T_{p}$ is a linear map the definition of which we now recall. Writing indices $i_{1},\cdots,i_{n}$ from left to right on the upper row of $p$ and indices $j_{1},\cdots,j_{k}$ from left to right on the lower row, we set $\delta_{p}(i_{1}\cdots i_{n},j_{1}\cdots j_{k})=1$ if all indices which are connected by strings of $p$ are equal, and $\delta_{p}(i_{1}\cdots i_{n},j_{1}\cdots j_{k})=0$ otherwise. Then, if $(e_{i})_{1\leqslant i\leqslant N}$ is the canonical basis of $\mathbb{C}^{N}$, we set
$$T_{p}(e_{i_{1}}\otimes\cdots\otimes e_{n})=\sum_{j_{1},\cdots,j_{k}=1}^{N}\delta_{p}(i_{1}\cdots i_{n},j_{1}\cdots j_{k})e_{j_{1}}\otimes\cdots\otimes e_{j_{k}}.$$
3. General theory
3.1. Definition and first examples
Free wreath products of pairs were introduced in [18] through the use of categories of coloured partitions. The definition can be seen as a generalization of the categories of partitions of free wreath products computed in [23], hence the name. However, free wreath products were defined in purely Hopf algebraic terms by J. Bichon in [3]. We will therefore start by providing a definition in the same spirit. At this level, working with arbitrary compact quantum groups does not entail any additional difficulty, hence we will use that setting for the moment.
Free wreath products can be defined as quotients of a free product involving a compact quantum group $\mathbb{G}$ and the quantum permutation group $S_{N}^{+}$ introduced by S. Wang in [34], whose definition we now recall.
Definition 3.1.
The CQG-algebra of the quantum permutation group $S_{N}^{+}$ is the universal unital $*$-algebra generated by elements $(p_{ij})_{1\leqslant i,j\leqslant N}$ such that
•
$p_{ij}^{2}=p_{ij}=p_{ij}^{*}$ for all $1\leqslant i,j\leqslant N$ ;
•
$\displaystyle\sum_{k=1}^{N}p_{ik}=1=\displaystyle\sum_{k=1}^{N}p_{kj}$ for all $1\leqslant i,j\leqslant N$ ;
•
$p_{ik}p_{jk}=\delta_{ij}p_{ik}$ and $p_{ki}p_{kj}=\delta_{ij}p_{kj}$ for all $1\leqslant i,j\leqslant N$ ;
endowed with the unique coproduct $\Delta:\mathcal{O}(S_{N}^{+})\to\mathcal{O}(S_{N}^{+})\otimes\mathcal{O}(S_{N}^{+})$ such that for all $1\leqslant i,j\leqslant N$,
$$\Delta(p_{ij})=\sum_{k=1}^{N}p_{ik}\otimes p_{kj}$$
and structure maps given by $\varepsilon(p_{ij})=\delta_{ij}$ and $S(p_{ij})=p_{ji}$.
We will follow the same path, and the novelty of the free wreath product with amalgamation construction can indeed be seen at this level simply as an additional amalgamation introduced in the free product. More precisely, we will consider the following $*$-algebra :
Definition 3.2.
Let $\mathbb{G}$ be a compact quantum group and let $\mathbb{H}$ be a dual quantum subgroup of $\mathbb{G}$. Let
$$A=\mathcal{O}(\mathbb{G})^{\ast_{\mathcal{O}(\mathbb{H})}^{N}}$$
be the iterated amalgamated free product and denote by $\nu_{i}:\mathcal{O}(\mathbb{G})\to A$ the canonical inclusion into the $i$-th factor. Then, the $*$-algebra $\mathcal{O}(\mathbb{G}\wr_{\ast,\mathbb{H}}S_{N}^{+})$ is defined as the quotient of $A$ by the relations
$$[\nu_{i}(x),p_{ij}]=0$$
for all $1\leqslant i,j\leqslant N$ and all $x\in\mathcal{O}(\mathbb{G})$.
Note that the CQG-algebra of the usual free wreath product $\mathbb{G}\wr_{\ast}S_{N}^{+}$ can be recovered as the special case where $\mathbb{H}=\{e\}$ is the trivial group (i.e. the inclusion $\mathbb{C}.1_{\mathcal{O}(\mathbb{G})}\subset\mathcal{O}(\mathbb{G})$). Moreover, we can in general see $\mathcal{O}(\mathbb{G}\wr_{\ast,\mathbb{H}}S_{N}^{+})$ as the quotient of $\mathcal{O}(\mathbb{G}\wr_{\ast}S_{N}^{+})$ by the ideal generated by $\nu_{i}(x)-\nu_{j}(x)$ for all $1\leqslant i\neq j\leqslant N$ and $x\in\mathcal{O}(\mathbb{H})$.
The main technical point in the definition of free wreath products in [3] is the coproduct, whose formula is not completely obvious. It is proven in [3, Thm 2.3] that, denoting by $\Delta_{\mathbb{G}}$ the coproduct on $\mathbb{G}$, there is a unique coproduct $\Delta$ on $\mathcal{O}(\mathbb{G}\wr_{\ast}S_{N}^{+})$ such that
$$\Delta(p_{ij})=\sum_{k=1}^{N}p_{ik}\otimes p_{kj}\quad\&\quad\Delta(\nu_{i}(x))=\sum_{k=1}^{N}(\nu_{i}\otimes\nu_{k})(\Delta_{\mathbb{G}}(x))(p_{ik}\otimes 1)$$
for all $1\leqslant i,j\leqslant N$ and all $x\in\mathcal{O}(\mathbb{G})$. The most natural thing to do is therefore to check that these formulæ again define a coproduct on $\mathcal{O}(\mathbb{G}\wr_{\ast,\mathbb{H}}S_{N}^{+})$.
Proposition 3.3.
Let $\mathbb{G}$ be a compact quantum group and let $\mathbb{H}$ be a dual quantum subgroup of $\mathbb{G}$. Then, the previous coproduct on $\mathcal{O}(\mathbb{G}\wr_{\ast}S_{N}^{+})$ factors through the quotient onto $\mathcal{O}(\mathbb{G}\wr_{\ast,\mathbb{H}}S_{N}^{+})$, endowing the latter with a compact quantum group structure.
Proof.
All that we have to do is to check that the $*$-ideal generated by the elements $\nu_{i}(x)-\nu_{j}(x)$ for all $x\in\mathcal{O}(\mathbb{H})$ is a Hopf ideal. Indeed, using Sweedler’s notation to write $\Delta_{\mathbb{G}}(x)=x_{1}\otimes x_{2}\in\mathcal{O}(\mathbb{H})\otimes\mathcal{O}(\mathbb{H})$,
$$\displaystyle\Delta(\nu_{i}(x)-\nu_{j}(x))$$
$$\displaystyle=\sum_{k=1}^{N}(\nu_{i}\otimes\nu_{k})(x_{1}\otimes x_{2})(p_{ik}\otimes 1)-(\nu_{j}\otimes\nu_{k})(x_{1}\otimes x_{2})(p_{jk}\otimes 1)$$
$$\displaystyle=\sum_{k,k^{\prime}=1}^{N}\nu_{i}(x_{1})p_{ik}\otimes(\nu_{k}-\nu_{k^{\prime}})(x_{2})+\sum_{k,k^{\prime}=1}^{N}\nu_{i}(x_{1})p_{ik}\otimes\nu_{k^{\prime}}(x_{2})$$
$$\displaystyle-\sum_{k,k^{\prime}=1}^{N}\nu_{j}(x_{1})p_{jk}\otimes(\nu_{k}-\nu_{k^{\prime}})(x_{2})-\sum_{k,k^{\prime}=1}^{N}\nu_{j}(x_{1})p_{jk}\otimes\nu_{k^{\prime}}(x_{2})$$
$$\displaystyle=\sum_{k,k^{\prime}=1}^{N}\nu_{i}(x_{1})p_{ik}\otimes(\nu_{k}-\nu_{k^{\prime}})(x_{2})+\sum_{k^{\prime}=1}^{N}\nu_{i}(x_{1})\otimes\nu_{k^{\prime}}(x_{2})$$
$$\displaystyle-\sum_{k,k^{\prime}=1}^{N}\nu_{j}(x_{1})p_{jk}\otimes(\nu_{k}-\nu_{k^{\prime}})(x_{2})-\sum_{k^{\prime}=1}^{N}\nu_{j}(x_{1})\otimes\nu_{k^{\prime}}(x_{2})$$
$$\displaystyle=\sum_{k,k^{\prime}=1}^{N}\nu_{i}(x_{1})p_{ik}\otimes(\nu_{k}-\nu_{k^{\prime}})(x_{2})-\sum_{k,k^{\prime}=1}^{N}\nu_{j}(x_{1})p_{jk}\otimes(\nu_{k}-\nu_{k^{\prime}})(a_{2})$$
$$\displaystyle+\sum_{k^{\prime}=1}^{N}(\nu_{i}(x_{1})-\nu_{j}(x_{1})\otimes\nu_{k^{\prime}}(x_{2})$$
proving our claim.
The fact that this yields a compact quantum group is then proven exactly as in the proof of [3, Thm 2.3].
∎
We are now ready for the definition of the main object of this work.
Definition 3.4.
The compact quantum group associated to the pair $(\mathcal{O}(\mathbb{G}\wr_{\ast,\mathbb{H}}S_{N}^{+}),\Delta)$ is called the free wreath product of $\mathbb{G}$ by $S_{N}^{+}$ with amalgamation over $\mathbb{H}$ and is denoted by $\mathbb{G}\wr_{\ast,\mathbb{H}}S_{N}^{+}$.
In order to illustrate the construction, let us show that it interpolates in a sense between the direct product and the free wreath product.
Proposition 3.5.
Let $\mathbb{G}$ be a compact quantum group and let $\mathbb{H}$ be a dual quantum subgroup. Then, for all $N\in\mathbf{N}$ there are inclusions
$$\mathbb{G}\times S_{N}^{+}<\mathbb{G}\wr_{\ast,\mathbb{H}}S_{N}^{+}<\mathbb{G}\wr_{\ast}S_{N}^{+}.$$
Proof.
Note first that if $\mathbb{K}$ is a dual quantum subgroup of $\mathbb{H}$, then there is a canonical surjection
$$\mathcal{O}(\mathbb{G})^{\ast_{\mathcal{O}(\mathbb{K})}^{N}}\to\mathcal{O}(\mathbb{G})^{\ast_{\mathcal{O}(\mathbb{H})}^{N}}$$
since $\{\nu_{i}(x)-\nu_{j}(x)\mid x\in\mathcal{O}(\mathbb{K})\}\subset\{\nu_{i}(x)-\nu_{j}(x)\mid x\in\mathcal{O}(\mathbb{H})\}$. Further quotienting by the commutation relations of Definition 3.2, yields a surjective $*$-homomorphisms
$$\mathcal{O}(\mathbb{G}\wr_{\ast,\mathbb{K}}S_{N}^{+})\to\mathcal{O}(\mathbb{G}\wr_{\ast,\mathbb{H}}S_{N}^{+})$$
and because the definition of the coproduct is given by the same formulæ of both sides, this yields an inclusion of compact quantum groups $\mathbb{G}\wr_{\ast,\mathbb{H}}S_{N}^{+}<\mathbb{G}\wr_{\ast,\mathbb{K}}S_{N}^{+}$. Applying this to $\mathbb{K}=\mathbb{G}$ and $\mathbb{K}=\{e\}$ yields
$$\mathbb{G}\wr_{\ast,\mathbb{G}}S_{N}^{+}<\mathbb{G}\wr_{\ast,\mathbb{H}}S_{N}^{+}<\mathbb{G}\wr_{\ast,\{e\}}S_{N}^{+}=\mathbb{G}\wr_{\ast}S_{N}^{+}.$$
To conclude, is suffices to check that $\mathbb{G}\wr_{\ast,\mathbb{G}}S_{N}^{+}=\mathbb{G}\times S_{N}^{+}$. Indeed, $\mathcal{O}(\mathbb{G})^{\ast_{\mathcal{O}(\mathbb{G})}^{N}}=\mathcal{O}(\mathbb{G})$ and the relations make that unique copy of $\mathcal{O}(\mathbb{G})$ commute with all the generators of $\mathcal{O}(S_{N}^{+})$.
∎
3.2. The group dual case
We will now restrict to the case where $\mathbb{G}$ is the dual of a discrete group $\Gamma$. Then, $\mathbb{H}$ is nothing but the dual of a subgroup $\Lambda$ of $\Gamma$. In that case, there is an alternative description of the free wreath product using explicit generators. Let us first recall how this works for free wreath products. Consider the universal $*$-algebra $\mathcal{O}(H_{N}^{+}(\Gamma))$ generated by elements $a_{ij}(g)$ for all $1\leqslant i,j\leqslant N$ and $g\in\Gamma$ subject to the relations.
$$\displaystyle a_{ij}(g)a_{ik}(h)$$
$$\displaystyle=\delta_{ik}a_{ij}(gh)$$
$$\displaystyle a_{ji}(g)a_{ki}(h)$$
$$\displaystyle=\delta_{jk}a_{ji}(gh)$$
$$\displaystyle\sum_{k=1}^{N}a_{ik}(e)$$
$$\displaystyle=1=\sum_{k=1}^{N}a_{kj}(e)$$
Using the universal property, it is easy to see that there exists a coproduct $\Delta$ satisfying
$$\Delta(a_{ij}(g))=\sum_{k=1}^{N}a_{ik}(g)\otimes a_{kj}(g)$$
and yielding a compact quantum group structure on $\mathcal{O}(H_{N}^{+}(\Gamma))$. Moreover, it is proven in [3, Ex 2.5] that the corresponding quantum group $H_{N}^{+}(\Gamma)$ is isomorphic to the free wreath product $\widehat{\Gamma}\wr_{\ast}S_{N}^{+}$.
We will now give an analogue of this result for pairs of discrete groups. Here is how the definition has to be modified.
Definition 3.6.
Let $\Gamma$ be a discrete group and let $\Lambda$ be a subgroup of $\Gamma$. The universal $*$-algebra $\mathcal{O}(H_{N}^{+}(\Gamma,\Lambda))$ is defined as the quotient of $\mathcal{O}(H_{N}^{+}(\Gamma))$ by the Hopf $*$-ideal generated by the relations
$$\sum_{k=1}^{N}a(g)_{ik}=\sum_{k=1}^{N}a(g)_{jk}$$
for all $1\leqslant i,j\leqslant N$ and all $g\in\Lambda$.
Remark 3.7.
Note that setting $s(g)$ to be the sum appearing in the definition (if it is independent of $i$), the set of elements $g$ such that $s(g)$ is well-defined must be a group : $s(e)=1$, $s(g^{-1})=S(s(g))$ (where $S$ denotes the antipode) and $s(gh)=s(g)s(h)$. Indeed,
$$\displaystyle s(g)\sum_{k=1}^{N}a_{ik}(h)$$
$$\displaystyle=\sum_{k,k^{\prime}=1}^{N}a_{ik^{\prime}}(g)a_{ik}(h)$$
$$\displaystyle=\sum_{k,k^{\prime}=1}^{N}\delta_{k,k^{\prime}}a_{ik}(gh)$$
$$\displaystyle=\sum_{k=1}^{N}a_{ik}(gh)$$
so that the sum in the last line does not depend on $i$.
It is not difficult to check that the coproduct on $\mathcal{O}(H_{N}^{+}(\Gamma))$ again defines a coproduct on the quotient and that this yields a compact quantum group. This will however be a consequence of the next result.
Proposition 3.8.
There exists a $*$-isomorphism
$$\Phi:\mathcal{O}(\widehat{\Gamma}\wr_{\ast,\widehat{\Lambda}}S_{N}^{+})\to\mathcal{O}(H_{N}^{+}(\Gamma,\Lambda))$$
such that push-forward of the coproduct maps $a_{ij}(g)$ to
$$\sum_{k=1}^{N}a_{ik}(g)\otimes a_{kj}(g).$$
Proof.
As in [3, Ex 2.5], set
$$\phi_{i}(g)=\sum_{k=1}^{N}a_{ik}(g)\quad\&\quad\phi_{N+1}(p_{ij})=a_{ij}(e)$$
to define by universality a $*$-homomorphism
$$\mathbb{C}[\Gamma]^{\ast N}\ast S_{N}^{+}\to\mathcal{O}(H_{N}^{+}(\Gamma,\Lambda)).$$
Observing that $\phi_{i}(g)=\phi_{j}(g)$ for all $1\leqslant i,j\leqslant N$ and all $g\in\Lambda$, we see that this map factors to a $*$-homomorphism
$$\phi:\mathcal{O}(\widehat{\Gamma}\wr_{\ast,\widehat{\Lambda}}S_{N}^{+})\to\mathcal{O}(H_{N}^{+}(\Gamma,\Lambda)).$$
Conversely, one easily checks that the elements
$$\psi_{ij}(g)=\nu_{i}(g)p_{ij}\in\mathcal{O}(\widehat{\Gamma}\wr_{\ast,\widehat{\Lambda}}S_{N}^{+}))$$
satisfy the defining relations of $\mathcal{O}(H_{N}^{+}(\Gamma))$. Moreover, for $g\in\Lambda$ and $1\leqslant i,j\leqslant N$,
$$\sum_{k=1}^{N}\psi_{ik}(g)=\nu_{i}(g)=\nu_{j}(g)=\sum_{k=1}^{N}\psi_{jk}(g)$$
so that these elements even satisfy the defining relations of $\mathcal{O}(H_{N}^{+}(\Gamma,\Lambda))$. By universality, this induces a $*$-homomorphism
$$\psi:\mathcal{O}(H_{N}^{+}(\Gamma,\Lambda))\to\mathcal{O}(\widehat{\Gamma}\wr_{\ast,\widehat{\Lambda}}S_{N}^{+})$$
which a straightforward computation shows to be inverse to $\phi$. The formula for the coproduct is then obtained by conjugating by these $*$-homomorphisms.
∎
When $\Gamma$ (hence also $\Lambda$) is abelian, another description of the amalgamated free wreath product is possible, involving the notion of glued direct product from [20] in a specific case. Because $\Gamma$ is assumed to be abelian, $\Lambda$ is normal and we can consider the usual free wreath product $H_{N}^{+}(\Gamma/\Lambda)=H_{N}^{+}(\Gamma/\Lambda,\{e\})$ and build its direct product in the sense of compact quantum groups with $\widehat{\Gamma}$. At the level of CQG-algebras, this yields
$$B=\mathcal{O}(H_{N}^{+}(\Gamma/\Lambda))\otimes\mathcal{O}(\widehat{\Gamma}).$$
Now, we denote by $\mathcal{O}(H_{N}^{+}(\Gamma/\Lambda)\widetilde{\times}\widehat{\Gamma})$ the $*$-subalgebra of $B$ generated by the elements $a(g)_{ij}g$ for all $g\in\Gamma$ and $1\leqslant i,j\leqslant N$. This is a CQG-algebra for the restriction of the coproduct, and the corresponding compact quantum group is called the glued direct product of $H_{N}^{+}(\Gamma/\Lambda)$ and $\widehat{\Gamma}$.
Remark 3.9.
The previous construction is slightly outside the setting of [20]. However, it can easily be made to fit in if $\Gamma$ is finitely generated. In that case, if $\SS$ is a finite generating set, then one can consider $H_{N}^{+}(\Gamma/\Lambda)$ and $\widehat{\Gamma}$ as compact matrix quantum groups with fundamental representations respectively $\oplus_{g\in\SS}a(g)$ and $\oplus_{g\in\SS}g$, and the glued product construction then coincides with ours.
The proof of the next theorem will make use of an explicit description of the representation theory of $H_{N}^{+}(\Gamma,\Lambda)$ obtained in [17, Thm 4.4]. Strictly speaking, this result concerns the partition quantum groups $\mathbb{G}_{N}(\mathcal{C}_{\Gamma,\Lambda,S})$, but we will prove that it coincides with $H_{N}^{+}(\Gamma,\Lambda)$ in Proposition 4.2. Let $F(\Gamma)$ be the free monoid on $\Gamma$ and consider the relation $\sim$ defined on it by
$$w_{1}\dots(w_{i}.\lambda)w_{i+1}\dots w_{n}\sim w_{1}\dots w_{i}(\lambda.w_{i+1})\dots w_{n}$$
for any $1\leqslant i\leqslant n-1$ and $\lambda\in\Lambda$. The quotient by the transitive closure of the relation will be denoted by $W(\Gamma,\Lambda)$. Then, the one-dimensional representations of $H_{N}^{+}(\Gamma,\Lambda)$ can be indexed by the elements of $\Lambda$ and all the other ones by elements of $W(\Gamma,\Lambda)$. Moreover, if $g_{1}\cdots g_{n}\in W(\Gamma,\Lambda)$ and $\lambda\in\Lambda$, then $\lambda\otimes(g_{1}\cdots g_{n})=(\lambda g_{1})g_{2}\cdots g_{n}$. We are now ready for the result.
Theorem 3.10.
If $\Gamma$ is abelian, then there is an isomorphism of compact quantum groups
$$H_{N}^{+}(\Gamma,\Lambda)\cong H_{N}^{+}(\Gamma/\Lambda)\widetilde{\times}\widehat{\Gamma}.$$
Proof.
Let us set, for $g\in\Gamma$, $b(g)=a([g])g\in\mathcal{O}(H_{N}^{+}(\Gamma/\Lambda)\widetilde{\times}\widehat{\Gamma})$, where $[g]$ denotes the image of $g$ in $\Gamma/\Lambda$. Then, for any $1\leqslant i,j\leqslant N$,
$$\displaystyle b(g)_{ij}b(h)_{ik}$$
$$\displaystyle=a([g])_{ij}ga([h])_{ik}h$$
$$\displaystyle=a([g])_{ij}a([h])_{ik}gh$$
$$\displaystyle=\delta_{jk}a([gh])_{ij}gh$$
$$\displaystyle=\delta_{jk}b(gh)_{ij}$$
and similarly $b(g)_{ji}b(g)_{ki}=\delta_{jk}b(gh)_{ji}$. Moreover, for $g\in\Lambda$ and $1\leqslant i\leqslant N$,
$$\sum_{k=1}^{N}b(g)_{ik}=\left(\sum_{k=1}^{N}a([g])_{ik}\right)g=\left(\sum_{k=1}^{N}a([e])_{ik}\right)g=g.$$
As a consequence, there is a surjective $*$-homomorphism
$$\Phi:\mathcal{O}(H_{N}^{+}(\Gamma,\Lambda))\to\mathcal{O}(H_{N}^{+}(\Gamma/\Lambda)\widetilde{\times}\widehat{\Gamma})$$
sending $a(g)$ to $b(g)$.
To prove injectivity, it is enough to prove that the image of any irreducible representation under $\Phi$ is irreducible. Let us first prove by induction on $n$ that if $g_{1},\cdots,g_{n}$ are elements of $\Gamma$ and $\lambda_{1},\cdots,\lambda_{n}$ are elements of $\Lambda$, then
$$\Phi(u^{g_{1}}\lambda_{1}\cdots u^{g_{n}}\lambda_{n})=\lambda_{1}\cdots\lambda_{n}u^{[g_{1}]}\cdots u^{[g_{n}]}g_{1}\cdots g_{n}.$$
For $n=1$, the result is clear. Assume that it holds for some $n$. If $g_{n}g_{n+1}\notin\Lambda$, then the fusion rules computed in [17, Prop 4.6] imply that
$$\displaystyle\Phi(u^{g_{1}}\lambda_{1}\cdots u^{g_{n}}\lambda_{n}u^{g_{n+1}}\lambda_{n+1})$$
$$\displaystyle=\Phi(\lambda_{1}\cdots\lambda_{n+1}u^{g_{1}}\cdots u^{g_{n}}\otimes u^{g_{n+1}})$$
$$\displaystyle=\lambda_{1}\cdots\lambda_{n+1}u^{[g_{1}]}\cdots u^{[g_{n}]}\otimes u^{[g_{n+1}]}g_{1}\cdots g_{n+1}$$
$$\displaystyle=\lambda_{1}\cdots\lambda_{n+1}u^{[g_{1}]}\cdots u^{[g_{n}]}u^{[g_{n+1}]}g_{1}\cdots g_{n+1}$$
Otherwise,
$$\displaystyle\Phi(u^{g_{1}}\lambda_{1}\cdots u^{g_{n}}\lambda_{n}g_{n+1}\lambda_{n+1})$$
$$\displaystyle=\Phi\left((\lambda_{1}\cdots\lambda_{n+1}\left(u^{g_{1}}\cdots u^{g_{n}}\otimes u^{g_{n+1}}-u^{g_{1}}\cdots u^{g_{n}g_{n+1}}\right)\right))$$
$$\displaystyle=\lambda_{1}\cdots\lambda_{n+1}\left(u^{[g_{1}]}\cdots u^{[g_{n}]}\otimes u^{[g_{n+1}]}-u^{[g_{1}]}\cdots u^{[g_{n}g_{n+1}]}\right)g_{1}\cdots g_{n+1}$$
$$\displaystyle=\lambda_{1}\cdots\lambda_{n+1}u^{[g_{1}]}\cdots u^{[g_{n}]}u^{[g_{n+1}]}g_{1}\cdots g_{n+1}$$
so that our claim is proven.
To conclude we have to check that the images are irreducible representations. Note that the irreducible representations of $H_{N}^{+}(\Gamma/\Lambda)\times\widehat{\Gamma}$ are, by [33, Thm 2.11], of the form $gu^{[g_{1}]}\cdots u^{[g_{n}]}$ for $g,g_{1},\cdots,g_{n}\in\Gamma$. Moreover, $\mathcal{O}(H_{N}^{+}(\Gamma/\Lambda)\widetilde{\times}\widehat{\Gamma})$ being a Hopf $*$-subalgebra of $\mathcal{O}(H_{N}^{+}(\Gamma/\Lambda)\times\widehat{\Gamma})$, its irreducible representations are exactly those of $H_{N}^{+}(\Gamma/\Lambda)\times\widehat{\Gamma}$ whose coefficients all lie in $\mathcal{O}(H_{N}^{+}(\Gamma/\Lambda)\widetilde{\times}\widehat{\Gamma})$. In other words, the images under $\Phi$ of irreducible representations are irreducible, hence $\Phi$ is an isomorphism.
∎
Applying Theorem 3.10 to cyclic groups, we deduce that if $d\mid k$, then
$$\left(\mathbf{Z}_{d}\wr_{\ast}S_{N}^{+}\right)\widetilde{\times}\mathbf{Z}_{k}\cong\mathbf{Z}_{k}\wr_{\ast,d\mathbf{Z}_{k}}S_{N}^{+},$$
thereby recovering a result by P. Tarrago and M. Weber [30].
Remark 3.11.
In the case of cyclic groups, the glued product was introduced earlier in [30] under the name of $k$-tensor complexification. These complexification were later studied by D. Gromada and M. Weber in [21] were they consider several variants of them and consider them as extensions of a given compact quantum group by $\mathbf{Z}_{2}$. The term “extension” was used there in a broad sense, but Proposition 5.2 will show that in our case, the glued direct product yields a genuine extension in the sense of Hopf algebras. It is nevertheless important to note that the extension does not involve $\mathbf{Z}_{k}$ itself but rather the subgroup $d\mathbf{Z}_{k}$.
3.3. The classical construction
As is usual with such constructions in compact quantum group theory, the free wreath product with amalgamation has a classical counterpart obtained by considering the abelianization of the CQG-algebra. To describe this in more details, first note that if $G$ is a classical compact group, then a dual quantum subgroup is simply given by a quotient compact group $H$.
Definition 3.12.
Let $G$ be a compact group and $\pi:G\to H$ a quotient map. The wreath product of $G$ by $S_{N}$ with amalgamation over $H$ is defined to be the classical compact group corresponding to the CQG-algebra $\mathcal{O}(G\wr_{\ast,H}S_{N}^{+})_{\text{ab}}$, where the index “ab” denotes abelianization.
In order to give alternate descriptions of this object, let us recall some elementary facts. First, the abelianization of the amalgamated free product $\mathcal{O}(G)^{\ast_{\mathcal{O}(H)}^{N}}$ is the amalgamated direct product
$$\mathcal{O}(G)^{\times_{\mathcal{O}(H)}^{N}}=\mathcal{O}(G^{\times_{H}^{N}}).$$
Second, the amalgamated direct product $G^{\times_{H}^{N}}$ is the subgroup of $G^{\times N}$ of elements whose image in $H^{\times N}$ is diagonal. Third, the set of “diagonal” elements in $H\wr S_{N}$ is a subgroup isomorphic to $H\times S_{N}$.
Proposition 3.13.
Let $G$ be a compact group and let $\pi:G\to H$ be a quotient map. The following compact groups are isomorphic to the wreath product of $G$ by $S_{N}$ with amalgamation over $H$ :
(1)
$G^{\times^{N}_{H}}\rtimes S_{N}$, where $S_{N}$ acts by permutation of the copies ;
(2)
The pre-image of $H\times S_{N}\subset H\wr S_{N}$ under the surjective group homomorphism $G\wr S_{N}\to H\wr S_{N}$ induced by $\pi$.
Proof.
(1)
The abelianization of $\mathcal{O}(G\wr_{\ast}S_{N}^{+})$ is $\mathcal{O}(G\wr S_{N})$, which is the CQG-algebra corresponding to the group $G^{\times N}\rtimes S_{N}$ with $S_{N}$ acting by permutation of the copies. The result therefore follows from the fact that $\mathcal{O}(G\wr_{\ast,H}S_{N}^{+})_{\text{ab}}$ is the quotient of $\mathcal{O}(G\wr S_{N})$ by the relations identifying the copies of $C(H)\subset C(G)$.
(2)
Recall that $\mathcal{O}(H)$ is seen as a subalgebra of $\mathcal{O}(G)$ through the map $f\mapsto f\circ\pi$. As a consequence, the amalgamated tensor product of $\mathcal{O}(G)$ over $\mathcal{O}(H)$ consists in quotienting $\mathcal{O}(G)^{\otimes N}$ by the relations $\pi(x_{i})=\pi(x_{j})$ for all $x_{1}\otimes\cdots\otimes x_{N}\in\mathcal{O}(G)^{\otimes N}$ and all $1\leqslant i,j\leqslant N$. At the level of groups, this is equivalent to restricting to the subgroup of tuples of elements of $G$ having the same image in $H$, hence the result.
∎
Using this result, we can give yet another characterization in the cyclic case. Recall that a matrix is called monomial if it has exactly one non-zero coefficient in each row and column. The reflection group $H_{N}^{k}=H_{N}(\mathbf{Z}_{k})$ can then be described as the group of all monomial matrices in $M_{N}(\mathbb{C})$ whose coefficients are $k$-th roots of unity.
Proposition 3.14.
Let $d,k\in\mathbf{N}$ with $d\mid k$. Then, the group $\mathbf{Z}_{k}\wr_{\mathbf{Z}_{d}}S_{N}$ is the group of all monomial matrices in $H_{N}^{k}$ such that all non-zero coefficients have equal $k/d$-th power.
Proof.
Set $\omega=e^{2i\pi/k}$ so that $\mathbf{Z}_{d}$ is seen as the subgroup generated by $\omega^{k/d}$. Let now $M\in H_{N}^{k}$ and let $(z_{1},\cdots,z_{N})$ be its non-zero coefficients. Assume that $z_{i}^{d}=z_{1}^{d}$ for all $2\leqslant i\leqslant N$. Then, there exists $j_{1},\cdots j_{N}$ such that $z_{i}=\omega^{kj_{i}/d}z_{1}$. In other words, $M$ is the product of a monomial matrix with coefficients $(\omega^{kj_{1}/d},\cdots,\omega^{kj_{N}/d})$ and the diagonal matrix with constant coefficient $z_{1}$. This is an element of $H_{N}(\mathbf{Z}_{k/d})\widetilde{\times}\mathbf{Z}_{k}$, which by Theorem 3.10 is isomorphic to $\mathbf{Z}_{k}\wr_{\mathbf{Z}_{d}}S_{N}$. We have therefore proven one inclusion, and the converse one is trivial.
∎
4. Coloured partitions and applications
We will now investigate the interplay between free wreath products with amalgamation for duals of discrete groups and coloured partitions in the sense of [16]. This will in particular clarify the connection between this work and the original definition from [18].
4.1. Free wreath products with amalgamation as partition quantum groups
As explained in the introduction, free wreath products with amalgamation originally appeared in [18], under the name of free wreath products of pairs, as a specific family of partition quantum groups. They were therefore defined through a category of partitions, which is denoted by $\mathcal{C}_{\Gamma,\Lambda,S}$. Let us recall that definition.
Definition 4.1.
Let $\Gamma$ be a discrete group with a symmetric generating set $\SS$ not containing the neutral element and let $\Lambda$ be a subgroup. We consider $\SS$ as a colour set with involution $g\to g^{-1}$. Then, $\mathcal{C}_{\Gamma,\SS}$ is the category of all non-crossing $\SS$-coloured partitions such that in each block, the product of the colours in the upper row (from left to right) equals the product of the colours in the lower row (also from left to right) as elements of $\Gamma$.
If now $g\in\Lambda$ can be written as a product $g=g_{1}\cdots g_{n}$ of elements of $\SS$, we set
$\dots$$\dots$$g_{1}$$g_{2}$$g_{n-1}$$g_{n}$$g_{1}$$g_{2}$$g_{n-1}$$g_{n}$$\beta_{g}=$
and let $\mathcal{C}_{\Gamma,\Lambda,\SS}$ be the category of partitions generated by $\mathcal{C}_{\Gamma,\SS}$ and $\beta_{g}$ for all $g\in\lambda$.
We will now prove, using the universal algebra picture, that this defines the same object as the two previous constructions. The idea is simply that adding a partition to a category of partitions is equivalent to forcing a linear a map to intertwine two representations, hence amounts to adding a polynomial relation between the generators (see for instance [18] for details and illustrations of this principle).
Proposition 4.2.
There is an isomorphism of compact quantum groups
$$\mathbb{G}_{N}(\mathcal{C}_{\Gamma,\Lambda,\SS})\cong H_{N}^{+}(\Gamma,\Lambda).$$
Proof.
We already know from [23] that $\mathbb{G}_{N}(\mathcal{C}_{\Gamma,\SS})\cong H_{N}^{+}(\Gamma)$ and that this isomorphism sends the representation corresponding to the $g$-identity partition to the representation $a(g)=(a(g)_{ij})_{1\leqslant i,j\leqslant N}$. Now, adding the partition $\beta_{g}$ is equivalent to adding the relations
$$T_{\beta_{g}}a(g)=a(g)T_{\beta_{g}}.$$
But $T_{\beta_{g}}$ is, up to multiplication by a scalar, the orthogonal projection onto the vector $\xi=\sum e_{i}$. Thus, adding it to the category of partitions amounts to imposing conditions on $a(g)$ making $\xi$ fixed. Because
$$a(g)\xi=\sum_{i,j=1}^{N}a_{ij}(g)\otimes e_{i},$$
$\xi$ is fixed if and only if the sum of the coefficients of $a(g)$ on each row is constant, which is exactly the relation defining $H_{N}^{+}(\Gamma,\Lambda)$.
∎
4.2. Topological generation and monoidal equivalence
We will now give two applications of the previous result to structural and approximation properties. Recall that given a compact quantum group $\mathbb{G}$ and a family of compact quantum subgroups $\mathcal{O}(\mathbb{H}_{i})$ for $i$ ranging in some set $I$, we say that $\mathbb{G}$ is topologically generated by these quantum subgroups if for any representations $u$ and $v$ of $\mathbb{G}$,
$$\operatorname{Mor}_{\mathbb{G}}(u,v)=\bigcap_{i\in I}\operatorname{Mor}_{\mathbb{H}_{i}}(u,v).$$
That property was first introduced by A. Chirvasitu in [10] to study residual finite-dimensionality of quantum group C*-algebras and we will exploit it here for the same purpose. Let us say that a compact quantum group $\mathbb{G}$ is residually finite if its CQG-algebra $\mathcal{O}(\mathbb{G})$ is residually finite-dimensional as a complex $*$-algebra, i.e. its finite-dimensional $*$-algebra representations separate the points.
Let $\Lambda<\Gamma$ be a pair of discrete groups. We have several natural quantum subgroups of $\widehat{\Gamma}\wr_{\ast,\widehat{\Lambda}}S_{N}^{+}$, and in particular we have $\widehat{\Gamma}\times S_{N}^{+}=\widehat{\Gamma}\wr_{\ast,\widehat{\Gamma}}S_{N}^{+}$ already mentioned in Proposition 3.5. Moreover, if $\overline{\Lambda}$ denotes the normal closure of $\Lambda$ (that is, the intersection of all normal subgroups of $\Gamma$ containing $\Lambda$), then $\widehat{\Gamma/\overline{\Lambda}}\wr_{\ast}S_{N}^{+}$ is a quantum subgroup of $\widehat{\Gamma}\wr_{\ast,\widehat{\Lambda}}S_{N}^{+}$. This in fact follows from the following result :
Lemma 4.3.
Let $\beta_{g}^{u}$ denote the upper part of the partition $\beta_{g}$ and set
$$\mathcal{C}=\left\langle\mathcal{C}_{\Gamma,\Lambda,\SS},\beta_{g}^{u}\mid g\in\Lambda\right\rangle.$$
Then,
$$\mathbb{G}_{N}(\mathcal{C})=\widehat{\Gamma/\overline{\Lambda}}\wr_{\ast}S_{N}^{+}$$
Proof.
Because $\mathcal{C}_{\Gamma,\Lambda,\SS}\subset\mathcal{C}$, we know by the proof of [18, Prop 3.17] that
$$\mathcal{C}=\mathcal{C}_{\Gamma/\Theta,\Lambda/(\Lambda\cap\Theta),\SS},$$
where $\Theta$ is the subgroup consisting in all elements $g$ such that $\beta_{g}^{u}\in\mathcal{C}$ (note that there is a slight mistake in the reference, since in general $\Theta$ is not contained in $\Lambda$, hence one has to consider $\Lambda/(\Lambda\cap\Theta)$ instead of $\Theta$), which turns out to be normal. By assumption, $\Lambda\subset\Theta$ so that by normality $\overline{\Lambda}\subset\Theta$. In particular, the quotient map $\mathcal{O}(\mathbb{G}_{N}(\mathcal{C}_{\Gamma,\Lambda,\SS}))\to\mathcal{O}(\mathbb{G}_{N}(\mathcal{C}))$ factors through $\mathcal{O}(\mathbb{G}_{N}(\mathcal{C}_{\Gamma/\overline{\Lambda},\SS}))$.
Moreover, all one-dimensional representations of $\mathbb{G}_{N}(\mathcal{C}_{\Gamma,\Lambda,\SS})$ are sent to $1$ in $\mathcal{O}(\mathbb{G}_{N}(\mathcal{C}_{\Gamma/\overline{\Lambda},\SS}))$ and the linear map corresponding to $\beta_{g}^{u}$ is an intertwiner if and only if the one-dimensional representation corresponding to $g$ is trivial. Thus, the map $\mathcal{O}(\mathbb{G}_{N}(\mathcal{C}_{\Gamma,\Lambda,\SS}))\to\mathcal{O}(\mathbb{G}_{N}(\mathcal{C}_{\Gamma/\overline{\Lambda},\SS}))$ factors through $\mathcal{O}(\mathbb{G}_{N}(\mathcal{C}))$ and the proof is complete.
∎
The previous result shows that it is difficult to distinguish in the construction between a subgroup and its normal closure. We will therefore have to assume in the next statement that $\Lambda$ is normal to avoid that issue.
Theorem 4.4.
Assume that $\Lambda$ is normal. Then, for $N\geqslant 4$, $\widehat{\Gamma}\wr_{\ast,\widehat{\Lambda}}S_{N}^{+}$ is topologically generated by $\widehat{\Gamma}\times S_{N}^{+}$ and $\widehat{\Gamma/\Lambda}\wr_{\ast}S_{N}^{+}$.
Proof.
It is easy to see that it is enough to prove that for two words $w$ and $w^{\prime}$ on $\SS$,
$$\operatorname{Mor}_{\widehat{\Gamma}\wr_{\ast,\widehat{\Lambda}}S_{N}^{+}}(u^{\otimes w},u^{\otimes w^{\prime}})=\operatorname{Mor}_{\widehat{\Gamma}\times S_{N}^{+}}(u^{\otimes w},u^{\otimes w^{\prime}})\cap\operatorname{Mor}_{\widehat{\Gamma/\Lambda}\wr_{\ast}S_{N}^{+}}(u^{\otimes w},u^{\otimes w^{\prime}}).$$
So let $x$ be an element of the right-hand side. Then, $x$ can be written in two ways as a linear combination of operators $T_{p}$ associated to non-crossing partitions. Because $N\geqslant 4$, these operators are linearly independent (see for instance [19, Lem 4.16]), hence the partitions appearing in both linear combinations must be the same. In other words, $x$ is a linear combination of operators associated to partitions in $\mathcal{C}_{\Gamma,\Gamma,\SS}\cap\mathcal{C}_{\Gamma/\Lambda,\SS}$. We therefore have to prove that this intersection reduces to $\mathcal{C}_{\Gamma,\Lambda,\SS}$. It is clear that it contains it, since it contains all its generators. Moreover, if follows from this and the proof of [18, Prop 3.18] that
$$\mathcal{C}_{\Gamma,\Gamma,\SS}\cap\mathcal{C}_{\Gamma/\Lambda,\SS}=\mathcal{C}_{\Gamma/\Lambda_{0},\Lambda^{\prime}/(\Lambda^{\prime}\cap\Lambda_{0}),\SS}$$
for some normal subgroup $\Lambda_{0}\subset\Gamma$ and a subgroup $\Lambda^{\prime}$ containing $\Lambda$. Because the intersection is contained in $\mathcal{C}_{\Gamma,\Gamma,\SS}$, we have that $\Lambda_{0}$ is trivial. Moreover, for any $g\in\Lambda^{\prime}\setminus\Lambda$, $u^{g}$ is irreducible in $\widehat{\Gamma/\Lambda}\wr_{\ast}S_{N}^{+}$, hence it is also irreducible in $\widehat{\Gamma}\wr_{\ast,\widehat{\Lambda}^{\prime}}S_{N}^{+}$. But by [17, Thm 4.4] $u^{g}$ cannot be irreducible if $g\notin\Lambda^{\prime}$. Thus, $\Lambda^{\prime}=\Lambda$ and the proof is complete.
∎
Corollary 4.5.
Let $\Gamma$ be a residually finite group and let $\Lambda$ be a normal subgroup such that $\Gamma/\Lambda$ is residually finite. Then, $\widehat{\Gamma}\wr_{\ast,\widehat{\Lambda}}S_{N}^{+}$ is residually finite.
Proof.
It is clear that a direct product of residually finite compact quantum groups is residually finite, and both $\widehat{\Gamma}$ and $S_{N}^{+}$ are residually finite, the first one by assumption and the second one by [8, Thm 3.6]. Moreover, it was proven in [8, Thm 3.11 and Rem 3.21] that the free wreath product of a residually finite discrete group by $S_{N}^{+}$ is residually finite. Eventually, A. Chirvasitu proved in [10, Cor 2.12] that if a compact quantum group is topologically generated by two residually finite ones, then it is itself residually finite, hence the result.
∎
The second structure result that we will deduce from the partition description concerns the notion of monoidal equivalence. Two compact quantum groups are said to be monoidally equivalent (see [7]) if their respective categories of representations are equivalent as monoidal categories. In particular, they have the same representation theory and using the techniques of [14, Sec 6] one can use monoidal equivalence to transfer results about approximation properties.
In [24], F. Lemeux and P. Tarrago proved that the free wreath product $\widehat{\Gamma}\wr_{\ast}S_{N}^{+}$ is monoidally equivalent to another compact quantum group which is easier to study explicitely. More precisely, consider the $*$-subalgebra $H$ of $\mathcal{O}(\widehat{\Gamma}\ast SU_{q}(2))$ (we refer the reader for instance to [37] for the definition of quantum $SU(2)$ and to [32] for the construction of the free product of two compact quantum groups), where $q+q^{-1}=\sqrt{N}$, generated by the coefficients of the representations $u^{1}gu^{1}$ where $u^{1}$ is the fundamental representation of $SU_{q}(2)$ and $g\in\Gamma$ is seen as an irreducible representation of $\widehat{\Gamma}$. It is easily checked that $H$ is a Hopf $*$-subalgebra and if $\mathbb{H}_{q}(\Gamma)$ denotes the corresponding dual quantum subgroup of $\widehat{\Gamma}\ast SU_{q}(2)$, then $\widehat{\Gamma}\wr_{\ast}S_{N}^{+}$ is monoidally equivalent to $\mathbb{H}_{q}(\Gamma)$.
We will now prove a refinement of this in our setting, but this requires an extra definition. Consider the $*$-subalgebra $\mathcal{A}\subset\mathcal{O}(SU_{q}(2))$ generated by the coefficients of $u\otimes u$. It is a CQG-algebra and, paralleling the classical case, the corresponding compact quantum group is denoted by $SO_{q}(3)$.
Proposition 4.6.
Let $\mathcal{O}(\mathbb{G}_{q})$ be the quotient of $\mathcal{O}(\widehat{\Gamma}\ast SU_{q}(2))$ by the relations making $\mathcal{O}(\widehat{\Lambda})$ commute with $\mathcal{O}(SO_{q}(3))$ and let $\mathbb{H}_{q}(\Gamma,\Lambda)$ be the image of $\mathcal{O}(\mathbb{H}_{q}(\Gamma))$ in $\mathcal{O}(\mathbb{G}_{q})$. Then, there is a monoidal equivalence
$$\widehat{\Gamma}\wr_{\ast,\widehat{\Lambda}}S_{N}^{+}\cong\mathbb{H}_{q}(\Gamma,\Lambda).$$
Proof.
First note that the monoidal equivalence constructed in [24] is defined at the level of all partitions and then simply restricted to the category of partitions of $\widehat{\Gamma}\wr_{\ast}S_{N}^{+}$. All we have to do is therefore to look at the image of the extra generators of $\mathcal{C}_{\Gamma,\Lambda,\SS}$, that is to say the partitions $\beta_{\lambda}$. Let therefore $\lambda=g_{1}\cdots g_{n}\in\Lambda$ and consider the fattening of the partition $\beta_{\lambda}$ as defined in [24, Prop 5.2]. It has the following form :
$\cdots$$\bullet$$g_{1}$$\bullet$$\bullet$$g_{2}$$g_{n-1}$$\bullet$$\bullet$$g_{n}$$\bullet$$\cdots$$\bullet$$g_{1}$$\bullet$$\bullet$$g_{2}$$g_{n-1}$$\bullet$$\bullet$$g_{n}$$\bullet$$p_{\lambda}=$
where the $\bullet$ symbols stand for the fundamental representation $u^{1}$ of $SU_{q}(2)$. Let $b_{\lambda}$ be the partition obtained by taking the upper row and removing the outermost pair partition. That partition encodes the representation $g_{1}\otimes\varepsilon_{SU_{q}(2)}\otimes g_{2}\otimes\cdots\otimes g_{n-1}\otimes\varepsilon_{SU_{q}(2)}\otimes g_{n}=\lambda$ of the free product $\widehat{\Gamma}\ast SU_{q}(2)$. Moreover, adding the intertwiner corresponding to $p_{\lambda}$ to the representation category of $\widehat{\Gamma}\ast SU_{q}(2)$ is equivalent to adding the intertwiner corresponding to the rotated version
$\bullet$$\bullet$$\bullet$$\bullet$$b_{\lambda}\>\otimes$$\otimes\>b_{\lambda}^{*}$
Adding this is in turn equivalent to requiring $\lambda$ to commute with all coefficients of $u\otimes u$, where $u$ is the fundamental representation of $SU_{q}(2)$. Because the subalgebra generated by the coefficients of $u\otimes u$ is exactly $\mathcal{O}(SO_{q}(3))$, we conclude that the essential image of the representation category of $\widehat{\Gamma}\wr_{\ast,\widehat{\Lambda}}S_{N}^{+}$ under the fully faithful functor of [24, Thm 5.11] is exactly the representation category of $\mathbb{H}_{q}(\Gamma,\Lambda)$.
∎
Remark 4.7.
There is a more general notion of a free wreath product of a compact quantum group by a quantum group acting on a finite-dimensional C*-algebra given by P. Tarrago and J. Wahl in [29]. In particular, the image of $\widehat{\Gamma}\wr_{\ast}S_{N}^{+}$ under the previous monoidal equivalence is in fact $\widehat{\Gamma}\wr_{\ast}SO_{q}(3)$. For that reason, it would be interesting to try to generalize [29] and check whether $\mathbb{H}_{q}(\Gamma,\Lambda)=(\widehat{\Gamma},\widehat{\Lambda})\wr_{\ast}SO_{q}(3)$ in some sense.
As a corollary, we can establish some approximation properties for free wreath products of pairs, provided we are able to prove them on $\mathbb{H}_{q}(\Gamma,\Lambda)$. Here is an example of such an instance concerning the Haagerup property, the definition of which we recall (see for instance [11] for details on that notion).
Definition 4.8.
A compact quantum group $\mathbb{G}$ is said to have the central Haagerup property if there exists a net $(\varphi_{i})_{i\in I}$ of functions on $\operatorname{Irr}(\mathbb{G})$ such that
•
$\varphi_{i}(\alpha)\displaystyle\underset{i}{\longrightarrow}\dim(\alpha)$ for all $\alpha\in\operatorname{Irr}(\mathbb{G})$ ;
•
For any $\epsilon>0$, there exists a finite subset $F\subset\operatorname{Irr}(\mathbb{G})$ such that for any $\alpha\notin F$, $|\varphi_{i}(\alpha)|<\epsilon\dim(\alpha)$ ;
•
The map $\varphi_{i}$ extend to a state (that is, a positive linear functional such that $\varphi_{i}(1)=1$) on $\mathcal{O}(\mathbb{G})$ for all $i\in I$.
Corollary 4.9.
Assume that $\Gamma$ has the Haagerup property and that $\Lambda$ is finite. Then, $\widehat{\Gamma}\wr_{\ast,\widehat{\Lambda}}S_{N}^{+}$ has the central Haagerup property.
Proof.
First, by averaging over $\Lambda$ we may assume that there is a net of functions $\varphi_{i}:\Gamma\to\mathbb{C}$, for $i\in I$, witnessing the central Haagerup property for $\widehat{\Gamma}$ and such that for any $g\in\Gamma$ and $\lambda\in\Lambda$, $\varphi_{i}(\lambda g)=\varphi_{i}(g)$. Let now $(\psi_{j})_{j\in J}$ be the net of functions on $\operatorname{Irr}(SU_{q}(2))$ witnessing the central Haagerup property constructed in [12, Sec 3]. We define a map $\eta_{ij}$ on $\operatorname{Irr}(\widehat{\Gamma}\ast SU_{q}(2))$ by the formula
$$\eta_{ij}\left(\prod_{l=1}^{k}g_{l}u^{n_{l}}\right)=\left(\prod_{l=1}^{k}\varphi_{i}(g_{l})\right)\left(\prod_{l=1}^{k}\psi_{j}(u^{n_{l}})\right).$$
The extension of these maps to $\mathcal{O}(\widehat{\Gamma}\ast SU_{q}(2))$ is nothing but the conditional free product of the extensions of $\varphi_{i}$ and $\psi_{j}$ (see for instance [11, Thm 7.7]) and is therefore known to be a state. The net $(\eta_{ij})_{ij\in I\times J}$ therefore witnesses the central Haagerup property for the free product and moreover has the following property : if $\lambda\in\Lambda$, then
$$\eta_{ij}\left(g_{1}u^{n_{1}}\cdots g_{i-1}u^{n_{i-1}}(\lambda g_{i})u^{n_{i}}\cdots g_{n}u^{n_{k}}\right)=\eta_{ij}\left(g_{1}u^{n_{1}}\cdots(g_{i-1}\lambda)u^{n_{i-1}}g_{i}u^{n_{i}}\cdots g_{n}u^{n_{k}}\right)$$
This follows directly from the equivariance property of $\varphi_{i}$. As a consequence, $\eta_{ij}$ factors through the quotient to $\mathbb{H}_{q}(\Gamma,\Lambda)$ and witnesses the central Haagerup property there, hence the result by 4.6 and [14, Sec 6].
∎
The same method applies to another approximation property called weak amenability. We will not introduce it but simply state the result for the records (see [14] for details).
Corollary 4.10.
Assume that $\Lambda_{cb}(\Gamma)=1$ and that $\Lambda$ is finite. Then,
$$\Lambda_{cb}\left(\widehat{\Gamma}\wr_{\ast,\widehat{\Lambda}}S_{N}^{+}\right)=1.$$
Proof.
We can average the mutlipliers witnessing weak amenability of $\Gamma$ to make them $\Lambda$-invariant just as above. Then, the corresponding multipliers implementing weak amenability on $\widehat{\Gamma}\ast SU_{q}(2)$ through the construction of [15] have the same invariance property as the multipliers $\eta_{ij}$ in the proof of Corollary 4.9 and the result follows.
∎
5. Decomposition results
As we have seen in Corollary 4.5, when $\Lambda$ is normal properties of $H_{N}^{+}(\Gamma,\Lambda)$ can have strong connections to properties of $\Lambda$ and of the quotient $\Gamma/\Lambda$. We will now describe that connection from another point of view. Let us start with an alternative expression of the normality of $\Lambda$, namely the fact that it fits into a short exact sequence
$$1\longrightarrow\Lambda\longrightarrow\Gamma\longrightarrow\Gamma/\Lambda\longrightarrow 1.$$
It is quite natural to wonder whether this induces a corresponding decomposition of $H_{N}^{+}(\Gamma,\Lambda)$, and answering that question first requires recalling some facts about exact sequences of compact quantum groups.
Let $\mathbb{G}$ be a compact quantum group and let $\mathbb{H}$ be a dual quantum subgroup of $\mathbb{G}$, so that $\mathcal{O}(\mathbb{H})\subset\mathcal{O}(\mathbb{G})$ is a Hopf $*$-subalgebra. If we denote by $I$ the kernel of the counit of $\mathcal{O}(\mathbb{H})$, then $I\mathcal{O}(\mathbb{G})$ and $\mathcal{O}(\mathbb{G})I$ are both ideals in $\mathcal{O}(\mathbb{G})$. If they turn out to be equal, then it is a Hopf $*$-ideal and the short sequence
$$\mathbb{C}\longrightarrow\mathcal{O}(\mathbb{H})\longrightarrow\mathcal{O}(\mathbb{G})\longrightarrow\mathcal{O}(\mathbb{G})/\mathcal{O}(\mathbb{G})I\longrightarrow\mathbb{C}$$
is then said to be exact.
Remark 5.1.
Our definition is seemingly weaker than the usual one for exact sequences of Hopf algebras in [1]. We are here taking advantage of the fact that CQG-algebras are co-semisimple, hence faithfully flat over their Hopf subalgebras as proven in [9], so that the condition above implies that $\mathcal{O}(\mathbb{H})$ coincides with both the left and right coinvariants of the surjection (see [28, Thm 1] for a proof).
Back to our initial situation, it was proven in [17, Lem 4.3] that the group of one-dimensional representations of $\mathcal{O}(H_{N}^{+}(\Gamma,\Lambda))$ naturally identifies with $\Lambda$. This provides us with a canonical copy of $\mathbb{C}[\Lambda]=\mathcal{O}(\widehat{\Lambda})$ inside $\mathcal{O}(H_{N}^{+}(\Gamma,\Lambda))$. As expected, this is the correct object to consider.
Proposition 5.2.
Assume that $\Lambda$ is normal in $\Gamma$. Then, there is an exact sequence of Hopf $*$-algebras
$$\mathbb{C}\longrightarrow\mathcal{O}(\widehat{\Lambda})\longrightarrow\mathcal{O}(H_{N}^{+}(\Gamma,\Lambda))\longrightarrow\mathcal{O}(H_{N}^{+}(\Gamma/\Lambda))\longrightarrow\mathbb{C}.$$
Proof.
Any element of $\mathcal{O}(H_{N}^{+}(\Gamma,\Lambda))$ being a linear combination of coefficients of irreducible representations, it is enough to consider one such coefficient. The representation theory of $H_{N}^{+}(\Gamma,\Lambda)$ computed in [17, Thm 4.4] (it is also recalled just before Theorem 3.10) shows that any irreducible representation of dimension strictly greater than one is of the form
$$u^{g_{1}}\cdots u^{g_{n}}$$
where $u^{g_{i}}=a(g_{i})$ if $g_{i}\notin\Lambda$ and is the complement of the unique non-trivial one-dimensional subrepresentation of $a(g_{i})$ otherwise. As a consequence, it is enough to check that $u^{g}_{ij}I=Iu^{g}_{ij}$. Note moreover that $I$ is the linear span of the elements $\lambda_{1}-\lambda_{2}$ for $\lambda_{1},\lambda_{2}\in\Lambda$.
Let us therefore consider an element of the form $x=(\lambda_{1}-\lambda_{2})u^{g}_{ij}\in I\mathcal{O}(H_{N}^{+}(\Gamma,\Lambda))$. By the fusion rules, this equals $u^{\lambda_{1}g}_{ij}-u^{\lambda_{2}g}_{ij}$. Now by normality of $\Lambda$, there exists $\lambda^{\prime}_{1},\lambda^{\prime}_{2}\in\Lambda$ such that $\lambda_{1}g=g\lambda_{1}^{\prime}$ and $\lambda_{2}g=g\lambda^{\prime}_{2}$. It follows that
$$x=u^{g\lambda^{\prime}_{1}}_{ij}-u^{g\lambda^{\prime}_{2}}_{ij}=u^{g}_{ij}(\lambda^{\prime}_{1}-\lambda^{\prime}_{2})\in\mathcal{O}(H_{N}^{+}(\Gamma,\Lambda))I$$
and the proof is complete.
∎
As a first consequence of this result, we can bound the cohomological dimension (in the sense of Hochschild cohomology, see for instance [35, Chap 4]) of $H_{N}^{+}(\Gamma,\Lambda)$.
Corollary 5.3.
If $\Lambda$ is normal in $\Gamma$, then
$$\max\left(3,\operatorname{cd}(\Gamma^{\ast_{\Lambda}^{N}}))\right)\leqslant\operatorname{cd}(H_{N}^{+}(\Gamma,\Lambda))\leqslant\max\left(3,\operatorname{cd}(\Gamma/\Lambda)\right)+\operatorname{cd}(\Lambda).$$
In particular, if $\Gamma/\Lambda$ is finite, then
$$\operatorname{cd}(H_{N}^{+}(\Gamma,\Lambda))=3.$$
Proof.
It follows from [4, Prop 3.2], noticing that because $\mathcal{O}(H_{N}^{+}(\Gamma,\Lambda))$ is co-semisimple, the exact sequence is in fact strict by [28, Thm 2], that
$$\operatorname{cd}(H_{N}^{+}(\Gamma,\Lambda))\leqslant\operatorname{cd}(H_{N}^{+}(\Gamma/\Lambda))+\operatorname{cd}(\Lambda).$$
On then combines this with [6, Thm 7.4] to get the upper bound.
As for the lower bound, first notice that both $\mathcal{O}(\widehat{\Gamma}^{\ast_{\Lambda}^{N}})$ and $\mathcal{O}(S_{N}^{+})$ are Hopf $*$-subalgebras of $\mathcal{O}(H_{N}^{+}(\Gamma,\Lambda))$. We then apply [4, Prop 3.2], using the facts that $\operatorname{cd}(S_{N}^{+})=3$ by [4, Thm 6.5].
As for the last statement, first recall that finite groups have cohomological dimension $0$ (note that we are considering the dimension of the complex group ring, not the integer one). Therefore, the right-hand side of the inequalities reduces to $3$, and the result follows.
∎
Remark 5.4.
It is know (see [5, Cor 5.3]) that for a CQG-algebra $A$, $\operatorname{cd}(A^{\ast N})=\operatorname{cd}(A)$. There does not however not seem to be a similar relation for iterated amalgamated free product. In particular, it is not clear how to relate $\operatorname{cd}(\Gamma^{\ast_{\Lambda}^{N}})$ to $\operatorname{cd}(\Gamma/\Lambda)$ in general.
There is no hope to get further information on the structure of $H_{N}^{+}(\Gamma,\Lambda)$ without strengthening the assumption on $\Lambda$. But it would be natural to consider now the case where the exact sequence is split, so that $\Gamma$ decomposes as a semi-direct product. We will see that a similar decomposition then holds for the free wreath products in terms of smash product of Hopf algebras. Instead of introducing the general theory, let us simply define the important object in our setting.
Assume that there exists a group homomorphism $\rho:\Gamma/\Lambda\to\Gamma$ such that $\pi\circ\rho=\operatorname{id}$, where $\pi:\Gamma\to\Gamma/\Lambda$ is the quotient map. Consider the vector space $H=\mathcal{O}(\widehat{\Gamma})\otimes\mathcal{O}(H_{N}^{+}(\Gamma/\Lambda))$ and equip it with the unique multiplication such that
$$(\lambda\otimes u^{g}_{ij})(\mu\otimes u^{h}_{i^{\prime}j^{\prime}})=(\lambda\rho(g)\mu\rho(g)^{-1})\otimes u^{g}_{ij}u^{h}_{i^{\prime}j^{\prime}}$$
and with the unique comultiplication such that
$$\Delta(\lambda\otimes u^{g}_{ij})=\sum_{k=1}^{N}\lambda\otimes u^{g}_{ik}\otimes\lambda\otimes u^{g}_{kj}.$$
Then, $H$ is a CQG-algebra and the corresponding compact quantum group will be denoted by $\widehat{\Gamma}\>\sharp\>H_{N}^{+}(\Gamma,\Lambda)$.
Theorem 5.5.
Let $\Gamma$ be a discrete group and let $\Lambda$ be a normal subgroup such that the corresponding short exact sequence is split. Then, there is an isomorphism
$$H_{N}^{+}(\Gamma,\Lambda)\cong\widehat{\Gamma}\>\sharp\>H_{N}^{+}(\Gamma/\Lambda).$$
Proof.
First observe that, in a way similar to the comment in the beginning of the proof of Proposition 3.5, if $\Lambda_{1}<\Gamma_{1}$ and $\Lambda_{2}<\Gamma_{2}$ are two pairs of discrete groups and $\rho:\Gamma_{1}\to\Gamma_{2}$ is a group homomorphism such that $\rho(\Lambda_{1})\subset\Lambda_{2}$, then there is a $*$-homomorphism $\mathcal{O}(H_{N}^{+}(\Gamma_{1},\Lambda_{1}))\to\mathcal{O}(H_{N}^{+}(\Gamma_{2},\Lambda_{2}))$ sending $a(g)_{ij}$ to $a(\rho(g))_{ij}$.
Let $\rho$ be the splitting of the exact sequence and consider the corresponding map
$$\gamma:\mathcal{O}(H_{N}^{+}(\Gamma/\Lambda))\to\mathcal{O}(H_{N}^{+}(\Gamma,\Lambda)).$$
Because it is a $*$-homomorphism, it is convolution invertible with inverse $\gamma^{-1}=f\circ S$ (where $S$ denotes the antipode). In other words, the extension of compact quantum groups is cleft with cleaving map $\gamma$ and it follows from the general theory (see for instance [26, Thm 7.2.3]) that $H_{N}^{+}(\Gamma,\Lambda)$ is a cocycle smash product. We will now prove that this cocycle smash product coincides with the definition above.
First observe that the corresponding cocycle is trivial because $\gamma$ is an algebra homomorphism. Furthermore, the measuring corresponding to the cleaving map $\gamma$ is given by
$$\displaystyle u^{g}_{ij}\triangleright\lambda$$
$$\displaystyle=\sum_{k=1}^{N}\gamma(u^{g}_{ik}\lambda\gamma^{-1}(u^{g}_{kj})$$
$$\displaystyle=\sum_{k=1}^{N}u^{\rho(g)\lambda}_{ik}u^{\rho(g)^{-1}}_{jk})$$
$$\displaystyle=\delta_{ij}\sum_{k=1}^{N}u^{\rho(g)\lambda\rho(g)^{-1}}_{ik}$$
$$\displaystyle=\delta_{ij}\rho(g)\lambda\rho(g)^{-1}.$$
Now, the algebra structure on $\mathcal{O}(\widehat{\Gamma})\otimes\mathcal{O}(H_{N}^{+}(\Gamma/\Lambda))$ is given by the formula
$$(\lambda\otimes u^{g}_{ij})(\mu\otimes u^{h}_{i^{\prime}j^{\prime}})=\sum_{k=1}^{N}\lambda\left(u_{ik}^{g}\triangleright\mu)\right)\otimes u^{g}_{kj}u^{h}_{i^{\prime}j^{\prime}}=\lambda\rho(g)\mu\rho(g)^{-1}\otimes u^{g}_{ij}u^{h}_{i^{\prime}j^{\prime}}.$$
∎
Even though Theorem 5.5 gives a decomposition in terms of $\Lambda$ and $\Gamma/\Lambda$, it cannot be used directly to improve results concerning approximation properties, because even for discrete groups it is know that the Haagerup property for instance does not pass to general semi-direct products. Let us therefore work out a special case where everything works smoothly, namely the case where the smash product reduces to a direct product.
Corollary 5.6.
There is an isomorphism
$$H_{N}^{+}(\Gamma,\Lambda)\cong\widehat{\Lambda}\times H_{N}^{+}(\Gamma/\Lambda)$$
if and only if $\Gamma$ is of the form $\Lambda\underset{\Lambda_{0}}{\times}\Gamma_{0}$.
Proof.
If the isomorphism holds, then by the definition of $\widehat{\Lambda}\>\sharp\>H_{N}^{+}(\Gamma,\Lambda)$, $\Lambda$ must commute with $\Gamma_{0}=\rho(\Gamma/\Lambda)$. Since these two groups generate $\Gamma$, it follows that $\Gamma$ is a quotient of $\Lambda\times\Gamma_{0}$. Let us denote by $\pi$ the quotient map. Because $\pi$ is injective on $\Lambda$, for any $\lambda\in\Lambda$ there exists at most one $\gamma\in\Gamma_{0}$ such that $(\lambda,\gamma)\in\ker(\pi)$, namely $\lambda^{-1}$. As a consequence, if
$$\Lambda_{0}=\{\lambda\in\Lambda\mid(\lambda,\gamma)\in\ker(\pi)\},$$
then the map $\lambda\mapsto\gamma^{-1}$ is well-defined isomorphism between $\Lambda_{0}$ and a subgroup $\Lambda_{1}\subset\Gamma_{0}$ and $\pi$ is the quotient map identifying these two isomorphic groups. The converse is clear.
∎
In particular, under the previous assumptions, if both $\Lambda$ and $\Gamma/\Lambda$ are residually finite (resp. have the Haagerup property, resp. are weakly amenable) then so does $H_{N}^{+}(\Gamma,\Lambda)$.
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Mixture of Bilateral-Projection Two-dimensional Probabilistic Principal Component Analysis
Fujiao Ju${}^{1}$, Yanfeng Sun${}^{1}$, Junbin Gao${}^{2}$, Simeng Liu${}^{1}$ and Yongli Hu${}^{1}$
${}^{1}$College of Metropolitan Transportation,
Beijing University of Technology, Beijing 100124, China
School of Computing and Mathematics, Charles Sturt University, Bathurst, NSW 2795, Australia
{jufujiao2013,smliu}@emails.bjut.edu.cn; {yfsun,huyongli}@bjut.edu.cn; jbgao@csu.edu.au
Abstract
The probabilistic principal component analysis (PPCA) is built upon a global linear mapping, with which it is insufficient to model complex data variation. This paper proposes a mixture of bilateral-projection probabilistic principal component analysis model (mixB2DPPCA) on 2D data. With multi-components in the mixture, this model can be seen as a ‘soft’ cluster algorithm and has capability of modeling data with complex structures. A Bayesian inference scheme has been proposed based on the variational EM (Expectation-Maximization) approach for learning model parameters. Experiments on some publicly available databases show that the performance of mixB2DPPCA has been largely improved, resulting in more accurate reconstruction errors and recognition rates than the existing PCA-based algorithms.
1 Introduction
Principle Component Analysis (PCA) [3] is one of popular dimensionality reduction methods widely used in image analysis [9, 11], pattern recognition [8, 14] and machine learning [13] for data analysis. It can be derived under algebraic framework. However, algebraic models don’t have flexibility of providing confidence information of the model when dealing with noisy data. This is due to the absence of an associated probability density or generative model in algebraic framework.
To compensate the algebraic PCA drawbacks, Tipping and Bishop [19] firstly proposed a probabilistic PCA model, called PPCA. Under the probabilistic framework, PPCA takes advantage of Bayesian learning and inference by combining the likelihood with appropriate priors. As a result, the observed data are regarded as random variables, generated from a set of latent random variables which follow the Gaussian distribution of zero mean and identity covariance, with additive noises following a Gaussian distribution with zero mean and an isotropic covariance.
Under such a probabilistic learning framework, the model parameters in PPCA can be easily solved by the maximum likelihood estimation (MLE). Much progress has been made based on PCA and PPCA in the last couple of decades [2, 5].
PPCA and standard PCA methods can be interpreted in many ways, one of which assumes that the observed high-dimensional data are generated from their low-dimensional factors through a linear model with the corruption of Gaussian noise. So those algorithms essentially use a linear model for representing the entire data in a low dimensional subspace. It may be insufficient to model data with large variation caused by, for example, pose, expression and lighting in face recognition. Thus the application scope of PPCA and PCA-based methods is necessarily somewhat limited by its global linearity assumption. An alternative improving paradigm is to model the complex manifold with a mixture of local linear PPCA sub-models. Thus the single PCA model could be extended to a mixture of such sub-models.
A number of ‘mixture of PPCA’ have been proposed in literature. The first work was done by Ghahramani and Hinton [7]. They presented an exact Expectation-Maximization (EM) algorithm for fitting the parameters of the mixture of factor analyzers. By constraining the error covariance to be a diagonal matrix whose elements are usually equal, the mixture of factor analyzers became the mixture of PPCA [20]. Bishop and Tipping [4] extended the mixture of PPCA model to achieve a hierarchical mixture model. Su and Dy [17] introduced an automated hierarchical mixture of PPCA algorithm, which utilizes the integrated classification likelihood as a criterion for splitting and stopping the addition of hierarchical levels. Kim et al. [12] proposed a fast and sub-optimal selection method of model order such as the number of mixture components and the number of PCA bases for the PCA mixture model, consisting of a combination of many PCAs. In addition, under the assumption of the Student-$t$ distribution, the related research includes the mixture model of Student-$t$ components [15], which actually is a generalized mixture of Gaussian model without considering subspace structures, and more recent work such as the robust subspace mixture model [16], in which both the likelihood and the latent variables were supposed to follow the Student-$t$ distribution and the EM algorithm was applied to the model. In 2005, Archambeau [1] discussed the robust models in the context of finite mixture models, and a similar work for the mixture of the robust Laplacians was presented in [6]. These mixture models are important as it enables one to model nonlinear relationships by aligning a collection of such local models.
The aforementioned models are concerned with vectorial data. In order to apply these methods to 2D data, a typical workaround way is to vectorize 2D data. Vectorizing 2D data not only results in very high-dimensional data, causing the problem of the curse of dimensionality [23], but also ignores valuable information on the spatial relationship among 2D data. Instead of using vectorization, PCA approaches for two-dimensional data (2DPCA) have been proposed [22, 24, 26], to generally extract features of 2D data under the assumption of Gaussian noises. Ju et al. [10] proposed a probabilistic 2DPCA model to deal with outlier noises by using Laplacian distribution. This model benefits outlier detection. Wang et al. [21] extended the probabilistic 2DPCA to a mixture of local probabilistic 2DPCA models (MP2DPCA). MP2DPCA offers a tempting prospect of being able to model data with complex variation.
MP2DPCA model regards each row vector of the 2D data as a observed sample and used all rows to train the mixture model, resulting in mean vectors from the mixture model. This is essentially a unilateral projection based scheme, where only one side multiplication is taken into account. The unilateral scheme usually preserves the correlation information among the row/column vectors of the images and more parameters are needed to well represent an image. To tackle these problems, a bilateral-projection scheme is favored. In this study, our intention is propose a mixture of bilateral-projection-based probabilistic 2DPCA (mixB2DPPCA) model. Different from MP2DPCA, we regard each 2D images as observed samples in their natural shape and reduce 2D dimensionality directly. The mixB2DPPCA has two major advantages: 1) The model makes use of structured information of 2D data and can be easily extended for high order tensorial data. All the algorithm derivations remain without major difficulties. 2) mix2DPPCA carries over all the advantages of the mixture of PPCA.
The remainder of the paper is organized as follows. In Section 2, the mixture of bilateral-projection two-dimensional probabilistic PCA model is introduced. The variational approximation approach for solving the model is presented in Section 3. In Section 4, some experimental results are conducted to evaluate the performance of the proposed model. Finally, conclusions are summarized in Section 5.
2 Mixture of Bilateral-Projection 2DPPCA Model (mixB2DPPCA)
In this section, we introduce the mixture of bilateral-projection probabilistic 2DPCA model. For the purpose, we introduce several notations. Let $\mathcal{X}=\{\mathbf{X}_{1},\mathbf{X}_{2},...,\mathbf{X}_{N}\}$ be $N$ independent and identical random samples with values in $\mathbb{R}^{p\times q}$. For $n=1,...,N$, we suppose that sample $\mathbf{X}_{n}$ is generated independently from a mixture of $K$ underlying components with unknown probabilities $\pi_{1},\pi_{2},...,\pi_{K}$,
$$\displaystyle p(\mathbf{X}_{n}|\mathbf{B}_{n})=\sum_{k=1}^{K}\pi_{k}\mathcal{N}(\mathbf{L}_{k}\mathbf{B}_{n}^{(k)}\mathbf{R}_{k}^{T}+\mathbf{M}_{k},\sigma_{k}\mathbf{I},\sigma_{k}\mathbf{I})$$
(1)
where $\mathbf{M}_{k}\in\mathbb{R}^{p\times q}$ is the mean matrix, $\pi_{k}$s satisfy $\pi_{k}>0$ and $\sum_{k=1}^{K}\pi_{k}=1$, and $\mathbf{L}_{k}\in\mathbb{R}^{p\times r}$ and $\mathbf{R}_{k}\in\mathbb{R}^{q\times c}$ are the row and column loading matrices with $r\leq p,c\leq q$. Note that $\mathbf{M}_{k}$, $\mathbf{L}_{k}$ and $\mathbf{R}_{k}$ are associated with each component of mixture model, respectively. $\mathbf{B}_{n}^{(k)}\in\mathbb{R}^{r\times c}$ is the latent variable core of $\mathbf{X}_{n}$ associated with $k$-th matrix-variate Gaussian component [18, Sec 3.3] with $\sigma_{k}^{2}$ as residual variance.
Like [3], we introduce a $K$-dimensional binary random variable $\mathbf{z}$ having a 1-of-$K$ representation in which a particular element $z_{k}$ is equal to 1 and all other elements are equal to 0. That is, $z_{k}\in\{0,1\}$ and $\sum_{k=1}^{K}z_{k}=1$.
The distribution of $\mathbf{z}$ is defined by
$$p(z_{k}=1):=\pi_{k},$$
which can be written as
$$p(\mathbf{z})=\prod_{k=1}^{K}\pi_{k}^{z_{k}}.$$
Thus the conditional distribution of $\mathbf{X}_{n}$ given a particular value for $\mathbf{z}_{n}$ and $\mathbf{B}_{n}^{(k)}$ is the matrix-variate Gaussian
$$\displaystyle p(\mathbf{X}_{n}|z_{nk}=1,\mathbf{B}_{n}^{(k)})=\mathcal{N}(\mathbf{X}_{n}|\mathbf{L}_{k}\mathbf{B}_{n}^{(k)}\mathbf{R}_{k}^{T}+\mathbf{M}_{k},\sigma_{k}\mathbf{I},\sigma_{k}\mathbf{I}).$$
Generally we have
$$p(\mathbf{X}_{n}|\mathbf{z}_{n},\mathbf{B}_{n}^{(k)})=\prod_{k=1}^{K}\mathcal{N}(\mathbf{X}_{n}|\mathbf{L}_{k}\mathbf{B}_{n}^{(k)}\mathbf{R}_{k}^{T}+\mathbf{M}_{k},\sigma_{k}\mathbf{I},\sigma_{k}\mathbf{I})^{z_{nk}}$$
In this model setting, the parameters are $\Theta=\{\pi_{k},\mathbf{M}_{k},\mathbf{L}_{k},\mathbf{R}_{k},\sigma^{2}_{k}\}(k=1,..,K)$, and the latent variables are $\mathbf{z}_{n}$ and $\mathbf{B}_{n}^{(k)}(n=1,...,N)$.
To develop a generative Bayesian model, we define a matrix-variate Gaussian prior $p(\mathbf{B}_{n}^{(k)})$ over the latent variable with zero-mean unit-covariance, defined as
$$p(\mathbf{B}_{n}^{(k)})=\mathcal{N}(0,\mathbf{I}_{r},\mathbf{I}_{c})=\bigg{(}\frac{1}{2\pi}\bigg{)}^{\frac{rc}{2}}\cdot\exp\{-\frac{1}{2}\mathbf{tr}(\mathbf{B}_{n}^{(k)T}\mathbf{B}_{n}^{(k)})\}$$
Hence the joint log-likelihood of the observed data set for such a mixture model is:
$$\displaystyle\mathcal{L}=\sum_{n=1}^{N}\sum_{k=1}^{K}z_{nk}\ln\{\pi_{k}p(\mathbf{X}_{n},\mathbf{B}_{n}^{(k)})\}.$$
3 Variational Approximation for mixB2DPPCA Model
We employ the Expectation Maximization (EM) algorithm to solve for model parameters $\Theta$. To maximize the log-likelihood of mixB2DPPCA, we take the expectation of $\mathcal{L}$ with respect to the posterior distribution of both $\mathbf{B}_{n}^{(k)}$ and $z_{nk}$, i.e.,
$$\displaystyle\langle\mathcal{L}\rangle$$
$$\displaystyle=\sum_{n=1}^{N}\sum_{k=1}^{K}\langle z_{nk}\rangle\{\ln\pi_{k}-\frac{pq}{2}\ln\sigma^{2}_{k}-\frac{1}{2}\text{tr}(\langle\mathbf{B}_{n}^{(k)T}\mathbf{B}_{n}^{(k)}\rangle)$$
$$\displaystyle-\frac{1}{2\sigma^{2}_{k}}\text{tr}(\mathbf{X}_{n}-\mathbf{M}_{k})^{T}(\mathbf{X}_{n}-\mathbf{M}_{k})$$
$$\displaystyle+\frac{1}{\sigma^{2}}\text{tr}((\mathbf{X}_{n}-\mathbf{M}_{k})^{T}\mathbf{L}_{k}\langle\mathbf{B}^{(k)}_{n}\rangle\mathbf{R}_{k}^{T})$$
$$\displaystyle-\frac{1}{2\sigma^{2}}\text{tr}(\langle\mathbf{B}^{(k)T}_{n}\mathbf{L}_{k}^{T}\mathbf{L}_{k}\mathbf{B}^{(k)}_{n}\rangle\mathbf{R}_{k}^{T}\mathbf{R}_{k})$$
(2)
where $\langle\cdot\rangle$ denotes the expectation.
In E-step, we update Q-distributions of all hidden variables $\mathbf{B}_{n}^{(k)}$ and $z_{n,k}$ with the current fixed parameter values for $\Theta$. In M-step, maximizing the function $\langle\mathcal{L}\rangle$ with respect to the model parameters $\Theta$, we can obtain ‘new’ values for these parameters.
3.1 Variational E-step
3.1.1 Update the Posterior Distribution of $z_{nk}$
Suppose $\gamma_{nk}:=\langle z_{nk}\rangle$ and it is actually the posterior probability of $k$-mixture generating data point $\mathbf{X}_{n}$. By using the same strategy for the mixture Gaussian model [3], we can obtain
$$\displaystyle\gamma_{nk}=\frac{\pi_{k}p(\mathbf{X}_{n}|k)}{p(\mathbf{X}_{n})},$$
(3)
where $p(\mathbf{X}_{n}|k)$ is the $k$-the component, representing the marginal distribution for the observed data $\mathbf{X}_{n}$ over the latent variable. In our case, the marginal distribution of $\mathbf{X}_{n}$ is obtained by integrating out the latent variable $\mathbf{B}_{n}^{(k)}$:
$$p(\mathbf{X}_{n}|k)=\int p(\mathbf{X}_{n}|\mathbf{B}_{n}^{(k)})p(\mathbf{B}_{n}^{(k)})d\mathbf{B}_{n}^{(k)}.$$
Different from the vectorial PPCA, we note that the marginal distribution of the observed data $\mathbf{X}_{n}$ is in general no longer a matrix-variate Gaussian. Thus it is difficult to work with $p(\mathbf{X}_{n}|k)$ directly. Let $\mathbf{x}_{n}:=\text{vec}(\mathbf{X}_{n})$, now we can work with $p(\mathbf{x}_{n}|k)$ instead of $p(\mathbf{X}_{n}|k)$. Fortunately, the marginal distribution of $\mathbf{x}_{n}$ is a multivariate Gaussian distribution when taking the special matrix-variate Gaussian prior $\mathbf{B}_{n}^{(k)}\sim\mathcal{N}(0,\mathbf{I},\mathbf{I})$. Let $\mathbf{m}_{k}=\text{vec}(\mathbf{M}_{k})$, we can obtain
$$p(\mathbf{x}_{n}|k)\sim\mathcal{N}(\mathbf{m}_{k},\mathbf{C}_{k})$$
where the observation covariance model is specified by $\mathbf{C}_{k}=(\mathbf{R}_{k}\mathbf{R}_{k}^{T})\otimes(\mathbf{L}_{k}\mathbf{L}_{k}^{T})+\sigma_{k}^{2}\mathbf{I}$. We refer readers to [3, 18] for more details. Then the denominator in (3)
becomes
$$\displaystyle p(\mathbf{x}_{n})=\sum_{k=1}^{K}\pi_{k}p(\mathbf{x}_{n}|k)$$
After getting $\gamma_{nk}$, we update the estimated mean matrices $\mathbf{M}_{k}$’s and mixing proportions $\pi_{k}$’s, respectively, by
$$\displaystyle\pi_{k}=\frac{1}{N}\sum_{n=1}^{N}\gamma_{nk}\quad\text{and}\quad\mathbf{M}_{k}=\frac{\sum_{n=1}^{N}\gamma_{nk}\mathbf{X}_{n}}{\sum_{n=1}^{N}\gamma_{nk}}$$
(4)
3.1.2 Update the Posterior Distribution of $\mathbf{B}_{n}^{(k)}$
In computing the posterior distribution of $\mathbf{B}_{n}^{(k)}$, we encounter a difficulty that
the posteriori distribution of $\mathbf{B}_{n}^{(k)}$ given $\mathbf{X}_{n}$
$$p(\mathbf{B}_{n}^{(k)}|\mathbf{X}_{n},\mathbf{L}_{k},\mathbf{R}_{k},\sigma^{2})\propto p(\mathbf{X}_{n}|\mathbf{B}_{n}^{(k)},\mathbf{L}_{k},\mathbf{R}_{k},\sigma^{2})p(\mathbf{B}_{n}^{(k)})$$
is also in general not a matrix-variate Gaussian. To get a tractable posterior in the variational EM, we restrict the approximated variational distribution to be a matrix-variate Gaussian $\mathcal{N}(\mathbf{B}_{n}^{(k)}\,|\,\mathbf{Q}_{n}^{(k)},\mathbf{T}_{n}^{(k)},\mathbf{S}_{n}^{(k)})$ to approximate the true posterior with the mean $\mathbf{Q}_{n}^{(k)}$ in size $r\times c$ and covariances $\mathbf{T}_{n}^{(k)}\succ 0$ of size $r\times r$ and $\mathbf{S}_{n}^{(k)}\succ 0$ of size $c\times c$, respectively. For mixB2DPPCA model, it follows as a natural extension of a single 2DPPCA. So the parameters $\mathbf{Q}_{n}^{(k)}$, $\mathbf{T}_{n}^{(k)}$ and $\mathbf{S}_{n}^{(k)}$ can be estimated through the maximization of a single likelihood function. Particularly, the derived formulas for estimating these parameters are given by, see more details in [26],
$$\displaystyle\mathbf{T}_{n}^{(k)}=c\sigma_{k}^{2}[\text{tr}(\mathbf{R}^{T}_{k}\mathbf{R}_{k}\mathbf{S}_{n}^{(k)})\mathbf{L}_{k}^{T}\mathbf{L}_{k}+\sigma_{k}^{2}\text{tr}(\mathbf{S}_{n}^{(k)})\mathbf{I}_{r}]^{-1}$$
$$\displaystyle\mathbf{S}_{n}^{(k)}=r\sigma_{k}^{2}[\text{tr}(\mathbf{L}_{k}^{T}\mathbf{L}_{k}\mathbf{T}_{n}^{(k)})\mathbf{R}^{T}_{k}\mathbf{R}_{k}+\sigma_{k}^{2}\text{tr}(\mathbf{T}_{n}^{(k)})\mathbf{I}_{c}]^{-1}$$
and each $\mathbf{Q}_{n}^{(k)}$ needs to satisfy
$$\mathbf{L}^{T}_{k}\mathbf{L}_{k}\mathbf{Q}_{n}^{(k)}\mathbf{R}_{k}^{T}\mathbf{R}_{k}+\sigma_{k}^{2}\mathbf{Q}_{n}^{(k)}=\mathbf{L}^{T}_{k}(\mathbf{X}_{n}-\mathbf{M}_{k})\mathbf{R}_{k}.$$
To solve this we need to make a vectorization on both sides and solve a linear equation
$$\displaystyle(\mathbf{R}_{k}^{T}\mathbf{R}_{k}\otimes\mathbf{L}^{T}_{k}\mathbf{L}_{k}+\sigma_{k}\mathbf{I}\otimes\sigma_{k}\mathbf{I})\text{vec}(\mathbf{Q}^{(k)}_{n})=\mathbf{y}^{(k)}_{n}$$
(5)
with respect to $\text{vec}(\mathbf{Q}_{n}^{(k)})$, where
$$\displaystyle\mathbf{y}^{(k)}_{n}=\text{vec}(\mathbf{L}^{T}_{k}(\mathbf{X}_{n}-\mathbf{M}_{k})\mathbf{R}_{k})$$
then reshape $\text{vec}(\mathbf{Q}_{n}^{(k)})$ back to get $\mathbf{Q}_{n}^{(k)}$.
As we assume the approximated posterior distribution of $\mathbf{B}_{n}^{(k)}$ is matrix-variate Gaussian, so we can get $\langle\mathbf{B}_{n}^{(k)}\rangle=\mathbf{Q}_{n}^{(k)}$ and the following second-order expectations:
$$\displaystyle\langle\mathbf{B}_{n}^{(k)T}\mathbf{B}_{n}^{(k)}\rangle=\mathbf{Q}_{n}^{(k)T}\mathbf{Q}_{n}^{(k)}+\mathbf{S}_{n}^{(k)}\text{tr}(\mathbf{T}_{n}^{(k)})$$
(6)
$$\displaystyle\langle\mathbf{B}^{(k)T}_{n}\mathbf{L}_{k}^{T}\mathbf{L}_{k}\mathbf{B}^{(k)}_{n}\rangle=\mathbf{Q}^{(k)T}_{n}\mathbf{L}_{k}^{T}\mathbf{L}_{k}\mathbf{Q}^{(k)}_{n}+\mathbf{S}_{n}^{(k)}\text{tr}(\mathbf{T}_{n}^{(k)}\mathbf{L}_{k}^{T}\mathbf{L}_{k})$$
(7)
$$\displaystyle\langle\mathbf{B}^{(k)}_{n}\mathbf{R}_{k}^{T}\mathbf{R}_{k}\mathbf{B}^{(k)T}_{n}\rangle=\mathbf{Q}^{(k)}_{n}\mathbf{R}_{k}^{T}\mathbf{R}_{k}\mathbf{Q}^{(k)T}_{n}+\mathbf{T}_{n}^{(k)}\text{tr}(\mathbf{S}_{n}^{(k)}\mathbf{R}_{k}^{T}\mathbf{R}_{k})$$
(8)
3.2 Variational M-step
In the M-step, we fix all the distributions over the hidden variables and gather all the terms containing parameters $\mathbf{L}_{k}$, $\mathbf{R}_{k}$ and $\sigma_{k}^{2}$ in (2) to
maximize them respectively. It turns out that:
$$\displaystyle\mathbf{L}_{k}=$$
$$\displaystyle[\sum_{n=1}^{N}\gamma_{nk}(\mathbf{X}_{n}-\mathbf{M}_{k})\mathbf{R}_{k}\langle\mathbf{B}_{n}^{(k)}\rangle^{T}]$$
$$\displaystyle\times[\sum_{n=1}^{N}\gamma_{nk}\langle\mathbf{B}_{n}^{(k)}\mathbf{R}^{T}_{k}\mathbf{R}_{k}\mathbf{B}_{n}^{(k)T}\rangle]^{-1}$$
(9)
$$\displaystyle\mathbf{R}_{k}=$$
$$\displaystyle[\sum_{n=1}^{N}\gamma_{nk}(\mathbf{X}_{n}-\mathbf{M}_{k})^{T}\mathbf{L}_{k}\langle\mathbf{B}_{n}^{(k)}\rangle]$$
$$\displaystyle\times[\sum_{n=1}^{N}\gamma_{nk}\mathbf{\langle}\mathbf{B}_{n}^{(k)T}\mathbf{L}^{T}_{k}\mathbf{L}_{k}\mathbf{B}_{n}^{(k)}\rangle]^{-1}$$
(10)
and
$$\displaystyle\sigma^{2}_{k}$$
$$\displaystyle=\frac{1}{pqN_{k}}\{\sum_{n=1}^{N}\gamma_{nk}\text{tr}(\mathbf{X}_{n}-\mathbf{M}_{k})^{T}(\mathbf{X}_{n}-\mathbf{M}_{k})$$
$$\displaystyle-2\sum_{n=1}^{N}\gamma_{nk}\text{tr}(\mathbf{R}_{k}\langle\mathbf{B}^{(k)}_{n}\rangle^{T}\mathbf{L}_{k}^{T}(\mathbf{X}_{n}-\mathbf{M}_{k}))$$
$$\displaystyle+\sum_{n=1}^{N}\gamma_{nk}\text{tr}(\langle\mathbf{B}^{(k)T}_{n}\mathbf{L}_{k}^{T}\mathbf{L}_{k}\mathbf{B}^{(k)}_{n}\rangle\mathbf{R}_{k}^{T}\mathbf{R}_{k})\}$$
(11)
where $N_{k}=\sum_{n}\gamma_{nk}$.
The overall variational EM algorithm is to alternate between E-step and M-step.
The final variational EM algorithm is summarized
in Algorithm 1.
Define the average reconstruction error
$$\displaystyle\mathbf{e}(t)=\sqrt{\frac{\sum_{n=1}^{N}\|\mathbf{X}_{n}-\widehat{\mathbf{X}}_{n}^{(t)}\|_{F}^{2}}{N}}$$
(12)
where $\widehat{\mathbf{X}}_{n}=\mathbf{L}_{k^{\prime}}\mathbf{B}^{(k^{\prime})}_{n}\mathbf{R}^{T}_{k^{\prime}}+\mathbf{M}_{k^{\prime}}$ with $k^{\prime}=\text{arg}\max_{k}\{\gamma_{nk}\}$ the reconstructed image.
Algorithm 1 may terminate either a given maximum iterative number $T$ is achieved or the following condition is satisfied,
$$\displaystyle|\mathbf{e}(t)-\mathbf{e}(t+1)|\leq\epsilon$$
(13)
where $\epsilon$ is a given error tolerance.
3.3 The Reduced-Dimensionality Representation for a New Sample
In order to obtain the reduced-dimensionality representation for a given sample, we should solve for the latent variable cores. From the probabilistic perspective, the posterior mean $\mathbf{Q}_{new}^{(k)}:=\langle\mathbf{B}_{new}^{(k)}|\mathbf{X}_{new}\rangle$ can be seen as the reduced-dimensionality representation, which is a $r\times c$ feature matrix and given by solving a linear equation
$$\displaystyle(\mathbf{R}_{k}^{T}\mathbf{R}_{k}\otimes\mathbf{L}^{T}_{k}\mathbf{L}_{k}+\sigma_{k}\mathbf{I}\otimes\sigma_{k}\mathbf{I})\text{vec}(\mathbf{Q}^{(k)}_{new})=\mathbf{y}_{new}^{(k)}$$
with respect to $\text{vec}(\mathbf{Q}_{new}^{(k)})$, where
$$\displaystyle\mathbf{y}_{new}^{(k)}=\text{vec}(\mathbf{L}^{T}_{k}(\mathbf{X}_{new}-\mathbf{M}_{k})\mathbf{R}_{k})$$
then reshape $\text{vec}(\mathbf{Q}_{new}^{(k)})$ back to get $\mathbf{Q}_{new}^{(k)}$. As the same time, we can compute the corresponding $\gamma_{new,k}$, i.e., the posterior probability of $k$-th component generating the new sample, given by
$$\gamma_{new,k}=\frac{p(\mathbf{X}_{new}|k)\pi_{k}}{p(\mathbf{X}_{new})}$$
We find the largest $\gamma_{new,k}$ ($k=1,...,K$) from which the most appropriate local 2DPPCA model can be identified for the new sample. That is, a natural choice is to assign the new sample to a cluster with the largest posterior probability.
4 Experimental Results and Analysis
In this section, we conduct several experiments on some public databases to assess the proposed mixB2DPPCA model. These experiments are designed to evaluate the performance of the proposed mix2DPPCA in reconstruction and recognition by comparing with existing models and algorithms.
The relevant PCA algorithms that can be fairly compared against our proposed mixB2DPPCA are GLRAM (Generalized Low Rank Approximations of Matrices) [25], PSOPCA (Probabilistic Second-Order PCA) [27], mixture of PPCA [20] with the code from http://www.science.uva.nl/j̃verbeek. Because the zero-noise PSOPCA model and GLRAM have the same stationary point [27], we only compare with GLRAM.
4.1 Data Preparation and Experiment Setting
All of the experiments are conducted on the following four public available datasets:
•
A subset of handwritten digits images from the MNIST database (http://yann.lecun.com/exdb/mnist).
•
The Yale face database (http://vision.ucsd.edu/content/yale-face-database).
•
The AR face database (http://rvl1.ecn.purdue.edu/aleix/aleix_face_DB.html).
•
The FERET face database (http://www.itl.nist.gov/iad/humanid/feret/feret_master.html).
The subset of handwritten digits images is selected from MNIST database, which contains 1000 digital images with 100 images of each digit. All images are in grayscale and have a uniform size of $28\times 28$ pixels.
The Yale face database contains 15 individuals, with 11 images for each individual. The images were captured under different illumination and expression conditions. The images are all $100\times 100$ pixels with 256 grey levels. In the experiments, we randomly select 6 images of each person as the training samples, and use the remaining images to form the testing sample set. All images are scaled to a resolution of $64\times 64$ pixels.
The AR face database contains over 4,000 color images corresponding to 126 subjects. There are variations of facial expressions, illumination conditions, and occlusions (sun glasses and scarf) with each person. Each individual consists of 26 frontal view images taken in two sessions (separated by 2 weeks), where each session has 13 images. Figure 1 shows the 26 images of one subject. In the experiments, we select 30 subjects (15 man and 15 women), and only use the non-occluded 14 images (i.e., the first seven face images of each row in Figure 1). The first seven of each subject are used for training and the last seven for testing. All images are cropped and resized to $50\times 40$ pixels.
FERET database includes 1400 images of 200 different subjects, with
7 images per subject. In the experiments, we select 50 subjects randomly. Five images of each subject are used for training and the remained images are used for testing. All images are cropped and resized to $32\times 32$ pixels.
In experiments, the initial mixing proportions are set to $\pi_{k}=1/K$ and the initial loading matrices $\mathbf{L}_{k}$ and $\mathbf{R}_{k}$ are given randomly. Besides, we choose randomly $K$ samples as mean matrices $\mathbf{M}_{k}$ of the mixture gaussian model and set all $\sigma_{k}^{2}=1$.
4.2 Reconstruction Performance
In this section, we test reconstruction error of the proposed mixB2DPPCA model (1). Applying the proposed model, all digital images can be softly grouped into $K$ clusters, each of which is modelled by a local B2DPPCA. From all the trained $\gamma_{nk}$, the most appropriate local B2DPPCA for a given sample can be found. Then we use the most appropriate local B2DPPCA to reconstruct the initial digit image, that is:
$$\widehat{\mathbf{X}}_{n}=\mathbf{L}_{k^{\prime}}*\mathbf{Q}_{n}^{(k^{\prime})}*\mathbf{R}_{k^{\prime}}^{T}+\mathbf{M}_{k^{\prime}}.$$
where $k^{{}^{\prime}}$ represents the $k^{{}^{\prime}}$-th local B2DPPCA which most appropriate to the sample $\mathbf{X}_{n}$. After obtaining all reconstructed digit images $\widehat{\mathbf{X}}_{n}$, we can using the equation (12) to compute the average reconstruction error.
Next we compare the reconstruction error of different algorithms on three databases. In all algorithms, we set the iterative number is $T=50$ and the reduced dimension is $r=c=4$.
4.2.1 Reconstruction Error on Digit Image Set
We use the given digital image subset in Section 4.1 as training set. In this phase, we compare the reconstruction error of the training set.
Figure 2 shows the average reconstruction error of the relevant algorithms. From left to right, the component number is $K=2$, $K=5$ and $K=10$ respectively. Firstly, from these three sub-figures, we can see that the reconstruction error of GLRAM algorithm has no change. This is because GLRAM has no relationship with $K$. Besides, GLRAM works by iteratively computing the leading eigenvectors of the left and right one-sided sample covariance matrices. Thus GLRAM convergent in five steps and the change of reconstruction error is not obvious in the figure. Secondly, fixing the same number of reduced dimension, the performance of our proposed mixB2DPPCA is better than GLRAM. From the view of compression, decoded images from our algorithm have higher quality for the compression ratio of $49:1$. It illustrates that mixB2DPPCA can correctly identify data according to clusters. When $K$ becomes larger, the mixB2DPPCA outperform the mixture of PPCA in terms of reconstruction errors.
The reconstructed images of different methods are shown in Fig. 3 with $K=10$. The first row shows three original images. The second, third and fourth rows are the reconstructed images by GLRAM, mixture of PPCA and mixB2DPPCA, respectively. It can be found that the proposed mixB2DPPCA has better reconstruction outcomes, while the results of other two methods show a litter degradation.
4.2.2 Reconstruction Error on Yale and AR Databases
In this experiment, we compare the reconstruction error on Yale and AR databases.
Figure 4 shows the average reconstruction error of all the algorithms: (a) on the Yale database and (b) on the AR database. The component number is $K=5$ and the reduced dimensionality is $(r,c)=(4,4)$. It is obvious that the reconstruction error of mixB2DPPCA on testing set has reduced greatly than other algorithms.
Figure 5 shows some reconstructed images of different algorithms on Yale database. The first row is four original images. The second, third, and fourth rows are the corresponding images reconstructed by mixture of PPCA, GLRAM and mixB2DPPCA. It can be shown that the results of our algorithm have better visual effect than that of GLRAM. Besides we can also see that although the face images reconstructed by mixture of PPCA are relatively clear, they don’t match the same original images visually. The reconstructed images on AR database are shown in Figure 6. The first row shows five original images in the test set and the last three rows are the reconstructed images from three models.
From the reconstruction experiments, we can conclude that the mixB2DPPCA generally outperforms global linear 2DPCA algorithms in terms of reconstruction errors. It demonstrates that the classification of training set in advanced is important for the performance of feature extraction.
4.3 Recognition Performance
In this section, we compare the recognition performances of GLRAM, mixture of PPCA and mixB2DPCA on Yale, AR and FERET face databases. These algorithms can be used for extracting features of facial images from the training samples, respectively, and then a nearest neighbor classifier (1-NN) is used to find the most-similar face from the training samples for a querying face. In our experiments, the distance measure between two sets of feature matrices $\mathbf{B}_{n_{1}}$ and $\mathbf{B}_{n_{2}}$, is defined as
$$\text{dist}=\sum^{K}_{k=1}\|\mathbf{B}^{(k)}_{n_{1}}-\mathbf{B}^{(k)}_{n_{2}}\|_{F}.$$
where $\mathbf{B}_{n}=[\mathbf{B}_{n}^{(1)},\mathbf{B}_{n}^{(2)},...,\mathbf{B}_{n}^{(K)}]$ represents the combination of $K$ latent variable cores related with $n$-th sample111A more accurate way is to use $\gamma_{n_{1}k}\gamma_{n_{2}k}$ to weight the individual distance.. In all algorithms, we set maximum iteration number is 50 and $\epsilon$ in (13) is 1E-3. We repeat the procedure 10 times, and the mean values and relevant variances are reported in Tables 1 to 3.
Table 1 shows the recognition rates of three feature extraction algorithms: GLRAM, mixture of PPCA and mixB2DPPCA training on Yale database. The mean values and relevant variances are reported for the cases of the reduced dimension $(r,c)=(2,2)$, $(4,4)$, $(6,6)$ and $(8,8)$. For the mixture of PPCA and mixB2DPPCA, we also computed the recognition rates for the different component number $K$ ($K=4,6,8$), shown in Table 1. Firstly, from the table we can see that the recognition rates of the mixture of PPCA and mixB2DPPCA have a little fluctuation compared with GLRAM. This may be caused by the uncertainty of probability. Secondly, compared with GLRAM, the mean recognition rates of mixB2DPPCA algorithm have obviously improved. The bold figures are the best results in the comparison.
Table 2 shows the recognition rates of the above three algorithms training on AR database. The reduced dimensions are $(r,c)=(4,4)$, $(6,6)$ and $(8,8)$ and component numbers are $K=6,8,10$, respectively. From the table we can see that the mean recognition rates of mixB2DPPCA algorithm have better improvement over the other two algorithms.
Table 3 shows the recognition rates on FERET database. The reduced dimensions are $(r,c)=(4,4)$, $(6,6)$, $(8,8)$ and $(10,10)$, and the component numbers are $K=6,8,10$, respectively. In this case, both the mixture of PPCA and the proposed mixB2DPPCA produce slightly larger variances, however the mean recognition rates have risen greatly. GLRAM is relatively more robust.
5 Conclusions
In this paper, we proposed a mixture of bilateral-projection probabilistic PCA model for feature extraction and dimensionality reduction for 2D data. Different from the standard PCA which is a global dimension reduction model, this model employs the mixture of matrix-variate Gaussian to model local linear sub-models. All the parameters in the resulting probabilistic model can be estimated through the maximization of the likelihood function. The new model not only makes good use of spatial (structural) information of 2D data but also can softly group data into a given number of clusters. The performance of feature extraction of the proposed method generally outperforms other existing 2D algorithms in terms of reconstruction error and recognition rate. The approach used in this paper can be readily extended to higher order tensorial data and other non-Gaussian noise models can also be integrated into the model such.
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[
[
Virgilio.Gomez@uclm.es
[
Dept. of Management, Economics and Quantitative Methods,
University of Bergamo,
Bergamo, IT
[
Dept. of Epidemiology and Biostatistics, Imperial College London,
London, UK
Abstract
In this paper we recast the problem of missing values in the covariates of a regression model as a
latent Gaussian Markov random field (GMRF) model in a fully Bayesian framework. Our
proposed approach is based on the definition of the covariate imputation sub-model
as a latent effect with a GMRF structure. We show how this formulation works
for continuous covariates and provide some insight on how this could be extended
to categorical covariates.
The resulting Bayesian hierarchical model naturally fits within the integrated
nested Laplace approximation (INLA) framework, which we use for
model fitting. Hence, our work fills an important gap in the INLA methodology
as it allows to treat models with missing values in the covariates.
As in any other fully Bayesian framework, by relying on INLA for model fitting it is possible to formulate a
joint model for the data, the imputed covariates and their missingness
mechanism. In this way, we are able to tackle the more general problem of
assessing the missingness mechanism by conducting a sensitivity analysis on the
different alternatives to model the non-observed covariates.
Finally, we illustrate the proposed approach with two examples on modeling
health risk factors and disease mapping. Here, we rely on two different
imputation mechanisms based on a typical multiple linear regression and a
spatial model, respectively. Given the speed of model fitting with
INLA we are able to fit joint models in a short time, and to easily conduct
sensitivity analyses.
keywords: imputation, missing values, GMRF, INLA, sensitivity analysis
Missing data analysis with latent GMRFs]Missing data analysis and imputation via latent Gaussian Markov random fields
Gómez-Rubio et al.]Virgilio Gómez-Rubio
\coaddressDepartment of Mathematics,
School of Industrial Engineering,
University of Castilla-La Mancha,
02071 Albacete, Spain.
Gómez-Rubio et al.]Michela Cameletti
Gómez-Rubio et al.]Marta Blangiardo
1 Introduction
In any statistical analysis missing data is one of the most important issues a
researcher needs to deal with; failing to properly account for it can result in
a reduction of statistical power, or even in biased statistical inference.
Consequently, countless methods have focused on this (see, for
example, Schafer, 1997; Little and Rubin, 2002; Craig K., 2010; van Buuren, 2002; Trivellore, 2015).
Missing data can occur for a number of reasons, as described in
Little and Rubin (2002). Sometimes, the missingness mechanism is ignorable and
inference can rely on the observed data alone, appropriately coupled with a suitable
imputation or data augmentation model if needed. When the missingness mechanism
is not ignorable, a joint approach is required to fit the analysis model,
impute the missing values and assess the missingness mechanism. Under
this scenario, it is recommended that a sensitivity analysis is carried out to assess the
impact of the missingness mechanism on the model parameters estimates (Mason et al., 2012).
The Bayesian paradigm has gained popularity for dealing with missing data,
making no distinction between parameters and missing data which are considered
as additional unknown parameter with a prior distribution. For these reasons
and differently from other ad-hoc methods (Nakagawa, 2015), with a full
Bayesian approach it is possible to combine together the analysis and
imputation model in a joint estimation framework (Erler et al., 2016). For
instance, Mason (2009) and Mason et al. (2012) developed a
fully Bayesian missing imputation framework in order to adjust for
several missing covariates in longitudinal or cross-sectional studies; each of
the missing covariates is assigned an imputation model, all
jointly modelled with the analysis model.
The approach we propose in this paper is based on recasting the imputation
sub-model to define it as a latent Gaussian Markov random field
(GMRF, Rue and Held, 2005) which is part of a larger Bayesian hierarchical
model. This fits naturally within the integrated nested Laplace approximation
(INLA, Rue et al., 2009) methodology, as an alternative to Markov chain
Monte Carlo (MCMC). This approach is suitable for continuous covariates and can
be also extended to impute categorical variables.
This makes model fitting with
missing covariates possible in INLA, and our new approach fills an important
gap, as INLA has always required the data in the latent GMRF that
defines the model to be fully observed. Here we focus on missing covariates only as
INLA can
easily fit models with missing data in the response variable, simply
computing the corresponding posterior predictive distribution derived from the
analysis model to be fit (Gómez-Rubio, 2020).
A previous attempt to solve the issue of missing values in the covariates in
the INLA framework can be found in Gómez-Rubio and Rue (2018). They adopt a Gaussian
prior for the imputation of the missing values in the covariates and sample
from the missing data posterior distribution through INLA within MCMC.
A different approach is proposed in Chapter 8 of Blangiardo and Cameletti (2015), where a
bivariate model for spatially misaligned data is estimated by adopting the
stochastic partial differential equations (SPDE) approach
(Lindgren et al., 2011). Covariate values are imputed (in new locations) by
assuming a spatial Gaussian field which is also included in the linear
predictor of the response model. Alternatively Gómez-Rubio (2020) proposes a multiple
imputation (MI) approach (Rubin, 1987, 1996; Carpenter and Kenward, 2012) so
that the covariates are imputed multiple times through resampling so that $N$ complete
datasets are used in the analysis model. All the results are then combined
to
obtain the final estimates of the model parameters
(see Rubin, 1987, for details).
We differ from the previous approaches in that we propose a joint framework, similarly to Mason et al. (2012). Through the joint model, the uncertainty about the imputation of the missing covariates
propagates throughout the model so that it also reflects on the model
parameter estimates in the analysis. At the same time, information from the outcome in the analysis model feedback on the imputation, making it un-necessary to include the outcome in the imputation model, as commonly done in the classic MI approach. Our new approach fits naturally within the
INLA framework, can be extended to consider different types of problems (i.e.,
not only spatial models) and can be easily fit with the associated R-INLA
package for the R programming language (Gómez-Rubio, 2020).
The paper is structured as follows. In Section 2 we
review methods for missing values, while in Section 3
we introduce a novel method for missing values imputation.
Section 4 presents a
brief summary of the INLA approach to Bayesian inference and how our novel
approach fits within this framework. Section 5 shows a two
examples for the application of our proposed method and
Section 6 presents discussion points.
2 Approaches to deal with missing data
In their seminal book, Little and Rubin (2002) identify three possible
mechanisms of missingness. If the probability of being missing is the same for
all the observations, we can assume that the missing data distribution does not
depend on any of the observed or missing variables. In this case the data are
said to be missing completely at random (MCAR). If the distribution
of the missing data depends on completely observed variables (and do not depend
on the missing ones), the data are called missing at random (MAR). An
example of MAR is that women are less likely to answer questions related to
their income than men, but this has nothing to do with the income itself.
Finally, if neither MCAR or MAR holds the missing not at random (MNAR)
case occurs and the missing values distribution depends on both missing and
observed variables. For instance, in a neurological questionnaire, a subject is less likely to answer questions related to the disease if this is severe.
Under MCAR or MAR,
the missing data mechanism is ignorable.
As reported in Seaman and White (2013) this means that inferences obtained from a
parametric model for the observed data alone are the same as inferences
obtained from a joint model for the data and missingness mechanism. On the
contrary, if the data are MNAR the missing data mechanism is not ignorable and
a model for the missingness mechanism is required. It is important to note that we cannot gather evidence from
the data at hand about the missing data mechanism (MCAR, MNAR or MAR). On the
basis of the knowledge regarding the data collection methods and the assumed relationship among the collected variables, it is
possible only to make assumptions about the reasons for missing data, choose
the best corresponding strategy for data analysis (Pigott, 2001) and
conduct a sensitivity analysis on these assumptions
(Mason et al., 2012).
The simplest and most popular ad-hoc methods to deal with missing information
consists in replacing the missing data with a plausible value, such as the mean
or median calculated over the observed cases (or the mode if the variable is
categorical) or to perform a complete cases analysis (i.e., removing the
observations with one or more missing values). However, while the first method
has the potential of distorting the data distribution and to underestimating
their variability, the second one has the major drawback of reducing the power
of the study (as the dataset for the analysis will have a reduced size) and of
producing biased estimates if the MCAR assumption is not valid. To overcome
this issue, inverse probability weighting was developed, based on the idea of
assigning different weights to the different complete cases based on specific
characteristics which are relevant for the missing data; in two reviews
Carpenter et al. (2006); Seaman and White (2013) showed advantages and drawbacks of such approach.
In the last three decades model-based methods have been preferred to account
for missing data in the case of an ignorable missing data mechanism; see for
instance the papers by Little 1992;
Little and Rubin 1987; Schafer and Graham 2002. Regression
mean imputation is the simplest of the model-based methods, where the variable
with missing data is predicted based on a regression model which includes the
other variables as regressors. To overcome the issue of unreasonably lack of
uncertainty for the imputed values, stochastic regression imputation was
proposed, to generate imputed values adding some random noise
(Nakagawa, 2015).
A well established and increasingly popular model-based approach to dealing with missing data occurring in more than one variable is multiple imputation (MI) proposed by Rubin (1987, 1996). Through Monte Carlo simulation, it produces several versions of the complete dataset which differ by the imputed missing values. Then, for each complete dataset the estimates of interest are computed (by fitting a substantive model) and the results are pooled together into a final estimate which takes into account the uncertainty of the imputed data. The imputation of the missing values can be done using mainly two strategies (van Buuren, 2002): i) joint modeling, when missing values are imputed by sampling from a multivariate model fitted to the data (usually the multivariate Gaussian is used); ii) fully conditional specification (also known as multiple imputation using chained equation, MICE, van Buuren and Groothuis-Oudshoorn, 2011), when conditional univariate distributions are used to impute the missing values iteratively through a variable-by-variable approach (see White et al. 2011 for a thorough review of this method).
2.1 Bayesian inference
Bayesian inference provides a suitable framework for dealing with missing data,
as it treats missing data similarly to model parameters, making no distinction
between them. For these reasons and differently from other methods, with a
full Bayesian approach, it is possible to include the main model for data
analysis, imputation of missing values and a missingness model in a joint
estimation framework (Erler et al., 2016).
Let $\bm{z}$ denote the complete set of data. We assume that $\bm{z}=(\bm{z}_{obs},\bm{z}_{mis})$, where $\bm{z}_{obs}$ denote the observed values
while $\bm{z}_{mis}$ refer to the missing values. Moreover, let $\bm{M}$ be the
missing data indicator variable, i.e. a vector or matrix with the same length
or dimension as $\bm{z}$ with values equal to 1 (or 0) if the corresponding
values of $\bm{z}$ is missing.
Following the selection model approach (Nakagawa, 2015) the joint
distribution of $\bm{z}$, $\bm{M}$, the model parameters $\bm{\theta}_{z}$ and the parameters
in the missingness model $\bm{\theta}_{M}$ can be expressed as
$$\pi(\bm{z},\bm{M},\bm{\theta}_{z},\bm{\theta}_{M})=\pi(\bm{z},\bm{\theta}_{z})%
\pi(\bm{M}\mid\bm{z},\bm{\theta}_{M}).$$
This formulation assumes that parameters $\bm{\theta}_{z}$ and $\bm{\theta}_{M}$ are
distinct and with independent priors.
Following this, $\pi(\bm{M}\mid\bm{z}_{obs},\bm{z}_{mis},\bm{\theta}_{M})$ depends on a set of parameters $\bm{\theta}_{M}$, and models the missing data mechanism for the three cases introduced above (Little and Rubin, 2002):
MCAR
if the distribution does not depend on any of the fully or partiallyobserved variables, i.e. $\pi(\bm{M}\mid\bm{z}_{obs},\bm{z}_{mis},\bm{\theta}_{M})=\pi(\bm{M}\mid\bm{%
\theta}_{M})$.
MAR
if the distribution depends only on fully observed variables, which
means that $\pi(\bm{M}\mid\bm{z}_{obs},\bm{z}_{mis},\bm{\theta}_{M})=\pi(\bm{M}\mid\bm{z}_%
{obs},\bm{\theta}_{M})$. This implies that, given the observed data, the missingness mechanism does not depend on the unobserved data.
MNAR
if the distribution $\pi(\bm{M}\mid\bm{z}_{obs},\bm{z}_{mis},\bm{\theta}_{M})$ depends on fully and partially observed variables.
If the data are MCAR or MAR and the parameters $\bm{\theta}_{M}$ are distinct of the
parameters of the data generating process, $\bm{\theta}_{z}$, and with independent
priors, then the missing data
mechanism is ignorable and and $\pi(\bm{M}\mid\bm{z}_{obs},\bm{z}_{mis},\bm{\theta}_{M})$ can
be omitted (Seaman and White, 2013). On the contrary if the data are MNAR the missing data mechanism is not ignorable and a model for missingness is required (i.e. a logistic model)
and has to be jointly estimated with the main model, that will
include an imputation sub-model for the missing values.
Note that we can not tell from the data at hand whether the missing observations are MCAR, MNAR or MAR and at the same time it is not trivial to specify a model of missingness. In this case, a sensitivity analysis needs to be carried out to assess the impact of different scenarios for the missing data on the estimates of the model parameters (Carpenter et al., 2007; Mason et al., 2012).
2.2 Missing data in the response variable
Let $\bm{z}=(\bm{y},\bm{x})$ be the set of data including the response $\bm{y}$
and the covariates $\bm{x}$. If we assume that the covariates are fully
observed, we have that $\bm{z}_{obs}=(\bm{y}_{obs},\bm{x})$ and $\bm{z}_{mis}=(\bm{y}_{mis})$. In this case the likelihood
$\pi(\bm{z}_{obs},\bm{z}_{mis}\mid\bm{\theta}_{z})$ corresponds to the
distribution of $\pi(\bm{y}_{obs},\bm{y}_{mis}\mid\bm{x},\bm{\theta}_{y})$, with
$\bm{\theta}_{y}$ the hyperparameters in the likelihood.
If we assume that the missing data mechanism is ignorable, the imputation of
the missing data values $\bm{y}_{mis}$ is simply done through the posterior
predictive distribution $p(\bm{y}_{mis}\mid\bm{y})$. In general, we will have the
observation model by defining an appropriate distribution for the likelihood.
In addition, the mean of observation $i$, $\phi_{i}$ ,will be linked to
a linear predictor on the covariates and other effects using an appropriate
link function $g(\cdot)$, i.e.,
$$g(\phi_{i})=\beta_{0}+\sum_{p=1}^{P}\beta_{p}x_{pi}+\sum_{l=1}^{L}f_{l}(z_{li})$$
(1)
Here, $\beta_{0}$ is an intercept, $\{\beta_{p}\}_{p=1}^{P}$ the coefficients of the
$P$ covariates available $\{\bm{x}_{p}\}_{p=1}^{P}$ and $\{f_{l}(\cdot)\}_{l=1}^{L}$ represent
$L$ different non linear effects on covariates $\{\bm{z}_{l}\}_{l=1}^{L}$.
If instead data are MNAR a missing mechanism model $\pi(\bm{M}\mid\bm{y},\bm{x},\bm{\theta}_{M})$ is
required, e.g.
$$\displaystyle M_{i}$$
$$\displaystyle\sim$$
$$\displaystyle Bernoulli(p_{i})$$
$$\displaystyle logit(p_{i})$$
$$\displaystyle=$$
$$\displaystyle\gamma_{0}+\sum_{r=1}^{R}\gamma_{r}x_{ri}+\delta y_{i}$$
(2)
where $\bm{\theta}_{M}=(\gamma_{1},\gamma_{1},\ldots,\gamma_{R},\delta)$ and $M_{i}$
is a missingness indicator for $y_{i}$. In addition,
an imputation model for the missing values will be required.
However, in this work we will assume that there are no missing observations in
the response or that the missingness mechanism is ignorable, which means that
posterior inference is based on the predictive distribution.
2.3 Missing data in the covariates
We now consider the case when $\bm{z}_{obs}=(\bm{y},\bm{x}_{obs})$ and $\bm{z}_{mis}=(\bm{x}_{mis})$,
with $\bm{x}_{obs}$ the observed values of the covariates and $\bm{x}_{mis}$ the missing
ones. Distribution $\pi(\bm{y},\bm{z}_{obs},\bm{z}_{mis}\mid\bm{\theta}_{z})$ can be written as
$$\pi(\bm{y},\bm{x}_{obs},\bm{x}_{mis}\mid\bm{\theta}_{z})=\pi(\bm{y}\mid\bm{x}_%
{obs},\bm{x}_{mis},\bm{\theta}_{y})\pi(\bm{x}_{obs},\bm{x}_{mis}\mid\bm{\theta%
}_{x})$$
assuming that $\bm{\theta}_{z}=(\bm{\theta}_{y},\bm{\theta}_{x})$ is the vector of conditionally
independent parameters. The distribution $\pi(\bm{x}_{obs},\bm{x}_{mis}\mid\bm{\theta}_{x})$
represents the joint distribution of observed and missing covariates and it
includes the imputation model. For example, the joint distribution can be a
multivariate normal distribution (taking into consideration correlation between
covariates) for continuous covariates, or a discrete distribution if the
covariate is categorical.
In general, we will have the observation model as in
equation (1) together with the imputation model and the
missingness model (described in Section 3) as in equation (2.2) but only if the
missing data are MNAR.
3 Imputation of continuous missing covariates
For simplicity, we will consider the imputation of a single covariate with
missing observations, but this approach can be easily extended to consider the
imputation of missing values in several continuous covariates using a
multivariate model.
We define a latent effect
$\bm{x}^{\prime}$ as a GMRF with mean $\bm{\mu}^{\prime}(\bm{\theta}_{I})$
and precision $\bm{Q}^{\prime}(\bm{\theta}_{I})$, with $\bm{\theta}_{I}$ being the set
of hyperparameters related to the imputation procedure. The latent effect is
split in two parts $\bm{x}^{\prime}=(\bm{x}^{\prime}_{mis},\bm{x}^{\prime}_{obs})$. The distribution of $\bm{x}^{\prime}_{obs}$ is
assumed to be as close as possible to the observed covariate data
$\bm{z}_{obs}$; to guarantee this, we set the mean equal to $\bm{z}_{obs}$ and
a high precision (e.g., $10^{10}$). An imputation model is built for
$\bm{x}^{\prime}_{mis}$ (with mean $\bm{\mu}_{c}$ and precision
$\bm{Q}_{c}$), depending on $\bm{\theta}_{I}$ and whose details will be given in
Section 3.1. Finally, we will also assume that
$\bm{x}_{obs}^{\prime}$ and $\bm{x}_{mis}^{\prime}$ are independent given the
latent effect hyperparameter $\bm{\theta}_{I}$. Consequently the joint
distribution of $\bm{x}^{\prime}$ is given by
$$\bm{x}^{\prime}\mid\bm{\theta}_{I}\sim\text{Normal}\Bigg{(}\left(\begin{array}%
[]{c}\bm{\mu}_{c}\\
\bm{z}_{obs}\end{array}\right),\left[\begin{array}[]{cc}\bm{Q}_{c}&\bm{0}\\
\bm{0}&10^{10}\bm{I}\end{array}\right]\Bigg{)}.$$
(3)
3.1 Imputation model
Differently from Section 2, let $\bm{z}=(\bm{z}_{mis},\bm{z}_{obs})$ denote the complete set of values of
the covariate and we will write the response values $\bm{y}y$ separately where needed.
The imputation model will provide the distribution of the missing
values $\bm{z}_{mis}$ given the observed data $\bm{z}_{obs}$ and the
hyperparameters $\bm{\theta}_{I}$ (that take values in the parameter space
$\Theta_{I}$). In a Bayesian framework, a
sub-model is specified, where $\bm{z}_{obs}$ can be regarded as the data while $\bm{z}_{mis}$ and $\bm{\theta}_{I}$ the parameters to
estimate. Hence, we have
$$\pi(\bm{z}_{mis}\mid\bm{z}_{obs})=\int_{\Theta_{I}}\pi(\bm{z}_{mis},\bm{\theta%
}_{I}\mid\bm{z}_{obs})d\bm{\theta}_{I}=\int_{\Theta_{I}}\pi(\bm{z}_{mis}\mid%
\bm{z}_{obs},\bm{\theta}_{I})\pi(\bm{\theta}_{I}\mid\bm{z}_{obs})d\bm{\theta}_%
{I}.$$
Here, $\pi(\bm{z}_{mis}\mid\bm{z}_{obs},\bm{\theta}_{I})$ is the
conditional distribution of the missing values given the observed data and the
hyperparameters of the imputation model. Also, $\pi(\bm{\theta}_{I}\mid\bm{z}_{obs})$ can be regarded as the posterior distribution of the
hyperparameters in the imputation sub-model given the observed data. Note
that this distribution is estimated only from the observed data $\bm{z}_{obs}$,
so it can be regarded as an informative prior for $\bm{\theta}_{I}$.
Moreover, it can be rewritten as
$$\pi(\bm{\theta}_{I}\mid\bm{z}_{obs})\propto\pi(\bm{z}_{obs}\mid\bm{\theta}_{I}%
)\pi(\bm{\theta}_{I})$$
where $\pi(\bm{z}_{obs}\mid\bm{\theta}_{I})$ is obtained by integrating
$\bm{z}_{mis}$ out in the distribution of $\bm{z}$.
Finally, the hyperparameters $\bm{\theta}_{I}$ are typically modeled as exchangeable a priori.
As stated above, to derive the distribution for the imputation model $\pi(\bm{z}_{mis}\mid\bm{z}_{obs},\bm{\theta}_{I})$, we first assume a multivariate Normal distribution for the joint distribution of the complete set of covariates $\bm{z}$:
$$\bm{z}\mid\bm{\theta}_{I}\sim\text{Normal}\Bigg{(}\left(\begin{array}[]{c}\bm{%
\mu}_{mis}\\
\bm{\mu}_{obs}\end{array}\right),\left[\begin{array}[]{cc}\bm{Q}_{mis,mis}&\bm%
{Q}_{mis,obs}\\
\bm{Q}_{obs,mis}&\bm{Q}_{obs,obs}\end{array}\right]\Bigg{)}=\text{Normal}\left%
(\bm{\mu},\bm{Q}\right),$$
(4)
where both the mean and the precision matrix can depend on $\bm{\theta}_{I}$. It follows that the imputation model is defined by the following conditional distribution (Rue and Held, 2005):
$$\bm{z}_{mis}\mid\bm{z}_{obs},\bm{\theta}_{I}\sim\text{Normal}\left(\bm{\mu}_{c%
},\bm{Q}_{c}\right)$$
where $\bm{\mu}_{c}=\bm{\mu}_{mis}-\bm{Q}_{mis,mis}^{-1}\bm{Q}_{mis,obs}\left(\bm{z}_%
{obs}-\bm{\mu}_{obs}\right)$ and $\bm{Q}_{c}=\bm{Q}_{mis,mis}$. Note that $\bm{\mu}_{c}$ and $\bm{Q}_{c}$ are necessary to define the distribution of the new latent effect defined by equation (3).
We now describe two particular examples of imputation with a
typical linear regression and spatial models (useful when the covariate is
spatially correlated). However, the principles presented below can be extended to a wide range
of models, including longitudinal data, time series and other smooth
terms.
3.2 Imputation with a linear regression model
The first imputation model that we describe is based on the linear
regression model. We assume that the mean of the multivariate Normal
distribution in equation (4) is defined, considering the $n$
observations, as $\bm{X}\bm{\beta}^{\top}$. Here, $\bm{X}$ is a matrix
of $p$ fully observed covariates (columnwise) with associated coefficient
vector $\bm{\beta}=(\beta_{0},\ldots,\beta_{P})$. To match the structure of
$\bm{z}=(\bm{z}_{mis},\bm{z}_{obs})$, matrix $\bm{X}$ can be rewritten as a block
matrix as
$$\bm{X}=\left[\begin{array}[]{c}\bm{X}_{mis}\\
\bm{X}_{obs}\\
\end{array}\right]$$
Under the linear regression model, we assume that the mean of $\bm{z}$ depends on a linear
combination of the fully observed covariates, i.e. $\bm{\mu}=E(\bm{z})=\bm{X}\bm{\beta}^{\top}$. By adopting the block notation, we thus assume the
following joint distribution:
$$\bm{z}\mid\bm{\theta}_{I}\sim\text{Normal}\Bigg{(}\left(\begin{array}[]{c}\bm{%
X}_{mis}\bm{\beta}^{\top}\\
\bm{X}_{obs}\bm{\beta}^{\top}\end{array}\right),\left[\begin{array}[]{cc}\tau%
\bm{I}_{mis}&\bm{0}\\
\bm{0}&\tau\bm{I}_{obs}\end{array}\right]\Bigg{)}$$
where $\tau$ is the precision hyperparameter and $\bm{I}_{mis}$ and $\bm{I}_{obs}$ are identity matrices whose dimensions depend on the number of missing and observed data in $\bm{z}$. In this case the vector of hyperparameters
is given by $\bm{\theta}_{I}=(\bm{\beta},\tau)$. Note that, given $\bm{\theta}_{I}$, observations
are assumed independent of each other, which simplifies the model.
Following the approach presented in Section 3.1, we obtain that the conditional distribution of $\bm{z}_{mis}\mid\bm{z}_{obs},\bm{\theta}_{I}$ (i.e. the imputation model) has
the following mean and precision:
$$\bm{\mu}_{c}=\bm{X}_{mis}\bm{\beta}^{\top}\qquad\qquad\bm{Q}_{c}=\tau\bm{I}_{mis}$$
As stated above, note that $\bm{\beta}$ and $\tau$ are informed by $\pi(\bm{\beta},\tau\mid\bm{z}_{obs})$, which is proportional to $\pi(\bm{z}_{obs}\mid\beta,\tau)\pi(\beta,\tau)$.
Finally, priors must be set on the hyperparameters. For simplicity, each of the
elements in $\bm{\beta}$ is assigned a Normal distribution with zero mean and
large precision. Parameter $\tau$ has a vague prior (e.g., a Gamma distribution
with large variance).
All hyperparameters are independent a priori, so that
$\pi(\bm{\theta}_{I})=\pi(\tau)\Pi_{i=0}^{P}\pi(\beta_{i})$. Note that other
priors could be easily considered here.
3.3 Imputation with a spatial model
When the covariate to be imputed is spatially correlated
we can assume a conditional
autoregressive specification (Gelfand, 2010, Chapter 13) so that
the mean is $\bm{\mu}=\bm{\alpha}^{\top}=(\alpha,\ldots,\alpha)^{\top}$ and
the precision is $\bm{Q}=\tau(\bm{I}-\rho\bm{W})$. Here, $\alpha$ is the intercept
of the linear predictor, $\rho$ is a spatial autocorrelation parameter, and
$\bm{W}$ is an adjacency matrix, defining the sets of neighbours. This is
often scaled
dividing it by its largest eigenvalue as this will allow us to take $\rho$ in
the $(0,1)$ interval.
Note that $\bm{W}$ can be rewritten as a block matrix
with four sub-matrices according to missing and observed values, as done with
$\bm{Q}$ in equation (4). The vector of hyperparameters is now given
by $\bm{\theta}_{I}=(\tau,\rho,\alpha)$.
Adopting the block notation, under the CAR specification for imputation we thus
assume the following joint distribution for $\bm{z}=(\bm{z}_{mis},\bm{z}_{obs})$:
$$\bm{z}\mid\bm{\theta}_{I}\sim\text{Normal}\Bigg{(}\left(\begin{array}[]{c}\bm{%
\alpha}_{mis}^{\top}\\
\bm{\alpha}_{obs}^{\top}\end{array}\right),\left[\begin{array}[]{cc}\tau(\bm{I%
}_{mis}-\rho\bm{W}_{mis,mis})&\bm{-}\tau\rho\bm{W}_{mis,obs}\\
\bm{-}\tau\rho\bm{W}_{obs,mis}&\tau(\bm{I}_{obs}-\rho\bm{W}_{obs,obs})\end{%
array}\right]\Bigg{)}.$$
It then follows that the conditional distribution of $\bm{z}_{mis}\mid\bm{z}_{obs},\bm{\theta}_{I}$ (i.e. the imputation model) is characterized by
the following mean and precision matrix:
$$\bm{\mu}_{c}=\bm{\alpha}_{mis}^{\top}-(\bm{I}_{mis}-\rho\bm{W}_{mis,mis})^{-1}%
(-\rho\bm{W}_{mis,obs})(\bm{z}^{\top}_{obs}-\bm{\alpha}_{obs}^{\top})$$
$$\bm{Q}_{c}=\tau\left(\bm{I}_{mis}-\rho\bm{W}_{mis,mis}\right)$$
Again, $\tau$, $\rho$ and $\alpha$ are informed by $\pi(\tau,\rho,\alpha\mid\bm{z}_{obs})$, which is proportional to the product $\pi(\bm{z}_{obs}\mid\tau,\rho,\alpha)\pi(\tau,\rho,\alpha)$.
Finally, $\alpha$ is given a Gaussian prior with zero mean and small precision,
$\tau$ is assigned a vague prior (e.g., a Gamma distribution with a small precision), while
$logit(\rho)$ is
assigned a Gaussian prior with zero mean and small precision (see, for
example, Gómez-Rubio, 2020, Chapter 5, for details on why this parameterization is used).
3.4 Extension to the imputation of categorical missing covariates
The imputation of the missing values in categorical variables does not fit into
the GMRF framework described in Section 3 as these
variables are defined in a discrete space. For this reason, a different
approach will be considered for defining the imputation model
$\pi(\bm{z}_{mis}\mid\bm{z}_{obs})$ and for estimating the model. In particular,
as imputation model we will consider a multinomial likelihood which can be
fit with INLA by using the multinomial-Poisson transformation
(Baker, 1994).
Note that in this case the procedure is similar to the multiple imputation
approach: the imputation model is specified where the categorical variables
with missing values are considered as the response variables, so that the
predictive distribution of the missing observations can be computed. Similarly
to the case of missing data in the response, values are sampled to fill the
missing values in the covariates. Then, the analysis model is run by using the
imputed covariates as completely known. This procedure is repeated by
simulating several samples and estimating the corresponding models; finally,
all the resulting models are averaged by using Bayesian model averaging (Gómez-Rubio and Rue, 2018). Note
that this approach does not produce feedback in the estimation of the
parameters of the imputation model as in the previous approach, given that it is done in two-stages rather than jointly. For this reason, and similarly to the classical MI, the outcome $\bm{y}$ should be included in the imputation model. Alternatively,
INLA within MCMC can be used to fit the joint model using a fully Bayesian
approach (see the example in Gómez-Rubio and Rue, 2018).
Inference on the model parameters when multiple imputation of a categorical
covariate can be summarized as follows. Considering the generic
parameter $\theta_{k}$ we can write its posterior marginal
distribution as:
$$\pi(\theta_{k}\mid\bm{z}_{obs},\bm{y})=\sum_{\bm{z}_{mis}\in\Theta_{mis}}\pi(%
\theta_{k},\bm{z}_{mis}\mid\bm{z}_{obs},\bm{y})=\sum_{\bm{z}_{mis}\in\Theta_{%
mis}}\pi(\theta_{k}\mid\bm{z}_{obs},\bm{z}_{mis},\bm{y})\pi(\bm{z}_{mis}\mid%
\bm{z}_{obs},\bm{y}).$$
Here, $\Theta_{mis}$ represents the parametric space of the missing values
of the categorical covarite, which in a Bayesian framework are considered
to be random variables.
Given $L$ samples $\{\bm{z}_{mis}^{(l)}\}_{l=1}^{L}$ from
$\pi(\bm{z}_{mis}\mid\bm{z}_{obs},\bm{y})$, the previous marginal can be
approximated as
$$\pi(\theta_{k}\mid\bm{z}_{obs},\bm{y})\simeq\frac{1}{L}\sum_{l=1}^{L}\pi(%
\theta_{k}\mid\bm{z}_{obs},\bm{z}^{(l)}_{mis},\bm{y}),$$
where $\pi(\theta_{k}\mid\bm{z}_{obs},\bm{z}^{(l)}_{mis})$ is the
marginal of $\theta_{k}$ obtained from fitting the original model with the
observed data and the imputed covariate $\bm{z}^{(l)}_{mis}$.
Note that when continuous covariates with missing values are also present both
approaches can be combined. For example, an imputation sub-model can be
combined for the continuous covariate which is part of the joint model that is
fit to every simulated dataset where only the missing values of the categorical
covariate are filled in. Furthermore, a missingness
sub-model for the categorical variables can be incorporated into the model
similarly to the one used for the continuous variables.
4 The Integrated Nested Laplace Approximation approach (INLA)
The approach presented in the previous sections overcome a major limitation in INLA, as at present it cannot cope with missing values in covariates. We present here an introduction to the INLA method and the computationa details; then we focus on how to implement our proposed framework.
INLA (Rue et al., 2017; Martino and Riebler, 2019; Gómez-Rubio, 2020) is a deterministic
approach for Bayesian inference. It is designed for the class of latent
Gaussian Markov random field models, where the response $y_{i}$ observed for the $i$-th unit is
assumed to belong to a distribution family (usually part of the exponential
family). This is often characterized by a parameter $\phi_{i}$ defined as a
function of a structured additive predictor $\eta_{i}$ through a link function
such that $g(\phi_{i})=\eta_{i}$ (e.g. the logarithm function is used for
Poisson data). The linear predictor is defined as follows
$$\eta_{i}=\beta_{0}+\sum_{j=1}^{n_{\beta}}\beta_{j}z_{ji}+\sum_{k=1}^{n_{f}}f^{%
(k)}(u_{ki}),\qquad i=1,\ldots,n$$
(5)
where $\beta_{0}$ is the intercept, the coefficients $\bm{\beta}=(\beta_{1},\ldots,\beta_{n_{\beta}})$ quantify the (linear) effect of some covariates
$\bm{z}=\{\bm{z}_{j}\}_{j=1}^{n_{\beta}}$ on the response, and $\bm{f}=\left\{f^{(1)}(\cdot),\ldots,f^{(n_{f})}(\cdot)\right\}$ is a set of functions
defined in terms of some covariates $\bm{u}=\{\bm{u}_{k}\}_{k=1}^{n_{f}}$.
Through functions $f(\cdot)$ it is possible to include in the model random
effects (perhaps indexed in space and time), smooth and non-linear effects of
the covariates. For this reason, the class of latent GMRF models can
accommodate a wide range of models, from standard generalized linear models
(GLM) to generalized linear mixed models (GLMM), including data for time
series, lattice data, point pattern and geostatistical data.
As stated, the vector of latent effects $\bm{\chi}=\{\bm{\eta},\beta_{0},\bm{\beta},\bm{f}\}$ is a latent GMRF in the model, which depends on some hyperparameters $\bm{\theta}_{2}$. Moreover, observations are assumed to be independent given the
latent effects $\bm{\chi}$ and the likelihood hyperparameters denoted by $\bm{\theta}_{1}$. For convenience in the following we will denote the vector of
hyperparameters with $\bm{\theta}=(\bm{\theta}_{1},\bm{\theta}_{2})$.
The objectives of Bayesian inference with INLA are the marginal posterior distributions
for each element of the latent effects and hyperparameters vector denoted by
$p(\chi\mid\bm{y})$ and $p(\theta\mid\bm{y})$, respectively. INLA
provides deterministically accurate approximations to these distributions in a
short computing time by using the Laplace approximation and numerical
integration.
Each latent GMRF model can be rewritten in a hierarchical fashion with three levels:
1.
The model for the observed data $\bm{y}=(y_{1},\ldots,y_{n})$ (i.e. the likelihood) defined as a function of some parameters $\bm{\chi}$ and hyperparameters $\bm{\theta}$:
$$\bm{y}\mid\bm{\chi},\bm{\theta}\sim\pi(\bm{y}\mid\bm{\chi},\bm{\theta})=\prod_%
{i\in n}\pi(y_{i}\mid\chi_{i},\bm{\theta}).$$
2.
The model for the latent effects $\bm{\chi}$
$$\bm{\chi}\mid\bm{\theta}\sim\text{Normal}\left(\bm{0},\bm{Q}(\bm{\theta})\right)$$
where $\bm{Q}(\bm{\theta})$ is a sparse precision matrix given the GMRF assumption.
3.
The model for the complete vector of hyperparameters: $\pi(\bm{\theta})$. As usually hyperparameters are assumed to be independent a priori, $\pi(\bm{\theta})$ will be defined as the product of different univariate prior distributions.
Given all these models and components the joint posterior distribution of the random effects and the hyperparameters is given by
$$\pi(\bm{\chi},\bm{\theta}\mid\bm{y})\propto\pi(\bm{y}\mid\bm{\chi},\bm{\theta}%
)\pi(\bm{\chi}\mid\bm{\theta})\pi(\bm{\theta}).$$
INLA computes the posterior marginals of the hyperparameters and latent effects
using that representation by means of numerical integration and the Laplace
approxiamtion (see Rue et al., 2009, for details).
4.1 Computational details
The INLA approach is implemented through an R package named
R-INLA, which is available from the INLA website
(http://www.r-inla.org/home). The model to be fit is defined by setting
an formula with all the additive latent effects in the model, which includes
fixed and random effects. The R-INLA package includes a good number of
implemented latent effects but others can be implemented as well (see,
for example Gómez-Rubio, 2020). Note that by default, when R-INLA finds
missing values in the covariates (which have the value NA in
R) they are replaced by zeros so that the effect of the covariate does
not affect the linear prediction of that subject. However,
this is an issue that could result in biased estimates of the coefficients
of the covariates. This is described in the
R-INLA list of frequently asked questions (FAQ) in the package
website. If the missing value is found in the response variable, the
predictive distribution is computed.
Generic latent effects can be implemented by defining their structure as a
latent GMRF. This means definining the mean, precision, hyperparameters and the
priors of the hyperparameters. These are known as rgeneric latent
effects in R-INLA (see, for example Gómez-Rubio, 2020, Chapter 11). Once a
new latent effect is defined, it can be easily incorporated as any other
additive effect in the model formula.
For the new latent effects described in this paper, note that it is
defined as in equation (3), so that the only
difference will be in how the mean $\bm{\mu}_{c}$ and precision $\bm{Q}_{c}$
of the block of the missing values is defined. Remember that the block
of the observed covariates is simply there to make those values of
the latent effect to be as close as possible to the observed values
and that it does not depend on any hyperparameter or other data.
Furthermore, the role of the prior on the hyperparameters of the imputation
model $\bm{\theta}_{I}$ is now taken by distribution $\pi(\bm{\theta}_{I}\mid\bm{z}_{obs})$. Hence, the actual prior used in the latent effects is taken as
$$\pi(\bm{\theta}_{I}\mid\bm{z}_{obs})\propto\pi(\bm{z}_{obs}\mid\bm{\theta}_{I}%
)\pi(\bm{\theta}_{I})$$
and the normalizing constant is ignored as it is not needed. In a typical
implementation of a latent effect, the prior of $\bm{\theta}_{I}$ would be a
typical distribution density that depends on a set of fixed hyperparameters,
but now the prior of $\bm{\theta}_{I}$ is made of the product of the two terms
above. For this reason, it can be regarded as an informative prior as it is
essentially estimated from a model fit to $z_{obs}$. This is what will allow
the latent effect to produce good estimates of the missing values (if the
imputation model is correct). In general, there is no way to assess this, but
the more covariates used in the imputation model the better (see Gelman and Hill, 2007, Chapter
25).
The actual prior of the model hyperparameters is $\pi(\bm{\theta}_{I})$ and this
can take different forms depending on the number and type of hyperameters
in the model. Usually, this will be split into the product of several univariate
prior distributions.
Note also that R-INLA works with unbounded hyperparameters, so that
the parameters in $\bm{\theta}_{I}$ may need to be transformed when the latent
effect is defined. This may also require to include additional terms
in the prior (see, for example Gómez-Rubio, 2020, Chapter 11).
A typical example is to use internally the log-precision instead of the
precision.
Once the imputation latent effect is included in the model formula
it will be part of the joint latent effect $\bm{\chi}$ and incorporated
into the Bayesian model, so that a full Bayesian approach is used to
estimate all the model parameters.
As stated in previous sections, a missingness sub-model can be
included (in addition to an imputation one) for the case in which
missingness is MAR or MNAR. Including a missingness model requires defining a model with two likelihoods:
one for the main model and a binomial model for the missingness indicator
variables. Note that under MCAR and MAR both models are independent, hence the latter is not needed; however, under MNAR it is necessary to explicitly include it and to make it dependent on the variables with imputed values.
Hence, there will be feedback between both models that may affect the imputation
process and the estimation of the other model parameters.
Full details about how to fit these models in R are provided in the
Supplementary materials together with the associated R code for the examples
developed in Section 5.
5 Examples
In this section we develop two examples to show how the imputation method
proposed above can be used with INLA under MCAR, MAR and MNAR. The first
example shows typical regression model in biostatistics with real missing
data. This is useful to show how a typical multiple linear regression can be
used for multiple imputation. The second one is based on spatially correlated
data to assess the performance of our proposal on a simulated study in which a
spatially correlated covariate is missing. Note that the aim is not to provide
a comprehensive analysis of the dataset with missing values but to illustrate
the methods described in this paper.
All models have been fit with INLA and its associated R package
R-INLA. The latent effects required to impute the missing values of
the covariates are implemented in a new R package called
MIINLA which is available from the Github repository at
https://github.com/becarioprecario/MIINLA.
The R code to run the examples is available from
https://github.com/becarioprecario/MIINLA_paper.
5.1 Imputation using linear models
The nhanes2 dataset (Schafer, 1997) in the mice R
package (van Buuren and Groothuis-Oudshoorn, 2011) records data on 25 participants in the National Health
and Nutrition Examination Survey (NHANES). Variables in the dataset include
body mass index, cholesterol level, age group and hypertensive status. The
dataset presents missing observations in body mass index, hypertensive status
and cholesterol level.
We will use this dataset to build a model to explain cholesterol level on age
group and body mass index, where this is imputed. The imputation model will be
based on a linear regresion on the age group. There are three age groups 20-39,
40-59 and 60+ years, and the first group will be set as the reference level.
It is worth noting that having missing values in the response variable (i.e.,
cholesterol level) is not a problem as the predictive distribution can be
easily computed with INLA. Hence, the output from fitting this model will
include the posterior distribution of the imputed values as well as
the predictive distribution for the missing responses.
The analysis model is the following:
$$chol_{i}=\alpha+\beta_{1}age^{40-59}_{i}+\beta_{2}age^{60+}_{i}+\beta_{3}bmi_{%
i}+\varepsilon_{i},\ i=1,\ldots,25$$
where $chol_{i}$ refers to the cholesterol level, $bmi_{i}$ to the body mass index,
$age^{40-59}_{i}$ and $age^{60+}_{i}$ are indicator variables of age for groups
40-59 and 60+, respectively, and $\varepsilon_{i}$ is a Gaussian error term with
zero mean and precision $\tau$.
Note that the missing values of $bmi_{i}$ are obtained from the imputation model
based on linear regression discussed above using as predictors variables
$age^{40-59}_{i}$ and $age^{60+}_{i}$. The imputation model is specified as
$$bmi_{i}=\alpha_{I}+\beta_{I1}age^{40-59}_{i}+\beta_{I2}age^{60+}_{i}+%
\varepsilon_{Ii},\ i\in\mathcal{I}.$$
Here, $\mathcal{I}$ represents the set of indices of the observations with
missing values of body mass index. Parameters $\alpha_{I}$, $\beta_{I1}$,
$\beta_{I2}$ represent the intercept and the covariate coefficients used in the imputation model, and $\varepsilon_{Ii}$ is a
Gaussian error with zero mean and precision $\tau_{I}$. Note that all the
parameters in the imputation model are mainly informed from the observed values
of the body mass index and age, and their prior distributions. Because the
imputation model is part of the joint model there is also feedback from all the
other parts of the model when estimating the imputation model parameters and
the imputed values of body mass index.
Finally, a logistic regression is used on the missingness status of $bmi_{i}$
using a logistic regression with a particular linear predictor. We tried
three different approaches to assess missingness under MCAR, MAR and MNAR.
Under MCAR, the linear predictor has simply an intercept term, under MAR it is
an intercept plus the covariate of age group, and under MNAR it is the
intercept plus covariate $bmi_{i}$ (that includes the imputed values).
The coefficient of $bmi_{i}$ would indicate whether the
values of $bmi_{i}$ have been missed completely at random or following a different
scheme.
This model can be represented as
$$\displaystyle M_{i}$$
$$\displaystyle\sim$$
$$\displaystyle Bernoulli(p_{i}),\ i=1\ldots,25$$
$$\displaystyle logit(p_{i})$$
$$\displaystyle=$$
$$\displaystyle\alpha_{M}+\beta_{M1}age^{40-59}_{i}+\beta_{M2}age^{60+}_{i}+%
\delta bmi_{i}$$
(6)
where $M_{i}$ is a missingness indicator for $bmi_{i}$ (0 for observed and 1 for
missing). The priors for the coefficients of the fixed effects are independent
Normal distributions with zero mean and precision 0.001. For the precision
parameters, a Gamma with parameters 1 and 0.00005 is used to provide a vague
prior. All parameters are considered to be independent a priori.
Table 1 shows the different estimates for all the models
considered. Regarding the main Gaussian sub-model, it seems that all three
covariates included in the model play a significant role when explaining
cholesterol level. In addition, point estimates are very similar across
different missingness mechanisms. In the imputation sub-model, we also observe
that point estimates are very similar across missingness mechanisms. Age also
plays an important role when imputing the missing values of body mass index.
Finally, the different sub-models for the missingness mechanism are not
directly comparable. Under MCAR, parameter $\alpha_{M}$ estimates the log-proportion of missing values in the covariate. Under MAR, $age^{40-59}$ helps
to explain why some values of body mass index are missing. Lastly, under
MNAR the missing values do not appear to depend on their actual values
as the estimate of $\delta$ is close to zero.
Cholesterol level seems to
increase with age. In addition, the imputation models points to that body mass
index seems to decrease with age. Although this is counterintuitive, we
believe that is due to the general pattern observed in the dataset, which
contains data on 25 people and only 13 of them have completely observed
covariates.
As a final remark, it is worth noting that fitting these models took a few
seconds. Hence, the sensitivity analysis could include other models than the
ones presented here. See, for example, Mason et al. (2012) for a
general discussion and alternative models for the sensitivity analysis.
Larger datasets may take longer to run, but INLA will be able to fit
these models faster than typical MCMC algorithms.
5.1.1 Imputation of categorical covariates with missing values
As we have mentioned in the description, this dataset includes an
indicator of hypertensive status of the subjects. This categorical covariate
also contains several missing values. To illustrate how missing values in
continuous and categorical covariates can be handled at the same time we
fit a model in which body mass index and hypertensive status are included. The
imputation of body mass index will be done within the joint model as previously described, but the imputation of hypertension will be done
using a multiple imputation approach; this means that an imputation model will be fit for
hypertension, values of hypertensive status sampled from this model and used to fill
the gaps in the original dataset. This will provide a number of complete
datasets to which the analysis model will be fit; then the results will be
pooled to obtain final estimates using Bayesian
model averaging with equal weights (Gómez-Rubio et al., 2019).
The analysis model becomes:
$$chol_{i}=\alpha+\beta_{1}age^{40-59}_{i}+\beta_{2}age^{60+}_{i}+\beta_{3}bmi_{%
i}+\beta_{4}hyp_{i}+\varepsilon_{i},\ i=1,\ldots,25$$
For simplicity, the missingness mechanism will not be asessed now. This
implies assuming MCAR, but we have already seen that the model estimates
will be close to model fit under MAR and MNAR for the case of body mass
index.
The imputation model for hypertensive status ($hyp_{i}$) will be a
multinomial model fit using the multinomial-Poisson transformation
(Baker, 1994). This will provide estimates of the
posterior probabilities of being hypertensive given the age group,
which will be used to impute the missing values according to the
age group of the patient. These posterior probabilities are
shown in Table 2. Note that in this particular case
a logistic regression would have been enough, but we have preferred
to use the multinomial-Poisson transformation because it is a more
general approach for the case of more than two categories.
We have drawn 100 samples to fill in the missing values of the hypertensive
status, so that 100 different completed datasets have been used to fit
the model. The resulting models have been pooled to obtained the posterior
marginals of the model parameters using Bayesian
model averaging with equal weights (Gómez-Rubio et al., 2019). These are shown in Table 3.
As expected, the estimates of the coefficients of age are close to the ones in
the previous models. The coefficient of hypertensive status is close to zero,
which indicates no association between cholesterol level and hypertensive
status. Furthermore, the imputation model for body mass index based on a linear
regression on age provides similar estimates to the imputation models fit
previously and with similar effects of age on body mass index.
5.2 Simulation study: Imputation of correlated data
The second example that we present is a simulation study based on the North
Carolina Sudden Infant Death Syndrome (SIDS) dataset. It records several data,
which includes the number of sudden infant deaths per county in the period 1974-78
($O_{i}$), the total number of births ($N_{i}$), as well as the number of non-white
births ($NW_{i}$). The expected number of cases in each county ($E_{i}$) can be
obtained using internal standardization, so that the standardized mortality
ratio (SMR) can be computed as $O_{i}/E_{i}$. Furthermore, several authors
(see, for example, Cressie, 2015) have described the strong spatial
pattern in the data, in the relative risk (estimated using the SMR, for
example) and its correlation with the proportion of non-white births.
The model of interest to be fit is simply a Poisson regression, as follows:
$$O_{i}\sim Po(\mu_{i});\mu_{i}=E_{i}\theta_{i},\ i=1,\ldots,100$$
$$\log(\theta_{i})=\alpha+\beta\;nwpropbirths_{i}$$
Here, the covariate $nwpropbirths_{i}$ is the logit of the proportion of
non-white births ($NW_{i}$), that has been re-centered and re-scaled so that it
is not bounded. This derived covariate has still a strong spatial pattern
and a high correlation with the SMR.
Figure 1 shows the SMR for the period 1974-78 and the
transformed proportion of non-white births ($nwpropbirths_{i}$). The SMR shows some
areas of high risk and a strong correlation with the proportion of non-white
births. Hence, this covariate can be useful when building models
to explain the spatial variation of SIDS in North Carolina.
The simulation study will remove 5%, 10%, 15%, 30% and 50% of the
covariate values (i.e., proportion of non-white births) using MCAR and MNAR
mechanisms. Note that MAR can be regarded as an extension to MCAR that
considers other observed covariates in the linear predictor of the logistic
regression in the imputation model. Although MAR may seem more
reasonable, it is simply a matter of including other covariates in the linear
predictor of the missingness model so it is computationally feasible but
it adds little to the comparisson. This is why we have not considered it.
The missing observations will be nested accross the five scenarios, i.e., the
observations removed in the 10% secenario will also be removed in the 15%
scenario and so on. Furthermore, the probability of being missing under the
MNAR mechanism $p_{i}$ is
$$logit(p_{i})=\alpha_{M}+5x_{i}$$
where $\alpha_{M}$ is set as the logit of 0.5 and $x_{i}$ represents the value
of the covariate with missing values.
This simulation is intended to compare mild to severe missingness under five
different scenarios for MCAR and MNAR. Models will be fit assuming MCAR and
MNAR missingness, so that we fit 20 models in total. Under MCAR, we only
fit the analysis and imputation model. Under MNAR, in addition we will
assess whether the joint approach including the missingness mechanism is able to capture the type of missingness.
Figure 2 shows the missing values of the proportion of
non-white births for three of the scenarios considered in this simulation
study. As it can be seen, when the percentage of missing values is 50% under
MNAR missing values concentrate in the counties with high values of the
covariate.
In addition, the imputation model proposed is based on the conditional autoregressive specification presented in Section 3.3, so that imputation is included within the main model. This imputation
model will have the following parameters: $\tau_{I}$ is the precision of the CAR
specification, $\rho_{I}$ the spatial autocorrelation and $\alpha_{I}$ the mean value of
the covariates.
Finally, a logistic regression on the missingness variable $m_{i}$ (0 for
observed and 1 for missing) is used to model the missingness mechanism
(under MNAR):
$$\displaystyle m_{i}$$
$$\displaystyle\sim$$
$$\displaystyle Bernoulli(p_{i});\ i=1,\ldots,100$$
$$\displaystyle logit(p_{i})$$
$$\displaystyle=$$
$$\displaystyle\alpha_{M}+\beta_{M}nwpropbirths_{i}$$
Note that the imputed values appear both in the Poisson regression and the
sub-model on the missingness mechanism. Non-zero values of $\beta_{M}$ indicate
that the probability of being missing depends on the actual values.
Table 4 summarizes the models fit to the data under MCAR.
Here, an imputation sub-model for body mass index has been included but not a
joint model for the missingness as under MCAR it is not necessary. In general,
there are not large differences between the different models fit to the
datasets regarding percentage of missing values and type of actual missingness.
However, these differences become larger as the proportion of missing values
increases, which was to be expected. These differences are noticeable for the
case of 50% of missing values both under MCAR and MNAR.
The estimates of the imputation models are quite similar as well,
across missingness type in the data and proportion of missing values.
However, some differences are observed for 30% and 50% of missing values.
In particular, the estimates of $\alpha_{I}$ differ.
Table 5 summarize the (joint) models fit to the data
considering a MNAR scenario. This includes the model fit to the complete
dataset, and the binomial sub-model in the joint model to assess the
missingness mechanism. First of all, the posterior distribution of $\beta_{M}$
helps to determine the missingness mechanism. Its posterior estimate is very
close to zero under MCAR, while it is above zero under MNAR (but for the case
of 5% of missing values). It is worth stating that we can assess this now
because this is simulated data and the true missingness mechanism is known.
Regarding the imputation model, the estimates are very similar across
scenarios. Finally, the estimates of the parameters in the Poisson model
are in general very close to the model fit to the full dataset.
It is worth noting that under MNAR with 50% of missing observations the point
estimates of the parameters in the Poisson sub-model show the largest departure
from the model fit to the full dataset. This is probably due to the fact that
the imputation model is not able to fully recover the values of the covariates
as missing values tend to have high values and there is not enough information
in the observed values as to recover this pattern.
To sum up, we believe that the imputation model behaves as expected and
provides a good performance in all cases. Most importatly, the joint model is
able to identify between MCAR and MNAR situations as well as imputing the
covariates and fit the model of interest to the data. Again, ths is possible
now because the missingness mechanism is known but in real applications
we would propose different models and conduct a sensitivity analysis.
When the models fit under MCAR (Table 4) and under MNAR
(Table 5) are compared, it should be mentioned that when
data under MCAR are analysed both models produce very similar results because
the missingness mechanism is independent of the observed data. For the
analysis of the data simulated under MNAR, differences can be observed because
now the missingness mechanism depends on the covariate (including the imputed
data) and the estimates of the parameters in the imputation sub-model are
different.
Finally, we have included the posterior distributions of some imputed values of
the covariate in Figure 3. In particular, we have considered
the dataset with 50% missing values under MNAR and taken nine counties with
missing values that have missing values also in the simulated data under MCAR.
This produces a set of counties with a wide variety in the posterior marginals
of the imputed values. The posterior marginals shown are for the imputation
model under MCAR in Table 4 (dashed line) and the imputation
model under MNAR in Table 5 (dotted line). The vertical
solid line shows the actual value of the missing covariate. Furthermore, we
have kept the same axes scale in all plots so that differences are appreciated
better.
In general, both marginals are close in all cases. Under MNAR (dotted
lines), the posterior mode seems to be closer to the actual value
for most of the counties in the plot. This should not be surprising as
this is the actual missingness mechanism in the data.
It is worth noting that when the missing data obtained under MCAR are considered
the posterior marginals of the analogous models look the same in the plots.
This shows that handling imputation of missing values with INLA is an
interesting way to conduct sensitivity analysis.
6 Discussion
We have shown how the general problem of dealing with missing observations
in the covariates and performing multiple imputation under different missingness
mechanisms can be recast within the framework on latent Gaussian Markov
random field models. This has the main advantadge that models expressed
as latent GMRFs can be fit with the INLA methodology for speed. Furthermore,
this fills an important gap in the INLA methodology as now models with
missing values in the covariates can be easily fit.
Imputation models for the covariates can also take many different forms when
defined as GMRFs. In this work we have only considered a linear regression model
and spatially correlated model for imputation, but other similar imputation
models could be easily developed. For example, these could tackle
missing observations in longitudinal data or time series.
Furthermore, the methods proposed can be extended to consider imputation
of more than one covariate at the same time by relying on multivariate
Gaussian models.
The implementation of the multiple imputation models take the form of new
latent effects for the R-INLA package and they are available within
the MIINLA package for the R programming language. These new
latent effects have been developed using the rgeneric framework for
latent effects development within the R-INLA package. Nonetheless,
this approach could be implemented in any other software packages for Bayesian
inference.
Although we have focused on imputation of continuous covariates, missing values
in categorical covariates can also be handled. However, as stated in the paper,
this case does not fit within the paradigm of latent GMRF models easily.
However, INLA can be used to propose an imputation model for the missing
categorical data and to fit the model of interest to these full datasets. The
fitted models can then be combined to account for the uncertainty of the imputed
values in the estimation of the model parameters using Bayesian model
averaging.
When the missing values of the categorical covariates index a latent effect
the imputation of missing values becomes more complex. This is the
case, for example, when random effects are estimated for different groups
in the data using multilevel models. However, this scenario could also be
handled using the multiple imputation methods described in this paper.
In addition to handling and imputing missing values, this new framework allows
us to consider the missingness mechanism using a joint model fit within the
INLA methodology. Hence, the analysis of data with missing observations can now
be completely carried out within the INLA framework.
Sensitivity analysis on the missingness mechanism, required when it is not
ignorable, can benefit from the the computational speed of the INLA method.
First of all, models are fit faster than with typical MCMC methods, which helps
to define the scenarios to test. Secondly, more scenarios can be tested as the
time required to fit the models is reduced.
Acknowledgements
V. Gómez-Rubio has been
supported by grant MTM2016-77501-P from the Spanish Ministry of Economy and Competitiveness co-financed with FEDER funds, and grant SBPLY/17/180501/000491 funded by Consejería de Educación, Cultura y Deportes (JCCM, Spain) and FEDER. Marta Blangiardo acknowledges partial support through the grant R01HD092580 funded by the National Institute of Health.
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A scalable optical detection scheme for matter wave interferometry
Alexander Stibor, André Stefanov, Fabienne Goldfarb${}^{1}$, Elisabeth
Reiger, and Markus Arndt
Institut für Experimentalphysik, Universität Wien, Boltzmanngasse 5, A-1090
Wien
${}^{1}$Laboratoire Aimé Cotton, F-91405 Orsay Cedex
markus.arndt@univie.ac.at
(December 4, 2020)
Abstract
Imaging of surface adsorbed molecules is investigated as a novel
detection method for matter wave interferometry with fluorescent
particles. Mechanically magnified fluorescence imaging turns out
to be an excellent tool for recording quantum interference
patterns. It has a good sensitivity and yields patterns of high
visibility. The spatial resolution of this technique is only
determined by the Talbot gratings and can exceed the optical
resolution limit by an order of magnitude. A unique advantage of
this approach is its scalability: for certain classes of nanosized
objects, the detection sensitivity will even increase
significantly with increasing size of the particle.
pacs: 03.65.Ta,03.65.-w,03.75.-b,39.20.+q,33.80.-b,42.50.-p
1 Introduction
Recent years have seen a tremendous progress in the development of
various matter wave interference experiments, using
electrons [1, 2], large ultra-cold atomic
ensembles [3], cold
clusters [4, 5] or hot
macromolecules [6, 7, 8].
Interferometry is also expected to lead to interesting
applications in molecule metrology and molecule lithography. In
particular quantum interference of complex systems is intriguing
as it opens new ways for testing fundamental decoherence
mechanisms [9].
Further progress along this line of research requires an efficient
source, a versatile interferometer and a scalable detection
scheme. Scalable sources represent still a significant
technological challenge, but an appropriate interferometer scheme
has already been suggested [10] and successfully
implemented [7] for large molecules. All coherence
experiments with clusters or molecules up to date finally employed
ion detection. However, most ionization schemes run into
efficiency limits when the mass and complexity of the particles
increases [11]. Surface adsorption in combination
with fluorescence detection is therefore a promising alternative.
Its high efficiency will reduce the intensity constraints on
future molecular beam sources for interferometry. It appears to be
an important prerequisite not only for experiments exploring
molecular coherence beyond 10,000 amu but also for decoherence
and dephasing experiments with various dyes below 1,000 amu.
In the present article we demonstrate the feasibility of optically
detecting matter wave interference fringes of dye molecules. Such
structures usually have periods between $100...1000$ nm and would
be hardly resolved in direct imaging. But we show that a
mechanical magnification step is a simple and very efficient
technique to circumvent the optical resolution limit for
interferograms. We can thus combine the high sensitivity of the
fluorescence method with the high spatial resolution of our
interferometer setup.
2 Experimental setup
2.1 Talbot-Lau interferometry with molecules
The idea behind the Talbot Lau interferometer has been previously
described for instance in [12, 13, 7, 8] and the modification by our new
detection scheme is shown in Fig. 1. Molecules,
which pass the device, reveal their quantum wave nature by forming
a regular density pattern at the location of the third grating.
The distances between the gratings are equal in the experiment and
corresponds to the Talbot length $L=0.38$ m of molecules with a
velocity of 250 m/s. It is chosen such that the period of the
molecular fringe system is the same as that of the third grating.
The regular interference pattern can then be visualized by
recording the total transmitted molecular flux as a function of
the position of the transversely scanning third grating.
The quantum wave nature of various large molecules was already
studied in a similar interferometer but using either laser
ionization [7] or electron impact ionization in
combination with quadrupole mass spectrometry for the detection of
the molecules [8]. In spite of their
success, both previous methods will probably be limited to masses
below 10,000 amu. It is therefore important to develop a
scalable detection scheme, such as fluorescence recording, which
does not degrade but rather improve with molecular mass.
Molecule interferometry often operates with low particle numbers.
Direct molecule counting in free flight therefore typically
exhibits too weak signals [14]. In contrast to
that, the light-exposure time of surface adsorbed molecules can
exceed that of free-flying particles by orders of magnitude. When
bound to a surface the molecules may also release part of their
internal energy to the substrate. This helps in limiting the
internal molecular temperature and in maximizing the number of
fluorescent cycles. In our experiment the molecular beam was
collected on a quartz plate behind the third grating. Several
studies were done with similar aromatic molecules on silica or
quartz surfaces. Due to their insulating properties the molecular
fluorescence yield exceeds that on simple metals or
semi-conductors [15].
2.2 Mechanically magnified fluorescence imaging
For demonstrating the feasibility of this novel detection method
we chose meso-Tetraphenylporphyrin (TPP, Porphyrin Systems
PO890001), a biodye with a mass of 614 amu. It exhibits
sufficiently strong fluorescence, and a sufficiently high vapor
pressure [14] to be evaporated in a thermal source
which was set to a temperature of about 420 ${}^{\circ}$C. Moreover
it was known to show quantum wave behavior in our
setup [8]. Quadrupole mass spectroscopy
allowed us to determine a mass purity of approximately 93%. A
small contribution (7%) of porphyrin molecules lacked one phenyl
ring. Smaller contaminations may contribute up to 2% to the mass
in the initial powder but not to the fluorescence on the surface.
The adsorbing quartz surface was mounted on a motorized
translation stage and it was shifted stepwise, parallel to the
third grating as shown in Fig. 1. A fixed slit
between the third grating and the quartz plate with a width of
170 $\mu$m limited the exposed area on the surface.
Molecules were deposited and accumulated over a time span of eight
minutes under stationary conditions. Then the third grating, with
a grating period of 991 nm, (about 400 nm open slits and a
thickness of 500 nm) was shifted by 100 nm, and the adsorber
plate was simultaneously displaced by 425 $\mu$m to an unexposed
spot.
By repeating this process more than 30 times the third grating
was moved over three periods and 30 stripes of fluorescent TPP
molecules were accumulated on the surface. This way the molecular
interference pattern was recorded with a mechanical magnification
factor of 4250. The large (260 $\mu$m) gap between two stripes
prevented any mixing of the molecules which could otherwise be
caused by surface diffusion between the stripes. We have verified
in independent diffusion experiments with TPP on quartz surfaces
that the molecules aggregate and get immobilized on the
300…400 nm scale at room temperature.
With our new method the resolution is only limited by the
dimensions of the gratings in the interferometer. These may have
openings down to 50 nm and periods as small as
100 nm [16] as already used in earlier molecule
interference experiments [6, 17].
2.3 Velocity selection
High contrast interferences fringes require that the molecular
velocity spread be not too large. Actually for TPP and our present
grating period of 990 nm a width of $\Delta v/v\simeq 10\%$ is
sufficient. As in earlier experiments [7], this is
done by selecting certain free-flight parabola of the molecules in
the Earth’s gravitational field using three horizontally oriented
slits. The first slit is provided by the oven aperture of
200 $\mu$m, another slit of 150 $\mu$m width is placed 1.2 m
away from the oven. The third point of the parabola is given by
the vertical position on the detecting surface, which is located
2.9 m behind the oven. Fast molecules arrive at the top of the
plate. Slow molecules, with a longer falling time, reach the
surface at a lower position. In principle our method therefore
provides the option to select the longitudinal coherence length a
posteriori, after the experiment is already finished. However, in
the present configuration the width of the velocity distribution
is essentially determined by the size of the first two slits,
since we integrate typically only over a much smaller position
interval of 33 $\mu$m on the surface.
2.4 Surface preparation and fluorescence readout
An important requirement for the experiment is to have a perfectly
clean substrate of low self-fluorescence. We used fused silica
(suprasil I) of 500 $\mu$m thickness. It was cleaned from dust
and organic solvents using the RCA-1 cleaning
procedure [18] followed by methanol sonication and
rinsing with ultrapure water. The clean surface was then bleached
by an expanded 16 W argon ion laser beam for 30 minutes with an
intensity of about 3 W/cm${}^{2}$. Owing to this preparation, no
further bleaching could be observed and the background
fluorescence was correspondingly low.
After depositing the molecules, the quartz plate was removed from
the vacuum chamber and put under a fluorescence microscope (Zeiss;
Axioskop 2 mot plus) in air, where a picture of each stripe was
taken with an optical magnification factor of 20 and an
integration time of 20 s. The irradiation intensity was
1.5 W/cm${}^{2}$. TPP absorbs well in the blue and emits in the red.
Correspondingly, we used a standard mercury lamp (HBO 100) with an
excitation filter transmitting wavelengths between 405 and
445 nm, a dichroic beam splitter with a pass band above 460 nm
and an emission filter which transmitted above 600 nm.
At the chosen optical magnification and because of the limited
size of the CCD camera (1 Megapixels) a single microscope image
covers a height of 340 $\mu$m. As the molecular beam is spread
over about 3000 $\mu$m due to its velocity distribution, a whole
picture matrix had to be recorded to image all velocity classes.
Four rows of this 9 x 30 matrix selected around the positions
with the highest molecular coverage are shown in
Fig. 2.
The high quality of the data could be obtained because of the
initial surface preparation and no smoothing was needed. The
single images of one matrix row were arranged a bit closer to each
other than they lie on the surface. Most of the empty gap between
the stripes was removed for presentation purposes and their upper
and lower ends were clipped to avoid regions of optical
aberration.
From available vapor pressure data for porphyrins
[14, 19], we estimate that around
0.1 monolayers of TPP reach the surface in eight minutes. Based
on the work by [15] we assume that the
fluorescence signal grows linearly with the deposition time within
our experimental parameter range. Hence the fluorescence signal is
proportional to the incident intensity, $I_{i}(x,y)$, the
molecular fluorescence efficiency $\eta$, a geometrical
collection factor $K\left(x,y\right)$ and to the molecular
surface number density $N(x,y)$, which we want to determine:
$$I_{f}(x,y)=\left[\eta N(x,y)+B\right]\cdot K(x,y)I_{i}(x,y)+I_{c}(x,y)$$
(1)
where $B$ is the background fluorescence emitted by the
illuminated substrate and $I_{c}(x,y)$ represents the intrinsic
detector noise. The surface sticking coefficient is assumed to be
independent of the molecular surface coverage in our density
regime and is included in N. The intensity of a reference image on
a clean portion of the surface without molecules is
$$I_{r}(x,y)=BK(x,y)I_{i}(x,y)+I_{c}(x,y)$$
(2)
Using Eq. (1) and Eq. (2) we can evaluate from the
experimental data the molecular surface density up to a constant
factor
$$\tilde{N}(x,y)=\frac{\eta}{B}N(x,y)=\frac{I_{f}(x,y)-I_{c}(x,y)}{I_{r}(x,y)-I_%
{c}(x,y)}-1$$
Fig. 2 shows the corrected intensity distribution
$\tilde{N}(x,y)$. For each vertical stripe the total signal
$\tilde{N}_{tot}(h)$ is computed by integrating $\tilde{N}(x,y)$
over a rectangle centered at position $h$ in the middle of the
stripe. The integration height is $33\,\mu$m and the width is
$100\,\mu$m. The resulting intensity cross sections for four
heights selected in Fig. 2 (A, B, C and D) are
shown in Fig. 3. An evaluation of altogether 43 such
interference curves allows to create a smooth plot of the
interference fringe visibility versus the molecular velocity, as
shown in Fig. 4. Note, that for TPP the velocity
class with the highest contrast (b in Fig. 2 at h=350 $\mu$m) is
very close but not equal to the most probable velocity (h=500
$\mu$m).
3 Results
The experimental fringe visibility (full squares in
Fig. 4) clearly varies in a non-monotonic and
quasi-periodic way with the vertical molecular position on the
detector, i.e. with the velocity or the de Broglie wavelength. The
classical model, shown as the falling green line, cannot even
qualitatively reproduce the velocity dependence of the fringe
contrast – even if we take into account the van der Waals
interaction with the grating walls as done here. The quantum
prediction (dashed curve) also includes the molecule-grating
interaction. It is computed by averaging the theoretical
visibility over the velocity distribution, which is obtained from
the geometry of our setup. The experimental contrast is well
reproduced for fast molecules (about 250 m/s) and falls below
the quantum model for velocities around 130 m/s. Slower
molecules are more sensitive to both laboratory noise causing
interferometer vibrations [20] and to collisional
decoherence [21]. The observed contrast
reduction for porphyrins at about 130 m/s is consistent with
these effects. This is however not a fundamental limit, as in
future experiments the present base pressure of $2\times 10^{-7}$ mbar in the interferometer chamber can certainly be
improved by about two orders of magnitude. And also mechanical
vibrations should be suppressed by a factor of ten in future
experiments with additional passive damping systems. The deviation
at medium velocities is ascribed to molecules which do not follow
a perfect free fall trajectory, because of scattering at edges
along the beam path. In independent velocity measurements we have
already observed before the effect of scattering, which deflects
molecules of a given speed into the trajectory of another
free-fall velocity class. The red continuous line shows the
results of a model, which allows about 20% of the molecules at
the most probable velocity to be spread out over the whole
detector area. The resulting curve then fits indeed all
experimental points, except those at low velocities, as discussed
above.
Our accumulation and imaging method requires a good mechanical
stability of the whole setup. From the good reproducibility of the
expected and observed fringe period we derive an upper limit for
the slow grating drift of 50 nm over four hours. A drift of
10 nm over this period is realistic in a second generation
experiment.
A clear advantage of our new detection scheme is that all velocity
classes are simultaneously recorded and encoded in the vertical
position on the screen. This ensures utmost mechanical stability
between the interferograms belonging to different velocities. The
simultaneous recording can therefore be used to measure a possible
phase shift between these interference fringes.
Ideally, we should not expect any velocity dependent phase shift
in a symmetric Talbot Lau interferometer, where the distance
between the first and second grating equals the distance between
the second and the third one [23]. But an
evaluation of Fig. 2 yields a phase variation in
the vertical direction of about 0.4$\pi/$mm in our experiment.
This effect can be traced back to a small angular misalignment
between the gratings around the molecular beam axis. Our
observation is for example consistent with a tilt as small as
200 $\mu$rad between the second and the third grating.
The present detection scheme is therefore a very sensitive method
for identifying the presence of such tilts, which will be
important for interferometry with very massive molecules.
Generally, the alignment requirements increase critically with
increasing mass of the interfering particles [20].
In contrast to the present setup, other grating configurations may
show additional non-classical effects, for instance the fractional
Talbot effect [22]. In particular we do expect a phase
jump of $\pi$ between fringes of certain velocity
classes [23] in an asymmetric Talbot Lau
configuration. This is a non-classical feature which can still be
observed in a regime where the classical Moiré effect and
quantum interference are expected to yield comparable fringe
visibilities. And the present experiment indicates that such
features should be stably recorded using our new detection method,
even in the presence of overall drifts of the interferometer.
4 Conclusion
Mechanically magnified fluorescence imaging offers several
advantages for future experiments aiming at recording
interferograms of nanometer-sized objects. The demonstrated method
scales favorably with the complexity of the observed particles:
organic molecules can be tagged with several dye molecules or
semiconductor nanocrystals [24] and large proteins,
such as GFP [25], or again
nanocrystals [26] will even exhibit a much
higher fluorescence quantum yield and a significantly smaller
bleaching rate than the molecules in our current experiments. At
present, the smallest commercially available fluorescent
nanocrystals have a mass around
3000 amu [27]111Evidenttech private
communication: The masses of the quantum dots were recently
measured by Evidenttech and turned out to be substantially
smaller than found in earlier publications. in the core and
roughly the same mass in the ligand shell. The high efficiency of
our optical detection method will also allow to study the
relevance of different electric and magnetic dipole moments in
interference with molecules of rather similar masses, such as for
example various porphyrin derivatives. Some of them have too low
vapor pressures for experiments with ionization detectors, but
will still be detectable in fluorescence. Mechanically magnified
fluorescence imaging is therefore expected to be a scalable method
for exploring the wave-particle duality of a large class of
nanosized materials. It is an enabling technique for a range of
dephasing and decoherence studies, which will also be useful in
molecule metrology.
Acknowledgments
This work has been supported by by the Austrian Science Fund
(FWF), within the projects START Y177 and F1505 and by the
European Commission under contract No. HPRN-CT-2002-00309
(QUACS). We acknowledge fruitful discussions with Klaus
Hornberger, Lucia Hackermüller and Sarayut Deachapunya.
References
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http://www.evidenttech.com/ |
Primal-Dual Optimization for Fluids
[
Abstract
We apply a novel optimization scheme from the image processing and machine learning areas, a fast Primal-Dual method, to achieve controllable and realistic fluid simulations.
While our method is generally applicable to many problems in fluid simulations, we focus on the two topics of fluid guiding and separating solid-wall boundary conditions.
Each problem is posed as an optimization problem and solved using our method, which contains acceleration schemes tailored to each problem.
In fluid guiding, we are interested in partially guiding fluid motion to exert control while preserving fluid characteristics.
With our method, we achieve explicit control over both large-scale motions and small-scale details which is valuable for many applications,
such as level-of-detail adjustment (after running the coarse simulation), spatially varying guiding strength, domain modification, and resimulation with different fluid parameters.
For the separating solid-wall boundary conditions problem, our method effectively eliminates unrealistic artifacts of fluid crawling up solid walls and sticking to ceilings, requiring few changes to existing implementations.
We demonstrate the fast convergence of our Primal-Dual method with a variety of test cases for both model problems.
\electronicVersion\PrintedOrElectronic
T. Inglis, M.-L. Eckert, J. Gregson & Nils Thuerey]
T. Inglis${}^{1}$,
M.-L. Eckert${}^{1}$,
J. Gregson${}^{2}$,
N. Thuerey${}^{1}$
${}^{1}$Technical University of Munich
${}^{2}$University of British Columbia
\teaser
Our modular Primal-Dual optimization method can be applied to fluid guiding (left, right) and to simulate liquids with separating boundary conditions (center).
1 Introduction
Advances in fluid simulation have had a tremendous effect in engineering and
graphics. Since fluids play an important role in our everyday lives, smoke and
liquid simulations now routinely appear as regular elements in feature films,
television and commercials. While engineering applications are primarily
concerned with accuracy, the focus in graphics lies in simulating realistic
behavior that can be controlled to tell a story. Therefore, it is important to
provide controllable and visually plausible fluid solvers. Visual plausibility
is crucial since people easily recognize unrealistic behavior due to
their daily interactions with a wide range of fluid phenomena, e.g., pouring a
glass of water, boiling a kettle, or driving past a chimney.
However, it is a recurring challenge to simultaneously achieve controllable and
realistic behavior. Fluids are usually chaotic at human scales; miniscule
perturbations regularly trigger large-scale behavior. It is almost
impossible for artists to add small-scale details to a low-resolution
simulation without changing the large-scale motion. Even more challenging
is to modify the domain or rerun a simulation with different fluid
parameters without affecting the large-scale motion.
These problems can be mitigated by guiding the velocity within an optimization framework.
We realize guiding by constraining the velocity at each time step to
be arbitrarily close to the current and the target velocity. At the same time,
we ensure that the resulting motion is divergence-free, and we allow for
spatially varying guiding parameters.
This framework also benefits animators in the special effects industry who
may wish to create fictitious but visually plausible fluid motion to increase
entertainment value.
Common fluid solvers typically feature boundary conditions (BCs) that lead to fluid
unnaturally crawling up walls and sticking to ceilings, diminishing its
visual plausibility. The culprit is the solid-wall BC that enforces a normal velocity of zero at obstacle
walls, thus preventing fluids from separating.
While it is physically correct to leave a thin film of fluid on the wall, its thickness in simulations is on the order of the discretization and usually far too big for realistic animations.
One proposed remedy by Batty et al. [BBB07] is to integrate
inequality constraints into the pressure solve to allow for positive normal
velocities at solid walls. Unfortunately, this greatly increases the complexity of
the pressure solve of a fluid solver, and typically requires Quadratic
Programming solvers.
Our goal is to implement flexible separating BCs, while reusing the popular preconditioned conjugate gradient method.
First, we classify boundary cells into separating and non-separating cells.
We then enforce solid-wall BCs for the velocity at non-separating cells while ensuring zero divergence.
To guide simulations and realize separating BCs, we take inspiration from convex optimization.
Gregson et al. [GITH14] already established a connection between fluid pressure correction and convex optimization.
This connection allows us to impose all required physical constraints on the velocity for both guiding and separating BCs via an efficient alternating minimization algorithm
that exploits problem structure, removing the need for a monolithic and slow framework.
We introduce a novel optimization scheme from the computer vision area, the fast, or first-order
Primal-Dual (PD) method proposed by Pock et al. [PCBC09]. It features an optimal convergence rate for non-smooth convex problems.
Note that a whole class of primal-dual methods exists, but we will hereafter use the abbreviation PD to refer to this particular instance,
which we will focus on due to its fast convergence properties.
Specifically, our contributions are
•
the introduction of a modular convex optimization approach for fluids based on the PD method;
•
a general method for handling the difficult problem of fluid guiding that involves spatially varying operators;
•
a fast method to approximately invert the linear system for flow guiding in arbitrary domains; and
•
a novel, practical way to handle separating BCs for liquids.
2 Previous Work
Fluid simulation has a long history in computer graphics [KM90]. One of the most widely used methods is the
stable fluid solver [Sta99], for which many extensions have been proposed over the years, e.g.,
the grid-particle hybrid FLIP (Fluid Implicit Particle) method [ZB05], which is particularly popular for liquid simulations. A good overview of these methods is given in Bridson’s book [Bri08].
A key component of incompressible fluid simulation is ensuring that the
time-evolved velocity is divergence-free (i.e., mass-preserving),
which is typically implemented via Chorin-like
pressure-projections, e.g. [FF01] or [BBB07] for variational approaches.
Convex Optimization
has become a powerful component of many computer vision algorithms.
Even before, the works of Boyd et al. [BPC${}^{*}$11]
have laid the foundations for many popular optimization algorithms.
Closer to computer graphics, Heide et al. [HRH${}^{*}$13]
have proposed using the PD method to correct the deficiencies of simple lenses.
Recently, Heide et al. [HDN${}^{*}$16] have solved the difficult task of choosing the optimal image priors and optimization algorithms for image processing tasks such as deconvolution, denoising or inpainting by applying PD as well.
Iterated Orthogonal Projection (IOP), the algorithm proposed by Molemaker et al. [MCPN08], is a specialized method in the convex optimization area requiring all operators to be orthogonal projections.
Approaches such as position-based fluids (PBF) [MM13] also can be seen as an iterative application of a set of constraint projections. As the constraints are directly applied to the degrees of freedom, PBF is more closely related to IOP than PD, which achieves improved convergence by projecting the convex conjugate.
We extend upon the ideas presented by Gregson et al. [GITH14], where the method Alternating Direction Methods of Multipliers (ADMM) is applied, by introducing a more efficient splitting algorithm for flow guiding and BCs.
Very recently, Narain et al. [NOB16] animated deformable objects with an ADMM algorithm combining a fast and robust nonlinear elasticity model with hard constraints.
Our method can be seen as preconditioned version of ADMM, and its convergence is accelerated by an improved update direction for each iteration.
O’Connor et al. [OV14] found ADMM outperforming PD for few iteration counts while using a restricted version of PD which does not exploit PD’s full potential of optimal update directions.
We demonstrate the advantages of our approach over both ADMM and IOP in Section 4.1.
Guiding
A limitation of fluid simulation for computer graphics is the issue of control.
Simply changing the resolution of a simulation can alter its behavior significantly.
A popular class of methods circumvents this problem by synthesizing smaller
scales in a decoupled fashion [BHN07, KTJG08, PTSG09].
As a consequence, the details are easy to fine-tune, but do not tightly integrate with the base flow,
which is left unmodified.
Fluid guiding is a challenging example of an inverse problem for fluids.
Adjoint methods have been used in engineering and graphics [GP00, MTPS04] but require differentiation of the entire fluid solver.
Guide shapes [NB11] are able to add detail to free surface flows by applying velocity conditions to high-resolution simulations distant to the interface but are limited in volumetric contexts.
Other approaches aiming for high-level control have proposed the use of
Lagrangian coherent structures [YCZ11] or sketches [PHT${}^{*}$13] to give users intuitive controls.
As our guiding technique takes an arbitrary flow field as input to calculate
plausible and tightly coupled detail, such approaches would be a good complement for our method.
While Gregson et al. [GITH14] also demonstrate preliminary results for guided flows, their approach
employed the fast Fourier transform for filtering low-frequencies and pressure-projection. This is a very efficient approach, but it becomes
impractical for more complex geometries and spatially varying guiding.
A simpler approach to guiding is via control
forces [SY05b, FL04] or
velocities [SY05a]. These approaches can result in
artificially viscous flows, as noted by Thuerey et al. [TKRP06], who
proposed a multi-scale approach based on control particles that controls
low-spatial frequencies. Possibly the most similar approach to ours is the one
by Nielsen et al. [NCZ${}^{*}$09, NC10] who use a
multi-scale formulation based on low pass filters. They arrive at a monolithic
system with four degrees of freedom per cell, requiring a specialized solver to
reduce run time. In contrast, our control scheme decouples into separate and
more easily solved subproblems via recent developments in non-smooth
optimization.
Furthermore, for practical purposes, we introduce an upres method into our workflow [KTJG08, YCZ11, HMK11, RLL${}^{*}$13, HK13] to facilitate the separation of low-resolution and high-resolution guiding.
Boundary Conditions
play a crucial role in fluid simulation, and as such have received a significant amount of attention.
Foster et al. [FF01] describe how to allow for tangential motions of liquids,
and are the first to note problems with liquid unnaturally sticking to domain boundaries.
Batty et al. [BBB07] propose to solve inequality constraints with Quadratic Programming,
while Chentanez et al. [CMF12] observe that they can incorporate these inequalities into their multigrid solver.
Methods such as the FFT-based one of R. Henderson [Hen12] are similar in spirit
to our method, as they separate the BCs from the divergence-free projection step,
however, without targeting separating boundaries.
Other approaches realize unilateral incompressibility to allow for separation effects [NGL10],[NGCL09], [GB13] but they also require complex
solvers to solve their proposed Quadratic Programming Problems. We will demonstrate how to solve for
separating motions with a regular conjugate gradient (CG) solver
based on our modular optimization framework.
3 Methodology
In graphics, fluids are typically simulated by solving the incompressible Euler equations,
written as
$$\displaystyle\frac{\partial\mathbf{u}}{\partial t}+\mathbf{u}\cdot\nabla%
\mathbf{u}$$
$$\displaystyle=$$
$$\displaystyle-\nabla p+\mathbf{f}_{\text{ext}}\ ,$$
(1)
$$\displaystyle\nabla\cdot\mathbf{u}$$
$$\displaystyle=$$
$$\displaystyle 0,$$
(2)
where $\mathbf{u}$ is the flow velocity, $p$
the pressure, and $\mathbf{f}_{\text{ext}}$ the external body forces.
Simulations commonly proceed via operator splitting to satisfy both constraints.
First, using all but the pressure term in Eq. (1), an intermediate velocity field is computed.
Then a pressure projection is applied to satisfy the divergence-free condition in Eq. (2).
A pressure field that exactly counteracts the divergence is computed to correct the velocity field.
Orthogonality of the curl-free and divergence-free components ensures that divergence-free components of the flow
are not affected and allows the pressure projection to be interpreted as an Euclidean projection onto the space of divergence-free velocity fields.
The splitting approach closely resembles convex optimization approaches originally developed for imaging inverse problems and machine learning involving non-smooth or constrained objective functions.
We show how several of these approaches can be adapted to solve difficult problems in fluids.
Convex Optimization
aims to solve problems of the form
$$\displaystyle\underset{\mathbf{x}}{\text{minimize}}$$
$$\displaystyle h(\mathbf{x}),$$
(3)
where $h$ is a convex function that may be non-smooth or even discontinuous.
If $h$ is the sum of two simpler functions (say $h=f+g$), then we have
$$\displaystyle\underset{\mathbf{x}}{\text{minimize}}$$
$$\displaystyle f(\mathbf{x})+g(\mathbf{x}).$$
(4)
A number of recently developed algorithms target this type of problem by employing an iterative divide-and-conquer approach.
These algorithms are known as proximal methods and are defined in terms of so-called proximal operators (we use $\mathbf{\xi}$ exclusively to denote the generic argument variable):
$$\displaystyle\mathrm{\mathbf{prox}}_{f,\sigma}(\mathbf{\xi})$$
$$\displaystyle:=\operatorname*{arg\,min}_{\mathbf{x}}\left(f(\mathbf{x})+\frac{%
\sigma}{2}\left\lVert\mathbf{x}-\mathbf{\xi}\right\rVert^{2}\right).$$
(5)
One such algorithm is the PD method [PCBC09], which solves a slightly more general problem:
$$\displaystyle\underset{\mathbf{x}}{\text{minimize}}$$
$$\displaystyle f(K\mathbf{x})+g(\mathbf{x})$$
(6)
for some linear operator $K$.
PD solves the problem iteratively by providing a series of variable updates that terminate when $\mathbf{z}$ converges to the solution.
The combination of $\mathbf{x},\mathbf{z}$ and $\mathbf{y}$ ensures that $\mathbf{z}$ converges to the optimal value of Eq. 6.
The updates are given by
$$\displaystyle\mathbf{x}^{k+1}$$
$$\displaystyle:=\mathrm{\mathbf{prox}}_{f^{*},1/\sigma}(\mathbf{x}^{k}+\sigma K%
\mathbf{y}^{k})$$
(7)
$$\displaystyle\mathbf{z}^{k+1}$$
$$\displaystyle:=\mathrm{\mathbf{prox}}_{g,1/\tau}(\mathbf{z}^{k}-\tau K^{*}%
\mathbf{x}^{k+1})$$
(8)
$$\displaystyle\mathbf{y}^{k+1}$$
$$\displaystyle:=\mathbf{z}^{k+1}+\theta(\mathbf{z}^{k+1}-\mathbf{z}^{k}),$$
(9)
where $\{\sigma,\tau,\theta\}$ are parameters that affect convergence, $f^{*}$ and $K^{*}$ are the convex conjugates of $f$ and $K$, respectively.
For our problem, $K$ is simply the identity, which leads to $K^{*}=K^{T}=I$. Note that if additionally $\sigma,\tau,\theta=1$, the iterative update scheme reduces to ADMM.
A more appropriate choice ($\sigma,\tau,\theta\neq 1$) leads to optimal control over convergence.
As for $f^{*}$, it is not necessary to compute it directly. The proximal operator can be transformed using Moreau’s identity:
$$\displaystyle\mathrm{\mathbf{prox}}_{f^{*},1/\sigma}(\mathbf{\xi})$$
$$\displaystyle=\mathbf{\xi}-\sigma\,\mathrm{\mathbf{prox}}_{f,\sigma}(\mathbf{%
\xi}/\sigma).$$
(10)
The variable updates are thus reduced to
$$\displaystyle\mathbf{x}^{k+1}$$
$$\displaystyle:=\mathbf{x}^{k}+\sigma\mathbf{y}^{k}-\sigma\,\mathrm{\mathbf{%
prox}}_{f,\sigma}(\tfrac{1}{\sigma}\mathbf{x}^{k}+\mathbf{y}^{k})$$
(11)
$$\displaystyle\mathbf{z}^{k+1}$$
$$\displaystyle:=\mathrm{\mathbf{prox}}_{g,1/\tau}(\mathbf{z}^{k}-\tau\mathbf{x}%
^{k+1})$$
(12)
$$\displaystyle\mathbf{y}^{k+1}$$
$$\displaystyle:=\mathbf{z}^{k+1}+\theta(\mathbf{z}^{k+1}-\mathbf{z}^{k}).$$
(13)
A more in-depth discussion of PD can be found in [CP11].
The advantage of using proximal methods is that the optimization can be performed separately for the two objective functions, allowing difficult optimizations to be split into more manageable components.
Also, depending on the form of $f$ and $g$, many special cases can be significantly simplified [BPC${}^{*}$11] by exploiting their mathematical structures.
To the best of our knowledge, PD has not yet been applied to fluid problems,
despite its provably optimal convergence properties for the class of problems of Eq. (6).
A pseudocode implementation of our PD-based optimization method is given in Algorithm (1) for one time step.
Comparing to previous methods, the improved
convergence is achieved with only a minimal increase in computational cost.
A few more vector additions
and multiplications are necessary, which typically are negligible compared to the cost of
the proximal operators.
We demonstrate the convergence of our method and compare it to other methods in Sections 4.1 and 5.1.
For reference, we briefly review IOP and ADMM in Appendix B.
Convex Optimization of Fluids
In fluid simulation, we often encounter problems of the form
$$\displaystyle\underset{\mathbf{x}}{\text{minimize}}$$
$$\displaystyle f(\mathbf{x})$$
(14)
subject to
$$\displaystyle\mathbf{x}\in C_{\mathrm{DIV}},$$
where $\mathbf{x}$ is the velocity field we seek, and $C_{\mathrm{DIV}}$ is the space of divergence-free velocity fields.
$f$ must be a convex function. Practically, this can be either a quantity we are trying to minimize, or a hard constraint that must be satisfied (in which case, $f$ would be an indicator function).
The second constraint requires $\mathbf{x}$ to be divergence-free.
Removing the divergent part of the flow can also be viewed as an orthogonal projection [Cho68, PB13].
Gregson et al. [GITH14] made the key observation that the proximal operator
for $C_{\mathrm{DIV}}$ can be easily computed via a pressure projection. In other words, we have
$$\mathrm{\mathbf{prox}}_{g,1/\tau}(\mathbf{\xi})=\Pi_{\mathrm{DIV}}(\mathbf{\xi%
}),$$
(15)
where $\Pi_{\mathrm{DIV}}$ denotes a projection onto $C_{\mathrm{DIV}}$ with a commonly used Poisson solver.
Hence, formulating a fluid problem this way allows the optimization algorithm to be easily integrated into a common fluid solver—we simply replace the call for a pressure projection subroutine
with a call to the PD optimization step outlined in Algorithm (1).
We check for convergence using a threshold parameter $\epsilon$, and stop the algorithm once the per-iteration change of $\mathbf{z}$ falls below this threshold (Algorithm (1), line 12).
Note that we define the pressure projection to include only the calculation of the pressure values,
and a subtraction of the pressure gradient from $\mathbf{x}$, excluding any optional modifications of the velocity field.
The pressure projection implemented by a CG solver is usually the most expensive part in regular fluid simulation, and its effect is magnified in our algorithm due to its iterative nature.
However, this iterative set-up gives us an opportunity to apply an adaptive scheme.
Let $\epsilon_{\mathrm{CG}}$ be the accuracy of the CG solver.
We choose its value starting with a high threshold (e.g., $\epsilon_{\mathrm{CG}}=10^{-2}$) and then adaptively decrease it over the PD iterations.
We decrease $\epsilon_{\mathrm{CG}}$ as soon as the per-iteration change of $\mathbf{z}$ is close to $\epsilon$; until $\epsilon_{\mathrm{CG}}$ reaches the desired final accuracy (e.g., $\epsilon_{\mathrm{CG}}=10^{-5}$).
Using this adaptive CG scheme greatly accelerates the performance by cutting down on CG iterations in the beginning of the optimization when the divergence-free constraint does not need to be strictly enforced.
Algorithm (1) summarizes the general framework of our method as it applies to fluid simulation.
Specific applications call for different definitions of $f$, which in turn affects how the $\mathbf{x}$-update is computed.
In the following sections, we discuss how to apply this method to the
fluid guiding and the separating BC problem.
For each application, we define the appropriate $f$ and discuss how to compute its proximal operator.
4 Fluid Guiding
In fluid guiding, the goal is to minimize the change applied to the current velocity field such that the resulting velocity field follows the large-scale motions of a given target velocity.
The objective function $f$ is given by
$$\displaystyle f(\mathbf{x})$$
$$\displaystyle=\left\lVert G(\mathbf{x}-\mathbf{u}_{t})\right\rVert^{2}+\left%
\lVert W(\mathbf{x}-\mathbf{u}_{c})\right\rVert^{2},$$
(16)
where $\mathbf{u}_{c}$ is the current velocity field (after advection and before pressure projection), $\mathbf{u}_{t}$ is the target velocity field, $\mathbf{x}$ is the guided velocity field,
$W$ is the guiding weights matrix, and $G$ is the Gaussian blur matrix.
The first term of this objective function is minimal for a solution that matches the target velocity when blurred by $G$, while the second term penalizes solutions far away from the current flow field.
In order to keep the application general, both matrices in our objective function are spatially varying (Nielsen and Christensen [NC10] also performed fluid guiding with spatially varying guiding weights,
but not with spatially varying blur).
$W=W(\mathbf{x})$ is a diagonal matrix containing spatially varying weights that control the guiding strength (larger entries denote weaker guiding),
and $G=G(\beta,\mathbf{x})$ is the Gaussian blur matrix that applies a blur of radius $\beta$ only to fluid cells so that we can handle boundaries and obstacles in our domain.
Our objective function in Eq. (16) is quadratic. That is, it can be expressed as
$$\displaystyle f(\mathbf{x})$$
$$\displaystyle=\tfrac{1}{2}\mathbf{x}^{T}A\mathbf{x}+\mathbf{b}^{T}\mathbf{x}+c,$$
(17)
where
$$\displaystyle A$$
$$\displaystyle=2(G^{T}G+W^{2})$$
(18)
$$\displaystyle\mathbf{b}$$
$$\displaystyle=-2(G^{T}G\mathbf{u}_{t}+W^{2}\mathbf{u}_{c})$$
(19)
$$\displaystyle c$$
$$\displaystyle=\mathbf{u}_{t}^{T}G^{T}G\mathbf{u}_{t}+\mathbf{u}_{c}^{T}W^{2}%
\mathbf{u}_{c}.$$
(20)
Note that $W^{T}W=W^{2}$ since $W$ is symmetric.
We can combine $f^{\prime}(\mathbf{x})=\mathbf{0}$ (three equations per cell in 3D) and the divergence-free constraint (one equation per cell) into a linear system $L\mathbf{x}=\mathbf{d}$.
But since this system is overconstrained, we solve it in the least-squares sense by considering
$$\displaystyle L^{T}L\mathbf{x}$$
$$\displaystyle=L^{T}\mathbf{d},$$
(21)
where $L=\left(\begin{smallmatrix}A\\
\nabla\cdot\end{smallmatrix}\right)$ and $\mathbf{d}=\left(\begin{smallmatrix}\mathbf{b}\\
\mathbf{0}\end{smallmatrix}\right)$.
This quadratic system can be solved using an iterative solver such as the CG solver.
However, this approach is infeasible since the number of elements in the matrix grows by $\mathcal{O}(N)$, where $N$ is the volume of the simulation domain.
Thus PD is still a good choice even for seemingly simple quadratic energies. We will see later how this direct CG solver compares to our algorithm in terms of performance.
Instead, we make use of the quadratic property of $f$ to simplify its proximal operator to
$$\displaystyle\mathrm{\mathbf{prox}}_{f,\sigma}(\mathbf{\xi})$$
$$\displaystyle=(A+\sigma I)^{-1}(\sigma\mathbf{\xi}-\mathbf{b}).$$
(22)
Factoring out $\mathbf{u}_{c}$, we obtain
$$\displaystyle\mathrm{\mathbf{prox}}_{f,\sigma}(\mathbf{\xi})=\mathbf{u}_{c}+M^%
{-1}(\sigma\mathbf{\xi}+\mathbf{q}),$$
(23)
where
$$\displaystyle M$$
$$\displaystyle=2G^{T}G+2W^{2}+\sigma I$$
(24)
$$\displaystyle\mathbf{q}$$
$$\displaystyle=2G^{T}G(\mathbf{u}_{t}-\mathbf{u}_{c})-\sigma\mathbf{u}_{c}.$$
(25)
Computing $M^{-1}(\sigma\mathbf{\xi}+\mathbf{q})$ for every iteration is slow.
Instead, we precompute $\mathbf{q}$ and $M^{-1}$ since they do not rely on previous iterations.
In fact, using the Sherman-Morrison-Woodbury formula, $M^{-1}$ can be approximated as
$$\displaystyle M^{-1}$$
$$\displaystyle\approx(2W^{2}+\sigma I)^{-1}-2(2W^{2}+\sigma I)^{-1}G^{T}G(2W^{2%
}+\sigma I)^{-1}.$$
(26)
See Appendix C for derivation details.
Lastly, spatially invariant Gaussian blurs are symmetric, so $G^{T}G=G^{2}$.
They are also separable, and can be applied independently in each dimension.
This combined with the matrix inversion approximation greatly speeds up the $\mathbf{x}$-update.
For Gaussian blurs with spatial variation, we still use a symmetric and hence separable Gaussian blur (with different blur radii) for each cell, but $G$ itself is no longer symmetric.
However, if the spatial variation is limited (e.g. different blur radii on the left and right sides of the simulation domain),
we approximate $G^{2}$ by $G^{T}G$, since $G^{T}G\approx G^{2}$ for regions of constant blur.
We found our approximation to work well in practice, and we will show examples using this approximation later.
Our PD guiding scheme is independent of the particular choice and implementation of the blur kernel.
Given a fast implementation for calculating $G^{T}$, it would be straight-forward to extend our method with a full evaluation of $G^{T}G$.
Additionally, our approach for applying the blur kernel easily allows for interior boundaries by using a blur radius of zero for obstacle cells.
This is a key difference from guiding scheme based on the fast Fourier transform [GITH14], which typically require periodic domains without internal boundaries.
Algorithm (2) summarizes how $\mathrm{\mathbf{prox}}_{f,\sigma}(\mathbf{\xi})$ is approximated.
Note that the two extra parameters $W$ and $\beta$ in Algorithm (2) as compared to Algorithm (1) are specific to the guiding application.
Evaluation
In this section, we will first demonstrate the efficacy of our method using 2D examples, followed by more impressive 3D results.
To simplify the notation, when the guiding weight is spatially invariant (say, $c$ everywhere), we will write $W=c$.
And when the guiding weight is spatially varying (say, $c_{1}$ and $c_{2}$ on the left and right side on the simulation domain, respectively), we will write $W=(c_{1},c_{2})$.
Similarly, spatially varying blur radii would be written as $\beta=(r_{1},r_{2})$.
We first compare our method to two naïve guiding methods, with a counterclockwise circular velocity field as the target $\mathbf{u}_{t}$, as shown in Figure 0(a).
Linear velocity
blend computes the new velocity as a linear combination of the current velocity and the target velocity: $\mathbf{u}_{\mathrm{new}}=r\mathbf{u}_{c}+(1-r)\mathbf{u}_{t}$,
where $0\leq r\leq 1$.
Detail-preserving guiding [TKRP06] subtracts the large-scale (i.e., blurred) motions from the current velocity before adding the target velocity: $\mathbf{u}_{\mathrm{new}}=\mathbf{u}_{c}-G\mathbf{u}_{c}+\mathbf{u}_{t}$.
The idea is to create a new velocity with large-scale motions from $\mathbf{u}_{t}$ and small-scale details from $\mathbf{u}_{c}$.
Both methods are followed by a pressure solve to ensure zero divergence.
With linear velocity blend, guiding strength is increased with smaller $r$.
The method’s main weakness is that small fluid details are smoothed out with strong guiding, and more details come at the expense of reduced trajectory control (see Figure 0(b)).
Detail-preserving blend allows guiding of large-scale motions towards the target velocity while preserving details.
However, the details are difficult to control and have an unnatural frosted glass appearance (see Figure 0(c)).
In contrast, our method has much more flexible motion control when applied to fluid guiding, with $W$ affecting the large-scale guiding strength and the blur radius $\beta$ controlling the small-scale details.
Figure 2 shows how the parameters affect guiding for a 2D smoke simulation with a circular target velocity.
Notice that larger $W$ values allow for more freedom to deviate from the target, while larger $\beta$ values lead to the formation of larger vortices.
We also compare our algorithm to wavelet turbulence [KTJG08], as shown in
Figure 3.
Although wavelet turbulence is fast—since it is purely a postprocessing technique—the resulting vortices do not couple as tightly and realistically as with our method.
Performance
Next, we examine our method in relation to other proximal methods, namely, ADMM and IOP (see Appendix B for details).
ADMM is fairly straightforward to apply due to its similarity to PD.
IOP can be interpreted as a fluid-specific version of the Projection onto Convex Sets (PoCS) algorithm [BB96],
which solves convex feasibility problems (locating an intersection point of convex sets) rather than the more general problem solved by ADMM and PD.
Applying IOP naïvely alternates between the minimizer of $f$ and its closest divergence-free neighbor but does not yield the divergence-free minimizer of $f$, as illustrated by the failure case in Figure 4.
In practice, we discovered instances for which IOP found plausible approximate solutions in a short time,
but its reliability is limited for general guiding applications. As such, we focus on only comparing our method to ADMM.
All performance measurements were done on PCs with Intel Xeon E5-1650 CPUs.
For fairness of comparison, we experimented with several test scenes to deduce as-optimal-as-possible parameter sets for both ADMM and our method.
These parameters, on average, produce the fastest convergence rates under various guiding weights and blur radii.
The optimal parameters are correlated with the guiding weight; we analyzed their relationship and define the parameters in terms of the average guiding weight,
$\bar{W}$, which is simply the mean of the diagonal terms of $W$.
For our method, we chose $(\tau,\sigma,\theta)=(0.58/\bar{W},2.44/\tau,0.3)$, and for ADMM, $\rho=1.4\bar{W}^{2}$.
Before comparing their performance, it is important to note that we apply one of our contributions crucial for performance (the matrix inverse approximation and the separable Gaussians) to both methods.
We compare the performance of ADMM and our method for 2D and 3D problems.
The 2D problem is a $256^{2}$ simulation guided by a circular target velocity, similar to the one shown in Figure 2, but with spatially varying weights and $\beta=1$.
Specifically, the guiding weight is fixed at 1 for the right side of the domain, while the left side takes values from $\{2,4,8,16\}$.
Figure 4(a) compares, for the two methods, the mean number of iterations required to reach convergence.
When the spatial variation is low, both methods perform decently. However, as the spatial variation increases, ADMM takes much longer to converge.
The 3D problem is a $120^{3}$ simulation with an obstacle, similar to the one in Figure 12, but with the same $W$ and $\beta$ parameters as the 2D problem.
The performance results are shown in Figure 4(b). Again, our method outperforms ADMM. In fact, the difference in convergence rates is even more apparent for the 3D example for large spatial variations.
The run times for both ADMM and our method very closely correlate with the mean iteration counts. For brevity, the run times are given in Table 1.
In addition to the comparison with ADMM, we also analyze how our method scales as the resolution increases.
In Figure 5(a), we run the guided 3D tornado simulation shown in Figure Primal-Dual Optimization for Fluids at five different resolutions and compare the mean run time per time step.
The scaling factor is with respect to the lowest resolution $40^{2}\times 60$; for instance, a scaling factor of 3 corresponds to the resolution $120^{2}\times 180$.
Notice that even at the largest resolution $200^{2}\times 300$, each time step takes less than four minutes to complete.
However, since the run times appear to scale exponentially with respect to resolution, we plot the mean run time per grid cell (see Figure 5(b)) to show that the scaling is in fact linear.
Previously, we mentioned that applying a CG solver also works for the guiding problem, although its poor performance renders it infeasible.
To test this, we ran a small $128^{2}$ example with a circular target velocity field and found the direct CG solver (see Eq. (4)) approach to be 4000 times slower than our method.
The performance tests above are all done with our generic guiding method, which handles spatially varying weights and blurs.
In the special case with spatially invariant weights, we can apply a non-iterative method to greatly speed up the simulation.
The method involves defining $f(\mathbf{x})$ in terms of the non-divergent components of the input velocities and then finding its minimizer.
Since differentiation commutes with convolution, the result will be automatically divergence-free for spatially invariant weights.
Although this non-iterative method offers potential speed-up, we focus on the general case of spatially varying operators that requires iterating.
4.1 Guiding Results
To realize interesting guided flows, we found an iterative up-res workflow that works well in practice. This is in line with previous work, where a low-resolution
input is refined in subsequent stages to yield a final result [KTJG08, YCZ11, HMK11, RLL${}^{*}$13, HK13].
Examples of this process are shown in Figure 6(a). We start with a $64^{2}$ simulation guided by a synthetic star-shaped velocity field with $W=1$ and $\beta=1$.
Then we run $256^{2}$ high-resolution simulations with various $\beta$ values.
If the same guiding is done without up-res, that is, if we run the $256^{2}$ guided simulation directly as shown in Figure 6(b),
then we lose the ability to guide the fluid to desirable shapes at multiple levels.
We further demonstrate this up-res process for 3D simulations.
Figure 10 shows a simple plume example. We first run a low-resolution ($50^{2}\times 100$) simulation to capture the velocities.
Then a $200^{2}\times 400$ resimulation is performed, guided by the upsampled velocity field. The blur radii can be kept spatially invariant or spatially varying for different effects.
In particularly, increasing the blur radius introduces more small-scale details.
In our spatially varying example, we use a blur kernel with a sharp transition between two blur radii to contrast between the two guided regions.
If desired, this obvious seam can be softened via a more gradual transition between the two blur radii.
Note that even though the simulation uses the approximation $G^{T}G=G^{2}$, it works quite well in practice.
Our second example in Figure 10 shows 3D simulations following star-shaped input velocities.
We use a $50^{3}$ target velocity as guide for a $250^{3}$ final resolution, with $\beta=2$.
Here we make use of spatially varying weights, with $W=1$ on the left side of the domain, and $W\in\{1,5,7\}$ on the right.
The precomputation time is negligible
compared to the overall run time: around $1.5$ seconds per time step for a $250^{3}$ simulation.
Next, we show a tornado simulation. We first run a low-resolution ($40^{2}\times 60$) simulation using a cylindrical target velocity with a small upward component.
Once the general shape is fixed, a $200^{2}\times 300$ resimulation is performed, guided by the upsampled velocity field.
The results in Figure 10 show the effectiveness of varying $\beta$ to achieve different levels of turbulence.
Finally, Figure 11 demonstrates an example with obstacle, for which we upsampled from $40^{3}$ to $200^{3}$ with $W=1$ and $\beta=5$.
This high-resolution version captures the input motion, but develops many interesting small-scale details.
Figure 12 shows a similar simulation with $\beta=1$ and spatially varying weights, where the left side has $W=1$ while the right side has $W=100$.
The result has little detail on the left side due to strong guiding from the low-resolution simulation, whereas the right side behaves like a regular smoke simulation due to weak guiding.
This guided simulation with an obstacle is a case that cannot be handled by previous Fourier-domain
guiding schemes [GITH14] due to the interior boundaries.
Arbitrary boundaries are easily handled by our separable approximation to the
fluid guiding proximal operator while remaining highly efficient and easily
parallelizable.
5 Separating Solid-Wall BCs
Another application of our PD-based method is the realization of separating solid-wall BCs.
As a motivational example, consider Figure 13, which shows a 2D breaking dam simulation.
When generated by a common CG pressure solver (top row), the fluid exhibits the
undesirable behavior of sticking to the ceiling and getting stuck in corners.
In contrast, our method (bottom row) features a clean separation of the
fluid from all solids. In many visual effect settings that aim for large-scale fluid flow, the separation is preferable due to its improved realism.
We achieve this behavior by implementing separating solid-wall BCs in the form of
an inequality constraint $\mathbf{u}\cdot\hat{n}\geq 0$
(with $\mathbf{u}$ and $\hat{n}$ denoting velocity and obstacle normal)
without transforming the linear system of equations for the pressure solve into
a more complicated problem.
As outlined in Section 3, we split the problem into two simpler objective functions, with $f$ controlling the BCs and $g$ enforcing zero divergence (see Eq. (15)).
With this setup, a regular CG solver is employed to
compute $g$, while $f$ is handled by an efficient projection and
classification scheme, as described next.
Velocity BCs
The proximal operator
$\mathrm{\mathbf{prox}}_{f,\sigma}(\mathbf{\xi})$ ensures that velocities at
obstacle surfaces never point into the solid (i.e., negative normal
velocity components at fluid-solid faces). Separating velocities (i.e., positive normal components), on the other hand, are allowed and thus left alone. We assume an obstacle is
static and has a velocity of zero at its surface.
BCs that form a linear constraint on the velocity field, like requiring velocity
components to be zero, can be expressed as orthogonal projections [MCPN08].
Therefore, $\mathrm{\mathbf{prox}}_{f,\sigma}(\mathbf{\xi})$ reduces to a projection
onto the space of velocities without flows into obstacles.
In its simplest form, this proximal operator sets normal velocity components to zero
wherever $\mathbf{u}\cdot\hat{n}<0$.
The pressure solver, on the other hand, is unaware of any obstacle
boundaries and uses free surface Dirichlet BCs at liquid surfaces.
Later, we present an accelerated version that makes the pressure solver partially aware of obstacles.
For now, all obstacles are fully handled by $f$.
Although this simple velocity projection works for exact solutions, it breaks down when solving up to finite accuracy. The pressure solver
can accumulate small positive velocities during iterations of our optimization procedure,
leading to small separating motions where the liquid should only be standing still or moving tangentially.
This can happen for hydrostatic cases, for instance, where the final velocity at the wall should be exactly zero.
We address this problem by introducing a classification step with hysteresis.
We assume by default that our cells are separating boundary cells with a
Dirichlet $p=0$ condition, and then create a list of cells that are explicitly not allowed to separate.
For the classification, in each iteration of our algorithm, we accumulate all motions into a wall
returned by the CG solver for a cell $i$ and store it in $m_{i}$. This allows for a solid cell classification with temporal coherence.
A non-separating cell only becomes separating if the magnitude of its velocity component
away from the wall $\mathbf{u}_{i}\cdot\hat{n}_{i}$ exceeds the magnitude of $m_{i}$.
Additionally, a change in the separation classification is only made if
the absolute value of $\mathbf{u}_{i}\cdot\hat{n}_{i}$ is greater than the accuracy of the CG solver
$\epsilon_{\mathrm{CG}}$. Otherwise, the cell keeps its previous state.
The last two steps effectively implement a hysteresis that prevents cells
from changing status due to numerical errors, which is crucial for a
reliable convergence of our method.
Figure 14 illustrates this process. It shows two possible
fluid behaviors at the face of a solid: non-separating (left) and separating
liquid (right). Both velocities fulfill $|\mathbf{u}_{i}\cdot\hat{n}_{i}|>\epsilon_{\mathrm{CG}}$ (black bars), and
can thus potentially change state.
On the left of Figure 14, the cell face has a large
accumulated value $m_{i}$ (red arrow).
Assuming that the face velocity $\mathbf{u}_{i}\cdot\hat{n}_{i}$ (green arrow) exceeds the threshold
$\epsilon_{\mathrm{CG}}$, the cell is classified as non-separating, and $\mathbf{u}_{i}\cdot\hat{n}_{i}$ is later on set to zero despite
its positive value.
In contrast, a positive normal velocity component
greater than $|m_{i}|$ indicates that fluid is currently separating from the solid
cell. The liquid boundary is treated as a free surface in this case, and its
velocity is left unmodified.
Algorithm Summary
Algorithm (3) shows pseudocode for the classification and the
proximal operator for separating BCs. The classification proceeds as
outlined above, updating the list of non-separating cells $S_{\mathrm{nsep}}$ based on the velocity field $\mathbf{u}$. The normal velocity components at non-separating cell faces
are set to zero in the proximal operator $\mathrm{\mathbf{prox}}_{f,\sigma}(\mathbf{\xi})$. The variable $\mathbf{\xi}$ is a generic argument variable being a combination of multiple velocity fields, including dual variables.
The PD algorithm solving for separating BCs follows the generic PD
implementation outlined in Algorithm (1). We exchange the generic
proximal operator by our BC specific projection from Algorithm (3).
Additionally, we call the classification function with $\mathbf{z}$ after line 6 of
Algorithm (1).
We adopt two smaller modifications from previous work that
improve convergence. The first is a Krylov method from IOP
[MCPN08], which we identified to work nicely within our BC solver
(the full algorithm can be found in Appendix B). Additionally, we use the
adaptive parameter scheme [CP11] for dynamic values of $\tau$,
$\sigma$ and $\theta$. The parameter updates are the following after choosing
a suitable value for $\gamma$, $\tau^{0}$ and $\sigma^{0}$ (we use $\gamma=200$, $\tau^{0}=150$, $\sigma^{0}=1/\tau^{0}$):
$\theta^{k}\leftarrow 1/\sqrt{1+2\tau^{k-1}\gamma}$, $\tau^{k}\leftarrow\tau^{k-1}\theta^{k}$, and $\sigma^{k}\leftarrow\sigma^{k-1}/\theta^{k}$.
Both methods reduce the overall run time, but are specific extensions for
proximal operators given by orthogonal projections. As such, we use the extensions
for all BC problems, but not for our guided simulations.
Accelerated BC Solver
So far,
the solid-wall BC handling is fully done in $f$ while $g$ simply ensures incompressibility.
This is in line with the standard splitting approach in other methods [MCPN08, Hen12],
and we demonstrate in the next section that our optimization scheme very efficiently
calculates the solution in this case.
However, in practice, the total number of iterations can be reduced
significantly by letting the pressure solver take care of all BCs that it is
capable of enforcing correctly: retaining the input normal velocity components at fluid-solid faces by enforcing a zero pressure gradient at walls.
For this version of our solver, we update the BCs of the
pressure solver in $g$ in accordance with our classification: Neumann BCs for
non-separating cells, and Dirichlet (free surface) BCs for
separating ones.
This effectively locks the classification after few iterations, leading to a stable solution within the next iteration.
However, it prevents non-separating cells from changing their state back to separating ones. Thus, the
accelerated BC solver does not yield the same result as the standard version, but
achieves much higher performance while featuring only negligible differences. A summary of our accelerated solver with pseudo-code can be found in Appendix D.
5.1 Evaluation and Results
We now evaluate the performance of our BC solver, and demonstrate
the importance of allowing liquid to separate from walls in high-resolution simulations.111
Thresholds for our algorithm are set to $\epsilon_{\mathrm{CG}}=10^{-5}$ (as liquids are more sensitive to mass loss) and $\epsilon_{\mathrm{abs}}=\epsilon_{\mathrm{rel}}=10^{-3}$.
To validate that our PD solver yields the correct results, we use our
method to calculate regular non-separating boundaries
for a $200\times 140$ breaking dam setup, which can also be handled by a
regular CG solver. In this case, we can achieve arbitrary accuracies depending
on the choice of parameters for CG and our scheme. These
experiments show that our method converges to the correct solution.
For separating boundaries, we can no longer compare to a regular CG solver.
Instead, we evaluate the performance of our method against methods from previous convex optimization work: IOP and ADMM.
IOP has been applied to enforce non-divergence and complex BCs simultaneously in [CMF12].
For the standard BC solver simulating a 2D breaking dam, our method converges six times faster than ADMM, and more than twice as fast as IOP, as shown in Figure 14(c).
Figure 14(a) and Figure 14(b) show the mean number of IOP/ADMM/PD and CG iterations.
Our method generally requires a lower number of PD iterations compared to ADMM and IOP.
Due to our adaptive CG accuracy scheme, the number of PD iterations is increased while the total amount of CG iterations is decreased.
Both numbers influence the performance, but the total number of CG iterations more strongly influences runtime as shown in Figure 14(c).
Similar behavior is observed for a 3D complex breaking dam simulation (see Figure 17 for a visual example).
The run time measurements are shown in Figure 16 where ADMM is omitted due to its impractical run time.
The accelerated solver speeds up all methods significantly. In this case, our PD-based method
performs on par with IOP, which we attribute to the low number of iterations required in this case;
the higher-order convergence of the PD method does not develop its full potential in such a setting.
Figure 17 highlights that regular non-separating walls often yield undesirable results
in practical settings. We simulate a complex 3D scene with liquid splashing onto multiple
obstacles with a resolution of $256\times 230\times 256$. A regular solver leads
to large amounts of liquid crawling along ceilings, and sticking to walls,
while our separating boundaries give a much more believable large scale look,
with liquid naturally separating from obstacle boundaries.
For this setting, our accelerated BC solver requires $38.7s$ on average per
frame. Comparing the overall time to generate this result (including surface
generation) to a version with a regular pressure solver, our method increases
run time by only $12\%$, with the added benefit of enabling separating BCs.
This is a very practical result, considering that it was achieved based on a
regular CG solver, without the need for specialized methods, such as Linear Complementarity Problems
solvers [GB13].
Although solving LCPs has been studied in detail, these methods still fall into complexity classes that make large scale solves infeasible. An indication for this can be found in [GB13], where solving times of ca. 25s per LCP solve were given for a $100^{3}$ example.
6 Discussion and Conclusion
We presented a general framework for incorporating the Primal-Dual method into
fluid simulation, and demonstrated two applications: fluid guiding and
separating BCs.
We proposed a generic version of the Primal-Dual based optimization scheme
with fast convergence for general proximal operator subproblems. In addition,
we discussed several extensions that are particularly well-suited for optimizations
with orthogonal projections as subproblems.
Additionally, we demonstrated a novel formulation for the flow guiding
problem, and an efficient approach for simulating liquids with separating
BCs.
Limitations
One limitation of our fluid guiding method is a lack of shape controls. Unlike
smoke, liquids can require shape constraints to achieve a desired outline or
shape. We have focused on velocity guiding in this work, so controlling the
shape of liquids will require extensions that take the position of
the liquid’s surface into account. Such constraints should integrate well into our overall
pipeline, and we plan to investigate this topic as future work.
Another area of improvement is bridging the gap between our optimization framework and an artist’s workflow.
Although we gave examples of different target velocities used for guiding, there is still a lot more to explore in terms of how to achieve various artistic visual effects intuitively.
A limitation of our accelerated separating BC solver is that it can lead to slight deviations from the
accurate, standard solution. We have not encountered visual artifacts resulting from these inaccuracies, and we believe that the
improved performance typically outweighs these slight deviations.
The specialized multigrid solvers [CMF12] could possibly outperform our BC solver.
We believe that the attractiveness of our BC solver stems from its modularity and fast convergence.
Adding support for wall separating BCs given an existing pressure solver requires little code with our method.
Outlook
Our method is a very generic approach applicable to a large range of problems in fluid simulation.
The price of this generality is that it may not be as fast as specialized methods tailored to specific problems.
However, the modularity of our approach makes it easy to incorporate into existing implementations.
As such, it has the potential to add powerful functionality, such as high-level
flow guiding, into existing solvers without the need for complicated
extensions.
Furthermore,
there is a large number of interesting avenues
to be explored with high-level optimizations of fluids flows.
For example, we are interested in exploring shape optimization to adapt
the geometry of obstacles with respect to their flow properties,
and partial resimulations of regions in a flow.
7 Acknowledgments
This work was funded by the ERC Starting Grant realFlow (StG-2015-637014) and the NSERC Postdoctoral Fellowships Program.
Appendix A Notation
For reference we provide a quick description of our notation.
More detailed descriptions can be found in the body of the paper.
•
$f$: first objective function in PD (application-dependent)
•
$g$: second objective function in PD for incompressibility
•
$\mathbf{\xi}$: general input velocity field for several algorithms, a combination of several velocity fields, possibly dual variables
•
$\mathbf{x}$, $\mathbf{z}$, $\mathbf{y}$: variables with iterative updates in PD
•
$\tau$, $\sigma$, $\theta$: PD parameters controlling convergence rate
•
$\alpha$, $\beta$, $G$, $\mathbf{q}$, $M$, $\mathbf{u}_{t}$, $\mathbf{u}_{c}$, $L$: variables specific to fluid guiding
•
$\mathbf{u}$: fluid velocity for the BC problem
•
$\hat{n}$: normal of a solid cell
•
$m$: memory velocity field
•
$S_{\mathrm{nsep}}$: set of non-separating fluid-solid faces
Appendix B ADMM and IOP
Iterated Orthogonal Projection (IOP) [MCPN08]—a method similar to von Neumann’s alternating projections [BPC${}^{*}$11]—requires both subproblems to be expressed as orthogonal projections
$$\displaystyle\mathbf{x}^{k+1}$$
$$\displaystyle=\Pi_{f}(\mathbf{z}^{k})$$
(27)
$$\displaystyle\mathbf{z}^{k+1}$$
$$\displaystyle=\Pi_{g}(\mathbf{x}^{k+1}).$$
(28)
The Krylov method can improve its convergence rate:
1:procedure krylov($\mathbf{z}^{k}$, $\mathbf{z}^{k-1}$, $k$, $\epsilon^{k-1}$)
2: $\epsilon^{k}=\text{error}(\mathbf{z}^{k})$
3: if $k\textgreater 1$ then
4: $\mathbf{z}_{\epsilon_{\mathrm{dif}}}=\mathbf{z}^{k}-\mathbf{z}^{k-1}$ // correction vector
5: $\epsilon_{\mathrm{ratio}}=\epsilon^{k}/\epsilon^{k-1}$
6: $\mathbf{z}_{\epsilon_{\mathrm{tmp}}}=\mathbf{z}^{k}-\epsilon_{\mathrm{ratio}}%
\mathbf{z}_{\epsilon_{\mathrm{dif}}}$
7: $\epsilon_{\epsilon_{\mathrm{tmp}}}=\text{error}(\mathbf{z}_{\epsilon_{\mathrm{%
tmp}}})$
8: if $\epsilon^{\epsilon_{\mathrm{tmp}}}\textless\epsilon^{k}$ then $\mathbf{z}^{k}=\mathbf{z}_{\epsilon_{\mathrm{tmp}}}$
The Alternating Direction Method of Multipliers (ADMM) [BPC${}^{*}$11, GOSB14] is a proximal method more general than IOP given by
$$\displaystyle\mathbf{x}^{k+1}$$
$$\displaystyle:=\mathrm{\mathbf{prox}}_{f,\rho}(\mathbf{z}^{k}-\mathbf{y}^{k})$$
(29)
$$\displaystyle\mathbf{z}^{k+1}$$
$$\displaystyle:=\mathrm{\mathbf{prox}}_{g,\rho}(\mathbf{x}^{k+1}+\mathbf{y}^{k})$$
(30)
$$\displaystyle\mathbf{y}^{k+1}$$
$$\displaystyle:=\mathbf{y}^{k}+\mathbf{x}^{k+1}-\mathbf{z}^{k+1}.$$
(31)
Instead of the three parameters $\{\tau,\sigma,\theta\}$ in PD that control convergence rate,
ADMM only has one such parameter, $\rho$.
Appendix C Inverse Matrix Approximation for Fluid Guiding
Here we present the details for deriving the inverse matrix approximation in Eq. (26).
We want to invert a matrix of the form $M=A+(2G^{T}G)$, where $A=2W^{2}+\sigma I$ contains the large diagonal terms and $2G^{T}G$ the small off-diagonal terms. By the Sherman-Morrison-Woodbury Formula,
$$\displaystyle M^{-1}$$
$$\displaystyle=A^{-1}-2A^{-1}G^{T}(I+2GA^{-1}G^{T})GA^{-1}.$$
(32)
Now since $G$ and $A^{-1}$ both contains small value entries, $2GA^{-1}G^{T}$ is approximately zero. Hence
$$\displaystyle M^{-1}$$
$$\displaystyle\approx A^{-1}-2A^{-1}G^{T}GA^{-1}$$
(33)
$$\displaystyle=(2W^{2}+\sigma I)^{-1}-2(2W^{2}+\sigma I)^{-1}G^{T}G(2W^{2}+%
\sigma I)^{-1}.$$
(34)
Note that the calculation of $A^{-1}$ is trivial since $A=2W^{2}+\sigma I$ is diagonal.
Appendix D Accelerated BC Solver
Below, we summarize our accelerated solver for separating solid-wall BCs.
Instead of fully handling the solid-wall BCs in a proximal operator outside of the pressure projection, our accelerated
solver employs the commonly used Neumann BCs during the pressure projection depending on the current classification of solid-wall cells.
First, all solid wall cells are classified as separating (i.e. Dirichlet BCs for the pressure solver).
Depending on the initial velocity field, solid wall cells with $\mathbf{u}\cdot\hat{n}<0$ are classified
as non-separating cells (Neumann BCs for the pressure solver).
We then iteratively set the velocity BCs, apply a regular CG pressure solver and re-classify solid cells with the classify procedure.
In our standard BC solver (Section 5), a memory field $m$ is used to determine whether a cell is allowed to change its state to separating.
In the accelerated case, $m$ is not used, instead the pressure solver enforces solid wall BCs for non-separating cells. This prevents
the normal velocity from becoming positive, and from changing its state back to separating.
In the following procedure, $m$ is replaced with the placeholder "$\cdot$".
1:procedure solvePressureWithSeparatingBCs($\mathbf{u}$, $\epsilon$)
2: $S_{\mathrm{nsep}}={0}$ // no cell is classified as non-separating yet
3: classify($\mathbf{u}$, $\cdot$, $\epsilon$, $S_{\mathrm{nsep}}$)
4: while $S_{\mathrm{nsep}}$ did not change in last call of classify do
5: $\textsc{proxF}_{BC}$($\mathbf{u}$, $S_{\mathrm{nsep}}$)
6: $\Pi_{\mathrm{DIV}}$($\mathbf{u}$, $\epsilon$)
7: classify($\mathbf{u}$, $\cdot$, $\epsilon$, $S_{\mathrm{nsep}}$)
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A Characterization of Norm Compactness in the Bochner Space $L^{p}\left(G;B\right)$
For an Arbitrary Locally Compact Group $G$
Josh Isralowitz
Department of Mathematical Sciences, New Jersey Institute of Technology,
Newark, NJ 07102-1982 U.S.A.
jbi2@buffalo.edu
Abstract
In this paper, we generalize a result of N. Dinculeanu
which characterizes
norm compactness in the Bochner space $L^{p}\left(G;B\right)$ in terms of an approximate identity
and translation operators,
where $G$ is a locally compact abelian group and $B$ is a Banach space.
Our characterization includes the case where $G$ is nonabelian,
and we weaken the hypotheses on the approximate identity used, providing
new results even for the case $B=\mathbb{C}$ and $G=\mathbb{R}^{n}.$
keywords:
Bochner Integral, Vector Valued Integration, Function Spaces,
Compactness
MSC: 46G10
††journal: Journal of Mathematical Analysis and Applications
††thanks: Current Address: Mathematics Department, University at Buffalo, Buffalo, NY 14260-2900
1 Introduction
The classical Kolmogorov-Riesz-Tamarkin theorem
characterizes norm compact subsets $\Gamma$ of $L^{p}\left(\mathbb{R}^{n}\right)$ in terms of the uniform convergence (in
the $L^{p}$ norm) of families of convolution and translation
operators, where a translation operator $T^{h}$ is defined by
$\left(T^{h}f\right)\left(x\right)=f\left(x-h\right)$ and a
convolution operator $J$ is given by $\left(Jf\right)\left(x\right)=\int_{\mathbb{R}^{n}}j\left(y\right)f\left(x-y%
\right)\,dy$ for
an appropriate kernel $j$ (for a brief history of this theorem see
[2] or [8]). Many authors have generalized this
theorem to $L^{p}\left(X\right)$ where $X$ is more general than
$\mathbb{R}^{n}$ and for function spaces that are more general than
the classical Lebesgue spaces $L^{p}$. One such generalization is the
following result, due to N. Dinculeanu [2, Sect. 5] :
Theorem 1
Let $G$ be a locally compact abelian group with Haar measure
$\mu$ and $B$ be a Banach space.
A bounded subset $\Gamma$ of the Bochner space $L^{p}\left(G;B\right),1\leq p<\infty,$ is relatively norm compact
iff
1) for any $A\subseteq G$ with $\mu\left(A\right)<\infty$, the
set $\{\int_{A}f\left(y\right)\,d\mu\left(y\right):f\in\Gamma\}\subseteq B$ is relatively norm compact in $B$,
2) for any $\epsilon>0$, there exists a compact $S\subseteq G$
such that
$$\forall f\in\Gamma,\ \ \ \left\|f\right\|_{p;G\backslash S}<\epsilon,$$
and one of the following is true:
3a) let $\mathfrak{V}$ be a basis of relatively compact
neighborhoods of the identity in $G,$ and for each $V\in\mathfrak{V}$ let $u_{V}$ be a positive, bounded, symmetric function
which vanishes outside $V$ where $\left\|u_{V}\right\|_{1;G}=1.$ Then $\lim_{V\in\mathfrak{V}}\left\|u_{V}\ast f-f\right\|_{p;G}=0,$
3b) for any $\epsilon>0$, there exists a neighborhood $\Theta$ of
the identity such that
$$\forall f\in\Gamma,\ \ \ h\in\Theta\Longrightarrow\left\|T^{h}f-f\right\|_{p;G%
}<\epsilon.$$
Here we let
$$\left\|f\right\|_{p;A}=\left(\int_{A}\left\|f\left(x\right)\right\|_{B}^{p}\,d%
\mu\left(x\right)\right)^{\frac{1}{p}}$$
where $\left\|\cdot\right\|_{B}$ is the
Banach space norm.
We generalize the above result in two ways. First, we
generalize the theorem to include the case where $G$ is nonabelian.
Second, we weaken the hypotheses in property
3a). More specifically, we show that property 3a) can be replaced by the following:
for any $\epsilon>0$ and compact $S\subseteq G$ provided by
property 2), there exists
$j\in L^{p^{\prime}}\left(G\right)$ such that $\forall f\in\Gamma,\left\|j\ast f-f\right\|_{p;S}<\epsilon$, where $p^{\prime}$ is the
conjugate exponent of $p$ and where $j$ has compact support if $G$
is nonabelian or $p=1$.
We note that our weakening of these hypotheses
appears to be new even for the classical cases $B=\mathbb{C}$ and $G=\mathbb{R}^{n}$.
2 Preliminary Results
In this section we provide some preliminary results the will be
needed to prove our main theorem. We start by introducing some
relevant definitions and notation. In all of the rest of the paper,
$G$ will denote a locally compact group with left invariant Haar
measure $\mu$. For $1\leq p<\infty$ and a Banach space $B,L^{p}\left(G;B\right)$ will denote the
space of $\mu$ measurable functions $f:G\rightarrow B$ satisfying $\left\|f\right\|_{B}^{p}\in L^{1}\left(G\right),$ where $L^{p}\left(G\right)$ denotes the standard
Lebesgue space of complex valued functions. The integral $\int_{A}f(x)\,d\mu\left(x\right)$ over a set $A$ will also be written as
$\int_{A}f(x)\,dx$. We refer the reader to [4, Chap. 3]
for further properties of the Bochner integral.
We let $\triangle$ denote the continuous homomorphism from $G$ into
$\mathbb{R}^{+}$ defined by $\triangle\left(x\right)\mu\left(A\right)=\mu\left(Ax^{-1}\right)$, which by the definition of the Bochner
integral implies that $\int_{G}f\left(yx\right)\,dy=\triangle\left(x\right)\int_{G}f\left(y\right)\,dy$.
If $j\in L^{p^{\prime}}(G)$ and $f\in L^{p}(G,B)$, then the convolution
product $\left(j\ast f\right)\left(x\right)$ is defined by $=\int_{G}j\left(xy\right)f\left(y^{-1}\right)\,dy.$ The
convolution can be defined equivalently as $=\int_{G}j\left(y\right)f\left(y^{-1}x\right)\,dy$, since the left
invariance of $\mu$ implies that $\int_{G}h\left(xy\right)\,dy=\int_{G}h\left(y\right)\,dy$ for any $x\in G$ and any Bochner
integrable function $h$. [2, Sect. 3]
If we require that $j$ have compact support when either $G$ is
nonabelian or $p=1$, then the continuity of $\triangle$ and
Hölder’s inequality implies that $j*f$ gives us a well defined
function from $G$ to $B$. Moreover, Lemma $7$ and the fact that $G$
is locally compact will imply that $j*f$ is continuous on $G$ so
that $j\ast f\in L^{r}_{\rm loc}\left(G;B\right)$ for any $r\geq 1$, where $L^{r}_{\rm loc}\left(G;B\right)$ denotes the space of
$\mu$ measurable functions $f:G\rightarrow B$ such that $\left\|f\right\|_{r;K}<\infty$ for every compact $K\subseteq G$.
The main component in the proof of our main result will be Lemma 7.
Before proving this lemma, however, we need an elementary result
regarding the integration of functions in $\Gamma$ over sets of
small measure.
Definition 2
A subset $\Gamma\subseteq L^{1}\left(G;B\right)$ is called
Uniformly Integrable on $S\subseteq G$ if for any
$\epsilon>0$, there exists $\delta>0$ such that
$$\displaystyle\forall f\in\Gamma,\ \ \ \ A\subseteq S\mathrm{\ and\ }\mu(A)<%
\delta\Longrightarrow\|f\|_{1;A}<\epsilon.$$
Lemma 3
Let $\Gamma\subseteq L^{1}\left(G;B\right)$ be bounded and let $S\subseteq G$ be compact. For any $\epsilon>0$, assume that there
exists $j\in L^{\infty}\left(G\right)$ with compact support $V$
such that
$$\forall f\in\Gamma,\ \ \ \left\|f-j\ast f\right\|_{1;S}<\epsilon.$$
Then $\Gamma^{-}$ is uniformly integrable on $S$ where $\Gamma^{-1}=\{f^{-}:f\in\Gamma\}$ and $f^{-}$ denotes the function
defined by $f^{-}\left(x\right)=f\left(x^{-1}\right).$
{@proof}
[Proof.]
First we show that for fixed $j$, the set
$j\ast{\Gamma}=\{j\ast f:f\in\Gamma\}$ is uniformly integrable on $G$.
Note that as $V$ is compact, there exists $m>0$ where
$\triangle\left(y^{-1}\right)>m$ for all $y\in V$ since
$\triangle$ has a (nonzero) minimum on $V^{-1}.$ Let $\left\|f\right\|_{\infty;V}$ denote the essential supremum
$\left\|f\right\|_{B}$ over $V$.
Thus, we have
$$\displaystyle\left\|j\ast f\right\|_{1;A}$$
$$\displaystyle\leq$$
$$\displaystyle\int_{A}\left(\int_{V}|j\left(xy\right)|\left\|f\left(y^{-1}%
\right)\right\|_{B}\,dy\right)\,dx$$
$$\displaystyle\leq$$
$$\displaystyle\left(\left\|j\right\|_{\infty;V}\right)\mu\left(A\right)\int_{V}%
\left\|f\left(y^{-1}\right)\right\|_{B}\,dy$$
$$\displaystyle=$$
$$\displaystyle\left(\left\|j\right\|_{\infty;V}\right)\mu\left(A\right)\int_{V}%
\left(\frac{\triangle\left(y^{-1}\right)}{\triangle\left(y^{-1}\right)}\right)%
\left\|f\left(y^{-1}\right)\right\|_{B}\,dy$$
$$\displaystyle<$$
$$\displaystyle\frac{1}{m}\left(\left\|j\right\|_{\infty;V}\right)\mu\left(A%
\right)\int_{V}\triangle\left(y^{-1}\right)\left\|f\left(y^{-1}\right)\right\|%
_{B}\,dy$$
$$\displaystyle\leq$$
$$\displaystyle\frac{1}{m}\left(\left\|j\right\|_{\infty;V}\right)\mu\left(A%
\right)\int_{G}\left\|f\left(y\right)\right\|_{B}\,dy$$
$$\displaystyle\leq$$
$$\displaystyle M^{\prime}\mu\left(A\right)$$
where $M^{\prime}$ does not depend on $f\in\Gamma$, and so
$j\ast{\Gamma}$ is uniformly integrable on $G$.
Now we show that $\Gamma$ is uniformly
integrable on $G$. By hypothesis, for any $\epsilon>0$, we can
choose $j\in L^{\infty}\left(G\right)$
with compact support, satisfying
$$\forall f\in\Gamma,\ \ \ \left\|f-j\ast f\right\|_{1;G}\leq\frac{\epsilon}{2}.$$
Since $\{j\ast f:f\in\Gamma\}$ is uniformly integrable
on $G$, choose $\delta>0$ so that
$$\mu\left(A\right)<\delta\Longrightarrow\forall f\in\Gamma,\ \ \ \left\|j\ast f%
\right\|_{1;A}<\frac{\epsilon}{2},$$
and so
$$\mu\left(A\right)<\delta\Longrightarrow\forall f\in\Gamma,\ \ \ \left\|f\right%
\|_{1;A}\leq\left\|f-j\ast f\right\|_{1;A}+\left\|j\ast f\right\|_{1;A}<\epsilon.$$
Finally, let $S\subseteq G$ be compact and let $A\subseteq S$. By compactness, there exist $m$ and $M$ such that $0<m<\triangle\left(y^{-1}\right)<M$ for all $y\in S$. Thus, if $f\in\Gamma,$ we have
$$\displaystyle\int_{A}\left\|f\left(y^{-1}\right)\right\|_{B}\,dy$$
$$\displaystyle<$$
$$\displaystyle\frac{1}{m}\int_{A}\triangle\left(y^{-1}\right)\left\|f\left(y^{-%
1}\right)\right\|_{B}\,dy$$
$$\displaystyle\leq$$
$$\displaystyle\frac{1}{m}\int_{G}\triangle\left(y^{-1}\right)\left(\chi_{A^{-1}%
}\left(y^{-1}\right)\left\|f\left(y^{-1}\right)\right\|_{B}\right)\,dy$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{m}\int_{G}\left(\chi_{A^{-1}}\left(y\right)\right)\left%
\|f\left(y\right)\right\|_{B}\ \,dy$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{m}\int_{A^{-1}}\left\|f\left(y\right)\right\|_{B}\,dy.$$
However,
$$\displaystyle\mu\left(A^{-1}\right)$$
$$\displaystyle=$$
$$\displaystyle\int_{G}\chi_{A^{-1}}\left(x\right)\,dx$$
$$\displaystyle=$$
$$\displaystyle\int_{G}\chi_{A^{-1}}\left(x^{-1}\right)\triangle\left(x^{-1}%
\right)dx$$
$$\displaystyle<$$
$$\displaystyle M\int_{G}\chi_{A^{-1}}\left(x^{-1}\right)\,dx$$
$$\displaystyle=$$
$$\displaystyle M\mu\left(A\right).$$
Therefore, since $\Gamma$ is
uniformly integrable on $G$, we have that $\Gamma^{-}$ is
uniformly integrable on $S$.
∎
Before we state and prove Lemma 7, which is crucial to our main
result, we will need two preliminary results, the first of which is
a norm compactness criteria for general normed spaces and was proven
in [3, Lemma 3].
Lemma 4
Let $S$ be any set, $F$ be a normed space with norm $\left\|\ \cdot\ \right\|_{F},$ and $\{f_{\alpha}\}$ be a generalized
sequence of functions with $f_{\alpha}:S\rightarrow F$.
If $f:S\rightarrow F$ is a function such that $\lim_{\alpha}\left\|f_{\alpha}\left(s\right)-f\left(s\right)\right\|_{F}=0$ uniformly for $s\in S$, and if each $f_{\alpha}\left(S\right)$ is
relatively norm compact in $F$, then $f\left(S\right)$ is relatively
norm compact in $F$.
The second preliminary result is the following form of the
Ascoli-Arzela theorem, which is a special case of the general
theorem [6, Sect. 47]. First we introduce some relevant
notation and definitions.
Let $S$ be a subspace of $G$. We let $C_{B}(S)$ denote the Banach
space of all continuous $B$ valued functions on $S$ endowed with the
“sup” norm $\mathrm{sup}_{x\in S}\|f(x)\|_{B}$.
Definition 5
If $S$ is a subspace of $G$, then a subset $K\subseteq C_{B}(S)$ is
said to be equi-continuous at x if for any $\epsilon>0$,
there exists a relative neighborhood $N_{x}\subseteq S$ of $x$ such
that
$$\displaystyle x^{\prime}\in N_{x}\Longrightarrow\forall f\in K,\hskip 14.22637%
8pt\|f(x)-f(x^{\prime})\|_{B}<\epsilon.$$
If $K$ is equi-continuous on all of $S$, we say $K$ is
equi-continuous on $S$
Lemma 6
Let $S$ be a compact Hausdorff space and $B$ a Banach space.
A subset $K\subseteq C_{B}\left(S\right)$ is relatively norm
compact iff
1) $K$ is equi-continuous on $S$,
2) for every $s\in S$, the set $K\left(s\right)=\{f\left(s\right):f\in K\}$ is relatively
norm compact in $B$.
Lemma 7
Let $\Gamma\subseteq L^{p}\left(G;B\right)$ be bounded and let $j\in L^{p^{\prime}}\left(G\right)$ where $j$ has compact support if either
$G$ is nonabelian or $p=1$. Moreover, assume that when $A\subseteq G$ and $\ \mu\left(A\right)<\infty$, then $\{\int_{A}f^{-}\left(y\right)\,dy:f\in\Gamma\}\subseteq B$ is relatively
norm compact in $B$.
Suppose that $S\subseteq G$ is compact and contains the
identity. Then $\left(j\ast\Gamma\right)|_{S}$ is relatively norm
compact in $C_{B}\left(S\right)$ where $\left(j\ast\Gamma\right)|_{S}=\{\left(j\ast f\right)|_{S}:f\in\Gamma\}$ and $\left(j\ast f\right)|_{S}$ is the restriction of $j\ast f$ to $S$.
{@proof}
[Proof.] We first remark that we will show
the existence of each $\{\int_{A}f^{-}\left(y\right)\,dy:f\in\Gamma\}$. We now check that $\left(j\ast\Gamma\right)$ is
equi-continuous on $S.$ Assume that $p>1$, $G$ is nonabelian, and
supp $j=V$ is compact . Let $SV$ denote the set $\{sv:s\in S,v\in V\}$ so that
$V\subseteq SV$ since $S$ contains the identity.
Since $\Gamma$ is bounded and $SV$ is compact, we can choose $M>0$ such that
$$\forall f\in\Gamma,\ \ \ \left(\int_{SV}\left\|f\left(y^{-1}\right)\right\|_{B%
}^{p}\,dy\right)^{\frac{1}{p}}<M.$$
Then using Hölder’s equality, we compute that
$$\displaystyle\left\|\left(j\ast f\right)\left(x\right)-\left(j\ast f\right)%
\left(x^{\prime}\right)\right\|_{B}$$
$$\displaystyle\leq$$
$$\displaystyle\int_{V}|j\left(xy\right)-j\left(x^{\prime}y\right)|\left\|f\left%
(y^{-1}\right)\right\|_{B}\,dy$$
$$\displaystyle\leq$$
$$\displaystyle\left(\int_{V}|j\left(xy\right)-j\left(x^{\prime}y\right)|^{p^{%
\prime}}\,dy\right)^{\frac{1}{p^{\prime}}}\left(\int_{V}\left\|f\left(y^{-1}%
\right)\right\|_{B}^{p}\,dy\right)^{\frac{1}{p}}$$
$$\displaystyle\leq$$
$$\displaystyle M\left(\int_{V}|j\left(xy\right)-j\left(x^{\prime}y\right)|^{p^{%
\prime}}\,dy\right)^{\frac{1}{p^{\prime}}}.$$
This clearly shows equi-continuity, since $j$ is translation
continuous in the $L^{p^{\prime}}$ norm. If $G$ is abelian and $p>1$, then unimodularity
implies that $j$ does not need compact support.
If $p=1$ and $G$ is not necessarily abelian, choose any
$\epsilon>0$. Since Lemma 1 states that
$\Gamma^{-}$ is uniformly integrable on $SV$, we can choose
$\delta>0$ so that
$$A\subseteq SV\ \mathrm{and}\ \mu\left(A\right)<\delta\Longrightarrow\forall f%
\in\Gamma,\ \ \ \left\|f^{-}\right\|_{1;A}<\frac{\epsilon}{4\left\|j\right\|_{%
\infty;A}}.$$
By Lusin’s theorem,
we may choose a compact $F\subseteq SV$
where
$$\mu\left(SV\right)-\mu\left(F\right)<\frac{\delta}{2}$$
and $j|_{F}$ is continuous.
Moreover, by the Tietze extension theorem, we may extend $j$ to a continuous function $g$ on $SV$,
which provides us with a continuous $g:SV\rightarrow\mathbb{C}$
and a compact $F\subseteq SV$ such that
$j=g$ on $F$, where
$$\mu\left(SV\right)-\mu\left(F\right)<\frac{\delta}{2}.$$
Since $g$ has compact support, for any $x\in G$, there
exists a relative neighborhood $N_{x}\subseteq S$ of $x$ such that
$$x^{\prime}\in N_{x}\Longrightarrow|j\left(xy\right)-j\left(x^{\prime}y\right)|%
=|g\left(xy\right)-g\left(x^{\prime}y\right)|<\frac{\epsilon}{2M}$$
for all $y\in\left(x^{-1}F\cap{x^{\prime}}^{-1}F\right)$.
Thus fixing $x$ and fixing any such $x^{\prime}$, we find that
$$\displaystyle\left\|\left(j\ast f\right)\left(x\right)-\left(j\ast f\right)%
\left(x^{\prime}\right)\right\|_{B}$$
$$\displaystyle\leq$$
$$\displaystyle\int_{V}|j\left(xy\right)-j\left(x^{\prime}y\right)|\left\|f\left%
(y^{-1}\right)\right\|_{B}\,dy$$
$$\displaystyle\leq$$
$$\displaystyle\int_{SV\backslash\left(x^{-1}F\cap{x^{\prime}}^{-1}F\right)}|j%
\left(xy\right)-j\left(x^{\prime}y\right)|\left\|f\left(y^{-1}\right)\right\|_%
{B}\,dy$$
$$\displaystyle+$$
$$\displaystyle\int_{\left(x^{-1}F\cap{x^{\prime}}^{-1}F\right)}|j\left(xy\right%
)-j\left(x^{\prime}y\right)|\left\|f\left(y^{-1}\right)\right\|_{B}\,dy$$
$$\displaystyle<$$
$$\displaystyle\frac{\epsilon}{2}+\int_{\left(x^{-1}F\cap{x^{\prime}}^{-1}F%
\right)}|j\left(xy\right)-j\left(x^{\prime}y\right)|\left\|f\left(y^{-1}\right%
)\right\|_{B}\,dy$$
$$\displaystyle<$$
$$\displaystyle\frac{\epsilon}{2}+\frac{\epsilon}{2M}\int_{\left(x^{-1}F\cap{x^{%
\prime}}^{-1}F\right)}\left\|f\left(y^{-1}\right)\right\|_{B}\,dy<\epsilon.$$
Therefore, we have
$$\forall f\in\Gamma,\ \ \ \left\|\left(j\ast f\right)\left(x\right)-\left(j\ast
f%
\right)\left(x^{\prime}\right)\right\|_{B}<\epsilon.$$
Since $x\in S$ was arbitrary, it follows that $j\ast\Gamma$ is
equi-continuous on
$S$.
Now we check that the set
$\left(j\ast\Gamma\right)\left(s\right)=\{\left(j\ast f\right)\left(s\right):f%
\in\Gamma\}$ is relatively norm compact
in $B$ for every $s\in S$. Although Dinculeanu proved this for $G$
abelian, and although the proof is nearly identical for $G$
non-abelian, we nevertheless present the proof for the sake of
completeness. Assume that $\mu\left(A\right)<\infty\Longrightarrow\{\int_{A}f^{-}\left(y\right)\,dy:f\in\Gamma\}$
is relatively norm compact in $B$ and that $G$ is not necessarily
abelian.
Let $j$ be
a simple function of the form $j\left(y\right)=\sum{\alpha}_{i}\chi_{A_{i}}\left(y\right)$ where each $A_{i}$ has finite measure and
where the sum is taken over finitely many $i$. Then we have
$$\left(j\ast f\right)\left(x\right)=\sum{\alpha}_{i}\int_{G}\chi_{A_{i}}\left(%
xy\right)f\left(y^{-1}\right)\,dy=\sum{\alpha}_{i}\int_{x^{-1}A_{i}}f\left(y^{%
-1}\right)\,dy.$$
Thus,
since each $\{\int_{x^{-1}A_{i}}f\left(y^{-1}\right)\,dy:f\in\Gamma\}$ is relatively norm compact in $B$, we have that
$\left(j\ast\Gamma\right)\left(x\right)=\{\sum{\alpha}_{i}\int_{x^{-1}A_{i}}f%
\left(y^{-1}\right)\,dy:f\in\Gamma\}$ is
relatively norm compact in $B$.
If $j$ is not a simple function, then let $j_{n}\rightarrow j$ in the $L^{p^{\prime}}$ norm where $j_{n}$ is a simple function. We show
that $\left(j_{n}\ast f\right)\left(x\right)\rightarrow\left(j\ast f\right)\left(x\right)$ uniformly for $f\in\Gamma.$ To that
end, again let $0<m<\triangle\left(y^{-1}\right)$ on $V$, so
from
Hölder’s inequality we have
$$\displaystyle\left\|\left(j_{n}\ast f\right)\left(x\right)-\left(j\ast f\right%
)\left(x\right)\right\|_{B}$$
$$\displaystyle\leq$$
$$\displaystyle\int_{G}|j_{n}\left(y\right)-j\left(y\right)|\left\|f\left(y^{-1}%
x\right)\right\|\,dy$$
$$\displaystyle\leq$$
$$\displaystyle\left\|j_{n}-j\right\|_{p^{\prime};V}\left(\int_{V}\left\|f\left(%
y^{-1}x\right)\right\|_{B}^{p}\,dy\right)^{\frac{1}{p}}$$
$$\displaystyle<$$
$$\displaystyle\frac{\left(\triangle\left(x\right)\right)^{\frac{1}{p}}}{m}\left%
\|j_{n}-j\right\|_{p^{\prime};V}\left(\int_{V}\left\|f\left(y\right)\right\|_{%
B}^{p}\,dy\right)^{\frac{1}{p}}$$
$$\displaystyle\leq$$
$$\displaystyle\frac{M}{m}\left({\triangle\left(x\right)}\right)^{\frac{1}{p}}%
\left\|j_{n}-j\right\|_{p^{\prime};V}$$
where $\Gamma$ is norm bounded by $M$.
Thus,
$\left(j_{n}\ast f\right)\left(x\right)\rightarrow\left(j\ast f\right)\left(x\right)$ uniformly for $f\in\Gamma.$ Note that if
$G$ is abelian, then unimodularity implies that $j$ does not need
compact support. Also note that if $j$ has compact support $V$, then
we may assume that $A\subseteq V$ so that $A$ is relatively
compact, and therefore each $\int_{A}f^{-}\left(y\right)\,dy$
exists.
Finally, by Lemma $4$, each $\left(j\ast\Gamma\right)\left(x\right)$
for $x\in K$ is norm compact in $B$. Therefore, by Lemma $6$, we
are done.
∎
We note that if $G$ is abelian or compact, then unimodularity
implies that the following two statements are equivalent:
1) $A\subseteq G\ \mathrm{and}\ \mu\left(A\right)<\infty\Longrightarrow\{\int_{A}f%
^{-}\left(y\right)\,dy:f\in\Gamma\}$
is relatively norm compact in $B$.
2) $A\subseteq G\ \mathrm{and}\ \mu\left(A\right)<\infty\Longrightarrow\{\int_{A}f%
\left(y\right)\,dy:f\in\Gamma\}$ is
relatively norm compact in $B$.
3 Main Result
Now we may state our principal theorem.
Theorem 8
Let $G$ be a locally compact group with left invariant Haar measure
$\mu$ and $B$ be a Banach space.
A bounded subset $\Gamma$ of the Bochner space $L^{p}\left(G;B\right)$ is relatively norm compact
iff
1) for any $A\subseteq G$ with $\ \mu\left(A\right)<\infty$,
the set $\{\int_{A}f^{-}\left(y\right)\,dy:f\in\Gamma\}\subseteq B$ is relatively
norm compact in $B$,
and one of the following is true:
2a) for any $\epsilon>0$, there exists a compact $S\subseteq G$
and $j\in L^{p^{\prime}}\left(G\right)$ such that
$$\forall f\in\Gamma,\ \ \ \left\|f\right\|_{p;G\backslash S}<\epsilon$$
and
$$\forall f\in\Gamma,\ \ \ \left\|f-j\ast f\right\|_{p;S}<\epsilon$$
where $j$
has compact support if either $G$ is nonabelian or $p=1$,
2b) for any $\epsilon>0$, there exists a compact $S\subseteq G$
and a neighborhood $\Theta$ of the identity such that
$$\forall f\in\Gamma,\ \ \ \left\|f\right\|_{p;G\backslash S}<\epsilon$$
and
$$\forall f\in\Gamma,\ \ \ h\in\Theta\Longrightarrow\left\|T^{h}f-f\right\|_{p;S%
}<\epsilon$$
where $T^{h}$ is the translation operator defined by
$\left(T^{h}f\right)\left(x\right)=f\left(h^{-1}x\right)$.
{@proof}
[Proof.] First we prove necessity for the general case of a locally compact
group and $1\leq p<\infty$. We show that both 2a) and 2b) hold.
To that end, let $\epsilon>0$. Since $\Gamma$ is totally bounded,
there exists $\{f_{1},\ldots,f_{N}\}\subseteq\Gamma$ such that
$$\Gamma\subseteq\bigcup_{i=1}^{N}B\left(f_{i},\frac{\epsilon}{3}\right)$$
where $B\left(f_{i},\frac{\epsilon}{3}\right)$ denotes a ball of
radius $\frac{\epsilon}{3}$ at $f_{i}$ in $L^{p}\left(G,B\right).$
Choose a compact $S\subseteq G$ such that
$$\left\|f_{i}\right\|_{p;G\backslash S}<\frac{\epsilon}{2}$$
for all
$i\in\{1,\ldots,N\}$.
Then, we have
$$\forall f\in\Gamma,\ \ \ \left\|f\right\|_{p;G\backslash S}\leq\left\|f-f_{k}%
\right\|_{p;G\backslash S}+\left\|f_{k}\right\|_{p;G\backslash S}<\epsilon,$$
where $k\in\{1,\ldots,N\}$ corresponds to $f$.
For property 2a), let $\mathfrak{V}$ be a basis of
relatively compact neighborhoods of the identity of $G$. For each $V\in\mathfrak{V}$, let $u_{V}=\mu(V)^{-1}\chi_{V}$.
We show that $\lim_{V\in\mathfrak{V}}\left\|u_{V}\ast f-f\right\|_{p;G}=0$
uniformly in $\Gamma$. This holds for any $f\in\{f_{1},\ldots,f_{N}\}$,
so choose $V^{\prime}$ such that
$$\forall f\in\{f_{1},\ldots,f_{N}\},\ \ \ V\subseteq V^{\prime}\Longrightarrow%
\left\|u_{V}\ast f-f\right\|_{p;G}<\frac{\epsilon}{3}.$$
Since $\left\|u_{V}\right\|_{1;G}=1$, a standard computation
using Hölder’s inequality and left invariance shows that
$\left\|u_{V}\ast f-u_{V}\ast f_{k}\right\|_{p;G}\leq\left\|f-f_{k}\right\|_{p;G}$. Therefore, we have
$$\begin{array}[]{lll}\forall f\in\{f_{1},...,f_{N}\},\ \ \ V\subseteq V^{\prime%
}&\Longrightarrow&\left\|u_{V}\ast f-f\right\|_{p;G}\\
&\leq&\left\|u_{V}\ast f-u_{V}\ast f_{k}\right\|_{p;G}+\left\|u_{V}\ast f_{k}-%
f_{k}\right\|_{p;G}\\
&&\qquad+\left\|f_{k}-f\right\|_{p;G}\\
&\leq&2\left\|f-f_{k}\right\|_{p;G}+\left\|f_{k}-j\ast f_{k}\right\|_{p;G}\\
&<&\epsilon,\end{array}$$
where $k\in\{1,\ldots,N\}$
corresponds to $f$.
For property 2b), choose a neighborhood $\Theta$ of the
identity with the property
$$\forall k\in\{1,\ldots,N\},\ \ \ y\in\Theta\Longrightarrow\int_{G}\left\|f_{k}%
\left(x\right)-f_{k}\left(y^{-1}x\right)\right\|_{B}^{p}\,dx<\left({\frac{%
\epsilon}{3}}\right)^{p}.$$
Thus, we have
$$\begin{array}[]{lll}\forall f\in\Gamma,\ \ \ y\in\Theta&\Longrightarrow&\left(%
\int_{G}\left\|f(x)-f(y^{-1}x)\right\|_{B}^{p}dx\right)^{1/p}\\
&\leq&\left(\int_{G}\left\|f(x)-f_{k}(x)\right\|_{B}^{p}dx\right)^{1/p}+\left(%
\int_{G}\left\|f_{k}(x)-f_{k}(y^{-1}x)\right\|_{B}^{p}dx\right)^{1/p}\\
&&\qquad+\left(\int_{G}\left\|f_{k}(x)-f(y^{-1}x)\right\|_{B}^{p}dx\right)^{1/%
p}\\
&<&\epsilon\end{array}$$
by left invariance.
Finally, if $\mu\left(A\right)<\infty$ and $G$ is abelian,
then the continuity of the map $f\mapsto\int_{A}f^{-}\left(x\right)\,dx$ of $L^{p}\left(G;B\right)$ into $B$ implies that $\{\int_{A}f^{-}\left(x\right)\,dx:f\in\Gamma\}\subseteq B$ is norm
compact in $B$. If $G$ is nonabelian, then as we noted in the proof
of Lemma 7, we only need to consider the case where $A$ is
relatively compact. Therefore, the continuity of the map $f\mapsto\int_{A}f^{-}\left(x\right)\,dx$ easily follows.
Now we prove sufficiency. Assume that properties 1) and 2a) hold.
Let $\{f_{n}\}\subseteq\Gamma$ and choose any $\epsilon>0$. Choose
a compact $S\subseteq G$ and $j\in L^{p^{\prime}}\left(G\right)$ so that
$$\forall f_{n}\in\Gamma,\ \ \ \left\|f_{n}\right\|_{p;G\backslash S}<\frac{%
\epsilon}{6}$$
and
$$\forall f_{n}\in\Gamma,\ \ \ \left\|f_{n}-j\ast{f_{n}}\right\|_{p;S}<\frac{%
\epsilon}{3}.$$
Morever, since $G$ is locally compact, we may assume that $S$ contains the identity. Therefore,
Lemma $7$ implies that $\left(j\ast\Gamma\right)|_{S}$ is
relatively compact in $C_{B}\left(S\right)$, and so passing to a
subsequence if necessary (and assuming without loss of generality
that each $j\ast{f_{n}}$ is restricted to $S$), we have that $j\ast{f_{n}}\rightarrow f$ in
$C_{B}\left(S\right)$ for some $f\in C_{B}\left(S\right)$. Define
$f$ to be zero on $G\backslash S$.
By our hypothesis and the fact that $j\ast{f_{n}}\rightarrow f$ in
$C_{B}\left(S\right)$, we have that for all large enough $n\in\mathbb{N}$
,
$$\begin{array}[]{lll}\forall f\in\Gamma,\ \ \ \left\|f_{n}-f\right\|_{p;G}&\leq%
&\left\|f_{n}-f\right\|_{p;G\setminus S}+\left\|f_{n}-f\right\|_{p;S}\\
&\leq&\epsilon/3+\left\|f_{n}-f\right\|_{p;S}\\
&\leq&\epsilon/3+\left\|f_{n}-j\ast f_{n}\right\|_{p;S}+\left\|j\ast f_{n}-f%
\right\|_{p;S}\\
&\leq&\epsilon\end{array}$$
Hence, $\Gamma$ is relatively norm compact.
Finally, we show that property 2b) implies property 2a).
Choose a neighborhood $\Theta^{\prime}$ of the identity such that
$$y\in\Theta^{\prime}\Longrightarrow\int_{S}\left\|f\left(x\right)-f\left(y^{-1}%
x\right)\right\|_{B}^{p}\,dx<{\epsilon}^{p}.$$
Since $G$ is locally compact, we may
assume $\Theta^{\prime}$ is relatively compact, and so $\Theta^{\prime}$ has nonzero
finite measure. Let $j=\mu\left(\Theta^{\prime}\right)^{-1}\chi_{\Theta^{\prime}}$.
Then for
any $f\in\Gamma$, we find, using Hölder’s inequality and Fubini’s theorem, that
$$\displaystyle\forall f\in\Gamma,\ \ \ \left\|f-j\ast f\right\|_{p;S}^{p}$$
$$\displaystyle=$$
$$\displaystyle\int_{S}\left\|f\left(x\right)-\int_{\Theta^{\prime}}j\left(y%
\right)f\left(y^{-1}x\right)\,dy\right\|_{B}^{p}\,dx$$
$$\displaystyle\leq$$
$$\displaystyle\int_{S}\left(\int_{\Theta^{\prime}}j\left(y\right)^{\frac{1}{p^{%
\prime}}}j\left(y\right)^{\frac{1}{p}}\left\|f\left(x\right)-f\left(y^{-1}x%
\right)\right\|_{B}\,dy\right)^{p}\,dx$$
$$\displaystyle\leq$$
$$\displaystyle\int_{S}\left(\int_{\Theta^{\prime}}j\left(y\right)\left\|f\left(%
x\right)-f\left(y^{-1}x\right)\right\|_{B}^{p}\,dy\right)dx$$
$$\displaystyle=$$
$$\displaystyle\int_{\Theta^{\prime}}\left(\int_{S}\left\|f\left(x\right)-f\left%
(y^{-1}x\right)\right\|_{B}^{p}\,dx\right)j\left(y\right)\,dy.$$
But for all $y\in\Theta^{\prime}$, we have $\int_{S}\left\|f\left(x\right)-f\left(y^{-1}x\right)\right\|_{B}^{p}\,dx<{%
\epsilon}^{p}$, and so $\left\|f-j\ast f\right\|_{p;S}<\epsilon$.
∎
4 Final Comments
We make two final comments. First, Theorem $1$ was proven
in [7] for the case where $B=\mathbb{C}$, $G$ is an arbitrary
locally compact group, and each $u_{V}=\mu\left(V\right)^{-1}\chi_{V}$ where $V$ is a nonempty neighborhood
of the identity, so that our
theorem generalizes this result in two different directions.
Also, in the case $B=\mathbb{C}$ and $G=\mathbb{R}^{n}$, Shurenkova and
Buldygin [1]
proved Theorem $1$ for the case when $j$ is a function
of the form $\omega_{h}$ for some $h>0$, where
$\{\omega_{h}:h\in\mathbb{R}^{+}\}$ is any family of real functions
in $\subseteq L^{1}\left(\mathbb{R}^{n}\right)\cap L^{p^{\prime}}\left(\mathbb{R}^%
{n}\right)$ satisfying the following:
1) $h>0,\left\|\omega_{h}\right\|_{1;\mathbb{R}^{n}}=1,\text{and }\omega_{h}$ is continuous
a.e. on $\mathbb{R}^{n}$,
2) for any $\delta>0$, $\lim_{h\rightarrow 0}\left\|\omega_{h}\right\|_{p;\{|x|>\delta\}}=0$,
3) $\left\|\omega_{h}\right\|_{1;\mathbb{R}^{n}}=1$
(here of course, we do not consider the property 3b) in the
statement of Theorem 1.)
Thus, considering the well-known fact that
$L^{p}\left(\mathbb{R}^{n}\right)$ functions can be arbitrarily
approximated (in the $L^{p}$ norm) by such functions [9, chap. 9,
Sect. 2], it is clear that our theorem is a significant
generalization of Shurenkova and Buldygin’s results.
The author
would like to thank the entire NJIT mathematics department, and
especially Professor Denis Blackmore, for their time and very
generous help.
References
[1]
Buldygin, V. and Shurenkova, A. On a Criterion of
Compactness in the Space $L_{p}$, Theory of Probability and
Mathematical Statistics, No. 52, 1996, 33 - 37.
[2]
Dinculeanu, N. On Kolmogorov-Tamarkin and M. Riesz
Compactness Criteria in Function Spaces Over a Locally Compact
Group, Journal of Mathematical Analysis and Applications, No. 89,
1982, 67 - 85.
[3]
Dinculeanu, N. Strong Additivity, Absolute Continuity,
and Compactness in Spaces of Measures, Journal of Mathematical
Analysis and Applications, No. 45, 1974, 172 - 188.
[4]
Hille, E. and Phillips, R.Functional Analysis and
Semi-Groups, American Mathematical Society., 1957
[5]
Loomis, L. An Introduction to Abstract Harmonic
Analysis, Van Nostrand Company, inc., 1953.
[6]
Munkres, J. Topology Prentice Hall, 1999.
[7]
Teleman, S. Sur Les Ensembles Compacts de Fonctions
Sommables, Rev. Math. Pures Appl. No. 6, 1961, 659?684.
[8]
Welland, R. and Goes, S. Compactness Criteria for
Köthe Spaces Math Ann. No. 188, 1970, 251 - 269.
[9]
Wheeden, R. and Zygmund, A. Measure and Integral,
Marcel Dekker Inc., 1977. |
Three-weight codes and the quintic construction
Yan Liu
School of Mathematical Sciences, Anhui University, Hefei, Anhui Province 230601, P.R. China
liuyan2612@126.com
Minjia Shi${}^{*}$111The author is supported by NNSF of China (61672036),
Technology Foundation for Selected Overseas Chinese Scholar, Ministry of Personnel of China (05015133), the Open Research Fund of National Mobile Communications Research Laboratory, Southeast University (2015D11) and Key projects of support program for outstanding young talents in Colleges and Universities (gxyqZD2016008).
smjwcl.good@163.com
Patrick Solé
sole@enst.fr
Key Laboratory of Intelligent Computing $\&$ Signal Processing,
Ministry of Education, Anhui University No. 3 Feixi Road, Hefei Anhui Province 230039, P. R. China, National Mobile Communications Research Laboratory, Southeast University and School of Mathematical Sciences of Anhui University,
Anhui, 230601, P. R. China
CNRS/LAGA, University Paris 8, 93 526 Saint-Denis, France
Abstract
We construct a class of three-Lee-weight and two infinite families of five-Lee-weight codes over the ring $R=\mathbb{F}_{2}+v\mathbb{F}_{2}+v^{2}\mathbb{F}_{2}+v^{3}\mathbb{F}_{2}+v^{4}%
\mathbb{F}_{2},$ where
$v^{5}=1.$ The same ring occurs in the quintic construction of binary quasi-cyclic codes.
They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using character sums.
Given a linear Gray map, we obtain three families of binary abelian codes with few weights. In particular, we obtain a class of three-weight codes which are optimal.
Finally, an application to secret sharing schemes is given.
keywords:
Three-weight codes; Quintic construction; Character sums; Griesmer bound; Secret sharing schemes
MSC: [2010] 94B25, 05 E30
††journal: Journal of LaTeX Templates
1 Introduction
Let $\mathbb{F}_{p}$ denote the finite field with $p$ elements. An $[n,k,d]$ linear code $C$ over $\mathbb{F}_{p}$ is an $k$-dimensional subspace of $\mathbb{F}_{p}^{n}$ with minimum Hamming distance $d.$ An $[n,k,d]$ linear code is called optimal if no $[n,k,d+1]$ code exists.
A classical construction of codes over finite fields called trace codes is documented in D ; DLLZ ; SLS1 ; SWLP . Many known codes DY ; HY1 ; ZD1 ; ZD2 ; ZDL
can be produced by this construction.
In a series of papers SLS1 ; SLS2 ; SLS3 ; SWLP , we have extended the notion of trace codes from fields to rings as follows. If $R$ is a finite ring, and $R_{m}$ an extension of $R$ of degree $m,$ $R_{m}^{*}$ denotes the group of units of $R_{m}$, we construct
a trace code with a defining set $\mathcal{L}=\{d_{1},d_{2},\dots,d_{n^{\prime}}\}\subseteq R_{m}^{*}$ by the formula
$$C_{\mathcal{L}}=\{(Tr_{m}(xd_{1}),Tr_{m}(xd_{2}),\dots,Tr_{m}(xd_{n^{\prime}})%
):x\in R_{m}\}=\{(Tr_{m}(xd))_{d\in\mathcal{L}}:x\in R_{m}\},$$
where $Tr_{m}()$ is a linear function from $R_{m}$ down to $R.$
By varying $\mathcal{L}$ and $R,$ various codes can be constructed. We can summarize this research program as shown below.
SLS1
$\mathcal{L}=R_{m}^{*},$ $R=\mathbb{F}_{2}+u\mathbb{F}_{2}$;
SLS2
$L=R_{m}^{*},$ $R=\mathbb{F}_{2}+u\mathbb{F}_{2}+v\mathbb{F}_{2}+uv\mathbb{F}_{2}$;
SWLP
$[R_{m}^{*}:\mathcal{L}]=2,$ $R=\mathbb{F}_{p}+u\mathbb{F}_{p}$;
SLS3
$\mathcal{L}=D+u\mathbb{F}_{p^{m}},$ $[R_{m}^{*}:\mathcal{L}]=(p-1)gcd(N,\frac{p^{m}-1}{p-1}),$ $R=\mathbb{F}_{p}+u\mathbb{F}_{p}$.
Note that all rings in the above list are chain rings, except the second one.
In the present paper, we will consider another semi-local ring $\mathbb{F}_{2}+v\mathbb{F}_{2}+v^{2}\mathbb{F}_{2}+v^{3}\mathbb{F}_{2}+v^{4}%
\mathbb{F}_{2}$, where $v^{5}=1.$ This ring occurs in the quintic construction of binary quasi-cyclic codes
BNS . While the binary codes we describe are not obtained by that construction, they are still quasi-cyclic of co-index $5.$
By the generic method of our research program, we obtain a family of optimal binary codes by using a linear Gray map.
Furthermore,
an application to secret sharing schemes is sketched out.
The rest of this paper is organized as follows. The next section describes the basic notations and defines a Gray map, which will be needed in Section 4. Section 3 shows that the codes are abelian. Section 4 gives the main results in this paper, the Lee weight distribution of our codes. Furthermore, we show that the Gray images of three-Lee-weight codes are optimal. Section 5 determines the minimum Lee distance of their dual codes. The codes we constructed have applications in secret sharing schemes in section 6. We will sum up all we have done throughout this paper in section 7, and make some conjectures for future research.
2 Preliminaries
2.1 Rings
Throughout this paper, we let $\mathbb{F}_{2}$ be a finite field with two elements, i.e., $\mathbb{F}_{2}=\{0,1\}$. Denote by $R$ the commutative ring $\mathbb{F}_{2}+v\mathbb{F}_{2}+v^{2}\mathbb{F}_{2}+v^{3}\mathbb{F}_{2}+v^{4}%
\mathbb{F}_{2}$,
constructed via $v^{5}=1$. $R$ is a ring of size $2^{5}$ with characteristic 2. Because the factorization of $v^{5}-1$ into irreducible factors is $(v-1)(1+v+v^{2}+v^{3}+v^{4}),$ the ring $R$ has two
maximal ideals, namely, $(1+v)=\{(1+v)(a_{0}+a_{1}v+a_{2}v^{2}+a_{3}v^{3}+a_{4}v^{4}):a_{i}\in\mathbb{F%
}_{2},i=0,1,2,3,4\}$ and $(1+v+v^{2}+v^{3}+v^{4})=\{0,1+v+v^{2}+v^{3}+v^{4}\}$. Thus it is a non-local, non-chain principal ideal ring.
Given a positive integer $m$, we can construct the ring extension $R_{m}=\mathbb{F}_{2^{m}}+v\mathbb{F}_{2^{m}}+v^{2}\mathbb{F}_{2^{m}}+v^{3}%
\mathbb{F}_{2^{m}}+v^{4}\mathbb{F}_{2^{m}}.$ Let $\epsilon\in\mathbb{F}_{16}$ be a root of the irreducible polynomial $1+v+v^{2}+v^{3}+v^{4}$ in $\mathbb{F}_{2}$, then by a simple calculation, we get the
factorization of $v^{5}-1$ as follow:
$$\displaystyle v^{5}-1$$
$$\displaystyle=$$
$$\displaystyle(1+v)(1+v+v^{2}+v^{3}+v^{4})~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}%
\mathrm{in}~{}\mathbb{F}_{2},$$
$$\displaystyle=$$
$$\displaystyle(1+v)(1+\omega^{2}v+v^{2})(1+\omega v+v^{2})~{}~{}~{}\mathrm{in}~%
{}\mathbb{F}_{4}=\{0,1,\omega,\omega^{2}\},$$
$$\displaystyle=$$
$$\displaystyle\prod_{i=0}^{4}(v+\epsilon^{i})~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~%
{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathrm{in}~{}%
\mathbb{F}_{16}=\{\nu_{0}+\nu_{1}\epsilon+\nu_{2}\epsilon^{2}+\nu_{3}\epsilon^%
{3}:\nu_{i}\in\mathbb{F}_{2},i=0,1,2,3\}.$$
Hence, by using Chinese Remainder Theorem, the ring $R_{m}$ is seen to be isomorphic to
$\mathbb{F}_{2^{m}}\bigoplus\mathbb{F}_{16^{m}}$ when $m$ is odd, and $\mathbb{F}_{2^{m}}\bigoplus\mathbb{F}_{4^{m}}\bigoplus\mathbb{F}_{4^{m}}$ when $m$ is singly-even, and
$\mathbb{F}_{2^{m}}\bigoplus\mathbb{F}_{2^{m}}\bigoplus\mathbb{F}_{2^{m}}%
\bigoplus\mathbb{F}_{2^{m}}\bigoplus\mathbb{F}_{2^{m}}$ when $m$ is doubly-even.
Here $R_{m}^{*}$ denotes the group of units in $R_{m}$, and $\mathbb{F}_{2^{m}}^{*}$ denotes the multiplicative cyclic group of nonzero elements of $\mathbb{F}_{2^{m}}.$ Likewise, we have $R_{m}^{*}\cong\mathbb{F}_{2^{m}}^{*}\bigoplus\mathbb{F}_{16^{m}}^{*}$ when $m$ is odd, and $R_{m}^{*}\cong\mathbb{F}_{2^{m}}^{*}\bigoplus\mathbb{F}_{4^{m}}^{*}\bigoplus%
\mathbb{F}_{4^{m}}^{*}$ when $m$ is singly-even,
and $R_{m}^{*}\cong\mathbb{F}_{2^{m}}^{*}\bigoplus\mathbb{F}_{2^{m}}^{*}\bigoplus%
\mathbb{F}_{2^{m}}^{*}\bigoplus\mathbb{F}_{2^{m}}^{*}\bigoplus\mathbb{F}_{2^{m%
}}^{*}$ when $m$ is doubly-even.
Let $tr_{m}()$ be the trace function from $\mathbb{F}_{2^{m}}$ to $\mathbb{F}_{2}$, namely, for any $\ell\in\mathbb{F}_{2^{m}},tr_{m}(\ell)=\ell+\ell^{2}+\dots+\ell^{2^{m-1}}$. Similar to the definition of $tr_{m}()$, we define the Trace function, denoted by $Tr_{m}()$, over $R_{m}.$
Definition 2.1 For any $a=a_{0}+a_{1}v+a_{2}v^{2}+a_{3}v^{3}+a_{4}v^{4}\in R_{m},$ where $a_{i}\in\mathbb{F}_{2^{m}},i=0,1,2,3,4,$ the trace $Tr_{m}(a)$ of $a$ over $R$ is defined by
$$Tr_{m}(a_{0}+a_{1}v+a_{2}v^{2}+a_{3}v^{3}+a_{4}v^{4})=tr_{m}(a_{0})+tr_{m}(a_{%
1})v+tr_{m}(a_{2})v^{2}+tr_{m}(a_{3})v^{3}+tr_{m}(a_{4})v^{4}.$$
It is well known that the trace function $tr_{m}()$ is a linear transformation from $\mathbb{F}_{2^{m}}$ onto $\mathbb{F}_{2}$. So it is immediate to check that $R$-linearity of $Tr_{m}()$ follows from the $\mathbb{F}_{2}$-linearity of $tr_{m}()$.
2.2 Codes and Gray map
A linear code $C$ over $R$ of length $n$ is an $R$-submodule of $R^{n}$. The elements of a such code are called its codewords. For $x=(x_{1},x_{2},\dots,x_{n}),y=(y_{1},y_{2},\dots,y_{n})\in R^{n}$, their standard inner product
is defined by $\langle x,y\rangle=\sum_{i=1}^{n}x_{i}y_{i}$, where the operation is performed in $R$.
The dual code $C^{\perp}$ of a linear code $C$, over $R$, consists of all vectors of $R^{n}$
which are orthogonal to every codeword in $C$, that is, $C^{\perp}=\{y\in R^{n}|\langle x,y\rangle=0,\forall x\in C\}.$
We define the following linear Gray map which takes a linear code over $R$ of length $n$ to a binary linear code of length $5n$.
Definition 2.2
The Gray map $\Phi$ from $R$ to $\mathbb{F}_{2}^{5}$ is defined as
$$\Phi(a_{0}+a_{1}v+a_{2}v^{2}+a_{3}v^{3}+a_{4}v^{4})=(a_{0},a_{1},a_{2},a_{3},a%
_{4}),$$
where $a_{i}\in\mathbb{F}_{2},i=0,1,2,3,4.$
This map $\Phi$ can be extended to $R^{n}$ in an obvious way. Define the Lee weight of an element $a_{0}+a_{1}v+a_{2}v^{2}+a_{3}v^{3}+a_{4}v^{4}$ of $R$ as $w_{L}(a_{0}+a_{1}v+a_{2}v^{2}+a_{3}v^{3}+a_{4}v^{4})=w_{H}(\Phi(a_{0}+a_{1}v+a%
_{2}v^{2}+a_{3}v^{3}+a_{4}v^{4}))=w_{H}((a_{0},a_{1},a_{2},a_{3},a_{4}))=\sum_%
{i=0}^{4}w_{H}(a_{i}),$ where $w_{H}$ denotes the usual Hamming weight. Then $\Phi$ is a distance preserving isometry from $(R^{n},d_{L^{\prime}})$ to $(\mathbb{F}_{2}^{5n},d_{H})$, where $d_{L^{\prime}}$ and $d_{H}$ denote the Lee and Hamming distance in $R^{n}$ and $\mathbb{F}_{2}^{5n}$, respectively. What is more, if $C$ is a linear code over $R$ with parameters $(n,2^{k},d)$, then $\Phi(C)$ is a linear code over $\mathbb{F}_{2}$ with parameters $[5n,k,d]$.
It is immediate that linear codes of length $n$ over $R$ are mapped into $n$-quasi-cyclic binary codes of length $5n.$
Given a finite abelian group $G,$ a code over $R$ is said to be abelian if it is an ideal of the group ring $R[G].$
Recall that the ring $R[G]$ is defined on functions from $G$ to $R$ with pointwise addition as addition, and convolution product as multiplication.
Concretely it is the set of all formal sums $f=\sum_{h\in G}f_{h}X^{h},$ where $X$ an undeterminate, with addition and multiplication defined as follows.
If $f,g$ are in $R[G],$ we write
$$f+g=\sum_{g\in G}f_{h}+g_{h}X^{h},$$
and
$$fg=\sum_{h\in G}(\sum_{r+s=h}f_{r}g_{s})X^{h}.$$
In other words, the coordinates of $C$ are indexed by elements of $G$, and $G$ acts regularly on this set.
In the special case when $G$ is cyclic, the code is a cyclic code in the usual sense MS .
3 Symmetry
First, for $a\in R_{m}$ define the vector $ev(a)$ by the evaluation map $ev(a)=(Tr_{m}(ax))_{x\in R_{m}^{*}}.$ Define the code $\mathcal{C}(m,2,L)$ of length $L$ by the formula $\mathcal{C}(m,2,L)=\{ev(a):a\in R_{m}\}$. Thus $\mathcal{C}(m,2,L)$ is a linear code over $R$ and $\Phi(\mathcal{C}(2,m,L))$ is a linear binary code of length $5L.$
Proposition 3.1 If $L=|R_{m}^{*}|$, then the group $R_{m}^{*}$ acts regularly on the coordinates of $\mathcal{C}(m,2,L).$
Proof.
For any $w,v\in L$ the change of variables $x\mapsto(v/w)x$ maps $w$ to $v.$ This transformation defines thus a transitive action of $L$ on itself.
Given an ordered pair $(w,v)$ this transformation is unique,
hence the action is regular.
∎
The code $\mathcal{C}(m,2,|R_{m}^{*}|)$ is thus an abelian code with respect to the group $R_{m}^{*}.$ In other words, it is an ideal of the group ring $R[R_{m}^{*}].$ As observed in the previous section $R_{m}^{*}$ is a not cyclic group, hence $\mathcal{C}(m,2,L)$ may be not cyclic.
4 The Lee Weight of $\mathcal{C}(m,2,L)$
For convenience, we let $s=5|R_{m}^{*}|$. If $y=(y_{1},y_{2},\dots,y_{s})\in\mathbb{F}_{2}^{s},$ let
$$\theta(y)=\sum_{j=1}^{s}(-1)^{y_{j}}.$$
For simplicity, we let $\Theta(a)=\theta(\Phi(ev(a))).$
In order to determine the Lee weight of the codewords of $\mathcal{C}(m,2,L)$, we first recall the following two lemmas.
Lemma 4.1 SWLP For all $y=(y_{1},y_{2},\dots,y_{s})\in\mathbb{F}_{2}^{s},$ we have
$$2w_{H}(y)=s-\sum_{j=1}^{s}(-1)^{y_{j}}.$$
Lemma 4.2 MS If $z\in\mathbb{F}_{2^{m}}^{*},$ then
$$\sum\limits_{x\in\mathbb{F}_{2^{m}}}(-1)^{tr_{m}(zx)}=0.$$
According to Lemma 4.1, for $ev(a)\in\mathcal{C}(m,2,L)$, by definition of the Gray map, we have
$$2w_{L}(ev(a))=2w_{H}(\Phi(ev(a)))=s-\Theta(a).$$
(1)
4.1 The first family of codes $\mathcal{C}(m,2,L_{1})$ when $m$ is odd
In the previous section, we know $R_{m}^{*}\cong\mathbb{F}_{2}^{*}\bigoplus\mathbb{F}_{16^{m}}^{*}$ when $m$ is odd.
A simple calculation shows that
$$R_{m}^{*}=\Bigg{\{}\sum_{i=0}^{4}x_{i}v^{i}:\sum_{i=0}^{4}x_{i}\neq 0,(x_{1}+x%
_{2}+x_{3}+x_{4},x_{0}+x_{1},x_{3}+x_{4},x_{0}+x_{1}+x_{2}+x_{3})\neq(0,0,0,0)%
\Bigg{\}}.$$
For convenience, we adopt the following notations unless otherwise stated
in this section. Set $I=\sum_{i=0}^{4}x_{i},I_{1}=x_{1}+x_{2}+x_{3}+x_{4},I_{2}=x_{0}+x_{1},I_{3}=x_%
{3}+x_{4},I_{4}=x_{0}+x_{1}+x_{2}+x_{3}$.
Then $R_{m}^{*}=\{\sum_{i=0}^{4}x_{i}v^{i}:I\neq 0,(I_{1},I_{2},I_{3},I_{4})\neq(0,0%
,0,0),I,I_{1},I_{2},I_{3},I_{4}\in\mathbb{F}_{2^{m}}\}.$
We are now ready to discuss the Lee weight of the
codewords of the abelian codes introduced above.
Theorem 4.3 Let $m$ be odd, then the set $\mathcal{C}(m,2,L_{1})$ is a three-Lee-weight linear code of length $L_{1}=(2^{m}-1)(2^{4m}-1)$ and its weight distribution is given in Table I.
$\mathrm{Table~{}I.}~{}~{}~{}\mathrm{weight~{}distribution~{}of}~{}\mathcal{C}(%
m,2,L_{1})$
Weight
Frequency
0
1
$$5\times(2^{5m-1}-2^{4m-1}-2^{m-1})$$
$$(2^{m}-1)(2^{4m}-1)$$
$$5\times(2^{5m-1}-2^{4m-1})$$
$$2^{4m}-1$$
$$5\times(2^{5m-1}-2^{m-1})$$
$$2^{m}-1$$
Proof.
Set $x=x_{0}+x_{1}v+x_{2}v^{2}+x_{3}v^{3}+x_{4}v^{4}\in R_{m}^{*}$, then $I\neq 0,(I_{1},I_{2},I_{3},I_{4})\neq(0,0,0,0)$.
For $a=a_{0}+a_{1}v+a_{2}v^{2}+a_{3}v^{3}+a_{4}v^{4}\in R_{m}$, we have
$$\displaystyle ax$$
$$\displaystyle=$$
$$\displaystyle[(a_{0}+a_{1}+a_{2}+a_{3}+a_{4})I+(a_{0}+a_{4})I_{1}+(a_{3}+a_{4}%
)I_{2}+(a_{2}+a_{3})I_{3}+(a_{1}+a_{2})I_{4}]$$
$$\displaystyle+[(a_{0}+a_{1}+a_{2}+a_{3}+a_{4})I+(a_{0}+a_{1})I_{1}+(a_{0}+a_{4%
})I_{2}+(a_{3}+a_{4})I_{3}+(a_{2}+a_{3})I_{4}]v$$
$$\displaystyle+[(a_{0}+a_{1}+a_{2}+a_{3}+a_{4})I+(a_{1}+a_{2})I_{1}+(a_{0}+a_{1%
})I_{2}+(a_{0}+a_{4})I_{3}+(a_{3}+a_{4})I_{4}]v^{2}$$
$$\displaystyle+[(a_{0}+a_{1}+a_{2}+a_{3}+a_{4})I+(a_{2}+a_{3})I_{1}+(a_{1}+a_{2%
})I_{2}+(a_{0}+a_{1})I_{3}+(a_{0}+a_{4})I_{4}]v^{3}$$
$$\displaystyle+[(a_{0}+a_{1}+a_{2}+a_{3}+a_{4})I+(a_{3}+a_{4})I_{1}+(a_{2}+a_{3%
})I_{2}+(a_{1}+a_{2})I_{3}+(a_{0}+a_{1})I_{4}]v^{4}$$
$$\displaystyle=:$$
$$\displaystyle A_{0}+A_{1}v+A_{2}v^{2}+A_{3}v^{3}+A_{4}v^{4}.$$
So
$$\Phi(Tr_{m}(ax))=(tr_{m}(A_{0}),tr_{m}(A_{1}),tr_{m}(A_{2}),tr_{m}(A_{3}),tr_{%
m}(A_{4})).$$
Taking character sums, yields
$$\Theta(a)=\sum_{i=0}^{4}\sum_{I\in\mathbb{F}_{2^{m}}^{*}}\sum_{(I_{1},I_{2},I_%
{3},I_{4})\neq(0,0,0,0),I_{1},I_{2},I_{3},I_{4}\in\mathbb{F}_{2^{m}}}(-1)^{tr_%
{m}(A_{i})}.$$
If $a=a_{0}+a_{1}v+a_{2}v^{2}+a_{3}v^{3}+a_{4}v^{4}\in R_{m}^{*}$, then $\sum_{i=0}^{4}a_{i}\neq 0$ and $(a_{1}+a_{2}+a_{3}+a_{4},a_{0}+a_{1},a_{3}+a_{4},a_{0}+a_{1}+a_{2}+a_{3})\neq(%
0,0,0,0)$, and $\Theta(a)=5.$
By Equation (1), we get $w_{L}(ev(a))=5\times(2^{5m-1}-2^{4m-1}-2^{m-1})$.
Note that $\sum_{i=0}^{4}a_{i}=0$ and $(a_{1}+a_{2}+a_{3}+a_{4},a_{0}+a_{1},a_{3}+a_{4},a_{0}+a_{1}+a_{2}+a_{3})=(0,0%
,0,0)$, i.e., $a=0$, corresponds to the zero codewords, and then its Lee weight is 0.
If $\sum_{i=0}^{4}a_{i}=0$ and $(a_{1}+a_{2}+a_{3}+a_{4},a_{0}+a_{1},a_{3}+a_{4},a_{0}+a_{1}+a_{2}+a_{3})\neq(%
0,0,0,0)$, then $a=a_{0}+a_{1}v+a_{2}v^{2}+a_{3}v^{3}+a_{4}v^{4}$ is a zero-divisor, and $\Theta(a)=-5\times(2^{m}-1).$
By Equation (1), we get $w_{L}(ev(a))=5\times(2^{5m-1}-2^{4m-1})$.
If $a=a_{0}+a_{1}v+a_{2}v^{2}+a_{3}v^{3}+a_{4}v^{4}$ satisfies $\sum_{i=0}^{4}a_{i}\neq 0$ and $(a_{1}+a_{2}+a_{3}+a_{4},a_{0}+a_{1},a_{3}+a_{4},a_{0}+a_{1}+a_{2}+a_{3})=(0,0%
,0,0)$, then we can claim $a=\alpha(1+v+v^{2}+v^{3}+v^{4})$, where $\alpha\in\mathbb{F}_{2^{m}}^{*}.$
By calculation, we know
$$\displaystyle\Theta(a)$$
$$\displaystyle=$$
$$\displaystyle 5\sum_{I\in\mathbb{F}_{2^{m}}^{*}}\sum_{(I_{1},I_{2},I_{3},I_{4}%
)\neq(0,0,0,0),I_{1},I_{2},I_{3},I_{4}\in\mathbb{F}_{2^{m}}}(-1)^{tr_{m}(%
\alpha I)}=-5\times(2^{4m}-1).$$
Hence we get $w_{L}(ev(a))=5\times(2^{5m-1}-2^{m-1})$ by Equation (1).
∎
According to Theorem 4.3, we have constructed a binary linear code of length $s=5\times(2^{m}-1)(2^{4m}-1)$, of dimension $5m$, with three nonzero weights $w_{1}<w_{2}<w_{3}$ of values
$$w_{1}=5\times(2^{5m-1}-2^{4m-1}-2^{m-1}),~{}~{}~{}w_{2}=5\times(2^{5m-1}-2^{4m%
-1}),~{}~{}~{}w_{3}=5\times(2^{5m-1}-2^{m-1}),$$
with respective frequencies
$$f_{1}=(2^{m}-1)(2^{4m}-1),~{}~{}~{}f_{2}=2^{4m}-1,~{}~{}~{}f_{3}=2^{m}-1.$$
4.2 The second family of codes $\mathcal{C}(m,2,L_{2})$ when $m$ is singly-even
We now consider $m$ is singly-even, which implies $R_{m}^{*}\cong\mathbb{F}_{2^{m}}^{*}\bigoplus\mathbb{F}_{4^{m}}^{*}\bigoplus%
\mathbb{F}_{4^{m}}^{*}.$
A simple calculation shows that
$$\displaystyle R_{m}^{*}$$
$$\displaystyle=$$
$$\displaystyle\Bigg{\{}\sum_{i=0}^{4}h_{i}v^{i}:\sum_{i=0}^{4}h_{i}\neq 0,(%
\omega^{2}h_{1}+\omega h_{2}+\omega h_{3}+\omega^{2}h_{4},\omega^{2}h_{0}+%
\omega h_{1}+\omega h_{2}+\omega^{2}h_{4})\neq(0,0),$$
$$\displaystyle(\omega h_{1}+\omega^{2}h_{2}+\omega^{2}h_{3}+\omega h_{4},\omega
h%
_{0}+\omega^{2}h_{1}+\omega^{2}h_{2}+\omega h_{4})\neq(0,0)\Bigg{\}}.$$
For convenience, we adopt the following notations unless otherwise stated
in this section. Set $H=\sum_{i=0}^{4}h_{i},H_{1}=\omega^{2}h_{1}+\omega h_{2}+\omega h_{3}+\omega^{%
2}h_{4},H_{2}=\omega^{2}h_{0}+\omega h_{1}+\omega h_{2}+\omega^{2}h_{4},H_{3}=%
\omega h_{1}+\omega^{2}h_{2}+\omega^{2}h_{3}+\omega h_{4},H_{4}=\omega h_{0}+%
\omega^{2}h_{1}+\omega^{2}h_{2}+\omega h_{4}$.
Then $R_{m}^{*}=\{\sum_{i=0}^{4}h_{i}v^{i}:H\neq 0,(H_{1},H_{2})\neq(0,0),(H_{3},H_{%
4})\neq(0,0),H,H_{1},H_{2},H_{3},H_{4}\in\mathbb{F}_{2^{m}}\}.$
We are now ready to discuss the Lee weight of the
codewords of the abelian codes introduced above.
Theorem 4.4 Let $m$ be singly-even, then the set $\mathcal{C}(m,2,L_{2})$ is a five-Lee-weight linear code of length $L_{2}=(2^{m}-1)(2^{2m}-1)^{2}$ and its weight distribution is given in Table II.
$\mathrm{Table~{}II.}~{}~{}~{}\mathrm{weight~{}distribution~{}of}~{}\mathcal{C}%
(m,2,L_{2})$
Weight
Frequency
0
1
$$5\times(2^{2m}-1)(2^{3m-1}-2^{2m-1}-2^{m-1})$$
$$2\times(2^{m}-1)(2^{2m}-1)$$
$$5\times(2^{m}-1)(2^{2m-1}-1)2^{2m}$$
$$(2^{2m}-1)^{2}$$
$$\frac{5\times[(2^{m}-1)(2^{2m}-1)^{2}+1]}{2}$$
$$(2^{m}-1)(2^{2m}-1)^{2}$$
$$5\times(2^{m}-1)(2^{2m}-1)2^{2m-1}$$
$$2\times(2^{2m}-1)$$
$$5\times(2^{2m}-1)^{2}2^{m-1}$$
$$2^{m}-1$$
Proof.
Set $h=h_{0}+h_{1}v+h_{2}v^{2}+h_{3}v^{3}+h_{4}v^{4}\in R_{m}^{*}$, then $H\neq 0,(H_{1},H_{2})\neq(0,0),(H_{3},H_{4})\neq(0,0)$.
For $b=b_{0}+b_{1}v+b_{2}v^{2}+b_{3}v^{3}+b_{4}v^{4}\in R_{m}$, we have
$$\displaystyle bh$$
$$\displaystyle=$$
$$\displaystyle[(b_{0}+b_{1}+b_{2}+b_{3}+b_{4})H+(b_{0}+b_{2}+\omega^{2}b_{3}+%
\omega^{2}b_{4})H_{1}+(b_{1}+b_{4}+\omega^{2}b_{2}+\omega^{2}b_{3})H_{2}$$
$$\displaystyle+(b_{0}+b_{2}+\omega b_{3}+\omega b_{4})H_{3}+(b_{1}+b_{4}+\omega
b%
_{2}+\omega b_{3})H_{4}]$$
$$\displaystyle+[(b_{0}+b_{1}+b_{2}+b_{3}+b_{4})H+(b_{1}+b_{3}+\omega^{2}b_{0}+%
\omega^{2}b_{4})H_{1}+(b_{0}+b_{2}+\omega^{2}b_{3}+\omega^{2}b_{4})H_{2}$$
$$\displaystyle+(b_{1}+b_{3}+\omega b_{0}+\omega b_{4})H_{3}+(b_{0}+b_{2}+\omega
b%
_{3}+\omega b_{4})H_{4}]v$$
$$\displaystyle+[(b_{0}+b_{1}+b_{2}+b_{3}+b_{4})H+(b_{2}+b_{4}+\omega^{2}b_{0}+%
\omega^{2}b_{1})H_{1}+(b_{1}+b_{3}+\omega^{2}b_{0}+\omega^{2}b_{4})H_{2}$$
$$\displaystyle+(b_{2}+b_{4}+\omega b_{0}+\omega b_{1})H_{3}+(b_{1}+b_{3}+\omega
b%
_{0}+\omega b_{4})H_{4}]v^{2}$$
$$\displaystyle+[(b_{0}+b_{1}+b_{2}+b_{3}+b_{4})H+(b_{0}+b_{3}+\omega^{2}b_{1}+%
\omega^{2}b_{2})H_{1}+(b_{2}+b_{4}+\omega^{2}b_{0}+\omega^{2}b_{1})H_{2}$$
$$\displaystyle+(b_{0}+b_{3}+\omega b_{1}+\omega b_{2})H_{3}+(b_{2}+b_{4}+\omega
b%
_{0}+\omega b_{1})H_{4}]v^{3}$$
$$\displaystyle+[(b_{0}+b_{1}+b_{2}+b_{3}+b_{4})H+(b_{1}+b_{4}+\omega^{2}b_{2}+%
\omega^{2}b_{3})H_{1}+(b_{0}+b_{3}+\omega^{2}b_{1}+\omega^{2}b_{2})H_{2}$$
$$\displaystyle+(b_{1}+b_{4}+\omega b_{2}+\omega b_{3})H_{3}+(b_{0}+b_{3}+\omega
b%
_{1}+\omega b_{2})H_{4}]v^{4}$$
$$\displaystyle=:$$
$$\displaystyle B_{0}+B_{1}v+B_{2}v^{2}+B_{3}v^{3}+B_{4}v^{4}.$$
So
$$\Phi(Tr_{m}(bh))=(tr_{m}(B_{0}),tr_{m}(B_{1}),tr_{m}(B_{2}),tr_{m}(B_{3}),tr_{%
m}(B_{4})).$$
Taking character sums, yields
$$\displaystyle\Theta(b)$$
$$\displaystyle=$$
$$\displaystyle\sum_{i=0}^{4}\sum_{H\in\mathbb{F}_{2^{m}}^{*}}\sum_{(H_{1},H_{2}%
)\neq(0,0),H_{1},H_{2}\in\mathbb{F}_{2^{m}}}\sum_{(H_{3},H_{4})\neq(0,0),H_{3}%
,H_{4}\in\mathbb{F}_{2^{m}}}(-1)^{tr_{m}(B_{i})}.$$
1) If $b=b_{0}+b_{1}v+b_{2}v^{2}+b_{3}v^{3}+b_{4}v^{4}\in R_{m}^{*}$, then $\sum_{i=0}^{4}b_{i}\neq 0$ and $(\omega^{2}b_{1}+\omega b_{2}+\omega b_{3}+\omega^{2}b_{4},\omega^{2}b_{0}+%
\omega b_{1}+\omega b_{2}+\omega^{2}b_{4})\neq(0,0),(\omega b_{1}+\omega^{2}b_%
{2}+\omega^{2}b_{3}+\omega b_{4},\omega b_{0}+\omega^{2}b_{1}+\omega^{2}b_{2}+%
\omega b_{4})\neq(0,0)$.
What is more, we obtain $\Theta(b)=-5.$ By Equation (1), we get
$$w_{L}(ev(b))=\frac{5\times[(2^{m}-1)(2^{2m}-1)^{2}+1]}{2}.$$
2) If $\sum_{i=0}^{4}b_{i}=0$ and
$(\omega^{2}b_{1}+\omega b_{2}+\omega b_{3}+\omega^{2}b_{4},\omega^{2}b_{0}+%
\omega b_{1}+\omega b_{2}+\omega^{2}b_{4})\neq(0,0),(\omega b_{1}+\omega^{2}b_%
{2}+\omega^{2}b_{3}+\omega b_{4},\omega b_{0}+\omega^{2}b_{1}+\omega^{2}b_{2}+%
\omega b_{4})\neq(0,0)$,
then $b=b_{0}+b_{1}v+b_{2}v^{2}+b_{3}v^{3}+b_{4}v^{4}$ is a zero-divisor. Furthermore, $\Theta(b)=5\times(2^{m}-1).$
By Equation (1), we get
$$w_{L}(ev(b))=5\times(2^{m}-1)(2^{2m-1}-1)2^{2m}.$$
3) If $\sum_{i=0}^{4}b_{i}\neq 0$ and $(\omega^{2}b_{1}+\omega b_{2}+\omega b_{3}+\omega^{2}b_{4},\omega^{2}b_{0}+%
\omega b_{1}+\omega b_{2}+\omega^{2}b_{4})=(0,0),(\omega b_{1}+\omega^{2}b_{2}%
+\omega^{2}b_{3}+\omega b_{4},\omega b_{0}+\omega^{2}b_{1}+\omega^{2}b_{2}+%
\omega b_{4})\neq(0,0),$
then $\Theta(b)=5\times(2^{2m}-1).$
When $\sum_{i=0}^{4}b_{i}\neq 0$ and $(\omega^{2}b_{1}+\omega b_{2}+\omega b_{3}+\omega^{2}b_{4},\omega^{2}b_{0}+%
\omega b_{1}+\omega b_{2}+\omega^{2}b_{4})\neq(0,0),(\omega b_{1}+\omega^{2}b_%
{2}+\omega^{2}b_{3}+\omega b_{4},\omega b_{0}+\omega^{2}b_{1}+\omega^{2}b_{2}+%
\omega b_{4})=(0,0)$,
then $\Theta(b)=5\times(2^{2m}-1).$
By Equation (1), we get
$$w_{L}(ev(b))=5\times(2^{2m}-1)(2^{3m-1}-2^{2m-1}-2^{m-1}).$$
4) If $\sum_{i=0}^{4}b_{i}=0$ and $(\omega^{2}b_{1}+\omega b_{2}+\omega b_{3}+\omega^{2}b_{4},\omega^{2}b_{0}+%
\omega b_{1}+\omega b_{2}+\omega^{2}b_{4})=(0,0),(\omega b_{1}+\omega^{2}b_{2}%
+\omega^{2}b_{3}+\omega b_{4},\omega b_{0}+\omega^{2}b_{1}+\omega^{2}b_{2}+%
\omega b_{4})\neq(0,0),$
then $\Theta(b)=-5\times(2^{m}-1)(2^{2m}-1).$
When $\sum_{i=0}^{4}b_{i}=0$ and $(\omega^{2}b_{1}+\omega b_{2}+\omega b_{3}+\omega^{2}b_{4},\omega^{2}b_{0}+%
\omega b_{1}+\omega b_{2}+\omega^{2}b_{4})\neq(0,0),(\omega b_{1}+\omega^{2}b_%
{2}+\omega^{2}b_{3}+\omega b_{4},\omega b_{0}+\omega^{2}b_{1}+\omega^{2}b_{2}+%
\omega b_{4})=(0,0)$,
then $\Theta(b)=-5\times(2^{m}-1)(2^{2m}-1).$
By Equation (1), we get
$$w_{L}(ev(b))=5\times(2^{m}-1)(2^{2m}-1)2^{2m-1}.$$
5) If $b=b_{0}+b_{1}v+b_{2}v^{2}+b_{3}v^{3}+b_{4}v^{4}$ is a zero-divisor, where $\sum_{i=0}^{4}b_{i}\neq 0$ and $(\omega^{2}b_{1}+\omega b_{2}+\omega b_{3}+\omega^{2}b_{4},\omega^{2}b_{0}+%
\omega b_{1}+\omega b_{2}+\omega^{2}b_{4})=(0,0),(\omega b_{1}+\omega^{2}b_{2}%
+\omega^{2}b_{3}+\omega b_{4},\omega b_{0}+\omega^{2}b_{1}+\omega^{2}b_{2}+%
\omega b_{4}=(0,0),$
then $b=\beta(1+v+v^{2}+v^{3}+v^{4})$, where $\beta\in\mathbb{F}_{2^{m}}^{*}$, and $bh=\beta H(1+v+v^{2}+v^{3}+v^{4})$.
So $\Phi(Tr_{m}(bh))=(tr_{m}(\beta H),tr_{m}(\beta H),tr_{m}(\beta H),tr_{m}(\beta
H%
),tr_{m}(\beta H))$.
Taking character sums, yields
$$\displaystyle\Theta(b)$$
$$\displaystyle=$$
$$\displaystyle 5\sum_{H\in\mathbb{F}_{2^{m}}^{*}}\sum_{(H_{1},H_{2})\neq(0,0),H%
_{1},H_{2}\in\mathbb{F}_{2^{m}}}\sum_{(H_{3},H_{4})\neq(0,0),H_{3},H_{4}\in%
\mathbb{F}_{2^{m}}}(-1)^{tr_{m}(\beta H)}=-5\times(2^{2m}-1)^{2}.$$
By Equation (1), we get
$$w_{L}(ev(b))=5\times(2^{2m}-1)^{2}2^{m-1}.$$
Note that $\sum_{i=0}^{4}b_{i}=0$ and $(\omega^{2}b_{1}+\omega b_{2}+\omega b_{3}+\omega^{2}b_{4},\omega^{2}b_{0}+%
\omega b_{1}+\omega b_{2}+\omega^{2}b_{4})=(\omega b_{1}+\omega^{2}b_{2}+%
\omega^{2}b_{3}+\omega b_{4},\omega b_{0}+\omega^{2}b_{1}+\omega^{2}b_{2}+%
\omega b_{4})=(0,0),$
i.e.,
$b=0,$ corresponds to the zero codewords, and then its Lee weight is 0.
∎
From Theorem 4.4, we have constructed a binary linear code of length $s=5\times(2^{m}-1)(2^{2m}-1)^{2}$, of dimension $5m$, with five nonzero weights $w_{1}<w_{2}<w_{3}<w_{4}<w_{5}$ of values
$$w_{1}=5\times(2^{2m}-1)(2^{3m-1}-2^{2m-1}-2^{m-1}),~{}~{}~{}w_{2}=5\times(2^{m%
}-1)(2^{2m-1}-1)2^{2m},~{}~{}~{}$$
$$w_{3}=\frac{5\times[(2^{m}-1)(2^{2m}-1)^{2}+1]}{2},~{}~{}~{}w_{4}=5\times(2^{m%
}-1)(2^{2m}-1)2^{2m-1},~{}~{}~{}w_{5}=5\times(2^{2m}-1)^{2}2^{m-1},$$
with respective frequencies
$$f_{1}=2\times(2^{m}-1)(2^{2m}-1),~{}~{}f_{2}=(2^{2m}-1)^{2},~{}~{}f_{3}=(2^{m}%
-1)(2^{2m}-1)^{2},~{}~{}f_{4}=2\times(2^{2m}-1),~{}~{}f_{5}=2^{m}-1.$$
4.3 The third family of codes $\mathcal{C}(m,2,L_{3})$ when $m$ is doubly-even
Writing $\eta_{0}=1+v+v^{2}+v^{3}+v^{4},~{}\eta_{1}=1+\epsilon^{4}v+\epsilon^{3}v^{2}+%
\epsilon^{2}v^{3}+\epsilon v^{4},~{}\eta_{2}=1+\epsilon^{3}v+\epsilon v^{2}+%
\epsilon^{4}v^{3}+\epsilon^{2}v^{4},~{}\eta_{3}=1+\epsilon^{2}v+\epsilon^{4}v^%
{2}+\epsilon v^{3}+\epsilon^{3}v^{4}$ and $\eta_{4}=1+\epsilon v+\epsilon^{2}v^{2}+\epsilon^{3}v^{3}+\epsilon^{4}v^{4},$ where $\epsilon^{5}=1$. According to Chinese Remainder Theorem, we then obtain $R_{m}=\eta_{0}\mathbb{F}_{2^{m}}\oplus\eta_{1}\mathbb{F}_{2^{m}}\oplus\eta_{2}%
\mathbb{F}_{2^{m}}\oplus\eta_{3}\mathbb{F}_{2^{m}}\oplus\eta_{4}\mathbb{F}_{2^%
{m}}$ in the case of $m$ is doubly-even. Hence, we have $R_{m}^{*}\cong\mathbb{F}_{2^{m}}^{*}\bigoplus\mathbb{F}_{2^{m}}^{*}\bigoplus%
\mathbb{F}_{2^{m}}^{*}\bigoplus\mathbb{F}_{2^{m}}^{*}\bigoplus\mathbb{F}_{2^{m%
}}^{*}$.
It means that $R_{m}^{*}=\{\eta_{0}r_{0}+\eta_{1}r_{1}+\eta_{2}r_{2}+\eta_{3}r_{3}+\eta_{4}r_%
{4}:r_{j}\in\mathbb{F}_{2^{m}}^{*},j=0,1,2,3,4\}$. Now, we investigate the Lee weight of codewords of $\mathcal{C}(m,2,L_{3})$ in this case.
Theorem 4.5 Let $m$ be doubly-even, then the set $\mathcal{C}(m,2,L_{3})$ is a five-Lee-weight linear code of length $L_{3}=(2^{m}-1)^{5}$ and its weight distribution is given in Table III.
$\mathrm{Table~{}III.}~{}~{}~{}\mathrm{weight~{}distribution~{}of}~{}\mathcal{C%
}(m,2,L_{3})$
Weight
Frequency
0
1
$$5\times(2^{m}-1)^{3}(2^{m}-2)2^{m-1}$$
$$10\times(2^{m}-1)^{2}$$
$$5\times(2^{m}-1)(2^{m}-2)(2^{2m}-2^{m+1}+2)2^{m-1}$$
$$5\times(2^{m}-1)^{4}$$
$$\frac{5\times[(2^{m}-1)^{5}+1]}{2}$$
$$(2^{m}-1)^{5}$$
$$5\times(2^{m}-1)^{2}(2^{3m-1}-2^{2m-1}+2^{m-1})$$
$$10\times(2^{m}-1)^{3}$$
$$5\times(2^{m}-1)^{4}2^{m-1}$$
$$5\times(2^{m}-1)$$
Proof.
Set $r=\eta_{0}r_{0}+\eta_{1}r_{1}+\eta_{2}r_{2}+\eta_{3}r_{3}+\eta_{4}r_{4}\in R_{%
m}^{*}$, where $r_{j}\in\mathbb{F}_{2^{m}}^{*},j=0,1,2,3,4$.
For $a=\eta_{0}a_{0}+\eta_{1}a_{1}+\eta_{2}a_{2}+\eta_{3}a_{3}+\eta_{4}a_{4}\in R_{m}$, where $a_{j}\in\mathbb{F}_{2^{m}},j=0,1,2,3,4$, then
$ar=\eta_{0}a_{0}r_{0}+\eta_{1}a_{1}r_{1}+\eta_{2}a_{2}r_{2}+\eta_{3}a_{3}r_{3}%
+\eta_{4}a_{4}r_{4}$ and
$$\displaystyle\Phi(Tr_{m}(ar))$$
$$\displaystyle=$$
$$\displaystyle(tr(a_{0}r_{0}+a_{1}r_{1}+a_{2}r_{2}+a_{3}r_{3}+a_{4}r_{4}),tr(a_%
{0}r_{0}+\epsilon^{4}a_{1}r_{1}+\epsilon^{3}a_{2}r_{2}+\epsilon^{2}a_{3}r_{3}+%
\epsilon a_{4}r_{4}),$$
$$\displaystyle tr(a_{0}r_{0}+\epsilon^{3}a_{1}r_{1}+\epsilon a_{2}r_{2}+%
\epsilon^{4}a_{3}r_{3}+\epsilon^{2}a_{4}r_{4}),tr(a_{0}r_{0}+\epsilon^{2}a_{1}%
r_{1}+\epsilon^{4}a_{2}r_{2}+\epsilon a_{3}r_{3}+\epsilon^{3}a_{4}r_{4}),$$
$$\displaystyle tr(a_{0}r_{0}+\epsilon a_{1}r_{1}+\epsilon^{2}a_{2}r_{2}+%
\epsilon^{3}a_{3}r_{3}+\epsilon^{4}a_{4}r_{4}))$$
$$\displaystyle=:$$
$$\displaystyle(A^{\prime}_{0},A^{\prime}_{1},A^{\prime}_{2},A^{\prime}_{3},A^{%
\prime}_{4}).$$
Taking character sums, yields
$$\Theta(a)=\sum_{i=0}^{4}\sum_{r_{0},r_{1},r_{2}\in\mathbb{F}_{2^{m}}^{*}}\sum_%
{r_{3},r_{4}\in\mathbb{F}_{2^{m}}^{*}}(-1)^{tr_{m}(A^{\prime}_{i})}.$$
If $a=\eta_{0}a_{0}+\eta_{1}a_{1}+\eta_{2}a_{2}+\eta_{3}a_{3}+\eta_{4}a_{4}\in R_{%
m}^{*}$, where $a_{j}\in\mathbb{F}_{2^{m}}^{*},j=0,1,2,3,4$, then $\Theta(a)=-5$. So $w_{L}(ev(a))=\frac{5\times[(2^{m}-1)^{5}+1]}{2}.$
Next, we consider $a\in R_{m}\backslash\{R_{m}^{*}\}$. It is easily check that $a=0$ corresponds to the zero codewords, and then its Lee weight is 0.
If $a=\eta_{i}a_{i},i\in\{0,1,2,3,4\}$, then $\Theta(a)=5\times(2^{m}-1)$.
Further, $w_{L}(ev(a))=5\times(2^{m}-1)(2^{m}-2)(2^{2m}-2^{m+1}+2)2^{m-1}$ by Equation (1). If $a=\eta_{i}a_{i}+\eta_{j}a_{j},i,j\in\{0,1,2,3,4\},i\neq j$, then $\Theta(a)=-5\times(2^{m}-1)^{2}$. By Equation (1), we have $w_{L}(ev(a))=5\times(2^{m}-1)^{2}(2^{3m-1}-2^{2m-1}+2^{m-1})$.
If $a=\eta_{i}a_{i}+\eta_{j}a_{j}+\eta_{k}a_{k},i,j,k\in\{0,1,2,3,4\},$ where $i,j,k$ are pairwise different, then $\Theta(a)=5\times(2^{m}-1)^{3}$. What is more, $w_{L}(ev(a))=5\times(2^{m}-1)^{3}(2^{m}-2)2^{m-1}$. If $a=\eta_{0}a_{0}+\eta_{1}a_{1}+\eta_{2}a_{2}+\eta_{3}a_{3}+\eta_{4}a_{4}$, where $a_{j}=0,a_{i}\in\mathbb{F}_{2^{m}}^{*},j\in\{0,1,2,3,4\},i\in\{0,1,2,3,4\}%
\backslash\{j\}$, then we can obtain $\Theta(a)=-5\times(2^{m}-1)^{4}.$
By Equation (1), we can easily get $w_{L}(ev(a))=5\times(2^{m}-1)^{4}2^{m-1}$.
∎
By Theorem 4.5, we have constructed a binary linear code of length $s=5\times(2^{m}-1)^{5}$, of dimension $5m$, with five nonzero weights $w_{1}<w_{2}<w_{3}<w_{4}<w_{5}$ of values
$$w_{1}=5\times(2^{m}-1)^{3}(2^{m}-2)2^{m-1},~{}~{}w_{2}=5\times(2^{m}-1)(2^{m}-%
2)(2^{2m}-2^{m+1}+2)2^{m-1},~{}~{}w_{3}=\frac{5\times[(2^{m}-1)^{5}+1]}{2}$$
$$w_{4}=5\times(2^{m}-1)^{2}(2^{3m-1}-2^{2m-1}+2^{m-1}),~{}w_{5}=5\times(2^{m}-1%
)^{4}2^{m-1},$$
with respective frequencies
$$f_{1}=10\times(2^{m}-1)^{2},~{}~{}f_{2}=5\times(2^{m}-1)^{4},~{}~{}f_{3}=(2^{m%
}-1)^{5},~{}~{}f_{4}=10\times(2^{m}-1)^{3},~{}~{}f_{5}=5\times(2^{m}-1).$$
Note that $L=L_{1}=L_{2}=L_{3}=|R^{*}_{m}|.$ Next, we study their optimality.
Theorem 4.6 The three-weight binary linear code $\Phi(\mathcal{C}(m,2,L_{1}))$, for $m>6$ and $m$ is odd, is optimal.
Proof.
Recall the Griesmer bound G . If $[N,K,d]$ are the parameters of a linear binary code, then
$$\sum_{j=0}^{K-1}\Big{\lceil}\frac{d}{2^{j}}\Big{\rceil}\leq N.$$
In our situation $N=5\times(2^{5m}-2^{4m}-2^{m}+1),K=5m,d=5\times(2^{5m-1}-2^{4m-1}-2^{m-1}).$
The ceiling
function takes the following values depending on the position of $j:$
•
$0\leq j\leq m-1\Rightarrow\lceil\frac{d+1}{2^{j}}\rceil=5\times(2^{4m}-2^{3m}-%
1)2^{m-1-i}+1,$
•
$j=m\Rightarrow\lceil\frac{d+1}{2^{j}}\rceil=5\times(2^{4m-1}-2^{3m-1})-2,$
•
$j=m+1\Rightarrow\lceil\frac{d+1}{2^{j}}\rceil=5\times(2^{4m-2}-2^{3m-2})-1,$
•
$m+2\leq j\leq 4m-1\Rightarrow\lceil\frac{d+1}{2^{j}}\rceil=5\times(2^{m}-1)2^{%
4m-1-i},$
•
$j=4m\Rightarrow\lceil\frac{d+1}{2^{j}}\rceil=5\times 2^{m-1}-2,$
•
$j=4m+1\Rightarrow\lceil\frac{d+1}{2^{j}}\rceil=5\times 2^{m-2}-1,$
•
$4m+2\leq j\leq 5m-1\Rightarrow\lceil\frac{d+1}{2^{j}}\rceil=5\times 2^{5m-1-i}.$
Thus,
$$\displaystyle\sum_{j=0}^{K-1}\Big{\lceil}\frac{d+1}{2^{j}}\Big{\rceil}$$
$$\displaystyle=$$
$$\displaystyle\sum_{j=0}^{5m-1}5\times 2^{5m-1-i}-\sum_{j=0}^{4m-1}5\times 2^{4%
m-1-i}-\sum_{j=0}^{m-1}5\times 2^{m-1-i}+m-6$$
$$\displaystyle=$$
$$\displaystyle 5\times(2^{5m}-1)-5\times(2^{4m}-1)-5\times(2^{m}-1)+m-6$$
$$\displaystyle=$$
$$\displaystyle 5\times 2^{5m}-5\times 2^{4m}-5\times 2^{m}+m-1.$$
By simply calculation, we have $5\times 2^{5m}-5\times 2^{4m}-5\times 2^{m}+m-1-N>0$ when $m>6$.
This proof is completed.
∎
Example 4.7
Let $m=2$. By Theorem 4.4, we obtain a five-weight binary linear code of length 1215, of dimension 10, with five nonzero weights 1650, 1680, 1690, 1800, 2250, and frequencies 90, 225, 675, 30 and 3, respectively.
Example 4.8 Taking $m=3$. Then we get a three-weight binary linear code of dimension 15, with three nonzero weights 71660, 71680, 81900, and frequencies 28665, 4095 and 7, respectively.
5 The dual code
We compute the dual distance of $\mathcal{C}(m,2,|R_{m}^{*}|).$ A property of the trace that we need is that it is nondegenerate. The proof of the following lemma is similar to that in SLS2 , so we omit it here.
Lemma 5.1
If for all $a\in R_{m},$ we have that $Tr_{m}(ax)=0,$ then $x=0.$
Combining the sphere-packing bound and Lemma 5.1, we can get the following results.
Theorem 5.2
The dual Lee distance $d^{\prime}$ of $\mathcal{C}(m,2,|R_{m}^{*}|)$ is $2.$
Proof.
First, we show that $d^{\prime}<3.$ If not, we can apply the sphere-packing bound to $\Phi(\mathcal{C}(m,2,|R_{m}^{*}|)^{\bot}),$ to obtain
$$2^{5m}\geq 1+s.$$
•
$m$ is odd, for $\Phi(\mathcal{C}(m,2,L_{1})^{\bot})$, then
$$2^{5m}\geq 1+s=1+5\times(2^{m}-1)(2^{4m}-1),$$
or, after expansion
$$5\times(2^{4m}+2^{m})\geq 6+2^{5m+2}.$$
Dropping the $6$ in the RHS, and dividing both sides by $2^{m},$ we find that this inequality would imply $5\times(2^{3m}+1)>2^{5m+2},$ i.e., $5>2^{3m}(2^{2m+2}-5)$. Contradiction with $m\geq 1$, which implies $d^{\prime}<3.$
•
$m$ is singly-even, for $\Phi(\mathcal{C}(m,2,L_{2})^{\bot})$, then
$$2^{5m}\geq 1+s=1+5\times(2^{m}-1)(2^{2m}-1)^{2}=1+5\times(2^{5m}-2^{4m}-2^{3m+%
1}+2^{2m+1}+2^{m}-1).$$
(2)
When $m=2$, then $2^{5m}<1+5\times(2^{5m}-2^{4m}-2^{3m+1}+2^{2m+1}+2^{m}-1).$ Now, we consider $m\geq 6$.
From Equation (2), we have
$$\displaystyle 0$$
$$\displaystyle\geq$$
$$\displaystyle 2^{5m+2}-5\times(2^{4m}+2^{3m+1})+5\times(2^{2m+1}+2^{m})-4$$
$$\displaystyle\geq$$
$$\displaystyle 2^{3m+1}(2^{2m+1}-5\times 2^{m-1}-5).$$
Contradiction with $2^{2m+1}-5\times 2^{m-1}-5=2^{m-1}(2^{m+2}-5)-5>0$ when $m\geq 6.$
So $d^{\prime}<3.$
•
$m$ is doubly-even, for $\Phi(\mathcal{C}(m,2,L_{3})^{\bot})$, then
$$\displaystyle 2^{5m}$$
$$\displaystyle\geq$$
$$\displaystyle 1+5\times(2^{m}-1)^{5}$$
$$\displaystyle=$$
$$\displaystyle 5\times(2^{5m}-2^{4m+2}-2^{4m}+2^{3m+3}+2^{3m+1}-2^{2m+3}-2^{2m+%
1}+2^{m+2}+2^{m})-4$$
$$\displaystyle>$$
$$\displaystyle 5\times(2^{5m}-2^{4m+2}-2^{4m}).$$
Note that $5\times(2^{5m}-2^{4m+2}-2^{4m})-2^{5m}=2^{4m}(2^{m+2}-25)>0$ for $m\geq 4.$
It is contradiction, which implies $d^{\prime}<3.$
Next, we check that $d^{\prime}=2,$ by showing that $\mathcal{C}(m,2,|R_{m}^{*}|)^{\bot},$ does not contain a word of Lee weight one.
If it does, let us assume first that it has value $\gamma v^{j}$ at some $x\in R_{m}^{*},$ where $\gamma\in\mathbb{F}_{2^{m}}^{*},j\in\{0,1,2,3,4\}.$
This implies that $\forall a\in R_{m},\gamma v^{j}Tr_{m}(ax)=0,$ then we must have $Tr_{m}(\gamma ax)=0$ since $Tr_{m}()$ is a linear map, and by using Lemma 5.1, we know $x=0.$ Contradiction with $x\in R_{m}^{*}.$ So $d^{\prime}=2.$
∎
6 Application to secret sharing schemes
Secret sharing is an important topic of cryptography, which has been studied for over thirty years. In this section, we will study the secret sharing schemes based on linear codes studied in this paper.
Originally secret sharing was motivated
by the problem of sharing a secret digital key. In order to keep the secret efficiently and
safely, Shamir and Blakley introduced secret sharing schemes (SSS) in 1979. An SSS based on error-correcting codes was introduced by Massey, and the minimal coalitions
in such a scheme were characterized as a function of the minimal vectors, to be defined next, of the dual code.
A minimal codeword of a linear code $C$ is a nonzero codeword that does not cover any other nonzero codeword. Recall
that a vector $x$ covers a vector $y$ if $s(x)$ contains $s(y)$, where $s(y)$, the support $s(y)$ of a vector $y\in\mathbb{F}_{p}^{s}$, is defined as the set of indices where it is nonzero.
Although the minimal codewords of a given linear code is hard to determine in general, there is a numerical condition derived in AB , bearing on the weights of the code,
that is easy to check, once the weight distribution is known. By recalling the following result of Ashikhmin and Barg (see AB ), it is shown that all
nonzero codewords of linear code are minimal.
Lemma 6.1 (Ashikhmin-Barg) Given a $p$-ary code $C,$ denote by $w_{0}$ and $w_{\infty}$ the minimum and maximum nonzero weights of $C$, respectively. If
$$\frac{w_{0}}{w_{\infty}}>\frac{p-1}{p},$$
then every nonzero codeword of $C$ is minimal.
Theorem 6.2 Let $m$ be odd,
then all the nonzero codewords of $\Phi(\mathcal{C}(m,2,L_{1}))$, for $m>1$, are minimal.
Proof.
By the preceding Lemma 6.1 with $w_{0}=5\times(2^{5m-1}-2^{4m-1}-2^{m-1}),$ and $w_{\infty}=5\times(2^{5m-1}-2^{m-1}).$ Rewriting the inequality of Lemma 6.1 as $2\omega_{1}>\omega_{2}$, and dividing both sides by $5\times 2^{m-1}$, we obtain
$$2^{4m}>2^{3m+1}+1.$$
The condition follows from the fact that $2^{m-1}>1$ when $m>1.$
Hence the result follows.
∎
Theorem 6.3 Let $m$ be singly-even,
then all the nonzero codewords of $\Phi(\mathcal{C}(m,2,L_{2}))$, for $m>1$, are minimal.
Proof.
We use Lemma 6.1 with $w_{0}=5\times(2^{2m}-1)(2^{3m-1}-2^{2m-1}-2^{m-1})$ and $w_{\infty}=5\times(2^{2m}-1)^{2}2^{m-1}.$ Rewriting the inequality of Lemma 6.1 as $2w_{0}>w_{\infty},$ and dividing both sides by $5\times(2^{m}-1)2^{m-1}$, we obtain
$$2\times(2^{2m}-1)>2^{2m}-2^{m}-1,$$
namely, $2^{2m}+2^{m}>1$. Note that $2^{2m}+2^{m}>1$ for a positive integer $m.$ The result follows.
∎
Theorem 6.4 Let $m$ be doubly-even,
then all the nonzero codewords of $\Phi(\mathcal{C}(m,2,L_{3}))$, for $m>1$, are minimal.
Proof.
We use Lemma 6.1 with $w_{0}=5\times(2^{m}-1)^{3}(2^{m}-2)2^{m-1},$ and $w_{\infty}=5\times(2^{m}-1)^{4}2^{m-1}.$ Rewriting the inequality of Lemma 6.1 as $2w_{0}>w_{\infty},$ and dividing both sides by $5\times(2^{m}-1)^{3}2^{m-1}$, we obtain
$$2\times(2^{m}-2)>2^{m}-1,$$
namely, $2^{m}>3$. Noting that $2^{m}>3$ when $m>1.$ So the result follows.
∎
The Massey’s scheme is a construction of an SSS using a code $C$ of length $s$ over $\mathbb{F}_{p}.$ In essence,
the secret is carried by the first coordinate of a codeword,
and the coalitions correspond to supports of codewords in
the dual code with a one in that coordinate. It is worth mentioning that in some special cases, that is, when all nonzero codewords are minimal, it was shown in DY2 that there is the following alternative, depending on $d^{\prime}$:
•
If $d^{\prime}\geq 3,$ then the SSS is “democratic”: every user belongs to the same number of coalitions,
•
If $d^{\prime}=2,$ then the SSS is “dictatorial”: some users belong to every coalition.
Depending on the application, one or the other situation might be more suitable.
By Theorems 4.3, 4.4, 4.5 and 5.2, we see that for some values of the parameters, a SSS built on $\Phi(\mathcal{C}(m,p,|R_{m}^{*}|))$ is dictatorial.
7 Conclusion
Trace codes over fields are a well-known source of constructions for few weights codes. In the present work, we have extended the notion of trace codes from fields to rings. On the basis of the linear Gray map we defined, we constructed a family of three-weight binary linear codes, which are optimal by using the Griesmer bound, and two families of five-weight binary
linear codes. These codes are abelian, and quasi-cyclic, but not visibly cyclic. Finally, an application to secret sharing schemes is given.
It is worth exploring more general constructions by varying the alphabet of the code, or the defining set of the trace code.
Compared with linear codes in DD ; DLLZ ; DY ; HY1 ; ZDL , the codes in this paper have different weight distribution.
References
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KAM Theory for secondary tori
L. Biasco & L. Chierchia
Dipartimento di Matematica e Fisica
Università degli Studi Roma Tre
Largo San L. Murialdo 1 - 00146 Roma, Italy
biasco@mat.uniroma3.it, luigi@mat.uniroma3.it
Abstract
(i)
In [4] (Rend. Lincei Mat. Appl. 26 (2015), 1–10; see also arXiv:1503.08145 [math.DS]) the following result has been announced:
Theorem Consider a real–analytic nearly–integrable mechanical system with potential $f$, namely,
a Hamiltonian system with real-analytic Hamiltonian
$$H(y,x)=\frac{1}{2}\sum_{i=1}^{n}y_{i}^{2}+\varepsilon f(x)\ ,$$
$(y,x)\in{\mathbb{R}}^{n}\times{\mathbb{T}}^{n}$ being standard action–angle variables.
For “general non–degenerate” potentials $f$’s there exists $\varepsilon_{0},a>0$ such that, if $0<\varepsilon<\varepsilon_{0}$, then the Liouville measure of the complementary of $H$–invariant tori is smaller than $\varepsilon|\log\varepsilon|^{a}$.
In this paper we provide a proof of such result.
(ii)
The class of “general non–degenerate” potentials ${\mathcal{P}}_{s}$ (defined in §1) is, for any given $s>0$, an open and dense subset of real–analytic functions on ${\mathbb{T}}^{n}$ having holomorphic extensions on $\{x\in{\mathbb{C}}^{n}|\,|\,{\rm Im}\,x_{i}|<s\}$, the topology being that induced by the weighted Fourier norm $|f|_{s}:=\sup_{k\in{\mathbb{Z}}^{n}}|f_{k}|e^{|k|s}$.
The class ${\mathcal{P}}_{s}$ is also of “full measure” in natural ways (compare Proposition 1.1 proven in Appendix A).
(iii)
The above Theorem is based on an extension of KAM Theory to a suitable $\varepsilon$–dependent neighborhood
of simple resonances $\{y\in B:y\cdot k=0\}$, with $|k|\leq K\sim|\log\varepsilon|^{b}$, for a suitable $b>0$ and any given ball $B\subset{\mathbb{R}}^{n}$. The main issue is giving a quantitative analytic description of the integrability structure of the averaged Hamiltonian at simple resonances suitable for application of KAM methods. Such analytic properties are summarized in the “Structure Theorem” of §4, whose proof occupies the main part of this paper, namely, Sect’s 5, 6 and Appendix C, where action–angle variables for generic parameter–depending systems are discussed.
(iv)
In view of the Structure Theorem one can then apply simultaneously (for $|k|\leq K$) explicit classical KAM measure estimates (as given, e.g., in [6]) and conclude the proof of the main Theorem.
Contents
1 Functional setting and Main Theorem
2 Geometry of resonances
3 Normal Forms
3.1 Normal form in $\Omega^{0}$ (non–resonant regime)
3.2 Normal form in $\Omega^{1}$ (simple resonances)
3.2.1 The effective potential
3.2.2 Rescalings
4 The nearly–integrable structure at simple resonances
4.1 A class of Morse non-degenerate functions
4.2 The Structure Theorem
5 Proof of Part I of the Structure Theorem
5.1 Critical points and critical energies of
the “unperturbed potential” $F^{0}$
5.2 The slow angle
5.3 The auxiliary Hamiltonian
5.4 A
special group
of symplectic transformations
5.5 An intermediate transformation
5.6 The integrating transformation
5.7 Properties of the actions as functions of the energy
5.8 The final canonical transformation
5.9 Conclusion of the
proof of part one of the Structure Theorem
6 Proof of Part II of the Structure Theorem
7 Proof of the Main Theorem
7.1 Application of the Structure Theorem
7.2 Application of the KAM theorem
A Properties of the class of non–degenerate potentials
B Proof of the Normal Form Lemma 3.1
C On action–angle variables for 1D mechanical systems with parameters
C.1 The “unperturbed case”
C.2 The action as a function of the angle at constant energy
C.3 The domains of definition of action angle variables
C.4 Definition of action variables
C.5 Properties of the actions as functions of the energy and viceversa
D Miscellanea
1 Functional setting and Main Theorem
In this paper we consider real--analytic functions, which are ‘‘non--degenerate’’ in a suitable sense111It would be easier to consider larger function spaces of smooth functions. However, the natural (both from the theoretical and applicative point of view) and most challenging setting is, we believe, that of real–analytic potentials..
Let $s>0$ and
consider the real–analytic functions on ${\mathbb{T}}^{n}$ having zero average and finite “sup–Fourier norm”
$$|f|_{s}:=\sup_{k\in{\mathbb{Z}}^{n}}|f_{k}|e^{|k|s}<\infty\ ,$$
(1)
where $f_{k}$ denotes Fourier coefficients and, as usual, $|k|$, for integer vectors, denotes the 1-norm $\sum|k_{j}|$. Denote by ${\mathcal{A}}_{s}^{n}$ the Banach space of such functions.
Let ${\mathbb{Z}}^{n}_{\sharp}$ denote the set of integer vectors $k\neq 0$ in ${\mathbb{Z}}^{n}$ such that the first non–null component is positive:
$${\mathbb{Z}}^{n}_{\sharp}:=\big{\{}k\in{\mathbb{Z}}^{n}:\ k\neq 0\ {\rm and}\ %
k_{j}>0\ {\rm where}\ j=\min\{i:k_{i}\neq 0\}\big{\}}\ ,$$
(2)
and denote by ${{\mathbb{Z}}}^{n}_{*}$ the generators of one–dimensional maximal lattices, namely, the set of vectors $k\in{\mathbb{Z}}^{n}_{\sharp}$ such that the greater common divisor of their components is 1, namely
$${\mathbb{Z}}^{n}_{*}:=\{k\in{\mathbb{Z}}^{n}_{\sharp}:\ {\rm gcd}(k_{1},\ldots%
,k_{n})=1\}\ ;$$
(3)
then, the list of one–dimensional maximal lattices is given by the sets ${\mathbb{Z}}k$ with $k\in{\mathbb{Z}}^{n}_{*}$.
We can, now, decompose the Fourier expansion of
any function $f\in{\mathcal{A}}_{s}^{n}$ as sum of real–analytic functions of one–variable, which are the projection of $f$ onto the one–dimensional maximal lattice ${\mathbb{Z}}k$ (with $k\in{\mathbb{Z}}^{n}_{*}$), as follows:
$$f(x)=\sum_{k\in{\mathbb{Z}}^{n}_{*}}F^{k}(k\cdot x)\ ,\quad{\rm where}\quad F^%
{k}(t):=\sum_{j\in{\mathbb{Z}}\backslash\{0\}}f_{jk}e^{ijt}$$
(4)
$f_{jk}$ being the Fourier coefficient of $f$ with Fourier index $jk\in{\mathbb{Z}}^{n}$.
Notice that, since $f\in{\mathcal{A}}_{s}^{n}$, the functions $F^{k}$ belong to ${\mathcal{A}}_{|k|s}^{1}$.
Definition 1.1
(The class ${\mathcal{P}}_{s}$ of non–degenerate potentials)
Let $s>0,0<\delta\leq 1$ and let
$$K_{s}(\delta):={\,c\,}\max\big{\{}1\ ,\ \frac{1}{s}\ ,\ \frac{1}{s}\ \log\frac%
{1}{s\,\delta}\big{\}}\,,$$
(5)
where ${\,c\,}>1$ is
a suitably large constant to be chosen
below (see (253)), depending only on $n$.
Let ${\mathcal{P}}_{s}(\delta)$ be the set of functions in ${\mathcal{A}}_{s}^{n}$
such that, for $k\in{\mathbb{Z}}^{n}_{*}$, the following holds
(P1)
$\displaystyle{|f_{k}|\geq\delta|k|^{-\frac{n+3}{2}}\ e^{-|k|s}\ \ {\sl if}\ \ %
|k|>K_{s}(\delta)\,;}$
(P2)
$\displaystyle{\ \min_{\xi\in{\mathbb{R}}}\ \big{(}|\partial_{\xi}F^{k}(t)|+|%
\partial^{2}_{\xi}F^{k}(t)|\big{)}>0\ \ {\sl if}\ \ |k|\leq K_{s}(\delta)\,;}$
(P3) $F^{k}(t_{1})\neq F^{k}(t_{2})$
for every $0\leq t_{1}<t_{2}<2\pi$ such that
$\partial_{t}F^{k}(t_{1})=\partial_{t}F^{k}(t_{2})=0$ and
$|k|\leq K_{s}(\delta)$.
Finally, $\displaystyle{\mathcal{P}}_{s}:=\bigcup_{\delta>0}{\mathcal{P}}_{s}(\delta)$.
An example of function $f\in{\cal P}_{s}(\delta)$, as it is immediate to verify, is given by
$$f(x)=2\delta\sum_{k\in{\mathbb{Z}}^{n}_{*}}e^{-|k|s}\,\cos(k\cdot x)\ ,\qquad{%
\rm i.e.,}\qquad f_{k}=\left\{\begin{array}[]{ll}{\displaystyle\delta e^{-|k|s%
}}&\mbox{ {\rm if} ${\pm k\in{\mathbb{Z}}^{n}_{*}}$}\\
{0}&\mbox{ {\rm otherwise}}\end{array}\right.\ .$$
(6)
The class ${\mathcal{P}}_{s}$ is “general” in several ways: from a probabilistic, topological and measure theoretical points of view.
To describe the probabilistic point of view, let us
denote by
$\ell_{\infty}^{n}$ the Banach space of complex sequences $z=\{z_{k}\}_{k\in{\mathbb{Z}}^{n}_{\sharp}}$ with finite sup–norm
$|z|_{\infty}:=\sup_{k\in{\mathbb{Z}}^{n}_{\sharp}}|z_{k}|$.
The map
$$j:f\in{\mathcal{A}}_{s}^{n}\to\big{\{}f_{k}e^{|k|s}\big{\}}_{k\in{\mathbb{Z}}^%
{n}_{\sharp}}\in\ell_{\infty}^{n}$$
(7)
is an isomorphism of Banach spaces222Recall that since the functions in ${\mathcal{A}}^{n}_{s}$ are real–analytic one has the reality condition $f_{k}=\bar{f}_{-k}$.,
which allows to identify functions in ${\mathcal{A}}_{s}^{n}$ with points in $\ell_{\infty}^{n}$ and the Borellians of ${\mathcal{A}}_{s}^{n}$ with those of $\ell_{\infty}^{n}$.
Now, consider the probability measure given by the standard normalized Lebesgue–product measure
on the unit closed ball of $\ell_{\infty}^{n}$, namely, the unique probability measure $\mu$ on the Borellians of $\{z\in\ell_{\infty}^{n}:|z|_{\infty}\leq 1\}$ such that,
given Lebesgue measurable sets $A_{k}$ in the unit complex disk $A_{k}\subseteq D:=\{w\in{\mathbb{C}}:\ |w|\leq 1\}$
with $A_{k}\neq D$ only for finitely many $k$, one has
$$\mu\Big{(}\prod_{k\in\mathbb{Z}^{n}_{\sharp}}A_{k}\Big{)}=\prod_{\{k\in{%
\mathbb{Z}}^{n}_{\sharp}:\,A_{k}\neq D\}}{\rm meas}(A_{k})\,$$
where “meas” denotes the normalized Lebesgue measure on the unit complex disk $D$. Denote by ${\mathbb{B}}$ the closed ball of radius one in ${\mathcal{B}}_{s}^{n}$ and by ${\mathcal{B}}$ the Borellians in ${\mathbb{B}}$.
Then, the isometry $j$ in (7)
naturally induces a measure
$\mu_{s}$ on the Borellians ${\mathcal{B}}$.
The properties of ${\mathcal{P}}_{s}$ are collected in the following proposition, whose simple proof is given in Appendix A.
Proposition 1.1
Let $s>0$. The set ${\mathcal{P}}_{s}\subseteq{\mathcal{A}}_{s}^{n}$ contains an open dense set, is prevalent, ${\mathcal{P}}_{s}\,\cap\,{\mathbb{B}}\in{\mathcal{B}}$ and $\mu_{s}({\mathcal{P}}_{s}\cap{\mathbb{B}})=1$.
Let $\|\cdot\|$ be the standard Euclidean norm on $\mathbb{R}^{n}.$ Then, one has the following
Theorem 1.1
Let $s>0$ and let $\Omega$ be a bounded region in ${\mathbb{R}}^{n}$ with $n\geq 2$.
Let $f\in{\mathcal{P}}_{s}$ and consider the Hamiltonian
$$H:=\frac{1}{2}\|y\|^{2}+\varepsilon f(x)\ ,$$
(8)
There exist $\varepsilon_{0}>0$ and $\kappa>0$ such that, for any $0<\varepsilon<\varepsilon_{0}$, the measure of the set of $H$–trajectories in $\Omega\times{\mathbb{T}}^{n}$, which do not lie on an invariant Lagrangian (Diophantine) torus, is bounded by $\varepsilon|\log\varepsilon|^{\kappa}$.
We remark that the constants $\varepsilon_{0}$ and $\kappa$ depend only on
$n$, $s$, and $F^{k}$ with $k\in{\mathbb{Z}}^{n}_{*}$, $|k|\leq K_{s}(\delta)$.
Remark 1.1
In proving Theorem 1.1 we will assume that
$$\Omega=B_{1}(0):=\{\|y\|<1\}\qquad\text{and}\qquad|f|_{s}=1\,.$$
(9)
This is not restrictive since we can always consider
a large enough ball $B_{R}(0)\supseteq\Omega$
and rescale the action and the time in order to obtain (9)
(suitably renaming $\varepsilon$ and $f$).
2 Geometry of resonances
In this section we construct a covering of ${\mathbb{R}}^{n}$ (thought of as frequency space) by three regions: a non–resonant region
$\Omega_{0}$, a neighborhood of simple resonances $\Omega_{1}$ and a region $\Omega_{2}$ of “small” measure containing all other resonances. The sets $\Omega^{1}$ and $\Omega^{2}$ will be described in terms of linear maps $L_{k}$, $k\in{\mathbb{Z}}^{n}$, that depend on a given resonance $\{y\cdot k=0\}$: such maps $L_{k}$ will later be associated to generating functions $S(J,x)=x\cdot L_{k}J$, whose corresponding symplectic maps have the role of “straighten out” the geometry.
Fix $k\in{\mathbb{Z}}^{n}\backslash\{0\}$ with gcd$(k_{1},\ldots,k_{n})=1$.
Then, there exists a matrix
$A_{k}\in\ {\rm Mat}_{n\times n}(\mathbb{Z})$ such that333Here, $k$ is a row vector. Normally we do not distinguish between row and column vectors since it will be clear from context. The notation $|M|_{\infty}$, with $M$ matrix or vector, denotes the maximum norm $\max_{ij}|M_{ij}|$ or, respectively, $\max_{i}|M_{i}|$.
$$A_{k}=\binom{\hat{A}_{k}}{k}\,,\ \ \ \hat{A}_{k}=\hat{A}_{k}\in{\rm Mat}_{(n-1%
)\times n}(\mathbb{Z})\,,\ \ \ \det A_{k}=1\,,\ \ \ |\hat{A}_{k}|_{\infty}\leq%
|k|_{\infty}\ ,$$
(10)
and444Recalling that
for any $n\times n$ matrix $M$, one always has
$|\det M|\leq n^{n/2}|M|_{\infty}^{n}$, (11) follows by the D’Alembert expansion of determinants (with $c=(n-1)^{(n-1)/2}$).
$$|A_{k}^{-1}|_{\infty}\leq{\,c\,}|k|_{\infty}^{n-1}\,;$$
(11)
the existence of such a matrix is guaranteed by an elementary result of linear algebra based on Bezout’s Lemma (see Lemma D.8 in appendix D).
We then define a linear map555Without further notice, we shall always identify linear maps with the associated matrices. $L_{k}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ by setting
$$L_{k}:J=(\hat{J},J_{n})\in{\mathbb{R}}^{n-1}\times{\mathbb{R}}\mapsto L_{k}J:=%
J_{n}k+\,{\mathtt{p}}^{\perp}_{k}\hat{A}^{T}_{k}\hat{J}\,,$$
(12)
where
$\,{\mathtt{p}}^{\perp}_{k}$
is the orthogonal
projection on the subspace perpendicular to666Explicitely,
for $y\in\mathbb{R}^{n}$,
$\,{\mathtt{p}}^{\perp}_{k}y:=y-\frac{1}{\kappa}(y\cdot k)k\,.$
(13)
$k$. Observe that $L_{k}$ can be also written as composition of two linear maps:
$$L_{k}=A_{k}^{T}U_{k}$$
(14)
where $U_{k}$ acts as the identity on the first $(n-1)$ components and:
$$U_{k}:(\hat{J},J_{n})\mapsto\big{(}\hat{J},J_{n}-\frac{1}{\kappa}(\hat{A}_{k}k%
)\cdot\hat{J}\big{)}\,,\qquad{\rm i.e.}\quad U_{k}=\left(\begin{matrix}{\rm I}%
_{n-1}&0\cr-\kappa^{-1}\hat{A}_{k}k&1\cr\end{matrix}\right)\ ,$$
(15)
where ${\rm I}_{m}$ denotes the $(m\times m)$–identity matrix and where
$$\kappa:=\|k\|^{2}=k_{1}^{2}+\cdots+k_{n}^{2}\in{\mathbb{N}}\ .$$
(16)
Note that, by (10), we
get777Here and in the following, we shall denote by “${\,c\,}$” suitable constants (which, in general, differ from formula to formula) depending only on $n$.
$$\|U_{k}\|\ ,\|U_{k}^{-1}\|\leq{\,c\,}\ .$$
(17)
Elementary properties of $L_{k}$ are the following888Eq. (18) follows from (10), (14) and (15).
Identities (19),
(20), (21) and
the first bound in (22)
follow directly from definition (12); the bound on $\|L_{k}^{-1}\|$ follows by observing that $\|L_{k}^{-1}\|=\|U_{k}^{-1}A_{k}^{-T}\|\leq\|U_{k}^{-1}\|\,\|A_{k}^{-T}\|$ and that $\|U_{k}^{-1}\|\leq{\,c\,}$ and that
$\|A_{k}^{-T}\|\leq{\,c\,}\|k\|^{n-1}$ (as it follows by bounding the norm of a matrix by a constant times the maximum of its entries
and using the co–factor representation for the inverse of $A_{k}$, and taking into account (10)).
:
$$\displaystyle\det L_{k}=1\ ,$$
(18)
$$\displaystyle L_{k}J\cdot k=\kappa J_{n}\ ,$$
(19)
$$\displaystyle\|L_{k}J\|^{2}=\kappa J_{n}^{2}+\|\,{\mathtt{p}}_{k}^{\perp}\hat{%
A}_{k}\hat{J}\|^{2}\ ,$$
(20)
$$\displaystyle\hat{\Omega}\subseteq{\mathbb{R}}^{n-1}\ ,\ a>0\ \implies\ \ L_{k%
}\big{(}\hat{\Omega}\times(-\frac{a}{\kappa},\frac{a}{\kappa})\big{)}=\{|y%
\cdot k|<a\}\cap\big{\{}y:\,{\mathtt{p}}_{k}^{\perp}y\in\,{\mathtt{p}}_{k}^{%
\perp}\hat{A}_{k}^{T}\hat{\Omega}\big{\}}\ ,$$
(21)
$$\displaystyle\|L_{k}\|\leq{\,c\,}\|k\|\ ,\qquad{\rm and}\qquad\|L_{k}^{-1}\|%
\leq{\,c\,}\|k\|^{n-1}\ .$$
(22)
Further more interesting properties of $L_{k}$ are given in the following simple
Lemma 2.1
(i) The map $\,{\mathtt{p}}^{\perp}_{k}\hat{A}_{k}:{\mathbb{R}}^{n-1}\to k^{\perp}$ is a linear isomorphism.
(ii) For any $a>0$,
$L_{k}\big{(}{\mathbb{R}}^{n-1}\times(-\frac{a}{\kappa},\frac{a}{\kappa})\big{)%
}=\{|y\cdot k|<a\}$.
Proof (i): Let $\hat{A}_{k}=\left(\begin{matrix}a^{1}\\
a^{2}\\
\vdots\\
a^{n-1}\end{matrix}\right)$, with $a^{i}\in{\mathbb{Z}}^{n}$. Then $\,{\mathtt{p}}_{k}^{\perp}\hat{A}_{k}\ {\mathbb{R}}^{n-1}={\rm span}\{\,{%
\mathtt{p}}_{k}^{\perp}a^{1},...,\,{\mathtt{p}}_{k}^{\perp}a^{n-1}\}$. But the vectors $\,{\mathtt{p}}_{k}^{\perp}a^{i}$ are linearly independent (if $0=\sum\lambda_{i}\,{\mathtt{p}}_{k}^{\perp}a^{i}=\,{\mathtt{p}}_{k}^{\perp}(%
\sum\lambda_{i}\ a^{i})$, then there exists $c$ such that $\sum\lambda_{i}\ a^{i}=ck$, which implies that $\lambda_{1}=\cdots=\lambda_{n-1}=c=0$ since $A_{k}={\hat{A}_{k}\choose k}$ has determinant one and hence the vectors $a^{1},...,a^{n-1},k$ are linearly independent). The claim then follows from the rank–nullity theorem of linear algebra.
(ii): Let $W:=L_{k}({\mathbb{R}}^{n-1}\times(-a/\kappa,a/\kappa))$. From (19) it follows that $W\subseteq\{y|\ |y\cdot k|<a\}$.
Now, let $y\in{\mathbb{R}}^{n}$ be such that $|y\cdot k|<a$ and define $J_{n}:=(y\cdot k)/\kappa$. Then, $|J_{n}|<a/\kappa$ and, furthermmore, $y-J_{n}k\in k^{\perp}$. Thus, by part (i) of this Lemma, there exists $\hat{J}\in{\mathbb{R}}^{n-1}$ such that $y-J_{n}k=\,{\mathtt{p}}_{k}^{\perp}\hat{A}_{k}\hat{J}$, hence, $y=L_{k}(\hat{J},J_{n})$, proving that
$\{y|\ |y\cdot k|<a\}\subseteq W$ and, thus, $\{y|\ |y\cdot k|<a\}=W$.
Set999Recall (3).
$$\mathbb{Z}^{n}_{*,K}:=\{k\in\mathbb{Z}^{n}_{*}\,,\ \ |k|\leq K\}\,.$$
(23)
To quantify neighborhoods of resonances, as standard, we introduce two $\varepsilon$–dependent Fourier cut-offs $K,{\mathtt{K}}$ and an $\varepsilon$–dependent “width” $\alpha$ of simple resonance, by setting:
$$K:=\log^{2}\frac{1}{\varepsilon}\,,\qquad\qquad{\mathtt{K}}:=K^{2}\,,\qquad%
\qquad\alpha:=\sqrt{\varepsilon}K^{\nu+1}\ ,$$
(24)
where
$$\nu>n+1$$
(25)
is a suitable constant to be fixed later.
Let us assume that $\varepsilon$ is so small that
(recall (22))
$$\|L_{k}^{-1}\|\leq K^{n}\ .$$
(26)
Set
$$\hat{D}:=B_{K^{n}}(0)$$
(27)
For $k\in\mathbb{Z}^{n}_{*,K}$, define
$$\displaystyle\hat{Z}_{k}:=\left\{\hat{J}\in\hat{D}\ :\ \min_{l\in\mathbb{Z}^{n%
}_{*,{\mathtt{K}}}\,,\ l\notin\mathbb{Z}k}\big{|}\big{(}\,{\mathtt{p}}_{k}^{%
\perp}\hat{A}_{k}^{T}\hat{J}\big{)}\cdot l\big{|}\geq 3\alpha{\mathtt{K}}\frac%
{\|l\|}{\|k\|}\right\}\,,\ \ Z_{k}^{\sharp}:=\hat{Z}_{k}\times(-\frac{\alpha}{%
2\kappa},\frac{\alpha}{2\kappa})\subseteq\mathbb{R}^{n}\,,$$
(28)
$$\displaystyle Z_{k}:=\hat{Z}_{k}\times(-\frac{\alpha}{\kappa},\frac{\alpha}{%
\kappa})\subseteq\mathbb{R}^{n}\,,\qquad{\rm and}\qquad Z_{k}^{\prime}:=\big{(%
}\hat{D}\setminus\hat{Z}_{k}\big{)}\times(-\frac{\alpha}{\kappa},\frac{\alpha}%
{\kappa})\subseteq\mathbb{R}^{n}\,,$$
(29)
$$\displaystyle\Omega^{0}:=\{\|y\|<1\,:\,\min_{k\in\mathbb{Z}^{n}_{*,K}}|y\cdot k%
|\geq\alpha/2\}\,,\ \ \ \Omega^{1}:=\!\!\bigcup_{k\in\mathbb{Z}^{n}_{*,K}}\!\!%
L_{k}Z_{k}^{\sharp}\,,\ \ \ \Omega^{2}:=\!\!\bigcup_{k\in\mathbb{Z}^{n}_{*,K}}%
\!\!L_{k}Z_{k}^{\prime}\,.$$
(30)
Remark 2.1
The set $\Omega^{0}$ is a non–resonant set. The set $\Omega^{1}$, by (21) and Lemma 2.1, is seen to be a suitable neighborhood of simple resonances $\{y\cdot k=0\}$ with $k\in\mathbb{Z}^{n}_{*,K}$; finally, $\Omega^{2}$ is a neighborhood of order–two or higher resonances. Next Lemma clarifies and quantifies these observations.
Proposition 2.1
(i)
$\Omega^{0}\cup\Omega^{1}\cup\Omega^{2}\supseteq B_{1}(0)$.
(ii)
The set $\Omega^{0}$ is $(\alpha/2,K)$ completely non--resonant101010We follow the terminology introduced in [15]: given $\alpha,K>0$ and a sublattice $\Lambda\subseteq{\mathbb{Z}}^{n}$, one says that $D\subseteq{\mathbb{R}}^{n}$ (or $D\subseteq{\mathbb{C}}^{n}$) is “$(\alpha,K)$ non–resonant modulo $\Lambda$” if $|y\cdot k|\geq\alpha$ for all $y\in D$ and $k\in{\mathbb{Z}}^{n}\,\backslash\,\Lambda$ with $|k|\leq K$; if
$\Lambda=\{0\}$ is the trivial lattice, then $D$ is said to be $(\alpha,K)$ completely non–resonant., i.e.,
$$y\in\Omega^{0}\qquad\Longrightarrow\qquad|y\cdot k|\geq\alpha/2\,,\ \ \ %
\forall\,0<|k|\leq K\,.$$
(31)
(iii) For each $k\in\mathbb{Z}^{n}_{*,K}$, the set $L_{k}Z_{k}$ is $(2\alpha{\mathtt{K}}/\|k\|,{\mathtt{K}})$ non–resonant modulo ${\mathbb{Z}}k$, i.e.,
$$y\in L_{k}Z_{k}\qquad\Longrightarrow\qquad|y\cdot l|\geq 2\alpha{\mathtt{K}}/%
\|k\|\,,\ \ \ \forall\,\ l\in{\mathbb{Z}}^{n}\ ,\ l\notin\mathbb{Z}k\ ,\ |l|%
\leq{\mathtt{K}}\ .$$
(32)
(iv) There exists a constant ${\,c\,}>0$
depending only on $n$ such that:
$${\rm\,meas\,}(\Omega^{2})\leq{\,c\,}\alpha^{2}K^{n^{2}-n-1}{\mathtt{K}}^{n+2}\ .$$
(33)
Remark 2.2
Since $\alpha$ has been chosen as $\sqrt{\varepsilon}K^{\,c\,}$ (see (24)), from (33) it follows that
the region of second or higher order resonances $\Omega^{2}$ has Lebesgue measure smaller than $({\,c\,}\varepsilon|\log\varepsilon|^{\,c\,})$ and therefore no further analysis on $\Omega^{2}$ is needed.
Remark 2.3
In the definition of $\hat{Z}_{k}$ in (28) one could also use a
smaller set $\hat{D}_{k}\subset\hat{D},$
such that Proposition 2.1 still holds true. We have chosen a unique $\hat{D}$ for every $k$, just for simplicity.
Remark 2.4
The geometry of resonances here
is different from the geometry of resonances (in the convex case) as discussed, e.g., in [15]. In fact, in [15]
more resonances are disregarded in the non–resonant set,
namely, the resonances with $|k|\leq(1/\varepsilon)^{a}$, $a>0$.
Furthermore,
the neighborhood of simple resonances in [15] has width $\varepsilon^{b},$ $0<b<1/2$,
and, as a consequence, the set of double resonances has measure greater than
$\varepsilon^{2b}$, which is a set not negligible for our purposes.
On the other hand, in Nekhoroshev’s
theorem one can average out the perturbation up to an exponentially
high order $e^{-{\rm const\,}(1/\varepsilon)^{a}},$ while we will get only $\varepsilon^{|\log\varepsilon|}$.
Proof of Proposition 2.1.
(i): If $y\notin\Omega^{0}$ with $\|y\|<1$, there exists a $k\in{\mathbb{Z}}^{n}_{*,K}$ such that $|y\cdot k|<\alpha/2$; but then, in view of Lemma 2.1, $y$ belongs to
$$\displaystyle\{y^{\prime}\in{\mathbb{R}}^{n}:|y^{\prime}\cdot k|<\alpha/2\}%
\cap B_{1}(0)=L_{k}\big{(}{\mathbb{R}}^{n-1}\times(-\alpha/2\kappa,\alpha/2%
\kappa)\big{)}\cap B_{1}(0)$$
$$\displaystyle=L_{k}\bigg{(}\Big{(}{\mathbb{R}}^{n-1}\times(-\alpha/2\kappa,%
\alpha/2\kappa)\Big{)}\cap L_{k}^{-1}B_{1}(0)\bigg{)}$$
$$\displaystyle\stackrel{{\scriptstyle\eqref{LPDquater}}}{{\subseteq}}L_{k}\bigg%
{(}\Big{(}{\mathbb{R}}^{n-1}\times(-\alpha/2\kappa,\alpha/2\kappa)\Big{)}\cap B%
_{K^{n}}(0)\bigg{)}$$
$$\displaystyle\stackrel{{\scriptstyle\eqref{arrosticini}}}{{\subseteq}}L_{k}%
\Big{(}\hat{D}\times(-\alpha/2\kappa,\alpha/2\kappa)\Big{)}$$
$$\displaystyle\subset L_{k}(Z_{k}^{\sharp}\cup Z^{\prime}_{k})\subset\Omega^{1}%
\cup\Omega^{2}\,.$$
(ii): Let $y\in\Omega^{0}$ and let $0<|k|\leq K$. Then, there exist $j\in\mathbb{Z}\setminus\{0\}$
and $k^{\prime}\in\mathbb{Z}^{n}_{*,K}$
with111111Indeed, $j=\pm\,{\rm gcd}\{k_{1},...,k_{n}\}$. $k=jk^{\prime}$, so that
$$|y\cdot k|=|j||y\cdot k^{\prime}|\geq|y\cdot k^{\prime}|\stackrel{{%
\scriptstyle\eqref{neva}}}{{\geq}}\alpha/2\,,$$
proving (31).
(iii): Let $y=L_{k}J=L_{k}(\hat{J},J_{n})$ for some $k\in{\mathbb{Z}}^{n}_{*,K}$, $\hat{J}\in\hat{Z}_{k}$ and $|J_{n}|<\alpha/\kappa$. Let, also, $l\in{\mathbb{Z}}^{n}$, $l\notin{\mathbb{Z}}k$ with $|l|\leq{\mathtt{K}}$. As above, there exists
$j\in\mathbb{Z}\setminus\{0\}$ and
$l^{\prime}\in\mathbb{Z}^{n}_{*,{\mathtt{K}}}$ such that $l=jl^{\prime}$. Then,
$$\displaystyle|y\cdot l|$$
$$\displaystyle=$$
$$\displaystyle|L_{k}J\cdot l|\stackrel{{\scriptstyle{\rm(\ref{LPD})}}}{{=}}\Big%
{|}J_{n}\,k\cdot l+\big{(}\,{\mathtt{p}}_{k}^{\perp}\hat{A}_{k}^{T}\hat{J}\big%
{)}\cdot l\Big{|}\geq\Big{|}\big{(}\,{\mathtt{p}}_{k}^{\perp}\hat{A}_{k}^{T}%
\hat{J}\big{)}\cdot l\Big{|}-|k\cdot l||J_{n}|$$
$$\displaystyle=$$
$$\displaystyle|j|\,\Big{|}\big{(}\,{\mathtt{p}}_{k}^{\perp}\hat{A}_{k}^{T}\hat{%
J}\big{)}\cdot l^{\prime}\Big{|}-|k\cdot l||J_{n}|$$
$$\displaystyle\stackrel{{\scriptstyle\eqref{shevket}}}{{\geq}}$$
$$\displaystyle 3\alpha{\mathtt{K}}\frac{\|l\|}{\|k\|}-\alpha\,\frac{\|l\|}{\|k%
\|}\geq 2\alpha{\mathtt{K}}\frac{\|l\|}{\|k\|}\geq 2\alpha\frac{{\mathtt{K}}}{%
\|k\|}\,,$$
proving (32).
(iv):
Then (denoting Lebesgue measure by “${\rm\,meas\,}$”), from the definition of $\Omega^{2}$ in (30) it follows:
$$\displaystyle{\rm\,meas\,}(\Omega^{2})$$
$$\displaystyle=$$
$$\displaystyle{\rm\,meas\,}\big{(}\bigcup_{k\in{\mathbb{Z}}^{n}_{*,K}}L_{k}Z^{%
\prime}_{k}\big{)}\leq\sum_{k\in{\mathbb{Z}}^{n}_{*,K}}{\rm\,meas\,}(L_{k}Z^{%
\prime}_{k})$$
(34)
$$\displaystyle=$$
$$\displaystyle\sum_{k\in{\mathbb{Z}}^{n}_{*,K}}|\det L_{k}|{\rm\,meas\,}(Z^{%
\prime}_{k})\stackrel{{\scriptstyle{\rm(\ref{diego})}}}{{=}}\sum_{k\in{\mathbb%
{Z}}^{n}_{*,K}}{\rm\,meas\,}(Z^{\prime}_{k})\ .$$
Moreover121212
Denoting Lebsgue measure on ${\mathbb{R}}^{n-1}$ again by “${\rm\,meas\,}$”.
$$\displaystyle{\rm\,meas\,}(Z^{\prime}_{k})\leq{\,c\,}\,\alpha\kappa^{-1}\,\sum%
_{l\in{\mathbb{Z}}^{n}_{*,{\mathtt{K}}},l\notin{\mathbb{Z}}k}{\rm\,meas\,}\Big%
{\{}\|\hat{J}\|\leq K^{n}\ :\ \big{|}(\,{\mathtt{p}}_{k}^{\perp}\hat{A}_{k}^{T%
}\hat{J})\cdot l\big{|}<3\alpha{\mathtt{K}}\frac{\|l\|}{\|k\|}\Big{\}}\,.$$
(35)
Now, denoting by
$$v_{k,l}:=\|k\|^{2}l-(l\cdot k)k=\|k\|^{2}\,{\mathtt{p}}_{k}^{\perp}l\ ,$$
(36)
we see that
$$|(\,{\mathtt{p}}_{k}^{\perp}\hat{A}_{k}^{T}\hat{J})\cdot l|<3\alpha{\mathtt{K}%
}\frac{\|l\|}{\|k\|}\qquad\Longleftrightarrow\qquad|\hat{J}\cdot\hat{A}_{k}v_{%
k,l}|<3\alpha\|l\|\,\|k\|{\mathtt{K}}\ ,$$
(37)
so that (35) reads
$${\rm\,meas\,}(Z^{\prime}_{k})\leq{\,c\,}\,\alpha\kappa^{-1}\,\sum_{l\in{%
\mathbb{Z}}^{n}_{*,{\mathtt{K}}},l\notin{\mathbb{Z}}k}{\rm\,meas\,}\Big{\{}\|%
\hat{J}\|\leq K^{n}\ :\ |\hat{J}\cdot\hat{A}_{k}v_{k,l}|<3\alpha\|l\|\,\|k\|{%
\mathtt{K}}\}\ .$$
(38)
Now, observe that $v_{k,l}\in{\mathbb{Z}}^{n}\,\backslash\,\{0\}$ (since $l\notin{\mathbb{Z}}k$) and that $v_{k,l}\in k^{\perp}$. But then
$\hat{A}_{k}v_{k,l}\neq 0$ (indeed, $\hat{A}_{k}v_{k,l}=0$ implies that $A_{k}v_{k,l}=\binom{\hat{A}_{k}v_{k,l}}{k\cdot v_{k,l}}=0$, contradicting the invertibility of $A_{k}$), hence (since $\hat{A}_{k}v_{k,l}\in{\mathbb{Z}}^{n}\,\backslash\,\{0\}$),
$\|\hat{A}_{k}v_{k,l}\|\geq 1$.
Thus, from (38) there follows131313In general, fixed a positive integer $m$, there exists a constant $c>0$ such that
for every $w\in{\mathbb{R}}^{m}\,\backslash\,\{0\}$ and $b>0$ one has: ${\rm\,meas\,}\{y\in{\mathbb{R}}^{m}:\|y\|\leq r\ ,\ {\rm and}\ |y\cdot w|<b\}%
\leq cr^{m-1}b/\|w\|$.
$${\rm\,meas\,}(Z^{\prime}_{k}\cap B_{R_{k}})\leq{\,c\,}\,\alpha\,\sum_{l\in{%
\mathbb{Z}}^{n}_{*,{\mathtt{K}}},l\notin{\mathbb{Z}}k}\alpha{\mathtt{K}}\frac{%
\|l\|}{\|k\|}K^{n(n-2)}\leq{\,c\,}\alpha^{2}K^{n(n-2)}{\mathtt{K}}^{n+2}\|k\|^%
{-1}\ ,$$
(39)
which, together with (34), yields
(33).
3 Normal Forms
In this section we describe a normal form lemma, which allows to average out non–resonant Fourier modes of the perturbation on suitable non–resonant regions, and then apply it on $\Omega^{0}$ and $\Omega^{1}$.
We remark that such normal form lemma is not standard as, for technical reasons which will be clarified later, we shall need estimates in a complex domain very close to the initial one.
Notation
Given a set $D\subseteq\mathbb{R}^{m},$ $r>0$
we denote by $D_{r}\subseteq\mathbb{C}^{m}$ the complex open neighborhood of $D$ formed by points $z\in{\mathbb{C}}^{m}$ such that $\|z-y\|<r$, for some $y\in D$.
Given $s>0,$ we denote by ${\mathbb{T}}^{n}_{s}$ the open complex neighborhood of ${\mathbb{T}}^{n}$ given by
$$\mathbb{T}^{n}_{s}:=\{x\in\mathbb{C}^{n}\ \ :\ \ \max_{1\leq j\leq n}|{\rm Im}%
x_{j}|<s\}/2\pi\mathbb{Z}^{n}$$
Given a real–analytic function
$f:D_{r}\times\mathbb{T}^{n}_{s}\to\mathbb{C}$,
$f(y,x)=\sum_{k\in\mathbb{Z}^{n}}f_{k}(y)e^{{\rm i}k\cdot x},$
we consider the weighted sup–norm
$$|f|_{r,s}:=\sup_{k\in\mathbb{Z}^{n}}\big{(}\sup_{y\in D_{r}}|f_{k}(y)|e^{|k|s}%
\big{)}\,;$$
(40)
if the (real) domain need to be specified, we let:
$$|f|_{D,r,s}:=|f|_{r,s}\ .$$
(41)
Given $f(y,x)=\sum_{k\in\mathbb{Z}^{n}}f_{k}(y)e^{{\rm i}k\cdot x}$ and a sublattice $\Lambda$ of $\mathbb{Z}^{n}$,
we denote by $\,{\mathtt{p}}_{\Lambda}$ the projection on the Fourier coefficients in $\Lambda,$ namely
$$\,{\mathtt{p}}_{\Lambda}f:=\sum_{k\in\Lambda}f_{k}(y)e^{{\rm i}k\cdot x}\,.$$
and by $\,{\mathtt{p}}_{\Lambda}^{\perp}$ its “orthogonal” operator (projection on the Fourier modes in ${\mathbb{Z}}^{n}\,\backslash\,\Lambda$):
$$\,{\mathtt{p}}_{\Lambda}^{\perp}f:=\sum_{k\notin\Lambda}f_{k}(y)e^{{\rm i}k%
\cdot x}\,.$$
Finally, given $N>0$, we introduce the following “truncation” and “high–mode” operators
$$T_{N}f:=\sum_{|k|\leq N}f_{k}(y)e^{{\rm i}k\cdot x}\,,\qquad T_{N}^{\perp}f:=%
\sum_{|k|>N}f_{k}(y)e^{{\rm i}k\cdot x}\,.$$
(42)
For later use, we point the following elementary decay property of analytic function with vanishing low modes:
$$T_{N}f=0\ ,\ 0<\sigma<s\qquad\Longrightarrow\qquad|f|_{r,s-\sigma}\leq e^{-N%
\sigma}|f|_{r,s}\,.$$
(43)
We are, now, ready to state the Normal Form Lemma we need. In order not to introduce too many symbols we shall denote by $H=h+f$ (but without $\varepsilon$) the Hamiltonian and by $\alpha$ and $K$ (which have been already fixed in (24)) the non–resonance parameters, however the lemma applies to arbitrary $H$, $\alpha$ and $K$.
Lemma 3.1
(Normal Form Lemma)
Let $r,s,\alpha$ be positive numbers, $K\geq 2$, $D\subseteq{\mathbb{R}}^{n}$, and let $\Lambda$ be a sublattice of ${\mathbb{Z}}^{n}$.
Let $H(y,x)=h(y)+f(y,x)$ be real–analytic on
$D_{r}\times\mathbb{T}^{n}_{s}$ with $|f|_{r,s}<\infty.$
Assume that $D_{r}$ is
($\alpha$,$K$) non--resonant modulo141414In case $\Lambda=\{0\}$, one also says that $D_{r}$ is completely $(\alpha,K)$ non–resonant. $\Lambda,$ namely
$$|h^{\prime}(y)\cdot k|\geq\alpha\,,\qquad\forall\,y\in D_{r}\,,\ k\notin%
\Lambda\,,\ |k|\leq K$$
(44)
and that
$$\vartheta_{*}:=\frac{2^{9}nK^{3}}{\alpha rs}\,|f|_{r,s}<1\,.$$
(45)
Then, there exists a real–analytic
symplectic change of variables
$$\Psi:D_{r_{*}}\times\mathbb{T}^{n}_{s_{*}}\to D_{r}\times\mathbb{T}^{n}_{s}\,%
\quad{\rm with}\qquad r_{*}:=r/2\,,\ \ s_{*}:=s(1-1/K)$$
(46)
such that
$$H\circ\Psi=h+f^{\flat}+f_{*}\,,\qquad f^{\flat}:=\,{\mathtt{p}}_{\Lambda}f+{T_%
{K}^{\perp}}\,{\mathtt{p}}_{\Lambda}^{\perp}f$$
(47)
with
$$|f_{*}|_{r_{*},s_{*}}\leq 2\vartheta_{*}|f|_{r,s}\,,\qquad|T_{K}\,{\mathtt{p}}%
_{\Lambda}^{\perp}f_{*}|_{r_{*},s_{*}}\leq(\vartheta_{*}/2)^{K}|f|_{r,s}\,.$$
(48)
Proof : See Appendix B.
Remark 3.1
(i) Having information on non–resonant Fourier modes up to order $K$, the best one can do is to average out
the non–resonant Fourier modes up to order $K$, namely, to “kill” the term $T_{K}\,{\mathtt{p}}_{\Lambda}^{\perp}f$ of the Fourier expansion of the perturbation. This explains the “flat” term $f^{\flat}=\,{\mathtt{p}}_{\Lambda}f+{T_{K}^{\perp}}\,{\mathtt{p}}_{\Lambda}^{%
\perp}f$ surviving in (47) and which cannot be removed in general.
Now, think of the remainder term $f_{*}$ as
$$f_{*}=\,{\mathtt{p}}_{\Lambda}f_{*}+\big{(}{T_{K}^{\perp}}\,{\mathtt{p}}_{%
\Lambda}^{\perp}f_{*}+T_{K}\,{\mathtt{p}}_{\Lambda}^{\perp}f_{*}\big{)}\ ;$$
then, $\,{\mathtt{p}}_{\Lambda}f_{*}$ is a $(\vartheta_{*}|f|_{r,s})$–perturbation of the part in normal form (i.e., with Fourier modes in $\Lambda$), while
${T_{K}^{\perp}}\,{\mathtt{p}}_{\Lambda}^{\perp}f_{*}$ is, by (43), a term exponentially small with $K$ (see also below)
and
$T_{K}\,{\mathtt{p}}_{\Lambda}^{\perp}f_{*}$ is a very small remainder bounded by $(\vartheta_{*}/2)^{K}|f|_{r,s}$.
(ii)
The “novelty” of this lemma is that the bounds in (48) hold on the large angle domain ${\mathbb{T}}^{n}_{s_{*}}$ with $s_{*}=s(1-1/K)$. In particular, it will be important in our analysis
(precisely in order to obtain (70) below) the first estimate in (48). The drawback of the gain in angle–analyticity strip is that the power of $K$ in the smallness condition
(45) is not optimal: for example in [15] the power of $K$ is one (and $s_{*}=s/6$).
(iii) To compare with more standard formulations, such as the Normal Form Lemma in § 2 of [15], write (47) as
$$H\circ\Psi=h+g+f_{**}\qquad{\rm with}\quad\,{\mathtt{p}}_{\Lambda}g=g\ ,\quad%
\,{\mathtt{p}}_{\Lambda}f_{**}=0\ .$$
(49)
Then, $g=\,{\mathtt{p}}_{\Lambda}f+\,{\mathtt{p}}_{\Lambda}f_{*}$,
$f_{**}={T_{K}^{\perp}}\,{\mathtt{p}}_{\Lambda}^{\perp}f+\,{\mathtt{p}}_{%
\Lambda}^{\perp}f_{*}=T_{K}\,{\mathtt{p}}_{\Lambda}^{\perp}f_{*}+{T_{K}^{\perp%
}}\,{\mathtt{p}}_{\Lambda}^{\perp}(f_{*}+f)$ and the following bounds hold
$$|g-\,{\mathtt{p}}_{\Lambda}f|_{r_{*},s_{*}}\leq 2\vartheta_{*}|f|_{r,s}\ ,%
\qquad|f_{**}|_{r_{*},s/2}\leq 2e^{-Ks/2}|f|_{r,s}\ ,$$
(50)
provided
$$\vartheta_{*}\leq e^{-s}/2\ ,\qquad\qquad K\geq 2$$
(51)
(which will be henceforth assumed).
To check (50), notice that by (48) and
(43) (used with $N=K$, $s$ replaced by $s_{*}$ and $\sigma=\frac{s}{2}-\frac{s}{K}$ so that $s_{*}-\sigma=s/2$ and $e^{-K\sigma}=e^{-Ks/2}\cdot e^{s}$), one gets
$$\displaystyle|f_{**}|_{r_{*},s/2}$$
$$\displaystyle\leq$$
$$\displaystyle|T_{K}\,{\mathtt{p}}_{\Lambda}^{\perp}f_{*}|_{r_{*},s_{*}}+|{T_{K%
}^{\perp}}\,{\mathtt{p}}_{\Lambda}^{\perp}(f_{*}+f)|_{r_{*},s_{*}-\sigma}$$
$$\displaystyle\leq$$
$$\displaystyle|T_{K}\,{\mathtt{p}}_{\Lambda}^{\perp}f_{*}|_{r_{*},s_{*}}+|{T_{K%
}^{\perp}}f_{*}|_{r_{*},s_{*}-\sigma}+|{T_{K}^{\perp}}f|_{r_{*},s-s/2}$$
$$\displaystyle\leq$$
$$\displaystyle\vartheta_{*}^{K}|f|_{r,s}+e^{-Ks/2}(e^{s}\vartheta_{*}+1)|f|_{r,s}$$
$$\displaystyle\stackrel{{\scriptstyle{\rm(\ref{francescone})}}}{{\leq}}$$
$$\displaystyle 2e^{-Ks/2}|f|_{r,s}\ .$$
(iv) In our applications $\alpha$ and $K$ (or ${\mathtt{K}}$) are as in (24),
$r\stackrel{{\scriptstyle\sim}}{{>}}\alpha/K$
and $f$ is replaced by $\varepsilon f$. Thus,
$$\vartheta_{*}\sim|\log\varepsilon|^{-4(\nu-1)}\qquad\stackrel{{\scriptstyle{%
\rm(\ref{gnu})}}}{{\implies}}\qquad\vartheta_{*}^{K}\ll\varepsilon^{|\log%
\varepsilon|}\ ,$$
(52)
which is smaller than any power of $\varepsilon$ (but not exponentially small with $1/\varepsilon$).
(v)
If a set $D\subseteq{\mathbb{R}}^{n}$ is $(\alpha,K)$ non–resonant (mod $\Lambda$) for $h=\|y\|^{2}/2$, then the complex domain $D_{r}$ is $(\alpha-rK,K)$ non–resonant (mod $\Lambda$), provided151515Indeed, if $y\in D_{r}$ there exists $y_{0}\in D$ such that $\|y-y_{0}\|<r$ and
$|y_{0}\cdot k|\geq\alpha$ for all $k\in{\mathbb{Z}}^{n}\,\backslash\,\Lambda$, $|k|\leq K$. Thus, $|y\cdot k|=|y_{0}\cdot k-(y_{0}-y)\cdot k|\geq|y_{0}\cdot k|-rK\geq\alpha-rK$.
$rK<\alpha$.
We now apply the Normal Form Lemma to the Hamiltonian
$H$ in (8) in the non-resonant and simple resonant regions.
3.1 Normal form in $\Omega^{0}$ (non–resonant regime)
Recalling the definition of $\alpha$ given in (24), we set
$$r_{\{0\}}:=\frac{\alpha}{4K}=\frac{1}{4}\sqrt{\varepsilon}K^{\nu}\,.$$
(53)
We can apply Lemma 3.1 to $H$ in
(8)
with161616The set $\Omega^{0}$ is defined in (30). By Remark 3.1–(v)
and (31),
the domain $\Omega^{0}_{r_{\{0\}}}$ is $(\alpha/4,K)$ completely non–resonant.:
$$\displaystyle f\rightsquigarrow\varepsilon f\ ,\quad D\rightsquigarrow\Omega^{%
0}\ ,\quad r\rightsquigarrow r_{\{0\}}\ ,\quad\Lambda\rightsquigarrow\{0\}\ ,%
\quad\alpha\rightsquigarrow\alpha/4\ ,$$
and
$$\vartheta_{*}\rightsquigarrow\vartheta_{\{0\}}:=2^{11}n\frac{K^{3}\varepsilon|%
f|_{s}}{\alpha r_{\{0\}}s}\stackrel{{\scriptstyle\eqref{bada},\eqref{%
lapparenza},\eqref{timone}}}{{=}}\frac{2^{13}n}{sK^{2\nu-2}}\,.$$
(54)
By (24), $\vartheta_{\{0\}}<1$, provided
$\varepsilon$ is small enough depending on $s$ and $n$
(recall that $\nu>n+1$).
Thus, there exists a symplectic change of variables
$$\Psi_{\{0\}}:\Omega^{0}_{r_{\{0\}}/2}\times\mathbb{T}^{n}_{s_{*}}\to\Omega^{0}%
_{r_{\{0\}}}\times\mathbb{T}^{n}_{s}\,,\qquad s_{*}:=s(1-1/K)$$
(55)
(recall (46))
such that $H$ is transformed in
$$H_{\{0\}}:=H\circ\Psi_{\{0\}}=\|I\|^{2}/2+\varepsilon g^{\{0\}}(I)+\varepsilon
f%
^{\{0\}}_{**}(I,\varphi)\,,\qquad{\rm with}\quad\langle f^{\{0\}}_{**}\rangle=%
0\,,$$
(56)
where $\langle\cdot\rangle=\,{\mathtt{p}}_{\{0\}}\cdot$ denotes the
average with respect to the angles $\varphi$; by (50) and (9), one has:
$$\sup_{\Omega^{0}_{r_{\{0\}}}}|g^{\{0\}}-\langle f\rangle|\leq 2\vartheta_{\{0%
\}}\,,\qquad|f^{\{0\}}_{**}|_{r_{\{0\}}/2,s/2}\leq 2e^{-Ks/2}\stackrel{{%
\scriptstyle\eqref{lapparenza}}}{{=}}2\,\varepsilon^{\frac{s}{2}|\log%
\varepsilon|}$$
(57)
provided $\varepsilon$ is
small enough (depending on $s$ and $n$)
so that (51) is satisfied.
3.2 Normal form in $\Omega^{1}$ (simple resonances)
In order to construct normal forms near simple resonances, recall that $\Omega^{1}$ is the union of sets171717Recall the definitions given in (12), (29), (30).
$L_{k}Z_{k}$,
with $k\in{\mathbb{Z}}^{n}_{*,K}$, which are $(2\alpha{\mathtt{K}}/\|k\|,{\mathtt{K}})$ non–resonant modulus the one–dimensional lattice ${\mathbb{Z}}k$;
compare Proposition 2.1, (iii). Therefore, fixed $k\in\mathbb{Z}^{n}_{*,K}$, we let
$$r_{k}:=\frac{\alpha}{\|k\|}=\frac{\sqrt{\varepsilon}K^{\nu+1}}{\|k\|}\,,$$
(58)
and apply the normal form Lemma 3.1
with181818
The symbol “$a\rightsquigarrow b$” reads “with $a$ replaced by $b$”.
$$f\rightsquigarrow\varepsilon f\ ,\quad D\rightsquigarrow D^{k}:=L_{k}Z_{k}\ ,%
\quad r\rightsquigarrow r_{k}\ ,\quad\alpha\rightsquigarrow\alpha{\mathtt{K}}/%
\|k\|\ ,\quad K\rightsquigarrow{\mathtt{K}}\ ,\quad\Lambda\rightsquigarrow%
\mathbb{Z}k$$
(59)
and
$$\vartheta_{*}\rightsquigarrow\vartheta_{k}\stackrel{{\scriptstyle\eqref{limone%
}}}{{:=}}2^{9}n\frac{{\mathtt{K}}^{2}\|k\|^{2}\varepsilon|f|_{s}}{\alpha^{2}s}%
\stackrel{{\scriptstyle\eqref{bada},\eqref{lapparenza}}}{{=}}\frac{2^{9}n{%
\mathtt{K}}^{2}\|k\|^{2}}{sK^{2\nu+2}}\leq\frac{2^{9}n}{sK^{2\nu-4}}\,;$$
(60)
Notice that by Remark 3.1–(v)
and (32),
the domain $D^{k}_{r_{k}}$ is $(2\alpha{\mathtt{K}}/\|k\|-r_{k}{\mathtt{K}},{\mathtt{K}})=(\alpha{\mathtt{K}}%
/\|k\|,{\mathtt{K}})$ non–resonant modulus ${\mathbb{Z}}k$.
Again, by (24), $\vartheta_{k}<1$, provided
$\varepsilon$ is small enough (depending on $s$ and $n$).
Thus, there exists a symplectic change of variables191919Recall (46) and
Rematk 3.1, (iii).
$$\Psi_{k}:D^{k}_{r_{k}/2}\times\mathbb{T}^{n}_{s_{*}}\to D^{k}_{r_{k}}\times%
\mathbb{T}^{n}_{s}\,,\qquad s_{*}:=s(1-1/{\mathtt{K}})$$
(61)
such that $H$ in (8) is transformed in
$$H\circ\Psi_{k}=\|I\|^{2}/2+\varepsilon g^{k}(I,\varphi)+\varepsilon f^{k}_{**}%
(I,\varphi)\,,$$
(62)
where
$$g^{k}=\,{\mathtt{p}}_{{}_{k\mathbb{Z}}}g^{k}\,,\qquad\,{\mathtt{p}}_{{}_{k%
\mathbb{Z}}}f^{k}_{**}=0\,,$$
(63)
with the following estimates holding
(recall (50) and (9)):
$$|g^{k}-\,{\mathtt{p}}_{{}_{k\mathbb{Z}}}f|_{r_{k}/2,s_{*}}\leq 2\vartheta\,,%
\qquad|f^{k}_{**}|_{r_{k}/2,s/2}\leq 2e^{-{\mathtt{K}}s/2}\stackrel{{%
\scriptstyle\eqref{lapparenza}}}{{=}}2\,\varepsilon^{\frac{s}{2}|\log%
\varepsilon|^{3}}\ ,$$
(64)
where
$$\vartheta:=\frac{2^{9}n}{sK^{2\nu-4}}\ ,\qquad(\nu>n+1)\ .$$
(65)
Note that
$g^{k}$ and $\,{\mathtt{p}}_{{}_{k\mathbb{Z}}}f$ depend, effectively, only on one angle $t\in\mathbb{T}^{1}$: more precisely, setting
$$F^{k}_{j}:=f_{jk}\,,\ \ \ G^{k}_{j}(I):=g^{k}_{jk}(I)\,,\quad{\rm and}\quad F^%
{k}(t):=\sum_{j\in\mathbb{Z}}F^{k}_{j}e^{{\rm i}jt}\,,\ \ \ G^{k}(I,t):=\sum_{%
j\in\mathbb{Z}}G^{k}_{j}(I)e^{{\rm i}jt}\,,$$
(66)
we have (recall (4))
$$\,{\mathtt{p}}_{{}_{k\mathbb{Z}}}f(\varphi)=F^{k}(k\cdot\varphi)\,,\qquad g^{k%
}(I,\varphi)=G^{k}(I,k\cdot\varphi)\,.$$
(67)
We also remark that
since $f\in{\mathcal{A}}_{s}^{n}$, the functions
$F^{h}$ belong to ${\mathcal{A}}_{|h|s}^{1}$ for every $h\in\mathbb{Z}^{n}_{*},$ with
$$|F^{h}|_{|h|s}\leq|f|_{s}\stackrel{{\scriptstyle{\rm(\ref{bada})}}}{{=}}1\,,$$
(68)
Analogously, by (64)
$$|G^{k}-F^{k}|_{r_{k}/2,|k|s_{*}}\leq 2\vartheta\,.$$
(69)
For later use (compare (72) and (75) below), we point out that
$$\displaystyle\frac{1}{|f_{k}|}|G^{k}_{1}-f_{k}|_{r_{k}/2}$$
$$\displaystyle\stackrel{{\scriptstyle{\rm(\ref{cristina})}}}{{\leq}}$$
$$\displaystyle\frac{2\vartheta e^{-|k|s_{*}}}{|f_{k}|}\stackrel{{\scriptstyle%
\eqref{canarino}}}{{=}}\frac{2e^{s|k|/{\mathtt{K}}}\vartheta e^{-|k|s}}{|f_{k}%
|}\leq\frac{2\vartheta e^{(1-|k|)s}}{|f_{k}|}$$
(70)
$$\displaystyle\stackrel{{\scriptstyle\rm(P1)}}{{\leq}}$$
$$\displaystyle\frac{2\vartheta e^{s}|k|^{\frac{n+3}{2}}}{\delta}\leq\frac{2%
\vartheta e^{s}K^{\frac{n+3}{2}}}{\delta}\stackrel{{\scriptstyle\eqref{tetta}}%
}{{=}}\frac{2^{10}ne^{s}}{\delta sK^{\frac{4\nu-n-11}{2}}}\,,$$
which is small if
$\varepsilon$ is small (recall that $n\geq 2$ and $\nu>n+1$).
3.2.1 The effective potential
We now show that for $|k|$ large, the “effective potential” $G^{k}$ (defined in (66)) behaves, essentially, as a cosine; compare, in particular,
Eq. (72) below.
Recalling the definition of $T_{h}$ given in(42),
we have that
$$T_{1}F^{k}(\psi_{n}^{\prime})=f_{k}e^{{\rm i}\psi_{n}^{\prime}}+f_{-k}e^{-{\rm
i%
}\psi_{n}^{\prime}}=2|f_{k}|\cos(\psi_{n}^{\prime}+\psi_{n}^{(k)})$$
for suitable constants $\psi_{n}^{(k)}$.
Remark 3.2
We can assume, up to translation, that $\psi_{n}^{(k)}=\pi$.
So, from now on, we assume that
$$T_{1}F^{k}(\psi_{n}^{\prime})=f_{k}e^{{\rm i}\psi_{n}^{\prime}}+f_{-k}e^{-{\rm
i%
}\psi_{n}^{\prime}}=-2|f_{k}|\cos(\psi_{n}^{\prime})\,.$$
(71)
Lemma 3.2
$$\displaystyle G^{k}(I,\psi^{\prime}_{n})=2|f_{k}|\Big{(}-\cos(\psi_{n}^{\prime%
})+R^{k}(I,\psi^{\prime}_{n})\Big{)}\,,$$
$$\displaystyle|R^{k}(\sqrt{2\varepsilon/\kappa}A_{k}^{T}J^{\prime},\psi^{\prime%
}_{n})|_{D^{k},r_{k}/2,|k|s/3}\leq\frac{2|k|^{\frac{n+3}{2}}e^{-|k|s/4}}{%
\delta}+\frac{2^{10}ne^{s}}{\delta sK^{\frac{4\nu-n-11}{2}}}\,.$$
(72)
Proof Indeed,
$$\displaystyle\frac{1}{|f_{k}|}|F^{k}(\psi_{n}^{\prime})+2|f_{k}|\cos(\psi_{n}^%
{\prime})|_{|k|s/3}\stackrel{{\scriptstyle{\rm(\ref{sumo})}}}{{=}}\frac{1}{|f_%
{k}|}|T^{\perp}_{1}F^{k}|_{|k|s/3}=\frac{1}{|f_{k}|}\sup_{|j|\geq 2}|f_{kj}|e^%
{|k|s|j|/3}$$
$$\displaystyle\stackrel{{\scriptstyle\eqref{bada}}}{{\leq}}\frac{1}{|f_{k}|}%
\sup_{|j|\geq 2}e^{-2|k|s|j|/3}=\frac{1}{|f_{k}|}e^{-4|k|s/3}\stackrel{{%
\scriptstyle(P1)}}{{\leq}}\frac{|k|^{\frac{n+3}{2}}e^{-|k|s/3}}{\delta}\,.$$
(73)
Also,
$$\displaystyle\frac{1}{|f_{k}|}\Big{|}T^{\perp}_{1}\big{(}G^{k}(I,\psi^{\prime}%
_{n})-F^{k}(\psi_{n}^{\prime})\big{)}\Big{|}_{D^{k},r_{k}/2,|k|s/3}\stackrel{{%
\scriptstyle\eqref{cristina}}}{{\leq}}\frac{2\vartheta}{|f_{k}|}\sup_{|j|\geq 2%
}e^{-|j||k|(s_{*}-s/3)}$$
$$\displaystyle=\frac{2\vartheta}{|f_{k}|}e^{-2|k|(s_{*}-s/3)}\stackrel{{%
\scriptstyle(P1)}}{{\leq}}\frac{2\vartheta|k|^{\frac{n+3}{2}}e^{-|k|(2s_{*}-5s%
/3)}}{\delta}\stackrel{{\scriptstyle\eqref{canarino}}}{{\leq}}\frac{2\vartheta%
|k|^{\frac{n+3}{2}}e^{-|k|s/4}}{\delta}\,,$$
(74)
provided ${\mathtt{K}}\geq 24$.
Then (72) follows by (70),(73) and
(3.2.1).
Moreover by (70), (3.2.1), we have also that
$$|f_{k}|^{-1}|G^{k}(I,\psi^{\prime}_{n})-F^{k}(\psi^{\prime}_{n})|_{D^{k},r_{k}%
/2,|k|s/3}\leq\frac{4\vartheta e^{s}K^{\frac{n+3}{2}}}{\delta}\stackrel{{%
\scriptstyle\eqref{tetta}}}{{=}}\frac{2^{11}ne^{s}}{\delta sK^{\frac{4\nu-n-11%
}{2}}}\,,$$
(75)
which is small if
$\varepsilon$ is small (recall that $n\geq 2$ and $\nu>n+1$).
3.2.2 Rescalings
Recalling the definition of $K_{s}(\delta)$ in (5), we set
$$\delta_{k}:=\left\{\begin{array}[]{ll}1&\ \ {\rm if}\ \ |k|\leq K_{s}(\delta)%
\\
2|f_{k}|&\ \ {\rm if}\ \ |k|>K_{s}(\delta)\end{array}\right.\,.$$
(76)
Note that by (9)
$$\delta_{k}\leq 1\,.$$
(77)
Define the conformally symplectic transformation
$$\Phi^{(0)}:(I^{\prime},\varphi^{\prime})\mapsto(I,\varphi)=(\varsigma_{k}I^{%
\prime},\varphi^{\prime})\,,$$
(78)
where
$$\varsigma_{k}:=\sqrt{\frac{2\delta_{k}\varepsilon}{\kappa}}=\sqrt{\frac{2%
\delta_{k}\varepsilon}{\|k\|^{2}}}\ .$$
(79)
Then, the flow of $H\circ\Psi_{k}$ (recall (62) and (67)) is equivalent to the flow of the Hamiltonian202020See Lemma D.2, (ii) in Appendix D).
$$\frac{1}{\varsigma_{k}}H\circ\Psi_{k}\circ\Phi^{(0)}(I^{\prime},\varphi^{%
\prime})=\sqrt{\frac{\delta_{k}\varepsilon}{2\kappa}}\|I^{\prime}\|^{2}+\sqrt{%
\frac{\kappa\varepsilon}{2\delta_{k}}}\big{(}G^{k}(\varsigma_{k}I^{\prime},k%
\cdot\varphi^{\prime})+f^{k}_{**}(\varsigma_{k}I^{\prime},\varphi^{\prime})%
\big{)}\,.$$
Dividing such Hamiltonian by $\sqrt{\delta_{k}\kappa\varepsilon/2}$ (which corresponds to a time rescaling212121See, again, Lemma D.2.),
we are lead to study the Hamiltonian
$$H_{k}(I^{\prime},\varphi^{\prime}):=\frac{1}{\kappa}\|I^{\prime}\|^{2}+\frac{1%
}{\delta_{k}}\left(G^{k}(\varsigma_{k}I^{\prime},k\cdot\varphi^{\prime})+f^{k}%
_{**}(\varsigma_{k}I^{\prime},\varphi^{\prime})\right)\,,$$
(80)
which is defined on the domain
$$\displaystyle D^{{}^{\prime}k}_{r^{\prime}_{k}}\times\mathbb{T}^{n}_{s_{*}}\,,%
\ \ \ {\rm with}\qquad r^{\prime}_{k}:=\frac{r_{k}}{2\varsigma_{k}}\stackrel{{%
\scriptstyle\eqref{limone}}}{{=}}\frac{K^{\nu+1}}{\sqrt{8\delta_{k}}}\qquad{%
\rm and}$$
(81)
$$\displaystyle D^{{}^{\prime}k}:=\frac{1}{\varsigma_{k}}D^{k}\stackrel{{%
\scriptstyle\eqref{boston}}}{{=}}\frac{1}{\varsigma_{k}}L_{k}Z_{k}\stackrel{{%
\scriptstyle\eqref{caspio},\eqref{lapparenza},\eqref{capocotta}}}{{=}}L_{k}%
\left(\frac{1}{\varsigma_{k}}\hat{Z}_{k}\times\Big{(}-\frac{K^{\nu+1}}{\sqrt{2%
\delta_{k}}\|k\|},\frac{K^{\nu+1}}{\sqrt{2\delta_{k}}\|k\|}\Big{)}\right)\,.$$
Note that, by (64)
$$|\delta_{k}^{-1}f^{k}_{**}(\varsigma_{k}I^{\prime},\varphi^{\prime})|_{D^{{}^{%
\prime}k},r^{\prime}_{k},s/2}\leq\frac{2}{\delta}|k|^{\frac{n+3}{2}}e^{|k|s}e^%
{-{\mathtt{K}}s/2}\stackrel{{\scriptstyle\eqref{lapparenza}}}{{\leq}}\frac{1}{%
\delta}e^{-{\mathtt{K}}s/4}=\frac{1}{\delta}\varepsilon^{\frac{s}{4}|\log%
\varepsilon|^{3}}$$
(82)
for $\varepsilon$ small enough.
Recalling (28) we set
$$D^{{}^{\prime}k}_{\sharp}:=\frac{1}{\varsigma_{k}}L_{k}Z_{k}^{\sharp}\,.$$
(83)
Note that
$$\Phi^{(0)}\big{(}D^{{}^{\prime}k}_{\sharp}\times\mathbb{T}^{n}\big{)}=L_{k}Z_{%
k}^{\sharp}\times\mathbb{T}^{n}\,.$$
(84)
Then by (30)
$$\Phi^{(0)}\bigg{(}\bigcup_{k\in\mathbb{Z}^{n}_{*,K}}D^{{}^{\prime}k}_{\sharp}%
\times\mathbb{T}^{n}\bigg{)}=\bigcup_{k\in\mathbb{Z}^{n}_{*,K}}\Phi^{(0)}\big{%
(}D^{{}^{\prime}k}_{\sharp}\times\mathbb{T}^{n}\big{)}=\Omega^{1}\times\mathbb%
{T}^{n}\,.$$
(85)
4 The nearly–integrable structure at simple resonances
Given a bounded holomorphic function
$f:D_{r}\times\mathbb{T}^{n}_{s}\to\mathbb{C},$ with $D\subseteq\mathbb{R}^{n^{\prime}}$
we set
$$\|f\|_{D,r,s}=\|f\|_{r,s}:=\sup_{D_{r}\times\mathbb{T}^{n}_{s}}|f|\,.$$
(86)
The following relation between the two norms
$|\cdot|$ and $\|\cdot\|$ holds:
for $\sigma>0$, we have222222
Since $\sum_{k\in\mathbb{Z}^{n}}e^{-|k|\sigma}=\Big{(}\sum_{k\in\mathbb{Z}}e^{-|k|%
\sigma}\Big{)}^{n}=\Big{(}1+2\sum_{j\geq 1}e^{-j\sigma}\Big{)}^{n}=\Big{(}%
\frac{e^{\sigma}+1}{e^{\sigma}-1}\Big{)}^{n}=\coth^{n}(\sigma/2).$
$$|f|_{r,s}\leq\|f\|_{r,s}\leq\coth^{n}(\sigma/2)|f|_{r,s+\sigma}\leq(1+2/\sigma%
)^{n}|f|_{r,s+\sigma}\,.$$
(87)
4.1 A class of Morse non-degenerate functions
Let $s_{0}>0$ and let us consider a bounded holomorphic function
$$F^{0}:\mathbb{T}_{s_{0}}\to\mathbb{C}\,,\qquad\text{with}\qquad\|F^{0}\|_{s_{0%
}}<\infty\,.$$
(88)
Definition 4.1
Let $\beta,M>0.$ We say that $F^{0}$ as in (88) is
$(\beta,M)$-Morse-non-degenerate if
$\|F^{0}\|_{s_{0}}\leq M$ and
$$\displaystyle\min_{x}\ \big{(}|(F^{0})^{\prime}(x)|+|(F^{0})^{\prime\prime}(x)%
|\big{)}$$
$$\displaystyle\geq$$
$$\displaystyle\beta\,,$$
(89)
$$\displaystyle\min_{1\leq i<j\leq 2N}|F^{0}(x^{0}_{i})-F^{0}(x^{0}_{j})|$$
$$\displaystyle\geq$$
$$\displaystyle\beta\,,$$
(90)
where $x^{0}_{i}$, $1\leq i\leq 2N$
are the critical points of $F^{0}$
in $(-\pi,\pi].$
We note that, by (89),
the function $F^{0}$ has only non-degenerate critical points:
let us say
$N$ minima: $x^{0}_{2j-1},$
and $N$ maxima: $x^{0}_{2j},$
in $(-\pi,\pi],$ for some integer $N\geq 1$ and $1\leq j\leq N.$
It is immediate to realize that
$N$
is uniformly bounded by a constant depending only on
$s_{0},M$
and the minimum appearing in (89).
The corresponding critical energies are
$$E^{0}_{i}:=F^{0}(x^{0}_{i})\,,\qquad 1\leq i\leq 2N\,.$$
(91)
By (90), $E^{0}_{i}$ are all different.
Definition 4.2
We say that $F^{0}$ as in (88) is
$\gamma$-cosine-like232323Actually we should say minus-cosine-like if
$$\|F^{0}(x)+\cos x\|_{s_{0}}\leq\gamma\,,\qquad\text{for some}\qquad 0<\gamma%
\leq\frac{1}{4}\min\{1,s_{0}^{2}\}\,.$$
Lemma 4.1
If $F^{0}$ is
$\gamma$-cosine-like, then it is also
$(\beta,M)$-Morse-non-degenerate
with
$$\beta=1/4\,,\qquad M=\gamma+\cosh s_{0}\leq\frac{1}{4}+\cosh s_{0}\,.$$
Moreover $F^{0}$
has only two non-degenerate critical points
(a maximum and a minimum).
Proof We have, by Cauchy estimates,
$$|(F^{0})^{\prime}(x)|+|(F^{0})^{\prime\prime}(x)|\geq|\sin x|+|\cos x|-\frac{%
\gamma}{s_{0}}-2\frac{\gamma}{s_{0}^{2}}\geq 1-\frac{\gamma}{s_{0}}-2\frac{%
\gamma}{s_{0}^{2}}\geq\frac{1}{4}\,.$$
We can choose $M$ as above since
$\|\cos x\|_{s_{0}}=\cosh s_{0}.$
Regarding the last sentence
we note that for $x\in(-\pi,\pi]$
we have only two critical points,
a minimum in $(-\pi/6,\pi/6)$
and a maximum in $(-\pi,-5\pi/6)\cup(5\pi/6,\pi].$ Indeed we have that,
setting $g(x):=F^{0}(x)+\cos x$,
$(F^{0})^{\prime}(x)=\sin x+g^{\prime}(x),$ so that
$$(F^{0})^{\prime}(x)=\sin x+g^{\prime}(x)\geq\sin x-\gamma/s_{0}\geq\sin x-1/4\,.$$
(92)
This implies that
$(F^{0})^{\prime}(\pi/6)\geq 1/4,$
$(F^{0})^{\prime}(-\pi/6)\leq-1/4.$
Then, by continuity, there exists a critical
point of $F^{0}$ in $(-\pi/6,\pi/6).$
Moreover such point is a minimum and there are no other critical points in $(-\pi/6,\pi/6)$ since there $F^{0}$
is strictly convex:
$$(F^{0})^{\prime\prime}(x)=\cos x+g^{\prime\prime}(x)\geq\sqrt{3}/2-2\gamma/s_{%
0}^{2}\geq\sqrt{3}/2-1/2>0\,.$$
Similarly in $(-\pi,-5\pi/6)\cup(5\pi/6,\pi]$ there is only one critical point,
which is a maximum.
Finally, by (92),
$(F^{0})^{\prime}(x)\geq 1/4$ for $x\in[\pi/6,5\pi/6]$
and, analogously,
$(F^{0})^{\prime}(x)\leq-1/4$ for
$x\in[-5\pi/6,-\pi/6];$
so that there are no other critical points.
4.2 The Structure Theorem
We start introducing a parameter
$$\theta\geq 0\,,$$
(93)
that will be chosen in Section 7 as a function of
$\varepsilon.$
We also say that a function $\theta\to E(\theta)\subseteq\mathbb{R}^{m}$
is decreasing w.r.t. $\theta$ if
$\theta\leq\theta^{\prime}$ implies $E(\theta)\supseteq E(\theta^{\prime}).$
In light of (80) we are now
going to study the
behavior of the “effective Hamiltonian”
close to a simple resonance identified by
a fixed $k\in\mathbb{Z}^{n}_{*},$ namely we
are considering Hamiltonian of the form
$${\mathcal{H}}(I^{\prime},\varphi^{\prime}):=\frac{1}{\kappa}\|I^{\prime}\|^{2}%
+\mathfrak{G}(I^{\prime},k\cdot\varphi^{\prime})\,.$$
(94)
Assumptions on the “effective potential” $\mathfrak{G}$
Consider the parameters
$$s_{0}\,,\,r_{0}>\,0\,,\ \ r^{\prime}:=cn|k|_{\infty}r_{0}\,,$$
(95)
$c>1$ being the constant defined in (17), which depends only on $n$.
We will make the following assumptions:
(A1) There exists
$$F^{0}\in{\mathcal{A}}^{1}_{s_{0}}$$
(96)
such that
$$\|\mathfrak{G}-F^{0}\|_{{\mathcal{D}},r^{\prime},s_{0}}\leq\eta_{*}\,,$$
(97)
with242424Recall the definition of
$L_{k}$ given in (12).
$${\mathcal{D}}=L_{k}\big{(}\hat{D}\times(-R_{0},R_{0})\big{)}\,,\ \ \ \hat{D}%
\subset\mathbb{R}^{n-1}\,,\qquad R_{0}\geq 4+\cosh s_{0}\,;$$
(98)
$$\displaystyle\textbf{(A2)}\ F^{0}\ \text{is}\ (\beta,M)\text{-Morse-non-%
degenerate with}\ \ \max\{2\sqrt{M},4\}\leq R_{0}\,.$$
(99)
In alternative to (A2)
we will assume, when it holds, the following
stronger (recall Lemma 4.1 and (98)) hypothesis:
$$\displaystyle\textbf{(A3)}\ F^{0}\ \text{is}\ \gamma\text{-cosine-like
with}\ \ \gamma\leq\mathfrak{c}(s_{0}):=\mathfrak{c}_{*}\min\{1,s_{0}^{4}\}\,,$$
(100)
where $0<\mathfrak{c}_{*}\leq 1/4$ is a suitably small positive
constant to be chosen below
(see Lemma C.5).
Recalling (98) we set
$${\mathcal{D}}_{\sharp}:=L_{k}\big{(}\hat{D}\times(-R_{0}/2,R_{0}/2)\big{)}\,,%
\ \ \ \hat{D}\subset\mathbb{R}^{n-1}\,.$$
(101)
Theorem 4.1 (Integrable structure at simple resonances)
$\phantom{.}$
Part I. Assume that $\mathfrak{G}$ in (94) satisfies (A1)
and (A2) or (A3).
Then there exist a suitably large constant $\mathtt{c}>1,$ which, when (A3) holds,
depends only on $n,s_{0},r_{0}$,
otherwise it depends also on252525In any case it is independent of $\hat{D}.$ $F^{0},$
such that if
$$\eta_{*}\leq 1/\mathtt{c}$$
(102)
the following holds.
For every262626$2N$ being the number of critical points of $F^{0}.$
Note that, when (A3) holds,
$N=1$
by Lemma 4.1.
$0\leq i\leq 2N$ there exist
i) disjoint open subsets272727The set $\bigcup_{i}\mathfrak{C}^{i}(0)$ contains ${\mathcal{D}}_{\sharp}\times\mathbb{T}^{n}$
up to the connected components of the critical energy level containing critical points.
$\mathfrak{C}^{i}(\theta)\subseteq{\mathcal{D}}\times\mathbb{T}^{n}$
decreasing w.r.t. $\theta$, with
$${\rm\,meas\,}\Big{(}\big{(}{\mathcal{D}}_{\sharp}\times\mathbb{T}^{n}\big{)}%
\setminus\bigcup_{0\leq i\leq 2N}\mathfrak{C}^{i}(\theta)\Big{)}\leq\mathtt{c}%
\theta|\ln\theta|\,;$$
(103)
ii) open subsets $\mathtt{B}^{i}(\theta)\subseteq\mathbb{R}^{n}$, decreasing w.r.t. $\theta$ with282828
Where $c$ is the constant defined in (17).
$${\rm diam}\big{(}\mathtt{B}^{i}(0)\big{)}\leq 2c\big{(}R_{0}+{\rm diam}(\hat{D%
})\big{)}\,,\qquad\forall\,0\leq i\leq 2N\,;$$
(104)
iii)
a symplectomorphism
$$\Psi^{i}\ :\ (p,q)\in\mathtt{B}^{i}(0)\times\mathbb{T}^{n}\ \to\ \mathfrak{C}^%
{i}(0)\ni(I^{\prime},\varphi^{\prime})\,,\qquad\mbox{with}\quad\Psi^{i}\Big{(}%
\mathtt{B}^{i}(\theta)\times\mathbb{T}^{n}\Big{)}=\mathfrak{C}^{i}(\theta)\,,%
\ \ \ \forall\,\theta\geq 0\,,$$
(105)
with holomorphic extension
$$\Psi^{i}\ :\ \big{(}\mathtt{B}^{i}(\theta)\big{)}_{\rho_{*}}\times\mathbb{T}^{%
n}_{\sigma_{*}}\ \to\ {\mathcal{D}}_{r^{\prime}}\times\mathbb{T}^{n}_{s_{0}}\,%
,\qquad\mbox{with}\quad\rho_{*}:=\frac{\theta}{\mathtt{c}|k|_{\infty}^{n-1}}\,%
,\quad\ \sigma_{*}:=\frac{1}{{\mathtt{c}}|k|_{\infty}^{n-1}|\log\theta|}\,,$$
(106)
such that
$${\mathcal{H}}\circ\Psi^{i}(p,q)=:h^{(i)}(p)\,.$$
(107)
Moreover
$$\|\partial_{pp}h^{(i)}\|_{\mathtt{B}^{i}(\theta),\rho_{*}}\leq\mathtt{c}/%
\theta\,,\qquad\text{for }\ \ \ 0\leq i\leq 2N\,.$$
(108)
Part II.
Assuming
$$0<\eta_{*}\leq 1/\mathtt{c}\|k\|^{2n}\,,$$
(109)
we have that
for every
$$0<\mu\leq 1/\mathtt{c}\|k\|^{2n}\,,$$
(110)
there exist
open subsets $\tilde{\mathtt{B}}^{i}(\mu)\subseteq\mathtt{B}^{i}(0)$, decreasing w.r.t. $\mu,$
such that
$${\rm\,meas\,}(\mathtt{B}^{i}(0)\,\backslash\,\tilde{\mathtt{B}}^{i}(\mu))\leq%
\mathtt{c}\|k\|^{4n}\mu^{1/\mathtt{c}}$$
(111)
and
$$\left|{\rm det}\left(\partial_{pp}h^{(i)}(p)\right)\right|>\mu\,,\qquad\forall%
\,\ 0\leq i\leq 2N\,,\ \ |k|\leq K\,,\qquad\forall\ p\in\tilde{\mathtt{B}}^{i}%
(\mu)\ .$$
(112)
The following two sections are devoted to the proof
of Theorem 4.1 part I and part
II,
respectively.
5 Proof of Part I of the Structure Theorem
In this section we will prove
Theorem 4.1 part I.
5.1 Critical points and critical energies of
the “unperturbed potential” $F^{0}$
We order the critical points of $F^{0}$
(recall (96))
in the following way (where292929Similarly we will set $E_{0}^{0}:=E_{2N}^{0}$ below.
$x^{0}_{0}:=x^{0}_{2N}-2\pi$)
$$x^{0}_{0}<x^{0}_{1}<x^{0}_{2}<\ldots<x^{0}_{2N-1}<x^{0}_{2N}\,,\qquad x^{0}_{2%
j-1}\ {\rm minimum}\,,\ \ \ x^{0}_{2j}\ {\rm maximum}\,,\ \ 1\leq j\leq N\,.$$
(113)
Fix $1\leq j\leq N$ and consider a minimum point $x^{0}_{2j-1},$
thanks to (89) the function $F^{0}$ is strictly increasing, resp. strictly decreasing, in the interval $[x^{0}_{2j-1},x^{0}_{2j}],$
resp. $[x^{0}_{2j-2},x^{0}_{2j-1}],$
then we can invert $F^{0}$ on the above intervals
obtaining two functions
$$X^{0}_{2j}:[E^{0}_{2j-1},E^{0}_{2j}]\to[x^{0}_{2j-1},x^{0}_{2j}]\qquad{\rm and%
}\qquad X^{0}_{2j-1}:[E^{0}_{2j-1},E^{0}_{2j-2}]\to[x^{0}_{2j-2},x^{0}_{2j-1}]$$
(114)
such that
$$F^{0}(X^{0}_{i}(E))=E\,,\qquad X^{0}_{i}(F^{0}(\psi_{n}))=\psi_{n}\,,\qquad%
\forall\,1\leq i\leq 2N\,.$$
Note that $X^{0}_{i}$ is increasing, resp. decreasing, if
$i$ is even, resp. odd.
Set
$$\displaystyle E^{(i),0}_{-}:=E^{0}_{i}\,,\qquad E^{(2j-1),0}_{+}:=\min\{E_{2j-%
2}^{0},E_{2j}^{0}\}\ \ {\rm for}\ \ 1\leq j\leq N\,,$$
$$\displaystyle E^{(2j),0}_{+}:=\min\{E^{0}_{2j_{-}},E^{0}_{2j_{+}}\}\ \ {\rm for%
}\ \ 1\leq j<N\,,\qquad E^{(2N),0}_{+}=E^{(0),0}_{+}=+\infty\,,$$
(115)
where
$$j_{-}:=\max\{i<j\ \ {\rm s.t.}\ \ E^{0}_{2i}>E^{0}_{2j}\}\,,\qquad j_{+}:=\min%
\{i>j\ \ {\rm s.t.}\ \ E^{0}_{2i}>E^{0}_{2j}\}\,.$$
(116)
5.2 The slow angle
Let us, now, perform the linear symplectic change of variables
$\Phi^{(1)}:(J^{\prime},\psi^{\prime})\mapsto(I^{\prime},\varphi^{\prime})$
generated by
$S(J^{\prime},\varphi^{\prime}):=A_{k}\varphi^{\prime}\cdot J^{\prime},$:
$$\Phi^{(1)}:(J^{\prime},\psi^{\prime})\mapsto(I^{\prime}\varphi^{\prime})=(A_{k%
}^{T}J^{\prime},A_{k}^{-1}\psi^{\prime})=(kJ^{\prime}_{n}+\hat{A}_{k}^{T}\hat{%
J}^{\prime},A_{k}^{-1}\psi^{\prime})\,.$$
(117)
Note that $\Phi^{(1)}$ does not mix actions with angles, its projection on the angles is a diffeomorphism of $\mathbb{T}^{n}$ onto $\mathbb{T}^{n}$,
and, most relevantly,
$$\psi^{\prime}_{n}=k\cdot\varphi^{\prime}$$
(118)
is the canonical angle associated to the one-dimensional “secular system” near the simple resonance $\{y\cdot k=0\}$
(i.e., the one-dimensional system governed by the Hamiltonian obtained disregarding the small term $f_{**}^{k}$ in (80)).
In the $(J^{\prime},\psi^{\prime})$–variables, we have:
$$\tilde{H}(J^{\prime},\psi^{\prime}):={\mathcal{H}}\circ\Phi^{(1)}(J^{\prime},%
\psi^{\prime})=\frac{1}{\kappa}\|A_{k}^{T}J^{\prime}\|^{2}+\mathfrak{G}(A_{k}^%
{T}J^{\prime},\psi^{\prime}_{n})\,.$$
(119)
As for the $(J^{\prime},\psi^{\prime})$–domain, we see that
the real $J^{\prime}$–domain is given by
$$\tilde{D}:=A_{k}^{-T}{\mathcal{D}}=A_{k}^{-T}L_{k}\big{(}\hat{D}\times(-R_{0},%
R_{0})\big{)}\stackrel{{\scriptstyle{\rm(\ref{LPDbis})}}}{{=}}U_{k}\big{(}\hat%
{D}\times(-R_{0},R_{0})\big{)}\,,$$
(120)
with $U_{k}$ in (15).
We also set
$$\tilde{D}_{\sharp}:=U_{k}\big{(}\hat{D}\times(-R_{0}/2,R_{0}/2)\big{)}\,.$$
(121)
Then, if we choose
$$\tilde{r}:=\frac{r^{\prime}}{n|k|_{\infty}}\ ,\qquad\tilde{s}:=\frac{s_{0}}{c|%
k|_{\infty}^{n-1}}\,,$$
(122)
(for a suitable $c$ depending only on $n$)
we see that $\Phi^{(1)}$ has holomorphic extension on the complex domain
(recall Lemma D.4)
$$\Phi^{(1)}:\tilde{D}_{\tilde{r}}\times\mathbb{T}^{n}_{\tilde{s}}\to{\mathcal{D%
}}_{r^{\prime}}\times\mathbb{T}^{n}_{s_{0}}$$
(123)
indeed: by (10), $\|A_{k}^{T}\|=\|A_{k}\|\leq n|k|_{\infty}$,
so that $\|A_{k}\|\tilde{r}\leq r^{\prime}$, while, for every $1\leq i\leq n,$
$$\sum_{1\leq j\leq n}|(A_{k}^{-1})_{ij}|\leq n|A_{k}^{-1}|_{\infty}\stackrel{{%
\scriptstyle\eqref{atlantide}}}{{\leq}}c|k|_{\infty}^{n-1}\,,$$
Note that by (97)
$$\|\mathfrak{G}(A_{k}^{T}J^{\prime},\psi^{\prime}_{n})-F^{0}(\psi^{\prime}_{n})%
\|_{\tilde{D},\tilde{r},s_{0}}\leq\eta_{*}\,.$$
(124)
5.3 The auxiliary Hamiltonian
A crucial role will be played by the
auxiliary Hamiltonian
$$H^{*}:=(J_{n}^{\prime\prime})^{2}+F^{*}(J^{\prime\prime},\psi^{\prime\prime}_{%
n})\,,\qquad\quad\text{where}\qquad F^{*}(J^{\prime\prime},\psi^{\prime\prime}%
_{n}):=\mathfrak{G}(L_{k}J^{\prime\prime},\psi^{\prime\prime}_{n})\,.$$
(125)
This Hamiltonian represents a
one dimensional mechanical system depending on the parameter $\hat{J}^{\prime\prime}.$
The relation between $\tilde{H}$
(defined in (119))
and $H^{*}$ is the following:
recalling (12),(14) and the change $J^{\prime}=U_{k}J^{\prime\prime}$, $U_{k}$ defined in (15),
it results
$$H^{*}(J^{\prime\prime},\psi^{\prime\prime})=\tilde{H}(U_{k}J^{\prime\prime},%
\psi_{n}^{\prime\prime})-\frac{1}{\kappa}\|\,{\mathtt{p}}_{k}^{\perp}\hat{A}_{%
k}^{T}\hat{J}^{\prime\prime}\|^{2}\,.$$
(126)
Recalling (120), the potential $F^{*}$
in (125) is defined
for
$$(J^{\prime\prime},\psi_{n}^{\prime\prime})\in D_{r_{0}}\times\mathbb{T}^{1}_{s%
_{0}}\,,\qquad\text{where}\qquad D:=\hat{D}\times(-R_{0},R_{0})\,,$$
(127)
(95),(122) and (17).
Note that, by (124),
$$\|F^{*}-F^{0}\|_{D,r_{0},s_{0}}\leq\eta_{*}\,.$$
(128)
5.4 A
special group
of symplectic transformations
In the following, symplectic transformations will have a special form, namely, they will belong to a special group
${\cal G}$, formed by symplectic transformations $\Phi$ satisfying
$$\hat{I}=\hat{J}\,,\quad I_{n}=I_{n}(J,\psi_{n})\,,\quad\hat{\varphi}=\hat{\psi%
}+\hat{\varphi}^{\prime}(J,\psi_{n})\,,\quad\varphi_{n}=\varphi_{n}(J,\psi_{n}%
)\,,$$
(129)
where, in general, $\varphi,\psi$ may belong either to $\mathbb{T}^{n}$ or to $\mathbb{R}^{n}$.
For a transformation $\Phi$ as in (129)
we let
$\check{\Phi}$
denote the map
$$\check{\Phi}(J,\psi_{n}):=\big{(}\hat{J},I_{n}(J,\psi_{n}),\varphi_{n}(J,\psi_%
{n})\big{)}\,.$$
(130)
Some general properties of ${\cal G}$ are discussed in Appendix D.
5.5 An intermediate transformation
To simplify geometry, we now introduce a symplectic transformation that removes the dependence upon
$J_{n}$ from the potential.
Since (102) holds303030
Note that this implies
(349).
, by Lemma D.10 in Appendix D,
one can find a symplectomorphism
$\Phi^{(2{\rm bis})}\in\mathcal{G}$
satisfying
$$\Phi^{(2{\rm bis})}:(J,\psi)\to(J^{\prime\prime},\psi^{\prime\prime})\,,\qquad%
\hat{J}^{\prime\prime}=\hat{J},\ \ J_{n}^{\prime\prime}=J_{n}+a_{*}(\hat{J},%
\psi_{n}),\ \ \hat{\psi}^{\prime\prime}=\hat{\psi}+b_{*}(\hat{J},\psi_{n}),\ %
\psi_{n}^{\prime\prime}=\psi_{n}\,,$$
(131)
with (taking $\mathtt{c}$ large enough)
$$\Phi^{(2{\rm bis})}:\ D_{r_{0}/2}\times\mathbb{T}^{n}_{s_{0}/2}\ \to\ D_{r_{0}%
}\times\mathbb{T}^{n}_{s_{0}}\,.$$
(132)
and
$$\|a_{*}\|_{\hat{D}_{k},r_{0},s_{0}}\leq 4\eta_{*}/r_{0}\,,\ \|b_{*}\|_{\hat{D}%
_{k},r_{0}/2,s_{0}}\leq(16\pi+8)\eta_{*}/r_{0}^{2}\,,$$
(133)
and such that
$$\displaystyle H_{\rm pend}(J,\psi_{n}):=H^{*}\circ\Phi^{(2{\rm bis})}=\big{(}1%
+b(J,\psi_{n})\big{)}\big{(}J_{n}-J_{n}^{*}(\hat{J})\big{)}^{2}+F(\hat{J},\psi%
_{n})\,,$$
(134)
$$\displaystyle F(\hat{J},\psi_{n})=F^{0}(\psi_{n})+G(\hat{J},\psi_{n})\,.$$
Furthermore (see (D.10) below)
$$\displaystyle\|J_{n}^{*}\|_{\hat{D},r_{0}}\leq 2\eta_{*}/r_{0}\leq\eta r_{0}\,%
,\qquad\|G\|_{\hat{D},r_{0},s_{0}}\leq\left(1+4/r_{0}^{2}\right)\eta_{*}\leq%
\eta\,,$$
$$\displaystyle\|(1+|J_{n}-J_{n}^{*}(\hat{J})|)b(J,\psi_{n})\|_{D,r_{0}/2,s_{0}}%
\leq\left(4+\frac{34}{r_{0}^{2}}\right)\eta_{*}\leq\eta\,,$$
(135)
$$\displaystyle\||J_{n}-J_{n}^{*}(\hat{J})|\partial_{J_{n}}b(J,\psi_{n})\|_{D,r_%
{0}/2,s_{0}}\leq\frac{48}{r_{0}^{2}}\eta_{*}\leq\eta\,,$$
where
$$\eta:=\left(4+\frac{48}{r_{0}^{2}}\right)\eta_{*}$$
(136)
Notations
For brevity we introduce the following notations.
$$\mathfrak{p}:=(n,r_{0},s_{0},\beta,M)\,.$$
(137)
We say that
$$a\lessdot b\ \ \ {\rm if}\ \ \ \exists\ C=C(\mathfrak{p})>0\ \ \ {\rm s.t.}\ %
\ \ a\leq Cb\,.$$
(138)
We also say that, given $F^{0}$
satisfying (A1) and (A2),
$$a\lessdot_{F^{0}}b\ \ \ {\rm if}\ \ \ \exists\ C=C(F^{0})>0\ \ \ {\rm s.t.}\ %
\ \ a\leq Cb\,.$$
(139)
Remark 5.1
Note that
$a\lessdot b$ implies
$a\lessdot_{F^{0}}b.$
Note also that if (A3) holds,
then, by Lemma 4.1,
$\mathfrak{p}$ reduces to $(n,r_{0},s_{0})$
Let us assume that
$$\eta\leq\eta_{0}=\eta_{0}(\mathfrak{p})\,,$$
(140)
for a suitable small $\eta_{0}.$
By (89), for $\eta_{0}$ small enough,
we can continue the critical points $x^{0}_{j}$ (defined in (113)),
resp. critical energies $E^{0}_{j},$ of $F^{0}$ obtaining
critical points
$x_{j}(\hat{J}),$
resp. critical energies $E_{j}(\hat{J}),$
of $F(\hat{J},\cdot),$
solving the implicit function equation313131To find $x_{j}=:x^{0}_{j}+\chi_{j}$ we have to solve (for every $\hat{J}$) the equation $\partial_{\psi_{n}}F(\hat{J},x^{0}_{j}+\chi_{j})=0.$ Since
$\partial_{\psi_{n}}F(\hat{J},x^{0}_{j}+\chi_{j})=\partial^{2}_{\psi_{n}\psi_{n%
}}F^{0}(x^{0}_{j})\chi_{j}+O(\chi_{j}^{2})+O(\eta_{0})$
the equation reduces, by (89), to find $\chi_{j}$ solving the fixed point
$\chi_{j}=O(\chi_{j}^{2})+O(\eta_{0})$.
Moreover since $F$ is an analytic function of $\hat{J}\in\hat{D}_{r_{0}}$
the same holds for $\chi_{j}(\hat{J}).$
$$\partial_{\psi_{n}}F(\hat{J},x_{j}(\hat{J}))=0$$
(141)
and then evaluating
$$F(\hat{J},x_{j}(\hat{J}))=:E_{j}(\hat{J})\,,$$
(142)
respectively.
Note that
$x_{j}(\hat{J}),$
and $E_{j}(\hat{J})$ are analytic functions of $\hat{J}\in\hat{D}_{r_{0}}.$
By (89)
$$\sup_{\hat{J}\in\hat{D}_{r_{0}}}|x_{j}(\hat{J})-x_{j}^{0}|\,,\ \sup_{\hat{J}%
\in\hat{D}_{r_{0}}}|E_{j}(\hat{J})-E_{j}^{0}|\ \lessdot\,\eta\,.$$
(143)
Therefore we note that
$x_{j}(\hat{J})$ and
$E_{j}(\hat{J})$ maintain the same order
of $x^{0}_{j}$ and $E^{0}_{j}.$
In particular, recalling 5.1
$$\displaystyle E^{(i)}_{-}(\hat{J}):=E_{i}(\hat{J})\,,\qquad E^{(2j-1)}_{+}(%
\hat{J}):=\min\{E_{2j-2}(\hat{J}),E_{2j}(\hat{J})\}\ \ {\rm for}\ \ 1\leq j%
\leq N\,,$$
$$\displaystyle E^{(2j)}_{+}(\hat{J}):=\min\{E_{2j_{-}}(\hat{J}),E_{2j_{+}}(\hat%
{J})\}\ \ {\rm for}\ \ 1\leq j<N\,,\qquad E^{(2N)}_{+}(\hat{J})=E^{(0)}_{+}(%
\hat{J})=+\infty\,,$$
(144)
where
$j_{\pm}$ were defined in (116).
By (89),(90) for $\eta_{0}$ small enough we get
$$\inf_{\hat{J}\in\hat{D}_{r_{0}}}\min_{\psi_{n}\in\mathbb{R}}\Big{(}|\partial_{%
\psi_{n}}F(\hat{J},\psi_{n})|+|\partial_{\psi_{n}\psi_{n}}F(\hat{J},\psi_{n})|%
\Big{)}\geq\frac{\beta}{2}\,,\qquad\inf_{\hat{J}\in\hat{D}_{r_{0}}}\min_{i\neq
j%
}|E_{j}(\hat{J})-E_{i}(\hat{J})|\geq\frac{\beta}{2}\,.$$
(145)
Reasoning as above, by (145), for
$\hat{J}\in\hat{D}$ (namely $\hat{J}$ real) and $\eta_{0}$ small enough,
we can “continue” also the functions $X^{0}_{2j},X^{0}_{2j-1},$
obtaining
$$\displaystyle X_{2j}(\cdot,\hat{J})$$
$$\displaystyle:$$
$$\displaystyle\big{[}E_{2j-1}(\hat{J}),E_{2j}(\hat{J})\big{]}\to\big{[}x_{2j-1}%
(\hat{J}),x_{2j}(\hat{J})\big{]}\,,$$
$$\displaystyle X_{2j-1}(\cdot,\hat{J})$$
$$\displaystyle:$$
$$\displaystyle\big{[}E_{2j-1}(\hat{J}),E_{2j-2}(\hat{J})\big{]}\to\big{[}x_{2j-%
2}(\hat{J}),x_{2j-1}(\hat{J})\big{]}\,,$$
(146)
solving the implicit function equations
$$F\big{(}\hat{J},X_{i}(E,\hat{J})\big{)}=E\,,\qquad X_{i}\big{(}F(\hat{J},\psi_%
{n}),\hat{J}\big{)}=\psi_{n}\,,\qquad\forall\,1\leq i\leq 2N\,.$$
(147)
Note that $X_{i}$ is increasing, resp. decreasing (as a function of $E$), if
$i$ is even, resp. odd.
Note also that
$$\partial_{E}X_{i}(E,\hat{J})=1/\partial_{\psi_{n}}F\big{(}\hat{J},X_{i}(E,\hat%
{J})\big{)}$$
(148)
and
$$\displaystyle X_{2j-1}(E_{2j-2}(\hat{J}),\hat{J})=x_{2j-2}(\hat{J})\,,\qquad X%
_{2j-1}(E_{2j-1}(\hat{J}),\hat{J})=x_{2j-1}(\hat{J})\,,$$
$$\displaystyle X_{2j}(E_{2j-1}(\hat{J}),\hat{J})=x_{2j-1}(\hat{J})\,,\qquad X_{%
2j}(E_{2j}(\hat{J}),\hat{J})=x_{2j}(\hat{J})\,.$$
(149)
5.6 The integrating transformation
Proposition 5.1
Let $H_{\rm pend}$ be as in (134), (5.5), (102).
There exist a suitably large constant $C>1,$ which, when (A3) holds,
depends only on $n,s_{0},r_{0}$, otherwise depend also on $F^{0}$
(introduced in (96))
such that, if
$$\eta\leq 1/C\,,\qquad\qquad 0\leq\theta\leq 1/C\,,$$
(150)
the following holds.
There exist:
i) disjoint open connected sets
$$\mathcal{C}^{i}(\theta)=\check{\mathcal{C}}^{i}(\theta)\times\mathbb{T}^{n-1}%
\,,\quad 0\leq i\leq 2N\,,$$
(151)
decreasing w.r.t. $\theta$ and
satisfying (recall (127))
$$\hat{D}\times(-R_{0}/2,R_{0}/2)\times\mathbb{T}^{n}\subset\bigcup_{0\leq i\leq
2%
N}\overline{\mathcal{C}^{i}(0)}\subset\hat{D}\times(-R_{0},R_{0})\times\mathbb%
{T}^{n}\,,$$
(152)
$${\rm\,meas\,}\left(\Big{(}\hat{D}\times(-R_{0}/2,R_{0}/2)\times\mathbb{T}^{n}%
\Big{)}\ \setminus\ \bigcup_{0\leq i\leq 2N}\mathcal{C}^{i}(\theta)\right)\leq
C%
\theta|\log\theta|\,;$$
(153)
ii) open connected sets $\mathfrak{P}^{i}(\theta)$ decreasing w.r.t. $\theta$
with
$${\rm diam}\,\big{(}\mathfrak{P}^{i}(0)\big{)}\leq 2\big{(}R_{0}+{\rm diam}(%
\hat{D})\big{)}\,,\qquad\forall\,0\leq i\leq 2N\,;$$
(154)
iii) symplectomorphisms
$$\mathfrak{F}^{i}:\mathfrak{P}^{i}(0)\times\mathbb{T}^{n}\ni(P,Q)\to(J,\psi)\in%
\mathcal{C}^{i}(0)$$
(155)
in $\mathcal{G}$
such that323232Recall the notation introduced in (130).
$$\mathfrak{F}^{i}(\mathfrak{P}^{i}(\theta)\times\mathbb{T}^{n})=\mathcal{C}^{i}%
(\theta)\ ,\qquad\check{\mathfrak{F}}^{i}(\mathfrak{P}^{i}(\theta)\times%
\mathbb{T}^{1})=\check{\mathcal{C}}^{i}(\theta)$$
(156)
with $\check{\mathfrak{F}}^{i}$ injective.
Furthermore, $\mathfrak{F}^{i}$ have the following form
$$\displaystyle\hat{J}=\hat{P}\,,\quad J_{n}=\mathtt{v}^{i}(P,Q_{n})\,,\quad\hat%
{\psi}=\hat{Q}+\mathtt{z}^{i}(P,Q_{n})\,,\quad\psi_{n}=\mathtt{u}^{i}(P,Q_{n})%
\,,\qquad\text{for}\ 1\leq i\leq 2N-1\,,$$
(157)
$$\displaystyle\hat{J}=\hat{P}\,,\quad J_{n}=\mathtt{v}^{i}(P,Q_{n})\,,\quad\hat%
{\psi}=\hat{Q}+\mathtt{z}^{i}(P,Q_{n})\,,\quad\psi_{n}=Q_{n}+\mathtt{u}^{i}(P,%
Q_{n})\,,\qquad\,\,\text{for}\ i=0,2N\,,$$
(158)
with $\mathtt{v}^{i},\mathtt{z}^{i},\mathtt{u}^{i}$,
$2\pi$-periodic in $Q_{n},$
$|\mathtt{z}^{i}|\leq C\eta$
and, for $i=0,2N,$
$\sup|\partial_{Q_{n}}\mathtt{u}^{i}|<1$;
for $\theta>0,$ $\mathfrak{F}^{i}$ have holomorphic
extension
$$\mathfrak{F}^{i}:\big{(}\mathfrak{P}^{i}(\theta)\big{)}_{\rho}\times\mathbb{T}%
^{n}_{\sigma}\ \to\ D_{r_{0}}\times\mathbb{T}^{n}_{s_{0}}\,,$$
(159)
with
$$\rho=\theta/C\,,\quad\sigma=1/C|\log\theta|\,.$$
(160)
Finally, $\mathfrak{F}^{i}$ “integrates” $H_{\rm pend}$,
namely333333The function $\mathtt{E}^{(i)}(P)$ is actually the inverse
of the action function $E\to P_{n}^{(i)}(E,\hat{P})$; see (162)
below and Appendix C.:
$$H_{\rm pend}\circ\mathfrak{F}^{i}(P,Q)=H_{\rm pend}\circ\check{\mathfrak{F}}^{%
i}(P,Q_{n})=:\mathtt{E}^{(i)}(P)\,.$$
(161)
Proposition 5.1
is proved in [5];
some more details are given in
Appendix C.
Remark 5.2
(i) Actually, as standard in the theory of integrable systems, one first introduces the action
$P_{n}^{(i)}$ through line integrals $\oint pdq$ as
function of energy $E$ and then defines the integrated Hamiltonian $\mathtt{E}^{(i)}$ inverting such function; in particular, one has
$$P_{n}^{(i)}\Big{(}\mathtt{E}^{(i)}(P_{n},\hat{P}),\hat{P}\Big{)}=P_{n}\,;$$
(162)
for more details, see Appendix C.
(ii) Recalling the definition of $E^{(i)}_{\pm}$ in
(5.5) and setting
$$\displaystyle a^{(2j-1)}_{-}:=0\,,\qquad a^{(2j-1)}_{+}:=P_{n}^{(2j-1)}\big{(}%
E^{(2j-1)}_{+}(\hat{P})-2\theta,\hat{P}\big{)}\,,\qquad 1\leq j\leq N\,,$$
$$\displaystyle a^{(2j)}_{-}:=P_{n}^{(2j)}\big{(}E^{(2j)}_{-}(\hat{P})+2\theta,%
\hat{P}\big{)}\,,\qquad a^{(2j)}_{+}:=P_{n}^{(2j)}\big{(}E^{(2j)}_{+}(\hat{P})%
-2\theta,\hat{P}\big{)}\,,\qquad 1\leq j<N\,,$$
$$\displaystyle a^{(0)}_{-}:=P_{n}^{(0)}\big{(}R_{0}^{2}-M-2\theta,\hat{P}\big{)%
}\,,\qquad a^{(0)}_{+}:=P_{n}^{(0)}\big{(}E^{(2N)}_{-}(\hat{P})+2\theta,\hat{P%
}\big{)}\,,$$
$$\displaystyle a^{(2N)}_{-}:=P_{n}^{(2N)}\big{(}E^{(2N)}_{-}(\hat{P})+2\theta,%
\hat{P}\big{)}\,,\qquad a^{(2N)}_{+}:=P_{n}^{(2N)}\big{(}R_{0}^{2}-M-2\theta,%
\hat{P}\big{)}\,,$$
(163)
one easily recognizes that
$$\mathfrak{P}^{i}(\theta):=\Big{\{}P=(\hat{P},P_{n})\ |\ \hat{P}\in\hat{D},\ \ %
a^{(i)}_{-}(\hat{P},\theta)<P_{n}<a^{(i)}_{+}(\hat{P},\theta)\Big{\}}\subseteq%
\hat{D}\times\mathbb{R}\subseteq\mathbb{R}^{n}\,.$$
(164)
5.7 Properties of the actions as functions of the energy
Next proposition, which is proved in [5]),
contains the fundamental properties
of $P_{n}^{(i)}$ (defined in (162)), which will be heavily exploited in the following.
Let $P_{n}^{(i),0}$ be the function in (162)
when $\eta=0,$ namely when
$b,J_{n}^{*},G$ in (134) vanish
(see (304) below).
Proposition 5.2
There exist suitably small, resp. large, constant $\mathtt{r}>0$, resp.
$C>1,$ which, when (A3) holds,
depend only on $n,s_{0},r_{0}$,
otherwise depend also on $F^{0}$
(introduced in (96))
such that, if
$\eta\leq 1/C$ then the following holds.
There exist
real-analytic functions $\phi^{(i)}_{\pm}(\zeta,\hat{J}),$ $\chi^{(i)}_{\pm}(\zeta,\hat{J})$,
defined for $|\zeta|<\mathtt{r}$, $\hat{J}\in\hat{D}_{r_{0}}$,
with
$$\sup_{|\zeta|<\mathtt{r},\,\hat{J}\in\hat{D}_{r_{0}}}\Big{(}|\phi^{(i)}_{\pm}|%
+|\chi^{(i)}_{\pm}|\Big{)}<C\,,\qquad\sup_{|\zeta|<\mathtt{r},\,\hat{J}\in\hat%
{D}_{r_{0}/2}}\Big{(}|\partial_{\hat{J}}\phi^{(i)}_{\pm}|+|\partial_{\hat{J}}%
\chi^{(i)}_{\pm}|\Big{)}<C\eta\,,$$
(165)
such that343434For $0\leq i\leq 2N$ except
$i=0,2N$ and the $+$ sign, since
$E_{+}^{(0)}(\hat{J})=E_{+}^{(2N)}(\hat{J})=+\infty$
(recall (5.5)).
$$P_{n}^{(i)}\Big{(}E_{\pm}^{(i)}(\hat{J})\mp\zeta,\,\hat{P}\Big{)}=\phi^{(i)}_{%
\pm}(\zeta,\hat{J})+\zeta\log\zeta\,\chi^{(i)}_{\pm}(\zeta,\hat{J})\,,\qquad%
\text{for}\ \ \ 0<\zeta<\mathtt{r}\,,\ \ \hat{J}\in\hat{D}\,.$$
(166)
Moreover
$$\chi^{(2j-1)}_{-}=0\,,\qquad{}$$
(167)
and353535Recall the definition of
$F$ in (134).
$$|\chi^{(i)}_{\pm}(0,\hat{J})|\geq\frac{1}{4\pi\sqrt{\|\partial_{\psi_{n}\psi_{%
n}}F\|_{\hat{D},r_{0},s_{0}}}}\geq 1/C>0\,.$$
(168)
Notice that (167) implies that
$P_{n}^{(2j-1)}(E,\hat{P})$
has holomorphic extension
on $\{|E-E_{-}^{(2j-1)}(\hat{J})|<\mathtt{r}\}\times\hat{D}_{r_{0}}$, as well as
$P_{n}^{(2j-1),0}(E)$
has holomorphic extension
on363636Recall (5.1). $\{|E-E_{-}^{(2j-1),0}|<\mathtt{r}\}.$
Furthermore
$$\hat{D}\times(E^{(i),0}_{-}+\mathfrak{r}/4,E^{(i),0}_{+}-\mathfrak{r}/4)%
\subset\{(\hat{P},E)\ {\rm s.t.}\ \hat{P}\in\hat{D},\ E^{(i)}_{-}(\hat{P})<E<E%
^{(i)}_{+}(\hat{P})\}$$
(169)
and
$$\sup_{\hat{D}_{r_{0}}\times\big{(}E_{-}^{(i),0}+(-1)^{i}\mathtt{r}/2,E_{+}^{(i%
),0}-\mathtt{r}/2\big{)}_{\mathtt{r}/4}}|P_{n}^{(i)}(\hat{P},E)-P_{n}^{(i),0}(%
E)|<C\eta\,.$$
(170)
Finally by Lemma C.4
we get
$$\|\partial_{PP}\mathtt{E}^{(i)}\|_{\mathfrak{P}^{i}(\theta),\rho}\leq C/\rho\,%
,\qquad{\rm for}\ \ \ 0\leq i\leq 2N\,.$$
(171)
5.8 The final canonical transformation
Let us define
$$\mathcal{C}^{i}_{*}(\theta):=\Phi^{(2{\rm bis})}\big{(}\mathcal{C}^{i}(\theta)%
\big{)}\,,\qquad\check{\mathcal{C}}^{i}_{*}(\theta):=\check{\Phi}^{(2{\rm bis}%
)}\big{(}\check{\mathcal{C}}^{i}(\theta)\big{)}\,.$$
(172)
We have that
$$\mathcal{C}^{i}_{*}(\theta)=\check{\mathcal{C}}^{i}_{*}(\theta)\times\mathbb{T%
}^{n-1}\,,$$
(173)
since
$$\mathcal{C}^{i}_{*}(\theta)=\Phi^{(2{\rm bis})}\big{(}\mathcal{C}^{i}(\theta)%
\big{)}\stackrel{{\scriptstyle{\rm(\ref{sax})}}}{{=}}\Phi^{(2{\rm bis})}\big{(%
}\check{\mathcal{C}}^{i}(\theta)\times\mathbb{T}^{n-1}\big{)}\stackrel{{%
\scriptstyle{\rm(\ref{Ventura})}}}{{=}}\check{\Phi}^{(2{\rm bis})}\big{(}%
\check{\mathcal{C}}^{i}(\theta)\big{)}\times\mathbb{T}^{n-1}\stackrel{{%
\scriptstyle{\rm(\ref{kyoto})}}}{{=}}\check{\mathcal{C}}^{i}_{*}(\theta)\times%
\mathbb{T}^{n-1}\,.$$
Let us also define
$$\mathfrak{F}^{i}_{*}:=\Phi^{(2{\rm bis})}\circ\mathfrak{F}^{i}\,.$$
(174)
By (157),(158)
we have that
$\mathfrak{F}^{i}_{*}$ has the form
$$\displaystyle\hat{J}=\hat{P}\,,\quad J_{n}=\mathtt{v}^{i}_{*}(P,Q_{n})\,,\quad%
\hat{\psi}=\hat{Q}+\mathtt{z}^{i}_{*}(P,Q_{n})\,,\quad\psi_{n}=\mathtt{u}^{i}(%
P,Q_{n})\,,\qquad\text{for}\ 1\leq i\leq 2N-1\,,$$
(175)
$$\displaystyle\hat{J}=\hat{P}\,,\quad J_{n}=\mathtt{v}^{i}_{*}(P,Q_{n})\,,\quad%
\hat{\psi}=\hat{Q}+\mathtt{z}^{i}_{*}(P,Q_{n})\,,\quad\psi_{n}=Q_{n}+\mathtt{u%
}^{i}(P,Q_{n})\,,\qquad\text{for}\ i=0,2N\,,$$
(176)
with $\mathtt{v}^{i}_{*},\mathtt{z}^{i}_{*},\mathtt{u}^{i}$,
$2\pi$-periodic in $Q_{n},$
$|\mathtt{z}^{i}_{*}|\leq C\eta$.
Let us define the linear symplectic transformation of the form in (129)
$\Phi_{\rm lin}:\mathbb{R}^{2n}\ni(J^{\prime\prime},\psi^{\prime\prime})\mapsto%
(J^{\prime},\psi^{\prime})\in\mathbb{R}^{2n}$ generated by the generating function
$\hat{J}^{\prime}\cdot\hat{\psi}^{\prime\prime}+\big{(}J_{n}^{\prime}+\frac{1}{%
\kappa}(\hat{A}k)\cdot\hat{J}^{\prime}\big{)}\psi_{n}^{\prime\prime}$ namely (recalling (15))
$$J^{\prime}=U_{k}J^{\prime\prime}\,,\quad{\rm with}\quad\hat{J}^{\prime}=\hat{J%
}^{\prime\prime}\,,\quad J^{\prime}_{n}=J_{n}^{\prime\prime}-\frac{1}{\kappa}(%
\hat{A}k)\cdot\hat{J}^{\prime\prime}\,,\quad\hat{\psi}^{\prime}=\hat{\psi}^{%
\prime\prime}+\frac{\psi_{n}^{\prime\prime}}{\kappa}\hat{A}k\,,\quad\psi^{%
\prime}_{n}=\psi_{n}^{\prime\prime}\,.$$
(177)
Remark 5.3
Note that such map is only $2\pi\kappa$-periodic in $\psi_{n}^{\prime\prime}.$
Note also that its inverse is
$$\Phi_{\rm lin}^{-1}\ \ \ \text{with}\ \ \ \hat{J}^{\prime\prime}=\hat{J}^{%
\prime}\,,\quad J^{\prime\prime}_{n}=J_{n}^{\prime}+\frac{1}{\kappa}(\hat{A}k)%
\cdot\hat{J}^{\prime}\,,\quad\hat{\psi}^{\prime\prime}=\hat{\psi}^{\prime}-%
\frac{\psi_{n}^{\prime}}{\kappa}\hat{A}k\,,\quad\psi^{\prime\prime}_{n}=\psi_{%
n}^{\prime}$$
(178)
and that
the operatorial norms of $\Phi_{\rm lin},\Phi_{\rm lin}^{-1}$ are bounded
by some constant $c(n)>0$
$$\|\Phi_{\rm lin}\|\,,\ \|\Phi_{\rm lin}^{-1}\|\ \leq\ c(n)\,.$$
(179)
Recalling the notation in (130)
we introduce the volume preserving map
$\check{\Phi}_{\rm lin}:\mathbb{R}^{n}\times\mathbb{T}^{1}.$
Recalling (126)
we have that $H^{*}$ defined in
(125)
satisfies
$$H^{*}(J^{\prime\prime},\psi_{n}^{\prime\prime})=(\tilde{H}\circ\check{\Phi}_{%
\rm lin})(J^{\prime\prime},\psi_{n}^{\prime\prime})-\frac{1}{\kappa}\|\,{%
\mathtt{p}}^{\perp}\hat{A}^{T}\hat{J}^{\prime\prime}\|^{2}\,,\qquad\text{(%
namely \ \eqref{spigolo})}$$
(180)
where $\tilde{H}$ was defined in (119).
Moreover, recalling (134),
we have
$$(\tilde{H}\circ\check{\Phi}_{\rm lin}\circ\check{\Phi}^{(2{\rm bis})})(J,\psi_%
{n})=H_{\rm pend}(J,\psi_{n})+\frac{1}{\kappa}\|\,{\mathtt{p}}^{\perp}\hat{A}^%
{T}\hat{J}\|^{2}$$
(181)
and, recalling (161)
$$(\tilde{H}\circ\check{\Phi}_{\rm lin}\circ\check{\Phi}^{(2{\rm bis})}\circ%
\check{\mathfrak{F}}^{i})(P,Q_{n})=\mathtt{E}^{(i)}(P)+\frac{1}{\kappa}\|\,{%
\mathtt{p}}^{\perp}\hat{A}^{T}\hat{P}\|^{2}$$
(182)
Recall also the definition of
${\mathcal{C}}^{i}_{*}(\theta)=\check{\mathcal{C}}^{i}_{*}(\theta)\times\mathbb%
{T}^{n-1}$
given in (172), (173)
and of $\mathfrak{P}^{i}(\theta)$ given in
Proposition 5.1.
Set
$${\mathtt{C}}^{i}(\theta):=\check{\mathtt{C}}^{i}(\theta)\times\mathbb{T}^{n-1}%
\,,\qquad\qquad\check{\mathtt{C}}^{i}(\theta):=\check{\Phi}_{\rm lin}\big{(}%
\check{\mathcal{C}}^{i}_{*}(\theta)\big{)}\times\mathbb{T}^{n-1}$$
(183)
and
$$\mathfrak{C}^{i}(\theta):=\Phi^{(1)}\big{(}{\mathtt{C}}^{i}(\theta)\big{)}\,.$$
(184)
By (153) we get373737
Note that $U_{k}\big{(}\hat{D}\times(-R_{0},R_{0})\big{)}=U_{k}\check{D}=\tilde{D}^{k}$ defined in
(120).
$${\rm\,meas\,}\Big{(}\Big{(}U_{k}\big{(}\hat{D}\times(-R_{0}/2,R_{0}/2)\big{)}%
\times\mathbb{T}^{n}\Big{)}\ \setminus\ \bigcup_{0\leq i\leq 2N}{\mathtt{C}}^{%
i}(\theta)\Big{)}\lessdot\theta|\log\theta|.$$
(185)
Recalling (164), one sees that one can define the sets $\mathtt{B}^{i}(\theta)$ appearing in 2) of Theorem 4.1 as
$$\mathtt{B}^{i}(\theta):=\left\{\begin{array}[]{ll}\mathfrak{P}^{i}(\theta)&\ %
\ {\rm if}\ \ 1\leq i\leq 2N-1\\
U_{k}\mathfrak{P}^{i}(\theta)&\ \ {\rm if}\ \ i=0,2N\end{array}\right.\,.$$
(186)
Note that (104) follows by (154)
and (17).
1) The oscillatory case: $1\leq i\leq 2N-1$
Fix
$1\leq i\leq 2N-1$.
Let us define the symplectomorphism of the form (129)
$$\displaystyle\Phi_{i}\ :\ \mathtt{B}^{i}(\theta)\times\mathbb{T}^{n}\ \to\ %
\mathtt{C}^{i}(\theta)\,,\qquad\text{defined \ as}$$
$$\displaystyle\hat{J}^{\prime}=\hat{p}\,,\qquad J_{n}^{\prime}=\mathtt{v}^{i}_{%
*}(p,q_{n})-\frac{1}{\kappa}(\hat{A}k)\cdot\hat{p}\,,$$
$$\displaystyle\hat{\psi}^{\prime}=\hat{q}+\mathtt{z}^{i}_{*}(p,q_{n})+\frac{%
\mathtt{u}^{i}(p,q_{n})}{\kappa}\hat{A}k\,,\qquad\psi_{n}^{\prime}=\mathtt{u}^%
{i}(p,q_{n})\,,$$
(187)
with
$\mathtt{v}^{i}_{*},\mathtt{z}^{i}_{*},\mathtt{u}^{i}$
defined in (175),(157).
The fact that it is symplectic can be seen directly by (187) but also noting that locally
$$\Phi_{i}=\Phi_{\rm lin}\circ\Phi^{{2\rm bis}}\circ\mathfrak{F}^{i}=\Phi_{\rm
lin%
}\circ\mathfrak{F}^{i}_{*}$$
(188)
with
$\Phi_{\rm lin}$
defined in (177),
$\Phi^{{2\rm bis}}$ defined in
(131)
and
$\mathfrak{F}^{i}$ defined in (155)
(recall also (157)).
By (183),(156) and (186),
$\Phi_{i}$ is surjective on
$\mathtt{C}^{i}(\theta),$
namely
$$\Phi_{i}\big{(}\mathtt{B}^{i}(\theta)\times\mathbb{T}^{n}\big{)}=\mathtt{C}^{i%
}(\theta)\,.$$
(189)
The injectivity is obvious.
Note that, by Lemma D.5,
(159),
(160), (179),
$\Phi_{i}$
has a holomorphic extension on
$$\Phi_{i}\ :\ \big{(}\mathtt{B}^{i}(\theta)\big{)}_{\rho_{0}}\times\mathbb{T}^{%
n}_{\sigma_{0}}\ \to\ D_{r_{0}}\times\mathbb{T}^{n}_{s_{0}}\,,\qquad\rho_{0}:=%
c\frac{\theta}{C}\,,\quad\ \sigma_{0}:=c\frac{1}{C\log\theta}\,,$$
(190)
where $c$ is a (small) constant
depending only on $n.$
Applying again Lemma D.5,
we also prove that
$$\Phi_{i}\ :\ \big{(}\mathtt{B}^{i}(\theta)\big{)}_{\rho_{*}}\times\mathbb{T}^{%
n}_{\sigma_{*}}\ \to\ D_{\tilde{r}}\times\mathbb{T}^{n}_{\tilde{s}}\,,$$
(191)
with $\rho_{*},\sigma_{*}$, resp. $\tilde{r},\tilde{s}$, defined in
(106), resp. (122),
taking $\mathtt{c}$ large enough.
2) The libration case: $i=0,2N$
Fix $i=0,2N$.
Let us define the symplectomorphism in $\mathcal{G}$ (recall (183) and (186))
$$\displaystyle\Phi^{i}\ :\ \mathtt{B}^{i}(\theta)\times\mathbb{T}^{n}\ \to\ %
\mathtt{C}^{i}(\theta)\,,\qquad\text{defined \ as}$$
$$\displaystyle\hat{J}^{\prime}=\hat{p}\,,\qquad J_{n}^{\prime}=\mathtt{v}^{i}_{%
*,k}(U_{k}^{-1}p,q_{n})-\frac{1}{\kappa}(\hat{A}k)\cdot\hat{p}\,,$$
$$\displaystyle\hat{\psi}^{\prime}=\hat{q}+\mathtt{z}^{i}_{*,k}(U_{k}^{-1}p,q_{n%
})+\frac{\mathtt{u}^{i}(U_{k}^{-1}p,q_{n})}{\kappa}\hat{A}k\,,\qquad\psi_{n}^{%
\prime}=q_{n}+\mathtt{u}^{i}(U_{k}^{-1}p,q_{n})\,,$$
(192)
$$\displaystyle{\rm where,\ recall\ \eqref{centocelle},}\qquad U_{k}^{-1}p=\Big{%
(}\hat{p},p_{n}+\frac{1}{\kappa}(\hat{A}k)\cdot\hat{p}\Big{)}\,,$$
with
$\mathtt{v}^{i}_{*},\mathtt{z}^{i}_{*},\mathtt{u}^{i}$
defined in (175),(157).
The fact that it is symplectic can be seen directly by (192) but also noting that locally
$$\Phi^{i}=\Phi_{\rm lin}\circ\Phi^{{2\rm bis}}\circ\mathfrak{F}^{i}\circ\Phi_{%
\rm lin}^{-1}=\Phi_{\rm lin}\circ\mathfrak{F}^{i}_{*}\circ\Phi_{\rm lin}^{-1}\,,$$
(193)
with
$\Phi_{\rm lin}$
defined in (177),
$\Phi^{{2\rm bis}}$ defined in
(131)
and
$\mathfrak{F}^{i}$ defined in (155)
(recall also (157)).
Note that $\Phi^{i}$ is injective, as it directly follows
by the fact that so are
$\Phi^{{2\rm bis}}$ and
$\mathfrak{F}^{i}$,
and also surjective, namely
(189) holds;
indeed
$$\displaystyle\Phi^{i}\big{(}\mathtt{B}^{i}(\theta)\times\mathbb{T}^{n}\big{)}%
\stackrel{{\scriptstyle{\rm(\ref{Ventura})}}}{{=}}\check{\Phi}^{i}\big{(}%
\mathtt{B}^{i}(\theta)\times\mathbb{T}^{1}\big{)}\times\mathbb{T}^{n-1}%
\stackrel{{\scriptstyle{\rm(\ref{peggylee})}}}{{=}}\check{\Phi}_{\rm lin}\big{%
(}\check{\Phi}^{{2\rm bis}}\big{(}\check{\mathfrak{F}}^{i}\big{(}\check{\Phi}_%
{\rm lin}^{-1}\big{(}\mathtt{B}^{i}(\theta)\times\mathbb{T}^{1}\big{)}\big{)}%
\big{)}\big{)}\times\mathbb{T}^{n-1}$$
$$\displaystyle\stackrel{{\scriptstyle{\rm(\ref{closetotheedge})}}}{{=}}\check{%
\Phi}_{\rm lin}\big{(}\check{\Phi}^{{2\rm bis}}\big{(}\check{\mathfrak{F}}^{i}%
\big{(}\mathfrak{P}^{i}(\theta)\times\mathbb{T}^{1}\big{)}\big{)}\big{)}\times%
\mathbb{T}^{n-1}\stackrel{{\scriptstyle{\rm(\ref{ofdelirium})}}}{{=}}\check{%
\Phi}_{\rm lin}\big{(}\check{\Phi}^{{2\rm bis}}\big{(}\mathcal{C}^{i}(\theta)%
\big{)}\big{)}\times\mathbb{T}^{n-1}$$
$$\displaystyle\stackrel{{\scriptstyle{\rm(\ref{kyoto})}}}{{=}}\check{\Phi}_{\rm
lin%
}\big{(}\check{\mathcal{C}}^{i}(\theta)\big{)}\times\mathbb{T}^{n-1}\stackrel{%
{\scriptstyle{\rm(\ref{cruel3})}}}{{=}}\check{\mathtt{C}}^{i}(\theta)\times%
\mathbb{T}^{n-1}=\mathtt{C}^{i}(\theta)$$
Reasoning as in the case
$1\leq i<2N,$ we get that
$\Phi^{i}$
has a holomorphic extension as in (190).
(191) holds as well.
5.9 Conclusion of the
proof of part one of the Structure Theorem
We set
$$\Psi^{i}:=\Phi^{(1)}\circ\Phi_{i}\,,$$
(194)
where $\Phi^{(1)}$
was defined in (117); therefore
(106) holds by (191)
and (123).
Then (103)
follows by (185)
recalling (14), (101)
and since $\Phi^{(1)}$
preserve volume being symplectic.
(105) follows
by (189).
Recalling (119)
and (182) we have that
$h^{(i)}:={\mathcal{H}}\circ\Phi^{(1)}\circ\Phi_{i}$
can be written as
$$h^{(i)}(p)=\left\{\begin{array}[]{ll}\mathtt{E}^{(i)}(p)+\hat{h}_{k}(\hat{p})&%
\ \ {\rm if}\ \ 1\leq i<2N\,,\\
(\mathtt{E}^{(i)}+\hat{h}_{k})(U_{k}^{-1}p)&\ \ {\rm if}\ \ i=0,2N\,.\end{%
array}\right.$$
(195)
where
$$\hat{h}_{k}(\hat{p}):=\kappa^{-1}\|\,{\mathtt{p}}^{\perp}\hat{A}_{k}^{T}p\|^{2%
}\,.$$
(196)
(107)
follows.
Finally (108) follows by
(171),
(195), (196),
(10), (17).
This concludes the proof of part
one of the Structure Theorem.
6 Proof of Part II of the Structure Theorem
A crucial fact is that the integrable Hamiltonian
$h^{(i)}$ defined in (195)
twists, namely (112) (toghether with the measure estimate (111)) holds.
This will be a direct consequence of the following
Proposition 6.1
Under the assumptions of Theorem 4.1
(in particular (109)),
for any $\mu$ satisfying (110), there exists a subset383838Decreasing w.r.t. $\mu.$ $\widetilde{\mathfrak{P}}^{i}(\mu)\subseteq\mathfrak{P}^{i}(0),$
satisfying393939$\mathtt{c}$ defined in
Theorem 4.1.
$${\rm\,meas\,}(\mathfrak{P}^{i}(0)\,\backslash\,\widetilde{\mathfrak{P}}^{i}(%
\mu))\leq\mathtt{c}\|k\|^{4n}\mu^{1/\mathtt{c}}\,,$$
(197)
such that
$$\left|{\rm det}\left[\partial_{PP}\left(\mathtt{E}^{(i)}(P)+\hat{h}_{k}(\hat{P%
})\right)\right]\right|>\mu\,,\qquad\forall\,\ 0\leq i\leq 2N\,,\qquad\forall%
\ P\in\widetilde{\mathfrak{P}}^{i}(\mu)\ .$$
(198)
Proof of (112).
One sees that, in analogy to
(186),
one can define the sets
$\tilde{\mathtt{B}}^{i}(\mu)$ appearing in 3) of Theorem 4.1 as
$$\tilde{\mathtt{B}}^{i}(\mu):=\left\{\begin{array}[]{ll}\tilde{\mathfrak{P}}^{i%
}(\mu)&\ \ {\rm if}\ \ 1\leq i\leq 2N-1\,,\\
U_{k}\tilde{\mathfrak{P}}^{i}(\mu)&\ \ {\rm if}\ \ i=0,2N\,.\end{array}\right.$$
(199)
Recalling (186)and (17) , by (197), we get (111).
Finally (112) follows by (198), Lemma
D.1 and noting that det$\,U_{k}=1$ by (15).
Proof of Proposition 6.1.
First we note that
$$\det\big{(}\partial_{\hat{P}\hat{P}}\hat{h}_{k}\big{)}=2^{n-1}\kappa^{-n}\,.$$
(200)
Indeed by (20)
$$\hat{h}_{k}(\hat{P})+(P_{n})^{2}=\kappa^{-1}\|L_{k}P\|^{2}\,,$$
(with $L_{k}$ defined in (12)) and,
by Lemma D.1 and (18), we get
$$\mathtt{d}_{k}:=\det\Big{(}\partial_{PP}\big{(}\hat{h}_{k}(\hat{P})+(P_{n})^{2%
}\big{)}\Big{)}=2^{n}\kappa^{-n}$$
(201)
and, therefore, (200) follows.
The “twist” determinant as a function of the energy
Let us fix $0\leq i\leq 2N$.
Consider the analytic
function
$$d^{i}_{k}(E,\hat{P}):={\rm det}\left[\partial_{PP}\hat{h}_{k}(\hat{P})+%
\partial_{PP}\mathtt{E}^{(i)}\big{(}\hat{P},P_{n}^{(i)}(\hat{P},E)\big{)}%
\right]\,,$$
(202)
where, $P_{n}^{(i)}(E,\hat{P})$ is, by
definition404040
This map can be obviously explicitely constructed, see
subsection C.4 below.,
the inverse map of $\mathtt{E}^{(i)}(\hat{P},P_{n})$
(recall (162)).
For brevity we will often omit to write the indexes ${}_{k}$ and/or ${}^{(i)}.$
The case $i$ odd; close to a maximum
of the potential
Instead of use the variable $E$
we use, in the case of odd $i=2j-1$,
$$\zeta:=E_{+}^{(2j-1)}(\hat{P})-E\,,$$
(203)
where $E_{+}^{(2j-1)}(\hat{P})$
was defined in (5.5).
In the variable $\zeta$, (note that $\partial_{E}$ is equal to
$\partial_{\zeta}$, up to sign), recalling (166),
$$P_{n}(E,\hat{P})=P_{n}\big{(}E_{+}^{(2j-1)}(\hat{P})-\zeta,\hat{P}\big{)}=1%
\oplus\zeta\log\zeta\,,$$
(204)
where by $g=g_{1}\oplus g_{2}$
we mean that there exist two functions
$\varphi_{1}(\zeta,\hat{P}),\varphi_{2}(\zeta,\hat{P})$, analytic in a complex neighborhood
of zero times $\hat{D}$ (depending only on $\mathfrak{p}$ defined in (137)), such that
$g(\zeta,\hat{P})=g_{1}(\zeta)\varphi_{1}(\zeta,\hat{P})+g_{2}(\zeta)\varphi_{2%
}(\zeta,\hat{P}).$
We get (for $i,j=1,\ldots,n-1$)
$$\displaystyle\partial_{E}P_{n}=1\oplus\log\zeta\,,\quad\partial_{\hat{P}_{i}}P%
_{n}=\eta(1\oplus\zeta\log\zeta)\,,\quad\partial_{EE}P_{n}=\log\zeta\oplus%
\zeta^{-1}\,,$$
$$\displaystyle\partial_{E\hat{P}_{i}}P_{n}=\eta(1\oplus\log\zeta)\,,\quad%
\partial_{\hat{P}_{i}\hat{P}_{j}}P_{n}=\eta(1\oplus\zeta\log\zeta)\,.$$
(205)
Recalling (162), by the chain rule we get414141By $\partial_{\hat{P}}$ we mean the row vector
$(\partial_{P_{1}},\ldots,\partial_{P_{n-1}})$, then $\partial_{\hat{P}}^{T}$ is a column vector.
$$\displaystyle\partial_{P_{n}}\mathtt{E}=\frac{1}{\partial_{E}P_{n}}\,,\qquad%
\partial_{\hat{P}}\mathtt{E}=-\frac{\partial_{\hat{P}}P_{n}}{\partial_{E}P_{n}%
}\,,\qquad\partial_{P_{n}P_{n}}\mathtt{E}=-\frac{\partial_{EE}P_{n}}{(\partial%
_{E}P_{n})^{3}}\,,$$
$$\displaystyle\partial_{P_{n}\hat{P}}\mathtt{E}=\frac{\partial_{EE}P_{n}%
\partial_{\hat{P}}P_{n}}{(\partial_{E}P_{n})^{3}}-\frac{\partial_{E\hat{P}}P_{%
n}}{(\partial_{E}P_{n})^{2}}\ \in\ \mathbb{R}^{n-1}\,,$$
$$\displaystyle\partial_{\hat{P}\hat{P}}\mathtt{E}=-\frac{\partial_{\hat{P}\hat{%
P}}P_{n}}{\partial_{E}P_{n}}+2\frac{\partial_{\hat{P}}^{T}P_{n}\ \partial_{%
\hat{P}}(\partial_{E}P_{n})}{(\partial_{E}P_{n})^{2}}-\frac{\partial_{EE}P_{n}%
\ \partial_{\hat{P}}^{T}P_{n}\ \partial_{\hat{P}}P_{n}}{(\partial_{E}P_{n})^{3%
}}\ \in\ {\rm Mat}_{(n-1)\times(n-1)}\,,$$
(206)
where $\mathtt{E}$ and $P_{n}$ are evaluated in $\big{(}P_{n}(E,\hat{P}),\hat{P}\big{)}$
and $(E,\hat{P}),$ respectively424242
Or, which is equivalent, in $P$ and
$\big{(}\mathtt{E}(P),\hat{P}\big{)},$
respectively..
By (6) and (206)
we get (for $i,j=1,\ldots,n-1$)
$$\displaystyle\zeta(\partial_{E}P_{n})^{3}\partial_{P_{n}P_{n}}\mathtt{E}$$
$$\displaystyle=$$
$$\displaystyle-\zeta\partial_{EE}P_{n}=1\oplus\zeta\log\zeta\,,$$
$$\displaystyle\zeta(\partial_{E}P_{n})^{3}\partial_{P_{n}\hat{P}_{i}}\mathtt{E}$$
$$\displaystyle=$$
$$\displaystyle\zeta\partial_{EE}P_{n}\partial_{\hat{P}_{i}}P_{n}-\zeta\partial_%
{E\hat{P}_{i}}P_{n}\partial_{E}P_{n}=\eta(1\oplus\zeta\log\zeta\oplus\zeta\log%
^{2}\zeta)\,,$$
$$\displaystyle\zeta(\partial_{E}P_{n})^{3}\partial_{\hat{P}_{i}\hat{P}_{j}}%
\mathtt{E}$$
$$\displaystyle=$$
$$\displaystyle-\zeta(\partial_{E}P_{n})^{2}\partial_{\hat{P}_{i}\hat{P}_{j}}P_{%
n}+2\zeta\partial_{E}P_{n}\partial_{\hat{P}_{i}}P_{n}\ \partial_{\hat{P}_{j}E}%
P_{n}-\zeta\partial_{EE}P_{n}\ \partial_{\hat{P}_{i}}P_{n}\ \partial_{\hat{P}_%
{j}}P_{n}$$
(207)
$$\displaystyle=$$
$$\displaystyle\eta(1\oplus\zeta\log\zeta\oplus\zeta\log^{2}\zeta\oplus\zeta^{2}%
\log^{3}\zeta)\,.$$
The “rescaled” determinant $\mathtt{f}$
Recalling that, by (203) we have
$$E=E_{+}^{(2j-1)}(\hat{P})-\zeta\,,$$
we set
$$\mathtt{f}(\zeta,\hat{P})=\mathtt{f}^{(i)}_{k}(\zeta,\hat{P}):=\zeta^{n}\big{(%
}\partial_{E}P_{n}(E_{+}^{(2j-1)}(\hat{P})-\zeta,\hat{P})\big{)}^{3n}d^{i}_{k}%
(E_{+}^{(2j-1)}(\hat{P})-\zeta,\hat{P})\,.$$
(208)
We will omit the dependence on $k,i$.
Note that
$$\displaystyle d^{i}_{k}$$
$$\displaystyle=$$
$$\displaystyle{\rm det}\left(\partial_{PP}\hat{h}_{k}+\partial_{PP}\mathtt{E}^{%
(i)}_{k}\right)=\text{det}\left(\begin{array}[]{cc}\partial_{\hat{P}\hat{P}}h+%
\partial_{\hat{P}\hat{P}}\mathtt{E}&\quad\partial_{\hat{P}}^{T}(\partial_{P_{n%
}}\mathtt{E})\\
\partial_{\hat{P}}(\partial_{P_{n}}\mathtt{E})&\partial_{P_{n}P_{n}}\mathtt{E}%
\\
\end{array}\right)$$
$$\displaystyle=$$
$$\displaystyle\partial_{P_{n}P_{n}}\mathtt{E}\ \text{det}(\partial_{\hat{P}\hat%
{P}}h+\partial_{\hat{P}\hat{P}}\mathtt{E})+\text{det}\left(\begin{array}[]{cc}%
\partial_{\hat{P}\hat{P}}h+\partial_{\hat{P}\hat{P}}\mathtt{E}&\quad\partial_{%
\hat{P}}^{T}(\partial_{P_{n}}\mathtt{E})\\
\partial_{\hat{P}}(\partial_{P_{n}}\mathtt{E})&0\\
\end{array}\right)$$
developing the determinat w.r.t. the last
column.
By (207) we get
$$\displaystyle\zeta^{n}(\partial_{E}P_{n})^{3n}\partial_{P_{n}P_{n}}\mathtt{E}%
\ \text{det}(\partial_{\hat{P}\hat{P}}h+\partial_{\hat{P}\hat{P}}\mathtt{E})$$
$$\displaystyle=\zeta^{n-1}\sum_{\ell=0}^{3n-3}\varphi_{\ell}^{(1)}(\zeta,\hat{P%
})\log^{\ell}\zeta\ +\ \eta\sum_{\ell=0}^{3n-5}\varphi_{\ell}^{(2)}(\zeta,\hat%
{P})\log^{\ell}\zeta\ +\ R^{(1)}\,,$$
where
$$\inf_{\hat{P}\in\hat{D}}|\varphi_{3n-3}^{(1)}(0,\hat{P})|\geq c(\mathfrak{p})%
\mathtt{d}_{k}>0\,,$$
(209)
with $\mathtt{d}=\mathtt{d}_{k}$ defined in (201)
and
$$R^{(1)}=O(\zeta^{n})=\zeta^{n}\sum_{\ell=0}^{3n-2}\varphi_{\ell}^{(3)}(\zeta,%
\hat{P})\log^{\ell}\zeta\,,$$
for suitable $\varphi_{\ell}^{(1)},\varphi_{\ell}^{(2)},\varphi_{\ell}^{(3)},$ analytic and uniformly
bounded434343Here and in the following of this section by “uniformly bounded”
we mean “bounded by a constant depending only on $\mathfrak{p}$ in in a uniform neighborhood of zero (and $\hat{P}\in\hat{D}$).
Remark 6.1
We stress that, as a consequence of (165),
all the estimates of this section are uniform
in $\hat{P}\in\hat{D}.$
Moreover
$$\displaystyle\zeta^{n}(\partial_{E}P_{n})^{3n}\ \text{det}\left(\begin{array}[%
]{cc}\partial_{\hat{P}\hat{P}}h+\partial_{\hat{P}\hat{P}}\mathtt{E}&\quad%
\partial_{\hat{P}}^{T}(\partial_{P_{n}}\mathtt{E})\\
\partial_{\hat{P}}(\partial_{P_{n}}\mathtt{E})&0\\
\end{array}\right)$$
$$\displaystyle=$$
$$\displaystyle\eta^{2}(1\oplus\zeta\log\zeta\oplus\zeta\log^{2}\zeta)^{2}(1%
\oplus\zeta\log\zeta\oplus\zeta\log^{2}\zeta\oplus\zeta\log^{3}\zeta)^{n-2}$$
$$\displaystyle=$$
$$\displaystyle\eta^{2}\sum_{\ell=0}^{3n-4}\varphi_{\ell}^{(4)}(\zeta,\hat{P})%
\log^{\ell}\zeta\ +\ R^{(2)}\,,$$
where
$$R^{(2)}=O(\zeta^{n})=\zeta^{n}\sum_{\ell=0}^{3n-2}\varphi_{\ell}^{(5)}(\zeta,%
\hat{P})\log^{\ell}\zeta\,,$$
for suitable
$\varphi_{\ell}^{(4)},\varphi_{\ell}^{(5)},$
analytic and uniformly bounded in a uniform neighborhood of zero (and $\hat{P}\in\hat{D}$).
Therefore
$$\mathtt{f}=\mathtt{f}^{(i)}_{k}=\zeta^{n-1}\sum_{\ell=0}^{3n-3}\varphi_{\ell}(%
\zeta,\hat{P})\log^{\ell}\zeta\ +\ \eta\sum_{\ell=0}^{3n-4}\chi_{\ell}(\zeta,%
\hat{P})\log^{\ell}\zeta\ +\ R$$
(210)
where, by (209),
$$\inf_{\hat{P}\in\hat{D}}|\varphi_{3n-3}(0,\hat{P})|\geq c(\mathfrak{p})\mathtt%
{d}_{k}>0\,,$$
(211)
and
$$R=O(\zeta^{n})=\zeta^{n}\sum_{\ell=0}^{3n-2}\psi_{\ell}(\zeta,\hat{P})\log^{%
\ell}\zeta\,,$$
(212)
for suitable $\varphi_{\ell},\chi_{\ell},\psi_{\ell}$ analytic and uniformly bounded in a uniform neighborhood of zero (and $\hat{P}\in\hat{D}$).
A class of useful linear operators
We now consider the linear operator
$$L:=\zeta\partial_{\zeta}$$
and, recursively, $L^{\ell}:=L\circ L^{\ell-1}.$
We have
$$\displaystyle L\left(\zeta^{h}\sum_{\ell=0}^{\ell_{0}}g_{\ell}(\zeta,\hat{P})%
\log^{\ell}\zeta\right)$$
$$\displaystyle=$$
$$\displaystyle\zeta^{h}\sum_{\ell=0}^{\ell_{0}}\tilde{g}_{\ell}(\zeta,\hat{P})%
\log^{\ell}\zeta\,,\qquad\text{with}\ \ \tilde{g}_{\ell_{0}}(0,\hat{P})=hg_{%
\ell_{0}}(0,\hat{P})\,,$$
(213)
$$\displaystyle L^{\ell_{0}}\left(\sum_{\ell=0}^{\ell_{0}}g_{\ell}(\zeta,\hat{P}%
)\log^{\ell}\zeta\right)$$
$$\displaystyle=$$
$$\displaystyle\ell_{0}!\,g_{\ell_{0}}(\zeta,\hat{P})\,+\,\zeta\sum_{\ell=0}^{%
\ell_{0}}\tilde{g}_{\ell}(\zeta,\hat{P})\log^{\ell}\zeta\,,$$
(214)
$$\displaystyle L^{\ell_{1}}\left(\sum_{\ell=0}^{\ell_{0}}g_{\ell}(\zeta,\hat{P}%
)\log^{\ell}\zeta\right)$$
$$\displaystyle=$$
$$\displaystyle\zeta\sum_{\ell=0}^{\ell_{0}}\tilde{g}_{\ell}(\zeta,\hat{P})\log^%
{\ell}\zeta\,,\qquad\text{when}\ \ \ell_{1}>\ell_{0}\,,$$
(215)
for (different) suitable $\tilde{g}_{\ell}.$
Moreover
$$\displaystyle\partial_{\zeta}\left(\zeta^{h}\sum_{\ell=0}^{\ell_{0}}g_{\ell}(%
\zeta,\hat{P})\log^{\ell}\zeta\right)\ =$$
(216)
$$\displaystyle hg_{\ell_{0}}(\zeta,\hat{P})\zeta^{h-1}\log^{\ell_{0}}\ +\ \zeta%
^{h-1}\sum_{\ell=0}^{\ell_{0}-1}\Big{(}hg_{\ell}(\zeta,\hat{P})+(\ell+1)g_{%
\ell+1}(\zeta,\hat{P})\Big{)}\log^{\ell}\zeta+\ \zeta^{h}\sum_{\ell=0}^{\ell_{%
0}}\tilde{g}_{\ell}(\zeta,\hat{P})\log^{\ell}\zeta\,.$$
In particular by (213),
(215),(216)
we get
$$(\partial_{\zeta}\circ L^{\ell_{1}})\left(\sum_{\ell=0}^{\ell_{0}}g_{\ell}(%
\zeta,\hat{P})\zeta^{h}\log^{\ell}\zeta\right)=\sum_{\ell=0}^{\ell_{0}}\tilde{%
g}_{\ell}(\zeta,\hat{P})\zeta^{h}\log^{\ell}\zeta\,,\qquad\text{when}\ \ \ell_%
{1}>\ell_{0}\,,$$
(217)
for suitable $\tilde{g}_{\ell}.$
Introduce the linear differential operator (w.r.t. $\zeta$) of order $3n^{2}-3n:$
$$\mathcal{L}:=L^{3n-3}(\partial_{\zeta}\circ L^{3n-3})^{n-1}\,.$$
(218)
Non-degeneracy of the derivatives of $\mathtt{f}$
Let us decompose $\mathtt{f}$
in (210) as
$$\mathtt{f}=\zeta^{n-1}\varphi_{3n-3}(0,\hat{P})\log^{3n-3}\zeta+\tilde{\mathtt%
{f}}+\tilde{R}\,,$$
with
$$\tilde{\mathtt{f}}:=\zeta^{n-1}\sum_{\ell=0}^{3n-4}\varphi_{\ell}(\zeta,\hat{P%
})\log^{\ell}\zeta\ +\ \eta\sum_{\ell=0}^{3n-4}\chi_{\ell}(\zeta,\hat{P})\log^%
{\ell}\zeta\,,$$
and444444Note that the function
$\frac{\varphi_{3n-3}(\zeta,\hat{P})-\varphi_{3n-3}(0,\hat{P})}{\zeta}$ is analytic.
$$\tilde{R}:=\zeta^{n}\left(\frac{\varphi_{3n-3}(\zeta,\hat{P})-\varphi_{3n-3}(0%
,\hat{P})}{\zeta}\right)\log^{3n-3}\zeta+R$$
We want to evaluate $\mathcal{L}\mathtt{f}.$
We have
$$\mathcal{L}\,\Big{(}\zeta^{n-1}\varphi_{3n-3}(0,\hat{P})\log^{3n-3}\zeta\Big{)%
}=(3n-3)!\big{(}(n-1)!\big{)}^{3n-2}\varphi_{3n-3}(0,\hat{P})+\zeta\sum_{\ell=%
0}^{3n-2}\tilde{\varphi}_{\ell}(\zeta,\hat{P})\log^{\ell}\zeta\,,$$
for suitable $\tilde{\varphi}_{\ell}.$
By (215) and
(217)
we get
$$\mathcal{L}\,\tilde{\mathtt{f}}\,=\,\zeta\sum_{\ell=0}^{3n-4}\tilde{\chi}_{%
\ell}(\zeta,\hat{P})\log^{\ell}\zeta\,,$$
for suitable $\tilde{\chi}_{\ell}.$
Finally, by (213) and
(216),
we have that
$$\mathcal{L}\tilde{R}=\zeta\sum_{\ell=0}^{3n-2}\tilde{\psi}_{\ell}(\zeta,\hat{P%
})\log^{\ell}\zeta\,,$$
(with $R$ defined in (212))
for suitable $\tilde{\psi}_{\ell}$.
Recollecting we get
$$(\mathcal{L}\mathtt{f})(\zeta,\hat{P})=\mathtt{c}_{k}(\hat{P})\ +\ \zeta\sum_{%
\ell=0}^{3n-2}\tilde{\mathtt{f}}_{\ell}(\zeta,\hat{P})\log^{\ell}\zeta\,,$$
(219)
for suitable
$\tilde{\mathtt{f}}_{\ell}(\zeta,\hat{P})$
analytic and uniformly bounded in a uniform neighborhood of zero (and $\hat{P}\in\hat{D}$)
and
where
$$\mathtt{c}_{k}(\hat{P}):=(3n-3)!\big{(}(n-1)!\big{)}^{3n-2}\varphi_{3n-3}(0,%
\hat{P})\,,$$
(220)
satisfies, by (211),
$$\inf_{\hat{P}\in\hat{D}}|\mathtt{c}_{k}(\hat{P})|\geq c(\mathfrak{p})\mathtt{d%
}_{k}>0\,.$$
(221)
By (219)
$$\inf_{0<\zeta\leq\zeta_{0}}\inf_{\hat{P}\in\hat{D}}\big{|}(\mathcal{L}\mathtt{%
f})(\zeta,\hat{P})\big{|}\ \geq\ c(\mathfrak{p})\mathtt{d}_{k}/2>0\,,\qquad{%
\rm for\ a\ suitable}\ \ \ \zeta_{0}=\zeta_{0}(\mathfrak{p})>0\,.$$
(222)
By (218) we have that
$$(\mathcal{L}\mathtt{f})(\zeta,\hat{P})=\sum_{d=0}^{m_{n}}a_{d}(\zeta)\partial_%
{\zeta}^{d}\mathtt{f}(\zeta,\hat{P})\,,\qquad\text{where}\ \ m_{n}:=3n^{2}-3n\,,$$
(223)
for suitable polynomials $a_{d}(\zeta).$
Then by (222), taking in case $\zeta_{0}$ smaller,
we get
$$\inf_{0<\zeta\leq\zeta_{0}}\inf_{\hat{P}\in\hat{D}}\max_{1\leq d\leq m_{n}}|%
\partial_{\zeta}^{d}\mathtt{f}(\zeta,\hat{P})|/d!\geq c(\mathfrak{p})\mathtt{d%
}_{k}>0$$
(224)
(with a smaller $c(\mathfrak{p})$).
The measure of sublevels of $\mathtt{f}$
We need the following result, which is proved in the appendix
Lemma 6.1
Let $f\in C^{m+1}([a,b])$
and assume that for some $m\geq 1$
$$\min_{x\in[a,b]}\max_{1\leq d\leq m}|\partial_{x}^{d}f(x)|/d!=:\xi_{m}>0\,.$$
(225)
Then for $0<\mu<1$
$${\rm meas}\big{(}\{x\in[a,b]\ \ :\ \ |f(x)|\leq\mu\}\big{)}\leq\frac{m(M+1)(b-%
a+2\mu^{1/m+1})}{\xi_{m}}\mu^{\frac{1}{m(m+1)}}\,,$$
with
$M:=\max_{x\in[a,b],\ 2\leq d\leq m+1}|\partial_{x}^{d}f(x)|/d!$
We apply Lemma 6.1 with
$$f\rightsquigarrow\mathtt{f}\,,\quad m\rightsquigarrow m_{n}\,,\quad a%
\rightsquigarrow\zeta_{1}\,,\quad b\rightsquigarrow\zeta_{0}\,,\quad\xi_{m}%
\rightsquigarrow c(\mathfrak{p})\mathtt{d}_{k}\ \ \text{(in\ \eqref{loosing})}%
\,,\quad x\rightsquigarrow\zeta\,,\quad$$
with $\zeta_{1}$ to be chosen later.
By (210) we have that454545Denoting,
as usual, the $\zeta_{1}/2$-complex-neighborhood of the real interval
$[\zeta_{1},\zeta_{0}]$ by $[\zeta_{1},\zeta_{0}]_{\zeta_{1}/2}$.
$$\sup_{\hat{P}\in\hat{D}}\sup_{[\zeta_{1},\zeta_{0}]_{\zeta_{1}/2}}|\mathtt{f}(%
\zeta,\hat{P})|\lessdot|\log^{2n-2}\zeta_{1}|\leq 1/\zeta_{1}$$
for $\zeta_{1}$ small enough.
Then, by Cauchy estimates, we get that $M$ in Lemma 6.1
satisfies
$$M\lessdot 1/\zeta_{1}^{m_{n}+2}\,.$$
By Lemma 6.1 we get,
for $0<\mu<1,$
$${\rm meas}\big{(}\{\zeta\in[\zeta_{1},\zeta_{0}]\ \ :\ \ |\mathtt{f}(\zeta,%
\hat{P})|\leq\mu\}\big{)}\lessdot\mu^{\frac{1}{m_{n}(m_{n}+1)}}/\mathtt{d}_{k}%
\zeta_{1}^{m_{n}+2}\,,$$
for every464646Note that the hidden constant
in $\lessdot$ is independent of $\hat{P}\in\hat{D}.$
$\hat{P}\in\hat{D}.$
We can optimize the choice of $\zeta_{1}$ taking
$$\zeta_{1}=\mu^{\frac{1}{m_{n}(m_{n}+1)}}/\zeta_{1}^{m_{n}+2}\,,\qquad\text{%
namely}\ \ \ \zeta_{1}:=\mu^{\frac{1}{m_{n}(m_{n}+1)(m_{n}+3)}}\,,$$
so that
$${\rm meas}\big{(}\{\zeta\in(0,\zeta_{0}]\ \ :\ \ |\mathtt{f}_{k}(\zeta,\hat{P}%
)|\leq\mu\}\big{)}\lessdot\mu^{\frac{1}{m_{n}(m_{n}+1)(m_{n}+3)}}/\mathtt{d}_{%
k}\,,$$
for every $\hat{P}\in\hat{D}.$
Since by the first equality in (6)
$$\zeta^{n}\big{(}\partial_{E}P_{n}(E_{+}^{(2j-1)}(\hat{P})-\zeta,\hat{P})\big{)%
}^{3n}\leq 1$$
for $\zeta_{0}$ small,
we get, recalling (208),
$${\rm meas}\big{(}\{\zeta\in(0,\zeta_{0}]\ \ :\ \ |d^{i}_{k}(E_{+}^{(2j-1)}(%
\hat{P})-\zeta,\hat{P})|\leq\mu\}\big{)}\lessdot\mu^{\frac{1}{m_{n}(m_{n}+1)(m%
_{n}+3)}}/\mathtt{d}_{k}\,,$$
(226)
for every $\hat{P}\in\hat{D},$
where $d_{k}^{i}$ was defined in (202).
Recalling (203), we get
$${\rm meas}\big{(}\{E\in\big{[}E_{+}^{(2j-1)}(\hat{P})-\zeta_{0},E_{+}^{(2j-1)}%
(\hat{P})\big{)}\ \ :\ \ |d^{i}_{k}(E,\hat{P})|\leq\mu\}\big{)}\lessdot\mu^{%
\frac{1}{m_{n}(m_{n}+1)(m_{n}+3)}}/\mathtt{d}_{k}\,,$$
(227)
for every
$\hat{P}\in\hat{D}$.
Conslusion of the proof in the case
$i$ odd, close to maxima of the potential
Recalling (202) and (161), we have
$${\rm det}\left[\partial_{PP}\big{(}\hat{h}_{k}(\hat{P})+\mathtt{E}^{(i)}_{k}(P%
)\big{)}\right]=d^{i}_{k}(\mathtt{E}^{(i)}_{k}(P),\hat{P})\,.$$
(228)
Recalling that, by (163), we have
$E_{+}^{(2j-1)}(\hat{P})=\mathtt{E}^{(2j-1)}_{k}(\hat{P},a_{+}^{(2j-1)}(\hat{P}%
))\,,$ we set
$$\mathtt{r}_{0}(\hat{P}):=a_{+}^{(2j-1)}(\hat{P})-P_{n}\Big{(}E_{+}^{(2j-1)}(%
\hat{P})-\zeta_{0},\hat{P}\Big{)}=P_{n}\Big{(}E_{+}^{(2j-1)}(\hat{P}),\hat{P}%
\Big{)}-P_{n}\Big{(}E_{+}^{(2j-1)}(\hat{P})-\zeta_{0},\hat{P}\Big{)}\,.$$
(229)
Note that by (229)
and (333)
(which relies on Proposition
5.2) there exists $\mathtt{r}_{0}>0$,
which, when (A3) holds,
depends only on $n,s_{0},r_{0}$, otherwise depend also on $F^{0}$,
such that
$$\mathtt{r}_{0}(\hat{P})\geq\mathtt{r}_{0}\,,\qquad\hat{P}\in\hat{D}\,.$$
(230)
For every $\hat{P}\in\hat{D},$
we now consider the change of variable
$P_{n}=P_{n}(E,\hat{P}).$
By (227) and (228),
and noting that (recall (204))
$$0<\partial_{E}P_{n}(E,\hat{P})\lessdot\big{|}\log(E_{+}^{(2j-1)}(\hat{P})-E)|\,,$$
(231)
we
get474747 Let
$\mathcal{E}(\hat{P})\subseteq\big{[}E_{+}^{(2j-1)}(\hat{P})-\zeta_{0},E_{+}^{(%
2j-1)}(\hat{P})\big{)}$
and
$\mathcal{P}(\hat{P})\subseteq\big{[}a_{+}^{(2j-1)}(\hat{P})-\mathtt{r}_{0},a_{%
+}^{(2j-1)}(\hat{P})\big{)}$ be
the sets whose measures are estimated in (227)
and (232); so that
$\mathcal{P}(\hat{P})=P_{n}(\mathcal{E}(\hat{P}),\hat{P}).$
Then, denoting by $\mathcal{Z}(\hat{P})\subseteq(0,\zeta_{0}]$ the
set whose measure
$\mu_{0}:={\rm meas}(\mathcal{Z}(\hat{P}))$
is estimated in (226), we have
$\displaystyle{\rm meas}(\mathcal{P}(\hat{P}))=\int_{\mathcal{P}(\hat{P})}dP_{n%
}=\int_{\mathcal{E}(\hat{P})}\partial_{E}P_{n}(E,\hat{P})dE\stackrel{{%
\scriptstyle\eqref{stracciatella}}}{{\lessdot}}\int_{\mathcal{E}(\hat{P})}|%
\log(E_{+}^{(2j-1)}(\hat{P})-E)|dE$
$\displaystyle\lessdot\int_{\mathcal{Z}(\hat{P})}|\log\zeta|d\zeta=\int_{%
\mathcal{Z}(\hat{P})\cap(0,\mu_{0}]}|\log\zeta|d\zeta+\int_{\mathcal{Z}(\hat{P%
})\cap(\mu_{0},\zeta_{0}]}|\log\zeta|d\zeta\leq\mu_{0}|\log\mu_{0}|+|\log\zeta%
_{0}|\mu_{0}\lessdot\mu_{0}|\log\mu_{0}|\,.$
Therefore by (226)
${\rm meas}(\mathcal{P}(\hat{P}))\lessdot\left(\mu^{\frac{1}{m_{n}(m_{n}+1)(m_{%
n}+3)}}/\mathtt{d}_{k}\right)\left|\log\left(\mu^{\frac{1}{m_{n}(m_{n}+1)(m_{n%
}+3)}}/\mathtt{d}_{k}\right)\right|\,.$
$$\displaystyle{\rm meas}\big{(}\{P_{n}\in\big{[}a_{+}^{(2j-1)}(\hat{P})-\mathtt%
{r}_{0},a_{+}^{(2j-1)}(\hat{P})\big{)}\ \ :\ \ |{\rm det}\left[\partial_{PP}%
\big{(}\hat{h}_{k}(\hat{P})+\mathtt{E}^{(2j-1)}(P)\big{)}\right]|\leq\mu\}\big%
{)}\lessdot\mu^{\mathtt{a}_{n}}\mathtt{d}_{k}^{-2}\,,$$
$$\displaystyle\mathtt{a}_{n}:=\frac{1}{27n^{6}}<{\frac{1}{m_{n}(m_{n}+1)(m_{n}+%
3)}}$$
(232)
(recall (223)) uniformly in $\hat{P}\in\hat{D}.$
Then by Fubini theorem we get
$${\rm meas}\big{(}\{\ P\ |\ P_{n}\in\big{[}a_{+}^{(2j-1)}(\hat{P})-\mathtt{r}_{%
0},a_{+}^{(2j-1)}(\hat{P})\big{)},\ \hat{P}\in\hat{D}\,,\ \ |{\rm det}[%
\partial_{PP}(\hat{h}_{k}+\mathtt{E}^{(2j-1)})]|\leq\mu\}\big{)}\lessdot\mu^{%
\mathtt{a}_{n}}\mathtt{d}_{k}^{-2}\,.$$
(233)
This conclude the proof close to the maxima of the odd case
$i=2j-1$.
The case $i$ even close to maxima
The proof is analogous to the odd case,
leading to
$${\rm meas}\big{(}\{\ P\ |\ P_{n}\in\big{[}a_{+}^{(2j)}(\hat{P})-\mathtt{r}_{0}%
,a_{+}^{(2j)}(\hat{P})\big{)},\ \hat{P}\in\hat{D}\,,\ \ |{\rm det}[\partial_{%
PP}(\hat{h}_{k}+\mathtt{E}^{(2j)})]|\leq\mu\}\big{)}\lessdot\mu^{\mathtt{a}_{n%
}}\mathtt{d}_{k}^{-2}\,,$$
(234)
$${\rm meas}\big{(}\{\ P\ |\ P_{n}\in\big{(}a_{-}^{(2j)}(\hat{P}),a_{-}^{(2j)}(%
\hat{P})+\mathtt{r}_{0}\big{]},\ \hat{P}\in\hat{D}\,,\ \ |{\rm det}[\partial_{%
PP}(\hat{h}_{k}+\mathtt{E}^{(2j)})]|\leq\mu\}\big{)}\lessdot\mu^{\mathtt{a}_{n%
}}\mathtt{d}_{k}^{-2}\,.$$
(235)
Far away from maxima
We now study the point far away from maxima, where we note that the second derivatives of
the functions $\mathtt{E}^{(i)}$ are uniformly bounded
(see Lemma C.3).
We have
$${\rm det}\left[\partial_{PP}\big{(}\hat{h}_{k}(\hat{P})+\mathtt{E}^{(i)}(P)%
\big{)}\right]={\rm det}\left[\partial_{\hat{P}\hat{P}}\hat{h}_{k}(\hat{P})%
\right]\cdot\partial_{P_{n}P_{n}}\mathtt{E}^{(i)}(P)\ +\ O(\vartheta)\stackrel%
{{\scriptstyle\eqref{lovebuzz}}}{{=}}2^{n-1}\kappa^{-n}\partial_{P_{n}P_{n}}%
\mathtt{E}^{(i)}(P)\ +\ O(\eta)\,,$$
(236)
valid in the sets484848Recall that $a_{-}^{(2j-1)}(\hat{P})=0$ as defined in(163).
$$\displaystyle\Big{\{}\ P=(\hat{P},P_{n})\ |\ P_{n}\in\big{(}0,a_{+}^{(2j-1)}(%
\hat{P})-\mathtt{r}_{0}/2\big{]}\,,\ \hat{P}\in\hat{D}\ \Big{\}}\,,$$
(237)
$$\displaystyle\Big{\{}\ P=(\hat{P},P_{n})\ |\ P_{n}\in\big{[}a_{-}^{(2j)}(\hat{%
P})+\mathtt{r}_{0}/2,a_{+}^{(2j)}(\hat{P})-\mathtt{r}_{0}/2\big{]}\,,\ \hat{P}%
\in\hat{D}\ \Big{\}}\,,$$
(238)
when $i$ is odd, respectively even.
We have to distinguish the cases
when (A2) or (A3) hold.
The case when (A2) holds
We note that, by Proposition 5.2, we can extend the Hamiltonian
$\mathtt{E}^{(2j-1)}(P)$
in a complex
$4\mathtt{r}_{0}$-neighborhood494949Reducing, in case, $\mathtt{r}_{0}=\mathtt{r}_{0}(\mathfrak{p})>0$. of zero so that $\mathtt{E}^{(2j-1)}(P)$ has holomorphic extension on the complex
domain
$$\hat{D}_{r_{0}}\,\times\,\big{[}-2\mathtt{r}_{0},\ a_{+}^{(2j-1),0}-\mathtt{r}%
_{0}/4\big{]}_{\mathtt{r}_{0}/8}\,,$$
(239)
Then
(236)
holds in the domain (239).
Let us define the intervals
$$\mathtt{I}^{(2j-1)}:=\big{[}-\mathtt{r}_{0}/4,a_{+}^{(2j-1),0}-\mathtt{r}_{0}/%
4\big{]}\,,\qquad\mathtt{I}^{(2j)}:=\big{[}a_{-}^{(2j),0}+3\mathtt{r}_{0}/4,a_%
{+}^{(2j)}(\hat{P})-3\mathtt{r}_{0}/4\big{]}$$
(240)
Since we are far away from hyperbolic equilibria, by Lemma C.3
we get that
$$\sup_{\hat{D}_{r_{0}}\times\mathtt{I}^{(i)}_{\mathtt{r}_{0}/8}}\|\partial_{PP}%
\mathtt{E}^{(i)}(P)\|\lessdot 1\,,\qquad\forall\,0\leq i\leq 2N\,.$$
(241)
Note that (236)
holds in
$\hat{D}\times\mathtt{I}^{(i)}$
for every $i$
and also that, for $\eta$ small enough,
$$\displaystyle\Big{(}\hat{D}\times\mathtt{I}^{(2j-1)}\Big{)}\ \cup\ \Big{\{}\ P%
\ |\ P_{n}\in\big{[}a_{+}^{(2j-1)}(\hat{P})-\mathtt{r}_{0},a_{+}^{(2j-1)}(\hat%
{P})\big{)},\ \hat{P}\in\hat{D}\Big{\}}\ \supseteq\ \mathfrak{P}^{(2j-1)}(0)$$
(242)
$$\displaystyle\Big{(}\hat{D}\times\mathtt{I}^{(2j)}\Big{)}\ \cup\ \Big{\{}\ P\ %
|\ P_{n}\in\big{(}a_{-}^{(2j)}(\hat{P}),a_{-}^{(2j)}(\hat{P})+\mathtt{r}_{0}%
\big{]}\cup\big{[}a_{+}^{(2j)}(\hat{P})-\mathtt{r}_{0},a_{+}^{(2j)}(\hat{P})%
\big{)},\ \hat{P}\in\hat{D}\Big{\}}\ =\ \mathfrak{P}^{(2j)}(0)$$
(defined in (164)).
Let us consider now the finitely many non constant analytic functions505050$\mathtt{E}^{(i),0}$
being the inverse of $P_{n}^{(i),0}.$
$$P_{n}\to\partial_{P_{n}P_{n}}\mathtt{E}^{(i),0}(P_{n})\,,\qquad 0\leq i\leq 2N\,.$$
By analyticity we have that there exist
$m=m(F^{0})\geq 1$ and $\xi=\xi(F^{0})>0,$
depending on the function
$F^{0}$ introduced in (96),
such that
$$\min_{P_{n}\in\mathtt{I}^{(i)}}\max_{1\leq d\leq m}|\partial_{P_{n}}^{d}%
\partial_{P_{n}P_{n}}\mathtt{E}^{(i),0}(P_{n})|/d!\geq 2\xi>0\,,\qquad\forall 0%
\leq i\leq 2N\,.$$
By Lemma C.3
$$\sup_{\hat{D}_{r_{0}}\times\mathtt{I}^{(i)}_{\mathtt{r}_{0}/8}}|\partial_{P_{n%
}}\mathtt{E}^{(i)}(P)-\partial_{P_{n}}\mathtt{E}^{(i),0}(P_{n})|\leq C\eta\,,%
\qquad\forall 0\leq i\leq 2N\,,$$
(243)
where $C>1$ was defined in Proposition
5.2.
By Cauchy
estimates
we get, for $\eta$ small enough depending on $F^{0}$,
$$\inf_{P\in\hat{D}\times\mathtt{I}^{(i)}}\max_{1\leq d\leq m}|\partial_{P_{n}}^%
{d}\partial_{P_{n}P_{n}}\mathtt{E}^{(i)}(P)|/d!\geq\xi>0\,,\qquad\forall 0\leq
i%
\leq 2N\,.$$
(244)
Recalling (236)
we have
$${\rm det}\left[\partial_{PP}\big{(}\hat{h}_{k}(\hat{P})+\mathtt{E}^{(i)}(P)%
\big{)}\right]=2^{n-1}\kappa^{-n}\partial_{P_{n}P_{n}}\mathtt{E}^{(i)}(P)\ +\ %
O(\vartheta)=2^{n-1}\kappa^{-n}\Big{(}\partial_{P_{n}P_{n}}\mathtt{E}^{(i)}(P)%
\ +\ O(\kappa^{n}\eta)\Big{)}\,.$$
By (109) we get that the term515151More precisely
$|O(\kappa^{n}\eta)|\leq C\kappa^{n}\eta.$ $O(\kappa^{n}\eta)$ above is negligible, together with its derivatives of any order; then by
(244) we obtain
$$\inf_{P\in\hat{D}\times\mathtt{I}^{(i)}}\max_{1\leq d\leq m}\left|\partial_{P_%
{n}}^{d}{\rm det}\left[\partial_{PP}\big{(}\hat{h}_{k}(\hat{P})+\mathtt{E}^{(i%
)}(P)\big{)}\right]\right|/d!\geq 2^{n-2}\kappa^{-n}\xi>0\,,\qquad\forall 0%
\leq i\leq 2N\,.$$
(245)
We can apply Lemma 6.1
uniformly in $\hat{P}\in\hat{D}$
with
$$f(\cdot)\rightsquigarrow{\rm det}\left[\partial_{PP}\big{(}\hat{h}_{k}(\hat{P}%
)+\mathtt{E}^{(i)}(\hat{P},\cdot)\big{)}\right]\,,\ \ x\rightsquigarrow P_{n}%
\,,\ \ [a,b]\rightsquigarrow\mathtt{I}^{(i)}\,,\ \ \xi_{m}\rightsquigarrow 2^{%
n-2}\kappa^{-n}\xi$$
and $M\lessdot 1$ (by (241)
and Cauchy estimates); then we get
$${\rm meas}\left(\left\{P_{n}\in\mathtt{I}^{(i)}\ |\ \left|{\rm det}\left[%
\partial_{PP}\big{(}\hat{h}_{k}(\hat{P})+\mathtt{E}^{(i)}(P)\big{)}\right]%
\right|\leq\mu\right\}\right)\leq C\kappa^{n}\mu^{\frac{1}{m(m+1)}}\,,$$
uniformly in $\hat{P}\in\hat{D}.$
Then by Fubini theorem we have
$${\rm meas}\left(\left\{P\in\hat{D}\times\mathtt{I}^{(i)}\ |\ \left|{\rm det}%
\left[\partial_{PP}\big{(}\hat{h}_{k}(\hat{P})+\mathtt{E}^{(i)}(P)\big{)}%
\right]\right|\leq\mu\right\}\right)\leq C\kappa^{n}\mu^{\frac{1}{m(m+1)}}\,.$$
(246)
Recalling the definition of $\mathtt{a}_{n}$ in (232)
we choose
$$\mathtt{c}\geq\max\{6^{3}n^{6},m(m+1)\}$$
Recalling
(242)
by (233),(234),
(235) and (246)
we get (197) when
(A2) holds.
The case when (A3) holds
We are in the cosine-like case (recall (100)); by Lemma C.5
below
(see (341)), (236) and
(109)
we have that
$$\kappa^{-n}\lessdot\left|{\rm det}\left[\partial_{PP}\big{(}\hat{h}_{k}(\hat{P%
})+\mathtt{E}^{(i)}(P)\big{)}\right]\right|$$
for all the values of $P$ in (237), (238) respectively.
Then, by (110)
$$\left|{\rm det}\left[\partial_{PP}\big{(}\hat{h}_{k}(\hat{P})+\mathtt{E}^{(i)}%
(P)\big{)}\right]\right|>\mu$$
(247)
again for all the values of $P$ in (237), (238) respectively.
Then
(232), (233),
(235), (247)
prove Proposition 6.1 in the case when
(A3) holds.
This concludes the proof of
Proposition 6.1.
7 Proof of the Main Theorem
In this final section we will show the existence of a high density (almost exponential) of Kolmogorov’s tori of different topologies in
all neighbourhoods of simple resonances.
7.1 Application of the Structure Theorem
Fix $k\in\mathbb{Z}^{n}_{*,K}.$
In this subsection we will apply
the Structure Theorem 4.1
to the “effective part” of the Hamiltonian $H_{k}$ in (80).
Namely we apply Theorem
4.1 with
$$\displaystyle{\mathcal{H}}(I^{\prime},\varphi^{\prime})\rightsquigarrow\frac{1%
}{\kappa}\|I^{\prime}\|^{2}+\frac{1}{\delta_{k}}G^{k}(\varsigma_{k}I^{\prime},%
k\cdot\varphi^{\prime})\,,\quad\mathfrak{G}(I^{\prime},x)\rightsquigarrow\frac%
{1}{\delta_{k}}G^{k}(\varsigma_{k}I^{\prime},x)\,,\quad F^{0}\rightsquigarrow%
\frac{1}{\delta_{k}}F^{k}\,,$$
$$\displaystyle{\mathcal{D}}\rightsquigarrow D^{{}^{\prime}k}\,,\quad\hat{D}%
\rightsquigarrow\frac{1}{\varsigma_{k}}\hat{Z}_{k}\,,\quad R_{0}%
\rightsquigarrow R_{0,k}:=\frac{K^{\nu+1}}{\sqrt{2\delta_{k}}\|k\|}\,,\quad r_%
{0}\rightsquigarrow 4\,,\quad s_{0}\rightsquigarrow s/4\,.$$
(248)
where $\delta_{k},$ $G^{k}$ and $F^{k},$
$\varsigma_{k},$
$D^{{}^{\prime}k},$
$\hat{Z}_{k},$
were defined in
(76),
(66),
(79),
(81),
(29).
We note that with such positions
we have that
$r^{\prime}_{k}$ defined in (81)
satisfies
$$r^{\prime}_{k}=K^{\nu+1}/\sqrt{8\delta_{k}}\geq r^{\prime}=cn|k|_{\infty}r_{0}\,,$$
(249)
where $r^{\prime}$ was defined in
(95).
Indeed (249) follows
by (77) taking $\varepsilon$
small enough (recall (24)).
We also note that
$s_{*}$ defined in (61)
satisfies
$$s_{*}\geq 3s_{0}\,,$$
(250)
taking $\varepsilon$ small enough
(recall again (24)).
Note that, by (87)
$$\|g\|_{s_{0}}\leq\coth^{n}\left(\frac{s^{\prime}-s_{0}}{2}\right)|g|_{s^{%
\prime}}\,,\qquad\forall\,g\in{\mathcal{A}}^{n}_{s^{\prime}}\,.$$
(251)
Remark 7.1
Here we take $\varepsilon\leq\varepsilon_{0}$
small enough. Obviously
$\varepsilon_{0}$ is small
uniformly
on $k.$
We now verify that the hypotheses of Theorem 4.1 hold.
Let us start with (A1).
We see that (97) follow
by525252Recall also (81) and (251)
with $n=1$.
(69), resp.
(75), if $|k|\leq K_{s}(\delta),$
resp. $K_{s}(\delta)<|k|\leq K,$
taking
$$\eta_{*}:=\coth(s/24)\frac{2^{10}ne^{s}}{\delta sK^{\frac{4\nu-n-11}{2}}}\,.$$
(252)
Moreover the inequality in (98) holds
(recall the definition of $R_{0}$ in (248))
taking $\varepsilon$
small enough (recall (24)).
We now check that (A3)
holds
when $K_{s}(\delta)<|k|\leq K,$
taking the constant ${\,c\,}$ in (5)
large enough.
Indeed we have that
by (72) and (251)
(with $n=1$, also recall (76))
$\delta_{k}^{-1}F^{k}$ is $\gamma$-cosine-like
taking
$$\gamma:=\coth(s/24)\left(\frac{2|k|^{\frac{n+3}{2}}e^{-|k|s/4}}{\delta}+\frac{%
2^{10}ne^{s}}{\delta sK^{\frac{4\nu-n-11}{2}}}\right)\,.$$
(253)
Then (100) holds taking
$c$ in (5)
large enough
depending on $n$ in order to adjust the first addendum and, then, $\varepsilon$ small enough
in order to adjust the second addendum.
We finally check that (A2)
holds
when $|k|\leq K_{s}(\delta).$
We first note that in this case
$\delta_{k}^{-1}F^{k}=F^{k}$ since
$\delta_{k}=1$ (recall (76)).
By (68) and (87) we get
$$\|F^{k}\|_{s_{0}}\leq\coth(3s/8)=:M\,.$$
Moreover every
$F^{k}$ is $(\beta_{k},M)$-Morse-non-degenerate for some $\beta_{k}>0$
by (P2) and (P3) of Definition
1.1.
Then we can take
$$\beta:=\min_{|k|\leq K_{s}(\delta)}\beta_{k}>0\,,$$
since we are taking the minimum over a finite set.
We conclude that
every
$F^{k}$ is $(\beta,M)$-Morse-non-degenerate.
Remark 7.2
A crucial fact is that we can choose the constant
$\mathtt{c}$ appearing in Theorem 4.1, uniformly
in $k\in Z^{n}_{*,K}$.
Indeed for all the cases
with $K_{s}(\delta)<|k|\leq K,$
$\mathtt{c}$ only depends
on $n,s_{0},r_{0},$ namely, in view of (248), on
$n$ and $s.$
In the finite number of cases
$|k|\leq K_{s}(\delta)$ the constant
$\mathtt{c}$ actually depends
on $k$ (since it depends on the particular for of $F^{k}$);
however we can the minimum
over all this finite number of constants obtaining a
constant $\mathtt{c}=\mathtt{c}(n,s)>0$
for which Theorem 4.1 holds
uniformly
in $k\in\mathbb{Z}^{n}_{*,K}.$
Finally we have that
(109) and, a fortiori
(102), are
simultaneously, namely for every $k\in\mathbb{Z}^{n}_{*,K},$ satisfied taking $\varepsilon$
small enough
(recall (24)).
Then, as a corollary of Theorem
4.1, we get the following
result (where we denote by $N_{k}$
the number of maxima/minima of
$F^{k}$).
Proposition 7.1
Let $\varepsilon$ be small enough.
Let $\theta>0$ and535353The constant $\mathtt{c}$ is defined in Remark 7.2.
$$0<\mu\leq 1/\mathtt{c}K^{2n}\,,$$
(254)
where $\mathtt{c}=\mathtt{c}(n,s)>0$
was defined in Remark 7.2.
For every $k\in\mathbb{Z}^{n}_{*,K}$ and
$0\leq i\leq 2N_{k}$ there exist
i) disjoint subsets
$\mathfrak{C}^{i}_{k}(\theta,\mu)\subseteq D^{{}^{\prime}k}\times\mathbb{T}^{n}$
decreasing w.r.t. $\theta$ and $\mu$, with545454$D^{{}^{\prime}k}_{\sharp}$ was defined in (83).
$${\rm\,meas\,}\Big{(}\big{(}D^{{}^{\prime}k}_{\sharp}\times\mathbb{T}^{n}\big{)%
}\setminus\bigcup_{0\leq i\leq 2N_{k}}\mathfrak{C}^{i}_{k}(\theta,\mu)\Big{)}%
\leq\mathtt{c}\big{(}\theta|\ln\theta|+K^{4\nu}\mu^{1/\mathtt{c}}\big{)}\,;$$
(255)
ii) $\mathtt{B}^{i}_{k}(\theta,\mu)\subseteq\mathbb{R}^{n}$, decreasing w.r.t. $\theta$
and $\mu$, with555555With
$R_{0,k}$ defined in (248) and $c$
in (17) respectively.
$${\rm diam}\big{(}\mathtt{B}^{i}_{k}(\theta,\mu)\big{)}\leq 2c\big{(}R_{0,k}+%
\varsigma_{k}^{-1}{\rm diam}(\hat{Z}_{k})\big{)}\,;$$
(256)
iii) holomorphic
symplectomorphisms565656
$r^{\prime}_{k}$ defined in (81).
$$\Psi^{i}_{k}\ :\ \big{(}\mathtt{B}^{i}_{k}(\theta,\mu)\big{)}_{\rho^{\prime}}%
\times\mathbb{T}^{n}_{\sigma^{\prime}}\ \to\ D^{{}^{\prime}k}_{r^{\prime}_{k}}%
\times\mathbb{T}^{n}_{s/4}\,,\qquad\mbox{with}\quad\rho^{\prime}:=\frac{\theta%
}{\mathtt{c}K^{n-1}}\,,\quad\ \sigma^{\prime}:=\frac{1}{{\mathtt{c}}K^{n-1}|%
\log\theta|}\,,$$
(257)
with
$$\Psi^{i}_{k}\Big{(}\mathtt{B}^{i}_{k}(\theta,\mu)\times\mathbb{T}^{n}\Big{)}=%
\mathfrak{C}^{i}_{k}(\theta,\mu)\,,$$
(258)
such that575757$H_{k}$
defined in (80).
$$H_{k}\circ\Psi^{i}(p,q)=:h^{(i)}_{k}(p)+f^{(i)}_{k}(p,q)\,.$$
(259)
with585858By (82)
and (251), taking $\varepsilon$ small enough,
depending on $s$ and $\delta$.
$$\|f^{(i)}_{k}\|_{\mathtt{B}^{i}_{k}(\theta,\mu),\rho^{\prime},\sigma^{\prime}}%
\leq\varepsilon^{\frac{s}{5}|\log\varepsilon|^{3}}$$
(260)
Furthermore
$$\left|{\rm det}\left(\partial_{pp}h^{(i)}_{k}(p)\right)\right|\geq\mu\,,\qquad%
\forall\,\ 0\leq i\leq 2N\,,\ \ |k|\leq K\,,\qquad\forall\ p\in{\mathtt{B}}^{i%
}_{k}(\theta,\mu)\ .$$
(261)
Finally595959Recall (108).
$$\|\partial_{pp}h^{(i)}\|_{\mathtt{B}^{i}_{k}(\theta,\mu),\rho^{\prime}}\leq%
\mathtt{c}/\theta\,,\qquad\text{for }\ \ \ 0\leq i\leq 2N\,.$$
(262)
7.2 Application of the KAM theorem
Let us start by stating a quantitative version, suitable for our purposes, of the classical KAM Theorem; for references, discussions and extensions we refer to [2] and references therein; see also [7] for a nice divulgative account of KAM theory.
Theorem 7.1
Fix $n\geq 2$ and $\tau>n-1$. Let
$\mathtt{D}$ be any non–empty, bounded subset of ${\mathbb{R}}^{n}$.
Let
$$\mathtt{H}(p,q):=h(p)+f(p,q)$$
real–analytic on $\mathtt{D}_{r_{0}}\times\mathbb{T}^{n}_{s},$
for some $r_{0}>0$ and $0<s\leq 1$, and having finite norms:
$$\mathtt{M}:=\|\partial_{pp}h\|_{\mathtt{D},r_{0}}\,,\qquad\qquad\|f\|_{\mathtt%
{D},r_{0},s}\,.$$
(263)
Assume that the frequency map $p\in\mathtt{D}\to\omega=\partial_{p}h$ is a local diffeomorphism, namely, assume:
$$d:=\inf_{\mathtt{D}}|\det\partial_{pp}h|>0\,.$$
(264)
Define
$$\mathtt{m}:=\frac{d}{\mathtt{M}^{n}}\leq 1\ .$$
(265)
Then there exists a positive constant $c<1$, depending only on $n$ and $\tau$, such that, if
$$\epsilon:=\frac{\|f\|_{\mathtt{D},r_{0},s}}{\mathtt{M}r_{0}^{2}}\leq c\,%
\mathtt{m}^{8}\ s^{4\tau+4}\ ,$$
(266)
then the following holds.
Define
$$\alpha:=\frac{c}{\mathtt{m}\,s^{3\tau+3}}\,(\mathtt{M}r_{0})\,\sqrt{\epsilon}%
\ ,\qquad\hat{r}:=\,\mathtt{m}^{2}r_{0}\ ,\qquad r_{\epsilon}:=\frac{1}{c\,%
\mathtt{m}}\,\sqrt{\epsilon}\,r_{0}\ .$$
(267)
Then, there exists a positive measure
set ${\cal T}_{\alpha}\subseteq(\mathtt{D}_{\hat{r}}\cap\mathbb{R}^{n})\times{%
\mathbb{T}}^{n}$ formed by “primary” Kolmogorov’s tori; more precisely,
for any point $(p,q)\in{\cal T}_{\alpha}$, $\phi^{t}_{\mathtt{H}}(p,q)$ covers densely an $\mathtt{H}$–invariant, analytic, Lagrangian torus, with $\mathtt{H}$–flow analytically conjugated to a linear flow with $(\alpha,\tau)$–Diophantine frequencies
$\omega=h_{p}(p_{0})$, for a suitable $p_{0}\in\mathtt{D}$; each of such tori is a graph over ${\mathbb{T}}^{n}$ $r_{\epsilon}$–close
to the unperturbed trivial graph $\{(p,\theta)=(p_{0},\theta)|\ \theta\in{\mathbb{T}}^{n}\}$.
Finally,
the Lebesgue outer measure of $(\mathtt{D}\times{\mathbb{T}}^{n})\,\backslash\,{\cal T}_{\alpha}$ is bounded by:
$${\rm\,meas\,}\big{(}(\mathtt{D}\times{\mathbb{T}}^{n})\,\backslash\,{\cal T}_{%
\alpha}\big{)}\leq C\,\sqrt{\epsilon}$$
(268)
with
$$C:=\big{(}\max\big{\{}\mathtt{m}^{2}r_{0}\,,\,{\rm diam}\,\mathtt{D}\big{\}}%
\big{)}^{n}\cdot\frac{1}{c\,\mathtt{m}^{n+5}\ s^{3\tau+3}}\,.$$
(269)
Remark 7.3
The statement of the above quantitative KAM theorem
is as Theorem 1 in [6] with the following minor simplification. In Theorem 1 of [6]
appear the quantity
$\lambda:=\mathtt{L}\mathtt{M}$, where $\mathtt{M}$ is defined in
(263) and $\mathtt{L}$ denotes a suitable uniform Lipschitz constant of the local complex inverse of the “frequency map” $p\mapsto\omega=\partial_{p}h(p)$ (compare formula (9) of [6]); since one can show that $1\leq\lambda\leq 2\cdot n!\,\mathtt{m}^{-1}$ (see formula
(14) of [6]), we substitute everywhere
$\lambda$ with $1$ or $2\cdot n!\,\mathtt{m}^{-1}$ in Theorem 7.1,
obtaining a slightly weaker formulation of Theorem 1 of [6].
KAM tori in $\Omega^{0}$
We now apply Theorem 7.1 to the Hamiltonian
$H_{\{0\}}$
in (56).
It is immediato to see, thanks to (57),
that KAM tori cover all $\Omega^{0}$ (defined in (30))
up to a set of measure
$\varepsilon^{\frac{s}{5}|\log\varepsilon|}.$
KAM tori in $\Omega^{1}$
We, now, want to apply Theorem 7.1 to the Hamiltonians $h^{(i)}_{k}(p)+f^{(i)}_{k}(p,q)$ defined in (259), for all $k\in{\mathbb{Z}}^{n}_{*,K},$
$0\leq i\leq 2N_{k}.$
The objects appearing in Theorem 7.1 have to be replaced by the following:
$$h\ \rightsquigarrow\ h^{(i)}_{k}\,,\qquad f\ \rightsquigarrow\ f^{(i)}_{k}\,,%
\qquad\mathtt{D}\ \rightsquigarrow\ \mathtt{B}^{i}_{k}(\theta,\mu)\,,\qquad r_%
{0}\ \rightsquigarrow\ \rho^{\prime}=\frac{\theta}{\mathtt{c}K^{n-1}}\,,\qquad
s%
\ \rightsquigarrow\ \sigma^{\prime}:=\frac{1}{{\mathtt{c}}K^{n-1}}\,,$$
(270)
(recall (257)).
By (260), it follows immediately that
$$\|f\|_{\mathtt{D},r_{0},s}\,\leq\varepsilon^{\frac{s}{5}|\log\varepsilon|^{3}}\ .$$
(271)
By (262)
we get
$$\mathtt{M}\,\leq{\,c\,}/\theta\,,$$
(272)
where, here and in the following,
$$c=c(n,s)\geq 1\,,$$
are suitably large (different) constants depending only on $n$ and $s.$
By (261) we have that
$d$ defined in (264)
satisfies
$$d\geq\mu\,.$$
So we get that $\mathtt{m}$ in (265)
satisfies
$$\mathtt{m}=\frac{d}{\mathtt{M}^{n}}\geq\mu\theta^{n}/{\,c\,}\,.$$
(273)
Now, we choose the parameters
$\mu$ and $\theta$ as follows606060Where
$\mathtt{c}$ is the constant defined in Proposition
7.1.
$$\mu=\theta:=\varepsilon^{|\log\varepsilon|^{2}}\,.$$
(274)
With such choices the condition of the KAM Theorem 7.1 are met:
in particular (265) follows by (273)
and also (266),
which is implied by the stronger condition
$$\|f\|_{\mathtt{D},r_{0},s}\leq c\,\mu^{8}\theta^{8n+1}K^{2-2n}s^{4\tau+4}\ ,$$
which holds by (271) (recall also
(24)), taking $\varepsilon$ small enough.
Noting that, by (256), (248)
and (79),
$${\rm diam}(\mathtt{D})\leq\frac{cK^{\nu+1}}{\sqrt{\delta_{k}}\|k\|}+\varsigma_%
{k}^{-1}{\rm diam}(\hat{Z}_{k})\leq c\frac{K^{\nu+1}\sqrt{\varepsilon}}{%
\varsigma_{k}}+\varsigma_{k}^{-1}{\rm diam}(\hat{D})\stackrel{{\scriptstyle%
\eqref{arrosticini}}}{{\leq}}c\frac{K^{\nu+1}\sqrt{\varepsilon}}{\varsigma_{k}%
}+\varsigma_{k}^{-1}K^{n}\,,$$
(275)
the maximum in (269)
can be estimated by
$c\frac{K^{\nu+1}}{\varsigma_{k}}.$
By (268) and (269)
we get that the measure of the non torus set
in every $\mathtt{B}^{i}_{k}(\theta,\mu)$
is bounded by
$$c\frac{K^{c}}{\varsigma_{k}^{n}}\sqrt{\|f\|_{\mathtt{D},r_{0},s}}\frac{\mathtt%
{M}^{n(n+5)-1/2}}{d^{n+5}s^{3\tau+3}}\leq\frac{c}{\varsigma_{k}^{n}\varepsilon%
^{c}}\varepsilon^{\frac{s}{10}|\log\varepsilon|^{3}}\leq\frac{1}{\varsigma_{k}%
^{n}}\varepsilon^{\frac{s}{11}|\log\varepsilon|^{3}}\,,$$
for $\varepsilon$ small enough.
Then by (255) (and (258))
we get that the measure of the non-torus
set in
$\big{(}D^{{}^{\prime}k}_{\sharp}\times\mathbb{T}^{n}\big{)}$
is estimated by616161Note that
$\varsigma_{k}\leq 1$, see (79).
$$\frac{1}{\varsigma_{k}^{n}}\varepsilon^{2|\log\varepsilon|}\,,$$
for $\varepsilon$ small enough.
Therefore
the measure of the non-torus
set in $\bigcup_{k\in\mathbb{Z}^{n}_{*,K}}D^{{}^{\prime}k}_{\sharp}\times\mathbb{T}^{n}$
is estimated by
$$\frac{1}{\varsigma_{k}^{n}}\varepsilon^{|\log\varepsilon|}\,,$$
for $\varepsilon$ small enough (recall (24)).
By (85), (78) and
(79)
we get that the measure of the non-torus
set in $\Omega^{1}\times\mathbb{T}^{n}$ is bounded by
$\varepsilon^{|\log\varepsilon|}.$
Acknowledgment.
We are indebted to V. Kaloshin, G. Loddi, A. Neishtadt and A. Sorrentino.
Appendix A Properties of the class of non–degenerate potentials
Proof of Proposition 1.1.
$\bullet$ ${\mathcal{P}}_{s}\,\cap\,{\mathbb{B}}_{s}^{n}\in{\mathcal{B}}$ and $\mu_{s}({\mathcal{P}}_{s}\cap{\mathbb{B}}_{s}^{n})=1$
We shall prove that, for every $\delta>0,$
the measure of the sets of
potentials $f$ that do not satisfy, respectively, (P1), (P2), (P3), (P4) is,
respectively, $O(\delta^{2}),0,0,0$, the result will follow letting $\delta\to 0$.
First, by the identification (7), the
measure of the set of
potentials $f$ that do not satisfy (P1)
with a given $\delta$
is bounded by $\delta^{2}\,\sum_{k\in{\mathbb{Z}}^{n}}|k|^{-n-3}$.
Next, recall that properties (P2), (P3) and (P4) concern only a finite number of $k$,
i.e., $k\in{\mathbb{Z}}^{n}_{*},\ |k|\leq K_{s}(\delta)$.
To show that the set of potentials that do not satisfy (P2) has $\mu_{s}$-measure zero it is enough to check that, for every $k\in{\mathbb{Z}}^{n}_{*},\ |k|\leq K_{s}(\delta)$, the set
${\cal E}^{(k)}$
of $f$’s
for which626262Recall the definition of $F^{k}$ in (4). $F^{k}$ has a degenerate critical point has zero $\mu_{s}$-measure.
Fix $k\in{\mathbb{Z}}^{n}_{*},\ |k|\leq K_{s}(\delta)$ and
denote points in ${\mathtt{E}}^{(k)}$ by $(\zeta,\varphi)$, where $\zeta=f_{k}$ and $\varphi=\{f_{h}\}_{h\neq k}$.
Write
$$F^{k}(\xi)=\zeta e^{{\rm i}\xi}+\bar{\zeta}e^{-{\rm i}\xi}+G(\xi)\,,\quad{\rm
where%
}\ \ \zeta:=f_{k}\ \ {\rm and}\ \ G(\xi):=\sum_{|j|\geq 2}f_{jk}e^{{\rm i}j\xi%
}\,.$$
(276)
Now, one checks immediately that $\partial_{\xi}F^{k}(\xi_{0})=0=\partial^{2}_{\xi}F^{k}(\xi_{0})$ is equivalent to
$\zeta=\zeta(\xi_{0},\varphi)=\frac{1}{2}e^{-{\rm i}\xi_{0}}\big{(}{\rm i}G^{%
\prime}(\xi_{0})+G^{\prime\prime}(\xi_{0})\big{)}$, which, as $\xi_{0}$ varies in ${\mathbb{T}}$, describes a smooth closed “critical” curve in ${\mathbb{C}}$,
as a side remark, notice that $\zeta$ depends on $\varphi$ only through the Fourier coefficients $f_{jk}$ with $|j|\geq 2$. Thus the section ${\mathtt{E}}^{(k)}_{\varphi}=\{\zeta\in D:(\zeta,\varphi)\in{\mathtt{E}}^{(k)}\}$ is (a piece of) a smooth curve in $D=\{z\in{\mathbb{C}}:|z|\leq 1\}$, hence meas$({\mathtt{E}}^{(k)}_{\varphi})=0$ for every $\varphi$ and by Fubini’s theorem $\mu_{s}({\mathtt{E}}^{(k)})=0$, as claimed.
An analogous result636363In this case the critical curve is given by
$\{\zeta=(-b(\xi)\pm\sqrt{b^{2}(\xi)-c(\xi)}+{\rm i}G^{\prime}(\xi))e^{-{\rm i}%
\xi}/2,\ \xi\in{\mathbb{R}}\,,b^{2}(\xi)\geq c(\xi)\},$
where $b(\xi):=(G^{\prime\prime\prime\prime}(\xi)-G^{\prime\prime}(\xi))/2$ and $c(\xi):=-G^{\prime\prime}(\xi)G^{\prime\prime\prime\prime}(\xi)+5(G^{\prime}(%
\xi)+G^{\prime\prime\prime}(\xi))^{2}/3$.
holds true for
(P3).
Regarding (P4) we have that
the three real equations
$$\partial_{\xi}F^{k}(\xi_{1})=\partial_{\xi}F^{k}(\xi_{2})=0\,,\qquad F^{k}(\xi%
_{1})-F^{k}(\xi_{2})=0\,,\qquad{\rm for}\ \ \ \xi_{1},\xi_{2}\in\mathbb{T}\,,$$
can be rewritten as (recall (276)) the complex
equation
$$\zeta=\zeta(\xi_{1},\xi_{2},\varphi)=\frac{{\rm i}}{2(e^{{\rm i}\xi_{1}}-e^{{%
\rm i}\xi_{2}})}\Big{(}G^{\prime}(\xi_{1})-G^{\prime}(\xi_{2})+{\rm i}G(\xi_{1%
})-{\rm i}G(\xi_{2})\Big{)}$$
(277)
and the real one
$$\displaystyle\frac{1}{2}(e^{{\rm i}(\xi_{1}-\xi_{2})}-e^{-{\rm i}(\xi_{1}-\xi_%
{2})})\Big{(}G^{\prime}(\xi_{1})-G^{\prime}(\xi_{2})+{\rm i}G(\xi_{1})-{\rm i}%
G(\xi_{2})\Big{)}-(e^{{\rm i}\xi_{1}}-e^{{\rm i}\xi_{2}})\Big{(}e^{-{\rm i}\xi%
_{2}}G^{\prime}(\xi_{1})-e^{-{\rm i}\xi_{1}}G^{\prime}(\xi_{2})\Big{)}$$
$$\displaystyle=:g(\xi_{1},\xi_{2},\varphi)=\big{(}1-\cos(\xi_{1}-\xi_{2})\big{)%
}\Big{(}G^{\prime}(\xi_{1})+G^{\prime}(\xi_{2})\Big{)}-\sin(\xi_{1}-\xi_{2})%
\Big{(}G(\xi_{1})-G(\xi_{2})\Big{)}=0\,.$$
(278)
We claim that, for every fixed $\varphi$, the analytic function
$(\xi_{1},\xi_{2})\mapsto g(\xi_{1},\xi_{2},\varphi)$
is not identically zero and, therefore,
the set
$R_{\varphi}$ of its zeros has zero measure.
Assume by contradiction that $g$ is identically zero.
Then $g(\xi_{2}+\varepsilon,\xi_{2},\varphi)\equiv 0$ for every $\xi_{2}$ and
$\varepsilon,$ in particular, evaluating the order fourth term
of the Taylor expansion in $\varepsilon$ around
$\varepsilon=0,$
we get
$\frac{1}{12}\Big{(}G^{\prime\prime\prime}(\xi_{2})+G^{\prime}(\xi_{2})\Big{)}=%
0\,,\ \forall\,\xi_{2}\,.$
The general (real) solution of the above equation
is $G(\xi_{2})=ce^{{\rm i}\xi_{2}}+\bar{c}e^{-{\rm i}\xi_{2}}+c_{0},$
with $c\in\mathbb{C},$ $c_{0}\in\mathbb{R},$
which contradicts the expression of $G$
in (276).
Therefore, for every fixed $\varphi,$
the image of the zero measure set $R_{\varphi}$
through the Lipschitz function $(\xi_{1},\xi_{2})\mapsto\zeta(\xi_{1},\xi_{2},\varphi)$ (defined in (277))
has zero measure in $D.$
Then we conclude as in the case (P2) above.
$\bullet$ ${\mathcal{P}}_{s}$ contains an open subset ${\mathcal{P}}_{s}^{\prime}$ which is dense in the unit ball of ${\mathcal{A}}_{s}^{n}$.
Let us define ${\mathcal{P}}_{s}^{\prime}$ as ${\mathcal{P}}_{s}$ but with the difference that (P1) is replaced by the
stronger condition646464Note that $\mu_{s}({\mathcal{P}}^{\prime}_{s})=0$.
(P$1^{\prime}$)
$\exists\,\delta>0$ s.t.
$\displaystyle{|f_{k}|\geq\delta\ e^{-|k|s}\,,\ \ \forall\,k\in{\mathbb{Z}}^{n}%
_{*},\ |k|>K_{s}(\delta)}$.
Let us first prove that ${\mathcal{P}}_{s}^{\prime}$ is open. Let $f\in{\mathcal{P}}_{s}^{\prime}$.
We have to show that there exists $\rho>0$ such that if $|g|_{s}<\rho,$ then $f+g\in{\mathcal{P}}_{s}^{\prime}$.
Fix $\delta>0$ such that (P$1^{\prime}$) holds and choose $\rho<\delta$ small enough such that
$[K_{s}(\delta)]>K_{s}(\delta^{\prime})-1\,,$ where $\delta^{\prime}:=\delta-\rho$
and $[\cdot]$ denotes integer part.
Then, it is immediate to verify that $|k|>K_{s}(\delta)\iff|k|>K_{s}(\delta^{\prime})$.
Moreover
$$|f_{k}+g_{k}|e^{|k|s}\geq|f_{k}|e^{|k|s}-|g|_{s}\geq\delta-\rho=\delta^{\prime%
}\,,\qquad\forall\,k\in{\mathbb{Z}}^{n}_{*},\ |k|>K_{s}(\delta^{\prime})\,,$$
namely $f+g$ satisfies (P$1^{\prime}$) (with $\delta^{\prime}$ instead of $\delta$).
Since (P2), (P3) and (P4) are “open” conditions and regard only a finite number of $k$
it is simple to see that they are satisfied also by $f+g$ for $\rho$ small enough.
Then $f+g\in{\mathcal{P}}_{s}^{\prime}$ for $\rho$ small enough.
Let us now show that ${\mathcal{P}}_{s}^{\prime}$ is dense in the unit ball of ${\mathcal{A}}_{s}^{n}$.
Take $f$ in the unit ball of ${\mathcal{A}}_{s}^{n}$ and $0<\theta<1$. We have to find $\tilde{f}\in{\mathcal{P}}_{s}^{\prime}$
with $|\tilde{f}-f|_{s}\leq\theta$.
Let $\delta:=\theta/4$ and denote by $f_{k}$ and $\tilde{f}_{k}$ (to be defined) be the Fourier coefficients of, respectively,
$f$ and $\tilde{f}$. We, then, let $\tilde{f}_{k}=f_{k}$ unless one of the following two cases occurs:
•
$k\in{\mathbb{Z}}^{n}_{*}$, $|k|>K_{s}(\delta)$ and $|f_{k}|e^{|k|s}<\delta$, in which case, $\tilde{f}_{k}=\delta e^{-|k|s}$,
•
$k\in{\mathbb{Z}}^{n}_{*}$, $|k|\leq K_{s}(\delta)$ and $F^{k}$ (defined as in (4)) does not satisfy either (P2), (P3) or (P4), in which
case, $\tilde{f}_{k}$ is chosen at a distance less than $\theta e^{-|k|s}$ from $f_{k}$ but outside the critical curves defined above.
At this point, it is easy to check that $\tilde{f}\in{\mathcal{P}}^{\prime}_{s}$ and is $\theta$–close to $f$.
$\bullet$ ${\mathcal{P}}_{s}$ is prevalent.
Consider the following compact subset of
$\ell_{\infty}^{n}$:
let $\mathcal{K}:=\{z=\{z_{k}\}_{k\in{\mathbb{Z}}^{n}_{\sharp}}:z_{k}\in D_{1/|k|}\},$
where $D_{1/|k|}:=\{w\in{\mathbb{C}}:\ |w|\leq 1/|k|\},$
and let $\nu$ be
the unique probability measure supported on $\mathcal{K}$ such that,
given Lebesgue measurable sets
$A_{k}\subseteq D_{1/|k|}$,
with $A_{k}\neq D_{1/|k|}$
only for finitely many $k$, one has
$$\nu\Big{(}\prod_{k\in\mathbb{Z}^{n}_{\sharp}}A_{k}\Big{)}:=\prod_{\{k\in{%
\mathbb{Z}}^{n}_{\sharp}:\,A_{k}\neq D_{1/|k|}\}}\frac{|k|^{2}}{\pi}{\rm meas}%
(A_{k})\,.$$
The isometry $j_{s}$ in (7)
naturally induces a probability measure
$\nu_{s}$ on ${\mathcal{A}}^{n}_{s}$ with support in the compact set $\mathcal{K}_{s}:=j_{s}^{-1}\mathcal{K}$.
Now, for $\delta>0,$ let ${\mathcal{P}}_{s,\delta}$ be the set of $f$’s
in the unit ball of ${\mathcal{A}}_{s}^{n}$
satisfying (P1)–(P4), so that ${\mathcal{P}}_{s}=\cup_{\delta>0}{\mathcal{P}}_{s,\delta}$.
Reasoning as in the proof of
$\mu_{s}({\mathcal{P}}_{s})=1,$
one can show that
$\nu_{s}({\mathcal{P}}_{s,\delta})\geq 1-{\rm const}\,\delta^{2}$.
It is also easy to check that, for every $g\in{\mathcal{A}}_{s}^{n}$, the translated
set ${\mathcal{P}}_{s,\delta}+g$ satisfies
$\nu_{s}({\mathcal{P}}_{s,\delta}+g)\geq\nu_{s}({\mathcal{P}}_{s,\delta})$.
Thus, one gets
$\nu_{s}({\mathcal{P}}_{s}+g)=\nu_{s}({\mathcal{P}}_{s})=1$, $\forall\,g\in{\mathcal{A}}_{s}^{n}$, which means that
${\mathcal{P}}_{s}$ is prevalent.
(recall footnote LABEL:nurzia)
Appendix B Proof of the Normal Form Lemma 3.1
Given a function $\phi$ we denote by $X_{\phi}^{t}$
the hamiltonian flow at time $t$ generated by $\phi$
and by “ad” the linear operator $u\mapsto{\rm ad}_{\phi}u:=\{u,\phi\}$ and ${\rm ad}^{\ell}$ its iterates:
$${\rm ad}^{0}_{\phi}u:=u\,,\qquad{\rm ad}^{\ell}_{\phi}u:=\{{\rm ad}^{\ell-1}_{%
\phi}u,\phi\}\,,\qquad\ell\geq 1\,,$$
as standard, $\{\cdot,\cdot\}$ denotes Poisson bracket656565Explicitly,
$\displaystyle\{u,v\}=\sum_{i=1}^{n}(u_{x_{i}}v_{y_{i}}-u_{y_{i}}v_{x_{i}})$..
Recall the identity (“Lie series expansion”)
$$u\circ X_{\phi}^{1}=\sum_{\ell\geq 0}\frac{1}{\ell!}{\rm ad}^{\ell}_{\phi}u=%
\sum_{\ell=0}^{\infty}\frac{\partial_{t}^{\ell}(u\circ X_{\phi}^{t})}{\ell!}%
\Big{|}_{t=0}\,,$$
(279)
valid for analytic functions and small $\phi$.
By standard Cauchy estimates,
we get (compare, e.g., Lemma B4 of [15])
Lemma B.1
For $0<r-\rho<r_{0},$ $0<s-\sigma<s_{0},$ $\rho,\sigma>0$
$$|\{f,g\}|_{r-\rho,s-\sigma}\leq\frac{n}{e}\left(\frac{1}{(r_{0}-r+\rho)\sigma}%
+\frac{1}{(s_{0}-s+\sigma)\rho}\right)|f|_{r_{0},s_{0}}|g|_{r,s}\,.$$
(280)
Summing the Lie series in (279) and using Lemma B5 of [15], we get, also,
Lemma B.2
Let $0<\rho<r\leq r_{0}-\rho$ and
$0<\sigma<s\leq s_{0}-\sigma.$
Assume that
$$\hat{\vartheta}:=\frac{4n|\phi|_{r_{0},s_{0}}}{\rho\sigma}\leq 1\,.$$
(281)
Then
$$\big{|}u\circ X_{\phi}^{1}-u\big{|}_{r-\rho,s-\sigma}\leq\sum_{\ell\geq 1}%
\frac{1}{\ell!}\big{|}{\rm ad}^{\ell}_{\phi}u\big{|}_{r-\rho,s-\sigma}\leq\hat%
{\vartheta}|u|_{r,s}\,.$$
(282)
Given $K\geq 2$ and a lattice $\Lambda$, recall the definition of $f^{\flat}$ in (47) and define
$$f^{K}:=f-f^{\flat}=T_{K}\,{\mathtt{p}}_{\Lambda}^{\perp}f\ ,$$
(283)
so that we have the decomposition (valid for any $f$):
$$f=f^{\flat}+f^{K}\,,\qquad f^{\flat}:=P_{\Lambda}f+{T_{K}^{\perp}}\,{\mathtt{p%
}}_{\Lambda}^{\perp}f\,,\qquad f^{K}:=T_{K}\,{\mathtt{p}}_{\Lambda}^{\perp}f\,.$$
(284)
Lemma B.3
Consider a real–analytic Hamiltonian
$$H=H(y,x)=h(y)+f(y,x)\qquad\mbox{analytic \ on \ }D_{r}\times\mathbb{T}^{n}_{s}\,.$$
(285)
Suppose that $D_{r}$ is ($\alpha$,$K$) non–resonant modulo $\Lambda$
for $h$ (with $K\geq 2$).
Assume that
$$\check{\vartheta}:=\frac{2^{5}nK^{3}}{\alpha rs}\,|f^{K}|_{r,s}\leq 1\,.$$
(286)
Then there exists a real–analytic
symplectic change of coordinates
$$\Psi:D_{r_{+}}\times\mathbb{T}^{n}_{s_{+}}\to D_{r}\times\mathbb{T}^{n}_{s}\,,%
\qquad r_{+}:=r(1-1/2K)\,,\ \ \ s_{+}:=s(1-1/K^{2})\,,$$
such that
$$H\circ\Psi=h(y)+f_{+}(y,x)\,,\qquad f_{+}:=f^{\flat}+f_{*}$$
(287)
with
$$|f_{*}|_{r_{+},s_{+}}\leq\ 2\check{\vartheta}|f|_{r,s}\,.$$
(288)
Notice that, by (284) and (288) (and the fact that $|f-f^{K}|_{r,s}\leq|f|_{r,s}$), one has
$$f_{+}^{K}=f_{*}^{K}\ ,\quad|f_{+}|_{r_{+},s_{+}}=|f_{*}+f-f^{K}|_{r_{+},s_{+}}%
\leq|f_{*}|_{r_{+},s_{+}}+|f|_{r,s}\leq(1+2\check{\vartheta})|f|_{r,s}\ .$$
(289)
Notice also that
$$f_{+}^{\flat}-f^{\flat}\stackrel{{\scriptstyle{\rm(\ref{olintobis})}}}{{=}}f_{%
*}^{\flat}\quad\Longrightarrow\quad|f_{+}^{\flat}-f^{\flat}|_{r_{+},s_{+}}\leq%
|f_{*}|_{r_{+}.s_{+}}\stackrel{{\scriptstyle{\rm(\ref{salamina})}}}{{\leq}}2%
\check{\vartheta}|f|_{r,s}\ .$$
(290)
Proof (of Lemma B.3)
Let us define
$$\phi=\phi(y,x):=\sum_{|m|\leq K,m\notin\Lambda}\frac{f_{m}(y)}{{\rm i}h^{%
\prime}(y)\cdot m}e^{{\rm i}m\cdot x}\,,\qquad\Psi:=X^{1}_{\phi}\,,$$
and note that $\phi$ solves the homological equation
$$\{h,\phi\}+f^{K}=0\,.$$
(291)
Since $D_{r}$ is ($\alpha$,$K$) non–resonant modulo $\Lambda$
$$|\phi|_{r,s}\leq|f^{K}|_{r,s}/\alpha\,.$$
(292)
Then, one has
$$H\circ\Psi=h+f^{\flat}+f_{*}$$
with
$$f_{*}=(h\circ\Psi-h-\{h,\phi\})+(f\circ\Psi-f)\,.$$
In order to estimate $f_{*}$
we now use Lemma B.2 with parameters
$$r_{0}\rightsquigarrow r\,,\ \ s_{0}\rightsquigarrow s\,,\ \ r\rightsquigarrow r%
(1-1/4K)\,,\ \ s\rightsquigarrow s(1-1/2K^{2})\,,\ \ \rho\rightsquigarrow r/4K%
\,,\ \ \sigma\rightsquigarrow s/2K^{2}\,.$$
With these choices it is
$\hat{\vartheta}=\check{\vartheta}$, and, by (286) $\check{\vartheta}\leq 1$.
Thus, (281) holds and Lemma B.2 applies.
By (282) we get
(288) noting that
$$h\circ\psi-h-\{h,\phi\}=\sum_{\ell\geq 2}\frac{1}{\ell!}{\rm ad}^{\ell}_{\phi}%
h=\sum_{\ell\geq 1}\frac{1}{(\ell+1)!}{\rm ad}^{\ell}_{\phi}\{h,\phi\}%
\stackrel{{\scriptstyle\eqref{tessalonica}}}{{=}}-\sum_{\ell\geq 1}\frac{1}{(%
\ell+1)!}{\rm ad}^{\ell}_{\phi}f^{K}\,,$$
which implies (again by (282))
that
$$|h\circ\Psi-h-\{h,\phi\}|_{r_{+},s_{+}}\leq\check{\vartheta}|f^{K}|_{r,s}\leq%
\check{\vartheta}|f|_{r,s}\ .$$
Finally, applying again Lemma B.2 with $u=f$, by (282), we get $|f\circ\Psi-f|_{r_{+},s_{+}}\leq\check{\vartheta}|f|_{r,s}$, concluding the proof of Lemma B.3.
Proof of the Normal Form Lemma 3.1
Denote by
$$\bar{K}:=\lceil K\rceil:=\min\{n\in{\mathbb{Z}}:\ n\geq K\}\ ,$$
(293)
the ceiling function of $K$.
The idea is to construct $\Psi$ by applying $\bar{K}$ times Lemma B.3.
To do this, fix $1\leq j<K$ and
make the following inductive assumption:
Let
$$\displaystyle f_{0}:=f\ ,\quad H_{0}:=h+f_{0}=H\,,\quad\rho:=\frac{r}{4\bar{K}%
}\,,\qquad\sigma:=\frac{s}{2K\bar{K}}\,,$$
$$\displaystyle r_{i}:=r-2i\rho\,,\qquad s_{i}:=s-2i\sigma\,,\qquad|\cdot|_{i}:=%
|\cdot|_{r_{i},s_{i}}\,,$$
(294)
and assume that there exist, for $1\leq i\leq j$, real–analytic symplectic transformations
$$\Psi_{i-1}\ :\ D_{r_{i}}\times\mathbb{T}^{n}_{s_{i}}\to D_{r_{i-1}}\times%
\mathbb{T}^{n}_{s_{i-1}}\,,$$
such that
$$H_{i}:=H_{i-1}\circ\Psi_{i-1}=:h+f_{i}$$
(295)
satisfies, for $1\leq i\leq j$, the estimates
$$\vartheta_{i}\leq(4\delta|f|_{r,s})^{i+1}\,,\qquad|f_{i}^{\flat}-f^{\flat}_{i-%
1}|_{i}\leq 2\vartheta_{i-1}\,|f_{i-1}|_{i-1}\ ,$$
(296)
where
$$\vartheta_{i}:=\delta|f_{i}^{K}|_{i}\qquad{\rm with}\qquad\delta:=\frac{2^{5}%
\,n\,K^{3}}{\alpha rs}\ .$$
(297)
Notice that, recalling (45), it is
$$\vartheta_{*}=2^{4}\delta|f|_{r,s}\quad\Longrightarrow\quad 4\delta|f|_{r,s}=%
\frac{\vartheta_{*}}{4}<\frac{1}{4}<1\ .$$
(298)
Let us first show that the inductive hypothesis is true for $j=1$. Indeed, by (298),
$\delta|f^{K}|_{0}\leq\delta|f|_{0}<1/16<1$, therefore, by the definition of $\delta$ and $\check{\vartheta}$ in, respectively, (297) and (286),
we see that we
can apply Lemma B.3 with $f=f_{0}$, being $\check{\vartheta}=\vartheta_{0}=\delta|f^{K}|_{0}$. Thus, we obtain the existence of $\Psi_{0}$ so that
$H_{1}:=H_{0}\circ\Psi_{0}=h+f_{1}$ and, by (289) and (288),
$$\vartheta_{1}=\delta|f_{1}^{K}|_{1}\leq\delta(2\vartheta_{0}|f|_{0})=2\,\delta%
^{2}|f^{K}|_{0}\,|f|_{0}\leq 2(\delta|f|_{0}|)^{2}\leq(4\delta|f|_{0})^{2}\ ,$$
showing that the first inequality in (296) holds for $i=1$, the second inequality follows from (290).
Now, let us assume that the inductive hypothesis holds true for $1\leq i\leq j<K$ and let us prove that it holds also for $i=j+1$.
First, let us check that
$$|f_{i}|_{i}\leq 2|f|_{r,s}\ ,\qquad\forall\ 1\leq i\leq j\ .$$
(299)
Indeed, by the estimate in (289), one has that $|f_{i}|_{i}\leq(1+2\vartheta_{i})|f_{i-1}|_{i-1}$, for all $1\leq 1\leq j$, which, iterated, yields
$$\displaystyle|f_{i}|_{i}$$
$$\displaystyle\leq$$
$$\displaystyle\displaystyle|f_{0}|_{0}\prod_{\ell=1}^{i}(1+2\vartheta_{\ell})=|%
f|_{r,s}\exp\big{(}\sum_{\ell=1}^{i}\log(1+2\vartheta_{\ell})\big{)}\leq|f|_{r%
,s}\exp\big{(}2\sum_{\ell=1}^{i}\vartheta_{\ell}\big{)}$$
$$\displaystyle\stackrel{{\scriptstyle\eqref{pontina}}}{{\leq}}$$
$$\displaystyle\displaystyle|f|_{r,s}\exp\big{(}{2\sum_{\ell=1}^{i}(4\delta|f|_{%
r,s})^{\ell}}\big{)}\stackrel{{\scriptstyle{\rm(\ref{cappuccino})}}}{{\leq}}|f%
|_{r,s}\exp\big{(}{2\sum_{\ell=1}^{\infty}2^{-2\ell}\big{)}}\leq 2|f|_{r,s}\,.$$
Now, by (297), (296) with $i=j$ (inductive assumption) and (298), we have that $\vartheta_{j}<1$. Thus, we can apply Lemma B.3 to $f_{j}$ (with $\check{\vartheta}=\vartheta_{j}$) and get a symplectic transformation $\Psi_{j}$ such that $H_{j+1}:=H_{j}\circ\Psi_{j}=h+f_{j+1}$ satisfies
$$\vartheta_{j+1}:=\delta|f^{K}_{j+1}|_{j+1}\stackrel{{\scriptstyle{\rm(\ref{%
salamina})}}}{{\leq}}\delta\,(2\vartheta_{j}|f_{j}|_{j})\stackrel{{%
\scriptstyle{\rm(\ref{ausoni})}}}{{\leq}}(4\delta|f|_{r,s})\,\vartheta_{j}%
\stackrel{{\scriptstyle{\rm(\ref{pontina})}_{j}}}{{\leq}}(4\delta|f|_{r,s}|)^{%
j+2}\ ,$$
which is the first inequality in (296) with $i=j+1$, the second inequality comes from (290). This completes the proof of the induction.
Now, we can conclude the proof of Lemma 3.1: recall (293) and define
$$\Psi:=\Psi_{0}\circ\cdots\circ\Psi_{\bar{K}-1}\ .$$
Notice that, by (B), $r_{\bar{K}}=r/2=r_{*}$ and $s_{\bar{K}}=s(1-1/K)=s_{*}$ and notice that, by the induction, it is
$$H\circ\Psi=H_{{\bar{K}}-1}\circ\Psi_{{\bar{K}}-1}\stackrel{{\scriptstyle{\rm(%
\ref{olintoj})}_{\bar{K}}}}{{=}}h+f_{\bar{K}}=:h+f^{\flat}+f_{*}\ .$$
(300)
But, then, since $T_{K}P_{\Lambda}^{\perp}f^{\flat}=(f^{\flat})^{K}=0$ (for any $f$), using that ${\bar{K}}\geq 2$, we have
$$\displaystyle|T_{K}P_{\Lambda}^{\perp}f_{*}|_{r_{*},s_{*}}$$
$$\displaystyle=$$
$$\displaystyle|f_{\bar{K}}^{K}|_{\bar{K}}\stackrel{{\scriptstyle{\rm(\ref{%
corcira})}}}{{=}}\delta^{-1}\vartheta_{\bar{K}}\stackrel{{\scriptstyle{\rm(%
\ref{pontina})}}}{{\leq}}\delta^{-1}(4\delta|f|_{0})^{{\bar{K}}+1}=4(4\delta|f%
|_{0})^{\bar{K}}\,|f|_{0}$$
(301)
$$\displaystyle\leq$$
$$\displaystyle(2^{3}\delta|f|_{0})^{\bar{K}}|f|_{0}\stackrel{{\scriptstyle{\rm(%
\ref{cappuccino})}}}{{=}}\big{(}2^{-1}\vartheta_{*}\big{)}^{\bar{K}}|f|_{r,s}%
\leq\big{(}2^{-1}\vartheta_{*}\big{)}^{K}|f|_{r,s}\ ,$$
proving the second estimates in (48).
Finally, (using again that ${\bar{K}}\geq 2$ and that $\vartheta_{*}<1$)
$$\displaystyle|f_{*}|_{r_{*},s_{*}}$$
$$\displaystyle\stackrel{{\scriptstyle{\rm(\ref{bic})}}}{{=}}$$
$$\displaystyle|f_{\bar{K}}-f^{\flat}|_{\bar{K}}\stackrel{{\scriptstyle{\rm(\ref%
{decomposizione})}}}{{=}}|f_{\bar{K}}^{K}+f_{\bar{K}}^{\flat}-f^{\flat}|_{\bar%
{K}}\leq|f_{\bar{K}}^{K}|_{\bar{K}}+|f_{\bar{K}}^{\flat}-f^{\flat}|_{\bar{K}}$$
$$\displaystyle\stackrel{{\scriptstyle{\rm(\ref{biro})}}}{{\leq}}$$
$$\displaystyle\frac{\vartheta_{*}}{4}\,|f|_{0}+\sum_{i=1}^{\bar{K}}|f_{i}^{%
\flat}-f_{i-1}^{\flat}|_{i}\stackrel{{\scriptstyle{\rm(\ref{pontina})}}}{{\leq%
}}\frac{\vartheta_{*}}{4}\,|f|_{0}+\sum_{i=1}^{\bar{K}}2\vartheta_{i-1}|f_{i-1%
}|_{i-1}$$
$$\displaystyle\stackrel{{\scriptstyle{\rm(\ref{ausoni})},{\rm(\ref{pontina})}}}%
{{\leq}}$$
$$\displaystyle\frac{\vartheta_{*}}{4}\,|f|_{0}+4|f|_{0}\sum_{i=1}^{\bar{K}}(4%
\delta|f|_{0})^{i}\stackrel{{\scriptstyle{\rm(\ref{cappuccino})}}}{{=}}\frac{%
\vartheta_{*}}{4}\,|f|_{0}+4|f|_{0}\sum_{i=1}^{\bar{K}}(\vartheta_{*}/4)^{i}%
\leq 2\vartheta_{*}\,|f|_{0}\ ,$$
which proves also the first estimate in (48).
Appendix C On action–angle variables for 1D mechanical systems with parameters
We will use the notations of sections
4 and 5, in particular
subsections 5.1 and 5.5.
C.1 The “unperturbed case”
Consider the “unperturbed case” when
$\eta=\eta_{*}=0$ (recall (136)).
Namely consider the one dimensional Hamiltonian
$$H_{\rm pend}^{0}(J_{n},\psi_{n})=J_{n}^{2}+F^{0}(\psi_{n})\,,\quad\text{with}%
\ F^{0}\ \text{satisfying }\ \eqref{legna}\,.$$
(302)
In the particular important case in which $F^{0}$
is minus cosine we can explicitly evaluate
$$F^{0}(x)=-\cos x\quad\Longrightarrow\quad M=\cosh s_{0}\,,\ \ N=1\,,\ \ x^{0}_%
{1}=0\,,\ x^{0}_{2}=\pi\,,\ \ E^{0}_{1}=-1\,,\ E^{0}_{2}=1\,,\ \ \beta=1\,.$$
(303)
For $E\in(E^{(i),0}_{-},E^{(i),0}_{+})$, let us define the functions $P_{n}^{(i),0}(E)$ as
$$\displaystyle P_{n}^{(2j-1),0}(E)$$
$$\displaystyle:=$$
$$\displaystyle\frac{1}{\pi}\int_{X_{2j-1}^{0}(E)}^{X_{2j}^{0}(E)}\sqrt{E-F^{0}(%
x)}\,dx\,,$$
$$\displaystyle P_{n}^{(2j),0}(E)$$
$$\displaystyle:=$$
$$\displaystyle\frac{1}{\pi}\int_{X_{2j_{-}+1}^{0}(E)}^{X_{2j_{+}}^{0}(E)}\sqrt{%
E-F^{0}(x)}\,dx\,,$$
$$\displaystyle P_{n}^{(2N),0}(E)$$
$$\displaystyle:=$$
$$\displaystyle\frac{1}{2\pi}\int_{-\pi}^{\pi}\sqrt{E-F^{0}(x)}\,dx\,,$$
$$\displaystyle P_{n}^{(0),0}(E)$$
$$\displaystyle:=$$
$$\displaystyle-\frac{1}{2\pi}\int_{-\pi}^{\pi}\sqrt{E-F^{0}(x)}\,dx\,.$$
(304)
In the following we will use the notations
$\mathfrak{p}$ and $\lessdot$
introduced in (137) and (138).
Lemma C.1
For real $E$, we have that
$$\min_{1\leq i\leq 2N-1}\inf_{E\in(E^{(i),0}_{-},E^{(i),0}_{+})}\partial_{E}P_{%
n}^{(i),0}(E)=:C_{F^{0}}>0\,.$$
(305)
In particular666666
In the special case in which $F^{0}(\psi_{n})=-\cos\psi_{n}$
(note that $N=1$), the minimum is $1/\sqrt{2}.$
$$\text{if}\ \ F^{0}\ \ \text{satisfies}\ \ \text{({\bf A3})}\ \ \text{then}\ \ %
C_{F^{0}}\geq 1/2\,.$$
(306)
Proof See [5].
C.2 The action as a function of the angle at constant energy
Let us consider now the Hamiltonian $H_{\rm pend}$
defined in
(134).
For $\eta$ small enough we can solve, w.r.t. $J_{n}$, the implicit function equation
$$J_{n}^{*}(\hat{J})+\frac{z}{\sqrt{1+b(J,\psi_{n})}}-J_{n}=0\,,$$
(307)
finding
$$J_{n}=\mathcal{J}_{n}(z,\psi_{n},\hat{J})\,,$$
where $\mathcal{J}_{n}$ is the analytic function $\mathcal{J}_{n}:(-R_{0},R_{0})_{r_{0}/4}\times\mathbb{T}_{s_{0}}\times\hat{D}_%
{r_{0}}$
$$\mathcal{J}_{n}(z,\psi_{n},\hat{J})=J_{n}^{*}(\hat{J})+z+\tilde{\mathcal{J}_{n%
}}(z,\psi_{n},\hat{J})\,$$
(308)
whit $\tilde{\mathcal{J}_{n}}$ solving
the fixed point equation
$$\tilde{\mathcal{J}_{n}}=\Phi(\tilde{\mathcal{J}_{n}};z,\psi_{n},\hat{J}):=%
\frac{1}{\sqrt{1+b\big{(}\hat{J},J_{n}^{*}(\hat{J})+z+\tilde{\mathcal{J}_{n}},%
\psi_{n}\big{)}}}-1\,.$$
(309)
We are going to solve
(309)
in the closed ball
$$\|\tilde{\mathcal{J}_{n}}\|_{\mathcal{B}}\leq\eta\,,$$
(310)
of the Banach space
$\mathcal{B}$
of analytic functions $\phi:(-R_{0},R_{0})_{r_{0}/4}\times\mathbb{T}_{s_{0}}\times\hat{D}_{r_{0}}\to%
\mathbb{C}$
endowed with the sup-norm
$$\|\phi\|_{\mathcal{B}}:=\sup_{z\in(-R_{0},R_{0})_{r_{0}/4}}\|\phi(z,\cdot,%
\cdot)\|_{\hat{D},r_{0},s_{0}}\,.$$
We first note that
by (308), (310), (5.5)
we get
$$\|\mathcal{J}_{n}(z,\cdot,\cdot)\|_{\mathcal{B}}\leq r_{0}\eta+R_{0}+r_{0}/4+%
\eta\leq R_{0}+3r_{0}/8\,,$$
(311)
assuming
$$\eta\leq\min\{1,r_{0}\}/32\,.$$
(312)
For $|t|\leq 1/4$,
we have that $\left|\frac{d}{dt}\frac{1}{\sqrt{1+t}}\right|\leq 1,$ then
we get by (5.5), (312),
(311) and Cauchy estimates
$$\|\Phi(\tilde{\mathcal{J}_{n}})\|_{\mathcal{B}}\leq\eta\,,\qquad\|D_{\tilde{%
\mathcal{J}_{n}}}\Phi(\tilde{\mathcal{J}_{n}})\|_{\mathcal{L}(\mathcal{B},%
\mathcal{B})}\leq\left\|\frac{\partial_{z}b(\hat{J},J_{n}^{*}+z+\tilde{%
\mathcal{J}_{n}},\psi_{n})}{2\big{(}1+b(\hat{J},J_{n}^{*}+z+\tilde{\mathcal{J}%
_{n}},\psi_{n})\big{)}^{3/2}}\right\|_{\mathcal{B}}\leq\frac{8\eta}{r_{0}}\leq%
\frac{1}{4}$$
(313)
in the closed ball of $\tilde{\mathcal{J}_{n}}$ satisfying (310).
Obviously676767For real values of $\hat{J},$ $\psi_{n},$ $E.$
$$J_{n}=\mathcal{J}_{n}\Big{(}\pm\sqrt{E-F(\hat{J},\psi_{n})},\psi_{n},\hat{J}%
\Big{)}\quad{\rm solves\ (w.r.t.}\ J_{n})\quad H_{\rm pend}(\hat{J},J_{n},\psi%
_{n})=E\,,$$
(314)
according to $\pm\big{(}J_{n}-J_{n}^{*}(\hat{J})\big{)}\geq 0,$
for every (real) $E$ such
that686868So that $\mathcal{J}_{n}\Big{(}\pm\sqrt{E-F(\hat{J},\psi_{n})},\psi_{n},\hat{J}\Big{)}$
is well defined. Recall (99).,
$$E+M<R_{0}^{2}\,.$$
(315)
By (309) we get
$$\partial_{z}\tilde{\mathcal{J}}_{n}=-\left(1+\frac{\partial_{z}b}{2(1+b)^{3/2}%
}\right)^{-1}\frac{\partial_{z}b}{2(1+b)^{3/2}}$$
so that, recalling (313),
$$\|\partial_{z}\tilde{\mathcal{J}}_{n}\|_{\mathcal{B}}\leq\frac{4}{3}\frac{8%
\eta}{r_{0}}\leq\frac{1}{3}\,.$$
Then $\mathcal{J}_{n}$ is an increasing function of (real) $z$,
indeed by (308) we obtain
$\partial_{z}\mathcal{J}_{n}=1+\partial_{z}\tilde{\mathcal{J}}_{n}$.
Remark C.1
In the following we will often omit the explicit dependence on $\hat{J}$, for brevity.
C.3 The domains of definition of action angle variables
Outside the zero measure set formed by the connected components in the set of critical energies $\{H_{\rm pend}=E_{i}\},$
$1\leq i\leq 2N,$ containing the critical points $x_{i}$,
the phase space $\mathbb{R}^{n}\times\mathbb{T}^{n}$
is composed by $2N+1$ open connected components
$\mathcal{C}^{i},$ $0\leq i\leq 2N,$
defined as
$$\mathcal{C}^{i}:=\check{\mathcal{C}}^{i}\times\mathbb{T}^{n-1}\,,$$
where
$$\check{\mathcal{C}}^{i}\subseteq\hat{D}\times\mathbb{R}\times\mathbb{T}^{1}%
\subseteq\mathbb{R}^{n}\times\mathbb{T}^{1}$$
are defined as follows696969Omitting to write, for brevity,
the explicit dependence of $\mathcal{J}_{n},F,X_{i}$ on
$\hat{J}$.
For $i=2j-1$ odd, $1\leq j\leq N,$
$\check{\mathcal{C}}^{2j-1}$ is a normal set with respect to the variable
$J_{n},$
$$\displaystyle\check{\mathcal{C}}_{2j-1}:=$$
(316)
$$\displaystyle\Big{\{}\mathcal{J}_{n}\Big{(}-\sqrt{E^{(2j-1)}_{+}(\hat{J})-F(%
\psi_{n})},\psi_{n}\Big{)}<J_{n}<\mathcal{J}_{n}\Big{(}\sqrt{E^{(2j-1)}_{+}(%
\hat{J})-F(\psi_{n})},\psi_{n}\Big{)}\,,$$
$$\displaystyle\qquad X_{2j-1}\big{(}E^{(2j-1)}_{+}(\hat{J})\big{)}<\psi_{n}<X_{%
2j}\big{(}E^{(2j-1)}_{+}(\hat{J})\big{)}\,,\quad\hat{J}\in\hat{D}\ \Big{\}}$$
$$\displaystyle\setminus\ \Big{\{}J_{n}=J_{n}^{*}\,,\psi_{n}=x_{2j-1}\Big{\}}\,.$$
For $i=2j$ even, $1\leq j\leq N-1,$
$\check{\mathcal{C}}^{2j}$ is still a normal set with respect to the variable $J_{n}$:
$$\displaystyle\check{\mathcal{C}}_{2j}:=$$
(317)
$$\displaystyle\Big{\{}\mathcal{J}_{n}\Big{(}-\sqrt{E^{(2j)}_{+}(\hat{J})-F(\psi%
_{n})},\psi_{n}\Big{)}<J_{n}<\mathcal{J}_{n}\Big{(}\sqrt{E^{(2j)}_{+}(\hat{J})%
-F(\psi_{n})},\psi_{n}\Big{)}\,,$$
$$\displaystyle\quad X_{2j_{-}+1}\big{(}E^{(2j)}_{+}(\hat{J})\big{)}<\psi_{n}<X_%
{2j_{+}}\big{(}E^{(2j)}_{+}(\hat{J})\big{)}\,,\quad\hat{J}\in\hat{D}\ \Big{\}}$$
$$\displaystyle\setminus\ \Big{\{}\mathcal{J}_{n}\Big{(}-\sqrt{E^{(2j)}_{-}(\hat%
{J})-F(\psi_{n})},\psi_{n}\Big{)}\leq J_{n}\leq\mathcal{J}_{n}\Big{(}\sqrt{E^{%
(2j)}_{-}(\hat{J})-F(\psi_{n})},\psi_{n}\Big{)}\,,$$
$$\displaystyle\qquad X_{2j_{-}+1}\big{(}E^{(2j)}_{-}(\hat{J})\big{)}\leq\psi_{n%
}\leq X_{2j_{+}}\big{(}E^{(2j)}_{-}(\hat{J})\big{)}\,,\quad\hat{J}\in\hat{D}\ %
\Big{\}}\,,$$
where $j_{-},j_{+}$ were defined in (116).
Finally
$$\displaystyle\check{\mathcal{C}}_{2N}$$
$$\displaystyle:=$$
$$\displaystyle\Big{\{}J_{n}>\mathcal{J}_{n}\Big{(}\sqrt{E^{(2N)}_{-}(\hat{J})-F%
(\psi_{n})},\psi_{n}\Big{)}\,,\ \ \ \psi_{n}\in\mathbb{T}\,,\quad\hat{J}\in%
\hat{D}\ \Big{\}}$$
(318)
$$\displaystyle\check{\mathcal{C}}_{0}$$
$$\displaystyle:=$$
$$\displaystyle\Big{\{}J_{n}<\mathcal{J}_{n}\Big{(}-\sqrt{E^{(2N)}_{-}(\hat{J})-%
F(\psi_{n})},\psi_{n}\Big{)}\,,\ \ \ \psi_{n}\in\mathbb{T}\,,\quad\hat{J}\in%
\hat{D}\ \Big{\}}$$
(319)
Note that actually in $\check{\mathcal{C}}_{i}$ with $1\leq i<2N,$
$\psi_{n}$ is not an angle!
Let us introduce the (small) parameter
$$\theta\geq 0\,.$$
(320)
Recalling (315), we define the following subsets of $\check{\mathcal{C}}^{i}$ (defined in (316),(317),(318), (319))
$$\displaystyle\check{\mathcal{C}}^{2j-1}(\theta)$$
$$\displaystyle:=$$
$$\displaystyle\check{\mathcal{C}}^{2j-1}\cap\{E^{(2j-1)}_{-}(\hat{J})<H_{\rm
pend%
}<E^{(2j-1)}_{+}(\hat{J})-2\theta\}\,,\qquad\text{for}\ \ \ 1\leq j\leq N\,,$$
$$\displaystyle\check{\mathcal{C}}^{2j}(\theta)$$
$$\displaystyle:=$$
$$\displaystyle\check{\mathcal{C}}^{2j}\cap\{E^{(2j)}_{-}(\hat{J})+2\theta<H_{%
\rm pend}<E^{(2j)}_{+}(\hat{J})-2\theta\}\,,\qquad\text{for}\ \ \ 1\leq j<N\,,$$
$$\displaystyle\check{\mathcal{C}}^{i}(\theta)$$
$$\displaystyle:=$$
$$\displaystyle\check{\mathcal{C}}^{i}\cap\{E^{(2N)}_{-}(\hat{J})+2\theta<H_{\rm
pend%
}<R_{0}^{2}-M-2\theta\}\,,\qquad\text{for}\ \ \ i=0,\,2N\,,$$
(321)
where
$H_{\rm pend}$ was defined in
(134). Note that $\check{\mathcal{C}}^{i}(0)=\check{\mathcal{C}}^{i}$ for $1\leq i<2N.$
Finally we define707070With a little abuse of notation we invert the order of
$\psi_{n}$ and $\hat{\psi}.$
$${\mathcal{C}}^{i}(\theta)=\check{\mathcal{C}}^{i}(\theta)\times\mathbb{T}^{n-1%
}\,,\qquad\check{\mathcal{C}}^{i}(\theta)\ni(J,\psi_{n})\,,\quad\mathbb{T}^{n-%
1}\ni\hat{\psi}\,.$$
(322)
The sets $\check{\mathcal{C}}^{i}(\theta)$ and, therefore, ${\mathcal{C}}^{i}(\theta)$ have different homotopy: for every fixed $\hat{J}$
the set
$$\check{\mathcal{C}}^{i}_{\hat{J}}(\theta):=\{(J_{n},\psi_{n})\ |\ (J,\psi_{n})%
\in\check{\mathcal{C}}^{i}(\theta)\}\subseteq\mathbb{R}^{1}\times\mathbb{T}^{1}$$
is contractible for $1\leq i\leq 2N-1$
and is not contractible for $i=0,2N.$
Note that, recalling (99),
$$\hat{D}\times(-R_{0}/2,R_{0}/2)\times\mathbb{T}^{n}\subset\bigcup_{0\leq i\leq
2%
N}\overline{\mathcal{C}^{i}(0)}\subset\hat{D}\times(-R_{0},R_{0})\times\mathbb%
{T}^{n}\,.$$
(323)
C.4 Definition of action variables
On the above connected components $\mathcal{C}_{i},$
$0\leq i\leq 2N,$
we want to define action angle variables integrating
$H_{\rm pend}.$
We first define the action variables
as a function of the energy $E$ and of the dummy
variable $\hat{J}$.
More precisely, for $0\leq i\leq 2N,$ we are going to define the functions
$$P_{n}^{(i)}\ :\ (E,\hat{J})\in\mathcal{E}^{i}\ \to\mathbb{R}\,,\qquad{\rm where%
}\ \ \ \mathcal{E}^{i}:=\mathcal{E}^{i}(0)$$
and
$$\displaystyle\mathcal{E}^{2j-1}(\theta)$$
$$\displaystyle:=$$
$$\displaystyle\{(E,\hat{J})\ \ \text{s.t.}\ \ E^{(2j-1)}_{-}(\hat{J})<E<E^{(2j-%
1)}_{+}(\hat{J})-2\theta\,,\ \hat{J}\in\hat{D}\}\,,\qquad 1\leq j\leq N\,,$$
$$\displaystyle\mathcal{E}^{2j}(\theta)$$
$$\displaystyle:=$$
$$\displaystyle\{(E,\hat{J})\ \ \text{s.t.}\ \ E^{(2j)}_{-}(\hat{J})+2\theta<E<E%
^{(2j)}_{+}(\hat{J})-2\theta\,,\ \hat{J}\in\hat{D}\}\,,\qquad 1\leq j<N\,,$$
$$\displaystyle\mathcal{E}^{2N}(\theta)=\mathcal{E}^{0}(\theta)$$
$$\displaystyle:=$$
$$\displaystyle\{(E,\hat{J})\ \ \text{s.t.}\ \ E^{(2N)}_{-}(\hat{J})+2\theta<E<R%
_{0}^{2}-M-2\theta\,,\ \hat{J}\in\hat{D}\}\,,$$
(324)
where the positive parameter $\theta$ was introduced in (320)
(and recall (315)).
We also introduce the complex $\theta$-neighborhoods717171Recall the notation on page 3.
$$\mathcal{E}^{i}_{\theta}(\theta):=\big{(}\mathcal{E}^{i}(\theta)\big{)}_{%
\theta}\subseteq\mathbb{C}^{n}\,.$$
(325)
The functions $P_{n}^{(i)}$ are defined as
follows727272Sometimes omitting, for brevity,
to write the explicit dependence on $\hat{J}$..
For $i=2j-1$ odd, $1\leq j\leq N,$
and $E^{(2j-1)}_{-}(\hat{J})<E<E^{(2j-1)}_{+}(\hat{J})$, we set
$$\displaystyle P_{n}^{(2j-1)}(E)=P_{n}^{(2j-1)}(E,\hat{J})$$
(326)
$$\displaystyle:=\frac{1}{2\pi}\int_{X_{2j-1}(E)}^{X_{2j}(E)}\Big{[}\mathcal{J}_%
{n}\Big{(}\sqrt{E-F(x)},x\Big{)}-\mathcal{J}_{n}\Big{(}-\sqrt{E-F(x)},x\Big{)}%
\Big{]}\,dx$$
$$\displaystyle=\frac{1}{\pi}\int_{X_{2j-1}(E)}^{X_{2j}(E)}\sqrt{E-F(x)}\Big{(}1%
+b_{\sharp}\big{(}\sqrt{E-F(x)},x\big{)}\Big{)}\,dx\,,$$
where the last equality holds recalling (307)
and defining $b_{\sharp}=b_{\sharp}(\hat{J},z,x)$ as follows:
$$2+2b_{\sharp}(\hat{J},z,x):=\frac{1}{\sqrt{1+b\big{(}\hat{J},\mathcal{J}_{n}(z%
,x,\hat{J}),x\big{)}}}+\frac{1}{\sqrt{1+b\big{(}\hat{J},\mathcal{J}_{n}(-z,x,%
\hat{J}),x\big{)}}}$$
(327)
($b$ defined in (134)).
Note that
$$b_{\sharp}\ \ \text{is even w.r.t.}\ z\ \ \text{and}\ \ \sup_{z\in(-R_{0},R_{0%
})_{r_{0}/2}}\|b_{\sharp}(\hat{J},z,x)\|_{\hat{D},r_{0},s_{0}}\lessdot\eta\,.$$
(328)
For $i=2j$ even, $1\leq j\leq N-1$
and $E^{(2j)}_{-}(\hat{J})<E<E^{(2j)}_{+}(\hat{J})$,
we set (recall (116))
$$\displaystyle P_{n}^{(2j)}(E)=P_{n}^{(2j)}(E,\hat{J})$$
(329)
$$\displaystyle:=\frac{1}{2\pi}\int_{X_{2j_{-}+1}(E)}^{X_{2j_{+}}(E)}\Big{[}%
\mathcal{J}_{n}\Big{(}\sqrt{E-F(x)},x\Big{)}-\mathcal{J}_{n}\Big{(}-\sqrt{E-F(%
x)},x\Big{)}\Big{]}\,dx$$
$$\displaystyle=\frac{1}{\pi}\int_{X_{2j_{-}+1}(E)}^{X_{2j_{+}}(E)}\sqrt{E-F(x)}%
\Big{(}1+b_{\sharp}\big{(}\sqrt{E-F(x)},x\big{)}\Big{)}\,dx\,,$$
where $j_{-},j_{+}$ were defined in (116).
Finally for $E>E^{(2N)}_{-}(\hat{J})$ we set
$$\displaystyle P_{n}^{(2N)}(E)=P_{n}^{(2N)}(E,\hat{J})$$
$$\displaystyle:=$$
$$\displaystyle\frac{1}{2\pi}\int_{-\pi}^{\pi}\mathcal{J}_{n}\Big{(}\sqrt{E-F(x)%
},x\Big{)}\,dx\,,$$
(330)
$$\displaystyle P_{n}^{(0)}(E)=P_{n}^{(0)}(E,\hat{J})$$
$$\displaystyle:=$$
$$\displaystyle\frac{1}{2\pi}\int_{-\pi}^{\pi}\mathcal{J}_{n}\Big{(}-\sqrt{E-F(x%
)},x\Big{)}\,dx\,.$$
(331)
C.5 Properties of the actions as functions of the energy and viceversa
Lemma C.2
Assume that
$$\frac{\eta}{c_{F^{0}}}\leq\eta_{0}(\mathfrak{p})\,,$$
(332)
with $c_{F^{0}}$ defined in (305) and $\eta_{0}=\eta_{0}(\mathfrak{p})$ small enough.
Then
for every $1\leq i\leq 2N-1$
$$\inf\partial_{E}P_{n}^{(i)}(E,\hat{J})\geq\ c_{F^{0}}/2>0\,,$$
(333)
while737373Recall (315)
$$\displaystyle\frac{1}{8\sqrt{E}}\leq\partial_{E}P_{n}^{(2N)}(E,\hat{J})\,,\ -%
\partial_{E}P_{n}^{(0)}(E,\hat{J})\leq\frac{2}{\sqrt{E}}\,,\qquad\forall\,2M%
\leq E\leq R_{0}^{2}-M\,,\ \ \hat{J}\in\hat{D}\,,$$
(334)
$$\displaystyle\partial_{E}P_{n}^{(2N)}(E,\hat{J})\,,\ -\partial_{E}P_{n}^{(0)}(%
E,\hat{J})\geq\frac{1}{8\sqrt{2M}}\,,\qquad\forall\,E_{2N}<E\leq 2M\ \ \hat{J}%
\in\hat{D}$$
(335)
($M$ defined in (302)).
Proof It essentially follows from
Proposition
5.2; see [5]
for details.
By (333) we have that, for every fixed $\hat{J}\in\hat{D},$
the function $E\mapsto P_{n}^{(i)}(E,\hat{J})$ is strictly monotone
and, therefore, invertible with inverse
$\mathtt{E}^{(i)}(\hat{J},P_{n})$ such that
$$\mathtt{E}^{(i)}(\hat{J},P_{n}^{(i)}(E,\hat{J}))=E\qquad\text{and}\qquad P_{n}%
^{(i)}(\mathtt{E}^{(i)}(\hat{J},P_{n}),\hat{J}))=P_{n}\,.$$
(336)
As a corollary of Proposition 5.2
and of the cain rule applied to (336), giving
$$\partial_{P_{n}}\mathtt{E}^{(i)}(P)=\frac{1}{\partial_{E}P_{n}}\,,\qquad%
\partial_{\hat{P}}\mathtt{E}=-\frac{\partial_{\hat{P}}P_{n}}{\partial_{E}P_{n}%
}\,,$$
by Lemma C.2
we get the following.
For every $1\leq i\leq 2N-1$
$$0<\partial_{P_{n}}\mathtt{E}^{(i)}\leq\ 2/c_{F^{0}}\,,$$
(337)
($c_{F^{0}}$ defined
in (305)).
Moreover
$$\displaystyle\frac{\sqrt{E}}{2}\leq\partial_{P_{n}}\mathtt{E}^{(2N)}\,,\ -%
\partial_{P_{n}}\mathtt{E}^{(0)}\leq 8\sqrt{E}\,,\qquad\forall\,2M\leq E\leq R%
_{0}^{2}-M\,,\ \ \hat{P}\in\hat{D}\,,$$
(338)
$$\displaystyle\partial_{P_{n}}\mathtt{E}^{(2N)}\,,\ -\partial_{P_{n}}\mathtt{E}%
^{(0)}\leq 8\sqrt{2M}\,,\qquad\forall\,E^{(0)}_{-}\,,\ E^{(2N)}_{-}<E\leq 2M\ %
\ \hat{P}\in\hat{D}$$
(339)
($M$ defined in (302)).
The proofs of the following lemmata
essentially follows from
Proposition
5.2; see [5]
for details.
Lemma C.3
Let $C>1$ as in Proposition 5.2.
$$\sup_{\hat{D}_{r_{0}}\times\mathtt{I}^{(i)}_{\mathtt{r}_{0}/8}}\|\partial_{PP}%
\mathtt{E}^{(i)}(P)\|\leq C\,,\qquad\sup_{\hat{D}_{r_{0}}\times\mathtt{I}^{(i)%
}_{\mathtt{r}_{0}/8}}|\partial_{P_{n}}\mathtt{E}^{(i)}(P)-\partial_{P_{n}}%
\mathtt{E}^{(i),0}(P_{n})|\leq C\eta\,,\qquad\forall 0\leq i\leq 2N\,,$$
(340)
where the intervals $\mathtt{I}^{(i)}$ where defined in
(240) and $\mathtt{r}_{0}>0$
in (230).
Lemma C.4
(171) holds.
Lemma C.5
Assume that $F$
is cosine-like according to Definition
4.2, with $\mathfrak{c}$ (namely $\mathfrak{c}_{*}$ defined in (100)) small enough.
Then
$$\inf_{E_{1}<E<E_{2},\,\hat{P}\in\hat{D}}\left|\partial_{P_{n}P_{n}}\mathtt{E}^%
{(1)}\right|\,\geq\,c_{\sharp}\,,\qquad\inf_{E_{2}<E<R_{0}^{2}-2,\,\hat{P}\in%
\hat{D}}\left|\partial_{P_{n}P_{n}}\mathtt{E}^{(2)}\right|\,\geq\,c_{\sharp}\,,$$
(341)
for a suitable (absolute constant)
$c_{\sharp}>0$.
Appendix D Miscellanea
Composition of maps
Lemma D.1
Let $y=Ly^{\prime},$ where $L:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is linear.
Then for every function $h:\mathbb{R}^{n}\to\mathbb{R}$ we have
$$\det\Big{(}\partial_{y^{\prime}y^{\prime}}\big{(}h\circ L(y^{\prime})\big{)}%
\Big{)}=(\det L)^{2}\det(\partial_{yy}h)_{|{y=Ly^{\prime}}}\ .$$
Lemma D.1 follows immediately observing that $\partial_{y^{\prime}y^{\prime}}\big{(}h(Ly^{\prime})\big{)}=L^{T}(\partial_{yy%
}h)_{|{y=Ly^{\prime}}}L$.
Given a Hamiltonian $H(J,\psi)$
we denote by $\Phi^{t}_{H}$ its flow at time $t.$
Lemma D.2
(i) [Time rescaling] Let
$c>0$. Then
$\Phi^{t}_{H}=\Phi^{ct}_{c^{-1}H}$.
(ii) [Action rescaling] Consider the conformally symplectic change of
variables
$$(J,\psi)=\Phi(\tilde{J},\tilde{\psi}):=(c\tilde{J},\tilde{\psi})$$
(342)
and set
$\tilde{H}:=H\circ\Phi$ and $\hat{H}:=c^{-1}H\circ\Phi$.
Then $\Phi\circ\Phi^{t/c}_{\tilde{H}}=\Phi^{t}_{H}\circ\Phi$ and
$\Phi\circ\Phi^{t}_{\hat{H}}=\Phi^{t}_{H}\circ\Phi$.
The proof is a straightforward check.
Lemma D.3
Consider $s_{1}>0,s>s_{2}>0$ and an holomorphic map
$\Phi:\mathbb{T}^{n}_{s_{1}}\to\mathbb{T}^{n}_{s_{2}}$ and an holomorphic function
$f$ with $|f|_{s}<\infty.$ Then
$$|f\circ\Phi|_{s_{1}}\leq\coth^{n}\left(\frac{s-s_{2}}{2}\right)|f|_{s}\leq%
\left(1+\frac{2}{s-s_{2}}\right)^{n}|f|_{s}\,.$$
Proof By (87) we get
$\displaystyle|f\circ\Phi|_{s_{1}}\leq\|f\circ\Phi\|_{s_{1}}\leq\|f\|_{s_{2}}%
\leq\coth^{n}\left(\frac{s-s_{2}}{2}\right)|f|_{s}\,.$
Lemma D.4
Given a matrix $M\in{\rm Mat}_{n\times n}(\mathbb{Z})$ with
$\det M=\pm 1,$ consider the symplectic linear map
$\Phi:\mathbb{R}^{n}\times\mathbb{T}^{n}\to\mathbb{R}^{n}\times\mathbb{T}^{n}$ defined as
$$(J^{\prime},\psi^{\prime})=\Phi(J,\psi):=(M^{T}J,M^{-1}\psi)\,.$$
Let $D\subseteq\mathbb{R}^{n}$ and $r,s>0.$ Set
$D^{\prime}:=(M^{T})^{-1}D.$ Then
$$\Phi(D^{\prime}_{r^{\prime}}\times\mathbb{T}^{n}_{s^{\prime}})\subseteq D_{r}%
\times\mathbb{T}^{n}_{s}\,,\qquad\text{where}\ \ \ r^{\prime}:=r/\|M^{T}\|\,,%
\ \ s^{\prime}:=s/\sup_{1\leq i\leq n}\sum_{1\leq j\leq n}|(M^{-1})_{ij}|\,.$$
Moreover, given a function $f:D_{r_{0}}\times\mathbb{T}^{n}_{s_{0}}$
with $r_{0}\geq r,$ $s_{0}>s,$ we have
$$|f\circ\Phi|_{D^{\prime},r^{\prime},s^{\prime}}\leq\coth^{n}\left(\frac{s_{0}-%
s}{2}\right)|f|_{s_{0}}\leq\left(1+\frac{2}{s_{0}-s}\right)^{n}|f|_{s_{0}}\,.$$
The first part is obvious; the second part follows from Lemma D.4.
Restrictions of maps
Lemma D.5
Let $\Phi:D^{\prime}\times\mathbb{T}^{n}\to D\times\mathbb{T}^{n}$ be a real analytic map with holomorphic
extension
$$\Phi:D^{\prime}_{r^{\prime}}\times\mathbb{T}^{n}_{s^{\prime}}\to D_{r}\times%
\mathbb{T}^{n}_{s}$$
for some $r,r^{\prime}s,s^{\prime}>0.$
There exists
a suitably small
constant $c$ depending only on $n$ such that
$$\Phi\left(D^{\prime}_{car^{\prime}}\times\mathbb{T}^{n}_{cas^{\prime}}\right)%
\ \ \subseteq\ D_{ar}\times\mathbb{T}^{n}_{as}\,,\qquad\forall\,0<a\leq 1\,.$$
(343)
Proof By
Cauchy estimates,
applied to the various components of $\Phi$.
A group of parameter–dependent symplectic transformations
Let us consider the group $\cal G$
introduced in
(129).
Lemma D.6
Given a symplectic transformation of the form
(129), we have that, for every fixed $\hat{J},$ the restriction
$$(J_{n},\psi_{n})\mapsto\big{(}I_{n}(J,\psi_{n}),\varphi_{n}(J,\psi_{n})\big{)}$$
is also symplectic.
Proof Note that by the conservation of the symplectic form
$dI\wedge d\varphi=dJ\wedge d\psi$ follows that
$\partial_{J_{n}}I_{n}\partial_{\psi_{n}}\varphi_{n}-\partial_{J_{n}}\varphi_{n%
}\partial_{\psi_{n}}I_{n}=1.$
Recall the definition given in (130) and note that
$${(\Phi_{1}\circ\Phi_{2})}\check{\phantom{A}}=\check{\Phi}_{1}\circ\check{\Phi}%
_{2}\,.$$
(344)
Furthermore, obviously, one has
$$\check{\phi}(E)\times{\mathbb{T}}^{n-1}=\phi(E\times{\mathbb{T}}^{n-1})\ ,%
\qquad\forall\phi\in{\cal G}\ ,\qquad\forall E\subseteq{\mathbb{R}}^{n}\times{%
\mathbb{T}}^{1}\ .$$
(345)
By Lemma D.6 we have the following
Lemma D.7
If $\Phi\in{\cal G}$, then $\check{\Phi}$ is volume-preserving.
An elementary result in linear algebra
Lemma D.8
Given $k\in\mathbb{Z}^{n},$ $k\neq 0$
there exists a matrix $A=(A_{ij})_{1\leq i,j\leq n}$ with integer entries such that
$A_{nj}=k_{j}$ $\,\forall\,1\leq j\leq n$,
$\det A=d:={\rm gcd}(k_{1},\ldots,k_{n})$, and
$|A|_{\infty}=|k|_{\infty}$.
Proof The argument is by induction over $n$.
For $n=1$ the lemma is obviously true. For $n=2$, it follows at once from747474The first statement in this formulation of Bezout’s Lemma is well known
and it can be found in any textbook on elementary number theory; the estimates on $x$ and $y$ are easily deduced from the well known fact that
given a solution $x_{0}$ and $y_{0}$ of the equation $ax+by=d$, all other solutions have the form $x=x_{0}+k(b/d)$ and $y=y_{0}-k(a/d)$ with $k\in{\mathbb{Z}}$
and by choosing $k$ so as to minimize $|x|$.
Bezout’s Lemma Given two integers $a$ and $b$ not both zero, there exist two integers $x$ and $y$ such that $ax+by=d:={\rm gcd}(a,b)$, and such that
$\max\{|x|,|y|\}\leq\max\{|a|/d,|b|/d\}$.
Indeed, if $x$ and $y$ are as in Bezout’s Lemma with $a=k_{1}$ and $b=k_{2}$ one can take $A=\begin{pmatrix}y&-x\\
k_{1}&k_{2}\end{pmatrix}$.
Now, assume, by induction for $n\geq 3$ that the claim holds true for $(n-1)$ and let us prove it for $n$.
Let $\bar{k}=(k_{1},...,k_{n-1})$ and $\bar{d}={\rm gcd}(k_{1},...,k_{n-1})$ and notice that ${\rm gcd}(\bar{d},k_{n})=d$. By the inductive assumption, there exists a matrix $\bar{A}=\begin{pmatrix}\tilde{A}\\
\bar{k}\end{pmatrix}\in{\rm Mat}_{(n-1)\times(n-1)}(\mathbb{Z})$ with $\tilde{A}\in{\rm Mat}_{(n-2)\times(n-1)}(\mathbb{Z})$, such that $\det\bar{A}=\bar{d}$ and $|\bar{A}|_{\infty}=|\bar{k}|_{\infty}$.
Now, let $x$ and $y$ be as in Bezout’s Lemma with $a=\bar{d}$, and $b=k_{n}$.
We claim that $A$ can be defined as follows:
$$A=\begin{pmatrix}&\tilde{k}&&\tilde{x}\\
&\bar{A}&&\begin{pmatrix}0\\
\vdots\\
0\\
k_{n}\end{pmatrix}\end{pmatrix}\ ,\qquad\tilde{k}=(-1)^{n}y\,\frac{\bar{k}}{%
\bar{d}}\ ,\qquad\tilde{x}:=(-1)^{n+1}x\ .$$
(346)
First, observe that since $\bar{d}$ divides $k_{j}$ for $j\leq(n-1)$, $\tilde{k}\in{\mathbb{Z}}^{n-1}$. Then, expanding the determinant of $A$ from last column, we get
$$\displaystyle\det A$$
$$\displaystyle=$$
$$\displaystyle(-1)^{n+1}\tilde{x}\det\bar{A}+k_{n}\det\begin{pmatrix}\tilde{k}%
\\
\tilde{A}\end{pmatrix}$$
$$\displaystyle=$$
$$\displaystyle(-1)^{n+1}\tilde{x}\,\bar{d}+k_{n}(-1)^{n-2}\det\begin{pmatrix}%
\tilde{A}\\
\tilde{k}\end{pmatrix}$$
$$\displaystyle=$$
$$\displaystyle(-1)^{n+1}\tilde{x}\,\bar{d}+k_{n}(-1)^{n-2}(-1)^{n}\frac{y}{\bar%
{d}}\det\bar{A}$$
$$\displaystyle=$$
$$\displaystyle x\bar{d}+k_{n}y=d\ .$$
Finally, by Bezout’s Lemma, we have that $\max\{|x|,|y|\}\leq\max\{\bar{d}/d,|k_{n}|/d\}$, so that
$$|\tilde{k}|_{\infty}=|y|\frac{|\bar{k}|_{\infty}}{\bar{d}}\leq\frac{|\bar{k}|_%
{\infty}}{d}\leq|k|_{\infty}\ ,\quad|\tilde{x}|=|x|\leq\frac{|k_{n}|}{d}\leq|k%
|_{\infty}\ ,$$
which, together with $|\bar{A}|_{\infty}=|\bar{k}|_{\infty}$, shows that $|A|_{\infty}=|k|_{\infty}$.
Measure of sub–levels of smooth non–degenerate functions
Here we prove Lemma 6.1.
We start by recalling an elementary result, whose proof can be found in [3]:
Lemma D.9
Let $g(x)$ a monic polynomial of degree $d$.
Then
$${\rm meas}\big{(}\{x\in\mathbb{R}\ \ :\ \ |g(x)|\leq\gamma\}\big{)}\leq 2d%
\gamma^{1/d}\,.$$
We now prove Lemma 6.1.
Let us divide the interval $[a,b]$ in disjoint intervals
of length $2r:=2\mu^{1/m+1}$.
Let $I$ one of such intervals and let $x_{0}$ is middle point.
By (225) let $1\leq d\leq m$ such that
$$|\partial_{x}^{d}f(x_{0})|/d!\geq\xi_{m}\,.$$
(347)
By the Taylor remainder formula
we get, for $x\in I$,
$$|f(x)-P^{d}_{x_{0}}(x)|\leq Mr^{d+1}=M\mu^{\frac{d+1}{m+1}}\,,$$
where $P^{d}_{x_{0}}(x):=\sum_{0\leq j\leq d}\frac{\partial_{x}^{j}f(x_{0})}{j!}(x-x_%
{0})^{j}$
is the Taylor polynomial of degree $d$.
Then we have that
$$\{x\in I\ \ :\ \ |f(x)|\leq\mu\}\ \subseteq\ \{x\in I\ \ :\ \ |P^{d}_{x_{0}}(x%
)|\leq(M+1)\mu^{\frac{d+1}{m+1}}\}\,.$$
(348)
We now apply Lemma D.9 to the monic polynomial
$g(x):=d!P^{d}_{x_{0}}(x)/\partial_{x}^{d}f(x_{0})$
with
$$\gamma:=(M+1)\mu^{\frac{d+1}{m+1}}/\xi_{m}\stackrel{{\scriptstyle\eqref{%
macinebis}}}{{\geq}}d!(M+1)\mu^{\frac{d+1}{m+1}}/|\partial_{x}^{d}f(x_{0})|\ .$$
By (348) and Lemma D.9 we get
(recall $1\leq d\leq m$)
$${\rm meas}\big{(}\{x\in I\ \ :\ \ |f(x)|\leq\mu\}\big{)}\leq 2d\gamma^{1/d}%
\leq\frac{2m(M+1)}{\xi_{m}}\mu^{1/m}$$
Since the number of disjoint intervals is smaller that
$\frac{b-a}{2\mu^{1/m+1}}+1,$
this concludes the proof of Lemma 6.1.
Canonical form of generalized pendula
The following lemma describes how to make independent of the action $J_{n}$ a pendulum depending on parameters.
Lemma D.10
Let
$$H^{*}(y,x):=y_{n}^{2}+F^{0}(x_{n})+G^{*}(y,x_{n})\,,$$
with $\|G^{*}\|_{D,r_{0},s_{0}}\leq\eta_{*}$ .
Assume that
$$\eta_{*}\leq r_{0}^{2}/16\,.$$
(349)
Then the fixed point equation
$$\mathtt{y}(\hat{Y},X_{n})=-\frac{1}{2}\partial_{Y_{n}}G^{*}\big{(}\hat{Y},%
\mathtt{y}(\hat{Y},X_{n}),X_{n}\big{)}$$
(350)
has
a unique solution
$\mathtt{y}=\mathtt{y}(\hat{Y},X_{n})$ with
$$\|\mathtt{y}\|_{\hat{D},r_{0},s_{0}}\leq 2\eta_{*}/r_{0}\leq r_{0}/8\,.$$
(351)
Set
$$J^{*}_{n}(\hat{Y}):=\langle\mathtt{y}(\hat{Y},X_{n})\rangle\,,\qquad a_{*}(%
\hat{Y},X_{n}):=\mathtt{y}(\hat{Y},X_{n})-\langle\mathtt{y}(\hat{Y},X_{n})%
\rangle\,,$$
where $\langle\cdot\rangle$ denotes the average w.r.t. $X_{n}.$ Let $\phi=\phi(\hat{Y},X_{n})$ the unique function
satisfying $a_{*}=\partial_{x_{n}}\phi$
with $\langle\phi\rangle=0.$
Consider the canonical transformation $\Psi$
$$y_{n}=Y_{n}+a_{*}(\hat{Y},X_{n})=Y_{n}-J^{*}_{n}(\hat{Y})+\mathtt{y}(\hat{Y},X%
_{n})\,,\qquad x_{n}=X_{n}\,,\qquad\hat{y}=\hat{Y}\,,\quad\hat{x}=\hat{X}+b_{*%
}(\hat{Y},X_{n}),$$
(352)
obtained by the
generating function
$Y_{n}x_{n}+\hat{Y}\hat{x}+\phi(\hat{Y},x_{n}),$
with
$b_{*}=-\partial_{\hat{Y}}\phi$
and
$$\|J^{*}_{n}\|_{\hat{D},r_{0}}\leq 2\eta_{*}/r_{0}\stackrel{{\scriptstyle\eqref%
{urea}}}{{\leq}}r_{0}/8\,,\quad\|a_{*}\|_{\hat{D},r_{0},s_{0}}\leq 4\eta_{*}/r%
_{0}\,,\quad\|b_{*}\|_{\hat{D},r_{0}/2,s_{0}}\leq(16\pi+8)\eta_{*}/r_{0}^{2}\,.$$
(353)
Note that
$$\Psi\ :\ D_{r_{0}/2}\times\mathbb{T}^{n-1}_{\hat{s}}\times\mathbb{T}_{s_{0}}\ %
\to\ D_{r_{0}}\times\mathbb{T}^{n-1}_{\hat{s}+(16\pi+8)\eta_{*}/r_{0}^{2}}%
\times\mathbb{T}_{s_{0}}\,.$$
(354)
Then
(352) casts $H^{*}$ into
$$\displaystyle\Big{(}Y_{n}-J^{*}_{n}(\hat{Y})+\mathtt{y}(\hat{Y},X_{n})\Big{)}^%
{2}+F^{0}(X_{n})+G^{*}(\hat{Y},Y_{n}-J^{*}_{n}(\hat{Y})+\mathtt{y}(\hat{Y},X_{%
n}),X_{n})$$
$$\displaystyle=\big{(}1+b(Y,X_{n})\big{)}\big{(}Y_{n}-J^{*}_{n}(\hat{Y})\big{)}%
^{2}+F(\hat{Y},X_{n})\,,$$
with
$$F=F^{0}+G\,,\qquad G:=G^{*}(\hat{Y},\mathtt{y}(\hat{Y},X_{n}),X_{n})+(\mathtt{%
y}(\hat{Y},X_{n}))^{2}$$
and757575Using (350).
omitting, for brevity, the dependence on
$\hat{Y},X_{n},$
$$b=\frac{G^{*}(\mathtt{y}+Y_{n}-J_{n}^{*})-G^{*}(\mathtt{y})-\partial_{Y_{n}}G^%
{*}(\mathtt{y})(Y_{n}-J_{n}^{*})}{(Y_{n}-J_{n}^{*})^{2}}=\int_{0}^{1}(1-t)%
\partial_{Y_{n}Y_{n}}G^{*}\big{(}\mathtt{y}+t(Y_{n}-J_{n}^{*})\big{)}dt\,.$$
(355)
Finally
$$\displaystyle\|G\|_{\hat{D},r_{0},s_{0}}\leq\left(1+4/r_{0}^{2}\right)\eta_{*}%
\,,\qquad\|(1+|Y_{n}-J_{n}^{*}|)b(Y,X_{n})\|_{D,r_{0}/2,s_{0}}\leq\left(4+%
\frac{34}{r_{0}^{2}}\right)\eta_{*}\,,$$
$$\displaystyle\||Y_{n}-J_{n}^{*}|\partial_{Y_{n}}b(Y,X_{n})\|_{D,r_{0}/2,s_{0}}%
\leq\frac{48}{r_{0}^{2}}\eta_{*}\,.$$
(356)
Proof (350) is solved by the standard Fixed Point Theorem for $\mathtt{y}$ in the ball in (351).
(353) and the first estimate in (D.10) follow by
(349), (351) and Cauchy estimates.
By (355),
(349), (351) and Cauchy estimates
we get
$$\|b\|_{D,r_{0}/2,s_{0}}\leq\frac{16}{r_{0}^{2}}\eta_{*}$$
(357)
and, therefore,
dividing the cases
$|Y_{n}-J_{n}^{*}|\leq 1$ and $|Y_{n}-J_{n}^{*}|>1,$
we get the second estimate in
(D.10).
Finally, by (355), we have
$$\partial_{Y_{n}}b=\frac{\partial_{Y_{n}}G^{*}(\mathtt{y}+Y_{n}-J_{n}^{*})-%
\partial_{Y_{n}}G^{*}(\mathtt{y})}{(Y_{n}-J_{n}^{*})^{2}}-\frac{2b}{Y_{n}-J_{n%
}^{*}}\,.$$
Then
$$|\partial_{Y_{n}}b||Y_{n}-J_{n}^{*}|\leq\frac{|\partial_{Y_{n}}G^{*}(\mathtt{y%
}+Y_{n}-J_{n}^{*})-\partial_{Y_{n}}G^{*}(\mathtt{y})|}{|Y_{n}-J_{n}^{*}|}+2|b|%
\leq\frac{48}{r_{0}^{2}}\eta_{*}\,.\hskip 14.226378pt\vrule width 4.836969pt h%
eight 9.958465pt depth 0.0pt$$
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\frontpagetrue\pubtype
T
SLAC-PUB-96-7104
SU-ITP-96-06
SCIPP 96-08
hep-ph/9601367
\title\seventeenbf
Experimental Signatures of Low Energy
\title\seventeenbf
Gauge Mediated Supersymmetry Breaking
\authorSavas Dimopoulos${}^{ab}$, Michael Dine${}^{c}$\footWork
supported by the Department of Energy.,
Stuart Raby${}^{d}$\footWork supported by the Department of
Energy under contract DOE-ER-01545-646.,
Scott Thomas${}^{e}$\footWork supported by the Department of
Energy under contract DE-AC03-76SF00515.
\address
${}^{a}$Physics Department, Stanford University, Stanford, CA 94309
\address${}^{b}$Theoretical Physics Division, CH-1211, Geneva 23,
Switzerland
\address${}^{c}$Santa Cruz Institute for Particle Physics,
University of California, Santa Cruz, CA 95064
\address${}^{d}$Physics Department, Ohio State University,
Columbus, OH 43210
\address${}^{e}$Stanford Linear Accelerator Center,
Stanford, CA 94309
Abstract
The experimental signatures for low energy gauge mediated supersymmetry
breaking are distinctive since the gravitino is naturally the LSP.
The next lightest supersymmetric particle (NLSP) can be a gaugino,
Higgsino, or right handed slepton.
For a significant range of parameters decay of the NLSP to its
partner plus the gravitino can be measured as a displaced vertex
or kink in a charged particle track.
In the case that the NLSP is mostly gaugino, we identify
the discovery modes as $e^{+}e^{-}\rightarrow\gamma\gamma+\not\!\!E$,
and $p\bar{p}\rightarrow l^{+}l^{-}\gamma\gamma+\not\!\!E_{T}$.
If the NLSP is a right handed slepton the discovery modes are
$e^{+}e^{-}\rightarrow l^{+}l^{-}+\not\!\!E$ and
$p\bar{p}\rightarrow l^{+}l^{-}+\not\!\!E_{T}$.
An NLSP which is mostly Higgsino is also considered.
\endpage\pagenumber
=1
\singlespace
\singlespace
\chapter
Introduction
\REF\lsgauge
M. Dine, W. Fischler, and M. Srednicki, Nucl. Phys. B
189 (1981) 575;
S. Dimopoulos and S. Raby, Nucl. Phys. B 192 (1981) 353;
M. Dine and W. Fischler, Phys. Lett. B 110 (1982) 227;
M. Dine and M. Srednicki, Nucl. Phys. B 202 (1982) 238;
L. Alvarez-Gaumé, M. Claudson, and M. Wise, Nucl. Phys. B
207 (1982) 96;
C. Nappi and B. Ovrut, Phys. Lett. B 113 (1982) 175.
\REF\hsgauge
M. Dine and W. Fischler, Nucl. Phys. B 204 (1982) 346;
S. Dimopoulos and S. Raby, Nucl. Phys. B 219 (1983) 479.
\REF\dnns
M. Dine, A.E. Nelson and Y. Shirman, Phys. Rev. D 51 (1995) 1362;
M. Dine, A.E. Nelson, Y. Nir and Y. Shirman,
Phys. Rev. D 53 (1996) 2658.
\REF\soft
S. Dimopoulos and H. Georgi, Nucl. Phys. B 193 (1981) 150.
\REF\dimsut
S. Dimopoulos and D. Sutter, Nucl. Phys. B
452 (1995) 496.
\REF\ilI.E. Ibanez and D. Lust, Nucl. Phys. B
382 (1992) 305.
\REF\kl
V. Kaplunovsky and J. Louis, Phys. Lett. B
306 (1993) 269.
\REF\niretal
M. Leurer, Y. Nir and N. Seiberg,
Phys. Lett. B 309 (1993) 337.
\REF\dklM. Dine, A. Kagan and R. Leigh,
Phys. Rev. D 48 (1993) 4269.
\REF\kaplanD.B. Kaplan and M. Schmalz,
Phys. Rev. D 49 (1994) 3741.
Low energy supersymmetry is widely viewed as a plausible
solution of the hierarchy problem. If nature is supersymmetric,
it is important to understand how supersymmetry is broken.
It is usually assumed that supersymmetry breaking is
communicated to ordinary fields and their
superpartners by supergravity. The breaking
scale is then necessarily of order $10^{11}$ GeV.
An alternative possibility,
which has been less thoroughly explored, is that supersymmetry
is broken at some lower energy scale, and that
the ordinary gauge interactions
act as the messengers of supersymmetry
breaking\refmark\lsgauge-\dnns. In this
case, the scale of supersymmetry breaking can be as low as
$10$’s of TeV\refmark\lsgauge,\dnns.
Independent of source and messenger, supersymmetry
breaking is represented among ordinary fields (the
visible sector) by soft supersymmetry breaking terms
\refmark\soft.
The most general soft-breaking Lagrangian is described
by $105$ parameters beyond those of the minimal
standard model\refmark\dimsut.
There are a number of constraints which these
parameters must satisfy, coming from direct experimental
searches for superpartners, electric dipole moments,
and the lack of flavor changing neutral currents.
Most model builders simply postulate a high degree of degeneracy among
squarks and sleptons at a high energy scale to deal with this
problem\refmark\soft.
In certain classes of
superstring theories, there are weak hints for
such a universality\refmark\il,\kl. Alternatively,
the various experimental constraints might
be satisfied as a result of flavor symmetries
or by other means\refmark\niretal-\kaplan.
With gauge mediated supersymmetry breaking
the entire soft breaking Lagrangian can
be calculated in terms of a small
number of parameters. In addition,
the regularities required
to avoid flavor changing neutral currents
0are automatically obtained since the ordinary gauge
interactions do not distinguish generations.
For these reasons, we believe the gauge-mediated
possibility should be taken seriously.
In this letter, we discuss some striking and distinctive
signatures of low energy gauge-mediated supersymmetry
breaking. The first is the spectrum of superpartner
masses.
These masses are functions of the gauge quantum numbers,
and are roughly in the ratio of the appropriate gauge couplings squared.
In the simplest models, definite relations
exist among these masses.
As a result, the lightest standard model superpartner is almost
inevitably either
a neutralino or a right-handed slepton.
The second important
signature arises from the fact that the
lightest supersymmetric particle (LSP) is the gravitino.
The lightest standard model superpartner is then the
next to lightest supersymmetric particle (NLSP).
Assuming
that $R$-parity is conserved, the principle decay of the NLSP
is then to its partner plus a gravitino. The longitudinal
component of the gravitino – the Goldstino – couples
to matter with strength proportional to $F^{-1}$, where
$F$ is the scale of supersymmetry breaking.
For a plausible range of $F$, the decay length can be
100’s of $\mu$m to meters.
The decays can therefore take place within a detector.
This leads to signatures for supersymmetry which are
distinct from the conventional minimal supersymmetric
standard model (MSSM), and with potentially
dramatic displaced vertices.
\chapter
Superpartner Spectrum
\REF\kenscott
K. Intriligator and S. Thomas,
SLAC-PUB-95-7041, hep-th/9603158.
In gauge mediated models, supersymmetry is broken in a messenger
sector which transforms under the standard model gauge group.
The matter fields in this sector are generally referred to as
messenger quarks and leptons.
Supersymmetry breaking is transmitted to the visible sector
by ordinary gauge interactions.
To preserve the successful supersymmetric prediction of the low
energy $\sin^{2}\theta_{W}$ it is sufficient that the messengers
form a GUT representation.
In the simplest versions, the messenger fields are weakly
coupled, and possess the quantum
numbers of a single ${\bf 5}+\bar{\bf 5}$ of $SU(5)$, i.e. there
are triplets, $q$ and $\bar{q}$, and doublets $\ell$ and
$\bar{\ell}$. They couple to a single gauge singlet field,
$S$, through a superpotential
$W=\lambda_{1}Sq\bar{q}+\lambda_{2}S\ell\bar{\ell}$.
The field $S$ has
non-zero expectation values for both scalar and auxiliary
components, $S$ and $F_{S}$.
Integrating out the messenger sector then gives rise to
gaugino masses at one loop.
For $F_{S}\ll S$,
these masses are given by \refmark\dnns
$$m_{\lambda_{i}}=c_{i}\ N{\alpha_{i}\over 4\pi}\ \Lambda\ .\eqn\gluinomass$$
where $c_{1}={5\over 3},c_{2}=c_{3}=1$,
$\Lambda=F_{S}/S$,
and
for a more general messenger sector $N$ is the
equivalent number of $SU(5)$ $\bf{5}+\bar{\bf 5}$ representations.
The scalar masses squared arise at two-loops \refmark\dnns
$$\tilde{m}^{2}={2\Lambda^{2}}N\left[C_{3}\left({\alpha_{3}\over 4\pi}\right)^{2%
}+C_{2}\left({\alpha_{2}\over 4\pi}\right)^{2}+{5\over 3}{\left(Y\over 2\right%
)^{2}}\left({\alpha_{1}\over 4\pi}\right)^{2}\right].\eqn\squarks$$
where $C_{3}={4\over 3}$
for color triplets and zero for singlets, $C_{2}={3\over 4}$ for
weak doublets and zero for singlets,
and $Y$ is the ordinary hypercharge normalized
as $Q=T_{3}+{1\over 2}Y$.
It should be stressed that $F_{S}$ is not necessarily
the intrinsic supersymmetry breaking scale, $F$,
since the gauge singlet field may not be coupled directly
to the supersymmetry breaking sector.
For example, in the model of Ref. \refmark\dnns,
$F\gg F_{S}$. However, it is also perfectly possible that
$F\sim F_{S}$\refmark\kenscott.
While $F_{S}$ determines the superpartner masses, it is $F$ which
determines the Goldstino coupling discussed in the next section.
These expressions for the masses possess a number of noteworthy features.
There is a hierarchy of masses, with colored particles
being the most massive, and $SU(3)\times SU(2)$ singlet
particles the lightest.
The gaugino masses are in the ratio $7:2:1$,
just as for supersymmetry breaking with universal
gaugino masses at a high scale.
For $N=1$ the squark, left handed slepton, right handed slepton,
and bino (partner of the hypercharge
gauge boson) masses are in the ratio $11.6:~{}2.5~{}:~{}1.1~{}:1$.
In this case the bino is the natural candidate for the NLSP.
The gaugino masses grow as $N$, while the scalar masses
grow as $\sqrt{N}$.
For $N=2$ the above masses are in the ratio
$10.6~{}:~{}2.3~{}:~{}1~{}:~{}1.3$.
In this case the right handed slepton is the candidate for the
NLSP.
In more general models the above relations among the masses
can be modified.
For example both
\gluinomass and \squarks are corrected at
${\cal O}(F/\lambda S^{2})$.
Additional modifications can arise with several gauge singlet
fields coupling to $q\bar{q}$ and $\ell\bar{\ell}$.
In the model with
one singlet, the couplings $\lambda_{1}$ and $\lambda_{2}$
cancel out in the expressions for the masses, but this is not true
of the more general case. As a result, both the ratios
of the squark and slepton masses and the ratio of these
masses to gaugino masses are modified.
More generally scalar masses require only supersymmetry
breaking, while gaugino masses require also that
$U(1)_{R}$ be broken to at most $R$-parity.
In principle $U(1)_{R}$ could effectively be broken at a lower scale
than supersymmetry, leading to gauginos which are much lighter
than the scalars.
Perhaps a more interesting possibility is that the messenger
sector is strongly coupled.
Gaugino masses can then arise directly from
non-perturbative dynamics in the messenger sector,
$m_{\lambda}\sim\alpha\Lambda$.
The scalar masses require one perturbative gauge loop,
$\tilde{m}^{2}\sim\alpha(\alpha/4\pi)\Lambda^{2}$.
So in this case the gauginos are much heavier than the scalars,
and the natural candidate for the NLSP is the
right handed slepton.
All of the possibilities given above for the messenger sector
have in common the feature that
masses for standard model superpartners
go roughly as gauge couplings squared, although
the relation of scalar to gaugino masses is model dependent.
The dimensionful terms
which must arise in the Higgs sector
$W=\mu H_{1}H_{2}$, and $V=m_{12}^{2}H_{1}H_{2}+h.c.$,
do not follow directly from the anzatz of
gauge mediated supersymmetry breaking, and are model dependent.
This is because these terms require that the Peccei-Quinn
symmetry between $H_{1}$ and $H_{2}$
be broken by non-gauge interactions.
Specific models with additional singlets and
vector quarks have been constructed in which
$\mu$ and $m_{12}^{2}$ do arise with reasonable magnitude \refmark\dnns.
Because the properties of the Higgs sector are not generic,
we leave open the possibility that the lightest
electroweak neutralino is a general mixture of
gaugino and Higgsino.
\chapter
Phenomenology
\REF\fayet
P. Fayet, Phys. Lett. B 84 (1979) 416;
Phys. Rept. 105 (1984) 21.
Perhaps the most dramatic consequence of low energy gauge
mediated supersymmetry breaking is that the gravitino
is the LSP.
In the global limit the Goldstone fermion, or Goldstino,
of supersymmetry breaking is massless.
In local supersymmetry, the Goldstino becomes the
longitudinal component of the gravitino,
giving a gravitino mass (assuming the cosmological
constant vanishes) of
$$m_{G}={F\over\sqrt{3}M_{p}}\simeq 2.5~{}\left({F\over(100~{}{\rm TeV})^{2}}%
\right)~{}{\rm eV}\eqn\gravmass$$
where $F$ is the supersymmetry breaking scale.
The lightest standard model supersymmetric particle is
then the NLSP, and can
decay to its partner and the gravitino.
The lowest order coupling of the Goldstino is
fixed by the supersymmetric
Goldberger-Treiman low energy theorem to
be given by \refmark\fayet
$${\cal L}=-{1\over F}j^{\alpha\mu}\partial_{\mu}G_{\alpha}~{}+~{}h.c.\eqn\goldcoupling$$
where $j^{\alpha\mu}$ is the supercurrent
and $G_{\alpha}$ is the spin ${1\over 2}$ longitudinal Goldstino
component of the gravitino.
The decay to the Goldstino component is then suppressed only
by $F$ rather than $M_{p}$.
In the case that the NLSP is mostly bino, $\tilde{B}$,
the coupling \goldcoupling
leads to a transition magnetic dipole moment between
the NLSP and gravitino,
$\cos\theta_{W}(m_{\tilde{B}}/2\sqrt{2}F)\tilde{B}\bar{\sigma}^{\mu}\sigma^{\nu%
}G~{}F_{\mu\nu}~{}+h.c.$, giving rise to a decay rate
$$\Gamma(\tilde{B}\rightarrow G+\gamma)={\cos^{2}\theta_{W}~{}m_{\tilde{B}}^{5}%
\over 16\pi F^{2}}\eqn\binorate$$
This translates to a decay length
$$c\tau\simeq 130\left(100~{}{\rm GeV}\over m_{\tilde{B}}\right)^{5}\left(\sqrt{%
F}\over 100~{}{\rm TeV}\right)^{4}~{}\mu{\rm m}\eqn\decaylength$$
So there is a range of $F$ and $m$ for which the decay
occurs within the detector, with the gravitino
carrying off missing energy.
For $m_{\tilde{B}}>m_{Z}^{0}$ there is also a non-negligible branching
fraction $\tilde{B}\rightarrow G+Z^{0}$
(Br$(\tilde{B}\rightarrow G+Z^{0})\rightarrow\sin^{2}\theta_{W}$
for $m_{\tilde{B}}\gg m_{Z}$).
In the case that the NLSP is a right handed slepton
it can decay by
$\tilde{l}_{R}\rightarrow G+l_{R}$ with a decay
length similar to \decaylength.
If the NLSP is mostly Higgsino,
it can decay by
$\tilde{H}^{0}\rightarrow G+h^{0}$ if $m_{h^{0}}<m_{\tilde{H}}$,
where $h^{0}$ is the lightest Higgs boson.
For $m_{h^{0}}>m_{\tilde{H}}$ decay
$\tilde{H}^{0}\rightarrow G+b\bar{b}$ is possible;
however for reasonable values of the parameters the
NLSP decays predominantly to $G+\gamma$
through its gaugino components.
Decay of the lightest standard model supersymmetric particle
to its partner plus the gravitino
within the detector gives signatures which are distinct from
the conventional MSSM.
Let us focus on the discovery modes at $e^{+}e^{-}$ and
hadron colliders.
Consider first the case in which the NLSP is mostly bino.
At $e^{+}e^{-}$ colliders
$e^{+}e^{-}\rightarrow\tilde{B}\tilde{B}\rightarrow\gamma\gamma+\not\!\!E$ is
dominated by $t$- and $u$-channel right handed selectron
exchange.
The production cross section for this process can be significant.
For example,
with $\sqrt{s}=2.2~{}m_{\tilde{B}}$, and assuming the spectrum
resulting from the simple model with $N=1$
given in the previous section,
$\sigma(e^{+}e^{-}\rightarrow\tilde{B}\tilde{B})\simeq.87~{}R$
where $R=4\pi\alpha^{2}/3s$ is the $e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}$
cross section.
In many models, since the bino and slepton masses
are related, the total cross section is related to the bino mass.
This process should show significant polarization dependence
since $\tilde{e}_{R}$ is lighter than $\tilde{e}_{L}$, and
the hypercharge of $\tilde{e}_{R}$ is twice that of
$\tilde{e}_{L}$.
For the parameters given above
$\sigma(e^{+}e^{-}_{L}\rightarrow\tilde{B}\tilde{B})/\sigma(e^{+}e^{-}_{R}%
\rightarrow\tilde{B}\tilde{B})\simeq.01$.
The bino decay is isotropic in the rest frame, implying that
the photons have a flat energy distribution in the lab frame.
Cuts on the $\gamma\gamma$ invariant mass can easily eliminate
the background from $e^{+}e^{-}\rightarrow\gamma\gamma Z^{0}$
with $Z^{0}\rightarrow\nu\bar{\nu}$.
The signature $\gamma\gamma+\not\!\!E$ can also arise in
the conventional MSSM
in some regions of parameter space
if the LSP is mostly Higgsino.
In this case the NLSP
is not much heavier than the LSP, is also mostly
Higgsino, and has a significant branching ratio
$\tilde{H}_{2}\rightarrow\tilde{H}_{1}+\gamma$.
$e^{+}e^{-}\rightarrow\tilde{H}^{0}_{2}\tilde{H}^{0}_{2}$ then gives rise
to this mode.
In the gauge mediated case however,
since $\not\!\!E$ is carried by the essentially massless
gravitinos, the photon energy is bounded by
${1\over 4}\sqrt{s}(1-\beta)\leq E_{\gamma}\leq{1\over 4}\sqrt{s}(1+\beta)$,
where $\beta=\sqrt{1-4m^{2}_{\tilde{B}}/s}$ is the bino
velocity.
In the conventional case since $\not\!\!E$ is carried
by the massive LSP the photon energy end points are
smaller by a factor
$(1-m_{\tilde{H}_{1}^{0}}^{2}/m_{\tilde{H}_{2}^{0}}^{2})$, where $\beta$
in this case
is the $\tilde{H}_{2}^{0}$ velocity.
This allows the decay to a gravitino to be distinguished
from decay to the LSP in the conventional MSSM. In addition, in this region of parameter space the lightest
chargino is just slightly heavier, is also mostly Higgsino, and
decays predominantly by
$\tilde{H}^{\pm}\rightarrow\tilde{H}^{0}W^{\pm*}$.
In the MSSM the additional signatures
$e^{+}e^{-}\rightarrow\tilde{H}^{+}\tilde{H}^{-}\rightarrow 4j+\not\!\!E$, $jjl+\not\!\!E$, and $l^{+}{l^{\prime}}^{-}+\not\!\!E$
are likely to be accessible at comparable $\sqrt{s}$.
This is in contrast to the gauge mediated case with
a mostly bino NLSP.
As discussed in the previous section, with a
weakly coupled messenger sector giving an NLSP
which is mostly bino, it is likely that the right
handed sleptons are not too much heavier than the NLSP.
In this case, in addition to bino pair production, slepton
pair production may be kinematically accessible.
Cascade decay through the bino then gives rise to
$e^{+}e^{-}\rightarrow\tilde{l}^{+}_{R}\tilde{l}^{-}_{R}\rightarrow l^{+}l^{-}%
\gamma\gamma+\not\!\!E$.
If the NLSP is a right handed slepton,
the discovery mode is
$e^{+}e^{-}\rightarrow\tilde{l}^{+}_{R}\tilde{l}^{-}_{R}\rightarrow l^{+}l^{-}+%
\not\!\!E$.
As for the decay to photons,
the leptons have a flat energy distribution,
with end points determined by $\sqrt{s}$ and $m_{\tilde{l}_{R}}$.
The final states with $e$, $\mu$, and $\tau$, should appear
with very nearly equal $m_{\tilde{l}_{R}}$.
Cuts on $\not\!\!E$ can easily eliminate the
background $e^{+}e^{-}\rightarrow Z^{0}l^{+}l^{-}$ with
$Z^{0}\rightarrow\nu\bar{\nu}$.
This signature can also arise in the conventional MSSM
where the missing energy is carried by the massive LSP.
However, the lepton energy endpoints again distinguish
this from an essentially massless gravitino.
It is interesting to note that if $\sqrt{F}$ is much larger
than a few 1000 TeV the decay of $\tilde{l}_{R}$
takes place well outside the detector.
The signature for supersymmetry is then massive charged particles,
rather than the traditional missing energy.
If the NLSP is mostly Higgsino, and $m_{\tilde{H}}>m_{h^{0}}$,
the discovery mode is
$e^{+}e^{-}\rightarrow\tilde{H}^{0}\tilde{H}^{0}\rightarrow 4b+\not\!\!E$, with
of course two pairs of $b$ jets reconstructing the Higgs mass.
In this part of parameter space
the next heaviest neutralino and lightest chargino are
mostly Higgsino, not much heavier than $\tilde{H}^{0}$,
and have the same decay modes to $\tilde{H}^{0}$ as in the MSSM.
The signatures $e^{+}e^{-}\rightarrow\tilde{H}^{+}\tilde{H}^{-}\rightarrow 4b4j+\not\!\!E$, $4bjjl+\not\!\!E$, and $4bl^{+}{l^{\prime}}^{-}+\not\!\!E$
should therefore also be accessible at comparable $\sqrt{s}$,
with the additional jets and leptons fairly soft.
\REF\park
S. Park representing the CDF Collaboration,
“Search for New Phenomena in CDF,” in 10th Topical
Workshop on Proton-Antiproton Collider Physics,
edited by R. Raha and J. Yoh, AIP Press, New York, 1995.
\REF\cdfreport
CDF Collaboration, report CDF-3456.
\REF\paige
H. Baer, C.-h Chen, F. Paige and X. Tata,
Phys. Rev. D49 (1994) 3283.
The discovery modes at hadron colliders
can be somewhat different than for $e^{+}e^{-}$ colliders.
If the NLSP is very nearly purely bino,
$p\bar{p}\rightarrow\tilde{B}\tilde{B}\rightarrow\gamma\gamma~{}~{}+\not\!\!E_{T}$ proceeds predominantly through
$t$- and $u$- channel squark exchange, and is therefore
highly suppressed because of the large squark masses.
However, sleptons can be pair
produced by the Drell-Yan process.
Cascade decay through the bino then leads to
$p\bar{p}\rightarrow\tilde{l}^{+}_{R}\tilde{l}^{-}_{R}\rightarrow l^{+}l^{-}%
\gamma\gamma+\not\!\!E_{T}$.
One such spectacular $ee\gamma\gamma$
event has in fact been observed
at the Tevatron by the CDF collaboration
(event 257646 in run 68739) \refmark\park.
The obvious background from $p\bar{p}\rightarrow WW\gamma\gamma$
has a very small production rate, and
would give rise to other decays modes which are not
observed \refmark\cdfreport.
In contrast, the production cross section for
$p\bar{p}\rightarrow\tilde{l}^{+}_{R}\tilde{l}^{-}_{R}$
with $m_{\tilde{l}_{R}}\simeq 95$ GeV is roughly $10^{-2}$ pb
\refmark\paige.
With $\sim$100 pb${}^{-1}$ of integrated luminosity,
the single observed event could be consistent with
right handed slepton pair production.
The kinematics of this event favor a fairly light bino,
implying that $e^{+}e^{-}\rightarrow\tilde{B}\tilde{B}\rightarrow\gamma\gamma+\not\!\!E$ is likely to be observed at LEPII.
\REF\trilepton
H. Baer, K. Hagiwara, and X. Tata, Phys. Rev. D
35 (1987) 1598;
P. Nath and R. Arnowitt, Mod. Phys. Lett. A 2 (1987) 331.
If the sleptons are much heavier than the gauginos,
and the NLSP is mostly bino,
pair production of winos becomes
the dominant production mechanism,
$p\bar{p}\rightarrow W^{*}\rightarrow\tilde{W}^{\pm}\tilde{W}^{0}$.
The dominant wino decay modes are
$\tilde{W}^{\pm}\rightarrow\tilde{B}W^{\pm*}$
and $\tilde{W}^{0}\rightarrow\tilde{B}Z^{0*}$ through
mixing with the Higgsino states, and
$\tilde{W}^{\pm}\rightarrow\tilde{B}l\nu$ and
$\tilde{W}^{0}\rightarrow\tilde{B}l^{+}l^{-}$ through off shell sleptons.
Cascade decays through the bino then lead to the signatures
$p\bar{p}\rightarrow\tilde{W}^{\pm}\tilde{W}^{0}\rightarrow 4j\gamma\gamma+\not%
\!\!E_{T}$, $jjl\gamma\gamma+\not\!\!E_{T}$, and
$l^{+}l^{-}l^{\prime}\gamma\gamma+\not\!\!E_{T}$.
The last one is similar to the standard
tri-lepton signature of chargino pair production \refmark\trilepton.
Here the additional hard photons significantly reduce the
background.
If the NLSP is mostly Higgsino or a right handed slepton, the
signatures at hadron colliders are similar to those at $e^{+}e^{-}$.
By far the most dramatic signature of low energy
supersymmetry breaking
is the possibility of measuring directly the decay of the NLSP
to its partner plus the gravitino.
If the NLSP is a neutralino this appears as a displaced
vertex, while for a slepton NLSP it appears as a kink
in a charged particle track.
Measurement of the decay
distribution would allow a direct determination
of the supersymmetry breaking scale.
For the decay of right handed sleptons
to leptons,
or the decay of Higgsinos to the lightest Higgs boson,
tracking of the resulting
charged particles in a silicon vertex detector
and central tracking region would allow
measurements of $c\tau$ between roughly
100 $\mu$m – 10 m.
In the case of decay to a photon, the tracking ability
for the displaced vertex is generally not good.
However if such a signal were established experimentally,
detectors could be optimized to convert
photons within the tracking region.
So depending on the specific decay modes of the NLSP,
displaced vertices for
$\sqrt{F}$ between roughly 100 – 1000’s of TeV
could be accessible to collider experiments.
\REF\raxion
A. Nelson and N. Seiberg, Nucl. Phys. B 416
(1994) 46; J. Bagger, E. Poppitz, and L. Randall,
Nucl. Phys. B 426 (1994) 3.
\REF\raffelt
G. Raffelt, Phys. Rept. 198 (1990) 1.
This range of experimentally accessible $\sqrt{F}$ is in
fact consistent with astrophysical and cosmological considerations.
Unless there is an inflation with low reheat temperature,
avoiding overclosure of the universe from relic gravitinos requires
$\sqrt{F}\lsim 2\times 10^{3}$ TeV.
In many theories a potentially
dangerous $R$-axion arises in the supersymmetry breaking
sector \refmark\raxion.
For $\sqrt{F}$ above a few TeV,
$R$-violating interactions suppressed by a single
power of the Planck scale make the $R$-axion
too heavy to be produced during helium ignition
in red giants \refmark\raffelt.
In addition, it is either trapped or too heavy to deplete
the neutrino pulse from SN1987A.
Finally, for weakly coupled models with a single additional scale,
such as the simple example in the previous section with
$F_{S}\sim F$, electroweak scale superpartners are obtained for
$\sqrt{F}\sim 100$ TeV.
\REF\cryo
P. Michelson, D. Osheroff, private communication;
J. Price, in Proceedings of the International Symposium
on Experimental Gravitational Physics, eds. P. Michelson,
H. En-ke, G. Pizzella (World Scientific, Singapore, 1987)
p. 436.
\REF\beams
M. Kasevitch, private communication.
A final possible consequence of these theories is that
scalar moduli with Planck suppressed couplings to matter
obtain masses of order or smaller than the gravitino
mass as the result of supersymmetry breaking.
These fields can mediate coherent forces in the sub-millimeter
range, which has not been explored experimentally.
New techniques employing small cryogenic mechanical oscillators
\refmark\cryo
or atomic beams \refmark\beams
may allow the detection of such short range gravitational
strength forces.
Low energy
gauge-mediated supersymmetry breaking clearly
makes distinct and dramatic predictions for future
experiments. The new particle
spectrum is predicted
in terms of a small number of parameters.
For a quite plausible
range of these parameters, it predicts signatures
distinctly different than those of the conventional
MSSM.
Most dramatic of these is the possibility of measuring
displaced vertices or kinks in charged particle tracks
from decays to the gravitino.
We would like to thank G. Anderson, R. Barbieri, G. Giudice,
H. Haber, L. Hall, M. Peskin, A. Pomarol,
and J. Wells for valuable discussions.
\refout |
The No-Triangle Hypothesis for ${\cal N}=8$ Supergravity
N. E. J. Bjerrum-Bohr${}^{1,2}$, David C. Dunbar${}^{1}$,
Harald Ita${}^{1}$,
Warren B. Perkins${}^{1}$ and Kasper Risager${}^{3}$
${}^{1}$
Department of Physics,
Swansea University, Swansea, SA2 8PP, UK
${}^{2}$
Institute for Advanced Study, Princeton,
NJ 08540, USA
${}^{3}$
Niels Bohr Institute,
Blegdamsvej 17, DK-2100,
Copenhagen,
Denmark
Abstract:
We study the perturbative expansion of ${\cal N}=8$
supergravity in four dimensions from the viewpoint of the
“no-triangle” hypothesis, which states that one-loop graviton
amplitudes in ${\cal N}=8$ supergravity only contain scalar box
integral functions. Our computations constitute a direct proof at
six-points and support the no-triangle conjecture for seven-point
amplitudes and beyond.
Models of Quantum Gravity, Supergravity Models
††preprint: hep-th/0610043
SWAT-06-473
1 Introduction
${\cal N}=8$ supergravity [1] is a remarkable theory,
being the maximally supersymmetric field theory containing gravity
that is consistent with unitarity. It is a beautiful but complicated
theory containing (massless) particles of all spins ($\leq 2$) whose
interactions are constrained by a large symmetry group.
This article explores the perturbative expansion of this theory. It
has been postulated that the perturbative expansion of this theory
is more akin to that of ${\cal N}=4$ super Yang-Mills theory than
expected from its known symmetries. In particular, it is
hypothesised that the one-loop amplitudes can be expressed as scalar
box functions with rational coefficients [2]. We provide
considerable evidence for this “no-triangle hypothesis” by
examining the behaviour of physical on-shell amplitudes.
This dramatic simplification of the one-loop amplitudes is
presumably a signature of an undiscovered symmetry or principle
present in ${\cal N}=8$ supergravity. These simplifications do not
occur on a “diagram by diagram” basis in any current expansion
scheme, instead they arise only when the diagrams are summed.
Theories of supergravity in four dimensions are one (and two) loop
finite [3]. Since the box functions are UV finite,
the simplifications we see are certainly consistent with these
arguments. However the cancellations are considerably stronger than they demand:
for example theories with ${\cal N}<8$
supergravity are UV finite at one-loop but the one-loop amplitudes are not
merely box functions.
In this article, we consider one-loop amplitudes and care must be used
in extending the implications beyond one-loop. However, we do expect
the higher loops to have a softer UV structure than previously
thought[4]. This opens the door to the possibility that
${\cal N}=8$ supergravity may, like ${\cal N}=4$ super Yang-Mills, be a
finite theory in four dimensions.
2 The No-triangle Hypothesis
2.1 Background: One-loop Amplitudes
First we review the general structure of one-loop amplitudes in
theories of massless particles. Consider the general form for an
$n$-point amplitude obtained from, for example, a Feynman diagram
calculation 111For simplicity we restrict ourselves to
covariant gauges with Feynman gauge-like propagators $\sim 1/p^{2}$,
$$M^{\rm 1\hbox{-}loop}_{n}(1,\cdots,n)=\sum_{\rm Feynman\,\,diagrams}I_{r}[P^{m%
}(l,\{k_{i},\epsilon_{i}\})]\,,$$
(2.1)2.1( 2.1 )
where each $I_{r}$ is a loop momentum integral with $r$
propagators in the loop and numerator $P^{m}(l,\{k_{i},\epsilon_{i}\})$. Here
$k_{i}$ denotes the external (massless) momenta, $\epsilon_{i}$ denotes the
polarisation tensors of the external states and $l$ denotes the loop
momentum. For clarity we suppress the $k_{i}$ and $\epsilon_{i}$ labels. In
general the numerator is a polynomial of degree $m$ in the loop
momentum. The value of $m$ depends on the theory under
consideration.
The summation is over all possible diagrams.
We choose to organise
the diagrams according to the number of propagators in the loop, $r$ . For
$r=n$ the integral will have only massless legs, while for $r<n$ at
least one of the legs attached to the loop will have momentum,
$K=k_{a}+\cdots k_{b}$, which is not null, $K^{2}\neq 0$. We will call
these massive legs (although it is a slight misnomer in a purely
massless theory).
An important technique for dealing with these integrals is that of
Passarino-Veltman Reduction [5]
which reduces any $r$-point integral to a sum of $(r-1)$ point
integrals ($r>4$),
$$I_{r}[P^{m}(l)]\longrightarrow\sum_{i}\,I_{r-1}^{i}[P^{m-1}(l)]\,.$$
(2.2)2.2( 2.2 )
We will be evaluating the loop momentum integrals by dimensional regularisation
in $D=4-2\epsilon$ and working to ${\cal O}(\epsilon)$.
In the reduction the degree of the loop momentum polynomial is also
reduced by $1$ from $m$ to $(m-1)$. The $(r-1)$ point functions
appearing are those which may be obtained from $I_{r}$ by contracting
one of the loop legs.
This process can be iterated until we obtain four point integrals,
$$I_{r}[P^{m}(l)]\longrightarrow\sum_{i}\,I_{4}^{i}[P^{m-(r-4)}(l)]\,.$$
(2.3)2.3( 2.3 )
The four point integrals reduce
$$I_{4}^{i}[P^{m^{\prime}}(l)]\longrightarrow c_{i}\,I_{4}^{i}[1]+\sum_{j}\,I_{3%
}^{j}[P^{m^{\prime}-1}(l)]\,,$$
(2.4)2.4( 2.4 )
where we now have the “scalar box functions”, $I_{4}[1]$, whose loop
momentum polynomials are just unity. The coefficients $c_{i}$ are
rational functions of the momentum invariants of the amplitude (By
rational we really mean non-logarithmic, since these coefficients
may contain Gram determinants.) Similarly, reduction of polynomial
triangles gives scalar triangles plus tensor bubble integral
functions,
$$I_{3}^{j}[P^{m}(l)]\longrightarrow d_{j}\,I_{3}^{j}[1]+\sum_{k}\,I_{2}^{k}[P^{%
m-1}(l)]\,.$$
(2.5)2.5( 2.5 )
Finally we can express the tensor bubbles as scalar bubble functions
plus rational terms,
$$I_{2}^{k}[P^{m}(l)]=e_{k}\,I_{2}^{k}[1]+R+O(\epsilon)$$
(2.6)2.6( 2.6 )
Consequently any one-loop amplitude can be reduced to the form,
$$M^{\rm 1\hbox{-}loop}_{n}(1,\cdots,n)=\sum_{i\in\cal C}\,c_{i}\,I_{4}^{i}+\sum%
_{j\in\cal D}\,d_{j}\,I_{3}^{j}+\sum_{k\in\cal E}\,e_{k}\,I_{2}^{k}+R\,+O(%
\epsilon),$$
(2.7)2.7( 2.7 )
where the amplitude has been split into a sum of integral functions
with rational coefficients and a rational part. The sums run over
bases of box, triangle and bubble integral functions: ${\cal C},\,{\cal D}$ and ${\cal E}$. Which integral functions appear in a
specific case will depend on the theory and process under
consideration, as will be discussed below.
2.2 ${\cal N}=4$ Super Yang-Mills Amplitudes
For Yang-Mills amplitudes the three-point vertex is linear in
momentum, so generically an $r$-point integral function has a loop
momentum polynomial of degree $r$. In general, a Passarino-Veltman
reduction gives one-loop amplitudes containing all possible integral
functions.
For supersymmetric theories cancellations between the different
types of particle circulating in the loop lead to a reduction in
the order of the loop momentum polynomial. For ${\cal N}=4$ super
Yang-Mills amplitudes, formalisms exist where four powers of loop
momentum cancel and the generic starting point for the reduction is
a polynomial of degree $m=(r-4)$. This implies that the amplitude
consists only of box and higher point integrals which, via a
Passarino-Veltman reduction (2.1), give a
very restricted set of
functions: namely scalar box-functions,
$$A^{\rm 1\hbox{-}loop}_{{\cal N}=4}\,=\,\sum_{i\in\cal C}\,c_{i}\,I_{4}^{i}\,.$$
(2.8)2.8( 2.8 )
These cancellations can be more or less transparent depending on
the gauge fixing and computational scheme employed. In general the manifest
diagram by diagram cancellation is less than the maximal four
powers. Schemes in which these cancellations are manifest include
the Bern-Kosower string based rules [6] (where
technically the cancellation occurs at the level of Feynman
parameter polynomials) and well chosen background field gauge
schemes [7]. In less favourable schemes cancellations
between diagrams occur relatively late in the calculation.
2.3 ${\cal N}=8$ Supergravity Amplitudes
Computation schemes for gravity calculations tend to be rather more
complicated than for Yang-Mills as the three-point vertex is
quadratic in momenta and so the loop momentum polynomial is of
degree $2r$ [8]. For maximal supergravity we expect to see
considerable cancellations.
In string theory, closed strings contain gravity and open strings
contain gauge theories, so the heuristic relation,
$$\hbox{closed string}\sim\hbox{(open-string)}\times\hbox{(open-string)}\,,$$
(2.9)2.9( 2.9 )
suggests a relationship between amplitudes of the form,
$$\hbox{gravity}\sim\hbox{(Yang-Mills)}\times\hbox{(Yang-Mills)}\,,$$
(2.10)2.10( 2.10 )
in the low energy limit. For tree amplitudes this relationship is
exhibited by the Kawai-Lewellen-Tye relations [9]. Even in
low energy effective field theories for
gravity [10] the KLT-relations can be seen to link
effective operators [11]. The KLT-relations also hold
regardless of massless matter content [12]. For
one-loop amplitudes we expect such relations for the integrands of one-loop amplitudes rather than the amplitudes
themselves. Indeed, the equivalent of the Bern-Kosower rules for
gravity [13, 14] give an initial loop momentum
polynomial of degree,
$$2r-8=2(r-4)\,.$$
(2.11)2.11( 2.11 )
This power counting is consistent with the heuristic expectation
of string theory.
Using this power counting, reduction for $r>4$ leads to a sum of tensor
box integrals with integrands of degree $r-4$ which would then reduce
to scalar boxes and triangle, bubble and rational functions,
$$M^{\rm 1\hbox{-}loop}_{{\cal N}=8}\,=\,\sum_{i\in\cal C}\,c_{i}\,I_{4}^{i}+%
\sum_{j\in\cal D}\,d_{j}\,I^{j}_{3}+\sum_{k\in\,\cal E}\,e_{k}\,I^{k}_{2}+R\,,$$
(2.12)2.12( 2.12 )
where we expect that the triangle functions $I_{3}$ are present for
$n\geq 5$, the bubble functions $I_{2}$ for $n\geq 6$ and the
rational terms for $n\geq 7$. Note that functions, other than the
scalar boxes, only appear after reduction.
2.4 The No-triangle Hypothesis
The “No-triangle hypothesis” states that any one-loop amplitude of
${\cal N}=8$ supergravity is a sum of box integral functions
multiplied by rational coefficients,
$$M^{\rm 1\hbox{-}loop}_{{\cal N}=8}=\sum_{i\in\cal C}\,c_{i}\,I_{4}^{i}\,.$$
(2.13)2.13( 2.13 )
The hypothesis originates from explicit computations which show that
despite the previous power counting arguments, one-loop amplitudes
for ${\cal N}=8$ supergravity have a form analogous to that of
one-loop ${\cal N}=4$ super Yang-Mills amplitudes.
The first definite calculation of a one-loop amplitude for both
${\cal N}=4$ super Yang-Mills and ${\cal N}=8$ supergravity was
performed by Green, Schwarz and Brink [15]. By taking the low
energy limit of string theory, they obtained the four point one-loop
amplitudes:
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{A^{\rm 1\hbox{-}loop}_{(1,2,3,%
4)}}$&$\displaystyle{{}=st\times A^{\rm tree}_{(1,2,3,4)}\times I_{4}(s,t)\,,}%
$\cr$\displaystyle{M^{\rm 1\hbox{-}loop}_{(1,2,3,4)}}$&$\displaystyle{{}=stu%
\times M^{\rm tree}_{(1,2,3,4)}\Bigl{(}I_{4}(s,t)+I_{4}(s,u)+I_{4}(t,u)\Bigr{)%
}\,.}$\cr$\displaystyle{}$}}\,$$
(2.14)2.14( 2.14 )
Here $I_{4}(s,t)$ denotes the scalar box integral with attached legs
in the order $1234$ and $s$, $t$ and $u$ are the usual Mandelstam
variables. The above Yang-Mills amplitude is the leading in colour
contribution. For gravity amplitudes we suppress factors of $\kappa$
( $\kappa^{n-2}$ for tree amplitudes and $\kappa^{n}$ for one-loop
amplitudes.) Although only composed of boxes, this gravity
amplitude is consistent with the power counting of $2(r-4)$ with
$r\leq n$.
Beyond four-point we expect to find contributions from other
integral functions in addition to the boxes. However in
ref. [16] the five and six-point MHV 222
Amplitudes are conveniently organised according to the number of
negative helicity external states. For amplitudes with
“all-positive” or “one negative the remaining positive” helicity
configurations the tree amplitudes vanish for any gravity theory and
the loop amplitudes vanish for any supergravity theory. The first
non-vanishing amplitudes are those with two negative helicity
gravitons, known as “Maximally Helicity Violating” or
MHV amplitudes. Amplitudes with three negative helicity gravitons
are “next-to-MHV” or NMHV amplitudes. Amplitudes with
exactly two positive helicity gravitons and the remaining negative
helicity can be obtained by conjugation and are known alternatively
as “googly amplitudes” or, as used by us, $\overline{\it MHV}$.
amplitudes were evaluated using unitarity techniques and shown to
consist solely of box integral functions. It was conjectured that
this behaviour continued to all MHV amplitudes and an all-$n$
ansatz consisting of box functions was presented. This ansatz was
also consistent with factorisation. In ref. [2] is was
postulated that this was a general feature of ${\cal N}=8$
amplitudes. In ref. [17] the hypothesis was explored
for the six-point NMHV amplitude and it was shown that the boxes
alone gave the correct IR behaviour of the amplitude.
In this paper we aim to present further evidence in favour of the
“no-triangle hypothesis”. While we fall short of presenting a
proof, we feel that the weight of evidence is compelling. The
evidence is based on IR structure, unitarity and factorisation. In
the six-point case this evidence does constitute a proof.
3 Evidence For The No-triangle Hypothesis
We use a range of techniques to study different parts of the
amplitude: unitarity, factorisation and the singularity structure of
the on-shell physical amplitudes. Our arguments are complete for $n\leq 6$ point amplitudes. Fortunately there has been considerable
progress in computing one-loop amplitudes inspired by the duality
with twistor space [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]: we will freely use many of these new
techniques.
We use arguments based on the IR divergences of the amplitude to
conclude that the one and two-mass triangles must vanish. We use a
study of the two-particle cuts to deduce that the bubble integrals
are absent and, by numerically examining triple cuts, we show that
the coefficients of three-mass triangles vanish. Finally we use
factorisation arguments to discuss the rational pieces of the
amplitude.
3.1 IR: Soft Divergences
The expected soft divergence of an $n$-point one-loop
graviton amplitude [32] is,
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{M^{\rm one-loop}_{(1,2,\ldots,%
n)}\Bigl{|}_{\rm soft}={i\kappa^{2}\over(4\pi)^{2}}\bigg{[}{\sum_{i<j}s_{ij}%
\ln[-s_{ij}]\over 2\epsilon}\bigg{]}\!\!\times\!M^{\rm tree}_{(1,2,\ldots,n)}%
\,.}$\cr$\displaystyle{}$}}\,$$
(3.1)3.1( 3.1 )
(The factors of $\kappa$ have been reinstated in the amplitudes
within this equation.) For a general amplitude the boxes with three
or fewer massive legs, the one and two mass triangles and the bubble
integrals all have $1/\epsilon$ singularities which can contribute to
the above.
A necessary condition for the no-triangle hypothesis is that the box
contributions alone yield the complete $1/\epsilon$ structure. In other
words,
$$\sum_{i\in\cal C}\,c_{i}\,I_{4}^{i}{\Big{|}}_{1/\epsilon}={i\over(4\pi)^{2}}%
\bigg{[}{\sum_{i<j}s_{ij}\ln[-s_{ij}]\over 2\epsilon}\bigg{]}\!\!\times\!M^{%
\rm tree}_{(1,2,\ldots,n)}\,.$$
(3.2)3.2( 3.2 )
If this condition is satisfied, it implies the vanishing of a large
number of the triangle coefficients, specifically that the one and
two-mass triangle functions are not present. The one- and two-mass
triangles are actually not an independent set of integral functions.
As shown in the appendix they can be replaced by a set of basis
functions,
$$G(-K^{2})={(-K^{2})^{-\epsilon}\over\epsilon^{2}}\;={1\over\epsilon^{2}}-{\ln(%
-K^{2})\over\epsilon}+{\rm finite}\,,$$
(3.3)3.3( 3.3 )
where the set of $G$’s runs over all the independent momentum
invariants, $K^{2}$, of the amplitude. These functions plus the boxes
then give the only $\ln(-K^{2})/\epsilon$ contributions to the amplitude
since the $1/\epsilon$ terms in bubbles do not contain logarithms. If
the boxes completely reproduce the required singularity, the
coefficients of the $G$ functions must be zero and consequently the
coefficients of the one- and two-mass triangles can be set to zero,
$$d_{1m,i}=d_{2m,i}=0\,.$$
(3.4)3.4( 3.4 )
Having the correct soft behaviour only imposes a single constraint
on the sum of the bubble coefficients,
$$\sum_{i}e_{i}=0\,,$$
(3.5)3.5( 3.5 )
and, importantly, places no constraint on the three-mass triangles
as they are IR finite.
To verify the IR behaviour, one must know the box coefficients.
Fortunately, there has been considerable progress in computing the
box coefficients in gauge theory. Box coefficients may be
determined using unitarity [33, 34]. In
ref. [35], Britto, Cachazo and Feng showed that
quadruple cuts can be used to algebraically obtain box coefficients
from the four tree amplitudes at the corners of the cut box.
Specifically, if we consider an amplitude containing a scalar box
integral function, the coefficient of this function is given by the
product of four tree amplitudes with on-shell cut
legs [35],
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{c={1\over 2}\sum_{h_{i}\in\,%
\cal S}}$&$\displaystyle{{}\biggl{(}M^{\rm tree}\big{(}(-\ell_{1})^{-h_{1}},i_%
{1},\ldots,i_{2},(\ell_{2})^{h_{2}}\big{)}\times M^{\rm tree}\big{(}(-\ell_{2}%
)^{-h_{2}},i_{3},\ldots,i_{4},(\ell_{3})^{h_{3}}\big{)}}$\cr$\displaystyle{}$&%
$\displaystyle{{}\times M^{\rm tree}\big{(}(-\ell_{3})^{-h_{3}},i_{5},\ldots,i%
_{6},(\ell_{4})^{h_{4}}\big{)}\times M^{\rm tree}\big{(}(-\ell_{4})^{-h_{4}},i%
_{7},\ldots,i_{8},(\ell_{1})^{h_{1}}\big{)}\biggr{)}\,.}$\cr$\displaystyle{}$}%
}\,\hskip-28.452756pt$$
(3.6)3.6( 3.6 )
Here ${\cal S}$ indicates the set of possible particle and helicity
configurations of the legs $\ell_{i}$ which give a non-vanishing product
of tree amplitudes We often denote the above coefficient by the clustering on the legs,
$c^{[\{i_{1}\cdots i_{2}\},\{i_{3}\cdots i_{4}\},\{i_{5}\cdots i_{6}\},\{i_{7}%
\cdots i_{8}\}]}$.
In the above the tree amplitudes at massless
corners require analytic continuation.
The box coefficients may also be obtained from the known box
coefficients for ${\cal N}=4$ Yang-Mills [34, 18, 19] by
squaring and summing [2]. For example for the three-mass
boxes within the seven-point NMHV amplitude we have,
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{c_{{\cal N}=8}^{[1^{-},\{4^{+}%
5^{+}\},\{2^{-}3^{-}\},\{6^{+}7^{+}\}]}}$&$\displaystyle{{}=2s_{23}s_{45}s_{67%
}\;c_{{\cal N}=4}^{[1^{-},\{4^{+}5^{+}\},\{2^{-}3^{-}\},\{6^{+}7^{+}\}]}\;c_{{%
\cal N}=4}^{[1^{-},\{5^{+}4^{+}\},\{3^{-}2^{-}\},\{7^{+}6^{+}\}]}}$\cr$%
\displaystyle{}$}}\,,$$
(3.7)3.7( 3.7 )
which allows us to obtain the ${\cal N}=8$ coefficients from
the ${\cal N}=4$ box coefficients.
We have computed the IR behaviour of the six and seven-point NMHV
amplitudes. The six-point box coefficients are given in
ref. [17] and the seven-point box coefficients are
given in appendix A. In both cases amplitudes were
constructed using these box-coefficients and, after some computer algebra,
the resultant amplitudes were found to
reproduce the complete IR behaviour. This allows us to conclude
that,
$$d_{2m,i}=d_{1m,i}=0\hbox{ for }n=6,7.$$
(3.8)3.8( 3.8 )
3.2 Two-Particle Cuts
A general unitarity cut of the amplitude $M_{n}(1,2,\ldots n)$ in the
channel carrying momentum $P=k_{i}+\ldots k_{j}$, is given by a sum of
phase space integrals of products of tree amplitudes,
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{C_{i\cdots j}=i\sum_{h_{1},h_{%
2}\in\,\cal S^{\prime}}\,\int d{\rm LIPS}(-l_{1},l_{2})}$&$\displaystyle{{}M^{%
\rm tree}\big{(}(-l_{1})^{-h_{1}},i,\cdots,j,(l_{2})^{h_{2}}\big{)}}$\cr$%
\displaystyle{}$&$\displaystyle{{} \times M^{\rm tree}\big{(}(-l_{2})^%
{-h_{2}},j+1,\cdots,i-1,(l_{1})^{h_{1}}\big{)}\,,}$}}\,\hskip-28.452756pt$$
(3.9)3.9( 3.9 )
where ${\cal S^{\prime}}$ denotes the helicities of the particles from the
${\cal N}=8$-multiplet that can run in the loop. This unitarity cut
is equal to the leading discontinuity of the loop amplitude,
$${\hskip-136.573228pt\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{}$&$%
\displaystyle{{}\sum_{i\in\cal C}\,c_{i}\,I_{4}^{i}+\sum_{j\in\,\cal D}\,d_{j}%
\,I^{j}_{3}+\sum_{k\in\,\cal E}\,e_{k}\,I^{k}_{2}\,{\Big{|}}_{\rm Disc}}$\cr$%
\displaystyle{}$&$\displaystyle{{}=i\,\int d{\rm LIPS}(-l_{1},l_{2})\left[\sum%
_{i\in\cal C^{\prime}}{c_{i}\over(l_{1}-K_{i,4})^{2}(l_{2}-K_{i,2})^{2}}+\sum_%
{j\in\,\cal D^{\prime}}{d_{j}\over(l_{1}-K_{j,3})^{2}}+e_{k^{\prime}}\right]\,%
.}$}}\,\hskip-113.811024pt$$
(3.10)3.10( 3.10 )
The sets of box and triangle functions that contribute to a given cut
are denoted by ${\cal C^{\prime}}$ and ${\cal D^{\prime}}$ respectively and the
single bubble function that contributes is labelled by $k^{\prime}$. In
principle the coefficients of all the integral functions can be
obtained by performing all possible two-particle cuts. In practice
it is often simpler to determine the box and triangle coefficients
by other means before using the two-particle cuts to determine the
bubble terms. The rational pieces of the amplitude are not
“cut-constructible” [33, 34].
To show that a given integral function is absent from the amplitude
we have to show that its contribution to the cut integral vanishes.
This test may be done by either evaluating the cut integral
explicitly or, equivalently, by algebraically reducing the integrand to
a sum of constant coefficients times specific products of
propagators, that are the signatures of the cuts of
specific integral functions.
3.3 Bubble Integrals from the two-particle cuts
In this section we will show, by explicit computation of the
two-particle cuts, that all bubble integrals in the
six-point amplitudes vanish.
These arguments can also be used to show
that bubble integrals are absent from all the cuts of all-n one-loop
MHV amplitudes as discussed in section (4.2).
Recently, the realisation that Yang-Mills amplitudes are dual to a
twistor string theory [36] has given
considerable impetus to gauge theory calculations. In particular, it
appears that the two-particle cuts can be efficiently calculated if
expressed in spinor or twistor variables [30].
Consider the two-particle cut,
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{C_{12}\!=}$&$\displaystyle{{}i%
\int d\mu\,M^{\rm tree}_{4}\big{(}(-l_{1})^{+},1^{-},2^{-},(l_{2})^{+}\big{)}%
\times M^{\rm tree}_{6}\big{(}(l_{1})^{-},3^{-},4^{+},5^{+},6^{+},(-l_{2})^{-}%
\big{)}\,,}$\cr$\displaystyle{}$&$\displaystyle{{}{\hbox{ where }}d\mu=d^{4}l_%
{1}d^{4}l_{2}\delta^{(+)}(l_{1}^{2})\delta^{(+)}(l_{2}^{2})\delta^{(4)}(l_{1}-%
l_{2}-k_{1}-k_{2}),}$\cr$\displaystyle{}$}}\,\hskip-28.452756pt$$
(3.11)3.11( 3.11 )
with a graviton running in the loop. We denote the integrand by
${\cal I}(l_{1},l_{2})$. Setting $l_{1}=t\ell$ with $\ell_{a\dot{a}}=\lambda_{a}\tilde{\lambda}_{\dot{a}}$, the measure
becomes [37, 30],
$$d^{4}l_{1}\delta^{(+)}(l_{1}^{2})=tdt\left\langle\lambda\,d\lambda\right%
\rangle\left[\tilde{\lambda}\,d\tilde{\lambda}\right]\,,$$
(3.12)3.12( 3.12 )
so that the cut becomes,
$${\hskip-150.799606pt\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{C_{12}}$&$%
\displaystyle{{}=i\!\int\!d\mu\,{\cal I}(l_{1},l_{2})=i\int_{0}^{\infty}\!\!\!%
tdt\int\!\left\langle\lambda\,d\lambda\right\rangle\left[\tilde{\lambda}\,d%
\tilde{\lambda}\right]\delta^{(+)}\big{(}P^{2}-tP_{a\dot{a}}\lambda^{a}\tilde{%
\lambda}^{\dot{a}}\big{)}\,{\cal I}\big{(}t\ell,-P-t\ell\big{)}}$\cr$%
\displaystyle{}$&$\displaystyle{{}=i\!\int\left\langle\lambda\,d\lambda\right%
\rangle\left[\tilde{\lambda}\,d\tilde{\lambda}\right]\frac{P^{2}}{(P_{a\dot{a}%
}\lambda^{a}\tilde{\lambda}^{\dot{a}})^{2}}\,{\cal I}\biggr{(}{P^{2}\over P_{a%
\dot{a}}\lambda^{a}\tilde{\lambda}^{\dot{a}}}\ell,-P-{P^{2}\over P_{a\dot{a}}%
\lambda^{a}\tilde{\lambda}^{\dot{a}}}\ell\biggr{)}\,,}$}}\,\hskip-142.26378pt$$
(3.13)3.13( 3.13 )
where $P$ denotes the total momentum on one side of the cut. In the
example above, $P=k_{1}+k_{2}$. Powers of $l_{1}$ within ${\cal I}$ give
rise to powers of $t$ which in turn give rise to extra powers of
$P^{2}/(P_{a\dot{a}}\lambda^{a}\tilde{\lambda}^{\dot{a}})$ due to the
$\delta(P^{2}-tP_{a\dot{a}}\lambda^{a}\tilde{\lambda}^{\dot{a}})$-function.
Thus in general the cut will be a sum of terms with different powers
of $(P_{a\dot{a}}\lambda^{a}\tilde{\lambda}^{\dot{a}})$,
$$C_{12}=\int\left\langle\lambda\,d\lambda\right\rangle\left[\tilde{\lambda}\,d%
\tilde{\lambda}\right]\sum_{n}{f_{n}(\lambda,\tilde{\lambda})\over(P_{a\dot{a}%
}\lambda^{a}\tilde{\lambda}^{\dot{a}})^{n}}.$$
(3.14)3.14( 3.14 )
The key observation of [30, 38] is that the
different classes of integral function that contribute to the cut
can be recognised by the powers of $(P_{a\dot{a}}\lambda^{a}\tilde{\lambda}^{\dot{a}})$ that are present. Generically,
any term containing, $1/(P_{a\dot{a}}\lambda^{a}\tilde{\lambda}^{\dot{a}})^{n}$ with $n<2$ in $C_{12}$ will not generate a contribution
to the coefficient of any bubble integral function. In terms of $t$,
such terms correspond to terms in ${\cal I}$ of the form $t^{m}$
with $m<1$. In the following we show that only terms of this
form arise in two-particle cuts of the six-point one-loop amplitudes
and hence that no bubble integral functions contribute to these
amplitudes.
The NMHV amplitude $M^{\rm 1\hbox{-}loop}(1^{-},2^{-},3^{-},4^{+},5^{+},6^{+})$ has four
inequivalent cuts up to relabelling of external legs; $C_{12}$,
$C_{34}$, $C_{123}$ and $C_{234}$. Of these $C_{12}$ and $C_{123}$
are what we call singlet cuts. These cuts vanish unless the two
outgoing cut legs have the same helicity, implying that these states
can only be gravitons. These singlet cuts are thus independent of
the matter content of the theory and the absence of bubble
functions is independent of the number of supersymmetries. For the
non-singlet cuts, the two outgoing cut legs have opposite helicity
and so the full ${\cal N}=8$ multiplet contributes. For these cuts,
bubble functions are only absent from the ${\cal N}=8$ amplitudes.
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\Text(100,-20)[]NON-SINGLET
We now examine the four distinct cuts in turn. First we consider
$C_{123}$, as this is the simplest: it is a singlet cut and the tree
amplitudes that appear are either MHV or $\overline{\it MHV}$
amplitudes. Explicitly the product of tree amplitudes is,
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{M^{\rm tree}_{\rm MHV}\big{(}(%
-l_{1})^{+},}$&$\displaystyle{{}1^{-},2^{-},3^{-},(l_{2})^{+}\big{)}\times M^{%
\rm tree}_{\rm MHV}\big{(}(-l_{2})^{-},4^{+},5^{+},6^{+},(l_{1})^{-}\big{)}.}$%
\cr$\displaystyle{}$&$\displaystyle{{}=-\left[l_{1}\,l_{2}\right]^{8}\frac{%
\left[3\,l_{1}\right]\left\langle l_{1}\,1\right\rangle\left[1\,2\right]\left%
\langle 2\,3\right\rangle-\left\langle 3\,l_{1}\right\rangle\left[l_{1}\,1%
\right]\left\langle 1\,2\right\rangle\left[2\,3\right]}{\left[l_{1}\,l_{2}%
\right]\left[l_{1}\,1\right]\left[l_{1}\,2\right]\left[l_{1}\,3\right]\left[l_%
{2}\,1\right]\left[l_{2}\,2\right]\left[l_{2}\,3\right]\left[1\,2\right]\left[%
1\,3\right]\left[2\,3\right]}}$\cr$\displaystyle{}$&$\displaystyle{{} %
\times\left\langle l_{1}\,l_{2}\right\rangle^{8}\frac{\left\langle 6\,l_{1}%
\right\rangle\left[l_{1}\,4\right]\left\langle 4\,5\right\rangle\left[5\,6%
\right]-\left[6\,l_{1}\right]\left\langle l_{1}\,4\right\rangle\left[4\,5%
\right]\left\langle 5\,6\right\rangle}{\left\langle l_{1}\,l_{2}\right\rangle%
\left\langle l_{1}\,4\right\rangle\left\langle l_{1}\,5\right\rangle\left%
\langle l_{1}\,6\right\rangle\left\langle l_{2}\,4\right\rangle\left\langle l_%
{2}\,5\right\rangle\left\langle l_{2}\,6\right\rangle\left\langle 4\,5\right%
\rangle\left\langle 4\,6\right\rangle\left\langle 5\,6\right\rangle}\,.}$}}\,%
\hskip-28.452756pt$$
(3.15)3.15( 3.15 )
This can be simplified to,
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{}$&$\displaystyle{{}-\frac{%
\big{(}P_{123}^{2}\big{)}^{10}}{\left[1\,2\right]\left[1\,3\right]\left[2\,3%
\right]\left\langle 4\,5\right\rangle\left\langle 4\,6\right\rangle\left%
\langle 5\,6\right\rangle}\times}$\cr$\displaystyle{}$&$\displaystyle{{}\frac{%
\big{(}\left[3\,l_{1}\right]\left\langle l_{1}\,1\right\rangle\left[1\,2\right%
]\left\langle 2\,3\right\rangle-\left\langle 3\,l_{1}\right\rangle\left[l_{1}%
\,1\right]\left\langle 1\,2\right\rangle\left[2\,3\right]\big{)}\big{(}\left%
\langle 6\,l_{1}\right\rangle\left[l_{1}\,4\right]\left\langle 4\,5\right%
\rangle\left[5\,6\right]-\left[6\,l_{1}\right]\left\langle l_{1}\,4\right%
\rangle\left[4\,5\right]\left\langle 5\,6\right\rangle\big{)}}{\left[l_{1}\,1%
\right]\left[l_{1}\,2\right]\left[l_{1}\,3\right]\left\langle l_{1}\,4\right%
\rangle\left\langle l_{1}\,5\right\rangle\left\langle l_{1}\,6\right\rangle%
\prod_{x=1,2,3}\left\langle l_{1}|P_{123}|x\right]\prod_{y=4,5,6}\left[l_{1}|P%
_{123}|y\right\rangle\ }}$}}\,$$
(3.16)3.16( 3.16 )
Substituting $l_{1}=tl$ into the above term we find a factor of
$1/t^{4}$ and hence there are no bubble contributions to this cut.
Next we consider $C_{234}$. Again the tree amplitudes are either
MHV or $\overline{\it MHV}$ amplitudes, but this is a non-singlet cut,
so we must include a summation over the super-multiplet.
MHV($\overline{\it MHV}$) tree amplitudes with a single pair of
non-graviton particles are related to the corresponding pure
graviton amplitude by simple factors, $X(h)$. The summed integrand is most
naturally expressed in terms of tree amplitudes with a scalar
circulating in the loop and a $\rho$-factor. Using a
superscript $s$ to denote a scalar in the loop, we have,
$${\hskip-7.113189pt\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{}$&$%
\displaystyle{{}\sum_{h\in S^{\prime}}M^{\rm tree}\big{(}(-l_{1})^{-h},2^{-},3%
^{-},4^{+},(l_{2})^{h}\big{)}\times M^{\rm tree}\big{(}(-l_{2})^{-h},5^{+},6^{%
+},1^{-},(l_{1})^{h}\big{)}}$\cr$\displaystyle{}$&$\displaystyle{{} =M^{\rm
tree%
}\big{(}(-l_{1})^{s},2^{-},3^{-},4^{+},(l_{2})^{s}\big{)}\times M^{\rm tree}%
\big{(}(-l_{2})^{s},5^{+},6^{+},1^{-},(l_{1})^{s}\big{)}\sum_{h\in S^{\prime}}%
X(h)}$\cr$\displaystyle{}$&$\displaystyle{{} =\rho\times M^{\rm tree}\big{(}(%
-l_{1})^{s},2^{-},3^{-},4^{+},(l_{2})^{s}\big{)}\times M^{\rm tree}\big{(}(-l_%
{2})^{s},5^{+},6^{+},1^{-},(l_{1})^{s}\big{)},}$\cr$\displaystyle{}$}}\,$$
(3.17)3.17( 3.17 )
where,
$$\rho=\sum_{h\in S^{\prime}}X(h)\!\!=\sum_{a=-4}^{a=4}{8!\over(4-a)!(4+a)!}%
\left({x\over y}\right)^{a}\!\!={(x+y)^{8}\over x^{4}y^{4}}=\frac{\left\langle
1%
|P_{234}|4\right]^{8}}{\big{(}\left[4\,l_{1}\right]\left[4\,l_{2}\right]\left%
\langle 1\,l_{1}\right\rangle\left\langle 1\,l_{2}\right\rangle\big{)}^{4}}\,.$$
(3.18)3.18( 3.18 )
The factor $n_{h}={8!/((4-a)!(4+a)!)}$ is the multiplicity within the
${\cal N}=8$ multiplet of the states of helicity $h=a/2$.
Rewriting the amplitude in terms of $l_{1}$ we can count the powers of $t$.
Overall the leading contributions are $O(t^{-4})$, just as in
the singlet case. Once again the cut receives no contributions from
bubble functions.
The remaining cuts are algebraically more complicated, but they
repeat the patterns seen above. $C_{12}$ is a singlet cut involving
the product of a four-point MHV amplitude, $M^{\rm tree}\big{(}(-l_{1})^{+},1^{-},2^{-},(l_{2})^{+}\big{)}$, and a six-point NMHV
amplitude,$M^{\rm tree}\big{(}(-l_{2})^{-},3^{-},4^{+},5^{+},6^{+},(l_{1})^{-}\big{)}$. The
six-point NMHV tree amplitude has only recently been calculated
using on-shell recursion [39, 40, 41].
An
explicit form for this amplitude as a sum of fourteen terms is given in
appendix C.333In general much less is
known about gravity tree amplitudes than Yang-Mills amplitudes. Traditional
Feynman diagram approaches tend to be excessively complicated as
evidenced by the computation by Sannon [8] of the
four-point tree amplitude. The KLT relations, which express the
gravity amplitudes as sums of permutations of products of two
Yang-Mills amplitudes [9], are an extremely useful
technique, however the factorisation structure is rather obscure and
the permutation sum grows quickly with the number of legs. Of the
new techniques, the BCF recursion readily extends to gravity
amplitudes [40, 41] giving useful compact
results. The MHV-vertex approach of Cachazo, Svrček and Witten
also extends to gravity [42] although the correct analytic
continuation of the MHV gravity vertices is only clear after using
the appropriate factorisation [43]. Currently, there is no
Lagrangian based proof of these techniques such as exists for
Yang-Mills [44], however we have numerically checked the
expressions for both MHV vertices and recursion against the KLT
expressions for amplitudes with seven or fewer points.
We will illustrate here how one of the terms gives a contribution that
vanishes at large $t$. The remaining thirteen terms will follow
analogously and thus we see term-by-term that this cut receives no
contributions from bubble functions. The singlet 12-cut reads,
$$C_{12}=i\int d\mu\,M_{4}\big{(}(-l_{1})^{+},1^{-},2^{-},(l_{2})^{+}\big{)}\,%
\times\,M_{6}\big{(}(-l_{2})^{-},3^{-},4^{+},5^{+},6^{+},(l_{1})^{-}\big{)}\,,$$
(3.19)3.19( 3.19 )
where the four-point amplitude is,
$$M_{4}\big{(}(-l_{1})^{+},1^{-},2^{-},(l_{2})^{+}\big{)}\,=\,\frac{i\left%
\langle 1\,2\right\rangle^{7}\left[1\,2\right]}{\left\langle 1\,l_{1}\right%
\rangle\left\langle 1\,l_{2}\right\rangle\left\langle 2\,l_{1}\right\rangle%
\left\langle 2\,l_{2}\right\rangle\left\langle l_{1}\,l_{2}\right\rangle^{2}}\,,$$
(3.20)3.20( 3.20 )
and the six-point amplitude is given in [17]. We
will in this example analyse the contribution to the cut given by
the term $G_{4}^{ns}[-l_{2},3,5,4,6,l_{1}]$ in the full amplitude,
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{M_{6}\big{(}(-l_{2})^{-},3^{-}%
,4^{+},5^{+},6^{+},(l_{1})^{-}\big{)}|_{G^{ns}_{4}[-l_{2},3,5,4,6,l_{1}]}}$&$%
\displaystyle{{}=}$\cr$\displaystyle{}$&$\displaystyle{{}\frac{is_{35}s_{46}s_%
{l_{1}l_{2}}[5\,|P_{l_{2}35}|l_{1}\rangle^{7}}{\left\langle 46\right\rangle^{2%
}\left\langle 4l_{1}\right\rangle\left\langle 6l_{1}\right\rangle\left[35%
\right]^{2}\left[3l_{2}\right]\left[5l_{2}\right][3\,|P_{l_{2}35}|l_{1}\rangle%
[l_{2}\,|P_{l_{2}35}|4\rangle[l_{2}\,|P_{l_{2}35}|6\rangle t_{35l_{2}}}\,,}$}}\,$$
(3.21)3.21( 3.21 )
so that the integrand of (3.3) is,
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{}$&$\displaystyle{{}\frac{-%
\left\langle 1\!\,2\right\rangle^{7}\left[1\!\,2\right]}{\left\langle 1\,\!l_{%
1}\right\rangle\!\left\langle 1\,\!l_{2}\right\rangle\!\left\langle 2\,\!l_{1}%
\right\rangle\!\left\langle 2\,\!l_{2}\right\rangle\!\left\langle l_{1}\,\!l_{%
2}\right\rangle^{2}}\times}$\cr$\displaystyle{}$&$\displaystyle{{} \frac{s_%
{35}s_{46}s_{l_{1}l_{2}}[5\,|P_{l_{2}35}|l_{1}\rangle^{7}}{\left\langle 46%
\right\rangle^{2}\left\langle 4l_{1}\right\rangle\left\langle 6l_{1}\right%
\rangle\left[35\right]^{2}\left[3l_{2}\right]\left[5l_{2}\right][3\!\,|P_{l_{2%
}35}|l_{1}\rangle[l_{2}\!\,|P_{l_{2}35}|4\rangle[l_{2}\!\,|P_{l_{2}35}|6%
\rangle t_{35l_{2}}}\,,}$}}\,$$
(3.22)3.22( 3.22 )
which can be written as,
$${\cal C}\times\frac{[5\,|P_{l_{2}35}|l_{1}\rangle^{7}}{\left\langle 1\,\!l_{2}%
\right\rangle\!\left\langle 1\,\!l_{1}\right\rangle\!\left\langle 2\,\!l_{2}%
\right\rangle\!\left\langle 2\,\!l_{1}\right\rangle\!\left\langle l_{2}\,\!l_{%
1}\right\rangle^{2}\!\left\langle 4l_{1}\right\rangle\!\left\langle 6l_{1}%
\right\rangle\left[3l_{2}\right]\!\left[5l_{2}\right][3\,|P_{l_{2}35}|l_{1}%
\rangle[l_{2}\!\,|P_{l_{2}35}|4\rangle[l_{2}\!\,|P_{l_{2}35}|6\rangle t_{35l_{%
2}}}\,,$$
(3.23)3.23( 3.23 )
where,
$$\displaystyle{\cal C}=\frac{s_{35}s_{46}s_{12}^{2}\left\langle 1\,2\right%
\rangle^{6}}{\left\langle 46\right\rangle^{2}\left[35\right]^{2}}\,.$$
(3.24)3.24( 3.24 )
Now transforming all $l_{2}$ into $l_{1}$ using
$\displaystyle[Xl_{2}]\rightarrow{[X|{P_{12}|l_{1}\rangle}\over{\left\langle l_%
{2}l_{1}\right\rangle}}$ and
$\displaystyle\langle Yl_{2}\rangle\rightarrow{\langle Y|{P_{12}|l_{1}]}\over{%
\left[l_{2}l_{1}\right]}}$
we get,
$$\frac{s_{12}^{2}\cal C}{\left\langle 1\,2\right\rangle^{2}}\times{\cal H}(|l_{%
1}\rangle)\times\frac{1}{\left[2\,l_{1}\right]\left[1\,l_{1}\right]t_{46l_{1}}%
}\,,$$
(3.25)3.25( 3.25 )
where,
$$\displaystyle{\cal H}(|l_{1}\rangle)=\frac{[5|P\!_{46}|l_{1}\rangle^{7}}{\left%
\langle 1l_{1}\right\rangle\left\langle 2l_{1}\right\rangle\left\langle 4l_{1}%
\right\rangle\left\langle 6l_{1}\right\rangle[3|P\!_{12}|l_{1}\rangle[5|P\!_{1%
2}|l_{1}\rangle[3|P\!_{46}|l_{1}\rangle\langle l_{1}|P\!_{12}P\!_{46}|4\rangle%
\langle l_{1}|P\!_{12}P\!_{46}|6\rangle}\,.$$
Now we have to count the number of factors of $t$. We get a total count of $1/t^{2}$ hence no bubbles integral functions are present in the cut.
The remaining $C_{34}$ cut is non-singlet and so we again need to
sum over the multiplet. Explicit forms for the relevant six-point
amplitudes involving an arbitrary pair of particles plus gravitons
are given in appendix C. These tree amplitudes
are each a sum of fourteen terms. As we change the non-graviton
particles, the individually terms in the amplitude each behave like
MHV amplitudes in that they collect simple multiplicative factors.
Performing the sum over the multiplet term-by-term we find a
$\rho$-factor for each term. Just as in the $C_{234}$ cut, these are
very important as they introduce large inverse powers of $t$. For
most terms, $\rho\sim 1/t^{8}$. Again we pick a sample term to
illustrate the process: the other thirteen terms follow analogously.
We will consider the cut,
$$C_{34}=i\int d\mu\,\sum_{h\in{\cal S}^{\prime}}M_{4}\big{(}(-l_{1})^{h},3^{-},%
4^{+},(l_{2})^{-h}\big{)}\,M_{6}\big{(}(-l_{2})^{h},5^{+},6^{+},1^{-},2^{-},(l%
_{1})^{-h}\big{)}\,.$$
(3.26)3.26( 3.26 )
The four-point tree amplitude
$M_{4}\big{(}(-l_{1})^{h},3^{-},4^{+},(l_{2})^{-h}\big{)}$ is given by,
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{}$&$\displaystyle{{}M_{4}\big{%
(}(-l_{1})^{h},3^{-},4^{+},(l_{2})^{-h}\big{)}=\frac{i\,\left\langle l_{2}3%
\right\rangle^{7}\,\left[l_{2}3\right]}{\left\langle 34\right\rangle\,\left%
\langle 3l_{1}\right\rangle\,{\left\langle 4l_{1}\right\rangle}^{2}\,\left%
\langle l_{2}4\right\rangle\,\left\langle l_{2}l_{1}\right\rangle}\left(\frac{%
\left\langle-l_{1}3\right\rangle}{\left\langle l_{2}3\right\rangle}\right)^{4-%
2h}\,.}$}}\,$$
(3.27)3.27( 3.27 )
For the six-point corner we consider a specific but representative
term from the fourteen in eq (C),
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{}$&$\displaystyle{{}M_{6}\big{%
(}(-l_{2})^{h},5^{+},6^{+},1^{-},2^{-},(l_{1})^{-h}\big{)}|_{T_{2}}}$\cr$%
\displaystyle{}$&$\displaystyle{{}=\left(-\frac{i\left\langle 1l_{2}\right%
\rangle\left[6l_{1}\right]}{\langle 6|P_{26l_{1}}|1]}\right)^{4-2h}\times}$\cr%
$\displaystyle{}$&$\displaystyle{{} \frac{-i\left\langle 2l_{1%
}\right\rangle\langle 1|P_{26l_{1}}|6]^{8}\left[5l_{2}\right]}{\left\langle 15%
\right\rangle\left\langle 1l_{2}\right\rangle\left\langle 5l_{2}\right\rangle%
\langle 1|P_{26l_{1}}|2]\langle 1|P_{26l_{1}}|l_{1}]\langle 5|P_{26l_{1}}|6]%
\langle l_{2}|P_{26l_{1}}|6]\left[26\right]\left[2l_{1}\right]\left[6l_{1}%
\right]t_{26l_{1}}}\,.}$}}\,\hskip-85.358268pt$$
(3.28)3.28( 3.28 )
The particle type dependent factors can be extracted and we find relative to the graviton amplitude,
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{\rho=}$&$\displaystyle{{}\sum_%
{h\in\,{\cal S}^{\prime}}\left(-\frac{i\left\langle 1l_{2}\right\rangle\left[6%
l_{1}\right]}{[6\,|P_{26l_{1}}|1\rangle}\frac{\left\langle-l_{1}3\right\rangle%
}{\left\langle l_{2}3\right\rangle}\right)^{4-2h}\,=\left(\frac{-\left\langle 1%
2\right\rangle\left[26\right]\left\langle 3l_{2}\right\rangle+\left\langle 13%
\right\rangle\left[6|P_{34}|l_{2}\right\rangle}{[6\,|P_{15-l_{2}}|1\rangle%
\left\langle l_{2}3\right\rangle}\right)^{8}\,.}$}}\,$$
(3.29)3.29( 3.29 )
Next we rewrite the cut in terms of the loop momenta $l_{2}$ using the
on-shell conditions and $l_{1}=l_{2}+k_{3}+k_{4}$. The $\rho$ factor already has
the correct form. The remaining contributions to the cut integral are
then,
$${\cal C}\times{\cal H}(|l_{2}\rangle)\times\frac{\left\langle 2|P_{34}|l_{2}%
\right]\left[5l_{2}\right]}{\left[3l_{2}\right]{\left[4l_{2}\right]}\langle 1|%
P_{15-l_{2}}|2]\langle 5|P_{15-l_{2}}|6]t_{15-l_{2}}}\,,$$
(3.30)3.30( 3.30 )
where
$${\cal C}=\frac{\left[34\right]^{2}}{\left\langle 34\right\rangle^{2}\left%
\langle 15\right\rangle\left[26\right]}\,,$$
(3.31)3.31( 3.31 )
and
$${\cal H}(|l_{2}\rangle)=\frac{(-\left\langle 12\right\rangle\left[26\right]%
\left\langle 3l_{2}\right\rangle+\left\langle 13\right\rangle\left[6|P_{34}|l_%
{2}\right\rangle)^{8}}{\left\langle l_{2}3\right\rangle\left\langle l_{2}4%
\right\rangle\left\langle 1l_{2}\right\rangle\left\langle 5l_{2}\right\rangle%
\left[6|P_{34}|l_{2}\right\rangle\left[2|P_{34}|l_{2}\right\rangle\left\langle
l%
_{2}|P_{15}|6\right]\left\langle 1|P_{26}P_{34}|l_{2}\right\rangle}\,.$$
(3.32)3.32( 3.32 )
We now replace $l_{2}$ by $l_{2}=t\,\ell=t\,\lambda\tilde{\lambda}$ and
do the $t$-integration. With the definitions,
$\left\langle\ell|Q_{1}|\ell\right]=\left\langle\ell|P_{34}|\ell\right]\left%
\langle 1|P_{15}|2\right]-s_{34}\left\langle\ell 1\right\rangle\left[2\ell\right]$,
$\left\langle\ell|Q_{2}|\ell\right]=\left\langle\ell|P_{34}|\ell\right]\left%
\langle 5|P_{15}|6\right]-s_{34}\left\langle\ell 5\right\rangle\left[6\ell\right]$,
$\left\langle\ell|Q_{3}|\ell\right]=\left\langle\ell|s_{15}P_{34}-s_{34}P_{15}|%
\ell\right]$,
we can rewrite the cut as,
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{}$&$\displaystyle{{}-{\cal C}%
\times s_{34}\times{\cal H}(|l\rangle)\times\left\langle\ell|P_{34}|\ell\right%
]\times\frac{i\left\langle 2|P_{34}|\ell\right]\left[5\ell\right]}{\left[3\ell%
\right]{\left[4\ell\right]}\left\langle\ell|Q_{1}|\ell\right]\left\langle\ell|%
Q_{2}|\ell\right]\left\langle\ell|Q_{3}|\ell\right]}}$}}\,\,.$$
(3.33)3.33( 3.33 )
It is important to notice that the $\rho$-factor contributes
$\left\langle\ell|P_{34}|\ell\right]^{8}$, while the product of the graviton
amplitudes gives rise to $1/\left\langle\ell|P_{34}|\ell\right]^{5}$, with a further factor of
$1/\left\langle\ell|P_{34}|\ell\right]^{2}$ coming from the integration measure. The powers
of $\left\langle\ell|P_{34}|\ell\right]$ are important in that they
indicate the type of integral functions that are present. For the
above term with only single poles in the denominator, bubbles can
only arise from terms carrying a factor of $1/\left\langle\ell|P_{34}|\ell\right]^{2}$. We
therefore conclude that no bubbles are present.
By considering all distinct two-particle cuts of the six-point
one-loop NMHV amplitude we have shown that the amplitude receives
no contributions from bubble integral functions.
3.4 Triple Cuts
Having verified that no one or two-mass triangles or bubble integral
functions are present in the amplitude, we now consider the
three-mass triangle integral function. These have no IR
singularities and so the previous arguments have nothing to say
regarding their absence or presence. In this section we illustrate
how the coefficients of three-mass triangles can be evaluated by
numerically integrating triple cuts of amplitudes. Note that MHV
amplitudes do not contain triple cuts for any gravity theory so this
is a previously untested class of functions.
Consider a physical triple cut in an amplitude where all three
corners are massive,
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\Text(0,10)[]$l_{2}$\Text(-30,65)[]$l_{3}$\Text(30,65)[]$l_{1}$
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{C_{3}=\sum_{h_{i}\in{\cal S}^{%
\prime}}\int d^{4}l_{1}\delta(l_{1}^{2})\delta(l_{2}^{2})\delta(l_{3}^{2})}$&$%
\displaystyle{{}M\big{(}(l_{1})^{h_{1}},i_{m},\cdots i_{j},(-l_{2})^{-h_{2}}%
\big{)}}$\cr$\displaystyle{}$&$\displaystyle{{}\times M\big{(}(l_{2})^{h_{2}},%
i_{j+1},\cdots i_{l},(-l_{3})^{-h_{3}}\big{)}\times M\big{(}(l_{3})^{h_{3}},i_%
{l+1},\cdots i_{m-1},(-l_{1})^{-h_{1}}\big{)}\,,}$\cr$\displaystyle{}$}}\,%
\hskip-28.452756pt$$
(3.34)3.34( 3.34 )
where the summation is over all possible intermediate states. As
the momentum invariants, $K_{1}=k_{i_{m}}+k_{i_{m}+1}+\cdots k_{i_{j}}$ etc, are all
non-null, there exist kinematic regimes is which the integration has
non-vanishing support for real loop momentum. In such cases the
remaining one dimensional integral can readily be evaluated
numerically. In the generic expression of an amplitude the only
integral functions contributing to the triple cut are box functions
and the specific three mass triangle for the cut,
$$C_{3}=\sum_{i}c_{i}(I_{4}^{i})_{\rm triple-cut}+d_{3m}(I_{3}^{3m})_{\rm triple%
-cut}\,.$$
(3.35)3.35( 3.35 )
The box functions which can contribute are the two-mass-hard,
three-mass and four mass. This equation can be inverted to express
the coefficient $d_{3m}$ in terms of $C_{3}$ and the box-coefficients.
For the six-point case the cut,
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{C_{3}=\sum_{h_{i}\in{\cal S}^{%
\prime}}\int d^{4}l_{i}}$&$\displaystyle{{}\delta(l_{1}^{2})\delta(l_{2}^{2})%
\delta(l_{3}^{2})M_{4}\big{(}(l_{1})^{h_{1}},1,2,(-l_{2})^{-h_{2}}\big{)}}$\cr%
$\displaystyle{}$&$\displaystyle{{}\times M_{4}\big{(}(l_{2})^{h_{2}},3,4,(-l_%
{3})^{-h_{3}}\big{)}\times M_{4}\big{(}(l_{3})^{h_{3}},5,6,(-l_{1})^{-h_{1}}%
\big{)}\,,}$}}\,$$
(3.36)3.36( 3.36 )
only receives contributions from two-mass-hard boxes,
such as $I_{4}^{2m\;h}[{2,\{3,4\},\{5,6\},1}]$, and the three mass triangle.
The triple cut of a two-mass hard box is,
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{(I_{4}^{2m\;h})_{triple-cut}}$%
&$\displaystyle{{}=\int{d^{4}p\over p^{2}(p-k_{2})^{2}(p-k_{2}-K_{3})^{2}(p+k_%
{1})^{2}}\bigl{|}_{\rm cut}}$\cr$\displaystyle{}$&$\displaystyle{{}=\int{d^{4}%
p\delta((p-k_{2})^{2})\delta((p-k_{2}-K_{3})^{2})\delta((p+k_{1})^{2})\over p^%
{2}}}$\cr$\displaystyle{}$&$\displaystyle{{}={\pi\over 2(k_{1}+k_{2})^{2}(k_{2%
}+K_{3})^{2}}\,,}$\cr$\displaystyle{}$}}\,$$
(3.37)3.37( 3.37 )
while the triple cut of the three mass triangle is,
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{(I_{3}^{3m})_{triple-cut}}$&$%
\displaystyle{{}=\int{d^{4}p\over p^{2}(p-K_{1})^{2}(p+K_{3})^{2}}\bigl{|}_{%
\rm cut}}$\cr$\displaystyle{}$&$\displaystyle{{}=\int{d^{4}p\delta(p^{2})%
\delta((p+K_{3})^{2})\delta((p-K_{1})^{2})}={\pi\over 2\sqrt{\Delta_{3}}}}$\cr%
$\displaystyle{}$&$\displaystyle{{}={\pi\over 2\sqrt{(K_{1}^{2})^{2}+(K_{2}^{2%
})^{2}+(K_{3}^{2})^{2}-2(K_{1}^{2}K_{2}^{2}+K_{1}^{2}K_{3}^{2}+K_{2}^{2}K_{3}^%
{2})}}\,.}$\cr$\displaystyle{}$}}\,$$
(3.38)3.38( 3.38 )
Thus we see that,
$${\pi\over 2\sqrt{\Delta_{3}}}d_{3}^{3m}=C_{3}-{\pi\over 2}\sum_{i}{c_{2m\;h,i}%
\over(k_{1}+k_{2})^{2}(k_{2}+K_{3})^{2}}\,.$$
(3.39)3.39( 3.39 )
The integral in $C_{3}$ is well behaved and can be determined
numerically from the tree amplitudes. Using the box-coefficients for
the six-point amplitude [17] we have verified
numerically that,
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{d_{3}^{3m}[\{1^{-},2^{-}\},\{3%
^{-},4^{+}\},\{5^{+},6^{+}\}]}$&$\displaystyle{{}=0\,,}$\cr$\displaystyle{d_{3%
}^{3m}[\{1^{-},4^{+}\},\{2^{-},5^{+}\},\{3^{-},6^{+}\}]}$&$\displaystyle{{}=0%
\,.}$\cr$\displaystyle{}$}}\,$$
(3.40)3.40( 3.40 )
The first zero is true for any (massless) gravity theory whilst the
second is true only for ${\cal N}=8$ supergravity.
For the seven-point amplitude we must also include three-mass boxes
in the triple cut. Using the seven-point box coefficients given in
the appendix we have verified that three mass triangles are absent
in the seven-point NMHV amplitude. Explicitly,
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{d_{3}^{3m}[\{1^{-},2^{-}\},\{3%
^{-},4^{+}\},\{5^{+},6^{+},7^{+}\}]}$&$\displaystyle{{}=0\,,}$\cr$%
\displaystyle{d_{3}^{3m}[\{1^{-},2^{-}\},\{3^{-},4^{+},5^{+}\},\{6^{+},7^{+}\}%
]}$&$\displaystyle{{}=0\,,}$\cr$\displaystyle{d_{3}^{3m}[\{1^{-},2^{-},4^{+}\}%
,\{3^{-},5^{+}\},\{6^{+},7^{+}\}]}$&$\displaystyle{{}=0\,,}$\cr$\displaystyle{%
d_{3}^{3m}[\{1^{-},4^{+}\},\{2^{-},5^{+}\},\{3^{-},6^{+},7^{+}\}]}$&$%
\displaystyle{{}=0\,,}$\cr$\displaystyle{}$}}\,$$
(3.41)3.41( 3.41 )
with the first three coefficients vanishing for any matter content
but the last only zero for ${\cal N}=8$ supergravity.
3.5 Factorisation
The unitarity constraints of the previous sections are sufficient to
show the absence of integral functions involving logarithms. This is
sufficient to prove the no-triangle hypothesis for six or fewer
gravitons. At seven-point and beyond, the amplitude may, in
principle, contain rational terms which do not appear in the
four-dimensional cuts. Unitarity can be used to obtain
these [45, 46, 47, 48]
but one must evaluate the cuts fully in $4-2\epsilon$ dimensions.
Recently, there has been much progress in determining the rational
parts of QCD one-loop amplitudes based on the physical
factorisations of the
amplitudes [49, 50, 51]. Gravity
amplitudes are also heavily constrained by factorisation so the
absence of terms other than boxes for six or fewer legs makes it
difficult to envisage their presence at higher points.
More explicitly, consider the multi-particle factorisations. From
general field theory considerations, amplitudes must factorise (up
to subtleties having to do with infrared singularities) on
multi-particle poles. For $K^{\mu}\equiv k_{i}^{\mu}+\ldots+k_{i+r+1}^{\mu}$ the amplitude factorises when $K$ becomes on shell.
Specifically, as $K^{2}\rightarrow 0$ the factorisation properties of
one-loop massless amplitudes are described
by [52],
$${\hskip-139.418504pt\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{M_{n}^{{1%
\mbox{-}\rm loop}}}$&$\displaystyle{{}\mathop{\longrightarrow}^{K^{2}%
\rightarrow 0}\hskip 4.267913pt\sum_{\lambda=\pm}\Biggl{[}M_{r+1}^{{1\mbox{-}%
\rm loop}}\big{(}k_{i},\ldots,k_{i+r-1},K^{\lambda}\big{)}\,{i\over K^{2}}\,M_%
{n-r+1}^{{\rm tree}}\big{(}(-K)^{-\lambda},k_{i+r},\ldots,k_{i-1}\big{)}}$\cr$%
\displaystyle{}$&$\displaystyle{{}\hbox{}\!+\!M_{r+1}^{{\rm tree}}\big{(}k_{i}%
,\ldots,k_{i+r-1},K^{\lambda}\big{)}{i\over K^{2}}M_{n-r+1}^{{1\mbox{-}\rm loop%
}}\big{(}(-K)^{-\lambda},k_{i+r},\ldots,k_{i-1}\big{)}}$\cr$\displaystyle{}$&$%
\displaystyle{{}\hbox{}\!+\!M_{r+1}^{{\rm tree}}\big{(}k_{i},\ldots,k_{i+r-1},%
K^{\lambda}\big{)}{i\over K^{2}}M_{n-r+1}^{{\rm tree}}\big{(}(-K)^{-\lambda},k%
_{i+r},\ldots,k_{i-1}\big{)}\,\hat{r}_{\Gamma}\,{\cal F}_{n}\big{(}K^{2};k_{1}%
,\ldots,k_{n}\big{)}\Biggr{]},}$\cr$\displaystyle{}$}}\,\hskip-142.26378pt$$
(3.42)3.42( 3.42 )
where the one-loop “factorisation function” ${\cal F}_{n}$ is
helicity independent.
Gravity one-loop amplitudes also have soft and collinear
factorisations. In ref. [16] it was shown that these have
a universal collinear behaviour given by,
$$\hskip-2.845276ptM_{n}^{\rm 1-loop}(\ldots,a^{\lambda_{a}},b^{\lambda_{b}},%
\ldots)\mathop{\longrightarrow}^{a\parallel b}\,\sum_{\lambda}\!{\rm Split^{%
\rm gravity}}(z,a^{\lambda_{a}},b^{\lambda_{b}})\!\times\!M_{n-1}^{\rm 1-loop}%
(\ldots,P^{\lambda},\ldots)\,,\hskip-28.452756pt$$
(3.43)3.43( 3.43 )
when $k_{a}$ and $k_{b}$ are collinear. The pure graviton splitting
amplitudes are,
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{{\rm Split^{\rm gravity}}_{+}(%
z,a^{+},b^{+})}$&$\displaystyle{{}=\ 0\,,}$\cr$\displaystyle{{\rm Split^{\rm
gravity%
}}_{-}(z,a^{+},b^{+})}$&$\displaystyle{{}=\ -{1\over z(1-z)}{\left[a\,b\right]%
\over\left\langle a\,b\right\rangle}\,,}$\cr$\displaystyle{{\rm Split^{\rm
gravity%
}}_{+}(z,a^{-},b^{+})}$&$\displaystyle{{}=\ -{z^{3}\over 1-z}{\left[a\,b\right%
]\over\left\langle a\,b\right\rangle}\,.}$\cr$\displaystyle{}$}}\,$$
(3.44)3.44( 3.44 )
There is also a universal soft behaviour given by,
$$M_{n}^{\rm 1-loop}(\ldots,a,s^{\pm},b,\ldots)\ \mathop{\longrightarrow}^{k_{s}%
\to 0}\ {\cal S}^{\rm gravity}(s^{\pm})\times M_{n-1}^{\rm 1-loop}(\ldots,a,b,%
\ldots)\,,$$
(3.45)3.45( 3.45 )
when $k_{s}$ becomes soft. For the limit $k_{n}\rightarrow 0$ in
$M_{n}^{\rm tree}(1,2,\ldots,n)$, the gravitational soft factor (for
positive helicity) is,
$${\hskip 42.679134pt\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{{\cal S}_{n}%
\ \equiv\ \mathop{{\cal S}^{\rm gravity}}\nolimits(n^{+})}$&$\displaystyle{{}=%
\ {-1\over\left\langle 1\,n\right\rangle\left\langle n,\,n-1\right\rangle}\sum%
_{i=2}^{n-2}{\left\langle 1\,i\right\rangle\left\langle i,\,n-1\right\rangle%
\left[i\,n\right]\over\left\langle i\,n\right\rangle}\,.}$\cr$\displaystyle{}$%
}}\,$$
(3.46)3.46( 3.46 )
Note that the collinear behaviour is only a “phase singularity”
for real momenta [16], however it should be regarded as a
genuine singularity when using complex momenta.
These factorisations place constraints on the rational terms $R_{n}$.
Since $R_{n}=0$ for $n\leq 6$ the natural solution is $R_{n}=0$ for all
$n$. For QCD the factorisation constraints have been turned into
recursion relations for the rational
terms [49, 50]. If this bootstrap also
applies to gravity amplitudes then we would be able to immediately
deduce that $R_{n}=0$ for ${\cal N}=8$ amplitudes. At present a direct
calculation of the rational terms beyond six-points seems unfeasible
although there has been recent progress in producing algorithms
focused on computing the rational terms in six-point QCD
amplitudes [53, 54].
4 Checking Bubble-cuts by Large-$z$ Shifts
In this section we look at a different way to test for
bubble integral functions in the two-particle cuts. This approach is
based on the scaling behaviour of amplitudes under specific shifts
of the loop momenta.
Starting from equations (3.2) and (3.2),
lifting the integral implies,
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{}$&$\displaystyle{{}M^{\rm tree%
}\big{(}(-l_{1})^{-},i,\cdots,j,(l_{2})^{-}\big{)}\times M^{\rm tree}\big{(}(-%
l_{2})^{+},j+1,\cdots,i-1,(l_{1})^{+}\big{)}}$\cr$\displaystyle{}$&$%
\displaystyle{{}=\sum_{i\in\cal C^{\prime}}{c_{i}\over(l_{1}-K_{i,4})^{2}(l_{2%
}-K_{i,2})^{2}}+\sum_{j\in\,\cal D^{\prime}}{d_{j}\over(l_{1}-K_{j,3})^{2}}+e_%
{k^{\prime}}+D(l_{1},l_{2})\,.}$}}\,\hskip-113.811024pt$$
(4.1)4.1( 4.1 )
Here $D(l_{1},l_{2})$ is a total derivative, $\int d{\rm LIPS}(-l_{1},l_{2})D(l_{1},l_{2})=0$, which may or may not be present.
Note that in the above a number of boxes and
triangles may contribute but only one bubble. Let us consider
(4) under the shift of the two-cut legs,
$$\lambda_{l_{1}}\longrightarrow\lambda_{l_{1}}+z\,\lambda_{l_{2}}\,,\;\;\;\;%
\tilde{\lambda}_{l_{2}}\longrightarrow\tilde{\lambda}_{l_{2}}-z\,\tilde{%
\lambda}_{l_{1}}\,.$$
(4.2)4.2( 4.2 )
This shift does not change the coefficients but it does enter the propagator
terms (and possibly the $D(l_{1},l_{2})$). In the large-$z$ limit the
propagators will vanish as $1/z$ leaving behind the
bubble coefficient $e_{k^{\prime}}$. This suggests a test for bubble terms: if,
$$\lim_{z\rightarrow\infty}M^{\rm tree}\big{(}(-l_{1})^{-h_{1}},i,\cdots,j,(l_{2%
})^{h_{2}}\big{)}\times M^{\rm tree}\big{(}(-l_{2})^{-h_{2}},j+1,\cdots,i-1,(l%
_{1})^{h_{1}}\big{)}{\longrightarrow 0}\,,$$
(4.3)4.3( 4.3 )
in the large-$z$ limit, then,
$$e_{k^{\prime}}=0\,,$$
(4.4)4.4( 4.4 )
under the assumption that the total derivative vanishes at
infinity. In the following section we discuss criteria for
when this test may be used.
This test is particularly useful as in many cases it follows from the
general behaviour of gravity tree amplitudes and may be tested numerically
when the tree amplitudes are known.
4.1 Relation to large $t$ behaviour
A key step is to relate the large $z$ behaviour to the large
$t$ behaviour of the cut parameterised as in the previous section.
In that section, following [30, 38], we
discussed how the integral functions that a given term in a unitarity cut contributes to are
determined by the power, $n$, of $1/(P_{a\dot{a}}\lambda^{a}\tilde{\lambda}^{\dot{a}})^{n}$. A term in the cut integral
(3.3) can be written as a rational
expression in holomorphic and anti-holomorphic spinors $\lambda^{a}$
and $\tilde{\lambda}^{\dot{a}}$ respectively (recall these spinors are
NOT the same as the $\lambda_{l_{i}}$ but are related via
$l_{1}=t\lambda\tilde{\lambda}$),
$$\int\frac{\left\langle\lambda\,d\lambda\right\rangle\left[\tilde{\lambda}\,d%
\tilde{\lambda}\right]}{(P_{a\dot{a}}\lambda^{a}\tilde{\lambda}^{\dot{a}})^{n}%
}\,\frac{n_{a_{1}\ldots a_{j},\dot{a}_{1}\ldots\dot{a}_{k}}\lambda^{a_{1}}%
\cdots\lambda^{a_{j}}\tilde{\lambda}^{\dot{a}_{1}}\cdots\tilde{\lambda}^{\dot{%
a}_{k}}}{d_{b_{1}\ldots b_{l},\dot{b}_{1}\ldots\dot{b}_{m}}\lambda^{b_{1}}%
\cdots\lambda^{b_{l}}\tilde{\lambda}^{\dot{b}_{1}}\cdots\tilde{\lambda}^{\dot{%
b}_{m}}},$$
(4.5)4.5( 4.5 )
where the tensors $n_{a_{1}\ldots a_{j},\dot{a}_{1}\ldots\dot{a}_{k}}$ and
$d_{b_{1}\ldots b_{l},\dot{b}_{1}\ldots\dot{b}_{m}}$ contain no factors of
$(P_{a\dot{a}}\lambda^{a}\tilde{\lambda}^{\dot{a}})$.
Since the integrand must carry spinor weight $-2$ in $\lambda$ and
$\tilde{\lambda}$, the counters $j,k,l,m$ and $n$ obey $j-l-n=-2$
and $k-m-n=-2$. The $n_{a_{1}...\dot{a}_{k}}$ are non-vanishing for the
contractions,
$$n_{a_{1}\ldots a_{j}}\,^{\dot{a}_{1}\ldots\dot{a}_{k}}\lambda^{a_{1}}\cdots%
\lambda^{a_{j}}\big{(}\lambda^{b_{1}}P_{b_{1}\dot{a}_{1}}\big{)}\cdots\big{(}%
\lambda^{b_{k}}P_{b_{k}\dot{a}_{k}}\big{)}\neq 0\,,$$
(4.6)4.6( 4.6 )
and similarly for $d_{b_{1}...\dot{b}_{m}}$. This can always be achieved: were
the above contraction to vanish for all values of
$\lambda$, then the spinor obtained by contracting all but one index
has to be parallel to $\lambda^{b_{k}}P_{b_{k}}\,^{\dot{a}_{k}}$, that is,
$$n_{a_{1}\ldots a_{j}}\,^{\dot{a}_{1}\ldots\dot{a}_{k}}\lambda^{a_{1}}\cdots%
\lambda^{a_{j}}\big{(}\lambda^{b_{1}}P_{b_{1}\dot{a}_{1}})\cdots\big{(}\lambda%
^{b_{k-1}}P_{b_{k-1}\dot{a}_{k-1}}\big{)}=n^{\prime}(\lambda,P\lambda)\big{(}%
\lambda^{b_{k}}P_{b_{k}}\,^{\dot{a}_{k}}\big{)},$$
(4.7)4.7( 4.7 )
with $n^{\prime}$ a tensor of lower rank in $\lambda$ and $\tilde{\lambda}$.
We would then be able to pull out a power of $(P_{a\dot{a}}\lambda^{a}\tilde{\lambda}^{\dot{a}})$ and write the contraction of $n$ as,
$$n(\lambda,\tilde{\lambda})=n^{\prime}(\lambda,\tilde{\lambda})\,(P_{a\dot{a}}%
\lambda^{a}\tilde{\lambda}^{\dot{a}}),$$
(4.8)4.8( 4.8 )
contrary to our condition that no such factors exist.
The central observation is that the power $n$ of $1/(P_{a\dot{a}}\lambda^{a}\tilde{\lambda}^{\dot{a}})^{n}$ is related to the leading
power in large-$z$ of $\lambda_{l_{i}}$. The shift in (4) translates into a shift on the
$\lambda$ and $\tilde{\lambda}$ of,
$$\lambda_{a}\longrightarrow\lambda_{a}\,,\quad\tilde{\lambda}_{\dot{a}}%
\rightarrow\tilde{\lambda}_{\dot{a}}+z\,\lambda^{a}P_{a\dot{a}}\,.$$
(4.9)4.9( 4.9 )
The terms $(P_{a\dot{a}}\lambda^{a}\tilde{\lambda}^{\dot{a}})$ are
invariant under the shift, so the leading term at large-$z$ is
given by,
$$z^{k-m}\frac{1}{(P_{a\dot{a}}\lambda^{a}\tilde{\lambda}^{\dot{a}})^{n}}\times%
\frac{n(\lambda,P\lambda)}{d(\lambda,P\lambda)}.$$
(4.10)4.10( 4.10 )
Using $n=(k-m)+2$ one finds that the large-$z$ scaling is,
$$\sim z^{n-2},$$
(4.11)4.11( 4.11 )
for a term with a $1/(P_{a\dot{a}}\lambda^{a}\tilde{\lambda}^{\dot{a}})^{n}$ factor. Consequently, if the product of the two tree
amplitudes vanishes as $z\longrightarrow\infty$ then this product can
only be composed of terms with $n\leq 1$. These terms do not
contribute to bubble functions and hence the coefficient of the bubble
corresponding to this cut must vanish. 444Note that since there is only
a single bubble in each cut, there is no possibility of
cancellation. It is not uncommon for cancellations to occur
amongst the box functions appearing in a cut. The propagators of a
single box vanish as $1/z^{2}$, however, as in many of the cases we
consider, the leading terms cancel amongst
the boxes leaving a $1/z^{4}$ behaviour as $z\longrightarrow\infty$.
4.2 Using the large-$z$ test for Gravity Amplitudes.
In this section we apply the test of the previous section to
the two-particle cuts for graviton scattering in ${\cal N}=8$
supergravity. We can use the behaviour of the gravity amplitudes
under the shift (4) to determine the behaviour of the
cut. We will need to consider two types of cut: singlet cuts where
only graviton amplitudes are needed and non-singlet cuts where
amplitudes involving other states in the supergravity multiplet contribute.
It is useful to briefly review the known results for the large-$z$
behaviour of gravity amplitudes under the shifts,
$$\lambda_{i}\rightarrow\lambda_{i}+z\,\lambda_{j}\,,\quad\tilde{\lambda}_{j}%
\rightarrow\tilde{\lambda}_{j}-z\tilde{\lambda}_{i}.$$
(4.12)4.12( 4.12 )
The scaling of a given tree amplitude depends on the helicity of the
two shifted legs and the helicity of the scattering gravitons. For
MHV-amplitudes there is an explicit all-$n$
representation of the tree amplitudes [55]. This can be used to show
that [40, 41, 42],
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{(h_{i},h_{j})=}$&$%
\displaystyle{{}(+,+),(-,-),(+,-):M^{\rm tree}|_{z\longrightarrow\infty}\sim{1%
\over z^{2}},}$\cr$\displaystyle{(h_{i},h_{j})=}$&$\displaystyle{{}(-,+)\quad%
\quad\quad\quad\quad\quad\;\;:M^{\rm tree}|_{z\longrightarrow\infty}\sim{z^{6}%
}.}$}}\,$$
(4.13)4.13( 4.13 )
Slightly more surprisingly this behaviour extends to NMHV
amplitudes also - at least up to seven points where we have checked
the result explicitly. It is tempting to conjecture that this is
true for all graviton tree amplitudes. We only need the behaviour up
to seven points to test for bubbles in the six and seven point
amplitudes.
We first
consider the MHV case for arbitrary $n$.
The singlet cuts are of the form,
$$M^{\rm tree}\big{(}(l_{1})^{+},1^{-},2^{-},3^{+},\cdots r^{+},(l_{2})^{+})%
\times M^{\rm tree}\big{(}(l_{1})^{-},(r+1)^{+},\cdots n^{+},(l_{2})^{-}).$$
(4.14)4.14( 4.14 )
When we shift this we find that each tree shifts as $1/z^{2}$, so the product
behaves as $1/z^{4}$ at large-$z$ and we can deduce that the bubble
integral function $I_{2}(K_{1\ldots r})$ has vanishing coefficient.
The non-singlet cut is more involved,
$${\hskip-5.690551pt\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{}$&$%
\displaystyle{{}\sum_{h}M^{\rm tree}\big{(}(-l_{1})^{-h},2^{-},3^{+},\cdots r^%
{+},(l_{2})^{h})\!\times\!M^{\rm tree}\big{(}(-l_{2})^{-h},(r+1)^{+},\cdots n^%
{+},1^{-},(l_{1})^{h}\big{)}=}$\cr$\displaystyle{}$&$\displaystyle{{}M^{\rm
tree%
}\big{(}(-l_{1})^{-},2^{-},3^{+},\cdots r^{+},(l_{2})^{+}\big{)}\times M^{\rm
tree%
}\big{(}(-l_{2})^{-},(r+1)^{+},\cdots n^{+},1^{-},(l_{1})^{+}\big{)}}$\cr$%
\displaystyle{}$&$\displaystyle{{}%
%
\times\sum_{h\in{\cal S}^{\prime}}\left({\left\langle 2\,l_{1}\right\rangle%
\left\langle 1\,l_{2}\right\rangle\over\left\langle 2\,l_{2}\right\rangle\left%
\langle 1\,l_{1}\right\rangle}\right)^{2h-4}}$\cr$\displaystyle{}$&$%
\displaystyle{{}=\rho\times M^{\rm tree}\big{(}(-l_{1})^{-},2^{-},3^{+},\cdots
r%
^{+},(l_{2})^{+}\big{)}\times M^{\rm tree}\big{(}(-l_{2})^{-},(r+1)^{+},\cdots
n%
^{+},1^{-},(l_{1})^{+}\big{)},}$\cr$\displaystyle{}$}}\,\hskip-113.811024pt$$
(4.15)4.15( 4.15 )
where,
$$\rho=\left({\left\langle 2\,l_{2}\right\rangle\left\langle 1\,l_{1}\right%
\rangle-\left\langle 2\,l_{1}\right\rangle\left\langle 1\,l_{2}\right\rangle%
\over\left\langle 2\,l_{1}\right\rangle\left\langle 1\,l_{2}\right\rangle}%
\right)^{8}=\left({\left\langle l_{1}\,l_{2}\right\rangle\left\langle 1\,2%
\right\rangle\over\left\langle 2\,l_{1}\right\rangle\left\langle 1\,l_{2}%
\right\rangle}\right)^{8}.$$
(4.16)4.16( 4.16 )
Under the shift the amplitudes scale as,
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{}$&$\displaystyle{{}M^{\rm tree%
}\big{(}(-l_{1})^{-},2^{-},3^{+},\cdots r^{+},(l_{2})^{+}\big{)}\,\sim\,z^{6}%
\,,}$\cr$\displaystyle{}$&$\displaystyle{{}M^{\rm tree}\big{(}(-l_{2})^{-},(r+%
1)^{+},\cdots n^{+},1^{-},(l_{1})^{+}\big{)}\,\sim\,1/z^{2}\,,}$}}\,$$
(4.17)4.17( 4.17 )
however the $\rho$-factor scales, noting that $\left\langle l_{1}\,l_{2}\right\rangle$ is unshifted, as,
$$\rho\,\sim\,{1\over z^{8}},$$
(4.18)4.18( 4.18 )
and we find the non-singlet cuts scale as $1/z^{4}$, exactly as in the
singlet case. Within the sum over the multiplet (4.2)
the product of tree amplitudes scales as $z^{4}$ for any given
state and the simplification only arises when the sum over the entire
${\cal N}=8$ multiplet is performed.
We will now discuss the possible cuts of
the six and seven-point NMHV amplitudes.
For any singlet cut,
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{M^{\rm tree}\big{(}(-l_{1})^{-%
},\ldots,(l_{2})^{-}\big{)}\times M^{\rm tree}\big{(}(-l_{2})^{+},\ldots,(l_{1%
})^{+}\big{)}\,,}$}}\,$$
(4.19)4.19( 4.19 )
the trees both vanish as $1/z^{2}$
under the shift (4.2)
and we deduce that bubble integral functions are
absent from these cuts.
Thus bubbles corresponding to singlet cuts are absent
up to seven-points.
The non-singlet cuts are more involved. For the six-point amplitude,
$M(1^{-},2^{-},3^{-},4^{+},5^{+},6^{+})$
there are two types of cut corresponding to the cuts $C_{234}$ and $C_{34}$.
For the $C_{234}$ cut the amplitudes are a product of an MHV and
a $\overline{\it MHV}$. Summing over the multiplet gives an overall $\rho$ factor just as in the ${\it MHV}$ case and we deduce the coefficient of this bubble function is absent. The $C_{34}$ cut
is given by,
$$\sum_{h\in\,{\cal S}^{\prime}}M^{\rm tree}\big{(}(-l_{1})^{-h},3^{-},4^{+},(l_%
{2})^{h}\big{)}\times M^{\rm tree}\big{(}(-l_{2})^{-h},1^{-},2^{-},5^{+},6^{+}%
,(l_{1})^{h}\big{)}\,.$$
(4.20)4.20( 4.20 )
The amplitude involving a state of helicity $h$ behaves as,
$$M^{\rm tree}\big{(}(-l_{1})^{-h},1^{-},2^{-},5^{+},6^{+},(l_{2})^{h}\big{)}\,%
\sim\,z^{2h+2}\,,$$
(4.21)4.21( 4.21 )
which is a natural refinement of (4.2) and can be checked using the
form of the amplitude in appendix C.
Thus
the product of the two tree amplitudes in the cut (which will have
states of $\pm h$) will always behave as $z^{4}$ and the corresponding
scattering amplitude will contain bubble functions (or boundary
terms). By explicit computation it can be seen that after
including all the states from the ${\cal N}=8$ supergravity multiplet
we have,
$$\hskip-19.916929pt\sum_{\rm\ \ \ \ \ {\cal N}=8\,multiplet}\!\!\!\!\!\!\!\!\!%
\!\!\!\!\!M^{\rm tree}\big{(}(-l_{1})^{-h},3^{-},4^{+},(l_{2})^{h}\big{)}%
\times M^{\rm tree}\big{(}(-l_{2})^{-h},1^{-},2^{-},5^{+},6^{+},(l_{1})^{h}%
\big{)}|_{z\longrightarrow\infty}\,\longrightarrow 0\,,$$
(4.22)4.22( 4.22 )
and the bubble functions drop out. This calculation shows how
the sum over the multiplet leads to the absence of bubble
functions in the six-point NMHV amplitude even though they
are present in the contribution from any single state in the multiplet.
For the seven-point amplitude
$M(1^{-},2^{-},3^{-},4^{+},5^{+},6^{+},7^{+})$
there are three types of cut: $C_{234}$, $C_{345}$
and $C_{34}$. Of these, the large $z$ behaviour of
$C_{234}$ and $C_{345}$ can be checked using the
six-point amplitudes verifying the absence of these bubbles.
5 Consequences and Conclusions
In this paper we have given further evidence that the one-loop
perturbative expansion for ${\cal N}=8$ supergravity is much closer
to that of ${\cal N}=4$ super Yang Mills than expected from power
counting arguments. We argue that the one-loop amplitudes are
composed entirely of box integral functions and contain
“no-triangle” (or bubble or rational) integral functions. We have
provided evidence rather than a proof for this “no-triangle
hypothesis”, but the evidence amounts to a proof for the six-point
amplitudes. The evidence for $n$-point amplitudes with $n\geq 7$
based on unitarity, factorisation and IR behaviour is, for us,
compelling.
The cancellation we observe is not “diagram-by-diagram” -
at least not in any computational framework we are aware of.
Individual diagrams appear to have loop momentum polynomials of
degree $2(r-4)$ and simplification only occurs when the diagrams are
summed. The simplification observed is quite dramatic: to yield only
boxes the simplification would be equivalent to a cancellation
between terms such that the leading $(r-4)$ terms in the loop
momentum polynomials cancel.
The “no-triangle hypothesis” applies strictly to one-loop
amplitudes only. However we expect it to have consequences for
higher loops. For ${\cal N}=8$ supergravity in $D=4$ the four point
amplitude is expected to diverge at five loops [4]. This
argument is based on power counting and the known symmetries of the
theory [56]. Specifically, the argument attempts to
estimate the power of the loop momentum integral of individual
higher loop diagrams and finds that they generically have twice the
power of the equivalent Yang-Mills diagram. Cancellations between
diagrams analogous to those occuring at one-loop would lead to a softer
UV behaviour than this prediction with the theory possibly even
being finite.
Presumably there is a symmetry underlying this simplification. We
are not aware of any potential candidates for this symmetry.
Although examining on-shell
amplitudes has many advantages, the nature of the underlying symmetry
is obscure in the amplitudes. The symmetries implied by the twistor
duality [36] are one potential source,
although originally the duality seemed to involve super-conformal
rather than Einstein gravity [57]. Recently
twistor strings involving Einstein gravity have been
constructed [58] and it would be interesting to
explore these for potential symmetries. If ${\cal N}=8$ were
“weak-weak” dual to a UV finite string theory then obviously the
finiteness of ${\cal N}=8$ supergravity would follow.
Acknowledgments
We thank Zvi Bern for many useful discussions. This research was
supported in part by the PPARC and the EPSRC of the UK and in part
by grant DE-FG02-90ER40542 of the US Department of Energy.
Appendix A Seven-Point Amplitude
The seven-point NMHV amplitude $M_{7}(1^{-},2^{-},3^{-},4^{+},5^{+},6^{+},7^{+})$ can be
expressed as a sum of scalar boxes together with rational
coefficients;
$$M^{\rm 1\hbox{-}loop}=\sum_{a}c_{a}I_{4}^{a}\,.$$
(A.1)A.1( roman_A .1 )
The scalar boxes can be of four types, three mass, two-mass-hard,
two-mass-easy and one mass shown below with our choice of labelling.
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\Text(60,10)[l]$g$\Text(-5,10)[l]$a$\Text(0,67)[l]$b$\Text(15,67)[l]$c$\Text(35,67)[l]$d$\Text(50,67)[l]$e$\Text(65,67)[l]$f$\SetWidth1
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\Text(60,10)[l]$g$\Text(15,0)[l]$a$\Text(-10,0)[l]$b$\Text(-8,10)[l]$c$\Text(0,67)[l]$d$\Text(42,67)[l]$e$\Text(60,58)[l]$f$\SetWidth1
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\Text(52,0)[l]$g$\Text(-5,10)[l]$a$\Text(-8,52)[l]$b$\Text(10,67)[l]$c$\Text(40,67)[l]$d$\Text(55,55)[l]$e$\Text(62,10)[l]$f$
The coefficients of the box-functions can be obtained by
unitarity [33, 34]. Recently, it was observed that the
box-coefficients can be efficiently obtained from the quadruple
cut [35],
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\Text(50,-5)[]$l_{1}$\Text(50,105)[]$l_{3}$\Text(-5,50)[]$l_{2}$\Text(110,50)[]$l_{4}$
which yields the coefficient of the corresponding integral function,
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{c={1\over 2}\sum_{h_{i}\in\,%
\cal S}\biggl{(}M^{\rm tree}\big{(}}$&$\displaystyle{{}(-\ell_{1})^{-h_{1}},i_%
{1},\ldots,i_{2},(\ell_{2})^{h_{2}}\big{)}\times M^{\rm tree}\big{(}(-\ell_{2}%
)^{-h_{2}},i_{3},\ldots,i_{4},(\ell_{3})^{h_{3}}\big{)}}$\cr$\displaystyle{}$&%
$\displaystyle{{}\times M^{\rm tree}\big{(}(-\ell_{3})^{-h_{3}},i_{5},\ldots,i%
_{6},(\ell_{4})^{h_{4}}\big{)}\times M^{\rm tree}\big{(}(-\ell_{4})^{-h_{4}},i%
_{7},\ldots,i_{8},(\ell_{1})^{h_{1}}\big{)}\biggr{)}\,.}$\cr$\displaystyle{}$}%
}\,\hskip-42.679134pt$$
(A.2)A.2( roman_A .2 )
In this expression the sum is over all possible states of the
${\cal N}=8$ multiplet and all possible helicity configurations for
which the four tree amplitudes are non-zero. The four cut momenta
are all on-shell, $l_{i}^{2}=0$. If the four tree amplitudes have four or
more legs then this is solved for real momenta whereas if a corner
has only three legs then the solution involves complex momenta.
Alternately the box-coefficients of ${\cal N}=8$ can be obtained from
those of ${\cal N}=4$ super Yang-Mills where for example, with the
above labelling, the coefficients of the three-mass boxes are
related by,
$${\hskip 28.452756pt\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{c_{N=8}^{[a,%
\{b,c\},\{d,e\},\{f,g\}]}=2s_{bc}s_{de}s_{fg}\times}$&$\displaystyle{{}c_{N=4}%
^{[a,\{b,c\},\{d,e\},\{f,g\}]}\times c_{N=4}^{[a,\{c,b\},\{e,d\},\{g,f\}]}.}$%
\cr$\displaystyle{}$}}\,$$
(A.3)A.3( roman_A .3 )
To implement the quadruple cuts requires a knowledge of the tree
amplitudes up to and including six-points, where two
particles are states other than gravitons. The three, four and five
points amplitudes are all MHV or $\overline{\it MHV}$ amplitudes and relatively simple.
For MHV amplitudes the amplitude with $n-2$ gravitons and two
non-graviton particles are related to the MHV amplitude by,
$$\hskip-1.422638ptM(1_{h}^{-},2^{-},3^{+},\cdots,(n-1)^{+},n^{+}_{h})=\left({%
\left\langle 2\,n\right\rangle\over\left\langle 2\,1\right\rangle}\right)^{2h-%
4}\!\!\!M(1^{-},2^{-},3^{+},\cdots,(n-1)^{+},n^{+}).\hskip-28.452756pt$$
(A.4)A.4( roman_A .4 )
For the six-point corners the tree may be MHV or NMHV. The
six-graviton NMHV tree amplitudes were computed
recently [41, 17]. To complete the calculation of
the box-coefficients we also need the six-point amplitudes with
two non-gravitons. These are presented in
appendix C.
A.1 Definitions
The coefficients of the boxes are expressed using spinor products.
We use the notation $\left\langle j\,l\right\rangle\equiv\langle j^{-}|l^{+}\rangle$, $\left[j\,l\right]\equiv\langle j^{+}|l^{-}\rangle$, with $|i^{\pm}\rangle$ being massless Weyl spinors with momentum
$k_{i}$ and chirality $\pm$ [59]. The
spinor products are related to momentum invariants by
$\left\langle i\,j\right\rangle\left[j\,i\right]=2k_{i}\cdot k_{j}\equiv s_{ij}$ .
As in twistor-space studies we use the notation,
$$\lambda_{i}\;=\;|i^{+}\rangle\;,\;\;\;\tilde{\lambda}_{i}\;=\;|i^{-}\rangle\,.$$
(A.5)A.5( roman_A .5 )
We also define spinor strings,
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{[k|{{K}_{i\ldots j}}|l\rangle}%
$&$\displaystyle{{}\equiv\;\langle k^{+}|{/\hskip-6.259606ptK_{i\ldots j}}|l^{%
+}\rangle\;\equiv\;\langle l^{-}|{/\hskip-6.259606ptK_{i\ldots j}}|k^{-}%
\rangle\;\equiv\;\langle l|{{K}_{i\ldots j}}|k]\;\equiv\;\sum_{a=i}^{j}\left[k%
\,a\right]\left\langle a\,l\right\rangle\,,}$\cr$\displaystyle{\langle k|K_{i%
\ldots j}K_{m\ldots n}|l\rangle}$&$\displaystyle{{}\equiv\;\langle k^{-}|{/%
\hskip-6.259606ptK_{i\ldots j}/\hskip-6.259606ptK_{m\ldots n}}|l^{+}\rangle=%
\sum_{a=i}^{j}\sum_{b=m}^{n}\left\langle k\,a\right\rangle\left[a\,b\right]%
\left\langle b\,l\right\rangle\,,}$\cr$\displaystyle{{[k|{{K}_{i\ldots j}{K}_{%
m\ldots n}}|l]}}$&$\displaystyle{{}\equiv\;{\langle k^{+}|{/\hskip-6.259606ptK%
_{i\ldots j}/\hskip-6.259606ptK_{m\ldots n}}|l^{-}\rangle}\;\equiv\;\sum_{a=i}%
^{j}\sum_{b=m}^{n}\left[k\,a\right]\left\langle a\,b\right\rangle\left[b\,l%
\right]\,,}$\cr$\displaystyle{}$}}\,$$
(A.6)A.6( roman_A .6 )
etc. We will often use the momentum invariants
$s_{ij}=(k_{i}+k_{j})^{2}$ and $t_{ijk}=(k_{i}+k_{j}+k_{k})^{2}$.
A.2 Three Mass Boxes
The three mass boxes have one graviton attached to one corner and
two gravitons to each of the others. The three-point corner is
$\overline{\it MHV}$ while the four-point corners are MHV. This means
that all four corners are relatively simple and that different
helicity configurations are also relatively simply related. In the
case of $L_{6}$ there is a summation over the full $\mathcal{N}=8$
multiplet running in the loop. We get,
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{}$&$\displaystyle{{}c{[a^{+},%
\{b^{+},c^{+}\},\{d^{-},e^{+}\},\{f^{-},g^{-}\}]}=L_{0},}$\cr$\displaystyle{}$%
&$\displaystyle{{}c{[a^{+},\{b^{+},c^{+}\},\{d^{-},e^{-}\},\{f^{-},g^{+}\}]}=L%
_{1}=\left({\langle f|K_{de}K_{bc}|a\rangle\over[e|{K}_{bc}|a\rangle\left%
\langle f\,g\right\rangle}\right)^{8}L_{0},}$\cr$\displaystyle{}$&$%
\displaystyle{{}c{[a^{+},\{b^{+},c^{-}\},\{d^{+},e^{+}\},\{f^{-},g^{-}\}]}=L_{%
2}=\left({\left\langle a\,c\right\rangle\left[d\,e\right]\over[e|{K}_{bc}|a%
\rangle}\right)^{8}L_{0}}$\cr$\displaystyle{}$&$\displaystyle{{}c{[a^{-},\{b^{%
+},c^{+}\},\{d^{-},e^{-}\},\{f^{+},g^{+}\}]}=L_{3}=\left({\langle a|{K}_{bc}K_%
{fg}|a\rangle\over[e|{K}_{bc}|a\rangle\left\langle f\,g\right\rangle}\right)^{%
8}L_{0},}$\cr$\displaystyle{}$&$\displaystyle{{}c{[a^{-},\{b^{-},c^{-}\},\{d^{%
+},e^{+}\},\{f^{+},g^{+}\}]}=0,}$\cr$\displaystyle{}$&$\displaystyle{{}c{[a^{-%
},\{b^{+},c^{-}\},\{d^{+},e^{+}\},\{f^{+},g^{-}\}]}=L_{4}=\left({\ \left%
\langle a\,c\right\rangle\left\langle a\,g\right\rangle\left[d\,e\right]\over[%
e|{K}_{bc}|a\rangle\left\langle f\,g\right\rangle}\right)^{8}L_{0},}$\cr$%
\displaystyle{}$&$\displaystyle{{}c{[a^{-},\{b^{-},c^{+}\},\{d^{-},e^{+}\},\{f%
^{+},g^{+}\}]}=L_{5}=\left({[e|{K}_{fg}|a\rangle\left\langle a\,b\right\rangle%
\over[e|{K}_{bc}|a\rangle\left\langle f\,g\right\rangle}\right)^{8}L_{0},}$\cr%
$\displaystyle{}$&$\displaystyle{{}c{[a^{+},\{b^{-},c^{+}\},\{d^{-},e^{+}\},\{%
f^{-},g^{+}\}]}=L_{6}}$\cr$\displaystyle{}$&$\displaystyle{{} %
=\left(\left\langle b\,a\right\rangle[e|K_{fg}|a\rangle\langle f|K_{de}K_{bc},%
a]-\left\langle f\,a\right\rangle[e|K_{bc}|a\rangle\langle b|K_{de}K_{fg},a]%
\over[e|{K}_{bc}|a\rangle\left\langle f\,g\right\rangle\langle a|K_{bc}K_{de}|%
a\rangle\right)^{8}L_{0},}$\cr$\displaystyle{}$}}\,\hskip-42.679134pt$$
(A.7)A.7( roman_A .7 )
where,
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{L_{0}=}$&$\displaystyle{{}{-s_%
{bc}s_{de}s_{fg}\left\langle g\,f\right\rangle^{6}[e|{K}_{bc}|a\rangle^{8}(t_{%
abc}t_{fga}-s_{bc}s_{fg})^{2}\over 2\left[d\,e\right]^{2}\left\langle b\,c%
\right\rangle^{2}{\prod\atop{x=b,c,g,f}}\left\langle a\,x\right\rangle{\prod%
\atop{y=d,e}}[y|{K}_{fg}|a\rangle[y|{K}_{bc}|a\rangle{\prod\atop{z=b,c}}%
\langle z|{K}_{de}K_{fg}|a\rangle{\prod\atop{w=f,g}}\langle w|{K}_{de}K_{bc}|a%
\rangle}.}$\cr$\displaystyle{}$}}\,$$
(A.8)A.8( roman_A .8 )
A.3 Two Mass Hard Boxes
The two mass hard boxes have two adjacent three-point corners, a
four-point corner and a five-point corner. The four- and five-point
corners are MHV and of the two three-point corners one is MHV
and the other is $\overline{\it MHV}$. The two ways of assigning these
give rise to the $G_{i}$ and $H_{i}$ terms below. Because all corners
are either MHV or $\overline{\it MHV}$, the different helicity
configurations are simply related. We get,
$${\hskip-14.226378pt\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{c[a^{-},\{b^%
{-},c^{-}\},\{d^{+},e^{+},f^{+}\},g^{+}]}$&$\displaystyle{{}=G_{0},}$\cr$%
\displaystyle{c[a^{-},\{b^{-},c^{+}\},\{d^{-},e^{+},f^{+}\},g^{+}]}$&$%
\displaystyle{{}=G_{1}+H_{1}=\left({[c|K_{abc}|d\rangle\over t_{abc}}\right)^{%
8}G_{0}+\left({\left\langle a\,b\right\rangle[g|K_{abc}|d\rangle\over\left%
\langle b\,c\right\rangle t_{def}}\right)^{8}H_{0},}$\cr$\displaystyle{c[a^{-}%
,\{b^{+},c^{+}\},\{d^{-},e^{-},f^{+}\},g^{+}]}$&$\displaystyle{{}=G_{2}+H_{2}=%
\left({\left[b\,c\right]\left\langle d\,e\right\rangle\over t_{abc}}\right)^{8%
}G_{0}+\left({\left\langle d\,e\right\rangle[g|K_{abc}|d\rangle\over\left%
\langle b\,c\right\rangle t_{def}}\right)^{8}H_{0},}$\cr$\displaystyle{c[a^{+}%
,\{b^{+},c^{+}\},\{d^{-},e^{-},f^{-}\},g^{+}]}$&$\displaystyle{{}=0,}$\cr$%
\displaystyle{c[a^{+},\{b^{-},c^{-}\},\{d^{-},e^{+},f^{+}\},g^{+}]}$&$%
\displaystyle{{}=G_{3}+H_{3}=\left({[a|K_{abc}|d\rangle\over t_{abc}}\right)^{%
8}G_{0}+\left({[g|K_{abc}|d\rangle\over t_{def}}\right)^{8}H_{0},}$\cr$%
\displaystyle{c[a^{+},\{b^{-},c^{+}\},\{d^{-},e^{-},f^{+}\},g^{+}]}$&$%
\displaystyle{{}=G_{4}+H_{4}=\left({\left[a\,c\right]\left\langle d\,e\right%
\rangle\over t_{abc}}\right)^{8}G_{0}+\left({\left\langle d\,e\right\rangle[g|%
K_{abc}|b\rangle\over\left\langle b\,c\right\rangle t_{def}}\right)^{8}H_{0},}%
$\cr$\displaystyle{c[a^{+},\{b^{-},c^{-}\},\{d^{+},e^{+},f^{+}\},g^{-}]}$&$%
\displaystyle{{}=G_{5}+H_{5}=\left({[a|K_{abc}|g\rangle\over t_{abc}}\right)^{%
8}G_{0}+H_{0},}$\cr$\displaystyle{c[a^{+},\{b^{-},c^{+}\},\{d^{-},e^{+},f^{+}%
\},g^{-}]}$&$\displaystyle{{}=G_{6}+H_{6}=\left({\left\langle d\,g\right%
\rangle\left[a\,b\right]\over t_{abc}}\right)^{8}G_{0}+\left({\langle d|K_{def%
}K_{abc}|b\rangle\over\left\langle b\,c\right\rangle t_{def}}\right)^{8}H_{0},%
}$\cr$\displaystyle{c[a^{+},\{b^{+},c^{+}\},\{d^{-},e^{-},f^{+}\},g^{-}]}$&$%
\displaystyle{{}=G_{7}+H_{7}=0.G_{0}+\left({t_{abc}\left\langle d\,e\right%
\rangle\over\left\langle b\,c\right\rangle t_{def}}\right)^{8}H_{0},}$\cr$%
\displaystyle{c[a^{-},\{b^{-},c^{+}\},\{d^{+},e^{+},f^{+}\},g^{-}]}$&$%
\displaystyle{{}=G_{8}+H_{8}=\left({[c|K_{abc}|g\rangle\over t_{abc}}\right)^{%
8}G_{0}+\left({\left\langle a\,b\right\rangle\over\left\langle c\,b\right%
\rangle}\right)^{8}H_{0},}$\cr$\displaystyle{c[a^{-},\{b^{+},c^{+}\},\{d^{-},e%
^{+},f^{+}\},g^{-}]}$&$\displaystyle{{}=G_{9}+H_{9}=\left({\left\langle g\,d%
\right\rangle\left[b\,c\right]\over t_{abc}}\right)^{8}G_{0}+\left({\langle a|%
K_{bc}K_{def}|d\rangle\over\left\langle b\,c\right\rangle t_{def}}\right)^{8}H%
_{0},}$\cr$\displaystyle{}$}}\,$$
(A.9)A.9( roman_A .9 )
where,
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{G_{0}=}$&$\displaystyle{{}{s_{%
ag}^{2}\left\langle b\,c\right\rangle t_{abc}^{8}(\left[d\,e\right]\left%
\langle e\,f\right\rangle[f|K_{abc}|g\rangle[a|K_{abc}|d\rangle-\left\langle d%
\,e\right\rangle\left[e\,f\right][d|K_{abc}|g\rangle[a|K_{abc}|f\rangle)\over 2%
\bar{N}(a,b,c)N(d,e,f,g)[c|K_{abc}|g\rangle[b|K_{abc}|g\rangle[a|K_{abc}|d%
\rangle[a|K_{abc}|e\rangle[a|K_{abc}|f\rangle},}$\cr$\displaystyle{}$}}\,$$
(A.10)A.10( roman_A .10 )
and,
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{H_{0}=}$&$\displaystyle{{}{s_{%
ag}^{2}s_{bc}^{7}t_{def}^{7}\left(\left\langle d\,e\right\rangle\left[e\,f%
\right]\langle f|K_{abc}|g]\langle a|K_{abc}|d]-\left[d\,e\right]\left\langle e%
\,f\right\rangle\langle d|K_{abc}|g]\langle a|K_{abc}|f]\right)\over 2\left[b%
\,c\right]^{2}N(d,e,f){\prod\atop{j=b,c}}[j|K_{abc}|a\rangle[j|K_{bc}K_{def}|g%
]{\prod\atop{i=d,e,f}}\langle i|K_{def}K_{bc}|a\rangle[g|K_{abc}|i\rangle}.}$%
\cr$\displaystyle{}$}}\,$$
(A.11)A.11( roman_A .11 )
Here,
$N(a,b,\cdots m)=\prod_{i<j,i,j\in\{a,b,\cdots m\}}\left\langle i\,j\right\rangle$
and
$\bar{N}(a,b,\cdots m)=\prod_{i<j,i,j\in\{a,b,\cdots m\}}\left[i\,j\right]$
A.4 Two Mass Easy Boxes
The two mass easy boxes have two opposite $\overline{\it MHV}$
three-point corners, an MHV four-point corner and a
$\overline{\it MHV}$ five-point corner. Again, the terms are relatively
simple and related. They are,
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{c[\{a^{-},b^{-},c^{-}\},d^{+},%
\{e^{+},f^{+}\},g^{+}]\equiv}$&$\displaystyle{{}W_{0},}$\cr$\displaystyle{c[\{%
a^{-},b^{-},c^{+}\},d^{+},\{e^{+},f^{-}\},g^{+}]\equiv}$&$\displaystyle{{}W_{1%
}=\left({[c|K_{abc}|f\rangle\over t_{abc}}\right)^{8}W_{0},}$\cr$\displaystyle%
{c[\{a^{-},b^{-},c^{+}\},d^{+},\{e^{+},f^{+}\},g^{-}]\equiv}$&$\displaystyle{{%
}W_{2}=\left({[c|K_{abc}|g\rangle\over t_{abc}}\right)^{8}W_{0},}$\cr$%
\displaystyle{c[\{a^{-},b^{+},c^{+}\},d^{+},\{e^{-},f^{-}\},g^{+}]\equiv}$&$%
\displaystyle{{}W_{3}=\left({\left\langle e\,f\right\rangle\left[b\,c\right]%
\over t_{abc}}\right)^{8}W_{0},}$\cr$\displaystyle{c[\{a^{-},b^{+},c^{+}\},d^{%
+},\{e^{-},f^{+}\},g^{-}]\equiv}$&$\displaystyle{{}W_{4}=\left({\left\langle e%
\,g\right\rangle\left[b\,c\right]\over t_{abc}}\right)^{8}W_{0},}$\cr$%
\displaystyle{c[\{a^{-},b^{+},c^{+}\},d^{-},\{e^{+},f^{+}\},g^{-}]\equiv}$&$%
\displaystyle{{}W_{5}=\left({\left\langle d\,g\right\rangle\left[b\,c\right]%
\over t_{abc}}\right)^{8}W_{0}}$\cr$\displaystyle{c[\{a^{+},b^{+},c^{+}\},d^{-%
},\{e^{-},f^{-}\},g^{+}]=}$&$\displaystyle{{}0,}$\cr$\displaystyle{c[\{a^{+},b%
^{+},c^{+}\},d^{-},\{e^{-},f^{+}\},g^{-}]=}$&$\displaystyle{{}0,}$\cr$%
\displaystyle{}$}}\,$$
(A.12)A.12( roman_A .12 )
where,
$$W_{0}\equiv{\left([g|K_{abc}|d\rangle[d|K_{abc}|g\rangle\right)^{2}\left[e\,f%
\right]^{2}(t_{abc})^{7}(\langle g|K_{abc}k_{a}k_{b}k_{c}|d\rangle-\langle g|K%
_{abc}k_{c}k_{b}k_{a}|d\rangle)\over 2\left[a\,b\right]\left[a\,c\right]\left[%
b\,c\right]\prod_{x=e,f}\left\langle x\,d\right\rangle\left\langle x\,g\right%
\rangle\prod_{x=a,b,c}[x|K_{abc}|g\rangle[x|K_{abc}|d\rangle s_{ef}}.$$
(A.13)A.13( roman_A .13 )
A.5 One Mass Boxes
The one mass boxes have three three-point corners and one massive
six-point corner. For each external helicity configuration there are
(one or) two internal helicity configurations which cause the
massive corner to be either MHV or NMHV. These give rise to the
$F_{i}$ and $P_{i}$ terms, respectively. Taking the second external
configuration below as an example, $F_{1}$ and $P_{1}$ come from,
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\Text(60,10)[l]$a^{-}$\Text(-5,10)[l]$b^{-}$\Text(8,62)[l]$c^{+}$\Text(30,62)[l]$d^{-}$\Text(54,67)[l]$e^{+}$\Text(66,52)[l]$f^{+}$\Text(60,30)[l]$g^{+}$\Text(-10,30)[r]$F_{1}$:
\Text(48,38)[r]$-$\Text(48,22)[r]$+$\Text(41,16)[r]$-$\Text(19,16)[l]$+$\Text(12,22)[l]$+$\Text(12,38)[l]$-$\Text(19,44)[l]$-$\Text(41,44)[r]$+$\SetWidth1
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\Text(60,10)[l]$a^{-}$\Text(-5,10)[l]$b^{-}$\Text(8,62)[l]$c^{+}$\Text(30,62)[l]$d^{-}$\Text(54,67)[l]$e^{+}$\Text(66,52)[l]$f^{+}$\Text(60,30)[l]$g^{+}$\Text(-10,30)[r]$P_{1}$:
\Text(48,38)[r]$-$\Text(48,22)[r]$+$\Text(41,16)[r]$+$\Text(19,16)[l]$-$\Text(12,22)[l]$+$\Text(12,38)[l]$-$\Text(19,44)[l]$+$\Text(41,44)[r]$-$
The $F_{i}$ terms have the same simplifications as noted above, while
the calculational approach for the resulting $P_{i}$ terms is
discussed below.
$${\hskip 28.452756pt\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{c[a^{-},b^{-%
},c^{-},\{d^{+},e^{+},f^{+},g^{+}\}]=}$&$\displaystyle{{}F_{0}+0,}$\cr$%
\displaystyle{c[a^{-},b^{-},c^{+},\{d^{-},e^{+},f^{+},g^{+}\}]=}$&$%
\displaystyle{{}F_{1}\;+P_{1}=\left({[c|K_{abc}|d\rangle\over t_{abc}}\right)^%
{8}F_{0}\;+P_{1},}$\cr$\displaystyle{c[a^{-},b^{+},c^{-},\{d^{-},e^{+},f^{+},g%
^{+}\}]=}$&$\displaystyle{{}F_{2}\;+P_{2}=\left({\left[b\,c\right]\left\langle
d%
\,b\right\rangle\over\left[a\,c\right]\left\langle a\,b\right\rangle}\right)^{%
8}F_{0}\;+P_{2},}$\cr$\displaystyle{c[a^{-},b^{+},c^{+},\{d^{-},e^{-},f^{+},g^%
{+}\}]=}$&$\displaystyle{{}F_{3}\;+P_{3}=\left({\left\langle d\,e\right\rangle%
\left[b\,c\right]\rangle\over t_{abc}}\right)^{8}F_{0}\;+P_{3},}$\cr$%
\displaystyle{c[a^{+},b^{-},c^{+},\{d^{-},e^{-},f^{+},g^{+}\}]=}$&$%
\displaystyle{{}F_{4}\;+P_{4}=\left({\left\langle d\,e\right\rangle\left[a\,c%
\right]\rangle\over t_{abc}}\right)^{8}F_{0}\;+P_{4},}$\cr$\displaystyle{c[a^{%
+},b^{+},c^{+},\{d^{-},e^{-},f^{-},g^{+}\}]=}$&$\displaystyle{{}0+P_{5},}$\cr$%
\displaystyle{}$}}\,$$
(A.14)A.14( roman_A .14 )
where,
$$F_{0}={t_{abc}^{6}\left\langle a\,b\right\rangle^{2}\left\langle c\,b\right%
\rangle^{2}\left[a\,g\right]\left[d\,e\right][f|K_{de}K_{abc}|a]\over 4\left[a%
\,c\right]\left\langle d\,e\right\rangle\left\langle e\,f\right\rangle\left%
\langle d\,f\right\rangle\left\langle f\,g\right\rangle[c|K_{abc}|g\rangle{%
\prod_{x=d,e,g}}[a|K_{abc}|x\rangle}+{\rm Perm}(d,e,f,g)\,.$$
(A.15)A.15( roman_A .15 )
For $P_{1}$ we use the form of the NMHV six-point tree amplitude
in appendix
C.
We then get,
$$P_{1}^{[a,b,c,d,e,f,g]}=\left({\cal T}^{1}_{1}\right)+\left({\cal T}^{2}_{1}+{%
\cal T}^{3}_{1}+{\cal T}^{4}_{1}+{\cal T}^{5}_{1}\right)|_{\{(efg)+(feg)+(gef)%
\}},$$
(A.16)A.16( roman_A .16 )
with,
$${\hskip-14.226378pt\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{{\cal T}_{1}%
^{1}}$&$\displaystyle{{}=M_{0}[a,b,c,d,e,f,g]}$\cr$\displaystyle{\equiv}$&$%
\displaystyle{{}{s_{ab}^{2}\left\langle c\,d\right\rangle\left\langle a\,b%
\right\rangle^{6}\left[b\,c\right]^{2}t_{efg}^{7}\Bigl{(}[d|K_{efg}|e\rangle%
\left[e\,f\right]\left\langle f\,g\right\rangle[g|K_{abc}|c\rangle-[d|K_{efg}|%
g\rangle\left[g\,f\right]\left\langle f\,e\right\rangle[e|K_{abc}|c\rangle%
\Bigr{)}\over 2t_{abc}^{2}\left\langle a\,c\right\rangle[d|K_{abc}|a\rangle%
\left\langle e\,f\right\rangle\left\langle f\,g\right\rangle\left\langle e\,g%
\right\rangle\prod_{x=e,f,g}\langle c|K_{abc}K_{efg}|x\rangle[d|K_{efg}|x%
\rangle},}$\cr$\displaystyle{{\cal T}^{2}_{1}}$&$\displaystyle{{}=\left({[e|K_%
{dfg}|d\rangle\over t_{dfg}}\right)^{8}M_{0}[a,b,c,e,d,f,g],}$\cr$%
\displaystyle{}$}}\,$$
$${\hskip-14.226378pt\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{{\cal T}^{3}%
_{1}}$&$\displaystyle{{}={{s_{ab}^{2}\left\langle a\,b\right\rangle^{6}\left[b%
\,c\right]^{2}\left[f\,g\right]^{8}\langle c|K_{abc}|e]\Bigl{(}\langle e|K_{%
dfg}|d]\left\langle d\,f\right\rangle\left[f\,g\right]\left\langle g\,c\right%
\rangle-\langle e|K_{dfg}|g]\left\langle g\,f\right\rangle\left[f\,d\right]%
\left\langle d\,c\right\rangle\Bigr{)}\over 2t_{dfg}\left\langle a\,c\right%
\rangle\left\langle a\,e\right\rangle\left[d\,f\right]\left[f\,g\right]\left[d%
\,g\right]\prod_{x=d,f,g}[x|K_{efg}|c\rangle[x|K_{efg}|e\rangle}},}$\cr$%
\displaystyle{{\cal T}^{4}_{1}}$&$\displaystyle{{}={-[e|K_{fg}|c\rangle^{7}%
\left[b\,c\right]^{2}\left\langle a\,b\right\rangle^{6}s_{de}s_{fg}s_{ab}^{2}%
\over 2\langle c|K_{fg}K_{ade}|a\rangle\left\langle a\,c\right\rangle\left[d\,%
e\right]^{2}{\prod\atop{y=d,e}}\langle c|K_{abc}|y]\left\langle f\,g\right%
\rangle^{2}\left\langle g\,c\right\rangle\left\langle f\,c\right\rangle{\prod%
\atop{x=f,g}}\langle c|K_{abc}K_{de}|x\rangle[d|K_{fg}|c\rangle},}$\cr$%
\displaystyle{{\cal T}^{5}_{1}}$&$\displaystyle{{}={\left\langle a\,b\right%
\rangle^{6}\left[b\,c\right]^{2}\left\langle c\,d\right\rangle^{7}\left[f\,g%
\right]^{6}s_{de}s_{fg}t_{abc}s_{ab}^{2}\over 2\left\langle a\,c\right\rangle%
\langle c|K_{abfg}K_{afg}|a\rangle\left\langle d\,e\right\rangle^{2}\left%
\langle c\,e\right\rangle{\prod_{y=f,g}}[y|K_{abc}|c\rangle[y|K_{de}|c\rangle%
\prod_{x=d,e}\langle x|K_{fg}K_{abc}|c\rangle}.}$\cr$\displaystyle{}$}}\,$$
(A.17)A.17( roman_A .17 )
$P_{1}$ and $P_{2}$ are related by,
$$P_{2}^{[a,b,c,d,e,f,g]}=\left(\frac{\langle ca\rangle}{\langle bc\rangle}%
\right)^{8}P_{1}^{[a,b,c,d,e,f,g]}.$$
(A.18)A.18( roman_A .18 )
$P_{3}$ is obtained by using the NMHV amplitude of Cachazo and Svrček
[41]. We then get,
$$P_{3}^{[a,b,c,d,e,f,g]}=\sum_{i=1}^{13}{\cal T}_{3}^{i},$$
(A.19)A.19( roman_A .19 )
where,
$${\hskip-39.833858pt\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{{\cal T}_{3}%
^{1}=}$&$\displaystyle{{}\frac{[ab]^{2}[bc]^{2}\langle de\rangle\langle a|K_{%
de}|f]^{7}\big{(}\langle a|K_{de}|f]\langle g|K_{ef}|d]\langle c|K_{ab}|g]-%
\langle c|K_{ab}|d][fg]\langle ga\rangle t_{def}\big{)}}{2\langle ag\rangle[de%
][ef]^{2}\langle gc\rangle t_{def}\langle a|K_{ef}|d]{\prod\atop{x=c,g}}{\prod%
\atop{y=d,f}}\langle x|K_{def}|y]},}$\cr$\displaystyle{}$}}\,$$
$${\hskip-14.226378pt\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{{\cal T}_{3}%
^{2}=}$&$\displaystyle{{}\frac{[ab]^{2}[bc]^{2}\langle ea\rangle\big{(}\langle
da%
\rangle\langle c|K_{ab}|f]+\langle ca\rangle\langle de\rangle[ef]\big{)}^{7}}{%
2\langle ca\rangle\langle dg\rangle\langle dc\rangle\langle c|K_{ab}|e][ef]^{2%
}\langle gc\rangle\langle c|K_{dg}|f]}}$\cr$\displaystyle{}$&$\displaystyle{{}%
\times\frac{\big{(}\langle da\rangle\langle c|K_{ab}|f]+\langle ca\rangle%
\langle de\rangle[ef]\big{)}\langle c|K_{ab}K_{ef}|g\rangle[gd]-\langle c|K_{%
ab}|d][fg]\langle gd\rangle\langle a|K_{ef}K_{dg}|c\rangle}{\langle a|K_{ef}K_%
{dg}|c\rangle\langle c|K_{ab}K_{ef}|d\rangle{\prod\atop{x=c,g}}\langle c|K_{ab%
}K_{ef}|x\rangle\big{(}\langle ga\rangle\langle c|K_{ab}|f]+\langle ca\rangle%
\langle ge\rangle[ef]\big{)}},}$\cr$\displaystyle{}$}}\,$$
$${\hskip-19.916929pt\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{{\cal T}_{3}%
^{3}=}$&$\displaystyle{{}\frac{[ab]^{2}[bc]^{2}[gf]\langle e|K_{gf}K_{bc}|a%
\rangle^{7}}{2\langle a|K_{bc}|d]t_{abc}\langle fg\rangle\langle fe\rangle^{2}%
\langle c|K_{ab}|d]t_{gfe}\langle g|K_{fe}K_{bc}|a\rangle}}$\cr$\displaystyle{%
}$&$\displaystyle{{} \times\frac{%
\langle e|K_{gf}K_{bc}|a\rangle\langle g|K_{fe}|d]\langle dc\rangle+\langle cg%
\rangle\langle ed\rangle\langle a|K_{bc}|d]t_{gfe}}{{\prod\atop{x=e,g}}\langle
x%
|K_{efg}|d]\langle c|K_{ab}K_{efg}|x\rangle},}$\cr$\displaystyle{}$}}\,$$
$${\hskip-184.942913pt\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{{\cal T}_{3%
}^{4}=}$&$\displaystyle{{}\frac{[ab]^{2}[bc]^{2}\langle a|K_{bc}|f]\big{(}%
\langle ce\rangle\langle a|K_{bc}|g]+\langle ca\rangle\langle ef\rangle[fg]%
\big{)}^{7}}{2\langle ca\rangle[gd]\langle cf\rangle\langle fe\rangle^{2}{%
\prod\atop{x=d,g}}\langle c|K_{ab}|x]\langle c|K_{fe}|x]}}$\cr$\displaystyle{}%
$}}\,$$
$${\hskip 42.679134pt\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{}$&$%
\displaystyle{{}\times\frac{\big{(}\langle ce\rangle\langle a|K_{bc}|g]+%
\langle ca\rangle\langle ef\rangle[fg]\big{)}\langle c|K_{fe}|d]\langle dg%
\rangle+\langle gc\rangle\langle ed\rangle[dg]\langle a|K_{dg}K_{ef}|c\rangle}%
{\langle a|K_{dg}K_{ef}|c\rangle\langle c|K_{ab}K_{fe}|c\rangle\langle c|K_{ab%
}K_{dg}|e\rangle\big{(}\langle ce\rangle\langle a|K_{bc}|d]+\langle ca\rangle%
\langle ef\rangle[fd]\big{)}},}$\cr$\displaystyle{}$}}\,$$
$${\hskip-210.550394pt\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{{\cal T}_{3%
}^{5}=}$&$\displaystyle{{}\frac{[ab]^{2}[bc]^{2}\langle ae\rangle^{7}\langle dg%
\rangle[fg]^{7}t_{abc}}{2[df][dg]\langle ec\rangle t_{dfg}{\prod\atop{x=a,c}}%
\langle x|K_{dg}|f]{\prod\atop{y=d,g}}\langle e|K_{dfg}|y]},}$\cr$%
\displaystyle{}$}}\,$$
$${\hskip-31.298031pt\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{{\cal T}_{3}%
^{6}=}$&$\displaystyle{{}\frac{[ab]^{2}[bc]^{2}\langle ca\rangle^{7}\langle de%
\rangle^{7}\langle ga\rangle[fg]^{7}\langle a|K_{bc}|d]}{2\langle dc\rangle{%
\prod\atop{x=f,g}}\langle c|K_{ab}|x]\langle ec\rangle\langle a|K_{fg}K_{de}|c%
\rangle\langle c|K_{de}|f]\langle c|K_{ab}K_{fg}|e\rangle}}$\cr$\displaystyle{%
}$&$\displaystyle{{} \times\frac{1}{\big{(}\langle da%
\rangle\langle c|K_{ab}|f]+\langle ca\rangle\langle dg\rangle[gf]\big{)}\big{(%
}\langle ce\rangle\langle a|K_{bc}|g]+\langle ca\rangle\langle ed\rangle[dg]%
\big{)}},}$\cr$\displaystyle{}$}}\,$$
$${\hskip-39.833858pt\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{{\cal T}_{3}%
^{7}=}$&$\displaystyle{{}\frac{[ab]^{2}[bc]^{2}\langle ae\rangle^{7}\langle cd%
\rangle\langle a|K_{bc}|f]^{7}\langle c|K_{ab}|g]}{2\langle ca\rangle\langle ag%
\rangle[df]\langle a|K_{bc}|d]\langle eg\rangle\langle a|K_{eg}K_{df}|c\rangle%
\langle a|K_{eg}|f]\langle e|K_{df}K_{bc}|a\rangle}}$\cr$\displaystyle{}$&$%
\displaystyle{{} \times\frac{1}{\big{(}\langle cg%
\rangle\langle a|K_{bc}|f]+\langle ca\rangle\langle gd\rangle[df]\big{)}\big{(%
}\langle ea\rangle\langle c|K_{ab}|d]+\langle ca\rangle\langle eg\rangle[gd]%
\big{)}},}$\cr$\displaystyle{}$}}\,$$
$${\hskip-147.954331pt\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{{\cal T}_{3%
}^{8}=}$&$\displaystyle{{}\frac{[ab]^{2}[bc]^{2}\langle de\rangle^{7}\langle a%
|K_{bc}|f]^{7}[dg]}{2\langle dg\rangle\langle c|K_{ab}|f]t_{abc}\langle ge%
\rangle t_{deg}{\prod\atop{x=d,g}}\langle x|K_{deg}|f]{\prod\atop{y=a,c}}%
\langle y|K_{abc}K_{dg}|e\rangle},}$\cr$\displaystyle{}$}}\,$$
$${\hskip-36.988583pt\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{{\cal T}_{3}%
^{9}=}$&$\displaystyle{{}\frac{[ab]^{2}[bc]^{2}\langle ae\rangle^{8}\langle c|%
K_{ab}|f]\langle a|K_{bc}|g]^{7}}{2\langle ca\rangle\langle af\rangle[dg]%
\langle a|K_{bc}|d]\langle ef\rangle^{2}\langle a|K_{ef}K_{dg}|c\rangle\langle
a%
|K_{ef}K_{bc}|a\rangle\langle e|K_{dg}K_{bc}|a\rangle}}$\cr$\displaystyle{}$&$%
\displaystyle{{} \times\frac{\langle de%
\rangle\langle gc\rangle\langle a|K_{bc}|d]\langle a|K_{ef}|g]-\langle eg%
\rangle\langle a|K_{bc}|g]\langle cd\rangle\langle a|K_{ef}|d]}{{\prod\atop{x=%
d,g}}\langle a|K_{ef}|x]\big{(}\langle ea\rangle\langle c|K_{ab}|x]+\langle ca%
\rangle\langle ef\rangle[fx]\big{)}},}$\cr$\displaystyle{}$}}\,$$
$${\hskip-65.441339pt\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{{\cal T}_{3}%
^{10}=}$&$\displaystyle{{}\frac{[ab]^{2}[bc]^{2}\langle de\rangle^{8}[df]%
\langle a|K_{bc}|g]^{7}}{2\langle df\rangle\langle c|K_{ab}|g]t_{abc}\langle ef%
\rangle^{2}t_{def}}\frac{\langle ae\rangle\langle cg\rangle t_{abc}\langle d|K%
_{ef}|g]+\langle eg\rangle\langle a|K_{bc}|g]\langle c|K_{ab}K_{ef}|d\rangle}{%
{\prod\atop{x=d,e}}{\prod\atop{y=a,c}}\langle x|K_{def}|g]\langle y|K_{abc}K_{%
def}|x\rangle},}$\cr$\displaystyle{}$}}\,$$
$${\hskip-65.441339pt\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{{\cal T}_{3}%
^{11}=}$&$\displaystyle{{}\frac{[ab]^{2}[bc]^{2}\langle a|K_{bc}|f]^{8}\langle
ce%
\rangle\langle da\rangle^{7}}{2\langle ca\rangle\langle a|K_{bc}|e]\langle gd%
\rangle\langle ga\rangle[fe]^{2}\langle a|K_{dg}K_{ef}|c\rangle\langle a|K_{fe%
}K_{bc}|a\rangle\langle a|K_{dg}|f]}}$\cr$\displaystyle{}$&$\displaystyle{{}%
\times\frac{[fg]\langle c|K_{ab}|d]\langle ag\rangle\langle d%
|K_{fe}K_{bc}|a\rangle+[fd]\langle da\rangle\langle c|K_{ab}|g]\langle g|K_{fe%
}K_{bc}|a\rangle}{{\prod\atop{x=d,g}}\langle x|K_{fe}K_{bc}|a\rangle\big{(}%
\langle cx\rangle\langle a|K_{bc}|f]+\langle ca\rangle\langle xe\rangle[ef]%
\big{)}},}$\cr$\displaystyle{}$}}\,$$
$${\hskip-62.596063pt\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{{\cal T}_{3}%
^{12}=}$&$\displaystyle{{}\frac{[ab]^{2}[bc]^{2}[gf]^{8}\langle ge\rangle%
\langle da\rangle^{7}\big{(}-\langle a|K_{bc}|f]\langle c|K_{ab}|d]\langle d|K%
_{fe}|g]+[fd]\langle da\rangle t_{abc}\langle c|K_{fe}|g]\big{)}}{2[ge]\langle
cd%
\rangle[fe]^{2}t_{gfe}{\prod\atop{x=a,c,d}}{\prod\atop{y=f,g}}\langle x|K_{efg%
}|y]},}$\cr$\displaystyle{}$}}\,$$
$${\hskip-48.369685pt\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{{\cal T}_{3}%
^{13}=}$&$\displaystyle{{}\frac{[ab]^{2}[bc]^{2}\langle ad\rangle\langle a|K_{%
bc}|g]\big{(}\langle ea\rangle\langle c|K_{ab}|f]+\langle ca\rangle\langle ed%
\rangle[df]\big{)}^{8}}{2\langle ca\rangle{\prod\atop{x=d,f}}\langle c|K_{ab}|%
x][df]\langle eg\rangle\langle ce\rangle\langle gc\rangle\langle a|K_{df}K_{eg%
}|c\rangle\langle c|K_{eg}|f]\langle c|K_{ab}K_{df}|e\rangle}}$\cr$%
\displaystyle{}$&$\displaystyle{{} \times\frac{1}{%
\big{(}\langle ga\rangle\langle c|K_{ab}|f]+\langle ca\rangle\langle gd\rangle%
[df]\big{)}\big{(}\langle ce\rangle\langle a|K_{bc}|d]+\langle ca\rangle%
\langle eg\rangle[gd]\big{)}}.}$}}\,$$
(A.20)A.20( roman_A .20 )
$P_{4}$ has the additional complication that we must sum over the full
$\mathcal{N}=8$ multiplet running in the loop.
We obtain a form based on $P_{3}$
with relative factors for
each ${\cal T}_{3}^{i}$.
We also obtain one
extra term which is not present in $P_{3}$. We get,
$$P_{4}^{[a,b,c,d,e,f,g]}=\sum_{i_{1}}^{13}\big{(}Y_{4}^{i})^{8}{\cal T}_{3}^{i}%
+{\cal T}_{4}^{14},$$
(A.21)A.21( roman_A .21 )
where,
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{Y_{4}^{1}=}$&$\displaystyle{{}%
-\frac{\langle b|K_{de}|f]}{\langle a|K_{de}|f]},}$\cr$\displaystyle{Y_{4}^{2}%
=}$&$\displaystyle{{}\frac{\langle bd\rangle\langle c|K_{ab}|f]+\langle bc%
\rangle\langle de\rangle[ef]}{\langle da\rangle\langle c|K_{ab}|f]+\langle ca%
\rangle\langle de\rangle[ef]},}$\cr$\displaystyle{Y_{4}^{3}=}$&$\displaystyle{%
{}-\frac{\langle b|K_{ac}K_{fg}|e\rangle}{\langle a|K_{bc}K_{fg}|e\rangle},}$%
\cr$\displaystyle{Y_{4}^{4}=}$&$\displaystyle{{}-\frac{\langle ce\rangle%
\langle b|K_{ac}|g]+\langle cb\rangle\langle ef\rangle[fg]}{\langle ce\rangle%
\langle a|K_{bc}|g]+\langle ca\rangle\langle ef\rangle[fg]},}$\cr$%
\displaystyle{Y_{4}^{5}=}$&$\displaystyle{{}\frac{\langle eb\rangle}{\langle ae%
\rangle},}$\cr$\displaystyle{Y_{4}^{6}=}$&$\displaystyle{{}\frac{\langle bc%
\rangle}{\langle ca\rangle},}$\cr$\displaystyle{Y_{4}^{7}=}$&$\displaystyle{{}%
\frac{\langle eb\rangle\langle a|K_{bc}|f]-\langle ba\rangle\langle ed\rangle[%
df]}{\langle ae\rangle\langle a|K_{bc}|f]},}$\cr$\displaystyle{Y_{4}^{8}=}$&$%
\displaystyle{{}-\frac{\langle b|K_{ac}|f]}{\langle a|K_{bc}|f]},}$\cr$%
\displaystyle{Y_{4}^{9}=}$&$\displaystyle{{}\frac{\langle be\rangle\langle a|K%
_{bc}|g]+\langle ba\rangle\langle ed\rangle[dg]}{\langle ea\rangle\langle a|K_%
{bc}|g]},}$\cr$\displaystyle{Y_{4}^{10}=}$&$\displaystyle{{}-\frac{\langle b|K%
_{ac}|g]}{\langle a|K_{bc}|g]},}$\cr$\displaystyle{Y_{4}^{11}=}$&$%
\displaystyle{{}\frac{\langle bd\rangle\langle a|K_{bc}|f]+\langle ba\rangle%
\langle de\rangle[ef]}{\langle da\rangle\langle a|K_{bc}|f]},}$\cr$%
\displaystyle{Y_{4}^{12}=}$&$\displaystyle{{}\frac{\langle db\rangle}{\langle
ad%
\rangle},}$\cr$\displaystyle{Y_{4}^{13}=}$&$\displaystyle{{}\frac{\langle be%
\rangle\langle c|K_{ab}|f]+\langle bc\rangle\langle ed\rangle[df]}{\langle ea%
\rangle\langle c|K_{ab}|f]+\langle ca\rangle\langle ed\rangle[df]},}$\cr$%
\displaystyle{{\cal T}_{4}^{14}=}$&$\displaystyle{{}\left(\frac{\langle ab%
\rangle}{\langle ca\rangle}\right)^{8}{\cal T}_{3}^{6}(a\leftrightarrow c).}$}}\,$$
(A.22)A.22( roman_A .22 )
Last comes $P_{5}$ which has been obtained from the amplitude of Cachazo
and Svrček by
letting 5 and 6 be the internal gravitons. We get,
$${\hskip 28.452756pt\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{P_{5}^{[a,b,%
c,d,e,f,g]}=}$&$\displaystyle{{}{\cal T}_{5}^{1}+{\cal T}_{5}^{1}(d%
\leftrightarrow e)+{\cal T}_{5}^{2}+{\cal T}_{5}^{2}(a\leftrightarrow c)}$\cr$%
\displaystyle{}$&$\displaystyle{{}+{\cal T}_{5}^{3}+{\cal T}_{5}^{3}(d%
\leftrightarrow e)+{\cal T}_{5}^{3}(a\leftrightarrow c)+{\cal T}_{3}^{c}(a%
\leftrightarrow c,d\leftrightarrow e)}$\cr$\displaystyle{}$&$\displaystyle{{}+%
{\cal T}_{5}^{4}+{\cal T}_{5}^{4}(d\leftrightarrow e)+{\cal T}_{5}^{5}+{\cal T%
}_{5}^{5}(a\leftrightarrow c)+{\cal T}_{5}^{6},}$}}\,$$
(A.23)A.23( roman_A .23 )
where,
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{{\cal T}_{5}^{1}=}$&$%
\displaystyle{{}\frac{[ab]^{2}[bc]^{2}\langle ef\rangle\langle d|K_{ef}|g]^{7}%
\big{(}\langle d|K_{ef}|g]\langle a|K_{fg}|e]\langle c|K_{ab}|d]-[de]\langle c%
|K_{ab}|g]\langle ad\rangle t_{efg}\big{)}}{2\langle da\rangle\langle cd%
\rangle[ef][fg]^{2}t_{efg}\langle d|K_{fg}|e]{\prod\atop{x=e,g}}\langle a|K_{%
efg}|x]\langle c|K_{efg}|x]},}$\cr$\displaystyle{{\cal T}_{5}^{2}=}$&$%
\displaystyle{{}\frac{[ab]^{2}[bc]^{2}\langle c|K_{ab}|g]\langle f|K_{de}K_{bc%
}|a\rangle^{7}}{2\langle ca\rangle\langle ag\rangle\langle gf\rangle^{2}[ed]%
\langle a|K_{gf}K_{de}|c\rangle\langle a|K_{fg}K_{bc}|a\rangle}}$\cr$%
\displaystyle{}$&$\displaystyle{{} \times\frac{\langle f|%
K_{de}K_{bc}|a\rangle\langle a|K_{gf}|e]\langle ec\rangle-\langle fe\rangle%
\langle a|K_{bc}|e]\langle a|K_{gf}K_{de}|c\rangle}{{\prod\atop{x=d,e}}\langle
a%
|K_{bc}|x]\langle a|K_{fg}|x]\big{(}\langle fa\rangle\langle c|K_{ab}|x]+%
\langle ca\rangle\langle fg\rangle[gx]\big{)}},}$\cr$\displaystyle{{\cal T}_{5%
}^{3}=}$&$\displaystyle{{}\frac{[ab]^{2}[bc]^{2}\langle df\rangle^{7}\langle ea%
\rangle\langle c|K_{ab}|g]^{7}\langle a|K_{bc}|d]}{2\langle ca\rangle\langle dc%
\rangle[eg]\langle c|K_{ab}|e]\langle fc\rangle\langle a|K_{eg}K_{df}|c\rangle%
\langle c|K_{df}|g]\langle f|K_{eg}K_{ab}|c\rangle}}$\cr$\displaystyle{}$&$%
\displaystyle{{} \times\frac{1}{\big{(}\langle da\rangle%
\langle c|K_{ab}|g]+\langle ca\rangle\langle de\rangle[eg]\big{)}\big{(}%
\langle cf\rangle\langle a|K_{bc}|e]+\langle ca\rangle\langle fd\rangle[de]%
\big{)}},}$\cr$\displaystyle{{\cal T}_{5}^{4}=}$&$\displaystyle{{}\frac{[ab]^{%
2}[bc]^{2}\langle df\rangle^{8}[dg]t_{abc}^{7}\big{(}\langle ef\rangle\langle a%
|K_{bc}|e]\langle c|K_{ab}K_{fg}|d\rangle+\langle fa\rangle t_{abc}\langle ce%
\rangle\langle d|K_{fg}|e]\big{)}}{2\langle gd\rangle\langle fg\rangle^{2}t_{%
dfg}{\prod\atop{x=d,f}}{\prod\atop{y=a,c}}\langle x|K_{dfg}|e]\langle y|K_{abc%
}|e]\langle y|K_{abc}K_{dfg}|x\rangle},}$\cr$\displaystyle{{\cal T}_{5}^{5}=}$%
&$\displaystyle{{}\frac{[ab]^{2}[bc]^{2}\langle a|K_{bc}|g]^{8}\langle cf%
\rangle\langle ed\rangle^{7}}{2\langle ca\rangle\langle a|K_{bc}|f]\langle ae%
\rangle\langle ad\rangle[gf]^{2}\langle a|K_{de}K_{gf}|c\rangle\langle a|K_{de%
}|g]}}$\cr$\displaystyle{}$&$\displaystyle{{} \times\frac%
{\langle c|K_{ab}|g][ed]\langle da\rangle\langle e|K_{fg}K_{bc}|a\rangle+[ge]%
\langle de\rangle\langle c|K_{ab}|d]\langle a|K_{fg}K_{bc}|a\rangle}{\langle a%
|K_{fg}K_{bc}|a\rangle{\prod\atop{x=d,e}}\big{(}\langle cx\rangle\langle a|K_{%
bc}|g]+\langle ca\rangle\langle xf\rangle[fg]\big{)}\langle x|K_{fg}K_{bc}|a%
\rangle},}$\cr$\displaystyle{{\cal T}_{5}^{6}=}$&$\displaystyle{{}\frac{[ab]^{%
2}[bc]^{2}\langle de\rangle t_{abc}\langle f|K_{de}|g]^{8}}{2[de][dg][eg]%
\langle fa\rangle\langle fc\rangle t_{deg}\langle a|K_{de}|g]\langle c|K_{de}|%
g]\langle f|K_{eg}|d]\langle f|K_{dg}|e]}.}$}}\,$$
(A.24)A.24( roman_A .24 )
Appendix B Relations Between Box Coefficients
The box-coefficients exhibit a large number of relations. As a
consequence of the IR structure many combinations can be used to
create expressions for the tree amplitudes. This has in fact been
used to obtain relatively compact formulae for tree
amplitudes [19, 60] and gave rise to the BCFW
recursion relations [39]. Since the IR relations are
satisfied, the box-coefficients are related to the tree amplitudes
and in fact yield a form of the tree amplitude which is equivalent
to that obtained via recursion. Before commencing it is convenient
to define scaled box-coefficients555The scaling factors are
essentially the momentum prefactors appearing in the integral
functions.,
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{\hat{c}^{1m}[a,b,c,\{d,e,f,g\}%
]}$&$\displaystyle{{}\equiv{c^{1m}[a,b,c,\{d,e,f,g\}]\over s_{ab}s_{bc}},}$\cr%
$\displaystyle{\hat{c}^{2m\;h}[a,\{b,c\},\{d,e,f\},g]}$&$\displaystyle{{}%
\equiv{c^{2m\;h}[a,\{b,c\},\{d,e,f\},g]\over s_{ga}t_{abc}},}$\cr$%
\displaystyle{\hat{c}^{2m\;e}[a,\{b,c\},d,\{e,f,g\}]}$&$\displaystyle{{}\equiv%
{c^{2m\;e}[a,\{b,c\},d,\{e,f,g\}]\over t_{abc}t_{bcd}-s_{bc}t_{efg}}.}$\cr$%
\displaystyle{}$}}\,$$
(B.1)B.1( roman_B .1 )
We will also use this notation to indicate the scaled functions
which define the box-coefficients.
B.1 Expressions for tree amplitudes
For the seven-point one-loop NMHV amplitude there are circa 1000
independent boxes with each box coefficient containing two or more
terms. We can extract the tree by looking at the coefficient of a
specific logarithm: there being three independent choices:
$\ln(-s_{12})$, $\ln(-s_{45})$ and $\ln(-s_{34})$. If we take the
coefficient of $\ln(-s_{12})$ then only a subset of boxes will
contribute to this. Contained within this is a further subset where
the legs $1$ and $2$ are massless and the boxes are the one-mass and
two-mass hard.
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{M_{7}^{\rm tree}=}$&$%
\displaystyle{{}\left(\hat{F}_{0}^{[1,2,3,4,5,6,7]}+\{1\leftrightarrow 2\}%
\right)}$\cr$\displaystyle{+}$&$\displaystyle{{}\left(\hat{F}_{1}^{[1,2,4,3,5,%
6,7]}+\hat{F}_{1}^{[1,2,5,3,4,6,7]}+\hat{F}_{1}^{[1,2,6,3,5,4,7]}+\hat{F}_{1}^%
{[1,2,7,3,5,6,4]}+\{1\leftrightarrow 2\}\right)}$\cr$\displaystyle{+}$&$%
\displaystyle{{}\left(\hat{P}_{1}^{[1,2,3,4,5,6,7]}+\hat{P}_{1}^{[1,2,3,5,4,6,%
7]}+\hat{P}_{1}^{[1,2,3,6,5,4,7]}+\hat{P}_{1}^{[1,2,3,7,5,6,4]}+\{1%
\leftrightarrow 2\}\right)}$\cr$\displaystyle{+}$&$\displaystyle{{}\left(\hat{%
G}_{8}^{[2,3,4,5,6,7,1]}+\hat{G}_{8}^{[2,3,5,4,6,7,1]}+\hat{G}_{8}^{[2,3,6,5,4%
,7,1]}+\hat{G}_{8}^{[2,3,7,5,6,4,1]}+\{1\leftrightarrow 2\}\right)}$\cr$%
\displaystyle{+}$&$\displaystyle{{}\left(\hat{H}_{8}^{[2,3,4,5,6,7,1]}+\hat{H}%
_{8}^{[2,3,5,4,6,7,1]}+\hat{H}_{8}^{[2,3,6,5,4,7,1]}+\hat{H}_{8}^{[2,3,7,5,6,4%
,1]}+\{1\leftrightarrow 2\}\right)}$\cr$\displaystyle{+}$&$\displaystyle{{}%
\Bigl{(}\hat{G}_{9}^{[2,4,5,3,6,7,1]}+\hat{G}_{9}^{[2,4,6,3,5,7,1]}+\hat{G}_{9%
}^{[2,4,7,3,6,5,1]}+\hat{G}_{9}^{[2,5,6,3,4,7,1]}}$\cr$\displaystyle{}$&$%
\displaystyle{{} +\hat{G}_{9}^{[2,5,7,3,4,6,%
1]}+\hat{G}_{9}^{[2,6,7,3,4,5,1]}+\{1\leftrightarrow 2\}\Bigr{)}}$\cr$%
\displaystyle{+}$&$\displaystyle{{}\Bigl{(}\hat{H}_{9}^{[2,4,5,3,6,7,1]}+\hat{%
H}_{9}^{[2,4,6,3,5,7,1]}+\hat{H}_{9}^{[2,4,7,3,6,5,1]}+\hat{H}_{9}^{[2,5,6,3,4%
,7,1]}}$\cr$\displaystyle{}$&$\displaystyle{{}%
+\hat{H}_{9}^{[2,5,7,3,4,6,1]}+\hat{H}_{9}^%
{[2,6,7,3,4,5,1]}+\{1\leftrightarrow 2\}\Bigr{)}.}$\cr$\displaystyle{}$}}\,$$
(B.2)B.2( roman_B .2 )
Within this set there are two subsets which each yield the tree, e.g.
$${\hskip-2.845276pt\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{\hat{F}_{0}^{%
[1,2,3,4,5,6,7]}+\biggl{(}\sum_{(a,b,c,d)\in{\cal S}_{1}}\!\!\!\!\!\!\hat{F}_{%
1}^{[1,2,a,3,b,c,d]}\biggr{)}+\biggl{(}\sum_{(a,b,c,d)\in{\cal S}_{1}}\!\!\!\!%
\!\!\hat{G}_{8}^{[1,3,a,b,c,d,2]}\biggr{)}+\biggl{(}\sum_{(a,b,c,d)\in{\cal S}%
_{2}}\!\!\!\!\!\!\hat{G}_{9}^{[1,a,b,3,c,d,2]}\biggr{)}}$\cr$\displaystyle{+%
\biggl{(}\sum_{(a,b,c,d)\in{\cal S}_{1}}\!\!\!\!\!\!\hat{H}_{8}^{[2,3,a,b,c,d,%
1]}\biggr{)}+\biggl{(}\sum_{(a,b,c,d)\in{\cal S}_{2}}\!\!\!\!\!\!\hat{H}_{9}^{%
[2,a,b,3,c,d,1]}\biggr{)}+\biggl{(}\sum_{(a,b,c,d)\in{\cal S}_{1}}\!\!\!\!\!\!%
\hat{P}_{1}^{[2,1,3,a,b,c,d]}\biggr{)},}$\cr$\displaystyle{}$}}\,$$
(B.3)B.3( roman_B .3 )
where,
$${\hskip 14.226378pt\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{{\cal S}_{1}%
}$&$\displaystyle{{}=\{(4,5,6,7),(5,4,6,7),(6,4,5,7),(7,4,5,6)\},}$\cr$%
\displaystyle{{\cal S}_{2}}$&$\displaystyle{{}=\{(4,5,6,7),(4,6,5,7),(4,7,5,6)%
,(5,6,4,7),(5,7,4,6),(6,7,4,5)\}.}$\cr$\displaystyle{}$}}\,$$
(B.4)B.4( roman_B .4 )
This provides a fairly compact realisation of the
seven-point tree amplitude containing twenty-nine individual terms.
This collection of terms corresponds exactly to the terms that
would be obtained from a recursive calculation using legs $1$ and
$2$ for the recursion. The above expression has all the necessary
symmetries although not all are manifest.
This subset of the box-coefficients corresponds to those terms where
legs $1$ and $2$ are isolated at massless corners and where these corners have the
helicity structure indicated.
\Line(30,30)(30,70)
\Line(70,30)(70,70)
\Line(30,30)(70,30)
\Line(70,70)(30,70)
\Line(30,30)(20,20)
\Line(70,30)(80,20)
\Line(30,70)(20,70)
\Line(30,70)(30,80)
\Line(30,70)(25,75)
\Line(70,70)(70,80)
\Line(70,70)(80,70)
\Text(13,15)[l]$1^{-}$\Text(78,15)[l]$2^{-}$\Text(12,72)[l]$a$\Text(18,80)[l]$b$\Text(27,88)[l]$c$\Text(67,87)[l]$d$\Text(83,72)[l]$e$\Text(35,40)[c]${}^{+}$\Text(66,40)[c]${}^{+}$\Text(60,33)[c]${}^{+}$\Text(42,33)[c]${}^{-}$
\DashLine(50,39)(50,27)2
\SetWidth1.5
\DashLine(10,50)(90,50)3
\Line(30,30)(30,70)
\Line(70,30)(70,70)
\Line(30,30)(70,30)
\Line(70,70)(30,70)
\Line(30,30)(20,20)
\Line(70,30)(80,20)
\Line(30,70)(20,70)
\Line(30,70)(30,80)
\Line(70,70)(70,80)
\Line(70,70)(80,70)
\Line(70,70)(75,75)
\Text(13,15)[l]$1^{-}$\Text(78,15)[l]$2^{-}$\Text(12,72)[l]$a$\Text(27,88)[l]$b$\Text(67,87)[l]$c$\Text(77,81)[l]$d$\Text(83,72)[l]$e$\Text(35,40)[c]${}^{+}$\Text(66,40)[c]${}^{+}$\Text(60,33)[c]${}^{+}$\Text(42,33)[c]${}^{-}$\DashLine(50,39)(50,27)2
\SetWidth1.5
\DashLine(10,50)(90,50)3
\Line(30,30)(30,70)
\Line(70,30)(70,70)
\Line(30,30)(70,30)
\Line(70,70)(30,70)
\Line(30,30)(20,20)
\Line(70,30)(80,20)
\Line(30,70)(20,70)
\Line(30,70)(25,80)
\Line(30,70)(18,76)
\Line(30,70)(30,80)
\Line(70,70)(80,80)
\Text(13,14)[l]$1^{-}$\Text(78,14)[l]$2^{-}$\Text(12,72)[l]$a$\Text(11,84)[l]$b$\Text(17,87)[l]$c$\Text(27,87)[l]$d$
\Text(83,88)[l]$e$\Text(35,40)[c]${}^{+}$\Text(66,40)[c]${}^{+}$\Text(60,33)[c]${}^{+}$\Text(42,33)[c]${}^{-}$\DashLine(50,39)(50,27)2
\SetWidth1.5
\DashLine(10,50)(90,50)3
\Line(30,30)(30,70)
\Line(70,30)(70,70)
\Line(30,30)(70,30)
\Line(70,70)(30,70)
\Line(30,30)(20,20)
\Line(70,30)(80,20)
\Line(30,70)(20,80)
\Line(70,70)(70,80)
\Line(70,70)(80,75)
\Line(70,70)(75,80)
\Line(70,70)(80,70)
\Text(13,14)[l]$1^{-}$\Text(78,14)[l]$2^{-}$\Text(12,87)[l]$a$\Text(67,88)[l]$b$\Text(78,86)[l]$c$\Text(84,82)[l]$d$\Text(83,70)[l]$e$
\Text(35,40)[c]${}^{+}$\Text(66,40)[c]${}^{+}$\Text(60,33)[c]${}^{+}$\Text(42,33)[c]${}^{-}$\DashLine(50,39)(50,27)2
\SetWidth1.5
\DashLine(10,50)(90,50)3
Alternate expressions may be obtained by examining the coefficients of
$\ln(s_{45})$ and $\ln(s_{34})$.
B.2 The coefficient of $\ln(-t_{123})$
A different type of relationship holds for the box-coefficients
which contribute to the soft divergence $\ln(-t_{123})/\epsilon$. These
soft divergences are absent so the box coefficients are
conspiring to make them cancel. There are three types of box giving
this divergence: two-mass easy, two-mass hard and one-mass.
Specifically we must have,
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{}$&$\displaystyle{{}\left(\sum%
_{Z_{(1,2,3)}}\sum_{Z_{(4,5,6,7)}}\hat{c}^{2mh}_{(1^{-},\{2^{-},3^{-}\},\{4^{+%
},5^{+},6^{+}\},7^{+})}\right)-\left(\sum_{Z_{(1,2,3)}}\hat{c}^{1m}_{(1^{-},2^%
{-},3^{-},\{4^{+},5^{+},6^{+},7^{+}\})}\right)}$\cr$\displaystyle{}$&$%
\displaystyle{{} -%
\left(\sum_{P_{(4,5,6,7)}}\hat{c}^{2me}_{(\{1^{-},2^{-},3^{-}\},4^{+},\{5^{+},%
6^{+}\},7^{+})}\right)=0,}$\cr$\displaystyle{}$}}\,\hskip-28.452756pt$$
(B.5)B.5( roman_B .5 )
where $Z$ denotes cyclic permutations and,
$$P_{(4,5,6,7)}=\{(4,5,6,7),(4,7,5,6),(4,6,7,5),(5,4,6,7),(5,4,7,6),(6,4,5,7)\}.$$
This relationship is indeed satisfied since,
$${\hskip-28.452756pt\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{\sum_{Z_{(1,%
2,3)}}\sum_{Z_{(4,5,6,7)}}\hat{c}^{2mh}_{(1^{-},\{2^{-},3^{-}\},\{4^{+},5^{+},%
6^{+}\},7^{+})}=2\sum_{Z_{(1,2,3)}}\hat{c}^{1m}_{(1^{-},2^{-},3^{-},\{4^{+},5^%
{+},6^{+},7^{+}\})},}$\cr$\displaystyle{\sum_{P_{(4,5,6,7)}}\hat{c}^{2me}_{(\{%
1^{-},2^{-},3^{-}\},4^{+},\{5^{+},6^{+}\},7^{+})}=\sum_{Z_{(1,2,3)}}\hat{c}^{1%
m}_{(1^{-},2^{-},3^{-},\{4^{+},5^{+},6^{+}\},7^{+})},}$\cr$\displaystyle{}$}}\,$$
(B.6)B.6( roman_B .6 )
although clearly these two constraints are considerably stronger than the single constraint (B.2).
Appendix C Six-Point Tree amplitudes involving non-gravitons
To calculate the cuts of the seven-point amplitude we need
the six-point NMHV amplitudes where one pair of particles is of arbitrary type.
The six-point amplitude is given in the form,
$$M(1^{-},2^{-},(l_{1})^{-}_{h},(l_{2})^{+}_{h},5^{+},6^{+})=\sum_{i=1}^{14}T_{i%
}(h)=\sum_{i=1}^{14}A_{i}(X_{i})^{2h},$$
(C.1)C.1( roman_C .1 )
where $h=2$ for a graviton, $h=3/2$ for a gravitino, $h=1$ for a
vector, $h=1/2$ for a Dirac fermion and $h=0$ for a scalar particle.
The expression is also valid for negative values of $h$ provided we recognise
that this corresponds to a particle of the opposite helicity e.g.
$1^{-}_{-2}\equiv 1^{+}_{+2}$.
The explicit forms of the $T_{i}$ are given by,
$${\hskip-36.988583pt\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{T_{1}}$&$%
\displaystyle{{}\!=\!\frac{-i\left\langle 12\right\rangle^{7}\left\langle 5l_{%
2}\right\rangle\left[2l_{1}\right]\left[56\right]^{7}}{\left\langle 1l_{1}%
\right\rangle\left\langle 2l_{1}\right\rangle\langle 1|P_{\!12l_{1}}|5]\langle
1%
|P_{\!12l_{1}}|l_{2}]\langle 2|P_{\!12l_{1}}|6]\langle l_{1}|P_{\!12l_{1}}|6]%
\left[5l_{2}\right]\left[6l_{2}\right]t_{12l_{1}}}\Big{[}\delta_{h,2}\Big{]},}%
$\cr$\displaystyle{T_{2}}$&$\displaystyle{{}\!=\!\frac{-i\left\langle 2l_{1}%
\right\rangle\langle 1|P_{\!26l_{1}}|6]^{8}\left[5l_{2}\right]}{\left\langle 1%
5\right\rangle\left\langle 1l_{2}\right\rangle\left\langle 5l_{2}\right\rangle%
\langle 1|P_{\!26l_{1}}|2]\langle 1|P_{\!26l_{1}}|l_{1}]\langle 5|P_{\!26l_{1}%
}|6]\langle l_{2}|P_{\!26l_{1}}|6]\left[26\right]\left[2l_{1}\right]\left[6l_{%
1}\right]t_{26l_{1}}}\!\bigg{[}\!\!\frac{-i\left\langle 1l_{2}\right\rangle%
\left[6l_{1}\right]}{\langle 1|P_{\!26l_{1}}|6]}\!\bigg{]}^{A},}$\cr$%
\displaystyle{T_{3}}$&$\displaystyle{{}\!=\!\frac{-i\left\langle 12\right%
\rangle^{7}\left\langle 5l_{1}\right\rangle\left[2l_{2}\right]\left[56\right]^%
{7}}{\left\langle 1l_{2}\right\rangle\left\langle 2l_{2}\right\rangle\langle 1%
|P_{\!56l_{1}}|5]\langle 1|P_{\!56l_{1}}|l_{1}]\langle 2|P_{\!56l_{1}}|6]%
\langle l_{2}|P_{\!56l_{1}}|6]\left[5l_{1}\right]\left[6l_{1}\right]t_{56l_{1}%
}}\big{[}\delta_{h,-2}\big{]},}$\cr$\displaystyle{T_{4}}$&$\displaystyle{{}\!=%
\!\frac{i\left\langle 1l_{1}\right\rangle^{7}\left\langle 25\right\rangle\left%
[56\right]^{7}\left[l_{1}l_{2}\right]}{\left\langle 1l_{2}\right\rangle\left%
\langle l_{1}l_{2}\right\rangle\langle 1|P_{\!256}|2]\langle 1|P_{\!256}|5]%
\langle l_{1}|P_{\!256}|6]\langle l_{2}|P_{\!256}|6]\left[25\right]\left[26%
\right]t_{256}}\!\left[\frac{i\left\langle 1l_{2}\right\rangle}{\left\langle 1%
l_{1}\right\rangle}\right]^{A},}$\cr$\displaystyle{T_{5}}$&$\displaystyle{{}\!%
=\!\frac{i\left\langle 12\right\rangle^{7}\left\langle l_{1}l_{2}\right\rangle%
\left[25\right]\left[6l_{2}\right]^{7}}{\left\langle 15\right\rangle\left%
\langle 25\right\rangle\langle 1|P_{\!125}|l_{1}]\langle 1|P_{\!125}|l_{2}]%
\langle 2|P_{\!125}|6]\langle 5|P_{\!125}|6]\left[6l_{1}\right]\left[l_{1}l_{2%
}\right]t_{125}}\!\left[\frac{i\left[6l_{1}\right]}{\left[6l_{2}\right]}\right%
]^{A},}$\cr$\displaystyle{T_{6}}$&$\displaystyle{{}\!=\!\frac{-i\left\langle 1%
l_{1}\right\rangle^{7}\left\langle 2l_{2}\right\rangle\left[5l_{1}\right]\left%
[6l_{2}\right]^{7}}{\left\langle 15\right\rangle\left\langle 5l_{1}\right%
\rangle\langle 1|P_{\!15l_{1}}|2]\langle 1|P_{\!15l_{1}}|l_{2}]\langle 5|P_{\!%
15l_{1}}|6]\langle l_{1}|P_{\!15l_{1}}|6]\left[26\right]\left[2l_{2}\right]t_{%
15l_{1}}}\!\left[\frac{-i\langle 1|P_{\!15l_{1}}|6]}{\left\langle 1l_{1}\right%
\rangle\left[6l_{2}\right]}\right]^{A},}$\cr$\displaystyle{T_{7}}$&$%
\displaystyle{{}\!=\!\frac{i\langle 1|P_{\!156}|l_{2}]^{7}(\left\langle 1l_{2}%
\right\rangle\left\langle 2l_{1}\right\rangle\langle 5|P_{\!156}|l_{1}]\left[2%
l_{2}\right]-\left\langle 1l_{1}\right\rangle\left\langle 2l_{2}\right\rangle%
\langle 5|P_{\!156}|l_{2}]\left[2l_{1}\right])\left[56\right]}{\left\langle 16%
\right\rangle^{2}\left\langle 56\right\rangle\langle 1|P_{\!156}|2]\langle 1|P%
_{\!156}|l_{1}]\langle 5|P_{\!156}|2]\langle 5|P_{\!156}|l_{1}]\langle 5|P_{\!%
156}|l_{2}]\left[2l_{1}\right]\left[2l_{2}\right]\left[l_{1}l_{2}\right]t_{156%
}}\!\left[\frac{i\langle 1|P_{\!156}|l_{1}]}{\langle 1|P_{\!156}|l_{2}]}\right%
]^{A},}$\cr$\displaystyle{T_{8}}$&$\displaystyle{{}\!=\!\frac{i\langle 1|P_{\!%
25l_{1}}|5]^{7}(\left\langle 15\right\rangle\left\langle 2l_{1}\right\rangle%
\langle l_{2}|P_{\!25l_{1}}|l_{1}]\left[25\right]-\left\langle 1l_{1}\right%
\rangle\left\langle 25\right\rangle\langle l_{2}|P_{\!25l_{1}}|5]\left[2l_{1}%
\right])\left[6l_{2}\right]}{\left\langle 16\right\rangle^{2}\left\langle 6l_{%
2}\right\rangle\langle 1|P_{\!25l_{1}}|2]\langle 1|P_{\!25l_{1}}|l_{1}]\langle
l%
_{2}|P_{\!25l_{1}}|2]\langle l_{2}|P_{\!25l_{1}}|5]\langle l_{2}|P_{\!25l_{1}}%
|l_{1}]\left[25\right]\left[2l_{1}\right]\left[5l_{1}\right]t_{25l_{1}}}\!%
\left[\frac{i\left\langle l_{2}1\right\rangle\left[5l_{1}\right]}{\langle 1|P_%
{\!25l_{1}}|5]}\right]^{A},}$\cr$\displaystyle{T_{9}}$&$\displaystyle{{}\!=\!%
\frac{i\left\langle 1l_{1}\right\rangle^{8}\left[5l_{2}\right]^{7}(\left%
\langle 1l_{2}\right\rangle\left\langle 25\right\rangle\langle l_{1}|P_{\!16l_%
{1}}|2]\left[5l_{2}\right]-\left\langle 12\right\rangle\left\langle 5l_{2}%
\right\rangle\langle l_{1}|P_{\!16l_{1}}|l_{2}]\left[25\right])\left[6l_{1}%
\right]}{\left\langle 16\right\rangle^{2}\hskip-5.0pt\left\langle 6l_{1}\right%
\rangle\hskip-2.0pt\langle 1|P_{\!16l_{1}}|2]\langle 1|P_{\!16l_{1}}|5]\langle
1%
|P_{\!16l_{1}}|l_{2}]\langle l_{1}|P_{\!16l_{1}}|2]\langle l_{1}|P_{\!16l_{1}}%
|5]\langle l_{1}|P_{\!16l_{1}}|l_{2}]\left[25\right]\hskip-1.0pt\left[2l_{2}%
\right]t_{16l_{1}}}\hskip-5.0pt\left[\!\hskip-1.0pt\frac{i\langle 1|P_{\!16l_{%
1}}|5]}{\left\langle l_{1}1\right\rangle\left[5l_{2}\right]}\!\right]^{A},}$%
\cr$\displaystyle{T_{10}}$&$\displaystyle{{}\!=\!\frac{i\left\langle 12\right%
\rangle^{8}\left[26\right]\left[5l_{2}\right]^{7}(\left\langle 1l_{2}\right%
\rangle\left\langle 5l_{1}\right\rangle\langle 2|P_{\!126}|l_{1}]\left[5l_{2}%
\right]-\left\langle 1l_{1}\right\rangle\left\langle 5l_{2}\right\rangle%
\langle 2|P_{\!126}|l_{2}]\left[5l_{1}\right])}{\left\langle 16\right\rangle^{%
2}\left\langle 26\right\rangle\langle 1|P_{\!126}|5]\langle 1|P_{\!126}|l_{1}]%
\langle 1|P_{\!126}|l_{2}]\langle 2|P_{\!126}|5]\langle 2|P_{\!126}|l_{1}]%
\langle 2|P_{\!126}|l_{2}]\left[5l_{1}\right]\left[l_{1}l_{2}\right]t_{126}}\!%
\left[\hskip-2.0pt\frac{i\left[5l_{1}\right]}{\left[5l_{2}\right]}\right]^{A},%
}$\cr$\displaystyle{T_{11}}$&$\displaystyle{{}\!=\!\frac{i\left\langle 15%
\right\rangle\left\langle 2l_{1}\right\rangle^{7}\left[56\right]^{8}(\left%
\langle 2l_{2}\right\rangle\langle l_{1}|P_{\!156}|5]\left[2l_{1}\right]\left[%
6l_{2}\right]-\left\langle 2l_{1}\right\rangle\langle l_{2}|P_{\!156}|5]\left[%
2l_{2}\right]\left[6l_{1}\right])}{\left\langle 2l_{2}\right\rangle\left%
\langle l_{1}l_{2}\right\rangle\langle 2|P_{\!156}|5]\langle 2|P_{\!156}|6]%
\langle l_{1}|P_{\!156}|5]\langle l_{1}|P_{\!156}|6]\langle l_{2}|P_{\!156}|5]%
\langle l_{2}|P_{\!156}|6]\left[15\right]\left[16\right]^{2}t_{156}}\!\left[%
\frac{i\left\langle 2l_{2}\right\rangle}{\left\langle 2l_{1}\right\rangle}%
\right]^{A},}$\cr$\displaystyle{T_{12}}$&$\displaystyle{{}\!=\hskip-2.0pt\frac%
{-i\left\langle 1l_{2}\right\rangle\left\langle 2l_{1}\right\rangle^{7}(\left%
\langle 25\right\rangle\langle l_{1}|P_{\!25l_{1}}|l_{2}]\left[2l_{1}\right]%
\left[56\right]+\left\langle 2l_{1}\right\rangle\langle 5|P_{\!25l_{1}}|l_{2}]%
\left[25\right]\left[6l_{1}\right])\left[6l_{2}\right]^{8}}{\left\langle 25%
\right\rangle\hskip-2.0pt\left\langle 5l_{1}\right\rangle\hskip-2.0pt\langle 2%
|P_{\!25l_{1}}|6]\langle 2|P_{\!25l_{1}}|l_{2}]\langle 5|P_{\!25l_{1}}|6]%
\langle 5|P_{\!25l_{1}}|l_{2}]\langle l_{1}|P_{\!25l_{1}}|6]\langle l_{1}|P_{%
\!25l_{1}}|l_{2}]\hskip-2.0pt\left[1l_{2}\right]\hskip-2.0pt\left[16\right]^{2%
}\hskip-3.0ptt_{25l_{1}}}\hskip-5.0pt\left[\frac{i\langle 2|P_{\!25l_{1}}|6]}{%
\left\langle l_{1}2\right\rangle\left[6l_{2}\right]}\!\right]^{A},}$\cr$%
\displaystyle{T_{13}}$&$\displaystyle{{}\!=\!\frac{-i\left\langle 1l_{1}\right%
\rangle\langle 2|P_{\!16l_{1}}|6]^{7}(\left\langle 25\right\rangle\langle l_{2%
}|P_{\!16l_{1}}|l_{1}]\left[26\right]\left[5l_{2}\right]+\left\langle 5l_{2}%
\right\rangle\langle 2|P_{\!16l_{1}}|l_{1}]\left[25\right]\left[6l_{2}\right])%
}{\left\langle 25\right\rangle\left\langle 2l_{2}\right\rangle\left\langle 5l_%
{2}\right\rangle\langle 2|P_{\!16l_{1}}|l_{1}]\langle 5|P_{\!16l_{1}}|6]%
\langle 5|P_{\!16l_{1}}|l_{1}]\langle l_{2}|P_{\!16l_{1}}|6]\langle l_{2}|P_{%
\!16l_{1}}|l_{1}]\left[16\right]^{2}\left[1l_{1}\right]t_{16l_{1}}}\!\left[\!%
\frac{i\left\langle l)22\right\rangle\left[6l_{1}\right]}{\langle 2|P_{\!16l_{%
1}}|6]}\!\right]^{A},}$\cr$\displaystyle{T_{14}}$&$\displaystyle{{}\!=\!\frac{%
i\left\langle 12\right\rangle\langle l_{1}|P_{\!126}|6]^{7}(\left\langle 5l_{2%
}\right\rangle\langle l_{1}|P_{\!126}|2]\left[5l_{1}\right]\left[6l_{2}\right]%
-\left\langle 5l_{1}\right\rangle\langle l_{2}|P_{\!126}|2]\left[5l_{2}\right]%
\left[6l_{1}\right])}{\left\langle 5l_{1}\right\rangle\left\langle 5l_{2}%
\right\rangle\left\langle l_{1}l_{2}\right\rangle\langle 5|P_{\!126}|2]\langle
5%
|P_{\!126}|6]\langle l_{1}|P_{\!126}|2]\langle l_{2}|P_{\!126}|2]\langle l_{2}%
|P_{\!126}|6]\left[12\right]\left[16\right]^{2}t_{126}}\!\left[\frac{i\langle l%
_{2}|P_{\!126}|6]}{\langle l_{1}|P_{\!126}|6]}\right]^{A},}$}}\,$$
(C.2)C.2( roman_C .2 )
where $A=4-2h$.
Appendix D Integral Functions
D.1 Box Functions
\Line(30,20)(70,20)
\Line(30,60)(70,60)
\Line(30,20)(30,60)
\Line(70,60)(70,20)
\Line(30,20)(15,5)
\Line(30,60)(15,75)
\Line(70,20)(85,5)
\Line(70,60)(85,75)
\Line(70,60)(70,75)
\Line(70,60)(85,60)
\Text(75,75)[c]$\bullet$\Text(85,65)[c]$\bullet$\Text(5,5)[l]2\Text(90,5)[l]1\Text(90,80)[l]${K_{4}}$\Text(5,80)[l]3\Text(40,40)[l]$I^{1m}_{4}$\Line(30,20)(70,20)
\Line(30,60)(70,60)
\Line(30,20)(30,60)
\Line(70,60)(70,20)
\Line(30,20)(15,5)
\Line(30,60)(15,75)
\Line(70,20)(85,5)
\Line(70,60)(85,75)
\Line(30,20)(30,5)
\Line(30,20)(15,20)
\Text(22,5)[c]$\bullet$\Text(15,12)[c]$\bullet$\Line(70,60)(70,75)
\Line(70,60)(85,60)
\Text(75,75)[c]$\bullet$\Text(85,65)[c]$\bullet$
\Text(-5,5)[l]${\rm K_{2}}$\Text(90,5)[l]1\Text(5,80)[l]3\Text(90,80)[l]${\rm K_{4}}$\Text(40,40)[l]$I^{2me}_{4}$\Line(30,20)(70,20)
\Line(30,60)(70,60)
\Line(30,20)(30,60)
\Line(70,60)(70,20)
\Line(30,20)(15,5)
\Line(30,60)(15,75)
\Line(70,20)(85,5)
\Line(70,60)(85,75)
\Line(30,60)(30,75)
\Line(30,60)(15,60)
\Text(15,65)[c]$\bullet$\Text(25,75)[c]$\bullet$\Line(70,60)(70,75)
\Line(70,60)(85,60)
\Text(75,75)[c]$\bullet$\Text(85,65)[c]$\bullet$\Text(5,5)[l]2\Text(90,5)[l]1\Text(90,80)[l]${\rm K_{4}}$\Text(-5,80)[l]${\rm K_{3}}$\Text(40,40)[l]$I^{2mh}_{4}$
\Line(30,20)(70,20)
\Line(30,60)(70,60)
\Line(30,20)(30,60)
\Line(70,60)(70,20)
\Line(30,20)(15,5)
\Line(30,60)(15,75)
\Line(70,20)(85,5)
\Line(70,60)(85,75)
\Line(30,20)(30,5)
\Line(30,20)(15,20)
\Text(22,5)[c]$\bullet$\Text(15,12)[c]$\bullet$\Line(30,60)(30,75)
\Line(30,60)(15,60)
\Text(15,65)[c]$\bullet$\Text(25,75)[c]$\bullet$\Line(70,60)(70,75)
\Line(70,60)(85,60)
\Text(75,75)[c]$\bullet$\Text(85,65)[c]$\bullet$\Text(-5,5)[l]${\rm K_{2}}$\Text(85,10)[l]${\rm 1}$\Text(90,75)[l]${\rm K_{4}}$\Text(-5,75)[l]${\rm K_{3}}$\Text(37,40)[l]$I^{3m}_{4}$\Line(30,20)(70,20)
\Line(30,60)(70,60)
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\Line(70,60)(85,60)
\Text(75,75)[c]$\bullet$\Text(85,65)[c]$\bullet$\Line(70,20)(70,5)
\Line(70,20)(85,20)
\Text(75,5)[c]$\bullet$\Text(85,15)[c]$\bullet$\Text(-5,5)[l]${\rm K_{2}}$\Text(90,5)[l]${\rm K_{1}}$\Text(90,75)[l]${\rm K_{4}}$\Text(-5,75)[l]${\rm K_{3}}$\Text(35,40)[l]$I^{4m}_{4}$
The scalar box integrals considered here have vanishing internal
masses, but may have up to four non-vanishing external masses. Again
by external masses we mean off-shell legs with $K^{2}\not=0$. These
integrals are defined and given in [61] (the
four-mass box was computed by Denner, Nierste, and
Scharf [62]) and are shown in the figures above.
The scalar box integral is,
$$I_{4}=-i\left(4\pi\right)^{2-\epsilon}\,\int{d^{4-2\epsilon}p\over\left(2\pi%
\right)^{4-2\epsilon}}\;{1\over p^{2}\left(p-K_{1}\right)^{2}\left(p-K_{1}-K_{%
2}\right)^{2}\left(p+K_{4}\right)^{2}}\;.$$
(D.1)D.1( roman_D .1 )
The external momentum arguments, $K_{i}$,
are sums of external momenta $k_{i}$.
In general the integrals are functions of the momentum invariants $K_{i}^{2}$ together with
$S\equiv(K_{1}+K_{2})^{2}$ and $T=(K_{2}+K_{3})^{2}$.
The no-mass box is, to
${\cal O}(\epsilon^{0})$ ,
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{I^{0\rm m}_{4}[1]}$&$%
\displaystyle{{}=\ r_{\Gamma}\,{1\over st}\biggl{\{}{2\over\epsilon^{2}}\Bigl{%
[}(-s)^{-\epsilon}+(-t)^{-\epsilon}\Bigr{]}-\ln^{2}\left({-s\over-t}\right)-%
\pi^{2}\biggr{\}}\ ,}$\cr$\displaystyle{}$}}\,$$
(D.2)D.2( roman_D .2 )
where $s=(k_{1}+k_{2})^{2}$ and $t=(k_{2}+k_{3})^{2}$ are the usual
Mandelstam variables. The factor $r_{\Gamma}$ arises within dimensional
regularisation and is,
$$r_{\Gamma}\;=\;{1\over(4\pi)^{2\,-\,\epsilon}}\,{\Gamma(1\,+\,\epsilon)\,%
\Gamma^{2}(1\,-\,\epsilon)\over\Gamma(1\,-\,2\,\epsilon)}\,.$$
(D.3)D.3( roman_D .3 )
This function appears only in four-point amplitudes with massless particles.
With the labelling of legs shown above, the scalar box integrals,
$I_{4}$, expanded to ${\cal O}(\epsilon^{0})$ for the different cases
reduce to,
$${\hskip-28.452756pt\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{I_{4}^{1{\rm
m%
}}}$&$\displaystyle{{}=\ {-2r_{\Gamma}\over ST}\biggl{\{}-{1\over\epsilon^{2}}%
\Bigl{[}(-S)^{-\epsilon}+(-T)^{-\epsilon}-(-K_{4}^{2})^{-\epsilon}\Bigr{]}}$%
\cr$\displaystyle{}$&$\displaystyle{{}\ +\mathop{\hbox{\rm Li}}\nolimits_{2}%
\left(1-{K_{4}^{2}\over S}\right)\ +\ \mathop{\hbox{\rm Li}}\nolimits_{2}\left%
(1-{K_{4}^{2}\over T}\right)\ +{1\over 2}\ln^{2}\left({S\over T}\right)\ +\ {%
\pi^{2}\over 6}\biggr{\}}\ ,}$\cr$\displaystyle{}$}}\,$$
(D.4)D.4( roman_D .4 )
$${\hskip-21.339567pt\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{I_{4}^{2{\rm
m%
}e}}$&$\displaystyle{{}=\ {-2r_{\Gamma}\over ST-K_{2}^{2}K_{4}^{2}}\biggl{\{}-%
{1\over\epsilon^{2}}\Bigl{[}(-S)^{-\epsilon}+(-T)^{-\epsilon}-(-K_{2}^{2})^{-%
\epsilon}-(-K_{4}^{2})^{-\epsilon}\Bigr{]}}$\cr$\displaystyle{}$&$%
\displaystyle{{}\ +\ \mathop{\hbox{\rm Li}}\nolimits_{2}\left(1-{K_{2}^{2}%
\over S}\right)\ +\ \mathop{\hbox{\rm Li}}\nolimits_{2}\left(1-{K_{2}^{2}\over
T%
}\right)\ +\ \mathop{\hbox{\rm Li}}\nolimits_{2}\left(1-{K_{4}^{2}\over S}%
\right)}$\cr$\displaystyle{}$&$\displaystyle{{}\ \ +\ \mathop{\hbox{\rm Li}}%
\nolimits_{2}\left(1-{K_{4}^{2}\over T}\right)-\ \mathop{\hbox{\rm Li}}%
\nolimits_{2}\left(1-{K_{2}^{2}K_{4}^{2}\over ST}\right)\ +\ {1\over 2}\ln^{2}%
\left({S\over T}\right)\biggr{\}}\ ,}$\cr$\displaystyle{}$}}\,$$
(D.5)D.5( roman_D .5 )
$${\hskip-8.535827pt\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{\!\!I_{4}^{2{%
\rm m}h}}$&$\displaystyle{{}=\ {-2r_{\Gamma}\over ST}\biggl{\{}-{1\over%
\epsilon^{2}}\Bigl{[}(-S)^{-\epsilon}+(-T)^{-\epsilon}-(-K_{3}^{2})^{-\epsilon%
}-(-K_{4}^{2})^{-\epsilon}\Bigr{]}}$\cr$\displaystyle{}$&$\displaystyle{{}\ -%
\ {1\over 2\epsilon^{2}}{(-K_{3}^{2})^{-\epsilon}(-K_{4}^{2})^{-\epsilon}\over%
(-S)^{-\epsilon}}\ +\ {1\over 2}\ln^{2}\left({S\over T}\right)\ +\ \mathop{%
\hbox{\rm Li}}\nolimits_{2}\left(1-{K_{3}^{2}\over T}\right)\ +\ \mathop{\hbox%
{\rm Li}}\nolimits_{2}\left(1-{K_{4}^{2}\over T}\right)\biggr{\}}\ ,}$\cr$%
\displaystyle{}$}}\,$$
(D.6)D.6( roman_D .6 )
$${\hskip-5.121496pt\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{I_{4}^{3{\rm m%
}}}$&$\displaystyle{{}=\ {-2r_{\Gamma}\over ST-K_{2}^{2}K_{4}^{2}}\biggl{\{}-{%
1\over\epsilon^{2}}\Bigl{[}(-S)^{-\epsilon}+(-T)^{-\epsilon}-(-K_{2}^{2})^{-%
\epsilon}-(-K_{3}^{2})^{-\epsilon}-(-K_{4}^{2})^{-\epsilon}\Bigr{]}}$\cr$%
\displaystyle{\hbox{}}$&$\displaystyle{{}\;\;\;\;\;\;\;\;\;\;\ -\ {1\over 2%
\epsilon^{2}}{(-K_{2}^{2})^{-\epsilon}(-K_{2}^{2})^{-\epsilon}\over(-T)^{-%
\epsilon}}\ -\ {1\over 2\epsilon^{2}}{(-K_{3}^{2})^{-\epsilon}(-K_{4}^{2})^{-%
\epsilon}\over(-T)^{-\epsilon}}\ +\ {1\over 2}\ln^{2}\left({S\over T}\right)}$%
\cr$\displaystyle{}$&$\displaystyle{{}\;\;\;\;\;\;\;\;\;\;+\ \mathop{\hbox{\rm
Li%
}}\nolimits_{2}\left(1-{K_{2}^{2}\over S}\right)\ +\ \mathop{\hbox{\rm Li}}%
\nolimits_{2}\left(1-{K_{4}^{2}\over T}\right)\ -\ \mathop{\hbox{\rm Li}}%
\nolimits_{2}\left(1-{K_{2}^{2}K_{4}^{2}\over ST}\right)\biggr{\}}\ ,}$\cr$%
\displaystyle{}$}}\,$$
(D.7)D.7( roman_D .7 )
$${\hskip-25.60748pt\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{I_{4}^{4{\rm m%
}}=}$&$\displaystyle{{}{-r_{\Gamma}\over S\;T\;\rho}\biggl{\{}-\mathop{\hbox{%
\rm Li}}\nolimits_{2}\left({\textstyle{1\over 2}}(1-\lambda_{1}+\lambda_{2}+%
\rho)\right)\ +\ \mathop{\hbox{\rm Li}}\nolimits_{2}\left({\textstyle{1\over 2%
}}(1-\lambda_{1}+\lambda_{2}-\rho)\right)}$\cr$\displaystyle{}$&$\displaystyle%
{{}\ -\ \mathop{\hbox{\rm Li}}\nolimits_{2}\left(\textstyle-{1\over 2\lambda_{%
1}}(1-\lambda_{1}-\lambda_{2}-\rho)\right)\ +\ \mathop{\hbox{\rm Li}}\nolimits%
_{2}\left(\textstyle-{1\over 2\lambda_{1}}(1-\lambda_{1}-\lambda_{2}+\rho)%
\right)}$\cr$\displaystyle{}$&$\displaystyle{{}\ -\ {1\over 2}\ln\left({%
\lambda_{1}\over\lambda_{2}^{2}}\right)\ln\left({1+\lambda_{1}-\lambda_{2}+%
\rho\over 1+\lambda_{1}-\lambda_{2}-\rho}\right)\biggr{\}}\ ,}$\cr$%
\displaystyle{}$}}\,$$
(D.8)D.8( roman_D .8 )
where,
$$\rho\ \equiv\ \sqrt{1-2\lambda_{1}-2\lambda_{2}+\lambda_{1}^{2}-2\lambda_{1}%
\lambda_{2}+\lambda_{2}^{2}}\ ,$$
(D.9)D.9( roman_D .9 )
and,
$$\lambda_{1}={K_{2}^{2}\;K_{4}^{2}\over S\;T}\;,\hskip 42.679134pt\lambda_{2}={%
K_{1}^{2}\;K_{3}^{2}\over S\;T}\ .$$
(D.10)D.10( roman_D .10 )
When checking the soft divergences of the seven-point amplitude we
need the $1/\epsilon$ singularities
arising from soft singularities in the loop integration.
For the boxes relevant to the seven-point amplitude these are,
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{I^{abc\{defg\}}|_{1/\epsilon}}%
$&$\displaystyle{{}=-{2\over s_{ab}s_{bc}(4\pi)^{2}}\Bigl{[}{\ln(-s_{ab})+\ln(%
-s_{bc})-\ln(-t_{abc})\over\epsilon}\Bigr{]}\,,}$\cr$\displaystyle{I^{a(bc)(%
def)g}|_{1/\epsilon}}$&$\displaystyle{{}=-{2\over s_{ag}t_{abc}(4\pi)^{2}}%
\Bigl{[}{\ln(-s_{ag})+2\ln(-t_{abc})-\ln(-s_{bc})-\ln(-t_{def})\over 2\epsilon%
}\Bigr{]}\,,}$\cr$\displaystyle{I^{a(bc)d(efg)}|_{1/\epsilon}}$&$\displaystyle%
{{}=-{2\over(t_{abc}t_{bcd}-s_{bc}t_{efg})(4\pi)^{2}}\Bigl{[}{\ln(-t_{abc})+%
\ln(-t_{bcd})-\ln(-s_{bc})-\ln(-t_{efg})\over\epsilon}\Bigr{]},}$\cr$%
\displaystyle{I^{a(bc)(de)(fg)}|_{1/\epsilon}}$&$\displaystyle{{}=-{2\over(t_{%
abc}t_{fga}-s_{bc}s_{fg})(4\pi)^{2}}\Bigl{[}{\ln(-t_{abc})+\ln(-t_{fga})-\ln(-%
s_{bc})-\ln(-s_{fg})\over 2\epsilon})\Bigr{]}.}$\cr$\displaystyle{}$}}\,$$
(D.11)D.11( roman_D .11 )
D.2 Triangle and Bubble integral Functions
Triangle integral functions may have one, two or three massless legs:
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\Line(-30,30)(-70,40)
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The one-mass triangle depends only on the momentum invariant of the
massive leg,
$$I_{3}^{1\rm m}={r_{\Gamma}\over\epsilon^{2}}(-K_{1}^{2})^{-1-\epsilon}\ .$$
(D.12)D.12( roman_D .12 )
The next integral function is
the two-mass triangle integral,
$$I_{3}^{2\rm m}={r_{\Gamma}\over\epsilon^{2}}{(-K_{1}^{2})^{-\epsilon}-(-K_{2}^%
{2})^{-\epsilon}\over(-K_{1}^{2})-(-K_{2}^{2})}\ .$$
(D.13)D.13( roman_D .13 )
Note that the one and two mass triangles are linear combinations of
the set of functions,
$$G(-K^{2})=r_{\Gamma}{(-K^{2})^{-\epsilon}\over\epsilon^{2}}\;,$$
(D.14)D.14( roman_D .14 )
with,
$$I^{1m}_{3}=G(-K_{1}^{2})\;\;,\;\;I^{2m}_{3}={1\over(-K_{1}^{2})-(-K_{2}^{2})}%
\left(G(-K_{1}^{2})-G(-K_{2}^{2})\right).$$
(D.15)D.15( roman_D .15 )
The $G(-K^{2})$ are labelled by the independent momentum invariants
$K^{2}$ and in fact form an independent basis of functions, unlike the
one and two-mass triangles which are not all independent. For
example, for six-point kinematics there are only twenty-five
independent options for $K^{2}$ corresponding to 15 independent
$s_{ij}$’s and 10 independent $t_{ijk}$’s, whereas there are 15
one-mass triangles and 60 two-mass triangles.
The final scalar triangle is the three-mass integral function.
The evaluation of this integral is more
involved,
and can be obtained from [63, 61],
$$I_{3}^{3\rm m}=\ {i\over\sqrt{\Delta_{3}}}\sum_{j=1}^{3}\left[\mathop{\hbox{%
\rm Li}}\nolimits_{2}\left(-\left({1+i\delta_{j}\over 1-i\delta_{j}}\right)%
\right)-\mathop{\hbox{\rm Li}}\nolimits_{2}\left(-\left({1-i\delta_{j}\over 1+%
i\delta_{j}}\right)\right)\right]\ +\ {\cal O}(\epsilon),$$
(D.16)D.16( roman_D .16 )
where,
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{\delta_{1}}$&$\displaystyle{{}%
={K_{1}^{2}-K_{2}^{2}-K_{3}^{2}\over\sqrt{\Delta_{3}}}\;,}$\cr$\displaystyle{%
\delta_{2}}$&$\displaystyle{{}={-K_{1}^{2}+K_{2}^{2}-K_{3}^{2}\over\sqrt{%
\Delta_{3}}}\;,}$\cr$\displaystyle{\delta_{3}}$&$\displaystyle{{}={-K_{1}^{2}-%
K_{2}^{2}+K_{3}^{2}\over\sqrt{\Delta_{3}}}\;,}$\cr$\displaystyle{}$}}\,$$
(D.17)D.17( roman_D .17 )
and
$$\Delta_{3}\equiv-(K_{1}^{2})^{2}-(K_{2}^{2})^{2}-(K_{3}^{2})^{2}+2(K_{1}^{2}K_%
{2}^{2}+K_{3}^{2}K_{1}^{2}+K_{2}^{2}K_{3}^{2}).$$
(D.18)D.18( roman_D .18 )
Finally, the bubble integral is,
$${\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{I_{2}(K^{2})}$&$\displaystyle{%
{}={r_{\Gamma}\over\epsilon(1-2\epsilon)}(-K^{2})^{-\epsilon}.}$\cr$%
\displaystyle{}$}}\,$$
(D.19)D.19( roman_D .19 )
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MEDL and MEDLA: Methods for Assessment of Scaling by Medians of Log-Squared Nondecimated Wavelet Coefficients
Minkyoung Kang and Brani Vidakovic
H. Milton Stewart School of Industrial & Systems Engineering, Georgia Institute of Technology, 765 Ferst Drive NW, Atlanta, Georgia 30332
Abstract
High-frequency measurements and images acquired from various sources in the real world often possess a degree of self-similarity and inherent regular scaling. When data look like a noise,
the scaling exponent may be the only informative feature that summarizes such data. Methods for the assessment of self-similarity by estimating Hurst exponent
often involve analysis of rate of decay in a spectrum defined in various multiresolution domains. When this spectrum is calculated
using discrete non-decimated wavelet transforms, due to increased autocorrelation in wavelet coefficients, the estimators of $H$ show increased bias compared to the estimators that use traditional orthogonal transforms.
At the same time, non-decimated transforms have a number of advantages when employed for calculation of wavelet spectra and estimation of Hurst exponents: the variance of the estimator is smaller, input signals and images could be of arbitrary size, and due to the shift-invariance, the local scaling can be assessed as well.
We propose two methods based on robust estimation and resampling that alleviate the effect of increased autocorrelation while maintaining all advantages of non-decimated wavelet transforms. The proposed methods extend the approaches in existing literature where the logarithmic transformation and pairing of wavelet coefficients are used for lowering the bias.
In a simulation study we use fractional Brownian motions with a range of theoretical Hurst exponents.
For such signals for which “true” $H$ is known, we demonstrate bias reduction and overall
reduction of the mean-squared error by the two proposed estimators.
For fractional Brownian motions, both proposed methods yield
estimators of $H$ that are asymptotically normal and unbiased.
keywords:
Non-decimated wavelet transform, scaling, and Hurst exponent
††journal: Journal of LaTeX Templates
1 Introduction
At first glance, data that scale look like noisy observations, and often
the large scale features
(basic descriptive statistics, trends, smoothed functional estimates, etc.)
carry no useful information.
For example, the pupil diameter in humans fluctuates at a high frequency
(hundreds of Hz), and prolonged monitoring leads to
massive data sets. Researchers found that the high-frequency dynamic of change
in the diameter is informative of eye pathologies, e.g., macular degeneration, Moloney et al. (2006).
Yet, the trends and traditional summaries of the data
are clinically irrelevant, for the magnitude of the diameter depends on the ambient light,
and not on the inherent eye pathology.
Our interest focuses on the analysis of self-similar objects.
Formally, a deterministic function $f(\mathbf{t})$ of a $d$-dimensional argument $\mathbf{t}$ is said to be
self-similar if $f(\lambda\mathbf{t})=\lambda^{-H}f(\lambda\mathbf{t}),$ for some choice of the exponent
$H$, and for all dilation factors $\lambda$. The notion of
self-similarity has been extended to random processes. Specifically, a
stochastic process $\{X(\mathbf{t}),\ \mathbf{t}\in R^{d}\}$ is self-similar with
scaling exponent (or Hurst exponent) $H$ if, for any
$\lambda\in R^{+}$,
$$\displaystyle X(\lambda\mathbf{t})\stackrel{{\scriptstyle d}}{{=}}\lambda^{H}X%
(\mathbf{t}),$$
(1)
where the relation “$\stackrel{{\scriptstyle d}}{{=}}$” is understood as the equality in all finite dimensional distributions.
In this paper, we are concerned with a precise estimation
of scaling exponent $H$ in one-dimensional setting. The results can be readily extended to self-similar objects of arbitrary number of dimensions.
A number of estimation methods for $H$ exist, including:
re-scaled range calculation ($R/S$), Fourier-spectra methods,
variance plots, quadratic variations, zero-level crossings, etc.
For a comprehensive
description, see Beran (1994), Doukhan et al. (2003), and Abry et al. (2013). Wavelet transforms are
especially suitable for modeling self-similar phenomena, as is reflected by vibrant research.
An overview is provided in Abry et al. (2000a).
If processes possess a stochastic structure (e.g. Gaussianity, stationary increments),
the scaling exponent $H$ becomes a parameter in a well-defined statistical model
and can be estimated as such.
Fractional Brownian motion (fBm)
is important and
well understood model for data that scale. Its importance follows from the fact that
fBm is a unique Gaussian process with stationary increments that
is self-similar, in the sense of (1).
A fBm has a
(pseudo)-spectrum of the form $S(\omega)\propto|\omega|^{-(2H+1)}$, and consequently the log-magnitudes
of detail coefficients at different resolutions in a wavelet decomposition exhibit a linear
relationship. Using non-decimated wavelet domains to leverage on this linearity constitutes the staple of this paper.
Each decomposition level in nondecimated wavelet transform (NDWT) contains the same number
of coefficients as the size of the original signal. This redundancy contributes to the accuracy of estimators of $H$. However, reducing the bias induced by level-wise correlation among the redundant
coefficients becomes an important issue.
The two estimators we propose are based on the so-called “logarithm-first” approach, connecting Hurst exponent with a robust location
and resampling techniques.
The rest of the paper consists of three additional sections and an appendix. Section 2 provides background of wavelet transforms as well as the properties of resulting wavelet coefficients. Section 3 presents distributional results on which the proposed methods are based. Section 4 provides the simulation results and compares the estimation performance of the proposed methods to some standardly used methods. The final Section is reserved for concluding remarks. Appendix contains all technical details for the results
presented in Section 3.
2 Orthogonal and non-decimated wavelet transforms
Discrete signals from an acquisition domain can be mapped to the wavelet domain in multiple ways. We overview two versions of discrete wavelet transform: orthogonal wavelet transform (DWT) and non-decimated wavelet transform (NDWT). We also describe algorithmic procedures in performing two versions of wavelet transform and obtaining the wavelet coefficients. Here we focus on functional representations of wavelet transform which is more critical for the subsequent derivations. Interested readers can refer to Nason and Silverman (1995), Vidakovic (1999), and Percival and Walden (2006) for alternative definitions.
Any square-integrable $L_{2}(\mathbb{R})$ function $f(x)$ can be represented in the wavelet domain as
$$\displaystyle f(x)=\sum_{k}c_{J_{0},k}\phi_{J_{0},k}(x)+\sum_{j\geq J_{0}}^{%
\infty}\sum_{k}d_{j,k}\psi_{j,k}(x),$$
where $c_{J_{0},k}$ indicates coarse coefficients, $d_{j,k}$ detail coefficients, $\phi_{J_{0},k}(x)$ scaling functions, and $\psi_{jk}(x)$ wavelet functions. We use different decomposing atom functions, as scaling and wavelet functions, depending on a version of wavelet transform.
For DWT, the atoms are
$$\displaystyle\phi_{J_{0},k}(x)$$
$$\displaystyle=$$
$$\displaystyle 2^{J_{0}/2}\phi(2^{J_{0}}x-k)$$
$$\displaystyle\psi_{jk}(x)$$
$$\displaystyle=$$
$$\displaystyle 2^{j/2}\psi(2^{j}x-k),$$
where $x\in\mathbb{R}$, $j$ is a resolution level, $J_{0}$ is the coarsest resolution level, and $k$ is the location of an atom.
For NDWT, atoms are
$$\displaystyle\phi_{J_{0},k}(x)$$
$$\displaystyle=$$
$$\displaystyle 2^{J_{0}/2}\phi(2^{J_{0}}(x-k))$$
$$\displaystyle\psi_{jk}(x)$$
$$\displaystyle=$$
$$\displaystyle 2^{j/2}\psi(2^{j}(x-k)).$$
Notice that atoms in NDWT have a constant location shift $k$ at all levels, which yields the maximal sampling rate at each level. Two types of coefficients, $c_{J_{0},k}$ and $d_{j,k}$, capture coarse and detail fluctuations of an input signal, respectively. These are obtained as
$$\displaystyle c_{J_{0},k}$$
$$\displaystyle=$$
$$\displaystyle\langle f(x),\phi_{J_{0},k}\rangle$$
$$\displaystyle d_{jk}$$
$$\displaystyle=$$
$$\displaystyle\langle f(x),\psi_{jk}\rangle.$$
In a $p$-depth decomposition of an input signal of size $m$, a NDWT yields $m\times(p+1)$ wavelet coefficients, while DWT yields $m$ wavelet coefficients independent of $p$. The redundant transform NDWT decreases the variance of the scaling estimators, but at the same time, increases the correlations among wavelet coefficients.
Since the estimators
of scaling are based on the second order properties of wavelet coefficients, the NDWT-based estimators can be biased.
Figure 1 illustrates the autocorrelation within wavelet coefficients in the level $J-4$
(the level of finest detail is $J-1$, so $J-4$ is 4th “most detailed” level) in DWT and NDWT. Haar wavelet was used on a Brownian motion path of size $2^{11}$. As we noted before, the coefficients from the NDWT are highly correlated while such correlation is not strong among the DWT coefficients.
The two methods introduced in the following section reduce the effect of correlation among the coefficients,
while maintaining redundancy and invariance as desirable threads of NDWT.
3 MEDL and MEDLA Methods
We start by an overview of properties of wavelet coefficients and a brief literature overview methods in literature based on which we develop the proposed methods.
For defining a wavelet spectrum, and subsequently, for estimating $H$ only detail wavelet coefficients are used. When an fBm with Hurst exponent $H$ is mapped to the wavelet domain by DWT, the resulting detail wavelet coefficients satisfy the following properties (Tewfik and Kim, 1992; Abry et al., 1995; Flandrin, 1992):
(i)
$d_{j}$, a detail wavelet coefficient at level $j$, follows the Gaussian distribution with mean 0 and variance $\sigma_{0}^{2}2^{-j(2H+1)}$, where $\sigma_{0}^{2}$ is the variance of a detail coefficient at level 0,
(ii)
a sequence of wavelet coefficients from level $j$ is stationary, and
(iii)
the covariance between two coefficients from any level of detail decreases exponentially as the distance between them increases; the rate of the decrease depends additionally on the number of vanishing moments of the decomposing wavelet.
From property (i), the relationship between detail wavelet coefficients and Hurst exponent $H$ is
$$\displaystyle\log_{2}\mathbb{E}\{d^{2}_{j}\}=-j(2H+1)+2\log_{2}\sigma_{0}.$$
Abry et al. (2000b) calculate sample variance of wavelet coefficients to estimate $\mathbb{E}\{d^{2}_{j}\}$ assuming i.i.d. Gaussianity of coefficients at level $j$. Frequently, a squared wavelet coefficient is
referred as an “energy.”
Empirically, we look at the levelwise average of squared coefficients (energies),
$$\displaystyle\overline{d^{2}_{j}}=\frac{1}{n_{j}}\sum_{i=1}^{n_{j}}d^{2}_{j,k},$$
where $n_{j}$ is the number of wavelet coefficients at level $j$.
The relationship between average energy $\overline{d^{2}_{j}}$ and $H$ is
$$\displaystyle\log_{2}\overline{d^{2}_{j}}\overset{d}{\approx}-(2H+1)j-\log_{2}%
C-\log\chi^{2}_{n_{j}}/\log 2,$$
where $\overset{d}{\approx}$ indicates approximate equality in distribution, $\chi^{2}_{n_{j}}$ follows a chi-square distribution with ${n_{j}}$ degrees of freedom, and $C$ is a constant. The method of Abry et al. (2000b) is affected by the non-normality of $\log_{2}\overline{d^{2}_{j}}$ and correlation among detail wavelet coefficients, which results in biases of weighted least squares estimates. To reduce the bias, Soltani et al. (2004) defined “mid-energies,” as
$$\displaystyle D_{j,k}=\frac{d_{j,k}^{2}+d_{j,k+n_{j}/2}^{2}}{2},k=1,...,n_{j}/2.$$
According to this approach, each multiresolution level is split on two equal parts and corresponding coefficients from each part are paired,
squared, and averaged. This produces a
quasi-decorrelation effect.
Soltani et al. (2004) show that level-wise averages of $\log_{2}D_{j,k}$ are
asymptotically normal with the mean $-(2H+1)j+C,$ which is used to estimate $H$ by regression.
The estimators in Soltani et al. (2004) consistently outperform the estimators
that use log-average energies, under various settings. Shen et al. (2007) show that the method of Soltani et al. (2004) yields more accurate estimators since it takes the logarithm
of a mid-energy, and then averages. Moreover, averaging logged squared wavelet coefficients, rather than taking logarithm of averaged squared wavelet coefficients, is theoretically justified and this approach will be pursued in this paper. For both proposed methods, MEDL and MEDLA, we first take the logarithm of a squared wavelet coefficient
or an average of two squared wavelet coefficients, then derive the distribution of such logarithms under the assumption of independence. Next, we use the median of the derived distribution instead of the mean.
The medians are more robust to potential outliers that can occur when logarithmic transform of a squared wavelet coefficient is taken and the magnitude of coefficient is close to zero. This
numerical instability may increase the bias and variance of sample means. However, since the logarithms are monotone, the variability of the sample medians will not be affected.
The first proposed method is based on the relationship between the median of the logarithm of squared wavelet coefficients and the Hurst exponent. We use acronym “MEDL” to refer to this method. In MEDL, the logarithmic transform reduces the autocorrelation, while the number of coefficients remains the same. The second method derives the relationship between the median of the logarithm of average of two squared wavelet coefficients and the Hurst exponent. We use acronym “MEDLA” to refer to this method. The MEDLA method is similar in concept to approach of Soltani et al. (2004) who paired
and averaged energies prior to taking logarithm. Then the mean of logarithms was connected to $H$.
Instead, we repeatedly sample with replacement $m$ random pairs keeping distance between them at least $q_{j}$.
Then, as in Soltani et al. (2004) we find the logarithm of pair’s average and connect the Hurst exponent with the median of the logarithms.
As we relax the constraints on the distance between energies in each pair, we obtain a larger amount of distinct samples and selecting only $N$ samples out of such sample population further reduces the correlation.
To illustrate the decorrelation effects of the proposed methods, in Figure 2, we compare the autocorrelation present in variables that are averaged: means of $d_{jk}^{2}$ for traditional method, means of $\log_{2}\bigg{[}(d_{jk}^{2}+d_{j,k+m/2}^{2})/2\bigg{]}$ for Soltani-like method, medians of $\log d_{jk}^{2}$ for MEDL, and
medians of sampled $\log\bigg{[}(d_{jk_{1}}^{2}+d_{jk_{2}}^{2})/2\bigg{]}$ for MEDLA method. The two default methods exhibit higher amount of autocorrelation that decreases at a slower rate. The MEDLA shows substantial reduction in correlation.
For formal distributional assessment of the two proposed methods,
we start with an arbitrary wavelet coefficient from decomposition level $j$ at location $k$, $d_{jk}$, resulting from a non-decimated wavelet transform of a one-dimensional fBm $B_{H}(\omega,t),t\in\mathbb{R}$,
$$\displaystyle d_{j}=\int_{\mathbb{R}}B_{H}(\omega,t)\psi_{jk}(t)dt,\text{for %
some fixed }k.$$
As Flandrin (1992) showed, the distribution of a single wavelet coefficient is
$$\displaystyle d_{j}\stackrel{{\scriptstyle d}}{{=}}2^{-(H+1/2)j}\sigma Z,$$
(2)
where $Z$ follows a standard normal distribution, and $\sigma^{2}$ is the variance of wavelet coefficients at level 0. We will use (2) repeatedly for the derivations that follow.
3.1 MEDL Method
For the median of the logarithm of squared wavelet coefficients (MEDL) method, we derive the relationship between the median of the logarithm on an arbitrary squared wavelet coefficient from decomposition level $j$ and Hurst exponent $H$. The following theorem serves as a basis for the MEDL estimator:
Theorem 3.1.
Let $y_{j}^{*}$ be the median of $\log d_{j}^{2}$, where $d_{j}$ is an arbitrary wavelet coefficient from level $j$ in a NDWT of a fBm with Hurst exponent $H$.
Then, the population median is
$$\displaystyle y^{*}_{j}=-\log 2\,(2H+1)j+C,$$
(3)
where $C$ is a constant independent of $j$.
The Hurst exponent can be estimated as
$$\displaystyle\widehat{H}=-\frac{\widehat{\beta}}{2\log 2}-\frac{1}{2},$$
(4)
where $\widehat{\beta}$ is the slope in ordinary least squares (OLS) linear regression on pairs $(j,{\hat{y}}_{j}^{*}),$
and ${\hat{y}^{*}_{j}}$ is the sample median.
The proof of Theorem 3.1 is deferred to Appendix A. We estimate $y_{j}^{*}$ by taking sample median of logged energies at each level. The use of OLS linear regression is justified by the fact that variances of the sample medians ${\hat{y}^{*}_{j}}$ are constant in $j$, that is,
Lemma 3.1.
The variance of sample median ${\hat{y}^{*}_{j}}$ at level $j$ is approximately
$$\displaystyle\frac{\pi e^{Q}}{2NQ},$$
where $N$ is the sample size and $Q=\left(\Phi^{-1}(3/4)\right)^{2}$.
The theorem is stating that the logarithm acts as a variance stabilizing operator;
the variance of the sample median is independent of level $j$, and ordinary regression to find slope $\beta$ in Theorem 3.1 is fully justified.
Note that the use of OLS regression is not adequate in DWT; the weighted regression is needed to account for
levelwise heteroscedasticity.
The levelwise variance is approximately $5.4418/N,$ independent of $H$ and $\sigma^{2}.$
The proof of Theorem 3.1 is deferred to Appendix A. In addition, for $\widehat{H}$ the normal approximation applies:
Theorem 3.2.
The MEDL estimator $\widehat{H}$ follows the asymptotic normal distribution
$$\displaystyle\widehat{H}\overset{approx}{\sim}{\cal N}\left(H,\frac{3A}{Nm(m^{%
2}-1)(\log 2)^{2}}\right),$$
where $A=\pi e^{Q}/(2Q)\cong 5.4418$, $N$ is the sample size, and $m$ is the number of levels used in the spectrum.
The proof of Theorem 3.2 is deferred to Appendix A. To illustrate Theorem 3.2, we perform an NDWT of depth 10 on simulated fBm’s with $H=0.3,0.5,$ and $0.7$. We use resulting wavelet coefficients from levels $J-7$ to $J-2$
inclusive (i.e., six levels) to estimate $H$ with MEDL. Following Theorem 3.2, $\widehat{H}$ of MEDL in the simulation follows a normal distribution with mean $H$ and variance $7.9007\times 10^{-5}$, which is illustrated in Figure 3.
3.2 MEDLA Method
For the median of the logarithm of averaged squared wavelet coefficients (MEDLA) method, we derive the relationship between logarithm of an average of two energies and Hurst exponent $H$. Soltani et al. (2004) proposed a method that quasi-decorrelates wavelet coefficients by splitting all wavelet coefficients from one level into left and right sections and pairing every coefficient in the left section with its counterpart in the right section, maintaining an equal distance to its pair (i.e., members in each pair are $m/2$ apart when $m$ is the number of wavelet coefficients on that level). Then, Soltani et al. (2004) averaged every pair of energies and took logarithm of each average. We follow similar idea except that instead of fixing the combinations of pairs, which amounts to $m/2$ pairs in Soltani et al. (2004), we randomly sample with replacement $m$ pairs whose members are at least $q_{j}$ apart. Based on sample autocorrelation graphs, we define $q_{j}=2^{J-j}$ that decrease with level $j$ because the finer the subspace (i.e., larger $j$), the lower the correlation among wavelet coefficients. Then, we propose an estimator of $H$ based on the following result.
Theorem 3.3.
Let $d_{jk_{1}}$ and $d_{jk_{2}}$ be two wavelet coefficients from level $j$, at positions $k_{1}$ and $k_{2}$, respectively, from a NDWT of a fBm
with Hurst exponent $H$. Assume that $|k_{1}-k_{2}|>q_{j},$ where $q_{j}$ is the minimum separation distance
that depends on level $j$ and selected wavelet base.
Let $y_{j}^{*}$ be the median of $\log\bigg{[}\frac{d_{jk_{1}}^{2}+d_{jk_{2}}^{2}}{2}\bigg{]}$. Then,
as in Theorem 3.1, results
(3) and (4) hold.
The proof of Theorem 3.3 is deferred to Appendix B. To estimate $y_{j}^{*}$, we first repeatedly sample $m$ pairs of wavelet coefficients with replacement from all pairs that are at least $q_{j}$ apart. Then, we take logarithm of pair’s average energy and take the median. As in Theorem 3.1, the variances of sample medians ${\hat{y}^{*}_{j}}$ are free of $j$.
Lemma 3.2.
The variance of the sample median $\hat{y^{*}_{j}}$ at level $j$ is approximated by
$$\displaystyle\frac{1}{N(\log 2)^{2}},$$
where $N$ is the sample size.
The proof is straightforward and given in Appendix B. Thus, the variance of ${\hat{y}^{*}_{j}}$ is constant over levels. We find that MEDLA estimator of $H$ indeed follows a approximately normal distribution with a mean and a variance given in the following theorem.
Theorem 3.4.
The estimator $\widehat{H}$ of MEDLA follows the asymptotic normal distribution
$$\displaystyle\widehat{H}\overset{approx}{\sim}{\cal N}\left(H,\frac{3}{Nm(m^{2%
}-1)(\log 2)^{4}}\right),$$
where $N$ is the sample size, and $m$ is the number of levels used in the spectrum.
The proof of Theorem 3.4 is deferred to Appendix B. To illustrate Theorem 3.4, we use the same wavelet coefficients from the simulation in section 3.1. Following Theorem 3.4, $\widehat{H}$ of MEDLA in the simulation follows an approximate normal distribution with mean $H$ and variance $7.9007\times 10^{-5}$, which is shown in Figure 4.
4 Simulations
Next, we assess the performance of MeDL and MEDLA in estimation of Hurst exponent,
We simulate three sets of three hundred one-dimensional fractional Brownian motion (1-D fBm) paths of size $2^{11}$ with Hurst exponents 0.3, 0.5, and 0.7 respectively. Then, we perform an NDWT of depth 10 with a Haar wavelet on each simulated signal and obtain wavelet coefficients to which we apply MEDL and MEDLA.
For all methods and estimations, we use wavelet coefficients from levels $J-7$ to $J-2$ in the regression. We compare the estimation performance of the proposed methods to two standard methods: a method of Veitch and Abry (1999) and a method of Soltani et al. (2004), both in the context of NDWT. We present the estimation performance in terms of mean, variance, bias-squared, and mean squared error, based on 300 simulations for each case. Table 1 and Figure 5 indicate that as $H$ increases, the proposed methods outperform the standard methods. For smaller $H$, the estimation performance of all methods is comparable.
5 Conclusions
We proposed two methods for robust estimation of Hurst exponent in one- and two-dimensional signals that scale. Unlike the standard methods, the proposed methods are based on NDWT. The motivation for using NDWT was its
redundancy and time-invariance. However, the redundancy, which was useful for
the stability of estimation, increases autocorrelations among the wavelet coefficients.
The proposed methods lower the present autocorrelation by (i) taking logarithm of the squared wavelet coefficients
prior to averaging, (ii) relating the Hurst exponent to the median of the model distribution, rather than the mean,
and (iii) resampling the coefficients.
The methods are compared to standard approaches and give estimators with smaller MSE for a range of
input conditions.
Instead of medians in (ii) we could employ any other quantile; the methodology is equivalent and will differ for
the intercept and variance in the regressions.
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Appendix
A. Derivation of MEDL
Proof of Theorem 3.1
A single wavelet coefficient in a non-decimated wavelet transform of fBm(H) is normally distributed, with variance depending on
its level $j$,
$$\displaystyle d_{j}$$
$$\displaystyle\overset{d}{=}$$
$$\displaystyle{\cal N}(0,2^{-(2H+1)j}\sigma^{2}).$$
Its rescaled energy is $\chi^{2}$ with one degree of freedom,
$$\displaystyle\delta=\frac{2^{(2H+1)j}}{\sigma^{2}}d_{j}^{2}\stackrel{{%
\scriptstyle d}}{{=}}\chi_{1}^{2},$$
with density
$$\displaystyle\frac{\delta^{1/2-1}(\frac{1}{2})^{1/2}}{\Gamma(\frac{1}{2})}e^{-%
\delta/2}=\frac{e^{-\delta/2}}{\sqrt{2\delta}\Gamma(\frac{1}{2})}.$$
The pdf of $d_{j}^{2}$ is
$$\displaystyle f(d_{j}^{2})=\frac{e^{-c_{j}d_{j}^{2}/2}}{\sqrt{2c_{j}d_{j}^{2}}%
\Gamma(1/2)}c_{j},$$
where $c_{j}=\frac{2^{(2H+1)j}}{\sigma^{2}}$.
Let $y=\log d_{j}^{2}$, then $d_{j}^{2}=e^{y}$ and $\frac{\partial d_{j}^{2}}{\partial y}=e^{y}$. The pdf of $y$ is
$$\displaystyle f(y)$$
$$\displaystyle=$$
$$\displaystyle\frac{c_{j}e^{\frac{-c_{j}e^{y}}{2}}}{\sqrt{2c_{j}e^{y}}\Gamma(1/%
2)}e^{y}=\frac{\sqrt{c_{j}}e^{-\frac{c_{j}e^{y}}{2}}e^{y/2}}{\sqrt{2}\Gamma(1/%
2)}=\sqrt{\frac{c_{j}}{2\pi}}e^{-\frac{c_{j}e^{y}}{2}}e^{y/2},$$
The cdf of $y$ is
$$\displaystyle F(y)=\int_{-\infty}^{y}f(t)dt=2\Phi\left(\sqrt{c_{j}}e^{y/2}%
\right)-1,$$
where $\Phi$ is the cdf of standard normal distribution.
Let $y^{*}$ be the median of the distribution of $y$. We obtain the expression of $y*$ by solving
$F(y^{*})=1/2$. This results in
$$\displaystyle y^{*}=2\log\left[\frac{1}{\sqrt{c_{j}}}\Phi^{-1}(3/4)\right]$$
From this equation, we can find a link between $y^{*}$ and the Hurst exponent $H$ by substituting $c_{j},$
$$\displaystyle y^{*}$$
$$\displaystyle=$$
$$\displaystyle 2\log[\Phi^{-1}(3/4)]-\log c_{j}$$
$$\displaystyle=$$
$$\displaystyle-\log 2\,(2H+1)j+\log\sigma^{2}+2\log[\Phi^{-1}(3/4)]$$
$$\displaystyle=$$
$$\displaystyle-\log 2\,(2H+1)j+C,$$
where $C$ is a constant independent on the level $j$.
Proof of Lemma 3.1
An approximation of variance of sample median ${\hat{y}}_{j}^{*}$ is obtained
using normal approximation to a quantile of absolutely continuous distributions,
$$\displaystyle Var({\hat{y}}_{j}^{*})\approx\frac{1}{4N(f(y^{*}_{j}))^{2}}.$$
After substituting the expression for $y^{*}$
we obtain Lemma 3.1
$$\displaystyle Var(\hat{y}_{j}^{*})\approx\frac{\pi e^{Q}}{2NQ},\leavevmode%
\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ %
\leavevmode\nobreak\ Q=\left[\Phi^{-1}(3/4)\right]^{2}\approx 0.4549.$$
Thus the variance of the sample median is approximately $5.4418/N.$
Proof of Theorem 3.2
An NDWT-based spectrum uses the pairs
$$\displaystyle\left(j,\leavevmode\nobreak\ \hat{y}^{*}_{j}\right),\leavevmode%
\nobreak\ j=J-m-a-1,\dots,J-a-1.$$
from $m$ decomposition levels, starting with a coarse $j=J-m-a$ level and ending with finer level
$j=J-1-a$. Here $a$ is an arbitrary integer between 0 and $J-3$. When $a=0$, the finest level $j=J-1$ until level $J-1-m$ are used.
Then the spectral slope is
$$\displaystyle\hat{\beta}=\frac{12}{m(m^{2}-1)}\sum_{j=J-m-a-1}^{J-1-a}(j-J-a-(%
m+1)/2)\leavevmode\nobreak\ {\hat{y}}^{*}_{j}.$$
The estimator $\widehat{\beta}$ is unbiased,
$$\displaystyle E\hat{\beta}$$
$$\displaystyle=$$
$$\displaystyle\frac{12}{m(m^{2}-1)}\sum_{j=J-m-a-1}^{J-1-a}(j-J-a-(m+1)/2)%
\leavevmode\nobreak\ \left(-\log 2\,(2H+1)j+C\right)$$
$$\displaystyle=$$
$$\displaystyle-\log 2\,(2H+1),$$
where $C$ is a constant and $H$ is the theoretical Hurst exponent.
By substituting $\mbox{Var}({\hat{y}}^{*}_{j})=A/N$ from Theorem 3.1
we find
$$\displaystyle\mbox{Var}(\widehat{\beta})=\frac{12A}{Nm(m^{2}-1)},\mbox{ and %
\leavevmode\nobreak\ \leavevmode\nobreak\ }\mbox{Var}(\widehat{H})=\frac{3A}{%
Nm(m^{2}-1)(\log 2)^{2}},$$
for $\widehat{H}=-{\widehat{\beta}}/{(2\log 2)}-1/2.$
Thus, the MEDL estimator $\widehat{H}$ is approximately normal with mean $H$ and variance $3A/(Nm(m^{2}-1)(\log 2)^{2}),$
where $A\cong 5.4418$, $N$ is the sample size, and $m$ is the number of levels used for the spectrum.
B. Derivation of MEDLA
Proof of Theorem 3.3.
We begin by selecting the pair of wavelet coefficients that follow a normal distribution with a zero mean and a variance dependent on level $j$, from which the wavelet coefficients are sampled.
$$\displaystyle d_{j,k_{1}},d_{j,k_{2}}$$
$$\displaystyle\sim$$
$$\displaystyle{\cal N}(0,2^{-(2H+1)j}\sigma^{2}),$$
where $\sigma$ is the standard deviation of wavelet coefficients from level 0, $k_{1}$ and $k_{2}$ are positions of wavelet coefficients in level $j$, and $H$ is the Hurst exponent. We also assume that coefficients $d_{j,k_{1}}$ and $d_{j,k_{2}}$ are independent, which is a reasonable assumption when
the distance $|k_{1}-k_{2}|>q_{j}=2^{J-j}$.
Then, we define $\delta$ as
$$\displaystyle\delta$$
$$\displaystyle=$$
$$\displaystyle\frac{2^{(2H+1)j}}{\sigma^{2}}(d_{j,k_{1}}^{2}+d_{j,k_{2}}^{2})=C%
_{j}\cdot a,$$
for $C_{j}=\frac{2\cdot 2^{(2H+1)j}}{\sigma^{2}}\text{ and }\;a=\frac{d^{2}_{1}+d^{%
2}_{2}}{2}$.
Since $\delta$ follow $\chi^{2}_{2}$ distribution, the pdf of the average of two squared wavelet coefficients $a$ is
$$\displaystyle f(a)$$
$$\displaystyle=$$
$$\displaystyle\frac{C_{j}}{2}e^{-\frac{C_{j}a}{2}}.$$
Denote $y=\log a$. The pdf and cdf of $y$ are
$$\displaystyle f(y)$$
$$\displaystyle=$$
$$\displaystyle\frac{C_{j}}{2}e^{-\frac{C_{j}e^{y}}{2}}e^{y}$$
$$\displaystyle F(y)$$
$$\displaystyle=$$
$$\displaystyle 1-e^{-C_{j}e^{y}/2}.$$
Let $y^{*}$ be the median of $y$. The expression for $y^{*}$ is obtained by
solving
$F(y^{*})=1-e^{-C_{j}e^{y^{*}}/2}=1/2,$
$$\displaystyle y^{*}=\log\,(\log 4)-\log C_{j}.$$
After replacing $C_{j}$ with $\frac{2\cdot 2^{(2H+1)j}}{\sigma^{2}},$ the median becomes
$$\displaystyle y^{*}$$
$$\displaystyle=-\log 2\,(2H+1)j+\log\sigma^{2}+\log\,(\log 2),$$
similarly as in (A. Derivation of MEDL) in the MEDL method.
Proof of Lemma 3.2
An approximation of variance of sample median is obtained as
$$\displaystyle Var(\hat{y}_{j}^{*})\approx\frac{1}{4n(f(y^{*}_{j}))^{2}}.$$
After plugging in the expression for $y^{*}$ into $\frac{1}{4n(f(y^{*}_{j}))^{2}}$,
we obtain
$$\displaystyle Var(\hat{y}_{j}^{*})\approx\frac{1}{N(\log 2)^{2}},$$
Thus the variance of the sample median in MEDLA method is approximately $2.08/N.$
Proof of Theorem 3.4
For the distribution of $\widehat{H}$ from MEDLA, we follow the same regression steps on pair $(j,\hat{y_{j}^{*}})$ as in Appendix A. By substituting $\mbox{Var}({\hat{y}}^{*}_{j})=\frac{1}{N(\log 2)^{2}}$ from (3.2) we find
$$\displaystyle\mbox{Var}(\widehat{\beta})=\frac{12}{Nm(m^{2}-1)(\log 2)^{2}},%
\mbox{ and \leavevmode\nobreak\ \leavevmode\nobreak\ }\mbox{Var}(\widehat{H})=%
\frac{3}{Nm(m^{2}-1)(\log 2)^{4}},$$
for $\widehat{H}=-{\hat{\beta}}/{(2\log 2})-1/2.$ Thus, the MEDLA estimator $\hat{H}$ is approximately normal with mean $H$ and variance $3/(Nm(m^{2}-1)(\log 2)^{2}),$ where $N$ is the sample size, and $m$ is the number of levels used for the spectrum. |
A Unifying Approach to Decide Relations for Timed Automata and their Game Characterization
Shibashis Guha 111The research of Shibashis Guha was supported by Microsoft Corporation and Microsoft Research India under the Microsoft Research India PhD Fellowship Award.
Shankara Narayanan Krishna
Indian Institute of Technology Delhi Indian Institute of Technology BombayIndian Institute of Technology Delhi Indian Institute of Technology Delhi
Chinmay Narayan S. Arun-Kumar
Indian Institute of Technology Delhi Indian Institute of Technology Delhi
Abstract
In this paper we present a unifying approach for deciding various bisimulations, simulation equivalences and preorders between two timed automata states. We propose a zone based method for deciding these relations in which we eliminate an explicit product construction of the region graphs or the zone graphs as in the classical methods. Our method is also generic and can be used to decide several timed relations. We also present a game characterization for these timed relations and show that the game hierarchy reflects the hierarchy of the timed relations. One can obtain an infinite game hierarchy and thus the game characterization further indicates the possibility of defining new timed relations which have not been studied yet. The game characterization also helps us to come up with a formula which encodes the separation between two states that are not timed bisimilar. Such distinguishing formulae can also be generated for many relations other than timed bisimilarity.
1 Introduction
Bisimulation [18] is one of the most important
notions used to study process equivalence in concurrency theory.
Given two processes (untimed/timed/probabilistic), deciding whether
they are equivalent in some way is a fundamental question of practical significance;
over the years, several researchers have contributed theory and techniques to answer this question.
In this paper, we are interested in checking various kinds of equivalences and preorders between timed systems.
Timed automata, introduced in [4] are one of the most popular formalisms for modelling timed systems.
It is known that given two timed automata,
checking whether they accept the same timed language is undecidable [4].
However, bisimulation equivalences between timed automata have been shown to be
decidable [2][5][17][23].
The decidability of timed bisimilarity between two timed automata
was proved in [5] via a product construction on region graphs.
[15] also uses regions as the basis of checking timed bisimilarity for timed automata.
To overcome the state space explosion in region graphs, [23] applies the product
construction on zone graphs. The article [22] proposes weaker equivalences (several
variants of time abstracted bisimulations), and
uses zone graphs for the same purpose of overcoming the state explosion in region graphs.
In this work, we propose a uniform way of deciding various timed and time abstracted relations
present in the literature using a zone based approach. The zone graph is constructed in such a way that
every zone is (i) convex, and (ii) intersects with exactly one hyperplane on elapsing time.
First, for deciding timed bisimilarity, we define corner point bisimulation and prove that
two timed automata states are corner point bisimilar iff they are timed bisimilar.
Apart from the fact that ours is a zone based approach, we also
do not compute a product of individual zone graphs, as done in [23].
Thus we expect our approach to save computation since it does not require the product zone
graph to be stored along with the individual zone graphs of the two timed automata.
Moreover, the product based approach cannot be used to check all possible relations, for instance,
it is not useful in checking timed performance prebisimulation [12]. Corresponding to each of the
bisimulation relations described above, we can consider a simulation relation and
our zone graph can be used to check all these relations in a uniform way,
Further, our method checks timed bisimulation between two states with arbitrary rational valuations; many of the existing approaches
[15], can only check for timed bisimulation between the initial states.
Next, we define a game semantics corresponding to the various timed relations;
this is an extension of Stirling’s bisimulation games for discrete time relations [19].
The game theoretic formulation obviates the need for tedious operational reasoning
which is required many a time to compare various timed relations:
the game formulation helps in obtaining a hierarchy among various timed relations in a very elegant and succinct way.
Playing these games on two timed automata which are not timed bisimilar, we synthesize a formula which captures the difference.
The technique of synthesizing distinguishing formulae on two structures using EF games
is known in the literature [21].
Given two timed automata $A$ and $B$, [15] builds a characteristic formula
$\psi_{A}$ that describes $A$ and checks if $B\models\psi_{A}$; $A$ and $B$ are
timed bisimilar iff $B\models\psi_{A}$. The distinguishing formula $\varphi$ we synthesize, only captures
the difference between $A$ and $B$; for many practical situations, $\varphi$
would hence be much more succinct than $\psi_{A}$.
Paper [11] also describes a method for constructing a distinguishing formula. However, there too the formula construction depends on the entire (branching) structure of a timed automaton, whereas in our method, the formula is synthesized based on the moves in the game and thus leads to a more succinct formula.
Given a specification $S$, and an implementation $I$, both modeled using timed automata,
our approach can be used to synthesize the distinguishing formula $\varphi$ (if it exists);
$\varphi$ can then be used to refine $I$ to obtain an implementation $J$ which satisfies $S$.
A prototype tool which constructs the zone graph as described above, and checks for various timed relations
is underway.
Our tool thus will be a unifying framework to check various timed and time abstracted relations; it
will also aid in system refinement by generating a distinguishing formula.
In section 2, we give a brief
introduction to timed automata, introduce several definitions required in the paper and describe
the way we construct the zone graph.
In section 3 we describe the various timed and time abstracted relations considered in
this work. In section 4, we present the methods for deciding these relations.
The game semantics is given in section 5. The zone graph construction used here acts as a common framework to decide several kinds of timed and time abstracted relations. Finally, we conclude in section 6.
2 Timed Automata
Timed automata, introduced in [4] are a very popular
formalism for modelling time critical systems.
These are finite state automata over which time constraints
are specified using real variables called clocks.
Given a finite set of clocks $C$, the set of constraints $\mathcal{B}(C)$ allowed are
given by the grammar $g::=\;x\smile c\>|\>g\wedge g$,
where $c\in\mathbb{N}$ and $x\in C$ and $\smile\>\in\>\{\leq,<,=,>,\geq\}$.
Formally a timed automaton is a tuple $A=(L,Act,l_{0},E,C)$ where
(i) $L$ is a finite set of locations, (ii) $Act$ is a finite set
of visible actions, (iii) $l_{0}\in L$ is the initial location,
and (iv)
$E\subseteq L\>\times\>\mathcal{B}(C)\>\times\>Act\>\times\>2^{C}\>\times\>L$ is a finite set of edges.
Given two locations $l,l^{\prime}$, a transition from $l$ to $l^{\prime}$ is of the form
$(l,g,a,R,l^{\prime})$: on action $a$, we can go from $l$ to $l^{\prime}$ if the constraints
specified by $g$ are satisfied; $R\subseteq C$ is a set of clocks which are reset to zero during
the transition.
2.1 Semantics
The semantics of a timed automaton can be described with a timed labeled
transition system (TLTS) [2]. Let $A=\>(L,Act,l_{0},E,C)$ be a timed automaton over a set of clocks
$C$ and a set of visible actions $Act$. The timed transition system $T(A)$ generated by $A$ can be defined as
$T(A)=(Q,Lab,Q_{0},\{\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}|\alpha%
\in Lab\})$, where
$Q\>=\>\{(l,v)\>|\>l\in L,v\in{\mathbb{R}_{\geq 0}}^{|C|}\}$ is the set of states; each state is of the form $(l,v)$, where $l$ is a location of the timed automaton
and $v$ is a valuation assigned to
the clocks of $A$. $Lab=Act\cup\mathbb{R}_{\geq 0}$ is the set of labels.
Let $v_{0}$ denote the valuation such that $v_{0}(x)=0$ for all
$x\in C$. $Q_{0}=(l_{0},v_{0})$ is the initial state of $T(A)$.
A transition happens in one of the following ways:
(i) Delay transitions : $(l,v)\stackrel{{\scriptstyle d}}{{\longrightarrow}}(l,v+d)$. Here, $d\in\mathbb{R}_{\geq 0}$ and $v+d$ is the valuation
in which the value of every clock is incremented by $d$.
(ii) Discrete transitions : $(l,v)\stackrel{{\scriptstyle a}}{{\longrightarrow}}(l^{\prime},v^{\prime})$ if for an edge
$e=(l,g,a,R,l^{\prime})\in\>E$, $v\models g,v^{\prime}=v_{[R\leftarrow 0]}$, where $v_{[R\leftarrow 0]}$ denotes that the valuation of every clock in $R$ has been reset to 0, while the remaining clocks are unchanged.
From a state $(l,v)$, we take an $a$-transition to reach a state $(l^{\prime},v^{\prime})$
if the valuation $v$ of the clocks satisfies $g$;
after this, the clocks in $R$ are reset while those in $C\backslash R$ remain unchanged.
For example, let $A$ be a timed automaton with two clocks $x$ and $y$. Consider a state $(l,v)$ of $T(A)$ with
$(v(x),v(y))=(0.3,1.6)$. Consider an edge $e=(l,x<1\wedge y>2,a,\{y\},l^{\prime})$.
Starting from $(l,(0.3,1.6))$, here is a sequence of transitions in $T(A)$ :
$(l,(0.3,1.6))\stackrel{{\scriptstyle 0.5}}{{\longrightarrow}}(l,(0.8,2.1))%
\stackrel{{\scriptstyle a{}}}{{\longrightarrow}}(l^{\prime},(0.8,0))$.
For simplicity, we do not consider annotating
locations with clock constraints (known as invariant conditions [13]). Our results extend in a
straightforward manner to timed automata with invariant conditions.
We now define various concepts that will be used in the paper.
Definition 1.
Let $A=(L,Act,l_{0},E,C)$ be a timed automaton, and $T(A)$ be the TLTS corresponding to $A$.
1.
Timed trace: A sequence of delays and visible actions $d_{1}a_{1}d_{2}a_{2}\dots d_{n}a_{n}$
is called a timed trace iff there is a sequence of transitions
$p_{0}\xrightarrow{d_{1}}p_{1}\xrightarrow{a_{1}}p_{1}^{\prime}\xrightarrow{d_{%
1}}p_{2}\xrightarrow{a_{2}}p_{2}^{\prime}\cdots\xrightarrow{d_{n}}p_{n}%
\xrightarrow{a_{n}}p^{\prime}$ in $T(A)$, with
$p_{0}{}$ being a state of the timed automaton.
For a timed trace $tr=d_{1}a_{1}d_{2}a_{2}\dots d_{n}a_{n}$, $untime(tr)=a_{1}a_{2}\dots a_{n}$ represents the sequence of visible actions in $tr$.
2.
Zone: A zone $z$ is a set of valuations $\{v\in\mathbb{R}_{\geq 0}^{|C|}\mid v\models\gamma\}$, where
$\gamma$ is of the form $\gamma::=\;x\smile c\>|\>x-y\smile c\>|\>g\wedge g$,
and $c\in\mathbb{Z}$, $x,y\in C$ and $\smile\>\in\>\{\leq,<,=,>,\geq\}$.
$z\uparrow$ denotes the future of the zone $z$.
$z\uparrow=\{v+d\mid v\in z,d\geq 0\}$ is the set of all valuations
reachable from $z$ by time elapse.
3.
Pre-stability: A zone $z_{1}$ is pre-stable with respect to another zone
$z_{2}$ if $z_{1}\subseteq preds(z_{2})$ or $z_{1}\cap preds(z_{2})=\emptyset$ where
$preds(z)\stackrel{{\scriptstyle def}}{{=}}\{v\in\mathbb{R}_{\geq 0}^{|C|}\>|\>%
\exists v^{\prime}\in z$ such that $v\xrightarrow{\alpha}v^{\prime}$,
$\alpha\in Act\cup\mathbb{R}_{\geq 0}$}.
4.
Canonical decomposition: Let $z$ be a zone, and let $g=\bigwedge_{i=1}^{n}g_{i}\in\mathcal{B}(C)$, where
each $g_{i}$ is of the form $x_{i}\smile c_{i}$.
A canonical decomposition of $z$ with respect to $g$
is obtained by splitting $z$ into a set of zones $z_{1},\dots,z_{m}$ such that
for each $1\leq i\leq m$, and $1\leq j\leq n$, for every valuation $v\in z_{i}$, either (i) $v\models g_{j}$, or (ii)
$v\nvDash g_{j}$.
For example, consider the zone $z=x\geq 0\wedge y\geq 0$ and the guard $x\leq 2\wedge y>1$.
$z$ is split with respect to $x\leq 2$, and then with respect to $y>1$, hence
into four zones :
$x\leq 2\wedge y\leq 1$, $x>2\wedge y\leq 1$, $x\leq 2\wedge y>1$ and $x>2\wedge y>1$.
Given a timed automaton $A$, a zone graph of $A$
is used to check reachability in $A$.
A node in the zone graph is a pair consisting of a location and a zone. The edges between nodes
are defined as follows.
$(l,z)\stackrel{{\scriptstyle a}}{{\rightarrow}}(l^{\prime},z^{\prime})$, where $a\in Act$, if for every $v$ in
$z$, $\exists v^{\prime}$ in $z^{\prime}$ such that $(l,v)\stackrel{{\scriptstyle a}}{{\rightarrow}}(l^{\prime},v^{\prime})$. If the
zones corresponding to $(l,v)$ and $(l,v^{\prime})$ are $z$ and $z^{\prime}$ respectively and there is a transition
in $T(A)$ such that $(l,v)\xrightarrow{d}(l,v^{\prime})$, then we have an edge $(l,z)\xrightarrow{\varepsilon}(l,z^{\prime})$ in the zone graph.
Every node has an $\varepsilon$ transition to itself and the $\varepsilon$ transitions are also transitive.
The zone $z^{\prime}$ is called a delay successor zone of zone $z$. Since $\varepsilon$ is reflexive, delay successor is also a reflexive relation.
For both $a$ and $\varepsilon$ transitions, if $z$ is a zone then $z^{\prime}$ is also a zone, i.e. $z^{\prime}$ is a convex set. A zone graph may be formally defined as a quadruple $(S,s_{0},Lep,\rightarrow)$, where $S$ is the set
of nodes of the zone graph, $s_{0}$ is the initial node, $Lep=Act\cup\{\varepsilon\}$ and $\rightarrow$ denotes the
set of transitions.
$Z_{(A,p)}$ denotes a zone graph corresponding to the state $p$, i.e. the initial state of $Z_{(A,p)}$ is $p$.
For a state $q\in T(A)$, $\mathcal{N}(q)$ represents the node
of the zone graph with the same location as that of $q$ such that the zone corresponding to $\mathcal{N}(q)$ includes
the valuation of $q$. We often say that a state $q$ is in node $s$ to indicate that $q$ is in the zone associated with node $s$. For two zone graphs,
$Z_{(A_{1},p)}=(S_{1},s_{p},Lep,\rightarrow_{1})$, $Z_{(A_{2},q)}=(S_{2},s_{q},Lep,\rightarrow_{2})$ and a relation
$\mathcal{R}\subseteq S_{1}\times S_{2}$, $Z_{(A_{1},p)}\>\mathcal{R}\>Z_{(A_{2},q)}$ iff $(s_{p},s_{q})\in\mathcal{R}$.
While checking $\mathcal{R}$, $\varepsilon$ is considered visible similar to an action in $Act$.
An $\varepsilon$ action represents a delay $d\in\mathbb{R}_{\geq 0}$.
The detailed algorithm for creating the zone graph has been described in algorithm 1
and consists of two phases, the first one being a forward analysis of the timed automaton while the second phase
ensures pre-stability in the zone graph. The set of valuations for every
location is initially split into zones based on the canonical decomposition of its outgoing transition. The forward analysis may cause a zone graph to become infinite [8]. Several kinds of abstractions
have been proposed in the literature [7][8][9]. We use location dependent maximal constants
abstraction [8] to ensure finiteness of the zone graph. In algorithm 1, $max_{x}^{l}$ denotes the maximum constant in location $l$ beyond which the value of clock $x$ is irrelevant.
After phase 2, pre-stability ensures the following: For a node $(l,z)$ in the zone graph, with $v\in z$,
for a timed trace $tr$,
if $(l,v)\xrightarrow{tr}(l^{\prime\prime},v^{\prime\prime})$, with $v^{\prime\prime}\in z^{\prime\prime}$, then
$\forall v^{\prime}\;{}\mbox{in}\;z$, $\exists tr^{\prime}.(l,v^{\prime})\xrightarrow{tr^{\prime}}(l^{\prime\prime},%
\tilde{v})$, with
$untime(tr^{\prime})=untime(tr)$ and $\tilde{v}\in z^{\prime\prime}$. According to the construction given
in algorithm 1, for a particular location of the timed automaton, the zones corresponding
to any two nodes are disjoint. Convexity of the zones and pre-stability property together ensure
that a zone with elapse of time is intercepted by a single hyperplane of the form $x=h$ as in the case of regions,
where $x\in C$ and $h\in\mathbb{N}$. Some approaches for preserving convexity and implementing pre-stability have been discussed in [22].
As an example consider the timed automaton in Figure 1. The zones corresponding to location $l_{1}$ as produced through algorithm 1 are shown in the right side of the figure.
A similar construction of zone graph has also been used in [12]. In the construction used in [12], in the final phase, the nodes corresponding to a particular location with zones that are time abstracted bisimilar to each other are merged as long as the merged zone is convex. Though this may reduce the number of zones in the final zone graph, the operation itself is exponential in the number of clocks of the timed automaton. Due to the absence of this merging phase in the algorithm described in this paper, while checking the existence of the relations following the method described here, one may need to consider more pairs of states, but we expect this overhead to be less compared to the expensive operation of merging the nodes with time abstracted bisimilar zones.
3 Equivalences for Timed Systems
In this section, we define the timed and the time abstracted
relations considered in this work. We only consider the strong form of these relations here. We enumerate a few clauses first using which we define $p_{1}\>\mathcal{R}\>p_{2}$ where $p_{1}$ and $p_{2}$ are two timed automata states and $\mathcal{R}$ is a timed or a time abstracted relation.
1.
$\forall a\in Act\wedge\forall p_{1}^{\prime}$, $p_{1}\stackrel{{\scriptstyle a}}{{\rightarrow}}p_{1}^{\prime}\Rightarrow[\>%
\exists p_{2}^{\prime}:p_{2}\stackrel{{\scriptstyle a}}{{\rightarrow}}p_{2}^{%
\prime}\wedge p_{1}^{\prime}\mathcal{R}p_{2}^{\prime}\>]$
2.
$\forall a\in Act\wedge\forall p_{2}^{\prime}$, $p_{2}\stackrel{{\scriptstyle a}}{{\rightarrow}}p_{2}^{\prime}\Rightarrow[\>%
\exists p_{1}^{\prime}:p_{1}\stackrel{{\scriptstyle a}}{{\rightarrow}}p_{1}^{%
\prime}\wedge p_{1}^{\prime}\mathcal{R}p_{2}^{\prime}\>]$
3.
$\forall a\in Act\wedge\forall p_{1}^{\prime}$, $p_{1}\stackrel{{\scriptstyle a}}{{\rightarrow}}p_{1}^{\prime}\Rightarrow[\>%
\exists p_{2}^{\prime}\>\exists d\in\mathbb{R}_{\geq 0}:p_{2}\stackrel{{%
\scriptstyle d}}{{\rightarrow}}\stackrel{{\scriptstyle a}}{{\rightarrow}}p_{2}%
^{\prime}\wedge p_{1}^{\prime}\mathcal{R}p_{2}^{\prime}\>]$
4.
$\forall a\in Act\wedge\forall p_{1}^{\prime}$, $p_{1}\stackrel{{\scriptstyle a}}{{\rightarrow}}p_{1}^{\prime}\Rightarrow[\>%
\exists p_{2}^{\prime}\>\exists d_{1},d_{2}\in\mathbb{R}_{\geq 0}:p_{2}%
\stackrel{{\scriptstyle d_{1}}}{{\rightarrow}}\stackrel{{\scriptstyle a}}{{%
\rightarrow}}\stackrel{{\scriptstyle d_{2}}}{{\rightarrow}}p_{2}^{\prime}%
\wedge p_{1}^{\prime}\mathcal{R}p_{2}^{\prime}\>]$
5.
$\forall d\in\mathbb{R}_{\geq 0}\wedge\forall p_{1}^{\prime}$, $p_{1}\stackrel{{\scriptstyle d}}{{\rightarrow}}p_{1}^{\prime}\Rightarrow[\>%
\exists p_{2}^{\prime}:p_{2}\stackrel{{\scriptstyle d}}{{\rightarrow}}p_{2}^{%
\prime}\wedge p_{1}^{\prime}\mathcal{R}p_{2}^{\prime}\>]$
6.
$\forall d\in\mathbb{R}_{\geq 0}\wedge\forall p_{1}^{\prime}$, $p_{1}\stackrel{{\scriptstyle d}}{{\rightarrow}}p_{1}^{\prime}\Rightarrow[\>%
\exists p_{2}^{\prime}\>\exists d^{\prime}\in\mathbb{R}_{\geq 0}:p_{2}%
\stackrel{{\scriptstyle d^{\prime}}}{{\rightarrow}}p_{2}^{\prime}\wedge p_{1}^%
{\prime}\mathcal{R}p_{2}^{\prime}\>]$
7.
$\forall d\in\mathbb{R}_{\geq 0}\wedge\forall p_{1}^{\prime}$, $p_{1}\stackrel{{\scriptstyle d}}{{\rightarrow}}p_{1}^{\prime}\Rightarrow[\>%
\exists p_{2}^{\prime}\>\exists d^{\prime}\in\mathbb{R}_{\geq 0}\>\wedge\>d%
\leq d^{\prime}\>:p_{2}\stackrel{{\scriptstyle d^{\prime}}}{{\rightarrow}}p_{2%
}^{\prime}\wedge p_{1}^{\prime}\mathcal{R}p_{2}^{\prime}\>]$
8.
$\forall d\in\mathbb{R}_{\geq 0}\wedge\forall p_{2}^{\prime}$, $p_{2}\stackrel{{\scriptstyle d}}{{\rightarrow}}p_{2}^{\prime}\Rightarrow[\>%
\exists p_{1}^{\prime}\>\exists d^{\prime}\in\mathbb{R}_{\geq 0}\>\wedge\>d%
\geq d^{\prime}\>:p_{1}\stackrel{{\scriptstyle d^{\prime}}}{{\rightarrow}}p_{1%
}^{\prime}\wedge p_{1}^{\prime}\mathcal{R}p_{2}^{\prime}\>]$
$\mathcal{R}$ is a timed simulation if the clauses 1 and 5 hold. For each $(p_{1},p_{2})\in\mathcal{R}$, $p_{2}$ time simulates $p_{1}$. $\mathcal{R}$ is a timed simulation equivalence if $p_{1}$ time simulates $p_{2}$ and $p_{2}$ time simulates $p_{1}$. A symmetric timed simulation is a timed bisimulation relation. A symmetric relation that satisfies clauses 1 and 6 is a time abstracted bisimulation. A relation that is symmetric and satisfies clauses 3 and 6 is a time abstracted delay bisimulation relation. A symmetric relation satisfying clauses 4 and 6 is a time abstracted observational bisimulation. A timed performance prebisimulation relation [12] satisfies the clauses 1, 2, 7 and 8.
The corresponding largest bisimulation relations are called bisimilarity relations and they are timed bisimilarity ($\sim_{t}$), time abstracted bisimilarity ($\sim_{u}$), time abstracted delay bisimilarity ($\sim_{y}$), time abstracted observational bisimilarity ($\sim_{o}$) whereas the largest prebisimulation relation is called timed performance prebisimilarity ($\precsim$). $p\precsim q$ denotes that $p$ is at least as fast as $q$. It is easy to see from the definitions that timed bisimilarity implies time-abstracted bisimulation whereas the converse is not true. Besides, the definitions imply $\sim_{u}\>\subseteq\>\sim_{y}\>\subseteq\>\sim_{o}$. Also the existence of a bisimulation relation between two states implies the existence of the corresponding simulation equivalence and timed performance prebisimilarity lies in between timed bisimulation and time abstracted bisimulation. Hence we have $\sim_{t}\>\subseteq\>\precsim\>\subseteq\>\sim_{u}\>\subseteq\>\sim_{y}\>%
\subseteq\>\sim_{o}$ and similar containment
relations also exist among the corresponding simulation equivalences.
4 Deciding Relations for Timed Automata
In this section, we present a unifying approach to decide several relations for timed automata using the zone graph constructed in algorithm 1.
4.1 Deciding Timed Bisimulation
Timed bisimulation has been proven to be decidable for timed automata [5]. A product construction technique on the region graphs has been used in [5] whereas in [23], a product construction is applied on zone graphs instead for deciding timed bisimulation. Though decidable,
timed bisimulation may have uncountably many equivalence classes[3].
We define corner point bisimulation relation and show that
corner point bisimulation coincides with timed bisimulation. With corner point bisimulation, only a finite
number of pairs of corner points are needed for bisimilarity checking. Further,
our method eliminates the product construction on zone graphs.
Let $A$, $B$ be timed automata having $C_{A},C_{B}$ as the respective maximum constants used in the constraints appearing in the two automata. Let
$p$ and $q$ be two states in $T(A)$ and $T(B)$ respectively. We show that
(i)
if $p$ and $q$ are initial states, or states where all clock valuations are integers, then timed bisimulation for $p,q$ can be decided by checking delays of the form $n$, $n+\ \delta$ or $n-\delta$, where $n\in\{0,1,\dots,C\}$, $C=max(C_{A},C_{B})$, and $\delta$ is a symbolic value for an infinitesimal positive quantity.
(ii)
If there is some clock $y$ having a non-zero rational fractional part, then along with the delays of the form mentioned above, we check delays of the form $f$, $f+\delta$ or $f-\delta$, with $f=1-frac(v(y))$, $frac(v(y))$ is the fractional part of the value of clock $y$.
Delays of the form mentioned above are called corner point delays or cp-delays.
We define corner point bisimulation formally below.
Definition 2.
1.
Corner point simulation (cp-simulation): A relation $\mathcal{R}$ is a corner point simulation relation, if for every pair of timed automata states $(p,q)\in\mathcal{R}$, the following conditions hold.
(i)
For every visible action $a\in Act$, if $p\xrightarrow{a}p^{\prime}$, then $\exists q^{\prime}$ such that $q\xrightarrow{a}q^{\prime}$ and $p^{\prime}\mathcal{R}q^{\prime}$
(ii)
Considering the maximum possible delay $d$ from $p$, if $p\xrightarrow{d}p^{\prime}$ and $p^{\prime}$ is in node $\mathcal{N}(p)$, then $\exists q^{\prime}$ such that $q\xrightarrow{d}q^{\prime}$ and $p^{\prime}\mathcal{R}q^{\prime}$
(iii)
For every node $\mathcal{N}(p^{\prime})\neq\mathcal{N}(p)$ such that $\mathcal{N}(p)\xrightarrow{\varepsilon}\mathcal{N}(p^{\prime})$, considering the minimum delay $d$ from $p$, if $p\xrightarrow{d}p^{\prime}$, then $\exists q^{\prime}$ such that $q\xrightarrow{d}q^{\prime}$ and $p^{\prime}\mathcal{R}q^{\prime}$
Here $q$ cp-simulates $p$. A symmetric corner point simulation relation is a corner point bisimulation (cp-bisimulation).
2.
Corner point trace: A timed trace from a state $p$ to $p^{\prime}$, where all the delays are cp-delays
is called a corner point trace.
Lemma 1.
For checking whether the timed automata states $p$ and $q$ are related through corner point simulation or corner point bisimulation relation, there are only finitely many pairs of states that need to be considered.
This is due to the fact that for any $(p,q)\in\mathcal{R}$, $\mathcal{R}$ being a cp-bisimulation relation, the valuations of all the clocks appearing in both $p$ and $q$ are of the form $n$, $n+\delta$ or $n-\delta$, where $n\in\{0,1,\dots,C_{A}\}$ or $n\in\{0,1,\dots,C_{B}\}$. If $p\in T(A)$ and $q\in T(B)$, then $C_{A}$ and $C_{B}$ are the maximum constants appearing in $A$ and $B$ respectively.
Theorem 1.
Corner point simulation and corner point bisimulation relations are decidable.
Theorem 2.
For two timed automata states $p$ and $q$,
1.
$p\sim_{t}q\Rightarrow p\mathcal{R}q$, where $\mathcal{R}$ is a corner point bisimulation relation.
2.
$p\mathcal{R}q\Rightarrow p\sim_{t}q$, where $\mathcal{R}$ is a corner point bisimulation relation.
3.
$p$ and $q$ are timed bisimilar if and only if $p$ and $q$ are cp-bisimilar.
Theorem 2 shows that the decidability of cp-bisimulation is sufficient for timed bisimulation.
Synthesis of Distinguishing Formulae.
Given two timed automata $A$ and $B$ which are not timed bisimilar,
we propose a technique that synthesizes a formula that captures
the differences between $A$ and $B$.
In [15], a characteristic formula for timed automata has been defined using a certain fragment of the $\mu$-calculus presented in [13]. Timed bisimilarity between two timed automata is decided by comparing one timed automaton with the characteristic formula of the other. A characteristic formula is a significantly complex formula describing the entire behaviour of the timed automaton. Here we describe how we can in general generate a simpler formula using a fragment of the logic described in [15]. The logic we use for generating the distinguishing formula has been described in [2] which is a timed extension of Hennessy-Milner logic and does not contain any recursion as opposed to the logic used in [15]. The set $\mathcal{M}_{t}$ of Hennessy-Milner logic formulae with time over a set of actions $Act$, set $D$ of formula clocks (distinct from the clocks of any timed automaton) is generated by the abstract syntax
$\phi::=\mbox{{t\!t}}\;|\;\mbox{{f\!f}}\;|\;\phi\wedge\psi\;|\;\phi\vee\psi\;|%
\;\langle a\rangle\phi\;|\;[a]\phi\;|\;\exists\!\!\!\!\exists\phi\;|\;\forall%
\!\!\!\!\forall\phi\;|\;x\>\mbox{{\text@underline{in}}}\>\phi\;|\;g$
where $a\in Act$, $x\in D$ and $g\in\mathcal{B}(D)$.
The logic used in [11] for constructing distinguishing formula uses an explicit negation rather than using the operators $[\;]$ and $\forall\!\!\!\!\forall$. Besides a distinguishing formula in [11] uses real delays whereas in our case, the formula clock values are compared with integers. Also the distinguishing formula synthesized in [11] considers the entire branching structure of the given automata whereas in our case, the formula is synthesized from the moves in a game and is thus more succinct.
Given a timed automaton $A$, $\mathcal{M}_{t}$ is interpreted over an extended state $\langle(l,v)u\rangle$, where $(l,v)$ is a state of $A$ and $u$ is a time assignment of $D$. Transitions between the extended states are defined by: $\langle(l,v)u\rangle\xrightarrow{d}\langle(l,v+d)u+d\rangle$ and $\langle(l,v)u\rangle\xrightarrow{a}\langle(l^{\prime},v^{\prime})u^{\prime}\rangle$ iff $\langle(l,v)\rangle\xrightarrow{a}\langle(l^{\prime},v^{\prime})\rangle$ and $u=u^{\prime}$.
$\exists\!\!\!\!\exists\phi$ holds in an extended state if there exists a delay transition leading to an extended state satisfying $\phi$. Similarly $\forall\!\!\!\!\forall$ denotes universal quantification over delay transitions, and $\langle a\rangle$ and $[a]$ respectively denote existential and universal quantification over $a$-transitions. The formula $x\>\mbox{{\text@underline{in}}}\>\phi$ introduces a formula clock $x$ and initializes it to 0, i.e. $\langle(l,v)u\rangle\models x\>\mbox{{\text@underline{in}}}\>\phi\implies%
\langle(l,v)u_{[x\leftarrow\bar{0}]}\rangle\models\phi$. The formula clocks are used in formulas of the form $g$ which is satisfied by an extended state if the values of the formula clocks used in $g$ satisfy the specified relationship. A formula is said to be closed if each occurrence of a formula clock $x$ is within the scope of an $x\>\mbox{{\text@underline{in}}}\>$ construct.
While checking cp-bisimulation, we describe below a method to generate a closed formula in $\mathcal{M}_{t}$ that distinguishes two timed automata that are not timed bisimilar. For constructing the formula, we consider a cp-bisimulation game between the initial states of the timed automata
which can be thought of as a bisimulation game (see [20]) for deciding timed bisimulation between two timed automata.
The game is played between two players, the challenger and the defender. Each round of the game consists of
the challenger choosing one of the zone graphs and making a move as defined in the definition of the cp-bisimulation relation. The defender tries to replicate the move in the other zone graph. The defender loses the game if after a finite sequence of rounds, the challenger makes a move on one zone graph which the defender cannot replicate on the other. If the defender loses the game, we look at the sequence of moves chosen by the challenger to construct the distinguishing formula as follows:
Given two timed automata $A$ and $B$ (with $C_{1}\cap C_{2}=\emptyset$ where $C_{1}$ and $C_{2}$ are clocks of $A$ and $B$ respectively) and their zone graphs being $Z_{A}$ and $Z_{B}$, let us suppose without loss of generality that the challenger makes a move on $Z_{A}$ in the first round. We derive a formula from the moves of the game which is satisfied by automaton $A$ and not by automaton $B$. The set of formula clocks are disjoint from the clocks in $A$ and $B$. The distinguishing formula $\zeta$ is initialized to $x_{1}\>\>\mbox{{\text@underline{in}}}\>\>()$ with the introduction of a formula clock $x_{1}$. Corresponding to every clock $y$ in $C_{1}\cup C_{2}$, there exists a clock $x\in D$ such that their valuations are the same, i.e. $v(y)=v(x)$. We can define a mapping $\eta:C_{1}\cup C_{2}\rightarrow D$. With the introduction of the formula clock $x_{1}$ mentioned above, we have $\forall y\in C_{1}\cup C_{2}$, $\eta(y)=x_{1}$. Whenever one or more clocks are reset either in $A$ or $B$ corresponding to the visible actions chosen by the challenger and the
defender, a new formula clock is introduced. If $U\subseteq C_{1}$ and $V\subseteq C_{2}$ be the subset of clocks reset for actions chosen in a particular round, then a new formula clock $x_{i}$ is introduced such that $\forall y\in U\cup V,\eta(y)=x_{i}$. Subformulas of $\zeta$ are always added to the scope of the innermost or the last added clock in $\zeta$. The subformulas of $\zeta$ are added based on the move of the challenger as described below.
-
The challenger performs action $a\in Act$ in $Z_{A}(Z_{B})$. If the defender can reply with $a$ in $Z_{B}(Z_{A})$ and in either of the moves at least one clock is reset in the corresponding timed automata, then add $\langle a\rangle\;x\;\>\mbox{{\text@underline{in}}}\>\;(\;)\;(\>[a]\;x\;\>%
\mbox{{\text@underline{in}}}\>\;(\;)\>)$ to the innermost scope of $\zeta$, where $x$ is a new formula clock. If no clock is being reset, then simply add $\langle a\rangle([a])$ to the innermost scope of $\zeta$. If the defender cannot reply to the move of the challenger, then append $\langle a\rangle\mbox{{t\!t}}\;(\>[a]\mbox{{f\!f}}\>)$ in the innermost scope and declare $\zeta$ to be the distinguishing formula.
-
The challenger performs a delay $d$ in $Z_{A}(Z_{B})$, where the delays are as defined in the cp-bisimulation relation, i.e. cp-delays and it reaches a state with clock valuation $v$. For every clock $y$ in $Z_{A}(Z_{B})$, we construct a subformula as follows: if $v(y)=n$ is an integer, then construct the subformula $x=n$, where $x$ is $\eta(y)$. If $v(y)$ is of the form $n+\delta$, where $n\in\mathbb{N}$, then construct the subformula $n<x<n+1$ and if $v(y)$ is of the form $n-\delta$, then construct the subformula $n-1<x<n$. Conjunct all these subformulas to obtain $\psi$ and append $\exists\!\!\!\!\exists(\psi)\;(\>\forall\!\!\!\!\forall(\psi)\>)$ to the innermost scope of $\zeta$.
Note that in $\zeta$, the subformulas of the form $g$ define the smallest set of extended states reachable along the trace followed in $Z_{A}$ that can be specified by subformulas of the form $g$.
In the two timed automata shown in Figure 2, the moves of the challenger are marked by a $\checkmark$ mark. The challenger starts making a move from the automaton on the left and hence we construct a formula in $\mathcal{M}_{t}$ that is satisfied by the automaton on the left but not by the automaton on the right. Following the steps mentioned above, we obtain the formula $x_{1}\;\>\mbox{{\text@underline{in}}}\>\;(\langle a\rangle[b]\;x_{2}\;\>\mbox{%
{\text@underline{in}}}\>\;(\langle c\rangle\;\exists\!\!\!\!\exists(1<x_{2}<2%
\wedge\langle d\rangle\mbox{{t\!t}})))$.
We note that though the cp-bisimulation relation can be decided between any two timed automata states with arbitrary clock valuation, the distinguishing formula is constructed only while checking the relation between the initial states of the two timed automata.
The technique described here to check for lack of timed bisimilarity can be
adapted to many of the other relations studied in this paper.
Our prototype tool implementation which is currently underway will
also incorporate them.
4.2 Deciding Timed Performance Prebisimulaton
In this section, we define corner point prebisimulation relation and show that this relation coincides with the timed performance prebisimulation relation. We use the zone graph constructed according to algorithm 1 for checking
corner point prebisimulation. Unlike the case of timed bisimulation, a product construction on zone graphs
is not useful for deciding timed performance prebisimulation relation : for example, consider two simple timed automata with clocks $x$ and $y$ respectively, each with two locations and one edge between them such that the edge in one automaton is labelled with $\langle x=2,a,\emptyset\rangle$ while the other is labelled with $\langle y=5,a,\emptyset\rangle$. These two timed automata are timed performance prebisimilar though a product on the region graphs of these two automata does not produce an action transition and thus does not have sufficient information to show that one of the automata can actually perform action $a$ following a lesser delay.
We define the corner point prebisimulation relation in terms of a two player game that is similar to the bisimulation game. The game is played between two players, challenger and defender
on the zone graphs (as constructed in algorithm 1) of two timed automata.
In each round, the challenger chooses a side and the defender chooses the other side. After selecting a side, the challenger can either perform a visible action or a delay action. Note that in the corner point prebisimulation relation,
the delays are cp-delays as given by Definition 2.
Two timed automata states $p$ and $q$ in $T(A)$ and $T(B)$ respectively are cp-prebisimilar, denoted $p\precsim_{cp}q$, if starting from $p$ and $q$, the defender wins and the cp-delay moves in $A$ are less than or equal to the corresponding cp-delay moves in $B$. We write $A\precsim_{cp}B$ if $p\precsim_{cp}q$, where $p$ and $q$ are respectively the initial states of $T(A)$ and $T(B)$.
We now explain the possible moves of the game on the respective zone graphs $Z_{(A,p)}$ and $Z_{(B,q)}$. Each move results in a new state in a possibly new zone from which the next move is made in the next round.
-
(Challenger chooses $T(A)$ (Move 1)): Performs a visible action $a\in Act$.
(Defender chooses $T(B)$): i) Performs the same action $a$.
-
(Challenger chooses $T(A)$ (Move 2)): Performs maximum delay $d$ and stays inside the same zone.
(Defender chooses $T(B)$): i) Performs delay $d$.
-
(Challenger chooses $T(A)$ (Move 3)): Performs the minimum delay $d$ and moves to the next zone.
(Defender chooses $T(B)$ and performs one of the following delays): i) delay $d$ or ii) cp-delays $d^{\prime}\geq d$ that take $q$ to the delay successor zones.
-
(Challenger chooses $T(B)$ (Move 1)): Performs a visible action $a\in Act$.
(Defender chooses $T(A)$): i) Performs the same action $a$.
-
(Challenger chooses $T(B)$ (Move 2)): Performs maximum delay $d$ and stays inside the same zone.
(Defender chooses $T(A)$ and performs one of the following delays): i) delay $d$ itself or ii) Consider cp-delay $d^{\prime}\leq d$, such that $p$ on elapsing $d^{\prime}$ reaches the end of the same zone or other delay successor zones.
-
(Challenger chooses $T(B)$ (Move 3)): Performs the minimum delay $d$ and moves to the next zone.
(Defender chooses $T(A)$ and performs one of the following delays): i) delay $d$ itself or ii) cp-delays $d^{\prime}\leq d$ such that $p$ on elapsing $d^{\prime}$ reaches the beginning of the delay successor zones or iii) cp-delays $d^{\prime}\leq d$ such that it reaches the end of the same or other delay successor zones.
Figure 3 illustrates how a corner point prebisimulation game is played between the challenger and the defender and shows each of the moves described above.
Note that in automata $A$ and $B$, for locations $l_{0}$ and $L_{0}$ respectively,
-
action $a$ is enabled at all delays.
-
actions $a$ and $b$, both are enabled when $x\leq 12$ and $y\leq 15$ respectively and
-
action $c$ is enabled in the interval $12<x\leq 16$ in $A$ and in the interval $15<y<22$ in $B$.
For these two automata, we can see that $(l_{0},x=0)\precsim(l_{1},y=0)$, i.e. the automaton $A$ is at least as fast as automaton $B$. In $A$, the zones created using the algorithm 1
corresponding to $l_{0}$ are $x\leq 4$, $4<x\leq 12$, $12<x\leq 16$ and $x>16$, whereas in $B$, the zones created are $y\leq 11$, $11<y\leq 15$, $15<y<20$, $20\leq y<22$ and $y\geq 22$. In Figure 3, we also show a representative diagram of these zones. The dots on the axis of the clock denote the boundary of a zone that does not signify any change in behaviour whereas the small vertical lines on the axis of the clock denote an actual change in behaviour. A formal definition of similarity in behaviour is given in the appendix.
Similar to corner point bisimulation, in cp-prebisimulation too, only finitely many pairs of states are considered for checking the relation.
Theorem 3.
Corner point prebisimilarity between two timed automata states is decidable.
Theorem 4.
Two timed automata states are timed performance prebisimilar if and only if they are corner point prebisimilar.
4.3 Deciding Time Abstracted Bisimulation
Time abstracted bisimulation between two timed automata has been shown to be decidable [2][17] using the region graph construction [4]. Two timed automata are timed abstracted bisimilar if and only if their region graphs are strongly bisimilar. We use the zone graph constructed in algorithm 1 instead of the region graph.
The size of the zone graph is independent of the constants with which the clocks are compared in the timed automaton guards. Let $Z_{(A,p)}$ denote the zone graph for state $p$ of timed automaton $A$. If there are two valuations $(l,v)$ and $(l,v^{\prime})$ such that they belong to the same node, then by construction of $Z_{(A,p)}$, $(l,v)$ and $(l,v^{\prime})$ are time abstracted bisimilar. Thus in the zone graph, it is the case that a state $(l,v)$ in $T(A)$ is time abstracted bisimilar to the zone $z$ to which $(l,v)$ belongs in the zone graph $Z_{(A,p)}$. The same holds for a timed state $(l_{2},v_{2})$ in $T(B)$. Thus checking whether two states $(l_{1},v_{1})$ and $(l_{2},v_{2})$ of two timed automata $A$ and $B$ are time abstracted bisimilar involves checking whether their corresponding nodes in the two zone graphs are strongly bisimilar.
The following theorems show how time abstracted delay bisimulation and time abstracted
observational bisimulation [22] too can be decided along with strong time abstracted bisimulation using our zone graph.
Theorem 5.
Let $\mathcal{R}\subseteq S_{1}\times S_{2}$ be a symmetric relation. Two nodes ($s_{1},s_{2}$) $\in\mathcal{R}$ if and only if
$\forall a\in Act,\forall s_{1}^{\prime}[s_{1}\stackrel{{\scriptstyle a}}{{%
\rightarrow}}s_{1}^{\prime}\Rightarrow\exists s_{2}^{\prime}\>.\>s_{2}%
\stackrel{{\scriptstyle\beta}}{{\rightarrow}}s_{2}^{\prime}$ and $(s_{1}^{\prime},s_{2}^{\prime})\in\mathcal{R}]$ and
$\forall s_{1}^{\prime},[s_{1}\stackrel{{\scriptstyle\varepsilon}}{{\rightarrow%
}}s_{1}^{\prime}\Rightarrow\exists s_{2}^{\prime}\>.\>s_{2}\stackrel{{%
\scriptstyle\varepsilon}}{{\rightarrow}}s_{2}^{\prime}$ and $(s_{1}^{\prime},s_{2}^{\prime})\in\mathcal{R}]$
Two states $p$ and $q$ are time abstracted bisimilar iff $Z_{(A,p)}\>\mathcal{R}\>Z_{(B,q)}$ and $\beta$ is the action $a$, $p$ and $q$ are time abstracted delay bisimilar iff $Z_{(A,p)}\>\mathcal{R}\>Z_{(B,q)}$ and $\beta$ is the sequence of actions $\varepsilon.a$ whereas $p$ and $q$ are time abstracted observational bisimilar iff $Z_{(A,p)}\>\mathcal{R}\>Z_{(B,q)}$ and $\beta$ is the sequence $\varepsilon.a.\varepsilon$.
4.4 Complexity
In our work, we decide the timed and the time abstracted relations using a zone graph approach. For a given location, the zones in the zone graph are disjoint. In the worst case, the size of the zone graph is limited by the size of the region graph and it is thus exponential in the number of clocks of the timed automaton. However, in most cases, the size of the zone graph is much smaller than the size of the region graph. Existing approaches for checking timed bisimulation involve a product construction on the region graphs or zone graphs
which characterizes the common behaviour of the two timed automata. The product, along with the individual region graphs or zone graphs is stored in order to check for timed simulation relation or timed bisimulation.
In our case, we do not use a product construction on the zones and use the individual zone graphs of the two timed automata directly for deciding the relations. Thus our method is more space efficient than the approaches that store the product of the region graphs or zone graphs for checking timed bisimulation. Deciding timed bisimulation and timed simulation is known to be EXPTIME-complete [16]. Thus our algorithm is not asymptotically better than existing approaches, however we expect that
we will obtain significant performance gains than existing approaches since our method eliminates the product construction which is an expensive operation. For time abstracted relations, the complexity is similar to the method used in [22] which too uses strong bisimulation on zone graph.
5 Game Characterization
Bisimulation games were defined in [19] for discrete processes. In [6], Game characteizations have been given for relations in the van Glabbeek spectrum [10]. We present here game characterizations for timed relations that is similar to bisimulation games and define the game semantics using our zone graph. As in the bisimulation game, the game is played in rounds on two graphs. The game may be played between the nodes of the zone graph (game for time abstracted relations) or between the timed states appearing in some node of the zone graph (game for timed relations). In each round, the challenger chooses a graph and the defender tries to make a corresponding move on the other graph where the correspondence of the moves is defined in subsection 5.1 in terms of the tuple $\alpha$. If the defender can always make a move in response to the challenger’s move, then it has a winning strategy implying that the two states are related through the relation that corresponds to the game. Otherwise it loses which implies that the two states are not related in which case the challenger is said to have a winning strategy. An alternation occurs if the challenger changes the graph between two consecutive rounds. Alternations are not allowed in simulation preorder and simulation equivalence games. A game always terminates due to the finiteness of the zone graph and due to the fact that the moves of the game are not repeated from a pair of points that have been visited earlier. In the games described in this section, a move is a visible action or a delay action or a sequence of actions where each action belongs to the set $Act\cup\{\varepsilon\}$.
5.1 Game Template
A timed game proposed in this work can be described as $n-\Gamma_{k}^{\alpha,\beta}$. The timed performance prebisimulation game (cp-prebisimulation game) consists of two parts where the second subgame is played if the defender loses in the first subgame. Either of the subgames can be played first and hence the two subgames are connected by a $\vee$. Each game is characterized by the following parameters:
-
$n$ : number of alternations. If not mentioned, then there is no restriction on the number of alternations.
-
$k\in\{{\mathbb{N}\cup\infty}\}$ : number of rounds; $n\leq k-1\text{ when }k\neq\infty$.
-
$\alpha$ : a tuple $\langle\alpha_{1},\alpha_{2}\rangle$. $\alpha_{1}$ denotes the move chosen by the challenger. Depending on the game for the timed relation, either $\alpha_{1}\in Lep$ or $\alpha_{1}\in Act\cup\mathbb{R}_{\geq 0}$ whereas $\alpha_{2}$ denotes the move chosen by the defender and may be the same as $\alpha_{1}$ or may be a sequence of the form $\varepsilon.\alpha_{1}$ or $\varepsilon.\alpha_{1}.\varepsilon$, as in the case of time abstracted relations. For example, for the pair $\langle a/\varepsilon,\varepsilon.a/\varepsilon\rangle$ where $a\in Act$, the challenger makes a move $a$ whereas the defender’s move consists of $\varepsilon$ followed by an $a$. In the case of timed bisimulation game (or cp-bisimulation game), $\alpha$ is assigned $\langle a/d,a/d\rangle$, which denotes that a visible action by the challenger has to be matched by the defender and a delay action $d$ by the challenger has to be matched with an exact delay $d$ move by the defender. In a timed performance prebisimulation game or (cp-prebisimulation game), $\alpha$ is assigned $\langle a/d_{1},a/d_{2}\rangle$ denoting that the delays performed by the challenger and the defender need not be the same.
-
$\beta$ : This is an extra condition which is used in the cp-prebisimulation game. When the game is played between the zone graphs $Z_{A}$ and $Z_{B}$ corresponding to the two timed automata $A$ and $B$, $Z_{A},\leq$ denotes that the delay moves made in $Z_{A}$ are no more than the delays made in $Z_{B}$. $\beta$ if not specified denotes that there is no extra condition.
\xyoption
curve
5.2 Hierarchy of Timed Games
A hierarchy among the timed relations discussed in this paper is captured in Figure 4(a). We show here several lemmas which capture this hierarchy through the game semantics. These lemmas also help us build an infinite game hierarchy which also suggests defining several new timed relations that do not exist in the literature. The arrow from a game $\Gamma_{1}$ to a game $\Gamma_{2}$ denotes that if the defender has a winning strategy for $\Gamma_{1}$, then it also has a winning strategy for $\Gamma_{2}$. Besides in each of the following lemmas, for each pair of games, if $\Gamma_{1}\longrightarrow\Gamma_{2}$, then $\Gamma_{2}\not\longrightarrow\Gamma_{1}$. Figure 4(b) shows the games corresponding to the relations shown in Figure 4(a). The game hierarchy reflects the hierarchy of the timed relations.
Lemma 2.
$\Gamma_{\infty}^{\alpha,\beta}\longrightarrow n\!-\!\Gamma_{\infty}^{\alpha,%
\beta}\longrightarrow(n\!-\!1)\!-\!\Gamma_{\infty}^{\alpha,\beta}$, for all $n>0$
$\Gamma_{k}^{\alpha,\beta}\longrightarrow n\!-\!\Gamma_{k}^{\alpha,\beta}%
\longrightarrow(n\!-\!1)\!-\!\Gamma_{k}^{\alpha,\beta}$, for all $k>0$, $n<k$
Other parameters remaining the same, if the defender has a winning strategy when the challenger is allowed more alternations, then the defender also wins the game where the challenger is allowed only a smaller number of alternations.
Lemma 3.
$\Gamma_{\infty}^{\alpha,\beta}\longrightarrow\Gamma_{k}^{\alpha,\beta}%
\longrightarrow\Gamma_{k-1}^{\alpha,\beta}$, for all $k>0$
$n\!-\!\Gamma_{\infty}^{\alpha,\beta}\longrightarrow n\!-\!\Gamma_{k}^{\alpha,%
\beta}\longrightarrow n\!-\!\Gamma_{k-1}^{\alpha,\beta}$, for all $k>0$, $n<k$
Other parameters remaining the same, if the defender wins the game with more number of rounds, then it also wins the game which has a smaller number of rounds in the game.
Lemma 4.
$n-\Gamma_{k}^{\langle a/d,a/d\rangle}\longrightarrow n-\Gamma_{k}^{\langle a/d%
_{1},a/d_{2}\rangle,(Z_{A},\leq)}$.
$n-\Gamma_{k}^{\langle a/d,a/d\rangle}\longrightarrow n-\Gamma_{k}^{\langle a/d%
_{1},a/d_{2}\rangle,(Z_{A},\leq)}\>\vee\>n-\Gamma_{k}^{\langle a/d_{1},a/d_{2}%
\rangle,(Z_{B},\leq)}$
The first half of the above lemma states that all the parameters remaining the same, if the defender can always reply with an exact delay, then the defender can reply with a delay $d_{2}$ in $Z_{A}$ such that $d_{2}\leq d_{1}$ and it can reply with a delay $d_{2}$ in $Z_{B}$ such that $d_{1}\leq d_{2}$. This also leads to the fact that all the parameters remaining the same, if the defender wins the cp-bisimulation game, then it also wins the cp-prebisimulation game.
Lemma 5.
$n-\Gamma_{k}^{\langle a/d,a/d\rangle}\longrightarrow n-\Gamma_{k}^{\langle a/%
\varepsilon,a/\varepsilon\rangle}\longrightarrow n-\Gamma_{k}^{\langle a/%
\varepsilon,\varepsilon.a/\varepsilon\rangle}\longrightarrow n-\Gamma_{k}^{%
\langle a/\varepsilon,\varepsilon.a.\varepsilon/\varepsilon\rangle}$
If the defender can match a delay action exactly as in the corner point bisimulation, then it can match an epsilon move of the challenger. Also if the defender can reply to a visible action of the challenger, then it can reply with an $\varepsilon.a$ or an $\varepsilon.a.\varepsilon$ move since $\varepsilon$ represents delay including zero delay.
5.3 Infinite Game Hierarchy
On assigning different values to the parameters $n$, $k$, $G$, $\alpha$ and $\beta$ in the game template and using the lemmas given in subsection 5.2, we can generate an infinite game hierarchy which is shown in Figure 4(c). The dashed lines in the figure denote that if the defender has a winning strategy for a game with infinitely many rounds or alternations, then it also wins a game with a finite number of rounds or alternations. Figure 4(b) shows the hierarchy of the games that correspond to the timed relations in Figure 4(a).
The diagram in Figure 4(b) is only a small part of the entire hierarchy of timed games and this leaves us with the scope of studying several timed relations that are not present in the existing literature.
6 Conclusion
In this paper, we present a unified zone based approach to decide various timed relations between two timed automata states. In our method, we do not need the product construction on regions or zones for deciding these relations as done in [5] or [23]. We also provide a game semantics for deciding these timed relations and show that the hierarchy among the games reflects the hierarchy among the relations. The advantage of a game-theoretic formulation is that it allows fairly general relationships between the parameters on $\Gamma$ to define the hierarchy. The fine-tuning and variations of these parameters allow formulations of many more equivalences and preorders than the ones present in the literature related to behavioural equivalences involving real time which otherwise may not be easily captured through operational definitions and reasoning.
Unlike existing approaches which check if two timed automata states are related through some relation, our game approach also allows generating a distinguishing formula that guides us to find a path in one of the zone graphs which was responsible for the relation not holding good between the corresponding states. Identifying this path helps us to refine appropriately an implementation that should conform to a given specification through the relation.
As further work, we plan to extend the game semantics to relations over probabilistic extensions to timed automata [14]. We would also like to investigate the applicability of our zone graph construction for deciding these relations.
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CondenseNet V2: Sparse Feature Reactivation for Deep Networks
Le Yang${}^{1}$ Haojun Jiang${}^{1}$11footnotemark: 1 Ruojin Cai${}^{2}$ Yulin Wang${}^{1}$ Shiji Song${}^{1}$ Gao Huang${}^{1}$ Qi Tian${}^{3}$
${}^{1}$Department of Automation
Equal contribution.Corresponding author.
Tsinghua University
Beijing
China
Beijing National Research Center for Information Science and Technology (BNRist)
${}^{2}$Cornell University
${}^{3}$Huawei Cloud $\mathbf{\&}$ AI
{yangle15, jhj20, wang-yl19}@mails.tsinghua.edu.cn
rc844@cornell.edu
{shijis, gaohuang}@tsinghua.edu.cn
tian.qi1@huawei.com
Abstract
Reusing features in deep networks through dense connectivity is an effective way to achieve high computational efficiency. The recent proposed CondenseNet [15] has shown that this mechanism can be further improved if redundant features are removed. In this paper, we propose an alternative approach named sparse feature reactivation (SFR), aiming at actively increasing the utility of features for reusing. In the proposed network, named CondenseNetV2, each layer can simultaneously learn to 1) selectively reuse a set of most important features from preceding layers; and 2) actively update a set of preceding features to increase their utility for later layers. Our experiments show that the proposed models achieve promising performance on image classification (ImageNet and CIFAR) and object detection (MS COCO) in terms of both theoretical efficiency and practical speed.
1 Introduction
Deep convolutional neural networks (CNNs) have achieved remarkable success in the past few years [34, 8, 16]. However, their state-of-the-art performance is usually fueled with sufficient computational resources, which hinders deploying deep models on low-compute platforms, e.g., mobile phones and Internet of Things (IoT) products. This issue has motivated a number of researchers on designing efficient CNN architectures [16, 3, 12, 45, 26, 35]. Among these efforts, DenseNet [16] is a promising architecture that improves the computational efficiency by reusing early features with dense connections.
Recently, it has been shown that dense connectivity may introduce a large number of redundancies when the network becomes deeper [15]. In a dense network, the output of a layer will never be modified once it is produced. Given that shallow features will be repeatedly processed by their following layers, directly exploiting them in deep layers might be inefficient or even redundant. CondenseNet [15] alleviates this problem via strategically pruning less important connections in DenseNet. ShuffleNetV2 [26] shares a similar spirit, where early features are dropped according to layer-distance, leading to an exponentially decaying of long-distance feature re-usage. Although both models show their effectiveness, we hypothesize that straightforwardly abandoning long connections is overly aggressive. These early features which are considered to be “obsolete ” at deeper layers may contain useful information, which can benefit network generalization ability, and potentially contribute to a more efficient model if properly utilized.
In this paper, instead of directly discarding obsolete features, we are interested in whether we can revive them to make obsolete features useful again. To this end, we develop a novel module to conduct feature reactivation, which learns to update shallow features and enables them to be more efficiently reused by deep layers. Our main idea is illustrated in Figure 1. Compared to DenseNet [16] and CondenseNet [15], where earlier features keep unchanged throughout the whole feed-forward process, we propose to allow the outputs of a layer to be reactivated by later layers. Such a way keeps features maps always “fresh” at each dense layer, and therefore the redundancy in dense connections can be largely reduced.
Although the feature reactivation procedure effectively reduces the redundancy in dense connections, naively reactivating all features will introduce excessive extra computation, which still hurts the overall efficiency. In fact, it is unnecessary to reactivate all features since a large number of them can be already effectively reused without any change in dense connections, resulting in that only sparse feature reactivation (SFR) is required. For this purpose, we develop a cost-efficient SFR module which actively and selectively reactivates early features at each layer, using the increments learned from the newly produced feature maps. Importantly, both the features to be updated and the updating formulas are determined automatically via learning. During the training process, we first assume all previous features require reactivating, and then gradually remove the reactivation that have less effect on feature re-usage. Moreover, the resulting SFR modules can be converted to efficient group convolutions at test time. As a consequence, the proposed method involves minimal extra computational cost or latency and keeps early features “fresh” even through very deep layers, which leads to a significant efficiency gain.
We implement SFR on the basis of the efficient CondenseNet [15], where the SFR along with the learned group convolutions (LGCs) [15] can be learned compatibly to improve the efficiency of dense networks. The resulting network, CondenseNetV2, are empirically evaluated on image classification benchmarks (ImageNet and CIFAR) and the COCO object detection task. The results demonstrate that SFR significantly boosts the performance by encouraging long-distance feature reusing, and that CondenseNetV2 compares favorably with even state-of-the-art light-weighted deep models. We also show that SFR can be plugged into any CNNs that adopt the concatenation based feature reusing mechanism to further improve their efficiency, such as ShuffleNetV2 [26].
2 Related Work
Efficient network architectures.
Designing better network architectures is an effective way to improve the computation efficiency of deep networks. Efficient building units are introduced for light-weighted CNN architectures. For instance, MobileNets [12, 35, 11] propose inverted residuals and linear bottlenecks to build network architectures. Sandglass blocks, which flip the inverted residuals, are developed in MobileNeXts [46]. Cheap
operations are developed for generating features in GhostNet [6]. In addition, shuffle layer and learned group convolution (LGC) are employed by ShuffleNets [45, 26] and CondenseNet [15], respectively. Recent study also shows that developing dynamic neural networks [7] can obviously improve the efficiency of deep models, such as [14, 42, 39]. In this paper, we follow the first line of the research and propose a novel efficient unit named SFR module. The proposed deep models with SFR module retains the simplicity of CondenseNet while significantly improves its accuracy on image classification and detection tasks for mobile applications.
Densely connected neural network.
Compared to ResNet [8] and it variants [41, 44], DenseNet architectures [16, 15, 14, 42] can achieve a higher computational efficiency by encouraging feature reuse. However, superfluous re-usage may introduce redundant connections. To address this problem, existing work mainly proposes to remove dense connections to feature maps that are less useful [15, 38], or to discard long-range connections according to a predefined probability [26]. However, as these seemingly redundant connections may have large potential in deep layers if properly utilized, we propose to conduct sparse feature reactivation to deal with the redundant connections rather than pruning them.
Filter pruning.
Although the sparsifying procedure in SFR module is related to filter pruning methods [40, 10, 19, 9, 24], our method differs greatly from filter pruning methods in the way dealing with the redundant connections. Instead of removing connections between layers where the feature re-usage is superfluous, our approach aims at building reactivation connections to revive obsolete features to increase their utility. Notably, our reactivation idea is orthogonal to filter pruning, and both are utilized for building CondenseNetV2. Additionally, compared with recent work [28], which proposes to graft new weights to the unimportant filters, the proposed SFR module reactivates obsolete features to improve efficiency.
3 Method
A recently confirmed inefficiency in DenseNet [16] architecture lies in the presence of long-distance connections [15], where the deeper layers seem to consider the early features as “obsolete” ones and ignore them during learning new representations. CondenseNet [15] and ShuffleNetV2 [26] alleviate this inefficiency through strategically pruning redundant connections and exponentially discarding cross-layer connections, respectively. In this paper, we postulate in this paper that directly abandoning shallow features can be an overly aggressive design. To be specific, we find that by involving a learnable sparse feature reactivation (SFR) module with a negligible computational cost at each layer, the originally “obsolete” features can be “reactivated” and hence effectively exploited by the later layers. In this section, we first describe the details of the proposed SFR module, and then implement it to build our light-weighted networks.
3.1 Sparse Feature Reactivation
Feature reuse mechanism.
We first formulate the feature reusing mechanism introduced in [16]. Assume that a standard network block of $L$ layers produces $L$ feature maps $\mathbf{x}_{1},\mathbf{x}_{2},\ldots,\mathbf{x}_{L}$, where $\mathbf{x}_{\ell}$ is the output of the $\ell$-th layer and $\mathbf{x}_{0}$ denotes the input feature. Since all previous layers are connected to the $\ell$-th layer via the dense connections, the composite function $H_{\ell}(\cdot)$ of the $\ell$-th layer will take all of $\mathbf{x}_{0},\ldots,\mathbf{x}_{\ell-1}$ as inputs:
$$\mathbf{x}_{\ell}=H_{\ell}([\mathbf{x}_{0},\mathbf{x}_{1},\ldots,\mathbf{x}_{\ell-1}]).$$
(1)
In CondenseNet [15], learned group convolutions (LGC) are employed in $H_{\ell}(\cdot)$ to automatically learn the input groupings and remove unimportant connections, while in ShuffleNetV2 [26], inputs of $H_{\ell}$ will be dropped according to their distance to layer $\ell$. Consequently, both of them remove the superfluous long-distance connections between layers, which are proven effective in term of efficiency. However, given that the output $\mathbf{x}_{\ell}$ will never change once it is produced, a side effect is that these seemingly less useful features from shallow layers tend to be permanently discarded by deeper layers. This static design may impede exploring more efficient feature reusing mechanisms. To this end, we propose a cost-efficient sparse feature reactivation module, enabling obsolete features to be cheaply revived.
Reactivating obsolete features.
We start by describing the details of feature reactivation. For $\ell$-th layer, we introduce a reactivation module denoted by $G_{\ell}(\cdot)$. The module takes $\mathbf{x}_{\ell}$ as input, and its output $\mathbf{y}_{\ell}$ is used to reactive features from preceding layers. In this paper, we define the reactivation operation, $U(\cdot,\cdot)$, as adding111Other reactivation schemes may also be considered, such as applying channel-wise attention or spatial attention. However, in this paper, we note that a straightforward sum has already achieved good performance. the increment $\mathbf{y}_{\ell}$. A dense layer with feature reactivation can be written as
$$\displaystyle\mathbf{x}^{\text{in}}_{\ell}\leftarrow[\mathbf{x}_{0},\mathbf{x}_{1},\ldots,\mathbf{x}_{\ell-1}],\quad\mathbf{x}_{\ell}\!=\!H_{\ell}(\mathbf{x}^{\text{in}}_{\ell}),$$
(2)
$$\displaystyle\mathbf{y}_{\ell}\!=\!G_{\ell}(\mathbf{x}_{\ell}),\quad\mathbf{x}^{\text{out}}_{\ell}\!=\!U(\mathbf{x}^{\text{in}}_{\ell},\mathbf{y}_{\ell}),$$
(3)
$$\displaystyle[\mathbf{x}_{0},\mathbf{x}_{1},\ldots,\mathbf{x}_{\ell-1}]\leftarrow\mathbf{x}^{\text{out}}_{\ell},$$
(4)
where $\mathbf{x}^{\text{out}}_{\ell}$ is the reactivated output feature. With $H_{\ell}(\cdot)$, the $\ell$-th layer learns to produce new feature $\mathbf{x}_{\ell}$. Additionally, previous representations ($\mathbf{{x}}_{i},i\!=\!1,..\ell\!\!-\!\!1$) will be reactivated to increase their utility.
Obviously, it is unnecessary to reactivate all features since a large number of them can be effectively reused without any change (shown in DenseNet [16]). We also empirically observe that dense reactivation will introduce much computation and degrade the overall efficiency of the network. Therefore, we seek to automatically find the features required to be reactivated and merely refresh them. In the following, this aim is formulated by a pruning based approach and can be achieved gradually during training (shown in Figure 2). The resulting architecture is named as sparse feature reactivation (SFR) module.
Sparse feature reactivation (SFR).
WLOG, we assume that the reactivation module $G_{\ell}(\cdot)$ consists of a regular 1$\times$1 convolutional layer followed by batch normalization (BN) [17] and a rectified linear unit (ReLU) [29]. The size of the filter weight matrix $\mathbf{F}$ in $G_{\ell}(\cdot)$ is represented as $(O,I)$, where $O$ and $I$ denote the number of output and input channels222We can apply max-pooling on the absolute value of the 4D weights, $\mathbf{F}\in\mathbb{R}^{O\times I\times k\times k}$, to generate the matrix with the size of $(O,I)$ when dealing with larger convolutional kernels.. In each $G_{\ell}(\cdot)$, we divide $\mathbf{x}_{\ell}$ into $G$ groups, and the $\mathbf{F}$ is split correspondingly among the input channel dimension to obtain $G$ groups $\mathbf{F}^{1},\ldots,\mathbf{F}^{G}$, and each of them has the size of $(O,I/G)$. To sparsify the reactivation connections, we further define a sparse factor $S$, which may differ from $G$, and allow each group to only select $\frac{O}{S}$ output channels to reactivate after training.
During training, in each $G_{\ell}(\cdot)$, the connection pattern is controlled by $G$ binary masks. Therefore, we aim at learning these binary masks $\mathbf{M}^{g}\in\{0,1\}^{O\times\frac{I}{G}},g\!=\!1,...,G$ to screen out unnecessary connections in $\mathbf{F}^{g}$ by zeroing the corresponding values. In other words, the weight of $g$-th group can be obtained by $\mathbf{M}^{g}\odot\mathbf{F}^{g}$, where $\odot$ denotes the element-wise multiplication.
We then introduce how to train a network with SFR modules in an end-to-end manner.
Inspired by [15], the whole training process consists of $S-1$ sparsification stages followed by an optimization stage. Assuming that $E$ denotes the total number of training epochs, we set the training epochs of each sparsification stage to $\frac{E}{2(S-1)}$ and optimization stage to $\frac{E}{2}$. During training, the SFR module first reactivates all features, and then gradually removes the superfluous connections. Therefore, at the beginning of training, we set all $\mathbf{M}^{g}$ to all-ones matrices, thus all input feature maps in $g$-th group are connected with all output features. During sparsification, the importance of reactivating $i$-th output within $g$-th group is measured by the L1-norm of the corresponding weights $\sum_{j=1}^{I/G}|\mathbf{F}_{i,j}^{g}|$. At the end of each sparsification stage, $\frac{O}{S}$ output features whose L1-norms are smaller than others are pruned in $g$-th group, and $\mathbf{M}_{i,j}^{g}$ is set to zero for all $j$ in $g$-th group for each pruned output feature map $i$. Note that, if $i$-th output feature map is pruned from every input group, the $i$-th feature map in $\mathbf{y}$ will equal to $\mathbf{0}$, which implies that the $i$-th feature map from previous layers do not need to reactivate. Therefore, after training, each input group will only update the outputs with a portion of $1/S$, which means that for final $\mathbf{M}^{g}$, we have $\sum_{i=1}^{O}\mathbf{M}^{g}_{i,j}\!=\!\frac{O}{S}$. The higher the value of $S$, the sparser the connection pattern is.
Convert to standard group convolution.
At test time, our SFR model can be implemented
using a standard group convolution and an index layer, allowing for efficient computation in practice. This is illustrated in Figure 3: the converted group convolution contains $G$ groups with output and input channels as $(\frac{OG}{S},I)$. After generating the intermediate features with the group convolution, the index layer is applied to rearrange the features by their indices to obtain the $\mathbf{y}_{\ell}$. Note that the intermediate features with the same index will first be summed and then arranged according to the index.
3.2 Architecture Design
Architecture of CondenseNetV2.
Based on the proposed SFR module, the new dense layer of CondenseNetV2 is shown in Figure 4 (right), which is designed on the basis of CondenseNet [15]. In the proposed architecture, the LGC first selects important connections and the new representations $\mathbf{x}_{\ell}$ are generated based on these selected features using Eq. (2). Then, the SFR module takes $\mathbf{x}_{\ell}$ as input and learns to reactivate the obsolete representations. The refreshed features can be derived by Eq. (3). Following [15], we shuffle the output channels of each convolutional layer to ensure communication between different groups. It is worth knowing that the CondenseNetV2 is essentially different from CondenseNet [15]: the outputs of each layer in CondenseNet [15] will never change once it is produced. Therefore, the potential re-usage of previous features can be blocked. In contrast, old features can be reactivated in each layer of CondenseNetV2, resulting in a more efficient and effective feature reuse mechanism.
The architecture of CondenseNetV2 follows the exponentially increasing growth rate and fully dense connectivity design principle [15]. Based on the newly designed SFR-DenseLayer, we develop our CondenseNetV2 as presented in Table 1. The squeeze and excite (SE) module [13] and hard-swish nonlinearity function (HS) are also applied following [11]. The presented architecture provides a basic design for reference, further hyper-parameters tuning or network architecture searching can further boost the performance.
3.3 Sparse Feature Reactivation in ShuffleNetV2
As SFR is able to be plugged into any CNNs with the feature reusing mechanism, we claim that ShuffleNetV2 [26] can also benefit from the proposed SFR module. The implementation details are illustrated in Figure 5. We refer to the modified ShuffleNetV2 with sparse feature reactivation as SFR-ShuffleNetV2. Note that in SFR-ShuffleNetV2, only basic units conduct the feature reactivation, and the units for spatial down sampling keep unchanged. The detailed architecture is provided in the Appendix.
4 Experiments
We empirically demonstrate the effectiveness of the proposed SFR module and CondenseNetV2 on image classification and object detection tasks, and compare with state-of-the-art light-weighted CNN architectures. Code is available at https://github.com/jianghaojun/CondenseNetV2.
Dataset.
Experiments are conducted on several benchmark visual datasets, including CIFAR-10 and CIFAR-100 [18], ImageNet (ILSVRC2012 [4]), and MS COCO object detection benchmark [22].
The CIFAR-10 and CIFAR-100 datasets consist of $32\!\times\!32$ RGB images, with 10 and 100 classes of natural scene objects, respectively. Both CIFAR datasets contain 50,000 training images and 10,000 test images. On the two CIFAR datasets, following [16], we apply a set of transformations to augment the training set. The ImageNet (ILSVRC2012 [4]) classification dataset contains 1.2 million training images and 50,000 validation images, with 1000 classes. Data augmentation schemes are applied at training and we adopt a $224\!\times\!224$ center crop at test time. On MS COCO dataset, following [33, 20], we use the trainval35k split as training data and report the results in mean Average Precision (mAP) on minival split with 5000 images.
4.1 Efficiency of Sparse Feature Reactivation
In this subsection, we conduct a series of experiments on densely connected networks with sparse feature reactivation to verify its effectiveness.
Reactivated features.
As the proposed SFR module is designed for reactivating the redundant features to improve the network efficiency, a natural question is whether the utility of these obsolete features indeed shows great importance at later layers after being reactivated. To investigate this question, Figure 6 and 7 visualize the learned weights for CondenseNet [15] and CondenseNetV2 to verify whether those reactivation will indeed encourage feature reuse.
Figure 6 shows detailed weight strength (averaged absolute value of non-pruned weights) between a filter group of a certain layer (corresponding to a column in the figure) and an input feature map (corresponding to a row in the figure). For each layer, there are four filter groups (consecutive columns). Red dots are connections with significant contributions and white dots are connections that have been pruned. One can observe that connections in CondenseNet are more concentrated in neighbor layers, while long-distance connections appear more frequently in CondenseNetV2 (shown by dense colored dots in the top-right corner). This implies that later layers make more use of feature maps produced by early layers in CondenseNetV2 than in CondenseNet.
Figure 7 shows the overall connection strength between two layers in the CondenseNet and CondenseNetV2. The color shows the L1-norm of weights between layers and red means large weight. We notice that the top-right parts of figures for CondenseNetV2 are more brilliant than these corresponding areas of figures for CondenseNet, which implies that the utility of early features largely increases in CondenseNetV2. As the performance of CondenseNetV2 is shown to be superior to CondenseNet, we can conclude that the dense network will benefit from sparse feature reactivation. This validates our hypothesize: although the features produced by early layers seem unimportant at deep layers in dense networks, they may have potential after being reactivated. Moreover, from the results in Figure 7 in (b) and (d), we further observe that early features are more frequently utilized at deep layers in the model trained on ImageNet than the model trained on CIFAR-10, from which we can infer that the reactivated early features show more importance in complicate tasks.
Necessity of sparse feature reactivation.
Although the dense feature reactivation can improve the network performance, the involved extra computation is much, which degrades the overall efficiency of the network. Therefore, it is necessary to make the reactivation sparse. We conduct experiments on CIFAR-10 using CondenseNetV2 with $S=1$ and $S>1$. The experimental results are shown in Figure 8 (b). From the results, we observe that the dense feature reactivation ($S=1$) becomes inefficient due to the heavy extra computational overheads. On the contrary, conducting sparse feature reactivation ($S>1$) can deal with the aforementioned problems effectively, and therefore, can boost the network performance with minor extra computation.
The hyper-parameters in SFR module.
We conduct the experiments with CondenseNetV2 on CIFAR-10 for evaluation. We fix the group number and the condense factor for LGC layers, and we only compare different settings for SFR module. In Figure 8 (a), the sparse factor is fixed to 4 and we show the effect of group number $G$ which actually does not affect the FLOPs of the network. One can observe that as $G$ increases, the performance improves gradually. This is due to that a finer-grained sparsification, which corresponds to a large $G$, is usually able to achieve higher efficiency. In Figure 8 (b), we compare CondenseNetV2s with varying sparse factors in SFR module. A network with a sparse factor $S$ of 1 means that all reactivation connections are preserved (dense reactivation). If $S$ is increased to $4$, then each layer only keeps a quarter of reactivation connections. The results show that $S=4$ outperforms other settings. Also, all the settings with $S>1$ perform better than the setting with $S=1$, indicates that removing a set of unnecessary connections is indeed important for building efficient CondenseNetV2.
4.2 Experiments on ImageNet
Implementation details.
Following the common practice [12, 45, 26], the proposed network has three levels of computational complexity. The CondenseNetV2 with different sizes are summarized in Table 2, where $d$ and $k$ are number of layers and growth rate of each dense block, respectively. In all experiments, we use the same condense factor ($C$), group number ($G$), and sparse factor ($S$) for all LGCs and SFR modules in the network. CondenseNetV2 are trained using the stochastic gradient descent (SGD) optimizer with an initial learning rate of 0.4, the cosine learning rate [25], and The batch size of 1024. To compare with SOTA baselines, we implement an Augmented Setting differing from the original setting in [15]. More details can be found in Appendix C.
Results on ImageNet.
We conduct experiments on ImageNet to evaluate the effectiveness of the proposed methods. As the SFR module can be deployed in both CondenseNet and ShuffleNetV2, we first compare the original model and the network with SFR module in Table 3. From the results, we can see that the latter clearly exceeds the former. The computational cost of CondenseNetV2-A is 18% lower than CondenseNet-A (46M v.s. 56M). Using SFR on ShuffleNet can also boost the efficiency with minor extra FLOPs (fewer than 3%). Moreover, we conduct ablation studies on ImageNet to show how each additional design benefit the original CondenseNet. The results are shown in Table 4, from which we observe that implementing the proposed SFR can boost the performance of CondenseNet by a large margin.
We further compare our networks with several efficient network architectures designed by handcraft, including MobileNetV2 [35], CondenseNet [15], MobileNeXt [46] and ShuffleNetV2[26]. The results are shown in Figure 9 (a). One can observe that the proposed CondenseNetV2 outperforms these models in terms of the computational efficiency.
In addition to the deep models in Figure 9, networks based on neural architecture search (NAS) are further compared with the proposed networks, including MobileNetV3 [11], RegNetX [31], ProxylessNAS[1] and MnasNet [37]. The results are summarized in Table 5. We group different models according to their computational costs. Importantly, the proposed CondenseNetV2 does not leverage any technique of NAS, however, it can outperform most of the competitive baselines with similar FLOPs.
To show how our method can benefit from NAS, we further deploy our SFR module on ShuffleNetV2+333https://github.com/megvii-model/ShuffleNet-Series/tree/master/ShuffleNetV2%2B, a strengthened version of ShuffleNetV2 obtained by one-shot NAS based on ShuffleNet Units. The proposed SFR-ShuffleNetV2+ outperforms both MobileNet V3 and the original ShuffleNetV2+, which confirms the effectiveness of the proposed SFR module. More experiments on ImageNet are provided in Appendix B.
Actual inference time.
Since the proposed CondenseNetV2 is designed for edge devices, we further measure the actual inference speed of CondenseNetV2 on an ARM processor444Quad-Core ARM Cortex-A57 MPCore combined with Dual-Core NVIDIA Denver 2 64-Bit CPU. and an iPhone XS Max (with Apple A12 Bionic). The single-thread mode with batch size 1 is used following [11] and we use a 224$\times$224 input image.
On the ARM processor, all models are implemented in PyTorch1.6.0. From Figure 9 (b), one can observe that the proposed CondenseNetV2 achieves faster runtime under the same error compared with other light-weighted deep models. Specifically, our model obtains about 0.5% lower top-1 error than MobileNetV3 with slightly lower latency. It is noteworthy that the power of the tested processor is lower than most smart mobile phones. We believe that such a speed test is necessary: although mobile phones and processors with high performance have been widely deployed and popularized nowadays, the computational resources of most edge devices, such as IoT products, are still highly limited. The proposed CondenseNetV2 outperforms most light-weighted networks in such a resource-limited scenario.
We further test the inference time on an iPhone XS Max, which can be considered as a high performance edge device. Our implementation is based on the Pytorch Mobile555https://pytorch.org/mobile/home/.. The results are presented in Figure 9 (c), from which we see that the SFR-ShuffleNetV2 outperforms other competitors. Here, only SFR-ShuffleNetV2 are tested because we found that the Pytorch Mobile might have poor support for group convolution operator: although CondenseNetV2-A only has 46M FLOPs, its latency on iPhone is still up to 34.6ms.
4.3 Experiments on CIFAR
Implementation details.
We apply SGD to train all the models with similar hyper-parameters setting as in [15]. We use the cosine learning rate annealing with an initial learning rate of 0.1. The training process lasts for 300 epochs with a mini-batch size of 64. Other training settings are the same as the experiments on ImageNet. CondenseNetV2s on CIFAR follow the configuration listed below. The network consists of three dense blocks with the same number of layers, and the resolutions of feature maps are $32\!\times\!32$, $16\!\times\!16$, and $8\!\times\!8$, respectively. The growth rates are set to 8, 16, 32 for each block. The $C$, $S$, and $G$ are all set to 4. We modify the number of blocks $d$ in each stage to change the computational complexity of CondenseNetV2. Moreover, we do not implement SE and HS in CondenseNetV2 for CIFAR models, and the last Conv2d 1$\times$1 is also removed.
Results on CIFAR.
We show the comparison results of CondenseNetV2s and other competitive baselines in Table 6. The baselines include several recently proposed network pruning algorithms. It can be observed that the CondenseNetV2s outperform all other approaches with lower error rates and less computational costs — indicating that the effectiveness of the proposed feature reuse mechanisms.
4.4 Experiments on MS COCO
MS COCO [22] is used for evaluating the generalization ability of our networks. Following [33], we use the trainval35k split as training data and report the results in mean Average Precision (mAP) on minival split. Faster R-CNN [33] with Feature Pyramid Networks (FPN) [20] and RetinaNet [21] are implemented as detection frameworks. Only backbone networks are replaced during experiments. Models are pretrained on ImageNet and then finetuned on the detection task. During finetuning, we train all models using SGD for 12 epochs. The input images are resized to a short side of 800 and a long side not exceed 1333.
The backbone FLOPs are calculated with 224$\times$224 input size following [26].
The detection results are shown in Table 7. As we can see, with comparable computational cost, our CondenseNetV2-C achieves higher mAP compared with ShuffleNetV2 and MobileNetV2, both on RetinaNet and Faster R-CNN frameworks.
5 Conclusion
In this paper, we proposed a novel sparse feature reactivation module, which can strategically reactivate a set of previous features to increase their utility for later layers. Importantly, the features to be reactivated are not pre-defined, but learned automatically during training. Due to the sparsity of the feature reactivation, this procedure can be highly computational-efficient. Therefore, the resulting model, CondenseNetV2, based on the proposed SFR module can achieve high efficiency during inference. Encouraging results have been obtained on the image classification tasks (ImageNet and CIFAR) and the COCO object detection task, without resorting to neural architecture search.
Acknowledgement
This work is supported in part by the National Key R$\&$D Program of China (2020AAA0105200), the National Natural Science Foundation of China (61906106, 62022048), the Institute for Guo Qiang of Tsinghua University and Beijing Academy of Artificial Intelligence.
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Appendix A Network Structure
A.1 Implementation details of SFR-ShuffleNetV2 and SFR-ShuffleNetV2+
In this section, we provide more details for building SFR-ShuffleNetV2 and SFR-ShuffleNetV2+ based on ShuffleNetV2 [26] and ShuffleNetV2+666https://github.com/megvii-model/ShuffleNet-Series/tree/master/ShuffleNetV2%2B, respectively. The general network architecture of SFR-ShuffleNetV2 and SFR-ShuffleNetV2+ are shown in Table 8 and Table 10. The ShuffleNet Unit with stride 2 in Table 8 is implemented following [26]. The details of SFR-ShuffleNet Unit is illustrated in Figure 5 of the main paper. For SFR-ShuffleNetV2+, we follow the same principle to integrate the SFR modules into different ShuffleNet Units. Moreover, different from CondenseNetV2, the SFR procedure is conducted by 3$\times$3 convolutions in SFR-ShuffleNet. There are four types of ShuffleNet Units in ShuffleNetV2+ (refer to the original implementation of ShuffleNetV2+ for more details). Note that we do not conduct the feature reactivation on the unit with a stride equal to 2. Additionally, the network configurations of SFR-ShuffleNetV2 and SFR-ShuffleNetV2+ are provided in Table 9 and 11.
A.2 Implementation details of CondenseNet
In CondenseNet [15], only the network architecture of CondenseNet-C with 300M FLOPs is provided. In order to conduct a more comprehensive comparison, we further design CondenseNet-A/B which are under another two computation levels(50M and 150M). The general network architecture and configurations of CondenseNet are provided in Table 12 and Table 13, respectively. Here, the DenseLayers in Table 12 are implemented with learned group convolutions.
Appendix B A comperhensive study of efficient deep learning models on ImageNet
In this section, we provide more comprehensive comparisons between the proposed network architectures and other state-of-the-art efficient deep learning models. These models are grouped into three levels of computational costs, including 50M, 150M and 300M FLOPs. Our comparisons include the most efficient network architectures: (1) Handcrafted Light-weighted CNN architectures, such as CondenseNet [15], MobileNetV1 [12], MobileNetV2 [35], ShuffleNetV1 [45], ShuffleNetV2 [26], IGCV3 [36], Xception [3] and ESPNetV2 [27], are shown in Table 14. 2) NAS based methods, such as NASNet [47], PNASNet [23], MnasNet [37], ProxylessNas [1], AmoebaNet [32], GhostNet [5], MobileNetV3 [11] and RegXNet [31], are shown in Table 15.
From the results in Table 14, we conclude that the proposed CondenseNetV2 are superior to many other handcraft-designed efficient deep CNNs significantly. We also observe that the proposed networks outperform the CondenseNets by a large margin, which demonstrates that the effectiveness of our SFR module.
As we can see, in Table 15, our CondenseNetV2-C surpass most of the efficient models based on NAS under the computational budget of $\sim$300M. Note that our CondenseNetV2-C’s FLOPs is only half of NASNet-A, AmoebaNet-C and PNASNet-5, however, CondenseNetV2-C still outperform these NAS based models. Although the EfficientNetB0 achieves the best performance when the computational budget is $\sim$300M, its FLOPs is much larger than CondenseNetV2-C (390M vs 309M). The MobileNetV3 outperforms our models with $\sim$50M and $\sim$150M FLOPs which can be due to the effectiveness of NAS. Since our implemented SFR-ShuffleNetV2+ Small can surpass the MobileNetV3 Large 0.75 $\times$ by 1.2 percent in terms of Top-1 Error, we believe that CondenseNetV2’s performance can be further boosted by NAS algorithms. Therefore, the future work will mainly focus on applying NAS methods on the proposed CondenseNetV2.
Appendix C Experimental Setup on ImageNet
In our experiments, all our models are conducted under Pytorch Implementation [30]. Our training is based on the open-source code777https://github.com/rwightman/pytorch-image-models/ which successfully reproduces the reported performance in MobileNetV3 [11]. We follow most of the training settings used in MobileNetV3 [11], except that we use the stochastic gradient descent (SGD) optimizer with an initial learning rate of 0.4, the cosine learning rate [25], a Nesterov momentum of weight 0.9 without dampening and a weight decay of $4\!\times\!10^{-5}$ when batch size is 1024. |
Experimental method for perching
flapping-wing aerial robots
††thanks: The research is funded by the European
Project GRIFFIN ERC Advanced Grant 2017, Action
788247.
Raphael Zufferey
GRVC Robotics Lab,
University of Seville
Seville, Spain, and
EPFL, Switzerland
raph.zufferey@gmail.com
Daniel Feliu-Talegón
GRVC Robotics Lab
University of Seville
Seville, Spain
danielfeliu@us.es
Saeed Rafee Nekoo
GRVC Robotics Lab
University of Seville
Seville, Spain
saerafee@yahoo.com
Jose-Angel Acosta
GRVC Robotics Lab
University of Seville
Seville, Spain
jaar@us.es
Anibal Ollero
GRVC Robotics Lab
University of Seville
Seville, Spain
aollero@us.es
Abstract
In this work, we present an experimental setup and guide to enable the perching of large flapping-wing robots. The combination of forward flight, limited payload, and flight oscillations imposes challenging conditions for localized perching. The described method details the different operations that are concurrently performed within the 4 second perching flight. We validate this experiment with a 700 g ornithopter and demonstrate the first autonomous perching flight of a flapping-wing robot on a branch. This work paves the way towards the application of flapping-wing robots for long-range missions, bird observation, manipulation, and outdoor flight.
I Introduction
Flapping-wing flight offers significant benefits over rotor-based propulsion ranging from noise reduction, increased safety, and potentially higher maneuverability and efficiencies [1]. While these traits could grant safe flight around structures, significant challenges remain to land large flapping-wing robots on perches and branches.
Indeed, the combination of forward flight requirement and vertical center of mass oscillations arising in large flapping-wing robots vastly increases the difficulty to perch on objects. Achieving this task - as we often see birds do seamlessly everyday - is complex in terms of the control required in stall conditions and in terms of grasping appendage capability. In a robot, this would require a vehicle capable of carrying a claw system that can hold the robot upright for manipulation, a vision system to identify the branch and a control algorithm that can reliably bring the robot there with minimal remaining velocity. We propose to solve the perching maneuver in a controlled context as a significant step towards vision-based outdoor perching. Here, we would like to show a flight experiment to demonstrate autonomous perching of flapping robots in controlled conditions. Complete information about the design, modeling and resulting flight performance of the robotic bird can be found in [2].
II Experiment
Demonstration and validation of this experimental method is performed with a large 700 g, 150 cm-wingspan ornithopter whose design stems from a previous robot class described in [3]. Successful perching flights were achieved through a series of state phases, visually represented in Fig. 1 and described in this work.
II-A Reaching flight velocity
Large ornithopters are not capable of hovering in windless conditions, due to unfavorable scaling. Such robots can only sustain flight with forward velocity and therefore require a process to reach this velocity initially. While one can certainly achieve this velocity via a hand throw, this results in a wide range of initial conditions. We instead propose an automated launcher system, which boasts sufficient precision for repeatable flights in space-constraints, indoor conditions, see Fig. 1.A.
The system features a 2 m long double rail which can accelerate a robot to 10 m/s. In our validation scenario, speeds of 4 m/s are sufficient to initiate flight without losing altitude and without excessive speed. The robotic birds are held at a perpendicular offset of 50 cm, sufficient to avoid any collisions between the wing, tail and structure. Launcher mounting of robot occurs at the trailing edge, to remain as close as possible to the center of mass. A 20 degree angle-of-attack at launch is selected. Results show that launches are consistent, both in direction and speed, allowing us to repeatedly perform controlled flights to a point location at a 14 m distance from the launcher.
II-B Controlled flapping flight
At the end of the rail, the robot’s position and attitude are captured by the motion capture system and start being wirelessly streamed to the robot’s flight companion computer. The ornithopter enters a controlled flapping flight phase, see Fig. 1.B. This is handled by the onboard triple loop pitch-yaw-altitude controller. This flight control system needs to handle stable flight with the appendage payload, misalignment compensation system and all the onboard electronics required for flight and perch.
The forward flight velocity is a key parameter for perching. It should be kept low to minimize impact forces and increase reaction time before impact. To achieve this, the pitch angle of the robot needs to be high. Experiments show that pitch angles above $40\text{\,}\mathrm{\SIUnitSymbolDegree}$ result in insufficient speed and consequently loss of lateral control. As such, a pitch angle of $30\text{\,}\mathrm{\SIUnitSymbolDegree}$is appropriate for perching flights and is sufficient to fly stably. This set-point is maintained by a proportional-integral (PI) controller. Consequently, when the pitch reaches $30\text{\,}\mathrm{\SIUnitSymbolDegree}$, the velocity drops to 2.5-3 m/s, kept until perching.
The altitude target value is achieved through regulated flapping frequency control action. Using this method, all flight trajectories reached a 2 m set-point within 8-12 m of flight distance. The flight controller performs adequately for different branch altitudes, with a maximum average vertical error of 16 cm and a lateral error of rarely exceeding 60 cm.
II-C Branch approach
The claw dimension is smaller than the flight position accuracy. As this is insufficient to reliably touch the target, we propose an correction method. The leg of the robotic bird is articulated at the elbow level, permitting semi-vertical motion of the claws. When the robot gets to within 150 cm of the branch, an optical detection system is enabled, see Fig. 1.C. The line detection system located on the claw feeds the relative position of the branch to the leg microcontroller. The leg servo then corrects the angle of the leg to align it with the branch, at 50 Hz. During this approach phase, the flapping motion is maintained and the flight controller remains active. The perching target, a 80 cm-black-painted branch is held at 2 m height at its center by a $45\text{\,}\mathrm{\SIUnitSymbolDegree}$ aluminum profile which reaches out through the safety net.
II-D Perching
At 20 cm from the branch, the flapping is stopped. At this point, the forward velocity of the robot is situated between 2.5 and 3 m/s. Grasping is made possible thanks to a pair of bistable claws which pivot from the open to the closed state within 25 ms and with a 2 Nm torque. A combination of soft silicone pads and spikes offers friction forces sufficient to compensate the torque resulting from center of mass offsets.
Experiments have demonstrated that we can reach the branch location repeatedly, within the tolerance range of the leg-claw system. The low forward flight velocity shortly before landing ensures that forces remain below 150 N, significantly reducing the likelihood of damage. We report that branch perching was achieved in 6 out of 9 perching flight tests, and that no damage occurred in any of the successful flights. The results were validated with a second robot which perched without re-tuning.
References
[1]
G. de Croon, “Flapping wing drones show off their skills,” Science Robotics, vol. 5, no. 44, p. eabd0233, 2020.
[2]
R. Zufferey, J. T. Barbero, D. F. Talegon, S. R. Nekoo, J. A. Acosta, and A. Ollero, “How ornithopters can perch autonomously on a branch,” Nature Communications 13 (1), 7713, 2022.
[3]
R. Zufferey, J. Tormo-Barbero, M. Mar Guzman, F. J. Maldonado, E. Sanchez-Laulhe, P. Grau, M. Perez, J. A. Acosta, and A. Ollero, “Design of the High-Payload Flapping Wing Robot E-Flap,” IEEE Robotics and Automation Letters, vol. 6, no. 2, pp. 3097–3104, 2021. |
Influence of Ising-anisotropy on the
zero-temperature phase transition in the square lattice
spin-$\frac{1}{2}$ $J-J^{\prime}$ model
R. Darradi†,
J. Richter†and S.E. Krüger‡
†Institut für Theoretische Physik, Universität Magdeburg,
P.O. Box 4120, D-39016 Magdeburg, Germany
‡IESK, Kognitive Systeme, Universität Magdeburg,
P.O. Box 4120, D-39016 Magdeburg, Germany
Abstract
We use a variational mean-field like approach,
the coupled cluster method (CCM) and exact diagonalization
to investigate the ground-state order-disorder transition
for the square
lattice spin-half XXZ model with two different nearest-neighbor
couplings $J$
and $J^{\prime}$. Increasing $J^{\prime}>J$ the model shows in the isotropic Heisenberg limit
a second-order
transition from semi-classical Néel order to a quantum paramagnetic
phase with enhanced local dimer correlations on the $J^{\prime}$ bonds at about
$J^{\prime}_{c}\sim 2.5\ldots 3J$.
This transition is driven by the quantum competition between $J^{\prime}$ and $J$.
Increasing the anisotropy parameter $\Delta>1$ we diminish the quantum
fluctuations and thus the degree of competition.
As a result the transition point $J^{\prime}_{c}$ is shifted to larger values.
We find indications for a linear increase of $J^{\prime}_{c}$ with $\Delta$,
i.e. the transition disappears in the Ising limit
$\Delta\to\infty$.
1 Introduction
The study of zero-temperature phase transitions driven by quantum fluctuations
has been a subject of great interest to
physicists during the last decade, see [1, 2, 3, 4]
and references therein.
For order-disorder quantum phase
transitions we basically need the interplay between the interparticle
interactions and fluctuations. A canonical model to study quantum
phase transitions is the spin-half Heisenberg
model with competing interactions in two dimensions.
Competition between bonds appears in frustrated systems.
Besides frustration,
there is another mechanism weakening the
ground-state Néel order in
Heisenberg antiferromagnets,
namely the competition of non-equivalent nearest-neighbor (NN)
bonds leading to the
formation of local singlets of two (or even four)
coupled spins.
By contrast to frustration, which yields
competition in quantum as well as in classical systems, this type of
competition is present only in quantum systems.
Recent experiments on
$\mathrm{SrCu}_{2}(\mathrm{BO}_{3})_{2}$
[5, 6] and on $\mathrm{CaV}_{4}\mathrm{O}_{9}$[7, 8]
demonstrate the
existence of gapped
quantum paramagnetic ground states in (quasi-)two-dimensional
Heisenberg systems and have stimulated various theoretical studies of
quantum spin lattices with competing interactions.
A famous example for competition in a frustrated Heisenberg antiferromagnet
is the spin-half
$J_{1}-J_{2}$ model on
the square lattice, where the frustrating $J_{2}$ bonds plus quantum
fluctuations lead to a second-order transition from Néel ordering to a
disordered quantum spin liquid, see e.g. [9, 10, 11, 12, 13].
On the other hand,
it has been predicted that frustration may lead
to a first-order transition in
quantum spin systems in contrast to a second-order transition in the
corresponding classical model [14, 15, 16, 17].
An example for competition without frustration is the ’melting’
of semi-classical Néel order by local singlet formation in
Heisenberg systems with two non-equivalent nearest-neighbor bonds
like the bilayer antiferromagnet [18, 19],
the $J-J^{\prime}$ antiferromagnet
on the square lattice [17, 20, 21] and
on the depleted square
(CaVO) lattice [22, 23].
In the above mentioned
papers[17, 18, 19, 20, 22, 23] the strength
of quantum fluctuations is tuned by variation of the exchange bonds.
Alternatively, the strength of quantum fluctuations can be tuned by
the anisotropy
$\Delta$ in an XXZ model.
In this paper we study the influence of the Ising anisotropy
on the zero-temperature magnetic order-disorder transition
for the $J-J^{\prime}$ spin-half XXZ antiferromagnet
on the square lattice.
We use a variational
mean-field like approach (MFA), the coupled cluster method (CCM) and exact
diagonalization (ED). We mention that the CCM, being
one of the most
powerful methods of quantum many-body theory, has previously been
applied to various quantum spin systems with much success
[12, 17, 21, 24, 25, 26, 27, 28].
2 Model
We consider an anisotropic spin-$\frac{1}{2}$ Heisenberg (XXZ) model on the
square lattice
with two kinds of nearest-neighbor bonds $J$ and $J^{\prime}$, as shown in
Fig.1:
$$\displaystyle H=J\sum_{<ij>_{1}}\left(s_{i}^{x}s_{j}^{x}+s_{i}^{y}s_{j}^{y}+%
\Delta s_{i}^{z}s_{j}^{z}\right)+J^{\prime}\sum_{<ij>_{2}}\left(s_{i}^{x}s_{j}%
^{x}+s_{i}^{y}s_{j}^{y}+\Delta s_{i}^{z}s_{j}^{z}\right).$$
(1)
The sums over $<ij>_{1}$ and $<ij>_{2}$ run over
the two kinds of nearest-neighbor bonds, respectively (cf. Fig.1).
Each square-lattice plaquette consists of three $J$ bonds and one $J^{\prime}$
bond. We consider antiferromagnetic bonds $J^{\prime}\geq J>0$, i.e. there is no
frustration in the model. In what follows we set $J=1$ and consider the
Ising anisotropy $\Delta\geq 1$ and $J^{\prime}$ as the parameters of the
model.
Since there is no frustration the classical ground state is the
two-sublattice Néel state.
3 Methods
3.1 Variational mean-field approach (MFA)
For the square-lattice antiferromagnet ($J=J^{\prime}$) the ground state is Néel
ordered. The corresponding uncorrelated mean-field state
is the Néel state
$|\phi_{MF_{1}}\rangle=|\hskip-3.698858pt\uparrow\rangle|\hskip-3.698858pt%
\downarrow\rangle|\hskip-3.698858pt\uparrow\rangle|\hskip-3.698858pt\downarrow\rangle\ldots$ . In the limit $J^{\prime}\to\infty$ and for finite $\Delta$ the ground
state approaches a rotationally invariant
product state of local pair singlets (valence-bond state)
$|\phi_{MF_{2}}\rangle=\prod_{i\in A}\left\{|\uparrow_{i}\rangle|\downarrow_{i+%
\hat{x}}\rangle-|\downarrow_{i}\rangle|\uparrow_{i+\hat{x}}\rangle\right\}/%
\sqrt{2}\;,\quad$
where
$i$ and $i+\hat{x}$ correspond to those sites which cover the $J^{\prime}$ bonds.
In order to describe the transition between both states, we consider
an uncorrelated
product state of the form[17, 19]
$$|\Psi_{var}\rangle=\prod_{i\in A}\frac{1}{\sqrt{1+t^{2}}}[|\uparrow_{i}%
\downarrow_{i+\hat{x}}\rangle-t|\downarrow_{i}\uparrow_{i+\hat{x}}\rangle].$$
(2)
The trial function $|\Psi_{var}\rangle$
depends on the variational parameter $t$ and
interpolates between the valence-bond state $\;|\phi_{MF_{2}}\rangle\;$
realized for
$t=1$ and the Néel state $\;|\phi_{MF_{1}}\rangle\;$ realized
for $t=0$. By minimizing $E_{var}=\langle\Psi_{var}|H|\Psi_{var}\rangle$
with respect to the variational parameter $t$ we obtain
$$\displaystyle\frac{E_{var}}{N}=\left\{\begin{array}[]{ll}-\frac{1}{24\Delta}%
\left(J^{\prime 2}+3\Delta^{2}J^{\prime}+9\Delta^{2}\right)&\;\textrm{ for $J^%
{\prime}\leq 3\Delta$}\\
-\frac{1}{8}J^{\prime}(\Delta+2)&\;\textrm{ for $J^{\prime}>3\Delta$ }.\end{%
array}\right.$$
(3)
The relevant order parameter describing the Néel order is the sublattice
magnetization
$$\displaystyle M_{s}=\langle\Psi_{var}|s^{z}_{i\in A}|\Psi_{var}\rangle=\left\{%
\begin{array}[]{ll}\frac{1}{2}\sqrt{1-(J^{\prime}/3\Delta)^{2}}&\;\textrm{ for%
$J^{\prime}\leq 3\Delta$}\\
0&\;\textrm{ for $J^{\prime}>3\Delta.$}\end{array}\right.$$
(4)
$M_{s}$ vanishes at the critical value $J^{\prime}_{c}=3\Delta J$. The
corresponding critical index is the mean-field index
$1/2$. Eq. (3) may be rewritten in terms of $M_{s}$ as
$$E_{var}/N=-\frac{1}{8}J^{\prime}\Delta-\frac{1}{4}J^{\prime}\sqrt{1-4M_{s}^{2}%
}-\frac{3}{2}\Delta M_{s}^{2}.$$
(5)
We expand $E_{var}$ up to the fourth order in $M_{s}$ near the
critical point and we find a Landau-type expression, given by
$$E_{var}/N=-\frac{1}{8}J^{\prime}(\Delta+2)+\frac{1}{2}(J^{\prime}-3\Delta)M_{s%
}^{2}+\frac{1}{2}J^{\prime}M_{s}^{4}.$$
(6)
3.2 Coupled cluster method (CCM)
The CCM formalism is now briefly considered, for
further details the interested reader is referred to Refs.
[27, 28].
The starting point for the calculation of the many-body ground state is
a normalized reference or model state $|\Phi\rangle$.
For our model the appropriate
reference state is the Néel state.
To treat each site equivalently we perform a rotation of the local axis of
the up spins such that all spins in the reference state align in the same
direction, namely along the negative $z$ axis.
After this transformation we have
$\;\;|\Phi\rangle=|\downarrow\rangle|\downarrow\rangle|\downarrow\rangle|%
\downarrow\rangle\ldots\;\;$.
Now we define a set of multi-spin creation operators $C_{I}^{+}=s_{r}^{+}\;,\;s_{r}^{+}s_{l}^{+}\;,\;s_{r}^{+}s_{l}^{+}s_{m}^{+}\;,%
\;\ldots\;$ .
The choice of the $C_{I}^{+}$ ensures that $\langle\Phi|C_{I}^{+}=0=C_{I}|\phi\rangle$, where $C_{I}$ is the Hermitian adjoint of $C_{I}^{+}$.
The CCM parametrizations of the ket and bra ground
states are then given by
$$|\Psi\rangle=e^{S}|\Phi\rangle,\quad S=\sum_{I\neq 0}{\cal S}_{I}C_{I}^{+},$$
(7)
$$\langle\tilde{\Psi}|=\langle\Phi|\tilde{S}e^{-S},\quad\tilde{S}=1+\sum_{I\neq 0%
}\tilde{\cal S}_{I}C_{I}.$$
(8)
The correlation operators $S$ and $\tilde{S}$ contain
the correlation coefficients
${\cal S}_{I}$ and $\tilde{\cal S}_{I}$ which have to be calculated.
Using the Schrödinger equation, $H|\Psi\rangle=E|\Psi\rangle$, we can now write
the ground-state energy as $E=\langle\Phi|e^{-S}He^{S}|\Phi\rangle$ .
The order parameter
is calculated by $M_{s}=-\langle\tilde{\Psi}|s_{i}^{z}|\Psi\rangle$ .
To find the correlation coefficients
${\cal S}_{I}$ and $\tilde{\cal S}_{I}$ we have
to require that the expectation value $\bar{H}=\langle\tilde{\Psi}|H|\Psi\rangle$
is a minimum with respect to
${\cal S}_{I}$ and $\tilde{\cal S}_{I}$.
This formalism is exact if we take into account all possible multispin
configurations in
the correlation operators $S$ and $\tilde{S}$.
Of course, this is impossible for our model.
Thus we have to use approximation schemes to truncate the expansion
of $S$ and $\tilde{S}$ in the Eqs. (7) and (8). The most common scheme
is the LSUB$n$ scheme, where we include only $n$ or fewer correlated spins
in all configurations (or lattice animals in the language of graph theory)
which span a range of no more than $n$ adjacent
lattice sites.
To improve the results it is useful to extrapolate the ’raw’ CCM-LSUB$n$
results to the limit $n\to\infty$.
Although no exact scaling theory for results of LSUB$n$ approximations is
available,
there are empirical indications[17, 25, 27, 28]
of scaling laws for the order parameter for antiferromagnetic
spin models.
In accordance with those findings we use
$M_{s}(n)=M_{s}(\infty)+a_{1}(1/n)+a_{2}(1/n)^{2}$ to extrapolate to $n\to\infty$.
Vanishing $M_{s}(\infty)$ determines the critical point $J^{\prime}_{c}$.
The values for $J^{\prime}_{c}$ obtained by extrapolation of the $LSUBn$ results
for $M_{s}$ are,
however, found to be slightly too large [17].
We may also consider the inflection
points of the $M_{s}(J^{\prime})$ curve for the $LSUBn$ approximation, assuming that the
true $M_{s}(J^{\prime})$ curve will have a negative curvature up to the critical point.
We might expect
that (for increasing $n$) the inflection point $J^{\prime}_{inf}$
approaches the critical point $J^{\prime}_{c}$.
Thus determining the
inflection points for the $LSUBn$ approximation
again we can extrapolate to the limit $n\to\infty$
using a scaling law $J^{\prime}_{inf}(n)=J^{\prime}_{inf}(\infty)+b_{1}(1/n)+b_{2}(1/n)^{2}$
and interpret $J^{\prime}_{inf}(\infty)$ as the critical value $J^{\prime}_{c}$.
3.3 Exact diagonalization (ED)
In addition to the variational mean-field approach and the CCM we use ED
to calculate the order parameter. We consider finite
square lattices of $N=8,10,16,18,20,26,32$ sites and employ periodic
boundary conditions. The relevant order parameter for finite systems is
the square of the sublattice magnetization $M_{s}^{2}$, here defined as
$M_{s}^{2}=\langle[\frac{1}{N}\sum^{N}_{i=1}\tau_{i}{\bf s}_{i}]^{2}\rangle$ with the staggered factor $\tau_{i\in A}=+1$, $\tau_{i\in B}=-1$.
For the finite-size
scaling of $M_{s}^{2}$
we use the standard three-parameter
formula[29, 30, 31, 32]
$M_{s}^{2}(N)=M_{s}^{2}(\infty)+c_{1}N^{-1/2}+c_{2}N^{-1}$.
The critical value $J^{\prime}_{c}$ is that point where
$M_{s}^{2}(\infty)$ vanishes.
4 Results
To illustrate the behavior of the oder parameter in dependence on
$J^{\prime}$ we present $M_{s}(J^{\prime})$ calculated by CCM (Fig.2) and by
ED (Fig.3) and the resulting extrapolated values for
a particular value of Ising anisotropy $\Delta=2$.
Notice that corresponding results for the isotropic Heisenberg case ($\Delta=1$) can
be found in Ref.[17].
From the extrapolated order parameters one gets the
critical values for $\Delta=2$: $J^{\prime}_{c}=6.46$ (CCM) and $J^{\prime}_{c}=4.97$ (ED). The
mean-field value is $J^{\prime}_{c}=6$. The inflection points of the $M_{s}(J^{\prime})$
curves in Fig.2 are
$J^{\prime}_{inf}(n)=5.57$ (LSUB2), $5.42$ (LSUB4), $5.26$ (LSUB6), $5.11$
(LSUB8) leading to an extrapolated value of $J^{\prime}_{c}=J^{\prime}_{inf}(\infty)=4.66$.
As mentioned above, the extrapolation of the CCM results of the
order parameter
tends to overestimate the critical value and yields the largest $J^{\prime}_{c}$.
This is connected with the change
of the curvature in the $M_{s}$-$J^{\prime}$ curve in the vicinity of the
critical point, cf. Fig. 2.
Therefore the critical value $J^{\prime}_{c}$ taken form the inflection points
seems to be more realistic.
Obviously, the difference in $J^{\prime}_{inf}$ between the LSUB$n$ approximations
is small and the extrapolated value is quite close to
the value for
LSUB8. This statement holds for all values of $\Delta$. E.g. for $\Delta=4$
one finds $J^{\prime}_{inf}=10.37$ (LSUB2), $10.29$ (LSUB4),
$10.13$ (LSUB6), $9.94$
(LSUB8) leading to an extrapolated value of $J^{\prime}_{c}=J^{\prime}_{inf}(\infty)=9.41$.
Our results for the critical point $J^{\prime}_{c}(\Delta)$ obtained by MFA, CCM and
ED are collected in Fig.4.
We find that
the CCM results obtained by the extrapolation of the order parameter are in good
agreement with the MFA data. On the other hand, there is an excellent
agreement between the CCM results obtained by the extrapolation of the
inflection points and
the ED results obtained by the extrapolation of the order parameter.
Clearly we see indications for a linear increase in $J^{\prime}_{c}$ as predicted by
mean-field theory.
We mention that the curves shown in Fig.4 cannot be extrapolated to
$\Delta<1$. Similar to the effect of the Ising anisotropy ($\Delta>1$)
one rather expects an increase of $J^{\prime}_{c}$ due to XY anisotropy, i.e. for
$0\leq\Delta<1$. Indeed for the pure XY $J-J^{\prime}$ model ($\Delta=0$)
the critical value
was estimated to $J^{\prime}_{c}=4.56J$[33].
5 Summary
We have studied the zero-temperature magnetic ordering in a square-lattice
spin-half anisotropic Heisenberg (XXZ) model with two kinds of
nearest-neighbor exchange bonds $J$ and $J^{\prime}$, see Fig.1.
In particular we discuss the influence of the Ising anisotropy $\Delta$
on the
position of the quantum critical
point $J^{\prime}_{c}$ separating the phase with
semi-classical Néel order ($J^{\prime}<J^{\prime}_{c}$) and the quantum
paramagnetic phase without magnetic long-range order ($J^{\prime}>J^{\prime}_{c}$).
For this we
calculate the order parameter within a variational mean-field approach,
the coupled cluster method and exact
diagonalization of finite lattices up to $N=32$ sites.
We find in good approximation a linear relation
$J^{\prime}_{c}(\Delta)\propto\alpha\Delta$ ($\Delta\geq 1$) with
$\alpha\sim 2.3\ldots 3.0$. This result can be attributed to the reduction of
quantum spin fluctuations with increasing Ising anisotropy. In the
pure Ising limit ($\Delta\to\infty$) the only
remaining $z-z$ terms in the
Hamiltonian (1) commute with each other, i.e. no quantum spin
fluctuations
are present and, consequently,
the critical point disappears in the pure Ising limit.
R. Darradi thanks the Land Sachsen-Anhalt for financial support.
The authors are indebted to J. Schulenburg for assistance in numerical
calculatuons.
References
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Differential Gerstenhaber Algebras
of Generalized Complex Structures
Daniele Grandini,
Yat-Sun Poon, Brian Rolle
Address:
Department of Mathematics, University of California at Riverside,
Riverside CA 92521, U.S.A. Address:
Department of Mathematics, University of California at Riverside,
Riverside CA 92521, U.S.A., Email: ypoon@ucr.edu. Partially
supported by NSF DMS-0906264 and Mathematical Sciences Center of Tsinghua
UniversityAddress:
Department of Mathematics, University of California at Riverside,
Riverside CA 92521, U.S.A.
( August 26, 2011)
Abstract
Associated to every generalized complex structure is a differential Gerstenhaber algebra ($\mathop{\mathrm{DGA}}\nolimits$).
When the generalized complex structure deforms, so does the associated $\mathop{\mathrm{DGA}}\nolimits$.
In this paper, we identify the infinitesimal conditions when the $\mathop{\mathrm{DGA}}\nolimits$ is invariant as the generalized complex
structure deforms. We prove that the infinitesimal condition is always integrable. When the underlying manifold is a holomorphic
Poisson nilmanifolds, or simply a group in the general, and the geometry is invariant, we find a general construction
to solve the infinitesimal conditions under some geometric conditions.
Examples and counterexamples of existence of solutions to the infinitesimal
conditions are given.
1 Introduction
A few years ago, the second author computed the weak Frobenius structure on the moduli space of
the Barannikov-Kontsevich’s extended deformation [2] of
the complex structure on a primary Kodaira surface [25].
Among other observations, one could see from [25, Table (45)] that the restriction of the weak
Frobenius structure to the even part of the extended moduli space is trivial. The parameter space of the
even part of the extended moduli at the unperturbed point is contained in
$$\oplus_{k={\mbox{even}}}H^{k}_{J},\quad\mbox{ where }\quad H^{k}_{J}=\oplus_{p%
+q=k}H^{q}(M,\wedge^{p}T^{1,0})$$
(1)
and $T^{1,0}$ is the holomorphic tangent bundle of the complex manifold $M$.
The computation in [25] dwells in the fact that the primary Kodaira surface was chosen to be
a nilmanifold and the
complex structure he worked with is invariant. Along the line of thoughts in [24] [15], the
Dolbeault cohomology could be computed by means of algebraic methods.
Thanks to the work of Hitchin [17] and Gualtieri [16], it is now well known that the degree-2
portion of the extended deformation is realized by deformation of generalized geometry. While we will
provide further details on generalized geometry in Section 2, at this stage we simply note
that the parameter
space of generalized deformation is the degree-2 portion of extended deformation.
$$H_{J}^{2}=H^{0}(M,\wedge^{2}T^{1,0})\oplus H^{1}(M,T^{1,0})\oplus H^{2}(M,{%
\cal O})$$
(2)
where ${\cal O}$ is the structure sheaf of the complex manifold $M$.
The key ingredient in constructing the weak Frobenius structure on extended deformation is a variation of the
exterior product structure when the concerned cohomology spaces vary. However, it is also known that the
differential geometric object controlling the extended deformations is the differential Gerstenhaber algebras
$(\mathop{\mathrm{DGA}}\nolimits)$
associated to each (extended) complex structure [5] [22] [23] [28]. We will provide necessary details on the construction of $\mathop{\mathrm{DGA}}\nolimits$s in Section 3.
This structure contains the exterior differential algebra as a sub-structure.
In this context, we could paraphrase a result of [25] in a context of generalized complex geometry, and
say that the exterior differential algebras along a generalized deformation of a
primary Kodaira surface is rigid, meaning that all the exterior differential algebras are quasi-isomorphic to the unperturbed one. From this perspective, we seek a general understanding of the rigidity of the full differential Gerstenhaber algebra structures.
Question 1
Suppose that $M$ is a manifold with generalized complex structure $J$. Let $\mathop{\mathrm{DGA}}\nolimits(0)$ be the
associated differential Gerstenhaber algebra. Suppose that $\Gamma(t)$ is a family of deformation of $J$
along generalized complex structure with parameter $t$,
with associated differential Gerstenhaber algebra $\mathop{\mathrm{DGA}}\nolimits(t)$. Under what condition
will $\mathop{\mathrm{DGA}}\nolimits(t)$ be quasi-isomorphic to $\mathop{\mathrm{DGA}}\nolimits(0)$?
The infinitesimal counter-part of $\Gamma(t)$ is $\Gamma_{1}$, which represents an element in the cohomology
space $H^{2}_{J}$. If there is a quasi-isomorphism $\Phi(t)$, depending on $t$, we consider its infinitesimal version $\phi$. The pair $\Gamma_{1}$ and $\phi$ will be addressed as compatible pair. Together, they have to satisfy
a set of constraints as given in Definition 1. The main result in Section 3 is Theorem
2, which states essentially that compatible pairs are always integrable. Therefore, answers to
Question 1 above are reduced to infinitesimal level.
In identity (2), we see that there are three special kinds of deformations to analyze. Those from
$H^{1}(M,T^{1,0})$ are due to classical complex deformation theory. Those from
$H^{2}(M,{\cal O})$ are due to $B$-field transformations if the underlying complex structure is Kählerian
[16]. Therefore, we focus on those in the component $H^{0}(M,\wedge^{2}T^{1,0})$. As we will explain later,
this class of deformation is due to holomorphic Poisson structures, objects under investigation
from various perspectives [14] [18] [19]. If the holomorphic Poisson structure
has full rank everywhere, it leads to a deformation from a classical complex structure $J$
to a symplectic structure $\Omega$. If the induced differential Gerstenhaber algebras along this deformation is
rigid, then $\mathop{\mathrm{DGA}}\nolimits(J)$ and $\mathop{\mathrm{DGA}}\nolimits(\Omega)$ are quasi-isomorphic. It presents the complex manifold
$(M,J)$ and the symplectic manifold $(M,\Omega)$ as a weak mirror pair in the sense of Merkulov
[23]. An investigation on such possibility also motivates this paper. Therefore, in Section
4 we refine our analysis in Section 3 to holomorphic Poisson manifolds, and illustrate our theory with a computation on a Hopf surface.
For nilmanifolds, i.e. the compact quotient of simply connected nilpotent Lie groups, it is known for
a very long time that
the DeRham cohomology is given by invariant elements [24]. From our current perspective,
the invarant $\mathop{\mathrm{DGA}}\nolimits$ with an invariant symplectic structure on a nilmanifold is quasi-isomorphic to the
full $\mathop{\mathrm{DGA}}\nolimits$ of the symplectic structure.
For a large class of nilmanifolds examples, we also know that the invariant $\mathop{\mathrm{DGA}}\nolimits$ theory for invariant complex
structures is quasi-isomorphic to the $\mathop{\mathrm{DGA}}\nolimits$ of the corresponding nilmanifolds [10] [15]
[25] [27]. Therefore, we reduce the theory
in the previous sections in terms of invariant objects on Lie algebras and develop a method to
construct compatible pairs on a class of holomorphic Poisson algebras in Section 5.
Finally, in Section 6 we analyze all non-trivial real four-dimensional examples.
Among other observations, we conclude that the differential Gerstenhaber algebra structures are rigid
when one deforms the complex structure on a Kodaira surface by a holomorphic Poisson structure.
It extends the results in [25] on weak Frobenius structures, at least along the degree-2 direction
of the extended moduli space. On the other hand, we also discover an example of holomorphic symplectic
algebra on which there is no compatible pair. Therefore, a solution to Question 1 is
non-trivial.
In this notes, we assume that readers are familiar with the concepts of
Lie algebroids and Lie bialgebroids. Otherwise,
[20] and [21] are our references. On Differential Gerstenhaber
algebras, we rely on [21] and [25] for their formal aspects.
For generalized complex structures, our references are [17] and [16]. Much of the computation
in Section 5 and Section 6 could be found in the third author’s thesis.
Therefore, our presentation will be relatively sketchy.
2 Generalized complex structures
Let $M$ be a smooth connected manifold without boundary. Denote its tangent and cotangent bundle
respectively by $T$ and $T^{*}$. If $V$ is a vector bundle on $M$, we denote its space of sections
by $C^{\infty}(V)$. Generic vector fields will be denoted by $X$ and $Y$. One-forms are denoted
by $\alpha$ and $\beta$. On the bundle $T\oplus T^{*}$, there is a natural pairing defined by
$$\langle X+\alpha,Y+\beta\rangle=\frac{1}{2}(\alpha(Y)+\beta(X)).$$
(3)
As this pairing is non-degenerate, it identifies the bundle $T\oplus T^{*}$ to its dual.
We choose the identification to be
$$\sigma:T\oplus T^{*}\to(T\oplus T^{*})^{*},\quad\sigma(X+\alpha)(Y+\beta)=2%
\langle X+\alpha,Y+\beta\rangle.$$
(4)
The Courant bracket [20] is the real bilinear map on $C^{\infty}(T\oplus T^{*})$ defined by
$$[\![X+\alpha,Y+\beta]\!]=[X,Y]+{{\cal L}}_{X}\beta-{{\cal L}}_{Y}\alpha-\frac{%
1}{2}d(\iota_{X}\beta-\iota_{Y}\alpha).$$
(5)
The Courant bracket, the non-degenerate pairing above, together with the natural projection on the tangent component
make $T\oplus T^{*}$ a standard example of a Courant algebroid
[11] [20].
An almost generalized complex structure is a real bundle map
$J:T\oplus T^{*}\to T\oplus T^{*}$ such that $J\circ J=-$identify and $J^{*}=-J$.
Let $L$ be the bundle of $+i$-eigenvectors with respect to $J$ and over the
complex numbers. With respect to the non-degenerate pairing, $L$ is maximal isotropic.
So is its conjugate bundle $\overline{L}$.
The choices of the tensorial object $J$ with the given prescription is equivalent to
the choice of maximal isotropic subbundle $L$ such that
$L\cap{\overline{L}}$ is trivial [16].
An almost generalized complex structure is said to be integrable if and only if the space
$C^{\infty}(L)$ is closed under the Courant bracket.
By complex conjugation, it is of course equivalent to $C^{\infty}(\overline{L})$ being closed.
In such case, the structure $J$, or equivalently, either the bundle $L$ or the bundle
$\overline{L}$ is said to be a
generalized complex structure. It is now well known that complex structures in the classical sense
are generalized complex. So are symplectic structures. For classical complex structure, the complexified
tangent bundle splits into the direct sum of type $(1,0)$ and type $(0,1)$ vectors. Their related
bundles are denoted by $T^{1,0}$ and $T^{0,1}$ respectively. Their dual bundles are denoted by
$T^{*(1,0)}$ and $T^{*(0,1)}$. Then the corresponding bundles $L$ and $L^{*}$ are
$$L=T^{1,0}\oplus T^{*(0,1)},\quad L^{*}\cong{\overline{L}}=T^{0,1}\oplus T^{*(1%
,0)}.$$
If $\omega$ is a symplectic form on the manifold $M$, then
$$L=\{X-i\iota_{X}\omega:X\in C^{\infty}(T)\},\quad L^{*}\cong{\overline{L}}=\{X%
+i\iota_{X}\omega:X\in C^{\infty}(T)\}$$
represent an example of a generalized complex structure.
Since $L$ is isotropic, the restriction of the Courant bracket on $L$
makes it a Lie algebroid whenever the generalized complex structure is integrable.
As such, it has a Lie algebroid differential acting on the exterior algebra of the dual bundle [21].
$$\partial:C^{\infty}(\wedge^{n}L^{*})\to C^{\infty}(\wedge^{n+1}L^{*}).$$
(6)
Using the identification as given in (4), we identify $\overline{L}=L^{*}$. Then
$$\partial:C^{\infty}(\wedge^{n}\overline{L})\to C^{\infty}(\wedge^{n+1}%
\overline{L}).$$
(7)
Similarly, $\overline{L}\cong L^{*}$ is also a Lie algebroid. Its Lie algebroid differential
is precisely the conjugation of the above operator:
$$\overline{\partial}:C^{\infty}(\wedge^{n}L)\to C^{\infty}(\wedge^{n+1}L).$$
(8)
As noted in [20, Theorem 2.6], $(L,{\overline{L}})$ forms a Lie bialgebroid. It means
that for any sections $\ell_{1}$ and $\ell_{2}$ of
the bundle $L$,
$$\overline{\partial}[\![\ell_{1},\ell_{2}]\!]=[\![\overline{\partial}\ell_{1},%
\ell_{2}]\!]-[\![\ell_{1},\overline{\partial}\ell_{2}]\!].$$
(9)
Making use of [21, Theorem 7.5.2], we deduce that the space of sections of the exterior
algebra generated by $L$, $C^{\infty}(\wedge^{\bullet}L)$ carries the structure of a differential
Gerstenhaber algebra structures, with the Courant bracket, exterior product and Lie algebroid
differential of $\overline{L}\cong L^{*}$.
We denote it by
$$DGA(J):=(C^{\infty}(\wedge^{\bullet}L),[\![-,-]\!],\wedge,\overline{\partial}).$$
(10)
In this context, the bracket on $C^{\infty}(\wedge^{\bullet}L)$ is known as Schouten bracket [21].
The integrability implies that the restriction of the Courant bracket on $C^{\infty}(L^{*})$ satisfies
the Jacobi identity. In terms of the operator
$\overline{\partial}$, it is equivalent to $\overline{\partial}\circ\overline{\partial}=0$. Therefore,
$\overline{\partial}:C^{\infty}(\wedge^{n}L)\to C^{\infty}(\wedge^{n+1}L)$ determines a differential
complex, and hence generates cohomology spaces. i.e. for all $k\geq 1$,
$$H^{k}_{J}=\frac{\ker\overline{\partial}:\wedge^{k}L\to\wedge^{k+1}L}{{\mbox{%
\rm Image }}\overline{\partial}:\wedge^{k-1}L\to\wedge^{k}L}.$$
Given the identity (9),
the cohomology spaces inherit a Gerstenhaber algebra structure.
When the generalized complex structure is classical, one could verify that if $\overline{\omega}$ is
a type $(0,k)$-form, then $\overline{\partial}\overline{\omega}$ is the classical $\overline{\partial}$-operator in complex analysis
on $\mathbb{C}^{n}$ [27]. On the other hand, if $Z$ is a $(1,0)$-vector field and $\overline{X}$ is
a $(0,1)$-vector field, then
$$\overline{\partial}_{\overline{X}}Z=[Z,{\overline{X}}]^{1,0}.$$
(11)
This is precisely the Cauchy-Riemann operator [13] [26].
The cohomology of degree-$k$ in this case is
$$H^{k}_{J}=\oplus_{p+q=k}H^{q}(M,\wedge^{p}T^{1,0}).$$
(12)
Using Dolbeault theory, the elements in these cohomology spaces are represented by $\overline{\partial}$-closed
$(0,q)$-forms with coefficients in holomorphic $(p,0)$-vector fields.
On the other hand, if a generalized complex structure is defined by a symplectic form $\omega$, then
$$\overline{\partial}({X}-i\iota_{X}\theta)=-2id\iota_{X}\theta$$
(13)
for all $X$ in $C^{\infty}(T_{\mathbb{C}})$ [26]. In particular, the k-th cohomology of this complex is the k-th
complexified deRham cohomology of the manifold $M$.
As a subbundle of $(T\oplus T^{*})_{\mathbb{C}}$, the bundle $L$ has a natural projection $\rho$
onto the direct summand $T_{\mathbb{C}}$.
The type of a generalized complex structure at a point of the manifold $M$ is defined to be
the complex co-dimension
of the projection of $L$ in $T_{\mathbb{C}}$ over the concerned point [16]. From the description above, one sees that
the type of a classical complex structure on a real $2n$-dimensional manifold is
equal to $n$. All symplectic structures are type-0 generalized complex structures.
3 Deformation of generalized complex structures
A deformation of a generalized complex structure is given by a section $\Gamma$
of $\wedge^{2}L$ [20] [16].
To be more precise,
$$L_{\overline{\Gamma}}=\{\ell+{\overline{\Gamma}}(\ell):\ell\in C^{\infty}(L)\}%
,\quad\mbox{ and }\quad{\overline{L}}_{\Gamma}=\{{\overline{\ell}}+\Gamma({%
\overline{\ell}}):{\overline{\ell}}\in C^{\infty}({\overline{L}})\}.$$
(14)
$L_{\overline{\Gamma}}\cap{\overline{L}}_{\Gamma}=\{0\}$ if and only
if $\Gamma\circ{\overline{\Gamma}}$ does not have non-trivial fixed points [26].
The deformed generalized complex structure $(L_{\overline{\Gamma}},{\overline{L}}_{\Gamma})$
is integrable if and only if
$\Gamma$ satisfies the Maurer-Cartan equation [20, Theorem 6.1]:
$$\overline{\partial}\Gamma+\frac{1}{2}[\![\Gamma,\Gamma]\!]=0.$$
(15)
The infinitesimal version of the Maurer-Cartan equation is simply $\overline{\partial}\Gamma_{1}=0$.
Therefore, it represents an element in the second cohomology
$H^{2}_{J}$ of the differential Gerstenhaber algebra of the unperturbed generalized complex structure $J$.
3.1 Deformation of associated $\mathop{\mathrm{DGA}}\nolimits$
Let $\overline{\delta}$ be the Lie algebroid differential of ${\overline{L}}_{\Gamma}$.
Due to our natural pairing (3), it acts on
the conjugate bundle $L_{\overline{\Gamma}}$. Therefore, we have the new differential Gerstenhaber algebra
$$DGA(J_{\Gamma})=(\wedge^{\bullet}L_{\overline{\Gamma}},[\![-,-]\!],\wedge,%
\overline{\delta}).$$
(16)
Meanwhile, for $\Gamma$ sufficiently close to zero, $L$ and ${\overline{L}}_{\Gamma}$
are also transversal in $(T\oplus T^{*})_{\mathbb{C}}$.
By [20, Theorem 2.6], $L$ and ${\overline{L}}_{\Gamma}$ form a Lie bialgebroid.
We could denote the Lie algebroid differential of
the Lie algebroid ${\overline{L}}_{\Gamma}$ acting on $L$ by $\overline{\partial}_{\Gamma}$.
Since $L_{\overline{\Gamma}}$ is simply the graph of the map $\overline{\Gamma}$, there is a natural
map from $L$ to $L_{\Gamma}$. It enables one to identify the differential $\overline{\partial}_{\Gamma}$. The
computation below is a consequence of [20, Theorem 2.6] and [20, Theorem 6.1].
It should be well known to experts. We outline a proof here
for completeness. A complete proof for a case most relevant to this paper could be found in
[26].
Proposition 1
The pair
$L$ and $\overline{L}_{\Gamma}$ forms a Lie bialgebroid. The Lie algebroid
differential $\overline{\partial}_{\Gamma}$ for the
deformed Lie algebroid $\overline{L}_{\Gamma}$ acting on $L$ is
given by $\overline{\partial}+[\![\Gamma,-]\!].$ i.e. for any
section $\ell$ of $L$,
$$\overline{\partial}_{\Gamma}\ell=\overline{\partial}\ell+[\![\Gamma,\ell]\!].$$
Proof: By definition, the vector bundle $\overline{L}_{\Gamma}$ is maximally isotropic
in the Courant algebroid $(T\oplus T^{*})_{\mathbb{C}}$. Since $\Gamma$
satisfies the Maurer-Cartan equation, it follows that the space of sections of $\overline{L}_{\Gamma}$
is closed with respect to the Courant bracket. To find a more precise description, we follow
the computation in [20].
For ${\overline{\sigma}}\in C^{\infty}({\overline{L}})$, $\ell_{1},\ell_{2}\in C^{\infty}(L)$, define a
Lie derivative by
$$({\cal L}_{\ell_{1}}{\overline{\sigma}})\ell_{2}=\rho(\ell_{1})({\overline{%
\sigma}}(\ell_{2}))-{\overline{\sigma}}([\![\ell_{1},\ell_{2}]\!]),$$
where $\rho$ is the natural projection from $(T\oplus T^{*})_{\mathbb{C}}$ onto $T_{\mathbb{C}}$.
The property of the Lie derivative in algebroid theory could be found in [21]. Follow
[20, Identity (23)], for any
$\overline{\ell}_{1},\overline{\ell}_{2}\in C^{\infty}(\overline{L})$ define
$$[\![\overline{\ell}_{1},\overline{\ell}_{2}]\!]_{\Gamma}={\mathcal{L}}_{\Gamma%
\overline{\ell}_{1}}\overline{\ell}_{2}-{\mathcal{L}}_{\Gamma\overline{\ell}_{%
2}}\overline{\ell}_{2}+\overline{\partial}(\overline{\ell}_{1}(\Gamma\overline%
{\ell}_{2})).$$
(17)
As noted in the proof of [20, Theorem 6.1], $\Gamma$ satisfies the
Maurer-Cartan equation if and only if
$$[\![\overline{\ell}_{1}+\Gamma\overline{\ell}_{1},\overline{\ell}_{2}+\Gamma%
\overline{\ell}_{2}]\!]=[\![\overline{\ell}_{1},\overline{\ell}_{2}]\!]+[\![%
\overline{\ell}_{1},\overline{\ell}_{2}]\!]_{\Gamma}+\Gamma\left([\![\overline%
{\ell}_{1},\overline{\ell}_{2}]\!]+[\![\overline{\ell}_{1},\overline{\ell}_{2}%
]\!]_{\Gamma}\right).$$
Now we are ready to compute the Lie algebroid differential of $\overline{L}_{\Gamma}$ with $L$ as its dual.
For every $\ell\in C^{\infty}(L)$ and $\ell_{1},\ell_{2}\in C^{\infty}(\overline{L})$,
$$\displaystyle\left(\overline{\partial}_{\Gamma}\ell\right)(\overline{\ell}_{1}%
+\Gamma\overline{\ell}_{1},\overline{\ell}_{2}+\Gamma\overline{\ell}_{2})$$
$$\displaystyle=$$
$$\displaystyle\rho(\overline{\ell}_{1}+\Gamma\overline{\ell}_{1})(\ell(%
\overline{\ell}_{2}))-\rho(\overline{\ell}_{2}+\Gamma\overline{\ell}_{2})(\ell%
(\overline{\ell}_{1}))-\ell([\![\overline{\ell}_{1}+\Gamma\overline{\ell}_{1},%
\overline{\ell}_{2}+\Gamma\overline{\ell}_{2}]\!])$$
$$\displaystyle=$$
$$\displaystyle\rho(\overline{\ell}_{1}+\Gamma\overline{\ell}_{1})(\ell(%
\overline{\ell}_{2}))-\rho(\overline{\ell}_{2}+\Gamma\overline{\ell}_{2})(\ell%
(\overline{\ell}_{1}))$$
$$\displaystyle-\ell\left([\![\overline{\ell}_{1},\overline{\ell}_{2}]\!]+[\![%
\overline{\ell}_{1},\overline{\ell}_{2}]\!]_{\Gamma}+\Gamma\left([\![\overline%
{\ell}_{1},\overline{\ell}_{2}]\!]+[\![\overline{\ell}_{1},\overline{\ell}_{2}%
]\!]_{\Gamma}\right)\right).$$
Since the image of a section in $\overline{L}$ under $\Gamma$ is a section
of $L$, and $L$ isotropic, the above is equal to
$$\displaystyle=$$
$$\displaystyle\rho(\overline{\ell}_{1}+\Gamma\overline{\ell}_{1})(\ell(%
\overline{\ell}_{2}))-\rho(\overline{\ell}_{2}+\Gamma\overline{\ell}_{2})(\ell%
(\overline{\ell}_{1}))-\ell\left(\left([\![\overline{\ell}_{1},\overline{\ell}%
_{2}]\!]+[\![\overline{\ell}_{1},\overline{\ell}_{2}]\!]_{\Gamma}\right)\right)$$
$$\displaystyle=$$
$$\displaystyle\left(\overline{\partial}{\ell}\right)(\overline{\ell}_{1},%
\overline{\ell}_{2})+\rho(\Gamma\overline{\ell}_{1})(\ell(\overline{\ell}_{2})%
)-\rho(\Gamma\overline{\ell}_{2})(\ell(\overline{\ell}_{1}))-\ell\left([\![%
\overline{\ell}_{1},\overline{\ell}_{2}]\!]_{\Gamma}\right).$$
The proof of this proposition is completed if we could show that
$$[\![\Gamma,\ell]\!](\overline{\ell}_{1},\overline{\ell}_{2})=\rho(\Gamma%
\overline{\ell}_{1})(\ell(\overline{\ell}_{2}))-\rho(\Gamma\overline{\ell}_{2}%
)(\ell(\overline{\ell}_{1}))-\ell\left([\![\overline{\ell}_{1},\overline{\ell}%
_{2}]\!]_{\Gamma}\right).$$
It is now a matter of definition of Lie derivative to show that the right hand side of the above is equal to
$$\overline{\ell}_{2}\left([\![\Gamma\overline{\ell}_{1},\ell]\!]\right)-%
\overline{\ell}_{1}\left([\![\Gamma\overline{\ell}_{2},\ell]\!]\right)-\rho(%
\ell)\left(\overline{\ell}_{1}(\Gamma\overline{\ell}_{2})\right).$$
Finally, the following identity always hold for any section $\Gamma$
of $\wedge^{2}L$, $\ell$ of $L$ and ${\overline{\ell}}_{1},{\overline{\ell}}_{2}$ of $\overline{L}$.
$$[\![\Gamma,\ell]\!](\overline{\ell}_{1},\overline{\ell}_{2})=\overline{\ell}_{%
2}\left([\![\Gamma\overline{\ell}_{1},\ell]\!]\right)-\overline{\ell}_{1}\left%
([\![\Gamma\overline{\ell}_{2},\ell]\!]\right)-\rho(\ell)\left(\overline{\ell}%
_{1}(\Gamma\overline{\ell}_{2})\right)$$
(18)
See [26] for a detailed proof for (18).
Therefore, the proof of the proposition is completed.
As far as analyzing the deformation of
associated differential Gerstenhaber algebras is concerned,
the above observation reduces an analysis to one on deformation of differentials.
As the bundle structure could remain constant, we now focus on the variation
from $\overline{\partial}$ to $\overline{\partial}_{\Gamma}$, and hence denote the respective
differential Gerstenhaber algebras by $\mathop{\mathrm{DGA}}\nolimits(\overline{\partial})$ and $\mathop{\mathrm{DGA}}\nolimits(\overline{\partial}_{\Gamma})$.
To compare $\mathop{\mathrm{DGA}}\nolimits(\overline{\partial}_{\Gamma})$ with $\mathop{\mathrm{DGA}}\nolimits(\overline{\partial})$, we identify the constraints for them to be homomorphic.
If the homomorphism induces an isomorphism at cohomology level, these two $\mathop{\mathrm{DGA}}\nolimits$s are said to be quasi-isomorphic.
In such case, the generalized complex structure $J$ and the deformed one are also said to form a weak mirror pair
[23], [6], [7]. Let
$$\Phi:L\to L$$
be a vector bundle homomorphism depending on $\Gamma$.
It induces a homomorphism of the exterior
algebra generated by $L$. That is
$$\Phi(A\wedge B):=\Phi(A)\wedge\Phi(B)$$
(19)
for any $A,B\in C^{\infty}(\wedge^{\bullet}L)$.
It is a homomorphism of graded Lie algebras if
for any $A,B$ in $C^{\infty}(\wedge^{\bullet}L)$,
$$[\![\Phi(A),\Phi(B)]\!]=\Phi([\![A,B]\!]).$$
(20)
If in addition, when the following diagram is commutative for all $0\leq k\leq n$,
$$\begin{array}[c]{ccc}\wedge^{k}L&\overset{\overline{\partial}_{\Gamma}}{%
\longrightarrow}&\wedge^{k+1}L\\
\Phi\downarrow&&\downarrow\Phi\\
\wedge^{k}L&\overset{\overline{\partial}}{\longrightarrow}&\wedge^{k+1}L,\end{array}$$
(21)
we have a
homomorphism of differential Gerstenhaber algebras
$$\Phi:(\wedge^{\bullet}L,[\![-,-]\!],\wedge,\overline{\partial}_{\Gamma})\to(%
\wedge^{\bullet}L,[\![-,-]\!],\wedge,\overline{\partial}).$$
In general, if the bundle homomorphism intertwines the differentials as in
(21) and satisfies (19), then the map $\Phi$ is an exterior
differential algebra homomorphism.
If the bundle homomorphism intertwines the differentials as in
(21) and satisfies (20), the map $\Phi$ is
a graded differential algebra homomorphism.
The diagram (21) above is equivalent to
$$\Phi\circ\overline{\partial}_{\Gamma}=\overline{\partial}\circ\Phi.$$
(22)
By Proposition 1, that means for any section $A$
of the bundle $\wedge^{\bullet}L$,
$$\Phi(\overline{\partial}A+[\![\Gamma,A]\!])=\overline{\partial}(\Phi A).$$
(23)
Suppose that $\Omega$ is a closed 2-form and $L$ is a generalized complex structure, then
$${\overline{L}}_{\Omega}=\{X+\alpha+\iota_{X}\Omega:X+\alpha\in{\overline{L}}\}$$
is again a generalized complex structure. This is because the closedness of $\Omega$
and its skew-symmetry imply that
the map
$$X+\alpha\mapsto X+\alpha+\iota_{X}\Omega$$
is an automorphism of the Courant bracket $[\![-,-]\!]$ in (5) and the
non-degenerate bilinear pairing
$\langle-,-\rangle$ in (3). This map is known as a $B$-field transformation
by the closed 2-form $\Omega$.
It naturally defines an isomorphism between the $\mathop{\mathrm{DGA}}\nolimits$ of the bundle $L$ and the $\mathop{\mathrm{DGA}}\nolimits$ of
the $B$-field transformation of $L$.
On infinitesimal level, deformation theory of generalized complex structures
is elliptic and a copy of Kuranishi theory follows [16]. In this spirit of Kuranishi theory,
let $\phi$ be the infinitesimal version of $\Phi$. It is now an endomorphism from the vector bundle
$L$ to $L$. The infinitesimal version of (23) becomes
$$\phi(\overline{\partial}A)+[\![\Gamma_{1},A]\!]=\overline{\partial}(\phi A),$$
(24)
where $\Gamma_{1}$ is the infinitesimal deformation, representing an element in the
second cohomology $H^{2}_{J}$.
The infinitesimal version of (20) is
$$[\![\phi A,B]\!]+[\![A,\phi B]\!]=\phi[\![A,B]\!].$$
(25)
In these expressions, the map $\phi$ is an infinitesimal version of
a homomorphism of exterior algebras. Therefore, it is an
endomorphism with following property.
$$\phi(A\wedge B)=(\phi A)\wedge B+A\wedge(\phi B).$$
(26)
Now we treat $\phi$ as an element in $L^{*}\otimes L=\mathop{\mathrm{End}}\nolimits(L)$. More
generally, it is a section of $\mathop{\mathrm{End}}\nolimits(\wedge^{n}L)$. On the other hand,
$\Gamma_{1}$ is a section of $\wedge^{2}L$.
Now we summarize the above discussion with a concept and its implication
to variation of the structure of associated differential Gerstenhaber algebras.
Definition 1
Suppose that $M$ is a manifold with a generalized
complex structure $J$, whose $+i$-eigenbundle is $L$.
A section
$\Gamma_{1}\in C^{\infty}(\wedge^{2}L)$ and a section $\phi\in C^{\infty}(L^{*}\otimes L)$ form
a compatible pair if $\overline{\partial}\Gamma_{1}=0$ and
$$\displaystyle\overline{\partial}(\phi A)-\phi(\overline{\partial}A)=[\![\Gamma%
_{1},A]\!];$$
(27)
$$\displaystyle[\![\phi A,B]\!]+[\![A,\phi B]\!]=\phi[\![A,B]\!];$$
(28)
$$\displaystyle\phi(A\wedge B)=(\phi A)\wedge B+A\wedge(\phi B).$$
(29)
Note that if $\Gamma_{1}$ is in the center of the Gerstenhaber algebra, i.e.
$[\![\Gamma_{1},A]\!]=0$ for all $A\in C^{\infty}(\wedge^{\bullet}L)$, then $\phi=0$ is
an obvious solution for the above three identities. For future reference, we note the following
Proposition 2
Suppose that $M$ is a manifold with a generalized
complex structure $J$, whose $+i$-eigenbundle is $L$.
Let
$\Gamma_{1}\in C^{\infty}(\wedge^{2}L)$ be a closed section: $\overline{\partial}\Gamma_{1}=0$.
Then $(\Gamma_{1},\phi=0)$ form a compatible pair if and only if $\Gamma_{1}$ is
central: $[\![\Gamma_{1},A]\!]=0$ for all $A\in C^{\infty}(\wedge^{\bullet}L)$.
Theorem 1
Suppose that $M$ is a manifold with a generalized
complex structure $J$, whose $+i$-eigenbundle is $L$. Suppose that
$\Gamma$ is an integrable deformation with infinitesimal
deformation $\Gamma_{1}$. If there is a homomorphism $\Phi$ of the bundle
$L$ to itself such that it generates a homomorphism from the
differential Gerstenhaber algebra of the deformation generalized
complex structure to the un-perturbed one, then there exists a
compatible pair $\Gamma_{1}$ and $\phi$ such that $\overline{\partial}\Gamma_{1}=0$, and up to
first order, $\Gamma$ is equal to $\Gamma_{1}$ and
$\Phi$ is equal to $1+\phi$.
3.2 Integrability of compatible pairs
Given a compatible pair $\Gamma_{1}$ and $\phi$, an immediate issue is whether they
actually come from a deformation $\Gamma$ and a homomorphism of $\mathop{\mathrm{DGA}}\nolimits$s. In
this section, we apply the principles of
Kuranishi’s recursive method to prove that
this is the case. We will divide the proof of the following
theorem in several steps.
Theorem 2
Suppose that $M$ is a manifold with a generalized
complex structure $J$, whose $+i$-eigenbundle is $L$. Let $\Gamma_{1}\in C^{\infty}(\wedge^{2}L)$
and $\phi\in C^{\infty}(L^{*}\otimes L)$
be a
compatible pair. Let $t$ be a real variable. Define
$$\Gamma(t)=\sum_{n=1}^{\infty}(-1)^{n-1}\frac{1}{n!}t^{n}\phi^{n-1}\Gamma_{1},%
\quad\Phi(t)=\sum_{n=1}^{\infty}\frac{1}{n!}t^{n}\phi^{n}.$$
(30)
Then $\Gamma(t)$ satisfies the Maurer-Cartan equation. Moveover, if $\mathop{\mathrm{DGA}}\nolimits(\Gamma(t))$
represents the Differential Gerstenhaber algebra
$(\wedge^{\bullet}L,[\![-,-]\!],\wedge,\overline{\partial}_{\Gamma(t)})$, then $\Phi(t)$ is a
homomorphism from $\mathop{\mathrm{DGA}}\nolimits(\Gamma(t))$ to $\mathop{\mathrm{DGA}}\nolimits(\Gamma(0))$.
The Maurer-Cartan equation at degree-1 with respect to the variable $t$ is
simply $\overline{\partial}\Gamma_{1}=0$. For $n\geq 2$, it is
$$(-1)^{n-1}\frac{1}{n!}\overline{\partial}(\phi^{n-1}\Gamma_{1})+\frac{1}{2}%
\sum_{j+k=n}[\![(-1)^{k-1}\frac{1}{k!}\phi^{k-1}\Gamma_{1},(-1)^{j-1}\frac{1}{%
j!}\phi^{j-1}\Gamma_{1}]\!]=0.$$
(31)
Let the binomial coefficients be
$$C^{n}_{k}=\frac{n!}{k!(n-k)!}.$$
Then the above equation is equivalent to
$$\overline{\partial}(\phi^{n-1}\Gamma_{1})=\frac{1}{2}\sum_{k=1}^{n-1}C^{n}_{k}%
[\![\phi^{k-1}\Gamma_{1},\phi^{n-k-1}\Gamma_{1}]\!].$$
(32)
Similarly, $\Phi(t)$ is a homomorphism of $[\![-,-]\!]$ and $\wedge$ if and
only if for degree $n$, and for any sections
$A$ and $B$ of $L$,
$$\frac{1}{n!}\phi^{n}[\![A,B]\!]=\sum_{k+j=n}[\![\frac{1}{k!}\phi^{k}A,\frac{1}%
{j!}\phi^{j}B]\!],\quad\frac{1}{n!}\phi^{n}({A}\wedge{B})=\sum_{k+j=n}({\frac{%
1}{k!}\phi^{k}A}\wedge{\frac{1}{j!}\phi^{j}B}).$$
(33)
It is equivalent to
$$\phi^{n}[\![A,B]\!]=\sum_{k=1}^{n}C^{n}_{k}[\![\phi^{k}A,\phi^{n-k}B]\!],\quad%
\phi^{n}({A}\wedge{B})=\sum_{k=1}^{n}C^{n}_{k}({\phi^{k}A}\wedge{\phi^{n-k}B}).$$
(34)
Finally, $\Phi(t)$ intertwines $\overline{\partial}_{\Gamma(t)}$ if and only if they satisfy the identity
(23). Assuming that $\Phi(t)$ is a homomorphism of the Courant bracket on $\wedge^{\bullet}L$,
we need to show that for any section $A$ of $\wedge^{\bullet}L$,
$$\Phi(t)(\overline{\partial}A)+[\![\Phi(t)\Gamma(t),\Phi(t)A]\!]=\overline{%
\partial}(\Phi(t)A).$$
(35)
Consider the infinite product $\Phi(t)\Gamma(t)$. Its degree-$n$ term is equal to
$$\sum_{k+j=n}\frac{1}{j!}\phi^{j}(-1)^{k-1}\frac{1}{k!}\phi^{k-1}\Gamma_{1}=%
\left(\sum_{k+j=n}(-1)^{k-1}\frac{1}{j!k!}\right)\phi^{n-1}\Gamma_{1}$$
On the other hand, consider the power series
$$g=\sum_{k=1}^{\infty}(-1)^{k}\frac{1}{k!}x^{k-1}\quad\mbox{ and }\quad e^{x}=%
\sum_{k=1}^{\infty}\frac{1}{k!}x^{k}.$$
We have
$1-xg=e^{-x}.$ Therefore, $e^{x}-xe^{x}g=1$. i.e. $e^{x}(xg)=e^{x}-1$. Equating the $n$-th
order terms for $n\geq 1$, we find that
$$\sum_{k+j=n}(-1)^{k-1}\frac{1}{j!k!}=\frac{1}{n!}.$$
Therefore, $\Phi(t)\Gamma(t)=\sum_{n\geq 1}\frac{1}{n!}t^{n}\phi^{n-1}\Gamma_{1}$.
Then the identity (35) at degree $n$ becomes
$$\frac{1}{n!}\phi^{n}\overline{\partial}A+\sum_{k+j=n}[\![\frac{1}{k!}\phi^{k-1%
}\Gamma_{1},\frac{1}{j!}\phi^{j}A]\!]=\frac{1}{n!}\overline{\partial}\phi^{n}A$$
Equivalently, it is
$$\overline{\partial}\phi^{n}A-\phi^{n}\overline{\partial}A=\sum_{k=1}^{n}C^{n}_%
{k}[\![\phi^{k-1}\Gamma_{1},\phi^{n-k}A]\!]$$
(36)
To complete a proof of Theorem 2, we need to prove that the identities
(32), (34) and (36) hold.
Lemma 1
Suppose that $\Gamma_{1}$ and $\phi$ form a compatible pair,
then identity (32) holds.
Proof: Using the fact that $\overline{\partial}\Gamma_{1}=0$ and equation
(27), we get a telescopic sum
$$\displaystyle\overline{\partial}\phi^{n-1}\Gamma_{1}$$
$$\displaystyle=$$
$$\displaystyle\overline{\partial}(\phi^{n-1}\Gamma_{1})-\phi\overline{\partial}%
\phi^{n-2}\Gamma_{1}$$
$$\displaystyle+$$
$$\displaystyle\phi\overline{\partial}\phi^{n-2}\Gamma_{1}-\phi^{2}\overline{%
\partial}\phi^{n-3}\Gamma_{1}+\phi^{2}\overline{\partial}\phi^{n-3}\Gamma_{1}-%
\phi^{3}\overline{\partial}\phi^{n-4}\Gamma_{1}$$
$$\displaystyle+$$
$$\displaystyle\ldots\ \ \dots\ \ \dots+\phi^{n-2}\overline{\partial}\phi\Gamma_%
{1}-\phi^{n-1}\overline{\partial}\Gamma_{1}$$
$$\displaystyle=$$
$$\displaystyle\sum_{h=0}^{n-2}\phi^{h}[\![\Gamma_{1},\phi^{n-2-h}\Gamma_{1}]\!].$$
Since $\phi$ satisfies (28) and the Schouten bracket is
commutative when restricted to section of $\Lambda^{2}L$, we rewrite the above identity as
$$\displaystyle\overline{\partial}\phi^{n-1}\Gamma_{1}$$
$$\displaystyle=$$
$$\displaystyle\sum_{h=0}^{n-2}\sum_{k=0}^{h}C^{h}_{k}[\![\phi^{k}\Gamma_{1},%
\phi^{n-2-k}\Gamma_{1}]\!]=\sum_{h=1}^{n-1}\sum_{k=1}^{h}C^{h-1}_{k-1}[\![\phi%
^{k-1}\Gamma_{1},\phi^{n-1-k}\Gamma_{1}]\!]$$
$$\displaystyle=$$
$$\displaystyle\sum_{k=1}^{n-1}\left(\sum_{h=k}^{n-1}C^{h-1}_{k-1}\right)[\![%
\phi^{k-1}\Gamma_{1},\phi^{n-1-k}\Gamma_{1}]\!]$$
$$\displaystyle=$$
$$\displaystyle\sum_{k=1}^{n-1}C^{n-1}_{k}[\![\phi^{k-1}\Gamma_{1},\phi^{n-1-k}%
\Gamma_{1}]\!].$$
Performing the index substitution $k\mapsto n-k$ and using the
commutativity of the Schouten bracket again, we get
$$\displaystyle\overline{\partial}\phi^{n-1}\Gamma_{1}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}\sum_{k=1}^{n-1}\left(C^{n-1}_{k}+C^{n-1}_{k-1}\right)%
[\![\phi^{k-1}\Gamma_{1},\phi^{n-1-k}\Gamma_{1}]\!]$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}\sum_{k=1}^{n-1}C^{n}_{k}[\![\phi^{k-1}\Gamma_{1},\phi%
^{n-1-k}\Gamma_{1}]\!].$$
Lemma 2
Suppose that $\Gamma_{1}$ and $\phi$ form a compatible pair, then the two identities
in (34) hold.
Proof: It is an elementary induction.
The proof for both cases are identical. We work only through the
case with Schouten bracket. When $n=1$, the equation (34) is
precisely the equation (28), which is satisfied by assumption. Assuming that the equation (34) holds for
all $k\leq n$. We next compute $\phi^{n+1}[\![A,B]\!]$, which we
take as $\phi(\phi^{n}[\![A,B]\!])$. By induction hypothesis, it is
equal to
$$\displaystyle\sum_{k=0}^{n}C_{k}^{n}\phi([\![\phi^{n-k}A,\phi^{k}B]\!])$$
$$\displaystyle=$$
$$\displaystyle\sum_{k=0}^{n}C_{k}^{n}([\![\phi^{n+1-k}A,\phi^{k}B]\!]+[\![\phi^%
{n-k}A,\phi^{k+1}B]\!])$$
$$\displaystyle=$$
$$\displaystyle[\![\phi^{n+1}A,B]\!]+\sum_{k=1}^{n}C_{k}^{n}[\![\phi^{n+1-k}A,%
\phi^{k}B]\!]$$
$$\displaystyle+\sum_{k=0}^{n-1}C_{k}^{n}[\![\phi^{n-k}A,\phi^{k+1}B]\!]+[\![A,%
\phi^{n+1}B]\!]$$
$$\displaystyle=$$
$$\displaystyle[\![\phi^{n+1}A,B]\!]+\sum_{k=1}^{n}(C_{k}^{n}+C_{k-1}^{n})[\![%
\phi^{n+1-k}A,\phi^{k}B]\!]+[\![A,\phi^{n+1}B]\!]$$
$$\displaystyle=$$
$$\displaystyle\sum_{k=0}^{n+1}C_{k}^{n+1}[\![\phi^{n-k}A,\phi^{k}B]\!].$$
Lemma 3
Suppose that $\Gamma_{1}$ and $\phi$ form a compatible pair,
then identity (36) holds
for all $n\geq 1$.
Proof: Since $\overline{\partial}\Gamma_{1}=0$, we substitute $A$ by $\Gamma_{1}$ in (27) to see that
identity
(36) holds when $n=1$. Assume that (36) holds for
all $k\leq n$. We next prove that it holds for $n+1$. Since
$$\displaystyle\overline{\partial}\phi^{n+1}A-\phi^{n+1}\overline{\partial}A$$
$$\displaystyle=$$
$$\displaystyle\overline{\partial}\phi^{n}(\phi A)-\phi^{n}(\overline{\partial}%
\phi A)+\phi^{n}\left((\overline{\partial}\phi A)-\phi\overline{\partial}A%
\right),$$
by induction hypothesis, the above is equal to
$$\sum_{k=1}^{n}C_{k}^{n}[\![\phi^{k-1}\Gamma_{1},\phi^{n+1-k}A]\!]+\phi^{n}[\![%
\Gamma_{1},A]\!].$$
By Lemma 2, it is equal to
$$\displaystyle\sum_{k=1}^{n}C_{k}^{n}[\![\phi^{k-1}\Gamma_{1},\phi^{n+1-k}A]\!]%
+\sum_{k=0}^{n}C_{k}^{n}[\![\phi^{k}\Gamma_{1},\phi^{n-k}A]\!]$$
$$\displaystyle=$$
$$\displaystyle\sum_{k=1}^{n}\left(C_{k}^{n}+C_{k-1}^{n}\right)[\![\phi^{k-1}%
\Gamma_{1},\phi^{n+1-k}A]\!]+[\![\phi^{n}\Gamma_{1},A]\!].$$
By Pascal Identity, it is equal to
$$\sum_{k=1}^{n}C_{k}^{n+1}[\![\phi^{k-1}\Gamma_{1},\phi^{n+1-k}A]\!]+[\![\phi^{%
n}\Gamma_{1},A]\!]=\sum_{k=1}^{n+1}C_{k}^{n+1}[\![\phi^{k-1}\Gamma_{1},\phi^{n%
+1-k}A]\!].$$
4 Holomorphic Poisson manifolds
On a complex manifold $(M,J)$,
$L=T^{1,0}\oplus T^{*(0,1)}$, and $\overline{L}=L^{*}=T^{0,1}\oplus T^{*(1,0)}.$
Therefore, the exterior bundle has a decomposition
$$\wedge^{\bullet}L=\oplus_{k}\left(\oplus_{p+q=k}\wedge^{p}T^{1,0}\otimes\wedge%
^{q}T^{*(0,1)}\right)$$
We will use the notations $T^{p,0}=\wedge^{p}T^{1,0}$ and
$T^{*(0,q)}=\wedge^{q}T^{*(0,1)}$. Sections of $T^{p,0}$ are addressed as $(p,0)$-vectors, more generally
polyvector fields.
4.1 Type decomposition of deformations
The cohomology of $\mathop{\mathrm{DGA}}\nolimits(J)$
decomposes accordingly into the direct sum of classical Dolbeault cohomology
with the sheaf of exterior product of the holomorphic tangent
bundle as coefficients.
$$H^{k}_{J}=\oplus_{p+q=k,p,q\geq 0}H^{q}(M,T^{p,0}).$$
(37)
If $\Gamma_{1}$ is in $H^{2}_{J}$ it has three components:
$$\Gamma_{1}=\Lambda+{\widehat{\Gamma}}_{1}+\Omega\in H^{0}(M,T^{2,0})\oplus H^{%
1}(M,T^{1,0})\oplus H^{2}(M,{\mathcal{O}}),$$
(38)
where $\Lambda$ is a (2,0)-bivector field,
$\Omega$ is a (0,2)-form, and
${\widehat{\Gamma}}_{1}$ is a classical infinitesimal complex deformation.
Similarly,
$$\displaystyle L^{*}\otimes L$$
$$\displaystyle=$$
$$\displaystyle\mathop{\mathrm{End}}\nolimits(L,L)$$
$$\displaystyle=$$
$$\displaystyle\mathop{\mathrm{End}}\nolimits(T^{1,0},T^{1,0})\oplus\mathop{%
\mathrm{End}}\nolimits(T^{*(0,1)},T^{*(0,1)})$$
$$\displaystyle\quad\quad\oplus\mathop{\mathrm{End}}\nolimits(T^{1,0},T^{*(0,1)}%
)\oplus\mathop{\mathrm{End}}\nolimits(T^{*(0,1)},T^{1,0}).$$
If $\phi$ is a section of $L^{*}\otimes L$, we represent its decomposition
by $\phi=\phi_{1}+\phi_{2}+\phi_{3}+\phi_{4}$ such that
$$\displaystyle\phi_{1}\in C^{\infty}(\mathop{\mathrm{End}}\nolimits(T^{1,0},T^{%
1,0})),\quad\phi_{2}\in C^{\infty}(\mathop{\mathrm{End}}\nolimits(T^{*(0,1)},T%
^{*(0,1)})),$$
$$\displaystyle\phi_{3}\in C^{\infty}(\mathop{\mathrm{End}}\nolimits(T^{1,0},T^{%
*(0,1)})),\quad\phi_{4}\in C^{\infty}(\mathop{\mathrm{End}}\nolimits(T^{*(0,1)%
},T^{1,0})).$$
Proposition 3
A pair $\Gamma_{1}\in C^{\infty}(M,\wedge^{2}L)$ and $\phi\in C^{\infty}(M,L^{*}\otimes L)$ is compatible if and only if the pairs $(\Lambda,\phi_{4})$, $(\Omega,\phi_{3})$ and $({\widehat{\Gamma}}_{1},\phi_{1}+\phi_{2})$ are
compatible.
Proof: This theorem is an inspection of type
decompositions. For example,
$$\overline{\partial}\Gamma_{1}=\overline{\partial}\Lambda+\overline{\partial}{%
\widehat{\Gamma}}_{1}+\overline{\partial}\Omega.$$
Since
$\overline{\partial}\Lambda\in$ $C^{\infty}(M,T^{2,0}\otimes T^{*(0,1)})$,
$\overline{\partial}{\widehat{\Gamma}}_{1}$ $\in C^{\infty}(M,T^{1,0}\otimes T^{*(0,2)})$, and $\overline{\partial}\Omega\in$
$C^{\infty}(M,T^{*(0,3)})$, each component has to vanish individually if
$\overline{\partial}\Gamma_{1}=0.$ i.e.
$$\overline{\partial}\Lambda=0,\quad\overline{\partial}{\widehat{\Gamma}}_{1}=0,%
\quad\overline{\partial}\Omega=0.$$
Next, for all $Z\in C^{\infty}(M,T^{1,0})$ and $\overline{\omega}\in C^{\infty}(M,T^{*(0,1)})$,
$$\displaystyle[\![\Lambda,Z]\!]\in C^{\infty}(T^{2,0}),\quad[\![\Lambda,%
\overline{\omega}]\!]\in C^{\infty}(T^{(1,0)}\otimes T^{*(0,1)});$$
$$\displaystyle[\![{\widehat{\Gamma}}_{1},Z]\!]\in C^{\infty}(T^{*(0,2)}),\quad[%
\![{\widehat{\Gamma}}_{1},\overline{\omega}]\!]=0;$$
$$\displaystyle[\![\Omega,Z]\!]\in C^{\infty}(T^{1,0}\otimes T^{*(0,1)}),\quad[%
\![\Omega,\overline{\omega}]\!]\in C^{\infty}(T^{*(0,2)}).$$
On the other hand,
$$\displaystyle\overline{\partial}(\phi_{1}(Z))-\phi_{1}(\overline{\partial}Z)%
\in C^{\infty}(T^{1,0}\otimes T^{*(0,1)}),\quad\overline{\partial}(\phi_{1}(%
\overline{\omega}))-\phi_{1}(\overline{\partial}\overline{\omega})=0,$$
$$\displaystyle\overline{\partial}(\phi_{2}(Z))-\phi_{2}(\overline{\partial}Z)=-%
\phi_{2}(\overline{\partial}Z)\in C^{\infty}(T^{1,0}\otimes T^{*(0,1)}),\quad%
\overline{\partial}(\phi_{2}(\overline{\omega}))-\phi_{2}(\overline{\partial}%
\overline{\omega})\in C^{\infty}(T^{*(0,2)}),$$
$$\displaystyle\overline{\partial}(\phi_{3}(Z))-\phi_{3}(\overline{\partial}Z)%
\in C^{\infty}(T^{*(0,2)}),\quad\overline{\partial}(\phi_{3}(\overline{\omega}%
))-\phi_{3}(\overline{\partial}\overline{\omega})=0,$$
$$\displaystyle\overline{\partial}(\phi_{4}(Z))-\phi_{4}(\overline{\partial}Z)=-%
\phi_{4}(\overline{\partial}Z)\in C^{\infty}(T^{2,0}),\quad\overline{\partial}%
(\phi_{4}(\overline{\omega}))-\phi_{4}(\overline{\partial}\overline{\omega})%
\in C^{\infty}(T^{1,0}\otimes T^{*(0,1)}).$$
By equating the types, we arrive at the conclusion of this proposition.
In view of the last proposition and the decomposition of $H^{2}_{J}$, one should
focus an initial analysis of deformations on
the simple types, namely those whose infinitesimal deformations are
contained in a unique summand of the decomposition
of $H^{2}_{J}$.
Infinitesimal deformations given by a $\overline{\partial}$-closed section ${\widehat{\Gamma}}_{1}$
of $T^{1,0}\otimes T^{*(0,1)}$ could always be
represented and analyzed as classical complex deformation theory.
If one considers a $\overline{\partial}$-closed 2-form representing an element in $H^{2}(M,{\cal O})$,
then by definition of Courant bracket $[\![\Omega,\Omega]\!]=0$.
Therefore, $\Omega$ satisfies the Maurer-Cartan equation, and
$$L_{\overline{\Omega}}=\{X+\alpha+\iota_{X}{\overline{\Omega}}:X+\alpha\in L\}$$
is a generalized complex structure. The issue of integrability is trivial.
However, this deformation does not change the type of the generalized complex
structure.
It is still type-$n$ where $n$ is the complex dimension of the manifold $M$.
If the $(0,2)$-form $\Omega$ is not only $\overline{\partial}$-closed but also closed, then
this deformation is trivial within the realm of
generalized complex structures because the deformation is only the result of a B-field transformation
[16].
4.2 Holomorphic bivector fields
Suppose that $\Gamma$ is a deformation whose first order term is a
bivector field $\Lambda$ with $\overline{\partial}\Lambda=0$.
Let $\Gamma_{2}$ be its second order term. As $\Gamma$ satisfies
the Maurer-Cartan equation, up to second order term, we have
$$\overline{\partial}(t\Lambda+t^{2}\Gamma_{2})+\frac{1}{2}[\![t\Lambda+t^{2}%
\Gamma_{2},t\Lambda+t^{2}\Gamma_{2}]\!]=0.$$
It yields
$$\overline{\partial}\Gamma_{2}+\frac{1}{2}[\![\Lambda,\Lambda]\!]=0.$$
Since $\Lambda$ is a bivector, $[\![\Lambda,\Lambda]\!]$ is a $(3,0)$-vector field.
On the other hand, $\overline{\partial}\Gamma_{2}$ must have a components with
$(0,1)$-forms. Therefore, the only solution is when $[\![\Lambda,\Lambda]\!]=0$.
It follows immediately that $\Lambda$ is a solution of the
Maurer-Cartan equation and $\Gamma_{2}$ could be chosen to be zero.
Therefore, a bivector field $\Lambda$ representing an element in $H^{0}(M,T^{2,0})$ is an
infinitesimal deformation of an integrable deformation if and only if $[\![\Lambda,\Lambda]\!]=0$.
Definition 2
A $(2,0)$-vector field is a holomorphic Poisson structure on a
complex manifold if $\overline{\partial}\Lambda=0$ and
$[\![\Lambda,\Lambda]\!]=0$. In such case, we call $\Lambda$ a holomorphic Poisson
vector field.
Given such a $\Lambda$, suppose that $\phi\in C^{\infty}(\mathop{\mathrm{End}}\nolimits(T^{*(0,1)},T^{1,0}))$ is
compatible with $\Lambda$. By Theorem 2,
$\Phi=\sum\frac{1}{n!}\phi^{n}$ is a $\mathop{\mathrm{DGA}}\nolimits$ homomorphism. However, as an endomorphism
from the bundle $L$ to $L$, its kernel contains at least
$T^{1,0}$. Therefore, $\phi\Lambda=0$ and $\phi\circ\phi=0$. Therefore, we could
conclude that the homomorphism $\Phi$ is simply $1+\phi$.
Furthermore, given a section $X+\overline{\alpha}$ of
$T^{1,0}\oplus T^{*(0,1)}$, $\Phi(X+{\overline{\alpha}})=X+\phi({\overline{\alpha}})+{\overline{\alpha}}$.
As the vector part is $X+\phi({\overline{\alpha}})$ and the form part is
${\overline{\alpha}}$, $X+{\overline{\alpha}}$ is in the kernel of
$\Phi$ if and only if it is identically zero. Therefore, $\Phi$ as a
bundle map from $L$ to $L$ is an isomorphism. It is extended to an
isomorphism from the exterior bundle $\wedge^{\bullet}L$ to $\wedge^{\bullet}L$.
Therefore, $\Phi$ is not only a $\mathop{\mathrm{DGA}}\nolimits$ homomorphism, but also an isomorphism.
We summarize our observation below.
Theorem 3
Let $M$ be a complex manifold with a holomorphic Poisson vector field $\Lambda$.
Suppose that $\phi$ is a section of $\mathop{\mathrm{End}}\nolimits(T^{*(0,1)},T^{1,0})$ compatible
with the $\Lambda$ in the sense of Definition 1. Then $\Lambda$ defines a family of
generalized complex deformation
of the complex structure on $M$ with $t\Lambda$. Moreover, if $\mathop{\mathrm{DGA}}\nolimits(t\Lambda)$ represents
the $\mathop{\mathrm{DGA}}\nolimits$ of the deformed complex structure, then they are all isomorphic to $\mathop{\mathrm{DGA}}\nolimits(0)$,
the differential Gerstenhaber algebra of the complex structure on the manifold $M$.
Although from the viewpoint of deformation of $\mathop{\mathrm{DGA}}\nolimits$s, the presence of a compatible pair
on a holomorphic Poisson manifold makes the deformation of $\mathop{\mathrm{DGA}}\nolimits$s trivial, on the geometric
level, it is non-trivial. Recall that
$$L_{\overline{\Lambda}}=\{X+{\overline{\alpha}}+\iota_{\overline{\alpha}}{%
\overline{\Lambda}}:X+{\overline{\alpha}}\in T^{1,0}\oplus T^{*(0,1)}\}.$$
As $\iota_{\overline{\alpha}}{\overline{\Lambda}}$ is a (0,1)-vector, the type of
the generalized complex structure
$L_{\overline{\Lambda}}$ is different from the un-deformed one $L$.
If $\Lambda$ as a bundle map from $T^{*(0,1)}$ to
$T^{1,0}$ is everywhere non-degenerate, then $L_{\overline{\Lambda}}$ is a
type-0 generalized complex structure. By a Gualtieri’s lemma
[16], there exists a symplectic structure $\Omega$ on the manifold $M$
such that the complexified $\mathop{\mathrm{DGA}}\nolimits$ of $\omega$ is isomorphic to that
of $\mathop{\mathrm{DGA}}\nolimits(\Lambda)$ via a B-field transformation. Since $\mathop{\mathrm{DGA}}\nolimits(\Lambda)$ is isomorphic
to $\mathop{\mathrm{DGA}}\nolimits(0)$. We obtain the following result.
Theorem 4
Let $M$ be a manifold with complex structure $J$. Denote its associated $\mathop{\mathrm{DGA}}\nolimits$
by $\mathop{\mathrm{DGA}}\nolimits(J)$. Suppose that $\Lambda$ is a non-degenerate holomorphic Poisson
structure. If there exists a section of $T^{1,0}\otimes T^{0,1}$ compatible with
$\Lambda$ in the sense of Definition 1, then there exists a
symplectic structure $\Omega$ in the deformation family of $J$ such that $\mathop{\mathrm{DGA}}\nolimits(\Omega)$ is isomorphic to
$\mathop{\mathrm{DGA}}\nolimits(J)$.
In the sense of Merkulov, the pair $(M,J)$ and $(M,\Omega)$ form a weak mirror pair
[6] [7] [23].
4.3 Rational surfaces
In this section, we compute the first cohomology of some well known holomorphic Poisson manifolds to demonstrate
that for many holomorphic Poisson structures, Theorem 3 does not have solution.
Assume that we have a compact holomorphic Poisson manifold. Denote the Poisson bivector field by $\Lambda$.
Consider $Z$ a section of $T^{1,0}$ and $\overline{\omega}$ a section of $T^{*(0,1)}$. Then $Z+\overline{\omega}$ is a section of
$L=T^{1,0}\oplus T^{*(0,1)}$. By Proposition 1, it represents an element of the first cohomology
of $\mathop{\mathrm{DGA}}\nolimits(\overline{\partial}_{\Lambda})$ if and only if
$\overline{\partial}_{\Lambda}(Z+\overline{\omega})=0$. That is
$$\displaystyle\overline{\partial}_{\Lambda}(Z+\overline{\omega})$$
$$\displaystyle=$$
$$\displaystyle\overline{\partial}Z+[\![\Lambda,Z]\!]+\overline{\partial}%
\overline{\omega}+[\![\Lambda,\overline{\omega}]\!]$$
$$\displaystyle=$$
$$\displaystyle[\![\Lambda,Z]\!]+\overline{\partial}Z+[\![\Lambda,\overline{%
\omega}]\!]+\overline{\partial}\overline{\omega}=0.$$
The terms above are sections of $\wedge^{2}L=T^{2,0}\oplus T^{1,0}\otimes T^{*(0,1)}\oplus T^{*(0,2)}$.
As each component in this decomposition has to vanish, we conclude that
$$[\![\Lambda,Z]\!]=0,\quad\overline{\partial}Z+[\![\Lambda,\overline{\omega}]\!%
]=0,\quad\overline{\partial}\overline{\omega}=0.$$
(39)
In particular the (0,1)-form $\overline{\omega}$ is $\overline{\partial}$-closed. To push this computation further,
assume that the Dolbeault cohomology $H^{1}(M,{\cal O})$ vanishes.
It follows that the (0,1)-form is $\overline{\partial}$-exact, and there is a smooth function $f$ on the manifold
$M$ such that $\overline{\omega}=\overline{\partial}f$. Consider the vector field $V=[\![\Lambda,f]\!]$. Since $\overline{\partial}\Lambda=0$,
$$\overline{\partial}V=\overline{\partial}[\![\Lambda,f]\!]=[\![\overline{%
\partial}\Lambda,f]\!]-[\![\Lambda,\overline{\partial}f]\!]=-[\![\Lambda,%
\overline{\omega}]\!].$$
With (39) above, we conclude that $\overline{\partial}(Z-V)=0$. Therefore, $Z-V$ is a holomorphic vector
field on the manifold $M$.
By Jacobi identity of Gerstenhaber algebras,
$$[\![\Lambda,[\![\Lambda,f]\!]]\!]+[\![\Lambda,[\![f,\Lambda]\!]]\!]+[\![f,[\![%
\Lambda,\Lambda]\!]]\!]=0.$$
Since $[\![\Lambda,\Lambda]\!]=0$, the above is reduced to
$$[\![\Lambda,V]\!]=[\![\Lambda,[\![\Lambda,f]\!]]\!]=0.$$
Combined with the first identity in (39), we conclude that
$$[\![\Lambda,Z-V]\!]=0.$$
Let $W=Z-V$, then $Z=W+V=W+[\![\Lambda,f]\!]$ such that $\overline{\partial}W=0$ and $[\![\Lambda,W]\!]=0$.
Moreover, the section
$$Z+\overline{\omega}=W+[\![\Lambda,f]\!]+\overline{\partial}f=W+\overline{%
\partial}_{\Lambda}f.$$
Since $\overline{\partial}_{\Lambda}f$ is $\overline{\partial}_{\Lambda}$-exact, $W$ and $Z+\overline{\omega}$ represent the same cohomology
class in $H^{1}_{\overline{\partial}_{\Lambda}}$.
Proposition 4
Suppose that $M$ is a holomorphic Poisson manifold with
Poisson vector field $\Lambda$. If $H^{1}(M,{\cal O})$ vanishes, then
$$H^{1}_{\overline{\partial}_{\Lambda}}=\{W\in H^{0}(M,T^{1,0}):[\![\Lambda,W]\!%
]=0\}.$$
On the other hand, the first cohomology of $\mathop{\mathrm{DGA}}\nolimits(\overline{\partial})$ is equal to
$$H^{0}(M,T^{1,0})\oplus H^{1}(M,{\cal O}).$$
Given the assumption of Proposition 4, it is equal to $H^{0}(M,T^{1,0})$.
It is easy to find example on which there exists non-trivial holomorphic Poisson structures but it does
not admit compatible pairs due to the
difference between $H^{0}(M,T^{1,0})$ and $H^{1}_{\overline{\partial}_{\Lambda}}$.
For instance, there is a classification of compact complex surfaces admitting holomorphic Poisson structures [3].
Among them, the minimal rational surfaces are all holomorphic Poisson manifolds with vanishing irregularity.
Except when the surface is a complex projective plane, they are rational ruled surfaces.
For the complex projective plane $\Lambda$ is an element in $H^{0}({\bf C}{\bf P}^{2},{\cal O}(3))$. It could be identified
to a homogeneous polynomial of degree-3 in the homogeneous coordinates of the
complex projective plane. Meanwhile the space of holomorphic vector fields $H^{0}({\bf C}{\bf P}^{2},T^{1,0})$
is the complex algebra $\mathfrak{sl}(3,\mathbb{C})$, treated as the set of $3\times 3$-matrices acting on
of $\mathbb{C}^{3}$ by natural matrix multiplications. From this perspective, for any $W$ in
$H^{0}({\bf C}{\bf P}^{2},T^{1,0})$, the action $[\![W,-]\!]$ on $H^{0}({\bf C}{\bf P}^{2},{\cal O}(3))$
is the induced representation of $\mathfrak{sl}(3,\mathbb{C})$ on the third symmetric product $S^{3}\mathbb{C}^{3}$.
Then for each $\Lambda\neq 0$, one could find a $W$ such that
$[\![W,\Lambda]\!]\neq 0$.
Therefore, for each holomorphic Poisson structure on the complex projective plane, $H^{1}_{\overline{\partial}_{\Lambda}}$ is strictly smaller then
$H^{1}_{\overline{\partial}}=H^{0}({\bf C}{\bf P}^{2},T^{1,0})=\mathfrak{sl}(3,%
\mathbb{C})$. It shows that $\mathop{\mathrm{DGA}}\nolimits({\bf C}{\bf P}^{2},\overline{\partial})$ and
$\mathop{\mathrm{DGA}}\nolimits({\bf C}{\bf P}^{2},\overline{\partial}_{\Lambda})$ for any holomorphic Poisson structure could never be quasi-isomorphic.
4.4 Hopf surfaces
In this section, we compute $H^{1}_{\overline{\partial}_{\Lambda}}$ when the underlying manifold
$M$ is the Hopf surface, and demonstrates
that this does admit compatible pairs.
Consider $\mathbb{C}^{2}$ with coordinates $z=(z_{1},z_{2})$. Let $\lambda>1$ be a real number. It generates a one-parameter
group of automorphism on $\mathbb{C}^{2}$. The quotient of $\mathbb{C}^{2}\backslash\{0\}$ with respect this group is diffeomorphic
to the Lie group $M=U(1)\times SU(2)$. The complex structure on $\mathbb{C}^{2}$ descends onto $M$ to define an
integrable complex structure, invariant of the left-action of the Lie group. In this section, by Hopf surface, we
mean this particular complex structure. The classical complex deformation theory of this
complex structure was analyzed by Dabrowski
[12]. We focus on the deformations generated by its holomorphic Poisson structures.
Consider
$$\displaystyle X_{0}=\frac{1}{2}(z_{1}\frac{\partial}{\partial z_{1}}+z_{2}%
\frac{\partial}{\partial z_{2}}),$$
$$\displaystyle X_{1}=\frac{i}{2}(z_{1}\frac{\partial}{\partial z_{1}}-z_{2}%
\frac{\partial}{\partial z_{2}})$$
$$\displaystyle X_{2}=\frac{i}{2}(z_{2}\frac{\partial}{\partial z_{1}}+z_{1}%
\frac{\partial}{\partial z_{2}}),$$
$$\displaystyle X_{3}=\frac{1}{2}(-z_{2}\frac{\partial}{\partial z_{1}}+z_{1}%
\frac{\partial}{\partial z_{2}}),$$
and
$$\overline{\sigma}=\overline{\partial}\ln|z|^{2}=\frac{z_{1}d{\overline{z}}_{1}%
+z_{2}d{\overline{z}}_{2}}{|z_{1}|^{2}+|z_{2}|^{2}}.$$
The cohomology spaces for the $\mathop{\mathrm{DGA}}\nolimits(J)$ are given below. The computation of these cohomology spaces
are not new. We do not present any details.
$$\displaystyle H^{1}(M,{\cal O})=\langle{\overline{\sigma}}\rangle,\quad H^{0}(%
M,T^{1,0})=\langle X_{0},X_{1},X_{2},X_{3}\rangle\cong{\mathfrak{u}}(1)\oplus{%
\mathfrak{sl}}(2),$$
(40)
$$\displaystyle H^{1}(M,T^{1,0})=\langle X_{0}\wedge{\overline{\sigma}},X_{1}%
\wedge{\overline{\sigma}},X_{2}\wedge{\overline{\sigma}},X_{3}\wedge{\overline%
{\sigma}}\rangle,$$
(41)
$$\displaystyle H^{0}(M,T^{2,0})=\langle X_{0}\wedge X_{1},X_{0}\wedge X_{2},X_{%
0}\wedge X_{3}\rangle,$$
(42)
$$\displaystyle H^{1}(M,T^{2,0})=\langle X_{0}\wedge X_{1}\wedge{\overline{%
\sigma}},X_{0}\wedge X_{2}\wedge{\overline{\sigma}},X_{0}\wedge X_{3}\wedge{%
\overline{\sigma}}\rangle.$$
(43)
In addition,
$$\displaystyle[\![X_{0},X_{1}]\!]=0,\quad[\![X_{0},X_{2}]\!]=0,\quad[\![X_{0},X%
_{3}]\!]=0,$$
(44)
$$\displaystyle[\![X_{1},X_{2}]\!]=-X_{3},\quad[\![X_{2},X_{3}]\!]=-X_{1},\quad[%
\![X_{3},X_{1}]\!]=-X_{2}.$$
(45)
Set $f=\ln|z|^{2}$, then ${\cal L}_{X_{0}}f=\frac{1}{2}$. For $j=1,2,3$, define $f_{j}={\cal L}_{X_{j}}f$, then
$$f_{1}=\frac{i}{2|z|^{2}}(z_{1}\overline{z}_{1}-z_{2}\overline{z}_{2}),\quad f_%
{2}=\frac{i}{2|z|^{2}}(z_{2}\overline{z}_{1}+z_{1}\overline{z}_{2}),\quad f_{3%
}=\frac{1}{2|z|^{2}}(-z_{2}\overline{z}_{1}+z_{1}\overline{z}_{2}).$$
The functions $f_{1},f_{2},f_{3}$ are invariant of the group of actions generated by
$(\lambda z_{1},\lambda z_{2})$, and hence they are globally defined on the quotient space $M$.
Then we have
$$[\![X_{0},\overline{\sigma}]\!]=0,\quad[\![X_{1},\overline{\sigma}]\!]=%
\overline{\partial}f_{1},\quad[\![X_{2},\overline{\sigma}]\!]=\overline{%
\partial}f_{2},\quad[\![X_{3},\overline{\sigma}]\!]=\overline{\partial}f_{3}.$$
(46)
Whenever $A=a_{1}X_{1}+a_{2}X_{2}+a_{3}X_{3}$ is a holomorphic vector field in the
${\mathfrak{sl}}(2)$ component of $H^{0}(M,T^{1,0})$,
$$[\![A,\overline{\sigma}]\!]=a_{1}[\![X_{1},\overline{\sigma}]\!]+a_{2}[\![X_{2%
},\overline{\sigma}]\!]+a_{3}[\![X_{3},\overline{\sigma}]\!]=\overline{%
\partial}(a_{1}f_{1}+a_{2}f_{2}+a_{3}f_{3}).$$
We use the notation $f_{A}$ to denote the function $a_{1}f_{1}+a_{2}f_{2}+a_{3}f_{3}$. By (46),
$$[\![A,\overline{\sigma}]\!]=\overline{\partial}f_{A}.$$
(47)
Since $X_{0}$ commutes with $X_{j}$ for $j=1,2,3$,
${\cal L}_{X_{0}}f_{j}={\cal L}_{X_{j}}{\cal L}_{X_{0}}f={\cal L}_{X_{j}}\frac{%
1}{2}=0.$
Then for all $A$
$${\cal L}_{X_{0}}f_{A}=0.$$
(48)
Given the above preparation, we begin to compute the first cohomology of $\mathop{\mathrm{DGA}}\nolimits(\overline{\partial}_{\Lambda})$ where $\Lambda$ is any holomorphic Poisson structure on $M$.
Let $A=a_{1}X_{1}+a_{2}X_{2}+a_{3}X_{3}$ be a holomorphic vector field, then $\Lambda=X_{0}\wedge A$ is a holomorphic
Poisson structure. As noted in (42),
by choosing the complex numbers $(a_{1},a_{2},a_{3})$, we exhaust all holomorphic Poisson structure.
Now we calculate the first cohomology with respect to $\overline{\partial}_{\Lambda}=\overline{\partial}+[\![\Lambda,-]\!]$.
Suppose that $Z$ is a smooth (1,0)-vector field and $\overline{\omega}$ is a smooth (0,1)-form.
$\overline{\partial}_{\Lambda}(Z+\overline{\omega})=0$ if and only if $Z$ and $\overline{\omega}$ satisfy the constraints
(39). Once again, they are
$$\overline{\partial}\overline{\omega}=0,\quad[\![\Lambda,Z]\!]=0,\quad\overline%
{\partial}Z+[\![\Lambda,\overline{\omega}]\!]=0.$$
(49)
Since the cohomology $H^{1}(X,{\cal O})$ is spanned by $\overline{\sigma}$, there exists a function $\psi$ and a
unique complex
number $a$ such that
$$\overline{\omega}=a\overline{\sigma}+\overline{\partial}\psi.$$
Let $V$ be the vector field $[\![\Lambda,\psi]\!]$. Since $\overline{\partial}\Lambda=0$,
$$\displaystyle\overline{\partial}V$$
$$\displaystyle=$$
$$\displaystyle\overline{\partial}[\![\Lambda,\psi]\!]=[\![\overline{\partial}%
\Lambda,\psi]\!]-[\![\Lambda,\overline{\partial}\psi]\!]=-[\![\Lambda,%
\overline{\partial}\psi]\!]$$
$$\displaystyle=$$
$$\displaystyle-[\![\Lambda,\overline{\omega}-a\overline{\sigma}]\!]=-[\![%
\Lambda,\overline{\omega}]\!]+a[\![\Lambda,\overline{\sigma}]\!].$$
By definition of $\Lambda$, (46) and (47), this is equal to
$$-[\![\Lambda,\overline{\omega}]\!]+aX_{0}\wedge[\![A,\overline{\sigma}]\!]=-[%
\![\Lambda,\overline{\omega}]\!]+aX_{0}\wedge\overline{\partial}f_{A}=-[\![%
\Lambda,\overline{\omega}]\!]-a\overline{\partial}(f_{A}X_{0}).$$
It follows from (49) that
$$\overline{\partial}(V-Z+af_{A}X_{0})=0.$$
(50)
Next, consider the Schouten bracket. By (49),
$$\displaystyle[\![\Lambda,V-Z+af_{A}X_{0}]\!]$$
$$\displaystyle=$$
$$\displaystyle[\![\Lambda,[\![\Lambda,\psi]\!]]\!]-[\![\Lambda,Z]\!]+[\![X_{0}%
\wedge A,af_{A}X_{0}]\!]$$
$$\displaystyle=$$
$$\displaystyle[\![\Lambda,[\![\Lambda,\psi]\!]]\!]+aX_{0}\wedge[\![A,f_{A}X_{0}%
]\!]-aA\wedge[\![X_{0},f_{A}X_{0}]\!]$$
$$\displaystyle=$$
$$\displaystyle[\![\Lambda,[\![\Lambda,\psi]\!]]\!]+aX_{0}\wedge[\![A,f_{A}]\!]X%
_{0}+af_{A}X_{0}\wedge[\![A,X_{0}]\!]-aA\wedge[\![X_{0},f_{A}]\!]X_{0}.$$
Due to (44) and (48), this is equal to
$[\![\Lambda,[\![\Lambda,\psi]\!]]\!].$
By the Jacobi identity for Gerstenhaber algebra and the fact that $[\![\Lambda,\Lambda]\!]=0$,
$[\![\Lambda,V-Z+af_{A}X_{0}]\!]=0$. Define
$$W=-V+Z-af_{A}X_{0}=-[\![X_{0}\wedge A,\psi]\!]+Z-af_{A}X_{0}.$$
(51)
Then $[\![\Lambda,W]\!]=0$. However, by (49) and the identity above,
$$[\![\Lambda,W]\!]=-[\![X_{0}\wedge A,W]\!]=-X_{0}\wedge[\![A,W]\!].$$
(52)
As $\overline{\partial}W=0$, it is a linear combination of $X_{0},X_{1},X_{2},X_{3}$.
Therefore, $[\![\Lambda,W]\!]$
is equal to zero if and only if there exist constants $b$ and $c$ such that
$W=bX_{0}+cA.$
Therefore,
$$Z=V+W+af_{A}X_{0}=[\![\Lambda,\psi]\!]+bX_{0}+cA+af_{A}X_{0}.$$
(53)
As we have already resolved the first
two constraints in (49), we could now substitute
$Z$ in the last constraint to check that it does not generate
additional conditions.
So, $Z+\overline{\omega}$ is $\overline{\partial}_{\Lambda}$-closed for $\Lambda=X_{0}\wedge A$
if and only if there exist a function $\psi$ and
constants $a,b,c$ such that
$$\displaystyle\overline{\omega}$$
$$\displaystyle=$$
$$\displaystyle a\overline{\sigma}+\overline{\partial}\psi,$$
$$\displaystyle Z$$
$$\displaystyle=$$
$$\displaystyle[\![\Lambda,\psi]\!]+bX_{0}+cA+af_{A}X_{0}=[\![\Lambda,\psi]\!]+W%
+af_{A}X_{0}.$$
Since
$$Z+\overline{\omega}=[\![\Lambda,\psi]\!]+bX_{0}+cA+af_{A}X_{0}+a\overline{%
\sigma}+\overline{\partial}\psi=bX_{0}+cA+af_{A}X_{0}+a\overline{\sigma}+%
\overline{\partial}_{\Lambda}\psi,$$
$Z+\overline{\omega}$ and
$bX_{0}+cA+af_{A}X_{0}+a\overline{\sigma}$ represent the same element in the first cohomology
space $H^{1}(M,\overline{\partial}_{X_{0}\wedge A})$.
Therefore, we have
$$H^{1}(X,\overline{\partial}_{X_{0}\wedge A})=\langle X_{0},A,f_{A}X_{0}+%
\overline{\sigma}\rangle\cong\mathbb{C}^{3}.$$
(54)
On the other hand, it is noted in (40) that the first cohomology of
$\mathop{\mathrm{DGA}}\nolimits(J)$ is a five-dimensional space.
$$H^{1}(M,\overline{\partial})=H^{1}(M,{\cal O})\oplus H^{0}(M,T^{1,0})=\langle{%
\overline{\sigma}}\rangle\oplus\langle X_{0},X_{1},X_{2},X_{3}\rangle.$$
Therefore, along the deformation given by holomorphic Poisson vector field
$\Lambda=X_{0}\wedge A$, the first cohomology jumps and hence $\Lambda$ could not be
part of any compatible pair.
5 Holomorphic symplectic algebras
In an explicit computation in [25], part of the result in Theorem 4 has
been observed on the Kodaira-Thurston surface. It was possible to do an explicit computation
due to the fact that the manifold is a low-dimension nilmanifold.
If $H$ is a simply-connected nilpotent Lie group and $K$ is a co-compact subgroup, then the
quotient manifold $M=H/K$ is said to be a nilmanifold.
Let $\mathfrak{h}$ be the Lie algebra of the group $H$,
the Chevalley-Eilenberg differential $d$ determines a complex
$$d:\wedge^{k}{\mathfrak{h}}^{*}\to\wedge^{k+1}{\mathfrak{h}}^{*}.$$
It is known for a long time that the inclusion $\mathfrak{h}^{*}$ as invariant section of
$T^{*}$ induces an isomorphism on the cohomology level
[24]. If the nilmanifold has an invariant symplectic structure $\Omega$,
one could therefore consider this inclusion as a quasi-isomorphism from the differential
Gerstenhaber algebra with invariant objects $\mathop{\mathrm{DGA}}\nolimits(\mathfrak{h},\Omega)$ to the manifold level
$\mathop{\mathrm{DGA}}\nolimits(H/K,\Omega)$.
There were a series of attempt to attain a similar result for Dolbeault cohomology
[8] [9] [10] [27]. This body of research generates a
collection of examples of nilmanifolds for which the inclusion of invariant sections
in the space of sections of the bundle $L=T^{1,0}\oplus T^{*(0,1)}$ induces a quasi-isomorphism
of $\mathop{\mathrm{DGA}}\nolimits$s. Kodaira-Thurston surfaces is a prominent example with small
dimension. To illustrate the theory of the past few chapters, we now focus on $\mathop{\mathrm{DGA}}\nolimits(\mathfrak{h},J)$
for some Lie algebra $\mathfrak{h}$.
In our subsequent computation, we do not restrict $\mathfrak{h}$ to being nilpotent, but will
construct algebras on which there is a good collection of geometric objects as in [7].
5.1 Pseudo-Kähler structures
Let $(\mathfrak{g},\omega)$ denote a real Lie algebra equipped
with a symplectic structure $\omega$. Let $V$ denote the underlying vector
space of $\mathfrak{g}$. We seek a linear map $\gamma:\mathfrak{g}\to{\mathop{\mathrm{End}}\nolimits}(V)$ such that for all $x,y,z\in\mathfrak{g}$,
$$\displaystyle\gamma(x)y-\gamma(y)x=[x,y];$$
(55)
$$\displaystyle\omega(\gamma(x)y,z)+\omega(y,\gamma(x)z)=0;$$
(56)
$$\displaystyle\gamma([x,y])=\gamma(x)\gamma(y)-\gamma(y)\gamma(x).$$
(57)
The last condition requires $\gamma$ to be a representation. The
second condition means that it is a
symplectic.
If one uses $\gamma$ as a operator of vector fields on the Lie group of
the algebra $\mathfrak{g}$, the last condition is equivalent to require $\gamma$ to be a flat connection.
Condition in (56) is to require the connection to be symplectic. The condition in (55)
is to require the connection to be torsion-free.
Given the representation $\gamma$, one obtains a semi-direct product Lie
algebra $\mathfrak{h}:=\mathfrak{g}\ltimes V$ with a Lie bracket defined by
$$[\![(x,0),(y,0)]\!]=([x,y],0)\quad[\![(x,0),(0,v)]\!]=(0,\gamma(x)v),$$
(58)
for all $x,y\in\mathfrak{g}$ and $v\in V$. Here we denote a generic element in $\mathfrak{g}\ltimes V$ in terms
of the decomposition $(x,u)\in\mathfrak{g}\oplus V$.
On the semi-direct product, consider the linear map.
$$J(x,y)=(-y,x).$$
(59)
This is an almost complex structure. The $(1,0)$ vectors are given by
$$\mathfrak{h}^{1,0}=\{(x,-ix)\in(\mathfrak{g}\oplus V)_{\mathbb{C}}:x\in%
\mathfrak{g}\}.$$
(60)
$J$ is an integrable complex structure due to (55) because
$$[\![x-iJx,y-iJy]\!]=[\![(x,-ix),(y,-iy)]\!]=([x,y],-i(\gamma(x)y-\gamma(y)x)).$$
The symplectic structure $\omega$ induces three different symplectic
forms on the semi-direct product $\mathfrak{h}$.
$$\displaystyle\Omega_{1}((x,u),(y,v)):=-\omega(x,v)-\omega(u,y),$$
(61)
$$\displaystyle\Omega_{2}((x,u),(y,v)):=\omega(x,y)-\omega(u,v),$$
(62)
$$\displaystyle\Omega_{3}((x,u),(y,v)):=\omega(x,y)+\omega(u,v).$$
(63)
With respect to the complex structure $J$, $\Omega_{c}=\Omega_{1}+i\Omega_{2}$ is a closed
(2,0)-form. It is non-degenerate in the sense that the contraction map
$$V\mapsto\Omega_{c}(V,{\ }),\quad\Omega_{c}:\mathfrak{h}^{1,0}\to\mathfrak{h}^{%
*(1,0)}$$
is non-degenerate.
The pair $(\Omega_{c},J)$ is called a complex symplectic structure on the algebra $\mathfrak{h}$.
Let $\Lambda$ be the inverse mapping of $\Omega_{c}$.
$$\Lambda:\mathfrak{h}^{*(1,0)}\to\mathfrak{h}^{1,0}.$$
It is a matter of definition that $\Lambda\in\wedge^{2}\mathfrak{h}^{1,0}=\mathfrak{h}^{2,0}$.
Therefore, it could play the role of an invariant holomorphic Poisson structure.
Indeed we have the following
Lemma 4
Let $\Lambda$ be the inverse of $\Omega_{c}$, then it satisfies the following.
•
For any $\alpha,\beta\in\mathfrak{h}^{*(1,0)}$,
$\Lambda(\alpha,\beta)=-\Omega_{c}(\Omega_{c}^{-1}(\alpha),\Omega_{c}^{-1}(%
\beta)).$
•
$[\![\Lambda,\Lambda]\!]=0.$
•
$\overline{\partial}\Lambda=0$.
Proof: Beyond tracing definitions, the first identity is an elementary application of the
algebraic properties of Gerstenhaber algebra. The second identity is equivalent to $d\Omega_{c}=0$.
The last is another application of the algebraic properties of Gerstenhaber algebra combined with
a type decomposition argument.
The last lemma leads to the next.
Lemma 5
Given a symplectic algebra $(\mathfrak{g},\omega)$ with a flat torsion-free symplectic
connection on the underlying vector space $V$ of $\mathfrak{g}$, then the semi-direct product
$\mathfrak{h}=\mathfrak{g}\ltimes V$ has a holomorphic Poisson structure $(J,\Lambda=\Omega_{c}^{-1})$.
Given the above holomorphic Poisson structure, we consider the generalized deformation generated by
the holomorphic Poisson vector field $\Lambda$. It yields
$$L_{\overline{\Lambda}}=\mathfrak{h}^{1,0}\oplus\{\overline{\zeta}+\overline{%
\Lambda}{}\overline{\zeta}:\overline{\zeta}\in\mathfrak{h}^{*(0,1)}\}.$$
(64)
Since $\overline{\Lambda}:\mathfrak{h}^{*(0,1)}\to\mathfrak{h}^{1,0}$ is an isomorphism with ${\overline{\Omega}}_{c}$
as its inverse,
$$\displaystyle L_{\overline{\Lambda}}$$
$$\displaystyle=$$
$$\displaystyle\mathfrak{h}^{1,0}\oplus\{{\overline{\Omega}}_{c}(\overline{Y})+%
\overline{Y}:\overline{Y}\in\mathfrak{h}^{(0,1)}\}.$$
Since ${\overline{\Omega}}_{c}$ is a (0,2)-form, for any (1,0)-vector $X$, ${\overline{\Omega}}_{c}(X)=0$.
Therefore, the above is equal to
$$\displaystyle=$$
$$\displaystyle\{X+{\overline{\Omega}}_{c}(X)+\overline{Y}+{\overline{\Omega}}_{%
c}(\overline{Y}):X\in\mathfrak{h}^{1,0},\overline{Y}\in\mathfrak{h}^{0,1}\}$$
$$\displaystyle=$$
$$\displaystyle\{V+\Omega_{c}(V):V\in\mathfrak{h}^{1,0}\oplus\mathfrak{h}^{0,1}%
\}=\{V+\Omega_{1}(V)-i\Omega_{2}(V):V\in\mathfrak{h}_{c}\}$$
$$\displaystyle=$$
$$\displaystyle e^{\Omega_{1}}\{V-i\Omega_{2}(V):V\in\mathfrak{h}_{c}\}.$$
The last equality means that the deformed generalized complex structure
$L_{\overline{\Lambda}}$ is the $B$-field transformation by the closed 2-form $\Omega_{1}$
of the generalized complex structure defined by the symplectic form $\Omega_{2}$.
In conclusion, we have
Proposition 5
Given a symplectic algebra $(\mathfrak{g},\omega)$ with a flat torsion-free symplectic
connection on the underlying vector space $V$ of $\mathfrak{g}$, then up to the $B$-field transformation with respect
to the closed 2-form $\Omega_{1}$, the generalized deformation of the
classical complex structure by holomorphic Poisson structure $\Lambda=(\Omega_{1}+i\Omega_{2})^{-1}$ is the
the symplectic structure $\Omega_{2}$. In particular,
$\mathop{\mathrm{DGA}}\nolimits(L_{\overline{\Lambda}})$ is isomorphic to $\mathop{\mathrm{DGA}}\nolimits(\Omega_{2})$.
5.2 Compatible pairs
A different perspective in understanding $\mathop{\mathrm{DGA}}\nolimits(L_{\overline{\Lambda}})$ is in terms of compatible pair.
That is to identify an element $\phi$ in $\mathfrak{h}^{0,1}\otimes\mathfrak{h}^{1,0}$
so that $(\Lambda,\phi)$ forms a compatible pair.
As $\Omega_{3}$ is a (1,1)-form and its contraction map is non-degenerate
$$\Omega_{3}:\mathfrak{h}^{1,0}\to\mathfrak{h}^{*(0,1)},$$
its inverse map
$$\Omega_{3}^{-1}:\mathfrak{h}^{*(0,1)}\to\mathfrak{h}^{1,0}$$
is a natural candidate to form a compatible pair with $\Lambda$.
On the other hand, if $g$ is a non-degenerate symmetric bilinear
form the algebra $\mathfrak{g}$, it induces a non-degenerate form on $\mathfrak{g}\ltimes V$ by
$$\Delta((x,u),(y,v))=g(x,y)+g(u,v).$$
Then its fundamental form is a (1,1)-form:
$$\Omega_{4}((x,u),(y,v))=\Delta(J(x,u),(y,v))=\Delta((-u,x),(y,v))=g(x,v)-g(y,u).$$
Therefore, $\Omega_{4}^{-1}$ is also a candidate to match with $\Lambda$ as a compatible pair.
It is a natural question to ask when $\Omega_{4}$ is closed. It amounts to asking the pair
$J$ and $\Delta$ to form a pseudo-Kähler structure.
Lemma 6
The pair $(J,\Delta)$ on $\mathfrak{h}$ forms a pseudo-Kähler structure if and only if
$$g(\gamma(x)y,w)-g(\gamma(y)x,w)-g(x,\gamma(y)w)+g(y,\gamma(x)w)=0$$
for all $x,y,w\in\mathfrak{g}$.
Proof: For any $(x,u),(y,v),(z,w)\in\mathfrak{g}\ltimes V$, expand
$d\Omega_{4}((x,u),(y,v),(z,w))$. Since $\gamma$ is torsion-free, it is equal to
$$\displaystyle-g(\gamma(x)y,w)+g(\gamma(y)x,w)+g(x,\gamma(y)w)-g(y,\gamma(x)w)$$
$$\displaystyle-g(\gamma(z)x,v)+g(\gamma(x)z,v)+g(z,\gamma(x)v)-g(x,\gamma(z)v)$$
$$\displaystyle-g(\gamma(y)z,u)+g(\gamma(z)y,u)+g(y,\gamma(z)u)-g(z,\gamma(y)u)$$
Since the last three lines are cyclic permutations of $(x,u)$, $(y,v)$ and $(z,w)$, if one of these lines is
equal to zero, all three equal to zero and therefore $d\Omega_{4}=0$. Conversely, if $d\Omega_{4}=0$,
set $z=u=v=0$. Then the last two lines equal to zero, and the lemma follows.
Suppose that $(\Omega_{c},J)$ is a holomorphic symplectic structure on the semi-direct product
$\mathfrak{h}=\mathfrak{g}\ltimes V$ as above. Let $\Omega_{3}$ and $\Delta$ be the natural
symplectic and pseudo-metric structure on $\mathfrak{h}$. Assume that $(\Delta,J)$ is pseudo-Kähler.
Both $\Omega^{-1}_{3}$ and $\Omega_{4}^{-1}$ are candidates to be compatible with $\Lambda=\Omega_{c}^{-1}$,
so are their linear combinations.
Below is a key technical result in this section.
Proposition 6
Suppose that $(\Omega_{c},J)$ is a holomorphic symplectic structure on the semi-direct product
$\mathfrak{h}=\mathfrak{g}\ltimes V$ as above. Let $\Omega_{3}$ and $\Delta$ be the natural
symplectic and (pseudo-)metric structure on $\mathfrak{h}$. Assume that $(\Delta,J)$ is pseudo-Kähler structure.
If there is a real number $\mu$ such that
$$(g^{-1}\omega)(\gamma(a)b)=-4\mu\gamma((g^{-1}\omega)(a))((g^{-1}\omega)(b))$$
(65)
for all $a,b\in\mathfrak{g}$, then
$$\phi=-\frac{i}{4}\Omega_{3}^{-1}+\mu\Omega_{4}^{-1}$$
(66)
and $\Lambda=\Omega_{c}^{-1}$ forms a compatible pair.
In the expression (65), we consider the contractions with $\omega$ and $g$ as
maps from the underlying vector
space $V$ of $\mathfrak{g}$ to its dual. Therefore, $\jmath=g^{-1}\omega$ is a map from $V$ to $V$.
The following are used frequently in our proof of Proposition 65 above.
Lemma 7
Recall that
$\mathfrak{h}^{1,0}=\{(a,-ia)\in(\mathfrak{g}\oplus V)_{\mathbb{C}}:a\in%
\mathfrak{g}\}.$
•
As (0,1)-forms, $\Omega_{3}((a,-ia),-)=-i\Omega_{4}((\jmath(a),-i\jmath(a)),-).$
•
As (1,0)-forms, $\Omega_{3}((a,ia),-)=i\Omega_{4}((\jmath(a),i\jmath(a)),-).$
•
As (1,0)-forms, $\Omega_{c}((a,-ia),-)=-2\Omega_{4}((\jmath(a),i\jmath(a)),-).$
•
$[\![(a,-ia),(b,ib)]\!]^{1,0}=(-\gamma(b)a,i\gamma(b)a)$.
•
$[\![(a,-ia),(b,ib)]\!]^{0,1}=(\gamma(a)b,i\gamma(a)b)$.
To prove Proposition 6, we consider a generic linear combination of $\Omega_{3}^{-1}$ and
$\Omega_{4}^{-1}$, $\phi=\lambda\Omega_{3}^{-1}+\mu\Omega^{-1}_{4}$.
Note that we first extend
both $\Omega_{3}^{-1}$ and
$\Omega_{4}^{-1}$ by zeroes on $\mathfrak{h}^{1,0}$. Then they are extended
as endomorphisms defined on $\mathfrak{h}^{1,0}\oplus\mathfrak{h}^{*(0,1)}$
to endomorphisms defined on the exterior
product
$\wedge^{\bullet}(\mathfrak{h}^{1,0}\oplus\mathfrak{h}^{*(0,1)})$ through the identity (29),
by linearity $\phi$ also satisfies (29). Therefore, we will determine the coefficients
$\lambda$ and $\mu$ by solving the non-trivial constraints in (27) and (28).
In the current context, the constraint (27) is equivalent to requiring that for all
$\ell_{1},\ell_{2}\in\mathfrak{h}^{1,0}\oplus\mathfrak{h}^{*(0,1)}$,
$$\phi([\![\ell_{1},\ell_{2}]\!])=[\![\phi\ell_{1},\ell_{2}]\!]+[\![\ell_{1},%
\phi\ell_{2}]\!].$$
(67)
Since $\mathfrak{h}^{1,0}$ is annihilated by $\phi$, and it is closed with respect
to Schouten bracket, if both $\ell_{1}$ and $\ell_{2}$ are in $\mathfrak{h}^{1,0}$, then the identity (67)
is trivially satisfied, and hence does not pose any constraint on $\lambda$ and $\mu$.
If $\ell_{1}\in\mathfrak{h}^{1,0}$, then there exists $a\in\mathfrak{g}$ such that $\ell_{1}=(a,-ia)$.
If $\ell_{2}\in\mathfrak{h}^{*(0,1)}$, then there exists $(b,-ib)\in\mathfrak{h}^{1,0}$ such that
$\ell_{2}=\Omega_{3}((b,-ib),-)$. By Lemma 7,
$$\ell_{2}=\Omega_{3}((b,-ib),-)=\Omega_{4}((-i\jmath(b),-\jmath(b)),-).$$
Since $\phi\ell_{1}=0$,
the constraint in (67) is reduced to
$\phi[\![\ell_{1},\ell_{2}]\!]=[\![\ell_{1},\phi\ell_{2}]\!]$. Since both sides of this identity are
$(1,0)$-vectors, to verify that they are identical, it suffices to show that the evaluation of
any $(1,0)$-forms on these two vectors are identical. Since $\Omega_{3}$ is non-degenerate, any
$(1,0)$-form has the form
$\Omega_{3}((n,in),-)$ for some $(0,1)$-vector $(n,in)$. Then a proof of
(67) is reduced to
check whether the following holds:
$$\Omega_{3}((n,in),\phi[\![\ell_{1},\ell_{2}]\!])-\Omega_{3}((n,in),[\![\ell_{1%
},\phi\ell_{2}]\!])=0.$$
Making use of various definitions and Lemma 7, we reduce the above identity to
$$\displaystyle\lambda\Omega_{3}((b,-ib),(\gamma(a)\jmath(n),-\gamma(a)\jmath(n)%
))-\lambda\Omega_{3}((n,in),([a,b],-i[a,b]))$$
$$\displaystyle+\mu\Omega_{4}((-i\jmath(b),-\jmath(b)),(i\gamma\jmath(n),-\gamma%
(a)\jmath(n)))$$
$$\displaystyle-\mu\Omega_{4}((i\jmath(n),-\jmath(n)),(-i[a,\jmath(b)],-[a,%
\jmath(b)]))=0.$$
Using definition of $\Omega_{3}$ and $\Omega_{4}$ in terms of $\omega$, the above is reduced to
$$-\lambda\omega(\gamma(b)n,a)+i\mu g(\gamma(\jmath(b))\jmath(n),a)=0.$$
It is equivalent to
$$\lambda\jmath(\gamma(b)n)=i\mu\gamma(\jmath(b))(\jmath(n))$$
(68)
for all $b,n\in\mathfrak{g}$. This identity is the first preliminary constraint on $\mu$ and $\lambda$.
Similarly, if $\ell_{1},\ell_{2}\in\mathfrak{h}^{*(0,1)}$, choose $(a,-ia)$ and $(b,-ib)$ such that
$$\ell_{1}=\Omega_{3}((a,-ia),-),\quad\ell_{2}=\Omega_{3}((b,-ib),-).$$
(69)
Since $[\![\ell_{1},\ell_{2}]\!]=0$,
(67) is reduced to
$$[\![\phi\ell_{1},\ell_{2}]\!]+[\![\ell_{1},\phi\ell_{2}]\!]=0.$$
(70)
As both terms in the above sum are (0,1)-forms, then its evaluation on any (0,1)-vector $(n,in)$ is equal to
zero. Substitute (69) into identity (70), evaluate on a (0,1)-vector $(n,in)$,
and make use of Lemma 7, we get
$$\displaystyle-\lambda\Omega_{3}((b,-ib),(\gamma(a)n,i\gamma(a)n))+\lambda%
\Omega_{3}((a,-ia),(\gamma(b)n,i\gamma(b)n))$$
$$\displaystyle-\mu\Omega_{4}((-i\jmath(b),-\jmath(b)),(-i\gamma(\jmath(a))n,%
\gamma(\jmath(a))n))$$
$$\displaystyle+\mu\Omega_{4}((-i\jmath(a),-\jmath(a)),(-i\gamma(\jmath(b))n,%
\gamma(\jmath(b))n))=0.$$
Using definitions of $\Omega_{3}$ and $\Omega_{4}$, together with Lemma 7, The above
identity is reduced to
$$\lambda\omega([a,b],n)-i\mu g([\jmath(a),\jmath(b)],n)$$
for all $n\in\mathfrak{g}$. That is
$$\lambda\jmath([a,b])=i\mu[\jmath(a),\jmath(b)].$$
(71)
Since $\gamma(a)b-\gamma(b)a=[a,b]$ for all $a,b$, the above is equivalent to
$$\lambda\jmath(\gamma(a)b)-\lambda\jmath(\gamma(b)a)=-\mu\gamma(\jmath(a))%
\jmath(b)+\mu\gamma(\jmath(b))\jmath(a).$$
This identity holds for all $a,b\in\mathfrak{g}$ so long as
(68) holds. Therefore, (68) is the only constraint
for solving (67).
Next, we need to find the constraints on $\lambda$ and $\mu$ to satisfy the
identify (28). This is equivalent to requiring
$$\lambda(\overline{\partial}\Omega_{3}^{-1}(\ell)-\Omega_{3}^{-1}\overline{%
\partial}\ell)+\mu(\overline{\partial}\Omega_{4}^{-1}(\ell)-\Omega_{4}^{-1}%
\overline{\partial}\ell)=[\![\Lambda,\ell]\!]$$
(72)
for all $\ell\in\mathfrak{h}^{1,0}\oplus\mathfrak{h}^{*(0,1)}$.
Since $\Omega_{3}^{-1}$ and $\Omega_{4}^{-1}$ are extended by zero on $\mathfrak{h}^{1,0}$, when
$\ell$ is an element in $\mathfrak{h}^{1,0}$, the constraint (72) is reduced to
$$-\lambda\Omega_{3}^{-1}\overline{\partial}\ell-\mu\Omega_{4}^{-1}\overline{%
\partial}\ell=[\![\Lambda,\ell]\!].$$
(73)
Let $A,B$ be elements in $\mathfrak{h}^{1,0}$, with identity (18) and the fact that
$d\Omega_{c}=0$, one could check that
$$[\![\Lambda,\ell]\!](\Omega_{c}A,\Omega_{c}B)=\Omega_{c}(\ell,[\![A,B]\!]).$$
If we set $\ell=(x,-ix),A=(a,-ia),B=(b,-ib)$ with $x,a,b\in\mathfrak{g}$, recall the definitions
of $\Omega_{c}$ in terms of $\omega$, then the above is
further simplified to
$$[\![\Lambda,\ell]\!](\Omega_{c}A,\Omega_{c}B)=4i\omega(x,[a,b]).$$
(74)
In view of (11), the first term on the left-hand-side of the identity in
(73) evaluated on
the ordered pair $\Omega_{c}A,\Omega_{c}B$ is simplified to
$$\displaystyle-\Omega_{3}^{-1}\overline{\partial}\ell(\Omega_{c}A,\Omega_{c}B)$$
$$\displaystyle=$$
$$\displaystyle-2i\left(\Omega_{c}B([\![(x,-ix),(a,ia)]\!])-\Omega_{c}A([\![(x,-%
ix),(b,ib)]\!]\right)$$
$$\displaystyle=$$
$$\displaystyle-2i\left(\Omega_{c}((b,-ib),[\![(x,-ix),(a,ia)]\!]^{1,0})-\Omega_%
{c}((a,-ia),[\![(x,-ix),(b,ib)]\!]^{1,0})\right).$$
With Lemma 7 and various definitions, one could show that
$$-\Omega_{3}^{-1}\overline{\partial}\ell(\Omega_{c}A,\Omega_{c}B)=-8\omega(x,[a%
,b]).$$
(75)
Similarly,
$$\displaystyle-\Omega_{4}^{-1}\overline{\partial}\ell(\Omega_{c}A,\Omega_{c}B)$$
$$\displaystyle=$$
$$\displaystyle-2\left(\Omega_{c}A(-\gamma(\jmath b)x,i\gamma(\jmath b)x)-\Omega%
_{c}B(-\gamma(\jmath a)x,i\gamma(\jmath a)x)\right).$$
By Lemma 7, it is equal to
$$2\left(\Omega_{4}((-2\jmath a,-2i\jmath a),(-\gamma(\jmath b)x,i\gamma(\jmath b%
)x))-\Omega_{4}((-2\jmath b,-2i\jmath b),(-\gamma(\jmath a)x,i\gamma(\jmath a)%
x))\right).$$
By definition of $\Omega_{4}$, we have
$$-\Omega_{4}^{-1}\overline{\partial}\ell(\Omega_{c}A,\Omega_{c}B)=8g(i[\jmath a%
,\jmath b],x).$$
Since (68) is satisfied, (71) holds. Therefore,
$$-\mu\Omega_{4}^{-1}\overline{\partial}\ell(\Omega_{c}A,\Omega_{c}B)=8\lambda g%
(\jmath[a,b],x)=-8\lambda\omega(x,[a,b]).$$
Combined the above identity with (75) and (74),
we obtain
$$-16\lambda\omega(x,[a,b])=4i\omega(x,[a,b])$$
for all $x,a,b\in\mathfrak{g}$.
Therefore, $\lambda=-\frac{i}{4}$.
Further and similar calculations demonstrate that this is the only constraint [26].
Substitute this constraint into (68), we find that $\mu$ is a real number and
for all $a,b\in\mathfrak{g}$,
$$\jmath(\gamma(a)b)=-4\mu\gamma(\jmath a)(\jmath b).$$
It concludes the proof of Proposition 6.
Let us analyze Proposition 6 further. If $\mu=0$, constraint (65) implies
that $\gamma(a)b=0$ for all $a,b\in\mathfrak{g}$. Therefore, $\gamma=0$. However, the connection $\gamma$ is
torsion-free. This implies that $[a,b]=0$. Therefore, the algebra $\mathfrak{h}=\mathfrak{g}\ltimes V$ is trivial.
In particular, $\Lambda$ is central in the Gerstenhaber algebra $(\wedge^{\bullet}\mathfrak{h},\wedge,[\![-,-]\!])$, and hence $(\Lambda,\phi=0)$ forms a compatible pair.
Therefore, whenever
$\mathfrak{h}$ is non-abelian, we may assume that $\mu\neq 0$. In such case, if one multiplies the non-degenerate
bilinear form $g$ on $\mathfrak{g}$ by the constant $-4\mu$, then the inhomogeneity in equation
(65) allows us to simply this identity to
$$(g^{-1}\omega)(\gamma(a)b)=\gamma((g^{-1}\omega)(a))((g^{-1}\omega)(b)).$$
(76)
Now we could apply Proposition 6 and
Theorem 2 to conclude the following.
Theorem 5
Let $\mathfrak{g}$ be a Lie algebra with an invariant symplectic structure $\omega$ and non-degenerate bilinear
form $g$. Let $V$ be its underlying vector space. Let $\gamma:\mathfrak{g}\to\mathop{\mathrm{End}}\nolimits(V)$ be a torsion-free
flat connection and $\mathfrak{h}=\mathfrak{g}\ltimes_{\gamma}V$ the associated semi-direct product.
Then $\mathfrak{h}$ has a natural complex structure $J$, a symplectic structure $\Omega$
and a pseudo-metric $\Delta$. If this triple forms a pseudo-Kähler structure and if
$$(g^{-1}\omega)(\gamma(a)b)=\gamma((g^{-1}\omega)(a))((g^{-1}\omega)(b)).$$
then there exists a deformation from the complex structure $J$ to a symplectic structure $\Omega_{2}$
such that $\mathop{\mathrm{DGA}}\nolimits(J)$ is isomorphic to $\mathop{\mathrm{DGA}}\nolimits(\Omega_{2})$.
6 Low-dimension examples
According to Andranda [1], there are three non-trivial four-dimensional complex symplectic algebras.
Let $e_{1},e_{2}$ be a basis of $\mathfrak{g}$ and
$v_{1},v_{2}$ be a basis for $V$ such that
$$Je_{1}=v_{1},\quad Je_{2}=v_{2}.$$
(77)
Let $e^{1},e^{2}$ and $v^{1},v^{2}$ be the dual bases.
We choose the symplectic structure $\omega$ and the pseudo-metric $g$ on the algebra $\mathfrak{g}$ to be
$$\omega=e^{1}\wedge e^{2},\quad g=e^{1}\otimes e^{2}+e^{2}\otimes e^{1}.$$
It follows that
$$\jmath=g^{-1}\omega=e^{1}\otimes e_{1}-e^{2}\otimes e_{2}.$$
The natural symplectic form and metric on $\mathfrak{g}\ltimes V$ are respectively
$$\Omega=e^{1}\wedge e^{2}+v^{1}\wedge v^{2},\quad\Delta=e^{1}\otimes e^{2}+e^{2%
}\otimes e^{1}+v^{1}\otimes v^{2}+v^{2}\otimes v^{1}.$$
Moreover, let $z_{1}=\frac{1}{2}(e_{1}-iv_{1})$ and $z_{2}=\frac{1}{2}(e_{2}-iv_{2})$, $z^{1}=e^{1}+iv^{1},$ and
$z^{2}=e^{2}+iv^{2}$, then
$$\displaystyle\Omega_{1}=-e^{1}\wedge v^{2}-v^{1}\wedge e^{2}=\frac{1}{2i}(z^{1%
}\wedge z^{2}-\overline{z}^{1}\wedge\overline{z}^{2})$$
(78)
$$\displaystyle\Omega_{2}=e^{1}\wedge e^{2}-v^{1}\wedge v^{2}=\frac{1}{2}(z^{1}%
\wedge z^{2}+\overline{z}^{1}\wedge\overline{z}^{2})$$
(79)
$$\displaystyle\Omega_{3}=e^{1}\wedge e^{2}+v^{1}\wedge v^{2}=\frac{1}{2}(z^{1}%
\wedge\overline{z}^{2}+\overline{z}^{1}\wedge z^{2})$$
(80)
$$\displaystyle\Omega_{4}=e^{1}\wedge v^{2}-v^{1}\wedge e^{2}=\frac{i}{2}(z^{1}%
\wedge\overline{z}^{2}-\overline{z}^{1}\wedge z^{2})$$
(81)
In particular,
$$\displaystyle\Omega_{c}=\Omega_{1}+i\Omega_{2}=iz^{1}\wedge z^{2},$$
$$\displaystyle\Lambda=\Omega_{c}^{-1}=iz_{1}\wedge z_{2},$$
$$\displaystyle\Omega_{3}^{-1}=2(z_{2}\wedge\overline{z}_{1}+\overline{z}_{2}%
\wedge z_{1}),$$
$$\displaystyle\Omega_{4}^{-1}=2i(z_{2}\wedge\overline{z}_{1}-\overline{z}_{2}%
\wedge z_{1}).$$
(82)
6.1 Example 1
When the two-dimensional Lie algebra $\mathfrak{g}$ is
abelian, the only non-trivial object in constructing a four-dimensional semi-direct product in this
case is the torsion-free flat
connection $\gamma$. It is determined by the identities,
$$\gamma(e_{1})v_{1}=v_{2},\quad\gamma(e_{1})v_{2}=0,\quad\gamma(e_{2})=0.$$
Equivalently, the only non-trivial structure equation for $\mathfrak{h}=\mathfrak{g}\ltimes V$ is
$$[\![e_{1},v_{1}]\!]=v_{2}.$$
The dual structure equation is $dv^{2}=-e^{1}\wedge v^{1}.$ Therefore, it is apparent that
$\Omega_{4}$ is closed, and hence $\mathfrak{h}$ has a natural pseudo-Kähler metric.
As $\jmath e_{1}=e_{1}$ and $\jmath e_{2}=-e_{2}$,
Proposition 6 is solved when $\mu=\frac{1}{4}$.
By the expressions in (82),
$$\phi=-\frac{i}{4}\Omega_{3}^{-1}+\frac{1}{4}\Omega_{4}^{-1}=\frac{1}{4}(\Omega%
_{4}^{-1}-i\Omega_{3}^{-1})=iz_{1}\wedge\overline{z}_{2}.$$
Therefore by Theorem 5,
for the complex structure $J$ in (77) and the symplectic structure
$\Omega_{2}$ in (79), $\mathop{\mathrm{DGA}}\nolimits(\Omega_{2})$ and $\mathop{\mathrm{DGA}}\nolimits(J)$ are isomorphic and they exist in one
generalized deformation class.
Indeed, for this particular example, the algebraic $\mathfrak{h}$ is the covering space of the Kodaira-Thurston surface.
It is known that all the concerned cohomology
spaces are given by invariant objects. Therefore, we may also apply
Theorem 4 on manifold level, and recovers a key result obtained by ad hoc computation in [25].
6.2 Example 2
In this example, the algebra $\mathfrak{g}$ is solvable, with structure equation
$[e_{1},e_{2}]=e_{2}$. The connection $\gamma$ is given by
$$\gamma(e_{1})v_{1}=-v_{1},\quad\gamma(e_{1})v_{2}=v_{2},\quad\gamma(e_{2})=0.$$
The structure equations for the semi-direct product $\mathfrak{h}$ are equivalently given by
$$de^{2}=-e^{1}\wedge e^{2},\quad dv^{1}=e^{1}\wedge v^{2},\quad dv^{2}=-e^{1}%
\wedge v^{2}.$$
It follows that $\Omega_{4}$ is closed.
Further, $\mu=-\frac{1}{4}$ solves the constraint in Proposition 6,
and $\phi=-iz_{2}\wedge\overline{z}_{1}$. Therefore, by Theorem 5
the complex structure $J$ is deformed to a $\Omega_{2}$ via a holomorphic Poisson structure, and
$\mathop{\mathrm{DGA}}\nolimits(J)$ is isomorphic to $\mathop{\mathrm{DGA}}\nolimits(\Omega_{2})$.
6.3 Example 3
In this example, the algebra $\mathfrak{g}$ is solvable: $[e_{1},e_{2}]=e_{2}$.
The connection $\gamma$ is given by
$$\gamma(e_{1})v_{1}=-\frac{1}{2}v_{1},\quad\gamma(e_{1})v_{2}=\frac{1}{2}v_{2},%
\quad\gamma(e_{2})v_{1}=-\frac{1}{2}v_{2},\quad\gamma(e_{2})v_{2}=0.$$
On the semi-direct product the non-trivial structure equations become
$$[e_{1},e_{2}]=e_{2},\quad[e_{1},v_{1}]=-\frac{1}{2}v_{1},\quad[e_{1},v_{2}]=%
\frac{1}{2}v_{2},\quad[e_{2},v_{1}]=-\frac{1}{2}v_{2}.$$
(83)
The dual equations are
$$de^{2}=-e^{1}\wedge e^{2},\quad dv^{1}=\frac{1}{2}e^{1}\wedge v^{1},\quad dv^{%
2}=-\frac{1}{2}e^{1}\wedge v^{2}+\frac{1}{2}e^{2}\wedge v^{1}.$$
(84)
It follows that $d\Omega_{4}=2v^{1}\wedge e^{1}\wedge e^{2}$. In particular, Proposition
6 and Theorem 5 are not applicable.
In terms of complex frames, we have
$$[\![z_{1},z_{2}]\!]=\frac{1}{2}z_{2},\quad dz^{1}=-\frac{1}{4}z^{1}\wedge%
\overline{z}^{1},\quad dz^{2}=-\frac{1}{4}(z^{1}+\overline{z}^{1})\wedge z^{2}%
-\frac{1}{4}z^{1}\wedge(z^{2}+\overline{z}^{2}).$$
From the differentials, we further obtain that
$$[\![z_{1},\overline{z}^{1}]\!]=\frac{1}{4}\overline{z}^{1},\quad[\![z_{1},%
\overline{z}^{2}]\!]=-\frac{1}{4}\overline{z}^{2},\quad[\![z_{2},\overline{z}^%
{2}]\!]=\frac{1}{4}\overline{z}^{1}.$$
(85)
Taking the complex conjugation, and then the dual expression is
$$\overline{\partial}z_{1}=-\frac{1}{4}\overline{z}^{1}\wedge z_{1}-\frac{1}{4}%
\overline{z}^{2}\wedge z_{2},\quad\overline{\partial}z_{2}=\frac{1}{4}%
\overline{z}^{1}\wedge z_{2}.$$
As an intermediate step, we put together the structure equation of $\mathop{\mathrm{DGA}}\nolimits(J)$ on this particular
algebra:
$$\displaystyle[\![z_{1},z_{2}]\!]=\frac{1}{2}z_{2},\quad[\![z_{1},\overline{z}^%
{1}]\!]=\frac{1}{4}\overline{z}^{1},\quad[\![z_{1},\overline{z}^{2}]\!]=-\frac%
{1}{4}\overline{z}^{2},\quad[\![z_{2},\overline{z}^{2}]\!]=\frac{1}{4}%
\overline{z}^{1}$$
(86)
$$\displaystyle\overline{\partial}z_{1}=-\frac{1}{4}\overline{z}^{1}\wedge z_{1}%
-\frac{1}{4}\overline{z}^{2}\wedge z_{2},\quad\overline{\partial}z_{2}=\frac{1%
}{4}\overline{z}^{1}\wedge z_{2},\quad\overline{\partial}\overline{z}^{2}=-%
\frac{1}{2}\overline{z}^{1}\wedge\overline{z}^{2}.$$
(87)
On the other hand,
$$\Omega_{2}(e_{1})=e^{2},\quad\Omega_{2}(e_{2})=-e^{1},\quad\Omega_{2}(v_{1})=-%
v^{2},\quad\Omega_{2}(v_{2})=v^{1}.$$
Then the linear isomorphism $\Omega_{2}$ take the Lie bracket on vectors in (83) to
a Lie bracket on forms.
$$[\![e^{1},e^{2}]\!]=-e^{1},\quad[\![e^{2},v^{2}]\!]=\frac{1}{2}v^{2},\quad[\![%
e^{2},v^{1}]\!]=\frac{1}{2}v^{1},\quad[\![e^{1},v^{2}]\!]=-\frac{1}{2}v^{1}.$$
With respect to these Lie algebra structures, the first derived subalgebra
$\mathfrak{h}^{1,0}\oplus\mathfrak{h}^{*(0,1)}$ is
the three-dimensional Heisenberg algebra spanned by $e^{1},v^{1},v^{2}$ with $v^{1}$ being its center.
In view of the exterior differential as given in (84), $v^{1}$ is not closed in
the differential Gerstenhaber algebra of the symplectic structure $\Omega_{2}$.
On the other hand, from (86), we find that the first derived subalgebra
in $\mathop{\mathrm{DGA}}\nolimits(J)$ is the three-dimensional Heisenberg algebra spanned by $z_{2},\overline{z}^{1},\overline{z}^{2}$ with
$\overline{z}^{1}$ being its center. In view of (87), $\overline{z}^{1}$ is $\overline{\partial}$-closed.
Since the center of the derived subalgebra of $\mathop{\mathrm{DGA}}\nolimits(J)$ is $\overline{\partial}$-closed and that of $\mathop{\mathrm{DGA}}\nolimits(\Omega_{2})$ is
not $d$-closed, then two $\mathop{\mathrm{DGA}}\nolimits$s could not be quasi-isomorphic
Remark
Given the definition of $\Omega_{1}$ in (78), it is apparent that $\mathfrak{g}$ and $V$ are Lagrangian with
respect to $\Omega_{1}$. As $J\mathfrak{g}=V$ and $JV=\mathfrak{g}$, the complex structure $J$ and the complex symplectic
structure is special Lagrangian in the sense of [7, Definition].
Let $\gamma^{*}$ be the dual representation of $\gamma$, then one obtains the dual semi-direct product
$\widehat{\mathfrak{h}}=\mathfrak{g}\ltimes_{\gamma^{*}}V^{*}$. Through this space as an intermediate object, it is provided
in [7, Theorem 5.2] that there is a natural isomorphism from $\mathop{\mathrm{DGA}}\nolimits(J)$ to $\mathop{\mathrm{DGA}}\nolimits(\Omega_{1})$. See also
[4].
Therefore, we have
$$\mathop{\mathrm{DGA}}\nolimits(\Omega_{1})\cong\mathop{\mathrm{DGA}}\nolimits(%
J)\cong\mathop{\mathrm{DGA}}\nolimits(\Omega_{2}).$$
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Beyond Gaussian : A Comment
Kazuyuki FUJII
Department of Mathematical Sciences
Yokohama City University
Yokohama, 236–0027
Japan
E-mail address : fujii@yokohama-cu.ac.jp
()
Abstract
In this paper we treat a non–Gaussian integral and give a
fundamental formula in terms of discriminant. We also present
some related problems.
This is a comment paper to arXiv:0903.2595 [math-ph] by
Morozov and Shakirov.
1 Introduction
Gaussian plays a fundamental role in Mathematics (including
Statistics) and Physics and et al. We want to (or we should)
overcome its high wall in this century, and so a try is
introduced.
In the paper [1] the following “formula” is listed :
$$\int\int e^{-\left(ax^{3}+bx^{2}y+cxy^{2}+dy^{3}\right)}dxdy=\frac{1}{\sqrt[6]%
{-D}}$$
(1)
where $D$ is the discriminant given by
$$D=-\left(27a^{2}d^{2}+4ac^{3}-18abcd-b^{2}c^{2}+4b^{3}d\right)$$
(2)
of the cubic equation
$$ax^{3}+bx^{2}+cx+d=0.$$
(3)
The equation (1) is of course non–Gaussian.
However, if we consider it in the framework of real category
then (1) is not correct because the left hand side
diverges. In this paper we treat only real category, and so $a,b,c,d,x,y$ are real numbers.
Formally, by performing the change of variable $x=t\rho,\ y=\rho$
for (1) we have
LHS of (1)
$$\displaystyle=$$
$$\displaystyle\int\int e^{-\rho^{3}\left(at^{3}+bt^{2}+ct+d\right)}|\rho|dtd\rho$$
$$\displaystyle=$$
$$\displaystyle\int\left\{\int e^{-\left(at^{3}+bt^{2}+ct+d\right)\rho^{3}}|\rho%
|d\rho\right\}dt$$
$$\displaystyle=$$
$$\displaystyle\int|\sigma|e^{-\sigma^{3}}d\sigma\int\frac{1}{\sqrt[3]{(at^{3}+%
bt^{2}+ct+d)^{2}}}dt$$
by the change of variable $\sigma=\sqrt[3]{at^{3}+bt^{2}+ct+d}\ \rho$.
Therefore we can conjecture that the formula may be
$$\int_{{\bf R}}\frac{1}{\sqrt[3]{(ax^{3}+bx^{2}+cx+d)^{2}}}dx=\frac{C}{\sqrt[6]%
{-D}}$$
(4)
under the change $t\rightarrow x$. Here $C$ is a constant.
In the paper we calculate the left hand side of (4)
directly.
2 Fundamental Formula
Before stating the result let us make some preparations.
The Gamma–function $\Gamma(p)$ is defined by
$$\Gamma(p)=\int_{0}^{\infty}e^{-x}x^{p-1}dx\quad(p>0)$$
(5)
and the Beta–function $B(p,q)$ is
$$B(p,q)=\int_{0}^{1}x^{p-1}(1-x)^{q-1}dx\quad(p,\ q>0).$$
(6)
Note that the Beta–function is rewritten as
$$B(p,q)=\int_{0}^{\infty}\frac{x^{p-1}}{(1+x)^{p+q}}dx.$$
See [2] in more detail. Now we are in a position to
state the result.
Integral Formula
(I) For $D<0$
$$\int_{{\bf R}}\frac{1}{\sqrt[3]{(ax^{3}+bx^{2}+cx+d)^{2}}}dx=\frac{C_{-}}{%
\sqrt[6]{-D}}$$
(7)
where
$$C_{-}=\sqrt[3]{2}B(\frac{1}{2},\frac{1}{6}).$$
(II) For $D>0$
$$\int_{{\bf R}}\frac{1}{\sqrt[3]{(ax^{3}+bx^{2}+cx+d)^{2}}}dx=\frac{C_{+}}{%
\sqrt[6]{D}}$$
(8)
where
$$C_{+}=3B(\frac{1}{3},\frac{1}{3}).$$
(III) $C_{-}$ and $C_{+}$ are related to $C_{+}=\sqrt{3}C_{-}$
by the equation
$$\sqrt{3}B(\frac{1}{3},\frac{1}{3})=\sqrt[3]{2}B(\frac{1}{2},\frac{1}{6}).$$
(9)
Our result shows that the integral depends on the sign of $D$, and so
our question is as follows.
Problem Can the result be derived from the method developed in
[1] ?
A comment is in order. If we treat the Gaussian case
(: $e^{-(ax^{2}+bxy+cy^{2})}$) then the integral is reduced to
$$\int_{{\bf R}}\frac{1}{ax^{2}+bx+c}dx=\frac{2\pi}{\sqrt{-D}}$$
(10)
if $a>0$ and $D=b^{2}-4ac<0$. Noting
$$\pi=\frac{\sqrt{\pi}\sqrt{\pi}}{1}=\frac{\Gamma(\frac{1}{2})\Gamma(\frac{1}{2}%
)}{\Gamma(1)}=B(\frac{1}{2},\frac{1}{2})$$
(10) should be read as
$$\int_{{\bf R}}\frac{1}{ax^{2}+bx+c}dx=\frac{2B(\frac{1}{2},\frac{1}{2})}{\sqrt%
{-D}}.$$
3 Discriminant
In this section we make some comments on the discriminant
(2). See [3] in more detail
([3] is strongly recommended).
For the equations
$$f(x)=ax^{3}+bx^{2}+cx+d,\quad f^{\prime}(x)=3ax^{2}+2bx+c$$
(11)
the resultant $R(f,f^{\prime})$ of $f$ and $f^{\prime}$ is given by
$$R(f,f^{\prime})=\left|\begin{array}[]{ccccc}a&b&c&d&0\\
0&a&b&c&d\\
3a&2b&c&0&0\\
0&3a&2b&c&0\\
0&0&3a&2b&c\end{array}\right|.$$
(12)
It is easy to calculate (12) and the result becomes
$$\frac{1}{a}R(f,f^{\prime})=27a^{2}d^{2}+4ac^{3}-18abcd-b^{2}c^{2}+4b^{3}d=-D.$$
(13)
On the other hand, if $\alpha$, $\beta$, $\gamma$ are three solutions
of $f(x)=0$ in (11), then the following relations
are well–known.
$$\left\{\begin{array}[]{ll}\alpha+\beta+\gamma=-\frac{b}{a}\\
\alpha\beta+\alpha\gamma+\beta\gamma=\frac{c}{a}\\
\alpha\beta\gamma=-\frac{d}{a}\end{array}\right.$$
(14)
If we set
$$\Delta=(\alpha-\beta)(\alpha-\gamma)(\beta-\gamma)$$
(15)
the discriminant $D$ is given by
$$D=a^{4}\Delta^{2}.$$
(16)
Let us calculate $\Delta^{2}$ directly. For the Vandermonde matrix
$$V=\left(\begin{array}[]{ccc}1&1&1\\
\alpha&\beta&\gamma\\
\alpha^{2}&\beta^{2}&\gamma^{2}\end{array}\right)\ \Longrightarrow\ |V|=-\Delta$$
we obtain
$$\displaystyle\Delta^{2}$$
$$\displaystyle=$$
$$\displaystyle(-|V|)^{2}=|V||V^{T}|=|VV^{T}|$$
(17)
$$\displaystyle=$$
$$\displaystyle\left|\begin{array}[]{ccc}3&\alpha+\beta+\gamma&\alpha^{2}+\beta^%
{2}+\gamma^{2}\\
\alpha+\beta+\gamma&\alpha^{2}+\beta^{2}+\gamma^{2}&\alpha^{3}+\beta^{3}+%
\gamma^{3}\\
\alpha^{2}+\beta^{2}+\gamma^{2}&\alpha^{3}+\beta^{3}+\gamma^{3}&\alpha^{4}+%
\beta^{4}+\gamma^{4}\end{array}\right|$$
$$\displaystyle=$$
$$\displaystyle\cdots$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{a^{4}}\frac{-1}{3}\left\{(bc-9ad)^{2}-4(b^{2}-3ac)(c^{2}%
-3bd)\right\}.$$
This result is very suggestive. In fact, from the cubic equation
$$ax^{3}+bx^{2}+cx+d=0$$
we have three data
$$A=b^{2}-3ac,\quad B=bc-9ad,\quad C=c^{2}-3bd$$
, and so if we consider the quadratic equation
$$AX^{2}+BX+C=0$$
then the discriminant is just $B^{2}-4AC$. This is very interesting.
Problem Make the meaning clear !
4 Concluding Remarks
In the paper we calculated the non–Gaussian integral
(4) in a direct manner.
Details of calculation will be published in [4].
In this stage we can consider the general case. For the general
equation
$$f(x)=a_{0}x^{n}+a_{1}x^{n-1}+\cdots+a_{n-1}x+a_{n}$$
(18)
the (non–Gaussian) integral becomes
$$\int_{{\bf R}}\frac{1}{\sqrt[n]{f(x)^{2}}}dx.$$
(19)
The discriminant $D$ of the equation $f(x)=0$ is given by
the resultant $R(f,f^{\prime})$ of $f$ and $f^{\prime}$ like
$$\frac{1}{a_{0}}R(f,f^{\prime})=(-1)^{\frac{n(n-1)}{2}}D\ \Longleftrightarrow\ %
D=(-1)^{\frac{n(n-1)}{2}}R(f,f^{\prime})/a_{0}$$
(20)
where
$$R(f,f^{\prime})=\left|\begin{array}[]{cccccccc}a_{0}&a_{1}&\cdots&a_{n-1}&a_{n%
}&&&\\
&a_{0}&a_{1}&\cdots&a_{n-1}&a_{n}&&\\
&&\ddots&&&\ddots&&\\
&&&a_{0}&a_{1}&\cdots&a_{n-1}&a_{n}\\
na_{0}&(n-1)a_{1}&\cdots&a_{n-1}&&&&\\
&na_{0}&(n-1)a_{1}&\cdots&a_{n-1}&&&\\
&&\ddots&&&\ddots&&\\
&&&&na_{0}&(n-1)a_{1}&\cdots&a_{n-1}\end{array}\right|,$$
see (11) and (12).
For example, if $n=4$ and $n=5$ then we have
$$\displaystyle D_{n=4}$$
$$\displaystyle=$$
$$\displaystyle 256a_{0}^{3}a_{4}^{3}-4a_{1}^{3}a_{3}^{3}-27a_{0}^{2}a_{3}^{4}-2%
7a_{1}^{4}a_{4}^{2}-128a_{0}^{2}a_{2}^{2}a_{4}^{2}+a_{1}^{2}a_{2}^{2}a_{3}^{2}%
+16a_{0}a_{2}^{4}a_{4}$$
$$\displaystyle-4a_{0}a_{2}^{3}a_{3}^{2}-4a_{1}^{2}a_{2}^{3}a_{4}+144a_{0}^{2}a_%
{2}a_{3}^{2}a_{4}-6a_{0}a_{1}^{2}a_{3}^{2}a_{4}+144a_{0}a_{1}^{2}a_{2}a_{4}^{2%
}-192a_{0}^{2}a_{1}a_{3}a_{4}^{2}$$
$$\displaystyle+18a_{0}a_{1}a_{2}a_{3}^{3}+18a_{1}^{3}a_{2}a_{3}a_{4}-80a_{0}a_{%
1}a_{2}^{2}a_{3}a_{4},$$
and
$$\displaystyle D_{n=5}$$
$$\displaystyle=$$
$$\displaystyle 3125a_{0}^{4}a_{5}^{4}-2500a_{0}^{3}a_{1}a_{4}a_{5}^{3}-3750a_{0%
}^{3}a_{2}a_{3}a_{5}^{3}+2000a_{0}^{3}a_{2}a_{4}^{2}a_{5}^{2}+2250a_{0}^{3}a_{%
3}^{2}a_{4}a_{5}^{2}$$
$$\displaystyle-1600a_{0}^{3}a_{3}a_{4}^{3}a_{5}+256a_{0}^{3}a_{4}^{5}+2000a_{0}%
^{2}a_{1}^{2}a_{3}a_{5}^{3}-50a_{0}^{2}a_{1}^{2}a_{4}^{2}a_{5}^{2}+2250a_{0}^{%
2}a_{1}a_{2}^{2}a_{5}^{3}$$
$$\displaystyle-2050a_{0}^{2}a_{1}a_{2}a_{3}a_{4}a_{5}^{2}+160a_{0}^{2}a_{1}a_{2%
}a_{4}^{3}a_{5}-900a_{0}^{2}a_{1}a_{3}^{3}a_{5}^{2}+1020a_{0}^{2}a_{1}a_{3}^{2%
}a_{4}^{2}a_{5}-192a_{0}^{2}a_{1}a_{3}a_{4}^{4}$$
$$\displaystyle-900a_{0}^{2}a_{2}^{3}a_{4}a_{5}^{2}+825a_{0}^{2}a_{2}^{2}a_{3}^{%
2}a_{5}^{2}+560a_{0}^{2}a_{2}^{2}a_{3}a_{4}^{2}a_{5}-128a_{0}^{2}a_{2}^{2}a_{4%
}^{4}-630a_{0}^{2}a_{2}a_{3}^{3}a_{4}a_{5}$$
$$\displaystyle+144a_{0}^{2}a_{2}a_{3}^{2}a_{4}^{3}+108a_{0}^{2}a_{3}^{5}a_{5}-2%
7a_{0}^{2}a_{3}^{4}a_{4}^{2}-1600a_{0}a_{1}^{3}a_{2}a_{5}^{3}+160a_{0}a_{1}^{3%
}a_{3}a_{4}a_{5}^{2}$$
$$\displaystyle-36a_{0}a_{1}^{3}a_{4}^{3}a_{5}+1020a_{0}a_{1}^{2}a_{2}^{2}a_{4}a%
_{5}^{2}+560a_{0}a_{1}^{2}a_{2}a_{3}^{2}a_{5}^{2}-746a_{0}a_{1}^{2}a_{2}a_{3}a%
_{4}^{2}a_{5}+144a_{0}a_{1}^{2}a_{2}a_{4}^{4}$$
$$\displaystyle+24a_{0}a_{1}^{2}a_{3}^{3}a_{4}a_{5}-6a_{0}a_{1}^{2}a_{3}^{2}a_{4%
}^{3}-630a_{0}a_{1}a_{2}^{3}a_{3}a_{5}^{2}+24a_{0}a_{1}a_{2}^{3}a_{4}^{2}a_{5}%
+356a_{0}a_{1}a_{2}^{2}a_{3}^{2}a_{4}a_{5}$$
$$\displaystyle-80a_{0}a_{1}a_{2}^{2}a_{3}a_{4}^{3}-72a_{0}a_{1}a_{2}a_{3}^{4}a_%
{5}+18a_{0}a_{1}a_{2}a_{3}^{3}a_{4}^{2}+108a_{0}a_{2}^{5}a_{5}^{2}-72a_{0}a_{2%
}^{4}a_{3}a_{4}a_{5}+16a_{0}a_{2}^{4}a_{4}^{3}$$
$$\displaystyle+16a_{0}a_{2}^{3}a_{3}^{3}a_{5}-4a_{0}a_{2}^{3}a_{3}^{2}a_{4}^{2}%
+256a_{1}^{5}a_{5}^{3}-192a_{1}^{4}a_{2}a_{4}a_{5}^{2}-128a_{1}^{4}a_{3}^{2}a_%
{5}^{2}+144a_{1}^{4}a_{3}a_{4}^{2}a_{5}$$
$$\displaystyle-27a_{1}^{4}a_{4}^{4}+144a_{1}^{3}a_{2}^{2}a_{3}a_{5}^{2}-6a_{1}^%
{3}a_{2}^{2}a_{4}^{2}a_{5}-80a_{1}^{3}a_{2}a_{3}^{2}a_{4}a_{5}+18a_{1}^{3}a_{2%
}a_{3}a_{4}^{3}+16a_{1}^{3}a_{3}^{4}a_{5}$$
$$\displaystyle-4a_{1}^{3}a_{3}^{3}a_{4}^{2}-27a_{1}^{2}a_{2}^{4}a_{5}^{2}+18a_{%
1}^{2}a_{2}^{3}a_{3}a_{4}a_{5}-4a_{1}^{2}a_{2}^{3}a_{4}^{3}-4a_{1}^{2}a_{2}^{2%
}a_{3}^{3}a_{5}+a_{1}^{2}a_{2}^{2}a_{3}^{2}a_{4}^{2}.$$
However, to write down the general case explicitly is not easy.
Problem Calculate (19) for $n=4$ (and
$n=5$) directly.
The wall called Gaussian is very high and not easy to overcome,
and therefore hard work will be needed.
References
[1]
A. Morozov and Sh. Shakirov :
Introduction to Integral Discriminants,
arXiv:0903.2595 [math-ph].
[2]
E. T. Whittaker and G. N. Watson :
A Course of MODERN ANALYSIS,
1990 (latest), Cambridge University Press.
[3]
I. Satake :
Linear Algebra (in Japanese),
1989 (latest), Shokabo, Tokyo.
As far as I know this is the best book on Elementary
Linear Algebra.
[4]
K. Fujii :
in preparation. |
Stability of Self-Configuring Large Multiport Interferometers
Ryan Hamerly${}^{1,2}$
Saumil Bandyopadhyay${}^{1}$
and Dirk Englund${}^{1}$
Abstract—Realistic multiport interferometers (beamsplitter meshes) are sensitive to component imperfections, and this sensitivity increases with size. Self-configuration techniques can be employed to correct these imperfections, but not all techniques are equal. This paper highlights the importance of algorithmic stability in self-configuration. Naïve approaches based on sequentially setting matrix elements are unstable and perform poorly for large meshes, while techniques based on power ratios perform well in all cases, even in the presence of large errors. Based on this insight, we propose a self-configuration scheme for triangular meshes that requires only external detectors and works without prior knowledge of the component imperfections. This scheme extends to the rectangular mesh by adding a single array of detectors along the diagonal.
${}^{1}$ Research Laboratory of Electronics, MIT, 50 Vassar Street, Cambridge, MA 02139, USA
${}^{2}$ NTT Research Inc., Physics and Informatics Laboratories, 940 Stewart Drive, Sunnyvale, CA 94085, USA
Photonic technologies increasingly rely on programmable and reconfigurable circuits. A central component in such circuits is the universal multiport interferometer: an optical device with $N>2$ inputs and outputs, whose linear input-output relation (transfer matrix) is set by the user. Such interferometers are indispensable in applications ranging from linear optical quantum computing [1, 2] and RF photonics [3, 4] to signal processing [5, 6] and machine learning acceleration [7, 8, 9, 10], and will play an important role in proposed photonic field-programmable gate arrays [5, 11]. Size (i.e. number of ports) is an important figure of merit for all of these applications, and scaling up multiport interferometers is an active field in research. Recent advances in silicon photonics are promising, allowing the scale-up from small proof-of-concept designs to large (and therefore technologically useful) systems [2, 12, 13].
Component imperfections are a major challenge to scaling the size of multiport interferometers. This is because all large devices are based on dense meshes of tunable beamsplitters, whose circuit depth grows with size. Most non-recirculating designs are variants of the triangular Reck [14] or rectangular Clements [15] beamsplitter mesh, both of which encode an $N\times N$ unitary transfer matrix into a compact mesh of programmable Mach-Zehnder interferometers (MZIs). These circuits have $O(N)$ depth, meaning that component errors cascade as light propagates down the mesh. The upshot is that scaling in size must be accompanied by scaling in precision to preserve the accuracy of the input-output map. This challenge is most acute for optical machine learning applications [10, 7], which rely on very large mesh sizes for performance [12, 13], where fabrication errors from even state-of-the-art technology are predicted to significantly degrade ONN accuracy in hardware [16].
Several self-configuration techniques can suppress the effect of component imprecisions. For machine learning applications, the MZI phase shifts can be learned by in situ training [17], but this requires extra hardware (inline power detectors [18]) and the learned weights are specific to the given device. Alternatively, if the chip has been pre-calibrated so the imperfections are known, global optimization can be used to find the phase shifts offline [19, 20]; however, this approach is time-consuming and requires that the hardware imperfections be known to high accuracy. MZI errors can also be eliminated by pairing MZIs, though this doubles the loss and chip area [21]. Finally, for triangular meshes, the MZIs of each diagonal can be configured sequentially [22, 23, 24]. This approach, however, also requires $O(N^{2})$ inline power detectors (or pre-calibrated MZIs that can be configured to a perfect “bar” or “cross” state). In short, all configuration schemes to date rely on either (i) additional hardware complexity, i.e. inline detectors or MZI pairing, or (ii) accurate pre-calibration of the mesh’s component errors.
In this article, we analyze self-configuration algorithms that require only external detectors and do not rely on prior calibration of the MZI mesh. Not all algorithms are created equal, and algorithmic stability distinguishes good algorithms from bad ones: for example, a straightforward approach based on sequentially matching matrix elements works in principle, but performs poorly in the presence of large errors. Based on this insight, we propose an algorithm based on orthogonality and power ratios that performs well in all cases, even in the presence of large errors. This scheme is directly applicable to triangular meshes, but can also be extended to a rectangular mesh with the addition of a single array of inline power detectors along the diagonal.
This paper is organized as follows: In Sec. 1, we introduce the Reck and Clements meshes, analyze their statistical properties in the presence of errors, and derive analytic estimates for the improvements possible by self-configuration. Sec. 2 covers the theory of self-configuring algorithms and introduces our proposed methods in the context of the Reck scheme. Sec. 3 analyzes the accuracy of self-configuration in the presence of component errors and highlights the importance of algorithmic stability. Finally, Sec. 4 extends our method to the rectangular Clements scheme by splitting the rectangle with a diagonal of internal monitors.
1 Statistics of Imperfect Meshes
The most common multiport interferometer designs are the Reck triangle [14] and the Clements rectangle [15]. In both cases, the circuit can be laid out on a regular grid of $2\times 2$ elements (Fig. 1(a)) without any waveguide crossings, a major advantage compared to competing designs [25, 26, 27, 28]. The input-output matrix of the mesh is accordingly a product:
$$U=\Bigl{(}\sum_{n}T_{n}\Bigr{)}D$$
(1)
where $D$ is a phase mask and the $T_{n}$ are $2\times 2$ block matrices representing a phase shifter cascaded into an MZI crossing (Fig. 1(b)):
$$\displaystyle T_{n}$$
$$\displaystyle\!=\!$$
$$\displaystyle\underbrace{\begin{bmatrix}1&0\\
0&e^{i\phi_{n}}\end{bmatrix}}_{P_{2}(\phi_{n})}\underbrace{\begin{bmatrix}\cos(\tfrac{\pi}{4}+\beta_{n})&i\sin(\tfrac{\pi}{4}+\beta_{n})\\
i\sin(\tfrac{\pi}{4}+\beta_{n})&\cos(\tfrac{\pi}{4}+\beta_{n})\end{bmatrix}}_{S(\tfrac{\pi}{4}+\beta_{n})}$$
(11)
$$\displaystyle\times\underbrace{\begin{bmatrix}e^{i\theta_{n}}&0\\
0&1\end{bmatrix}}_{P_{1}(\theta_{n})}\underbrace{\begin{bmatrix}\cos(\tfrac{\pi}{4}+\alpha_{n})&i\sin(\tfrac{\pi}{4}+\alpha_{n})\\
i\sin(\tfrac{\pi}{4}+\alpha_{n})&\cos(\tfrac{\pi}{4}+\alpha_{n})\end{bmatrix}}_{S(\tfrac{\pi}{4}+\alpha_{n})}$$
Here $(\theta_{n},\phi_{n})$ are the phases programmed by the user, e.g. through thermo-optic [29], or MEMS [13] phase shifters, while $(\tfrac{\pi}{4}+\alpha_{n},\tfrac{\pi}{4}+\beta_{n})$ are the coupler angles, a property of the circuit and its imperfections ($\alpha_{n},\beta_{n}$). These angles are $\pi/4$ in an ideal MZI, which enables perfect contrast on each MZI output. In such a device, the phase shifts can be found by a procedure that diagonalizes $U$ with a sequence of $2\times 2$ rotations [14, 15].
(An equally valid convention is to place the phase mask at the end, $U=D\prod_{n}T_{n}$, and put the phase shifter $\phi_{n}$ at the beginning of the unit cell: $T_{n}=S(\tfrac{\pi}{4}+\beta_{n})P_{1}(\theta_{n})S(\tfrac{\pi}{4}+\alpha_{n})P_{1}(\phi_{n})$; however, the algorithm described in Sec. 2 is easiest to adapt to the convention in Eqs. (1-11)).
1.1 Component Errors
Component errors (deviations from design) will perturb the input-output matrix. We are primarily interested in the magnitude of this perturbation $\Delta U$, quantified by the Frobenius norm $\lVert\Delta U\rVert^{2}=\sum_{ij}|\Delta U_{ij}|^{2}$ and normalized to define an error measure:
$$\mathcal{E}=\frac{1}{\sqrt{N}}\langle\lVert\Delta U\rVert\rangle_{\rm rms}\in[0,2]$$
(12)
This metric can be interpreted as an average relative error per entry in the matrix $U$; for small $\mathcal{E}$, the quantity $(1-\mathcal{E})$ plays a role similar to the fidelity of a quantum operation.
To first order, the effect of component errors is linear:
$$\Delta U=\sum_{n}\Bigl{[}\Bigl{(}\frac{\partial U}{\partial\alpha_{n}}\Bigr{)}\alpha_{n}+\Bigl{(}\frac{\partial U}{\partial\beta_{n}}\Bigr{)}\beta_{n}\Bigr{]}$$
(13)
Applying Eqs. (1-11), we find $\partial U/\partial\alpha_{n}=U_{\rm pre}S^{\prime}(\pi/4)U_{\rm post}$ (and likewise for $\partial U/\partial\beta_{n}$). Since we are interested here in the magnitude $\lVert\Delta U\rVert$ and matrices $(U_{\rm pre},U_{\rm post})$ are unitary, it follows that:
$$\Bigl{\lVert}\frac{\partial U}{\partial\alpha_{n}}\Bigr{\rVert}=\Bigl{\lVert}\frac{\partial U}{\partial\beta_{n}}\Bigr{\rVert}=\lVert S^{\prime}(\pi/4)\rVert=\sqrt{2}$$
(14)
Thus, the mesh is equally sensitive to all beamsplitters, irrespective of geometry.
At this point it becomes necessary to introduce an error model, since perturbations from nearby crossings may lead to correlated errors in $U$. While real imperfections are correlated, this adds significant complexity to the math that obfuscates the insights. Moreover, while correlations may affect the error measure of a particular matrix, when considering ensembles of matrices, they are expected to average to zero (see Appendix A). Therefore, for the remainder of this paper, we assume an uncorrelated error model where $\langle\alpha_{n}\rangle_{\rm rms}=\langle\beta_{n}\rangle_{\rm rms}=\sigma$ for a given error amplitude $\sigma$.
Under the uncorrelated model, the error terms in Eq. (13) add in quadrature over the $N(N-1)$ couplers to give:
$$\mathcal{E}=\sqrt{2(N-1)}\,\sigma\sim(2N)^{1/2}\sigma$$
(15)
Since the depth of the circuit is $O(N)$, and independent errors in each layer add in quadrature, it is not surprising that $\mathcal{E}\propto N^{1/2}$. Component precision therefore must increase as meshes are scaled up in order to maintain a desired matrix accuracy.
1.2 Error Correction
As mentioned earlier, if the imperfections $(\alpha_{n},\beta_{n})$ are known, correction schemes can be used to program a unitary to, in most cases, better accuracy than Eq. (15). Recently, we presented a straightforward “local” scheme to correct for imperfections at each MZI separately [30]. The method is based on the observation that $2\times 2$ unitaries with the same power splitting ratio are equivalent up to external phase shifts. In the presence of imperfections, the MZI does not achieve perfect contrast in both output ports. The range of splitting angles is truncated to:
$$2|\alpha+\beta|\leq\theta\leq\pi-2|\alpha-\beta|$$
(16)
Provided Eq. (16) is satisfied, a perfect MZI $T(\theta_{n},\phi_{n})$ can be replaced by an imperfect MZI with external phase shifts:
$$\displaystyle\exists\ \theta^{\prime}_{n},\phi^{\prime}_{n},\chi^{\prime}_{n},\psi^{\prime}_{n}:$$
$$\displaystyle T(\theta_{n},\phi_{n})=T(\theta^{\prime}_{n},\phi^{\prime}_{n}|\alpha_{n},\beta_{n})\begin{bmatrix}e^{i\chi^{\prime}_{n}}&0\\
0&e^{i\psi^{\prime}_{n}}\end{bmatrix}$$
(19)
(The extra phase shifts can be absorbed into the neighboring MZIs so that the number of physical phase shifters on the mesh does not increase.) Provided that Eq. (16) is satisfied for all MZIs in the ideal Reck / Clements decomposition, this procedure in Ref. [30] leads to a perfect representation of the matrix. The fraction of unitary matrices realizable by this imperfect mesh is called the coverage, $\text{cov}(N,\sigma)$. If some MZIs do not satisfy Eq. (16), we pick the closest possible $\hat{\theta}_{n}\in\{\theta_{\rm min},\theta_{\rm max}\}$, which leads to an error in the matrix:
$$\displaystyle\lVert\Delta U\rVert_{\rm MZI}$$
$$\displaystyle=$$
$$\displaystyle\biggl{\lVert}\begin{bmatrix}\cos(\theta_{n}/2)&i\sin(\theta_{n}/2)\\
i\sin(\theta_{n}/2)&\cos(\theta_{n}/2)\end{bmatrix}$$
(25)
$$\displaystyle\ \ -\begin{bmatrix}\cos(\hat{\theta}_{n}/2)&i\sin(\hat{\theta}_{n}/2)\\
i\sin(\hat{\theta}_{n}/2)&\cos(\hat{\theta}_{n}/2)\end{bmatrix}\biggr{\rVert}$$
$$\displaystyle=$$
$$\displaystyle 2^{-1/2}|\theta_{n}-\hat{\theta}_{n}|+O\bigl{(}(\theta_{n}-\hat{\theta}_{n})^{3}\bigr{)}$$
(26)
Not all unitaries are equally easy to express on an MZI mesh. For this reason, when analyzing the efficiency of an error correction scheme, one must specify the probability distribution of $U$. Here, we consider random unitaries under the Haar measure, a distribution that samples uniformly from the space of unitary matrices [31, 32]. Under this measure, the phase shifts $\theta_{n}\in[0,\pi)$ and $\phi_{n}\in[0,2\pi)$ are uncorrelated and distributed according to [33]:
$$\displaystyle P(\phi)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2\pi}\ \ \ \text{(uniform over $[0,2\pi)$)}$$
(27)
$$\displaystyle P(\theta|k)$$
$$\displaystyle=$$
$$\displaystyle k\sin(\theta/2)\cos(\theta/2)^{2k-1}$$
(28)
where $k$ is the row index of the Reck mesh, starting from $k=1$ at the bottom (Fig. 1(a)). The phase shifts of the Clements mesh follow the same distribution with a different layout of crossings [33]. The distribution Eq. (28) clusters tightly around $\theta=0$ for MZIs with large $k$ [33, 19, 20] (Fig. 1(c)). Therefore, coverage and accuracy in large meshes is primarily limited by the small $\theta$ values. $P(\theta|k)$ can be linearized for small $\theta$, so the probability of a single MZI breaking the bound Eq. (16) is approximately
$$p_{\rm unsat}(k)=\int_{0}^{2|\alpha+\beta|}P(\theta|k){\rm d}\theta\approx k(\alpha+\beta)^{2}$$
(29)
For an $N\times N$ unitary, there are $(N-k)$ MZIs of rank k. The coverage is equal to the probability, under the Haar measure, that all $\theta_{n}$ are realizable. Since the $\theta_{n}$ are uncorrelated, this is a product of the probabilities for each MZI:
$$\displaystyle\text{cov}(N)=\prod_{k}\bigl{(}1-p_{\rm unsat}(k)\bigr{)}^{N-k}$$
$$\displaystyle\ \ \approx\exp\Bigl{[}-\underbrace{\sum_{k}k(N-k)\langle(\alpha+\beta)^{2}}_{n_{\rm unsat}}\rangle\Bigr{]}\approx e^{-N^{3}\sigma^{2}/3}$$
(30)
This is vanishingly small for large MZI meshes: for example, taking a reasonable value of $\sigma=0.02$, even a $32\times 32$ mesh has a coverage around 1%. In general, the number $n_{\rm unsat}$ of “unsatisfiable” MZIs that break condition (16) increases rapidly with error and problem size (Fig 2(a)).
Even if most unitaries cannot be realized exactly, they can be approximated to much better accuracy than the uncorrected result Eq. (15). Each MZI with an unrealizable $\theta_{n}$ will lead to a matrix error per Eq. (26). Over the Haar measure, the average error induced by a particular MZI is thus:
$$\displaystyle\langle\lVert\Delta U\rVert^{2}\rangle_{{\rm MZI},k}=\int{P(\theta|k)\lVert\Delta U(\theta)\rVert^{2}{\rm d}\theta}$$
$$\displaystyle\approx\int_{0}^{2|\alpha+\beta|}{\frac{k\theta}{2}\frac{(2|\alpha+\beta|-\theta)^{2}}{2}{\rm d}\theta}=\frac{k}{3}(\alpha+\beta)^{4}$$
(31)
Assuming the errors are uncorrelated (see Appendix A for the correlated case), they add in quadrature: $\langle\lVert\Delta U\rVert^{2}\rangle=\sum_{k}(N-k)\langle\lVert\Delta U\rVert^{2}\rangle_{k}\approx\tfrac{1}{18}N^{3}\langle(\alpha+\beta)^{4}\rangle=\frac{2}{3}N^{3}\sigma^{4}$. Following Eq. (12), the corrected normalized error is therefore:
$$\mathcal{E}_{\rm corr}=\sqrt{2/3}\,N\sigma^{2}$$
(32)
This is plotted in Fig. 2(b). Recall from Eq. (15) that the uncorrected error scales as $\mathcal{E}\propto\sqrt{N}\,\sigma$. This means that $\mathcal{E}_{\rm corr}\propto\mathcal{E}^{2}$, i.e. correction allows for an effective “squaring” of the error. If the errors are very large to begin with, error correction will not provide much benefit. However, for most fabricated circuits the uncorrected $\mathcal{E}$ is reasonably small (though not small enough for many applications), and error correction can give a significant boost in accuracy.
2 Self-Configuring Algorithms
In many circumstances, the correction procedure in Sec. 1.2 cannot be applied because the errors in an MZI mesh are known to sufficient accuracy. Nevertheless, for triangular meshes, “progressive” self-configuration strategies can still be used. As noted earlier, existing strategies rely on inline photodetectors or pre-calibration, which limits their usefulness in many systems [22, 23, 24, 34]. This section introduces two schemes that do not rely on these assumptions and can be run on uncalibrated hardware with only external detectors: a simple “direct” method based on sequentially setting matrix elements and a “ratio” method based on setting power ratios. While both schemes work in principle, only the ratio method is robust in the presence of large errors. This distinction highlights the importance of algorithmic stability when configuring large multiport interferometers.
2.1 Direct Method
The simplest way to program a triangular mesh is to set the MZI phases one at a time to match the target matrix elements $\hat{U}_{ij}$. This is easiest to understand by considering first the case of a tunable 1:N splitter and later generalizing to an $N\times N$ unitary. Fig. 3(a) shows the “direct” method for programming a 1:N splitter, which we wish to set to a target splitting vector $\hat{u}$, $\lVert\hat{u}\rVert=1$. Given a coherent input $E_{\rm in}$, the procedure consists of $N$ steps, starting from the input of the splitter and working towards the final output. In the first $N-1$ steps, a pair of phases $(\theta,\phi)$ (corresponding to an MZI and phase shifter, Fig. 1(a)) are tuned to set the $m^{\rm th}$ (complex) output amplitude to $\hat{u}_{m}E_{\rm in}$. The two degrees of freedom are sufficient to independently set the real and imaginary components of $\hat{u}_{m}$. In the final step, there is only a single degree of freedom (a phase shift $\phi$); however, since $\hat{u}$ has unit norm, at this stage its amplitude is already constrained and only the phase is free; therefore, provided the preceding elements $\hat{u}_{m}$ are set properly, only single phase shift is needed to set $\hat{u}_{N}$.
Fig. 3(b) shows the direct method for programming a Reck triangle, which can be divided into $N$ diagonals, each functioning as a tunable one-to-many splitter, with the output of each diagonal fed into the inputs of its upper-right neighbor. The triangle is configured from the top diagonal to the bottom, working down each diagonal. Each MZI element $(\theta_{mn},\phi_{mn})$ (the $n^{\rm th}$ element of the $m^{\rm th}$ diagonal, starting from the top) is configured to set the matrix element $\hat{U}_{nm}$ between output $n$ and input $m$. In this way, the lower diagonal of $\hat{U}$ is correctly configured – which, given the unitarity of $\hat{U}$, correctly configures the entire matrix.
The matrix must be triangular in order for this procedure to work. Triangularity guarantees that tuning steps do not disturb the matrix elements that have already been set, provided that the order in Eq. 3(b) is followed. Thus, the direct method cannot configure the Clements matrix, although we show in Sec. 4 that Clements can be divided into two triangles, which can be separately configured.
2.2 Ratio Method
One can also configure the mesh by a method based on power ratios. As before, it is easiest to describe this method in the case of a 1:N splitter (Fig. 4(a)) and then generalize to the Reck triangle. In this case, the splitter phases are configured in reverse order to set the power ratios (and relative phases) of the outputs. As before, let $\hat{u}$ be the target vector and $\vec{u}$ be the output of the physical splitter. The configuration steps are as follows:
•
Step 1: Set splitter angle $\theta$ to match the power ratio $|u_{N-1}/u_{N}|=|\hat{u}_{N-1}/\hat{u}_{N}|$. Next, set phase shift $\phi$ to match the relative phase $\text{arg}(u_{N-1}/u_{N})=\text{arg}(\hat{u}_{N-1}/\hat{u}_{N})$ between the last two outputs.
•
Intermediate Steps: Here, we configure the phase shifts corresponding to the $n^{\rm th}$ output, $1\leq n<N$. These are set to align the partial output vectors: $\vec{u}_{n:N}\parallel\hat{u}_{n:N}$ (here $\vec{a}_{n:N}=[a_{n},\ldots,a_{N}]$ denotes the slice of a vector $\vec{a}$ over a given index set, red outputs in Fig. 4(a)). The splitter angle $\theta$ is set to match the power ratios $P_{n}/(P_{n+1}+\ldots+P_{N})$, while the phase shift is used to compensate any relative phases. Overall, this corresponds to maximizing the inner product $\text{max}_{\theta,\phi}\bigl{|}\langle\vec{u}_{n:N}|\hat{u}_{n:N}\rangle\bigr{|}$.
Without errors, this is equivalent to matching the amplitude and phase of $u_{n}/u_{n+1}$ as all downstream ratios have already been configured; however, using all the outputs in the configuration is more robust to errors (especially when $u_{n+1}$ is small).
•
Step N: Set the final phase shift to align the phase of $\vec{u}$ with $\hat{u}$.
The Reck triangle is configured one diagonal at a time in the order shown in Fig. 4(b). Here, we have indexed the each MZI $(m,n)$ according to its diagonal ($m$) and position relative to the triangle base ($n$). When configuring an MZI along the $m^{\rm th}$ diagonal, light enters port $m$ so that only the top port of the MZI is excited. All MZIs downstream from $(m,n)$ have been tuned, while upstream MZIs are untuned, and a spacelike separator $S_{mn}$ (purple line in figure) divides the configured and unconfigured parts of the mesh. We can write the unitary of this circuit as:
$$U=U_{\rm post}TU_{\rm pre}$$
(33)
where $T=\bigl{[}[T_{11},T_{12}],[T_{21},T_{22}]\bigr{]}$ is the MZI transfer function, which depends on the phases $(\theta_{mn},\phi_{mn})$. The output field is a sum of 3 contributions (Fig. 4(c)):
1.
Light that bypasses the MZI $(m,n)$. At surface $S_{mn}$, this is denoted by $\vec{\alpha}$, and at the output it is $\vec{a}=U_{\rm post}\vec{\alpha}$.
2.
Light that enters $(m,n)$ and leaves through its top port. The input light has an unknown amplitude $s\,e^{i\psi}$ ($\psi$ set by the upstream phase shifter, purple), but only relative amplitudes matter when configuring $(m,n)$. The output to the top port is $s\,e^{i\psi}T_{11}$. At surface $S_{mn}$, the field is denoted by the vector $s\,e^{i\psi}T_{11}\vec{\beta}$, where $\vec{\beta}=\hat{e}_{N-n}$ is the one-hot vector for waveguide $(N-n)$ (the top output of $(m,n)$). Thus the output field is $s\,e^{i\psi}T_{11}\vec{b}$, where $b=U_{\rm post}\vec{\beta}$.
3.
Light that enters $(m,n)$ and leaves through the bottom port. Analogous to the top port, we have $s\,e^{i\psi}T_{21}\vec{\gamma}$ at $S_{mn}$ ($\vec{\gamma}=\hat{e}_{N-n+1}$), and $s\,e^{i\psi}T_{21}\vec{c}$ at the output ($\vec{c}=U_{\rm post}\vec{\gamma}$).
Summing these terms, the output from port $m$ is:
$$\vec{u}_{m}=\vec{a}+s\,e^{i\psi}(T_{11}\vec{b}+T_{21}\vec{c})$$
(34)
The goal is to configure the MZI so that this output best approximates $\hat{u}_{m}$, the $m^{\rm th}$ column of target matrix $\hat{U}$. This is done by minimizing the $L_{2}$ norm $\lVert\vec{u}_{m}-\hat{u}_{m}\rVert$. Since $T$ and $U_{\rm post}$ are unitary, we have $\vec{a}\perp\vec{b}\perp\vec{c}$ and $|T_{11}|^{2}+|T_{21}|^{2}=1$; applying these relations we find:
$$\displaystyle\lVert\vec{u}_{m}-\hat{u}_{m}\rVert^{2}=\underbrace{\lVert\hat{u}_{m}-\vec{a}\rVert^{2}+|s|^{2}}_{\rm const}$$
$$\displaystyle\qquad\quad-2\,\text{Re}\bigl{[}s\,e^{i\psi}\bigl{(}T_{11}\langle\hat{u}_{m}|\vec{b}\rangle+T_{21}\langle\hat{u}_{m}|\vec{c}\rangle\bigr{)}\bigr{]}$$
(35)
The first two terms drop out as constants since they do not depend on the optimization variables $\theta_{mn},\phi_{mn}$ (which determine $(T_{11},T_{21})$). Since each MZI is preceded by a phase shifter, the phase of $\psi$ is also freely tunable; we therefore wish to perform the following maximization:
$$\displaystyle\text{max}_{\theta,\phi}\text{max}_{\psi}\text{Re}\bigl{[}s\,e^{i\psi}\bigl{(}T_{11}\langle\hat{u}_{m}|\vec{b}\rangle+T_{21}\langle\hat{u}_{m}|\vec{c}\rangle\bigr{)}\bigr{]}$$
$$\displaystyle\quad\propto\text{max}_{\theta,\phi}\bigl{|}T_{11}\langle\hat{u}_{m}|\vec{b}\rangle+T_{21}\langle\hat{u}_{m}|\vec{c}\rangle\bigr{|}$$
(36)
subject to the constraint $|T_{11}|^{2}+|T_{21}|^{2}=1$. This is just optimizing a dot product, which amounts to setting the amplitude ratio:
$$\frac{T_{11}}{T_{21}}=\frac{\langle\vec{b}|\hat{u}_{m}\rangle}{\langle\vec{c}|\hat{u}_{m}\rangle}$$
(37)
We cannot measure $T_{11},T_{21},\vec{b}$, or $\vec{c}$ directly in an experiment. Instead, we proceed as follows: first sweep the value of $\psi$ to obtain $\vec{a}$, which is the value of $u_{i}$ averaged over opposite phases $\psi$:
$$\vec{a}=\frac{\vec{u}_{m}(\psi=0)+\vec{u}_{m}(\psi=\pi)}{2}=\frac{1}{2\pi}\int{\vec{u}_{m}(\psi){\rm d}\psi}$$
(38)
Next, once $\vec{a}$ is found, set $(\theta,\phi)$ to maximize the quantity:
$$\text{max}_{\theta,\phi}\bigl{|}\langle\hat{u}_{m}|\vec{u}_{m}(\theta,\phi)-\vec{a}\rangle\bigr{|}$$
(39)
which is mathematically the same as Eq. (36) and independent of $\psi$, but only relies on external output measurements.
3 Performance Comparison
To compare the strategies, we simulate the self-calibration of Reck meshes in the presence of component errors. Here, the target unitaries $\hat{U}$ are sampled from the Haar measure, with random, Gaussian-distributed errors in the beamsplitter angles (i.e. $\langle\alpha\rangle_{\rm rms}=\langle\beta\rangle_{\rm rms}=\sigma$). We consider mesh sizes in the range $8\leq N\leq 64$ to analyze the scaling of the algorithms with mesh size. Along the lines of Sec. 1.2, we expect that error correction should allow perfect configuration when errors are low enough (coverage is order unity), and an error reduction of $\mathcal{E}_{\rm corr}\propto\mathcal{E}^{2}$ in the uncorrectable case.
Fig. 5(a) shows the scaling of error metric $\mathcal{E}$ with $\sigma$ for a $64\times 64$ Reck mesh. As expected, the uncorrected error increases linearly with $\sigma$, following Eq. (15). For sufficiently small $\sigma$, the corrected error diverges to zero for both methods. However, the direct method suffers a hard performance cutoff around $\sigma=0.005$. For realistic $\sigma\gtrsim 0.01$, the direct method actually performs worse than no correction at all! In contrast, the ratio method performs well at both small and large $\sigma$. Above the cutoff, it roughly follows the trend $\mathcal{E}=\sqrt{2/3}\,N\sigma^{2}$, the same relation derived for the local scheme in Eq. (32). Unlike the local scheme, the ratio method requires no prior knowledge of the MZI imperfections, and can be configured using only output detectors.
By combining the two fits in Fig. 5(a), we arrive at the following expression:
$$\mathcal{E}_{\rm corr}=\frac{1}{\sqrt{6}}\mathcal{E}^{2}$$
(40)
This relation is independent of $N$, and can be used to test the scaling of the algorithms as the matrix dimension increases. Fig. 5(b) plots $\mathcal{E}_{\rm corr}$ against $\mathcal{E}$ for $8\leq N\leq 64$. Predictably, there is a sharp drop for both schemes corresponding to perfect correction, but the threshold for such perfect correction decreases with $N$. This is a serious challenge for the direct method, since $\mathcal{E}_{\rm corr}>\mathcal{E}$ in the imperfect correction regime. On the contrary, the ratio method shows an improvement for the whole range of $N$, with the data asymptoting to Eq. (40), suggesting that the approach is scalable.
Fig. 5(c) plots the same data in $(N,\sigma)$ space. For both methods, we observe a transition when the coverage of the mesh drops below unity. Since $\text{cov}(N)\sim e^{-N^{3}\sigma^{2}/3}$, this transition occurs roughly at $N^{3}\sigma^{2}=3$ (white curve). Both methods work in the exact regime, but only the ratio method is successful when errors are large enough that $U$ cannot be represented exactly.
Why does the direct method perform poorly in the uncorrectable error regime? The structure of the error matrix $|U-\hat{U}|$ (Fig. 6) sheds light on the problem. The direct method only guarantees $U_{mn}=\hat{U}_{mn}$ for the upper-left triangle of entries $m+n<N$. If these are exactly satisfied, the matrices will be equal. However, even small errors are pushed to the lower-right triangle, where they cascade as the mesh is configured column by column. This instability leads, in general, to a matrix that is only well configured for at most half of its entries.
When following the ratio method, errors do not build up. To understand the stability of the ratio method, it is helpful to relate the method to the $2\times 2$ block decomposition of a unitary matrix [14] (Fig. 7). Given a target matrix $\hat{U}$, we wish to find $2\times 2$ blocks $U_{mn}$ such that $U_{N1}^{\dagger}\ldots U_{12}^{\dagger}U_{11}^{\dagger}\hat{U}=I$. We start by configuring block $U_{11}$, which mixes the last two rows to zero out the lower-right element of the matrix. Next, $U_{12}$ is configured to zero the element directly above. The procedure is repeated until all elements in the lower diagonal have been zeroed, at which point the matrix equals the identity.
Because of MZI errors, not all off-diagonal terms can be zeroed. If a term cannot be zeroed, it leaves a residual term $(|\alpha_{mn}-\beta_{mn}|-\theta_{mn})$ below the diagonal, where $\theta_{mn}$ is the target splitting ratio for MZI $(m,n)$, which is unrealizable since $|\alpha_{mn}-\beta_{mn}|>\theta_{mn}$. Let $V^{(mn)}=U_{mn}^{\dagger}\ldots U_{12}^{\dagger}U_{11}^{\dagger}\hat{U}$ be the matrix after configuring $U_{mn}$ and define
$$\epsilon_{mn}=\sum_{i=1}^{m-1}\sum_{j=i+1}^{N}|V^{(mn)}_{ij}|^{2}+\sum_{j=N-n}^{N}|V^{(mn)}_{mj}|^{2}$$
(41)
which is the sum of squares of all elements $V^{(mn)}_{ij}$ in the zero region below the diagonal (white and green in Fig. 7). Each imperfect configuration step adds a new element to this region, incrementing $\epsilon_{mn}$ by $(|\alpha_{mn}-\beta_{mn}|-\theta_{mn})^{2}$, the norm of the new element added. The existing matrix elements are mixed around, but the norm for them does not change because the mixing is unitary. Therefore, errors do not grow in the ratio scheme; they just get mixed into other matrix elements. This is the critical difference between the direct and ratio schemes. The final matrix will be close to the identity and therefore takes the form $V^{(N1)}\approx I+iH$ for some Hermitian $H$. Therefore, $\lVert U-\hat{U}\rVert^{2}\approx 2\epsilon_{N1}$, and we have:
$$\displaystyle\mathcal{E}_{\rm corr}^{2}=\frac{2\langle\epsilon_{N1}\rangle}{N}$$
$$\displaystyle=\frac{2}{N}\sum_{mn}\langle\text{max}(|\alpha_{mn}-\beta_{mn}|-\theta_{mn},0)^{2}\rangle=\frac{2N^{2}\sigma^{4}}{3}$$
(42)
which is the same result as for the local scheme Eq. (32).
4 Rectangular Mesh
Compared to the Reck triangle, the rectangular Clements mesh has the advantages of increased compactness, reduced circuit depth, and relative insensitivity to fixed component losses [15]. However, it cannot be written as a cascade of diagonals, so the self-configuration techniques presented above cannot be used. However, a simple modification to Clements—placing a diagonal of tunable drop ports and detectors—suffices to make the mesh programmable (Fig. 8(a)). The diagonal elements effectively split the mesh into two triangles, each which can be programmed independently.
First, following the Clements representation of an ideal MZI mesh [15], the target matrix is decomposed into two components $\hat{U}=\hat{U}_{2}\hat{U_{1}}$ for the left and right triangles. Next, the diagonal ports are set to the “cross” state to collect all of the light along the diagonal. This allows the left triangle to be programmed to $\hat{U}_{1}$ (Fig. 8(b)), achieved by a reciprocal form of the ratio method described below. Finally, the diagonal switches are used to isolate the inputs of the right triangle; this allows it to be programmed to $\hat{U}_{2}$ (up to an input phase) by the conventional ratio method (Sec. 2.2).
In the reciprocal form of the ratio method, instead of sending light into a single port and matching the output vector to a column of $\hat{U}$, we send in a column $u_{m}^{*}$ of $\hat{U}^{\dagger}$ as input and try to direct all the power to a single output. This is analogous to the Reverse Local Light Interference Method (RELLIM) [24], but does not require internal detectors. The MZIs are programmed along falling diagonals, but compared to Sec. 2.2, the order is reversed (bottom to top, down each diagonal).
When configuring MZI $(m,n)$, all upstream components have been configured, while downstream components have not. The input-output relation is the product $v=U_{\rm post}TU_{\rm pre}\hat{u}_{m}^{*}$, where $U_{\rm pre}$ has been configured but $U_{\rm post}$ has not. The light at output $m$ can take one of three paths: (1) bypassing the MZI, (2) entering the top port of the MZI, or (3) entering the bottom port of the MZI (Fig. 8(c)), leading to a sum:
$$v_{m}=a+s\,e^{i\psi}(T_{21}\beta+T_{22}\gamma)$$
(43)
The MZI must be configured to direct all its output power to the bottom port ($T_{21}/T_{22}=(\beta/\gamma)^{*}$). In RELLIM [24], this is accomplished with the use of internal detectors. However, external detectors can also be used, even if the downstream MZIs have not been calibrated to a “cross” state. As in Sec. 2.2, first the phase $\psi$ is swept and the value of $a$ is extracted from the average:
$$a=\frac{v_{m}(\psi=0)+v_{m}(\psi=\pi)}{2}=\frac{1}{2\pi}\int{v_{m}(\psi){\rm d}\psi}$$
(44)
Next, the phases of the MZI are set to maximize:
$$\text{max}_{\theta,\phi}|v_{m}-a|$$
(45)
As with the Reck scheme, we can prove that this configuration method is resistant to errors by visualizing how each configuration step zeroes out entries in the target matrix. Each matrix has $\approx\tfrac{3}{4}N^{2}$ free entries and $\approx\tfrac{1}{4}N^{2}$ zeroes below the diagonal. Each mesh is self-configured to eliminate the remaining nonzero elements in the lower triangle, which takes $\approx\tfrac{1}{4}N^{2}$ steps, half the number of steps as the Reck triangle. In each step in the reciprocal ratio method (Fig. 9(a)), the target matrix is right-multiplied by a $2\times 2$ block to zero out a matrix entry; after $\approx\tfrac{1}{4}N^{2}$ steps, the upper triangle of Fig. 8(b) is configured and $\hat{U}_{1}$ has been diagonalized. Likewise, the conventional ratio method configures the lower triangle and diagonalizes $\hat{U}_{2}$ (Fig. 9(b)). The remaining phase shifts along the diagonal can be set by inspection.
The accuracy of the self-configured meshes is plotted in Fig. 10. Again, we see that the ratio method successfully corrects MZI errors and follows the same relation $\mathcal{E}=\sqrt{2/3}\,N\sigma^{2}$ observed for the Reck triangle (Fig. 10(a)). The direct method can also be used to configure the sub-triangles in Fig. 8(b), but shows poor performance in the large-$\sigma$ regime where MZI errors cannot be exactly corrected. The boundary between the exact and inexact correction regime is the same (Fig. 10(b)) owing to the fact that the MZI splitting angles in Reck and Clements meshes have the same statistics over the Haar measure (Sec. 1). Visualizing the matrix imperfections (Fig. 10(c)) again reveals the harmful cascading of errors in the direct method, which cause certain parts of the matrix to be well-configured while other parts are not. This cascading effect is avoided in the ratio method for the same reasons applicable to Reck (Sec. 3).
5 Discussion
Component imprecision will limit practical performance as multiport interferometers grow larger—barring a breakthrough in fabrication accuracy, some form of self-calibration or error correction will need to be employed. In this paper, we have presented a method based on power ratios that can operate without any internal detectors or knowledge of the component imperfections. This algorithm is applicable to any triangular mesh, but can be extended to rectangular meshes by adding a single diagonal of drop ports, a small amount of additional complexity as the mesh size grows large. The accuracy of our algorithm is guaranteed by the algorithmic stability of unitary matrix diagonalization, and follows the asymptotic form $\mathcal{E}_{\rm corr}\propto N\sigma^{2}$ over the Haar measure with independent Gaussian component errors. Employing this algorithm suppresses matrix errors by a quadratic factor: $\mathcal{E}_{\rm corr}=\mathcal{E}^{2}/\sqrt{6}$, allowing MZI meshes to scale to large sizes ($N>64$) without unreasonable demands on fabrication tolerance.
Two limitations to our algorithm merit future work. First, it relies heavily on unitarity and will fail to correct non-unitary errors. Non-unitary mesh architectures and calibration are important topics for future study, as many ultra-compact or energy-efficient component designs [29, 35, 36, 37] involve some loss (often phase-dependent). Second, the algorithm will only be effective if uncorrected errors are reasonably small to begin with. For large enough meshes ($N\gtrsim 1/\sigma^{2}$), $\mathcal{E}\sim 1$ and errors will be uncorrectable. This may lead to a fundamental limit on size, closely tied to the transition between ballistic and diffusive transport of light [20]. For such extreme sizes, entirely new crossing designs [21] or mesh architectures [28, 25, 26, 27, 38] may be required.
Acknowledgements
S.B. is supported by a National Science Foundation (NSF) Graduate Research Fellowship (grant no. 1745302). D.E. acknowledges funding from the Air Force Office of Scientific Research (AFOSR) (grant no. FA9550-20-1-0113 and FA9550-16-1-0391).
Source code implementing the algorithms described in this paper is available in the Meshes package [39].
Appendix A Correlated Errors
In this paper, we have assumed an uncorrelated error model, where $\alpha_{n}$ and $\beta_{n}$ are independent random variables sampled from a zero-mean Gaussian with standard deviation $\sigma$. In practice, the fabrication imperfections that lead to splitter errors have a long correlation length, so errors will be strongly correlated. In this section, we show that:
1.
Averaged over the Haar measure, inter-MZI correlations cancel out. Therefore, only correlations within each MZI (i.e. between $\alpha_{n}$ and $\beta_{n}$) need be considered.
2.
For most unitaries, the effect of the symmetric error $\alpha_{n}+\beta_{n}$ is dominant.
The error metric Eq. (12) can be expanded to second order, accounting for all correlations:
$$\displaystyle\mathcal{E}^{2}=\frac{1}{N}\sum_{mn}\Bigl{(}\Bigl{\langle}\frac{\partial U}{\partial\alpha_{m}}\Bigr{|}\frac{\partial U}{\partial\alpha_{n}}\Bigr{\rangle}\langle\alpha_{m}\alpha_{n}\rangle+\Bigl{\langle}\frac{\partial U}{\partial\alpha_{m}}\Bigr{|}\frac{\partial U}{\partial\beta_{n}}\Bigr{\rangle}\langle\alpha_{m}\beta_{n}\rangle$$
$$\displaystyle\qquad\quad+\,\Bigl{\langle}\frac{\partial U}{\partial\beta_{m}}\Bigr{|}\frac{\partial U}{\partial\alpha_{n}}\Bigr{\rangle}\langle\beta_{m}\alpha_{n}\rangle+\Bigl{\langle}\frac{\partial U}{\partial\beta_{m}}\Bigr{|}\frac{\partial U}{\partial\beta_{n}}\Bigr{\rangle}\langle\beta_{m}\beta_{n}\rangle\Bigr{)}$$
(46)
with the matrix inner product $\langle V|W\rangle=\text{tr}(V^{\dagger}W)$.
Correlations can be classified into two types: intra-MZI and inter-MZI (Fig. 11(a)). We first show that, averaged over the Haar measure, inter-MZI correlations are zero or at least very small. Consider an arbitrary inter-MZI pair of splitters $(p\in\{\alpha_{m},\beta_{m}\},q\in\{\alpha_{n},\beta_{n}\})$. The unitary takes the form:
$$\displaystyle U$$
$$\displaystyle=U_{\rm post}S(\tfrac{\pi}{4}\!+\!q)U_{q}\!\begin{bmatrix}e^{i\phi}&0\\
0&1\end{bmatrix}\!U_{\rm int}\!\begin{bmatrix}1&0\\
0&e^{i\phi^{\prime}}\end{bmatrix}\!U_{p}S(\tfrac{\pi}{4}\!+\!p)U_{\rm pre}$$
(51)
$$\displaystyle\ \ \ \ +P(p)+Q(q)$$
(52)
where $S$ is the symmetric splitter matrix (Eq. (11))
$$\displaystyle S(\psi)$$
$$\displaystyle=$$
$$\displaystyle e^{i\psi\sigma_{x}}=\begin{bmatrix}\cos\psi&i\sin\psi\\
i\sin\psi&\cos\psi\end{bmatrix}$$
(55)
$$\displaystyle S^{\prime}(\psi)$$
$$\displaystyle=$$
$$\displaystyle iS(\psi)\sigma_{x}=i\sigma_{x}S(\psi)$$
(56)
and $\sigma_{x}=[[0,1],[1,0]]$ is the Pauli matrix.
The terms $P(p)$ and $Q(q)$ in Eq. (52) correspond to paths of light that pass through at most one splitter. These are mutually orthogonal and do not contribute to the correlation in Eq. (46). Error correlations only arise from the first term, corresponding to paths of light that pass through both splitters $p$ and $q$. The resulting inner product is independent of $U_{\rm pre}$ and $U_{\rm post}$ and takes the form (at $p=q=0$):
$$\displaystyle\Bigl{\langle}\frac{\partial U}{\partial p}\Bigr{|}\frac{\partial U}{\partial q}\Bigr{\rangle}$$
$$\displaystyle=\Bigl{\langle}S(\tfrac{\pi}{4})U_{q}\!\begin{bmatrix}e^{i\phi}&0\\
0&1\end{bmatrix}\!U_{\rm int}\!\begin{bmatrix}1&0\\
0&e^{i\phi^{\prime}}\end{bmatrix}\!U_{p}S(\tfrac{\pi}{4})\sigma_{x}\Bigr{|}$$
(61)
$$\displaystyle\qquad\Bigl{|}\sigma_{x}S(\tfrac{\pi}{4})U_{q}\!\begin{bmatrix}e^{i\phi}&0\\
0&1\end{bmatrix}\!U_{\rm int}\!\begin{bmatrix}1&0\\
0&e^{i\phi^{\prime}}\end{bmatrix}\!U_{p}S(\tfrac{\pi}{4})\Bigr{\rangle}$$
(66)
In general, this quantity is nonzero. However, the phases $(\phi,\phi^{\prime})$ are uniformly distributed over $[0,2\pi)$ for Haar-random unitaries. Therefore, in the ensemble average over the Haar measure, the phase-dependent terms in Eq. (66) cancel out. This means that each path from splitter $p$ to $q$, which has a separate phase dependence, can be considered separately in the inner product. Consider the path from output $i\in\{1,2\}$ to input $j\in\{1,2\}$ ($i=j=1$ shown in Fig. 11(b)). Focusing on a single path, we can ignore ($\phi,\phi^{\prime})$ and replace $U_{\rm int}\rightarrow\hat{e}_{j}\hat{e}_{i}^{T}$:
$$\displaystyle\Bigl{\langle}\frac{\partial U}{\partial p}\Bigr{|}\frac{\partial U}{\partial q}\Bigr{\rangle}_{ij}$$
$$\displaystyle\propto\bigl{\langle}S(\tfrac{\pi}{4})U_{q}\hat{e}_{j}\hat{e}_{i}^{T}U_{p}S(\tfrac{\pi}{4})\sigma_{x}\bigr{|}$$
$$\displaystyle\qquad\bigl{|}\sigma_{x}S(\tfrac{\pi}{4})U_{q}\hat{e}_{j}\hat{e}_{i}^{T}U_{p}S(\tfrac{\pi}{4})\bigr{\rangle}$$
$$\displaystyle=(\hat{e}_{j}^{T}U_{q}^{\dagger}\sigma U_{q}\hat{e}_{j})(\hat{e}_{i}^{T}U_{p}\sigma U_{p}^{\dagger}\hat{e}_{i})$$
(67)
There are four cases to be considered (Fig. 11(b)). If $p$ is directly adjacent to the connecting path (cases I-II), then $U_{p}$ is the identity and $\hat{e}_{i}^{T}U_{p}\sigma U_{p}^{\dagger}\hat{e}_{i}=\hat{e}_{i}^{T}\sigma\hat{e}_{i}=0$. Likewise, if $q$ is adjacent to the path (cases I and III), $\hat{e}_{j}^{T}U_{q}^{\dagger}\sigma U_{q}\hat{e}_{j}=0$. Only in case IV does Eq. (67) lead to a nontrivial inner product:
$$\Bigl{\langle}\frac{\partial U}{\partial p}\Bigr{|}\frac{\partial U}{\partial q}\Bigr{\rangle}_{ij}\propto\sin(\theta_{m})\sin(\theta_{n})$$
(68)
This is very small for the majority of MZIs, where the splitter angles ($\theta_{m}$, $\theta_{n}$) cluster tightly around zero. Moreover, while it is always possible to find a matrix decomposition with only positive $\theta$ (e.g. distribution of Eqs. (27-28)), one can also sample from the Haar measure employing both positive and negative $\theta$ with equal probability; in this case, under the ensemble average, Eq. (68) vanishes and all inter-MZI correlations are zero.
On the other hand, correlations within an MZI, i.e. between $\alpha_{n}$ and $\beta_{n}$, always matter. The matrix error of a single MZI is:
$$\displaystyle\lVert\Delta U\rVert^{2}$$
$$\displaystyle=$$
$$\displaystyle 2\bigl{[}\langle\alpha_{n}^{2}\rangle+\langle\beta_{n}^{2}\rangle+2\cos(\theta_{n})\langle\alpha_{n}\beta_{n}\rangle\bigr{]}$$
(69)
$$\displaystyle=$$
$$\displaystyle 2\bigl{[}\cos^{2}(\theta_{n}/2)\langle(\alpha_{n}+\beta_{n})^{2}\rangle$$
$$\displaystyle\ \ \ +\sin^{2}(\theta_{n}/2)\langle(\alpha_{n}-\beta_{n})^{2}\rangle\bigr{]}$$
For the large majority of MZIs, $\theta_{n}\approx 0$ and the symmetric error dominates $\lVert\Delta U\rVert^{2}$. The normalized error for the whole mesh is approximately:
$$\mathcal{E}\approx\sqrt{N}\langle(\alpha+\beta)\rangle_{\rm rms}$$
(70)
which in the case of uncorrelated $(\alpha,\beta)$ reduces to the form derived in the main text: $\mathcal{E}=\sqrt{2N}\,\sigma$.
For completeness, we now consider the case of fully correlated splitter errors, i.e. $\alpha_{n}=\beta_{n}=\mu$, where $\mu$ is a constant. Systematic effects such as imperfect coupler design or fabrication errors with long correlation length will lead to this situation, as will operating the mesh away from the coupler design wavelength. The uncorrected error amplitude is (compare Eq. (15)):
$$\mathcal{E}=2\sqrt{N}\mu$$
(71)
Following the analysis of Eq. (27-30), the coverage of unitary space is found to be (compare Eq. (30)):
$$\text{cov}(N)=e^{-(2/3)N^{3}\mu^{2}}$$
(72)
Finally, in the presence of error correction, the matrix error becomes (compare Eqs. (32, 40)):
$$\mathcal{E}_{\rm corr}=\sqrt{8/9}N\mu^{2}=\frac{1}{\sqrt{18}}\mathcal{E}^{2}$$
(73)
Eqs. (71-73) are plotted against numerical data in Fig. 12.
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A Distributed Cubic-Regularized Newton Method for Smooth Convex Optimization over Networks
César A. Uribe and Ali Jadbabaie
Laboratory for Information, and Decisions Systems
Institute for Data, Systems, and Society
Massachusetts Institute of Technology
Cambridge, MA 02139
{cauribe,jadbabai}@mit.edu
Abstract
We propose a distributed, cubic-regularized Newton method for large-scale convex optimization over networks. The proposed method requires only local computations and communications and is suitable for federated learning applications over arbitrary network topologies. We show a $O(k^{{-}3})$ convergence rate when the cost function is convex with Lipschitz gradient and Hessian, with $k$ being the number of iterations. We further provide network-dependent bounds for the communication required in each step of the algorithm. We provide numerical experiments that validate our theoretical results.
1 Introduction
Newton’s method for minimizing smooth strongly convex functions has a longstanding history in optimization and scientific computing Bertsekas1999 ; Bennett592 . The main reason for its popularity is its fast convergence rate. However, the Newton step’s computational cost has often limited its applicability to modern large-scale machine learning problems. Despite these computational challenges, there has been a resurgence of interest in Newton-type algorithms from a theoretical perspective. Over the past two decades, a series of papers by Nesterov and coauthors nesterov2006cubic ; nesterov2006cubic2 have shown that with appropriate higher-order (e.g., cubic) regularization, such methods achieve provably-fast global convergence rates cartis2011adaptive ; nes83 ; nesterov2013introductory ; nemirovski2004interior .
Additionally, fast higher-order methods have been driven by new insights into their accelerated convergence rates, fundamental limits, and complexity bounds monteiro2013accelerated ; agarwal2018adaptive ; pmlr-v99-gasnikov19a ; gasnikov2019near , leading to a series of implementable practical algorithms nesterov2018implementable ; kamzolov2020near ; nesterov2020superfast . Nevertheless, as mentioned earlier, the impact in modern machine learning applications has been limited zhou2019stochastic . Specially as increasing amounts of data and distributed storage technologies have now driven the need for distributed and federated architectures wang2018giant that split computational cost among many nodes hendrikx2020optimal , e.g., Peer-to-peer federating learning lalitha2019peer ; pilet2019simple , distributed optimization methods sca17 ; scaman2018optimal ; lan17 ; Uribe2018a ; Rasul19 ; Mokhtari2016 ; Zargham2013 ; ye2020multi ; li2018sharp , MapReduce dean2008mapreduce , Apache Spark yang2013trading , and Parameter Server li2014scaling .
Several second-order distributed methods have been proposed in the literature for smooth, strongly convex functions Rasul19 ; wei2013distributed ; jadbabaie2009distributed ; Zargham2013 ; mokhtari2016decentralized . Nevertheless, such approaches do not provide global convergence rates Rasul19 and require strong convexity assumptions to guarantee some linear convergence rate, or require specific master/worker architectures Shamir2014 ; pmlr-v37-zhangb15 . Other approaches use Quasi-Newton/BFGS-like approaches to compute approximations to the Hessian inverse eisen2017decentralized efficiently, but exact non-asymptotic convergence rates are not available.
The goal of this paper is to address the existing gap in the literature between cubic regularization and distributed optimization. Specifically, motivated buy Empirical Risk Minimization in machine learning applications, we consider the following finite sum minimization problem
$$\displaystyle\min_{x\in Q}\left\{f(x)\triangleq\sum_{i{=}1}^{m}f^{i}(x)\right\},$$
(1)
where $f^{i}$ is the local empirical risk of a subset of data points stored locally by an agent $i$, which means that each agent $i$ has access to the function $f^{i}(x)$ only. Moreover, we assume the computing units/agents are connected over a network that allows for sparse communication between them. Thus, the proposed solution needs to be executed locally at each agent, using local information only, and achieve the convergence rate as if they had access to the complete dataset.
The key innovation in our solution is to provide a novel analysis for the inexact constrained cubic-regularized Newton method developed in baes2009estimate , to carefully control the errors induced by the disagreement among the nodes in the network, without sacrificing the convergence rate.
To summarize, the main contributions of this paper are as follows:
•
We propose a provably-correct (and globally convergent) distributed algorithm based on cubic-regularization. We take into account distributed storage and sparse communications and obtain a convergence rate of $O(k^{{-}3})$. To the best of the authors’ knowledge, this is the first, fully distributed cubic-regularized second-order method that achieves $O(k^{{-}3})$.
•
We characterize the communication complexity of the proposed algorithm and relate the corresponding approximation error, induced by the sparse communication, to guarantee the desired convergence rate.
•
We propose a primal-dual distributed method for the minimization of non-separable cubic-regularized second-order functions.
This paper is organized as follows. Section 2 introduces the distributed optimization problem and assumptions and presents the proposed algorithm and its convergence rate analysis. Section 3 describes the distributed approximate solution of the cubic model minimization. Section 6 shows some experimental results. Section 7 discusses open problems on high-order methods in distributed optimization. Finally, conclusions are presented in Section 8.
Notation: Nodes/agents are indexed from $1$ through $m$ (no actual enumeration is needed in the execution of the proposed algorithms). Superscripts $i$ or $j$ denote agent indices and the subscript $k$ denotes the iteration index of an algorithm. $[A]_{ij}$ denotes the entry of the matrix $A$ in its $i$-th row and $j$-th column. $\mathbf{I}_{n}$ denotes the identity matrix of size $n$. For a symmetric non-negative matrix $W$, $\lambda_{\max}(W)$ denotes its largest eigenvalue
and $\lambda_{\min}^{+}(W)$ its smallest positive eigenvalue. The condition number of $W$ is denoted as $\chi(W)=\lambda_{\max}{(W)}/\lambda_{\min}^{+}{(W)}$. The Euclidean norm is denoted as $\|\cdot\|$. $\boldsymbol{1}_{n}$ is a vector of ones of size $n$, $\otimes$ is the Kronecker product.
2 Problem Statement, Algorithm, and Main Result
Consider a network of $m$ agents, modeled as a fixed, connected, and undirected graph $\mathcal{G}{=}(V,E)$, where $V{=}(1,\cdots,m)$, and $E\subseteq V\times V$ is a set of edges such that $(j,i)\in E$ if and only if agent $j$ is connected to agent $i$. Agents try to jointly solve (1), but an agent $i\in V$ has access to $f^{i}(x)$, $\nabla f^{i}(x)$, and $\nabla^{2}f^{i}(x)$ only. However, agents are allowed to exchange information over the network $\mathcal{G}$ with its neighbors. We assume each $f^{i}:Q\to\mathbb{R}$ is convex with Lipschitz continuous gradient and Hessian, defined in a nonempty, convex, and compact set $Q\subset\mathbb{R}^{n}$. We further assume without loss of generality that $f$ attains its minimum $f^{*}$ in the interior of $Q$.
We can write (1) to introduce the graph $\mathcal{G}$ into the problem formulation sca17 ; lan17 ; Uribe2018a . Consider the Laplacian $W_{\mathcal{G}}\in\mathbb{R}^{m\times m}$ of the graph $\mathcal{G}$, defined as a matrix with entries $[W_{\mathcal{G}}]_{ij}{=}{-}1$ if $(j,i)\in E$, $[W_{\mathcal{G}}]_{ij}=\text{deg}(i)$ if $i{=}j$, and $[W_{\mathcal{G}}]_{ij}=0$ otherwise, where $\text{deg}(i)$ is the degree of the node $i$, i.e., the number of neighbors of the node. The matrix $W_{\mathcal{G}}$ is symmetric and positive semi-definite, and $\boldsymbol{1}_{m}$ is the unique (up to a scaling factor) eigenvector associated with the eigenvalue
$\lambda_{W}^{1}{=}0$. Thus, for a vector $z\in\mathbb{R}^{m}$ it holds that $W_{\mathcal{G}}{z}{=}0$ if and only if $z_{1}{=}\ldots{=}z_{m}$. If each agent holds a local copy $x^{i}\in\mathbb{R}^{n}$ of the decision variable, we obtain the optimization problem:
$$\displaystyle\min_{\begin{subarray}{c}\mathbf{x}\in Q^{m}\\
\sqrt{\mathbf{W}}\mathbf{x}{=}\boldsymbol{0}_{nm}\end{subarray}}\left\{F(%
\mathbf{x})\triangleq\sum_{i{=}1}^{m}f^{i}(x^{i})\right\},$$
(2)
where $\mathbf{W}\triangleq W_{\mathcal{G}}\otimes\mathbf{I}_{n}$ and $Q^{m}{=}\{\mathbf{x}\in\mathbb{R}^{nm}\mid\mathbf{x}^{\intercal}{=}[(x^{1})^{%
\intercal},\cdots,(x^{m})^{\intercal}]^{\intercal},x^{i}\in Q\ \forall i\in V\}$.
Problem $\eqref{eq:main2}$ is a reformulation of Problem $\eqref{eq:main1}$, as the constraint $\sqrt{\mathbf{W}}\mathbf{x}{=}\boldsymbol{0}_{m}$ implies $x^{1}{=}\cdots{=}x^{m}$. Thus, an optimal point of $\eqref{eq:main2}$ is such that $\mathbf{x}^{*}{=}\boldsymbol{1}_{m}\otimes x^{*}$, where $x^{*}$ is an optimal point of $\eqref{eq:main1}$.
Our goal is to find approximate distributed solutions to Problem $\eqref{eq:main2}$ defined as follows:
Definition 2.1 ((lan17, , Definition $1$)).
A point $\hat{\mathbf{x}}$ is called an $(\varepsilon,\tilde{\varepsilon})$-solution of (2) if $F(\hat{\mathbf{x}}){-}F^{*}\leq\varepsilon$, and $\|\sqrt{\mathbf{W}}\hat{\mathbf{x}}\|_{2}\leq\tilde{\varepsilon}$,
where $F^{*}$ denotes the optimal value of (2).
Additionally, we define an inexact solution of a constrained optimization problem as:
Definition 2.2.
We define a point $\hat{x}\approx_{\delta}\operatorname*{arg\,min}_{x\in Q}f(x)$ as a point in $X$ such that $f(\hat{x}){-}f^{*}\leq\delta$, where $f^{*}$ is the minimum value of the function $f(x)$ over the set $X$.
For analysis purposes we define the set $\mathcal{Q}_{\tilde{\varepsilon}}{=}\{\mathbf{x}\in\mathbb{R}^{nm}\mid\|\sqrt{%
\mathbf{W}}\hat{\mathbf{x}}\|_{2}\leq\tilde{\varepsilon}\}$, which will come handy in later sections. Furthermore, we assume the following conditions are satisfied.
Assumption 2.3 (Lipschitz gradient).
Each function $f^{i}(x)$ is differentiable and has $M^{i}_{1}$-Lipschitz continuous gradients over the set $Q$, i.e., for any $x,y\in Q$, $\|\nabla f^{i}(x){-}\nabla f^{i}(y)\|\leq M^{i}_{1}\|x{-}y\|$.
Assumption 2.4 (Lipschitz Hessian).
Each function $f^{i}(x)$ is twice differentiable and has $M^{i}_{2}$-Lipschitz continuous Hessian over the set $Q$, i.e., for any $x,y\in Q$, $\|\nabla^{2}f^{i}(x){-}\nabla^{2}f^{i}(y)\|\leq M^{i}_{2}\|x{-}y\|$. Note that $F(\mathbf{x})$ has $M_{1}$-Lipschitz gradient and $M_{2}$-Lipschitz Hessian, with $M_{1}{=}\max_{i\in V}M_{1}^{i}$, and $M_{2}{=}\max_{i\in V}M_{2}^{i}$.
Assumption 2.5.
The diameter of the compact set $Q$ is upper bounded by a constant $D_{Q}$, i.e., $\max_{x,y\in Q}\|x{-}y\|\leq D_{Q}$.
Next, we state our main result. In what follows we show that the distributed Algorithm in 1 guarantees that agents jointly construct a $(\varepsilon,\tilde{\varepsilon})$-solution to (2) with a convergence rate of $O(k^{-3})$. Algorithm 1 follows the same structure as the constrained cubic regularized Newton method proposed in baes2009estimate . We define the cubic regularized second order approximation of the function $F(\mathbf{x})$ at a point $\mathbf{z}$ as follows:
$$\displaystyle\hat{F}(\mathbf{z},\mathbf{x})\triangleq F(\mathbf{z}){+}\langle%
\nabla F(\mathbf{z}),\mathbf{x}{-}\mathbf{z}\rangle+\frac{1}{2}\langle\nabla^{%
2}F(\mathbf{z})(\mathbf{x}{-}\mathbf{z}),\mathbf{x}{-}\mathbf{z}\rangle{+}%
\frac{N}{6}\|\mathbf{x}{-}\mathbf{z}\|^{3}.$$
(3)
Theorem 2.6 (Main Result).
Let Assumptions 2.3, 2.4 and 2.5 hold, $\varepsilon>0$ be a desired accuracy, and $\gamma\in(0,1)$. Moreover, set the number of iterations Algorithm 1 to $K\geq\lceil{12\varepsilon^{-1/3}\left(F(\mathbf{x}_{0}){-}F^{*}{+}\frac{M_{2}}%
{6}\|\mathbf{x}_{0}{-}\mathbf{x}^{*}\|^{3}\right)^{1/3}}\rceil$, where $\mathbf{x}^{*}$ maximizes $R=O(\|\boldsymbol{x}_{0}-\boldsymbol{x}^{*}\|)$, and at every $k\geq 1$, set the accuracy of the auxiliary sub-problems (Lines $10$ and $13$) as
$$\displaystyle 0$$
$$\displaystyle\leq\delta^{\phi}_{k}\leq\min\left\{1,\left(\frac{\left(\frac{%
\alpha_{k}\gamma}{1{-}\alpha_{k}}\right)\varepsilon}{1{+}D_{Q}L_{0}\big{(}%
\frac{6\lambda_{k}^{2}}{\sigma_{0}}\big{)}^{1/3}}\right)^{3}\right\},\ \ 0\leq%
\delta^{F}_{k}\leq\min\left\{1,\left(\frac{\left((1{-}\gamma)\alpha_{k}{+}%
\frac{1}{2}\right)\varepsilon}{1{+}D_{Q}L_{0}\big{(}\frac{3}{\sigma_{0}}\big{)%
}^{1/3}}\right)^{3}\right\}.$$
Then, the output of Algorithm 1, i.e., $\mathbf{x}_{K}$, is an $(\varepsilon,\varepsilon/R)$- approximate solution of Problem (2).
Theorem 2.6 states that with the appropriate selection of inexactness of the subproblems in Algorithm 1, Lines $10$ and $13$, it is possible to obtain a fast convergence rate of $O(k^{-3})$ in a fully distributed manner. Section 3 shows that such an approximate solution can be computed in a distributed manner via Algorithm 2.
Proof Sketch (Theorem 2.6) In baes2009estimate , the authors provide an inexact cubic regularized Newton method with an oracle complexity of $O(\varepsilon^{-1/3})$ for constrained convex problems with Lipschitz Hessian.
We exploit the dual representation of the cubic terms to build a separable problem amenable to distributed computation. Thus, we bound the communication complexity of the algorithm by primal-dual analysis of the subproblems (Algorithm 1, Lines $10$, and $13$). Technically, we show that with appropriate selection of $\delta_{k}^{F}$ and $\delta_{k}^{\phi}$, if for some $k\geq 1$ it holds that $\min_{\mathbf{x}\in Q^{m}\bigcap\mathcal{Q}_{\tilde{\varepsilon}}}\phi_{k}(%
\mathbf{x})\geq F(\hat{\mathbf{x}}_{k})-\varepsilon$, then this also holds for $k+1$. Once the inexactness bounds are computed, the oracle complexity of Algorithm 1 follows from the analysis of the estimating sequences for the particular problem.$\blacksquare$
We recognize that the result on $O(\varepsilon^{-1/3})$ oracle calls obtained in Theorem 2.6 is not optimal. Second-order methods have been shown to have a lower complexity bound of $O(\varepsilon^{-2/7})$ agarwal2018adaptive ; monteiro2013accelerated . However, as pointed out in (Nesterov2018, , Section 4.3.3), the gain by achieving the optimal rate is bounded by a factor of $O(\varepsilon^{-1/21})$. Therefore, for values of $\varepsilon$ used in practical applications, e.g., $10^{-12}$, the gain is an absolute constant less than $4$. Nevertheless, from a conceptual point of view, getting near-optimal rates remains a valuable open problem. The main difficulty lies in the implementation of distributed line-search procedures, which is an open question in distributed optimization and out of the scope of this paper.
In the next section, we describe the technical details of the proposed approach for the approximate distributed minimization of a cubic regularized second-order model (3).
3 Distributed Approximate Minimization of Cubic functions
In this section, we study how Algorithm 2 approximately solves (c.f. Definition 2.2) a cubic regularized second-order approximation (3) in a distributed manner over a network. We focus on optimization problems of the form
$$\displaystyle\min_{\begin{subarray}{c}\mathbf{h}\in\mathcal{H}\subset\mathbb{R%
}^{nm}\\
A\mathbf{h}{=}\boldsymbol{0}_{nm}\end{subarray}}\left\{\boldsymbol{\Phi}(%
\mathbf{h}){\triangleq}\langle\mathbf{g},\mathbf{h}\rangle{{+}}\frac{1}{2}%
\langle\mathbf{H}\mathbf{h},\mathbf{h}\rangle{{+}}\frac{N}{6}\|\mathbf{h}\|^{3%
}\right\},$$
(4)
where $A$ is a generic matrix whose null space is the consensus subspace, i.e., $Ax{=}0\iff x_{i}{=}x_{j}$. Note the subproblems in Lines $10$ and $13$ of Algorithm 1, have the form (4).
Later in this section, we will see the specific details when for a set of points $(z^{1}_{k},\cdots,z^{m}_{k})$ where each $z^{i}_{k}$ is stored locally by an agent $i$, we have $\mathbf{g}^{\intercal}{=}[\mathbf{g}^{\intercal}_{1},\cdots,\mathbf{g}^{%
\intercal}_{m}]$, where $\mathbf{g}_{i}{=}\nabla f^{i}(z^{i}_{k})$ for $i\in V$, and $\mathbf{H}{=}{blkdiag}(\mathbf{H}_{1},\cdots,\mathbf{H}_{m})$111The function ${blkdiag}(A,\cdots,B)$ generates a block diagonal matrix whose elements are each of the input arguments. where $\mathbf{H}_{i}{=}\nabla^{2}f^{i}(z^{i}_{k})$ for $i\in V$, $\mathcal{H}{=}Q^{m}$, $A{=}\sqrt{\mathbf{W}}$ and $\mathbf{h}{=}\mathbf{x}{-}\mathbf{z}$.
Finding a distributed solution to separable problems with linear constraints has been extensively studied in recent literature, due to the flexibility of such an approach in incorporating limited storage and sparse computations gas16b ; lan17 ; sca17 ; Uribe2018a . However, the main requirement is for the cost function to be separable, i.e., write it as a finite sum of functions. This is true for the first two terms in (4) by construction (i.e., the linear and quadratic terms), where one can write $\langle\mathbf{g},\mathbf{h}\rangle{+}\frac{1}{2}\langle\mathbf{H}\mathbf{h},%
\mathbf{h}\rangle{=}\sum_{i{=}1}^{m}\mathbf{g}_{i}^{\intercal}\mathbf{h}_{i}{+%
}\frac{1}{2}\sum_{i{=}1}^{m}\mathbf{h}_{i}^{\intercal}\mathbf{H}_{i}\mathbf{h}%
_{i}$. Unfortunately, this is not the case for the cubic term $\|\mathbf{h}\|^{3}$.
Our first task is to exploit the dual structure of the cubic term in (4) to construct a surrogate cost function amenable to distributed optimization algorithms. We provide this dual structure for the cubic term in the next proposition.
Proposition 3.1.
Given some $\mathbf{x}^{\intercal}{=}[\mathbf{x}^{\intercal}_{1},\cdots,\mathbf{x}^{%
\intercal}_{m}]$ where $\mathbf{x}_{i}\in\mathbb{R}^{n}$ for all $i\in V$. Then,
$$\displaystyle\frac{1}{3}\|\mathbf{x}\|^{3}$$
$$\displaystyle{=}\max_{\begin{subarray}{c}{\tau}_{i}\geq 0,\ \tau_{i}{=}\tau_{j%
}\\
i,j\in V\end{subarray}}\left\{\sum_{i{=}1}^{m}\|\mathbf{x}_{i}\|^{2}\tau_{i}{-%
}\frac{4}{3m}\sum_{i{=}1}^{m}\tau_{i}^{3}\right\}.$$
Proof.
Projecting on the consensus subspace where $\tau_{i}{=}\tau_{j}$, we have $\sum_{i{=}1}^{m}\|\mathbf{x}_{i}\|^{2}\bar{\tau}{-}(4/3)\bar{\tau}^{3}$,
and by first order optimality conditions $\sum_{i{=}1}^{m}\|\mathbf{x}_{i}\|^{2}{-}4\bar{\tau}^{2}{=}0$. Solving for $\bar{\tau}$ completes the proof.
∎
With Proposition 3.1 at hand, we can rewrite (4) as
$$\displaystyle\min_{\begin{subarray}{c}\mathbf{h}\in\mathcal{H}\subset\mathbb{R%
}^{nm}\\
A\mathbf{h}{=}\boldsymbol{0}_{nm}\end{subarray}}\max_{\begin{subarray}{c}{\tau%
}_{i}\geq 0\ i\in(1,\cdots,m)\\
B\tau{=}\boldsymbol{0}_{m}\end{subarray}}\left\{\langle\mathbf{g},\mathbf{h}%
\rangle{+}\frac{1}{2}\langle\mathbf{H}\mathbf{h},\mathbf{h}\rangle{+}\frac{N}{%
2}\sum_{i{=}1}^{m}\|\mathbf{h}_{i}\|^{2}\tau_{i}{-}\frac{2N}{3m}\sum_{i{=}1}^{%
m}\tau_{i}^{3}\right\},$$
(5)
where, we have written the consensus constraints on $\tau_{i}$ by introducing a vector $\tau{=}[\tau_{1},\cdots,\tau_{m}]$ and a generic matrix $B\in\mathbb{R}^{m\times m}$ with $B\tau{=}0\iff\tau_{i}{=}\tau_{j}$ for $i\in(1,\cdots,m)$.
First, in the next lemma we show that the constraint $B\tau{=}\boldsymbol{0}_{m}$ is not required in (5), as the structure of the problem will guarantee a feasible optimal point in $A\mathbf{h}{=}\boldsymbol{0}_{nm}$ will also be in $B\tau{=}\boldsymbol{0}_{m}$. This will simplify the analysis for the design of the distributed approximate solver of (4).
Lemma 3.2.
An optimal solution pair $(\mathbf{h}^{*},\tau^{*})$ of
$$\displaystyle\min_{\begin{subarray}{c}\mathbf{h}\in\mathcal{H}\subset\mathbb{R%
}^{nm}\\
A\mathbf{h}{=}\boldsymbol{0}_{nm}\end{subarray}}\max_{\begin{subarray}{c}{\tau%
}_{i}\geq 0\\
i\in(1,\cdots,m)\end{subarray}}\left\{\langle\mathbf{g},\mathbf{h}\rangle{+}%
\frac{1}{2}\langle\mathbf{H}\mathbf{h},\mathbf{h}\rangle{+}\frac{N}{2}\sum_{i{%
=}1}^{m}\|\mathbf{h}_{i}\|^{2}\tau_{i}{-}\frac{2N}{3m}\sum_{i{=}1}^{m}\tau_{i}%
^{3}\right\},$$
(6)
is also an optimal pair for (5).
Proof.
We can build the Lagrangian function of (5) as
$$\displaystyle\max_{\mathbf{y}}\min_{\mathbf{h}\in\mathcal{H}}\min_{\boldsymbol%
{\eta}}\max_{\begin{subarray}{c}{\tau}_{i}\geq 0\\
i\in(1,\cdots,m)\end{subarray}}\left\{\langle\mathbf{g},\mathbf{h}\rangle{+}%
\frac{1}{2}\langle\mathbf{H}\mathbf{h},\mathbf{h}\rangle{+}\frac{N}{2}\sum_{i{%
=}1}^{m}\|\mathbf{h}_{i}\|^{2}\tau_{i}{-}\frac{2N}{3m}\sum_{i{=}1}^{m}\tau_{i}%
^{3}{-}\langle\mathbf{y},A\mathbf{h}\rangle{+}\langle\boldsymbol{\eta},B%
\boldsymbol{\tau}\rangle\right\}.$$
Thus, the first order optimality conditions are
$$\displaystyle A\mathbf{h}$$
$$\displaystyle{=}0,$$
(7a)
$$\displaystyle B\boldsymbol{\tau}$$
$$\displaystyle{=}0,$$
(7b)
$$\displaystyle\mathbf{g}{+}\left(\mathbf{H}{+}M\mathbf{T}\right)\mathbf{h}{-}A^%
{\intercal}\mathbf{y}$$
$$\displaystyle{=}0,$$
(7c)
$$\displaystyle\frac{n}{4}\|\mathbf{h}_{i}\|^{2}{-}\tau_{i}^{2}{+}A^{\intercal}%
\boldsymbol{\eta}$$
$$\displaystyle{=}0.$$
(7d)
Initially, note that (7a) and (7b) guarantee that all entries of both $\mathbf{h}$ and $\boldsymbol{\tau}$ are equal respectively. This fact, along side (7d) implies that all the entries of $A^{\intercal}\boldsymbol{\eta}$ are equal as well. Thus, it follows that $A^{\intercal}\boldsymbol{\eta}{=}\alpha\boldsymbol{1}$
for some value of $\alpha$. It is enough to show that this is true if and only if $\alpha{=}0$.
If $\alpha{=}0$, then it implies that all entries of $\boldsymbol{\eta}$ are equal to each other, and the solution to both problems are equivalent. Now assume $\alpha\neq 0$. Initially, we can write the matrix $A$ as $A{=}V\Lambda V^{\intercal}$ as its eigenvalue decomposition. Thus,
$$\displaystyle V\Lambda V^{\intercal}\boldsymbol{\eta}{=}\alpha\boldsymbol{1}%
\quad\text{and},\quad\Lambda V^{\intercal}\boldsymbol{\eta}{=}\alpha V%
\boldsymbol{1}.$$
Given that the vector $\boldsymbol{1}$ is the corresponding eigenvector for the eigenvalue $\Lambda_{1,1}{=}0$, it follows that $\Lambda V^{\intercal}\boldsymbol{\eta}{=}\alpha\boldsymbol{e}_{1}$
where $\boldsymbol{e}_{1}$ is the zeroes vector with entry $1$ in its position $i{=}1$. Which implies that $\Lambda_{1,1}[V^{\intercal}\boldsymbol{\eta}]_{1}{=}\alpha$, and since $\Lambda_{1,1}{=}0$, the only solution is $\alpha{=}0$, which is a contradiction.
∎
Now, we are ready to focus on the design of a distributed algorithm for Problem (6). First, we can define the Lagrangian dual function, for the consensus constraints in $\mathbf{h}$ as
$$\displaystyle\varphi(\mathbf{y})$$
$$\displaystyle{=}\min_{\mathbf{h}\in\mathcal{H}}\max_{\begin{subarray}{c}{\tau}%
_{i}\geq 0\\
i\in(1,\cdots,m)\end{subarray}}\left\{\langle\mathbf{g},\mathbf{h}\rangle{+}%
\frac{1}{2}\langle(\mathbf{H}{+}N\mathbf{T}\mathbf{h},\mathbf{h}\rangle{-}%
\frac{2N}{3m}\sum_{i{=}1}^{m}\tau_{i}^{3}{-}\langle\mathbf{y},A\mathbf{h}%
\rangle\right\},$$
where $\mathbf{T}{=}blkdiag(\tau_{1}\mathbf{I}_{n},\cdots,\tau_{m}\mathbf{I}_{n})$, and the dual problem is defined as $\max_{\mathbf{y}}\varphi(\mathbf{y})$.
The dual problem has a number of important properties whose structure we can exploit. For example, since the function $F$ has $M_{1}$-Lipschitz gradient, then it follows that the dual function $\varphi(\mathbf{y})$ is $\mu_{\varphi}$-strongly convex on $\ker(A^{T})^{\perp}$ where $\mu_{\varphi}{=}(\lambda^{+}_{\text{min}}(A^{\intercal}A)/M_{1})$ (Beck2014, , Lemma $3.1$), (rockafellar2011variational, , Proposition $12.60$), (nes05, , Theorem $1$), (Kakade2009a, , Theorem $6$).
Moreover, it follows from Demyanov-Danskin’s theorem (Bertsekas2003, , Proposition $4.5.1$),
that $\nabla\varphi(\mathbf{y}){=}A\mathbf{h}^{*}(A^{T}\mathbf{y})$ where $\mathbf{h}^{*}(A^{T}y)$ denotes the unique solution
of the inner maximization problem
$$\displaystyle\mathbf{h}^{*}(A^{\intercal}\mathbf{y}){=}\operatorname*{arg\,min%
}_{\mathbf{h}\in\mathcal{H}}\max_{\begin{subarray}{c}{\tau}_{i}\geq 0\\
i\in(1,\cdots,m)\end{subarray}}\left\{\langle\mathbf{g},\mathbf{h}\rangle{+}%
\frac{1}{2}\big{\langle}\left(\mathbf{H}{+}N\mathbf{T}\right)\mathbf{h},%
\mathbf{h}\big{\rangle}{-}\frac{2N}{3m}\sum_{i{=}1}^{n}\tau_{i}^{3}{-}\langle%
\mathbf{y},A\mathbf{h}\rangle\right\}.$$
(8)
3.1 Primal-Dual Properties for Distributed Implementation over Networks
At this point, we observe some properties that make the reformulation (6) amenable for a distributed implementation over a network.
The solution of (8) can be computed using local information only at each node, i.e., $\mathbf{h}^{*}(A^{T}\mathbf{y})^{\intercal}{=}[\mathbf{h}^{*}_{1}([A^{T}%
\mathbf{y}]_{1})^{\intercal},\cdots,\mathbf{h}^{*}_{m}([A^{T}\mathbf{y}]_{m})^%
{\intercal}]$ where
$$\displaystyle\mathbf{h}^{*}_{i}([A^{\intercal}\mathbf{y}]_{i}){=}\operatorname%
*{arg\,min}_{\bar{\mathbf{h}}\in\bar{\mathcal{H}}}\max_{{\tau}_{i}\geq 0}\left%
\{\langle\mathbf{g}_{i},\bar{\mathbf{h}}\rangle{-}\frac{2N}{3m}\tau_{i}^{3}{+}%
\frac{1}{2}\big{\langle}\left(\mathbf{H}_{i}{+}N\tau_{i}\mathbf{I}_{n}\right)%
\bar{\mathbf{h}},\bar{\mathbf{h}}\big{\rangle}{-}\langle\mathbf{\mathbf{y}_{i}%
},[A\bar{\mathbf{h}}]_{i}\rangle\right\},$$
(9)
with the set $\bar{\mathcal{H}}$ is the corresponding marginal set for a single agent only.
The gradient $A\mathbf{h}^{*}(A^{T}\mathbf{y})$ can be computed distributively if the matrix $A$ has the same sparsity pattern as the network. Suppose that $[A]_{ij}\neq 0$ if and only if $(j,i)\in E$. Then, each entry $[A\mathbf{h}^{*}(A^{T}\mathbf{y})]_{i}$ to be used by an agent $i\in V$, corresponds to a weighted sum of the $\mathbf{h}^{*}_{j}([A^{T}\mathbf{y}]_{j})$ for all other nodes $j\in V$ such that $(j,i)\in E$. That is, the information an agent requires to take gradient steps is available to him via network communications. The dual function gradient computation corresponds to a communication round over the network.
Recall that a function is called dual-friendly (Uribe2018a, , Definition $2$), if we can “efficiently” compute (in a closed form or by polynomial time algorithms) a solution to (9). In this subsection, we show that our cubic regularized second-order approximation (3) is indeed dual-friendly.
Initially, let us write the optimality conditions of $\eqref{eq:prop1}$. The optimal point $\mathbf{h}^{*}_{i}([A^{T}\mathbf{y}]_{i})$ is a solution to the following systems of nonlinear equations $\mathbf{g}_{i}{+}\left(\mathbf{H}_{i}{+}N\tau_{i}\mathbf{I}_{n}\right)\mathbf{%
h}_{i}{-}[A\bar{\mathbf{h}}]_{i}$ and ${m}/{4}\|\mathbf{h}_{i}\|^{2}{-}\tau_{i}^{2}{=}0$. It follows that $\mathbf{h}_{i}{=}\left(\mathbf{H}_{i}{+}N\tau_{i}\mathbf{I}_{n}\right)^{{-}1}%
\left([A\bar{\mathbf{h}}]_{i}{-}\mathbf{g}_{i}\right)$. Moreover, suppose that the matrix $\mathbf{H}_{i}$ has an eigendecomposition $\mathbf{H}_{i}{=}U_{i}^{\intercal}\Lambda_{i}U_{i}$,
where $\Lambda_{i}$ is a diagonal matrix of eigenvalues $s_{1}\leq\ldots\leq s_{d}$ and $U_{i}$ is an orthonormal matrix of associated eigenvectors. Then $\mathbf{h}_{i}{=}U_{i}^{\intercal}\left(\Lambda_{i}{+}N\tau_{i}\mathbf{I}_{n}%
\right)^{{-}1}U_{i}\left([A\bar{\mathbf{h}}]_{i}{-}\mathbf{g}_{i}\right)$. Furthermore, we have $\|\mathbf{h}_{i}\|^{2}{=}\|U_{i}^{\intercal}\left(\Lambda_{i}{+}N\tau_{i}%
\mathbf{I}_{n}\right)^{{-}1}U_{i}\left([A\bar{\mathbf{h}}]_{i}{-}\mathbf{g}_{i%
}\right)\|^{2}$ and $\|\mathbf{h}_{i}\|^{2}{=}\sum_{j{=}1}^{d}{\gamma_{j}^{2}}/{(s_{j}{+}N\tau_{i})%
^{2}}$, where $\gamma_{j}=[U_{i}\left([A\bar{\mathbf{h}}]_{i}{-}\mathbf{g}_{i}\right)]_{j}$. Therefore, each agent needs to solve the following nonlinear equation: ${\sqrt{m}}/{2}\|\mathbf{h}_{i}\|{-}\tau_{i}{=}0$.
However, conn2000trust suggest that a simpler approach is to solve the secular equation $({2}/{\sqrt{m}}){1}/{\|\mathbf{h}_{i}\|}{-}{1}/{\tau_{i}}{=}0$.
A comprehensive account of how to efficiently solve the above equation can be found in (conn2000trust, , Chapter $7$, Algorithm $7.3.1$), or in carmon2018analysis . Thus, we assume that each agent can locally and efficiently find a solution.
The dual function $\varphi(\mathbf{y})$ is strongly convex on a defined subspace, but it is non-smooth. One can use traditional approaches for non-smooth minimizationgoffin1977convergence ; shor2012minimization . However, we make the design choice of exploiting the max structure of the function by using Nesterov’s dual smoothing approach nes05 which has been shown optimal for the problem class of non-smooth minimization, specially for dual-friendly problems. To do so, we define a regularized problem
$$\displaystyle\hat{\varphi}(\mathbf{y})$$
$$\displaystyle{=}\min_{\mathbf{h}\in\mathcal{H}}\max_{\begin{subarray}{c}{\tau}%
_{i}\geq 0\\
i\in(1,\cdots,m)\end{subarray}}\left\{\langle\mathbf{g},\mathbf{h}\rangle{+}%
\frac{1}{2}\langle(\mathbf{H}{+}N\mathbf{T}\mathbf{h},\mathbf{h}\rangle{-}%
\frac{2N}{3m}\sum_{i{=}1}^{m}\tau_{i}^{3}{-}\langle\mathbf{y},A\mathbf{h}%
\rangle{+}\frac{\hat{\mu}}{2}\|\mathbf{h}\|^{2}\right\},$$
(10)
where we have added a quadratic term to our cost function to induce smoothness in the dual space. Moreover, an appropriate selection of $\hat{\mu}$ can provide bounds that relate to the original non-regularized function, see (Uribe2018a, , Proposition $5.2$), and (gas16b, , Lemma $3$). Particularly, if $\hat{\mu}\leq\delta/(2R^{2})$, where $R{=}\|\mathbf{h}^{*}(A^{\intercal}\mathbf{y}^{*})\|$, and $\mathbf{y}^{*}$ denotes the smallest norm solution of the non regularized problem, and $\delta>0$ is the desired accuracy. Then, an approximate solution point $\hat{\mathbf{y}}$ such that $\hat{\varphi}(\hat{\mathbf{y}}){-}\hat{\varphi}^{*}\leq\delta/2$ implies $\varphi(\hat{\mathbf{y}}){-}\varphi^{*}\leq\delta$, where $\hat{\varphi}^{*}$ and $\varphi^{*}$ are the optimal values of the regularized and non-regularized functions respectively. Therefore, the smoothed dual function $\hat{\varphi}(\mathbf{y})$ is $\mu_{\hat{\varphi}}$-strongly concave and has $M_{\hat{\varphi}}$-Lipschitz continuous gradients, where $\mu_{\hat{\varphi}}{=}{\lambda_{\min}^{{+}}(A^{\intercal}A)}/{(L_{0}{+}\hat{%
\mu})}$ and $M_{\hat{\varphi}}{=}{\lambda_{\max}(A^{\intercal}A)}/{\hat{\mu}}$. Having a strongly convex function with Lipschitz gradient allows for the use of traditional Fast Gradient Methods nes13 . More importantly, this regularization approach do not affect the decentralization properties.
3.2 Communication Complexity of the Cubic Approximate Solver
In this subsection, build upon recently develop dual-based optimal algorithms for dual-friendly functions Uribe2018a to provide an approximate solution to the auxiliary Subproblem (10).
Theorem 3.3.
Let Assumptions 2.3, and 2.4 hold. For any $\delta>0$, set the number of iterations in Algorithm 2 as
$$\displaystyle T{\geq}2\sqrt{\left(\frac{2M_{1}R^{2}_{\varphi}}{\delta}{{+}}1%
\right)\chi(W)}\log\left(\frac{8\sqrt{2}\lambda_{{\max}}(W)R^{2}_{\varphi}R_{%
\mathbf{h}}^{2}}{\delta^{2}}\right),$$
where $\chi(W){=}\lambda_{\max}{(W)}/\lambda_{\min}^{{{+}}}{(W)}$, $R_{\mathbf{h}}=O(\|\mathbf{h}^{*}-\mathbf{h}^{*}(0)\|)$, $R_{\varphi}=O(\|\mathbf{w}^{*}\|)$ are bounds on the distance to the optimal solution and the initial point for the primal and dual variables. Then, the output $\mathbf{h}^{*}(\tilde{\mathbf{w}}_{T})$ of Algorithm 2 is an $(2\delta,\delta/R_{\varphi})$-approximate solution of (4).
The result in Theorem 3.3 shows that $\tilde{O}(\sqrt{\chi(W)/\delta})$ communication rounds on the network are needed to reach an $(\delta,\delta/R)$ approximate solution of (3). Moreover, this can be done in a fully distributed manner.
Remark 3.4.
Note that we have followed one particular approach in Uribe2018a to solve the smooth inner problem. However, there are other algorithms with similar convergence rate guarantees, for example, lan2018communication ; ye2020multi ; li2018sharp . It follows from (nesterov2015universal, , Lemma 1), (dvurechensky2017gradient, , Lemma 1), or (bauschke2011convex, , Corollary 18.14) that uniform convexity of the function $\boldsymbol{\Phi}(\cdot)$ implies Hölder continuity of the dual function $\varphi(\cdot)$, with order $\nu=1/2$ and parameter $M_{\nu}=1/\sqrt{\sigma_{0}}$. Therefore, one can use more sophisticated methods yashtini2016global to improve the communication complexity for the solution of the sub-problem (4). For example, the recently proposed Universal Intermediate Gradient Method Kamzolov2020 .
4 Proof of Theorem 2.6: Inexactness in the Estimate Sequence Approach for Cubic Regularization
Our goal in this section is to prove Theorem 2.6, we extend the results of estimate sequences of Baes baes2009estimate to take into account inexactness coming from approximate solutions of the auxiliary subproblems and provide a communication complexity to Algorithm 1. To do so, we start with Algorithm 3, which is a modified version of Baes’ Cubic Regularized method. The main difference between Algorithm 3 and Baes’ constrained cubic regularized Newton’s method (baes2009estimate, , Algorithm 4.1) is that we define inexactness in both subproblems according to Definition 2.2. That is, in terms of distance to optimality measured by function value. We are allowed to make such analysis due to the specific structure induced by the problems we are required to solve and the algorithms we have available for computing such an approximate solution.
The idea of estimate sequences was first introduced by nes83 ; nes05 and later extended in Auslender2006 . Baes baes2009estimate shed some light on the use of estimate sequences for the design of high-order optimization algorithms that generalized first-order methods. We follow the estimate sequence approach in baes2009estimate to prove the convergence rate properties of Algorithm 1.
For simplicity of notation, we will consider the generic problem $\min_{x\in Q}f(x)$ for a compact, convex and bounded set, and a convex function $f$ with $M_{1}$-Lipschitz gradient and $M_{2}$-Lipschitz Hessian. To do so, we provide a slightly modified cubic regularized Newton method based on estimated sequences, introduced in Algorithm 3. Later on, we will provide the specific result Problem (2).
Note that Algorithm 3 is different from the cubic regularized Newton method proposed in baes2009estimate because it is stated in terms of function value suboptimality in both the subproblems.
Recall a couple of definitions and properties for estimate sequences.
Definition 4.1 (Chapter $2$ in nes13 ).
An estimate sequence for the function $f(x)$ is a sequence of convex functions $(\phi_{k})_{k\geq 0}$ and a sequence of positive numbers $(\lambda_{k})_{k\geq 0}$ satisfying: $\lim_{k\to 0}\lambda_{k}{{=}}0$ and $\phi_{k}(x)\leq(1{{-}}\lambda_{k})f(x){{+}}\lambda_{k}\phi_{0}(x)$ ,
for all $x\in Q$ for $k\geq 1$.
Estimate sequences provide an understanding of the convergence rate of a sequence of iterates generated by some arbitrary algorithm, as described in the next proposition.
Proposition 4.2 (Adapted from Proposition $2.1$ in baes2009estimate ).
Suppose that a sequence of iterates $(x_{k})_{k\geq 0}$ in $Q$ satisfies $f(x_{k})-\varepsilon\leq\min_{x\in Q}\phi_{k}(x)$, and $\varepsilon>0$. Then, $f(x_{k}){-}f^{*}\leq\lambda_{k}(\phi_{0}(x^{*}){-}f^{*})+\varepsilon$ for $k\geq 1$.
Proof.
It follows from the Definition 4.1 that
$$\displaystyle f(x_{k}){-}\varepsilon$$
$$\displaystyle\leq\min_{x\in Q}\phi_{k}(x)\leq\min_{x\in Q}f(x){+}\lambda_{k}(%
\phi_{0}(x){-}f(x))$$
$$\displaystyle\leq f^{*}+\lambda_{k}(\phi_{0}(x^{*}){-}f(x^{*})).$$
∎
For example, if Assumption 2.4 holds, one useful way to construct an estimate sequence is:
$$\displaystyle\phi_{0}(x)$$
$$\displaystyle{=}f(x_{0}){+}{M_{2}}/{6}\|x{-}x_{0}\|^{3},\quad\text{and}\quad%
\phi_{k}(x){=}(1{-}\alpha_{k})\phi_{k}{+}\alpha\left(f(y_{k}){+}\langle\nabla f%
(y_{k}),x{-}y_{k}\rangle\right)$$
(11)
for a given starting point $x_{0}\in Q$ and an appropriate choice of $(\alpha_{k})_{k\geq 0}$ and $(y_{k})_{k\geq 0}$. Additionally, $\lambda_{0}=1$, and $\lambda_{k+1}=\lambda_{k}(1{-}\alpha_{k})$ for a sequence $(\alpha_{k})_{k\geq 0}$ whose sum diverges.
Proposition 4.2 provides an insight, which as pointed out in baes2009estimate , indicates that one key element in the use of estimate sequences is for an algorithm to be able to construct a sequence $(x_{k})_{k\geq 0}$ for which $f(x_{k})\leq\min_{x\in Q}\phi_{k}(x)$ holds or $f(x_{k})-\varepsilon\leq\min_{x\in Q}\phi_{k}(x)$ in the inexact case.
Proposition 4.3.
Let $\boldsymbol{\Phi}(\mathbf{h})$ be defined in (4), then it holds that for all $x,y\in\mathcal{H}$
$$\displaystyle\frac{L_{0}+M_{1}}{2}\|y-x\|^{2}\geq\boldsymbol{\Phi}(y)-%
\boldsymbol{\Phi}(x)-\langle\boldsymbol{\Phi}^{\prime}(x),y-x\rangle\geq\frac{%
\sigma_{0}}{3}\|y-x\|^{3},$$
(12)
where $L_{0}=M_{2}D_{Q}$, and $\sigma_{0}=M_{2}/{6}$.
Proof.
From (baes2009estimate, , Lemma 8.2) with $p=3$ in our case, we have that
$$\displaystyle\phi(x)=\|x-x_{0}\|^{p}$$
then
$$\displaystyle\phi(y)-\phi(x)-\langle\phi^{\prime}(x),y-x\rangle\geq c_{p}\|x-y%
\|^{p}$$
where
$$\displaystyle c_{p}=\frac{p-1}{\big{(}(2p-3)^{\frac{1}{p-2}}+1\big{)}^{p-2}}.$$
Moreover, from (baes2009estimate, , Lemma 5.1) we have
$$\displaystyle\phi_{0}(x)=f(x_{0})+\frac{M}{p!}\|x-x_{0}\|^{p}$$
then
$$\displaystyle L_{0}\|y-x\|^{2}\geq\langle\phi^{\prime}_{0}(y)-\phi^{\prime}_{0%
}(x),y-x\rangle\geq\sigma_{0}\|y-x\|^{p},$$
where
$$\displaystyle L_{0}=M\frac{D_{Q}^{p-2}}{(p-2)!}\qquad\text{and}\qquad\sigma_{0%
}=2M\frac{c_{p}}{p!}.$$
Finally, it follows from (4), that $\boldsymbol{\Phi}(x)$ is uniformly strongly convex of order $p=3$. Moreover, since we assume $F(x)$ is $M_{1}$ smooth, we have that $\boldsymbol{\Phi}(x)$ is $M_{1}+L_{0}$ smooth.
∎
Next, we show that Algorithm 3 builds an estimate sequence, and furthermore, one can appropriately chose the accuracy of each of the subproblems, such that the error does not accumulate and we can apply Proposition 4.2 for the convergence rate analysis. In particular, and considering Problem (2), following a construction of an estimate sequence as suggested in (11), we seek to inductively prove for the output sequence $(\mathbf{x}_{k})_{k\geq 0}$ of Algorithm 1, if $\min_{\mathbf{x}\in Q^{m}\bigcap\mathcal{Q}_{\tilde{\varepsilon}}}\phi_{k}(%
\mathbf{x})\geq F(\mathbf{x}_{k})-\varepsilon$, then $\min_{\mathbf{x}\in Q^{m}\bigcap\mathcal{Q}_{\tilde{\varepsilon}}}\phi_{k+1}(%
\mathbf{x})\geq F(\mathbf{x}_{k+1})-\varepsilon$.
The next lemma provides bounds for the accuracy of solving each of the subproblems in Algorithm 3 such that we can apply Proposition 4.2.
Lemma 4.4.
Let Assumptions 2.3, 2.4 and 2.5 hold. Let $\varepsilon>0$ and $\gamma\in(0,1)$, and assume that for a fixed $k\geq 0$ and points $\hat{x}_{k},\hat{\nu}_{k}\in Q$: $\min_{x\in Q}\phi_{k}(x)\geq f(\hat{x}_{k})-\varepsilon$,
with
$$\displaystyle 0$$
$$\displaystyle\leq\delta^{\phi}_{k}\leq\min\left\{1,\left(\frac{\left(\frac{%
\alpha_{k}\gamma}{1{-}\alpha_{k}}\right)\varepsilon}{1{+}D_{Q}L_{0}\big{(}%
\frac{6\lambda_{k}^{2}}{\sigma_{0}}\big{)}^{1/3}}\right)^{3}\right\},\qquad 0%
\leq\delta^{f}_{k}\leq\min\left\{1,\left(\frac{\left((1{-}\gamma)\alpha_{k}{+}%
\frac{1}{2}\right)\varepsilon}{1{+}D_{Q}L_{0}\big{(}\frac{3}{\sigma_{0}}\big{)%
}^{1/3}}\right)^{3}\right\}.$$
Then $\min_{x\in Q}\phi_{k+1}(x)\geq f(\hat{x}_{k+1})-\varepsilon$.
Proof.
Initially, by definition of the sequence $(\phi_{k}(x))_{k\geq 0}$ in (11),
$$\displaystyle\min_{x\in Q}\phi_{k+1}(x)=\min_{x\in Q}\{(1-\alpha_{k})\phi_{k}(%
x)+\alpha_{k}\big{(}f(\hat{x}_{k+1})+\langle\nabla f(\hat{x}_{k+1}),x-\hat{x}_%
{k+1}\rangle\big{)}\}$$
$$\displaystyle\geq\min_{x\in Q}\{(1-\alpha_{k})\big{(}\phi_{k}(\hat{\nu}_{k})+%
\langle\nabla\phi_{k}(\hat{\nu}_{k}),x-\hat{\nu}_{k}\rangle+\lambda_{k}\frac{M%
_{2}}{6}\|x-\hat{\nu}_{k}\|^{3}\big{)}+$$
$$\displaystyle\qquad+\alpha_{k}\big{(}f(\hat{x}_{k+1})+\langle\nabla f(\hat{x}_%
{k+1}),x-\hat{x}_{k+1}\rangle\big{)}\}$$
$$\displaystyle=\min_{x\in Q}\{(1-\alpha_{k})\big{(}\phi_{k}(\hat{\nu}_{k})+%
\langle\nabla\phi_{k}(\hat{\nu}_{k}),x-\hat{\nu}_{k}\rangle\big{)}+\lambda_{k+%
1}\frac{M_{2}}{6}\|x-\hat{\nu}_{k}\|^{3}+$$
$$\displaystyle\qquad\alpha_{k}\big{(}f(\hat{x}_{k+1})+\langle\nabla f(\hat{x}_{%
k+1}),x-\hat{x}_{k+1}\rangle\big{)}\},$$
(13)
where the inequality in the second line follows from Lemma $8.2$ in baes2009estimate , that shows that $\phi_{k}(y)\geq\phi_{k}(x)+\langle\nabla\phi_{k}(x),y-x\rangle+\lambda_{k}{M_{%
2}}/{6}\|y-x\|^{3}$ for all $x,y\in Q$ and $k\geq 0$, and the last equality from the definition of $\lambda_{k+1}$.
Now, we focus on bounding the term $\langle\nabla\phi_{k}(\hat{\nu}_{k}),x-\hat{\nu}_{k}\rangle$. Initially, by adding and subtracting $\nabla\phi_{k}({\nu}_{k})$ we have
$$\displaystyle\min_{x\in Q}\langle\nabla\phi_{k}(\hat{\nu}_{k}),x-\hat{\nu}_{k}%
\rangle=\min_{x\in Q}\langle\nabla\phi_{k}({\nu}_{k}),x-\hat{\nu}_{k}\rangle-%
\langle\nabla\phi_{k}({\nu}_{k})-\nabla\phi_{k}(\hat{\nu}_{k}),x-\hat{\nu}_{k}\rangle$$
$$\displaystyle\qquad\geq\min_{x\in Q}\{\langle\nabla\phi_{k}({\nu}_{k}),x-\hat{%
\nu}_{k}\rangle-\|\nabla\phi_{k}(\hat{\nu}_{k})-\nabla\phi_{k}({\nu}_{k})\|\|x%
-\hat{\nu}_{k}\|\}$$
$$\displaystyle\qquad\geq\min_{x\in Q}\{\langle\nabla\phi_{k}({\nu}_{k}),x-{\nu}%
_{k}\rangle+\langle\nabla\phi_{k}({\nu}_{k}),\nu_{k}-\hat{\nu}_{k}\rangle-\|%
\nabla\phi_{k}(\hat{\nu}_{k})-\nabla\phi_{k}({\nu}_{k})\|\|x-\hat{\nu}_{k}\|\},$$
where the first inequality follows from Cauchy–Schwarz inequality, and the second one by adding and subtracting $\nu_{k}$. Next, given that the function $\phi_{0}$ has Lipschitz gradients with constant $M_{2}D_{Q}$ (see Lemma $5.1$ in baes2009estimate ) where $D_{Q}$ is the diameter of the set $Q$, it holds that
$$\displaystyle\min_{x\in Q}\langle\nabla\phi_{k}(\hat{\nu}_{k}),x-\hat{\nu}_{k}\rangle$$
$$\displaystyle\geq\min_{x\in Q}\{\langle\nabla\phi_{k}({\nu}_{k}),x-{\nu}_{k}%
\rangle+\langle\nabla\phi_{k}({\nu}_{k}),\nu_{k}-\hat{\nu}_{k}\rangle-M_{2}D_{%
Q}\lambda_{k}\|\hat{\nu}_{k}-\nu_{k}\|D_{Q}\},$$
$$\displaystyle\qquad\geq\langle\nabla\phi_{k}({\nu}_{k}),\nu_{k}-\hat{\nu}_{k}%
\rangle-M_{2}D_{Q}\lambda_{k}\|\hat{\nu}_{k}-\nu_{k}\|D_{Q}$$
$$\displaystyle\qquad\geq\phi_{k}(\nu_{k})-\phi_{k}(\hat{\nu}_{k})+\lambda_{k}%
\frac{M_{2}}{6}\|\nu_{k}-\hat{\nu}_{k}\|^{3}-M_{2}D_{Q}\lambda_{k}\|\hat{\nu}_%
{k}-\nu_{k}\|D_{Q},$$
where the second inequality follows from the constrained optimality conditions for the function $\phi_{k}$, recall that $\nu_{k}$ is defined as the minimizer of $\phi_{k}$ on $Q$. Thus, the first-order optimality condition reads as $\langle\nabla\phi_{k}({\nu}_{k}),x-\hat{\nu}_{k}\rangle\geq 0$ for all $x\in Q$. The third inequality follows again from (baes2009estimate, , Lemma $8.2$).
Assuming the accuracy of the approximate solution $\hat{\nu}_{k}$ is such that $\phi_{k}(\hat{\nu}_{k})-\phi_{k}(\nu_{k})\leq\delta^{\phi}_{k}$. Then,
$$\displaystyle\min_{x\in Q}\langle\nabla\phi_{k}(\hat{\nu}_{k}),x-\hat{\nu}_{k}\rangle$$
$$\displaystyle\geq-\delta^{\phi}_{k}-M_{2}D_{Q}\lambda_{k}\|\hat{\nu}_{k}-\nu_{%
k}\|D_{Q},$$
where we have removed the positive term in the upper bound. Moreover, we can express the last term $\|\hat{\nu}_{k}-\nu_{k}\|$ in terms of the accuracy $\delta^{\phi}_{k}$ since it follows from (baes2009estimate, , Lemma $5.1$) that:
$$\displaystyle\delta^{\phi}_{k}\geq\phi_{k}(\hat{\nu}_{k})-\phi_{k}(\nu_{k})%
\geq\frac{\sigma_{0}\lambda_{k}}{p}\|\hat{\nu}_{k}-\nu_{k}\|^{3},$$
from which we obtain the bound:
$$\displaystyle\min_{x\in Q}\langle\nabla\phi_{k}(\hat{\nu}_{k}),x-\hat{\nu}_{k}\rangle$$
$$\displaystyle\geq-\delta^{\phi}_{k}-L_{0}D_{Q}\lambda_{k}\left(\frac{p%
\varepsilon_{\phi}}{\sigma_{0}\lambda_{k}}\right)^{1/3}.$$
Furthermore, assuming $\varepsilon_{\phi}\leq 1$ without loss of generality, we have that
$$\displaystyle\min_{x\in Q}\langle\nabla\phi_{k}(\hat{\nu}_{k}),x-\hat{\nu}_{k}\rangle$$
$$\displaystyle\geq-(\delta^{\phi}_{k})^{1/3}\left(1+L_{0}D_{Q}\lambda_{k}\left(%
\frac{p}{\sigma_{0}\lambda_{k}}\right)^{1/3}\right)\geq-\bar{\delta}^{\phi}_{k},$$
(14)
for an appropriate selection of the error $\delta^{\phi}_{k}$.
Lets recall (4), and use the bound (14), then
$$\displaystyle\min_{x\in Q}\phi_{k+1}(x)\geq\min_{x\in Q}\{(1-\alpha_{k})\big{(%
}\phi_{k}(\hat{\nu}_{k})-\bar{\delta}^{\phi}_{k}\big{)}+\lambda_{k+1}\frac{M_{%
2}}{6}\|x-\hat{\nu}_{k}\|^{3}+$$
$$\displaystyle\qquad+\alpha_{k}\big{(}f(\hat{x}_{k+1})+\langle\nabla f(\hat{x}_%
{k+1}),x-\hat{x}_{k+1}\rangle\big{)}\}$$
$$\displaystyle\geq(1-\alpha_{k})\big{(}\phi_{k}(\hat{\nu}_{k})-\bar{\delta}^{%
\phi}_{k}\big{)}+\min_{x\in Q}\{\lambda_{k+1}\frac{M_{2}}{6}\|x-\hat{\nu}_{k}%
\|^{3}+\alpha_{k}\big{(}f(\hat{x}_{k+1})+\langle\nabla f(\hat{x}_{k+1}),x-\hat%
{x}_{k+1}\rangle\big{)}\}$$
$$\displaystyle\geq(1-\alpha_{k})\big{(}\phi_{k}(\hat{\nu}_{k})-\bar{\delta}^{%
\phi}_{k}\big{)}+\alpha_{k}\big{(}f(\hat{x}_{k+1})+\langle\nabla f(\hat{x}_{k+%
1}),\hat{\nu}_{k}-\hat{x}_{k+1}\rangle\big{)}+$$
$$\displaystyle\qquad+\min_{x\in Q}\{\lambda_{k+1}\frac{M_{2}}{6}\|x-\hat{\nu}_{%
k}\|^{3}+\alpha_{k}\langle\nabla f(\hat{x}_{k+1}),x-\hat{\nu}_{k}\rangle\},$$
where in the last inequality we have added and subtracted $\hat{\nu}_{k}$. Now, from the hypotheses that $\min_{x\in Q}\phi_{k}(x)\geq f(\hat{x}_{k})-\varepsilon$, we have
$$\displaystyle\min_{x\in Q}\phi_{k+1}(x)\geq(1-\alpha_{k})\big{(}f(\hat{x}_{k})%
-\varepsilon-\bar{\delta}^{\phi}_{k}\big{)}+\alpha_{k}\big{(}f(\hat{x}_{k+1})+%
\langle\nabla f(\hat{x}_{k+1}),\hat{\nu}_{k}-\hat{x}_{k+1}\rangle\big{)}$$
$$\displaystyle\qquad+\min_{x\in Q}\{\lambda_{k+1}\frac{M_{2}}{6}\|x-\hat{\nu}_{%
k}\|^{3}+\alpha_{k}\langle\nabla f(\hat{x}_{k+1}),x-\hat{\nu}_{k}\rangle\},$$
$$\displaystyle\geq(1-\alpha_{k})\big{(}(f(\hat{x}_{k+1})+\langle\nabla f(\hat{x%
}_{k+1}),\hat{x}_{k}-\hat{x}_{k+1}\rangle)-\varepsilon-\bar{\delta}^{\phi}_{k}%
\big{)}+\alpha_{k}\big{(}f(\hat{x}_{k+1})+\langle\nabla f(\hat{x}_{k+1}),\hat{%
\nu}_{k}-\hat{x}_{k+1}\rangle\big{)}$$
$$\displaystyle\qquad\qquad+\min_{x\in Q}\{\lambda_{k+1}\frac{M_{2}}{6}\|x-\hat{%
\nu}_{k}\|^{3}+\alpha_{k}\langle\nabla f(\hat{x}_{k+1}),x-\hat{\nu}_{k}\rangle\},$$
$$\displaystyle\geq f(\hat{x}_{k+1})+(1-\alpha_{k})\big{(}\langle\nabla f(\hat{x%
}_{k+1}),\hat{x}_{k}-\hat{x}_{k+1}\rangle-\varepsilon-\bar{\delta}^{\phi}_{k}%
\big{)}+\alpha_{k}\langle\nabla f(\hat{x}_{k+1}),\hat{\nu}_{k}-\hat{x}_{k+1}\rangle$$
$$\displaystyle\qquad\qquad+\min_{x\in Q}\{\lambda_{k+1}\frac{M_{2}}{6}\|x-\hat{%
\nu}_{k}\|^{3}+\alpha_{k}\langle\nabla f(\hat{x}_{k+1}),x-\hat{\nu}_{k}\rangle\},$$
where the second to last inequality follows form a linear lower bound of the function $f$ at the point $f(\hat{x}_{k})$, and the last one follows form eliminating common terms. Rearranging some terms, and defining $z_{k}=(1-\alpha_{k})\hat{x}_{k}+\alpha_{k}\hat{\nu}_{k}$, we obtain:
$$\displaystyle\min_{x\in Q}\phi_{k+1}(x)\geq f(\hat{x}_{k+1})+\langle\nabla f(%
\hat{x}_{k+1}),z_{k}-\hat{x}_{k+1}\rangle-(1-\alpha_{k})\big{(}\varepsilon+%
\bar{\delta}^{\phi}_{k}\big{)}$$
$$\displaystyle\qquad\qquad+\min_{x\in Q}\{\lambda_{k+1}\frac{M_{2}}{6}\|x-\hat{%
\nu}_{k}\|^{3}+\alpha_{k}\langle\nabla f(\hat{x}_{k+1}),x-\hat{\nu}_{k}\rangle\}.$$
Next, by Lemma $4.3$ in baes2009estimate , we have
$$\displaystyle\min_{x\in Q}\phi_{k+1}(x)\geq f(\hat{x}_{k+1})+\langle\nabla f(%
\hat{x}_{k+1}),z_{k}-\hat{x}_{k+1}\rangle-(1-\alpha_{k})\big{(}\varepsilon+%
\bar{\delta}^{\phi}_{k}\big{)}$$
$$\displaystyle\qquad\qquad+\min_{x\in Q}\{\frac{\lambda_{k+1}}{\alpha^{3}_{k}}%
\frac{M_{2}}{6}\|x-z_{k}\|^{3}+\langle\nabla f(\hat{x}_{k+1}),x-z_{k}\rangle\},$$
$$\displaystyle\geq f(\hat{x}_{k+1})-(1-\alpha_{k})\big{(}\varepsilon+\bar{%
\delta}^{\phi}_{k}\big{)}+\min_{x\in Q}\{\frac{\lambda_{k+1}}{\alpha^{3}_{k}}%
\frac{M_{2}}{6}\|x-z_{k}\|^{3}+\langle\nabla f(\hat{x}_{k+1}),x-\hat{x}_{k+1}%
\rangle\},$$
(15)
where the last inequality follows by adding and subtracting $\hat{x}_{k+1}$.
The next step is to bound the last term in the above relation, i.e., $\langle\nabla f(\hat{x}_{k+1}),x-\hat{x}_{k+1}\rangle$, from the fact that $\hat{x}_{k+1}$ is an approximate solution to the auxiliary sub problem
$$\displaystyle\operatorname*{arg\,min}_{x\in Q}\left\{\varphi(x)\triangleq%
\langle\nabla f(z),x-z\rangle+\frac{1}{2}\langle\nabla^{2}f(z)(x-z),x-z\rangle%
+\frac{N}{6}\|x-z\|^{3}\right\}.$$
However, note that the optimality condition $\langle\nabla\varphi(x_{k+1}),y-x_{k+1}\rangle\geq 0$ for all $y\in Q$ holds for an optimal point $x_{k+1}$, but we only have access to inexact solvers which return a point $\hat{x}_{k+1}\in Q$ such that $\varphi(\hat{x}_{k+1})-\varphi({x}_{k+1})\leq\delta^{f}_{k}$. So we will see what is the effect of this error.
We will use Proposition 4.3 to relate error in the function value to error in the gradient.
Assume we are able to optain a point $\tilde{x}$ such that
$$\displaystyle\varphi(\tilde{x})-\min_{x\in Q}\varphi(x)\leq\delta^{f}_{k}.$$
$\delta^{f}_{k}$ is the accuracy by which we solve the subproblem of the cubic regularization method.
$$\displaystyle\langle\varphi^{\prime}(\tilde{x}),x-\tilde{x}\rangle$$
$$\displaystyle=\langle\varphi^{\prime}(x^{*}),x-\tilde{x}\rangle-\langle\varphi%
^{\prime}(x^{*})-\varphi^{\prime}(\tilde{x}),x-\tilde{x}\rangle$$
$$\displaystyle\geq\langle\varphi^{\prime}(x^{*}),x-\tilde{x}\rangle-\|\varphi^{%
\prime}(x^{*})-\varphi^{\prime}(\tilde{x})\|\|x-\tilde{x}\|\qquad\text{By %
Cauchy-Schwarz}$$
$$\displaystyle\geq\underbrace{\langle\varphi^{\prime}(x^{*}),x-x^{*}\rangle}_{%
\geq 0\ \ \text{by optimality}}+\langle\varphi^{\prime}(x^{*}),x^{*}-\tilde{x}%
\rangle-\|\varphi^{\prime}(x^{*})-\varphi^{\prime}(\tilde{x})\|\|x-\tilde{x}\|%
\qquad\text{Add and subtract $x^{*}$}$$
$$\displaystyle\geq\underbrace{\varphi(x^{*})-\varphi(\tilde{x})}_{-\delta^{f}_{%
k}}+\underbrace{\frac{\sigma_{0}}{p}\|x^{*}-\tilde{x}\|^{p}}_{\geq 0}-%
\underbrace{\|\varphi^{\prime}(x^{*})-\varphi^{\prime}(\tilde{x})\|}_{\ \leq(L%
_{0}+M_{1})\|x^{*}-\tilde{x}\|}\underbrace{\|x-\tilde{x}\|}_{\leq D_{Q}}\qquad%
\text{Using\leavevmode\nobreak\ \eqref{eq:prop_subproblem}}$$
$$\displaystyle\geq-\delta^{f}_{k}-(L_{0}+M_{1})\|x^{*}-\tilde{x}\|D_{Q}$$
(16)
Also from (12), we have:
$$\displaystyle\delta^{f}_{k}\geq\varphi(\tilde{x})-\varphi(x^{*})\geq\frac{%
\sigma_{0}}{p}\|\tilde{x}-x^{*}\|^{p}\qquad\text{then}\qquad\|\tilde{x}-x^{*}%
\|\leq\left(\frac{\delta^{f}_{k}\cdot p}{\sigma_{0}}\right)^{1/p}$$
Therefore, from (4), we obtain:
$$\displaystyle\langle\varphi^{\prime}(\tilde{x}),x-\tilde{x}\rangle$$
$$\displaystyle\geq-\delta^{f}_{k}-(L_{0}+M_{1})\|x^{*}-\tilde{x}\|D_{Q}$$
$$\displaystyle\geq-\delta^{f}_{k}-(L_{0}+M_{1})D_{Q}\left(\frac{\delta^{f}_{k}%
\cdot p}{\sigma_{0}}\right)^{1/p}$$
$$\displaystyle\geq-(\delta^{f}_{k})^{1/p}-(L_{0}+M_{1})D_{Q}\left(\frac{\delta^%
{f}_{k}\cdot p}{\sigma_{0}}\right)^{1/p}\qquad\text{assuming $\delta^{f}_{k}<1$}$$
$$\displaystyle=-(\delta^{f}_{k})^{1/p}\left(1+(L_{0}+M_{1})\left(\frac{p}{%
\sigma_{0}}\right)^{1/p}D_{Q}\right)$$
$$\displaystyle=-\tilde{\delta}^{f}_{k}$$
(17)
The result in (4) allows us to control the error in the gradient from the error in the function value.
Recall that we need to prove
$$\displaystyle\min_{x\in Q}\phi_{k+1}(x)\geq f(\hat{x}_{k+1})-\varepsilon.$$
And so far we have
$$\displaystyle\min_{x\in Q}\phi_{k+1}(x)$$
$$\displaystyle\geq f(\hat{x}_{k+1})-(1-\alpha_{k})\big{(}\varepsilon+\bar{%
\delta}^{\phi}_{k}\big{)}+\min_{x\in Q}\{\frac{\lambda_{k+1}}{\alpha^{3}_{k}}%
\frac{M_{2}}{6}\|x-z_{k}\|^{3}+\langle\nabla f(\hat{x}_{k+1}),x-\hat{x}_{k+1}%
\rangle\}.$$
(18)
So we continue this proof by bounding the last term above
$$\displaystyle\langle\nabla f(\hat{x}_{k+1}),x-\hat{x}_{k+1}\rangle.$$
It follows from (4)
$$\displaystyle-\tilde{\delta}^{f}_{k}$$
$$\displaystyle\leq\langle\varphi^{\prime}(\hat{x}_{k+1}),x-\hat{x}_{k+1}\rangle$$
$$\displaystyle\leq\langle\nabla f(z)+\nabla^{2}f(z)(\hat{x}_{k+1}-z),x-\hat{x}_%
{k+1}\rangle+\frac{N\|z-\hat{x}_{k+1}\|}{2}\langle\hat{x}_{k+1}-z,x-\hat{x}_{k%
+1}\rangle$$
$$\displaystyle\leq\langle\nabla f(z)+\nabla^{2}f(z)(\hat{x}_{k+1}-z),x-\hat{x}_%
{k+1}\rangle+\frac{N\|z-\hat{x}_{k+1}\|}{2}\big{(}\|z-\hat{x}_{k+1}\|\|x-z\|-%
\|z-\hat{x}_{k+1}\|^{2}\big{)}$$
(19)
On the other hand, from Hessian Lipschitz continuity, it follows that
$$\displaystyle\langle\nabla f(z)+\nabla^{2}f(z)(\hat{x}_{k+1}-z),x-\hat{x}_{k+1}\rangle$$
$$\displaystyle\leq\|\nabla f(z)+\nabla^{2}f(z)(\hat{x}_{k+1}-z)-\nabla f(\hat{x%
}_{k+1})\|\|x-\hat{x}_{k+1}\|+\langle\nabla f(\hat{x}_{k+1}),x-\hat{x}_{k+1}\rangle$$
$$\displaystyle\leq\frac{M_{2}}{2}\|z-\hat{x}_{k+1}\|^{2}\|x-\hat{x}_{k+1}\|+%
\langle\nabla f(\hat{x}_{k+1}),x-\hat{x}_{k+1}\rangle$$
$$\displaystyle\leq\frac{M_{2}}{2}\|z-\hat{x}_{k+1}\|^{2}(\|x-z\|+\|z-\hat{x}_{k%
+1}\|)+\langle\nabla f(\hat{x}_{k+1}),x-\hat{x}_{k+1}\rangle,$$
which implies
$$\displaystyle\frac{M_{2}}{2}\|z-\hat{x}_{k+1}\|^{2}(\|x-z\|+\|z-\hat{x}_{k+1}%
\|)+\langle\nabla f(\hat{x}_{k+1}),x-\hat{x}_{k+1}\rangle-$$
(20)
$$\displaystyle\qquad\langle\nabla f(z)+\nabla^{2}f(z)(\hat{x}_{k+1}-z),x-\hat{x%
}_{k+1}\rangle\geq 0.$$
(21)
Adding (4) and (20) we obtain:
$$\displaystyle\frac{N+M_{2}}{2}\|z-\hat{x}_{k+1}\|\|x-z\|+\frac{M-N}{2}\|z-\hat%
{x}_{k+1}\|^{3}+\langle\nabla f(\hat{x}_{k+1}),x-\hat{x}_{k+1}\rangle\geq-%
\tilde{\delta}^{f}_{k}.$$
(22)
Lets recall (18), and replacing in (22) we have:
$$\displaystyle\min_{x\in Q}\phi_{k+1}(x)$$
$$\displaystyle\geq f(\hat{x}_{k+1})-(1-\alpha_{k})\big{(}\varepsilon+\bar{%
\delta}^{\phi}_{k}\big{)}+\min_{x\in Q}\{\frac{\lambda_{k+1}}{\alpha^{3}_{k}}%
\frac{M_{2}}{6}\|x-z_{k}\|^{3}+\langle\nabla f(\hat{x}_{k+1}),x-\hat{x}_{k+1}%
\rangle\}.$$
Now, lets focus on the second term above,
$$\displaystyle\min_{x\in Q}\{\frac{\lambda_{k+1}}{\alpha^{3}_{k}}\frac{M_{2}}{6%
}\|x-z_{k}\|^{3}+\langle\nabla f(\hat{x}_{k+1}),x-\hat{x}_{k+1}\rangle\}$$
$$\displaystyle\geq\min_{x\in Q}\{\frac{\lambda_{k+1}}{\alpha^{3}_{k}}\frac{M_{2%
}}{6}\|x-z_{k}\|^{3}-\tilde{\delta}^{f}_{k}-\frac{N+M_{2}}{2}\|z-\hat{x}_{k+1}%
\|\|x-z\|+\frac{N-M}{2}\|z-\hat{x}_{k+1}\|^{3}\}$$
$$\displaystyle\geq-\tilde{\delta}^{f}_{k}+\underbrace{\min_{x\in Q}\{\frac{%
\lambda_{k+1}}{\alpha^{3}_{k}}\frac{M_{2}}{6}\|x-z_{k}\|^{3}-\frac{N+M_{2}}{2}%
\|z-\hat{x}_{k+1}\|\|x-z\|+\frac{N-M}{2}\|z-\hat{x}_{k+1}\|^{3}\}}_{\geq 0\ \ %
\text{when}\ \ N=5M}$$
Moreover,the choice of $N=5M$ guarantees that
$$\displaystyle\frac{1}{12}\geq\frac{\alpha_{k}^{3}}{\lambda_{k+1}}.$$
(23)
As a final step, we need to control the errors
$$\displaystyle-(1-\alpha_{k})\big{(}\varepsilon+\bar{\delta}^{\phi}_{k}\big{)}-%
\tilde{\delta}^{f}_{k}\leq\varepsilon.$$
Recall that
$$\displaystyle\tilde{\delta}^{f}_{k}$$
$$\displaystyle=\delta_{f}^{1/3}\left(1+(L_{0}+M_{1})D_{Q}\left(\frac{3}{\sigma_%
{0}}\right)^{1/3}\right)$$
$$\displaystyle\bar{\delta}^{\phi}_{k}$$
$$\displaystyle=\delta_{\phi}^{1/3}\left(1+L_{0}\lambda_{k}D_{Q}\left(\frac{3}{%
\sigma_{0}\lambda_{k}}\right)^{1/3}\right)$$
Therefore it is enough to select $\delta_{f}$ and $\varepsilon_{\phi}$ as follows
$$\displaystyle 0\leq\delta^{\phi}_{k}\leq\min\left\{1,\left(\frac{\alpha_{k}%
\gamma}{1-\alpha_{k}}\right)^{3}\left(\frac{\varepsilon}{1+D_{Q}L_{0}\big{(}3%
\lambda_{k}^{2}/\sigma_{0}\big{)}^{1/3}}\right)^{3}\right\}$$
$$\displaystyle 0\leq\delta^{f}_{k}\leq\min\left\{1,\left((1-\gamma)\alpha_{k}+%
\frac{1}{2}\right)^{3}\left(\frac{\varepsilon}{1+D_{Q}L_{0}\big{(}3/\sigma_{0}%
\big{)}^{1/3}}\right)^{3}\right\}.$$
With the two above choices of ac accuracy for the subproblem, we obtain the final desire result
$$\displaystyle\min_{x\in Q}\phi_{k+1}(x)\geq f(\hat{x}_{k+1})-\varepsilon.$$
∎
With Lemma 4.4 at hand, we are finally ready to state and prove our main result. But first, lets recall an auxiliary results form baes2009estimate that will allow us to bound the rate of convergence.
Lemma 4.5 (Lemma 8.1 in baes2009estimate ).
Consider a sequence $\{\alpha_{k}:k\geq 0\}$ in $(0,1)$, and define $\lambda_{0}=1$, $\lambda_{k+1}=(1-\alpha_{k})\lambda_{k}$ for every $k\geq 0$. If there exists a constant $\beta>0$ and an integer $p>0$ for which $\alpha_{k}^{p}/\lambda_{k+1}\geq\beta$ for every $k\geq 0$, then, for all $K\geq 1$, $\lambda_{K}\leq\big{(}{p}/\big{(}{p+N\big{(}\beta\big{)}^{1/p}\big{)}}\big{)}^%
{p}$.
Next, we state our main auxiliary result about the complexity of Algorithm 3, which we use for the proof of convergence rate of Algorithm 1.
Theorem 4.6.
Let Assumptions 2.3, 2.4 and 2.5 hold. Let $\varepsilon>0$ and $\gamma\in(0,1)$. Moreover, set $\delta_{\phi}$ and $\delta_{f}$ according to Lemma 4.4 and,
$$\displaystyle K\geq\left\lceil{12\left(\frac{1}{\varepsilon}\right)^{1/3}\left%
(f(x_{0})-f^{*}+\frac{M}{6}\|x_{0}-x^{*}\|^{3}\right)^{1/3}}\right\rceil.$$
Then, the output of Algorithm 3 has the following property: $f(\hat{x}_{K})-f^{*}\leq\varepsilon$.
Theorem 4.6 states that if one is allowed to solve the subproblems of Algorithm 3 with the prescribed accuracy, then the oracle complexity of Algorithm 3 is $O(1/\varepsilon^{1/3})$ to reach an solution that is $\varepsilon$ away to the optimal function value.
Proof.
Initially, from Lemma 4.4 we have that
$$\displaystyle\min_{x\in Q}\phi_{k}(x)$$
$$\displaystyle\geq f(\hat{x}_{k})-\varepsilon\qquad\forall k\geq 1.$$
Then, from Proposition 4.2, it follows that
$$\displaystyle f(\hat{x}_{k}){-}f^{*}\leq\lambda_{k}(\phi_{0}(x^{*}){-}f^{*})+\varepsilon.$$
which in combination with Lemma 4.5 and (23) provides:
$$\displaystyle f(\hat{x}_{k}){-}f^{*}\leq\left(\frac{3}{3+k\big{(}1/12\big{)}^{%
1/3}}\right)^{3}(\phi_{0}(x^{*}){-}f^{*})+\varepsilon.$$
and the desired result follows.
∎
Now that we have Theorem 4.6 at hand, we can prove our main result.
Proof.
(Theorem 2.6)
The proof follows the same arguments as proof of Theorem 4.6.
Recall that in Algorithm 1 we define:
$$\displaystyle\phi_{k{+}1}(\mathbf{x}){=}(1{-}\alpha_{k})\phi_{k}(\mathbf{x})+%
\alpha_{k}\big{(}F(\mathbf{x}_{k{+}1})+\langle\nabla F(\mathbf{x}_{k{+}1}),%
\mathbf{x}{-}\mathbf{x}_{k{+}1}\rangle\big{)}$$
Then, from Lemma 4.4 it follows that
$$\displaystyle\min_{\mathbf{x}\in Q^{m}\bigcap\mathcal{Q}_{\tilde{\varepsilon}}%
}\phi_{k}(\mathbf{x})$$
$$\displaystyle\geq F(\hat{\mathbf{x}}_{k})-\varepsilon\qquad\forall k\geq 1.$$
Then, from Proposition 4.2, it follows that
$$\displaystyle F(\hat{\mathbf{x}}_{k}){-}F^{*}\leq\lambda_{k}(\phi_{0}(\mathbf{%
x}^{*}){-}F^{*})+\varepsilon.$$
which in combination with Lemma 4.5 and (23) provides:
$$\displaystyle F(\hat{\mathbf{x}}_{k}){-}F^{*}\leq\left(\frac{3}{3+k\big{(}1/12%
\big{)}^{1/3}}\right)^{3}(\phi_{0}(\mathbf{x}^{*}){-}F^{*})+\varepsilon.$$
and the desired result follows by finding $k\geq 1$ such that the first term above is less than $\varepsilon$.
∎
5 Proof of Theorem 3.3: Communication Complexity of the Decentralized Cubic Regularized Newton Method
In this section, we study the communication complexity of Algorithm 1. Recall that at each iteration, we are required to solve two auxiliary subproblems (Lines 10 and 13) in a distributed manner via Algorithm 2.
Note that we have defined our consensus constraints as $Ax=0$, with $A$ being the graph Laplacian obtained from the network.
Lets consider the sets $\mathcal{A}{=}\{x\mid Ax{=}0\}$, and $\mathcal{B}{=}\{x\mid\|Ax\|{=}0\}$. Then $\mathcal{A}{=}\mathcal{B}$. Thus, we compare the following two problems
$$\displaystyle\min_{\|Ax\|\leq 0}f(x)$$
(24)
$$\displaystyle\min_{\|Ax\|\leq\varepsilon}f(x)$$
(25)
Proposition 5.1.
Denote as $f_{0}^{*}$ as the optimal value of the optimization problem (24), and $f_{\varepsilon}^{*}$ as the optimal value of the optimization problem (25). Moreover assume $f_{0}^{*}$ and $f_{\varepsilon}^{*}$ are finite, and define as $y_{0}^{*}$ and $y_{\varepsilon}^{*}$ the corresponding dual optimal solutions and that there is no duality gap. Then
$$\displaystyle y^{*}_{\varepsilon}\varepsilon\leq f^{*}_{0}{-}f^{*}_{%
\varepsilon}\leq y^{*}_{0}\varepsilon.$$
(26)
Proof.
The proof of (26) follows from the fact that:
$$\displaystyle f^{*}_{0}$$
$$\displaystyle{=}\inf_{x}\{f(x){+}y^{*}_{0}\|Ax\|\}$$
$$\displaystyle f^{*}_{\varepsilon}$$
$$\displaystyle{=}\inf_{x}\{f(x){+}y^{*}_{\varepsilon}(\|Ax\|{-}\varepsilon)\}$$
moreover, denote $q^{*}_{0}(y){=}\inf_{x}\{f(x){+}y\|Ax\|\}.$
Then
$$\displaystyle f^{*}_{0}{{-}}f^{*}_{\varepsilon}$$
$$\displaystyle{{=}}\inf_{x}\{f(x){{+}}y^{*}_{0}\|Ax\|\}{{-}}\inf_{x}\{f(x){{+}}%
y^{*}_{\varepsilon}(\|Ax\|{{-}}\varepsilon)\}$$
$$\displaystyle{{=}}\inf_{x}\{f(x){{+}}y^{*}_{0}\|Ax\|\}{{-}}\inf_{x}\{f(x){{+}}%
y^{*}_{\varepsilon}\|Ax\|\}{{+}}y^{*}_{\varepsilon}\varepsilon$$
$$\displaystyle{=}q^{*}_{0}(y^{*}_{0}){-}q^{*}_{0}(y^{*}_{\varepsilon}){+}y^{*}_%
{\varepsilon}\varepsilon$$
$$\displaystyle\geq y^{*}_{\varepsilon}\varepsilon,$$
because by definition $y^{*}_{0}$ maximizes $q^{*}_{0}$. The other direction follows similarly. We can conclude that if we have a point $\hat{x}$ such that for $\delta>0$ it holds that
$$\displaystyle f(\hat{x}){-}f^{*}_{0}\leq\delta.$$
Then, it follows from (26) that
$$\displaystyle f(\hat{x}){-}f^{*}_{\varepsilon}{+}f^{*}_{\varepsilon}{-}f^{*}_{%
0}\leq\delta$$
$$\displaystyle f(\hat{x}){-}f^{*}_{\varepsilon}\leq\delta{+}f^{*}_{0}{-}f^{*}_{%
\varepsilon}\leq\delta{+}y_{0}^{*}\varepsilon,$$
(27)
where recall that $\varepsilon$ is the upper bound in the approximate consensus constraint, and $\delta$ is the optimality gap by which the exact consensus problem has been solved.
∎
Proposition 5.1 shows that if we obtain a point $\hat{x}$ that is $\delta$ away from optimality in terms of function value with respect to the constraint $Ax=0$. Then, it will at most $2\delta$ away from optimality in terms of function value with respect to the constraint $\|Ax\|\leq\delta/R$. This result will be important in analyzing the communication complexity of the proposed algorithm as we analyze our algorithm with respect to the set $\mathcal{Q}_{\tilde{\varepsilon}}{=}\{\mathbf{x}\in\mathbb{R}^{nm}\mid\|\sqrt{%
\mathbf{W}}\hat{\mathbf{x}}\|_{2}\leq\tilde{\varepsilon}\}$.
Proof.
(Theorem 3.3)
The main idea of this proof is to exploit the fact that the subproblem of minimizing the cubic regularized approximation in (3) is dual-friendly, as shown in Section 3. Algorithm 2 is an adaptation to subproblem (4) in (Uribe2018a, , Algorithm $5$), whose communication complexity is explicitly available in (Uribe2018a, , Theorem $5.3$). However, there are some technical aspects we have to take care of first.
Initially, (Uribe2018a, , Theorem $5.3$) guarantees at the end of required number of iterations $T$ we obtain an approximate solution $\mathbf{h}^{*}(\tilde{\mathbf{w}}_{T})$ such that
$$\displaystyle\boldsymbol{\Phi}(\mathbf{h}^{*}(\tilde{\mathbf{w}}_{T})){-}%
\boldsymbol{\Phi}_{0}^{*}\leq\delta\quad\text{and}\quad\|A\mathbf{h}^{*}(%
\tilde{\mathbf{w}}_{T})\|\leq\delta/R.$$
But it is important to note that $\boldsymbol{\Phi}^{*}_{0}$ is the optimal value for the function $\eqref{eq:generic_subproblem}$ with the linear constraint $A\mathbf{h}{=}\boldsymbol{0}_{nm}$. Whereas we need an approximate solution with respect to $\|A\mathbf{h}\|\leq\tilde{\varepsilon}$. It follows from Proposition 5.1 that the point $\mathbf{h}^{*}(\tilde{\mathbf{w}}_{T})$ has the following property:
$$\displaystyle\boldsymbol{\Phi}(\mathbf{h}^{*}(\tilde{\mathbf{w}}_{T})){{-}}%
\boldsymbol{\Phi}_{\tilde{\varepsilon}}^{*}\leq\delta{{+}}\tilde{\varepsilon}%
\|\mathbf{y}^{*}_{0}\|\ \text{and}\ \|A\mathbf{h}^{*}(\tilde{\mathbf{w}}_{T})%
\|\leq\delta/R,$$
where $\mathbf{y}^{*}_{0}$ is the optimal value of the dual function (10). The desired result follows by setting $\tilde{\varepsilon}=\delta/R$.
∎
6 Experimental Results
In this section, we present numerical experiments for the implementation of Algorithm 1 applied to the logistic regression problem:
$$\displaystyle\min_{x\in\mathbb{R}^{n}}\frac{1}{d}\sum\limits_{i=1}^{d}\ln\Bigl%
{(}1+\exp\bigl{(}-y_{i}\langle w_{i},x\rangle\bigr{)}\Bigr{)}.$$
(28)
We given a set of $d$ data pairs $\{y_{i},w_{i}\}$ for $1\leq i\leq d$, where $y_{i}\in\{1,-1\}$ is the class label of object $i$, and $w_{i}\in\mathbb{R}^{n}$ is the set of features of object $i$. Moreover, we assume the data points are uniformly split among $m$ nodes, connected over a network, such that each node has $d^{i}$ data points, and $d=m\cdot d^{i}$.
We compare the performance of Algorithm 1 (DecAccCubic), centralized gradient method (CenGM), distributed Newton method (DecNewton), distributed non-accelerated cubic method (DecCubic), and distributed accelerated gradient method (DecAccGM). Next, we describe each of these methods.
Lets recall the two main optimization problems:
$$\displaystyle\min_{x\in Q}\left\{f(x)\triangleq\sum_{i{=}1}^{m}f^{i}(x)\right%
\}\ \ \text{and}\ \ \min_{\begin{subarray}{c}\mathbf{x}\in Q^{m}\\
\sqrt{\mathbf{W}}\mathbf{x}{=}\boldsymbol{0}_{nm}\end{subarray}}\left\{F(%
\mathbf{x})\triangleq\sum_{i{=}1}^{m}f^{i}(x^{i})\right\},$$
Centralized gradient method (CenGM): Gradient descent when all the data points are stored at the same location, i.e.,
$$\displaystyle x_{k+1}=x_{k}-\alpha\sum_{i{=}1}^{m}\nabla f^{i}(x_{k}).$$
Distributed Newton method (DecNewton): Netwon method with consensus contraints in the subproblem, i.e.,
$$\displaystyle\mathbf{x}_{k+1}=\operatorname*{arg\,min}_{A\mathbf{x}=0}\left\{F%
(\mathbf{x}_{k}){+}\langle\nabla F(\mathbf{x_{k}}),\mathbf{x}{-}\mathbf{x_{k}}%
\rangle+\frac{1}{2}\langle\nabla^{2}F(\mathbf{x_{k}})(\mathbf{x}{-}\mathbf{x_{%
k}}),\mathbf{x}{-}\mathbf{x_{k}}\rangle\right\}$$
Distributed non-accelerated cubic (DecCubic): Constrained Cubic Regularized Newton method with no acceleration
$$\displaystyle\mathbf{x}_{k+1}=\operatorname*{arg\,min}_{A\mathbf{x}=0}\left\{F%
(\mathbf{x}_{k}){+}\langle\nabla F(\mathbf{x_{k}}),\mathbf{x}{-}\mathbf{x_{k}}%
\rangle+\frac{1}{2}\langle\nabla^{2}F(\mathbf{x_{k}})(\mathbf{x}{-}\mathbf{x_{%
k}}),\mathbf{x}{-}\mathbf{x_{k}}\rangle+\frac{N}{5}\|\mathbf{x}{-}\mathbf{x_{k%
}}\|^{3}\right\}$$
Distributed accelerated gradient method (DecAccGM): Accelerated Gradient Method with consensus constraints
Figure 1 shows the oracle complexity of Algorithm 1 in terms of the optimality gap of the generated iterations for different network topologies (Complete, Erdös-Rényi, and Cycle graphs) and various problem parameters (number of agents, number of data points and dimensions). In all scenarios we explore, the proposed approach has the best performance with respect to the oracle complexity. Figure 2 shows the communication complexity of solving the auxiliary subproblems with Algorithm 2. The top row shows the consensus gap, which indicates that the agreement among the agents on a solution increases as the number of communication rounds increases. The bottom row shows the agreement is on a solution to the auxiliary subproblem.
7 Discussion and Open Problems
Related Work: Cubic-regularized Newton’s method in the centralized setup has been extensively studied for large problem classes of convex and non-convex problems, e.g., Riemannian Manifolds in zhang2018cubic ; agarwal2018adaptive , where it was shown that the proposed algorithm reaches a second-order $\varepsilon$-stationary point within $O(\varepsilon^{-3/2})$ under certain smoothness conditions, which is optimal for the function classes. A stochastic variance-reduced cubic regularized newton methods was proposed in zhou2018stochastic , where it was shown that the proposed algorithm converges to an $(\varepsilon,\sqrt{\varepsilon}$)-approximate local minimum within $\tilde{O}(n^{4/5}\varepsilon^{-3/2})$. In wang2018stochastic an iteration complexity of $O(n^{2/3}\varepsilon^{-3/2})$, and randomized blocks in doikov2018randomized . See Jiang2017 ; cartis2011adaptive ; cartis2011adaptive2 ; cartis2020concise for a extensive treatment of Cubic regularization. Additionally, inexactness in cubic regularization has also been explored, ghadimi2017second ; wang2018note explores inexact Hessian, song2019inexact assumes geometric convergence rates in the approximate solution of the subproblem or inexact subproblem computation in the unconstrained case Cartis2012 . In cartis2011adaptive ; cartis2011adaptive2 , the authors explored adaptive methods to handle unknown Lipschitz constants and inexact problem solution, but only a $O(k^{-2})$ convergence rate was shown. In Grapiglia2019 ; Grapiglia2019a inexact solutions of high-order unconstrained problems and Hölder continuity were also explored.
Inexact gradient and Hessian: We assumed that each agent could compute the Hessian of the function stored locally in its memory, i.e., $\nabla^{2}f^{i}(x)$. In the case where $f^{i}(x)=\sum_{l=1}^{d}\ell_{l}(x)$ where $\ell_{l}$ is the loss function at a point $l$, this reduces the computation of the full Hessian $\nabla^{2}f(x)$, as each node can compute its local Hessian in parallel. However, as studied in ghadimi2017second ; wang2018note ; Jiang2017 , even in such a reduced setting, the computational of the local Hessian might be practically or computationally intractable. Thus, effective ways to incorporate such inexactness should be studied. Studying the effects of an inexact gradient, Hessian, or function evaluation in distributed cubic regularized methods remains an open problem.
Distributed implementation beyond strong convexity: A main technical result in obtaining a distributed algorithm was described in Section 3.1. In short, the structure of the problem allows for distributed computation of the gradient of the dual function using local information only. However, only a sublinear convergence rate was achieved in the solution of the subproblem in Theorem 3.3. Note that if the regularization term was quadratic, then linear rates could be achieved. The study of the primal-dual relationship between uniform convexity and Hölder continuity requires further study (nesterov2015universal, , Lemma 1), (dvurechensky2017gradient, , Lemma 1), or (bauschke2011convex, , Corollary 18.14).
Reaching optimality in the distributed setting: The convergence rate obtained by Algorithm 1 is not optimal for the class of functions with Lipschitz Hessian. From first-order methods, it is known that optimal bounds are proportional to centralized lower bounds times a measure of connectivity of the network Uribe2018a , usually $O(m)$. However, optimal rates in high-order methods strongly depend on online search procedures monteiro2013accelerated ; pmlr-v99-gasnikov19a . Such line search methods are not currently available for distributed methods.
Distributed high-order methods: Recently, implementable high-order methods have been proposed kamzolov2020near ; nesterov2020superfast , where third-order information is approximated by second and first-order generating methods with very fast convergence rates, e.g. $O(k^{-5})$. The study of decentralization and inexactness for such methods require further study.
8 Conclusions
In this paper, we developed a second-order Newton-type method based on cubic regularization to minimize convex, finite-sum minimization problems over networks. With the additional assumption that the objective function has a Lipschitz Hessian, the convergence rate is shown to be ${O}(k^{-3})$, which improves on first-order distributed methods ${O}(k^{-2})$. The proposed algorithm extends the inexact cubic regularized Newton method baes2009estimate to the distributed setup, and shows that the auxiliary subproblems can be solved cooperatively and in a distributed manner over an arbitrary network by exploiting the primal-dual structure of the cubic terms. Compared to centralized approaches, the achieved convergence rate is slightly sub-optimal, as lower bounds for second-order methods are known to be ${O}(k^{-7/2})$. However, the proposed algorithm is suitable for applications with distributed storage and computation capabilities spread over arbitrary networks. It is an open question of whether optimal rates can be achieved in a distributed setup. Moreover, further analysis of the proposed method’s communication complexity is required, as we have focused on improving the oracle complexity (computations of gradients and Hessians) while guaranteeing a distributed, nearest-neighbor based implementation.
Broader Impact
This work does not present any foreseeable societal consequence.
{ack}
This work was supported by the MIT-IBM AI grant and a Vannevar Bush Fellowship. The authors would like to thank Pavel Dvurechensky, and Alexander Gasnikov for fruitful discussions and comments.
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Frustration by population trapping with polar molecules
O. Dutta
omjyoti@gmail.com
Instytut Fizyki imienia Mariana Smoluchowskiego, Uniwersytet Jagielloński, ulica Reymonta 4, PL-30-059 Kraków, Poland
M. Lewenstein
ICFO —Institut de Ciencies Fotoniques, Av. Carl Friedrich Gauss, num. 3, 08860 Castelldefels (Barcelona), Spain
ICREA – Institució Catalana de Recerca i Estudis Avançats, Lluis Companys 23, E-08010 Barcelona, Spain
J. Zakrzewski
Instytut Fizyki imienia Mariana Smoluchowskiego, Uniwersytet Jagielloński, ulica Reymonta 4, PL-30-059 Kraków, Poland
(November 19, 2020)
Abstract
One of the most important fundamental problems in contemporary physics is to understand the effects of frustration in quantum many-body systems. Typically, frustration is induced by lattice geometries, disorder and/or strong interactions mila . The resulting states show strong correlations and have exotic properties with possible applications for condensed matter physics, material science, and quantum information technologies, etc. A paradigmatic system that exhibits frustration is a gas of ultracold polar molecules trapped in an optical lattice macbook . Notably, such system is extremely difficult to model theoretically due to proliferation of dipole induced multi-band excitations. In this article we develop for the first time a consistent theoretical model of the polar molecules in a lattice by applying the concepts and ideas of ionization theory hand . Our approach is necessary for correct account of strong dipolar interaction. Additionally, by merging concepts from quantum optics (population transfer), we show that one can induce frustration and engineer exotic states, such as Majumdar-Ghosh state, or vector-chiral states in such situation.
This forms the first attempt to bridge the seemingly two distant areas of research, and opens up a new direction to engineer different strongly-correlated many-body quantum states.
I Introduction
In recent years the ultracold gases have been used as a tool to quantum engineer various novel states of matter with an unprecedented precision and control. In this regard, particularly challenging is the engineering of frustrated systems for ultracold gases trapped in optical lattices. Frustration can either be induced by the lattice geometry, which can lead to kinetic frustration, or by higher order exchange processes due to strong interactions macbook . Polar molecules are particularly interesting in this context, as they can interact via long-range dipolar forces, which can induce yet another kind of frustration. In particular, dipolar lattice gases have been proposed to simulate various quantum phases and exotic phenomena, such as supersolidity maciej1 ; guido , quantum magnetism gorshkov1 , topological states gorshkov2 ; dutta1 , exotic pair-superfluidity dutta2 , etc. Experimental progress towards creation of quantum degenerate gas of ground state polar molecules has been spectacular over the last years Ni ; Ospel ; Aikawa ; Deiglmayr , leading, for instance, to realization of quantum spin models using
fermionic molecules Ye or dipolar Chromium atoms Sant .
One of the important properties of the polar molecules is that their dipole moment can be tuned by applying an electric field. The more polarized these molecules get, the stronger becomes the dipolar interaction between them. Theoretically it is a challenge to investigate the properties of these strongly interacting molecules trapped in an optical lattice. Standard approach based on Bose-Hubbard models limited to the lowest Bloch band becomes inapplicable due to strong interaction induced coupling between the bands. In this paper, we provide a novel route to describe such strongly interacting systems. Specifically, we consider bosonic polar molecules trapped in an one dimensional optical lattice. We find that the system can be modeled with effective couplings between the localized states at lattices sites and the continuum of highly excited states. This connects our approach to the extensive studies of strong laser field induced ionization of atoms and molecules. In particular we find analogies to auto-ionization processes, in which multi-configuration interactions couple discrete
states with continua, as in the celebrated Fano model Fano . Usually, due to the coupling to the continuum, the electrons in atoms or molecules are transferred from the bound states to the continuum, which leads in the long-time limit to the irreversible decay of bound state population. Strong laser field, however, enables efficient couplings between different ionization paths leading to various interference phenomena. For strong field auto-ionization it may lead to the so called confluence of coherence Rza ; Lamb , which slows down very efficiently the ionization process.
Similarly, if several (at least two) bound states are coupled to a common continuum, a phenomenon of coherent population trapping may occur – the ionization is incomplete and a significant part of the system population is trapped in the bound subspace night . The resulting stable bound configuration is a superposition of original bound states with properties depending on the details of the coupling to the continua.
The population trapping phenomenon appears also for multi-level discrete systems when coherent driving may create non-absorbing states (often called “dark states”) – for a review of coherent population trapping see arimondo . Most importantly, in our system of polar molecules, we find that similar phenomenon can give rise to frustration in lattice systems as the population trapping can involve particles trapped in different sites of the optical lattice. Specifically we find that for the half-filling, the many-body population trapped state is a dimer state known in the condensed-matter physics as Majumdar-Ghosh state mg . Majumdar-Ghosh state is a paradigmatic example in the study of frustrated models, since it retains basic properties of spin-liquid phases, such as fractional excitations diep . For lower filling we find that the effective model can be written as a $J_{1}-J_{2}$ Hamiltonian with nearest and next-nearest neighbour tunneling, along with long range dipolar interaction. Such models, but restricted only to nearest and next-nearest sites, have been investigated for long in connection with various magnetic materials diep . But, in solid-state materials faze as well as in optical lattices macbook , such the next-nearest neighbour tunneling can only come from higher order exchange processes, which makes it considerably weaker than the nearest neighbour tunneling. The corresponding temperature is thus very low. Amazingly, the temperature scale associated with population trapped frustration remains comparable to the characteristic temperature scale of the system. An alternative way to achieve long range "tunneling" in spin models is offered in ultracold ions setting Porras-Cirac ; Hauke ; Maik , but such systems are not easily scalable to macro- or even mezo-scopic sizes.
II The model
We consider bosonic polar molecules trapped in an optical potential inducing a one-dimensional lattice geometry,
$$V_{\rm latt}=V_{0}\sin^{2}\frac{\pi x}{a}+\frac{1}{2}m\Omega^{2}(y^{2}+z^{2}),$$
where $V_{0}$ denotes the lattice depth and $a$ is the lattice constant. $\Omega$ denotes the harmonic (strong) trapping frequency along the $y$ and $z$ direction. The molecules are polarized by an electric field along the $z$ axis. To describe this system we make two assumptions: i) along the trapping directions, only the lowest harmonic oscillator eigenstate is occupied, and ii) at time $t<0$, repulsively bound pairs of molecules in the limit of weak dipolar strength are prepared by tuning the lattice depth Winkler , or by applying a weak electric field. The molecules of the pair repel each other and cannot separate due to the energy conservation - a separation would imply populating single particle states in the band gap. Then at $t=0$, we switch on a strong polarizing electric field to induce a strong dipolar interaction between the molecules. The strength of the dipolar interaction in dimensionless units is denoted by $D=m_{b}d^{2}/2\epsilon_{0}\hbar^{2}a$, where $d$ is the dipole moment, $\epsilon_{0}$ is the vacuum permittivity and $m_{b}$ is the mass of the molecules. The dipolar interaction is given by $V_{\rm dd}(\mathbf{r})=D\left[1-3z^{2}/r^{2}\right]/r^{3}$.
To describe the strongly interacting regime for our system, one needs to go beyond the simple single band tight-binding approximations Jaksch . The single-particle motion in a periodic potential results in energy bands, known as Bloch bands which can be expressed in terms of quasi-momentum $q$. For each such a band, one constructs localized basis states or orbitals [the so called Wannier functions(WF)] from the Bloch states Kohn . By taking into account only the lowest energy Wannier states, one arrives at a Hamiltonian containing density-density
interactions terms (both on-site and long-range) and nearest neighbour tunneling processes along the $x$ direction. In the presence of a strong interactions such an approximation breaks down due to two primary reasons: i) the interaction mixes different bands or orbitals and different sites (specially for the higher orbitals), and ii) for higher orbitals, one has to take into account long-range tunneling matrix elements.
These problems have been partially addressed taking into account higher excited bands in the tight-binding approximation and considering only the onsite interactions larson2009 ; johnson2009 ; will2010 ; dutta2011 ; jurgensen2012 ; luhmann2012 ; lacki2013 . For strong interactions though, serious complications appear due to the lack of convergence of results as a function of number of bands taken into consideration. Subsequently, standard approaches become questionable and impractical. For such strong interactions an entirely new approach is needed. The method initiated in this work paves the way for efficient description of such systems.
The essential observation, forming the core of our approach, is that for typical optical lattice depths, only few lowest bands are separated from each other energetically with forbidden gaps in between. The higher bands, in reality, form a continuum of energies. The simplest situation occurs
for relatively small lattice depths say of few recoil energies, as shown in
Fig. 1(a) for $V_{0}=7E_{R}$. Here two lowest, $s$ and $p$, bands are separated from the continuum formed by other bands. The Wannier states (called often orbitals) of the first two bands are relatively well localized, and the mean energy calculated for them is lower than the optical lattice depth $V_{0}$. In this situation it is natural to express the motion of the particles in a mixed basis, where only the low-energy motion is expressed in terms of localized Wannier orbitals. Instead of using Wannier basis for the other bands too, as in the standard approaches, the remaining higher energy states will be treated by continuous Bloch functions. For much deeper lattices a natural generalization of our approach will be to take more than two discrete bands into account; in this work we limit ourselves to the simplest situation. Therefore, we may write down the field operator in the chosen mixed Wannier-Bloch basis as:
$$\displaystyle\Phi(\mathbf{r})$$
$$\displaystyle=$$
$$\displaystyle\sum_{i}\left[\hat{s}_{i}\omega^{s}_{i}(x)+\hat{p}_{i}\omega^{p}_%
{i}(x)\right]\phi_{0}(z)\phi_{0}(y)$$
(1)
$$\displaystyle+$$
$$\displaystyle\sum_{q,n>n_{0}}\mathcal{U}_{n}(q)\hat{a}_{nq}\phi_{0}(z)\phi_{0}%
(y)$$
where $\omega^{\mathcal{\alpha}}_{i}(x)$ is the localized Wannier function at site $i$ corresponding to $\alpha=s$ or $\alpha=p$ orbital while $\phi_{0}$ is the lowest harmonic oscillator eigenfunction for trapping directions. $\hat{s}^{\dagger}_{i},\hat{s}_{i},\hat{p}^{\dagger}_{i},\hat{p}_{i}$ are the creation and annihilation operators for the bosons in the $s$- and $p$-orbitals. $\mathcal{U}_{n}(q)$ denotes the Bloch functions for band $n$ with quasi-momentum $q$ ($n>n_{0}=2$ the latter counts the number of bands treated using Wannier basis). Consequently, $\hat{a}^{\dagger}_{nq},\hat{a}_{nq}$ denote the boson creation and annihilation operators in the bands considered in the Bloch basis with a quasi-momentum $q$. To define dimensionless quantities, we first rescale the distance $\pi x/a\rightarrow x$, that defines the scale for the energy $E_{R}=\pi^{2}\hbar^{2}/2m_{b}a^{2}$. So in the limit of $E_{n_{0}}\gg V_{0}$,the simplified Hamiltonian in the Wannier-Bloch basis is given by,
$$\displaystyle H$$
$$\displaystyle=$$
$$\displaystyle\sum_{i}E_{p}\hat{p}^{\dagger}_{i}\hat{p}_{i}`+\sum_{<i,j>}[-t_{s%
}\hat{s}^{\dagger}_{i}\hat{s}_{j}+t_{p}\hat{p}^{\dagger}_{i}\hat{p}_{j}]$$
(2)
$$\displaystyle+$$
$$\displaystyle H_{\rm Int}+H_{\rm Bloch}+H_{\rm WB},$$
with
$$\displaystyle H_{\rm Int}$$
$$\displaystyle=$$
$$\displaystyle\sum_{i,\sigma=s,p}\frac{U_{\sigma\sigma}}{2}\hat{n}_{\sigma i}(%
\hat{n}_{\sigma i}-1)+U_{\rm ps}\sum_{i}\hat{n}_{si}\hat{n}_{pi}$$
$$\displaystyle+\frac{T_{\rm ps}}{2}$$
$$\displaystyle\sum_{i}$$
$$\displaystyle\left[\hat{p}^{\dagger}_{i}\hat{p}^{\dagger}_{i}\hat{s}_{i}\hat{s%
}_{i}+H.c\right]+\frac{D}{2\pi^{3}}\sum_{\sigma,\sigma^{\prime},i\neq j}\frac{%
\hat{n}_{\sigma i}\hat{n}_{\sigma^{\prime}j}}{|i-j|^{3}}$$
(3)
and
$$\displaystyle H_{\rm WB}$$
$$\displaystyle=$$
$$\displaystyle\sum_{i}\sum_{q_{1}q_{2};n}\left[\frac{1}{2}P^{n}_{i,ss}(q_{1}q_{%
2})\hat{a}^{\dagger}_{nq_{1}}\hat{a}^{\dagger}_{nq_{2}}\hat{s}_{i}\hat{s}_{i}+%
P^{n}_{i,sp}(q_{1}q_{2})\right.$$
$$\displaystyle\times$$
$$\displaystyle\left.\hat{a}^{\dagger}_{nq_{1}}\hat{a}^{\dagger}_{nq_{2}}\hat{p}%
_{i}\hat{s}_{i}+\frac{1}{2}P^{n}_{i,pp}(q_{1}q_{2})\hat{a}^{\dagger}_{nq_{1}}%
\hat{a}^{\dagger}_{nq_{2}}\hat{p}_{i}\hat{p}_{i}+H.c\right],$$
where $E_{p}$ gives the single particle energy of the $p$-orbital ($E_{s}=0$ is assumed), $t_{s}$ and $t_{p}$ are the nearest neighbor tunnelings for the particles in $s$- and $p$-orbitals respectively. The interaction between particles in localized orbitals $H_{\rm Int}$ contains (with $\sigma,\sigma^{\prime}=s,p$ denoting the orbitals) the onsite intra-orbital interactions $U_{\rm\sigma\sigma^{\prime}}$, the possible transitions of a pair between orbitals with the strength $T_{\rm ps}$ and the long range dipolar interaction. $H_{\rm Bloch}$ in Eq.(2) contains the kinetic energy of the molecules in the continuous band, $\sum_{q,n>n_{0}}E_{n}(q)\hat{a}^{\dagger}_{nq}\hat{a}_{nq}$ as well as interaction between particles within the continuum (see Methods section). The Wannier-Bloch Hamiltonian part $H_{\rm WB}$ describes the coupling between Wannier-described sites with two particles and the Bloch continuum. $P^{n}_{i,ss}(q_{1}q_{2}),P^{n}_{i,sp}(q_{1}q_{2})$ and $P^{n}_{i,pp}(q_{1}q_{2})$ are the corresponding coupling
constants of two particles at site $i$ and the continuum for
the $ss$-, $sp$- and $pp$-orbitals respectively. A cartoon of these various transition processes is shown in Fig.1(a). In the following we neglect the single particle tunneling as the pairs can only tunnel through second-order processes. Moreover, due to bond-charge tunneling as discussed in Ref. dutta2 ,
the effective single-particle tunneling in each orbital decreases considerably. For these reasons from now on we assume $t_{s}=t_{p}=0$. To simplify the notation we denote the basis states for zero or two particles on a site as
$$\left|00\right\rangle\rightarrow\left|0\right\rangle,\left|20\right\rangle%
\rightarrow\left|1\right\rangle,\left|11\right\rangle\rightarrow\left|2\right%
\rangle,\left|02\right\rangle\rightarrow\left|3\right\rangle,$$
where the state $\left|n_{1}n_{2}\right>$ denotes $n_{1}$ particles in the $s$-orbital and $n_{2}$ particles at $p$-orbital. We refer to these states as Wannier states in the following sections.
III Possible configurations
A single site coupled to a continuum: We first look into the case of a pair in a single site coupled to the continuum. Such situation may be realized in the insulating states where the particles are localized in single sites. Then, the state of the system can be expressed as
$$\displaystyle\left|\Phi\right\rangle_{i}$$
$$\displaystyle=$$
$$\displaystyle\sum^{3}_{l=1}C^{0}_{l}\left|l\right\rangle_{i}\left|\mathbf{0}%
\right\rangle+\sum_{n_{1}n_{2};q_{1}q_{2}}\alpha^{n_{1}n_{2}}_{q_{1}q_{2}}%
\left|0\right\rangle_{i}\left|\mathbf{1}_{q_{1}}\mathbf{1}_{q_{2}}\right\rangle,$$
where the state $\left|\mathbf{0}\right\rangle$ denotes the vacuum for the continuum and $\left|\mathbf{1}_{q_{1}}\mathbf{1}_{q_{2}}\right\rangle$ denotes the state with both particles in the continuum corresponding to the quantum numbers $n_{1}q_{1}$ and $n_{2}q_{2}$. $C^{0}_{l}$’s denote the probability amplitudes of different possible combinations of bosons at the $s$- and $p$-orbitals for a site $i$. The probability amplitudes for the continuum state are denoted by $\alpha^{n_{1}n_{2}}_{q_{1}q_{2}}$. We find that due to the orthogonality of the continuum states and $E_{n_{0}}\gg 1$, the transition amplitudes are small unless $n_{1}=n_{2}=n$. Next, we write the corresponding time-dependent Schrödinger equation,
$$\displaystyle i\mathbf{\dot{C}}^{0}$$
$$\displaystyle=$$
$$\displaystyle\mathbf{U}\mathbf{C}^{0}+\sum_{n,q_{1}q_{2}}\left[\mathbf{P}^{n}_%
{q_{1}q_{2}}\right]\alpha^{nn}_{q_{1}q_{2}},$$
$$\displaystyle i{\dot{\alpha}}^{nn}_{q_{1}q_{2}}$$
$$\displaystyle=$$
$$\displaystyle\left[E_{n}(q_{1})+E_{n}(q_{2})\right]\alpha^{nn}_{q_{1}q_{2}}+%
\left[\mathbf{P}^{n}_{q_{1}q_{2}}\right]^{\dagger}\mathbf{C}^{0},$$
(5)
where the array $\mathbf{C}^{0}=(C^{0}_{1},C^{0}_{2},C^{0}_{3})^{T}$, and $\mathbf{U}$ is the corresponding $3\times 3$ matrix denoting the interaction strength and transition amplitudes between the Wannier states due to dipolar interaction. The discrete-continuum coupling array $\mathbf{P}^{n}_{i,q_{1}q_{2}}=[P^{n}_{i,ss}(q_{1}q_{2}),P^{n}_{i,sp}(q_{1}q_{2%
}),P^{n}_{i,pp}(q_{1}q_{2})]^{T}$ denotes the corresponding coupling from the Wannier states to the continuum Bloch states. Following the standard procedure of eliminating the continuum within the Markov approximation Barnett we obtain effective coupled equations for the Wannier states, $\mathbf{\dot{C}}^{0}=\mathcal{M}(0)\mathbf{C}^{0}$. The coupling matrix $\mathcal{M}(0)$ is expressed as
$$\displaystyle\mathcal{M}(0)$$
$$\displaystyle=$$
$$\displaystyle-\left[i\mathbf{U}+\frac{2}{\pi}\left[\frac{2D}{3\Omega_{\rm eff}%
}\right]^{2}\mathbf{\Gamma}(0)\right],$$
(6)
$$\displaystyle\mathbf{\Gamma}(0)$$
$$\displaystyle=$$
$$\displaystyle\sum_{n}\int\int dq_{1}dq_{2}\left[\mathbf{P}^{n}_{i,q_{1}q_{2}}%
\right]\left[\mathbf{P}^{n}_{i,q_{1}q_{2}}\right]^{\dagger},$$
(7)
where we have introduced a single-site decay matrix $\mathbf{\Gamma}(0)$.
One immediately notices that in the absence of interference effects, the decay rate of each channel will be proportional to $D^{2}$ [compare Eq.(6)]. The full time dependent solution for the coefficient $\mathbf{C}^{0}(t)$ is
$$\mathbf{{C}}^{0}(t)=\sum^{3}_{l=1}c^{0}_{l}\exp[-\Gamma_{l}(0)t-i\epsilon_{l}(%
0)t]\mathbf{u}_{l}(0)$$
where $\mathbf{u}_{l}(0)$’s are the eigenvectors of the matrix $\mathcal{M}(0)$, $\Gamma_{l}(0)$ denotes the decay rate and $\epsilon_{l}(0)$ is the energy of the respective eigenstates. The coefficients $c^{0}_{l}$ can be derived from the initial conditions. Only when $\Gamma_{l}(0)\ll\epsilon_{l}(0)$, the corresponding eigenstate $\mathbf{u}_{l}(0)$ can be considered as the physical metastable eigenstate with energy $\epsilon_{l}$.
In Fig. 2(a) we present the decay rates corresponding to different eigenvalues.
For high dipolar strength $D$, all three decay rates are relatively large indicating fast decaying states. The channels represented by blue and red lines show decay rates proportional to $D^{2}$. The most interesting channel for our purpose is the state with the lowest decay rate (the black line in Fig.2(a)). This state shows a decay rate scaling slower than the expected $D^{2}$. It is expressed in terms of the $s$ and $p$ orbitals as,
$$\left|\phi\right\rangle_{i}=\left[\beta_{1}(\hat{s}^{\dagger}_{i})^{2}+\beta_{%
2}(\hat{p}^{\dagger}_{i})^{2}\right]\left|0\right\rangle$$
(8)
For this state we find that the ratio between the decay rate and the energy lies in the range, $\Gamma_{\phi}(0)/\epsilon_{\phi}(0)=0.1\rightarrow 1$ as the dipolar strength changes from $10\rightarrow 50$.
Thus in that range of $D$, the state $\left|\phi\right\rangle$ has a meaning as a quasi-bound eigenstate, but with a strong decay constant. The strong decay rates for all the eigenstates indicate that insulating states where the particles are localized to single sites are unstable. Then due to the strong Quantum-Zeno like effect, one can not reach a state with two or more particles per site due to the blockade of single-particle tunneling essentially ensuring a hardcore limit.
Two sites coupled to a continuum : We consider now two neighbouring sites
$i$ and $j$ with a single pair localized in either of the sites. Due to the action of $H_{WB}$, states in the neighbouring sites will be coupled via transitions to the common continuum. Following similar arguments as in the single-site case, the state of a pair distributed among sites $i$ and $j$ is written as,
$$\displaystyle\left|\Phi\right\rangle_{ij}$$
$$\displaystyle=$$
$$\displaystyle\sum^{3}_{l=1}C^{|i-j|}_{l}\left|l\right\rangle_{i}\left|0\right%
\rangle_{j}\left|\mathbf{0}\right\rangle+\sum^{6}_{l=4}C^{|i-j|}_{l}\left|0%
\right\rangle_{i}\left|l-3\right\rangle_{j}\left|\mathbf{0}\right\rangle$$
(9)
$$\displaystyle+$$
$$\displaystyle\sum_{n_{1}n_{2};q_{1}q_{2}}\alpha^{n_{1}n_{2}}_{q_{1}q_{2}}\left%
|0\right\rangle_{i}\left|0\right\rangle_{j}\left|\mathbf{1}_{q_{1}}\mathbf{1}_%
{q_{2}}\right\rangle.$$
Following the method outlined for a single site, by eliminating the continuum, we get a set of coupled equations for the Wannier states coefficients, $\mathbf{\dot{C}}^{|i-j|}=\mathcal{M}(|i-j|)\mathbf{C}^{|i-j|}$, where the coupling matrix $\mathcal{M}(|i-j|)$ is
$$\displaystyle\mathcal{M}(|i-j|)$$
$$\displaystyle=$$
$$\displaystyle-\left[i\mathbf{U}_{|i-j|}+\frac{2}{\pi}\left[\frac{2D}{3\Omega_{%
\rm eff}}\right]^{2}\mathbf{\Gamma}(|i-j|)\right],$$
(10)
$$\displaystyle\mathbf{\Gamma}(|i-j|)$$
$$\displaystyle=$$
$$\displaystyle\sum_{n}\int\int dq_{1}dq_{2}\left[\mathbf{P}^{n}_{ij,q_{1}q_{2}}%
\right]\left[\mathbf{P}^{n}_{ij,q_{1}q_{2}}\right]^{\dagger},$$
(11)
with the two-site decay matrix $\mathbf{\Gamma}(|i-j|)$.
Due to a lack of the direct coupling between the Wannier states at different sites, $\mathbf{U}$ is block diagonal. The non-zero elements linking different sites of the discrete-continuum coupling array $\mathbf{P}^{n}_{ij,q_{1}q_{2}}=[\mathbf{P}^{n}_{i,q_{1}q_{2}},\mathbf{P}^{n}_{%
j,q_{1}q_{2}}]^{T}$
will induce an additional effective hopping terms for the pairs from site $i$ to site $j$.
The full time dependent solution of the problem now reads
$\mathbf{{C}}^{|i-j|}(t)=\sum^{6}_{l=1}c^{ij}_{l}\exp\left[-\Gamma_{l}(|i-j|)t-%
i\epsilon_{l}(|i-j|)\right]\mathbf{u}_{l}(|i-j|)$
where $\mathbf{u}_{l}$ is the eigenvector of the matrix $\mathcal{M}$, $\Gamma_{l}(|i-j|)$ and $\epsilon_{l}(|i-j|)$ denote the decay rate and the energy of the lth eigenstate for the pair of sites $ij$. In Fig. 2(b) we plot the decays rates for two neighbouring sites $|i-j|=1$. Let us concentrate on the states with the low decay rates (the black line and the black-circled line). All the other channels (denoted by red and blue curves) have decay rates proportional to $D^{2}$, which points towards absence of interference effects. The states with low decay rates show a much different and slower scaling as a function of $D$. The corresponding states can be approximately expressed as symmetric and anti-symmetric combinations of the single-site eigenstates $\left|\pm\right\rangle_{ij}=\left(\left|\phi\right\rangle_{i}\pm\left|\phi%
\right\rangle_{j}\right)/\sqrt{2}$ with energies $\epsilon_{\pm}$ with $\epsilon_{+}<\epsilon_{-}$. The states $\left|\phi\right\rangle_{i,j}$ have the form shown in
Eq. (8). The overlap of these states with the exact eigenstates: $\left|\left\langle\sigma\right|\sigma^{\prime}\rangle_{exact}\right|_{ij}%
\approx\mathcal{F}\delta_{\sigma\sigma^{\prime}}$ is large with $\mathcal{F}\sim 0.95$, where $\sigma,\sigma^{\prime}=\pm$. The deviation from the perfect overlap is due to the fact that there is an additional continuum induced off-site transition between states with opposite parity, $\left|\phi\right\rangle_{i}\leftrightarrow sign(i-j)\hat{s}^{\dagger}_{j}\hat{%
p}^{\dagger}_{j}\left|0\right\rangle$.
The state with the lowest decay rate (the black line in Fig.2b) corresponds to the state $\left|-\right\rangle_{ij}$ with higher energy. When comparing to the single-site case we find that the decay rate of the $\left|-\right\rangle_{ij}$ is smaller by more that one order of magnitude. For this state we find that the ratio between the decay rate and the energy lies in the range, $\Gamma_{-}(1)/\epsilon_{-}(1)=0.01\rightarrow.05$ as the dipolar strength changes from $10\rightarrow 50$. On the other hand, for the state $\left|+\right\rangle_{ij}$, for the same dipolar range, $\Gamma_{+}(1)/\epsilon_{+}(1)=0.05\rightarrow 0.1$.
It follows that on the timescale of $\sim 1/\Gamma_{+}(1)\approx 10/E_{R}$, only the $\left|-\right\rangle_{ij}$ survives and will be populated. Therefore, we find a surprising situation in which a state is stabilized by delocalizing to the neighbouring site due to continuum-induced tunneling. This state shows apparently the phenomenon of population trapping arimondo . While the physics seems to be quite similar to a strong laser field induced trapping arimondo let us stress that the “dark state” now entangles two distinct lattice sites. We like to point out that in our scenario both the decay and delocalization is induced by strong coupling to the continuum. We also find that the effect of the continuum-assisted coupling for sites with $|i-j|>1$ is much smaller within the regime of dipolar strengths studied.
Continuum-assisted creation of dimer states: Next we discuss the creation of dimer states due to population trapping for the half-filling of the pairs. It is known that strong dipolar interaction induces a density-wave phase where the pairs arrange in a checkerboard pattern batrouni . As such pairs are pinned to the sites, the checkerboard configuration will not be stable as each pair occupied site will decay rapidly to the continuum. The stable configuration can only have states containing the delocalized state $\left|-\right\rangle_{ij}$. Then, in the limit of strong interaction and for half-filling of pairs, $\left|-\right\rangle_{ij}$ will cover the whole region of lattice sites. The resulting many-body state is a checkerboard state of nearest-neighbour dimers, $\left|\Psi\right\rangle=\Pi_{i}\left|-\right\rangle_{2i,2i+1}$ or $\Pi_{i}\left|-\right\rangle_{2i-1,2i}$. These dimer states are the ground states of the celebrated Majumder-Ghosh (MG) model mg . This paradigmatic model consists
of a frustrated one-dimensional spin chain consisting of nearest and next-nearest neighbour hopping with a particular ratio. The dimer state is characterized by an absence of long-range correlations, $\left\langle\hat{b}^{\dagger}_{i}\hat{b}_{j}\right\rangle_{\Psi}=\left\langle%
\hat{n}_{i}\hat{n}_{j}\right\rangle=0$ for $|i-j|>1$. This dimerized state can be thought of as the simplest form of the valance-bond solid with short-range correlations and with double the period of the original lattice. This doubling of periodicity can form an experimental signature in the time of flight image due to the reduction of the Brillouin zone. The required temperature to reach this phase depends on the delocalization energy which is given by the energy difference $\delta E$ between the single-site state $\left|\phi\right\rangle_{i}$ and the dimer state $\left|-\right\rangle_{ij}$. For a dipolar strength of $D\sim 20$ (near the lowest decay rate in Fig.2) this energy difference is of the order of $0.4E_{R}$. For RbCs
molecules, these parameters correspond to a dipole moment of $\sim 0.7$Debye with a lattice constant $\sim 500$nm. Then the relevant temperature scale to observe this phase is $\sim 50$nK. Such a temperature is much larger than the one needed to reach the super-exchange regime for the ultracold atoms, and thus it is much easier to access experimentally. The price to pay in our present case is the meta-stability of the dimerized state with lifetime $\sim 10$ms. Though one way to increase the stability is by switching off the electric field within the decay time, which makes all the interaction term vanish . Then $\left|\Psi\right\rangle$ becomes a highly excited state for a new free particle Hamiltonian. The normal tunneling rate for lattice depth of $7E_{R}$ is on the order of $\sim 0.3E_{R}$(in the p-band), which is much lower than the energy $\epsilon_{-}\sim 8E_{R}$. Subsequently, the system can not relax to the ground state by releasing the energy, making the dimer state stable for much longer time. Such way of stabilizing an excited state
has already been
demonstrated by preparing repulsive bound pairs in Ref.Winkler .
IV Constructing effective Hamiltonian for low filling
In this section we discuss a possible way to construct an effective Hamiltonian in terms of local operators for low density of the pairs. To do this, we consider a simple system where one pair of atoms is moving in three sites coupled to the continuum.
Following the same procedure as before we derive the full coupling matrix $\mathcal{M}(i,i+1,i+2)$ for three sites. Studying eigenstates related to low decay rates we find, as before,
that population trapping occurs due to coupling of the neighbouring sites via $\left|\pm\right\rangle_{ij}$ states. Subsequently, a tunneling Hamiltonian in terms of the states $\left|\pm\right\rangle_{ij}$ is given by,
$H_{i,i+1,i+2}=-J_{\rm eff}(\left|-\right\rangle_{i,i+1}\left\langle-\right|_{i%
+1,i+2}+\alpha\left|+\right\rangle_{i,i+1}\left\langle+\right|_{i+1,i+2}+H.c.)$, where
$\alpha$ can be extracted from the eigenvalues of the effective coupling matrix $\mathcal{M}(i,i+1,i+2)$. As the states $\left|-\right\rangle_{i,i+1},\left|\pm\right\rangle_{i+1,i+2}$ are not orthogonal, it is convenient to rewrite $H_{i,i+1,i+2}$ in terms of local orthogonal operators. To do that we define a pair local operator, $\left|\phi\right\rangle_{i}=b^{\dagger}_{i}\left|0\right\rangle$ which creates a pair at site $i$ in the lowest decay state. Then the composite operators satisfy the bosonic commutation relations $[b_{i},b^{\dagger}_{j}]=\delta_{ij}$. In terms of these local operators we can rewrite the states $\left|\pm\right\rangle_{ij}=\frac{1}{\sqrt{2}}\left[b^{\dagger}_{i}\pm b^{%
\dagger}_{j}\right]\left|0\right\rangle$. Subsequently, the Hamiltonian $H_{i,i+1,i+2}$ is re-expressed as,
$$\displaystyle\frac{H_{i,i+1,i+2}}{J_{\rm eff}}$$
$$\displaystyle=$$
$$\displaystyle-\frac{1+\alpha}{2}\left[b^{\dagger}_{i}b_{i+1}+b^{\dagger}_{i+1}%
b_{i+2}+h.c\right]$$
$$\displaystyle+$$
$$\displaystyle(1-\alpha)b^{\dagger}_{i+1}b_{i+1}+\frac{1-\alpha}{2}\left[b^{%
\dagger}_{i}b_{i+2}+b^{\dagger}_{i+2}b_{i}\right]$$
The value of $J_{\rm eff}$ and $\alpha$ is derived by comparing the energies of Hamiltonian in (IV) and the energies of the states with three lowest decay rates derived from the full coupling matrix $\mathcal{M}(i,i+1,i+2)$. For low value of dipolar strength $D$, we find that $\alpha\approx 1$, thus the long-range tunneling is small and one recovers the usual picture with only nearest-neighbour tunneling. But, for higher dipolar strengths, due to the different decay rates between $\left|\pm\right\rangle$ states, $\alpha\neq 1$. Thus, we see from Eq.(IV) that such values of $\alpha$ result in an effective model with next-nearest neighbour tunneling inducing frustration. We like to point out that, in the present situation, the origin of such frustration is entirely different from the usual origin of such terms due to the higher order processes in solid-state systems faze .
At this point, we write down the effective many-body Hamiltonian including long-range dipolar interaction and involving all sites as,
$$\displaystyle H_{\rm eff}$$
$$\displaystyle=$$
$$\displaystyle\sum_{i}H_{i,i+1,i+2}+\frac{D}{\pi^{3}}\sum_{ij}\frac{n_{i}n_{j}}%
{|i-j|^{3}}-\mu\sum_{i}n_{i}$$
(13)
$$\displaystyle=$$
$$\displaystyle-J_{\rm eff}(1+\alpha)\sum_{<ij>}b^{\dagger}_{i}b_{j}+J_{\rm eff}%
\frac{1-\alpha}{2}\sum_{<<ij>>}b^{\dagger}_{i}b_{j}$$
$$\displaystyle+$$
$$\displaystyle\frac{2D}{\pi^{3}}\sum_{i\neq j}\frac{n_{i}n_{j}}{|i-j|^{3}}-\mu%
\sum_{i}n_{i},$$
where we have introduced the chemical potential $\mu$ for the pairs and $<<ij>>$
is a shorthand for next nearest neighbour summation index. The Hamiltonian in Eq.(13) contains two sources of frustration: i) the effective next-nearest neighbour tunneling, and ii) long-range dipolar interaction. For our present system, the deviation of dipolar interaction from the cubic power law is negligible Carr . Such Hamiltonian is a generalization of the $J_{1}-J_{2}$ model where the interaction is present only to the next-nearest neighbour. These models are prototypes for studying the effect of frustration and emergence of various proposed exotic phases in magnetic materials mila . The single particle dispersion relation for this Hamiltonian is given by
$\epsilon_{q}=J_{\rm eff}(1+\alpha)\cos qa+J_{\rm eff}\frac{1-\alpha}{2}\cos 2qa$. For $\frac{1-\alpha}{1+\alpha}>1/2$ it shows two minima at wavevectors $\pm Qa=cos^{-1}\left[-\frac{1+\alpha}{2(1-\alpha)}\right]$. In our case, the two-minima limit corresponds to $D>18$. In the low-density limit, one way to treat the problem is by going to the two-component homogenous Bose gas limit vekua with the effective Hamiltonian,
$$H_{\rm eff}=\int\left[\frac{1}{2}T_{1}(\rho^{2}_{1}+\rho^{2}_{2})+T_{12}\rho_{%
1}\rho_{2}-\mu(\rho_{1}+\rho_{2})\right]dx,$$
(14)
where $\rho_{1,2}$ are the densities of the two component Bose gas centered around the the minima $\pm Q$ and $T_{1},T_{12}$ are the renormalized intra-component and inter-component interaction. A detailed discussion of the Hamiltonian (14) is presented in the methods section. For a short-range $J_{1}-J_{2}$ model, the phase diagram from such a procedure shows qualitative agreement with more involved Density-Matrix Renormalization Group simulations vekua . When $T_{1}<T_{12}$, the mean-field ground state solution is given by the phase-separated state $\rho_{1}\neq 0,\rho_{2}=0$ or $\rho_{1}=0,\rho_{2}\neq 0$. Choosing one of the ground state will break the discrete symmetry which will result in true long-range order (LRO) even in one-dimension. The nature of this phase can readily be observed by writing the wavefunction in phase space, $\psi_{s}=\sqrt{\rho_{s}}\exp[-i\theta_{s}]$, with $s=1,2$. When $\rho_{1}=0$, we see that $\langle\hat{b}^{\dagger}_{i}\rangle=\sqrt{\rho_{1}}\exp[-iQx+\theta]$. Such a "cone" phase is identified as a the vector-chiral (VC) phase which breaks the $\mathcal{Z}_{2}$ symmetry. In contrast when $T_{1}>T_{12}>0$, we have a mixed state with equal density from both components. This homogeneous solution with $\rho_{1}=\rho_{2}$ is known as the two-component Tomonaga-Luttinger (TLL${}_{2}$) liquid. There can be another possibility when the effective inter-species interaction is attractive $T_{1}>0,T_{12}<0$. In this situation, intra-component bound states with emerge with center of mass momentum $\sim 2Q$. Such bound states with finite momenta are usually not present in the anti-ferromagnetic model vekua . In present case, these bound states are direct consequence of the long-range nature of the dipolar interaction which can induce resonances chiara . The quasi-condensate of such bound pairs can give rise to a spin-nematic phase zhito , or spin-density wave phase hikihara , a detailed discussion of which is beyond
the scope of
current article. In the Fig. 3, we have plotted the
the resulting phase diagram in the $D-\mu$ parameter space for vanishingly small $\mu$. We find that the vector-chiral phase is stable for smaller and larger values of the dipolar strength $D$. In between the homogeneous TLL${}_{2}$ phase is the ground state. For larger values of chemical potential $\mu$, one finds that there is a bound state phase due to $T_{12}<0$.
V Conclusions
Summarizing, in the present article we have demonstrated a novel approach to theoretically model strongly interacting molecules in optical lattices.
Our approach is a new alternative to the multiband expansions that suffer serious convergence problems larson2009 ; johnson2009 ; will2010 ; dutta2011 ; jurgensen2012 ; luhmann2012 ; lacki2013 .
We have explored a mathematical analogy between the system studied and
strong bound-continuum couplings present in the
theory of strong field ionization. We have found that the phenomenon of population trapping, well established in quantum optics interference effect, in the present context is responsible for frustration in the system in a form of dimerization and next-nearest neighbour tunneling. One strong point of our proposal is that the required temperature scale is much higher than the one corresponding to the usual super-exchange regime. Our results can be generalized to higher dimensions, where one can
look for simulation of spin liquids, and valance bond crystals diep . Our method can also be extended to other strongly interacting systems, such as atoms in optical lattices, strongly-coupled cavity-QED systems jin , recently proposed nano-plasmonic lattices chang , and possible lattice geometries for the indirect excitons with strong dipolar interactions rem . We hope that further progress can be obtained in studies of strongly inteacting systems by exploring analogies with strognly coupled quantum optics problems in general, and strong field ionisation theory in particular.
VI Acknowledgements
We thank L. Barbiero, O. Jürgensen, D.-S. Lühmann, and C. Menotti for enlightening discussions. The work of O.D. and J.Z. has been supported by Polish
National Center for Science within project No. DEC-2012/04/A/ST2/00088.
M.L. acknowledge financial support from Spanish
Government Grant TOQATA (FIS2008-01236), EU IP SIQS, EU STREP EQuaM, and ERC Advanced
Grants QUAGATUA and OSYRIS.
VII Methods
Microscopic model :
The many-body Hamiltonian in terms of the field operators is given by $H=H_{\rm kin}+H_{\rm int}$,
with $H_{\rm kin}=\int d\mathbf{r}\left[-\Phi^{\dagger}(\mathbf{r})\frac{\hbar^{2}%
\nabla^{2}}{2m_{b}}\Phi(\mathbf{r})\right]$ and $H_{\rm int}=\frac{1}{2}\int d\mathbf{r}d\mathbf{r^{\prime}}\left[\Phi^{\dagger%
}(\mathbf{r})\Phi^{\dagger}(\mathbf{r^{\prime}})V_{\rm dd}(\mathbf{r-r^{\prime%
}})\Phi(\mathbf{r})\Phi(\mathbf{r})\right]$. Representing the field operators $\Phi(\mathbf{r})$ by local site operators as in Eq. (1) and performing appropriate integrations one gets the Hamiltonian in Eq.(2). Before finding the parameters of the Hamiltonian (2), we look into the property of the Bloch states. Let us denote the Bloch band $n_{0}$ as the start of the continuous bands with $E_{n_{0}}$ as the minimum of energy and $E_{n_{0}}\gg 1$. Then the energy of the Bloch band $n=n_{0}+m$ can be written as, $E_{n=n_{0}+m}(q)=E_{n_{0}}+2n_{0}m+m^{2}+2(n_{0}+m)|q|+q^{2}$ for even $m$ and one can get similar results for odd $m$. Moreover, we write the Bloch wavefunctions in the $E_{n_{0}}\gg 1$ limit as Mathieu ,
$\mathcal{U}_{nq}(x)\approx\sqrt{\frac{2}{L}}\exp[iqx]\cos\left[\sqrt{E_{n}-q.^%
{2}-V/2}x\right],$
for even $n$ and $\mathcal{U}_{nq}(x)\approx\sqrt{\frac{2}{L}}\exp[iqx]\sin\left[\sqrt{E_{n}-q.^%
{2}-V/2}x\right]$ for $n$ odd. As these functions are eigenstates, they are also orthogonal, $\int U^{*}_{nq}(x)U^{*}_{mq^{\prime}}(x)dx=\delta_{n,m}\delta_{q,q^{\prime}}.$
Now we write down the discrete-continuum coupling matrix elements as,
$$\displaystyle P^{n}_{i,p_{1}p_{2}}(q_{1}q_{2})$$
$$\displaystyle=$$
$$\displaystyle\int U^{*}_{nq_{1}}(x)U^{*}_{nq_{2}}(x^{\prime})V_{\rm dd}(%
\mathbf{r-r^{\prime}})\omega^{p_{1}}_{i}(x^{\prime})\omega^{p_{2}}_{i}(x)$$
$$\displaystyle\times$$
$$\displaystyle|\phi_{0}(z)\phi_{0}(y)|^{2}|\phi_{0}(z^{\prime})\phi_{0}(y^{%
\prime})|^{2}d\mathbf{r}d\mathbf{r^{\prime}},$$
$$\displaystyle\approx$$
$$\displaystyle\frac{1}{4}\int dkV_{\rm dd}(k)\left[\mathcal{W}^{\sigma_{1}}_{i}%
(k-q_{1}+\sqrt{E_{n_{1}}(q_{1})})\right.$$
$$\displaystyle\times$$
$$\displaystyle\left.\mathcal{W}^{\sigma_{2}}_{i}(-k-q_{2}-\sqrt{E_{n_{2}}(q_{1}%
)})+\right.$$
$$\displaystyle\left.\mathcal{W}^{\sigma_{1}}_{i}(k-q_{1}-\sqrt{E_{n_{1}}(q_{1})%
})\right.$$
$$\displaystyle\times$$
$$\displaystyle\left.\mathcal{W}^{\sigma_{2}}_{i}(-k-q_{2}+\sqrt{E_{n_{2}}(q_{1}%
)})\right],$$
where $\mathcal{W}^{\sigma}_{i}(k)$ is the Fourier transform of the Wannier function $\omega^{\sigma}_{i}(x)$.
In deriving the above form, we have used the orthogonality condition between the Bloch functions and assumed that $E_{n}\gg 1$.
Additionally, in the Hamiltonian (2), we have neglected terms corresponding to processes like $\hat{a}^{\dagger}_{nq_{1}}\hat{s}^{\dagger}_{i}\hat{s}_{i}\hat{s}_{i}$ where one particle is coupled to the continuum. The transition amplitudes for such processes contains
convolution sums of the form
$$S_{n}\sim\int\mathcal{W}^{\sigma_{1}}(k+q+\sqrt{E_{n}})\mathcal{W}^{\sigma_{2}%
}(-k)V_{\rm dd}(k)dk.$$
As $E_{n}\gg 1$, such terms are negligibly small. Thus we ignored them in comparison to the leading two-particle transition amplitudes.
While deriving the Hamiltonian (2), we have taken into account the interaction with the continuum using the Markov approximation. We find that the interaction between the molecules in continua is also strong with strength $\propto D$. In this limit, the kinetic energy of each such pair with momenta $q_{1},q_{2}$ at band $n$ can be written as $E_{n}(q_{1})+E_{n}(q_{2})-2fD$, where the constant $f\sim O(1)$. In the limit of $D>>1$, this reduces the minimum of the energy of the pairs in the continuum below that of the discrete states which enforces the Markov approximation while integrating out the continuum.
In term of Wannier states the various interaction parameters in (2) are expressed as,
$$\displaystyle U_{\rm ss}=\int\int$$
$$\displaystyle d\mathbf{r}d\mathbf{r^{\prime}}$$
$$\displaystyle|\omega^{s}_{i}(x)\omega^{s}_{i}(x^{\prime})|^{2}V_{\rm dd}(%
\mathbf{r-r^{\prime}})|\phi_{00}(y,z,y^{\prime},z^{\prime})|^{2}$$
$$\displaystyle U_{\rm ps}=\int\int$$
$$\displaystyle d\mathbf{r}d\mathbf{r^{\prime}}$$
$$\displaystyle|\omega^{p}_{i}(x)\omega^{s}_{i}(x^{\prime})|^{2}V_{\rm dd}(%
\mathbf{r-r^{\prime}})|\phi_{00}(y,z,y^{\prime},z^{\prime})|^{2}$$
$$\displaystyle U_{\rm pp}=\int\int$$
$$\displaystyle d\mathbf{r}d\mathbf{r^{\prime}}$$
$$\displaystyle|\omega^{p}_{i}(x)\omega^{p}_{i}(x^{\prime})|^{2}V_{\rm dd}(%
\mathbf{r-r^{\prime}})|\phi_{00}(y,z,y^{\prime},z^{\prime})|^{2}$$
$$\displaystyle T_{\rm ps}=\int\int$$
$$\displaystyle d\mathbf{r}d\mathbf{r^{\prime}}$$
$$\displaystyle\omega^{p}_{i}(x)\omega^{p}_{i}(x^{\prime})\omega^{s}_{i}(x)%
\omega^{s}_{i}(x^{\prime})V_{\rm dd}(\mathbf{r-r^{\prime}})$$
$$\displaystyle\times|\phi_{00}(y,z,y^{\prime},z^{\prime})|^{2}$$
$$\displaystyle\phi_{00}(y,z,y^{\prime},z^{\prime})$$
$$\displaystyle=$$
$$\displaystyle\phi_{0}(y)\phi_{0}(y^{\prime})\phi_{0}(z)\phi_{0}(z^{\prime})$$
(15)
We can then rewrite the coupling matrix between the Wannier states as,
$$\mathbf{U}=\left(\begin{array}[]{ccc}U_{\rm ss}&0&T_{\rm ps}\\
0&E_{1}+U_{\rm ps}&0\\
T_{\rm ps}&0&2E_{1}+U_{\rm pp}\end{array}\right)$$
The same coupling matrix for two site $i$ and $j$ can be expressed as,
$$\mathbf{U}_{|i-j|}=\left(\begin{array}[]{cc}\mathbf{U}&\mathbf{0}\\
\mathbf{0}&\mathbf{U}\end{array}\right)$$
.
Two-component Bose gas limit of Eq.(13) :
We rewrite our Hamiltonian (13) in the conventional $J_{1}-J_{2}$ form as,
$$\displaystyle H_{\rm eff}$$
$$\displaystyle=$$
$$\displaystyle J_{1}\sum_{<ij>}b^{\dagger}_{i}b_{j}+J_{2}\sum_{<<ij>>}b^{%
\dagger}_{i}b_{j}$$
(16)
$$\displaystyle+$$
$$\displaystyle V\sum_{ij}\frac{n_{i}n_{j}}{|i-j|^{3}}-\mu\sum_{i}n_{i},$$
where $J_{1},J_{2}$ are the nearest and next-nearest neighbour tunneling and $V$ is the strength of the long-range interaction. In the dilute limit, such a system, with nearest and next-nearest neighbour interaction only, has been solved qualitatively by mapping the problem to a two-component Bose gas modelvekua . Here we extend this treatment to include long-range dipolar interaction. To do that we transform the Hamiltonian to the momentum space,
$$H_{\rm eff}=\sum_{q}\epsilon_{q}b^{\dagger}_{q}b_{q}+\sum_{k,k^{\prime},q}V(q)%
b^{\dagger}_{k+q}b^{\dagger}_{k^{\prime}-q}b_{k^{\prime}}b_{k}-\mu\sum_{i}b^{%
\dagger}_{q}b_{q},$$
(17)
where the dispersion relation is given by $\epsilon_{q}=2J_{1}\cos qa+2J_{2}\cos 2qa$ and the interaction energy in momentum space is given by, $V(q)=U+2V\sum^{\infty}_{n=1}\cos nqa/n^{3}$, where the hard-core constraint is given by $U\rightarrow\infty$. We only consider the dilute limit, $\mu\rightarrow 0$. When $J_{2}>J_{1}/4$, the dispersion relation has two minima at wavevectors, $Qa=\cos^{-1}\left[-J_{1}/4J_{2}\right]$. Around these minima, we can write the dispersion relation as, $\epsilon_{Q+k}=\epsilon_{Q}+\hbar^{2}k^{2}/2m^{*}$, where $m^{*}$ is the effective mass. Then we expand the boson operator near the two minima, $b_{k}=\phi_{1,Q+k}+\phi_{2,-Q+k}+\phi_{k}$, where $\phi_{1}$ and $\phi_{2}$ are the two-component Bose gas centered around momentum $\pm Q$ respectively, while $\phi_{k}$ denotes the high momentum contribution, which is integrated out. Then one can re-express the Hamiltonian (17) in terms of the $\phi_{1,2}$ which in position space reads,
$$\displaystyle H_{\rm eff}$$
$$\displaystyle=$$
$$\displaystyle\int dx\left[\sum_{\sigma=1,2}\left[-\phi^{\dagger}_{\sigma}\frac%
{\hbar^{2}}{2m^{*}}\nabla^{2}_{x}\right]\phi_{\sigma}\right.$$
(18)
$$\displaystyle+$$
$$\displaystyle\left.\frac{1}{2}T_{1}(\rho^{2}_{1}+\rho^{2}_{2})+T_{12}\rho_{1}%
\rho_{2}-\mu(\rho_{1}+\rho_{2})\right],$$
where $T_{1}$ and $T_{12}$ are renormalized interactions. To find these renormalized interactions, we first write down the full Bethe-Salpeter equation,
$$T(k,k^{\prime};q)=V(q)-\int\frac{V(p-q)T(k,k^{\prime};p)}{\epsilon_{k+p}+%
\epsilon_{k^{\prime}-p}+\Omega}\frac{dp}{2\pi}.$$
(19)
In the dilute limit we can substitute $\Omega=2\mu$. Then the respective renormalized interaction is given by, $T_{1}=T(Q,Q,0)$ and $T_{12}=T(Q,-Q;0)+T(Q,-Q;2Q)$. Imposing the hard-core constraint with $U\rightarrow\infty$, we get an additional equation,
$$\int\frac{T(k,k^{\prime};p)}{\epsilon_{k+p}+\epsilon_{k^{\prime}-p}+\Omega}%
\frac{dp}{2\pi}=1.$$
Due to the form of the interaction $V(q)$, we expand the full renormalized interaction as, $T(k,k^{\prime};q)=A_{0}+\sum_{n}A_{n}\cos nqa$, where the coefficients $A_{0},A_{n}$ depends on $k,k^{\prime}$. Putting this ansatz in Eq.(19), we get a set of coupled equations for $m>0$,
$$A_{m}=\frac{2V}{m^{3}}-\frac{2V}{m^{3}}\sum^{\infty}_{m^{\prime}=0}A_{m^{%
\prime}}\int\frac{\cos mpa\cos m^{\prime}pa}{\hbar^{2}p^{2}/m^{*}+\Omega}\frac%
{dp}{2\pi}.$$
and from the constraint condition,
$$A_{0}=\sum^{\infty}_{m=0}\int\frac{A_{m}\cos mpa}{\hbar^{2}p^{2}/m^{*}+\Omega}%
\frac{dp}{2\pi}.$$
We found that in the limit of $\Omega\rightarrow 0$, the magnitude of the integral like $\int\frac{\cos mpa}{\hbar^{2}p^{2}/m^{*}+\Omega}\frac{dp}{2\pi}$ falls off when $m>1$. Then we get the following relation,
$$A_{m}=\frac{2V}{m^{3}}-\frac{2V}{m^{3}}\sum^{m+1}_{m^{\prime}=m-1}A_{m^{\prime%
}}\int\frac{\cos(m-m^{\prime})pa}{\hbar^{2}p^{2}/m^{*}+\Omega}\frac{dp}{2\pi}.$$
We numerically find convergent solution for the $A_{m}$ by taking $m_{max}=100$.
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Commentary on Guyll
et al. (2023): Misuse of Statistical Method Results in Highly Biased Interpretation of Forensic Evidence
Michael Rosenblum111Department of Biostatistics, Johns Hopkins University, Baltimore, MD 222Corresponding author: Michael Rosenblum mrosen@jhu.edu, Elizabeth T. Chin${}^{*}$, Elizabeth L. Ogburn${}^{*}$,
Akihiko Nishimura${}^{*}$, Daniel Westreich333Department of Epidemiology, UNC-Chapel Hill, Chapel Hill, NC, Abhirup Datta${}^{*}$, Susan Vanderplas444Department of Statistics, University of Nebraska Lincoln, Lincoln, NE,
Maria Cuellar555Departments of Criminology; Statistics and Data Science, University of Pennsylvania, Philadelphia, PA, William C. Thompson666Departments of Criminology, Law, and Society; Psychology and Social Behavior; and Law, University of California, Irvine, CA.
1 Summary
Since the National Academy of Sciences released their report outlining paths for improving reliability, standards, and policies in the forensic sciences NAS (2009), there has been heightened interest in evaluating and improving the scientific validity within forensic science disciplines. Guyll
et al. (2023) seek to evaluate the validity of forensic cartridge-case comparisons. However, they make a serious statistical error that leads to highly inflated claims about the probability that a cartridge case from a crime scene was fired from a reference gun, typically a gun found in the possession of a defendant. It is urgent to address this error since these claims, which are generally biased against defendants, are being presented by the prosecution in an ongoing homicide case where the defendant faces the possibility of a lengthy prison sentence (DC Superior
Court, 2023).
2 Error in Statistical Reasoning: Equiprobability Bias
Firearms examiners try to determine whether a cartridge case from a crime scene was fired from a reference gun.
They do this by comparing surface contour patterns on the crime scene cartridge case to those on cartridge cases fired from the reference gun. The two possible ground truth states are called “same source” and “different source”, meaning that the crime scene cartridge case was fired from the reference gun or some different gun, respectively.
A key goal of Guyll
et al. (2023) is to
estimate the conditional odds (called “posttest odds”) that a crime scene cartridge case was fired from a reference gun given the firearms examiner’s decision. Using Bayes rule, they represent the posttest odds as the product of (i) an assumed prior odds (called “pretest odds”) of same vs. different source ground truth, and (ii) the likelihood ratio (LR)111Guyll
et al. (2023) estimate the LR using data from an experiment to assess how frequently forensic examiners make a particular decision when the ground truth is same source or different source. We focus in this manuscript on the pretest odds, but refer the reader to Cuellar
et al. (2022) for information about potential sources of bias in the LR estimate. of the examiner’s decision given same vs. different source.
Next,
Guyll
et al. (2023, pp.3-4)
describe how the above Bayesian procedure can be used by
“triers of fact in the legal
system, such as judges and juries, who are tasked with evaluating
forensic decisions”; specifically, they state that such triers of fact can set “pretest odds equal 1” in (i) above to represent “a situation corresponding to being initially
unbiased and withholding all judgment as to a comparison’s ground-truth status”.222A related issue not addressed here is whether it is appropriate for forensic scientists (rather than triers of fact such as a judge or jury) to make determinations about prior beliefs/odds. See e.g., Thompson et al. (2013); Lund and
Iyer (2017) for discussions of this issue.
This claim is incorrect: in fact, such an assumption is substantially biased.
By definition, “pretest odds equal 1” represents the belief that, a priori (i.e., before the firearms examiner’s decision is known), it is equally likely that the ground truth is same source or different source. And while there are indeed two possible ground truths, there is no reason to assume that they are equally likely. This becomes more clear when we consider that the ground truth same source requires that a single gun—the reference gun—fired the cartridge case, while the alternative ground truth different source requires that the cartridge case was fired by any gun other than the reference gun.333More precisely, different source means that the crime scene cartridge case was fired by a “compatible” gun, i.e., one with class characteristics (e.g., the caliber) matching those of the crime scene cartridge case. In other words, the compatible guns are all the guns that are consistent with the gross characteristics of the crime scene cartridge case and are therefore candidates for having fired it.
If there are more than 2 possible guns that could have fired the crime scene cartridge case, then allocating 50% prior probability to the defendant’s gun and the remaining 50% prior probability to be divided in some manner across all the other guns is, in effect, biasing the pre-test odds toward the reference gun compared to any other individual gun.
The fallacy underlying the above error is the false claim that being unbiased about two possible ground truths (i.e., same, different source) implies that one should believe these ground truths are equally likely (Figure 1).
This fallacy is an example of what is sometimes called equiprobability bias (Gauvrit and
Morsanyi, 2014, p.119).
We further illustrate this bias using the following analogy. Suppose an expert is asked to determine, without knowing anything about your birthday besides that it is one of the 365 calendar days “compatible” with being a birthday, the probability that you have the same birthday as George Washington. What is an unbiased prior probability for having the same birthday vs. different birthdays? The probability of having the same birthday is approximately $1/365=0.3\%$ and that of having a different birthday is $364/365\approx 99.7\%$. But the logic of Guyll et al. implies that, instead of considering each possible birth date to be equally likely, an unbiased observer should instead consider the chances of the two scenarios “same birthday” and “different birthday” occurring to be equally likely at $50\%$ each.
This illustrates, intuitively, the problem with the above logic: the number of possible birthdays is critically important but is ignored, just as the number of candidate guns that could have produced the crime scene evidence is ignored.
Unlike birthdays, the number of candidate guns will typically be unknown, which makes determining the pretest odds challenging. Attempting to estimate an “unbiased” pretest odds is non-trivial, subjective, and likely sensitive to assumptions. It is out of the scope of our expertise to posit realistic but unbiased priors. However, Guyll
et al. (2023)’s assertion that an unbiased probability is universally 50% for every reference gun should be rejected out of hand.
3 Impact of Error on Claims about Probative Value of Firearms Examiner Decisions
Concretely, the erroneous claim of Guyll
et al. (2023, pp.3-4) of “being initially
unbiased and withholding all judgment as to a comparison’s ground-truth status” is equivalent to assuming a 50% prior probability that the reference gun (and no other gun) fired the crime scene cartridge case.
Multiplying the prior by the aforementioned LR using Equation 4 of Guyll
et al. (2023, p.3) propagates this error. In the upper left of Guyll et al.’s Table 3, for example, this approach leads to post-test odds of 177.458:1 (equivalent to 99.4% probability).
Dr. Guyll presented the above argument in an ongoing homicide case. Specifically, he asserted that if the firearms examiner made an “identification” decision (i.e., a “match”), then an initially unbiased trier of fact should now believe that there is a 99.4% probability that the crime scene cartridge case was fired from the reference gun (DC Superior
Court, 2023, p.67).
He goes on to state: “I would consider that to be
extreme [sic] strong support for making the judgment in line with
the forensic decision.” (DC Superior
Court, 2023, pp.69–70).
The argument is incorrect because it relies on the erroneous claim in the first sentence of this section.
Since the posttest odds are highly dependent on the pretest odds, the error of Guyll
et al. (2023, pp.3-4) is not innocuous; to the contrary, it can result in highly inflated estimates of the posttest odds, which could lead judges and jurors in criminal trials to grossly misinterpret the forensic evidence.
Consider the case where a firearms examiner’s decision is an “identification”. As described above, combining Guyll
et al. (2023)’s likelihood ratio estimate of $177.458$ with their 50%/50% prior on same vs. different source ground truth results in the posterior probability of same source $99.4$%. However, if one uses smaller priors, the posterior probability decreases rapidly. For simplicity, consider the case where there are $n$ possible guns and where the prior probability of same source ground truth is set to $1/n$.444In all of our examples, we use the framework of equiprobable events to calculate probabilities simply by enumerating the number of, in this case guns, comprising an event. In this case the event “same source” includes just one gun and the event “different source” includes many. One possible refinement would be to move away from the equiprobable events framework and, for example, weight guns as having more prior probability of having fired the bullet if, e.g., they were known to be used in previous similar crimes. As we increase $n$, the posterior probability of same source decreases from $99.4$% (n=2, equivalent to Guyll
et al. (2023)’s prior) to $64.2$% (n=100) to $15.1$% (n=1000) to $1.7$% (n=10,000) (Figure 2). In an urban area, of course, the number of possible firearms is likely quite large. We are not suggesting to base the prior odds solely on the number of possible firearms, rather using it illustratively to show the sensitivity of the posterior to the choice of prior.
Guyll
et al. (2023) misconstrue the experimental performance of forensic decisions with the probability that a forensic decision reflects the ground truth. In the Abstract, Guyll
et al. (2023, p.1) state, “Considering probative value, which is a decision’s usefulness for determining a comparison’s ground-truth state, conclusive decisions predicted their corresponding ground-truth states with near perfection.” As we showed above, the posterior, which Guyll
et al. (2023, p.4) equates to the probative value, is highly dependent on the prior. Guyll
et al. (2023)’s arbitrary and biased prior odds of 1 (which they also built into their experimental design by assigning same and different source test cases each with 50% probability) was used to compute the posttest odds, rendering the latter arbitrary and biased as well.
Guyll
et al. (2023) acknowledged that the prevalence in casework of same source ground truth (i.e., the marginal probability of same source ground truth) is unknown and therefore one should consider different possible values of it when computing the positive predictive value (posterior probability) of same source ground truth given a firearms examiner’s decision. However, this did not stop them from asserting that an unbiased trier of fact believes equal prior probabilities for same and different source ground truth, which as we argued above is incorrect. Guyll
et al. (2023)’s prior is not unbiased, rather an assumption that results in an estimated positive predictive value close to the upper bound of 100%.
Given the high reliance of the posterior probability on the choice of prior, we urge careful consideration of the assumptions that underlie priors, resulting uncertainty in the posterior probability, and interpretation of results in the legal context.
4 Conclusion
Guyll
et al. (2023) made a serious statistical error that could lead judges and jurors in criminal trials to grossly misunderstand how to interpret forensic evidence. The error should be acknowledged and immediately corrected.
5 Acknowledgments and Disclosures
M.R., E.T.C., and E.O. were supported in this research by a Nexus Award from Johns Hopkins University.
The opinions expressed herein are those of the authors and do not necessarily reflect the views of The Johns Hopkins University, the D.C. Public Defender Service (PDS), nor anyone else.
We mention the PDS because
M.R. is an expert witness for it in a homicide case where Dr. Guyll is an expert witness for the prosecution; each is paid for their work on this case, but no such funding was used to support the work on this Comment.
WT is an expert witness for the Innocence Project, which is involved in the same case as Amicus Curiae.
D.W. and A.D. report no conflicts and had no funding support for this work.
We thank Dr. Charles Poole of UNC-Chapel Hill for his helpful input.
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DBSN: Measuring Uncertainty through Bayesian Learning of Deep Neural Network Structures
Zhijie Deng, Yucen Luo, Jun Zhu, Bo Zhang
Department of Computer Science & Technology, Institute for Artificial Intelligence,
State Key Lab for Intell. Tech. & Sys., BNRist Center, THBI Lab, Tsinghua University
{dzj17,luoyc15}@mails.tsinghua.edu.cn, {dcszj,dcszb}@tsinghua.edu.cn
corresponding author
Abstract
Bayesian neural networks (BNNs) introduce uncertainty estimation to deep networks by performing Bayesian inference on network weights. However, such models bring the challenges of inference, and further BNNs with weight uncertainty rarely achieve superior performance to standard models. In this paper, we investigate a new line of Bayesian deep learning by performing Bayesian reasoning on the structure of deep neural networks. Drawing inspiration from the neural architecture search, we define the network structure as gating weights on the redundant operations between computational nodes, and apply stochastic variational inference techniques to learn the structure distributions of networks. Empirically, the proposed method substantially surpasses the advanced deep neural networks across a range of classification and segmentation tasks. More importantly, our approach also preserves benefits of Bayesian principles, producing improved uncertainty estimation than the strong baselines including MC dropout and variational BNNs algorithms (e.g. noisy EK-FAC).
DBSN: Measuring Uncertainty through Bayesian Learning of Deep Neural Network Structures
Zhijie Deng, Yucen Luo, Jun Zhu††thanks: corresponding author, Bo Zhang
Department of Computer Science & Technology, Institute for Artificial Intelligence,
State Key Lab for Intell. Tech. & Sys., BNRist Center, THBI Lab, Tsinghua University
{dzj17,luoyc15}@mails.tsinghua.edu.cn, {dcszj,dcszb}@tsinghua.edu.cn
1 Introduction
Bayesian deep learning aims at equipping the flexible and expressive deep neural networks with appropriate uncertainty quantification (MacKay, 1992; Neal, 1995; Hinton & Van Camp, 1993; Graves, 2011; Blundell et al., 2015; Gal & Ghahramani, 2016). Traditionally, Bayesian neural networks (BNNs) introduce uncertainty in the network weights, addressing the over-fitting issue which standard neural networks (NNs) are prone to. Besides, the predictive uncertainty derived from the weight uncertainty is also of central importance in practical applications, e.g., medical analysis, automatic driving, and financial tasks.
Modeling the uncertainty on network weights is plausible and well-evaluated (Blundell et al., 2015; Ghosh et al., 2018). However, BNNs usually preserve benefits of Bayesian principles such as well-calibrated predictions at the expense of compromising performance and hence are impractical in real-world applications (Osawa et al., 2019), due to various reasons.
On one hand, specifying a sensible prior for networks weights is difficult (Sun et al., 2019; Pearce et al., 2019).
On the other hand, the flexible variational posterior of BNNs comes with inference challenges (Louizos & Welling, 2017; Zhang et al., 2018; Shi et al., 2018). Recently, the efficient particle-based variational methods (Liu & Wang, 2016) have been developed with promise, but they still suffer from the particle collapsing and degrading issues for BNNs due to the high dimension of the weights and the over-parameterization nature of such models (Zhuo et al., 2018; Wang et al., 2019).
In this work, we investigate a new direction of Bayesian deep learning that performs Bayesian reasoning on the structure of neural networks while keeping the weights as point estimates. We propose an approach, named Deep Bayesian Structure Networks (DBSN). Specifically, in the spirit of differentiable neural architecture search (NAS) (Liu et al., 2019; Xie et al., 2019), DBSN builds a deep network by repeatedly stacking a computational cell in which any two nodes (i.e. tensors) are connected by redundant transformations (see Figure 1). The network structure is defined as the gating weights on these transformations, whose distribution is much easier to capture than those of the high-dimensional network weights. To jointly optimize the network weights and the parameterized distribution of the network structure, we adopt a stochastic variational inference paradigm (Blundell et al., 2015) and use the reparameterization trick (Kingma & Welling, 2013). One technical challenge is driving DBSN to achieve satisfying convergence, since the network weights can hardly fit all the structures sampled from the structure distribution. To overcome this challenge, we propose two techniques. First, we advocate reducing the variance of the sampled structures with a simple modification of the sampling procedure. Second, we suggest using a more compact structure learning space than that of NAS, to make the training more feasible and more efficient.
There are at least two motivations that make DBSN an appealing choice:
1) DBSN bypasses the frustrating difficulties of characterizing weight uncertainty and enables the performance-enhancing structure learning (Zoph & Le, 2016; Liu et al., 2019), so DBSN shall have better predictive performance than classic BNNs. 2) Previous analysis (Wang et al., 2019) shows that due to the over-parametrization nature of BNNs, the state-of-the-art inference algorithms for weight uncertainty can suffer from mode collapsing, as multiple configurations of weights with a fixed structure correspond to one single function. In contrast, DBSN compactly models the uncertainty of structure and performs inference in a much lower-dimensional space, avoiding this issue and hence being able to exhibit more calibrated predictive uncertainty. Moreover, in the perspective of NAS, DBSN is also promising as it provides another principled way to learn network structures by resorting to the Bayesian formalism instead of the widely used meta-learning formalism in differentiable NAS.
To empirically validate these hypotheses, we evaluate DBSN with extensive experiments. We first testify the data fitting and structure learning ability of DBSN on challenging classification and segmentation tasks. Then, we compare the quality of predictive uncertainty estimates via calibration, which is a common concern in the community. We further evaluate the predictive uncertainty on adversarial examples and out-of-distribution samples, drawn from shifted distributions from the training data, to verify whether the model knows what it knows.
At last, we perform an experiment to validate a promising application of DBSN in the one-shot NAS (Bender et al., 2018; Guo et al., 2019).
Surprisingly, across all the tasks, DBSN consistently achieves comparable or even better results than the strong baselines.
2 Background
We first review the necessary background for DBSN and then elaborate DBSN in the next section.
2.1 Stochastic Variational Inference for BNNs
Let $\mathcal{D}=\{(x_{i},y_{i})\}_{i=1}^{N}$ be a set of $N$ data points. BNNs are typically defined by placing a prior $p({\bm{v}})$ on some variables of interest (e.g., network weights or network structure) and the likelihood is $p(\mathcal{D}|{\bm{v}})$.
Directly inferring the posterior distribution $p({\bm{v}}|\mathcal{D})$ is intractable because it is hard to integrate w.r.t. ${\bm{v}}$ exactly.
Instead, variational BNNs (Hinton & Van Camp, 1993; Graves, 2011; Blundell et al., 2015) suggest approximating $p({\bm{v}}|\mathcal{D})$ with a $\bm{\theta}$-parameterized distribution $q({\bm{v}}|\bm{\theta})$ by minimizing the Kullback-Leibler (KL) divergence between them:
$$\begin{split}\displaystyle\min_{\bm{\theta}}D_{\mathrm{KL}}(q({\bm{v}}|\bm{%
\theta})\|p({\bm{v}}|\mathcal{D}))=-\mathbb{E}_{q({\bm{v}}|\bm{\theta})}[\log p%
(\mathcal{D}|{\bm{v}})]+D_{\mathrm{KL}}(q({\bm{v}}|\bm{\theta})\|p({\bm{v}}))+%
\log p(\mathcal{D}),\end{split}$$
(1)
where $\log p(\mathcal{D})$ is a constant w.r.t. $\bm{\theta}$ and usually omitted in the minimization. To solve problem (1), the most commonly used method is the low-variance reparameterization trick (Kingma & Welling, 2013; Blundell et al., 2015), which replaces the sampling procedure ${\bm{v}}\sim q({\bm{v}}|\bm{\theta})$ with the corresponding deterministic transformation ${\bm{v}}=t(\bm{\theta},\epsilon)$ with a sample of
parameter-free noise $\epsilon$, to enable the direct gradient back-propagation through $\bm{\theta}$.
2.2 Cell-based Differentiable Neural Architecture Search (NAS)
Cell-based NAS has shown promise (Zoph et al., 2018; Pham et al., 2018) and been developed to be differentiable for better scalability (Liu et al., 2019; Xie et al., 2019; Weng et al., 2019).
Generally, the network in cell-based differentiable NAS111We will refer to the cell-based differentiable NAS as NAS for short if there is no misleading. is composed of a sequence of cells (e.g., modules) which have the same internal structure and are separated by upsampling or downsampling modules. Every cell contains $B$ sequential nodes (i.e., tensors): ${\bm{N}}^{1},\dots,{\bm{N}}^{B}$. Each node ${\bm{N}}^{j}$ is connected to all of its predecessors ${\bm{N}}^{i}$ so long as $i<j$ by $K$ possible redundant operations $o^{(i,j)}_{1},\dots,o^{(i,j)}_{K}$, e.g., convolution, skip connection, pooling.
The network structure is defined as $\bm{\alpha}=\{\bm{\alpha}^{(i,j)}|1\leq i<j\leq B\}$ where $\bm{\alpha}^{(i,j)}\in\Delta^{K-1}$ corresponds to the gating weights on the $K$ available operations from ${\bm{N}}^{i}$ to ${\bm{N}}^{j}$. Therefore, the information gathered from ${\bm{N}}^{i}$ to ${\bm{N}}^{j}$ is a weighted sum of the outputs from $K$ different operations on ${\bm{N}}^{i}$ (we denote the set including the parameters of all the operations in the network as ${\bm{w}}$):
$${\bm{N}}^{(i,j)}=\sum_{k=1}^{K}\bm{\alpha}^{(i,j)}_{k}\cdot o^{(i,j)}_{k}({\bm%
{N}}^{i};{\bm{w}}).$$
(2)
Then, the node ${\bm{N}}^{j}$ is calculated by summing all the information from its predecessors:
$${\bm{N}}^{j}=\sum_{i<j}{\bm{N}}^{(i,j)}.$$
(3)
Meta-learning-like gradient descent is adopted for optimization to reduce the prohibitive computational cost needed by RL or evolution (Liu et al., 2019; Xie et al., 2019). However, the goal of the optimization is the network structure instead of the model performance. Thus, after training, this kind of NAS needs to prune the searched structure and re-train a new network model with the compact structure for performance comparison, which is labor-intensive and is avoided in our work.
3 Deep Bayesian Structure Networks
In this work, we propose a novel Bayesian structure learning approach for the deep neural networks. Concretely, we follow the network design of NAS but we view $\bm{\alpha}$ as Bayesian variables and ${\bm{w}}$ as point estimates (see the graphical model in Figure 1). To infer the posterior distribution $p(\bm{\alpha}|\mathcal{D},{\bm{w}})=\frac{p(\bm{\alpha})p(\mathcal{D}|\bm{%
\alpha},{\bm{w}})}{p(\mathcal{D})}$, where $p(\bm{\alpha})$ is the prior (we omit its dependency on the hyper-parameter $\bm{\theta_{0}}$ here), we adopt the techniques in Section 2.1.
We assume both the prior and the introduced variational are fully factorizable categorical distributions, namely, $p(\bm{\alpha})=\prod_{i<j}p(\bm{\alpha}^{(i,j)})$ and $q(\bm{\alpha}|\bm{\theta})=\prod_{i<j}q(\bm{\alpha}^{(i,j)}|\bm{\theta}^{(i,j)})$, where $\bm{\theta}=\{\bm{\theta}^{(i,j)}\in{\mathbb{R}}^{K}|1\leq i<j\leq B\}$ denotes the trainable categorical logits. We rewrite Eq. (1) and obtain the negative evidence lower bound (ELBO):
$$\begin{split}\displaystyle\mathcal{L}(\bm{\theta},{\bm{w}})=-\mathbb{E}_{q(\bm%
{\alpha}|\bm{\theta})}[\log p(\mathcal{D}|\bm{\alpha},{\bm{w}})]+D_{\mathrm{KL%
}}(q(\bm{\alpha}|\bm{\theta})\|p(\bm{\alpha})).\end{split}$$
(4)
Notably, minimizing $\mathcal{L}$ w.r.t. $\bm{\theta}$ and ${\bm{w}}$ corresponds to Bayesian inference on $\bm{\alpha}$ and maximum a posteriori (MAP) estimation of ${\bm{w}}$222This is because we use regularizor on weights, e.g., weight decay, to alleviate over-fitting., respectively. Thus, the optimization of the network structure and network weights can be unified as $\min_{\bm{\theta},{\bm{w}}}\mathcal{L}(\bm{\theta},{\bm{w}})$. To resolve this, we relax both $p(\bm{\alpha}^{(i,j)})$ and $q(\bm{\alpha}^{(i,j)}|\bm{\theta}^{(i,j)})$ to be the concrete distributions (Maddison et al., 2016). Then, samples $\bm{\alpha}$ from $q(\bm{\alpha}|\bm{\theta})$ are generated via the softmax transformation:
$$\bm{\alpha}=g(\bm{\theta},\bm{\epsilon})=\{\mathrm{softmax}({(\bm{\theta}^{(i,%
j)}+\bm{\epsilon}^{(i,j)})}/{\tau})\},$$
(5)
where $\bm{\epsilon}=\{\bm{\epsilon}^{(i,j)}\in\mathbb{R}^{K}|\bm{\epsilon}^{(i,j)}_{%
k}\sim\mathrm{Gumbel}\;\mathrm{i.i.d.}\}$ are the Gumbel variables and $\tau\in\mathbb{R}_{+}$ is the temperature. Then we derive the following gradient estimators:
$$\nabla_{\bm{\theta}}\mathcal{L}(\bm{\theta},{\bm{w}})=\mathbb{E}_{\bm{\epsilon%
}}[-\nabla_{\bm{\theta}}\log p(\mathcal{D}|g(\bm{\theta},\bm{\epsilon}),{\bm{w%
}})+\nabla_{\bm{\theta}}\log q(g(\bm{\theta},\bm{\epsilon})|\bm{\theta})-%
\nabla_{\bm{\theta}}\log p(g(\bm{\theta},\bm{\epsilon}))],$$
(6)
$$\nabla_{{\bm{w}}}\mathcal{L}(\bm{\theta},{\bm{w}})=\mathbb{E}_{\bm{\epsilon}}[%
-\nabla_{{\bm{w}}}\log p(\mathcal{D}|g(\bm{\theta},\bm{\epsilon}),{\bm{w}})].$$
(7)
The first term in Eq. (6) corresponds to the gradient of the negative log likelihood and we leave how to estimate the last two terms (i.e. log densities) in the next section.
In practice, we approximate the expectation in Eq. (6) and Eq. (7) with $T$ Monte Carlo (MC) samples, and update the structure and the weights ${\bm{w}}$ simultaneously.
After training, we gain the following predictive distribution:
$$p(y|x_{new},\bm{w}^{*})=\mathbb{E}_{q(\bm{\alpha}|\bm{\theta}^{*})}[p(y|x_{new%
},\bm{\alpha},\bm{w}^{*})],$$
(8)
where $\bm{\theta}^{*}$ and $\bm{w}^{*}$ denote the converged parameters. Eq. (8) implies that the model predicts by ensembling the predictions of the networks whose structures are randomly sampled.
3.1 Adaptive Concrete Distribution
The weight sharing mechanism in DBSN is a non-trivial contribution for Bayesian structure learning, enabling computationally efficient optimization. But it also causes unignorable training challenges. Specifically, because of the limited capacity of the shared weights ${\bm{w}}$, we have challenges to train it sufficiently well to be suitable for all the structures. The under-fitting of ${\bm{w}}$ then brings bias in the learning of $\bm{\alpha}$’s variational posterior and results in unsatisfying convergence of the whole model.
We note that an analogous phenomenon was also observed by Mackay et al. (2019) in the gradient-based hyper-parameter optimization scenario.
Therefore, to facilitate ${\bm{w}}$ to fit the structure distribution better and eventually benefit the Bayesian structure learning,
we expect to reduce the variance of the structure distribution. Specifically, we analyze the reparameterization procedure of the concrete distribution, and propose to multiply a tunable scalar $\bm{\beta}^{(i,j)}$ with $\bm{\epsilon}^{(i,j)}$ in the sampling:
$$\bm{\alpha}^{(i,j)}=g(\bm{\theta}^{(i,j)},\bm{\beta}^{(i,j)},\bm{\epsilon}^{(i%
,j)})=\mathrm{softmax}((\bm{\theta}^{(i,j)}+\bm{\beta}^{(i,j)}\bm{\epsilon}^{(%
i,j)})/\tau).$$
(9)
Accordingly, we derive the log probability density of the adaptive concrete distribution which is slightly different from that of the concrete distribution (see the detailed derivation in Appendix A):
$$\begin{split}\displaystyle\log p(&\displaystyle\bm{\alpha}^{(i,j)}|\bm{\theta}%
^{(i,j)},\bm{\beta}^{(i,j)})=\log((K-1)!)+(K-1)\log\tau-(K-1)\log\bm{\beta}^{(%
i,j)}\\
&\displaystyle-\sum_{k=1}^{K}\log\bm{\alpha}_{k}^{(i,j)}+\sum_{k=1}^{K}\left[%
\frac{\bm{\theta}^{(i,j)}_{k}-\tau\log\bm{\alpha}^{(i,j)}_{k}}{\bm{\beta}^{(i,%
j)}}\right]-K*\overset{K}{\underset{k=1}{\mathrm{L\Sigma E}}}\left[\frac{\bm{%
\theta}^{(i,j)}_{k}-\tau\log\bm{\alpha}^{(i,j)}_{k}}{\bm{\beta}^{(i,j)}}\right%
],\end{split}$$
(10)
where $\mathrm{L\Sigma E}$ represents the log-sum-exp operation. With this, the last two terms of Eq. (6) can be estimated exactly.
Obviously, the adaptive concrete distribution degrades to the concrete distribution when $\bm{\beta}^{(i,j)}=1$. As shown in Figure 2, sliding $\bm{\beta}^{(i,j)}$ from 1 to 0
decreases the diversity of the sampled structures gradually. Therefore, we should also keep $\bm{\beta}^{(i,j)}$ from being too small to avoid the over-fitting issue which the point-estimate structure (i.e., $\bm{\beta}^{(i,j)}=0$) may suffer from. In practice, we choose to gradually reduce the sample variance along with the convergence of the weights, by decaying $\bm{\beta}^{(i,j)}$ from 1 to 0.5 with a linear schedule in the training.
3.2 Practical Improvements of the Structure Learning Space
In order to make the training more stable and more efficient, we modify some changes to the structure learning space (i.e., the support of the structure distribution) commonly adopted in NAS.
Overall modification. To facilitate more effective information flow in the cell, we let the input of a cell (i.e., the output of the previous cell) be fixedly connected to all the internal nodes by 1$\times$1/3$\times$3 convolutions in the classification/segmentation tasks. We only learn the connections between the $B$ internal nodes, as shown in Appendix F. The resulted nodes are concatenated along with the input to get the cell’s output. In spirit of DenseNet (Huang et al., 2017) and FC-DenseNet (Jégou et al., 2017), we constrain the downsampling/upsampling modules to be the typical BN-ReLU-Conv-Pooling/ConvTranspose operations, to ease the learning of the network structure.
Batch normalization. NAS usually adopts the order of ReLU-Conv-BN in operations. However, in the searching stage, the learnable affine transformations in batch normalizations are always disabled to avoid the output rescaling issue (Liu et al., 2019). NAS does not suffer from this since it trains another network with learnable batch normalizations in the extra re-training stage. Instead, DBSN has to fix the issue because we do not re-train the model. Thus, we propose to put a complete batch normalization in the front of the next layer. Namely, we adopt the BN-ReLU-Conv-BN convolutional layers, where the first BN has learnable affine parameters while the second one does not.
Candidate operations.
In order to make the training more efficient, we remove the operations which are popular in NAS but unnecessary in DBSN, including all the 5$\times$5 convolutions that can be replaced by stacked 3$\times$3 convolutions, and all the pooling layers which are mainly used for the downsampling module. Then, the candidate operations in DBSN are: 3$\times$3 separable convolutions, 3$\times$3 dilated separable convolutions, identity and zero. We follow Liu et al. (2019) for the detailed settings of these operations.
Group operation. To obtain the $j^{th}$ node in a cell, there are $(j-1)K$ operations from its predecessors to calculate, which can be organized into $K$ groups according to the operation type. Note that the operations in a group are independent, so we advocate replacing them with a group operation (e.g., group convolution), which improves the efficiency significantly.
3.3 Discussion
One may concern that the practical choice of weight sharing could push the structure distribution toward the most likely point for the weights and result in a Dirac structure distribution. However, the prior keeps the variational posterior from collapsing via a KL regularization (last term of Eq. (4)). Besides, recall that ${\bm{w}}$ is a set including the parameters of all the redundant operations. Then, in fact, different network structures adjust w.r.t. different subsets of ${\bm{w}}$, further alleviating the structure collapsing issue. The widely used technique of MC Dropout (Gal & Ghahramani, 2016; Gal et al., 2017) can also be seen as using the same weights for different structures. Their empirical results also prove that this kind of model choice is reasonable. Nevertheless, capturing the dependency of ${\bm{w}}$ on $\bm{\alpha}$ may indeed bring more accurate modeling and we leave this as future work.
We also emphasize that using point estimates for the weights benefits the whole model’s learning significantly. On one hand, as stated in the introduction, there are still frustrating difficulties to achieve scalable Bayesian inference on the high-dimensional network weights, which is also proven by the results in Table 1, Table 3, and Appendix C. On the other hand, DBSN deploys weight decay regularizor on weights, which implicitly imposes a Gaussian prior on ${\bm{w}}$. Then, DBSN performs maximum a posteriori (MAP) estimation of ${\bm{w}}$, namely, estimating the mode of ${\bm{w}}$’s posterior distribution $p({\bm{w}}|\mathcal{D})$, which can be viewed as doing approximate Bayesian inference on ${\bm{w}}$.
4 Related Work
Learning flexible Bayesian models has long been the goal of the community (MacKay, 1992; Neal, 1995; Balan et al., 2015; Wang & Yeung, 2016). The stochastic variational inference methods for Bayesian neural networks are particularly appealing owing to their analogy to the ordinary back-propagation (Graves, 2011; Blundell et al., 2015). More expressive distributions, such as matrix-variate Gaussians (Sun et al., 2017) or multiplicative normalizing flows (Louizos & Welling, 2017), have also been introduced to represent the posterior dependencies, but they are hard to train without heavy approximations. Recently, there is an increasing interest in developing Adam-like optimizers to perform natural-gradient variational inference for BNNs (Zhang et al., 2018; Bae et al., 2018; Khan et al., 2018). Despite enabling the scalability, these methods seem to demonstrate compromising performance compared to the state-of-the-art deep models. Interpreting the stochastic techniques of the deep models as Bayesian inference is also insightful (Gal & Ghahramani, 2016; Kingma et al., 2015; Teye et al., 2018; Mandt et al., 2017; Lakshminarayanan et al., 2017), but these methods still have relatively restricted and inflexible posterior approximations. Dikov & Bayer (2019) propose a unified Bayesian framework to infer the posterior of both the network weights and the structure, which is most similar to DBSN, but the network structure considered by them, namely layer size and network depth, is essentially impractical for complicated deep models. Instead, we inherit the design of the structure learning space for NAS, and provide insightful techniques to improve the convergence, thus enabling effective Bayesian structure learning for deep neural networks.
Neural architecture search (NAS) has drawn tremendous attention, where reinforcement learning (Zoph & Le, 2016; Zoph et al., 2018; Pham et al., 2018), evolution (Real et al., 2019) and Bayesian optimization (Kandasamy et al., 2018) have all been introduced to solve it. More recently, differentiable NAS (Liu et al., 2019; Xie et al., 2019; Cai et al., 2019; Wu et al., 2019) is attractive because it reduces the prohibitive computational cost immensely.
However, existing differentiable NAS methods search the network structure in a meta-learning way (Finn et al., 2017), and need to re-train another network with the pruned compact structure after the searching. In contrast, DBSN unifies the learning of weights and structure in one training stage, alleviating the mismatch of structures during the search and re-training, as well as inefficiency issues suffered by differentiable NAS.
5 Experiments
To validate the structure learning ability and the predictive performance of DBSN,
we first evaluate it on image classification and segmentation tasks. For the estimation of the predictive uncertainty, we concern model calibration and generalization of the predictive uncertainty to adversarial examples as well as out-of-distribution samples, following existing work. We show that DBSN outperforms strong baselines in these tasks, shedding light on practical Bayesian deep learning.
5.1 Image Classification on CIFAR-10 and CIFAR-100
Setup. We set $B=7$, $T=4$ and $K=4$, thus, $\bm{\alpha}$ consists of $7\times 6/2=21$ sub-variables. The whole network is composed of 12 cells and 2 downsampling modules which have a channel compression factor of 0.4 and are located at the 1/3 and 2/3 depth. We employ a 3$\times$3 convolution before the first cell and put a global average pooling followed by a fully connected (FC) layer after the last cell. The redundant operations all have 16 output channels. We initialize ${\bm{w}}$ and $\bm{\theta}$ following He et al. (2015) and Liu et al. (2019), respectively. The prior distributions of $\bm{\alpha}^{(i,j)}$ are set to be concrete distributions with uniform class probabilities. A momentum SGD with initial learning rate 0.1 (divided by 10 at 50% and 75% of the training procedure following (Huang et al., 2017)), momentum $0.9$ and weight decay $10^{-4}$ is used to train the weights ${\bm{w}}$. An Adam optimizer with learning rate $3\times 10^{-4}$, momentum ($0.5$, $0.999$) is used to learn $\bm{\theta}$. We deploy the standard data augmentation scheme (mirroring/shifting) and normalize the data with the channel statistics. The whole training set is used for optimization. We train DBSN for 100 epochs with batch size 64, which takes one day on 4 GTX 1080-Tis. The implementation depends on PyTorch (Paszke et al., 2017) and the codes are available online at https://github.com/anonymousest/DBSN.
Baselines. Besides comparison to the advanced deep models, we also design a series of baselines for fair comparisons.
1) DBSN*: we substitute the concrete distribution for the adaptive concrete distribution. 2) DBSN-1: we use $T=1$ sample in the gradient estimation. 3) Fixed $\bm{\alpha}$: we fix the structure of the network by setting the weight of every operation to be $1/K$. 4) Dropout: based on Fixed $\bm{\alpha}$, we further add dropout on every computational node with a drop rate of 0.2. 5) Drop-path: based on Fixed $\bm{\alpha}$, we further apply drop-path (Larsson et al., 2016) regularisation on the convolutional redundant operations with a path drop rate of 0.3. 6) Random $\bm{\alpha}$: we fix the distributions of $\bm{\alpha}^{(i,j)}$ as concrete distributions with uniform class probabilities and only train ${\bm{w}}$ with randomly sampled $\bm{\alpha}$. 7) PE: we view the structure as point estimates and train it as well as ${\bm{w}}$ simultaneously. 8) DARTS: we view the structure as point estimates but we train it on half of the training set while train ${\bm{w}}$ on the other half, resembling the first order DARTS (Liu et al., 2019). 9) NEK-FAC: we train a VGG16 network with weight uncertainty using the noisy EK-FAC (Bae et al., 2018) and the corresponding default settings. 10) BNN-LS: we replace all the convolutional and fully connected layers in PE with their Bayesian counterparts to build a BNN with Learnable Structure. 11) Fully Bayesian DBSN: we replace all the convolutional and fully connected layers in DBSN with their Bayesian counterparts to build a Fully Bayesian neural network. In BNN-LS and Fully Bayesian DBSN, we employ fully factorized Gaussian distributions on weights and adopt BBB (Blundell et al., 2015) for inference. When testing DBSN, DBSN*, DBSN-1, Random $\bm{\alpha}$, NEK-FAC, Dropout, Drop-path, BNN-LS and Fully Bayesian DBSN, we ensemble the predictive probabilities from 100 random runs (we adopt this strategy in all the following experiments, unless stated otherwise).
We repeat every experiment 3 times and report the averaged error rate and standard deviation in Table 1.
Notably, DBSN demonstrates comparable performance with state-of-the-art deep neural networks. DBSN outperforms the powerful ResNet (He et al., 2016a) and DenseNet (Huang et al., 2017) with statistical evidence, and only presents modestly higher error rates than those of DenseNet-BC (Huang et al., 2017), which probably results from the usage of the expressive and efficient bottleneck layer in DenseNet-BC. This comparison highlights the practical value of DBSN.
Comparisons between DBSN and the baselines designed by ourselves are more insightful and convincing. 1) DBSN surpasses DBSN*, revealing the effectiveness of the adaptive concrete distribution. 2) DBSN-1 is remarkably worse than DBSN owing to the higher variance of the estimated gradients with only one sample. 3) Comparison of DBSN and Fixed $\bm{\alpha}$ validates that adapting the network structure w.r.t. the data distribution benefits the fitting of the model, resulting in substantially enhanced performance. 4) Random $\bm{\alpha}$, Dropout, and Drop-path train the networks with manually-designed untunable randomness, and hence are inferior to DBSN. 5) NEK-FAC gains rather compromising performance, with the powerful VGG16 architecture and one of the most advanced variational BNNs algorithms, suggesting us to prefer DBSN instead of the classic BNNs in the scenarios where the performance is a major concern. 6) BNN-LS and Fully Bayesian DBSN both have poor performance, due to the fundamental difficulties of modeling distributions over high dimensional weights. 7) PE and DARTS are two methods to learn the point-estimate network structure, both of which fall behind in terms of the test error. In particular, DARTS is much worse as it only trains the weights on half of the training set. This shows that DBSN is an appealing choice for effective neural structure learning with only one-stage training.
5.2 Semantic Segmentation on CamVid
To further verify that learning the network structure w.r.t. the data helps DBSN to obtain better performance than the standard NNs and BNNs, we apply DBSN to the challenging segmentation benchmark CamVid (Brostow et al., 2008). Our implementation is based on the brief FC-DenseNet framework (Jégou et al., 2017). Specifically, we only replace the original dense blocks with the structure-learnable cells, without introducing further advanced techniques from the semantic segmentation community, to figure out the performance gain only resulted from the learnable network structure.
For the setup, we set $B=5$ (same as the number of layers in every dense block of FC-DenseNet67) and $T=1$, and learn two cell structures for the downsampling path and upsampling path, respectively. We use a momentum SGD with initial learning rate 0.01 (which decays linearly after 350 epochs), momentum 0.9 and weight decay $10^{-4}$ instead of the original RMSprop for better results. The other settings follow Jégou et al. (2017) and the classification experiments above. We also implement FC-DenseNet67 as a baseline. We present the results in Table 2 and Figure 3.
It is evident that DBSN surpasses the competing FC-DenseNet67 by a large margin while using fewer parameters. DBSN also demonstrates significantly better performance than the classic Bayesian SegNet which adopts MC dropout for uncertainty estimation. We emphasize this experiment shows that the proposed approach is generally applicable. It is also worth noting that the uncertainty produced by DBSN is interpretable (see Figure 3): the edges of the objects and the regions which contain overlapping have substantially higher uncertainty than the other parts.
5.3 Estimation of Predictive Uncertainty
To validate that DBSN can provide promising predictive uncertainty, we evaluate it via calibration. We further examine the predictive uncertainty on adversarial examples and out-of-distribution (OOD) samples to test if the model knows what it knows. We also pay particular attention to the comparison between Drop-path and Dropout to double-check if more structured randomness (Larsson et al., 2016) benefits predictive uncertainty more.
Calibration is orthogonal to the accuracy (Lakshminarayanan et al., 2017) and can be well estimated by the Expected Calibration Error (ECE) (Guo et al., 2017). Thus, we evaluate the trained models on the test set of CIFAR-10 and CIFAR-100 and calculate their ECE, as shown in Table 3. We also plot some reliability diagrams (Guo et al., 2017) in Appendix D, to provide a direct explanation of calibration. Unsurprisingly, DBSN achieves state-of-the-art calibration. DBSN outperforms the strong baselines, Dropout and NEK-FAC. NEK-FAC, BNN-LS and Fully Bayesian DBSN all have much worse ECE than DBSN, implying structure uncertainty’s superiority over weight uncertainty. We also notice that Drop-path is better than Dropout in terms of ECE, supporting our hypothesis that more structured randomness is more beneficial to the predictive uncertainty.
To test the predictive uncertainty on the adversarial examples, we apply the fast gradient sign method (FGSM) (Goodfellow et al., 2014) to attack the trained models on CIFAR-10 and CIFAR-100 using the corresponding test samples333For DBSN, DBSN*, Random $\bm{\alpha}$, NEK-FAC, Dropout, and Drop-path, we attack using the ensemble of predictions from 30 stochastic runs and then we test the manipulated adversarial examples with 30 runs as well.. Then we calculate the predictive entropy of the generated adversarial examples and depict the average entropy in Figure 4. As expected, the entropy of DBSN grows rapidly as the perturbation size increases, implying DBSN becomes pretty uncertain when encountering adversarial perturbations. By contrast, the change in entropy of Dropout and NEK-FAC is relatively moderate, which means that these methods are not as sensitive as DBSN to the adversarial examples. Besides, Drop-path is still better than Dropout, consistent with the conclusion above. We also note that Random $\bm{\alpha}$ has the highest predictive entropy. We speculate that this is because Random $\bm{\alpha}$ adopts the most diverse network structures (which results from the uniform class probabilities), and the ensemble of predictions from the corresponding networks is easier to be uniform. We further attack with more powerful algorithms, e.g., the Basic Iterative Method (BIM) (Kurakin et al., 2016), and provide the results in Appendix E.
Moreover, we look into the entropy of the predictive distributions on OOD samples, to adequately evaluate the quality of uncertainty estimation. We use the trained models on CIFAR-10 and CIFAR-100, and take the samples from the test set of SVHN as OOD samples. We calculate their predictive entropy and draw the empirical CDF of the entropy in Figure 5, following Louizos & Welling (2017). The curve close to the bottom right corner is expected as it means most OOD samples have relatively large entropy (i.e., low prediction confidence). Obviously, DBSN demonstrates comparable or even better results than the competing methods like Dropout and NEK-FAC. In addition, Drop-path attains substantially improved results than Dropout. Analogous to the experiments on adversarial examples, Random $\bm{\alpha}$ provides impressive predictive uncertainty on the OOD samples.
In conclusion, DBSN consistently delivers state-of-the-art predictive uncertainty in various scenarios, validating the effectiveness of structure uncertainty.
5.4 Rethinking of the One-shot NAS
One-shot NAS (Bender et al., 2018; Guo et al., 2019) first trains the weights of a super network and then searches for a good structure given the weights. This avoids the bias induced by the gradient-based joint optimization of the differentiable NAS. However, we argue that the super network trained with the fixed (Bender et al., 2018) or uniformly sampled (Guo et al., 2019) network structures cannot flexibly focus its capacity on the most crucial operations, harming the subsequent searching. To this end, we have conducted a set of experiments to check whether dynamically adjusting the network structure at the stage of weight training helps to find better network structures eventually. Observing that DBSN trains a super network with adaptive network structures and Random $\bm{\alpha}$ trains a super network with unadjustable structures (similar to the uniform sampling used by Guo et al. (2019)), we choose to search for the optimal structure distributions based on the trained weights from DBSN and Random $\bm{\alpha}$444We initialize $\bm{\theta}^{(i,j)}$ randomly and initialize $\bm{\beta}^{(i,j)}$ with $1$. Given the fixed network weights, we optimize $\bm{\theta}^{(i,j)}$ and $\bm{\beta}^{(i,j)}$ by gradient descent. The searching lasts for 20 epochs.. After searching, we train new networks with the searched structure distributions (fixed in the training) from scratch, and then test their performance. The results are shown in Table 4. The searched structure distribution based on the weights trained by DBSN outperforms the other one significantly, supporting our hypotheses. Therefore, we propose to reasonably adapt the structure in the weight-training stage of one-shot NAS, which drives the most useful operations to be optimized thoroughly and eventually yields more powerful network structures.
5.5 Visualization of the Learned Structure Distributions
We visualize the learned structure distributions in Appendix F. The structure distributions for different tasks look quite different, which implies that the structures are learned in a way that accounts for the specific characteristics in the data.
6 Conclusion
In this work, we have introduced a novel Bayesian structure learning approach for deep neural networks. The proposed DBSN draws the inspiration from the network design of NAS and models the network structure as Bayesian variables. Stochastic variational inference is employed to jointly learn the network weights and the distribution of the network structure. We further develop the adaptive concrete distribution and improve the structure learning space to facilitate the convergence of the whole model. Empirically, DBSN has revealed impressive performance on the discriminative learning tasks, surpassing the advanced deep models, and presented state-of-the-art predictive uncertainty in various scenarios. In conclusion, DBSN provides a more practical way for Bayesian deep learning, without compromise between the predictive performance and the Bayesian uncertainty.
There are two major directions for future work. On one hand, the current DBSN is not efficient enough,
so some strategies need to be discovered to make DBSN more efficient. On the other hand, DBSN still has a relatively restricted structure learning space. Thus, more operations can be introduced and more global network structures can be learned in future work.
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Appendix A Derivation of the Log Probability Density of the Adaptive Concrete Distribution
For clear expression, We simply denote $\bm{\alpha}^{(i,j)}$, $\bm{\theta}^{(i,j)}$, $\bm{\beta}^{(i,j)}$ and $\bm{\epsilon}^{(i,j)}$ as $\bm{\alpha}$, $\bm{\theta}$, $\beta$ and $\bm{\epsilon}$, respectively. Let ${\bm{p}}=\mathrm{softmax}(\bm{\theta})$. Consider
$$\bm{\alpha}_{k}=\frac{\exp((\bm{\theta}_{k}+\beta\bm{\epsilon}_{k})/\tau)}{%
\sum_{i=1}^{K}\exp((\bm{\theta}^{i}+\beta\bm{\epsilon}^{i})/\tau)}=\frac{\exp(%
(\log{\bm{p}}_{k}+\beta\bm{\epsilon}_{k})/\tau)}{\sum_{i=1}^{K}\exp((\log{\bm{%
p}}^{i}+\beta\bm{\epsilon}^{i})/\tau)}.$$
Let ${\bm{z}}_{k}=\log{\bm{p}}_{k}+\beta\bm{\epsilon}_{k}=\log{\bm{p}}_{k}-\beta%
\log(-\log({\bm{u}}_{k}))$, where ${\bm{u}}_{k}\sim\mathcal{U}(0,1)\;\mathrm{i.i.d.}$. It has density
$$\frac{1}{\beta}{\bm{p}}_{k}^{1/\beta}\exp(-\frac{{\bm{z}}_{k}}{\beta})\exp(-{%
\bm{p}}_{k}^{1/\beta}\exp(-\frac{{\bm{z}}_{k}}{\beta})).$$
We denote $c=\sum_{i=1}^{K}\exp({\bm{z}}_{i}/\tau)$, then $\bm{\alpha}_{k}=\exp({\bm{z}}_{k}/\tau)/c$. We consider this transformation:
$$F({\bm{z}}_{1},\dots,{\bm{z}}_{K})=(\bm{\alpha}_{1},\dots,\bm{\alpha}_{K-1},c),$$
which has the following inverse transformation:
$$F^{-1}(\bm{\alpha}_{1},\dots,\bm{\alpha}_{K-1},c)=(\tau(\log\bm{\alpha}_{1}+%
\log c),\dots,\tau(\log\bm{\alpha}_{K}+\log c)),$$
whose Jacobian has the determinant (refer to the derivation of the concrete distribution (Maddison et al., 2016)):
$$\frac{\tau^{K}}{c\prod_{i=1}^{K}\bm{\alpha}_{i}}.$$
Multiply this with the density of ${\bm{z}}$, we get the density
$$\frac{\tau^{K}\prod_{i=1}^{K}\frac{1}{\beta}{\bm{p}}_{i}^{1/\beta}\exp(-\frac{%
\tau(\log\bm{\alpha}_{i}+\log c)}{\beta})\exp(-{\bm{p}}_{i}^{1/\beta}\exp(-%
\frac{\tau(\log\bm{\alpha}_{i}+\log c)}{\beta}))}{c\prod_{i=1}^{K}\bm{\alpha}_%
{i}}.$$
Let $r=\log c$, then apply the change of variables formula, we obtain the density:
$$\frac{\tau^{K}\prod_{i=1}^{K}{\bm{p}}_{i}^{1/\beta}}{\beta^{K}\prod_{i=1}^{K}%
\bm{\alpha}_{i}^{(1+\tau/\beta)}}\exp(-\frac{K\tau r}{\beta})\exp(-\sum_{i=1}^%
{K}({\bm{p}}_{i}\bm{\alpha}_{i}^{-\tau})^{1/\beta}\exp(-\frac{\tau r}{\beta})).$$
We use $\gamma$ to substitute $\log\sum_{i=1}^{K}({\bm{p}}_{i}\bm{\alpha}_{i}^{-\tau})^{1/\beta}$, then get:
$$\frac{\tau^{K}\prod_{i=1}^{K}{\bm{p}}_{i}^{1/\beta}}{\exp(\gamma)\beta^{K}%
\prod_{i=1}^{K}\bm{\alpha}_{i}^{(1+\tau/\beta)}}\exp(\gamma-\frac{K\tau r}{%
\beta})\exp(-\exp(\gamma-\frac{\tau r}{\beta})).$$
Naturally, we can integrate out $r$, and get:
$$\begin{split}&\displaystyle\frac{\tau^{K}\prod_{i=1}^{K}{\bm{p}}_{i}^{1/\beta}%
}{\exp(\gamma)\beta^{K}\prod_{i=1}^{K}\bm{\alpha}_{i}^{(1+\tau/\beta)}}\left[%
\frac{\beta}{\tau}\exp(\gamma-K\gamma)\Gamma(K)\right]\\
\displaystyle=&\displaystyle\frac{\tau^{K-1}\prod_{i=1}^{K}{\bm{p}}_{i}^{1/%
\beta}}{\beta^{K-1}\prod_{i=1}^{K}\bm{\alpha}_{i}^{(1+\tau/\beta)}}\exp(-K%
\gamma)\Gamma(K)\\
\displaystyle=&\displaystyle\frac{((K-1)!)\tau^{K-1}}{\beta^{K-1}\prod_{i=1}^{%
K}\bm{\alpha}_{i}}\times\frac{\prod_{i=1}^{K}({\bm{p}}_{i}\bm{\alpha}_{i}^{-%
\tau})^{1/\beta}}{(\sum_{i=1}^{K}({\bm{p}}_{i}\bm{\alpha}_{i}^{-\tau})^{1/%
\beta})^{K}}.\end{split}$$
Then, the log density is:
$$\begin{split}&\displaystyle\log((K-1)!)+(K-1)\log\frac{\tau}{\beta}-\sum_{i=1}%
^{K}\log\bm{\alpha}_{i}+\sum_{i=1}^{K}\frac{\log{\bm{p}}_{i}-\tau\log\bm{%
\alpha}_{i}}{\beta}-K*\overset{K}{\underset{i=1}{\mathrm{L\Sigma E}}}\frac{%
\log{\bm{p}}_{i}-\tau\log\bm{\alpha}_{i}}{\beta}\\
\displaystyle=&\displaystyle\log((K-1)!)+(K-1)\log\frac{\tau}{\beta}-\sum_{i=1%
}^{K}\log\bm{\alpha}_{i}+\sum_{i=1}^{K}\frac{\bm{\theta}_{i}-\tau\log\bm{%
\alpha}_{i}}{\beta}-K*\overset{K}{\underset{i=1}{\mathrm{L\Sigma E}}}\frac{\bm%
{\theta}_{i}-\tau\log\bm{\alpha}_{i}}{\beta},\end{split}$$
which is equal to Eq. (10).
Appendix B The Effects of the Number of MC Samples in Test Phase
We draw the change of test loss, test error rate and test ECE with respect to the number of MC samples used for testing DBSN in Figure 6 (CIFAR-10) and Figure 7 (CIFAR-100). It is clear that ensembling the predictions from models with various sampled network structures enhances the final predictive performance and calibration significantly. This is in marked contrast to the situation of classic variational BNNs, where using more MC samples does not necessarily bring improvement over using the most likely sample. As shown in the plots, we would better utilize 20+ MC samples to predict the unseen data, for adequately exploiting the learned structure distribution. Indeed, we use 100 MC samples in all the experiments, except the adversarial attack experiments where we use 30 MC samples for attacking and evaluation.
Appendix C More Comparisons between DBSN and Competing Baselines with Weight Uncertainty
We realized the BBB method used for modeling weight uncertainty in BNN-LS and Fully Bayesian DBSN may be restrictive, resulting in such weakness. Therefore, we further implemented these two baselines with a most-recently proposed mean-field natural-gradient variational inference method, called Variational Online Gauss-Newton (VOGN) (Khan et al., 2018; Osawa et al., 2019). VOGN is known to work well with advanced techniques, e.g., momentum, batch normalisation, data augmentation. As claimed by Osawa et al. (2019), VOGN demonstrates comparable results to Adam. Then, we replaced the used BBB (Blundell et al., 2015) in BNN-LS and Fully Bayesian DBSN with VOGN, based on VOGN’s official repository (https://github.com/team-approx-bayes/dl-with-bayes). With the original network size ($B=7$, 12 cells), the baselines trained with VOGN needed more than one hour for one epoch. Thus we adopted smaller networks ($B=4$, 3 cells), which have almost 41K parameters, for the two baselines. We also trained a DBSN in the same setting. The detailed parameters to initialize VOGN are here (https://github.com/anonymousest/DBSN/blob/master/dbsn/train_bnn_torchsso.py#L220). The experiments were conducted on CIFAR-10 and the results are provided in Table 5. The predictive performance and uncertainty gaps between DBSN and the two baselines are very huge, which possibly results from the under-fitting of the high-dim weight distributions in BNN-LS and Fully Bayesian DBSN. We believe that our implementation is correct because our results are consistent with the original results in Table 1 of Osawa et al. (2019) (VOGN has 75.48% and 84.27% validation accuracy even with even larger 2.5M AlexNet and 11.1M ResNet-18 architectures). Further, DBSN is much more efficient than them. These comparisons strongly reveal the benefits of modeling structure uncertainty over modeling weight uncertainty, highlighting the practical value of DBSN.
Appendix D More Results for Calibration
We plot the reliability diagrams of 4 typical methods, which represent the deep BNN with structure uncertainty, the classic BNN with weight uncertainty, the deterministic NN with MC dropout and the standard NN, respectively, in Figure 8. Obviously, DBSN has better reliability diagrams than NEK-FAC and Dropout, proving the effectiveness of the uncertainty on network structure.
Appendix E Attack with BIM
We perform an adversarial attack using BIM algorithm. Concretely, we set the number of iteration to be 3 and set the perturbation size in every step to be 1/3 of the whole perturbation size. The experiments mainly focus on the models trained on CIFAR-10. Figure 9 shows the results. Random $\bm{\alpha}$, DBSN* and DBSN have increasing entropy when the perturbation size changes from 0 to 0.01, but all the other approaches are attacked successfully with entropy dropping. However, strictly, only the Random $\bm{\alpha}$ at perturbation size 0.01 provides useful predictive uncertainty, and we can use the entropy to reject the predictions. Therefore, we have to agree that BIM is powerful enough to break all the methods, including DBSN. So we advise adjusting DBSN accordingly (e.g., employing adversarial training, using more robust loss) if we want to use DBSN to defend the adversarial attacks.
Appendix F Visualization of the Learned Structures
We visualize the learned structure distributions on different tasks in Figure 10, Figure 11, Figure 12 and Figure 13 (we do not draw the zero operation). The structure distributions learned on different tasks look different, validating that DBSN can adapt the network structure according to the data distribution flexibly. |
Anomalies and Decoupling of charginos and neutralinos
in the MSSM
J. Lorenzo Diaz-Cruz
Instituto de Fisica, BUAP,
Ap. Postal J-48, 72500 Puebla, Pue., Mexico
Abstract
We study the contribution of charginos and neutralinos
of the Minimal SUSY extension of the Standard model (MSSM)
to the 1-loop vertices $ZAA$, $ZZA$, $ZZZ$,
and examine the related cancellation of anomalies.
It is found that when the SUSY parameter $\mu$
satisfies $|\mu|>>M,M^{\prime},m_{W}$, the couplings
of charginos and neutralinos with the gauge bosons
become purely vectorial,
and then their contribution to the amplitudes for
$ZAA$, $ZZA$ and $ZZZ$
vanish, which implies that this sector of
the MSSM does not generate a Wess-Zumino term.
We evaluate also the contribution of
charginos and neutralinos to the $\rho$ parameter,
and find that $\rho=0$ in the large-$\mu$ limit.
The minimal SUSY standard model (MSSM) [1, 2]
has become one the most preferred extension of the
standard model (SM).
The success of the MSSM is not only due to its ability
to imitate the SM agreement with most high energy data,
but also because it gives a plausible explanation
of the new results that seem to be in conflict with
the SM (e.g. the large $Z\to b\bar{b}$ width [3],
the $ee\gamma\gamma$ event observed at FNAL [4]).
Moreover, the model predicts new signatures
associated to the superpartners that are expected
to appear in the future colliders (LHC,NLC), or
even in the present ones (FNAL/LEP).
However, if the superpartners are not light,
it will become relevant to search for
any indirect physical effect that could be left
by them, and to verify by explicit
calculations their expected decoupling.
Since gauge invariant
masses are allowed in vector-like theories,
the effects of heavy fermion decouple
from the corresponding low-energy
effective lagrangian [5].
However, in chiral theories heavy particles
do not decouple in general. For instance,
if the Higgs mechanism of spontaneous
symmetry breaking (SSB) is used to generate masses,
the associated v.e.v. ($v$) is fixed by the scale
of the interactions, then in order to generate a heavy
fermion mass ($m>>v$), a large Yukawa coupling is required,
which induces strong effects that prevent
the decoupling of the fermion.
Chiral fermions may also pose another problem, namely
the appearance of anomalies in gauge currents.
An interesting problem arises when the anomaly
cancellation occurs between fermions with very
different masses [6, 7]. In this case,
integrating out the heavy fermion
leaves an anomalous effective theory,
which is the signal of its non-decoupling [8].
In the MSSM there are two sources of anomalies:
the ones due to quarks and leptons, and the
ones due to the fermionic partners of
the Higgs bosons, the higgsinos, which
cancel separately to make
the model anomaly-free.
In fact, the higgsinos are not mass eigenstates,
they mix with the superpartners of the
gauge bosons (gauginos),
and the resulting charged and neutral eigenstates are
known as chargino and neutralino, respectively.
Gauginos do not contribute to the
anomaly because their couplings to the gauge bosons
are vectorial.
In this letter we are interested in studying the
effects of heavy charginos and neutralinos,
and to understand the role that the anomalies
can play for their decoupling. In particular, we evaluate
the contribution of charginos and neutralinos
to the vertex $ZAA$, $ZZA$, $ZZZ$
and to the $\rho$ parameter, focusing in the
limit when the SUSY parameter $\mu$ satisfies
$|\mu|>>M,M^{\prime},m_{W}$, where $M,M^{\prime}$ denote the
gaugino soft-breaking masses, and the W-boson
mass $m_{W}$ is used to
characterize the electroweak scale.
111The case when a complete
supermultiplet is integrated out was
discussed in [9], however in this paper
we are interested in the limit when
only the superpartners are heavy.
Although the MSSM is anomaly-free, it is
relevant to understand the conditions under
which charginos and neutralinos participate
in the cancellation of anomalies, since
this can play an important role for their
decoupling. For instance, if it were possible
for a heavy higgsino to contribute to the anomaly,
the cancellation of anomalies would take place between
different scales, which could prevent its decoupling.
In order to identify the origin of anomalies we shall
work with 4-component spinors for the
gaugino and higgsino fields,
however, the analysis of the large-$\mu$ limit
and the calculations of interest will be performed in the
mass-eigenstate basis, namely in terms of charginos
and neutralinos.
We shall review first the lagrangian of the model,
focusing mainly in the interaction of
the gauge bosons ($A_{\mu},W^{\pm}_{\mu},Z_{\mu}$),
with the charged
($\tilde{H}$) and neutral ($\tilde{H}_{1},\tilde{H}_{2}$)
higgsinos, and with the wino ($\tilde{W}$),
photino ($\tilde{A}$) and zino ($\tilde{Z}$) fields.
The lagrangian for the mass and mixing
terms of gauginos and higgsinos in the MSSM is given by:
$$\displaystyle{\cal{L}}$$
$$\displaystyle=$$
$$\displaystyle M_{\tilde{W}}\bar{\tilde{W}}\tilde{W}+\frac{M_{\tilde{A}}}{2}%
\bar{\tilde{A}}\tilde{A}+\frac{M_{\tilde{Z}}}{2}\bar{\tilde{Z}}\tilde{Z}+\mu%
\bar{\tilde{H}}\tilde{H}$$
(1)
$$\displaystyle+\frac{M_{\tilde{Z}}-M_{\tilde{A}}}{2}\tan 2\theta_{W}\bar{\tilde%
{A}}\tilde{Z}-\frac{\mu}{2}[\bar{\tilde{H_{1}}}\tilde{H_{2}}+\bar{\tilde{H_{2}%
}}\tilde{H_{1}}]$$
$$\displaystyle-\frac{g}{\sqrt{2}}[\bar{\tilde{Z}}P_{R}\tilde{H_{1}}H^{1}_{1}-%
\bar{\tilde{H_{2}}}P_{R}\tilde{Z}H^{2}_{2}+h.c.]$$
$$\displaystyle-g[\bar{\tilde{W}}P_{R}\tilde{H}H^{1}_{1}+\bar{\tilde{H}}P_{R}%
\tilde{W}H^{2}_{2}+h.c.]$$
which includes the interaction of gauginos and higgsinos with
the neutral components of the scalar Higgs doublets
($H^{1}_{1},H^{2}_{2}$).
The zino and photino masses can be expressed
in terms of the soft-breaking masses $M$ and $M^{\prime}$,
as follows:
$M_{\tilde{Z}}=M\cos^{2}\theta_{W}+M^{\prime}\sin^{2}\theta_{W}$,
$M_{\tilde{A}}=M\sin^{2}\theta_{W}+M^{\prime}\cos^{2}\theta_{W}$.
After SSB the Higgs scalars acquire v.e.v.’s
($<H^{1}_{1}>=v_{1}$ and $<H^{2}_{2}>=v_{2}$), and
the trilinear terms in eq. (1) generate a mixing among
the gauginos and higgsinos. The resulting
mass-mixing matrices need to be
diagonalized; the mass-eigenstates and the
diagonalizing matrices depend in general
on the parameters $M,M^{\prime},\mu$ and $\tan\beta\,(=v_{2}/v_{1})$.
The diagonalizing matrices ($U,V$) for the charged case
can be found in [10], whereas the $(4\times 4)$
matrix corresponding to the neutral case ($Z$)
is evaluated numerically.
The charginos and neutralinos are denoted by:
$\tilde{\chi}^{+}_{i}\;(i=1,2)$ and
$\tilde{\chi}^{0}_{j}\;(j=1-4)$,
respectively.
The interaction of gauginos and higgsinos
with the gauge bosons of the model
are described by the lagrangian:
$$\displaystyle{\cal{L}}$$
$$\displaystyle=$$
$$\displaystyle e[\bar{\tilde{W}}\gamma_{\mu}\tilde{W}+\bar{\tilde{H}}\gamma_{%
\mu}\tilde{H}]A^{\mu}-e[\bar{\tilde{A}}\gamma^{\mu}\tilde{W}W^{+}_{\mu}+\bar{%
\tilde{W}}\gamma^{\mu}\tilde{A}W^{-}_{\mu}]$$
(2)
$$\displaystyle-g\cos\theta_{W}[\bar{\tilde{Z}}\gamma^{\mu}\tilde{W}W^{-}_{\mu}+%
\bar{\tilde{W}}\gamma^{\mu}\tilde{Z}W^{+}_{\mu}-\bar{\tilde{W}}\gamma_{\mu}%
\tilde{W}Z^{\mu}]$$
$$\displaystyle+\frac{g}{2cos\theta_{W}}[\cos 2\theta_{W}\bar{\tilde{H}}\gamma_{%
\mu}\tilde{H}-\frac{1}{2}(\bar{\tilde{H_{1}}}\gamma_{\mu}\gamma_{5}\tilde{H_{1%
}}-\bar{\tilde{H_{2}}}\gamma_{\mu}\gamma_{5}\tilde{H_{2}})]Z^{\mu}$$
We can discuss now the origin of anomalies in the higgsino
sector. Before SSB the charged higssino couplings to the
neutral gauge bosons are of vector type, and at this stage
it does not contribute to the gauge anomaly.
However, after SSB the mixing
treats in a different way the L- and
R-handed components of the charged higgsinos and winos,
which induces an axial-vector part for their couplings,
then each chargino contributes to the anomaly,
but with opposite signs for the MSSM
to remain anomaly-free. However,
the coupling becomes again vector-like
for $tan\beta=1$ $(v_{1}=v_{2})$.
On the other hand, the neutral higgsinos
contribute to the anomaly, because their
couplings have an axial-vector part even before SSB.
However, these axial-vector couplings also vanish
in the large $\mu$ limit.
The complete Feynman rules for the interaction of the charginos
and neutralinos are summarized in [1, 10, 11].
For the purpose of evaluating the 3-point vertex functions
$Z_{\mu}A_{\nu}A_{\rho},Z_{\mu}Z_{\nu}A_{\rho}$
and $Z_{\mu}Z_{\nu}Z_{\rho}$, we only need to specify the
following vertices:
$$\displaystyle\chi^{+}_{i}\chi^{-}_{i}A^{\mu}$$
$$\displaystyle:$$
$$\displaystyle-ie\gamma^{\mu}$$
(3)
$$\displaystyle\chi^{+}_{i}\chi^{0}_{j}W^{-\mu}$$
$$\displaystyle:$$
$$\displaystyle+ig\gamma^{\mu}(O^{L}_{ij}P_{L}+O^{R}_{ij}P_{R})$$
(4)
$$\displaystyle\chi^{+}_{i}\chi^{-}_{j}Z^{\mu}$$
$$\displaystyle:$$
$$\displaystyle-i\frac{g}{cos\theta_{W}}\gamma^{\mu}(O^{\prime L}_{ij}P_{L}+O^{%
\prime R}_{ij}P_{R})$$
(5)
$$\displaystyle\chi^{0}_{i}\chi^{0}_{j}Z^{\mu}$$
$$\displaystyle:$$
$$\displaystyle-i\frac{g}{cos\theta_{W}}\gamma^{\mu}(O^{\prime\prime L}_{ij}P_{L%
}+O^{\prime\prime R}_{ij}P_{R})$$
(6)
where $P_{R,L}=(1\pm\gamma_{5})/2$, and:
$$\displaystyle O^{L}_{ij}$$
$$\displaystyle=$$
$$\displaystyle Z_{i2}V^{*}_{j1}-\frac{1}{\sqrt{2}}Z_{i4}V^{*}_{j2}$$
(7)
$$\displaystyle O^{R}_{ij}$$
$$\displaystyle=$$
$$\displaystyle Z^{*}_{i2}U_{j1}+\frac{1}{\sqrt{2}}Z^{*}_{i3}U_{j2}$$
(8)
$$\displaystyle O^{\prime L}_{ij}$$
$$\displaystyle=$$
$$\displaystyle-V_{i1}V^{*}_{j1}-\frac{1}{2}V_{i2}V^{*}_{j2}+\delta_{ij}sin^{2}%
\theta_{W}$$
(9)
$$\displaystyle O^{\prime R}_{ij}$$
$$\displaystyle=$$
$$\displaystyle-U_{j1}V^{*}_{i1}-\frac{1}{2}U_{j2}U^{*}_{i2}+\delta_{ij}sin^{2}%
\theta_{W}$$
(10)
$$\displaystyle O^{\prime\prime L}_{ij}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}(Z_{i4}Z^{*}_{j4}-Z_{i3}Z^{*}_{j3})$$
(11)
$$\displaystyle O^{\prime\prime R}_{ij}$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{2}(Z_{i4}Z^{*}_{j4}-Z_{i3}Z^{*}_{j3})$$
(12)
We can evaluate now the SUSY contribution to the
1-loop vertex $Z_{\mu}A_{\nu}A_{\rho}$.
Using the CP-properties of the vector (V) and axial-vector (A)
currents, it can be shown that the 3-point functions
$VVV$ and $VAA$ vanish in general. These CP-properties can be
used also to show that sfermions, goldstone, Higgs and gauge bosons
do not contribute to the vertex $Z_{\mu}A_{\nu}A_{\rho}$.
Thus, the amplitude can only arise from the triangle
graphs with charged fermions inside the loop.
The contribution from each chargino ($\tilde{\chi}^{+}_{i}$)
to the amplitude can for the vertex
$Z_{\mu}(q)A_{\nu}(k_{1})A_{\rho}(k_{2})$
can be obtained from the results of [12],
and it is written as follows,
$$\displaystyle T^{\mu\nu\rho}_{i}$$
$$\displaystyle=$$
$$\displaystyle\frac{4\alpha ga^{\prime}_{ii}Q^{2}_{i}}{\pi cos\theta_{W}}\{%
\varepsilon^{\mu\nu\rho\alpha}[k_{1\alpha}(f_{1}k^{2}_{2}-f_{3}k_{1}.k_{2})-k_%
{2\alpha}(f_{2}k^{2}_{1}-f_{3}k_{1}.k_{2})]+$$
(13)
$$\displaystyle\varepsilon^{\alpha\mu\beta\nu}k_{1\alpha}k_{2\beta}[f_{2}k^{\rho%
}_{1}+f_{3}k^{\rho}_{2}]+\varepsilon^{\alpha\mu\beta\rho}k_{1\alpha}k_{2\beta}%
[f_{1}k^{\nu}_{2}+f_{3}k^{\nu}_{1}]\},$$
where the axial coupling is given by:
$a^{\prime}_{ij}=O^{\prime R}_{ij}-O^{\prime L}_{ij}$, and the functions $f_{i}$ are
defined by:
$$\displaystyle f_{1}$$
$$\displaystyle=$$
$$\displaystyle\int^{1}_{0}dx_{1}\int^{1-x_{1}}_{0}dx_{2}\frac{x_{1}(x_{1}-1)}{D},$$
(14)
$$\displaystyle f_{2}$$
$$\displaystyle=$$
$$\displaystyle\int^{1}_{0}dx_{1}\int^{1-x_{1}}_{0}dx_{2}\frac{x_{2}(x_{2}-1)}{D},$$
(15)
$$\displaystyle f_{3}$$
$$\displaystyle=$$
$$\displaystyle\int^{1}_{0}dx_{1}\int^{1-x_{1}}_{0}dx_{2}\frac{x_{1}x_{2}}{D},$$
(16)
with $D=M^{2}_{\chi^{+}_{i}}+x_{2}(x_{2}-1)k^{2}_{1}+x_{1}(x_{1}-1)k^{2}_{2}-2k_{1}.%
k_{2}x_{1}x_{2}$.
Then, the condition for the non-conservation of
the axial-current is written as:
$$\displaystyle q_{\mu}T^{\mu\nu\rho}$$
$$\displaystyle=$$
$$\displaystyle\sum_{i}\frac{4\alpha ga_{ii}Q^{2}_{i}}{\pi cos\theta_{W}}%
\varepsilon^{\mu\nu\rho\alpha}k_{1\alpha}k_{2\mu}(f_{1}k^{2}_{2}+f_{2}k^{2}_{1%
}-2f_{3}k_{1}.k_{2})$$
(17)
$$\displaystyle=$$
$$\displaystyle\sum_{i}\frac{4\alpha ga_{ii}Q^{2}_{i}}{\pi cos\theta_{W}}%
\varepsilon^{\mu\nu\rho\alpha}k_{1\alpha}k_{2\mu}[\frac{1}{2}-M^{2}_{\chi^{+}_%
{i}}f_{0}]$$
where
$$f_{0}=\int^{1}_{0}dx_{1}\int^{1-x_{1}}_{0}dx_{2}\frac{1}{D},$$
(18)
When both charginos are taken into account the
mass independent term (i.e. the anomaly)
should cancel ($\Sigma_{i}q_{\mu}T^{\mu\nu\rho}_{i}=0$),
as we have verified by the direct substitution of the elements
of $U,V$ in eq. (17), namely: $\sum_{i}Q^{2}_{i}a^{\prime}_{ii}=0$.
In order to study the limit when the mass parameters
are very large ($>>m_{W}$), one can use the
results of [13], which presents an analytical
expression for the diagonalizing matrices $U,V,Z$,
under the assumption that the couplings are CP-invariant,
and with the mass parameters satisfying
the conditions: $|M\pm\mu|,|M^{\prime}\pm\mu|>>m_{W}$,
and $|M\mu|>>m^{2}_{W}\sin 2\beta$. Then, the
matrices $U,V$ are given by:
$$U=\left(\begin{array}[]{ll}\;1&\sqrt{2}m_{W}\frac{Mc_{\beta}+\mu s_{\beta}}{M^%
{2}-\mu^{2}}\\
-\sqrt{2}m_{W}\frac{Mc_{\beta}+\mu s_{\beta}}{M^{2}-\mu^{2}}&\;1\end{array}\right)$$
(19)
$$V=\left(\begin{array}[]{ll}\;1&\sqrt{2}m_{W}\frac{Ms_{\beta}+\mu c_{\beta}}{M^%
{2}-\mu^{2}}\\
-\sqrt{2}m_{W}\frac{Ms_{\beta}+\mu c_{\beta}}{M^{2}-\mu^{2}}&\;sign(\mu)\end{%
array}\right),$$
(20)
whereas the neutralino Z matrix
takes the form:
$$Z=\left(\begin{array}[]{ll}A&B\\
C&D\end{array}\right)$$
(21)
with:
$$A=\left(\begin{array}[]{ll}\;\;1&-\frac{m^{2}_{Z}s_{2W}(M^{\prime}+\mu s_{2%
\beta})}{2(M^{\prime}-M)(M^{\prime 2}-\mu^{2})}\\
-\frac{m^{2}_{Z}s_{2W}(M+\mu s_{2\beta})}{2(M^{\prime}-M)(M^{\prime 2}-\mu^{2}%
)}&\;\;1\end{array}\right)$$
(22)
$$B=\left(\begin{array}[]{ll}\frac{-m_{Z}s_{W}(M^{\prime}c_{\beta}+\mu s_{\beta}%
)}{M^{\prime 2}-\mu^{2}}&\frac{m_{Z}s_{W}(M^{\prime}c_{\beta}+\mu s_{\beta})}{%
M^{\prime 2}-\mu^{2}}\\
\frac{m_{Z}c_{W}(Mc_{\beta}+\mu s_{\beta})}{M^{2}-\mu^{2}}&\frac{-m_{Z}c_{W}(%
Ms_{\beta}+\mu c_{\beta})}{M^{2}-\mu^{2}}\end{array}\right)$$
(23)
$$C=\left(\begin{array}[]{ll}\frac{-m_{Z}s_{W}(s_{\beta}-c_{\beta})}{\sqrt{2}(M^%
{\prime}+\mu)}&\frac{m_{Z}c_{W}(s_{\beta}-c_{\beta})}{\sqrt{2}(M+\mu)}\\
\frac{m_{Z}s_{W}(s_{\beta}+c_{\beta})}{\sqrt{2}(M^{\prime}-\mu)}&-\frac{m_{Z}c%
_{W}(s_{\beta}+c_{\beta})}{\sqrt{2}(M-\mu)}\end{array}\right)$$
(24)
$$D=\left(\begin{array}[]{ll}\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\
\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\end{array}\right)$$
(25)
and where: $s_{W}=\sin\theta_{W},s_{2W}=\sin 2\theta_{W}$,
$s_{\beta}=\sin\beta,c_{\beta}=\cos\beta,s_{2\beta}=\sin 2\beta$.
Our results will be presented assuming also
that $|\mu|>>M,M^{\prime},m_{W}$, and keeping
only the leading terms in $1/\mu$, in whose case the
mass eigenstates are given by:
$M_{\chi^{+}_{2}}=\mu$, $M_{\chi^{+}_{1}}=M$,
$M_{\chi^{0}_{1}}=M^{\prime}$, $M_{\chi^{0}_{2}}=M$,
$M_{\chi^{0}_{3}}=\mu$, $M_{\chi^{0}_{4}}=\mu$,
The chargino mixing matrices take the values
$U=V=1$, whereas the expression for $Z$
also reduces considerably.
In this limit one obtains
$(M^{2}_{\chi^{+}_{i}}/D)\to 1$, and consequently
$(M^{2}_{\chi^{+}_{i}}f_{0})\to 1/2$, which appears as if an
anomalous term would remain.
However, a careful analysis of the couplings
shows that in this limit the lightest chargino
becomes a pure gaugino ($\tilde{\chi}^{+}_{1}=\tilde{W}$),
which does not have axial couplings, i.e. $a^{\prime}_{11}=0$,
and the heavy chargino becomes a pure higgsino,
which also has $a^{\prime}_{22}=0$, thus
$T^{\mu\nu\rho}=0$, and charginos decouple
from this function.
Similarly, we can evaluate the contribution of
charginos to the vertex $Z(q_{1})Z(q_{2})A(k)$,
which we denote by $R^{\mu\nu\rho}$.
In this case the condition for
the conservation of the axial-vector current
is written as:
$$\displaystyle q_{\mu}R^{\mu\nu\rho}$$
$$\displaystyle=$$
$$\displaystyle-\sum_{i}\frac{4\alpha ga^{\prime}_{ii}v^{\prime}_{ii}Q^{2}_{i}}{%
\pi cos^{2}\theta_{W}}\varepsilon^{\mu\nu\rho\alpha}q_{2\alpha}k_{\mu}$$
(26)
which must be zero because of the anomaly cancellation.
The vector coupling is given by $v^{\prime}_{ij}=O^{\prime R}_{ij}+O^{\prime L}_{ij}$.
Moreover, the vertex $R^{\mu\nu\rho}$ itself
also vanishes in the large $\mu$ limit,
because the chargino couplings $a^{\prime}_{ij}$ vanishes.
The amplitude for the 1-loop vertex $Z(q_{1})Z(q_{2})Z(q_{3})$
($=S^{\mu\nu\rho}$), receives contributions from charginos
and neutralinos, and the conservation of the currents
takes the form:
$$\displaystyle q_{\mu}S^{\mu\nu\rho}$$
$$\displaystyle=$$
$$\displaystyle-\sum_{ij}\frac{g^{3}}{2\pi^{2}cos^{3}\theta_{W}}(v^{\prime\prime
2%
}_{ii}+\frac{1}{3}a^{\prime\prime 2}_{ii})a^{\prime\prime}_{ii}\varepsilon^{%
\mu\nu\rho\alpha}q_{2\alpha}q_{3\mu}$$
(27)
which also vanishes because of the anomaly cancellation.
Charginos do not contribute to the
the vertex $S^{\mu\nu\rho}$ itself,
because their axial-vector coupling vanish
in the limit $\mu>>M,M^{\prime},m_{W}$, and
the 3-point function with vector couplings only
($VVV$) vanishes.
Moreover, since the axial-vector couplings of the neutralinos
vanish in the large-$\mu$ limit,
its contribution to $S^{\mu\nu\rho}$ itself
vanishes too.
Thus, integrating out the charginos and neutralinos
does not leave a mass-independent Wess-Zumino
term in the low-energy effective lagrangian.
Another process that illustrates this decoupling result
is the (1-loop) Higgs amplitude $hA_{\mu}A_{\nu}$.
For the light Higgs scalar ($h^{0}$) of the MSSM,
the contribution of charginos to the
amplitude is proportional to the function [11]:
$$I(m_{h},m_{\chi^{+}_{i}})=\frac{4m_{W}m_{\chi^{+}_{i}}A_{ii}}{sin\beta}\int^{1%
}_{0}dx\int^{1-x}_{0}dy\frac{1-4xy}{m^{2}_{\chi^{+}_{i}}-xym^{2}_{h}}$$
(28)
where $A_{ij}$ is a dimensionless coefficient that depends
on the elements of the matrices $U,V$.
The amplitude behaves like $m_{W}/M_{\chi^{+}_{i}}$,
and it vanishes in the large $\mu$ limit.
On the other hand, the contribution of the top quark
to the amplitude becomes a constant in the
limit of a large top mass
222In order to derive systematically the full
effective lagrangian that remains
after the charginos/neutralinos are integrated out,
one could use the method of ref. [14],
as it was done in [15] for the
integration of the top quark in the SM, however the
specific cases discussed in the present paper,
allows us to understand the limit of
heavy charginos and neutralinos..
Finally, we evaluate the contributions
of heavy charginos and neutralinos to
Veltman’s $\rho$ parameter.
Complete calculations of radiative corrections for the
MSSM have been performed in the literature [16],
however the results for the chargino/neutralino sector
are presented only in numerical form, which does not help
to clarify the subtleties associated
with the heavy-mass limits.
The definitions of the $\rho$ parameter in terms
of the self-energies of the gauge bosons
($\Pi_{ZZ},\Pi_{WW}$) is the following [17]:
$$\displaystyle\rho$$
$$\displaystyle=$$
$$\displaystyle\frac{\Pi_{ZZ}(0)}{m^{2}_{Z}}-\frac{\Pi_{WW}(0)}{m^{2}_{W}}$$
(29)
where $\Pi(0)^{\prime}s$ are obtained from the expansions
$\Pi_{ij}(q^{2})=\Pi_{ij}(0)+\Pi^{\prime}_{ij}(0)q^{2}$.
The total contribution of chargino and neutralino
to the self-energies, keeping only the leading
terms in the limit $|\mu|>>M,M^{\prime},m_{W}$, is:
$$\displaystyle{\Pi}_{WW}(0)$$
$$\displaystyle=$$
$$\displaystyle\frac{g^{2}}{8\pi^{2}}[\mu^{2}+A_{0}(\mu)]$$
(30)
$$\displaystyle{\Pi}_{ZZ}(0)$$
$$\displaystyle=$$
$$\displaystyle\frac{g^{2}}{8\pi^{2}c^{2}_{W}}[\mu^{2}+A_{0}(\mu)]$$
(31)
where $A_{0}(\mu)=-\mu^{2}+2\mu^{2}\log(\mu/\mu_{0})$, and
$\mu_{0}$ is the mass-scale that arises in the
$\bar{MS}$-scheme with dimensional regularization.
Thus, it follows eqs. (39) and (31) that
$\rho=0$.
This result can be understood if we remind
that $\rho$ is associated with the breaking of
isospin, and
the large-$\mu$ limit does not induce a
mass-splitting among the charginos and neutralinos
that interact with the gauge bosons,
i.e. $M_{\chi^{0}_{3,4}}=M_{\chi^{+}_{2}}$;
thus $\rho$ must vanish in this limit.
In conclusion, we have studied the
effects that remain after taking
the large $\mu$ limit in the MSSM.
It is found that despite the fact that charginos and
neutralinos contribute to the anomaly, they
do not induce a Wess-Zumino terms in the effective lagrangian,
unlike other cases studied in the literature
[18]. This result can be explained by
the form of the soft-SUSY breaking mass terms,
which do not allow a large mass-splitting among
the higgsinos; moreover, the large-$\mu$ limit
in the gaugino-higsino sector of the MSSM is obtained
by rendering large a dimensional parameter,
associated to gauge invariant mass terms,
which does not produce strong interaction
effects. A large mass splitting among higgsinos
could be obtained if it were possible to include a
mass term for each higgsino in the lagrangian, however
this type of term is not soft [1]. Moreover,
we also found that the effects of charginos-neutralinos
to the $\rho$ parameter vanish
in the large $\mu$ limit. Thus, they decouple
in all quantities studied in this paper.
We have also reviewed the pattern of anomaly
cancellation due to higgsinos in the Minimal SUSY extension
of the Standard model (MSSM), and found that they
have different characteristics as compared with the
ones due to quarks and leptons. For instance,
when $tan\beta=1$ it happens that
the charged higgsinos do not contribute to the
anomaly, whereas the neutralinos do not contribute
to the anomaly when $\mu$ is large.
This result suggest an alternative
mechanism to obtain anomaly-free theories.
In the usual approach, it is assumed that the chirality
of the fermions is fixed, then the representations
are adjusted in order to cancel the anomalies.
However, in extended SUSY models, new particles
are predicted, whose chiralities are not
known yet, and if their fixing depends on
some unknown parameters, then it may be possible that
those new parameters have values that make the
theory anomaly-free.
Acknowledgment.- Valuable discussions with G. Kane,
M.J. Herrero, J.J. Toscano, M. Hernandez and correspondence
with E. D’Hoker are acknowledged.
This work was supported by CONACYT and SNI (México).
References
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U. of Colorado, SCIPP-92/93.
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Phys. Rev. Lett. 76 (1996) 869, and references therein.
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J. Preskill, Ann. Phys. (N.Y.) 210 (1991) 323.
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J.F. Gunion and H. Haber,
Nucl. Phys. B272 (1986) 1; J. Rosiek,
Phys. Rev. D41 (1990) 3464.
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S. Dawson et al., The Higgs Hunter Guide,
(Adison Wesley, 1992).
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A. Barroso et al., Z. Phys. C33 (1986) 243.
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J.F. Gunion and H. Haber,
Phys. Rev. D37 (1988) 2515.
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I.J.R. Aitchison and C.M. Fraser,
Phys. Rev. D31 (1985) 2605.
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Guy-lin Liu, H. Steger and Y.P. Yao,
Phys. Rev. D44 (1991) 2139.
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W. de Boer et al., preprint IEKP-KA/96-08;
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R. Ball, Phys. Rep. 182 (1989) 1. |
On Recognizable Tree Languages
Beyond the Borel Hierarchy
Olivier Finkel
Equipe de Logique Mathématique
CNRS et Université Paris 7
France.
finkel@logique.jussieu.fr
Pierre Simonnet
Systèmes physiques pour l’environnement
Faculté des Sciences
Université de Corse
Quartier Grossetti BP52 20250, Corte, France
simonnet@univ-corse.fr
Abstract
We investigate the topological complexity of non Borel recognizable tree languages with regard to the difference hierarchy of analytic sets.
We show that, for each integer $n\geq 1$, there is a $D_{\omega^{n}}({\bf\Sigma}^{1}_{1})$-complete tree language $\mathcal{L}_{n}$
accepted by a (non deterministic) Muller tree automaton.
On the other hand, we prove that a tree language accepted by an unambiguous Büchi tree automaton must be Borel.
Then we consider the game tree languages $W_{(\iota,\kappa)}$, for Mostowski-Rabin indices $(\iota,\kappa)$. We prove that
the $D_{\omega^{n}}({\bf\Sigma}^{1}_{1})$-complete tree languages $\mathcal{L}_{n}$ are Wadge reducible to the game tree language
$W_{(\iota,\kappa)}$ for $\kappa-\iota\geq 2$.
In particular these languages $W_{(\iota,\kappa)}$ are not in any class $D_{\alpha}({\bf\Sigma}^{1}_{1})$
for $\alpha<\omega^{\omega}$.
keywords:
Infinite trees; tree automaton; regular tree language; Cantor topology: topological complexity; Borel hierarchy; difference hierarchy of
analytic sets; complete sets; unambiguous tree automaton; game tree language.
††issue: (2009)
On Recognizable Tree Languages Beyond the Borel Hierarchy
1 Introduction
A way to study the complexity of languages of infinite words or infinite trees accepted by various kinds of automata is to study their topological
complexity, and firstly to locate them with regard to the Borel and the projective hierarchies.
It is well known that every $\omega$-language accepted by a deterministic Büchi automaton is a ${\bf\Pi}^{0}_{2}$-set. This implies that
any $\omega$-language accepted by a deterministic Muller automaton is a
boolean combination of ${\bf\Pi}^{0}_{2}$-sets hence a
${\bf\Delta}^{0}_{3}$-set. [Tho90, Sta97, PP04]. But then it follows from Mc Naughton’s Theorem, that
all regular $\omega$-languages, which are accepted by deterministic Muller automata, are also
${\bf\Delta}^{0}_{3}$-sets.
The Borel hierarchy of regular $\omega$-languages is then determined. Moreover Wagner determined a much more refined hierarchy on regular
$\omega$-languages, which is in fact the
trace of the Wadge hierarchy on regular $\omega$-languages, now called the Wagner hierarchy.
On the other hand, many questions remain open about the topological
complexity of regular languages of infinite trees. We know that they can be much more complex than regular sets of infinite words.
Skurczynski proved that for every integer $n\geq 1$, there are some ${\bf\Pi}^{0}_{n}$-complete and some ${\bf\Sigma}^{0}_{n}$-complete
regular tree languages, [Sku93]. Notice that it is an open question to know whether there exist some regular sets of trees
which are Borel sets of infinite rank. But there exist some regular sets of trees which are not Borel. Niwinski showed that there are some
${\bf\Sigma}^{1}_{1}$-complete regular sets of trees accepted by Büchi tree automata, and some
${\bf\Pi}^{1}_{1}$-complete regular sets of trees accepted by deterministic Muller tree automata,
[Niw85]. Every set of trees accepted by a Büchi tree
automaton is a ${\bf\Sigma}^{1}_{1}$-set and every set of trees accepted by a deterministic Muller tree automaton is a
${\bf\Pi}^{1}_{1}$-set. Niwinski and Walukiewicz proved that a tree language which is accepted by a deterministic Muller tree automaton
is either in the class ${\bf\Pi}^{0}_{3}$ or ${\bf\Pi}^{1}_{1}$-complete, [NW03]. More recent results of Duparc and Murlak,
on the Wadge hierarchy of recognizable tree languages, may be found in [Mur08, ADMN07].
It follows from the definition of acceptance by non deterministic Muller or Rabin automata and from Rabin’s complementation Theorem that
every regular set of trees is a ${\bf\Delta}^{1}_{2}$-set, see [Rab69, PP04, Tho90, LT94].
But there are only few known results on the complexity of non Borel regular tree languages.
The second author gave examples of $D_{\omega^{n}}({\bf\Sigma}^{1}_{1})$-complete regular tree languages in [Sim92].
Arnold and Niwinski showed in [AN08] that the game tree languages $W_{(\iota,\kappa)}$ form a infinite hierarchy of non Borel
regular sets of trees with regard to the Wadge
reducibility.
In this paper, we investigate the topological complexity of non Borel recognizable tree languages with regard to the difference hierarchy of analytic sets.
We show that, for each integer $n\geq 1$, there is a $D_{\omega^{n}}({\bf\Sigma}^{1}_{1})$-complete tree language $\mathcal{L}_{n}$
accepted by a (non deterministic) Muller tree automaton.
On the other hand, we prove that non Borel recognizable tree languages accepted by Büchi tree automata have the maximum degree of ambiguity.
In particular, a tree language recognized by an unambiguous Büchi tree automaton must be Borel.
Then we consider the game tree languages $W_{(\iota,\kappa)}$, for Mostowski-Rabin indices $(\iota,\kappa)$. We prove that
the $D_{\omega^{n}}({\bf\Sigma}^{1}_{1})$-complete tree languages $\mathcal{L}_{n}$ are Wadge reducible to the game tree language
$W_{(\iota,\kappa)}$ for $\kappa-\iota\geq 2$.
In particular, these languages $W_{(\iota,\kappa)}$ are not in any class $D_{\alpha}({\bf\Sigma}^{1}_{1})$
for $\alpha<\omega^{\omega}$.
The paper is organized as follows. In Section 2 we recall the notions of Büchi or Muller tree automata and of regular tree languages. The notions of topology,
including the definition of the difference hierarchy of analytic sets, are recalled in Section 3. We show in Section 4 that there are
$D_{\omega^{n}}({\bf\Sigma}^{1}_{1})$-complete tree languages $\mathcal{L}_{n}$
accepted by Muller tree automata. We consider the complexity of game tree languages in Section 5.
2 Recognizable tree languages
We recall now usual notations of formal language theory.
When $\Sigma$ is a finite alphabet, a non-empty finite word over $\Sigma$ is any
sequence $x=a_{1}\cdots a_{k}$, where $a_{i}\in\Sigma$
for $i=1,\ldots,k$ , and $k$ is an integer $\geq 1$. The length
of $x$ is $k$, denoted by $|x|$.
The empty word has no letter and is denoted by $\lambda$; its length is $0$.
$\Sigma^{\star}$ is the set of finite words (including the empty word) over $\Sigma$. A finitary language
over the alphabet $\Sigma$ is a subset of
$\Sigma^{\star}$.
The first infinite ordinal is $\omega$.
An $\omega$-word over $\Sigma$ is an $\omega$ -sequence $a_{1}\cdots a_{n}\cdots$, where for all
integers $i\geq 1$,
$a_{i}\in\Sigma$. When $\sigma$ is an $\omega$-word over $\Sigma$, we write
$\sigma=\sigma(1)\sigma(2)\cdots\sigma(n)\cdots$, where for all $i$, $\sigma(i)\in\Sigma$,
and $\sigma[n]=\sigma(1)\sigma(2)\cdots\sigma(n)$ for all $n\geq 1$ and $\sigma[0]=\lambda$.
The usual concatenation product of two finite words $u$ and $v$ is
denoted $u\cdot v$ (and sometimes just $uv$). This product is extended to the product of a
finite word $u$ and an $\omega$-word $v$: the infinite word $u\cdot v$ is then the $\omega$-word such that:
$(u\cdot v)(k)=u(k)$ if $k\leq|u|$ , and
$(u\cdot v)(k)=v(k-|u|)$ if $k>|u|$.
The prefix relation is denoted $\sqsubseteq$: a finite word $u$ is a prefix
of a finite word $v$ (respectively, an infinite word $v$), denoted $u\sqsubseteq v$,
if and only if there exists a finite word $w$
(respectively, an infinite word $w$), such that $v=u\cdot w$.
The set of $\omega$-words over the alphabet $\Sigma$ is denoted by $\Sigma^{\omega}$.
An $\omega$-language over an alphabet $\Sigma$ is a subset of $\Sigma^{\omega}$.
We introduce now languages of infinite binary trees whose nodes
are labelled in a finite alphabet $\Sigma$.
A node of an infinite binary tree is represented by a finite word over
the alphabet $\{l,r\}$ where $r$ means “right” and $l$ means “left”. Then an
infinite binary tree whose nodes are labelled in $\Sigma$ is identified with a function
$t:\{l,r\}^{\star}\rightarrow\Sigma$. The set of infinite binary trees labelled in $\Sigma$ will be
denoted $T_{\Sigma}^{\omega}$.
Let $t$ be a tree. A branch $B$ of $t$ is a subset of the set of nodes of $t$ which
is linearly ordered by the tree partial order $\sqsubseteq$ and which
is closed under prefix relation,
i.e. if $x$ and $y$ are nodes of $t$ such that $y\in B$ and $x\sqsubseteq y$ then $x\in B$.
A branch $B$ of a tree is said to be maximal iff there is not any other branch of $t$
which strictly contains $B$.
Let $t$ be an infinite binary tree in $T_{\Sigma}^{\omega}$. If $B$ is a maximal branch of $t$,
then this branch is infinite. Let $(u_{i})_{i\geq 0}$ be the enumeration of the nodes in $B$
which is strictly increasing for the prefix order.
The infinite sequence of labels of the nodes of such a maximal
branch $B$, i.e. $t(u_{0})t(u_{1})\cdots t(u_{n})\cdots$ is called a path. It is an $\omega$-word
over the alphabet $\Sigma$.
Let then $L\subseteq\Sigma^{\omega}$ be an $\omega$-language over $\Sigma$. Then we denote $\exists\mathrm{Path}(L)$ the set of
infinite trees $t$ in $T_{\Sigma}^{\omega}$ such that $t$ has (at least) one path in $L$.
We are now going to define tree automata and recognizable tree languages.
Definition 2.1
A (nondeterministic topdown) tree automaton is a quadruple $\mathcal{A}=(K,\Sigma,\Delta,q_{0})$, where $K$
is a finite set of states, $\Sigma$ is a finite input alphabet, $q_{0}\in K$ is the initial state
and $\Delta\subseteq K\times\Sigma\times K\times K$ is the transition relation. The tree automaton $\mathcal{A}$ is said to be
deterministic if the relation $\Delta$ is a functional one, i.e. if for each $(q,a)\in K\times\Sigma$ there is at most one pair of states $(q^{\prime},q^{\prime\prime})$
such that $(q,a,q^{\prime},q^{\prime\prime})\in\Delta$.
A run of the tree automaton $\mathcal{A}$ on an infinite binary tree $t\in T_{\Sigma}^{\omega}$ is a infinite binary tree $\rho\in T_{K}^{\omega}$ such that:
(a) $\rho(\lambda)=q_{0}$ and (b) for each $u\in\{l,r\}^{\star}$, $(\rho(u),t(u),\rho(u.l),\rho(u.r))\in\Delta$.
Definition 2.2
A Büchi (nondeterministic topdown) tree automaton is a 5-tuple $\mathcal{A}=(K,\Sigma,\Delta,q_{0},F)$, where $(K,\Sigma,\Delta,q_{0})$ is a
tree automaton and $F\subseteq K$ is the set of accepting states.
A run $\rho$ of the Büchi tree automaton $\mathcal{A}$ on an infinite binary tree $t\in T_{\Sigma}^{\omega}$ is said to be accepting if
for each path of $\rho$ there is some accepting state appearing infinitely often on this path.
The tree language $L(\mathcal{A})$ accepted by the Büchi tree automaton $\mathcal{A}$ is the set of infinite binary trees $t\in T_{\Sigma}^{\omega}$
such that there is (at least) one accepting run of $\mathcal{A}$ on $t$.
Definition 2.3
A Muller (nondeterministic topdown) tree automaton is a 5-tuple $\mathcal{A}=(K,\Sigma,\Delta,q_{0},\mathcal{F})$, where $(K,\Sigma,\Delta,q_{0})$ is a
tree automaton and $\mathcal{F}\subseteq 2^{K}$ is the collection of designated state sets.
A run $\rho$ of the Muller tree automaton $\mathcal{A}$ on an infinite binary tree $t\in T_{\Sigma}^{\omega}$ is said to be accepting if
for each path $p$ of $\rho$, the set of states appearing infinitely often on this path is in $\mathcal{F}$.
The tree language $L(\mathcal{A})$ accepted by the Muller tree automaton $\mathcal{A}$ is the set of infinite binary trees $t\in T_{\Sigma}^{\omega}$
such that there is (at least) one accepting run of $\mathcal{A}$ on $t$.
The class $REG$ of regular, or recognizable, tree languages is the class of tree languages accepted by some Muller automaton.
Remark 2.4
Each tree language accepted by some (deterministic) Büchi automaton is also accepted by some (deterministic)
Muller automaton. A tree language is accepted by a Muller tree automaton iff it is accepted by some Rabin tree automaton. We refer for instance to
[Tho90, PP04] for the definition of Rabin tree automaton.
Example 2.5
Let $L\subseteq\Sigma^{\omega}$ be a regular $\omega$-language
(see [PP04] about regular $\omega$-languages which are the $\omega$-languages accepted by Büchi or Muller automata).
Then the set $\exists\mathrm{Path}(L)\subseteq T_{\Sigma}^{\omega}$ is accepted by a
Büchi tree automaton, hence also by a Muller tree automaton.
The set of infinite binary trees $t\in T_{\Sigma}^{\omega}$ having all their paths in $L$, denoted $\forall\mathrm{Path}(L)$,
is accepted by a deterministic Muller tree automaton. It is in fact the
complement of the set $\exists\mathrm{Path}(\Sigma^{\omega}-L)$.
3 Topology
We assume the reader to be familiar with basic notions of topology which
may be found in [Mos80, LT94, Kec95, Sta97, PP04].
There is a natural metric on the set $\Sigma^{\omega}$ of infinite words
over a finite alphabet
$\Sigma$ containing at least two letters which is called the prefix metric and defined as follows. For $u,v\in\Sigma^{\omega}$ and
$u\neq v$ let $\delta(u,v)=2^{-l_{\mathrm{pref}(u,v)}}$ where $l_{\mathrm{pref}(u,v)}$
is the first integer $n$
such that the $(n+1)^{st}$ letter of $u$ is different from the $(n+1)^{st}$ letter of $v$.
This metric induces on $\Sigma^{\omega}$ the usual Cantor topology for which open subsets of
$\Sigma^{\omega}$ are in the form $W\cdot\Sigma^{\omega}$, where $W\subseteq\Sigma^{\star}$.
A set $L\subseteq\Sigma^{\omega}$ is a closed set iff its complement $\Sigma^{\omega}-L$
is an open set.
There is also a natural topology on the set $T_{\Sigma}^{\omega}$ [Mos80, LT94, Kec95].
It is defined
by the following distance. Let $t$ and $s$ be two distinct infinite trees in $T_{\Sigma}^{\omega}$.
Then the distance between $t$ and $s$ is $\frac{1}{2^{n}}$ where $n$ is the smallest integer
such that $t(x)\neq s(x)$ for some word $x\in\{l,r\}^{\star}$ of length $n$.
The open sets are then in the form $T_{0}\cdot T_{\Sigma}^{\omega}$ where $T_{0}$ is a set of finite labelled
trees. $T_{0}\cdot T_{\Sigma}^{\omega}$ is the set of infinite binary trees
which extend some finite labelled binary tree $t_{0}\in T_{0}$, $t_{0}$ is here a sort of prefix,
an “initial subtree”
of a tree in $t_{0}\cdot T_{\Sigma}^{\omega}$.
It is well known that the set $T_{\Sigma}^{\omega}$, equipped with this topology, is homeomorphic to the Cantor set $2^{\omega}$, hence also to the topological spaces
$\Sigma^{\omega}$, where $\Sigma$ is an alphabet having at least two letters.
We now define the Borel Hierarchy of subsets of $\Sigma^{\omega}$. It is defined similarly on the space $T_{\Sigma}^{\omega}$.
Definition 3.1
For a non-null countable ordinal $\alpha$, the classes ${\bf\Sigma}^{0}_{\alpha}$
and ${\bf\Pi}^{0}_{\alpha}$ of the Borel Hierarchy on the topological space $\Sigma^{\omega}$
are defined as follows:
${\bf\Sigma}^{0}_{1}$ is the class of open subsets of $\Sigma^{\omega}$,
${\bf\Pi}^{0}_{1}$ is the class of closed subsets of $\Sigma^{\omega}$,
and for any countable ordinal $\alpha\geq 2$:
${\bf\Sigma}^{0}_{\alpha}$ is the class of countable unions of subsets of $\Sigma^{\omega}$ in
$\bigcup_{\gamma<\alpha}{\bf\Pi}^{0}_{\gamma}$.
${\bf\Pi}^{0}_{\alpha}$ is the class of countable intersections of subsets of $\Sigma^{\omega}$ in
$\bigcup_{\gamma<\alpha}{\bf\Sigma}^{0}_{\gamma}$.
For
a countable ordinal $\alpha$, a subset of $\Sigma^{\omega}$ is a Borel set of rank $\alpha$ iff
it is in ${\bf\Sigma}^{0}_{\alpha}\cup{\bf\Pi}^{0}_{\alpha}$ but not in
$\bigcup_{\gamma<\alpha}({\bf\Sigma}^{0}_{\gamma}\cup{\bf\Pi}^{0}_{\gamma})$.
There exists another hierarchy beyond the Borel hierarchy, which is called the
projective hierarchy. The classes ${\bf\Sigma}^{1}_{n}$ and ${\bf\Pi}^{1}_{n}$, for integers $n\geq 1$,
of the projective hierarchy are obtained from the Borel hierarchy by
successive applications of operations of projection and complementation.
The first level of the projective hierarchy is formed by the class ${\bf\Sigma}^{1}_{1}$ of analytic sets and the
class ${\bf\Pi}^{1}_{1}$ of co-analytic sets which are complements of
analytic sets.
In particular,
the class of Borel subsets of $\Sigma^{\omega}$ is strictly included in
the class ${\bf\Sigma}^{1}_{1}$ of analytic sets which are
obtained by projection of Borel sets.
Definition 3.2
A subset $A$ of $\Sigma^{\omega}$ is in the class ${\bf\Sigma}^{1}_{1}$ of analytic sets
iff there exists another finite set $Y$ and a Borel subset $B$ of $(\Sigma\times Y)^{\omega}$
such that $x\in A\leftrightarrow\exists y\in Y^{\omega}$ such that $(x,y)\in B$,
where $(x,y)$ is the infinite word over the alphabet $\Sigma\times Y$ such that
$(x,y)(i)=(x(i),y(i))$ for each integer $i\geq 1$.
Remark 3.3
In the above definition we could take $B$ in the class ${\bf\Pi}^{0}_{2}$. Moreover
analytic subsets of $\Sigma^{\omega}$ are the projections of ${\bf\Pi}^{0}_{1}$-subsets of
$\Sigma^{\omega}\times\omega^{\omega}$, where $\omega^{\omega}$ is the Baire space, [Mos80].
We now define the notion of Wadge reducibility via the reduction by continuous functions.
Let $X$, $Y$ be two finite alphabets.
For $L\subseteq X^{\omega}$ and $L^{\prime}\subseteq Y^{\omega}$, $L$ is said to be Wadge reducible to $L^{\prime}$, denoted by $L\leq_{W}L^{\prime}$,
iff there exists a continuous function $f:X^{\omega}\rightarrow Y^{\omega}$, such that
$L=f^{-1}(L^{\prime})$.
We now define completeness with regard to reduction by continuous functions.
For a countable ordinal $\alpha\geq 1$, and an integer $n\geq 1$, a set $F\subseteq\Sigma^{\omega}$ is said to be
a ${\bf\Sigma}^{0}_{\alpha}$
(respectively, ${\bf\Pi}^{0}_{\alpha}$, ${\bf\Sigma}^{1}_{n}$, ${\bf\Pi}^{1}_{n}$)-complete set
iff for any set $E\subseteq Y^{\omega}$ (with $Y$ a finite alphabet):
$E\in{\bf\Sigma}^{0}_{\alpha}$ (respectively, $E\in{\bf\Pi}^{0}_{\alpha}$, $E\in{\bf\Sigma}^{1}_{n}$, $E\in{\bf\Pi}^{1}_{n}$)
iff $E\leq_{W}F$.
${\bf\Sigma}^{0}_{n}$
(respectively ${\bf\Pi}^{0}_{n}$)-complete sets, with $n$ an integer $\geq 1$,
are thoroughly characterized in [Sta86].
The Borel hierarchy and the projective hierarchy on $T_{\Sigma}^{\omega}$ are defined from open
sets in the same manner as in the case of the topological space $\Sigma^{\omega}$.
The $\omega$-language $\mathcal{R}=(0^{\star}\cdot 1)^{\omega}$
is a well known example of
${\bf\Pi}^{0}_{2}$-complete subset of $\{0,1\}^{\omega}$. It is the set of
$\omega$-words over $\{0,1\}$ having infinitely many occurrences of the letter $1$.
Its complement
$\{0,1\}^{\omega}-(0^{\star}\cdot 1)^{\omega}$ is a
${\bf\Sigma}^{0}_{2}$-complete subset of $\{0,1\}^{\omega}$.
The set of infinite trees in $T_{\Sigma}^{\omega}$, where $\Sigma=\{0,1\}$, having at least one path in the $\omega$-language $\mathcal{R}=(0^{\star}\cdot 1)^{\omega}$ is
${\bf\Sigma}^{1}_{1}$-complete. Its complement is the set of trees in $T_{\Sigma}^{\omega}$ having all their paths in $\{0,1\}^{\omega}-(0^{\star}\cdot 1)^{\omega}$; it is
${\bf\Pi}^{1}_{1}$-complete.
We now recall the notion of difference hierarchy of analytic sets.
Let $\eta\!<\!\omega_{1}$ (where $\omega_{1}$ is the first uncountable ordinal)
be an ordinal and $(A_{\theta})_{\theta<\eta}$ be an increasing sequence of subsets of some space $X$,
then the set
$D_{\eta}[(A_{\theta})_{\theta<\eta}]$ is the set of elements $x\in X$ such that $x\!\in\!A_{\theta}\!\setminus\!\bigcup_{\theta^{\prime}<\theta}\ A_{\theta^{%
\prime}}$ for some
$\theta\!<\!\eta$ whose parity is opposite to that of $\eta$. (Recall that a countable ordinal $\gamma$ is said to be even iff it can be written in the form
$\gamma=\alpha+n$, where $\alpha$ is a limit ordinal and $n$ is an even non-negative integer; otherwise the ordinal $\gamma$ is said to be odd; notice that all
limit ordinals, like the ordinals $\omega^{n}$, $n\geq 1$, or $\omega^{\omega}$, are even ordinals.)
We can now define the class of $\eta$-differences of analytic subsets of $X$, where $X=\Sigma^{\omega}$ or $X=T_{\Sigma}^{\omega}$.
$D_{\eta}({\bf\Sigma}^{1}_{1})\!:=\!\{D_{\eta}[(A_{\theta})_{\theta<\eta}]\mid%
\mbox{ for each ordinal }\theta<\eta~{}~{}A_{\theta}\mbox{ is a }{\bf\Sigma}^{%
1}_{1}\mbox{-set }\}$
It is well known that the hierarchy of differences of analytic sets is strict, i.e. that for all countable ordinals $\alpha<\beta<\omega_{1}$, it holds that
$D_{\alpha}({\bf\Sigma}^{1}_{1})\subset D_{\beta}({\bf\Sigma}^{1}_{1})$. This is considered as a folklore result of descriptive set theory which follows from the
existence of universal sets for each class $D_{\alpha}({\bf\Sigma}^{1}_{1})$. Indeed we know first that the class ${\bf\Sigma}^{1}_{1}$ of analytic sets admits a universal
set, see [Kec95, page 205]or [Mos80, page 43].
Then, using classical methods of descriptive set theory, one can show that, for each countable ordinal $\alpha$,
the class $D_{\alpha}({\bf\Sigma}^{1}_{1})$ admits also a universal set, see [Kan97, page 443].
This implies, as in the case of the Borel hierarchy in [Kec95, page 168],
that the difference hierarchy of analytic sets is strict. As a universal set for the class $D_{\alpha}({\bf\Sigma}^{1}_{1})$ is also a
$D_{\alpha}({\bf\Sigma}^{1}_{1})$-complete set for reduction
by continuous functions, this implies also that there exists a $D_{\alpha}({\bf\Sigma}^{1}_{1})$-complete set.
Notice that in the sequel we shall only consider the classes $D_{\alpha}({\bf\Sigma}^{1}_{1})$, for ordinals $\alpha<\omega^{\omega}$, and that we shall
reprove that there exists some $D_{\alpha}({\bf\Sigma}^{1}_{1})$-complete subsets of $T_{\Sigma}^{\omega}$, giving examples which are regular sets of trees.
Another folklore result of descriptive set theory is that the union $\bigcup_{\alpha<\omega_{1}}D_{\alpha}({\bf\Sigma}^{1}_{1})$
represents only a small part of the class ${\bf\Delta}_{2}^{1}$. It is quoted for instance in [Ste82] or [Kan97, page 443]. (It is noticed
in [Ste82] that the union $\bigcup_{\alpha<\omega_{1}}D_{\alpha}({\bf\Sigma}^{1}_{1})$ is strictly included in the class
$\mathcal{A}({\bf\Pi}^{1}_{1})$ which is the closure of the class ${\bf\Pi}^{1}_{1}$ under Souslin’s operation. The class $\mathcal{A}({\bf\Pi}^{1}_{1})$ is
included in the class ${\bf\Delta}_{2}^{1}$ by [Mos80, 2.B.5 page 75]).
Notice however that this result is not necessary
in the sequel.
4 $D_{\omega^{n}}({\bf\Sigma}^{1}_{1})$-complete recognizable languages
It follows from the definition of the Büchi acceptance condition for infinite trees that each tree language recognized by a (non deterministic)
Büchi tree automaton is an analytic set.
Niwinski showed that some Büchi recognized tree languages are actually ${\bf\Sigma}^{1}_{1}$-complete sets.
An example is any tree language $T\subseteq T_{\Sigma}^{\omega}$ in the form $\exists\mathrm{Path}(L)$,
where $L\subseteq\Sigma^{\omega}$ is a regular $\omega$-language
which is a
${\bf\Pi}^{0}_{2}$-complete subset of $\Sigma^{\omega}$.
In particular, the tree language $\mathcal{L}=\exists\mathrm{Path}(\mathcal{R})$, where
$\mathcal{R}=(0^{\star}\cdot 1)^{\omega}$, is ${\bf\Sigma}^{1}_{1}$-complete
hence non Borel [Niw85, PP04, Sim92].
Notice that its complement $\mathcal{L}^{-}=\forall\mathrm{Path}(\{0,1\}^{\omega}-(0^{\star}\cdot 1)^{%
\omega})$ is a ${\bf\Pi}^{1}_{1}$-complete set.
It cannot be accepted by any
Büchi tree automaton because it is not a ${\bf\Sigma}^{1}_{1}$-set. On the other hand, it can be easily seen that it is accepted by a deterministic
Muller tree automaton.
The tree languages $\mathcal{L}$ and $\mathcal{L}^{-}$
have been used by the second author in [Sim92] to give examples of
$D_{\omega^{n}}({\bf\Sigma}^{1}_{1})$-complete recognizable tree languages, for integers $n\geq 1$.
We now give first the construction of a $D_{\omega}({\bf\Sigma}^{1}_{1})$-complete set.
For a tree $t\in T_{\Sigma}^{\omega}$ and $u\in\{l,r\}^{\star}$, we shall denote $t_{u}:\{l,r\}^{\star}\rightarrow\Sigma$ the subtree defined by
$t_{u}(v)=t(u\cdot v)$ for all $v\in\{l,r\}^{\star}$. It is in fact the subtree of $t$ which is rooted in $u$.
Now we can define a $D_{\omega}({\bf\Sigma}^{1}_{1})$-complete tree language $\mathcal{L}_{1}$.
$\mathcal{L}_{1}=\{t\in T_{\{0,1\}}^{\omega}\mid\exists n\geq 0~{}~{}t_{l^{n}%
\cdot r}\in\mathcal{L}\mbox{ and min}\{n\geq 0\mid t_{l^{n}\cdot r}\in\mathcal%
{L}\}\mbox{ is odd }\}$.
Proposition 4.1
The tree language $\mathcal{L}_{1}$ is $D_{\omega}({\bf\Sigma}^{1}_{1})$-complete.
Proof. We first show that the language $\mathcal{L}_{1}$ is in the class $D_{\omega}({\bf\Sigma}^{1}_{1})$.
Consider firstly, for some integer $k\geq 0$, the set $T_{k}=\{t\in T_{\{0,1\}}^{\omega}\mid~{}t_{l^{k}\cdot r}\in\mathcal{L}\}$.
It is clear that this set is in the class
${\bf\Sigma}^{1}_{1}$ because the function $F_{k}:T_{\{0,1\}}^{\omega}\rightarrow T_{\{0,1\}}^{\omega}$ defined by $F_{k}(t)=t_{l^{k}\cdot r}$ is continuous
and $T_{k}=F_{k}^{-1}(\mathcal{L})$ and the class ${\bf\Sigma}^{1}_{1}$ is closed under inverses of continuous functions.
Let now $H_{n}=\{t\in T_{\{0,1\}}^{\omega}\mid\exists k\leq n~{}t_{l^{k}\cdot r}\in%
\mathcal{L}\}$. This set is also in the class
${\bf\Sigma}^{1}_{1}$ because the class ${\bf\Sigma}^{1}_{1}$ is closed under finite (and even countable) union and $H_{n}=\bigcup_{k\leq n}T_{k}$.
The sets $H_{n}$ form an increasing sequence of ${\bf\Sigma}^{1}_{1}$-sets, and we can check that
$$\mathcal{L}_{1}=D_{\omega}[(H_{n})_{n<\omega}]$$
We now prove that $\mathcal{L}_{1}$ is $D_{\omega}({\bf\Sigma}^{1}_{1})$-complete.
Let $L\subseteq\Sigma^{\omega}$ be a $D_{\omega}({\bf\Sigma}^{1}_{1})$-subset of $\Sigma^{\omega}$, where $\Sigma$ is an alphabet having at least two letters. Then
there is an increasing sequence $(A_{n})_{n\in\omega}$ of ${\bf\Sigma}^{1}_{1}$-subsets of $\Sigma^{\omega}$ such that $L=D_{\omega}[(A_{n})_{n<\omega}]$.
On the other hand, we know that the tree language $\mathcal{L}$ is ${\bf\Sigma}^{1}_{1}$-complete. Thus for each integer $n\geq 0$ there exists a continuous
function $f_{n}:\Sigma^{\omega}\rightarrow T_{\{0,1\}}^{\omega}$ such that $A_{n}=f_{n}^{-1}(\mathcal{L})$.
We now define a function $F:\Sigma^{\omega}\rightarrow T_{\{0,1\}}^{\omega}$ by : for all $x\in\Sigma^{\omega}$, for all integers $k\geq 0$, $F(x)(l^{k})=0$
and $F(x)_{l^{k}\cdot r}=f_{k}(x)$.
It is clear that the function $F$ is continuous because each function $f_{k}$ is continuous.
We can now check that for every $x\in\Sigma^{\omega}$, $x$ is in the set $L=D_{\omega}[(A_{n})_{n<\omega}]$ iff there is an odd integer $n$ such that
$x\in A_{n}\ \setminus\ \bigcup_{k<n}\ A_{k}$ iff there is an odd integer $n$ such that $f_{n}(x)\in\mathcal{L}$ and for all $k<n$
$f_{k}(x)\in\mathcal{L}^{-}$.
This means that $x\in L=D_{\omega}[(A_{n})_{n<\omega}]$ iff $F(x)\in\mathcal{L}_{1}$.
Finally we have shown, using the reduction $F$, that $L=D_{\omega}[(A_{n})_{n<\omega}]\leq_{W}\mathcal{L}_{1}$ and so
the tree language $\mathcal{L}_{1}$ is $D_{\omega}({\bf\Sigma}^{1}_{1})$-complete.
$\square$
We can now generalize this construction to obtain some $D_{\omega^{n}}({\bf\Sigma}^{1}_{1})$-complete tree languages, for every integer $n\geq 1$.
Recall first that an ordinal $\alpha$ is strictly smaller than the ordinal $\omega^{n}$, where $n\geq 2$ is an integer, if and only if it admits a Cantor Normal Form
$$\alpha=\omega^{n-1}\cdot a_{n-1}+\omega^{n-2}\cdot a_{n-2}+\ldots+\omega\cdot a%
_{1}+a_{0}$$
where $a_{n-1},a_{n-2},\ldots,a_{0},$ are non-negative integers. In that case we shall denote
$\mathrm{Ord}(a_{n-1},a_{n-2},\ldots,a_{0})=\omega^{n-1}\cdot a_{n-1}+\omega^{n%
-2}\cdot a_{n-2}+\ldots+\omega\cdot a_{1}+a_{0}$.
Recall also that if $\alpha=\mathrm{Ord}(a_{n-1},a_{n-2},\ldots,a_{0})$ and $\beta=\mathrm{Ord}(b_{n-1},b_{n-2},\ldots,b_{0})$,
then $\alpha<\beta$ if and only
if there is an integer $k$ such that $0\leq k\leq n-1$ and $a_{j}=b_{j}$ for $n-1\geq j>k$ and $a_{k}<b_{k}$.
We now define the tree language $\mathcal{L}_{n}$, for $n\geq 2$, as the set of trees $t\in T_{\{0,1\}}^{\omega}$ for which
there exist some integers $a_{n-1},a_{n-2},\ldots,a_{0}\geq 0$ such that:
1.
$t_{l^{a_{n-1}}\cdot r\cdot l^{a_{n-2}}\cdot r\cdots l^{a_{0}}\cdot r}$ is in $\mathcal{L}$
and the parity of $\mathrm{Ord}(a_{n-1},a_{n-2},\ldots,a_{0})$ is odd,
2.
If $\mathrm{Ord}(b_{n-1},b_{n-2},\ldots,b_{0})<\mathrm{Ord}(a_{n-1},a_{n-2},\ldots%
,a_{0})$ then the tree
$t_{l^{b_{n-1}}\cdot r\cdot l^{b_{n-2}}.r\cdots l^{b_{0}}\cdot r}$ is not in $\mathcal{L}$.
Proposition 4.2
For each integer $n\geq 2$, the tree language $\mathcal{L}_{n}$ is $D_{\omega^{n}}({\bf\Sigma}^{1}_{1})$-complete.
Proof. The proof is a simple generalization of the proof of Proposition 4.1. Notice that we have to use
the closure of the class ${\bf\Sigma}^{1}_{1}$ under countable (and not only under finite) union. Details are here left to the reader.
$\square$
The tree languages $\mathcal{L}_{n}$ can not be accepted by any Büchi tree automaton
because each tree language accepted by a (non deterministic)
Büchi tree automaton is an analytic set and $D_{\omega^{n}}({\bf\Sigma}^{1}_{1})$-complete sets, for $n\geq 1$, are not in the class ${\bf\Sigma}^{1}_{1}$.
We are going to see that the tree languages $\mathcal{L}_{n}$
are accepted by Muller tree automata.
We now recall the following result proved by Niwinski in [Niw85], see also for instance [PP04, Tho90].
Lemma 4.3
The language $\mathcal{L}^{-}=\forall\mathrm{Path}(\{0,1\}^{\omega}-(0^{\star}.1)^{\omega})$ is a ${\bf\Pi}^{1}_{1}$-complete set
accepted by a deterministic Muller tree automaton.
On the other hand, the tree language $\mathcal{L}$ is a ${\bf\Sigma}^{1}_{1}$-complete set. Thus it is not a ${\bf\Pi}^{1}_{1}$-set otherwise it would be
in the class ${\bf\Delta}^{1}_{1}={\bf\Sigma}^{1}_{1}\cap{\bf\Pi}^{1}_{1}$ which is the class of Borel sets by Suslin’s Theorem.
But every tree language which is recognizable by a deterministic Muller tree automaton is a ${\bf\Pi}^{1}_{1}$-set therefore
the tree language $\mathcal{L}$ can not be accepted by any deterministic Muller tree automaton.
However we can now state the following result.
Lemma 4.4
The language $\mathcal{L}$ is a ${\bf\Sigma}^{1}_{1}$-complete set accepted by a non deterministic Büchi tree automaton,
hence also by a non deterministic Muller tree automaton.
Proof. We recall informally how we can define a non-deterministic Büchi tree automaton $\mathcal{A}$ accepting the language $\mathcal{L}$.
When reading a tree $t\in\mathcal{L}$, the automaton $\mathcal{A}$, using the non determinism, guesses an infinite branch of the tree. Then the
automaton checks, using the Büchi acceptance condition, that the sequence of labels of nodes on this branch forms an $\omega$-word in $(0^{\star}.1)^{\omega}$,
i.e. contains an infinite number of letters $1$.
$\square$
Lemma 4.5
For each integer $n\geq 1$, the language $\mathcal{L}_{n}$ is
accepted by a (non deterministic) Muller tree automaton.
Proof. We first construct a non deterministic Muller tree automaton $\mathcal{A}_{1}$ accepting
the language $\mathcal{L}_{1}$.
Recall that, for each tree $t\in\mathcal{L}_{1}$, there exists a least integer $n\geq 0$ such that $t_{l^{n}\cdot r}\in\mathcal{L}$.
This (odd) integer is defined in a unique way. One can now construct, from
Muller tree automata $\mathcal{A}^{-}$ and $\mathcal{A}^{+}$ accepting the tree languages $\mathcal{L}^{-}$ and $\mathcal{L}$, a
Muller tree automaton $\mathcal{A}_{1}$ accepting the tree language $\mathcal{L}_{1}$.
Using the non-determinism, the automaton $\mathcal{A}_{1}$ will guess the (odd) integer $n\geq 0$ and then,
using the behaviour of $\mathcal{A}^{-}$ and $\mathcal{A}^{+}$, it will check that $t_{l^{n}\cdot r}\in\mathcal{L}$ and that, for every integer
$k<n$, $t_{l^{k}\cdot r}\notin\mathcal{L}$.
We now give the exact construction of the non deterministic Muller tree automaton $\mathcal{A}_{1}$.
Let $\Sigma=\{0,1\}$ and $\mathcal{A}^{-}=(K,\Sigma,\Delta,q_{0},\mathcal{F})$ be a (deterministic) Muller tree automaton accepting
the tree language $\mathcal{L}^{-}$.
And let
$\mathcal{A}^{+}=(K^{\prime},\Sigma,\Delta^{\prime},q^{\prime}_{0},\mathcal{F}^%
{\prime})$ be a (non deterministic) Muller tree automaton accepting
the tree language $\mathcal{L}$. We assume that $K\cap K^{\prime}=\emptyset$.
Then it is easy to see that the tree language $\mathcal{L}_{1}$ is accepted by the Muller tree automaton
$\mathcal{A}_{1}=(K^{1},\Sigma,\Delta^{1},$ $q^{1}_{0},\mathcal{F}^{1})$, where
$K^{1}=K\cup K^{\prime}\cup\{q^{1}_{0},q^{1}_{1},q_{f}\}$,
$\Delta^{1}=\Delta\cup\Delta^{\prime}\cup\{(q^{1}_{0},a,q^{1}_{1},q_{0}),(q^{1}%
_{1},a,q_{f},q^{\prime}_{0}),(q_{f},a,q_{f},q_{f}),(q^{1}_{1},a,q^{1}_{0},q_{0%
})\mid a\in\{0,1\}\}$,
$\mathcal{F}^{1}=\mathcal{F}\cup\mathcal{F}^{\prime}\cup\{q_{f}\}$.
For every integer $n>1$, we can construct in a similar way a Muller tree automaton $\mathcal{A}_{n}$ accepting the tree language
$\mathcal{L}_{n}$.
Recall that for each tree $t\in\mathcal{L}_{n}$ there exists a least ordinal $\alpha=\mathrm{Ord}(a_{n-1},a_{n-2},\ldots,a_{0})<\omega^{n}$ such that
$t_{l^{a_{n-1}}\cdot r\cdot l^{a_{n-2}}\cdot r\cdots l^{a_{0}}\cdot r}$ is in $\mathcal{L}$. This (odd) ordinal is defined in a unique way.
One can now construct, from the
Muller tree automata $\mathcal{A}^{-}$ and $\mathcal{A}^{+}$ accepting the tree languages $\mathcal{L}^{-}$ and $\mathcal{L}$, a
Muller tree automaton $\mathcal{A}_{n}$ accepting the tree language $\mathcal{L}_{n}$.
Using the non-determinism, the automaton $\mathcal{A}_{n}$ will guess the (odd) ordinal $\alpha=\mathrm{Ord}(a_{n-1},a_{n-2},\ldots,a_{0})<\omega^{n}$
and then, using the behaviour of $\mathcal{A}^{-}$ and $\mathcal{A}^{+}$, it will check that
$t_{l^{a_{n-1}}\cdot r\cdot l^{a_{n-2}}\cdot r\cdots l^{a_{0}}\cdot r}$ is in $\mathcal{L}$
and that for each ordinal $\beta=\mathrm{Ord}(b_{n-1},b_{n-2},\ldots,b_{0})<\mathrm{Ord}(a_{n-1},a_{n-2},%
\ldots,a_{0})$ the tree language
$t_{l^{b_{n-1}}\cdot r\cdot l^{b_{n-2}}.r\cdots l^{b_{0}}\cdot r}$ is not in $\mathcal{L}$.
$\square$
We can now summarize the above results in the following theorem.
Theorem 4.6
For each integer $n\geq 1$, the language $\mathcal{L}_{n}$ is a $D_{\omega^{n}}({\bf\Sigma}^{1}_{1})$-complete set
accepted by a (non deterministic) Muller tree automaton.
Corollary 4.7
The class of tree languages recognized by Muller tree automata is not included into the boolean closure of the class
of tree languages recognized by Büchi tree automata.
Proof. We know that every tree language recognized by a Büchi tree automaton is a ${\bf\Sigma}^{1}_{1}$-set. But a tree language which is a boolean
combination of ${\bf\Sigma}^{1}_{1}$-sets is in the class $D_{\omega}({\bf\Sigma}^{1}_{1})$ which does not contain all
tree languages recognized by (non deterministic) Muller tree automata.
$\square$
Remark 4.8
We have given above examples of $D_{\omega^{n}}({\bf\Sigma}^{1}_{1})$-complete tree languages accepted by Muller tree automata.
In a similar way it is easy to construct, for each ordinal $\alpha<\omega^{\omega}$, a $D_{\alpha}({\bf\Sigma}^{1}_{1})$-complete tree language
accepted by a Muller tree automaton.
Each ordinal $\alpha<\omega^{\omega}$ may be written in the form $\alpha=\mathrm{Ord}(a_{n-1},a_{n-2},\ldots,a_{0})<\omega^{n}$ for some
integer $n\geq 1$ and where $a_{n-1},a_{n-2},\ldots,a_{0},$ are non-negative integers with $a_{n-1}\neq 0$.
The tree language $\mathcal{T}_{\alpha}$ is then the set of trees $t\in T_{\{0,1\}}^{\omega}$ for which
there exist some integers $b_{n-1},b_{n-2},\ldots,b_{0}\geq 0$ such that:
1.
$\mathrm{Ord}(b_{n-1},b_{n-2},\ldots,b_{0})<\mathrm{Ord}(a_{n-1},a_{n-2},\ldots%
,a_{0})$.
2.
$t_{l^{b_{n-1}}\cdot r\cdot l^{b_{n-2}}.r\cdots l^{b_{0}}\cdot r}$ is in $\mathcal{L}$
and the parity of $\mathrm{Ord}(b_{n-1},b_{n-2},\ldots,b_{0})$ is odd iff the parity of
$\mathrm{Ord}(a_{n-1},a_{n-2},\ldots,a_{0})$ is even.
3.
If $\mathrm{Ord}(c_{n-1},c_{n-2},\ldots,c_{0})<\mathrm{Ord}(b_{n-1},b_{n-2},\ldots%
,b_{0})$ then the tree
$t_{l^{c_{n-1}}\cdot r\cdot l^{c_{n-2}}.r\cdots l^{c_{0}}\cdot r}$ is not in $\mathcal{L}$.
The tree language $\mathcal{T}_{\alpha}$ is $D_{\alpha}({\bf\Sigma}^{1}_{1})$-complete and it is accepted by a (non deterministic)
Muller tree automaton.
The above results show that the topological complexity of tree languages recognized by non deterministic
Muller tree automata is much greater than that of tree languages accepted by deterministic Muller tree automata.
Recall that a Büchi (respectively, Muller) tree automaton $\mathcal{A}$, reading trees labelled in the alphabet $\Sigma$,
is said to be unambiguous if and only if
each tree $t\in T_{\Sigma}^{\omega}$ admits at most one accepting run of $\mathcal{A}$.
A natural question is whether the tree languages $\mathcal{L}_{n}$ could be accepted by unambiguous Muller tree automata.
A first step would be to prove that the
tree language $\mathcal{L}$ is accepted by an unambiguous Muller tree automaton.
But this is not possible. We have learned by
personal communication from Damian Niwinski that the
language $\mathcal{L}$ is inherently ambiguous, [Niw09].
We consider now the notion of ambiguity for Büchi tree automata and we shall prove in particular that
a tree language accepted by an unambiguous Büchi tree automaton must be Borel. We shall indicate also why our methods do not
work in the case of Muller automata.
We first recall some notations and a lemma proved in [FS03].
For two finite alphabets $\Sigma$ and $X$,
if $B\subseteq\Sigma^{\omega}\times X^{\omega}$ and $\alpha\in\Sigma^{\omega}$, we denote
$B_{\alpha}=\{\beta\in X^{\omega}\mid(\alpha,\beta)\in B\}$ and
$\mathrm{PROJ}_{\Sigma^{\omega}}(B)=\{\alpha\in\Sigma^{\omega}\mid B_{\alpha}%
\neq\emptyset\}$.
The cardinal of the continuum will be denoted by $2^{\aleph_{0}}$; it is also the cardinal
of every set $\Sigma^{\omega}$ or $T_{\Sigma}^{\omega}$, where $\Sigma$ is an alphabet having at least two letters.
Lemma 4.9 ([FS03])
Let $\Sigma$ and $X$ be two finite alphabets having at least two letters and
$B$ be a Borel subset of
$\Sigma^{\omega}\times X^{\omega}$ such that $\mathrm{PROJ}_{\Sigma^{\omega}}(B)$ is not a Borel subset of $\Sigma^{\omega}$.
Then there are $2^{\aleph_{0}}$ $\omega$-words $\alpha\in\Sigma^{\omega}$ such that the section $B_{\alpha}$
has cardinality $2^{\aleph_{0}}$.
Proof. Let $\Sigma$ and $X$ be two finite alphabets having at least two letters and
$B$ be a Borel subset of
$\Sigma^{\omega}\times X^{\omega}$ such that $\mathrm{PROJ}_{\Sigma^{\omega}}(B)$ is not Borel.
In a first step we prove that
there are uncountably many $\alpha\in\Sigma^{\omega}$ such that the section $B_{\alpha}$
is uncountable.
Recall that by a Theorem of Lusin and Novikov, see [Kec95, page 123], if for
all $\alpha\in\Sigma^{\omega}$, the section $B_{\alpha}$ of the Borel set $B$ was countable,
then $\mathrm{PROJ}_{\Sigma^{\omega}}(B)$ would be a Borel subset of $\Sigma^{\omega}$.
Thus there exists at least one $\alpha\in\Sigma^{\omega}$ such that $B_{\alpha}$ is uncountable.
In fact we have not only one $\alpha$ such that $B_{\alpha}$ is uncountable.
For $\alpha\in\Sigma^{\omega}$ we have
$\{\alpha\}\times B_{\alpha}=B\cap[\{\alpha\}\times X^{\omega}]$.
But $\{\alpha\}\times X^{\omega}$ is a closed
hence Borel subset of $\Sigma^{\omega}\times X^{\omega}$ thus $\{\alpha\}\times B_{\alpha}$
is Borel as intersection of two Borel sets.
If there was only one $\alpha\in\Sigma^{\omega}$ such that $B_{\alpha}$ is uncountable, then
$C=\{\alpha\}\times B_{\alpha}$ would be Borel so $D=B-C$ would be borel
because the class of Borel sets is closed
under boolean operations.
But all sections of $D$ would be countable thus
$\mathrm{PROJ}_{\Sigma^{\omega}}(D)$ would be Borel by Lusin and Novikov’s Theorem.
Then $\mathrm{PROJ}_{\Sigma^{\omega}}(B)=\{\alpha\}\cup\mathrm{PROJ}_{\Sigma^{\omega%
}}(D)$
would be also Borel as union of two Borel sets, and this would lead to a
contradiction.
In a similar manner we can prove that the set $U=\{\alpha\in\Sigma^{\omega}\mid B_{\alpha}\mbox{ is uncountable }\}$ is uncountable, otherwise
$U=\{\alpha_{0},\alpha_{1},\ldots\alpha_{n},\ldots\}$ would be Borel as the countable union
of the closed sets $\{\alpha_{i}\}$, $i\geq 0$.
For each $n\geq 0$ the set $\{\alpha_{n}\}\times B_{\alpha_{n}}$ would be Borel,
and
$C=\cup_{n\in\omega}\{\alpha_{n}\}\times B_{\alpha_{n}}$ would be Borel as a
countable union of Borel sets.
So $D=B-C$ would be borel too.
But all sections of $D$ would be countable thus
$\mathrm{PROJ}_{\Sigma^{\omega}}(D)$ would be Borel by Lusin and Novikov’s Theorem.
Then $\mathrm{PROJ}_{\Sigma^{\omega}}(B)=U\cup\mathrm{PROJ}_{\Sigma^{\omega}}(D)$ would be
also Borel as union of two Borel sets, and this would lead to a
contradiction.
So we have proved that the set
$\{\alpha\in\Sigma^{\omega}\mid B_{\alpha}\mbox{ is uncountable }\}$
is uncountable.
On the other hand we know from
another Theorem of Descriptive Set Theory that the set
$\{\alpha\in\Sigma^{\omega}\mid B_{\alpha}\mbox{ is countable }\}$ is a ${\bf\Pi^{1}_{1}}$-subset of
$\Sigma^{\omega}$, see [Kec95, page 123].
Thus its complement $\{\alpha\in\Sigma^{\omega}\mid B_{\alpha}\mbox{ is uncountable }\}$
is analytic.
But by Suslin’s Theorem an analytic subset of $\Sigma^{\omega}$ is either countable
or has cardinality $2^{\aleph_{0}}$, [Kec95, p. 88]. Therefore the set
$\{\alpha\in\Sigma^{\omega}\mid B_{\alpha}\mbox{ is uncountable }\}$
has cardinality $2^{\aleph_{0}}$.
Recall now that we have already seen that, for each $\alpha\in\Sigma^{\omega}$, the set
$\{\alpha\}\times B_{\alpha}$ is Borel.
Thus $B_{\alpha}$
itself is Borel and by Suslin’s Theorem $B_{\alpha}$ is either countable or has cardinality $2^{\aleph_{0}}$.
From this we deduce that
$\{\alpha\in\Sigma^{\omega}\mid B_{\alpha}\mbox{ is uncountable }\}=\{\alpha\in%
\Sigma^{\omega}\mid B_{\alpha}\mbox{ has cardinality }2^{\aleph_{0}}\}$
has cardinality $2^{\aleph_{0}}$. $\square$
This Lemma was used in [FS03] to
prove that analytic but non Borel
context-free $\omega$-languages have a maximum degree of ambiguity.
Theorem 4.10 ([FS03])
Let $L(\mathcal{A})$ be a context-free $\omega$-language accepted by a Büchi pushdown automaton $\mathcal{A}$ such that $L(\mathcal{A})$
is an analytic but non Borel set. Then the set of $\omega$-words,
which have $2^{\aleph_{0}}$ accepting runs by $\mathcal{A}$, has cardinality $2^{\aleph_{0}}$.
Reasoning in a very similar way as in the proof of Theorem 4.10 in [FS03], we can now state that
analytic but non Borel tree languages accepted by Büchi tree automata have a maximum degree of ambiguity.
If $\Sigma$ is an alphabet having at least two letters, the topological space $T_{\Sigma}^{\omega}$ is homeomorphic to the topological space $\Sigma^{\omega}$, so we can first
state Lemma 4.9 in the following equivalent form.
Lemma 4.11
Let $\Sigma$ and $K$ be two finite alphabets having at least two letters and
$B$ be a Borel subset of
$T_{\Sigma}^{\omega}\times T_{K}^{\omega}$ such that $\mathrm{PROJ}_{T_{\Sigma}^{\omega}}(B)$ is not a Borel subset of $T_{\Sigma}^{\omega}$.
Then there are $2^{\aleph_{0}}$ infinite trees $t\in T_{\Sigma}^{\omega}$ such that the section $B_{t}$
has cardinality $2^{\aleph_{0}}$.
We can now state the following result.
Theorem 4.12
Let $L(\mathcal{A})\subseteq T_{\Sigma}^{\omega}$ be a regular tree language accepted by a Büchi tree automaton $\mathcal{A}$ such that $L(\mathcal{A})$
is an analytic but non Borel set. Then the set of trees $t\in T_{\Sigma}^{\omega}$
which have $2^{\aleph_{0}}$ accepting runs by $\mathcal{A}$, has cardinality $2^{\aleph_{0}}$.
Proof. Let $\mathcal{A}=(K,\Sigma,\Delta,q_{0},F)$ be a Büchi tree automaton accepting a non Borel tree language
$L(\mathcal{A})\subseteq T_{\Sigma}^{\omega}$, and let
$R\subseteq T_{\Sigma}^{\omega}\times T_{K}^{\omega}$ be defined by :
$$R=\{(t,\rho)\mid t\in T_{\Sigma}^{\omega}\mbox{ and }\rho\in T_{K}^{\omega}%
\mbox{ is an accepting run of }\mathcal{A}\mbox{ on the tree }t\}.$$
The set $R$ can be seen as a tree language over the product alphabet $\Sigma\times K$. Then it is easy to see that $R$ is accepted by a
deterministic Büchi tree automaton. But every tree language which is accepted by a deterministic Büchi tree automaton is a
${\bf\Pi}_{2}^{0}$-set, see [Mur05].
Thus the tree language $R$ is a ${\bf\Pi}_{2}^{0}$-subset of the space $T_{(\Sigma\times K)^{\omega}}$ which is
identified to the topological space $T_{\Sigma}^{\omega}\times T_{K}^{\omega}$. In particular, $R$ is a Borel subset of $T_{\Sigma}^{\omega}\times T_{K}^{\omega}$.
But by definition of $R$ it turns out that $\mathrm{PROJ}_{T_{\Sigma}^{\omega}}(R)=L(\mathcal{A})$. Thus
$\mathrm{PROJ}_{T_{\Sigma}^{\omega}}(R)$ is not Borel and Lemma 4.11 implies that
there are $2^{\aleph_{0}}$ trees $t\in T_{\Sigma}^{\omega}$ such that $R_{t}$ has cardinality
$2^{\aleph_{0}}$. This means that these trees have $2^{\aleph_{0}}$
accepting runs by the Büchi tree automaton $\mathcal{A}$.
$\square$
Remark 4.13
The above proof is no longer valid if we replace “Büchi tree automaton” by “Muller tree automaton”. Indeed if
$L(\mathcal{A})\subseteq T_{\Sigma}^{\omega}$ is a regular tree language accepted by a Muller tree automaton $\mathcal{A}=(K,\Sigma,\Delta,q_{0},\mathcal{F})$,
then the set $R\subseteq T_{\Sigma}^{\omega}\times T_{K}^{\omega}$ defined by :
$$R=\{(t,\rho)\mid t\in T_{\Sigma}^{\omega}\mbox{ and }\rho\in T_{K}^{\omega}%
\mbox{ is an accepting run of }\mathcal{A}\mbox{ on the tree }t\}.$$
is now accepted by a
deterministic Muller tree automaton. Thus we can now only say that $R$ is a ${\bf\Pi}_{1}^{1}$-set, and we cannot use the fact that $R$ is Borel, which
was crucial in the proof of Theorem 4.12.
In particular, Theorem 4.12 implies the following important result.
Corollary 4.14
Let $L(\mathcal{A})\subseteq T_{\Sigma}^{\omega}$ be a regular tree language accepted by an unambiguous Büchi tree automaton. Then the tree language
$L(\mathcal{A})$ is a Borel subset of $T_{\Sigma}^{\omega}$.
Remark 4.15
The result given by Corollary 4.14 is weaker than the result given by Theorem 4.12. This weaker result can be proved
by a simpler argument. We give now this proof which is also interesting.
Proof. Let $L(\mathcal{A})\subseteq T_{\Sigma}^{\omega}$ be a regular tree language accepted by an unambiguous
Büchi tree automaton $\mathcal{A}=(K,\Sigma,\Delta,q_{0},F)$.
Let $R$ be defined as in the proof of Theorem 4.12 by:
$$R=\{(t,\rho)\mid t\in T_{\Sigma}^{\omega}\mbox{ and }\rho\in T_{K}^{\omega}%
\mbox{ is an accepting run of }\mathcal{A}\mbox{ on the tree }t\}.$$
The set $R$ is accepted by a
deterministic Büchi tree automaton so it is a ${\bf\Pi}_{2}^{0}$-subset of the space $T_{(\Sigma\times K)^{\omega}}$.
Consider now the projection $\mathrm{PROJ}_{T_{\Sigma}^{\omega}}:~{}T_{\Sigma}^{\omega}\times T_{K}^{\omega%
}\rightarrow T_{\Sigma}^{\omega}$ defined by
$\mathrm{PROJ}_{T_{\Sigma}^{\omega}}(t,\rho)=t$ for all $(t,\rho)\in T_{\Sigma}^{\omega}\times T_{K}^{\omega}$. This projection is a continuous function
and it is injective on the Borel set $R$ because the automaton $\mathcal{A}$ is unambiguous.
By a Theorem of Lusin and Souslin, see [Kec95, Theorem 15.1 page 89], the injective
image of $R$ by the continuous function $\mathrm{PROJ}_{T_{\Sigma}^{\omega}}$ is then Borel.
Thus the tree language $L(\mathcal{A})=\mathrm{PROJ}_{T_{\Sigma}^{\omega}}(R)$ is a Borel subset of $T_{\Sigma}^{\omega}$.
$\square$
Remark 4.16
The above result given by Corollary 4.14
is of course false in the case of Muller automata because we already know an example of non Borel regular tree language
accepted by a deterministic hence unambiguous Muller tree automaton. By Lemma 4.3, the tree language
$\mathcal{L}^{-}=\forall\mathrm{Path}(\{0,1\}^{\omega}-(0^{\star}.1)^{\omega})$ is a ${\bf\Pi}^{1}_{1}$-complete set
accepted by a deterministic Muller tree automaton.
5 Game tree languages
Game tree languages are particular recognizable tree languages which are defined by the use of parity games. So we now recall the definition
of these games, as introduced in [AN08, ADMN07].
A parity game is a game with perfect information between two players named Eve and Adam, as in [AN08, ADMN07].
The game is defined by
a tuple $G=(V_{\exists},V_{\forall},\mathrm{Move},p_{0},\mathrm{rank})$.
The sets $V_{\exists}$ and $V_{\forall}$ are disjoint sets of positions of Eve and Adam, respectively. We denote
$V=V_{\exists}\cup V_{\forall}$ the set of positions.
The relation $\mathrm{Move}\subseteq V\times V$ is the relation of possible moves. The initial position in a play is $p_{0}\in V$. The ranking function is
$\mathrm{rank}:V\rightarrow\omega$ and the number of values taken by this function is finite.
At the beginning of a play there is a token at the initial position $p_{0}$ where the play starts.
The players move the token according to the relation $\mathrm{Move}$, always to a successor of the current position.
The move is done by Eve if the current position is an element of $V_{\exists}$, otherwise Adam moves the token.
This way the two players form a path in the graph $(V,\mathrm{Move})$. If at some moment a player cannot move then she or he looses. Otherwise the
two players construct an infinite path in the graph, $v_{0},v_{1},v_{2},\ldots$
In this case Eve wins the play if $\lim\sup_{n\rightarrow\infty}\mathrm{rank}(v_{n})$ is even, otherwise
Adam wins the play.
Eve (respectively, Adam) wins the game $G$ if she (respectively, he) has a winning strategy. It is well known that parity games are determined,
i. e., that one of the players has a winning strategy. Moreover any position is winning for one of the players and she or he has a positional strategy from
this position, see [GTW02] for more details.
We now recall the definition of game languages $W_{(\iota,\kappa)}$.
A Mostowski-Rabin index is a pair $(\iota,\kappa)$, where $\iota\in\{0,1\}$ and $\iota\leq\kappa<\omega$. For such an index, we define the alphabet
$\Sigma_{(\iota,\kappa)}=\{\exists,\forall\}\times\{\iota,\ldots,\kappa\}$.
For a letter $a\in\Sigma_{(\iota,\kappa)}$ we denote $a=(a_{1},a_{2})$, where $a_{1}\in\{\exists,\forall\}$ and $a_{2}\in\{\iota,\ldots,\kappa\}$.
For each tree $t\in T_{\Sigma_{(\iota,\kappa)}}^{\omega}$
we associate a parity game $G(t)=(V_{\exists},V_{\forall},\mathrm{Move},p_{0},\mathrm{rank})$, where
•
$V_{\exists}=\{v\in\{l,r\}^{\star}\mid t(v)_{1}=\exists\}$,
•
$V_{\forall}=\{v\in\{l,r\}^{\star}\mid t(v)_{1}=\forall\}$,
•
$\mathrm{Move}=\{(w,wi)\mid w\in\{l,r\}^{\star}\mbox{ and }i\in\{l,r\}\}$,
•
$p_{0}=\lambda$ is the root of the tree,
•
$\mathrm{rank}(v)=t(v)_{2}$, for each $v\in\{l,r\}^{\star}$.
The set $W_{(\iota,\kappa)}\subseteq T_{\Sigma_{(\iota,\kappa)}}^{\omega}$ is the set of infinite binary trees $t$ labelled in the alphabet
$\Sigma_{(\iota,\kappa)}$ such that Eve wins the associated game $G(t)$.
The recognizable tree language $W_{(\iota,\kappa)}$ is accepted by an alternating parity tree automaton of index $(\iota,\kappa)$. This notion
will be useful in the sequel so we recall it now, as presented in [ADMN07].
Definition 5.1
An alternating parity tree automaton is a tuple $\mathcal{A}=(\Sigma,Q_{\exists},$ $Q_{\forall},q_{0},\delta,\mathrm{rank})$,
where the set of states $Q$ is partitioned in $Q_{\exists}$
and $Q_{\forall}$. The set $Q_{\exists}$ is the set of existential states and the set $Q_{\forall}$ is the set of universal states. The transition relation is
$\delta\subseteq Q\times\Sigma\times\{l,r,\lambda\}\times Q$ and $\mathrm{rank}:Q\rightarrow\omega$ is the rank function.
A tree $t\in T_{\Sigma}^{\omega}$ is accepted by
the automaton $\mathcal{A}$ iff Eve has a winning strategy in the parity game
$(Q_{\exists}\times\{l,r\}^{\star},Q_{\forall}\times\{l,r\}^{\star},(q_{0},%
\lambda),\mathrm{Move},\Omega)$, where $\mathrm{Move}=\{((p,v),(q,vd))\mid v\in\mathrm{dom}(t),~{}~{}(p,t(v),$ $d,q)\in\delta\}$ and $\Omega(q,v)=\mathrm{rank}(q)$.
Notice that it can be assumed without lost of generality
that $\mathrm{min}~{}\mathrm{rank}(Q)$ is equal to $0$ or $1$. The pair $(\mathrm{min}~{}\mathrm{rank}(Q),\mathrm{max}~{}\mathrm{rank}(Q))$
is called the Mostowski-Rabin index of the automaton.
It follows from [Rab69] that any alternating parity tree automaton can be simulated by a non deterministic Muller automaton,
see also [GTW02].
There is a usual partial order on Mostowski-Rabin indices: $(\iota,\kappa)\sqsubseteq(\iota^{\prime},\kappa^{\prime})$
if either $\iota^{\prime}\leq\iota$ and $\kappa\leq\kappa^{\prime}$ (i.e. $\{\iota,\ldots,\kappa\}\subseteq\{\iota^{\prime},\ldots,\kappa^{\prime}\}$),
or $\iota=0,\iota^{\prime}=1$, and $\kappa+2\leq\kappa^{\prime}$ (i.e. $\{\iota+2,\ldots,\kappa+2\}\subseteq\{\iota^{\prime},\ldots,\kappa^{\prime}\}$).
The indices $(1,n)$ and $(0,n-1)$ are called dual and $\overline{(\iota,\kappa)}$ denotes the index dual to $(\iota,\kappa)$.
It is easy to see
that each tree language $W_{(\iota,\kappa)}$ is accepted by an alternating parity tree automaton of index $(\iota,\kappa)$.
Moreover the set $W_{(\iota,\kappa)}$ is in some sense of the greatest possible topological complexity among tree languages accepted by
alternating parity tree automata of index $(\iota,\kappa)$. This is expressed by the following lemma.
Lemma 5.2 ( see [ADMN07] )
If a set of trees $T$ is recognized by an alternating parity tree automaton of index $(\iota,\kappa)$, then $T\leq_{W}W_{(\iota,\kappa)}$.
In order to use this result to get a lower bound on the topological complexity of the game tree languages $W_{(\iota,\kappa)}$, we first
construct some alternating parity tree automata accepting the tree languages $\mathcal{L}$ and $\mathcal{L}^{-}$ defined in the preceding section.
Lemma 5.3
The tree language $\mathcal{L}$ is accepted by an alternating parity tree automaton of index $(1,2)$.
Proof. Recall that $\mathcal{L}=\exists\mathrm{Path}(\mathcal{R})$, where $\mathcal{R}=(0^{\star}.1)^{\omega}$.
The tree language $\mathcal{L}$ is then accepted by the
alternating parity tree automaton $\mathcal{A}=(\Sigma,Q_{\exists},$ $Q_{\forall},q_{0},\delta,\mathrm{rank})$, where
$\Sigma=\{0,1\}$,
$Q_{\exists}=Q=\{q_{0},q_{1}\}$,
$Q_{\forall}=\emptyset$,
$\delta=\{(q,1,d,q_{1}),(q,0,d,q_{0})\mid q\in Q\mbox{ and }d\in\{l,r\}\}$,
$\mathrm{rank}(q_{0})=1$ and $\mathrm{rank}(q_{1})=2$.
$\square$
Notice that in the above automaton $\mathcal{A}$ all states are existential.
Lemma 5.4
The tree language $\mathcal{L}^{-}$ is accepted by an alternating parity tree automaton of index $(0,1)$.
Proof. Recall that $\mathcal{L}^{-}=T_{\Sigma}^{\omega}-\mathcal{L}=\forall\mathrm{Path}(\{0,1\}^{%
\omega}-(0^{\star}.1)^{\omega})$.
The tree language $\mathcal{L}^{-}$ is then accepted by the
alternating parity tree automaton $\mathcal{A}^{\prime}=(\Sigma,Q^{\prime}_{\exists},$ $Q^{\prime}_{\forall},q^{\prime}_{0},\delta^{\prime},\mathrm{rank}^{\prime})$, where
$\Sigma=\{0,1\}$,
$Q^{\prime}_{\exists}=\emptyset$,
$Q^{\prime}_{\forall}=Q^{\prime}=\{q^{\prime}_{0},q^{\prime}_{1}\}$,
$\delta^{\prime}=\{(q^{\prime},1,d,q^{\prime}_{1}),(q^{\prime},0,d,q^{\prime}_{%
0})\mid q^{\prime}\in Q^{\prime}\mbox{ and }d\in\{l,r\}\}$,
$\mathrm{rank}^{\prime}(q^{\prime}_{0})=0$ and $\mathrm{rank}^{\prime}(q^{\prime}_{1})=1$.
$\square$
Notice that in the above automaton $\mathcal{A}^{\prime}$ all states are universal.
Remark 5.5
The ${\bf\Sigma}^{1}_{1}$-complete tree language $\mathcal{L}$ is accepted by an alternating parity tree automaton of index $(1,2)$ and the
${\bf\Pi}^{1}_{1}$-complete tree language $\mathcal{L}^{-}$ is accepted by an alternating parity tree automaton of index $(0,1)$. In fact for
every tree language $T$ accepted by an alternating parity tree automaton of index $(1,2)$ (respectively, $(0,1)$) it holds that
$T$ is in the class ${\bf\Sigma}^{1}_{1}$ (respectively, ${\bf\Pi}^{1}_{1}$), see [ADMN07, Theorem 3.6].
Recall now the definition of the $D_{\omega}({\bf\Sigma}^{1}_{1})$-complete tree language $\mathcal{L}_{1}$.
$\mathcal{L}_{1}=\{t\in T_{\{0,1\}}^{\omega}\mid\exists n\geq 0~{}~{}t_{l^{n}%
\cdot r}\in\mathcal{L}\mbox{ and min}\{n\geq 0\mid t_{l^{n}\cdot r}\in\mathcal%
{L}\}\mbox{ is odd }\}$.
We can now state the following result.
Lemma 5.6
The tree language $\mathcal{L}_{1}$ is accepted by an alternating parity tree automaton of index $(0,2)$.
Proof. Let, as in the proofs of the two previous lemmas, $\mathcal{A}=(\Sigma,Q_{\exists},$ $Q_{\forall},q_{0},\delta,\mathrm{rank})$ be an
alternating parity tree automaton of index $(1,2)$ accepting the tree language $\mathcal{L}=\exists\mathrm{Path}(\mathcal{R})$, and
$\mathcal{A}^{\prime}=(\Sigma,Q^{\prime}_{\exists},$ $Q^{\prime}_{\forall},q^{\prime}_{0},\delta^{\prime},\mathrm{rank}^{\prime})$ be an
alternating parity tree automaton of index $(0,1)$ accepting the tree language $\mathcal{L}^{-}$. We assume that
$Q\cap Q^{\prime}=\emptyset$, where $Q=Q_{\exists}\cup Q_{\forall}=Q_{\exists}$ and $Q^{\prime}=Q^{\prime}_{\exists}\cup Q^{\prime}_{\forall}=Q^{\prime}_{\forall}$.
It is then easy to see that the tree language $\mathcal{L}_{1}$
is accepted by the
alternating parity tree automaton $\mathcal{A}^{1}=(\Sigma,Q^{1}_{\exists},$ $Q^{1}_{\forall},q^{1}_{0},\delta^{1},\mathrm{rank}^{1})$, where
$\Sigma=\{0,1\}$,
$Q^{1}_{\exists}=Q_{\exists}\cup Q^{\prime}_{\exists}\cup\{q_{\exists}\}=Q_{%
\exists}\cup\{q_{\exists}\}$,
$Q^{1}_{\forall}=Q_{\forall}\cup Q^{\prime}_{\forall}\cup\{q^{1}_{0},q^{1}_{1}%
\}=Q^{\prime}_{\forall}\cup\{q^{1}_{0},q^{1}_{1}\}$,
$\delta^{1}=\delta\cup\delta^{\prime}\cup\{(q^{1}_{0},a,l,q_{\exists}),(q^{1}_{%
0},a,r,q^{\prime}_{0}),(q_{\exists},a,r,q_{0}),(q_{\exists},a,\lambda,q^{1}_{1%
}),(q^{1}_{1},a,r,q^{\prime}_{0}),(q^{1}_{1},a,l,q^{1}_{0})\mid a\in\{0,1\}\}$,
$\mathrm{rank}^{1}(q)=\mathrm{rank}(q)$ for $q\in Q$,
$\mathrm{rank}^{1}(q^{\prime})=\mathrm{rank}^{\prime}(q^{\prime})$ for $q^{\prime}\in Q^{\prime}$,
$\mathrm{rank}^{1}(q^{1}_{0})=0$, $\mathrm{rank}^{1}(q^{1}_{1})=1$.
$\square$
Notice that in the above construction of the alternating automaton $\mathcal{A}^{1}$ the universal states $q^{1}_{0},q^{1}_{1}$ and the existential state
$q_{\exists}$ are used to choose, when reading a tree $t\in\mathcal{L}_{1}$, the least integer $n$ such that $t_{l^{n}\cdot r}\in\mathcal{L}$ and to check
that this integer is really the least (and odd) one with this property.
In a very similar manner, for each integer $n\geq 1$,
we can define an alternating parity tree automaton $\mathcal{A}^{n}$ of index $(0,2)$ accepting the language $\mathcal{L}_{n}$.
The complete description would be tedious but the idea is that now the additional universal or existential states not in $Q\cup Q^{\prime}$
are used to choose, for a given tree $t\in\mathcal{L}_{n}$,
the least ordinal $\alpha=\omega^{n-1}\cdot a_{n-1}+\omega^{n-2}\cdot a_{n-2}+\ldots+\omega\cdot a%
_{1}+a_{0}$
such that $t_{l^{a_{n-1}}\cdot r\cdot l^{a_{n-2}}\cdot r\cdots l^{a_{0}}\cdot r}$ is in $\mathcal{L}$ and to check that $\alpha$ is odd and that for any
smaller ordinal $\beta=\mathrm{Ord}(b_{n-1},b_{n-2},\ldots,b_{0})<\alpha$, the tree
$t_{l^{b_{n-1}}\cdot r\cdot l^{b_{n-2}}\cdot r\cdots l^{b_{0}}\cdot r}$ is not in $\mathcal{L}$.
We can then state the following result.
Proposition 5.7
For each integer $n\geq 1$, the tree language $\mathcal{L}_{n}$ is accepted by an alternating parity tree automaton of index $(0,2)$.
We can now infer from Theorem 4.6, Proposition 5.7, and Lemma 5.2, the following result.
Theorem 5.8
For each integer $n\geq 1$, the $D_{\omega^{n}}({\bf\Sigma}^{1}_{1})$-complete tree language $\mathcal{L}_{n}$ is Wadge reducible to the game tree language
$W_{(0,2)}$, i.e. $\mathcal{L}_{n}\leq_{W}W_{(0,2)}$. In particular the language $W_{(0,2)}$ is not in any class $D_{\alpha}({\bf\Sigma}^{1}_{1})$
for $\alpha<\omega^{\omega}$.
On the other hand, Arnold and Niwinski proved in [AN08] that the game tree languages form a hierarchy with regard to the Wadge
reducibility.
Theorem 5.9 ([AN08])
For all Mostowski-Rabin indices $(\iota,\kappa)$ and $(\iota^{\prime},\kappa^{\prime})$, it holds that :
$$(\iota,\kappa)\sqsubseteq(\iota^{\prime},\kappa^{\prime})~{}~{}\mbox{ if and %
only if }~{}~{}W_{(\iota,\kappa)}\leq_{W}W_{(\iota^{\prime},\kappa^{\prime})}$$
Then we can state the following result.
Theorem 5.10
For each integer $n\geq 1$ and each Mostowski-Rabin index $(\iota,\kappa)$
such that $(0,2)\sqsubseteq(\iota,\kappa)$ or $(\iota,\kappa)=(1,3)=\overline{(0,2)}$,
the $D_{\omega^{n}}({\bf\Sigma}^{1}_{1})$-complete tree language $\mathcal{L}_{n}$ is Wadge reducible to the game tree language
$W_{(\iota,\kappa)}$, i.e. $\mathcal{L}_{n}\leq_{W}W_{(\iota,\kappa)}$.
In particular the language $W_{(\iota,\kappa)}$ is not in any class $D_{\alpha}({\bf\Sigma}^{1}_{1})$
for $\alpha<\omega^{\omega}$.
Proof. The result follows directly from Theorems 5.8 and 5.9 in the case $(0,2)\sqsubseteq(\iota,\kappa)$.
What remains is the case of the index $(1,3)$ which is the dual of the index $(0,2)$. But it is proved in
[AN08, Lemma 1] that $W_{\overline{(\iota,\kappa)}}$
coincide with $\overline{W_{(\iota,\kappa)}}=T_{\Sigma_{(\iota,\kappa)}}^{\omega}-W_{(\iota,%
\kappa)}$ up to renaming of symbols. On the other hand,
we know from
Theorem 5.8 that for each integer $n\geq 1$,
the $D_{\omega^{n+1}}({\bf\Sigma}^{1}_{1})$-complete tree language $\mathcal{L}_{n+1}$ is Wadge reducible to the game tree language
$W_{(0,2)}$, i.e. $\mathcal{L}_{n+1}\leq_{W}W_{(0,2)}$. This is easily seen to be equivalent to
$T_{\{0,1\}}^{\omega}-\mathcal{L}_{n+1}\leq_{W}\overline{W_{(0,2)}}$, i.e. $T_{\{0,1\}}^{\omega}-\mathcal{L}_{n+1}\leq_{W}W_{(1,3)}$.
But $\mathcal{L}_{n}$ is $D_{\omega^{n}}({\bf\Sigma}^{1}_{1})$-complete and $\mathcal{L}_{n+1}$ is $D_{\omega^{n+1}}({\bf\Sigma}^{1}_{1})$-complete
so it follows from the properties of the difference hierarchy of analytic sets that $\mathcal{L}_{n}\leq_{W}T_{\{0,1\}}^{\omega}-\mathcal{L}_{n+1}$ and so
$\mathcal{L}_{n}\leq_{W}W_{(1,3)}$ by transitivity of the relation $\leq_{W}$.
$\square$
6 Concluding remarks
We have got some new results on
the topological complexity of non Borel recognizable tree languages with regard to the difference hierarchy of analytic sets. In
particular, we have showed that the game tree language $W_{(0,2)}$ is not in any class $D_{\alpha}({\bf\Sigma}^{1}_{1})$
for $\alpha<\omega^{\omega}$.
The great challenge in the study of the topological complexity of recognizable tree languages is to determine the Wadge hierarchy of
tree languages accepted by non deterministic Muller or Rabin tree automata. Notice that the case of deterministic Muller or Rabin tree automata
have been solved recently by Murlak, [Mur08].
It would be interesting to locate in a more precise way the game tree languages with regard to the difference hierarchy of analytic sets.
We already know that $W_{(0,2)}$ is not in any class $D_{\alpha}({\bf\Sigma}^{1}_{1})$ for $\alpha<\omega^{\omega}$. Is there an ordinal $\alpha$ such that
$W_{(0,2)}$ is in $D_{\alpha}({\bf\Sigma}^{1}_{1})$ and then what is the smallest such ordinal $\alpha$? The same question may be asked for the other
game tree
languages $W_{(\iota,\kappa)}$.
On the other hand, there are some sets in the class ${\bf\Delta}^{1}_{2}$ which does not belong to the
$\sigma$-algebra generated by the analytic sets, see [Kec95, Exercise 37.8]. Could we expect that $W_{(0,2)}$ or another game tree language
$W_{(\iota,\kappa)}$ is such an example?
Acknowledgements. We thank the anonymous referees for their very helpful comments which have led to a great improvement of our paper.
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Molecules of the Euler problem of two fixed centers and its applications
Seongchan Kim
Universität Augsburg, Universitätsstrasse 14, D-86159 Augsburg, Germany
seongchan.kim@math.uni-augsburg.de
Abstract.
We study the molecules of negative energy hypersurfaces of the Euler problem. As an application, we determine the knot types of periodic orbits. More precisely, we show that for energies below the critical Jacobi energy, every periodic orbits is a torus knot. Moreover, we consider homoclinic orbits and prove that in the Euler problem homoclinic orbits exist for all mass ratios and for all energies at which the Lyapunov orbit exist. In particular, the invariant manifolds, i.e., the unstable and the stable manifolds, of the Lyapunov orbit coincide. Moreover, by means of the elliptic coordinates, we show that every homoclinic orbit rotates around one of the primaries precisely once.
Contents
1 Introduction
2 The Molecule Theory
2.1 2-atoms
2.2 3-atoms
3 The Euler problem of two fixed centers
4 Molecules of the Euler problem
5 Knot types of periodic orbits
6 Homoclinic orbits in the Euler problem
A Knot types of periodic orbits in the rotating Kepler problem
1. Introduction
A Hamiltonian system $(M^{2n},\omega,H)$ is called completely integrable if there exist $n$ independent smooth functions $f_{1}=H,\cdots,f_{n}\in C^{\infty}(M,{\mathbb{R}})$ which are in involution, i.e., $\left\{f_{i},f_{j}\right\}=0$ for all $i,j$. We define the energy momentum mapping $F:M\rightarrow{\mathbb{R}}^{n}$ by
(1)
$$F(q,p)=(f_{1}(q,p),\cdots,f_{n}(q,p)).$$
By the Arnold-Liouville theorem, see for example the appendix in [10], each compact connected component of the preimage of a regular value of the energy momentum mapping is an $n$-dimensional torus and the nearby Liouville foliations on an energy hypersurface is trivial, i.e., the direct product of a torus and an interval. However, one cannot apply this theorem to the critical fibers. In particular, the topology of a critical fiber is not just the topology of a torus.
Bolsinov-Fomenko study integrable Hamiltonian systems with two degrees of freedom. They introduce the molecule theory to describe topologies of energy hypersurfaces of a given Hamiltonian system. A molecule is a combination of atoms, where each atom corresponds to a critical fiber of the energy momentum mapping. In this article, we consider a specific example: the Euler problem of two fixed centers. After computing the molecule structures of the Euler problem for negative energies, we apply this to determine the knot types of periodic orbits and to find homoclinic orbits.
The Euler problem of two fixed centers describes the motion of a point-mass under the influence of a Newtonian potential with two fixed attracting points. We refer to the two attracting bodies as the Earth and the Moon and the point-mass as the satellite. This problem can be obtained from the restricted three-body problem by ignoring the centrifugal and Coriolis terms. That this system is completely integrable was discovered by Euler in 1760 using the elliptic coordinates.
We denote by $\mu\in(0,1)$ the mass ratio of the two massive bodies and locate the Earth and the Moon at $\text{E}=(0,0)$ and $\text{M}=(1,0)$, respectively. The describing Hamiltonian $H:({\mathbb{R}}^{2}\setminus\left\{\text{E, M}\right\})\times{\mathbb{R}}^{2}%
\rightarrow{\mathbb{R}}$ is given by
(2)
$$H(q,p)=\frac{1}{2}|p|^{2}-\frac{1-\mu}{|q-\text{E}|}-\frac{\mu}{|q-\text{M}|}.$$
The Hamiltonian has a unique critical point $L=(l,0,0,0)$, $l\in(0,1)$, of Morse index 1. It corresponds to the saddle point of the potential. The integral is given by
(3)
$$B(q,p)=-(q_{1}p_{2}-q_{2}p_{1})^{2}+(q_{1}p_{2}-q_{2}p_{1})p_{2}-\frac{(1-\mu)%
q_{1}}{|q-\text{E}|}-\frac{\mu(1-q_{1})}{|q-\text{M}|}.$$
For an energy $c$, we define the Hill’s region to be
(4)
$$\mathcal{K}_{c}=\pi(H^{-1}(c))\subset{\mathbb{R}}^{2}\setminus\left\{\text{E, %
M}\right\},$$
where $\pi:({\mathbb{R}}^{2}\setminus\left\{\text{E, M}\right\})\times{\mathbb{R}}^{2%
}\rightarrow{\mathbb{R}}^{2}\setminus\left\{\text{E, M}\right\}$ is the projection along ${\mathbb{R}}^{2}$. In what follows, we only consider negative energies. Note that for a negative energy $c<0$ the Hill’s region is bounded. We denote by $c_{J}$ the value $H(L)=-1-2\sqrt{\mu(1-\mu)}$ and call it the critical Jacobi energy. Note that $c_{J}$ depends on $\mu$. For $c<c_{J}$, the Hill’s region consists of two connected components, where one of them is a neighborhood of the Earth and the other is a neighborhood of the Moon. We denote them by $\mathcal{K}_{c}^{E}$ and $\mathcal{K}_{c}^{M}$, respectively. Note that the satellite moves either near the Earth or near the Moon. For $c>c_{J}$, these two components become connected. The satellite can move from the Earth to the Moon and vice versa. There are two further distinguished energy levels, which are denoted by $c_{e}$ and $c_{h}$, at which the Liouville foliation changes.
Critical values of the energy momentum mapping give rise to the five critical orbits: the exterior collision orbit, the interior collision orbit, the double-collision orbit, the hyperbolic orbit and the elliptic orbit. Each of them corresponds to an atom of the molecule, given in Fig. 1.
Remark 1.1.
In [18] Waalkens-Dullin-Richter drawed the similar picture, which they call the Fomenko diagram. The only difference is that they expressed the vertices of the molecule not by the atoms, but by sketches of their projections to the configuration space. On the other hand, Vozmischeva [17] computed the molecules of the Euler problem on the sphere.
By the $T_{k,l}$-torus we mean the Liouville torus whose rotation number is $R=k/l$, where $k,l\in{\mathbb{Z}}$ are relatively prime. Then the first application of the molecules is the following.
Theorem 1.2.
Every periodic orbit on the $T_{k,l}$-torus is a $(k,l)$-torus knot.
Next we consider homoclinic orbits in the Euler problem. A homoclinic orbit, whose concept was introduced by Poincaré, is an orbit which is asymptotic to an unstable periodic orbit in both forward and backward time. In view of the Poincaré section map, an unstable periodic orbit corresponds to a hyperbolic fixed point of this map and a homoclinic orbit is represented by a point whose orbit is doubly asymptotic to the the hyperbolic fixed point.
Lyapunov [13] showed that near each collinear Lagrangian point of the restricted three-body problem there exists a family of unstable periodic orbits, which are called the Lyapunov orbits. The family of the Lyapunov orbits converges uniformly to the corresponding Lagrangian point as the energy goes to the associated critical energy from above. In the Euler problem, the hyperbolic orbits form such a family. We also call these orbits the Lyapunov orbits.
For the restricted three body problem, Conley [7] showed that if the energy is sufficiently close to $H(L_{1})$ from above and if the mass ratio is sufficiently small, then homoclinic orbits exist in a neighborhood of the heavier mass, where $L_{1}$ is the first Lagrangian point. McGehee [14] modified the Conley’s argument and proved that the similar result is also valid for the second Lagrangian point. In particular, he showed that every homoclinic orbit rotates around the heavier mass in the same direction. Note that Conley and McGehee used perturbative methods, which are applicable if the given system is close to a completely integrable system: if the second primary is sufficiently small, then the restricted three-body problem can be considered as a perturbation of the rotating Kepler problem. Similarly, for the mass ratio small enough the Euler problem is a perturbation of the (inertial) Kepler problem. Then by the similar perturbation method one can conclude that for small ranges of the mass ratio and energy, homoclinic orbit also exists in the Euler problem.
The goal of this article is to extend this result to larger ranges using a global method by means of the integrability. By the definition of a homoclinic orbit, it lies on the intersection of the unstable and the stable manifolds of the Lyapunov orbit. For the problem of the restricted three bodies, Llibre-Martínez-Simó [12] proved analytically that under appropriate conditions the unstable and the stable manifolds of the Lyapunov orbit around the first Lagrangian point intersect transversally. Recall that the existence of a tranverse intersection of the invariant manifolds implies the existence of Smale-horseshoe chaos, see for instance [19]. However, this is not the case for the Euler problem. Indeed, the presence of a transverse intersection prevents the existence of any real analytic integral. Instead, the following theorem says that in the Euler problem the invariant manifolds coincide.
Theorem 1.3.
For any $\mu\in(0,1)$ and for any $c\in(c_{J},c_{h})$, the unstable and stable manifolds of the Lyapunov orbit coincide. In particular, homoclinic orbits exist for any mass ratio and for any energy at which the Lyapunov orbit exists. Moreover, every homoclinic orbit rotates one of the primaries exactly once.
Outline of the paper: We start by providing the basic description of the molecule theory in Section 2. In Section 3 we recall some results about the Euler problem. Then using the results in the previous two sections we describe the molecules of the Euler problem in Section 4. We deal with two applications of molecules in the remaining two sections. In Section 5 we determine the knot types of periodic orbits, i.e., we prove Theorem 1.2. In Section 6 we consider homoclinic orbits and prove Theorem 1.3.
Acknowledgments: I wish to express my deepest gratitude to Professor Urs Frauenfelder for his encouragement. I would also like to thank to the Institute for Mathematics of University of Augsburg for providing a supportive research environment. This work is supported by DFG-CI-45/6-1.
2. The Molecule Theory
We recall basic definitions and facts about the molecule theory which we need to prove Theorems 1.2 and 1.3. In what follows, we assume that every manifold is compact and orientable.
2.1. 2-atoms
Let $X$ be a surface. A Morse function on $X$ determines a foliation whose fibers are connected components of level sets. We only consider simple Morse functions, i.e., each critical level set contains precisely one critical point. For complicated cases, see [4] and the literature cited therein. Near each regular fiber the foliaion is trivial, i.e., the direct product of a circle by an interval, while near a singular fiber the foliation might be much more complicated.
Definition 2.1.
The Morse functions $f$ and $g$ on surfaces $X$ and $Y$ are called fiberwise equivalent if there exists a diffeomorphism $\varphi:X\rightarrow Y$ which maps connected components of level sets of $f$ into those of $g$.
Definition 2.2.
A neighborhood of a singular fiber of a Morse function, up to the fiberwise equivalence, on a surface is called a 2-atom.
Let $f:X\rightarrow{\mathbb{R}}$ be a Morse function and $c$ be a critical value. Denote by $L=f^{-1}(c)$ the corresponding singular fiber. Then the 2-atom associated with the value $c$ is represented by
$$U(L)=\left\{c-\epsilon\leq f\leq c+\epsilon\right\}$$
for some $\epsilon>0$. By the Morse Lemma, a critical point corresponds to either the maximum, the minimum, or a saddle. For the first two cases, a singular fiber consists of a point: $L=\left\{p\right\}$ and for the other case it is homeomorphic to the figure eight. Each regular fiber near the singularity is a circle. It follows that the 2-atom for the maximum or the minimum can be identified with a disk whose center is a critical point and the 2-atom for the saddle is identified with a thickend figure eight.
We can describe a 2-atom in a visual manner as follows. Consider the cases of the maximum and the minimum. We represent each regular fiber as one point which is located on the level $a$. As $a$ changes this point moves along a segment. As $a\rightarrow c$, the point approaches to one of the endpoint of the segment and the corresponding regular fiber, i.e., a circle, shrinks into a point which corresponds to a singular fiber $L$. We denote this event by the letter $A$ with a segment connected with it, see Fig. 2.
We now consider the saddle case. Let $a$ be a regular value close to $c$ whose level set is represented by two circles. As $a$ tends to $c$, the two circles are getting closer and touch at a point when $a=c$. If $a$ passes by $c$, then the level set becomes a ”bigger” circle. As in the previous case, we represent each regular circle by a point along a segment. This situation is denoted by the letter $B$ with three segments connected with it, see Fig. 3.
Definition 2.3.
The atom in the extreme case is called the atom A and the saddle one is called the atom B.
In view of the Morse Lemma we have the following.
Lemma 2.4.
(Bolsinov-Fomenko) A simple atom on a compact orientable surface has either the type $A$ or $B$.
As mentioned in the introduction, a molecule is defined by a combination of atoms. The following figure provides some examples of molecules.
The Morse Lemma proves the following theorem which tells us why molecule theory is a good tool to describe topologies of surfaces.
Theorem 2.5.
(Bolsinov-Fomenko) Let $W(X,f)$ and $W(Y,g)$ be molecules corresponding to Morse functions $f$ and $g$ on surfaces $X$ and $Y$. If the molecules coincide, then $X$ and $Y$ are diffeomorphic, and the functions $f$ and $g$ are fiberwise equivalent.
As shown in the figures 2 and 3, the atom can be realized as a surface with a connected graph. With this in mind we give the following equivalent geometrical definition of a 2-atom.
Definition 2.6.
A pair $(P,K)$ is called a 2-atom if $P$ is a connected compact surface with boundary and $K\subset P$ is a connected graph satisfying the following:
$\bullet$ either $K$ consists of a single point or all the vertices of $K$ have degree 4,
$\bullet$ each connected component of $P\setminus K$ is homeomorphic to an annulus $S^{1}\times(0,1]$,
$\bullet$ the set of the annuli can be divided into two classes in such a way that for each edge of $K$ there is exactly one positive anuulus and exactly one negative annulus incident to the edge.
Obviously, the graph $K$ in the above definition corresponds to a critical fiber of a foliation determined by a Morse function. Note that for the atom A the connected graph $K$ and the complement $P\setminus K$ are comprised of a single point and an annulus, while for the atom B, $K$ is a figure eight and $P\setminus K$ consists of three annuli.
For later use, we introduce the following notion. Let $c$ be a critical value of a Morse function $f$ and consider a 2-atom $(P,K)$, where $K=f^{-1}(c)$. Choose some interior points, which are denoted by stars, on edges of the graph $K$. We declare them to be new vertices of $K$, see Fig. 5.
Definition 2.7.
A 2-atom $(P,K)$ which contains at least one star is called a 2-atom with stars.
2.2. 3-atoms
We now turn to three-dimensional case. Let $M$ be a four-dimensional symplectic manifold. Suppose that the Hamiltonian system associated with the Hamiltonian $H:M\rightarrow{\mathbb{R}}$ is integrable: there is an integral $F:M\rightarrow{\mathbb{R}}$. Choose a regular level $h$ of $H$ and denote by $\Sigma:=H^{-1}(h)$ the level set which is a three-dimensional manifold. We also assume that $\Sigma$ is compact and orientable. We use the same letter $F$ for its restriction onto $\Sigma$.
Recall that in the previous case, i.e., two-dimensional case, we work with a Morse function. However, now the integral $F$ cannot be a Morse since there is the symmetry associated to $F$. More precisely, the Hamiltonian flow $\varphi_{F}^{t}$ of $F$ satisfies $H(\varphi_{F}^{t}(x))=H(x)$ for all $x\in M$. Instead, we assume that the integral $F$ is Morse-Bott, i.e., each connected component of $\text{crit}F$, which is the set of critical points of $F$, is a submanifold of $\Sigma$ and for each $x\in\text{crit}F$ we have $T_{x}(\text{crit}F)=\ker H_{F}(x)$, where $H_{F}$ is the Hessian of $F$.
Example 2.8.
Let $S^{2}=\left\{x=(x_{1},x_{2},x_{3})\in{\mathbb{R}}^{2}:||x||=1\right\}$ be a two-sphere and consider the function $f:S^{2}\rightarrow{\mathbb{R}}$ which is defined by $f(x)=x_{3}^{2}$. The function $f$ is a Morse-Bott function with
$$\text{crit}f=\left\{(0,0,\pm 1)\right\}\cup\left\{(x_{1},x_{2},0):x_{1}^{2}+x_%
{2}^{2}=1\right\}.$$
By the Arnold-Liouville theorem each regular level set of $F$ is a two-dimensional torus. It is well known that the set of critical points of $F$ is either one-dimensional or two-dimensional submanifold in $\Sigma$. Moreover, each connected critical submanifold of $F$ in $\Sigma$ is diffeomorphic to either a circle, a two-dimensional torus, or a Klein bottle. We assume that only critial circle appears.
Definition 2.9.
Let $L$ and $L^{\prime}$ be singular leaves of Liouville foliations determined by Morse-Bott integrals. Small neighborhoods $U(L)$ and $U(L^{\prime})$ of $L$ and $L^{\prime}$ are called fiberwise equivalent if
$\bullet$ there exists an orientation-preserving diffeomorphism $\varphi:U(L)\rightarrow U(L^{\prime})$ which maps the leaves into the leaves,
$\bullet$ the diffeomorphism $\varphi$ preserves the orientation on the critical circles defined by the Hamiltonian flows.
Definition 2.10.
Let $L$ be a singular fiber of the Liouville fibration. The fiberwise equivalent class of the three-dimensional manifold $U(L)$ is called a 3-atom.
The number of critical circles in a 3-atom is called its complexity. But as in the two-dimensional case, we only consider 3-atoms with complexity one.
3-atoms a priori seem very difficult to visualize. However, this can be done easily in terms of 2-atoms. To formulate its relation, we introduce the notion of a Seifert fibration. Consider the cylinder $D^{2}\times I$ and glue the endpoints by rotation through the angle $2\pi\beta/\alpha$, where $\alpha>1$ and $0<\beta<\alpha$. Here, the two positive integers $\alpha$ and $\beta$ are relatively prime. The obtained three-dimensional manifold is a solid torus which is foliated into circles $\left\{\text{pt}\right\}\times I$, called fibers. The core of the torus, i.e., $\left\{0\right\}\times I$, goes along the torus precisely once. All the other fibers go along the torus $\alpha$-times. The core is called the singular fiber. We call such a solid torus a nontrivially fibered solid torus. If $\alpha=\beta=1$, then every fiber goes along the torus once and the solid torus is called a trivially fibered solid torus.
Definition 2.11.
A compact orientable three-dimensional manifold is called a Seifert manifold if the following are satisfied:
$\bullet$ it is foliated into non-intersecting simple closed curves which are called fibers,
$\bullet$ each fibers has a neighborhood which consists of the whole fibers and is homeomorphic to a fibered solid torus.
Let $Q$ be a Seifert manifold. We say two points $x,y\in Q$ are equivalent if and only if they belong to the same fiber. Denote by $P$ the quotient space of the Seifert manifold $Q$ by this equivalence relation. We call $P$ the base of the Seifert fibration.
Lemma 2.12.
(Fomenko-Matveev, [9]) The base $P$ of a Seifert fibration is a compact surface.
Let $U(L)$ be a sufficiently small tubular neighborhood of a singular leaf $L$ of the Liouville foliation on $Q$.
Theorem 2.13.
(Bolsinov-Fomenko) The three-dimensional manifold $U(L)$ is a Seifert manifold. If it is nontrivially fibered, then it has the type $\alpha=2$ and $\beta=1$. If its fibration is trivial, then the manifold $U(L)$ is the direct product of the base $P$ and a circle-fiber $S^{1}$. The Morse-Bott integral is constant on the fibers of the Seifert fibration.
Consider a 3-atom $U(L)$ with a Seifert fibration structure. Denote by $P$ the base of a Seifert fibration and let $\pi:U(L)\rightarrow P$ be the projection.
Theorem 2.14.
(Bolsinov-Fomenko) The projection $\pi:U(L)\rightarrow P$ maps bijectively the 3-atom $U(L)$ to the 2-atom $(P,K)$ with or without stars. Moreover, if $U(L)$ is nontrivially fibered, then the singular fibers of the Seifert fibration on the 3-atom are in one-to-one correspondence with the stars of the 2-atom.
In view of the classification of 3-atoms by Bolsinov-Fomenko, a 3-atom with complexity 1 has the type either $A$, $B$ or $A^{*}$, which are described in the following. In particular, we see how 3-atoms $A$, $B$ and $A^{*}$ and 2-atoms $A$ and $B$ are related.
Let $S$ be a critical circle of a Morse-Bott integral $F$ on an energy hypersurface $\Sigma$ and $D$ be a transversal disk. Then in a suitable coordinate system, the integral $F$ on $D$ can be written as $F=x^{2}+y^{2}$, $F=-x^{2}-y^{2}$ and $F=x^{2}-y^{2}$ for the minimum, the maximum and a saddle. Since $F$ is an integral, the level sets of $\left.\begin{matrix}F\end{matrix}\right|_{D}$ defines a foliaion on $D$ with one singularity at the center of $D$. This foliaion is preserved by the Hamiltonian flow and if the disk $D$ is sufficiently small, then the Hamiltonian flow defines the Poincaré section map on the disk. Thus, we obtain a small tubular neighborhood of the critical circle $S$ which is foliated into two-dimensional leaves and singularity along $S$ determined by the Hamiltonian flow. In view of Theorem 2.13, all possible situations are described in the following figure.
(a) The 3-atom A.
In Figure 6 (a), a neighborhood of the critical circle is a solid torus foliated into concentric tori, which shrink into the core. Namely, it is the direct product of a circle and a disk which is foliated into concentric circles. Since the projection maps each concentric circle to a single point, the 3-atom $A$ turns into the 2-atom $A$ under this projection, see Fig. 7.
(b) The 3-atom B.
In Figure 6 (b) a neighborhood is the direct produce of the cross and the circle. It is also trivially fibered. In particular, the singular leaf is the union of two two-dimensional tori which intersect along the critical circle. Thus, under the projection the 3-atom $B$ turns into the 2-atom $B$, see Fig. 7.
(ii) The 3-atom $A^{*}$.
Consider the siatuation described in Figure 6 (c). As in the case $(b)$, the singular fiber is the union of two tori intersecting along the critical circle. We take a transversal section of a small neighborhood of the singular leaf, which is homeomorphic to a thickend figure eight. We denote this surface by $\widehat{P}$. In view of Fig. 6 (c), we have the involution $I$ on $\widehat{P}$. Note that the involution $I$ has precisely one fixed point which corresponds to the critical circle, see Fig. 8.
Thus, one can represent the 3-atom $U(L)$ by the cylinder $\widehat{P}\times[0,1]$ such that $(x,0)$ and $(I(x),1)$ are identified. Consequently, we obtain an orientable three-dimensional manifold $U(L)$ with boundary. Note that the manifold $U(L)$ is a fiber bundle over a circle with fibers homeomorphic to $\widehat{P}$. Under the projection the 3-atom $A^{*}$ turns into the 2-atom with one star. Note that the marked point, which corresponds to the critical circle, maps to the star, see Fig. 9
In the two-dimensional case, it is relatively easy to see topologies of underlying surfaces by given molecules. However, in the three-dimensional case it is not that simple. For example, if a compact orientable three-manifold $Q$ has a molecule $A-A$, then $Q$ can be obtained by gluing two solid torus, and hence it is homeomorhic to either $S^{3}$, ${\mathbb{R}}P^{3}$, $S^{1}\times S^{2}$, or lens spaces. To know the precise topology of $Q$, one needs additional data, see [4].
3. The Euler problem of two fixed centers
The Euler problem describes the motion of the massless body under the influence of two fixed primaries according to Newton’s law of gravitation. We refer to the two primaries as the Earth and the Moon and the massless body as the satellite. To introduce the elliptic coordinates it is convenient to apply the translation $(q_{1},q_{2},p_{1},p_{2})\mapsto(q_{1}+1/2,q_{2},p_{1},p_{2})$ which is a symplectomorphicm. Then the Earth and the Moon are located at $E=(-1/2,0)$ and $M=(1/2,0)$, respectively, and the Hamiltonian is given by
(5)
$$H:\big{(}{\mathbb{R}}^{2}\setminus\left\{{E,M}\right\}\big{)}\times{\mathbb{R}%
}^{2}\rightarrow{\mathbb{R}},\;\;\;(q,p)\mapsto\frac{1}{2}|p|^{2}-\frac{\mu}{|%
q-M|}-\frac{1-\mu}{|q-E|},$$
where $\mu\in(0,1)$ is the mass ratio of the two primaries. Without loss of generality, we may assume $\mu\leq 1/2$, i.e., the Earth is stronger. There is a unique critical point $L=(l,0,0,0)$ of the Hamiltonian, where
(6)
$$l=\begin{cases}\displaystyle\frac{1-2\sqrt{\mu(1-\mu)}}{2(1-2\mu)}&\;\;\mu\neq
1%
/2\\
0&\;\;\mu=1/2.\end{cases}$$
Note that in the configuration space, i.e., $(q_{1},q_{2})$-plane, the projection of the critical point lies on the line segment joining the Earth and the Moon. A direct computation shows that the critical point $L$ is of Morse index 1. The corresponding energy value is $c_{J}:=-1-2\sqrt{\mu(1-\mu)}$, which we call the critical Jacobi energy.
Since the two primaries are fixed, they can be regarded as the foci of a set of ellipses and hyperbolas. Thus, one can introduce the elliptic coordinates system:
(7)
$$\xi=|q-E|+|q-M|\in[1,\infty),\;\;\;\eta=|q-E|-|q-M|\in[-1,1].$$
Note that in the $(q_{1},q_{2})$-plane $\xi\equiv C$ or $\eta\equiv C$ represents an ellipse or a hyperbola, respectively. The momentum variable $p_{\xi}$ and $p_{\eta}$ are determined by the relation $p_{1}dq_{1}+p_{2}dq_{2}=p_{\xi}d\xi+p_{\eta}d\eta$. In $(\xi,\eta,p_{\xi},p_{\eta})$-coordinates the Hamiltonian becomes
(8)
$$H=\frac{H_{\xi}+H_{\eta}}{\xi^{2}-\eta^{2}},$$
where $H_{\xi}=2(\xi^{2}-1)p_{\xi}^{2}-2\xi$ and $H_{\eta}=2(1-\eta^{2})p_{\eta}^{2}+2(1-2\mu)\eta$.
It is well known that the Euler problem is completely integrable, i.e., there exists the first integral which commutes with the total energy $H$. Following the convention of Strand-Reinhardt [16] we choose the first integral by $G=-H-2B$, which is given by
(9)
$$G=-\frac{\eta^{2}H_{\xi}+\xi^{2}H_{\eta}}{\xi^{2}-\eta^{2}},$$
where $B$ is given in the introduction.
Given $(G,H)=(g,c)$, the momentum variables are expressed by
(10)
$$p_{\xi}^{2}=\frac{c\xi^{2}+2\xi+g}{2(\xi^{2}-1)},\;\;\;\;\;p_{\eta}^{2}=\frac{%
c\eta^{2}+2(1-2\mu)\eta+g}{2(\eta^{2}-1)}.$$
The classically allowed regions of the configuration space are those in which the momentum variables are real, and the classical turning points are located at which one of them or both vanish. Define the functions
(11)
$$f(\xi)=(c\xi^{2}+2\xi+g)(\xi^{2}-1)$$
and
(12)
$$h(\eta)=(c\eta^{2}+2(1-2\mu)\eta+g)(\eta^{2}-1).$$
The function $f$ or $h$ has four roots $\pm 1$ and $\xi_{1,2}=\displaystyle\frac{-1\pm\sqrt{1-gc}}{c}$ or $\eta_{1,2}=\displaystyle\frac{-(1-2\mu)\pm\sqrt{(1-2\mu)^{2}-gc}}{c}$, respectively. These functions are related with the momentum variables by the relations
(13)
$$(\xi^{2}-1)p_{\xi}=\pm\sqrt{\frac{f(\xi)}{2}},\;\;\;\;\;(\eta^{2}-1)p_{\eta}=%
\pm\sqrt{\frac{h(\eta)}{2}}.$$
In view of the relations (13) the transition of the motion occurs when the functions $f$ or $h$ has double roots. Note that these functions should be positive. In the following we describe the process to get the bifurcation diagram. For references, see [16], [18].
We first consider the $\xi$-case. The discriminant of $f$ is given by
(14)
$$\Delta=16(c+g+2)^{2}(c+g-2)^{2}(gc-1)^{2}.$$
Double roots of $f=0$ appear when $\Delta=0$, namely a point $(g,c)$ lies on the curve either $c=-g-2$, $c=-g+2$ or $gc=1$. These curves divide the lower half $(g,c)$-plane into five regions. Note that the curve $gc=1$ and the line $c=-g-2$ intersect at the point $(-1,-1)$.
i) Below the curve $gc=1$.
For points below the curve $gc=1$ we have $gc>1$ and hence the two roots $\xi_{1,2}$ are complex. The graph of the function $f$ for this case is depicted in Fig. 10 (a). It follows from $\xi\in[1,\infty)$ that there is no value of $\xi$ at which the function $f$ is positive. This shows that the region below the curve $gc=1$ is not classically accessible.
ii) Between the curve $gc=1$ and the line $c=-g-2$.
Since $gc<1$, the roots $\xi_{1}<\xi_{2}$ are real. If $c<-1$, then we have $-1<\xi_{1}<\xi_{2}<1$ and hence by the same reason with i) this region is not classically accessible, see Fig. 10 (b-1). On the other hand, since $-1<1<\xi_{1}<\xi_{2}$ for $c>-1$, we have $\xi\in[\xi_{1},\xi_{2}]$, see Fig 10 (b-2). The satellite moves between the ellipses $\xi=\xi_{1}$ and $\xi=\xi_{2}$.
iii) Between the lines $c=-g-2$ and $c=-g+2$.
We have $-1<\xi_{1}<1<\xi_{2}$. Thus, we have $\xi\in[1,\xi_{2}]$, see Fig. 10 (c-1). The satellite moves inside the ellipse $\xi=\xi_{2}$.
iv) Above the line $c=-g+2$.
Since $\xi_{1}<-1<1<\xi_{2}$, we have the same range as in iii), i.e., $\xi\in[1,\xi_{2}]$, see Fig 10 (c-2).
We now consider boundary cases: (v) $gc=1$ with $c>-1$, (vi) $c=-g-2$ with $c<-1$, (vii) $c=-g-2$ with $c>-1$ and (viii) $c=-g+2$.
(v) We have $\xi_{1}=\xi_{2}=-1/c>1$, see Fig. 10 (d). The only possible case is $\xi=\xi_{1}$ and hence the satellite moves along the ellipse $\xi=-1/c$.
(vi) It follows from $-1<\xi_{1}<1=\xi_{2}$ that the satellite moves on the line segment joining the Earth and the Moon, see Fig. 10 (e).
(vii) Since $-1<1=\xi_{1}<\xi_{2}$, the satellite moves inside the ellipse $\xi=\xi_{2}$, see Fig. 10 (f).
(viii) We have $-1=\xi_{1}<1<\xi_{2}$, see Fig. 10 (g). The satellite moves as in (vii).
We now consider the $\eta$-case. The discriminant of the function $h$ is given by
(15)
$$\Delta=16(c+g+2(1-2\mu))^{2}(c+g-2(1-2\mu))^{2}(gc-(1-2\mu)^{2})^{2}$$
and we obtain the curves $c=-g-2(1-2\mu)$, $c=-g+2(1-2\mu)$, and $gc=(1-2\mu)^{2}$ which divide the lower half $(g,c)$-plane into five regions. The bifurcation scheme as in the $\xi$-case, but with $\eta\in[-1,1]$, gives the similar results. The argument so far is summarized in Fig. 11.
The combination of the results for $\xi$- and $\eta$-motions yields the bifurcation diagram, see Fig. 12. There are five critical curves which bound the classically accessible regions in the lower half $(g,c)$-plane:
(16)
$$l_{1,2}:c=-g\pm 2(1-2\mu),\;\;l_{3}:c=-g-2,\;\;l_{4}:gc=(1-2\mu)^{2},~{}c_{J}<%
c<c_{h},\;\;l_{5}:gc=1,~{}c_{e}<c.$$
Here, $c_{e}$ or $c_{h}$ are the energy levels at which the line $l_{3}$ and the curve $l_{5}$ or the line $l_{2}$ and the curve $l_{4}$ intersect, respectively(the letters $e$ and $h$ stand for elliptic and hyperbolic). Note that at these points the Liouville foliation changes. A point on the critical curves corresponds to a critical value of the energy momentum mapping $(\xi,\eta)\mapsto(G(\xi,\eta),H(\xi,\eta))$, while an interior point is a regular value. In particular, the preimages represent Liouville tori. Following the notations from [6], [15], the regions are labeled by ${S^{\prime}}$, ${S}$(satellite), ${L}$(lemniscate), and ${P}$(planetary). The ranges of the variables $\xi$ and $\eta$ for each region are summarised in the following.
We now investigate the critical orbits, i.e., the periodic orbits which correspond to the points on one of the critical curves, see Fig. 12.
$\bullet$ On $l_{1}:c=-g+2(1-2\mu)$
The satellite moves along the ray $\eta=-1$, but the motion is bounded by the ellipse $\xi=\xi_{2}$. We call this orbit the exterior collision orbit in the Earth component.
$\bullet$ On $l_{2}:c=-g-2(1-2\mu)$
Each point on the line $l_{2}$ represents an orbit in $\mathcal{K}_{c}^{E}$ or $\mathcal{K}_{c}^{M}$. Assume the satellite moves near the Earth, which occurs only for $c<c_{h}$. The variables $(\xi,\eta)$ move in $[1,\xi_{2}]\times[-1,\eta_{2}]$. In particular, the motion is regular. Consider the orbit near the Moon, i.e., in $\mathcal{K}_{c}^{M}$. Then the satellite moves along the ray $\eta=1$, but bounded by the ellipse $\xi=\xi_{2}$. This orbit is called the exterior collision orbit in the Moon component.
$\bullet$ On $l_{3}:c=-g-2$
On the line $l_{3}$ we have $\xi=1$, i.e, the satellite moves along the line segment joining the two masses. If $c<c_{J}$, the variable $\eta$ lies in $[-1,\eta_{2}]\cup[\eta_{1},1]$, where the first or the second interval corresponds the motions in the Earth or the Moon component. For $\eta\in[-1,\eta_{2}]$, the satellite moves on the line segment $\left\{(q_{1},0):0\leq q_{1}\leq b_{c}^{E}\right\}$, where $(b_{c}^{E},0)\in\partial\mathcal{K}_{c}^{E}$ and $b_{c}^{E}>0$. Similarly, in the Moon component we have $\left\{(q_{1},0):b_{c}^{M}\leq q_{1}\leq 1\right\}$, where $(b_{c}^{M},0)\in\partial\mathcal{K}_{c}^{M}$ and $b_{c}^{E}<1$. We refer to such an orbit as the interior collision orbit in the Earth or the Moon component. As $c$ passes by $c_{J}$, the two components of the Hill’s region become connected. The two interior collision orbits also become connected so that the satellite can move from the Earth to the Moon and vice versa, see Fig. 13 (a). Indeed, there is no restriction on $\eta$, i.e., $\eta\in[-1,1]$ for $c>c_{J}$. This orbit is refered to as the double-collision orbit.
$\bullet$ On $l_{4}:gc=(1-2\mu)^{2}$
We have $\eta\in[-1,\eta_{1})$, $\eta\in(\eta_{1},1]$, or $\eta=\eta_{1}$. For the first two cases, orbits are not critical. For the remaining case, i.e., $\eta=\eta_{1}$, the satellite moves along the hyperbola $\eta=\eta_{1}$ within the ellipse $\xi=\xi_{2}$. We call this orbit the hyperbolic orbit.
As mentioned in the introduction, the family $\gamma_{\text{hyp}}^{c}$ of the hyperbolic orbits, $c\in(c_{J},c_{h})$, are Lyapunov orbits. To this end, we need to show that the family $\gamma_{\text{hyp}}^{c}$ converges uniformly to $L=(l,0,0,0)$ as $c$ tends to $c_{J}$, i.e, it emanates from the critical point. Recall that along the hyperbolic orbits the equation $c\eta^{2}+2(1-2\mu)\eta+g=0$ has double roots $-(1-2\mu)/c$. In other words, the hyperbola along which the satellite moves is $|q+(1/2,0)|-|q-(1/2,0)|=-(1-2\mu)/c$. In particular, for the symmetric case, i.e., $\mu=1/2$, we have $\eta=0$, the $q_{2}$-axis. We observe that the hyperbola, for $\mu\neq 1/2$, is closer to the Moon than to the Earth since $-(1-2\mu)/c>0$. We claim that the hyperbola $\eta=-(1-2\mu)/c$ consists point $(l,0)$ at $c=c_{J}$, namely
(17)
$$\displaystyle\left\{(q_{1},q_{2}):|q+(1/2,0)|-|q-(1/2,0)|=-\frac{1-2\mu}{c_{J}%
}\right\}\cap\mathcal{K}_{c_{J}}=\left\{(l,0)\right\}.$$
Since $\xi\rightarrow 1$ as $c\rightarrow c_{J}$, it suffices to show that the vertex of the hyperbola $\eta=-(1-2\mu)/c_{J}$ is $(l,0)$. We compute that
$$\frac{1-2\mu}{-c_{J}}=\frac{1-2\mu}{1+2\sqrt{\mu(1-\mu)}}=\frac{1-2\sqrt{\mu(1%
-\mu)}}{1-2\mu}.$$
Then the vertex of the hyperbola is given by
$$(q_{1},q_{2})=\bigg{(}\frac{1-2\sqrt{\mu(1-\mu)}}{2(1-2\mu)},0\bigg{)}=(l,0).$$
This completes the proof of the claim. On the other hand, as $c\rightarrow c_{h}$ the hyperbolic orbits degenerates to the exterior collision orbit in the Moon component, see Fig. 13 (b).
$\bullet$ On $l_{5}:gc=1$
We have no restriction on $\eta$, but $\xi$ can take only $\xi=\xi_{1}=\xi_{2}$. This implies that the satellite moves along the ellipse $\xi=\xi_{1}$. This orbit is refered to as the elliptic orbit. As $c\rightarrow c_{e}$, the ellipse $\xi=\xi_{1}$ degenerates to the line segment $\xi=1$ joining the two masses, namely the eccentricity of the ellipse converges to 1, see Fig. 13 (c).
Before finishing this section, we recall the concept of the rotation number and some results which will be needed in Section 5. Since the Hamiltonian $H$ has the two singularities at the Earth and the Moon, an energy hypersurface is not compact. However, one can regularize such a two-body collision by means of a suitable time rescaling as follows.
Fix $c<0$. We now show how to regularize the dynamics on the energy level set $H^{-1}(c)$. Define the new Hamiltonian
(18)
$$K=(\xi^{2}-\eta^{2})(H-c)=H_{\xi}+H_{\eta}-c(\xi^{2}-\eta^{2}).$$
For points at $K=0$ we have $\partial_{\sigma}K=(\xi^{2}-\eta^{2})\partial_{\sigma}H$ for each $\sigma=\xi,\;\eta,\;p_{\xi},$ or $p_{\eta}$. With the time scaling
(19)
$$dt=(\xi^{2}-\eta^{2})d\tau,$$
the orbits of $H$ with energy $c$ and time parameter $t$ correspond to orbits of $K$ with energy $0$ and time parameter $\tau$. Note that the energy hypersurface $H^{-1}(c)$ is compactified to $K^{-1}(0)$. The equations of the momenta (10) and the Hamiltonian equations of $K$ on $K^{-1}(0)$ give rise to
(20)
$$\begin{cases}\dot{\xi}=4(\xi^{2}-1)p_{\xi}=2\sqrt{2}\sqrt{f(\xi)}\\
\dot{\eta}=4(1-\eta^{2})p_{\eta}=2\sqrt{2}\sqrt{h(\eta)},\end{cases}$$
where the dot denotes the differentiation with respect to $\tau$ and the functions $f$ and $h$ are defined as in (11) and (12). It follows that
(21)
$$\displaystyle\int_{\xi_{0}}^{\xi(\tau)}\frac{d\xi}{2\sqrt{2}\sqrt{f(\xi)}}=%
\tau-\tau_{0}=\int_{\eta_{0}}^{\eta(\tau)}\frac{d\eta}{2\sqrt{2}\sqrt{h(\eta)}}.$$
One can compute $\xi$- and $\eta$-periods, which are denoted by $\tau_{\xi}$ and $\tau_{\eta}$, of the regularized motions by means of the complete elliptic integrals of the first kind, see for instance [8], [11].
Given a regular level $(g,c)$ of the energy momentum mapping, we define the rotation number of the corresponding Liouville torus by the ratio $R=(\tau_{\eta}/\tau_{\xi})(g,c)$. We also define the rotation number of an orbit by the same formula. Since the rotation number depends only on the value $(g,c)$, all orbits on the same torus have the same rotation number. Moreover, an orbit is periodic if and only if the rotation number is rational. For this reason, a Liouville torus on which periodic orbits lie is called rational. Fixing an energy level $c$ and varying the integral $g$ defines the rotation function $R_{c}:=R(\cdot,c)$ of the energy hypersurface $K^{-1}(0)$. On the other hand, given a rotation number $R$ the equation $R=R(g,c)$ defines a smooth family of Liouville tori of the fixed rotation number $R$. If $R=k/l$, where $k,l\in{\mathbb{Z}}$ are relatively prime, then the family of Liouville tori is called the $T_{k,l}$-torus family. The $T_{k,l}$-family plays an important role in calculation of the Conley-Zehnder indices of periodic orbits in [11]. Each family draws a curve in the lower-half $(g,c)$-plane, see Fig. 14. Obviously, any two families with different rotation numbers never intersect.
Remark 3.1.
Note that no torus family bifurcates from the hyperbolic orbit, the double-collision orbit with $c>c_{e}$, and the exterior collision orbit in the Moon component with $c>c_{h}$. In fact, they are unstable periodic orbits, see [18].
Lemma 3.2.
Assume that $c<c_{J}$. Then along the lines $l_{1}$, $l_{2}$ and $l_{3}$ the rotation function strictly increases from $1$ as the energy increases from $-\infty$ to $c_{J}$. In particular, along $l_{3}$ it goes to $\infty$.
Proof.
See [11].
∎
4. Molecules of the Euler problem
The results in the previous section imply that each singular fiber of the Morse-Bott integral $G$ contains exactly one critical circle, i.e., one of the critical orbits. Therefore, there exist only simple atoms. In view of Section 2 they are either $A$, $B$ or $A^{*}$.
We consider first the Earth component for $c<c_{J}$. There are only two critical orbits, the exterior and the interior collision orbits, which correspond to the maximum and the minimum, respectively Therefore, the molecule is of the simplest form $A-A$. The result is also valid for the Moon component. Since we have more $g$-values for orbits in the Earth component, we draw the molecule for the Earth component longer than that for the Moon component.
Assume that $c_{J}<c<c_{e}$. There are the four critical orbits : the double-collision orbit, the hyperbolic orbit, and the exterior collision orbits in the both components. The double-collision orbit corresponds to the minimum and the exterior collision orbits to maxima. Therefore, all three atoms are of type $A$. Since the hyperbolic orbit is saddle, it is of type $B$ or $A^{*}$. Note that the atom associated with the hyperbolic orbit needs three edges, where each of them is connected with one of the three $A$. We conclude that the hyperbolic orbit has the atom $B$.
For $c_{e}<c<c_{h}$, we have the elliptic orbits with both orientations, the double-collision orbit, the hyperbolic orbit, the exterior collision orbits in both components. As before, the elliptic orbits and the exterior collision orbits have the type $A$. Then the two saddles, the double-collision orbit and the hyperbolic orbit, need three edges and hence they are of type $B$.
Finally, for $c>c_{h}$ we have all the orbits in the case $c_{e}<c<c_{h}$ but the hyperbolic orbit. Again, the double-collision orbit needs three edges which are connected with the elliptic orbits and the other saddle orbit, i.e., the exterior collision orbit in the Moon component. Thus, it is of type $B$. On the other hand, the atom corresponding to the exterior collision orbit in the Moon component has two edges which are connected with the hyperbolic orbit and the exterior collision orbit in the Earth component. This implies that the corresponding atom is $A^{*}$.
The above descriptions of molecules of the Euler problem are illustrated in Fig. 1.
Remark 4.1.
In page 12 of [18] Waalkens-Dullin-Richter draw similar pictures. They call them Fomenko graphs. The difference is that they represent the critical orbits not by alphabets, i.e., not by atoms, but by sketches of their projections to configuration space.
5. Knot types of periodic orbits
As the first application of molecules, we determine the knot types of periodic orbits. We assume that $c<c_{J}$.
We introduce the following coordinates [18]:
(22)
$$q_{1}=\frac{1}{2}\cosh\lambda\cos\nu\;\;\;\;\;\text{and}\;\;\;\;\;q_{2}=\frac{%
1}{2}\sinh\lambda\sin\nu,$$
where $(\lambda,\nu)\in{\mathbb{R}}\times[-\pi,\pi]$. Note that $(\lambda,\nu)$-coordinates are related with the $(\xi,\eta)$-coordinates in the relation
(23)
$$\xi=\cosh\lambda\;\;\;\;\ \text{and}\;\;\;\;\;\eta=\cos\nu.$$
Moreover, with the new coordinates we work in the double covering. The Hamiltonian and the integral become
(24)
$$H=\frac{H_{\lambda}+H_{\nu}}{\cosh^{2}\lambda-\cos^{2}\nu}\;\;\;\;\;\text{and}%
\;\;\;\;\;G=-\frac{H_{\lambda}\cos^{2}\nu+H_{\nu}\cosh^{2}\lambda}{\cosh^{2}%
\lambda-\cos^{2}\nu},$$
where $H_{\lambda}=2p_{\lambda}^{2}-2\cosh\lambda$ and $H_{\nu}=2p_{\nu}^{2}+2(1-2\mu)\cos\nu$. For $(G,H)=(g,h)$ the momentum variables are given by
(25)
$$p_{\lambda}^{2}=\frac{c\cosh^{2}\lambda+2\cosh\lambda+g}{2}\;\;\;\;\;\text{and%
}\;\;\;\;\;p_{\nu}^{2}=\frac{-c\cos^{2}\nu-2(1-2\mu)\cos\nu-g}{2}.$$
Note that $p_{\lambda}$ and $p_{\nu}$ have no singularities, cf. Eq. (10). Fix an energy level $c<0$. With the time rescaling $dt=(\cosh^{2}-\cos\nu^{2})d\tau$, the regularized Hamiltonian is given by
(26)
$$Q=(H-c)(\cosh^{2}\lambda-\cos^{2}\nu)=Q_{\lambda}+Q_{\nu},$$
where
$$Q_{\lambda}=2p_{\lambda}^{2}-2\cosh\lambda-c\cosh^{2}\lambda\;\;\;\;\;\text{%
and}\;\;\;\;\;Q_{\nu}=2p_{\nu}^{2}+2(1-2\mu)\cos\nu+c\cos^{2}\nu.$$
The regularized energy hypersurface is diffeomorphic to the disjoint union of two $S^{3}$, which is the double cover(or the universal cover) of ${\mathbb{R}}P^{3}$, for $c<c_{J}$ and is diffeomorphic to $S^{1}\times S^{2}$, which is the double cover of the connected sum ${\mathbb{R}}P^{3}\sharp{\mathbb{R}}P^{3}$, for $c>c_{J}$. In particular, since we have assumed $c<c_{J}$, every periodic orbit can be seen as a knot in the three-dimensional closed manifold $S^{3}$. Together with the molecule structures described in the previous section, this allows us to apply the following theorem.
Theorem 5.1.
(Casasayas-Martinez Alfaro-Nunes, [5]) Consider a Hamiltonian system $H$ with two degrees of freedom and let $Q$ be a regular energy hypersurface diffeomorphic to $S^{3}$. Suppose that $H$ is completely integrable on $Q$ by means of a Bott integral $f$. Assume further that there are only two critical values of the integral $f$, i.e., the corresponding molecule is $A-A$. Then all the periodic orbits of $H$ in $Q$ are torus knots.
Remark 5.2.
The original statement of the theorem does not contain the assumption on the critical values of the integral(or on the molecule), see [5].
It follows that every periodic orbit in the regularized Euler problem for $c<c_{J}$ is a torus knot. Recall that a $(k,l)$-torus knot is a knot on a torus which winds around the core of a torus in a such way that it meets $k,l$-times the meridian and the longitude, respectively. Here, the integers $k$ and $l$ are relatively prime.
Now we will determine the knot types of periodic orbits. Since the Euler problem is completely integrable, by the Arnold-Liouville theorem the energy hypersurface is foliated by Liouville tori. In the action-angle variables $(J,\theta)$, the Hamiltonian equations on an energy hypersurface become
(27)
$$\dot{J}=\frac{\partial F}{\partial\theta}(\theta)\;\;\;\text{ and }\;\;\;\dot{%
\theta}=0,$$
where $F$ is the transformed Hamiltonian. The solution is expressed in terms of the antion-angle variables by
(28)
$$J(t)=J(0)+t\frac{\partial F}{\partial\theta}(\theta(0))\;\;\;\text{ and }\;\;%
\;\theta(t)=\theta(0).$$
In particular, all the orbits on a given Liouville torus have the same angle values $\theta(0)$ and the frequencies $\frac{\partial F}{\partial\theta}(\theta(0))$ of the action variables. We denote by $\widetilde{R}$ the ratio of the frequencies, which we call the rotation number. If the rotation number is rational, or equivalently the frequencies of the motion are in resonance, then the motion is periodic. It is obvious that the rotation number determines the knot types of periodic orbits.
Note that we have two rotation numbers: $R$ which is defined by means of the periods in Section 3 and $\widetilde{R}$ defined in the above. We claim that they in fact coincide. In particular, the claim implies that any periodic orbit on the $T_{k,l}$-torus is a $(k,l)$-torus knot.
Recall that one can compute the action variables by integrating the Liouville differential $pdq$ along independent fundamental cycles on the Liouville tori. Since the Euler problem is separable, it is natural to choose fundamental cycles as the coordinate lines, i.e, $\xi$ and $\eta$ are constant. Then the action variables are given by
(29)
$$J_{\xi}=\frac{1}{2\pi}\oint p_{\xi}d\xi\;\;\;\;\;\text{and}\;\;\;\;J_{\eta}=%
\frac{1}{2\pi}\oint p_{\eta}d\eta.$$
In view of (10) we compute that
$$\displaystyle J_{\xi}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2\pi}\oint p_{\xi}d\xi=\frac{1}{\sqrt{2}\pi}\oint\sqrt{%
\frac{h\xi^{2}+2\xi+g}{\xi^{2}-1}}d\xi=\frac{-\sqrt{2}h}{\pi}\int_{\xi_{-}}^{%
\xi_{+}}\frac{(\xi_{+}-\xi)(\xi-\xi_{-})}{\sqrt{f(\xi)}}d\xi$$
$$\displaystyle J_{\eta}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2\pi}\oint p_{\eta}d\eta=\frac{1}{\sqrt{2}\pi}\oint\sqrt%
{\frac{h\eta^{2}+2(1-2\mu)\eta+g}{\eta^{2}-1}}d\eta=\frac{-\sqrt{2}h}{\pi}\int%
_{\eta_{-}}^{\eta_{+}}\frac{(\eta_{+}-\eta)(\eta-\eta_{-})}{\sqrt{h(\eta)}}d\eta.$$
Recall that $\xi_{-}=1$, $\xi_{+}=\xi_{2}$ and ${\eta_{-}=-1,}$ ${\eta_{+}=\eta_{1}}$ for orbits in the Earth component and ${\xi_{-}=1,}$ ${\xi_{+}=\xi_{2}}$ and ${\eta_{-}=\eta_{2},}$ ${\eta_{+}=1}$ for orbits in the Moon component in the case of $c<c_{J}$. Since the transformed Hamiltonian $F$ depends only on the action variables, we have
(30)
$$X_{F}=\frac{\partial F}{\partial J_{\xi}}\partial_{\theta_{1}}+\frac{\partial F%
}{\partial J_{\eta}}\partial_{\theta_{2}}.$$
This implies that $\widetilde{R}$ equals to $(\partial F/\partial J_{\xi})/(\partial F/\partial J_{\eta})$. A direct computation shows that
(31)
$$\widetilde{R}=\frac{\partial F/\partial J_{\xi}}{\partial F/\partial J_{\eta}}%
=-\frac{dJ_{\eta}}{dJ_{\xi}}=-\frac{\partial J_{\eta}/\partial g}{\partial J_{%
\xi}/\partial g}=\bigg{(}\int_{-1}^{\eta_{2}}\frac{d\eta}{\sqrt{h(\eta)}}\bigg%
{)}\bigg{/}\bigg{(}\displaystyle\int_{1}^{\xi_{1}}\frac{d\xi}{\sqrt{f(\xi)}}%
\bigg{)}=\frac{\tau_{\eta}}{\tau_{\xi}}=R.$$
This completes the proof of the claim and hence Theorem 1.2.
Remark 5.3.
Similar result also holds for the rotating Kepler problem, see Appendix A.
6. Homoclinic orbits in the Euler problem
A homoclinic orbit is an orbit which is doubly asymptotic to the Lyapunov orbit. Obviously, homoclinic orbits (if exist) take the same integral value with the Lyapunov orbit. In view of the argument in Sec. 3, the corresponding value $(g,c)$ lies on the curve $l_{4}$. Moreover, such orbits lie in a neighborhood of the Earth or the Moon in the case $\eta\in[-1,-(1-2\mu)/c)$ or $\eta\in(-(1-2\mu)/c,1]$, respectively. Note that $\eta=-(1-2\mu)/c$ represents the Lyapunov orbit.
As mentioned in the introduction the existence of homoclinic orbits in the Euler problem for small ranges of the energy and the mass ratio can be shown by a perturbation method of Conley and McGehee. In this section we will answer to the question if such ranges can be extended by a global method, i.e., we prove Theorem 1.3.
The definition of a homoclinic orbit implies that it lies on the intersection of the unstable and the stable manifolds of the Lyapunov orbit, $W^{u}(\gamma_{\text{Lya}})$ and $W^{s}(\gamma_{\text{Lya}})$, which are locally cylinders. Then a homoclinic orbit exists if and only if the unstable and stable manifolds of the Lyapunov orbit are not disjoint.
Put $g_{c}=(1-2\mu)^{2}/c<0$ so that the Lyapunov orbit and its invariant manifolds lie in the intersection
$$\Gamma:=H^{-1}(c)\cap G^{-1}(g_{c}),$$
which is the preimage of $(g_{c},c)$ under the energy momentum mapping. Since $(g_{c},c)$ is a critial value, $\Gamma$ is not a manifold: it has singularities along the Lyapunov orbit. Equivalently, the derivatives $dH$ and $dG$ are linearly dependent on $\Gamma$ except along $\gamma_{\text{hyp}}$. It follows that
(32)
$$W^{u}(\gamma_{\text{Lya}}),\;W^{s}(\gamma_{\text{Lya}})\subset\Gamma\setminus%
\gamma_{\text{Lya}}.$$
We denote by $L=(\left.\begin{matrix}G\end{matrix}\right|_{H^{-1}(c)})^{-1}(g_{c})$ the singular fiber, which is homeomorphic to the figure eight, and by $U(L)=(\left.\begin{matrix}G\end{matrix}\right|_{H^{-1}(c)})^{-1}(g_{c}-%
\epsilon,g_{c}+\epsilon)$ a small neighborhood, which is homeomorphic to a thickened figure eight. Recall that the atom corresponding to the Lyapunov orbit is the type $B$, which is described by $U(L)\times S^{1}$, where $\left\{\text{vertex}\right\}\times S^{1}$ represents the Lyapunov orbit. This description by atom implies that $(U(L)\setminus L)\times S^{1}$ consists of the three connected components $\Sigma_{E}$, $\Sigma_{M}$, and $\Sigma_{T}$, where on $\Sigma_{E}$ or $\Sigma_{M}$ the satellite moves near the Earth or near the Moon and on $\Sigma_{T}$ it can transfer from near the Earth to near the Moon and vice versa. In view of the integral values, the three components can be represented by
$$\displaystyle\Sigma_{E}\cup\Sigma_{M}$$
$$\displaystyle=$$
$$\displaystyle\left\{(q,p):H(q,p)=c,~{}G(q,p)\in(g_{c},g_{c}+\epsilon)\right\}$$
$$\displaystyle\Sigma_{T}$$
$$\displaystyle=$$
$$\displaystyle\left\{(q,p):H(q,p)=c,~{}G(q,p)\in(g_{c}-\epsilon,g_{c}))\right\},$$
for some sufficiently small $\epsilon>0$, see Fig. 15.
For each $g\in(g_{c},g_{c}+\epsilon)$ the preimage is the disjoint union of two torus while for $g\in(g_{c}-\epsilon,g_{c})$ it consists of precisely one torus. Note that for $g=g_{c}$ the preimage is homeomorphic two torus attached along the Lyapunov orbit. By varying $g$ this describes the foliation of the energy hypersurface $H^{-1}(c)$ near the Lyapunov orbit: at $g>g_{c}$ we have two tori and they are getting closer as $g$ goes to $g_{c}$. At $g=g_{c}$, the two tori intersect along the circle, which corresponds to the Lyapunov orbit, and for $g<g_{c}$ the leaf becomes a ”bigger” torus. Consequently, $\Gamma\setminus\gamma_{\text{Lya}}$ is comprised of two connected components: one is around the Earth and the other is around the Moon. Each of them is ${\mathbb{T}}^{2}\setminus\gamma$, where $\gamma$ is a periodic orbit corresponding to the Lyapunov orbit, see Fig. 16.
In view of the symmetry there is no periodic orbit on ${\mathbb{T}}^{2}\setminus\gamma$ and hence all the orbits have irrational slopes. Moreover, the unstable and the stable manifolds of the Lyapunov orbit should coincide and they are homeomorphic to ${\mathbb{T}}^{2}\setminus\gamma$. In particular, since the argument does not depend on the energy and the mass ratio, homoclinic orbits exist for all $\mu\in(0,1)$ and for all $c\in(c_{J},c_{h})$.
We now see how the satellite moves along homoclinic orbits. From (25) we obtain
(33)
$$\begin{cases}\dot{\lambda}=\displaystyle\pm\sqrt{\frac{g_{c}+c(\cosh^{2}%
\lambda)+2\cosh\lambda}{2}}\\
\dot{\nu}=\displaystyle\pm\sqrt{\frac{-g_{c}-c(\cos^{2}\nu)-2(1-2\mu)\cos\nu}{%
2}}\end{cases}$$
along a homoclinic orbit. It follows from the first equality that $-\lambda_{1}\leq\lambda\leq\lambda_{1}$, and $\dot{\lambda}=0\;\Leftrightarrow\;\lambda=\lambda_{1}$, where $\cosh\lambda_{1}=c_{J}/c$. On the other hand, we observe that
$$\dot{\nu}=0\;\;\Leftrightarrow\;\;c(\cos^{2}\nu)+2(1-2\mu)\cos\nu+g_{c}=0\;\;%
\Leftrightarrow\;\;\cos\nu=\frac{1-2\mu}{-c}$$
and $\nu_{1}\leq\nu\leq 2\pi-\nu_{1}$ for the Earth component and $\nu\in[0,\nu_{1}]\cup[2\pi-\nu_{1},2\pi]$ for the Moon component, where $\cos\nu_{1}=-(1-2\mu)/c$. Summarizing along a homoclinic orbit the variables $\lambda$ and $\nu$ are either strictly increasing or strictly decreasing. This implies that a homoclinic orbit rotates one of the primaries once and tends to the Lyapunov orbit as $t\rightarrow\pm\infty$. For example, assume that we have an orbit with $\dot{\nu}>0$ in the Earth component. It follows that $\nu\rightarrow\nu_{1}$ and $\nu\rightarrow 2\pi-\nu_{1}$ as $t\rightarrow-\infty$ and $t\rightarrow\infty$, respectively. The satellite moves from the vicinity of the Lyapunov orbit to the Earth, rotates around it, and goes back to the vicinity. This proves Theorem 1.3.
Remark 6.1.
In McGehee’s result [14] every homoclinic orbit rotates in the same direction. This difference follows from the fact that in the Euler problem we have additional anti-symplectic involution $(q,p)\mapsto(q,-p)$.
Appendix A Knot types of periodic orbits in the rotating Kepler problem
In this appendix, we determine the knot types of periodic orbits in the rotating Kepler problem.
The rotating Kepler problem is the Kepler problem in a rotating coordinate system whose Hamiltonian is given by
(34)
$$H(q,p)=\frac{1}{2}|p|^{2}-\frac{1}{|q|}+q_{1}p_{2}-q_{2}p_{1}.$$
This system is completely integrable with the first integral $L(q,p)=q_{1}p_{2}-q_{2}p_{1}$ which generates counter-clockwise rotation around the origin with constant unit angular speed. The Hamiltonian has a unique critical value $-3/2$. For energies below the critical energy, the Hill’s region consists of two connected components: one is bounded and the other is unbounded. We denote by $\Sigma=\Sigma_{h}$ the bounded connected component, where $h<-3/2$ is an energy level. For energies above the critical energy, the Hill’s region is the whole plane except for the origin $(0,0)$. On $\Sigma$, there are precisely two critical orbits: the retrograde and the direct circular orbits. It follows that the molecule is $A-A$.
In view of the Moser regularization the regularized energy hypersurface $\overline{\Sigma}$ coincides with the three-dimensional real projective space ${\mathbb{R}}P^{3}$, see for example [3]. Under the Levi-Civita regularization, $\Sigma$ lifts to the double cover which is diffeomorphic to $S^{3}$, see for instance [1].
Definition A.1.
(Albers-Fish-Frauenfelder-van Koert, [2]) A $T$-periodic orbit $\alpha:{\mathbb{R}}/T{\mathbb{Z}}\rightarrow{\mathbb{R}}^{4}$ of the rotating Kepler problem is a $k$-fold covered ellipse in an $l$-fold covered coordinate system provided the following hold.
$\bullet$ there exists positive $l\in{\mathbb{N}}$ such that $T=2\pi l$,
$\bullet$ the corresponding trajectory in the inertial coordinate system given by $\gamma(t):=\Phi^{-1}_{t}\alpha(t)$ is a $k$-fold covered ellipse of the inertial Kepler problem, where $\Phi_{t}:{\mathbb{R}}^{4}\rightarrow{\mathbb{R}}^{4}$ denotes the time-dependent change of coordinates from the inertial problem to the rotating problem.
Ellipses of positive eccentricity in an inertial system form $T^{2}$-families of periodic orbits under the $S^{1}$-action which rotates the coordinate system. The torus comprised of $k$-fold ellipses in an $l$-fold covered rotating coordinate system is denoted by $T_{k,l}$. If we consider the double covering, in view of Theorem 5.1 the previous arguments imply that any periodic orbit on $T_{k,l}$ is a torus knot. Note that the rotation number(and hence the knot type) does not change when we consider the double covering.
To determine the knot types, we introduce the polar coordinates $(r,\theta)$. The corresponding momenta $(p_{r},p_{\theta})$ are determined by the relation $p_{r}dr+p_{\theta}d\theta=p_{1}dq_{1}+p_{2}dq_{2}$. The Hamiltonian of the rotating Kepler problem in the polar coordinates is given by
$$\displaystyle H(r,\theta,p_{r},p_{\theta})=\frac{1}{2}\begin{pmatrix}%
\displaystyle p_{r}^{2}+\frac{p_{\theta}^{2}}{r^{2}}\end{pmatrix}-\frac{1}{r}+%
p_{\theta}.$$
Note that the variable ${\theta}$ is cyclic, i.e., ${\partial H/\partial\theta=0}$ and ${p_{\theta}=L}$. To compute the action variables, we put
$$\displaystyle W=W_{r}(r,h)+W_{\theta}(\theta,h)$$
and
$$\displaystyle p_{r}=\frac{\partial W}{\partial r},~{}\text{ }~{}p_{\theta}=%
\frac{\partial W}{\partial\theta}=L$$
so that
$$\displaystyle H=\frac{1}{2}\begin{bmatrix}\displaystyle\begin{pmatrix}%
\displaystyle\frac{\partial W}{\partial r}\end{pmatrix}^{2}+\frac{L^{2}}{r^{2}%
}\end{bmatrix}-\frac{1}{r}+L.$$
It follows that
$$\displaystyle\begin{pmatrix}\displaystyle\frac{\partial W}{\partial r}\end{%
pmatrix}^{2}+\frac{L^{2}}{r^{2}}+2L=2H+\frac{2}{r}.$$
A direct computation gives rise to
$$\displaystyle J_{r}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2\pi}\oint p_{r}dr=\frac{1}{2\pi}\oint\frac{\partial W}{%
\partial r}dr=\frac{1}{2\pi}\oint\sqrt{2(H-L)-\frac{L^{2}}{r^{2}}+\frac{2}{r}}dr$$
$$\displaystyle J_{\theta}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2\pi}\oint p_{\theta}d\theta=\frac{1}{2\pi}\oint Ld%
\theta=L.$$
Since we are only interested in the energy below the critical level, we may put ${-c=E=H-L}$, where ${c>0}$. We obtain that
$$\displaystyle J_{r}=\frac{1}{2\pi}\oint\frac{1}{r}\sqrt{-2cr^{2}+2r-L^{2}}dr.$$
To compute this integral, we differentiate both sides with respect to $c$
$$\displaystyle\frac{\partial J_{r}}{\partial c}=\frac{-1}{2\pi}\oint\frac{rdr}{%
\sqrt{-2cr^{2}+2r-L^{2}}}=\frac{-2}{2\pi\sqrt{2c}}\int_{r_{1}}^{r_{2}}\frac{%
rdr}{\sqrt{(r-r_{1})(r_{2}-r)}}.$$
Introducing the new variable ${\theta}$ which is defined by the relation
$$\displaystyle r=\frac{r_{1}+r_{2}}{2}-\frac{r_{1}-r_{2}}{2}\cos\theta$$
yields
$$\displaystyle\frac{\partial J_{r}}{\partial c}=\frac{-2}{2\pi\sqrt{2c}}\int_{0%
}^{\pi}\frac{r_{1}+r_{2}}{2}-\frac{r_{1}-r_{2}}{2}\cos\theta d\theta=-\frac{r_%
{1}+r_{2}}{2\sqrt{2c}}=\frac{-1}{\sqrt{2c}^{3}}.$$
Integrating both sides gives rise to
$$\displaystyle J_{r}=\frac{1}{\sqrt{2c}}+C,$$
where ${C}$ is a constant of integration. To determine the constant ${C}$, consider a circular orbit, i.e., ${r=r_{0}}$ and ${p_{r}=0}$. It follows from the Hamiltonian equation that ${L^{2}=p_{\theta}^{2}=r_{0}}$. Also,
$$\displaystyle-c=H-L=\frac{1}{2}\begin{pmatrix}\displaystyle 0+\frac{L^{2}}{r_{%
0}^{2}}\end{pmatrix}-\frac{1}{r_{0}}+L-L=-\frac{1}{2L^{2}}.$$
By definition of ${J_{r}}$, for a circular orbit we have ${J_{r}=0}$. Then the previous argument gives rise to
$$\displaystyle 0=C+\frac{1}{\sqrt{1/L^{2}}}=C+L.$$
Therefore,
$$\displaystyle J_{r}=-L+\frac{1}{\sqrt{2c}}.$$
By definition of rotation function, we conclude that for a fixed energy ${c}$
$$\displaystyle\rho=\frac{\partial J_{r}/\partial l}{\partial J_{\theta}/%
\partial l}=1+\frac{1}{\sqrt{2(L-H)}^{3}}=1+\frac{1}{\sqrt{-2E}^{3}}.$$
In particular, an orbit is periodic if and only if $\sqrt[3]{-2E}$ is rational, which is already described in [2] using Kepler’s laws. Let $\sqrt{-2E}^{3}=k/l$ for some ${k,l\in{\mathbb{N}}}$ which are relatively prime, or equivalently
$$\displaystyle E_{k,l}=-\frac{1}{2}\begin{pmatrix}\displaystyle\frac{k}{l}\end{%
pmatrix}^{\frac{2}{3}}.$$
Note that $E_{k,l}$ is the Kepler energy of the $T_{k,l}$-family, see [2]. It follows that the rotation number of the $T_{k,l}$-family is given by $(l+k)/k$. We have proven the following.
Theorem A.2.
In the rotating Kepler problem, every periodic orbit on the $T_{k,l}$-torus is a $(l+k,k)$-torus knot.
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Torques felt by solid accreting planets
Zs. Regály${}^{1}$
${}^{1}$Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, 1121, Budapest, Konkoly Thege Miklós út 15-17, Hungary
Contact e-mail: regaly@konkoly.hu
Abstract
The solid material of protoplanetary discs forms an asymmetric pattern around a low-mass planet ($M_{\mathrm{p}}\leq 10\,M_{\oplus}$) due to the combined effect of dust-gas interaction and the gravitational attraction of the planet. Recently, it has been shown that although the total solid mass is negligible compared to that of gas in protoplanetary discs, a positive torque can be emerged by a certain size solid species. The torque magnitude can overcome that of gas which may result in outward planetary migration. In this study, we show that the accretion of solid species by the planet strengthens the magnitude of solid torque being either positive or negative. We run two-dimensional, high-resolution ($1.5\rm{K}\times 3\rm{K}$) global hydrodynamic simulations of an embedded low-mass planet in a protoplanetary disc. The solid material is handled as a pressureless fluid. Strong accretion of well-coupled solid species by a $M_{\mathrm{p}}\lesssim 0.3\,M_{\oplus}$ protoplanet results in the formation of such a strongly asymmetric solid pattern close to the planet that the positive solid torque can overcome that of gas by two times. However, the accretion of solids in the pebble regime results in increased magnitude negative torque felt by protoplanets and strengthened positive torque for Earth-mass planets. For $M_{\mathrm{p}}\geq 3\,M_{\oplus}$ planets the magnitude of the solid torque is positive, however, independent of the accretion strength investigated. We conclude that the migration of solid accreting planets can be substantially departed from the canonical type-I prediction.
keywords:
accretion, accretion discs — hydrodynamics — methods: numerical — protoplanetary discs
††pubyear: 2020††pagerange: Torques felt by solid accreting planets–A
1 Introduction
Planets gravitationally interacting with their natal gaseous discs create two-armed spiral waves in the gas. The inner and outer spiral wakes being stationary in the planetary reference frame exert positive and negative torques on the planet, respectively (Ogilvie & Lubow, 2002). In an isothermal disc assuming that the surface mass density distribution of gas follows a power law of $r^{-1}$ the total torque exerted by spiral waves is negative (Ward, 1997). As a result, the planet loses angular moment and migrates inwards in the so-called type I regime. Since the migration speed is linearly proportional to the planet-to-star mass ratio (Goldreich & Tremaine, 1980), migration of growing planetary embryos (about at Mars mass) are negligibly slow (Tanaka, Takeuchi, & Ward, 2002). However, type I migration of grown low-mass planets (about an Earth-mass, $M_{\oplus}$) can be fatal, i.e. they lost to the central star within a hundred thousand years, within about $6$ Myr typical disc lifetime (Hernández et al., 2007).
Overall, no mechanism has been found that globally reduce the radial migration rate of low-mass planets (Morbidelli & Raymond, 2016). Thus, to reproduce the statistics of observed planetary systems, planet synthesis models had to reduce the speed of type I migration by about an order of magnitude (see, e.g., Ida & Lin, 2004a, b; Alibert et al., 2005; Miguel, Guilera, & Brunini, 2011).
Masset et al. (2006) suggested that type I migration trap may exist in places where the density gradients are positive, e.g., at disc inner edge, the boundaries of MRI active and inactive regions so-called dead zones or opacity transitions (Masset, 2011). However, two-dimensional hydrodynamic simulations showed that trapping of low-mass planets at viscosity transitions occur only if the dead zone edges are sharp enough to form a large-scale anticyclonic vortex Regály et al. (2013). Nonetheless, the existence of such planetary traps might explain the abundance of close-in super-Earths.
Another promising mechanism to slow down or even reverse the direction of type I migration in non-isothermal discs is associated with sharp temperature gradients (Paardekooper & Mellema, 2006; Baruteau & Masset, 2008; Paardekooper & Papaloizou, 2008). Two and three-dimensional numerical simulations in non-isothermal radiative discs have shown that migration of low mass planets ($5-35\,M_{\oplus}$) can even reverse (Kley & Crida, 2008; Kley, Bitsch, & Klahr, 2009; Lega et al., 2014). Type I torque depends on various physical parameters of the disc, such as viscosity and opacity, which are varying during the disc evolution. In an evolving disc by applying non-isothermal type I torque formulae (see, e.g., Masset & Casoli, 2010; Paardekooper et al., 2010; Paardekooper, Baruteau, & Kley, 2011) the outward migration of low-mass planets is found to be confined in a radial range of the disc at about 3-9 au (Hasegawa & Pudritz, 2011; Hellary & Nelson, 2012; Bitsch et al., 2015).
Radiative effects of a solid accreting low-mass planet can significantly modify the torque felt by the planet (Benítez-Llambay et al., 2015). The so-called heating torque is positive and caused by the formation of asymmetric gaseous density lobes in the vicinity of the planet. The heat released by a vigorously accreting $3\,M_{\oplus}$ planet with a mass doubling time less than about $5000$ orbits results in migration reversal. Studies considered neglect the solid torque exerted on low-mass planets. Since the dust-to-gas mass ratio is only about one percent in canonical protoplanetary discs (see, e.g., Williams & Cieza, 2011), it is plausible to assume that its gravitational effect dwarfed by that of gas. However, the asymmetric distribution of solid matter in the vicinity of the planet might significantly modify the torque felt by that planet, similarly to the heating torque caused by the asymmetric gas distribution inside the planetary Hill sphere. Indeed due to the combined effect of dust-gas interaction and the gravitational attraction of the planet, highly asymmetric distribution of solid material can form near low-mass planets. Benítez-Llambay & Pessah (2018) reveald that the spatial distribution of solid material can be extremely asymmetric such that it changes the speed or even alters the direction of type I migration.
To study the dynamics of solid material Benítez-Llambay & Pessah (2018) have taken into account the gravitational forces of the central star and the planet, and the aerodynamical drag force between gas and solid species. Planetary accretion, however, modifies the spatial distribution of solids by the removal of solid material from the vicinity of the planet. In this study, we present an investigation of how planetary accretion of solids modifies the spatial distribution of various size solid species and solid torque felt by low-mass planets. We present hydrodynamical simulations of planet-disc interactions assuming that the solid content of the disc is a pressureless fluid. We investigate the effect of planet mass in the range of $0.1-10\,M_{\oplus}$, size of solid species, the initial slope of gas profile, accretion strength on the solid torque.
The paper is organised as follows: In Section 2, the applied hydrodynamical model and the initial conditions of gas and solids are presented. Section 3 presenting our results show a numerical convergence study and the effect of accretion strength on the total torque felt by low-mass planets. Section 4 discusses the torque profiles of different solid specie, the effect of solid accretion and the smoothing strength of planetary potential on the spatial distribution of solid material in the proximity of the planet. The paper closes with our summary and conclusions in Section 5. In Appendix A, test simulations on the dust solver is presented.
2 Hydrodynamical model
2.1 Governing equations
We investigate the effect of protoplanetary discs’ solid material on the torque felt by an embedded low-mass planet using two-dimensional hydrodynamical simulations. The solid material is assumed to be a pressureless fluid whose dynamic is affected by the aerodynamic drag force arising due to the velocity difference between the gas and solid parcels. We use GFARGO2 for this investigation, which is our extension to GFARGO, a GPU supported version of the FARGO code (Masset, 2000).
The dynamics of gas and solid fluids perturbed by the embedded planet is described by the following equations:
$$\displaystyle\frac{\partial\Sigma_{\mathrm{g}}}{\partial t}+\nabla\cdot(\Sigma%
_{\mathrm{g}}{\bm{v}})=0,$$
(1)
$$\displaystyle\frac{\partial\bm{v}}{\partial t}+(\bm{v}\cdot\nabla)\bm{v}=-%
\frac{1}{\Sigma_{\mathrm{g}}}\nabla P+\nabla\cdot\bm{T}-\nabla\Phi,$$
(2)
$$\displaystyle\frac{\partial\Sigma_{\mathrm{d}}}{\partial t}+\nabla\cdot(\Sigma%
_{\mathrm{d}}\bm{u})=-\dot{\Sigma}_{\mathrm{acc}},$$
(3)
$$\displaystyle\frac{\partial\bm{u}}{\partial t}+(\bm{u}\cdot\nabla)\bm{u}=-%
\nabla\Phi+\bm{f}_{\mathrm{drag}},$$
(4)
where $\Sigma_{\mathrm{g}}$, $\Sigma_{\mathrm{d}}$ and $\bm{v}$, $\bm{u}$ are the surface mass density and velocity of gas and solid (being either dust particles or pebbles), respectively. The gas pressure is given by assuming a locally isothemral equation of state, for which case $P=c_{\mathrm{s}}^{2}\Sigma$, where $c_{\mathrm{s}}$ is the local sound speed. A flat disc approximation is used for which case the disc pressure scale-height is defined as $H=hR$, where the aspect ratio is set to $h=0.05$.
In Equation (2) $\bm{T}$ is the viscous stress tensor calculated as
$$\bm{T}=\nu\left(\nabla\bm{v}+\nabla\bm{v}^{T}-\frac{2}{3}\nabla\cdot\bm{v}\bm{%
I}\right),$$
(5)
where $\nu$ is the disc viscosity and $\bm{I}$ is the two-dimensional unit tensor, see details in Masset (2002) for calculating $\bm{T}$ in cylindrical coordinate system. We use the $\alpha$ prescription of Shakura & Sunyaev (1973) for the disc viscosity in which case $\nu=\alpha c_{\mathrm{s}}H$. $\alpha=10^{-4}$ is applied which is in the order of the smallest effective viscosity presumably present in protoplanetary discs, e.g. arising due to vertical shear instability (Stoll & Kley, 2016). Note that the diffusion of solid species is neglected, which can only be done in a low-viscosity limit applied in this study (see, e.g., Youdin & Lithwick, 2007).
The gravitational potential of the system, $\Phi$, is calculated as
$$\Phi=-G\frac{M_{*}}{R}-G\frac{M_{\mathrm{p}}}{\sqrt{R^{2}+R_{\mathrm{p}}^{2}-2%
RR_{\mathrm{p}}\cos(\phi-\phi_{\mathrm{p}})+(\epsilon H)^{2}}}+\Phi_{\mathrm{%
ind}},$$
(6)
where $G$ is the gravitational constant, $R$, $\phi$ and $R_{\mathrm{p}}$, $\phi_{\mathrm{p}}$ are the radial and azimuthal coordinates of a given numerical grid cell and the planet, respectively. $M_{*}$ and $M_{\mathrm{p}}$ is the star and planet mass, respectively. The indirect potential, $\Phi_{\mathrm{ind}}$, is taken into account as a non-inertial frame, which co-rotates with the planet is used for the simulation. Note, however, that the effect of $\Phi_{\mathrm{ind}}$ presumably negligible in our case as the investigated planet-to-star mass ratio is small, $M_{\mathrm{p}}/M_{*}\leq 10^{-5}$ and no significant large-scale asymmetries develop in the disc (see more details in Regály & Vorobyov, 2017). Both the gas and solid self-gravity are neglected. The planetary potential is smoothed by a factor of $\varepsilon H$ assuming $\varepsilon=0.6$, which is found to be appropriate for two-dimensional simulations (Kley et al., 2012; Müller, Kley, & Meru, 2012). Note that the vertical scale-height of solids may differ from that of gas due to size-dependent speed of sedimentation to the disc midplane (Dullemond & Dominik, 2004). However, applying a planetary potential that depends on the Stokes number of solids has no solid ground and requires further three-dimensional investigations. Thus, both solids and gas experience the same smoothing of planetary potential in our simulations. To shed light on the effect of smoothing strength on the solid torques we run additional simulations with $\varepsilon=0.3$ and $0.9$, see details in Section 4.4.
In Equation (4) the drag force exerted by the gas on the solid is
$$\bm{f}_{\mathrm{drag}}=\frac{\bm{v}-\bm{u}}{\mathrm{St}/\Omega},$$
(7)
where $\mathrm{St}$ is the Stokes number of the given solid species and $\Omega=(GM_{*})^{1/2}R^{-3/2}$ is the local Keplerian angular velocity. Note that diffusion of solid material and the back-reaction of the solid onto the gas are neglected.
Equation (4) is solved by a two-step numerical method. First, the source term, i.e., the right-hand side of Equation (4) is calculated then it is followed by the conventional advection calculation. For the source term we use a fully implicit scheme (see details in Stoyanovskaya, Snytnikov, & Vorobyov, 2017, 2018). The solid velocity is updated at every step as
$$\bm{u}^{n+1}=\bm{v}^{n}+\Delta t\bm{a}^{n}_{\mathrm{g}}-\frac{(\bm{v}^{n}-\bm{%
u}^{n})+\Delta t(\bm{a}^{n}_{\mathrm{g}}-\bm{a}^{n}_{\mathrm{d}})}{1+\Delta t/%
\tau_{\mathrm{s}}},$$
(8)
where $\tau_{\mathrm{s}}=\mathrm{St}/\Omega$ is the stopping time of the solid species, $\Delta t$ is the time-step applied, and $\bm{a}^{n}_{\mathrm{g}}$ and $\bm{a}^{n}_{\mathrm{d}}$ are the acceleration of gas and solid parcels due to the pressure gradient and gravitational forces:
$$\displaystyle\bm{a}^{n}_{\mathrm{g}}=-\frac{1}{\Sigma_{\mathrm{g}}^{n}}\nabla P%
^{n}-\nabla\Phi^{n},$$
(9)
$$\displaystyle\bm{a}^{n}_{\mathrm{d}}=-\nabla\Phi^{n}.$$
(10)
The above expressions of solid velocity are equivalent to Equations (23)-(25) in Stoyanovskaya, Snytnikov, & Vorobyov (2018) by assuming that $\epsilon=0$, which corresponds to vanishing solid-to-gas mass ratio, i.e. $\Sigma_{\mathrm{d}}/\Sigma_{\mathrm{g}}\simeq 0$. This assumption is valid as long as there is no significant solid enhancement. In this case the back-reaction of solid onto the gas (a term $-(\Sigma_{\mathrm{d}}/\Sigma_{\mathrm{g}})\bm{f}_{\mathrm{drag}}$ in the right-hand-side of equation (2)) can be neglected. With the above scheme, the effect of aerodynamic drag can be modelled for solid species that have stopping time that is much smaller than the time-step ($\Delta t\ll\tau_{\mathrm{s}}$). For pebbles that have large stopping time ($\Delta t\gg\tau_{\mathrm{s}}$), the method described above is only applicable if the orbit of solids are not crossing.
The accretion of solid material onto the planet represented by $\dot{\Sigma}_{\mathrm{acc}}$) in Equation (3) is modelled by a reduction of the solid density inside the planetary Hill sphere. We use a scheme similar to the prescription of gas accretion given in Kley (1999). At each time step the solid density reduced by $1-\eta\Delta t$ inside the planet’s Hill sphere with the radius of $R_{\mathrm{Hill}}=a(M_{\mathrm{p}}/3M_{*})^{(1/3)}$, where $\eta$ is the strength of accretion. With the above prescription the half-emptying time is $\log(2)/\eta$, i.e. about 2/3 orbital time for $\eta=1$. Two accreting scenarios are investigated $\eta=1$ and $0.1$ referred to as strong and weak accreting scenarios. The solid density reduction is done in two steps: first, one-third of the density is removed from the inner part of the Hill sphere ($\Delta R\leq 0.75R_{\mathrm{Hill}}$), then in the second step, two-third of the density is removed from the innermost part of the Hill sphere ($\Delta R\leq 0.45R_{\mathrm{Hill}}$). For simplicity, the planet mass is kept constant, i.e., the removed mass (and the momentum) is not added to the planet. As a result, the total solid mass and moment are strictly not conserved in the system. Note that strong accretion results in an accretion rate in the same order of magnitude applied by Benítez-Llambay et al. (2015) to model the effect of heating torque. The total mass accreted by the planet is negligible compared to the planet mass for all solid species.
2.2 Initial and boundary conditions
We handled solid material with fix Stokes number throughout the simulation domain. We modelled multiple solid species in nine bins: $\mathrm{St}=0.01,\,0.1,1,\,2,\,3,\,4,\,5,$ and $30$. The back-reaction of the solids to the gas is neglected, thus multiple species of solids can be modelled in one simulation. According to Supulver & Lin (2000) the Stokes number of a solid species having a physical size of $s$ can be given as
$$\mathrm{St}=\frac{\rho_{\mathrm{i}}}{\rho_{\mathrm{g}}}\frac{s}{(1-f)\sqrt{8/%
\pi}c_{\mathrm{s}}+fC_{\mathrm{D}}/2v_{\mathrm{rel}}}\Omega,$$
(11)
where $\rho_{\mathrm{i}}$ is the volume density of the solid material. Parameter $f$ describes the transition between Epstein and Stokes regimes as $f=s/(s+\lambda)$. Assuming that protoplanetary discs are dominated by hydrogen molecule, the mean free path is $\lambda=3.3458910^{-9}/\rho_{\mathrm{g}}$. In the Stokes regime $C_{\mathrm{D}}-0.44$ according to (Whipple, 1972). By assuming a vertical pressure balance, the gas density in the disc midplane can be given as $\rho_{\mathrm{g}}=(1/2\pi)(\Sigma/H)$. The relative velocity between the gas and solid, $v_{\mathrm{rel}}$ can be given as
$$v_{\mathrm{rel}}=\sqrt{\left(v_{R}-u_{R}\right)^{2}+\left(v_{\phi}-u_{\phi}%
\right)^{2}}.$$
(12)
Taking into account Equations,(13)-(14), the relative velocity between the gas and dust is $v_{\mathrm{rel}}\leq 0.01$ in a Minimum Solar Mass Nebula (MMSN) model (Hayashi, 1981). The solid physical size as a function of the Stokes number, $s(\mathrm{St})$, is shown in Fig. 1 at three distances (0.1, 1, and 10 au) from the star. The solid bins investigated roughly represent the following regimes in the planet forming region at about several astronomical units distance from the star: $\mathrm{St}0.01$ is the mm-sized dust regime, $\mathrm{St}=0.1-1$ is the cm-sized pebble regime, $\mathrm{St}=1-10$ is the m-sized boulder regime, and $\mathrm{St}>10$ is the asteroid regime.
The mass of planets in our models are in the range of $q=3\times 10^{-7}-3\times 10^{-5}$, which correspond to $0.1M_{\oplus}-10M_{\oplus}$. We investigate five models in which planet mass in logarithmic bins is $M_{\mathrm{p}}=0.1,\,0.3,\,1,\,3$ and $10\,M_{\oplus}$. The orbital distance of the planet is set to unity, and the planet is kept on a circular orbit, i.e., no migration is allowed.
Initially $\Sigma_{\mathrm{g}}=\Sigma_{0}R^{-p}$ and $\Sigma_{\mathrm{d}}=\epsilon\Sigma_{\mathrm{g}}$ assuming $\Sigma_{0}=6.45\times 10^{-4}$ and solid-to-gas mass ratio $\xi=10^{-2}$. We investigate three different initial gas and solid density profile steepness: $p=0.5,\,1.0$ and $1.5$ assuming fixed $\Sigma_{0}$. Emphasise that the absolute value of $\Sigma_{\mathrm{g}}$ does not affect our results for the following reasons: 1) disc self-gravity is neglected; 2) the amount of solid contained by the disc is scaled with the gas mass; 3) the dust back-reaction is neglected.
The initial velocity components of gas ($v_{R}$ and $v_{\phi}$) are defined as
$$\displaystyle v_{R}=-3\alpha h^{2}(1-p)\Omega R,$$
(13)
$$\displaystyle v_{\phi}=\sqrt{1-h^{2}(1+p)}\Omega R.$$
(14)
The above equations satisfy the
steady-state solution to the viscous evolution of the surface mass density in $\alpha$ discs for $p=0.5$ and $p=1$. The initial velocity components of solid material ($u_{R}$ and $u_{\phi}$) are given by the analytic solution of an unperturbed disc, which reads
$$\displaystyle u_{R}=\frac{v_{\mathrm{R}}\mathrm{St}^{-1}-h^{2}(1+p)}{\mathrm{%
St}+\mathrm{St}^{-1}}\Omega R,$$
(15)
$$\displaystyle u_{\phi}=\sqrt{1-h^{2}(1+p)}\Omega R-\frac{1}{2}u_{R}\mathrm{St},$$
(16)
(see, e.g., Nakagawa, Sekiya & Hayashi, 1986; Takeuchi & Lin, 2002).
We use a computational domain whose extension is $0.48\leq R\leq 2.08$ in code units, which contains 1536 logarithmically distributed radial and 3072 equidistant azimuthal grid cells. With these settings, the numerical resolution is about $0.02H$ at all distances. We confirmed that simulations with that resolution are in the numerically convergent regime (see Section 3.1).
At the inner and outer boundaries a wave damping boundary condition is applied for the gas (see, e.g., de Val-Borro et al., 2006). An open inner boundary condition is applied for the solid material. However, solids are replenishing at the outer boundary due to the applied damping boundary condition there. As a result, the solid disc is not depleted.
We assume 1 au for the unit of length, and the stellar mass for the unit of mass and the gravitational constant, G, is defined 1. With these assumptions, the orbital period is 2$\pi$ at 1 au.
3 Results
3.1 Resolution required for numerical convergency
First, we test that the hydrodynamical solutions are in the numerically convergent regime. We run non-accreting and strong accreting models assuming $q=3\times 10^{-6}$ with different numerical resolutions: $N_{R}\times N_{\phi}=[512\times 1024\,,768\times 1536\,,1024\times 2048\,,1536%
\times 3072\,,1920\times 3820\,,2048\times 4096]$.
Both the solid and gas torques are calculated as
$$\Gamma=\Gamma_{0}\sum_{i,j=1}^{Nr,N\phi}\left(x_{\mathrm{p}}\frac{\Sigma_{i,j}%
}{\Delta R_{i,j}^{3}}-y_{\mathrm{p}}\frac{\Sigma_{i,j}}{\Delta R_{i,j}^{3}}%
\right),$$
(17)
where
$$\displaystyle\Delta R_{i,j}=\sqrt{\Delta x_{i,j}^{2}+\Delta y_{i,j}^{2}},$$
(18)
$$\displaystyle\Delta x_{i,j}=R_{i,j}\cos(\phi_{i,j})-x_{\mathrm{p}},$$
(19)
$$\displaystyle\Delta y_{i,j}=R_{i,j}\sin(\phi_{i,j})-y_{\mathrm{p}},$$
(20)
where $R_{i,j}$ and $\phi_{i,j}$ are the cylindrical coordinates of cell $i,j$, $\Sigma_{i,j}$ is the surface mass density of gas or the given solid species inside that cell. $x_{\mathrm{p}}$, $y_{\mathrm{p}}$ are the Cartesian coordinate of the planet. To compare torques felt by different mass planets it is useful to normalise the torques. The change in the semi-major axis, $a$, of a planet due to the torque, $\Gamma$, exerted by the gas and solid species can be given as $da/dt=2\Gamma/(qa\Omega(a))$. Thus, $\Gamma_{0}=2/(qa\Omega(a))$ is used as a normalisation factor throughout this paper. Since $a=1$ in our models, the normalisation constant is $\Gamma_{0}=2/q$.
The evolution of normalised torque of solid species, $\Gamma_{d}/\Gamma_{0}$, is shown in Figure 2. The torque exerted by the gas saturates at $\Gamma_{\mathrm{g}}/\Gamma_{0}\simeq-0.8$ within 10 orbits. The gas torque is found to be independent of the numerical resolution therefore it is not shown in the Figure. However, the saturation value of the solid torque depends on the Stokes number and the numerical resolution too. In non-accreting models, torque magnitudes are significant and sensitive to the numerical resolution for $1\leq\mathrm{St}\leq 10$. However, in the strong accreting models solid torques become sensitive to the numerical resolution for the well-coupled species ($\mathrm{St}\leq 0.01$ ) too. Although the effect of solid accretion (see details in the next section) can be identified in model that uses the lowest numerical resolution, a relatively high numerical resolution is required for proper torque values.
We found that the magnitude of solid torques is converged in models that use numerical resolution above $1024\times 2048$. Therefore, a relatively fine resolution $N_{R}\times N_{\phi}=1536\times 3072$ is used in this study. With this numerical resolution the planetary Hill sphere is resolved by about 72 cells for the smallest mass planet modelled.
3.2 Effect of solid accretion
In this section, the effect of solid accretion on the torque felt by the planet is investigated. Fig. 3 shows the total torques measured at the end of simulation, after 200 orbits of the planet. Torques are normalised with the absolute value of the gas torque, $(\Gamma_{\mathrm{g}}+\Gamma_{\mathrm{d}})/|\Gamma_{\mathrm{g}}|$. Five different planet mass ($0.1,\,0.3,\,1,\,3$ and $10\,M_{\oplus}$) are investigated with three different strength of accretion, $\eta=0,\,0.1$ and $1$. The three panels show our results assuming three different steepness ($p=0.5,\,1.0$ and $1.5$) of the initial gas profile.
Based on the value of the normalised total torque three different regimes can be defined: $(\Gamma_{\mathrm{g}}+\Gamma_{\mathrm{d}})/|\Gamma_{\mathrm{g}}|>0$ shaded with red, $-1<(\Gamma_{\mathrm{g}}+\Gamma_{\mathrm{d}})/|\Gamma_{\mathrm{g}}|<0$ shaded with blue, and $(\Gamma_{\mathrm{g}}+\Gamma_{\mathrm{d}})/|\Gamma_{\mathrm{g}}|<-1$ shaded with green colours. It is appreciable that solid accretion increases the magnitude of solid torque (for both positive or negative torque values) independent of $p$.
Let’s investigate the model where $M_{\mathrm{p}}=1\,M_{\oplus}$ and $p=0.5$ , see upper panel of Fig. 3. In all cases, except $\mathrm{St}=3$, the solid torques are positive. For $3\leq\mathrm{St}\leq 5$, the solid torque is positive and its magnitude exceeds that of gas, which results in positive total torque. For $\mathrm{St}=3$, the solid torque is negative, therefore the planet feels stringer negative total torque with increasing accretion strength. For $\mathrm{St}>5$ and $\mathrm{St}<0.1$, the magnitude of solid torques are positive and has a small amplitude (except cases of strong accretion), therefore solid material slightly decreases the magnitude of negative total torque.
For $M_{\mathrm{p}}\geq 3\,M_{\oplus}$ planets the total torque amplitude is nearly independent of $\eta$, i.e., the effect of solid accretion is negligible. However, for smaller planet mass ($M_{\mathrm{p}}\leq 1\,M_{\oplus}$), solid accretion can have a severe effect on planetary torque: planet can feel two times larger magnitude torque compared to that of conventional type I regime. For low-mass planet ($M_{\mathrm{p}}\leq 0.3\,M_{\oplus}$) positive total torque can be observed if it strongly accretes well-coupled $\mathrm{St}=0.01$ solid species.
Two additional panels of Fig. 3 show models in which different slopes for the gas density profile, $p=1$ and $1.5$ are used. The steeper the initial density profile, the more negative the normalised total torque. Emphasise that $M_{\mathrm{p}}\leq 0.3\,M_{\oplus}$ planets that strongly accrete $\mathrm{St}\leq 0.1$ solids experience positive torque. Another effect of the steep initial gas profile is the moderate growth of negative torque amplitude (about 1.5 times) due to solid accretion.
In summary, solid accretion generally increases the magnitude (either positive or negative) of the total torque felt by the planet. The total torque can be positive if the accretion of the well-coupled solid species is strong. For about an Earth-mass planet, the pebble accretion ($3\leq\mathrm{St}\leq 5$), independent of accretion strength, can also lead to positive torque (for $p=0.5$) or strongly reduced (for $p=1$ or $p=1.5$) negative torque. The effect of accretion strength on the total torque felt by the planet weakens with steeper radial density profile of gas.
4 Discussion
4.1 Torque profiles
First, let’s investigate the radial profiles of torques exerted by the gas and different solid species. Fig. 4 shows the azimuthally averaged radial torque profiles in the three accretion regimes as a function of the radial distance from the planet measured in units of the radius of planetary Hill sphere. Two models assuming $0.1\,M_{\oplus}$ and $1\,M_{\oplus}$ planets are shown on the top and bottom panels of Fig. 4, respectively. Note that the spatial region covered by the plots are the same for all panels.
The gas torque arises from a region that has an extent of about $\pm 10R_{\mathrm{Hill}}$ and it is independent on the applied accretion strength. The latter statement is valid as long as the solid back-reaction and the self-gravity of the disc are neglected. The inner and outer gas disc exerts exclusively positive and negative torques, respectively. Since $\Gamma_{\mathrm{g}}$ profile is only slightly asymmetric a small non-negligible negative gas torque is exerted on the planet as the theory predicts (see, e.g., Ward, 1997).
On the contrary, solid torque profiles are highly non-symmetric and their shapes are sensitive to the Stokes number of the given species and the accretion strength too. Torques of solid species generally vanish beyond $\pm 6R_{\mathrm{Hill}}$. Independent of the planet mass, the solid torques of the well-coupled solid species ($\mathrm{St}\leq 0.1$) arising from a region of $\left[-3\Delta R_{\mathrm{Hill}},1\Delta R_{\mathrm{Hill}}\right]$. This means that the physical distance from within the well-coupled solid species can exert torque on the planet is roughly bound to the planetary Hill sphere, namely it depends on the planet mass. It is also appreciable that the torque profile of $\mathrm{St}=0.01$ becomes positive as the strength of solid accretion increases. This explains why the torque exerted by the well-coupled solid species can even overcome that of gas.
Solid torques for $\mathrm{St}\gg 1$ are non-vanishing up to a distance of $\pm 6\Delta R_{\mathrm{Hill}}$ and $\pm 3\Delta R_{\mathrm{Hill}}$ in the case of $0.1\,M_{\oplus}$ and $1\,M_{\oplus}$ planets, respectively. Thus, the extent of the influencing region of the less coupled solid species is generally independent of the planet mass.
The torque profile can be significantly affected by the accretion of less coupled solid species, $\mathrm{ST}>1$. The general trend is that the positive part of the torque profile diminishes with increasing $\eta$. As a result, solids tend to increase the magnitude of negative torque. This phenomenon is prominent for $M_{\mathrm{p}}=1\,M_{\oplus}$ planets up to $\mathrm{St}=10$, while suppressed for larger mass planets. Consequently, the accretion profiles are insensitive to the accretion of $\mathrm{St}>1$ solids for larger mass planets ($M_{\mathrm{p}}\geq 1\,M_{\oplus}$), which explains the decreasing amplitude of solid torques for larger mass planets (see Fig. 3). Note, however, that the torque profile of pebbles ($1<\mathrm{St}\leq 5$) is mainly positive for larger mass planets.
4.2 Distribution of solids in the vicinity of planet
Based on the fact that the density of solids determines the torque magnitude, see Equations (17), it is worth investigating its spatial distribution. Fig. 5 shows the solid density distribution of all investigated solid species in the vicinity of $1\,M_{\oplus}$ planet in $p=0.5$ disc at the end of the simulation, when a quasi-steady state has been developed. It can be seen that solids form a spiral-like overdense pattern. Emphasise that these patterns do not coincide with the gas spiral, except for the well-coupled solid species, $\mathrm{St}=0.01$. The opening angle of spiral patterns is increasing with the Stokes number of solid species.
Another prominent feature of the solid distribution is the development of a strong density depletion behind the planet (concerning planetary orbital motion) for the non-coupled solid species ($\mathrm{St}>1$). The size of the solid depleted region grows with both the Stokes number and $\eta$. Emphasise that the solid depleted region can be developed for the well-coupled dust species ($\mathrm{St}\leq 0.1$) too in accreting models (see panels $\eta=1$, $\mathrm{St}=0.01$ and $\mathrm{St}=0.1$ of Fig. 5). Due to the formation of these overdense and depleted solid patterns, the solid distribution can become highly non-symmetric. As a result, the disc region beyond and behind the planet can exert different magnitude and sign torques.
Now, let’s investigate the change in the spatial distributions of solid species due to their accretion by calculating the value of $[\Sigma_{d}(\eta=1)-\Sigma_{d}(\eta=0)]/\Sigma_{d}(\eta=0)$ in the vicinity of the planet. In Fig. 6 red colour corresponds to regions where the distribution of solids is unaltered by the solid accretion, while blue coloured regions reveal strong depletion due to solid accretion. In the strong accreting model, the solid depleted region formed beyond the planet is narrower than behind the planet for the well-coupled solid species, $\mathrm{St}\leq 0.1$. Therefore, in case of strong accretion the positive torque arising from the disc region beyond the planet overcomes the negative torque exerted by the disc region behind the planet. The bright vertical structures inside the planetary orbit appear due to a solid depleted in-spiralling pattern present in the strong accreting model. For $\mathrm{St}=1$, only a small depletion developed very close to the planet as a result of solid accretion, which causes a negative net solid torque (see panel $\mathrm{St}=1$ on Fig.4). Solid species with $2\leq\mathrm{St}\leq 5$ generally form a depleted region behind the planet which is wider in the strong accreting regime. As a result, a positive solid torque appears, whose magnitude increases with accretion strength. For the decoupled solid species ($\mathrm{St}\geq 10$) the accretion results in symmetric solid removal inside the planetary Hill sphere. Hence the torque profiles are damped and do not change significantly due to accretion (see panels $\mathrm{St}=10$ and $\mathrm{St}=30$ on Fig.4)).
By examining the solid distribution in models where the initial gas density profile is steeper, we found no significant departure to that of $p=0.5$ model. As a result, the solid torques have a similar dependence on $\eta$ and magnitudes independent of $p$. However, due to the fact that the gas torque increases with $p$ ($\Gamma_{\mathrm{g}}/\Gamma_{0}=0.75,\,0.95$, and $1.18$ for $p=0.5,\,1.0$, and $1.5$, respectively). solids exert lowered torque on the planet in models that have steeper gas density slope.
4.3 Comparison to previous work
To date, only one work addressed the solid torque felt by low-mass ($0.3\,M_{\oplus}\leq M_{\mathrm{p}}\leq 10\,M_{\oplus}$) non-accreting plants. Benítez-Llambay & Pessah (2018) investigated the effect of solids in a disc with $p=0.5$ and similar spatial extensions. They used a somewhat more viscous disc, $\alpha=3\times 10^{-3}$, however, the dust diffusion was also neglected in their simulations. They stated that the majority of simulations reach steady-state, which was confirmed by our runs. Note, however, that to reach steady-state requires 200 orbits for the well-coupled solid species, see Fig. 2.
According to Figure 2 of Benítez-Llambay & Pessah (2018) planets generally feels positive total torque for $\mathrm{St}\gtrsim 0.1$ solid species as $\log(\Gamma_{\mathrm{d}}/|\Gamma_{\mathrm{g}}|)\geq 0$ for those cases. Some exceptions, however, can be identified for $0.1<\mathrm{St}<1$ and $M_{\mathrm{p}}\leq 1\,M_{\oplus}$ cases. These findings are practically confirmed by our $M_{\mathrm{p}}\geq 0.3\,M_{\oplus}$ simulations as the normalised total torques are in the blue or red shaded region for all non-accreting models upper panel of Fig. 3. Note, however, that for our $M_{\mathrm{p}}=0.3\,M_{\oplus}$ model, solids in the pebble regime ($2\leq\mathrm{St}\leq 4$) provide negative torque. The discrepancy can be attributed to the difference in torque calculation in the two studies or the difference in the applied magnitude of viscosity. The latter requires further study as diffusion of solids (especially with small Stokes number) should be taken into account in non-inviscid disc.
Benítez-Llambay & Pessah (2018) completely neglect torques arising from the inner half of the planetary Hill sphere, while we take in to account the entire planetary Hill sphere. Note that the idea of torque cut-off was introduced by Crida, et al. (2009) to calculate disc torques exerted on massive planets to account their circumplanetary discs in non-self-gravitating models. Crida, et al. (2009) conclude that removing the half of the material of the planetary Hill sphere is appropriate. Since the circumplanetary disc does not form around low mass ($M_{\mathrm{p}}\lesssim 30\,M_{\oplus}$) planets (Masset et al., 2006), the exclusion of material in the planetary Hill sphere has no physical argument.
4.4 Effect of smoothing of planetary potential
Due to numerical issues and for approximating three-dimensional effects it is necessary to smooth the planetary potential in two-dimensional simulations. In this study, we use a conventional method of potential smoothing with $\varepsilon=0.6$, see Equation (6). In this method, the smoothing length, $\varepsilon H$, is proportional to the local pressure scale-height. However, solids are subject to sediment to the disk midplane. As a result, the vertical scale-height of solids differs from that of gas (see, e.g., Dullemond & Dominik, 2004). In a simple approach, the solid scale height can be given as
$$H_{\mathrm{d}}=H\sqrt{\frac{\alpha}{St+\alpha}},$$
(21)
in which case the well-coupled solid species have the same scale-height as the gas, while decoupled solid particles sink to the disk midplane (Birnstiel, Fang & Johansen, 2016). This process is affected by the vertical turbulent mixing, which depends on the magnitude of viscosity via $\alpha$. The appropriate method of smoothing the planetary potential for solid species in two-dimensional simulations is unknown yet. However, the importance of smoothing strength can be revealed by changing the strength of smoothing.
We investigated the effect of smoothing strength by applying $\varepsilon=0.3$ and $0.9$ for a $M_{\mathrm{p}}=1\,M_{\oplus}$ planet embedded in a disc having $p=1.0$ slope of initial density profiles in the three accretion regimes, see Fig. 7. It is found that the gas torque depends on $\varepsilon$: $\Gamma_{\mathrm{g}}/\Gamma_{0}=1.82$ and $0.7$ for $\varepsilon=0.3$ and $0.9$, respectively. A similar trend can be observed for solid torques. For the well-coupled solids ($\mathrm{St}=0.01$ and $0.1$). the change in smoothing strength has a week effect. However, the magnitude of solid torques decreases with increasing $\varepsilon$ relative to the gas torque for $\mathrm{St}\geq 1$. Interestingly, the effect of accretion strength on solid torques becomes very strong in models that use $\varepsilon=0.9$. For weak smoothing ($\varepsilon=0.3$), the effect of solids is so strong that even torque reversal can be observed for the less coupled solid species ( $\mathrm{St}\geq 4$).
Fig. 8 compares the density distribution of $\mathrm{St}=1$ solid in the vicinity of $1~{}M_{\oplus}$ planet in non-accreting and strong accreting models assuming $\varepsilon=0.3$ and 0.9. It is appreciable that the dust depletion beyond the planet is prominent in $\varepsilon=0.3$ and completely missing in $\varepsilon=0.9$ model. Note that this solid depletion is also missing in our standard $\varepsilon=0.6$ models, see Fig. 6 for comparison. The same can be observed for $\mathrm{St}\geq 1$ species. Generally, the strength of solid asymmetry weakens with increasing strength of planetary potential smoothing. Therefore the total torques exerted by solids also decreasing with $\varepsilon$.
5 Summary and Conclusion
We investigated the effect of solid accretion on the torque felt by a low-mass planet with the mass in the range $0.1M_{\oplus}-10M_{\oplus}$, embedded in a protoplanetary disc using two-dimensional grid-based locally isothermal hydrodynamic simulations. We used $\alpha$-prescription for the disc viscosity in a low-viscosity regime $\alpha=10^{-4}$. The disc self-gravity and solid back-reaction are neglected. For simplicity, we modelled solids assuming constant Stokes numbers in the range $0.01-30$. The gas-to-solid mass ratio assumed to be the canonical value of 0.01. We found that the accretion of solids can be important with regards to the magnitude or even the sign of the torque felt by the low-mass planet. Our key findings are the followings:
(i) As a result of solid accretion, the spatial asymmetry developed in the solid distribution in the vicinity of the planet strengthens. The magnitude of asymmetry depends on the mass of the accreting planet, the Stokes number of solids, and the accretion strength. We found that the magnitude of solid torques (either being positive or negative) increases with accretion strength.
(ii) The effect of solid accretion on the total torque is significant for low-mass, $M_{\mathrm{p}}\lesssim 1\,M_{\oplus}$, planets. The solid torque magnitude can overcome that of gas for vigorously accreting low-mass planets, which can cause either total torque reversal or strengthened negative total torque. Planets with $M_{\mathrm{p}}1\,M_{\oplus}$ generally experience reduced negative total torque independent of solid accretion strength.
(iii) The steepness of radial profiles of gas, $p$, affects the total torque felt by the planet. Planets generally feel more negative total torque in case of steeper profiles due to the larger magnitude of the negative gas torque.
(iv) Accretion of well-coupled solids ($\mathrm{St}\leq 0.1$) can cause a very strong positive solid torque for a $M_{\mathrm{p}}<1\,M_{\oplus}$ planet independent of $p$. In contrast, accretion of larger solids ($\mathrm{St}\geq 0.1$) generally causes increased magnitude negative total torque. For an Earth-mass planet, accretion of $2\leq\mathrm{St}\leq 5$ solid material causes large positive solid torque, which results in positive total torque in $p=0.5$ discs. However, in $p\geq 1.0$ discs, the accretion of same sized solid species results in weaker but negative total torque for an Earth-mass planet. Solid torques are found to be insensitive to accretion strength for several Earth-mass, $M_{\mathrm{p}}\geq 3\,M_{\oplus}$, planets. As a result of positive solid torque, those planets feel weaker negative total torque compared to the analytical prediction of the canonical type-I approximation.
(v) Care must be taken in two-dimensional simulations as the effect of the smoothing of planetary potential can be significant on the gas and solid torques felt by the planet. We found that the weaker the smoothing (i.e., the smaller the value of $\varepsilon$) is, the stronger is the effect of solid on the planetary torque.
Let us consider a scenario, in which the accretion of solids is strong, i.e., $\eta=1$ and the protoplanetary disc has a shallow initial radial profile ($p=0.5$). At the beginning of planet formation, it is plausible to assume that the growing planet is in the low-mass regime (e.g. $M_{\mathrm{p}}\lesssim 0.1\,M_{\oplus}$), while the majority of solid material is well-coupled to the gas (i.e, $\mathrm{St}\simeq 0.01$) at the planet-forming region (see Fig.1). As we have shown, a solid accreting low-mass planet feels positive total torque (see the upper panel of Fig.3). As a result, a small mass planet migrates outward, which continues as long as $M_{\mathrm{p}}\simeq 0.3\,M_{\oplus}$ and solids are not grown above $\mathrm{St}\simeq 0.1$. Meanwhile, both the planet and solid material grow. Assuming that the planet has grown to Earth-mass and solids are in the pebble regime ($\mathrm{St}\simeq 1$), the migration is still directed outward. By the time the planet has grown above several Earth-mass and solids size reaches the boulder regime ($2\leq\mathrm{St}\leq 5$), its migration reverses. Emphasise that the rate of inward migration is under the prediction of the canonical type I regime duo to the positive solid torque. By assuming a steeper initial radial density profile for the disc, $p\geq 1.0$, the migration reversal can only occur below Earth-mass.
An important effect of the solid accretion is that low-mass protoplanets ($<1~{}M_{\oplus}$) migrate faster if the solid material is in the pebble-sized regime. Thus, the survival of a solid accreting migrating protoplanet might be uncertain in an evolved disc, in which the majority of solids are in the pebble-sized regime. In other words, plant formation might be compromised in evolved discs due to solid accretion. Finally, we emphasise that we used a canonical value of for the solid-to-gas mass ratio $epsilon=0.01$. Since the solid torque magnitudes are linearly scaled with $\epsilon$, low-mass planets outward or the fast inward migration is slower in solid depleted discs. In contrast, if the solid content of the disc is above the canonical value (by only several times, e.g. $\epsilon=0.05$) the migration speed increases significantly.
The above-mentioned hypothetical migration history, however, can be altered by the disc model applied being two-, or three-dimensional and the details of solid accretion phenomenon. In this study, the solid accretion was prescribed in a simplistic approach. By analysing the flow pattern around a low-mass planet, Ormel (2013) has shown that the planetary atmosphere is asymmetric and the accretion rate of well-coupled dust can be reduced. Thus, to better understand the effect of solid accretion on the torque felt by low-mass planets, it is worth reproducing our simulations by handling solid material as Lagrangian particles like in Morbidelli & Nesvorny (2012).
Due to the necessity of high numerical resolution, our investigation has been done in two-dimensional thin disc approximation. Although some of the three-dimensional effects are taken into account (e.g., by applying gravity softening), the gas and dust flow around the planet is a three-dimensional phenomenon (see, e.g., D’Angelo, Kley, & Henning, 2003; Bitsch & Kley, 2011; Lega et al., 2014; Fung, Artymowicz, & Wu, 2015). Moreover, as the appropriate magnitude of smoothing of the gravitational potential of the planet for solid species is unknown yet, it would be important to investigate the dynamics of solids and accretion in three-dimension.
For simplicity, we applied a simple disc thermodynamics, i.e. locally isothermal approximation. Since the effect of negative entropy gradients in adiabatic disc or the accretion heating may result in outward migration (see, e.g., Paardekooper & Mellema, 2006; Paardekooper et al., 2010; Masset & Casoli, 2010; Benítez-Llambay et al., 2015) it is worth study the effect of solid accretion in adiabatic discs too.
Diffusion of solid species was neglected in this study. Therefore, we used nearly inviscid models, for which case it is plausible to assume that diffusion is negligible. Since diffusion may smear out the asymmetric distribution of solids, it is worth investigating the effect of viscosity and solid diffusion on the total torque felt by solid accreting planets.
Finally, we note that the inclusion of the solid feed-back might also be important. However, its effect can be minor as the maximum of the solid-to-gas mass ratio measured in the our simulations is about $\epsilon\simeq 0.04$.
Acknowledgements
This project was supported by the Hungarian OTKA Grant No. 119993 and by OeAD-OMAA program through project 95öu13. I gratefully acknowledge the support of NVIDIA Corporation with the donation of the Tesla 2075 and K40 GPUs. We thank for the usage of MTA Cloud (https://cloud.mta.hu/) which significantly helped us achieving the results published in this paper. Discussion on the numerical solution of solid dynamics with E. Vorobyov is acknowledged.
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Appendix A Dust solver test
In order to test the numerical solver for the solid species (see Equations (8)-(10), the analytical solutions of velocity component of solids (see Equations (15)-(16)) are compared to the numerical solutions. In these simulations there is no embedded planet in the disc, i.e. the disc is unperturbed. Initially solid species have zero radial velocity and Keplerian azimuthal velocities.
Top and bottom panels on Figure 9 compare the radial and azimuthal velocity (Keplerian velocity is subtracted) components resulted after 50 orbits of the planet and the analytical solution, respectively. One can see that the numerical solution perfectly fits the analytical one independent of the solid’s Stokes number. |
QCD Equations of State and the QGP Liquid Model
Jean Letessier
Laboratoire de Physique Théorique et Hautes Energies
Université Paris 7, 2 place Jussieu, F–75251 Cedex 05.
Johann Rafelski
Department of Physics, University of Arizona, Tucson, Arizona, 85721, USA
(January 14, 2003)
Abstract
Recent advances in the study of equations of state of thermal lattice
Quantum Chromodynamics obtained at non-zero baryon density
allow validation of the
quark-gluon plasma (QGP) liquid model equations of state (EoS). We study here
the properties of the QGP-EoS near to the phase transformation
boundary at finite baryon density and show a close agreement with
the lattice results.
pacs: 12.38.Mh
††preprint:
The ab-initio exploration of Quantum Chromodynamics (QCD) on the lattice
at finite temperature and baryon density has seen
rapid recent progress Fod02 ; Fod02b ; All02 .
It is already known that for vanishing baryon density lattice results
can be described in terms of ideal quantum gases allowing only for lowest
order perturbative interactions, provided that a non-perturbative
temperature dependence of the coupling constant is introduced,
along with vacuum latent heat Ham00 . We will
refine this quark-gluon plasma model, and
demonstrate that it agrees extremely well with the finite
baryon density lattice results,
except in a region of temperature in vicinity of the phase
transition/transformation domain.
A strong motivation to have an understanding of the thermal QCD
lattice results in
terms of a set of more intuitive degrees of freedom originates in the need to
treat a fast evolving system
created in relativistic heavy ion collisions. At the onset of
the QCD thermal matter formation,
e.g., at $\tau_{0}\simeq 0.5$ fm, one cannot expect a full
chemical equilibrium to have been reached, i.e., the
phase space occupancy of quark and/or gluon gases has not
approached unity. In a model as we will present these
situations can be easily incorporated, allowing study of
initial temperature, phase space occupancy evolution,
and other physical conditions
in the realistic environment of
high energy relativistic heavy ion collisions.
More generally,
a handy model of QCD matter equations of state employing established
physical concepts allows to study many physical properties which are
only difficult to infer from the numerical lattice results. But perhaps
the most important question we pursue is if one can
already in the temperature domain explored by present
day experiments describe thermal QCD matter in terms of nearly
free quark and gluon degrees of freedom. If this is
confirmed, we would be able to verify if the physical system
formed in these reactions is the deconfined quark-gluon plasma.
A systematic perturbative expansion within the framework of
thermal field theory of the interacting quark-gluon gas
converges poorly Kaj02 . The situation is illustrated for
the case of a pure gauge SU(3) case in figure 1, where
we see that the ratio of the computed perturbative pressure $P$ to
the Stefan-Boltzmann pressure $P_{0}$
is oscillating around unity depending on the order $n$ of the expansion
in the coupling constant $g$ considered. Convergence seems to occur only
in the asymptotic freedom
limit of relatively high temperature $T$, beyond the experimental
reach. Better agreement with the
lattice results (solid dots) can be achieved (solid line)
at a relatively low $T$, here
expressing the temperature in units of the renormalization
scale $\Lambda_{\overline{\rm MS}}$, provided that the unknown
relative coefficient of the $g^{6}$ term is optimized to the value 0.7 .
We advocate a very different approach, combining
the perturbative result with key nonperturbative
features.
Our QGP liquid model arises from the empirical observation that the
lowest order thermal perturbation contribution evaluation of the QCD matter
properties, combined with a non-perturbative temperature dependent strong
coupling constant agrees with the QCD thermal lattice results, once
the other often used non-perturbative effect, e.g. bag constant, has been
incorporated.
The QGP-liquid partition
function is assumed to have the form,
$$\displaystyle\frac{T}{V}\ln{\cal Z}_{\mathrm{Q}GP}\equiv P_{\mathrm{Q}GP}=-{%
\cal B}+\frac{8}{45\pi^{2}}c_{1}(\pi T)^{4}$$
(1)
$$\displaystyle+\sum_{i=q,s}\frac{n_{i}}{15\pi^{2}}\left[\frac{7}{4}c_{2}(\pi T)%
^{4}+\frac{15}{2}c_{3}\left(\mu_{i}^{2}(\pi T)^{2}+\frac{1}{2}\mu_{i}^{4}%
\right)\right]$$
where $n_{q}=2$, $n_{s}=1$, $\mu_{s}=0$ and:
$$\displaystyle c_{1}$$
$$\displaystyle=$$
$$\displaystyle 1-\frac{15\alpha_{s}}{4\pi}\,,$$
$$\displaystyle c_{2}$$
$$\displaystyle=$$
$$\displaystyle 1-\frac{50\alpha_{s}}{21\pi}\,,\qquad c_{3}=1-\frac{2\alpha_{s}}%
{\pi}\,.$$
(2)
We recall that $\mu_{b}=3\mu_{q}$ and $\lambda_{q}=e^{\mu_{q}/T}$. A value
${\cal B}=(0.211\,\mbox{GeV})^{4}$ fits best the lattice results
we consider here.
The temperature dependence $\alpha_{s}(T)$ is
obtained from $\alpha_{s}(x)$ setting the energy scale $x$
to be:
$$x=2\pi\beta^{-1}\sqrt{1+\frac{1}{\pi^{2}}\ln^{2}\lambda_{\mathrm{q}}}=2\sqrt{(%
\pi T)^{2}+\mu_{\mathrm{q}}^{2}}\,.$$
(3)
$\alpha_{s}(x)$ is obtained integrating the renormalization group equation,
incorporating physical thresholds for heavy flavor. Use of semi-analytical
formulas with fixed active number of quarks introduces
an unacceptable error in the value of $\alpha_{s}$ and
has lead to false conclusions in some earlier work. We evaluate:
$$x\frac{\partial\alpha_{s}}{\partial x}=-b_{0}\alpha_{s}^{2}-b_{1}\alpha_{s}^{3%
}+\ldots\equiv\beta^{\mbox{\scriptsize pert}}_{2}\,.$$
(4)
$\beta^{\mbox{\scriptsize pert}}_{2}$ is
the beta-function of the renormalization group
in two loop approximation, and
$$b_{0}=\frac{11-2n_{\mathrm{f}}/3}{2\pi}\,,\quad b_{1}=\frac{51-19n_{\mathrm{f}%
}/3}{4\pi^{2}}\,.$$
$\beta^{\mbox{\scriptsize pert}}_{2}$
does not depend on the renormalization scheme,
and solutions of Eq. (4) differ from higher
order renormalization scheme
dependent results by less than the error introduced by the experimental
uncertainty in the measured value of $\alpha_{s}(\mu=M_{Z})=0.118+0.001-0.0016$,
used as the initial value in numerical integration of Eq. (4).
We note for $\mu_{\mathrm{q}}\to 0$ the empirical form:
$$\alpha_{s}(T/T_{c})\simeq{0.47\over{1+0.72\ln(T/T_{c})}}\,,\quad T_{c}=172\,%
\mbox{MeV}\,.$$
(5)
When we explore the effect of the finite
quark masses, we introduce the correction
$\delta_{m}\ln Z^{q}_{\rm QGP}$
to the partition function,
$$\displaystyle\frac{1}{T^{3}V}\delta_{m}\ln Z^{q}_{\rm QGP}$$
$$\displaystyle=$$
$$\displaystyle n_{i}{m_{i}^{3}\over T^{3}}{3\over\pi^{2}}$$
(6)
$$\displaystyle\times\int_{0}^{\infty}\!\!dxx^{2}\!\left[\ln\!\left(\!\frac{1+%
\lambda_{i}e^{-\sqrt{1+x^{2}}{m_{i}\over T}}}{1+\lambda_{i}e^{-x{m_{i}\over T}%
}}\!\right)\!+\!\left(\!\lambda_{i}\to{1\over\lambda_{i}}\!\right)\right],$$
where $i=q,s$. We use $m_{s}=170$ MeV. Note that we
did not allow for the QCD-thermal $\alpha_{s}$ correction
in $\delta_{m}\ln Z^{q}_{\rm QGP}$. When we compare withe lattice results, we use
the values $m_{q}=65$ MeV and $m_{s}=135$ MeV, as reported to have
been used in lattice simulations we compare with.
Strange quarks enter below only when we consider the absolute
pressure. Similarly, when we explore how an effective gluon mass
alters the pressure, we consider a correction
$$\frac{\delta_{m}\ln Z^{G}_{\rm QGP}}{T^{3}V}=-{m_{G}^{3}\over T^{3}}{8\over\pi%
^{2}}\int_{0}^{\infty}\!\!\!dxx^{2}\ln\!\left(\!\frac{1-e^{-\sqrt{1+x^{2}}{m_{%
G}\over T}}}{1-e^{-x{m_{G}\over T}}}\right)\!.$$
(7)
We need to introduce a finite thermal glue
mass $m_{G}\simeq 0.2$ GeV in order to reach for $T>T_{c}$ a line width
agreement with lattice results. This is not the thermal gluon
mass which, as we have discussed elsewhere, can be used as an alternative
way to express the perturbative QCD effect Raf02 .
In order to be able to consider our results in currently ongoing
experiments, it is important that
the QGP-liquid equations of state are verified at chemical
potential which is relatively small, up to $\mu_{b}\simeq 100$ MeV at initial
conditions. However, we extend this study to all values considered
in the lattice simulation Fod02b , $\mu_{b}=100,210,330,410$ and 530 MeV,
which aside of RHIC, also encompasses the physical reach of CERN-SPS experiments.
Our objective is to test if for $T>1.5T_{c}$ the model and lattice-‘experiment’ agree,
while near to the critical temperature domain significant modifications
due to the non-perturbative features of QCD must be expected.
Physical properties we consider will be rendered dimensionless by considering a
suitable ratio with either $T^{4}$ (pressure $P$) or $T^{3}$ (baryon density $n_{b}$).
To asses if the liquid model has the right density dependence
we consider in turn the pressure $P$ modification at finite
baryon density, $\Delta P\equiv P(T,\mu_{b})-P(T,\mu_{b}=0)$,
the baryon density $n_{b}$, the pressure $P(T,\mu_{b}=0)$ and
finally $\epsilon(T,\mu_{b})-3P(T,\mu_{b})$, which vanishes
for ideal gases. In the following figures all ‘experimental’
points are taken directly from the postscript
format of the figures of Ref.Fod02b , into which
we add our results.
The reader should note that for $T<T_{c}$
equations of state with hadrons must be employed. Hence we do not here
include results for this domain of lattice results.
The study of this confined phase domain involves also modeling of excluded volume
effect, and/or within the bootstrap model, the understanding of the
singularity of the hadron mass spectrum. Given the availability of
lattice results we hope to return to
discuss these intricate matters in near future.
The change on $\Delta P$ at finite baryochemical
potential is shown in figure 2. We note that the
agreement with the lattice results at large $T$ is very satisfactory.
This result depends on the empirical choice
made for the dependence on chemical potential
of the scale of $\alpha_{s}$, see Eq. (3).
Had we omitted $\mu_{q}$, or doubled the coefficient
of $\mu_{q}$, there would be a well visible deviation
in figure 2. Thus the empirical choice of the Matsubara
frequency as the combination of temperature and chemical potential
is qualitatively verified
by lattice results. For small chemical potential
$\mu_{b}=100$ MeV, we see that there is agreement for $T\geq 1.1T_{c}$, at a
baryochemical potential $\mu_{b}=530$ MeV, agreement with lattice data is
assured for $T\geq 1.5T_{c}$.
The practically invisible dashed lines (hidden mostly under solid lines)
show the here negligible effect of finite quark masses. By definition,
$\Delta P$ depends only on light quark degrees of freedom. The agreement
we see is thus not testing strange quark, or glue behavior.
A similarly remarkable agreement is obtained for the baryon density,
shown in figure 3. Here a finite quark mass used in
lattice simulations becomes visible (dashed lines). Once we allow for it, we see
agreement within the lattice data error for $T\geq 1.2T_{c}$
for all baryochemical potentials. For very small baryon density we can expect
agreement down to near the critical temperature.
Since both the pressure difference $\Delta P$ and baryon density $n_{B}$
show satisfactory
behavior as function of chemical potential, it is expected that
the change of energy density, and entropy, with chemical potential is
well described, both being derived of $\Delta P$ with respect
to $\beta=1/T$ and $T$ and suitable linear combination with $n_{B}$.
However, in order to fully certify the liquid model, we need now to
fine-tune the behavior of pressure $P$ at zero chemical potential.
In figure 4, we show the pressure
$P(T,\mu_{b}=0)/T^{4}$ as function of temperature $T/T_{c}$.
We find that there is some difference in the lattice result shown
in Fod02b and earlier work Kar00 , which practically agreed
with our solid line result in figure 4. To obtain the QGP
model agreement with the more recent lattice
data Fod02b we need to:
a) change the value of the
Bag constant, from our earlier value $(0.195\,\mbox{GeV})^{4}$ to
${\cal B}=(0.211\,\mbox{GeV})^{4}$, in response to somewhat higher
value of $T_{c}$, see solid line in figure 4,
b) introduce an effective gluon mass $m_{G}=0.2$ GeV
in order to reduce the number of effective degrees of freedom as is shown
by the dashed line in figure 4.
This procedure does indeed
produce the expected agreement within the line-width.
We do not have a lattice
entropy figure to compare with, but we are assured of a valid result
by our ability to reproduce the shape of the pressure functions, see
figures 2 and 4. A sensitive test of this
assertion is obtained considering $(\epsilon-3P)/T^{4}$ as function
of $T/T_{c}$ in figure 5. All finite chemical curves we
plot coincide and hence only one is visible in the diagram.
This agreement with lattice results for $T>1.15T_{c}$ confirms
that we have obtained a remarkably
precise representation of the behavior of equations of state
of deconfined QCD matter, except in direct
vicinity of the critical temperature.
We have shown here that QCD equations of state at nonzero baryon density
for $T>1.5T_{c}$, but even very near to $T=T_{c}$ for
small chemical potentials, behave in a way expected for the quark-gluon
plasma phase. The remarkable agreement has been made possible by
introduction of the first order thermal interactions which comprise
precise nonperturbative thermal coupling $\alpha_{s}(T,\mu_{q})$ and the vacuum
latent heat $\cal B$. In the glue sector, effective gluon mass
$m_{G}\simeq 200$ MeV is also required to obtain exactly the
expected number of degrees of freedom, required to model the
absolute magnitude of the pressure to within line-width.
It is important to realize that even without such detailed
refinements which may indeed change again as the lattice
results evolve, the approach we advocate does not suffer from the
convergence problem illustrated in figure 1 and
it provides a very good and natural description of the wealth of
the lattice results.
We conclude that thermal lattice QCD matter
behaves just like quark gluon plasma, and thus a
naive use of the QGP model is appropriate, provided that
suitable coupling strength $\alpha_{s}(T,\mu_{q})$
and vacuum bag constant $\cal B$ is introduced.
Moreover, considering that at RHIC the
following initial conditions have been reached Raf03 ,
$T>1.5T_{c}\simeq 260$ MeV for $\lambda_{q}=1.09$, corresponding to the
initial baryochemical potential
$\mu_{b}=3T\ln\lambda_{q}\simeq 45$ MeV,
the results presented confirm that the QGP state is
established in these reactions. Moreover, we are
reassured that we can explore in detail
the RHIC initial conditions, allowing for chemical composition
dynamics. This should lead us to understanding of QGP
properties and conditions in RHIC reactions.
Acknowledgments:
Work supported in part by a grant from the U.S. Department of
Energy, DE-FG03-95ER40937 . LPTHE, Univ. Paris 6 et 7 is:
Unité mixte de Recherche du CNRS, UMR7589.
References
(1)
Z. Fodor, S.D. Katz, and K.K. Szabo,
“The QCD equation of state at nonzero densities: Lattice result”,
e-Print Archive: hep-lat/0208078.
(2)
Z. Fodor, and S.D. Katz,
JHEP 0203:014, (2002).
(3)
C.R. Allton, S. Ejiri, S.J. Hands, O. Kaczmarek, F. Karsch,
E. Laermann, C. Schmidt, and L. Scorzato,
Phys. Rev. D 66,
074507 (2002).
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S. Hamieh, J. Letessier, and J. Rafelski,
Phys. Rev. C 62,
064901 (2000).
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K. Kajantie, M. Laine, K. Rummukainen, and Y. Schroder,
“The pressure of hot QCD up to $g^{6}ln(1/g)$”,
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proceedings of Quark Matter Conference
held 18-24 July 2002, in Nantes, France. |
Optimal Codes Correcting Localized Deletions
Rawad Bitar1,
Serge Kas Hanna1,
Nikita Polyanskii12,
and Ilya Vorobyev2
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 801434) and from the Technical University of Munich - Institute for Advanced Studies, funded by the German Excellence Initiative
and European Union Seventh Framework Programme under Grant Agreement
No. 291763. N. Polyanskii’s work was supported by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) under Grant No. WA3907/1-1. Ilya Vorobyev was supported in part by the Russian Foundation for Basic Research through grant no. 20-01-00559.
1Technical University of Munich,
Munich, Germany,
{rawad.bitar, serge.k.hanna, nikita.polianskii}@tum.de
2Skolkovo Insitute of Science and Technology,
Moscow, Russia,
vorobyev.i.v@yandex.ru
Abstract
We consider the problem of constructing codes that can correct deletions that are localized within a certain part of the codeword that is unknown a priori. Namely, the model that we study is when at most $k$ deletions occur in a window of size $k$, where the positions of the deletions within this window are not necessarily consecutive. Localized deletions are thus a generalization of burst deletions that occur in consecutive positions. We present novel explicit codes that are efficiently encodable and decodable and can correct up to $k$ localized deletions. Furthermore, these codes have $\log n+\mathcal{O}(k\log^{2}(k\log n))$ redundancy, where $n$ is the length of the information message, which is asymptotically optimal in $n$ for $k=o(\log n/(\log\log n)^{2})$.
I Introduction
Localized deletions correspond to a specific class of errors in which symbols are deleted in certain parts of a string that are unknown a priori. In this setting, the deletions are localized in a certain window in the string but these deletions do not necessarily occur in consecutive positions. Therefore, localized deletions are a generalization of burst deletions which were studied in [1, 2, 3, 4, 5]. Localized and burst deletions are experienced in various applications such as DNA-based storage and file synchronization.
The study of constructing codes that can correct deletions goes back to the 1960s. In 1966, Levenshtein [6] showed that any code that can correct $k$ deletions can also correct $k$ insertions, and vice-versa. Moreover, he showed that such codes can also correct any combination of at most $k$ insertions and deletions. Levenshtein also derived fundamental limits which show that the optimal number of redundant bits needed to correct $k$ deletions that are arbitrarily located in a binary string of length $N$ is $\Theta(k\log(N/k))$ [6]. In addition, he showed that the codes constructed by Varshamov and Tenengolts (VT codes) [7] are capable of correcting a single deletion and have asymptotically optimal redundancy111We say that the redundancy of a code construction is asymptotically optimal if the ratio between the redundancy of the construction and some lower bound, e.g., sphere-packing bound, on the redundancy approaches $1$ as the code length goes to infinity.. Several previous works studied the classical problem of constructing binary codes that correct $k>1$ deletions that are arbitrarily located in a string [8, 9, 10, 11, 12, 13, 14, 15, 16]. For the classical setting, the state-of-the-art results in [14, 15, 16] give codes with $\mathcal{O}(k\log N)$ redundancy. The results in [15] and [16] apply for constant $k$, while the result in [14] applies for $k\leq N^{1-\alpha}$ with $0<\alpha<1$. Furthermore, the code presented in [16] is systematic, whereas the codes in [14] and [15] are non-systematic.
The document exchange problem is a related problem that can also provide useful insights for correcting deletions and insertions. In the document exchange setting, a sender (Alice) holds a string $x$ and a receiver (Bob) holds a string $y$, where the edit distance between $x$ and $y$ is assumed to be bounded by $k$. The goal is to design an efficient document exchange protocol by which Alice can send Bob a small sketch based on its string $x$, which allows Bob to recover $x$. The state-of-the-art results on document exchange in [13] and [14] give deterministic protocols with sketch size $\mathcal{O}(k\log^{2}(N/k))$ for any $k<N$.
A separate line of work has focused on the problem of correcting deletions that occur in a single burst, i.e., bits are deleted in consecutive positions. For this setting, Levenshtein showed that at least $\log N+k-1$ redundant bits are required to correct $k$ deletions that occur in consecutive positions [1]. In [1], Levenshtein constructed asymptotically optimal codes that can correct a burst of at most two deletions. Cheng et al. [3] provided three constructions of codes that can correct a burst of exactly $k>2$ deletions. The lowest redundancy achieved by the codes in [3] is $k\log(N/k+1)$. The fact that the number of deletions in the burst is exactly $k$, as opposed to at most $k$, is a crucial factor in the code constructions of [3]. Schoeny et al. [4] proved the existence of codes that can correct a burst of exactly $k$ deletions and have at most $\log N+(k-1)\log\log N+k-1$ redundancy, for sufficiently large $N$. The authors of [4] also constructed codes that can correct a single burst of at most $k$ deletions. The redundancy for the latter case is at most $(k-1)\log N+\big{(}\binom{k}{2}-1\big{)}\log\log N+\binom{k}{2}+\log\log k$ which improves on a previous result by Bours in [2]. The construction from [4] has been further improved to a redundancy of $\lceil\log k\rceil\log n+\left(k(k+1)/2-1\right)\log\log n+c_{k}^{\prime\prime}$ in [17]. Note that in the aforementioned results [3, 4, 2, 17], the size of the burst $k$ is assumed to be a constant, i.e., $k$ does not grow with $N$. Recently, the authors of [5] constructed optimal codes for the case of correcting a single burst of at most $k=o((\log N)^{1/3})$ deletions. The redundancy of these codes is $\log N+\binom{k+1}{2}\log\log N+c_{k}$ for some constant $c_{k}$ that only depends on $k$.
In this paper, we study a more general model, in which at most $k$ deletions are localized in a single window of size $k$, where the position of this window is unknown a priori. Codes for correcting localized errors have been previously studied for other types of errors such as substitutions [18, 19], where the term “burst-error-correcting code" is commonly used. In our work, we consider a model where deletions are restricted to a window of size $k$, but do not necessarily occur at consecutive positions within this window. This model for deletions was first studied in [4] under the name of “bursts of non-consecutive deletions". The authors of [4] proved the existence of codes for the particular cases where $k=3$ and $k=4$, with redundancy at most $4\log N+2\log\log N+6$ for $k=3$, and at most $7\log N+2\log\log N+4$ for $k=4$. Localized deletions were also studied in [20] for $k=o(n)$, where $n$ is the length of the information message. The authors in [20] presented explicit codes that have a non-zero probability of failure which vanishes asymptotically in $n$ for a uniform i.i.d. message. These codes have redundancy at most $4\log n+1$ for $k=o(\log n)$, and at most $4k+1$ for $\Omega(\log n)\leq k\leq o(n)$; furthermore, the encoding/decoding complexity of these codes is $\mathcal{O}(n^{2})$.
The main contribution of this paper is constructing an efficiently encodable and decodable zero-error code that can correct up to $k=\mathcal{O}(n/\log^{2}n)$ localized deletions with $N-n=\log n+\mathcal{O}(k\log^{2}(k\log n))$ redundancy. It follows from the converse bounds in [1, 4] that the redundancy of our code is asymptotically optimal in $n$ for $k=o(\log n/(\log\log n)^{2})$. Furthermore, the encoding and decoding complexity is $n\cdot{\rm poly}(k\log n)$. A comparison to the state-of-the-art results on codes for consecutive deletions and classical $k$-deletion correcting codes is given in Section III.
The rest of the paper is organized as follows. In Section II we introduce important definitions and notations that are used throughout the paper. We present our code construction and explain all the components of our code in Section III. Section IV concludes the paper and lists some open problems for future research.
II Preliminaries
For simplicity of presentation, hereafter one-based numbering is used and log $n$ stands for the base-two logarithm of $n$. A vector is denoted by bold lowercase letters, such as $\boldsymbol{x}$,
and the $i$th entry of the vector $\boldsymbol{x}$ is referred to as $x_{i}$. The length of a vector $\boldsymbol{x}$ is denoted by $|\boldsymbol{x}|$. The set of integers from $i$ to $j-1$, $1\leq i<j$, is abbreviated by $[i,j)$ or $[i,j-1]$.
Given a set of indices $I$ and a vector $\boldsymbol{x}$, we define the vector $\boldsymbol{x}_{I}$ of length $|I|$ to be the restriction of $\boldsymbol{x}$ to coordinates from $I$. For a string $\boldsymbol{x}\in\{0,1\}^{n}$, by $B_{k}(\boldsymbol{x})$ denote the set of all possible strings that can be obtained from $\boldsymbol{x}$ by deleting bits indexed by $I=\{i_{1},\ldots,i_{k^{\prime}}\}$ with $1\leq k^{\prime}\leq k$ such that the difference between the maximal element and the minimal element of $I$ is at most $k-1$. Further, we call $B_{k}(\boldsymbol{x})$ the $k$-ball centered in $\boldsymbol{x}$. For an integer $\ell$, we define the vector $\mathbf{1}^{\ell}$ to be the all one vector of length $\ell$. We define $\mathbf{0}^{\ell}$ similarly. We call the vector $\mathbf{1}^{r}$ as a run of $1$’s of length $r$. We say a vector $\boldsymbol{x}\in\{0,1\}^{n}$ contains a run of $1$’s of length $r$ if there exists $i\in[1,n-r+1]$ such that $(x_{i},x_{i+1},\dots,x_{r+i-1})=\mathbf{1}^{r}$.
Definition 1.
We say that a code $\mathcal{C}\subseteq\{0,1\}^{n}$ corrects deletions localized in a window of length $k$ if for any two distinct codewords $\boldsymbol{x}_{1},\boldsymbol{x}_{2}\in\mathcal{C}$, the corresponding $k$-balls have the empty intersection, i.e., $B_{k}(\boldsymbol{x}_{1})\cap B_{k}(\boldsymbol{x}_{2})=\emptyset$.
Given a string $\boldsymbol{a}\in\mathbb{Z}^{n}$, we define the Varshamov-Tenengolts (VT) check sum as follows
$$\mathsf{VT}(\boldsymbol{a}):=\sum_{i=1}^{n}ia_{i}.$$
Definition 2.
Let $\mathcal{P}\subset\{0,1\}^{m}$ be a family of strings of length $m>0$. For a positive integer $\Delta$, $m<\Delta\leq n$, we say that a binary string $\boldsymbol{x}$ of length $n$ is $(\mathcal{P},\Delta)$-dense, if each interval of length $\Delta$ in $\boldsymbol{x}$ contains at least one string from $\mathcal{P}$, i.e., for each $i\in[1,n-\Delta+1]$ there exist $j\in[i,i+\Delta-m]$ and $\boldsymbol{p}\in\mathcal{P}$ such that $\boldsymbol{p}=\boldsymbol{x}_{[j,j+m)}$.
Definition 3.
Let $\boldsymbol{x}$ and $\mathcal{P}$ be a binary string of length $n$ and a family of strings of length $m$, respectively. Then, we define the indicator vector $\mathbbm{1}_{\mathcal{P}}(\boldsymbol{x})$ of the family $\mathcal{P}$ in $\boldsymbol{x}$ to be a vector of length $n$ with entries
$$\mathbbm{1}_{\mathcal{P}}(\boldsymbol{x})_{i}:=\begin{cases}1,\quad\text{if }\boldsymbol{x}_{[i,i+m)}=\boldsymbol{p}\text{ for some }\boldsymbol{p}\in\mathcal{P}\text{ and }i\leq n-m+1,\\
0,\quad\text{otherwise}\end{cases}$$
Further let $n_{\mathcal{P}}(\boldsymbol{x})$ be the number of ones in $\mathbbm{1}_{\mathcal{P}}(\boldsymbol{x})$. We define $\boldsymbol{a}_{\mathcal{P}}(\boldsymbol{x})$ to be a vector of length $n_{\mathcal{P}}(\boldsymbol{x})+1$ whose $i$-th entry is the distance between positions of the $i$-th and $(i+1)$-st $1$ in the string $(1,\mathbbm{1}_{\mathcal{P}}(\boldsymbol{x}),1)$.
Definition 4.
We consider $\mathcal{P}\subset\{0,1\}^{m}$ with $m=k+\left\lceil\log k\right\rceil+4$ to be the family of strings that end with a run of $1$’s of length $\left\lceil\log k\right\rceil+4$ and do not have any run of $1$’s of length $\left\lceil\log k\right\rceil+4$ in its first $k$ entries. We refer to the string $\mathbf{1}^{\left\lceil\log k\right\rceil+4}$ as the marker and to any string $\boldsymbol{p}\in\mathcal{P}$ as the pattern.
Property 1 (Marker-rich strings).
A string $\boldsymbol{x}$ is said to be marker rich if it contains at least one marker $\mathbf{1}^{\left\lceil\log k\right\rceil+4}$ per window of length
$$B:=(\left\lceil\log k\right\rceil+4)2^{\left\lceil\log k\right\rceil+8}\lceil\log n\rceil.$$
Property 2 (String with distanced markers).
A string $\boldsymbol{x}$ is said to have distanced markers if every substring of $\boldsymbol{x}$ of length
$$R:=(k+\left\lceil\log k\right\rceil+4)(\left\lceil\log n\right\rceil+\left\lceil\log k\right\rceil+8)$$
contains at least one substring of length $m=k+\left\lceil\log k\right\rceil+4$ that does not contain any copy of the marker $\mathbf{1}^{\left\lceil\log k\right\rceil+4}$.
In Proposition 1, we shall show that any string $\boldsymbol{x}$ satisfying Property 1 and Property 2 is a $(\mathcal{P},\Delta)$-dense string with $\Delta:=R+B-m$.
III Code construction
Our main result is summarized in the following statement.
Theorem 1.
For $k=\mathcal{O}(n/\log^{2}n)$, there exists an efficiently encodable and decodable code that maps an arbitrary string of length $n$ into a string of length $N$ and is capable of correcting deletions localized in a window of length $k$. The encoding and decoding
complexity of the proposed code is
$n\cdot{\rm poly}(k\log n)$. Furthermore, for $n\to\infty$ the redundancy of this code is
$$N-n=\log n+\mathcal{O}(k\log^{2}(k\log n)).$$
We have several remarks and comments illustrating the contribution of our paper.
•
From the converse bounds [1, 4], it follows that the redundancy of such a code, written as $N-n$, has to be at least $\log N+k-1$. This implies that our construction is asymptotically optimal when $k=o(\log n/(\log\log n)^{2})$, e.g., it holds if $k$ is an absolute constant or $k=\sqrt{\log n}$.
•
For the case of consecutive deletions, our construction is asymptotically optimal as the one from [5], but better in terms of complexity and limitations of $k$, namely: $k=o((\log n)^{1/3})$ in [5] and $k=o(\log n/(\log\log n)^{2})$ in the proposed code.
•
We observe that a $k$-deletion correcting code can be naively used for correcting deletions localized in a window of length $k$. For the regime of sub-polynomial $k=2^{o(\sqrt{\log n})}$, the best known efficient $k$-deletion correcting code [13, 16] has redundancy $\mathcal{O}(k\log n)$ and a ${\rm poly}(n)$ time encoding and decoding algorithms, which are worse than the redundancy and complexities of the suggested construction.
Our construction consists of four components. Specifically, we show how to encode a string $\boldsymbol{u}\in\{0,1\}^{n}$ into a string $\widetilde{\boldsymbol{x}}\in\{0,1\}^{N}$ with $N=n+\log n(1+o(1))+\mathcal{O}(k\log^{2}(k\log n))$, where
$$\widetilde{\boldsymbol{x}}:=(\boldsymbol{x},H_{\text{sep}},H_{\text{loc}}(\boldsymbol{x}),H_{\text{cor}}(\boldsymbol{x})),\quad\boldsymbol{x}:=E(\boldsymbol{u}).$$
Suppose that $\widetilde{\boldsymbol{y}}$ is the channel output if $\widetilde{\boldsymbol{x}}$ is transmitted over the channel, i.e., $\widetilde{\boldsymbol{y}}\in B_{k}(\widetilde{\boldsymbol{x}})$.
Let $k^{\prime}:=|\widetilde{\boldsymbol{x}}|-|\widetilde{\boldsymbol{y}}|$ be the actual number of deletions. Based on $\widetilde{\boldsymbol{y}}$, we will show that it is possible to reconstruct $\boldsymbol{u}$.
The injective map $E(\boldsymbol{u}):\{0,1\}^{n}\to\{0,1\}^{n+k+3\left\lceil\log k\right\rceil+12}$ that encodes a string $\boldsymbol{u}$ into a $(\mathcal{P},\Delta)$ string $\boldsymbol{x}=E(\boldsymbol{u})$ will be discussed in Section III-A.
The vector $H_{\text{sep}}\in\{0,1\}^{k+1}$ makes it possible to deduce whether at least one deletion occurs in the part $\boldsymbol{x}$.
If at least one deletion occurs in the part $\boldsymbol{x}$, then the part $(H_{\text{loc}}(\boldsymbol{x}),H_{\text{cor}}(\boldsymbol{x}))$ is error-free. More details on $H_{\text{sep}}$ will be given in Section III-B. The hash function $H_{\text{loc}}(\boldsymbol{x})$ that requires $\log n+\mathcal{O}(1)$ bits and allows us to locate deletions in the erroneous part $\boldsymbol{x}$ up to an interval of length $\mathcal{O}(\Delta^{2})$ will be presented in Section III-C.
Finally, in Section III-D, we describe the function $H_{\text{cor}}(\boldsymbol{x})$ correcting deletions occurred in $\boldsymbol{x}$. This part requires a small amount of redundancy $\mathcal{O}(k\log^{2}(k\log n))$ since the location of deletions is known up to a small interval. Note that based on $\boldsymbol{x}=E(\boldsymbol{u})$, we can extract the information string $\boldsymbol{u}$.
III-A Encoding into $(\mathcal{P},\Delta)$-dense strings
We explain the injective map $E(\boldsymbol{u}):\{0,1\}^{n}\to\{0,1\}^{n+k+3\left\lceil\log k\right\rceil+12}$ that encodes $\mathbf{u}\in\{0,1\}^{n}$ into a $(\mathcal{P},\Delta)$-dense string $\boldsymbol{x}$. The encoding follows the same procedures from [15] and is replicated here for completeness. Let $\mathcal{P}\subset\{0,1\}^{m}$ to be the family of strings as defined in Definition 4. Before we explain the construction we prove the following proposition.
Proposition 1.
A string $\boldsymbol{x}\in\{0,1\}^{n}$ that satisfies Property 1 and Property 2 is $(\mathcal{P},\Delta=R+B-m)$-dense.
Proof.
Denote by $t_{1},<t_{2}\dots<t_{J}$ the locations where $\mathbbm{1}_{\mathcal{P}}(\boldsymbol{x})$ is equal to $1$. Define $t_{0}=0$ and $t_{J+1}=n+1$. We need to show that for all $i\in[0,J]$, the following holds
$$t_{i+1}-t_{i}\leq B+R-m+1=\Delta+1.$$
Since $\boldsymbol{x}$ satisfies Property 2, then there exists an index $j^{\star}\in[t_{i},t_{i}+R-k-\left\lceil\log k\right\rceil-3]$ such that for every $\ell\in[j^{\star},j^{\star}+k-1]$ we have $(x_{\ell},\dots,x_{\ell+\left\lceil\log k\right\rceil+3})\neq\mathbf{1}^{\left\lceil\log k\right\rceil+4}$. Following Property 1, there exists an integer $a\in[j^{\star}+1,j^{\star}+B]$ such that $(x_{a},\dots,x_{a+\left\lceil\log k\right\rceil+3})=\mathbf{1}^{\left\lceil\log k\right\rceil+4}$. Let $w=\min\{a\geq j^{\star}:(x_{a},\dots,x_{a+\left\lceil\log k\right\rceil+3})=\mathbf{1}^{\left\lceil\log k\right\rceil+4}\}$. We now have that $w\neq j^{\star}$ and $w\leq a\leq j^{\star}+B$. Therefore, $w\in[j^{\star}+1,j^{\star}+B]$. In addition, for every $\ell\in[j^{\star},w-1]$ we have $(x_{\ell},\dots,x_{\ell+\left\lceil\log k\right\rceil+3})\neq\mathbf{1}^{\left\lceil\log k\right\rceil+4}$. By definition of $j^{\star}$ and $w$, we have that $w-j^{\star}\geq k$. Since $(x_{w},\dots,x_{w+\left\lceil\log k\right\rceil+3})=\mathbf{1}^{\left\lceil\log k\right\rceil+4}$, we know that $\mathbbm{1}_{\mathcal{P}}(\boldsymbol{x})_{w}=1$. We can now write
$$\displaystyle t_{i+1}-t_{i}$$
$$\displaystyle\leq w-t_{i}$$
$$\displaystyle\leq j^{\star}+B-t_{i}$$
$$\displaystyle\leq R+B-m+1$$
$$\displaystyle=\Delta+1.$$
∎
We are now ready to explain the construction. We split the explanation into three parts. In the first part, we give an encoding that ensures that the string $\boldsymbol{x}$ is marker-rich, i.e., satisfies Property 1. In the second part we give an intermediate encoding that would be needed to distance the markers. In the third part we give an encoder that ensures that the markers in the string $\boldsymbol{x}$ are distanced so that the string satisfies Property 2. We show at the end that the resulting string $\boldsymbol{x}$ satisfies both properties and is thus a $(\mathcal{P},\Delta=R+B-m)$-dense string.
Marker enriching
This stage consists of identifying windows of size $B$ in the input string $\boldsymbol{u}$ that do not contain a marker. Every such window of bits (substring) is deleted from $\boldsymbol{u}$. Then, a compressed version of the deleted substring is appended to the end of the input string together with a marker. We first show that all substrings of length $B$ that do not contain a marker can be compressed.
Observation 1.
Let $\mathcal{S}$ be the set of strings of length $B$ that do not contain a run of $1$’s of length greater than or equal to $\left\lceil\log k\right\rceil+4$. Then, every string $\mathbf{s}\in\mathcal{S}$ can be compressed to a string $\widetilde{\mathbf{s}}:=\phi(\mathbf{s})\in\{0,1\}^{B-\left\lceil\log n\right\rceil-2\left\lceil\log k\right\rceil-10}$. Here $\phi$ is an invertible map such that $\phi$ and $\phi^{-1}$ can be computed in $O(B^{2})$ time.
The compression works as follows. Divide every string into $2^{\lceil\log k\rceil+8}\lceil\log n\rceil$ strings each of length $\lceil\log k\rceil+4$. Since every substring cannot be the all one vector, then it can be represented using a symbol from an alphabet of size $2^{\left\lceil\log k\right\rceil+4}-1$. Therefore, the string $\mathbf{s}$ can be represented by a string $\mathbf{v}$ consisting of $2^{\lceil\log k\rceil+8}\lceil\log n\rceil$ symbols each from an alphabet of size $2^{\left\lceil\log k\right\rceil+4}-1$. The number of bits $n_{v}$ required to represent $\mathbf{v}$ using a binary sequence is given by
$$\displaystyle n_{v}$$
$$\displaystyle=\left\lceil\log\left(2^{\left\lceil\log k\right\rceil+4}-1\right)^{2^{\left\lceil\log k\right\rceil+8}\left\lceil\log n\right\rceil}\right\rceil$$
$$\displaystyle=\left\lceil\log\left(1-2^{-\left\lceil\log k\right\rceil-4}\right)^{2^{\left\lceil\log k\right\rceil+8}\left\lceil\log n\right\rceil}\right\rceil+\left((\left\lceil\log k\right\rceil+4)2^{\left\lceil\log k\right\rceil+8}\left\lceil\log n\right\rceil\right)$$
$$\displaystyle\stackrel{{\scriptstyle(a)}}{{\leq}}16\left\lceil\log n\right\rceil\log\left(\frac{1}{e}\right)+1+B$$
$$\displaystyle\leq B-23\left\lceil\log n\right\rceil+1$$
$$\displaystyle\stackrel{{\scriptstyle(b)}}{{\leq}}B-\left\lceil\log n\right\rceil-2\left\lceil\log k\right\rceil-10.$$
Here, (a) follows from the fact that for $x>1$, the function $\left(1-\frac{1}{x}\right)^{x}$ is increasing in $x$ and $\lim_{x\to\infty}\left(1-\frac{1}{x}\right)^{x}=\frac{1}{e}$. And (b) holds because for large values of $n$ the following holds
$$23\left\lceil\log n\right\rceil\geq\left\lceil\log n\right\rceil+2\left\lceil\log k\right\rceil+11.$$
Encoding: We now present the details of the marker enriching process $T_{e}:\{0,1\}^{n}\to\{0,1\}^{n+2\left\lceil\log k\right\rceil+8}$. For a string $\boldsymbol{u}\in\{0,1\}^{n}$ let $T_{e}(\boldsymbol{u})$ be the result of the encoding applied on $\boldsymbol{u}$ to make it rich with markers. The encoding works as follows. First we set $T_{e}(\boldsymbol{u})=\boldsymbol{u}$ and use a dummy variable $n^{\prime}=n$. Then we append two markers, i.e., $\mathbf{1}^{2\left\lceil\log k\right\rceil+8}$, at the end of $T_{e}(\boldsymbol{u})$. We now check the string $T_{e}(\boldsymbol{u})$ for substrings of length $B$ that do not contain a marker. In other words, we look for an integer $i\in[1,n^{\prime}]$ such that for every $j\in[i,i+B-\left\lceil\log k\right\rceil-4]$ it holds that
$$(T_{e}(\boldsymbol{u})_{j},T_{e}(\boldsymbol{u})_{j+1},\dots,T_{e}(\boldsymbol{u})_{j+\left\lceil\log k\right\rceil+3})\neq\mathbf{1}^{\left\lceil\log k\right\rceil+4}.$$
If we find such an integer $i$ such that $i\leq n^{\prime}-B+1$, then we delete the substring $(T_{e}(\boldsymbol{u})_{i},\dots,T_{e}(\boldsymbol{u})_{i+B-1})$ from $T_{e}(\boldsymbol{u})$ and append $(i,\phi((T_{e}(\boldsymbol{u})_{i},\dots,T_{e}(\boldsymbol{u})_{i+B-1}),0,\mathbf{1}^{2\left\lceil\log k\right\rceil+8},0)$ to the end of $T_{e}(\boldsymbol{u})$. The appended index $i$ is encoded using a $\left\lceil\log n\right\rceil$ binary string. We set $n^{\prime}=n^{\prime}-B$.
If we find such an integer $i$ such that $i>n^{\prime}-B+1$, then we delete the substring $(T_{e}(\boldsymbol{u})_{i},\dots,T_{e}(\boldsymbol{u})_{n^{\prime}})$ from $T_{e}(\boldsymbol{u})$ and append $(i,\phi((T_{e}(\boldsymbol{u})_{i},\dots,T_{e}(\boldsymbol{u})_{n^{\prime}},\mathbf{0}^{i+B-n^{\prime}-1}),0,\mathbf{1}^{2\left\lceil\log k\right\rceil+8-(i+B-n^{\prime}-1)},0)$ to the end of $T_{e}(\boldsymbol{u})$. The appended index $i$ is encoded using a $\left\lceil\log n\right\rceil$ binary string. We set $n^{\prime}=i-1$. In this case the deleted substring has length $n^{\prime}-i+1$ so that we do not delete any of the appended bits. The quantity $2{\left\lceil\log k\right\rceil+8}-(i+B-n^{\prime}-1)$ is always positive due to appending a run of $1$’s of length $2{\left\lceil\log k\right\rceil}+8$ in the initialization step. This guarantees that $i+B-n^{\prime}-1\leq\left\lceil\log k\right\rceil+4$, otherwise the considered substring contains $(T_{e}(\boldsymbol{u})_{n^{\prime}+1},\dots,T_{e}(\boldsymbol{u})_{n^{\prime}+\left\lceil\log k\right\rceil+4})$ which is a run of $1$’s of length $\left\lceil\log k\right\rceil+4$ and therefore is not deleted by the algorithm.
We keep looking for values of $i$ that satisfy the aforementioned constraints and applying the explained transformations until deleting all substrings of $\boldsymbol{u}$ that do not contain a marker.
Redundancy: Appending the run of $1$’s adds $2\left\lceil\log k\right\rceil+8$ redundant bits. All other operations do not increase the length of the string. Therefore, the overall redundancy is at most $2\left\lceil\log k\right\rceil+8$ bits. To see this, recall from Observation 1 that for any string $\boldsymbol{s}\in\mathcal{S}$, $\widetilde{\boldsymbol{s}}=\phi(\boldsymbol{s})$ is expressed using $B-\left\lceil\log n\right\rceil-2\left\lceil\log k\right\rceil-10$ bits. The two appended markers and the two $0$’s take $2\left\lceil\log k\right\rceil+10$ bits. The index $i$ is expressed using $\left\lceil\log n\right\rceil$ bits. When deleting a sequence that starts at position $i\leq n^{\prime}-B+1$, then the appended sequence has a length exactly $B$. When deleting a sequence that starts at position $i>n^{\prime}-B+1$, then the appended sequence has a length $B-(i+B-n^{\prime}-1)\leq B$.
Correctness: We show that the encoding ensures that any string of length $B$ in $T_{e}(\boldsymbol{u})$ contains the marker $\mathbf{1}^{\left\lceil\log k\right\rceil+4}$. First we note that $n^{\prime}$ is always the split index between the original string and the appended parts. Thus, no bits from the added strings are deleted throughout the process. Having $n^{\prime}$ be the split index is guaranteed by appending the run of $1$’s in the beginning and only deleting bits with index less than $n^{\prime}$. In addition, $n^{\prime}$ is decreased every time a string is deleted. Since $n^{\prime}$ is always the split index, then all strings before $n^{\prime}$ satisfy the desired property, otherwise the encoding keeps going. In the appended sequences, if $i\leq n^{\prime}-B+1$ a marker of desired length is added. If $i>n^{\prime}-B+1$, then $2\left\lceil\log k\right\rceil+8-(i+B-n^{\prime}-1)\geq\left\lceil\log k\right\rceil+4$ and a marker of the desired length is added. Moreover, the start index of any two markers in the added strings is at most $B-\left\lceil\log k\right\rceil-4$ bits apart. Therefore, the string $T_{e}(\boldsymbol{u})$ satisfy the desired properties.
Decoding strategy: The decoding follows a reverse procedure of the encoder. We show that when decoding $T_{e}(\boldsymbol{u})$ the output of the decoder is unique and corresponds to the input string $\boldsymbol{u}$. The decoder looks at the last bit of the input string. All appended strings in the encoding end with a $0$. Therefore, if the last bit of $T_{e}(\boldsymbol{u})$ is a $0$, there is an appended string. The decoder counts the length of the run of $1$’s before that $0$ to determine the length of the originally deleted string. The length of the deleted string is equal to $B$ if the run is of length $2\left\lceil\log k\right\rceil+8$. Otherwise, the length of the deleted string can be determined using $2\left\lceil\log k\right\rceil+8$ minus the length of the run of $1$’s. The decoder deletes the $0$ and the run of $1$’s. Now the decoder can invert the compressed version of the deleted string and insert the original string at position $i$ read from the header of the appended sequence. The decoder deletes the remaining part of this appended string. The decoder repeats this procedure until the last bit of the modified string is $1$. At that point the decoder has reached the run of $1$’s of length $2\left\lceil\log k\right\rceil+8$ appended in the first step of the encoding. The decoder deletes this run and outputs $\boldsymbol{u}$. The output is guaranteed to be unique because all appended sequences are encoded using the bijective function $\phi$. The recovered indices of the deleted sequences is computed given the current value of $n$ and is guaranteed to be correct. The decoder ends if and only if the last bit of the string is $1$ which is designed by the encoder to be the correct stopping criteria.
Complexity: We claim that the total complexity of the encoding process can be reduced to $\mathcal{O}(nB)$. Let a string $\boldsymbol{s}$ be equal to $(\boldsymbol{u},\mathbf{1}^{2\left\lceil\log k\right\rceil+8})$. We write the encoded sequence into strings $\boldsymbol{t}_{1}$ and $\boldsymbol{t}_{2}$. The string $\boldsymbol{t}_{1}$ will store the string $\boldsymbol{s}$ without the deleted substrings. The string $\boldsymbol{t}_{2}$ will store compressed versions of the deleted substrings. The resulting string $\boldsymbol{x}$ is equal to the concatenation of $\boldsymbol{t}_{1}$ and $\boldsymbol{t}_{2}$.
Scan the string $\boldsymbol{s}$ from left to right, copy its symbols into $\boldsymbol{t}_{1}$ and count the number of consecutive ones in the string $\boldsymbol{t}_{1}$. If we have at least $\left\lceil\log k\right\rceil+4$ consecutive ones in some position, it means that we have found a marker. Save this position to a special variable. Update this position every time a new marker is found.
If in some position the distance to the end of the last marker is bigger than $B$, then we have found a substring in $\boldsymbol{s}$ of length $B$ without a marker. Cut the last $B$ symbols from the string $\boldsymbol{t}_{1}$, compress them using the map $\phi$ and append the result to the string $\boldsymbol{t}_{2}$. Continue scanning of the string $s$.
It is easy to see that the described algorithm implements the encoding process. The complexity of this procedure is $\mathcal{O}(n)$ since each symbol can be scanned, inserted in $\boldsymbol{t}_{1}$, used in compression and appended to $\boldsymbol{t}_{2}$ at most one time.
To decode the sequence we can find $n^{\prime}$ and the partition of $\boldsymbol{x}$ into $\boldsymbol{t}_{1}$ and $\boldsymbol{t}_{2}$. Then we write the string $\boldsymbol{s}$ from right to left by scanning the string $t_{1}$ from right to left and inserting decompressed parts from $\boldsymbol{t}_{2}$ in the proper moments. The decoding also has a linear complexity.
Intermediate encoding
We describe an intermediate step that takes a string of length $k+\left\lceil\log k\right\rceil+4$ that contains some copies of the markers and deletes all the marker. The presented encoding must be invertible. Thus, we present next the encoding and decoding procedure.
Encoding: Consider a string $\boldsymbol{c}\in\{0,1\}^{k+\left\lceil\log k\right\rceil+4}$ that contains at least one copy of the marker. The encoding $T_{d}:\{0,1\}^{k+\left\lceil\log k\right\rceil+4}\to\{0,1\}^{k+\left\lceil\log k\right\rceil+3}$ explained next deletes all the markers, i.e., encodes $\boldsymbol{c}$ into $T_{d}(\boldsymbol{c})\in\{0,1\}^{k+\left\lceil\log k\right\rceil+3}$ such that $T_{d}(\boldsymbol{c})$ does not contain any copy of the marker. The encoding is invertible, i.e., given the encoding $T_{d}(\boldsymbol{c})$ one can recover $\boldsymbol{c}$.
Similarly to the first presented encoding, $T_{d}$ consists of a series of deleting and appending strings. First, the encoders assigns $T_{d}(\boldsymbol{c})=\boldsymbol{c}$. Then, the encoder appends a $0$ to the end of $T_{d}(\boldsymbol{c})$. Now, the encoder finds the smallest value of $i\in[1,k]$ such that $T_{d}(\boldsymbol{c})_{i}=\dots=T_{d}(\boldsymbol{c})_{i+\left\lceil\log k\right\rceil+3}=1$. The encoder deletes $(T_{d}(\boldsymbol{c})_{i},\dots,T_{d}(\boldsymbol{c})_{i+\left\lceil\log k+3\right\rceil})$ from $T_{d}(\boldsymbol{c})$ and appends $(i,0,0)$ to the end of $T_{d}(\boldsymbol{c})$. The value of $i$ is encoded by $\left\lceil\log k\right\rceil$ bits. The encoder sets $n^{\prime}=k$. We refer to this as the initialization step.
Subsequently, the encoder looks for the smallest value of $i\leq n^{\prime}$ such that $T_{d}(\boldsymbol{c})_{i}=\dots=T_{d}(\boldsymbol{c})_{i+\left\lceil\log k\right\rceil+3}=1$. If such an integer $i$ exists, the encoder deletes $(T_{d}(\boldsymbol{c})_{i},\dots,T_{d}(\boldsymbol{c})_{i+\left\lceil\log k\right\rceil+3})$ from $T_{d}(\boldsymbol{c})$ and appends $(i,0,0,0,1)$ to the end of $T_{d}(\boldsymbol{c})$. The encoder changes the value of $n^{\prime}$ to $n^{\prime}=n^{\prime}-\left\lceil\log k\right\rceil-4$. The encoder keeps repeating this process until $n^{\prime}\leq\left\lceil\log k\right\rceil+3$ or no values of $i\leq n^{\prime}$ exist for which $i$ is the start of a marker.
This encoding procedure indeed results in a string that is one bit shorter than the original string. The initialization step appends a $0$. Then, it deletes a string and appends another one that is $2$ bits shorter. All other deleting and appending processes delete and append a string of the same length. One can verify that the appended strings are not modified by the encoding due to decreasing $n^{\prime}$ and the appended $0$ at the initialization step. In addition, the resulting string does not contain any copy of the marker. A string that contains the marker is always deleted. All appended strings are guaranteed to have the bit $\left\lceil\log k\right\rceil+1$ equal to $0$ which ensures that no marker exists.
Decoding of $T_{d}(\boldsymbol{c})$: The decoding is straightforward. The appended string in the initialization step ends with a $0$. All other appended strings end with a $1$. The decoder looks at the last bit of the string input string $T_{d}(\boldsymbol{c})$. If the last bit is a $1$, it skips the $1$ and the appended $0$’s, and gets the integer value of $i$ from the $\left\lceil\log k\right\rceil$ bits representing $i$. The decoder deletes the appended string and adds a marker at position $i$. This process is repeated until the last bit is $0$. In this case, the decoder skips the appended $0$’s and gets the integer value of $i$ from the $\left\lceil\log k\right\rceil$ bits representing $i$. The decoder deletes the appended string and adds a marker at position $i$.
The encoding and decoding processes can be implemented effectively in the same manner as in the marker enriching step. The total complexities of decoding and encoding are linear, i.e. $O(k)$.
Encoding into $(\mathcal{P},\Delta=R+B-m)$-dense strings
We present the encoder
$$T:\{0,1\}^{n+2\left\lceil\log k\right\rceil+8}\to\{0,1\}^{n+k+3\left\lceil\log k\right\rceil+12}$$
that takes $T_{e}(\boldsymbol{u})$ as input and outputs $T(\boldsymbol{u})$ that is $(\mathcal{P},\Delta)$-dense. To do so, the encoding ensures that every substring of $T(\boldsymbol{u})$ of length
$$R=(k+\left\lceil\log k\right\rceil+4)(\left\lceil\log n\right\rceil+\left\lceil\log k\right\rceil+8)$$
contains at least one substring of length $m=k+\left\lceil\log k\right\rceil+4$ that does not contain any copy of the marker $\mathbf{1}^{\left\lceil\log k\right\rceil+4}$.
The encoding is as follows. Let $T(\boldsymbol{u})=T_{e}(\boldsymbol{u})$. The encoder appends $(\mathbf{0}^{k},\mathbf{1}^{\left\lceil\log k\right\rceil+4})$ to the end of $T(\boldsymbol{u})$. Let $n^{\prime}=n+2\left\lceil\log k\right\rceil+8$ (the length of $T_{e}(\boldsymbol{u})$). The encoder searches for an integer $i\leq\min\{n^{\prime},n+k+3\left\lceil\log k\right\rceil+13-R\}$ such that for every $j\in[i,i+R-k-\left\lceil\log k\right\rceil-3]$, there exists an integer $\ell\in[j,j+k-1]$ satisfying $(T(\boldsymbol{u})_{\ell},\dots,T(\boldsymbol{u})_{\ell+\left\lceil\log k\right\rceil+3})=\mathbf{1}^{\left\lceil\log k\right\rceil+4}$. If such an integer exists, the encoder splits $(T(\boldsymbol{u})_{i},\dots,T(\boldsymbol{u})_{i+R-1})$ into $(\left\lceil\log n\right\rceil+\left\lceil\log k\right\rceil+8)$ blocks $\boldsymbol{b}_{1},\dots,\boldsymbol{b}_{\left\lceil\log n\right\rceil+\left\lceil\log k\right\rceil+8}$ of length $k+\left\lceil\log k\right\rceil+4$ each. The encoder deletes $\boldsymbol{b}_{2},\dots,\boldsymbol{b}_{\left\lceil\log n\right\rceil+\left\lceil\log k\right\rceil+7}$ from $T(\boldsymbol{u})$ and appends
$$(0,T_{d}(\boldsymbol{b}_{2}),\dots,T_{d}(\boldsymbol{b}_{\left\lceil\log n\right\rceil+\left\lceil\log k\right\rceil+7}),i+k+\left\lceil\log k\right\rceil+3,\mathbf{1}^{\left\lceil\log k\right\rceil+4},0)$$
to the end of $T(\boldsymbol{u})$. The appended number $i+k+\left\lceil\log k\right\rceil+3$ is represented with a binary string of length $\left\lceil\log n\right\rceil$. The encoder sets $n^{\prime}=n^{\prime}-R+2k+2\left\lceil\log k\right\rceil+8$. The encoder repeats the same process until there is no more values of $i$ that satisfy the desired properties.
In terms of redundancy, the encoding only adds $m$ bits. Every deleted string has the same length of the appended string. This is because the intermediate encoding $T_{d}$ reduces one bit per encoded sequence. We have $\left\lceil\log n\right\rceil+\left\lceil\log k\right\rceil+6$ encoded sequences. The encoder uses $\left\lceil\log n\right\rceil$ bits to append $i+k+\left\lceil\log k\right\rceil+3$. The added marker takes $+\left\lceil\log k\right\rceil+4$ bits and the remaining $2$ bits are taken by the added $0$’s. Therefore, the only addition made by this encoding is the initial addition of the string $(\mathbf{0}^{k},\mathbf{1}^{\left\lceil\log k\right\rceil+4})$.
Similarly to previous encoding algorithms, the appended sequence in the initialization and the decrease of $n^{\prime}$ at every round (deleting and appending) guarantee that the encoder does not modify the appended strings.
Correctness: We now prove that the resulting string $T(\boldsymbol{u})$ satisfies Property 1 and Property 2. We start by showing by induction over the number of rounds $r$ (searching for an index $i$, deleting and appending a sequence) that $T(\boldsymbol{u})$ satisfies Property 1. At the initial step, $r=0$, the string $T(\boldsymbol{u})=(T_{e}(\boldsymbol{u}),\mathbf{0}^{k},\mathbf{1}^{\left\lceil\log k\right\rceil+4})$ satisfies Property 1 by construction of $T_{e}$. Hence, the hypothesis holds for $r=0$. Assume that after $r>0$ rounds $T(\boldsymbol{u})$ satisfies the desired property. At round $r+1$, after a string is deleted, the blocks $\boldsymbol{b}_{1}$ and $\boldsymbol{b}_{k+\left\lceil\log k\right\rceil+8}$ are not deleted. Those blocks both contain the marker. Therefore, after the deletion step the string $T(\boldsymbol{u})$ still satisfies Property 1. All appended strings contain the marker $\mathbf{1}^{\left\lceil\log k\right\rceil+4}$. The index distance between any two markers in the appended sequences is at most $R-2k-2\left\lceil\log k\right\rceil-8\leq B-\left\lceil\log k\right\rceil-4$. Thus, the appended strings satisfy Property 1 and so does the whole string $T(\boldsymbol{u})$.
Now we prove that $T(\boldsymbol{u})$ satisfies Property 2. Due to the properties of the encoder, for any value $i\in[1,\min\{n^{\prime},n+k+3\left\lceil\log k\right\rceil+13-R\}]$, there exists some $j\in[i,i+R-k-\left\lceil\log k\right\rceil-3]$ such that for every $\ell\in[j,j+k-1]$ the following holds $(T(\boldsymbol{u})_{\ell},\dots,T(\boldsymbol{u})_{\ell+\left\lceil\log k\right\rceil+3})\neq\mathbf{1}^{\left\lceil\log k\right\rceil+4}$. Recall that all appended strings satisfy Property 2 and are not deleted by the algorithm.
We only have to prove the property for $i\in[n^{\prime}+1,n+k+3\left\lceil\log k\right\rceil+13-R]$. Since the algorithm does not delete the appended strings, the interval $[i,i+R-1]$ contains the string $(0,T_{d}(\boldsymbol{b}_{2}))$ of length $m$ that comes from an appended string. According to the intermediate encoding $T_{d}$, the string $T_{d}(\boldsymbol{b}_{2})$ does not contain the marker. Hence, for $i\in[n^{\prime}+1,n+k+3\left\lceil\log k\right\rceil+13-R]$ there exists a string of length $m$ that does not contain the marker. Thus, $T(\boldsymbol{u})$ satisfies Property 2 as well.
Decoding of $T(\boldsymbol{u})$: We show that the decoder outputs $T_{e}(\boldsymbol{u})$ from which we have previously shown that $\boldsymbol{u}$ can be uniquely determined. The decoding procedure follows the reverse steps done by the encoding. The decoder looks at the last bit of $T(\boldsymbol{u})$. If this bit is $1$, then only the pattern $(\mathbf{0}^{k},\mathbf{1}^{\left\lceil\log k\right\rceil+4})$ has been added and no intermediate encoding has been done. The decoder deletes the additional pattern and obtains $T_{e}(\boldsymbol{u})$. Otherwise, the last string of length $m(\left\lceil\log n\right\rceil+\left\lceil\log k\right\rceil+6)$ is encoded using the intermediate encoding $T_{d}(\boldsymbol{c})$. The decoder deletes this string from the input string, then determines the strings $T_{d}(\boldsymbol{b}_{2}),\dots,T_{d}(\boldsymbol{b}_{\left\lceil\log n\right\rceil+\left\lceil\log k\right\rceil+7})$ encoded using $T_{d}(\boldsymbol{c})$ and inverts them. Then, the encoder reads the index $i+m-1$ and inserts $\boldsymbol{b}_{2},\dots,T_{d}(\boldsymbol{b}_{\left\lceil\log n\right\rceil+\left\lceil\log k\right\rceil+7}$ at this position. This step is repeated until the last bit of the input string is $1$.
Complexity: The encoding and decoding can be implemented effectively in linear time $\mathcal{O}(n)$ in the similar manner as in the marker enrichment step. Indeed, let a string $\boldsymbol{s}$ be equal to $(T_{e}(\boldsymbol{u}),\mathbf{0}^{k},\mathbf{1}^{\left\lceil\log k\right\rceil+4})$. We write the encoded sequence into strings $\boldsymbol{t}_{1}$ and $\boldsymbol{t}_{2}$. The string $\boldsymbol{t}_{1}$ will store the string $\boldsymbol{s}$ without the deleted substrings. The string $\boldsymbol{t}_{2}$ will store compressed versions of the deleted substrings. The resulting string $\boldsymbol{x}$ is equal to the concatenation of $\boldsymbol{t}_{1}$ and $\boldsymbol{t}_{2}$.
Scan the string $\boldsymbol{s}$ from left to right, copy its symbols into $\boldsymbol{t}_{1}$ and
compute the distance to the last marker and the distance to the last substring of length $m$ without a marker in the similar manner as in the marker enrichment step. If at some moment the distance to the start of the last substring of length $m$ without a marker is bigger than $R$,
then we delete $R$ last symbols from $\boldsymbol{t}_{1}$, append encoding of deleted part to $\boldsymbol{t}_{2}$, append the first and the last blocks of the deleted part to $\boldsymbol{t}_{1}$, compute the actual distances to the last marker and the last string of length $m$ without a marker. Continue scanning of the string $\boldsymbol{s}$.
One can see that this algorithm implements the encoding process and works in linear time.
The decoding can be done in linear time analogously to the decoding in the marker enrichment step.
III-B Protecting the hashes
As previously mentioned, we use the hashes $H_{\text{loc}}(\boldsymbol{x})$ and $H_{\text{cor}}(\boldsymbol{x})$ to locate and correct potential deletion errors that may occur in $\boldsymbol{x}$.
Since the deletions could also affect $H_{\text{loc}}(\boldsymbol{x})$ and $H_{\text{cor}}(\boldsymbol{x})$, we need to protect these hashes and enable their recovery at the decoder in order to be able to correct deletions in $\boldsymbol{x}$. One way to this is to encode $(H_{\text{loc}}(\boldsymbol{x}),H_{\text{cor}}(\boldsymbol{x}))$ using a deletion correcting code. However, given the localized nature of the deletions in our problem, we propose the following simpler solution.
To protect the hashes, we insert a buffer $H_{\text{sep}}$ of size $k+1$ that separates $(H_{\text{loc}}(\boldsymbol{x}),H_{\text{cor}}(\boldsymbol{x}))$ from $\boldsymbol{x}$. Since the $k^{\prime}\leq k$ deletions are localized in a window of size at most $k$, a consequence of inserting the buffer of size $k+1$ is that the deletions now cannot affect $\boldsymbol{x}$ and $(H_{\text{loc}}(\boldsymbol{x}),H_{\text{cor}}(\boldsymbol{x}))$ simultaneously. We design $H_{\text{sep}}$ in a way such that we can detect whether the deletions have affected $\boldsymbol{x}$ or not. Therefore, if we detect that the deletions have affected $\boldsymbol{x}$, we know that the hashes are error-free so we use them to proceed with our decoding methods explained in the subsequent sections. Otherwise, we detect that no deletions have affected $\boldsymbol{x}$, then we know that the deletions have only affected $(H_{\text{sep}},H_{\text{loc}}(\boldsymbol{x}),H_{\text{cor}}(\boldsymbol{x}))$, so we immediately recover $\boldsymbol{x}$. Next, we explain how we design this buffer and detect whether the deletions have affected $\boldsymbol{x}$ or not.
We set the buffer of length $k+1$ to $H_{\text{sep}}\coloneqq(\mathbf{0}^{k},1)$. Consider the transmitted sequence $\widetilde{\boldsymbol{x}}$ that is affected by $k^{\prime}\leq k$ localized deletions resulting in the received sequence $\widetilde{\boldsymbol{y}}$ of length $N-k^{\prime}$. The buffer $H_{\text{sep}}$ is composed of $k$ zeros followed by a single one, and its position in $\widetilde{\boldsymbol{x}}$ ranges from the $(|\boldsymbol{x}|+1)^{st}$ bit to the $(|\boldsymbol{x}|+k+1)^{st}$ bit. Let $\widetilde{y}_{\alpha}$ be the bit in position $\alpha$ in the received string, where $\alpha\coloneqq|\boldsymbol{x}|+k-k^{\prime}+1$.
The decoder observes $\widetilde{y}_{\alpha}$:
1.
If $\widetilde{y}_{\alpha}=1$, then this means that the one in the buffer has shifted $k^{\prime}$ positions to the left because of the deletions, i.e., all the deletions occurred to the left of the one in the buffer. In this case, we consider that the deletions have affected $\boldsymbol{x}$ and we detect that $(H_{\text{loc}}(\boldsymbol{x}),H_{\text{cor}}(\boldsymbol{x}))$ are error-free.
2.
If $\widetilde{y}_{\alpha}=0$, then this indicates that the $k^{\prime}$ deletions occurred to the right of the first zero in the buffer, i.e., $\boldsymbol{x}$ was unaffected. In this case, the decoder simply outputs the first $|\boldsymbol{x}|$ bits of the received string.
III-C Locating deletions
To determine the function $H_{\text{loc}}(\boldsymbol{x})$, we use the VT check sum, which is defined over integers, similar to that in [1, 5]. A key observation of this section is given in the following statement.
Observation 2 (Localized deletions and $(\mathcal{P},\Delta)$-dense strings).
Let $\boldsymbol{x}$ be a $(\mathcal{P},\Delta)$-dense string with $\mathcal{P}$ being as stated in Definition 4. Then, deletions localized in a window of length $k$ do not destroy nor create more than two patterns from $\mathcal{P}$ in $\boldsymbol{x}$.
Proof.
We note that the distance between 1’s in the vector $\mathbbm{1}_{\mathcal{P}}(\boldsymbol{x})$ is at least $k+1$, i.e., patterns from $\mathcal{P}$ are located in $\boldsymbol{x}$ at distance at least $k+1$. Therefore, deletions localized in a window of length $k$ occur in at most two patterns in $\boldsymbol{x}$ and, thus, do not destroy more than two patterns. Similar arguments work when localized deletions result in creating new patterns in $\boldsymbol{x}$.
∎
Now we proceed by describing the encoding and decoding algorithm of the locating procedure.
Encoding: Let us define $c_{1}:=n_{\mathcal{P}}(\boldsymbol{x})\pmod{5}$, $c_{1}\in[0,4]$, and $c_{2}:=\mathsf{VT}(\boldsymbol{a}_{\mathcal{P}}(\boldsymbol{x}))\pmod{6n}$, $c_{2}\in[0,6n-1]$. Assume that there is a standard map from non-negative integers at most $n_{1}$ to their binary representations of length $\lceil\log n_{1}\rceil$. Then we define the function $H_{\text{loc}}(\boldsymbol{x}):=\left(c_{1},c_{2}\right)$.
Decoding strategy:
Utilizing the arguments in Section III-B, we can assume that $c_{1}$ and $c_{2}$ are known before we start the decoding procedure. Define $\boldsymbol{y}:=\widetilde{\boldsymbol{y}}_{[1,|\boldsymbol{x}|-k^{\prime}]}$ and note that $\boldsymbol{y}\in B_{k^{\prime}}(\boldsymbol{x})$.
We want to locate the positions of the localized deletions up to an interval of length $\mathcal{O}(\Delta^{2})$.
Recall that $a_{\mathcal{P}}(\boldsymbol{x})_{i}\leq\Delta$ as $\boldsymbol{x}$ is $(\mathcal{P},\Delta)$-dense. By Observation 2, we obtain that $a_{\mathcal{P}}(\boldsymbol{y})_{i}\leq 3\Delta$ and $n_{\mathcal{P}}(\boldsymbol{x})-2\leq n_{\mathcal{P}}(\boldsymbol{y})\leq n_{\mathcal{P}}(\boldsymbol{x})+2$.
Compare the vectors $\boldsymbol{a}_{\mathcal{P}}(\boldsymbol{x})$ and $\boldsymbol{a}_{\mathcal{P}}(\boldsymbol{y})$. We claim that the second vector can be obtained from the first by replacing the substring $\boldsymbol{s}_{1}$ by the substring $\boldsymbol{s}_{2}$, $0\leq|\boldsymbol{s}_{1}|,|\boldsymbol{s}_{2}|\leq 3$, $||\boldsymbol{s}_{1}|-|\boldsymbol{s}_{2}||\leq 2$. Let $\boldsymbol{s}_{1}$ and $\boldsymbol{s}_{2}$ be equal $\boldsymbol{a}_{\boldsymbol{p}}(\boldsymbol{x})_{[b,e_{1})}$ and $\boldsymbol{a}_{\boldsymbol{p}}(\boldsymbol{y})_{[b,e_{2})}$ with $e_{1}:=b+|\boldsymbol{s}_{1}|$ and $e_{2}:=b+|\boldsymbol{s}_{2}|$. In other words, the vectors $\boldsymbol{a}_{\mathcal{P}}(\boldsymbol{x})$ and $\boldsymbol{a}_{\mathcal{P}}(\boldsymbol{y})$ can be represented as concatenations in the following way
$$\displaystyle\boldsymbol{a}_{\mathcal{P}}(\boldsymbol{x})$$
$$\displaystyle=(\boldsymbol{a}_{\mathcal{P}}(\boldsymbol{x})_{[1,b)},\boldsymbol{s}_{1},\boldsymbol{a}_{\mathcal{P}}(\boldsymbol{x})_{[e_{1},n_{\mathcal{P}}(\boldsymbol{x})]})$$
$$\displaystyle=(\boldsymbol{a}_{\mathcal{P}}(\boldsymbol{y})_{[1,b)},\boldsymbol{s}_{1},\boldsymbol{a}_{\mathcal{P}}(\boldsymbol{y})_{[e_{2},n_{\mathcal{P}}(\boldsymbol{y})]}),$$
$$\displaystyle\boldsymbol{a}_{\mathcal{P}}(\boldsymbol{y})$$
$$\displaystyle=(\boldsymbol{a}_{\mathcal{P}}(\boldsymbol{x})_{[1,b)},\boldsymbol{s}_{2},\boldsymbol{a}_{\mathcal{P}}(\boldsymbol{x})_{[e_{1},n_{\mathcal{P}}(\boldsymbol{x})]})$$
$$\displaystyle=(\boldsymbol{a}_{\mathcal{P}}(\boldsymbol{y})_{[1,b)},\boldsymbol{s}_{2},\boldsymbol{a}_{\mathcal{P}}(\boldsymbol{y})_{[e_{2},n_{\mathcal{P}}(\boldsymbol{y})]}).$$
(1)
Let $d$ be equal to $|\boldsymbol{s}_{2}|-|\boldsymbol{s}_{1}|$. Observe that the value $d$ can be computed based on $\boldsymbol{y}$ as we require $c_{1}=n_{\mathcal{P}}(\boldsymbol{x})\pmod{5}$ and, thus, $d=n_{\mathcal{P}}(\boldsymbol{y})-c_{1}\pmod{5}$.
Using the property (1) and the fact $\sum_{i=b}^{e_{1}-1}a_{\mathcal{P}}(\boldsymbol{x})_{i}-\sum_{i=b}^{e_{2}-1}a_{\mathcal{P}}(\boldsymbol{y})_{i}=k^{\prime}$, we compute the difference between the VT checksums for vectors $\boldsymbol{a}_{\mathcal{P}}(\boldsymbol{y})$ and $\boldsymbol{a}_{\mathcal{P}}(\boldsymbol{x})$:
$$\displaystyle D$$
$$\displaystyle:=\mathsf{VT}(\boldsymbol{a}_{\mathcal{P}}(\boldsymbol{y}))-\mathsf{VT}(\boldsymbol{a}_{\mathcal{P}}(\boldsymbol{x}))$$
$$\displaystyle=d\sum\limits_{i=e_{2}}^{n_{\mathcal{P}}(\boldsymbol{y})}a_{\mathcal{P}}(\boldsymbol{y})_{i}-\sum\limits_{i=b}^{e_{1}-1}ia_{\mathcal{P}}(\boldsymbol{x})_{i}+\sum\limits_{i=b}^{e_{2}-1}ia_{\mathcal{P}}(\boldsymbol{y})_{i}$$
$$\displaystyle=d\sum\limits_{i=e_{2}}^{n_{\mathcal{P}}(\boldsymbol{y})}a_{\mathcal{P}}(\boldsymbol{y})_{i}-b\left(\sum\limits_{i=b}^{e_{1}-1}a_{\mathcal{P}}(\boldsymbol{x})_{i}+\sum\limits_{i=b}^{e_{2}-1}a_{\mathcal{P}}(\boldsymbol{y})_{i}\right)$$
$$\displaystyle\quad-\sum\limits_{i=b+1}^{e_{1}-1}(i-b)a_{\mathcal{P}}(\boldsymbol{x})_{i}+\sum\limits_{i=b+1}^{e_{2}-1}(i-b)a_{\mathcal{P}}(\boldsymbol{y})_{i}$$
$$\displaystyle=d\sum\limits_{i=b+3}^{n_{\mathcal{P}}(\boldsymbol{y})}a_{\mathcal{P}}(\boldsymbol{y})_{i}-bk^{\prime}+E,$$
where $E$ is defined by
$$E:=d\sum\limits_{i=e_{2}}^{b+2}a_{\mathcal{P}}(\boldsymbol{y})_{i}-\sum\limits_{i=b+1}^{e_{1}-1}(i-b)a_{\mathcal{P}}(\boldsymbol{x})_{i}+\sum\limits_{i=b+1}^{e_{2}-1}(i-b)a_{\mathcal{P}}(\boldsymbol{y})_{i}.$$
We observe that $e_{1},e_{2}\leq b+3$ and $|d|\leq 2$. Therefore, the first and the last summations in the definition of $E$
has at most three addends with the coefficient $\leq\max(|d|,e_{2}-b-1)$.
Therefore, $|E|$ can be upper bounded as $18\Delta$. Finally, we derive that
$$D=d\sum\limits_{i=b+2}^{n_{\mathcal{P}}(\boldsymbol{y})}\boldsymbol{a}_{\mathcal{P}}(\boldsymbol{y})_{i}-bk^{\prime}+E,\quad|E|\leq 18\Delta.$$
Note that $-3n-18\Delta\leq D\leq 2n+18\Delta$. Thus, for $n>36\Delta$, we have $6n>5n+36\Delta$ and it suffices to know $c_{2}=\mathsf{VT}(\boldsymbol{a}_{\mathcal{P}}(\boldsymbol{x}))\pmod{6n}$ to compute $D$. Define the function
$$G(s):=d\sum\limits_{i=s+2}^{n_{\mathcal{P}}(\boldsymbol{y})}\boldsymbol{a}_{\mathcal{P}}(\boldsymbol{y})_{i}-sk^{\prime}.$$
We observe that $D-E=G(b)$ and the function $G(s)$ is a strictly monotone function of $s$. Indeed,
if $d\geq 0$ then $G(s)$ is a strictly decreasing function of $s$.
If $d<0$ then using the fact $a_{\mathcal{P}}(\boldsymbol{y})_{i}\geq m>k\geq k^{\prime}$ we conclude that $G(s)$ is a strictly increasing function of $s$. Since it is easy to compute $E$ based on $\boldsymbol{y}$, we make use of the bound $|E|\leq 18\Delta$ and define the set
$$F:=\left\{s:\ G(s)\in[D-18\Delta,D+18\Delta]\right\}.$$
Note that the difference between two elements from $F$ cannot be larger than $36\Delta$.
Since each $a_{\mathcal{P}}(\boldsymbol{y})\leq 3\Delta$, we can locate a segment of length at most $108\Delta^{2}+3\Delta=\mathcal{O}(\Delta^{2})$, where all deletions are located. Note that computing the vector $\boldsymbol{a}_{\mathcal{P}}(\boldsymbol{y})$, finding the set $F$ and determining the required interval of length $\mathcal{O}(\Delta^{2})$ can be done in $\mathcal{O}(n)$ time.
III-D Correcting deletions
In this step, we make use of the following result from [14].
Theorem 2 (Theorem 1.3, Remark 1.5 from [14]).
For any positive integers $n$ and $k$ with $k\leq n/4$, there exists an explicit code which maps any
message $\boldsymbol{x}$ of length $n$ into a codeword $(\boldsymbol{x},h(\boldsymbol{x}))$ of length $n+\mathcal{O}(k\log^{2}(n/k))$ and
corrects up to $k$ deletions and insertions. The complexities of encoding, i.e. computing of function $h$, and decoding algorithms are polynomial in $n$.
Remark 1.
We emphasize that Theorem 1.3 from [14] allows us to encode in a systematic way, i.e., each codeword contains an original message. This property is important for our further analysis.
Although for our range of parameters Theorem 1.4 from [14] gives smaller redundancy, the encoding is not systematic.
We note that codes with the same redundancy as in Theorem 2 were obtained in [13].
By the argument in Section III-C, we can find the segment of length at most $108\Delta^{2}+3\Delta$ such that all deletions are situated inside this segment. Let $M:=108\Delta^{2}+3\Delta$. Recall that $\Delta=\mathcal{O}(k\log k\log n)$ as $n\to\infty$. Now we show how to define $H_{\text{cor}}(\boldsymbol{x})$.
Encoding: Append the smallest amount of zeroes to the end of $\boldsymbol{x}$ to obtain the vector $\hat{\boldsymbol{x}}$, which length $n^{\prime}$ is divisible by $M$. Partition a codeword $\hat{\boldsymbol{x}}$ into $L=n^{\prime}/M$ parts of length $M$, $\hat{\boldsymbol{x}}=(\boldsymbol{x}_{1},\boldsymbol{x}_{2},\ldots,\boldsymbol{x}_{L}).$
For each message $\boldsymbol{x}_{i}$, we use the construction from Theorem 2 to compute parity bits $\boldsymbol{h}_{i}=h(\boldsymbol{x}_{i})$ to protect them against $2k$ deletions. Note that the length
of $\boldsymbol{h}_{i}$ is $\mathcal{O}(k\log^{2}(M/(2k)))=\mathcal{O}(k\log^{2}k+k(\log\log n)^{2})$.
Define $\boldsymbol{h}_{even}$ and $\boldsymbol{h}_{odd}$ to be the following bit-wise sums
$$\displaystyle\boldsymbol{h}_{\text{even}}$$
$$\displaystyle=\sum\limits_{i=1}^{\lfloor L/2\rfloor}\boldsymbol{h}_{2i},$$
$$\displaystyle\boldsymbol{h}_{\text{odd}}$$
$$\displaystyle=\sum\limits_{i=1}^{\lceil L/2\rceil}\boldsymbol{h}_{2i-1}.$$
Finally, we let $H_{\text{cor}}(\boldsymbol{x})$ to be $(\boldsymbol{h}_{\text{even}},\boldsymbol{h}_{\text{odd}})$.
Decoding strategy: Since the position of deletions is already located, we can recover $\boldsymbol{x}_{j}$ for all $j$ except possibly two consecutive blocks indexed by $l$ and $l+1$ for some integer $l$.
Append $LM-|\boldsymbol{x}|$ zeroes to the end of the string $\boldsymbol{y}$ to obtain $\hat{\boldsymbol{y}}$. Divide $\hat{\boldsymbol{y}}$ into blocks $(\boldsymbol{y}_{1},\boldsymbol{y}_{2},\ldots,\boldsymbol{y}_{L})$ such that $|\boldsymbol{y}_{j}|=M$ for $j\neq l$ and $|\boldsymbol{y}_{l}|=M-k^{\prime}$. Clearly, $\boldsymbol{x}_{j}=\boldsymbol{y}_{j}$ for all $j$ except $j=l,l+1$. For $j=l,l+1$, the string $\boldsymbol{y}_{j}$ is obtained from $\boldsymbol{x}_{j}$ by at most $k$ deletions and at most $k$ insertions.
Compute $\boldsymbol{h}_{j}=h(\boldsymbol{y}_{j})=h(\boldsymbol{x}_{j})$ for $j\neq l,l+1$. Using $\boldsymbol{h}_{\text{even}}$ and $\boldsymbol{h}_{\text{odd}}$, we find $\boldsymbol{h}_{l}$ and $\boldsymbol{h}_{l+1}$. Thus, applying the decoding algorithm from [13, 14], we can reconstruct $\boldsymbol{x}_{l}$ and $\boldsymbol{x}_{l+1}$.
Complexity: We claim that the encoding complexity of this step is $n\cdot{\rm poly}(k\log n)$. Indeed, the complexity of computing hash $h_{i}$ is polynomial in $M$ by Theorem 2, and the total number of such hashes is at most $n$. The decoding complexity
is equal to the complexity of computing of hashes and the complexity of decoding of two codes of length ${\rm poly}(k\log n)$, that results in $n\cdot{\rm poly}(k\log n)$ total complexity.
IV Conclusion
In this paper, we propose a novel efficient code of length $n$ capable of correcting deletions occurred in a window of length $k$, where $k=\mathcal{O}(n/\log^{2}n)$. The encoding and decoding algorithms of the proposed construction run in $n\cdot{\rm poly}(k\log n)$ time. For $k=o(\log/(\log\log n)^{2})$ and $n\to\infty$, the redundancy of this code is $\log n(1+o(1))$, i.e., it is asymptotically optimal. An interesting question is whether this construction can be generalized for edit errors.
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Wide-Aperture Dense Plasma Fluxes Production Based on ECR Discharge
in a Single Solenoid Magnetic Field
V.A. Skalyga
[
S.V. Golubev
I.V. Izotov
R.A. Shaposhnikov
shaposhnikov-roma@mail.ru
S.V. Razin
A.V. Sidorov
[
A.F. Bokhanov
M.Yu. Kazakov
R.L. Lapin
S.S. Vybin
Institute of Applied Physics, Russian Academy of Sciences, 603950 Nizhny Novgorod, Russia
(November 25, 2020)
Abstract
Results of experimental investigation of the ECR discharge in a single coil magnetic field as an alternative to RF and helicon discharges for wide-aperture dense plasma fluxes production are presented. A possibility of obtaining wide-aperture high density hydrogen plasma fluxes with homogenous transverse distribution was demonstrated in such a system. The prospects of using this system for obtaining high current ion beams are discussed.
††preprint: RSI/Shaposhnikov_1
Also at ]Lobachevsky State University
of Nizhny Novgorod, 603155 Nizhny Novgorod, Russia.
Also at ]Lobachevsky State University
of Nizhny Novgorod, 603155 Nizhny Novgorod, Russia.
I Introduction
One of the most widespread methods of plasma heating in systems of nuclear fusion is a high-energy neutral beam injection Grisham et al. (2012). Neutral beam injectors development is a complex and a multistage task. At the first step it is necessary to create a high-density plasma flux for the subsequent ion beam extraction. The beam current has to reach values of tens of Amperes to satisfy requirements of modern fusion facilities, and this could be realized only with using a large-scale multi-aperture extraction system, which implies a necessity of a spatially wide plasma fluxes production with a homogenous transverse distribution. Plasma sources, based on RF and helicon discharges, are mainly used for this purpose (Ed.) (1989); Shikhvotsev et al. (2016).
An alternative approach based on the ECR discharge sustained by powerful millimetre wavelength gyrotron radiation was proposed in the Institute of Applied Physics of Russian Academy of Sciences (IAP RAS). The ECR discharge with a quasi-gasdynamic plasma confinement in a simple mirror trap was studied in the very first experiments Golubev et al. (2004). A possibility to obtain 100%-ionized plasma with low content of molecular ions (at a level of several percent), density on the level of 10${}^{13}$ – 10${}^{14}$ cm${}^{-3}$, and electron temperature of 10 – 100 eV was demonstrated Skalyga et al. (2014, 2016, 2017). Combination of plasma parameters mentioned above seems to be optimal for light ion production. The ECR discharge has advantages of lower operating pressure, consequently, higher ionization degree, and the electron temperature being closer to the optimal one for production of pure atomic beams when compared to RF discharges. Additionally, plasma heating with microwaves allows to use quasi-optical methods of its injection into a discharge, which eliminates the need of biasing the microwave generator, thus reducing electric power consumption at a high-voltage platform of an ion source. This factor favourably distinguishes the ECR discharge from other plasma production methods. The novelty of the present work is the use of a single solenoid field (magnetic field is produced by the only magnetic coil) instead of a simple mirror trap, when compared to what has been done at IAP RAS recently. Unlike simple mirror traps, the longitudinal plasma confinement in a single coil field is provided only by the ambipolar potential distribution, which accelerates ions and slows down electrons resulting in lower plasma lifetime. One could also expect lower ionization degree in this case, however it was demonstrated that energy input provided by powerful gyrotron is enough for sufficient hydrogen dissociation and ionization even under conditions of a poor confinement. The numerical simulations based on the theoretical model of plasma expanding along the magnetic field lines to a metal wall Abramov et al. (2019) demonstrates that existing gyrotron systems are capable of providing the necessary heating power for a large-scale plasma production and dense proton fluxes formation with the parameters required in modern neutral injection systems. Also such a simple magnetic configuration is convenient for further scaling and looks technically and technologically attractive.
II Experimental facility
Experiments were conducted at the IAP RAS using SMIS 37 facility Skalyga et al. (2014, 2016, 2017) (Fig.1), modified for described experiments. The gyrotron radiation with frequency 37.5 GHz, power up to 100 kW and pulse duration of 1.5 ms was used for discharge ignition and plasma heating. Electromagnetic radiation was focused by a special quasi-optical system to the centre of the discharge chamber with a diameter of 68 mm and a length of 250 mm placed inside a pulsed magnetic coil.
The wedge-like coupling system was located inside the discharge chamber in order to prevent plasma flux reaching the quartz vacuum window. The microwave injection part has an inner diameter of 38 mm, whereas the plasma chamber – 68 mm (see Fig.1). An influence of that fact on obtained result is discussed in the conclusion section of the present paper. A metal grid with a transparency of 70% was installed at the farther end of the discharge chamber, opposite to the injection side, in order to create a microwave cavity to improve the efficiency of plasma heating. In the presented research the metal grid was a part of the discharge chamber and, accordingly, they were under the same potential, namely, they were grounded. The magnetic field in the centre of the coil was varied from 1 to 3 T, while the resonance field value for the radiation frequency of 37.5 GHz is 1.34 T.
The neutral gas (hydrogen) was injected into the discharge chamber along the field axis through the gas supply line integrated into the microwave coupling system. The gas injection system was arranged as follows. A special tank filled by the neutral gas was connected by a tube to a pulsed valve placed on the discharge chamber. Pressure in the discharge chamber was determined by two parameters: gas pressure above the valve and time delay between the moment of the valve opening and the beginning of the microwave pulse. Pressure measurements in the discharge chamber were conducted using a gas discharge lamp before the experiments start in order to calibrate the gas feeding system. Thus, during the experiments the gas pressure in the discharge chamber was controlled by varying of that set of the gas inlet system parameters. The gas pressure was not measured in case of the discharge ignition. An example of gas pressure time dependence during the gas injection pulse is shown in Fig.2. The microwave pulse triggering time was tuned to ignite the plasma at the rising edge of the pressure whilst it was in the range of $2\times 10^{-4}$ – $8\times 10^{-4}$ Torr. The residual gas pressure in the diagnostic chamber was on the order of $10^{-6}-10^{-7}$ Torr.
The plasma flux was studied with a single flat Langmuir probe with 1 mm${}^{2}$ square, placed on a 3D movable rod mounted on the back flange of the diagnostic chamber providing measurement both in longitudinal and transverse directions with respect to the magnetic field. The probe was biased negatively, thus measuring the ion saturation current.
A series of experiments on the optical spectroscopy of the ECR plasma were performed to determine the plasma density in the discharge. An MS5204i monochromator-spectrograph manufactured by SOL Instruments was used to register the plasma emission spectrum. The spectrograph was equipped with removable diffraction gratings (1800 grooves/mm, blaze wavelength 270 nm; 1200 grooves/mm, blaze wavelength 400 nm) and was capable of registering radiation in the range of 200 – 900 nm. A CCD camera connected to a PC was used as a radiation detector having resolution of 14$\times$2048 pixels, pixel size -– 14$\times$14 $\mu$m. The instrumental function of the spectrograph was equal to 0.16 nm for an entrance slit width of 60 $\mu$m. Plasma emission in optical range going through the optical flange, specially mounted for these studies on the right end of the diagnostic chamber, was collected with a quartz lens into an entrance of a fiber, whose output was coupled with the entrance slit of the spectrograph. Thus, we analysed the plasma emission received along the system axis.
III Plasma density measurements
As it was mentioned above, plasma density in the ECR discharge is one of the key parameters which determines the maximum value of extracted ion beam current, and it could be enhanced by means of increase of the frequency of microwaves heating the plasma. Similar to Skalyga et al. (2017) a series of experiments on spectroscopy were performed to determine the plasma density according to Stark broadening of the lines of Balmer series of atomic hydrogen Griem (1974). Analysis of hydrogen lines of Balmer series (transition from highly excited electron levels to the level with principal quantum number $n=2$) was performed, as this series lies in visible and near-ultraviolet range. In case of 37.5 GHz ECR discharge, plasma density is expected to be on the level of $N_{e}=(1-5)\cdot 10^{13}$ cm${}^{-3}$. H${}_{\alpha}$ and H${}_{\beta}$ lines broadening is hardly detectable for a given plasma density. Therefore, analysis of shorter wavelength lines (H${}_{\gamma}$ and H${}_{\delta}$) is needed, as Stark splitting of electron level in hydrogen atom, which leads to a line broadening, is proportional to $n^{2}\cdot N_{e}^{3/2}$ ($n=3$ for H${}_{\gamma}$, $n=6$ for H${}_{\delta}$).
Fig.3 shows a part of plasma emission spectrum containing H${}_{\gamma}$ (434.1 nm) and H${}_{\delta}$ (410.1 nm) lines. These lines had a noticeable broadening, which differed from instrumental function significantly at certain experimental conditions. Obtained full width at half maximum (FWHM) were 0.175 nm, and 0.185 nm for H${}_{\gamma}$ and H${}_{\delta}$ lines respectively. Real broadening was estimated as $\Delta\lambda$=($\Delta\lambda^{2}_{meas}$ - $\Delta\lambda^{2}_{inst}$)${}^{0.5}$, where $\Delta\lambda_{meas}$ –- measured FWHM of a line, and $\Delta\lambda_{inst}$ -– instrumental function equal to 0.16 nm. Stark broadening of these lines is the main effect, since other effects are negligible at our experimental conditions. For example, Doppler broadening $\Delta\lambda/\lambda$ is on the order of 10${}^{-5}$ even at gas temperature of 2000 K (see Ochkin (2009)), whereas observed broadening is on the level of 10${}^{-4}$ and could be induced only by Stark effect. Calculations and various experiments Ochkin (2009) show that for hydrogen atom in case of Stark effect lines full widths at half maximum (FWHM) depends on plasma density as $\Delta\lambda=C\cdot N_{e}^{2/3}$, where C is a constant slightly depending on electron density and temperature (C varies only 20 – 30% in 5000 -– 20000 K temperature range and in 10${}^{14}$ -– 10${}^{17}$ cm${}^{-3}$ density range).
This method allows to estimate the plasma density accurate to the order of magnitude that is why it was difficult to establish dependencies of plasma density on system parameters. In performed experiments plasma density according to this data varied in the range of 10${}^{13}$ -– 10${}^{14}$ cm${}^{-3}$. The obtained values coincide in the order of magnitude with the expected ECR plasma density for gyrotron frequency used in experiments.
IV Plasma flux transverse profile measurements
Plasma flux transverse profile measurements were provided with a Langmuir probe mounted on 3D movable rod as it shown in Fig.1, allowed to obtain flux distribution at various distances from the centre of the discharge. The voltage U = -40 V corresponding to the ion saturation current was applied to the probe to collect a signal proportional to the plasma flux density.
To study the discharge properties under various experimental conditions plasma flux density was measured for different values of microwave heating power and magnetic field at the system axis close to the metallic grid separating plasma and diagnostic chambers, i. e. 7.4 cm downstream the magnetic field maximum. These results obtained with a gas pressure equal to $3\times 10^{-4}$ Torr (measured at the leading edge of the microwave pulse) are shown in Fig.4.
It is clearly seen that the plasma flux density could reach 750 mA/cm${}^{2}$ level. At the same time it should be mentioned that dependence on the magnetic field looks quite complicated and probably it could be explained by significant change in microwave-to plasma coupling efficiency together with the ECR surface shift along the plasma chamber. The effect of the ECR surface position would be studied in further experiments.
The dependence of the flux density on the longitudinal coordinate was studied, performing measurements at a several distances from the coil centre at experimental conditions corresponding to the maximum flux density. The measurements are shown in Fig.5 together with the magnitude of the magnetic field. It may be argued from the Fig.5 that the plasma flux follows the magnetic field lines, which means that for real application plasma flux profile could be efficiently shaped by magnetic field distribution in expanding region.
The data shown in the Fig.5 could be additionally used for as one more way of plasma density estimation. As far as the plasma flux density varies in proportion with the magnetic field value, it could be recalculation into the coil center, and it corresponds to the value of 2 A/cm${}^{2}$. We can estimate the plasma density assuming that the flux has a velocity close to the ion sound and the electron temperature in the discharge 10 – 30 eV, typical for SMIS 37 conditions. Such approach gives us the plasma density value of $2-4\times 10^{12}$ cm${}^{-3}$ and that corresponds to the ionization degree on the level of 50%. These numbers are in a good agreement with the data presented above.
To demonstrate the homogeneity of the plasma flux, the Fig.6 shows its transversal distribution in the diagnostic chamber close to the grid (see Fig.1). While the maximum density was 750 mA/cm${}^{2}$, the corresponding total ion current reached 5 A, evaluated as $\int j(x)\cdot 2\pi xdx$, where j(x) is transverse flux density profile, x is transverse coordinate. Also it could be seen that plasma flux FWHM is equal to 3 cm, while the plasma chamber is 68 mm in diameter.
That may be explained by the structure of magnetic field lines, which are shown in Fig.7 together with the plasma chamber geometry. It is seen in Fig.7 that the plasma flux is likely constrained by the diameter of microwave coupling system. This fact additionally underlines the importance of microwave injection scheme for the discharge parameters optimization. More studies on microwave coupling scheme are needed even in such a simple configuration.
V Conclusion
Wide-aperture plasma fluxes from the ECR discharge in the magnetic field of a single solenoid were obtained. It was found that the flux density increases with the microwave power and magnetic field. Maximum value of flux density was j${}_{max}$ = 750 mA/cm${}^{2}$, and corresponding total ion current was equal to 5 A. It was also found that the plasma follows magnetic field lines while expanding into the diagnostic chamber.
The obtained results may be supposedly improved with the increase of the frequency and the power of heating microwaves. An increase in the flux aperture could be provided by the use of discharge chambers of larger diameters simultaneously with a specific design of the microwave injection, optimised for more uniform electromagnetic field distribution in the volume.
Obviously, a question about possibility of a continuous wave (CW) operation of such ion source arises. In our opinion, the CW operation could be realised after a number of the experimental facility upgrades. The first one is a change of the gyrotron system from pulsed to the CW one. This is not really a problem as far as a numerous number of such devices are produced commercially and widely used for plasma heating. It would be also necessary to provide a CW magnetic field, and that could be easily dome using permanent magnets. Probably the main difficulty on the way to the CW operation would be the plasma chamber and metallic grid cooling. The plasma chamber surface used in the described experiments has a square about 400 cm${}^{2}$, thus taking into account that a regular water cooling could be used at power deposition level below 1 kW/cm${}^{2}$, we can state that there is no any problem hear. The cooling of the metallic grid looks more complicated, but using of refractory materials (like molybdenum) which could stay under the temperatures high enough for efficient heat removal by radiation losses is a kind of usual technique.
The obtained results demonstrate perspectives of the proposed system based on a single coil for wide-aperture plasma flux formation, including its probable application for neutral injectors development.
The further investigations will be dedicated to the obtaining of plasma fluxes with apertures of 100 cm${}^{2}$ and flux density of 1 A/cm${}^{2}$. We are also planning to optimize the gas inlet system. In the described experiments the gas inlet was carried out along the axis of the system. In this case the plasma flux had the density maximum on the axis. In the next experiments we are also going to use several tubes placed as a circle at a certain distance from the axis (a kind of a “ring” gas injection). This configuration could allow to provide a better control of transverse plasma flux profile and make it more homogenous. Another step in the future experiment would be devoted to a wide ion beam formation using this plasma source.
Acknowledgements.
The reported study was funded by RFBR, project number 19-32-90079 and supported by Presidential Grants Foundation (Grant # MD-2745.2019.2).
References
Grisham et al. (2012)
L. R. Grisham, P. Agostinetti, G. Barrera, P. Blatchford,
D. Boilson, J. Chareyre, et al., Fusion Eng Des 87, 1805 (2012).
(Ed.) (1989)
I. G. B. (Ed.), The physics
and technology of ion sources (John Wiley & Sons,
Inc., 1989).
Shikhvotsev et al. (2016)
I. V. Shikhvotsev, V. I. Davydenko, A. A. Ivanov, I. A. Kotelnikov, E. I. Kuzmin, A. Kreter,
et al., AIP Conf Proc 1771, 070006 (2016).
Golubev et al. (2004)
S. V. Golubev, S. V. Razin,
A. V. Sidorov, V. A. Skalyga, A. V. Vodopyanov, and V. G. Zorin, Rev Sci Instrum 75, 1675 (2004).
Skalyga et al. (2014)
V. Skalyga, I. Izotov,
S. Razin, A. Sidorov, S. Golubev, T. Kalvas, H. Koivisto, and O. Tarvainen, Rev Sci Instrum 85, 02A702 (2014).
Skalyga et al. (2016)
V. Skalyga, I. Izotov,
S. Golubev, A. Sidorov, S. Razin, A. Vodopyanov, O. Tarvainen, H. Koivisto, and T. Kalvas, Rev Sci Instrum 87, 02A716 (2016).
Skalyga et al. (2017)
V. A. Skalyga, I. V. Izotov,
A. V. Sidorov, S. V. Golubev, and S. V. Razin, Rev Sci Instrum 88, 033503 (2017).
Abramov et al. (2019)
I. S. Abramov, E. D. Gospodchikov, R. A. Shaposhnikov, and A. G. Shalashov, Nucl Fusion 59, 106004 (2019).
Griem (1974)
H. R. Griem, Spectral Line Broadening
by Plasma (Academic Press, New York, 1974).
Ochkin (2009)
V. N. Ochkin, Spectroscopy of Low
Temperature Plasma (Wiley-VCH, 2009). |
The Aleph Cosmological Principle${}^{1}$
D.S.L. Soares
Departamento de Física, ICEx, UFMG
— C.P. 702
30161-970, Belo Horizonte — Brazil
(November 26, 2020)
Abstract
A general cosmological principle — Aleph — is proposed as a
substitute to the Anthropic principle. Furthermore, the universe,
conceived as a world ensemble, is
characterized by many (possibly infinite) X-Life world
principles. The only known X-Life world principle recovers much of the
Anthropic conjecture. The inescapable final conclusion is the formulation
of the Strong Copernicus principle.
‘I saw the Aleph from every point and angle,
and in the Aleph I saw the earth and in the earth the Aleph.’
The Aleph, Jorge Luis Borges, 1945
11footnotetext: For more short comments on modern cosmology check
at the following address:
http://www.fisica.ufmg.br/ dsoares/notices.htm
1 Introduction
The Anthropic cosmological principle (Carter 1974, Barrow & Tipler 1986)
has been criticized, and eventually rejected as inadequate by some authors,
for being heavily inspired on unproved cosmological models, namely,
those known as Hot Big Bang models (Soares 2004a).
Carter presented his Anthropic principle in two versions, weak and strong,
whilst Barrow & Tipler described other versions. The great novelty lies
in the weak version. The discussion that is done here focuses,
therefore, upon the weak version of the principle.
In fact — and it is worth-stressing —, the different versions of the
Anthropic Principle are not different versions of the same principle
but rather are independent principles by themselves, which is
totally opposed to the view expressed mainly by Barrow & Tipler. Such
a thesis is further elaborated elsewhere (Soares 2004b).
Towards a broader and unprejudiced view, one can depart from the idea of
a universe as a world ensemble (e.g., Carter 1974), except that
in a different perspective from what is usually found in the literature,
namely, that of a multiverse (see details in Stoeger et al. 2004
and references therein). Let each world vector — i.e.,
each element in the ensemble —, in fact,
belong to the same universe, not being
a universe by itself, with its own cosmology, as assumed in the
usual world-ensemble approach. That
is to say, the total mass-energy content of the universe is given by
adding up the mass-energy content of each world vector. Furthermore, each
world vector is assumed as potentially suitable for the existence — or
development — of life. In other words, the overall conditions in that —
and all — world element are such that live organisms are bound to emerge.
It is thus characterized by a X-Life world principle, which simply
states that world is as it is because of restrictions imposed by X-Life being
the way it is. These are essentially the same words in Carter’s formulation
of his Anthropic principle. Here they are used in the context of a much
broader cosmological view as it will be apparent below.
The Anthropic principle has been used in many ways since its proposition.
A sort of strange devotion sometimes characterizes those dealing
with the principle. This resulted into an exaggerated bending
of the bow towards one direction. The situation, comprehensibly,
led Soares (2004a) to use irony on the whole issue, in an attempt to bend
the bow to the opposite direction, eventually reaching a state of
reasonable equilibrium. That is also the spirit pervading the present
essay, except that now with a grave approach.
2 The E-Life world principle
Ours — the only presently applicable X-Life world principle —,
conveniently, could be termed E-Life world principle, where
”E” stands obviously for ”Earth”. Since the DNA-molecule
is the unifying feature of terrestrial life, the principle
is thus stated as constraints derived from DNA-based life upon the world
vector properties. Very much so, it is the present general idea pervading the
Anthropic principle.
Except for the cosmological implications, much of the
conclusions derived from the Anthropic principle (Barrow & Tipler 1986)
surely still holds. For example, the prediction by Fred Hoyle concerning
the 7.7 MeV excited state of ${}^{12}$C, which was necessary in order to
increase the probability of the reaction between helium and beryllium
to produce carbon, might be considered as a genuine E-Life world principle
prediction. In 1952, from the evident abundance of carbon —
namely, E-Life —, Hoyle predicted the existence of a resonance
of ${}^{12}$C, in nuclear reactions, at around 7.7 MeV, and almost
immediately, in 1953, D.N.F. Dunbar, R.F. Pixley, W.A. Wenzel &
W. Whaling (1953), at Kellogg Radiation Laboratory, Caltech,
discovered a state with the correct properties, at 7.68$\pm$0.03 MeV
excitation energy. E-Life would not exist without the 7.7-MeV excited
state of ${}^{12}$C. Such a prediction is, of course, often mentioned
in classic Anthropic discussions (see Barrow & Tipler, p. 252).
The reason why general cosmological implications are not valid is that a given
cosmology must be applied to the whole world ensemble and not to a sole
element of it. Aleph is the applicable principle here (see below).
Cosmological predictions are always biased when based in a X-Life world
principle.
Intelligent life is always an issue whenever one speaks of life.
Intelligence, another variable in the general cosmological equation, is
not considered in the present discussion. Irrespective of
its prevalence, communications between world-ensemble elements, e.g.,
between a particular X-based organism and a DNA-based one, may or may
not be possible.
In any case, whether or not two elements of the world ensemble
are or may be connected in one or other way — communication being one
of them — is entirely irrelevant here.
3 A hypothetical X-Life world principle
Sagan & Salpeter (1976) discuss many aspects of a possible Jovian
biology, an investigation motivated mainly by the fact that contemporary
Jovian atmosphere has many similarities to the primitive terrestrial
atmosphere. They hypothesized the characteristics of Jovian live organisms —
in the form of sinkers and floaters, understandable in a gaseous
environment — departing from chemical composition, temperature,
density, pressure and other known features of the planet atmosphere.
Fundamental for the origin of life in Jupiter is the time-scale taken
by synthesized complex molecules to move towards large depths, as a
result of convective streaming. The time-scale should be short enough
to avoid reaching pyrolytic depths, which would severe restrict
the possibility of biological evolution.
Now, take the Sagan-Salpeter problem in the reverse order. Assume
sinkers and floaters are abundant in the Jovian atmosphere.
One may then formulate the J-Life world principle —
”J” for ”Jupiter”. With such a life principle, properties of the Jovian
atmosphere might be obtained in the same way E-Life — Anthropic —
predictions are made.
Incidentally, Jovian live balloons would be, in principle, totally
disconnected from DNA-based terrestrial life. In other words, J-Life
world and E-Life world would be disconnected from each other.
I am assuming here that Jovian organisms are not DNA-based, which may
not be true. But that does not invalidate the example.
4 Aleph and Copernicus principles
The Aleph Cosmological Principle is the underlying principle to the world
ensemble, i.e., to the universe. Predictions about the formation, evolution
and structure of the universe — cosmology — are related to the Aleph
principle. It is much more general in scope than each X-Life world principle.
Strictly speaking, X-Life world principles do not need intelligent
life to hold. In particular, E-Life world principle would still hold
even in the absence of mankind, of human beings. This is the essence
of what could be termed the Strong Copernicus principle. As long
as any sort of DNA-based life do exist, the E-Life principle would still
be there. Useless, due to the absence of intelligent life. But still there.
The Strong Copernicus principle does not require the existence of human
beings. Man is not central in the universe, it may even not exist. Putting
it in another form, amongst all DNA-based forms of life, man is not special.
Consciousness makes mankind different — not special — from other
E-life organisms, in the sense that mankind is dotted with moral
and ethical values. These are fundamental aspects of human life but do not
change the biological status of human life. In conclusion,
the Strong Copernicus principle is an imperative scientific
principle.
The philosophical implications herein are innumerable and will not be
treated here.
5 Conclusion
At this point in time there is only one X-Life principle known.
Conceivably, the Aleph cosmological principle is a matter of speculation.
Conceptually, however, it leads to a broader scenario for the knowledge of
the universe we live.
Soares (2001) suggests that the arrow of time is given by the prominence of
life, that is, the universe evolves towards life. In an eternal universe,
that would lead to the startling conclusion that the universe itself
is alive! The Aleph principle is, then, at a certain point, vindicated.
The Aleph cosmological principle is the Aleph-Life principle. It is of course
prompted to speculation what is the nature of Aleph-Life, in a way or other,
the live universe. Highly speculative matter, on the other hand,
scientifically unavoidable.
6 References
Barrow, J.D. & Tipler, F.J. 1986, The Anthropic cosmological
principle (Oxford University Press, Oxford)
Carter, B. 1974, Large number coincidences and the Anthropic principle
in cosmology, in Confrontation of cosmological theories with observational
data, IAU Symposium No. 63, Krakow, Poland, September 10-12, 1973.
ed. M.S. Longair, D. Reidel Publishing, Dordrecht, p. 291-298
Dunbar, D.N.F., Pixley, R.E., Wenzel, W.A. & Whaling, W. 1953,
The 7.68-Mev State in C${}^{12}$. Phys. Rev., 92, 649-650
Sagan, C. & Salpeter, E.E. 1976, Particles, environments, and
possible ecologies in the Jovian atmosphere. Astrophys. J. Supp., 32, 737-755.
Soares, D.S.L. 2001, Time is life,
http://arXiv.org/abs/astro-ph/0108180
Soares, D.S.L. 2004a, The Anthropic Fake Principle,
http://www.fisica.ufmg.br/˝dsoares/antr/fake.htm
Soares, D.S.L. 2004b, In preparation
Stoeger, W.R., Ellis, G.F.R. & Kirchner, U. 2004, Multiverses and Cosmology:
Philosophical Issues, http://arXiv.org/abs/astro-ph/0407329 |
Gauge Theories on Sphere and Killing Vectors
Rabin
Banerjee111On leave from S.N.Bose Natl. Ctr. for Basic Sciences,
Calcutta, India; e-mail:rabin@newton.skku.
ac.kr; rabin@bose.res.in
BK21 Physics Research Division and Institute
of Basic Science,
SungKyunKwan
University,
Suwon 440- 746, Republic of Korea
Abstract
We provide a general method for studying
manifestly $O(n+1)$ covariant formulation of $p$-form gauge
theories by stereographically projecting these theories,
defined in flat Euclidean space, onto the
surface of a hypersphere. The gauge fields in the two descriptions are mapped by conformal
Killing vectors while conformal Killing spinors are necessary for the matter fields, allowing
for a very transparent analysis and compact
presentation of results. General expressions for these Killing
vectors and spinors are given. The familiar results for a vector gauge
theory are reproduced.
1 Introduction
Stereographic projection has important applications in both physics and mathematics. In general
terms, streographic projection from an $n$-dimensional sphere $(S)$, embedded in $(n+1)$- dimensional
flat Euclidean space, onto a plane tangent to $S$ at $x$
is the map that projects each
point $P$ on
$S$, to the intersection $P^{\prime}$ of the line $Px^{\prime}$ with the plane, where $x^{\prime}$ is the
point opposite to $x$ on $S$.
In mathematics, it gives a picture of $S^{3}$ as $R^{3}\cup\{\infty\}$, that makes $S^{3}$ intuitively
more understandable than directly visualising it as the unit hypersphere $(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=1)$. Likewise. several models of hypebolic space are derived from more familiar models
(e.g. Poincare ball) by streographic projection [1].
In the realm of physics, manifestly $O(n+1)$ covariant formulation of vector gauge theories
is done by stereographically projecting the usual gauge theory, defined on the Euclidean
plane, onto
the hypersphere[2, 3]. It provides a deep understanding of the topological properties of a
gauge theory[4, 5] including, for instance, the connection between the axial anomaly and the
index theorem[6].
Streographic projection of a vector gauge theory is usually presented as a set of formulas
relating the gauge as well as matter sectors in the usual flat space and hyperspherical
formulations[7]. Once this dictionary is established, the rest are technical details.
The derivation and significance of these formulas are, however, not particularly illuminating.
Also, their connection with the precise map among the coordinates of the plane and the sphere,
obtained from a geometrical construction, is lost. Moreover, explicit results are given
only for vector gauge theories in two and four dimensions; so it is not clear how far such
results are general and whether it is possible to analyse $p$-form gauge theories or
include higher dimensions. Indeed since antisymmetric tensor fields do occur in various contexts,
it is desirable, if not essential, to have a manifestly covariant formulation of such
theories for the precise reasons that led to the original treatment of the vector gauge
theory; namely, to avoid the ambiguities inherent in a flat space treatment by working
on a compact manifold.
In this paper we discuss a method, relying on first principles,
for obtaining the various formulas and expressions providing the transition from the flat
space to the hypersphere. It is
based on the observation that a stereographic projection is a conformal transformation.
We show that quantities in the gauge sector (like gauge fields, field strengths etc.)
in the two descriptions are related by rules similar to usual tensor analysis, with the
conformal Killing vectors playing the role of the metric. The explicit structures of these vectors
is derived by solving the Cartan-Killing equation. Using this formalism, results for
a vector gauge theory are economically reproduced, apart from the fact that
they become quite transparent. It is then extended to obtain new
results for a $p$-form gauge theory. Although the two form case has been actually done
in details, the treatment for higher forms parallels this example and is
an obvious generalisation.
The analysis is done for arbitrary dimensions and nonabelian gauge groups have been
considered.
The inclusion of matter sector is also possible within this scheme. Just as gauge fields
are related by conformal Killing vectors, the matter fields in the flat and hyperspherical
surfaces are connected by conformal Killing spinors. We derive an equation where these Killing
vectors are expressed as bilinear combinations of Killing spinors. The solution to this
equation yields the structure of these spinors. In this manner conformal
Killing spinors on $S^{n}$
are given by a compact formula.
The paper is organised as follows: section 2 analyses the connection between stereographic
projection and conformal Killing vectors, including a derivation of the latter from the
Cartan-Killing equation; sections 3 and 4 treat the covariant formulation of a vector
and a two form gauge theory, respectively, pointing out the differences
between the two; also, a new gauge symmetry for a two form theory on a hypersphere is
noted that does not have any analogue in the flat space; section 5 discusses the group contraction
of $SO(3)$ to $E(2)$ by means of the Killing vectors found here; section 6 contains
concluding remarks.
2 Stereographic Projection and Killing Vectors: A First Principle Analysis
A manifestly $O(n+1)$-covariant formulation of a gauge theory is attained
by stereographically projecting $n$-dimensional Euclidean
space onto the surface of a unit hypersphere embedded in $(n+1)$-dimensional
Euclidean space. Introducing the $(n+1)$-dimensional coordinates on the
unit hypersphere by $r_{a}(a=1,2,......(n+1))$ with $r_{a}r_{a}=1$,
and the $n$-dimensional coordinates on the hyperplane by $x_{\mu}(\mu=1,2......n)$,
then a mapping from the south pole yields the familiar relations222Latin labels run from 1 to $(n+1)$, while Greek labels run from 1 to $n$,
$$r_{\mu}=\frac{2x_{\mu}}{1+x^{2}}$$
(1)
$$r_{(n+1)}=\frac{1-x^{2}}{1+x^{2}}$$
(2)
The gauge potentials on the sphere (denoted by a caret) and the
conventional ones are likewise related by [2, 3, 4],
$$\hat{A}_{\mu}=\frac{1+x^{2}}{2}A_{\mu}-x_{\mu}x_{\nu}A^{\nu}$$
(3)
$$\hat{A}_{(n+1)}=-x_{\mu}A^{\mu}$$
(4)
It is easy to see that the inverse map is provided by,
$$x_{\mu}=\frac{r_{\mu}}{1+r_{3}}$$
(5)
and,
$$\frac{1+x^{2}}{2}A_{\mu}=\hat{A}_{\mu}-x_{\mu}\hat{A}_{(n+1)}$$
(6)
While (1) and (2) are a consequence of a straightforward geometrical exercise, the mapping among the potentials seems somewhat obscure, relegating it to a matter of definition. As far as degrees of freedom are concerned, these are preserved. The $(n+1)$-dimensional $\hat{A}$ variables are mapped to the $n$-dimensional $A$ variables, subjected to the constraint,
$$r_{a}\hat{A}_{a}=0$$
(7)
implying that the $\hat{A}$-fields live on the tangent space of the hypersphere.
We now provide a systematic
derivation of (3) and (4), based on the symmetries of the problem.
There is a mapping among the symmetries of the plane and the sphere
(e.g. translations on the plane correspond to rotations on the sphere)
that is captured by the relevant Killing vectors.
Moreover, stereographic projection is known to be a
conformal transformation. Even geometrically it can be proved
that stereographic projection from a sphere to a plane
is identical to an inversion
(which is a discrete transformation of the conformal
group) in a sphere of twice the radius [1].
Variables on the sphere and plane should thus be related by
conformal Killing vectors. We may write this relation as,
$$\hat{A}_{a}=K_{a}^{\mu}A_{\mu}+r_{a}\phi$$
(8)
where the conformal Killing vectors satisfy the transversality condition,
$$r_{a}K_{a}^{\mu}=0$$
(9)
and an additional scalar field $\phi$, which is just the normal component of $\hat{A}_{a}$, is introduced,
$$\phi=r_{a}\hat{A}_{a}$$
(10)
The $(n+1)$ components of $\hat{A}$ are expressed in terms of the $n$ components of $A$ plus a scalar degree of freedom. To simplify the analysis the scalar field is put to zero. It is straightforward to resurrect it by using the above equations. With the scalar field gone, $\hat{A}$ is now given by,
$$\hat{A}_{a}=K_{a}^{\mu}A_{\mu}$$
(11)
and satisfies the condition (7), enabling a suitable comparison.
The conformal Killing vectors $K_{a}^{\mu}$ are now determined. These should satisfy the Cartan-Killing equation which, specialised to a flat $n$-dimensional manifold, is given by,
$$\partial^{\nu}K_{a}^{\mu}+\partial^{\mu}K_{a}^{\nu}=\frac{2}{n}\partial^{%
\lambda}K_{a}^{\lambda}\delta_{\mu\nu}$$
(12)
The most general solution for this equation (except for $n=2$) is given by [8],
$$K_{a}^{\mu}=t_{a}^{\mu}+\epsilon_{a}x^{\mu}+\omega_{a}^{\mu\nu}x_{\nu}+\lambda%
_{a}^{\mu}x^{2}-2\lambda_{a}^{\sigma}x_{\sigma}x^{\mu}$$
(13)
where $\omega^{\mu\nu}=-\omega^{\nu\mu}$.
The various transformations of the conformal group are characterised by the parameters appearing in the above equation; translations by $t$, dilitations by $\epsilon$, rotations by $\omega$ and inversions (or the special conformal transformations) by $\lambda$. Imposing the condition (9) and equating coefficients of terms with distinct powers of $x$, we find,
$$\displaystyle t_{(n+1)}^{\mu}$$
$$\displaystyle=$$
$$\displaystyle 0$$
(14)
$$\displaystyle 2x_{\nu}t_{\nu}^{\mu}+\omega^{\mu\nu}_{(n+1)}x_{\nu}+\epsilon_{(%
n+1)}x^{\mu}$$
$$\displaystyle=$$
$$\displaystyle 0$$
(15)
$$\displaystyle-x^{2}t_{(n+1)}^{\mu}+2ix_{\nu}\epsilon_{\nu}x^{\mu}+2x_{\nu}%
\omega_{\nu\sigma}^{\mu}x^{\sigma}+\lambda^{\mu}_{(n+1)}x^{2}-2\lambda^{\sigma%
}_{(n+1)}x_{\sigma}x^{\mu}$$
$$\displaystyle=$$
$$\displaystyle 0$$
(16)
$$\displaystyle 2x_{\nu}\lambda_{\nu}^{\mu}x^{2}-4\lambda_{\nu}^{\sigma}x_{\nu}x%
_{\sigma}x^{\mu}-x^{2}\epsilon_{(n+1)}x^{\mu}-\omega_{(n+1)\nu}^{\mu}x^{\nu}x^%
{2}$$
$$\displaystyle=$$
$$\displaystyle 0$$
(17)
$$\displaystyle\lambda_{(n+1)}^{\mu}x^{2}-2\lambda_{(n+1)}^{\sigma}x_{\sigma}x^{\mu}$$
$$\displaystyle=$$
$$\displaystyle 0$$
(18)
Contracting the above equations (except the first one) by $x_{\mu}$ yields,
$$\displaystyle 2x_{\mu}x_{\nu}t_{\nu}^{\mu}+\epsilon_{(n+1)}x^{2}$$
$$\displaystyle=$$
$$\displaystyle 0$$
(19)
$$\displaystyle 2\epsilon_{\mu}x_{\mu}-\lambda_{(n+1)}^{\mu}x_{\mu}$$
$$\displaystyle=$$
$$\displaystyle 0$$
(20)
$$\displaystyle 2\lambda_{\nu}^{\sigma}x_{\sigma}x^{\nu}+x^{2}\epsilon_{(n+1)}$$
$$\displaystyle=$$
$$\displaystyle 0$$
(21)
$$\displaystyle x_{\mu}x^{2}\lambda_{(n+1)}^{\mu}$$
$$\displaystyle=$$
$$\displaystyle 0$$
(22)
Since $x_{\mu}$ is completely arbitrary, we find that $\lambda_{(n+1)}^{\mu}=\epsilon^{\mu}=0$. Using this in (16)
it is seen that $\omega_{\nu\sigma}^{\mu}=0$. Equations (19) and (21) determine the symmetric parts of $t$ and $\lambda$, respectively, as,
$$(t_{\nu}^{\mu})_{symm.}=(\lambda_{\nu}^{\mu})_{symm.}=-\frac{1}{2}\delta_{\nu}%
^{\mu}\epsilon_{(n+1)}$$
(23)
Their antisymmetric parts are determined in terms of the antisymmetric tensor $\omega$, on using (15) and (17),
so that the complete solution has the form,
$$\displaystyle t_{\nu}^{\mu}$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{2}(\delta_{\nu}^{\mu}\epsilon_{(n+1)}+\omega^{\mu}_{(n+%
1)\nu})$$
(24)
$$\displaystyle\lambda_{\nu}^{\mu}$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{2}(\delta_{\nu}^{\mu}\epsilon_{(n+1)}-\omega^{\mu}_{(n+%
1)\nu})$$
(25)
The Cartan-Killing equation (12) is linear in the Killing vectors, which can thus be appropriately scaled. This freedom is exploited in setting the parameter of the scale transformations $\epsilon_{(n+1)}=-1$.
That leaves only $\omega^{\mu}_{(n+1)\nu}$ undetermined. This is set to zero by requiring that the Killing vectors are symmetric under interchange of $\mu$ and $\nu$, so that the rotational symmetry of the problem is preserved. Thus all the parameters have been fixed. The explicit structures of the Killing vectors, isolating the $(n+1)-th$ component are now written,
$$K_{\nu}^{\mu}=\frac{1+x^{2}}{2}\delta_{\nu}^{\mu}-x_{\mu}x_{\nu}$$
(26)
$$K_{(n+1)}^{\mu}=-x_{\mu}$$
(27)
Susequently we shall show that the standard forms of the Killing vectors, either on the sphere or on the plane, are recovered from this result.
With the above solution for the Killing vectors, eq. (11) reproduces the defining relations for the potentials (3) and (4), thereby completing a systematic derivation for them.
There are two useful relations satisfied by these Killing vectors, namely;
$$K_{a}^{\mu}K_{a}^{\nu}=\Big{(}\frac{1+x^{2}}{2}\Big{)}^{2}\delta^{\mu\nu}$$
(28)
and,
$$K_{a}^{\mu}K_{b}^{\mu}=\Big{(}\frac{1+x^{2}}{2}\Big{)}^{2}(\delta_{ab}-r_{a}r_%
{b})$$
(29)
Note that the conformal factor that relates the volume element on the hypersphere with that in the $n$-dimensional flat manifold,
$$d^{n}x=\Big{(}\frac{1+x^{2}}{2}\Big{)}^{n}d\Omega$$
(30)
naturally emerges in (28) and (29). Relation (28) hows that the product of the Killing vectors with repeated ‘a’ indices yields, up to the conformal factor, the induced metric. It is exactly the induced metric only in
$n=2$, which is a special case. The other relation can be interpreted as the transversality condition emanating from (9).
For computing derivatives involving Killing vectors, a particularly useful identity is given by,
$$K_{a}^{\mu}\partial_{\mu}K_{a}^{\nu}=(2-n)\Big{(}{{1+x^{2}}\over 2}\Big{)}x_{\nu}$$
(31)
The relation (27) shows that the $(n+1)$-th component
is just given by the dilatation (scaling), while the other
components (given by (26) involve the special conformal transformations and the translations.
3 Covariant Formulation of Vector Gauge Theory
In this section we discuss the manifestly $O(n+1)$-covariant formulation of a nonabelian vector gauge theory.
The theory is obtained by stereographically projecting the usual theory defined on the $n$-dimensional Euclidean plane
on to the unit hypersphere embedded in $(n+1)$-dimensional Euclidean space.
Results are known for the specific cases
of two and four dimensions [2, 3, 4, 6].
We present the analysis for any $n$-dimensions. This is the first use of the
generalisation effected by working in terms of the Killing vectors.
The pure Yang-Mills theory on the Euclidean space is governed by the standard Lagrangian,
$${\cal L}=-{1\over 4}F_{\mu\nu}^{i}F_{\mu\nu}^{i}$$
(32)
where the field tensor is given by,
$$F_{\mu\nu}^{i}=\partial_{\mu}A_{\nu}^{i}-\partial_{\nu}A_{\mu}^{i}+f^{ijk}A_{%
\mu}^{j}A_{\nu}^{k}$$
(33)
and $f^{ijk}$ are the structure constants of the gauge group.
To define this theory on the hypersphere, the map among the potentials is first stated,
$$\hat{A}_{a}^{i}=K_{a}^{\mu}A_{\mu}^{i}+r_{a}\phi^{i}$$
(34)
which is a generalisation of (8). The scalar field (with $i$-multiplets) is included for the sake
of completeness, but will be discarded subsequently. These fields are the normal components of
the potentials,
$$\phi^{i}=r_{a}\hat{A}_{a}^{i}$$
(35)
and are the analogues of (10).
They have a role in defining the shift transformations on the hypersphere. In order to see this, note
that the usual gauge tranformations are given by,
$$\delta A_{\mu}^{i}=\partial_{\mu}\lambda^{i}+f^{ijk}A_{\mu}^{j}\lambda^{k}$$
(36)
The gauge transformations of the fields $(\hat{A}_{a}^{i})$ on the hypersphere
are thus derivable from (34),
$$\delta\hat{A}_{a}^{i}=K_{a}^{\mu}\partial_{\mu}\lambda^{i}+f^{ijk}(\hat{A}_{a}%
^{j}-r_{a}r_{b}A_{b}^{j})\lambda^{k}+r_{a}\Lambda^{i}$$
(37)
where the definition (35) for the scalar fields has been used and we have set $\delta\phi^{i}=\Lambda^{i}$.
It can be put in a more tractable form by introducing the angular
momentum operator,
$$L_{ab}=r_{a}p_{b}-r_{b}p_{a}=-i(r_{a}\partial_{b}-r_{b}\partial_{a})$$
(38)
By using (1) and (2) it is possible to express the derivatives in terms of
those occurring in the plane and it is found that,
$$L_{ab}=-i(r_{a}K_{b}^{\mu}-r_{b}K_{a}^{\mu})\partial_{\mu}$$
(39)
Contracting by $r_{a}$ and using the tranversality condition (9)
yields,
$$r_{a}L_{ab}=-iK_{b}^{\mu}\partial_{\mu}$$
(40)
The transformation law (37) is thus expressed as,
$$\delta\hat{A}_{a}^{i}=ir_{b}L_{ba}\lambda^{i}+f^{ijk}(\hat{A}_{a}^{j}-r_{a}r_{%
b}A_{b}^{j})\lambda^{k}+r_{a}\Lambda^{i}$$
(41)
The transformation of the scalars is completely arbitrary since the defining relation
(35) does not constrain $\Lambda^{i}$ in any way, as may be easily seen by taking
the variations on both sides of that equation. In fact it corresponds to a shift symmetry,
while the gauge symmetry is governed by the parameter $\lambda^{i}$. To compare with the
transformation law given in the literature, it is necessary to consider the potentials
defined on the tangent plane of the hypersphere which, as has already been discussed,
corresponds to setting the scalar field (35) to zero. Then the gauge transformation
is given by the first two terms in (41), which agrees with the known form [4].
We have however provided a systematic derivation of this gauge transformation.
Although the infinitesimal gauge transformations were considered, it is quite simple to
construct the finite version. If the ordinary potential transforms as,
$$A_{\mu}^{\prime}=U^{-1}(A_{\mu}+\partial_{\mu})U$$
(42)
then the projected potential transforms as,
$$\hat{A}_{a}^{\prime}=K_{a}^{\mu}A_{\mu}^{\prime}=U^{-1}(\hat{A}_{a}+ir_{b}L_{%
ba})U$$
(43)
obtained by using (34) and (40).
The next issue concerns a suitable mapping of the field tensors so that the
appropriate actions may be obtained. Also, from now on we set the scalar field
to zero to facilitate comparison with existing results. Since translations on a plane are
equivalent to rotations on a sphere, usual derivatives should be replaced by angular
derivatives. The field tensor on the sphere is therefore a three index object.
It should thus be mapped to the field tensor on the flat space by the following
operation involving the Killing vectors,
$$\hat{F}_{abc}^{i}=\Big{(}r_{a}K_{b}^{\mu}K_{c}^{\nu}+r_{b}K_{c}^{\mu}K_{a}^{%
\nu}+r_{c}K_{a}^{\mu}K_{b}^{\nu}\Big{)}F_{\mu\nu}^{i}$$
(44)
so that symmetry properties under exchange of the indices is correctly preserved.
To show that this definition is equivalent to that followed in the literature [4],
$$\hat{F}_{abc}^{i}=\Big{(}iL_{ab}\hat{A}_{c}^{i}+f^{ijk}r_{a}\hat{A}_{b}^{j}%
\hat{A}_{c}^{k}\Big{)}+c.p.$$
(45)
(where $c.p.$ stands for the other pair of terms involving cyclic permutations in $a,b,c$), equations
(34) and (39) are used to simplify (45),
$$\hat{F}_{abc}^{i}=\Big{(}r_{a}K_{b}^{\mu}-r_{b}K_{a}^{\mu}\Big{)}\partial_{\mu%
}\Big{(}K_{c}^{\nu}A_{\nu}^{i}\Big{)}+f^{ijk}r_{a}\Big{(}K_{b}^{\nu}A_{\nu}^{j%
}\Big{)}\Big{(}K_{c}^{\mu}A_{\mu}^{k}\Big{)}+c.p.$$
(46)
The derivatives acting on the Killing vectors sum up to zero on account of the identity,
$$\Big{(}r_{a}K_{b}^{\mu}-r_{b}K_{a}^{\mu}\Big{)}\partial_{\mu}K_{c}^{\nu}+c.p.=0$$
(47)
The derivatives acting on the potentials, together with the other pieces, combine to
reproduce (44), thereby completing the proof.
The components of the field tensor are easily read-off from (44),
$$\hat{F}_{\mu\nu\sigma}^{i}={{1+x^{2}}\over 2}\Big{(}x_{\mu}F_{\nu\sigma}^{i}+x%
_{\nu}F_{\sigma\mu}^{i}+x_{\sigma}F_{\mu\nu}^{i}\Big{)}$$
(48)
and,
$$\hat{F}_{\mu\nu(n+1)}^{i}={{1+x^{2}}\over 2}\Big{(}{{1-x^{2}}\over 2}F_{\mu\nu%
}^{i}-x_{\mu}x_{\rho}F_{\nu\rho}^{i}-x_{\nu}x_{\rho}F_{\rho\mu}^{i}\Big{)}$$
(49)
which is valid for any dimensions. For the particular examples in two and four
dimensions only, these were given in [7]. Indeed
the two dimensional case is special where only the
second component survives, which simplifies to,
$$\hat{F}_{123}^{i}=({{1+x^{2}}\over 2})^{2}F_{12}^{i}$$
(50)
Having obtained the map of the field tensor, it is straightforward to obtain the
action. Taking the repeated product of the field tensor (44) and using the
transversality of the Killing vectors, we get,
$$\hat{F}_{abc}^{i}\hat{F}_{abc}^{i}=3\Big{(}K_{a}^{\mu}K_{a}^{\nu}K_{b}^{%
\lambda}K_{b}^{\rho}\Big{)}F_{\mu\nu}^{i}F_{\lambda\rho}^{i}$$
(51)
Finally, using (28), we obtain,
$$\hat{F}_{abc}^{i}\hat{F}_{abc}^{i}=3\Big{(}{{1+x^{2}}\over 2}\Big{)}^{4}F_{\mu%
\nu}^{i}F_{\mu\nu}^{i}$$
(52)
a result that is valid in any dimensions. In particular, on $S^{4}$, the conformal factor exactly cancels
and the actions on the flat space and the hypersphere are identified as,
$$S=-{1\over 4}\int d^{4}xF_{\mu\nu}^{i}F_{\mu\nu}^{i}=-{1\over 12}\int d\Omega%
\hat{F}_{abc}^{i}\hat{F}_{abc}^{i}$$
(53)
The lagrangian following from this action,
$${\cal L}_{\Omega}=-{1\over 12}\hat{F}_{abc}^{i}\hat{F}_{abc}^{i}$$
(54)
is taken as the starting point of all
computations on the hypersphere.
The inverse relations are obtained on using the properties of the Killing vectors.
For example, multiplying (34) by $K_{a}^{\nu}$ and using (28), it follows,
$$A_{\nu}^{i}=\Big{(}{2\over{1+x^{2}}}\Big{)}^{2}K_{a}^{\nu}\hat{A}_{a}^{i}$$
(55)
It is possible to show, after some amount of algebra, that this relation is equivalent
to (6).
Likewise, contracting (44) by $r_{a}$ and using the transversality of the Killing vectors, we obtain,
$$r_{a}\hat{F}_{abc}^{i}=\Big{(}K_{b}^{\mu}K_{c}^{\nu}\Big{)}F_{\mu\nu}^{i}$$
(56)
from which, proceeding as before in deriving (55), we obtain the final form,
$$F_{\mu\nu}^{i}=\Big{(}{2\over{1+x^{2}}}\Big{)}^{4}r_{a}K_{b}^{\mu}K_{c}^{\nu}%
\hat{F}_{abc}^{i}$$
(57)
3.1 Matter Fields and Killing Spinors
The inclusion of the matter sector is also done with the help of the Killing vectors.
Form invariance of the interaction requires that,
$$\int dx(j_{\mu}^{i}A_{\mu}^{i})=\int d\Omega(\hat{j}_{a}^{i}\hat{A}_{a}^{i})$$
(58)
where $j_{\mu}$ and $\hat{j}_{a}$ are the currents in the two decsriptions.
It is clear therefore that the currents are also mapped by a relation similar to
(34). However since the measure is given by (30), the currents will
involve the conformal factor, depending on the dimensions. For the case of two dimensions,
it is exactly identical to (34),
$$\hat{j}_{a}^{i}=K_{a}^{\mu}j_{\mu}^{i}$$
(59)
while for four dimensions, the conformal factor appears,
$$\hat{j}_{a}^{i}=\Big{(}{{1+x^{2}}\over 2}\Big{)}^{2}K_{a}^{\mu}j_{\mu}^{i}$$
(60)
Higher powers of the conformal factor occur for higher dimensions. Likewise,
the axial vector current is also defined. These forms
of the fermionic current have been postulated in the literature [2], particularly in
the discussion of anomalies and their connection with index theorems [6].
In the language of the Killing vectors the anomaly equation
can be expressed in a very economical manner,
as analysed below.
An essential ingredient in the discussion on anomalies is the completely antisymmetric
tensor $\epsilon_{\mu\nu\lambda\rho....}$ whose value is the same in all systems. It is necessary
to define the analogue of this tensor $\epsilon_{abcd...}$ on the hypersphere.
We adopt the same rule (44) used for defining the antisymmetric field tensor.
However there is a slight subtlety. Strictly speaking, this Levi-Civita epsilon is a tensor density.
Hence its transformation law is modified by appropriate weight factors. For two dimensions,
it is given by,
$$\epsilon_{abc}=\Big{(}{2\over{1+x^{2}}}\Big{)}^{2}\Big{(}r_{a}K_{b}^{\mu}K_{c}%
^{\nu}+r_{b}K_{c}^{\mu}K_{a}^{\nu}+r_{c}K_{a}^{\mu}K_{b}^{\nu}\Big{)}\epsilon_%
{\mu\nu}$$
(61)
while in four dimensions it is given by,
$$\epsilon_{abcde}=\Big{(}{2\over{1+x^{2}}}\Big{)}^{4}\Big{(}r_{a}K_{b}^{\mu}K_{%
c}^{\nu}K_{d}^{\lambda}K_{e}^{\rho}+cyclic\,\,\,\,permutations\,\,\,in\,\,\,(a%
,b,c,d,e)\Big{)}\epsilon_{\mu\nu\lambda\rho}$$
(62)
It is possible to verify the above relation by an explicit calculation, taking the convention
that both the epsilons are $+1(-1)$ for any even (odd) permutation of distinct entries $(1,2,3,...)$ in that order.
Similar extension to other higher dimensions is straightforward.
The axial $U(1)$-anomaly in two and four dimensions is known to be given by,
$$\partial_{\mu}j_{\mu 5}={1\over 2\pi}\epsilon_{\mu\nu}F_{\mu\nu}$$
(63)
$$\partial_{\mu}j_{\mu 5}={1\over{16\pi^{2}}}\epsilon_{\mu\nu\lambda\rho}F_{\mu%
\nu}F_{\lambda\rho}$$
(64)
Using (40) and the definition of the currents (59), (60), it is possible to obtain the
identification,
$$ir_{a}L_{ab}\hat{j}_{b5}=\Big{(}{{1+x^{2}}\over 2}\Big{)}^{2}\partial_{\mu}j_{%
\mu 5}$$
(65)
for two dimensions and,
$$ir_{a}L_{ab}\hat{j}_{b5}=\Big{(}{{1+x^{2}}\over 2}\Big{)}^{4}\partial_{\mu}j_{%
\mu 5}$$
(66)
for four dimensions. In getting at the final result, use was made of the identity
(31). It is easy to extend this identification for higher dimensions, with
appropraite increase in the powers of the weight factor.
The explicit expressions for the anomaly are also identified with the minimum of effort.
For two dimensions it trivially follows from (50),
$$\epsilon_{abc}\hat{F}_{abc}=6\hat{F}_{123}=3\Big{(}{{1+x^{2}}\over 2}\Big{)}^{%
2}\epsilon_{\mu\nu}F_{\mu\nu}$$
(67)
For four dimensions, using (44) and (62), it follows that,
$$r_{a}\epsilon_{bcdef}\hat{F}_{abc}\hat{F}_{def}=3\Big{(}{{1+x^{2}}\over 2}\Big%
{)}^{4}\epsilon_{\mu\nu\lambda\rho}F_{\mu\nu}F_{\lambda\rho}$$
(68)
The structural similarity among
the results in two and four dimensions is obvious.
Once again, extension to other dimensions is equally
clear and can be done naturally.
Inclusion of the nonabelian gauge group also poses no problems and is done by
inserting suitable indices. The weight factors cancel out from both sides of the
anomaly equation, which now takes its familiar form [6, 9],
$$ir_{a}L_{ab}\hat{j}_{b5}={1\over{48\pi^{2}}}r_{a}\epsilon_{bcdef}\hat{F}_{abc}%
\hat{F}_{def}$$
(69)
Before concluding this section, we derive the map among the basic fermion fields.
From the transversality of the Killing
vectors it is clear that the currents, as defined in (59) or (60)
(including other dimensions), must satisfy,
$$r_{a}\hat{j}_{a}=0$$
(70)
Thus, modulo a normalisation, the fermionic current on the hypersphere must be given by,
$$\hat{j}_{a}=\hat{\psi}^{\dagger}(\delta_{ab}-r_{a}r_{b})\sigma_{b}\hat{\psi}$$
(71)
where $\sigma_{b}$ are a set of hermitian matrices generating the Clifford algebra,
$\{\sigma_{a},\sigma_{b}\}=2\delta_{ab}$. For $S^{2}$ embedded in $R^{3}$, these are the standard Pauli matrices.
Also note that since the map between the fermion variables will be linear, the normalisation
may be absorbed and thus set to unity.
Let us now perform the analysis for $S^{2}$. The examples of $S^{4}$ or other hyperspheres may be done
analogously. Using (59) and the identity (29), we obtain,
$$\Big{(}{2\over{1+x^{2}}}\Big{)}^{2}\hat{\psi}^{\dagger}K_{a}^{\lambda}K_{b}^{%
\lambda}\sigma_{b}\hat{\psi}=\psi^{\dagger}K_{a}^{\mu}\gamma_{\mu}\psi$$
(72)
where the Euclidean gamma matrices are given by $\gamma_{\mu}=\epsilon_{\alpha\mu}\sigma_{\alpha}.$
This may be simplified as,
$$\hat{\psi}^{\dagger}K_{a}^{\lambda}\sigma_{a}\hat{\psi}=\Big{(}{{1+x^{2}}\over
2%
}\Big{)}^{2}\psi^{\dagger}\gamma_{\lambda}\psi$$
(73)
Now defining the map between the fermi fields by,
$$\psi=\Big{(}{2\over{1+x^{2}}}\Big{)}W\hat{\psi}$$
(74)
where $W$ is the transformation matrix, we get the following condition,
$$K_{a}^{\lambda}\sigma_{a}=W^{\dagger}\gamma_{\lambda}W$$
(75)
To solve this equation it becomes convenient to square both sides,
$$2\Big{(}\frac{1+x^{2}}{2}\Big{)}^{2}=(W^{\dagger}\gamma_{\lambda}W)(W^{\dagger%
}\gamma_{\lambda}W)$$
(76)
obtained on using the identity (29) and properties of the Pauli matrices.
Since $\gamma_{\lambda}\gamma_{\lambda}=2$, it leads to the following simplification,
$$WW^{\dagger}=W^{\dagger}W=\Big{(}\frac{1+x^{2}}{2}\Big{)}$$
(77)
This suggests the following solution,
$$W={1\over{\sqrt{2}}}(1-i\gamma_{\mu}x_{\mu})$$
(78)
which is checked from (75) by using
the known forms for the Killing vectors.
Putting this in (74), the result for the mapping of the fermion
fields quoted in the literature [2, 4, 6] is reproduced. For four
dimensions, expectedly the only change comes in the weight factor, which now occurs
with a power of 2,
$$\psi=\Big{(}{2\over{1+x^{2}}}\Big{)}^{2}W\hat{\psi}$$
(79)
Similar features hold in other higher dimensions.
The matrix $W$, on the other hand, is fundamental, as is now elucidated.
Equation (75) shows that the conformal Killing vectors are expressed as a bilinear
combination of $W^{\prime}s$. Consequently these are related to the conformal Killing spinors.
As is known [10], the Killing spinors on $S^{n}$ are expressed as the product of a
matrix with an arbitrary constant spinor. In the case of $S^{2}$, for example,
the Killing spinor is given by
$\epsilon=\Omega\left(\begin{array}[]{c}a\\
b\end{array}\right)$. A bilinear combination yields the Killing vector $K^{\mu}=\epsilon^{\prime\dagger}\gamma^{\mu}\epsilon$, where $\epsilon^{\prime}=\Omega\left(\begin{array}[]{c}a^{\prime}\\
b^{\prime}\end{array}\right)$. The transformation matrix $\Omega$ multiplying the constant spinor is
found by solving a consistency condition. The formalism here naturally
yields the conformal Killing spinors with $W$ replacing the matrix $\Omega$.
Usually these matrices are given in terms of the Euler angles, but that can be
trivially obtained here by using the formulas related to stereographic projection
and finally passing over to the polar variables. In particular, using the parametrisation
(117) used later, the structure for the matrix turns out to be,
$$\sqrt{2}W=\left(\begin{array}[]{clcr}1&e^{-i\phi}tan\frac{\theta}{2}\\
-e^{i\phi}tan\frac{\theta}{2}&1\end{array}\right)$$
(80)
For Killing spinors on $S^{n}$, the matrix $\Omega$ turns out to be unitary333Although this
was not discussed in [10], it can be proved from their results. Hence, using (77),
it transpires that the matrices $W$ and $\Omega$ are, up to a conformal factor, unitarily
equivalent.
Relation (74) shows that the mapping among the matter fields is effected by
the conformal Killing spinors. This complements the mapping in the gauge sector done by the
conformal Killing vectors.
4 Covariant Formulation of Antisymmetric Tensor Gauge Theory
The general formalism developed so far is particularly suited for obtaining a
covariant formulation of $p$-form gauge theories. Here we discuss it for the
second rank antisymmetric tensor gauge theory. Also, there are some features
which distinguish it from the analysis for the vector gauge theory.
The extension for higher forms
is obvious. Both abelian and nonabelian theories will be considered. To set
up the formulation it is convenient to begin with the abelian case which can
be subsequently generalised to the nonabelian version. The action for a free
2-form gauge theory in flat $n$-dimensional Euclidean space is given by [11],
$$S=-{1\over{12}}\int d^{n}xF_{\mu\nu\rho}F_{\mu\nu\rho}$$
(81)
where the field strength is defined in terms of the basic field as,
$$F_{\mu\nu\rho}=\partial_{\mu}B_{\nu\rho}+\partial_{\nu}B_{\rho\mu}+\partial_{%
\rho}B_{\mu\nu}$$
(82)
The infinitesimal gauge symmetry is given by the transformation,
$$\delta B_{\mu\nu}=\partial_{\mu}\Lambda_{\nu}-\partial_{\nu}\Lambda_{\mu}$$
(83)
which is reducible since it trivialises for the choice $\Lambda_{\mu}=\partial_{\mu}\lambda$.
It is sometimes useful to express the action (or the lagrangian) in a first order form
by introducing an extra field,
$${\cal L}=-{1\over{8}}\epsilon_{\mu\nu\rho\sigma}F_{\mu\nu}B_{\rho\sigma}+{1%
\over 8}A_{\mu}A_{\mu}$$
(84)
where the $B\wedge F$ term involves the field tensor,
$$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$$
(85)
Eliminating the auxiliary $A_{\mu}$ field by using its equation of motion, the previous
form (81) is reproduced. The gauge symmetry is given by (83) together with
$\delta A_{\mu}=0$. The first order form is ideal for analysing the nonabelian theory.
To express the theory on the hypersphere, the mapping of the tensor field is first given.
From the previous analysis, it is simply given by,
$$\hat{B}_{ab}=K_{a}^{\mu}K_{b}^{\nu}B_{\mu\nu}+r_{a}\phi_{b}-r_{b}\phi_{a}$$
(86)
The tensor field with the latin indices is defined on the hypersphere while those with
the greek symbols are the usual one on the flat space. There are additional vector fields
$\phi_{a}$, which are the analogues of the scalar field $\phi$, given in (34).
However, there is a point of distinction
from the vector theory, which is explained by counting the degrees of freedom. The
vector field on $S^{n}$ embedded in $R^{n+1}$ had $(n+1)$-components that were expressed in
terms of the $n$-components of the usual vector on $R^{n}$ plus one extra scalar degree of
freedom. Subsequently by constraining the vector field to lie on the tangent plane of the
hypersphere, the scalar field could be set to zero. For a rank 2 tensor on the hypersphere
there are ${{(n+1)n}\over 2}$ components, while on the flat space it has ${{n(n-1)}\over 2}$
components. Their difference is $n$. Since $\phi_{a}$ has $(n+1)$ components, all of them cannot
be independent. We have to impose an additional constraint; their normal component is set to
zero,
$$r_{a}\phi_{a}=0$$
(87)
so that there is a correct matching of the degrees of freedom.
By contracting (86) with $r_{a}$, using the above equation and the transversality of the
Killing vectors, we get,
$$r_{a}\hat{B}_{ab}=\phi_{b}$$
(88)
showing that $\phi_{a}$ is the normal component of the tensor field, expectedly
satisfying (87). In analogue with the
vector theory it is possible to do away with this field completely by requiring that
the tensor field resides on the tangent plane of the hypersphere, in which case it is given by,
$$\hat{B}_{ab}=K_{a}^{\mu}K_{b}^{\nu}B_{\mu\nu}$$
(89)
This is written in component notation by using the explicit form for the Killing vectors
given in (26) and (27),
$$\hat{B}_{\mu\nu}=\frac{1+x^{2}}{2}\Big{(}\frac{1+x^{2}}{2}B_{\mu\nu}-x_{\rho}x%
_{\nu}B_{\mu\rho}-x_{\rho}x_{\mu}B_{\rho\nu}\Big{)}$$
(90)
and,
$$\hat{B}_{\mu(n+1)}=-\frac{1+x^{2}}{2}x_{\rho}B_{\mu\rho}$$
(91)
These are the analogues of (3) and (4). The inverse relation is given by,
$$\Big{(}\frac{1+x^{2}}{2}\Big{)}^{4}B_{\mu\nu}=K_{a}^{\mu}K_{b}^{\nu}\hat{B}_{ab}$$
(92)
which may also be put in the form,
$$\Big{(}\frac{1+x^{2}}{2}\Big{)}^{2}B_{\mu\nu}=\hat{B}_{\mu\nu}+x_{\mu}\hat{B}_%
{\nu(n+1)}-x_{\nu}\hat{B}_{\mu(n+1)}$$
(93)
which is the direct analogue of (6).
Next, the gauge transformations are discussed. From (83), the defining relation (89) and the
angular momentum operator (40), infinitesimal transformations are given by
444If the additional $\phi_{a}$-fields had been retained, the shift symmetry
$\delta_{s}\hat{B}_{ab}=r_{a}\delta\phi_{b}-r_{b}\delta\phi_{a}$ would have been found,
exactly as happened in the previous example.,
$$\delta\hat{B}_{ab}=ir_{c}\Big{(}K_{b}^{\mu}L_{ca}-K_{a}^{\mu}L_{cb}\Big{)}%
\Lambda_{\mu}$$
(94)
In this form the expression is not manifestly covariant. This may be contrasted with
(41) which has this desirable feature. The point is that an appropriate map
of the gauge parameter is necessary. In the previous example the gauge parameter was a scalar
which retained its form. Here, since it is a vector, the required map is provided by a relation
like (34), so that,
$$\hat{\Lambda}_{a}=K_{a}^{\mu}\Lambda_{\mu}$$
(95)
Pushing the Killing vectors through the angular momentum operator and using the above
map yields,
$$\delta\hat{B}_{ab}=ir_{c}\Big{(}L_{ca}\hat{\Lambda}_{b}-L_{cb}\hat{\Lambda}_{a%
}\Big{)}+ir_{c}\Lambda_{\mu}\Big{(}L_{cb}K_{a}^{\mu}-L_{ca}K_{b}^{\mu}\Big{)}$$
(96)
The last bracket can be simplified leading to the cherished expression,
$$\delta\hat{B}_{ab}=ir_{c}\Big{(}L_{ca}\hat{\Lambda}_{b}-L_{cb}\hat{\Lambda}_{a%
}\Big{)}-r_{a}\hat{\Lambda}_{b}+r_{b}\hat{\Lambda}_{a}$$
(97)
This transformation could have been deduced from other arguments too. On grounds of covariance and
symmetry, the expression within the brackets is expected. The remainder is necessary to satisfy
the tranversality condition $\delta(r_{a}\hat{B}_{ab})=0$, which is inherent in the construction.
This condition follows from (97) on use of the identity,
$$\hat{\Lambda}_{a}+ir_{b}r_{c}L_{ba}\hat{\Lambda}_{c}=0$$
(98)
obtained on repeated use of $r_{a}\hat{\Lambda}_{a}=0$ and relations derived by successively
differentiating it with operators like $\partial_{a},\partial_{b}etc$.
It is also reassuring to note that (97) manifests the reducibility of the
gauge transformations. Since $\Lambda_{\mu}=\partial_{\mu}\lambda$ leads to a trivial gauge
transformation in
flat space, it follows from (95) that the corresponding feature should be present
in the hyperspherical formulation when,
$$\hat{\Lambda}_{a}=ir_{c}L_{ca}\lambda$$
(99)
It is easy to check that with this choice, the gauge transformation (97)
trivialises; i.e. $\delta\hat{B}_{ab}=0$.
The field tensor on the hypersphere is constructed from the usual one given in (82).
Since the Killing vectors play the role of the metric in connecting the two surfaces, this
expression is given by a natural extension of (44),
$$\hat{F}_{abcd}=\Big{(}r_{a}K_{b}^{\mu}K_{c}^{\nu}K_{d}^{\rho}+r_{b}K_{c}^{\mu}%
K_{a}^{\nu}K_{d}^{\rho}+r_{c}K_{d}^{\mu}K_{a}^{\nu}K_{b}^{\rho}+r_{d}K_{a}^{%
\mu}K_{c}^{\nu}K_{b}^{\rho}\Big{)}F_{\mu\nu\rho}$$
(100)
Note that cyclic permutations have to taken carefully since there is an even number of
indices.
In terms of the basic variables, the field tensor is expressed as,
$$\hat{F}_{abcd}=i\Big{(}L_{ab}\hat{B}_{cd}+L_{bc}\hat{B}_{ad}+L_{bd}\hat{B}_{ca%
}+L_{ca}\hat{B}_{bd}+L_{da}\hat{B}_{cb}+L_{cd}\hat{B}_{ab}\Big{)}$$
(101)
To show that (100) is
equivalent to (101), the same strategy as before, is
adopted. Using the definition of the angular
momentum (39), (101) is simplified
as,
$$\hat{F}_{abcd}=\Big{(}r_{a}K_{b}^{\mu}-r_{b}K_{a}^{\mu}\Big{)}\partial_{\mu}%
\Big{(}K_{c}^{\nu}K_{d}^{\sigma}B_{\nu\sigma}\Big{)}+............$$
(102)
where the carets denote the inclusion of other similar
(cyclically permuted) terms. Now there are two types
of contributions. Those where the derivatives act on the Killing vectors and those where
they act on the fields. The first class of terms cancel out as a consequence of an identity
that is an extension of (47). The other class combines to reproduce (100).
The action on the hypersphere is now obtained by first taking a repeated product of the field
tensor (100). Using the properties of the Killing vectors, this yields,
$$\hat{F}_{abcd}\hat{F}_{abcd}=4\Big{(}\frac{1+x^{2}}{2}\Big{)}^{6}F_{\mu\nu\rho%
}F_{\mu\nu\rho}$$
(103)
From the definition of the flat space action (81) and the volume element (30),
it follows that the above identification leads to the hyperspherical action (for $n=4$),
$$S_{\Omega}=-\frac{1}{48}\int d\Omega\Big{(}\frac{2}{1+x^{2}}\Big{)}^{2}\hat{F}%
_{abcd}\hat{F}_{abcd}$$
(104)
Thus, up to a conformal factor, the corresponding lagrangian is given by,
$${\cal L}_{\Omega}=-\frac{1}{48}\hat{F}_{abcd}\hat{F}_{abcd}$$
(105)
Only in six dimensions, the conformal factor does not occur. This would be the analogue of
the vector theory in four dimensions.
By its very construction this lagrangian would be invariant under the gauge transformation
(97). There is however another type of gauge symmetry
which does not seem to have any analogue in the flat space.
To envisage such a possibility, consider a transformation of the
type 555Recently such a transformation was considered in [12],
though only on $S^{4}$,
$$\delta\hat{B}_{ab}=L_{ab}\lambda$$
(106)
which could be a meaningful gauge symmetry
operation on the hypersphere. However, in flat space, it
leads to a trivial gauge transformation. To see this explicitly, consider the effect of
(106) on (92),
$$\Big{(}\frac{1+x^{2}}{2}\Big{)}^{4}\delta B_{\mu\nu}=K_{a}^{\mu}K_{b}^{\nu}L_{%
ab}\lambda$$
(107)
Inserting the expression for the angular momentum from (38) and
exploiting the transversality (9) of the Killing vectors, it follows that,
$$\delta B_{\mu\nu}=0$$
(108)
thereby proving the statement. To reveal that (106)
indeed leaves the lagrangian (105) invariant, it is desirable
to recast it in the form,
$${\cal L}_{\Omega}=\frac{1}{32}\hat{\Sigma}_{a}\hat{\Sigma}_{a}$$
(109)
where,
$$\hat{\Sigma}_{a}=\epsilon_{abcde}L_{bc}\hat{B}_{de}$$
(110)
Under the gauge transformation (106), a simple algebra shows that $\delta\hat{\Sigma}_{a}=0$ and hence
the lagrangian remanins invariant.
The inclusion of a nonabelian gauge group is feasible. Results follow logically
from the abelian theory with suitable insertion of the nonabelian indices. As
remarked earlier it is useful to consider the first order form (84). The
lagrangian is given by its generalisation [13],
$${\cal L}=-{1\over{8}}\epsilon_{\mu\nu\rho\sigma}F_{\mu\nu}^{i}B_{\rho\sigma}^{%
i}+{1\over 8}A_{\mu}^{i}A_{\mu}^{i}$$
(111)
where the nonabelian field strength has already been defined in (33). It is
gauge invariant under the nonabelian generalisation of (83) with the
ordinary derivatives replaced by the covariant derivatives with respect to the potential
$A_{\mu}$, and $\delta A_{\mu}^{i}=0$. By the help of our equations it is straightforward to
project this lagrangian on the hypersphere. For instance, the corresponding gauge
transformations look like,
$$\delta\hat{B}_{ab}^{i}=ir_{c}\Big{(}L_{ca}\hat{\Lambda}_{b}^{i}-L_{cb}\hat{%
\Lambda}_{a}^{i}\Big{)}-r_{a}\hat{\Lambda}_{b}^{i}+r_{b}\hat{\Lambda}_{a}^{i}+%
f^{ijk}(\hat{A}_{a}^{j}\hat{\Lambda}_{b}^{k}-\hat{A}_{b}^{j}\hat{\Lambda}_{a}^%
{k})$$
(112)
and so on.
Matter fields may be likewise defined. The fermion current $j_{\mu\nu}$ will
be defined just as the two form field, except that conformal weight factors appear,
so that form invariance of the interaction is preserved,
$$\int dx(j_{\mu\nu}^{i}B_{\mu\nu}^{i})=\int d\Omega(\hat{j}_{ab}^{i}\hat{B}_{ab%
}^{i})$$
(113)
quite akin to (58).
5 Killing Vectors and Group Contraction
Now we consider the case of $S^{2}$ in some details. Since $SO(3)$ as a group of
transformation is the symmetry group of a surface of a sphere, the Killing vectors discussed
here must be equivalent to the angular momentum operators, which are the generators of this group.
This will be shown explicitly. Furthermore, by using the technique of group contraction,
two of these Killing vectors reduce to the translation generators on the plane, while the third
remains invariant. Together these constitute the generators of $E_{2}$, which is the symmetry
group of the plane. Thus the familiar contraction of $SO(3)$ to $E_{2}$ is revealed.
Since we have specialised to $S^{2}$ it is convenient to recast the three components of
the angular momentum (39) as,
$$L_{a}=\frac{1}{2}\epsilon_{abc}L_{bc}$$
(114)
By using an identity,
$$\epsilon_{abc}r_{b}K_{c}^{\mu}=\epsilon_{\mu\nu}K_{a}^{\nu}$$
(115)
it is possible to write (114) directly in terms of the rotated Killing vectors,
$$L_{a}=-i\epsilon_{\mu\nu}K_{a}^{\nu}\partial_{\mu}$$
(116)
The spherical coordinates are expressed in terms of the polar angles,
$$r_{1}=\sin\theta\cos\phi,\,\,\,r_{2}=\sin\theta\sin\phi,\,\,\,r_{3}=\cos\theta$$
(117)
Using this transformation, the explicit forms for the Killing vectors (26), (27),
and the relation (5), the following result is derived from (116),
$$\displaystyle L_{1}$$
$$\displaystyle=$$
$$\displaystyle i\sin\phi\,\,\partial_{\theta}+i\cot\theta\,\,\cos\phi\,\,%
\partial_{\phi}$$
(118)
$$\displaystyle L_{2}$$
$$\displaystyle=$$
$$\displaystyle-i\cos\phi\,\,\partial_{\theta}+i\cot\theta\,\,\sin\phi\,\,%
\partial_{\phi}$$
(119)
$$\displaystyle L_{3}$$
$$\displaystyle=$$
$$\displaystyle-i\partial_{\phi}$$
(120)
which are the familiar angular momentum operators generating the $SO(3)$ symmetry.
To discuss the contraction of $SO(3)$ to $E_{2}$, the $L_{a}$-operators in (116) are written
in terms of the planar coordinates,
$$\displaystyle L_{1}$$
$$\displaystyle=$$
$$\displaystyle i\frac{x_{1}x_{2}}{R}\partial_{1}+i\Big{(}\frac{R^{2}+x^{2}}{2R}%
-\frac{x_{1}^{2}}{R}\Big{)}\partial_{2}$$
(121)
$$\displaystyle L_{2}$$
$$\displaystyle=$$
$$\displaystyle-i\frac{x_{1}x_{2}}{R}\partial_{2}-i\Big{(}\frac{R^{2}+x^{2}}{2R}%
-\frac{x_{2}^{2}}{R}\Big{)}\partial_{1}$$
(122)
$$\displaystyle L_{3}$$
$$\displaystyle=$$
$$\displaystyle ix_{2}\partial_{1}-ix_{1}\partial_{2}$$
(123)
where the radial coordinate has been explicitly inserted. Taking the large radius limit so that
the sphere approximates to the plane (around the 3-axis), these identifications are obtained,
$$Lt_{R\rightarrow\infty}\frac{2L_{1}}{R}=i\partial_{2}=-P_{2}$$
(124)
and,
$$Lt_{R\rightarrow\infty}\frac{2L_{2}}{R}=-i\partial_{1}=P_{1}$$
(125)
while $L_{3}$ remains invariant. Here $P_{1}$ and $P_{2}$ are the usual translation generators on the plane.
Along with $L_{3}$ it comprises the $E_{2}$ group.
It is easy to verify that the Lie algebra is preserved in the large $R$-limit. Thus,
$$[P_{1},P_{2}]=\frac{4}{R^{2}}[L_{2},L_{1}]=0$$
(126)
$$[P_{1},L_{3}]=\frac{2}{R}[L_{2},L_{3}]=i\frac{2L_{1}}{R}=-iP_{2}$$
(127)
$$[P_{2},L_{3}]=-\frac{2}{R}[L_{1},L_{3}]=i\frac{2L_{2}}{R}=iP_{1}$$
(128)
thereby reproducing the algebra of $E(2)$; $[P_{\alpha},P_{\beta}]=0,[P_{\alpha},L_{3}]=-i\epsilon_{\alpha\beta}P_{\beta}$.
This completes the demonstration of the group contraction.
6 Conclusions
We have discussed a manifestly $O(n+1)$ covariant formulation of vector and tensor
gauge theories. It was done by mapping the usual forms of these theories, defined
on the Euclidean flat surface, onto a unit hypersphere by the method of stereographic
projection. A distinctive feature was to provide a first principle
analysis of this mapping to abstract the relevant conformal
Killing vectors by solving the Cartan-Killing equation. The importance of these
Killing vectors lay in the fact that tensor forms constructed by taking their
products acted like a metric connecting the results between the flat space
and the hypersphere. This essentially new ingredient was crucial for generalisations
to include higher form gauge theories or arbitrary dimensions.
Furthermore, in this formalism the results of a vector gauge theory
became illuminating and were compactly
reproduced.
As stated above, an extension of the analysis to include the two (and higher) forms was possible. The
two form tensor gauge theory was worked out in details. Once again the transition from the
flat surface to the hypersphere was completely encapsuled in the conformal Killing vectors.
A gauge symmetry was found in the hyperspherical formulation that did not
have any analogue in the flat case.
Matter fields were suitably inducted in the formalism. Just as the mapping in the gauge sector
was done through conformal Killing vectors, that among the basic matter fields was
realised through conformal Killing spinors. An equation was found, revealing these Killing
vectors as bilinear combinations of these Killing spinors. From a knowledge of the Killing
vectors,
an explicit structure for the conformal
Killing spinors was obtained. It may be mentioned that there exist results for the Killing spinors
on $S^{n}$ [10].
However these have to be calculated case by case and their explicit forms are known only
in a few low dimensional examples (like, $n=2,3,4$). Here a compact formula
(78) for the conformal
Killing spinors was given, valid in any dimensions. Also, it was shown that the transformation
matrices defining the conformal Killing spinors and usual Killing spinors were unitarily
equivalent, modulo the conformal factor.
Finally, the Killing vectors obtained here were shown to induce the contraction of the $SO(3)$
group to $E(2)$, in the limit where the radius of $S^{2}$ was taken very large.
Regarding future prospects we presume that, apart from its use in discussing the
quantisation of such theories, it may find applications in the recent
discussions on noncommutative gauge theories defined on a fuzzy sphere
[14, 15, 16]. A possible approach
is to stereographically project the models on the noncommutative plane onto the
fuzzy sphere. On account of the noncommutativity, there is an ordering problem which
makes it difficult to extract closed form
expressions and recourse is sometimes taken to coherent states [15, 16].
Although some results
exist for the projection of the coordinates, that for the gauge or matter fields remains
unknown. Based on a systematic approach using Killing vectors (and spinors) it may be
feasible to construct such a mapping.
Acknowledgments
This work was supported by a grant from the Physics Department,
SungKyunKwan University, Korea. I thank members of this department for their gracious
hospitality. Also, I express my thanks to Satoshi Iso and Keiichi Nagao for discussions
during my stay at KEK, Japan, where this work was initiated.
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Theory of Cation Solvation and Ionic Association in Non-Aqueous Solvent Mixtures
Zachary A. H. Goodwin
zachary.goodwin13@imperial.ac.uk
Department of Materials, Imperial College of London, South Kensington Campus, London SW7 2AZ, UK
John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA
Michael McEldrew
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA
Boris Kozinsky
John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA
Robert Bosch LLC Research and Technology Center, Cambridge, MA 02139, USA
Martin Z. Bazant
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA
Abstract
Conventional lithium-ion batteries, and many next-generation technologies, rely on organic electrolytes with multiple solvents to achieve the desired physicochemical and interfacial properties. The complex interplay between these physicochemical and interfacial properties can often be elucidated via the coordination environment of the cation. We develop a theory for the coordination shell of cations in non-aqueous solvent mixtures that can be applied with high fidelity, up to very high salt concentrations. Our theory can naturally explain simulation and literature values of cation solvation in “classical” non-aqueous electrolytes. Moreover, we utilise our theory to understand general design principles of emerging classes of non-aqueous electrolyte mixtures. It is hoped that this theory provides a systematic framework to understand simulations and experiments which engineer the solvation structure and ionic associations of concentrated non-aqueous electrolytes.
I Introduction
Lithium-ion batteries (LIBs) have transformed our everyday lives through myriad portable electronic devices Xu (2004, 2014); Zheng et al. (2017); Li et al. (2020a); Tian et al. (2021); Xu (2022); Piao et al. (2022); Cheng et al. (2022). Although current LIBs are tremendously successful, further development is necessary to meet safety and performance requirements for widespread use of electric vehicles and grid-level energy storage of renewable energy Li et al. (2020a); Tian et al. (2021); Xu (2022); Piao et al. (2022); Cheng et al. (2022). Thus, much of electrochemical energy storage research is focused on improving the materials beyond commerical LIBs, including high voltage and high capacity cathode materials Li et al. (2020b), silicon anodes Chen et al. (2020), lithium metal anodes Wang et al. (2022); Yu et al. (2020, 2022a), and even non-lithium based battery chemistries Qin et al. (2019); Zheng et al. (2018); Chen et al. (2021a). A requirement of any next-generation battery chemistry is an electrolyte that can operate safely, efficiently and reversibly over the course of the battery’s lifetime Xu (2004, 2014); Zheng et al. (2017); Li et al. (2020a); Tian et al. (2021); Xu (2022); Piao et al. (2022); Cheng et al. (2022).
One of the primary design concepts in electrolyte engineering is balancing desired, often seemingly conflicting, physicochemical properties, such as low viscosity and high conductivity, with interfacial properties, such as low charge transfer resistance and electrode passivation Li et al. (2020a). Commercial LIBs accomplish this design concept, to some extent, with blends of linear carbonates, such as dimethyl carbonate (DMC) for low viscosity, and cyclic carbonates, such as ethylene carbonate (EC) to passivate carbonaceous anodes Xu (2004, 2014); Zheng et al. (2017); Li et al. (2020a); Tian et al. (2021); Xu (2022); Piao et al. (2022); Cheng et al. (2022). Although carbonate blends provide a serviceable solution, electrolyte design for LIBs is far from optimized Li et al. (2020a); Tian et al. (2021); Xu (2022); Piao et al. (2022); Cheng et al. (2022).
An important advance in electrolyte engineering over the past decade has been the link of ion solvation chemistry to electrolyte properties Xu et al. (2007); von Wald Cresce et al. (2012); Borodin et al. (2020); Andersson et al. (2020); Cheng et al. (2022); Wang et al. (2021); Kim et al. (2021). On one hand, at low salt concentrations, solvent dominated coordination shells ensure the complete dissociation of salt, enabling efficient ion transport. However, even at low concentrations, when multiple types of solvent are used, a cation’s solvation structure can be tuned via the bulk solvent ratios to produce very different solid-electrolyte interphases (SEIs) to passivate the working electrodes von Wald Cresce et al. (2012). This is in large part due to the increased reactivity, from reduced LUMO levels, of solvent molecules that are in direct contact with lithium. At higher salt concentrations, the introduction of anions to the coordination shell tends to decrease ionic conductivity and increase concentration overpotentials Xu (2004, 2014). However, in some cases, highly passivating inorganic anion-derived (as opposed to solvent-derived) SEIs can be formed, resulting in an overall increase in cell performance Suo et al. (2013). Furthermore, across all salt concentrations and solvent ratios, the solvation/desolvation dynamics governing charge transfer resistance will be subject to large changes depending on the species comprising the coordination shells of cations in the electrolyte Cheng et al. (2022); Piao et al. (2022).
In fact, next-generation electrolytes are engineered to tune the solvation shell of cations, and therefore physicochemical and interfacial properties of the electrolytes Li et al. (2020a); Tian et al. (2021); Xu (2022); Piao et al. (2022); Cheng et al. (2022); Chen et al. (2021b); Yu et al. (2022a). For example, localized high-concentration electrolytes (LHCE) Chen et al. (2018a); Zheng et al. (2018); Chen et al. (2021a); Ren et al. (2018); Qin et al. (2019); Cao et al. (2021); Liu et al. (2022) use typical salts, such as fluoro-sulfonamides, in high ratios with ether solvents which historically showed promising compatibility with lithium metal anodes, such as 1,2-dimethoxyethane, but which are notoriously limited in terms of oxidative stability Koch and Young (1978); Koch et al. (1982); Foos and Stolki (1988); Qian et al. (2015). The singular aspect of LHCE’s is their use of diluents, generally in the form of fluorinated ethers, which are non-solvating, but highly oxidatively stable Chen et al. (2018a); Zheng et al. (2018); Chen et al. (2021a); Ren et al. (2018); Qin et al. (2019); Cao et al. (2021); Liu et al. (2022). Thus, the non-fluorinated solvent and anion is mostly retained in the lithium coordination shell, where it has enhanced oxidative stability, while the fluorinated ether diluent strictly appears outside of the lithium coordination shell, serving to lubricate the electrolyte (decrease viscosity) while remaining stable against both electrodes Chen et al. (2018a); Zheng et al. (2018); Chen et al. (2021a); Ren et al. (2018); Qin et al. (2019); Cao et al. (2021); Liu et al. (2022). However, a major issue in engineering such next-generation electrolytes is knowing exactly how the solvation shell of the cations will be effected by the mixtures which are created to balance the physicochemical and interfacial properties, as they are often not just an average of their neat properties.
To probe the solvation structure of electrolytes, spectroscopic techniques Xu (2004, 2014); Cheng et al. (2022) such as fourier-transform infrared Seo et al. (2015), Raman Cresce et al. (2017), nuclear magnetic resonance Seo et al. (2015) and mass spectrometry von Wald Cresce et al. (2012); Zhang et al. (2018), have been employed to give some insight. However, atomistic simulations, including molecular dynamics (MD) Postupna et al. (2011); von Wald Cresce et al. (2012); Borodin and Bedrov (2014); Li et al. (2015); Skarmoutsos et al. (2015); Borodin et al. (2017); Han (2017); Ravikumar et al. (2018); Shim (2018); Han (2019); Piao et al. (2020); Hou et al. (2021); Wu et al. (2022) and density functional theory (DFT) Cresce et al. (2017); Borodin et al. (2017); Beltran et al. (2020); Hou et al. (2021); Wu et al. (2022), are typically relied on to give detailed insight into ion solvation. An advantage of DFT is predicting HOMO/LUMO levels of different species, and therefore, their expected reactivity; in addition to understanding the spectra from different techniques Cresce et al. (2017). However, DFT is computationally expensive, and often limited to short length and time scales (when performing ab initio MD) Beltran et al. (2020). To reach larger scales, MD simulations are indispensable in understanding solvation structures and dynamics of ions Postupna et al. (2011); von Wald Cresce et al. (2012); Borodin and Bedrov (2014); Li et al. (2015); Skarmoutsos et al. (2015); Borodin et al. (2017); Han (2017); Ravikumar et al. (2018); Shim (2018); Han (2019); Piao et al. (2020); Hou et al. (2021); Wu et al. (2022). However, a systematic way to report solvation structures has not emerged, with practically every study reporting results in a different way, making it difficult to compare different works systematically Xu (2022). Moreover, both MD and DFT can make predictions of specific systems, but they are not able to provide a general understanding that a simple, analytical theory often does.
In this paper, we develop a theory for the solvation structure and ionic associations of concentrated non-aqueous electrolytes. This theory is based on McEldrew et al.’s McEldrew et al. (2020, 2021a, 2021b, 2021c) recent generalisation of the reversible aggregation and gelation in polymer systems works of Flory Flory (1941a, b, 1942a, 1942b, 1953, 1956), Stockmayer Stockmayer (1943, 1944, 1952) and Tanaka Tanaka (1989, 1990); Tanaka and Stockmayer (1994); Ishida and Tanaka (1997); Tanaka and Ishida (1995); Tanaka (1998, 2002); Tanaka and Ishida (1999); Tanaka (2011), to concentrated electrolytes. Our theory only has a handful of physically transparent parameters, all of which can be obtained directly from atomistic simulations. To demonstrate this parameterization, we perform molecular dynamics (MD) simulations of a “classical” non-aqueous electrolyte, 1M LiPF${}_{6}$ in a mixture of ethylene carbonate and propylene carbonate and compare the results to our theory. Overall, our theory is able to naturally explain the solvation structure of this electrolyte for all the studied mixtures. Having benchmarked our theory against a well studied system, we utilise “toy models” of our theory to understand the design principles of emerging next-generation LIB electrolytes. The theory is able to provide a depth of understanding, not possible with MD or DFT simulations, which is indispensable in aiding design principles of these electrolytes. Moreover, the theory provides a systematic framework for such simulation techniques to report their results, which we hope will be adopted to bring more transparency to the field.
II Theory
Initially, we base our theory on a “classical” non-aqueous electrolyte - 1M LiPF${}_{6}$ dissolved in a mixture of ethylene carbonate (EC) and propylene carbonate (PC) - because of its importance in LIBs Xu (2004, 2014). While this is used as an example here, the theory is not limited to specifically this system, and can be adapted for a wide variety of electrolytes, up to extremely high salt concentrations. As we already know it works well for ionic liquids McEldrew et al. (2021a), water-in-salt electrolytes McEldrew et al. (2021b) and salt-in-ionic liquids McEldrew et al. (2021c). Later, we explore the theory for emerging classes of non-aqueous electrolytes which are gaining interested because of their improved performance, to see if their design principles are supported by our theory.
There are $N_{+}$ Li${}^{+}$ cations and $N_{-}$ [PF${}_{6}$]${}^{-}$ anions, and from electroneutrality in the bulk $N_{+}=N_{-}$. This salt is dissolved in a mixture of EC and PC, of which there are $N_{x}$ and $N_{y}$, respectively. It is assumed that the cations can bind with the anions and both solvent molecules, but no other species bind with each other, i.e. no binding between anion and the solvents (although this can be taken into account McEldrew et al. (2020)) or between solvent molecules (which has been considered elsewhere Choi et al. (2018)). This non-aqueous electrolyte is considered to form a polydisperse mixture of ionic clusters of rank $lmsq$, containing $l$ cations, $m$ anions, $s$ EC solvent molecules and $q$ PC solvent molecules, of which there are $N_{lmsq}$ McEldrew et al. (2020).
The cations can form a maximum of $f_{+}$ associations and the anions a maximum of $f_{-}$ associations, referred to as their functionality. The solvent molecules are assumed to have a functionality of 1. The cations and anions are able to form extended clusters (because $f_{\pm}>2$ for LiPF${}_{6}$) Yu et al. (2022b). However, as the concentration of the salt is relatively low and the interactions between the solvents and Li cations strong, a percolating ionic network (gel) should not form at ambient conditions McEldrew et al. (2020). Therefore, for simplicity, we neglect the formation of the gel for 1M LiPF${}_{6}$ in EC/PC, and refer readers to Refs. 49; 50; 51; 52 to see how it is included.
The electrolyte is assumed to form Cayley tree clusters McEldrew et al. (2020, 2021c); Goodwin et al. (2022). The cations and anions form the “backbone of the branched aggregates”, and the “dangling association sites” from this backbone can either be left empty or can be decorated with the solvent molecules, or the strong solvent-cation interactions can break apart the ionic backbone. These Cayley tree clusters have no intra-cluster loops, with the number of each type of bond being uniquely determined by the rank of the cluster, $lmsq$. In Fig. 1 a schematic of some example clusters are displayed, where the cations and anions have a functionality of $3$. The cluster of rank $1300$ is comprised of $1$ cation and $3$ anions, which has an overall $-2$ negative charge, cannot bind to any solvent molecules from the lack of open cation association sites. For $1011$, the cation is bound to both solvent molecules, and there is a dangling association site which could bind to an anion or another solvent molecule to create a cluster of a larger rank. Finally, the cluster of rank $1120$ is a cation-anion ion pair, where the cation is also bound to two of the $s$ solvent molecules. These are just some examples of cluster to demonstrate the possible allowed clusters.
The theory assumes the electrolyte to be an incompressible lattice fluid, with a single lattice site having the volume of a Li cation, $v_{+}$ McEldrew et al. (2020). Anions occupy $\xi_{-}=v_{-}/v_{+}$ lattice sites, and the solvents occupy $\xi_{x/y}=v_{x/y}/v_{+}$ lattice sites. The total number of lattice sites is given by
$$\displaystyle\Omega=\sum_{lmsq}(l+\xi_{-}m+\xi_{x}s+\xi_{y}q)N_{lmsq}.$$
(1)
Dividing through by the total number of lattice sites gives
$$1=\sum_{lmsq}(l+\xi_{-}m+\xi_{x}s+\xi_{y}q)c_{lmsq}=\sum_{lmsq}\phi_{lmsq}.$$
(2)
Here $c_{lmsq}=N_{lmsq}/\Omega$ is the dimensionless concentration of a $lmsq$ cluster (the number of $lmsq$ clusters per lattice site), and $\phi_{lmsq}$ is the volume fraction of clusters of rank $lmsq$. The volume fraction of each species is determined through
$$\phi_{i}=\sum_{lmsq}\xi_{i}jc_{lmsq},$$
(3)
where $j=l,m,s,q$ for $i=+,-,x,y$, respectively, and $\xi_{+}=1$. Therefore, the incompressibility condition is also given by
$$\displaystyle 1=\phi_{+}+\phi_{-}+\phi_{x}+\phi_{y}.$$
(4)
Based on the works of Flory Flory (1941a, b, 1942a, 1942b, 1953, 1956), Stockmayer Stockmayer (1943, 1944, 1952) and Tanaka Tanaka (1989, 1990); Tanaka and Stockmayer (1994); Ishida and Tanaka (1997); Tanaka and Ishida (1995); Tanaka (1998, 2002); Tanaka and Ishida (1999); Tanaka (2011), and our previous application of this theory to concentrated electrolytes McEldrew et al. (2020, 2021a, 2021b, 2021c); Goodwin et al. (2022); Goodwin and Kornyshev (2022), the free energy is taken to be
$$\mathcal{F}=\sum_{lmsq}\left[N_{lmsq}k_{B}T\ln\left(\phi_{lmsq}\right)+N_{lmsq}\Delta_{lmsq}\right],$$
(5)
where the first term is the ideal entropy of each cluster of rank $lmsq$, and the second term is the free energy of forming those clusters, with $\Delta_{lmsq}$ denoting the free energy of formation for a cluster of rank $lmsq$McEldrew et al. (2020), which has three contributions
$$\displaystyle\Delta_{lmsq}=\Delta_{lmsq}^{bind}+\Delta_{lmsq}^{comb}+\Delta_{lmsq}^{conf},$$
(6)
where $\Delta_{lmsq}^{bind}$ is the binding energy, $\Delta_{lmsq}^{comb}$ is the combinatorial entropy, and $\Delta_{lmsq}^{conf}$ is the configurational entropy.
The binding energy of a cluster, $\Delta^{bind}_{lmsq}$, is uniquely defined by the cluster rank $lmsq$, owing to the assumption of Cayley tree clusters. For $l>0$, the binding energy of a cluster is
$$\displaystyle\Delta^{bind}_{lmsq}=(l+m-1)\Delta u_{+-}+s\Delta u_{+x}+q\Delta u_{+y},$$
(7)
where $\Delta u_{ii^{\prime}}=\Delta u_{i^{\prime}i}$ is the energy of an association between $i$ and $i^{\prime}$. When $l=0$, the binding energy of a cluster is $0$.
Physically, the cation-anion associations are driven by their strong electrostatic interactions McEldrew et al. (2020). These associations are a representation of short-range electrostatic correlations beyond mean-field McEldrew et al. (2020, 2021a, 2021b, 2021c); Goodwin and Kornyshev (2017); Chen et al. (2018b); Feng et al. (2019), with Levy et al. Levy et al. (2019) showing that only retaining short-ranged correlations in concentrated systems is sufficient to reproduce the spatial arrangements of concentrated electrolytes. The fact that we have neglected long-ranged electrostatic interactions between different clusters, relies on the assumption that the majority of the electrostatic energy is incorporated in the formation of the ionic clusters (via this binding contribution), as opposed to between different clusters. Moreover, the electrolyte is at a concentration where the activity coefficient of the salt is typically larger than $1$ Xu (2014), which motivates the exclusion of further electrostatic energy terms, with our theory predicting such values of activity coefficients for concentrated electrolytes McEldrew et al. (2020, 2021b). The cation-solvent associations are driven by the strong interactions between Li and the carbonyl oxygen. It is assumed that cation-solvent interactions are the dominant contribution to the free energy of solvation, instead of, say, the Born solvation energy.
The combinatorial entropy describes the number of ways in which a cluster of rank $lmsq$ can be arranged, as seen by
$$\Delta_{lmsq}^{comb}=-k_{B}T\ln\left\{f_{+}^{l}f_{-}^{m}W_{lmsq}\right\},$$
(8)
where
$$W_{lmsq}=\dfrac{(f_{+}l-l)!(f_{-}m-m)!}{l!m!s!q!(f_{+}l-l-m-s-q+1)!}.$$
(9)
The configurational entropy is associated with building the aggregate from its constituent parts on a lattice. This contribution is not universal, and can depend on how “fexible” the associations are McEldrew et al. (2021a). Importantly, this contribution produces a term that depends on the number of associations in a cluster, which means it contributes to the entropy change of an association, as seen in the association constant later. We refer the reader to Ref. 50 for further details of the configurational entropy, where the temperature dependence was studied.
Establishing chemical equilibria, as shown in Ref. 49, the dimensionless concentration of a cluster of rank $lmsq$, with $l>0$, is given by
$$\displaystyle c_{lmsq}=\frac{W_{lmsq}}{\Lambda_{-}}\left(\psi_{l}\Lambda_{-}\right)^{l}\left(\psi_{m}\Lambda_{-}\right)^{m}(\psi_{s}\Lambda_{x})^{s}(\psi_{q}\Lambda_{y})^{q},$$
(10)
where $\psi_{l}=f_{+}\phi_{1000}/\xi_{+}$ and $\psi_{m}=f_{+}\phi_{0100}/\xi_{+}$ are number of association sites per lattice site for free cations and free anions, respectively, and $\psi_{s}=\phi_{0010}/\xi_{x}$ and $\psi_{q}=\phi_{0001}/\xi_{y}$ are number of association sites per lattice site for the free solvent $x$ and $y$, respectively. The association constant between species $i$ and cations is given by
$$\displaystyle\Lambda_{i}=e^{-\beta\Delta f_{+i}}=e^{-\beta\left[\Delta u_{+i}-T\Delta s_{+i}\right]},$$
(11)
where $\beta$ is inverse thermal energy, and $\Delta s_{+i}$ is the entropy change of an association, as determined by the configurational entropy McEldrew et al. (2020, 2021a). Note $i\neq+$ in these equations, as no cation-cation associations are considered. When $l=0$ and $m=1$ (or $m=0$ and $l=0$), and $s=q=0$, Eq. (10) yields the correct limit for free anions (cations). For $l=0$, the free volume fractions of $x$ and $y$ are separately defined, as $\Delta^{bind}_{0010}=\Delta^{bind}_{0001}=0$, which is not a limit of Eq. (7).
The volume fractions of each free species is (usually) not a known input, however. In fact, these quantities are often what we aim to determine from the theory. To overcome this seemingly circular problem, the free volume fraction of each species can be related to the total volume fraction of that species, the number of bonds that species can make to the other species, and the probability of the associations McEldrew et al. (2020). The probability of species $i$ being associated to species $j$ is given by $p_{ij}$, the number of associations was previously defined as the functionality, and the total volume fractions of each species is known. For free cations $\phi_{1000}=\phi_{+}(1-\sum_{i}p_{+i})^{f_{+}}$, for free anions $\phi_{0100}=\phi_{-}(1-p_{-+})^{f_{-}}$, and for free solvent we have $\phi_{0010}=\phi_{x}(1-p_{x+})$ and $\phi_{0001}=\phi_{y}(1-p_{y+})$.
The association probabilities are related through the conservation of associations and mass action laws. The conservation of associations states
$$\displaystyle\psi_{+}p_{+i}=\psi_{i}p_{i+}=\Gamma_{i},$$
(12)
where $\Gamma_{i}$ is the number of $+i$ associations per lattice site, and $\psi_{i}=f_{i}\phi_{i}/\xi_{i}=f_{i}c_{i}$ is the number of association sites of species $i$ per lattice site. Again, note $i\neq+$ in these equations, which means for the 4-component system being discussed, there are 3 sets of these equations. Similarly, the set of 3 mass action laws for the associations are generally given by
$$\displaystyle\Lambda_{i}\Gamma_{i}=\frac{p_{i+}p_{+i}}{(1-\sum_{i^{\prime}}p_{+i^{\prime}})(1-p_{i+})}.$$
(13)
By solving the set of equations described by Eqs. (12) and (13), one can determine the association probabilities, and therefore the volume fraction of free species and the cluster distribution. For the 4-component system, numerical solutions are required. For the assumption that there is no gel, $(f_{+}-1)(f_{-}-1)>p_{+-}p_{-+}$ must hold, as this inequality determines the percolation point on a Bethe lattice McEldrew et al. (2020).
Having outlined the basic equations, before studying the example of non-aqueous electrolytes in detail, we believe it is prudent that some intuition is outlined from limiting cases. It is suggested that the reader also goes through our general theory McEldrew et al. (2020), and special cases for ionic liquids McEldrew et al. (2021a); Goodwin et al. (2022); Goodwin and Kornyshev (2022), water-in-salt electrolytes McEldrew et al. (2021b) and salt-in-ionic liquids McEldrew et al. (2021c), as these contain a lot of details of the ionic associations not shown here. Let us take $\Delta f_{+i}=0$ for all $i$, which would correspond to $\Lambda_{i}=1$. In such a case, despite the free energy of an association being zero, the association probabilities are finite, which means there is a distribution of clusters of various ranks. This occurs because of the ideal entropy of mixing and the combinatorial entropy each cluster drives the formation of more types of clusters with more different components in them, as this increases the entropy McEldrew et al. (2020); Goodwin et al. (2022). We know, however, from Flory’s lattice expression for the entropy of disorientation Flory (1942b, 1953), that $\Delta s_{+i}>0$. At low temperature, the negative binding energies dominate, and $\Lambda_{i}>1$, but at high temperatures the entropy term dominates and $\Lambda_{i}\approx 0$. In this latter limit, the clusters dissociate, and we are left with only free species, with the ideal entropy of each species reducing down to the entropy of mixing of different species on a lattice McEldrew et al. (2020); Goodwin et al. (2022).
III Cation Solvation in Classical Non-Aqueous Electrolytes
To test our theories ability to understand cation solvation, we perform atomistic MD simulations using LAMMPS Plimpton (1995); Thompson et al. (2022) of LiPF${}_{6}$ in mixtures of EC and PC. For each simulation, we have 83 Li cations and 83 PF${}_{6}$ anions. In addition, there are (83, 167, 250, 333, 417, 500, 583, 667, 750) EC and (750, 667, 583, 500, 417, 333, 250, 167, 83) PC solvent molecules. The initial configurations for all simulations were generated using PACKMOL Martínez et al. (2009). An initial energy minimisation was performed before equilibrating the bulk, periodic system. The production run used to analyse the associations in this system were run at 300 K in an NVT ensemble (the volume of the simulation box was determined from NPT equilibration at 1 bar) for 10 ns with a 1 fs time step.
We employed the CL$\&$P force field for Li cations and PF${}_{6}$ anions Lopes and Pádua (2012), with the van der Waals radius of Li cations set to $1.44~{}\textrm{\AA}$ Ravikumar et al. (2018). For EC and PC, the OPLS/AA force fields were utilised Jorgensen et al. (1996); Jorgensen and Tirado-Rives (2005); Dodda et al. (2017a, b), which are commonly used for simulations of these electrolytes. Long range electrostatic interactions were computed using the particle-particle particle-mesh solver (with a cut-off length of 12 Å).
It has been shown numerous times, and as found here, that the pair correlation function (or cumulative coordination number) of Li-O (carbonyl), has a large peak (sharp increase) at $\sim 2~{}\textrm{\AA}$, with a pronounced minimum (long plateau) at $\sim 3~{}\textrm{\AA}$ until a moderate second peak at $4-5~{}\textrm{\AA}$ Ravikumar et al. (2018); Han (2017, 2019). The Li-O (ether), in contrast, has its first large peak at $\sim 3~{}\textrm{\AA}$. Therefore, it is natural to define a EC/PC solvent molecule to be in the coordination shell if the Li-O (carbonyl) separation is less than 2.5 Å. Whereas, the Li-F pair correlation function (cumulative coordination number) often has much smaller peak(s) (much steadier increase) at $3-4~{}\textrm{\AA}$, despite the van der Waals radius of F (in the employed MD simulation) only being only slightly larger than O Han (2017, 2019); Xie et al. (2023). Therefore, we shall focus solely on the solvents in the cation coordination shell, not delving into the information about cation-anion association in this section.
Using this cut-off distance to define an association between Li and EC/PC, we computed the ensemble average coordination number of Li cations, $\mathcal{N}_{i}$ where $i$ is one of these solvent molecules, for the various mixtures of EC/PC. The results of this simulation analysis are shown in Fig. 2, plotted as a function of the mole fraction of EC to the total mole fraction of solvent, $n_{EC}$. We find that the $\mathcal{N}_{EC}$ increases monotonically with $n_{EC}$ and $\mathcal{N}_{PC}$ decreases monotonically with $n_{EC}$, as might be expected from an increasing proportion of EC. Their dependence on $n_{EC}$ is non-linear, but the sum of the solvents in the coordination shell of Li remains to be $\sim 4$ for all $n_{EC}$.
The composition of the first coordination shell of Li is a quantity which is often calculated from simulations. It provides insight into the interactions in the system, and is often correlated with certain behaviours of the electrolyte, such as the conductivity or SEI forming abilities. Moreover, within our theory, the ensemble average coordination numbers is a key quantity in determining the association probabilities, $\mathcal{N}_{i}=f_{+}p_{+i}$, i.e. the functionality (maximum coordination number) of Li multiplied by the probability that Li is associated to $i$ gives the coordination number of that species. In Fig. 2, it is clear that $f_{+}=4$. Therefore, the association probabilities of cations binding to other species, $p_{+i}$, can be readily determined, and from Eq. (12) the association probabilities of species binding to Li, $p_{i+}$, can be found.
To progress further, we make a few simplifications. Firstly, we assume that the volume of EC and PC molecules are the same, and therefore, the volume fraction of LiPF${}_{6}$ is constant (note this is not a necessary assumption, and doesn’t change the results significantly, but it is done for convenience). Secondly, the total Li coordination number was consistently 4, which means Li practically had no “dangling bonds” because the interactions between Li and the solvents are so energetically favourable. This means the probability of Li binding to solvents is equal to unity, $1=p_{+x}+p_{+y}$, which we refer to as the “sticky cation” approximation McEldrew et al. (2021b). From inspecting Eq. (13), with only solvent molecules included in the summation over $i^{\prime}$, it should be clear there is singular behaviour because the probability of an open cation site, which is $0$ in the sticky cation approximation, sits in the denominator of the equation. This singular behaviour needs to be removed from the system of equations. By dividing the mass action laws between Li and the two different solvents, we arrive at
$$\dfrac{\Lambda_{x}}{\Lambda_{y}}=\tilde{\Lambda}=\dfrac{p_{x+}(1-p_{y+})}{p_{y+}(1-p_{x+})},$$
(14)
which is no longer singular. Using this equation, in addition to $1=p_{+x}+p_{+y}$, permits an analytical solution to the association probabilities
$$\displaystyle\psi_{+}p_{+x}=\psi_{x}p_{x+}=\frac{\psi_{y}-\psi_{+}+\tilde{\Lambda}(\psi_{+}+\psi_{x})}{2(\tilde{\Lambda}-1)}$$
$$\displaystyle-\frac{\sqrt{4\psi_{y}\psi_{+}(\tilde{\Lambda}-1)+[\tilde{\Lambda}(\psi_{x}-\psi_{+})+\psi_{+}+\psi_{y}]^{2}}}{2(\tilde{\Lambda}-1)},$$
(15)
$$\displaystyle\psi_{+}p_{+y}=\psi_{y}p_{y+}=\frac{\psi_{y}+\psi_{+}+\tilde{\Lambda}(\psi_{x}-\psi_{+})}{2(1-\tilde{\Lambda})}$$
$$\displaystyle-\frac{\sqrt{4\psi_{y}\psi_{+}(\tilde{\Lambda}-1)+[\tilde{\Lambda}(\psi_{x}-\psi_{+})+\psi_{+}+\psi_{y}]^{2}}}{2(1-\tilde{\Lambda})}.$$
(16)
When $\tilde{\Lambda}>1$, the associations between Li and EC is favoured over Li and PC ($\Lambda_{x}>\Lambda_{y}$), and when $\tilde{\Lambda}<1$ the associations between Li and PC is favoured over Li and EC ($\Lambda_{y}>\Lambda_{x}$). The other factor in determining the association probabilities is the amount of each species.
In Fig. 3 we show the probabilities computed from the values of $\mathcal{N}_{i}/f_{+}$ obtained from our MD simulations as a function of $n_{EC}$. Using these probabilities we can compute $\tilde{\Lambda}$ using Eq. (14) at each $n_{EC}$ and take the average of the obtained values. We find $\tilde{\Lambda}=0.135$, which indicates that PC has more favourable associations with Li than EC (which can also be seen in Fig. 3 from $p_{y+}>p_{x+}$ for all $n_{EC}$). Now that $\tilde{\Lambda}$ has been determined, Eqs. (15) and (16) can be used to predict the association probabilities for any value of $n_{EC}$. In Fig. 3 we show that these probabilities match the simulated values extremely well. Therefore, our simple, analytical theory can allow us to naturally understand the solvation structure competition of concentrated Li electrolytes. Having understood the association probabilities, we can now turn to the distributions of solvation structures.
In Fig. (4) we display the probability of $s$ EC solvent molecules in the coordination shell of the Li from the simulations, for $s=0,...,4$ as a function of $n_{EC}$. For $s=0$, i.e. Li only being solvated by PC, the probability of this solvation structure is initially close to 1, and it monotonically decreases with increasing $n_{EC}$. Whereas, $s=1,2,3$, i.e. mixed EC and PC solvation structures, all have a non-monotonic dependence on $n_{EC}$, where they go through a maximum at intermediate $n_{EC}$. Finally, $s=4$, i.e. only EC associated to Li, monotonically increases with $n_{EC}$, where it is the largest component in the coordination shell at the largest $n_{EC}$.
To investigate the solvation distribution from the theory, the singular behaviour needs to be removed from Eq. (10) in the sticky cation approximation. Using $1=p_{+x}+p_{+y}$, $f_{+}=s+q$, and Eq. (14), we can arrive at
$$\dfrac{c_{s}}{\phi_{+}}=\dfrac{f_{+}!}{s!(f_{+}-s)!}p_{+x}^{s}(1-p_{+x})^{f_{+}-s},$$
(17)
which is the probability of $s$ EC solvent molecules in the coordination shell of the Li, with $c_{s}$ being used as a shorthand for $c_{10sq}$. An equivalent expression for the probability of $q$ PC solvent molecules in the coordination shell of the Li, $c_{p}$, can be derived, which is also simply $c_{p}/\phi_{+}=1-c_{s}/\phi_{+}$. Equation (17) is clearly the number of ways of arranging a solvation shell with $s$ EC and $q$ PC solvent molecules, multiplied by the probability that $s$ solvent molecules are bound to the Li and the probability of the remaining $f_{+}-s$ PC molecules being bound to Li, i.e. it is simply a binomial distribution of the two solvent molecules.
In Fig. (4) we also plot the probability of $s$ EC molecules in the coordination shell of Li from the theory, using the value of $\tilde{\Lambda}$ extracted from the simulations. Overall, we find good agreement between the theory and simulation. The theory allows us to understand this dependence of $c_{s}/\phi_{+}$ on $n_{EC}$. For $s=0$, Eq. (17) reduces to $p_{+y}^{f_{+}}$, and for $s=f_{+}=4$, we arrive at $p_{+x}^{f_{+}}$, which are clearly the probability of $f_{+}$ PC and EC molecules binding to the Li cation, respectively. As can be understood from Fig. 3, these functions, $p_{+y}^{f_{+}}$ and $p_{+x}^{f_{+}}$, are monotonically decreasing and monotonically increasing, respectively. For all values of $s$ between these limits, both $p_{+x}$ and $1-p_{+x}$ occur in the expressions, which requires that $c_{s}$ goes to zero at the limits of $n_{EC}$, and that a maximum occurs between these limits. If $\tilde{\Lambda}=1$, the $s=2$ case would peak at $n_{EC}=1/2$, but because $\tilde{\Lambda}<1$ the peak occurs at $n_{EC}>1/2$ as clusters with PC is favoured and larger concentrations of EC are required to reach this point. This demonstrates that the binomial distribution is skewed based on the relative interactions of the solvents with Li.
III.1 Experimental Comparison
In Ref. 18 mass spectra of 1M LiPF${}_{6}$ in mixtures of non-aqueous solvents were measured. In Fig. 5 we reproduce Ref. 18 values for the fraction of EC in the solvation shell, i.e. $\mathcal{N}_{EC}/f_{+}$, for EC/PC and EC/DMC mixtures, as a function of the mole fraction of EC relative to the mole fraction of total solvent. In addition, we reproduce the values simulated from the previous section.
For EC/PC mixtures, the values of $\mathcal{N}_{EC}/f_{+}$ from simulations and experiments are quite similar. In the experiments, it was noted that only up to 3 solvent molecules coordinating were found. The discrepancy between simulations and experiments as to whether a maximum of 3 or 4 solvent molecules can coordinate lithium in these carbonate blends was discussed in Ref. 18 and thought to be an attribute of the electrospray ionization technique only retaining the most tightly bound species. Therefore, for the experimental data we use $f_{+}=3$, and we found that $f_{+}=4$ the fitted curves significantly worse than $f_{+}=3$. Based on this functionality, we calculated $\tilde{\Lambda}=0.0978$ for the experiments, which is similar to the value obtained from the simulations. This value of $\tilde{\Lambda}$ indicates that the PC-Li associations are more favourable than the EC-Li associations, which is evident from the curves being below the diagonal dotted line which corresponds to equal numbers of EC/PC in the coordination shell of Li.
On the other hand, for the EC/DMC mixtures, the EC-Li associations are favoured over the DMC-Li associations, and so $\mathcal{N}_{EC}/f_{+}$ resides above the diagonal. The value of $\tilde{\Lambda}=26.7$ extracted from the experimental data is now larger than 1, demonstrating the more favourable EC-Li associations. Therefore, the value of $\tilde{\Lambda}$ is a clear, physical parameter which describes the competition between the solvation structure of Li electrolytes. Again, $f_{+}=3$ did a significantly better job at reproducing the data than $f_{+}=4$.
There are many examples of simulations which show similar trends to those reported in Refs. 18. While there can be some differences in the reported experimental or predicted values from simulations, with and without associations between cations-anions, we wish to stress that our theory should be able to capture the behaviour of all these systems, provided the assumptions of the theory are valid. What changes is the exact volume fractions of species, association constants, functionalities, etc.
IV Beyond Classical Non-aqueous Electrolytes
In the previous sections, we focused on cation solvation in common non-aqueous electrolytes that are studied primarily for use in commercial LIBs. We chose this electrolyte because it has been extensively studied, and the coordination structure is qualitatively agreed upon. While these electrolytes are commonly and successfully used in commercial LIBs, moderately concentrated carbonate blend electrolytes generally do not perform well in many next-generation battery technologies Li et al. (2020a); Tian et al. (2021); Xu (2022); Piao et al. (2022); Cheng et al. (2022). For such applications, there are many emerging classes of electrolyte currently being explored in literature, where our developed theory can be adapted for Li et al. (2020a); Tian et al. (2021); Xu (2022); Piao et al. (2022); Cheng et al. (2022); Yu et al. (2022b).
In this section, we utilise our theory to understand some “design principles” of several classes of next-generation electrolytes currently being investigated. Note we have only picked a couple of examples to discuss here to demonstrate the theories ability, with an exhaustive comparison being left for future work. We focus on “toy models” of the theory for these electrolytes, details of which are shown in the Supplementary Material (SM), and we do not perform simulations to back up these findings, but hope these results inspire further investigation of these systems. For brevity, we will not review exactly why these next-generation electrolytes are of interest or their exact chemistries, but refer readers to where this information can be found.
IV.1 Localized high-concentration electrolytes
The design of localized high-concentration electrolytes (LHCE) – the reader is referred to Ref. 27 for a review – starts from finding a promising highly concentrated electrolyte (HCE), typically just a salt dissolved in a solvent that solvates the cation strongly. This HCE is then diluted with a solvent which does not solvate the cation, known as the diluent, in an attempt to preserve the coordination structure of the cation (see Introduction for why this is important). Therefore, the lithium coordination shell – likely invaded by anions – resembles one that could typically be obtained at only high salt concentrations in 2-component mixtures. From the perspective of the lithium ions in solution, the electrolyte is “locally” concentrated.
From our toy model, shown in the SM, we find that the design principle of LHCE is supported by our theory, under certain approximations. In the sticky cation approximation, the addition of a diluent, which does not interact with any other species, leaves the coordination shell of the cations unchanged, despite the dilution effect. If the sticky cation approximation is not employed, the predictions of the theory sensitively depend on the assumed association constants, and therefore, we do not make any predictions for this case. Using our theory, it should be possible to reveal the necessary components to include in a model that accurately describes these promising electrolytes.
IV.1.1 High entropy electrolytes
Within the chemistries of LHCE’s, it has been proposed that by adding more types of solvents, without changing the concentration of salt and the solvation enthalpy, the average cluster size can decreases while retaining roughly the same number of anions in the coordination shell of Li Kim et al. (2022). This is thought to occur because of the increased entropy of mixing, similar to high entropy alloys, which is why these systems have been referred to as high entropy electrolytes (HEE) Kim et al. (2022). These HEE have been reported to have significantly better transport properties, without diminishing the stability of the electrolyte Kim et al. (2022).
The toy model we utilise to investigate this system is shown in the SM. We find that our theory does not support the design principle of HEE. More types of solvent, which interact identically with the cation, leave the average cation-anion cluster size unaltered. In the SM, we have given some possible explanations as to why this could be, but reserve making final, physical conclusions until a detailed comparison can be made between our theory and simulations of these electrolytes. We believe it is essential to perform this comparison to understand the origin of the electrolytes promising transport properties Kim et al. (2022).
IV.2 Chelating Agents
Solvents which can bind to cations from multiple points, i.e. multi-dentate ligands, have been proposed to be beneficial in reverting the negative transference numbers of salt-in-ILs Molinari et al. (2019a, b); McEldrew et al. (2021c). A limiting case of such electrolytes is when the solvent completely encapsulates the cation, preventing it from binding to any other species, be it anions or other solvent molecules. The “design principle” of the addition of encapsulating chelating agents is to break up the cation-anion associations and prevent anions from binding strongly to the cations Molinari and Kozinsky (2020). Here we shall develop an approach to include these multi-dentate, encapsulating solvents, to see if this principle is supported by our theory.
Our developed theory extension for this type of solvent, shown in the SM, yields predictions which are consistent with that of Ref. 91, i.e. dissociation of larger clusters. Therefore, the design principle of adding chelating agents which encapsulate the cations is supported by our simple theory. Further work should compare how the cluster distribution is altered by the presence of these chelating agents in detail.
V Conclusions
In summary, we have developed a theory for the solvation structure of non-aqueous electrolytes. We have tested this theory against molecular dynamics simulations of a “classical” non-aqueous electrolyte, 1 M LiPF${}_{6}$ in mixtures of carbonates. Overall, we found excellent agreement between our theory and the classical molecular dynamics simulations, and our theory also worked well in comparison to already published experimental results for the competition between EC and PC in the coordination shell of Li. Confident in the theory, we investigated non-aqueous electrolytes of interest in the next generation of LIBs. Overall, it is hoped that this theory will provide a framework for understanding the solvation shell of electrolytes in a more systematic way (in the Supplementary Material, we provide some notes on this), and aid the design of next-generation chemistries.
While we have focused on the solvation structure of the cations here, which is known to be crucial in understanding the properties of these electrolytes, our theory can be directly used to calculate physical observables, which can therefore quantify the connection between solvation structure and properties of interest. For example, we have developed a consistent treatment of ionic transport based on vehicular motion of species, which could be used to understand conductivities and transference numbers of different mixtures McEldrew et al. (2021a, c); we can calculate activity coefficients from our theory McEldrew et al. (2021b), which can provide indications of if the electrolyte is expected to form a stable SEI; and we recently developed the theory to predict the composition of electrolytes at charged interfaces Goodwin et al. (2022); Goodwin and Kornyshev (2022), where desolvation of cations could be a result of the breakdown of the cluster distribution at an interface.
Moreover, looking forward, our theory could be developed to understand the kinetics of SEI formation Huang et al. (2019); Pinson and Bazant (2013); Das et al. (2019) in more detail with explicit account of the solvation shell, and electron transfer reactions of solvated cations at interfaces Huang et al. (2021); Fraggedakis et al. (2021), where the reorganisation energy could be linked to the free energy of forming those clusters. Here we focused on non-aqueous electrolytes, and previously we have studied ionic liquids McEldrew et al. (2021a), water-in-salt electrolytes McEldrew et al. (2021b) and salt-in-ionic liquids McEldrew et al. (2021c), but the theory is quite general, and could be applied to a wide variety of electrolytes, from high concentration to low concentration (provided long-ranged electrostatics are introduced McEldrew et al. (2020)), for various temperatures McEldrew et al. (2021a), and for various types of solvent (aqueous and non-aqueous) and salt compositions, provided Cayley tree clusters are formed.
VI Acknowledgements
We thank Nicola Molinari, Julia Yang, Sang Cheol Kim and Jingyang Wang for stimulating discussions. We acknowledge the Imperial College London Research Computing Service (DOI:10.14469/hpc/2232) for the computational resources used in carrying out this work. MM and MZB acknowledge support from a Amar G. Bose Research Grant.
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Is there an algorithm which takes as input a Diophantine equation,
returns an integer, and this integer is greater than the number
of integer (non-negative integer) solutions if the solution set is finite?
Apoloniusz Tyszka
()
Abstract
Let $E_{n}=\{x_{i}=1,~{}x_{i}+x_{j}=x_{k},~{}x_{i}\cdot x_{j}=x_{k}:i,j,k\in\{1,%
\ldots,n\}\}$.
For a positive integer $n$, let $f(n)$ denote the greatest finite total
number of solutions of a subsystem of $E_{n}$ in integers $x_{1},\ldots,x_{n}$.
We prove: (1) the function $f$ is strictly increasing, (2) if a non-decreasing
function $g$ from positive integers to positive integers satisfies $f(n)\leq g(n)$
for any $n$, then a finite-fold Diophantine representation of $g$ does not exist,
(3) if the question of the title has a positive answer, then there is a computable
strictly increasing function $g$ from positive integers to positive integers such that
$f(n)\leq g(n)$ for any $n$ and a finite-fold Diophantine representation
of $g$ does not exist.
Key words: Davis-Putnam-Robinson-Matiyasevich theorem, finite-fold Diophantine representation.
2010 Mathematics Subject Classification: 03D25, 11U05.
The Davis-Putnam-Robinson-Matiyasevich theorem states that every recursively enumerable
set ${\cal M}\subseteq{\mathbb{N}}^{n}$ has a Diophantine representation, that is
$$(a_{1},\ldots,a_{n})\in{\cal M}\Longleftrightarrow\exists x_{1},\ldots,x_{m}%
\in\mathbb{N}~{}~{}W(a_{1},\ldots,a_{n},x_{1},\ldots,x_{m})=0$$
(R)
for some polynomial $W$ with integer coefficients, see [3] and [2].
The polynomial $W$ can be computed, if we know a Turing machine $M$
such that, for all $(a_{1},\ldots,a_{n})\in{\mathbb{N}}^{n}$, $M$ halts on $(a_{1},\ldots,a_{n})$ if and only if
$(a_{1},\ldots,a_{n})\in{\cal M}$, see [3] and [2].
The representation (R) is said to be finite-fold if for any $a_{1},\ldots,a_{n}\in\mathbb{N}$
the equation $W(a_{1},\ldots,a_{n},x_{1},\ldots,x_{m})=0$ has only finitely many solutions
$(x_{1},\ldots,x_{m})\in{\mathbb{N}}^{m}$.
Open Problem ([1, pp. 341–342], [4, p. 42], [5, p. 79]).
Does each recursively enumerable set ${\cal M}\subseteq{\mathbb{N}}^{n}$ has a finite-fold Diophantine representation?
Let $\cal{R}$ng denote the class of all rings K that extend $\mathbb{Z}$.
Th. Skolem proved that any Diophantine equation can be algorithmically transformed
into an equivalent system of Diophantine equations of degree at most $2$,
see [6, pp. 2–3] and [3, pp. 3–4].
Let
$$E_{n}=\{x_{i}=1,~{}x_{i}+x_{j}=x_{k},~{}x_{i}\cdot x_{j}=x_{k}:i,j,k\in\{1,%
\ldots,n\}\}$$
The following result strengthens Skolem’s theorem.
Lemma 1.
Let $D(x_{1},\ldots,x_{p})\in{\mathbb{Z}}[x_{1},\ldots,x_{p}]$.
Assume that $d_{i}={\rm deg}(D,x_{i})\geq 1$ for each $i\in\{1,\ldots,p\}$. We can compute a positive
integer $n>p$ and a system $T\subseteq E_{n}$ which satisfies the following two conditions:
(4)
If $\textbf{{K}}\in{\cal R}{\sl ng}\cup\{\mathbb{N}\}$, then
$$\forall\tilde{x}_{1},\ldots,\tilde{x}_{p}\in\textbf{{K}}~{}\Bigl{(}D(\tilde{x}%
_{1},\ldots,\tilde{x}_{p})=0\Longleftrightarrow$$
$$\exists\tilde{x}_{p+1},\ldots,\tilde{x}_{n}\in\textbf{{K}}~{}(\tilde{x}_{1},%
\ldots,\tilde{x}_{p},\tilde{x}_{p+1},\ldots,\tilde{x}_{n})~{}solves~{}T\Bigr{)}$$
(5)
If $\textbf{{K}}\in{\cal R}{\sl ng}\cup\{\mathbb{N}\}$, then
for each $\tilde{x}_{1},\ldots,\tilde{x}_{p}\in\textbf{{K}}$ with $D(\tilde{x}_{1},\ldots,\tilde{x}_{p})=0$,
there exists a unique tuple $(\tilde{x}_{p+1},\ldots,\tilde{x}_{n})\in{\textbf{{K}}}^{n-p}$ such that the tuple
$(\tilde{x}_{1},\ldots,\tilde{x}_{p},\tilde{x}_{p+1},\ldots,\tilde{x}_{n})$ solves $T$.
Conditions (4) and (5) imply that for each $\textbf{{K}}\in{\cal R}{\sl ng}\cup\{\mathbb{N}\}$, the
equation $D(x_{1},\ldots,x_{p})=0$ and the system $T$ have the same number of solutions in K.
Proof.
Let
$$D(x_{1},\ldots,x_{p})=\sum a(i_{1},\ldots,i_{p})\cdot x_{1}^{\textstyle i_{1}}%
\cdot\ldots\cdot x_{p}^{\textstyle i_{p}}$$
where $a(i_{1},\ldots,i_{p})$ denote non-zero integers, and let $M$ denote the maximum of the absolute values
of the coefficients of $D(x_{1},\ldots,x_{p})$. Let ${\cal T}$ denote the set of all
polynomials $W(x_{1},\ldots,x_{p})\in{\mathbb{Z}}[x_{1},\ldots,x_{p}]$ such that their coefficients
belong to the interval $[0,M]$ and ${\rm deg}(W,x_{i})\leq d_{i}$ for each $i\in\{1,\ldots,p\}$.
Let $n$ denote the cardinality of ${\cal T}$. It is easy to check that
$$n=(M+1)^{\textstyle(d_{1}+1)\cdot\ldots\cdot(d_{p}+1)}\geq 2^{\textstyle 2^{p}%
}>p$$
We define:
$$A(x_{1},\ldots,x_{p})=\sum_{a(i_{1},\ldots,i_{p})>0}a(i_{1},\ldots,i_{p})\cdot
x%
_{1}^{\textstyle i_{1}}\cdot\ldots\cdot x_{p}^{\textstyle i_{p}}$$
$$B(x_{1},\ldots,x_{p})=\sum_{a(i_{1},\ldots,i_{p})<0}-a(i_{1},\ldots,i_{p})%
\cdot x_{1}^{\textstyle i_{1}}\cdot\ldots\cdot x_{p}^{\textstyle i_{p}}$$
The equation $D(x_{1},\ldots,x_{p})=0$ is equivalent to $0+A(x_{1},\ldots,x_{p})=B(x_{1},\ldots,x_{p})$,
where $0,~{}~{}A(x_{1},\ldots,x_{p}),~{}~{}B(x_{1},\ldots,x_{p})\in{\cal T}$.
We choose any bijection $\tau:\{1,\ldots,n\}\longrightarrow{\cal T}$ such that
$\tau(1)=x_{1},~{}~{}\ldots,~{}~{}\tau(p)=x_{p}$, and $\tau(p+1)=0$. Let ${\cal H}$ denote the set of all
equations from $E_{n}$ which are identities in ${\mathbb{Z}}[x_{1},\ldots,x_{p}]$, if $x_{i}=\tau(i)$
for each $i\in\{1,\ldots,n\}$. Since $\tau(p+1)=0$, the equation $x_{p+1}+x_{p+1}=x_{p+1}$
belongs to ${\cal H}$. We define $T$ as ${\cal H}\cup\{x_{p+1}+x_{s}=x_{t}\}$, where
$s={\tau}^{-1}(A(x_{1},\ldots,x_{p}))$ and $t={\tau}^{-1}(B(x_{1},\ldots,x_{p}))$.
For each $\tilde{x}_{1},\ldots,\tilde{x}_{p}\in\textbf{{K}}$ with $D(\tilde{x}_{1},\ldots,\tilde{x}_{p})=0$,
the sought-for elements $\tilde{x}_{p+1},\ldots,\tilde{x}_{n}\in\textbf{{K}}$ exist, are unique, and satisfy
$$\forall i\in\{p+1,\ldots,n\}~{}~{}\tilde{x}_{i}=\tau(i)[x_{1}\mapsto\tilde{x}_%
{1},\ldots,x_{p}\mapsto\tilde{x}_{p}]$$
∎
A stronger form of Lemma 1 is proved in [7], where conditions (4) and (5)
are formulated for any $\textbf{{K}}\in{\cal R}{\sl ng}\cup\{\mathbb{N},~{}\mathbb{N}\setminus\{0\}\}$.
For $\textbf{{K}}\in{\cal R}{\sl ng}$, Lemma 1 is proved in [8].
For a positive integer $n$, let $f(n)$ denote the greatest finite total number of solutions of
a subsystem of $E_{n}$ in integers $x_{1},\ldots,x_{n}$. Obviously, $f(1)=2$ as the equation
$x_{1}\cdot x_{1}=x_{1}$ has exactly two integer solutions.
Lemma 2.
For each positive integer $n$, $f(n+1)\geq 2\cdot f(n)>f(n)$.
Proof.
If $r$ is a positive integer and a system $S\subseteq E_{n}$ has exactly $r$
solutions in integers $x_{1},\ldots,x_{n}$, then the system
$S\cup\{x_{n+1}\cdot x_{n+1}=x_{n+1}\}\subseteq E_{n+1}$ has exactly
$2r$ solutions in integers $x_{1},\ldots,x_{n+1}$.
∎
Corollary. The function $f$ is strictly increasing.
A function $\beta:\mathbb{N}\setminus\{0\}\to\mathbb{N}\setminus\{0\}$ is said to majorize a function
$\alpha:\mathbb{N}\setminus\{0\}\to\mathbb{N}\setminus\{0\}$ provided $\alpha(n)\leq\beta(n)$
for any $n$.
Theorem 1.
If a non-decreasing function $g:\mathbb{N}\setminus\{0\}\to\mathbb{N}\setminus\{0\}$ majorizes $f$,
then a finite-fold Diophantine representation of $g$ does not exist.
Proof.
Assume, on the contrary, that there is a finite-fold Diophantine representation of $g$.
It means that there is a polynomial $W(x_{1},x_{2},x_{3},\ldots,x_{m})$ with integer coefficients
such that
(6)
for any non-negative integers $x_{1},x_{2}$,
$$(x_{1},x_{2})\in g\Longleftrightarrow\exists x_{3},\ldots,x_{m}\in\mathbb{N}~{%
}~{}W(x_{1},x_{2},x_{3},\ldots,x_{m})=0$$
and for each non-negative integers $x_{1},x_{2}$ at most finitely many tuples
$(x_{3},\ldots,x_{m})\in{\mathbb{N}}^{m-2}$ satisfy $W(x_{1},x_{2},x_{3},\ldots,x_{m})=0$.
By Lemma 1, there is a formula $\Phi(x_{1},x_{2},x_{3},\ldots,x_{s})$ such that
(7)
$s\geq{\rm max}(m,3)$ and $\Phi(x_{1},x_{2},x_{3},\ldots,x_{s})$ is a conjunction of formulae of the forms
$x_{i}=1$, $x_{i}+x_{j}=x_{k}$, $x_{i}\cdot x_{j}=x_{k}$ $(i,j,k\in\{1,\ldots,s\})$ which equivalently
expresses that $W(x_{1},x_{2},x_{3},\ldots,x_{m})=0$ and each $x_{i}~{}(i=1,\ldots,m)$ is a sum of four squares.
Let $S$ denote the following system
$$\left\{\begin{array}[]{rcl}a\cdot a&=&A\\
b\cdot b&=&B\\
c\cdot c&=&C\\
d\cdot d&=&D\\
A+B&=&u_{1}\\
C+D&=&u_{2}\\
u_{1}+u_{2}&=&u_{3}\\
\tilde{a}\cdot\tilde{a}&=&\tilde{A}\\
\tilde{b}\cdot\tilde{b}&=&\tilde{B}\\
\tilde{c}\cdot\tilde{c}&=&\tilde{C}\\
\tilde{d}\cdot\tilde{d}&=&\tilde{D}\\
\tilde{A}+\tilde{B}&=&\tilde{u}_{1}\\
\tilde{C}+\tilde{D}&=&\tilde{u}_{2}\\
\tilde{u}_{1}+\tilde{u}_{2}&=&\tilde{u}_{3}\\
u_{3}+\tilde{u}_{3}&=&x_{2}\\
t_{1}&=&1\\
t_{1}+t_{1}&=&t_{2}\\
t_{2}\cdot t_{2}&=&t_{3}\\
t_{3}\cdot t_{3}&=&t_{4}\\
&\ldots&\\
t_{s-1}\cdot t_{s-1}&=&t_{s}\\
t_{s}\cdot t_{s}&=&t_{s+1}\\
t_{s+1}\cdot t_{s+1}&=&x_{1}\\
{\rm all~{}equations~{}occurring~{}in~{}}\Phi(x_{1},x_{2},x_{3},\ldots,x_{s})%
\\
\end{array}\right.$$
with $2s+23$ variables. The system $S$ equivalently expresses the following conjunction:
$$\Biggl{(}\left(a^{2}+b^{2}+c^{2}+d^{2}\right)+\left({\tilde{a}}^{2}+{\tilde{b}%
}^{2}+{\tilde{c}}^{2}+{\tilde{d}}^{2}\right)=x_{2}\Biggr{)}\wedge\Biggl{(}x_{1%
}=2^{\textstyle 2^{s}}\Biggr{)}\wedge\Phi(x_{1},x_{2},x_{3},\ldots,x_{s})$$
Conditions (6)-(7) and Lagrange’s four-square theorem imply that the system $S$ is satisfiable over
integers and has only finitely many integer solutions. Let $L$ denote the number of integer solutions to $S$.
If an integer tuple solves $S$, then $x_{1}=2^{\textstyle 2^{s}}$ and $x_{2}=g(x_{1})=g\left(2^{\textstyle 2^{s}}\right)$.
Since the equation $u_{3}+\tilde{u}_{3}=x_{2}$ belongs to $S$ and Lagrange’s four-square theorem holds,
$L\geq g\left(2^{\textstyle 2^{s}}\right)+1$. The definition of $f$ implies that
$$L\leq f\left(2s+23\right)$$
(8)
Since $g$ majorizes $f$,
$$f\left(2s+23\right)<g\left(2s+23\right)+1$$
(9)
Since $s\geq 3$ and $g$ is non-decreasing,
$$g\left(2s+23\right)+1\leq g\left(2^{\textstyle 2^{s}}\right)+1$$
(10)
Inequalities (8)-(10) imply that $L<g\left(2^{\textstyle 2^{s}}\right)+1$, a contradiction.
∎
Theorem 2.
If the question of the title has a positive answer, then there is a computable strictly increasing
function $g:\mathbb{N}\setminus\{0\}\to\mathbb{N}\setminus\{0\}$ such that $g$ majorizes $f$ and
a finite-fold Diophantine representation of $g$ does not exist.
Proof.
For each positive integer $r$, there are only finitely many Diophantine equations whose
lengths are not greater than $r$, and these equations can be algorithmically constructed.
This and the assumption that the question of the title has a positive answer imply that
there exists a computable function $\delta:\mathbb{N}\setminus\{0\}\to\mathbb{N}\setminus\{0\}$
such that for each positive integer $r$ and for each Diophantine equation whose length is not
greater than $r$, $\delta(r)$ is greater than the number of integer solutions if the solution set is finite.
There is a computable function $\psi:\mathbb{N}\setminus\{0\}\to\mathbb{N}\setminus\{0\}$
such that each subsystem of $E_{n}$ is equivalent to a Diophantine equation whose length
is not greater than $\psi(n)$. The function
$$\mathbb{N}\setminus\{0\}\ni n\stackrel{{\scriptstyle\textstyle h}}{{%
\longmapsto}}\delta(\psi(n))\in\mathbb{N}\setminus\{0\}$$
is computable. The definition of $f$ implies that $h$ majorizes $f$. The function
$$\mathbb{N}\setminus\{0\}\ni n\stackrel{{\scriptstyle\textstyle g}}{{%
\longmapsto}}\sum_{i=1}^{n}h(i)\in\mathbb{N}\setminus\{0\}$$
is computable and strictly increasing. Since $g$ majorizes $h$ and $h$ majorizes $f$, $g$ majorizes $f$.
By Theorem 1, a finite-fold Diophantine representation of $g$ does not exist.
∎
The analogous results for solutions in non-negative integers can be found in [9].
References
[1]
M. Davis, Yu. Matiyasevich, J. Robinson,
Hilbert’s tenth problem. Diophantine equations: positive aspects of a negative solution,
in: Mathematical developments arising from Hilbert problems (ed. F. E. Browder),
Proc. Sympos. Pure Math., vol. 28, Part 2, Amer. Math. Soc., 1976, 323–378;
reprinted in: The collected works of Julia Robinson (ed. S. Feferman), Amer. Math. Soc., 1996, 269–324.
[2]
L. B. Kuijer,
Creating a Diophantine description of a r.e. set and on the complexity of such a description,
MSc thesis, Faculty of Mathematics and Natural Sciences, University of Groningen, 2010,
http://irs.ub.rug.nl/dbi/4b87adf513823.
[3]
Yu. Matiyasevich,
Hilbert’s tenth problem,
MIT Press, Cambridge, MA, 1993.
[4]
Yu. Matiyasevich,
Hilbert’s tenth problem: what was done and what is to be done.
Hilbert’s tenth problem: relations with arithmetic and algebraic geometry (Ghent, 1999), 1–47,
Contemp. Math. 270, Amer. Math. Soc., Providence, RI, 2000.
[5]
Yu. Matiyasevich,
Towards finite-fold Diophantine representations,
Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 377 (2010), 78–90,
ftp://ftp.pdmi.ras.ru/pub/publicat/znsl/v377/p078.pdf.
[6]
Th. Skolem,
Diophantische Gleichungen,
Julius Springer, Berlin, 1938.
[7]
A. Tyszka,
Conjecturally computable functions which unconditionally do not have any finite-fold Diophantine representation,
Inform. Process. Lett. 113 (2013), no. 19-21, 719–722.
[8]
A. Tyszka,
Does there exist an algorithm which to each Diophantine equation assigns an integer which
is greater than the modulus of integer solutions, if these solutions form a finite set?
Fund. Inform. 125(1): 95–99, 2013.
[9]
A. Tyszka,
Is there an algorithm which takes as input a Diophantine equation,
returns an integer, and this integer is greater than the number
of solutions in non-negative integers if the solution set is finite?
http://www.cyf-kr.edu.pl/~rttyszka/sol˙in˙N.pdf.
Apoloniusz Tyszka
Faculty of Production and Power Engineering
University of Agriculture
Balicka 116B, 30-149 Kraków, Poland
E-mail: rttyszka@cyf-kr.edu.pl |
LPOP: Challenges and Advances in Logic and Practice of Programming
David S. Warren Yanhong A. Liu
Computer Science Department, Stony Brook University
()
Abstract
This article describes the work presented at the first Logic and Practice of Programming (LPOP) Workshop,
which was held in Oxford, UK, on July 18, 2018, in conjunction with the Federated Logic Conference (FLoC) 2018.
Its focus is challenges and advances in logic and practice of programming.
The workshop was organized around a challenge problem that specifies issues in role-based access control (RBAC),
with many participants proposing combined imperative and declarative solutions expressed
in the languages of their choice.
\includepdfset
pagecommand=
1 Introduction
The focus of the 2018 Logic and Practice of Programming workshop was on logic and declarative languages for the practice of programming. Of particular interest were languages (1) that have a clear semantic foundation, so that they can be used for concise modeling of complex application problems, facilitating formal proofs and automated analysis, and (2) that are also implementable, so that the implementations can run as specified, as part of real applications. Also of interest were (a) the design of declarative languages, libraries, and tools that facilitate the construction of complex systems and applications, (b) approaches to integrate declarative and procedural programming, and (c) the use of declarative languages to facilitate other programming paradigms, e.g., distributed programming. The target audience for these languages was students who wish to model complex application problems, and practitioners who want to use them.
The goal of the workshop was to bring together the best people and best languages, tools, and ideas to help improve logic languages for the practice of programming and to improve the practice of programming with logic and declarative programming. We prepared to organize the workshop around a number of ”challenge problems”, including in particular expressing a set of system components and functionalities clearly and precisely using a chosen description language. To that end, we created an extensive challenge for this purpose in the general area of role-based access control. We also organized invited talks and additional presentations by the proponents of some well-known description methods. We grouped presentations of description methods by the kind of problems that they address, and tried to allow ample time to understand the strengths of the various approaches and how they might be combined.
Potential workshop participants were invited to submit position papers (1 or 2 pages in PDF format), and to state whether they wished to present a talk at the workshop, explaining how they would express the challenge problem. Because we intended to bring together researchers from many parts of logic and declarative languages and practice of programming communities, it was essential that all talks be accessible to non-specialists.
The program committee invited attendees based on their position paper submissions and attempted to accommodate presentation requests in ways that fit with the broader organizational goals outlined above.
1.1 Program
The schedule for the presentation of contributed position papers that describe solutions to the challenge problem follows.
Session 1: Logic and Practice of Programming
Session Chair: Marc Denecker
09:00 Marc Denecker. Opening and introduction.
09:10 Invited Talk: Michael Leuschel. Practical uses of Logic, Formal Methods, B and ProB.
09:50 Invited Talk: Nicola Leone, Bernardo Cuteri, Marco Manna, Kristian Reale and Francesco Ricca.
On the Development of Industrial Applications with ASP.
Session 2: Security Policies as Challenge Problems
Session Chair: Annie Liu
11:00 Annie Liu. Introduction: Role-Based Access Control as a Programming Challenge.
11:10 Thom Fruehwirth (in spirit). Discussions on RBAC and
"Security Policies in Constraint Handling Rules".
11:20 David S. Warren. LPOP2018 XSB Position Paper.
11:30 Roberta Costabile, Alessio Fiorentino, Nicola Leone, Marco Manna, Kristian Reale
and Francesco Ricca. Role-Based Access Control via JASP.
11:40 Marc Denecker. The RBAC challenge in the Knowledge Base Paradigm.
11:50 Tuncay Tekle. Role-Based Access Control via LogicBlox.
12:00 Joost Vennekens. Logic-based Methods for Software Engineers and Business People.
12:10 Yanhong A. Liu and Scott Stoller. Easier Rules and Constraints for Programming.
12:20 All Workshop Participants. Questions about RBAC challenge solutions.
Session 3: Challenge Solutions and Constraint Solving
Session Chair: K. Tuncay Tekle
14:00 Panel: Practice of Modeling and Programming.
Panel Chair: Peter Van Roy. Panelists: All Morning Speakers.
14:30 Invited Talk: John Hooker. A Modeling Language Based on Semantic Typing.
15:10 Neng-Fa Zhou and Hkan Kjellerstrand.
A Picat-based XCSP Solver - from Parsing, Modeling, to SAT Encoding.
15:20 Paul Fodor. Role-Based Access Control as a LP/CP/Prolog Programming Challenge.
Session 4: Logic and Constraints in Applications
Session Chair: David Warren
16:00 Invited Talk: Rustan Leino. The Young Software Engineers Guide to Using
Formal Methods.
16:40 Torsten Schaub. How to upgrade ASP for true dynamic modelling and solving?
16:50 Peter Van Roy. A software system should be declarative except where it interacts
with the real world.
17:00 All Workshop Participants. Questions about logic and constraints in real-world applications.
17:10 Panel: Future of Programming with Logic and Knowledge.
Panel Chair: David Warren. Panelists: All Afternoon Speakers
17:40 David Warren and Annie Liu. Future of LPOP.
17:50 Tuncay Tekle and Marc Denecker. Closing.
1.2 Organization
The organizers and others responsible for the workshop were:
Chairs
David Warren
Stony Brook University
Annie Liu
Stony Brook University
Program Committee Chairs
Marc Denecker
KU Leuven
Tuncay Tekle
Stony Brook University
Program Committee
Molham Aref
Relational AI
Manuel Carro
IMDEA Software
Thomas Eiter
Technical University of Vienna
Jacob Feldman
OpenRules
Thom Frühwirth
University of Ulm
Michael Kifer
Stony Brook University
Mark Miller
Google
Enrico Pontelli
New Mexico State University
Francesco Ricca
University of Calabria
Peter Van Roy
Université catholique de Louvain
Joost Vennekens
Katholieke Universiteit Leuven
Jan Wielemaker
Vrije Universiteit Amsterdam
Neng-Fa Zhou
City University of New York
1.3 Homepage
http://lpop.cs.stonybrook.edu/
It contains the full workshop program with links to the presentation slides.
2 Invited Talks
Four invited speakers gave excellent talks:
John Hooker
Carnegie Mellon University
Rustan Leino
Amazon Web Services
Nicola Leone
University of Calabria
Michael Leuschel
University of Dusseldorf
2.1 A Modeling Language Based on Semantic Typing
Speaker: John Hooker, Carnegie Mellon University
Abstract:
A growing trend in modeling is the construction of high-level modeling languages that invoke a suite of solvers. This requires automatic reformulation of parts of the problem to suit different solvers, a process that typically introduces many auxiliary variables. We show how semantic typing can manage relationships between variables created by different parts of the problem. These relationships must be revealed to the solvers if efficient solution is to be possible. The key is to view variables as defined by predicates, and declaration of variables as analogous to querying a relational database that instantiates the predicates. The modeling language that results is self-documenting and self-checks for a number of modeling errors.
(Joint work with André Ciré and Tallys Yunes.)
Slides:
https://drive.google.com/file/d/1emvbNY9bp3AWn6h4EHI7y3VphaZZ7gYL/
2.2 The Young Software Engineerâs Guide to Using Formal Methods
Speaker: Rustan Leino, Amazon Web Services
Abstract:
If programming was ever a hermit-like activity, those days are in the past. Like other internet-aided social processes, software engineers connect and learn online. Open-source repositories exemplify common coding patterns and best practices, videos and interactive tutorials teach foundations and pass on insight, and online forums invite and answer technical questions. These knowledge-sharing facilities make it easier for engineers to pick up new techniques, coding practices, languages, and libraries. This is good news in a world where software quality is as important as ever, where logic specification can be used to declare intent, and where formal verification tools have become practically feasible.
In this talk, I give one view of the future of software engineering, especially with an eye toward software quality. I will survey some techniques, look at the history of tools, and inspire with some examples of what can be daily routine in the lives of next-generation software engineers.
Slides:
https://drive.google.com/file/d/0B9ffoWLQuWUXRTRtRElodFliMW5uaVhYMGQtb1FiLTJXLTJZ/
2.3 On the Development of Industrial Applications with ASP
Speaker: Nicola Leone, University of Calabria
Asbtract:
Answer Set Programming (ASP) is a powerful rule-based language for knowledge representation and reasoning that has been developed in the field of logic programming and nonmonotonic reasoning. After many years of basic research, the ASP technology has become mature for the development of significant real-world applications. In particular, the well-known ASP system DLV has undergone an industrial exploitation by a spin-off company called DLVSYSTEM srl, which has led to its successful usage in a number of industry-level applications. The success of DLV for applications development is due also to its endowment with powerful development tools, supporting researchers and software developers that simplify the integration of ASP in real-world applications which usually require to combine logic-based modules within a complete system featuring user interfaces, services etc. In this talk, we first recall the basics of the ASP language. Then, we overview our advanced development tools, and we report on the recent implementation of some challenging industry-level applications of our system.
(Joint work with Bernardo Cuteri, Marco Manna, Francesco Ricca)
Slides:
https://drive.google.com/file/d/1GGWtDzsIVnh43_kLpPjA8h7P1kDUTkYA/
A paper describing this work is included in Appendix LABEL:leone-invited.
2.4 Practical Uses of Logic, Formal Methods, B and ProB
Speaker: Michael Leuschel, University of Dusseldorf
Abstract:
The B method is quite popular for developing provably correct software for safety critical railway systems, particularly for driverless trains. In recent years, the B method has also been used successfully for data validation (http://www.data-validation.fr). There, the B language has proven to be a compact way to express complex validation rules, and tools such as predicateB, Ovado or ProB can be used to provide high assurance validation engines, where a secondary toolchain validates the result of the primary toolchain.
This talk will give an overview of our experience in using logic-based formal methods in general and B in particular for industrial applications. We will also touch subjects such as training and readability and the implementation of ProB in Prolog. We will examine which features of B make it well suited for, e.g., the railway domain, but also point out some weaknesses and suggestions for future developments. We will also touch upon other formal methods such as Alloy or TLA+, as well as other constraint solving backends for B, not based on Prolog (SAT via Kodkod/Alloy and SMT via Z3 and CVC4).
Slides:
https://drive.google.com/file/d/1Q19wdAQJiXBTRGiiYM_YjkYqyvEaSvqU/
A Jupyter notebook can be found at:
https://drive.google.com/file/d/11UNiLAIlHLHTAmMH__d2JEqrm8kAzc6Z/
3 The Challenge Problem
The domain and the specific functions and components of the challenge problem were selected to give participants the opportunity to demonstrate the best features of their (preferred) logic language. Those features may be from a broad spectrum: elegance, naturalness, compactness, modularity of expression, broadness of the functionality of the logic tools (e.g., a strong point would be if tools are available to prove correctness of your solutions), reuse of the specification to solve different parts of the problem, efficiency, etc.
Participants were free to select only a subset of the functions and components, or to implement variants of them, as long as their solutions showed the utility of their logic approach.
The domain of the challenge was Role-Based Access Control (RBAC). This is a security policy framework for controlling user access to resources based on roles. The challenge included functions and components for several well-known variants and extensions of RBAC, each involving its own set of constraints.
Participants were free, indeed encouraged, to present solutions for other components that show specific strengths of their logic, e.g., such as the aforementioned proof of correctness of logic solutions. We were interested as well in new challenges for logic systems, tasks that cannot yet be solved by existing systems but that pose an interesting research goal.
The RBAC programming challenge is included in Appendix LABEL:liu-challenge.
The slides for the presentation are in the first half of those at:
https://drive.google.com/file/d/1kzfE_CTYfAYgGSLg75ZJojhF1fEk4BSC/
Participants were encouraged to include programs, specifications, and other related materials in appendices to their position papers.
These papers appear as appendices.
4 Solutions to the Challenge
We summarize each proposed solution to the RBAC challenge in the following sections.
4.1 Answer Set Programming with Java: JASP
Nicola Leone presents joint work with Roberta Costabile, Alessio Fiorentino, Marco Manna, Kristian Reale, and Francesco Ricca that attacks the RBAC challenge problem using Answer Set Programming (ASP), as implemented in the JASP system. JASP is an extension of the Java programming language with an ASP solver, allowing a programmer to use Java for procedural, state-changing operations and use ASP for declarative query solving.
To solve the RBAC challenge problem, Java is used to update an external relational database and to read the state of the database, generate the necessary facts and rules in the correct form required for the ASP solver, invoke the ASP solver to compute query answers declaratively, and finally update the external database based on the query results when necessary.
This approach separates the procedural aspects of the problem from the query aspects by implementing the procedural aspects in the procedural language Java and the query aspects in the declarative ASP framework.
Leone’s presentation describes in detail the issues around implementing the RBAC function GetRolesShortestPlan, which he says is ”the hardest function” to implement in this framework. It is mentioned that other tasks of the challenge can be solved in a similar way but no specifics are given.
The slides for the presentation of Leone, et al. can be found at:
https://drive.google.com/file/d/1cBOHB4Vj3QS21iwp_-fLB36-KUv8ZEGF/
4.2 Prolog with Tabling: XSB
David S. Warren approaches the RBAC challenge problem using classical Prolog, in particular, the version implemented in the XSB system [2], taking advantage of particular features of that implementation.
The traditional approach would be to use Prolog’s assert and retract operations to update RBAC facts stored in Prolog’s global internal database.
But this approach is non-declarative.
So instead Warren uses a data structure to represent a database state that is explicitly passed through all defined update operations and query operations required to solve the challenge problem.
This makes them all purely declarative.
The procedural aspects are integrated into the declarative logical framework by making all update predicates depend on input and output database arguments.
This can be seen as a primitive implementation of a kernel portion of Transaction Logic [1].
Integrity constraint checking can be done before or after database update, with Prolog’s standard backtracking naturally handling ”transaction rollback” if a check fails.
Warren notes that Prolog’s DCG notation can be used to avoid having to explicitly pass the database parameter through all update operations.
The main challenge with this approach is that of efficiency, i.e., whether this Prolog data structure can compete in efficiency with Prolog’s native assert and retract and whether tabling, which is fundamental to XSB’s evaluation strategy, can be made efficient when applied to predicates containing the database as argument(s).
These problems are attacked by the use of a data structure defined in a new XSB package that supports update and query operations on a set of Prolog rules stored in a complex, trie-based Prolog term, but no detailed discussion of performance is provided.
Warren’s RBAC implementation solved all the update and direct query and aggregation problems proposed in the challenge; he did not attempt the more complex optimization problems.
Warren’s paper and RBAC solutions are included in Appendix LABEL:warren.
His presentation slides are available at:
https://drive.google.com/file/d/1DhgLh4LkUCs3JcrieqcPTdb_hBL8yoO4/
4.3 Knowledge Base Paradigm: IDP
Marc Denecker, in joint work with Jo Deviendt, describes an approach to solving the RBAC challenge problem in the framework of IDP, an implementation of a knowledge base paradigm.
IDP uses first-order logic combined with inductive definitions to specify declarative knowledge, and then applies a variety of inference mechanisms to this static data to solve various knowledge problems.
The difficult aspect of the RBAC tasks for this framework is how to incorporate the database update operations within this purely logical paradigm.
This presentation is a theoretical exploration of how this might be done in the IDB framework; no actual code for any RBAC task is provided.
The approach taken here is to add an explicit temporal argument to each predicate that describes the RBAC state.
Thus the procedural aspects of the problem are handled by using a temporal logic and explicitly reasoning with time.
Then the framework needs ”boiler-plate” frame axioms that describe the important properties of time, and also axioms that describe the properties of time for predicates that contain a temporal argument.
Finally, one must explore how the axioms can be efficiently processed by the inference mechanisms of IDP to ensure that this approach will lead to practical solutions to the various tasks of the RBAC challenge.
The paper is provided in Appendix LABEL:denecker.
The slides of Denecker’s presentation are available at:
https://drive.google.com/open?id=1Q5JHPuAwWPhBIBdbO_JyUikIeDlMswfy/
4.4 Datalog Extensions and Scripting Blocks: LogicBlox
Tuncay Tekle presented a solution to the RBAC challenge using LogicBlox, a
commercial system for developing enterprise transactions and analytics
applications. The solution was enabled by LogicBlox’s powerful query language,
LogiQL, which extends Datalog with constraint checking, aggregates, and
updates.
Tekle summarizes LogicBlox as ”a state-based system with a persistent
database that can be manipulated, where one can add facts to the database,
and rules and constraints to the state, and query the database at any point
in time”. LogicBlox also uses command line scripting to execute blocks of rules, facts, etc.
For the RBAC challenge, specifications of sets and relations, and
constraints over them are easily written in LogiQL. So are relational
queries over them, including recursive queries, all expressed easily using
Datalog rules. Aggregations such as count are expressed use special, extended forms
of rules, less direct than can be expressed using SQL. Updates are
directly expressed with notations + and - in the conclusions of rules.
For the two optimization problems in administrative RBAC, a restricted
version of one of them could be expressed in the LogicBlox framework with
some rewrite. The other optimization problem and the two planning problems
could not be solved using LogicBlox.
Tekle’s paper is included in Appendix LABEL:tekle.
The slides are available at:
https://drive.google.com/open?id=1sp5poNjknmNVbkNhHYhyvupPY9VTeGyn/
4.5 Logic with Interface: IDP with Python API, or DMN
Joost Vennekens illustrated solving the RBAC challenge using an approach
he had recently proposed.
In this approach, a relation is represented as a list of tuples, directly
written as so in the Python programming language, and a relational query is expressed using a Python
generator expression, such as ”all” for universal quantification. This way,
programmers need to know only the programming constructs in Python, not those in
logic programming systems.
These programming constructs are taken as an interface to a logic
programming systems, where the data and queries could be interpreted
with a more general meaning, e.g., as constraints relating the data,
instead of queries of some derived data from given data. This general
meaning allows some desired derived data be given and some other data be
inferred.
Vennekens had developed such a Python API for the IDP system. He
expresses in Python two example relations and an example query from core RBAC in the RBAC
challenge, but not the rest of the functions and components.
Vennekens also uses the recent Decision Model and Notation (DMN) standard
to support the general argument that more familiar notations to domain
experts can help increase the impact of logic-based methods to business people.
This paper is provided in Appendix LABEL:vennekens.
The slides can be found at:
https://drive.google.com/open?id=1WYQjkfS1blOU5zvM5ZAzDm12akSGAoBk/
4.6 Rules and Constraints Extending Python: DistAlgo Extensions
Annie Liu presented in joined work with Scott Stoller a solution to the RBAC
challenge in a high-level language that extends the Python programming
language. This work starts with DistAlgo, an extension of Python for
distributed programming especially with high-level set and logical queries,
and proposes to add rules, constraint optimizations, and backtracking.
With DistAlgo, their solution specifies the hierarchical component
structure of the challenge RBAC explicitly, as required in the challenge
and as in the ANSI standard. This includes core RBAC, hierarchical RBAC,
core RBAC with constraints, hierarchical RBAC with constraints, and
Administrative RBAC, as in the main challenge, as well as distributed RBAC
as an optional component in the challenge.
Each component includes the definitions of sets and relations, in addition
to those inherited from the parent components if any, as well as all query
and update operations. For computing transitive closures in hierarchical
RBAC, they gave an implementation that uses high-level set queries and an
alternative implementation that uses Datalog rules.
All components and operations are fully executable in DistAlgo except for
administrative RBAC, which needs the extensions for constraint optimization
and backtracking, and the alternative implementation of transitive closure
using rules.
This paper including the solution program is provided in Appendix LABEL:liu.
The slides for the presentation are in the second half of those at:
https://drive.google.com/file/d/1kzfE_CTYfAYgGSLg75ZJojhF1fEk4BSC/
4.7 RBAC Role Minimization as a LP/CP Programming Contest Challenge
Paul Fodor presented different logic programming solutions to the
problem of minimum role assignments with hierarchy in
Administrative RBAC. This is formulated as a constraint optimization problem, and as such
does not address the issues of state update and imperative programming.
Fodor used this problem as the first problem in the Logic
Programming and Constraint Programming Contest at ICLP
2018 (https://sites.google.com/site/prologcontest2018/).
He presented the best four solutions, two in ASP, one in Prolog, and one in Picat.
The two ASP solutions both used the #minimize operator, but defined the predicate to
be optimized in somewhat different ways.
The Prolog solution used tabling to find all possible solutions, findall to collect them, and then
explicit comparisons to find the optimal one.
The Picat solution formulated the problem as a constraint problem,
similar in concept to the ASP solutions, but using Picat’s syntax and primitives.
This does not have a paper. The slides can be found at:
https://drive.google.com/file/d/1aszplEMEUdUyaUqNU8_GhFtzbLN42dmf/
4.8 Security Policies in Constraint Handling Rules (CHR)
Thom Fruhwirth provided a position paper on the use of CHR
for the representation of security policies, but he did not provide a solution to the RBAC challenge. He was unable to attend the workshop to give a presentation.
His paper can be found in Appendix LABEL:fruhwirth.
5 Additional presentations
Some authors and presenters did not address the RBAC challenge but discussed methods, tools, and ideas for integrating different programming paradigms. We summary each of these below.
5.1 Upgrading ASP for True Dynamic Modelling and Solving
In his presentation ”How to upgrade ASP for true dynamic modelling and solving” Torsten Schaub discusses the issues involved in extending ASP concepts and implementations in ways that support the solving of dynamic problems, i.e., problems that involve data that change over time.
He discusses three important aspects of extending ASP in this direction: modeling, encoding and solving, and bench-marking.
Modeling issues involve what formal extension to the logic of ASP is appropriate for specifying dynamic systems.
Schaub proposes Temporal Equilibrium Logic, which combines the ideas of the logic of Here-and-There with Linear Temporal Logic.
Encoding involves how to represent a problem in the modeling language in such a way that it can be efficiently solved by ASP solvers and their extensions.
And finally Schaub emphasizes the importance of good, scalable, realistic benchmarks that allow various systems to be effectively compared.
He proposes that benchmarks be developed to address the real-world problem of controlling warehouse operations that use robot vehicles to retrieve items from mobile shelves.
He argues that this provides an excellent domain for exploring many aspects of using an ASP framework for dynamic systems.
This paper is available in Appendix LABEL:schaub.
The slides for this talk can be found at:
https://drive.google.com/file/d/1GUQC4qXYkt9lK3te0Uc6ZrEFgYNDLQuX/view
5.2 A Picat-Based XCSP Solver
Neng-Fa Zhou presented joint work with Hakan Kjellerstrand in a presentation titled ”A Picat-based XCSP Solver - from Parsing, Modeling, to SAT Encoding.”
The presentation provides an overview of a Picat-based XCSP3 solver, named PicatSAT, which demonstrates the strengths of Picat, a logic-based language, in parsing, modeling, and encoding constraints into SAT.
XCSP3 is an XML-based language for specifying constraint satisfaction problems, and PicatSAT uses Picat to process these specifications.
The presentation included a brief description of parsing the XCSP3 language, the advantages of using specialized Picat constructs to compactly implement a variety of constraints, and issues involved in encoding SAT problems in Picat.
This paper is provided as Appendix LABEL:zhou.
The slides are available at:
https://drive.google.com/file/d/1-F0RwPQVISeqzR1yr1_wkdvGcsT_i4Hq/
5.3 Declarative Programming for All Except Interaction with the Real World
Peter Van Roy proposes a principle for combining declarative programming
and imperative programming when building software systems.
While declarative programming supports ease of reasoning for
analysis, verification, optimization, and maintenance, it cannot
express interaction with the real world, because it does not
support common real-world concepts such as physical time and
named state, which are supported by imperative programming.
Therefore, the principle is: a software system should be
declarative except where it interacts with the real world.
Examples such as the client-server model from distributed
computing are used as motivation, and a formal argument is
outlined using lambda-calculus and an extension.
Van Roy’s paper is provided as Appendix LABEL:vanroy.
The slides for the talk can be viewed at:
https://drive.google.com/file/d/1qVrRwsO3b9LJv8OdhCF_Ali98Gpz443t/
6 Conclusion
The workshop was deemed a success, with the panel discussions and audience participation that followed invited talks and paper presentations being particular noteworthy.
The intention of the organizers is to hold LPOP every two years.
LPOP 2020 was initially intended to be held in conjunction with LICS 2020 in Beijing, but due to travel complexities
will instead be held in conjunction with SPLASH 2020.
References
[1]
Anthony J. Bonner and Michael Kifer.
Transaction logic programming.
In ICLP, 1993.
[2]
David S. Warren, Terrance Swift, and Konstantinos F. Sagonas.
The XSB programmer’s manual, Version 2.7.1.
Technical report, Department of Computer Science, State University of
New York at Stony Brook, Stony Brook, New York, 11794-4400, Mar 2007.
The XSB System is available from xsb.sourceforge.net, and the
system and manual is continually updated.
See pages 1-1 of papers/leone_invited_proceedings_paper_803.pdf
See pages 2-3 of papers/leone_invited_proceedings_paper_803.pdf
See pages 1-1 of papers/liu_Challenge.pdf
See pages 2-4 of papers/liu_Challenge.pdf
See pages 1 of papers/warren_proceedings_paper_808.pdf
See pages 2-6 of papers/warren_proceedings_paper_808.pdf
See pages 1 of papers/leone_proceedings_paper_799.pdf
See pages 2-3 of papers/leone_proceedings_paper_799.pdf
See pages 1 of papers/denecker_proceedings_paper_1010.pdf
See pages 2-17 of papers/denecker_proceedings_paper_1010.pdf
See pages 1 of papers/tekle_proceedings_paper_1002.pdf
See pages 2-5 of papers/tekle_proceedings_paper_1002.pdf
See pages 1 of papers/vennekens_proceedings_paper_807.pdf
See pages 2-3 of papers/vennekens_proceedings_paper_807.pdf
See pages 1 of papers/liu_RBACsolution.pdf
See pages 2-9 of papers/liu_RBACsolution.pdf
See pages 1 of papers/fruhwirth_proceedings_paper_801.pdf
See pages 2-3 of papers/fruhwirth_proceedings_paper_801.pdf
See pages 1 of papers/schaub_proceedings_paper_805.pdf
See pages 2-6 of papers/schaub_proceedings_paper_805.pdf
See pages 1 of papers/zhou_proceedings_paper_809.pdf
See pages 2-3 of papers/zhou_proceedings_paper_809.pdf
See pages 1 of papers/vanroy_proceedings_paper_806.pdf
See pages 2 of papers/vanroy_proceedings_paper_806.pdf |
Remembering Sergio Fubini111Fubini Memorial, CERN, Geneva, May 2005
R. Jackiw
Center for Theoretical Physics
Massachusetts Institute of Technology
Cambridge, MA 02139
MIT-CTP-3628
Abstract
Abstract
The author recollects Sergio Fubini’s impact on field theory
(radial quantization, merons, conformal quantum mechanics) and on MIT.
Sergio Fubini
1928-2005
Sergio Fubini was in Boston in 1967, delivering the Loeb lectures at Harvard. I was a postdoc there and so I met him for the first time. In the same period Steven Weinberg also visited and lectured. It must have been clear that both were interested in staying in the area (at least for a while) because Viki Weisskopf, the very astute chairman of MIT’s physics department, succeeded in hiring them – a fortunate development that caused MIT to become the fountainhead of the principal themes in fundamental physics for the remainder of the 20th century. This was a direct consequence of the research that the two accomplished in their years at MIT: Weinberg reviving quantum field theory, unifying particle physics forces and discussing cosmology; Fubini developing an operator formalism based on the Veneziano amplitude, going beyond field-theoretic descriptions of Nature and leading to today’s string theory.
Here we are commemorating Fubini, and Gabriele Veneziano already described their seminal string theory work at MIT. I shall speak about our joint work on field theory, and also about related investigations that Sergio subsequently carried out.
Urged by Fubini and Weinberg, and wishing to be their colleague, I joined the MIT physics department in 1969. In my research I was finding new phenomena within current algebra and symmetry behavior in quantum field theory. These were subjects for which Fubini and his collaborators established fundamental results: approximate symmetries described by chiral currents; procedures for deriving physically useful sum rules from equal-time current commutators; the infinite momentum frame. fub1
At that time, the findings by MIT-SLAC experimentalists of scaling in high energy (deep inelastic) electron scattering called attention to scale and conformal symmetries fub2 . Moreover, the observed scattering amplitudes become determined at high energies by the behavior of relevant currents near the light cone fub3 . These observations led to the technique of “light-cone” quantization, first discussed by Paul Dirac fub4 . In this approach, which is equivalent to the usual equal-time quantization, canonical commutation relations are posited on light-like surfaces, rather than at equal times. My collaborators and I took these ideas to the further step of deriving and postulating light-cone commutators for currents fub5 . This interested Sergio very much, because it turned out that the light-cone technique gives an operator formulation for his infinite momentum frame and results in a more efficient and accurate derivation of the sum rules, which he first discussed on the basis of equal-time current commutators.
I expect that this confluence of our research streams led to our collaboration. We took another quantization idea from Dirac, fub4 and developed a canonical formalism for fields by positing canonical commutation relations on surfaces of fixed $X^{\mu}X_{\mu}$, with dynamical evolution proceeding in the direction normal to these surfaces fub6 . This manifestly Lorentz invariant formulation is especially convenient in Euclidean space, where fixed $X^{\mu}X_{\mu}$ defines a sphere, and evolution in the radial direction, normal to the sphere, is governed by the dilation generator $D$. Thus scale and conformally invariant theories fit very nicely into this approach, since there the dilation generator $D$ is a constant of motion (like the Hamiltonian $H$ in usual equal-time quantization of time-independent systems).
Initially our results appeared to be only of methodological, rather than practical, interest. Because in radial Euclidean quantization the kinematical space at fixed radius is finite, we could give well-defined formulas for the Virasoro generators as moments of the energy-momentum tensor in 2-dimensional, conformally invariant theories. Moreover, the important center in that algebra was identified with the anomalous Schwinger terms in the “equal-radius” commutators of the energy-momentum tensor. In this way another bridge appeared between our earlier researches: linking chiral anomalies on the one side with scale/conformal anomalies on the other. Both symmetries rely on absence of mass terms for the fields. But Nature’s evident abhorrence of masslessness is encoded in quantum field theory by anomalies.
Years later, when string theory revived, our radial quantization procedure became generally adopted as the preferred method for quantizing 2-dimensional, conformally symmetric models, which are at the core of string theory fub7 .
After this joint work, Sergio left for CERN in 1973, and our direct collaboration ended. However, I was fortunate that a virtual collaboration continued, with both of us pursuing further topics of contemporaneous and mutual interest. The 1970’s were a time for exploring field theoretic classical solutions and establishing their quantum meaning. The most important of these for physics was the Euclidean Yang-Mills instanton, found by a soviet of physicists fub8 . The conformal invariance of Yang-Mills theory enlarges its ISO(3,1) Poincaré symmetry to an O(4,2) conformal symmetry, which becomes O(5,1) for the Euclidean theory. Claudio Rebbi and I showed that the instanton is O(5) invariant fub9 . Fubini, with Vittorio de Alfaro and Giuseppe Furlan, considered the Lorentzian theory and found regular solutions that preserve the O(4) $\times$ O(2) subgroup of O(4,2) fub10 . Thus both solutions are symmetric under the maximal compact group on their respective spaces. Moreover, in the Euclidean version, the Fubini et al. solution acquires localized singularities and in minimal form carries half of the instanton’s topological quantum number.
The result lay fallow until the Princeton group fub11 suggested that the new solution should be called a “meron” (after the Greek word for “portion” [of an instanton]) and that merons cause color confinement. This idea has not been universally accepted, although it continues generating research to this day fub12 .
An alternative quantization in a conformally invariant model was put forward by Fubini within particle quantum dynamics, which can be viewed as a “field theory” in one time and zero space dimensions. When the dynamics is also conformally invariant, its time translation “ISO(1,0)” symmetry becomes enlarged to O(2,1), with three generators. Compact rotations are generated by one of them, R: the time translation generator $H$ summed with the conformal generator $K,R\propto H+K$. [The third generator is the dilation $D$.] In the quantum theory one may choose to diagonolize $R$, which has a discrete spectrum. This choice replaces conventional diagonalization of $H$ and/or radial quantization with its diagonalization of $D$. With de Alfaro and Furlan, Fubini applied this method to the quantum mechanics of the $1/r^{2}$ potential, known to be scale/conformal invariant fub2 . A very elegant group theoretical analysis ensued, and “conformal quantum mechanics” was born fub13 .
But its utility at first remained obscure. It was established that interactions with magnetic point monopoles and vortices also preserve conformal symmetry and could fit into conformal quantum mechanics fub14 . Moreover, it was shown that the higher O(2,1) symmetry of the time-dependent Schrödinger equation allows separating variables in ways other than the conventional time/space factorization. Thus conformal quantum mechanics arises from such an unconventional separation of variables in the O(2,1) invariant, time dependent Schrödinger equation fub14 . [A familiar analogous situation arises with the 1/r potential, which in addition to rotational invariance, enjoys an O(4) symmetry; correspondingly the Schrödinger equation can be separated in spherical and parabolic coordinates.] Fubini also demonstrated that similar conformal quantization can be carried out for conformally invariant field theories on space-time, but space-translation invariance is lost, to be replaced by a statistical conservation law for spatial momentum fub15 . Although this mechanism has not found favor, it anticipates contemporary interest in violations of space-time symmetries.
The full impact of conformal quantum mechanics came only after string theorist realized that dynamics of a particle near a black hole may enjoy an O(2,1) symmetry, and the Fubini et al. choice of $R$ as the evolution operator corresponds to a redefinition of the time variable near the black hole fub16 .
Thus it turns out that no matter how far Sergio’s research strays from string theory, its final relevance reverts to string theory. Indeed an overview of his physics career shows that in early days he approached fearlessly dynamics of elementary particles, even where fundamental laws were unknown. This characterizes his work on the multiperipheral model, his sum rules and superconvergence relations, and of course the dual resonance model. But it seems to me that the mathematical depths that he plumbed in developing the latter into string theory awakened in him an interest in mathematical formalism, for which he surely had a genetic predisposition from his forbears. Nevertheless, Sergio’s physical intuition prevailed and his explorations of formal mathematical physics, unified by the themes of conformal symmetry and alternative quantization procedures, found unanticipated applications.
Fubini left MIT in 1973, but his impact on our physics department persisted, and continues to this day. He raised the visibility of the particle physics group by centering at MIT initial string theory research. Also one of the conditions of his employment was an agreement for an exchange program with Turin University. This brought to MIT countless Italian students, post-docs and senior visitors – not only from Turin – who were eagerly participating in the rich physics activity generated for the nascent string theory by Fubini and Veneziano, and also turning to other areas of fundamental physics. On the faculty level Fubini encouraged and supported his compatriots Rebbi and Veneziano on term appointments and Bruno Coppi in a permanent position at MIT.
The flow of people continued informally after he left, but later it was formalized by a new INFN-MIT agreement, which established the “Bruno Rossi” exchange program, named after an earlier Italian, who made his professional home at MIT and, like Sergio, did wonderful physics.
I last saw Sergio during a visit to CERN, where we met for lunch. During the meal, he became ill, so I insisted on driving him home to St. Cergue. He was very moved by my offer, and said that I treat him like a brother. I realize that for many years he was like an older brother, advising me professionally and encouraging me to enter new activities, like his Middle East peace initiative. That effort, similar to some of his physics researches, awaits success only in the future, even while today it puts into vivid evidence his great talents in the service of humane pursuits.
References
(1)
A guide to this research is V. de Alfaro, S. Fubini, G. Furlan and C. Rossetti, Currents in Hadron Physics (North Holland, Amsterdam, 1973).
(2)
R. Jackiw, “Introducing Scale Symmetry,” Physics Today, 25 (1), 23 (1972).
(3)
R. Jackiw, R. Van Royen and G. B. West,
“Measuring Light Cone Singularities,”
Phys. Rev. D 2, 2473 (1970).
(4)
P.A.M. Dirac, “Forms of Relativistic Dynamics,”
Rev. Mod. Phys. 21, 392, (1949).
(5)
J. M. Cornwall and R. Jackiw,
“Canonical Light Cone Commutators,”
Phys. Rev. D 4, 367 (1971);
D. A. Dicus, R. Jackiw and V. L. Teplitz,
“Tests of Light Cone Commutators: Fixed Mass Sum Rules,”
Phys. Rev. D 4, 1733 (1971).
(6)
S. Fubini, A. J. Hanson and R. Jackiw,
“New Approach to Field Theory,”
Phys. Rev. D 7, 1732 (1973).
(7)
For a review, see P. Ginsparg, “Applied Conformal Field Theory,” in Les Houches Summer School (1988), [hep-th/9108028].
(8)
A. A. Belavin, A. M. Polyakov, A. S. Shvarts and Y. S. Tyupkin,
“Pseudoparticle Solutions of the Yang-Mills Equations,”
Phys. Lett. B 59, 85 (1975).
(9)
R. Jackiw and C. Rebbi,
“Conformal Properties of a Yang-Mills Pseudoparticle,”
Phys. Rev. D 14, 517 (1976).
(10)
V. de Alfaro, S. Fubini and G. Furlan,
“A New Classical Solution of the Yang-Mills Field Equations,”
Phys. Lett. B 65, 163 (1976);
“Properties of O(4) x O(2) Symmetric Solutions of the Yang-Mills Field
Equations,”
Phys. Lett. B 72, 203 (1977).
(11)
C. G. Callan, R. F. Dashen and D. J. Gross,
“A Mechanism for Quark Confinement,”
Phys. Lett. B 66, 375 (1977);
“Toward a Theory of the Strong Interactions,”
Phys. Rev. D 17, 2717 (1978).
(12)
F. Lenz, J. Negele and M. Thies,
“Confinement form Merons,”
Phys. Rev. D 69, 074009 (2004).
(13)
V. de Alfaro, S. Fubini and G. Furlan,
“Conformal Invariance in Quantum Mechanics,”
Nuovo Cim. A 34, 569 (1976).
(14)
R. Jackiw,
“Dynamical Symmetry of the Magnetic Monopole,”
Annals Phys. 129, 183 (1980);
“Dynamical Symmetry of the Magnetic Vortex,”
Annals Phys. 201, 83 (1990).
(15)
S. Fubini,
“A New Approach to Conformal Invariant Field Theories,”
Nuovo Cim. A 34, 521 (1976).
(16)
For a review, see R. Kallosh “Black Holes and Quantum Mechanics” in Novelties in String Theory,
L. Brink and R. Marnelius editors (World Scientific, Singapore, 1999). |
Context-Aware Parse Trees
Fangke Ye${}^{*1,2}$
Shengtian Zhou${}^{*1}$
Anand Venkat${}^{1}$
Ryan Marcus${}^{1,3}$
Paul Petersen${}^{1}$
Jesmin Jahan Tithi${}^{1}$
Tim Mattson${}^{1}$
Tim Kraska${}^{3}$
Pradeep Dubey${}^{1}$
Vivek Sarkar${}^{2}$
Justin Gottschlich${}^{1,4}$
${}^{1}$Intel Labs ${}^{2}$Georgia Tech ${}^{3}$MIT CSAIL ${}^{4}$University of Pennsylvania
(2018)
Abstract.
The simplified parse tree (SPT) presented in Aroma, a state-of-the-art code recommendation system, is a tree-structured representation used to infer code semantics by capturing program structure rather than program syntax. This is a departure from the classical abstract syntax tree, which is principally driven by programming language syntax. While we believe a semantics-driven representation is desirable, the specifics of an SPT’s construction can impact its performance. We analyze these nuances and present a new tree structure, heavily influenced by Aroma’s SPT, called a context-aware parse tree (CAPT). CAPT enhances SPT by providing a richer level of semantic representation. Specifically, CAPT provides additional binding support for language-specific techniques for adding semantically-salient features, and language-agnostic techniques for removing syntactically-present but semantically-irrelevant features. Our research quantitatively demonstrates the value of our proposed semantically-salient features, enabling a specific CAPT configuration to be 39% more accurate than SPT across the 48,610 programs we analyzed.
* Lead Authors.
code similarity, program synthesis, machine programming, software development, software maintenance
††copyright: none††doi: ††journalyear: 2018††isbn: –††conference: Preprint; March; 2020
1. Introduction
Machine programming (MP), as defined by Gottschlich et al. in ”The Three Pillars of Machine Programming,” is any system that automates some aspect of software development (Gottschlich
et al., 2018). An open research challenge in MP is how to build effective automated code similarity systems. The potential use cases for such code similarity systems ranges from code recommendation to automated software bug patching, to name a few (Dinella et al., 2020; Luan
et al., 2019; Allamanis et al., 2018b; Pradel and Sen, 2018; Bhatia
et al., 2018; Bader
et al., 2019; Barman
et al., 2016). Yet, as others have noted, the correct structural representation for such a code similarity system remains unclear (Luan
et al., 2019; Odena and Sutton, 2020; Ben-Nun
et al., 2018; Alon
et al., 2018, 2019b; Tufano et al., 2018; Alon
et al., 2019a; Zhang
et al., 2019; Allamanis et al., 2018a).
In this work, we present a new structural representation, a context-aware parse tree (CAPT). In contrast to syntactic representations, such as an abstract syntax tree (AST), CAPT is designed to capture the semantic meaning of the user’s code. That is, CAPT provides information relevant to whether two code snippets 111For the purposes of this work, we precisely define code snippet as a C/C++ function, discussed in more detail in Section 3. are semantically convergent, even if they are syntactically divergent. Code similarity is informally defined as the process of determining whether two code snippets are semantically similar. In this work, we show that CAPTs are competitive with simplified parse trees (SPTs), the representation used by the Aroma (Luan
et al., 2019) state-of-the-art code recommendation system. From the point of view of machine programming (Gottschlich
et al., 2018), this work seeks to improve our ability to automatically recognize a user’s intent from code. Thus, it principally falls in the ”intention pillar.” (Gottschlich
et al., 2018) In addition, once such intention is understood it may be augmented or transferred from language to language, thereby advancing the “adaptation pillar.”
Why Not An AST?
While the abstract syntax tree (AST) has had tremendous, demonstrable value in cases where syntactic structure is of primary importance (e.g., source code compilation (Aho
et al., 2006)), the utility of the AST in the space of extracting semantic meaning from code (i.e., lifting intention (Kamil et al., 2016; Ginsbach et al., 2018; Ahmad
et al., 2019)) may be less clear. The AST contains a complete syntactic representation of the program, which can contain many details relevant to program compilation, but that may be less salient for semantic analysis. For example, an AST for a single line of C/C++ code like,
int x = (y+3);
may contain separate nodes for a variable declaration, a compound statement, and three implicit casts.222https://clang.llvm.org/docs/IntroductionToTheClangAST.html#examining-the-ast While such details are critical for correctly implementing a compiler, they may not be relevant for semantic analysis.
For example,
int x = y;
may be considered semantically similar to the prior code snippet, even if the second operation does not have implicit casts or compound statements. Unlike an AST, a CAPT can omit these details to help improve similarity analysis. We note that the Aroma authors have previously illustrated some of these AST limitations as well (Luan
et al., 2019).
Why Not An SPT?
The Aroma team (Luan
et al., 2019), inspired, at least partially, by weaknesses in the AST (as stated directly by the authors), introduced the simplified parse tree (SPT). An SPT is a new structural representation for code similarity, which intentionally departs from the AST. By design, the SPT reduces the syntactic information collected from source code: each node in the SPT is strictly a token from the original program (no other nodes are introduced). The Aroma authors demonstrate that this reduction, or in some cases elimination, of lower-level syntactic information can be helpful for code similarity systems. Such a reduction may be especially salient in the context of type 3 and 4 similarity, where code may be syntactically different, but is semantically similar (Roy
et al., 2009).
While the AST may contain too much low-level syntactic information, the SPT may omit semantically-relevant information. For example, because the SPT contains only tokens, the SPT effectively omits whether or not a token binds to a local or a global variable. On the other hand, the SPT may also include too much specificity. For example, the exact name of a variable or function is not always relevant to its semantic meaning (e.g., two global variables with the same name but from different programs might not imply similar semantics). In light of this, we identified two areas where an SPT can be modified such that the resulting tree yields improved code similarity accuracy. Those areas are: (i) language-agnostic modifications, which remove potentially irrelevant syntactic information, and (ii) language-specific modifications, which introduces new syntactic information that may be semantically salient. We believe CAPTs can capture the essence of these two design elements. Our paper provides the following technical contributions:
•
We introduce the context-aware parse tree (CAPT), a novel modification of the simplified parse tree (SPT) intended to improve code similarity analysis.
•
We illustrate and discuss the two flexibility enhancements CAPT has compared to Aroma’s simplified parse tree: (i) language-agnostic and (ii) language-specific.
•
Our research quantitatively demonstrates the value of our proposed semantically-salient features, enabling a specific CAPT configuration to be 39% more accurate than SPT across the 48,610 programs we analyzed.
2. System Design
Before discussing the specifics of our approach, we first provide some background on how both CAPTs and SPTs are used in code similarity systems. Figure 2 presents an abbreviated overview of our code similarity system, MISIM (we illustrate MISIM using CAPTs, but any process that transforms a code snippet into a feature vector could be used, including SPTs). Figure 2 illustrates the process of transforming source code (e.g., a C function) to a feature vector. Once a feature vector is generated, a code similarity measurement (e.g., vector dot product (Lipschutz, 1968), cosine similarity (Singhal, 2001), machine-learned similarity (Zhao and Huang, 2018)) calculates the similarity score between the input program and other programs stored in a database.
333In the context of this work, we perform code similarity analysis on an entire C/C++ program, where we differentiate code snippets by uniquely defined C/C++ function bodies. Although the current system only supports C/C++ code, our design is agnostic to the underlying programming language. While we have built a prototype of the entire system, CAPT is the emphasis of this paper. As such, we omit a deeper dissection of other components of the system as we consider them outside of the scope of this paper.
To generate a CAPT, the code is first parsed into a language-agnostic parse tree. Next, the system performs language-specific transformations in constructing an initial intermediate form of CAPT by adding pertinent information used to disambiguate code. The system then performs language-agnostic transformations (e.g., abstracting the number of code statements in a function) by potentially pruning or modifying the CAPT’s nodes. Subsequently, a CAPT is featurized into a vector using the same procedure as SPT’s featurization process (Luan
et al., 2019).
3. The Context-Aware Parse Tree (CAPT)
In this section, we describe the fundamental design of CAPT, including the key differences between CAPT and Aroma’s simplified parse tree (SPT). Some of these details are visually illustrated in Figure 1. Fundamentally, a CAPT is the result of transforming an SPT in specific ways according to a configuration, options that give CAPT a greater degree of flexibility. Different configurations may result in better performance in some domains, but worse performance in others. Here, we focus on describing the intuition behind these options, and evaluating every possible configuration in one domain. In this preliminary work, we do not address figuring out which configuration to use in a particular domain, but plan to investigate this in future work.
CAPT Configuration Categories (see Table 1)
CAPT’s configuration categories come in two general forms: language-specific configurations and language-agnostic configurations, listed in Table 1. We next give an intuitive overview of both.
Table 1 lists the current types and options for the language-specific and language-agnostic categories in CAPT. 444While our exploration into CAPT is still early, we believe our categories may be exhaustive (that is, fully encompassing). Yet, we do not believe our configuration types or options are exhaustive. Each configuration type has multiple options associated with it to afford the user the flexibility in exploring a number of CAPT configurations. For all configuration types, option 0 always corresponds to the Aroma system’s original SPT. Each of the types in Table 1 are described in greater detail in the following sections.
Language-specific configurations, described in Section 3.1 are designed to resolve syntactic ambiguity present in the SPT. For example, in Figure 1, the SPT treats the parenthetical expression (global1 + global2) identically to the parenthetical expression init(global1), whereas the CAPT configuration shown disambiguates these two terms (the first is a parenthetical expression, the second is an argument list). Such a disambiguation may be useful to a code similarity system, as the CAPT representation makes the presence of a function call more clear.
Language-agnostic configurations, described in Section 3.2, can improve code similarity analysis by unbinding overly-specific semantics that may be present in the original SPT structure. For example, in Figure 1, the SPT includes the literal names global1, global2, etc. The CAPT variant, on the other hand, unbinds these names and replaces them with a generic string (#GVAR). This could improve code similarity analysis if the exact token names are irrelevant, and the semantically-salient feature is simply that there is a global variable.
We note that that these examples are not universal. One specific CAPT configuration is unlikely to work in all scenarios: sometimes, disambiguating parenthetical expressions may be good, other times, it may be bad. This work seeks to explore and analyze these possible configurations. We provide a formalization and concrete examples of both language-agnostic and language-specific configurations later in this section.
3.1. Language-Specific Configurations
Language-specific configurations are meant to capture semantic meaning by resolving ambiguity and introducing specificity related to the specific underlying programming language. Intuitively, these configurations can be thought of as syntax-binding, capturing semantic information that are bound to the particular syntactical structure of the program. In some cases, these specifications may capture relevant semantic information, whereas in other cases these specifications may capture irrelevant details.
Node Annotations.
We define a node annotation as a modification to a tree’s node to incorporate more information. Node annotations are generally used to facilitate disambiguation caused by language-specific syntax ambiguity. These ambiguous scenarios tend to arise when certain code constructs and/or operators have been overloaded in a specific language. In such cases, the original SPT structure may be insufficient to properly disambiguate between them, potentially reducing its ability to evaluate code semantic similarity (see Figure 3). CAPTâs node annotation options are meant to help resolve this.
As we incorporate more language-specific syntax into CAPT nodes, we run the chance of overloading the tree with syntactic details. This could potentially undo the general reasoning behind Aroma’s SPT and our CAPT structure. We discuss this in greater detail in Section 3.3.
3.1.1. C and C++ Node Annotations
For our first embodiment of CAPT, we have focused solely on C and C++ programs. We have found that programs in C/C++ present at least two interesting challenges.555We do not claim that these challenges are unique to C/++: these challenges may be present in other languages as well.
(Lack of) Compilation.
We have found that, unlike programs in higher-level programming languages (e.g., Python (Van Rossum and
Drake, 2009), JavaScript (Flanagan, 2006)), C/C++ programs found ”in the wild” tend to not immediately compile from a source repository (e.g., GitHub (Cosentino et al., 2017)). Thus, code similarity analysis may need to be performed without relying on successful compilation.
Many Solutions.
The C and C++ programming languages provide multiple diverse ways to solve the same problem (e.g., searching a list with a for loop vs. using std::find). Because of this, C/C++ enables programmers to create semantically similar programs that are syntactically divergent. In extreme cases, such semantically similar (or identical) solutions may differ in computation time by orders of magnitude (Satish
et al., 2012). This requires that code similarity techniques to be robust in their ability to identify semantic similarity in the presence of syntactic dissimilarity (i.e. a type 4 code similarity exercise (Roy
et al., 2009)).
We believe that analytically deriving the optimal selection of node annotations across all C/C++ code may be untenable. To accommodate this, we currently provide two levels of granularity for C/C++ node annotations in CAPT. 666This is still early work and we expect to identify further refinement options in C/C++ and other languages as the research progresses.
•
Option 0: original Aroma SPT configuration.
•
Option 1: annotation of all nodes with their language-specific node type.
•
Option 2: annotation of all nodes containing parentheticals with their language-specific node type.
Option 1 corresponds to an extreme case of a concrete syntax embedding (e.g., every node contains syntactic information, and all syntactic information is represented in some node). Since such an embedding may ”overload” the code similarity system with irrelevant syntactic details, Option 2 can be used to annotate only parentheticals, which we have empirically identified to often have notably divergent semantic meaning based on context.
An example is shown in Figure 1. In one case the parentheses is applied as a mathematical precedence operator, in the other it is used as a function call. If left unresolved, such ambiguity would cause the subtree rooted at node 7 of function f1 to be classified identically to the subtree rooted at node 5 of function f2. The intended purpose of the parenthesis operator is context sensitive and is disambiguated by encoding the contextual information into the two distinct node annotations, i.e. the parenthesized expression and the argument list respectively.
3.2. Language-Agnostic Configurations
Unlike language-specific configurations, language-agnostic configurations are not meant to be restricted to the specific syntax of a specific language. Instead, they are meant to be applied generally across multiple languages. Intuitively, these configurations can be thought of as syntax-unbinding in nature: they generally abstract (or, in some cases, entirely eliminate) syntactical information in the attempt to improve its ability to derive semantic meaning from the code.
Compound Statements.
The compound statements configuration is a language-agnostic option that enables the user to control how much non-terminal node information is incorporated into the CAPT. Again, Option 0 corresponds to the original Aroma SPT. Option 1 omits separate features for compound statements altogether. Option 2 does not discriminate between compound statements of different lengths and specifies a special label to denote the presence of a compound statement. For example, the for loop construct in C/C++ is represented with a single label with this option instead of constructing three separate labels for the loop initialization, test condition and increment.
Global Variables.
The global variables configuration specifies the degree of global variable-specific information contained in a CAPT. In addition to Aroma’s original configuration (Option 0), which annotates nodes by including the precise global variable name, CAPT provides three additional configurations. Option 1 specifies the extreme case of eliding all information on global variables. Option 2 annotates all global variables with the special label â#GVARâ, omitting the names of the global variable identifiers. Option 3 designates global variables with the label â#VARâ rendering them indistinguishable from the usage of local variables.
Intuitively, including the precise global variable names (Option 0) may be appropriate if code similarity is being performed on a single code-base, where two references to a global variable with the same name necessarily refer to the same global variable. Options 1 through 3, which remove global variable information to varying degrees, may be appropriate when performing code similarity between unrelated code-bases, where two different global variables named (for example) foo are most likely unrelated.
Global Functions.
The global functions configuration serves the dual purpose of (i) controlling the amount of function-specific information to featurize and (ii) to disambiguate between the usage of global functions and global variables in CAPT, a feature that is curiously absent in the original SPT design: the SPT shown in Figure 1 has no distinction between init (a function) and global1 (a variable). Option 1 removes all features pertaining to global functions. Option 2 annotates all global function references with the special label â#EXFUNCâ while eliminating the function identifier. Intuitively, these options behave similarly to the global variable options. Our current prototype, which handles only single C/C++ functions, does not differentiate between external functions. In future work, we plan to investigate CAPT variants that differentiate between local, global, and library functions.
3.3. Discussion
We believe there is no silver bullet solution for code similarity for all programs and programming languages. Based on this belief, a key intuition of CAPT’s design is to provide a structure that is semantically rich based on structure, with heavy inspiration from Aroma’s SPT, while simultaneously providing a range of customizable parameters to accommodate a wide variety of scenarios. CAPT’s language-agnostic and language-specific configurations and their associated options serve for exploration of a series of tree variants, each differing in their granularity of detail of abstractions.
For instance, the compound statements configuration provides three levels of abstraction. Option 0 is Aroma’s baseline configuration and is the finest level of abstraction, as it featurizes the number of constituents in a compound statement node. Option 2 reduces compound statements to a single token and represents a slightly higher level of abstraction. Option 1 eliminates all features related to compound statements and is the coarsest level of abstraction. The same trend applies to the global variables and global functions configurations. It is our belief, based on early evidence, that the appropriate level of abstraction in CAPT is likely based on many factors such as (i) code similarity purpose, (ii) programming language expressiveness, and (iii) application domain.
Aroma’s original SPT seems to work well for a common code base where global variables have consistent semantics and global functions are standard API calls also with consistent semantics (e.g., a single code-base). However, for cases outside of such spaces, some question about applicability arise. For example, assumptions about consistent semantics for global variables and functions may not hold in cases of non-common code-bases or non-standardized global function names (Wulf and Shaw, 1973; Gellenbeck and
Cook, 1991; Feitelson et al., 2020). The capacity to differentiate between these cases, and others, is a key motivation for CAPT.
We do not believe that CAPT’s current structure is exhaustive. With this in mind, we have designed CAPT to be extensible, enabling a seamless mechanism to add new configurations and options (described in Section 4). Our intention with this paper is to present initial findings in exploring CAPT’s structure. Based on our early experimental analysis, presented in Section 4, CAPT seems to be a promising research direction for code similarity.
An Important Weakness
While CAPT provides added flexibility over SPT, such flexibility may be misused. With CAPT, system developers are free to add or remove as much syntactic differentiation detail they choose for a given language or given code body. Such overspecification (or underspecification), may result in syntactic overload (or underload) which may cause reduced code similarity accuracy over the original SPT design, as we illustrate in Section 4.
4. Experimental Results
In this section, we discuss our experimental setup and analyze the performance of CAPT compared to Aroma’s simplified parse tree (SPT). In Section 4.1, we describe the code corpus used with CAPT that includes hundreds of unique programming solutions for 104 different programming problems. In Section 4.2, we explain the dataset grouping and enumeration for our experiments. We also discuss the metrics used to quantitatively rank the different CAPT configurations and those chosen for evaluation of code similarity. Section 4.3 demonstrates that, a code similarity system built using CAPT (i) has a greater frequency of improved accuracy for the total number of problems and (ii) is, on average, more accurate than SPT. For completeness, we also include cases where CAPT configurations perform poorly.
4.1. Dataset
Our experiments use the POJ-104 dataset. The POJ-104 dataset is the result of educationally-inspired programming questions, which consist of student written programs to 104 problems (Mou
et al., 2016). Each problem has 500 unique solutions written in C/C++. Each solution has been validated for correctness. We categorize all solutions for a given POJ-104 problem as being in the same semantic similarity equivalence class. We make no claims about the semantic similarity or dissimilarity of solutions to two or more different POJ-104 problems.
Using this approach, we treat the problem of code similarity analysis as a classification problem. We classify two programs as semantically similar if they originate from the same equivalence class (i.e., the same POJ-104 problem). Using this approach, the labels for these classifications can be implicitly lifted using the problem’s unique identifier.
Eliminated Programs From POJ-104.
Some of the coding solutions in the POJ-104 dataset have been marked as illegal by the parser we used 777Tree-sitter: http://tree-sitter.github.io/tree-sitter. After investigating, we found this to be due to the code using non-standard coding conventions (e.g., unspecified return types, lack of semicolons at the end of structure definitions, etc.). Because they could not be properly parsed, we have pruned them from the dataset. We also eliminated all of the solutions that we could find that had hard-coded answers to problems. 888Unfortunately, due to the size of the dataset, we cannot guarantee all such programs were eliminated. The resulting dataset consists of 48,610 programming solutions with 370 to 499 uniquely coded solutions per problem.
4.2. Experimental Setup
In this section, we describe our experimental setup. At the highest level, we compare the performance of various configurations of CAPT to Aroma’s SPT. The list of possible CAPT configurations are shown in Table 1.
Problem Group Selection.
Given that POJ-104 consists of 104 unique problems and 48,610 programs, depending on how we analyze the data, we might face intractability problems in both computational and combinatorial complexity. With this in mind, our initial approach is to construct 1000 sets of five unique, pseudo-randomly selected problems for code similarity analysis. Using this approach, we evaluate every configuration of CAPT and Aroma’s original SPT on each pair of solutions for each problem set. We then aggregate the results across all the groups to estimate their overall performance. While this approach is not exhaustive of possible combinations (in set size or set combinations), we aim for it to be a reasonable starting point. As our research with CAPT matures, we plan to explore a broader variety of set sizes and a more exhaustive number of combinations.
Code Similarity Performance Evaluation.
For each problem group, we exhaustively calculate code similarity scores for all unique solution pairs, including pairs constructed from the same program solution (i.e., program $A$ compared to program $A$). We use $G$ to refer to the set of groups and $g$ to indicate a particular group in $G$. We denote $|G|$ as the number of groups in $G$ (i.e. cardinality) and —g— as the number of solutions in group $g$. For $g$ = $G_{i}$, where $i=\{1,2,\ldots,1000\}$, the total unique program pairs (denoted by $g_{P}$) in $G_{i}$ is $\mathit{|g_{P}|}=\frac{1}{2}|g|(|g|+1)$.
To compute the similarity score of a solution pair, we use Aroma’s approach. This includes calculating the dot product of two feature vectors (i.e., a program pair), each of which is generated from a CAPT or SPT structure. The larger the magnitude of the dot product, the greater the similarity.
We evaluate the quality of the recommendation based on average precision. Precision is the ratio of true positives to the sum of true positives and false positives. Here, true positives denote solution pairs correctly classified as similar and false positives refer to solution pairs incorrectly classified as similar. Recall is the ratio of true positives to the sum of true positives and false negatives, where false negatives are solution pairs incorrectly classified as different. As we monotonically increase the threshold from the minimum value to the maximum value, precision generally increases while recall generally decreases. The average precision (AP) summarizes the performance of a binary classifier under different thresholds for categorizing whether the solutions are from the same equivalence class (i.e., the same POJ-104 problem) (Liu, 2009). AP is calculated using the following formula over all thresholds.
(1)
All unique values from the $M$ similarity scores, corresponding to the solution pairs, are gathered and sorted in descending order. Let $N$ be the number of unique scores and $s_{1},s_{2},\ldots,s_{N}$ be the sorted list of such scores.
(2)
For $i$ in $\{1,2,\ldots,N\}$, the precision $p_{i}$ and recall $r_{i}$ for the classifier with the threshold being $s_{i}$ is computed.
(3)
Let $r_{0}=0$. The average precision is computed as:
$$AP=\sum_{i=1}^{N}(r_{i}-r_{i-1})p_{i}$$
Configuration Identifier.
In the following sections, we refer to a configuration of CAPT by its unique identifier (ID). A configuration ID is formatted as A-B-C-D. Each of the four letters corresponds to a configuration type in the second column of Table 1, and will be replaced by an option number specified in the third column of the table. Configuration 0-0-0-0 corresponds to Aroma’s SPT.
4.3. Results
Figure 3(a) depicts the number of problem groups where a particular CAPT variant performed better (blue) or worse (orange) than SPT. For example, the CAPT configuration 2-0-0-0 outperformed SPT in 774 of 1000 problem groups, and underperformed in 226 problem groups. This equates to a 54.8% accuracy improvement of CAPT over SPT. Figure 3(a) shows the two best (2-0-0-0 and 2-1-2-2), the median (1-1-0-1), and the two worst (1-0-1-0 and 1-2-1-0) configurations with respect to SPT. 2-1-2-2 demonstrates a CAPT configuration where all its options are exercised and not tunable in SPT. This configuration performs better than SPT on 695 of the 1000 problem groups, i.e. on $\approx 39\%$ of the problem groups. Although it is less performant than the 2-0-0-0 configuration, it exercises all of CAPT’s unique tunable parameters. We speculate that these configuration results may vary based on programming language, domain, and problem type, amongst other parameters.
Figure 3(b) shows the group containing the problems for which CAPT achieved the best performance relative to SPT, among all 1000 problem groups. In other words, Figure 3(b) shows the performance of SPT and CAPT for the single problem group with the greatest difference between a CAPT configuration and SPT. In this single group, CAPT achieves the maximum improvement of more than 30% over SPT for this problem group on two of its configurations. We note that, since we tested 108 CAPT configurations across 1000 different problem groups, there is a reasonable chance of observing such a large difference even if CAPT performed identically to SPT in expectation. We do not intend for this result to demonstrate statistical significance, but simply to illustrate the outcome of our experiments.
Figure 3(c) compares the mean of AP over all 1000 problem groups. In it, the blue bars, moving left to right, depict the CAPT configurations that are (i) the two best, (ii) the median, and (iii) the two worst in terms of average precision. Aroma’s baseline SPT configuration is highlighted in orange. The best two CAPT configurations show an average improvement of more than 1% over SPT, while the others degraded performance relative to the baseline SPT configuration.
These results illustrate that certain CAPT configurations can outperform the SPT on average by a small margin, and can outperform the SPT on specific problem groups by a large margin. However, we also note that choosing a good CAPT configuration for a domain is essential. We leave automating this configuration selection to future work.
4.3.1. Analysis of Configurations
Figures 4(a)-4(d) serve to illustrate the performance variation for individual configurations. Figure 4(a) shows the effect of varying the options for the node annotation configuration. Applying the annotations for the parentheses operator (option 2) results in the best overall performance while annotating every internal node (option 1) results in a concrete syntax tree and the worst overall performance. This underscores the trade-offs in incorporating syntax-binding transformations in CAPT. In Figure 4(b) we observe that removing all features relevant to compound statements (option 1) leads to the best overall performance when compared with other options. This indicates that adding separate features for compound statements obscures the code’s intended semantics when the constituent statements are also individually featurized.
Figure 4(c) shows that removing all features relevant to global variables (option 1) degrades performance. We also observe that eliminating the global variable identifiers and assigning a tag to signal their presence (option 2) performs best overall, possibly because global variables appearing in similar contexts may not use the same variable identifiers. Further, option 2 performs better than the case where global variables are indistinguishable from local variables (option 3). Figure 4(d) indicates that removing features relevant to identifiers of global functions, but flagging their presence with a special tag as done in option 2, generally gives the best performance. This result is consistent with the intuitions for eliminating features of function identifiers in CAPT as discussed in Section 3.3.
A Subtle Observation.
A more nuanced and subtle observation is that our results seem to indicate that for each CAPT configuration the optimal granularity of abstraction detail is different. For compound statements the best option seems to corresponds to the coarsest level of abstraction detail, while for node annotation, global variables, and global functions the best option seems to corresponds to one of the intermediate levels of abstraction detail. For our future work, we aim to perform a deeper analysis on this and hopefully learn such configurations, to reduce (or eliminate) the overhead necessary of trying to manually discover such configurations.
5. Related Work
Research into code representations in the space of code similarity is still in its infancy, yet, there is a large and growing body of work to consider. A classical approach to infer code semantics is to utilize an intermediate representation (IR), such as LLVM (Lattner and Adve, 2004). However, these representations were originally designed with the purpose of mapping efficiently to low-level instruction set architectures (ISAs). As such, they might not be ideal candidates for code similarity.
Still, advances in ML seem to have stimulated a number of approaches (Ben-Nun
et al., 2018; Zhao and Huang, 2018) that rely on such IRs to infer high-level semantics for the purpose of code similarity. Nevertheless, such approaches suffer from the disadvantage of requiring compilation to determine the validity of the input code, hence limiting their applicability. Other research avoids this reliance on compilation by representing a program using its dynamic execution trace (Wang
et al., 2018). While such approaches enable the encoding of concrete details of the program semantics, the collection of dynamic traces can be costly (program execution is required).
The idea of utilizing the compiler’s IR has been extended further by more recent ML-based approaches (Alon
et al., 2019b, a; Zhang
et al., 2019), that use an abstract syntax tree (AST), which is at a higher level of abstraction than some IRs, such as LLVM. These AST approaches tend to rely on featurizing the AST to include some of its meta-properties, such as paths in the AST, to discover structural similarities. A key intuition for these approaches is that structural similarity of an AST may correlate to code similarity. Other recent research has focused on constructing code representations from raw source code tokens or sequences (Sachdev et al., 2018; Sajnani et al., 2016) with some success. However, these rely on certain strict assumptions on the input code and might be challenging to generalize.
To our knowledge, Aroma’s simplified parse tree (SPT) represents a state-of-the-art structural representation for code similarity (Luan
et al., 2019). Aroma extends all the aforementioned approaches in that it uses a customized parse tree representation, SPT, which encapsulates high-level code semantics. The SPT is at higher level of abstraction than previous AST-based approaches as it avoids representing irrelevant syntactic information. Our work is inspired by Aroma’s SPT and aims to take it one step further by allowing for systematic exploration of a range of customizable configuration parameters that control CAPT’s construction.
6. Future Work and Conclusion
In this paper, we presented the context-aware parse tree (CAPT), a novel tree structure that we have developed principally for the purpose of code similarity analysis. CAPT is heavily inspired by Aroma’s simplified parse tree (SPT).
Our research quantitatively demonstrates the value of our proposed semantically-salient features, enabling a specific CAPT configuration to be 39% more accurate than SPT across the 48,610 programs we analyzed. We believe CAPT is able to produce improved code similarity accuracy because it provides a more flexible semantic configuration across (i) language-specific ambiguity resolution and (ii) unbinding support via language-agnostic techniques for removal of syntactic features that are semantically irrelevant. Our exploration into CAPT is still in its infancy.
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Resistance distances in corona and neighborhood corona graphs with Laplacian generalized inverse approach
Jia-Bao Liu${}^{a,b}$, Xiang-Feng Pan${}^{a,}$ , Fu-Tao Hu${}^{a}$
${}^{a}$ School of Mathematical Sciences, Anhui University, Hefei 230601, P. R. China
${}^{b}$ Department of Public Courses, Anhui Xinhua
University, Hefei 230088, P. R. China
Corresponding author. Tel:+86-551-63861313. E-mail:liujiabaoad@163.com (J.Liu),
xfpan@ahu.edu.cn(X.Pan), hufu@mail.ustc.edu.cn(F.Hu).
()
Abstract
Let $G_{1}$ and $G_{2}$ be two graphs on disjoint sets of $n_{1}$ and
$n_{2}$ vertices, respectively. The corona of graphs $G_{1}$ and
$G_{2}$, denoted by $G_{1}\circ G_{2}$, is the graph formed from one
copy of $G_{1}$ and $n_{1}$ copies of $G_{2}$ where the $i$-th vertex of
$G_{1}$ is adjacent to every vertex in the $i$-th copy of $G_{2}$. The
neighborhood corona of $G_{1}$ and $G_{2}$, denoted by $G_{1}\diamond G_{2}$, is the graph obtained by taking one copy of $G_{1}$ and $n_{1}$
copies of $G_{2}$ and joining every neighbor of the $i$-th vertex of
$G_{1}$ to every vertex in the $i$-th copy of $G_{2}$ by a new edge.
In this paper, the Laplacian generalized inverse for the
graphs $G_{1}\circ G_{2}$ and $G_{1}\diamond G_{2}$ are investigated, based on which the resistance distances of
any two vertices in $G_{1}\circ G_{2}$ and $G_{1}\diamond G_{2}$ can be
obtained. Moreover, some examples as applications are presented,
which illustrate the correction and efficiency of the proposed
method.
Keywords: Laplacian matrix; Generalized inverse;
Moore-Penrose inverse; Schur
complement; Resistance distance
1 Introduction
All graphs considered in this paper are simple and undirected. Let
$G=(V(G),E(G))$ be a graph with vertex set $V(E)=\{v_{1},v_{2},\dots,v_{n}\}$ and edge set $E(G)=\{e_{1},e_{2},\dots,e_{m}\}$.
The adjacency matrix of $G$, denoted by $A(G)$, is the $n\times n$ matrix whose $(i,j)$-entry is $1$ if $v_{i}$ and $v_{j}$ are
adjacent in $G$ and $0$ otherwise. Denote $D(G)$ to be the
diagonal matrix with diagonal entries $d_{G}(v_{1}),d_{G}(v_{2}),\dots,d_{G}(v_{n}).$ The Laplacian matrix of $G$ is defined as $L(G)=D(G)-A(G)$. For other undefined notations and terminology from graph
theory, the readers may refer to [1] and the
references therein.
The conventional distance between vertices $v_{i}$ and $v_{j}$, denoted by
$d_{i,j}$, is the length of a shortest path between them.
Klein and Randić [2] introduced a new distance function named
resistance distance based on electrical network theory, the
resistance distance between vertices $i$ and $j$, denoted by
$r_{ij}$, is defined to be the effective electrical resistance
between them if each edge of $G$ is replaced by a unit
resistor [2].
For more information on resistance distance
of graphs, the readers are referred to the most recent papers
[3, 5, 6, 7, 8, 15, 16, 17, 18].
Until now, many graph operations such as the Cartesian product,
the Kronecker product, the corona and neighborhood corona graphs
have been introduced
in [10, 11, 12, 9, 13].
Let $G_{1}$ and $G_{2}$ be two vertex disjoint graphs. The following
definition comes from [11].
Definition 1.1
(see [11])
Let $G_{1}$ and $G_{2}$ be two graphs on disjoint sets of $n_{1}$ and
$n_{2}$ vertices, respectively. The corona of two graphs $G_{1}$ and
$G_{2}$ is the graph $G=G_{1}\circ G_{2}$ formed from one copy of
$G_{1}$ and $n_{1}$ copies of $G_{2}$ where the $i$-th vertex of $G_{1}$
is adjacent to every vertex in the $i$-th copy of $G_{2}$.
The neighborhood corona, which is a variant of the corona
operation, was recently introduced in [13].
Definition 1.2
(see [13])
Let $G_{1}$ and $G_{2}$ be two graphs on disjoint sets of $n_{1}$ and
$n_{2}$ vertices, respectively. The neighborhood corona of $G_{1}$ and
$G_{2}$, denoted by $G_{1}\diamond G_{2}$, is the graph obtained by
taking one copy of $G_{1}$ and $n_{1}$ copies of $G_{2}$ and joining
every neighbor of the $i$-th vertex of $G_{1}$ to every vertex in
the $i$-th copy of $G_{2}$ by a new edge.
Let $P_{n}$ and $C_{n}$ denote a path and cycle with $n$ vertices,
respectively. From the definitions, Figure 1 shows the graphs
$C_{3}\circ P_{3}$ and $C_{4}\diamond P_{2}$.
Bu et al. investigated
resistance distances in subdivision-vertex join and subdivision-edge join of graphs [3].
Motivated by the
results, in this paper, we further explored the Laplacian
generalized inverse for the corona and neighborhood corona graphs,
based on which all the resistance distances between arbitrary two
vertices can be directly obtained via simple calculations.
2 Preliminaries and Lemmas
At the beginning of this section, we review some concepts in
matrix theory.
Let $A$ be a matrix, $X$ is called the
$\{1\}$ inverse of $A$ and denoted by $A^{\{1\}}$, if $X$
satisfies the following condition: $AXA=A.$ Given a square matrix
$A$, the group inverse of $A$, denoted by $A^{\#}$, is the unique
matrix $X$ that satisfies matrix equations [3] $(I).AXA=A,~{}(II).XAX=X,~{}(III).AX=XA.$
If $A$ is real symmetric, then $A^{\#}$ exists and $A^{\#}$ is a
symmetric $\{1\}$-inverse of $A$. In fact, $A^{\#}$ is equal to
the Moore-Penrose inverse of $A$ since $A$ is symmetric
[3].
The Kronecker product $A\bigotimes B$ of two matrices $A=(a_{ij})_{m\times n}$ and $B=(b_{ij})_{p\times q}$ is the
$mp\times nq$ matrix obtained from $A$ by replacing each element
$a_{ij}$ by $a_{ij}B$. The reader is referred
to [14] for other properties of the Kronecker product
not mentioned here.
It is known that resistance distances in a connected graph $G$
can be obtained from any $\{1\}$-inverse of $L(G)$ according to
the following lemma (see [3]).
Lemma 2.1
(see [3]) Let $G$ be a connected graph, and $(L_{G})_{ij}$ denote the ${(i,j)}$-entry of $L_{G}$.
Then
$$r_{ij}(G)=(L_{G}^{(1)})_{ii}+(L_{G}^{(1)})_{jj}-(L_{G}^{(1)})_{ij}-(L_{G}^{(1)%
})_{ji}=(L_{G}^{\#})_{ii}+(L_{G}^{\#})_{jj}-2(L_{G}^{\#})_{ij}.$$
Lemma 2.2
(see [3]) Let $M=\left[\begin{array}[]{cc}A&B\\
C&D\\
\end{array}\right]$ be a nonsingular matrix. If $A$ and $D$ are nonsingular,
then
$$M^{-1}=\left[\begin{array}[]{cc}A^{-1}+A^{-1}BS^{-1}CA^{-1}&~{}~{}~{}~{}-A^{-1%
}BS^{-1}\\
-S^{-1}CA^{-1}&~{}~{}~{}~{}S^{-1}\\
\end{array}\right],\\
$$
where $S=D-CA^{-1}B$ is the Schur complement of $A$ in $M$.
The following similar result holds for Laplacian matrix of a
connected graph.
Lemma 2.3
(see [4]) Let $L=\left[\begin{array}[]{cc}L_{1}&L_{2}\\
L_{2}^{T}&L_{3}\\
\end{array}\right]$ be the Laplacian matrix of a connected graph. If $L_{1}$ is
nonsingular, then
$X=\left[\begin{array}[]{cc}L_{1}^{-1}+L_{1}^{-1}L_{2}S^{\#}L_{2}^{T}L_{1}^{-1}%
&~{}~{}~{}-L_{1}^{-1}L_{2}S^{\#}\\
-S^{\#}L_{2}^{T}L_{1}^{-1}&~{}~{}~{}S^{\#}\\
\end{array}\right]$ is a symmetric $\{1\}$-inverse of $L$, where
$S=L_{3}-L_{2}^{T}L_{1}^{-1}L_{2}.$
3 The Laplacian generalized inverse for
graphs $G_{1}\circ G_{2}$ and $G_{1}\diamond G_{2}$
3.1 The Laplacian generalized inverse for
graph $G_{1}\circ G_{2}$
Let $\mathbf{1}_{n}$ and $J_{n\times n}$ be all-one column vector of
dimensions $n$ and all-one $n\times n$ matrix, respectively.
Theorem 3.1 Let $G_{1}$ be an $r_{1}$-regular graph with $n_{1}$
vertices and $m_{1}$ edges, and $G_{2}$ an arbitrary graph with $n_{2}$
vertices, then the following matrix
$$\left[\begin{array}[]{c|lr}L_{1}^{-1}+L_{1}^{-1}L_{2}S^{\#}L_{2}^{T}L_{1}^{-1}%
&~{}~{}-L_{1}^{-1}L_{2}S^{\#}\\
\hline-S^{\#}L_{2}^{T}L_{1}^{-1}&~{}~{}~{}~{}~{}~{}~{}S^{\#}\\
\end{array}\right]$$
is a symmetric $\{1\}$-inverse of $L_{(G_{1}\circ G_{2})}$, where
$L_{1}=[L_{(G_{1})}+n_{2}I_{n_{1}}],L_{2}=[-\mathbf{1}^{T}_{n_{2}}\otimes I_{n_%
{1}}],L_{3}=(L(G_{2})+I_{n_{2}})\otimes I_{n_{1}},S=\Big{[}L_{3}-J_{n_{2}%
\times n_{2}}\otimes L_{1}^{-1}\Big{]}.$
Proof. Let $G_{1}$ be an arbitrary $r_{1}$-regular graphs with
$n_{1}$ vertices and $m_{1}$ edges, and $G_{2}$ an arbitrary graphs with
$n_{2}$ vertices, respectively. Label the vertices of $G_{1}\circ G_{2}$
as follows. Let $V(G_{1})=\{v_{1},v_{2},\dots,v_{n_{1}}\}$ and
$V(G_{2})=\{w_{1},w_{2},\dots,w_{n_{2}}\}$. For $i=1,2,\dots,n_{1}$, let $w^{i}_{1},w^{i}_{2},\dots,w_{n_{2}}^{i}$ denote the vertices of
the $i$-th copy of $G_{2}$, with the understanding that $w^{i}_{j}$ is
the copy of $w_{j}$ for each $j$. Denote $W_{j}=\{w^{1}_{j},w^{2}_{j},\dots,w^{n_{1}}_{j}\}$, for $j=1,2,\dots,n_{2}.$ Then
$$V(G_{1})\bigcup\big{[}W_{1}\bigcup W_{2}\bigcup\dots\bigcup W_{n_{2}}\big{]}$$
(1)
is a partition
of $V(G_{1}\circ G_{2})$. Obviously, the degrees of the vertices of
$G_{1}\circ G_{2}$ are:
$d_{G_{1}\circ G_{2}}(e_{i})=2$, for $i=1,2,\dots,m_{1}$, $d_{G_{1}\circ G_{2}}(v_{i})=n_{2}+d_{G_{1}}(v_{i})$, for $i=1,2,\dots,n_{1}$, and $d_{G_{1}\circ G_{2}}(w_{j}^{i})=d_{G_{2}}(w_{j})+1$, for $i=1,2,\dots,n_{1},j=1,2,\dots,n_{2}$.
Since $G_{1}$ is an
$r_{1}$-regular graph, we have $D(G_{1})=r_{1}I_{n_{1}}$. With respect
to the partition (1), then the Laplacian matrix of $G_{1}\circ G_{2}$
can be written as
$$L(G_{1}\circ G_{2})=\left[\begin{array}[]{c|lr}L(G_{1})+n_{2}I_{n_{1}}&~{}~{}~%
{}-\mathbf{1}^{T}_{n_{2}}\otimes I_{n_{1}}\\
\hline-\mathbf{1}_{n_{2}}\otimes I_{n_{1}}&~{}~{}(L(G_{2})+I_{n_{2}})\otimes I%
_{n_{1}}\\
\end{array}\right].$$
We begin with the calculation $S$. For convenience, let
$L_{1}=[L_{(G_{1})}+n_{2}I_{n_{1}}],L_{2}=[-\mathbf{1}^{T}_{n_{2}}\otimes I_{n_%
{1}}],$ $L_{2}^{T}=[-\mathbf{1}_{n_{2}}\otimes I_{n_{1}}],L_{3}=(L(G_{2})+I_{n_{2}})%
\otimes I_{n_{1}}.$
By Lemma 2.3, we have
$$\displaystyle S$$
$$\displaystyle=\Big{[}L(G_{2})+I_{n_{2}}\Big{]}\otimes I_{n_{1}}-\left[\begin{%
array}[]{c}-\mathbf{1}_{n_{2}}\otimes I_{n_{1}}\\
\end{array}\right]\left[\begin{array}[]{c}L(G_{1})+n_{2}I_{n_{1}}\\
\end{array}\right]^{-1}\left[\begin{array}[]{c}-\mathbf{1}^{T}_{n_{2}}\otimes I%
_{n_{1}}\end{array}\right]$$
$$\displaystyle=\Big{[}L(G_{2})+I_{n_{2}}\Big{]}\otimes I_{n_{1}}-J_{n_{2}\times
n%
_{2}}\otimes\left[\begin{array}[]{c}L(G_{1})+n_{2}I_{n_{1}}\\
\end{array}\right]^{-1}$$
$$\displaystyle=L_{3}-J_{n_{2}\times n_{2}}\otimes L_{1}^{-1}.$$
Based on Lemma 2.3, the following matrix
$$\left[\begin{array}[]{c|lr}L_{1}^{-1}+L_{1}^{-1}L_{2}S^{\#}L_{2}^{T}L_{1}^{-1}%
&~{}-L_{1}^{-1}L_{2}S^{\#}\\
\hline-S^{\#}L_{2}^{T}L_{1}^{-1}&~{}~{}~{}~{}~{}~{}~{}~{}S^{\#}\\
\end{array}\right]$$
is a symmetric $\{1\}$-inverse of $L_{(G_{1}\circ G_{2})}$,
where
$L_{1}=[L_{(G_{1})}+n_{2}I_{n_{1}}],L_{2}=[-\mathbf{1}^{T}_{n_{2}}\otimes I_{n_%
{1}}],L_{3}=(L(G_{2})+I_{n_{2}})\otimes I_{n_{1}},S=\Big{[}L_{3}-J_{n_{2}%
\times n_{2}}\otimes L_{1}^{-1}\Big{]}.$ $\blacksquare$
3.2 The Laplacian generalized inverse for
graph $G_{1}\diamond G_{2}$
When $G_{1}$ is a regular graph, we obtain the Laplacian generalized
inverse for graph
$G_{1}\diamond G_{2}$ as follows.
Theorem 3.2 Let $G_{1}$ be an $r_{1}$-regular graph with $n_{1}$
vertices and $m_{1}$ edges, and $G_{2}$ an arbitrary graph with $n_{2}$
vertices, then the following matrix
$$\left[\begin{array}[]{c|lr}L_{1}^{-1}+L_{1}^{-1}L_{2}S^{\#}L_{2}^{T}L_{1}^{-1}%
&~{}-L_{1}^{-1}L_{2}S^{\#}\\
\hline-S^{\#}L_{2}^{T}L_{1}^{-1}&~{}~{}~{}~{}~{}~{}~{}~{}S^{\#}\\
\end{array}\right]$$
is a symmetric $\{1\}$-inverse of $L_{(G_{1}\diamond G_{2})}$, where
$L_{1}=L(G_{1})+n_{2}D(G_{1}),L_{2}=-\mathbf{1}_{n_{2}}^{T}\otimes A(G_{1}),$
$L_{3}=L(G_{2})\otimes I_{n_{1}}+I_{n_{2}}\otimes D(G_{1}),S=L_{3}-J_{n_{2}%
\times n_{2}}\otimes\big{[}A(G_{1})^{T}L_{1}^{-1}A(G_{1})\big{]}.$
Proof. We label the vertices of $G_{1}\diamond G_{2}$ as
follows. Let $V(G_{1})=\{v_{1},v_{2},\dots,v_{n_{1}}\}$, and $V(G_{2})=\{w_{1},w_{2},\dots,w_{n_{2}}\}$. For $i=1,2,\dots,n_{1}$, let
$w^{i}_{1},w^{i}_{2},\dots,w_{n_{2}}^{i}$ denote the vertices of the
$i$-th copy of $G_{2}$, with the understanding that $w^{i}_{j}$ is the
copy of $w_{j}$ for each $j$. Denote $U_{j}=\{w^{1}_{j},w^{2}_{j},\dots,w^{n_{1}}_{j}\}$, for $j=1,2,\dots,n_{2}.$ Then
$$V(G_{1})\bigcup\Big{[}U_{1}\bigcup U_{2}\bigcup\dots\bigcup U_{n_{2}}\Big{]}$$
(2)
is a partition of $V(G_{1}\diamond G_{2})$. Clearly, the degrees of the vertices of
$G_{1}\diamond G_{2}$ are:
$d_{G_{1}\diamond G_{2}}(v_{i})=(n_{2}+1)d_{G_{1}}(v_{i}),$ for $i=1,2,\dots,n_{1}$, and
$d_{G_{1}\diamond G_{2}}(w_{j}^{i})=d_{G_{2}}(w_{j})+d_{G_{1}}(v_{i}),$ for
$i=1,2,\dots,m_{1},j=1,2,\dots,n_{2}$.
Since $G_{1}$ is an $r_{1}$-regular graph, we have $D(G_{1})=r_{1}I_{n_{1}}$. With respect to the partition (2), then the
Laplacian matrix of $G_{1}\diamond G_{2}$ can be written as
$$L(G_{1}\diamond G_{2})=\left[\begin{array}[]{c|lr}~{}L(G_{1})+n_{2}D(G_{1})&~{%
}~{}~{}~{}~{}~{}~{}~{}~{}-\mathbf{1}_{n_{2}}^{T}\otimes A(G_{1})\\
\hline~{}-\mathbf{1}_{n_{2}}\otimes A(G_{1})^{T}&~{}~{}L(G_{2})\otimes I_{n_{1%
}}+I_{n_{2}}\otimes D(G_{1})\\
\end{array}\right].$$
For convenience, let $L_{1}=L(G_{1})+n_{2}D(G_{1}),L_{2}=-\mathbf{1}_{n_{2}}^{T}\otimes A(G_{1}),L_{%
2}^{T}=-\mathbf{1}_{n_{2}}\otimes A(G_{1})^{T},$
$L_{3}=L(G_{2})\otimes I_{n_{1}}+I_{n_{2}}\otimes D(G_{1}).$
Similarly, by Lemma 2.3, we have
$$\displaystyle S$$
$$\displaystyle=\big{[}L(G_{2})\otimes I_{n_{1}}+I_{n_{2}}\otimes D(G_{1})\big{]%
}-\left[-\mathbf{1}_{n_{2}}\otimes A(G_{1})^{T}\right]\left[\begin{array}[]{c}%
L(G_{1})+n_{2}D(G_{1})\end{array}\right]^{-1}\left[\begin{array}[]{c}-\mathbf{%
1}_{n_{2}}^{T}\otimes A(G_{1})\end{array}\right]$$
$$\displaystyle=L_{3}-J_{n_{2}\times n_{2}}\otimes\big{[}A(G_{1})^{T}L_{1}^{-1}A%
(G_{1})\big{]}.$$
Based on Lemma 2.3, the following matrix
$$\left[\begin{array}[]{c|lr}L_{1}^{-1}+L_{1}^{-1}L_{2}S^{\#}L_{2}^{T}L_{1}^{-1}%
&~{}-L_{1}^{-1}L_{2}S^{\#}\\
\hline-S^{\#}L_{2}^{T}L_{1}^{-1}&~{}~{}~{}~{}~{}~{}~{}~{}S^{\#}\\
\end{array}\right]$$
is a symmetric $\{1\}$-inverse of $L_{(G_{1}\circ G_{2})}$,
where $L_{1}=L(G_{1})+n_{2}D(G_{1}),L_{2}=-\mathbf{1}_{n_{2}}^{T}\otimes A(G_{1}),$
$L_{3}=L(G_{2})\otimes I_{n_{1}}+I_{n_{2}}\otimes D(G_{1}),S=L_{3}-J_{n_{2}%
\times n_{2}}\otimes\big{[}A(G_{1})^{T}L_{1}^{-1}A(G_{1})\big{]}.$
$\blacksquare$
4 Applications and some examples
As an application of the proposed theorems, we present some examples to show all the resistance distances of
any two vertices in graphs $G_{1}\circ G_{2}$ and $G_{1}\diamond G_{2}$ can be obtained by the proposed method.
Example 4.1
Laplacian generalized inverse for $C_{3}\circ P_{3}$ and resistance
distances matrix.
The Laplacian matrix
$L_{(C_{3}\circ P_{3})}=\left[\begin{array}[]{c|lr}L(C_{3})+3I_{3}&~{}~{}~{}~{}%
-\mathbf{1}^{T}_{3}\otimes I_{3}\\
\hline-\mathbf{1}_{3}\otimes I_{3}&~{}(L(P_{3})+I_{n_{2}})\otimes I_{3}\\
\end{array}\right].$
Based on Theorem 3.1, we can obtain that
$$L^{\{1\}}_{(C_{3}\circ P_{3})}=\left[\begin{array}[]{cccccccccccc}\frac{1}{3}&%
0&0&\frac{2}{9}&\frac{-1}{9}&\frac{-1}{9}&\frac{2}{9}&\frac{-1}{9}&\frac{-1}{9%
}&\frac{2}{9}&\frac{-1}{9}&\frac{-1}{9}\\
0&\frac{1}{3}&0&\frac{-1}{9}&\frac{2}{9}&\frac{-1}{9}&\frac{-1}{9}&\frac{2}{9}%
&\frac{-1}{9}&\frac{-1}{9}&\frac{2}{9}&\frac{-1}{9}\\
0&0&\frac{1}{3}&\frac{-1}{9}&\frac{-1}{9}&\frac{2}{9}&\frac{-1}{9}&\frac{-1}{9%
}&\frac{2}{9}&\frac{-1}{9}&\frac{-1}{9}&\frac{2}{9}\\
\frac{2}{9}&\frac{-1}{9}&\frac{-1}{9}&\frac{53}{72}&\frac{-2}{9}&\frac{-2}{9}&%
\frac{13}{36}&\frac{-2}{9}&\frac{-2}{9}&\frac{17}{72}&\frac{-2}{9}&\frac{-2}{9%
}\\
\frac{-1}{9}&\frac{2}{9}&\frac{-1}{9}&\frac{-2}{9}&\frac{53}{72}&\frac{-2}{9}&%
\frac{-2}{9}&\frac{13}{36}&\frac{-2}{9}&\frac{-2}{9}&\frac{17}{72}&\frac{-2}{9%
}\\
\frac{-1}{9}&\frac{-1}{9}&\frac{2}{9}&\frac{-2}{9}&\frac{-2}{9}&\frac{53}{72}&%
\frac{-2}{9}&\frac{-2}{9}&\frac{13}{36}&\frac{-2}{9}&\frac{-2}{9}&\frac{17}{72%
}\\
\frac{2}{9}&\frac{-1}{9}&\frac{-1}{9}&\frac{13}{36}&\frac{-2}{9}&\frac{-2}{9}&%
\frac{11}{18}&\frac{-2}{9}&\frac{-2}{9}&\frac{13}{36}&\frac{-2}{9}&\frac{-2}{9%
}\\
\frac{-1}{9}&\frac{2}{9}&\frac{-1}{9}&\frac{-2}{9}&\frac{13}{36}&\frac{-2}{9}&%
\frac{-2}{9}&\frac{11}{18}&\frac{-2}{9}&\frac{-2}{9}&\frac{13}{36}&\frac{-2}{9%
}\\
\frac{-1}{9}&\frac{-1}{9}&\frac{2}{9}&\frac{-2}{9}&\frac{-2}{9}&\frac{13}{36}&%
\frac{-2}{9}&\frac{-2}{9}&\frac{11}{18}&\frac{-2}{9}&\frac{-2}{9}&\frac{13}{36%
}\\
\frac{2}{9}&\frac{-1}{9}&\frac{-1}{9}&\frac{17}{72}&\frac{-2}{9}&\frac{-2}{9}&%
\frac{13}{36}&\frac{-2}{9}&\frac{-2}{9}&\frac{53}{72}&\frac{-2}{9}&\frac{-2}{9%
}\\
\frac{-1}{9}&\frac{2}{9}&\frac{-1}{9}&\frac{-2}{9}&\frac{17}{72}&\frac{-2}{9}&%
\frac{-2}{9}&\frac{13}{36}&\frac{-2}{9}&\frac{-2}{9}&\frac{53}{72}&\frac{-2}{9%
}\\
\frac{-1}{9}&\frac{-1}{9}&\frac{2}{9}&\frac{-2}{9}&\frac{-2}{9}&\frac{17}{72}&%
\frac{-2}{9}&\frac{-2}{9}&\frac{13}{36}&\frac{-2}{9}&\frac{-2}{9}&\frac{53}{72%
}\\
\end{array}\right].$$
By Lemma 2.1 and $L^{\{1\}}_{(C_{3}\circ P_{3})}$, the resistance
distances matrix of $C_{3}\circ P_{3}$ is
$$R_{(C_{3}\circ P_{3})}=\left[\begin{array}[]{cccccccccccc}0&\frac{2}{3}&\frac{%
2}{3}&\frac{5}{8}&\frac{31}{24}&\frac{31}{24}&\frac{1}{2}&\frac{7}{6}&\frac{7}%
{6}&\frac{5}{8}&\frac{31}{24}&\frac{31}{24}\\
\frac{2}{3}&0&\frac{2}{3}&\frac{31}{24}&\frac{5}{8}&\frac{31}{24}&\frac{7}{6}&%
\frac{1}{2}&\frac{7}{6}&\frac{31}{24}&\frac{5}{8}&\frac{31}{24}\\
\frac{2}{3}&\frac{2}{3}&0&\frac{31}{24}&\frac{31}{24}&\frac{5}{8}&\frac{7}{6}&%
\frac{7}{6}&\frac{1}{2}&\frac{31}{24}&\frac{31}{24}&\frac{5}{8}\\
\frac{5}{8}&\frac{31}{24}&\frac{31}{24}&0&\frac{23}{12}&\frac{23}{12}&\frac{5}%
{8}&\frac{43}{24}&\frac{43}{24}&1&\frac{23}{12}&\frac{23}{12}\\
\frac{31}{24}&\frac{5}{8}&\frac{31}{24}&\frac{23}{12}&0&\frac{23}{12}&\frac{43%
}{24}&\frac{5}{8}&\frac{43}{24}&\frac{23}{12}&1&\frac{23}{12}\\
\frac{31}{24}&\frac{31}{24}&\frac{5}{8}&\frac{23}{12}&\frac{23}{12}&0&\frac{43%
}{24}&\frac{43}{24}&\frac{5}{8}&\frac{23}{12}&\frac{23}{12}&1\\
\frac{1}{2}&\frac{7}{6}&\frac{7}{6}&\frac{5}{8}&\frac{43}{24}&\frac{43}{24}&0&%
\frac{5}{3}&\frac{5}{3}&\frac{5}{8}&\frac{43}{24}&\frac{43}{24}\\
\frac{7}{6}&\frac{1}{2}&\frac{7}{6}&\frac{43}{24}&\frac{5}{8}&\frac{43}{24}&%
\frac{5}{3}&0&\frac{5}{3}&\frac{43}{24}&\frac{5}{8}&\frac{43}{24}\\
\frac{7}{6}&\frac{7}{6}&\frac{1}{2}&\frac{43}{24}&\frac{43}{24}&\frac{5}{8}&%
\frac{5}{3}&\frac{5}{3}&0&\frac{43}{24}&\frac{43}{24}&\frac{5}{8}\\
\frac{5}{8}&\frac{31}{24}&\frac{31}{24}&1&\frac{23}{12}&\frac{23}{12}&\frac{5}%
{8}&\frac{43}{24}&\frac{43}{24}&0&\frac{23}{12}&\frac{23}{12}\\
\frac{31}{24}&\frac{5}{8}&\frac{31}{24}&\frac{23}{12}&1&\frac{23}{12}&\frac{43%
}{24}&\frac{5}{8}&\frac{43}{24}&\frac{23}{12}&0&\frac{23}{12}\\
\frac{31}{24}&\frac{31}{24}&\frac{5}{8}&\frac{23}{12}&\frac{23}{12}&1&\frac{43%
}{24}&\frac{43}{24}&\frac{5}{8}&\frac{23}{12}&\frac{23}{12}&0\\
\end{array}\right],$$
where $r_{ij}$ denotes resistance distance of two
vertices between $i$ and $j.$ $\blacksquare$
Example 4.2
Laplacian generalized inverse for $C_{4}\diamond P_{2}$ and
resistance distances matrix.
Completely similar deduction by Theorem 3.2, we can obtain
$$L^{\{1\}}_{(C_{4}\diamond P_{2})}=\left[\begin{array}[]{cccccccccccc}\frac{5}{%
24}&0&\frac{1}{24}&0&\frac{-1}{16}&\frac{1}{16}&\frac{-1}{16}&\frac{1}{16}&%
\frac{-1}{16}&\frac{1}{16}&\frac{-1}{16}&\frac{1}{16}\\
0&\frac{5}{24}&0&\frac{1}{24}&\frac{1}{16}&\frac{-1}{16}&\frac{1}{16}&\frac{-1%
}{16}&\frac{1}{16}&\frac{-1}{16}&\frac{1}{16}&\frac{-1}{16}\\
\frac{1}{24}&0&\frac{5}{24}&0&\frac{-1}{16}&\frac{1}{16}&\frac{-1}{16}&\frac{1%
}{16}&\frac{-1}{16}&\frac{1}{16}&\frac{-1}{16}&\frac{1}{16}\\
0&\frac{1}{24}&0&\frac{5}{24}&\frac{1}{16}&\frac{-1}{16}&\frac{1}{16}&\frac{-1%
}{16}&\frac{1}{16}&\frac{-1}{16}&\frac{1}{16}&\frac{-1}{16}\\
\frac{-1}{16}&\frac{1}{16}&\frac{-1}{16}&\frac{1}{16}&\frac{3}{8}&\frac{-1}{8}%
&0&\frac{-1}{8}&\frac{1}{8}&\frac{-1}{8}&0&\frac{-1}{8}\\
\frac{1}{16}&\frac{-1}{16}&\frac{1}{16}&\frac{-1}{16}&\frac{-1}{8}&\frac{3}{8}%
&\frac{-1}{8}&0&\frac{-1}{8}&\frac{1}{8}&\frac{-1}{8}&0\\
\frac{-1}{16}&\frac{1}{16}&\frac{-1}{16}&\frac{1}{16}&0&\frac{-1}{8}&\frac{3}{%
8}&\frac{-1}{8}&0&\frac{-1}{8}&\frac{1}{8}&\frac{-1}{8}\\
\frac{1}{16}&\frac{-1}{16}&\frac{1}{16}&\frac{-1}{16}&\frac{-1}{8}&0&\frac{-1}%
{8}&\frac{3}{8}&\frac{-1}{8}&0&\frac{-1}{8}&\frac{1}{8}\\
\frac{-1}{16}&\frac{1}{16}&\frac{-1}{16}&\frac{1}{16}&\frac{1}{8}&\frac{-1}{8}%
&0&\frac{-1}{8}&\frac{3}{8}&\frac{-1}{8}&0&\frac{-1}{8}\\
\frac{1}{16}&\frac{-1}{16}&\frac{1}{16}&\frac{-1}{16}&\frac{-1}{8}&\frac{1}{8}%
&\frac{-1}{8}&0&\frac{-1}{8}&\frac{3}{8}&\frac{-1}{8}&0\\
\frac{-1}{16}&\frac{1}{16}&\frac{-1}{16}&\frac{1}{16}&0&\frac{-1}{8}&\frac{1}{%
8}&\frac{-1}{8}&0&\frac{-1}{8}&\frac{3}{8}&\frac{-1}{8}\\
\frac{1}{16}&\frac{-1}{16}&\frac{1}{16}&\frac{-1}{16}&\frac{-1}{8}&0&\frac{-1}%
{8}&\frac{1}{8}&\frac{-1}{8}&0&\frac{-1}{8}&\frac{3}{8}\\
\end{array}\right].$$
By Lemma 2.1 and $L^{\{1\}}_{(C_{4}\diamond P_{2})}$, the resistance
distances matrix of $C_{4}\diamond P_{2}$ is
$$R_{(C_{4}\diamond P_{2})}=\left[\begin{array}[]{cccccccccccc}0&\frac{5}{12}&%
\frac{1}{3}&\frac{5}{12}&\frac{17}{24}&\frac{11}{24}&\frac{17}{24}&\frac{11}{2%
4}&\frac{17}{24}&\frac{11}{24}&\frac{17}{24}&\frac{11}{24}\\
\frac{5}{12}&0&\frac{5}{12}&\frac{1}{3}&\frac{11}{24}&\frac{11}{24}&\frac{17}{%
24}&\frac{11}{24}&\frac{17}{24}&\frac{17}{24}&\frac{11}{24}&\frac{17}{24}\\
\frac{1}{3}&\frac{5}{12}&0&\frac{5}{12}&\frac{17}{24}&\frac{11}{24}&\frac{17}{%
24}&\frac{11}{24}&\frac{17}{24}&\frac{11}{24}&\frac{17}{24}&\frac{11}{24}\\
\frac{5}{12}&\frac{1}{3}&\frac{5}{12}&0&\frac{11}{24}&\frac{17}{24}&\frac{11}{%
24}&\frac{17}{24}&\frac{11}{24}&\frac{17}{24}&\frac{11}{24}&\frac{17}{24}\\
\frac{17}{24}&\frac{11}{24}&\frac{17}{24}&\frac{11}{24}&0&1&\frac{3}{4}&1&%
\frac{1}{2}&1&\frac{3}{4}&1\\
\frac{11}{24}&\frac{17}{24}&\frac{11}{24}&\frac{17}{24}&1&0&1&\frac{3}{4}&1&%
\frac{1}{2}&1&\frac{3}{4}\\
\frac{17}{24}&\frac{11}{24}&\frac{17}{24}&\frac{11}{24}&\frac{3}{4}&1&0&1&%
\frac{3}{4}&1&\frac{1}{2}&1\\
\frac{11}{24}&\frac{17}{24}&\frac{11}{24}&\frac{17}{24}&1&\frac{3}{4}&1&0&1&%
\frac{3}{4}&1&\frac{1}{2}\\
\frac{17}{24}&\frac{11}{24}&\frac{17}{24}&\frac{11}{24}&\frac{1}{2}&1&\frac{3}%
{4}&1&0&1&\frac{3}{4}&1\\
\frac{11}{24}&\frac{17}{24}&\frac{11}{24}&\frac{17}{24}&1&\frac{1}{2}&1&\frac{%
3}{4}&1&0&1&\frac{3}{4}\\
\frac{17}{24}&\frac{11}{24}&\frac{17}{24}&\frac{11}{24}&\frac{3}{4}&1&\frac{1}%
{2}&1&\frac{3}{4}&1&0&1\\
\frac{11}{24}&\frac{17}{24}&\frac{11}{24}&\frac{17}{24}&1&\frac{3}{4}&1&\frac{%
1}{2}&1&\frac{3}{4}&1&0\\
\end{array}\right],$$
where $r_{ij}$ denotes resistance distance of two
vertices between $i$ and $j.$
$\blacksquare$
Acknowledgments
Partially supported by National Natural Science Foundation of
China (Nos. 11401004 and 11471016); Natural Science Foundation of
Anhui Province of China (No. KJ2013B105), Anhui Provincial Natural
Science Foundation (No. 1408085QA03).
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Kurepa trees and spectra of $\mathcal{L}_{\omega_{1},\omega}$-sentences
Dima Sinapova
University of Illinois at Chicago, Mathematics Department, 851 S Morgan St, Chicago, IL
60607
sinapova@math.uic.edu
and
Ioannis Souldatos
University of Detroit Mercy, Mathematics Department, 4001 McNichols Ave, Detroit, MI 48221
souldaio@udmercy.edu
(Date:: November 19, 2020)
Abstract.
We construct a single $\mathcal{L}_{\omega_{1},\omega}$-sentence $\psi$ that codes Kurepa trees to prove the consistency of
the following:
(1)
The spectrum of $\psi$ is consistently equal to $[\aleph_{0},\aleph_{\omega_{1}}]$ and also consistently
equal to $[\aleph_{0},2^{\aleph_{1}})$, where $2^{\aleph_{1}}$ is weakly inaccessible.
(2)
The amalgamation spectrum of $\psi$ is consistently equal to $[\aleph_{1},\aleph_{\omega_{1}}]$ and
$[\aleph_{1},2^{\aleph_{1}})$, where again $2^{\aleph_{1}}$ is weakly inaccessible.
This is the first example of an $\mathcal{L}_{\omega_{1},\omega}$-sentence whose spectrum and
amalgamation spectrum are consistently both right-open and right-closed. It also provides a positive answer to a question
in [14].
(3)
Consistently, $\psi$ has maximal models in finite, countable, and
uncountable many cardinalities. This complements the examples given in [1] and [2] of sentences with maximal
models in countably many cardinalities.
(4)
$2^{\aleph_{0}}<\aleph_{\omega_{1}}<2^{\aleph_{1}}$ and there exists an $\mathcal{L}_{\omega_{1},\omega}$-sentence with
models in $\aleph_{\omega_{1}}$, but no models in $2^{\aleph_{1}}$.
This relates to a conjecture by Shelah that if $\aleph_{\omega_{1}}<2^{\aleph_{0}}$, then any $\mathcal{L}_{\omega_{1},\omega}$-sentence
with a model of size $\aleph_{\omega_{1}}$ also has a model of size $2^{\aleph_{0}}$. Our result proves that $2^{\aleph_{0}}$ can not be
replaced by $2^{\aleph_{1}}$, even if $2^{\aleph_{0}}<\aleph_{\omega_{1}}$.
Key words and phrases:Kurepa trees, Infinitary Logic, Abstract Elementary Classes, Spectra, Amalgamation, Maximal Models
2010 Mathematics Subject Classification: Primary 03E75, 03C55 , Secondary 03E35, 03C75, 03C48, 03C52
1. Introduction
Definition 1.1.
For an $\mathcal{L}_{\omega_{1},\omega}$-sentence $\phi$, the spectrum of $\phi$ is the set
$$Spec(\phi)=\{\kappa|\exists M\models\phi\text{ and }|M|=\kappa\}.$$
If $Spec(\phi)=[\aleph_{0},\kappa]$, we say that $\phi$ characterizes $\kappa$.
The amalgamation spectrum of $\phi$, AP-$Spec(\phi)$ for short, is the set of all
cardinals $\kappa$ so that the models of $\phi$ of size $\kappa$ satisfy the amalgamation property. Similarly define JEP-$Spec(\phi)$ the joint embedding spectrum of $\phi$.
The maximal models spectrum of $\phi$ is the set
$$\text{MM-$Spec(\phi)$ =}\{\kappa|\exists M\models\phi\text{ and $M$ is maximal%
}\}.$$
Morley and López-Escobar independently established that all cardinals that are characterized by an $\mathcal{L}_{\omega_{1},\omega}$- sentence are
smaller than $\beth_{\omega_{1}}$.111See [9] for more details.
Theorem 1.2 (Morley, López-Escobar).
Let $\Gamma$ be a countable set of sentences of $\mathcal{L}_{\omega_{1},\omega}$. If $\Gamma$ has models of
cardinality $\beth_{\alpha}$ for all
$\alpha<\omega_{1}$, then it has models of all infinite cardinalities.
In 2002, Hjorth ([5]) proved the following.
Theorem 1.3 (Hjorth).
For all $\alpha<\omega_{1}$, $\aleph_{\alpha}$ is
characterized by a complete
$\mathcal{L}_{\omega_{1},\omega}$-sentence.
Combining Theorems 1.2 and 1.3 we get under GCH that $\aleph_{\alpha}$ is characterized by a complete
$\mathcal{L}_{\omega_{1},\omega}$-sentence if and only if $\alpha<\omega_{1}$.
Given these results, one might ask if it is consistent under the failure of GCH that $\aleph_{\omega_{1}}$ is characterizable. The
answer is positive because one can force $\aleph_{\omega_{1}}=2^{\aleph_{0}}$ and by [12], $2^{\aleph_{0}}$ can
be characterized by a complete $\mathcal{L}_{\omega_{1},\omega}$-sentence. Further, in
[14] the question was raised if any
cardinal outside the smallest set which contains $\aleph_{0}$ and which is closed under successors,
countable unions, countable products and powerset, can be characterized by an $\mathcal{L}_{\omega_{1},\omega}$-sentence. Lemma 3.2
proves $\aleph_{\omega_{1}}$ to be such an example, thus providing a
positive answer.
In this paper we work with a single $\mathcal{L}_{\omega_{1},\omega}$-sentence $\psi$ that codes a Kurepa tree and we investigate the effects of
set-theory on the spectra of $\psi$. In particular, we prove the consistency of the following.
(1)
$2^{\aleph_{0}}<\aleph_{\omega_{1}}$ and $Spec(\psi)=[\aleph_{0},\aleph_{\omega_{1}}]$.
(2)
$2^{\aleph_{0}}<2^{\aleph_{1}}$, $2^{\aleph_{1}}$ is weakly inaccessible and $Spec(\psi)=[\aleph_{0},2^{\aleph_{1}})$.
The only previously known examples of $\mathcal{L}_{\omega_{1},\omega}$-sentences with a right-open spectra were of the form
$[\aleph_{0},\kappa)$,
with $\kappa$ of countable cofinality. If $\kappa=\sup_{n\in\omega}\kappa_{n}$ and $\phi_{n}$ characterizes $\kappa_{n}$, then
$\bigvee_{n}\phi_{n}$ has spectrum $[\aleph_{0},\kappa)$.
Lemma 2.2 proves that our methods can not be used to establish the consistency of
$Spec(\psi)=[\aleph_{0},\aleph_{\omega_{1}})$, which remains open. It also open to find a complete sentence with a right-open
spectrum.
Moreover, $\psi$ is the first example of an $\mathcal{L}_{\omega_{1},\omega}$-sentence whose spectrum is consistently both right-open and
right-closed. The same observation holds true for the amalgamation spectrum too.
(3)
$2^{\aleph_{0}}<\aleph_{\omega_{1}}$ and AP-$Spec(\psi)$ =$[\aleph_{1},\aleph_{\omega_{1}}]$.
(4)
$2^{\aleph_{0}}<2^{\aleph_{1}}$, $2^{\aleph_{1}}$ is weakly inaccessible and AP-$Spec(\psi)$ =$[\aleph_{1},2^{\aleph_{1}})$.
This is the first example of an $\mathcal{L}_{\omega_{1},\omega}$-sentence whose amalgamation spectrum is consistently both right-open and
right-closed.
Moreover, by manipulating the size of $2^{\aleph_{1}}$, Corollary 2.15 implies that for
$\kappa$ regular with $\aleph_{2}\leq\kappa\leq 2^{\aleph_{1}}$, $\kappa$-amalgamation for $\mathcal{L}_{\omega_{1},\omega}$-sentences is non-absolute for
models of ZFC.
(5)
MM-$Spec(\psi)$ is the set of all cardinals $\kappa$ such that $\kappa$ is either equal to $\aleph_{1}$, or equal to $2^{\aleph_{0}}$,
or there exists a Kurepa tree with exactly $\kappa$-many branches. Manipulating the cardinalities on which there are Kurepa
trees, we can prove that $\psi$ consistently has maximal models in finite, countable, and
uncountable many cardinalities.
Our example complements the examples of $\mathcal{L}_{\omega_{1},\omega}$-sentences with maximal models
in countably many cardinalities found in [1] and [2].
(6)
$2^{\aleph_{0}}<\aleph_{\omega_{1}}<2^{\aleph_{1}}$ and there exists an $\mathcal{L}_{\omega_{1},\omega}$-sentence with
models in $\aleph_{\omega_{1}}$, but no models in $2^{\aleph_{1}}$.
Shelah conjectured that if $\aleph_{\omega_{1}}<2^{\aleph_{0}}$, then any $\mathcal{L}_{\omega_{1},\omega}$-sentence
with a model of size $\aleph_{\omega_{1}}$ also has a model of size $2^{\aleph_{0}}$. In [13], Shelah proves the
consistency of the conjecture. Our result proves that $2^{\aleph_{0}}$ can not be replaced by $2^{\aleph_{1}}$, even if
$2^{\aleph_{0}}<\aleph_{\omega_{1}}$. Our example can not
be used to refute Shelah’s conjecture.
Section 2 contains the description of the sentence $\psi$ and the results about the model-theoretic properties of
$\psi$.
Section 3 contains the consistency results.
2. Kurepa Trees and $\mathcal{L}_{\omega_{1},\omega}$
The reader can consult [6] about trees. The following definition summarizes all
that we
will use in this paper.
Definition 2.1.
Assume $\kappa$ is an infinite cardinal. A
$\kappa$-tree has height $\kappa$ and each level has at most $<\kappa$ elements.
A $\kappa$-Kurepa tree is a $\kappa$-tree with at least $\kappa^{+}$ branches of
height $\kappa$. Kurepa trees, with no $\kappa$ specified, refer to $\aleph_{1}$-Kurepa trees.
For this paper we will assume that $\kappa$-Kurepa trees are pruned,
i.e. all maximal branches have height $\kappa^{+}$.
If $\lambda\geq\kappa^{+}$, a $(\kappa,\lambda)$-Kurepa tree is a $\kappa$-Kurepa tree with
exactly $\lambda$ branches of height $\kappa$.
$KH(\kappa,\lambda)$ is the statement that there exists a $(\kappa,\lambda)$-Kurepa tree.
Define $\mathcal{B}(\kappa)=\sup\{\lambda|KH(\kappa,\lambda)\text{ holds}\}$ and $\mathcal{B}=\mathcal{B}(\aleph_{1})$.
If $\kappa$-Kurepa trees exist, it is immediate that $\mathcal{B}(\kappa)$
is always between $\kappa^{+}$ and $2^{\kappa}$. We are interested in the case where $\mathcal{B}$ is a supremum but not a maximum. The next
lemma proves some restrictions when $\mathcal{B}$ is not a maximum. In Section 3 we prove that it is consistent with
ZFC that $\mathcal{B}$ is not a maximum.
Lemma 2.2.
If $\mathcal{B}(\kappa)$ is not a maximum, then
$cf(\mathcal{B}(\kappa))\geq\kappa^{+}$.
Proof.
Towards contradiction assume that $cf(\mathcal{B}(\kappa))=\mu\leq\kappa$. Let $\mathcal{B}(\kappa)=sup_{i\in\mu}\kappa_{i}$ and let
$(T_{i})_{i\in\mu}$ be a collection of $\kappa$-Kurepa trees, where each $T_{i}$ has exactly $\kappa_{i}$-many cofinal branches. Create
new
$\kappa$-Kurepa trees $S_{i}$ by induction on $i\leq\mu$: $S_{0}$ equals $T_{0}$; $S_{i+1}$ equals the disjoint union of $S_{i}$
together with a
copy of $T_{i+1}$, arranged so that the $j^{th}$ level $T_{i+1}$ coincides with the $(j+i)^{th}$ level of $S_{i}$. At limit
stages take unions. We leave the verification to the reader that $S_{\mu}$ is
a $\kappa$-Kurepa tree with exactly $sup_{i\in\mu}\kappa_{i}=\kappa$ cofinal branches, contradicting the fact that $\mathcal{B}(\kappa)$ is
not a maximum.
∎
Definition 2.3.
Let $\kappa\leq\lambda$ be infinite cardinals.
A sentence $\sigma$ in a language with a unary predicate $P$ admits $(\lambda,\kappa)$, if $\sigma$
has a model $M$ such that $|M|=\lambda$ and $|P^{M}|=\kappa$. In this case, we will say that $M$ is
of type $(\lambda,\kappa)$.
From [3], theorem $7.2.13$, we know
Theorem 2.4.
There is a (first-order) sentence $\sigma$ such that for all
infinite cardinals
$\kappa$, $\sigma$ admits $(\kappa^{++},\kappa)$ iff $KH(\kappa^{+},\kappa^{++})$.
We describe here the construction behind Theorem 2.4 in order to use it later.
The vocabulary $\tau$ consists of the unary symbols $P,L$, the binary symbols $V,T,<,\prec,H$, and
the ternary symbols $F,G$. The idea is to build a $\kappa$-tree. $P,L,V$ are disjoint and their
union is the universe. $L$ is a set that corresponds to the “levels” of the tree. $L$
is linearly ordered by $<$ and it has a minimum and maximum element. Every element $a\in L$
that is not the maximum element has a
successor, which we will freely denote by $a+1$. The maximum element is not a successor.
For every
$a\in L$, $V(a,\cdot)$ is the set of
nodes at level $a$ and we assume that $V(a,\cdot)$ is disjoint from $L$. If $V(a,x)$, we will say
that $x$ is at level $a$. If $M$ is the maximum
element of $L$, $V(M,\cdot)$ is the set of maximal branches through the tree. $T$ is a tree
ordering
on $V=\bigcup_{a\in L}V(a,\cdot)$. If $T(x,y)$, then $x$ is at some level strictly less than the
level of $y$. If $a$ is a limit, that is neither a successor nor the least element in $L$, then
two distinct elements in $V(a,\cdot)$ can not have the same predecessors. Both “the
height of $T$” and “the height of $L$” refer to the order type of $(L,<)$. Although it is not
necessary for Theorem 2.4, we can stipulate that the Kurepa tree is pruned.
We use the predicate $P$ to bound the size of every initial segment of
$L$ of the form $L_{\leq a}=\{b\in L|b\leq a\}$, where $a$ is not the maximum element of $L$. We
also bound the size of each level $V(a,\cdot)$. For every $a\in L\setminus\{M\}$, where $M$ is the maximum element of $L$, there is a surjection
$F(a,\cdot,\cdot)$ from $P$ to $L_{\leq a}$ and another surjection $G(a,\cdot,\cdot)$ from $P$ to
$V(a,\cdot)$.
We linearly order the set of maximal branches $V(M,\cdot)$ by $\prec$ so that there is no maximum
element. $H$ is a surjection from $L$ to each initial segment of $V(M,\cdot)$ of the form
$\{x\in V(M,\cdot)|x\preceq y\}$.
Call $\sigma$ the (first-order) sentence that stipulates all the above.
In all models of $\sigma$, if $P$ has size $\kappa$, then $L$ has size at most $\kappa^{+}$ and
$V(M,\cdot)$ has size at most $\kappa^{++}$. In the case that $|V(M,\cdot)|=\kappa^{++}$,
then also $|L|=\kappa^{+}$. So, all models of $\sigma$ where $P$ is infinite and
$|V(M,\cdot)|=|P|^{++}$ code a Kurepa tree. This proves Theorem 2.4.
We want to emphasize here the fact that since well-orderings can not be characterized by an
$\mathcal{L}_{\omega_{1},\omega}$-sentence, it is unavoidable that we will be working with non-well-ordered trees.
However, using an $\mathcal{L}_{\omega_{1},\omega}$-sentence we can express the fact that $P$ is countably infinite.222There are many ways to
do this. The simplest way is to introduce countably many new constants $(c_{n})_{n\in\omega}$ and require that $\forall x,\;P(x)\rightarrow\bigvee_{n}x=c_{n}$.
Let $\phi$ be the conjunction of $\sigma$ together with the requirement that $P$ is countably
infinite. Then $\phi$ has models of size $\aleph_{2}$ iff there exist a Kurepa tree of size
$\aleph_{2}$ iff $KH(\aleph_{1},\aleph_{2})$.
Fix some $n\geq 2$. Then the above construction of $\sigma$ (and $\phi$) can be modified to
produce a first-order sentence $\sigma_{n}$ and the corresponding $\mathcal{L}_{\omega_{1},\omega}$-sentence $\phi_{n}$ so
that $\phi_{n}$ has a model of size $\aleph_{n}$ iff there exist a Kurepa tree of size
$\aleph_{n}$ iff $KH(\aleph_{1},\aleph_{n})$. The argument breaks down at $\aleph_{\omega}$. Since
we will be dealing with Kurepa trees of size potentially larger than $\aleph_{\omega}$, we must make some modifications.
Let $\tau^{\prime}$ be equal to $\tau$ with the symbols $\prec,H$ removed. Let $\sigma^{\prime}$ be equal to
$\sigma$ with all requirements that refer to $\prec,H$ removed. Let $\psi$ be the conjunction of
$\sigma^{\prime}$ and the requirement that $P$ is countably infinite. For any $\lambda\geq\aleph_{2}$, any
$(\aleph_{1},\lambda)$-Kurepa tree gives rise to a model of $\psi$, but unfortunately, there are
models of $\psi$ of size $2^{\aleph_{0}}$, that do
not code a Kurepa tree. For instance consider the tree $(\omega^{\leq\omega},\subset)$
which has countable height, but contains $2^{\aleph_{0}}$ many maximal branches.
Notice also that both $\phi$ and $\psi$ are not complete sentences.
The dividing line for models of $\psi$ to code Kurepa trees is the size of $L$. By definition
$L$
is $\aleph_{1}$-like, i.e. every initial segment has countable size. If in addition $L$ is
uncountable, then we can embed $\omega_{1}$ cofinally into $L$.333The embedding is not
necessarily continuous, i.e. it may not respect limits. Hence, every model of $\psi$
of size $\geq\aleph_{2}$ and for which $L$ is uncountable, codes a Kurepa tree. Otherwise, the model
does not code a Kurepa tree.
Let $K$ be the collection of all models of $\psi$, equipped with the substructure relation. I.e.
for $M,N\in\mbox{\boldmath$K$}$, $M\prec_{\mbox{\scriptsize\boldmath$K$}}N$ if $M\subset N$.
Observation 2.5.
If $M\prec_{\mbox{\scriptsize\boldmath$K$}}N$, the following follow from the definition:
(1)
$L^{M}$ is an initial segment of $L^{N}$. Towards contradiction, assume that $L^{N}$ contains some
point $x$ and there exists some $y\in L^{M}$ such that $x<y$. Then the function
$F^{M}(y,\cdot,\cdot)$ defined on $M$ disagrees with the function $F^{N}(y,\cdot,\cdot)$ defined on
$N$. Contradiction.
(2)
for every non-maximal $a\in L^{M}$, $V^{M}(a,\cdot)$ equals $V^{N}(a,\cdot)$. The argument is
similar to the argument for $(1)$, using the functions $G^{M}(y,\cdot,\cdot)$ and
$G^{N}(y,\cdot,\cdot)$
this time.
(3)
the tree ordering is preserved.
We will express $(1)-(3)$ by saying that “(the tree defined by ) $M$ is an initial segment
of (the
tree defined by) $N$”.
Corollary 2.6.
Assume $M$ is an initial segment of $N$. Then:
•
If $L^{N}=L^{M}$, then $N$ differs from $M$ only in the maximal branches it
contains.
•
If $L^{N}$ is a strict end-extension of $L^{M}$, then $M$ must be countable.
Proof.
Part (1) is immediate from Observation 2.5 (2). Part (2) follows from the
requirement that all models of $\psi$ have countable levels.
∎
Convention 2.7.
For the rest of the paper, when we talk about “the models of $\psi$” we will mean
$(\mbox{\boldmath$K$},\prec_{\mbox{\scriptsize\boldmath$K$}})$.
The next theorem characterizes the spectrum and the
maximal models spectrum of $\psi$.
Theorem 2.8.
The spectrum of $\psi$ is characterized by the following
properties:
(1)
$[\aleph_{0},2^{\aleph_{0}}]\subset Spec(\psi)$;
(2)
if there exists a Kurepa tree with $\kappa$ many branches, then
$[\aleph_{0},\kappa]\subset Spec(\psi)$;
(3)
no cardinal belongs to $Spec(\psi)$ except
those required by (1) and (2). I.e. if $\psi$ has a model of size $\kappa$, then either
$\kappa\leq 2^{\aleph_{0}}$, or there exists a Kurepa tree which $\kappa$ many branches.
The maximal models spectrum of $\psi$ is characterized by the following:
(4)
$\psi$ has maximal models in cardinalities $2^{\aleph_{0}}$ and $\aleph_{1}$;
(5)
if there exists a Kurepa tree with exactly $\kappa$ many branches, then $\psi$ has a
maximal model in $\kappa$;
(6)
$\psi$ has maximal models only in those cardinalities required by (4) and (5).
Proof.
For (1) and (2), we observed already that $(\omega^{\leq\omega},\subset)$ is a model
of $\psi$ and that every Kurepa tree gives rise to a model of $\psi$.
For (3), let $N$ be a model of $\psi$ of size $\kappa$. If $L^{N}$ is countable, then $N$ has size
$\leq 2^{\aleph_{0}}$. If $L^{N}$ is uncountable, then $N$ codes a $\kappa$-tree with
$|N|$-many branches. By the proof of theorem 2.4, this is a Kurepa tree, assuming that $|N|\geq\aleph_{2}$.
Otherwise, $|N|\leq\aleph_{1}\leq 2^{\aleph_{0}}$.
To prove (4), notice that $(\omega^{\leq\omega},\subset)$ is a maximal model of
$\psi$, and it is easy to construct trees with height $\omega_{1}$ and
$\leq\aleph_{1}$ many branches.
Now, let $N$ code a Kurepa tree with exactly $\kappa$ many
branches. By Corollary 2.6, $N$ is maximal. This proves (5).
For (6), let $N$ be a maximal model. If $N$ was countable, we could end-extend it and
it would not be maximal. So, $N$ must be uncountable. We split into two cases depending on
the size of $L^{N}$. If $L^{N}$ is countable, assume without loss of generality, that it has
height $\omega$ (otherwise consider a cofinal subset of order type $\omega$). So, $(V^{N},T^{N})$ is a
pruned tree which is a subtree of $\omega^{\leq\omega}$. The set of maximal branches through
$(V^{N},T^{N})$ is a closed subset of the Baire space (cf. [8], Proposition 2.4).
Since closed subsets of $\omega^{\omega}$ have size either $\aleph_{0}$ or $2^{\aleph_{0}}$, we conclude
that $N$ has size $2^{\aleph_{0}}$. The second case is when $L^{N}$ is uncountable. If $N$ has size $\aleph_{1}$, we are done. If $|N|\geq\aleph_{2}$, by Corollary 2.6 and maximality of $N$, the tree defined by $N$ contains exactly $|N|$ many maximal
branches. Therefore, $N$ defines a Kurepa tree.
∎
Recall that $\mathcal{B}=\mathcal{B}(\aleph_{1})$ is the supremum of the size of Kurepa trees.
Corollary 2.9.
(1)
If there are no Kurepa trees, then $Spec(\psi)$ equals $[\aleph_{0},2^{\aleph_{0}}]$ and MM-$Spec(\psi)$ equals
$\{\aleph_{1},2^{\aleph_{0}}\}$.
(2)
If $\mathcal{B}$ is a maximum, i.e. there is a Kurepa tree of size
$\mathcal{B}$, then $\psi$ characterizes $\max\{2^{\aleph_{0}},\mathcal{B}\}$.
(3)
If $\mathcal{B}$ is not a maximum, then $Spec(\psi)$ equals either $[\aleph_{0},2^{\aleph_{0}}]$ or $[\aleph_{0},\mathcal{B})$, whichever is
greater. Moreover $\psi$ has maximal models in $\aleph_{1}$, $2^{\aleph_{0}}$ and in cofinally many cardinalities
below $\mathcal{B}$.
Proof.
(1) and (2) follow immediately from Theorem 2.8. We only establish (3).
If $\mathcal{B}$ is not a maximum, then $[\aleph_{0},\mathcal{B})\subset Spec(\psi)$ and $Spec(\psi)$
equals either $[\aleph_{0},2^{\aleph_{0}}]$ or $[\aleph_{0},\mathcal{B})$, whichever is greater. For the the last
assertion, assume $\mathcal{B}=\sup_{i}\kappa_{i}$ and for each $i$, there is a Kurepa tree with $\kappa_{i}$ many branches. Then
each $\kappa_{i}$ is in the MM-$Spec(\psi)$ by Theorem 2.8 (5).
∎
In Section 3 we prove the following consistency results.
Theorem 2.10.
The following are consistent with ZFC:
(i)
ZFC+ ($2^{\aleph_{0}}<\aleph_{\omega_{1}}=\mathcal{B}<2^{\aleph_{1}}$) +“$\mathcal{B}$ is a maximum“, i.e. there
exists a Kurepa
tree of size $\aleph_{\omega_{1}}$”.
(ii)
ZFC+ ($\aleph_{\omega_{1}}=\mathcal{B}<2^{\aleph_{0}}$)+ “$\mathcal{B}$ is a maximum”.
(iii)
ZFC+ ($2^{\aleph_{0}}<\mathcal{B}=2^{\aleph_{1}}$) + “$2^{\aleph_{1}}$ is weakly inaccessible
+ “for every $\kappa<2^{\aleph_{1}}$ there is a Kurepa tree with exactly $\kappa$-many maximal branches, but no Kurepa tree
has exactly $2^{\aleph_{1}}$-many branches.”
Moreover, in (i) and (ii) we can replace $\aleph_{\omega_{1}}$ by most cardinals below or
equal to $2^{\aleph_{1}}$ and $2^{\aleph_{0}}$ respectively.
From [7] we know the consistency of the following:
Theorem 2.11 (R. Jin).
Assume the existence of two strongly inaccessible cardinals. It is consistent with $CH$ (or
$\neg CH$ ) plus $2^{\aleph_{1}}>\aleph_{2}$ that there exists a
Kurepa tree with $2^{\aleph_{1}}$ many branches and no $\omega_{1}$-trees have $\lambda$-many branches
for some $\lambda$ strictly between $\aleph_{1}$ and $2^{\aleph_{1}}$. In particular, no Kurepa trees
have less than $2^{\aleph_{1}}$ many branches.
It follows from the proof of Theorem 2.11 that if $\lambda$ is regular cardinal
above $\aleph_{2}$, we can force the size of $2^{\aleph_{1}}$ to equal $\lambda$.
Corollary 2.12.
There exists an $\mathcal{L}_{\omega_{1},\omega}$-sentence $\psi$ such that it is consistent with ZFC that
(1)
$\psi$ characterizes $2^{\aleph_{0}}$;
(2)
CH (or $\neg CH$ ), $2^{\aleph_{1}}$ is a regular cardinal greater than $\aleph_{2}$ and $\psi$ characterizes
$2^{\aleph_{1}}$;
(3)
$2^{\aleph_{0}}<\aleph_{\omega_{1}}$ and $\psi$ characterizes $\aleph_{\omega_{1}}$; and
(4)
$2^{\aleph_{0}}<2^{\aleph_{1}}$, $2^{\aleph_{1}}$ is weakly inaccessible and $Spec(\psi)=[\aleph_{0},2^{\aleph_{1}})$.
For the same $\psi$ it is consistent with ZFC that
(5)
MM-$Spec(\psi)$ =$\{\aleph_{1},2^{\aleph_{0}}\}$
(6)
MM-$Spec(\psi)$ =$\{\aleph_{1},2^{\aleph_{0}},2^{\aleph_{1}}\}$
(7)
$2^{\aleph_{0}}<2^{\aleph_{1}}$, $2^{\aleph_{1}}$ is weakly inaccessible and
MM-$Spec(\psi)$ =$[\aleph_{1},2^{\aleph_{1}})$.
Proof.
The result follows from Theorem 2.8 and Corollary 2.9 using the appropriate model of ZFC each time. For
(1) consider a model with no Kurepa trees, or a model where $\mathcal{B}<2^{\aleph_{0}}$, e.g. case (ii) of Theorem 2.10.
For (2) and (6) use Theorem 2.11. For (3),(4), use Theorem 2.10 cases (i),(iii) respectively. For (5)
consider a model of ZFC with no Kurepa trees. For (7) use Theorem 2.10
case (iii) again.
∎
Corollary 2.13.
It is consistent with ZFC that
$2^{\aleph_{0}}<\aleph_{\omega_{1}}<2^{\aleph_{1}}$ and there exists an $\mathcal{L}_{\omega_{1},\omega}$-sentence with
models in $\aleph_{\omega_{1}}$, but no models in $2^{\aleph_{1}}$.
2.1. Amalgamation and Joint Embedding Spectra
In this section we provide the amalgamation and joint embedding spectrum of models of $\psi$.
The following characterizes JEP-$Spec(\psi)$ and AP-$Spec(\psi)$.
Theorem 2.14.
(1)
$(\mbox{\boldmath$K$},\prec_{\mbox{\scriptsize\boldmath$K$}})$ fails JEP in all cardinals;
(2)
$(\mbox{\boldmath$K$},\prec_{\mbox{\scriptsize\boldmath$K$}})$ satisfies AP for all uncountable cardinals that belong to $Spec(\psi)$, but fails AP in
$\aleph_{0}$.
Proof.
The first observation is that in all cardinalities there exists two linear orders
$L^{M}$, $L^{N}$ none of which is an initial segment of the other. By Observation 2.5(1), $M,N$
can not be be jointly embedded to some larger structure in $K$. So, JEP fails in all
cardinals.
A similar argument to JEP proves that there exist three countable linear orders
$L^{M_{0}}$, $L^{M_{1}}$, $L^{M_{2}}$ such that $L^{M_{0}}$ is an initial segment of both $L^{M_{1}}$ and
$L^{M_{2}}$, and the triple $(L^{M_{0}},L^{M_{1}},L^{M_{2}})$ can not be amalgamated. This proves that
amalgamation fails in $\aleph_{0}$.
Now, assume that $M,N$ are uncountable models of $\psi$ and $M\prec_{\mbox{\scriptsize\boldmath$K$}}N$. By Corollary
2.6, $L^{M}=L^{N}$ and $M,N$ agree on all levels, except $N$ may contain more maximal
branches. We use this observation to prove amalgamation.
Let $M_{0},M_{1},M_{2}$ be uncountable models in $K$ with $M_{0}\prec_{\mbox{\scriptsize\boldmath$K$}}M_{1},M_{2}$. Since
$L^{M_{0}}=L^{M_{1}}=L^{M_{2}}$ and $M_{0},M_{1},M_{2}$ agree on all levels, except possible the maximal
level, define the amalgam $N$ of $(M_{0},M_{1},M_{2})$ to be the union of $M_{0}$ together with all maximal branches in $M_{1}$
and $M_{2}$. If two maximal branches have exactly the same predecessors, we identify them. It follows
that $N$ is a structure in $K$ and $M_{i}\prec_{\mbox{\scriptsize\boldmath$K$}}N$, $i=1,2,3$.
∎
Notice that, in general, the amalgamation is not disjoint, since both $M_{1}$ and $M_{2}$ may contain
the same maximal branch.
Corollary 2.15.
The following are consistent:
(1)
AP-$Spec(\psi)=[\aleph_{1},2^{\aleph_{0}}]$;
(2)
AP-$Spec(\psi)=[\aleph_{1},2^{\aleph_{1}}]$;
(3)
$2^{\aleph_{0}}<\aleph_{\omega_{1}}$ and AP-$Spec(\psi)=[\aleph_{1},\aleph_{\omega_{1}}]$; and
(4)
$2^{\aleph_{0}}<2^{\aleph_{1}}$, $2^{\aleph_{1}}$ is weakly inaccessible and AP-$Spec(\psi)=[\aleph_{1},2^{\aleph_{1}})$.
Proof.
The result follows from Theorem 2.14 and Corollary 2.12.
∎
It follows from Corollary 2.15 that if $\aleph_{2}\leq\kappa\leq 2^{\aleph_{1}}$ is a regular cardinal, then the
$\kappa$-amalgamation property for $\mathcal{L}_{\omega_{1},\omega}$-sentences is not absolute for models of ZFC. The result is useful especially
under the failure of GCH, since GCH implies that $2^{\aleph_{1}}=\aleph_{2}$. Assuming GCH the absoluteness question remains open
for $\kappa\geq\aleph_{3}$.
In addition, it is an easy application of Shoenfield’s absoluteness that $\aleph_{0}$-amalgamation is an absolute property
for models of ZFC. The question for $\aleph_{1}$-amalgamation remains open. Recall that by
[4], model-existence in $\aleph_{1}$ for $\mathcal{L}_{\omega_{1},\omega}$-sentences is
an absolute property for models of ZFC.
Open Questions 2.16.
(1)
Is $\aleph_{1}$-amalgamation for $\mathcal{L}_{\omega_{1},\omega}$-sentences absolute for models of ZFC?
(2)
Let $3\leq\alpha<\omega_{1}$. Does the non-absoluteness of $\aleph_{\alpha}$-amalgamation hold if we assume GCH?
3. Consistency Results
In this section we prove the consistency results announced by Theorem 2.10. Recall that $\mathcal{B}$ is the supremum of
$\{\lambda|\text{there exists a Kurepa tree with $\lambda$-many branches}\}$.
Theorem 3.1.
It is consistent with ZFC that:
(1)
$2^{\aleph_{0}}<\aleph_{\omega_{1}}=\mathcal{B}<2^{\aleph_{1}}$ and there exists a Kurepa tree of size $\aleph_{\omega_{1}}$.
(2)
$\aleph_{\omega_{1}}=\mathcal{B}<2^{\aleph_{0}}$ and there exists a Kurepa tree of size $\aleph_{\omega_{1}}$.
We start with a model $V_{0}$ of ZFC+GCH. Let $\mathbb{P}$ be the standard $\sigma$-closed poset for adding a Kurepa tree $K$
with $\aleph_{\omega_{1}}$-many $\omega_{1}$-branches. More precisely, conditions in $\mathbb{P}$ are of the form $(t,f)$, where:
•
$t$ is a tree of height $\beta+1$ for some $\beta<\omega_{1}$ and countable levels;
•
$f$ is a function with $\mathop{\mathrm{dom}}\nolimits(f)\subset\aleph_{\omega_{1}}$, $|\mathop{\mathrm{dom}}\nolimits(f)|=\omega$, and $\mathop{\mathrm{ran}}\nolimits(f)=t_{\beta}$, where $t_{\beta}$ is the
$\beta$-th level of $t$.
Intuitively, $t$ is an initial segment of the generically added tree, and each $f(\delta)$ determines where the $\delta$-th branch
intersects the tree at level $\beta$.
The order is defined as follows: $(u,g)\leq(t,f)$ if,
•
$t$ is an initial segment of $u$,
$\mathop{\mathrm{dom}}\nolimits(f)\subset\mathop{\mathrm{dom}}\nolimits(g)$,
•
for every $\delta\in\mathop{\mathrm{dom}}\nolimits(f)$, either $f(\delta)=g(\delta)$ (if $t$ and $u$ have the same height) or $f(\delta)<_{u}g(\delta)$.
We have that $\mathbb{P}$ is countably closed and has the $\aleph_{2}$-chain condition.
Suppose that $H$ is $\mathbb{P}$-generic over $V_{0}$. Then $\bigcup_{(t,f)\in H}t$ is a Kurepa tree with $\aleph_{\omega_{1}}$-many
branches, where for $\delta<\aleph_{\omega_{1}}$, the $\delta$-th branch is given by $\bigcup_{(t,f)\in H,\delta\in\mathop{\mathrm{dom}}\nolimits(f)}f(\delta)$. These branches are distinct by standards density arguments. Also, note that since $|B|$ cannot exceed $2^{\aleph_{1}}$
and in this model $\aleph_{\omega_{1}}=|B|$, we have that $\aleph_{\omega_{1}}<2^{\aleph_{1}}$.
The model $V_{0}[H]$ proves part (1) of
Theorem 3.1.
The same model also answers positively a question raised in
[14]. The question was whether any
cardinal outside the smallest set which contains $\aleph_{0}$ and which is closed under successors,
countable unions, countable products and powerset, can be characterized by an $\mathcal{L}_{\omega_{1},\omega}$-sentence. $\aleph_{\omega_{1}}$ is
consistently such an example.
Lemma 3.2.
Let $C$ be the smallest set of cardinals that contains $\aleph_{0}$ and is closed under
successors, countable unions, countable products and powerset. In $V_{0}[H]$, the set $C$ does not
contain $\aleph_{\omega_{1}}$.
Proof.
Since $2^{\omega}<\aleph_{\omega_{1}}<2^{\omega_{1}}$, it is enough to show that $\aleph_{\omega_{1}}$ is
not the countable product of countable cardinals. Suppose $\langle\alpha_{n}\mid n<\omega\rangle$ is
an increasing sequence of countable ordinals and let $\alpha=\sup_{n}\alpha_{n}+1$. Then
$\prod_{n}\aleph_{\alpha_{n}}=(\aleph_{\alpha})^{\omega}=\aleph_{\alpha+1}$.
∎
Let $\mathbb{C}=Add(\omega,\aleph_{\omega_{1}+1})$ denote the standard poset for adding $\aleph_{\omega_{1}+1}$-many Cohen reals. Suppose
$G$ is $\mathbb{C}$-generic over $V:=V_{0}[H]$. (Note that $\mathbb{C}$ is interpreted the same in $V_{0}$ and in $V$ and that actually genericity
over $V_{0}$ implies genericity over $V$ by the ccc.)
We claim that the forcing extension $V_{0}[H\times G]=V[G]$ satisfies $2^{\aleph_{0}}>\aleph_{\omega_{1}}=\mathcal{B}$.
Let $T$ be a Kurepa tree in $V[G]$. Denote $\mathbb{C}_{\omega_{1}}:=Add(\omega,\omega_{1})$, i.e. the Cohen poset
for adding $\omega_{1}$ many reals. The following fact is standard and can be found in [6], but we give the proof
for completeness.
Lemma 3.3.
There is a generic $\bar{G}$ for $\mathbb{C}_{\omega_{1}}$, such that $V[\bar{G}]\subset V[G]$ and $T\in V[\bar{G}]$.
Proof.
Since $T$ is a tree of height $\omega_{1}$ and countable levels, we can index the nodes of $T$ by $\langle\alpha,n\rangle$, for
$\alpha<\omega_{1}$ and $n<\omega$, where the first coordinate denotes the level of the node. In particular each level
$T_{\alpha}=\{\alpha\}\times\omega$. Note that this is in the ground model (although of course the relation $<_{T}$ may not be).
Working in $V[G]$, for every $\alpha<\beta<\omega_{1}$ and $n,m$, let $p_{\alpha,\beta,m,n}\in G$ decide the statement
$\langle\alpha,m\rangle<_{\dot{T}}\langle\beta,n\rangle$. Let $d_{\alpha,\beta,m,n}=\mathop{\mathrm{dom}}\nolimits(p_{\alpha,\beta,m,n})$; this is a
finite subset of $\aleph_{\omega_{1}+1}\times\omega$. Now let $d^{*}=\bigcup_{\alpha,\beta,m,n}d_{\alpha,\beta,m,n}$ and
$d=\{i<\aleph_{\omega_{1}+1}\mid(\exists k)\langle i,k\rangle\in d^{*}\}$. Then $d$ has size at most $\omega_{1}$. By increasing $d$
if necessary, assume that $|d|=\omega_{1}$.
Write $G$ as $\prod_{i\in\aleph_{\omega_{1}+1}}G_{i}$, where every $G_{i}$ is $Add(\omega,1)$-generic, and let $\bar{G}=\prod_{i\in d}G_{i}$. Then $\bar{G}$ is $\mathbb{C}_{\omega_{1}}$-generic, containing every $p_{\alpha,\beta,m,n}$. And so $T\in V[\bar{G}]\subset V[G]$.
∎
Fix $\bar{G}$ as in the above lemma.
Lemma 3.4.
All $\omega_{1}$-branches of $T$ in $V[G]$ are already in the forcing extension $V[\bar{G}]$.
Proof.
This is because $\mathbb{C}/\bar{G}$, i.e. the forcing to get from $V[\bar{G}]$ to $V[G]$ is Knaster:
Suppose $\dot{b}$ is forced to be an $\omega_{1}$-branch of $T$. For $\alpha<\omega$ let $p_{\alpha}\Vdash_{\mathbb{C}/\bar{G}}u_{\alpha}\in\dot{b}\cap T_{\alpha}$, where $T_{\alpha}$ is the $\alpha$-th level. Then there is an unbounded $I\subset\omega_{1}$ such
that for all $\alpha,\beta\in I$, $p_{\alpha}$ and $p_{\beta}$ are compatible, and so $\{u_{\alpha}\mid\alpha\in I\}$ generate the
branch in $V[\bar{G}]$.
∎
For every $\alpha<\aleph_{\omega_{1}}$, let $\mathbb{P}_{\alpha}:=\{(t,f\upharpoonright{\aleph_{\alpha}})\mid(t,f)\in%
\mathbb{P}\}$. Then clearly the
poset $\mathbb{P}$ is the
union of the sequence $\langle\mathbb{P}_{\alpha}\mid\alpha<\omega_{1}\rangle$ and each $\mathbb{P}_{\alpha}$ is a regular $\aleph_{2}$-cc
subordering of $\mathbb{P}$ that adds $\aleph_{\alpha}$ many branches to the generic Kurepa tree. Let $H_{\alpha}$ be the generic filter for
$\mathbb{P}_{\alpha}$ obtained from $H$. Also, for a condition $p\in\mathbb{P}$, we use the notation $p=(t^{p},f^{p})$.
Claim 3.5.
For every $\omega_{1}$ branch $b$ of $T$, there is some $\alpha<\omega_{1}$, such that $b\in V_{0}[H_{\alpha}\times\bar{G}]$.
Proof.
As before, we index the nodes of $T$ by $\langle\alpha,n\rangle$, for
$\alpha<\omega_{1}$ and $n<\omega$, where the first coordinate denotes the level of the node. Similarly to the arguments in Lemma
3.3, we can find a set $d\subset\aleph_{\omega_{1}}$ of size $\omega_{1}$, such that $T$ is in $V_{0}[\bar{G}\times\bar{H}]$, where $\bar{H}=\{(t,f\upharpoonright d)\mid(t,f)\in H\}$. Then $\bar{H}$ is actually a generic filter for
$\mathbb{P}_{1}$, and we view $V_{0}[\bar{G}\times H]=V_{0}[\bar{G}\times\bar{H}][H^{\prime}]$, where $H^{\prime}$ is
$\mathbb{P}/\bar{H}:=\{(t,f)\in\mathbb{P}\mid(t,f\upharpoonright\aleph_{1})\in%
\bar{H}\}$ -generic.
Suppose $\dot{b}$ is a $\mathbb{P}$-name for a cofinal branch through $T$, which is not in $V_{0}[\bar{G}\times\bar{H}]$. We say
$p\Vdash\dot{b}(\alpha)=n$ to mean that $p$ forces that $\dot{b}\cap\dot{T}_{\alpha}=\{\langle\alpha,n\rangle\}$. In
$V_{0}[\bar{G}\times\bar{H}]$ let $\mathbb{Q}:=\{(t,f\upharpoonright\aleph_{1}+1)\mid(t,f)\in\mathbb{P}/\bar{H}\}$. Define a
$\mathbb{Q}$-name $\tau$, by setting
$$\tau=\{(\langle\alpha,n\rangle,q)\mid\alpha<\omega_{1},n<\omega,f^{q}(\omega_{%
1})=n,\exists p\in\mathbb{P}/\bar{H},t^{p}\upharpoonright(\alpha+1)=t^{q},p%
\Vdash\dot{b}(\alpha)=n\}.$$
Also let $K=\{(t,f)\mid\exists\alpha<\omega_{1},\exists p\in H,t^{p}\upharpoonright%
\alpha+1=t,\omega_{1}\in\mathop{\mathrm{dom}}\nolimits(f),p\Vdash\dot{b}(%
\alpha)=f(\omega_{1})\}$. Since $b$ is not in $V_{0}[\bar{G}\times\bar{H}]$, it is straightforward to check that $K$ is
$\mathbb{Q}$ -generic, and also that $\dot{b}_{H}=\tau_{K}$.
Finally, since $\bar{H}*K\in V_{0}[\bar{G}\times H]$ is generic for the suborder $\{(t,f\upharpoonright\aleph_{1}+1)\mid(t,f)\in\mathbb{P}\}$, $K$ must be in $V_{0}[\bar{G}\times H_{\alpha}]$, for some $\alpha<\omega_{1}$. Then $b\in V_{0}[\bar{G}\times H_{\alpha}]$.
∎
Lemma 3.6.
If $\mathbb{P}\times\mathbb{C}_{\omega_{1}}$ adds more than $\aleph_{\omega_{1}}$-many $\omega_{1}$-branches to $T$
then there is some $\alpha<\omega_{1}$, so that $\mathbb{P}_{\alpha}\times\mathbb{C}_{\omega_{1}}$ adds
more than $\aleph_{\omega_{1}}$ many $\omega_{1}$-branches to $T$.
Proof.
For every $\alpha<\omega_{1}$, let $H_{\alpha}$ be $\mathbb{P}_{\alpha}$-generic over $V_{0}$, induced by $H$. Then $V_{0}[H_{\alpha}\times\bar{G}]\subset V_{0}[H\times\bar{G}]=V[\bar{G}]$.
Suppose that for some $\lambda>\aleph_{\omega_{1}}$, $T$ has $\lambda$-many branches, enumerate them by $\langle b_{i}\mid i<\lambda\rangle$.
For every $i<\lambda$, let $\alpha_{i}<\omega_{1}$ be such that $b_{i}\in V_{0}[H_{\alpha_{i}}\times\bar{G}]$ given by Claim 3.5.
Then for some $\alpha<\omega_{1}$ there is an unbounded
$I\subset\lambda$, such that for all $i\in I$, $\alpha_{i}=\alpha$.
But that implies that in $V_{0}[H_{\alpha}\times\bar{G}]$, $2^{\omega_{1}}>\aleph_{\omega_{1}}$, which is a contradiction since we
started with $V_{0}\models GCH$.
So the forcing extension of
$\mathbb{P}\times\mathbb{C}$ has at most $\aleph_{\omega_{1}}$ many $\omega_{1}$-branches of $T$, i.e
$\mathcal{B}=\aleph_{\omega_{1}}$ in this forcing extension.
∎
$V[G]=V_{0}[H][G]$ proves part (2) of Theorem 3.1.
Theorem 3.7.
It is consistent with ZFC that $2^{\aleph_{0}}<\mathcal{B}=2^{\aleph_{1}}$, for every $\kappa<2^{\aleph_{1}}$
there is a Kurepa tree with exactly $\kappa$-many maximal branches, but no Kurepa tree has $2^{\aleph_{1}}$-many maximal branches.
Proof.
The proof uses the forcing axiom principle $GMA$ defined by Shelah. $GMA_{\kappa}$ states that for every $\kappa$-closed,
stationary $\kappa^{+}$-linked, well met poset $\mathbb{P}$ with greatest lower bound if $\kappa$ is regular and for every
collection of less than $2^{\kappa}$ many dense sets there is a filter for $\mathbb{P}$ meeting them.
For an exact definition of stationary $\kappa^{+}$-linked, see section 4 of [10].
We take a model constructed in [10], section 4. More precisely, following the arguments in that
section, from some fairly mild large cardinals (a Mahlo cardinal will suffice), we get a model $V$, where the following holds:
(1)
$GMA_{\omega_{1}}$,
(2)
$CH$,
(3)
$2^{\omega_{1}}$ is weakly inaccessible,
(4)
every $\Sigma_{1}^{1}$-subset of $\omega_{1}^{\omega_{1}}$ of cardinality $2^{\omega_{1}}$ contains a perfect set.
We claim that this is the desired model. Let $\omega_{1}<\kappa<2^{\omega_{1}}$. To see that there is a Kurepa tree with exactly
$\kappa$-many maximal branches, let $\mathbb{P}$ be the standard poset to add such a tree (i.e. we take the poset from earlier but
with $\kappa$ in place of $\aleph_{\omega_{1}}$). Then $\mathbb{P}$ satisfies the hypothesis of $GMA_{\omega_{1}}$, and there are only
$\kappa$-many dense sets to meet in order to get a Kurepa tree with $\kappa$-many branches.
Also, the last item of the properties listed above implies that there are no Kurepa trees with $2^{\omega_{1}}$-many branches; for
details see the discussion of page 22 of [11].
∎
Although we will not give the details here, the results presented here can be extended to
$\kappa$-Kurepa trees with $\kappa\geq\aleph_{2}$.
Acknowledgements
The ideas of Lemma 2.2 and Theorem 3.7 were communicated to the
authors by Philipp Lücke. The consistency results of Section 3 were based on early ideas of Stevo Todorcevic.
The authors would also like to thank John Baldwin for his useful feedback on an earlier version of the manuscript that
helped improve the exposition.
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Notre Dame Journal of Formal Logic, 55(4):533–551, 2014. |
10-qubit entanglement and parallel logic operations
with a superconducting circuit
Chao Song${}^{1,2}$
Kai Xu${}^{1,2}$
Wuxin Liu${}^{1}$, Chuiping Yang${}^{3}$
Shi-Biao Zheng${}^{4}$
t96034@fzu.edu.cn
Hui Deng${}^{5}$, Qiwei Xie${}^{6}$, Keqiang Huang${}^{5}$, Qiujiang Guo${}^{1}$, Libo Zhang${}^{1}$, Pengfei Zhang${}^{1}$, Da Xu${}^{1}$, Dongning Zheng${}^{5}$
Xiaobo Zhu${}^{2}$
xbzhu16@ustc.edu.cn
H. Wang${}^{1,2}$
hhwang@zju.edu.cn
Y.-A. Chen${}^{2}$, C.-Y. Lu${}^{2}$, Siyuan Han${}^{7}$
J.-W. Pan${}^{2}$
${}^{1}$Department of Physics, Zhejiang University, Hangzhou, Zhejiang 310027, China,
${}^{2}$Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China,
${}^{3}$Department of Physics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, China,
${}^{4}$Fujian Key Laboratory of Quantum Information and Quantum Optics, College of Physics and Information Engineering, Fuzhou University, Fuzhou, Fujian 350116, China,
${}^{5}$Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China,
${}^{6}$Institute of Automation, Chinese Academy of Sciences, Beijing 100190, China,
${}^{7}$Department of Physics and Astronomy, University of Kansas, Lawrence, Kansas 66045, USA
(December 2, 2020)
Abstract
Here we report on the production and tomography of genuinely entangled Greenberger-Horne-Zeilinger states
with up to 10 qubits connecting to a bus resonator in a superconducting circuit, where
the resonator-mediated qubit-qubit interactions are used to controllably
entangle multiple qubits and to operate on different pairs of qubits in parallel.
The resulting 10-qubit density matrix is unambiguously probed, with a fidelity of $0.668\pm 0.025$.
Our results demonstrate the largest entanglement created
so far in solid-state architectures, and pave the way to large-scale quantum computation.
pacs:
††thanks: C. S. and K. X. contributed equally to this work.††thanks: C. S. and K. X. contributed equally to this work.
Entanglement is one of the most counter-intuitive features of quantum mechanics.
The creation of increasingly large number of entangled quantum bits (qubits) is central
for measurement-based quantum computation Raussendorf2001 ,
quantum error correction Calderbank1996 ; Knill2005 , quantum
simulation Lloyd1996 , and foundational studies of nonlocality Greenberger1990 ; Ansmann2009
and quantum-to-classical transition Leggett2008 .
A significant experimental challenge for engineering multiqubit
entanglement Wang2016 ; Monz2011 ; Barends2014 has been noise control.
With solid-state platforms, the largest
number of entangled qubits reported so far is five Barends2014 ,
and further scaling up would be difficult as constrained
by the qubit coherence and the employed sequential-gate method.
Superconducting circuits are a promising solid-state platform for quantum
state manipulation and quantum computing owing to the microfabrication
technology scalability, individual qubit addressability, and ever-increasing
qubit coherence time. The past decade has witnessed significant progresses
in quantum information processing and entanglement engineering with
superconducting qubits: preparation of three-qubit entangled states DiCarlo2010 ; Neeley2010 ,
demonstration of elementary quantum algorithms DiCarlo2009 ; Mariantoni2011 , realization of
three-qubit Toffoli gates and quantum error correction Fedorov2012 ; Reed2012 . In
particular, a recent experiment has achieved a two-qubit controlled-phase
gate with a fidelity above 99 percent with a superconducting quantum
processor Barends2014 , where five transmon qubits with nearest-neighbor coupling
are arranged in a linear array. Based on this gate, a five-qubit Greenberger-Horne-Zeilinger (GHZ) state
was produced step by step; the number of entangled qubits is increased by one at a time.
In this letter we demonstrate a versatile superconducting quantum processor featuring
high connectivity with programmable qubit-qubit couplings mediated
by a bus resonator, and experimentally produce GHZ states with up to 10
qubits using this quantum processor. The
resonator-induced qubit-qubit couplings result in a phase shift that is
quadratically proportional to the total qubit excitation number, evolving
the participating qubits from an initially product state to the GHZ state
after a single collective interaction, irrespective of the number of the
entangled qubits Zheng2001 . We characterize the multipartite entanglement by quantum state tomography achieved by synchronized local manipulations and detections
of the entangled qubits, and measure a fidelity of
$0.668\pm 0.025$ for the 10-qubit GHZ state, which confirms the genuine tenpartite
entanglement Guhne2009 with 6.7 standard deviations ($\sigma$).
We also implement parallel entangling operations mediated by the resonator,
simultaneously generating three Einstein-Podolsky-Rosen (EPR) pairs;
this feature was previously suggested in the
context of ion traps Sorensen1999 and quantum dots coupled to an optical cavity Imamoglu1999 ,
but experimental demonstrations are still lacking.
The superconducting quantum processor is illustrated in Fig. 1(a), which is constructed as
10 transmon qubits ($Q_{j}$ for $j$ = 1 to 10), with resonant frequencies
$\omega_{j}/2\pi$ tunable from 5 to 6 GHz, symmetrically
coupled to a central resonator ($B$), whose resonant frequency is fixed at
$\omega_{B}/2\pi\approx$ 5.795 GHz Lucero2012 . Measured qubit-resonator ($Q_{j}$-$B$) coupling strengths
$g_{j}/2\pi$ range from 14 to 20 MHz
(see Supplemental Material supp for details on device, operation, and readout).
The central resonator serves as a multipurpose actuator,
enabling controlled long-range logic operations, scalable multiqubit entanglement, and quantum state transfer.
In the rotating-wave approximation and ignoring the crosstalks between qubits (see Supplemental Material supp ),
the Hamiltonian of the system is given by
$$H/\hbar=\omega_{B}a^{+}a+\sum_{j=1}^{10}{\left[\omega_{j}|1_{j}\rangle\langle 1%
_{j}|+g_{j}(\sigma_{j}^{+}a+\sigma_{j}^{-}a^{+})\right]},$$
(1)
where $\sigma_{j}^{+}$ ($\sigma_{j}^{-}$) is the raising (lowering)
operator of $Q_{j}$ and $a^{+}$ ($a$) is the creation (annihilation)
operator of $B$.
The qubit-qubit coupling can be realized through the superexchange (SE)
interaction Trotzky2008 mediated by the bus resonator $B$ Zheng2000 ; Osnaghi2001 ; Majer2007 .
With multiplexing we can further arrange multiple qubit pairs
at different frequencies to turn on the intra-pair SE interactions simultaneously.
To illustrate this feature, we consider three qubit pairs, $Q_{k}$-$Q_{k^{\prime}}$,
$Q_{l}$-$Q_{l^{\prime}}$, and $Q_{m}$-$Q_{m^{\prime}}$,
detuned from resonator $B$ by $\Delta_{j}$ ($\equiv\omega_{j}-\omega_{B}$, and $\omega_{j}=\omega_{j^{\prime}}$)
for $j=k$, $l$, and $m$, respectively,
while all other qubits are far detuned and can be neglected for now.
In the dispersive regime and when the resonator $B$ is initially in the
ground state, it will remain so throughout the procedure and the effective Hamiltonian for the qubit pairs is
$$\displaystyle H_{1}/\hbar=\sum_{j\in\{k,l,m\}}{\lambda_{j}\left(\sigma_{j}^{-}%
\sigma_{j^{\prime}}^{+}+\sigma_{j}^{+}\sigma_{j^{\prime}}^{-}\right)}\\
\displaystyle+\sum_{j\in\{k,l,m\}}\left[\frac{g_{j}^{2}}{\Delta_{j}}\left|1_{j%
}\right\rangle\left\langle 1_{j}\right|+\frac{g_{j^{\prime}}^{2}}{\Delta_{j}}%
\left|1_{j^{\prime}}\right\rangle\left\langle 1_{j^{\prime}}\right|\right],$$
(2)
where $\lambda_{j}=\frac{g_{j}g_{j^{\prime}}}{\Delta_{j}}$, $|\Delta_{j}|\gg g_{j},\,g_{j}^{\prime}$,
and $|\Delta_{j_{1}}-\Delta_{j_{2}}|\gg\lambda_{j_{1}},\,\lambda_{j_{1}}^{\prime},%
\,\lambda_{j_{2}},\,\lambda_{j_{2}}^{\prime}$ for $j_{1},\,j_{2}\in\{k,l,m\}$ and $j_{1}\neq j_{2}$.
With this setting, the resonator $B$ is simultaneously used for three
intra-pair SE processes; the inter-pair couplings are effectively switched off due to large detunings between different pairs.
With the fast z control on each qubit, coupling between any two qubits can be dynamically turned on and off
by matching (intra-pair) and detuning (inter-pair), respectively, their frequencies,
i.e., we can reconfigure the coupling structure in-situ
without modifying the physical wiring of the circuit.
For example, by arranging $\Delta_{k}$, $\Delta_{l}$, and $\Delta_{m}$ in Eq. (2) at three
distinct frequencies, we create three qubit pairs
($Q_{2}$-$Q_{9}$, $Q_{3}$-$Q_{8}$, and $Q_{5}$-$Q_{6}$) featuring programmable
intra-pair SE interactions with negligible inter-pair crosstalks,
enabling parallel couplings as demonstrated in Fig. 1(b). According to the probability evolutions shown in Fig. 1(b),
a characteristic gate time, $t_{\sqrt{\text{iSWAP}}}$, for each qubit pair can be be identified Ansmann2009 .
Operating multiple pairs in parallel
naturally produces multiple EPR pairs Zheng2000 ; Osnaghi2001 .
As the pulse sequence shows in Fig. 2(a),
three EPR pairs are produced after the completion of all three SE-$\sqrt{\text{iSWAP}}$ gates,
with the 6-qubit quantum state tomography
measuring an overall state fidelity of $0.904\pm 0.018$. The inferred density matrix $\rho$ is validated by satisfying the physical
constraints of Hermitian, unit trace, and positive semi-definite.
We further perform partial trace on $\rho$ to obtain three 2-qubit reduced density matrices,
each corresponding to a EPR pair with a fidelity above 0.93 (Fig. 2(b)).
Remarkably, our architecture allows high-efficiency generation of
multiqubit GHZ states. In contrast to the previous approach
where GHZ states are generated by a series of controlled-NOT (CNOT) gates Barends2014 ,
here all the qubits connected to the bus resonator can be entangled
with a single collective qubit-resonator interaction.
In the theoretical proposal Zheng2001 ; Agarwal1997 , $N$ qubits are assumed to be equally coupled to the resonator
and are detuned from the resonator by the same amount $\Delta$ that is much larger than the qubit-resonator coupling.
When all qubits are initialized in the same equal superpositions of $0\rangle$
and $|1\rangle$, e.g., $\left(\ket{0}-i\ket{1}\right)/\sqrt{2}$,
the SE interaction does not induce any energy exchange between qubits; instead, it produces a dynamic
phase that nonlinearly depends upon the collective qubit excitation
number $k$ as $k(N+1-k)\theta$,
where $\theta$ is determined by the effective qubit-qubit coupling strength
and the interaction time. With the choice $\theta=\pi/2$, this
gives rise to the GHZ state
$\left(|+_{1},+_{2},...,+_{N}\rangle+i|-_{1},-_{2},...,-_{N}\rangle\right)/%
\sqrt{2}$,
where $|\pm_{j}\rangle=\left(\ket{0_{j}}\pm i^{N}\ket{1_{j}}\right)/\sqrt{2}$ Zheng2001 .
Here we apply this proposal to our experiment.
We find that, though the qubit-resonator couplings are
not uniform and unwanted crosstalk couplings exist in our circuit, we can optimize
each qubit’s detuning and the overall interaction time
to achieve GHZ states with high fidelities as guided by numerical simulation.
The pulse sequence is shown in Fig. 3(a). We
start with initializing the chosen $N$ qubits in $\left(\ket{0}-i\ket{1}\right)/\sqrt{2}$ by applying $\pi/2$ pulses
at their respective idle frequencies, following which we bias them
to nearby $\Delta/2\pi\approx-140$ MHz for an optimized duration of approximately twice $t_{\sqrt{\text{iSWAP}}}$. The phase of each qubit’s xy drive is calibrated according to the rotating frame with respect to $\Delta$,
ensuring that all $N$ qubits are in the same initial state just
when their SE interactions are switched on Zhong2016 ; supp .
After the optimized interaction time, these qubits approximately evolve to the GHZ state
$|\Psi_{1}\rangle=\left(|+_{1},+_{2},...,+_{N}\rangle+e^{i\varphi}|-_{1},-_{2},%
...,-_{N}\rangle\right)/\sqrt{2}$,
where $\varphi$ may not be equal to $\pi/2$ as in the ideal case with uniform qubit-qubit interactions;
however, this phase variation does not affect entanglement.
Later on we bias these $N$ qubits back to their idle frequencies;
during the process a dynamical phase $\phi_{j}$ is accumulated between $|0\rangle$ and $|1\rangle$ of $Q_{j}$.
Re-defining $|\pm_{j}\rangle=\left(\ket{0_{j}}\pm i^{N}e^{i\phi_{j}}\ket{1_{j}}\right)/%
\sqrt{2}$
ensures that the above-mentioned formulation of $|\Psi_{1}\rangle$ remains invariant, which is
equivalent to a $z$-axis rotation of the $x$-$y$-$z$ reference frame, i.e., $x\rightarrow x^{\prime}$ and $y\rightarrow y^{\prime}$.
Tracking the new axes is important for characterization of the produced GHZ states.
Tomography of the produced states requires individually measuring the qubits
in bases formed by the eigenvectors of the Pauli operators $X$, $Y$, and $Z$,
respectively. Measurement in the $Z$ basis can be directly performed.
For each state preparation and measurement event,
we record the 0 or 1 outcomes of each qubit and do so for $N$ qubits simultaneously;
repeating the state preparation and measurement event thousands of times we count $2^{N}$ probabilities
of {$P_{00...0}$, $P_{00...1}$, …., $P_{11...1}$}. Measurement in the
$X$ ($Y$) basis is achieved by inserting a Pauli $Y$ ($X$) rotation on each qubit before readout.
All together, the $3^{N}$ tomographic operations and the $2^{N}$ probabilities for each operation allow
us to unambiguously reconstruct all elements of the density matrix $\rho$ (see Supplemental Material supp for various aspects of our tomography technique including measurement stability,
reliability with reduced sampling size, and pre-processing for minimizing the computational cost).
The resulting 10-qubit GHZ density matrix is partially illustrated in Fig. 3(b), with a fidelity of $0.668\pm 0.025$,
and the $N$-qubit GHZ fidelity as function of $N$ is plotted in Fig. 3(b) inset.
The achieved fidelities are well above the threshold for genuine multipartite
entanglement Guhne2009 .
The full tomography technique, though general and accurate, is costly when $N$ is large.
The produced GHZ states can also be characterized by a shortcut,
since the ideal GHZ density matrix consists of only four non-zero elements in a suitably chosen basis.
To do so, we apply to each qubit a $\pi/2$ rotation around its $y^{\prime}$ or $x^{\prime}$ axis,
transforming $|\Psi_{1}\rangle$ to
$|\Psi_{2}\rangle=\left(|00...0\rangle+e^{i\varphi}|11...1\rangle\right)/\sqrt{2}$
(here and below we omit the subscripts of the qubit index
for clarity). The diagonal elements $\rho_{00...0}$ and $\rho_{11...1}$
can be directly measured; the off-diagonal elements $\rho_{00...0,\,11...1}$
and $\rho_{11...1,\,00...0}$ can be obtained by
measuring the system parity, defined as the expectation value of the operator
$P(\gamma)=\otimes_{j=1}^{N}(\cos{\gamma}Y^{\prime}_{j}+\sin{\gamma}X^{\prime}_%
{j})$, which is given by
$\langle P(\gamma)\rangle=2\left|\rho_{00...0,\,11...1}\right|\cos(N\gamma+\varphi)$
for $|\Psi_{2}\rangle$ Monz2011 .
The oscillation patterns of the measured parity as functions of $\gamma$
confirm the existence of coherence between the states $|00...0\rangle$ and $|11...1\rangle$ (Fig. 3(c)).
The fidelity of the $N$-qubit GHZ state $|\Psi_{2}\rangle$ can be estimated using the four non-zero elements,
which is $0.660\pm 0.020$ for $N=10$. This value agrees
with that of the GHZ state $|\Psi_{1}\rangle$ obtained by full state tomography.
A key advantage of the present protocol for generating GHZ states is its high scalability as demonstrated in Fig. 3(b).
If limited by decoherence, the achieved fidelity based on the
sequential-CNOT approach, $F_{\text{N,C}}$, scales approximately as
$F_{\text{N,C}}\propto F_{2,C}^{N^{2}/2}$ at large $N$ (see the red dashed line in Fig. 3(b) inset), while that based on
our protocol scales as $F_{\text{N}}\propto F_{2}^{N}$ (blue dashed line).
Here $F_{2,C}$ ($F_{2}$) is quoted as the decoherence-limited fidelity of the CNOT gate (present protocol) involving two qubits.
The falling of the experimental data (blue dots) below the scaling line when $N\geq 6$ is due
to the inhomogeneity of $g_{j}$ and the crosstalk couplings. One can see that, even with the two-qubit gate fidelity as high as 0.994, the coherence
performance of the device used in Ref. Barends2014 does not allow generation of
10-qubit GHZ state with fidelity above the genuine entanglement threshold
using the sequential-CNOT approach.
In summary, our experiment demonstrates the viability of the multiqubit-resonator-bus
architecture for scalable quantum information processing, with essential functions including
high-efficiency entanglement generation and parallel logic operations.
We deterministically generate the 10-qubit GHZ state,
the largest multiqubit entanglement ever created in solid-state systems, which is
verified by quantum state tomography for the first time as well.
In addition, our approach allows instant in situ rewiring of the qubits. These unique features
show the great potential of the demonstrated approach for scalable quantum information processing.
Acknowledgments.
This work was supported by the National Basic Research Program
of China (Grant No. 2014CB921201), the
National Natural Science Foundations of China (Grants No. 11434008,
No. 11374054, No. 11574380, and No. 11404386),
the Fundamental Research Funds for the Central Universities
of China (Grant No. 2016XZZX002-01),
and the Ministry of Science and Technology of China (Grant No. 2016YFA0301802).
S.H. was supported by NSF (Grant No. PHY-1314861).
Devices were made at the Nanofabrication
Facilities at Institute of Physics in Beijing, University
of Science and Technology in Hefei, and National
Center for Nanoscience and Technology in Beijing.
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Supplementary Material for
“10-qubit entanglement and parallel logic operations
with a superconducting circuit”
1 Device, control, and measurement information
1.1 Device
Our sample is a superconducting circuit consisting of 10 Xmon qubits interconnected by
a bus resonator, fabricated with two steps of aluminum deposition
as described elsewhere Song2017 .
There is no crossover layer to connect segments of grounding planes on the circuit chip.
Instead, aluminum bonding wires are manually applied as many as possible to
reduce the impact of parasitic slotline modes.
The bus resonator $B$ is a half-wavelength coplanar waveguide resonator with
a resonant frequency fixed at $\omega_{B}/2\pi\approx 5.795$ GHz, which is measured with all coupling qubits
staying in $|0\rangle$ at their respective idle frequencies (see below).
The resonator has 10 side arms, each is capacitively coupled to an Xmon qubit
with the coupling strength $g_{j}$ listed in Tab. S1.
The Xmon qubit is a variant of the transmon qubit, each with an individual
flux line for dynamically tuning its frequency and a microwave drive ($Q_{2}$, $Q_{4}$, and $Q_{6}$
share other qubits’ microwave lines in this experiment) for controllably
exciting its $|0\rangle\leftrightarrow|1\rangle$ transition.
The Xmon qubit reaches its maximum resonant frequency $\omega_{j}^{0}$ at the sweetpoint, where it is insensitive
to flux noise and exhibits the longest phase coherence time.
$\omega_{j}^{0}$ for $j=1$ to 10 in our device are around or slightly above $\omega_{B}$,
whose values are roughly estimated through the flux-biased spectroscopy measurement.
In this experiment all qubits are initialized to the ground
state $|0\rangle$ at their respective idle frequencies $\omega_{j}/2\pi$ that
spread in the range from 4.96 to 5.66 GHz, corresponding to 840 to 140 MHz below $\omega_{B}/2\pi$ (Tab. S1),
where single-qubit rotations and the qubit-state measurement are performed.
For entangling operations we dynamically bias all target qubits from
their idle frequencies $\omega_{j}$ to the interaction frequency $\omega_{I}$
for a specified interaction period,
following which we bias all these qubits back to their $\omega_{j}$ for measurement.
Qubit coherence performance at $\omega_{I}$ can be found in Tab. S1 and in Fig. S1.
1.2 XY control
Our instrument has 7 (expendable to more) independent xy signal channels controlled by digital analog converters (DACs):
3 channels are selected to output two tones
per channel and the rest 4 channels are programmed to output a single tone per channel,
for a total of xy controls with 10 tones targeting 10 qubits.
The 10 tones are generated with 10 sidebands mixed with a continuous microwave
whose carry frequency is $\omega_{c}/2\pi=5.324$ GHz.
Microwave leakage is minimized with this standard mixing method as calibrated by the room temperature electronics.
In our setup, $Q_{1}$ and $Q_{2}$ share $Q_{1}$’s on-chip xy line connecting to the 1st two-tone xy channel,
$Q_{3}$ and $Q_{4}$ share $Q_{3}$’s on-chip xy line connecting to the 2nd two-tone xy channel,
and $Q_{6}$ and $Q_{7}$ share $Q_{7}$’s on-chip xy line connecting to the 3rd two-tone xy channel.
For the two-tone xy signals that are supposed to simultaneously act on the two qubits,
we sequentially place two rotation pulses, each with a single tone
targeting one of the two qubits. Each $\pi/2$-rotation ($\pi$-rotation) pulse is 30 (60) ns-long
with a full width half maximum of 15 (30) ns.
We perform randomized benchmark (RB) on the two qubits simultaneously, and verify that
overlapping in time the two rotation pulses through the two-tone xy channel
is not a problem except for $Q_{6}$ and $Q_{7}$, which might be due to the large sideband used for $Q_{7}$.
The idle frequencies $\omega_{j}$ of the 10 qubits are detuned from each other to minimize
the microwave crosstalk during single-qubit rotations.
For two qubits sharing the same xy line, the cross-resonance interaction
reported elsewhere Chow2011 is not a major factor due to the large detuning.
For each qubit, we optimize the quadrature correction term with DRAG coefficient $\alpha$
to minimize leakage to higher levels Kelly2014 , yielding the $X/2$ ($\pi/2$ rotation around the $x$ axis)
gate fidelities no less than 0.998 for all 10 qubits as verified
by RB on each qubit (Tab. S1 and in Fig. S1).
We also select eight qubits and simultaneously perform RBs on them (see pulse sequences in Fig. S2(a)),
finding that the $X/2$ gate fidelities remain reasonably high, no less than 0.993
($Q_{7}$ is not selected due to the above-mentioned large sideband issue;
$Q_{8}$’s DAC control has a smaller memory, with a maximum sequence length only half of the others’).
We also perform RB simultaneously with shorter pulse sequences (smaller $m$-Number of gates) on $Q_{8}$ and $Q_{9}$,
two neighboring qubits with the idle frequencies being very close,
and find that the $X/2$ gate fidelities remain no less than 0.997.
We note that further optimization of the single-qubit gates are possible
by shortening the gate time and introducing a slight detuning to the xy pulse to minimize the phase error Kelly2014 .
We carry out the optimization procedure on $Q_{5}$ and obtain fidelities of
the typical single-qubit gates, $X$, $Y$, $\pm X/2$, and $\pm Y/2$, all above 0.999 (Fig. S2(b)).
We note that although single-qubit gates are optimized at the qubits’
idle frequencies $\omega_{j}$, the gate performance may slightly degrade after a big square pulse used to
tune the qubit frequency due to the finite rise-up time
on the order of a few to a few tens of nanoseconds for an ideally sharp step-edge.
In our experiment, after biasing the qubits
from the interaction frequency $\omega_{I}$ back to their idle frequencies $\omega_{j}$,
we wait for another 10 ns before applying single-qubit gates.
We also use the GHZ tomography data to benchmark the gate fidelities of our $\pi/2$ rotations.
For $N=7$, the density matrix of $|\Psi_{1}\rangle$ with four major elements in the $|\pm\rangle$ basis
has a fidelity of $0.796\pm 0.021$. After applying a $\pi/2$ rotation to each qubit we transform $|\Psi_{1}\rangle$ to
$|\Psi_{2}\rangle$ in the $|0\rangle$ and $|1\rangle$ basis, whose density
matrix is measured to exhibit a fidelity of $0.771\pm 0.022$ (each 7-qubit full tomography is done within 40 minutes). Therefore we estimate
that our $\pi/2$ rotation after the big square pulse for entanglement has an average fidelity around $(0.771/0.796)^{1/7}>0.995$,
while a more detailed numerical simulation suggests that the average $\pi/2$ gate fidelity is $>0.993$.
1.3 Z control
Our instrument has 10 (expendable to more) independent z signal channels controlled by DACs,
which give us the full capability of simultaneously
tuning all 10 qubits’ resonant frequencies.
To correct for the finite rise-up time of
an ideally sharp edge of a square pulse,
we generate a step-edge output from the DAC and
capture the waveform with a high-speed sampling oscilloscope.
The measured response of the step-edge gives two time constants
describing the response of the room-temperature wirings,
based on which we use de-convolution to correct for desired step-edge pulses.
Imperfection due to the cryogenic wirings are partially compensated
using the qubit’s transition frequency response influenced
by a step-edge z bias as a caliber Hofheinz2009 .
The effective z bias of a qubit due to a unitary bias applied to other qubits’ z lines is calibrated,
which yields the z-crosstalk matrix
$\tilde{M}_{z}$ as
$$\left[{\tiny\begin{smallmatrix}1&-0.023&-0.001&0.008&0.028&0.020&0.009&0.011&0%
.003&-0.008\\
-0.018&1&0.050&0.024&0.008&0.003&0.000&-0.002&0.005&-0.016\\
-0.021&-0.081&1&0.054&0.022&0.012&0.003&0.001&-0.004&-0.017\\
0.017&0.055&0.080&1&-0.024&-0.012&-0.003&0&0.003&0.013\\
0.016&0.019&0.009&-0.003&1&-0.015&-0.002&0.005&0.009&0.014\\
0.001&-0.002&-0.004&-0.005&-0.025&1&0.009&0.022&0.013&0.004\\
0.001&0&-0.003&-0.006&-0.028&-0.046&1&0.078&0.035&0.008\\
-0.004&0&0.001&0.003&0.013&0.020&0.025&1&-0.029&-0.006\\
-0.012&-0.006&-0.002&0&0.008&0.015&0.016&0.065&1&-0.014\\
0.002&0.011&0.015&0.015&0.029&0.023&0.010&0.008&-0.011&1\end{smallmatrix}}%
\right].$$
(S1)
With the z biases applied to the 10 qubits
written in a column format as $\tilde{z}_{\textrm{applied}}$
and the actual z biases sensed by these 10 qubits
written as $\tilde{z}_{\textrm{actual}}$, we have the mapping relation of $\tilde{z}_{\textrm{actual}}=\tilde{M}_{z}\cdot\tilde{z}_{\textrm{applied}}$.
The z crosstalks reach maximum at about 8% between two neighboring qubits.
We note that the z crosstalks may not contribute to the GHZ state errors
as we iteratively fine-tune the z bias of each qubit within a small range for optimal GHZ entanglement.
1.4 Qubit readout
Besides the above-mentioned 7 xy signal channels for the qubit control,
our instrument has an xy signal channel that can output a
readout pulse with multiple tones achieved by sideband mixing;
this readout pulse is captured by a room-temperature analog digital converter (ADC),
which simultaneously demodulates the multiple tones and returns a pair of $I$ and $Q$ values for each tone.
An impedance-transformed Josephson parametric amplifier (JPA) operating at 20 mK is used
before the ADC to enhance the signal-to-noise ratio.
The JPA was fabricated as described elsewhere Song2017 ,
whose signal-line impedance is continuously varied,
in a manner of the Klopfenstein taper, to transform
the environmental characteristic impedance from 50 to 15 $\Omega$ for
a bandwidth of more than 200 MHz centered around 6.72 GHz.
The JPA can be switched “ON” and “OFF” by turning on and off, respectively, an appropriate
pump tone that is about twice the signal frequency.
The signal transmission spectra
with the JPA in the states “ON” (red line) and “OFF” (blue line) are displayed in Fig. S3,
where the 10 dips correspond to the 10 readout resonators.
The amplification band of the JPA, identified by the vertical difference between
the red and blue lines, is tunable with a DC bias applied to the JPA.
The readout pulse is 1 $\mu$s-long, with the input tones and the power at each tone optimized for high-fidelity readout.
The $j$-th tone of the readout pulse, where $j$ is up to 10 in this experiment, populates $Q_{j}$’s readout resonator $R_{j}$ with
an average photon number of $n_{j}^{r}$ in 1 $\mu$s, which dispersively
interacts with $Q_{j}$ with the coupling strength $g_{j}^{r}$ and shifts $Q_{j}$’s frequency
downwards by an amount of $\delta\omega_{j}^{m}$.
Reversely, the qubit state affects the state of its readout resonator, which is
encoded in the $I$-$Q$ values at the tone $j$ of the transmitted readout pulse.
At the end of 1 $\mu$s, photons in $Q_{j}$’s readout resonator
leak into the circumferential transmission line at the
rate of $\kappa_{j}^{r}$, and the readout resonator returns to the ground state before the next sequence cycle starts.
Repeated readout signals amplified by the JPA are demodulated at room temperature, yielding
the $I$-$Q$ points at each tone on the complex plane
forming two blobs to differentiate the states $|0\rangle$ and $|1\rangle$ of each qubit
(see Tab. S1 and in Fig. S1).
The probabilities of correctly reading out each qubit
in $|0\rangle$ and $|1\rangle$ are listed in Tab. S1.
We note that the readout resonators of $Q_{3}$ and $Q_{4}$
are very close in frequency, and so are those of $Q_{6}$ and $Q_{7}$. We carefully
choose the readout tones and powers to minimize the readout crosstalk if $Q_{3}$ and $Q_{4}$ are both being measured,
which has a slight side-effect that the readout visibility of $Q_{3}$ drops a little bit compared with the case when $Q_{4}$ is not being measured (Fig. S3).
Nevertheless, our readout choice for minimizing the crosstalk is fully verified by preparing various product states
of $Q_{3}$, $Q_{4}$, and $Q_{5}$ with high-fidelity single-qubit gates and performing single- and two-qubit state tomography,
with the fidelities of all reconstructed density matrices being around or above 0.99.
1.5 $XX$-type crosstalk coupling
Due to insufficient crossover bonding wires to tie the ground segments on-chip,
we experience unwanted microwave crosstalk coupling between nearest-neighbor qubits.
The crosstalk coupling is calibrated by measuring the qubit-qubit energy swap process
around the interaction frequency $\omega_{I}$.
To understand the crosstalk coupling, we measure in detail
the energy swap process of $Q_{8}$ and $Q_{9}$ as a function of the qubit detuning from the resonator $\omega_{B}$,
with the result shown in Fig. S4(a) ($Q_{8}$ and $Q_{9}$ are chosen since they
are the nearest-neighbor qubits with two highest sweetpoint frequencies $\omega_{8}^{0}$ and $\omega_{9}^{0}$):
We excite $Q_{8}$ to $\ket{1}$ and then detune both $Q_{8}$ and $Q_{9}$ simultaneously to the same
detuning $\Delta$ from the resonator, with $\Delta$ being varied;
by subsequently monitoring the $|1\rangle$-state population of $Q_{8}$, $P_{1}$, as a function of the interaction time,
we obtain the energy-swap dynamics of the system at various detunings.
The qubit-qubit interaction strength can be inferred from the oscillation period of $P_{1}$.
We find that, in addition to the resonator mediated SE coupling $\lambda$ (see Eq. (2) of the main text) which changes sign across $\omega_{B}$,
a direct $XX$-type coupling with a magnitude of $\approx 2\pi\times 2.1$ MHz,
named as $\lambda_{8,9}^{c}$ for $Q_{8}$-$Q_{9}$, must be taken into account to explain our experimental data.
Figure S4(b) shows the Fourier transform of
the data in Fig. S4(a) along the $y$ axis,
based on which the net qubit-qubit interaction
as a function of $\Delta$ can be inferred, as shown in Fig. S4(d).
Overall the experimental data agree well with the numerical
simulation taking into account the extra $XX$-type coupling $\lambda^{c}$ (Figs. S4(c) and (d)).
The $XX$-type crosstalk couplings $\lambda_{j,j^{\prime}}^{c}$ between other nearest-neighbor qubits ($Q_{j}$ and $Q_{j^{\prime}}$) are roughly
estimated as the differences between the measured qubit-qubit coupling strengths and theoretical
SE interaction strengths, as given in Tab. S2.
Here numerical simulation is based on the Hamiltonian shown in Eq. (1) of the main text with the
additional nearest-neighbor crosstalk coupling term $\sum_{j,j^{\prime}}{\lambda_{j,j^{\prime}}^{c}\left(\sigma_{j}^{-}\sigma_{j^{%
\prime}}^{+}+\sigma_{j}^{+}\sigma_{j^{\prime}}^{-}\right)}$,
while the decoherence impact is included using the Lindblad master equation
taking into account the Markovian environment.Two characteristic decay times, the energy relaxation
time $T_{1}$ and the pure dephasing time $T_{\varphi}$, are used
for each qubit. However, the non-Markovian $1/f$ character of the phase noise
prevents us from directly using the $T_{2,j}^{\ast}$ values listed in
Tab. S1. Empirically we find that using $T_{\varphi}\geq 10T_{2,j}^{\ast}$
ensures that the numerical results agree reasonably well with the data in the first few hundred nanoseconds
(note that our GHZ sequences times are less than 150 ns).
The estimated $XX$-type nearest-neighbor couplings are included during the numerical optimization
of the parameters for generating GHZ states. It is find that
the introduction of the crosstalk couplings lowers the GHZ fidelities for $N>6$
but actually raises the fidelity for $N=10$.
2 Multiqubit GHZ phase calibration
For the multiqubit GHZ entanglement, it is critical that the phase of each qubit’s xy
drive is calibrated according to the rotating frame at $\omega_{I}$, after taking into account the
extra dynamical phases accumulated during the frequency adjustment of all qubits.
Here we follow the approach as done previously Zhong2016a : For simplicity we consider the product state of two qubits as
$\frac{1}{\sqrt{2}}\left(|0\rangle+|1\rangle\right)\otimes\frac{1}{\sqrt{2}}%
\left(|0\rangle+e^{i\varphi}|1\rangle\right)$,
where the extra $\varphi$ on the second qubit is seen right after the two qubits are placed on-resonance at $\omega_{I}$;
we intend to find a way to adjust $\varphi$ to be zero.
With the interaction Hamiltonian as $\hbar\lambda(|01\rangle\langle 10|+|10\rangle\langle 01|)$ (see Eq. (2) of the main text),
the amplitudes of $|01\rangle$ and $|10\rangle$
then oscillate in time, as described (in the rotating frame) by the unitary transformation
$U_{\textrm{int}}=\left[\begin{smallmatrix}1&0&0&0\\
0&\cos(\lambda t)&-i\sin(\lambda t)&0\\
0&-i\sin(\lambda t)&\cos(\lambda t)&0\\
0&0&0&1\\
\end{smallmatrix}\right]$.
At $t=\pi/4|\lambda|$, where $\lambda$ is negative, the two-qubit state evolves to
$|00\rangle/2+(e^{i\varphi}+i)/(2\sqrt{2})|01\rangle+(ie^{i\varphi}+1)/(2\sqrt{%
2})|10\rangle+e^{i\phi}|11\rangle/2$,
which gives equal probabilities for the four two-qubit computational states with $\varphi=0$.
Experimentally we choose $Q_{3}$ as the reference and adjust the phase of the other qubit’s microwave;
we perform the check pairwisely, with the data after all phase calibrations shown in Fig. S5.
3 10-qubit tomography
3.1 Effect of reduced sample size
The $N$-qubit tomography takes $3^{N}$ tomographic operations, and for each operation
$2^{N}$ occupation probabilities of the $N$-qubit computational states are measured.
Once a GHZ state is generated, a tomographic operation is appended, following which
the single-shot measurement yields a binary outcome for each qubit and for $N$ qubits simultaneously;
running the whole pulse sequence, which includes the state preparation, the tomographic operation, and the measurement,
once is one sampling event. Therefore
a sufficiently large sample size, i.e., repeating the same pulse sequence many times
for many synchronized binary outcomes of all $N$ qubits, is necessary to precisely count all $2^{N}$ probabilities,
which would significantly slow down the measurement.
For the multiqubit tomography, we maintain a fixed sample size of 3000, which is only about 3 times
$2^{N}$ when $N=10$; even in this case the full tomography measurement takes about 40 hours if uninterrupted,
and we have to constantly monitor our measurement to ensure that the system performance is reasonably stable (Fig. S1).
We note that for $N=10$ a huge number ($3^{10}$) of operations are involved, resulting in a set of over-constrained
equations to solve for the system’s density matrix, which may overcome the shortage of an insufficient sample size.
Furthermore, we carry out a test by performing the $N$-qubit tomography with a variable sample size for $N=6$ to 9.
The results show that, with the sample size around $3\times 2^{N}$, the reconstructed density matrix has a fidelity very close to
that with a sufficiently large sample size (Fig. S6).
3.2 Reducing the computation complexity
Here we use the unitary matrix $U^{j}$ to describe the $j$-th tomographic operation on the $N$-qubit system whose density matrix is $\rho$,
where both $U^{j}$ and $\rho$ are of size $2^{N}\times 2^{N}$.
The measured probability of the $k$-th computational state after the tomographic operation is therefore
$$P_{k}^{j}=\langle k|U^{j}\rho\left(U^{j}\right)^{\dagger}|k\rangle=\sum_{l,m=1%
}^{2^{N}}{U_{kl}^{j}\left(U_{km}^{j}\right)^{\ast}\rho_{lm}},$$
(S2)
where $k=1$ to $2^{N}$ indexes the $N$-qubit computational states.
Vectorization of the matrix $\rho$ by stacking its columns into a single column vector $\tilde{\rho}$,
we have $\tilde{P}^{j}=\tilde{U}^{j}\tilde{\rho}$, where $\tilde{P}^{j}$ is the vector format of $\{P_{1}^{j},\,P_{2}^{j},\,\ldots,\,P_{N}^{j}\}$ and
$\tilde{U}^{j}$ is a $2^{N}\times 4^{N}$ matrix replacing the summation terms of $U_{kl}^{j}$ and $(U_{km}^{j})^{\ast}$ in Eq. (S2).
Stacking all tomographic operation matrices $\tilde{U^{j}}$ and all measured probabilities
$\tilde{P}^{j}$ for $j=1$ to $3^{N}$ into $\tilde{U}$ and $\tilde{P}$, respectively,
we obtain the linear equations of $\tilde{U}\tilde{\rho}=\tilde{P}$, which is used to solve for $\tilde{\rho}$ given $\tilde{P}$ and $\tilde{U}$.
$\tilde{U}$ is a column full rank matrix of size $6^{N}\times 4^{N}$. When $N$ approaches 10,
it becomes extremely difficult to fully load $\tilde{U}$ into a computer’s memory and solve for $\tilde{\rho}$.
Fortunately, only a small fraction of $\tilde{U}$’s elements are
non-zero, so that we can use sparse matrix for storage and employ the pseudo-inverse method.
With $\tilde{U}^{\dagger}$ as the Hermitian conjugate of $\tilde{U}$, we have
$\tilde{U}^{\dagger}\tilde{U}\tilde{\rho}=\tilde{U}^{\dagger}\tilde{P}$,
where $\tilde{U}^{\dagger}\tilde{U}$ is a symmetric and positive definite matrix of size $4^{N}\times 4^{N}$.
$\tilde{U}^{\dagger}\tilde{U}$ is not only smaller in size, but also more sparse than $\tilde{U}$,
which greatly reduce the complexity when solving the equations.
Here we quote the time complexity to quantitatively describe the advantage of using $\tilde{U}^{\dagger}\tilde{U}$.
For a general full rank matrix of size $4^{N}\times 4^{N}$, the time complexity involved in computing
the inverse operation is $O\left(\left(4^{N}\right)^{3}\right)$.
For comparison, $\tilde{U}^{\dagger}\tilde{U}$ has non-zero elements only at the indices of
$\left[(k-1)\times 2^{N}+k+l,\,1+l\right]$ and $\left[1+l,\,(k-1)\times 2^{N}+k+l\right]$,
where $k=1,\,2,\,\ldots,\,2^{N}$
and $l$ can be any non-negative integers for the indices to be valid.
The number of non-zero elements in each row of $\tilde{U}^{\dagger}\tilde{U}$ is less than $2^{N}$, and thus
the number of column elementary operations needed for each row is less than $2^{N}$ during matrix inversion;
the total number of operations for the $4^{N}$ rows is less than $2^{N}\times 2^{N}\times 4^{N}$.
We conclude that the time complexity of solving the inverse matrix of $\tilde{U}^{\dagger}\tilde{U}$ is $O\left(\left(4^{N}\right)^{2}\right)$.
3.3 10-qubit $\rho$ in the $|0\rangle$ and $|1\rangle$ basis
The GHZ density matrix shown in Fig. 3 of the main text has four major elements in the $|\pm\rangle$ basis, while our
measurement is in the $|0\rangle$ and $|1\rangle$ basis. Here we show the partial matrix elements
for the 10-qubit GHZ density matrix in the $|0\rangle$ and $|1\rangle$ basis. It is seen that all matrix elements of $\rho$
are no higher than 0.003 in amplitude (Fig. S7).
References
(S1)
C. Song et al., submitted (2017).
(S2)
M. Steffen et al., Science 313, 1423 (2006).
(S3)
J. M. Chow et al., Phys. Rev. Lett. 107, 080502 (2011).
(S4)
J. Kelly et al., Phys. Rev. Lett. 112, 240504 (2014).
(S5)
E. Jeffrey et al., Phys. Rev. Lett. 112, 190504 (2014).
(S6)
M. Hofheinz et al., Nature 459, 546 (2009).
(S7)
Y. P. Zhong et al., Phys. Rev. Lett. 117, 110501 (2016). |
Algebraic non-hyperbolicity of hyperkähler manifolds
with Picard rank greater than one
Ljudmila Kamenova, Misha Verbitsky111Partially supported
by RSCF grant 14-21-00053 within AG Laboratory NRU-HSE.
Abstract.
A projective manifold is algebraically hyperbolic if the degree of any curve
is bounded from above by its genus times a constant, which is independent
from the curve.
This is a property which follows from Kobayashi hyperbolicity.
We prove that hyperkähler manifolds are not algebraically
hyperbolic when the Picard rank is at least 3, or if
the Picard rank is 2 and the SYZ conjecture on existence
of Lagrangian fibrations is true.
We also prove that if the automorphism group of a hyperkähler
manifold is infinite then it is algebraically non-hyperbolic.
1 Introduction
In [V1] M. Verbitsky proved that all hyperkähler
manifolds are Kobayashi non-hyperbolic. It is interesting to inquire if
projective hyperkähler manifolds are also algebraically
non-hyperbolic (2).
For a given projective manifold algebraic non-hyperbolicity
implies Kobayashi non-hyperbolicity. We prove algebraic non-hyperbolicity
for projective hyperkähler manifolds with infinite group of
automorphisms.
Theorem 1.1: Let $M$ be a projective hyperkähler manifold with infinite automorphism
group. Then $M$ is algebraically non-hyperbolic.
If a projective hyperkähler manifold has Picard rank at least three,
we show that it is algebraically non-hyperbolic. For the case when the
Picard rank equals to two we need an extra assumption in order to prove
algebraic non-hyperbolicity. The SYZ conjecture states that a nef
parabolic line bundle on a hyperkähler manifold gives rise to a
Lagrangian fibration (2).
Theorem 1.2: Let $M$ be a projective hyperkähler manifold with Picard rank $\rho$.
Assume that either $\rho>2$, or $\rho=2$ and
the SYZ conjecture holds.
Then $M$ is algebraically non-hyperbolic.
2 Basic notions
Definition 2.1: A hyperkähler manifold of maximal holonomy
(or irreducible holomorphic symplectic)
manifold
$M$ is a compact complex Kähler manifold with $\pi_{1}(M)=0$ and
$H^{2,0}(M)={\mathbb{C}}\sigma$, where $\sigma$ is everywhere non-degenerate.
From now on we would tacitly assume that hyperkähler manifolds are
of maximal holonomy.
Due to results of Matsushita, holomorphic maps from hyperkähler
manifolds are quite restricted.
Theorem 2.2: (Matsushita, [Mat])
Let $M$ be a hyperkähler manifold and $f\colon M\rightarrow B$ a
proper surjective morphism with a smooth base $B$. Assume that $f$ has
connected fibers and $0<\dim B<\dim M$. Then $f$ is Lagrangian
and $\dim_{{\mathbb{C}}}B=n$, where $\dim_{{\mathbb{C}}}M=2n$.
Following 2, we call the surjective morphism
$f\colon M\rightarrow B$ a Lagrangian fibration on the
hyperkähler manifold $M$.
A dominant map $f\colon M\dashrightarrow B$
is a rational Lagrangian fibration if there exists a
birational map $\varphi\colon M\dasharrow M^{\prime}$ between hyperkähler
manifolds such that the composition
$f\circ\varphi^{-1}\colon M^{\prime}\rightarrow B$ is
a Lagrangian fibration. J.-M. Hwang proved that if the base $B$ of a
hyperkähler Lagrangian fibration is smooth, then $B\cong{\mathbb{P}}^{n}$
(see [Hw]).
Definition 2.3: Given a hyperkähler manifold $M$, there is a non-degenerate primitive
form $q$ on $H^{2}(M,{\mathbb{Z}})$, called the Beauville-Bogomolov-Fujiki form
(or the “BBF form” for short),
of signature $(3,b_{2}-3)$, and satisfying the Fujiki relation
$$\int_{M}\alpha^{2n}=c\cdot q(\alpha)^{n}\qquad\text{for }\alpha\in H^{2}(M,{\mathbb{Z}}),$$
with $c>0$ a constant depending on the topological type of $M$.
This form generalizes the intersection pairing on K3 surfaces.
A detailed description of the form can be found in
[Be], [Bog] and [F].
Notice that given a Lagrangian fibration $f\colon M\rightarrow{\mathbb{P}}^{n}$,
if $h$ is the hyperplane class on $\mathbb{P}^{n}$, and $\alpha=f^{*}h$,
then $\alpha$ belongs to the birational Kähler cone of $M$ and
$q(\alpha)=0$. The following SYZ conjecture states that the converse
is also true.
Conjecture 2.4: (Tyurin, Bogomolov, Hassett-Tschinkel,
Huybrechts, Sawon)
If $L$ is a line bundle on a hyperkähler manifold $M$ with $q(L)=0$,
and such that $c_{1}(L)$ belongs to the birational Kähler cone of $M$,
then $L$ defines a rational Lagrangian fibration.
For more reference on this conjecture, please see
[HT],
[Saw], [Hu3] and [V2].
This conjecture is known for deformations of Hilbert schemes of points on
K3 surfaces (Bayer–Macrì [BM]; Markman [Mar]),
and for deformations of the generalized Kummer varieties $K_{n}(A)$
(Yoshioka [Y]).
In [V1] M. Verbitsky proved that all hyperkähler
manifolds are Kobayashi non-hyperbolic. In [KLV] together with
S. Lu we proved that the Kobayashi pseudometric vanishes
identically for K3 surfaces and for hyperkähler manifolds deformation
equivalent to Lagrangian fibrations under some mild assumptions.
In [De] Demailly introduced the following notion.
Definition 2.5:
A projective manifold $M$ is algebraically hyperbolic
if for any Hermitian metric $h$ on $M$ there exists a constant $A>0$
such that for any holomorphic map $\varphi\colon C\rightarrow M$
from a curve of genus $g$ to $M$ we have that
$2g-2\geqslant A\int_{C}\varphi^{\ast}\omega_{h},$ where $\omega_{h}$ is the Kähler
form of $h$.
In this paper all varieties we
consider are smooth and projective. For projective varieties,
Kobayashi hyperbolicity implies algebraic hyperbolicity ([De]).
Here we explore non-hyperbolic
properties of projective hyperkähler manifolds. Algebraic non-hyperbolicity
implies Kobayashi non-hyperbolicity.
3 Main Results
Proposition 3.1:
Let $M$ be a hyperkähler manifold admitting a (rational) Lagrangian
fibration. Then $M$ is algebraically non-hyperbolic.
Proof:
We use the fact that the fibers of a Lagrangian fibrations are
abelian varieties ([Mat]). The isogeny self-maps
on an abelian variety provide curves of fixed genus and arbirary large
degrees, and therefore they are algebraically non-hyperbolic.
An alternative way of proving this proposition is by using the
following result whose proof was suggested by Prof. K. Oguiso.
Lemma 3.2:
If a hyperkähler manifold $M$ admits a Lagrangian fibration,
then there exists a rational curve on $M$.
Indeed,
in [HO] J.-M. Hwang and K. Oguiso give a Kodaira-type classification
of the general singular fibers of a holomorphic Lagrangian fibration.
All of the general singular fibers are covered by rational curves.
The locus of singular fibers is non-empty (e.g., Proposition 4.1 in
[Hw]), and therefore there is a rational curve on $M$.
According to 3, $M$ contains a rational curve, and
therefore, $M$ is algebraically non-hyperbolic.
This finishes the proof of 3.
Lemma 3.3:
Let $M$ be a projective hyperkähler manifold with infinite automorphism
group $\Gamma$. Consider the natural map $f:\Gamma{\>\longrightarrow\>}\operatorname{Aut}(H^{1,1}(M))$.
Then the elements of the Kähler cone have infinite orbits with respect to
$f(\Gamma)$.
Proof:
See the discussion in section 2 of [O2].
Lemma 3.4:
Let $M$ be a projective hyperkähler manifold, and $\Gamma$ its
automorphism group. Consider the natural map
$g:\Gamma{\>\longrightarrow\>}\operatorname{Aut}(H_{tr}^{2}(M))\times\operatorname{Aut}(H^{1,1}(M))$. Then $g(\Gamma)$
is finite in the first component $\operatorname{Aut}(H_{tr}^{2}(M))$.
Proof:
This has been proven by Oguiso, see [O1].
The idea is that the BBF form restricted to the transcendental part
$H_{tr}^{2}(M)$ is of K3-type. Then we can apply Zarhin’s theorem
(Theorem 1.1.1 in [Z]) to deduce that $g(\Gamma)\subset\operatorname{Aut}(H_{tr}^{2}(M))$ is finite.
Theorem 3.5:
Let $M$ be a projective hyperkähler manifold with infinite automorphism
group. Then $M$ is algebraically non-hyperbolic.
Proof:
In the notations introduced above,
for any Kähler class $w$ on $M$, its $f(\Gamma)$-orbit is infinite by
3.
Fix a polarization $w$ on $M$ with normalization $q(w)=1$.
For a given constant $C>0$ consider the set
$${\cal D}_{C}=\{x\in H^{1,1}(M,{\mathbb{Z}})\ \ |\ \ q(x)\geqslant 0,\ \ q(x,w)\leqslant C\}.$$
Notice that ${\cal D}_{C}$ is compact.
Indeed, $y=x-q(x,w)w$ is orthogonal to $w$ with respect to the BBF
form $q$. The quadratic form $q$ is of type $(1,\rho-1)$ on
$H^{1,1}(M,{\mathbb{Z}})$ and since $q(w)>0$, the restriction
$q|_{w^{\perp}}$ is negative-definite. A direct computation shows that
$q(y)=q(x)-2q(x,w)^{2}+q(x,w)^{2}q(w)=q(x)-q(x,w)^{2}\geqslant-C^{2}$.
The set ${\cal D}_{C}$ is equivalent to the set of elements
$\{y\in w^{\perp}|q(y)\geqslant-C^{2}\}$, which is compact because
$q|_{w^{\perp}}$ is negative-definite.
Since the set ${\cal D}_{C}$ is compact,
$\sup_{x\in\Gamma\cdot\eta}\deg x=\infty$, which means
there is a class of a curve $\eta$ with $q(\eta)>0$. However, all
curves in the orbit $\Gamma\cdot\eta$ have constant genus.
Since their degrees could be arbitrarily high, then $M$ is algebraically
non-hyperbolic.
Lemma 3.6:
Let $M$ be a hyperkähler manifold such that the positive cone
does not coincide with the Kähler cone. Then $M$ contains a
rational curve.
Proof:
This is a classical result that Boucksom and Huybrechts knew in the
early 2000’s
[Bou, Hu2].
Theorem 3.7: Let $M$ be a hyperkähler manifold with Picard rank $\rho$.
Assume that either $\rho>2$ or $\rho=2$ and the SYZ conjecture holds.
Then $M$ is algebraically non-hyperbolic.
Proof:
Notice that the Hodge lattice $H^{1,1}(M,{\mathbb{Z}})$ of a hyperkähler manifold
has signature $(1,k)$. Therefore, for $\rho\geqslant 2$, the Hodge lattice
contains a vector with positive square, and $M$ is projective
([Hu1]).
First, consider the case when $\rho>2$. If the Kähler cone
coincides with the positive cone, then the automorphism group
$\operatorname{Aut}(M)$ is commensurable with the group of
isometries $SO(H^{2}(M,{\mathbb{Z}}))$ (Theorem 2.17 in [AV])
preserving the Hodge type. By 3, this group
is commensurable with the group of isometries of the Hodge lattice
$H^{1,1}(M,{\mathbb{Z}})$.
By Borel and Harish-Chandra’s theorem ([BHC]), if $\rho>2$,
any arithmetic subgroup of $SO(1,\rho-1)$ is a lattice.
However, Borel density theorem implies that any lattice in a non-compact simple Lie group
is Zariski dense ([Bor]).
Therefore, for $\rho>2$, $SO(H^{1,1}(M,{\mathbb{Z}}))$ is infinite.
In this case $\operatorname{Aut}(M)$ is also infinite and
we can apply 3. On the other hand, if the Kähler cone does not
coincide with the positive cone, then by 3 there
is a rational curve on $M$. Therefore, $M$ is algebraically non-hyperbolic.
Now let $\rho=2$. Assume the positive cone and the Kähler cone
coincide. If there is no $\eta\in H^{1,1}(M,{\mathbb{Z}})$ with $q(\eta)=0$,
then by Theorem 87 in
[Di], $SO(H^{1,1}(M,{\mathbb{Z}}))$ is isomorphic to ${\mathbb{Z}}\times{\mathbb{Z}}/2{\mathbb{Z}}$.
Therefore, both $SO(H^{1,1}(M,{\mathbb{Z}}))$ and $\operatorname{Aut}(M)$ are infinite and
we can apply 3.
If there is $\eta\in H^{1,1}(M,{\mathbb{Z}})$ with $q(\eta)=0$, then
the SYZ conjecture implies that $\eta$ defines a rational fibration
on $M$ and we could apply 3.
If $\rho=2$ and the positive and the Kähler cones are
different (i.e., the positive cone is divided into Kähler chambers),
then there is a nef class $\eta\in H^{1,1}(M,{\mathbb{Z}})$ with
$q(\eta)=0$. Since we assumed that the SYZ conjecture holds, the class
$\eta$ defines a Lagrangian fibration on $M$. Applying
3 we conclude that $M$ is
algebraically non-hyperbolic.
Remark 3.8: We conjecture that all projective hyperkähler manifolds
are algebraically non-hyperbolic. However, our proof fails for manifolds
with Picard rank 1.
Acknowledgments. This work was inspired by a question of Erwan
Rousseau about algebraic non-hyperbolicity of hyperkähler manifolds.
The paper was started while the first-named author was visiting
Université libre de Bruxelles and she is grateful to Joel Fine for
his hospitality. It was finished at the SCGP during the second-named author’s
stay there. We are grateful to the SCGP for making this possible.
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Ljudmila Kamenova
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Misha Verbitsky
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Université libre de Bruxelles, CP 218,
Bd du Triomphe, 1050 Brussels, Belgium |
The Minimal Polynomial over $\mathbb{F}_{q}$ of
Linear Recurring Sequence over $\mathbb{F}_{q^{m}}$
††thanks: This research is supported in part by the National Natural Science Foundation of
China under the Grant 60872025.
Zhi-Han Gao
and Fang-Wei Fu
Z.-H. Gao is with the Chern Institute of
Mathematics, Nankai University, Tianjin 300071, P.R. China. E-mail: gaulwy@mail.nankai.edu.cnF.-W. Fu is with the Chern Institute of
Mathematics and the Key Laboratory of Pure Mathematics and
Combinatorics, Nankai University, Tianjin 300071, P.R. China. Email:
fwfu@nankai.edu.cn
()
Abstract
Recently, motivated by the study of vectorized stream cipher
systems, the joint linear complexity and joint minimal polynomial of
multisequences have been investigated. Let $\mathcal{S}$ be a linear
recurring sequence over finite field $\mathbb{F}_{q^{m}}$ with minimal
polynomial $h(x)$ over $\mathbb{F}_{q^{m}}$. Since $\mathbb{F}_{q^{m}}$
and $\mathbb{F}_{q}^{m}$ are isomorphic vector spaces over the finite
field $\mathbb{F}_{q}$, $\mathcal{S}$ is identified with an $m$-fold
multisequence ${\bf S}^{(m)}$ over the finite field $\mathbb{F}_{q}$.
The joint minimal polynomial and joint linear complexity of the
$m$-fold multisequence ${\bf S}^{(m)}$ are the minimal polynomial
and linear complexity over $\mathbb{F}_{q}$ of $\mathcal{S}$
respectively. In this paper, we study the minimal polynomial and
linear complexity over $\mathbb{F}_{q}$ of a linear recurring sequence
$\mathcal{S}$ over $\mathbb{F}_{q^{m}}$ with minimal polynomial $h(x)$
over $\mathbb{F}_{q^{m}}$. If the canonical factorization of $h(x)$ in
$\mathbb{F}_{q^{m}}[x]$ is known, we determine the minimal polynomial
and linear complexity over $\mathbb{F}_{q}$ of the linear recurring
sequence $\mathcal{S}$ over $\mathbb{F}_{q^{m}}$.
Keywords: Linear recurring sequences, minimal polynomial,
linear complexity, multisequences, joint minimal polynomial, joint
linear complexity.
AMS Classifications: 94A55, 94A60
1 Introduction
Let $\mathbb{F}_{q^{m}}$ be a finite field with $q^{m}$ elements, which
contains a subfield $\mathbb{F}_{q}$ with $q$ elements. Let
$\mathcal{S}=(s_{0},s_{1},\ldots,s_{n},\ldots)$ be a linear recurring
sequence over $\mathbb{F}_{q^{m}}$. The monic polynomial
$f(x)=a_{0}+a_{1}x+\cdots+a_{n-1}x^{n-1}+x^{n}\in\mathbb{F}_{q^{m}}[x]$ is
called a characteristic polynomial over $\mathbb{F}_{q^{m}}$ of
$\mathcal{S}$ if
$$a_{0}s_{k}+a_{1}s_{k+1}+a_{2}s_{k+2}+\cdots+a_{n-1}s_{k+n-1}+s_{k+n}=0,\ \ \ %
\mbox{for all}\ k\geq 0.$$
If the characteristic polynomial $f(x)$
is a polynomial over $\mathbb{F}_{q}$, that is, all $a_{i}\in\mathbb{F}_{q}$, we call $f(x)$ a characteristic polynomial over
$\mathbb{F}_{q}$ of $\mathcal{S}$. Since the linear recurring sequence
$\mathcal{S}$ over $\mathbb{F}_{q^{m}}$ is ultimately periodic, a
characteristic polynomial over $\mathbb{F}_{q}$ of $\mathcal{S}$ does
exist. The minimal polynomial over $\mathbb{F}_{q^{m}}$ (resp.
$\mathbb{F}_{q}$) of $\mathcal{S}$ is the uniquely determined
characteristic polynomial over $\mathbb{F}_{q^{m}}$ (resp.
$\mathbb{F}_{q}$) of $\mathcal{S}$ with least degree. The linear
complexity over $\mathbb{F}_{q^{m}}$ (resp. $\mathbb{F}_{q}$) of
$\mathcal{S}$ is the degree of the minimal polynomial over
$\mathbb{F}_{q^{m}}$ (resp. $\mathbb{F}_{q}$) of $\mathcal{S}$. Let
$h(x)$ be the minimal polynomial over $\mathbb{F}_{q^{m}}$ of
$\mathcal{S}$. It is known that $h(x)|f(x)$ for any characteristic
polynomial $f(x)$ over $\mathbb{F}_{q^{m}}$ of $\mathcal{S}$.
Similarly, let $H(x)$ be the minimal polynomial over
$\mathbb{F}_{q}$ of $\mathcal{S}$, we have $H(x)|f(x)$ for any
characteristic polynomial $f(x)$ over $\mathbb{F}_{q}$ of
$\mathcal{S}$. Note that a characteristic polynomial $f(x)$ over
$\mathbb{F}_{q}$ of $\mathcal{S}$ is also a characteristic
polynomial over $\mathbb{F}_{q^{m}}$ of $\mathcal{S}$. Hence,
$h(x)|f(x)$ for any characteristic polynomial $f(x)$ over
$\mathbb{F}_{q}$ of $\mathcal{S}$. In particular, $h(x)|H(x)$.
Similarly, for any $m$-fold multisequence ${\bf S}^{(m)}=(S_{1},S_{2},\ldots,S_{m})$ over $\mathbb{F}_{q}$, the monic
polynomial $g(x)\in\mathbb{F}_{q}[x]$ is called a joint characteristic
polynomial of ${\bf S}^{(m)}$ if $g(x)$ is a characteristic
polynomial of $S_{j}$ for each $1\leq j\leq m$. The joint minimal
polynomial of ${\bf S}^{(m)}$ is the uniquely determined joint
characteristic polynomial of ${\bf S}^{(m)}$ with least degree, and
the joint linear complexity of ${\bf S}^{(m)}$ is the degree of the
joint minimal polynomial of ${\bf S}^{(m)}$. Since
$\mathbb{F}_{q^{m}}$ and $\mathbb{F}_{q}^{m}$ are isomorphic vector
spaces over the finite field $\mathbb{F}_{q}$, a linear recurring
sequence $\mathcal{S}$ over $\mathbb{F}_{q^{m}}$ is identified with an
$m$-fold multisequence ${\bf S}^{(m)}$ over $\mathbb{F}_{q}$. It is
well known that the joint minimal polynomial and joint linear
complexity of the $m$-fold multisequence ${\bf S}^{(m)}$ are the
minimal polynomial and linear complexity over $\mathbb{F}_{q}$ of
$\mathcal{S}$ respectively.
The linear complexity of sequences is one of the important security
measures for stream cipher systems (see [2], [5],
[26], [27]). For a general introduction to the theory of
linear feedback shift register sequences, we refer the reader to
[13, Chapter 8] and the references therein. The linear
complexity of sequences has been extensively studied by many
researchers. For a recent survey paper, see Niederreiter [21].
The notion of linear complexity over $\mathbb{F}_{q}$ of linear
recurring sequences over $\mathbb{F}_{q^{m}}$ was introduced by Ding,
Xiao and Shan in [5], and discussed by some authors, for
example, see [1], [12], [14]-[18],
[20], [21], [23]. Recently, in the study of
vectorized stream cipher systems, the joint linear complexity of
multisequences has been extensively investigated (see [3],
[4], [6]-[11], [14]-[25],
[28]-[30]).
In this paper, we study the minimal polynomial and linear complexity
over $\mathbb{F}_{q}$ of a linear recurring sequence $\mathcal{S}$
over $\mathbb{F}_{q^{m}}$ with minimal polynomial $h(x)$ over
$\mathbb{F}_{q^{m}}$. If the canonical factorization of $h(x)$ in
$\mathbb{F}_{q^{m}}[x]$ is known, we determine the minimal polynomial
and linear complexity over $\mathbb{F}_{q}$ of the linear recurring
sequence $\mathcal{S}$ over $\mathbb{F}_{q^{m}}$. The rest of the
paper is organized as follows. In Section 2 we introduce and
give some results on linear recurring sequences that will be used in
this paper. In Section 3 we introduce a ring automorphism of
the polynomial ring $\mathbb{F}_{q^{m}}[x]$. We derive some results on
this polynomial ring automorphism that are crucial to establish the
main results in this paper. In Section 4 we determine the
minimal polynomial and linear complexity over $\mathbb{F}_{q}$ of a
linear recurring sequence $\mathcal{S}$ over $\mathbb{F}_{q^{m}}$ with
minimal polynomial $h(x)$ over $\mathbb{F}_{q^{m}}$. In Section
5 we give a new proof for the lower bound of Meidl and
Özbudak [17] on the linear complexity over
$\mathbb{F}_{q^{m}}$ of linear recurring sequence $\mathcal{S}$ over
$\mathbb{F}_{q^{m}}$ with given minimal polynomial $g(x)$ over
$\mathbb{F}_{q}$. We show that this lower bound is tight if and only
if the minimal polynomial over $\mathbb{F}_{q^{m}}$ of $\mathcal{S}$
is in certain form.
2 Linear Recurring Sequences
Let $f(x)$ be a monic polynomial over $\mathbb{F}_{q}$. Denote
$\mathcal{M}(f(x))$ the set of all linear recurring sequences over
$\mathbb{F}_{q}$ with characteristic polynomial $f(x)$. Note that
$\mathcal{M}(f(x))$ is a vector space over $\mathbb{F}_{q}$ with
dimension $\mbox{deg}(f(x))$. We need the following results on
linear recurring sequences from [13]:
Theorem 1
[13, Theorem 8.55] Let $f_{1}(x),\ldots,f_{k}(x)$ be monic polynomials over $\mathbb{F}_{q}$. If $f_{1}(x),\ldots,f_{k}(x)$ are pairwise relatively prime, then the vector space
$\mathcal{M}(f_{1}(x)\cdots f_{k}(x))$ is the direct sum of the
subspaces $\mathcal{M}(f_{1}(x)),\cdots,\mathcal{M}(f_{k}(x))$, that is
$$\mathcal{M}(f_{1}(x)\cdots f_{k}(x))=\mathcal{M}(f_{1}(x))\dotplus\cdots%
\dotplus\mathcal{M}(f_{k}(x)).$$
Theorem 2
[13, Theorem 8.57] Let $S_{1},S_{2},\ldots,S_{k}$ be linear recurring sequences over $\mathbb{F}_{q}$.
The minimal polynomials over $\mathbb{F}_{q}$ of $S_{1},S_{2},\ldots,S_{k}$ are $h_{1}(x),h_{2}(x),\ldots,h_{k}(x)$ respectively. If $h_{1}(x),h_{2}(x),\ldots,h_{k}(x)$ are pairwise relatively prime, then the
minimal polynomial over $\mathbb{F}_{q}$ of $\sum_{i=1}^{k}S_{i}$ is
the product of $h_{1}(x),h_{2}(x),\ldots,h_{k}(x)$.
It is easy to extend this result to the following case:
Lemma 1
Let $\mathcal{S}_{1},\mathcal{S}_{2},\ldots,\mathcal{S}_{k}$ be linear recurring sequences over
$\mathbb{F}_{q^{m}}$. The minimal polynomials over $\mathbb{F}_{q}$ of
$\mathcal{S}_{1},\mathcal{S}_{2},\ldots,\mathcal{S}_{k}$ are $H_{1}(x),H_{2}(x),\ldots,H_{k}(x)$ respectively. If $H_{1}(x),H_{2}(x),\ldots,H_{k}(x)$ are pairwise relatively prime over $\mathbb{F}_{q}$, then
the minimal polynomial over $\mathbb{F}_{q}$ of
$\sum_{i=1}^{k}\mathcal{S}_{i}$ is the product of
$H_{1}(x),H_{2}(x),\ldots,H_{k}(x)$.
Now we establish the following lemma which will be used in this paper:
Lemma 2
Let $S$ be a linear recurring sequence over
$\mathbb{F}_{q}$. The minimal polynomial over $\mathbb{F}_{q}$ of
$S$ is given by $h(x)=h_{1}(x)h_{2}(x)\cdots h_{k}(x)$ where $h_{1}(x),h_{2}(x),\ldots,h_{k}(x)$ are monic polynomials over $\mathbb{F}_{q}$.
If $h_{1}(x),h_{2}(x),\ldots,h_{k}(x)$ are pairwise relatively prime,
then there uniquely exist sequences $S_{1},S_{2},\ldots,S_{k}$ over
$\mathbb{F}_{q}$ such that
$$S=S_{1}+S_{2}+\cdots+S_{k}$$
and the minimal polynomials over $\mathbb{F}_{q}$ of
$S_{1},S_{2},\ldots,S_{k}$ are $h_{1}(x),h_{2}(x),\ldots,h_{k}(x)$
respectively.
Proof:
By Theorem 1, we have
$$\mathcal{M}(h(x))=\mathcal{M}(h_{1}(x))\dotplus\cdots\dotplus\mathcal{M}(h_{k}%
(x)).$$
Then, there uniquely exist sequences $S_{1},S_{2},\ldots,S_{k}$ over
$\mathbb{F}_{q}$ such that $S_{j}\in\mathcal{M}(h_{j}(x))$ and
$$S=S_{1}+S_{2}+\cdots+S_{k}.$$
Assume that the minimal polynomial over $\mathbb{F}_{q}$ of $S_{j}$ is
$h_{j}^{{}^{\prime}}(x)$ which is a divisor of $h_{j}(x)$ for $1\leq j\leq k$. By
Theorem 2, the minimal polynomial over $\mathbb{F}_{q}$ of
$S$ is $\prod_{j=1}^{k}h_{j}^{{}^{\prime}}(x)$. Thus,
$$h_{1}^{{}^{\prime}}(x)h_{2}^{{}^{\prime}}(x)\cdots h_{k}^{{}^{\prime}}(x)=h_{1%
}(x)h_{2}(x)\cdots h_{k}(x).$$
Since $h_{j}^{{}^{\prime}}(x)|h_{j}(x)$ for $1\leq j\leq k$, we have
$$h_{j}^{{}^{\prime}}(x)=h_{j}(x),\;\;1\leq j\leq k,$$
which completes the proof.
3 Polynomial Ring Automorphism
We define $\sigma$ to be a mapping from the polynomial ring
$\mathbb{F}_{q^{m}}[x]$ to itself as follows: For
$f(x)=a_{0}+a_{1}x+\cdots+a_{n}x^{n}\in\mathbb{F}_{q^{m}}[x]$,
$$\sigma:\mathbb{F}_{q^{m}}[x]\longrightarrow\mathbb{F}_{q^{m}}[x],$$
$$f(x)\longrightarrow\sigma(f(x))$$
where $\sigma(f(x))=a_{0}^{q}+a_{1}^{q}x+\cdots+a_{n}^{q}x^{n}$. It is easy to see
that $\sigma$ is a ring automorphism of $\mathbb{F}_{q^{m}}[x]$.
Throughout the paper, we will use the fact that
$$\sigma(f(x)g(x))=\sigma(f(x))\sigma(g(x)),\;\;\mbox{for any}\;f(x),g(x)\in%
\mathbb{F}_{q^{m}}[x].$$
Denote $\sigma^{(k)}$ the $k$th usual
composition of $\sigma$. Note that $\sigma^{(0)}$ is the identity
mapping. Since $a^{q^{m}}=a$ for any $a\in\mathbb{F}_{q^{m}}$, we have
$\sigma^{(m)}(f(x))=f(x)$. Denote $k(f)$ the minimum positive
integer $k$ such that $\sigma^{(k)}(f(x))=f(x)$.
Lemma 3
For any $f(x)\in\mathbb{F}_{q^{m}}[x]$ and positive integer $l$,
$\sigma^{(l)}(f(x))=f(x)$ if and only if $k(f)|l$.
Proof:
It is easy to see that $\sigma^{(l)}(f(x))=f(x)$ if $k(f)|l$.
On the other hand, if $\sigma^{(l)}(f(x))=f(x)$, we assume that
$l=k(f)w+r$ and $0\leq r<k(f)$. Then
$$f(x)=\sigma^{(l)}(f(x))=\sigma^{(r)}(\sigma^{(k(f)w)}(f(x)))=\sigma^{(r)}(f(x)).$$
Hence, $r=0$ by the definition of $k(f)$. Therefore, $k(f)|l$.
Now we define an equivalence relation $\stackrel{{\scriptstyle\sigma}}{{\sim}}$ on
$\mathbb{F}_{q^{m}}[x]$: $f(x)\stackrel{{\scriptstyle\sigma}}{{\sim}}g(x)$ if and
only if there exists positive integer $j$ such that
$\sigma^{(j)}(f(x))=g(x)$. The equivalence classes induced by this
equivalence relation $\stackrel{{\scriptstyle\sigma}}{{\sim}}$ are called
$\sigma$-equivalence classes.
Lemma 4
Let $f(x)$ be a polynomial over $\mathbb{F}_{q^{m}}$.
Then $\sigma(f(x))$ is irreducible over $\mathbb{F}_{q^{m}}$ if and
only if $f(x)$ is irreducible over $\mathbb{F}_{q^{m}}$.
Proof:
Since $f(x)\in\mathbb{F}_{q^{m}}[x]$, we have
$f(x)=\sigma^{(m)}(f(x))$. Then, we only need to prove that
$\sigma(f(x))$ is irreducible over $\mathbb{F}_{q^{m}}$ if $f(x)$ is
irreducible over $\mathbb{F}_{q^{m}}$. Assume that $\sigma(f(x))$ is
not irreducible over $\mathbb{F}_{q^{m}}$, that is to say there exist
two nonconstant polynomials $r_{1}(x),r_{2}(x)$ in $\mathbb{F}_{q^{m}}[x]$
such that $\sigma(f(x))=r_{1}(x)r_{2}(x)$. Therefore,
$$f(x)=\sigma^{(m)}(f(x))=\sigma^{(m-1)}(\sigma(f(x)))=\sigma^{(m-1)}(r_{1}(x))%
\sigma^{(m-1)}(r_{2}(x))$$
where $\sigma^{(m-1)}(r_{1}(x)),\sigma^{(m-1)}(r_{2}(x))$ are
nonconstant polynomials over $\mathbb{F}_{q^{m}}$, which contradicts
to the fact that $f(x)$ is irreducible over $\mathbb{F}_{q^{m}}$.
Hence, $\sigma(f(x))$ is irreducible over $\mathbb{F}_{q^{m}}$.
The following theorem is crucial to establish the main results in
this paper.
Theorem 3
Let $f(x)$ be an irreducible polynomial in
$\mathbb{F}_{q^{m}}[x]$, then the product
$$f(x)\sigma(f(x))\sigma^{(2)}(f(x))\cdots\sigma^{(k(f)-1)}(f(x))$$
is an irreducible polynomial in $\mathbb{F}_{q}[x]$.
Proof:
Let $\mbox{deg}(f(x))=n$. Then, by [13, Chapter 2, Theorem
2.14] there exits $\alpha\in\mathbb{F}_{q^{mn}}$ such that
$$\displaystyle f(x)=(x-\alpha)(x-\alpha^{q^{m}})(x-\alpha^{q^{2m}})\cdots(x-%
\alpha^{q^{(n-1)m}})$$
(1)
where $\alpha,\alpha^{q^{m}},\ldots,\alpha^{q^{(n-1)m}}$ are different
roots of $f(x)$. Let $g(x)$ be the minimal polynomial of
$\alpha\in\mathbb{F}_{q^{mn}}$ over $\mathbb{F}_{q}$. By [13, Chapter
2, Theorem 2.14], $g(x)$ is an irreducible polynomial over
$\mathbb{F}_{q}$ and
$$\displaystyle g(x)=(x-\alpha)(x-\alpha^{q})(x-\alpha^{q^{2}})\cdots(x-\alpha^{%
q^{d-1}})$$
(2)
where $d$ is the least positive integer such that
$\alpha^{q^{d}}=\alpha$. Since $\alpha^{q^{mn}}=\alpha$ and
$\alpha,\alpha^{q^{m}},\ldots,\alpha^{q^{(n-1)m}}$ are distinct, we
have $d\mid mn$ but $d\nmid im$ for $1\leq i\leq n-1$. Then, we
claim that $d$ must be a multiple of $n$. Otherwise, we have $\gcd(d,n)<n$. Since $d\mid mn$, then we have $\frac{d}{\gcd(d,n)}\mid\frac{mn}{\gcd(d,n)}$. Since $\frac{d}{\gcd(d,n)}$
and $\frac{n}{\gcd(d,n)}$ are relatively prime, we have $\frac{d}{\gcd(d,n)}\mid m$. Then, $d\mid\gcd(d,n)m$. This gives a
contradiction since $\gcd(d,n)<n$. Therefore, $d$ is a multiple of
$n$. Let $k$ be the positive integer such that $d=nk$. Since $d\mid mn$, then $k\mid m$. Let $s$ be the positive integer such that
$m=sk$. Then, we claim that $s$ and $n$ are relatively prime.
Otherwise, we have $\frac{n}{\gcd(n,s)}<n$. Since $n\mid\frac{ns}{\gcd(n,s)}$, then $kn\mid\frac{kns}{\gcd(n,s)}$, that is
$d\mid m\frac{n}{\gcd(n,s)}$. This gives a contradiction since
$\frac{n}{\gcd(n,s)}<n$. Therefore, $s$ and $n$ are relatively
prime. Thus, $\{js|j=0,1,\ldots,n-1\}$ is a complete residue
system modulo $n$, i.e., there exits $(i_{0},i_{1},\ldots,i_{n-1})$, a
permutation of $(0,1,2,\ldots,n-1)$, such that $js\equiv i_{j}\;\;({\rm mod}\ n)$. So we have $kjs\equiv ki_{j}\;\;({\rm mod}\ kn)$, i.e., $jm\equiv ki_{j}\;\;({\rm mod}\ d)$. Hence,
$\alpha^{q^{jm}}=\alpha^{q^{ki_{j}}}$ for $0\leq j\leq n-1$.
Therefore, it follows from (1) that
$$\displaystyle f(x)$$
$$\displaystyle=$$
$$\displaystyle(x-\alpha^{q^{ki_{0}}})(x-\alpha^{q^{ki_{1}}})(x-\alpha^{q^{ki_{2%
}}})\cdots(x-\alpha^{q^{ki_{n-1}}})$$
(3)
$$\displaystyle=$$
$$\displaystyle(x-\alpha)(x-\alpha^{q^{k}})(x-\alpha^{q^{2k}})\cdots(x-\alpha^{q%
^{(n-1)k}}).$$
By (3) and the definition of $\sigma$, we have
$$\displaystyle\sigma^{(i)}(f(x))=(x-\alpha^{q^{i}})(x-\alpha^{q^{k+i}})(x-%
\alpha^{q^{2k+i}})\cdots(x-\alpha^{q^{(n-1)k+i}}).$$
(4)
By (2), (3), (4) and note that $d=nk$, we
have
$$g(x)=f(x)\sigma(f(x))\ldots\sigma^{(k-1)}(f(x))$$
and
$$\sigma^{(k)}(f(x))=(x-\alpha^{q^{k}})(x-\alpha^{q^{2k}})(x-\alpha^{q^{3k}})%
\ldots(x-\alpha^{q^{nk}})=f(x).$$
Since $d$ is the least positive integer such that
$\alpha^{q^{d}}=\alpha$ and $d=nk$, we have that
$f(x),\sigma(f(x)),\ldots,\sigma^{(k-1)}(f(x))$ are different from
each other. Hence, $k=k(f)$. Therefore,
$$g(x)=f(x)\sigma(f(x))\cdots\sigma^{(k(f)-1)}(f(x)).$$
Note that $g(x)$ is an irreducible polynomial over $\mathbb{F}_{q}$,
we complete the proof.
Let $f(x)$ be an irreducible polynomial in $\mathbb{F}_{q^{m}}[x]$. It
is known from Lemma 4 that $f(x),\sigma(f(x)),\ldots,\sigma^{(k(f)-1)}(f(x))$ are irreducible polynomials in
$\mathbb{F}_{q^{m}}[x]$. Denote
$$R(f(x))=f(x)\sigma(f(x))\cdots\sigma^{(k(f)-1)}(f(x)).$$
By Theorem 3, $R(f(x))$ is irreducible in
$\mathbb{F}_{q}[x]$. Note that $R(f(x))$ is a multiple of $f(x)$ in
$\mathbb{F}_{q^{m}}[x]$. Using Theorem 3, we could give an
refined version of [13, Chapter 3, Theorem
3.46] as follows:
Theorem 4
Let $f(x)$ be a monic irreducible polynomial over
$\mathbb{F}_{q}$ and $n=\deg(f(x))$. Let $m$ be a positive integer.
Denote $u={\gcd}(n,m)$. Then the canonical factorization of $f(x)$
into monic irreducibles over $\mathbb{F}_{q^{m}}$ is given by
$$f(x)=h(x)\sigma(h(x))\cdots\sigma^{(k(h)-1)}(h(x))$$
where $h(x)$ is a monic irreducible polynomial over
$\mathbb{F}_{q^{m}}$ and $k(h)=u$.
Proof:
By [13, Chapter 3, Theorem
3.46], the canonical factorization of $f(x)$
into monic irreducibles over $\mathbb{F}_{q^{m}}$ is given by
$$f(x)=f_{1}(x)f_{2}(x)\cdots f_{u}(x)$$
where $f_{1}(x),f_{2}(x),\ldots,f_{u}(x)\in\mathbb{F}_{q^{m}}[x]$ are
distinct irreducible polynomials with the same degree. Let
$h(x)=f_{1}(x)$. By Theorem 3, $R(h(x))$ is an irreducible
polynomial in $\mathbb{F}_{q}[x]$. Since $f(x)$ and $R(h(x))$ have a
common factor $h(x)$ in $\mathbb{F}_{q^{m}}[x]$, $f(x)$ and $R(h(x))$
are not relatively prime in $\mathbb{F}_{q}[x]$. Note that $f(x)$
and $R(h(x))$ are monic irreducible polynomials in
$\mathbb{F}_{q}[x]$. So, $f(x)=R(h(x))$. By Lemma 4,
$h(x),\sigma(h(x)),\ldots,\sigma^{(k(h)-1)}(h(x))$ are all
irreducible polynomials over $\mathbb{F}_{q^{m}}$. Therefore, the
canonical factorization of $f(x)$ into monic irreducibles over
$\mathbb{F}_{q^{m}}$ is given by
$$f(x)=h(x)\sigma(h(x))\cdots\sigma^{(k(h)-1)}(h(x))$$
and $k(h)=u$.
In certain sense, Theorem 4 could be considered as a converse
procedure of Theorem 3.
4 Minimal Polynomials over $\mathbb{F}_{q}$ and $\mathbb{F}_{q^{m}}$
Now we determine the minimal polynomial and linear complexity over
$\mathbb{F}_{q}$ of a linear recurring sequence $\mathcal{S}$ over
$\mathbb{F}_{q^{m}}$ with minimal polynomial $h(x)\in\mathbb{F}_{q^{m}}[x]$.
Theorem 5
Let $\mathcal{S}$ be a linear recurring sequence over
$\mathbb{F}_{q^{m}}$ with minimal polynomial $h(x)\in\mathbb{F}_{q^{m}}[x]$. Assume that the canonical factorization of
$h(x)$ in $\mathbb{F}_{q^{m}}[x]$ is given by
$$h(x)=\prod_{j=1}^{l}P_{j0}^{e_{j0}}P_{j1}^{e_{j1}}\cdots P_{ji_{j}}^{e_{ji_{j}}}$$
where $\{P_{uv}\}$ are distinct monic irreducible polynomials in
$\mathbb{F}_{q^{m}}[x]$, $P_{j0},P_{j1},\ldots,P_{ji_{j}}$ are in the
same $\sigma$-equivalence class and $P_{uv}$, $P_{tw}$ are in the
different $\sigma$-equivalence classes when $u\neq t$. Then the
minimal polynomial over $\mathbb{F}_{q}$ of $\mathcal{S}$ is given by
$$H(x)=\prod_{j=1}^{l}R(P_{j0})^{e_{j}}$$
where $e_{j}=\max\{e_{j0},e_{j1},\ldots,e_{ji_{j}}\}$ for $1\leq j\leq l$.
Proof:
By Lemma 2, there uniquely exist sequences
$\mathcal{S}_{1},\mathcal{S}_{2},\ldots,\mathcal{S}_{l}$ over
$\mathbb{F}_{q^{m}}$ such that
$$\mathcal{S}=\mathcal{S}_{1}+\mathcal{S}_{2}+\cdots+\mathcal{S}_{l}$$
and the minimal polynomial over $\mathbb{F}_{q^{m}}$ of $\mathcal{S}_{j}$ is $P_{j0}^{e_{j0}}P_{j1}^{e_{j1}}\cdots P_{ji_{j}}^{e_{ji_{j}}}$ for $1\leq j\leq l$. Let $H_{j}(x)$ be the
minimal polynomial over $\mathbb{F}_{q}$ of $\mathcal{S}_{j}$. Since
$P_{j0},P_{j1},\ldots,P_{ji_{j}}$ are in the same
$\sigma$-equivalence class, then $R(P_{j0})^{e_{j}}$ is a multiple of
$P_{j0}^{e_{j0}}P_{j1}^{e_{j1}}\cdots P_{ji_{j}}^{e_{ji_{j}}}$. So, by
Theorem 3, $R(P_{j0})^{e_{j}}$ is a characteristic polynomial
over $\mathbb{F}_{q}$ of $\mathcal{S}_{j}$. Hence, $H_{j}(x)$ divides
$R(P_{j0})^{e_{j}}$ in $\mathbb{F}_{q}[x]$. Since, by Theorem 3,
$R(P_{j0})$ is irreducible over $\mathbb{F}_{q}$, we have
$H_{j}(x)=R(P_{j0})^{e^{\prime}_{j}}$ where $e^{\prime}_{j}\leq e_{j}$. By the definition of
$e_{j}$, there exists $e_{ju_{j}}$ such that $e_{ju_{j}}=e_{j}$ where $0\leq u_{j}\leq i_{j}$. If $e^{\prime}_{j}<e_{j}$, then $P_{ju_{j}}^{e_{ju_{j}}}$ can’t divide
$H_{j}(x)$. However, $H_{j}(x)$ is a multiple of
$P_{j0}^{e_{j0}}P_{j1}^{e_{j1}}\cdots P_{ji_{j}}^{e_{ji_{j}}}$ in
$\mathbb{F}_{q^{m}}[x]$ since $H_{j}(x)$ is also a characteristic
polynomial over $\mathbb{F}_{q^{m}}$ of $\mathcal{S}_{j}$. This gives a
contradiction. Therefore, $e^{\prime}_{j}=e_{j}$, i.e.,
$H_{j}(x)=R(P_{j0})^{e_{j}}$. For any $0\leq u\neq v\leq l$, we claim
that $R(P_{u0})^{e_{u}}$ and $R(P_{v0})^{e_{v}}$ are relatively prime.
Suppose on the contrary that there exist $R(P_{u0})^{e_{u}}$ and
$R(P_{v0})^{e_{v}}$, where $u\neq v$, which are not relatively prime.
Since $R(P_{u0})$ and $R(P_{v0})$ are monic irreducible polynomials
over $\mathbb{F}_{q}$, then we have $R(P_{u0})=R(P_{v0})$. Hence,
$P_{u0}$ divides $R(P_{v0})$ in $\mathbb{F}_{q^{m}}[x]$. By Theorem
4, the canonical factorization of $R(P_{v0})$ in
$\mathbb{F}_{q^{m}}[x]$ is given by
$$R(P_{v0})=P_{v0}\sigma(P_{v0})\cdots\sigma^{(k(P_{v0})-1)}(P_{v0}).$$
Since $P_{u0}$ is irreducible over $\mathbb{F}_{q^{m}}$, there exists
a positive integer $j$ such that $P_{u0}=\sigma^{(j)}(P_{v0})$. This
contradicts to the fact that $P_{u0}$ and $P_{v0}$ are in the
different $\sigma$-equivalence classes. Therefore, $R(P_{u0})^{e_{u}}$
and $R(P_{v0})^{e_{v}}$ are relatively prime. Then,
$H_{1}(x),H_{2}(x),\ldots,H_{l}(x)$ are pairwise relatively prime. By
Lemma 1, the minimal polynomial over $\mathbb{F}_{q}$ of
$\mathcal{S}=\sum_{j=1}^{l}\mathcal{S}_{j}$ is the product of
$H_{1}(x),H_{2}(x),\ldots,H_{l}(x)$. Therefore, we have
$$H(x)=\prod_{j=1}^{l}R(P_{j0})^{e_{j}}$$
which completes the proof.
Corollary 1
Under the notation of Theorem 5, the linear
complexity over $\mathbb{F}_{q}$ of $\mathcal{S}$ is given by
$$L_{\mathbb{F}_{q}}(\mathcal{S})=\sum_{j=1}^{l}e_{j}k(P_{j0})\deg(P_{j0})$$
where $k(f)$ is defined in Section 3.
Using Theorem 5, we could also give a refinement of
[18, Proposition 2.1]:
Theorem 6
Let $f(x)$ be a polynomial over $\mathbb{F}_{q}$ with
$\deg(f)\geq 1$. Suppose that
$$\displaystyle f=r_{1}^{e_{1}}r_{2}^{e_{2}}\cdots r_{l}^{e_{l}},\mbox{~{}~{}~{}%
}e_{1},e_{2},\ldots,e_{l}>0$$
(5)
is the canonical factorization of $f$ into monic irreducibles over
$\mathbb{F}_{q}$. Denote $n_{i}=\deg(r_{i})$. Suppose by Theorem
4 that the canonical factorization of $r_{i}(x)$ into monic
irreducibles over $\mathbb{F}_{q^{m}}$ is given by
$$\displaystyle r_{i}(x)=P_{i}(x)\sigma^{(1)}(P_{i}(x))\cdots\sigma^{(u_{i}-1)}(%
P_{i}(x))$$
(6)
where $u_{i}=\gcd(n_{i},m)=k(P_{i}(x))$. Let $\mathcal{S}$ be a linear
recurring sequence over $\mathbb{F}_{q^{m}}$. Then, the minimal
polynomial over $\mathbb{F}_{q}$ of $\mathcal{S}$ is $f(x)$ if and
only if the minimal polynomial $h(x)$ over $\mathbb{F}_{q^{m}}$ of
$\mathcal{S}$ is of the following form:
$$h(x)=\prod_{i=1}^{l}P_{i}^{e_{i0}}\sigma^{(1)}(P_{i})^{e_{i1}}\cdots\sigma^{({%
u_{i}-1})}(P_{i})^{e_{iu_{i}-1}}$$
(7)
where $0\leq e_{ij}\leq e_{i}$ and
$\max\{e_{i0},e_{i1},\ldots,e_{iu_{i}-1}\}=e_{i}$ for every
$i=1,2,\ldots,l$.
Proof:
It follows from Theorem 5 that the minimal polynomial over
$\mathbb{F}_{q}$ of $\mathcal{S}$ is $f(x)$ if the minimal
polynomial $h(x)$ over $\mathbb{F}_{q^{m}}$ of $\mathcal{S}$ is given
by (7).
Conversely, suppose that the minimal polynomials over
$\mathbb{F}_{q}$ of $\mathcal{S}$ is $f(x)$. Then, $h(x)$ is a
factor of $f(x)$ in $\mathbb{F}_{q^{m}}[x]$ since $f(x)$ is also a
characteristic polynomial over $\mathbb{F}_{q^{m}}$ of $\mathcal{S}$.
By (5) and (6), the canonical factorization of
$f(x)$ into monic irreducibles over $\mathbb{F}_{q^{m}}$ is given by
$$f(x)=\prod_{i=1}^{l}P_{i}^{e_{i}}\sigma^{(1)}(P_{i})^{e_{i}}\cdots\sigma^{({u_%
{i}-1})}(P_{i})^{e_{i}}.$$
So $h(x)$ must be of the form
$$h(x)=\prod_{i=1}^{l}P_{i}^{e_{i0}}\sigma^{(1)}(P_{i})^{e_{i1}}\cdots\sigma^{({%
u_{i}-1})}(P_{i})^{e_{iu_{i}-1}}$$
where $0\leq e_{ij}\leq e_{i}$ for every $i=1,2,\ldots,l$. By Theorem
5, the minimal polynomial over $\mathbb{F}_{q}$ of
$\mathcal{S}$ is given by
$$H(x)=\prod_{i=1}^{l}R(P_{i})^{e^{\prime}_{i}}=\prod_{i=1}^{l}r_{i}(x)^{e^{%
\prime}_{i}}$$
where $e^{\prime}_{i}=\max\{e_{i0},e_{i1},\ldots,e_{iu_{i}-1}\}$. Due to the
uniqueness of the minimal polynomial over $\mathbb{F}_{q}$ of
$\mathcal{S}$, we have $H(x)=f(x)$. Hence, $e^{\prime}_{i}=e_{i}$. Therefore,
the minimal polynomial $h(x)$ over $\mathbb{F}_{q^{m}}$ of
$\mathcal{S}$ is of the form (7). This completes the proof.
At the end of this section, we give an example to illustrate Theorem 5 and Corollary 1.
Example 1
Let $\mathbb{F}_{2}\subseteq\mathbb{F}_{4}$ and let $\alpha$ be a root
of $x^{2}+x+1$ in $\mathbb{F}_{4}$. So, $\mathbb{F}_{4}=\{0,1,\alpha,1+\alpha\}$. Let $\mathcal{S}$ be a periodic sequence over
$\mathbb{F}_{4}$ with the least period $15$. The first period terms of
$\mathcal{S}$ are given by
$$\alpha^{2},\alpha,\alpha,\alpha^{2},\alpha^{2},\alpha^{2},0,\alpha,\alpha^{2},%
\alpha,0,\alpha,0,0,1.$$
The minimal polynomial over $\mathbb{F}_{4}$ of $\mathcal{S}$ is
$x^{3}+\alpha^{2}x^{2}+\alpha^{2}$. We first factor
$x^{3}+\alpha^{2}x^{2}+\alpha^{2}$ into irreducible polynomials over
$\mathbb{F}_{4}$:
$$x^{3}+\alpha^{2}x^{2}+\alpha^{2}=(x+\alpha)(x^{2}+x+\alpha).$$
Note that
$$\sigma(x+\alpha)=x+\alpha^{2},\;\;\sigma^{(2)}(x+\alpha)=x+\alpha,$$
$$\sigma(x^{2}+x+\alpha)=x^{2}+x+\alpha^{2},\;\;\sigma^{(2)}(x^{2}+x+\alpha)=x^{%
2}+x+\alpha.$$
So we have
$$k(x+\alpha)=2,\;\;k(x^{2}+x+\alpha)=2.$$
Then, by Theorem 5 and Corollary 1, the minimal polynomial over $\mathbb{F}_{2}$ of $\mathcal{S}$ is
$$\displaystyle(x+\alpha)\sigma(x+\alpha)(x^{2}+x+\alpha)\sigma(x^{2}+x+\alpha)$$
(8)
$$\displaystyle=$$
$$\displaystyle(x^{2}+x+1)(x^{4}+x+1)=x^{6}+x^{5}+x^{4}+x^{3}+1$$
(9)
and the linear complexity over $\mathbb{F}_{2}$ of $\mathcal{S}$ is
$$L=1\times k(x+\alpha)\times\deg(x+\alpha)+1\times k(x^{2}+x+\alpha)\times\deg(%
x^{2}+x+\alpha)=2+2\times 2=6.$$
5 Remarks on the Lower Bound of
Meidl and Özbudak
Meidl and Özbudak [17] derived a lower bound on the linear
complexity over $\mathbb{F}_{q^{m}}$ of a linear recurring sequence
$\mathcal{S}$ over $\mathbb{F}_{q^{m}}$ with given minimal polynomial
$g(x)$ over $\mathbb{F}_{q}$. In this section, using Theorem 6
we give a new proof for the lower bound of Meidl and Özbudak and
show that this lower bound is tight if and only if the minimal
polynomial over $\mathbb{F}_{q^{m}}$ of $\mathcal{S}$ is in certain
form.
Corollary 2
Let $f(x)$ be a monic polynomial in $\mathbb{F}_{q}[x]$
with the canonical factorization into irreducible polynomials over
$\mathbb{F}_{q}$ given by
$$\displaystyle f=r_{1}^{e_{1}}r_{2}^{e_{2}}\ldots r_{k}^{e_{k}},\;\;\;e_{1},e_{%
2},\ldots,e_{k}>0.$$
(10)
Suppose that $\mathcal{S}$ is a linear recurring sequence over
$\mathbb{F}_{q^{m}}$ and the minimal polynomial over $\mathbb{F}_{q}$
of $\mathcal{S}$ is $f(x)$. Then, the linear complexity
$L_{\mathbb{F}_{q^{m}}}(\mathcal{S})$ over $\mathbb{F}_{q^{m}}$ of
$\mathcal{S}$ is lower bounded by
$$\displaystyle L_{\mathbb{F}_{q^{m}}}(\mathcal{S})\geq\sum_{i=1}^{k}e_{i}\frac{%
n_{i}}{\gcd(n_{i},m)}$$
(11)
where $n_{i}=\deg(r_{i})$ for $i=1,2,\ldots,k$. Furthermore, suppose by
Theorem 4 that the canonical factorization of $r_{i}(x)$ into
monic irreducibles over $\mathbb{F}_{q^{m}}$ is given by
$$\displaystyle r_{i}(x)=P_{i}(x)\sigma^{(1)}(P_{i}(x))\ldots\sigma^{(u_{i}-1)}(%
P_{i}(x))$$
(12)
where $u_{i}=\gcd(n_{i},m)$ for $i=1,2,\ldots,k$. Then, the lower
bound is tight if and only if the minimal polynomial $h(x)$ over
$\mathbb{F}_{q^{m}}$ of $\mathcal{S}$ is of the following form:
$$h(x)=\prod_{i=1}^{k}\sigma^{(j_{i})}(P_{i})^{e_{i}}$$
where $0\leq j_{i}\leq u_{i}-1$ for $i=1,2,\ldots,k$.
Proof:
It follows from (10) and (12) and Theorem 6
that the minimal polynomial $h(x)$ over $\mathbb{F}_{q^{m}}$ of
$\mathcal{S}$ is of the form:
$$h(x)=\prod_{i=1}^{k}P_{i}^{e_{i0}}\sigma^{(1)}(P_{i})^{e_{i1}}\cdots\sigma^{({%
u_{i}-1})}(P_{i})^{e_{iu_{i}-1}}$$
(13)
where $0\leq e_{ij}\leq e_{i}$ and
$\max\{e_{i0},e_{i1},\ldots,e_{iu_{i}-1}\}=e_{i}$ for every
$i=1,2,\ldots,k$. Note from (12) that
$\deg(P_{i}(x))=n_{i}/u_{i}$. Hence, by (13),
$$\displaystyle L_{\mathbb{F}_{q^{m}}}(\mathcal{S})=\deg(h(x))\geq\sum_{i=1}^{k}%
e_{i}\deg(P_{i}(x))=\sum_{i=1}^{k}e_{i}\frac{n_{i}}{\mbox{gcd}(n_{i},m)}$$
and the equality holds if and only if
$$h(x)=\prod_{i=1}^{k}\sigma^{(j_{i})}(P_{i})^{e_{i}}$$
where $0\leq j_{i}\leq u_{i}-1$ for $i=1,2,\ldots,k$. This completes
the proof.
Remark 1
Meidl and Özbudak [17, Proposition 3] showed that there
exists a linear recurring sequence over $\mathbb{F}_{q^{m}}$ such that
the lower bound (11) is tight. We give in Corollary
2 the necessary and sufficient condition under which the
lower bound (11) is tight.
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Peng’s Maximum Principle for a Stochastic Control Problem Driven by a Fractional and a Standard Brownian Motion
BUCKDAHN Rainer${}^{a,b}$ and
JING Shuai${}^{c}$††Corresponding author. E-mail: shuaijingsj@gmail.com. Supported by National Natural Science Foundation of China (Project No. 11301560).
${}^{a}$ Département de Mathématiques, Université de Bretagne Occidentale, 29285, Brest, France
${}^{b}$ School of Mathematics, Shandong University, 250100, Jinan, P.R.China
${}^{c}$ School of Management Science and Engineering, Central University
of Finance and Economics,
100081,
Beijing, P.R.China
Abstract
We study a stochastic control system involving both a standard and a fractional Brownian motion with Hurst parameter less than 1/2. We apply an anticipative Girsanov transformation to transform the system into another one, driven only by the standard Brownian motion with coefficients depending on both the fractional Brownian motion and the standard Brownian motion. We derive a maximum principle and the associated stochastic variational inequality, which both are generalizations of the classical case.
Keywords: fractional Brownian motion, stochastic control system, backward stochastic differential equation, variational inequality, maximum principle, Girsanov transformation, Galtchouk-Kunita-Watanabe decomposition.
AMS Subject Classification: 60H05, 60G22, 93E20
1 Introduction
We study a control problem which controlled state process is driven by both a standard Brownian motion and a fractional Brownian motion with Hurst parameter $H\in(0,1/2)$, and we derive the stochastic maximum principle and the associated variational inequality. To be more precise, we consider the state process governed by the following controlled stochastic differential equation
$$\left\{\begin{array}[]{l}\mathrm{d}X^{u}(t)=\sigma(t)X^{u}(t)\mathrm{d}B^{H}(t)+\beta(t,X^{u}(t),u(t))\mathrm{d}W(t)+b(t,X^{u}(t),u(t))\mathrm{d}t,\quad t\in[0,T],\\
X^{u}(0)=x_{0},\end{array}\right.$$
(1.1)
where the functions $\sigma$, $\beta$ and $b$ are introduced in Section 2, and the control process $u$ takes values in a metric space $U$. Thus, in our framework, the diffusion part consists of two parts: one is represented by a stochastic integral with respect to the fractional Brownian motion $B^{H}$, which integrand is linear in the state process, and the other by an Itô integral with respect to the Brownian motion $W$, which integrand is nonlinear in the state process. The control problem consists in minimizing the cost functional defined as follows
$$J(u)=\mathbb{E}\left[\Phi(X^{u}(T))+\int^{T}_{0}f(t,X^{u}(t),u(t))\mathrm{d}t\right],$$
where the functions $\Phi$ and $f$ are introduced in Section 2.
Stochastic differential equations driven by both a standard Brownian motion and a fractional Brownian motion have been studied by several authors, for example, for the case $H\in(1/2,1)$ by Guerra and Nualart [9], Mishura and Shevchenko [17]. The properties of the fractional Brownian motion with Hurst parameter $H\in(1/2,1)$ and $H\in(0,1/2)$ are quite different. Besides a pathwise definition of the integral, the classical divergence operator is widely used to define the stochastic integral when $H\in(1/2,1)$. However, in the case $H\in(0,1/2)$, the domain of the divergence operator becomes too small. For instance, Cheridito and Nualart [8] showed that even the fractional Brownian motion itself is not included in the domain. To overcome this difficulty, Cheridito and Nualart [8] and León and Nualart [15] defined a new type of operator and called it the extended divergence operator. By using the extended divergence operator, León and San Martín [16] studied linear stochastic differential equations driven by a fractional Brownian motion with $H\in(0,1/2)$ with the help of the chaos decomposition. Jien and Ma [12] worked on stochastic differential equations driven by fractional Brownian motions by applying anticipative Girsanov transformations developed by Buckdahn [4], while Jing and León [13] also made use of the Girsanov transformation method to deal with semilinear backward doubly stochastic differential equations driven by a Brownian motion and a fractional Brownian motion with $H\in(0,1/2)$ and the associated stochastic partial differential equations driven by the fractional Brownian motion.
The variational inequality and stochastic maximum principle for controlled systems driven by a Brownian motion have been investigated by many authors. Without being exhaustive, let us mention among them, for instance, Bismut [3], Bensoussan [1], Peng [19] and Buckdahn et al. [5]. However, for controlled systems involving fractional Brownian motions, there are only very few works. Biagini et al. [2] studied a stochastic maximum principle for processes driven only by an $m$-dimensional fractional Brownian motion with $H\in(1/2,1)^{m}$ and derived an adjoint linear fractional backward stochastic differential equation. Hu and Zhou [11] considered an optimal control problem of stochastic linear systems involving a fractional Brownian motion with Hurst parameter $H\in(0,1/2)$, and they introduced a Riccati equation which is a backward stochastic differential equation driven by the fractional Brownian motion and a classical Brownian motion. It is worth noting that this Brownian motion is the one that generates the fractional Brownian motion, hence they are not independent. Han et al. [10] obtained a stochastic maximum principle for a stochastic control problem defined through a general controlled system driven by a fractional Brownian motion with $H>1/2$. Similar to [11], their adjoint backward stochastic differential equation is driven by the fractional Brownian motion and its underlying Brownian motion.
Here, in our framework, the controlled system involves both a standard and a fractional Brownian motion with $H\in(0,1/2)$. We use the extended divergence operator to define the stochastic integral with respect to the fractional Brownian motion. The linearity of the integrand of the integral with respect to $B^{H}$ in the state process allows, similarly to [13], to apply the anticipative Girsanov transformation to transform the original controlled system into another one driven only by the standard Brownian motion $W$, but with coefficients depending on the paths of both $W$ and $B^{H}$. Our adjoint backward stochastic differential equation involves, besides the Brownian martingale, also an orthogonal martingale, which is a Brownian martingale in the classical case (see Section 4). This orthogonal martingale comes from the Galtchouk-Kunita-Watanabe decomposition. Such backward stochastic differential equations were employed by Buckdahn and Ichihara [6] and Buckdahn et al. [7] to study optimal control systems and associated Hamilton-Jacobi-Bellman equations. In our work here, we compare our main result with the classical characterization of an optimal control and we show that, if we replace the fractional Brownian motion with a standard Brownian motion, i.e., if we apply our Girsanov transformation in the classical, Brownian framework, we get the same result. Hence, our result indeed generalizes the classical one.
In this paper we deal only with the case $H\in(0,1/2)$ since we use the extended divergence operator as stochastic integral with respect to the fractional Brownian motion. Nevertheless, when using the divergence operator in the case $H\in(1/2,1)$, our method is still valid and the computations are even easier. The key difference between the two cases relies mainly on the distinct definitions of the divergence operator and the extended divergence operator.
The paper is organized as follows: In Section 2 we recall some preliminaries, i.e., some basic settings and some basics on the fractional Brownian motion, the extended divergence operator and the Girsanov transformation. Our main results, the variational inequality and the stochastic maximum principle, are stated in Section 3. Finally, in Section 4 we compare our result with Peng’s criterion for the optimality of a stochastic control in the Brownian setting [19]. The proofs of the results in Section 3 are given in the Appendix to improve the readability.
2 Preliminaries
2.1 General Setting and Fractional Brownian Motion
Let $T>0$ be a fixed time horizon. Let $\{W(s),s\in[0,T]\}$ be a standard Brownian motion
on a complete probability space $(\Omega_{1},\mathcal{F}_{1},\mathbb{P}_{1})$ and $\{B^{H}(s),s\in[0,T]\}$ be a
fractional Brownian motion with Hurst parameter $H\in(0,1/2)$ defined
on another complete probability space $(\Omega_{2},\mathcal{F}_{2},$ $\mathbb{P}_{2})$.
We introduce $(\Omega,\mathcal{F},\mathbb{P})$ as the product space $(\Omega,\mathcal{F},\mathbb{P})=(\Omega_{1},\mathcal{F}_{1},\mathbb{P}_{1})\otimes(\Omega_{2},\mathcal{F}_{2},\mathbb{P}_{2})=(\Omega_{1}\times\Omega_{2},\mathcal{F}_{1}\otimes\mathcal{F}_{2},\mathbb{P}_{1}\otimes\mathbb{P}_{2})$ which we suppose to be completed.
The processes $W$ and $B^{H}$ are canonically extended from
$(\Omega_{1},\mathcal{F}_{1},$ $\mathbb{P}_{1})$ and $(\Omega_{2},\mathcal{F}_{2},$ $\mathbb{P}_{2})$, respectively,
to the product space $(\Omega,\mathcal{F},\mathbb{P})$.
We define three filtrations: one is generated by the Brownian motion: $\mathbb{F}^{W}=\{\mathcal{F}_{t}^{W}=\sigma\{W(s),0\leq s\leq t\}\vee{\cal N},t\in[0,T]\}$, one is generated by the fractional Brownian motion: $\mathbb{F}^{B}=\{\mathcal{F}_{t}^{B}=\sigma\{B^{H}(s),0\leq s\leq t\}\vee{\cal N},t\in[0,T]\},$ and another one is generated by the Brownian motion $W$ and the fractional Brownian motion $B^{H}$ over the time interval $[0,T]$ : $\mathbb{H}=\{\mathcal{H}_{t}=\mathcal{F}_{t}^{W}\vee\mathcal{F}_{t}^{B}\vee\mathcal{N},{t\in[0,T]}\}$. Here $\mathcal{N}$ denotes the set of all $\mathbb{P}$-null sets.
For $p>1$, we denote by $L^{p}_{\mathbb{H}}(\Omega\times[0,T])$ the space of real valued $\mathbb{H}$-adapted processes such that
$$\|\varphi\|_{L^{p}_{\mathbb{H}}}=\left(\mathbb{E}\left[\int^{T}_{0}\left|\varphi(t)\right|^{p}\mathrm{d}t\right]\right)^{1/p}<+\infty.$$
It is well-known that, for $H\in(0,1/2)$, there exists another canonical Wiener process $W^{0}$ on $(\Omega_{2},\mathcal{F}_{2},$ $\mathbb{P}_{2})$ such that we have the following representation:
$$B^{H}(t)=\int^{t}_{0}K_{H}(t,s)\mathrm{d}W^{0}(s),$$
where
$$K_{H}(t,s)=C_{H}\left[\left(\frac{t}{s}\right)^{H-1/2}t^{H-1/2}-(H-1/2)s^{1/2-H}\int^{t}_{s}u^{H-3/2}(u-s)^{H-1/2}\mathrm{d}u\right],$$
and
$$C_{H}=\sqrt{\frac{2H}{(1-2H)\beta(1-2H,H+1/2)}}.$$
Hence, the process $B^{H}$ is a centered Gaussian process with covariance function
$$R_{H}(t,s)=\mathbb{E}\left[B^{H}(t)B^{H}(s)\right]=\frac{1}{2}\left(t^{2H}+s^{2H}-|t-s|^{2H}\right).$$
2.2 Extended Divergence Operator
We briefly recall the definition of the extended divergence operator as the stochastic integral with respect to the fractional Brownian motion $B^{H}$; for more details, we refer to [13]. The extended divergence operator was first studied by Cheridito and Nualart [8] and further investigated by León and Nualart [15].
To this end, we define a Hilbert space $\mathcal{H}_{H}$ as the completion of the space of step functions over $[0,T]$ with respect to the inner product
$$\langle I_{[0,t]},I_{[0,s]}\rangle_{\mathcal{H}_{H}}=R_{H}(t,s),\ t,s\in[0,T].$$
On the space $\mathcal{H}_{H}$, an isometry $\mathcal{H}_{H}\ni\varphi\to B^{H}(\varphi)\in L^{2}(\Omega,\mathcal{F},\mathbb{P})$ is defined by extending the map $I_{[0,t]}\to B^{H}_{t}$. Moreover, by the transfer principle (see Nualart [18]), one has the existence of an operator $\mathcal{K}:\mathcal{H}_{H}\to L^{2}([0,T])$ such that
$$B^{H}(\varphi)=\int^{T}_{0}(\mathcal{K}\varphi)(s)\mathrm{d}W^{0}(s),\ \varphi\in\mathcal{H}_{H},\ \ \textrm{and}\ \ (\mathcal{K}I_{[0,t]})(s)=K_{H}(t,s),\ s,t\in[0,T].$$
We denote by $\mathcal{K}^{\ast}$ its adjoint operator.
Let $\mathcal{S}_{\mathcal{K}}$ be the class of all smooth functionals of the form
$$F=f\left(B^{H}(\varphi_{1}),\cdots,B^{H}(\varphi_{m}),W(\psi_{1}),\cdots,W(\psi_{n})\right),\ \ m,n\geq 1,$$
where $\varphi_{1},\cdots,\varphi_{m}$ are elements of $\mathcal{H}_{H}$, $\psi_{1},\cdots,\psi_{n}\in L^{2}([0,T])$, $W(\psi_{1}),\cdots,W(\psi_{n})$ are Wiener integrals of $\psi_{1},\cdots,\psi_{n}$ with respect to $W$, and $f\in C^{\infty}_{p}(\mathbb{R}^{m+n})$ - the space of all $C^{\infty}$ function over $\mathbb{R}^{m+n}$, which together with all their derivatives are of polynomial growth.
A smooth functional $F\in\mathcal{S}_{\mathcal{K}}$ of above form has Malliavin derivatives with respect to $B^{H}$ and $W$ defined as follows:
$$D^{B}F=\sum^{m}_{i=1}\frac{\partial f}{\partial x_{i}}\left(B^{H}(\varphi_{1}),\cdots,B^{H}(\varphi_{m}),W(\psi_{1}),\cdots,W(\psi_{n})\right)\varphi_{i},$$
and
$$D^{W}F=\sum^{n}_{i=1}\frac{\partial f}{\partial x_{m+i}}\left(B^{H}(\varphi_{1}),\cdots,B^{H}(\varphi_{m}),W(\psi_{1}),\cdots,W(\psi_{n})\right)\psi_{i}.$$
We remark that both $D^{B}F$ and $D^{W}F$ are in $L^{p}(\Omega\times[0,T];\mathcal{H}_{H})$, for all $p\geq 2$.
For $u\in L^{2}(\Omega\times[0,T])$, we define the following stochastic integrals with respect to $B^{H}$ and $W$, respectively.
Definition 2.1.
Let $u\in L^{2}(\Omega\times[0,T])$.
If there exists a random variable $\delta^{B}(u)\in L^{2}(\Omega,\mathcal{F},\mathbb{P})$ such that
$$\mathbb{E}\left[\langle\mathcal{K}^{\ast}\mathcal{K}D^{B}F,u\rangle_{L^{2}{([0,T])}}\right]=\mathbb{E}\left[F\delta^{B}(u)\right],\ \ \textrm{for all}\ \ F\in\mathcal{S}_{\mathcal{K}},$$
(2.2)
we say $u\in Dom\ \delta^{B}$ and call $\delta^{B}(u)$ the extended divergence operator of $u$ with respect to $B^{H}$.
Definition 2.2.
Let $u\in L^{2}(\Omega\times[0,T])$.
If there exists a random variable $\delta^{W}(u)\in L^{2}(\Omega,\mathcal{F},\mathbb{P})$ such that
$$\mathbb{E}\left[\langle D^{W}F,u\rangle_{L^{2}{([0,T])}}\right]=\mathbb{E}\left[F\delta^{W}(u)\right],\ \ \textrm{for all}\ \ F\in\mathcal{S}_{\mathcal{K}},$$
(2.3)
we say $u\in Dom\ \delta^{W}$ and call $\delta^{W}(u)$ the Skorohod integral of $u$ with respect to $W$.
Remark 2.3.
1. Given $u\in L^{2}(\Omega\times[0,T])$ and $t\in[0,T]$ such that $uI_{[0,t]}\in Dom\ \delta^{B}$, we write $\int^{t}_{0}u(s)\mathrm{d}B^{H}(s)$ for $\delta^{B}(uI_{[0,t]})$.
2. If $u\in L^{2}(\Omega\times[0,T])$ is $\mathbb{H}$-adapted, then the Skorohod integral $\delta^{W}(u)$ exists and it coincides with the Itô integral $\int^{T}_{0}u(s)\mathrm{d}W(s)$ (Recall that $W$ is an $\mathbb{H}$-Brownian motion).
2.3 Girsanov Transformations
The Girsanov transformation with respect to the fractional Brownian motion constitutes an essential tool in our approach for our stochastic control problem.
Throughout this paper we use the following hypothesis.
(H1) Let $\sigma:[0,T]\to\mathbb{R}$ be a square integrable Borel function such that $\sigma I_{[0,t]}$ belongs to $\mathcal{H}_{H}$, for every $t\in[0,T]$, and $\sup_{0\leq t\leq T}\int^{t}_{0}((\mathcal{K}\sigma I_{[0,t]})(r))^{2}\mathrm{d}r<+\infty.$
Recall that hypothesis (H1) is in particular satisfied if $\sigma(t)=\sigma,t\in[0,T]$, for some constant $\sigma\in\mathbb{R}$, and $(\mathcal{K}\sigma I_{[0,t]})(r)=K_{H}(t,r)$, $(t,r)\in[0,T]^{2}$.
For $t\in[0,T]$, we consider the following transformations on $\Omega_{2}$:
$$\mathcal{T}_{t}(\omega_{2})=\omega_{2}+\int^{\cdot\wedge t}_{0}(\mathcal{K}\sigma I_{[0,t]})(r)\mathrm{d}r,\ \ \omega_{2}\in\Omega_{2},\ \ t\in[0,T],$$
and
$$\mathcal{A}_{t}(\omega_{2})=\omega_{2}-\int^{\cdot\wedge t}_{0}(\mathcal{K}\sigma I_{[0,t]})(r)\mathrm{d}r,\ \ \omega_{2}\in\Omega_{2},\ \ t\in[0,T].$$
The Girsanov Theorem (see for example Buckdahn [4]) gives
that for any square integrable random variable $F$, we have
$$\mathbb{E}[F]=\mathbb{E}[F(\mathcal{A}_{t})\kappa_{t}]=\mathbb{E}[F(\mathcal{T}_{t})\kappa_{t}^{-1}(\mathcal{T}_{t})],$$
(2.4)
where
$$\kappa_{t}=\exp\left(\int^{t}_{0}\sigma(r)\mathrm{d}B^{H}(r)-\frac{1}{2}\int^{t}_{0}((\mathcal{K}\sigma I_{[0,t]})(r))^{2}\mathrm{d}r\right).$$
(2.5)
From Lemma 2.4 in [13] we have that
$$E\left[\sup_{0\leq t\leq T}\kappa^{p}_{t}\right]<+\infty,\ \ E\left[\sup_{0\leq t\leq T}\kappa^{p}_{t}(\mathcal{T}_{t})\right]<+\infty,\ \mbox{for\ all}\ p\in\mathbb{R}.$$
(2.6)
3 Variational Inequality and the Maximum Principle
3.1 The Stochastic Control Problem
Let $U$ be a nonempty subset of $\mathbb{R}^{k}$. Let
$\{u(s),0\leq s\leq T\}$ be an admissible control process,
which takes values in $U$ and is $\mathbb{H}$-adapted,
such that
$$\mathop{\mathrm{esssup}}_{0\leq t\leq T}\mathbb{E}[|u(t)|^{p}]<+\infty,\quad\textrm{for}\ \ p\geq 1.$$
The set of admissible control processes is denoted by
$\mathcal{U}_{ad}$. From (H1) and (2.6)
we get that if $\{u(s),0\leq s\leq T\}$ is an admissible control, then both
$\{u(s,\mathcal{T}_{s}),0\leq s\leq T\}$ and $\{u(s,\mathcal{A}_{s}),0\leq s\leq T\}$
are admissible controls. In particular, we have
$$\mathop{\mathrm{esssup}}_{0\leq t\leq T}\mathbb{E}[|u(t,\mathcal{T}_{t})|^{p}]<+\infty\ \ \textrm{and}\ \ \mathop{\mathrm{esssup}}_{0\leq t\leq T}\mathbb{E}[|u(t,\mathcal{A}_{t})|^{p}]<+\infty,\quad p\geq 1.$$
(3.7)
We consider the following stochastic control system:
$$\left\{\begin{array}[]{l}\mathrm{d}X^{u}(t)=\sigma(t)X^{u}(t)\mathrm{d}B^{H}(t)+\beta(t,X^{u}(t),u(t))\mathrm{d}W(t)+b(t,X^{u}(t),u(t))\mathrm{d}t,\quad t\in[0,T],\\
X^{u}(0)=x_{0}.\end{array}\right.$$
(3.8)
Notice that only the coefficients $\beta$ and $b$ depend on the control, but not $\sigma$.
Moreover, the stochastic integral with respect to the fractional Brownian motion is linear in $X^{u}$
and is interpreted in the extended divergence sense.
The cost functional is defined by
$$J(u)=\mathbb{E}\left[\Phi(X^{u}(T))+\int^{T}_{0}f(t,X^{u}(t),u(t))\mathrm{d}t\right].$$
(3.9)
Our control problem consists in minimizing the cost functional $J(u)$ over $\mathcal{U}_{ad}$.
Now we state the assumptions on the coefficients:
$$\beta,b,f:[0,T]\times\mathbb{R}\times\mathbb{R}^{k}\to\mathbb{R},\ \ \Phi:\mathbb{R}\to\mathbb{R}.$$
(H2) The functions $\beta,b,f,\Phi$ are twice differentiable with respect to $x$.
Moreover, $\beta,b,f,\Phi$ and their derivatives
$\beta_{x},b_{x},f_{x},\Phi_{x}$ $\beta_{xx},b_{xx},f_{xx},\Phi_{xx}$
are continuous in $(x,u)$ and bounded, uniformly with respect to $(t,u)\in[0,T]\times U$.
3.2 Main Results
In this subsection we state our main results, i.e., the variational inequality and the maximum principle.
First we state the following important theorem. It helps us to establish a link between the semilinear stochastic differential equation (3.8), driven by both the standard Brownian motion $W$ and the fractional Brownian motion $B^{H}$, and a stochastic differential equation driven only by the standard Brownian motion $W$, with coefficients depending on the fractional Brownian motion.
Theorem 3.1.
The process $X^{u}:=\left\{X^{u}(t)=\zeta^{u}(t,\mathcal{A}_{t})\kappa_{t},t\in[0,T]\right\}$ is the unique solution of equation (3.8) in $L^{2}_{\mathbb{H}}(\Omega\times[0,T])$, where $\zeta^{u}$ is the unique solution of the pathwise stochastic differential equation
$$\left\{\begin{array}[]{l}\mathrm{d}\zeta^{u}(t)=\kappa_{t}^{-1}(\mathcal{T}_{t})\beta(t,\zeta^{u}(t)\kappa_{t}(\mathcal{T}_{t}),u(t,\mathcal{T}_{t}))\mathrm{d}W(t)+\kappa_{t}^{-1}(\mathcal{T}_{t})b(t,\zeta^{u}(t)\kappa_{t}(\mathcal{T}_{t}),u(t,\mathcal{T}_{t}))\mathrm{d}t,\ t\in[0,T];\\
\zeta^{u}(0)=x_{0}.\end{array}\right.$$
(3.10)
For the reader’s convenience we give the proof; it is shifted to the Appendix.
The above theorem allows to rewrite the cost functional (3.9) as follows:
$$J(u)=\mathbb{E}\left[\Phi\big{(}\zeta^{u}(T)\kappa_{T}(\mathcal{T}_{T})\big{)}\kappa_{T}^{-1}(\mathcal{T}_{T})+\int^{T}_{0}f\big{(}t,\zeta^{u}(t)\kappa_{t}(\mathcal{T}_{t}),u(t,\mathcal{T}_{t})\big{)}\kappa_{t}^{-1}(\mathcal{T}_{t})\mathrm{d}t\right].$$
(3.11)
We have transformed our stochastic control problem into a formally classical control problem which contains the fractional Brownian motion implicitly.
Since the control process $u(t,\mathcal{T}_{t})$ appearing in (3.10) and (3.11) contains always the transformation $\mathcal{T}_{t}$, for the simplicity of notations, we denote it by $v(t)$, i.e., $v(t)=u(t,\mathcal{T}_{t})$. From (3.7) we know that both $u$ and $v$ are admissible controls.
Let us now suppose that $(y(\cdot),v(\cdot))$ is an optimal solution of the control problem, i.e.,
$$\left\{\begin{array}[]{l}\mathrm{d}y(t)=\kappa_{t}^{-1}(\mathcal{T}_{t})\beta(t,y(t)\kappa_{t}(\mathcal{T}_{t}),v(t))\mathrm{d}W(t)+\kappa_{t}^{-1}(\mathcal{T}_{t})b(t,y(t)\kappa_{t}(\mathcal{T}_{t}),v(t))\mathrm{d}t,\ t\in[0,T],\\
y(0)=x_{0},\end{array}\right.$$
(3.12)
and
$$J(v)=\inf_{u\in\mathcal{U}_{ad}}J(u).$$
Following Peng’s approach [19], we construct a perturbed admissible control as follows:
$$v^{\varepsilon}(t)=\left\{\begin{array}[]{ll}\tilde{v}(t)&\tau-\varepsilon\leq t\leq\tau+\varepsilon,\\
v(t)&\textrm{otherwise},\end{array}\right.$$
where $0<\tau<T$ is arbitrarily fixed, $\varepsilon>0$ is arbitrarily chosen such that $[\tau-\varepsilon,\tau+\varepsilon]\subset[0,T]$, and $\tilde{v}$ is an arbitrary bounded admissible control from $\mathcal{U}_{ad}$. Let $y^{\varepsilon}(\cdot)$ be the solution of (3.12) with $v^{\varepsilon}$ at the place of $v$. Then from the setting of the control problem, we have
$$J(v^{\varepsilon})-J(u)\geq 0.$$
Let $y_{1}(\cdot)$ and $y_{2}(\cdot)\in L^{\infty,-}_{\mathbb{H}}([0,T])(:=\bigcap_{p\geq 2}L^{p}_{\mathbb{H}}(\Omega\times[0,T]))$ be the solutions of the equations
$$\displaystyle y_{1}(t)=$$
$$\displaystyle\int^{t}_{0}\bigg{[}b_{x}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))y_{1}(s)$$
(3.13)
$$\displaystyle\qquad+\kappa_{s}^{-1}(\mathcal{T}_{s})\bigg{(}b(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v^{\varepsilon}(s))-b(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))\bigg{)}\bigg{]}\mathrm{d}s$$
$$\displaystyle+\int^{t}_{0}\bigg{[}\beta_{x}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))y_{1}(s)$$
$$\displaystyle\qquad\quad+\kappa_{s}^{-1}(\mathcal{T}_{s})\bigg{(}\beta(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v^{\varepsilon}(s))-\beta(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))\bigg{)}\bigg{]}\mathrm{d}W(s),$$
and
$$\displaystyle y_{2}(t)=$$
$$\displaystyle\int^{t}_{0}\bigg{[}b_{x}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))y_{2}(s)+\frac{1}{2}\kappa_{s}(\mathcal{T}_{s})b_{xx}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v^{\varepsilon}(s))y_{1}^{2}(s)\bigg{]}\mathrm{d}s$$
(3.14)
$$\displaystyle+\int^{t}_{0}\bigg{[}\beta_{x}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))y_{2}(s)+\frac{1}{2}\kappa_{s}(\mathcal{T}_{s})\beta_{xx}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v^{\varepsilon}(s))y_{1}^{2}(s)\bigg{]}\mathrm{d}W(s)$$
$$\displaystyle+\int^{t}_{0}\bigg{(}b_{x}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v^{\varepsilon}(s))-b_{x}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))\bigg{)}y_{1}(s)\mathrm{d}s$$
$$\displaystyle+\int^{t}_{0}\bigg{(}\beta_{x}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v^{\varepsilon}(s))-\beta_{x}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))\bigg{)}y_{1}(s)\mathrm{d}W(s).$$
We need the following estimates for $y_{1}$ and $y_{2}$.
Lemma 3.2.
Under our hypotheses (H1) and (H2), for any $p\geq 2$, there is some $C_{p}\in\mathbb{R}_{+}$ independent of $\varepsilon$ such that
$$\mathbb{E}\left[\sup_{0\leq t\leq T}|y_{1}(t)|^{p}\right]\leq C_{p}\varepsilon^{p/2},$$
(3.15)
$$\mathbb{E}\left[\sup_{0\leq t\leq T}|y_{2}(t)|^{p}\right]\leq C_{p}\varepsilon^{p}.$$
(3.16)
Proof: First we prove inequality (3.15). From equation (3.13), using (H2) and the Buckhölder-Davis-Gundy inequality, we obtain that, for $p\geq 2$,
$$\displaystyle\mathbb{E}\left[\sup_{0\leq r\leq t}|y_{1}(r)|^{p}\right]$$
$$\displaystyle\leq C_{p}\mathbb{E}\left[\sup_{0\leq r\leq t}\int^{r}_{0}|y_{1}(s)|^{p}\mathrm{d}s\right]+C_{p}\mathbb{E}\left[\left(\int^{t}_{0}I_{[\tau-\varepsilon,\tau+\varepsilon]}\kappa_{s}^{-2}(\mathcal{T}_{s})\mathrm{d}s\right)^{p/2}\right]$$
(3.17)
$$\displaystyle\leq C_{p}\mathbb{E}\left[\int^{t}_{0}\sup_{0\leq r\leq s}|y_{1}(r)|^{p}\mathrm{d}s\right]+C_{p}\varepsilon^{p/2}\mathbb{E}\left[\sup_{0\leq s\leq T}\kappa_{s}^{-p}(\mathcal{T}_{s})\right]$$
$$\displaystyle\leq C_{p}\mathbb{E}\left[\int^{t}_{0}\sup_{0\leq r\leq s}|y_{1}(r)|^{p}\mathrm{d}s\right]+C_{p}\varepsilon^{p/2},$$
where the constant $C_{p}$ can be chosen independent of $t$. By the Gronwall inequality we get that
$$\mathbb{E}\left[\sup_{0\leq t\leq T}|y_{1}(t)|^{p}\right]\leq C_{p}\varepsilon^{p/2}.$$
Now we prove (3.16). From equation (3.14) and (H2), applying Cauchy-Schwarz inequality and Buckhölder-Davis-Gundy inequality, we have
$$\displaystyle\mathbb{E}\left[\sup_{0\leq r\leq t}|y_{2}(r)|^{p}\right]\leq$$
$$\displaystyle C_{p}\mathbb{E}\left[\int^{t}_{0}\left(\sup_{0\leq r\leq s}|y_{2}(r)|^{p}+\sup_{0\leq r\leq s}\kappa_{r}^{p}(\mathcal{T}_{r})|y_{1}(r)|^{2p}\right)\mathrm{d}s\right]$$
$$\displaystyle+C_{p}\mathbb{E}\left[\left|\int^{t}_{0}|I_{[\tau-\varepsilon,\tau+\varepsilon]}(s)y_{1}(s)|\mathrm{d}s\right|^{p}\right]+C_{p}\mathbb{E}\left[\int^{t}_{0}I_{[\tau-\varepsilon,\tau+\varepsilon]}(s)\left|y_{1}(s)\right|^{2}\mathrm{d}s\right]^{\frac{p}{2}}$$
$$\displaystyle\leq$$
$$\displaystyle C_{p}\mathbb{E}\left[\int^{t}_{0}\sup_{0\leq r\leq s}|y_{2}(r)|^{p}\mathrm{d}s\right]+C_{p}\left(\mathbb{E}\left[\sup_{0\leq r\leq T}\kappa_{r}^{2p}(\mathcal{T}_{r})\right]\right)^{\frac{1}{2}}\left(\mathbb{E}\left[\sup_{0\leq r\leq T}y_{1}^{4p}(r)\right]\right)^{1/2}$$
$$\displaystyle+C_{p}\varepsilon^{p}\mathbb{E}\left[\sup_{0\leq t\leq T}|y_{1}(t)|^{p}\right]+C_{p}\varepsilon^{p/2}\mathbb{E}\left[\sup_{0\leq t\leq T}|y_{1}(t)|^{p}\right].$$
Hence, from (3.15) and the Gronwall inequality, we obtain
$$\mathbb{E}\left[\sup_{0\leq t\leq T}|y_{2}(t)|^{p}\right]\leq C_{p}\varepsilon^{p}.$$
The proof is complete. $\Box$
Set $y_{3}=y_{1}+y_{2}$. To derive our variational inequality, it is necessary to prove the following estimate.
Lemma 3.3.
Under the hypothesis (H2), for any $p\geq 2$, we have
$$\sup_{0\leq t\leq T}\mathbb{E}\left[|y^{\varepsilon}(t)-y(t)-y_{3}(t)|^{p}\right]=o(\varepsilon^{p}).$$
(3.18)
For convenience of the reader the proof is given in the Appendix.
The next lemma plays an important role in deriving the variational inequality.
Lemma 3.4.
Under the hypothesis (H2) we have
$$\displaystyle-\mathbb{E}\left[\int^{T}_{0}\left(f_{x}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))y_{3}(s)+\frac{1}{2}f_{xx}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))y_{1}^{2}(s)\kappa_{s}(\mathcal{T}_{s})\right)\mathrm{d}s\right]$$
(3.19)
$$\displaystyle-\mathbb{E}\left[\int^{T}_{0}\left(f(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v^{\varepsilon}(s))-f(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))\right)\kappa_{s}^{-1}(\mathcal{T}_{s})\mathrm{d}s\right]$$
$$\displaystyle-\mathbb{E}\left[\Phi_{x}(y(T)\kappa_{T}(\mathcal{T}_{T}))y_{3}(T)+\frac{1}{2}\Phi_{xx}(y(T)\kappa_{T}(\mathcal{T}_{T}))y_{1}^{2}(T)\kappa_{T}(\mathcal{T}_{T})\right]\leq o(\varepsilon).$$
The proof of this lemma is given in the Appendix.
For a pair of processes $(\varphi(\cdot),\psi(\cdot))$ in $L^{2}_{\mathbb{H}}(\Omega\times[0,T])\times L^{2}_{\mathbb{H}}(\Omega\times[0,T])$, we consider the following stochastic system:
$$\left\{\begin{array}[]{l}\mathrm{d}z(t)=\Big{(}b_{x}(t,y(t)\kappa_{t}(\mathcal{T}_{t}),v(t))z(t)+\varphi(t)\Big{)}\mathrm{d}t+\Big{(}\beta_{x}(t,y(t)\kappa_{t}(\mathcal{T}_{t}),v(t))z(t)+\psi(t)\Big{)}\mathrm{d}W(t),\\
z(0)=0.\end{array}\right.$$
(3.20)
With the help of this equation we define a linear functional
$$I(\varphi(\cdot),\psi(\cdot))=\mathbb{E}\left[\int^{T}_{0}f_{x}(t,y(t)\kappa_{t}(\mathcal{T}_{t}),v(t))z(t)\mathrm{d}t+\Phi_{x}(y(T)\kappa_{T}(\mathcal{T}_{T}))z(T)\right],$$
(3.21)
which is continuous in $L^{2}_{\mathbb{H}}(\Omega\times[0,T])\times L^{2}_{\mathbb{H}}(\Omega\times[0,T])$. The Riesz representation theorem yields that
there exists a unique pair of processes $(p(\cdot),K(\cdot))\in L^{2}_{\mathbb{H}}(\Omega\times[0,T])\times L^{2}_{\mathbb{H}}(\Omega\times[0,T])$ such that
$$I(\varphi(\cdot),\psi(\cdot))=\mathbb{E}\left[\int^{T}_{0}(p(t)\varphi(t)+K(t)\psi(t))\mathrm{d}t\right],\ (\varphi,\psi)\in L^{2}_{\mathbb{H}}(\Omega\times[0,T])\times L^{2}_{\mathbb{H}}(\Omega\times[0,T]).$$
(3.22)
Notice that the processes of $p$ and $K$ do not depend on $(\varphi(\cdot),\psi(\cdot))$. By applying the above
representation result to the definition of $y_{1}$ in (3.13), we get that
$$\displaystyle\mathbb{E}\left[\int^{T}_{0}f_{x}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))y_{1}(s)\mathrm{d}s+\Phi_{x}(y(T)\kappa_{T}(\mathcal{T}_{T}))y_{1}(T)\right]$$
(3.23)
$$\displaystyle=$$
$$\displaystyle E\left[\int^{T}_{0}p(s)\underbrace{\Big{(}b(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v^{\varepsilon}(s))-b(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))\Big{)}\kappa_{s}^{-1}(\mathcal{T}_{s})}_{=\varphi(s)}\mathrm{d}s\right]$$
$$\displaystyle+\mathbb{E}\left[\int^{T}_{0}K(s)\underbrace{\Big{(}\beta(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v^{\varepsilon}(s))-\beta(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))\Big{)}\kappa_{s}^{-1}(\mathcal{T}_{s})}_{=\psi(s)}\mathrm{d}s\right],$$
and with the above indicated choice of $\varphi$ and $\psi$, $z$ defined by (3.20) coincides with $y_{1}$ defined by (3.13). Thus, (3.21) and (3.22) yield (3.23). With similar argument applied to (3.14), we have
$$\displaystyle\mathbb{E}\left[\int^{T}_{0}f_{x}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))y_{2}(s)\mathrm{d}s+\Phi_{x}(y(T)\kappa_{T}(\mathcal{T}_{T}))y_{2}(T)\right]$$
(3.24)
$$\displaystyle=$$
$$\displaystyle E\left[\frac{1}{2}\int^{T}_{0}\Big{(}p(s)b_{xx}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))+K(s)\beta_{xx}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))\Big{)}y_{1}^{2}(s)\kappa_{s}(\mathcal{T}_{s})\mathrm{d}s\right]$$
$$\displaystyle+\mathbb{E}\left[\int^{T}_{0}p(s)\Big{(}b_{x}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v^{\varepsilon}(s))-b_{x}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))\Big{)}y_{1}(s)\mathrm{d}s\right]$$
$$\displaystyle+\mathbb{E}\left[\int^{T}_{0}K(s)\Big{(}\beta_{x}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v^{\varepsilon}(s))-\beta_{x}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))\Big{)}y_{1}(s)\mathrm{d}s\right].$$
We define a new (random) function $H$ by putting
$$H(s,x,v,p,K)=\Big{(}f(s,x\kappa_{s}(\mathcal{T}_{s}),v)+pb(s,x\kappa_{s}(\mathcal{T}_{s}),v)+K\beta(s,x\kappa_{s}(\mathcal{T}_{s}),v)\Big{)}\kappa_{s}^{-1}(\mathcal{T}_{s}).$$
Then, using (3.23) and (3.24), we can rewrite inequality (3.19) as
$$\displaystyle-\mathbb{E}\left[\int^{T}_{0}\Big{(}H(s,y(s),v^{\varepsilon}(s),p(s),K(s))-H(s,y(s),v(s),p(s),K(s))\Big{)}\mathrm{d}s\right]$$
(3.25)
$$\displaystyle-\mathbb{E}\left[\frac{1}{2}\int^{T}_{0}H_{xx}(s,y(s),v(s),p(s),K(s))y^{2}_{1}(s)\mathrm{d}s+\frac{1}{2}\Phi_{xx}\big{(}y(T)\kappa_{T}(\mathcal{T}_{T})\big{)}y_{1}^{2}(T)\kappa_{T}(\mathcal{T}_{T})\right]$$
$$\displaystyle\leq$$
$$\displaystyle o(\varepsilon).$$
Now we deal with the quadratic term. Let $Y(s):=y_{1}^{2}(s)$. Applying the Itô formula to $Y(s)$, we get
$$\displaystyle\mathrm{d}Y(s)=$$
$$\displaystyle\left[2Y(s)\Big{(}b_{x}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))+\frac{1}{2}\beta^{2}_{x}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))\Big{)}+\Xi^{\varepsilon}(s)\right]\mathrm{d}s$$
(3.26)
$$\displaystyle+\Big{[}2Y(s)\beta_{x}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))+\Psi^{\varepsilon}(s)\Big{]}\mathrm{d}W(s),$$
where
$$\displaystyle\Xi^{\varepsilon}(s)=$$
$$\displaystyle 2y_{1}(s)\varepsilon^{-1}_{s}(\mathcal{T}_{s})\Big{(}b(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v^{\varepsilon}(s))-b(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))$$
$$\displaystyle+\beta_{x}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))\big{(}\beta(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v^{\varepsilon}(s))-\beta(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))\big{)}\Big{)}$$
$$\displaystyle+\varepsilon^{-2}_{s}(\mathcal{T}_{s})\left(\beta(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v^{\varepsilon}(s))-\beta(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))\right)^{2}$$
and
$$\displaystyle\Psi^{\varepsilon}(s)=2y_{1}(s)\kappa_{s}^{-1}(\mathcal{T}_{s})\big{(}\beta(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v^{\varepsilon}(s))-\beta(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))\big{)}.$$
For any $(\Xi(\cdot),\Psi(\cdot))$ in $L^{2}_{\mathbb{H}}(\Omega\times[0,T])\times L^{2}_{\mathbb{H}}(\Omega\times[0,T])$, we consider the following stochastic system:
$$\left\{\begin{array}[]{l}\mathrm{d}Z(s)=\left[2Z(s)\Big{(}b_{x}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))+\beta^{2}_{x}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))\Big{)}+\Xi(s)\right]\mathrm{d}s\\
\qquad\qquad+\Big{[}2Z(s)\beta_{x}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))+\Psi(s)\Big{]}\mathrm{d}W(s),\\
Z(0)=0.\end{array}\right.$$
(3.27)
We define a new linear functional
$$L(\Xi(\cdot),\Psi(\cdot))=\mathbb{E}\left[\int^{T}_{0}Z(s)H_{xx}(s,y(s),v(s),p(s),K(s))\mathrm{d}s+Z(T)\Phi_{xx}\big{(}y(T)\kappa_{T}(\mathcal{T}_{T})\big{)}\kappa_{T}(\mathcal{T}_{T})\right],$$
(3.28)
which too is continuous on $L^{2}_{\mathbb{H}}(\Omega\times[0,T])\times L^{2}_{\mathbb{H}}(\Omega\times[0,T])$. Using the same argument as above we see that there exists a unique pair of
$(P(\cdot),Q(\cdot))$ in $L^{2}_{\mathbb{H}}(\Omega\times[0,T])\times L^{2}_{\mathbb{H}}(\Omega\times[0,T])$ such that
$$L(\Xi(\cdot),\Psi(\cdot))=\mathbb{E}\left[\int^{T}_{0}\Big{(}P(s)\Xi(s)+Q(s)\Psi(s)\Big{)}\mathrm{d}s\right].$$
(3.29)
Now we apply the above result with $Z(s):=Y(s)=y_{1}^{2}(s),\Xi(s):=\Xi^{\varepsilon}(s),\Psi(s):=\Psi^{\varepsilon}(s)$, $s\in[0,T]$.
By using the estimates in Lemma 3.2, we obtain that, for $1<p,q<\infty$ with $\frac{1}{p}+\frac{1}{q}=1$,
$$\displaystyle\mathbb{E}\left[\int^{T}_{0}y_{1}(s)\kappa_{s}^{-1}(\mathcal{T}_{s})\big{(}b(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v^{\varepsilon}(s))-b(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))\big{)}P(s)\mathrm{d}s\right]$$
$$\displaystyle\leq$$
$$\displaystyle C\mathbb{E}\left[\sup_{s\in[0,T]}|y_{1}(s)|\sup_{s\in[0,T]}(\kappa_{s}^{-1}(\mathcal{T}_{s}))\varepsilon^{1/2}\left(\int^{T}_{0}|P(s)|^{2}1_{[\tau-\varepsilon,\tau+\varepsilon]}(s)\mathrm{d}s\right)^{1/2}\right]$$
$$\displaystyle\leq$$
$$\displaystyle C\left(\mathbb{E}\left[\sup_{s\in[0,T]}|y_{1}(s)|^{2p}\right]\right)^{1/2p}\left(\mathbb{E}\left[\sup_{s\in[0,T]}|\kappa_{s}(\mathcal{T}_{s})|^{-2q}\right]\right)^{1/2q}\varepsilon^{1/2}\left(\mathbb{E}\left[\left(\int^{T}_{0}|P(s)|^{2}1_{[\tau-\varepsilon,\tau+\varepsilon]}(s)\mathrm{d}s\right)\right]\right)^{1/2}$$
$$\displaystyle\leq$$
$$\displaystyle C\varepsilon h(\varepsilon),$$
where, from the Dominated Convergence Theorem
$$h(\varepsilon)=\left(\mathbb{E}\left[\left(\int^{T}_{0}|P(s)|^{2}1_{[\tau-\varepsilon,\tau+\varepsilon]}(s)\mathrm{d}s\right)\right]\right)^{1/2}\to 0,\ \ \textrm{when}\ \varepsilon\to 0.$$
Hence, we have
$$\mathbb{E}\left[\int^{T}_{0}y_{1}(s)\kappa_{s}^{-1}(\mathcal{T}_{s})\big{(}b(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v^{\varepsilon}(s))-b(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))\big{)}P(s)\mathrm{d}s\right]=o(\varepsilon).$$
Similarly, we get
$$\mathbb{E}\left[\int^{T}_{0}2y_{1}(s)\kappa_{s}^{-1}(\mathcal{T}_{s})\big{(}\beta(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v^{\varepsilon}(s))-\beta(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))\big{)}Q(s)\mathrm{d}s\right]=o(\varepsilon).$$
Therefore, the relations (3.26), (3.27), (3.28) and (3.29) allow to rewrite inequality (3.25) as
$$\displaystyle-\mathbb{E}\left[\int^{T}_{0}\Big{(}H(s,y(s),v^{\varepsilon}(s),p(s),K(s))-H(s,y(s),v(s),p(s),K(s))\Big{)}\mathrm{d}s\right]$$
(3.30)
$$\displaystyle-\frac{1}{2}\mathbb{E}\left[\int^{T}_{0}\kappa_{s}^{-2}(\mathcal{T}_{s})\Big{(}\beta(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v^{\varepsilon}(s))-\beta(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))\Big{)}^{2}P(s)\mathrm{d}s\right]\leq o(\varepsilon).$$
Hence, by letting $\varepsilon$ tend to zero, we deduce that
$$\displaystyle H(\tau,y(\tau),v,p(\tau),K(\tau))-H(\tau,y(\tau),v(\tau),p(\tau),K(\tau))$$
(3.31)
$$\displaystyle+\frac{1}{2}\kappa_{\tau}^{-2}(\mathcal{T}_{\tau})\Big{(}\beta(\tau,y(\tau)\kappa_{\tau}(\mathcal{T}_{\tau}),v)-\beta(\tau,y(\tau)\kappa_{\tau}(\mathcal{T}_{\tau}),v(\tau))\Big{)}^{2}P(\tau)\geq 0$$
holds for any $U$-valued $\mathcal{F}^{B}_{\tau}$-measurable random variable $v$, $\mathrm{d}\tau$-a.e., a.s.,
where we recall
$$H(s,x,v,p,K)=\Big{(}f(s,x\kappa_{s}(\mathcal{T}_{s}),v)+pb(s,x\kappa_{s}(\mathcal{T}_{s}),v)+K\beta(s,x\kappa_{s}(\mathcal{T}_{s}),v)\Big{)}\kappa_{s}^{-1}(\mathcal{T}_{s}).$$
Inequality (3.31) is the stochastic variational inequality of our control problem.
Since in our case the variational inequality is different from those in Peng [19] and Buckdahn et al. [5], we prefer to give a detailed proof of deriving (3.31) from (3.30) in the Appendix.
Following similar arguments as the classical results of Bensoussan [1] and Peng [19], the pair of processes $(p(\cdot),K(\cdot))$ is determined by an adjoint backward stochastic differential equation, i.e., $(p(\cdot),K(\cdot))$ is the unique solution of
$$\left\{\begin{array}[]{rl}-\mathrm{d}p(s)&=\big{[}b_{x}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))p(s)+\beta_{x}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))K(s)+f_{x}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))\big{]}\mathrm{d}s\\
&\qquad-K(s)\mathrm{d}W(s)-\mathrm{d}N(s),\ \ s\in[0,T],\\
p(T)&=\Phi_{x}(y(T)\kappa_{T}(\mathcal{T}_{T})),\end{array}\right.$$
(3.32)
and $(P(\cdot),Q(\cdot))$ is the unique solution of the following adjoint backward stochastic differential equation:
$$\left\{\begin{array}[]{rl}-\mathrm{d}P(s)&=\big{[}2b_{x}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))P(s)+\beta_{x}^{2}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))P(s)\\
&\qquad\qquad+2\beta_{x}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))Q(s)+H_{xx}(s,y(s),v(s),p(s),K(s))\big{]}\mathrm{d}s\\
&\qquad-Q(s)\mathrm{d}W(s)-\mathrm{d}M(s),\ \ s\in[0,T],\\
P(T)&=\Phi_{xx}(y(T)\kappa_{T}(\mathcal{T}_{T}))\kappa_{T}(\mathcal{T}_{T}),\end{array}\right.$$
(3.33)
where $N(\cdot)$ and $M(\cdot)$ are $\mathbb{H}$-adapted square integrable martingales orthogonal to $W$.
One can easily verify that the solutions $(p(\cdot),K(\cdot))$ and $(P(\cdot),Q(\cdot))$ satisfy (3.23) and (3.24).
Remark 3.5.
The two martingales $N(\cdot)$ and $M(\cdot)$ are introduced here from the Galtchouk-Kunita-Watanabe decomposition (we refer to [14]); this allows to guarantee the adaptedness of $(p(\cdot),K(\cdot))$ and $(P(\cdot),$ $Q(\cdot))$ with respect to $\mathbb{H}$. Such backward stochastic differential equations with respect to a non-Brownian filtration have been well studied, and they were also employed to study control problems, for instance, in Buckdahn and Ichihara [6] and Buckdahn et al. [7].
As a consequence, we obtain the maximum principle theorem.
Theorem 3.6.
Let (H1) and (H2) hold. If $(y(\cdot),v(\cdot))$ is the optimal solution of the control problem (3.10) and (3.11), then we have
$$(p(\cdot),K(\cdot))\in L^{2}_{\mathbb{H}}(\Omega\times[0,T])\times L^{2}_{\mathbb{H}}(\Omega\times[0,T])\ \ \textrm{and}\ \ (P(\cdot),Q(\cdot))\in L^{2}_{\mathbb{H}}(\Omega\times[0,T])\times L^{2}_{\mathbb{H}}(\Omega\times[0,T]),$$
are the solutions of backward stochastic differential equations (3.32) and (3.33) respectively, such that the (stochastic) variational inequality (3.31) holds.
4 Comparison with the Classical Case and Conclusion
In this part we compare our result with the classical case, i.e., Peng’s result [19].
First, if $\sigma\equiv 0$, i.e.,if there is no fractional Brownian motion part, then obviously our result reduces to Peng’s. Second, if $\sigma\neq 0$ but $H=1/2$, i.e., the fractional Brownian motion $B^{H}$ is nothing else but a standard Brownian motion $B$, we show that our result coincides with Peng’s characterisation of the optimal control. Here we only show that from equation (19) in Peng [19] we can obtain $(\ref{eq_BSDE1})$. With our notations, equation (19) in Peng [19] becomes
$$\left\{\begin{array}[]{rl}-\mathrm{d}p(s)&=\big{[}b_{x}(s,X^{u}(s),u(s))p(s)+\beta_{x}(s,X^{u}(s),u(s))K(s)+\sigma(s)K_{1}(s)+f_{x}(s,X^{u}(s),u(s))\big{]}\mathrm{d}s\\
&\qquad-K(s)\mathrm{d}W(s)-K_{1}(s)\mathrm{d}B(s),\ \ s\in[0,T],\\
p(T)&=\Phi_{x}(X^{u}(T)).\end{array}\right.$$
(4.34)
We notice that in the classical case $H=1/2$, (2.5) yields $\kappa_{s}=\exp\left\{\int^{s}_{0}\sigma(r)\mathrm{d}B(r)-\frac{1}{2}\int^{s}_{0}\sigma^{2}(r)\mathrm{d}r\right\}$. We put $\overline{p}(s):=p(s,\mathcal{T}_{s})$, $\overline{K}_{1}(s)=K_{1}(s,\mathcal{T}_{s})$ and $\overline{K}(s)=K(s,\mathcal{T}_{s})$. Applying now standard arguments as above, and recalling the definition of $(y(\cdot,),v(\cdot))$ through (3.12),
we deduce that
$$\left\{\begin{array}[]{rl}-\mathrm{d}\overline{p}(s)&=\big{[}b_{x}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))\overline{p}(s)+\beta_{x}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))\overline{K}(s)+f_{x}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))\big{]}\mathrm{d}s\\
&\qquad-\overline{K}(s)\mathrm{d}W(s)-\overline{K}_{1}(s)\mathrm{d}B(s),\ \ s\in[0,T],\\
\overline{p}(T)&=\Phi_{x}(y(T)\kappa_{T}(\mathcal{T}_{T})).\end{array}\right.$$
(4.35)
Equation (4.35) coincides with equation (3.32) with $N(t)=\int^{t}_{0}\overline{K}_{1}(s)\mathrm{d}B(s)$.
Hence our result is really a generalization of the classical one.
5 Appendix
In the Appendix we state some proofs of the results in Section 3.
Proof of Theorem 3.1: The proof is a bit technical and we split it into 3 steps.
Step 1: First we prove the existence and uniqueness of $\zeta^{u}$ in $L^{p}_{\mathbb{H}}(\Omega\times[0,T]),$ $p\geq 2$. To this end, we define $\hat{\beta}$ and $\hat{b}$ as
$$\hat{\beta}(t,x,u):=\kappa^{-1}_{t}(\mathcal{T}_{t})\beta(t,x\kappa_{t}(\mathcal{T}_{t}),u),\ \ \hat{b}(t,x,u):=\kappa^{-1}_{t}(\mathcal{T}_{t})b(t,x\kappa_{t}(\mathcal{T}_{t}),u),\ \ (t,x,u)\in[0,T]\times\mathbb{R}\times U.$$
Then, from (H2) and (2.6), $\hat{\beta}(\cdot,0,0)$ and $\hat{b}(\cdot,0,0)\in L^{p}_{\mathbb{H}}(\Omega\times[0,T]),$ $p\geq 2$. Furthermore, from (H2), we know that there exists a constant $C>0$, such that
$$|\hat{\beta}(t,x_{1},u)-\hat{\beta}(t,x_{2},u)|\leq C|x_{1}-x_{2}|,\ |\hat{b}(t,x_{1},u)-\hat{b}(t,x_{2},u)|\leq C|x_{1}-x_{2}|,\ x_{1},x_{2}\in\mathbb{R},(t,u)\in[0,T]\times U.$$
Hence, equation (3.10) admits a unique solution $\zeta^{u}\in L^{p}_{\mathbb{H}}(\Omega\times[0,T])$, $p\geq 2$.
Step 2:
Next we prove that $X^{u}$ is a solution of equation (3.8). Observe, that from the definition of $X^{u}$ and the above property of $\zeta^{u}$, it follows that $X^{u}\in L^{2}_{\mathbb{H}}(\Omega\times[0,T])$. Let us
choose an arbitrary $F\in\mathcal{S}_{\mathcal{K}}$. Then we have from (2.4), for $t\in[0,T]$,
$$\displaystyle\mathbb{E}\left[FX^{u}(t)-Fx_{0}\right]=\mathbb{E}\left[F(\mathcal{T}_{t})\zeta^{u}(t)-F\zeta^{u}(0)\right]$$
$$\displaystyle=$$
$$\displaystyle\mathbb{E}\left[F(\mathcal{T}_{t})x_{0}-Fx_{0}+F(\mathcal{T}_{t})\int^{t}_{0}\kappa_{s}^{-1}(\mathcal{T}_{s})\beta(s,\zeta^{u}(s)\kappa_{s}(\mathcal{T}_{s}),u(s,\mathcal{T}_{s}))\mathrm{d}W(s)\right]$$
$$\displaystyle+\mathbb{E}\left[F(\mathcal{T}_{t})\int^{t}_{0}\kappa_{s}^{-1}(\mathcal{T}_{s})b(s,\zeta^{u}(s)\kappa_{s}(\mathcal{T}_{s}),u(s,\mathcal{T}_{s}))\mathrm{d}s\right].$$
We recall that, from (2.3),
$$\displaystyle\mathbb{E}\left[F(\mathcal{T}_{t})\int^{t}_{0}\kappa_{s}^{-1}(\mathcal{T}_{s})\beta(s,\zeta^{u}(s)\kappa_{s}(\mathcal{T}_{s}),u(s,\mathcal{T}_{s}))\mathrm{d}W(s)\right]$$
$$\displaystyle=$$
$$\displaystyle\mathbb{E}\left[\int^{t}_{0}\kappa_{s}^{-1}(\mathcal{T}_{s})\beta(s,\zeta^{u}(s)\kappa_{s}(\mathcal{T}_{s}),u(s,\mathcal{T}_{s}))D^{W}_{s}F(\mathcal{T}_{t})\mathrm{d}s\right].$$
From the fact that $F\in\mathcal{S}_{\mathcal{K}}$ and the definition of $\mathcal{T}_{t}$, we deduce that (see, Jing and León [13] Page 7),
$$\frac{\mathrm{d}F(\mathcal{T}_{t})}{\mathrm{d}t}=\sigma(t)(\mathcal{K}^{*}\mathcal{K}D^{B}F)(\mathcal{T}_{t},t).$$
Using the above result, we obtain
$$\displaystyle\mathbb{E}\left[FX^{u}(t)-Fx_{0}\right]$$
(5.36)
$$\displaystyle=$$
$$\displaystyle\mathbb{E}\left[x_{0}\int^{t}_{0}\sigma(s)(\mathcal{K}^{*}\mathcal{K}D^{B}F)(\mathcal{T}_{s},s)\mathrm{d}s\right]$$
$$\displaystyle+\mathbb{E}\left[\int^{t}_{0}\kappa_{s}^{-1}(\mathcal{T}_{s})\beta(s,\zeta^{u}(s)\kappa_{s}(\mathcal{T}_{s}),u(s,\mathcal{T}_{s}))D^{W}_{s}F(\mathcal{T}_{s})\mathrm{d}s\right]$$
$$\displaystyle+\mathbb{E}\left[\int^{t}_{0}\left(\int^{t}_{s}D^{W}_{s}(\sigma(r)(\mathcal{K}^{*}\mathcal{K}D^{B}F)(\mathcal{T}_{r},r))\mathrm{d}r\right)\kappa_{s}^{-1}(\mathcal{T}_{s})\beta(s,\zeta^{u}(s)\kappa_{s}(\mathcal{T}_{s}),u(s,\mathcal{T}_{s}))\mathrm{d}s\right]$$
$$\displaystyle+\mathbb{E}\left[\int^{t}_{0}\kappa_{s}^{-1}(\mathcal{T}_{s})b(s,\zeta^{u}(s)\kappa_{s}(\mathcal{T}_{s}),u(s,\mathcal{T}_{s}))F(\mathcal{T}_{s})\mathrm{d}s\right]$$
$$\displaystyle+\mathbb{E}\left[\int^{t}_{0}\left(\int^{t}_{s}\sigma(r)(\mathcal{K}^{*}\mathcal{K}D^{B}F)(\mathcal{T}_{r},r)\mathrm{d}r\right)\kappa_{s}^{-1}(\mathcal{T}_{s})b(s,\zeta^{u}(s)\kappa_{s}(\mathcal{T}_{s}),u(s,\mathcal{T}_{s}))\mathrm{d}s\right].$$
By applying the Fubini theorem, we get
$$\displaystyle I_{1}=$$
$$\displaystyle\mathbb{E}\left[\int^{t}_{0}\left(\int^{t}_{s}D^{W}_{s}\left(\sigma(r)(\mathcal{K}^{*}\mathcal{K}D^{B}F)(\mathcal{T}_{r},r)\right)\mathrm{d}r\right)\kappa_{s}^{-1}(\mathcal{T}_{s})\beta(s,\zeta^{u}(s)\kappa_{s}(\mathcal{T}_{s}),u(s,\mathcal{T}_{s}))\mathrm{d}s\right]$$
$$\displaystyle=$$
$$\displaystyle\int^{t}_{0}\mathbb{E}\left[\int^{r}_{0}D^{W}_{s}(\sigma(r)(\mathcal{K}^{*}\mathcal{K}D^{B}F)(\mathcal{T}_{r},r))\kappa_{s}^{-1}(\mathcal{T}_{s})\beta(s,\zeta^{u}(s)\kappa_{s}(\mathcal{T}_{s}),u(s,\mathcal{T}_{s}))\mathrm{d}s\right]\mathrm{d}r.$$
Thus, taking into account that $\sigma(r)(\mathcal{K}^{*}\mathcal{K}D^{B}F)(\mathcal{T}_{r},r)\in\mathcal{S}_{\mathcal{K}}$, we conclude from Remark 2.3 that
$$\displaystyle I_{1}=$$
$$\displaystyle\int^{t}_{0}\mathbb{E}\left[\sigma(r)(\mathcal{K}^{*}\mathcal{K}D^{B}F)(\mathcal{T}_{r},r)\int^{r}_{0}\kappa_{s}^{-1}(\mathcal{T}_{s})\beta(s,\zeta^{u}(s)\kappa_{s}(\mathcal{T}_{s}),u(s,\mathcal{T}_{s}))\mathrm{d}W(s)\right]\mathrm{d}r$$
$$\displaystyle=$$
$$\displaystyle\mathbb{E}\left[\int^{t}_{0}\sigma(r)(\mathcal{K}^{*}\mathcal{K}D^{B}F)(\mathcal{T}_{r},r)\int^{r}_{0}\kappa_{s}^{-1}(\mathcal{T}_{s})\beta(s,\zeta^{u}(s)\kappa_{s}(\mathcal{T}_{s}),u(s,\mathcal{T}_{s}))\mathrm{d}W(s)\mathrm{d}r\right].$$
Consequently, using the Fubini Theorem now also for the latter double integral in (5.36), we get
$$\displaystyle\mathbb{E}\left[FX^{u}(t)-Fx_{0}\right]=$$
$$\displaystyle\mathbb{E}\bigg{[}\int^{t}_{0}\sigma(r)(\mathcal{K}^{*}\mathcal{K}D^{B}F)(\mathcal{T}_{r},r)\bigg{\{}x_{0}+\int^{r}_{0}\kappa_{s}^{-1}(\mathcal{T}_{s})\beta(s,\zeta^{u}(s)\kappa_{s}(\mathcal{T}_{s}),u(s,\mathcal{T}_{s}))\mathrm{d}W(s)$$
$$\displaystyle\qquad\qquad\qquad+\int^{r}_{0}\kappa_{s}^{-1}(\mathcal{T}_{s})b(s,\zeta^{u}(s)\kappa_{s}(\mathcal{T}_{s}),u(s,\mathcal{T}_{s}))\mathrm{d}s\bigg{\}}\bigg{]}$$
$$\displaystyle+\mathbb{E}\left[\int^{t}_{0}\kappa_{s}^{-1}(\mathcal{T}_{s})\beta(s,\zeta^{u}(s)\kappa_{s}(\mathcal{T}_{s}),u(s,\mathcal{T}_{s}))D^{W}_{s}F(\mathcal{T}_{s})\mathrm{d}s\right]$$
$$\displaystyle+\mathbb{E}\left[\int^{t}_{0}\kappa_{s}^{-1}(\mathcal{T}_{s})b(s,\zeta^{u}(s)\kappa_{s}(\mathcal{T}_{s}),u(s,\mathcal{T}_{s}))F(\mathcal{T}_{s})\mathrm{d}s\right].$$
Hence, from (3.10) and by applying the Girsanov Theorem again, we get
$$\displaystyle\mathbb{E}\left[FX^{u}(t)-Fx_{0}\right]$$
$$\displaystyle=$$
$$\displaystyle\mathbb{E}\left[\int^{t}_{0}(\mathcal{K}^{*}\mathcal{K}D^{B}F)(\mathcal{T}_{s},s)\sigma(s)\zeta^{u}(s)\mathrm{d}s\right]+\mathbb{E}\left[\int^{t}_{0}\kappa_{s}^{-1}(\mathcal{T}_{s})\beta(s,\zeta^{u}(s)\kappa_{s}(\mathcal{T}_{s}),u(s,\mathcal{T}_{s}))D^{W}_{s}F(\mathcal{T}_{s})\mathrm{d}s\right]$$
$$\displaystyle+\mathbb{E}\left[\int^{t}_{0}\kappa_{s}^{-1}(\mathcal{T}_{s})b(s,\zeta^{u}(s)\kappa_{s}(\mathcal{T}_{s}),u(s,\mathcal{T}_{s}))F(\mathcal{T}_{s})\mathrm{d}s\right]$$
$$\displaystyle=$$
$$\displaystyle\mathbb{E}\left[\int^{t}_{0}(\mathcal{K}^{*}\mathcal{K}D^{B}F)(s)\sigma(s)X^{u}(s)\mathrm{d}s\right]+\mathbb{E}\left[\int^{t}_{0}\beta(s,X^{u}(s),u(s))D^{W}_{s}F\mathrm{d}s\right]$$
$$\displaystyle+\mathbb{E}\left[F\int^{t}_{0}b(s,X^{u}(s),u(s))\mathrm{d}s\right].$$
Since $\beta(\cdot,X^{u},u)I_{[0,t]}$ is $\mathbb{H}$-adapted and square integrable, its Skorohod integral with respect to $W$ is well defined and coincides with the Itô integral. Thus, from (2.3), we have
$$\mathbb{E}\left[\int^{t}_{0}\beta(s,X^{u}(s),u(s))D^{W}_{s}F\mathrm{d}s\right]=\mathbb{E}\left[F\int^{t}_{0}\beta(s,X^{u}(s),u(s))\mathrm{d}W(s)\right].$$
Consequently,
$$\mathbb{E}\left[\int^{t}_{0}(\mathcal{K}^{*}\mathcal{K}D^{B}F)(s)\sigma(s)X^{u}(s)\mathrm{d}s\right]=\mathbb{E}\left[FG(t)\right],$$
where
$$G(t)=X^{u}(t)-\left(x_{0}+\int^{t}_{0}\beta(s,X^{u}(s),u(s))\mathrm{d}W(s)+\int^{t}_{0}b(s,X^{u}(s),u(s))\mathrm{d}s\right).$$
Observing that $\sigma X^{u}\in L^{2}_{\mathbb{H}}(\Omega\times[0,T])$ and $G(t)\in L^{2}(\Omega)$, we conclude from (2.2) that
$\sigma X^{u}I_{[0,t]}\in Dom\ \delta^{B}$ and
$$\int^{t}_{0}\sigma(s)X^{u}(s)\mathrm{d}B^{H}(s)=\delta(\sigma X^{u}I_{[0,t]})=G(t).$$
This proves $X^{u}\in L^{2}_{\mathbb{H}}(\Omega\times[0,T])$ is a solution of (3.8).
Step 3: Now we prove the uniqueness. Suppose $X^{u}\in L^{2}_{\mathbb{H}}(\Omega\times[0,T])$ is a solution of (3.8) such that $\sigma X^{u}I_{[0,t]}\in\ Dom\ \delta^{B}$, for every $t\in[0,T]$. Define $\eta$ as
$\eta(t)=X^{u}(t,\mathcal{T}_{t})\kappa_{t}^{-1}(\mathcal{T}_{t})$, $t\in[0,T]$. Then we have $\eta\in L^{q}_{\mathbb{H}}(\Omega\times[0,T])$, $1<q<2.$
For any $F\in\mathcal{S}_{\mathcal{K}}$, we have
$$\displaystyle\mathbb{E}\left[F\eta(t)-FX^{u}_{0}\right]=\mathbb{E}\left[F(A_{t})X^{u}_{t}-FX_{0}^{u}\right]$$
$$\displaystyle=$$
$$\displaystyle\mathbb{E}\left[F(\mathcal{A}_{t})\left\{X^{u}_{0}+\int^{t}_{0}\sigma(s)X^{u}(s)\mathrm{d}B(s)+\int^{t}_{0}b(s,X^{u}(s),u(s))\mathrm{d}s+\int^{t}_{0}\beta(s,X^{u}(s),u(s))\mathrm{d}W(s)\right\}\right]$$
$$\displaystyle-\mathbb{E}\left[FX^{u}_{0}\right].$$
From the fact that
$$\frac{\mathrm{d}F(\mathcal{A}_{t})}{\mathrm{d}t}=-\sigma(t)(\mathcal{K}^{*}\mathcal{K}D^{B}F)(\mathcal{A}_{t},t),$$
and applying the same method as in Step 2, we deduce
$$\displaystyle\mathbb{E}\left[F\eta(t)-Fx_{0}\right]$$
$$\displaystyle=$$
$$\displaystyle\mathbb{E}\bigg{[}-x_{0}\int^{t}_{0}\sigma(s)(\mathcal{K}^{*}\mathcal{K}D^{B}F(\mathcal{A}_{s}))(s)\mathrm{d}s+\int^{t}_{0}(\mathcal{K}^{*}\mathcal{K}D^{B}F(\mathcal{A}_{s}))(s)\sigma(s)X^{u}(s)\mathrm{d}s$$
$$\displaystyle\qquad-\int^{t}_{0}\int^{r}_{0}\sigma(r)\mathcal{K}^{*}\mathcal{K}D^{B}(\mathcal{K}^{*}\mathcal{K}D^{B}F(\mathcal{A}_{r})(r))(s)\sigma(s)X^{u}(s)\mathrm{d}s\mathrm{d}r$$
$$\displaystyle\qquad+\int^{t}_{0}F(\mathcal{A}_{s})b(s,X^{u}(s),u(s))\mathrm{d}s-\int^{t}_{0}\int^{r}_{0}\sigma(r)\mathcal{K}^{*}\mathcal{K}D^{B}F(\mathcal{A}_{r})(r)b(s,X^{u}(s),u(s))\mathrm{d}s\mathrm{d}r$$
$$\displaystyle\qquad+\int^{t}_{0}(D^{W}F(\mathcal{A}_{s}))(s)\beta(s,X^{u}(s),u(s))\mathrm{d}s$$
$$\displaystyle\qquad-\int^{t}_{0}\int^{r}_{0}\sigma(r)D^{W}(\mathcal{K}^{*}\mathcal{K}D^{B}F(\mathcal{A}_{r})(r))\beta(s,X^{u}(s),u(s))\mathrm{d}s\mathrm{d}r\bigg{]}.$$
Since $X^{u}$ is a solution of (3.8), we derive that
$$\mathbb{E}\left[F\eta(t)-Fx_{0}\right]=\mathbb{E}\left[\int^{t}_{0}F(\mathcal{A}_{s})b(s,X^{u}(s),u(s))\mathrm{d}s+\int^{t}_{0}(D^{W}F(\mathcal{A}_{s}))(s)\beta(s,X^{u}(s),u(s))\mathrm{d}s\right].$$
We apply again the Girsanov Theorem and (2.3). Then
$$\displaystyle\mathbb{E}\left[F\eta(t)-Fx_{0}\right]$$
$$\displaystyle=$$
$$\displaystyle\mathbb{E}\left[F\int^{t}_{0}b(s,\eta(s)\kappa_{s}(\mathcal{T}_{s}),u(s,\mathcal{T}_{s}))\kappa_{s}^{-1}(\mathcal{T}_{s})\mathrm{d}s+F\int^{t}_{0}\beta(s,\eta(s)\kappa_{s}(\mathcal{T}_{s}),u(s,\mathcal{T}_{s}))\kappa_{s}^{-1}(\mathcal{T}_{s})\mathrm{d}W(s)\right].$$
From the arbitrariness of $F\in\mathcal{S}_{\mathcal{K}}$, we get
$$\eta(t)=x_{0}+\int^{t}_{0}b(s,\eta(s)\kappa_{s}(\mathcal{T}_{s}),u(s,\mathcal{T}_{s}))\kappa_{s}^{-1}(\mathcal{T}_{s})\mathrm{d}s+\int^{t}_{0}\beta(s,\eta(s)\kappa_{s}(\mathcal{T}_{s}),u(s,\mathcal{T}_{s}))\kappa_{s}^{-1}(\mathcal{T}_{s})\mathrm{d}W(s),$$
$t\in[0,T]$.
But the $\mathbb{H}$-adapted continuous solution of this equation is unique and standard estimates show that it belongs to $S^{2}_{\mathbb{H}}$. Hence, $\eta\in L^{2}_{\mathbb{H}}(\Omega\times[0,T])$ is a solution of (3.10). Since equation (3.10) admits a unique solution, we have proved the uniqueness.
$\Box$
Let us present now the
Proof of Lemma 3.3: In this proof, for simplicity of notations, we make the conventions that
$V^{\varepsilon}(s):=(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v^{\varepsilon}(s))$ and $V(s):=(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))$.
Putting
$$I=\int^{t}_{0}b\big{(}s,(y(s)+y_{3}(s))\kappa_{s}(\mathcal{T}_{s}),v^{\varepsilon}(s)\big{)}\kappa_{s}^{-1}(\mathcal{T}_{s})\mathrm{d}s+\int^{t}_{0}\beta\big{(}s,(y(s)+y_{3}(s))\kappa_{s}(\mathcal{T}_{s}),v^{\varepsilon}(s)\big{)}\kappa_{s}^{-1}(\mathcal{T}_{s})\mathrm{d}W(s),$$
we have, from the Taylor expansion, that
$$\displaystyle I=$$
$$\displaystyle\int^{t}_{0}\bigg{[}b\big{(}V^{\varepsilon}(s)\big{)}\kappa_{s}^{-1}(\mathcal{T}_{s})+b_{x}\big{(}V^{\varepsilon}(s)\big{)}y_{3}(s)$$
$$\displaystyle\qquad+\left(\int^{1}_{0}\int^{1}_{0}\lambda b_{xx}\big{(}s,y(s)\kappa_{s}(\mathcal{T}_{s})+\lambda\mu y_{3}(s)\kappa_{s}(\mathcal{T}_{s}),v^{\varepsilon}(s)\big{)}\mathrm{d}\lambda\mathrm{d}\mu\right)y^{2}_{3}(s)\kappa_{s}(\mathcal{T}_{s})\bigg{]}\mathrm{d}s$$
$$\displaystyle+\int^{t}_{0}\bigg{[}\beta\big{(}V^{\varepsilon}(s)\big{)}\kappa_{s}^{-1}(\mathcal{T}_{s})+\beta_{x}\big{(}V^{\varepsilon}(s)\big{)}y_{3}(s)$$
$$\displaystyle\qquad\quad+\left(\int^{1}_{0}\int^{1}_{0}\lambda\beta_{xx}\big{(}s,y(s)\kappa_{s}(\mathcal{T}_{s})+\lambda\mu y_{3}(s)\kappa_{s}(\mathcal{T}_{s}),v^{\varepsilon}(s)\big{)}\mathrm{d}\lambda\mathrm{d}\mu\right)y^{2}_{3}(s)\kappa_{s}(\mathcal{T}_{s})\bigg{]}\mathrm{d}W(s),$$
which can be rewritten as
$$\displaystyle I$$
$$\displaystyle=$$
$$\displaystyle\int^{t}_{0}b\big{(}V(s)\big{)}\kappa_{s}^{-1}(\mathcal{T}_{s})\mathrm{d}s+\int^{t}_{0}\beta\big{(}V(s)\big{)}\kappa_{s}^{-1}(\mathcal{T}_{s})\mathrm{d}W(s)$$
$$\displaystyle+\int^{t}_{0}b_{x}\big{(}V(s)\big{)}y_{3}(s)\mathrm{d}s+\int^{t}_{0}\beta_{x}\big{(}V(s)\big{)}y_{3}(s)\mathrm{d}W(s)$$
$$\displaystyle+\int^{t}_{0}\left(b\big{(}V^{\varepsilon}(s)\big{)}-b\big{(}V(s)\big{)}\right)\kappa_{s}^{-1}(\mathcal{T}_{s})\mathrm{d}s+\int^{t}_{0}\left(\beta\big{(}V^{\varepsilon}(s)\big{)}-\beta\big{(}V(s)\big{)}\right)\kappa_{s}^{-1}(\mathcal{T}_{s})\mathrm{d}W(s)$$
$$\displaystyle+\frac{1}{2}\int^{t}_{0}b_{xx}\left(V^{\varepsilon}(s)\right)y_{3}^{2}(s)\kappa_{s}(\mathcal{T}_{s})\mathrm{d}s+\frac{1}{2}\int^{t}_{0}\beta_{xx}\big{(}V^{\varepsilon}(s)\big{)}y_{3}^{2}(s)\kappa_{s}(\mathcal{T}_{s})\mathrm{d}W(s)$$
$$\displaystyle+\int^{t}_{0}\left(b_{x}\big{(}V^{\varepsilon}(s)\big{)}-b_{x}\big{(}V(s)\big{)}\right)y_{3}(s)\mathrm{d}s+\int^{t}_{0}\left(\beta_{x}\big{(}V^{\varepsilon}(s)\big{)}-\beta_{x}\big{(}V(s)\big{)}\right)y_{3}(s)\mathrm{d}W(s)$$
$$\displaystyle+\int^{t}_{0}\bigg{(}\int^{1}_{0}\int^{1}_{0}\lambda\Big{(}b_{xx}\big{(}s,y(s)\kappa_{s}(\mathcal{T}_{s})+\lambda\mu y_{3}(s)\kappa_{s}(\mathcal{T}_{s}),v^{\varepsilon}(s)\big{)}-b_{xx}\big{(}V(s)\big{)}\Big{)}\mathrm{d}\lambda\mathrm{d}\mu\bigg{)}y^{2}_{3}(s)\kappa_{s}(\mathcal{T}_{s})\mathrm{d}s$$
$$\displaystyle+\int^{t}_{0}\bigg{(}\int^{1}_{0}\int^{1}_{0}\lambda\Big{(}\beta_{xx}\big{(}s,y(s)\kappa_{s}(\mathcal{T}_{s})+\lambda\mu y_{3}(s)\kappa_{s}(\mathcal{T}_{s}),v^{\varepsilon}(s)\big{)}$$
$$\displaystyle\qquad\qquad\qquad\quad-\beta_{xx}\big{(}V(s)\big{)}\Big{)}\mathrm{d}\lambda\mathrm{d}\mu\bigg{)}y^{2}_{3}(s)\kappa_{s}(\mathcal{T}_{s})\mathrm{d}W(s).$$
Consequently, according to the definitions of $y_{1}$ and $y_{2}$, we get
$$I=y(t)+y_{3}(t)+x_{0}+\int^{t}_{0}G^{\varepsilon}(s)\mathrm{d}s+\int^{t}_{0}\Lambda^{\varepsilon}(s)\mathrm{d}W(s),$$
(5.37)
where
$$\displaystyle G^{\varepsilon}(s)=$$
$$\displaystyle\frac{1}{2}b_{xx}\big{(}V^{\varepsilon}(s)\big{)}\left(y_{2}^{2}(s)+2y_{1}(s)y_{2}(s)\right)\kappa_{s}(\mathcal{T}_{s})+\Big{(}b_{x}\big{(}V^{\varepsilon}(s)\big{)}-b_{x}\big{(}V(s)\big{)}\Big{)}y_{2}(s)$$
(5.38)
$$\displaystyle+\bigg{(}\int^{1}_{0}\int^{1}_{0}\lambda\Big{(}b_{xx}\big{(}s,y(s)\kappa_{s}(\mathcal{T}_{s})+\lambda\mu y_{3}(s)\kappa_{s}(\mathcal{T}_{s}),v^{\varepsilon}(s)\big{)}-b_{xx}\big{(}V(s)\big{)}\Big{)}\mathrm{d}\lambda\mathrm{d}\mu\bigg{)}y^{2}_{3}(s)\kappa_{s}(\mathcal{T}_{s})$$
and
$$\displaystyle\Lambda^{\varepsilon}(s)=$$
$$\displaystyle\frac{1}{2}\beta_{xx}\big{(}V^{\varepsilon}(s)\big{)}\left(y_{2}^{2}(s)+2y_{1}(s)y_{2}(s)\right)\kappa_{s}(\mathcal{T}_{s})+\Big{(}\beta_{x}\big{(}V^{\varepsilon}(s)\big{)}-\beta_{x}\big{(}V(s)\big{)}\Big{)}y_{2}(s)$$
(5.39)
$$\displaystyle+\bigg{(}\int^{1}_{0}\int^{1}_{0}\lambda\Big{(}\beta_{xx}\big{(}s,y(s)\kappa_{s}(\mathcal{T}_{s})+\lambda\mu y_{3}(s)\kappa_{s}(\mathcal{T}_{s}),v^{\varepsilon}(s)\big{)}-\beta_{xx}\big{(}V(s)\big{)}\Big{)}\mathrm{d}\lambda\mathrm{d}\mu\bigg{)}y^{2}_{3}(s)\kappa_{s}(\mathcal{T}_{s}).$$
We consider now the estimate of $\sup_{0\leq t\leq T}\mathbb{E}\left[\left|\int^{t}_{0}\Lambda^{\varepsilon}(s)\mathrm{d}W(s)\right|^{2}\right]$. From the Burkhölder-Davis-Gundy inequality we have
$$\displaystyle\sup_{0\leq t\leq T}\mathbb{E}\left[\left|\int^{t}_{0}\Lambda^{\varepsilon}(s)\mathrm{d}W(s)\right|^{2}\right]\leq C\mathbb{E}\left[\int^{T}_{0}\left|\Lambda^{\varepsilon}(s)\right|^{2}\mathrm{d}s\right]$$
$$\displaystyle\leq$$
$$\displaystyle C\mathbb{E}\bigg{[}\int^{T}_{0}\left(y^{2}_{2}(s)+2y_{1}(s)y_{2}(s)\right)^{2}\kappa_{s}^{2}(\mathcal{T}_{s})\mathrm{d}s+\int^{T}_{0}\Big{(}\beta_{x}\big{(}V^{\varepsilon}(s)\big{)}-\beta_{x}\big{(}V(s)\big{)}\Big{)}^{2}y_{2}^{2}(s)\mathrm{d}s\bigg{]}$$
$$\displaystyle+\int^{T}_{0}y^{4}_{3}(s)\kappa_{s}^{2}(\mathcal{T}_{s})\left(\int^{1}_{0}\left(\left|\beta_{xx}\big{(}s,y(s)\right|\kappa_{s}(\mathcal{T}_{s})+\theta\left|y_{3}(s)\right|\kappa_{s}(\mathcal{T}_{s}),v^{\varepsilon}(s)\big{)}+\left|\beta_{xx}\big{(}V(s)\big{)}\right|\right)\mathrm{d}\theta\right)^{2}\mathrm{d}s.$$
By using the estimates in Lemma 3.2 and applying the Dominated Convergence Theorem, we obtain
$$\sup_{0\leq t\leq T}\mathbb{E}\left[\left|\int^{t}_{0}\Lambda^{\varepsilon}(s)\mathrm{d}W(s)\right|^{2}\right]=o(\varepsilon^{2}).$$
Similar arguments can be applied to estimate $\sup_{0\leq t\leq T}\mathbb{E}\left[\left|\int^{t}_{0}G^{\varepsilon}(s)\mathrm{d}s\right|^{2}\right]$. Hence we have
$$\sup_{0\leq t\leq T}\mathbb{E}\left[\left|\int^{t}_{0}G^{\varepsilon}(s)\mathrm{d}s\right|^{2}+\left|\int^{t}_{0}\Lambda^{\varepsilon}(s)\mathrm{d}W(s)\right|^{2}\right]=o(\varepsilon^{2}).$$
Therefore, from (5.37) and the definition of $y^{\varepsilon}$, we get that
$$\displaystyle y^{\varepsilon}(t)-y(t)-y_{3}(t)=$$
$$\displaystyle\int^{t}_{0}A^{\varepsilon}(s)(y^{\varepsilon}(s)-y(s)-y_{3}(s))\mathrm{d}s+\int^{t}_{0}\Theta^{\varepsilon}(s)(y^{\varepsilon}(s)-y(s)-y_{3}(s))\mathrm{d}W(s)$$
$$\displaystyle+\int^{t}_{0}G^{\varepsilon}(s)\mathrm{d}s+\int^{t}_{0}\Lambda^{\varepsilon}(s)\mathrm{d}W(s),$$
with the both factors
$$A^{\varepsilon}(s)=b_{x}(s,(y^{\varepsilon}(s)-\theta(y(s)+y_{3}(s))),v^{\varepsilon}(s)),\ \theta\in[0,1],$$
and
$$\Theta^{\varepsilon}(s)=\beta_{x}(s,(y^{\varepsilon}(s)-\lambda(y(s)+y_{3}(s))),v^{\varepsilon}(s)),\ \lambda\in[0,1],$$
which are being uniformly bounded according to (H2). Finally, we can derive our estimate by applying standard arguments. $\Box$
Proof of Lemma 3.4: From the optimality of $(y(\cdot),v(\cdot))$, we have
$$\displaystyle 0\leq J(v^{\varepsilon})-J(v)=$$
$$\displaystyle\mathbb{E}\left[\int^{T}_{0}f(s,y^{\varepsilon}(s)\kappa_{s}(\mathcal{T}_{s}),v^{\varepsilon}(s))\varepsilon^{-1}_{s}(\mathcal{T}_{s})\mathrm{d}s+\Phi(y^{\varepsilon}(T)\kappa_{T}(\mathcal{T}_{T}))\kappa_{T}^{-1}(\mathcal{T}_{T})\right]$$
(5.40)
$$\displaystyle-$$
$$\displaystyle\mathbb{E}\left[\int^{T}_{0}f(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))\varepsilon^{-1}_{s}(\mathcal{T}_{s})\mathrm{d}s+\Phi(y(T)\kappa_{T}(\mathcal{T}_{T}))\kappa_{T}^{-1}(\mathcal{T}_{T})\right].$$
The Lemmata 3.2 and 3.3 lead to
$$\displaystyle 0$$
$$\displaystyle\leq$$
$$\displaystyle\mathbb{E}\left[\int^{T}_{0}\Big{(}f(s,(y(s)+y_{3}(s))\kappa_{s}(\mathcal{T}_{s}),v^{\varepsilon}(s))-f(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))\Big{)}\kappa_{s}^{-1}(\mathcal{T}_{s})\mathrm{d}s\right]$$
$$\displaystyle+\mathbb{E}\left[\Big{(}\Phi((y(T)+y_{3}(T))\kappa_{T}(\mathcal{T}_{T}))-\Phi(y(T)\kappa_{T}(\mathcal{T}_{T}))\Big{)}\kappa_{T}^{-1}(\mathcal{T}_{T})\right]+o(\varepsilon)$$
$$\displaystyle=$$
$$\displaystyle\mathbb{E}\left[\int^{T}_{0}\Big{(}f(s,(y(s)+y_{3}(s))\kappa_{s}(\mathcal{T}_{s}),v(s))-f(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))\Big{)}\kappa_{s}^{-1}(\mathcal{T}_{s})\mathrm{d}s\right]$$
$$\displaystyle+\mathbb{E}\left[\Big{(}\Phi((y(T)+y_{3}(T))\kappa_{T}(\mathcal{T}_{T}))-\Phi(y(T)\kappa_{T}(\mathcal{T}_{T}))\Big{)}\kappa_{T}^{-1}(\mathcal{T}_{T})\mathrm{d}s\right]$$
$$\displaystyle+\mathbb{E}\left[\int^{T}_{0}\Big{(}f(s,(y(s)+y_{3}(s))\kappa_{s}(\mathcal{T}_{s}),v^{\varepsilon}(s))-f(s,(y(s)+y_{3}(s))\kappa_{s}(\mathcal{T}_{s}),v(s))\Big{)}\kappa_{s}^{-1}(\mathcal{T}_{s})\mathrm{d}s\right]+o(\varepsilon).$$
Hence, by applying Taylor’s expansion up to the second order, we obtain
$$\displaystyle 0$$
$$\displaystyle\leq$$
$$\displaystyle\mathbb{E}\left[\int^{T}_{0}\Big{(}f_{x}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))y_{3}(s)+\frac{1}{2}f_{xx}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))\Big{)}y_{3}^{2}(s)\kappa_{s}(\mathcal{T}_{s})\mathrm{d}s\right]$$
$$\displaystyle+\mathbb{E}\left[\int^{T}_{0}\Big{(}f(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v^{\varepsilon}(s))-f(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))\Big{)}\kappa_{s}^{-1}(\mathcal{T}_{s})\mathrm{d}s\right]$$
$$\displaystyle+\mathbb{E}\left[\int^{T}_{0}\Big{(}f_{x}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v^{\varepsilon}(s))-f_{x}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))\Big{)}y_{3}(s)\mathrm{d}s\right]$$
$$\displaystyle+\mathbb{E}\left[\frac{1}{2}\int^{T}_{0}\Big{(}f_{xx}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v^{\varepsilon}(s))-f_{xx}(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))\Big{)}y_{3}^{2}(s)\kappa_{s}(\mathcal{T}_{s})\mathrm{d}s\right]$$
$$\displaystyle+\mathbb{E}\left[\Phi_{x}(y(T)\kappa_{T}(\mathcal{T}_{T}))y_{3}(T)+\frac{1}{2}\Phi_{xx}(y(T)\kappa_{T}(\mathcal{T}_{T}))y_{3}^{2}(T)\kappa_{T}(\mathcal{T}_{T})\right]+o(\varepsilon).$$
The desired inequality is obtained by using the hypothesis (H2) and Lemma 3.2. $\Box$
Proof $($of $($3.31$)$$)$: For any $\tilde{v}\in\mathcal{U}_{ad}$, we define a new admissible control
$$\tilde{v}^{\varepsilon}(t)=\left\{\begin{array}[]{ll}\tilde{v}(t),&t\in[\tau-\varepsilon,\tau+\varepsilon];\\
v(t),&t\in[0,T]\backslash[\tau-\varepsilon,\tau+\varepsilon].\end{array}\right.$$
Let us put
$$\displaystyle\Theta^{\tilde{v}}(s):=$$
$$\displaystyle H(s,y(s),\tilde{v}^{\varepsilon}(s),p(s),K(s))-H(s,y(s),v(s),p(s),K(s))$$
$$\displaystyle+\frac{1}{2}\kappa^{-2}_{s}(\mathcal{T}_{s})\Big{(}\beta(s,y(s)\kappa_{s}(\mathcal{T}_{s}),\tilde{v}^{\varepsilon}(s))-\beta(s,y(s)\kappa_{s}(\mathcal{T}_{s}),v(s))\Big{)}^{2}P(s),$$
and assume that (3.31) does not hold. Then there exist $\delta>0$ and an admissible control $\tilde{v}$ such that
the set $\Lambda^{\tilde{v}}:=\{(s,\omega):\Theta^{\tilde{v}}(s)(\omega)\leq-\delta\}$ satisfies
$$\mathbb{E}\left[\lambda\left(\Lambda^{\tilde{v}}\right)\right]\geq\delta>0,$$
(5.41)
where $\lambda\left(\Lambda^{\tilde{v}}\right)=\int^{T}_{0}I_{\Lambda^{\tilde{v}}}(s)\mathrm{d}s$. We derive from (5.41) that
$$\mathbb{E}\left[\lambda\left(\Lambda^{\tilde{v}}\cap[0,T/2]\right)\right]\geq\delta/2\ \ \textrm{or}\ \ \mathbb{E}\left[\lambda\left(\Lambda^{\tilde{v}}\cap[T/2,T]\right)\right]\geq\delta/2.$$
(5.42)
Hence there exists $\tau_{1}\in[0,T/2]$ such that
$$\mathbb{E}\left[\lambda\left(\Lambda^{\tilde{v}}\cap[\tau_{1},\tau_{1}+T/2]\right)\right]\geq\delta/2.$$
(5.43)
Similarly, from (5.43), we get that there exists $\tau_{2}\in[\tau_{1},\tau_{1}+T/4]$ such that
$$\mathbb{E}\left[\lambda\left(\Lambda^{\tilde{v}}\cap[\tau_{2},\tau_{2}+T/4]\right)\right]\geq\delta/4,$$
(5.44)
etc. Consequently, for $n\geq 2$, there exists $\tau_{n}\in[\tau_{n-1},\tau_{n-1}+T/2^{n}]$ such that
$$\mathbb{E}\left[\lambda\left(\Lambda^{\tilde{v}}\cap[\tau_{n},\tau_{n}+T/2^{n}]\right)\right]\geq\delta/2^{n}.$$
(5.45)
Furthermore, there exists $\tau\in[0,T]$ with $\tau_{n}\to\tau$ $(n\to\infty)$, and $|\tau_{n}-\tau|\leq T/2^{n}$, $n\geq 1.$ Hence, we have
$$\mathbb{E}\left[\lambda\left(\Lambda^{\tilde{v}}\cap[\tau-\varepsilon_{n},\tau+\varepsilon_{n}]\right)\right]\geq\mathbb{E}\left[\lambda\left(\Lambda^{\tilde{v}}\cap[\tau_{n},\tau_{n}+\varepsilon_{n}/2]\right)\right]\geq\delta\varepsilon_{n}/2T,$$
(5.46)
where $\varepsilon_{n}=2T/2^{n}$, $n\geq 1$.
We define
$$v^{\varepsilon_{n}}(t)=\left\{\begin{array}[]{ll}\overline{v}(t),&t\in[\tau-\varepsilon_{n},\tau+\varepsilon_{n}];\\
v(t),&t\in[0,T]\backslash[\tau-\varepsilon_{n},\tau+\varepsilon_{n}],\end{array}\right.$$
where $\overline{v}(t):=\tilde{v}(t)I_{\left\{\Theta^{\tilde{v}}\leq-\delta,\ t\in[\tau-\varepsilon_{n},\tau+\varepsilon_{n}]\right\}}+v(t)I_{\left\{\Theta^{\tilde{v}}>-\delta\ \textrm{or}\ t\notin[\tau-\varepsilon_{n},\tau+\varepsilon_{n}]\right\}}$. It follows that
$$\Theta^{\overline{v}}(t)=\Theta^{\tilde{v}}(t)I_{\left\{\Theta^{\tilde{v}}\leq-\delta,\ t\in[\tau-\varepsilon_{n},\tau+\varepsilon_{n}]\right\}}(t).$$
From (3.30), we derive that
$$\displaystyle o(\varepsilon_{n})\leq$$
$$\displaystyle\mathbb{E}\left[\int^{\tau+\varepsilon_{n}}_{\tau-\varepsilon_{n}}\Theta^{\overline{v}}(t)\mathrm{d}t\right]=\mathbb{E}\left[\int^{\tau+\varepsilon_{n}}_{\tau-\varepsilon_{n}}\Theta^{\overline{v}}(t)I_{\left\{\Theta^{\tilde{v}}\leq-\delta\right\}}\mathrm{d}t\right]$$
$$\displaystyle\leq$$
$$\displaystyle-\delta\mathbb{E}\left[\lambda\left(\Lambda^{\tilde{v}}\cap[\tau-\varepsilon_{n},\tau+\varepsilon_{n}]\right)\right]\leq-\delta\varepsilon_{n}/2T,\ n\geq 1.$$
This leads to contradiction. Consequently, $\Theta^{\tilde{v}}\geq 0$, a.s., $\mathrm{d}s$-a.e., for any $\tilde{v}\in\mathcal{U}_{ad}$, in particular, for $\tilde{v}\equiv v$, an $\mathcal{F}_{\tau}^{B}$-measurable random variable. $\Box$
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Rotation as a source of asymmetry
in the double-peak
lightcurves
of the bright EGRET pulsars
J. Dyks
1Copernicus Astronomical Center, Toruń, Poland
1
B. Rudak
1Copernicus Astronomical Center, Toruń, Poland
12Dept. of Astronomy and Astrophysics, Nicholas Copernicus University, Toruń, Poland
2
jinx@ncac.torun.pl
(Received …; accepted …)
Key Words.:
stars: rotation – pulsars: general
††offprints: J. Dyks,
We investigate the role of rotational effects
in inducing asymmetry present above $\sim 5$ GeV in the double-peak lightcurves
of the bright EGRET pulsars:
Vela, Crab, and Geminga. According to Thompson (thompson2001 ),
the trailing peak dominates over the leading peak above $\sim 5$ GeV
consistently for all three pulsars, even though this is not the case
over the entire energy range of EGRET,
i.e. above $\sim 100$ MeV.
We present the results of
Monte Carlo simulations of electromagnetic cascades
in a pulsar magnetosphere
within a single-polar-cap scenario
with rotationally-induced propagation effects of the order
of $\beta$ (where
$\beta$ is the dimensionless local corotation velocity).
We find that even in the case
of nearly aligned rotators with spin periods of $P\sim 0.1$ s
rotation may lead to asymmetric (with respect to the magnetic axis)
magnetic photon absorption which in turn leads to
asymmetric gamma-ray pulse profiles.
The resulting features - softer spectrum of the leading peak and the
dominance of the trailing peak above $\sim 5$ GeV -
agree qualitatively with the EGRET data of the bright gamma-ray pulsars.
1 Introduction
Good quality gamma-ray data
for three pulsars - Vela, Crab, and Geminga - acquired with EGRET
aboard the CGRO tempts to
analyse the properties of pulsar high-energy radiation as a function
of photon energy and phase of rotation.
Gamma-ray spectra of pulsed radiation from these sources
(as well as from three other EGRET pulsars:
B1706-44, B1951+32, and B1055-52)
extend up to $\la 10$ GeV.
All three pulsars feature gamma lightcurves characterised
by two strong peaks separated
by 0.4 to 0.5 in rotational phase.
These double-peak pulses are asymmetrical
and their profiles change with energy.
Above $\sim 100$ MeV the leading peak (LP)
is stronger than the trailing peak (TP) in the case of the Vela and the Crab pulsars,
and the opposite is true for Geminga.
However, for all three pulsars
their leading peaks exhibit lower
energy cutoffs - around $\sim 5$ GeV - than the trailing peaks (TP).
In other words, the trailing peaks dominate over
the leading peaks above $\sim 5$ GeV (Thompson thompson2001 ).
In the case of the Vela pulsar and Geminga, this effect is accompanied by
the softening of the spectrum of the leading peak
(Fierro et al. fmnt , Kanbach kanbach99 ).
Potential importance of the double-peak pulse asymmetry in the case
of Vela was already acknowledged – at the time when the COS-B data
became available – by
Morini morini who attempted to explain the asymmetry
with a hybrid model, with two different mechanisms responsible
for the formation of the leading and the trailing peak.
High-energy cutoffs in pulsar spectra
are interpreted within polar cap models as due to one-photon absorption
of gamma-rays in strong magnetic field with subsequent $e^{\pm}$-pair creation.
A piece of observational support for such an interpretation comes from
a strong correlation between the inferred ‘spin-down’ magnetic field
strength and the position of the high-energy cutoff (Baring & Harding
bh2000 , Baring b2001 ).
This in turn opens a possibility that the observed asymmetry
between LP and TP, i.e.
the dominance of LP over TP above $\sim 5$ GeV,
is a direct consequence of propagation effects
(which eventually lead to stronger magnetic photon absorption
for photons forming LP than TP)
rather than due to some inherent property of the gamma-ray emission
region itself.
The aim of this paper is to investigate
the role of pulsar rotation in a built-up of such asymmetry
in the double-peak pulse profiles.
We consider purely rotational effects:
due to presence of rotation-induced electric
field $\@vec{E}$, aberration of photon direction and
slippage of magnetosphere under the photon’s path.
In section 2 we compare them with some other effects which may be responsible
for the asymmetry (like various distortions of the magnetic field structure).
In section 3 we show that the rotation effects
result in an asymmetric pair production rate for the leading and the
trailing part of the magnetosphere
even in the case when the magnetic
field structure and the population of radiating particles are symmetric
around the magnetic pole.
In section 4 asymmetric pulse profiles are calculated
as a function of photon energy and then
the model predictions of the ratio of fluxes in the
leading and trailing peaks
are compared with the inferred
ratio for Vela at different energy bins.
In section 5 we address the significance of rotation-driven asymmetry
across the pulsar parameter space.
Our main results are discussed in Section 6.
2 Symmetric features of the model
It has been argued in many studies of radio properties of pulsars
(eg. Blaskiewicz et al. bcw91 , Gangadhara & Gupta gg2001 )
that a rigidly rotating static-like dipole
can be used as a good approximation of dipolar magnetic field
as long as only
most important rotation effects, of the order of $\beta=v/c$ (where
$v$ is the local corotation velocity and $c$ is the speed of light), are to be
considered.
According to order-of-magnitude estimates,
small distortions of the dipolar magnetosphere induced by
rotationally-driven currents can be neglected:
longitudinal currents suspected to flow within the open field line region
cannot modify $\@vec{B}$ by a factor exceeding $\beta^{3/2}$ whereas
toroidal currents due to plasma corotation
change $\@vec{B}$ barely by a value of the order of $\beta^{2}$.
A more comprehensive discussion of the influence of currents on the
magnetospheric structure
can be found in Beskin (beskin ) and references therein.
Below we follow the approximation of a rigidly rotating static-like dipole:
at any instant the magnetic field has the shape of a static dipole
in the frame which corotates with a neutron star.
Moreover, the magnetosphere is assumed to be filled out everywhere
with the Goldreich-Julian charge density, so that a rotation-induced
electric field $\@vec{E}=-\@vec{\beta}\times\@vec{B}$
is present in the OF, whereas $\@vec{E}^{\prime}=0$
in the frame corotating with the star.
We neglect deviations from this corotational electric field which are present
in the charge-deficient polar gap region – they do not exceed a factor of $\beta^{3/2}$.
In our Monte Carlo simulations (section 4) development of gamma-ray radiation is based on
the model of Daugherty & Harding dh82 , with
primary electrons being
injected along magnetic field lines at an altitude of a few neutron star radii,
at the magnetic colatitude corresponding to
the last open magnetic field lines, and uniformly in the magnetic azimuth.
The electrons are assumed to accelerate instantly to the energy
$E_{0}\sim 10^{7}$ MeV
and subsequently to cool down emitting curvature photons. Some of the photons
induce in turn electromagnetic
cascades which propagate outwards in a form of a hollow cone beam
(see Dyks & Rudak dr2000 for detailed description of directionality aspects
of the casades as well as viewing geometry).
3 Rotation-driven asymmetry
In this section we present in detail the mechanisms which lead to rotation-induced
asymmetry in the (otherwise axially symmetric) hollow-cone gamma-ray beam.
For the sake of better demonstration of the effects we consider
an exaggerated case: propagation of a photon in the equatorial
plane of a fast orthogonal rotator.
Fig. 1 shows the case for the rotation period $P=1.5$ ms:
photons of the same energy in the corotating frame (CF)
are emitted from two opposite points on the outer rim of the polar cap
tangentially to the magnetic field lines in the CF.
To follow their straight-line propagation in the inertial observer frame (OF)
three effects have to be taken into account:
1) In the OF the photons propagate at an aberrated direction
(dotted lines in Fig. 1) and differ in energy.
2) The rotation-induced electric field $\@vec{E}=-\@vec{\beta}\times\@vec{B}$,
which is present in the OF, modifies the rate of the magnetic photon
absorption in a different way for photons forming the LP than
for photons forming the TP.
3) The electromagnetic field in the OF is time-dependent: because of
the rotation the photons propagate through different parts of the
magnetosphere.
We find that the second effect - due to rotationally induced $\@vec{E}$ - plays a dominant role
in generating the asymmetry in the magnetic absorption rate $R$ between
photons of the LP and the TP. An importance of a weak electric field $\@vec{E}\perp\@vec{B}$
for the rate $R$ was for the first time recognized by Daugherty & Lerche (dl75 ) who
presented also its quantitative treatment.
A consequence of its presence is that
the rate $R$ does not vanish along the direction of local $\@vec{B}$ any more, and
becomes non-axisymmetric with respect to it as well.
Instead, the rate $R$ vanishes along two new directions which lie in the plane
perpendicular to $\@vec{E}$ and deviate from $\@vec{B}$ by angle
$\sim E/B$ in such a way that
in a local coordinate frame with $\hat{z}\parallel\@vec{B}$,
$\hat{y}\parallel\@vec{E}$, and $\hat{x}\parallel\@vec{E}\times\@vec{B}$ the
“free propagation” direction $\hat{\eta}^{\scriptscriptstyle FP}=[\eta^{\scriptscriptstyle FP}_{x},\eta^{%
\scriptscriptstyle FP}_{y},\eta^{\scriptscriptstyle FP}_{z}]$
has two solutions: $[E/B,0,\pm(1-E^{2}/B^{2})^{1/2}]$.
For definiteness, hereafter we will consider
photons which propagate outwards within the regions
above the northern magnetic pole (i.e.
with propagation vectors $\hat{\eta}$ satisfying $\hat{\eta}\cdot\@vec{B}>0$)
which corresponds to the case of
$\eta^{\scriptscriptstyle FP}_{z}=+(1-E^{2}/B^{2})^{1/2}$.
Fig. 2 shows the mean free path $\lambda_{\rm mfp}=c/R$
for the magnetic photon absorption for different directions
in the plane perpendicular to $\@vec{E}$.
The rate $R$ was calculated by performing Lorentz transformation
to a frame in which $\@vec{E}^{\prime}=0$ and then applying a purely magnetic
formula.
The formula of Erber (er66 )
with a modification of Daugherty & Harding (dh83 ) correcting its near-threshold inaccuracy
is used throughout the paper:
$$\displaystyle R^{\prime}(\varepsilon^{\prime},\@vec{B}^{\prime},\sin\theta_{B}%
^{\prime})=\hbox{}$$
(1)
$$\displaystyle\hbox{}=c_{1}\sin\theta_{B}^{\prime}B^{\prime}\exp{[-c_{2}f/(%
\varepsilon^{\prime}\sin\theta_{B}^{\prime}B^{\prime})]},$$
where $\theta_{B}^{\prime}=\angle(\hat{\eta}^{\prime},\@vec{B}^{\prime})$,
$\varepsilon^{\prime}$ is the photon energy; $c_{1}$ and $c_{2}$ are constant quantities,
while $f=f(\varepsilon^{\prime},\@vec{B}^{\prime})$ is the near-threshold correction of Daugherty & Harding dh83 .
Six lines in Fig. 2 correspond to six values of the ratio $E/B$: $0.01$, $0.1$,
$0.2$, $0.5$, $0.9$, and $0.999$.
Note that the free propagation direction deviates from the local $\@vec{B}$
by angle $\theta^{\scriptscriptstyle FP}$ which increases with increasing contribution of the electric
field: $\theta^{\scriptscriptstyle FP}_{z}\equiv\arccos(\eta^{\scriptscriptstyle FP}_{%
z})=\arcsin(E/B)\simeq E/B$.
Moreover, $R$ increases ($\lambda_{\rm mfp}$ decreases) monotonically with
increasing angle $\angle(\hat{\eta},\hat{\eta}^{\scriptscriptstyle FP})$. As long as $E\ll B$, magnetic photon absorption rate $R$
remains approximately symmetric around the free propagation direction $\hat{\eta}^{\scriptscriptstyle FP}$.
The directions of $\hat{\eta}^{\scriptscriptstyle FP}$ at various points within the magnetosphere
are shown in Fig. 1 in the OF (with solid arrows).
At the points of photon emission the direction $\hat{\eta}^{\scriptscriptstyle FP}$
overlaps with the photon direction $\hat{\eta}$ since
this is where the angle $\theta_{B}$ between the photon direction and the magnetic field
line equals
$\theta_{B}\simeq E/B$ and consequently
$R=0$ (Harding et al. hte78 , Zheng et al. zzq98 ).
Initially, therefore, the rate $R$
is the same for photons emitted from two opposite sites of the polar cap rim
which in this picture give rise to the leading peak and the trailing peak.
But as the photons propagate outward
their $\hat{\eta}$ begin to deviate from local
free propagation directions $\hat{\eta}^{\scriptscriptstyle FP}$ opening thus a possibility for magnetic photon absorption
and electron-positron pair creation. Three effects are responsible for it to happen:
(1) curvature of the magnetic field lines, (2) increase of $E/B$ with
altitude, and (3) slippage of magnetic field lines under the photon’s path.
For a static-like dipole (assumed in section 2)
the curvature of lines is symmetric with respect
to the magnetic axis. Consequently, symmetry is expected between the rates of magnetic absorption
for LP photons and TP photons.
Any asymmetry between the LP and the TP may occur via the effects (2) and (3) only.
Whenever the curvature of magnetic
field lines happens to be relatively large
(which is the case at the polar cap rims of pulsars with $P\sim 0.1$ s)
the effect (1) significantly dominates over (2) and (3), which
means that the resulting asymmetry is quite subtle.
However, for a relatively small curvature of magnetic
field lines (and appreciable local corotation velocities $\beta$)
the resulting asymmetry becomes pronounced
[note that these ’favourable’ conditions are fulfilled within
outer-gap accelerators].
We find that the effect (2) – the presence of electric
field – is crucial for generating the asymmetry.
The way in which this effect operates can be most clearly assessed
by inspecting Fig. 2.
Let us consider a photon propagation vector $\hat{\eta}$ anchored
at the origin of frame in the figure.
For photons emitted close to the star, both in the LP and the TP,
$\hat{\eta}$ initially points along the $\hat{\eta}^{\scriptscriptstyle FP}$ direction which differs only slightly
from the direction of $\@vec{B}$ in Fig. 2
since $E\ll B$.
As the LP photon moves away from emission point,
its propagation vector rotates clockwise in Fig. 2
which reflects the fact that $\@vec{B}$ diverges from $\hat{\eta}$
due to the magnetic field line curvature.
At the same time, however, the directional pattern of $\lambda_{\rm mfp}$
rotates counterclockwise due to increase in $E/B$
which additionally enhances the absorption rate.
In the case of the TP photon, however, both
its propagation vector $\hat{\eta}$ and the directional pattern
of $\lambda_{\rm mfp}$ rotate in the same direction
(counterclockwise) with respect to $\@vec{B}$ in Fig. 2
so that the absorption rate is weakened.
Thus, for photons within the LP the effects (1) and (2)
cummulate, whereas for the TP they effectively tend to cancel out
each other, and
the expected asymmetry between the peaks is due to stronger absorption suffered by
photons within the LP than by photons
of the same energy within the TP.
Accordingly,
the high energy cutoff in the LP spectrum will occur at
a slightly lower energy than the cutoff in the TP spectrum.
The slippage (3) affects this picture in the way which depends on
both the rotation period and the photon position within the magnetosphere,
but an overall picture remains unchanged. For most rotation periods ($P>$ a few ms),
the slippage reduces the asymmetry only marginally.
For the fastest rotators ($P\sim 1.5$ ms) it enhances
the asymmetry by making photons of the TP to propagate
along the free propagation direction (or, equivalently, along the
magnetic field lines in the CF).
The latter case is presented in Fig. 1 where the
photon propagation direction $\hat{\eta}$ as seen in the OF (dotted lines) and the local
free propagation direction $\hat{\eta}^{\scriptscriptstyle FP}$ in the OF (solid arrows) are shown for
several positions along the photon trajectory in the CF.
Strong absorption within
the LP is anticipated, whereas within the TP $\hat{\eta}$ nearly coincides with $\hat{\eta}^{\scriptscriptstyle FP}$.
Another way to understand this asymmetry is to
follow photon trajectories in a reference frame (with primed quantities) where the condition
$$\@vec{E}^{\prime}=0$$
(2)
is fulfilled.
In such a frame, an asymmetry in $R^{\prime}$ (cf. eq.(1))
for the LP and TP photons arises from transformation properties of
$\theta_{B}^{\prime}$ (aberration), $\varepsilon^{\prime}$ (red- or blue-shift),
and $B^{\prime}$.
One of the reference frames satisfying the condition (2) is a reference frame of local $\@vec{E}\times\@vec{B}$ drift.
Denoting dimensionless drift velocity $\@vec{\beta}_{\scriptscriptstyle D}=\@vec{E}\times\@vec{B}/B^{2}$,
the transformations read:
$$\displaystyle\varepsilon^{\prime}=\varepsilon\,\gamma_{\scriptscriptstyle D}(1%
-\eta_{x}\beta_{\scriptscriptstyle D}),\ \ \ \ \ \ \ \ \ \ \@vec{B}^{\prime}=%
\frac{\@vec{B}}{\gamma_{\scriptscriptstyle D}},$$
$$\displaystyle\sin\theta_{B}^{\prime}=\frac{[(\eta_{x}-\beta_{%
\scriptscriptstyle D})^{2}+\eta_{y}^{2}(1-\beta_{\scriptscriptstyle D}^{2})]^{%
1/2}}{(1-\eta_{x}\beta_{\scriptscriptstyle D})}.$$
(3)
In the equatorial plane of orthogonal rotator $\eta_{y}=0$
so that $\eta_{x}=\mp\sin\theta_{B}$, where the signs ‘minus’ and ‘plus’
correspond to the leading and the trailing peak, respectively.
The transformation of propagation angle reduces then to
$\sin\theta_{B}^{\prime}=|\eta_{x}-\beta_{\scriptscriptstyle D}|/(1-\eta_{x}%
\beta_{\scriptscriptstyle D})$.
The Taylor expansion of $\sin\theta_{B}^{\prime}$, $\varepsilon^{\prime}$,
and $B^{\prime}$ around $\beta_{\scriptscriptstyle D}=0$ reveals that
a difference between the magnetic absorption rates in the locally drifting frame and in the OF
results primarily from the aberration of photon direction, whereas the changes
in $\varepsilon$ and $\@vec{B}$ are second order effects.
Obviously, the aberration is asymmetric for the LP and the TP ($\eta_{x}<0$ and $\eta_{x}>0$, respectively).
Fig. 1 presents this ‘aberration effect’
in the rigidly corotating frame, where $\@vec{E}^{\prime}=0$ is assumed.
Photon trajectories in this frame
(dashed curves) indicate clearly that photons of the leading
peak encounter larger $B^{\prime}_{\perp}$
than photons of the trailing peak of the same energy.
4 Numerical modelling of the gamma-ray data
We performed Monte Carlo simulations of curvature-radiation-induced electromagnetic cascades
developing
above a polar cap. The cascade development due to magnetic photon absorption accompanied by $e^{\pm}$-pair creation
and then synchrotron emission was followed in a 3D space in order
to analyse pulse properties.
As an example we choose a model with basic parameters of the Vela pulsar:
$B_{\rm pc}=6.8$ TG, $P=0.0893$ s.
In order to meet observational restrictions for the Vela, both spectral and temporal,
the following general requirements within
polar-cap scenarios had to be satisfied:
1) a polar-cap accelerator should be placed a few stellar radii above pulsar’s surface (Dyks et al. alic );
2) an inclination angle $\alpha$ of the magnetic dipole with respect to the spin axis
must not be large, and the pulsar has to be a nearly-aligned rotator (Daugherty & Harding dh94 ).
Recently Harding & Muslimov (hm98 ) proposed a physical mechanism
for lifting the polar cap accelerator up to $0.5-1\ R_{\rm NS}$
above the surface. However, this altitude is still too low
to explain the $10$ GeV radiation emerging the Vela magnetosphere unattenuated.
Therefore, we placed the polar-cap accelerator at the
altitude of $h_{0}=4\ R_{\rm NS}$ to ensure that the magnetosphere
is not entirely opaque to curvature photons of
energy $\la 10$ GeV (see Dyks et al. alic for the detailed model spectral fitting for the Vela pulsar).
Similarly, Miyazaki & Takahara (mt97 )
achieved the best agreement between the observed and their
modelled pulse profiles
of the Crab pulsar
placing the accelerator at $h_{0}=4\ R_{\rm ns}$.
To reproduce
the observed peak-to-peak separation $\Delta^{\rm peak}\simeq 0.42$ (Kanbach et al. kab94 )
we assumed (after Dyks & Rudak dr2000 ) for the inclination angle
$\alpha$ and the observer’s angle $\zeta_{\rm obs}$ (an angle between the line-of-sight and
the spin axis) that $\alpha=\zeta_{\rm obs}=7.6^{\circ}$.
Our numerical results are presented in Fig. 3 (a + b).
The three columns of Fig. 3
show (from left to right): 1) mapping onto the parameter space $\zeta_{\rm obs}$ vs. $\phi$ of
outgoing photons with energy $\varepsilon>\varepsilon_{\rm limit}$ (where $\phi$ denotes a phase of rotation),
2) double-peak pulse profile due to these photons when $\zeta_{\rm obs}=7.6^{\circ}$, and 3) phase-integrated
energy spectrum of these photons, with the position of $\varepsilon_{\rm limit}$ indicated
with dotted vertical line.
The eight rows correspond to 8 different values of
$\varepsilon_{\rm limit}$: $1$, $10$, $10^{2}$, $10^{3}$, $4\cdot 10^{3}$, $6\cdot 10^{3}$,
$8\cdot 10^{3}$, and $10^{4}$ MeV (top to bottom).
An asymmetry in the double-peak profiles is noticable even
though the rotator is nearly aligned: at the highest energies, above
$\sim 6$ GeV, the leading peak LP is less intense than the trailing peak TP
(three lowermost panels in the middle column in Fig. 3 b).
This is a direct result of stronger magnetic absorption
of the LP photons comparing to the TP photons.
The distribution of these photons in the corresponding panels of
$\zeta_{\rm obs}$ vs. $\phi$ (the left column)
shows that at viewing angles $\zeta_{\rm obs}$
larger than $7.6^{\circ}$ (not allowed due to the fixed peak-to-peak separation of 0.42)
the asymmetry in pulse profile would be even stronger.
This demonstrates an increasing role of rotational effects
as the distance from the spin axis increases.
In the course of magnetic absorption high-energy curvature photons
are converted into electron-positron pairs which in turn emit low-energy synchrotron photons.
Asymmetry in the absorption rate as
discussed above means, therefore, an identical asymmetry in the $e^{\pm}$ pair production rate.
Consequently, higher number of low-energy synchrotron photons emerges at the LP than at the TP.
This is the reason for a dominance of the LP over the TP below $\sim 100$ MeV,
noticable in Fig. 3 a.
Combining the results from both energy domains,
a characteristic inversion in the relative strentgh of the LP and the TP occurs across the gamma-ray energy space.
A qualitatively similar inversion of peak intensities takes place in the gamma-ray double-pulse of the Vela pulsar
(Thompson thompson2001 ).
The beam of synchrotron radiation in our cascades occupies a very narrow range
of magnetic colatitudes; in other words - it is highly anisotropic.
The reasons for this include a very limited
range of altitudes at which the $e^{\pm}$ pairs are created
and the effects of relativistic beaming.
By comparison, curvature radiation below $\sim 100$ MeV is much less anisotropic.
Therefore, the prominent peaks visible at $\varepsilon<100$ MeV
(two uppermost panels of Fig. 3 a)
consist almost entirely of synchrotron radiation (SR) photons, whereas the apparently flat wings outside the peaks
(i.e. within the “offpulse” region corresponding to high altitudes) are composed of curvature radiation (CR) photons.
A close-up view of the double-peak pulse profile
for $\varepsilon>1$ MeV shown in Fig. 4 reveals that the CR wings
are not flat - in fact their intensity decreases with increasing phase $|\phi|$;
moreover, their shapes can be reproduced with analytical means:
Spectral power of curvature radiation
$\frac{{\rm d}P_{\rm cr}}{{\rm d}\varepsilon}$
well below a characteristic photon energy $\varepsilon_{\rm crit}\propto\frac{\gamma^{3}}{\rho_{\rm cr}}$ does not depend on the energy of radiating particles $\gamma$
but on the curvature radius $\rho_{\rm cr}$ of magnetic field lines solely.
Since primary electrons reside within a pulsar magnetosphere for a limited period of time
$\varepsilon_{\rm crit}$ has a lower limit which equals
roughly $\la 100$ MeV (see Rudak & Dyks (rd99 ) for details).
Therefore, the wings in the pulse profiles below $100$ MeV fall off
due exclusively to an increase in the curvature radius $\rho_{\rm cr}$ of magnetic field lines:
this proceeds according to the following relation
$$\frac{{\rm d}P_{\rm cr}}{{\rm d}\varepsilon}\propto\rho_{\rm cr}^{-2/3},$$
(4)
which then leads to the smooth solid lines in Fig. 4.
As the photon energy increases
and the strength of the synchrotron peaks decreases the curvature wings become more and more pronounced.
They are most noticeable near $100$ MeV.
Above $100$ MeV the wings gradually disappear (see the middle column in Fig. 3 b)
because radiating electrons are not energetic enough at high altitudes.
At the same time the peaks become narrower -
an effect noticed by Kanbach et al. (kab94 ) in the EGRET data for the Vela pulsar.
As noted by Daugherty & Harding (dh96 ) the wings within the offpulse region must not
be too strong within the entire energy range of EGRET if the theoretical pulse profiles are to
resemble those of the Vela pulsar.
We find that the intensity of wings relative to the intensity of peaks
depends sensitively on the richness of the cascades, i.e. on the multiplicity ${\cal M}$ (the number
of created pairs per primary electron).
The results discussed above and presented in Fig. 3 had been obtained for the initial
energy of primary electrons $E_{0}=10^{7}$ MeV which
yielded ${\cal M}=73$.
By increasing the initial energy $E_{0}$ up to $2\cdot 10^{7}$ MeV
the multiplicity reaches ${\cal M}=830$ and the corresponding pulse profile at $100$ MeV
(left panel of Fig. 5)
changes notably with respect to its counterpart of Fig. 3 a.
It reveals now a much lower level of wings outside the peaks.
Equally important is the change in the shape
of the phase-averaged energy spectrum which becomes much softer by gaining more power in the low-energy range
(right panel in Fig. 5).
Both new features are in rough agreement with the data for the Vela pulsar, contrary to the case with $E_{0}=10^{7}$ MeV.
It is interesting to note that the association of the broad peaks at 100 MeV
with the relatively hard spectrum (Fig. 3 a)
on one hand, and
of the narrow peaks with the soft spectrum (Fig. 5) on the other hand
do resemble qualitatively the observed characteristics
of Geminga and the Vela pulsar, respectively.
We may now test our model of the double-peak asymmetry by
confronting the numerical
results obtained for specific pulsar parameters
with the data for real objects. Since the effect is induced by magnetic
absorption the expected
weakening of the leading peak with respect to the trailing peak
occurs only in the vicinity of the high-energy spectral cutoff.
Therefore, it is essential
to have a good photon statistics also at the highest energy bins, i.e.
above
$1$ GeV. As far as the EGRET data are concerned this requirement is
barely satisfied even for Vela.
With these limitation in mind, we consider Vela as the only appropriate
case to provide the test.
We used the
EGRET data for Vela to calculate the ratio (denoted as $P2/P1$)
of the photon counts
in the LP and the TP (denoted as $P1$ and $P2$, respectively).
For each energy bin (the energy bins
cover the range between $\sim 30$ MeV and $\sim 10$ GeV) we calculated
$P1$ ($P2$)
by summing up all photons
within the range $\phi_{\rm lp}\pm 0.05$ ($\phi_{\rm tp}\pm 0.05$) in
phase,
where $\phi_{\rm lp}$ ($\phi_{\rm tp}$) is the phase of maximum in the
LP (the TP)
at $100$ MeV.
Fig. 6
shows the observational points as well as their estimated
errors111Detailed analysis of the data
will be presented elsewhere (Woźna et al. 2002, in preparation).
along with
the results of model calculations performed for
three different altitudes: $h_{0}=2R_{\rm NS}$, $3R_{\rm NS}$, and $4R_{\rm NS}$.
The overall qualitative and quantitative behaviour of $P2/P1$ for the
EGRET data
is very similar to the dependence presented by Kanbach et
al. kanbach80 for the COS-B data.
The data points certainly can acommodate our model.
However, to answer the question of whether it would be inevitable to
invoke any additional processes to reproduce
the increase in $P2/P1$ inferred from the data
requires better photon statistics at the spectral cut-off and careful
statistical analysis.
5 Rotational asymmetry as a function of pulsar parameters
The asymmetry effects are marginal for nearly aligned pulsars with periods
$P\sim 0.1$ s. For the above-described model of Vela, they are noticeable
only because of the high altitude of the accelerator ($h_{0}=4R_{\rm NS}$).
However, for highly inclined ($\alpha\ga 45^{\circ}$) and fast pulsars
($P\la 10$ ms) the rotational effects result in asymmetry of
considerable magnitude.
If detected by GLAST, GeV emission from such objects would provide
powerful diagnostics of the polar cap model. Below we present the magnitude of
such asymmetry predicted for a wide range of parameters for fast pulsars.
As a measure of rotational effects we consider the escape energy
$\varepsilon_{\rm esc}$ which is defined as a
maximum energy of a photon (and the photon is emitted tangentially to its ‘parent’
magnetic field line; a footpoint of this line has magnetic colatitude $\theta$)
for which the magnetosphere is still
transparent, i.e. for which the optical depth integrated along the photon
trajectory is less than $1$.
This energy was calculated with our numerical code and the results are shown below.
For the sake of comparison with the case of no rotational effects we recall
a simplified, yet quite accurate222For
presentation of detailed numerical results see e.g. Harding et al. hbg1997 . for magnetic
fields weaker than $\sim 0.1B_{\rm cr}$, analytic formula which (after e.g. Bulik et al. bulik ) reads
$$\varepsilon_{\rm esc}(\theta)=\frac{\theta_{\rm pc}}{\theta}\cdot\varepsilon_{%
\rm esc}(\theta_{\rm pc}),$$
(5)
where
$$\displaystyle\varepsilon_{\rm esc}(\theta_{\rm pc})\approx 1\ {\rm GeV}\ \left%
(\frac{B_{\rm pc}}{0.1\,B_{\rm cr}}\right)^{-1}\left(\frac{\theta_{\rm pc}}{0.%
01\,{\rm rad}}\right)^{-1}$$
$$\displaystyle\times\left(\frac{r_{\rm m}}{R_{\rm NS}}\right)^{5/2}.$$
(6)
Here $r_{\rm m}$ is the radial coordinate of the emission point
(with spherical coordinates ($r_{\rm m}$, $\theta_{\rm m}$, $\phi_{\rm m}$)
in the right-handed frame with $\hat{z}$-axis along the dipole
axis)
and $\theta\ll 1$ rad is the magnetic colatitude of a footpoint of the parent magnetic field line
at the neutron star
surface.
For field lines originating at the polar-cap rim one should take then
$\theta=\theta_{\rm pc}$,
where $\theta_{\rm pc}=\arcsin(\sqrt{(}R_{\rm NS}/R_{\rm lc}))\approx 1.45\cdot 10^{-%
2}/\sqrt{P}$ radians,
and $R_{\rm lc}=cP/2\pi$ is the light cylinder.
Note, that the escape energy of eq.(5) is symmetrical with respect to the dipole axis
and goes to infinity at the magnetic pole ($\theta=0$).
Such a behaviour is not the case, however, when the rotational effects discussed in section 3
are taken into account.
In Figs. 7, 8, and 9 we present the values of
$\varepsilon_{\rm esc}$ obtained numerically
for fast rotators
with $B_{\rm pc}=1$ TG, and emission points located on the polar cap surface,
i.e. $r_{\rm em}=R_{\rm NS}$ was assumed everywhere.
Let us begin with the case of orthogonal rotators (i.e. $\alpha=90^{\circ}$) - these are
shown in Figs. 7 and 8. Here we consider emission points lying
along the crossection of polar cap surface
with the equatorial plane of rotation (hence, the results would be relevant for
observers located at $\zeta_{\rm obs}=90^{\circ}$).
Fig. 7
shows how $\varepsilon_{\rm esc}$ varies with location of the emission point across the polar cap.
The location of each point is determined by the normalized magnetic colatitude $\theta/\theta_{\rm pc}$
in the range $[0,\,1]$ and the magnetic azimuth $\phi_{\rm m}$ equal either to
$\pi/2$ (for the leading half of the polar cap)
or
$-\pi/2$ (for the trailing half).
The spin periods of $0.1$ s, $10$ ms, and $1.5$ ms were considered.
In order to quantify the asymmetry in $\varepsilon_{\rm esc}$ between the leading and the trailing parts
of the polar cap we introduce the following parameter:
$${\cal R}_{\rm esc}(\theta)=\frac{\varepsilon_{\rm esc}^{\rm tp}}{\varepsilon_{%
\rm esc}^{\rm lp}},$$
(7)
where
$$\displaystyle\varepsilon_{\rm esc}^{\rm lp}\equiv\varepsilon_{\rm esc}(\phi_{%
\rm m}=\pi/2,\ \theta),$$
$$\displaystyle\varepsilon_{\rm esc}^{\rm tp}\equiv\varepsilon_{\rm esc}(\phi_{%
\rm m}=-\pi/2,\ \theta).$$
For example, ${\cal R}_{\rm esc}(\theta_{\rm pc})$
gives the asymmetry between two opposite points located on the outer rim of the polar cap.
For spin periods around $0.1$ s, typical for the known gamma-ray pulsars,
${\cal R}_{\rm esc}(\theta_{\rm pc})$ remains
close to unity; e.g. for the case of $P=0.1$ s in Fig.7, ${\cal R}_{\rm esc}(\theta_{\rm pc})\simeq 1.3$. However, for $P$ smaller than $0.01$ s the parameter
${\cal R}_{\rm esc}(\theta_{\rm pc})$ becomes definitely larger:
$\sim 2.5$ for $P=0.01$ s and $\sim 30$ for $P=1.5$ ms.
Thus, for pulsars with $P\la 0.01$ s and large inclination angles $\alpha$
we predict a notable difference (in excess of half a decade in photon energy)
between the positions of high-energy spectral cutoff
in the gamma-ray emission from the leading and the trailing part of the
polar cap.
Another interesting implication of fast rotation is that
$\varepsilon_{\rm esc}$ has
finite values for any $\theta$, including $\theta=0$
(the magnetic pole), in contrast to the ‘static’ case of eq. (5).
This can understood in the following way:
consider
emission points with decreasing
colatitude $\theta$ in the leading part of the polar cap.
As we approach the dipole axis ($\theta\rightarrow 0$),
$\varepsilon_{\rm esc}$ increases because
the decreasing curvature of magnetic field lines leads to smaller angles
between $\@vec{B}^{\prime}$ and the photon propagation direction $\hat{\eta}^{\prime}$
in the corotating frame CF. However, photon
trajectory bends backwards in the CF (see dashed lines in Fig. 1),
which implies that
also photons emitted at $\theta=0$
along the straight dipolar axis will
quickly encounter $B^{\prime}_{\perp}\neq 0$, thus being subject to the magnetic absorption.
Entering now the trailing part of the polar cap leads to
further increase
of $\varepsilon_{\rm esc}$. This is
because magnetic field lines start to bend in the same direction
as the photon trajectory in the CF
(in other words - the efficiency
of absorption decreases for emission points in the trailing
part of the polar cap). Eventually, at some point (we denote it as $\theta_{0}$) the escape energy
reaches a maximum. This is the point where
the magnetic field slippage along with the aberration of photon
direction ensure small angles between
$\@vec{B}^{\prime}$ and $\hat{\eta}^{\prime}$ over
large distances in the photon trajectory. Therefore,
the faster is the rotation,
the larger is the colatitude $\theta_{0}$ of that point.
For example, $\theta_{0}/\theta_{\rm pc}\simeq 0.15$
for $P=0.1$ s whereas
$\theta_{0}/\theta_{\rm pc}\simeq 0.42$ for $P=0.01$ s (see Fig. 7).
For $P=1.5$ ms this maximum occurs 333It is easy to reproduce the behaviour of $\theta_{0}/\theta_{\rm pc}$
as a function of $P$ in Fig. 7 with analytic formula:
The maximal value of
$B_{\perp}(r)/B_{\rm pc}$
encountered in a static dipolar field by a photon emitted along the local field line at $\theta_{0}$
is approximately equal to $0.1\,\theta_{0}$ (Sturrock sturrock , also Fig.1
in Rudak & Ritter rr94 ). This occurs always
at $r_{0}=4/3\,R_{\rm NS}$,
regardless the value of $\theta_{0}$. The angle $\psi$ between the photon propagation direction and
the local field line at $r_{0}$ is
$\psi_{0}\approx B_{\perp}(r_{0})/B(r_{0})$
and therefore $\psi_{0}\approx 0.1\,\theta_{0}(4/3)^{3}$.
To minimize $B_{\perp}^{\prime}(r_{0})$ as much as possible in the case of rotation,
the aberration angle due to local linear velocity $\beta(r_{0})=4/3\,R_{\rm NS}/R_{\rm lc}$ should be
close to
$\psi_{0}$. Since $\beta(r_{0})\ll 1$, the aberration angle is $\sim\beta(r_{0})$.
Therefore, we obtain the condition
$0.1\,(4/3)^{2}\,\theta_{0}\approx\theta_{\rm pc}^{2}$ which
gives
$\frac{\theta_{0}}{\theta_{\rm pc}}\approx 0.08\,P^{-1/2}.$
(9)
This formula overestimates the locations of maxima in Fig. 7
by a factor $\sim 2$ only.
close to the outer rim (in the
trailing part) and therefore huge asymmetry
with respect to the leading rim is predicted:
${\cal R}_{\rm esc}(\theta_{\rm pc})\simeq 30$.
With further increase of $\theta$
(towards the trailing rim of the polar cap)
local magnetic field lines
start to bend stronger than the photon trajectory in the CF,
and this is why
$\varepsilon_{\rm esc}$ should now decrease.
However, this decrease never compensates the asymmetry
in $\varepsilon_{\rm esc}$ at $\theta_{\rm pc}$
with respect to $\theta=0$,
i.e. one always ends up with ${\cal R}_{\rm esc}(\theta_{\rm pc})>1$.
Figure 8 presents escape energy $\varepsilon_{\rm esc}^{\rm lp}$ and
$\varepsilon_{\rm esc}^{\rm tp}$
in function of spin period $P$.
This energy was calculated for a fixed position $\theta_{\rm fxd}$ of emission point,
in order to highlight its dependence
on rotation.
We chose three pairs of oppositely located emission points at:
$\theta_{\rm fxd}=2^{\circ}.6$, $8^{\circ}.3$, and $22^{\circ}$,
which corespond to $\theta_{\rm pc}$ for $P=0.1$ s, $10$
ms, and $1.5$ ms, respectively. As in Fig. 7,
the emission points were placed at the neutron star surface,
in the equatorial plane of orthogonal rotator.
In the case of slow rotation ($P\sim 1$ s), the values of $\varepsilon_{\rm esc}^{\rm lp}(\theta_{\rm fxd})$ for the leading point and
$\varepsilon_{\rm esc}^{\rm tp}(\theta_{\rm fxd})$ for the trailing point
are practically identical,
and well approximated by eq.(5).
As rotation becomes faster ($P$ around $\sim 0.1$ s)
$\varepsilon_{\rm esc}^{\rm lp}(\theta_{\rm fxd})$ and
$\varepsilon_{\rm esc}^{\rm tp}(\theta_{\rm fxd})$
start to diverge due to the asymmetric influence of $\@vec{E}$.
At even shorter periods, below $\sim 0.03$ s,
the maxima in $\varepsilon_{\rm esc}^{\rm tp}$ are reached
(at the values of $P$, which can be reproduced by solving eq.(9)
with $\theta_{\rm fxd}$ in place of $\theta_{0}$)
because slippage of field lines starts to be important (see previous paragraph).
For increasing $\theta_{\rm fxd}$ ($2^{\circ}.6$, $8^{\circ}.3$, and $22^{\circ}$ in Fig. 8),
the asymmetry parameter ${\cal R}_{\rm esc}(\theta_{\rm fxd})$ decreases
for slow rotators ($P\ga 0.1$ s), and it increases for fast (millisecond) rotators.
In Fig. 9 we present $\varepsilon_{\rm esc}^{\rm lp}(\theta_{\rm pc})$
and $\varepsilon_{\rm esc}^{\rm tp}(\theta_{\rm pc})$ as a function of spin period $P$
for a set of dipole inclinations $\alpha$.
Note, that unlike in Fig. 8, the emission points are now placed at $\theta=\theta_{\rm pc}$, i.e. at the rim of polar cap corresponding to $P$.
In the case of small inclinations
(eg. $\alpha=10^{\circ}$),
the resulting ratio ${\cal R}_{\rm esc}(\theta_{\rm pc})$ remains close to unity
even for millisecond periods.
The general increase in $\varepsilon_{\rm esc}$ with $P$ increasing,
noticeable in Fig. 9, reflects
the approach of emission points to the dipole axis,
where the curvature of magnetic field lines is small.
The trend is well described by $\varepsilon_{\rm esc}\propto\sqrt{P}$
as given by eq. (6).
In the range of spin periods below $\sim 0.1$ s and with
$\alpha\ga 45^{\circ}$, the difference between $\varepsilon_{\rm esc}^{\rm lp}$ and $\varepsilon_{\rm esc}^{\rm tp}$ becomes pronounced, especially
for highly inclined millisecond pulsars.
If detected by GLAST, high-energy emission
from such objects would provide an ideal test of the polar cap model.
However, for the asymmetry effects to be detectable,
an additional condition (apart from short period
and large inclination) must be fulfilled: the viewing
geometry must be of the ‘on-beam’ type, i.e. the observer’s line of sight
must cross
the narrow beam of radiation at the high-energy cutoff - where
the magnetic absorption operates.
If this is the case,
the asymmetry in the absorption
may be noticeable even in the phase-averaged spectra.
As an example, we show in Fig. 10 the phase-averaged spectrum
calculated for a millisecond pulsar with
$P=2.3$ ms, $B_{\rm pc}=10^{9}$ G, $\alpha=60^{\circ}$, and for $\zeta_{\rm obs}=60^{\circ}$. For this rotator we obtain
$\varepsilon_{\rm esc}^{\rm lp}\simeq 10^{5}$ MeV
and $\varepsilon_{\rm esc}^{\rm tp}\simeq 5\cdot 10^{5}$ MeV
at the rim of the polar cap.
As a result of this rotationally induced asymmetry in magnetic absorption
for the leading and the trailing peak
the spectrum at its high-energy cutoff assumes a step-like shape:
below $\simeq 10^{5}$ MeV the spectrum consists of photons from
both the leading
and the trailing peak, whereas between $\sim 10^{5}$ MeV
and $\sim 5\cdot 10^{5}$ MeV
only photons of the trailing peak contribute to the spectrum;
at $\varepsilon\simeq\varepsilon_{\rm esc}^{\rm lp}$ the level of
the spectrum drops by a factor of $\sim 2$.
In these particular calculations of the spectrum, we assumed that
the density distribution of primary electrons over the polar cap
is dominated by an outer-rim component (see the captions to Fig. 10 for details of the distribution).
If, however, an emission from the inner part
of the polar cap were to contribute considerably to the outer-rim emission, the step-like
shape shown in Fig. 10 would be smoothed out, because of contribution
of many spectra with different values of $\varepsilon_{\rm esc}$.
In such a case
the polar cap origin of the observed radiation
could be easily revealed by noting
strong differences between the high-energy spectral cutoffs
in different ranges of the rotational phase (i.e. phase-resolved spectra would have
to be obtained).
The first obvious candidate to check for this effect (e.g. with GLAST) seems to be
J0218$+$4232 – the only gamma-ray pulsar among all millisecond pulsars
(Kuiper et al. k2000 ). However, this pulsar appears to be
a candidate for an ‘off-beam’ case (see Dyks & Rudak dr2002 for details).
The strength of magnetic field $B_{\rm pc}$ practically does not affect
the shapes of curves shown in Figs. 7 – 9.
The magnetic field only acts as a scaling factor: $\varepsilon_{\rm esc}\propto B^{-1}$; cf. eq. (6).
6 Discussion
We have shown that pulsar rotation induces an asymmetry in the
magnetic absorption rate with respect to the magnetic dipole axis.
Its consequences are potentially interesting in constraining
the phase-space of parameters in the polar cap models of high-energy radiation,
provided that very high quality gamma-ray
data (e.g. as expected from GLAST) are at hand.
Its magnitude depends mainly on the linear velocity $\beta$ of the magnetosphere at
sites of particle acceleration and magnetic photon absorption.
When the region of electron acceleration is placed just above the neutron star surface
rotation does not produce any detectable effects even for relatively
fast rotating young gamma-ray pulsars.
However, it has been argued that at least in the case of the Vela pulsar, such a situation
is difficult to reconcile with the spectral high-energy cutoff at about 10 GeV (e.g. Dyks et al. alic ).
We find then that raising the accelerator up to $\sim 4$ neutron star radii
(in the spirit of Harding & Muslimov hm98 )
above its polar cap produces asymmetric gamma-ray pulse profiles even in the case
of nearly aligned rotators with a spin period of $P\sim 0.1$ s.
The resulting features - softer spectrum of the leading peak and the
dominance of the trailing peak above $\sim 5$ GeV - do
agree qualitatively with the EGRET data of the bright gamma-ray pulsars (Thompson thompson2001 ).
We are far from concluding that the rotation effects alone
can account for the observed asymmetry
in the double peaks of the bright EGRET pulsars.
On the contrary - some axial asymmetry intrinsic to the region of electron acceleration is
inevitable in order to explain the double-peak properties at $\>100$ MeV
of Geminga and B1706-44, where the leading peak is weaker than the trailing peak.
Strong deviations of the actual magnetic field structure
from the pure dipole at the stellar surface
(eg. Gil et al. gmm2002 )
might be responsible for maintaining
axial asymmetry at the site of electron acceleration (unlike the symmetric initial conditions introduced in Section 2).
This in turn would lead to electromagnetic cascades whose properties vary with magnetic azimuth.
It is important, however, that the propagation effects due to rotation
work in the right direction, i.e. they explain qualitatively
the observed weakening of the leading peak with respect to the trailing peak.
We emphasize that this weakening
occurs only in the vicinity of the (phase-averaged) high-energy spectral cutoff,
where the flux level decreases significantly.
Another consequence of the magnetic absorption of high energy photons
is a noticeable change in the separation $\Delta^{\rm peak}$ between the two peaks in the pulse,
taking place near the high-energy spectral cutoff (Dyks & Rudak dr2000 ).
In the model discussed above, with electrons ejected
only from a rim of the polar cap, the higher energy of photons requires
higher emission altitudes to avoid absorption.
Therefore, a slight increase in $\Delta^{\rm peak}$ is visible
in the three lowermost pulse profiles in Fig. 3 b.
However, if the emission from the interior of the polar cap were included,
just the opposite behaviour would occur: $\Delta^{\rm peak}$ would decrease
near the high-energy cutoff in the spectrum.
This is because in this case of a “filled polar cap tube”,
the highest energy
non-absorbed photons are emitted closer to the magnetic dipole
axis (see Fig. 2 in Dyks & Rudak dr2000 ).
The latter case agrees qualitatively with the marginal decrease in peak
separation found in the EGRET data for Vela
(Kanbach kanbach99 ).
Stimulated by high-quality observations of
gamma-ray pulsars anticipated with GLAST we analysed in Sect. 5 the importance
of rotation-driven asymmetry in magnetic absorption for a broad range of pulsar parameters.
A decline in gamma-ray flux at high-energy spectral cutoff
should inevitably be accompanied by strong changes in pulse profiles: whereas at lower
photon energies the profile is determined by the density
distribution of primary electrons over the polar cap and the efficiency
of photon emission mechanism, in the vicinity of the cutoff
it becomes additionally constrained by likely high values of the asymmetry parameter
${\cal R}_{\rm esc}(\theta_{\rm pc})$ - the situation anticipated for
fastly rotating ($P<0.01$ s), and highly inclined ($\alpha\ga 45^{\circ}$)
pulsars.
Acknowledgements.
We thank V.S. Beskin and A.K. Harding for useful comments
on the issue of magnetospheric distortions. We are grateful to Gottfried Kanbach
for providing us with the EGRET data on Vela, and to Aga Woźna
for calculating the P2/P1 ratios used in Fig.6.
We acknowledge comments and stimulating suggestions made by the anonymous referee.
JD appreciates Young Researcher Scholarship of Foundation
for Polish Science.
This work was supported by KBN (grants 2P03D02117 and 5P03D02420) and NCU
(grant 405A).
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Self-Representation on Twitter Using Emoji Skin Color Modifiers
Alexander Robertson, Walid Magdy, Sharon Goldwater
Institute for Language, Computation and Cognition
School of Informatics, University of Edinburgh
alexander.robertson@ed.ac.uk, {wmagdy, sgwater}@inf.ed.ac.uk
Abstract
Since 2015, it has been possible to modify certain emoji with a skin tone. The five different skin tones were introduced with the aim of representing more human diversity, but some commentators feared they might be used as a way to negatively represent other users/groups. This paper presents a quantitative analysis of the use of skin tone modifiers on emoji on Twitter, showing that users with darker-skinned profile photos employ them more often than users with lighter-skinned profile photos, and the vast majority of skin tone usage matches the color of a user’s profile photo—i.e., tones represent the self, rather than the other. In the few cases where users do use opposite-toned emoji, we find no evidence of negative racial sentiment. Thus, the introduction of skin tones seems to have met the goal of better representing human diversity.
Self-Representation on Twitter Using Emoji Skin Color Modifiers
Alexander Robertson, Walid Magdy, Sharon Goldwater
Institute for Language, Computation and Cognition
School of Informatics, University of Edinburgh
alexander.robertson@ed.ac.uk, {wmagdy, sgwater}@inf.ed.ac.uk
Introduction
Emoji—icons used to represent emotions, ideas, or objects—became a formally recognized component of the Unicode Standard in 2010. At that time, all emoji depicting humans were rendered with the same yellow skin tone. However, in 2015, special modifier codes were introduced to allow users to change the skin tone of certain emoji. Unicode Technical Standard #51 (?), justified this on the grounds that “people all over the world want to have emoji that reflect more human diversity.” Nevertheless, the change was not uncontroversial. Although skin tone modifiers can be used to make emoji more personal to reflect the identity of the user, in principle they can also be used to represent others. This possibility sparked fears in the mainstream media that users might engage in “digital blackface”, negatively using dark emoji for entertainment purposes (?; ?)—if people would even use them at all (?). But despite this media and popular attention, and considerable academic investigation of emoji in general, we are not aware of any quantitative studies of how people actually use skin tone modifiers.
In this paper, we present an analysis of how skin tone modifiers are used in emoji on Twitter. In a sample of 0.6 billion tweets, we find that 42% of the 13 million tone-modifiable emoji (TME) tweeted in 2017 included skin tone modifiers (henceforth, we refer to these modified TME as TME+, in contrast to unmodified TME-) and the majority of these TME+ were light colors. By annotating the profile photos of 4,099 users and analyzing their tweet histories, we show that the overall prevalence of lighter skin tone modifiers is likely due to the prevalence of lighter-skinned users. Indeed, users with darker-skinned profile photos are more likely to use TME, and more likely to modify them. Moreover, the vast majority of TME+ are similar in tone to the user’s profile photo, suggesting that TME+ are used overwhelmingly for self-representation. Finally, we analyze the sentiment of the small proportion of tweets containing opposite-toned emoji, and find that positive tweets outweigh negative. Overall, our findings suggest that the Unicode Consortium has successfully met a real demand for diversity within emoji, with few negative consequences.
Emoji and Skin Tones
Recent years have seen significant academic interest in emoji. Studies show that emotional interpretation of some emoji is subject to ambiguity (?), that addition of emoji to messages (particularly neutral ones) can convey positive/negative tone and sentiment (?), and that popularity differences between emoji may be due to their relation to popular words (?). Resources commonly available for traditional language such as sense inventories have been compiled (?) and emoji usage has been linked to personality traits (?). But despite three years since their introduction, TME have been unstudied until now.
Not all emoji can have a skin tone. The traditional emoji faces (e.g. ) cannot be modified. The current Unicode Standard defines 102 TME, some of which can be combined to create compound emoji (e.g. ) which can also be modified for skin color. Five skin tones, based on the six tones of the Fitzpatrick phototyping scale (?), are available. We will refer to the five-scale tones as T1 (the lightest) to T5 (the darkest). Figure 1 illustrates the 102 available TME, showing how the tones affect them.
Emoji are rendered differently across platforms, which can cause ambiguity in semantic interpretation—e.g., writers/readers may perceive the expressed action or emotion differently (?). Since we are only concerned with skin color, we do not expect these ambiguities to be an issue in our study. However, there is slight variation in the rendering of specific skin tones across platforms. We therefore cannot be sure of the precise tonal values observed by a user, although all platforms have exactly 5-scale tones, where T1 is lighter than T2, T2 lighter than T3, and so forth.
Skin Tone Usage, Overall and By Region
For our initial analysis we used the Twitter API (1% sample) to create Dataset A, from 1.04 billion tweets made between February and December 2017. This dataset contains 595m original (non-retweet) tweets made by 769m unique users. Of these, 83.3m (14%) contain at least one emoji and 13.1m (2.2%) contain at least one TME. 5.5m (42%) of TME were TME+. Lighter tones dominate: T1 and T2 account for 68% of all TME+.
To gain a global view of TME usage by location, we used the Google Maps API to validate the location field of the authors of the 5.5m TME+ tweets 111Though this field is not necessarily reliable (users may travel or give a false location), our aim was to get a broad overview before the main analysis.. We located 169k users, then grouped them on the basis of broad geographic region (figure 2). Most regions follow the overall trend of using far more light tones. The exception is North America, where the distribution of tones is more uniform—though T5 still has relatively little usage, and is the least common tone worldwide.
This raises the question of why lighter TME+ are used so much more than darker ones in all regions outside North America. These results would seem to contradict the hypothesis that TME+ are used for self-representation, since we might naïvely expect more dark-skinned users in Africa and Asia, whereas the TME+ being used are mainly light.
However, many countries (particularly in Africa) have lower levels of internet access compared to highly developed countries (such as those in North America)222World Bank, World Telecommunication Report, http://data.worldbank.org/indicator/IT.NET.BBND.P2, and access to the internet or modern devices can be even more limited for some ethnic groups due to political oppression (?) or poverty. So, it could simply be that in most regions, lighter-skinned internet users predominate.
Another possible hypothesis for North America’s more uniform TME+ distribution (although the demographics do not have similar distributions) is that people use TME+ to refer to both others and themselves. We investigate this hypothesis in the next section, looking more closely at TME usage on Twitter by comparing users’ actual skin tone to that seen in their TME+.
Skin Tone Modifiers for Self-Identification
Since our goal is to better understand how TME relate to user identity, we randomly selected 10k users from Dataset A, subject to the condition that they had used at least one TME. Although we cannot perfectly determine the skin color of these users, we estimate it from their profile photos. We assume that photos of a single individual are self-portraits of the user. This might not be true in all cases, since users can use photos of others. We assume this practice is rare, with a negligible effect on our analysis. However, our annotation process was designed with this possibility in mind.
We collected the Twitter profile pictures of the selected 10k users. These pictures were then annotated using Crowdflower333www.crowdflower.com. We instructed annotators to label as “invalid” images that might not represent the Twitter user. “Invalid” images include: non-human (e.g. animals, nature), celebrities, group photos, gray-scale images. Instructions were given beforehand, with examples of valid/invalid photos provided. If a picture was annotated as “valid”, annotators compared it to an array of five TME+, as shown in Figure 3. We selected the example emoji for its overall lack of distracting features (e.g. little hair covering the face) and consistent appearance (all versions have the same hair/eye color).
Each photo was annotated by at least three annotators. A set of 50 pre-annotated pictures was used to control the quality of the annotators. Only workers achieving over 90% accuracy on the screening pictures were accepted.
After the annotation process, we retained only those pictures annotated as valid where at least two annotators agreed on the same skin tone. The result was 4,099 annotated pictures. The inter-annotator agreements for these pictures were 98% for validity, 88% for user skin tone.
TME Usage Analysis
We collected the historical tweets of the 4,099 annotated users using the Twitter API, to create Dataset B. This contains the tweets of these users from January 2018 back to April 2015—the time when TME were enabled. 5.86m original (non-retweet) tweets were collected in total. 2.46m tweets contained emoji, of which 479k contained TME.
Before looking at the relationship between a user’s skin tone and their use of TME, we first examined the overall usage of TME by user. The histogram in Figure 6 shows that, of the Dataset B users, roughly a quarter only ever use TME-, with about the same proportion always using TME+. The remaining users tend towards one end or the other.
Next, we looked at how TME+ usage varies across users according to their annotated skin color. Figure 6 shows the proportion of users in each of the five skin color groups who tweeted at least one TME+. A clear trend is visible, where users with darker skin are more likely to use TME+ than those with lighter skin. In fact, 80+% of users with skin tone 4 and 5 use TME+ in their tweets, which shows the importance of this feature to them.
However, not all TME are TME+. Thus, we examined users who use TME and computed the proportion of those tweets which are TME+. Figure 6 shows the percentage of TME+ vs TME- for each user group. Again, the same trend appears: darker skinned users tend to modify the skin tone of TME in their tweets much more than users with lighter skin. For instance, 82% of TME tweets by users with T4 skin change the skin tone of the emoji compared to only 52% of the white (T1 skin) users. This emphasizes how users with darker skin are keen to modify the appearance of their emoji, presumably in order to better reflect their own identity.
TME+ for Self-Reference vs Referencing Others
Finally, we looked at whether users’ choice of modifiers reflects their own skin color or others. For each tone user, we determined the TME+ tone used most often by that user. Figure 7 shows a heatmap of the distribution of the most common skin tones used by each user group in Dataset B.
Overall, our results suggest that by far the most common scenario is for users to choose skin tone modifiers that roughly match their own complexion, accounting for the variability in rendering of these tones across different platforms. There seems to be some tendency for users to choose modifiers towards the lighter end of the available spectrum; for example, users with skin tone 2 mostly select skin tone 1 in their emoji, and users of skin tone 5 mostly select skin tone 4 in their emoji.
The (Non)-Prevalence of Digital Blackface
The previous analysis shows that for nearly all users, their most common choice of skin tone matches their own skin. However, it still may be the case that users occasionally choose emoji of a different tone. We looked at how often this occurs, and whether there is any noticeable negative sentiment associated with this behavior.
For this analysis, we aggregated Dataset B users into two groups based on skin tone: tones 4/5 (dark) and tones 1/2 (light). First, we analysed the tweets of each group that use TME+ with the skin tone of the other group. These comprise 1229 tweets for the dark-skinned users and 1,729 tweets for the light-skinned users—a very small proportion of each group’s tweets: 0.14% (light group) and 0.24% (dark group). Next, we examined the opposite-tone tweets in English and computed their sentiment using Sentistrength (?),
which is
designed specifically for analyzing short online messages such as Twitter posts.
The majority of these tweets were neutral (almost 50% of the tweets of both groups). The distribution of the non-neutral sentiment of tweets for both groups are shown in Figure 8, where 4 is the most positive and -4 is the most negative444Sentiment was rated on a -5:+5 scale but no tweets at the extremes of this range were found in this subset of the data.. As shown, the distributions for both groups are almost the same, and positive tweets outnumber negative ones. Inspection of tweets with negative sentiment revealed tweets on generally negative topics, rather than anything specifically about race. Overall, we find no evidence to justify fears of widespread “digital blackface” or its black-against-white counterpart.
Conclusion
In this paper we presented the first quantitative study on the usage of tone-modifiable emoji (TME) on Twitter. We showed that although lighter-toned emoji are more common overall, different populations of users (based on the skin color of their profile picture) show markedly different characteristics in how they use TME and to what extent. The overall picture is one where users take advantage of emoji skin tone modifiers to represent an important aspect of their identity, and do so differentially depending on their own skin tone: compared to light-skinned users (the majority on Twitter), a higher proportion of dark-skinned users use skin tone modifiers, and they use them more frequently. Here we grouped all geographic regions together in looking at user skin tone, but an interesting question for future work will be to examine the relationship between a user’s majority/minority status within their real-life community (i.e., their geographic region) and their use of skin tone modifiers.
Acknowledgements
This work was supported in part by the EPSRC Centre for Doctoral Training in Data Science, funded by the UK Engineering and Physical Sciences Research Council (grant EP/L016427/1) and the University of Edinburgh.
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A poor man’s method to tune the flat-band energy
Tomonari Mizoguchi
Department of Physics, University of Tsukuba, 1-1-1 Thennoudai, Tsukuba, Ibaraki 305-8571, Japan
mizoguchi@rhodia.ph.tsukuba.ac.jp
Masafumi Udagawa
Department of Physics, Gakushuin University, 1-5-1 Mejiro, Toshima-ku, Tokyo 171-8588, Japan
Abstract
In typical flat-band models, defined as nearest-neighbor tight-binding models,
flat bands are usually pinned to
the special energies, such as top or bottom of dispersive bands, or band crossing points.
In this paper, we propose a simple method to tune
the energy of flat bands without losing the exact flatness of the bands.
The main idea is to add farther-neighbor hoppings to the original nearest-neighbor models, in such a way that the transfer integral depends only on the Manhattan distance.
We apply this method to several lattice models including two-dimensional kagome lattice and three dimensional pyrochlore lattice,
as well as their breathing lattices and non-line graphs.
The proposed method will be useful for engineering flat bands to generate desirable properties,
such as enhancement of $T_{c}$ of superconductors and non-trivial topological orders.
I Introduction
Diversity of materials may be attributed to the diversity of band structures.
Variety of band structures associated with lattice structures and orbital characters is a source of
rich phenomena in condensed matter systems, such as
spin and orbital magnetism Stoner1938 ; Kanamori1963 ; Ogata2015 , superconductivity Sigrist1991 ; Mackenzie2003 ; Stewart2011 , topological insulators Hasan2010 ; Qi2011 , and topological Dirac/Weyl semimetals Vafek2014 ; Armitage2018 .
Among characteristic band structures,
a completely dispersionless band, in entire Brillouin zone, is called a flat band.
One of the remarkable consequences of this “quench” of kinetic energy is the emergence of ferromagnetic ground state
when introducing the Hubbard interactions, and there has been a long history of study in that context Lieb1989 ; Mielke1991 ; Mielke1991_2 ; Tasaki1992 ; Mielke1993 ; Kusakabe1994 ; Aoki1994 ; Tasaki1998 ; Katsura2010 .
Topological physics in exact and nearly flat-band systems also attracts considerable interests Katsura2010 ; Tang2011 ; Sun2011 ; Neupert2011 ; Sheng2011 ; Wang2011 ; Liu2012 ; Bergholtz2015 ; Peotta2015 ; Liang2017 ; Misumi2017 .
To study such intriguing physics associated with
the flat-band systems,
number of tight-binding Hamiltonians, which mostly consider the nearest-neighbor (NN) hoppings, have been proposed.
Quite recently, the possibility of flat-band-assited superconductivity has been revisited
in the correlated electron systems, where the interband scattering between dispersive and flat bands plays an essential role Kuroki2005 ; Kobayashi2016 ; Matsumoto2018 .
In particular, this mechanism is
thought of as a one of the possible origins of
enhancement of $T_{c}$ in a twisted bilayer graphene with so-called “magic angles” Cao2018 ; Cao2018_2 ; Volovik2018 ; Yuan2018 ; Koshino2018 ; Ochi2018 ; Zou2018 .
There, it has been pointed out that the preferable band structure for such mechanism is
(i) the flat band is located slightly above or below the Fermi level, and (ii) the dispersive band has a large density of states (DOS) nearby the flat band.
Therefore, for further development of this mechanism for the high-$T_{c}$ superconductivity,
it is desirable to have an engineering method not only to realize flat band
but also to tune its energy.
In the present paper, we propose a simple guiding principle to tune the energy of flat bands.
It may sound surprising, since a flat band is extremely fragile; Infinitesimal amount of perturbation is enough to destroy its flatness.
Nevertheless, we will show that it is possible to systematically control its energy while keeping its exact flatness.
The main idea is to add farther-neighbor hoppings to the usual NN models with flat band(s),
in such a way that the transfer integrals depend only on the Manhattan distance.
After this modulation, the resulting Hamiltonian is expressed by the polynomial of the original
NN Hamiltonian.
As a result, the eigenfunctions remain exactly the same as the original ones and only the dispersion relations and the flat-band energy are modulated.
Our method, due to its simplicity, has two prominent advantages:
(i) the flat bands retain exact flatness after the modulation of the Hamiltonian,
and (ii) we only need a few parameters to control a flat-band energy.
The rest of this paper is organized as follows.
In Sec. II, we explain the basic mechanism of our method.
Then, in Sec. III, we apply this method to the line graphs in two- and three-dimensions, where
the existence of flat band(s) in the NN hopping models is guaranteed Mielke1991 .
In Sec. IV we apply this method with slight modifications
to the breathing lattices and a class of Lieb lattices, which have the site/bond inhomogeneity.
Finally, our conclusion is presented in Sec. V.
Some analytical formulas for the dispersions relations are shown in the Appendix.
II Formulation
In this section, we outline our method to tune the flat band energy, and
clarify the condition for this method to work.
We consider a tight-binding Hamiltonian for spinless fermions with NN hoppings:
$$\mathcal{H}_{1}=t_{1}\sum_{\langle i,j\rangle_{\rm NN}}c^{\dagger}_{i}c_{j}+c^%
{\dagger}_{j}c_{i},$$
(1)
where $t_{1}$ denotes the NN hopping integral.
For future use, we also write down this Hamiltonian by using the incident
matrix of the lattice:
$$\mathcal{H}_{1}=t_{1}\sum_{i,j}c_{i}^{\dagger}[\hat{h}_{1}]_{i,j}c_{j},$$
(2)
where the incident matrix, $\hat{h}_{1}$, satisfies $[\hat{h}_{1}]_{i,j}=1$ if $\langle i,j\rangle\in$ NN, and otherwise $[\hat{h}_{1}]_{i,j}=0$.
We note that, throughout this paper, $h$ denotes dimensionless matrices in either real or momentum spaces.
Their eigenvalues are denoted by $\lambda$, while energy eigenvalues of Hamiltonians are denoted by $\varepsilon$.
Suppose that the model is defined on a lattice with the number of sublattices $N_{\rm sub}$,
and that all sublattices have the same coordination number $z$.
We label sites as $i=(n,\alpha)$ where $n$ denotes a label of a unit cell and
$\alpha$ labels
sublattice.
By performing the Fourier transformation, we obtain
$$\mathcal{H}_{1}=t_{1}\sum_{\bm{k},\alpha,\beta}c^{\dagger}_{\bm{k},\alpha}[h_{%
1}(\bm{k})]_{\alpha\beta}c_{\bm{k},\beta},$$
(3)
where $c_{\bm{k},\alpha}=\sum_{n}c_{n,\alpha}e^{-i\bm{k}\cdot(\bm{R}_{n}+\bm{r}_{%
\alpha})}$;
$\bm{R}_{n}$ is the position of the unit cell
and $\bm{r}_{\alpha}$ is the position of the sublattice $\alpha$ inside the unit cell.
Let us assume that $h_{1}(\bm{k})$ has $N_{f}(<N_{\rm sub})$ flat bands and $N_{\rm sub}-N_{f}$ dispersive bands.
We label wavefunctions of flat [dispersive] bands at each $\bm{k}$ as $\psi^{\rm(f)}_{p}(\bm{k})$ [$\psi^{\rm(d)}_{q}(\bm{k})$]
and its eigenvalue $\lambda_{p}$ [$\lambda_{q}(\bm{k})$] with $p=1,\cdots N_{f}$ [$q=1,\cdots N-N_{f}$].
The corresponding eigenvalue equations are written as
$$h_{1}(\bm{k})\psi^{\rm(f)}_{p}(\bm{k})=\lambda_{p}\psi^{\rm(f)}_{p}(\bm{k}),$$
(4)
and
$$h_{1}(\bm{k})\psi^{\rm(d)}_{q}(\bm{k})=\lambda_{q}(\bm{k})\psi^{\rm(d)}_{q}(%
\bm{k}).$$
(5)
Under this setup, we now introduce our main idea for tuning the flat-band energy, that is,
we utilize the fact that if $\psi(\bm{k})$ is an eigenfunction of
$h_{1}(\bm{k})$, so it is of $[h_{1}(\bm{k})]^{m}$
for $m$ being arbitrary positive integer.
More generally, $\psi(\bm{k})$ is an eigenfunction for any polynomial of $h_{1}(\bm{k})$.
For instance, if we consider a generic quadratic form of $h_{1}(\bm{k})$ with real coefficients $a,b$ and $c$,
one obtains the eigenvalue equations as
$$\displaystyle\{a[h_{1}(\bm{k})]^{2}+bh_{1}(\bm{k})+c\hat{I}_{N_{\rm sub}}\}%
\psi^{\rm(f)}_{p}(\bm{k})$$
$$\displaystyle=[a(\lambda_{p})^{2}+b\lambda_{p}+c]\psi^{\rm(f)}_{p}(\bm{k}),$$
(6)
and
$$\displaystyle\{a[h_{1}(\bm{k})]^{2}+bh_{1}(\bm{k})+c\hat{I}_{N_{\rm sub}}\}%
\psi^{\rm(d)}_{q}(\bm{k})$$
$$\displaystyle=\{a[\lambda_{q}(\bm{k})]^{2}+b[\lambda_{q}(\bm{k})]+c\}\psi^{\rm%
(d)}_{q}(\bm{k}),$$
(7)
where $\hat{I}_{N_{\rm sub}}$ denotes $N_{\rm sub}\times N_{\rm sub}$ identity matrix.
Then, the new eigenvalues $a(\lambda_{p})^{2}+b\lambda_{p}+c$ and $a[\lambda_{q}(\bm{k})]^{2}+b[\lambda_{q}(\bm{k})]+c$
can intersect on some lines/surfaces in the Brillouin zone, even if the original eigenvalues, $\lambda_{p}$ and $\lambda_{q}(\bm{k})$, do not.
How can we implement a polynomial of $h_{1}(\bm{k})$ in the tight-binding models?
To see this, let us come back to the real-space representation, in which the square and the higher powers of $\hat{h}_{1}$ have a simple interpretation. $[\hat{h}_{1}^{2}]_{ij}=\sum_{k}[\hat{h}_{1}]_{ik}[\hat{h}_{1}]_{kj}$ is finite, only if there is a site $k$ neighboring both site $i$ and $j$, i.e., if the site $j$ can be reached from the site $i$ by two successive NN hoppings. Generalizing it, the $m$-th power of $\hat{h}_{1}$, $\hat{h}_{1}^{m}$, has finite matrix element, $[\hat{h}_{1}^{m}]_{ij}$, only if the sites $i$ and $j$ are $m$ NN hoppings away. To discuss the structure of $\hat{h}_{1}^{m}$ in a systematic way, it is convenient to introduce Manhattan distance of graph.
A Manhattan distance between two sites, say $i$ and $j$, is defined as minimum number of NN bonds one has to go through when
moving from $i$ to $j$ along the bonds.
For instance, if the Manhattan distance between $i$ and $j$ is two, it means that there exists a site
$k$ such that both $i$ and $j$ are connected to $k$ and $j$ is not the NN of $i$ [Fig. 1(a)].
At first sight, the above argument implies $\hat{h}_{1}^{2}$ is proportional to the incident matrix of the Manhattan distance two, i.e., $[\hat{h}_{1}^{2}]_{ij}$ is finite only if $i$ and $j$ are separate by Manhattan distance, two.
Indeed, if $i$ and $j$ are two Manhattan distance away, we have a finite matrix element, $[\hat{h}_{1}^{2}]_{i,j}=x$
where $x$ is a number of sites neighboring both $i$ and $j$.
However, we have to keep in mind that, if you move twice along NN bonds, there are two other possibilities,
other than reaching a site of two Manhattan distance away.
The first possibility is coming back to the original site, which occurs when going through the same bond twice [Fig. 1(b)].
The second possibility is reaching the NN site [Fig. 1(c)].
Let us assume that, for every NN pair, say $i$ and $j$,
there are $y$ distinct paths going from $i$ to $j$ with passing two NN bonds.
In other words, there exist sites $\ell_{1},\cdots\ell_{y}\neq i,j$,
such that $\langle i,\ell_{n}\rangle\in$ NN and $\langle j,\ell_{n}\rangle\in$ NN for $n=1,\cdots y$.
Under this assumption, we obtain the incident
matrix for a Manhattan distance two as
$$[(\hat{h}_{1})^{2}]_{i,j}=x[\hat{h}_{2}]_{i,j}+y\hat{h}_{1}+z\delta_{i,j},$$
(8)
where $\delta_{i,j}$ is the Kronecker delta.
Alternatively, in the momentum space representation, we obtain
$$[h_{1}(\bm{k})]^{2}=xh_{2}(\bm{k})+y\hat{h}_{1}(\bm{k})+z\hat{I}_{N_{\rm sub}},$$
(9)
where $h_{2}(\bm{k})$ is a (dimensionless) hopping matrix for “second-neighbor” hoppings.
Therefore, if we introduce the second-neighbor hoppings with a transfer integral $t_{2}$,
we obtain the quadratic form of $h_{1}(\bm{k})$ as
$$\displaystyle\mathcal{H}=\sum_{\bm{k},\alpha,\beta}c^{\dagger}_{\bm{k},\alpha}%
[t_{1}h_{1}(\bm{k})+t_{2}h_{2}(\bm{k})]_{\alpha\beta}c_{\bm{k},\beta}=\sum_{%
\bm{k},\alpha,\beta}c^{\dagger}_{\bm{k},\alpha}\left\{t_{2}\frac{1}{x}[h_{1}(%
\bm{k})]^{2}+(t_{1}-t_{2}\frac{y}{x})h_{1}(\bm{k})-t_{2}\frac{z}{x}\hat{I}_{N_%
{\rm sub}}\right\}_{\alpha\beta}c_{\bm{k},\beta}.$$
Consequently, the eigenenergies of this Hamiltonian are
$f(\lambda_{p/q})$ with $f(\lambda)=t_{2}\frac{1}{x}\lambda^{2}+(t_{1}-t_{2}\frac{y}{x})\lambda-t_{2}%
\frac{z}{x}$.
In the next two sections, we demonstrate how this idea works through the analyses of specific models.
We will first show canonical examples in kagome and pyrochlore models in Sec. III.
In these lattices, aforementioned lattice parameters, e.g. $x$, $y$ and $z$, are sublattice-independent,
thus these lattices are “homogeneous”.
In Sec. IV, we will discuss the applications to breathing lattices and a class of Lieb lattices,
in which existence of inequivalent sites/bonds modifies the simple polynomial expression mentioned above.
Before closing this section, we remark that
higher-order polynomials of $h_{1}(\bm{k})$ can be obtained by introducing
the “farther-neighbor” Manhattan-distance-dependent hoppings.
However, in light of material realization, short-ranged hoppings are favorable. Moreover, remarkable tunability of band structure is available, even within the model with second Manhattan distance, as we will show below.
III Canonical examples: kagome and pyrocholore lattices
We apply the idea we discussed in the previous section
to the kagome and pyrochlore lattices, which are the textbook examples of flat-band models.
In the previous study,
the authors investigated the band structures on these models in the context of magnetic mode analysis Mizoguchi2018 .
In this paper, we discuss their band structures, focusing on the quantities relevant to electronic systems, such as the DOS.
III.1 Kagome lattice
We first show the results for a kagome lattice.
We take the lattice vectors as $\bm{a}_{1}^{\rm(K)}=(1,0)$, $\bm{a}_{2}^{\rm(K)}=\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)$
and the sublattices’ coordinates as $\bm{r}_{1}^{\rm(K)}=(0,0)$, $\bm{r}_{2}^{\rm(K)}=\left(\frac{1}{4},\frac{\sqrt{3}}{4}\right)$,
$\bm{r}_{3}^{\rm(K)}=\left(\frac{1}{2},0\right)$.
Then, the NN hopping matrix in the momentum space is given by
$$h^{\rm(K)}_{1}(\bm{k})=\left(\begin{array}[]{ccc}0&h_{12}^{\rm(K,1)}(\bm{k})&h%
_{13}^{\rm(K,1)}(\bm{k})\\
h_{12}^{\rm(K,1)}(\bm{k})&0&h_{23}^{\rm(K,1)}(\bm{k})\\
h_{13}^{\rm(K,1)}(\bm{k})&h_{23}^{\rm(K,1)}(\bm{k})&0\\
\end{array}\right),$$
(11)
with $h_{12}^{\rm(K,1)}(\bm{k})=2\cos\left(\frac{k_{x}+\sqrt{3}k_{y}}{4}\right)$,
$h_{13}^{\rm(K,1)}(\bm{k})=2\cos\left(\frac{k_{x}}{2}\right)$,
and $h_{23}^{\rm(K,1)}(\bm{k})=2\cos\left(\frac{k_{x}-\sqrt{3}k_{y}}{4}\right)$.
Due to the nature of a line graph, $h^{\rm(K)}_{1}(\bm{k})$
has a $\bm{k}$-independent eigenvalue $\lambda^{\rm(K,f)}=-2$.
The other two bands are dispersive and their dispersion relations are given as
$$\displaystyle\lambda^{\rm(K,d)}_{1}(\bm{k})=1+\sqrt{2\left[\cos k_{x}+\cos%
\left(\frac{k_{x}+\sqrt{3}k_{y}}{2}\right)+\cos\left(\frac{k_{x}-\sqrt{3}k_{y}%
}{2}\right)\right]+3},$$
$$\displaystyle\lambda^{\rm(K,d)}_{2}(\bm{k})=1-\sqrt{2\left[\cos k_{x}+\cos%
\left(\frac{k_{x}+\sqrt{3}k_{y}}{2}\right)+\cos\left(\frac{k_{x}-\sqrt{3}k_{y}%
}{2}\right)\right]+3}.$$
The corresponding eigenfunctions are given by
$$\displaystyle\psi^{\rm(K,f)}(\bm{k})=\frac{1}{\mathcal{N}^{\rm(K,f)}(\bm{k})}%
\left[\sin\left(\varphi_{2}(\bm{k})-\varphi_{3}(\bm{k})\right),\sin\left(%
\varphi_{3}(\bm{k})-\varphi_{1}(\bm{k})\right),\sin\left(\varphi_{1}(\bm{k})-%
\varphi_{2}(\bm{k})\right)\right]^{\mathrm{T}},$$
$$\displaystyle\psi^{\rm(K,d)}_{1}(\bm{k})=\frac{1}{\mathcal{N}_{1}^{\rm(K,d)}(%
\bm{k})}\left[\cos\left(\varphi_{1}(\bm{k})+\theta(\bm{k})\right),\cos\left(%
\varphi_{2}(\bm{k})+\theta(\bm{k})\right),\cos\left(\varphi_{3}(\bm{k})+\theta%
(\bm{k})\right)\right]^{\mathrm{T}},$$
and
$$\psi^{\rm(K,d)}_{2}(\bm{k})=\frac{1}{\mathcal{N}_{2}^{\rm(K,d)}(\bm{k})}\left[%
\sin\left(\varphi_{1}(\bm{k})+\theta(\bm{k})\right),\sin\left(\varphi_{2}(\bm{%
k})+\theta(\bm{k})\right),\sin\left(\varphi_{3}(\bm{k})+\theta(\bm{k})\right)%
\right]^{\mathrm{T}},$$
(16)
where
$\mathcal{N}^{\rm(K,f)}(\bm{k})$, $\mathcal{N}_{1}^{\rm(K,d)}(\bm{k})$ and $\mathcal{N}_{2}^{\rm(K,d)}(\bm{k})$ are the normalization factors,
$\varphi_{1}(\bm{k})=\frac{k_{x}}{4}+\frac{k_{y}}{4\sqrt{3}}$, $\varphi_{2}(\bm{k})=-\frac{k_{y}}{2\sqrt{3}}$, $\varphi_{3}(\bm{k})=-\frac{k_{x}}{4}+\frac{k_{y}}{4\sqrt{3}}$,
and $\theta(\bm{k})=\frac{1}{2}\mathrm{arg}\left[e^{i\frac{k_{y}}{\sqrt{3}}}+2\cos%
\frac{k_{x}}{2}e^{-i\frac{k_{y}}{2\sqrt{3}}}\right]$
are the phase factors arising from the geometry of the lattice.
We show the band structure of the NN Hamiltonian with $t_{1}=-1$ in Fig 2(b).
Now, let us tune the flat-band energy.
To this end, we introduce the second-neighbor hoppings:
$$h^{\rm(K)}_{2}(\bm{k})=\left(\begin{array}[]{ccc}h_{11}^{\rm(K,2)}(\bm{k})&h_{%
12}^{\rm(K,2)}(\bm{k})&h_{13}^{\rm(K,2)}(\bm{k})\\
h_{12}^{\rm(K,2)}(\bm{k})&h_{22}^{\rm(K,2)}(\bm{k})&h_{23}^{\rm(K,2)}(\bm{k})%
\\
h_{13}^{\rm(K,2)}(\bm{k})&h_{23}^{\rm(K,2)}(\bm{k})&h_{33}^{\rm(K,2)}(\bm{k})%
\\
\end{array}\right),$$
(17)
where $h_{11}^{\rm(K,2)}(\bm{k})=2\left(\cos k_{x}+\cos\frac{k_{x}+\sqrt{3}k_{y}}{2}\right)$,
$h_{12}^{\rm(K,2)}(\bm{k})=2\cos\left(\frac{3k_{x}-\sqrt{3}k_{y}}{4}\right)$,
$h_{13}^{\rm(K,2)}(\bm{k})=2\cos\left(\cos\frac{\sqrt{3}k_{y}}{2}\right)$,
$h_{22}^{\rm(K,2)}(\bm{k})=2\left(\cos\frac{k_{x}+\sqrt{3}k_{y}}{2}+\cos\frac{k%
_{x}-\sqrt{3}k_{y}}{2}\right)$,
$h_{23}^{\rm(K,2)}(\bm{k})=2\cos\left(\frac{3k_{x}+\sqrt{3}k_{y}}{4}\right)$,
and
$h_{33}^{\rm(K,2)}(\bm{k})=2\left(\cos k_{x}+\cos\frac{k_{x}-\sqrt{3}k_{y}}{2}\right)$.
Constructed as such, $h^{\rm(K)}_{2}(\bm{k})$ is expressed by a quadratic form of $h^{\rm(K)}_{1}(\bm{k})$, as we have seen in the previous section.
Indeed, one can show that
$h^{\rm(K)}_{2}(\bm{k})$ can be written by using $h^{\rm(K)}_{1}(\bm{k})$ as
$$h^{\rm(K)}_{2}(\bm{k})=\left[h^{\rm(K,1)}_{1}(\bm{k})\right]^{2}-h^{\rm(K,1)}_%
{1}(\bm{k})-4\hat{I}_{3}.$$
(18)
since $(x,y,z)=(1,1,4)$ for a kagome lattice.
Then, let us consider the Hamiltonian
$$\mathcal{H}^{\rm(K)}=\sum_{\bm{k}}\hat{c}_{\bm{k}}^{\dagger}\left[t_{1}h_{1}^{%
\rm(K)}(\bm{k})+t_{2}h_{2}^{\rm(K)}(\bm{k})\right].$$
(19)
The band dispersion of $\mathcal{H}^{\rm(K)}$ is obtained by using Eq. (18) as
$$\varepsilon^{\rm(K,f)}=-2(t_{1}-t_{2}),$$
(20)
$$\displaystyle\varepsilon^{\rm(K,d)}_{1}(\bm{k})=$$
$$\displaystyle f(\lambda^{\rm(K,d)}_{1}(\bm{k}))$$
$$\displaystyle\equiv$$
$$\displaystyle t_{2}[\lambda^{\rm(K,d)}_{1}(\bm{k})]^{2}+(t_{1}-t_{2})\lambda^{%
\rm(K,d)}_{1}(\bm{k})-4t_{2},$$
$$\displaystyle\varepsilon^{\rm(K,d)}_{2}(\bm{k})=$$
$$\displaystyle f(\lambda^{\rm(K,d)}_{2}(\bm{k}))$$
$$\displaystyle\equiv$$
$$\displaystyle t_{2}[\lambda^{\rm(K,d)}_{2}(\bm{k})]^{2}+(t_{1}-t_{2})\lambda^{%
\rm(K,d)}_{2}(\bm{k})-4t_{2}.$$
Notice that, although the flat and dispersive bands touch at $t_{2}=0$ Bergman2008 ,
the intersection among these band does not occur as soon as finite $t_{2}$ is introduced.
Indeed, in the previous study, the authors have shown that this occurs when $t_{1}$ and $t_{2}$ have the same sign and they satisfy
$|t_{2}|>|t_{1}|/5$ Mizoguchi2018 .
The intersection of bands leads to the divergence of partial DOS contributed from dispersive bands.
As the introduction of $t_{2}$, the partial DOS, $\rho^{0}_{q}(\varepsilon)$, contributed from the original dispersive bands, $\lambda^{\rm(K,d)}_{q}(\bm{k})$, are deformed as
$$\displaystyle\rho_{q}(\varepsilon)=\frac{1}{|f^{\prime}(\varepsilon)|}\rho^{0}%
_{q}(f^{-1}(\varepsilon)).$$
(23)
As an example of the band intersection, we plot a band structure for $(t_{1},t_{2})=(-1,-0.3)$ in Fig. 2(c).
We also show the DOS, $\rho(\omega)$ for the same parameter in Fig. 2(e).
Here the DOS is computed numerically as
$$\displaystyle\rho(\omega)=$$
$$\displaystyle\frac{1}{N_{\mathrm{m}}}\sum_{\bm{k},n}\Theta\left(\varepsilon_{n%
}(\bm{k})-\left(\omega-\frac{\Delta\omega}{2}\right)\right)$$
$$\displaystyle\times$$
$$\displaystyle\Theta\left(\left(\omega+\frac{\Delta\omega}{2}\right)-%
\varepsilon_{n}(\bm{k})\right),$$
where
$n$ is the label of bands, $\Delta\omega$
is a unit of discretized energy set as $0.08$,
$N_{\mathrm{m}}$ is a number of mesh in the momentum space
set as $N_{\mathrm{m}}=200\times 200$,
and
$\Theta\left(x\right)$ is a Heaviside step function.
We see that, other than the contribution from the flat band, there is large DOS at the band top.
This is due to the fact that the band maxima form a line in the two-dimensional Brillouin zone, rather than discrete points,
meaning that it has a sub-extensive degeneracy Mizoguchi2018 .
This causes the divergence of the DOS at the band top.
In fact, from the relation between the original and modified dispersive bands, our method generally leads to the $d-1$ dimensional degenerate surface at the band top, giving rise to the strongly divergent DOS proportional to $\varepsilon^{-1/2}$, irrespective of the system dimension, $d$.
Since the flat band is relatively close to the band top,
the obtained band structure is potentially suitable for obtaining
high-$T_{c}$ superconductivity due to the interband scattering.
For comparison, we also show the results for $(t_{1},t_{2})=(-1,-0.7)$ in Figs. 2(d) and 2(f).
Although the penetration of flat band occurs as well, the DOS of dispersive band is more or less small near the flat band.
This is due to the fact that the flat-band energy is far away form the band top, where upper dispersive band has a large DOS.
III.2 Pyrochlore lattice
We can apply the same method to a pyrochlore lattice.
The lattice vectors are $\bm{a}_{1}^{\rm(P)}=(0,1/2,1/2)$, $\bm{a}_{2}^{\rm(P)}=(1/2,0,1/2)$, and $\bm{a}_{3}^{\rm(P)}=(1/2,1/2,0)$.
The coordinates of sublattices are $\bm{r}_{1}^{\rm(P)}=(0,0,0)$, $\bm{r}_{2}^{\rm(P)}=(0,1/4,1/4)$, $\bm{r}_{3}^{\rm(P)}=(1/4,0,1/4)$, and $\bm{r}_{4}^{\rm(P)}=(1/4,1/4,0)$.
The NN Hamiltonian in the momentum space is then given by
$$h^{\rm(P)}_{1}(\bm{k})=\left(\begin{array}[]{cccc}0&h^{\rm(P,1)}_{12}(\bm{k})&%
h^{\rm(P,1)}_{13}(\bm{k})&h^{\rm(P,1)}_{14}(\bm{k})\\
h^{\rm(P,1)}_{12}(\bm{k})&0&h^{\rm(P,1)}_{23}(\bm{k})&h^{\rm(P,1)}_{24}(\bm{k}%
)\\
h^{\rm(P,1)}_{13}(\bm{k})&h^{\rm(P,1)}_{23}(\bm{k})&0&h^{\rm(P,1)}_{34}(\bm{k}%
)\\
h^{\rm(P,1)}_{14}(\bm{k})&h^{\rm(P,1)}_{24}(\bm{k})&h^{\rm(P,1)}_{34}(\bm{k})&%
0\\
\end{array}\right)$$
(25)
with $h^{\rm(P)}_{12}(\bm{k})=2\cos\left(\frac{k_{y}+k_{z}}{4}\right)$,
$h^{\rm(P)}_{13}(\bm{k})=2\cos\left(\frac{k_{x}+k_{z}}{4}\right)$,
$h^{\rm(P)}_{14}(\bm{k})=2\cos\left(\frac{k_{x}+k_{y}}{4}\right)$,
$h^{\rm(P)}_{23}(\bm{k})=2\cos\left(\frac{k_{x}-k_{y}}{4}\right)$,
$h^{\rm(P)}_{24}(\bm{k})=2\cos\left(\frac{k_{x}-k_{z}}{4}\right)$,
and $h^{\rm(P)}_{34}(\bm{k})=2\cos\left(\frac{k_{y}-k_{z}}{4}\right)$.
$h^{\rm(P)}_{1}(\bm{k})$ has two flat eigenvalues, $\lambda_{1}^{\rm(P)}=\lambda_{2}^{\rm(P)}=-2$,
and the other two eigenvalues are
$$\lambda^{\mathrm{(P)}}_{1}(\bm{k})=2+\sqrt{4+F^{\rm(P)}(\bm{k})},$$
(26)
$$\lambda^{\mathrm{(P)}}_{2}(\bm{k})=2-\sqrt{4+F^{\rm(P)}(\bm{k})},$$
(27)
with
$$\displaystyle F^{\rm(P)}(\bm{k})\equiv 2\left[\cos\left(\frac{k_{x}+k_{y}}{2}%
\right)+\cos\left(\frac{k_{y}+k_{z}}{2}\right)+\cos\left(\frac{k_{z}+k_{x}}{2}%
\right)+\cos\left(\frac{k_{x}-k_{y}}{2}\right)+\cos\left(\frac{k_{y}-k_{z}}{2}%
\right)+\cos\left(\frac{k_{z}-k_{x}}{2}\right)\right].$$
As we did for kagome,
we introduce the second-neighbor hoppings as
$$h_{2}(\bm{k})=\left(\begin{array}[]{cccc}h^{\rm(P,2)}_{11}(\bm{k})&h^{\rm(P,2)%
}_{12}(\bm{k})&h^{\rm(P,2)}_{13}(\bm{k})&h^{\rm(P,2)}_{14}(\bm{k})\\
h^{\rm(P,2)}_{12}(\bm{k})&h^{\rm(P,2)}_{22}(\bm{k})&h^{\rm(P,2)}_{23}(\bm{k})&%
h^{\rm(P,2)}_{24}(\bm{k})\\
h^{\rm(P,2)}_{13}(\bm{k})&h^{\rm(P,2)}_{23}(\bm{k})&h^{\rm(P,2)}_{33}(\bm{k})&%
h^{\rm(P,2)}_{34}(\bm{k})\\
h^{\rm(P,2)}_{14}(\bm{k})&h^{\rm(P,2)}_{24}(\bm{k})&h^{\rm(P,2)}_{34}(\bm{k})&%
h^{\rm(P,2)}_{44}(\bm{k})\\
\end{array}\right),$$
(29)
with $h^{\rm(P,2)}_{11}(\bm{k})=2\left[\cos\left(\frac{k_{x}+k_{y}}{2}\right)+\cos%
\left(\frac{k_{z}+k_{x}}{2}\right)+\cos\left(\frac{k_{y}+k_{z}}{2}\right)\right]$,
$h^{\rm(P,2)}_{12}(\bm{k})=4\cos\left(\frac{k_{x}}{2}\right)\cos\left(\frac{k_{%
y}-k_{z}}{4}\right)$,
$h^{\rm(P,2)}_{13}(\bm{k})=4\cos\left(\frac{k_{y}}{2}\right)\cos\left(\frac{k_{%
x}-k_{z}}{4}\right)$,
$h^{\rm(P,2)}_{14}(\bm{k})=4\cos\left(\frac{k_{z}}{2}\right)\cos\left(\frac{k_{%
x}-k_{y}}{4}\right)$,
$h^{\rm(P,2)}_{22}(\bm{k})=2\left[\cos\left(\frac{k_{x}-k_{y}}{2}\right)+\cos%
\left(\frac{k_{z}-k_{x}}{2}\right)+\cos\left(\frac{k_{y}+k_{z}}{2}\right)\right]$,
$h^{\rm(P,2)}_{23}(\bm{k})=4\cos\left(\frac{k_{z}}{2}\right)\cos\left(\frac{k_{%
x}+k_{y}}{4}\right)$,
$h^{\rm(P,2)}_{24}(\bm{k})=4\cos\left(\frac{k_{y}}{2}\right)\cos\left(\frac{k_{%
x}+k_{z}}{4}\right)$,
$h^{\rm(P,2)}_{33}(\bm{k})=2\left[\cos\left(\frac{k_{x}-k_{y}}{2}\right)+\cos%
\left(\frac{k_{z}+k_{x}}{2}\right)+\cos\left(\frac{k_{y}-k_{z}}{2}\right)\right]$,
$h^{\rm(P,2)}_{34}(\bm{k})=4\cos\left(\frac{k_{x}}{2}\right)\cos\left(\frac{k_{%
y}+k_{z}}{4}\right)$,
and $h^{\rm(P,2)}_{44}(\bm{k})=2\left[\cos\left(\frac{k_{x}+k_{y}}{2}\right)+\cos%
\left(\frac{k_{z}-k_{x}}{2}\right)+\cos\left(\frac{k_{y}-k_{z}}{2}\right)\right]$.
Since the lattice parameters are given as $(x,y,z)=(1,2,6)$, $h_{2}(\bm{k})$ satisfies
$$h^{\rm(P)}_{2}(\bm{k})=\left[h^{\rm(P,1)}_{1}(\bm{k})\right]^{2}-2h^{\rm(P,1)}%
_{1}(\bm{k})-6\hat{I}_{4}.$$
(30)
Now let us consider the Hamiltonian
$$\mathcal{H}^{\rm(P)}=\sum_{\bm{k}}\hat{c}_{\bm{k}}^{\dagger}\left[t_{1}h_{1}^{%
\rm(P)}(\bm{k})+t_{2}h_{2}^{\rm(P)}(\bm{k})\right].$$
(31)
Then, if $t_{1}$ and $t_{2}$ have the same sign and $|t_{2}|/|t_{1}|>1/6$, the flat bands penetrate the dispersive band Mizoguchi2018 .
We show the band structure and DOS for $(t_{1},t_{2})=(-1,-0.25)$ in Figs. 3(c) and 3(d), respectively.
Here we use $32\times 32\times 32$ meshes in the Brillouin zone for the summation over $\bm{k}$.
Again, the upper dispersive band has relatively large DOS near the band top, which is penetrated by the flat band.
IV Extensions to inhomogeneous models
Our method is also applicable to the models with site/bond inequivalency.
We first consider “breathing” lattices of kagome and pyrochlore,
where the bond inhomogeneity is introduced to the original kagome/pyrochlore lattices.
These lattices are recently of interests particularly in the contest of higher-order topological insulators Hatsugai2011 ; Xu2017 ; Kunst2018 ; Romhanyi2018 ; Ezawa2018 , as well as frustrated magnetism Okamoto2013 ; Benton2015 ; Shaffer2017 ; Orain2017 ; Tsunetugu2017 ; Essafi2017 .
We also consider non-line-graph lattices, such as a Lieb lattice Lieb1989 and a dice lattice Sutherland1986 ,
as examples of site-inhomogeneous lattices.
IV.1 Breathing kagome and prochlore lattices
In the breathing kagome (pyrochlore) lattice, the transfer integrals are modulated
from the original models in such a way
that the transfer integral on upward triangles (tetrahedra), $t_{1}^{\rm U}$, is not equal to that on downward ones, $t_{1}^{\rm D}$.
Our method works even in breathing lattices despite the presence of bond inequivalency, because,
the eigenfunctions for a flat band does not change even if we introduce the breathing-type modulation Hatsugai2011 .
At the NN model,
the “position” of flat band(s) is sensitive to the relative sign between $t_{1}^{\rm U}$ and $t_{1}^{\rm D}$ Hatsugai2011 ; Essafi2017 ; Ezawa2018 .
If these two have the same sign, the flat band resides in the band top or bottom.
If they are opposite, on the other hand, it is located in the middle of two dispersive bands, keeping touching points with
either upper or lower bands.
When we introduce the second-neighbor hoppings, the flat-band penetration occurs in both cases for sufficiently large $|t_{2}|$,
but in a quite different manner.
First, let us see the case where both $t_{1}^{\rm U}$ and $t_{1}^{\rm D}$ have a negative sign.
In this case, the flat band penetrates the upper band for both $|t_{1}^{\rm U}|>|t_{1}^{\rm D}|$ and $|t_{1}^{\rm U}|<|t_{1}^{\rm D}|$
[Figs. 4(b) and 4(c) for a breathing kagome, and Figs. 5(b) and 5(c) for a breathing pyrochlore],
as is in the case of the original kagome/pyrochlore lattices.
Next, we consider the case of opposite sign, in particular, the case with $t_{1}^{\rm U}=-1$ and $t_{1}^{\rm D}=1$.
In the absence of $t_{2}(<0)$, the flat band intersects the line node
of the dispersive band at $\Gamma$ point Essafi2017 .
This line node reminds us of a Dirac cone, however, the structure of the eigenfunction comprising this line node structure is rather
close to the bosonic magnon mode associated with antiferromagnetic ordering, i.e., it is a fermionic realization of Goldstone mode Essafi2017 .
When we introduce small but finite $t_{2}$,
we first see that the line node
is gapped out, and the flat band stays
touched with the upper dispersive band at $\Gamma$ point
[Fig. 4(d) for a breathing kagome, and Fig. 5(d) for a breathing pyrochlore].
As increasing $|t_{2}|$, we see that the flat band penetrates the lower dispersive band,
with retaining a band-touching point with the upper dispersive band
[Fig. 4(e) for a breathing kagome, and Fig. 5(e) for a breathing pyrochlore].
The evolution of the aforementioned band structures is tracked by the analytical formulae of the dispersion relations given in the Appendix.
IV.2 Lieb lattice
In the following two subsections, we consider a class of Lieb lattices, as examples of site-inhomogeneous lattices.
We first study a (conventional) Lieb lattice Lieb1989 .
We take the lattice vectors as $\bm{a}_{1}^{\rm(L)}=(1,0)$, $\bm{a}_{2}^{\rm(L)}=(0,1)$,
and the coordinates of the sublattices are $\bm{r}_{1}^{\rm(L)}=(1/2,0)$, $\bm{r}_{2}^{\rm(L)}=(0,1/2)$, $\bm{r}_{3}^{\rm(L)}=(0,0)$.
The lattice has a site-dependent coordination numbers as
$z_{1}=z_{2}=2$ and $z_{3}=4$.
($z_{\alpha}$ is the coordination number of the sublattice $\alpha$.)
Notice that $x$ and $y$ are not sublattice-dependent, and are equal to one and zero, respectively.
We explicitly show that we can tune the flat-band energy even on this lattice.
To begin with, we consider the NN Hamiltonian given by
$$h^{\rm(L)}_{1}(\bm{k})=t_{1}\left(\begin{array}[]{ccc}0&0&h^{\mathrm{(L,1)}}_{%
13}(\bm{k})\\
0&0&h^{\mathrm{(L,1)}}_{23}(\bm{k})\\
h^{\rm(L,1)}_{13}(\bm{k})&h^{\rm(L,1)}_{23}(\bm{k})&0\\
\end{array}\right),$$
(32)
with $h_{13}(\bm{k})=2\cos\frac{k_{x}}{2}$ and $h_{23}(\bm{k})=2\cos\frac{k_{y}}{2}$.
The Hamiltonian has a flat eigenvalue $\lambda^{\rm(L,f)}=0$
and the corresponding eigenfunction is
$$\displaystyle\psi^{\rm(L,f)}(\bm{k})=\left(-\frac{\cos\frac{k_{y}}{2}}{%
\mathcal{N}^{(\mathrm{L})}(\bm{k})},\frac{\cos\frac{k_{x}}{2}}{\mathcal{N}^{(%
\mathrm{L})}(\bm{k})},0\right)^{\rm T},$$
with $\mathcal{N}^{(\mathrm{L})}(\bm{k})=\sqrt{\cos^{2}\frac{k_{x}}{2}+\cos^{2}\frac%
{k_{y}}{2}}$.
The other two eigenvalues are given by
$$\lambda_{1}^{\rm(L,d)}(\bm{k})=+2\sqrt{\cos^{2}\frac{k_{x}}{2}+\cos^{2}\frac{k%
_{y}}{2}},$$
(34)
and
$$\lambda_{2}^{\rm(L,d)}(\bm{k})=-2\sqrt{\cos^{2}\frac{k_{x}}{2}+\cos^{2}\frac{k%
_{y}}{2}},$$
(35)
thus they form a Dirac cone at M point, where they have a point contact with the flat band.
Let us introduce the second-neighbor hopping, as
$$h_{2}^{\rm(L)}(\bm{k})=\left(\begin{array}[]{ccc}h^{\rm(L,2)}_{11}(\bm{k})&h^{%
\rm(L,2)}_{12}(\bm{k})&0\\
h^{\rm(L,2)}_{12}(\bm{k})&h^{\rm(L,2)}_{22}(\bm{k})&0\\
0&0&h^{\rm(L,2)}_{33}(\bm{k})\\
\end{array}\right),$$
(36)
with
$h^{\rm(L,2)}_{11}(\bm{k})=2\cos k_{x}$,
$h^{\rm(L,2)}_{12}(\bm{k})=2\left[\cos\left(\frac{k_{x}}{2}+\frac{k_{y}}{2}%
\right)+\cos\left(\frac{k_{x}}{2}-\frac{k_{y}}{2}\right)\right]$,
$h^{\rm(L,2)}_{22}(\bm{k})=2\cos k_{y}$,
and $h^{\rm(L,2)}_{33}(\bm{k})=2(\cos k_{x}+\cos k_{y})$.
It is interesting to notice that, the block matrix for sublattice 1 and 2 is identical with
the NN hopping matrix on a checkerboard lattice, which is a line graph.
This indicates that there exists a flat mode of $h_{2}^{\rm(L)}(\bm{k})$, which is, as we will see, identical with
$\psi^{\rm(L,f)}(\bm{k})$ in Eq. (LABEL:eq:Lieb_flat).
$h_{2}^{\rm(L)}(\bm{k})$ is not expressed by the quadratic form of $h_{1}^{\rm(L)}(\bm{k})$;
rather, it is expressed as
$$h_{2}^{\rm(L)}(\bm{k})=\left[h_{1}^{\rm(L)}(\bm{k})\right]^{2}-\left(\begin{%
array}[]{ccc}2&&\\
&2&\\
&&4\\
\end{array}\right),$$
(37)
which reflects the fact that sublattice 3 has larger coordination number than the other two sublattices.
Nevertheless, the eigenvector of flat mode of
$h_{1}^{\rm(L)}(\bm{k})$, i.e. $\psi^{\rm(L,f)}(\bm{k})$, is also an eigenvector of $h_{2}^{\rm(L)}(\bm{k})$ with eigenvalue $-2$,
since it does not have a weight on sublattice 3.
Note that $\psi^{\rm(L,d)}_{1}(\bm{k})$ and $\psi^{\rm(L,d)}_{2}(\bm{k})$ are no longer eigenvectors after introducing the second-neighbor hopping.
We show the band structures for several values of $t_{1}$ and $t_{2}$ in Fig. 6.
The analytical formula of the dispersion relations is presented in the Appendix.
As is in the case of kagome and pyrochlore lattices, the band crossing does not occur for arbitrary $t_{2}$.
Indeed, for $|t_{2}|\leq\frac{|t_{1}|}{\sqrt{6}}$, the dispersive bands acquire the gap but they do not intersect the flat band:
Instead, the lower dispersive band retains the touching point with the flat band at M point [Fig. 6(b)].
Meanwhile, for $|t_{2}|>\frac{|t_{1}|}{\sqrt{6}}$, the upper dispersive band intersects the flat band [Fig. 6(e)].
IV.3 Dice lattice
We next study a dice lattice Sutherland1986 , which has a trigonal symmetry.
The lattice is constructed such that we add sites at the centers of hexagonal plaquettes on a honeycomb lattice;
each newly-added site has a finite hopping integral between only one of the two sublattices of an original honeycomb lattice, (say, 2).
Due to this choice of NN hopping, the coordination numbers differ from one sublattice to the others as $z_{1}=z_{3}=3$, and $z_{2}=6$.
Further, $x$ is also sublattice-dependent, as $x_{11}=x_{33}=x_{13}=1$, and $x_{22}=2$.
($x_{\alpha\beta}$ is a number of the second-neighbor-hopping paths between sublattices $\alpha$ and $\beta$.)
We take the lattice vectors as $\bm{a}_{1}^{\rm(D)}=\left(\frac{1}{2},-\frac{\sqrt{3}}{2}\right)$,
$\bm{a}_{2}^{\rm(D)}=\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)$,
and the coordinates of the sublattices as
$\bm{r}_{1}^{\rm(D)}=\left(\frac{1}{2},-\frac{1}{2\sqrt{3}}\right)$,
$\bm{r}_{2}^{\rm(D)}=\left(\frac{1}{2},\frac{1}{2\sqrt{3}}\right)$,
$\bm{r}_{3}^{\rm(D)}=\left(1,0\right)$.
Then, the NN hopping matrix on this lattice is given as
$$h_{1}^{\rm(D)}(\bm{k})=\left(\begin{array}[]{ccc}0&h^{\rm(D,1)}_{12}(\bm{k})&0%
\\
h^{\ast\mathrm{(D,1)}}_{12}(\bm{k})&0&h^{\rm(D,1)}_{12}(\bm{k})\\
0&h^{\ast\mathrm{(D,1)}}_{12}(\bm{k})&0\\
\end{array}\right),$$
(38)
with $h^{\rm(D,1)}_{12}(\bm{k})=e^{i\frac{k_{y}}{\sqrt{3}}}+2e^{-i\frac{k_{y}}{2%
\sqrt{3}}}\cos\frac{k_{x}}{2}$.
$h_{1}^{\rm(D)}(\bm{k})$ has a flat eigenvalue $\lambda^{\rm(D,f)}=0$, and
the corresponding eigenfunction is
$$\displaystyle\psi^{\rm(D,f)}(\bm{k})=\left(\frac{h^{\rm(D,1)}_{12}(\bm{k})}{%
\sqrt{2}|h^{\rm(D,1)}_{12}(\bm{k})|},0,-\frac{h^{\ast\mathrm{(D,1)}}_{12}(\bm{%
k})}{\sqrt{2}|h^{\rm(D,1)}_{12}(\bm{k})|}\right)^{\rm T},$$
which does not have a weight on sublattice 2.
The other two bands are dispersive and given as,
$$\lambda_{1}^{\rm(D,d)}(\bm{k})=2\sqrt{1+4\cos^{2}\frac{k_{x}}{2}+4\cos\frac{k_%
{x}}{2}\cos\sqrt{3}k_{y}}$$
(40)
$$\lambda_{2}^{\rm(D,d)}(\bm{k})=-2\sqrt{1+4\cos^{2}\frac{k_{x}}{2}+4\cos\frac{k%
_{x}}{2}\cos\sqrt{3}k_{y}}.$$
(41)
As is in the case of Lieb lattice, they form a Dirac cone, where they touch the flat band.
Now let us introduce the second-neighbor hoppings.
On this lattice, one needs a trick to obtain a desirable Hamiltonian, that is,
the second-neighbor hopping between neighboring sublattice 2 is twice as large as other second-neighbor hopping.
This reflects the inhomogeneity of $x$.
As a result, the second-neighbor hopping matrix we consider is given by
$$h_{2}^{\rm(D)}(\bm{k})=\left(\begin{array}[]{ccc}h^{\rm(D,2)}_{11}(\bm{k})&0&h%
^{\rm(D,2)}_{13}(\bm{k})\\
0&2h^{\rm(D,2)}_{11}(\bm{k})&0\\
h^{\ast\mathrm{(D,2)}}_{13}(\bm{k})&0&h^{\rm(D,2)}_{11}(\bm{k})\\
\end{array}\right),$$
(42)
where $h^{\rm(D,2)}_{11}(\bm{k})=2\left[\cos k_{x}+\cos\frac{k_{x}+\sqrt{3}k_{y}}{2}+%
\cos\frac{k_{x}-\sqrt{3}k_{y}}{2}\right]$,
and $h^{\rm(D,2)}_{13}(\bm{k})=e^{i\frac{\sqrt{3}k_{x}+k_{y}}{2\sqrt{3}}}+e^{i\frac%
{-\sqrt{3}k_{x}+k_{y}}{2\sqrt{3}}}+e^{-i\frac{k_{y}}{\sqrt{3}}}+e^{i\frac{2k_{%
y}}{\sqrt{3}}}+e^{-i\frac{\sqrt{3}k_{x}+k_{y}}{\sqrt{3}}}+e^{i\frac{\sqrt{3}k_%
{x}-k_{y}}{\sqrt{3}}}$.
This satisfies the relation
$$h_{2}^{\rm(D)}(\bm{k})=[h_{1}^{\rm(D)}(\bm{k})]^{2}-\left(\begin{array}[]{ccc}%
3&&\\
&6&\\
&&3\\
\end{array}\right).$$
(43)
Again, since the eigenvector of the flat-band does not have a weight at sublattice 2,
it is also an eigenvector of $h_{2}^{\rm(D)}(\bm{k})$ with the eigenvalue $-3$.
We plot the band structures for several values of $(t_{1},t_{2})$ in Fig. 7.
The analytical formula of the dispersion relations is presented in the Appendix.
As is in the previous examples, there is a critical value of $|t_{2}|/|t_{1}|$ above which the band crossing between flat and dispersive band occurs.
To be specific, the band crossing occurs for $|t_{2}|/|t_{1}|>\frac{1}{\sqrt{15}}$ [see Fig. 7(e)].
V Conclusion
We have introduced a simple idea to tune the flat band energy by using farther-neighbor hoppings
whose amplitude is dependent on the Manhattan distance, instead of the real distance.
Mathematically, this idea is based on the fact that, for a given matrix, polynomials of that matrix have the same eigenvectors as the original one
and its eigenvalues are given by the polynomial of the original ones.
The merit of this method is that we do not need to fine-tune many parameters to obtain the suitable band structure,
and that flat bands do not acquire a dispersion by the deformation of Hamiltonian.
We have also demonstrated that the proposed method is applicable to various lattices,
including kagome and pyrochlore lattices, their breathing lattices, and a class of Lieb lattices.
We expect that this method has broad potential applications to design suitable flat-band models,
and studying the intriguing properties of those models, such as correlation effects, superconductivity, and topological physics,
will be an interesting future problem.
Acknowledgements.We thank Yasuhiro Hatsugai for fruitful discussions.
This work is supposed by Grants-in-Aid for Scientific Research, KAKENHI, JP17H06138 (T. M.),
and JP15H05852 and JP16H04026 (M. U.), MEXT, Japan.
Appendix A Analytical formulas for dispersion relations
We summarize the dispersion relations for breathing kagome/pyrochlore lattices,
a Lieb lattice and a dice lattice in the presence of the second-neighbor hopping.
Breathing kagome lattice
In the NN hopping model, the eigenvalues and eigenvectors of dispersive bands can be obtained by using either
a “molecular orbital” method Hatsugai2011 or a line-graph correspondence Mizoguchi2018 ; Essafi2017 .
Here we employ the latter, which is also applicable to the case with $t_{2}$.
We first introduce the incident matrix between the original kagome lattice
and the dual honeycomb lattice:
$$\tilde{T}(\bm{k})=\left(\begin{array}[]{ccc}e^{i\varphi_{1}(\bm{k})}&e^{i%
\varphi_{2}(\bm{k})}&e^{i\varphi_{3}(\bm{k})}\\
e^{-i\varphi_{1}(\bm{k})}&e^{-i\varphi_{2}(\bm{k})}&e^{-i\varphi_{3}(\bm{k})}%
\\
\end{array}\right),$$
(44)
where $\varphi_{1}(\bm{k})$-$\varphi_{3}(\bm{k})$ are defined in Sec. III.1.
Then, both the NN term and the second-neighbor term are expressed by $\tilde{T}(\bm{k})$ Mizoguchi2018 ; Essafi2017 :
$$H=\sum_{\bm{k}}(c^{\dagger}_{\bm{k},1},c^{\dagger}_{\bm{k},2},c^{\dagger}_{\bm%
{k},3})\left[\tilde{T}^{\dagger}(\bm{k})\mathcal{D}(\bm{k})\tilde{T}(\bm{k})-(%
t_{1}^{\rm U}+t_{1}^{\rm D}-2t_{2})\hat{I}_{3}\right]\left(\begin{array}[]{c}c%
_{\bm{k},1}\\
c_{\bm{k},2}\\
c_{\bm{k},3}\\
\end{array}\right),$$
(45)
where
$$\mathcal{D}(\bm{k})=\left(\begin{array}[]{cc}t_{1}^{\rm U}-2t_{2}&t_{2}F^{\rm(%
H)}(\bm{k})\\
t_{2}F^{\mathrm{(H)}\ast}(\bm{k})&t_{1}^{\rm D}-2t_{2}\\
\end{array}\right).$$
(46)
$F^{\rm(H)}(\bm{k})=e^{i\frac{k_{y}}{\sqrt{3}}}+2\cos\frac{k_{x}}{2}e^{-i\frac{%
k_{y}}{2\sqrt{3}}}$ is
the Fourier transformation of the NN hoppings on the dual honeycomb lattice.
Now, an eigenvalue equation to solve is
$$\tilde{T}^{\dagger}(\bm{k})\mathcal{D}(\bm{k})\tilde{T}(\bm{k})\psi(\bm{k})=(%
\varepsilon+t_{1}^{\rm U}+t_{1}^{\rm D}-2t_{2})(\bm{k})\psi(\bm{k}).$$
(47)
To solve this, we define a two-component vector $\phi(\bm{k})$ such that
$$\phi(\bm{k})=\tilde{T}(\bm{k})\psi(\bm{k}).$$
(48)
Then, by multiplying $\tilde{T}(\bm{k})$ from left to Eq. (47),
we obtain an eigenvalue equation for $\phi(\bm{k})$ as
$$\tilde{T}(\bm{k})\tilde{T}^{\dagger}(\bm{k})\mathcal{D}(\bm{k})\phi(\bm{k})=(%
\varepsilon+t_{1}^{\rm U}+t_{1}^{\rm D}-2t_{2})\phi(\bm{k}).$$
(49)
Note that $\tilde{T}(\bm{k})\tilde{T}^{\dagger}(\bm{k})$ is a $2\times 2$ matrix which is given as
$$\tilde{T}(\bm{k})\tilde{T}^{\dagger}(\bm{k})=\left(\begin{array}[]{cc}3&F^{\rm%
(H)}(\bm{k})\\
F^{\mathrm{(H)}\ast}(\bm{k})&3\\
\end{array}\right).$$
(50)
To obtain (50), we use the fact that $e^{2i\varphi_{1}(\bm{k})}+e^{2i\varphi_{2}(\bm{k})}+e^{2i\varphi_{3}(\bm{k})}=%
F^{\rm(H)}(\bm{k})$.
From(50), we see that the eigenvalue
of the original problem, $\varepsilon^{\rm(BK)}(\bm{k})$, can be obtained by solving an eigenvalue equation of
the $2\times 2$ matrix.
By doing this, we finally obtain the dispersion relations as
$$\displaystyle\varepsilon^{\rm(BK)}_{\pm}(\bm{k})=\frac{(t_{1}^{\rm U}+t_{1}^{%
\rm D})\pm\sqrt{9(t_{1}^{\rm U}-t_{1}^{\rm D})^{2}+|F^{\rm(H)}(\bm{k})|^{2}(t_%
{1}^{\rm U}+t_{2})(t_{1}^{\rm D}+t_{2})}}{2}+(|F^{\rm(H)}(\bm{k})|^{2}-4)t_{2}.$$
Breathing pyrochlore lattice
We can apply the same method to the breathing pyrochlore lattice with $t_{2}$.
Here we show the resulting eigenvalues of dispersive bands:
$$\displaystyle\varepsilon^{\rm(BP)}_{\pm}(\bm{k})=(t_{1}^{\rm U}+t_{1}^{\rm D})%
\pm\sqrt{4(t_{1}^{\rm U}-t_{1}^{\rm D})^{2}+|F^{\rm(D)}(\bm{k})|^{2}(t_{1}^{%
\rm U}+2t_{2})(t_{1}^{\rm D}+2t_{2})}+(|F^{\rm(D)}(\bm{k})|^{2}-6)t_{2},$$
with $F^{\rm(D)}(\bm{k})=e^{-i\frac{k_{x}+k_{y}+k_{z}}{8}}+e^{i\frac{-k_{x}+k_{y}+k_%
{z}}{8}}+e^{i\frac{k_{x}-k_{y}+k_{z}}{8}}+e^{i\frac{k_{x}+k_{y}-k_{z}}{8}}$
being the Fourier transformation of the NN hoppings on the dual diamond lattice.
Lieb lattice
In the following two cases, we obtain the eigenenergies
by explicitly solving eigenvalue equations in two-dimensional space
spanned by two dispersive modes of the NN Hamiltonian,
where we utilize that fact that the flat mode is unchanged when introducing the second-neighbor term.
For the Lieb lattice, the dispersive bands have following dispersion relations:
$$\varepsilon_{1}^{\rm(L)}(\bm{k})=t_{2}\left[4\left(\cos^{2}\frac{k_{x}}{2}+%
\cos^{2}\frac{k_{y}}{2}\right)-3\right]+\sqrt{4t_{1}^{2}\left(\cos^{2}\frac{k_%
{x}}{2}+\cos^{2}\frac{k_{y}}{2}\right)+t_{2}^{2}},$$
(53)
and
$$\varepsilon_{1}^{\rm(L)}(\bm{k})=t_{2}\left[4\left(\cos^{2}\frac{k_{x}}{2}+%
\cos^{2}\frac{k_{y}}{2}\right)-3\right]-\sqrt{4t_{1}^{2}\left(\cos^{2}\frac{k_%
{x}}{2}+\cos^{2}\frac{k_{y}}{2}\right)+t_{2}^{2}}.$$
(54)
The upper dispersive band corresponds to $\varepsilon_{1}^{\rm(L)}(\bm{k})$.
The critical value for $t_{2}$ at which the intersection between flat and dispersive band occurs is determined by the condition
$$\varepsilon_{1}^{\rm(L)}(\bm{k}=0;t_{1},t^{c}_{2})=-2t_{2}^{c},$$
(55)
which, as described in the main text, leads to $|t_{2}^{c}|=\frac{|t_{1}|}{\sqrt{6}}$.
Dice lattice
Next, we consider the dice lattice.
The dispersive bands have following dispersion relations:
$$\varepsilon_{1}^{\rm(D)}(\bm{k})=t_{2}\left(2|h_{12}^{\rm(D,1)}(\bm{k})|^{2}-%
\frac{9}{2}\right)+\sqrt{2t_{1}^{2}|h_{12}^{\rm(D,1)}(\bm{k})|^{2}+\frac{9}{4}%
t_{2}^{2}},$$
(56)
and
$$\varepsilon_{2}^{\rm(D)}(\bm{k})=t_{2}\left(2|h_{12}^{\rm(D,1)}(\bm{k})|^{2}-%
\frac{9}{2}\right)-\sqrt{2t_{1}^{2}|h_{12}^{\rm(D,1)}(\bm{k})|^{2}+\frac{9}{4}%
t_{2}^{2}},$$
(57)
where $h_{12}^{\rm(D,1)}(\bm{k})$ is given in the main text.
The upper dispersive band corresponds to $\varepsilon_{1}^{\rm(D)}(\bm{k})$.
The critical value for $t_{2}$ at which the intersection between flat and dispersive band occurs is determined by the condition
$$\varepsilon_{1}^{\rm(D)}(\bm{k}=0;t_{1},t^{c}_{2})=-3t_{2}^{c},$$
(58)
which, as described in the main text, leads to $|t_{2}^{c}|=\frac{|t_{1}|}{\sqrt{15}}$.
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Uncertainty in Contrastive Learning: On the Predictability of Downstream Performance
Shervin Ardeshir
Netflix
shervina@netflix.com
&Navid Azizan
Massachusetts Institute of Technology
azizan@mit.edu
Abstract
The superior performance of some of today’s state-of-the-art deep learning models is to some extent owed to extensive (self-)supervised contrastive pretraining on large-scale datasets. In contrastive learning, the network is presented with pairs of positive (similar) and negative (dissimilar) datapoints and is trained to find an embedding vector for each datapoint, i.e., a representation, which can be further fine-tuned for various downstream tasks. In order to safely deploy these models in critical decision-making systems, it is crucial to equip them with a measure of their uncertainty or reliability. However, due to the pairwise nature of training a contrastive model, and the lack of absolute labels on the output (an abstract embedding vector), adapting conventional uncertainty estimation techniques to such models is non-trivial.
In this work, we study whether the uncertainty of such a representation can be quantified for a single datapoint in a meaningful way. In other words, we explore if the downstream performance on a given datapoint is predictable, directly from its pre-trained embedding.
We show that this goal can be achieved by directly estimating the distribution of the training data in the embedding space and accounting for the local consistency of the representations. Our experiments show that this notion of uncertainty for an embedding vector often strongly correlates with its downstream accuracy.
1 Introduction
Uncertainty estimation is an imperative for safe deployment of deep neural networks in critical decision-making systems. While deep learning approaches are capable of finding useful representations that have demonstrably enabled breakthroughs in a wide variety of tasks, one cannot wishfully assume that their predictions will always be accurate when queried on various inputs. There have been many examples of these systems making wrong predictions, which in some cases have led to fatal accidents (NTS, 2017; Varshney and Alemzadeh, 2017) and unacceptable errors (Guynn, 2015). Many such failures may be prevented if the system could supplement its predictions with a level of uncertainty or confidence in those predictions (Dietterich, 2017), which is crucial for building societal trust in such systems. Besides safety, uncertainty estimation is also needed as a part of certain learning algorithms (Hüllermeier and Waegeman, 2021), e.g., for uncertainty reduction in active learning (Aggarwal et al., 2014).
To that end, a large body of work on uncertainty estimation for deep neural networks has emerged over the past few years (see, e.g., (Blundell et al., 2015; Gal and Ghahramani, 2016; Lakshminarayanan et al., 2017; Loquercio et al., 2020; Sharma et al., 2021; Osband et al., 2021)). While there are many sources of uncertainty, in Bayesian modeling, they are often categorized into two types: aleatoric and epistemic111Aleatoric uncertainty relates to chance (Latin: alea $\leftrightarrow$ dice) and epistemic uncertainty relates to knowledge (ancient Greek: episteme $\leftrightarrow$ knowledge) Osband et al. (2021) (Kiureghian and Ditlevsen, 2009). Distinguishing between these two types of uncertainties in deep learning has been recently advocated for in the literature (Kendall and Gal, 2017). In particular, aleatoric uncertainty refers to the noise inherent in the observations, while epistemic uncertainty captures the uncertainty in the model. Aleatoric uncertainty cannot be reduced even if more data is collected (e.g., sensor noise), while epistemic uncertainty, which accounts for the model’s ignorance, can, in principle, be reduced with more data (Hüllermeier and Waegeman, 2021; Kendall and Gal, 2017).
Contrastive learning is a powerful approach to representation learning, which is responsible for the success of many state-of-the-art deep neural networks (Chen et al., 2020; Khosla et al., 2020; Wu et al., 2018; Hénaff et al., 2020; Oord et al., 2018; Tian et al., 2020; Hjelm et al., 2018; He et al., 2020). During a typical training regime of a contrastive model, pairs of datapoints are provided as positives or negatives; the contrastive objective then aims to find a data representation in which the positive pairs “attract,” (i.e., fall close to each other with an appropriate notion of distance) and the negative pairs “repulse” each other in the embedding space. This approach tends to yield rich and robust representations of the data, which may be further used or fine-tuned for downstream tasks. Depending on the availability of labeled data, the contrastive pre-training phase may be performed in a supervised (Khosla et al., 2020) or self-supervised (Chen et al., 2020) fashion.
Despite the recent developments in uncertainty quantification, the majority of the literature has been focused on supervised settings, in which a single input is mapped to an absolute target value. Such approaches are not readily applicable to contrastive models, as the model’s prediction on a single datapoint is an abstract embedding vector. Nonetheless, such approaches are applicable for measuring the (un)certainty of the model on a pair of datapoints, treating contrastive models as binary classifiers. To that end, a Bayesian metric learning framework was proposed by Wang and Tan (2017) to construct a robust estimation of the distance, given a pair of datapoints. Measuring the uncertainty of a metric learning model, given a pair of datapoints, was further explored by Qian et al. (2018). A relatively close work to ours is that of Oh et al. (2018), in which the aleatoric embedding uncertainty is evaluated in terms of instance retrieval, which is the objective the model is directly trained for. Unlike these works, we aim to measure the uncertainty/reliability of the embedding for a single datapoint, in such a way that is predictive of its downstream performance. The closest works to ours are recent works of Zhang et al. (2021); Wu and Goodman (2020), both of which tackle the problem of uncertainty estimation in the case of a contrastive objective. The key difference between our work and theirs is two folds: First, unlike Zhang et al. (2021); Wu and Goodman (2020), in our setup, a pre-trained model is given to us as a black box, for which the uncertainty of a datapoint should be estimated. In the aforementioned work, the model is being pre-trained using a modified objective function, to incorporate the notion of uncertainty. Our work, on the contrary, is a post-processing step on a pre-trained black-box contrastive model. Second, in our work, the notion of certainty for a datapoint is defined as its downstream performance, as opposed to the work of Zhang et al. (2021); Wu and Goodman (2020), in which a small variance of the embedding is indicative of certainty.
In this paper, we explore the possibility of capturing the reliability of an embedding resulting from a (black box) pre-trained contrastive model, in terms of predicting downstream performance.
Figure 1 shows an overview of our setup. A back-box model $f$ is pre-trained with a contrastive objective, on a training dataset, resulting in an embedding vector representing each datapoint. The goal is to study whether there are any notions of uncertainty for an embedding vector which is indicative of its reliability, i.e. how it would later perform downstream. Given the non-triviality of predicting downstream performance solely from a pre-trained embedding vector, we explore the possibility of such prediction using a few intuitive measures. To that end, given an input and a pre-trained model, we measure the reliability of the resulting embedding in three aspects: (1) How certain the model is about the location of an embedding vector. This is computed by introducing variations to the input datapoint and measuring variations in its embedding vector. (2) How familiar the model is with that area of the embedding space. In other words, has the model seen training examples with similar embeddings. This notion is computed by directly estimating the distribution of the embedding vectors of the training data. (3) How well does the model perform in that region of the embedding space. This is measured by calculating the local retrieval performance of the model. We study whether these intuitive notions meaningfully correlate with the downstream performance on a given input.
2 Framework and Proposed Method
In this section, we describe our framework and how our various notions of uncertainty/reliability are constructed. Let us consider a model $f:\mathbb{R}^{n}\to\mathcal{S}^{m-1}$, which maps an $n$-dimensional input datapoint (e.g., an image) $x$ to the $\ell_{2}$-normalized $m$-dimensional feature vector $f(x)$ (on the unit hypersphere). Given the model $f$, we aim to measure the reliability of the embedding vector $f(x)$ for any given input $x$. As discussed earlier, we do so based on quantifying the uncertainty in the location of the point in the embedding space as well as the consistency of the model’s prediction in that region.
2.1 Per-Sample Feature Variation: $\delta$
The first notion of uncertainty that we define aims to capture how certain the model is about the location of an embedding vector. Given a set of variations/transformations222These are often referred to as data “augmentation” techniques because they are used for augmenting the dataset. Geometric transformations, flipping, color modification, cropping, rotation, noise injection and random erasing are among the common ones (Shorten and Khoshgoftaar, 2019). $\{T_{1},T_{2},\dots,T_{l}\}$ used in the training of the contrastive model, we measure the variation across $\{z_{1},z_{2},\dots,z_{l}\}$, where $z_{i}=f(T_{i}(x))$ is the embedding vector corresponding to the $i$-th transformation of the input $x$. More specifically, we define $\delta({x})$ as the sum of the variances for different dimensions, i.e., the trace of the sample covariance matrix for the observation vectors $\{z_{1},z_{2},\dots,z_{l}\}$. Our experiments show that this simple quantity often meaningfully predicts the reliability of the embedding, as measured by the performance in a downstream task. An important characteristic of this metric is that it does not require access to the training data and would work on any black-box model. Note that the underlying assumption here is that the downstream task is invariant to the pre-training data transformations (augmentations).
2.2 Embedding Distribution Estimation
This notion is based on estimating the distribution of the embedding, which we refer to as $p_{\mathrm{emb}}$. This probability distribution captures the two key features of density and consistency over the unit hypersphere embedding space.
2.2.1 Density: $p_{\mathrm{emb}}$
The density of the embedding space at a point $z$ would intuitively capture how much data has the model observed around $z$ during training, which is the transformation of the training data distribution under $f$. To estimate this distribution, we fit a Gaussian333To be precise, one has to use a Fisher–Bingham (or Kent) distribution (Jupp and Mardia, 2009; Kent, 1982) over the $(m-1)$-sphere, which is the analogue of a Gaussian on hypersphere. mixture model (GMM) to the $m$-dimensional embeddings of the training data. Computing this density function requires access to the pre-trained model and an unsupervised training dataset, i.e., only the input datapoints and not the labels.
2.2.2 Consistency: $p^{k,\tau}_{\mathrm{emb}}$
The consistency of the model at $z$ measures whether the training datapoints mapped closest to $z$ have consistent labels. This notion would capture how accurate the model is at $z$, based on the fact that a more accurate contrastive model should have a more pure local correspondence. Note that unlike the density-only distribution mentioned above, estimating this distribution requires access to both training data and training labels (correspondences). For each training datapoint, we calculate the fraction of its $k$ nearest neighbors ($k$-NN) in the embedding space whose class labels are consistent with that datapoint. We then filter out the datapoints based on their $k$-NN accuracy with a threshold $\tau$, and fit a Gaussian mixture model to the datapoints whose $k$-NN consistency is above the threshold $\tau$. We denote this distribution by $p^{k,\tau}_{\mathrm{emb}}$. This notion would require access to the model and a supervised training dataset, and is thus only applicable to the supervised contrastive learning setup (Khosla et al., 2020).
It is worth noting that setting the threshold $\tau$ to zero yields $p^{k,0}_{\mathrm{emb}}(\cdot)=p_{\mathrm{emb}}(\cdot)$, which would solely capture the density of each datapoint in the training data.
Intuitively, the two notions defined above could lead to the following scenarios:
•
High $p_{\mathrm{emb}}(z)$ and high $p_{\mathrm{emb}}^{k,\tau}(z)$: The model has seen many consistent examples like $z$ (low uncertainty).
•
High $p_{\mathrm{emb}}(z)$ and low $p^{k,\tau}_{\mathrm{emb}}(z)$: The model has seen samples similar to $z$ during training, but has not been consistent for them. This could be due to the fact that these are hard examples, thus implying low epistemic uncertainty but high aleatoric uncertainty.
•
Low $p_{\mathrm{emb}}(z)$: The model has not seen samples similar to $z$. This implies the sample is likely out-of-distribution with respect to the training set, and thus has a high epistemic uncertainty.
2.3 Per-Sample Feature Variance + Embedding Distribution: $p^{k,\tau}_{\text{emb-ens}}$
One could also combine the two notions of per-sample variance and embedding distribution, which has the interpretation of a stochastic embedding (Wang and Isola, 2020). More specifically, we have an ensemble of probabilities through the $l$ transformations $\{T_{1},T_{2},...,T_{l}\}$, and using the law of total probability we have
$$p^{k,\tau}_{\text{emb-ens}}=\sum_{i=1}^{l}p^{k,\tau}_{\mathrm{emb}}(f(T_{i}(x)))p(T_{i})=\frac{1}{l}\sum_{i=1}^{l}p^{k,\tau}_{\mathrm{emb}}(f(T_{i}(x))).$$
The measures mentioned above have different requirements, ranging from access to the black-box model only (feature-variation measure), to requiring access to a fully supervised training dataset (consistency measure). Table 1 summarizes the requirements for each measure. Note that the last two measures (entropy and max score), which are explained in Section 3, require the full observation of the downstream task and are solely defined as a baseline.
3 Experimental Results
We pre-train self-supervised (SimCLR) (Chen et al., 2020) and supervised (SupCon) (Khosla et al., 2020) contrastive models with ResNet18 (He et al., 2016) backbones, and on the training set of CIFAR10 or CIFAR100 (Krizhevsky et al., 2009) datasets. We then perform inference on their test sets, alongside test sets of CUBS2011 (Wah et al., 2011) and SVHN (Netzer et al., 2011) as other out-of-distribution datasets. We follow the pre-training and linear fine-tuning protocols in accordance to Khosla et al. (2020).
3.1 Uncertainty Measures
As discussed earlier, our different uncertainty measures make different assumptions about access to the data and models, according to which we categorize them into several groups.
Pre-trained model only. The feature-variation measure ($\delta$), described in Section 2.1, only requires access to the pre-trained model. Our quantitative results indicate that this very simple approach (directly applicable at inference time) already allows for measuring uncertainty in many scenarios.
Pre-trained model + unsupervised training data. $p_{\mathrm{emb}}$ and $p_{\text{emb-ens}}$ (described in Section 2.2)
would require access to the model and the training dataset without supervision (i.e., training images only), making them applicable to both self-supervised and supervised setups. The training data is only used for a single forward pass.
Pre-trained model + supervised training data. $p^{k,\tau}_{\mathrm{emb}}$ and $p^{k,\tau}_{\text{emb-ens}}$ require access to the labeled training dataset, as they incorporate the consistency notion mentioned in Section 2.2.2. Thus, these measures are only applicable to the supervised contrastive (Khosla et al., 2020) setup. In our experiments, we define consistency as having top 1% $k$-NN accuracy threshold of 50%. We also study the impact of $k$ (the number of neighbors) and $\tau$ (the threshold) on different types of uncertainty estimation metrics in Section 3.3.
Fine-tuned model.
The following two measures are not computable in our scenario, as they are only measurable after a downstream classifier is fine-tuned on the pre-trained features. Thus, our approach would not be comparable to these measures. Regardless, we report the measurements to put our quantitative measurements in context. Our experiments indicate that in some scenarios, our measures achieve competitive, or sometimes even slightly better, performance compared with these measures.
Entropy: Entropy of a classifier is often used as a measure of uncertainty. We measure the entropy of the downstream fine-tuned classifier on each sample and use that as a measure of certainty.
Max score: The maximum score (confidence) of the downstream classifier is used as a measure of certainty.
A summary of the requirements for each of the measures is provided in Table 1.
3.2 Evaluation
We evaluate different notions of uncertainty to cover different aspects of downstream predictability. To evaluate the different metrics, we treat them as retrieval instances and compute their AUROC (Area Under the Receiver Operating Characteristic curve). The ground-truth label of the retrieval instance could be derived as a function of the downstream accuracy of a datapoint and whether the model has been exposed to the datapoint’s semantic class during pre-training. In what follows, we discuss the details of this evaluation for each uncertainty notion. Table 2 summarizes what each uncertainty notion is capturing.
3.2.1 Aleatoric Uncertainty
Aleatoric uncertainty is often defined as the “noise inherent in the data,” which leads to difficulty of understanding a sample datapoint. We use downstream performance of a datapoint as a proxy for measuring its difficulty.
To quantify such notion, we evaluate our proposed uncertainty measures on in-distribution test-set datapoints, and in terms of their capability in retrieving samples which are correctly classified in a downstream linear classifier. Table 3 shows this metric for our different uncertainty measures.
3.2.2 Epistemic Uncertainty
We evaluate our uncertainty estimation measures on images from the in-distribution (pre-training dataset) and an out-of-distribution dataset and quantify their performance in terms of retrieving the in-distribution embeddings. In other words, a model pre-trained (supervised or self-supervised) on the training set of dataset A is fed test datapoints from datasets A and B. Then, the effectiveness of the uncertainty measures are evaluated in terms of distinguishing datapoints of dataset A from those of B. Table 4 contains the performance of our different measures on this task. It can be observed that in most cases, $p_{\mathrm{emb}}$ has the best performance, whose definition is also more consistent with out-of-distribution detection tasks, as it directly estimates the embedding distribution that comes from the training data. Another observation would be the failure of the feature variation measure in detecting out-of-distribution samples of SVHN in the self-supervised setups. We hypothesize this could be due to the fact that SVHN is a less diverse dataset, which results in its images being mapped close to one another in a CIFAR10 or CIFAR100 pre-trained model. As a result, feature variation would not be a good notion for distinguishing such samples. On the other hand, the probability-based measures result in very high AUROC scores, alluding that these measures capture complementary notions of reliability. Another explanation could be that measuring the effect of data transformation could be interpreted as mainly a notion of data uncertainty. This would suggest that this measure is a better fit for aleatoric uncertainty estimation.
On another note, we observe that our probability measures have lower discriminative power distinguishing between CIFAR100 and CIFAR10, as opposed to between CIFAR examples and non-CIFAR examples. This observation is consistent across both datasets, and across both supervised (SupCon) and self-supervised (SimCLR) setups. Also, detecting CIFAR10 samples as out-of-distribution, given a CIFAR100 pre-trained $p_{\mathrm{emb}}$ model, is noticeably more difficult than distinguishing CIFAR100 samples using a CIFAR10 pre-trained $p_{\mathrm{emb}}$.
Here we followed standard practice for evaluating epistemic uncertainty. However, we argue that the assumption of all the datapoints in the test set of dataset A being “in-distribution” to the model may not necessarily hold. To address that, we define the following alternative, which captures such nuances.
3.2.3 Overall Uncertainty
Here we introduce a hybrid definition of uncertainty, taking into account both aleatoric and epistemic uncertainties. Given a model trained on dataset A, we evaluate its uncertainty measures on datasets A and B. We then evaluate how well the uncertainty measure retrieves datapoints in A which are correctly classified by the downstream classifier. In other words, the model should not be certain about all the datapoints in A, but only the ones that are going to be correctly classified downstream. The quantitative measures using this metric are reported in Table 5. Comparing values in this table with their corresponding values in the epistemic uncertainty evaluation (Table 4), we generally observe higher values across all measures.
3.3 Ablation Study
In this section, we analyze the effect of different parameters on the performance of our approach.
Effect of number of GMM components.
We evaluate the effect of the number of GMM components ($n_{\text{comp}}$), by evaluating the metrics, while sweeping $n_{\text{comp}}$ from 2 to 150 components. Figures 2 and 3 show the aleatoric and epistemic uncertainty measures (y-axis), respectively, using different number of components (x-axis). Interestingly, except for extremely small values ($n_{\text{comp}}<10$), we observe relatively stable performance across all setups. This observation seems to be consistent across both supervised and self-supervised setups. The overall uncertainty has a very similar trend as the epistemic uncertainty. For the sake of brevity, we relegated the result to the Appendix.
Effect of threshold and number of nearest neighbors. Higher thresholds and number of nearest neighbors would result in maintaining highly consistent points, at the cost of losing information (having less remaining datapoints for estimating the GMMs). The diminishing returns of such parameters can be seen in Figure 4. It can be observed that peak-performance for epistemic uncertainty (middle), is at threshold of 0 ($p_{\mathrm{emb}}$), which is consistent with its definition of directly estimating the training data (in-distribution likelihood). On the contrary, consistency does improve the estimation of aleatoric uncertainty, as the peak of the distribution does occur at ($\tau=0.4$, $k/N=0.025$), which means that a consistent datapoint is defined as one whose top 2.5% of nearest neighbors are more than 40% consistent with its semantic label. Overall, we observe the effect of consistency to be marginal in our experiments. We hypothesize that such behavior could be due to a highly accurate pre-training, resulting in a high correlation between consistency and density. This correlation could be measured at larger scale and across different setups and datasets (with different accuracies) to validate this hypothesis.
Effect of number of transformations. Figure 5 shows the effect of number of augmentations on aleatoric uncertainty estimation. It can be observed that after 2 augmentations, the results are relatively stable. On the other hand, for epistemic uncertainty, shown in Figure 6, more improvement (yet marginal) could be achieved with more augmentations. The overall uncertainty has a very similar behavior to the epistemic uncertainty, shown in Figure 7.
4 Conclusion
In this paper, we explored the possibility of estimating a reliability/uncertainty measure for the abstract embeddings of contrastive models. We show that our uncertainty measures not only are able to meaningfully detect out-of-distribution samples but also are predictive of performance in downstream tasks.
We believe that having such notion of reliability/uncertainty can particularly be insightful, e.g., for deciding between different options of pre-trained models, or for deciding on specific sample weighting policies in downstream fine-tuning tasks.
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Appendix A Appendix
Effect of number of transformations (for overall uncertainty)
As mentioned in Section 3.3, we provide the effect of number of data transformations (image augmentations) on the overall uncertainty estimation, shown in Figure 7. We observe the trends to be very similar to those of epistemic uncertainty. We observe a monotonically increasing performance as a function of the number of transformations. However, in many scenarios such as SupCon on CIFAR10 vs. SVHN, and CIFAR100 vs. CUBS2011, the amount of improvement becomes marginal beyond 2 augmentations.
Effect of number of GMM components (for overall uncertainty)
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A generalized approach to model the spectra and radiation dose rate of
solar particle events on the surface of Mars
Jingnan Guo11affiliation: Institute of Experimental and Applied Physics, Christian-Albrechts-University, Kiel, Germany ,
Cary Zeitlin 22affiliation: Leidos, Houston, Texas, USA ,
Robert F. Wimmer-Schweingruber11affiliation: Institute of Experimental and Applied Physics, Christian-Albrechts-University, Kiel, Germany ,
Thoren McDole11affiliation: Institute of Experimental and Applied Physics, Christian-Albrechts-University, Kiel, Germany ,
Patrick Kühl11affiliation: Institute of Experimental and Applied Physics, Christian-Albrechts-University, Kiel, Germany ,
Jan C. Appel11affiliation: Institute of Experimental and Applied Physics, Christian-Albrechts-University, Kiel, Germany ,
Johannes Krauss 11affiliation: Institute of Experimental and Applied Physics, Christian-Albrechts-University, Kiel, Germany ,
Jan Köhler 11affiliation: Institute of Experimental and Applied Physics, Christian-Albrechts-University, Kiel, Germany
guo@physik.uni-kiel.de
Abstract
For future human missions to Mars, it is important to study the surface radiation environment during extreme and elevated conditions.
In the long term, it is mainly Galactic Cosmic Rays (GCRs) modulated by solar activity that contributes to the radiation on the surface of Mars, but intense solar energetic particle (SEP) events may induce acute health effects.
Such events may enhance the radiation level significantly and should be detected as immediately as possible to prevent severe damage to humans and equipment.
However, the energetic particle environment on the Martian surface is significantly different from that in deep space due to the influence of the Martian atmosphere, and, to a lesser extent, the regolith.
Depending on the intensity and shape of the original solar particle spectra as well as particle types, the surface spectra may induce entirely different radiation effects. For instance, an intense SEP event with a soft spectrum that would be hazardous on the lunar surface may, in contrast, induce only low levels of radiation on the Martian surface that would be well within human health tolerances.
In order to give immediate and accurate alerts while avoiding unnecessary ones, it is important to model and well understand the atmospheric effect on the incoming SEPs including both protons and helium ions.
In this paper, we have developed a generalized approach to quickly model the surface response of any given incoming proton/helium ion spectra and have applied it to a set of historical large solar events, thus providing insights into the possible variety of surface radiation environments that may be induced during SEP events.
Sun: solar energetic particles – Mars: atmophere – GCR radiation – Manned mission to Mars – Predictions of radiation dose rate
1 Introduction and Motivation
In order to plan future human missions to Mars, the assessment of the radiation environment on and near the surface of Mars is necessary and fundamental for the safety of astronauts. There are two types of primary particles reaching the top of the atmosphere of Mars: galactic cosmic rays (GCRs) and solar energetic particles (SEPs).
GCRs, mainly composed of protons and helium ions, are modulated by heliospheric magnetic fields which evolve dynamically as solar activity varies in time and space, with a well-known 11-year cycle (e.g., Parker, 1958).
SEP events, consisting mainly of protons, are sporadic and highly variable in terms of their intensities and energy spectra. They take place much more frequently during solar maximum periods and they may enhance the radiation level significantly, and therefore should be detected as quickly as possible to minimize risks to humans and equipment on the Martian surface.
However, SEP measurements at Mars are very scarce and within a limited energy range. The radiation assessment detector (RAD) onboard the Mars Science Laboratory (landed on Mars in Aug 2012) has measured only 5 events in the course of 4 years during the declining phase of the past solar maximum (Hassler et al., 2014). The SEP instrument onboard the Mars Atmosphere and Volatile EvolutionN (MAVEN/SEP, Larson et al., 2015) spacecraft orbiting Mars since October 2014 only directly measures protons with energies $\leq 6$ MeV which do not contribute to the surface radiation enhancement as will be shown in this study.
At near-Earth environment, SEPs are measured much more frequently by particle detectors on various spacecraft such as the Solar and Heliospheric Observatory (SOHO), the Advanced Composition Explorer (ACE), the Geostationary Operational Environmental Satellite (GOES) and so on.
To derive the particle spectra at Mars location from these measurements is however very challenging.
This is because the propagation of coronal mass ejections (CMEs) and the associated shocks (which are believed to be a major accelerator for such highly-energetic particles) through the heliosphere may result in totally different particle spectral intensities and shapes at Mars compared to Earth (Li et al., 2003). Besides, the observed SEP spectra and intensity also depend on different magnetic connections of the planets/spacecraft to the acceleration locations.
The current paper will not address the above issues when considering the SEP induced radiation environment on the surface of Mars. Alternatively we focus on how the primary energy spectra are influenced and modified by the Martian atmosphere considering the presence of some SEP events at Mars which were have been observed at near-Earth locations.
The energetic particle environment on the Martian surface is different from that in deep space due to the presence of the Martian atmosphere.
Depending on the intensity and shape of the original solar particle spectra as well as the distribution of particle types, different SEP events may induce entirely different radiation effects on the surface.
This is because primary particles passing through the Martian atmosphere may undergo inelastic interactions with the ambient atomic nuclei creating secondary particles (via spallation and fragmentation processes), which may also interact while propagating further and finally result in very complex spectra including both primaries and secondaries at the surface of Mars (e.g., Saganti et al., 2002; Guo et al., 2015a).
Primary particles with small energies do not have sufficient range to reach the ground, but the exact energy cutoff is a strong function of elevation on Mars. Therefore, an intense but soft-spectrum SEP spectra could be well within biological tolerance seen on the surface of Mars, particularly in low-lying places such as Gale Crater, Hellas Planitia, Valles Marineris, etc., where atmospheric shielding is substantially greater than the global average.
In order to give immediate and precise alerts while avoiding unnecessary ones, it is important to model and well understand the atmospheric effect on the incoming SEPs.
There are various particle transport codes such as HZETRN (Slaba et al., 2016; Wilson et al., 2016), PHITS (Sato et al., 2013) and GEANT4/PLANETOCOSMICS (Desorgher et al., 2006) which can be employed for studying the particle spectra and radiation through the Martian atmosphere.
Gronoff et al. (2015) have applied both PLANETOCOSMICS and HZETRN to calculate the GCR radiation environment on the surface of Mars and found highly consistent results from both simulations.
In this paper we use the PLANETOCOSMICS transport code and develop a generalized approach to quickly model the surface response of any given incoming proton spectrum under different atmospheric depths.
We have applied the method to a set of significant solar events which took place in the last several decades, thus providing insights into the possible variety of surface particle spectra and induced radiation environment during SEP events.
2 A generalized model: PLANETOMATRIX
PLANETOCOSMICS is a simulation tool (Version g4.10 has been used herein) developed in order to simulate particles going through planetary atmospheres and magnetic fields (Desorgher et al., 2006).
It is based on GEANT4, a Monte Carlo approach for simulating the interactions of particles as they traverse matter (Agostinelli et al., 2003).
Different settings and features, e.g. the composition and depth of the atmosphere and the soil, can be used in the simulations.
Employing PLANETOCOSMICS to model of the radiation environment on the surface of Mars has been carried out in various studies (e.g., Dartnell et al., 2007; Gronoff et al., 2015; Matthiä et al., 2016; Ehresmann et al., 2011) and has been validated when compared to spectra of the proton, helium ion and heavier ion spectra on the surface of Mars (Ehresmann et al., 2014) measured by the radiation assessment detector (RAD) onboard the Mars Science Laboratory (MSL).
In order to provide a more realistic atmospheric environment for the simulations, we use the Mars Climate Database (MCD) which has been created using different Martian atmospheric circulation models which are further compared and modified by the observation results from past and current Mars missions (Lewis et al., 1999).
It offers the possibility to access atmospheric properties, such as temperature, pressure and composition, for different altitudes, seasons and even the time of the day on Mars.
A full PLANETOCOSMICS simulation can be highly time-consuming and in principle needs to be run for each different input spectra. To reduce the computational burden, we developed an alternative approach which we refer to as the PLANETOMATRIX method, which folds the complicated nuclear interaction process into a two-dimensional matrix $\rm{\bar{A}(E_{0},E)}$ where $E_{0}$ is the energy of a particle above the Martian atmosphere and $E$ is the particle energy on the Martian surface.
It is constructed in the following way: First, a primary particle spectrum $f_{m}$ in the range of a single energy bin $E_{0m}$ (e.g. from 200 MeV to 210 MeV) is fed into the PLANETOCOSMICS code to generate the surface spectrum which is different from the original due to the production of secondaries. This surface spectrum can be described by a histogram with N bins and the flux in each bin $n$ is $a_{mn}$ (normalized to the input flux $f_{m}$).
Second, this process is repeated M times for M different input energy bins (covering 1 to 10${}^{6}$ MeV of primary particle energy) and the resulting scaled fluxes in each $(m,n)$ bin are $a_{mn}$.
Thus, under a given atmospheric composition and column depth $\sigma$ setup, the matrix $\rm{\bar{A}(E_{0},E)}$ (with a shape of $\rm{M}\times N$) can be constructed by running M simulations of PLANETOCOSMICS.
Finally with an input spectrum $f(E_{0})$ at the top of the atmosphere, the surface spectrum can be calculated as $\rm{F(E)}=\rm{\bar{A}(E_{0},E)}\cdot f(E_{0})$.
PLANETOMATRIX has been tested extensively based on different physics lists in PLANETOCOSIMICS and can be treated as a ’black box’ where all interactions of primary fluxes and generations of secondaries take place, and a given input spectrum is modified to produce an output spectrum after passing through this box.
It is a statistical description, i.e., both f (with M bins) and F (with N bins) are energy-dependent distribution histograms and each element in the matrix $\rm{\bar{A}(E_{0},E)}$ represents the probability of a primary particle with energy $E_{0}$ resulting in a particle on the surface with energy $E$.
Although the construction of each matrix is time-consuming, the multiplication of different input spectra with such a matrix to generate different surface spectra is very much simplified.
Furthermore, with measurements of surface spectra $\rm{F(E)}$ by, e.g, MSL/RAD, an inversion technique can, in principle, be applied to the matrices in order to recover $f(E_{0})$ at the top of the atmosphere similar to the technique described in (e.g., Böhm et al., 2007). This is however a very challenging task due to both the ill-posed nature of the matrix inversion and the limited energy range of the measurement. (Development of a robust inversion method is in progress but not yet complete.)
To study the evolution of the particle spectra while propagating through the atmosphere, we construct different matrices $\rm{\bar{A}}_{\sigma}$ under different atmospheric column depths $\sigma$ from 1 g/cm${}^{2}$ thickness down to the surface where the column density is about 22.5 g/cm${}^{2}$ corresponding to pressures of 830 Pascals in a hydrostatic equilibrium state111This is the average surface pressure value through one Martian year at Gale Crater recently measured by the Rover Environmental Monitoring Station (REMS, Gómez-Elvira et al., 2012) onboard MSL. This is about 5-6 g/cm${}^{2}$ greater than the column depth at the mean surface elevation, since the altitude of Gale Crater is about - 4.4 km MOLA (Mars Orbiter Laser Altimeter), but less than the column depths found in some other locations..
In addition to SEP protons, which typically dominate, we have also considered primary ${}^{4}$He ions as input. The dominant secondary particle types (types $j$) include protons, ${}^{4}$He ions, deuterons, tritons, neutrons, gammas, muons, electrons and positrons.
For each primary and a given secondary type, we generate a matrix $\rm{\bar{A}}_{\sigma ij}$.
Furthermore, the particle flux reaching the surface may also produce backscattered particles, i.e., so-called albedo particles. These are produced by nuclear interactions in the regolith. Backscattered neutrons have been observed from orbit missions (e.g., Boynton et al., 2004) and in situ by the DAN instrument aboard Curiosity in its “passive” mode (Jun et al., 2013).
Since the energy spectra of upward- and downward-traveling particles are dissimilar, we have separately constructed the upward and downward directed matrices for each primary-secondary case as $\rm{\bar{A}}_{\sigma ij-{up}}$ and $\rm{\bar{A}}_{\sigma ij-{dn}}$ respectively. Therefore the total downward spectra of particle type j generated by different primary particle types at the depth of $\sigma$ is:
$$\displaystyle\rm{F}_{\sigma j-{dn}}(E_{j})=\sum_{i}\rm{\bar{A}}_{\sigma ij-{dn%
}}(E_{0},E)\cdot f_{i}(E_{0})$$
(1)
Panels (a) and (b) of Figure 1 show the matrices of primary protons generating secondary downward and upward protons respectively.
The atmospheric depth in this case is about 20 g/cm${}^{2}$. Primary protons with energies less than about 150 MeV, indicated by a vertical line in (a), lack the range to reach the surface; secondary particles with up to 150 MeV energy are from primaries with higher energies.
A similar cutoff energy for protons has also been found by Gronoff et al. (2015).
Figure 1(c) and (d) show the example of primary protons generating secondary downward and upward neutrons.
In most solar events, protons are a large majority of the primary particles reaching the top of the Martian atmosphere.
In some SEP events, significant numbers of helium ions are accelerated, and (energy-dependent) ${}^{4}$He/${}^{1}$H flux ratios from a few percent up to 10% have been observed (Bertsch et al., 1972; Benck et al., 2016). A ratio as large as 10% may be considered a reasonable upper limit for the ratio of time- and energy-integrated fluxes (Torsti et al., 1995).
Fig. 2 shows primary Helium ions induced secondary (a) ${}^{4}$He and (b) protons near the surface of Mars at a depth of 21 g/cm${}^{2}$.
Since ${}^{4}$He ions obey the same range-energy relationship as protons, the ${}^{4}$He-${}^{4}$He matrix, like the ${}^{1}$H matrix, shows a cutoff energy for incoming particles at about 150 MeV/nuc.
The diagonal line shows the primaries which reach this depth without losing energy in the atmosphere. Very few high energy ${}^{4}$He secondaries (larger than 2 GeV/nuc) have been generated in the atmosphere.
However, many secondary protons are generated by primary ${}^{4}$He particles as shown in (b).
Based on the matrices of primary proton and ${}^{4}$He induced secondaries, we have modeled the surface spectra and radiation environment induced by primary GCRs and SEPs.
We have ignored heavier primary ions since they contribute only $\sim$ 1% of the GCR flux (Simpson, 1983) and even less of the SEPs.
3 Radiation dose rates
Radiation dose rate is a key quantity used to evaluate the energetic particle environment. Both charged and neutral particles deposit energy while going through target materials such as skin, bones and internal organs.
Dose is defined as the energy deposited by radiation per unit mass, integrated over time, with a unit of J/kg (or Gy). The dose rate in space is often expressed in units of µGy/day.
Dose rate is one of the essential factors to be considered for future crewed missions to deep space and to Mars.
It is therefore very important to measure and model the GCR- and SEP-induced dose rate in the interplanetary (IP) space and on the surface of Mars.
For any given particle spectrum, the radiation dose rate can be calculated by the following logic (e.g., Guo et al., 2015a):
$$\displaystyle D=\sum\limits_{j}\sum\limits_{area}{\iint\limits_{0}^{\infty}%
\lambda_{j}(E,\epsilon)F_{j}(E)dEd\epsilon}/{m},$$
(2)
where $j$ is the particle type, $F_{j}(E)$ (in the unit of counts/MeV/sec/cm${}^{2}$/sr) is the particle spectrum, $m$ (kg) is the mass of the material (biological bodies or detectors) and $\epsilon$ is the energy deposited by the particle in the material.
This energy transfer process, included as a yield factor, $\lambda_{j}(E,\epsilon)$, can be accurately estimated using either the Bethe-Bloch equation (Bethe, 1932) (for charged particle ionization energy loss in an infinite volume) or with more sophisticated Monte Carlo models such as GEANT4 (Matthiä et al., 2016) accounting for the probability distribution of $\epsilon$ in finite volumes. Finally $D$ is the corresponding dose rate integrated over the entire collecting volume and all the detected particle species, per unit time, with units of MeV/kg/sec (sometimes expressed as µGy/day).
The dose rate on the surface of Mars is - apart from a negligible natural background -
mainly determined by the GCR fluxes of both primaries and secondaries during solar quiet times and it may be enhanced significantly during SEP events.
As the interactions of particles through the atmosphere depend on the particle type, energy, and the depth of the atmosphere, we model SEP-induced spectra with a variety of spectra and a range of elevations on Mars. The resulting induced dose rates can be compared with radiation dose during solar quiet times.
4 Interplanetary GCR and the induced spectra on the surface of Mars
The GCRs are modulated by solar activity: during solar maximum the increased solar and heliospheric magnetic fields are relatively efficient at preventing lower-energy GCRs from entering the inner heliosphere (e.g., Heber et al., 2007; Wibberenz et al., 2002) compared to solar minimum when the interplanetary magnetic field strength is reduced (Goelzer et al., 2013; Smith et al., 2013; Connick et al., 2011).
That is, the GCR flux is most intense during solar minimum (e.g., Mewaldt et al., 2010; Schwadron et al., 2012).
In order to compare the SEP spectra and induced dose rates with those during solar quiet periods, we have employed the 2010 version of the Badwahr-O’Neill model (BON10, O’Neill, 2010) to estimate GCR proton and ${}^{4}$He spectra under different modulation potentials, $\Phi$. The corresponding secondary spectra on the surface of Mars are obtained following Eq. 1.
Fig.3 shows the GCR proton flux between modulation extremes in a gray area. The lower-energy end of the spectra spans nearly two orders of magnitude as the modulation potential varies from 1500 MV (solar maximum) to 400 MV (solar minimum).
The long-term solar modulation of ${}^{4}$He ions is also shown in Fig. 6 (b) in gray shaded areas.
The secondary spectra on the surface of Mars under different modulation potentials are shown in pink shaded areas through Figs. 4 to 6.
In each panel, the surface dose rates (calculated following Eq. 2) are shown in the legends on the right side.
For instance, Fig.4(a) shows the GCR proton dose rate as 25.6 µGy/day at $\Phi$ = 1500 MV and 171.5 µGy/day at $\Phi$ = 400 MV. The GCR-induced surface downward proton has a dose rate value from 18.7 to 83.6 µGy/day during solar quiet periods.
The figure also shows that the surface GCR spectra and dose rates are much less affected by modulation than they are in interplanetary space.
This is because the Martian atmosphere filters out lower-energy primary particles which are most affected by solar modulation.
This effect has been supported by measurements on the surface of Mars compared to those in deep space, and the correlation between dose rate and solar modulation potential is (as expected) found to be smaller on the surface than in a spacecraft in deep space (Guo et al., 2015b, a).
5 Carrington event and 1989 Autumn events on the surface of Mars
To simulate the large variability and effects of extreme SEP events, three historic SEP events with different spectral shape, spectral hardness, and integral proton fluence were chosen. The famous Carrington solar flare occurred in 1859 and such extreme events are believed to occur very rarely. The associated geomagnetic storm lasted for several days, and it is estimated that the fluence of SEP protons with energies $>$ 30 MeV would have reached 1.88 $\times$ 10 ${}^{10}$ cm${}^{-2}$ (McCracken et al., 2001).
A previous study by Townsend et al. (2006) has scaled the flux of the August 1972 event (which had a soft spectrum) to the estimated level of the Carrington event. (Of course, no particle data exist for this event, so a spectral shape of some sort must be assumed.)
The assumed Carrington spectra is shown in both Figs. 3 and 4.
In addition, we also use the October 22nd 1989 (Oct89) event spectrum reconstructed using a Weibull distribution following Xapsos et al. (2000) as shown Figs. 3 and 5.
Finally, for the September 1989 (Sep89) event spectrum, we use that derived by Duldig (1998) from ground level Earth neutron monitors with different rigidity cutoffs; the spectrum is shown in Figs. 3 and 6.
In order to compare the properties of these events with power-law fitted spectra in the next section, we have also used single power-law fits to these spectra, apart from the assumed spectrum of the Carrington event, which is not well-represented by a power law.
The resulting fit parameters are shown in Fig. 3.
5.1 Carrington event
Fig. 4 shows the energetic particle spectra and dose rates induced by the primary proton flux associated with the Cartington event for various atmospheric depths on Mars.
Left (right) panels show downward (upward) spectra of protons (top) and electrons (bottom).
The primary SEP spectra are marked by black dashed lines in each panel, and the induced secondary particle spectra are obtained by multiplying the primary SEP spectra with the corresponding PLANETOMATRIX $\rm{\bar{A}}_{\sigma ij}$ at different depths $\sigma$.
For instance, Fig. 4 (a) and (b) show respectively the downward and upward proton spectra at $\sigma$ of 2, 4, 6, 8, 10, 12, 14, 16, 18 and 20 g/cm${}^{2}$ in color-dotted lines.
The surface spectra predicted at Gale Crater (the MSL landing site, with average atmospheric column depth of 22.5 g/cm${}^{2}$) are plotted in thick red dashed lines.
The proton fluxes, both upward and downward, decrease as the depth increases, particularly at lower energy. This is because the atmosphere stops a large share of the incoming protons; secondary production repopulates this part of the spectrum, but at levels far below the incident flux.
Using our modeled spectra at different depths of the atmosphere as well as on the surface of Mars (red dashed lines), we have also derived the corresponding upward or downward dose rate over a geometric angle of $2\pi$. This is the appropriate normalization since the spectra are averaged over only half the hemispheric angle.
The dose rate values for each spectra at different depths are recorded in the legends on the right side of each panel.
For instance, the surface downward proton dose rate of the assumed Carrington event spectrum is 3.64 $\times 10^{4}$ µGy/day, which is $\sim 10^{3}$ times larger than the downward proton dose rate during quiet time (18.7-83.6 µGy/day) as shown in Fig. 4(a).
The surface upward proton dose rate is 232.2 µGy/day for the event, much larger than the 4.8-13.9 µGy/day predicted during quiet time.
To calculate the unshielded deep space dose rate, we integrate over $4\pi$ steradians, implicitly assuming isotropic fluxes for both GCRs and SEPs.
The total dose rate induced by primary GCR protons in interplanetary space ranges between 25.6 - 171.5 µGy/day for $\Phi$ between 1500 and 400 MV and the GCR spectra are shown in gray areas in Fig. 4 (a) and (b).
The total dose rate induced by the Carrington SEPs in deep space over a $4\pi$ geometric angle is estimated to be about 1.19 $\times 10^{9}$ µGy/day, also shown in Table 1, which is $\sim 10^{7}$ times higher than solar quiet time.
This enhancement ratio during the event compared to quiet time in deep space is much larger than on the surface. This is mainly because the atmosphere stops most of the SEPs, especially the low-energy ones, which greatly contribute to the unshielded deep space dose but not to the surface dose.
Secondary ${}^{4}$He particles induced by the Carrington protons contribute very little to the increase of surface dose rate: 2.7 and 0.4 µGy/day for downward and upward directions respectively.
This is due to the very low probability of protons generating secondary ${}^{4}$He ions in the atmosphere; with the assumed soft spectrum, protons are highly likely to stop before interacting and producing ${}^{4}$He ions.
In comparison, the ${}^{4}$He dose rate at $\Phi$ = 400 MV on the surface is about 12.0 µGy/day mainly contributed by particles from primary GCR ${}^{4}$He ions, i.e., through matrix Fig. 2 (a).
Fig. 4(c) and (d) show the Carrington proton induced secondary downward and upward electrons respectively.
The surface electron dose rates are 2.33 $\times 10^{3}$ (downward) and 1.09 $\times 10^{3}$ (upward) µGy/day, both significantly larger than the background GCR values mainly due to the abundant increase of the lower-energy electrons during the event.
The generation of secondary electrons from primary protons is an efficient process, and it largely compensates the atmospheric shielding of electrons, so that the resulting spectra, unlike the proton spectra, decrease only slightly with increasing atmospheric depth.
The upward electron spectra and dose rate are not much smaller than the downward ones since electrons due to their small mass are more likely generated as backward albedo particles.
5.2 Oct89 event
Fig. 5 shows the modeled spectra and dose rate results of the Oct89 event.
The SEP dose rate in unshielded deep space at 1 AU is 1.03 $\times 10^{6}$ µGy/day, much smaller than that of the Carrington event as shown in Table 1.
Similar to the Carrington event, the Oct89 event results in proton fluxes, shown in Fig. 5(a), that are much higher than during solar quiet periods.
The downward electrons, shown in Fig. 5(b), are not enhanced as much as those produced by the assumed spectrum of the Carrington event, shown in Fig. 4(c), which has a much higher primary flux for generating secondary electrons.
Fig. 5(c) and (d) also show the secondary downward and upward neutrons.
The proton flux decreases as atmospheric depth increases while most other secondary (e.g., downward electrons and neutrons) fluxes increase as the column depth increases since they build up as the SEPs penetrate into the atmosphere.
The upward neutron flux is mostly produced by SEP protons that reach the surface, and as a result it follows the opposite trend: at points deeper atmosphere, the intensity is slightly smaller. For all SEP secondaries, the spectral shapes differ from GCR-induced secondary spectra in that they lack highly energetic particles above a few GeV.
5.3 Sep89 event
The Sep89 event is well-represented by a single power-law spectral shape (between 100 and 800 MeV) as shown in Figs. 3 and 6.
Its spectral shape is very different from that assumed for the Carrington event, which is very soft with a sharp drop-off above 500 MeV.
The Sep89 event has a quasi-power-law spectra at energies up to a few GeV, so that at higher energies ($\geq$ 300 MeV), the proton flux has a higher intensity than that of the Carrington event.
Because of the hard spectrum, the Sep89 event produces significant enhancements on the Martian surface for all different types of secondaries, as shown in Fig.6.
A summary table of the total dose rates (both upward and downward directions for the above 3 different events) from interplanetary deep space through the atmosphere down to the surface are shown in Table 1.
During the Carrington event, the total surface dose rate including both downward and upward secondaries is about 5 $\times 10^{4}$ µGy/day, only about 4 $\times 10^{-5}$ times compared to the SEP dose rate in unshielded deep space. During the other two events, this ratio is much higher. This is because the assumed Carrington event spectrum is much softer than the other two examples, so that the shielding by the Martian atmosphere is highly effective.
6 Twenty years of significant events modeled on the surface of Mars
Kühl et al. (2017) have studied a set of SEP events with the Electron Proton Helium Instrument (EPHIN) instrument onboard SOHO based on a newly developed optimization technique that exploits the response function of the penetrating protons through the detector sets and thus extends the usable energy range of the instrument from 5 to 50 MeV up to above 800 MeV (Kühl et al., 2015).
The studied SEP events are between December 1995 and December 2015 with protons accelerated to energies greater than 500 MeV.
A total of 42 events has been found, including all GLEs during the SOHO age, excluding one GLE during which EPHIN had a data gap. Due to the long lifetime of the instrument, its highly efficient operation during the mission, and the fact that observations spanned different phases of the solar cycle, the total number of events is likely typical for a 20 year time period. Supporting this supposition, we note that the range of monthly-average sunspot number during this period is in line with historical records during different solar activity levels.
For 33 of the events, the onset time is based on an energy channel covering from 100 MeV to 1 GeV, and proton spectra from 100 MeV up to 800 MeV were derived.
A statistical analysis was performed by calculating spectra in two-hour intervals starting 30 minutes after the event onset. Thus the spectra may not represent the maximum intensity in every single energy bin individually.
A single power law function was applied to fit each spectrum according to:
$$\displaystyle f(E_{0})=I_{\epsilon_{0}}\cdot(E_{0}/\epsilon_{0})^{\gamma}$$
(3)
where the SEP proton energy $E_{0}$ is in MeV, flux $f(E_{0})$ and $I_{0}$ are in particles/sec/cm${}^{2}$/sr/MeV and $I_{\epsilon_{0}}$ is the scaled intensity at $\epsilon_{0}$ MeV.
Section 5 focused on individual historical events and the modeling of each SEP spectra in a wide energy range from 1 up to 10${}^{4}$ MeV.
Here, we obtain an empirical correlation between deep space and Martian surface dose rates with the general properties of the SEP spectra as represented by $I_{\epsilon_{0}}$ and the power-law index $\gamma$.
Based on the fitted power-law parameters of the 33 events by Kühl et al. (2017), we use PLANETOMATRIX to forward-model the power-law fitted SEP spectra $f(E_{0})$ through the Martian atmosphere, and obtain the induced surface secondary spectra $F_{i}(E)$ of different particle species.
For each event, the deep space and Martian-surface dose rates have been calculated following Eq. 2 for two ranges of the primary SEP spectra: (a) the primary proton energy range of 100-800 MeV and (b) an extended primary proton energy range of 15-1000 MeV.
The two-range study is motivated by these considerations: case (a) is the trustworthy energy range for each single power-law fit spectrum (Kühl et al., 2017), while case (b) extrapolates the spectra to a wider energy range, which may yield more reliable estimates of the interplanetary dose rates contributed by events with complete energy spectra.
To avoid over-estimation of the total dose rate, we did not extrapolate the spectra to a much wider energy range since the power-law shape generally flattens out at low energies around 10-30 MeV depending on individual events (Band et al., 1993) and drops off quickly at high energies $\geq$ 1000 MeV as also shown in Fig. 3 for the Sep89 and Oct89 events.
The low-energy end of 15 MeV in case (b) has also been chosen following Wilson et al. (2006) which suggested little dose contribution from protons below $\sim$20 MeV for astronauts wearing a space suit during Extra-Vehicular Activities (EVA). The interplanetary-space dose rates could therefore be considered to represent an exposure scenario in which an astronaut is doing extra-vehicular activities when a SEP event occurs.
In both cases (a) and (b), we expect the induced Martian surface dose rates to have similar values, since the atmosphere stops low energy primaries (energies $\leq 140$ to $\sim 160$ MeV depending on elevation).
Fig. 7 summarizes the result of case (a) where 100-800 MeV primary protons were considered for each SEP spectrum with varying intensities and power-law spectral indices.
The calculated dose rates in deep space (in black with scales on left axes) and on the Martian surface (in red and right axes) are plotted versus the SEP flux $I_{\epsilon_{0}}$ (panel a for $\epsilon_{0}=300$ MeV and c for $\epsilon_{0}=200$ MeV) and spectral index $\gamma$ (panel b) as well as the integrated flux (panel d) of the 100-800 MeV proton spectra.
It is clear from panels (a) and (c) that both the deep-space and Martian surface dose rates correlate very well with $I_{\epsilon_{0}}$, with no evident distinction between values of 200 and 300 MeV for ${\epsilon_{0}}$.
The calculated dose rates clearly show a quasi-linear dependence on $I_{\epsilon_{0}}$ with the fitted logarithmic linear function plotted in dashed lines and shown as legends in the plots.
A very good correlation (R${}^{2}\geq$ 0.96) has been found between total dose rate and spectral fluxes, but no clear correlation is seen between dose rate and the power-law index $\gamma$.
There is also a good correlation between dose rate and $I_{\epsilon_{0}}$ for other energies, which is expected since a power-law can be readily defined with two fixed points.
This suggests that the induced dose rate correlates with the total integrated (over energy) flux of the power-law spectra in Eq. 3, and this is supported by the plot shown in Fig. 7(d).
As the integrated function of a power-law spectra has a non-linear dependence on the spectral index $\gamma$, it is not surprising that no simple linear correlations exist between the dose rates and the power-law index $\gamma$.
Fig. 8 summarizes the results of case (b), where the ranges of the primary SEP protons were expanded to 15-1000 MeV.
The calculated dose rates in deep space (in black and left axes) and on Martian surface (in red and right axes) are plotted versus the SEP flux $I_{\epsilon_{0}}$ at ${\epsilon_{0}}=200$ MeV (panel a) and the integrated flux (panel b) of the power-law SEP spectra.
Panel (a) shows a weaker correlation between deep space dose rate and $I_{\epsilon_{0}}$ (R${}^{2}$ is 0.82) in comparison to Fig. 7(c). This is likely due mostly to the inclusion of lower-energy protons which contribute significantly to the free-space dose due to their high $dE/dx$, which goes as $\sim 1/v^{2}$ ($v$ is the proton velocity). A smaller contribution comes from the higher-energy protons ($\geq 800$ MeV), which approach the minimum ionizing part of the $dE/dx$ curve.
The fitted parameters indicate that with an SEP power-law spectra of 15-1000 MeV, the expected proton dose rate in deep space is nearly 2 orders of magnitudes( $\frac{2.37\times 10^{7}}{3.33\times 10^{5}}=71$) larger than that from 100-800 MeV protons only.
The dose rate on the Martian surface continues to correlate well with $I_{\epsilon_{0}}$ (R${}^{2}=$ 0.99), since the lower-energy protons stop in the atmosphere, and the fitted parameters are within 6% of those in Fig. 7(c). This means that the surface dose rate depends mostly on primary protons in the energy range of 100-800 MeV since the lower energy protons ($\leq 100$ MeV) barely contribute to the surface dose, while the higher energy part contributes little in a power-law distribution.
The correlation between the surface dose rate and the integrated flux of the 15-1000 MeV SEP spectra in panel (b) is worse in comparison to that shown in Fig. 7(d) for the same reason: a large share of the 15-1000 MeV protons makes a negligible contribution to the surface radiation environment.
To validate the correlation derived from power-law SEPs, we compare these results with that from the historical events presented in Section 5.
To do so, we fitted the Oct89 and Sep89 events with power-law spectra in the energy range of 100-800 MeV, using the fitted $I_{0}$ and $\gamma$ shown in Fig.3.
The assumed Carrington event spectrum cannot be fitted with a power-law in this energy range, and is therefore excluded from this part of the analysis. (The spectrum is believed to differ from a power law because it arose from a very different acceleration process inside the flare (McCracken et al., 2001).)
For the other two events, we re-calculated the deep space and surface dose rates in the energy range of (a) 100-800 MeV and (b) 15-1000 MeV and plotted them versus the corresponding $I_{\epsilon_{0}}$ at 200 and 300 MeV.
These results are also shown in Figs. 7 and 8 with squares indicating the Oct89 event and diamonds standing for the Sep89 event.
Both events are highly consistent with the events observed by EPHIN, showing the quasi-linear correlation of dose rate and $I_{\epsilon_{0}}$ validating the correlations derived above. The values of dose rates calculated from these events (including the Carrington event) in cases (a) and (b) are also shown in Table 1.
It is interesting to note that when comparing the surface dose rate induced by 100-800 MeV protons (a) and 15-1000 MeV ones (b) to that from the full SEP spectra (f), the difference is fairly small for Oct89 and Sep89 events. However, for the Carrington event the surface dose rate from case (a) is only about 83% of that from case (f). This is because the Carrington event has a very soft spectra (based on the model used here) with a very intense low-energy part. Although protons with energies lower than about 150 MeV stop before reaching the surface, their secondaries, especially electrons and neutrons produced in the atmosphere, can travel downward and contribute to the surface dose rates.
The empirical correlation shown in Figs. 7 and 8
can be used for quick estimations of the expected dose rates both in deep space and on the surface of Mars upon the onset of a sudden solar particle event whose spectra has roughly a power-law distribution in the concerned energy range.
7 The potential extra contribution by ${}^{4}$He ions
Although protons are the large majority of the primary particles reaching the top of the Martian atmosphere, energetic helium ions can also propagate into deep space and a flux ratio of He/p to be about 10% has been estimated to be a reasonable worst-case scenario based on SOHO/ERNE measurements (Torsti et al., 1995). Based on this assumption, we have also scaled the EPHIN proton power-law fitted spectra to 1 order of magnitude smaller representing the ${}^{4}$He spectra which is then used to (1) calculate the deep space induced dose rates and (2) multiply the matrices for deriving surface spectra and dose rate from all secondaries induced by these primary He particles. This is likely an unrealistic assumption since the charge-to-mass ratio of ${}^{4}$He ions makes them more difficult to accelerate than ${}^{1}$H. They will therefore tend to have softer energy spectra than protons accelerated by the same mechanism, so a simple scaling of the proton flux is likely to overestimate the contributions of ${}^{4}$He.
A summary figure similar to Figs. 7 and 8 is shown in Fig. 9; the fitted parameters are again labeled.
To give a direct comparison of the fitted parameters, we plotted and fitted the ${}^{4}$He-induced dose rate to the same $I_{\epsilon_{0}}$ of the proton spectra. For unshielded deep space, the proton-induced dose rate is about 2.5 times larger than He-induced dose rate in both cases (a) and (b) as shown in by the fitted parameters labeled in black.
This is exactly a trivial result: scaling down the proton spectrum by a factor of 10 is partially compensated by the factor of 4 higher $dE/dx$ of a ${}^{4}$He ion at the same velocity ($Z^{2}$ scaling) yields a factor of 2.5 in the ratio of doses.
The Martian surface case is more complicated owing to the different transport properties of ${}^{1}$H and ${}^{4}$He.
The primary solar proton and helium induced dose ratio is about 2.74 and 2.62 for cases (a) and (b) respectively, slightly larger than, but still close to, 2.5.
This is because a fraction of the helium ions will undergo nuclear fragmentation in the atmosphere, reducing their contribution to the surface dose.
As noted above, simply scaling the proton spectrum to the ${}^{4}$He spectrum may lead to significant over-estimation of the ${}^{4}$He contributions. Precise measurements of He spectra by, e.g., PAMELA (Picozza et al., 2007) are needed for better estimations of the He induced dose rates.
8 Discussion and Conclusion
In terms of biological effectiveness associated with radiation exposures on human beings, the dose equivalent (in units of Sv) is often more referred to for evaluating the deep space exploration risks (Sievert & Failla, 1959). It can be computed using the linear energy transfer (LET) dependent quality factor, Q(L), from ICRP (1991). For LET less than 10 keV/$\mu$m in water, Q is identically 1; this value applies to the large majority of SEP protons, so the dose rates reported here are in most cases close to the corresponding dose equivalent rates.
We integrated such calculated dose equivalent rate for two hours for each event since the EPHIN event spectra was calculated in two-hour intervals shortly after the event onset (see Section 6). Fig. 10 (lower-left panel) shows the dose equivalent for deep space case (y-axis, from primary protons of 100-800 MeV energy range) versus dose equivalent on surface of Mars (x-axis, all secondaries induced by primary protons of 100-800 MeV energy range).
The black dots represent the dose equivalent from all EPHIN events and the correlation coefficient between the deep space and surface dose equivalents is 0.98 and they depend on each other roughly following a simple linear relationship which indicates the dose equivalent rate of such events on the surface is generally 8-9 times smaller than that (from 100-800 MeV protons) in deep space.
A similar fitting for case (b) where 15-1000 MeV primary protons were considered shows that the deep space dose equivalent rate is about 90 times larger than that on the Martian surface.
We have omitted the contributions by ${}^{4}$He ions here, since the intensity and spectra we modeled above are speculative; in a worst-case scenario, we might expect an additional 40% contribution from these ions.
To assess the differences between dose equivalents induced by the full spectra and the energy-limited power-law spectra, we adopted the three historical events and compared the modeled results between full spectra and 100-800 MeV range (case a) shown in red in Fig.10. The squares stand for the results from the full spectra (f) while the circles represent those from case (a).
For deep space case, dose equivalents from (f) and (a) differ significantly, by as much as 2 orders of magnitude.
Similar differences were also found in dose rate ratios, as listed in Table 1 in the row ”(ad)/(fd) ratio” where, e.g., the dose rate in deep space resulted from 100-800 MeV protons is only about 0.69% of the total dose rate for the Sep89 event.
The table also shows the dose rates in case (b) where primary protons with energies from 15 to 1000 MeV are considered.
Such ratios shown in the row ”(bd)/(fd) ratio” become 45%, 29% and 15% for the Carrington, Oct89, and Sep89 events respectively.
These values are much larger than in case (a), mainly due to the contributions of low energy protons.
For the Martian surface, the dose or dose equivalent rates do not depend significantly on the full primary spectra.
As shown in Fig. 10, the induced surface dose equivalents (x-axis) are very similar from the two different primary spectra (a) and (f).
The values of dose rates for Martian surface scenario are also shown in Table 1 and the surface dose rate resulted from 100-800 MeV protons is about 93% of the total surface dose rate for both the Oct89 and Sep89 events listed in the row ”(as)/(fs) ratio”.
Because of the atmospheric shielding, the contributions of 100-800 MeV protons dominate the SEP-induced environment on the surface of Mars, despite the fact that these protons contribute little to the free-space dose equivalent.
Typical human exposures [µSv] expected in daily life and defined in regulations for special cases are marked in the lower right and the upper left panels of Fig. 10 as a reference for possible potential biological effects of the SEPs studied here.
For SEP events encountered in deep space without any shielding around, the accumulated dose equivalent for e.g., the Sep89 event after two hours would be 1.64 $\times 10^{6}$ µSv, a value higher than the astronaut career limit $10^{6}$ µSv.
The consequence for a Carrington event is far worse: a total of $10^{8}$ µSv (for 2-hour time duration) is over the determined fatal value.
However this is an unrealistic estimation since at least the space suit shielding should be present for worse-case extravehicular activities (EVA).
Since the current paper is mainly focused on consequences of the extreme events for the Martian surface case considering the Martian atmospheric shielding, we will not go into details discussing about the deep space scenarios.
Interested readers are pointed to previous studies by e.g., (Wilson et al., 2006) who have carried out more detailed investigations of the dose and dose equivalent responses as a function of primary proton energies considering scenarios of EVA and within spacecraft shielding conditions.
On the surface of Mars, the dose equivalents induced by most of these events for the duration of two hours are below 10${}^{4}$ µSv, a value well below the limit of radiation worker annual limits.
Exposer to the Sep89 event for 2 hours would have an effect of approximately a head CT scan.
These values are calculated for the surface of Mars at -4.4 km elevation (Gale Crater), where the atmospheric column depth averages about 22 g cm${}^{-2}$.
A habitat covered by $\sim$ 10 cm of Martian soil would provide important additional shielding against energetic particles reaching the surface.
Alternatively, spacesuits would already provide a slight protection against low-energy particles.
Detailed studies would involve further modeling of the shielding response function (by a similar matrix-set) of the spacesuit and shelter materials and will be carried out in our future work.
Nevertheless, the current study has provided some benchmark and convenient formulas for estimating the Martian surface radiation environment induced by power-law shaped SEPs.
The results highlight the need for future astronauts on the surface of Mars to receive space weather forecasts, and to carry alarming dosimeters (NASA, 2014) so that they can seek an emergency shelter should a hard-spectrum event reach Mars.
For better space weather forecasts and predicting the arrival of such hazardous events, we emphasize the importance of a space weather monitoring package including a particle detector to be embarked in all planetary and astronomical missions as a basic payload requirement.
The work is supported by DLR and DLR’s Space Administration grant numbers 50QM0501 and 50QM1201 to the Christian Albrechts University, Kiel.
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EXPANDING THE TRANSFER ENTROPY TO IDENTIFY INFORMATION CIRCUITS IN COMPLEX SYSTEMS
S. Stramaglia${}^{1,2}$, Guo-Rong Wu${}^{3,4}$, M. Pellicoro${}^{1,2}$,
and D. Marinazzo${}^{3}$
${}^{1}$ Istituto Nazionale di
Fisica Nucleare, Sezione di Bari, Italy
${}^{2}$
Dipartimento di Fisica, University of Bari,
Italy
${}^{3}$ Faculty of Psychology and Educational
Sciences, Department of Data Analysis, Ghent University, Henri
Dunantlaan 1, B-9000 Ghent, Belgium
${}^{4}$ Key Laboratory for
NeuroInformation of Ministry of Education, School of Life Science
and Technology, University of Electronic Science and Technology of
China, Chengdu 610054, China.
(November 19, 2020)
Abstract
We propose a formal expansion of the transfer entropy to put in
evidence irreducible sets of variables which provide information for
the future state of each assigned target. Multiplets characterized
by a large contribution to the expansion are associated to
informational circuits present in the system, with an informational
character which can be associated to the sign of the contribution.
For the sake of computational complexity, we adopt the assumption of
Gaussianity and use the corresponding exact formula for the
conditional mutual information. We report the application of the
proposed methodology on two EEG data sets.
pacs: 05.45.Tp,87.19.L-
I INTRODUCTION
The inference of couplings between dynamical subsystems, from data,
is a topic of general interest. Transfer entropy te , which is
related to the concept of Granger causality granger , has been
proposed to distinguish effectively driving and responding elements
and to detect asymmetry in the interaction of subsystems. By
appropriate conditioning of transition probabilities this quantity
has been shown to be superior to the standard time delayed mutual
information, which fails to distinguish information that is actually
exchanged from shared information due to common history and input
signals lehnertz ; wibral . On the other hand, Granger
formalized the notion that, if the prediction of one time
series could be improved by incorporating the knowledge of past
values of a second one, then the latter is said to have a causal influence on the former. Initially developed for econometric
applications, Granger causality has gained popularity also in
neuroscience (see, e.g.,
blinoska ; smirnov ; dingprl ; noiprl ; faes ). A discussion about the
practical estimation of information theoretic indexes for signals of
limited length can be found in porta . Transfer entropy and
Granger causality are equivalent in the case of Gaussian stochastic
variables barnett : they measure the information flow between
variables hla . Recently it has been shown that the presence
of redundant variables influences the estimate of the information
flow from data, and that maximization of the total causality is
connected to the detection of groups of redundant variables
noired .
In recent years, information theoretic treatment of groups of
correlated degrees of freedom have been used to reveal their
functional roles as memory structures or those capable of processing
information borst . Information theory suggests quantities
that reveal if a group of variables is mutually redundant or
synergetic sch ; bett . Most approaches for the identification
of functional relations among nodes of a complex networks rely on
the statistics of motifs, subgraphs of k nodes that appear
more abundantly than expected in randomized networks with the same
number of nodes and degree of connectivity milo ; yeger .
An interesting approach to identify functional subgraphs in complex
networks, relying on an exact expansion of the mutual information
with a group of variables, has been presented in bettencourt .
In this work we generalize these results to show a formal expansion
of the transfer entropy which puts in evidence irreducible sets of
variables which provide information for the future state of the
target. Multiplets of variables characterized by an high value,
unjustifiable by chance, will be associated to informational
circuits present in the system.
Additionally, in applications where linear models are sufficient to
explain the phenomenology, we propose to use the exact formula for
the conditioned mutual information among Gaussian variables so as to
get a computationally efficient approach. An approximate procedure
is also developed, to find informational circuits of variables
starting from few variables of the multiplet by means of a greedy
search. We illustrate the application of the proposed expansion to a
toy model and two real EEG data sets.
The paper is organized as follows. In the next section we describe
the expansion and motivate our approach. In section III we report
the applications of the approach and describe our greedy search
algorithm. In section IV we draw our conclusions.
II Expansion
We start describing the work in bettencourt . Given a
stochastic variable $X$ and a family of stochastic variables
$\{Y_{k}\}_{k=1}^{n}$, the following expansion for the mutual
information, analogous to a Taylor series, has been derived there:
$$\displaystyle\begin{array}[]{l}S\left(X|\{Y\}\right)-S(X)=-I\left(X;\{Y\}%
\right)=\\
\sum_{i}{\Delta S(X)\over\Delta Y_{i}}+\sum_{i>j}{\Delta^{2}S(X)\over\Delta Y_%
{i}\Delta Y_{j}}+\cdots+{\Delta^{n}S(X)\over\Delta Y_{i}\cdots\Delta Y_{n}},\\
\end{array}$$
(1)
where the variational operators are defined as
$${\Delta S(X)\over\Delta Y_{i}}=S\left(X|Y_{i}\right)-S(X)=-I\left(X;Y_{i}%
\right),$$
(2)
$${\Delta^{2}S(X)\over\Delta Y_{i}\Delta Y_{j}}=-{\Delta I\left(X;Y_{i}\right)%
\over\Delta Y_{j}}=I\left(X;Y_{i}\right)-I\left(X;Y_{i}\right|Y_{j}),$$
(3)
$${\Delta^{3}S(X)\over\Delta Y_{i}\Delta Y_{j}\Delta Y_{k}}=I\left(X;Y_{i}|Y_{k}%
\right)-I\left(X;Y_{i}\right|Y_{j},Y_{k})-I\left(X;Y_{i}\right)+I\left(X;Y_{i}%
\right|Y_{j}),$$
(4)
and so on.
Now, let us consider $n+1$ time series $\{x_{\alpha}(t)\}_{\alpha=0,\ldots,n}$. The lagged state vectors are denoted
$$Y_{\alpha}(t)=\left(x_{\alpha}(t-m),\ldots,x_{\alpha}(t-1)\right),$$
$m$ being the window length.
Firstly we may use the expansion (1) to model the statistical
dependencies among the $x$ variables at equal times. We take $x_{0}$
as the target time series, and the first terms of the expansion are
$$W_{i}^{0}=-I\left(x_{0};x_{i}\right)$$
(5)
for the first order;
$$Z_{ij}^{0}=I\left(x_{0};x_{i}\right)-I\left(x_{0};x_{i}|x_{j}\right)$$
(6)
for the second order; and so on. We note that
$$Z_{ij}^{0}=-{\cal I}\left(x_{0};x_{i};x_{j}\right),$$
where ${\cal I}\left(x_{0};x_{i};x_{j}\right)$ is the interaction information, a
well known information measure for sets of three variables
mcgill ; it expresses the amount of information (redundancy or
synergy) bound up in a set of variables, beyond that which is
present in any subset of those variables. Unlike the mutual
information, the interaction information can be either positive or
negative. Common-cause structures lead to negative interaction
information . As a typical example of positive interaction
information one may consider the three variables of the following
system: the output of an XOR gate with two independent random inputs
(however some difficulties may arise in the interpretation of the
interaction information, see bell ). It follows that positive
(negative) $Z_{ij}^{0}$ corresponds to redundancy (synergy) among the
three variables $x_{0}$, $x_{i}$ and $x_{j}$.
In order to go beyond equal time correlations, here we propose to
consider the flow of information from multiplets of variables to a
given target. Accordingly, we consider
$$S\left(x_{0}|\{Y_{k}\}_{k=1}^{n}\right)-S(x_{0})=-I\left(x_{0};\{Y_{k}\}_{k=1}%
^{n}\right),$$
(7)
which measures to what extent all the remaining variables contribute
to specifying the future state of $x_{0}$. This quantity can be
expanded according to (1):
$$\displaystyle\begin{array}[]{l}S\left(x_{0}|\{Y_{k}\}_{k=1}^{n}\right)-S(x_{0}%
)=\\
\sum_{i}{\Delta S(x_{0})\over\Delta Y_{i}}+\sum_{i>j}{\Delta^{2}S(x_{0})\over%
\Delta Y_{i}\Delta Y_{j}}+\cdots+{\Delta^{n}S(x_{0})\over\Delta Y_{i}\cdots%
\Delta Y_{n}}.\\
\end{array}$$
(8)
A drawback of the expansion (7) is that it does not remove
shared information due to common history and input signals;
therefore we choose to condition it on the past of $x_{0}$, i.e.
$Y_{0}$. To this aim we introduce the conditioning operator ${\cal C}_{Y_{0}}$:
$${\cal C}_{Y_{0}}S(X)=S(X|Y_{0}),$$
and observe that ${\cal C}_{Y_{0}}$ and the variational operators (2) commute.
It follows that we can condition the expansion (8) term by
term, thus obtaining
$$\displaystyle\begin{array}[]{l}S\left(x_{0}|\{Y_{k}\}_{k=1}^{n},Y_{0}\right)-S%
(x_{0}|Y_{0})=-I\left(x_{0};\{Y\}_{k=1}^{n}|Y_{0}\right)=\\
\sum_{i}{\Delta S(x_{0}|Y_{0})\over\Delta Y_{i}}+\sum_{i>j}{\Delta^{2}S(x_{0}|%
Y_{0})\over\Delta Y_{i}\Delta Y_{j}}+\cdots+{\Delta^{n}S(x_{0}|Y_{0})\over%
\Delta Y_{i}\cdots\Delta Y_{n}}.\\
\end{array}$$
(9)
The first order terms in the expansion are given by:
$$A_{i}^{0}={\Delta S(x_{0}|Y_{0})\over\Delta Y_{i}}=-I\left(x_{0};Y_{i}|Y_{0}%
\right),$$
(10)
and coincide with the bivariate transfer entropies $i\to 0$ (times
-1). The second order terms are
$$B_{ij}^{0}=I\left(x_{0};Y_{i}|Y_{0}\right)-I\left(x_{0};Y_{i}|Y_{j},Y_{0}%
\right),$$
(11)
and may be seen as a generalization of the interaction information
${\cal I}$; hence a positive (negative) $B_{ij}^{0}$ corresponds to a
redundant (synergetic) flow of information $\{i,j\}\to 0$. The
typical examples of synergy and redundancy, in the present framework
of network analysis, are the same as in the static case, plus a
delay for the flow of information towards the target. The third
order terms are
$$\displaystyle\begin{array}[]{ll}C_{ijk}^{0}=&I\left(x_{0};Y_{i}|Y_{j},Y_{0}%
\right)+I\left(x_{0};Y_{i}|Y_{k},Y_{0}\right)\\
&-I\left(x_{0};Y_{i}|Y_{0}\right)-I\left(x_{0};Y_{i}|Y_{j},Y_{k},Y_{0}\right),%
\end{array}$$
(12)
and so on.
The generic term in the expansion (9),
$$\Omega_{k}={\Delta^{k}S(x_{0}|Y_{0})\over\Delta Y_{i}\cdots\Delta Y_{k}},$$
(13)
is symmetrical under permutations of the $Y_{i}$ and, remarkably,
statistical independence among any of the $Y_{i}$ results in vanishing
contribution to that order. Therefore each nonvanishing accounts for
an irreducible set of variables providing information for the
specification of the target: the search for for informational
multiplets is thus equivalent to search for terms (13) which
are significantly different from zero. Another property of
(9) is that the sign of each term is connected to the
informational character of the corresponding set of variables, see
bettencourt ).
For practical applications, a reliable estimate of conditional
mutual information from data should be used. Non parametric methods
are recommendable when nonlinear effects are relevant. However, a
conspicuous amount of phenomenology in brain can be explained by
linear models: therefore, for the sake of computational load, In
this work we adopt the assumption of Gaussianity and use the exact
expression that holds in this case barnett , which reads as
follows. Given multivariate Gaussian random variables $X$, $W$ and
$Z$, the conditioned mutual information is
$$I\left(X;W|Z\right)=\frac{1}{2}\ln{|\Sigma(X|Z)|\over|\Sigma(X|W\oplus Z)|},$$
(14)
where $|\cdot|$ denotes the determinant, and the partial covariance
matrix is defined
$$\Sigma(X|Z)=\Sigma(X)-\Sigma(X,Z)\Sigma(Z)^{-1}\Sigma(X,Z)^{\top},$$
(15)
in terms of the covariance matrix $\Sigma(X)$ and the cross
covariance matrix $\Sigma(X,Z)$; the definition of $\Sigma(X|W\oplus Z)$ is analogous.
The statistical significance of (13) can be assessed by
observing that it is the sum of terms like (14) which, under
the null hypothesis $I\left(X;W|Z\right)=0$, have a $\chi^{2}$
distribution. Alternatively, statistical testing may be done using
surrogate data obtained by random temporal shuffling of the target
vector $x_{0}$; the latter strategy is the one we use in this work.
III APPLICATIONS
III.1 Second order terms
In this subsection we show the application of the proposed
expansion, truncated at the second order. To this aim we turn to
real electroencephalogram (EEG) data, the window length $m$ being
fixed by cross validation. Firstly we consider recordings obtained
at rest from 10 healthy subjects. During the experiment, which
lasted for 15 min, the subjects were instructed to relax and keep
their eyes closed. To avoid drowsiness, every minute the subjects
were asked to open their eyes for 5 s. EEG was measured with a
standard 10-20 system consisting of 19 channels nolte . Data
were analyzed using the linked mastoids reference, and are available
from website_nolte .
For each subject we consider several epochs of 4 seconds in which
the subjects kept their eyes closed. For each epoch we compute the
second order terms at equal times $Z^{0}_{ij}$ and the lagged ones
$B^{0}_{ij}$; then we average the results over epochs. In order to
visualize these results, for each target electrode we plot a on a
topographic scalp map the pairs of electrodes which are redundant or
synergetic with respect to it. Both quantities are distributed with
a clear pattern across the scalp. Interactions at equal times are
one order of magnitude higher than the lagged interactions, and are
dominated by the effect of spatial proximity, see fig. 1. On the
other hand, $B^{0}_{ij}$ show a richer dynamics, such as
interhemispheric communications and predominance redundancy to and
from the occipital channels, see fig. 2, reflecting the prominence
of the occipital rhythms when the subjects rest with their eyes
closed.
As another example we consider intracranial EEG recordings from a
patient with drug-resistant epilepsy and which has thus been
implanted with an array of $8\times 8$ cortical electrodes and two
depth electrodes with six contacts. The data are available at
kol_web and described in kol_paper . For each seizure
data are recorded from the preictal period, the 10 seconds preceding
the clinical onset of the seizure, and the ictal period, 10 seconds
from the clinical onset of the seizure. We analyze data
corresponding to eight seizures and average the corresponding
results.
For each electrode we compute the lagged influences $B^{0}_{ij}$,
obtaining for each electrode the pair of other electrodes with
redundant or synergetic contribution to its future. The patient has
a putative epileptic focus in a deep hippocampal region, with the
seizure that then spreads to the cortical areas. In fig. 3 we
report the values of coefficients $B$ taking as the target a
cortical electrode located on the putative cortical focus: we report
the values of $B^{0}_{ij}$ corresponding to
all the couple of the electrodes, as well as their sum over
electrode $j$. It is clear how the redundancy increases during the
seizure. On the other hand, for sensors from 70 to 76, corresponding
to a depth electrode, the redundancy is higher in the preictal
period, reflecting the fact that the seizure is already active in
its primary focus even if not yet clinically observable. The values
of $B$ corresponding to this electrode are reported in fig. 4.
III.2 Greedy search of multiplets
Given a target variable, the time required for the exhaustive search
of all the subsets of variables, with a statistically significant
information flow (13), is exponential in the size of the
system. It follows that the exact search for large multiplets is
computationally unfeasible, hence we adopt the following approximate
strategy. We start from
a pair of variables with non-vanishing second order term $B$ w.r.t. the given target. We consider these two variables as a seed, and aggregate other variables to them so as to construct a multiplet.
The third variable of the subset is selected among the remaining
ones as those that, jointly with the previously chosen variable,
maximize the modulus $|C|$ of the corresponding third order term.
Then, one keeps adding the rest of the variables by iterating this
procedure. Calling $Z_{k-1}$ the selected set of k - 1 variables,
the set $Z_{k}$ is obtained adding, to $Z_{k-1}$, the variable, among
the remaining ones, with the greatest modulus of $\Omega_{k}$. These
iterations stop when $\Omega_{k}$, corresponding to $Z_{k}$, is not
significantly different from zero (the Bonferroni correction for
multiple comparisons is to be applied at each iteration); $Z_{k-1}$
is then recognized as the multiplet originated by the initial pair
of variables chosen as the seed.
We apply this strategy to the following toy model
$$\displaystyle\begin{array}[]{lll}x_{0}(t)&=a\;\eta(t-1)+\sigma\xi_{0}(t),&\\
x_{\alpha}(t)&=b_{\alpha}\;\eta(t)+\sigma_{1}\xi_{\alpha}(t),&\alpha=1,\ldots,%
m\\
x_{\beta}(t)&=\sigma_{2}\xi_{\beta}(t),&\beta=m+1,\ldots,m+M\\
\end{array}$$
(16)
where $\xi$ and $\eta$ are i.i.d. unit variance Gaussian variables.
In this model the target $x_{0}$ is
influenced by the process $\eta$; variables $x_{\alpha}$,
$\alpha=1,\ldots,m$, are a mixture of $\eta$ and noise $\xi$, whilst
the remaining $M$ variables are pure noise. Estimates of $\Omega_{k}$
are based on time series, generated from (16) and 1000
samples long. The results are displayed in figure (5).
Firstly we consider the case $m=20$ and $M=0$, with all the twenty
variables driving the target with equal couplings $b_{\alpha}$; in
figure (5)-A we depict the term $\Omega_{k}$
corresponding to the $k$-th iteration of the greedy search. We note
that $\Omega_{k}$ has alternating sign and its modulus decreases with
$k$. In figure (5)-B we consider another situation,
with $m=10$ and $M=10$, the ten non-zero couplings $b_{\alpha}$ being
non-uniform. $\Omega_{k}$ still shows alternating sign, and $\Omega_{k}$
vanishes for $k>9$; hence the multiplet of ten variables is
correctly identified. The order of selection is related to the
strength of the coupling: variables with stronger coupling are
selected first.
In figure (6) we consider again the EEG data from healthy
subjects with closed eyes website_nolte , and apply the greedy
search taking C3 as the target and $\{C4,C6\}$ as the seed. We
find a subset of 9 variables influencing the target. The fact that
the sign of $\Omega_{k}$ is alternating, as in the previous model,
suggests that the channels in this set correspond to a single source
which is responsible for the inter-hemispheric communication towards
the target electrode C3. In figure (7) we take O1 as the
target and $\{F3,C5\}$ as the seed. A subset of 11 variables is
found which describes the information flow from the frontal to the
occipital cortex.
IV CONCLUSIONS
Summarizing, we have proposed to describe the flow of information,
in a system, by means of multiplets of variables which send
information to each assigned target node. We used a recently
proposed expansion of the mutual information, between a stochastic
variable and a set of other variables, to measure the character and
the strength of multiplets of variables. Indeed, terms of the
proposed expansion put in evidence irreducible sets of variables
which provide information for the future state of the target
channel. The sign of the contributions are related to their
informational character; for the second order terms, synergy and
redundancy correspond to negative and positive sign, respectively.
For higher orders, we have shown that groups of variables, related
to the same source of information, lead to contributions with
alternating signs as the number of variables is increased. A
decomposition with similarities to the present work have been
reported in beer , where for multiple sources the distinction
between unique, redundant, and synergistic transfer has been
proposed; in lizier the inference of an effective network
structure, given a multivariate time series, using incrementally
conditioned transfer entropy measurements, has been discussed.
The main purpose of this paper is to introduce an information based decomposition, and we did that in a framework unifying Granger causality and Transfer entropy, thus using a formula which is exact for linear models. In cases in which a nonlinear model is required, the entropy has to be computed, requiring a high enough number of time points for statistical validation; nonetheless the expansion that we proposed remains valid and exact in both cases.
We have reported the results of the applications to two EEG
examples. The first data set is from resting brains and we
found signatures of inter-hemispherical communications and frontal
to occipital flow of information. Concerning a data set from an
epileptic subject, our analysis puts in evidence that the seizure is
already active, close to the primary lesion, before it is clinically
observable.
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MPI MIS Preprint 25/2012. |
A Minimax Regret Model for Hub Location under Uncertain Demand and Cost
Iman Kazemian
I. Kazemian Department of Industrial Engineering, Sharif University of Technology, Tehran, PO Box 11365â11155, Iran
Tel.: +98-913-1965513
22email: kazemian.ie@gmail.comS. Aref Department of Computer Science, University of Auckland, Auckland, Private Bag 92019, New Zealand
Te Punaha Matatini Centre for Complex Systems and Networks, New Zealand
44email: sare618@aucklanduni.ac.nz
Samin Aref
I. Kazemian Department of Industrial Engineering, Sharif University of Technology, Tehran, PO Box 11365â11155, Iran
Tel.: +98-913-1965513
22email: kazemian.ie@gmail.comS. Aref Department of Computer Science, University of Auckland, Auckland, Private Bag 92019, New Zealand
Te Punaha Matatini Centre for Complex Systems and Networks, New Zealand
44email: sare618@aucklanduni.ac.nz
(This is an e-print version that may be subject to further changes
$($To cite this document, please contact the authors$)$)
Abstract
Previous formulation methods offered for capacitated hub location problem, including deterministic models and seasonal optimization, were ineffective under uncertainty. The former might to lead to sub-optimal solutions in case of significant fluctuations in problem parameters and the latter was not practical due to offering different networks while hub location might be a strategic problem requiring one firm solution. In this paper multiple allocation capacitated hub location problem was formulated by a minimax regret model, considering uncertainty factors in setup cost and demand. A computational analysis was performed to investigate the impacts of uncertainty on location of the hubs. The results indicated the importance of the formulation technique in modeling hub location problems from a realistic viewpoint by incorporating different sources of uncertainty. Analyzing a case study derived from an industrial food production company demonstrated the efficiency of the formulation techniques to meet the challenge of companies with seasonal demands and uncertain setup cost by providing a firm strategic solution to be implemented throughout the year.
Keywords:Hub location Uncertain demand Uncertain setup cost Capacitated Multiple allocation Minimax Regret model
∎
1 Introduction
Hub networks are one of the most common types of logistic systems serving urban transportation, airline networks, communication systems, and cargo networks. The basic characteristic of hub networks is that products are routed through a subset of the connections between nodes instead of being routed through direct connections from the origin to the destination.
Hub networks configuration suggest using a set of hubs and spokes for connecting different origins and destinations. Different industries make use of hubs to deal with logistics activities in a more productive way by reducing direct transportation paths. Drawing an analogy might be helpful to clarify the essentiality of hub networks. A complete graph with $K$ nodes has $K\times(K-1)$ edges, while all the nodes can be connected to each other by having a central node (hub) being connected to all the other peripheral nodes (spokes) which reduces the number of edges to $2\times(K-1)$. So the connectivity can be achieved by utilizing fewer resources more productively. That is where the hub location problem originated from; the challenge of deciding upon location of the hubs for obtaining an efficient logistic network.
The communication industry seems to be the first platform for using hubs, while decades has passed since hubs networks were institutionalized in logistics systems, transportation industry, air cargos, and postal services. Nowadays, hub network design is a common practice for wholesalers, distribution companies, and food production industries whose main objective is enhancing transportation and logistics activities.
The main objective in a common hub location problem is to minimize the total costs of establishing hubs and transporting products between hubs and spokes. Hub location problems are categorized into two subsets of capacitated and uncapacitated problems such that the former embodies most of the real-world problems. After hub location problem was introduced, subsequent problems like p-median, p-hub center, and hub covering problems were emerged to address different challenges of industry in terms of facility location. The principal purpose of p-median is to locate a number of hubs in the network so that the total transportation cost is minimized. The second problem, p-hub center, aims to optimize location of the hubs and allocation of nodes such that the major routes in the network are minimized. In the hub covering problem, minimizing the total cost by finding the optimal location of the nodes and their corresponding allocation shapes the question while the number of hubs is not known. Such a problem introduces limits of coverage as the number of nodes that are connected to a hub is limited. Equally relevant to the problem type, main objective, and the decision variables are the questions of single and multiple network allocation patterns i.e. whether the spokes are to be connected to one hub or multiple hubs.
The hub location problem (HLP) was first posed by OâKelly in 1986 okelly86 . The author introduced Single-HLP concerning assignment of the appropriate location to the hub and its connection to the spokes while there was no cost for hub establishment and it had infinite capacity. He then used a mathematical model to formolise another location problem on hub airline network okelly87 . The basic model made progress towards P-HLP, a quadratic model, incorporating a number of hubs with direct transportation routes aykin1990quadratic . The proceeding linear model was then developed as p-median location problem by Campbell in 1991 in which each spoke could be connected to more than one hub campbell90 . A more complex mathematical model for multi allocation hub location problem was also presented by Campbell campbell94 embodying real world assumptions such as fixed cost for connecting spokes to hubs, minimum flow, and capacity of nodes. Although most of the hub location models developed assumed potential locations for the hubs in a discrete space, there were early research studies relaxing this assumption and considering a continuous space okelly86active ; aykin92 .
As the logistic operations got more complicated, new problems emerged with different objective functions, mathematical modeling, and solution techniques. Costa et al. developed hub location problem with infinite capacity, a multi-objective hub network design to minimize the total costs of product flow among the nodes as well as the product processing time in the hubs dagra . The taxonomy of HLP offered four ramifications of hub location problems including Capacitated p-Median Problem shamsi , HLP with star network structure yaman2008 ; yaman2012star , p-hub center problem campbell2005hub ; campbell2005hub2 , and p-hub covering problem kara2003single . Different solution methods were developed to handle HLPs with distinctive computational advantages like zero-one quadratic programming by which large problems can be simply linearized sherali2007improved ; he2012improved . For a more detailed review of HLP literature one may refer to two highly cited papers alumur2008network ; farahani2013hub .
In the pioneering articles the parameters were assumed to be deterministic though it was not a realistic approach. Some examples of research on HLP with a more realistic approach toward the model definition are as it follows. Serra and Marianov developed a model for locating hubs in the network of air transportation by formulating a M/D/c queuing system marianov2003location . The same research field is also investigated by Yang who provided a two-stage stochastic programming model for air transportation with uncertainty in demand yang2009stochastic . In the same year Sim suggested a model considering the travel time between nodes in a stochastic environment sim2009stochastic . Moreover, Contreras et al. designed a model for multiple allocation hub location problem in which uncertainty in both demands and transportation costs are accounted for contreras2011stochastic . More recently, Alumur presented multiple allocation and single allocation hub location problem considering uncertainty factors in demand and setup costs alumur2012hub .
In the real world problems capacity constraints are an indispensable component of hub location mathematical modeling. However, to the best of our knowledge there was no published research on the investigation of uncertainty impact on hub location problem with capacity constraints. In this study, a novel mathematical model was proposed to question this hypothesis whether deterministic modeling and seasonal optimization can be sound measures for obtaining the optimal location of the hubs or not.
As stated earlier hub location is an essential part of the strategic planning for distribution companies having far-reaching effects on their operational issues and productivity. On the other hand, logistic activities are changing within time while the data used in hub network designed might be outdated by the time of network utilization. Therefore, some of the parameters required for designing the network may not be accurately determined. The most common uncertain parameters are costs, distances, demands and the like. Inobservance to the uncertainty of parameters may lead to obtaining sub-optimal network designs when the determinant input parameters are changed. The reasons behind uncertainty of such parameters are as follows. The volatility of costs for initial procurement such as land, industrial equipment, and raw materials makes the setup cost uncertain. Although, the demand can be predicted by a simple market research, the lag time between designing the network and its actual utilization makes any prediction outdated especially for the case of time-dependent demands like seasonal goods. This issue indicates that uncertainty should also be considered for demand parameters. Sometimes uncertain parameters follow a familiar probability distribution which needs stochastic optimization and there are times when the data is not fit to familiar distributions which require robust programming to take uncertainty into consideration. In both cases considering different scenarios in a discrete probabilistic space would take the uncertainty of parameters into account alumur2012hub . This research aimed to investigate the effect of uncertainty on the solution obtained from different modeling techniques previously proposed by the contemporary researchers. It basically concerns different approaches toward the formulation of a typical hub location problem with comprehensive uncertainty factors and capacity constraints.
The structure of rest of the paper is as it follows. Section 2 presents a deterministic model to introduce the mathematical modeling foundation. Section 3 suggests a more sophisticated optimization model with uncertain parameters. The efficiency of optimization model is evaluated in the 4th Section by analyzing a numerical example. Finally, a practical case is discussed in Section 5 to demonstrate the effectiveness of the proposed model in solving real world problems.
2 Basic Model
Optimizing the location of the hubs in a logistic system has prospective impact on total cost making it a crucial strategic decision making process. As an attempt to deal with lack of precise information on the operational parameters of the logistic network, minimax regret model can be deployed.
A quick review on the current literature of hub location revealed the presence of research studies on hub location problems with capacity constraints ebery2000capacitated ; correia2011hub ; correia2014multi . However, the capacitated hub location literature suffered from lack of realistic views towards uncertainty that can be obtained by developing stochastic programming and robust optimization models. This study suggested a novel modeling approach with four linear constraints to offer a more realistic view towards uncertainty in demand and setup costs. As mentioned earlier, the foundation of optimization model was first introduced by replicating a deterministic model originally developed by Campbell campbell94 and then the main minimax regret model was introduced to analyze multiple allocation capacitated hub location problem in a complete graph where there was no direct links between the spokes. The total cost was structured in a way that it included setup costs for hubs and three types of transportation costs including collection costs for spoke-to-hub, distribution costs for hub-to-hub, and transfer costs for hub-to-spoke. Adapting from a deterministic model proposed in a highly cited study campbell94 , the basic model notation was stated in Table 1. Accordingly, the transportation cost is formulated in Eq. (1).
$$C_{ijkm}=\beta d_{ik}+\alpha d_{km}+\delta d_{mj}\qquad\forall i,j,k,m\in N$$
(1)
Thus, the multiple allocation capacitated hub location problem is formulated as it follows in (2) to (7):
$$\min\sum\limits_{i}\sum\limits_{j}\sum\limits_{k}\sum\limits_{m}W_{ij}C_{ijkm}%
x_{ijkm}+\sum\limits_{k}F_{k}y_{k}$$
(2)
S.t.
$$\sum\limits_{m}x_{ijkm}\leq y_{k}\qquad\forall i,j,k\in N$$
(3)
$$\sum\limits_{k}x_{ijkm}\leq y_{m}\qquad\forall i,j,m\in N$$
(4)
$$\sum\limits_{i}{\sum\limits_{j}}W_{ij}\sum\limits_{m}x_{ijkm}\leq\Gamma_{k}y_{%
k}\qquad\forall k\in N$$
(5)
$$y_{k}\in\{0,1\}\qquad\forall k\in N$$
(6)
$$0\leq x_{ijkm}\leq 1\qquad\forall i,j,k,m\in N$$
(7)
In this formulation, $x_{ijkm}$ stud for the decision variable and $y_{k}$ represented a binary variable showing whether node $k$ is a hub by taking one, or it is a spoke by taking zero. Eq. (3) and (4) ensured that the flow passes through the hubs. Eq. (5) took the capacity of the hubs into account. Finally, domain constraints were formulated in (6) and (7).
3 Minimax Regret Model
As already mentioned, network design is associated with uncertainty. It was assumed in this paper that the uncertainty in demand can be described by considering a limited number of scenarios. It was also assumed that in each scenario demand parameters are certain values. Moreover, uncertain behavior of setup costs is assumed to be interpretable by considering different scenarios. Deploying such scenarios alongside minimax regret programming as the modeling technique, the model will be able to tackle real world problems in an uncertain environment alumur2012hub . The minimax regret model notation was stated in Table 2.
$$\min\sum\limits_{s}P_{s}\sum\limits_{i}\sum\limits_{j}\sum\limits_{k}\sum%
\limits_{m}W_{ij}^{s}C_{ijkm}x_{ijkm}+\sum\limits_{k}F_{s^{\prime}}^{k}y_{k}$$
(8)
S.t.
$$\sum\limits_{m}x_{ijkm}\leq y_{k}\qquad\forall i,j,k\in N$$
(9)
$$\sum\limits_{k}x_{ijkm}\leq y_{m}\qquad\forall i,j,m\in N$$
(10)
$$\sum\limits_{i}{\sum\limits_{j}}W^{s}_{ij}\sum\limits_{m}x_{ijkm}\leq\Gamma_{k%
}y_{k}\qquad\forall k,s\in N$$
(11)
$$y_{k}\in\{0,1\}\qquad\forall k\in N$$
(12)
$$0\leq x_{ijkm}\leq 1\qquad\forall i,j,k,m\in N$$
(13)
If $Z_{s^{\prime}}^{*}$ is the optimum solution of the model above, as the exact scenario that will occur is not known, a minmax regret model can be considered as it follows in Eq. 14 to Eq. 20 in which the maximum regret is to be minimized as the objective function. In this mathematical model, the representation of $y_{k}$ was the same as before. The above model can be easily linearized by defining variable $R$ and setting it greater than or equal to all other $R_{s}^{\prime}$ for all $s^{\prime}\in S^{\prime}_{f}$.
$$\min\max_{s^{\prime}\in S^{\prime}_{f}}=R_{s}^{\prime}$$
(14)
S.t.
$$\sum\limits_{m}x_{ijkm}\leq y_{k}\qquad\forall i,j,k\in N$$
(15)
$$\sum\limits_{k}x_{ijkm}\leq y_{m}\qquad\forall i,j,m\in N$$
(16)
$$\sum\limits_{i}{\sum\limits_{j}}W^{s}_{ij}\sum\limits_{m}x_{ijkm}\leq\Gamma_{k%
}y_{k}\qquad\forall k,s\in N$$
(17)
$$\begin{split}\displaystyle R_{s}^{\prime}=\sum\limits_{s}P_{s}\sum\limits_{i}%
\sum\limits_{j}\sum\limits_{k}\sum\limits_{m}W_{ij}^{s}C_{ijkm}x_{ijkm}+\\
\displaystyle\sum\limits_{k}F_{s^{\prime}}^{k}y_{k}-Z_{s^{\prime}}^{*}\qquad%
\forall s^{\prime}\in S^{\prime}_{f}\end{split}$$
(18)
$$y_{k}\in\{0,1\}\qquad\forall k\in N$$
(19)
$$0\leq x_{ijkm}\leq 1\qquad\forall i,j,k,m\in N$$
(20)
4 Computational Analysis
To test the model proposed in 3, the theoretical data related to air transportation for five different cities was used and the other parameters were considered as $\beta=\delta=1$ and $\alpha\in\{0.2,0.4,0.6,0.8\}$.
Five different scenarios for uncertainty in the setup cost were designed in which $F_{s^{\prime}}^{k}$ was selected randomly. Moreover, for analyzing the uncertainty in demands four different scenarios were selected. Probability of each scenario occurrence was $0.25$. GAMS software was used to solve the numerical example. Optimum solution in different modes was represented in Table 3 in which basic deterministic model abbreviated to BDM was shown in the first row, four scenarios of the stochastic model were presented in the next rows according to four different values assumed for cost of land represented by $s_{f}$, and finally minimax regret model abbreviated to MRM was demonstrated in the lowest row. The problems were solved separately based on each scenario considering minmax regret models. The solutions were compared with the solution obtained from the basic deterministic model in which setup costs and demands were set to the mean values.
The second column form left side of Table 3 showed the total cost and the third column represented the optimal location of the hubs. Note that as the objective function in the represented the minmax regret solution, total cost function could not be compared with the costs of scenarios. Hence, the row for value of the minimax regret cost was left empty.
As was evident in Table 3, the optimum location of the hubs in the minimax regret model differed from the other scenarios. It could be concluded from the observation that it was better to use the minimax regret model instead of estimating costs and demands or using a deterministic scenario.
No relationship was observed between the costs of setting up a hub and selecting a location for the hub. For example, node 4 had the highest setup cost, but in some problems it was selected as the optimal hub. This observation indicated that in addition to the setup cost, the demand and geographical location were also determinant factors in hub location problem.
5 Real-life Example
The application of the proposed model was evaluated using the data from an Iranian industrial food production company. The case of chocolate production in Shirin Asal Tabriz Co. was a known local hub location example, analyzed and referred to by other hub location researchers rostami . Rostami and associates analyzed the impact of estimated demand on the configuration of hub networks in different scenarios. According to their model, the location of hubs should change seasonally. In contrast, they left the costs of changing role of the nodes unsaid i.e. they did not consider the costs of degrading hubs into spokes and upgrading spokes into hubs as their model required so. As already discussed, HLP is a strategic decision making process and it requires an unchangeable solution as proposed by this research.
As mentioned earlier, the fluctuation of prices makes hub establishment an uncertain activity in terms of monetary issues. Moreover, according to the data reported by the company, the demand for chocolate was seasonal and the setup cost was highly variable by time, making it an appropriate case to be analyzed by the proposed model of this study emphasizing on the fluctuations in demand and variability of the setup costs. The scenarios were designed by dividing the year into four seasons with equal length for demand and considering five scenarios for setup costs with $0.7F_{k}$ to $1.3F_{k}$.
The geographical structure of Shirin Asal Tabriz Co. market was as it follows. The main factory was located in the city of Tabriz in the north-west of Iran supporting 36 distribution points with different demand across the country. The national market can be divided into three regions namely: the west, the center and the east. The data presented in this study were related to the demand of the west side of the country with 14 nodes. The company management sought to establish hubs among these cities (14 destinations). Table 4 demonstrated the demands in each scenario. Likewise, Table 4 illustrated the capacities and the fixed costs for establishing hubs in each city. The problem was to find the best location for the hubs according to nodesâ capacities and uncertainty in demand and setup costs. According to the studies performed for this particular example, $\alpha=0.4,\beta=1$ and $\delta=1$ were calculated rostami . The modeling and solution method were performed by GAMS software on a personal computer with 2.4 GHz Intel Core i5 CPU and 4 GB of RAM.
To demonstrate the practical application of the proposed minimax regret model, a deterministic model (setup costs and demands are set to the mean values) was first solved and then the minimax regret model was used to analyze the practical case with respect to significant sources of uncertainties discussed. The result of solving the deterministic model was shown in the left side of Fig. 1 suggesting building two hubs in Ardebil and Kermanshah when basing the decision on mean values and ignoring the uncertainty. Accordingly, the minimax regret model offered to establish hubs in three cities of Ghazvin, Zanjan and Arak as the solution was illustrated on the map in the right side of Fig. 1. The substantial difference between the two models showed the substantial impact of uncertainty on hub location problems. The differences in hub network designs let us argue that deterministic analysis of hub location problems suffered from capability of industrial practice in cases where there are source of uncertainty. Deterministic analysis might lead to sub-optimal hub location solutions that are counterproductive as they imposed additional costs to the company in the long term. It is also noteworthy that deploying this prospective approach leads to substantial savings by obviating the seasonal spoke-to-hub upgrades and hub-to-spoke degrades suggested by Rostami and associates.
6 Conclusion
In this paper, multi allocation capacitated hub location problem was investigated in a setting where setup costs and demands are uncertain. A generic model was developed for considering such sources of uncertainty. The modeling technique was continued by performing a computational analysis to investigate the possible changes in the optimal location of the hub in an uncertain environment.
The result showed that the optimal solution changes when the model is associated with uncertain parameters. According to the numerical example, ignoring the uncertainty may change the whole hub location solution drastically. Moreover, the industrial application of the proposed method was addressed to by discussing a case from an industrial food production company. The results obtained from solving the case study confirmed the previously drawn conclusion about significance of uncertainty and how ignoring it may lead to sub-optimal practices.
There are a number of research directions that are hoped to be investigated in the near future. Firstly, one may introduce a more sophisticated problem with pervasive sources of uncertainty by parameters not necessarily following a familiar distribution function to be solved by stochastic programming and compared with the technique used in this paper. It would contribute to such an idea if the research is associated with a critical industrial-sized case of an uncertain environment. Secondly, it is suggested for further research to tackle intractable hub location problems with uncertain parameters by tailored evolutionary algorithms and analyze a practical case with large data set such as international couriers.
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Conical emission, pulse splitting and X-wave parametric amplification
in nonlinear dynamics of ultrashort light pulses
Daniele Faccio
daniele.faccio@uninsubria.it
INFM and Department of Physics & Mathematics, University
of Insubria, Via Valleggio 11, 22100 Como, Italy
Miguel A. Porras
Departamento de Fisíca Aplicada, ETSIM, Universidad Politécnica de Madrid, Rios Rosas
21, 28003 Madrid, Spain
Audrius Dubietis
Department of Quantum Electronics, Vilnius University, Sauletekio Ave. 9, 10222 Vilnius,
Lithuania
Francesca Bragheri
Department of Electronics, University of Pavia, Via Ferrata 1, 27100 Pavia, Italy
Arnaud Couairon
Centre de Physique Théorique, École Polytechnique, CNRS UMR 7644, 91128 Palaiseau
Cedex, France
Paolo Di Trapani
INFM and Department of Physics & Mathematics, University of Insubria, Via Valleggio 11, 22100
Como, Italy
Abstract
The precise observation of the angle-frequency spectrum of light filaments in water
reveals a scenario incompatible with current models of conical emission (CE). Its
description in terms of linear X-wave modes leads us to understand filamentation
dynamics requiring a phase- and group-matched, Kerr-driven four-wave-mixing
process that involves two highly localized pumps and two X-waves. CE and temporal
splitting arise naturally as two manifestations of this process.
pacs: 190.5940, 320.2250
Filamentation of intense light pulses in nonlinear media has attracted much interest
ever since first experimental evidences in the early ’60’s
(Marburger (1975) and references therein). Owing to the very high
intensities reached during the process, several nonlinear phenomena, e.g.,
multi-photon absorption, plasma formation, saturable nonlinear response, stimulated
Raman scattering etc., occur in addition to the optical Kerr effect. Indeed, the
filament regime is enriched by peculiar phenomena like pulse splitting,
self-steepening, shock-wave formation, super-continuum generation, and conical
emission (CE) Gaeta (2003). In media with normal group velocity dispersion (GVD), no matter if of
solid, liquid or gaseous nature, CE accompanies filamentation, producing
radiation at angles that increase with increasing detuning from the carrier
frequency Nibbering et al. (1996); Faccio et al. (2005). In spite of the generality of the process, a clear understanding of the
interplay between CE and filament dynamics is still missing. Only recently, Kolesik
et al. have proposed an interpretation of filamentation dynamics in water on
the basis of pulse splitting and dynamic nonlinear X waves at the far field
Kolesik et al. (2004), in which the double X-like structure observed in simulated
angle-frequency spectra arises from the scattering of an incident field at the two
main peaks of the split material response wave.
Originally, CE in light filaments was interpreted in terms of the modulation
instability (MI) angle-frequency gain pattern of the plane and monochromatic (PM)
modes of the nonlinear Schrödinger equation (NSE)
Luther et al. (1994); Liou et al. (1992). Measurements at large angles and detunings from
the carrier frequency gave in fact results fairly compatible with this interpretation
Alfano and Shapiro (1970); Xing et al. (1993). In the present work, owing to the use of a novel
imaging spectrograph technique Faccio et al. (2005), we have been able to observe
for the first time the CE in the region of small angles and detunings. The results
clearly indicate a scenario not compatible with the MI analysis of PM modes. Our
description by means of the spectra of the stationary linear X-waves supported
by the medium, indicates that the strong localization of the self-focused field plays
a crucial role in the substantial modification experienced by the MI pattern. We
propose a simple picture in which the latter results from the parametric
amplification of two weak X-waves by the strong, highly localized pump. Supporting
this interpretation, we are able to derive, from the matching condition among the
interacting waves, a simple analytical expression [Eq. (4)] that accurately
determines the overall CE structure, and predicts also the splitting velocity of the
pump wave as a function of the peak intensity reached at collapse.
The experiment was carried out using a 3 cm long cell filled with water as a bulk
nonlinear Kerr medium. Beam filamentation was induced by launching 200 fs pulses at
carrier wave length 527 nm, delivered from a mode-locked, chirped-pulse
regeneratively amplified, Nd:glass laser with a 10 Hz repetition rate (Twinkle; Light
Conversion Ltd.). The pulse energy was controlled by use of a half-wave plate and a
Glan-Taylor polarizer. A spatial filter guarantees high beam quality before focusing
with a $f=50$ cm lens placed at 48 cm from the cell input facet, with a beam
diameter at half-maximum equal to 100 $\mu$m. The output facet is then imaged with a
4-$f$ telescope onto the rear focal plane of the lens ($f_{F}$ = 15 cm) used to
obtain the spatial Fourier transform of the filament generated in the Kerr sample.
The input slit of the imaging spectrograph (MS260i, Lot-Oriel) with a high resolution
1200 lines/mm diffraction grating is placed at a distance $f_{F}$ from the Fourier
lens so as to reconstruct the angle-wavelength ($\theta$–$\lambda$) spectrum of the
filament on a commercial CCD 8-bit camera (Canon), placed at the output plane of the
spectrograph. With the change $\lambda=2\pi c/(\omega_{0}+\Omega)$, the
angle-frequency ($\theta$–$\Omega$) spectrum can also be obtained, $\Omega$ being
the detuning from the carrier frequency $\omega_{0}$. More details of the experimental
layout may be found elsewhere Faccio et al. (2005). We recorded only single-shot
spectra in order to avoid averaging effects due to possible shot-to-shot fluctuations
in the input pulse energy. Moreover we highly saturated the central peak of the
spectrum so as to highlight the less visible surrounding structure. For input
energies $E_{\rm in}\lesssim 1.8\,\mu$J, no CE, or clear X-like features were
observed. Figure 1 shows an example of $\theta$–$\lambda$ spectrum at
$E_{\rm in}=2\,\mu$J. The CE appears as distinctly separated red- and blue-shifted
X-shaped tails. This pattern remains very similar with increasing input energy up to
$E_{\rm in}\sim 4$ $\mu$J, while further increase produces a slowly deteriorating
picture, with a modulated intensity pattern that extends to nearly all recorded
values of $\theta$ and $\lambda$. We interpret this deterioration as due to the onset
of local breakdown in the water sample.
CE emission is commonly accepted to arise from the interplay of diffraction, dispersion and nonlinear
material response, the simplest model that accounts for it being the cubic NSE with normal GVD
Luther et al. (1994); Liou et al. (1992)
$$\displaystyle\frac{\partial A}{\partial z}=\frac{i}{2k_{0}}\nabla_{\bot}^{2}A-%
\frac{ik_{0}^{\prime\prime}}{2}\frac{\partial^{2}A}{\partial\tau^{2}}+i\frac{%
\omega_{0}n_{2}}{c}|A|^{2}A\,.$$
(1)
Here, $A(x,y,\tau,z)$ is the complex envelope of the wave packet $E=A\exp(ik_{0}z-i\omega_{0}t)$ of
carrier frequency $\omega_{0}$, $\nabla^{2}_{\bot}=\partial^{2}/\partial x^{2}+\partial^{2}/\partial y^{2}$ is the two-dimensional Laplace operator perpendicular to the
propagation direction $z$, $\tau=t-k^{\prime}_{0}z$ is the local time, $c$ the speed of light in vacuum, $n_{2}$
the nonlinear refractive index, and $k_{0}^{(m)}\equiv[d^{m}k(\omega)/d\omega^{m}]_{\omega_{0}}$, where
is $k(\omega)$ the frequency-dependent propagation constant. Mathematically, the $\theta$–$\Omega$ spectrum
where CE is observable, is directly related to the Fourier spectrum of the envelope. If for instance,
$A(r,\tau,z)$ [$r=(x^{2}+y^{2})^{1/2}$] is a cylindrical symmetric complex envelope, and
$\hat{A}(K_{\perp},\Omega,z)$ [$K_{\perp}=(k_{x}^{2}+k_{y}^{2})^{1/2}]$ is its spatiotemporal Fourier transform, then
the $\theta$–$\Omega$ spectrum is given by $\hat{A}(k_{0}\theta,\Omega,z)$, where
$\theta=K_{\perp}/k_{0}$ is the propagation angle with respect to the $z$ axis.
An accepted approach for understanding the structure of the CE relies upon the evaluation of the
MI gain profile of the PM modes of the NSE
Liou et al. (1992). In Kerr self-focusing media with normal dispersion, the perturbations to a PM
mode that grow at maximum rate are those whose spatiotemporal frequencies $(K_{\perp},\Omega)$ are related by
$$K_{\perp}(\Omega)=\sqrt{k_{0}k_{0}^{\prime\prime}\Omega^{2}+2k_{0}\tilde{\beta%
}},\quad\mbox{($\tilde{\beta}>0$)},$$
(2)
i.e., lie on a hyperbola on the $K_{\perp}$–$\Omega$ plane featuring an angle gap ($K_{\perp}$ gap)
Liou et al. (1992). In this respect, Luther et al. proposed an intuitive picture that assumes the
largest MI gain to occur at angles and frequencies fulfilling the linear phase-matching condition of the
four-wave mixing (FWM) process supported by the Kerr response Luther et al. (1994). Under this
hypothesis, the asymptotic linear approximation $K_{\perp}(\Omega)\simeq\sqrt{k_{0}k_{0}^{\prime\prime}}|\Omega|$
was re-obtained, and the observed discrepancies were attributed to the
nonlinear phase shift produced by the PM mode on the weak perturbation.
Preceding experimental observations of CE in borosilicate glass Alfano and Shapiro (1970) and ethylene glycol
Xing et al. (1993) were indeed interpreted to present a hyperbolic structure with an angle gap, which was
also attributed to the nonlinear phase shift. This trend is also visible in our measurements (Fig.
1) as an intersection of the two X arms at a non-zero angle. If we attempt to fit
Eq. (2) (taking $\tilde{\beta}$ and $k_{0}^{\prime\prime}$ as free parameters) to the experimental
data [Fig.2(a), dotted line and closed circles, respectively] we obtain
$k^{\prime\prime}_{0}=0.053\pm 0.03$ fs${}^{2}/\mu$m, which firmly supports the predictions
Luther et al. (1994) regarding the asymptotic slope ($k^{\prime\prime}_{0}=0.055$ fs${}^{2}/\mu$m at 527 nm
Van Engen et al. (1998)); however, the angle-gap fitting strongly departs from experimental data in the non-asymptotic
region, which strongly calls for a fitting to a hyperbola with opposite curvature, that is, to a hyperbola
with a frequency gap. This observation advocates a novel interpretation of the origin of CE.
To understand our proposal, note first that the usual MI interpretation of the
seemingly angle-gap hyperbolic CE can be equivalently reread in terms of the
excitation of a weak, linear X-wave mode by a strong, nonlinear PM pump.
Indeed, Eq. (2) represents also the $K_{\perp}$–$\Omega$ spectrum of an
X-wave mode of the medium Porras and Di Trapani (2004), that is, a diffraction- and
dispersion-free pulsed Bessel beam in which cone-angle-induced dispersion
[$\theta(\Omega)\simeq K_{\perp}(\Omega)/k_{0}$] and material GVD balance mutually. The
spectrum in Eq. (2) belongs to a wave-mode of carrier frequency $\Omega_{0}$
and shortened axial wave vector $k_{0}-\tilde{\beta}$. In this reading, the excitation of
a X-wave mode with angle-gap ($\tilde{\beta}>0$) is a consequence of the nonlinear
phase matching between PM pump and X-wave for maximum efficiency of the interaction:
taking into account the nonlinear corrections to the wave vectors of both pump and
X-wave, phase matching imposes $k_{0}+\Delta k=k_{0}-\tilde{\beta}+2\Delta k$, where
$\Delta k=\omega_{0}n_{2}I/c$ is the positive nonlinear phase shift (in
self-focusing Kerr medium) for the PM pump of intensity $I$, and where $2\Delta k$ is
the corresponding nonlinear correction to the weak perturbation
Alfano and Shapiro (1970); Xing et al. (1993), leading immediately to $\tilde{\beta}=\omega_{0}n_{2}I/c>0$ (as predicted by the standard MI analysis Liou et al. (1992)).
We also note that a medium with GVD
[$k(\Omega)=k_{0}+k^{\prime}_{0}\Omega+k_{0}^{\prime\prime}\Omega^{2}/2$] supports more general X-wave modes with
strengthened and shortened wave vectors ($\tilde{\beta}$ negative and positive), as well as shifted carrier
frequency $\omega_{0}+\tilde{\Omega}$, their most general $K_{\perp}$–$\Omega$ spectrum being expressed by
Porras and Di Trapani (2004)
$$K_{\perp}(\Omega)=\sqrt{k_{0}k_{0}^{\prime\prime}(\Omega-\tilde{\Omega})^{2}+2%
k_{0}\tilde{\beta}},\quad\mbox{($\tilde{\beta}\gtreqqless 0$).}$$
(3)
Their phase and group velocities are given by $v^{(p)}=(\omega_{0}+\tilde{\Omega})/[k(\tilde{\Omega})-\tilde{\beta}]$
and $v^{(g)}=1/k^{\prime}(\tilde{\Omega})$, respectively, which can take both subluminal and superluminal values.
Equation (3) describes a two-parameter family of hyperbolas, sharing the same asymptotic
slope $\sqrt{k_{0}k_{0}^{\prime\prime}}$, but admitting angle gaps ($\tilde{\beta}>0$) and frequency gaps
($\tilde{\beta}<0$), as well as positive and negative frequency shifts $\tilde{\Omega}$. X-wave spectra appear
then as a suitable tool for the description of the CE features.
The apparent frequency-gap hyperbolic form of the observed CE, and the actual absence of such a gap,
suggest that the CE cannot be described in terms of a single X-wave. This leads us to interpret the two
observed X tails as belonging to two different X waves both featuring frequency gaps, and proceed to fit Eq.
(3) with two independent sets of parameters $\tilde{\Omega}$ and $\tilde{\beta}$ (and $k_{0}^{\prime\prime}=0.056$ fs${}^{2}/\mu$m) to the experimental data. This procedure also finds motivation in
the X-X structure of the numerically evaluated angular spectra of light filaments in water, which was
interpreted to be a consequence of pulse splitting on X-wave generation Kolesik et al. (2004). Fittings
with $\tilde{\Omega}=+0.33$, $-0.33$ fs${}^{-1}$ and $\tilde{\beta}=-2.2,-2.2\,$mm${}^{-1}$ (dashed and continuous
lines, respectively) reproduce accurately the structure of the CE [Fig. 2(a)]. The interpretation
in term of X-waves leads moreover to suspect the existence of two additional, not yet observed X arms in the
$K$–$\Omega$ spectrum at wavelengths 450 and 635 nm [Fig. 2(a)].
To verify this prediction, we performed additional measurements with a lower resolution, 300 lines/mm
diffraction grating in order to cover a wider spectral region, as shown in Fig. 2(b).
Alongside the central tails (around 527 nm), we clearly observe the generation of new frequencies in the
vicinity of 450 and and 635 nm, as expected. The reduced visibility of the red tail at 635 nm can be
attributed to the stronger linear absorption at this wavelength. In fact, we had to increase the input
pulse energy up to $E_{\rm in}=3\,\mu$J in order to enhance the overall intensity of the new tails.
The question then arises of the mechanism responsible for the described spectral structure.
Here we propose that the double X-like CE is a result of a nonlinear FWM interaction —the
lowest order process supported by the nonlinear Kerr response—, in which two signal and idler X-waves
experience a parametric amplification by a strong, highly localized (i.e., non PM) pump. If as
consistently with the self-focusing dynamics, we consider a strongly localized pump rather than a PM wave,
it is expected that nonlinear phase modulation does not influence significantly an extended object such an
X-wave mode, whose axial wave vector is solely determined by its cone angle, that is, by its weak,
longstanding tails far from the pump. In fact, in the X-wave the energy does not flow along the axial
direction, but along a conical surface, which prevents pump-induced cross-phase accumulation to take place.
Under this assumption, phase matching between a localized pump and a single X-wave at same frequency
$\omega_{0}$ would require $k_{0}+\Delta k=k_{0}-\tilde{\beta}$, a condition that contrarily to the case of a PM
pump, is satisfied by a frequency-gap X-wave mode ($\tilde{\beta}=-\Delta k<0$). More generally, if
$k=k_{0}+\Delta k$ is the nonlinear pump wave number, and $k_{s,i}=k(\tilde{\Omega}_{s,i})-\tilde{\beta}_{s,i}$ are two, signal and idler, X wave numbers, where
$k(\tilde{\Omega}_{s,i})=k_{0}+k^{\prime}_{0}\Omega_{s,i}+k_{0}^{\prime\prime}%
\Omega_{s,i}^{2}/2$, and $\Omega_{s,i}$ are
their carrier frequency shifts, the conditions of energy and momentum conservation, $\tilde{\Omega}_{s}=-\tilde{\Omega}_{i}$ and $2k=k_{s}+k_{i}$, for maximum efficiency of a FWM process involving two (degenerate)
highly localized pumps and the signal and idler X waves, leads to the relation $\tilde{\beta}_{s}+\tilde{\beta}_{i}-k_{0}^{\prime\prime}\Omega_{s,i}^{2}=-2\Delta
k$. Among all possible couples of X-wave modes
satisfying this relation, those whose spectra cross the pump, located around $\Omega=0,K_{\perp}=0$, are
the most energetically favored, since these X-waves will not need to grow from noise, but from the more
effective pump self-phase modulation. This condition was also found in Conti (2004) to result in the
largest gain in the case of a single X wave excited by a travelling pump. The X-wave–pump crossing
condition leads, from Eq. (3), to $\tilde{\beta}_{s,i}=-k_{0}^{\prime\prime}\tilde{\Omega}_{s,i}^{2}/2$,
and then, on account that $\Delta k=\omega_{0}n_{2}I/c$, to
$$\tilde{\beta}_{s,i}=-\omega_{0}n_{2}I/2c\,.$$
(4)
Accordingly, the two X-waves will present symmetrically shifted carrier frequencies $\tilde{\Omega}_{i,s}=\pm\sqrt{\omega_{0}n_{2}I/ck_{0}^{\prime\prime}}$, and frequency-gaps of width $2|\tilde{\Omega}_{i,s}|$, from
$\Omega=0$ towards the Stokes and anti-Stokes bands, as observed in the experiment.
Eq. (4) predicts also a precise dependence of the whole CE structure on the
pump intensity $I$, whose validity we have tested numerically. We solved the NSE
(1) with $k_{0}^{\prime\prime}=0.056$ fs${}^{2}/\mu$m, $n=1.33$ and $n_{2}=1.6\times 10^{-16}$ cm${}^{2}/$W, for an input 200 fs-long, 100 $\mu$m-wide (FWHM)
Gaussian wave packet. Since CE is seen to develop explosively at the collapse,
we identified the pump intensity $I$ with
the absolute peak intensity reached during propagation. In order to attain different
values of this intensity without changing the rest of parameters, we added to the
second member of the NSE (1) the nonlinear dissipation term
$-\beta^{(K)}|A|^{K}A/2$, with $K=3$ (three-photon absorption at 527 nm in water), and
with $\beta^{(K)}$ ranging from $1.2\times 10^{-23}$ to $8\times 10^{-25}$
cm${}^{3}$/W${}^{2}$. The $K_{\perp}-\Omega$ spectra obtained numerically are symmetric with
respect to $\Omega=0$ due to the approximations involved in the NSE, and present a
nearly invariant X-X structure beyond the collapse point (in spite of the quickly
decreasing peak intensity $I$), with one tail of each X wave nearly passing through
$\Omega=0$, as observed experimentally. The spectrum at $3$ cm from the input plane
was then easily fitted with two, signal and idler, material X-waves modes crossing
$\Omega=0$ [i.e., with $k_{0}^{\prime\prime}=0.056$ fs${}^{2}/\mu$m, and with
$\tilde{\Omega}_{s,i}=\pm(-2\tilde{\beta}_{s,i}/k_{0}^{\prime\prime})^{1/2}$],
$\tilde{\beta}_{s,i}$ being then the only one free parameter. Figure 3
shows the values of $\tilde{\beta}_{s,i}$, obtained from the best fits to the spectra,
versus pump intensity $I$ (open squares). The excellent agreement with the predicted
$\tilde{\beta}_{s,i}$–$I$ dependence in Eq. (4) (solid line) strongly
supports the FWM analysis.
Extending further our interpretation, we conjecture that the well-known phenomenon of
pulse temporal splitting in filamentation with normal GVD, usually described from a purely temporal
perspective, is a consequence of higher-order matching among the interacting waves —in our case, of
group matching. Indeed,
the zero-order, phase-matching condition (4) entails that the two X-waves must travel at
different group velocities $v^{(g)}_{s,i}=1/k^{\prime}(\Omega_{s,i})=1/[k^{\prime}_{0}+k_{0}^{\prime\prime%
}\Omega_{s,i}]$,
and therefore split apart in time one from another with a group velocity mismatch (GVM)
$1/v^{(g)}_{s}-1/v^{(g)}_{i}=2k_{0}^{\prime\prime}|\tilde{\Omega}_{s,i}|=2\sqrt%
{k_{0}^{\prime\prime}\omega_{0}n_{2}I/c}$ proportional to the square root of the pump intensity. Group matching among the
interacting waves is then better attained if the two pump waves, breaking their initial degeneracy, split
also to co-propagate with the X-waves. In this view, pulse splitting is not a mere
collapse-arresting mechanism Luther et al. (1994), which further determines, taken it for granted, the
generation of two X-waves Kolesik et al. (2004); instead, pulse splitting emerges here as a particular
feature of the phase and group matched wave configuration most favored by the FWM nonlinear interaction
inherent to the Kerr NSE dynamics. In other words, X-wave instability or parametric amplification, and
splitting instability are not independent phenomena, but two aspects of the same process. To sustain this
hypothesis, we have evaluated the GVM of the split pumps in the same numerical simulations as in the
preceding paragraph. To do so, we plotted the temporal profiles at different propagation distances
beyond collapse, and obtained the GVM as the slope of a linear fitting to the
delay between the two temporal peaks versus propagation distance. Figure 3 shows that the
pump GVM (open circles) does not depart more than 10 percent from the predicted X-wave GVM
(solid curve), and that both follow a similar dependence with the square root of the intensity $I$
at collapse.
In summary, our experiments demonstrate that CE emission in the angular spectrum of filaments is not
interpretable in the frame of MI of PM modes of the NSE. Strong localization of the self-focusing pulse
substantially modifies the MI pattern, which finds accurate description in terms of linear X-wave modes of
the medium, and simple explanation as a result of a dominant, phase-matched FWM mixing process supported by
the NSE dynamics between two highly localized, strong pump waves and two amplifying weak X-waves. Pulse
temporal splitting emerges in this model as the necessary temporal dynamics for preserving group matching
among the interacting waves.
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Long time behavior of the volume of the Wiener sausage on Dirichlet spaces
Kazuki Okamura
School of General Education, Shinshu University, 3-1-1, Asahi, Matsumoto, Nagano, 390-8621, JAPAN.
kazukio@shinshu-u.ac.jp
Abstract.
In the present paper, we consider long time behaviors of the volume of the Wiener sausage on Dirichlet spaces.
We focus on the volume of the Wiener sausage for diffusion processes on metric measure spaces other than the Euclid space equipped with the Lebesgue measure.
We obtain the growth rate of the expectations and almost sure behaviors of the volumes of the Wiener sausages on metric measure Dirichlet spaces satisfying Ahlfors regularity and sub-Gaussian heat kernel estimates.
We show that the growth rate of the expectations on a “bounded” modification of the Euclidian space is identical with the one on the Euclidian space equipped with the Lebesgue measure.
We give an example of a metric measure Dirichlet space on which a scaled of the means fluctuates.
2010 Mathematics Subject Classification: 60J60, 31C25
1. Introduction and Main results
In the present paper, we consider the long time behavior of the volume of the Wiener sausage on metric measure Dirichlet spaces.
There have been many results for the standard Brownian motion on the Euclid spaces.
Here we focus on the volume of the Wiener sausage for diffusion processes on metric measure spaces other than the Euclid space equipped with the Lebesgue measure.
We review several known results for diffusion processes on metric measure spaces other than the Euclid space equipped with the Lebesgue measure.
Chavel-Feldman [CF86-1, CF86-2, CF86-3] considered the volume of the Wiener sausage for Brownian motion on Riemannian manifolds.
[CF86-1] shows radial asymptotic results (i.e. radius $\epsilon\to 0$) on hyperbolic 3-spaces, and a time asymptotic result (i.e. time $t\to\infty$) on Riemannian symmetric spaces of non-positive curvature.
[CF86-2] shows radial asymptotic results on complete Riemannian manifolds for the dimension $d\geq 3$.
[CF86-3] shows radial asymptotic results for the Wiener sausage of reflected Brownian motion on a domain in $\mathbb{R}^{d},d\geq 2$.
Sznitman [Sz89] obtained a time asymptotic result of negative exponentials of Brownian bridge on hyperbolic space which is similar to the result by Donsker-Varadhan [DV75].
Furthermore, in [Sz90] he obtains a Donsker-Varadhan type result for Brownian motion on nilpotent Lie groups.
Chavel-Feldman-Rosen [CFR91] obtained a second order radial asymptotic result for $2$-dimensional Riemannian manifold, extending Le Gall’s expansion [Le88, Theorem 2.1] in $\mathbb{R}^{2}$.
Gibson-Pivarski [GP15] obtained a time asymptotic result similar to [DV75] for diffusions on local Dirichlet spaces satisfying a sub-Gaussian heat kernel estimate.
Bass-Kumagai [BK00] showed the law of the iterated logarithms (LILs) of the ranges of a class of symmetric jump processes on metric measure spaces.
Recently, Kim-Kumagai-Wang [KKW17] showed the LILs of the ranges of a class of symmetric jump processes on metric measure spaces.
Here the range is the volume of the trace of a sample path up to a fixed time.
The results of [BK00, KKW17, GP15] are based on heat kernel estimates.
Our results are time asymptotics for the volume of the Wiener sausage on metric measure Dirichlet spaces.
First, we obtain growth rate of the means and almost sure behaviors on metric measure Dirichlet spaces satisfying Ahlfors regularity and sub-Gaussian heat kernel estimates.
Second, we show that the growth rate of the expectations on a modification of the Euclidian space equipped with the Lebesgue measure is identical with those of the Euclidian space equipped with the Lebesgue measure.
Third, we give an example of a metric measure Dirichlet space on which a scaled sequence of the expectations fluctuates.
For the Brownian motion on the Euclid spaces, by using the Brownian scaling, time asymptotic results can be derived from radial asymptotic results.
However, in this case, we cannot use the spatial homogeneity and the scaling of the Euclid space and the Brownian motion, which are used to show the time asymptotic results in Spitzer [Sp64] and [CF86-1].
Our proofs are based on two estimates for the probability for the hitting time to an open ball appearing in [CF86-2, (10) and (11)], which are stated in Lemma 2.1 below.
The range of random walk is the discrete object corresponding to the volume of the Wiener sausage.
There are differences between the ranges of random walks on graphs and the volumes of the Wiener sausages of diffusions on metric measure Dirichlet spaces.
Results for range of a simple random walk on infinite connected simple graphs were obtained by [O14, O] corresponds to the results in the present paper, however, our proofs are different from those of [O14, O].
The last exit decomposition in [O14, O], whose papers deal with the discrete case, is not applicable to this framework at least in direct manners.
For each $n$, the range of random walk up to time $n$ is always smaller than or equal to $n+1$, however, $V_{\epsilon}(t)$ is unbounded for each $t>0$.
1.1. Framework and main results
Let $(M,d)$ be a non-compact connected complete separable metric space such that every open ball is relatively compact.
Let $\mu$ be a Borel measure on $M$ such that for every relatively compact open subset $U$ of $M$, $0<\mu(U)\leq\mu(\overline{U})<+\infty$.
Here $\overline{U}$ is the closure of $U$.
Let $B(x,r):=\{y\in M:d(x,y)<r\}$, $\overline{B}(x,r):=\{y\in M:d(x,y)\leq r\}$ and $\overline{\partial}B(x,r):=\{y\in M:d(x,y)=r\}$.
We assume that
$$\overline{\partial}B(x,r)\neq\emptyset,\textup{ for every $x\in M$ and every $%
r\geq 0$.}$$
(1.1)
and furthermore
$$\mu(\overline{\partial}B(x,r))=0,\textup{ for every $x\in M$ and every $r\geq 0%
$.}$$
(1.2)
Let $(\mathcal{E},\mathcal{F})$ be a strongly-local regular symmetric conservative
Dirichlet form on $L^{2}(M,\mu)$ and $(X_{t},P^{x})$ be the associated Hunt process.
Assume that there exists heat kernel $p(t,x,y)$ which is jointly-continuous with respect to $(t,x,y)$.
We call a pair $(M,d,\mu,\mathcal{E},\mathcal{F})$ a metric measure Dirichlet space.
For a Borel measurable subset $B$ of $M$, let $T_{B}:=\inf\{t>0:X_{t}\in B\}$.
Let $\tau_{D}$ be the first exit time from a Borel measurable subset $D\subset M$, that is,
$\tau_{D}:=T_{M\setminus D}$.
Let the volume of the Wiener sausage:
$$V_{\epsilon}(t):=\mu\left(\bigcup_{s\in[0,t]}B(X_{s},\epsilon)\right).$$
We write $f\asymp g$ if there are two constants $c$ and $C$ such that $cg(x)\leq f(x)\leq Cg(x)$ for every $x$.
Theorem 1.1 (Growth rates for means).
Let $\epsilon>0$ and $a\in(0,1)$.
Assume that the following two conditions hold:
(i) There is a constant $c_{0}>0$ such that
$$\sup_{x\in M}\mu(B(x,a\epsilon))\leq c_{0}\inf_{x\in M}\mu(B(x,a\epsilon)).$$
(1.3)
(ii) There are an increasing function $f(t)$ and constants $c_{1},c_{2}>0$ such that
$$\lim_{t\to\infty}\frac{f(t)}{t}=0.$$
$$c_{1}\leq\liminf_{t\to\infty}\frac{1}{f(t)}\int_{0}^{t}p(s,x,y)ds\leq\limsup_{%
t\to\infty}\frac{1}{f(t)}\int_{0}^{t}p(s,x,y)ds\leq c_{2}.$$
(1.4)
holds for every $t\geq 1$ and every $x,y\in M$ satisfying that $(1-a)\epsilon\leq d(x,y)\leq(1+a)\epsilon$.
Then, we have that for every $x\in M$,
$$\frac{c_{0}}{c_{2}}\leq\liminf_{t\to\infty}\frac{E^{x}[V_{\epsilon}(t)]}{t/f(t%
)}\leq\limsup_{t\to\infty}\frac{E^{x}[V_{\epsilon}(t)]}{t/f(t)}\leq\frac{c_{0}%
}{c_{1}}.$$
The constants $c_{i},i=0,1,2$, and the function $f$ depend on $\epsilon>0$ and $a\in(0,1)$ both.
However in several cases we can choose $f$ as a function independent from $\epsilon>0$ and $a\in(0,1)$ both and $\lim_{a\to 0}c_{1}(a)=\lim_{a\to 0}c_{2}(a)$.
Assume that $M=\mathbb{R}^{d}$ and $(X_{t})_{t}$ is the standard Brownian motion.
If $d\geq 3$, then, we can let $f(t)=1$, $c_{0}(a)=1$ and $\lim_{a\to 0}c_{1}(a)=\lim_{a\to 0}c_{2}(a)=G(x,y)$, where $x$ and $y$ are points such that $d(x,y)=\epsilon$.
Therefore,
$$\lim_{t\to\infty}\frac{E^{x}[V_{\epsilon}(t)]}{t}=\frac{1}{G(x,y)}=\textup{Cap%
}(B(0,\epsilon)).$$
If $d=2$, then, we can let $f(t)=\log t$, $c_{0}(a)=1$ and $\lim_{a\to 0}c_{1}(a)=\lim_{a\to 0}c_{2}(a)=1/(2\pi)$.
$$\lim_{t\to\infty}\frac{E^{x}[V_{\epsilon}(t)]}{t/\log t}=2\pi.$$
If $d=1$, then, we can let $f(t)=t^{1/2}$, $c_{0}(a)=1$ and $\lim_{a\to 0}c_{1}(a)=\lim_{a\to 0}c_{2}(a)=(2\pi)^{-1/2}$.
$$\lim_{t\to\infty}\frac{E^{x}[V_{\epsilon}(t)]}{t^{1/2}}=\sqrt{2\pi}.$$
Assumptions (i) and (ii) in the above result are satisfied if the following conditions $\textup{Vol}(\alpha)$ and HK$(\beta)$ hold for some $\alpha,\beta>1$:
•
(Ahlfors regularity $\textup{Vol}(\alpha)$):
There exist positive constants $C_{1}$, $C_{2}$ such that for every $x\in M$ and $r\in(0,\epsilon)$,
$$C_{1}r^{\alpha}\leq\mu(B(x,r))\leq C_{2}r^{\alpha}.$$
•
(full heat kernel estimate HK$(\beta)$):
There exist positive constants $C_{i}$, $3\leq i\leq 6$, such that for every $t>0,x,y\in M$,
$$\frac{C_{3}}{\mu(B(x,t^{1/\beta}))}\exp\left(-C_{4}\left(\frac{d(x,y)^{\beta}}%
{t}\right)^{1/(\beta-1)}\right)\leq p(t,x,y)$$
$$\leq\frac{C_{5}}{\mu(B(x,t^{1/\beta}))}\exp\left(-C_{6}\left(\frac{d(x,y)^{%
\beta}}{t}\right)^{1/(\beta-1)}\right).$$
If $\textup{Vol}(\alpha)$ and HK$(\beta)$ hold for some $\alpha,\beta>1$,
then, we can choose the increasing function $f(t)$ in Theorem 1.1 as
$$f(t)=\left\{\begin{array}[]{l}1\ \ \ \ \ \ \ \ \ \ \alpha>\beta,\\
\log t\ \ \ \ \ \ \alpha=\beta,\\
t^{1-\alpha/\beta}\ \ \ \alpha<\beta.\end{array}\right.$$
(1.5)
Multiplying a positive constant to $f(t)$ is allowed.
For almost sure behaviors of $V_{\epsilon}(t)$, we have the following.
Theorem 1.2 (Almost sure behaviors).
Assume that Ahlfors regularity $\textup{Vol}(\alpha)$ and full heat kernel estimate HK$(\beta)$ hold. Then,
(i) If $\alpha\geq\beta$, then, there exist two constants $c_{1}\in[0,\infty)$ and $c_{2}\in(0,\infty)$ depending on $\epsilon$ such that for every $x\in M$,
$$c_{1}=\liminf_{t\to\infty}\frac{V_{\epsilon}(t)}{t/f(t)}\leq\limsup_{t\to%
\infty}\frac{V_{\epsilon}(t)}{t/f(t)}=c_{2},\textup{ $P^{x}$-a.s.}$$
(1.6)
(ii) If $\alpha<\beta$, then, there exist two constants $c_{3},c_{4}\in(0,\infty)$ depending on $\epsilon$ such that for every $x\in M$,
$$\liminf_{t\to\infty}\frac{V_{\epsilon}(t)}{(t/\log\log t)^{\alpha/\beta}}=c_{3%
},\textup{ $P^{x}$-a.s.,}$$
(1.7)
and,
$$\limsup_{t\to\infty}\frac{V_{\epsilon}(t)}{t^{\alpha/\beta}(\log\log t)^{1-%
\alpha/\beta}}=c_{4},\textup{ $P^{x}$-a.s.}$$
(1.8)
We remark that $c_{i},1\leq i\leq 4$, in the above theorem do not depend on $x\in M$.
We conjecture that $c_{1}>0$.
Theorems 1.1 and 1.2 are applicable to several classes of diffusions on fractal graphs and Riemannian manifolds.
Definition 1.3 (Bounded modification).
We say that $(\mathbb{R}^{d},d,\widetilde{\mu},\widetilde{X}_{t},\widetilde{P}^{x})$ is a bounded modification of a metric measure Dirichlet space $(\mathbb{R}^{d},d,\mu,X_{t},P^{x})$
if there exists a domain $D$ of $\mathbb{R}^{d}$ such that
(M1) For every Borel subset $A$ of $\mathbb{R}^{d}\setminus D$, $\widetilde{\mu}(A)=\mu(A)$.
(M2) For every $x\in\mathbb{R}^{d}\setminus D$, the law of $\widetilde{X}_{\cdot\wedge T_{D}}$ under $\widetilde{P}^{x}$ is identical with the law of $X_{\cdot\wedge T_{D}}$ under $P^{x}$.
Theorem 1.4 (Behaviors on bounded modifications).
(i) Let $d\geq 3$.
Assume that $(\mathbb{R}^{d},d,\mu,X_{t},P^{x})$ is a bounded modification of the pair of $\mathbb{R}^{d}$, $d$, the Lebesgue measure,
and the standard Brownian motion.
We furthermore assume that HK$(2)$ holds for $(\mathbb{R}^{d},d,\mu,X_{t},P^{x})$.
Then, for every $x\in M$,
$$\lim_{t\to\infty}\frac{E^{x}[V_{\epsilon}(t)]}{t}=\textup{Cap}_{\mathbb{R}^{d}%
}\left(\overline{B}(0,\epsilon)\right).$$
Here $\textup{Cap}_{\mathbb{R}^{d}}$ is Newtonian capacity.
(ii) If $(\mathbb{R}^{2},d,\mu,X_{t},P^{x})$ is a bounded modification of the pair of $\mathbb{R}^{2}$, the Euclid distance $d$, the Lebesgue measure and the standard Brownian motion.
We furthermore assume HK$(2)$ holds for $(\mathbb{R}^{2},d,\mu,X_{t},P^{x})$.
then, for every $x\in\mathbb{R}^{2}$,
$$\lim_{t\to\infty}\frac{E^{x}[V_{\epsilon}(t)]}{t/\log t}=2\pi.$$
The estimate HK$(2)$ is stable under rough isometries between metric measure Dirichlet spaces, given suitable local regularity of the two spaces.
(In the context of Riemannian manifolds, the notion of rough isometries is introduced by Kanai [Ka85]. For the definition of rough isometries, see Barlow-Grigor’yan-Kumagai [BGK12, Definition 2.20].)
The assumption HK$(2)$ prevents “singular” behaviors of $(X_{t})_{t}$ when it enters $D$.
Theorem 1.5 (Fluctuations).
Fix $\epsilon>0$.
Then, there is a metric measure Dirichlet space $(\mathbb{R}^{d},d,\mu,\mathcal{E},\mathcal{F})$ such that the following conditions (a) and (b):
(a) $(\mathbb{R}^{d},d,\mu)$ is roughly isometric to the Euclid space equipped with the Lebesgue measure.
(b)
$$\liminf_{t\to\infty}\frac{E^{0}[V_{\epsilon}(t)]}{t}\leq\textup{Cap}_{\mathbb{%
R}^{d}}\left(\overline{B}(0,\epsilon)\right)<2^{(d-2)/2}\textup{Cap}_{\mathbb{%
R}^{d}}\left(\overline{B}(0,\epsilon)\right)\leq\limsup_{t\to\infty}\frac{E^{0%
}[V_{\epsilon}(t)]}{t}.$$
We will show this by using Theorem 1.4.
The following is a more detailed result for $(V_{\epsilon}(t))_{t}$ of bounded modifications of $\mathbb{R}^{d}$ in high dimensions.
Theorem 1.6 (Behaviors on bounded modifications).
Let $d\geq 6$.
Assume that $\mathcal{M}=(\mathbb{R}^{d},d,\mu,X_{t},P^{x})$ is a bounded modification of the pair of $\mathbb{R}^{d}$, the Euclid distance $d$, the Lebesgue measure and the standard Brownian motion.
We furthermore assume that HK$(2)$ holds and there exists a constant $C$ such that
$$\sup_{x\in\mathbb{R}^{d}}\mu(B(x,R))\leq CR^{d},\forall R>0.$$
Then, for every $x\in\mathbb{R}^{d}$,
$$\lim_{t\to\infty}E_{\mathcal{M}}^{x}[V_{\epsilon}(t)]-t\textup{Cap}_{\mathbb{R%
}^{d}}\left(\overline{B}(0,\epsilon)\right)\textup{ exists and is finite.}$$
We do not know the value of the following limit
$$\lim_{t\to\infty}E_{\mathcal{M}}^{x}[V_{\epsilon}(t)]-E_{\textup{BM}}^{x}[V_{%
\epsilon}(t)].$$
The organization of the rest of the present paper is as follows.
In Section 2, we deal with growth rates of means and almost sure behaviors and show Theorems 1.1 and 1.2.
In Section 3, we consider the behavior of process on bounded modifications and show Theorem 1.4.
In Section 4, we deal with the case that a scaled sequence of the means fluctuates and show Theorem 1.5.
In Section 5, we consider more detailed behavior of process on bounded modifications and show Theorem 1.6.
2. Growth rates of means and almost sure behaviors
We first remark that by Fubini’s theorem,
$$E^{x}[V_{\epsilon}(t)]=\int_{M}P^{x}(T_{B(y,\epsilon)}\leq t)d\mu(y).$$
(2.1)
Contrary to the case that Brownian motion on Euclid spaces, we cannot expect in general that
$$P^{x}(T_{B(y,\epsilon)}\leq t)=P^{y}(T_{B(x,\epsilon)}\leq t).$$
We give upper and lower bounds for $P^{x}(T_{B(y,\epsilon)}\leq t)$.
Lemma 2.1 ([CF86-2, (10) and (11)]).
Let $t,T,\epsilon>0$.
Let $d(x,y)>\epsilon$.
Then,
(i) (upper bound)
For every $\eta>0$,
$$\int_{0}^{t+T}P^{x}(X_{s}\in B(y,\eta))ds\geq P^{x}(T_{B(y,\epsilon)}\leq t)%
\inf_{w\in\overline{\partial}B(y,\epsilon)}\int_{0}^{T}P^{w}(X_{s}\in B(y,\eta%
))ds.$$
(ii) (lower bound)
For every $a\in(0,1)$,
$$\int_{0}^{t}P^{x}(X_{s}\in B(y,a\epsilon))ds\leq P^{x}(T_{B(y,\epsilon)}\leq t%
)\sup_{w\in\overline{\partial}B(y,\epsilon)}\int_{0}^{t}P^{w}(X_{s}\in B(y,a%
\epsilon))ds.$$
This is easily seen by using the strong Markov property.
Proof of Theorem 1.1.
By (1.3), it suffices to show the following:
Lemma 2.2.
Let $c_{1}$ and $c_{2}$ be constants in (1.4).
Then, for every $\epsilon>0$ and every $o\in M$,
(i)
$$\limsup_{t\to\infty}\frac{E^{o}[V_{\epsilon}(t)]}{t/f(t)}\leq\frac{1}{c_{1}}%
\frac{\sup_{x\in M}\mu(B(x,a\epsilon))}{\inf_{x\in M}\mu(B(x,a\epsilon))}.$$
(ii)
$$\liminf_{t\to\infty}\frac{E^{o}[V_{\epsilon}(t)]}{t/f(t)}\geq\frac{1}{c_{2}}%
\frac{\inf_{x\in M}\mu(B(x,a\epsilon))}{\sup_{x\in M}\mu(B(x,a\epsilon))}.$$
Proof.
(i) Let $\delta\in(0,1)$.
Recall (1.1).
Applying Lemma 2.1 (i) to the case that $\eta=a\epsilon$ and $T=\delta t$, we have that
$$\displaystyle E_{M}^{o}[V_{\epsilon}(t)]$$
$$\displaystyle=\int_{M}P^{o}(T_{B(y,\epsilon)}\leq t)d\mu(y)$$
$$\displaystyle\leq\int_{M}\frac{\int_{0}^{(1+\delta)t}P^{o}(X_{s}\in B(y,a%
\epsilon))ds}{\inf_{w\in\overline{\partial}B(y,\epsilon)}\int_{0}^{t}P^{w}(X_{%
s}\in B(y,a\epsilon))ds}d\mu(y)$$
$$\displaystyle\leq\frac{\int_{M}\int_{0}^{(1+\delta)t}P^{o}(X_{s}\in B(y,a%
\epsilon))dsd\mu(y)}{\inf_{y,w\in M;d(y,w)=\epsilon}\int_{0}^{t}P^{w}(X_{s}\in
B%
(y,a\epsilon))ds}$$
$$\displaystyle\leq\frac{\int_{0}^{(1+\delta)t}\mu\left(B(X_{s},a\epsilon)\right%
)ds}{\inf_{y,w\in M;d(y,w)=\epsilon}\int_{0}^{t}P^{w}(X_{s}\in B(y,a\epsilon))%
ds}.$$
$$\displaystyle\leq\frac{(1+\delta)t\sup_{z\in M}\mu(B(z,a\epsilon))}{\inf_{y,w%
\in M;d(y,w)=\epsilon}\int_{0}^{t}P^{w}(X_{s}\in B(y,a\epsilon))ds}.$$
$$\displaystyle\leq(1+\delta)\frac{\sup_{z\in M}\mu(B(z,a\epsilon))}{\inf_{z\in M%
}\mu(B(z,a\epsilon))}\sup_{w,z\in M;(1-a)\epsilon\leq d(w,z)\leq(1+a)\epsilon}%
\frac{t}{\int_{0}^{t}p(s,w,z)ds}.$$
$$\displaystyle\leq\frac{1+\delta}{c_{0}}\sup_{w,z\in M;(1-a)\epsilon\leq d(w,z)%
\leq(1+a)\epsilon}\frac{t}{\int_{0}^{t}p(s,w,z)ds}.$$
Recall (2.1).
Since $\delta$ is taken arbitrarily, it holds that
$$\displaystyle E_{M}^{o}[V_{\epsilon}(t)]\leq\frac{1}{c_{0}}\sup_{w,z\in M;(1-a%
)\epsilon\leq d(w,z)\leq(1+a)\epsilon}\frac{t}{\int_{0}^{t}p(s,w,z)ds}.$$
(2.2)
On the other hand, by (1.4),
$$\limsup_{t\to\infty}\sup_{w,z\in M;(1-a)\epsilon\leq d(w,z)\leq(1+a)\epsilon}%
\frac{f(t)}{\int_{0}^{t}p(s,w,z)ds}\leq\frac{1}{c_{1}}.$$
By this and (2.2),
we have assertion (i).
(ii) By (1.1), (2.1) and Lemma 2.1,
$$\displaystyle E^{o}[V_{\epsilon}(t)]$$
$$\displaystyle\geq\int_{M}P^{o}(T_{B(y,\epsilon)}\leq t)d\mu(y)$$
$$\displaystyle\geq\int_{M}\frac{\int_{0}^{t}P^{x}(X_{s}\in B(y,a\epsilon))ds}{%
\sup_{w\in\overline{\partial}B(y,\epsilon)}\int_{0}^{t}P^{w}(X_{s}\in B(y,a%
\epsilon))ds}d\mu(y)$$
$$\displaystyle\geq\frac{\int_{M}\int_{0}^{t}\mu\left(B(z,a\epsilon)\right)p(s,x%
,z)d\mu(z)ds}{\sup_{w,y\in M;d(w,y)=\epsilon}\int_{0}^{t}P^{w}(X_{s}\in B(y,a%
\epsilon))ds}$$
$$\displaystyle\geq\frac{t\inf_{z\in M}\mu(B(z,a\epsilon))}{\sup_{w,y\in M;d(w,y%
)=\epsilon}\int_{0}^{t}P^{w}(X_{s}\in B(y,a\epsilon))ds}$$
$$\displaystyle\geq\frac{\inf_{z\in M}\mu(B(z,a\epsilon))}{\sup_{z\in M}\mu(B(z,%
a\epsilon))}\inf_{w,z\in M;(1-a)\epsilon\leq d(w,z)\leq(1+a)\epsilon}\frac{t}{%
\int_{0}^{t}p(s,w,z)ds},$$
(2.3)
where in the fourth inequality we have used the assumption that $(\mathcal{E},\mathcal{F})$ is conservative.
By (1.4),
$$\liminf_{t\to\infty}\inf_{w,z\in M;(1-a)\epsilon\leq d(w,z)\leq(1+a)\epsilon}%
\frac{f(t)}{\int_{0}^{t}p(s,w,z)ds}\geq\frac{1}{c_{2}}.$$
By this and (2),
we have assertion (ii).
∎
∎
Let
$$X[0,t]:=\{X_{s}:s\in[0,t]\},\ t>0.$$
Let
$$V_{0+}(t):=\mu\left(X[0,t]\right)=\lim_{\epsilon\to 0}V_{\epsilon}(t).$$
This is the volume of the range of $X$.
Proposition 2.3.
Assume that $\textup{Vol}(\alpha)$ and HK$(\beta)$ hold.
Then, the following hold for each $t>0$ and every $x\in M$:
(i) If $\alpha\geq\beta$, then, $V_{0+}(t)=0$, $P^{x}$-a.s.
(ii) If $\alpha<\beta$, then, $V_{0+}(t)>0$, $P^{x}$-a.s.
Proof.
(i) Let $\epsilon>0$.
By $\textup{Vol}(\alpha)$ and HK$(\beta)$, the following holds for $w,z\in M$ satisfying that $d(w,z)\leq 3\epsilon/2$,
$$\int_{0}^{t}p(s,w,z)ds\geq\int_{0}^{t}\frac{c}{s^{\alpha/\beta}}\exp\left(-c%
\left(\frac{\epsilon^{\beta}}{s}\right)^{1/(\beta-1)}\right)ds.$$
Hence, by the monotone convergence theorem,
$$\lim_{\epsilon\to 0}\inf_{w,z\in M;d(w,z)\leq 3\epsilon/2}\int_{0}^{t}p(s,w,z)%
ds=+\infty.$$
By using this and applying (2.2) to the case that $a=1/2$,
it follows from the Lebesgue convergence theorem that for every $x\in M$,
$$E^{x}[V_{0+}(t)]=\lim_{\epsilon\to 0+}E_{M}^{x}[V_{\epsilon}(t)]=0.$$
(ii) By $\textup{Vol}(\alpha)$, HK$(\beta)$ and the assumption that $\alpha\leq\beta$,
there exists a positive constant $\lambda>0$ such that for every $x\in M$,
$$\int_{0}^{\infty}\exp(-\lambda t)p(t,x,x)dt<+\infty.$$
By Marcus-Rosen [MR92, Theorem 3.2], the local time of $X$ exists, specifically, there exists a random field $\ell(x,t)(\omega)$ such that
$\ell(x,t)(\omega)$ is jointly measurable with respect to $(t,x,\omega)$, and
for every $T>0$ and every Borel measurable function $h$ on $M$,
$$\int_{0}^{t}h(X_{s})ds=\int_{M}h(x)\ell(x,t)\mu(dx).$$
Hence,
$$0<t=\int_{0}^{t}1_{X[0,t]}(X_{s})ds=\int_{X[0,t]}\ell(x,t)\mu(dx).$$
Hence, $\mu(X[0,t])>0$.
∎
Proof of Theorem 1.2.
We first show that $(V_{\epsilon}(t))_{t}$ is a diffusion.
Lemma 2.4.
For every $x\in M$, $V_{\epsilon}(t)$ is continuous with respect to $t$, $P^{x}$-a.s.
Proof.
For every $\eta>0$,
$$V_{\epsilon}(t+\eta)-V_{\epsilon}(t-\eta)$$
$$\leq\mu\left(B\left(X_{t},\ \epsilon+\sup_{s\in[t-\eta,t+\eta]}d_{M}(X_{t},X_{%
s})\right)\setminus B\left(X_{t},\epsilon\right)\right).$$
Since $(X_{t})_{t}$ is a diffusion,
$$\limsup_{\eta\to 0}\mu\left(B\left(X_{t},\ \epsilon+\sup_{s\in[t,t+\eta]}d_{M}%
(X_{t},X_{s})\right)\setminus B\left(X_{t},\epsilon-\sup_{s\in[t-\eta,t]}d_{M}%
(X_{t},X_{s})\right)\right)$$
$$\leq\mu\left(\overline{\partial}B\left(X_{t},\epsilon\right)\right),\textup{ $%
P^{x}$-a.s.}$$
By recalling (1.2),
$$\mu\left(\overline{\partial}B\left(X_{t},\epsilon\right)\right)=0.$$
Hence,
$$\limsup_{\eta\to 0}V_{\epsilon}(t+\eta)-V_{\epsilon}(t-\eta)\leq 0.$$
By noting that $V_{\epsilon}(t)$ is non-decreasing with respect to $t$, we have the assertion.
∎
Proposition 2.5 ([KKW17, Proposition A.2]).
Let $X$ be a strong Markov process on $(M,d,\mu)$.
Assume that $(H_{t})_{t}$ is a continuous adapted non-decreasing functional of $X$ satisfying the following conditions.
(1) There exists an increasing function $\varphi$ on $(0,\infty)$ and a constant $c>1$ such that
$$\lim_{b\to\infty}\sup_{x\in M,t>0}P^{x}\left(H_{t}\geq b\varphi(t)\right)=0,$$
and,
$$\varphi(2r)\leq c\varphi(r),\ \forall r>0.$$
(2)
$$H_{t}\leq H_{s}+H_{t-s}\circ\theta_{s},\ t\geq s>0.$$
Then, there exists a constant $C\in(0,\infty)$ such that for every $x\in M$,
$$\limsup_{t\to\infty}\frac{H_{t}}{\varphi(t/\log\log t)\log\log t}\leq C,%
\textup{$P^{x}$-a.s.}$$
Proof.
The proof is same as in the proof of [BK00, Theorem 3.1].
Since we use a formula in the proof of [BK00, Theorem 3.1] below, we write down the proof.
We denote the largest integer of a real number $z$ which is less than or equal to $z$ by $[z]$.
For $t>0$ and $s\in[0,1]$, we let
$$B_{t,s}:=\frac{H_{st/[\log\log t]}}{\varphi(t/\log\log t)}.$$
By assumption (1), there exists a constant $b>0$ such that
$$\sup_{t>0,x\in M}P^{x}(B_{t,1}>b)\leq\frac{1}{2}.$$
For $n\geq 1$, we let
$$T_{n}:=\inf\left\{s\geq 0:B_{t,s}\geq bn\right\}.$$
Then by the continuity of $F_{t}$,
we have that $B_{T_{n}}=bn$ if $T_{n}<+\infty$,
$$P^{x}(B_{t,1}>b(n+1))=P^{x}(B_{t,1}-B_{t,T_{n}}>b,T_{n}<1).$$
By the strong Markov property of $X$ and the assumption that $F_{t}$ is adapted and non-decreasing,
$$P^{x}(B_{t,1}-B_{t,T_{n}}>b,T_{n}<1)\leq E^{x}\left[P^{X_{T_{n}}}(B_{t,1}>b),T%
_{n}<1\right]\leq\frac{1}{2}\sup_{y\in M}P^{y}(B_{t,1}>bn).$$
Hence, for each $n\geq 1$,
$$\sup_{t>0,x\in M}P^{x}(B_{t,1}>bn)\leq\frac{1}{2^{n}}.$$
Hence there exists a constant $a>0$ such that
$$\sup_{x\in M,t>0}E^{x}\left[\exp(aB_{t,1})\right]<+\infty.$$
By the Markov property of $X$, we have that
$$P^{x}\left(\frac{H_{t}}{\varphi(t/\log\log t)[\log\log t]}\geq\lambda\right)$$
$$=P^{x}\left(\sum_{i=1}^{[\log\log t]}\frac{H_{it/[\log\log t]}-H_{(i-1)t/[\log%
\log t]}}{[\log\log t]}\geq\varphi(t/\log\log t)\lambda\right)$$
$$\leq\left(\exp(-a\lambda)\sup_{x\in M,t>0}E^{x}[\exp(aB_{t,1})]\right)^{[\log%
\log t]}.$$
(2.4)
If we take a sufficiently large $\lambda_{0}>0$, then, there exists a constant $p>1$ such that for every $x$ and every $t$,
$$P^{x}\left(\frac{H_{t}}{\varphi(t/\log\log t)[\log\log t]}\geq\lambda_{0}%
\right)\leq\exp(-p[\log\log t]).$$
By this and the Borel-Cantelli lemma, we have that for every $x$,
$$\limsup_{k\to\infty}\frac{H_{\exp(k)}}{\varphi(\exp(k)/\log k)[\log k]}\leq%
\lambda_{0},\textup{ $P^{x}$-a.s.}$$
By this and the assumption that $H_{t}$ is non-decreasing, we have the assertion.
∎
By Lemma 2.4, $\{V_{\epsilon}(t)\}_{t}$ is a diffusion.
In this proof we let $f$ be the function given by (1.5).
Let
$$\phi(t)=\left\{\begin{array}[]{l}f(1+t)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ %
\alpha\geq\beta,\\
t^{\alpha/\beta}(\log\log t)^{1-\alpha/\beta}\ \ \ \alpha<\beta,\end{array}\right.$$
We furthermore let $\varphi(t)=f(1+t)$.
By Theorem 1.1,
we can apply Proposition 2.5, and hence it holds that
there exists a positive constant $C_{0}$ such that
$$\limsup_{t\to\infty}\frac{V_{\epsilon}(t)}{\phi(t)}\leq C_{0},\textup{ $P^{x}$%
-a.s., }$$
The zero-one law of [KKW17, Theorem 2.10] is applicable to our case,
and we have (1.6) and (1.8) for some non-negative constants $c_{2}$ and $c_{4}$ respectively.
We show that $c_{2}>0$.
Assume that $c_{2}=0$.
Then,
$$\lim_{t\to\infty}\frac{V_{\epsilon}(t)}{\varphi(t/\log\log t)\log\log t}=0,%
\textup{ $P^{x}$-a.s.}$$
(2.5)
By (2.4) and the Lebesgue convergence theorem,
we have that for sufficiently large $\lambda_{1}>0$,
$$\lim_{t\to\infty}E^{x}\left[\frac{V_{\epsilon}(t)}{\varphi(t/\log\log t)\log%
\log t},\ \frac{V_{\epsilon}(t)}{\varphi(t/\log\log t)[\log\log t]}\geq\lambda%
_{1}\right]=0.$$
(2.6)
For every $\delta\in(0,\lambda_{1})$,
$$E^{x}\left[\frac{V_{\epsilon}(t)}{\varphi(t/\log\log t)\log\log t},\ \frac{V_{%
\epsilon}(t)}{\varphi(t/\log\log t)[\log\log t]}<\lambda_{1}\right]$$
$$\leq\delta+\lambda_{1}P^{x}\left(\frac{F_{t}}{\varphi(t/\log\log t)[\log\log t%
]}\geq\delta\right)$$
By this and (2.5),
$$\limsup_{t\to\infty}E^{x}\left[\frac{V_{\epsilon}(t)}{\varphi(t/\log\log t)%
\log\log t},\ \frac{V_{\epsilon}(t)}{\varphi(t/\log\log t)[\log\log t]}<%
\lambda_{1}\right]\leq\delta.$$
Since $\delta$ can be taken arbitrarily close to $0$,
$$\limsup_{t\to\infty}E^{x}\left[\frac{V_{\epsilon}(t)}{\varphi(t/\log\log t)%
\log\log t},\ \frac{V_{\epsilon}(t)}{\varphi(t/\log\log t)[\log\log t]}<%
\lambda_{1}\right]=0.$$
By this and (2.6),
$$\lim_{t\to\infty}\frac{E^{x}\left[V_{\epsilon}(t)\right]}{\varphi(t/\log\log t%
)\log\log t}=0.$$
This contradicts Theorem 1.1.
Hence $c_{2}>0$.
We show (1.7).
By $\textup{Vol}(\alpha)$ and [KKW17, Theorem 3.8],
there exist two positive constants $C_{1}$ and $C_{2}$ such that
$$\liminf_{t\to\infty}\frac{V_{\epsilon}(t)}{t^{\alpha/\beta}(\log\log t)^{-%
\alpha/\beta}}\leq\liminf_{t\to\infty}\frac{V\left(x,\epsilon+\sup_{s\leq t}d(%
x,X_{s})\right)}{t^{\alpha/\beta}(\log\log t)^{-\alpha/\beta}}$$
$$\leq\liminf_{t\to\infty}\frac{C_{1}\epsilon^{\alpha}\left(\sup_{s\leq t}d(x,X_%
{s})\right)^{\alpha}}{t^{\alpha/\beta}(\log\log t)^{-\alpha/\beta}}\leq C_{2}<%
+\infty.\textup{ $P^{x}$-a.s.}$$
By [KKW17, Theorem 2.10], there exists a non-negative constant $c_{3}$ such that
$$\liminf_{t\to\infty}\frac{V_{\epsilon}(t)}{t^{\alpha/\beta}(\log\log t)^{-%
\alpha/\beta}}=c_{3},\textup{ $P^{x}$-a.s.}$$
By [KKW17, Theorem 4.16], there exists a positive constant $c_{3}^{\prime}$ such that
$$\liminf_{t\to\infty}\frac{\mu(X[0,t])}{t^{\alpha/\beta}(\log\log t)^{1-\alpha/%
\beta}}=c_{3}^{\prime},\textup{ $P^{x}$-a.s.}$$
Hence $c_{3}\geq c_{3}^{\prime}>0$.
Finally we show $c_{4}>0$.
$$c_{4}=\limsup_{t\to\infty}\frac{V_{\epsilon}(t)}{t^{\alpha/\beta}(\log\log t)^%
{1-\alpha/\beta}}\geq\limsup_{t\to\infty}\frac{\mu(X[0,t])}{t^{\alpha/\beta}(%
\log\log t)^{1-\alpha/\beta}},\textup{ $P^{x}$-a.s.}$$
By [KKW17, Theorem 4.16], there exists a positive constant $c_{4}^{\prime}$ such that
$$\limsup_{t\to\infty}\frac{\mu(X[0,t])}{t^{\alpha/\beta}(\log\log t)^{1-\alpha/%
\beta}}=c_{4}^{\prime},\textup{ $P^{x}$-a.s.}$$
Hence $c_{4}>0$.
Thus the proof of 1.2 is completed.
∎
Remark 2.6.
We are not sure whether Lemma 2.4 holds without (1.2).
3. Processes on bounded modifications
Let $\mathcal{M}=(\mathbb{R}^{d},d,\mu,X_{t},P^{x})$ be a bounded modification of the pair of $\mathbb{R}^{d}$, $d$, the Lebesgue measure and the standard Brownian motion.
We denote by $p_{\mathcal{M}}(t,x,y)$ the heat kernel of $\mathcal{M}=(\mathbb{R}^{d},d,\mu,X_{t},P^{x})$.
For $y\in\mathbb{R}^{d}$ and $\epsilon_{2}>\epsilon_{1}>0$, let
$$A\left(y,\epsilon_{1},\epsilon_{2}\right):=\left\{z\in\mathbb{R}^{d}:\epsilon_%
{1}\leq d(z,y)\leq\epsilon_{2}\right\}.$$
By [BGK12, Theorem 3.2],
HK$(2)$ is equivalent to the parabolic Harnack inequality (the precise definition of this inequality is long in the framework of metric measure spaces, so we omit it here. see Barlow-Bass-Kumagai [BBK06, Remark 2.2] and [BGK12]),
arguing as in the proof of [BGK12, Lemma 4.6],
for every $z_{1},z_{2}\in A\left(y,\epsilon_{1},\epsilon_{2}\right)$,
$$\left|p_{\mathcal{M}}(s,y,z_{1})-p_{\mathcal{M}}(s,y,z_{2})\right|\leq 1\wedge
C%
_{\mathcal{M}}\left(\frac{d(z_{1},z_{2})}{s}\right)^{2}.$$
(3.1)
First we consider the case that $d\geq 3$.
Let the Green function
$$G^{\mathcal{M}}(x,y):=\int_{0}^{\infty}p_{\mathcal{M}}(t,x,y)dt.$$
Proof of Theorem 1.4 (i).
By HK$(2)$,
it holds that for every $a\in(0,1)$,
$$\lim_{T\to\infty}\sup_{x,y\in\mathbb{R}^{d};(1-a)\epsilon\leq d(x,y)\leq(1+a)%
\epsilon}\left|\left(\int_{0}^{T}p_{\mathcal{M}}(t,x,y)dt\right)^{-1}-\frac{1}%
{G^{\mathcal{M}}(x,y)}\right|=0,$$
Therefore, we can show the following by modifying the proof of Lemma 2.2 a little.
Lemma 3.1.
For every $\epsilon>0$, $a\in(0,1)$, and for each integer $n\geq 1$,
(i)
$$\limsup_{t\to\infty}\frac{E^{0}[V_{\epsilon}(t)]}{t}$$
$$\leq\frac{\sup_{x\in\mathbb{R}^{d}\setminus B(0,n)}\mu(B(x,a\epsilon))}{\inf_{%
x\in\mathbb{R}^{d}\setminus B(0,n)}\mu(B(x,a\epsilon))}\sup_{y,z\in\mathbb{R}^%
{d}\setminus B(0,n),(1-a)\epsilon\leq d(y,z)\leq(1+a)\epsilon}\frac{1}{G(x,y)}.$$
(ii)
$$\liminf_{t\to\infty}\frac{E^{0}[V_{\epsilon}(t)]}{t}$$
$$\geq\frac{\inf_{x\in\mathbb{R}^{d}\setminus B(0,n)}\mu(B(x,a\epsilon))}{\sup_{%
x\in\mathbb{R}^{d}\setminus B(0,n)}\mu(B(x,a\epsilon))}\inf_{y,z\in\mathbb{R}^%
{d}\setminus B(0,n),(1-a)\epsilon\leq d(y,z)\leq(1+a)\epsilon}\frac{1}{G(y,z)}.$$
Let $x\in\mathbb{R}^{d}$.
By Lemma 3.1, for every $a\in(0,1)$,
$$\inf_{y,z\in\mathbb{R}^{d}\setminus B(0,n),(1-a)\epsilon\leq d(y,z)\leq(1+a)%
\epsilon}\frac{1}{G^{\mathcal{M}}(x,y)}\leq\liminf_{t\to\infty}\frac{E_{%
\mathcal{M}}^{0}[V_{\epsilon}(t)]}{t}$$
$$\leq\limsup_{t\to\infty}\frac{E_{\mathcal{M}}^{o}[V_{\epsilon}(t)]}{t}\leq\sup%
_{y,z\in\mathbb{R}^{d}\setminus B(0,n),(1-a)\epsilon\leq d(y,z)\leq(1+a)%
\epsilon}\frac{1}{G^{\mathcal{M}}(x,y)}.$$
(3.2)
If $d(x,y)$ is sufficiently large, then, $B(y,d(x,y)/2)\cap D=\emptyset$.
Since (3.2) holds for every $a\in(0,1)$,
it suffices to show that for every $x\in\mathbb{R}^{d}$,
$$\lim_{y;d(x,y)\to\infty}\sup_{z\in A\left(y,\epsilon_{1},\epsilon_{2}\right)}%
\left|G^{\mathcal{M}}(y,z)-G^{\textup{BM}}(y,z)\right|=0.$$
(3.3)
Let $\mu_{\textup{BM}}$ be the Lebesgue measure on $\mathbb{R}^{d}$.
Fix $y\in\mathbb{R}^{d}$.
Let $u$ be a non-negative bounded continuous function supported on $A\left(y,\epsilon_{1},\epsilon_{2}\right)$.
By Definition 1.3,
$$\int_{\mathbb{R}^{d}}u(z)G^{\mathcal{M}}(y,z)\mu_{\mathcal{M}}(dz)-\int_{%
\mathbb{R}^{d}}u(z)G^{\textup{BM}}(y,z)\mu_{\textup{BM}}(dz)$$
$$=\int_{0}^{\infty}E^{y}_{\mathcal{M}}\left[u(X^{\mathcal{M}}_{t})\right]-E^{y}%
_{\textup{BM}}\left[u(X^{\mathbb{R}^{d}}_{t})\right]dt$$
$$=\int_{0}^{\infty}E^{y}_{\mathcal{M}}\left[u(X^{\mathcal{M}}_{t})-u\left(X^{%
\mathcal{M}}_{t\wedge\tau_{B(y,d(x,y)/2)}}\right)\right]-E^{y}_{\textup{BM}}%
\left[u(X^{\mathbb{R}^{d}}_{t})-u\left(X^{\mathbb{R}^{d}}_{t\wedge\tau_{B(y,d(%
x,y)/2)}}\right)\right]dt$$
Using $\textup{supp}(u)\subset A\left(y,\epsilon_{1},\epsilon_{2}\right)$ and HK$(2)$,
$$\left|\int_{0}^{\infty}E^{y}_{\mathcal{M}}\left[u\left(X^{\mathcal{M}}_{t}%
\right)-u\left(X^{\mathcal{M}}_{t\wedge\tau_{B(y,d(x,y)/2)}}\right)\right]dt%
\right|=\left|\int_{0}^{\infty}E^{y}_{\mathcal{M}}\left[u\left(X^{\mathcal{M}}%
_{t}\right),\ t\geq\tau_{B(y,d(x,y)/2)}\right]dt\right|$$
$$\leq\|u\|_{\infty}\int_{0}^{\infty}P_{\mathcal{M}}^{y}\left(X^{\mathcal{M}}_{t%
}\in A\left(y,\epsilon_{1},\epsilon_{2}\right),\,t\geq\tau_{B(y,d(x,y)/2)}%
\right)dt$$
$$\leq\|u\|_{\infty}\left(t_{0}P_{\mathcal{M}}^{y}\left(t_{0}\geq\tau_{B(y,d(x,y%
)/2)}\right)+C_{\mathcal{M}}t_{0}^{1-d/2}\right)$$
$$=\|u\|_{\infty}\left(t_{0}P_{\mathbb{R}^{d}}^{0}\left(t_{0}\geq\tau_{B(0,d(x,y%
)/2)}\right)+C_{\mathcal{M}}t_{0}^{1-d/2}\right)$$
Since we have the almost same estimate also for $\mathbb{R}^{d}$,
$$\lim_{y;d(x,y)\to\infty}\sup_{u}\left|\int_{\mathbb{R}^{d}}u(z)G^{\mathcal{M}}%
(y,z)\mu_{\mathcal{M}}(dz)-\int_{\mathbb{R}^{d}}u(z)G^{\textup{BM}}(y,z)\mu_{%
\textup{BM}}(dz)\right|=0.$$
where the supremum runs on the space of non-negative bounded continuous functions supported on $A\left(y,\epsilon_{1},\epsilon_{2}\right)$ whose supremum norm is smaller than or equal to one.
By (3.1),
$$\left|G^{\mathcal{M}}(y,z_{1})-G^{\mathcal{M}}(y,z_{2})\right|\leq C_{\mathcal%
{M},\epsilon_{1},\epsilon_{2}}d(z_{1},z_{2})^{\delta_{\mathcal{M},\epsilon_{1}%
,\epsilon_{2}}}$$
(3.4)
holds for $z_{1},z_{2}\in A\left(y,\epsilon_{1},\epsilon_{2}\right)$.
We have the almost same estimate for $\mathbb{R}^{d}$.
Recall that if $B(x,r)\cap D=\emptyset$, then, $\mu(B(x,r))\asymp r^{d}$.
If (3.3) fails, then, by (3.4),
we can construct an $L^{\infty}$-bounded function $u$ which is supported on $A\left(y,\epsilon_{1},\epsilon_{2}\right)$ and
sstisfies that
$$\limsup_{y;d(x,y)\to\infty}\left|\int_{\mathbb{R}^{d}}u(z)G^{\mathcal{M}}(y,z)%
\mu_{\mathcal{M}}(dz)-\int_{\mathbb{R}^{d}}u(z)G^{\textup{BM}}(y,z)\mu_{%
\textup{BM}}(dz)\right|>0.$$
Now (3.3) follows.
∎
Second we consider the case that $d=2$.
For $x\in\mathbb{R}^{2}$ and $A\subset\mathbb{R}^{2}$,
$$d(x,A):=\inf_{y\in A}d(x,y).$$
Proof of Theorem 1.4 (ii).
It follows from Lemma 2.1 that for $E\subset\mathbb{R}^{2}$ and $T,t>0$,
$$\int_{E}P^{x}(T_{B(y,\epsilon)}\leq t)\mu(dy)\leq\int_{E}\frac{\int_{0}^{t+T}P%
^{x}\left(X_{s}\in B(y,a\epsilon)\right)ds}{\inf_{w\in\overline{\partial}B(y,%
\epsilon)}\int_{0}^{T}P^{w}(X_{s}\in B(y,a\epsilon))ds}\mu(dy)$$
$$\leq\frac{\int_{E}\int_{0}^{t+T}P^{x}\left(X_{s}\in B(y,a\epsilon)\right)ds\mu%
(dy)}{\inf_{y\in E}\mu(B(y,a\epsilon))\inf_{z,w\in\mathbb{R}^{2};(1-a)\epsilon%
\leq d(z,w)\leq(1+a)\epsilon}\int_{0}^{T}p(s,w,z)ds}$$
$$\leq\frac{\sup_{y\in\mathbb{R}^{2}}\mu\left(B(y,a\epsilon)\cap E\right)(t+T)}{%
\inf_{y\in E}\mu(B(y,a\epsilon))\inf_{z,w\in\mathbb{R}^{2};d(w,E)\leq\epsilon,%
(1-a)\epsilon\leq d(z,w)\leq(1+a)\epsilon}\int_{0}^{T}p(s,w,z)ds},$$
(3.5)
and,
$$\int_{E}P^{x}(T_{B(y,\epsilon)}\leq t)\mu(dy)$$
$$\geq\frac{\inf_{d(z,\mathbb{R}^{2}\setminus E)\geq\epsilon}\mu(B(z,a\epsilon))%
\int_{0}^{t}\int_{d(z,\mathbb{R}^{2}\setminus E)\geq\epsilon}p(s,x,z)\mu(dz)ds%
}{\sup_{y\in E}\mu(B(y,a\epsilon))\sup_{d(w,E)\leq\epsilon,(1-a)\epsilon\leq d%
(z,w)\leq(1+a)\epsilon}\int_{0}^{t}p(s,w,z)ds}.$$
(3.6)
Let $x\in\mathbb{R}^{2}$.
Let $m$ be a large integer.
Let
$$F_{m}(t):=\frac{t^{m/(m+1)}}{(\log t)^{3}}.$$
Let
$$S_{x}\left(t,\epsilon_{1},\epsilon_{2}\right):=\left\{(y,z)\in(\mathbb{R}^{d})%
^{2}:d(x,y)\geq t^{1/2}/\log t,z\in A\left(y,\epsilon_{1},\epsilon_{2}\right)%
\right\}.$$
We show that
Lemma 3.2.
$$\lim_{t\to\infty}\sup_{(y,z)\in S_{x}\left(t,\epsilon_{1},\epsilon_{2}\right)}%
\left|\left(\int_{0}^{F_{m}(t)}p_{\mathcal{M}}(s,y,z)ds\right)^{-1}-\left(\int%
_{0}^{F_{m}(t)}p_{\textup{BM}}(s,y,z)ds\right)^{-1}\right|\log t=0.$$
Proof.
Thanks to HK$(2)$, it suffices to show that
$$\lim_{t\to\infty}\sup_{(y,z)\in S_{x}(t,\epsilon_{1},\epsilon_{2})}\left|\int_%
{0}^{F_{m}(t)}p_{\mathcal{M}}(s,y,z)-p_{\textup{BM}}(s,y,z)ds\right|\frac{1}{%
\log t}=0.$$
(3.7)
Fix $y\in\mathbb{R}^{2}$.
Let $u$ be a non-negative bounded continuous function supported on $A\left(y,\epsilon_{1},\epsilon_{2}\right)$.
Since $A\left(y,\epsilon_{1},\epsilon_{2}\right)\cap D=\emptyset$,
$u$ is well-defined also on $\mathbb{R}^{2}$.
In the same manner as in the case that $d\geq 3$,
by using the Burkholder-Davis-Gundy inequality for degree $m\geq 1$,
$$\left|\int_{0}^{F_{m}(t)}\left(\int_{\mathbb{R}^{2}}u(z)p_{\mathcal{M}}(s,y,z)%
\mu_{\mathcal{M}}(dz)-\int_{\mathbb{R}^{2}}u(z)p_{\textup{BM}}(s,y,z)\mu_{%
\textup{BM}}(dz)\right)ds\right|$$
$$\leq\|u\|_{\infty}\int_{0}^{F_{m}(t)}P_{\textup{BM}}^{0}\left(s\geq T_{\mathbb%
{R}^{2}\setminus B(0,t^{1/2}/\log t)}\right)ds$$
$$\leq 2\|u\|_{\infty}\int_{0}^{F_{m}(t)}P_{\textup{BM}}^{0}\left(\max_{0\leq s^%
{\prime}\leq s}\left|X_{s^{\prime}}\right|\geq t^{1/2}/2\log t\right)ds$$
$$\leq C_{m}\int_{0}^{F_{m}(t)}s^{m}ds\frac{1}{t^{m}/(\log t)^{2m}}\leq\frac{C_{%
m}\|u\|_{\infty}}{(\log t)^{m+3}}.$$
In the third display, $(X_{t})_{t}$ denotes the one-dimensional Brownian motion and $P_{\textup{BM}}^{0}$ denotes the law of the one-dimensional Brownian motion starting at the origin.
By (3.1),
$$\sup_{z_{1},z_{2}\in A\left(y,\epsilon_{1},\epsilon_{2}\right)}\int_{0}^{F_{m}%
(t)}\left|p_{\mathcal{M}}(s,y,z_{1})-p_{\mathcal{M}}(s,y,z_{2})\right|ds\leq C%
_{\mathcal{M}}\epsilon_{2}.$$
(3.8)
Moreover, by HK$(2)$,
$$\int_{F_{m}(t)}^{t}p_{\mathcal{M}}(s,y,z)ds\leq C_{\mathcal{M}}\frac{\log t}{m%
+1}.$$
(3.9)
The same estimates hold for the standard Brownian motion, that is, .
$$\int_{F_{m}(t)}^{t}p_{\textup{BM}}(s,y,z)ds\leq C_{\textup{BM}}\frac{\log t}{m%
+1}.$$
By using (3.8) and (3.9),
it holds that
$$\limsup_{t\to\infty}\sup_{(y,z)\in S_{x}(t,\epsilon_{1},\epsilon_{2})}\left|%
\int_{0}^{t}p_{\mathcal{M}}(s,y,z)-p_{\textup{BM}}(s,y,z)ds\right|\frac{1}{%
\log t}\leq\frac{C}{m+1}$$
holds for every $m$, and hence, (3.7) follows.
∎
Now we show the upper bound.
We remark that
$$\lim_{t\to\infty}\frac{\log t}{t}\mu_{\mathcal{M}}\left(B(x,t^{1/2}/\log t)%
\right)=0.$$
By applying Lemma 3.2 and (3.5) to the case that $E=\mathbb{R}^{d}\setminus B(x,t^{1/2}/\log t)$ and $T=F_{m}(t)$,
it holds that for every $m$,
$$\limsup_{t\to\infty}\frac{\log t}{t}E_{\mathcal{M}}^{x}\left[V_{\epsilon}(t)\right]$$
$$\leq\limsup_{t\to\infty}\frac{\log t}{t}\frac{t+F_{m}(t)}{\inf_{(y,z)\in S_{x}%
(t,\epsilon_{1},\epsilon_{2})}\int_{0}^{F_{m}(t)}p_{\mathcal{M}}(s,y,z)ds}$$
$$\leq\limsup_{t\to\infty}\frac{\log t}{t}\frac{t+F_{m}(t)}{\inf_{\epsilon_{1}%
\leq d(y,z)\leq\epsilon_{2}}\int_{0}^{F_{m}(t)}p_{\textup{BM}}(s,y,z)ds}\leq%
\frac{m+1}{m}2\pi.$$
Since $m$ is taken arbitrarily,
$$\limsup_{t\to\infty}\frac{\log t}{t}E_{\mathcal{M}}^{x}\left[V_{\epsilon}(t)%
\right]\leq 2\pi.$$
Now we show the lower bound.
By HK$(2)$ and conservativeness, it is easy to see that
$$\lim_{t\to\infty}\frac{1}{t}\int_{0}^{t}\int_{\mathbb{R}^{2}\setminus B(x,t^{1%
/2}/\log t)}p_{\mathcal{M}}(s,x,z)\mu(dz)ds=1.$$
By applying this, Lemma 3.2 and (3.6) to the case that $E=\mathbb{R}^{2}\setminus B(x,t^{1/2}/\log t)$,
$$\liminf_{t\to\infty}\frac{\log t}{t}E_{\mathcal{M}}^{x}\left[V_{\epsilon}(t)\right]$$
$$\geq\frac{\liminf_{t\to\infty}\int_{0}^{t}\int_{\mathbb{R}^{2}\setminus B(x,t^%
{1/2}/\log t)}p_{\mathcal{M}}(s,x,z)\mu(dz)ds}{\sup_{(w,z)\in S_{x}(t,\epsilon%
_{1},\epsilon_{2})}\int_{0}^{t}p_{\mathcal{M}}(s,w,z)ds}=2\pi.$$
Thus we have the assertion.
∎
Remark 3.3.
If $d\geq 3$,
$G(x,y)$ depends on the value of $d(x,y)$.
If $d=2$, then, for every $x,y\in\mathbb{R}^{2}$,
$$p(t,x,y)\sim 2\pi\log t,\ t\to\infty.$$
In this sense, in the case that $d=2$, it is not important how to take $\epsilon_{1}$ and $\epsilon_{2}$.
4. Fluctuation results
Before we state the proof, we prepare notation.
Definition 4.1 (Diffusion on Riemannian manifolds).
Let $\mathcal{M}=(M,g)$ be a connected Riemannian manifold and $\mu_{\mathcal{M}}$ be the Riemannian volume.
Let $\nabla_{\mathcal{M}}f$ be the weak gradient of $f$.
Let
$$W^{1}:=\left\{f\in L^{2}(M,\mu_{\mathcal{M}}):\nabla_{\mathcal{M}}f\in L^{2}(M%
,\mu_{\mathcal{M}})\right\}.$$
For $f,g\in W^{1}$, let
$$\mathcal{E}_{\mathcal{M}}(f,g):=\int_{M}\left(\nabla_{\mathcal{M}}f,\nabla_{%
\mathcal{M}}g\right)d\mu_{\mathcal{M}}.$$
Here $(,)$ is the canonical inner product on $\mathbb{R}^{d}$.
Let $\mathcal{F}_{\mathcal{M}}$ be the closure of $C^{\infty}(M)\textup{ under }W^{1}$.
Then, $(\mathcal{E}_{\mathcal{M}},\mathcal{F}_{\mathcal{M}})$ is a strongly-local regular Dirichlet form in $L^{2}(M,\mu_{\mathcal{M}})$.
Let $\left((X^{\mathcal{M}}_{t})_{t\geq 0},(P_{\mathcal{M}}^{x})_{x\in M}\right)$ be an associated diffusion.
If we consider a Riemannian manifold $M$, then we may simply write $\mathcal{M}=(M,g,\mathcal{E}_{\mathcal{M}},\mathcal{F}_{\mathcal{M}})$.
Let $p_{\mathcal{M}}(t,x,y)$ be the minimal positive fundamental solution of the heat equation on $M$.
Then, for every $h\in L^{2}(M,\mu_{\mathcal{M}})$, every $t\geq 0$ and every $x\in M$,
$$\int_{M}h(y)p_{\mathcal{M}}(t,x,y)\mu_{\mathcal{M}}(dy)=E_{\mathcal{M}}^{x}%
\left[h(X^{\mathcal{M}}_{t})\right].$$
In the following, we deal with the case that $M=\mathbb{R}^{d}$ but $g$ is not the Euclid metric.
However we consider a class of metric measure spaces which consist of the Euclid space $\mathbb{R}^{d}$, the Euclid distance $d$, the Riemannian volume $\mu_{\mathcal{M}}$, and $(\mathcal{E}_{\mathcal{M}},\mathcal{F}_{\mathcal{M}})$ for a Riemannian manifold $\mathcal{M}$.
We emphasize that we consider the Euclid distance instead of the Riemannian distance $d_{\mathcal{M}}$ defined by the Riemannian metric of $\mathcal{M}=(\mathbb{R}^{d},g)$.
Informally speaking we do not consider “singular” $\mathcal{M}$ in this section.
We consider only the case that the Riemannian distance $d_{\mathcal{M}}$ of $\mathcal{M}$ is equivalent to the Euclid distance, and hence the topologies of $\mathbb{R}^{d}$ induced by these two distances are identical with each other.
In this case $(\mathcal{E}_{\mathcal{M}},\mathcal{F}_{\mathcal{M}})$ and an associated diffusion $\left((X^{\mathcal{M}}_{t})_{t\geq 0},(P_{\mathcal{M}}^{x})_{x\in M}\right)$ are well-defined.
We can consider a metric measure Dirichlet space $\left(\mathbb{R}^{d},d,\mu_{\mathcal{M}},\mathcal{E}_{\mathcal{M}},\mathcal{F}%
_{\mathcal{M}}\right)$ and an associated diffusion of it.
Proof of Theorem 1.5.
Definition 4.2.
(i) For an infinite sequence
$0=R_{0}<R_{1}<R_{2}<R_{3}<\cdots$
and for $k\geq 0$, we let
$A_{k}:=[R_{2k},R_{2k+1}-1)$,
$C_{k,1}:=[R_{2k+1}-1,R_{2k+1})$,
$B_{k}:=[R_{2k+1},R_{2k+2}-1)$, and
$C_{k,2}:=[R_{2k+2}-1,R_{2k+2})$.
(ii) Let $k\geq 0$.
Let $G_{2k}$ be a smooth non-negative function on $[0,+\infty)$
such that
(a) $G_{2k}=1$ on $[R_{2k},\infty)$
(b) for every $j<k$,
$G_{2k}=1$ on $A_{j}$, $G_{2k}=4$ on $B_{j}$, and $1\leq G_{2k}\leq 4$ on $C_{j,1}\cup C_{j,2}$.
Let $G_{2k+1}$ be a smooth non-negative function on $[0,+\infty)$
such that
(a) $G_{2k+1}=4$ on $[R_{2k+1},\infty)$
(b) for every $j<k$,
$G_{2k+1}=1$ on $A_{j}$, $G_{2k+1}=4$ on $B_{j}$, and $1\leq G_{2k+1}\leq 4$ on $C_{j,1}\cup C_{j,2}$.
Moreover,
$G_{2k+1}=1$ on $A_{k}$, and $1\leq G_{2k+1}\leq 4$ on $C_{k,1}$.
Then we have
$$G_{\infty}:=\lim_{k\to\infty}G_{k}$$
exists, and
is a smooth non-negative function on $[0,+\infty)$ such that for every $j\geq 0$,
$G_{\infty}=1$ on $A_{j}$, $G_{\infty}=4$ on $B_{j}$, and $1\leq G_{\infty}\leq 4$ on $C_{j,1}\cup C_{j,2}$.
We now define a sequence of radially-symmetric Riemannian manifolds.
Definition 4.3.
(i) Let a family of Riemannian manifolds $\mathcal{M}_{k}:=(\mathbb{R}^{d},g_{k})$, $0\leq k\leq\infty$, as follows:
$$g_{k}(x):=G_{k}(d(0,x))I_{d}.$$
Here $I_{d}$ is the $d\times d$ identity matrix.
(ii) Let $\mathcal{A}=(\mathbb{R}^{d},g_{A})$ and $\mathcal{B}=(\mathbb{R}^{d},g_{B})$ be Riemannian manifolds such that their Riemannian metrics are given by $g_{A}\equiv I_{d}$ and $g_{B}\equiv 4I_{d}$ on each point of $\mathbb{R}^{d}$.
We remark that $\mathcal{M}_{0}$ is the Euclid space.
$\mathcal{M}_{k}$ depends only on the choice of the sequence $R_{1}<\cdots<R_{k}$.
For $x\in\mathbb{R}^{d}$, $t,\epsilon>0$,
let $B^{\mathcal{M}_{j}}_{\textup{Riem}}(x,\epsilon)$ be open balls with center $x$ and radius $\epsilon$ with respect to the Riemannian distance of $\mathcal{M}_{j}$.
We have that for every $0\leq j\leq+\infty$,
$$d(x,y)\leq d^{\mathcal{M}_{j}}_{\textup{Riem}}(x,y)\leq 2d(x,y).$$
(4.1)
This does not depend on the choices of $\{R_{k}\}_{k}$.
By (4.1), each $\mathcal{M}_{j}$ is complete with respect to $d_{\textup{Riem}}$.
By the Hopf-Rinow theorem, each $\mathcal{M}_{j}$ is geodesically complete.
Since it holds that $1\leq G_{k}\leq 4$ for every $k$,
we can show that
there exists a positive constant $C_{d}$ such that for every $r>0$, $x\in\mathbb{R}^{d}$ and every $j\geq 0$,
$$\mu_{\mathcal{M}_{j}}\left(B^{\mathcal{M}_{j}}_{\textup{Riem}}(x,r)\right)\leq%
\mu_{\mathcal{M}_{j}}\left(B(x,2r)\right)\leq C_{d}(4r)^{d}.$$
By this and Grigorýan [G99, Theorem 9.1],
we have the stochastic completeness, that is,
for each $t>0$, every $x\in\mathbb{R}^{d}$ and every $j\geq 0$,
$$\int_{\mathbb{R}^{d}}p_{\mathcal{M}_{j}}(t,x,y)\mu_{\mathcal{M}_{j}}(dy)=1.$$
Recall Definition 1.3.
Lemma 4.4.
If $k$ is odd, then, a metric measure space $(\mathbb{R}^{d},d,\mu_{\mathcal{M}_{k}},\mathcal{E}_{\mathcal{M}_{k}},\mathcal%
{F}_{\mathcal{M}_{k}})$ is a bounded modification of a metric measure space $(\mathbb{R}^{d},d,\mu_{A},\mathcal{E}_{A},\mathcal{F}_{A})$.
If $k$ is even, then, a metric measure space $(\mathbb{R}^{d},d,\mu_{\mathcal{M}_{k}},\mathcal{E}_{\mathcal{M}_{k}},\mathcal%
{F}_{\mathcal{M}_{k}})$ is a bounded modification of a metric measure space $(\mathbb{R}^{d},d,\mu_{B},\mathcal{E}_{B},\mathcal{F}_{B})$.
Proof.
We give a proof in the case that $k$ is odd.
(M1) is obvious.
(M2) follows from the definitions of the Dirichlet forms of $\mathcal{M}_{2k+1}$ and $\mathcal{A}$, and Shigekawa-Taniguchi [ST92, Lemma 6.3].
In the case that $k$ is even, we can show the assertion in the same manner.
∎
Lemma 4.5.
For each $k$, $(\mathbb{R}^{d},d,\mu_{\mathcal{M}_{k}},\mathcal{E}_{\mathcal{M}_{k}},\mathcal%
{F}_{\mathcal{M}_{k}})$ satisfies HK$(2)$.
Proof.
By the definition of Riemannian metric $g$ on $\mathcal{M}_{j}$, $|g|$, $|\partial_{i}g|$ and $|\partial^{2}_{ik}g|$, $1\leq i,k\leq d$, are uniformly bounded on $\mathcal{M}_{j}$.
Therefore, the Ricci curvature is bounded below.
Therefore, by the Li-Yau estimates ([LY86]),
$\mathcal{M}_{k}$ satisfies a local version of the Parabolic Harnack inequality with parameter $2$.
(See Barlow [B04] for the definition of the Parabolic Harnack inequality with parameter $2$.)
By the definition, as a Riemannian manifold, $\mathcal{M}_{k}$ is rough-isometric to $\mathbb{R}^{d}$ with the Euclid metric $d$.
By using Hebisch and Saloff-Coste [HSC01, Theorem 2.7] and [B04, Theorem 5.4], $(\mathcal{M}_{k},\mathcal{E})$ also satisfies the Parabolic Harnack inequality with parameter $2$, which is equivalent to HK$(2)$.
Now we replace the Riemannian metric with the Euclid metric $d$ and use (4.1).
∎
Lemma 4.6 (convergence results).
For $\mathcal{M}=\mathcal{A}\textup{ or }\mathcal{B}$, let
$$c(\mathcal{M},\epsilon):=\inf_{x,y\in\mathbb{R}^{d},d(x,y)\leq\epsilon}\frac{1%
}{G^{\mathcal{M}}(x,y)}=\inf_{x,y\in\mathbb{R}^{d},d(x,y)\leq\epsilon}\frac{1}%
{\int_{0}^{\infty}p_{\mathcal{M}}(t,x,y)dt}.$$
Then we have
(i)
$$\lim_{n\to\infty}\frac{E^{0}_{\mathcal{M}_{2k-1}}[V_{\epsilon}(t)]}{t}=c(%
\mathcal{A},\epsilon).$$
(ii)
$$\lim_{n\to\infty}\frac{E^{0}_{\mathcal{M}_{2k}}[V_{\epsilon}(t)]}{t}=c(%
\mathcal{B},\epsilon).$$
These convergences do not depend on choices of $\{R_{k}\}_{k}$.
Proof.
(i) follows from Lemmas 4.4, 4.5, and Theorem 1.4.
Now we show (ii).
Let $\widetilde{\mathcal{B}}=(\mathbb{R}^{d},d_{\textup{Riem}}^{\mathcal{B}},\mu_{B%
},\mathcal{E}_{B},\mathcal{F}_{B})$ and
let $(X_{t},P^{x})$ be a diffusion associated with $\widetilde{\mathcal{B}}$.
Then the law is identical with the standard Brownian motion.
Hence,
$$\lim_{n\to\infty}\frac{E^{0}_{\widetilde{\mathcal{B}}}[V_{\epsilon}(t)]}{t}=c(%
\mathcal{A},\epsilon).$$
Let $\widetilde{\mathcal{M}_{2k}}=(\mathbb{R}^{d},d_{\textup{Riem}}^{\mathcal{B}},%
\mu_{\mathcal{M}_{k}},\mathcal{E}_{\mathcal{M}_{k}},\mathcal{F}_{\mathcal{M}_{%
k}})$.
Then, $\widetilde{\mathcal{M}_{2k}}$ is a bounded modification of $\widetilde{\mathcal{B}}$.
By Theorem 1.4,
$$\lim_{n\to\infty}\frac{E^{0}_{\widetilde{\mathcal{M}}_{2k}}[V_{\epsilon}(t)]}{%
t}=c(\mathcal{A},\epsilon).$$
Since
$$E^{0}_{\widetilde{\mathcal{M}_{2k}}}[V_{2\epsilon}(t)]=E^{0}_{\mathcal{M}_{2k}%
}[V_{\epsilon}(t)],$$
it holds that
$$\lim_{n\to\infty}\frac{E^{0}_{\mathcal{M}_{2k}}[V_{\epsilon}(t)]}{t}=c(%
\mathcal{A},2\epsilon)=2^{(d-2)/2}c(\mathcal{A},\epsilon).$$
(4.2)
Since the two Riemannian manifolds $\mathcal{A}$ and $\mathcal{B}$ are obtained by multiplying positive constants to the Euclid metrics, and $X^{\mathcal{A}}$ and $X^{\mathcal{B}}$ are Brownian motion on $\mathcal{A}$ and $\mathcal{B}$ respectively,
$$p_{\mathcal{M}}(t,x,y)=\left(\frac{1}{2\pi t}\right)^{d/2}\exp\left(-\frac{d^{%
\mathcal{M}}_{\textup{Riem}}(x,y)}{2t}\right),$$
for $t>0$ and $x,y\in\mathbb{R}^{d}$, $\mathcal{M}=\mathcal{A}\textup{ or }\mathcal{B}$.
Since
$$d^{\mathcal{A}}_{\textup{Riem}}(x,y)=d(x,y)=\frac{d^{\mathcal{B}}_{\textup{%
Riem}}(x,y)}{2},$$
we have that
$$p_{\mathcal{A}}(t,x,y)=\left(\frac{1}{2\pi t}\right)^{d/2}\exp\left(-\frac{d(x%
,y)}{2t}\right)$$
and
$$p_{\mathcal{B}}(t,x,y)=\left(\frac{1}{2\pi t}\right)^{d/2}\exp\left(-\frac{d(x%
,y)}{t}\right)$$
By integrating these quantities with respect to $t$, we have that
$$G^{\mathcal{A}}(x,y)=2^{(d-2)/2}G^{\mathcal{B}}(x,y).$$
Thus we have
$$2^{(d-2)/2}c(\mathcal{A},\epsilon)=c(\mathcal{B},\epsilon).$$
This and (4.2) complete the proof of assertion (ii).
∎
We also have that
Lemma 4.7 (Uniform upper Gaussian heat kernel estimates).
There exist two constants $c_{1}$ and $c_{2}$ such that for every $0\leq j\leq+\infty$ and $x,y\in\mathbb{R}^{d}$,
$$p_{\mathcal{M}_{j}}(t,x,y)\leq\frac{c_{1}}{t^{d/2}}\exp\left(-c_{2}\frac{d^{%
\mathcal{M}_{j}}_{\textup{Riem}}(x,y)^{2}}{t}\right).$$
Proof.
By the definition of $\mathcal{M}_{j}$,
there is a general constant $c$ such that for every $0\leq j\leq+\infty$,
$$\left(\int_{\mathbb{R}^{d}}|f|^{2}d\mu_{\mathcal{M}_{j}}\right)^{1+2/d}\leq c%
\int_{\mathbb{R}^{d}}|\nabla f|^{2}d\mu_{\mathcal{M}_{j}}\left(\int_{\mathbb{R%
}^{d}}|f|d\mu_{\mathcal{M}_{j}}\right)^{4/d}.$$
By the Carlen-Kusuoka-Stroock [CKS87],
there is a general constant $c$ such that for every $j$ and $x\in\mathbb{R}^{d}$,
$$\sup_{t>0}t^{d/2}p_{\mathcal{M}_{j}}(t,x,x)\leq c.$$
The assertion now follows from this and Grigorýan [G97, Theorems 3.1 and 3.2].
∎
Lemma 4.8.
$$\lim_{k\to\infty}\sup_{j\geq k}\int_{\mathbb{R}^{d}\setminus B\left(0,\epsilon%
+R_{\mathcal{M}_{k}}(t)\right)}P^{0}_{\mathcal{M}_{j}}\left(T_{B(x,\epsilon)}%
\leq t\right)d\mu_{\mathcal{M}_{j}}(x)=0.$$
Proof.
By Lemma 4.7 and (4.1),
$$\displaystyle p_{\mathcal{M}_{j}}(t,x,y)$$
$$\displaystyle\leq\frac{c}{t^{d/2}}\exp\left(-\frac{d^{\mathcal{M}_{j}}_{%
\textup{Riem}}(x,y)}{2t}\right)$$
$$\displaystyle\leq\frac{c}{t^{d/2}}\exp\left(-\frac{d(x,y)}{2t}\right).$$
By this and Lemma 2.1,
it holds that if we take $R_{\mathcal{M}_{k}}(t)$ sufficiently large, then,
$$\sup_{j\geq k}\int_{\mathbb{R}^{d}\setminus B(0,\epsilon+R_{\mathcal{M}_{k}}(t%
))}P^{0}_{\mathcal{M}_{j}}\left(T_{B(x,\epsilon)}\leq t\right)d\mu_{\mathcal{M%
}_{j}}(x)\leq 2^{-k}.$$
This completes the proof.
∎
Lemma 4.9.
We have that for each $t>0$,
$$\lim_{R\to\infty}R^{d}\sup_{0\leq j\leq+\infty}P^{0}_{\mathcal{M}_{j}}\left(%
\tau_{B(0,R)}\leq t\right)=0.$$
Proof.
In this proof, we regard each $\mathcal{M}_{j}$ as a Riemannian manifold.
Let
$$K_{R}:=\overline{B}(0,R+1)\setminus B(0,R)$$
$$\subset O_{R}:=B(0,R+d)\setminus\overline{B}(0,R-d).$$
We follow Grigorýan and Saloff-Coste [GSC01, (1.9)] for the definition of $\textup{Cap}_{\mathcal{M}_{j}}(K_{R},O_{R})$, specifically,
$$\textup{Cap}_{\mathcal{M}_{j}}(K_{R},O_{R}):=\inf\left\{\int_{O_{R}}\left|%
\nabla\phi\right|d\mu_{\mathcal{M}_{j}}:\phi=1\textup{ on }K_{R},\ \phi\textup%
{ has a compact support in }O_{R}\right\}.$$
Here the gradient $\nabla\phi$ is taken with respect to the Riemannian metric on $\mathcal{M}_{j}$, but it differs from the corresponding gradient taken with respect to the Euclid metric on $\mathbb{R}^{d}$ only by positive constants.
Since
$$\textup{Cap}_{\mathcal{M}_{j}}(K_{R},O_{R})\leq C_{d,1}\mu_{\mathcal{M}_{j}}(O%
_{R})d^{\mathcal{M}_{j}}_{\text{Riem}}\left(K_{R},\mathcal{M}\setminus O_{R}%
\right)^{2},$$
we have that
$$\sup_{0\leq j\leq+\infty}\textup{Cap}_{\mathcal{M}_{j}}(K_{R},O_{R})\leq C_{d,%
2}R^{d}.$$
Here $C_{d,i},i=1,2$, are constants depending only on $d$, and does not depend on $0\leq j\leq+\infty$.
Since $K_{R}$ and $O_{R}$ are annuli defined with respect to the Euclid distance,
we can show that
$$\textup{Cap}_{\mathcal{M}_{j}}(K_{R},O_{R})>0$$
in the same manner as in the proof of
$$\textup{Cap}(K_{R},O_{R})>0,$$
where the capacity is defined on the Euclid space equipped with the Euclid distance and the Lebesgue measure.
Therefore we can apply [GSC01, Theorem 3.7] to this case and it holds that
$$P^{0}_{\mathcal{M}_{j}}\left(\tau_{B(0,R)}\leq t\right)\leq\textup{Cap}_{%
\mathcal{M}_{j}}(K_{R},O_{R})\int_{0}^{t}\sup_{y\in O_{R}\setminus K_{R}}p_{%
\mathcal{M}_{j},B(0,R)}(s,0,y)ds.$$
Now it suffices to give a uniform upper bound for $p_{\mathcal{M}_{j},B(0,R)}(s,0,y)$, $y\in O_{R}\setminus K_{R}$.
By Lemma 4.7 (see also [GSC01, Remark 3.8]), we have that
$$p_{\mathcal{M}_{j},B(0,R)}(s,0,y)\leq p_{\mathcal{M}_{j}}(s,0,y)\leq\frac{C_{d%
}}{s^{d/2}}\exp\left(-\frac{(R-1)^{2}}{2s}\right).$$
This leads the assertion.
∎
Lemma 4.10.
For each fixed $k\geq 1$ and each $t>0$,
there is a constant $R_{\mathcal{M}_{k}}(t)$ such that the following hold:
$$E_{\mathcal{M}_{k}}^{0}\left[V_{\epsilon}\left(t\wedge\tau_{B\left(0,R_{%
\mathcal{M}_{k}}(t)\right)}\right)\right]\geq(1-\exp(-t))E_{\mathcal{M}_{k}}^{%
0}\left[V_{\epsilon}(t)\right].$$
$$\sup_{j\geq k}E_{\mathcal{M}_{j}}^{0}\left[V_{\epsilon}(t),\ t>\tau_{B(0,R_{%
\mathcal{M}_{k}}(t))}\right]\leq 2^{-k}.$$
Proof.
Since
$$\lim_{R\to\infty}P^{0}_{\mathcal{M}_{k}}\left(\tau_{B(0,R)}\geq t\right)=0,$$
it holds that if we take $R_{\mathcal{M}_{k}}(t)$ sufficiently large,
$$\displaystyle E^{0}_{\mathcal{M}_{k}}\left[V_{\epsilon}(t\wedge\tau_{B(0,R(t))%
})\right]$$
$$\displaystyle\geq E_{\mathcal{M}_{k}}^{0}\left[V_{\epsilon}(t),\tau_{B(0,R(t))%
}\geq t\right]$$
$$\displaystyle\geq(1-\exp(-t))E^{0}_{\mathcal{M}_{k}}\left[V_{\epsilon}(t)%
\right].$$
Furthermore, it holds that
$$\displaystyle E_{\mathcal{M}_{j}}^{0}\left[V_{\epsilon}(t),\ t\geq\tau_{B(0,R_%
{\mathcal{M}_{k}}(t))}\right]$$
$$\displaystyle=\int_{\mathbb{R}^{d}}P^{0}_{\mathcal{M}_{j}}\left(T_{B(x,%
\epsilon)}\vee\tau_{B(0,R_{\mathcal{M}_{k}}(t))}\leq t\right)d\mu_{\mathcal{M}%
_{j}}(x)$$
$$\displaystyle\leq\mu_{\mathcal{M}_{j}}(B(0,R_{\mathcal{M}_{k}}(t)))P^{0}_{%
\mathcal{M}_{j}}\left(\tau_{B(0,R_{\mathcal{M}_{k}}(t))}\leq t\right)$$
$$\displaystyle\ \ +\int_{\mathbb{R}^{d}\setminus B(0,\epsilon+R_{\mathcal{M}_{k%
}}(t))}P^{0}_{\mathcal{M}_{j}}\left(T_{B(x,\epsilon)}\leq t\right)d\mu_{%
\mathcal{M}_{j}}(x).$$
By Lemmas 4.8 and 4.9,
we have that if $R_{\mathcal{M}_{k}}(t)$ is sufficiently large,
$$\sup_{j\geq k}E_{\mathcal{M}_{j}}^{0}\left[V_{\epsilon}(t),\ t\geq\tau_{B(0,R_%
{\mathcal{M}_{k}}(t))}\right]\leq 2^{-k}$$
∎
Lemma 4.11 (Specifying $t_{k}$).
Let $R_{\mathcal{M}_{k}}(t)$ as in the above lemma. Then,
(i) If $k$ is even, then, we can take $t_{k}$ such that
$$\frac{E_{\mathcal{M}_{k}}^{0}\left[V_{\epsilon}\left(t_{k}\wedge\tau_{B(0,R_{%
\mathcal{M}_{k}}(t_{k}))}\right)\right]}{t_{k}}\geq(1-\exp(-k))\left(c(%
\mathcal{B},\epsilon)-2^{-k}\right).$$
(ii) If $k$ is odd, then, we can take $t_{k}$ sufficiently large such that
$$\frac{E_{\mathcal{M}_{k}}^{0}\left[V_{\epsilon}(t_{k})\right]}{t_{k}}\leq c(%
\mathcal{A},\epsilon)+2^{-k}.$$
Proof.
This follows from Lemmas 4.6 and 4.10.
∎
We now define $R_{k}:=R_{\mathcal{M}_{k}}(t_{k})$.
Thus $\{\mathcal{M}_{j}\}_{j}$ are explicitly defined for every $0\leq j\leq+\infty$.
Proposition 4.12 (stability).
$$\frac{E^{0}_{\mathcal{M}_{j}}\left[V_{\epsilon}(t_{2k-1})\right]}{t_{2k-1}}%
\leq c(\mathcal{A},\epsilon)+2^{-(2k-1)},\ \forall j\geq 2k-1.$$
$$\frac{E^{0}_{\mathcal{M}_{j}}\left[V_{\epsilon}(t_{2k})\right]}{t_{2k}}\geq c(%
\mathcal{B},\epsilon)-2^{-2k},\ \forall j\geq 2k.$$
We remark that $j$ can take $+\infty$.
Proof.
We remark that
for every $j\geq k$,
$$E^{0}_{\mathcal{M}_{j}}\left[V_{\epsilon}(t_{k}\wedge\tau_{B(0,R_{k})})\right]%
=E^{0}_{\mathcal{M}_{j}}\left[\mu\left(\bigcup_{0\leq s\leq t_{k}\wedge\tau_{B%
(0,R_{k})}}B(X_{s},\epsilon)\right)\right].$$
(4.3)
This value does not depend on $j$.
Assume that $k$ is odd.
By Lemma 4.10, (4.3) and Lemma 4.11,
$$\displaystyle E^{0}_{\mathcal{M}_{j}}[V_{\epsilon}(t_{k})]$$
$$\displaystyle=E^{0}_{\mathcal{M}_{j}}\left[V_{\epsilon}(t_{k}),t_{k}>\tau_{B(0%
,R_{k})}\right]+E^{0}_{\mathcal{M}_{j}}\left[V_{\epsilon}(t_{k}),t_{k}\leq\tau%
_{B(0,R_{k})}\right].$$
$$\displaystyle\leq 2^{-k}+E^{0}_{\mathcal{M}_{j}}\left[V_{\epsilon}\left(t_{k}%
\wedge\tau_{B(0,R_{k})}\right)\right]$$
$$\displaystyle=2^{-k}+E^{0}_{\mathcal{M}_{k}}\left[V_{\epsilon}\left(t_{k}%
\wedge\tau_{B(0,R_{k})}\right)\right]$$
$$\displaystyle\leq 2^{-k}+E^{0}_{\mathcal{M}_{k}}\left[V_{\epsilon}(t_{k})\right]$$
$$\displaystyle\leq 2^{-k}(1+t_{k})+t_{k}c(\mathcal{A},\epsilon).$$
Assume that $k$ is even.
By (4.3) and Lemma 4.11,
$$\displaystyle E^{0}_{\mathcal{M}_{j}}\left[V_{\epsilon}(t_{k})\right]$$
$$\displaystyle\geq E^{0}_{\mathcal{M}_{j}}\left[V_{\epsilon}(t_{k}),t_{k}\leq%
\tau_{B(0,R_{k})}\right]$$
$$\displaystyle=E^{0}_{\mathcal{M}_{k}}\left[V_{\epsilon}\left(t_{k}\wedge\tau_{%
B(0,R_{k})}\right),t_{k}\leq\tau_{B(0,R_{k})}\right]$$
$$\displaystyle=E^{0}_{\mathcal{M}_{k}}\left[V_{\epsilon}(t_{k})\right]-E^{0}_{%
\mathcal{M}_{k}}\left[V_{\epsilon}(t_{k}),t_{k}>\tau_{B(0,R_{k})}\right]$$
$$\displaystyle\geq t_{k}(1-\exp(-k))(c(\mathcal{B},\epsilon)-2^{-k}).$$
∎
Hence,
$$\liminf_{t\to\infty}\frac{E^{0}_{\mathcal{M}_{\infty}}\left[V_{\epsilon}(t)%
\right]}{t}\leq c(\mathcal{A},\epsilon)<c(\mathcal{B},\epsilon)\leq\limsup_{t%
\to\infty}\frac{E^{0}_{\mathcal{M}_{\infty}}\left[V_{\epsilon}(t)\right]}{t}.$$
Thus the proof of Theorem 1.5 is completed.
∎
Remark 4.13.
We are not sure whether the above proof is applicable to the two-dimensional case with small modifications.
5. Further results for processes on bounded modifications
Lemma 5.1.
Let $\widetilde{\epsilon}>0$.
Then,
$$\limsup_{t\to\infty}t^{1+\widetilde{\epsilon}}\ \sup_{x\in\mathbb{R}^{d}%
\setminus B(0,t^{(1+\widetilde{\epsilon})/(d-2)})}P^{x}_{\textup{BM}}\left(T_{%
B(0,R_{0})}\leq t\right)<+\infty.$$
(5.1)
Proof.
By HK(2) for $\mathbb{R}^{d}$,
$$\displaystyle P^{x}_{\textup{BM}}(T_{B(0,R_{0})}\leq t)$$
$$\displaystyle\leq\frac{\int_{0}^{t+1}P^{x}(X_{s}\in B(0,R_{0}/2))ds}{\inf_{w%
\in\overline{\partial}B(0,R_{0})}\int_{0}^{1}P^{w}(X_{s}\in B(0,R_{0}/2))ds}$$
$$\displaystyle\leq C\int_{0}^{t}s^{-d/2}\exp\left(-\frac{(|x|-R_{0}/2)^{2}}{s}%
\right)ds.$$
By changing of variable $u=(|x|-R_{0}/2)/s^{1/2}$,
then, we have that
$$ds=-2(|x|-R_{0}/2)^{2}u^{-3}du$$
and
$$s^{-d/2}=u^{d}(|x|-R_{0}/2)^{-d}.$$
It holds that
$$\int_{0}^{t}s^{-d/2}\exp\left(-\frac{(|x|-R_{0}/2)^{2}}{s}\right)ds\leq 2\int_%
{0}^{\infty}u^{d-3}\exp(-u^{2})du\left(|x|-R_{0}/2\right)^{2-d}.$$
By recalling $|x|\geq t^{(1+\widetilde{\epsilon})/(d-2)}$, we have the assertion.
∎
Proof of Theorem 1.6.
Let $d\geq 6$.
Assume that $D\subset B(0,R_{0})$.
Let
$$W_{s,t}:=\bigcup_{u\in[s,t]}B(X_{u},\epsilon),\ 0\leq s\leq t.$$
We remark that by Lemma 2.1 (i) and the heat kernel estimates,
$$\sup_{y\in\mathbb{R}^{d}}E^{y}_{\mathcal{M}}\left[\left|W_{0,1}\right|\right]<%
+\infty.$$
(5.2)
Let
$$g(t):=t^{(1+\widetilde{\epsilon})/(d-2)}.$$
and
$$\widetilde{D}(t):=B(0,g(t)),\ t>0.$$
By the assumption,
there exists a constant $C$ such that $\mu_{\mathcal{M}}(B(0,r))\leq Cr^{d}$ holds for every $r>0$.
Using this, $d\geq 6$, HK$(2)$, and (5.1),
it holds that
if we choose sufficiently small $\widetilde{\epsilon}>0$ in Lemma 5.1,
then, for some $a>0$,
$$P_{\mathcal{M}}^{0}\left(\tau_{\widetilde{D}(t)}>t/2\right)\leq P^{0}_{%
\mathcal{M}}\left(X_{t/2}\in B(0,g(t))\right)\leq O(t^{-1-a}),$$
(5.3)
and,
$$\displaystyle\sup_{y\in\mathbb{R}^{d}\setminus\widetilde{D}(t)}P_{\mathcal{M}}%
^{y}\left(T_{D}<+\infty\right)$$
$$\displaystyle=\sup_{y\in\mathbb{R}^{d}\setminus\widetilde{D}(t)}P_{\textup{BM}%
}^{y}\left(T_{D}<+\infty\right)$$
$$\displaystyle\leq O(g(t)^{2-d})=O(t^{-1-a}).$$
(5.4)
Henceforth we fix $h\in(0,1)$.
For $y\in\mathbb{R}^{d}\setminus D$, let
$$F_{\mathcal{M}}\left(y,s\right):=E_{\mathcal{M}}^{y}\left[\left|W_{s,s+h}%
\setminus W_{0,s}\right|,\ T_{D}>s\right].$$
We also define this quantity by replacing the case that we deal with $\mathcal{M}$ with the case that we deal with the standard Brownian motion.
$$F_{\textup{BM}}\left(y,s\right):=E_{\textup{BM}}^{y}\left[\left|W_{s,s+h}%
\setminus W_{0,s}\right|,\ T_{D}>s\right].$$
Furthermore we let
$$F_{\textup{BM}}(s):=E_{\textup{BM}}^{0}\left[\left|W_{s,s+h}\setminus W_{0,s}%
\right|\right].$$
We will show that
Lemma 5.2.
$$\left|\sum_{n=0}^{\infty}E^{0}_{\mathcal{M}}\left[\left|W_{nh,(n+1)h}\setminus
W%
_{0,nh}\right|\right]-F_{\textup{BM}}(nh)\right|<+\infty.$$
(5.5)
Proof.
It holds that
$$E^{0}_{\mathcal{M}}\left[\left|W_{t,t+h}\setminus W_{0,t}\right|\right]$$
$$=E^{0}_{\mathcal{M}}\left[\left|W_{t,t+h}\setminus W_{0,t}\right|,\tau_{%
\widetilde{D}(t)}>t\right]+E^{0}_{\mathcal{M}}\left[\left|W_{t,t+h}\setminus W%
_{0,t}\right|,\tau_{\widetilde{D}(t)}\leq t\right]$$
By (5.2) and (5.3),
we have that
$$E^{0}_{\mathcal{M}}\left[\left|W_{t,t+h}\setminus W_{0,t}\right|,\tau_{%
\widetilde{D}(t)}>t\right]\leq\sup_{y\in\mathbb{R}^{d}}E^{y}_{\mathcal{M}}%
\left[\left|W_{0,h}\right|\right]O(t^{-1-a}).$$
(5.6)
Let
$$T_{D,\widetilde{D}(t)}:=\inf\left\{s>\tau_{\widetilde{D}(t)}:X_{s}\in D\right\}.$$
Then it holds that
$$E^{0}_{\mathcal{M}}\left[\left|W_{t,t+h}\setminus W_{0,t}\right|,\ \tau_{%
\widetilde{D}(t)}\leq t\right]$$
$$=E^{0}_{\mathcal{M}}\left[\left|W_{t,t+h}\setminus W_{0,t}\right|,\ \tau_{%
\widetilde{D}(t)}\leq t<\tau_{\widetilde{D}(t)}+T_{D,\widetilde{D}(t)}\right]$$
$$+E^{0}_{\mathcal{M}}\left[\left|W_{t,t+h}\setminus W_{0,t}\right|,\ \tau_{%
\widetilde{D}(t)}+T_{D,\widetilde{D}(t)}\leq t\right].$$
(5.7)
By the Markov property and (5),
$$E^{0}_{\mathcal{M}}\left[\left|W_{t,t+h}\setminus W_{0,t}\right|,\ \tau_{%
\widetilde{D}(t)}+T_{D,\widetilde{D}(t)}\leq t\right].$$
$$\leq\sup_{y\in\mathbb{R}^{d}}E^{y}_{\mathcal{M}}\left[\left|W_{0,h}\right|%
\right]P^{0}_{M}\left(\tau_{\widetilde{D}(t)}+T_{D,\widetilde{D}(t)}\leq t\right)$$
$$\leq\sup_{y\in\mathbb{R}^{d}}E^{y}_{\mathcal{M}}\left[\left|W_{0,h}\right|%
\right]\sup_{y\in\overline{\partial}\widetilde{D}(t)}P_{\mathcal{M}}^{y}(T_{D}%
\leq t)=O(t^{-1-a}).$$
(5.8)
By the strong Markov property,
we have that
$$E^{0}_{\mathcal{M}}\left[\left|W_{t,t+h}\setminus W_{0,t}\right|,\ \tau_{%
\widetilde{D}(t)}\leq t<\tau_{\widetilde{D}(t)}+T_{D,\widetilde{D}(t)}\right]$$
$$=E^{0}_{\mathcal{M}}\left[\left|W_{t,t+h}\setminus W_{\tau_{\widetilde{D}(t)},%
t}\right|,\ \tau_{\widetilde{D}(t)}\leq t<\tau_{\widetilde{D}(t)}+T_{D,%
\widetilde{D}(t)}\right]$$
$$=E^{0}_{\mathcal{M}}\left[F_{\mathcal{M}}\left(X_{\tau_{\widetilde{D}(t)}},t-%
\tau_{\widetilde{D}(t)}\right),\ \tau_{\widetilde{D}(t)}\leq t\right].$$
By Definition 1.3,
$$E^{0}_{\mathcal{M}}\left[F_{\mathcal{M}}\left(X_{\tau_{\widetilde{D}(t)}},t-%
\tau_{\widetilde{D}(t)}\right),\ \tau_{\widetilde{D}(t)}\leq t\right]=E^{0}_{%
\mathcal{M}}\left[F_{\textup{BM}}\left(X_{\tau_{\widetilde{D}(t)}},t-\tau_{%
\widetilde{D}(t)}\right),\ \tau_{\widetilde{D}(t)}\leq t\right].$$
(5.9)
By (5.1),
$$\sup_{s\in(0,t]}\left|F_{\textup{BM}}(y,s)-F_{\textup{BM}}(s)\right|=\sup_{y%
\in\mathbb{R}^{d}\setminus\widetilde{D}(t)}P^{y}\left(T_{D}\leq t\right)=O(t^{%
-1-a}).$$
Therefore,
$$E^{0}_{\mathcal{M}}\left[\left|F_{\textup{BM}}\left(X_{\tau_{\widetilde{D}(t)}%
},t-\tau_{\widetilde{D}(t)}\right)-F_{\textup{BM}}\left(t-\tau_{\widetilde{D}(%
t)}\right)\right|,\ \tau_{\widetilde{D}(t)}\leq t\right]\leq O(t^{-1-a}).$$
By this, (5.6), (5.7), (5.8) and (5.9),
$$\left|E^{0}_{\mathcal{M}}\left[\left|W_{t,t+h}\setminus W_{0,t}\right|\right]-%
E^{0}_{\mathcal{M}}\left[F_{\textup{BM}}\left(t-\tau_{\widetilde{D}(t)}\right)%
,\ \tau_{\widetilde{D}(t)}<t\right]\right|=O(t^{-1-a}).$$
By this and (5.3),
$$\left|E^{0}_{\mathcal{M}}\left[\left|W_{t,t+h}\setminus W_{0,t}\right|\right]-%
E^{0}_{\mathcal{M}}\left[F_{\textup{BM}}\left(t-\tau_{\widetilde{D}(t)}\right)%
,\ t/2\leq\tau_{\widetilde{D}(t)}<t\right]\right|=O(t^{-1-a}).$$
(5.10)
Using (5.3),
$$\displaystyle F_{\textup{BM}}(t)-O(t^{-1-a})$$
$$\displaystyle\leq E^{0}_{\mathcal{M}}\left[F_{\textup{BM}}(t),\ t/2\leq\tau_{%
\widetilde{D}(t)}<t\right]$$
$$\displaystyle\leq E^{0}_{\mathcal{M}}\left[F_{\textup{BM}}\left(t-\tau_{%
\widetilde{D}(t)}\right),\ t/2\leq\tau_{\widetilde{D}(t)}<t\right]$$
$$\displaystyle\leq E^{0}_{\mathcal{M}}\left[F_{\textup{BM}}(t/2),t/2\leq\tau_{%
\widetilde{D}(t)}<t\right]$$
$$\displaystyle\leq F_{\textup{BM}}(t/2)+O(t^{-1-a}).$$
(5.11)
By [Sp64] and the assumption that $d\geq 6$,
it holds that $F_{\textup{BM}}(s)$ is non-increasing, and,
$$\left|\sum_{n=0}^{\infty}h\textup{Cap}(\overline{B}(0,\epsilon))-F_{\textup{BM%
}}(nh)\right|<+\infty.$$
(5.12)
Since
$$\displaystyle\int_{0}^{\infty}F_{\textup{BM}}(t/2)-F_{\textup{BM}}(t)dt$$
$$\displaystyle\leq\int_{0}^{\infty}F_{\textup{BM}}(t/2)-h\textup{Cap}\left(%
\overline{B}(0,\epsilon)\right)dt$$
$$\displaystyle=2\int_{0}^{\infty}F_{\textup{BM}}(t)-h\textup{Cap}\left(%
\overline{B}(0,\epsilon)\right)dt<+\infty,$$
we have that
$$\int_{0}^{\infty}F_{\textup{BM}}(t/2)-h\textup{Cap}\left(\overline{B}(0,%
\epsilon)\right)dt<+\infty.$$
Now (5.5) follows from this, (5.10), (5) and (5.12).
∎
By (5.5),
$$\lim_{n\to\infty}E_{\mathcal{M}}^{0}\left[\left|W_{nh}\right|\right]-nh\textup%
{Cap}\left(\overline{B}(0,\epsilon)\right)\textup{ exists and is finite.}$$
Since $|W_{t}|$ is non-decreasing with respect to $t$,
$$\limsup_{t\to\infty}\left\{E_{\mathcal{M}}^{0}\left[\left|W_{t}\right|\right]-%
t\textup{Cap}\left(\overline{B}(0,\epsilon)\right)\right\}-\liminf_{t\to\infty%
}\left\{E_{\mathcal{M}}^{0}\left[\left|W_{t}\right|\right]-t\textup{Cap}\left(%
\overline{B}(0,\epsilon)\right)\right\}$$
$$\leq h\textup{Cap}\left(\overline{B}(0,\epsilon)\right)+\limsup_{n\to\infty}E^%
{0}_{\mathcal{M}}\left[\left|W_{nh,(n+1)h}\right|\right].$$
Therefore it suffices to show that
$$\lim_{h\to 0+}\limsup_{n\to\infty}E^{0}_{\mathcal{M}}\left[\left|W_{nh,(n+1)h}%
\setminus W_{0,nh}\right|\right]=0.$$
In order to show this, it suffices to show that
$$\limsup_{n\to\infty}E^{0}_{\mathcal{M}}\left[\left|W_{nh,(n+1)h}\setminus W_{0%
,nh}\right|\right]\leq E^{0}_{\textup{BM}}\left[\left|W_{0,h}\setminus W_{0,0}%
\right|\right].$$
(5.13)
Here we remark that $W_{0,0}=B(0,\epsilon)$.
Let $\delta>0$.
Then, there exists $R=R(\delta,h)>0$ such that
$$P^{0}_{\textup{BM}}\left(T_{\mathbb{R}^{d}\setminus B(0,R)}\leq h\right)\leq\delta.$$
By HK$(2)$, if $n$ is sufficiently large, then,
$$P^{0}_{\mathcal{M}}\left(X_{nh}\in B(0,2R+R_{0})\right)\leq\delta.$$
(Recall the definition of $R_{0}$.)
Therefore,
$$E^{0}_{\mathcal{M}}\left[\left|W_{nh,(n+1)h}\setminus W_{0,nh}\right|\right]%
\leq 2\delta\sup_{y\in\mathbb{R}^{d}}E^{y}_{\mathcal{M}}\left[\left|W_{0,h}%
\right|\right]+E^{0}_{\textup{BM}}\left[\left|W_{0,h}\setminus W_{0,0}\right|%
\right].$$
Since $\delta>0$ is taken arbitrarily,
we have (5.13).
Thus we have the assertion.
∎
Acknowledgements.
The author wishes to give his thanks to Kazumasa Kuwada for comments on the modification of a metric space.
This work was supported by Grant-in-Aid for JSPS Research Fellows (16J04213) and by the Research Institute for Mathematical Sciences, a Joint Usage/Research Center located in Kyoto University.
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First 3D Morpho-Kinematic model of Supernova Remnants.
The case of VRO 42.05.01 (G 166.0+4.3)
S. Derlopa,${}^{1,2}$
P. Boumis,${}^{1}$
A. Chiotellis,${}^{1}$
W. Steffen,${}^{3}$
S. Akras,${}^{4}$
${}^{1}$Institute for Astronomy, Astrophysics, Space Applications
and Remote Sensing, National Observatory of Athens,
GR 15236 Penteli, Greece
${}^{2}$Department of Physics, National and Kapodistrian University of Athens, Panepistimiopolis, GR 15783 Zografos, Greece
${}^{3}$Instituto de Astronomía, Universidad Nacional Autónoma de México, Ensenada 22800, Baja California, Mexico
${}^{4}$Instituto de Matemática, Estatística e Física, Universidade Federal do Rio Grande, Av. Italia km 8, 96203-900, Rio Grande, Brazil
E-mail: sophia.derlopa$@$noa.grE-mail: ptb$@$astro.noa.gr
(Accepted 2020 August 03. Received 2020 August 02; in original form 2020 June 17)
Abstract
We present the first three dimensional (3D) Morpho-Kinematic (MK) model of a supernova remnant (SNR), using as a case study the Galactic SNR VRO 42.05.01. We employed the astrophysical code SHAPE in which wide field imaging and high resolution spectroscopic data were utilized, to reconstruct its 3D morphology and kinematics. We found that the remnant consists of three basic distinctive components that we call: a “shell”, a “wing” and a “hat”. With respect to their kinematical behaviour, we found that the “wing” and the “shell” have similar expansion velocities ($V_{\rm exp}$ = 115$\pm 5~{}$ km s${}^{-1}$). The “hat” presents the lowest expansion velocity of the remnant ($V_{\rm exp}$ = 90$\pm 20~{}$ km s${}^{-1}$), while the upper part of the “shell” presents the highest velocity with respect to the rest of the remnant ($V_{\rm exp}$ = 155$\pm 15~{}$ km s${}^{-1}$). Furthermore, the whole nebula has an inclination of $\sim$3$\degr$ - $5$° with respect to the plane of the sky and a systemic velocity of $V_{\rm sys}=$ -17$\pm 3~{}$ km s${}^{-1}$. We discuss the interpretation of our model results regarding the origin and evolution of the SNR and we suggest that VRO 42.05.01 had an interaction history with an inhomogeneous ambient medium most likely shaped by the mass outflows of its progenitor star.
keywords:
Supernova Remnants: general – Individual objects: VRO 42.05.01 (G 166.0$+$4.3) – ISM: kinematicals and dynamics, 3D modelling
††pubyear: 2020††pagerange: First 3D Morpho-Kinematic model of Supernova Remnants. The case of VRO 42.05.01 (G 166.0+4.3)–First 3D Morpho-Kinematic model of Supernova Remnants. The case of VRO 42.05.01 (G 166.0+4.3)
1 Introduction
Massive stars (M $\geq$ 8 M${}_{\sun}$) and carbon oxygen white dwarfs members of interacting binaries (i.e. Filippenko 1997) may undergo an explosively violent death (supernova explosion, SN). The SN ejecta expand and sweep up the ambient medium. The resulting structure is progressively transformed into a beautiful gaseous nebula which is called supernova remnant (SNR). SNRs chemically enrich the host galaxy, they influence its dynamics, while the shock waves generated after the explosion are efficient cosmic ray accelerators. Moreover, the large-scale of asymmetries and complex structures that SNRs usually display, reveal inhomogeneities present in the ambient medium where they evolve in, and provide clues about the progenitor star, since SNRs interact with the material expelled during the progenitor’s evolution (McKee 1988; Chiotellis et al. 2012). The importance and utility of probing SNRs relies on providing answers for the above crucial astrophysical topics.
Valuable information about the physical processes that dominate in SNRs are gained through imaging and spectroscopic data. Nevertheless, regardless of the details that state-of-the-art astronomical instruments can depict, the fact that these data are two dimensional (2D) restricts the range of our knowledge for these objects, due to the absence of the information in the third dimension along the light of sight. Consequently, the benefits from a three dimensional (3D) study of a SNR aim at a deeper interpretation of the collected observational data.
Up to date, there are two main important tools for gaining information on the missing third dimension of SNRs: (a) the 3D (magneto) hydrodynamic (MHD) models (Toledo-Roy et al. 2014; Bolte
et al. 2015; Abellán
et al. 2017; Potter et al. 2014; Orlando
et al. 2019), which reproduce the 3D physical properties by comparing the 2D projection of the models with the observational data, and (b) the 3D velocity-maps (DeLaney
et al. 2010; Alarie
et al. 2014; Milisavljevic &
Fesen 2013; Williams
et al. 2017) which are created by the proper motion and Doppler shifted velocities of different parts of the remnant.
In this paper we take a third approach, the so-called "Morpho-Kinematic modeling" which, up to now, it has been applied successfully in Planetary Nebulae (Akras &
Steffen 2012; Akras &
López 2012; Clyne et al. 2015; Akras et al. 2016; Fang et al. 2018; Derlopa et al. 2019; Gómez-Gordillo
et al. 2020), but never in SNRs. This method reconstructs the 3D morphology of the object by using imaging and high-resolution spectroscopic data. The lack of such a 3D model in the field of SNRs was our motivation to proceed in the creation of the first 3D Morpho-Kinematical (MK) model of a SNR. For our 3D MK model the astrophysical software SHAPE was employed (Steffen &
Koning, 2017), while as a case study we used the Galactic SNR VRO 42.05.01 (hereafter VRO) for which no 3D model has been constructed before.
The paper is organised as follows. The VRO properties are presented in Section 2. The observations and data analysis are described in Section 3. The 3D MK modelling and its results are presented in Section 4 and 5, respectively. In Section 6 we discuss the interpretation of our results and we end with our conclusions in Section 7.
2 SNR VRO 42.05.01
VRO 42.05.01 (G 166.0$+$4.3, Dickel
et al. 1965) is a well studied Galactic, mixed-morphology SNR (Boumis et al. 2016; 2020 in prep; Arias
et al. 2019a; Arias et al. 2019b and references therein). This remnant was chosen for the 3D MK model due to its intriguing morphology which basically consists of two main parts: i) a hemisphere at the northeastern region called the “shell” and ii) a larger, bow-shaped shell at the southwestern region, called the “wing” (Landecker et al., 1982) (see Fig.1a). Due to its complex morphology, it has drawn the attention of the scientific community attempting to explain its overall shape. According to Pineault et al. (1987), the initial explosion of the VRO occurred in a region characterized by a density discontinuity. The part of the remnant that evolved into the denser region created the “shell” component, while the rest diffused into a hotter and tenuous medium and shaped the “wing” component. However, according to the recent results presented by Arias et al. (2019b), there is no physical proof of an interaction of the remnant with the surrounding molecular clouds and they attributed the almost triangular shape of the “wing” to a Mach cone cavity which was created by a supersonically moving progenitor star and was filled out by the SN ejecta. Finally, Chiotellis et al. (2019) modeled the observed morphology of VRO with 2D hydrodynamic simulations, suggesting that the remnant is currently interacting with the density wall of a wind bubble sculptured by the equatorial confined mass outflows of a supersonically moving progenitor star.
The interpretation of the remnant’s morphology still remains an open issue. Our aim is, using the physical results presented below that deduced from the 3D MK model, to contribute to the clarification of unanswered questions with respect to this intriguingly complicated SNR.
3 Observations
Wide-field and high-resolution imaging:
Wide-field optical images covering the whole area of VRO (55 $\times$ 35 arcmin${}^{2}$, with an image scale of 4 arcsec pixel${}^{-1}$) were obtained with the $0.3$ m telescope at Skinakas Observatory (Greece) in 2000 and 2001 with the use of H$\alpha+$[N ii] 6548, 6584 Å, [O iii] 5007 Å and [S ii] 6716, 6731 Å interference filters (Boumis et al. 2012, 2016, 2020 in prep.). For the needs of the presented 3D model, the image in H$\alpha+$[N ii] filter was used due to the high brightness of the remnant in this emission line (see Fig.1a). Furthermore, high-resolution H$\alpha+$[N ii] images of selected areas of VRO have been obtained with the $2.3$ m Aristarchos telescope at Helmos Observatory (Greece) between 2011 and 2019 (Boumis et al. 2020 in prep.). Although these images were not digitized into the model constraints, they were used for the 3D visualization of VRO as they depict in great detail the filamentary structures of the remnant.
High-dispersion long-slit spectroscopy: High-resolution long-slit echelle spectra were obtained in H$\alpha+$[N ii] 6548, 6584 Å, [O iii] 5007 Å and [S ii] 6716, 6731 Å between the years 2010 and 2019 (Boumis et al. 2020 in prep.) at the 2.1 m telescope in San Pedro Martir Observatory, Mexico, with the Manchester Echelle Spectrometer (MES-SPM; Meaburn
et al. 2003). A 2048 $\times$ 2048 (13.5$\mu$m pixel size) CCD was used with a two times binning in both the spatial and spectral dimensions, resulting in a 0.35 arcsec pixel${}^{-1}$ spatial scale. The slit length corresponds to 5.5 arcmin on the sky, while the slit width used was 300 $\mu$m (20 km s${}^{-1}$ or 3.9 arcsec wide or 0.44 Å). In total, 26 long-slit spectra were obtained with the purpose to cover the key areas of the remnant. From these, 21 spectra were obtained with slit Position Angle (P.A.) of 45$\degr$ with respect to the North and the rest 5 at a P.A. of 90$\degr$ (Fig.1a). The spectra were finally calibrated in heliocentric radial velocity ($V_{hel}$) to $\pm 2.6$ km s${}^{-1}$ accuracy against spectra of a Th-Ar lamp. The data were analyzed in the standard way using the iraf software package. From the spectra analysis, the kinematical information was obtained which was necessary for the needs of the 3D model.
4 3D Morpho-Kinematical (MK) Modelling
A 3D study of extended emission line sources like PNe, H ii regions and SNRs is essential to provide new insights to their formation and evolution. Although PNe have been extensively studied via 3D MK modeling, there is no similar work for SNRs. The reasons for this are i) SNRs are usually very extended sources, which means that, apart from the imaging data, a great amount of spectroscopic data is also required for a full coverage of the remnant, ii) the asymmetries that SNRs usually display in their shape denote a great complexity in morphology and kinematics which is an additional factor of difficulty in the 3D visualization and iii) many SNRs are thin shells with low central emission along the line of sight, therefore most of the optical emission is tangent to the observer and hence radial velocities are close to zero. Below, we present the first 3D MK model for a SNR (VRO 42.05.01), using SHAPE and briefly outline the method.
For the reconstruction of the 3D structure of VRO, H$\alpha$+[N ii] images were used to constrain its 2D morphology projected on the plane of the sky. For the third dimension along the line of sight, the position-velocity (PV) diagrams of high-dispersion H$\alpha$ spectra from 26 regions were employed. We started by building the remnant from the “wing” component considering a spherical structure, which we gradually deformed applying a number of geometric tools in SHAPE (size, squeeze, bump). The other two components, “shell” and “hat”, were simulated as spherical shell structures with a finite thickness.
A key assumption for 3D MK modelling is the velocity field of a structure that allows us to constrain its size in the direction perpendicular to the plane of the sky. The homologous expansion law $\overrightarrow{V}$ = $B(\frac{\overrightarrow{r}}{Ro})$, where $r$ is the distance in arcmin of a given point from the centre of the field, $Ro$ is the distance in arcmin at which the velocity is equal to $B$ (km s${}^{-1}$), was used for all the components of VRO, with expansion centres for the “shell” and “hat” that are offset from that of the “wing” (see Fig.1c) and with different $B$ coefficients. The model provides us with a 3D-snapshot in time of the changing structure, while it is distance independent.
In case of PNe models, the centre of the velocity field coincides with the geometric centre of the nebula. However, in complex SNRs such as VRO with three different structures is more difficult to define the centre, given the fact that the position of the explosion is uncertain. Therefore in our model, we assumed that the expansion velocity field centre coincides with the geometrical center of each component. The expansion velocity laws of the “shell” ($B$=115 km s${}^{-1}$, $Ro$=15.5 arcmin) and its upper part (green region in Fig.1c) ($B$=155 km s${}^{-1}$, $Ro$=15.5 arcmin), and of the “hat” ($B$=90 km s${}^{-1}$, $Ro$=10 arcmin) structures were determined by matching the model with the observations from various slit positions. Due to the complex morphology of the “wing”, the velocity law was constrained using only the PV diagram of slit 25 (Fig.1a). Because of the position of slit 25, the angle between the expansion and radial velocities is small enough to minimize the effect of the inclination and provides a less uncertain velocity field ($B$=115 km s${}^{-1}$, $Ro$=39 arcmin). For the remaining slits, we consider the same velocity law and we deformed the shape and size of the structure until a satisfactory matching was obtained between the model and the observations.
Apart from the kinematics, the filamentary structures that VRO presents were also intriguing. As shown in Fig.1a, the remnant shows a filamentary structure, but especially in the “shell” there is network of filaments, crossing the entire surface. The most intense of them were reproduced as indentations on the surface of the component (see Fig.1b). Pineault et al. (1987) had also characterized the“shell” as “surface spherical in grand design but indented in detail”. Furthermore, in the model we used the wing’s outer edges as closed-ended, even though in Fig.1a it seems that they are open-ended. A higher contrast version of this image, shows that even they are fainter and not continuous, they are close-ended, but the possibility of having shock break-out features there cannot be ruled out.
5 Results
Fig.1a illustrates the observational H$\alpha+$[N ii] image of VRO along with the slits’ positions, while in Fig.1b the 3D model of VRO is presented. Apart from the “shell” and the “wing” components, a third component has also been added in the southwestern region of the remnant, that we term it as the “hat”. The reason for representing this part as a separated structure was that, according to the observational data and the model, this lower part of the “wing” protrudes with respect to the rest “wing”, and also shows a different kinematic behaviour. In Fig.1c, the 3D visualization is illustrated but in a mesh representation, overlaid upon Fig.1a without the slits positions. The different colours correspond to the distinct components of the remnant, each one of which is characterized by its own morphology and velocity field. Fig.2 also presents the mesh representation of VRO, as seen from different directions.
We found that the whole remnant is tilted by approximately $\sim$3$\degr$-$5$° with respect to the plane of the sky. That means that the “shell” goes inwards the page, while the “hat” component goes outwards from the page. Concerning the “wing” component, the model showed that its northern part is bent with respect to its eastern counterpart, implying a possible interaction with a denser ambient medium at this part of the remnant (see also Arias et al. 2019b). In addition, a part of the “wing” penetrates the central region of the “shell” at its front side (see Fig.1a near slit $26$), but also at its back side as well as showed by the model. Furthermore, the straight filament that crosses the shell in the positions of slits 11 and 22, is attributed - based on our model - to the northern back side of the “wing” component. The systemic velocity of the SNR was calculated to be $V_{\rm sys}$ = -17$\pm 3~{}$ km s${}^{-1}$, lower than the value of $V_{\rm sys}$ = $-34~{}$ km s${}^{-1}$ proposed by Landecker et al. (1989).
With regards to the expansion velocities of each component, the model showed that VRO is not characterized by a uniform velocity law. The “shell” appears to expand at a velocity of $V_{\rm exp}$ = 115$\pm 5~{}$ km s${}^{-1}$. However, the upper part of the “shell” (green region in Fig.1c) presents an expansion velocity towards north-east of $V_{\rm exp}$ = 155$\pm 15$ km s${}^{-1}$, which is higher than that of the rest “shell”, and also corresponds to the the highest velocity of the remnant in total. The “wing” appears to have $V_{\rm exp}$ = 115$\pm 5~{}$ km s${}^{-1}$, same as the “shell” counterpart. On the opposite side of the remnant in the south-west region, the “hat” was found to expand at a velocity of $V_{\rm exp}$ = 90$\pm 20~{}$ km s${}^{-1}$, which is lower than that of the “wing” component. The black arrows in Fig.1c point to the direction of the radial expansion of the green region and the “hat” of VRO. The velocities of our [O iii] spectra in these regions, i.e. slits 5 and 8 (V$\sim$120-130 km s${}^{-1}$) and 18 (V$\sim$70-100 km s${}^{-1}$), agree with those deduced from our model, which in turn are consistent with the velocities of theoretical shock models (Hartigan et al., 1987).
Apart from the morphological resemblance of the reproduced model with the observational image, our guide in order to check the validity of our model was the overall agreement between the observational PV and the synthetic PV diagrams produced with SHAPE. In Fig.3, six PV observational diagrams (black lines in H$\alpha$) from six characteristic regions of VRO are presented along with the synthetic coloured PVs reproduced with SHAPE. These spectra correspond to regions of the “shell” (slits $5$, $10$), the “wing” (slit $14$), the “hat” (slit $18$), and the contact regions between the “shell” and the “wing” (slits $24$ and $26$). The blue and red lines correspond to the blue-and red- shifted part of the remnant, respectively. At the points where the slit’s position covers the regions of two components, see for example slits $24$ and $18$ in Fig.1a, the contribution in the synthetic spectra comes from both components. This is why there are two pairs of blue-red lines in the synthetic PVs of these slits. Similarly, in the synthetic PV of slit $26$, both the “wing” and the back side of the “shell” contribute to the reproduced, synthetic PV diagram. The matching is quite sufficient, and it has been achieved for all the $26$ spectra obtained for VRO. The goal was the overall fitting between observational and synthetic spectra, neglecting at this point individual substructures (blobs etc.) that the spectra may illustrate. Therefore, the model results are consistent with the observations in both imaging and spectroscopic data.
6 Discussion
Due to its peculiar morphology, VRO has become the subject of investigation for many years, in an attempt for its morphology to be correctly interpreted.
According to our model, VRO consists of three basic components: a “shell”, a “wing” and a “hat”. This distinction is on the basis of their morphology and kinematics. The first two structures were adopted from the already known literature and proved to have the same velocities range ($V_{\rm exp}$ = 115$\pm 5~{}$ km s${}^{-1}$), while the third structure was added in the model due to its different kinematics ($V_{\rm exp,hat}$ =90$\pm 20~{}$ km s${}^{-1}$) and the protrusion it presents with respect to the “wing” component. Finally, we found that, although the “shell”seems to be morphologically unified, its upper part (green region in Fig.1c) expands with a higher velocity of $V_{\rm exp}$ = 155$\pm 15~{}$ km s${}^{-1}$.
Our 3D MK model showed that the remnantâs morphology displays a roughly axial symmetry in the azimuthal and polar dimension. This result advocates that VRO most likely was shaped under an axis or central symmetric mechanism linked to the nature and evolution of the progenitor system. From this perspective our results are aligned to the wind blown medium around the VRO remnant suggested by Chiotellis et al. (2019). Within the framework of this model, the similar velocities that the “shell” and the “wing” display âdespite their different shapes and sizes- can be attributed to the deceleration of the remnant caused by the collision of the SN blast wave with the density walls of the wind bubble. In the fast expanding upper part of the “shell” (green region) we may witness a shock breakout, where the blast wave penetrated the CSM density wall and is currently propagating in the lower density ambient medium. Finally, the “hat” component coincides with the region of the bow shaped CSM where the stagnation point is lying. The high circumstellar densities expected in the area of the stagnation point (e.g Chiotellis et al., 2012) is aligned with the low velocities we gain from the “hat” component of the remnant.
An ISM density discontinuity suggested by Pineault et al. (1987) could also be possible to explain the VRO properties as extracted by our 3D MK modeling. Within this model the “wing” had evolved in the low density region of the ISM and thus, it gained its extended size compared to the “shell”. Currently, one may say that the “shell” and the “wing” have swept up about the same mass and as a result they display similar expansion velocities. However, an extra ISM density gradient toward the NE and SW direction is required in order to explain the high and low velocity of the upper “shell” and the “hat”, respectively.
7 Conclusion
We present for the first time a 3D Morpho-Kinematic model of a SNR, VRO 42.05.01. The principal conclusions from this study are:
1.
VRO can be represented by three basic distinct components, i.e. a “shell”, a “wing”, and a “hat”, each one of which presents specific morphological and kinematical characteristics.
2.
The “shell” and the “wing” reveal similar expansion velocities of $V_{\rm exp}$ = 115$\pm 5~{}$ km s${}^{-1}$ while the “hat” is expanding with $V_{\rm exp}$ = 90$\pm 20~{}$ km s${}^{-1}$. Finally, the upper part of the “shell” displays the higher expansion velocity of the SNR equal to $V_{\rm exp}=$ 155$\pm 15~{}$ km s${}^{-1}$.
3.
The remnant has an inclination of $\sim$3$\degr$ - $5$°with respect to the plane of the sky and a systemic velocity of $V_{\rm sys}$ = -17$\pm 3~{}$ km s${}^{-1}$.
4.
The northern part of the “wing” component is tilted with respect to its eastern counterpart, due to a possible interaction with a denser ambient medium in this region of the SNR.
5.
Our results are in line with the wind-bubble interaction model suggested by Chiotellis et al. (2019), however, a local ISM discontinuity in the vicinity of VRO suggested by Pineault et al. (1987) cannot be excluded.
Acknowledgements
The authors would like to thank the referee for his/her thorough comments that improved the manuscript. S.D. and A.C. acknowledge the support of this work by the PROTEAS
II project (MIS 5002515), which is implemented under the
âReinforcement of the Research and Innovation Infrastructureâ action,
funded by the âCompetitiveness, Entrepreneurship and Innovationâ
operational programme (NSRF 2014-2020) and co-financed
by Greece and the European Union (European Regional Development
Fund). S.D acknowledges the Operational Programme âHuman
Resources Development, Education and Lifelong Learningâ
in the context of the project âStrengthening Human Resources Research
Potential via Doctorate Researchâ (MIS-5000432), implemented
by the State Scholarships Foundation (IKY) and co-financed
by Greece and the European Union (European Social Fund- ESF).
P.B and A.C. acknowledge the support of this work by the Operational
Program âHuman Resources Development, Education and
Lifelong Learning 2014-2020â co-financed by Greece and the European
Union (European Social Fund-ESF) in the context of the
project âOn the interaction of Type Ia Supernovae with Planetary
Nebulaeâ (MIS 5049922). W.S. was supported by UNAM DGAPA
PASPA. We would like to thank J. Dickel for informing us about the
origin of the name of VRO 42.05.01, from its detection at the Vermilion
River Observatory (Dickel
et al., 1965). This paper is based
on observations carried out at the OAN-SPM (México), Skinakas
Observatory (Crete, Greece) and Aristarchos telescope (Helmos,
Greece).
Data Availability Statement
The data underlying this article will be shared on reasonable request to the corresponding author.
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On generalized Erdős-Ginzburg-Ziv constants of $C_{n}^{r}$
Dongchun Han
Department of Mathematics, Southwest Jiaotong University, Chengdu 610000, P.R. China
han-qingfeng@163.com
and
Hanbin Zhang
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing
100190, P.R. China
zhanghanbin@amss.ac.cn
Abstract.
Let $G$ be an additive finite abelian group with exponent $\exp(G)=m$. For any positive integer $k$, the $k$-th generalized Erdős-Ginzburg-Ziv constant $\mathsf{s}_{km}(G)$ is defined as the smallest positive integer $t$ such that every sequence $S$ in $G$ of length at least $t$ has a zero-sum subsequence of length $km$. It is easy to see that $\mathsf{s}_{kn}(C_{n}^{r})\geq(k+r)n-r$ where $n,r\in\mathbb{N}$. Kubertin conjectured that the equality holds for any $k\geq r$.
In this paper, for any $n\in\mathbb{N}$, we mainly prove the following results:
(1)
For every positive integer $k\geq 6$,
$$\mathsf{s}_{kn}(C_{n}^{3})=(k+3)n+O(\frac{n}{\ln n}).$$
(2)
For every positive integer $k\geq 18$,
$$\mathsf{s}_{kn}(C_{n}^{4})=(k+4)n+O(\frac{n}{\ln n}).$$
(3)
Assume that $p^{a}$ is the largest prime power divisor of $n$ for some $a\in\mathbb{N}$. For every $k,r\in\mathbb{N}$, if $p^{t}\geq r$ for some $t\in\mathbb{N}$,
$$\mathsf{s}_{kp^{t}n}(C_{n}^{r})=(kp^{t}+r)n+O_{r}(\frac{n}{\ln n}),$$
where $O_{r}$ depends on $r$.
Key words and phrases:
1. Introduction
Let $G$ be an additive finite abelian group with exponent $\exp(G)=m$. Let $S=g_{1}\boldsymbol{\cdot}\ldots\boldsymbol{\cdot}g_{k}$ be a sequence over $G$ (repetition is allowed), where $g_{i}\in G$ for $1\leq i\leq k$, $k$ is called the length of the sequence $S$. We call $S$ a zero-sum sequence if $\sum^{k}_{i=1}g_{i}=0$. The classical direct zero-sum problem studies conditions (mainly refer to lengths) which ensure that given sequences have non-empty zero-sum subsequences with prescribed properties. For example, the Davenport constant, denoted by $\mathsf{D}(G)$, is the smallest positive integer $t$ such that every sequence $S$ over $G$ of length $|S|\geq t$ has a nonempty zero-sum subsequence. It is easy to know that $\mathsf{D}(C_{n})=n$, where $C_{n}$ is the cyclic group of order $n$. For any positive integer $k$, the $k$-th generalized Erdős-Ginzburg-Ziv constant $\mathsf{s}_{km}(G)$ is defined as the smallest positive integer $t$ such that every sequence $S$ over $G$ of length at least $t$ has a zero-sum subsequence of length $km$. In particular, for $k=1$, $\mathsf{s}_{m}(G)$ is called the Erdős-Ginzburg-Ziv constant, which is a classical invariant in combinatorial number theory. In 1961, Erdős, Ginzburg and Ziv [6] proved that $\mathsf{s}_{n}(C_{n})=2n-1$ which is usually regarded as a starting point of zero-sum theory (see [2] for other different proofs of this result). We refer to [10] for a survey of zero-sum problem. In this paper, we will focus on $\mathsf{s}_{km}(G)$.
Let $G=C_{n}^{r}=\langle e_{1}\rangle\oplus\cdots\oplus\langle e_{r}\rangle$. Assume that $T$ consists of $n-1$ copies of $e_{i}$ for $1\leq i\leq r$. Let $S$ consist of $kn-1$ copies of $0$ and $T$, then it is easy to show that $S$ is a sequence over $C_{n}^{r}$ of length $(k+r)n-r-1$ and $S$ contains no zero-sum subsequences of length $kn$. Consequently we have
(1.1)
$$\mathsf{s}_{kn}(C_{n}^{r})\geq(k+r)n-r.$$
For general abelian group $G$ with $\exp(G)=m$, similar construction can be used to show that $\mathsf{s}_{km}(G)\geq km+\mathsf{D}(G)-1$ holds for every finite abelian group $G$, where $k\geq 1$. In 1996, Gao [9] proved that $\mathsf{s}_{km}(G)=km+\mathsf{D}(G)-1$, provided that $km\geq|G|$. In [13], Gao and Thangadurai proved that if $km<\mathsf{D}(G)$ then $\mathsf{s}_{km}(G)>km+\mathsf{D}(G)-1$. Define $l(G)$ as the smallest integer $t$ such that $\mathsf{s}_{km}(G)=km+\mathsf{D}(G)-1$ holds for every $k\geq t$. From the above we know that
$$\frac{\mathsf{D}(G)}{m}\leq l(G)\leq\frac{|G|}{m}.$$
Recently, Gao, Han, Peng and Sun conjectured ([11], Conjecture 4.7) that
$$l(G)=\lceil\frac{\mathsf{D}(G)}{m}\rceil.$$
For cyclic groups $G$, clearly we have $l(G)=1$ by the Erdős-Ginzburg-Ziv theorem. For finite abelian groups $G$ of rank two, $l(G)=2$ (see [11]). Let $p$ be a prime and $q$ be a power of $p$, the above conjecture was verified for $C_{q}^{r}$ where $1\leq r\leq 4$ (also more generally for abelian $p$-group $G$ with $\mathsf{D}(G)\leq 4m$) except for some cases when $p$ is rather small, see [13, 19, 22]. For the studies of $l(G)$ for the general cases, we refer to [11, 20, 22].
Recall (1.1) that $\mathsf{s}_{kn}(C_{n}^{r})\geq(k+r)n-r$, in [22], Kubertin conjectured that the equality actually holds for any $k\geq r$.
Conjecture 1.1.
For any positive integers $k,n$ with $k\geq r$, we have
$$\mathsf{s}_{kn}(C_{n}^{r})=(k+r)n-r.$$
By the results [13, 19, 22] cited above, Conjecture 1.1 has been verified for $r\leq 4$ except for some cases when $p$ is rather small ($p\leq 3$). Recently, Sidorenko [23, 24] verified Conjecture 1.1 for $C_{2}^{r}$. He [23] also applied his results to prove new bounds for the codegree Turán density of complete $r$-graphs. Moreover, he [24] established connections between $\mathsf{s}_{2k}(C_{2}^{r})$ and linear binary codes. Actually, he showed that the problem of determining $\mathsf{s}_{2k}(C_{2}^{r})$ is essentially equivalent to finding the lowest redundancy of a linear binary code of given length which does not contain words of Hamming weight $2k$.
Towards Conjecture 1.1, Kubertin [22] proved that
$$\mathsf{s}_{kq}(C_{q}^{r})\leq(k+\frac{3}{8}r^{2}+\frac{3}{2}r-\frac{3}{8})q-r,$$
where $p>\min\{2k,2r\}$ be a prime and $q$ be a power of $p$.
By extending the method of Kubertin, X. He [20] improved the above upper bound and obtained that
$$\mathsf{s}_{kq}(C_{q}^{r})\leq(k+5r-2)q-3r$$
when $2p\geq 7r-3$ and $k\geq r$. He also proved that $\mathsf{s}_{kn}(C_{n}^{r})\leq 6kn$ for $n$ with
large prime factors and $k$ sufficiently large. More precisely, he showed that for $r,l>0$, $n=p_{1}^{\alpha_{1}}\cdots p_{l}^{\alpha_{l}}$ with distinct prime factors $p_{1},\ldots,p_{l}\geq\frac{7}{2}r-3$ and $k=a_{1}\cdots a_{l}$ a product of positive integers $a_{1},\ldots,a_{l}\geq r$, $\mathsf{s}_{kn}(C_{n}^{r})\leq 6kn$. We also refer to [3, 14] for some recent results on the lower bound of $\mathsf{s}_{kn}(C_{n}^{r})$ when $k$ is much smaller than the rank $r$.
In this paper, we prove the following results.
Theorem 1.2.
Let $n,k\in\mathbb{N}$. We have
(1)
for every $k\geq 6$,
$$\mathsf{s}_{kn}(C_{n}^{3})=(k+3)n+O(\frac{n}{\ln n});$$
(2)
for every $k\geq 18$,
$$\mathsf{s}_{kn}(C_{n}^{4})=(k+4)n+O(\frac{n}{\ln n}).$$
Theorem 1.3.
Let $n\in\mathbb{N}$ and assume that $p^{a}$ is the largest prime power divisor of $n$ for some $a\in\mathbb{N}$. For every $k,r\in\mathbb{N}$, if $p^{t}\geq r$ for some $t\in\mathbb{N}$, we have
$$\mathsf{s}_{kp^{t}n}(C_{n}^{r})=(kp^{t}+r)n+O_{r}(\frac{n}{\ln n}),$$
where $O_{r}$ depends on $r$.
Note that the main terms in the above theorems are consistent with that in Conjecture 1.1. Moreover, the error term can be improved in some cases. Let
$$\mathsf{M}(n)=\max\{p^{k}\text{ }|\text{ with }p^{k}|n\text{ where $p$ is a %
prime and k}\in\mathbb{N}\}$$
be the largest prime power divisor of $n$, for convenience we let $\mathsf{M}(1)=1$. By some analytical studies of $\mathsf{M}(n)$, roughly speaking, for any real number $A\geq 1$, we can improve the error term $O_{r}(\frac{n}{\ln n})$ to $O_{r}(\frac{n}{(\ln n)^{A}})$ for almost every $n\geq 1$. Furthermore, when the number of distinct prime divisors of $n$ is a given integer $m$, we can even improve the error term to $O_{r}(n^{1-\frac{1}{m}})$.
The following sections are organized as follows. In Section 2, we shall introduce some notations and preliminary results. In Section 3, we will prove our main results. In Section 4, we will provide further studies on $\mathsf{M}(n)$ and then apply these results to improve our main results.
2. Preliminaries
This section will provide more rigorous definitions and notations. We also introduce some preliminary results that will be used repeatedly below.
Let $\mathbb{N}$ denote the set of positive integers and $\mathbb{N}_{0}=\mathbb{N}\cup\{0\}$. Let $f$ and $g$ be real valued functions, both defined on $\mathbb{N}$, such that $g(x)$ is strictly positive for all large enough values of $x$. Then we denote
$f(x)=O(g(x))$ if and only if there exists a positive real number $M$ and a positive integer $x_{0}$ such that
$$|f(x)|\leq M|g(x)|\qquad{\text{ for all }}x\geq x_{0}.$$
We also use the notations $O_{r}$ and $O_{A,\epsilon}$ which mean that the above $M$ depends on $r$ or $A$ and $\epsilon$, where $r\in\mathbb{N}_{0}$, $A,\epsilon\in\mathbb{R}$.
Similarly, we denote $f(x)=o(g(x))$ if and only if for every positive constant $\varepsilon$, there exists a positive integer $x_{0}$ such that
$$|f(x)|\leq\varepsilon g(x)\qquad{\text{for all }}x\geq x_{0}.$$
Let $G$ be an additive finite abelian group. By the fundamental theorem of finite abelian groups we have
$$G\cong C_{n_{1}}\oplus\cdots\oplus C_{n_{r}}$$
where $r=\mathsf{r}(G)\in\mathbb{N}_{0}$ is the rank of $G$, $n_{1}|\cdots|n_{r}\in\mathbb{N}$ are positive integers. Moreover, $n_{1},\ldots,n_{r}$ are uniquely determined by $G$, and $n_{r}=\exp(G)$ is the $exponent$ of $G$.
We define a $sequence$ over $G$ to be an element of the free abelian monoid $\big{(}\mathcal{F}(G),\boldsymbol{\cdot}\big{)}$, see Chapter 5 of [17] for detailed explanation. Our notations of sequences follows the notations in the papers [15]. In particular, in order to avoid confusion between exponentiation of the group operation in $G$ and exponentiation of the sequence operation $\boldsymbol{\cdot}$ in $\mathcal{F}(G)$, we define:
$$g^{[k]}=\underset{k}{\underbrace{g\boldsymbol{\cdot}\ldots\boldsymbol{\cdot}g}%
}\in\mathcal{F}(G)\quad\text{and}\quad T^{[k]}=\underset{k}{\underbrace{T%
\boldsymbol{\cdot}\ldots\boldsymbol{\cdot}T}}\in\mathcal{F}(G)\,,$$
for $g\in G$, $T\in\mathcal{F}(G)$ and $k\in\mathbb{N}_{0}$.
We write a sequence $S$ in the form
$$S=\prod_{g\in G}g^{\textsf{v}_{g}(S)}\text{ with }\textsf{v}_{g}(S)\in\mathbb{%
N}_{0}\text{ for all }g\in G.$$
We call
•
$\textsf{v}_{g}(S)$ the $multiplicity$ of $g$ in $S$,
•
$|S|=l=\sum_{g\in G}\textsf{v}_{g}(S)\in\mathbb{N}_{0}$ the $length$ of $S$,
•
$T=\prod_{g\in G}g^{\textsf{v}_{g}(T)}$ a $subsequence$ of $S$ if $\textsf{v}_{g}(T)\leq\textsf{v}_{g}(S)$ for all $g\in G$, and denote by $T|S$,
•
$\sigma(S)=\sum\limits_{i=1}\limits^{l}g_{i}=\sum_{g\in G}\textsf{v}_{g}(S)g\in
G$ the $sum$ of $S$,
•
$S$ a $zero$-$sum$ $sequence$ if $\sigma(S)=0$,
•
$S$ a $zero$-$sum$ $free$ $sequence$ if $\sigma(T)\neq 0$ for every $T|S$,
•
$S$ a $short$ $zero$-$sum$ $sequence$ if it is a zero-sum sequence of length $|S|\in[1,\text{exp}(G)]$.
Using these concepts, we can define
•
$\mathsf{D}(G)$ the smallest integer $l\in\mathbb{N}$ such that every sequence $S$ over $G$ of length $|S|\geq l$ has a non-empty zero-sum subsequence. We call $\mathsf{D}(G)$ the $Davenport$ $constant$ of $G$.
•
$\mathsf{s}_{k\exp(G)}(G)$ the smallest integer $l\in\mathbb{N}$ such that every sequence $S$ over $G$ of length $|S|\geq l$ has a non-empty zero-sum subsequence $T$ of length $|T|=k\exp(G)$, where $k\in\mathbb{N}$. In particular, we call $\mathsf{s}(G):=\mathsf{s}_{\exp(G)}(G)$ the Erdős-Ginzburg-Ziv constant.
Lemma 2.1.
([17], Theorem 5.5.9)
Let $G$ be a finite abelian $p$-group and $G=C_{p^{n_{1}}}\oplus\cdots\oplus C_{p^{n_{r}}}$, then
$$\mathsf{D}(G)=\sum\limits_{i=1}\limits^{r}(p^{n_{i}}-1)+1.$$
Lemma 2.2.
Let $G$ be a finite abelian group with $\exp(G)=m$, then
$$\mathsf{s}_{km}(G)\geq km+\mathsf{D}(G)-1$$
holds for every $k\geq 1$.
Proof.
By the definition of $\mathsf{D}(G)$, there exists a zero-sum free sequence $T$ of length $|T|=\mathsf{D}(G)-1$. Let $S=T\boldsymbol{\cdot}0^{[km-1]}$. It is easy to know that $S$ is a sequence over $G$ of length $|S|=km+\mathsf{D}(G)-2$ and $S$ contains no zero-sum subsequence of length $km$. This completes the proof.
∎
Lemma 2.3.
([8], Theorem 3.2)
Let $G$ be a finite abelian $p$-group and $\exp(G)=p^{n_{r}}$. If $p^{m+n_{r}}\geq\mathsf{D}(G)$ for some $m\in\mathbb{N}$, then
$$\mathsf{s}_{kp^{m}p^{n_{r}}}=k\cdot p^{m+n_{r}}+\mathsf{D}(G)-1,$$
holds for any $k\in\mathbb{N}$.
The following classical result of Alon and Dubiner is crucial in our proof.
Lemma 2.4 ([1], Theorem 1.1).
There exists an absolute constant $c>0$ such that
$$\mathsf{s}(C_{n}^{r})\leq(cr\log_{2}r)^{r}n.$$
Although the precise values of $\mathsf{s}(G)$ for general $C_{n}^{r}$ are not known, some cases (when $n$ is a power of a small prime) have been determined. We list some of these results which are very useful in our proof.
Lemma 2.5.
Let $n\in\mathbb{N}$.
(1)
$\mathsf{s}(C_{2^{n}}^{3})=8\cdot 2^{n}-7$;
(2)
$\mathsf{s}(C_{3^{n}}^{3})=9\cdot 3^{n}-8$;
(3)
$\mathsf{s}(C_{2^{n}}^{4})=16\cdot 2^{n}-15$;
(4)
$\mathsf{s}(C_{3^{n}}^{4})=20\cdot 3^{n}-19$.
Proof.
(1) See [4], Corollary 4.4. (2) See [12], Theorem 1.7. (3) See [4], Corollary 4.4. (4) See [4], Theorem 1.3, 1.4 and Section 5.
∎
In the rest of this section, we provide some results about $\mathsf{M}(n)$ which are useful in this paper. Recall that, for any $n\in\mathbb{N}$, let
$$\mathsf{M}(n)=\max\{p^{k}\text{ }|\text{ with }p^{k}|n\text{ where $p$ is a %
prime and k}\in\mathbb{N}\}$$
be the largest prime power divisor of $n$, for convenience we let $\mathsf{M}(1)=1$. For example, we have $\mathsf{M}(20)=\mathsf{M}(2^{2}5)=5$, $\mathsf{M}(40)=\mathsf{M}(2^{3}5)=2^{3}$ and $\mathsf{M}(200)=\mathsf{M}(2^{3}5^{2})=5^{2}$. Unlike the widely studied largest prime divisor function
$$\mathsf{p}(n)=\max\{p\text{ }|\text{ with }p|n\text{ and $p$ is a prime}\},$$
as far as we know, $\mathsf{M}(n)$ has not received much attention. As $\mathsf{M}(p)=p$ where $p$ is a prime, certainly we have $\limsup\limits_{n\rightarrow\infty}\frac{\mathsf{M}(n)}{n}=1$. It is known and easy to prove that
(2.1)
$$\liminf\limits_{n\rightarrow\infty}\frac{\mathsf{M}(n)}{\ln n}=1,$$
consequently
(2.2)
$$\lim\limits_{n\rightarrow\infty}\mathsf{M}(n)=\infty.$$
Recently, Girard [18] used (2.2) to show that $\mathsf{D}(C_{n}^{r})=rn+o(n)$, which is an important result in zero-sum theorey and also can be regarded as an example of application of $\mathsf{M}(n)$. In this paper, we will continue to employ the estimates of $\mathsf{M}(n)$ to the zero-sum problems. Although the proof of (2.1) is simple and elementary, it is hard to find this result in literatures or standard textbooks. So we decide to provide the proof in the following for the convenience of the reader.
Let
$$\pi(x)=\#\{p\text{ $|$ }p\leq x\}$$
be the prime-counting function and
$$\vartheta(x)=\sum_{p\leq x}\ln p$$
the Chebyshev $\vartheta$ function. The result in the following lemma is very classical and can be easily found in [25].
Lemma 2.6.
For any $x\geq 2$, we have
$$\pi(x)\leq 2\frac{x}{\ln x}.$$
Proof.
This result is an easy consequence of Theorem 3, Page 11 in [25].
∎
Lemma 2.7.
For any $n\in\mathbb{N}$, we have
$$\mathsf{M}(n)\geq\frac{1}{2}\ln n.$$
Proof.
When $n=p^{m}$ is a prime power, the result is obvious. If $n$ is not a prime power, we assume that
$$n=q_{1}^{r_{1}}\cdots q_{k}^{r_{k}}p^{m},$$
where $q_{1}<\cdots<q_{k}$ and $p$ are distinct prime numbers, $r_{1},\ldots,r_{k},m\in\mathbb{N}$ with $\mathsf{M}(n)=p^{m}$. By the definition of $\mathsf{M}(n)$, clearly we have $n\leq p^{m}p^{km}$. Moreover we have $k<\pi(p^{m})$. For otherwise if $k\geq\pi(p^{m})$, then we have
$$k=\pi(p_{k})\geq\pi(p^{m})\geq\pi(p)$$
and consequently $p_{k}\geq p$. As $q_{1}<\cdots<q_{k}$ and $p$ are distinct prime numbers, we have $q_{k}>p_{k}$. Therefore,
$$\pi(q_{k})>\pi(p_{k})=k\geq\pi(p^{m})$$
and consequently $q_{k}>p^{m}$. By the definition of $\mathsf{M}(n)$, we have $\mathsf{M}(n)\geq q_{k}^{r_{k}}>p^{m}$, but this contradicts $\mathsf{M}(n)=p^{m}$.
Therefore $n\leq p^{m\pi(p^{m})}$, and by Lemma 2.6 we have
$$\frac{\mathsf{M}(n)}{\ln n}=\frac{p^{m}}{\ln n}\geq\frac{p^{m}}{\ln p^{m\pi(p^%
{m})}}=\frac{p^{m}}{{\pi(p^{m})\ln p^{m}}}\geq\frac{1}{2}.$$
This completes the proof.
∎
For sufficiently large number $n$, $\mathsf{p}(n)$ may be rather small, for example $\mathsf{p}(2^{m})=2$ for any $m\in\mathbb{N}$. However, Lemma 2.7 means that $\mathsf{M}(n)$ cannot be too small for sufficiently large $n$. We shall use Lemma 2.7 to prove our main result in the next section.
In the following, we will prove (2.1) which shows that actually $\ln n$ is the minimal order of $\mathsf{M}(n)$, as follows.
Let $p_{k}$ denote the $k$-th prime and $n_{k}=p_{1}\cdots p_{k}\in\mathbb{N}$. Clearly, we have $\mathsf{M}(n_{k})=p_{k}$ and $\ln n_{k}=\vartheta(p_{k})$. By the Prime Number Theorem, for any $\epsilon>0$ there exists $k_{0}(\epsilon)>0$ such that for all $k>k_{0}(\epsilon)$ we have
$$\frac{\mathsf{M}(n_{k})}{\ln n_{k}}=\frac{p_{k}}{\vartheta(p_{k})}\leq(1+%
\epsilon).$$
Therefore we have $\liminf\limits_{n\rightarrow\infty}\frac{\mathsf{M}(n)}{\ln n}\leq 1$.
Similarly, from Lemma 2.7 together with the help of the Prime Number Theorem, for any $\epsilon>0$ there exists $n_{0}(\epsilon)>0$ such that for all $n>n_{0}(\epsilon)$ we have
$$\frac{\mathsf{M}(n)}{\ln n}\geq\frac{p^{m}}{{\pi(p^{m})\ln p^{m}}}\geq(1-%
\epsilon).$$
Therefore we have $\liminf\limits_{n\rightarrow\infty}\frac{\mathsf{M}(n)}{\ln n}\geq 1$ and actually we have
$$\liminf\limits_{n\rightarrow\infty}\frac{\mathsf{M}(n)}{\ln n}=1.$$
This completes the proof of (2.1).
3. Proof of the main results
In this section, we shall prove our main results, Theorem 1.2 and 1.3. Firstly, we have to verify Conjecture 1.1 for some small primes which are the remaining cases in [13, 19, 22].
Lemma 3.1.
For any $n\in\mathbb{N}$, we have
(1)
$\mathsf{s}_{k2^{n}}(C_{2^{n}}^{3})=(k+3)2^{n}-3$, holds for $k\geq 4$;
(2)
$\mathsf{s}_{k3^{n}}(C_{3^{n}}^{3})=(k+3)3^{n}-3$, holds for $k\geq 6$;
(3)
$\mathsf{s}_{k2^{n}}(C_{2^{n}}^{4})=(k+4)2^{n}-4$, holds for $k\geq 12$;
(4)
$\mathsf{s}_{k3^{n}}(C_{3^{n}}^{4})=(k+4)3^{n}-4$, holds for $k\geq 18$.
Proof.
By Lemma 2.2, it suffices to prove $\mathsf{s}_{k\exp(G)}\leq k\exp(G)+\mathsf{D}(G)-1$ for each case.
(1) Since $2^{2+n}=4\cdot 2^{n}\geq\mathsf{D}(C_{2^{n}}^{3})=3\cdot 2^{n}-2$ by Lemma 2.3 we have
$\mathsf{s}_{4\cdot 2^{n}}(C_{2^{n}}^{3})=7\cdot 2^{n}-3$. Let $S$ be any sequence over $C_{2^{n}}^{3}$ of length
$$|S|=(k+3)2^{n}-3$$
where $k\geq 5$. By Lemma 2.5.(1) and the fact that $|S|\geq 8\cdot 2^{n}-7$, we have $S$ contains a zero-sum subsequence $T$ of length $|T|=2^{n}$. Since
$$|S\boldsymbol{\cdot}T^{-1}|=(k+2)2^{n}-3\geq\mathsf{s}_{(k-1)\cdot 2^{n}}(C_{2%
^{n}}^{3}),$$
we have $S\boldsymbol{\cdot}T^{-1}$ contains a zero-sum subsequence $U$ of length $|U|=(k-1)2^{n}$. Consequently, $T\boldsymbol{\cdot}U$ is a zero-sum subsequence of $S$ of length $|T\boldsymbol{\cdot}U|=k\cdot 2^{n}$. Therefore we have $\mathsf{s}_{k\cdot 2^{n}}(C_{2^{n}}^{3})\leq(k+3)2^{n}-3$, holds for $k\geq 4$.
(2) Since $3^{1+n}=3\cdot 3^{n}\geq\mathsf{D}(C_{3^{n}}^{3})=3\cdot 3^{n}-2$ by Lemma 2.3 we have
$\mathsf{s}_{3\cdot 3^{n}}(C_{3^{n}}^{3})=6\cdot 3^{n}-3$ and $\mathsf{s}_{6\cdot 3^{n}}(C_{3^{n}}^{3})=9\cdot 3^{n}-3$. Let $S$ be any sequence over $C_{3^{n}}^{3}$ of length
$$|S|=(k+3)3^{n}-3$$
where $k\geq 7$. By Lemma 2.5.(2) and the fact that $|S|\geq 9\cdot 3^{n}-8$, we have $S$ contains a zero-sum subsequence $T$ of length $|T|=3^{n}$. Since
$$|S\boldsymbol{\cdot}T^{-1}|=(k+2)3^{n}-3\geq\mathsf{s}_{(k-1)\cdot 3^{n}}(C_{3%
^{n}}^{3}),$$
we have $S\boldsymbol{\cdot}T^{-1}$ contains a zero-sum subsequence $U$ of length $|U|=(k-1)3^{n}$. Consequently, $T\boldsymbol{\cdot}U$ is a zero-sum subsequence of $S$ of length $|T\boldsymbol{\cdot}U|=k\cdot 3^{n}$. Therefore we have $\mathsf{s}_{k\cdot 3^{n}}(C_{3^{n}}^{3})\leq(k+3)3^{n}-3$, holds for $k\geq 6$.
(3) Since $2^{2+n}=4\cdot 2^{n}\geq\mathsf{D}(C_{2^{n}}^{4})=4\cdot 2^{n}-3$ by Lemma 2.3 we have
$\mathsf{s}_{4\cdot 2^{n}}(C_{2^{n}}^{4})=8\cdot 2^{n}-4$ and $\mathsf{s}_{12\cdot 2^{n}}(C_{2^{n}}^{4})=16\cdot 2^{n}-4$. Let $S$ be any sequence over $C_{2^{n}}^{4}$ of length
$$|S|=(k+4)2^{n}-4$$
where $k\geq 13$. By Lemma 2.5.(3) and the fact that $|S|\geq 16\cdot 2^{n}-15$, we have $S$ contains a zero-sum subsequence $T$ of length $|T|=2^{n}$. Since
$$|S\boldsymbol{\cdot}T^{-1}|=(k+3)2^{n}-4\geq\mathsf{s}_{(k-1)\cdot 2^{n}}(C_{2%
^{n}}^{4}),$$
we have $S\boldsymbol{\cdot}T^{-1}$ contains a zero-sum subsequence $U$ of length $|U|=(k-1)2^{n}$. Consequently, $T\boldsymbol{\cdot}U$ is a zero-sum subsequence of $S$ of length $|T\boldsymbol{\cdot}U|=k\cdot 2^{n}$. Therefore we have $\mathsf{s}_{k\cdot 2^{n}}(C_{2^{n}}^{4})\leq(k+4)2^{n}-4$, holds for $k\geq 12$.
(4) Since $3^{2+n}=9\cdot 3^{n}\geq\mathsf{D}(C_{3^{n}}^{4})=4\cdot 3^{n}-4$ by Lemma 2.3 we have
$\mathsf{s}_{9\cdot 3^{n}}(C_{3^{n}}^{4})=13\cdot 3^{n}-4$ and $\mathsf{s}_{18\cdot 3^{n}}(C_{3^{n}}^{4})=22\cdot 3^{n}-4$. Let $S$ be any sequence over $C_{3^{n}}^{4}$ of length
$$|S|=(k+4)3^{n}-4$$
where $k\geq 19$. By Lemma 2.5.(4) and the fact that $|S|\geq 20\cdot 3^{n}-19$, we have $S$ contains a zero-sum subsequence $T$ of length $|T|=3^{n}$. Since
$$|S\boldsymbol{\cdot}T^{-1}|=(k+3)3^{n}-4\geq\mathsf{s}_{(k-1)\cdot 3^{n}}(C_{3%
^{n}}^{4}),$$
we have $S\boldsymbol{\cdot}T^{-1}$ contains a zero-sum subsequence $U$ of length $|U|=(k-1)3^{n}$. Consequently, $T\boldsymbol{\cdot}U$ is a zero-sum subsequence of $S$ of length $|T\boldsymbol{\cdot}U|=k\cdot 3^{n}$. Therefore we have $\mathsf{s}_{k\cdot 3^{n}}(C_{3^{n}}^{4})\leq(k+4)3^{n}-4$, holds for $k\geq 18$.
∎
Corollary 3.2.
Let $n,m\in\mathbb{N}$ and $p$ be any prime. we have
(1)
$\mathsf{s}_{kp^{m}}(C_{p^{m}}^{3})=(k+3)p^{m}-3$ holds for $k\geq 6$;
(2)
$\mathsf{s}_{kp^{m}}(C_{p^{m}}^{4})=(k+4)p^{m}-4$ holds for $k\geq 18$;
Proof.
By Lemma 3.1, (1) and (2) hold for the cases $p=2,3$. For $p\geq 5$, see Theorem 1.(3) in [22] and Theorem 1.2.(3) in [19].
∎
The following crucial lemma is based on a standard argument in zero-sum theory (we refer to [17], Proposition 5.7.11).
Lemma 3.3.
Let $n,m,p,k,r\in\mathbb{N}$ with $p$ is a prime, assume that $\mathsf{s}_{kp^{m}}(C_{p^{m}}^{r})=(k+r)p^{m}-r$.
Then we have
$$\mathsf{s}_{knp^{m}}(C_{np^{m}}^{r})=(k+r)np^{m}+O_{r}(n),$$
where $O_{r}$ depends on $r$.
Proof.
Let $S$ be a sequence of length $|S|=((k+r)p^{m}-r)n+\mathsf{s}(C_{n}^{r})$ over $C_{n}^{r}$.
Consider the following map:
$$\varphi:C_{np^{m}}^{r}\rightarrow C_{n}^{r}.$$
Then $\varphi(S)$ is a sequence over $C_{n}^{r}$ of length $((k+r)p^{m}-r)n+\mathsf{s}(C_{n}^{r})$. By definition of $\mathsf{s}(C_{n}^{r})$, we have $\varphi(S)$ contains at least $(k+r)p^{m}-r$ zero-sum subsequences $S_{1},\ldots,S_{(k+r)p^{m}-r}$ over $C_{n}^{r}$ with $|S_{i}|=n$ for $1\leq i\leq(k+r)p^{m}-r$. This means that
$$\sigma(S_{1}),\ldots,\sigma(S_{(k+r)p^{m}-r})\in\ker(\varphi)=C_{p^{m}}^{r}.$$
By the assumption that $\mathsf{s}_{kp^{m}}(C_{p^{m}}^{r})=(k+r)p^{m}-r$, there exist
$$\{i_{1},\ldots,i_{kp^{m}}\}\subset\{1,\ldots,(k+r)p^{m}-r\}$$
such that $\sigma(S_{i_{1}})+\ldots+\sigma(S_{i_{kp^{m}}})=0$ and this implies that $S_{i_{1}}\boldsymbol{\cdot}\ldots\boldsymbol{\cdot}S_{i_{kp^{m}}}$ is a zero-sum subsequence of $S$ over $C_{np^{m}}^{r}$ of length $knp^{m}$. Therefore
$$\mathsf{s}_{knp^{m}}(C_{np^{m}}^{r})\leq((k+r)p^{m}-r)n+\mathsf{s}(C_{n}^{r}).$$
Moreover, by Lemma 2.4, there exists an absolute constant $c$ such that
$$((k+r)p^{m}-r)n+\mathsf{s}(C_{n}^{r})\leq((k+r)p^{m}-r)n+(cr\log_{2}r)^{r}n.$$
Let $c_{r}=(cr\log_{2}r)^{r}-r+1$, we have $\mathsf{s}_{knp^{m}}(C_{np^{m}}^{r})\leq(k+r)np^{m}+c_{r}n$.
This completes the proof.
∎
Corollary 3.4.
Let $n,k,r\in\mathbb{N}$, assume that $\mathsf{M}(n)=p^{m}$ and
$$\mathsf{s}_{kp^{m}}(C_{p^{m}}^{r})=(k+r)p^{m}-r.$$
Then we have
$$\mathsf{s}_{kn}(C_{n}^{r})=(k+r)n+O_{r}(\frac{n}{\mathsf{M}(n)}).$$
By Corollary 3.4, to prove the main results, it suffices to combine the results regarding $\mathsf{M}(n)$ in Section 2.
Proof of the Theorem 1.2.
(1) By Corollary 3.2(1) and Lemma 2.7, for $k\geq 6$, we have
$$\mathsf{s}_{kn}(C_{n}^{3})=(k+3)n+O(\frac{n}{\mathsf{M}(n)}).$$
By lemma 2.7, for $k\geq 6$, actually we have
$$\mathsf{s}_{kn}(C_{n}^{3})=(k+3)n+O(\frac{n}{\ln n}).$$
This completes the proof.
(2) By Corollary 3.2(2) and Lemma 2.7, for $k\geq 18$, we have
$$\mathsf{s}_{kn}(C_{n}^{4})=(k+4)n+O(\frac{n}{\mathsf{M}(n)}).$$
By lemma 2.7, for $k\geq 18$, actually we have
$$\mathsf{s}_{kn}(C_{n}^{4})=(k+4)n+O(\frac{n}{\ln n}).$$
This completes the proof.∎
Proof of the Theorem 1.3.
Since $\mathsf{M}(n)=p^{a}$ and $p^{t}\geq r$, by Lemma 2.3 and Lemma 2.7, for any $k\in\mathbb{N}$, we have
$$\mathsf{s}_{kp^{t}n}(C_{n}^{r})=(kp^{t}+r)n+O_{r}(\frac{n}{\mathsf{M}(n)}),$$
where $O_{r}$ depends on $r$. By lemma 2.7, actually we have
$$\mathsf{s}_{kp^{t}n}(C_{n}^{r})=(kp^{t}+r)n+O_{r}(\frac{n}{\ln n}).$$
This completes the proof.∎
4. Further studies about $\mathsf{M}(n)$ and some improvements
In this section, we will provide some further estimates for $\mathsf{M}(n)$ in some special cases. With these further estimates, we can improve our main results. All these results can be seen as some applications of $\mathsf{M}(n)$.
Note that, by (2.1), it is impossible to improve the order of the lower bound in Lemma 2.7 of $\mathsf{M}(n)$ for every $n$ any more. However, we can get some better estimates in some special cases.
We denote
$$\mathsf{E}(x,y)=\{n\leq x\text{ }|\text{ }\mathsf{M}(n)\leq y\}$$
and $\overline{\mathsf{E}(x,y)}=\{n\leq x\text{ }|\text{ }n\notin\mathsf{E}(x,y)\}$.
We shall consider
$$\mathsf{E}(x,(\ln x)^{A})=\{n\leq x\text{ }|\text{ }\mathsf{M}(n)\leq(\ln x)^{%
A}\},$$
where $A\geq 1$.
Actually, we have the following lemma.
Lemma 4.1.
For any positive integer $A\geq 1$ and $\epsilon>0$, we have
$$|\mathsf{E}(x,(\ln x)^{A})|=O_{A,\epsilon}(x^{1-\frac{1}{A}+\epsilon}).$$
Proof.
Clearly we have $|\mathsf{E}(x,(\ln x)^{A})|=\sum\limits_{n\leq x\atop\mathsf{M}(n)\leq(\ln x)^%
{A}}1$, then for any $\delta>0$,
$$\sum\limits_{n\leq x\atop\mathsf{M}(n)\leq(\ln x)^{A}}1\leq\sum\limits_{n\leq x%
\atop\mathsf{M}(n)\leq(\ln x)^{A}}\big{(}\frac{x}{n}\big{)}^{\delta},$$
similar to the Euler product of the Riemann zeta function, by the fundamental theorem of arithmetic, we have
$$\displaystyle\sum\limits_{n\leq x\atop\mathsf{M}(n)\leq(\ln x)^{A}}\big{(}%
\frac{x}{n}\big{)}^{\delta}$$
$$\displaystyle\leq x^{\delta}\prod_{p\leq(\ln x)^{A}}(1-\frac{1}{p^{\delta}})^{%
-1}$$
$$\displaystyle=x^{\delta}\prod_{p\leq(\ln x)^{A}}(1+\frac{1}{p^{\delta}-1}).$$
If we take $c_{\delta}=\frac{2^{\delta}}{2^{\delta}-1}$, then $\frac{1}{p^{\delta}-1}\leq\frac{c_{\delta}}{p^{\delta}}$ and we have
$$x^{\delta}\prod_{p\leq(\ln x)^{A}}(1+\frac{1}{p^{\delta}-1})\leq x^{\delta}%
\prod_{p\leq(\ln x)^{A}}(1+\frac{c_{\delta}}{p^{\delta}}),$$
as $1+x\leq e^{x}$ for any $x\geq 0$, we have
$$\displaystyle x^{\delta}\prod_{p\leq(\ln x)^{A}}(1+\frac{c_{\delta}}{p^{\delta%
}})$$
$$\displaystyle\leq x^{\delta}\prod_{p\leq(\ln x)^{A}}\exp(\frac{c_{\delta}}{p^{%
\delta}})$$
$$\displaystyle=x^{\delta}\exp(\sum_{p\leq(\ln x)^{A}}\frac{c_{\delta}}{p^{%
\delta}}).$$
To estimate the last sum, we employ the relation between the sum and the integral,
$$\displaystyle x^{\delta}\exp(\sum_{p\leq(\ln x)^{A}}\frac{c_{\delta}}{p^{%
\delta}})$$
$$\displaystyle\leq x^{\delta}\exp(\sum_{2\leq n\leq(\ln x)^{A}}\frac{c_{\delta}%
}{n^{\delta}})$$
$$\displaystyle\leq x^{\delta}\exp(c_{\delta}\int_{1}^{(\ln x)^{A}}\frac{1}{t^{%
\delta}}dt).$$
Therefore
$$\displaystyle x^{\delta}\exp(c_{\delta}\int_{1}^{(\ln x)^{A}}\frac{1}{t^{%
\delta}}dt)$$
$$\displaystyle=x^{\delta}\exp\big{(}\frac{c_{\delta}}{1-\delta}((\ln x)^{A(1-%
\delta)}-1)\big{)}$$
$$\displaystyle=\exp(\frac{c_{\delta}}{\delta-1})x^{\delta}\exp\big{(}\frac{c_{%
\delta}}{1-\delta}((\ln x)^{A(1-\delta)})\big{)}.$$
Now let $\delta=1-\frac{1}{A}+\frac{\epsilon}{2}$, as
$$A(1-\delta)=1-\frac{A\epsilon}{2}<1$$
and
$$\exp(\frac{c_{\delta}}{1-\delta}((\ln x)^{A(1-\delta)}))=O_{A,\epsilon}(x^{%
\frac{\epsilon}{2}}),$$
we have
$$\exp(\frac{c_{\delta}}{\delta-1})x^{\delta}\exp(\frac{c_{\delta}}{1-\delta}((%
\ln x)^{A(1-\delta)}))=O_{A,\epsilon}(x^{1-\frac{1}{A}+\epsilon}).$$
This completes the proof.
∎
For any $A\geq 1$, let
$$\mathbb{S}_{k}^{r}(x,A)=\{n\leq x\text{ }|\text{ }\mathsf{s}_{kn}(C_{n}^{r})=(%
k+r)n+O_{r}(\frac{n}{(\ln n)^{A}})\}$$
and
$$\overline{\mathbb{S}_{k}^{r}(x,A)}=\{n\leq x\text{ }|\text{ }n\notin\mathbb{S}%
_{kn}(x,A)\}.$$
Theorem 4.2.
For any $A\geq 1$ and $\epsilon>0$, we have
(1)
$|\overline{\mathbb{S}_{k}^{3}(x,A)}|=O_{A,\epsilon}(x^{1-\frac{1}{A}+\epsilon}),$
holds for $k\geq 6$;
(2)
$|\overline{\mathbb{S}_{k}^{4}(x,A)}|=O_{A,\epsilon}(x^{1-\frac{1}{A}+\epsilon}),$
holds for $k\geq 18$;
In particular,
(1)
for $k\geq 6$, we have
$$\lim_{x\rightarrow\infty}\frac{|\overline{\mathbb{S}_{k}^{3}(x,A)}|}{x}=0;$$
(2)
for $k\geq 18$, we have
$$\lim_{x\rightarrow\infty}\frac{|\overline{\mathbb{S}_{k}^{4}(x,A)}|}{x}=0.$$
Proof.
By the definition of $\overline{\mathsf{E}(x,(\ln x)^{A})}$, it is easy to see that
$$\overline{\mathsf{E}(x,(\ln x)^{A})}\subset\mathbb{S}_{k}^{3}(x,A).$$
Therefore, we have
$$\overline{\mathbb{S}_{k}^{3}(x,A)}\subset\mathsf{E}(x,(\ln x)^{A}).$$
The desired result follows from Lemma 4.1. In particular, let $\epsilon=\frac{1}{2A}$ in the above result, we have
$$\lim_{x\rightarrow\infty}\frac{|\overline{\mathbb{S}_{k}^{3}(x,A)}|}{x}\leq%
\lim_{x\rightarrow\infty}\frac{|\mathsf{E}(x,(\ln x)^{A})|}{x}=\lim_{x%
\rightarrow\infty}\frac{O(x^{1-\frac{1}{2A}})}{x}=0.$$
This completes the proof of (1). The proof of (2) is similar.
∎
According to Theorem 4.2, roughly speaking, for any $A\geq 1$ and for almost every $n\geq 1$ we have
(1)
for $k\geq 6$,
$$\mathsf{s}_{kn}(C_{n}^{3})=(k+3)n+O(\frac{n}{(\ln n)^{A}});$$
(2)
for $k\geq 18$,
$$\mathsf{s}_{kn}(C_{n}^{4})=(k+4)n+O(\frac{n}{(\ln n)^{A}}).$$
Remark 4.3.
For any $n,r\in\mathbb{N}$, we define
$$p(n,r)=\min\{p^{t}\text{ }|\text{ }\mathsf{M}(n)=p^{a}\text{ and }p^{t}\geq r\}.$$
Using this notation, Theorem 1.3 also can be improved with the above similar arguments. That is, for any $A\geq 1$, roughly speaking, for almost every $n\geq 1$ we have
$$\mathsf{s}_{kp(n,r)n}(C_{n}^{r})=(kp(n,r)+r)n+O(\frac{n}{(\ln n)^{A}}),$$
where $k\geq 1$.
Let $\omega(n)$ denote the number of distinct prime divisors of $n$. In the following, we can improve the error terms for some $n\in\mathbb{N}$ when $\omega(n)$ is a given integer $m$.
Lemma 4.4.
For any $n\in\mathbb{N}$, we have
$$\mathsf{M}(n)\geq n^{\frac{1}{\omega(n)}}.$$
Proof.
For any $n\in\mathbb{N}$, we assume that $n=q_{1}^{r_{1}}\cdots q_{m}^{r_{m}},$
where $q_{1}^{r_{1}}<\cdots<q_{m}^{r_{m}}$ and $q_{1},\ldots,q_{m}$ are distinct prime numbers, $r_{1},\ldots,r_{m}\in\mathbb{N}$. Then by the definition of $\mathsf{M}(n)$, we have $\mathsf{M}(n)=q_{m}^{r_{m}}$.
Clearly we have $\omega(n)=m$ and
$$n=q_{1}^{r_{1}}\cdots q_{m}^{r_{m}}\leq q_{m}^{mr_{m}}=\mathsf{M}(n)^{\omega(n%
)}.$$
Consequently $\mathsf{M}(n)\geq n^{\frac{1}{\omega(n)}}$.
∎
Theorem 4.5.
Let $n,r,m\in\mathbb{N}$ with $\omega(n)=m$, we have
(1)
for $k\geq 6$, $\mathsf{s}_{kn}(C_{n}^{3})=(k+3)n+O(n^{1-\frac{1}{m}});$
(2)
for $k\geq 18$, $\mathsf{s}_{kn}(C_{n}^{4})=(k+4)n+O(n^{1-\frac{1}{m}});$
(3)
for $k\geq 1$, $\mathsf{s}_{kp(n,r)n}(C_{n}^{r})=(kp(n,r)+r)n+O_{r}(n^{1-\frac{1}{m}});$
Proof.
(1) By Corollary 3.2 and 3.4, Lemma 4.4 and $\omega(n)=m$, for $k\geq 6$ we have
$$\displaystyle\mathsf{s}_{kn}(C_{n}^{3})=(k+3)n+O(\frac{n}{\mathsf{M}(n)})=(k+3%
)n+O(\frac{n}{n^{\frac{1}{\omega(n)}}})$$
$$\displaystyle=(k+3)n+O(\frac{n}{n^{\frac{1}{m}}})=(k+3)n+O(n^{1-\frac{1}{m}}).$$
This completes the proof.
The proof of (2) and (3) are similar.
∎
Remark 4.6.
Compared with the previous error terms $O_{r}(\frac{n}{\ln n})$ and $O_{r}(\frac{n}{(\ln n)^{A}})$, the error term $O_{r}(n^{1-\frac{1}{m}})$ is a large improvement and it is valid for every $n\in\mathbb{N}$ with $\omega(n)=m$.
Acknowledgments
D.C. Han was supported by the National Science Foundation of China Grant No.11601448 and the Fundamental Research Funds for the Central Universities Grant No.2682016CX121. H.B. Zhang was supported by the National Science Foundation of China Grant No.11671218 and China Postdoctoral Science Foundation Grant No. 2017M620936. The authors would like to thank Prof. Weidong Gao for many useful comments and corrections.
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On the origin of ionising photons emitted by T Tauri stars
R.D. Alexander,
C.J. Clarke
and J.E. Pringle
Institute of Astronomy, Madingley Road, Cambridge, CB3 0HA
email: rda@ast.cam.ac.uk
Abstract
We address the issue of the production of Lyman continuum photons by T Tauri stars, in an attempt to provide constraints on theoretical models of disc photoionisation. By treating the accretion shock as a hotspot on the stellar surface we show that Lyman continuum photons are produced at a rate approximately three orders of magnitude lower than that produced by a corresponding black body, and that a strong Lyman continuum is only emitted for high mass accretion rates. When our models are extended to include a column of material accreting on to the hotspot we find that the accretion column is extremely optically thick to Lyman continuum photons. Further, we find that radiative recombination of hydrogen atoms within the column is not an efficient means of producing photons with energies greater than 13.6eV, and find that an accretion column of any conceivable height suppresses the emission of Lyman continuum photons to a level below or comparable to that expected from the stellar photosphere. The photospheric Lyman continuum is itself much too weak to affect disc evolution significantly, and we find that the Lyman continuum emitted by an accretion shock is similarly unable to influence disc evolution significantly. This result has important consequences for models which use photoionisation as a mechanism to drive the dispersal of circumstellar discs, essentially proving that an additional source of Lyman continuum photons must exist if disc photoionisation is to be significant.
keywords:
accretion, accretion discs - circumstellar matter - planetary systems: protoplanetary discs - stars: pre-main-sequence
††pagerange: On the origin of ionising photons emitted by T Tauri stars–References††pubyear: 2003
1 Introduction
The evolution and eventual dispersal of discs around young stars is an important area of study, as it provides constraints on theories of both star and planet formation. It is now well established that the majority of young stars at ages of $\sim 10^{6}$yr have circumstellar discs which are optically thick at optical and infrared wavelengths (Strom et al., 1989; Kenyon & Hartmann, 1995; Haisch, Lada & Lada, 2001). These discs are relatively massive, with masses of the order of a few percent of a solar mass (Beckwith et al., 1990; Eisner & Carpenter, 2003). However by an age of $\sim 10^{7}$yr most stars are no longer seen to have such massive discs, although low-mass “debris discs” may remain (eg. Mannings & Sargent 1997; Wyatt, Dent & Greaves 2003). The mechanism by which these discs are dispersed remains an important unsolved question.
One fact which is clear, however, is that a large fraction of the mass from the disc will eventually be accreted on to the central (proto)star.
The inner edge of the accretion disc is typically truncated by the magnetosphere at a radius of around 5$R_{*}$ (Meyer, Calvet & Hillenbrand, 1997; Calvet & Gullbring, 1998) and so material falling from the disc on to the central star can attain an extremely high velocity, resulting in a so-called “accretion shock” when this material impacts upon the stellar surface. Existing models of the accretion shock (Calvet & Gullbring, 1998; Gullbring et al., 1998; Lamzin, 1998; Gullbring et al., 2000) have paid a great deal of attention to the emission in the ultraviolet (1000-3000Å) and visible (3500-7000Å) wavebands, comparing theoretical predictions to observed spectra. Extremely good models have been constructed, and these emission spectra are now well understood. However, very little attention has been paid to the emission shortward of the Lyman break ($<$ 912Å), primarily because absorption by interstellar Hi makes it impossible to observe young stars in this wavelength regime. Recent theoretical studies (Clarke, Gendrin & Sotomayor, 2001; Matsuyama, Johnstone & Hartmann, 2003; Armitage, Clarke & Palla, 2003) have suggested that photoionisation by the central object may play an important role in disc dispersal, and so the emission shortward of the Lyman break has become important.
Currently the origin of photoionising radiation from young stars such as T Tauri stars is unclear, and even the magnitude of such emission is poorly constrained (Gahm et al., 1979; Imhoff & Appenzeller, 1987). To date, models of disc photoionisation have used either a constant ionising flux (eg. Hollenbach et al. 1994; Clarke et al. 2001), assumed to be chromospheric in origin, or modelled the accretion-driven flux simply as a constant temperature hotspot on the stellar surface, emitting as a blackbody (Matsuyama et al., 2003). The latter produces an ionising flux which is proportional to the mass accretion rate, thus decreasing dramatically with time as the accretion rate falls, and so naturally the models of Clarke et al. (2001) and Matsuyama et al. (2003) have produced markedly different results. These models (Hollenbach et al., 1994; Richling & Yorke, 1997; Hollenbach, Yorke & Johnstone, 2000) also find that the mass-loss rate scales approximately with the square root of the ionising flux, and so very high ionising fluxes are required to influence disc evolution significantly: an ionising flux of $10^{41}$photons s${}^{-1}$ will drive mass-loss from the disc at around the $10^{-10}$$M_{\odot}\mathrm{yr}^{-1}$ level.
In this paper we study the issue of the ionising flux generated by an accretion shock. The simplified model of the accretion shock adopted by Matsuyama et al. (2003) neglects two key points, which we address in turn. Firstly, it seems likely that the Lyman continuum emission from such a hotspot would resemble a stellar atmosphere rather than a blackbody; whilst the two are almost identical at longer wavelengths, photo-absorption by Hi provides a strong suppression of the flux shortward of 912Å. Secondly, the photons emitted by the accretion shock must pass through the column of material accreting on to it in order to interact with material in the disc, and again we would expect photo-absorption by Hi in the column to supress the Lyman continuum significantly. In order to address these issues we have modelled this process in some detail. In section 2 we investigate the effect of replacing the blackbody hotspot with a more realistic stellar atmosphere. In section 3 we investigate the effect of passing these photons, from both the blackbody and the stellar atmosphere, through a column of accreting material. In section 4 we discuss our results and the limitations of our analysis, and in section 5 we summarise our conclusions.
2 Stellar atmospheres
The photoionisation models of Matsuyama et al. (2003) model the ionising photons as follows. They assume that the flux from the accretion shock can be modelled as a constant temperature hotspot, and adopt a blackbody spectral energy distribution at a temperature of $T=15,000$K. They assume that half the accretion luminosity is radiated by this hotspot, and so for a star of mass $M_{*}$ and radius $R_{*}$ the accretion shock luminosity $L$ is given by:
$$L=\frac{GM_{*}\dot{M}_{\mathrm{d}}}{2R_{*}}=A\sigma_{\mathrm{SB}}T^{4}$$
(1)
where $\dot{M}_{\mathrm{d}}$ is the rate of mass accretion from the disc, $A$ is the area of the hotspot, and $\sigma_{\mathrm{SB}}$ is the Stefan-Boltzmann constant. As the temperature is constant the rate of ionising photons $\Phi_{\mathrm{a}}$ is simply proportional to the accretion rate $\dot{M}_{\mathrm{d}}$. In addition, Matsuyama et al. (2003) add a further contribution to the ionising flux from the stellar photosphere, neglecting the poorly-constrained chromospheric contribution, at a constant rate of $\Phi_{\mathrm{p}}=1.29\times 10^{31}$photons s${}^{-1}$, with the total ionising photon rate given by $\Phi_{\mathrm{a}}+\Phi_{\mathrm{p}}$. Our assertion is that, given the high density of atomic hydrogen, it is extremely unlikely that such a hotspot would radiate as a blackbody. A spectrum akin to a stellar atmosphere, showing a significant “Lyman edge”, seems much more likely. As a result, we first consider the strength of this effect.
2.1 Constant temperature
Our first models simply involve substituting model stellar atmospheres in place of the blackbody emission in equation 1. We have adopted the same stellar parameters as Matsuyama et al. (2003) ($R=1R_{\odot}$, $M=1M_{\odot}$), and similarly adopted a constant hotspot temperature of 15,000K. The luminosity $L$ scales with $\dot{M}_{\mathrm{d}}$ in the same manner as in equation 1. We have utilised the Kurucz model atmospheres (Kurucz, 1992) (which have been incorporated into the cloudy code) for this temperature and surface gravity. The model atmospheres do not deviate significantly from the blackbody at longer wavelengths, but are some 3 orders of magnitude less luminous than the corresponding blackbody at wavelengths shortward of the Lyman limit at 912Å, due to absorption by Hi. Fig. 1 compares the blackbody and Kurucz spectra. (The apparent lack of emission lines in Fig 1 is an artefact of the relatively large bin-width used within the code. The use of such large wavelength bins results in line fluxes which are negligible in comparison to the continuum.) Fig.2 plots the ionising fluxes as a function of accretion rate, and we see that the stellar atmosphere hotspot produces ionising photons at a rate that is a factor of 1100 less than that obtained from the blackbody model.
2.2 Constant area
Another consideration is that of the hotspot area. The Matsuyama et al. (2003) blackbody formulation described in equation 1 uses a hotspot temperature which remains constant for different mass accretion rates, implying a hotspot size which decreases as the accretion rate drops. The accreting material is thought to be channelled on to the magnetic poles as it falls on to the stellar surface, and unless the topology of the magnetic field varies systematically wth the accretion rate, the hotspot area should remain approximately constant with time. If the hotspot area $A$ is constant then we would expect, from equation 1, the hotspot temperature to vary as $T\propto\dot{M}_{\mathrm{d}}^{1/4}$ The total luminosity of a model atmosphere is very similar to that of a blackbody, and so we adopt this relationship for the stellar atmospheres also. We re-evaluated the model atmospheres described in section 2.1, keeping the scaling luminosity proportional to $\dot{M}_{\mathrm{d}}$, but now with the temperature given by:
$$T=\left(\frac{GM_{*}}{2R_{*}A\sigma_{\mathrm{SB}}}\dot{M}_{\mathrm{d}}\right)^%
{1/4}$$
(2)
The result of this is shown in Fig.2; the drop-off in the ionising photon rate is much more precipitous than in the constant temperature case, with only very high mass accretion rates, greater than $10^{-7}$$M_{\odot}$yr${}^{-1}$, producing ionising photons at greater than the photospheric rate.
In fact, the relationship between the hotspot area and the mass accretion rate is not well understood, and Calvet & Gullbring (1998) even found observational evidence for a hotspot area which increases with $\dot{M}_{\mathrm{d}}$ (which would imply an even steeper decline in the ionising flux). However, we go on to show that the presence of an accretion column above the hotspot is by far the dominant factor in controlling the ionising photon rate, and so the exact details of the hotspot area are not of great significance.
3 Accretion Columns
The other issue which affects the emitted ionising flux is the assumed presence of a column of accreting material directly above the hotspot. This material will absorb Lyman continuum photons through photoionisation of Hi, and so a large attenuation of the ionising flux is expected. Adopting the photoionisation cross-section from Cox (2000) of $\sigma_{\mathrm{13.6eV}}=6.3$$\times$$10^{-18}$cm${}^{2}$, indicates that any column density greater than 5$\times$$10^{18}$cm${}^{-2}$ will result in an attenuation of the incident flux by a factor of $>10^{13}$, enough to reduce any incident ionising photon rate to below photospheric levels. The density of the infalling material is of order 5$\times$$10^{12}$cm${}^{-3}$ (Calvet & Gullbring, 1998) and so this results in an attenuation length of order 10${}^{6}$cm (10${}^{-5}$$R_{\odot}$). As a result, the only Lyman continuum photons which can be emitted, at any significant rate, by an accretion column must be due to radiative recombination of hydrogen in the column, the so-called diffuse continuum.
In order to investigate this effect further we have constructed models of the accretion column using the cloudy photoionisation code (Ferland, 1996). These models consist of a uniform central source, radiating either as a blackbody or a stellar atmosphere, and an accretion column. The central source emits in the radial direction only, and the accretion column covers a constant solid angle. Thus, by a simple linear subtraction of the emission not incident on the column we can treat the column as if it were illuminated solely by a hotspot at its base with a flux in the radial direction. In reality such a hotspot would produce some lateral component of flux near to its edges: this is discussed in section 4. The accretion column has a solar chemical composition (although as the dominant effect is photoionisation of Hi there is almost no dependence on chemical composition) and covers a small fraction $f$ of the stellar surface. Following Calvet & Gullbring (1998) we adopt a “free-fall” scaling density of the form:
$$\displaystyle n_{\mathrm{H}}(0)=5.2\times 10^{12}\mathrm{cm}^{-3}\left(\frac{%
\dot{M}_{\mathrm{d}}}{10^{-8}M_{\odot}\mathrm{yr}^{-1}}\right)\left(\frac{M_{*%
}}{0.5M_{\odot}}\right)^{-1/2}$$
$$\displaystyle\times\left(\frac{R_{*}}{2R_{\odot}}\right)^{-3/2}\left(\frac{f}{%
0.01}\right)^{-1}$$
(3)
The radial behaviour of the density is obtained by assuming that the material falls along magnetic field lines, in a manner consistent with standard magnetospheric accretion models (eg. Ghosh & Lamb 1978). Therefore at a given radius the product of the field strength and the column cross-sectional area is a constant. Assuming a dipole magnetic field we have $B\propto R^{-3}$, and therefore the cross-sectional area is proportional to $R^{3}$. For a column with cross-sectional area $A^{\prime}$ and mass density $\rho$, mass conservation requires that:
$$\rho A^{\prime}v_{\mathrm{ff}}=\mathrm{constant}$$
(4)
The number density $n_{\mathrm{H}}\propto\rho$, and the free-fall velocity $v_{\mathrm{ff}}\propto R^{-1/2}$, and so we have a density scaling law of:
$$n_{\mathrm{H}}\propto R^{-5/2}$$
(5)
with the condition in equation 3 used to fix the scaling constant.
cloudy allows us to evaluate the continuum and line emission from the top of the accretion column, which is a combination of the continuum incident on the bottom of the column, attenuated by the column, and the diffuse emission from the heated column. It does not allow direct evaluation of the emission from the “sides” of the accretion column, which may be significant and is discussed in section 4.
Again, we have adopted $R=1R_{\odot}$ and $M=1M_{\odot}$, and have constructed models of these accretion columns for a broad range of accretion rates, hotspot areas and column heights. Initially both the blackbody and stellar atmosphere hotspot formulations were used, with the “constant temperature” formalism used as it provides the greatest ionising flux to the column and can be treated as a limiting case. However, as discussed above, the incident Lyman continua from both are extinguished over a very short length scale, and so the only Lyman continuum emission which emerges from the columns is the so-called “diffuse” emission due to the radiative recombination of atomic hydrogen. As seen in Fig.1, the Lyman continua emitted by the columns are identical in both cases, and depend only the nature of the column rather than the spectrum of the illuminating hotspot, as both hotspot spectra have the same bolometric luminosity. As a result, only the more realistic stellar atmosphere models were used for the remainder of the cases.
3.1 Results
The results of the simulations described above are presented in Figs.3 & 4. Fig.3 shows the incident and transmitted spectra for a typical case at a variety of different column heights, and Fig.4 shows the dependence of the emergent ionising flux on column height. cloudy is only valid for temperatures greater than $\simeq$3000K - for temperatures lower than this the thermal solutions are no longer unique - and for large column heights or cases with little heating (ie. low accretion rates) the temperature dropped below this critical value. The models were not pursued beyond this point, with 3000K used as the temperature limit of the calculations. As seen in Fig.4, the emergent ionising flux, as expected, decreases with both decreasing accretion rate and with increasing column height. Smaller hotspots result in higher density of material in the accretion column for a given accretion rate, and so tend to produce slightly larger ionising fluxes. However these variations are small relative to those due to variations in column height or accretion rate.
The most important point, however, is that the emergent ionising photon rates for all of the columns are less than the photospheric value of 10${}^{31}$photons s${}^{-1}$. Further, the photospheric value is itself some 10 orders of magnitude lower than the rate required to influence disc evolution significantly. This means that the accretion columns we have modelled cannot emit ionising photons at a rate that will be significant in disc photoionisation models, for any choice of parameters.
4 Discussion
There are obvious caveats to these models. The first, and most significant, is that the cloudy code is only completely reliable at densities less than $10^{13}$cm${}^{-3}$; it is prone to numerical problems at higher densities, and for cases of high $\dot{M}_{\mathrm{d}}$ and small $f$ the density in our models can exceed this value. However, the main uncertainties at these densities are regarding the treatment of heavy elements. The dominant effects in the regime in which we are interested are photoionisation and recombination of atomic hydrogen, and these processes are treated reliably by the code. However in order to compensate for this numerical problem we were forced to limit the density artificially to have a maximum value of $5\times 10^{13}$cm${}^{-3}$. This affected the 3 models with the highest densities ($\dot{M}_{\mathrm{d}}=1\times 10^{-7}$$M_{\odot}\mathrm{yr}^{-1}$, $f=0.01,0.05$ and $\dot{M}_{\mathrm{d}}=5\times 10^{-8}$$M_{\odot}\mathrm{yr}^{-1}$, $f=0.01$), and the reduction in density results in these models under-estimating the ionising flux somewhat. This does introduce some uncertainty into our models, but we consider the effect to be small relative to the gross effects which dominate the calculations.
A further caveat regards the issue of non-radial emission, both from the hotspot and from the “sides” of the column, as mentioned in section 3. The ionising photon rates from our models are those emitted from the top of the accretion columns only, and neglect emission from the sides of the columns. Further, the incident flux from the hotspot is assumed to be purely radial. The Lyman continuum emission in which we are interested arises from the radiative recombination of hydrogen atoms, an intrinsically isotropic emission process, and so emission from the sides of the column could be significant. However, as the recombination process is isotropic it is reasonable to assume that the Lyman continuum emitted from the sides of a column will be comparable to that emitted from the top of a column that is truncated at a height equal to its diameter. Fig.4 shows that the diffuse Lyman continuum from the top of the column decreases dramatically as the column height increases (and the density decreases). Consequently the Lyman continuum emission from the sides of the column will only be significant over a distance comparable to the hotspot diameter: the sides of the column only produce a significant Lyman continuum near to the stellar surface.
As noted in section 3, Lyman continuum photons incident on the columns will be attenuated by a factor of $10^{13}$ over a distance of approximately 10${}^{-5}$$R_{\odot}$. As a result of this, the majority of the non-radial Lyman continuum photons emitted by an isotropically emitting hotspot will be absorbed by the column. The only such photons not absorbed will be those emitted within a fraction of an attenuation length of the edge of the hotspot, in a direction away from the centre of the hotspot. A hotspot covering 1% of the stellar surface has a radius of 0.2$R_{\odot}$, so less than $10^{-5}$ of the hotspot photons are emitted within 10${}^{-5}$$R_{\odot}$ of the hotspot edge. Given this fact, and also the behaviour of the “raw” hotspot emission described in Fig.2, we neglect this effect.
The net result of these two simplifications is that in reality the emission from the bottom of any accretion column will dominate the Lyman continuum emitted by the column, and accretion columns of any height will emit ionising photons at a rate comparable to that provided by the lower part of the column (the left-hand end of the curves in Fig.4). This will increase the largest calculated ionising photon rates by a small geometric factor but still cannot increase their flux to significantly greater than 10${}^{31}$photons s${}^{-1}$; the photospheric emission will still dominate the overall Lyman continuum. More importantly, much higher ionising fluxes (of the order of 10${}^{41}$photons s${}^{-1}$) are required to have a significant effect on disc evolution (Clarke et al., 2001; Matsuyama et al., 2003), and the uncertainties caused by the approximations we have made are negligible compared to this 10 orders-of-magnitude difference.
Further simplifications used in our models regard the geometry of the accretion column. Our models use a column with a constant covering factor - essentially a truncated radial cone - and so the area of the outer surface at a given radius $R$ is proportional to $R^{2}$. However in reality the column is channelled by the magnetosphere and, as explained in section 3, has an area proportional to $R^{3}$; our model accretion columns are somewhat less flared than we would expect to see in reality. However the difference between the two is only significant at large radii; small column heights, which provide the highest ionising fluxes, will not show significant deviation between the two cases. Again, the net result of this is that we probably under-estimate the ionising flux by a small factor, but not by enough to alter the results significantly.
As our accretion columns are radial cones they do not bend to follow the magnetosphere, as expected in more realistic magnetospheric accretion models (eg. Ghosh & Lamb 1978). In such a model the column would bend over to meet the accretion disc, with a curvature dependent on both the latitude of the hotspot and the strength of the magnetosphere. However, as discussed above, the emission from the bottom part of the accretion column dominates over that from the upper parts (those affected by this curvature), so this simplification will not affect our results significantly.
There is also the issue of an infall velocity, which our models do not address. In reality the accretion column will be falling towards the stellar surface at close to the free-fall velocity, which can be several hundred kms${}^{-1}$, and so this could modify the absorbing effect of the cloud. However the infall velocity is much less than the speed of light, and there are no strong emission lines near to the Lyman break, and so we consider the impact of this effect on the emitted Lyman continuum to be negligible.
It should be noted that our model makes no predictions as to the behaviour of the photons emitted at wavelengths longward of the Lyman break. As shown in Fig.1, the emission from the top of the column longward of the Lyman break is essentially identical to that from the hotspot at the base of the column. We approximate the accretion shock crudely, and so are not able to fit our models to observed spectra in a manner similar to Calvet & Gullbring (1998) or Johns-Krull, Valenti & Linsky (2000). However the Lyman continuum emitted by our columns is insensitive to variations in the hotspot spectrum, due to the high optical depth of the columns to Lyman continuum photons. Consequently we find that any reasonable accretion-shock model will produce a similar Lyman continuum, and that this Lyman continuum is independent of the emission at longer wavelengths.
We have adopted stellar parameters of $R=1R_{\odot}$ and $M=1M_{\odot}$ to provide direct comparisons with the models of Clarke et al. (2001) and Matsuyama et al. (2003). However in the case of T Tauri stars a radius of $2R_{\odot}$ and a mass of $0.5M_{\odot}$ would be more realistic (Gullbring et al., 1998). The result of this will be that our models over-estimate the ionising flux somewhat, due to both reduction in the energy released by accretion on to the stellar surface and also due to a reduction in the star’s surface gravity. Once again, however, it is unlikely that these factors are significant in comparison to the gross effects we have already considered.
Similarly, the use of the “constant temperature” hotspot to heat the column probably over-estimates both the ionising flux and heating provided by the hotspot, and thus over-estimates the diffuse Lyman continuum. In effect we have constructed a “best-case” model, designed to produce the maximum ionising flux, and still found the ionising flux to be less than that emitted by the stellar photosphere. It seems extremely unlikely that any conceivable accretion column could produce ionising photons at a rate significantly greater than this.
5 Summary
In an attempt to provide some constraints on the nature and magnitude of the ionising continuum emitted by T Tauri stars, we have constructed models which treat the accretion shock as a hotspot on the stellar surface beneath a column of accreting material. We have modelled these columns for a variety of different accretion rates, hotspot sizes and column heights, and have found that:
•
A hotspot radiating like a stellar atmosphere radiates ionising photons at a rate some 3 orders of magnitude less than the corresponding blackbody.
•
A constant area hotspot radiating like a stellar atmosphere can only emit ionising photons at greater than photospheric rates for mass accretion rates greater than $10^{-7}$$M_{\odot}$yr${}^{-1}$. Such accretion rates are near the upper limit of the rates derived from observations (Hartmann et al., 1998; Johns-Krull et al., 2000).
•
Photoionisation of neutral hydrogen in the accretion column attenuates the Lyman continuum from any hotspot to zero over a very short length scale. The ionising photons which do emerge are due to radiative recombination of hydrogen atoms in the column, and the rate of ionising photon emission is less than the photospheric level for all of the accretion columns we have modelled.
In short, we find that accretion shocks and columns are extremely unlikely to produce Lyman continuum photons at a rate significantly greater than that expected from the stellar photosphere. The photospheric level itself is some 10 orders of magnitude below the rates required for photoionisation to affect disc evolution significantly, and so it seems that the Lyman continuum emitted by an accretion shock will not be large enough to be significant in disc photoionisation models. These models have provided an attractive explanation of some observed disc properties (eg. Clarke et al. 2001; Armitage et al. 2003) but we have shown, as suggested by Clarke et al. (2001), that they must be powered by something other than the accretion-shock emission. In order to produce ionising photons at a rate large enough to enable photoionisation models to provide a realistic means of disc dispersal, the central objects must sustain a source of ionising photons that is not driven by accretion from the disc.
Acknowledgments
We thank Bob Carswell for useful discussions and advice on the use of the cloudy code. RDA acknowledges the support of a PPARC PhD studentship. CJC gratefully acknowledges support from the Leverhulme Trust in the form of a Philip Leverhulme Prize.
We thank an anonymous referee for helpful comments which improved the clarity of the paper.
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Transport properties of Lévy walks: an analysis in terms of multistate processes
Giampaolo Cristadoro
Dipartimento di Matematica, Università di Bologna,
Piazza di Porta S. Donato 5, 40126 Bologna, Italy
Thomas Gilbert
Center for Nonlinear Phenomena and Complex Systems,
Université Libre de Bruxelles, C. P. 231, Campus Plaine, B-1050
Brussels, Belgium
Marco Lenci
Dipartimento di Matematica, Università di Bologna,
Piazza di Porta S. Donato 5, 40126 Bologna, Italy
Istituto Nazionale di Fisica Nucleare, Sezione di
Bologna, Via Irnerio 46, 40126 Bologna, Italy
David P. Sanders
Departamento de Física, Facultad de Ciencias, Universidad
Nacional Autónoma de México, Ciudad Universitaria,
04510 México D.F.,
Mexico
Abstract
Continuous time random walks combining diffusive and ballistic
regimes are introduced to describe a class of Lévy walks on
lattices. By including exponentially-distributed waiting times
separating the successive jump events of a walker, we are led to
a description of such Lévy walks in terms of multistate processes
whose time-evolution is shown to obey a set of coupled delay differential
equations.
Using simple arguments, we obtain asymptotic solutions to
these equations and rederive the scaling laws for the mean
squared displacement of such processes.
Our calculation
includes the computation of all relevant transport coefficients in
terms of the parameters of the models.
pacs: 05.40.Fb, 05.60.-k, 02.50.-r, 02.30.Ks, 02.70.-c
§
Random walks described by Lévy flights give rise to complex
diffusive processes Haus and Kehr (1987); Weiss (1994); Krapivsky et al. (2010); Klafter and Sokolov (2011) and have found many
applications in physics and beyond Shlesinger et al. (1995); Klages et al. (2008); Denisov et al. (2012); Méndez et al. (2013).
Whereas the random walks associated with Brownian motion are
characterized by Gaussian propagators whose variance grows linearly in
time, the propagators of Lévy flights have infinite variance
Shlesinger et al. (1993); Klafter et al. (1996); Metzler and Klafter (2000);
they occur in models of random walks such that the
probability of a long jump decays slowly with its length
Weiss and Rubin (1983).
By considering the propagation time between
the two ends of a jump, one
obtains a class of models known as Lévy walks
Geisel et al. (1985); Shlesinger and Klafter (1985); Shlesinger et al. (1987); Klafter et al. (1987); Blumen et al. (1989); Zumofen and Klafter (1993); Shlesinger et al. (1999). A Lévy
walker thus follows a continuous path between the
two end points of every jump, performing each in a
finite time; instead of having an infinite mean squared displacement, as
happens in a Lévy flight whose jumps take place instantaneously, a
Lévy walker moves with finite velocity and, ipso facto, has a finite
mean squared displacement, although it may increase faster
than linearly in time.
A Lévy flight is characterized by its probability density of jump
lengths $x$,
or free paths, which we denote $\phi(x)$. It is assumed to have
the asymptotic scaling, $\phi(x)\sim x^{-\alpha-1}$, whose exponent,
$\alpha>0$, determines whether the moments of the
displacement are finite. In particular, for $\alpha\leq 2$, the variance
diverges.
In the framework of continuous time random walks
(Shlesinger et al., 1995, Chs. 10 & 13), a
probability distribution $\Phi(\mathbf{r},t)$ of making a
displacement $\mathbf{r}$ in a time $t$ is introduced, such that, for
instance, in the so-called velocity picture,
$\Phi(\mathbf{r},t)=\phi(|\mathbf{r}|)\delta_{\mathrm{D}}(t-|\mathbf{r}|/v)$,
where $v$ denotes the constant speed of the particle and $\delta_{\mathrm{D}}(.)$
is the Dirac delta function. Considering the Fourier-Laplace
transform of the propagator of this process, one obtains, in terms of
the parameter $\alpha$, the following scaling laws for the
mean squared displacement after time $t$ Geisel et al. (1985); Wang (1992),
$$\langle r^{2}\rangle_{t}\sim\begin{cases}t^{2},&0<\alpha<1,\\
t^{2}/\log t,&\alpha=1,\\
t^{3-\alpha},&1<\alpha<2,\\
t\log t,&\alpha=2,\\
t,&\alpha>2.\end{cases}$$
(1)
In this Letter, we consider Lévy walks on lattices and generalize the
above description, according to which a new jump event takes place as
soon as the previous one is completed, to include an
exponentially-distributed waiting
time which separates successive jumps. This induces a
distinction between the states of particles which are in the process
of completing a jump and those that are waiting to start a new
one. As shown below, such considerations lead to a theoretical
formulation of the model as a multistate process
Landman et al. (1977), which translates into a
set of coupled delay differential equations for the corresponding
distributions.
The physical motivation for the inclusion of an
exponentially-distributed waiting time between successive jump
events is clear. For instance, in the framework of chaotic scattering,
it corresponds to the time required to escape from a fractal repeller
Ott and Tél (1993); Gaspard (1998), or, more generally, to the
time spent in a chaotic transient Kantz and Grassberger (1985).
We show below that, inasmuch as the mean squared displacement is
concerned, a complete characterization of the process can be
obtained, which reproduces the scaling laws (1), as
well as yields the corresponding transport coefficients, whether
normal or anomalous.
Lévy walks as multistate processes.
We call propagating the state of a particle which is in the
process of completing a jump. In contrast, the state of a particle
waiting to start a new jump is called scattering.
Whereas particles switch from propagating to scattering states as they
complete a jump, particles in a scattering state can make
transitions to both scattering and propagating states; as soon as their
waiting time has elapsed, they move on to a neighboring site and,
doing so, may switch to a propagating state and carry their motion on to
the next site, or start anew in a scattering state.
We consider a $d$-dimensional cubic lattice of individual cells
$\mathbf{n}\in\mathbb{Z}^{d}$. The state of a walker at position
$\mathbf{n}$ and time $t$ can take on a countable number of different
values, specified by two integers, $k\geq 0$ and $j\in\{1,\dots,z\}$, where $z\equiv 2d$ is the coordination number of the
lattice. Scattering states are labeled by the state $k=0$ and propagating
states by the pair $(k,j)$, such that $k\geq 1$ counts the remaining
number of lattice sites the particle has to travel in direction $j$
to complete its jump.
Time-evolution proceeds as follows. After a
random waiting time $t$, exponentially-distributed with mean
$\tau_{\mathrm{R}}$, a particle in the scattering state $k=0$ changes its state to
$(k,j)$ with probability $\rho_{k}/z$, moving its location from site
$\mathbf{n}$ to site $\mathbf{n}+\mathbf{e}_{j}$. Conversely,
particles which are at site $\mathbf{n}$ in a propagating state $(k,j)$, $k\geq 1$, jump to site $\mathbf{n}+\mathbf{e}_{j}$, in time
$\tau_{\mathrm{B}}$, changing their state to
$(k-1,j)$.
The waiting time density of the process is the
function
$$\psi_{k}(t)=\begin{cases}\tau_{\mathrm{R}}^{-1}e^{-t/\tau_{\mathrm{R}}},&k=0,%
\\
\delta_{\mathrm{D}}(t-\tau_{\mathrm{B}}),&k\neq 0.\end{cases}$$
(2)
When a step takes place, the
transition probability to go from state $(k,j)$ to state $(k^{\prime},j^{\prime})$ is
$$\mathsf{p}_{(k,j),(k^{\prime},j^{\prime})}=\begin{cases}\rho_{k^{\prime}}/z,&k%
=0,\\
\delta_{k-1,k^{\prime}}\delta_{j,j^{\prime}}&k\neq 0,\end{cases}$$
(3)
where $\delta_{.,.}$ is the Kronecker symbol.
For definiteness, we consider below the following simple
parameterization of the transition probabilities,
$$\rho_{k}=\begin{cases}1-\epsilon,&k=0,\\
\epsilon\left[k^{-\alpha}-(k+1)^{-\alpha}\right],&k\geq 1,\end{cases}$$
(4)
in terms of the parameters $0<\epsilon<1$, which weights scattering
states relative to propagating ones, and $\alpha>0$, the asymptotic
scaling parameter of free path lengths.
Master equation.
The probability distribution of particles at site $\mathbf{n}$ and
time $t$, $P(\mathbf{n},t)$, is a sum of the distributions over the
scattering states, $P_{0}(\mathbf{n},t)$, and propagating states,
$P_{k,j}(\mathbf{n},t)$, $k\geq 1$ and $1\leq j\leq z$. According to
Eqs. (2)-(3), changes in the distribution of
$(k,j)$-states, $k\geq 1$, in cell $\mathbf{n}$ arise from particles
located at cell $\mathbf{n}-\mathbf{e}_{j}$ which make
a transition from either state $0$ or state $(k+1,j)$. Since the
latter transitions can be traced back to changes in the distribution
of $(k+1,j)$-states in cell $\mathbf{n}-\mathbf{e}_{j}$
at time $\tau_{\mathrm{B}}$ earlier, we can write
$\partial_{t}P_{k,j}(\mathbf{n},t)-\partial_{t}P_{k+1,j}(\mathbf{n}-\mathbf{e}_%
{j},t-\tau_{\mathrm{B}})=\rho_{k}/(z\tau_{\mathrm{R}})[P_{0}(\mathbf{n}-%
\mathbf{e}_{j},t)-P_{0}(\mathbf{n}-\mathbf{e}_{j},t-\tau_{\mathrm{B}})]$,
which accounts for the fact that a positive $0$-state contribution at
time $t$ becomes a negative one at time $t+\tau_{\mathrm{B}}$
111The possible addition of source terms into this expression
will not be considered here..
Applying this relation recursively, we have
$$\displaystyle\partial_{t}P_{k,j}(\mathbf{n},t)$$
$$\displaystyle=\frac{1}{z\tau_{\mathrm{R}}}\sum_{k^{\prime}=1}^{\infty}\rho_{k+%
k^{\prime}-1}\Big{[}P_{0}(\mathbf{n}-k^{\prime}\mathbf{e}_{j},t-(k^{\prime}-1)%
\tau_{\mathrm{B}})-P_{0}(\mathbf{n}-k^{\prime}\mathbf{e}_{j},t-k^{\prime}\tau_%
{\mathrm{B}})\Big{]}.$$
(5)
Terms lost by $(1,j)$-states in cells $\mathbf{n}-\mathbf{e}_{j}$, $j=1,\dots,z$, are gained by the $0$-state in cell $\mathbf{n}$, which also
gains contributions from $0$-state transitions. Since the scattering state
also loses particles at exponential rate $1/\tau_{\mathrm{R}}$, we have
$$\displaystyle\partial_{t}P_{0}(\mathbf{n},t)$$
$$\displaystyle=\frac{1}{z\tau_{\mathrm{R}}}\sum_{j=1}^{z}\sum_{k=0}^{\infty}%
\rho_{k}P_{0}(\mathbf{n}-(k+1)\mathbf{e}_{j},t-k\tau_{\mathrm{B}})-\frac{1}{%
\tau_{\mathrm{R}}}P_{0}(\mathbf{n},t).$$
(6)
It is straightforward to check that
Eqs. (5)-(6) are consistent with conservation
of probability, $\sum_{\mathbf{n}}P(\mathbf{n},t)=1$
222A simplification occurs if one considers the distribution of
propagating states in direction $j$, $P_{j}(\mathbf{n},t)=\sum_{k=1}^{\infty}P_{k,j}(\mathbf{n},t)$, whose
time-evolution is, using Eq. (4), $\partial_{t}P_{j}(\mathbf{n},t)=\epsilon/(z\tau_{\mathrm{R}})\sum_{k=1}^{%
\infty}k^{-\alpha}[P_{0}(\mathbf{n}-k\mathbf{e}_{j},t-(k-1)\tau_{\mathrm{B}})-%
P_{0}(\mathbf{n}-k\mathbf{e}_{j},t-k\tau_{\mathrm{B}})]$..
Fraction of scattering particles.
As discussed below, an important role is played by the overall
fraction of particles in the scattering state,
$S_{0}(t)\equiv\sum_{\mathbf{n}\in\mathbb{Z}^{d}}P_{0}(\mathbf{n},t)$.
From Eq. (6), this quantity is found to obey the following
linear delay differential equation,
$$\tau_{\mathrm{R}}\dot{S}_{0}(t)=\sum_{k=1}^{\infty}\rho_{k}S_{0}(t-k\tau_{%
\mathrm{B}})-\epsilon S_{0}(t).$$
(7)
Given initial conditions, e.g. $S_{0}(t)=0$, $t<0$, and $S_{0}(0)=1$
(all particles start in a scattering state), this equation can be
solved by the method of steps
Driver (1977).
Because the sum of the coefficients on the right-hand side of
Eq. (7) is zero, the solutions are asymptotically
constant and can be classified in terms of the parameter $\alpha$.
For $\alpha>1$, the average return time to the $0$-state,
$\sum_{k=0}^{\infty}\rho_{k}(\tau_{\mathrm{R}}+k\tau_{\mathrm{B}})$, is finite and given in
terms of the Riemann zeta function, since $\sum_{k=0}^{\infty}k\rho_{k}=\epsilon\zeta(\alpha)$. The process is thus
positive-recurrent and we have
$$\lim_{t\to\infty}S_{0}(t)=\frac{\tau_{\mathrm{R}}}{\tau_{\mathrm{R}}+\epsilon%
\tau_{\mathrm{B}}\zeta(\alpha)}\qquad(\alpha>1).$$
(8)
In the remaining range of parameter values, $0<\alpha\leq 1$, the
process is null-recurrent: the average return time to the $0$-state
diverges and $\lim_{t\to\infty}S_{0}(t)=0$. If $\alpha\neq 1$, the decay is
algebraic,
$$\lim_{t\to\infty}(t/\tau_{\mathrm{B}})^{1-\alpha}S_{0}(t)=\frac{\sin(\pi\alpha%
)}{\pi\epsilon}\frac{\tau_{\mathrm{R}}}{\tau_{\mathrm{B}}}\qquad(0<\alpha<1),$$
(9)
which can be obtained from a result due to Dynkin
Dynkin (1961);
see also Refs. (Feller, 1968, Vol. 2, § XIV.3) and
(Bardou et al., 2001, § 4.4).
The case $\alpha=1$ is a singular limit with logarithmic decay,
$$\lim_{t\to\infty}\log(t/\tau_{\mathrm{B}})S_{0}(t)=\frac{1}{\epsilon}\frac{%
\tau_{\mathrm{R}}}{\tau_{\mathrm{B}}}\qquad(\alpha=1).$$
(10)
Mean squared displacement.
Assuming an initial position at the origin, the second moment of the
displacement is $\langle n^{2}\rangle_{t}=\sum_{\mathbf{n}\in\mathbb{Z}^{d}}n^{2}P(\mathbf{n},t)$. Its
time-evolution is obtained by differentiating this expression with
respect to time and substituting Eqs. (5)-(6),
$$\displaystyle\tau_{\mathrm{R}}\frac{\mathrm{d}}{\mathrm{d}t}\langle n^{2}%
\rangle_{t}$$
$$\displaystyle=S_{0}(t)+\epsilon\sum_{k=1}^{\infty}\frac{2k+1}{k^{\alpha}}S_{0}%
(t-k\tau_{\mathrm{B}}),$$
(11)
where, using Eq. (4), we made use of the identity $\sum_{j=k}^{\infty}\rho_{j}=1$ for $k=0$ and $\epsilon k^{-\alpha}$
otherwise.
The time-evolution of the second moment is thus obtained
by integrating the fraction of $0$-state particles,
$$\tau_{\mathrm{R}}\langle n^{2}\rangle_{t}=\int_{0}^{t}\!\mathrm{d}s\,S_{0}(s)+%
\epsilon\sum_{k=1}^{{\lfloor\hskip-0.711319ptt/\tau_{\mathrm{B}}\hskip-0.71131%
9pt\rfloor}}\frac{2k+1}{k^{\alpha}}\int_{0}^{t-k\tau_{\mathrm{B}}}\!\mathrm{d}%
s\,S_{0}(s),$$
(12)
where, assuming the process starts at $t=0$, we set $S_{0}(t)=0$ for
$t<0$.
Now substituting the asymptotic expressions
(8)-(10), into
Eq. (12), we retrieve the regimes described by
Eq. (1) and obtain the corresponding coefficients.
Starting with the positive-recurrent regime, $\alpha>1$,
Eq. (8), we have the three asymptotic regimes,
$t\gg\tau_{\mathrm{B}}$,
$$\displaystyle\langle n^{2}\rangle_{t}$$
$$\displaystyle\simeq\frac{t}{\tau_{\mathrm{R}}+\epsilon\tau_{\mathrm{B}}\zeta(%
\alpha)}$$
(13)
$$\displaystyle\quad\times\begin{cases}1+\epsilon[\zeta(\alpha)+2\zeta(\alpha-1)%
],&\alpha>2,\\
2\epsilon\log(t/\tau_{\mathrm{B}}),&\alpha=2,\\
\frac{2\epsilon}{(2-\alpha)(3-\alpha)}(t/\tau_{\mathrm{B}})^{2-\alpha},&1<%
\alpha<2.\end{cases}$$
Whereas the first regime, $\alpha>2$, yields normal diffusion, the
other two correspond, for $\alpha=2$, to a weak form of
super-diffusion, and, for $1<\alpha<2$, to super-diffusion, such that
the mean squared displacement grows with a power of time $3-\alpha>1$, faster than linear 333 Equation
(13) assumes $\epsilon>0$. If
one takes the limit $\epsilon\to 0$, sub-leading terms may become
relevant. In particular, when $\epsilon=0$, normal diffusion is
recovered and the right-hand side of (13) is
$t/\tau_{\mathrm{R}}$ for all $\alpha$..
Ballistic diffusion occurs in the null-recurrent regime of the parameter,
$0<\alpha\leq 1$. Using Eqs. (9) and
(10), we find
$$\langle n^{2}\rangle_{t}\simeq\frac{t^{2}}{\tau_{\mathrm{B}}^{2}}\begin{cases}%
1/\log(t/\tau_{\mathrm{B}}),&\alpha=1,\\
1-\alpha,&0<\alpha<1.\end{cases}$$
(14)
In Figs. 1 and 2,
the asymptotic results
(13)-(14) are compared
to numerical measurements of the mean squared displacement of the
process defined by Eqs. (2)-(3) and the
transition probabilities (4). Timescales were set to
$\tau_{\mathrm{R}}\equiv\tau_{\mathrm{B}}\equiv 1$ and the lattice dimension to $d=1$. The
algorithm is based on a classic
kinetic Monte Carlo algorithm Gillespie (1976), which
incorporates the possibility of a ballistic propagation of particles
after they undergo a transition from a scattering to a propagating state.
For each realization, the initial state is taken to be
scattering. Positions are measured at regular intervals on a
logarithmic time scale for times up to $t=10^{4}\tau_{\mathrm{R}}$. Averages are
performed over sets of $10^{8}$ trajectories.
Concluding remarks.
The specificity of our approach to Lévy walks lies in the inclusion
of exponentially-distributed waiting times that separate
successive jumps. This additional feature induces a natural
description of the process in terms of multiple propagating and scattering
states whose distributions evolve according to a set of coupled delay
differential equations.
The mean squared displacement of the process depends on the
distribution of free paths and boils down to a simple
expression involving time-integrals of the fraction of
scattering states. Using straightforward arguments, precise asymptotic
expressions were obtained for this quantity, which reproduce the
expected scaling regimes Geisel et al. (1985); Wang (1992), and provide values
of the diffusion coefficients, whether normal or anomalous.
Although the results discussed in this Letter are limited to diffusive
and super-diffusive regimes, our formalism can be easily extended
to include the possibility of waiting times with power law
distributions such as considered in
Ref. Portillo et al. (2011). Such processes are
known to allow for sub-diffusive transport regimes
Metzler and Klafter (2000). The combination of two power
law scaling parameters, one for the waiting time and the other for the
duration of flights, indeed yields a richer set of scaling regimes, which can
be studied within our framework.
Our results can on the other hand be readily applied to the regime
$\tau_{\mathrm{R}}/\tau_{\mathrm{B}}\ll 1$, i.e., such that the waiting times in the scattering state are
typically negligible compared to the ballistic timescale. This is the
regime commonly studied in reference to Lévy walks.
Our investigation simultaneously opens up new avenues for future
work. Among results to be discussed elsewhere, our formalism can
be used to obtain exact solutions of the mean squared displacement as
a function of time. This is particularly useful to study transient
regimes, such as can be observed when the distribution of free paths
has a cut-off or, more generally, when it crosses over from one regime
to another, e.g. from a power law for small lengths to exponential
decay for large ones, or when the anomalous regime is masked by
normal sub-leading contributions which may nonetheless dominate over
time scales accessible to numerical computations
Cristadoro et al. (2014).
Another interesting regime occurs when, in the positive-recurrent
range of the scaling parameter, $\alpha>1$, the likelihood of a
transition from a scattering to a propagating state is small, $\epsilon\ll 1$. A
similar perturbative regime arises in the infinite horizon Lorentz gas
in the limit of narrow corridors Bouchaud and Le Doussal (1985). As is
well-known Bleher (1992), the scaling parameter of the
distribution of free paths has the marginal value $\alpha=2$, such
that the mean squared displacement asymptotically grows with $t\log t$. Although it has long been acknowledged that the infinite horizon
Lorentz gas exhibits features
similar to a Lévy walk Levitz (1997); Barkai and Fleurov (1997), we
argue that a consistent treatment of this model in such terms is not
possible unless exponentially-distributed waiting times are taken into
account that separate successive jumps. Indeed, the parameter
$\epsilon$, which weights the likelihood of a transition from scattering
to propagating states, is the same parameter that separates the average
relaxation time of the scattering state from the ballistic timescale,
i.e., $\tau_{\mathrm{B}}/\tau_{\mathrm{R}}\propto\epsilon\ll 1$. This will be the subject
of a separate publication.
Acknowledgements.
This work was partially supported by FIRB-project RBFR08UH60 Anomalous transport of light in complex systems (MIUR, Italy), by
SEP-CONACYT grant CB-101246 and DGAPA-UNAM PAPIIT grant IN117214
(Mexico), and by FRFC convention 2,4592.11 (Belgium). TG is
financially supported by the (Belgian) FRS-FNRS.
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Approximate Privacy:
Foundations and Quantification
Joan Feigenbaum
Dept. of Computer Science
Yale University
joan.feigenbaum@yale.edu
Supported in part by NSF grants
0331548 and 0534052 and IARPA grant FA8750-07-0031.
Aaron D. Jaggard
DIMACS
Rutgers University
adj@dimacs.rutgers.edu
Supported in part by
NSF grants 0751674 and 0753492.
Michael
Schapira
Depts. of Computer Science
Yale University and UC Berkeley
michael.schapira@yale.edu
Supported by NSF grant 0331548.
()
Abstract
Increasing use of computers and networks in business, government,
recreation, and almost all aspects of daily life has led to a
proliferation of online sensitive data about individuals and
organizations. Consequently, concern about the privacy of
these data has become a top priority, particularly those data that
are created and used in electronic commerce. There have been many
formulations of privacy and, unfortunately, many negative results
about the feasibility of maintaining privacy of sensitive data in
realistic networked environments. We formulate
communication-complexity-based definitions, both worst-case and
average-case, of a problem’s privacy-approximation ratio. We
use our definitions to investigate the extent to which approximate
privacy is achievable in two standard problems: the $2^{nd}$-price
Vickrey auction [22] and the millionaires problem of
Yao [24].
For both the $2^{nd}$-price Vickrey auction and the millionaires
problem, we show that not only is perfect privacy impossible or
infeasibly costly to achieve, but even close approximations
of perfect privacy suffer from the same lower bounds. By contrast,
we show that, if the values of the parties are drawn uniformly at
random from $\{0,\ldots,2^{k}-1\}$, then, for both problems, simple
and natural communication protocols have privacy-approximation
ratios that are linear in $k$ (i.e., logarithmic in the size
of the space of possible inputs). We conjecture that this improved
privacy-approximation ratio is achievable for any probability
distribution.
1 Introduction
Increasing use of computers and networks in business, government,
recreation, and almost all aspects of daily life has led to a
proliferation of online sensitive data about individuals and
organizations. Consequently, the study of privacy has become a top
priority in many disciplines. Computer scientists have contributed
many formulations of the notion of privacy-preserving computation
that have opened new avenues of investigation and shed new light on
some well studied problems.
One good example of a new avenue of investigation opened by concern
about privacy can be found in auction design, which was our original
motivation for this work. Traditional auction theory is a central
research area in Economics, and one of its main questions is how to
incent bidders to behave truthfully, i.e., to reveal
private information that auctioneers need in order to compute optimal
outcomes. More recently, attention has turned to the complementary
goal of enabling bidders not to reveal private information that
auctioneers do not need in order to compute optimal outcomes.
The importance of bidders’ privacy, like that of algorithmic efficiency,
has become clear now that many auctions are conducted online, and
Computer Science has become at least as relevant as Economics.
Our approach to privacy is based on communication complexity.
Although originally motivated by agents’ privacy in mechanism
design, our definitions and tools can be applied to distributed
function computation in general. Because perfect privacy can be
impossible or infeasibly costly to achieve, we investigate
approximate privacy. Specifically, we formulate both worst-case and
average-case versions of the privacy-approximation ratio of a
function $f$ in order to quantify the amount of privacy that can be
maintained by parties who supply sensitive inputs to a distributed
computation of $f$. We also study the tradeoff between privacy
preservation and communication complexity.
Our points of departure are the work of Chor and
Kushilevitz [8] on characterization of privately computable
functions and that of Kushilevitz [17] on the communication
complexity of private computation. Starting from the same place,
Bar-Yehuda et al. [2] also provided a framework in
which to quantify the amount of privacy that can be maintained in
the computation of a function and the communication cost of
achieving it. Their definitions and results are significantly different from the
ones we present here (see discussion in Appendix A); as explained in
Section 6 below, a precise characterization
of the relationship between their formulation and ours is an
interesting direction for future work.
1.1 Our Approach
Consider an auction of a Bluetooth headset with $2$ bidders, $1$ and
$2$, in which the auctioneer accepts bids ranging from $0 to $7
in $1 increments. Each bidder $i$ has a private value
$x_{i}\in\{0,\ldots,7\}$ that is the maximum he is willing to pay for
the headset. The item is sold in a $2^{nd}$-price Vickrey auction,
i.e., the higher bidder gets the item (with ties
broken in favor of bidder $1$), and the price he pays is the
lower bid. The demand for privacy arises naturally
in such scenarios [19]: In a straightforward protocol, the
auctioneer receives sealed bids from both bidders and computes the
outcome based on this information. Say, e.g., that bidder $1$
bids $3, and bidder $2$ bids $6. The auctioneer sells the headset
to bidder $2$ for $3. It would not be at all surprising however if,
in subsequent auctions of headsets in which bidder $2$ participates,
the same auctioneer set a reservation price of $5. This could be
avoided if the auction protocol allowed the auctioneer to learn the
fact that bidder $2$ was the highest bidder (something he needs to
know in order to determine the outcome) but did not entail the full
revelation of $2$’s private value for the headset.
Observe that, in some cases, revelation of the exact private
information of the highest bidder is necessary. For example,
if $x_{1}=6$, then bidder $2$ will win only if $x_{2}=7$. In other
cases, the revelation of a lot of information is necessary,
e.g., if bidder $1$’s bid is $5$, and bidder $2$ outbids him,
then $x_{2}$ must be either $6$ or $7$. An auction protocol is said
to achieve perfect objective privacy if the auctioneer learns
nothing about the private information of the bidders that is not
needed in order to compute the result of the auction.
Figure 1 illustrates the information the auctioneer
must learn in order to determine the outcome of the
$2^{nd}$-price auction described above. Observe that the auctioneer’s failure to
distinguish between two potential pairs of inputs that belong to
different rectangles in Fig. 1 implies his
inability to determine the winner or the price the winner must pay.
Also observe, however, that the auctioneer need not be able to
distinguish between two pairs of inputs that belong to the
same rectangle.
Using the “minimal knowledge requirements” described in
Fig. 1, we can now characterize a perfectly
(objective) privacy-preserving auction protocol as one that induces
this exact partition of the space of possible inputs into
subspaces in which the inputs are indistinguishable to the
auctioneer. Unfortunately, perfect privacy is often hard or even
impossible to achieve. For $2^{nd}$-price auctions, Brandt and
Sandholm [6] show that every perfectly private auction
protocol has exponential communication complexity. This provides the
motivation for our definition of privacy-approximation ratio:
We are interested in whether there is an auction protocol that
achieves “good” privacy guarantees without paying such a high
price in computational efficiency. We no longer insist that the
auction protocol induce a partition of inputs exactly as in
Fig. 1 but rather that it “approximate” the
optimal partition well. We define two kinds of privacy-approximation
ratio (PAR): worst-case PAR and average-case PAR.
The worst-case PAR of a protocol $P$ for the $2^{nd}$-price auction
is defined as the maximum ratio between the size of a set $S$ of
indistinguishable inputs in Fig. 1 and the size
of a set of indistinguishable inputs induced by $P$ that is
contained in $S$. If a protocol is perfectly privacy preserving,
these sets are always the same size, and so the worst-case PAR is
$1$. If, however, a protocol fails to achieve perfect privacy, then at
least one “ideal” set of indistinguishable inputs strictly
contains a set of indistinguishable inputs induced by the protocol.
In such cases, the worst-case PAR will be strictly higher than $1$.
Consider, e.g., the sealed-bid auction protocol in which both
bidders reveal their private information to the auctioneer, who then
computes the outcome. Obviously, this naive protocol enables the
auctioneer to distinguish between every two pairs of private
inputs, and so each set of indistinguishable inputs induced by the
protocol contains exactly one element. The worst-case PAR of
this protocol is therefore $\frac{8}{1}=8$. (If bidder $2$’s value
is $0$, then in Fig. 1 the auctioneer is unable to
determine which value in $\{0,\ldots,7\}$ is $x_{1}$. In the sealed
bid auction protocol, however, the auctioneer learns the exact value
of $x_{1}$.) The average-case PAR is a natural Bayesian variant
of this definition: We now assume that the auctioneer has knowledge
of some market statistics, in the form of a probability
distribution over the possible private information of the bidders.
PAR in this case is defined as the average ratio and not as
the maximum ratio as before.
Thus, intuitively, PAR captures the effect of a protocol on the
privacy (in the sense of indistinguishability from other
inputs) afforded to protocol participants—it indicates the
factor by which, in the worst case or on average, using the
protocol to compute the function, instead of just being told
the output, reduces the number of inputs from which a given
input cannot be distinguished. To formalize and generalize the above intuitive definitions of PAR, we make
use of machinery from communication-complexity theory. Specifically,
we use the concepts of monochromaticity and tilings to
make formal the notions of sets of indistinguishable inputs and of
the approximability of privacy. We discuss other notions of
approximate privacy in Section 6.
1.2 Our Findings
We present both upper and lower bounds on the privacy-approximation
ratio for both the millionaires problem and $2^{nd}$-price auctions
with $2$ bidders. Our analysis of these two environments takes place
within Yao’s $2$-party communication model [23], in which the
private information of each party is a $k$-bit string, representing
a value in $\{0,\ldots,2^{k}-1\}$. In the millionaires problem, the
two parties (the millionaires) wish to keep their private
information hidden from each other. We refer to this
goal as the preservation of subjective privacy. In
electronic-commerce environments, each party (bidder) often
communicates with the auctioneer via a secure channel, and so the
aim in the $2^{nd}$-price auction is to prevent a third
party (the auctioneer), who is unfamiliar with any of the parties’
private inputs, from learning “too much” about the bidders. This
goal is referred to, in this paper, as the preservation of objective
privacy.
Informally, for both the $2^{nd}$-price Vickrey auction and the
millionaires problem, we obtain the following results: We show that
not only is perfect privacy impossible or infeasibly costly to
achieve, but even close approximations of perfect privacy suffer
from the same lower bounds. By contrast, we show that, if the values
of the parties are drawn uniformly at random from
$\{0,\ldots,2^{k}-1\}$, then, for both problems, simple and natural
communication protocols have privacy-approximation ratios that are
linear in $k$ (i.e., logarithmic in the size of the space of
possible inputs). We conjecture
that this improved PAR is achievable for any probability
distribution. The correctness of this conjecture would imply that,
no matter what beliefs the protocol designer may have about
the parties’ private values, a protocol that achieves reasonable
privacy guarantees exists.
Importantly, our results for the $2^{nd}$-price Vickrey auction are
obtained by proving a more general result for a large family of
protocols for single-item auctions, termed “bounded-bisection
auctions”, that contains both the celebrated ascending-price
English auction and the class of bisection
auctions [14, 15].
We show that our results for the millionaires problem also extend to
the classic economic problem of provisioning a public good,
by observing that, in terms of privacy-approximation ratios, the two
problems are, in fact, equivalent.
1.3 Related Work: Defining Privacy-Preserving
Computation
1.3.1 Communication-Complexity-Based Privacy Formulations
As explained above, the privacy work of Bar-Yehuda et
al. [2] and the work presented in this paper have common
ancestors in [8, 17]. Similarly, the work of Brandt and
Sandholm [6] uses Kushilevitz’s formulation to prove an
exponential lower bound on the communication complexity of
privacy-preserving $2^{nd}$-price Vickrey auctions. We elaborate on
the relation of our work to that of Bar-Yehuda et
al. [2] in Appendix A.
Similarly to [2, 8, 17], our work focuses on the two-party
deterministic communication model. We view our results as first step in a more
general research agenda, outlined in Sec. 6.
There are many formulations of privacy-preserving computation, both exact and approximate, that are not based on the
definitions and tools in [8, 17]. We now briefly review some of them and explain how they differ
from ours.
1.3.2 Secure, Multiparty Function Evaluation
The most extensively developed approach to privacy in distributed
computation is that of secure, multiparty function evaluation
(SMFE). Indeed, to achieve agent privacy in algorithmic mechanism
design, which was our original motivation, one could, in principle,
simply start with a strategyproof mechanism and then have the agents
themselves compute the outcome and payments using an SMFE protocol.
However, as observed by Brandt and Sandholm [6], these
protocols fall into two main categories, and both have inherent
disadvantages from the point of view of mechanism design:
•
Information-theoretically private protocols, the study of which was
initiated by Ben-Or, Goldwasser, and Wigderson [4] and
Chaum, Crépeau, and Damgaard [7], rely on the assumption
that a constant fraction of the agents are “honest” (or
“obedient” in the terminology of distributed algorithmic mechanism
design [12]), i.e., that they follow the protocol
perfectly even if they know that doing so will lead to an outcome
that is not as desirable to them as one that would result from their
deviating from the protocol; clearly, this assumption is
antithetical to the main premise of mechanism design, which is that
all agents will behave strategically, deviating from protocols when
and only when doing so will improve the outcome from their points of
view;
•
Multiparty protocols that use cryptography to achieve
privacy, the study of which was initiated by
Yao [24, 25], rely on (plausible but
currently unprovable) complexity-theoretic assumptions. Often, they
are also very communication-intensive (see, e.g., [6] for an explanation
of why some of the deficiencies of the Vickrey auction cannot be solved via cryptography).
Moreover, sometimes the deployment cryptographic machinery is infeasible (over the years,
many cryptographic variants of the current interdomain routing protocol, BGP, were proposed, but
not deployed due to the infeasibility of deploying a global Internet-wide PKI infrastructure
and the real-time computational cost of verifying signatures). For some mechanisms of interest,
efficient cryptographic protocols have been obtained (see, e.g., [9, 19]).
In certain scenarios, the demand for perfect privacy preservation
cannot be relaxed. In such cases, if the function cannot be computed in a
privacy-preserving manner without the use of cryptography, there is no choice
but to resort to a cryptographic protocol. There is an extensive body of work
on cryptography-based identity protocols, and we are not offering our notion of PAR
as an extension of that work. (In fact, the framework described here might be applied to SMFE protocols by replacing indistinguishability by computational indistinguishability. However, this does not appear to yield any new insights.)
However, in other cases, we argue that privacy preservation
should be regarded as one of several design goals, alongside
low computational/communication complexity, protocol
simplicity, incentive-compatibility, and more. Therefore, it is
necessary to be able to quantify privacy preservation in order to
understand the tradeoffs among the different design goals, and
obtain “reasonable” (but not necessarily perfect) privacy guarantees.
Our PAR approach continues the long line of research about information-theoretic
notions of privacy, initiated by Ben-Or et al. and by Chaum et al. Regardless of the above argument, we believe that information-theoretic formulations of
privacy and approximate privacy are also natural to consider in
their own right.
1.3.3 Private Approximations and Approximate Privacy
In this paper, we consider protocols that compute exact results but
preserve privacy only approximately. One can also ask what it means
for a protocol to compute approximate results in a
privacy-preserving manner; indeed, this question has also been
studied [3, 11, 16], but it is unrelated to the
questions we ask here. Similarly, definitions and techniques from
differential privacy [10] (see also [13]), in which the goal is to add noise
to the result of a database query in such a way as to preserve the
privacy of the individual database records (and hence protect the
data subjects) but still have the result convey nontrivial
information, are inapplicable to the problems that we study here.
1.4 Paper Outline
In the next section, we review and expand upon the connection
between perfect privacy and communication complexity. We present our
formulations of approximate privacy, both worst case and
average case, in Section 3; we present our
main results in Sections 4
and 5. Discussion and future directions can be
found in Section 6.
2 Perfect Privacy and Communication Complexity
We now briefly review Yao’s model of two-party communication and
notions of objective and subjective perfect privacy; see Kushilevitz
and Nisan [18] for a comprehensive overview of communication
complexity theory. Note that we only deal with deterministic
communication protocols. Our definitions can be extended to
randomized protocols.
2.1 Two-Party Communication Model
There are two parties, $1$ and $2$, each holding a $k$-bit
input string. The input of party $i$, $x_{i}\in\{0,1\}^{k}$, is
the private information of $i$. The parties communicate with
each other in order to compute the value of a function
$f:\{0,1\}^{k}\times\{0,1\}^{k}\rightarrow\{0,1\}^{t}$. The two parties
alternately send messages to each other. In communication round $j$, one of the parties sends a bit $q_{j}$ that is a
function of that party’s input and the history $(q_{1},\ldots,q_{j-1})$ of previously sent messages. We say that a bit is meaningful if it is not a constant function of this input and history and if, for every meaningful bit transmitted previously, there some combination of input and history for which the bit differs from the earlier meaningful bit. Non-meaningful bits (e.g., those sent as part of protocol-message headers) are irrelevant to our work here and will be ignored. A communication
protocol dictates, for each party, when it is that party’s turn to
transmit a message and what message he should transmit, based on the
history of messages and his value.
A communication protocol $P$ is said to compute $f$ if, for every
pair of inputs $(x_{1},x_{2})$, it holds that $P(x_{1},x_{2})=f(x_{1},x_{2})$. As
in [17], the last message sent in a protocol $P$ is assumed to
contain the value $f(x_{1},x_{2})$ and therefore may require up to $t$
bits. The communication complexity of a protocol $P$ is the
maximum, over all input pairs, of the number of bits transmitted during the execution of $P$.
Any function $f:\{0,1\}^{k}\times\{0,1\}^{k}\rightarrow\{0,1\}^{t}$ can
be visualized as a $2^{k}\times 2^{k}$ matrix with entries in
$\{0,1\}^{t}$, in which the rows represent the possible inputs of
party $1$, the columns represent the possible inputs of party $2$,
and each entry contains the value of $f$ associated with its row and
column inputs. This matrix is denoted by $A(f)$.
Definition 1 (Regions, partitions)
A region in a matrix $A$ is any subset of entries in $A$ (not
necessarily a submatrix of $A$). A partition of $A$ is a collection
of disjoint regions in $A$ whose union equals $A$.
Definition 2 (Monochromaticity)
A region $R$ in a matrix $A$ is called monochromatic if all
entries in $R$ contain the same value. A monochromatic
partition of $A$ is a partition all of whose regions are
monochromatic.
Of special interest in communication complexity are specific kinds
of regions and partitions called rectangles, and tilings,
respectively:
Definition 3 (Rectangles, Tilings)
A rectangle in a matrix $A$ is a submatrix of $A$. A
tiling of a matrix $A$ is a partition of $A$ into rectangles.
Definition 4 (Refinements)
A tiling $T_{1}(f)$ of a matrix $A(f)$ is said to be a
refinement of another tiling $T_{2}(f)$ of $A(f)$ if every
rectangle in $T_{1}(f)$ is contained in some rectangle in $T_{2}(f)$.
Monochromatic rectangles and tilings are an important concept in
communication-complexity theory, because they are linked to the
execution of communication protocols. Every communication protocol
$P$ for a function $f$ can be thought of as follows:
1.
Let $R$ and $C$ be the sets of row and column indices of $A(f)$, respectively. For $R^{\prime}\subseteq R$ and $C^{\prime}\subseteq C$, we will abuse notation and write $R^{\prime}\times C^{\prime}$ to denote the submatrix of $A(f)$ obtained by deleting the rows not in $R^{\prime}$ and the columns not in $C^{\prime}$.
2.
While $R\times C$ is not monochromatic:
•
One party $i\in\{0,1\}$ sends a single bit $q$ (whose value
is based on $x_{i}$ and the history of communication).
•
If $i=1$, $q$ indicates whether $1$’s value is in one of
two disjoint sets $R_{1},R_{2}$ whose union equals $R$. If $x_{1}\in R_{1}$,
both parties set $R=R_{1}$. If $x_{1}\in R_{2}$, both parties set $R=R_{2}$.
•
If $i=2$, $q$ indicates whether $2$’s value is in one of
two disjoint sets $C_{1},C_{2}$ whose union equals $C$. If $x_{2}\in C_{1}$,
both parties set $C=C_{1}$. If $x_{2}\in C_{2}$, both parties set $C=C_{2}$.
3.
One of the parties sends a last message (consisting of up to $t$ bits) containing the value in all
entries of the monochromatic rectangle $R\times C$.
Observe that, for every pair of private inputs $(x_{1},x_{2})$, $P$
terminates at some monochromatic rectangle in $A(f)$ that contains
$(x_{1},x_{2})$. We refer to this rectangle as “the monochromatic
rectangle induced by $P$ for $(x_{1},x_{2})$”. We refer to the tiling
that consists of all rectangles induced by $P$ (for all pairs of
inputs) as “the monochromatic tiling induced by $P$”.
Remark 2.1
There are monochromatic tilings that cannot be induced by communication
protocols. For example, observe that the tiling in
Fig. 2 (which is essentially an example from [17]) has this property.
2.2 Perfect Privacy
Informally, we say that a two-party protocol is perfectly
privacy-preserving if the two parties (or a third party observing
the communication between them) cannot learn more from the execution
of the protocol than the value of the function the protocol
computes. (These definition can be extended naturally to protocols involving more than two participants.)
Formally, let $P$ be a communication protocol for a function $f$.
The communication string passed in $P$ is the concatenation
of all the messages $(q_{1},q_{2}\ldots)$ sent in the course of the
execution of $P$. Let $s_{(x_{1},x_{2})}$ denote the communication
string passed in $P$ if the inputs of the parties are $(x_{1},x_{2})$.
We are now ready to define perfect privacy. The following two
definitions handle privacy from the point of view of a party $i$
that does not want the other party (who is, of course, familiar not
only with the communication string, but also with his own
value) to learn more than necessary about $i$’s private information.
We say that a protocol is perfectly private with respect to party
$1$ if $1$ never learns more about party $2$’s private information
than necessary to compute the outcome.
Definition 5 (Perfect privacy with respect to 1)
[8, 17]
$P$ is perfectly private with respect to party $1$ if, for
every $x_{2},x^{\prime}_{2}$ such that $f(x_{1},x_{2})=f(x_{1},x^{\prime}_{2})$, it holds that
$s_{(x_{1},x_{2})}=s_{(x_{1},x^{\prime}_{2})}$.
Informally, Def. 5 says that party 1’s knowledge of the communication
string passed in the protocol and his knowledge of $x_{1}$ do not
aid him in distinguishing between two possible inputs of $2$.
Similarly:
Definition 6 (Perfect privacy with respect to 2)
[8, 17]
$P$ is perfectly private with respect to party $2$ if, for
every $x_{1},x^{\prime}_{1}$ such that $f(x_{1},x_{2})=f(x^{\prime}_{1},x_{2})$, it holds that
$s_{(x_{1},x_{2})}=s_{(x^{\prime}_{1},x_{2})}$.
Observation 2.2
For any function $f$, the protocol in which party $i$ reveals $x_{i}$
and the other party computes the outcome of the function is
perfectly private with respect to $i$.
Definition 7 (Perfect subjective privacy)
$P$ achieves perfect subjective privacy if it is perfectly
private with respect to both parties.
The following definition considers a different form of privacy—privacy from a third party that observes the communication
string but has no a priori knowledge about the private
information of the two communicating parties. We refer to this
notion as “objective privacy”.
Definition 8 (Perfect objective privacy)
$P$ achieves perfect objective privacy if, for every two
pairs of inputs $(x_{1},x_{2})$ and $(x^{\prime}_{1},x^{\prime}_{2})$ such that
$f(x_{1},x_{2})=f(x^{\prime}_{1},x^{\prime}_{2})$, it holds that
$s_{(x_{1},x_{2})}=s_{(x^{\prime}_{1},x^{\prime}_{2})}$.
Kushilevitz [17] was the first to point out the interesting
connections between perfect privacy and communication-complexity theory. Intuitively, we can think of any monochromatic
rectangle $R$ in the tiling induced by a protocol $P$ as a set of
inputs that are indistinguishable to a third party. This is
because, by definition of $R$, for any two pairs of inputs in $R$, the
communication string passed in $P$ must be the same. Hence we can
think of the privacy of the protocol in terms of the tiling induced
by that protocol.
Ideally, every two pairs of inputs that are assigned the same
outcome by a function $f$ will belong to the same monochromatic
rectangle in the tiling induced by a protocol for $f$. This
observation enables a simple characterization of perfect
privacy-preserving mechanisms.
Definition 9 (Ideal monochromatic partitions)
A monochromatic region in a matrix $A$ is said to be a maximal
monochromatic region if no monochromatic region in $A$ properly
contains it. The ideal monochromatic partition of $A$ is made
up of the maximal monochromatic regions.
Observation 2.3
For every possible value in a matrix $A$, the maximal monochromatic
region that corresponds to this value is unique. This implies the
uniqueness of the ideal monochromatic partition for $A$.
Observation 2.4
(A characterization of perfectly privacy-preserving protocols)
A communication protocol $P$ for $f$ is perfectly privacy-preserving
iff the monochromatic tiling induced by $P$ is the ideal
monochromatic partition of $A(f)$. This holds for all of the above
notions of privacy.
3 Privacy-Approximation Ratios
Unfortunately, perfect privacy should not be taken for granted. As
shown by our results, in many environments,
perfect privacy can be either impossible or very costly (in terms of
communication complexity) to obtain.
To measure a protocol’s effect on privacy, relative to the
ideal—but perhaps impossible to implement—computation of the
outcome of a problem, we introduce the notion of
privacy-approximation ratios (PARs).
3.1 Worst-Case PARs
For any communication protocol $P$ for a function $f$, we denote by
$R^{P}(x_{1},x_{2})$ the monochromatic rectangle induced by $P$ for
$(x_{1},x_{2})$. We denote by $R^{I}(x_{1},x_{2})$ the monochromatic region
containing $A(f)_{(x_{1},x_{2})}$ in the ideal monochromatic partition
of $A(f)$. Intuitively, $R^{P}(x_{1},x_{2})$ is the set of inputs that are
indistinguishable from $(x_{1},x_{2})$ to $P$. $R^{I}(x_{1},x_{2})$ is the set
of inputs that would be indistinguishable from $(x_{1},x_{2})$ if
perfect privacy were preserved. We wish to asses how far one is from
the other. The size of a region $R$, denoted by $|R|$, is the
cardinality of $R$, i.e., the number of inputs in $R$.
We can now define worst-case objective PAR as follows:
Definition 10 (Worst-case objective PAR of $P$)
The worst-case objective privacy-approximation ratio of
communication protocol $P$ for function $f$ is
$$\alpha=\max_{(x_{1},x_{2})}\ \frac{|R^{I}(x_{1},x_{2})|}{|R^{P}(x_{1},x_{2})|}.$$
We say that $P$ is $\alpha$-objective-privacy-preserving in
the worst case.
Definition 11 ($i$-partitions)
The $1$-partition of a region $R$ in a matrix $A$ is the set of
disjoint rectangles $R_{x_{1}}=\{x_{1}\}\times\{x_{2}\ s.t.\ (x_{1},x_{2})\in R\}$ (over all possible inputs $x_{1}$).
$2$-partitions are defined analogously.
Intuitively, given any region $R$ in the matrix $A(f)$, if party
$i$’s actual private information is $x_{i}$, then $i$ can use this
knowledge to eliminate all the parts of $R$ other than $R_{x_{i}}$.
Hence, the other party should be concerned not with $R$ but rather
with the $i$-partition of $R$.
Definition 12 ($i$-induced tilings)
The $i$-induced tiling of a protocol $P$ is the refinement of the
tiling induced by $P$ obtained by $i$-partitioning each rectangle in
it.
Definition 13 ($i$-ideal monochromatic partitions)
The $i$-ideal monochromatic partition is the refinement of the ideal
monochromatic partition obtained by $i$-partitioning each region in
it.
For any communication protocol $P$ for a function $f$, we use
$R_{i}^{P}(x_{1},x_{2})$ to denote the monochromatic rectangle containing
$A(f)_{(x_{1},x_{2})}$ in the $i$-induced tiling for $P$. We denote by
$R_{i}^{I}(x_{1},x_{2})$ the monochromatic rectangle containing
$A(f)_{(x_{1},x_{2})}$ in the $i$-ideal monochromatic partition of
$A(f)$.
Definition 14 (Worst-case PAR of $P$ with respect to $i$)
The worst-case privacy-approximation ratio with respect to
$i$ of communication protocol $P$ for function $f$ is
$$\alpha=\max_{(x_{1},x_{2})}\ \frac{|R_{i}^{I}(x_{1},x_{2})|}{|R^{P}_{i}(x_{1},%
x_{2})|}.$$
We say that $P$ is $\alpha$-privacy-preserving with respect to
$i$ in the worst case.
Definition 15 (Worst-case subjective PAR of $P$)
The worst-case subjective privacy-approximation ratio of
communication protocol $P$ for function $f$ is the maximum of the
worst-case privacy-approximation ratio with respect to each party.
Definition 16 (Worst-case PAR)
The worst-case objective (subjective) PAR for a function $f$
is the minimum, over all protocols $P$ for $f$, of the worst-case objective (subjective) PAR of $P$.
3.2 Average-Case PARs
As we shall see below, it is also useful to define an average-case
version of PAR. As the name suggests, the average-case objective PAR
is the average ratio between the size of the monochromatic
rectangle containing the private inputs and the corresponding region
in the ideal monochromatic partition.
Definition 17 (Average-case objective PAR of $P$)
Let $D$ be a probability distribution over the space of inputs. The
average-case objective privacy-approximation ratio of
communication protocol $P$ for function $f$ is
$$\alpha=E_{D}\ [\frac{|R^{I}(x_{1},x_{2})|}{|R^{P}(x_{1},x_{2})|}].$$
We say that $P$ is $\alpha$-objective privacy-preserving in
the average case with distribution $D$ (or with respect to
$D$).
We define average-case PAR with respect to $i$ analogously, and average-case subjective PAR as the maximum over all players $i$ of the average-case PAR with respect to $i$.
We define the average-case objective (subjective) PAR for a function
$f$ as the minimum, over all protocols $P$ for $f$, of the average-case objective (subjective) PAR of $P$.
4 The Millionaires Problem and Public Goods: Bounds on PARs
In this section, we prove upper and lower bounds on the
privacy-approximation ratios for two classic problems: Yao’s
millionaires problem and the provision of a public good.
4.1 Problem Specifications
The millionaires problem.
Two millionaires want to know which one is richer.
Each millionaire’s wealth is private information known only to him,
and the millionaire wishes to keep it that way. The goal is to
discover the identity of the richer millionaire while preserving the
(subjective) privacy of both parties.
Definition 18 (The Millionaires Problem${}_{k}$)
Input: $x_{1},x_{2}\in\{0,\ldots,2^{k}-1\}$ (each represented by a $k$-bit string)
Output: the identity of the party with the
higher value, i.e., $\arg\max_{i\in\{0,1\}}x_{i}$ (breaking
ties lexicographically).
There cannot be a perfectly privacy-preserving communication protocol for The Millionaires Problem${}_{k}$ [17]. Hence, we are interested in the PARs for this
well studied problem.
The public-good problem.
There are two agents, each
with a private value in $\{0,\ldots,2^{k}-1\}$ that represents his
benefit from the construction of a public project (public good),
e.g., a bridge.111This is a discretization of the
classic public good problem, in which the private values are taken
from an interval of reals, as in [5, 1]. The goal of
the social planner is to build the public project only if the sum of
the agents’ values is at least its cost, where, as in [1],
the cost is set to be $2^{k}-1$.
Definition 19 (Public Good${}_{k}$)
Input: $x_{1},x_{2}\in\{0,\ldots,2^{k}-1\}$ (each represented by a $k$-bit string)
Output: “Build” if $x_{1}+x_{2}\geq 2^{k}-1$, “Do
Not Build” otherwise.
It is easy to show (via
Observation 2.4) that for Public Good${}_{k}$, as for The Millionaires Problem${}_{k}$,
no perfectly privacy-preserving communication protocol exists.
Therefore, we are interested in the PARs for this problem.
4.2 The Millionaires Problem
The following theorem shows that not only is perfect subjective
privacy unattainable for The Millionaires Problem${}_{k}$, but a stronger result holds:
Theorem 4.1 (A worst-case lower bound on subjective PAR)
No communication protocol for The Millionaires Problem${}_{k}$ has a worst-case subjective
PAR less than $2^{\frac{k}{2}}$.
Proof:
Consider a communication protocol $P$ for The Millionaires Problem${}_{k}$. Let $R$ represent
the space of possible inputs of millionaire $1$, and let $C$
represent the space of possible inputs of millionaire $2$. In the
beginning, $R=C=\{0,\ldots,2^{k}-1\}$. Consider the first (meaningful)
bit $q$ transmitted in course of $P$’s execution. Let us assume that
this bit is transmitted by millionaire $1$. This bit indicates
whether $1$’s value belongs to one of two disjoint subsets of $R$,
$R_{1}$ and $R_{2}$, whose union equals $R$. Because we are interested in the worst case, we can choose adversarially to
which of these subsets $1$’s input belongs. Without loss of
generality, let $0\in R_{1}$. We decide adversarially that $1$’s value
is in $R_{1}$ and set $R=R_{1}$. Similarly, if $q$ is transmitted by
millionaire $2$, then we set $C$ to be the subset of $C$ containing
$0$ in the partition of $2$’s inputs induced by $q$. We continue
this process recursively for each bit transmitted in $P$.
Observe that, as long as both $R$ and $C$ contain at least two values,
$P$ is incapable of computing The Millionaires Problem${}_{k}$. This is because $0$ belongs to
both $R$ and $C$, and so $P$ cannot eliminate, for either of the
millionaires, the possibility that that millionaire has a value of $0$
and the other millionaire has a positive value. Hence, this process
will go on until $P$ determines that the value of one of the
millionaires is exactly $0$, i.e., until either $R=\{0\}$ or
$C=\{0\}$. Let us examine these two cases:
•
Case I: $R=\{0\}$. Consider the subcase in which
$x_{2}$ equals $0$. Recall that $0\in C$, and so this is possible.
Observe that, in this case, $P$ determines the exact value of $x_{1}$,
despite the fact that, in the $2$-ideal-monochromatic partition, all $2^{k}$
possible values of $x_{1}$ are in the same monochromatic rectangle when
$x_{2}=0$ (because for all these values $1$ wins). Hence, we get a
lower bound of $2^{k}$ on the subjective privacy-approximation ratio.
•
Case II: $C=\{0\}$. Let $m$ denote the highest input in $R$.
We consider two subcases. If $m\leq 2^{\frac{k}{2}}$, then observe
that the worst-case subjective privacy-approximation ratio is at
least $2^{\frac{k}{2}}$. In the $2$-ideal-monochromatic partition,
all $2^{k}$ possible values of $x_{1}$ are in the same monochromatic
rectangle if $x_{2}=0$, and the fact that $m\leq 2^{\frac{k}{2}}$
implies that $|R|\leq 2^{\frac{k}{2}}$.
If, on the other hand, $m>2^{\frac{k}{2}}$, then consider the case
in which $x_{1}=m$ and $x_{2}=0$. Observe that, in the
$1$-ideal-monochromatic partition, all values of millionaire $2$ in
$\{0,\ldots,m-1\}$ are in the same monochromatic rectangle if
$x_{1}=m$. However, $P$ will enable millionaire $1$ to determine that millionaire $2$’s value
is exactly $0$. This implies a lower bound of $m$ on the subjective
privacy-approximation. We now use the fact that $m>2^{\frac{k}{2}}$
to conclude the proof.
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By contrast, we show that fairly good privacy guarantees can be
obtained in the average case. We define the Bisection Protocol for The Millionaires Problem${}_{k}$ as
follows: Ask each millionaire whether his value lies in
$[0,2^{k-1})$ or in $[2^{k-1},2^{k})$; continue this binary search
until the millionaires’ answers differ, at which point we know which
millionaire has the higher value. If the answers never differ the
tie is broken in favor of millionaire $1$.
We may exactly compute the average-case subjective PAR with respect to the uniform distribution for the Bisection Protocol applied to The Millionaires Problem${}_{k}$. Figure 3 illustrates the approach. The far left of the figure shows the ideal partition (for $k=3$) of the value space for The Millionaires Problem${}_{k}$; these regions are indicated with heavy lines in all parts of the figure. The center-left shows the $1$-partition of the regions in the ideal partition; the center-right shows the $1$-induced tiling that is induced by the Bisection Protocol. The far right illustrates how we may rearrange the tiles that partition the bottom-left region in the ideal partition (by reflecting them across the dashed line) to obtain a tiling of the value space that is the same as the tiling induced by applying the Bisection Auction to 2nd-Price Auction${}_{k}$.
Theorem 4.2
The average-case subjective PAR of the bisection protocol
The average-case subjective PAR with respect
to the uniform distribution for the Bisection Protocol applied to The Millionaires Problem${}_{k}$ is
$\frac{k}{2}+1$.
Proof:
Given a value of $i$, consider the $i$-induced-tiling
obtained by running the Bisection Protocol for The Millionaires Problem${}_{k}$ (as in the center-right of Fig. 3 for $i=1$).
Rearrange the rectangles in which player $i$ wins by reflecting them
across the line running from the bottom-left corner to the top-right
corner (the dashed line in the far right of Fig. 3). This produces a tiling of the value space in which the
region in which player $1$ wins is tiled by tiles of width $1$, and
the region in which player $2$ wins is tiled by tiles of height $1$;
in computing the average-case-approximate-privacy with respect to
$i$, the tile-size ratios that we use are the heights (widths) of
the tiles to the height (width) of the tile containing all values in
that column (row) for which player $1$ ($2$) wins. This tiling and
the tile-size ratios in question are exactly as in the computation
of the average-case objective privacy for 2nd-Price Auction${}_{k}$; the
argument used in Thm. 5.8 (for $g(k)=k$) below completes the proof.
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Consider the case in which a third party is observing the
interaction of the two millionaires. How much can this observer
learn about the private information of the two millionaires? We show
that, unlike the case of subjective privacy, good PARs are
unattainable even in the average case.
Because the values $(i,i)$ (in which case player $1$ wins) and the values $(i,i+1)$ (in which player $2$ wins) must all appear in different tiles in any tiling that refines the ideal partition of the value space for The Millionaires Problem${}_{k}$, any such tiling must include at least $2^{k}$ tiles in which player $1$ wins
and $2^{k}-1$ tiles in which player $2$ wins.
The total contribution of a tile in which player $1$ wins is the
number of values in that tile times the ratio of the ideal region
containing the tile to the size of the tile, divided by the total
number ($2^{2k}$) of values in the space. Each tile in which player
$1$ wins thus contributes $\frac{(1+2^{k})2^{k}}{2^{2k+1}}$ to the
average-case PAR under the uniform
distribution; similarly, each tile in which player $2$ wins
contributes $\frac{2^{k}(2^{k}-1)}{2^{2k+1}}$ to this quantity. This leads
directly to the following result.
Proposition 4.3 (A lower bound on average-case objective PAR)
The average-case objective PAR for The Millionaires Problem${}_{k}$ with respect to the uniform distribution is at least $2^{k}-\frac{1}{2}+2^{-(k+1)}$.
There are numerous different tilings of the value space that achieve
this ratio and that can be realized by communication protocols. For
the Bisection Protocol, we obtain the same exponential (in $k$) growth rate but
with a larger constant factor.
Proposition 4.4
(The average-case objective PAR of the bisection protocol)
The Bisection Protocol for The Millionaires Problem${}_{k}$ obtains an average-case objective
PAR of $3\cdot 2^{k-1}-\frac{1}{2}$ with
respect to the uniform distribution.
Proof:
The bisection mechanism induces a tiling that refines the ideal
partition and that has $2^{k+1}-1$ tiles in which the player $1$
wins and $2^{k}-1$ tiles in which the player $2$ wins. The
contributions of each of these tiles is as noted above, from which
the result follows.
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Finally, Table 1 summarizes our average-case PAR results (with respect to the uniform distribution) for The Millionaires Problem${}_{k}$.
4.3 The Public-Good Problem
The government is considering the construction of a bridge (a public good) at cost $c$. Each taxpayer has a $k$-bit private value that is the utility he would gain from the bridge if it were built. The government wants to build the bridge if and only if the sum of the taxpayers’ private values is at least $c$. In the case that $c=2^{k}-1$, we observe that $\hat{x}_{2}=c-x_{2}$ is again a $k$-bit value and that $x_{1}+x_{2}\geq c$ if and only if $x_{1}\geq\hat{x}_{2}$; from the perspective of PAR, this problem is equivalent to solving The Millionaires Problem${}_{k}$ on inputs $x_{1}$ and $\hat{x}_{2}$. We may apply our results for The Millionaires Problem${}_{k}$ to see that the public-good problem with $c=2^{k}-1$ has exponential average-case objective PAR with respect to the uniform distribution. Appendix B discusses average-case objective PAR for a truthful version of the public-good problem.
5 $2^{nd}$-Price Auctions: Bounds on PARs
In this section, we present upper and lower bounds on the
privacy-approximation ratios for the $2^{nd}$-price Vickrey auction.
5.1 Problem Specification
$2^{\mathrm{nd}}$-price Vickrey auction.
A single item
is offered to $2$ bidders, each with a private value for the item.
The auctioneer’s goal is to allocate the item to the bidder with the
highest value. The fundamental technique in mechanism design for
inducing truthful behavior in single-item auctions is Vickrey’s
$2^{nd}$-price auction [22]: Allocate the item to the
highest bidder, and charge him the second-highest bid.
Definition 20 (2nd-Price Auction${}_{k}$)
Input: $x_{1},x_{2}\in\{0,\ldots,2^{k}-1\}$ (each represented by a $k$-bit string)
Output: the identity of the party with the
higher value, i.e., $\arg\max_{i\in\{0,1\}}x_{i}$ (breaking
ties lexicographically), and the private information of the of the
other party.
Brandt and Sandholm [6] show that a perfectly
privacy-preserving communication protocol exists for 2nd-Price Auction${}_{k}$.
Specifically, perfect privacy is obtained via the
ascending-price English auction: Start with a price of $p=0$
for the item. In each time step, increase $p$ by $1$ until one of
the bidders indicates that his value for the item is less than $p$ (in each step first asking bidder $1$ and then, if necessary, asking bidder $2$). At that point, allocate the item to the other bidder for a price of
$p-1$. If $p$ reaches a value of $2^{k}-1$ (that is, the values of
both bidders are $2^{k}-1$) allocate the item to bidder $1$ for a
price of $2^{k}-1$.
Moreover, it is shown in [6] that the English auction is
essentially the only perfectly privacy-preserving protocol
for 2nd-Price Auction${}_{k}$. Thus, perfect privacy requires, in the worst-case, the
transmission of $\Omega(2^{k})$ bits. $2k$ bits suffice, because
bidders can simply reveal their inputs. Can we obtain “good”
privacy without paying such a high price in communication?
5.2 Objective Privacy PARs
We now consider objective privacy for 2nd-Price Auction${}_{k}$ (i.e.,
privacy with respect to the auctioneer). Bisection
auctions [14, 15] for 2nd-Price Auction${}_{k}$ are defined similarly to
the Bisection Protocol for The Millionaires Problem${}_{k}$: Use binary search to find a value $c$ that lies
between the two bidders’ values, and let the bidder with the higher
value be bidder $j$. (If the values do not differ, we will also
discover this; in this case, award the item to bidder $1$, who must
pay the common value.) Use binary search on the interval that
contains the value of the lower bidder in order to find the value of
the lower bidder. Bisection auctions are incentive-compatible in ex-post Nash [14, 15].
More generally, we refer to an auction protocol as a $c$-bisection
auction, for a constant $c\in(0,1)$, if in each step the interval
$R$ is partitioned into two disjoint subintervals: a lower
subinterval of size $c|R|$ and an upper subinterval of size
$(1-c)|R|$. Hence, the Bisection Auction is a $c$-bisection auction with
$c=\frac{1}{2}$. We prove that no $c$-bisection auction for The Millionaires Problem${}_{k}$ obtains a subexponential objective PAR:
Theorem 5.1 (A worst-case lower bound for $c$-bisection auctions)
For any constant $c>\frac{1}{2^{k}}$, the $c$-bisection auction for
2nd-Price Auction${}_{k}$ has a worst-case PAR of at least
$2^{\frac{k}{2}}$.
Proof:
Consider the ideal monochromatic partition of 2nd-Price Auction${}_{k}$ depicted for $k=3$ in
Fig. 1. Observe that, for perfect privacy to be
preserved, it must be that bidder $2$ transmits the first
(meaningful) bit, and that this bit partitions the space of inputs
into the leftmost shaded rectangle (the set $\{0,\ldots,2^{k}-1\}\times\{0\}$) and the rest of the value space (ignoring the rectangles depicted that further refine $\{0,\ldots,2^{k}-1\}\times\{1,\ldots,2^{k}-1\}$). What if the first bit is
transmitted by player $2$ and does not partition the space into
rectangles in that way? We observe that any other partition of the
space into two rectangles is such that, in the worst case, the
privacy-approximation ratio is at least $2^{\frac{k}{2}}$ (for any
value of $c$): If $c\leq 1-2^{-\frac{k}{2}}$, then the case in which
$x_{1}=c2^{k}-1$ gives us the lower bound. If, on the other hand, $c>1-2^{-\frac{k}{2}}$, then the case that $x_{1}=0$ gives us the lower
bound. Observe that such a bad PAR is also
the result of bidder $1$’s transmitting the first (meaningful) bit.
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By contrast, as for The Millionaires Problem${}_{k}$, reasonable privacy
guarantees are achievable in the average case:
Theorem 5.2
(The average-case objective PAR of the bisection auction) The average-case objective PAR of the Bisection Auction is
$\frac{k}{2}+1$ with respect to the uniform distribution.
Proof:
This follows by taking $g(k)=k$ in Thm. 5.8.
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We note that the worst-possible approximation of objective privacy
comes when the each value in the space is in a distinct tile; this
is the tiling induced by the sealed-bid auction. The resulting
average-case privacy-approximation ratio is exponential in $k$.
Proposition 5.3 (Largest possible objective PAR)
The largest possible (for any protocol) average-case objective
PAR with respect to the uniform distribution
for 2nd-Price Auction${}_{k}$ is
$$\frac{1}{2^{2k}}\left[\sum_{j=0}^{2^{k}-1}j^{2}+\sum_{j=1}^{2^{k}-1}j^{2}%
\right]=\frac{2}{3}2^{k}+\frac{1}{3}2^{-k}$$
5.3 Subjective Privacy PARs
We now look briefly at subjective
privacy for 2nd-Price Auction${}_{k}$. For subjective privacy with respect to $1$, we
start with the $1$-partition for 2nd-Price Auction${}_{k}$;
Fig. 4 shows the refinement of the
$1$-partition induced by the Bisection Auction for $k=4$. Separately considering
the refinement of the $2$-partition for 2nd-Price Auction${}_{k}$ by the Bisection Auction, we have
the following results.
Theorem 5.4
(The average-case PAR with respect to $1$ of the bisection auction) The average-case PAR with respect to $1$ of the Bisection Auction is
$$\frac{k+3}{4}-\frac{k-1}{2^{k+2}}$$
with respect to the uniform distribution.
Proof:
This follows by taking $g(k)=k$ in Thm. 5.11.
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Theorem 5.5
(The average-case PAR with respect to $2$ of the bisection auction) The average-case PAR with respect to $2$ of the Bisection Auction is
$$\frac{k+5}{4}+\frac{k-1}{2^{k+2}}$$
with respect to the uniform distribution.
Proof:
This follows by taking $g(k)=k$ in Thm. 5.12.
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Corollary 5.6
(The average-case subjective PAR of the bisection auction) The average-case subjective PAR of the Bisection Auction with respect to the uniform distribution is
$$\frac{k+5}{4}+\frac{k-1}{2^{k+2}}.$$
As for objective privacy, the sealed-bid auction gives the largest possible average-case subjective PAR.
Proposition 5.7 (Largest possible subjective PAR)
The largest possible (for any protocol) average-case subjective PAR with respect to the uniform distribution for 2nd-Price Auction${}_{k}$ is
$$\frac{2^{k}}{3}+1-\frac{1}{3\cdot 2^{k}}.$$
Proof:
For the sealed-bid auction, the average-case PAR with respect to $1$ is
$$\frac{1}{2^{2k}}\left[\sum_{j=1}^{2^{k}}j+\sum_{j=1}^{2^{k}-1}j^{2}\right]=%
\frac{2^{k}}{3}+\frac{1}{3\cdot 2^{k-1}}.$$
For the sealed-bid auction, the average-case PAR with respect to $2$ is
$$\frac{1}{2^{2k}}\left[\sum_{j=1}^{2^{k}}j^{2}+\sum_{j=1}^{2^{k}-1}j\right]=%
\frac{2^{k}}{3}+1-\frac{1}{3\cdot 2^{k}}.$$
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5.4 Bounded-Bisection Auctions
We now present a middle ground between the perfectly-private yet
highly inefficient (in terms of communication) ascending English
auction and the communication-efficient Bisection Auction whose average-case
objective PAR is linear in $k$ (and is thus unbounded as $k$ goes to
infinity): We bound the number of bisections, using an ascending
English auction to determine the outcome if it is not resolved by
the limited number of bisections.
We define the Bisection Auction${}_{g(k)}$ as follows: Given an instance of 2nd-Price Auction${}_{k}$, and a
integer-valued function $g(k)$ such that $0\leq g(k)\leq k$, run the Bisection Auction as above but
do at most $g(k)$ bisection operations. (Note that we will never do more than $k$ bisections.) If the outcome is
undetermined after $g(k)$ bisection operations, so that both
players’ values lie in an interval $I$ of size $2^{k-g(k)}$, apply
the ascending-price English auction to this interval to determine
the identity of the winning bidder and the value of the losing
bidder.
As $g(k)$ ranges from $0$ to $k$, the Bisection Auction${}_{g(k)}$ ranges from the ascending-price
English auction to the Bisection Auction. If we allow a fixed, positive number of
bisections ($g(k)=c>0$), computations show that for $c=1,2,3$ we obtain examples of protocols that do not
provide perfect privacy but that do have bounded average-case
objective PARs with respect to the uniform distribution. We wish to see if this holds for all positive $c$, determine the average-case objective PAR for general $g(k)$, and connect the amount of
communication needed with the approximation of privacy in this family of protocols. The following theorem allows us to do these things.
Theorem 5.8
For the Bisection Auction${}_{g(k)}$, the average-case objective PAR with respect to the uniform distribution equals
$$\frac{g(k)+3}{2}-\frac{2^{g(k)}}{2^{k+1}}+\frac{1}{2^{k+1}}-\frac{1}{2^{g(k)+1%
}}.$$
Proof:
Fix $k$, the number of bits used for bidding, and let $c=g(k)$ be the number of bisections; we have $0\leq c\leq k$, and we let $i=k-c$. Figure 5 illustrates this tiling for $k=4$, $c=2$, and $i=2$; note that the upper-left and lower-right quadrants have identical structure and that the lower-left and upper-right quadrants have no structure other than that of the ideal partition and the quadrant boundaries (which are induced by the first bisection operation performed).
Our general approach is the following. The average-case objective PAR with respect to the uniform distribution equals
$$\mathsf{PAR}=\frac{1}{2^{2k}}\sum_{(x_{1},x_{2})}\frac{|R^{I}(x_{1},x_{2})|}{|%
R^{P}(x_{1},x_{2})|},$$
where the sum is over all pairs $(x_{1},x_{2})$ in the value space; recall that $R^{I}(x_{1},x_{2})$ is the region in the ideal partition that contains $(x_{1},x_{2})$, and $R^{P}(x_{1},x_{2})$ is the rectangle in the tiling induced by the protocol that contains $(x_{1},x_{2})$. We may combine all of the terms corresponding to points in the same protocol-induced rectangle to obtain
$$\mathsf{PAR}=\frac{1}{2^{2k}}\sum_{R}|R|\frac{|R^{I}(R)|}{|R|}=\frac{1}{2^{2k}%
}\sum_{R}|R^{I}(R)|,$$
(1)
where the sums are now over protocol-induced rectangles $R$ (we will simplify notation and write $R$ instead of $R^{P}$), and $R^{I}(R)$ denotes the ideal region that contains the protocol-induced rectangle $R$. Each ideal region in which bidder $1$ wins is a rectangle of width $1$ and height at most $2^{k}$; each ideal region in which bidder $2$ wins is a rectangle of height $1$ and width strictly less than $2^{k}$. For a protocol-induced rectangle $R$, let $j_{R}=2^{k}-|R^{I}(R)|$. Let $a_{c,i}$ be the total number of tiles that appear in the tiling of the $k$-bit value space induced by the Bisection Auction${}_{g(k)}$ with $g(k)=c$, and let $b_{c,i}=\sum_{R}j_{R}$ (with this sum being over the protocol-induced tiles in this same partition). Then we may rewrite (1) as
$$\mathsf{PAR}_{c,i}=\frac{1}{2^{2k}}\sum_{R}(2^{k}-j_{R})=\frac{a_{c,i}2^{k}-b_%
{c,i}}{2^{2k}}.$$
(2)
(Note that (1) holds for general protocols; we now add the subscripts “$c,i$” to indicate the particular PAR we are computing.) We now determine $a_{c,i}$ and $b_{c,i}$.
Considering the tiling induced by $c+1$ bisections of a $c+i+1$-bit space (which has $a_{c+1,i}$ total tiles), the upper-left and lower-right quadrants each contain $a_{c,i}$ tiles while the lower-left and upper-right quadrants (as depicted in Fig. 5) each contribute $2^{c+i}$ tiles, so $a_{c+1,i}=2a_{c,i}+2^{c+i+1}$. When there are no bisections, the $i$-bit value space has $a_{0,i}=2^{i+1}-1$ tiles, from which we obtain $a_{c,i}=2^{c}\left(2^{i}(c+2)-1\right)$. The sum of $j_{R}$ over protocol-induced rectangles $R$ in the upper-left quadrant is $b_{c,i}$. For a rectangle $R$ in the lower-right quadrant, $j_{R}$ equals $2^{c+i}$ plus $j_{R^{\prime}}$, where $R^{\prime}$ is the corresponding rectangle in the upper-left quadrant; there are $a_{c,i}$ such $R$, so the sum of $j_{R}$ over protocol-induced rectangles $R$ in the upper-left quadrant is $b_{c,i}+a_{c,i}2^{c+i}$. Finally, the sum of $j_{R}$ over $R$ in the lower-left quadrant equals $\sum_{h=0}^{2^{c+i}-1}h$ and the sum over $R$ in the top-right quadrant equals $\sum_{h=1}^{2^{c+i}}h$. Thus, $b_{c+1,i}=2b_{c,i}+a_{c,i}2^{c+i}+2^{2(c+i)}$; with $b_{0,i}=\sum_{h=0}^{2^{i}-1}h+\sum_{h=1}^{2^{i}-1}h$, we obtain $b_{c,i}=2^{c+i-1}\left(\left(1+2^{c}\right)\left(-1+2^{i}\right)+2^{c+i}c\right)$. Rewriting (2), we obtain
$$\mathsf{PAR}_{c,i}=\frac{c+3}{2}-\frac{2^{c}}{2^{c+i+1}}+\frac{1}{2^{c+i+1}}-%
\frac{1}{2^{c+1}}.$$
Recalling that $k=c+i$, the proof is complete.
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For the protocols corresponding to values of $g(k)$
ranging from $0$ to $k$ (ranging from the ascending-price
English auction to the Bisection Auction), we may thus relate the amount of
communication saved (relative to the English auction) to the effect of this
on the PAR.
Corollary 5.9
Let $g$ be a function that maps non-negative integers to non-negative
integers. Then the average-case objective PAR with respect to the
uniform distribution for the Bisection Auction${}_{g(k)}$ is bounded
if $g$ is bounded and is unbounded if $g$ is unbounded.
We then have that the Bisection Auction${}_{g(k)}$ may require the exchange of $\Theta(k+2^{k-g(k)})$
bits, and it has an average-case objective PAR of $\Theta(1+g(k))$.
Remark 5.10
Some of the sequences that appear in the proof above also appear in other settings. For example, the sequences $\{a_{0,i}\}_{i}$, $\{a_{1,i}\}_{i}$, and $\{a_{2,i}\}_{i}$ are slightly shifted versions of sequences A000225, A033484, and A028399, respectively, in the OEIS [21], which notes other combinatorial interpretations of them.
5.4.1 Subjective privacy for bounded-bisection auctions
Theorem 5.11
(The average-case PAR w.r.t. $1$ of the bounded-bisection auction) The average-case PAR with respect to $1$ of the Bisection Auction${}_{g(k)}$ is
$$\frac{g(k)+5}{4}-\frac{1}{2^{g(k)+2}}-\frac{1}{2^{k-g(k)+1}}-\frac{g(k)-2}{2^{%
k+2}}$$
with respect to the uniform distribution.
Proof:
The approach is similar to that in the proof of Thm. 5.8. We start by specializing (1) to the present case.
Each ideal region in which bidder $1$ wins is a rectangle of size $1$; each ideal region in which bidder $2$ wins is a rectangle of height $1$ and width strictly less than $2^{k}$. For a protocol-induced rectangle $R$, let $j_{R}=2^{k}-|R^{I}(R)|$. Let $c=g(k)$ and let $i=k-c\geq 0$. Let $T_{c,i}^{1}$ be the refinement of the 2nd-Price-Auction${}_{k}$ $1$-partition of the $k$-bit value space induced by the Bisection-Auction${}_{g(k)}$. Let $x_{c,i}$ be the number of rectangles in $T_{c,i}^{1}$ in which bidder $2$ (the column player) wins, and let $y_{c,i}$ be the sum, over all rectangles $R$ in which bidder $2$ wins, of the quantity $2^{c+i}-|R^{I}(R)|$. Let $z_{c,i}$ be the number of rectangles $R$ in which bidder $1$ (the row player) wins.
Using $\mathsf{PAR}^{1}_{c,i}$ to denote the PAR w.r.t. bidder $1$ in this case ($c$ bisections and $i=k-c$), we may rewrite (1) as
$$\displaystyle\mathsf{PAR}^{1}_{c,i}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2^{2(c+i)}}\left[\left(\sum_{\begin{subarray}{c}R^{P}%
\mathrm{\ in\ which}\\
1\mathrm{\ wins}\end{subarray}}|R^{I}(R^{P})|\right)+\left(\sum_{\begin{%
subarray}{c}R^{P}\mathrm{\ in\ which}\\
2\mathrm{\ wins}\end{subarray}}|R^{I}(R^{P})|\right)\right]$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2^{2(c+i)}}\left[\left(z_{c,i}\right)+\left(2^{c+i}x_{c,%
i}-y_{c,i}\right)\right].$$
We now turn to the computation of $x_{c,i}$, $y_{c,i}$, and $z_{c,i}$.
Following the same approach as in the proof of Thm. 5.8, we have $x_{c+1,i}=2x_{c,i}+2^{c+i}$, $y_{c+1,i}=2y_{c,i}+\sum_{j=1}^{2^{c+i}}j+2^{c+i}x_{c,i}$, and $z_{c+1,i}=2z_{c,i}+2^{2(c+i)}$. With $x_{0,i}=2^{i}-1$, $y_{0,i}=\sum_{j=1}^{2^{i}-1}j$, and $z_{0,i}=\sum_{j=1}^{2^{c+1}}j$, we obtain
$$\displaystyle x_{c,i}$$
$$\displaystyle=$$
$$\displaystyle 2^{c-1}\left(2^{i}c+2^{i+1}-2\right),$$
$$\displaystyle y_{c,i}$$
$$\displaystyle=$$
$$\displaystyle 2^{c+i-2}\left(2^{c+i}c+2^{c+i}+2^{i}-2^{c+1}+c\right),\mathrm{%
\ and}$$
$$\displaystyle z_{c,i}$$
$$\displaystyle=$$
$$\displaystyle 2^{c+i-1}(2^{c+i}+1).$$
Using these in our expression for $\mathsf{PAR}^{1}_{c,i}$, we obtain
$$\mathsf{PAR}^{1}_{c,i}=\frac{c+5}{4}+\frac{2-c}{2^{c+i+2}}-\frac{1}{2^{i+1}}-%
\frac{1}{2^{c+2}}.$$
Recalling that $k=c+i$ and $g(k)=c$ completes the proof.
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Theorem 5.12
(The average-case PAR w.r.t. $2$ of the bounded-bisection auction) The average-case PAR with respect to $1$ of the Bisection Auction${}_{g(k)}$ is
$$\frac{g(k)+5}{4}-\frac{1}{2^{g(k)+2}}+\frac{g(k)}{2^{k+2}}$$
with respect to the uniform distribution.
Proof:
The approach is essentially the same as in the proof of Thm. 5.11, although the induced partition differs slightly.
Let $c=g(k)$ and let $i=k-c\geq 0$. Let $T_{c,i}^{2}$ be the refinement of the 2nd-Price-Auction${}_{k}$ $2$-partition of the $k$-bit value space induced by the Bisection-Auction${}_{g(k)}$. Let $u_{c,i}$ be the number of rectangles in $T_{c,i}^{2}$ in which bidder $1$ (the row player) wins, and let $v_{c,i}$ be the sum, over all rectangles $R$ in which bidder $1$ wins, of the quantity $2^{c+i}-|R^{I}(R)|$. Let $w_{c,i}$ be the number of rectangles $R$ in which bidder $2$ (the column player) wins. Using $\mathsf{PAR}^{2}_{c,i}$ to denote the PAR w.r.t. bidder $2$ in this case ($c$ bisections and $i=k-c$), we may rewrite (1) as
$$\mathsf{PAR}^{2}_{c,i}=\frac{1}{2^{2(c+i)}}\left[\left(2^{c+i}u_{c,i}-v_{c,i}%
\right)+\left(w_{c,i}\right)\right].$$
Mirroring the approach of the proof of Thm. 5.11, we have $u_{c+1,i}=2u_{c,i}+2^{c+i}$, $v_{c+1,i}=2v_{c,i}+2^{c+i-1}(2^{c+i}-1)+2^{c+i}u_{c,i}$, and $w_{c,i}=2^{c+i-1}(2^{c+i}-1)$. With $u_{0,i}=2^{i}$ and $v_{0,i}=2^{i-1}(2^{i}-1)$, we obtain
$$\displaystyle u_{c,i}$$
$$\displaystyle=$$
$$\displaystyle 2^{c+i-1}(c+2),$$
$$\displaystyle v_{c,i}$$
$$\displaystyle=$$
$$\displaystyle 2^{c+i-2}\left(2^{c+i}(c+1)+2^{i}-c-2\right),\mathrm{\ and}$$
$$\displaystyle w_{c,i}$$
$$\displaystyle=$$
$$\displaystyle 2^{c+i-1}(2^{c+i}-1).$$
Using these in our expression for $\mathsf{PAR}^{2}_{c,i}$, we obtain
$$\mathsf{PAR}^{2}_{c,i}=\frac{c+5}{4}-\frac{1}{2^{c+2}}+\frac{c}{2^{c+i+2}}.$$
Recalling that $k=c+i$ and $g(k)=c$ completes the proof.
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Because $g(k)\geq 0$, the average-case PAR with respect to $2$ is at least as large as the average-case PAR with respect to $1$; this gives the average-case subjective PAR of the Bisection Auction${}_{g(k)}$ as follows.
Corollary 5.13
(Average-case subjective PAR of the bounded-bisection auction) The average-case subjective PAR of the Bisection Auction${}_{g(k)}$ is
$$\frac{g(k)+5}{4}-\frac{1}{2^{g(k)+2}}+\frac{g(k)}{2^{k+2}}$$
with respect to the uniform distribution.
Finally, Table 2 summarizes the average-case PAR results (with respect to the uniform distribution) for 2nd-Price Auction${}_{k}$.
6 Discussion and Future Directions
6.1 Other Notions of Approximate Privacy
By our definitions, the worst-case/average-case PARs of a protocol
are determined by the worst-case/expected value of the expression
$\frac{|R^{I}(\mathbf{x})|}{|R^{P}(\mathbf{x})|}$, where $R^{P}(\mathbf{x})$ is the
monochromatic rectangle induced by $P$ for input $\mathbf{x}$, and
$R^{I}(\mathbf{x})$ is the monochromatic region containing
$A(f)_{\mathbf{x}}$ in the ideal monochromatic partition of $A(f)$.
That is, informally, we are interested in the ratio of the
size of the ideal monochromatic region for a specific pair of inputs to the size of the monochromatic rectangle induced by the protocol for that pair. More generally, we can define worst-case/average-case PARs with respect to a function $g$ by considering the ratio
$\frac{g(R^{I}(\mathbf{x}),\mathbf{x})}{g(R^{P}(\mathbf{x}),\mathbf{x})}$. Our definitions of PARs set $g(R,\mathbf{x})$ to be the cardinality of $R$. This captures the intuitive notion of the
indistinguishability of inputs that is natural to consider in the
context of privacy preservation. Other definitions of PARs may be
appropriate in analyzing other notions of privacy. We suggest a few here; further
investigation of these and other definitions provides many interesting avenues for future work.
Probability mass. Given a probability distribution $D$ over the parties’ inputs, a
seemingly natural choice of $g$ is the probability mass. That is,
for any region $R$, $g(R)=Pr_{D}(R)$, the probability (according to
$D$) that the input corresponds to an entry in $R$. However, a
simple example illustrates that this intuitive choice of
$g$ is problematic: Consider a problem for which $\{0,\ldots,n\}\times\{i\}$
is a maximal monochromatic region for $0\leq i\leq n-1$ as illustrated in the left part of Fig. 6. Let $P$ be the communication protocol consisting
of a single round in which party $1$ reveals whether or not his value is $0$;
this induces the monochromatic tiling with tiles $\{(0,i)\}$ and $\{(1,i),\ldots,(n,i)\}$ for each $i$ as illustrated in the right part of Fig. 6. Now, let $D_{1}$ and $D_{2}$ be the probability distributions over the inputs $\mathbf{x}=(x_{1},x_{2})$ such that, for $0\leq i\leq n-1$ and $1\leq j\leq n$, $Pr_{D_{1}}[(x_{1},x_{2})=(0,i)]=\frac{\epsilon}{n}$, $Pr_{D_{1}}[(x_{1},x_{2})=(j,i)]=\frac{1-\epsilon}{n^{2}}$, $Pr_{D_{2}}[(x_{1},x_{2})=(0,i)]=\frac{1-\epsilon}{n}$, and $Pr_{D_{2}}[(x_{1},x_{2})=(j,i)]=\frac{\epsilon}{n^{2}}$ for some small $\epsilon>0$. Intuitively, any
reasonable definition of PAR should imply that, for $D_{1}$, $P$
provides “bad” privacy guarantees (because w.h.p. it reveals the
value of $x_{1}$),
and, for $D_{2}$, $P$ provides “good”
privacy (because w.h.p. it reveals little
about $x_{1}$).
In sharp contrast, choosing $g$ to be the probability mass results
in the same average-case PAR in both cases.
Other additive functions. In our definition of PAR and in the probability-mass approach, each input $\mathbf{x}$ in a rectangle contributes to $g(R,\mathbf{x})$ in a way that is independent of the other inputs in $R$. Below, we discuss some natural approaches that violate this condition, but we start by noting that other functions that satisfy this condition may be of interest. For example, taking $g(R,\mathbf{x})=1+\sum_{\mathbf{y}\in R\setminus\mathbf{x}}d(\mathbf{x},%
\mathbf{y})$, where $d$ is some distance defined on the input space, gives our original definition of PAR when $d(x,y)=1-\delta_{\mathbf{x},\mathbf{y}}$ and might capture other interesting definitions (in which indistinguishable inputs that are farther away from $\mathbf{x}$ contribute more to the privacy for $\mathbf{x}$). (The addition of $1$ ensures that the ratio $g(R^{I},\mathbf{x})/g(R^{P},\mathbf{x})$ is defined, but that can be accomplished in other ways if needed.) Importantly, here and below, the notion of distance that is used might not be a Euclidean metric on the $n$-player input space $[0,2^{k}-1]^{n}$. It could instead (and likely would) focus on the problem-specific interpretation of the input space. Of course, there are may possible variations on this (e.g., also accounting for the probability mass).
Maximum distance. We might take the view that a protocol does not reveal much about an input $\mathbf{x}$ if there is another input that is “very different” from $\mathbf{x}$ that the protocol cannot distinguish from $\mathbf{x}$ (even if the total number of things that are indistinguishable from $\mathbf{x}$ under the protocol is relatively small). For some distance $d$ on the input space, we might than take $g$ to be something like $1+\max_{\mathbf{y}\in R\setminus\{\mathbf{x}\}}d(\mathbf{y},\mathbf{x})$.
Plausible deniability. One drawback to the maximum-distance approach is that it does not account for the probability associated with inputs that are far from $\mathbf{x}$ (according to a distance $d$) and that are indistinguishable from $\mathbf{x}$ under the protocol. While there might be an input $\mathbf{y}$ that is far away from $\mathbf{x}$ and indistinguishable from $\mathbf{x}$, the probability of $\mathbf{y}$ might be so small that the observer feels comfortable assuming that $\mathbf{y}$ does not occur. A more realistic approach might be one of “plausible deniability.” This makes use of a plausibility threshold—intuitively, the minimum probability that the “far away” inputs(s) (which is/are indistinguishable from $\mathbf{x}$) must be assigned in order to “distract” the observer from the true input $\mathbf{x}$. This threshold might correspond to, e.g., “reasonable doubt” or other levels of certainty. We then consider how far we can move away from $\mathbf{x}$ while still having “enough” mass (i.e., more than the plausibility threshold) associated with the elements indistinguishable from $\mathbf{x}$ that are still farther away. We could then take $g$ to be something like $1+\max\{d_{0}|Pr_{D}(\{\mathbf{y}\in R|d(\mathbf{y},\mathbf{x})\geq d_{0}\})/%
Pr_{D}(R)\geq t\}$; other variations might focus on mass that is concentrated in a particular direction from $\mathbf{x}$. (In quantifying privacy, we would expect to only consider those $R$ with positive probability, in which case dividing by $Pr_{D}(R)$ would not be problematic.) Here we use $Pr_{D}(R)$ to normalize the weight that is far away from $x$ before comparing it to the threshold $t$; intuitively, an observer would know that the value is in the same region as $x$, and so this seems to make the most sense.
Relative rectangle size. One observation is that a bidder likely has a very different view of an auctioneer’s being able to tell (when some particular protocol is used) whether his bid lies between $995$ and $1005$ than he does of the auctioneer’s being able to tell whether his bid lies between $5$ and $15$. In each case, however, the bids in the relevant range are indistinguishable under the protocol from $11$ possible bids. In particular, the privacy gained from an input’s being distinguishable from a fixed number of other inputs may (or may not) depend on the context of the problem and the intended interpretation of the values in the input space. This might lead to a choice of $g$ such as $diam_{d}(R)/|\mathbf{x}|$, where $diam_{d}$ is the diameter of $R$ with respect to some distance $d$ and $|\mathbf{x}|$ is some (problem-specific) measure of the size of $\mathbf{x}$ (e.g., bid value in an auction). Numerous variations on this are natural and may be worth investigating.
Information-theoretic approaches. Information-theoretic approaches using conditional entropy are also natural to consider when studying privacy, and these have been used in various settings. Most relevantly, Bar-Yehuda et al. [2] defined multiple measures based on the conditional mutual information about one player’s value (viewed as a random variable) revealed by the protocol trace and knowledge of the other player’s value. It would also be natural to study objective-PAR versions using the entropy of the random variable corresponding to the (multi-player) input conditioned only on the protocol output (and not the input of any player). Such approaches might facilitate the comparison of privacy between different problems.
6.2 Open Questions
There are many interesting directions for future research:
•
As discussed in the previous subsection, the definition and
exploration of other notions of PARs is a challenging and intriguing
direction for future work.
•
We have shown that, for both The Millionaires Problem${}_{k}$ and 2nd-Price Auction${}_{k}$, reasonable average-case PARs
with respect to the uniform distribution are achievable. We
conjecture that our upper bounds for these problems extend to
all possible distributions over inputs.
•
An interesting open question is proving lower bounds on the
average-case PARs for The Millionaires Problem${}_{k}$ and 2nd-Price Auction${}_{k}$.
•
It would be interesting to apply the PAR framework presented in this paper
to other functions.
•
The extension of our PAR framework to the $n$-party communication
model is a challenging direction for future research.
•
Starting from the same place that we did,
namely [8, 17], Bar-Yehuda et al. [2] provided
three definitions of approximate privacy. The one that seems most
relevant to the study of privacy-approximation ratios is their
notion of h-privacy. It would be interesting to know exactly
when and how it is possible to express PARs in terms
of $h$-privacy and vice versa.
Acknowledgements
We are grateful to audiences at University Residential Centre of Bertinoro, Boston University, DIMACS, the University of Massachusetts, Northwestern, Princeton, and Rutgers for helpful questions and feedback.
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Appendix A Relation to the Work of Bar-Yehuda et al. [2]
While there are certainly some parallels between the work here and the
Bar-Yehuda et al. work [2], there are significant differences in
what the two frameworks capture. Specifically:
1.
The results in [2] deal with what can be learned by a party who
knows one of the inputs. By contrast, our notion of objective
PAR captures the effect of a protocol on privacy with respect
to an external observer who does not know any of the players
private values.
2.
More importantly, the framework of [2] does not address the size of
monochromatic regions. As illustrated by the following example,
the ability to do so is necessary to capture the effects of
protocols on interesting aspects of privacy that are captured by
our definitions of PAR.
Consider the function $f:\{0,\ldots,2^{n}-1\}\times\{0,\ldots,2^{n}-1\}\rightarrow\{0,\ldots,2^{n-2}\}$
defined by $f(x,y)=floor(\frac{x}{2})$ if $x<2^{n-1}$ and $f(x,y)=2^{n-2}$ otherwise. Consider the following two protocols for $f$:
in $P$, player $1$ announces his value $x$ if $x<2^{n-1}$ and
otherwise sends $2^{n-1}$ (which indicates that $f(x,y)=2^{n-2}$); in $Q$, player $1$ announces $floor(\frac{x}{2})$ if $x<2^{n}-1$
and $x$ if $x=2^{n}-1$. Observe that each protocol induces
$2^{n-1}+1$ rectangles.
Intuitively, the effect on privacy of these two protocols is
different. For half of the inputs, $P$ reduces by a factor of $2$
the number of inputs from which they are indistinguishable
while not affecting the indistinguishability of the other
inputs. $Q$ does not affect the indistinguishability of the
inputs affected by $P$, but it does reduce the number of inputs
indistinguishable from a given input with $x\geq 2^{n-1}$ by at
least a factor of $2^{n-2}$.
Our notion of PAR is able to capture the different effects on
privacy of the protocols $P$ and $Q$. (The average-case objective
PARs are constant and exponential in $n$, respectively.) By
contrast, the three quantifications of privacy from [2]—$I_{c}$,
$I_{i}$, and $I_{c-i}$—do not distinguish between these two
protocols; we now sketch the arguments for this claim.
For each protocol, any function $h$ for which the protocol is
weakly $h$-private must take at least $2^{n-1}+1$ different
values. This bound is tight for both $P$ and $Q$. Thus, $I_{c}$ cannot
distinguish between the effects of $P$ and $Q$ on $f$.
The number of rectangles induced by $P$ that intersect each row
and column equals the number induced by $Q$. Considering the
geometric interpretation of $I_{P}$ and $I_{Q}$, as well as the
discussion in Sec. VII.A of [2], we see that $I_{i}$ and $I_{c-i}$
(applied to protocols) cannot distinguish between the effects
of $P$ and $Q$ on $f$.
Appendix B Truthful Public-Good Problem
B.1 Problem
As in Sec. 4.3, the government is considering the construction of a bridge at cost $c$. Each taxpayer has a private value that is the utility he would gain from the bridge if it were built, and the government wants to build the bridge if and only if the sum of the taxpayers’ private values is at least $c$. Now, in addition to determining whether to build the bridge, the government incentivizes truthful disclosure of the private values by requiring taxpayer $i$ to pay $c-\sum_{j\neq i}x_{j}$ if $\sum_{j\neq i}x_{j}<c$ but $\sum_{i}x_{i}\geq c$ (see, e.g., [20] for a discussion of this type of approach). The government should thus learn whether or not to build the bridge and how much, if anything, each taxpayer should pay. The formal description of the function is as follows; the corresponding ideal partition of the value space is shown in Fig. 7, in which regions for which the output is “Build” are just labeled with the appropriate value of $(t_{1},t_{2})$.
Definition 21 (Truthful Public Good${}_{k,c}$)
Input: $c,x_{1},x_{2}\in\{0,\ldots,2^{k}-1\}$ (each represented by a $k$-bit string)
Output: “Do Not Build” if $x_{1}+x_{2}<c$; “Build” and $(t_{1},t_{2})$ if $x_{1}+x_{2}\geq c$, where $t_{i}=c-x_{3-i}$ if $x_{3-i}<c$ and $x_{1}+x_{2}\geq c$, and $t_{i}=0$ otherwise.
B.2 Results
Proposition B.1
(Average-case objective PAR of Truthful Public Good${}_{k,c}$) The average-case objective PAR of Truthful Public Good${}_{k,c}$ with respect to the uniform distribution is
$$1+\frac{c^{3}}{2^{2k+1}}(1-\frac{1}{c^{2}}).$$
Proof:
We may rewrite Eq. 1 as (adding subscripts for the values of $k$ and $c$ in this problem):
$$\mathsf{PAR}_{k,c}=\frac{1}{2^{2k}}\left[\sum_{R_{\mathsf{DNB}}}|R^{I}(R_{%
\mathsf{DNB}})|+\sum_{R_{\mathsf{B}}}|R^{I}(R_{\mathsf{B}})|\right],$$
where the first sum is taken over rectangles $R_{\mathsf{DNB}}$ for which the output is “Do Not Build” and the second sum is taken over rectangles $R_{\mathsf{B}}$ for which the output is “Build” together with some $(t_{1},t_{2})$. Using the same argument as for The Millionaires Problem${}_{k}$, the first sum must be taken over at least $c$ rectangles; the ideal region containing these rectangles has size $\sum_{i=1}^{c}i=c(c+1)/2$. Considering the second sum, each of the ideal regions containing a protocol-induced rectangle is in fact a rectangle. If the protocol did not further partition these rectangles (and it is easy to see that such protocols exist) then the total contribution of the second sum is just the total number of inputs for which the output is “Do Not Build” together with some pair $(t_{1},t_{2})$, i.e., this contribution is $4^{k}-c(c+1)/2$. We may thus rewrite $\mathsf{PAR}_{k,c}$ as
$$\mathsf{PAR}_{k,c}=\frac{1}{2^{2k}}\left[c\frac{c(c+1)}{2}+4^{k}-\frac{c(c+1)}%
{2}\right]=1+\frac{c^{3}}{2^{2k+1}}(1-\frac{1}{c^{2}})$$
$\hskip 0.0pt\vbox{\hrule width 100% height 1px\hbox{\vrule width 1px\hbox to 6%
.67pt{\hfill\vbox to 6.67pt{\vfill}}\vrule width 1px}\hrule width 100% height %
1px}\vskip 3.0pt plus 1.0pt minus 1.0pt$
Unsurprisingly, if we take $c=2^{k}-1$ (as in Public Good${}_{k}$ in Sec. 4.3), we obtain $\mathsf{PAR}_{k,2^{k}-1}=2^{k-1}-\frac{1}{2}+\frac{1}{2^{k}}$, which is essentially half of the average-case PAR for The Millionaires Problem${}_{k}$. |
Group cohomology and control of $p$–fusion
David J. Benson
D. J. Benson
Institute of Mathematics, University of Aberdeen, Aberdeen AB24 3UE,
United Kingdom
J. Grodal and E. Henke
Department of Mathematical Sciences, University of
Copenhagen, Copenhagen, Denmark
44email: jg@math.ku.dk , henke@math.ku.dk
Jesper Grodal
D. J. Benson
Institute of Mathematics, University of Aberdeen, Aberdeen AB24 3UE,
United Kingdom
J. Grodal and E. Henke
Department of Mathematical Sciences, University of
Copenhagen, Copenhagen, Denmark
44email: jg@math.ku.dk , henke@math.ku.dk
Ellen Henke
All three authors were supported by the Danish National Research
Foundation through the Centre for Symmetry and
Deformation (DNRF92). The second author was also supported by a
European Science Foundation EURYI grant.
D. J. Benson
Institute of Mathematics, University of Aberdeen, Aberdeen AB24 3UE,
United Kingdom
J. Grodal and E. Henke
Department of Mathematical Sciences, University of
Copenhagen, Copenhagen, Denmark
44email: jg@math.ku.dk , henke@math.ku.dk
Abstract
We show that if an inclusion of finite groups $H\leq G$ of index
prime to $p$ induces a homeomorphism of mod $p$ cohomology varieties,
or equivalently an $F$–isomorphism in mod $p$ cohomology, then $H$ controls $p$–fusion
in $G$, if $p$ is odd. This generalizes classical results of Quillen
who proved this when $H$ is a Sylow $p$–subgroup, and furthermore
implies a hitherto difficult result of Mislin about cohomology
isomorphisms. For $p=2$ we give analogous
results, at the cost of replacing mod $p$ cohomology with higher
chromatic cohomology theories.
The results are consequences of a general algebraic
theorem we prove, that says that
isomorphisms between $p$–fusion systems over the same finite $p$–group
are detected on elementary
abelian $p$–groups if $p$ odd and abelian $2$–groups of exponent at most
$4$ if $p=2$.
Keywords:group cohomology $p$–fusion $F$–isomorphism HKR characters
MSC: 20J06 20D20 20J05
1 Introduction
The variety of the mod $p$ cohomology ring of a finite group was first
studied by
Quillen in his fundamental 1971 paper
Quillen:1971b+c , and has been a central tool in group
cohomology since then. The variety describes the mod $p$ group cohomology ring
up to $F$–isomorphism, i.e., a ring homomorphism
with nilpotent kernel and where every element in the target
raised to a $p^{k}$th power lies in the image; see
(Quillen:1971b+c, , Prop. B.8-9) and Remark 4.1.
Quillen’s first application of the theory was to show in Quillen:1971a that if the
Sylow $p$–subgroup inclusion $S\leq G$ induces an $F$–isomorphism
on mod $p$ cohomology
then $S$ controls
$p$–fusion in $G$, if $p$ is odd, which in this case means that
$G$ is $p$–nilpotent.
Quillen’s result has subsequently been revisited
in a number of contexts Henn:1990a ; Brunetti:1998a ; Gonzalez-Sanchez:2010a ; Isaacs/Navarro:2010a ; arXiv:1107.5158v2 , however all retaining the
hypothesis that $S$ is a Sylow $p$–subgroup in $G$.
The main goal of this paper is to considerably strengthen Quillen’s
result by replacing $S$ by an arbitrary subgroup $H$ of $G$ containing
$S$, thereby moving past $p$–nilpotent groups to all finite groups.
We recall that for $S\leq H\leq G$, $H$ is said to control $p$–fusion in $G$, if pairs of
tuples of elements of $S$ are conjugate in $H$ if they are conjugate
in $G$, or equivalently if for
all $p$–subgroups $P,Q\leq S$, $N_{H}(P,Q)/C_{H}(P)$ equals
$N_{G}(P,Q)/C_{G}(P)$ as homomorphisms from $P$ to $Q$.
Theorem A ($F$–isomorphism implies control of $p$–fusion, $p$
odd).
Let $\iota\colon\thinspace H\leq G$ be an inclusion of finite groups of
index prime to $p$, $p$ an odd prime, and consider the induced map
on mod $p$ group cohomology $\iota^{*}\colon\thinspace H^{*}(G;\mathbb{F}_{p})\to H^{*}(H;\mathbb{F}_{p})$. If for each $x\in H^{*}(H;\mathbb{F}_{p})$, $x^{p^{k}}\in\operatorname{im}(\iota^{*})$ for some $k\geq 0$,
then $H$ controls $p$–fusion in $G$.
Recall that $\iota^{*}$ is injective by an easy transfer
argument (Evens:1991a, , Prop. 4.2.5), since $p\nmid|G:H|$. Hence, the condition above that for each $x\in H^{*}(H,\mathbb{F}_{p})$
there exists $k\geq 0$ with $x^{p^{k}}\in\operatorname{im}(\iota^{*})$, is in fact
equivalent to $\iota^{*}$ being an $F$-isomorphism. Note that by the classical 1956 Cartan–Eilenberg stable elements formula
(Cartan/Eilenberg:1956a, , XII.10.1), $\iota^{*}$ is an (actual)
isomorphism if $H$ controls $p$–fusion in $G$, so the converse also holds.
The assumption in Theorem A that $H$ and $G$ share a common
Sylow $p$–subgroup is necessary as the inclusion $C_{p}\to C_{p^{2}}$
shows. Likewise the assumption that $p$ is odd is necessary, as
Quillen’s original example $Q_{8}<2A_{4}=Q_{8}\rtimes C_{3}$ shows. Stronger yet, we show in
Example 4.2 that for any $n$
there exists an inclusion $H\leq G$ of odd index with different
$2$–fusion but which
induces a mod $2$ cohomology isomorphism modulo
the class of
$n$–nilpotent unstable modules $\mathcal{N}\!\mathit{il}_{n}$
(Schwartz:1994a, , Ch. 6); $F$–isomorphism
means isomorphism modulo the largest class $\mathcal{N}\!\mathit{il}_{1}$.
Our proof of Theorem A is purely algebraic: By
(Quillen:1971b+c, , Prop. 10.9(ii)$\Rightarrow$(i)) (or the
algebraic reference Alperin:2006a ) $F$-isomorphism in mod $p$ group cohomology implies control of fusion on elementary abelian subgroups. Thus, Theorem A follows from the following group
theoretic statement, which is of
independent interest. For $p$ odd it says that if $H$ controls
$p$–fusion in $G$ on elementary abelian $p$–subgroups then it in fact
controls $p$–fusion. We formulate and prove the statement in terms of
fusion systems, and refer the reader for example to
Aschbacher/Kessar/Oliver:2011a for definitions and information
about these—we also recap the essential definitions in Section 2.
Theorem B (Small exponent abelian $p$–subgroups control $p$–fusion).
Let $\mathcal{G}\leq\mathcal{F}$ be two saturated fusion systems on the same
finite $p$–group $S$.
Suppose that
$\operatorname{Hom}_{\mathcal{G}}(A,B)=\operatorname{Hom}_{\mathcal{F}}(A,B)$ for all $A,B\leq S$ with $A,B$
elementary abelian if $p$ is odd, and abelian of exponent at
most $4$ if $p=2$. Then $\mathcal{G}=\mathcal{F}$.
Our proof of this theorem is rather short. In outline we use Alperin’s Fusion Theorem to reduce to a situation where we can apply
results of J. G. Thompson on $p^{\prime}$–automorphisms of $p$–groups
(Gorenstein:1968a, , Ch. 5.3). Consequently, our proof of Theorem A is also relatively elementary. In particular, at odd primes, we obtain a comparatively simple algebraic proof of Mislin’s Theorem. This theorem states that, for a homomorphism $\varphi\colon\thinspace H\to G$ of finite groups, which induces an isomorphism in mod $p$ group cohomology, $|\ker(\varphi)|$ and
$|G:\varphi(H)|$ are coprime to $p$ and $\varphi(H)$ controls $p$–fusion
in $G$. Here the first part is a 1978 theorem of Jackowski
(Jackowski:1978a, , Thm. 1.3). (Jackowski gave a topological
argument, but a short algebraic proof exists via Tate
cohomology; see (Benson:1991b, , Thm. 5.16.1) with ${\mathbb{Z}}$
replaced by ${\mathbb{Z}}_{(p)}$.) So the proof of Mislin’s Theorem reduces quickly to the situation that $\varphi$ is an inclusion of finite groups of index prime to $p$, where the statement follows from Theorem A if $p$ is odd. Mislin’s original proof of his theorem uses the
Dwyer–Zabrodsky theorem Dwyer/Zabrodsky:1987a in algebraic
topology, whose proof again relies on Lannes’ theory Lannes:1992a , extending Miller’s proof of the
Sullivan conjecture Miller:1984a .
In the early 1990s, for example at the 1994 Banff
conference on representation theory, Alperin made the highly publicized challenge
to find a purely algebraic proof of Mislin’s theorem, and this was pursued
by many authors.
Symonds Symonds:2004a ,
following an idea of Robinson (Robinson:1998a, , §7),
provided an algebraic reduction of the problem to a statement about
cohomology of trivial source modules, which he then proved
topologically. Algebraic proofs were finally completed
independently by Hida Hida:2007a and
Okuyama Okuyama:2006a , who gave algebraic proofs of
Symonds’ statement, through
quite delicate arguments in modular representation theory.
(See also e.g., Alperin:2006a and Symonds:2007b .)
We now come to a further application of Theorem B. As remarked above, the assumptions in Theorem A that $p$ is odd and $|G:H|$ is prime to $p$ are both in fact necessary. Switching from mod $p$ cohomology to generalized cohomology theories, we can
however combine the methods of Theorem B with Hopkins–Kuhn–Ravenel (HKR) generalized
character theory Hopkins/Kuhn/Ravenel:1992a ; Hopkins/Kuhn/Ravenel:2000a to obtain
a statement that holds for all primes, and that also avoids the assumption that $H$
and $G$ share a common Sylow $p$–subgroup.
Theorem C (Chromatic group cohomology isomorphism implies control
of $p$–fusion).
Let $\varphi\colon\thinspace H\to G$ be a homomorphism of finite groups, and let $E(n)$ denote height $n$ Morava $E$–theory at
a fixed prime $p$. Suppose that $\varphi$ induces an isomorphism
$$\varphi^{*}\colon\thinspace E(n)^{*}(BG)[\frac{1}{p}]\xrightarrow{\hskip 5.690%
551pt{\sim}\hskip 5.690551pt}E(n)^{*}(BH)[\frac{1}{p}]$$
for some $n\geq{\operatorname{rk}_{p}}(G)$. Then
$|\ker(\varphi)|$ and $|G:\varphi(H)|$ are prime to $p$, and $\varphi(H)$ controls $p$–fusion
in $G$.
In fact our proof works not just for $E(n)$ but for
any height $n$ cohomology theory satisfying the assumptions listed in
(Hopkins/Kuhn/Ravenel:2000a, , Thm. C). We
recall that for height $n$ Morava $E$–theory, $E(n)^{*}(pt)=W(\mathbb{F}_{p^{n}})\llbracket w_{1},\ldots,w_{n-1}\rrbracket[u,%
u^{-1}]$, with $W(\mathbb{F}_{p^{n}})$ the
unramified extension of degree $n$ of the $p$–adic integers, $|w_{i}|=0$ and $|u|=-2$.
As usual, the notation $[\frac{1}{p}]$ means that we invert
$p$ after taking cohomology, producing a $\mathbb{Q}_{p}$–algebra.
The converse to Theorem C is clear, e.g., by the standard
Cartan–Eilenberg stable elements formula and the fact that a mod $p$ cohomology isomorphism of spaces induces an $E(n)^{*}$–isomorphism. Theorem C also provides a new proof
of a strong form of Mislin’s
theorem, assuming only isomorphism in large degrees. This proof is valid at all
primes, but replaces the reliance on Quillen’s
variety theory by the (currently) less algebraic HKR character
theory; indeed the proof mirrors that of Atiyah’s 1961 $p$–nilpotence
criterion Quillen:1971a ; Atiyah:1961a , replacing
$K$–theory by higher chromatic $E(n)$–theories; see Remark 4.3.
A $p$–rank restriction in Theorem C is indeed necessary:
$\mathbb{F}_{p^{2}}\rtimes\mathbb{F}_{p^{2}}^{\times}<(\mathbb{F}_{p^{2}}%
\rtimes\mathbb{F}_{p^{2}}^{\times})\rtimes\operatorname{Aut}(\mathbb{F}_{p^{2}})$ is an example of an inclusion of groups of
index prime to $p$, for $p$ odd, which is an $E(1)^{*}[\frac{1}{p}]$–equivalence, by
HKR character theory (3.3), but with different
$p$–fusion; the same example with $\mathbb{F}_{p^{2}}$ replaced by $\mathbb{F}_{2^{3}}$
works for
$p=2$. We speculate that the bound $n\geq{\operatorname{rk}_{p}}(G)$ we give may be
close to optimal, but we currently do not know an example to this effect.
Finally, we remark that isomorphism on $E(n)^{*}$ is equivalent to
isomorphism on $n$th Morava $K$–theory $K(n)^{*}$, whereas an
$E(n)^{*}[\frac{1}{p}]$–isomorphism is a priori significantly weaker—see
Remark 4.4 for a variety interpretation of
Theorem C and
Remark 4.5 for the connection to other stable homotopy theory results.
To prove Theorem C we need the following variant of
Theorem B, where we drop the assumption of a common Sylow
$p$-subgroup, but on the other hand assume the same fusion on all
abelian $p$–subgroups—it again appears to be new, even in special
cases.
Theorem D (Abelian $p$–subgroups control fusion).
Assume that a finite group homomorphism
$\varphi\colon\thinspace H\to G$ induces a bijection
$$\operatorname{Rep}(A,H)\xrightarrow{\hskip 5.690551pt{\sim}\hskip 5.690551pt}%
\operatorname{Rep}(A,G)$$
for all finite abelian $p$–groups $A$ with ${\operatorname{rk}_{p}}(A)\leq{\operatorname{rk}_{p}}(G)$. Then
$|\ker(\varphi)|$ and $|G:\varphi(H)|$ are prime to $p$, and $\varphi(H)$ controls $p$–fusion
in $G$.
More generally, suppose that $\mathcal{F}$ and $\mathcal{G}$ are saturated fusion
systems on finite $p$–groups $S$ and $T$ respectively,
and that $\varphi\colon\thinspace T\to S$ is a fusion preserving homomorphism inducing a bijection $\operatorname{Rep}(A,\mathcal{G})\xrightarrow{\sim}\operatorname{Rep}(A,%
\mathcal{F})$ for
any finite abelian $p$–group $A$ with ${\operatorname{rk}_{p}}(A)\leq{\operatorname{rk}_{p}}(S)$. Then
$\varphi$ induces an isomorphism from $T$ to $S$ and $\mathcal{G}$ to $\mathcal{F}$.
Here $\operatorname{Rep}(A,G)$ denotes the quotient of $\operatorname{Hom}(A,G)$ where we identify
$\varphi$ with $c_{g}\circ\varphi$ for all $g\in G$, and likewise $\operatorname{Rep}(A,\mathcal{F})$ is the quotient of $\operatorname{Hom}(A,S)$,
identifying two morphisms if they differ by a morphism in $\mathcal{F}$; we
spell out what the assumptions of the theorem mean in
Lemma 2.6.
Finally, we remark that Theorems A and C can be formulated in
terms of the fusions systems of the groups, and they should hold for
abstract fusion systems as well. Indeed, as is clear from our proofs, the
only missing piece is a reference for the Quillen
stratification and the HKR character theorem in that context—we will however not pursue this here.
2 $p^{\prime}$–automorphisms of $p$–groups and proofs of Theorems B and D
The goal of this section is to prove Theorems B and D
by group theoretic methods, combining manipulations with fusion
systems with results of J. G. Thompson on $p^{\prime}$–automorphisms of
$p$–groups, which can by now be found in textbooks.
Thompson’s critical subgroup
theorem (Feit/Thompson:1963a, , Lem. 2.8.2) (see also the textbook
reference (Gorenstein:1968a, , Thm. 5.3.11)) says that for any
finite $p$–group $P$ there exists a
characteristic subgroup $C$ of $P$ such that $C/Z(C)$ is elementary
abelian, $[P,C]\leq Z(C)$, $C_{P}(C)=Z(C)$, and every nontrivial
$p^{\prime}$–automorphism of $P$ restricts to a non-trivial
$p^{\prime}$–automorphism of $C$. Our main classical group theoretic
tool in this paper is a variant of that theorem, where instead of a
critical subgroup we use a certain characteristic subgroup of $P$ of small exponent and consider its maximal abelian subgroups.
Theorem 2.1 (Small exponent abelian subgroups detect $p^{\prime}$–automorphisms).
Let $P$ be a finite $p$–group. There exists a characteristic subgroup
$D$ of $P$, of exponent $p$ if $p$ is odd and exponent $4$ if
$p=2$, such that $[D,P]\leq Z(D)$, and such that every non-trivial
$p^{\prime}$–automorphism of $P$ restricts to a non-trivial automorphism of
$D$. Furthermore, for any such $D$ and any
maximal (with respect to inclusion) abelian
subgroup $A$ of $D$ we have $A\unlhd P$ and $C_{\operatorname{Aut}(P)}(A)$ is a $p$–group.
Note that the example of the extra-special group $p^{1+2}_{+}$ shows that an
abelian characteristic subgroup that detects $p^{\prime}$–automorphisms need not exist.
Proof of Theorem 2.1.
Taking $D=\Omega_{1}(C)$, the subgroup generated by elements of
order $p$ of a critical subgroup $C$, produces such a subgroup $D$ as
in the theorem, for
$p$ odd, as proved in (Gorenstein:1968a, , Thm. 5.3.13). For
$p=2$ the claim holds for $D=\Omega_{2}(C)$, the
subgroup of $C$ generated by elements of order at most $2^{2}$; we establish this fact
in Lemma 2.2 below.
For the last part, let $A$ be a maximal abelian subgroup of $D$ with
respect to inclusion. Since $[A,P]\leq Z(D)\leq A$ it follows that $A\unlhd P$. Furthermore, if $B\leq C_{\operatorname{Aut}(P)}(A)$ is a $p^{\prime}$–group, then $A\times B$ acts on
$P$ and thus on $D$. Since $A$ is maximal abelian it follows that $C_{D}(A)=A$, and in particular $B$ acts trivially
on $C_{D}(A)$; Thompson’s $A\times B$–lemma
(Gorenstein:1968a, , Thm. 5.3.4) now says that $[D,B]=1$ and so $B=1$, and we
conclude that $C_{\operatorname{Aut}(P)}(A)$ is a $p$–group as wanted.
∎
We now provide a proof of the postponed lemma for $p=2$.
Lemma 2.2.
Let $P$ be a $2$–group such that $P/Z(P)$ is elementary abelian. Then
for all $x,y\in P$, $(xy)^{4}=x^{4}y^{4}$ and in particular $\Omega_{2}(P)$
is of exponent at most $4$. Furthermore if $B$ is a $p^{\prime}$–group
of automorphisms of $P$ with $[\Omega_{2}(P),B]=1$, then $B=1$.
Proof.
Note that $(xy)^{2}=x^{2}(x^{-1}yxy^{-1})y^{2}$ with all three factors in
$Z(P)$, so
$$\begin{split}\displaystyle(xy)^{4}=x^{4}y^{4}(x^{-1}yxy^{-1})^{2}&%
\displaystyle=x^{4}y^{4}(x^{-1}(x^{-1}yxy^{-1})yxy^{-1})\\
&\displaystyle=x^{4}y^{4}(x^{-2}yx^{2}y^{-1})=x^{4}y^{4}.\end{split}$$
For the last statement about $p^{\prime}$–automorphisms we follow (Gorenstein:1968a, , Thm. 5.3.10).
Let $P$ be a minimal counterexample. If $Q$ is a
proper $B$–invariant subgroup of $P$ then $Q/(Q\cap Z(P))$ is
elementary abelian and $Q\cap Z(P)\leq Z(Q)$, so $Q/Z(Q)$ is
elementary abelian. Moreover, $\Omega_{2}(Q)\leq\Omega_{2}(P)$ and thus
$[\Omega_{2}(Q),B]=1$. So, as $P$ is a minimal counterexample,
$[Q,B]=1$. By (Gorenstein:1968a, , Thm. 5.2.4), $P$ is
non-abelian. So in particular, $Z(P)$ is a proper characteristic
subgroup of $P$ and thus $[Z(P),B]=1$.
We now show that $[P,B]\leq\Omega_{2}(P)$: Suppose $x\in P$ and $b\in B$, and note that $x^{4}\in Z(P)$, as $P/Z(P)$ is elementary
abelian, and thus $(x^{4})^{b}=x^{4}$, as $[Z(P),B]=1$. Hence
$[x,b]^{4}=(x^{-1}x^{b})^{4}=x^{-4}(x^{4})^{b}=x^{-4}x^{4}=1$ as wanted,
where we also used the first part of the lemma.
By assumption $[\Omega_{2}(P),B]=1$, so in particular $[[P,B],B]=1$ by
the above, and we conclude that $[P,B]=1$, by
(Gorenstein:1968a, , Thm. 5.3.6).
∎
In the case where $\mathcal{F}$ is the fusion system of $G=S\rtimes K$, with $p\nmid|K|$, Theorem B follows
directly from Theorem 2.1, as the action of elements of $K$
on $S$ is detected by small abelian subgroups of $S$, but the proof of the
general statement requires more work, and here fusion systems
enter in a more prominent way. The arguments can be translated into
the special case of ordinary finite groups, but doing so provides
no essential simplifications, and indeed, from our perspective, the
arguments are considerably shorter and more transparent in the setup of fusion systems.
Recall that a saturated fusion system $\mathcal{F}$ on a finite
$p$–group $S$ (Broto/Levi/Oliver:2003a, , Def. 1.2)(Aschbacher/Kessar/Oliver:2011a, , Prop. I.2.5)
is a category whose objects are the subgroups of $S$, and morphisms
are group monomorphims satisfying axioms which mimic those satisfied
by morphisms induced by conjugation in
some ambient group $G$. More precisely, conjugation by elements in $S$ need to be in
the category, every map needs to factor as an isomorphism followed by
an inclusion, and furthermore two non-trivial conditions need to be
satisfied, called the Sylow and extension axiom, which we recall below
together with some terminology. We refer to
Aschbacher/Kessar/Oliver:2011a and
Broto/Levi/Oliver:2003a for detailed information, and also
direct the reader to Puig’s original work Puig:2006a , where
terminology however differs.
A subgroup $Q\leq S$ is called fully $\mathcal{F}$–normalized if $|N_{S}(Q)|$ is
maximal among $\mathcal{F}$–conjugates of $Q$, it is called fully
$\mathcal{F}$–centralized if the corresponding property holds for the
centralizer, and it is called $\mathcal{F}$–centric if $C_{S}(Q^{\prime})=Z(Q^{\prime})$ for all
$\mathcal{F}$–conjugates $Q^{\prime}$ of $Q$. The Sylow axiom says that if $Q$ is
fully
$\mathcal{F}$–normalized then it is fully $\mathcal{F}$–centralized and $\operatorname{Aut}_{S}(Q)$ is
a Sylow $p$–subgroup of $\operatorname{Aut}_{\mathcal{F}}(Q)$. (Here $\operatorname{Aut}_{S}(Q)$ means the
automorphisms of $Q$ induced by elements in $S$.) The extension
axiom says that any morphism $\varphi\colon\thinspace Q\to S$ with $\varphi(Q)$
fully $\mathcal{F}$–centralized extends to
$$N_{\varphi}=\{g\in N_{S}(Q)|{}^{\varphi}(c_{g}|_{Q})\in\operatorname{Aut}_{S}(%
\varphi(Q))\}.$$
The first tool we need is the following variant of the extension axiom.
Lemma 2.3.
Fix a saturated fusion system $\mathcal{F}$ on $S$ and
let
$\varphi\colon\thinspace P\to S$ be any monomorphism (not necessarily in $\mathcal{F}$). For $Q\unlhd P$ and $\psi=\varphi|_{Q}$ the following hold.
1.
$N_{\psi}\geq P$ and ${}^{\psi}\!\operatorname{Aut}_{P}(Q)=\operatorname{Aut}_{\varphi(P)}(\varphi(Q))$.
2.
If $\psi\in\mathcal{F}$ and $\varphi(Q)$ is fully
$\mathcal{F}$–centralized then $\psi$ extends to $\hat{\psi}\in$
$\operatorname{Hom}_{\mathcal{F}}(P,\varphi(P)C_{S}(\varphi(Q)))$.
Proof.
For (1) we calculate, for any $g\in P$ and $x\in\varphi(Q)$,
$$({}^{\psi}c_{g})(x)=\psi\circ c_{g}\circ\psi^{-1}(x)=\psi(g\psi^{-1}(x)g^{-1})%
=\varphi(g)x\varphi(g^{-1})=c_{\varphi(g)}(x),$$
from which it is clear that
$N_{\psi}\geq P$ and ${}^{\psi}\operatorname{Aut}_{P}(Q)=\operatorname{Aut}_{\varphi(P)}(\varphi(Q))$.
For (2) note that the extension axioms imply that
$\psi$ extends to $\hat{\psi}\in\operatorname{Hom}_{\mathcal{F}}(P,S)$. And, since
$\operatorname{Aut}_{\varphi(P)}(\varphi(Q))={{}^{\psi}\operatorname{Aut}_{P}(Q%
)}=\operatorname{Aut}_{\hat{\psi}(P)}(\varphi(Q))$, where the last
equality is by applying (1) with $\hat{\psi}$
in place of $\varphi$, we conclude that $\hat{\psi}(P)\leq\varphi(P)C_{S}(\varphi(Q))$ as wanted.
∎
For the purpose of the next proof, recall that a
proper subgroup $H$ of a finite group $G$ is called strongly
$p$-embedded if $p$ divides the order of $H$ and, for all $g\in G\backslash H$, $H\cap{{}^{g}H}$ has order prime to $p$. Provided $p$ divides $|G|$, one easily shows
that $H$ is strongly $p$–embedded in $G$ if and only if $H$ contains a
Sylow $p$–subgroup $S$ of $G$ such that $N_{G}(R)\leq H$ for every
$1\neq R\leq S$ (see for example
(Gorenstein/Lyons/Solomon:1996a, , Lem. 17.10) or (Quillen:1978a, , Prop. 5.2)); in particular an
overgroup of a strongly $p$–embedded subgroup is again strongly
$p$–embedded, if it is a proper subgroup. (Groups with strongly
embedded subgroups play a central role in many aspects of local group theory, and in
particular they show up in connection with Alperin’s fusion theorem (Aschbacher/Kessar/Oliver:2011a, , Thm. I.3.6), though
we shall only indirectly need them in that capacity here.)
We now give the key step in deducing Theorem
B from Theorem 2.1, providing a way to show
that the fusion in $\mathcal{F}$ and $\mathcal{G}$ agree on all subgroups $P$ by
downward induction starting with $S$.
Main Lemma 2.4.
Let $\mathcal{G}\leq\mathcal{F}$ be two saturated fusion
systems on the same finite $p$–group $S$, and $P\leq S$ an
$\mathcal{F}$–centric and fully $\mathcal{F}$–normalized subgroup,
with $\operatorname{Aut}_{\mathcal{F}}(R)=\operatorname{Aut}_{\mathcal{G}}(R)$ for every $P<R\leq N_{S}(P)$.
Suppose that there exists a subgroup $Q\unlhd P$ with $\operatorname{Hom}_{\mathcal{F}}(Q,S)=\operatorname{Hom}_{\mathcal{G}}(Q,S)$. Then $\operatorname{Aut}_{\mathcal{F}}(P)=\langle\operatorname{Aut}_{\mathcal{G}}(P)%
,C_{\operatorname{Aut}_{\mathcal{F}}(P)}(Q)\rangle$.
Proof.
To ease the notation set $G=\operatorname{Aut}_{\mathcal{F}}(P)$, $H=\operatorname{Aut}_{\mathcal{G}}(P)$, and $\overline{G}=G/\operatorname{Inn}(P)$, and denote by
$\overline{U}$ the image in $\overline{G}$ of any subgroup $U\leq G$. We want to
show that $G=\langle H,C_{G}(Q)\rangle$.
Step 1: We first assume in addition that
$$C_{S}(\xi(Q))\leq P\mbox{ for all }\xi\in H$$
($$*$$)
and show that $G=HC_{G}(Q)$. Let $\gamma\in G$ be arbitrary; set
$\psi=(\gamma^{-1})|_{\gamma(Q)}\in\operatorname{Hom}_{\mathcal{F}}(\gamma(Q),Q)$. Then $\psi\in\operatorname{Hom}_{\mathcal{G}}(\gamma(Q),Q)$ by assumption. We claim
that $Q$ is fully centralized in $\mathcal{G}$, and postpone the proof to
Lemma 2.5 below, since it is a general statement. Granted
this, Lemma 2.3(2), applied to $\gamma^{-1}$ and $\mathcal{G}$ in
the roles of $\varphi$ and $\mathcal{F}$, implies that we can extend $\psi\colon\thinspace\gamma(Q)\to Q$ to $\hat{\psi}\colon\thinspace P\to PC_{S}(Q)$ in $\mathcal{G}$, and, as $C_{S}(Q)\leq P$
by assumption ($*$ ‣ 2), we conclude that $\hat{\psi}\in H$.
Since $\gamma\circ\hat{\psi}\in C_{G}(Q)$, we have $\gamma\in C_{G}(Q)H$, and, as $\gamma$ was arbitrary, this
yields $G=HC_{G}(Q)$ as required.
Step 2: If $P=S$ assumption ($*$ ‣ 2) is automatically
satisfied and the lemma follows from Step 1; likewise we are done if $H=G$.
In this step we
show that if $P<S$ and $H<G$ then
$\bar{H}$ is strongly $p$–embedded in $\bar{G}$.
Consider $P<R\leq N_{S}(P)$. For $\varphi\in N_{G}(\operatorname{Aut}_{R}(P))$ it follows from the extension axiom that $\varphi$
extends to $\hat{\varphi}\in\operatorname{Hom}_{\mathcal{F}}(R,S)$, since $P$ is fully
$\mathcal{F}$–normalized and $R\leq N_{\varphi}$, cf. Lemma 2.3. Furthermore, by Lemma 2.3(1),
$\operatorname{Aut}_{R}(P)={{}^{\varphi}\operatorname{Aut}_{R}(P)}=%
\operatorname{Aut}_{\hat{\varphi}(R)}(P)$, so since
$C_{S}(P)\leq P$ by $\mathcal{F}$–centricity of $P$, we have
$\hat{\varphi}(R)=R$. It then follows from
our hypothesis that $\hat{\varphi}\in\operatorname{Aut}_{\mathcal{G}}(R)$ and thus
$\varphi\in H$. We conclude that
$\overline{H}$ is strongly
$p$–embedded in $\overline{G}$.
Step 3: Finally set $H_{0}=\langle H,C_{G}(Q)\rangle$, and suppose for contradiction that there exists $\chi\in G\setminus H_{0}$.
Then by Step 2, $\overline{H}_{0}$ is a
strongly $p$–embedded subgroup of $\overline{G}$, so in particular $\operatorname{Aut}_{S}(P)\cap{}^{\chi}H_{0}=\operatorname{Inn}(P)$ as $\operatorname{Aut}_{S}(P)\leq H_{0}$. Note that $C_{G}(\chi(Q))={}^{\chi}C_{G}(Q)\leq{}^{\chi}H_{0}$, so $C_{\operatorname{Aut}_{S}(P)}(\chi(Q))\leq\operatorname{Aut}_{S}(P)\cap{}^{%
\chi}H_{0}=\operatorname{Inn}(P)$. Hence $N_{S}(P)\cap C_{S}(\chi(Q))\leq P$, using that $P$ is centric. Now, as $C_{S}(\chi(Q))P$ is a
$p$–group, $C_{S}(\chi(Q))\leq P$ (see
(Gorenstein:1968a, , Thm. 2.3.4) for this elementary property of
finite $p$–groups).
Note that $\xi\circ\chi\in G\backslash H_{0}$ for any $\xi\in H$; so as $\chi$ was arbitrary the
argument actually shows that $C_{S}(\xi(\chi((Q))))\leq P$ for any $\xi\in H$. But now ($*$ ‣ 2) holds with $\chi(Q)$ in place of $Q$. Observe also that $\operatorname{Hom}_{\mathcal{F}}(\chi(Q),S)=\operatorname{Hom}_{\mathcal{G}}(%
\chi(Q),S)$. For if $\varphi\in\operatorname{Hom}_{\mathcal{F}}(\chi(Q),S)$ then $\varphi\circ\chi$ and $\chi^{-1}$ are in $\operatorname{Hom}_{\mathcal{F}}(Q,S)=\operatorname{Hom}_{\mathcal{G}}(Q,S)$, so $\varphi=(\varphi\circ\chi)\circ\chi^{-1}\in\operatorname{Hom}_{\mathcal{G}}(%
\chi(Q),S)$.
Therefore Step 1 gives that $G=HC_{G}(\chi(Q))$. As $C_{G}(\chi(Q))$ is conjugate to $C_{G}(Q)$ in $G$, it follows that $C_{G}(\chi(Q))$ is conjugate to $C_{G}(Q)$ by an element of $H$ and thus $G=HC_{G}(Q)$. This is a contradiction, and we conclude
that $G=H_{0}$ as wanted.
∎
We next prove the postponed lemma.
Lemma 2.5.
Let $\mathcal{F}$ be a saturated fusion system on $S$
and suppose that $Q\unlhd P\leq S$, with $P$ fully $\mathcal{F}$–normalized, and $C_{S}(\xi(Q))\leq P$ for all $\xi\in\operatorname{Aut}_{\mathcal{F}}(P)$. Then $Q$ is fully $\mathcal{F}$–centralized.
Proof.
By (Linckelmann:2007a, , Lem. 2.6) we may choose $\alpha\colon\thinspace N_{S}(Q)\to S$ in $\mathcal{F}$ such that $\alpha(Q)$ is fully
normalized. Furthermore, as $P$ is fully normalized, again by (Linckelmann:2007a, , Lem. 2.6), there is $\beta\in\operatorname{Hom}_{\mathcal{F}}(N_{S}(\alpha(P)),N_{S}(P))$ such that
$\beta(\alpha(P))=P$.
Then
$$\beta(C_{S}(\alpha(Q))\cap N_{S}(\alpha(P)))\leq C_{S}(\beta(\alpha(Q)))\leq P%
=\beta(\alpha(P))$$
where the second inclusion follows by assumption as $\beta\circ\alpha$
restricts to an element of $\operatorname{Aut}_{\mathcal{F}}(P)$. This yields
$C_{S}(\alpha(Q))\cap N_{S}(\alpha(P))\leq\alpha(P)$, so
$$N_{C_{S}(\alpha(Q))\alpha(P)}(\alpha(P))=(C_{S}(\alpha(Q))\cap N_{S}(\alpha(P)%
))\alpha(P)=\alpha(P).$$
Thus, as
$C_{S}(\alpha(Q))\alpha(P)$ is a $p$–group, it follows from (Gorenstein:1968a, , Thm. 2.3.3(iii) and Thm. 2.3.4) that $C_{S}(\alpha(Q))\leq\alpha(P)$. Hence, $C_{S}(\alpha(Q))=C_{\alpha(P)}(\alpha(Q))=\alpha(C_{P}(Q))=\alpha(C_{S}(Q))$ where the last equality holds since our assumption gives $C_{S}(Q)\leq P$. It follows $|C_{S}(\alpha(Q))|=|\alpha(C_{S}(Q))|=|C_{S}(Q)|$; so $Q$ is fully $\mathcal{F}$–centralized as $\alpha(Q)$ is fully $\mathcal{F}$–centralized.
∎
Proof of Theorem B.
By Alperin’s fusion theorem, $\mathcal{F}$ is generated by $\mathcal{F}$–automorphisms of fully $\mathcal{F}$–normalized and
$\mathcal{F}$–centric subgroups; see
(Aschbacher/Kessar/Oliver:2011a, , Thm. I.3.6) (in fact we only
need
“$\mathcal{F}$–essential” subgroups and $S$). We want to show that
$\operatorname{Aut}_{\mathcal{G}}(P)=\operatorname{Aut}_{\mathcal{F}}(P)$ for all $P\leq S$; by downward induction
on the order we can assume that $\operatorname{Aut}_{\mathcal{G}}(R)=\operatorname{Aut}_{\mathcal{F}}(R)$ for all subgroups $R\leq S$
with $|R|>|P|$, and by the fusion theorem we can furthermore assume
that $P$ is $\mathcal{F}$–centric and fully $\mathcal{F}$–normalized.
Now choose a characteristic subgroup $D$ of $P$ as described in
Theorem 2.1, and a maximal abelian subgroup $A$ of
$D$, and recall that the theorem tells us that $A\unlhd P$ and
that $C_{\operatorname{Aut}_{\mathcal{F}}(P)}(A)$ is a $p$–group. As $P$ is
fully $\mathcal{F}$–normalized, $\operatorname{Aut}_{S}(P)$ is a Sylow $p$–subgroup of
$\operatorname{Aut}_{\mathcal{F}}(P)$,
so if we replace $A$ by a conjugate of $A$ under
$\operatorname{Aut}_{\mathcal{F}}(P)$, we can arrange that $C_{\operatorname{Aut}_{\mathcal{F}}(P)}(A)\leq\operatorname{Aut}_{S}(P)\leq%
\operatorname{Aut}_{\mathcal{G}}(P)$.
But $A$ also satisfies the assumptions on $Q$ in Lemma 2.4,
so $\operatorname{Aut}_{\mathcal{F}}(P)=\langle\operatorname{Aut}_{\mathcal{G}}(P)%
,C_{\operatorname{Aut}_{\mathcal{F}}(P)}(A)\rangle$, and we
conclude that $\operatorname{Aut}_{\mathcal{G}}(P)=\operatorname{Aut}_{\mathcal{F}}(P)$ as wanted.
∎
We now head towards a proof of Theorem D. Recall
that for $Q$ a group and $\mathcal{F}$ a fusion system on $S$ we define
$\operatorname{Rep}(Q,\mathcal{F})=\operatorname{Hom}(Q,S)/\mathcal{F}$ as the quotient of $\operatorname{Hom}(Q,S)$ under $\mathcal{F}$–conjugation,
i.e., where we identify $\varphi\in\operatorname{Hom}(Q,S)$ with $\alpha\circ\varphi$,
for all $\alpha\in\operatorname{Hom}_{\mathcal{F}}(\varphi(Q),S)$. The proof of Theorem D reduces quickly to the case that $\mathcal{G}$ is a subsystem of $\mathcal{F}$. We first make explicit what
the assumption in Theorem D then means, and state
this as a lemma.
Lemma 2.6.
Let $\mathcal{F}$ be a fusion system on a finite $p$-group $S$ and let $\mathcal{G}$ be
a sub-fusion system of $\mathcal{F}$ on $T\leq S$.
Suppose $Q$ is a
finite $p$–group (not necessarily a subgroup of $S$). The induced map $\operatorname{Rep}(Q,\mathcal{G})\rightarrow\operatorname{Rep}(Q,\mathcal{F})$
is surjective if and only if every epimorphic image of $Q$ in
$S$ is $\mathcal{F}$–conjugate to a subgroup of $T$. It is injective, if and
only if $\mathcal{G}$ controls fusion on the epimorphic images of $Q$ in $T$,
i.e., for any epimorphic image $Q^{\prime}\leq T$ of $Q$ we have $\operatorname{Hom}_{\mathcal{F}}(Q^{\prime},T)=\operatorname{Hom}_{\mathcal{G}%
}(Q^{\prime},T)$. ∎
The next lemma, together with
Theorem B, will easily imply Theorem D.
Lemma 2.7.
Let $\mathcal{F}$ be a saturated fusion system on a finite $p$-group $S$ and
let $\mathcal{G}$ be a saturated subsystem of $\mathcal{F}$ on $T\leq S$.
Suppose that
there exists an $\mathcal{F}$–centric subgroup $Q\leq T$ with
$\operatorname{Hom}_{\mathcal{F}}(Q,T)=\operatorname{Hom}_{\mathcal{G}}(Q,T)$. Then $\operatorname{Aut}_{\mathcal{F}}(T)=\operatorname{Aut}_{\mathcal{G}}(T)$ and
$T=S$.
Proof.
As $T$ is a finite $p$–group, there is a finite chain
$$Q=T_{0}\vartriangleleft T_{1}\vartriangleleft\cdots\vartriangleleft T_{n}=T$$
with $T_{i+1}=N_{T}(T_{i})$ for $0\leq i<n$. Note that, as $Q$ is $\mathcal{F}$–centric, every $T_{i}$ is $\mathcal{F}$–centric and thus also $\mathcal{G}$–centric.
We want to show that
$\operatorname{Aut}_{\mathcal{F}}(T)=\operatorname{Aut}_{\mathcal{G}}(T)$ by proving that
$$\operatorname{Hom}_{\mathcal{F}}(T_{i},T)=\operatorname{Hom}_{\mathcal{G}}(T_{%
i},T)\mbox{, for all }0\leq i\leq n,$$
($$**$$)
by induction on $i$. For $i=0$ the claim is true by assumption. Let now $0\leq i<n$ such
that $\operatorname{Hom}_{\mathcal{F}}(T_{i},T)=\operatorname{Hom}_{\mathcal{G}}(T_{%
i},T)$. Let $\gamma\in\operatorname{Hom}_{\mathcal{F}}(T_{i+1},T)$. Then $\psi=\gamma|_{T_{i}}\in\operatorname{Hom}_{\mathcal{F}}(T_{i},T)=\operatorname%
{Hom}_{\mathcal{G}}(T_{i},T)$. As $T_{i}$ is $\mathcal{G}$–centric, $\gamma(T_{i})$ is fully $\mathcal{G}$–centralized and $C_{T}(\gamma(T_{i}))\leq\gamma(T_{i})$; so by
Lemma 2.3(2), applied to $\gamma$ and $\mathcal{G}$ in the roles
of $\varphi$ and $\mathcal{F}$, $\psi$ extends to $\hat{\psi}\in\operatorname{Hom}_{\mathcal{G}}(T_{i+1},\gamma(T_{i+1}))$. Then $\hat{\psi}^{-1}\circ\gamma\in C_{\operatorname{Aut}_{\mathcal{F}}(T_{i+1})}(T_%
{i})$ and, by (Broto/Levi/Oliver:2003a, , Prop. A.8) or (Aschbacher/Kessar/Oliver:2011a, , Lem. I.5.6),
$C_{\operatorname{Aut}_{\mathcal{F}}(T_{i+1})}(T_{i})=\operatorname{Aut}_{Z(T_{%
i})}(T_{i+1})$.
We conclude that $\hat{\psi}^{-1}\circ\gamma\in\operatorname{Aut}_{Z(T_{i})}(T_{i+1})\leq%
\operatorname{Aut}_{\mathcal{G}}(T_{i+1})$
and thus $\gamma\in\operatorname{Hom}_{\mathcal{G}}(T_{i+1},T)$, i.e., ($**$ ‣ 2)
holds. So
$$\operatorname{Aut}_{\mathcal{G}}(T)=\operatorname{Hom}_{\mathcal{G}}(T,T)=%
\operatorname{Hom}_{\mathcal{F}}(T,T)=\operatorname{Aut}_{\mathcal{F}}(T).$$
If $\operatorname{Aut}_{\mathcal{F}}(T)=\operatorname{Aut}_{\mathcal{G}}(T)$
then, in particular, $\operatorname{Aut}_{\mathcal{F}}(T)/\operatorname{Inn}(T)$ has order prime to $p$, by
the Sylow axiom for $\mathcal{G}$,
and so
$\operatorname{Aut}_{S}(T)=\operatorname{Inn}(T)$. Since $Q\leq T$ is $\mathcal{F}$–centric, this implies that
$N_{S}(T)=T$, and thus $S=T$.
∎
Proof of Theorem D.
We only prove the claim about fusion systems, as the claim about
groups is a special case. First, it is obvious that $T\to S$ has to be a monomorphism, since if an element is conjugate to the
trivial element, it is trivial. Hence, we may consider $\mathcal{G}$ as a subsystem
of $\mathcal{F}$. Choose a subgroup $A\leq T$ such that $A$ is of maximal order
among the abelian subgroups of $T$. The assumptions, together with Lemma 2.6,
imply that
every abelian subgroup of $S$ is $\mathcal{F}$–conjugate to a subgroup of $T$,
so $A$ is of maximal order among the abelian subgroups of
$S$, and hence $\mathcal{F}$–centric. Moreover, again by Lemma 2.6, $\operatorname{Hom}_{\mathcal{G}}(A,T)=\operatorname{Hom}_{\mathcal{F}}(A,T)$.
Lemma 2.7
now shows that $T=S$. This reduces us to a special case of the
setup of Theorem B, and the result follows.
∎
3 Proofs of Theorems A and C
Proof of Theorem A.
By Theorem B we just need to verify that an $F$–isomorphism on
cohomology rings implies that $H$ controls fusion in $G$ on elementary
abelian $p$–groups. However this is the statement of
(Quillen:1971b+c, , Prop. 10.9(ii)$\Rightarrow$(i)) (see also Alperin:2006a ).
∎
Before proving Theorem C we state a lemma
explaining the condition on $n$, whose proof is elementary and seems
best left to the reader. Below $\mathbb{Z}_{p}$ denotes the $p$–adic integers.
Lemma 3.1.
For a homomorphism $\varphi\colon\thinspace H\to G$ of
finite groups, and a fixed natural number $n$, $\operatorname{Rep}(\mathbb{Z}_{p}^{n},H)\xrightarrow{\sim}\operatorname{Rep}(%
\mathbb{Z}_{p}^{n},G)$ if and only if
$\operatorname{Rep}(A,H)\xrightarrow{\sim}\operatorname{Rep}(A,G)$ for all finite abelian
$p$–groups $A$ with ${\operatorname{rk}_{p}}(A)=n$.
Furthermore, isomorphism for a fixed positive $n\geq\min\{{\operatorname{rk}_{p}}(G),{\operatorname{rk}_{p}}(H)+1\}$ implies ${\operatorname{rk}_{p}}(G)={\operatorname{rk}_{p}}(H)$ and
isomorphism for all $n$. ∎
In further preparation for the proof of Theorem C, we briefly recall the
HKR character theorem (Hopkins/Kuhn/Ravenel:2000a, , Thm C):
For any multiplicative cohomology theory $E$ and finite group $G$, taking $E^{*}$–cohomology induces a
map
$$\operatorname{Rep}(\mathbb{Z}_{p}^{n},G)\xrightarrow{\hskip 5.690551pt{}\hskip
5%
.690551pt}\operatorname{Hom}_{E^{*}\mbox{-}\mathrm{alg}}(E^{*}(BG),E^{*}_{%
\mathrm{cont}}(B\mathbb{Z}_{p}^{n}))$$
with $E^{*}_{\mathrm{cont}}(B\mathbb{Z}_{p}^{n})=\operatorname{colim}_{r}E^{*}(B(%
\mathbb{Z}/p^{r})^{n})$, since any
$\alpha\colon\thinspace\mathbb{Z}_{p}^{n}\to G$ factors canonically through $(\mathbb{Z}/p^{r})^{n}$ for $r$ large.
By adjunction we can view this as an
$E^{*}$–algebra homomorphism
$$E^{*}(BG)\xrightarrow{\hskip 5.690551pt{}\hskip 5.690551pt}\prod_{%
\operatorname{Rep}(\mathbb{Z}_{p}^{n},G)}E^{*}_{\mathrm{cont}}(B\mathbb{Z}_{p}%
^{n})$$
(3.2)
where the right-hand side is $E^{*}_{\mathrm{cont}}(B\mathbb{Z}_{p}^{n})$–valued
functions on the finite set $\operatorname{Rep}(\mathbb{Z}_{p}^{n},G)$, with point-wise multiplication.
The map (3.2) is the $n$–character map and the HKR character theorem
(Hopkins/Kuhn/Ravenel:2000a, , Thm. C) says that, for certain $E$, this becomes an
isomorphism after suitable localization.
More precisely, assume that $E=E(n)$, so $E^{*}(BS^{1})\cong E^{*}\llbracket x\rrbracket$, $|x|=2$, and define $L(E^{*})$ to be the
ring of fractions of $E^{*}_{\mathrm{cont}}(B\mathbb{Z}_{p}^{n})$ obtained by inverting $\alpha^{*}(x)$ for all
for all non-zero $\alpha\in\operatorname{Hom}_{\mathrm{cont}}(\mathbb{Z}_{p}^{n},S^{1})\cong(%
\mathbb{Z}/p^{\infty})^{n}$. Then, by (Hopkins/Kuhn/Ravenel:2000a, , Thm. C),
$L(E^{*})$ is faithfully flat over $E(n)^{*}[\frac{1}{p}]$ (and in
particular non-zero) and (3.2) induces an isomorphism
$$L(E^{*})\otimes_{E^{*}[\frac{1}{p}]}E^{*}(BG)[\frac{1}{p}]\xrightarrow{\hskip 5%
.690551pt{\sim}\hskip 5.690551pt}\prod_{\operatorname{Rep}(\mathbb{Z}_{p}^{n},%
G)}L(E^{*})$$
(3.3)
Proof of Theorem C.
By the assumption of the theorem and the HKR character isomorphism
(3.3) we have an isomorphism
$$\prod_{\operatorname{Rep}(\mathbb{Z}_{p}^{n},G)}L(E^{*})\xrightarrow{\hskip 5.%
690551pt{\sim}\hskip 5.690551pt}\prod_{\operatorname{Rep}(\mathbb{Z}_{p}^{n},H%
)}L(E^{*})$$
(3.4)
given by precomposing with the natural map $\operatorname{Rep}(\mathbb{Z}_{p}^{n},H)\to\operatorname{Rep}(\mathbb{Z}_{p}^{%
n},G)$. Since $L(E^{*})\neq 0$ we conclude that
$\operatorname{Rep}(\mathbb{Z}_{p}^{n},H)\to\operatorname{Rep}(\mathbb{Z}_{p}^{%
n},G)$ is an isomorphism. By
the assumption on $n$ and Lemma 3.1 this implies that
$\operatorname{Rep}(A,H)\to\operatorname{Rep}(A,G)$ is an isomorphism for all finite abelian
groups, and Theorem C now follows from
Theorem D.
∎
4 Variations on the results and further comments
In this final section we elaborate on some supplementary results alluded to in the introduction.
Remark 4.1 (A variety version of
Theorem A).
In Theorem A we can replace the assumption of
$F$–isomorphism by the assumption that the map $\iota^{*}\colon\thinspace H^{*}(G;\bar{\mathbb{F}}_{p})\to H^{*}(H;\bar{%
\mathbb{F}}_{p})$ induces a
bijection of maximal ideals, by referencing
(Quillen:1971b+c, , Prop. 10.9(iii)$\Rightarrow$(i)), and noting that the
maximal ideal spectrum of $H^{*}(G;\bar{\mathbb{F}}_{p})$ identifies with $\operatorname{Hom}_{\bar{\mathbb{F}}_{p}\mbox{-}\mathrm{alg}}(H^{*}(G;\bar{%
\mathbb{F}}_{p}),\bar{\mathbb{F}}_{p})=\operatorname{Hom}_{\mathrm{rings}}(H^{%
*}(G;\mathbb{F}_{p}),\bar{\mathbb{F}}_{p})$.
In general a finite morphism $f:A\to B$ of
finitely generated $\mathbb{F}_{p}$–algebras is an $F$–isomorphism if and
only if it induces a variety
isomorphism, i.e., a bijection $\operatorname{Hom}_{\mathrm{rings}}(B,\Omega)\xrightarrow{\sim}\operatorname{%
Hom}_{\mathrm{rings}}(A,\Omega)$ for all algebraically closed fields
$\Omega$ (Quillen:1971b+c, , Prop. B.8-9). But to get the same fusion on elementary
abelian $p$-subgroups we in fact just need a bijection on $\operatorname{Hom}_{\mathrm{rings}}(-,\Omega)$, for some proper
field extension $\Omega$ of $\mathbb{F}_{p}$, by properties of the Quillen
stratification; see (Quillen:1971b+c, , §9-10) and also (Evens:1991a, , §9.1).
Example 4.2 (An isomorphism
modulo $\mathcal{N}\!\mathit{il}_{n}$ for $p=2$ which does not control
$p$–fusion).
For any $n$, let $G_{n}=(2A_{4})^{n}$, $P_{n}=(Q_{8})^{n}$ and $H_{n}=\ker(\psi)$, where
$\psi\colon\thinspace G_{n}\to G_{n}/P_{n}\cong(C_{3})^{n}\to C_{3}$ is given by $(g_{1},\dots,g_{n})\mapsto g_{1}\dots g_{n}$. Note that
$H_{n}$ does not control $p$–fusion in $G_{n}$.
We however claim that the restriction $H^{*}(G_{n};\mathbb{F}_{2})\to H^{*}(H_{n};\mathbb{F}_{2})$ is an
isomorphism modulo $\mathcal{N}\!\mathit{il}_{n}$, as defined in (Schwartz:1994a, , Ch. 6), hence showing that Theorem A fails
severely for $p=2$ ($F$–isomorphism is equivalent to isomorphism
modulo $\mathcal{N}\!\mathit{il}_{1}$):
Recall that $H^{*}(Q_{8};\mathbb{F}_{2})\cong H^{<4}(Q_{8};\mathbb{F}_{2})\otimes\mathbb{F}%
_{2}[z]$, with $|z|=4$, where the action of
$2A_{4}/Q_{8}\cong C_{3}$ on $\mathbb{F}_{2}[z]$ is trivial, while on
$H^{<4}(Q_{8};\mathbb{F}_{2})$ it is trivial in degrees 0 and 3, and degrees 1 and 2
consists of the two dimensional irreducible
$\mathbb{F}_{2}C_{3}$–module $V$.
Since $G_{n}$ and $H_{n}$ both have Sylow $2$–subgroup $P_{n}$, the
restriction map $H^{*}(G_{n};\mathbb{F}_{2})\to H^{*}(H_{n};\mathbb{F}_{2})$ is injective, and
the cokernel is a tensor product of $\mathbb{F}_{2}[z_{1},\dots,z_{n}]$ with a certain
finite module $M$, given as the sum of the non-trivial irreducible
$G_{n}/P_{n}$–representations on $H^{<4}(Q_{8};\mathbb{F}_{2})^{\otimes n}$ which restrict trivially to $H_{n}/P_{n}$.
Using the definition of $\mathcal{N}\!\mathit{il}_{m}$ (Schwartz:1994a, , Ch. 6), the largest $m$ for
which the restriction map is an isomorphism modulo $\mathcal{N}\!\mathit{il}_{m}$
therefore is the first degree where $M$ is non-zero.
To determine this degree we extend coefficients to $\mathbb{F}_{4}$ and use Frobenius reciprocity
$\operatorname{Hom}_{\mathbb{F}_{4}(H_{n}/P_{n})}(\mathbb{F}_{4},-)\cong%
\operatorname{Hom}_{\mathbb{F}_{4}(G_{n}/P_{n})}((\mathbb{F}_{4}){\uparrow_{H_%
{n}}^{G_{n}}},-)$, and note that
$(\mathbb{F}_{4}){\uparrow_{H_{n}}^{G_{n}}}\cong(\mathbb{F}_{4}\otimes\dots%
\otimes\mathbb{F}_{4})\oplus(\omega\otimes\dots\otimes\omega)\oplus(\bar{%
\omega}\otimes\dots\otimes\bar{\omega})$,
for $\mathbb{F}_{4}$, $\omega$ and $\bar{\omega}$ the three 1-dimensional
$\mathbb{F}_{4}C_{3}$–modules.
In this notation, we have to locate the first copy of
$\omega\otimes\dots\otimes\omega$ or $\bar{\omega}\otimes\dots\otimes\bar{\omega}$
in $(H^{<4}(Q_{8};\mathbb{F}_{4}))^{\otimes n}$.
This occurs for the first time in degree $n$, where there is a summand $V\otimes\dots\otimes V$,
which over $\mathbb{F}_{4}$ is $(\omega\oplus\bar{\omega})\otimes\dots\otimes(\omega\oplus\bar{\omega})$ completing
the proof of the claim.
Remark 4.3 (A generalization of Mislin’s theorem via Theorem C).
A notion of equivalence stronger than $F$–isomorphism, and in fact
also than that of Example 4.2, is isomorphism in large degrees.
If a homomorphism $\varphi\colon\thinspace H\to G$ induces an isomorphism
in mod $p$ cohomology in large degrees, we can use
Theorem C to see that $|\ker(\varphi)|$ and $|G:\varphi(H)|$
are coprime to $p$ and that $\varphi(H)$ controls $p$–fusion in $G$,
providing a new proof of a strengthening of Mislin’s theorem first obtained in (Mislin:1993a, , Cor. 3.4) (cf. also
(Benson/Carlson/Robinson:1990a, , Thm. 1.1)):
By finiteness of group
cohomology the induced map between
$E^{2}$–terms of $E(n)^{*}$–Atiyah–Hirzebruch spectral
sequences (Boardman:1999a, , Thm. 12.2) has kernel and cokernel a finite
$p$–group in each total degree. We therefore deduce an
isomorphism $E(n)^{*}(BG)[\frac{1}{p}]\xrightarrow{\sim}E(n)^{*}(BH)[\frac{1}{p}]$ by spectral sequence comparison
(Boardman:1999a, , Thm. 7.2), and the claim now follows from Theorem C.
It is perhaps
interesting to note that this proof structurally mirrors Atiyah’s 1961
proof of his $p$–nilpotency criterion
Atiyah:1961a (Quillen:1971a, , Thm. 1.3), which says that a
Sylow inclusion $S<G$ controls $p$–fusion if it induces an isomorphism in mod $p$ cohomology in
sufficiently high dimension: Reinterpreting
(Quillen:1971a, , p. 362), Atiyah uses his version of the Atiyah–Hirzebruch
spectral sequence (Atiyah:1961a, , Thm. 5.1) to conclude that $K^{*}(BG;\mathbb{Z}_{p})[\frac{1}{p}]\xrightarrow{\sim}K^{*}(BS;\mathbb{Z}_{p}%
)[\frac{1}{p}]$. It now follows from the Atiyah–Segal
completion theorem (Atiyah:1961a, , Thm. 7.2) that $S$ and $G$ have the same fusion on cyclic
$p$–subgroups, and hence the same $p$–fusion by (Huppert:1967a, , Satz IV.4.9).
Remark 4.4 (A variety version of Theorem C
and the role of inverting $p$).
Also in Theorem C it is enough to
assume a variety isomorphism: If $\varphi^{*}\colon\thinspace E(n)^{*}(BG)[\frac{1}{p}]\to E(n)^{*}(BH)[\frac{1}%
{p}]$ induces a
bijection on $\operatorname{Hom}_{\mathrm{rings}}(-,\Omega)$ for all algebraically closed
fields $\Omega$, then the same holds after
extending scalars along $E(n)^{*}[\frac{1}{p}]\to L(E^{*})$. Hence (3.3) shows that (3.4)
induces a bijection
$\coprod_{\operatorname{Rep}(\mathbb{Z}_{p}^{n},H)}\operatorname{Hom}_{\mathrm{%
rings}}(L(E^{*}),\Omega)\xrightarrow{\sim}\coprod_{\operatorname{Rep}(\mathbb{%
Z}_{p}^{n},G)}\operatorname{Hom}_{\mathrm{rings}}(L(E^{*}),\Omega)$
for any algebraically closed field $\Omega$, so $\operatorname{Rep}(\mathbb{Z}^{n}_{p},H)\xrightarrow{\sim}\operatorname{Rep}(%
\mathbb{Z}_{p}^{n},G)$, and
Theorem C follows from Theorem D as above.
In (Greenlees/Strickland:1999a, , §3)
Greenlees–Strickland
explain how the variety of $E(n)^{*}(BG)[\frac{1}{p}]$
constitutes a ‘zeroth pure stratum’ of a chromatic stratification
of the formal spectrum of $E(n)^{*}(BG)$. Hence
having an isomorphism on $E(n)^{*}(-)[\frac{1}{p}]$ is a
priori significantly weaker than having isomorphism on
$E(n)^{*}(-)$ or the formal spectrum ${\operatorname{Spf}}(E(n)^{*}(-))$.
Remark 4.5 (Theorem C in relationship to other results in stable
homotopy theory).
To illuminate the
assumptions in Theorem C we note
that a map induces an isomorphism on $E(n)$ (without inverting $p$) if and only if
it induces isomorphism on the corresponding uncompleted Johnson–Wilson
theory, or isomorphism on $K(i)$ for all $i\leq n$ (see (Ravenel:1984a, , Thm. 2.1) and
(Lurie:2010a, , Lec. 23)). This in turn happens if and only if it
induces isomorphism on just $K(n)$, by a result of Bousfield (Bousfield:1999a, , Thm. 1.1).
Homotopy theorists may wonder if there exists a ‘purely homotopic’
proof of Theorem C. We do not know such a proof, but
combining Mislin’s original theorem Mislin:1990a with some
deep results in homotopy theory, one can get a weaker statement that $E(n)^{*}$–isomorphism for a quite large $n$ (and without inverting $p$) implies that $H$ controls $p$–fusion in
$G$. We briefly explain this: Bousfield proved in 1982 a ‘$K(n)$–Whitehead theorem’
stating that a map between spaces which is an isomorphism on $K(n)^{*}$
also induces an isomorphism on $H^{i}(-;\mathbb{F}_{p})$ for $i\leq n$ (see
(Bousfield:1982a, , Ex. 8.4) and
(Bousfield:1999a, , Thm. 1.4)).
The claim now follows since it is possible to give a
large constant $n$, depending on the Sylow subgroup, such that isomorphism in $H^{i}(-;\mathbb{F}_{p})$ for $i\leq n$ implies
isomorphism on $H^{*}(-;\mathbb{F}_{p})$, e.g., using results of Symonds
(Symonds:2010a, , Prop. 10.2) that say that the generators and relations of group cohomology are at most in degree
$2k^{2}$, where $k$ is the minimal dimension of a
faithful complex representation of $G$. Observe the bound needs to
depend on more than the $p$–rank: For any $n$ we can pick $p$ such
that $2n\mid p-1$. In this case $\mathbb{F}_{p}\rtimes C_{n}<\mathbb{F}_{p}\rtimes C_{2n}$ induces an
isomorphism on $H^{i}(-;\mathbb{F}_{p})$ for $i<n$ without controlling
$p$–fusion. This is in
contrast to the
Huppert–Thompson–Tate $p$–nilpotency criterion
Tate:1964a , which states that an inclusion of a Sylow $p$–subgroup that induces isomorphism on
$H^{1}(-;\mathbb{F}_{p})$
controls $p$–fusion.
Acknowledgements.
Our interest
in Theorem A was piqued by a discussion during the problem session at
the August 2011 workshop on Homotopical Approaches to Group Actions
in Copenhagen with Peter Symonds and others, and also stimulated by
arXiv:1107.5158v2 .
We thank Mike Hopkins for pointers concerning the relationship between Theorem C
and classical stable homotopy theory, explained in
Remark 4.5, and Lucho Avramov, Nick Kuhn, and Neil
Strickland for other literature references.
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Kramers degeneracy in a magnetic field
and Zeeman spin-orbit coupling in antiferromagnetic
conductors
Revaz Ramazashvili
ENS, LPTMS, UMR8626, Bât. 100,
Université Paris-Sud, 91405 Orsay, France
(November 22, 2020)
Abstract
In this article, I analyze the symmetries
and degeneracies of electron eigenstates
in a commensurate collinear antiferromagnet.
In a magnetic field transverse to the
staggered magnetization, a hidden anti-unitary
symmetry protects double degeneracy of the Bloch
eigenstates at a special set of momenta.
In addition to this ‘Kramers degeneracy’ subset,
the manifold of momenta, labeling the doubly
degenerate Bloch states in the Brillouin zone,
may also contain an ‘accidental degeneracy’
subset, that is not protected by symmetry and
that may change its shape under perturbation.
These degeneracies give rise to a substantial
momentum dependence of the transverse $g$-factor
in the Zeeman coupling, turning the latter into
a spin-orbit interaction.
I discuss a number of materials, where Zeeman
spin-orbit coupling is likely to be present,
and outline the simplest properties and
experimental consequences of this interaction,
that may be relevant to systems from chromium
to borocarbides, cuprates, hexaborides, iron
pnictides, as well as organic and heavy fermion
conductors.
pacs: 75.50.Ee
††thanks: Present address: Department of Physics
and Astronomy, University of South Carolina,
Columbia, SC 29208, USA.\excludeversion
details
I I. Introduction
Antiferromagnetism is widespread in materials with
interesting electron properties. Chromium fawcett_1
and its alloys fawcett_2 ; kulikov , numerous
borocarbides mueller , electron- and hole-doped
cuprates birgeneau ; tranquada , iron pnictides
cruz , various organic chaikin and heavy
fermion kuramoto ; robinson ; flouquet compounds
all have an antiferromagnetic state present in their
phase diagram. The physics of these antiferromagnetic
phases has been a subject of active research.
In this article, I study the response of electron
Bloch eigenstates in an antiferromagnet to a weak
magnetic field.
I concentrate on the simplest case: a centrosymmetric
doubly commensurate collinear antiferromagnet, shown
schematically in Fig. 1, where
the magnetization density at any point in space
is parallel or anti-parallel to a single
fixed direction ${\bf n}$ of the staggered magnetization,
and changes sign upon primitive translation of the
underlying lattice.
{details}
Below, I show that, in a magnetic field transverse
to the staggered magnetization, a hidden anti-unitary
symmetry protects the Kramers degeneracy of Bloch
eigenstates at a special set of momenta. This
degeneracy gives rise to a peculiar spin-orbit
coupling, whose emergence and basic properties,
along with the degeneracy itself, are the main
results of this work.
In a paramagnet, the double degeneracy of the Bloch
eigenstates is commonly attributed to symmetry under
time reversal $\theta$ – and, indeed, perturbations
that break time reversal symmetry (such as ferromagnetism
or a magnetic field) do tend to remove the degeneracy.
Yet violation of $\theta$ alone does not preclude
degeneracy: in a commensurate centrosymmetric
Néel antiferromagnet, as in a paramagnet,
all Bloch eigenstates enjoy a Kramers degeneracy
herring1 in spite of time reversal symmetry
being broken in the former, but not in the latter.
In an antiferromagnet, the staggered magnetization sets
a special direction ${\bf n}$ in electron spin space,
making it anisotropic. A magnetic field along ${\bf n}$
removes the degeneracy of all Bloch eigenstates,
as it does in a paramagnet. By contrast, in a
transverse field, a hidden anti-unitary symmetry
protects the Kramers degeneracy of Bloch
eigenstates at a special set of momenta.
Generally, in $d$ dimensions, the manifold
of momenta, corresponding to doubly degenerate
Bloch states in a transverse field is
$(d-1)$-dimensional; within a subset
of this manifold, the degeneracy is dictated
by symmetry. This is in marked contrast
with what happens in a paramagnet, where
an arbitrary magnetic field lifts the
degeneracy of all Bloch eigenstates.
For brevity, in this article I often refer
to the manifold of momenta, labeling the
degenerate Bloch states in the Brillouin zone,
as to the ‘degeneracy manifold’.
As a consequence of the Kramers degeneracy of the
special Bloch states in a transverse field, the
transverse component $g_{\perp}$ of the electron
$g$-tensor vanishes for such states. Not being
identically equal to zero, $g_{\perp}$ must, therefore,
carry a substantial momentum dependence, and the
Zeeman coupling $\mathcal{H}_{ZSO}$ must take the
form
$$\mathcal{H}_{ZSO}=-\mu_{B}\left[g_{\|}({\bf H_{\|}\cdot\bm{\sigma}})+g_{\perp}%
({\bf p})({\bf H_{\perp}\cdot\bm{\sigma}})\right],$$
(1)
where
${\bf H}_{\|}=({\bf H}\cdot{\bf n}){\bf n}$
and
${\bf H}_{\perp}={\bf H}-{\bf H}_{\|}$
are the longitudinal and transverse components
of the magnetic field with respect to the unit
vector ${\bf n}$ of the staggered magnetization,
$\mu_{B}$
is the Bohr magneton, while $g_{\|}$ and
$g_{\perp}({\bf p})$ are the longitudinal
and transverse components of the $g$-tensor.
This significant momentum dependence of
$g_{\perp}({\bf p})$ turns the common Zeeman
coupling into a kind of spin-orbit interaction
$\mathcal{H}_{ZSO}$ (1), whose
appearance and key properties are the focus
of this work. Zeeman spin-orbit coupling may
manifest itself spectacularly in a number
of ways, which will be mentioned below and
discussed in detail elsewhere.
The symmetry properties of wave functions in magnetic
crystals have been studied by Dimmock and Wheeler
dimmock2 , who pointed out, among other things,
that magnetism not only lifts degeneracies by obviously
lowering the symmetry, but also may introduce new ones.
This may happen at the magnetic Brillouin zone (MBZ)
boundary, under the necessary condition that the
magnetic unit cell be larger than the paramagnetic
one dimmock2 .
For a Néel antiferromagnet on a lattice of square
symmetry, the response of the electron states to
a magnetic field was studied in Ref. braluk
using symmetry arguments, and in Ref. bralura
within a weak coupling model. The present work
is a detailed presentation of recent results
rr_sym . It revisits Ref. braluk ,
extends it to an arbitrary crystal symmetry and
to a finite as opposed to infinitesimal magnetic
field, and uncovers a rich interplay between the
symmetry of magnetic structure and that of the
underlying crystal lattice. At the same time,
the present work extends Ref. dimmock2
by allowing for an external magnetic field –
to show how, at special momenta, the Kramers
degeneracy in an antiferromagnet may persist
even in a transverse magnetic field.
This work treats antiferromagnetic order as static,
neglecting both its classical and quantum fluctuations.
This excludes from consideration strongly fluctuating
antiferromagnetic states such as those near a continuous
phase transition, be it a finite-temperature Néel
transition or a quantum ($T=0$) critical point. At
the same time, the single-electron Bloch eigenstates,
considered hereafter, must be well-defined. As in a
normal Fermi liquid state, this does not rule out
strong interaction between electrons, but simply
requires temperatures well below the Fermi energy.
Finally, to justify the neglect of quantum
fluctuations, the ordered magnetic moment must be
of the order of or greater than the Bohr magneton.
As a consequence, the present theory applies
to antiferromagnets (i) deep inside
a commensurate long-range antiferromagnetic
state, and far enough from any continuous
Néel transition, finite-temperature or quantum,
(ii) with an ordered moment noticeable
on the scale of the Bohr magneton, and
(iii) far below both the Néel and the
effective Fermi temperatures. All materials
mentioned in Section IV are meant to be
considered under these conditions.
The article is organized as follows.
Section II opens with a reminder of how, in spite of
broken time reversal symmetry, all Bloch eigenstates
in a commensurate collinear antiferromagnet retain
Kramers degeneracy, provided there is
an inversion center herring1 .
Then I show how, even in a transverse magnetic field,
a hidden symmetry of antiferromagnetic order may protect
the Kramers degeneracy for certain Bloch states.
Section III establishes several properties of the
single-electron spectrum in a weakly coupled
antiferromagnet, subject to a transverse magnetic field.
Section IV contains the analysis of simple examples,
that may be relevant to specific materials from
chromium to organic conductors, from borocarbides
to underdoped cuprates and to various heavy fermion
metals. The Discussion section reviews the findings,
and examines them in the light of earlier work,
while the Appendices present various technical
details.
II II. General arguments
It is convenient to begin by describing the crystal
host symmetry in the absence of magnetism, with
the average magnetization density notionally set
to zero dimmock2 . I refer to this as to
the paramagnetic state symmetry, even though the
symmetry of the actual paramagnetic state may be
different, for instance due to a lattice distortion upon
transition. Unitary symmetries of the paramagnetic state
form a group, $h$, which includes a set of elementary
translations ${\bf T_{a}}$ by primitive translation vectors
$\bf{a}$. Time reversal $\theta$ being indeed a symmetry
of the paramagnetic state, the full symmetry group $g$
of the paramagnetic state includes $h$, and products
of $\theta$ with each element of $h$:
$g=h+\theta\cdot h$; put otherwise, $h$
is an invariant unitary subgroup of $g$.
Antiferromagnetic order couples to the
electron spin $\bm{\sigma}$ via the exchange term
$({\bf\Delta_{r}}\cdot\bm{\sigma})$, where
${\bf\Delta_{r}}$ is proportional to the average
microscopic magnetization at point ${\bf r}$.
In keeping with the arguments of the Introduction,
fluctuations of ${\bf\Delta_{r}}$ are neglected.
Being of relativistic origin, spin-orbit couplings
of the crystal lattice to the electron spin and to
the magnetization density are also neglected. This
is a good approximation in a broad range of problems,
at the very least at temperatures above the scale
set by the spin-orbit coupling (see Section V for
details). In this ‘exchange symmetry’ approximation
andreev , magnetization density and electron
spin are assigned to a separate space, independent
of the real space of the crystal; this makes
coordinate rotations and other point symmetries
inert with respect to ${\bf\Delta_{r}}$ and
$\bm{\sigma}$.
A nonzero ${\bf\Delta_{r}}$ changes sign under time
reversal $\theta$, and removes the symmetry under
primitive translations ${\bf T_{a}}$, thus reducing
the symmetry with respect to that of paramagnetic state.
In a doubly commensurate collinear antiferromagnet,
${\bf\Delta_{r}}$ changes sign upon ${\bf T_{a}}$:
${\bf\Delta_{r+a}=-\Delta_{r}}$, while
${\bf T}_{\bf a}^{2}$ leaves ${\bf\Delta_{r}}$ intact:
${\bf\Delta}_{{\bf r}+2{\bf a}}={\bf\Delta_{r}}$.
Even though neither $\theta$ nor ${\bf T_{a}}$
remain a symmetry, their product $\theta{\bf T_{a}}$
does (see Fig. 1).
In a system with inversion center,
so does $\theta{\bf T_{a}}\mathcal{I}$,
where $\mathcal{I}$ is inversion. The importance of
the combined symmetry $\theta{\bf T_{a}}\mathcal{I}$
will become clear in the next subsection.
Together with the uniaxial character
(${\bf\Delta_{r}}$ at any point ${\bf r}$ pointing
along or against the single direction ${\bf n}$
of staggered magnetization), these relations define
a commensurate collinear Néel antiferromagnet via
transformation properties of its microscopic
magnetization density.
II.1 Kramers degeneracy in zero field
The combined anti-unitary symmetry
$\theta{\bf T_{a}}\mathcal{I}$ gives rise
to a Kramers degeneracy herring1 :
If $|{\bf p}\rangle$ is a Bloch eigenstate
at momentum ${\bf p}$, then
$\theta{\bf T_{a}}\mathcal{I}|{\bf p}\rangle$
is degenerate with $|{\bf p}\rangle$.
Since $\theta$ and $\mathcal{I}$ both invert
the momentum, both $|{\bf p}\rangle$ and
$\theta{\bf T_{a}}\mathcal{I}|{\bf p}\rangle$ carry the
same momentum label ${\bf p}$. Formally, this is verified
by the action of any translation ${\bf T}_{\bf b}$, that
remains a symmetry of the antiferromagnetic state:
$$\displaystyle{\bf T}_{\bf b}\theta{\bf T_{a}}\mathcal{I}|{\bf p}\rangle=\theta%
{\bf T_{a}}{\bf T}_{\bf b}\mathcal{I}|{\bf p}\rangle=\theta{\bf T_{a}}\mathcal%
{I}{\bf T}_{-\bf b}|{\bf p}\rangle=$$
(2)
$$\displaystyle=$$
$$\displaystyle\theta{\bf T_{a}}\mathcal{I}e^{-i\bf p\cdot b}|{\bf p}\rangle=e^{%
i\bf p\cdot b}\theta{\bf T_{a}}\mathcal{I}|{\bf p}\rangle.$$
At the same time, $|{\bf p}\rangle$ and
$\theta{\bf T_{a}}\mathcal{I}|{\bf p}\rangle$
are orthogonal.
This follows from Eqn. (22) of Appendix A
as soon as one chooses
$\mathcal{O}={\bf T_{a}}\mathcal{I}$,
$|\psi\rangle=|{\bf p}\rangle$, and
$|\phi\rangle={\bf T_{a}}\mathcal{I}\theta|{\bf p}\rangle$.
Recalling that $({\bf T_{a}}\mathcal{I})^{2}=-\theta^{2}=1$,
and hence $(\theta{\bf T_{a}}\mathcal{I})^{2}=-1$, one finds
$$\langle\bf{p}|\mathcal{I}{\bf T_{a}}\theta|\bf{p}\rangle=-\langle\bf{p}|%
\mathcal{I}{\bf T_{a}}\theta|\bf{p}\rangle.$$
(3)
Thus, in spite of broken time reversal symmetry,
in a centrosymmetric commensurate Néel antiferromagnet
all Bloch states retain a Kramers degeneracy.
II.2 Kramers degeneracy in a transverse field
Generally, a magnetic field ${\bf H}$ lifts this
degeneracy. However, in a transverse field,
a hidden anti-unitary symmetry may protect the
degeneracy at a special set of points in the
Brillouin zone, as I show below.
In an antiferromagnet, subject to a magnetic
field ${\bf H}$, the single-electron Hamiltonian
takes the form
$$\mathcal{H}=\mathcal{H}_{0}+({\bf\Delta_{r}}\cdot\bm{\sigma})-({\bf H}\cdot\bm%
{\sigma}),$$
(4)
where the ‘paramagnetic’ part $\mathcal{H}_{0}$ is
invariant under independent action of ${\bf T_{a}}$
and $\theta$, and $g\mu_{B}$ is set to unity.
In the absence of the field, all Bloch eigenstates
of Hamiltonian (4) are
doubly degenerate by virtue of Eqn. (3).
Consider the symmetries of Hamiltonian
(4), involving a combination
of an elementary translation ${\bf T_{a}}$, time reversal
$\theta$, or a spin rotation ${\bf U_{m}}(\phi)$ around
an axis ${\bf m}$ by an angle $\phi$. These symmetries
are listed in Table 1; the relative
orientation of ${\bf\Delta_{r}}$, ${\bf H}_{\|}$ and
${\bf H}_{\perp}$ is shown in Fig. 2.
The transverse field ${\bf H}_{\perp}$ breaks the
symmetries ${\bf U_{n}}(\phi)$ and ${\bf T_{a}}\theta$,
but preserves their combination at $\phi=\pi$,
i.e. ${\bf U_{n}}(\pi){\bf T_{a}}\theta$.
Acting on an exact Bloch state
$|{\bf p}\rangle$ at momentum ${\bf p}$,
this combined anti-unitary operator
creates a degenerate partner eigenstate
${\bf U_{n}}(\pi){\bf T_{a}}\theta|{\bf p}\rangle$,
which is orthogonal to $|{\bf p}\rangle$ everywhere in the
Brillouin zone, unless $({\bf p\cdot a})$ is an integer
multiple of $\pi$ (in other words, unless ${\bf p}$
lies at a paramagnetic Brillouin zone boundary):
$$\langle{\bf p}|{\bf U_{n}}(\pi){\bf T_{a}}\theta|{\bf p}\rangle=e^{-2i({\bf p%
\cdot a})}\langle{\bf p}|{\bf U_{n}}(\pi){\bf T_{a}}\theta|{\bf p}\rangle.$$
(5)
Equation (5) follows from Eqn.
(22) of Appendix A for
$\mathcal{O}={\bf U_{n}}(\pi){\bf T_{a}}$,
$|\psi\rangle=|{\bf p}\rangle$, and
$|\phi\rangle={\bf U_{n}}(\pi){\bf T_{a}}\theta|{\bf p}\rangle$
as soon as one observes, that
$\left[{\bf U_{n}}(\pi){\bf T_{a}}\theta\right]^{2}={\bf T}_{\bf a}^{2}={\bf T}%
_{2{\bf a}}$.
In a magnetic field, double translation
${\bf T}_{2{\bf a}}$ remains a symmetry;
according to the Bloch theorem, it acts
on $|{\bf p}\rangle$ as per
${\bf T}_{2{\bf a}}|{\bf p}\rangle=e^{2i({\bf p\cdot a})}|{\bf p}\rangle$,
thus leading to (5).
Notice, however, that the eigenstate
${\bf U_{n}}(\pi){\bf T_{a}}\theta|{\bf p}\rangle$
carries momentum label $-{\bf p}$ rather than ${\bf p}$.
By contrast with the case of zero field, combining
${\bf U_{n}}(\pi){\bf T_{a}}\theta$ with inversion
$\mathcal{I}$ no longer helps to produce a degenerate
partner eigenstate at the original momentum ${\bf p}$:
since $\theta$, ${\bf U_{n}}(\pi)$, and
${\bf T_{a}}\mathcal{I}$ all commute, and since
$\left[\mathcal{I}{\bf U_{n}}(\pi){\bf T_{a}}\theta\right]^{2}=1$,
equation (22) of Appendix A
for $\mathcal{O}=\mathcal{I}{\bf U_{n}}(\pi){\bf T_{a}}$,
$\psi=|{\bf p}\rangle$, and
$\phi=\mathcal{I}{\bf U_{n}}(\pi){\bf T_{a}}\theta|{\bf p}\rangle$ only confirms,
that $\langle{\bf p}|\mathcal{I}{\bf U_{n}}(\pi){\bf T_{a}}\theta|{\bf p}\rangle$
equals itself.
Thus, for an exact Bloch state $|{\bf p}\rangle$
at momentum ${\bf p}$, the anti-unitary symmetry
${\bf U_{n}}(\pi){\bf T_{a}}\theta$ produces an
orthogonal degenerate eigenstate
${\bf U_{n}}(\pi){\bf T_{a}}\theta|{\bf p}\rangle$ at momentum $-{\bf p}$.
The two momenta ${\bf p}$ and $-{\bf p}$
are different, with one key exception.
It occurs for ${\bf p}$ at the magnetic
Brillouin zone boundary, given a unitary symmetry
$\mathcal{U}$, that transforms $-{\bf p}$ into
a momentum, equivalent to ${\bf p}$ up to a
reciprocal lattice vector ${\bf Q}$ of the
antiferromagnetic state dimmock2 :
$$-\mathcal{U}{\bf p}={\bf p}+{\bf Q}.$$
(6)
In this case, the eigenstate
$\mathcal{U}{\bf U_{n}}(\pi){\bf T_{a}}\theta|{\bf p}\rangle$ carries momentum
label ${\bf p+Q\equiv p}$, is degenerate with
$|{\bf p}\rangle$ and orthogonal to it, thus
explicitly demonstrating Kramers degeneracy at
momentum ${\bf p}$ in a transverse field.
This result is general: combined with any
momentum-inverting anti-unitary symmetry, equation
(6) leads to Kramers
degeneracy at momentum ${\bf p}$ dimmock2 .
The simplest illustration, where $\mathcal{U}$
is the unity operator, is given by
${\bf p}={\bf Q}/2$ and shown in Figs.
5 and 6(a)
for two particular cases. These and other
examples are described in Section IV.
Notice that, at ${\bf p}={\bf Q}/2$
(with $\mathcal{U}={\bf 1}$), the degeneracy
in a transverse field is guaranteed even for a
low crystal symmetry, provided an inversion center.
Also notice, once more, that $\mathcal{U}$,
$\mathcal{I}$ and other point symmetries above
are inert with respect to spin as a
consequence of the ‘exchange symmetry’
approximation andreev .
Hamiltonian (4) and the subsequent
analysis ignored the response of the antiferromagnetic order
to the transverse field ${\bf H}_{\perp}$. This, however, does
not affect the set of points, where Kramers degeneracy in a
transverse field is protected by the anti-unitary symmetry
$\mathcal{U}{\bf U_{n}}(\pi){\bf T_{a}}\theta$.
Upon application of ${\bf H}_{\perp}$, the Néel sublattices
tilt towards the field, making it convenient to present
${\bf\Delta_{r}}$ as
$${\bf\Delta_{r}}(H_{\perp})={\bf\Delta}_{\bf r}^{\perp}(H_{\perp})+{\bf\Delta}_%
{\bf r}^{\|}(H_{\perp}),$$
(7)
where $\bm{\Delta}_{\bf r}^{\perp}(H_{\perp})$
points along ${\bf H}_{\perp}$, and
$\bm{\Delta}_{\bf r}^{\|}(H_{\perp})$
points along ${\bf n}$, as shown in Fig.
3.
Since $\bm{\Delta}_{\bf r+a}^{\perp}(H_{\perp})=\bm{\Delta}_{\bf r}^{\perp}(H_{\perp})$ and
$\bm{\Delta}_{\bf r+a}^{\|}(H_{\perp})=-\bm{\Delta}_{\bf r}^{\|}(H_{\perp})$, the second
column of Table 1 remains
intact upon replacing ${\bf\Delta_{r}}$
in Hamiltonian (4)
by ${\bf\Delta_{r}}(H_{\perp})$
of Eqn. (7).
If the antiferromagnetic unit cell is a multiple of its
paramagnetic counterpart, the magnetic Brillouin zone
boundary contains a set of points, that do not belong
to the paramagnetic Brillouin zone boundary (for example,
see Figs. 5 and 6).
In the paramagnetic state, no two points of this set,
separated by antiferromagnetic reciprocal lattice vector
${\bf Q}$ and satisfying condition
(6),
can be declared equivalent. As a curious consequence,
the magnetic group of such a wave vector is not
a subgroup of its paramagnetic counterpart dimmock2 .
Hence the degeneracy, if present,
does hinge on magnetic order.
III III. Clues from weak coupling
Additional insight into the locus of states, that
remain degenerate in a transverse magnetic field,
is afforded by a weak-coupling single-electron
Hamiltonian in a doubly commensurate collinear
antiferromagnet. Let ${\bf Q}$ be the
antiferromagnetic ordering wave vector (see the
examples below); ${\bf\Delta_{r}}$ creates a matrix
element $({\bf\Delta}\cdot\bm{\sigma})$ between
the Bloch states at momenta ${\bf p}$ and ${\bf p+Q}$
(for simplicity, I neglect its possible dependence
on ${\bf p}$). Sublattice canting in a transverse
field is taken into account in Appendix B.
In magnetic field ${\bf H}$, and at weak coupling,
Hamiltonian (4)
takes the form bralura
$$\mathcal{H}=\left[\begin{array}[]{cc}\epsilon_{\bf p}-({\bf H}\cdot\bm{\sigma}%
)&({\bf\Delta}\cdot\bm{\sigma})\\
&\\
({\bf\Delta}\cdot\bm{\sigma})&\epsilon_{\bf p+Q}-({\bf H}\cdot\bm{\sigma})\end%
{array}\right],$$
(8)
where
$\epsilon_{\bf p}$ and $\epsilon_{\bf p+Q}$
are single-particle energies of $\mathcal{H}_{0}$
in (4) at momenta
${\bf p}$ and ${\bf p+Q}$, and the ‘bare’
$g$-tensor in $({\bf H}\cdot\bm{\sigma})$
is omitted for brevity.
In a purely transverse field ${\bf H}_{\perp}$,
this Hamiltonian can be diagonalized simply by
choosing the $\hat{z}$-axis in spin space along
${\bf H}_{\perp}$, and the $\hat{x}$-axis along
$\bm{\Delta}$. As a result, Hamiltonian
(8) splits
into two decoupled pieces:
$\mathcal{H}_{1}({\bf p},{\bf H}_{\perp})$
for the amplitudes $|\bf{p};\uparrow\rangle$
and $|\bf{p+Q};\downarrow\rangle$, and
$\mathcal{H}_{2}({\bf p},{\bf H}_{\perp})$
for the amplitudes
$|\bf{p};\downarrow\rangle$ and
$|\bf{p+Q};\uparrow\rangle$:
$$\mathcal{H}_{1(2)}({\bf p},{\bf H}_{\perp})=\left[\begin{array}[]{cc}\epsilon_%
{\bf p}\mp H_{\perp}&\Delta\\
&\\
\Delta&\epsilon_{\bf p+Q}\pm H_{\perp}\end{array}\right].$$
(9)
The spectra $\mathcal{E}_{1(2)}({\bf p})$
of $\mathcal{H}_{1(2)}$ are given by
$$\mathcal{E}_{1(2)}({\bf p},{\bf H}_{\perp})=\eta_{\bf p}\pm\sqrt{\Delta^{2}+%
\left[\zeta_{\bf p}\mp({\bf H}_{\perp}\cdot\bm{\sigma})\right]^{2}},$$
(10)
with $({\bf H}_{\perp}\cdot\bm{\sigma})=H_{\perp}$
corresponding to $\mathcal{H}_{1}$, and
$({\bf H}_{\perp}\cdot\bm{\sigma})=-H_{\perp}$ to
$\mathcal{H}_{2}$, and with
$\eta_{\bf p}\equiv\frac{\epsilon_{\bf p}+\epsilon_{\bf p+Q}}{2}$,
and
$\zeta_{\bf p}\equiv\frac{\epsilon_{\bf p}-\epsilon_{\bf p+Q}}{2}$.
The same spectrum can be obtained by excluding,
say, $|\bf{p+Q};\sigma\rangle$ from the eigenvalue
equation for (8),
but it is important to keep in mind that
$\bm{\sigma}$ in (10)
no longer describes spin, but rather pseudospin:
since $({\bf H}_{\perp}\cdot\bm{\sigma})$ does
not commute with the Hamiltonian, the eigenstates
of $\mathcal{H}_{1(2)}$ are superpositions of
spin-up and spin-down states.
Equation (10) illustrates
a number of points. Firstly, the electron spectrum
acquires a gap of size $2\Delta$. Secondly, in the
absence of magnetic field, each eigenstate is indeed
doubly degenerate, in agreement with the arguments,
encapsulated in Eqn. (3).
Thirdly, Eqn. (10) shows,
that the degeneracy persists in a transverse field
(and, therefore, $g_{\perp}({\bf p})$
in Eqn. (1) vanishes)
whenever $\zeta_{\bf p}=0$.
Barring a special situation, this equation
defines a surface in three dimensions, a line
in two, and a set of points in one. This result
for the dimensionality of the manifold of
degenerate states hinges solely on the symmetry
of the antiferromagnetic state and holds beyond
weak coupling, as shown in Appendix C.
Furthermore, as shown above, this manifold
must contain all points, satisfying
Eqn. (6):
the points, where the degeneracy is enforced
by symmetry. Finally, expansion of Eqn.
(10) to first order in
$(\bf{H}_{\perp}\cdot\bm{\sigma})$ yields the expression
for $g_{\perp}({\bf p})$ in (1) within the weak
coupling model (8):
$$g_{\perp}({\bf p})=\frac{\zeta_{\bf p}}{\sqrt{\Delta^{2}+\zeta_{\bf p}^{2}}}.$$
(11)
At the end of the preceding Subsection, I showed that
tilting of the Néel sublattices in a transverse field
does not affect the set of points, where Kramers degeneracy
in a transverse field is protected by the anti-unitary
symmetry $\mathcal{U}{\bf U_{n}}(\pi){\bf T_{a}}\theta$.
However, generally, the rest of the degeneracy manifold
is not protected by symmetry, and may change shape upon
crystal deformation, or under another perturbation.
For instance, while leaving intact the
symmetry-protected set of degeneracy points,
the sublattice canting may change the shape
of the degeneracy manifold $g_{\perp}({\bf p})=0$
compared with $\zeta_{\bf p}=0$.
This effect is discussed in Appendix B.
Put otherwise, the degeneracy manifold may be
divided into two parts. The first part is the
‘Kramers degeneracy’ subset of special momenta,
fixed by conspiracy between the anti-unitary symmetry
${\bf U_{n}}(\pi){\bf T_{a}}\theta$ and the crystal
symmetry.
This ‘Kramers’ subset is insensitive to perturbations
that leave intact the crystal symmetry of the material.
The rest is an ‘accidental’ degeneracy subset, whose
geometry, by contrast, may vary under perturbations,
that do not affect the crystal symmetry, but only
alter the microscopic parameters of the system.
The division of the degeneracy manifold into the
‘Kramers’ and the ‘accidental’ degeneracy subsets
is well illustrated by the examples of two-dimensional
rectangular and square symmetry antiferromagnets
in Section IV.
III.1 Spectral symmetries in momentum space
The spectrum of Hamiltonian
(8) enjoys a
number of symmetries. Firstly, inversion symmetry
makes the spectrum even under inversion. At the same time,
$g_{\perp}({\bf p})({\bf H}_{\perp}\cdot\bm{\sigma})$
must also be even under inversion, which implies
$$g_{\perp}(-{\bf p})=g_{\perp}({\bf p}).$$
(12)
The anti-unitary symmetry ${\bf U_{l}}(\pi)\theta$
in the last line of Table 1
is another reason for $g_{\perp}({\bf p})$ to be
even under inversion, as ${\bf U_{l}}(\pi)\theta$
turns $g_{\perp}({\bf p})({\bf H}_{\perp}\cdot\bm{\sigma})$
into $g_{\perp}(-{\bf p})({\bf H}_{\perp}\cdot\bm{\sigma})$.
Periodicity doubling due to antiferromagnetism
manifests itself more interestingly. The
ordering wave vector ${\bf Q}$ being a reciprocal
lattice vector, any Bloch eigenstate at momentum
${\bf p}$ must have a degenerate partner eigenstate
at momentum ${\bf p+Q}$. Usually, this implies
${\bf Q}$-periodicity of any given band:
$\epsilon({\bf p})=\epsilon({\bf p+Q})$,
which is the case, for instance, in a longitudinal
field ${\bf H}_{\|}$, where
$\mathcal{E}({\bf p})$ undergoes the common Zeeman
splitting
$\mathcal{E}({\bf p})\rightarrow\mathcal{E}({\bf p})\pm H_{\|}$.
In a Néel antiferromagnet in a transverse field,
this is not the case: for a general ${\bf p}$,
$\mathcal{E}_{1(2)}({\bf p+Q})\neq\mathcal{E}_{1(2)}({\bf p})$.
Instead,
$$\mathcal{E}_{1}({\bf p+Q},{\bf H}_{\perp})=\mathcal{E}_{2}({\bf p},{\bf H}_{%
\perp}),$$
(13)
while both $\mathcal{E}_{1}({\bf p})$ and
$\mathcal{E}_{2}({\bf p})$
of Eqn. (10)
are invariant under
momentum shift ${\bf p}\rightarrow{\bf p}+2{\bf Q}$.
These properties are illustrated in
Fig. 4,
showing the splitting of a one-dimensional
conduction band in a transverse field.
The reason behind (13)
is that, as long as $H_{\perp}\neq 0$, neither
$\mathcal{H}_{1}$ nor $\mathcal{H}_{2}$ in Eqn.
(9)
is invariant under the momentum boost
$\bf{p\rightarrow p+Q}$, in spite of the
Hamiltonians (4)
and (8)
both having doubled periodicity. Rather,
$$\mathcal{H}_{1}({\bf p+Q},{\bf H}_{\perp})=\mathcal{H}_{2}({\bf p},{\bf H}_{%
\perp}),$$
(14)
which is made explicit by subsequent
exchange of the diagonal matrix elements,
and leads to Eqn. (13).
Zeeman splitting corresponds to the difference
$\mathcal{E}_{1}({\bf p},{\bf H}_{\perp})-\mathcal{E}_{2}({\bf p},{\bf H}_{%
\perp})$;
hence it changes sign upon momentum shift by $\bf{Q}$.
At the same time, direct inspection shows that
Hamiltonians $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ in
Eqn. (9)
turn into one another upon inversion of $\bf{H}_{\perp}$:
$\mathcal{H}_{1}({\bf p},-{\bf H}_{\perp})=\mathcal{H}_{2}({\bf p},{\bf H}_{%
\perp})$. Combining
this with Eqn. (14), one
finds that momentum boost by $\bf{Q}$ accompanied
by inversion of $\bf{H}_{\perp}$ is a symmetry
of both $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$:
$$\mathcal{H}_{1(2)}({\bf p+Q},-{\bf H}_{\perp})=\mathcal{H}_{1(2)}({\bf p},{\bf
H%
}_{\perp}).$$
(15)
For the transverse Zeeman term
$g_{\perp}({\bf p})({\bf H}_{\perp}\cdot\bm{\sigma})$,
this yields
$$g_{\perp}({\bf p+Q})=-g_{\perp}({\bf p}).$$
(16)
Combined, Eqns. (12)
and (16) lead to
$$g_{\perp}(\frac{\bf Q}{2}+{\bf p})=-g_{\perp}(\frac{\bf Q}{2}-{\bf p}).$$
(17)
This implies not only that $g_{\perp}({\bf p})$ must
vanish at ${\bf p}=\frac{\bf Q}{2}$, but also
that $g_{\perp}(\frac{\bf Q}{2}+{\bf p})$ is an
odd function of ${\bf p}$. The conclusions of this
subsection hold after the sublattice tilting
is taken into account (see Appendix B).
IV IV. Examples
In this section, I describe the manifolds of
degenerate states for a number of concrete
examples and thus show, that the Zeeman
spin-orbit coupling (1) is at work
in many materials of great interest. It gives
rise to various interesting phenomena,
some of which are outlined in the subsection
‘Experimental signatures’ of Section V.
Kramers degeneracy in a transverse field and the
Zeeman spin-orbit coupling (1) will
manifest themselves whenever carriers are
present at or near the manifold of degenerate
states $g_{\perp}({\bf p})=0$. In a weakly
doped antiferromagnetic insulator, this will
happen whenever the relevant band extremum
falls at or near the manifold of degenerate
states. In an antiferromagnetic metal, this
occurs when the Fermi surface crosses this
manifold. Hence, for metals, I mention the
Fermi surface geometry whenever known.
Between these two limiting cases of a weakly
doped antiferromagnetic insulator and an
antiferromagnetic metal with a large Fermi
surface, the experimental manifestations of
the Zeeman spin-orbit coupling will be
quantitatively different. On top of this,
certain effects will be sensitive to the
geometry of the degeneracy manifold and
its intersection with the Fermi surface,
as well as the orientation of the staggered
magnetization with respect to the crystal
axes. A detailed discussion of these effects
will be presented elsewhere.
When selecting the examples below, the preference
was given to materials, available in high-purity
samples, where de Haas-van Alphen oscillations
were observed, and where magnetic structure was
unambiguously characterized by neutron
scattering. As explained in the Introduction,
the results of this work apply to materials well
inside a long-range antiferromagnetic phase, and
far enough from any critical point, quantum or
classical. For both quantum and thermal
fluctuations of antiferromagnetic order to
be negligible, the ordered moment shall be
noticeable on the scale of the Bohr magneton,
and the sample shall be kept well below
both the Néel and the effective Fermi
temperatures.
IV.1 One dimension
In one dimension, the magnetic Brillouin zone
boundary reduces to two points
${\bf p}=\pm\frac{\pi}{2a}$, which in fact
coincide up to the antiferromagnetic wave
vector ${\bf Q}=\frac{\pi}{a}$, that is also
the reciprocal lattice vector of the antiferromagnetic
state (see Fig. 5). In terms of the
general condition (6),
this is the simplest case: $\mathcal{U}={\bf 1}$.
As a result, at ${\bf p}=\pm\frac{\pi}{2a}$,
the two exact Bloch states in a transverse field,
$|{\bf p}\rangle$ and
$\theta{\bf T_{a}U_{n}}(\pi)|{\bf p}\rangle$,
correspond to the same momentum ${\bf p}$, and
are degenerate by virtue of $\theta{\bf T_{a}U_{n}}(\pi)$
being a symmetry. Equation (5)
guarantees their orthogonality, thus protecting
Kramers degeneracy at momentum ${\bf p}=\pm\frac{\pi}{2a}$
against transverse magnetic field.
IV.2 Two dimensions, rectangular and square symmetries
Now consider a two-dimensional antiferromagnet
on a lattice of rectangular or square symmetry,
with the ordering wave vector ${\bf Q}=(\pi,\pi)$.
In a transverse magnetic field, the degeneracy
persists on a line in the Brillouin zone, by virtue
of Eqn. (10).
I will show that, in the rectangular case, the
degeneracy line must contain the point $\Sigma$
at the center of the magnetic Brillouin zone (MBZ)
boundary (i.e. the star of ${\bf p}={\bf Q}/2$),
shown in Fig. 6(a).
In the square symmetry case, the degeneracy persists at
the entire MBZ boundary (Fig. 6(b)).
The MBZ in Fig. 6 is the reciprocal
space counterpart of the Wigner-Seitz cell of the magnetic state (Fig. 1), and the ordering
wave vector ${\bf Q}=(\pi,\pi)$ connects points $X$ and
$Y$ in Figs. 6(a) and (b).
Consider a Bloch state $|{\bf p}\rangle$
at momentum ${\bf p}$ in a transverse field.
As discussed in Section II, the eigenstate
$\theta{\bf T_{a}U_{n}}(\pi)|{\bf p}\rangle$ at
momentum $-{\bf p}$ is degenerate with $|{\bf p}\rangle$
and, according to Eqn. (5), must be
orthogonal to it unless $({\bf p}\cdot{\bf a})$
is an integer multiple of $\pi$ – put otherwise,
unless ${\bf p}$ belongs to the paramagnetic
Brillouin zone boundary. At points $\Sigma$, X,
and Y, momenta ${\bf p}$ and $-{\bf p}$ coincide up
to a reciprocal lattice vector of the antiferromagnetic
state. However, at points X and Y (as well as at the
entire vertical segment of the MBZ boundary
in Fig. 6(a)),
$({\bf p}\cdot{\bf a})$ is an integer multiple
of $\pi$; hence $|{\bf p}\rangle$ and
$\theta{\bf T_{a}U_{n}}(\pi)|{\bf p}\rangle$
are not obliged to be orthogonal there as per
Eqn. (5). Thus, $\Sigma$
is the only point at the MBZ boundary, where the
two degenerate states $|{\bf p}\rangle$ and
$\theta{\bf T_{a}U_{n}}(\pi)|{\bf p}\rangle$
are orthogonal and correspond to the same
momentum: Dashed arrows in figure
6(a) show, that, for a
generic point ${\bf p}^{\prime}$ at the MBZ boundary,
no symmetry operation relates $-{\bf p}^{\prime}$ to a vector,
equivalent to ${\bf p}^{\prime}$. Therefore, it is only at
point $\Sigma$, that the symmetry protects Kramers
degeneracy against transverse magnetic field.
As in the one-dimensional example of the previous
subsection, in terms of Eqn.
(6) this corresponds
to the simplest case of $\mathcal{U}={\bf 1}$.
This can be illustrated by a nearest-neighbor
hopping spectrum
$$\epsilon_{\bf p}=t\left[\cos p_{x}+\eta\cos p_{y}\right]$$
(18)
in the weak-coupling example of the previous subsection:
for rectangular symmetry ($\eta\neq 1$), spectrum
(10) in a transverse field
remains degenerate at a thick (red) line, sketched in
Fig. 6(a). Upon variation
of $\eta\neq 1$, the line changes its shape,
but remains pinned at the star of wave vector
${\bf p=Q}/2$ (i.e. at point $\Sigma$)
in Fig. 6(a). In terms of the
preceding subsection, the star of ${\bf p=Q}/2$ is
the ‘Kramers’ subset of the degeneracy manifold,
while the rest of the degeneracy line in Fig.
6(a) is the ‘accidental’
degeneracy subset.
Promotion from the rectangular symmetry
($\eta\neq 1$) to that of a square
($\eta=1$) brings along invariance
under reflections $\sigma_{1,2}$ in
either of the two diagonal axes $1$ and $2$,
passing through point $\Gamma$
in Fig. 6(b).
As a result, the eigenstate
$\sigma_{1}\theta{\bf T_{a}U_{n}}(\pi)|{\bf p}\rangle$
at momentum $\sigma_{2}{\bf p}$
(Fig. 6(b)) is also degenerate
with $|{\bf p}\rangle$ and orthogonal to it, as one
can show analogously to the examples above. In terms
of the general condition (6),
this means $\mathcal{U}=\sigma_{1,2}$.
A momentum ${\bf p}$ at the MBZ boundary in
Fig. 6(b) differs from
$\sigma_{2}{\bf p}$ by a reciprocal lattice vector;
thus the two momenta coincide in the nomenclature
of the antiferromagnetic Brillouin zone. Hence, for
a square-symmetry lattice in a transverse field,
the degeneracy is of a ‘Kramers’ (i.e. symmetry-protected)
nature at the entire MBZ boundary, as shown in Fig.
6(b). In this case, barring
a particularly pathological band structure,
the degeneracy manifold is exhausted by its
‘Kramers’ subset.
In accordance with the symmetry arguments above,
for the toy nearest-neighbor hopping spectrum
(18) at the square symmetry point
$\eta=1$, the degeneracy line of Eqn. (10) coincides
with the MBZ boundary, as shown in Fig.
6(b).
By contrast, for rectangular symmetry, it is Eqn. (10) that restricts
the degeneracy in a transverse field to a line in momentum
space, and it is the symmetry that pins this line at point
$\Sigma$ at the middle of the MBZ boundary, as shown in
Fig. 6(a).
Now, $g_{\perp}({\bf p})$ can be expanded
in a vicinity of the degeneracy line
$g_{\perp}({\bf p})=0$. With the exception
of higher-symmetry points, such as point $X$
in Fig. 6(b), the leading
term of the expansion is linear in momentum
deviation ${\bf\delta p}$ from the degeneracy
line:
$$g_{\perp}({\bf p})\approx\frac{{\bf\Xi}_{\bf p}\cdot\delta{\bf p}}{\hbar},$$
(19)
where ${\bf\Xi}_{\bf p}/\hbar$ is the momentum
gradient of $g_{\perp}({\bf p})$ at point ${\bf p}$
on the degeneracy line. As mentioned in the previous
Section, inversion symmetry makes $g_{\perp}({\bf p})$
even under inversion. Therefore, ${\bf\Xi_{\bf p}}$
changes sign upon inversion, which is consistent with Eqns.
(13) and (16),
that require $g_{\perp}({\bf p})$ to change sign upon
momentum shift by ${\bf Q}$.
As shown in the last subsection of Section III,
$g_{\perp}({\bf p})$ is an odd function of the
deviation $\delta{\bf p}$ from point
$\Sigma$ (the star of ${\bf p}={\bf Q}/2$)
in Figs. 6(a) and (b).
Therefore, expansion of $g_{\perp}({\bf p})$
around point $\Sigma$ cannot contain an even
power of $\delta{\bf p}$.
IV.3 Chromium
This subsection is devoted to commensurate
antiferromagnetism in chromium – the simplest
of magnetic orders, occurring in this textbook
spin density wave metal. Chromium crystallizes
into a b.c.c. lattice, and undergoes various
magnetic and structural transitions upon
variation of temperature, pressure, or alloying fawcett_1 ; fawcett_2 ; kulikov .
Below the Néel temperature $T_{N}$ of about 311 K
at ambient pressure, chromium develops weakly
incommensurate antiferromagnetism with ordered
moment of about 0.5 $\mu_{B}$ per atom at 4.2K.
However, strain – or doping with some 0.1 to
0.3$\%$ of a transition metal (such as Mn, Re,
Rh, Ru, Ir, Os or Pt fawcett_2 )
– eliminate incommensurability in favor of
commensurate order with wave vector $\left[001\right]$,
shown in Fig. 7. Commensurate
order has also been observed and much studied in
thin films of chromium, often with an enhanced
Néel temperature and ordered moment zabel .
This article neglects fluctuations of magnetic
order and, conveniently, the high Néel
temperature of chromium facilitates experimental
access to $T\ll T_{N}$, where thermal fluctuations
are suppressed.
The paramagnetic and the antiferromagnetic Brillouin
zones for bulk commensurate antiferromagnetic chromium
are shown in Fig. 8(a).
An arbitrary momentum at the MBZ boundary becomes
equivalent to its opposite upon reflection in a
properly chosen plane.
Similarly to the two-dimensional square-symmetry
example above, this equivalence is up to a primitive
wave vector of the antiferromagnetic reciprocal space.
Hence, in a transverse magnetic field, the Kramers
degeneracy survives at the entire magnetic Brillouin
zone boundary in Fig. 8(a).
The disappearance of $g_{\perp}({\bf p})$ affects
electrons at two different sheets of the Fermi
surface, sketched in Fig. 8(b):
those at the nearly spherical electron parts, centered
at points $X$ in the middle of each MBZ face, and those
at the hole ellipsoids, centered at points $N$ in the
middle of each MBZ edge.
For the former, the leading term of the expansion is linear
in the momentum deviation $\delta p_{\perp}$ from the flat
face of the MBZ boundary. For the latter, the leading
term of the expansion is quadratic near each MBZ edge,
since $g_{\perp}({\bf p})$ vanishes at each of the two
intersecting faces of the MBZ boundary.
IV.4 CeIn${}_{3}$, UIn${}_{3}$, UGa${}_{3}$ …
A number of cerium and uranium binary intermetallics of
simple cubic Cu${}_{3}$Au structure, such as CeIn${}_{3}$, CeTl${}_{3}$,
UIn${}_{3}$, UGa${}_{3}$, UTl${}_{3}$ and UPb${}_{3}$, turn antiferromagnetic
at low temperatures. High-purity samples of CeIn${}_{3}$,
UIn${}_{3}$, and UGa${}_{3}$ have made it possible to
characterize magnetic order and electron properties
of these materials rather comprehensively.
Some of the basic properties of the samples are shown
in Table 2.
At low temperature, all three develop a type-II
antiferromagnetic structure with wave vector
${\bf Q}=\left[\frac{1}{2}\frac{1}{2}\frac{1}{2}\right]$,
shown in Fig. 9(a) for CeIn${}_{3}$. The materials
remain normal metals down to the lowest temperatures probed,
with the Sommerfeld coefficient substantially enhanced by
comparison with that of a simple metal (see the fourth
column of Table 2 versus about 0.65
mJ/K${}^{2}\cdot$mol for Ag).
Of the three materials, CeIn${}_{3}$ has been scrutinized
the most. Its early studies were driven by interest in
valence lawrence1 and magnetic lawrence2 fluctuations, in the nature of its magnetic order
lawrence2 , in large mass enhancement nasu
and related questions. Subsequent research focused on the
reduction of $T_{N}$ under pressure, and on superconductivity,
discovered near the critical pressure $p_{c}$, where the Néel
temperature is about to vanish – as well as on marked
departure from Landau Fermi liquid behavior, found in
the normal state near $p_{c}$ walker ; mathur ; grosche1 .
The most recent work included de Haas-van Alphen oscillation
measurements ebihara1 ; endo , electron-positron
annihilation experiments biasini , and
interpretation of the former gorkov .
According to Fig. 9(b), the Magnetic Brillouin
Zone of the three metals enjoys full cubic symmetry. Its
square faces belong to the paramagnetic Brillouin zone
boundary $({\bf p\cdot a})=\pm\pi$, where Eqn.
(5) does not enforce degeneracy; however,
$g_{\perp}({\bf p})$ does vanish at the hexagonal MBZ
faces, marked by darker shading in Fig. 9(b).
According to de Haas-van Alphen measurements
ebihara1 ; ebihara2 and to calculations rusz ,
one sheet of the Fermi surface of CeIn${}_{3}$ is
nearly spherical, and has radius of about
$\frac{\pi}{a}\frac{\sqrt{3}}{2}$, where $a$
is the lattice constant. Hence this sheet comes
close to the point L in Fig. 9(b),
which is the very same distance
$\frac{\pi}{a}\frac{\sqrt{3}}{2}$ away from the
Brillouin zone center. Disappearance of $g_{\perp}({\bf p})$
necessarily affects the dynamics of an electron on this
sheet in a transverse field.
Near a generic point at an MBZ face, far from its edges,
leading terms of the expansion of $g_{\perp}({\bf p})$ are
linear in transverse deviation of momentum from the MBZ
face as per Eqn. (19), with ${\bf\Xi_{p}}$
normal to the MBZ boundary. Near the edges, joining the
neighboring hexagonal faces in Fig. 9(b)
– for instance, near the points $\Sigma$ and $W$
– the leading terms become quadratic.
IV.5 Uranium nitride
Uranium nitride (UN) presents another example of interest.
This heavy fermion metal has a face-centered cubic lattice
of NaCl type, shown in Fig. 10(a). Below 53K,
it develops type-I antiferromagnetic order, with ordered
moment of about 0.75$\mu_{B}$ per uranium atom curry ,
and the Sommerfeld coefficient of 50 mJ/K${}^{2}\cdot$mol
nakashima2 . The Neéel temperature of UN drops
under pressure, vanishing at about 3.5 GPa.
Recent experiments nakashima2 studied the
low-temperature resistivity near the critical pressure
on samples with residual resistivity $\rho_{0}$ of about
2.3 $\mu\Omega$cm, and the residual resistivity ratio
$\rho(300K)/\rho_{0}$ of the order of $10^{2}$.
The real-space sketch of magnetic structure of UN
is shown in Fig. 10 together with its
paramagnetic and antiferromagnetic Brillouin zone
boundaries. The MBZ has full tetragonal symmetry
and, in a transverse field, all the states at its
boundary retain Kramers degeneracy. The leading terms
in the expansion of $g_{\perp}({\bf p})$ are linear near
the MBZ faces, quadratic near the edges, and cubic
near the vertices.
IV.6 CePd${}_{2}$Si${}_{2}$ and CeRh${}_{2}$Si${}_{2}$
The heavy fermion metal CePd${}_{2}$Si${}_{2}$ has a body-centered
tetragonal structure of ThCr${}_{2}$Si${}_{2}$ type, shown in Fig.
11(a). It is isostructural to CeCu${}_{2}$Si${}_{2}$
– the first discovered heavy fermion superconductor
steglich – and CeCu${}_{2}$Ge${}_{2}$, an incommensurate
antiferromagnet knopp , that becomes superconducting
above 70 kbar in a pressure cell jaccard .
Below about 10 K, CePd${}_{2}$Si${}_{2}$ orders antiferromagnetically
as shown in Fig. 11(a), with wave vector
${\bf Q}=\left[\frac{1}{2}\frac{1}{2}0\right]$,
and a low-temperature ordered moment of about 0.7 $\mu_{B}$
per Ce atom. Its Sommerfeld coefficient is enhanced
to about 100 mJ/K${}^{2}\cdot$mol.
Samples of the present generation show residual
resistivity in the $\mu\Omega\cdot$cm range
grosche1 . Under hydrostatic pressure of
26 kbar, the Néel temperature drops to under
1 K and, in a pressure window of $\pm$5 kbar
around this value, superconductivity appears,
with a maximum transition temperature of about
0.4 K grosche . Curiously enough, normal state
resistivity near this pressure follows a temperature dependence, that does not fit the
$\rho(T)=\rho_{0}+AT^{2}$ temperature dependence
of the Landau Fermi liquid theory, but instead
behaves as $\rho(T)\sim T^{1.2}$ over more than a decade
in temperature, between about 1 and 40 K grosche .
The unit cell of CePd${}_{2}$Si${}_{2}$, and its paramagnetic
and antiferromagnetic Brillouin zone boundaries are
shown in Fig. 11. By symmetry, the
degeneracy manifold in a transverse field includes
the two hexagonal faces of the MBZ boundary,
one of which is shown by darker shading in Fig.
11(b), and the four segments, two
of which are shown in black. Along these segments,
which are a three-dimensional analogue of point
$\Sigma$ in Fig. 6(a),
another sheet of the degeneracy surface crosses
the side faces of the MBZ.
According to de Haas-van Alphen (dHvA) experiments
sheikin , several Fermi surface sheets cross
the degeneracy surface. The leading term in the
expansion of $g_{\perp}({\bf p})$ around the hexagonal
MBZ faces is linear.
CeRh${}_{2}$Si${}_{2}$ is an isostructural relative of
CePd${}_{2}$Si${}_{2}$, with a modestly enhanced Sommerfeld
coefficient of about 23 mJ/K${}^{2}\cdot$mol. Between
$T_{N1}\approx 36$K and $T_{N2}\approx 25$K,
it develops Néel order with
${\bf Q}=\left[\frac{1}{2}\frac{1}{2}0\right]$
grier ; kawarazaki . Magnetic structure below
$T_{N2}$ has not yet been established unambiguously
grier ; kawarazaki . Both $T_{N1}$ and $T_{N2}$ drop
under pressure ohashi and, in an extended pressure
window above 5 kbar, CeRh${}_{2}$Si${}_{2}$ becomes superconducting
at a $T_{c}$ with a maximum of about 0.5K movshovich .
Antiferromagnetic structure of CeRh${}_{2}$Si${}_{2}$ between $T_{N1}$
and $T_{N2}$ coincides with that of CePd${}_{2}$Si${}_{2}$, as does
the degeneracy surface in Fig. 11(b).
According to araki , at least one sheet of the Fermi
surface of CeRh${}_{2}$Si${}_{2}$ crosses the degeneracy surface
or comes close to it.
IV.7 Neodymium hexaboride
Rare earth hexaborides RB${}_{6}$ are an interesting family,
whose members show diverse electron and magnetic
properties. Of the (relatively) simple ones, LaB${}_{6}$ is a
diamagnetic metal, and SmB${}_{6}$ is a mixed valence
semiconductor. Of the ordered materials, EuB${}_{6}$ is a
ferromagnetic semi-metal, and CeB${}_{6}$ is a heavy fermion
metal with at least two ordered phases, whose nature
remains to be elucidated after nearly forty years of research.
Three members of the family: NdB${}_{6}$, GdB${}_{6}$, and
PrB${}_{6}$, are antiferromagnetic at low temperature.
In PrB${}_{6}$ burlet and in GdB${}_{6}$
mcmorrow ; galera alike, two different
low-temperature antiferromagnetic
states have been found.
Neodymium hexaboride NdB${}_{6}$ presents a simpler
picture: below about 8 K, it is a collinear
type-I antiferromagnet with ordering vector
${\bf Q}=\left[00\frac{1}{2}\right]$ and an
ordered moment of about 1.74 $\mu_{B}$ mccarthy ;
antiferromagnetism doubles its cubic unit cell in the
[0 0 1] direction, as shown in Fig. 12(a).
Thus the cubic magnetic Brillouin zone reduces
by half in the [0 0 1] direction, while keeping
its other two dimensions intact, as shown
in Fig. 12(b).
In a transverse field, the Kramers degeneracy
is protected at the two faces of the MBZ boundary,
one of which is shown by darker shading in
Fig. 12(b).
According to de Haas-van Alphen measurements
onuki ; goodrich and to calculations kubo ,
at least one sheet of the Fermi surface crosses the
degeneracy surface. Recently studied samples had
residual resistivities well below $\mu\Omega\cdot$cm,
and residual resistivity ratios of over a 100
onuki ; stankiewicz .
IV.8 Other materials of interest
This subsection contains a brief discussion of other
antiferromagnets, where symmetry may protect the
degeneracy of special electron states against
transverse magnetic field, giving rise to Zeeman
spin-orbit coupling (1).
Cuprate superconductors: Electron-doped cuprates
such as Nd${}_{2-x}$Ce${}_{x}$CuO${}_{4\pm\delta}$ develop
commensurate antiferromagnetic order in a wide range
of doping motoyama , albeit with a modest
staggered moment matsuda . For such materials,
Fig. 6(b) describes the
paramagnetic and antiferromagnetic Brillouin zone boundaries.
Angle-resolved photoemission experiments armitage
on Nd${}_{2-x}$Ce${}_{x}$CuO${}_{4\pm\delta}$ have found carriers
in a vicinity of the MBZ boundary. In a transverse magnetic
field, these carriers are subject to Zeeman spin-orbit
coupling (1), provided antiferromagnetism
in the sample is developed well enough.
Recent observation doiron of magnetic oscillations
in YBa${}_{2}$Cu${}_{3}$O${}_{6.5}$ testifies to great progress
in sample quality of cuprates. And the fact that this and
other underdoped cuprates are, at the very least, close
to commensurate antiferromagnetism, makes them an
interesting opportunity to examine the effects of
Zeeman spin-orbit coupling.
Borocarbides RT${}_{2}$B${}_{2}$C with
R = Sc, Y, La, Th, Dy, Ho, Er, Tm or Lu
and T = Ni, Ru, Pd or Pt have been a subject
of active research, driven by interest in
interplay between antiferromagnetism and
superconductivity mueller . At low
temperatures, commensurate antiferromagnetism
develops in a number of borocarbides
(for instance, in $R$Ni${}_{2}$B${}_{2}$C with
$R=$ Pr, Dy or Ho), often with a large
staggered moment ($\approx$ 8.5 $\mu_{B}$
for Dy and Ho) mueller .
Zeeman spin-orbit coupling (1)
is present whenever a sheet of the Fermi
surface crosses the degeneracy manifold,
and successful growth of high-quality
single crystals cava makes these
materials an interesting case to study.
Organic conductors are an immense and ever
growing class of quasi-low-dimensional materials, that
show virtually all known types of electron states,
found in condensed matter physics chaikin .
Antiferromagnetism appears in several families of
organic conductors, and manifestations of Zeeman
spin-orbit coupling (1) are likely
to be found in some of them.
Unfortunately, so far nearly all of the
information on magnetic structure of organic
antiferromagnets has been coming from indirect
probes such as magnetic susceptibility
measurements iwasa ; uozaki and resonance
spectroscopies coulon ; miyagawa ; wzietek .
Neutron diffraction studies are hampered by a typically
small ordered moment, and by the difficulties of growing
large enough single-crystalline samples. At the moment
of writing, I am aware of only a single cycle of neutron
scattering experiments foury ; pouget ; pouget2
on an organic conductor. Moreover, in families
such as (TMTSF)${}_{2}$ Bechgaard salts chaikin
and $\kappa$-(BEDT-TTF)${}_{2}$X salts lefebvre ; kagawa ,
antiferromagnetic states are insulating, and their
controlled doping remains a challenge kanoda .
With this word of caution, a number of organic conductors
may deserve attention. Semi-metallic Bechgaard salt
(TMTSF)${}_{2}$NO${}_{3}$ biskup , developing a spin
density wave state below about 9 K, may be one interesting
case. Recently synthesized
ethylenedioxytetrathiafulvalenoquinone-1,3-diselenolemethide
(EDO-TTFVODS), that appears to turn antiferromagnetic below
about 4.5 K, and remains normal down to the lowest studied
temperature of 0.45 K xiao , may be another.
Finally, recent studies suzuki ; zhou ; hara
of [Au(tmdt)${}_{2}$], where tmdt denotes trimethylenetetrathiafulvalenedithiolate, draw
attention to this organic conductor. Albeit the material
is not yet fully characterized, and its large single
crystals remain difficult to grow, it appears to have
a Néel temperature of about 110 K zhou ; hara ,
which is anomalously high for an organic material
– and shows normal conduction down to at least 10 K.
Heavy fermion materials: Several heavy
fermion antiferromagnets were reviewed in detail
above. A number of other interesting examples
may be found in robinson .
Gadolinium antiferromagnets
(see barandiaran ; granado , and Table I
in rotter ) offer two important advantages
for an experimental study of the Zeeman spin-orbit
coupling. Firstly, their often elevated Néel
temperature $T_{N}$ (such as 134 K for GdAg, or 150 K
for GdCu) facilitates experimental access to
temperatures well below $T_{N}$, where thermal
fluctuations of antiferromagnetic order are frozen
out. Secondly, large ordered moment of these materials
(about 7.5$\mu_{B}$ for GdAg, and about 7.2$\mu_{B}$
for GdCu${}_{2}$Si${}_{2}$) quenches quantum fluctuations.
Therefore, gadolinium antiferromagnets fit well
into the present framework with its neglect
of both quantum and classical fluctuations
– and shall be convenient for a study of
various effects of the Zeeman spin-orbit coupling.
Iron pnictides have been attracting immense
attention grant due to appearance of commensurate
antiferromagnetism cruz and high-temperature
superconductivity takahashi in this copper-free
family of materials. Combination of commensurate
antiferromagnetism cruz with essentially metallic
normal state conduction takahashi ; kamihara
not only contrasts iron pnictides with the cuprates
(that are believed to be Mott insulators), but also
makes the former materials likely to manifest a
substantial momentum dependence of the $g$-tensor.
V V. Discussion
V.1 Effects of relativistic spin-orbit coupling
The arguments above appealed to the exchange symmetry
approximation andreev : the point symmetry
operations of the electron Hamiltonian in an
antiferromagnet were considered inert with respect
to spin, and the common relativistic spin-orbit
coupling, that appears in the absence of an external
magnetic field, was thus neglected.
I will now examine the effects it may have.
Firstly, this spin-orbit interaction generates magnetic
anisotropy, that creates a preferential orientation
of the staggered magnetization ${\bf n}$ with respect
to the crystal axes. In an experiment, this allows
one to vary the magnetic field orientation with
respect to ${\bf n}$ as long as the field remains
below the reorientation threshold.
At the same time, the spin-orbit coupling may eliminate
those spatial symmetries, that rotate the magnetization
density with respect to the lattice. For instance,
certain spin rotations and spatial transformations,
that were independent symmetries within the exchange
symmetry approximation, may survive only when combined.
I will now illustrate this by two examples of Section IV.
A simple case of the spin-orbit coupling affecting
the Kramers degeneracy manifold in a transverse field
is given by a two-dimensional antiferromagnet
on a square-symmetry lattice as in Fig.
6(b). Here, the Kramers
degeneracy at the antiferromagnetic
Brillouin zone boundary relies, everywhere
except for points $\Sigma$, on the symmetry
with respect to reflections in diagonal planes 1 and 2.
If either of these reflections changes the orientation
of $\bm{\Delta}_{\bf r}$ with respect to the crystal
axes, a spin-orbit coupling may lift the degeneracy
at a relevant part of the magnetic Brillouin zone
boundary, except for points $\Sigma$.
However, if the magnetization density points along
one of these diagonal axes, the degeneracy survives
at the two faces of the MBZ boundary, that are
normal to this axis.
In the case of commensurate order in chromium,
consider a single-domain sample with magnetic
structure shown in Fig. 7.
For a Bloch state $|{\bf p}\rangle$ with a momentum
${\bf p}$ at one of the two horizontal faces of the
magnetic Brillouin zone in Fig 8(a), the
degenerate partner state
$\theta{T_{a}U_{n}}(\pi)|{\bf p}\rangle$
has momentum $-{\bf p}$ at the other horizontal
face of the MBZ. Coordinate rotation by $\pi$ around
the vertical symmetry axis, passing through the center
$\Gamma$ of the Brillouin zone, transforms the momentum
${\bf p}$ into ${\bf p+Q}$, equivalent to ${\bf p}$
up to reciprocal lattice vector ${\bf Q}$
of the antiferromagnetic state.
By contrast, for a momentum ${\bf p}$ at one of the
vertical faces of the MBZ, coordinate rotation by $\pi$
around a horizontal axis is required, and such a rotation
inverts ${\bf\Delta_{r}}$ once the latter is attached to
the crystal axes. Thus spin-orbit coupling tends to lift
the degeneracy at the vertical faces of the MBZ, leaving
it intact at the two horizontal faces. The other examples
of Section III can be analyzed similarly.
Finally, those spin-orbit coupling terms, that
act directly on the electron spin and tend to lift
the double degeneracy of Bloch eigenstates even
in the absence of magnetic field, were neglected
here altogether.
V.2 Relation to earlier work
When symmetries of a system involve time reversal
– alone or in combination with other operations –
a proper treatment must involve non-unitary symmetry
groups: those containing unitary as well as anti-unitary
elements. In this case, construction of irreducible
representations is complicated by the fact that
anti-unitary elements involve complex conjugation.
In a group representation, combination of two unitary
elements $u_{1}$ and $u_{2}$ is represented by
the product of the corresponding matrices
${\bf D}(u_{1})$ and ${\bf D}(u_{2})$:
${\bf D}(u_{1}u_{2})={\bf D}(u_{1}){\bf D}(u_{2})$.
By contrast, combination of an anti-unitary element
$a$ with a unitary element $u$ involves complex
conjugation: ${\bf D}(au)={\bf D}(a){\bf D}^{*}(u)$.
As a result, irreducible representations of a non-unitary
group must include a unitary representation and its
complex conjugate on an equal footing. Discussion of
such representations (called co-representations) was
given by Wigner, along with the analysis of arising
possibilities with the help of the Frobenius-Schur
criterion wigner . Later, Herring studied
spectral degeneracies, emerging in crystals due to time
reversal symmetry and, among other things, extended this
criterion to space groups herring2 . In a subsequent
work, Dimmock and Wheeler generalized the criterion further,
to magnetic crystals, and pointed out the sufficient
condition (6) for the
appearance of extra degeneracies dimmock2 .
The present work identifies the symmetry, that protects
the Kramers degeneracy in a Néel antiferromagnet
against transverse magnetic field, as a conspiracy
between the anti-unitary symmetry
${\bf U_{n}}(\pi){\bf T_{a}}\theta$, inherent to any
collinear commensurate antiferromagnet in a transverse
field, and the crystal symmetry of those special
momenta at the MBZ boundary, that are defined by
Eqn. (6).
Formally, the present work is an extension of
dimmock2 , since one may think of the last
two terms in (4) as of
the exchange field of a fictitious magnetic crystal
in zero field.
However, Kramers degeneracy in a magnetic field
has rather special and remarkable experimental
signatures, some of which are outlined at the
end of this section.
Last but not the least, Ref. braluk
was an important source of inspiration for
the present work. Its authors studied the
electron eigenstates in a Néel antiferromagnet
on a lattice of square symmetry and, for this
particular case, pointed out the
disappearance of $g_{\perp}({\bf p})$ at the MBZ
boundary, as well as the ensuing substantial momentum
dependence of $g_{\perp}$ in the Zeeman coupling
(1). The present article builds on
Ref. braluk by elucidating the structure
of the manifold of degenerate states for an
arbitrary crystal symmetry, and for an arbitrary
transverse field that can be sustained by the
antiferromagnet before its sublattices collapse.
This is to be contrasted with the analysis of
Ref. braluk , performed to the linear
order in the field. Several other aspects of
Ref. braluk are discussed in Appendix D.
V.3 Experimental signatures
The Kramers degeneracy at special momenta on the
MBZ boundary and the resultant Zeeman spin-orbit
coupling have a number of interesting consequences.
For instance, a substantial momentum dependence
of $g_{\perp}({\bf p})$ in Eqn. (1)
means that, generally, the Electron Spin
Resonance (ESR) frequency of a carrier in the
vicinity of the degeneracy manifold varies
along the quasiclassical trajectory in momentum space.
For a weakly-doped antiferromagnetic insulator with
a conduction band minimum on the degeneracy manifold,
this leads to an inherent broadening of the ESR line
with doping and, eventually, complete loss
of the ESR signal. In fact, this may well be
the reason behind the long-known ‘ESR silence’ shengelaya of the cuprates. Suppression
of Pauli paramagnetism in the transverse direction
with respect to staggered magnetization is another
simple consequence of vanishing $g_{\perp}({\bf p})$.
At the same time, a momentum dependence of
$g_{\perp}({\bf p})$ allows excitation of spin
resonance transitions by AC electric rather
than magnetic field rr ; zedr – a vivid effect
of great promise for controlled spin manipulation,
currently much sought after in spin electronics.
Its absorption matrix elements are defined by
${\bf\Xi}_{\bf p}$ of Eqn. (19).
Comparison with Eqn. (11)
shows that, within the weak-coupling model
(8),
$\Xi_{\bf p}/\hbar$ is of the order
of the antiferromagnetic coherence length
$\xi\sim\frac{\hbar v_{F}}{\Delta}$, and
may be of the order of the lattice period
or much greater. By contrast, the ESR matrix
elements are defined by the Compton length
$\lambda_{C}=\frac{\hbar}{mc}\approx 0.4$ pm.
Thus, matrix elements of electrically excited
spin transitions exceed those of ESR by about
$\frac{\hbar c}{e^{2}}\cdot\frac{\epsilon_{F}}{\Delta}\approx 137\cdot\frac{%
\epsilon_{F}}{\Delta}$,
or at least by two orders of magnitude. Being
proportional to the square of the appropriate
transition matrix element, resonance absorption
due to electric excitation of spin transitions
exceeds that of ESR at least by four orders of
magnitude.
Last but not the least – according to
Eqn. (19), resonance absorption
in this phenomenon shows a non-trivial dependence
on the orientation of the AC electric field with
respect to the crystal axes, and on the orientation
of the DC magnetic field with respect to the
staggered magnetization.
The Zeeman spin-orbit coupling may also manifest itself
in other experiments on antiferromagnetic conductors.
In particular, de Haas-van Alphen oscillations
kabanov and magneto-optical response may
be modified. In various types of electron response,
interesting effects may arise due to an extra term
${\bf v}_{ZSO}$ in the electron velocity
operator, emerging due to a substantial
momentum dependence of $g_{\perp}({\bf p})$
in Eqn. (1):
$${\bf v}_{ZSO}=\nabla_{\bf p}\mathcal{H}_{ZSO}=-\mu_{B}\nabla_{\bf p}g_{\perp}(%
{\bf p})({\bf H_{\perp}\cdot\bm{\sigma}}).$$
(20)
This term describes spin current. However,
$g_{\perp}({\bf p})$ is even in ${\bf p}$
due to inversion symmetry and thus, in
equilibrium, the net spin current must
vanish. This may change, if the system
were tilted, say, by electric current
or otherwise – however, the resulting
effect would be proportional to the ‘tilt’
and, in addition to this, would be small
in the measure of $H_{\perp}/\Delta$.
V.4 Conclusions
In this work, I studied the degeneracy of electron
Bloch states in a Néel antiferromagnet, subject
to a transverse magnetic field – and described
the special points in momentum space, where the
degeneracy is protected by a hidden anti-unitary
symmetry.
I discussed the simplest properties and some of the
manifestations of the Zeeman spin-orbit coupling,
arising in a magnetic field due to this degeneracy,
and outlined several examples of interesting
materials, where such a coupling may be present.
Finally, I reviewed the results and their
relation to earlier work.
The degeneracy of special Bloch states in a transverse field
hinges only on the symmetry of the antiferromagnetic state,
and thus holds in weakly coupled and strongly correlated
materials alike – provided long-range antiferromagnetic
order and well-defined electron quasiparticles are present.
Under these conditions, thermal and quantum fluctuations
of the antiferromagnetic order primarily renormalize the
sublattice magnetization, leaving intact the degeneracy
of special electron states in a transverse field
– certainly in the leading order in fluctuations.
Detailed account of fluctuations is outside
the scope of this article.
VI Acknowledgments
I am indebted to S. Brazovskii and G. Shlyapnikov
for inviting me to Orsay, to LPTMS for the kind
hospitality, and to IFRAF for generous
support. I am grateful to S. Brazovskii,
M. Kartsovnik and K. Kanoda for references
on organic antiferromagnets, and to C. Capan for
drawing my attention to gadolinium compounds.
It is my pleasure to thank S. Carr, N. Cooper,
N. Shannon, and M. Zhitomirsky for their
helpful comments on the manuscript,
and A. Chubukov and G. Volovik
for enlightening discussions.
VII Appendix A: orthogonality relation
This Appendix proves the relation
$$\langle\phi|\left[\mathcal{O}\theta\right]^{+}|\left[\mathcal{O}\theta\right]|%
\psi\rangle=\langle\psi|\phi\rangle,$$
(21)
where $|\phi\rangle$ and $|\psi\rangle$
are arbitrary states, $\mathcal{O}$
is an arbitrary unitary operator, and $\theta$
is time reversal. In the main text, this relation
is used for
$|\phi\rangle=\mathcal{O}\theta|\psi\rangle$;
in this case, when read right to left,
Eqn. (21) yields
$$\langle\psi|\mathcal{O}\theta|\psi\rangle=\langle\psi|[\left(\mathcal{O}\theta%
\right)^{+}]^{2}|(\mathcal{O}\theta)|\psi\rangle.$$
(22)
Whenever $|\psi\rangle$ is an
eigenvector of the linear operator
$[\mathcal{O}\theta]^{2}$ with an
eigenvalue different from unity,
Eqn. (22) proves
orthogonality of $|\psi\rangle$ and
$\mathcal{O}\theta|\psi\rangle$.
The proof of Eqn. (21) is based
on the obvious relation
$(\mathcal{C}\phi,\mathcal{C}\psi)=(\psi,\phi)$ for arbitrary complex vectors
$\phi$ and $\psi$, where
$(\psi,\phi)\equiv\sum_{i}\psi_{i}^{*}\phi_{i}$
denotes scalar product, and $\mathcal{C}$
is complex conjugation.
Hence, for an arbitrary unitary operator
$\mathcal{O}$, one finds
$(\mathcal{O}\mathcal{C}\phi,\mathcal{O}\mathcal{C}\psi)=(\psi,\phi)$, due to invariance of scalar product
under unitary transformation. Time reversal $\theta$
can be presented as a product of $\mathcal{C}$ and a
unitary operator wigner :
$\theta=\mathcal{V}\mathcal{C}$
, thus
$\mathcal{C}=\mathcal{V}^{-1}\theta$ and, therefore,
$(\mathcal{O}\theta\phi,\mathcal{O}\theta\psi)=(\psi,\phi)$.
As a result, for arbitrary states
$|\psi\rangle$ and $|\phi\rangle$, one finds
$\langle\phi|\left[\mathcal{O}\theta\right]^{+}|\left[\mathcal{O}\theta\right]|%
\psi\rangle=\langle\psi|\phi\rangle$,
which indeed amounts to (21).
VIII Appendix B: canting of the sublattices
Canting of the two sublattices by transverse
field ${\bf H}_{\perp}$ induces a component
$\bm{\Delta}_{\bf r}^{\perp}$
of the magnetization density along the field,
with the periodicity of the underlying lattice:
$\bm{\Delta}_{\bf r+a}^{\perp}(H_{\perp})=\bm{\Delta}_{\bf r}^{\perp}(H_{\perp})$,
as shown in Fig. 3.
As a result, the diagonal part of Hamiltonian
(8) acquires
an additional term
$(\bm{\Delta}_{\bf p}^{\perp}\cdot\bm{\sigma})$,
and Hamiltonian (8)
thus takes the form
$$\mathcal{H}=\left[\begin{array}[]{cc}\epsilon_{\bf p}-({\bf\tilde{\Delta}_{p}}%
^{\perp}\cdot\bm{\sigma})&({\bf\Delta}_{\|}\cdot\bm{\sigma})\\
&\\
({\bf\Delta}_{\|}\cdot\bm{\sigma})&\epsilon_{\bf p+Q}-({\bf\tilde{\Delta}_{p+Q%
}}^{\perp}\cdot\bm{\sigma})\end{array}\right],$$
(23)
where ${\bf\tilde{\Delta}_{p}}^{\perp}\equiv\bf{H}_{\perp}+\bm{\Delta}_{\bf p}^{\perp}$.
The same choice of spin axes as in Section III splits
Hamiltonian (23)
into two independent pieces
$$\mathcal{H}_{1(2)}=\left[\begin{array}[]{cc}\epsilon_{\bf p}\mp\tilde{\Delta}_%
{\bf p}^{\perp}&\Delta_{\|}\\
&\\
\Delta_{\|}&\epsilon_{\bf p+Q}\pm\tilde{\Delta}_{\bf p+Q}^{\perp}\end{array}%
\right].$$
(24)
As in Section III, momentum shift by ${\bf Q}$
maps $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ onto
each other, and spectral symmetries of Hamiltonian
(23)
coincide with those discussed in the second
subsection of Section III. Thus all of the
conclusions of Section III remain valid after
sublattice canting is accounted for.
However, while degeneracy at special points is
protected by symmetry, the shape of the manifold
of degenerate states may change under various
perturbations. For instance, sublattice canting
in a transverse field modifies the equation,
describing this manifold and, for the conduction
band, turns it into
$$\psi_{\bf p}+\frac{\phi_{\bf p}\zeta_{\bf p}}{\sqrt{\Delta^{2}_{\|}+\phi_{\bf p%
}^{2}+\zeta_{\bf p}^{2}}},$$
(25)
where $\phi_{\bf p}\equiv\frac{1}{2}[\tilde{\Delta}_{\bf p}^{\perp}+\tilde{\Delta}_{%
\bf p+Q}^{\perp}]$
and
$\psi_{\bf p}\equiv\frac{1}{2}[\tilde{\Delta}_{\bf p}^{\perp}-\tilde{\Delta}_{%
\bf p+Q}^{\perp}]$.
Since $\bm{\Delta}_{\bf r}^{\perp}$
has the real-space periodicity of the paramagnetic
state, $\tilde{\Delta}_{\bf p}^{\perp}$ enjoys the same
reciprocal space symmetry as $\epsilon_{\bf p}$. In
particular, $\tilde{\Delta}_{{\bf p}+2{\bf Q}}^{\perp}=\tilde{\Delta}_{\bf p}^{\perp}$, and
$\tilde{\Delta}_{\bf p}^{\perp}=\tilde{\Delta}_{-\bf p}^{\perp}$ (the latter property
is also protected by the ${\bf U_{l}}\theta$ symmetry).
At the same time, $\psi_{\bf p+Q}=-\psi_{\bf p}$,
and $\phi_{\bf p+Q}=\phi_{\bf p}$; thus
the symmetry-dictated degeneracy points such
as ${\bf p}=\frac{\bf Q}{2}$ explicitly
belong to the manifold of Eqn.
(25), as they should.
In the limit of vanishing $H_{\perp}$,
$\tilde{\Delta}_{\bf p}^{\perp}$ is linear in the field:
$\tilde{\Delta}_{\bf p}^{\perp}=H_{\perp}[1+\chi_{\bf p}^{\perp}]$,
where $\chi_{\bf p}^{\perp}$ describes microscopic
transverse susceptibility of the antiferromagnet.
Now one may expand Eqn. (25)
to linear order in the field to obtain the
following equation for the degeneracy manifold:
$$\chi_{\bf p}^{-}+\frac{\chi_{\bf p}^{+}\zeta_{\bf p}}{\sqrt{\Delta^{2}_{\|}+%
\zeta_{\bf p}^{2}}}=0,$$
(26)
where
$\chi_{\bf p}^{\pm}\equiv\chi_{\bf p}\pm\chi_{\bf p+Q}$.
Compared with equation $\zeta_{\bf p}=0$ of Section III,
the sublattice canting affects the degeneracy
manifold already in zeroeth order in $H_{\perp}$.
IX Appendix C: Dimensionality of the degeneracy manifold
The dimensionality of the degeneracy manifold in
a transverse field is one less than that of the
momentum space for simple reasons, that rely
only on the symmetry of the antiferromagnetic
state. According to Eqn.
(3), zero-field Bloch eigenstates
$|1\rangle\equiv|{\bf p}\rangle$ and
$|2\rangle\equiv\mathcal{I}{\bf T_{a}}\theta|{\bf p}\rangle$
form a Kramers doublet at momentum ${\bf p}$.
Its splitting $\delta\mathcal{E}({\bf p})$ in
a transverse field ${\bf H}_{\perp}$ is given by
$$\delta\mathcal{E}({\bf p})=2\sqrt{|V_{12}({\bf p})|^{2}+\frac{1}{4}\left[V_{11%
}({\bf p})-V_{22}({\bf p})\right]^{2}},$$
(27)
where $V_{ij}({\bf p})\equiv\langle i|({\bf H}_{\perp}\cdot{\bm{\sigma}})|j\rangle$.
Magnetic field being uniform,
$({\bf H}_{\perp}\cdot{\bm{\sigma}})$ commutes
with $\mathcal{I}{\bf T_{a}}$; it also changes
sign under time reversal.
Thus, $V_{22}({\bf p})=-V_{11}({\bf p})$.
At the same time, the off-diagonal matrix element
$V_{12}({\bf p})$ vanishes identically:
$$\displaystyle\langle{\bf p}|({\bf H}_{\perp}\cdot{\bm{\sigma}})\mathcal{I}{\bf
T%
_{a}}\theta|{\bf p}\rangle=\sum_{\bf q}\langle{\bf p}|({\bf H}_{\perp}\cdot{%
\bm{\sigma}})|{\bf q}\rangle\langle{\bf q}|\mathcal{I}{\bf T_{a}}\theta|{\bf p}\rangle$$
$$\displaystyle=$$
$$\displaystyle=\sum_{\bf q}V_{11}({\bf p})\delta_{\bf pq}\langle{\bf q}|%
\mathcal{I}{\bf T_{a}}\theta|{\bf p}\rangle=V_{11}({\bf p})\langle{\bf p}|%
\mathcal{I}{\bf T_{a}}\theta|{\bf p}\rangle\equiv 0,$$
(28)
where insertion of unity ${\bf 1}=\sum_{\bf q}|{\bf q}\rangle\langle{\bf q}|$ was used in
the first line, uniformity of ${\bf H}_{\perp}$
in the second, and the final equality followed
from Eqn. (3). Therefore,
$$\delta\mathcal{E}({\bf p})=2|V_{11}({\bf p})|,$$
(29)
and, barring a special case, equation
$\delta\mathcal{E}({\bf p})=0$ defines a
$(d-1)$-dimensional surface of zero $g_{\perp}({\bf p})$
in $d$-dimensional momentum space. The Kramers
degeneracy subset contains, at the very least,
the star of the momentum ${\bf p}={\bf Q}/2$
[see Eqn.(6) and the
subsequent discussion], and the $({\bf k\cdot p})$
expansion kittel around these points shows,
that they are not isolated, but rather belong
to a $(d-1)$-dimensional manifold. The latter
is continuous, with the obvious exception of $d=1$.
Finally, notice that, according to (29),
$\delta\mathcal{E}({\bf p})$ is periodic with the
antiferromagnetic ordering wave vector ${\bf Q}$:
$$\delta\mathcal{E}({\bf p}+{\bf Q})=\delta\mathcal{E}({\bf p}),$$
(30)
thanks to ${\bf Q}$ being a reciprocal lattice
vector in the antiferromagnetic state. Therefore,
properties (16) and
(17) are indeed model-independent,
as opposed to hinging on an approximation of the
weak-coupling model (8).
X Appendix D: revisiting braluk
In Ref. braluk , Brazovskii and Lukyanchuk
stated that operator ${\bf\Lambda}=({\bf n}\cdot\bm{\sigma})$
exchanges the momenta ${\bf p}$ and ${\bf p+Q}$
in (8), and
thus represents the momentum boost by the
ordering wave vector ${\bf Q}$ in reciprocal space.
In a commensurate antiferromagnetic state, ${\bf Q}$
becomes a reciprocal lattice vector, and thus
${\bf\Lambda}$ must be a symmetry of the Hamiltonian.
With the assumption of the effective Zeeman coupling
(1), this lead the authors of Ref.
braluk to the relation
$g_{\perp}({\bf p+Q})=-g_{\perp}({\bf p})$
(Eqn. (16) of the present work),
and to the conclusion that $g_{\perp}({\bf p})=0$
at the MBZ boundary. Unfortunately, while this
beautiful result is indeed correct for a lattice
of square symmetry, several circumstances
prevent one from embracing these arguments.
Most importantly, they hinge solely on the
symmetry under translation by ${\bf Q}$,
put otherwise – on commensurability
of magnetic order with the crystal lattice.
If correct, this would imply that, in an
arbitrary commensurate Néel antiferromagnet,
Kramers degeneracy takes place at the entire
MBZ boundary regardless of the underlying
crystal symmetry. The toy example
(18) with $\eta\neq 1$ shows,
that this is not at all necessarily the case.
Indeed, for a generic crystal symmetry, the condition
$g_{\perp}({\bf p+Q})=-g_{\perp}({\bf p})$
(see braluk and Eqn. (16)
of the present work) does not, by itself,
restrict the manifold $g_{\perp}({\bf p})=0$
to the MBZ boundary. However, the ‘Kramers’ subset
of the manifold of degenerate states can be obtained
by combining Eqn. (16) with the
crystal symmetries (see the subsection on spectral
symmetries in momentum space in Section III).
For instance, combined with the inversion symmetry
$g_{\perp}(-{\bf p})=g_{\perp}({\bf p})$,
Eqn. (16) stipulates that
$g_{\perp}({\bf Q}/2)=0$. Similarly,
disappearance of $g_{\perp}({\bf p})$ at the
entire MBZ boundary for the square symmetry
case can be obtained by using
Eqn. (16) and the
point symmetries of the square lattice.
For a finite as opposed to infinitesimal field,
these results were established in Section II
and in the first two examples in Section IV.
On a more technical level, the operator ${\bf\Lambda}$
is equivalent to ${\bf U_{n}}(\pi)$ and thus inverts the
sign of the transverse component of the field. Hence,
in a field with non-zero transverse component
${\bf H}_{\perp}$, ${\bf\Lambda}$ ceases to be a symmetry
of the Hamiltonian, in agreement with the first line of
Table 1 – and thus can no longer
represent the momentum boost by ${\bf Q}$.
{details}
In spite of this, the authors of Ref. braluk
obtained the correct relation
$g_{\perp}({\bf p+Q})=-g_{\perp}({\bf p})$
(Eqn. (16) of the present work):
while alluding to the non-existent symmetry operator
${\bf\Lambda}$, in fact they used essentially the
Eqn. (15), even if without derivation.
Th beautiful result is indeed correct for a lattice
of square symmetry, but only due to the conspiracy
between Eqn. (16) and the square
symmetry of the lattice.
As we saw above, this beautiful result
is indeed correct for a lattice of square
symmetry. Unfortunately, several circumstances
prevent one from embracing this argument.
Most importantly, it hinges solely on the
symmetry under translation by ${\bf Q}$,
put otherwise – on commensurability of
magnetic order with the crystal lattice.
If correct, this argument would imply that, in
an arbitrary commensurate Néel antiferromagnet,
Kramers degeneracy takes place at the entire
MBZ boundary regardless of the underlying
crystal symmetry. The toy example
(18) with $\eta\neq 1$ shows,
that this is not at all necessarily the case:
the condition
$g_{\perp}({\bf p+Q})=-g_{\perp}({\bf p})$
does not, by itself, restrict the
manifold $g_{\perp}({\bf p})=0$ to the MBZ
boundary (see the subsection on spectral
symmetries in momentum space).
Technically, the error may be
traced back to the following.
Firstly, the way it is written in braluk ,
operator ${\bf\Lambda}$ does not represent
translation by wave vector ${\bf Q}$.
Rather, it is
$${\bf\Sigma}=\left[\begin{array}[]{cc}0&({\bf n}\cdot\bm{\sigma})\\
&\\
({\bf n}\cdot\bm{\sigma})&0\end{array}\right]$$
(31)
that, in a purely longitudinal magnetic field,
exchanges the diagonal matrix elements of Hamiltonian
(8), and thus is
equivalent to translation by ${\bf Q}$ in reciprocal space.
Secondly, ${\bf\Sigma}$ changes the sign of
the transverse component of the field, since
${\bf(n\cdot\bm{\sigma})(H\cdot\bm{\sigma})(n\cdot\bm{\sigma})=(H_{\|}\cdot\bm{%
\sigma})-(H_{\perp}\cdot\bm{\sigma})}$.
Hence, in a field with non-zero transverse component
${\bf H}_{\perp}$, ${\bf\Sigma}$ ceases to be a symmetry of
the Hamiltonian, in agreement with the first line of Table
1 – and thus no longer represents
translation by ${\bf Q}$.
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Formation of Low-Mass X-Ray Binaries. II.
Common Envelope Evolution of Primordial Binaries with Extreme Mass
Ratios
Vassiliki Kalogera and Ronald F. Webbink
Astronomy Department, University of Illinois at Urbana-Champaign,
1002 West Green St., Urbana, IL 61801.
e-mail: vicky,
webbink@astro.uiuc.edu
Abstract
We study the formation of low-mass X-ray binaries (LMXBs) through helium
star supernovae in binary systems that have each emerged from a
common-envelope phase. LMXB progenitors must satisfy a large number of
evolutionary and structural constraints, including : survival through
common-envelope evolution, through the post-common-envelope phase, where
the precursor of the neutron star becomes a Wolf-Rayet star, and survival
through the supernova event. Furthermore, the binaries that survive the
explosion must reach interaction within a Hubble time and must satisfy
stability criteria for mass-transfer. These constraints, imposed under
the assumption of a symmetric supernova explosion, prohibit the formation
of short-period LMXBs transferring mass at sub-Eddington rates through
any channel in which the intermediate progenitor of the neutron
star is not completely degenerate. Barring accretion-induced collapse,
the existence of such systems therefore requires that natal kicks be
imparted to neutron stars.
We use an analytical method to synthesize the distribution of nascent
LMXBs over donor masses and orbital periods, and evaluate their birth rate
and systemic velocity dispersion. Within the limitations imposed by
observational incompleteness and selection effects, and our neglect of
secular evolution in the LMXB state, we compare our results with
observations. However, our principal objective is to evaluate how basic
model parameters (common-envelope ejection efficiency, r.m.s. kick
velocity, primordial mass ratio distribution) influence these results. We
conclude that the characteristics of newborn LMXBs are primarily
determined by age and stability constraints and the efficiency of magnetic
braking, and are largely independent of the primordial binary population
and the evolutionary history of LMXB progenitors (except for extreme
values of the average kick magnitude or of the common-envelope ejection
efficiency). Theoretical estimates of total LMXB birth rates are not
credible, since they strongly depend on the observationally indeterminate
frequency of primordial binaries with extreme mass ratios in long-period
orbits.
keywords: binaries: close – stars: evolution – X-Rays: stars
Accepted for publication in The Astrophysical Journal
1 INTRODUCTION
The existence of Low-Mass X-ray Binaries (LMXBs) poses critical questions
to the theories for the evolution of close binaries. They are believed to
be accreting neutron stars or possibly black holes with low-mass
companions (for recent reviews see Bhattacharya & van den
Heuvel 1991; Verbunt 1993). The major problem
concerning their origin is that their orbits are now so small that they
could not accommodate the advanced evolution of the progenitor of the
compact object. A similar question was originally posed for cataclysmic
binaries and a solution was suggested by
Paczy$\acute{\mbox{n}}$ski (1976) : a common envelope is
formed around the binary and the spiral-in of the secondary into the
primary causes the envelope to be ejected and the orbit to contract
substantially, while exposing the degenerate core of the primary as a
newly-formed white dwarf. A common envelope phase is adequate to solve
the puzzle of LMXBs, as well, since not only can it account for the
shrinkage of the orbit, but it also reduces the primary mass, so that the
disruptive effect of mass loss at supernova is weakened, increasing the
chance for survival of LMXB progenitors.
Several scenarios have been proposed for the formation of LMXBs in the
galactic disk and three out of four invoke a common-envelope phase. One
involves accretion-induced collapse (AIC) of an accreting white dwarf. The
process was first discussed by Whelan & Iben (1973),
although in a context other than LMXB formation. A second scenario
proposes that a massive helium core, exposed in a small orbit by spiral-in
evolution, collapses to form a neutron star or a black hole (van den
Heuvel 1983). A variant of this evolutionary path,
involving extensive wind mass loss in place of common-envelope evolution,
has been suggested (Romani 1992) as an avenue for producing
black-hole LMXBs. More recently, triple-star evolution has been put
forward for LMXB formation with either a black hole or a neutron star, and
involves the formation of a Thorne-$\dot{\mbox{Z}}$ytkow star by merger of
a massive X-ray binary, and engulfment of the third component in a
common-envelope phase (Eggleton & Verbunt 1986). A fourth
scenario has been proposed, the direct-supernova mechanism
(Kalogera 1997), which obviates the need for a common
envelope phase and relies solely on natal kicks imparted to neutron stars
to keep the systems bound and also decrease the orbital separation.
All of these scenarios present plausible formation channels for LMXBs.
However, quantitative analysis of these evolutionary channels has been
hampered by our limited understanding of the details of the various
physical processes involved (e.g., spiral-in process, Wolf-Rayet mass
loss, asymmetric supernova explosion). It is possible to tailor an
evolutionary model to reproduce the properties of an isolated LMXB, but
this exercise provides little perspective on whether the putative initial
conditions and subsequent tailoring are plausible. A more useful approach
is to model the evolution of an entire ensemble of primordial binaries
under a common set of assumptions, and analyze the statistical properties
of the LMXB population. Such an approach has been taken in the past for
the study of other binary populations (e.g., Lipunov &
Postnov 1988; de Kool 1992; Kolb
1993; Tutukov & Yungel’son 1993; Politano
1996), and more recently for LMXBs (Romani 1992; Iben, Tutukov, &
Yungel’son 1995; Terman, Taam, & Savage
1996).
Our purpose here is to model the evolution of a primordial binary
population through a sequence of stages involving, among others, a
common-envelope phase and the supernova explosion of a helium star, and
leading to the formation of LMXBs. Although a direct result of our
calculations if the birth frequency of LMXBs, we focus more on identifying
the properties of LMXB progenitors and on investigating the dependence of
the final population characteristics on the uncertain model parameters. We
also examine the possibility of comparing our results to observations and
constraining the observationally undetermined properties of primordial
binaries feeding LMXB formation. Although we study one evolutionary
channel here, our techniques can be straightforwardly applied to other
channels, and some of our conclusions hold for all the LMXB formation
paths that invoke a common-envelope phase.
In § 2 the evolutionary scenario is described in some detail. The
relevant constraints which binaries must satisfy at various instances
throughout their evolution and the resulting limits on the LMXB-progenitor
parameter space are identified in § 3. We find that asymmetric
supernova explosions are needed to explain LMXB formation via the He-star
SN mechanism, and describe the method to incorporate their effect in a
synthesis calculation in § 4. We discuss our assumptions for the parent
population and the synthesis method in § 5. The results of the
population synthesis calculations in comparison to observations as well as
their dependence on the input parameters are discussed in § 6. Our
conclusions are stated in § 7. Finally, the set of analytic
approximations employed in our model is given in an Appendix.
2 DESCRIPTION OF THE EVOLUTIONARY CHANNEL
Low-Mass X-ray Binaries have donors of mass $\lesssim 1$ M${}_{\odot}$. As
elaborated below, these donors were probably always of low mass. The
primary of a LMXB-progenitor, however, must be massive enough to produce a
neutron star. Its helium core, exposed at the end of the common-envelope
phase, must therefore have been massive enough to reach core collapse. For
these reasons, we need to consider a primordial binary system with an
extreme mass ratio. The more massive star evolves much faster than its
companion and is the first to fill its Roche lobe. The fact that
initially the system had an extreme mass ratio affects its evolution in
two ways: (a) The time scale for nuclear evolution of the primary is so
much smaller than that of its companion that, when mass transfer begins,
the secondary is practically still on the Zero-Age Main Sequence (ZAMS);
and (b) as its mass increases the secondary relaxes toward thermal
equilibrium on its own thermal time scale, which is long compared to the
mass transfer time scale, dictated by the thermal or dynamical time scale
of the massive donor. Consequently, the transferred material cannot cool
as it is accreted and the secondary swells up and fills its Roche lobe.
In this way, a common envelope (CE) is created that engulfs the binary.
Even before the formation of the common envelope, when the massive primary
approaches its Roche lobe radius, spiral-in of the secondary is initiated,
as the primary’s angular momentum at synchronism exceeds one third of that
of the orbit and the Darwin tidal instability sets in (Darwin
1879). With the formation of the common envelope the secondary further
spirals toward the core of the primary due to frictional dissipation of
the orbital energy. The details of the physical processes involved are
not well understood, but it is generally accepted that, as energy is
dissipated in the common envelope, the envelope expands and is eventually
expelled. The orbital energy is assumed to be deposited in the envelope
with an efficiency $\alpha_{CE}$ (common-envelope efficiency). If the
orbital energy is sufficient, the binary system emerges with the secondary
and the core of the primary orbiting each other. The post-common-envelope
orbit is considerably smaller than the initial one due to the typically
large ratio of the envelope mass to secondary mass (eq. [A8]).
Numerical calculations of the common-envelope phase (for a review, see
Iben & Livio 1993) show that its duration is orders of
magnitude smaller than the nuclear time scales of both the donor and the
accretor. Furthermore, Hjellming & Taam (1991) showed that
the secondary remains practically unaffected at the end of the process and
the increase (or decrease) of its mass is insignificant ($\lesssim 1$%).
Accordingly, we may assume that at the end of the CE phase the secondary
preserves its mass and is still on the ZAMS. In addition, model
calculations show that, as a rule, mass transfer once started will
continue until the donor star is stripped down to a composition boundary
(Paczy$\acute{\mbox{n}}$ski 1971). We may therefore assume
that the mass of the post-CE primary is equal to the mass within its
nuclear-burning core (or within the outermost nuclear-burning shell) at
the moment it filled its Roche lobe. In this evolutionary scenario, the
binary emerging from the common envelope evolves “quietly” as a detached
system until the remnant core explodes as a supernova.
It should be noted that the binary evolution both before and after the
CE-phase is not conservative. The primaries of interest are so massive
that wind mass loss is expected to take place before the primary fills its
Roche lobe. This mass loss affects the structure and evolution of the
primary as well as the orbital characteristics of the system. Moreover,
the core of the primary emerging from the CE is still massive enough to
suffer substantial wind mass loss in a way analogous to that of a
Wolf-Rayet star. Once again the evolution of both the star and the orbit
is affected.
The supernova explosion is a crucial event in the evolution of the
LMXB-progenitors. Most systems are disrupted, but some fraction of them
must survive if they are to evolve further to become LMXBs. We will show
later that both the survival fraction and the characteristic properties of
the newly formed systems depend strongly on the existence and mean
magnitude of a kick velocity imparted to the newborn neutron star.
The systems that survive the supernova event can come into contact via two
physical processes: nuclear evolution of the secondary, and shrinkage of
the orbit (and hence of the Roche lobe) due to angular momentum losses.
Depending on the nature of the secondary, the physical mechanism
responsible for angular momentum losses may be gravitational radiation
and/or a magnetic stellar wind. Either way, the system comes into contact
and mass starts flowing from the secondary towards the neutron star. At
the time of contact we call the system a Zero-Age Low-Mass X-ray Binary
(ZALMXB).
3 CONSTRAINTS AND LIMITS ON THE LMXB-PROGENITORS
3.1 Structural and Evolutionary Constraints
Only a very small fraction of all binary systems follow the evolutionary
channel described above. By demanding that a system survive all
evolutionary stages in this specific sequence, we are able to constrain
the characteristics and physical parameters of the initial binaries, the
LMXB-progenitors.
A number of constraints are imposed by this scenario (Webbink &
Kalogera 1994) :
1.
The primary must fill its Roche lobe before it explodes as a
supernova. The orbit of the progenitor cannot be arbitrarily large, since
the system must reach interaction, and enter common-envelope evolution
before the primary becomes a neutron star.
2.
The system must
remain detached following the CE phase until the primary becomes a neutron
star. This is a two-fold constraint: a) The orbit at the end of the
CE-phase must be wide enough to accommodate the low-mass companion; b) it
must also be wide enough not to abort evolution of the remnant core prior
to its supernova explosion. The post-CE primary is a helium star (He-star)
losing mass in a copious Wolf-Rayet (WR) wind. Woosley, Langer, &
Weaver (1995) have evolved mass-losing He-stars with masses
from $4$ M${}_{\odot}$ to $20$ M${}_{\odot}$, and found that they produce iron
cores barely massive enough to collapse to a neutron star. We expect that
an episode of mass transfer occurring early or midway in the evolution of
the He-star will arrest the growth of the iron core, (by completely
stripping away the helium envelope feeding it), thus preventing the
formation of a neutron star.
3.
The system must remain bound after the supernova event. Under
the assumption of a symmetric supernova, there is an absolute limit on the
amount of mass lost in the event, for the binary to survive
(Boersma 1961). If we take into account a kick velocity
imparted to the newborn neutron star due to an asymmetric core collapse,
then survival depends on the magnitude and the direction of the kick.
4.
The mass transfer phase following the formation of the neutron
star must be appreciably long-lived. In order for the system to become a
LMXB with an appreciable lifetime, the companion to the neutron star must
remain in equilibrium and the mass transfer rate must not exceed the
Eddington limit ($\dot{M}_{Edd}\sim 10^{-8}$ M${}_{\odot}$ yr${}^{-1}$).
However, we will entertain the possibility that a system initially
transferring mass at super-Eddington rates may find the mass transfer rate
subsiding below that limit if the companion remains in thermal and
hydrostatic equilibrium.
5.
The post-SN system must reach interaction in a Hubble time. In
order for a system to be included in the LMXB population, it must become a
luminous X-ray source within a Hubble time. This means that the post-SN
orbit must be small enough so that the secondary will fill its Roche lobe
in $\sim 10^{10}$ yr, either due to its own evolution or due to the
shrinkage of the orbit caused by angular momentum losses.
3.2 Limits on the Parameter Space of LMXB-Progenitors
A binary system is characterized primarily by three parameters: the masses
of the two stars, $M_{1}$ and $M_{2}$ and their orbital separation $A$.
Eccentricity is another characteristic, but we will neglect it here,
assuming that tidal dissipation is efficient enough to destroy any initial
eccentricity prior to actual mass transfer. For a scale-less distribution
in orbital separation, as we will assume (§ 5.1), the distribution of
separations of circularized orbits will be identical to that of the
initial (eccentric) orbits, so long as the distribution of eccentricities
does not itself vary significantly with separation over the range of
interest. We can therefore assume equivalently that all the progenitors
are formed with circular orbits. The constraints described qualitatively
above substantially limit the range of values that $M_{1}$, $M_{2}$ and $A$
can cover and yet produce LMXBs. In the calculation of these limits we
use a number of approximate relations described in detail in the Appendix.
For specified masses111In this paper, radii and orbital separations
are expressed in terms of R${}_{\odot}$, masses in M${}_{\odot}$, orbital periods
in days, and time in years. of the primary and its companion, the first
of the constraints listed above sets an upper limit on the orbital
separations of the progenitors. This limit corresponds to the primaries
that first fill their Roche lobe just before core collapse. If we choose a
value for $\alpha_{CE}$, we can find the corresponding upper limit on the
post-CE orbital separations.
The second of the constraints sets two lower limits on the orbital
separations of the post-CE systems. One corresponds to the secondary just
filling its Roche lobe at the end of the CE phase and the other to the
He-star primary filling its Roche lobe just prior to core collapse. During
their evolution, He-stars lose mass in a strong WR wind and experience a
rapid growth in radius, which is more severe as the stellar mass decreases
(see Habets 1985; Woosley, Langer, & Weaver 1995). The
radii just prior to core collapse are considerably larger than those of
the low-mass companions at ZAMS, so that the second of the constraints
obviates the first one. The expansion of the secondary due to its own
nuclear evolution prior to the supernova is invariably negligible, since
the lifetime of the post-CE neutron star progenitor varies from $10^{5}$ to
$10^{6}$ yr (depending on its composition at the end of the CE phase),
which is orders of magnitude smaller than the evolutionary time scale of
the low-mass companion.
The evolutionary sequences of mass losing stars ($M<40$ M${}_{\odot}$)
presented by Schaller et al. (1992) show that massive stars
suffer most of their mass loss only during the nuclear-burning phases of
the core (H and He), when there is little or no radius expansion. In
contrast, rapid growth in radius occurs between core hydrogen exhaustion
and core helium ignition and again after helium exhaustion. During these
phases of rapid expansion, the stellar mass is nearly constant (Figure 1).
If mass is lost to infinity from one or both components of a binary, and
carries with it a specific angular momentum equal to the orbital angular
momentum per unit mass of its source component(s), then the binary
separation varies as the inverse of the total mass of the binary (Jeans
mode of mass loss). During core He-burning slow expansion but extensive
mass loss characterizes massive stars and we find that the rate of
Roche-lobe expansion due to systemic mass loss invariably exceeds the
evolutionary rate of stellar expansion. Therefore, the primary can only
fill its Roche lobe either (i) before central He-ignition or (ii) after
central He-exhaustion. In the first case, the post-CE primary will be a
helium star with a lifetime of $\sim 10^{6}$ yr (Habets 1985) losing mass
in a Wolf-Rayet wind. These stars apparently lose most of their mass
during this phase, leading to some orbital expansion, but they also
develop denser cores and much more extended envelopes at lower masses than
would otherwise be the case. The net effect is to demand a much larger
post-CE binary separation to accommodate the evolutionary expansion of the
core He-burning primary than would be the case if it evolved at constant
mass. In the second case, where the common envelope is formed after
central He exhaustion in the massive progenitor, the post-CE primary is
again a helium star but has a C-O core. It is also more massive (by about
1.1 M${}_{\odot}$) than the helium star in case (i) because
of core growth during the hydrogen-shell burning phase experienced by the
primary before CE formation. Furthermore, since helium has already been
exhausted in the center, the helium-star has also a shorter lifetime
($\sim 10^{5}$ yr) (Habets 1985), and therefore suffers minor further mass
loss, which can be ignored (Woosley, Langer, & Weaver 1995). Therefore
they remain massive enough so that the growth in radius is mild and hence
the limit on the orbital separation is lower. The relation between the limits is
depicted in Figure 2, from which it becomes evident that LMXB-progenitors
survive post-CE evolution up to the point of SN explosion only in
the case that the common envelope is formed after central
He-exhaustion, at which point the initial primary has already lost a
significant amount of its envelope due to its own wind.
In the event of a symmetric core collapse and a circular pre-SN orbit, the
system will remain bound (constraint 3) only if less than half of its
initial total mass is lost in the explosion. The assumption of a circular
orbit before the explosion is well justified, since the system has emerged
out of a common envelope, a highly dissipative process. Given a symmetric
collapse (in the frame of the primary), the binary will remain bound only
if:
$$\displaystyle(M_{He}-M_{NS})$$
$$\displaystyle<$$
$$\displaystyle\frac{M_{He}+M_{2}}{2}~{}~{}~{}~{}\mbox{or}$$
$$\displaystyle M_{He}$$
$$\displaystyle<$$
$$\displaystyle M_{2}+2M_{NS}$$
(1)
where $M_{He}$, $M_{2}$ and $M_{NS}$ are the (gravitational) masses of the
neutron star progenitor, the secondary and the neutron star respectively.
The limits imposed on masses and radii of LMXB-donors by the final two
constraints listed above have already been studied in detail by Kalogera
& Webbink (1996), hereafter Paper I. Here, we summarize
their results:
In the case of conservative mass transfer, main-sequence donors less
massive than $\sim 1.5$ M${}_{\odot}$ are stable against thermal time scale
mass transfer, while those crossing the Hertzsprung gap are stable if
their masses do not exceed $\sim 1.3$ M${}_{\odot}$. Donors that have evolved
beyond the base of the giant branch are stable against mass transfer on a
dynamical time scale and drive sub-Eddington mass transfer only if their
masses are smaller than $\sim 1$ M${}_{\odot}$. However, the population of
these donors is diminished by the constraint that their age must not
exceed the galactic disk age, $T$. For $T=10^{10}$ years the parameter
space ($\log M_{2}$ - $\log R_{2}$) occupied by donors first filling their
Roche lobes beyond the base of the giant branch and transferring mass at
sub-Eddington rates is extremely small (see Figure 9a in Paper I), and
vanishes altogether if angular momentum losses due to magnetic stellar
winds are significant 222Magnetic stellar wind losses were
inadvertently neglected in our estimates of initial mass transfer rates in
Paper I. Only for giant branch donors is the division between sub- and
super-Eddington systems measurably affected; none of the stability limits
is affected.. If super-Eddington mass transfer rates are allowed, but
still with the constraint that donors remain in dynamical and thermal
equilibrium, the limits on donor masses are extended to $\sim 2$ M${}_{\odot}$ on the main sequence, and to $\sim 1.5$ M${}_{\odot}$ on the
giant branch. However, it is not clear whether these systems will actually
emerge as X-ray sources. Finally, there are two additional groups of
systems, with donors first filling their Roche lobes while on the main
sequence or while crossing the Hertzsprung gap, that experience thermal
time scale mass transfer but eventually recover equilibrium and enter a
long-lived mass transfer phase. Those with donors filling their lobes in
the Hertzsprung gap all subside to sub-Eddington rates and emerge as
systems with giant branch donors. However, only a portion of those with
the main-sequence donors will drive mass transfer at rates below the
Eddington limit after recovering thermal equilibrium (see Figure 6 in
Paper I).
All relevant limits imposed on the post-CE orbital characteristics are
illustrated in Figure 3 for $M_{2}=1.0$ M${}_{\odot}$ and $\alpha_{CE}=1$
under the assumption of a symmetric supernova. Indeed, if we adhere to the
requirement that mass transfer be sub-Eddington, we find no combination of
limits that leaves viable sub-Eddington LMXB progenitors. We conclude that
binaries could not form short-period LMXBs via this evolutionary channel
if supernovae were symmetric, regardless of the rest of their
characteristics, because the only systems which can survive mass loss in
the supernova event are so wide (in order to accommodate the evolution of
the core) that they will subsequently reach mass transfer only as the
secondary ascends the giant branch. This process will take more than
$10^{10}$ yr (if $M_{2}\lesssim 1$ M${}_{\odot}$), or will result in
super-Eddington mass transfer rates (if $1$ M${}_{\odot}\lesssim M_{2}\lesssim 1.5$ M${}_{\odot}$), or will lead to dynamical instability (if $M_{2}\gtrsim 1.5$ M${}_{\odot}$). The existence of short-period LMXBs therefore
demand that one or more of the constraints be relaxed.
4 ASYMMETRIC SUPERNOVA EXPLOSIONS
Studies of the pulsar population (e.g., Harrison, Lyne &
Anderson 1993) show that it is characterized by a large
scale height and high space velocities, providing observational evidence
that, at their birth, pulsars are given a kick velocity, due to an
asymmetry associated with the supernova explosion. The magnitude of the
kick is large enough to influence the kinematics of the pulsar population
and certainly the orbital dynamics of a binary system hosting a neutron
star progenitor. The constraints discussed in the previous section imply
that, unless a kick velocity is imparted to the newborn compact star, it
is essentially impossible to form short-period LMXBs via the evolutionary
path considered here. Models attempting to explain the pulsar velocity
distribution and the putative velocity-magnetic moment correlation (Dewey
& Cordes 1987; Bailes 1989) require kick
velocities with mean magnitudes of $\sim 100-200$ km s${}^{-1}$. However,
a more recent study (Lyne & Lorimer 1994) of the pulsar
population takes into account a selection effect against high velocity
pulsars, and concludes that the mean pulsar velocity is $\sim 450$ km s${}^{-1}$. Additional evidence from supernova remnants and
associated pulsar positions (Caraveo 1993; Frail, Goss, &
Whiteoak 1994) supports the conclusion of high kick
velocities. Although pulsar velocities do not directly reflect the birth
velocities, these recent estimates do point towards high kick magnitudes.
Any correlation between kick direction or magnitude and orbital axis or
orbital velocity in a binary is at present purely conjectural, and hence
we will assume that kick velocities are isotropically oriented in the
center of mass frame of the collapsing component with a Maxwellian
distribution in magnitude.
The interplay between the different limits discussed in the previous
section changes dramatically if we relax the assumption of a symmetric
supernova explosion. An asymmetric core collapse, imparting a kick
velocity to the neutron star, breaks the one-to-one link between pre- and
post-SN orbital parameters. Those constraints in Figure 3 which reflect
post-SN conditions no longer sharply delimit possible LMXB progenitors.
Systems which in the case of symmetric supernovae would have certainly
been disrupted may now survive (if by chance the kick velocity has the
right direction and magnitude), and, conversely, systems which would have
survived may now be disrupted. Moreover, post-supernova orbits may now
become smaller than the pre-supernova ones (which can never be the case in
a symmetric core collapse), allowing the formation of short-period LMXBs.
Thus, for the case of an asymmetric collapse the limits imposed on the
progenitors, after the ejection of the common envelope, are only the ones
shown in Figure 4. In that case, a non-vanishing part of the parameter
space may be populated by LMXB progenitors. The post-CE progenitors are
Wolf-Rayet binaries, and for a $1$ M${}_{\odot}$ secondary they have
primaries with masses $\sim 3.5-8$ M${}_{\odot}$, orbital separations $\sim 8-25$ R${}_{\odot}$, and orbital periods $\sim 1-5$ d. The
corresponding limits on the primordial binaries are also shown in Figure
4; these O,B primaries have masses $\sim 13-25$ M${}_{\odot}$, orbital
separations $\sim 800-1800$ R${}_{\odot}$, and orbital periods $\sim 1.5-5$ yr.
The inclusion of a kick velocity imparted to the neutron star forces one
to follow the evolution of an initial population of binaries and not of a
single system. The stochastic element in this problem, of finding the
distribution of binaries after an asymmetric supernova explosion, has been
already addressed by Kalogera (1996). Assuming an
isotropic Maxwellian distribution of kick velocities, she developed an
analytical method of calculating the distribution of post-SN binary
systems over eccentricity, orbital separations (before and after
circularization) and systemic velocities. Here, we are interested only in
the distribution of orbital separations of post-SN circularized orbits.
Following the notation of Kalogera (1996), the distribution of systems
over of the dimensionless separation $\alpha_{c}\equiv A_{c}/A_{i}$, where
$A_{c}$ and $A_{i}$ are the circularized and pre-SN orbital separations,
respectively, is given by:
$${\cal H}(\alpha_{c})~{}=~{}\left(\frac{\beta}{2\xi^{2}}\right)~{}\exp\left(%
\frac{-(\beta\alpha_{c}+1)}{2\xi^{2}}\right)~{}I_{o}\left(\frac{\sqrt{\beta%
\alpha_{c}}}{\xi^{2}}\right)~{}\mbox{erf}\left(z_{o}\sqrt{\frac{\beta}{2\xi^{2%
}}}\right),$$
(2)
where
$$\displaystyle\mbox{erf}(x_{o})$$
$$\displaystyle\equiv$$
$$\displaystyle\frac{2}{\sqrt{\pi}}\int_{0}^{x_{o}}\mbox{e}^{-x^{2}}~{}dx\mbox{,}$$
$$\displaystyle z_{o}$$
$$\displaystyle=$$
$$\displaystyle\sqrt{2-\alpha_{c}-\frac{2c-\alpha_{c}}{c^{2}}}\mbox{,}~{}~{}~{}~%
{}~{}~{}~{}~{}~{}\frac{2c}{1+c}<\alpha_{c}<2c$$
$$\displaystyle=$$
$$\displaystyle\sqrt{2-\alpha_{c}}\mbox{,}\hskip 105.275197pt2c\leq\alpha_{c}<2$$
$$\displaystyle\beta$$
$$\displaystyle=$$
$$\displaystyle\frac{M_{NS}+M_{2}}{M_{c}+M_{2}},$$
$$\displaystyle\xi$$
$$\displaystyle=$$
$$\displaystyle\frac{\sigma}{V_{r}},$$
$I_{o}$ is the zeroth order Bessel function, $\sigma=\langle V_{k}^{2}/3\rangle^{1/2}$, $V_{r}$ is the relative orbital velocity of the two
stars in the pre-SN binary, and $c$ is the ratio of the radius of the
secondary to the pre-SN orbital separation.
Convolving the above distribution with that of the pre-SN binaries over
masses and orbital separations, as defined by the limits already
discussed, enables us to map precisely the distribution of post-SN
binaries and synthesize the population of nascent LMXBs.
5 POPULATION SYNTHESIS
5.1 Parent Binary Evolution
Having described the criteria which select LMXB progenitors from a parent
binary population, we require a statistical description of this primordial
population to produce quantitative results. We therefore assume that the
primordial binaries can be characterized by three parameters : the mass of
the primary $M_{1}$, the mass ratio $q\equiv M_{2}/M_{1}$ ($M_{2}$ being the mass
of the secondary star), and the orbital separation of the system $A$. In
selecting an initial distribution of binaries over these parameters, we
are guided by the results of a detailed analysis by Hogeveen
(1991), but with some important differences at small mass ratios, where
observational constraints are virtually non-existent.
We have adopted the field star Initial Mass Function (IMF) derived by
Scalo (1986) as a good representation of the primary mass
distribution. Based on his results we are able to fit the IMF of stars
more massive than $0.3$ M${}_{\odot}$ with a single power law of the form :
$$\Xi(M)=\Xi_{o}M^{-2.7}~{}~{}\mbox{stars}~{}\mbox{pc}^{-2}\mbox{yr}^{-1}\mbox{M%
}_{\odot}^{-1},~{}~{}~{}\Xi_{o}\simeq 6.83\times 10^{-10}$$
(3)
If we assume that the galactic disk has an exponential surface density
with a scale length of $4$ kpc, and that the distance of the Sun from the
galactic center is $8$ kpc, then we estimate the effective radius of the
galactic disk to be $15$ kpc. The birth rate of primaries per unit
logarithm of mass, integrated over the entire galactic disk, is then :
$$f_{1}(\log M_{1})\simeq 1.112~{}M_{1}^{-1.7}~{}\mbox{yr}^{-1}~{}(\log\mbox{M}_%
{\odot})^{-1}$$
(4)
The distribution of orbital separations is assumed to be inversely
proportional to A (Abt 1983), normalized to a wide range of
initial separations up to $10^{6}$ R${}_{\odot}$. This assumption may appear
inconsistent with more recent results regarding the orbital period
distribution of solar-type binaries (Duquennoy & Mayor
1991). However, we note that the range of orbital separations, hence
orbital periods, of interest to us is extremely narrow, from $\sim 2$ yr
to $\sim 5$ yr, so that our results are not sensitive to the specific
shape of the broader distribution. Furthermore, our choice of the
functional form and normalization is consistent with the one used by
Hogeveen (1991) in his study of the mass ratio distribution, the results
of which we have chosen to adopt.
The mass ratio distribution of unevolved binaries of interest to us is
quite uncertain. It is empirically known only in the limit of
approximately equal component masses and for relatively close binaries.
Results obtained by Hogeveen (1991) show that for $q\gtrsim 0.35$ the mass
ratio distribution at long orbital periods is described by an IMF-like
power law ($\propto q^{-2.7}$). However, we need to extrapolate to very
small values of $q$ ($<0.1$). For this range of values it is often
assumed that the distribution flattens, but this is in truth an ad hoc
assumption, because the contribution of such extreme mass ratio systems to
the observed distribution of spectroscopic or eclipsing binaries at long
periods ($>1$ yr) is negligible. Instead we have chosen to adopt an
IMF-like $q$-distribution, even for very small values of $q$. By making
this assumption, and demanding that the normalization accords with
observation as $q\rightarrow 1$, we must explicitly allow for the
possibility that our primordial systems are not only binary, but multiple.
In doing so, we recognize that the presence of additional stellar
components modifies our pool of progenitor binaries in two ways : (i) an
inner binary may abort evolution of the primary by mass exchange,
thwarting its expansion to a common-envelope stage involving the secondary
component of interest to our scenario; and (ii) triple systems are
dynamically stable only if the period ratio between outer and inner orbits
exceeds some critical value. Regarding the first of these two elements, an
inner binary with a secondary component less massive than the outer
one of interest to us is very unlikely to be of any consequence : the
inner binary will succumb to common-envelope evolution, but it is
incapable of extracting enough energy to eject the envelope before merging
– the outer binaries of interest to us typically only barely manage to
survive. We therefore exclude from our progenitor pool only those
multiples in which the inner binary contains a more massive secondary than
the outer. Similarly, in regard to the second element, dynamical
instability of a triple star typically leads to ejection of the least
massive component (Harrington 1975). We therefore exclude
from our progenitor pool only those multiples in which a third component,
more massive than the secondary of interest to us, lies within a critical
period (or separation) ratio of the secondary orbit. Following Kiseleva,
Eggleton, & Anosova (1994), we adopt a critical period
ratio of $6.3$ (separation ratio $\simeq 3.4$) for the extreme mass ratios
of interest here. All systems containing third components more massive
than our secondary are therefore excluded, from a maximum orbital period
of $6.3$ times that of interest down to a minimum physically allowable
separation, which we take (for simplicity) to be twice the primary radius.
Assuming that binary and multiple stars are chosen from a parent
population according to Poisson statistics (i.e., that they are
independent, uncorrelated events), we modify our simple inverse
distribution in $A$ and power-law distribution in $q$ by a factor
representing the Poisson probability that neither of the above
strictures is violated :
$$g(q,A)~{}=~{}\frac{0.075}{A}~{}0.04q^{-2.7}~{}\exp\left(-\int_{2R(M_{1})}^{A%
\cdot 6.3^{2/3}}\int_{q}^{1}~{}0.075A^{\prime-1}~{}0.04q^{\prime-2.7}~{}dA^{%
\prime}dq^{\prime}\right).$$
(5)
A plot of this assumed distribution over mass ratio, $q$, for specified
primary mass, $M_{1}$, and orbital separation, $A$, is shown in Figure 5. It
bears re-emphasizing that this distribution is unverifiable by current
observation for $q\lesssim 0.35$. The adoption of equation (5) is
motivated by three factors : (1) it is consistent with observed rates of
duplicity and mass ratio, where these are detectable, for binary
separations of interest to us; (2) it is a logical extrapolation of that
observable part of the distribution to the extreme mass ratios of interest
to us, without the invocation of ad hoc breaks or cut-offs; and (3) it
provides a consistent formalism for future modeling of LMXB formation by
triple star evolution.
We can transform equation (5) to a distribution over $\log M_{2}$ and $\log A$, $h_{in}(\log M_{2},\log A)$, using the definition of $q$. The
distribution representing the primordial binary population then becomes :
$$F_{in}(\log M_{1},\log M_{2},\log A)~{}=~{}f_{1}(\log M_{1})\cdot h_{in}(\log M%
_{2},\log A)$$
(6)
The range of values covered by the three parameters is dictated by the
evolutionary selection criteria already discussed.
5.2 Method
Having defined the parent binary population, we are able to follow its
transformation as the systems evolve through the various evolutionary
stages. This is done by identifying the system parameters at the end of
each stage and their dependence on the corresponding parameters at the
beginning of each phase, and by transforming the distribution function
according to these dependences. These transformations are performed
analytically, so that at each stage prior to the explosion the
distribution function of binaries can be expressed explicitly. At the
supernova stage the pre-SN function is convolved with the distribution
over post-SN circularized separations (eq. [2]), and the product is
integrated numerically now over pre-SN helium-star masses and orbital
separations. This method offers major advantages over Monte Carlo
techniques as it is free of any statistical errors and in principle allows
us to have an infinite resolution in the final LMXB parameters. This high
resolution reveals even the most subtle features in the nascent LMXB
distribution and permits us to trace back the origin of these features. In
what follows, we briefly describe the procedure for each evolutionary
stage of interest.
From all the systems represented by $F_{in}$, we are interested only in
those that experience a common-envelope phase. The post-CE systems are
characterized by the secondary mass $M_{2}$ (assumed unchanged by CE
evolution), the orbital separation $A_{post-CE}$, and the mass of the
remnant core $M_{He}$, which depends only on the primary mass. Using the
relations connecting the pre- and post-CE binary parameters we can find
analytically the transformed post-CE distribution function :
$$F_{CE}(\log M_{He},\log M_{2},\log A_{post-CE})=F_{in}~{}\cdot~{}J\left(\frac{%
\log M_{1},\log M_{2},\log A}{\log M_{He},\log M_{2},\log A_{post-CE}}\right).$$
(7)
Since $\partial\log A/\partial\log A_{post-CE}=1$ (eq. [A8]), $M_{2}$ is
unchanged, and $M_{He}$ is a function only of $M_{1}$ (eq. [A3]), the
distribution of post-CE orbital separations and secondary masses for a specific choice
of $M_{He}$ is simply a homologous transformation of their pre-CE
distribution at the corresponding value of $M_{1}$.
The post-CE primary, $M_{He}$, has already exhausted helium in its core,
since the initial primary entered common-envelope evolution after core-He
exhaustion. The time scale for nuclear evolution of the C-O core until
collapse is $\sim 10^{5}$ yr (Habets 1985), and is so short that the helium
star is essentially unaffected by wind mass loss (Woosley, Langer, &
Weaver 1995). Therefore the pre-SN distribution of binaries is identical
with the one just after the CE phase. The secondary is still on the main
sequence when the supernova occurs.
By convolving the pre-SN distribution with the survival probability
distribution for the supernova explosion, ${\cal H}(\alpha_{c})$ (eq. [2]),
we can obtain the distribution function, $Z(\log M_{2},\log A_{post-SN})$, of post-SN
circularized orbital separations $A_{post-SN}$ and secondary masses $M_{2}$ by integrating
over $M_{He}$ and $A_{pre-SN}$. In performing this transformation, we
assume that all He stars leave a remnant neutron star of the same
gravitational mass (see also Woosley, Langer, & Weaver 1995) of
$1.4$ M${}_{\odot}$. The post-SN distribution thus becomes a two-variable
function of $M_{2}$ and $A_{post-SN}$:
$$Z(\log M_{2},\log A_{post-SN})=\int_{\log M_{He}^{min}}^{\log M_{He}^{max}}~{}%
\int_{\log A_{pre-SN}^{min}}^{\log A_{pre-SN}^{max}}\zeta~{}d\log A_{pre-SN}~{%
}d\log M_{He},$$
(8)
where
$$\zeta\equiv F_{CE}\cdot{\cal H}(\alpha_{c})\cdot\alpha_{c}\ln 10~{},$$
and $\alpha_{c}\ln 10$ is the Jacobian corresponding to the variable
transformation from
$\alpha_{c}=A_{post-SN}/A_{pre-SN}$ to $\log A_{post-SN}$.
The limits of the integration over $\log A_{pre-SN}$ depend on both $M_{He}$
and $M_{2}$; those for
the integration over $M_{He}$ depend on $M_{2}$, according to the constraints
discussed in § 3.
We have assumed here that both synchronization and circularization of
the binary occurs relatively soon and certainly prior to the time the
secondary overflows its Roche lobe. The assumption is well justified
since the time scales for both processes for detached systems are
significantly shorter than the evolutionary time scale of the secondary
as well as the time scale for angular momentum losses due to magnetic
braking. As the binary approaches Roche lobe overflow the time scales
rapidly decrease down to tens to thousands of years (e.g., for
$R_{L}/R_{2}\simeq 2$; see Zahn 1977, 1989).
Systems surviving the supernova event do not all form LMXBs. Binaries
must still evolve further towards Roche lobe overflow of the secondary for
mass transfer to be initiated. At this stage binary evolution is driven by
nuclear evolution of the secondary and loss of angular momentum , and hence shrinkage
of the Roche lobe around the secondary. We consider two mechanisms
responsible for the loss of angular momentum: gravitational radiation (eq. [A11]) and
magnetic braking (eq. [A13]). In the latter process, a wind from the
secondary, locked onto the stellar magnetic field, drives angular momentum away from
the star. Assuming that the companion is maintained in synchronization
with the orbit by tidal dissipation, it follows that the binary loses angular momentum(Verbunt & Zwaan 1981). This angular momentum loss affects the
orbital
characteristics considerably, whereas the mass loss rate is assumed
negligible. For very low-mass secondaries ($M_{2}\leq 0.37$ M${}_{\odot}$)
that are fully convective, we assume that magnetic braking is negligible,
in accordance with arguments advanced to explain the $2^{\mbox{h}}-3^{\mbox{h}}$ gap in the orbital period distribution of cataclysmic
variables (Rappaport, Verbunt & Joss 1983). For these
masses, angular momentum loss due to gravitational radiation alone is considered.
It should be noted that studies of the magnetic braking mechanism rely
upon measurements of rotational velocities of solar-type stars (Verbunt &
Zwaan 1981). More massive stars develop radiative envelopes which are
expected to diminish the dynamo generation of magnetic fields and hence
the effect of magnetic braking. In accordance to this, massive stars
appear to rotate much faster than low-mass stars.
We have adopted the functional form used by Rappaport et al. (1983)
(with their index $\gamma=2$), but modifying the braking efficiency for
stars more massive than the Sun by introducing a cutoff factor, $b$,
dependent only on stellar mass.
Using observed mean rotational
velocities for main sequence stars, we were able to estimate the efficiency
factor,
$b(M_{2})$:
$$\displaystyle b(M_{2})$$
$$\displaystyle=$$
$$\displaystyle 0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}%
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}%
~{}~{}~{}~{}~{}~{}~{}~{}M_{2}\leq 0.37\mbox{\,M\,}_{\odot},$$
(9)
$$\displaystyle=$$
$$\displaystyle 1~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}%
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}0.37\mbox{\,M\,}_{%
\odot}<M_{2}\leq 1.03\mbox{\,M\,}_{\odot},$$
$$\displaystyle=$$
$$\displaystyle\mbox{exp}~{}\left[-4.15~{}(M_{2}-1.03)\right]~{}~{}~{}~{}~{}~{}~%
{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}M_{2}>1.03\mbox{\,M\,}_{%
\odot}.$$
This expression for the magnetic braking efficiency reproduces the
rotation velocities of main sequence stars of spectral types F5 and F0
(Allen 1973) assuming that they are born at rotational
break-up and neglecting evolutionary changes in mass and radius. Main
sequence stars of earlier spectral type show no evidence of magnetic
braking. Using more recent data (e.g., Fukuda 1982;
Kawaler 1987) leads to somewhat different expressions for
$b(M_{2})$, but has no qualitative effect on our results.
Because of the assumption of initial maximum rotation the above
estimate is actually an upper limit to the magnetic braking efficiency
factor.
The last step in evolving the distribution function $Z$ is to transform the post-SN systems
to nascent LMXBs. We set the radius of the secondary (eq. [A9] in Paper
I) equal to its Roche lobe radius (eq. [A7]) and eliminate the time by
using either equation (A12) or equation (A14). The resulting equation can
be solved numerically for the orbital separation, $A_{X}$, at the onset of
the mass transfer phase. In this way we are able to find the distribution
over orbital separation, $A_{X}$, and donor mass, $M_{2}$, of the LMXB
progenitors:
$$\Phi_{A}(\log M_{2},\log A_{X})=Z~{}\cdot~{}\left|\frac{\partial\log A_{post-%
SN}}{\partial\log A_{X}}\right|$$
(10)
The derivative in the above equation is calculated analytically. With one
last transformation we obtain the distribution over donor mass and orbital
period, $\Phi_{P}(\log M_{2},\log P_{X})$.
6 RESULTS
6.1 A Reference Model
Results from our population synthesis calculations are illustrated in
Figures 6a and 6b for a prototypical choice of input parameters, which
we shall deem our reference case. The two frames of this figure show
zero-age LMXB distributions, $\Phi(\log M_{2},\log P_{X})$, for systems
initiating sub-Eddington mass transfer only (Figure 6a), and for both
sub-Eddington and super-Eddington systems (Figure 6b). The constraints
delineating these regions were discussed in Paper I, and are illustrated
again here in Figure 7, where the regions are labeled $S_{E}$ and $S^{E}$,
respectively. Our choices of values for free parameters in this
reference case have been made in such a way as (i) to define a plausible
extreme, or (ii) to characterize the model distribution at the threshold
value of a specific parameter, that is, at a value where its influence
on the resulting models changes character. Thus, for example, our
choice of mass ratio distribution (eq. [5]) defines a plausible upper
limit to the frequency of the massive binaries with extreme mass ratios
which feed our evolutionary channel, since the Poisson cutoff invoked
in equation (5) (an upper limit to the number of close companions a
massive star may accommodate within the limits of dynamical stability)
is taking effect in just the range of companion masses of interest (see
Figure 5). For the common envelope ejection efficiency we choose
$\alpha_{CE}=0.3$, because below this value the survival window (the
region bounded by thick and thin solid lines in Figure 3) disappears
rapidly below the lower limits to post-supernova binary separation
imposed by the need to accommodate both the helium-star core of the
primary (the dotted line in Figure 3) and its companion (the thin
dashed line in Figure 3). Our choice of r.m.s. kick velocity for the
reference case, $\langle V_{k}^{2}\rangle^{1/2}=300$ km s${}^{-1}$, equates
approximately to the maximum pre-SN relative orbital velocities, and
therefore lies very near the peak in their survival probability in the
zero-age LMXB population.
Within the age and stability limits set by Figure 7, the general
features seen in Figures 6a and 6b are the result primarily of a
competition between nuclear evolution of the donor stars and angular
momentum loss from the binary. The prominent ridge extending towards
low companion masses and low orbital periods is due to systems with
essentially zero-age donors, brought to Roche lobe contact due to loss
of angular momentum. This ridge along the ZAMS disappears for donors
more massive than $\sim 1.4$ M${}_{\odot}$, because at these masses angular
momentum losses due to magnetic braking become inefficient (eq. [9]).
For donors more massive than $\sim 1$ M${}_{\odot}$, nuclear evolution
becomes increasingly important, and not all post-SN systems experience
orbital shrinkage. As a result, a minimum appears in the distribution
at orbital periods of about one day. Systems with donors on the giant
branch appear only in the super- Eddington population. They form the
broad peak at long periods between donor masses $\sim 1$ M${}_{\odot}$ and
$\sim 1.5$ M${}_{\odot}$, and have reached contact because of the advanced
nuclear evolution of the donor.
The competition between angular momentum losses and nuclear evolution is also
evident in the distribution over orbital periods, $\Psi_{P}(\log P_{X})$,
obtained by integrating $\Phi_{P}$ over $\log M_{2}$, and plotted in Figure 8.
The first peak at $\sim 0.3^{d}$ arises from the peak in the mass ratio
distribution (cf. Figure 5), whereas the peak at $\sim 0.5^{d}$ is the result
of the flattening of the ZAMS radius-mass relation above $\sim 1.3$ M${}_{\odot}$, which compresses a relatively wide range of donor masses
into a narrow range of periods. The valley at $\sim 1^{d}$ is a result of
magnetic braking evacuating this range. Systems with evolved donors that
transfer mass at super-Eddington rates populate the ”bump” at longer periods.
These systems may not at first appear as luminous X-ray sources, as we
anticipate that their dense super-Eddington outflows will quench X-ray
emission. Nevertheless, as the donor mass decreases, mass transfer may
subside to sub-Eddington rates, and the systems will then appear as LMXBs
with donors on the giant branch.
We note in passing that Figure 8 also bears witness to the power of the
analytical technique used for these synthesis calculations to reveal features
which are very difficult and computationally expensive to identify in Monte
Carlo approaches. A case in point is the inflection point visible at $\sim 0.23^{d}$, below the shortest-period maximum. This feature is in fact an
artifact of the ZAMS radius-mass relation we have adopted in this work
(eq. [A1] in Paper I), which is discontinuous in its first derivative at
$M_{2}\simeq 0.8$ M${}_{\odot}$. With an analytic approach, we have the power in
principle to increase resolution within a limited range of parameter space,
as desired, without being obliged to do so everywhere, and without suffering
the Poisson noise inherent in Monte Carlo calculations.
6.2 Observable Properties of the LMXB Population
Despite three decades’ effort in X-ray astronomy, our knowledge of the
underlying structural properties of LMXBs is still extremely limited and
fragmentary. Orbital periods, for example, are known only for a small
minority of systems, a large fraction of LMXBs lack optical counterparts
(because of low intrinsic optical luminosity and heavy interstellar
extinction), and dynamical mass estimates from spectroscopic orbits are
nearly absent outside that collection of soft X-ray transients which
evidently contain black hole accretors of mass $>3$ M${}_{\odot}$ (and
which cannot originate through the formation channel modeled here).
Nevertheless, there are several bases, summarized here in Table 1, on
which a comparison may be made between global observational properties
and the results of population synthesis models. The origin of the
observational estimates contained in Table 1 is described below;
theoretical estimates are listed separately for those systems which
transfer mass initially at sub-Eddington rates (regions $S_{E}$, which we
expect to remain LMXBs throughout this phase of interaction) and those
initially super-Eddington (regions $S^{E}$, which we expect to contribute
to the observed LMXB population only later during interaction, if at
all). It must be emphasized here that the values of free parameters
defining our reference model, from which results are extracted in Table
1, were chosen to aid in characterizing the dependence of model results
on those parameters; they have not been chosen to optimize agreement
between model and observation. The reader may glean some sense of the
adjustments required from the discussion of parameter dependences which
will follow below.
Some explanations are warranted for the entries in Table 1:
Birth rate. We estimate the birth rate of the observed population from
the catalogs of galactic LMXBs by van Paradijs (1995) and Bradt & McClintock
(1993). Black hole candidates and LMXBs in globular clusters have been
excluded. Distance estimates and mean X-ray luminosities of individual
systems were drawn, where available, from those catalogs. The birth rate in
steady state then follows from summing the observed mean X-ray luminosities,
and dividing by an average initial donor star mass (assumed to be $1.2$ M${}_{\odot}$,
as suggested by the synthesis results), and assuming an X-ray production rate
of $1.86\times 10^{20}$ erg g${}^{-1}$ of accreted matter. The theoretical
birth rates quoted here exclude any contribution from possible LMXB
progenitors which may emerge from thermal time scale mass transfer, regions
MS${}_{2}$ and HG${}_{2}$ in Figure 5 of Paper I; the birth rates for their
immediate progenitors are, respectively, $2\times 10^{-6}$ yr${}^{-1}$ for
region MS${}_{1}$ and $1\times 10^{-6}$ yr${}^{-1}$ for region MS${}_{2}$, in our
reference model.
Total X-ray luminosity. For comparison, we also include in Table 1
estimates of the observed and theoretical total X-ray luminosity for Galactic
disk LMXBs. We derive a statistical (Poisson) uncertainty in the observed
luminosity of $\pm 30\%$, but expect the true uncertainty to be substantially
greater due to systematic errors (from spectral fittings and distance
estimate errors). Since the deduced estimate of the birth rate of observed
LMXBs follows directly from their total X-ray luminosity, this entry does not
in reality provide a new benchmark for comparison, but it does strip away
some of the assumptions applied above to deduce an observed birth rate. We
apply the same assumptions instead to the synthesis models to convert birth
rates to total X-ray luminosity, but now employ the actual donor mass
distribution produced by those models, instead of an average value.
Fraction of short-period systems. Secular evolution among LMXBs
produces a natural bifurcation in their evolution, with short-period systems
($P_{X}\lesssim 20^{\mbox{h}}$) driven to shorter orbital periods by angular
momentum loss, and long-period systems driven to longer periods by nuclear
evolution of the donor star (Taam, Flannery & Faulkner 1980; Pylyser &
Savonije 1989). This behavior provides a basis for comparison between theory
and observation, even though our synthesis models do not address secular
evolution in the LMXB state. Unfortunately, orbital periods are known for
only 30% of galactic LMXBs; of the 24 systems with known periods, 18 fall
into the short-period group. The observational upper limit quoted in Table 1
reflects our expectation that the higher optical/infrared luminosities of
donors in longer-period systems favor detection of their orbital periods, so
that LMXBs with undetected periods are more likely to belong to the
short-period group. It is important to note as well that the theoretical
estimates listed for our reference case are probably lower limits, in that
they reflect relative birth rates of short- and long-period systems, and do
not account for the shorter lifetimes expected among longer orbital period
systems.
Fraction of neutron star accretors. A significant fraction of the
neutron stars in our model populations (at least among those transferring
mass at sub-Eddington rates) may be driven to gravitational collapse during
their X-ray lifetime, and become stellar black holes. An observational lower
limit to the fraction of LMXBs containing neutron stars, quoted in Table 1 is
set by those showing X-ray pulsations or classical X-ray bursts (see van
Paradijs 1995). To obtain a theoretical estimate for this fraction, we adopt
the equation of state (AV14/UVII) developed by Wiringa, Fiks & Fabrocini
(1988), which represents the most complete microscopic calculations available
at present; this equation of state predicts maximum gravitational and
baryonic (non-rotating) neutron star masses of 2.13 M${}_{\odot}$ and
2.64 M${}_{\odot}$, respectively (Cook, Shapiro & Teukolsky 1994). Model
systems with total baryonic mass exceeding 2.64 M${}_{\odot}$ are considered to
contain black hole accretors only once the accretor mass passes that
threshold.
We must emphasize that black hole formation through accretion-induced neutron
star collapse is incapable of explaining the existence of the low-mass
black-hole soft X-ray transients A 0620-00 (V616 Mon), GS 2023+338 (V404
Cyg), GS 1124-684 (GU Mus), GRO J1655-40, GS 2000+25 (QZ Vul), and H 1705-250
(V2107 Oph) (Cowley 1994; Bailyn et al. 1995;
Charles & Casares 1995; Remillard et al. 1996).
In each of these systems, lower limits to the masses of their compact
components, derived dynamically from the reflex orbital motion of their donor
stars, clearly exceed the maximum total mass of any of our modeled systems:
1.4 M${}_{\odot}+1.5$ M${}_{\odot}=2.9$ M${}_{\odot}$. At least one other
evolutionary channel is required (Eggleton & Verbunt 1986; Romani 1992).
Systemic velocities. We have derived an observed velocity dispersion
from the tabulation by Johnston (1992) of heliocentric radial velocities of
15 LMXBs, correcting for solar motion and for differential galactic rotation,
using her distance estimates and the galactic rotation model of
Clemens (1985), and assuming isotropic peculiar velocities with
respect to uniform rotation on cylinders. Neither the rotation model nor the
assumption of isotropic peculiar velocities can be strictly valid, but the
observed velocity dispersion is more seriously suspect because of distance
errors, since differential rotation corrections are large, and because of
small-number statistics. The theoretical velocity dispersions in Table 1
reflect one-dimensional peculiar velocities at birth; virialization within
the galactic potential should reduce them by a factor of $\sqrt{2}$, since
the hiatus between supernova explosion and the onset of mass transfer as an
LMXB significantly exceeds a galactic dynamical time scale for the
overwhelming majority of model systems.
6.3 Parameter Studies
Although one should treat the observed quantities listed in Table 1 with some
caution, for reasons outlined above, it is instructive to explore how the
theoretical quantities listed there respond to variations in the principal
input parameters to our population models: (i) the efficiency of common
envelope ejection, $\alpha_{CE}$; (ii) the r.m.s. kick velocity imparted to
a newborn neutron star, $\langle V_{k}^{2}\rangle^{1/2}$; (iii) the initial mass
ratio distribution, and (iv) the maximum neutron star mass. These
dependencies are summarized semi-quantitatively in Table 2, and discussed
physically below.
Common envelope efficiency. As illustrated in Figure 3,
progenitor systems of given donor star mass populate only a narrow range
of post-common-envelope orbital separations. That range shifts to
smaller separations for smaller companion masses (less orbital energy
available for envelope ejection) or for small ejections efficiencies,
$\alpha_{CE}$ (less efficient use of available orbital energy). Since
the lower limits to binary separations are fixed by Roche lobe
constraints, reductions in $\alpha_{CE}$ therefore result in (i)
progressive loss of the lowest-mass companions from the pool of donor
stars, and (ii) progressive loss of the longest-period component of the
survivor pool. The loss of low-mass donors suppresses the short-period
extreme of the LMXB orbital period distribution. Likewise, since
asymmetric supernovae cannot produce circularized post-supernova
separations exceeding twice the pre-supernova separation (Kalogera
1996), small values of $\alpha_{CE}$ also suppress the long-period
extreme in this distribution (see Figure 10). For $\alpha_{CE}\lesssim 0.3$, the peak of the donor mass distribution no longer survives, and
the birth rate falls precipitously (Figure 9). The slow increase in
systemic velocity dispersion of survivors as $\alpha_{CE}$ decreases
reflects (i) the selection of survivor systems, crudely, according to
whether the supernova kick by chance imparts to the neutron star a space
velocity closely matching the orbital velocity its companion at the
instant of the explosion, and (ii) the closing of the window in
separation spanned by companion stars of different masses.
Average kick velocity. The dynamical consequences of supernova
kicks are described in some detail by Kalogera (1996). Aside from a nearly
uniform suppression of survival probabilities, r.m.s. kick velocities
exceeding the largest pre-supernova relative orbital velocities ($\sim 300$ km s${}^{-1}$) exercise very little influence on either the mass- and
orbital period-distribution of survivors, or on their space velocities,
since survivors then come only from the low-velocity tail of the
Maxwellian kick distribution. However, when kick velocities are small,
they are capable only of binding relatively wide systems, which have
correspondingly small pre-supernova relative orbital velocities, and
consequently acquire only small post- supernova space velocities. (These
wide systems only survive common-envelope evolution if $\alpha_{CE}\gtrsim 0.5$.) Small kick velocities therefore suppress birth rates
(Figure 11), most severely among short-period systems (Figure 12).
Mass ratio distribution. As noted above, the range in primary
masses ($\sim 15-25$ M${}_{\odot}$), secondary masses ($\sim 0.5-1.5$ M${}_{\odot}$) and orbital periods ($\sim 2-5$ yr) from which
progenitor binaries are drawn (see Figure 4) is far beyond exploration by
current observational techniques. We consider that our adopted mass ratio
distribution represents a plausible maximum frequency to such systems,
consistent with constraints of dynamical stability. The birth rates we
derive must therefore be considered upper limits. Alternative choices of
mass ratio distribution produce lower birth rates; to the extent that
they differ greatly in function form within the mass ratio window of
interest ($q\sim 0.04-0.1$), they may also alter the character of the
LMXB distribution with respect to structural parameters. For example,
Figure 13 illustrates the period distribution derived for a mass ratio
distribution which is independent of $q$ (apart from a very weak
dependence introduced by retention of the Poisson cutoff parameter) below
a critical mass ratio, $q_{c}=0.35$. (Such a distribution closely resembles
those used for example by Pols et al. 1991, and Dalton &
Sarazin 1995) In this case, the total birth rate decreases
by a factor of $\simeq 27$, and there is a relative shift among surviving
systems from the period range $0.2-0.5$ days to the range $0.5-1$ days, a
consequence of the flattening in mass ratio distribution in the range of
interest, $q\sim 0.04-1$. For values of $\alpha_{CE}$ close to unity
(not shown), a relative excess of short-period systems appears below
$\sim 0.2$ days, but these systems do not survive common envelope
evolution in our reference case. Unfortunately, these variations tend to
be confined largely to the short-period ($P_{X}<20^{\mbox{h}}$) part of
the orbital period distribution, where they are easily masked by secular
evolution. The number ratio of long-period to short-period systems,
which is the principal factor influencing systemic velocities as well, is
only weakly dependent on the distribution of donor stars in mass (cf. Figure 6), so long as most of those donors are massive enough ($\gtrsim 1.0$ M${}_{\odot}$) to evolve to interaction.
Maximum neutron star mass. Given our observationally-motivated
assumption that neutron stars are born with uniform gravitational masses of
1.4 M${}_{\odot}$, this factor enters only into the estimate of the fraction of LMXB
accretors which may evolve to collapse to a black hole. Estimates of this
fraction for a range of equations of state (Cook et al. 1994), along with
the observational limit (Table 1) demand that the equation of state be
relatively stiff and the maximum baryonic mass for neutron stars exceed $\sim 1.9$ M${}_{\odot}$.
7 CONCLUSIONS
On undertaking this study, we hoped that the population synthesis
calculations described here would identify some feature or features
among observable parameters of LMXBs which might be unique artifacts of
their primordial distribution and of the evolutionary pathways leading
to the LMXB state. The analytic technique we have used to execute our
synthesis calculations offers enormous advantages for this purpose over
Monte Carlo approaches, as it is free of statistical noise, and can in
principle yield arbitrarily high resolution in the distribution of final
parameters (or of intermediate parameters), should it be warranted, at
minimal additional computational cost. Our initial hopes have been
confounded by the realization that supernova kicks must play a pivotal
role in the formation of LMXBs, one which severely limits our ability to
probe their origins on the basis of their observed properties. We see
three important conclusions emerging from this study:
(1) In the absence of supernova kicks, no LMXBs are formed at
short (${\boldmath P_{X}\lesssim 1^{\mbox{d}}}$) orbital periods.
Stellar winds from the helium star component during the
post-common-envelope, pre-supernova phase are capable of removing enough
mass to reduce many pre-supernova systems to less than twice the mass of
the post-supernova remnant (companion plus neutron star), a necessary
condition for the binary condition to remain bound under instantaneous
mass loss. However, short-period systems cannot then accommodate the
much greater pre-supernova expansion of the low-mass helium star.
Unless moderately large natal kicks are imparted to neutron stars (i.e.,
kicks averaging a substantial fraction of the relative orbital velocity
of the binary at the supernova stage), only sufficiently long-period
systems survive, and then only if $\alpha_{CE}$ is large ($\alpha_{CE}>0.6$). These long-period systems all contain giant branch donors, and
transfer mass at super-Eddington rates.
This conclusion in fact applies not only to the evolutionary channel
explored here, but to any putative formation channel in which the
neutron star progenitor has a non-degenerate envelope. Stars with
massive degenerate cores and hydrogen-rich envelopes, either in place of
or in addition to helium envelopes, become red supergiants, and could
leave only extremely long-period neutron star binaries. Only
accretion-induced collapse, in which the neutron star progenitor is
virtually completely degenerate, could allow pre-SN systems close enough
(and with little enough gravitational mass lost in the collapse) to
produce short-period LMXBs in the absence of supernova kicks. However,
whether accretion-induced collapse is a viable neutron star formation
mechanism remains an unresolved issue: We are not aware of any
plausible model which would feed accreted matter through a
hydrogen-burning shell fast enough to stabilize helium burning (and
thereby avoid mass loss during helium runaways) on a massive degenerate
core; on the contrary, evolutionary models of luminous asymptotic giant
branch stars invariably display thermally-pulsing helium shells (Iben &
Renzini 1983).
(2) The characteristics of newborn LMXBs are almost entirely
independent of the history of their progenitors. The ranges in donor
masses and orbital periods allowed to LMXBs are dictated by age and
stability constraints at the onset of the mass transfer phase. The
distribution of systems over these parameters is influenced primarily by
the efficiency of magnetic braking, which separates short- from
long-period LMXBs. To a much smaller extent, it is also affected (i) by
the average magnitude of the supernova kick, the effect being more
evident when this average tends to very small values (i.e.,
disappearance of short-period LMXBs in the absence of kicks); and (ii)
by the common envelope efficiency, values of $\alpha_{CE}<0.1$
precluding LMXB formation altogether. Apart from these extreme
circumstances, supernova kicks obliterate any memory of how binaries
arrived at the supernova stage; the LMXB distribution carries virtually
no information about their evolutionary history. As a result,
alternative formation mechanisms are indistinguishable, except where an
evolutionary channel leads to pre-SN binaries dramatically different
from those relevant to the present study, e.g., the direct-SN mechanism
(Kalogera 1997). Common envelope evolution, which
characterizes all other LMXB formation channels proposed to date,
inevitably leads to similar distributions of short-period pre-SN
binaries, sharing as their most prominent feature a short-period cutoff
dictated by the dimensions of donor and pre-SN components.
(3) Except as upper limits, theoretical estimates of galactic LMXB
birth rates are not credible. These estimates depend one-for-one on the
birth frequency of primordial binaries with suitable initial properties
(in our case, $M_{1}\sim 12-25$ M${}_{\odot}$, $M_{2}\sim 0.5-2$ M${}_{\odot}$, and $P\sim 2-5$ yr ($A\sim 800-1800$ R${}_{\odot}$). While details may vary somewhat, all LMXB formation
channels (including those proceeding through accretion-induced collapse)
appeal to a primordial population of massive stars ($M_{1}\gtrsim 10$ M${}_{\odot}$) with low-mass companions ($M_{2}\lesssim 2$ M${}_{\odot}$) in long-period orbits ($P>1$ yr). The true frequency
of such systems is observationally indeterminate, and constrained in the
number density of low-mass companions a massive star may retain
consistent with dynamical stability. In our case, we have pushed the
binary frequency to this limit, and so treat our birth rate estimates as
upper limits. We have found, moreover, that even variations among
possible mass ratio distributions within the range of interest are
probably obscured in their effect on LMXB properties by secular
evolution in that state.
Our conclusions regarding the role of supernova kicks in LMXB formation
support and extend those reached independently by Terman, Taam & Savage
(1996; hereafter TTS96). In contrast, Iben, Tutukov, & Yungel’son
(1995; hereafter ITY95) found such kicks unnecessary. This difference
appears to have its origin in several factors. One is the definition of
common-envelope efficiency. That which we use is identical with that
employed by TTS96; as previously noted by Han, Podsiadlowski &
Eggleton (1995) and again by TTS96, the expression used by ITY95
understates the binding energy of the envelope by a factor of two or
three, whereas detailed numerical simulations presented by Rasio &
Livio (1996) are consistent with our expression (eq. [A8]). ITY95
thus find wider post-common envelope systems, capable of accommodating
the radial expansion of the helium star progenitors of neutron stars.
Interestingly, in this regard, their models with assumed efficiency
$\alpha_{CE}=0.5$, corresponding roughly to our $\alpha_{CE}=1$,
produce no LMXBs with main sequence donors (see Table 1 in ITY95), in
agreement with our results. A second major difference concerns the
extent and consequences of wind mass loss from helium stars. In
contrast to our models and to TTS96, ITY95 find significant
contributions to the total LMXB birth rate from systems undergoing case
B mass transfer, which leave post-common envelope core helium burning
primaries. We find that the extensive mass loss suffered by helium
stars during core helium burning (eq. [A9]) greatly expands the range
of initial helium star masses and separations for which Roche lobe
overflow will abort evolution prior to core collapse (cf. Figure 2,
eq. [A10]), eliminating such stars as viable LMXB progenitors.
A final word is on order regarding angular momentum loss rates due to
magnetic braking. We have not explored the dependence of our results on
variants of our adopted braking rate. Qualitatively, stronger braking
will enable wider post-supernova systems to form short-period LMXBs. For
example, King & Kolb (1997) were able to produce
short-period LMXBs with donors more massive than $1.3$ M${}_{\odot}$ without
invoking kicks (these are not included by ITY95 or TTS96), because they
assume a magnetic braking law stronger than ours by about an order of
magnitude. However, our interpretation of braking rates among single stars
indicates that magnetic braking is strongly suppressed at masses this
large (cf. eq. [9]).
Acknowledgements.It is a pleasure to thank an anonymous referee for comments that helped us
improve the focus of our paper. We are grateful to D. Psaltis for
numerous valuable discussions and for carefully reading the manuscript. We
also thank I. Iben, A. V. Tutukov, and L. R. Yungel’son for
stimulating and enlightening discussions, and their help in identifying
the root causes for differences in our results; and U. Kolb for help in
resolving the effect of the efficiency of magnetic braking. This work was
supported by National Science Foundation under grant AST92-18074 and the
Graduate College of the University of Illinois under a Dissertation
Completion Fellowship.
Appendix A ANALYTIC APPROXIMATIONS USED IN THE MODEL
Following are the basic analytic relationships employed in our population
synthesis models for the formation of LMXBs. They are grouped in roughly
the sequence in which they enter consideration along the evolutionary path
from primordial binary to ZALMXB. References identify the sources of the
relationships used here, or (for stellar models) the detailed calculations
which we here analytically approximate. The stellar models in each case
assumed solar composition. The units used throughout are: masses
$(M)$ in M${}_{\odot}$; radii $(R)$ and orbital separations $(A)$
in R${}_{\odot}$; orbital periods $(P)$ in days; orbital angular
frequencies $(\omega)$ in $Hz$; and evolutionary times $(t)$ in
years. Natural logarithms are written “ln”, decimal logarithms “log”,
and arguments of trigonometric functions are in radians.
Massive stars (Schaller et al. 1992; Woosley &
Weaver 1986):
Total stellar mass, reduced by stellar wind losses, of a star at core
helium ignition, $M_{1,i}$, and at core helium exhaustion $M_{1,e}$, as a
function of its initial mass, $10$ M${}_{\odot}<M_{1,o}<40$ M${}_{\odot}$:
$$\log M_{1,i}=0.9454\log M_{1,o}+0.0533\hskip 256.074803pt$$
(A1)
$$\displaystyle\log M_{1,e}$$
$$\displaystyle=$$
$$\displaystyle 0.81\log M_{1,o}+0.174\hskip 184.942913ptM_{1,o}\leq 20\mbox{\,M%
}_{\odot}$$
(A2)
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}\left[(0.81\log M_{1,o}+0.174)(1-\sin\phi)+\right.$$
$$\displaystyle\left.0.9095(1+\sin\phi)\right]\hskip 147.954331pt20\mbox{\,M}_{%
\odot}<M_{1,o}<40\mbox{\,M}_{\odot}$$
where $\phi=10(\log M_{1,o}-\log(20)-\pi/20)$.
Mass of the helium core, $M_{He}$, produced by a star of initial mass
$M_{1,o}$ before central He ignition:
$$\log M_{He}=1.589\log M_{1,o}-1.393.\hskip 256.074803pt$$
(A3)
If the massive star evolves through the core He burning phase, the He-core
mass grows in mass by $\simeq$1.1 M${}_{\odot}$ because of shell-hydrogen
burning. The helium core is subsequently exposed by common envelope
evolution, becoming the primary component mass in the next evolutionary
phase.
Radii of stars at core helium ignition, $R_{1,i}$, at core helium exhaustion,
$R_{1,e}$, and at core collapse, $R_{1,SN}$:
$$\displaystyle\log R_{1,i}$$
$$\displaystyle=$$
$$\displaystyle 1.0785\log M_{1,o}+1.5123\hskip 170.716535ptM_{1,o}\leq 20\mbox{%
\,M}_{\odot}$$
(A4)
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}\left[(1.0785\log M_{1,o}+1.5123)(1-\sin\phi)+\right.$$
$$\displaystyle\left.(1.053\log M_{1,o}+1.111)(1+\sin\phi)\right]\hskip 65.44133%
9pt20\mbox{\,M}_{\odot}<M_{1,o}<40\mbox{\,M}_{\odot}$$
where $\phi=15(\log M_{1,o}-\log(20)-\pi/30)$,
$$\displaystyle\log R_{1,e}$$
$$\displaystyle=$$
$$\displaystyle 1.5745\log M_{1,o}+0.97125\hskip 159.335433ptM_{1,o}\leq 20\mbox%
{\,M}_{\odot}$$
(A5)
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}\left[(1.5745\log M_{1,o}+0.97125)(1-\sin\phi)+\right.$$
$$\displaystyle\left.0.74(1+\sin\phi)\right]\hskip 159.335433pt20\mbox{\,M}_{%
\odot}<M_{1,o}<40\mbox{\,M}_{\odot}$$
where $\phi=12(\log M_{1,o}-\log(20)-\pi/24)$,
$$\displaystyle\log R_{1,SN}$$
$$\displaystyle=$$
$$\displaystyle 1.148\log M_{1,o}+1.5888\hskip 159.335433ptM_{1,o}\leq 20\mbox{%
\,M}_{\odot}$$
(A6)
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}\left[(1.148\log M_{1,o}+1.5888)(1-\sin\phi)+\right.$$
$$\displaystyle\left.0.65(1+\sin\phi)\right]\hskip 147.954331pt20\mbox{\,M}_{%
\odot}<M_{1,o}<40\mbox{\,M}_{\odot}$$
where $\phi=12(\log M_{1,o}-\log(20)-\pi/24)$.
Roche geometry (Eggleton 1983):
Dimensionless radius of the Roche lobe of component 1 $(r_{L_{1}}\equiv R_{L_{1}}/A)$
as a function of binary mass ratio $(q_{1}\equiv M_{1}/M_{2})$:
$$r_{L_{1}}=\frac{0.49q_{1}^{2/3}}{0.6q_{1}^{2/3}+\ln(1+q_{1}^{1/3})}.$$
(A7)
Component indices are interchangeable in this expression.
Common envelope evolution (Webbink 1984):
Ratio of post-common envelope binary separation, $A_{f}$, to pre-common envelope
separation, $A_{i}$:
$$\frac{A_{f}}{A_{i}}=\frac{\alpha_{CE}r_{L_{1}}}{2}\left(\frac{M_{2}}{M_{1}}%
\right)\left(\frac{M_{He}}{(M_{1}-M_{He})+\frac{1}{2}\alpha_{CE}r_{L_{1}}M_{2}%
}\right).$$
(A8)
Helium stars (Habets 1985; Woosley, Langer, & Weaver 1995):
Helium stars experience mass loss in a wind and their masses can decrease
significantly during the central-He burning phase. The
final mass of a helium star, $M_{He,f}$, at supernova as a function of its
mass,
$M_{He}$ at core helium ignition is approximated by:
$$M_{He,f}=3.64-6.42\mbox{exp}\left[-\frac{(M_{He}-3.43)^{0.33}}{0.55}\right]~{}%
~{}~{}~{}4\mbox{\,M}_{\odot}<M_{He}<20\mbox{\,M}_{\odot}$$
(A9)
If the helium star is exposed after central He exhaustion then it is not
affected by mass loss and its mass at supernova is equal to its mass at
the end of the time of its exposure.
Radius of helium star at supernova,
$R_{He,f}$:
$$\displaystyle R_{He,f}$$
$$\displaystyle=$$
$$\displaystyle 3.0965-2.013\log M_{He,f}~{}~{}\hskip 91.048819ptM_{He,f}\leq 2.%
5\mbox{\,M}_{\odot}$$
(A10)
$$\displaystyle=$$
$$\displaystyle 0.0557\left[(\log M_{He,f}-0.172)^{-2.5}\right]~{}~{}~{}~{}~{}~{%
}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}M_{He,f}>2.5\mbox{\,M}_{\odot}$$
Angular momentum loss:
Loss rate from gravitational radiation for a circular orbit
(Landau & Lifshitz 1951):
$$\dot{J}_{GR}=-\frac{32}{5}\frac{G}{c^{5}}~{}\left(\frac{M_{NS}M_{2}}{M_{NS}+M_%
{2}}\right)^{2}~{}A^{4}~{}\omega^{5},$$
(A11)
where $G$ is the gravitational constant, $c$ is the speed of light,
and $\omega$
is the orbital frequence. We neglect
the enhancement of gravitational radiation losses in eccentric orbits
(Peters & Mathews 1963).
The above equation can be integrated over a time interval
$\Delta t$ required for a circular orbit to decay from orbital period $P_{i}$ to
$P_{f}$:
$$P_{f}^{8/3}-P_{i}^{8/3}+8~{}A_{GR}~{}\Delta t=0,$$
(A12)
where
$$\displaystyle A_{GR}$$
$$\displaystyle=$$
$$\displaystyle\frac{q~{}(1+q)^{-1/3}~{}M_{NS}^{5/3}}{3.75\times 10^{11}}~{}%
\mbox{yr}^{-1}~{}\mbox{day}^{8/3}$$
$$\displaystyle q$$
$$\displaystyle=$$
$$\displaystyle\frac{M_{2}}{M_{NS}}$$
Loss rate from the magnetic stellar wind of a synchronously-rotating secondary
(cf. Rappaport, Verbunt, & Joss 1983):
$$\dot{J}_{MB}=-1.8\times 10^{47}~{}b(M_{2})~{}M_{2}~{}R_{2}^{2}~{}\omega^{3},$$
(A13)
where $\dot{J}_{MB}$ is in cgs units (dyne cm s${}^{-1}$) and $b(M_{2})$ is the
magnetic braking efficiency (eq. [9]), which becomes equal to zero for fully
convective stars ($M_{2}\leq 0.37$ M${}_{\odot}$).
For stars with radiative cores ($M_{2}>0.37$ M${}_{\odot}$),
we neglect the evolutionary expansion of the secondary with time and find
that during a time interval $\Delta t$ a circular orbit decays from
orbital period $P_{i}$ to $P_{f}$:
$$\displaystyle\frac{a}{4}\left(P_{f}^{8/3}-P_{i}^{8/3}\right)-\frac{a^{2}}{3}%
\left(P_{f}^{2}-P_{i}^{2}\right)+\frac{a^{3}}{2}\left(P_{f}^{4/3}-P_{i}^{4/3}%
\right)-a^{4}\left(P_{f}^{2/3}-P_{i}^{2/3}\right)$$
$$\displaystyle +a^{5}\ln\frac{1+P_{f}^{2/3}/a%
}{1+P_{i}^{2/3}/a}+2A_{MB}\Delta t=0,$$
(A14)
where
$$\displaystyle A_{MB}$$
$$\displaystyle=$$
$$\displaystyle b(M_{2})\frac{q^{2}~{}(1+q)^{1/3}~{}M_{NS}^{4/3}}{5.78\times 10^%
{9}}~{}\mbox{yr}^{-1}~{}\mbox{day}^{10/3}~{},$$
$$\displaystyle a$$
$$\displaystyle=$$
$$\displaystyle\frac{A_{MB}}{A_{GR}}$$
Low-mass stars:
Radii at ZAMS, terminal main sequence, and at the base of the giant branch,
along with the time evolution of the stellar radius have been given
Paper I.
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Abstract GSOS Rules and a Modular Treatment of Recursive Definitions
Stefan Milius\rsupera
\lsuperaLehrstuhl für Theoretische Informatik, Friedrich-Alexander
Universität Erlangen-Nürnberg, Germany
mail@stefan-milius.eu
,\
Lawrence S. Moss\rsuperb
\lsuperbDepartment of Mathematics, Indiana University, Bloomington,
IN, USA
lsm@cs.indiana.edu
\ and\
Daniel Schwencke\rsuperc
\lsupercInstitute of Transportation Systems, German Aerospace Center (DLR), Braunschweig, Germany
daniel.schwencke@dlr.de
Abstract.
Terminal coalgebras for a functor serve as semantic domains for
state-based systems of various types. For example, behaviors of CCS
processes, streams, infinite trees, formal languages and
non-well-founded sets form terminal coalgebras. We present a uniform
account of the semantics of recursive definitions in terminal
coalgebras by combining two ideas: (1) abstract GSOS rules
$\ell$ specify additional algebraic operations on a terminal
coalgebra; (2) terminal coalgebras are also initial completely
iterative algebras (cias). We also show that an abstract GSOS
rule leads to new extended cia structures on the terminal
coalgebra. Then we formalize recursive function definitions
involving given operations specified by $\ell$ as recursive program
schemes for $\ell$, and we prove that unique solutions exist in the
extended cias. From our results it follows that the solutions of
recursive (function) definitions in terminal coalgebras may be used
in subsequent recursive definitions which still have unique
solutions. We call this principle modularity. We illustrate
our results by the five concrete terminal coalgebras mentioned
above, e. g., a finite stream circuit defines a unique stream
function.
Key words and phrases:recursion, semantics, completely iterative algebra, coalgebra, distributive law
\lsuperbThis work was partially supported by a grant from the Simons Foundation (#245591 to Lawrence Moss).
\lmcsheading
9(3:28)2013
1–52
Dec. 08, 2010
Sep. 27, 2013
\ACMCCS
[Theory of computiation]: Semantics and
reasoning—Program semantics—Categorical semantics; Semantics and
reasoning—Program reasoning—Program specifications
1. Introduction
Recursive definitions are a useful tool to specify infinite system
behavior. For example, Milner [milner] proved that in his
calculus CCS, one may specify a process uniquely by the equation
$$P=a.(P|c)+b.0\,\text{.}$$
(1.1)
In other words, the process $P$ is the unique solution of the recursive
equation $x=a.(x|c)+b.0$.
Another example is the shuffle product on streams of
real numbers defined uniquely by a behavioral differential equation
[rutten_stream]:
$$(\sigma\otimes\tau)_{0}=\sigma_{0}\cdot\tau_{0}\qquad\qquad(\sigma\otimes\tau)%
^{\prime}=\sigma\otimes\tau^{\prime}+\sigma^{\prime}\otimes\tau\,\text{.}$$
(1.2)
Here the real number $\sigma_{0}$ is the head of $\sigma$, and $\sigma^{\prime}$ is
the tail; the operation $+$ is the componentwise addition of infinite streams.
Besides these and other examples in the theory of computation, we
shall mention below recursive specifications which are also important in
other realms; we shall consider non-well-founded sets [aczel, bm], a
framework originating as a semantic basis for circular definitions.
Operations on non-well-founded sets can be specified uniquely by
recursive function definitions. For example, we prove that the equation
$$x\|y=\{\,x\|y^{\prime}\mid y^{\prime}\in y\,\}\cup\{\,x^{\prime}\|y\mid x^{%
\prime}\in x\,\}\cup\{\,x^{\prime}\|y^{\prime}\mid x^{\prime}\in x,y^{\prime}%
\in y\,\}$$
(1.3)
has a unique solution viz. a binary operation $\|$ on the class of all
non-well-founded sets, which is reminiscent of parallel composition in
process calculi. It is the aim of this paper to develop abstract tools
and results that explain why there exist unique solutions to all the
aforementioned equations.
Terminal coalgebras.
The key observation is that process behaviors, streams and
non-well-founded sets constitute terminal coalgebras
$(C,c:C\to HC)$ for certain endofunctors $H$ on appropriate categories. The functor $H$
describes the type of behavior of a class of state-based systems—the
$H$-coalgebras—and the terminal coalgebra serves as the semantic
domain for the behavior of states of systems of type $H$.
All of the theory and examples in this paper pertain to terminal coalgebras.
Abstract GSOS rules.
Let us return to the case of CCS operations to situate the work in a
historical context.
In the case of CCS one obtains algebraic operations on processes
using structural operational semantics (sos) [afv01]. Operations
are specified by operational rules such as
$$\frac{x\stackrel{{\scriptstyle a}}{{\to}}x^{\prime}\qquad y\stackrel{{%
\scriptstyle a}}{{\to}}y^{\prime}}{x|y\stackrel{{\scriptstyle a}}{{\to}}x^{%
\prime}|y^{\prime}}$$
exhibiting the interplay between the operation and the behavior
type. Syntactic restrictions on the rule format then ensure nice
algebraic properties of the specified operations, e. g., GSOS
rules [bim] ensure that bisimilarity is a congruence. Turi and
Plotkin gave in their seminal paper [tp] a categorical
formulation of GSOS rules, and they show how an abstract GSOS rule
gives rise to a distributive law of a monad over a comonad,
where the monad describes the signature of the desired algebraic
operations and the comonad arises from the “behavior” functor
$H$. Later Lenisa et al. [lpw04] proved that abstract GSOS rules
correspond precisely to distributive laws of a free monad $M$ over the
cofree copointed functor on the behavior functor $H$. In this paper
we shall not need the original formulation of GSOS rules, and so the
reader who is unfamiliar with this formulation will not be at a loss. What will be
important are the categorical generalizations which we call abstract
GSOS rules, and we begin with that topic in the next section. The
theme of this paper is that abstract GSOS rules, either in a form
close to the original one or in a more general one that we introduce,
account for many interesting recursive definitions on terminal
coalgebras in a uniform way. These include all of the examples we
mentioned above, and many more.
Modularity.
An important methodological goal in this paper is that our results
should be modular. What we mean is that we want results which,
given a class of algebras for endofunctors, enable us to expand the
algebraic structure of an algebra from that class by a recursively
defined operation, and stay inside the class. Thus the results will
iterate.
We mentioned above that our results all had to do with
terminal coalgebras, and so the reader might wonder
how this idea of modularity could possibly apply. The answer is that a classical result due to
Lambek [lambek] implies that the structure $c:C\to HC$ of a terminal
coalgebra is an isomorphism. The corresponding inverse $c^{-1}:HC\to C$ turns $C$ into an $H$-algebra. Moreover, we are typically interested to start
with an algebraic structure on $C$ that is already extended by additional operations.
For example, in the case of the shuffle product $\sigma\otimes\tau$
of streams, the definition of $\otimes$ uses the stream addition
operation $+$ on the right-hand side. So the definition is made on
the algebra
$$(C,\quad(\sigma_{0},\sigma^{\prime})\mapsto\sigma,\quad\sigma,\tau\mapsto%
\sigma+\tau)$$
consisting of the set of streams of reals, its structure $c^{-1}$ as an $H$-algebra,
and its structure as an algebra for $X\times X$ given by componentwise addition.
Returning to modularity, we aim to isolate an appropriateness
condition on algebra structures expanding the inverse $c^{-1}$ of the
terminal coalgebra structure with the property that
given an appropriate structure and a definition in a certain format,
the definition specifies a new operation in a unique way,
and if we add this operation to the given algebra structure, the
resulting algebra structure on $C$ is again appropriate. This is what
we will call modularity, and precise formulations of
appropriateness may be found in the Summaries LABEL:sum:1 and LABEL:sum:2.
Completely iterative algebras.
The desired class of algebras mentioned in the previous paragraph is
formed by completely iterative algebras (cias).
Complete iterativity means that recursive equations involving algebraic
operations corresponding to the $H$-algebra structure $c^{-1}$ can be
solved uniquely (see Definition LABEL:def:cia). For example, let
$H_{\Sigma}$ be a polynomial endofunctor on ${\mathsf{Set}}$ arising from a signature
of operation symbols with prescribed arity. In this case a cia is a
$\Sigma$-algebra $A$ in which every system of recursive equations
$$x_{i}=t_{i}\qquad(i\in I),$$
(1.4)
where $t_{i}$ is either a single operation symbol applied to recursion
variables (i. e., $t_{i}=\sigma(x_{i_{1}},\ldots,x_{i_{n}})$ for an
$n$-ary $\sigma$ from $\Sigma$) or $t_{i}=a\in A$, has a unique
solution. It can then be proved that more general equation systems
involving (almost) arbitrary $\Sigma$-terms on the right-hand side and
even recursive function definitions have unique solutions in a
cia [m_cia, mm].
Continuing, it was shown in [m_cia] that the inverse $c^{-1}:HC\to C$ of the structure of the terminal coalgebra turns $C$ into the
initial cia for $H$.
However, as we mentioned previously, cia structures for the
“behavior” functor $H$ are not sufficient to yield the
existence and uniqueness of solutions in our motivating examples;
these involve additional algebraic operations not captured by
$H$. These operations are $|$ and $+$ in the CCS process definition above, the stream addition $+$ in (1.2),
and union $\cup$ in the example (1.3) from non-well-founded set
theory.
Extended cia structures
We will show how the abstract GSOS rules of Turi and Plotkin extend the
structure of an initial cia for $H$ (alias terminal
$H$-coalgebra). This then allows us to equip the initial cia with the
desired additional operations such that circular definitions of
elements of the carrier admit a unique solution, e. g., the
equation $\sigma=1.(\sigma+\sigma)$ (for streams) defining the
stream of powers of $2$ or our first example (1.1) for CCS
processes above.
The first steps in this direction were taken
in Bartels’ thesis [bartels_thesis] (see also [bartels]); he systematically studies
definition formats giving rise to distributive
laws and shows how to solve parameter-free first order recursive equations involving
operations presented by a distributive law; Uustalu et al. [uvp] present
the dual of this result.
We review some basic material on abstract GSOS rules and the
solution theorem of Bartels in Section 2.
In Section LABEL:sec:cias we extend these solution theorems to
equations with parameters, thereby combining them with our previous work
on cias in [aamv, m_cia]. More concretely, given an abstract GSOS rule
(equivalently, a distributive law of the free monad $M$ over the
cofree copointed functor $H\times\mathrm{Id}$) we prove in Theorems LABEL:thm:distcia
and LABEL:thm:sandwich that the terminal $H$-coalgebra carries the structure of
a cia for $HM$ and for $MHM$, respectively. These results show how to construct new
structures of cias on $C$ using an abstract GSOS rule.
This improves Bartels’ result in the sense that recursive equations
may employ constant parameters in the terminal coalgebra.
In Section LABEL:sec:solthms we obtain new ways to provide the
semantics of recursive definitions by applying the existing solution
theorems from [aamv, m_cia, mm] to the new cia structures. For
example, we consider solutions of recursive program schemes. Classical
recursive program schemes [guessarian] are function definitions
such as
$$f(x)=F(x,G(f(x)))$$
(1.5)
defining a new function $f$ recursively in terms of given ones $F,G$. In [mm] is was shown how recursive program schemes can be
formalized categorically. It was proved that every
guarded111Guardedness is a mild syntactic restriction
stating that terms on right-hand sides of equations do not have a
newly defined function at their head. recursive program scheme has
a unique (interpreted) solution in every cia $HA\to A$, where the
functor $H$ captures the signature of given operations. This solution
is an algebra structure $VA\to A$, where $V$ captures the signature
of recursively defined operations. As a new result we now prove in
Theorem LABEL:thm:extcia that the unique solution extends the structure of
the given cia for $H$ to a cia for the sum $H+V$; this yields
modularity of unique solutions of recursive program schemes.
Extended formats of function definition. The
recursive program schemes from [mm] do not capture recursive
function definitions like the one of the shuffle product
in (1.2) above or the one for the operation
$\|$ from (1.3) above on non-well-founded
sets. So in Section LABEL:sec:lambdarps we provide results that do
capture those examples. We introduce for an abstract GSOS rule $\ell$
two formats of recursive program schemes w. r. t. $\ell$: $\ell$-rps
and a variant called sandwiched $\ell$-rps ($\ell$-srps, for
short). They give a categorical formulation of recursive function
definitions such as (1.2)
and (1.3) that refer not only to the
operations provided by the behavior functor $H$ but also to additional
given operations specified by the abstract GSOS rule $\ell$. We will also see
that $\ell$-srps’s allow specifications which go beyond the format of
abstract GSOS rules. In Theorems LABEL:thm:ellrps
and LABEL:thm:sandwich_rps we prove that every $\ell$-(s)rps has a
unique solution in the terminal coalgebra $C$. Moreover, we show that
this solution extends the cia structure on $C$. This again gives rise
to a modularity principle: operations defined uniquely by
$\ell$-(s)rps’s can be used as givens in subsequent (sandwiched) rps’s
(we make this precise in the Summary LABEL:sum:2).
This modularity of taking solutions of recursive specifications of
operations does not appear in any previous work in this generality.
A set of examples.
Finally, in Section LABEL:sec:app we demonstrate the value of our results by
instantiating them in five different concrete applications: (1)
CCS-processes—we explain how Milner’s solution theorem from [milner]
arises as a special case of Theorem LABEL:thm:sandwich, and we also show how
to define new process combinators recursively from given ones; (2) streams of
real numbers—here we prove that every finite stream circuit defines a
unique stream function, we obtain the result from [rutten_stream] that
behavioral differential equations specify operations on streams in a unique way
as a
special instance of our Theorem LABEL:thm:ellrps, and we show how to solve
recursive equations uniquely that cannot be captured by behavioral differential
equations by applying Theorem LABEL:thm:sandwich_rps; (3) infinite trees—we
obtain the result from [rs10] that behavioral differential
equations have unique solutions as a special case of Theorem LABEL:thm:ellrps;
(4) formal languages—here we show how
operations on formal languages like union, concatenation, complement,
etc. arise step-by-step using the modularity of unique solutions
of $\ell$-rps’s, and how languages generated by grammars arise as the
unique solutions of flat equation morphisms in cias; (5) non-well-founded sets—we prove
that operations on non-well-founded sets are uniquely determined as
solutions of $\ell$-(s)rps’s.
Related Work. As already mentioned, Turi’s and Plotkin’s work [tp] was taken further by Lenisa,
Power and Watanabe in [lpw00, lpw04]. Fiore and Turi [ft01]
applied the mathematical operational semantics to provide semantics of message-passing process calculi such as
Milner’s $\pi$-calculus. But these papers do not consider the
semantics of recursive definitions.
Turi [turi97] gives a treatment of guarded recursion. He does not
isolate the notion of a solution of a recursive specification and
whence does not prove that a solution exists uniquely. In addition
that paper does not deal with recursive function definitions as we do
here.
Jacobs [jacobs] shows
how to apply Bartels’ result to obtain the (first order) solution
theorems from [aamv, m_cia]. Capretta et al. [cuv] work in a
dual setting and generalize the results of [bartels] beyond
terminal coalgebras and they also obtain the (dual of) the solution
theorem from [aamv, m_cia] by an application of their general
results. Our Theorems LABEL:thm:distcia and LABEL:thm:sandwich are similar to
results in [cuv], but in the later sections we extend the work in [bartels] in a different
direction by considering parameters in recursive definitions. So our
results in the present paper go beyond what can be accomplished with
previous work.
Modularity in mathematical operational semantics has been studied
before in [pow03, lpw04]. These papers show how to combine two
different specifications of operations over the same behavior by
performing constructions at the level of distributive laws. This gives an abstract
explanation of adding operations to a process calculus. At the heart
of our proof of Theorem LABEL:thm:ellrps lies a construction similar
to the combination of two distributive laws that arises by taking the coproduct of
the corresponding monads. However, while in the coproduct construction of [pow03, lpw04] the two
distributive laws are independent of each other, our case is different
because the operations specified by the second distributive law interact with
the operations specified by the first one.
Much less related to our work is the work of Kick and
Power [kp04] who show how to combine distributive laws of one
monad over two sorts of behavior possibly interacting with one another.
To the best of our knowledge the results on modularity of solutions of
recursive specifications we present are new.
The present paper is a completely revised version containing full proofs of
the conference paper [mms].
2. Abstract GSOS Rules and Distributive Laws
We shall assume some familiarity with basic notions from category
theory such as functors, (initial) algebras and (terminal) coalgebras,
and monads, see e. g. [maclane, rutten, adamek_survey].
Suppose we are given an endofunctor $H$ on some category ${\mathcal{A}}$
describing the behavior type of a class of systems. In our work we
shall be interested in additional algebraic operations on the
terminal coalgebra $C$ for $H$. The type of these algebraic operations is
given by an endofunctor $K$ on ${\mathcal{A}}$, and the algebraic operations are given
by an abstract GSOS rule (cf. [tp, lpw00, lpw04]). Our goal
is to provide a setting in which recursive equations with operations specified by abstract
GSOS rules have unique solutions. We now review the necessary preliminaries.
Assumption \thethm.
Throughout the rest of this paper we assume that ${\mathcal{A}}$ is a category with
binary products and coproducts, $H:{\mathcal{A}}\to{\mathcal{A}}$ is a functor, and that $c:C\to HC$ is
its terminal coalgebra. We also assume that $K:{\mathcal{A}}\to{\mathcal{A}}$ is a functor
such that for every object $X$ of ${\mathcal{A}}$ there exists a free
$K$-algebra $MX$ on $X$.
{rem}
(1) We denote by $\varphi_{X}:KMX\to MX$ and $\eta_{X}:X\to MX$ the structure and universal morphism of the free $K$-algebra
$MX$. Recall that the corresponding universal property states that for every $K$-algebra $a:KA\to A$ and every morphism $f:X\to A$ there exists a unique $K$-algebra homomorphism $h:(MX,\varphi_{X})\to(A,a)$ such that $h\cdot\eta_{X}=f$.
(2) Free algebras for the functor $K$ exist under mild assumptions on $K$.
For example, whenever $K$ is an accessible endofunctor on ${\mathsf{Set}}$ it has all
free algebras $MX$ (see e. g. [at]).
(3) As proved by Barr [barr] (see also Kelly [kelly80]), the existence of free $K$-algebras as
stated in Assumption 2 implies that there is a free monad on $K$.
Indeed, $M$ is the object assignment of a
monad with the unit given by $\eta_{X}$ from item (1) and the multiplication
$\mu_{X}:MMX\to MX$ given as the unique homomorphic extension of
$\mathrm{id}_{MX}$. Also $\varphi:KM\to M$ is a natural transformation
and
$$\kappa=(\xymatrix@ 1{K{}{}{}{}{}{}{}{}{}{}\xy@@ix@{{\hbox{}}}}$$ |
Tel Aviv University, Israelyanivsadeh@mail.tau.ac.ilhttps://orcid.org/0000-0002-5712-1028 Tel Aviv University, Israelhaimk@tau.ac.ilhttps://orcid.org/0000-0001-9586-8002
\CopyrightYaniv Sadeh and Haim Kaplan {CCSXML}
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<concept>
<concept_id>10003752.10003809.10010047</concept_id>
<concept_desc>Theory of computation Online algorithms</concept_desc>
<concept_significance>300</concept_significance>
</concept>
<concept>
<concept_id>10003752.10003809.10010031.10010033</concept_id>
<concept_desc>Theory of computation Sorting and searching</concept_desc>
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\ccsdesc[300]Theory of computation Online algorithms
\ccsdesc[500]Theory of computation Sorting and searching
\fundingThe work of the authors is partially supported by Israel Science Foundation (ISF) grant number 1595-19, German Science Foundation (GIF) grant number 1367 and the Blavatnik research fund at Tel Aviv University. \EventEditorsPetra Berenbrink, Mamadou Moustapha Kanté, Patricia Bouyer, and Anuj Dawar
\EventNoEds4
\EventLongTitle40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)
\EventShortTitleSTACS 2023
\EventAcronymSTACS
\EventYear2023
\EventDateMarch 7–9, 2023
\EventLocationHamburg, Germany
\EventLogo
\SeriesVolume254
\ArticleNo39
Dynamic Binary Search Trees: Improved Lower Bounds for the Greedy-Future Algorithm
Yaniv Sadeh
Haim Kaplan
Abstract
Binary search trees (BSTs) are one of the most basic and widely used data structures. The best static tree for serving a sequence of queries (searches) can be computed by dynamic programming. In contrast, when the BSTs are allowed to be dynamic (i.e. change by rotations between searches), we still do not know how to compute the optimal algorithm (OPT) for a given sequence. One of the candidate algorithms whose serving cost is suspected to be optimal up-to a (multiplicative) constant factor is known by the name Greedy Future (GF). In an equivalent geometric way of representing queries on BSTs, GF is in fact equivalent to another algorithm called Geometric Greedy (GG). Most of the results on GF are obtained using the geometric model and the study of GG. Despite this intensive recent fruitful research, the best lower bound we have on the competitive ratio of GF is $\frac{4}{3}$. Furthermore, it has been conjectured that the additive gap between the cost of GF and OPT is only linear in the number of queries. In this paper we prove a lower bound of $2$ on the competitive ratio of GF, and we prove that the additive gap between the cost of GF and OPT can be $\Omega(m\cdot\log\log n)$ where $n$ is the number of items in the tree and $m$ is the number of queries.
keywords: Binary Search Trees, Greedy Future, Geometric Greedy, Lower Bounds, Dynamic Optimality Conjecture
1 Introduction
Binary search trees (BSTs) are one of the most basic and widely used data-structures. They are used to store a sorted set of keys
from a totally ordered universe. Traversing BSTs is usually done by using a single pointer, initially pointing to the root, and moving to the left or right child according to the order of the searched key and the key of the item at the current node. Therefore, we typically define the cost111Our cost model is formally defined in Definition 2.1, in Section 2. of a search to be the length of the search path. The data structure itself may be static, or change dynamically throughout time, in response to insertions and deletions of items, and possibly even restructured during queries.
Static BSTs are well understood. One can guarantee that the longest path from the root to a leaf is of length $O(\log n)$ if the number of keys is $n$, by using a balanced tree. If the access sequence is known in advance (in fact only the frequency of accesses of each key matters) then an $O(n^{2})$ time algorithm computing the optimal static tree for the particular set of frequencies was given by Knuth [13]. It is also notable that the lower bound on the cost when the known frequencies are $\vec{f}=[f_{1},f_{2},\ldots,f_{n}]$ and the number of queries is $m$, is $\Omega(m\cdot H(\vec{f}))$ where $H(\vec{f})=\sum_{i=1}^{n}{f_{i}\log\frac{1}{f_{i}}}$ is the entropy function. A simple way with $O(n\log n)$ running time to construct a near-optimal static (centroid) tree whose cost is $O(m\cdot H(\vec{f}))$, has been described by Mehlhorn [17]. The running time has been improved to $O(n)$ by Fredman [10].
In contrast to the static case, the dynamic case is less understood. One can, of course, serve the sequence with a static tree. But, for many sequences we must change the structure of the tree as we make the searches in order to be efficient. For example, the requested items may be different in different parts of the sequence so a different set of items has to be placed near the root during different parts of the sequence. Restructuring is done by rotations that maintain the symmetric order. When rotations are allowed, the cost is defined to be the size of the subtree that contains the search path and all edges which we rotate.
Here, we assume that the set of values stored in the tree does not change (no insertions or deletions), yet restructuring the tree is allowed to speed up future searches. One famous dynamic algorithm for doing this is the Splay algorithm of Sleator and Tarjan [20]. After each query, the splay algorithm moves the queried item to the root of the tree, according to three simple rules called zig-zag, zig-zig and zig. The splay algorithm is efficient in the sense that it is able to exploit the structure of many families of sequences. In particular splay is proven to be as good as the static optimum (up to a constant factor), which also implies that the cost of splay on any given sequence is at most $O(\log n)$ times the (dynamic) optimum cost. Sleator and Tarjan conjectured that splay is in fact dynamically-optimal, meaning that its cost is like the cost of an optimal algorithm that knows the whole sequence of queries in advance, up to some constant factor. However, this dynamic-optimality conjecture of splay is still open. In fact, it is open whether there is any dynamically-optimal online binary search tree algorithm. The best competitive ratio achievable to date is $O(\log\log n)$, and it is obtained by Tango [8], Multi-splay [21] and Chain-splay [11] trees, and a geometric divide-and-conquer approach of [1].
While seeking for (better) guaranteed competitiveness, other dynamic algorithms were considered. A promising candidate was independently proposed by Lucas [16] and Munro [18], which is now commonly referred to as Greedy Future, henceforth: $GF$ in short. As its name suggests, $GF$ is a greedy algorithm that rearranges the nodes on the path from the root to the current queried item as a treap whose priorities are according to the future accesses222Each item in a treap has two keys: value and priority. The treap is a binary search tree with respect to the values of the items and a heap with respect to their priorities. That is, the priority of an item is no larger than the priorities of its children. In our case, the priorities are deterministically defined by future requests in a way that we define precisely in Algorithm 1. (as this paper deals with analyzing $GF$, we detail it formally in Algorithm 1). Note that unlike splay, $GF$, by definition, is required to know the future in order to restructure the tree. Surprisingly however, Demaine et al. [7] showed that one can simulate $GF$ without knowing the future by a hierarchy of split-trees while losing only a constant factor in performance.
Additionally, [7] presented a geometric view of an algorithm serving queries by a dynamic binary search tree using a two dimensional grid on which we mark the sequence as well as the items accessed by the algorithm. In this presentation there is yet another natural promising candidate for dynamic optimality, which is commonly known as Geometric Greedy and sometimes simply Greedy, which we shall refer to as $GG$. [7] showed that $GG$ is in fact the same algorithm as $GF$.
The geometric view proved useful to obtain new results regarding $GG$ and hence $GF$. Fox [9] proved that an access-lemma that is analogues to the so called access-lemma of splay trees holds for $GG$. From this follows that most of the nice properties that hold for splay also hold for $GF$. In particular, it follows that $GF$ is $O(\log n)$ competitive. Chalermsook et al. [3] analyzed upper bounds on the cost of $GG$ for access patterns which are permutations, and in particular found that for highly structured permutations, which they called $k$-decomposable, the cost is $n\cdot 2^{\alpha(n)^{O(k)}}$ where $\alpha(n)$ is the inverse-Ackermann function. Chalermsook et al. [5] study special access patterns that belong to a broader family of pattern-avoiding permutations. See [4] for a survey of currently known properties of greedy and splay.
Our Contributions:
1.
It is known that $GF$ is not exactly optimal, but it is conjectured, like splay, to be optimal up to a constant factor. In fact, it has been even more strongly conjectured by Demaine et al. [7] to be optimal up to an additive $O(m)$ term, and possibly even exactly $m$. Kozma [14] refuted the second part and gave a specific sequence for which this additive gap is $m+1$. In this paper we refute the linear gap conjecture and show a family of sequences for which the additive gap is at least $\Omega(m\log\log n)$.
2.
The largest lower bound on the competitive ratio of $GF$ is $\frac{4}{3}$ by Demaine et al. [7].
They show a family of sequences on which after an initial query, the optimum pays $1.5$ on average per query while $GF$ pays $2$.333Reddmann [19] found an example in which the cost ratio between $GF$ and the optimum is $\frac{26}{17}\approx 1.53$. But this is for one particular sequence of a fixed length so it does not rule out any competitive ratio if we allow an additive constant.
We describe a technique that allows us to improve this lower bound to $2$.
We note that the best known lower bound on the competitive ratio of splay is $2$ (see [15, Section 2.5]). In both cases, the construction requires a rather large number of items (large $n$).
3.
Based on the multiplicative lower bound described above we show the following two interesting properties of $GF$: (1) There are sequences $X$ such that the cost of $GF$ on the reverse sequence is twice larger than the cost of $GF$ on $X$. (2) There are sequences $X$ such that we can remove some queries from them and get a subsequence $X^{\prime}$, such that the cost of $GF$ on $X^{\prime}$ is twice larger than the cost of $GF$ on $X$.
4.
Based on the additive lower bound described above, the two properties of $GF$ in the previous bullet also hold if we replace the (multiplicative) relation of “costs twice more” by the (additive) relation “costs $\Omega(m\log\log n)$ more”.
We study subsequences and reversal (contributions 3-4) since any dynamically-optimal algorithm $A$ must have a “nice” behavior in these cases. Concretely, $A$ must satisfy the approximately-monotone property (Definition 3.5) which states that there is a fixed constant $c$ such that the cost of $A$ on any subsequence of any sequence is never more than $c$ times the cost on the whole sequence. As for reversal, the optimum can process a sequence and its reversal with similar costs up to a difference of $n$, thus any dynamically-optimal algorithm must be able to do so with costs that differ by at most a constant factor. We discuss this motivation in more detail in Section 3 (right after stating Theorem 3.4).
Our contributions are all based on the same technique, which is quite simple. We enforce $GF$ to maintain a static tree and only query the leaves of this tree. Although being dynamic in general, there are some access-patterns that cause $GF$ not to change the tree. By studying these patterns, we can study $GF$ on a static tree, and the analysis of its cost simplifies to the weighted-average of the depth of the queries (weighted by frequency). To lower-bound the gap between $GF$ and $OPT$, we analyze the average cost that can be saved by promoting the items in the leaves to locations closer to the root. Note that any other item can be placed further away from the root since it is never queried by the sequence.
2 Model
In this section we describe the model which we use, and define our notations. First, we note that throughout the paper $\lg x$ is used to denote the base two logarithm of $x$.
We consider a totally ordered universe of (fixed size) $n$ items. For simplicity, one may think of the values $\{1,\ldots,n\}$. The items are organized in some initial BST which we denote by $T_{0}$. Then, a sequence of queries, denoted by $X=[x_{1},x_{2},\ldots,x_{m}]$, is given, one query at a time. We reserve $m$ to denote the length of the sequence. The tree before serving $x_{t}$ is denoted by $T_{t-1}$. An algorithm has to find the queried value $x_{t}$, by traversing $T_{t-1}$ from its root. After finding $x_{t}$, the algorithm is allowed to re-structure $T_{t-1}$ to get $T_{t}$. We define the cost of the algorithm at time $t$ to be the total number of nodes that were touched at time $t$, both on the path to $x_{t}$ and for restructuring. The cost of an algorithm for the whole sequence is simply the sum of its costs over all times. We define it formally below.
Definition 2.1 (Cost).
Let $X$ be a sequence of queries, and let $T_{0}$ be an initial tree.
Let $A$ be an algorithm that serves $X$ and let $T_{t}$ be the tree that $A$ has after serving $x_{t}$. Let $P_{t}$ be the set of nodes on the path from the root to $x_{t}$ in $T_{t-1}$ and let $U_{t}$ be the set of nodes of the minimal subtree that contains all the edges that were rotated by $A$ to transform $T_{t-1}$ to $T_{t}$. Then the cost of $A$ for serving $X$ at time $t$ is $|P_{t}\cup U_{t}|$, and the cost of $A$ for serving $X$ is the sum of costs over $t=1,\ldots,m$. We denote the cost of $A$ to serve $X$ starting with $T_{0}$ by $cost(A,X,T_{0})$. We denote the average cost per query by $\hat{c}(A,X,T_{0})=\frac{cost(A,X,T_{0})}{m}$. When $T_{0}$ is clear from the context, or immaterial, we write $cost(A,X)$ and $\hat{c}(A,X)$.
Definition 2.2 (Depth).
Let $T$ be a tree.
The depth of a node
$v\in T$, denoted by $d(v)$,
is the number of edges in the path from the root to $v$ (in particular $d(root)=0$). Note that the cost of querying $v$ (without restructuring) is $d(v)+1$. We also define the depth of the tree, denoted by $d(T)$, as the maximum depth of a node in $T$, that is $d(T)=\max_{v\in T}{d(v)}$.
Definition 2.3 (Competitiveness).
We say that an algorithm $A$ is $(\alpha,\beta)$-competitive for initial tree $T_{0}$ if for any sequence of queries $X$, it holds that $cost(A,X,T_{0})\leq\alpha\cdot cost(OPT,X,T_{0})+\beta$ where $OPT$ is a best algorithm to serve $X$ given $T_{0}$ (with full knowledge of $X$). When we do not specify $T_{0}$ we mean that the relation holds for all initial trees. We refer to $\alpha$ as the multiplicative term and to $\beta$ as the additive term. For ease of language, we regard the multiplicative term as the competitive ratio, and also write “the competitive ratio of” instead of “the multiplicative term of the competitiveness of”. In such cases, we assume that the additive term is $o(m)$. It is easiest to think of $\beta=O(n)$ while assuming that $m=\omega(n)$.
To conclude this section, we give a precise description of the $GF$ algorithm, in Algorithm 1.
We emphasize that its implementation is complex and probably would not be good in practice. However, its main benefit is its theoretical value, as a candidate for dynamic optimality. Should it be proven to be dynamically-optimal, then we would get a better understanding of the problem and also a stepping-stone to analyze simpler algorithms, such as splay, in comparison to $GF$ rather than against some “vague” optimum that depends on the sequence.
3 Stable Sequences and Lower Bounds
In this section we properly define the family of stable sequences (Definition 3.8) for which the tree maintained by $GF$ is never changed (i.e. the access path of the current query is a treap with respect to the suffix of the sequence). To prove our lower bounds we use such sequences in which only the items at the leaves of $GF$ are requested, and the internal nodes cause some extra cost that $OPT$ avoids. We use a natural way to represent such sequences as trees, and use this representation to prove the following lower bounds, which are the main results of this section.
Theorem 3.1.
If $GF$ is $(c,d)$-competitive where the additive term $d$ is sublinear in the length of the sequence, i.e. $d=o(m)$, then $c\geq 2$.
Theorem 3.2.
For every $n\geq 2$ there exist sequences $X\in[n]^{m}$ such that $cost(GF,X)=cost(OPT,X)+\Omega(m\cdot\lg\lg n)$.
Among these sequences, there exists a sequence whose length is $m=n^{\Theta(\frac{\lg\lg n}{\lg\lg\lg n})}$. (There exist other longer sequences too.)
Theorems 3.1 and 3.2 enable us to prove the following two theorems, proven in Appendix A.2.
Theorem 3.3.
For any $\epsilon>0$ there exists a sequence $X$ with a subsequence (not necessarily consecutive) $X^{\prime}\subseteq X$ such that $cost(GF,X^{\prime})\geq(2-\epsilon)\cdot cost(GF,X)$.
There exists a sequence $Y$ with a subsequence (not necessarily consecutive) $Y^{\prime}\subseteq Y$ such that $cost(GF,Y^{\prime})-cost(GF,Y)=\Omega(m\cdot\lg\lg n)$.
Theorem 3.4.
Let $S$ be a sequence, we define $rev(S)$ to be the sequence $S$ in reverse. For any $\epsilon>0$ there exists a sequence $X$ such that $cost(GF,rev(X))\geq(2-\epsilon)\cdot cost(GF,X)$.
There exists a sequence $Y$ such that $cost(GF,rev(Y))-cost(GF,Y)=\Omega(m\cdot\lg\lg n)$.
The motivation for studying subsequences (Theorem 3.3) is the fact that $OPT$ always saves costs when queries are removed from its sequence. Formally, if $X^{\prime}\subseteq X$, then $cost(OPT,X^{\prime})\leq cost(OPT,X)$. Indeed, $OPT$ can serve $X^{\prime}$ by simulating a run on $X$. More generally, this relation of costs when comparing a sequence to a subsequence of it, is an important property which even has a name:
Definition 3.5 (Approximate-monotonicity [12, 15]).
An algorithm $A$ is approximately-monotone with a constant $c$ if for any sequence $X$, subsequence $X^{\prime}\subseteq X$, and initial tree $T$, it holds that $cost(A,X^{\prime},T)\leq c\cdot cost(A,X,T)$.
Corollary 3.6.
If $GF$ is approximately-monotone with a constant $c$, then $c\geq 2$.
As noted, $OPT$ is approximately-monotone with $c=1$ (strictly monotone). The reason that approximate-monotonicity is of interest, in particular for $GF$, is because it is one of two properties that together are necessary and sufficient for any dynamically-optimal algorithm. The complementing property, which $GF$ is known to satisfy, is simulation-embedding:
Definition 3.7 (Simulation-Embedding [15]).
An algorithm $A$ has the simulation-embedding property with a constant $c$ if for any algorithm $B$ and any sequence $X$, there exists a supersequence $Y\supseteq X$ such that $cost(A,Y)\leq c\cdot cost(B,X)$. ($X$ is a subsequence of $Y$, not necessarily of consecutive queries.)
An algorithm $A$ which is approximately-monotone with a constant $c_{1}$ and has the simulation-embedding property with a constant $c_{2}$ is dynamically-optimal with a constant $c_{1}\cdot c_{2}$. Indeed, for any sequence $X$, there is some supersequence $Y(X)\supseteq X$ such that $cost(A,X)\leq c_{1}\cdot cost(A,Y(X))\leq c_{1}\cdot c_{2}\cdot cost(OPT,X)$. Harmon [12] proved that $GG$, and hence $GF$, has the simulation-embedding property, hence $GF$ is dynamically-optimal if and only if it is approximately-monotone. An alternative indirect proof was given by [6], proving that $GG$ is $O(1)$-competitive versus the move-to-root algorithm, therefore inheriting the property from move-to-root.
The motivation for studying reversal (Theorem 3.4) is that $OPT$ is oblivious to reversing the sequence of queries, up to an additive difference of $n$. Indeed, to serve a sequence $X$ in reverse, we can pay $n$ to restructure the initial tree $T_{0}$ to the final tree $T_{m}$, and then “reverse the arrow of time”: when serving query $x_{t}$, also modify the tree from $T_{t}$ to $T_{t-1}$ where $T_{i}$ is the tree that $OPT$ would get by the end of processing the $i$-th query of $X$, in order. This means that any dynamically-optimal algorithm must be able to serve a sequence of requests and its reverse with the same cost up to a constant factor. Theorem 3.4 does not disprove dynamic-optimality for $GF$, but gives some insight of how reversal affects $GF$.
3.1 Maintaining a Static Tree for GF
In this section we describe the basic “tool” which we use to fix a tree structure for $GF$ despite its dynamic nature. That is, we describe a class of sequences which we call mixed-stable sequences such that $GF$ never restructures its tree when serving a sequence in this class. For the sake of simplicity, we assume that the initial tree is structured as we need it to be. Appendix A.1 explains how to enforce a specific “initial” tree given an arbitrary initial tree, and also argues why this minor issue does not affect the competitive ratio of $GF$.
As noted, our objective is to produce a sequence that “tricks” $GF$ into having unnecessary nodes in the core of the tree, such that the requested values are only at the leaves. As an example, consider the classic sequence of queries $X=[1,3,1,3,\ldots]$ with an initial tree
containing $2$ at the root, $1$ as a left child of the root and $3$ as a right child of the root. Because of the alternating pattern, $GF$ never re-structures the tree, and the cost per query is $2$ rather than $1.5$ on average (e.g. when $1$ is in the root, and $3$ is its right child).
Definition 3.8 (Stable Nodes and Sequences).
Let $T$ be a full binary search tree, and let $X$ be a sequence of queries over the items in the leaves of $T$. We define the stability of nodes as follows, see also Figure 1.
We say that an inner node $v$ in $T$ is strongly-stable if it has two children, and
the subsequence of $X$ consisting only of the items in the subtree of $v$, alternates between accesses to the left and right subtrees of $v$.
We say that an inner node $v$
with a left child $u$ in $T$ is weakly-stable with a left-bias if both $v$ and $u$ have two children,
and the subsequence of $X$ consisting only of the items in the subtree of $v$,
repeats the following $3$-cycle. First it accesses the left-subtree of $u$, then the right subtree of $u$, and finally right subtree of $v$. (It is left-biased because $\frac{2}{3}$ of the accesses are to the left of $v$). Symmetrically, we say that $v$ is weakly-stable with a right-bias if $v$ has two children, its right child $u$ has two children, and the restriction of $X$ to accesses in the subtree of $v$ repeats a $3$-cycle consisting of an access to the right subtree of $u$, the left subtree of $u$, and the left subtree of $v$. Notice that $u$ is a strongly-stable node by definition, and we refer to it as the favored-child of $v$.
We regard the sequence $X$ as being induced by the tree $T$ with stability “attached” to its inner nodes. We assume that every node is stable, and refer to $X$ as a mixed-stable sequence and to $T$ as a mixed-stable tree. We distinguish two special cases: If all inner nodes are strongly-stable then we refer to $X$ and $T$ as strongly-stable, and if exactly half of the inner nodes of $T$ are weakly-stable then we refer to $X$ and $T$ as weakly-stable (recall that each weakly-stable node has a strongly-stable favored-child).
To motivate Definition 3.8 a little, note that the sequence $X=[1,3,1,3,\ldots]$ is a strongly-stable sequence that corresponds to a tree over the items $\{1,2,3\}$ where $2$ is in the root. $X$ yields a lower-bound of $\frac{4}{3}$ on the competitive ratio of $GF$. Similarly, the sequence $X^{\prime}=[5,3,1,5,3,1,\ldots]$ is a weakly-stable sequence that corresponds to the tree over $\{1,2,3,4,5\}$ with $2$ at the root and $4$ its right-child. $X^{\prime}$ yields a lower-bound of $\frac{8}{5}$ on the competitive ratio of $GF$, which is already an improvement over the best known lower bound, see also Figure 2. The distinction between strongly-stable and weakly-stable nodes is that $GF$ may modify the structure of the tree when a weakly-stable node is considered, but only temporarily and without affecting the cost. In our example with $X^{\prime}$, after querying $5$, $GF$ may put $4$ in the root instead of $2$, but following the query of $3$ this change will be reverted.
Motivated by the power of stable sequences over small trees, we proceed to a more general analysis of stable sequences.
Definition 3.9 (Atomic Sequence).
A tree $T$, along with stability type (weak/strong) for each node, and a subtree of each node to be accessed initially, induce a stable sequence. This sequence is unique up to its length, which can be extended indefinitely. We define the “atomic unit” of this sequence as the shortest sequence $X$ such that any repetition of $X$ is also a stable sequence that corresponds to $T$.
Throughout the paper we work with whole multiples of the atomic sequence. Moreover, unless stated otherwise, we work with the atomic sequence itself (a single repetition).
Lemma 3.10.
Let $X$ be a mixed-stable sequence with respect to a tree $T$. Then every leaf $u$ is visited once every $2^{a(u)}\cdot 3^{b(u)}$ queries where $a(u)$ and $b(u)$ are non-negative integers. In particular, the atomic length of $X$ is $2^{\max_{leaf\,u}{a(u)}}\cdot 3^{\max_{leaf\,u}{b(u)}}$ (the lcm). Moreover, if $X$ is strongly-stable then $\forall u:b(u)=0$, and if $X$ is weakly-stable then $\forall u:a(u)=0$.
Proof 3.11.
Consider a leaf $u$. Define the frequency of visiting an ancestor $w$ of $u$ to be the frequency of accessing a leaf in the subtree of $w$. If $w$ is a strongly-stable ancestor then the frequency of visiting a child of $w$ is
$\frac{1}{2}$ of the frequency of visiting $w$.
If $w$ is weakly-stable, $v$ is its favored-child, and $x$ is a child of $v$ then the frequency of visiting $x$ is $\frac{1}{3}$ of the frequency of visiting $w$. Similarly if
$w$ is weakly-stable, $v$ is its non-favored-child then the frequency of visiting $v$ is $\frac{1}{3}$ of the frequency of visiting $w$.
It follows that
$u$ is visited exactly once every $2^{a(u)}\cdot 3^{b(u)}$ queries where $a(u)$ is the number of strongly-stable nodes that are not favored-children (there are no such nodes if $X$ is weakly-stable), and $b(u)$ is the number of weakly-stable nodes (no such nodes if $X$ is strongly-stable), on the path to $u$. Finally, since every leaf $u$ is visited with a specific period, the whole sequence has a period which is the lcm of all periods.
Lemma 3.12.
Let $X$ be a mixed-stable sequence with respect to a tree $T$. If $GF$ serves $X$ with $T$ as initial tree, and breaks ties in favor of nodes of smaller-depth, then it never restructures $T$.
Proof 3.13.
The proof is by induction on the size of the tree. If $T$ has a single node, then it is trivial. Otherwise, the root $r$ is an inner-node, and we prove that it always remains the root. It then follows, by restricting the access sequence to values within each subtree, that the rest of the tree remains fixed as well. We use the notations of $\tau(v)$ and $v_{i}$ as in Algorithm 1.
First, consider the case that $r$ is a strongly-stable node (Definition 3.8). Given an access to some value $x$ in the left subtree of $r$, by definition, the next access would be to a value in the right subtree of $r$, hence $\tau(r)<\tau(v_{i})$ for any $v_{i}\neq r$ on the path from $r$ to $x$, and therefore $GF$ will keep $r$ in the root. The same argument holds if $x$ is in the right subtree of $r$, and the next access is in the left subtree.
Next, consider the case that $r$ is a weakly-stable node. Without loss of generality, assume that it is left-biased, and denote its favored-child (left child) by $u$. Denote the left and right subtrees of $u$ by $A$ and $B$ respectively, and the right subtree of $r$ by $C$. The access pattern of subtrees is $ABC(ABC\ldots)$.
•
If the current access was to some $x\in A$, both $r$ and $u$ have been touched. The next access queries in $B$, so $\tau(u)=\tau(r)<\tau(v_{i})$ for any $v_{i}\neq u,r$ on the access path to $x$. Since $GF$ tie-breaks in favor of smaller-depth, it will keep $r$ in the root.444This is the reason we defined this kind of access pattern as weakly-stable, because the stability can be chosen, but is not forced. We emphasize that putting $u$ as a parent of $r$ will not make the next access cheaper as both $u$ and $r$ will be touched anyway, and then $r$ will be reinstated as the root.
•
If the current access was to some $x\in B$, then both $r$ and $u$ have been touched. The next access touches $C$, so $\tau(r)<\tau(v_{i})$ for any $v_{i}\neq r$ on the access path to $x$, including $u$, thus $r$ must remain the root.
•
If the current access was to some $x\in C$, since the next access touches $A$, $\tau(r)<\tau(v_{i})$ for any $v_{i}\neq r$ on the access path to $x$, thus $r$ must remain the root. In this case $u$ was not touched, but nonetheless it remains the left child of $r$.
Lemma 3.14.
If $X$ is a mixed-stable sequence, the frequency of accessing $x\in X$ is in the range of $[\frac{1}{3^{d(x)}},\frac{1}{3^{d(x)/2}}]$.
In particular, if $X$ is strongly-stable then the frequency equals $\frac{1}{2^{d(x)}}$.
Proof 3.15.
The frequency of visiting a node depends on the path to it. The frequency is multiplied by $\frac{1}{2}$ when passing through a strongly-stable node, and multiplied by either $\frac{1}{3}$ or $\frac{2}{3}$ when passing through a weakly-stable node. Every factor of $\frac{2}{3}$ is followed by $\frac{1}{2}$, due to the strongly-stable favored-child of the weakly-stable node. Thus the frequency is bounded between $\frac{1}{3^{d(x)}}$ and $\frac{1}{2^{d(x)/2}}\cdot\big{(}\frac{2}{3}\big{)}^{d(x)/2}=\frac{1}{3^{d(x)/2}}$.
Corollary 3.16.
Let $X$ be a strongly-stable sequence, then: $\hat{c}(GF,X)=\sum_{x\in X}{\frac{d(x)+1}{2^{d(x)}}}$.
3.2 Promotions and Recursive Trees
The way in which we show our lower bounds relies on the fact that serving the leaves of a static tree is sub-optimal, since a trivial static optimization is to move the leaves closer to the root. We refer to this operation as a promotion of the leaf that we move. We emphasize that for the purpose of our result, we analyze the improvement one gets from promotions, but the actual $OPT$, which is dynamic, may be able to reduce the cost further.
Definition 3.17 (Promotion).
Consider trees $T$ and $T^{\prime}$. We say that a node $x$ was promoted in $T^{\prime}$ by $h$ (with respect to $T$), if $d_{T}(x)-d_{T^{\prime}}(x)=h$. Given a mixed-stable sequence $X$, the average promotion of $T$ to $T^{\prime}$ is the weighted average promotion in $T^{\prime}$ of the nodes of $T$, weighted by the query frequencies of the nodes.
By definition, static optimization of a tree $T$ to $T^{\prime}$ for a mixed-stable sequence $X$, implies a cost improvement for $OPT$ which is at least the average promotion of $T$ to $T^{\prime}$, per query. Intuitively, promoting leaves that are closer to the root contributes more to the average promotion than promoting deeper leaves since the access frequencies decrease exponentially with depth. That being said, our promotion scheme will be relatively uniform, promoting most leaves by roughly the same amount, as in the following example.
Example 3.18.
To clarify promotions, consider Figure 3. There, we can safely promote every node by one, except for one of the deepest nodes. Therefore, we immediately conclude that for the corresponding strongly-stable sequence $X$, we have:
$\hat{c}(GF,X)\geq\hat{c}(OPT,X)+(1-\frac{1}{2^{n}})$.
We define our trees using recursive structures.
Definition 3.19.
A recursive tree, $T_{r}$, of depth $r$ is defined by a specific full binary tree $T$ (independent of $r$) such that at least one of its leaves is an actual leaf, and some of its leaves are roots of recursive trees, $T_{r-1}$, of depth $r-1$. We refer to the inner nodes of $T$ as the trunk of $T_{r}$, and define
$T_{0}$ to be a single node. See Figure 4 for two examples.555The name of the pattern $F$ in Figure 4, stands for Fibonacci: One can verify that for $r\geq 2$, the number of leaves at depth $1\leq d\leq r-1$ is the $(d-1)$th Fibonnaci number $F_{d-1}$ (we define $F_{0}=0$). Moreover, this can be used to prove the nice equation: $\sum_{d=0}^{\infty}{\frac{F_{d}}{2^{d}}}=2$.
3.3 Multiplicative Lower Bound for GF
In this section we prove Theorem 3.1. We do it by describing a concrete weakly-stable sequence, whose average cost per query is $6$ while an average promotion of $3$ is possible, resulting in an optimal cost of at most $3$. We start by stating a purely mathematical lemma that will be used in the analysis.
Lemma 3.20.
Let $b_{r}$ be a sequence
defined by an initial value $b_{0}$ and the relation $b_{r}=\alpha\cdot b_{r-1}+\beta+\gamma\cdot\frac{r}{2^{r}}$ for some constants $\alpha,\beta,\gamma$ where $\alpha\neq\frac{1}{2},1$. Then $b_{r}=\frac{\beta}{1-\alpha}(1-\alpha^{r})+\alpha^{r}\cdot b_{0}+\frac{2\alpha\gamma}{(2\alpha-1)^{2}}\cdot(\alpha^{r}-\frac{1}{2^{r}})-\frac{\gamma}{(2\alpha-1)}\cdot\frac{r}{2^{r}}$.
In particular, when $\gamma=0$ then $b_{r}=\frac{\beta}{1-\alpha}(1-\alpha^{r})+\alpha^{r}\cdot b_{0}$.
Proof 3.21 (Proof Sketch).
Either use induction, or “guess” that a geometric sequence $y_{r}$ with a multiplier of $\alpha$ satisfies $y_{r}=p\cdot\frac{r}{2^{r}}+q\cdot\frac{1}{2^{r}}+s+b_{r}$, and determine the fixed coefficients $p,q,s$.
Lemma 3.22.
Let $X$ be a weakly-stable sequence implied by the recursive tree $F_{r}$ in Figure 4, where the root is a weakly-stable node with a right-bias. Then for any $\epsilon>0$, there is a sufficiently large recursive depth $r$ such that (1) $\hat{c}(GF,X)>6-\epsilon$, (2) a static optimization of the tree saves an average cost of at least $3-\epsilon$, and (3) regardless of $r$, $\hat{c}(OPT,X)<3$.
Proof 3.23.
Let $c_{r}$ denote the average cost of serving $X$ with $F_{r}$. Then $c_{0}=1$ and $c_{r}=\frac{1}{3}(c_{r-1}+1)+\frac{1}{3}\cdot 3+\frac{1}{3}(c_{r-1}+2)=\frac{2}{3}c_{r-1}+2$, which yields by Lemma 3.20 that $c_{r}=\frac{2}{1-2/3}(1-(2/3)^{r})+(2/3)^{r}\cdot 1=6\cdot(1-(2/3)^{r})+(2/3)^{r}$. To analyze the average promotion, we re-structure $F_{r}$ to a new static structure $F^{\prime}_{r}$ as follows, see Figure 5. The leaf is moved to the root, whose children are the recursive subtrees, optimized themselves by the same logic. The old root is moved to be a right child of the maximal value in the new left subtree, and the old right-child (of the old-root) is moved to be a left child of the minimal value in the new right subtree. $F^{\prime}_{r}$ maintains the order of values as was in $F_{r}$. The demotions of the old root and its right child do not affect the cost, because $X$ does not query these values. Denote by $p_{r}$ the average promotion of $F_{r}$ to $F^{\prime}_{r}$. Then $p_{0}=0$ since nothing is promoted for a singleton, and $p_{r}=\frac{1}{3}p_{r-1}+\frac{1}{3}\cdot 2+\frac{1}{3}(p_{r-1}+1)=\frac{2}{3}p_{r-1}+1$. Again by Lemma 3.20 we get that $p_{r}=\frac{1}{1-2/3}(1-(2/3)^{r})+(2/3)^{r}\cdot 0=3\cdot(1-(2/3)^{r})$.
Observe that for $r\to\infty$ we get that $c_{r}\to 6$ and $p_{r}\to 3$, thus parts (1) and (2) of the claim follow. For part (3), observe that $c_{r}-p_{r}=6\cdot(1-(2/3)^{r})+(2/3)^{r}-3\cdot(1-(2/3)^{r})=3-2\cdot(2/3)^{r}<3$.
See 3.1
Proof 3.24.
Assume by contradiction that $GF$ is $(2-\delta,f(m))$-competitive for some $\delta>0$ and a function $f(m)=o(m)$.
Let $X^{\prime}$ be a sequence that consists of $s$ repetitions of the atomic weakly-stable sequence that corresponds to the recursive tree $F_{r}$. It follows that $\hat{c}(GF,X^{\prime})\leq(2-\delta)\cdot\hat{c}(OPT,X^{\prime})+\frac{f(|X^{\prime}|)}{|X^{\prime}|}$.
By Lemma 3.22, we can choose $r$ large enough such that $\hat{c}(GF,X^{\prime})>6-\delta$, and regardless of $r$, $\hat{c}(OPT,X^{\prime})<3$. Then, since $f$ is sub-linear, we can choose the number of repetitions $s$ to be large enough such that $\frac{f(|X^{\prime}|)}{|X^{\prime}|}<2\delta$. But then we also get that $\hat{c}(GF,X^{\prime})<(2-\delta)\cdot 3+2\delta=6-\delta$, which is a contradiction.
By
Analyzing mixed-stable sequences we proved a lower bound of $2$ on the competitve ratio of $GF$. Theorem 3.25 gives an upper bound.
Theorem 3.25.
Let $X$ be a mixed-stable sequence and let $T$ be the tree that corresponds to it. Then $cost(GF,X,T)<c\cdot cost(OPT,X,T)$ for $c=\frac{5}{2}$.
If $X$ is strongly-stable, then $c=2$.
We defer the proof of Theorem 3.25 to Appendix A.2.
The upper-bound in Theorem 3.25 is clearly not tight, since in the proof of Theorem 3.25 we neglected a term using the inequality $\hat{c}(GF,X)\leq\frac{2}{\alpha}\cdot\hat{c}(OPT,X)-\frac{1}{\alpha}\big{(}1-\frac{n-1}{2m}\big{)}<\frac{2}{\alpha}\cdot\hat{c}(OPT,X)$, for a constant $\alpha$. The lack of tightness is more prominent when $\hat{c}(OPT,X)$ is small, like in the sequence studied in Lemma 3.22 (for Theorem 3.1). We suspect that the lower bound in Theorem 3.1 is tight, and more strongly, that the $F$-tree pattern is the best pattern to use. This is based on studying several other recursive patterns, including those in Figure 4 and Figure 6: None was stronger, and it also seems that patterns with large costs do not “compensate” with large enough promotions.
As a closing remark to the multiplicative results, we note that by the static optimality theorem for $GG$ [9], competitive analysis against a static
algorithm (i.e. an algorithm that does not change its initial tree)
cannot show a super-constant lower bound.
Concretely, the theorem states that $cost(GF,X)\equiv cost(GG,X)=O(m+\sum_{i=1}^{n}{n_{i}\lg\frac{m}{n_{i}}})$
and one can verify that the actual constants are $5m+6\sum_{i=1}^{n}{n_{i}\lg{\frac{m}{n_{i}}}}$. This bound can be re-written as $5m+6m\cdot H_{2}(X)$ where $H_{2}(X)=\sum_{i=1}^{n}{\frac{n_{i}}{m}\lg\frac{m}{n_{i}}}$ is the base-$2$ entropy of the frequencies of the values in $X$. By [17], $cost(OPT^{s},X)\geq m\cdot\frac{H_{2}(X)}{\lg 3}$ where $OPT^{s}$ is the static optimum, and therefore $cost(GF,X)\leq(5+6\lg 3)\cdot cost(OPT^{s},X)$.
Thus, no static argument can show a lower bound larger than $\approx 11.59$.
3.4 Additive Lower Bounds for GF
In this section we move on to analyze the additive gap between $GF$ and $OPT$. For this, we construct and analyze more elaborate patterns of recursively-defined trees, in order to get a large average promotion when optimizing the structure of the trees. The analysis is more involved since we cannot simply assume that the depth of the recurrence, $r$, approaches infinity. Here $n$ is a function of $r$ and the difference of cost can be meaningful in terms of $n$ only if $n$ is finite.
Definition 3.26.
For $k\geq 2$, and $r\geq 0$ we define a $(k,r)$-tree $T_{r}$ as follows. The tree is recursive of depth $r$ (as in Definition 3.19), such that its trunk is composed of a root and a left-chain of length $k-1$ that starts in the right-child of the root. The left child of the deepest node of the trunk is an actual leaf, and the rest of the leaves are $T_{r-1}$ subtrees. $T_{0}$ is a single node. See Figure 6. When $k$ is clear from the context, we also refer to the tree as $T_{r}$.
Observe that the tree $F_{r}$ that was used to prove Theorem 3.1 is in fact a $(k,r)$-tree with $k=2$. When we conclude the analysis, we will get the two ends of a “tradeoff” such that on the one end we have a relatively high cost ratio, and on the other a relatively high cost difference. Moreover, we will show that the higher the difference of costs on a sequence induced by $(k,r)$-tree, the closer the cost ratio is to $1$ (comparing $GF$ to $OPT$).
Lemma 3.27.
The depth of a $(k,r)$-tree is $k\cdot r$, and its left-most node is at depth $r$.
Proof 3.28.
Trivial by induction: For $r=0$, the deepest node is the root, at depth $0$. For $r\geq 1$, observe that the deepest node belongs to the deepest subtree $T_{r-1}$, which is rooted at depth $k$ since the path to it includes $k$ trunk nodes. Similarly, the depth of the left-most node is increased by $1$ per recursive level of the tree.
Lemma 3.29.
Let $T_{r}$ be a $(k,r)$-tree. Then $|T_{r}|=(2+\frac{2}{k-1})k^{r}-(1+\frac{2}{k-1})$ where $|T_{r}|$ is the number of nodes in $T_{r}$. In rougher terms, $|T_{r}|=\Theta(k^{r})$.
Proof 3.30.
Denote $n_{r}=|T_{r}|$. By definition, $n_{0}=1$ and $n_{r}=(k+1)+k\cdot n_{r-1}$. Hence, by Lemma 3.20 (with $\gamma=0$): $n_{r}=\frac{k+1}{1-k}(1-k^{r})+k^{r}=(2+\frac{2}{k-1})k^{r}-(1+\frac{2}{k-1})$.
Lemma 3.31.
Let $X$ be any mixed-stable sequence corresponding to a $(k,r)$-tree $T_{r}$.
Denote the average weighted promotion possible in $T_{r}$ by $p_{r}$, where weighting is according to the frequency of querying each leaf. Then $p_{r}>k\cdot(1-\alpha^{r})$ for $\alpha=1-\frac{1}{3^{k}}$. In particular, if $X$ is a strongly-stable sequence, then $p_{r}=(k+1)\cdot(1-\alpha^{r})+\delta$ for $\alpha=1-\frac{1}{2^{k}}$ and $0\leq\delta<\alpha^{r}$.
Proof 3.32.
We can promote by $k$ every explicit leaf in every $T_{r^{\prime}}$ for all recursive levels $1\leq r^{\prime}\leq r$, from its location to the root of $T_{r^{\prime}}$. Only nodes that are $T_{0}$ leaves do not contribute an explicit promotion of at least $k$, therefore $p_{r}>k\cdot(1-f)$ where $f$ is the sum of query-frequencies of all $T_{0}$ leaves (the inequality is strict due to unaccounted subtree promotions). To conclude, we argue that $f\leq(1-\frac{1}{3^{k}})^{r}$. The frequency of accessing the explicit leaf of $T_{r}$ is at least $\frac{1}{3^{k}}$ by Lemma 3.14, hence with frequency of at most $1-\frac{1}{3^{k}}$ we query a value in some $T_{r-1}$ subtree. Similarly, within the chosen subtree there is again a relative frequency of at most $1-\frac{1}{3^{k}}$ to query within some $T_{r-2}$ subtree. Overall, since there are $r$ levels of recursion, we conclude that $f\leq(1-\frac{1}{3^{k}})^{r}$.
Proving the second part of the claim required a more careful analysis. We define the following method of promotion, depicted in Figure 6. In the $(k,r)$-tree we promote the (only) explicit leaf to the root, and promote its sibling subtree by $1$.
Then we apply similar promotions recursively within every $(k,r-1)$-subtree. Finally, we promote the left-most node within each $(k,r-1)$-subtree that hangs as a right-subtree from the trunk to the parent of this subtree. Denote the total average (weighted) promotion by $p_{r}$.
Note that it does not matter if we promote the left-most nodes of the right subtrees before or after the recursive promotions, because the total order on the items guarantees that there is only one value that can be put instead of every demoted trunk node, and the recursive promotions within a specific subtree do not change the depth of its leftmost leaf.
The promotion of the explicit leaf of $T_{r}$ saves a cost of $k$ weighted by a factor (query frequency) of $\frac{1}{2^{k}}$. The promotion of the sibling subtree saves $1$ weighted by a factor of $\frac{1}{2^{k}}$. The recursive promotions are $p_{r-1}$ weighted by $\sum_{i=1}^{k}{\frac{1}{2^{i}}}$ (for all the $k$ subtrees), and finally the last promotions are technically negligible (as seen in the analysis below), but for the sake of completeness we consider them in the analysis as well: promoting the left-most node from each subtree saves $(r-1)+1=r$ since the leaf that we promote last is at depth $r-1$ within the recursive subtree, and this promotion is weighted by $\frac{1}{2^{r}}\cdot\sum_{i=2}^{k-1}{\frac{1}{2^{i}}}$ (factor of $\frac{1}{2^{r}}$ follows from Lemma 3.27). We get that:
$p_{r}=\frac{k+1}{2^{k}}+p_{r-1}\cdot\big{(}1-\frac{1}{2^{k}}\big{)}+\frac{r}{2^{r}}\cdot\frac{1}{2}\big{(}1-\frac{1}{2^{k-2}}\big{)}$.
Then by Lemma 3.20, with $\alpha=1-\frac{1}{2^{k}}$ and $\gamma=\frac{1}{2}(1-\frac{1}{2^{k-2}})$, we get:
$$p_{r}=(k+1)\cdot(1-\alpha^{r})+\delta\ \ \ ,\ \ \ \delta\equiv\alpha^{r}\cdot p_{0}+\frac{2\alpha\gamma}{{(2\alpha-1)}^{2}}\cdot\Big{(}\alpha^{r}-\frac{1}{2^{r}}\Big{)}-\frac{\gamma}{(2\alpha-1)}\cdot\frac{r}{2^{r}}$$
It remains to show that $0\leq\delta<\alpha^{r}$. It is simple to see that $\delta=0$ for $k=2$, because then $\gamma=0$ and $p_{0}=0$ is the average weighted promotion in a tree with a single node. For $k\geq 3$, by the definition of $\alpha$ and $\gamma$ we have that
$\frac{2\alpha\gamma}{{(2\alpha-1)}^{2}}=\frac{(2^{k}-1)(2^{k}-4)}{(2^{k}-2)^{2}}=1-\frac{1}{2^{k}-4+\frac{4}{2^{k}}}\in(\frac{3}{4},1)$ and $\frac{\gamma}{2\alpha-1}=\frac{1}{2}-\frac{1}{2^{k}-2}\in[\frac{1}{3},\frac{1}{2})$. Substituting these bounds
and $p_{0}=0$ into the formula for $\delta$
gives $\delta<\alpha^{r}$. Moreover, $\delta$ is positive since
$\delta>\frac{3}{4}(\alpha^{r}-\frac{1}{2^{r}})-\frac{1}{2}\cdot\frac{r}{2^{r}}=\frac{3}{4}(\alpha-\frac{1}{2})\cdot\sum_{i=0}^{r-1}{\alpha^{i}\cdot\big{(}\frac{1}{2}\big{)}^{r-1-i}}-\frac{r}{2^{r+1}}=\frac{3(\alpha-\frac{1}{2})}{2^{r+1}}\cdot\sum_{i=0}^{r-1}{(2\alpha)^{i}}-\frac{r}{2^{r+1}}>\frac{3(\alpha-\frac{1}{2})}{2^{r+1}}\cdot r-\frac{r}{2^{r+1}}=(3(\frac{1}{2}-\frac{1}{2^{k}})-1)\cdot\frac{r}{2^{r+1}}>0$ for $k\geq 3$.
Note that indeed the gain from promoting the left-most node of each subtree is negligible, since the effect is merely having $\gamma\neq 0$, which only contributes $0\leq\delta<\alpha^{r}<1$.
Corollary 3.33.
The average cost-per-query of $GF$ on a strongly-stable sequence induced by a $(k,r)$-tree is larger than the optimal cost by at least $(k+1)\cdot\big{(}1-\big{(}1-\frac{1}{2^{k}}\big{)}^{r}\big{)}$. On any mixed-stable sequence, the difference is at least $k\cdot\big{(}1-\big{(}1-\frac{1}{3^{k}}\big{)}^{r}\big{)}$.
We are ready to prove Theorem 3.2.
See 3.2
Proof 3.34.
Let $X$ be the strongly-stable sequence induced by a $(k,r)$-tree $T_{r}$, and for simplicity assume that the initial tree is $T_{r}$.666We remove this assumption in Remark A.3. By Lemma 3.29,
$n=(2+\frac{2}{k-1})k^{r}-(1+\frac{2}{k-1})$ therefore $\lg\lg n=\lg r+\lg\lg k+O(1)$.777$k\geq 2\Rightarrow k^{r}\leq n<4k^{r}\Rightarrow\lg n=r\lg k+c$ for $c\in[0,2)$, and so $\lg\lg n=\lg r+\lg\lg k+O(1)$.
By Corollary 3.33,
$\hat{c}(GF,X)-\hat{c}(OPT,X)\geq\Delta\equiv(k+1)\cdot(1-(1-\frac{1}{2^{k}})^{r})$. By choosing $r=2^{k}$ we get that $\Delta=(k+1)\cdot(1-(1-\frac{1}{2^{k}})^{2^{k}})\approx(1-\frac{1}{e})\cdot(k+1)$.888The approximation is off by less than $10\%$ for $k\geq 2$. ($60\%$ and $20\%$ for $k=0,1$ respectively.) We also get that $\lg\lg n=k+\lg\lg k+O(1)$, therefore $\Delta\approx(1-\frac{1}{e})\lg\lg n$ and we conclude that
$\hat{c}(GF,X)-\hat{c}(OPT,X)\geq\Omega(\lg\lg n)$.
By Lemma 3.10 the length of the atomic strongly-stable sequence of $T_{r}$ is $m=2^{d(T_{r})}$, hence $m=2^{rk}$ by Lemma 3.27. By Lemma 3.29, $\frac{n+(1+2/(k-1))}{2+2/(k-1)}=k^{r}=2^{r\lg k}$. Together we get that $m=2^{rk}=2^{(r\lg k)\cdot(k/\lg k)}=\Big{(}\frac{n+(1+2/(k-1))}{2+2/(k-1)}\Big{)}^{(k/\lg k)}=n^{\Theta(\frac{\lg\lg n}{\lg\lg\lg n})}$.
Remark 3.35.
In the proof of Theorem 3.2, the sequence $X$ does not have to be strongly-stable, and any mixed-stable sequence $X$ induced by a $(k,r)$-tree $T_{r}$ works as well. Indeed, Corollary 3.33 guarantees that $\hat{c}(GF,X)-\hat{c}(OPT,X)\geq\Delta$ for $\Delta=k\cdot\big{(}1-\big{(}1-\frac{1}{3^{k}}\big{)}^{r}\big{)}$, and then by choosing $r=3^{k}$ we get that $\Delta=\Theta(k)$, and $k=\Theta(\lg\lg n)$, and $m=2^{\Theta(rk)}=n^{\Theta(\frac{\lg\lg n}{\lg\lg\lg n})}$.
Remark 3.36.
The choice of $r=2^{k}$ in the proof of Theorem 3.2 maximizes our lower bound on the additive gap $cost(GF,X)-cost(OPT,X)$ (up to constants) for our $(k,r)$-trees. Indeed, revisiting the proof, we have that $\Delta$ and $n$ are both functions of $k$ and $r$, and we need to choose $r$ and $k$ to maximize $\Delta$ as a function of $n$. Note that $\Delta=O(k)$ regardless of $r$, and $\lg\lg(n)=\lg r+\lg\lg k+O(1)$. To simplify and eliminate a parameter we define $r=2^{k}\cdot f(k)$ for some monotone function $f$. Now we get simplified relations: $\Delta=(k+1)\cdot(1-(1-\frac{1}{2^{k}})^{2^{k}f(k)})\approx(k+1)\cdot(1-e^{-f(k)})$ and $\lg\lg n=k+\lg f(k)+\lg\lg k+O(1)$. Consider the following two cases.
•
If $f(k)=\Omega(1)$: then $\exists c\in\mathbb{R}$ such that $\forall k\geq 1:\lg f(k)\geq c$, and therefore $\lg\lg n=\Omega(k)$, written differently $k=O(\lg\lg n)$, which yields $\Delta=O(k)=O(\lg\lg n)$.
•
If $f(k)=o(1)$: Being $o(1)$ means that $\lim_{k\to\infty}{f(k)}=0$, so for sufficiently large values of $k$ we can use the approximation $e^{x}\approx 1+x$ (that holds for small $x$) to get: $\Delta\approx(k+1)\cdot f(k)=\frac{k+1}{1/f(k)}$. If $\frac{1}{f(k)}$ grows faster than $(k+1)$, we get $\Delta=O(1)$ which does not even grow with $n$. Therefore $\frac{1}{f(k)}$ is increasing, but at a sub-linear rate. Recall that $\lg\lg n=k-\lg\frac{1}{f(k)}+\lg k+O(1)$. Since $\frac{1}{f(k)}$ is sub-linear, we get that
$k=\Theta(\lg\lg n)$, which yields $\Delta=O(k)=O(\lg\lg n)$.
Corollary 3.37.
$GF$ is not $(1,O(m))$-competitive. If the multiplicative term is $1$, then the additive term is at least $\Omega(m\cdot\lg\lg n)$.
We have yet to analyze the cost of $OPT$ on a strongly-stable sequence $X$ corresponding to a $(k,r)$-tree that produces the gap of $\Omega(m\cdot\lg\lg n)$ in Theorem 3.2. Allegedly, if the cost is cheap, say linear, we would get a large competitive ratio as well. However, by Theorem 3.25 we expect a competitive ratio of at most $2$, and therefore we can conclude without further analysis, that $cost(OPT,X)=\Omega(m\cdot\lg\lg n)$.
In fact, we prove that $cost(OPT,X)=\Theta(m\cdot\frac{\lg n}{\lg\lg\lg n})$.
It follows that the competitive ratio deteriorates when the additive gap increases.
Lemma 3.38.
Define the constants $\alpha\equiv 1-\frac{1}{2^{k}}$ and $\beta\equiv\sum_{j=1}^{k}{\frac{j}{2^{j}}}+\frac{k+1}{2^{k}}$. Let $X$ be a strongly-stable sequence induced by a $(k,r)$-tree. Then $\hat{c}(GF,X)=2^{k}\cdot\beta\cdot(1-\alpha^{r})+\alpha^{r}$. In asymptotic terms: $\hat{c}(GF,X)=\Theta(2^{k}\cdot(1-\alpha^{r}))$.
Proof 3.39.
We write a recurrence for the average cost, $c_{r}$, of $GF$ on the strongly-stable sequence induced by $T_{r}$. We have $c_{0}=1$, and
$$c_{r+1}=\frac{1+k}{2^{k}}+\sum_{j=1}^{k}{\frac{j+c_{r}}{2^{j}}}=\Big{(}1-\frac{1}{2^{k}}\Big{)}c_{r}+\sum_{j=1}^{k}{\frac{j}{2^{j}}}+\frac{1+k}{2^{k}}\equiv\alpha\cdot c_{r}+\beta$$
($\frac{1+k}{2^{k}}$ is due to the actual leaf, and the summation is the contribution of all the $T_{r}$ subtrees.) By Lemma 3.20 (with $\gamma=0$), $c_{r}=\frac{\beta}{1-\alpha}(1-\alpha^{r})+\alpha^{r}\cdot c_{0}=2^{k}\cdot\beta(1-\alpha^{r})+\alpha^{r}$. Since $\alpha=1-\frac{1}{2^{k}}\in[\frac{3}{4},1)$ clearly $\alpha^{r}<1$.
Furthermore, $\beta=\Theta(1)$.
To see this note that $\beta$ only depends on $k$. Denote $\beta=\beta(k)$ and observe that: $\beta(k+1)-\beta(k)=\big{(}\frac{k+1}{2^{k+1}}+\frac{k+2}{2^{k+1}}\big{)}-\frac{k+1}{2^{k}}=\frac{1}{2^{k+1}}$.
Therefore, $\beta(k)=\beta(2)+\sum_{i=3}^{k}{\Big{(}\beta(i)-\beta(i-1)\Big{)}}=\Big{(}\frac{1}{2}+\frac{2}{4}+\frac{3}{4}\Big{)}+\sum_{i=3}^{k}{\frac{1}{2^{i}}}=2-\frac{1}{2^{k}}$, and $\beta(k)\in[\frac{7}{4},2)\Rightarrow\beta=\Theta(1)$. Because $2^{k}\cdot(1-\alpha^{r})\geq 2^{k}\cdot(1-\alpha)=1>\alpha^{r}$, we conclude that $c_{r}=\Theta(2^{k}\cdot(1-\alpha^{r}))$.
Lemma 3.40.
Let $X$ be a sequence from the family of sequences in Theorem 3.2, then $cost(OPT,X)=\Theta(m\cdot\frac{\lg n}{\lg\lg\lg n})$.
Proof 3.41.
Let $X$ be a strongly-stable sequence induced by querying a $(k,r)$-tree. We know that $\frac{1}{2}\hat{c}(GF,X)<\hat{c}(OPT,X)\leq\hat{c}(GF,X)$ where the lower-bound is by Theorem 3.25. Therefore, $\hat{c}(OPT,X)=\Theta(2^{k}\cdot(1-\alpha^{r}))$ by Lemma 3.38. By Lemma 3.29, $\lg n=r\cdot\lg k+O(1)$, or $r=\frac{\lg n-O(1)}{\lg k}$. When we substitute $r=2^{k}$ as in the proof of Theorem 3.2, we get that $(1-\alpha^{r})=\Theta(1)$ and $2^{k}=r=\frac{\lg n-O(1)}{\lg k}=\Theta(\frac{\lg n}{\lg\lg\lg n})$. Therefore, $cost(OPT,X)=\Theta(m\cdot\frac{\lg n}{\lg\lg\lg n})$.
As a concluding remark, we recall that the $F_{r}$-tree is a $(k,r)$-tree for $k=2$.
If we substitute $k=2$ in the formula of Lemma 3.38 we get that $\alpha=\frac{3}{4}$, $\beta=\frac{7}{4}$, and $\hat{c}(GF,X)=7\cdot(1-(3/4)^{r})+(3/4)^{r}$. By Lemma 3.31, the average promotion is $3\cdot(1-(3/4)^{r})$ (for $k=2$, we have $\delta=0$). These values are the strongly-stable analogues of Lemma 3.22, and can be used to show a weaker lower bound of $\frac{7}{4}$, on the competitive ratio of $GF$.
4 Conclusions and Open Questions
In this paper we gave improved lower bounds on the competitiveness of the Greedy Future ($GF$) algorithm for serving a sequence of queries by a dynamic binary search tree (BST). In contrast to many of the previous results on $GF$ that are obtained using the geometric-view by studying the equivalent Geometric Greedy ($GG$) algorithm, we used the
standard “tree-view” and the treap-based definition of $GF$. We showed that the competitive ratio of $GF$ is at least $2$, and that there are sequences $X\in[n]^{m}$ for which the cost difference (additive gap) between $GF$ and $OPT$ is $\Omega(m\cdot\lg\lg n)$. These lower bounds enabled us to show that if $GF$ is approximately-monotone (Definition 3.5) with some constant $c$ then $c\geq 2$. Also, the lower bounds show that
the cost of $GF$ on a sequence
compared to its cost on its reverse, may differ by a factor as close as we like to $2$, or by a difference of $\Omega(m\cdot\lg\lg n)$. In contrast,
the cost of $OPT$ on a sequence compared to its reverse may differ by at most $n$.
Our results give new insights on the “tradeoff” between the additive term and the multiplicative term in the competitiveness of $GF$, showing that the multiplicative term is typically larger when the total cost of the algorithm on the sequence is smaller. Indeed, our best multiplicative term is achieved for a sequence whose average cost per query is $6$. This tradeoff is not surprising since a fixed difference implies a larger ratio when the quantities are small. It may be interesting to figure out if this tradeoff hints of some underlying property of $GF$, or is just an artifact of our technique that requires high costs on average per query in order to increase the additive gap between $GF$ and $OPT$.
Clearly, these improved lower bounds still don’t settle the deeper question of whether $GF$ (and $GG$) is dynamically-optimal. Our techniques focused on a smaller family of sequences which we named mixed-stable sequences, whereas “most” sequences are not stable.
While it is possible that an improved lower bound (larger than $2$) can be found by a more clever pattern of mixed-stable sequences, it seems more likely to be found by analyzing sequences for which the tree maintained by $GF$ is not static. In addition, we note that $GF$ was not investigated too deeply directly, as most of the work has been done in the geometric view with respect to its counterpart $GG$. Therefore, studying other problems in tree-view may give complementing insights. One such problem is the deque conjecture, which has been partially settled for $GG$, in the case when deletions are only allowed on the minimum item [2].
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Appendix A Appendix: Deferred Proofs and Discussions
A.1 Enforcing a Stable Tree for GF
We describe how to restructure any initial tree, to a desired tree, when $GF$ is considered. The initial tree cannot simply be re-organized since $GF$ updates the tree in a specific way following each query.
Let $P\circ X$ denote the concatenation of the sequences
$P$ and $X$.
Note that even if $P$ enforces a desired tree when served alone, serving $P\circ X$ may give a different tree following $P$ when $X$ starts. The reason for this is that $GF$ restructures the tree while serving $P$ according to future queries, therefore the existence of $X$ may affect its decisions while serving $P$. Nevertheless, we present a simple technique to enforce a tree for $GF$.
Theorem A.1.
For any tree $T$ there is a sequence $S(T)$ such that: (1) $|S(T)|=O(n\cdot d(T))$, and more precisely, $|S(T)|\leq\min\Big{(}3n(d(T)-\lfloor\lg n\rfloor+1),\frac{3}{2}n(n-1)\Big{)}$, and, (2) for any suffix of queries $Y$, when $GF$ serves $S(T)\circ Y$, its tree when is it done with the last query of $S(T)$ is $T$. We say that $S(T)$ enforces $T$.
Proof A.2.
We enforce the structure of $T$ bottom-up, by first ensuring the position of the leaves, and continuing recursively upwards towards the root. Define $T^{[0]}\equiv T$ and $T^{[i+1]}$ is the tree $T^{[i]}$ stripped of all of its leaves, until the final tree $T^{[h]}$ contains only the root. Observe that $h=d(T)$ (the depth of $T$). For each tree $T^{[i]}$ we define $R_{i}$ to be the set of non-leaf nodes in $T^{[i]}$. Note that a non-leaf node may be binary or unary.
We construct $S(T)$ in steps. In step $i\geq 0$ we query $R_{i}$ in two monotonic phases, where the first phase queries every item twice, and the second phase queries every item once. Denote the queries of this step by $S_{i}$. For example, if $R_{0}=\{1,3,5\}$ then we query in the first step: $S_{0}=[1,1,3,3,5,5,1,3,5]$. Note that $S_{h}=\emptyset$ since $R_{h}=\emptyset$ by definition. The resulting sequence $S(T)$ is the concatenation of the queries in all the (non-empty) steps, that is, $S(T)=S_{0}\circ S_{1}\circ\ldots\circ S_{h}$ ($\circ$ for concatenation).
First we analyze $|S(T)|$. Observe that we strip leaves between steps, $|R_{i+1}|\leq|R_{i}|-1$. Initially $|R_{1}|\leq n-1$ hence $|R_{i}|\leq n-i$. Also, $h<n$. Therefore we get that: $|S(T)|=\sum_{i=1}^{h}{|S_{i}|}=\sum_{i=1}^{h}{3|R_{i}|}\leq 3\sum_{i=1}^{n}{(n-i)}=\frac{3}{2}n(n-1)$.
To get the other bound, observe that $|R_{h-i}|\leq 2^{i}-1$, which is meaningful for the last few steps, i.e., for $i\leq\lfloor\lg n\rfloor$. With that in mind:
$|S(T)|=3\sum_{i=1}^{h-\lfloor\lg n\rfloor}{|R_{i}|}+3\sum_{i=h-\lfloor\lg n\rfloor+1}^{h}{|R_{i}|}<3\sum_{i=1}^{h-\lfloor\lg n\rfloor}{n}+3\sum_{j=0}^{\lfloor\lg n\rfloor-1}{2^{j}}<3n(h-\lfloor\lg n\rfloor)+3n$. In conclusion, $|S(T)|<3n(d(T)-\lfloor\lg n\rfloor+1)$.
Next we prove that the tree of $GF$ is exactly $T$ when it finishes serving $S(T)$ regardless of any suffix of queries. The proof is by induction on $d(T)=h$. The base case is for $h=0$. In this case $S(T)=\emptyset$ (because $S_{0}=\emptyset$), while also $T^{[h]}=T^{[0]}=T$ contains only the root. Querying nothing is not a problem since there is a unique tree with a single node, and we are done.
Now, assume that the claim holds for $h=k$. Consider a tree $T$ of depth $k+1$. First we show that when $GF$ finishes processing $S_{0}$, all the leaves of $T$ are leaves of the tree of $GF$, and will never be touched (accessed or otherwise), therefore they remain leaves until $GF$ finishes processing $S(T)$. See Figure 7 for a visualization. Indeed, when we query a value $u$ twice in a row, $GF$ brings $u$ to the root when re-ordering the tree after the first query of the couple. Note that since we query $R_{0}$ monotonically, by the end of the first phase of $S_{0}$ (the queries in pairs), $GF$ has a tree whose left-spine is exactly the items of $R_{0}$. Indeed, each item, in turn, is brought to the root and demotes the previous root to the left. Moreover, no value of $T^{[0]}\setminus R_{0}$ is part of this spine, because of the second phase of queries (monotonously querying the values of $R_{0}$). To see this, note that if on the first phase we touch $v\notin R_{0}$ when $r$ is the root and the pair of queries is to $u$, such that $r<v<u$, and $u$ is the successor of $r$ in $R_{0}$, then the next access to $r$ is closer in the future than that of $v$, so $r$ will indeed be placed as the left-child of the new root $u$ (and $v$ will be the right child of $r$). If $r<u<v$ then $v$ does not interfere with $r$ being the left child of $u$.
Now that we know that all the leaves of $T=T^{[0]}$ are fixed as leaves of the tree of $GF$ when it finishes processing $S_{0}$, we can conclude by induction: $S^{\prime}\equiv S_{1}\circ\ldots\circ S_{k+1}$ is exactly the sequence that enforces $T^{[1]}$, and by our inductive assumption, $T^{[1]}$ is enforced correctly. Since the leaves of $T^{[0]}$ are not touched at all during $S^{\prime}$, we conclude that they must remain hanging off $T^{[1]}$ as “subtrees”. The location of each leaf is uniquely determined, and thus we conclude that $S(T)$ indeed enforces $T=T^{[0]}$.
Adding a prefix to our sequence may affect the competitive ratio. However, once we fixed the stable tree, we can repeat the corresponding stable sequence to “amplify” the original competitive ratio making the effect of the prefix negligible.
One difficulty raised by repetitions is when we care about the length of the sequence in our claim. This is the case in Theorem 3.2 where we claim the existence of a sequence of length $n^{\Theta(\frac{\lg\lg n}{\lg\lg\lg n})}$. In the proof of this theorem we assumed for simplicity that we can choose the initial tree. The following remark shows that indeed we can start with an arbitrary initial tree without weakening the theorem.
Remark A.3.
Let $X$ be an atomic mixed-stable sequence used to prove Theorem 3.2. Consider the sequence $Z=S\circ{X^{n}}$, where $S$ is the prefix (guaranteed by Theorem A.1) that is enforcing the desired “initial” tree $T$ that corresponds to $X$, $X^{n}$ are $n$ repetitions of $X$, and $\circ$ represents concatenation. There are no unary nodes in $T$ and $X$ queries every leaf at least once so $\frac{n}{2}<\frac{n+1}{2}\leq|X|$. Together with Theorem A.1 we get that $n|X|\leq|Z|<n(|X|+\frac{3n}{2})<4n|X|$, therefore we have: $|Z|=\Theta(n|X|)=n^{\Theta(\frac{\lg\lg n}{\lg\lg\lg n})}$ (the second equality is by Theorem 3.2). Since after processing $S$ the tree of $GF$ is fixed: $cost(GF,Z,T_{0})-cost(OPT,Z,T_{0})\geq n\cdot(cost(GF,X,T)-cost(OPT,X,T))$. By the proof of Theorem 3.2 and Remark 3.35: $cost(GF,X,T)-cost(OPT,X,T)=\Omega(|X|\cdot\lg\lg n)$, and putting everything together we get that: $cost(GF,Z,T_{0})-cost(OPT,Z,T_{0})=\Omega(|Z|\cdot\lg\lg n)$. Note that if $|X|=\Omega(n^{2})$ then it suffices to define $Z=S\circ X$ without repetitions and we get that $|Z|=\Theta(|X|)$, and the rest of the arguments remain the same. There may be additional cases where repetitions are not required, e.g., when the depth of $T$ is $\lg n+O(1)$ (in this case, $|S|=O(n)$).
A.2 Omitted Proofs
In this subsection we restate and prove Lemmas and Theorems that were omitted from the main text.
For convenience, we restate the claims
in their original numbering.
The proof of Theorem 3.25 makes use of Wilber’s first bound [22]. We use the original presentation of this bound which is a bit tighter than later simplified versions such as [14].
Definition A.4 (Wilber’s First Bound [22]).
Let $X$ be a sequence of queries, and let $T$ be a static reference tree such that every query of $X$ is in a leaf of $T$. An alternation at an inner node $u$ of $T$ is defined to be two queries closest in time such that one accesses either the left or right subtree of $u$ and the other accesses the other subtree of $u$. Define $ALT(u)$ to be the number of alternations at node $u$.
Then: $cost(OPT,X)\geq m+\frac{1}{2}\sum_{\text{inner\ }u\in T}{ALT(u)}$.
See 3.25
Proof A.5.
We use the tree that corresponds to the mixed-stable sequence as the reference tree for Wilber’s first bound. Arithmetic manipulations will yield an expression that we can tie to the cost of $GF$, according to the claim.
Let $X$ be a mixed-stable sequence, with a corresponding tree $T$. Let $S$ be the set of values that are in the leaves of $T$, and let $U$ be the set of inner nodes, $|U|=\frac{n-1}{2}$. We also denote by $A(i)$ the set of proper ancestors of $i$. By the definition of the cost of a static tree, we know that $\hat{c}(GF,X)=\sum_{i\in S}{(d(i)+1)\cdot f(i)}$ where $d(i)$ is the depth of $i$ and $f(i)$ is the frequency of accessing $i$. We extend $f(u)$ to refer to the frequency of visiting any node $u$. Note that $f(u)=\sum_{i\in S\wedge u\in A(i)}{f(i)}$ and that $\sum_{i\in S}{f(i)}=1$.
Now consider Wilber’s bound for $X$, with $T$ as the reference tree. We can use $T$ as the reference tree since $X$ only accesses leaves of $T$, by definition. We also denote $\alpha_{u}\equiv\frac{ALT(u)+1}{f(u)\cdot m}$ ($ALT(u)$ is defined in Defintion A.4, and note that $0\leq ALT(u)\leq f(u)\cdot m-1$). We have $\alpha_{u}\in(0,1]$, where $\alpha_{u}=1$ corresponds to fully alternating accesses to the subtree rooted at $u$.
The lower bound is $cost(OPT,X)\geq m+\frac{1}{2}\sum_{u\in U}{(ALT(u)+1)}-\frac{|U|}{2}=\frac{m}{2}+\frac{m}{2}(1+\sum_{u\in U}{\alpha_{u}\cdot f(u)})-\frac{n-1}{4}=\big{(}\frac{m}{2}-\frac{n-1}{4}\big{)}+\frac{m}{2}\sum_{i\in S}{(1+\sum_{u\in A(i)}{\alpha_{u}})f(i)}$.
Let $\alpha\leq\min_{u\in U}{\alpha_{u}}$, we get that $\hat{c}(OPT,X)\geq\big{(}\frac{1}{2}-\frac{n-1}{4m}\big{)}+\frac{\alpha}{2}\sum_{i\in S}{(d(i)+1)\cdot f(i)}=\frac{\alpha}{2}\hat{c}(GF,X)+\big{(}\frac{1}{2}-\frac{n-1}{4m}\big{)}$ where the equality holds since $GF$ maintains a static tree. Thus $\hat{c}(GF,X)\leq\frac{2}{\alpha}\cdot\hat{c}(OPT,X)-\frac{1}{\alpha}\big{(}1-\frac{n-1}{2m}\big{)}<\frac{2}{\alpha}\cdot\hat{c}(OPT,X)$.
In order to choose a suitable $\alpha$, recall that a strongly-stable node $u$ has a coefficient of $\alpha_{u}=1$, which means that for strongly-stable sequences, in which all inner nodes are stable, we can pick $\alpha=1$ and conclude that $\hat{c}(GF,X)<2\cdot\hat{c}(OPT,X)$. If $u$ is a weakly-stable node, then its coefficient is $\alpha_{u}=\frac{2}{3}$.
So for a mixed-stable sequence we can naively pick $\alpha=\frac{2}{3}$, resulting in $\hat{c}(GF,X)<3\cdot\hat{c}(OPT,X)$.
In order to improve from $3$ to $\frac{5}{2}$, we observe that by definition, every weakly-stable node has a strongly-stable child. Let $u$ be a weakly-stable node and let $w$ be its (strongly-stable) favored-child (recall Definition 3.8). Since $ALT(u)=ALT(w)$ (by definition of the access pattern in $u$), we can present Wilber’s bound differently, summing $(ALT(u)+1)\cdot(1+\beta)+(ALT(w)+1)\cdot(1-\beta)$ instead of $(ALT(u)+1)+(ALT(w)+1)$. We get modified coefficients $\alpha^{\prime}_{u}=\frac{(ALT(u)+1)\cdot(1+\beta)}{m\cdot f(u)}=\alpha_{u}\cdot(1+\beta)=\frac{2(1+\beta)}{3}$ and similarly $\alpha^{\prime}_{w}=\alpha_{w}(1-\beta)=(1-\beta)$. Choosing $\beta=\frac{1}{5}$ balances the coefficients: $\alpha^{\prime}_{u}=\alpha^{\prime}_{w}=\frac{4}{5}$. Now we can choose $\alpha=\frac{4}{5}$, and get $\hat{c}(GF,X)<\frac{5}{2}\cdot\hat{c}(OPT,X)$ for mixed-stable sequences.
See 3.3
Proof A.6.
Denote the initial tree by $T_{0}$. Let $Z$ be the weakly-stable sequence used for proving Theorem 3.1. Let $T_{P}$ be the tree that corresponds to $Z$ and $T_{Q}$ the optimized tree, in which the leaves are promoted as in Lemma 3.22. Let $P$ and $Q$ be the sequences that enforce $T_{P}$ and $T_{Q}$ by Theorem A.1, respectively. Note that $\epsilon$ determines $Z$, $P$ and $Q$ since it tells us how close to a ratio of $2$ we need to get.
Revisit Figure 5 to see the (recursive) structures of $T_{P}$ (on the left) and $T_{Q}$ (on the right, post-promotions). Observe that $T_{P}$ remains static when $GF$ serves $Z$ with it, by definition. Moreover, $T_{Q}$ remains static when $GF$ serves $Z$ with it. Indeed, let $r$ be the root of $T_{Q}$. $Z$ queries the item in $r$ every third access and the other accesses are alternating between its left and right subtrees, hence $r$ remains the root of $T_{Q}$. The rest of $T_{Q}$ remains static recursively.
Define $X=P\circ Q\circ Z^{k}$ for a large $k$, and $X^{\prime}=P\circ Z^{k}\subset X$ ($\circ$ for concatenation). Since $GF$ does not change $T_{P}$ and $T_{Q}$ while serving $Z$ we get that
$\frac{cost(GF,X^{\prime})}{cost(GF,X)}=\frac{cost(GF,P,T_{0})+k\cdot cost(GF,Z,T_{P})}{cost(GF,P\circ Q,T_{0})+k\cdot cost(GF,Z,T_{Q})}$. This ratio approaches $\frac{cost(GF,Z,T_{P})}{cost(GF,Z,T_{Q})}$ for large enough $k$, and since $T_{p}$ and $T_{Q}$ are exactly the trees used in the proof of Lemma 3.22, we conclude that we can make the resulting ratio as close to $2$ as we like (choosing $Z,P,Q$ according to the desired $\epsilon$).
The proof for $Y$ and $Y^{\prime}$ is similar. We define $Z$ to be the strongly-stable sequence used in Theorem 3.2, and define the appropriate tree-enforcing sequences $P$ and $Q$ by Theorem A.1. Revisit Figure 6 to see the (recursive) structures of $T_{P}$ (on the left) and $T_{Q}$ (on the right, post-promotions). We set $Y=P\circ Q\circ Z^{k}$ and $Y^{\prime}=P\circ Z^{k}$. The main difference is in the argument of why $GF$ does not change $T_{Q}$ when serving $Z$ ($T_{P}$ is static by definition). For this, observe that the root of $T_{Q}$, denote it $r$, is the left-most leaf in the right subtree of $T_{P}$. This means that the access pattern at $r$ is alternating between its left subtree and its right subtree including itself, thus again we conclude that $r$ remains at the root of $T_{Q}$, and the rest of $T_{Q}$ remains static recursively. Therefore, $cost(GF,Y^{\prime})-cost(GF,Y)\geq k\cdot\big{(}cost(GF,Z,T_{P})-cost(GF,Z,T_{Q}))-cost(GF,P\circ Q,T_{0})=\Omega(m\cdot\lg\lg n)$ for large enough $k$.
See 3.4
Proof A.7.
The proof is similar to that of Theorem 3.3, and we define $Z$, $T_{0}$, $P$, $T_{P}$, $Q$ and $T_{Q}$ the same way. Here we define $X=Q\circ(rev(Z))^{k+1}\circ rev(P)$ and $Y=Q\circ(rev(Z))^{k+1}\circ rev(P)$, for a large $k$. Recall that $Z$ is different between $X$ (by Theorem 3.1) and $Y$ (by Theorem 3.2).
We claim that $T_{Q}$ remains static when $GF$ serves $rev(Z)$ over it, rather than $Z$, by the same argument as in the proof of Theorem 3.3, because the interleaving pattern in the root is preserved under reversal. Moreover, $cost(GF,rev(Z),T_{Q})=cost(GF,Z,T_{Q})$ because the cost on a static tree depends only on the access frequencies. Putting everything together, we get: $\frac{cost(GF,rev(X))}{cost(GF,X)}=\frac{cost(GF,P,T_{0})+k\cdot cost(GF,Z,T_{P})+cost(GF,Z\circ rev(Q),T_{P})}{cost(GF,Q,T_{0})+k\cdot cost(GF,rev(Z),T_{Q})+cost(GF,rev(Z)\circ rev(P),T_{Q})}$. Note that the suffix contains one repetition of $Z$ so that the rest of it ($rev(P)$ or $rev(Q)$) does not affect the restructuring decisions of $GF$ during the earlier repetitions of $Z$. The limit of this ratio for large $k$ is $\frac{cost(GF,Z,T_{P})}{cost(GF,Z,T_{Q})}$. We finish the argument as in the proof of Theorem 3.3.
In the case of $Y$, we get that $cost(GF,Y^{\prime})-cost(GF,Y)\geq k\cdot\big{(}cost(GF,Z,T_{P})-cost(GF,Z,T_{Q}))-\big{(}cost(GF,Q,T_{0})+cost(GF,rev(Z)\circ rev(P),T_{Q})\big{)}=\Omega(m\cdot\lg\lg n)$ for large enough $k$. |
Pluripotential theory on Teichmüller space I
– Pluricomplex Green function –
Hideki Miyachi
Division of Mathematical and Physical Sciences,
Graduate School of Natural Science & Technology,
Kanazawa University,
Kakuma-machi, Kanazawa,
Ishikawa, 920-1192, Japan
miyachi@se.kanazawa-u.ac.jp
(Date:: December 2, 2020)
Abstract.
This is the first paper in a series of investigations of the pluripotential theory on Teichmüller space.
The main purpose of this paper is to give an alternative approach to the Krushkal formula of the pluricomplex Green function on Teichmüller space.
We also show that Teichmüller space carries a natural stratified structure of real-analytic submanifolds defined from the structure of singularities of the initial differentials of the Teichmüller mappings from a given point.
We will also give a description of the Levi form of the pluricomplex Green function using the Thurston symplectic form via the Dumas symplectic structure on the space of holomorphic quadratic differentials.
Key words and phrases:Singular Euclidean structures, Teichmüller space, Teichmüller distance, Levi forms, Pluricomplex Green functions
2010 Mathematics Subject Classification: 32G05, 57M50, 32G15,32U35
2010 Mathematics Subject Classification: 30F60, 30F40, 30F25, 32G15, 31B05, 31B10, 32U05, 32U35
This work is partially supported by JSPS KAKENHI Grant Numbers
16K05202,
16H03933,
17H02843
1. Introduction
This is the first paper in a series of investigations of the pluripotential theory on Teichmüller space.
The main purpose of this paper is to give a charaterization of the pluricomplex Green function on Teichmüller space.
The characterzation given here is first discussed by Krushkal in [28].
We prove the characerization from a different approach.
In the second paper,
we will establish the Poisson integral formula for pluriharmonic functions on Teichmüller space which are continuous on the Bers compactification.
The characterization of the pluricomplex Green function given here plays a crucial rule in the second paper.
This result is announced in [34].
Let $\mathcal{T}_{g,m}$ be the Teichmüller space of Riemann surfaces of analytically finite type $(g,m)$, and $d_{T}$ the Teichmüller distance on $\mathcal{T}_{g,m}$.
Teichmüller space $\mathcal{T}_{g,m}$ admits a natural complex structure.
Royden [39] proved that the Teichmüller distance coincides with the Kobayashi distance under the complex structure.
Krushkal [27] showed that Teichmüller space is hyperconvex.
By Bers’ theorem and Nehari-Kraus’ theorem,
$\mathcal{T}_{g,m}$ is biholomorphic to a bounded domain in the complex Euclidean space (cf. [19, Theorem 6.6]).
Therefore, from Demailly’s theory [6], $\mathcal{T}_{g,m}$ admits a unique pluricomplex Green function $g_{\mathcal{T}_{g,m}}(x,y)$ (cf. §7.1).
1.1. The Krushkal formula
The aim of this paper is
to give an alternate proof of the Krushkal formula
as follows.
Theorem 1 (Pluricomplex Green function on $\mathcal{T}_{g,m}$).
The pluricomplex Green function
on $\mathcal{T}_{g,m}$ satisfies
(1.1)
$$g_{\mathcal{T}_{g,m}}(x,y)=\log\tanh d_{T}(x,y)$$
for $x,y\in\mathcal{T}_{g,m}$.
From Klimek’s work [22],
it suffices to show that
the right-hand side of (1.1) is plurisubharmonic
(cf. §7.1).
To show this,
Krushkal applied Poletskii’s characterization of the pluricomplex Green function
(cf. [37]).
Our strategy is a more direct method.
Indeed,
we calculate the Levi form of
the log-tanh of the Teichmüller distance
at generic points,
and to check the non-negative definiteness
(§7.6).
From the calculation,
we deduce that
the Levi-form of the pluricomplex Green function is described by the Thurston symplectic
form on the space $\mathcal{MF}$ of measured foliations
via the Dumas symplectic structure (cf. [9])
on the space of holomorphic quadratic differentials.
This description induces a condition for deformations of Teichmüller mappings from a fixed basepoint
to complex-analytically varying targets
from the topological aspect in Teichmüller theory
(cf. §7.7).
1.2. Stratification of Teichmüller space and Removable singlarities
Dumas [9] gave a complex-analytic stratification in the space $\mathcal{Q}_{x_{0}}$
of (non-zero) holomorphic quadratic differentials
on $x_{0}\in\mathcal{T}_{g,m}$
in terms of the structure of singularities.
(cf. §6.1).
Sending the stratification on $\mathcal{Q}_{x_{0}}$ by the Teichmüller homeomorphism
(§2.2.2),
we obtain a topological stratification on $\mathcal{T}_{g,m}-\{x_{0}\}$.
The top stratum $\mathcal{T}_{\infty}$ consists of $x\in\mathcal{T}_{g,m}-\{x_{0}\}$
such that the initial differential of the Teichmüller mapping from $x_{0}$
to $x$ is generic.
We will show that the induced stratification on
$\mathcal{T}_{g,m}-\{x_{0}\}$ is a real-analytic statification
in the sense that each stratum is a real-analytic submanifold
(cf. Theorem 5).
Applying the stratification,
we shall show the following,
which is crucial in our proof of Theorem 1
(cf. §6.3).
Theorem 2 (Non-generic strata are removable).
A function of class $C^{1}$ on $\mathcal{T}_{g,m}-\{x_{0}\}$ is plurisubharmonic on $\mathcal{T}_{g,m}$ if
it is plurisubharmonic on the top stratum $\mathcal{T}_{\infty}$
and bounded above around $x_{0}$.
1.3. About the paper
This paper is organized as follows.
From §2 to §4,
we recall the basic notion and properties in Teichmüller theory.
In §5,
we discuss the deformation of singular Euclidean structures
associated to the Teichmüller deformations from a fixed point $x_{0}\in\mathcal{T}_{g,m}$.
In §6,
we will give the stratification on $\mathcal{T}_{g,m}-\{x_{0}\}$.
We show Theorem 1
and discuss the topological description of the Levi form in §7.
Acknowledgements
The author thanks Professor Ken’ichi Ohshika for fruitful discussions.
The author also thanks Professor Masanori Adachi for indicating him
to Blanchet’s and Chirka’s papers [2] and [5],
and Professor Hiroshi Yamaguchi for his warm advices and encouragements.
2. Teichmüller theory
Let $\Sigma_{g,m}$ be a closed orientable surface
of genus $g$ with $m$-marked points with $2g-2+m>0$ (possibly $m=0$).
In this section,
we recall basics in Teichmüller theory.
For reference,
see [7],
[12] ,
[18],
[19],
and [35]
for instance.
2.1. Teichmüller space
Teichmüller space $\mathcal{T}_{g,m}$
is the set of equvalence classes
of marked Riemann surfaces of type $(g,m)$.
A marked Riemann surface $(M,f)$ of type $(g,m)$
is a pair of a Riemann surface $M$ of analytically finite type $(g,m)$
and an orientation preserving homeomorphism
$f\colon\Sigma_{g,m}\to M$.
Two marked Riemann surfaces $(M_{1},f_{1})$ and $(M_{2},f_{2})$
of type $(g,m)$ are
(Teichmüller) equivalent if there is a conformal mapping
$h\colon M_{1}\to M_{2}$ such that $h\circ f_{1}$ is homotopic to $f_{2}$.
The Teichmüller distance $d_{T}$ is a complete distance
on $\mathcal{T}_{g,m}$ defined by
$$d_{T}(x_{1},x_{2})=\frac{1}{2}\log\inf_{h}K(h)$$
for $x_{i}=(M_{i},f_{i})$ ($i=1,2$),
where the infimum runs over all quasiconformal mapping
$h\colon M_{1}\to M_{2}$ homotopic to $f_{2}\circ f_{1}^{-1}$,
and $K(h)$ is the maximal dilatation of a quasiconformal mapping $h$.
2.2. Quadratic differentials and Infinitesimal complex structure on $\mathcal{T}_{g,m}$
For $x=(M,f)\in\mathcal{T}_{g,m}$,
we denote by $\mathcal{Q}_{x}$ be the complex Banach space
of holomorphic quadratic differentials $q=q(z)dz^{2}$
on $M$ with $L^{1}$-norm
$$\|q\|=\int_{M}|q(z)|\frac{\sqrt{-1}}{2}dz\wedge d\overline{z}<\infty.$$
From the Riemann-Roch theorem,
the space $\mathcal{Q}_{x}$ is isomorphic to $\mathbb{C}^{3g-3+m}$.
Let
$$\Pi\colon\mathcal{Q}_{g,m}=\cup_{x\in\mathcal{T}_{g,m}}\mathcal{Q}_{x}\to%
\mathcal{T}_{g,m}$$
be the complex vector
bundle of quadratic differentials over $\mathcal{T}_{g,m}$.
A differential $q\in\mathcal{Q}_{g,m}$ is said to be generic
if all zeros are simple and all marked points of the underlying surface
are simple poles of $q$.
Generic differentials are open and dense subset in $\mathcal{Q}_{g,m}$
and in each fiber $\mathcal{Q}_{x}$ for $x\in\mathcal{T}_{g,m}$.
2.2.1. Infinitesimal complex structure
The Teichmüller space $\mathcal{T}_{g,m}$ is a complex manifold of dimension
$3g-3+m$.
The infinitesimal complex structure is described as follows:
Let $x=(M,f)\in\mathcal{T}_{g,m}$.
Let $L^{\infty}(M)$ be the Banach space of measurable $(-1,1)$-forms
$\mu=\mu(z)d\overline{z}/dz$ on $M$
with
$$\|\mu\|_{\infty}={\rm ess.sup}_{p\in M}|\mu(p)|<\infty.$$
The holomorphic tangent space $T_{x}\mathcal{T}_{g,m}$ at $x$ of $\mathcal{T}_{g,m}$
is described as the quotient space
$$L^{\infty}(M)/\{\mu\in L^{\infty}(M)\mid\langle\mu,\varphi\rangle=0,\forall%
\varphi\in\mathcal{Q}_{x}\},$$
where
$$\langle\mu,\varphi\rangle=\int_{M}\mu(z)\varphi(z)\frac{\sqrt{-1}}{2}dz\wedge d%
\overline{z}.$$
For $v=[\mu]\in T_{x}\mathcal{T}_{g,m}$
and $\varphi\in\mathcal{Q}_{x}$,
the canonical pairing between $T_{x}\mathcal{T}_{g,m}$
and $\mathcal{Q}_{x}$
is defined by
$$\langle v,\varphi\rangle=\langle\mu,\varphi\rangle$$
and,
it induces
an identification between $\mathcal{Q}_{x}$
and the holomorphic cotangent space $T_{x}^{*}\mathcal{T}_{g,m}$.
2.2.2. The Teichmüller homeomorphism
Let $\mathcal{UQ}_{x}$ be the unit ball in $\mathcal{Q}_{x}$.
For $q\in\mathcal{UQ}_{x}$,
we define a quasiconformal mapping $f^{q}$ on $M$
from the Beltrami differential $\|q\|(\overline{q}/|q|)\in L^{\infty}(M)$.
Then,
$\mathcal{T}_{g,m}$ is homeomorphic to $\mathcal{UQ}_{x}$ with
(2.1)
$$\Xi=\Xi_{x}\colon\mathcal{UQ}_{x}\ni q\mapsto(f^{q}(M),f^{q}\circ f)\in%
\mathcal{T}_{g,m}.$$
We call the homeomorphism (2.1)
the Teichmüller homeomorphism.
The Teichmüller homeomorphism gives a useful representation of the Teichmüller distance
as
(2.2)
$$d_{T}(x,\Xi_{x}(q))=\frac{1}{2}\log\frac{1+\|q\|}{1-\|q\|}=\tanh^{-1}(\|q\|)$$
for $q\in\mathcal{UQ}_{x}$,
2.3. Measured foliations
Let $\mathcal{S}$ be the set of homotopy classes of
non-trivial and non-peripheral simple closed curves on $\Sigma_{g,m}$.
Let $i(\alpha,\beta)$ denote the geometric intersection number
for simple closed curves $\alpha,\beta\in\mathcal{S}$.
Let $\mathcal{WS}=\{t\alpha\mid t\geq 0,\alpha\in\mathcal{S}\}$
be the set of weighted simple closed curves.
The set $\mathcal{S}$
is canonically identified with
a subset of $\mathcal{WS}$
as weight $1$ curves.
We consider an embedding
$$\mathcal{WS}\ni t\alpha\mapsto[\mathcal{S}\ni\beta\mapsto t\,i(\alpha,\beta)]%
\in\mathbb{R}_{\geq 0}^{\mathcal{S}}.$$
We topologize the function space
$\mathbb{R}_{\geq 0}^{\mathcal{S}}$
with the topology of pointwise convergence.
The closure $\mathcal{MF}$ of the image of the embedding is called
the space of measured foliations on $\Sigma_{g,m}$.
The space $\mathcal{MF}$
is homeomorphic
to $\mathbb{R}^{6g-6+2m}$,
and
contains the weighted simple closed curves $\mathcal{WS}$
as a dense subset.
The intersection number on $\mathcal{WS}$
is defined by $i(t\alpha,s\beta)=ts\,i(\alpha,\beta)$
for $t\alpha,s\beta\in\mathcal{WS}$.
The intersection number extends continuously
as a non-negative function $i(\,\cdot\,,\,\cdot\,)$
on $\mathcal{MF}\times\mathcal{MF}$
with $i(F,F)=0$ and $F(\alpha)=i(F,\alpha)$
for $F\in\mathcal{MF}\subset\mathbb{R}_{\geq 0}^{\mathcal{S}}$
and $\alpha\in\mathcal{S}$.
2.4. Hubbard-Masur differentials and Extremal length
Let $x=(M,f)\in\mathcal{T}_{g,m}$.
For $q\in\mathcal{Q}_{x}$,
the vertcal foliation $v(q)$ of $q$ is a measured foliation
defined by
$$i(v(q),\alpha)=\inf_{\alpha^{\prime}\in f(\alpha)}\int_{\alpha^{\prime}}\left|%
{\rm Re}(\sqrt{q})\right|$$
for $\alpha\in\mathcal{S}$.
Hubbard and Masur observed that
the correspondence,
which we call the Hubbard-Masur homeomorphism,
(2.3)
$$\mathcal{V}_{x}\colon\mathcal{Q}_{x}\ni q\mapsto v(q)\in\mathcal{MF}$$
is a homeomorphism (cf. [17] and Remark 9).
For $F\in\mathcal{MF}$,
the Hubbard-Masur differential $q_{F,x}$ for $F$ at $x$ is defined to
satisfy $v(q_{F,x})=F$.
By definition,
$q_{tF,x}=t^{2}q_{F,x}$ for $F\in\mathcal{MF}$ and $t\geq 0$.
For $F\in\mathcal{MF}$,
the extremal length of $F$ at $x$ is defined by
$${\rm Ext}_{x}(F)=\|q_{F,x}\|.$$
Kerckhoff [21] observed that
the Teichmüller distance is expressed as
(2.4)
$$d_{T}(x,y)=\frac{1}{2}\log\sup_{\alpha\in\mathcal{S}}\frac{{\rm Ext}_{x}(%
\alpha)}{{\rm Ext}_{y}(\alpha)}$$
for $x,y\in\mathcal{T}_{g,m}$.
This expression is called the Kerckhoff formula of the Teichmüller distance.
3. Double covering spaces associated to quadratic differentials
3.1. Branched covering spaces
Let $x_{0}=(M_{0},f_{0})\in\mathcal{T}_{g,m}$.
Let $\Sigma_{m}=\Sigma_{m}(x_{0})$ be the marked points of $M_{0}$.
Let $q_{0}\in\mathcal{Q}_{x_{0}}\subset\mathcal{Q}_{g,m}$.
Let
$\Sigma_{s}(q_{0})$ be
the set
of singularities
of $q_{0}$.
In accordance with [32],
a singular point of $q_{0}$ is called orientable if it is of even order,
and non-orientable otherwise.
Any orientable singular point is a zero of $q_{0}$.
Let $\Sigma_{o}=\Sigma_{o}(q_{0})$ (resp. $\Sigma_{e}=\Sigma_{e}(q_{0})$)
be the set of non-orientable (resp. orientable)
zeros of $q_{0}$ (“$o$” and “$e$” stand for “odd” and “even”).
The number of non-orientable singularities is always even.
By definition,
$\Sigma_{s}(q_{0})=\Sigma_{o}(q_{0})\cup\Sigma_{e}(q_{0})$.
We define
$$\displaystyle\Sigma(q_{0})$$
$$\displaystyle=\Sigma_{e}(q_{0})\cup\Sigma_{o}(q_{0})\cup\Sigma_{m}(x_{0}),$$
$$\displaystyle\Sigma_{sm}(q_{0})$$
$$\displaystyle=\Sigma_{s}(q_{0})\cap\Sigma_{m}(x_{0}),$$
$$\displaystyle\Sigma_{s\setminus m}(q_{0})$$
$$\displaystyle=\Sigma_{s}(q_{0})\setminus\Sigma_{m}(x_{0}),$$
$$\displaystyle\Sigma_{m\setminus s}(q_{0})$$
$$\displaystyle=\Sigma_{m}(x_{0})\setminus\Sigma_{s}(q_{0}),$$
$$\displaystyle\Sigma_{ub}(q_{0})$$
$$\displaystyle=\Sigma_{e}(q_{0})\sqcup\Sigma_{m\setminus s}(q_{0}).$$
We call a marked point in $\Sigma_{m\setminus s}(q_{0})$ free.
The set $\Sigma(q_{0})$ is the totality of marked points
caused by $q_{0}$,
and it is represented as the disjoint unions
$$\displaystyle\Sigma(q_{0})$$
$$\displaystyle=\Sigma_{ub}(q_{0})\sqcup\Sigma_{o}(q_{0}).$$
Consider the double branched covering space
$\pi_{q_{0}}\colon\tilde{M}_{q_{0}}\to M_{0}$ of the square root $\sqrt{q_{0}}$
(cf. Figure 1).
For $p\in\Sigma(q_{0})$,
the preimage $\pi_{q_{0}}^{-1}(p)$ consists of two points
if and only if $p\in\Sigma_{ub}(q_{0})$ (“ub” stands for “unbranched”).
The projection
$\pi_{q_{0}}$ is double-branched over points in $\tilde{\Sigma}_{o}(q_{0})$.
The surface $\tilde{M}_{q_{0}}$ is a closed surface
of genus
$$\tilde{g}(q_{0})=\begin{cases}{\displaystyle 2g-1+\frac{1}{2}{}^{\#}\Sigma_{o}%
(q_{0})}&(\mbox{$q_{0}$ is not square})\\
2g&(\mbox{otherwise}).\end{cases}$$
(cf. [17, §2]).
In particular,
the surface $\tilde{M}_{q_{0}}$ is a closed Riemann surface
of genus $4g-3+m$ when $q_{0}$ is generic.
The square root $\sqrt{q_{0}}$ on $M_{0}$ is lifted as the
Abelian differential $\omega_{q_{0}}$
on $\tilde{M}_{q_{0}}$.
The covering transformation $i_{q_{0}}$
of the covering is a conformal involution on $\tilde{M}_{q_{0}}$
which satisfies $\pi_{q_{0}}\circ i_{q_{0}}=\pi_{q_{0}}$
and $i_{q_{0}}^{*}\omega_{q_{0}}=-\omega_{q_{0}}$.
For each set $\Sigma_{\bullet}(q_{0})$ defined above,
we denote by $\tilde{\Sigma}_{\bullet}(q_{0})$
the preimage of $\Sigma_{\bullet}(q_{0})$.
When $q_{0}$ is square in the sense that $q_{0}=\omega^{2}$
for some Abelian differential $\omega$ on $M_{0}$,
$\tilde{M}_{q_{0}}$ consists of two copies of $M_{0}$.
We consider the pair $(\tilde{M}_{q_{0}},\tilde{\Sigma}_{ub}(q_{0}))$
as a Riemann surface with marked points.
Convention Let $V$ be a vector space $V$ with an involution.
We denote by $V^{\pm}$ the eigenspace in $V$ of the eigenvalue $\pm 1$
of the action of the involution.∎
We also remark the following elementary fact:
For vector spaces $V_{i}$ with an involution ($i=1,2,3$),
an exact sequence $0\to V_{1}\to V_{2}\to V_{3}\to 0$
commuting the involutions
induces an exact sequence $0\to V_{1}^{-}\to V_{2}^{-}\to V_{3}^{-}\to 0$.
3.2. A subspace in quadratic differentials
For $q_{0}\in\mathcal{Q}_{x_{0}}$ and $x_{0}=(M_{0},f_{0})$,
we define
$$\displaystyle\mathcal{Q}^{T}_{x_{0}}(q_{0})$$
$$\displaystyle=\left\{\psi\in\mathcal{Q}_{x_{0}}\mid(\psi)\geq\prod_{p\in\Sigma%
_{s\setminus m}(q_{0})}p^{\boldsymbol{o}_{p}(q_{0})-1}\prod_{p\in\Sigma_{sm}(q%
_{0})}p^{\boldsymbol{o}_{p}(q_{0})}\right\}.$$
where $\boldsymbol{o}_{p}(q_{0})$ is the order of $q_{0}$ at $p\in M_{0}$,
and $(\psi)$ is the divisor of $\psi$.
The symbol “$T$” stands for “tangent”.
When $M_{0}$ has no marked point (i.e. $m=0$),
$\psi\in\mathcal{Q}^{T}_{x_{0}}(q_{0})$
is equivalent to the condition that
$\psi/q_{0}$ has at most simple poles on $M_{0}$
(cf. [9, Lemma 5.2].
See also Proposition 6.1 below).
Notice that
$\mathcal{Q}_{x_{0}}^{T}(q_{0})=\mathcal{Q}_{x_{0}}$
if $q_{0}$ is generic.
3.3. The $q$-realizations of tangent vectors
Let $x_{0}=(M_{0},f_{0})\in\mathcal{T}_{g,m}$ and $q_{0}\in\mathcal{Q}_{x_{0}}-\{0\}$
be a generic differential.
For $v\in T_{x_{0}}\mathcal{T}_{g,m}$,
a holomorphic quadratic differential
$\eta_{v}\in\mathcal{Q}_{x_{0}}$
said to be the $q_{0}$-realization of $v$
if it satisfies
(3.1)
$$\langle v,\psi\rangle_{x_{0}}=\int_{M_{0}}\frac{\overline{\eta_{v}}}{|q_{0}|}\psi$$
for all
$\psi\in\mathcal{Q}_{x_{0}}$
where $x_{0}=(M_{0},f_{0})\in\mathcal{T}_{g,m}$
(cf. [33]).
Notice that
$\eta_{v}$ in (3.1)
does exist because
(3.2)
$$(\psi_{1},\psi_{2})\mapsto\int_{M_{0}}\frac{\psi_{1}\overline{\psi_{2}}}{|q_{0%
}|}$$
is a non-degenerate Hermitian inner product on $\mathcal{Q}_{x_{0}}$
(cf. [9, §5]).
The correspondence
$$T_{x_{0}}\mathcal{T}_{g,m}\ni v\mapsto\eta_{v}\in\mathcal{Q}_{x_{0}}$$
is an anti-complex linear isomorphism.
The Hermitian form
(3.2) is calculated as
(3.3)
$$\displaystyle\int_{M_{0}}\frac{\psi_{1}\overline{\psi_{2}}}{|q_{0}|}$$
$$\displaystyle=\frac{\sqrt{-1}}{4}\int_{\tilde{M}_{q_{0}}}\frac{\pi_{q_{0}}^{*}%
(\psi_{1})}{\omega_{q_{0}}}\wedge\overline{\left(\frac{\pi_{q_{0}}^{*}(\psi_{2%
})}{\omega_{q_{0}}}\right)}$$
$$\displaystyle=\frac{1}{2}\int_{\tilde{M}_{q_{0}}}{\rm Re}\left(\frac{\pi_{q_{0%
}}^{*}(\psi_{1})}{\omega_{q_{0}}}\right)\wedge{\rm Im}\left(\frac{\pi_{q_{0}}^%
{*}(\psi_{2})}{\omega_{q_{0}}}\right)$$
$$\displaystyle\qquad+\frac{\sqrt{-1}}{2}\int_{\tilde{M}_{q_{0}}}{\rm Re}\left(%
\frac{\pi_{q_{0}}^{*}(\psi_{1})}{\omega_{q_{0}}}\right)\wedge{\rm Re}\left(%
\frac{\pi_{q_{0}}^{*}(\psi_{2})}{\omega_{q_{0}}}\right)$$
for $\psi_{1}$,
$\psi_{2}\in\mathcal{Q}_{x_{0}}$.
4. Stratifications on $\mathcal{Q}_{g,m}$
4.1. Stratification
Following Dumas [9],
we recall the definiton of stratifications on manifolds.
Let $Z$ be a manifold.
A stratification of $Z$ is a locally finite collection
of locally closed submanifolds $\{Z_{i}\}_{i\in I}$ of $Z$, the strata,
indexed by a set $I$ such that
(1)
$Z=\cup_{j\in I}Z_{j}$
(2)
$Z_{j}\cap\overline{Z_{k}}\neq\emptyset$
if and only if $Z_{j}\subset\overline{Z_{k}}$.
From the second condition,
$Z_{i}\cap Z_{j}\neq\emptyset$ if and only if $Z_{i}=Z_{j}$
because each $Z_{i}$ is locally closed.
A stratification of a complex manifold $Z$ a
complex-analytic stratification if
the closure $\overline{Z_{j}}$ and
the boundary $\overline{Z_{j}}\setminus Z_{j}$
of each stratum $Z_{j}$ are complex-analytic sets.
4.2. Strata in $\mathcal{Q}_{g,m}$
Our strata and symbol are slightly different from that treated by Masur-Smillie [32]
and Veech [44].
We consider here the deformation of
quadratic differentials with marked points
for our purpose
(see also §4.3 below).
If any marked point of given quadratic differential is a singular point,
our strata are coincides with their strata
(cf. [44, §1]).
A symbol of
$q\in\mathcal{Q}_{g,m}-\{0\}$ is a quadruple
$\boldsymbol{\pi}=(\boldsymbol{m},\boldsymbol{n}(-1),\boldsymbol{n}(\cdot),\varepsilon)$
where $\boldsymbol{m}$
is the number of free marked points,
$\boldsymbol{n}(-1)$ is the number of poles,
$\boldsymbol{n}(l)$ is the number of zeros of order $l\geq 1$,
and $\varepsilon=\pm 1$ according to whether
$q$ is square ($\varepsilon=1$)
or not ($\varepsilon=-1$).
We set $\boldsymbol{n}(0)=0$ for simplicity.
Notice that $\sum_{l\geq-1}l\cdot\boldsymbol{n}(l)=4g-4$.
Let $\mathcal{Q}(\boldsymbol{\pi})=\mathcal{Q}_{g,m}(\boldsymbol{\pi})\subset%
\mathcal{Q}_{g,m}$
be the set of holomorphic quadratic differentials in $\mathcal{Q}_{g,m}$
whose symbol is $\boldsymbol{\pi}$.
As we discuss in Proposition 4.1 below,
each component of $\mathcal{Q}(\boldsymbol{\pi})$ is a complex manifold of dimension
(4.1)
$$\displaystyle\dim_{\mathbb{C}}\mathcal{Q}(\boldsymbol{\pi})$$
$$\displaystyle=2g+\frac{\varepsilon-3}{2}+\boldsymbol{m}+\sum_{l\geq-1}%
\boldsymbol{n}(l).$$
Let ${\boldsymbol{\pi}}(q)=(\boldsymbol{m}_{q},\boldsymbol{n}_{q}(-1),\boldsymbol{n%
}_{q}(\cdot),\varepsilon_{q})$
be the symbol of $q\in\mathcal{Q}_{g,m}$.
If $\Sigma_{s}(q)=\emptyset$,
we have $g=1$ and $q$ is square.
We set ${\boldsymbol{\pi}}(q)=(m,0,\{0,\cdots\},1)$ in this case.
Since
$\boldsymbol{m}_{q}+\sum_{l\geq-1}\boldsymbol{n}_{q}(l)={}^{\#}\Sigma(q)={}^{\#%
}\Sigma_{0}(q)+{}^{\#}\Sigma_{ub}(q)$
for $q\in\mathcal{Q}_{g,m}$,
from
(4.1),
we can check the following.
$$\displaystyle\dim_{\mathbb{C}}\mathcal{Q}(\boldsymbol{\pi}(q))=\dim_{\mathbb{C%
}}{\rm Hom}(H_{1}(\tilde{M}_{q},\tilde{\Sigma}_{ub}(q),\mathbb{R})^{-},\mathbb%
{C}).$$
4.3. Remark on the stratification on $\mathcal{Q}_{g,m}$
Our stratification of $\mathcal{Q}_{g,m}$ is slightly different from
Masur-Smillie-Veech’s one
in the following sense:
We have mainly two differences from their stratification:
(1)
If a free marked point and a singular point collide
in a moving of quadratic differentials,
we recognize the quadratic differentials to be degenerating into the other stratum.
Two free marked points can not collide
because we consider the deformation on $\mathcal{T}_{g,m}$;
and
(2)
if a singular point of a quadratic differential in a stratum lies at a marked point,
the singular point stays on the marked point in deforming on the stratum
for the quadratic differential.
All of these phenomena can be handled by standard arguments
with complex analysis
(for instance, [32] and [43]).
4.4. Masur-Smillie-Veech charts of the strata in $\mathcal{Q}_{g,m}$
For $q_{0}\in\mathcal{Q}({\boldsymbol{\pi}}(q_{0}))$,
the union $\cup_{q}\tilde{M}_{q}$
is regarded as a trivial bundle
over a small contractible neighborhood of $q_{0}$
whose fiber is a (possibly disconnected) surface with marked points.
For each $q\in\mathcal{Q}({\boldsymbol{\pi}}(q_{0}))$
which is sufficiently close to $q_{0}$,
the surface $\tilde{M}_{q}$
admits a marking inherited from
the product structure of the bundle.
Hence,
we can identify $H_{1}(\tilde{M}_{q},\Sigma_{ub}(q),\mathbb{R})^{-}$
and $H_{1}(\tilde{M}_{q},\Sigma_{ub}(q),\mathbb{R})$
with
$H_{1}(\tilde{M}_{q_{0}},\Sigma_{ub}(q_{0}),\mathbb{R})^{-}$
and $H_{1}(\tilde{M}_{q_{0}},\Sigma_{ub}(q_{0}),\mathbb{R})$
for $q\in\mathcal{Q}({\boldsymbol{\pi}}(q_{0}))$
near $q_{0}$ in the canonical manner.
The following is well-known
(e.g. [32],
[31],
[43] and [44]).
Proposition 4.1 (Local chart).
There is a neighborhood $V_{0}$ of $q_{0}$ in
$\mathcal{Q}({\boldsymbol{\pi}}(q_{0}))$
such that
the mapping
$$\Phi_{0}\colon V_{0}\ni q\mapsto\left[C\mapsto\int_{C}\omega_{q}\right]\in{\rm
Hom%
}(H_{1}(\tilde{M}_{q_{0}},\Sigma_{ub}(q_{0}),\mathbb{R})^{-},\mathbb{C})$$
is a holomorphic local chart around $q_{0}$.
5. Deformations of quadratic differentials
Henceforth,
we set
${\rm Hom}(q_{0})={\rm Hom}(H_{1}(\tilde{M}_{q_{0}},\tilde{\Sigma}_{ub}(q_{0}),%
\mathbb{R})^{-},\mathbb{C})$
for the simplicity.
From Proposition 4.1,
$T_{q_{0}}\mathcal{Q}(\boldsymbol{\pi}(q_{0}))$
is isomorphic to ${\rm Hom}(q_{0})$ as $\mathbb{C}$-vector spaces.
In this section,
we consider a $\Delta$-complex structure on $\tilde{M}_{q_{0}}$
for given $q_{0}\in\mathcal{Q}_{g,m}$,
and describe the infinitesimal deformations along
elements in $T_{q_{0}}\mathcal{Q}(\boldsymbol{\pi}(q_{0}))\cong{\rm Hom}(q_{0})$
by piecewise affine deformations.
5.1. $\Delta$-complex structure
A $\Delta$-complex structure on a space $X$ is a
collection of a singular simplex $\sigma_{\alpha}\colon\Delta^{n}\to X$
($\Delta^{n}$ is the standard $n$-simplex),
with $n=n(\alpha)$ such that
(1)
the restriction $\sigma_{\alpha}$ to the interior of $\Delta^{n}$
is injective,
and each point of $X$ is in the image of exactly one such restriction;
(2)
each restriction of $\sigma_{\alpha}$ to a face of $\Delta^{n}$ is one of the maps $\sigma_{\beta}\colon\Delta^{n-1}\to X$.
Here, we are identifying the face of $\Delta^{n}$ with $\Delta^{n-1}$ by the canonical linear homeomorphism between them
that preserves the ordering of the vertices;
and
(3)
a set $A\subset X$ is open if and only if $\sigma^{-1}_{\alpha}(A)$ is open in $\Delta^{n}$ for each $\sigma_{\alpha}$
(cf. [16, §2.1]).
A $\Delta$-complex structure on a surface gives a kind of triangulations.
The (relative) (co)homology group defined by a $\Delta$-complex structure
on a space $X$ coincides with the (relative) (co)homology group of $X$
(cf. [16]).
5.2. Singular Euclidean structure on $\tilde{M}_{q_{0}}$
Let $x_{0}=(M_{0},f_{0})\in\mathcal{T}_{g,m}$
and $q_{0}\in\mathcal{Q}_{x_{0}}$.
Consider a $\Delta$-complex structure $\Delta$ on $M_{0}$
such that the $0$-skeleton $\Delta^{(0)}$ contains $\Sigma(q_{0})$,
each $1$-simplex is a straight segment
with respect to the $|q_{0}|$-metric,
and each $2$-simplex is a non-degenerate triangle.
Such a $\Delta$-complex exists.
For instance,
we can take it as a refinement (subdivision) of
the Delaunay triangulation with respect to the singularities of $q_{0}$
(cf. [32, §4]).
Let $\tilde{\Delta}$ be the lift of $\Delta$.
$\tilde{\Delta}$ is a $\Delta$-complex structure on $\tilde{M}_{q_{0}}$.
The covering transformation $i_{q_{0}}$ acts on
the $1$-chain group $C_{1}(\tilde{\Delta},\tilde{\Sigma}_{ub}(q_{0}),\mathbb{R})$.
We define $\widetilde{{\bf u}}[q_{0}]\in{\rm Hom}(C_{1}(\tilde{\Delta},\tilde{\Sigma}_{ub%
}(q_{0}))^{-},\mathbb{C})$
by
$$\widetilde{{\bf u}}[q_{0}](e)=\int_{e}\omega_{q_{0}}$$
for $e\in C_{1}(\tilde{\Delta},\Sigma_{ub}(q_{0}))^{-}$.
The restriction of $\widetilde{{\bf u}}[q_{0}]$
to the cycles $Z_{1}(\tilde{\Delta},\Sigma_{ub}(q_{0}),\mathbb{R})^{-}$
descends to
a homomorphism ${\bf u}[q_{0}]\in{\rm Hom}(q_{0})$
such that $\Phi_{0}(q_{0})={\bf u}[q_{0}]$
(cf. Proposition 4.1).
5.3. Piecewise affine deformations
Let $\sigma$ be a $2$-simplex in $\Delta$.
Let $\partial\sigma=e_{1}+e_{2}+e_{3}$ as $1$-chains.
The developing mapping $\sigma\ni p\mapsto z(p)=\int^{p}\omega_{q_{0}}$ maps $\sigma$
to a Euclidean triangle $\sigma^{\prime}$ in the complex plane $\mathbb{C}$
with (oriented) edges $\widetilde{{\bf u}}[q_{0}](e_{i})$.
Notice that $dz=\omega_{q_{0}}$ on $\sigma^{\prime}$
(cf. Figure 2).
For ${\bf v}\in{\rm Hom}(q_{0})\cong T_{q_{0}}\mathcal{Q}(\boldsymbol{\pi}(q_{0}))$,
the infinitesimal deformation along ${\bf v}$
of the singular Euclidean structure associated to $q_{0}$
is described by an assortment of the affine deformation of the triangle $\sigma$
along the lift $\widetilde{{\bf v}}\in{\rm Hom}(C_{1}(\tilde{\Delta},\tilde{\Sigma}_{ub}(q_{0}%
),\mathbb{R})^{-},\mathbb{C})$ of ${\bf v}$.
Here,
we define
the lift $\widetilde{{\bf v}}$ as follows:
We first take the pullback of ${\bf v}$
on $Z_{1}(\tilde{\Delta},\tilde{\Sigma}_{ub}(q_{0}),\mathbb{R})^{-}$
by precomposing the projection
from $Z_{1}(\tilde{\Delta},\tilde{\Sigma}_{ub}(q_{0}),\mathbb{R})^{-}$
to
$H_{1}(\tilde{M}_{q_{0}},\tilde{\Sigma}_{ub}(q_{0}),\mathbb{R})^{-}$,
and
set $\widetilde{{\bf v}}\equiv 0$ on a complementary space of
$Z_{1}(\tilde{\Delta},\tilde{\Sigma}_{ub}(q_{0}),\mathbb{R})^{-}$
in $C_{1}(\tilde{\Delta},\tilde{\Sigma}_{ub}(q_{0}),\mathbb{R})^{-}$.
To be more precise,
fix a norm on
${\rm Hom}(C_{1}(\tilde{\Delta},\tilde{\Sigma}_{ub}(q_{0}),\mathbb{R})^{-},%
\mathbb{C})$.
Since each $2$-simplex of $\Delta$ is a non-degenerate triangle,
vectors
$\{(\widetilde{{\bf u}}[q_{0}]+\widetilde{{\bf v}})(e_{i})\}_{i=1}^{3}$
also span a non-degenerate triangle when
$\widetilde{{\bf v}}$ is sufficiently short.
Collecting such new triangles defined from all $2$-simplices of $\tilde{\Delta}$,
and gluing them according to the combinatorial structure $\tilde{\Delta}$,
we get a new singular Euclidean surface $\tilde{M}_{q_{0}}[{\bf v}]$
which is homeomorphic to $\tilde{M}_{q_{0}}$ by a piecewise affine mapping
$\tilde{F}_{{\bf v}}\colon\tilde{M}_{q_{0}}\to\tilde{M}_{q_{0}}[{\bf v}]$
defined by assembling the affine deformations on the $2$-simplicies of $\tilde{\Delta}$.
Since $i_{q_{0}}^{*}(\widetilde{{\bf v}})=-\widetilde{{\bf v}}$,
$\tilde{M}_{q_{0}}[{\bf v}]$
admits an involution $i_{q_{0}}[{\bf v}]$
satisfies $i_{q_{0}}[{\bf v}]\circ\tilde{F}_{{\bf v}}=\tilde{F}_{{\bf v}}\circ i_{q_{0}}$,
and
the piecewise affine mapping $\tilde{F}_{{\bf v}}$ descends to
a quasiconformal mapping (a piecewise affine mapping)
$F_{{\bf v}}$ on $M_{0}$
to a Riemann surface $M_{0}[{\bf v}]$.
The surface $\tilde{M}_{q_{0}}[{\bf v}]$ admits a $\Delta$-complex structure $\tilde{\Delta}[{\bf v}]$
inherited form $\tilde{\Delta}$ on $\tilde{M}_{q_{0}}$
which is equivariant under the action of the involution $i_{q_{0}}[{\bf v}]$.
The $\Delta$-complex structure $\tilde{\Delta}[{\bf v}]$
descends to a $\Delta$-complex structure $\Delta[{\bf v}]$
on $M_{0}[{\bf v}]$.
Denote by $w$ the flat coordinate for $\tilde{M}_{q_{0}}[{\bf v}]$
(defined on each $2$-simplex of $\tilde{\Delta}[{\bf v}]$).
The holomorphic $1$-form $dw$ on each $2$-simplex for
$\tilde{M}_{q_{0}}[{\bf v}]$
defines a holomorphic $1$-form $\boldsymbol{\omega}_{q_{0}}[{\bf v}]$
on $\tilde{M}_{q_{0}}[{\bf v}]$.
The square $\boldsymbol{\omega}_{q_{0}}[{\bf v}]^{2}$
descends to a holomorphic quadratic differential
${\bf q}[q_{0},{\bf v}]$
on $M_{0}[{\bf v}]$.
Let $x_{{\bf v}}=(M_{0}[{\bf v}],F_{{\bf v}}\circ f_{0})\in\mathcal{T}_{g,m}$.
Since $\tilde{F}_{{\bf v}}$ sends the vertices of $\tilde{\Delta}$ to
the vertices of $\tilde{\Delta}[{\bf v}]$,
we can see that
${\bf q}[q_{0},{\bf v}]\in\mathcal{Q}(\boldsymbol{\pi}(q_{0}))\cap\mathcal{Q}_{%
x_{{\bf v}}}$ when $\widetilde{{\bf v}}$ is sufficiently short,
${\bf q}[q_{0},0]=q_{0}$
and
(5.1)
$$\displaystyle{\bf u}[{\bf q}[q_{0},{\bf v}]](C)=\int_{(\tilde{F}_{{\bf v}})_{*%
}(C)}\boldsymbol{\omega}_{q_{0}}[{\bf v}]=({\bf u}[q_{0}]+{\bf v})(C)$$
for all $C\in H_{1}(\tilde{M}_{q_{0}},\tilde{\Sigma}_{ub},\mathbb{R})^{-}$.
Summarizing the above argument,
we get a commuting diagram
$$\begin{CD}\tilde{M}_{q_{0}}@>{\tilde{F}_{{\bf v}}}>{}>\tilde{M}_{{\bf q}[q_{0}%
,{\bf v}]}\cong\tilde{M}_{q_{0}}[{\bf v}]\\
@V{}V{}V@V{}V{}V\\
M_{0}@>{F_{{\bf v}}}>{}>M_{0}[{\bf v}]\end{CD}$$
when the lift $\widetilde{{\bf v}}\in{\rm Hom}(C_{1}(\tilde{\Delta},\tilde{\Sigma}_{ub}(q_{0}%
),\mathbb{R})^{-},\mathbb{C})$ of ${\bf v}\in{\rm Hom}(q_{0})\cong T_{q_{0}}\mathcal{Q}(\boldsymbol{\pi}(q_{0}))$ is sufficiently short,
where the vertical directions are double-branched coverings
with covering involutions $i_{q_{0}}$ and $i_{q_{0}}[{\bf v}]$.
5.4. Teichmüller deformations
Let $x_{0}=(M_{0},f_{0})\in\mathcal{T}_{g,m}$
and $q_{0}\in\mathcal{Q}_{x_{0}}$.
For $t\geq 0$,
let $h_{t}\colon M_{0}\to M_{t}$ be the Teichmüller mapping
associated to the Betrami differntial $\tanh(t)\overline{q_{0}}/|q_{0}|$
(cf. §2.2.2).
Let $x_{t,q_{0}}=(M_{t},h_{t}\circ f_{0})\in\mathcal{T}_{g,m}$ and
Let ${\bf Q}[t,q_{0}]\in\mathcal{Q}_{x_{t,q_{0}}}$ be the terminal differential
(e.g. [19]).
For our purpose,
we assume that the Teichmüller mapping $h_{t}$ is represented
as an affine mapping associated to the matrix $\begin{pmatrix}1&0\\
0&e^{-2t}\end{pmatrix}$ in terms of the natural coordinates
(distinguished parameters) of the initial and terminal differentials
(cf. [42, Chapter II]).
In particular,
$\|{\bf Q}[t,q_{0}]\|=e^{-2t}\|q_{0}\|$
and $v({\bf Q}[t,q_{0}])=v(q_{0})$
from our assumption
(cf. [17, Lemma 4.3] and (5.2) below).
It is known that ${\bf Q}[t,q_{0}]\in\mathcal{Q}(\boldsymbol{\pi}(q_{0}))$
for $t\geq 0$
(e.g. [29]).
The Teichmüller mapping $h_{t}$ lifts as a quasiconformal mapping
$\tilde{h}_{t}\colon\tilde{M}_{q_{0}}\to\tilde{M}_{q_{t,q_{0}}}$
which is equivariant under the action of the involutions.
The lift gives the identification
$H_{1}(\tilde{M}_{{\bf Q}[t,q_{0}]},\tilde{\Sigma}_{ub}({\bf Q}[t,q_{0}]),%
\mathbb{R})^{-}\cong H_{1}(\tilde{M}_{q_{0}},\tilde{\Sigma}_{ub}(q_{0}),%
\mathbb{R})^{-}$.
By the analytic continuation along a continuous path
$t\mapsto{\bf Q}[t,q_{0}]\in\mathcal{Q}(\boldsymbol{\pi}(q_{0}))$
from initial point $q_{0}$,
the chart given in Proposition 4.1 extends
a neighborhood of the path.
The image of ${\bf Q}[t,q_{0}]$ by the chart satisfies
(5.2)
$$\displaystyle{\bf u}[{\bf Q}[t,q_{0}]](C)$$
$$\displaystyle={\rm Re}\left({\bf u}[q_{0}](C)\right)+\sqrt{-1}e^{-2t}{\rm Im}%
\left({\bf u}[q_{0}](C)\right)$$
$$\displaystyle=\frac{1+e^{-2t}}{2}{\bf u}[q_{0}](C)+\frac{1-e^{-2t}}{2}%
\overline{{\bf u}[q_{0}](C)}$$
for $C\in H_{1}(\tilde{M}_{q_{0}},\tilde{\Sigma}_{ub}(q_{0}),\mathbb{R})^{-}$.
5.5. Piecewise affine deformations around Teichmüller geodesics
Let ${\bf v}\in{\rm Hom}(q_{0})\cong T_{q_{0}}\mathcal{Q}(\boldsymbol{\pi}(q_{0}))$
and $t>0$.
When the lift $\widetilde{{\bf v}}$ is sufficiently short as §5.3,
we defined
$x_{{\bf v}}\in\mathcal{T}_{g,m}$
and ${\bf q}[q_{0},{\bf v}]\in\mathcal{Q}_{x_{\bf v}}$
associated to ${\bf v}$
(cf. §5.3).
Consider the Teichmüller deformation
on $x_{{\bf v}}$ associated to
${\bf q}[q_{0},{\bf v}]$ and $t$.
From the discussion in §5.4,
the chart in Proposition 4.1
is defined around
the terminal differential ${\bf Q}[t,{\bf q}[q_{0},{\bf v}]]$
when the lift of ${\bf v}$ is sufficiently short.
From (5.1)
and (5.2),
${\bf Q}[t,{\bf q}[q_{0},0]]={\bf Q}[t,q_{0}]$,
${\bf Q}[0,{\bf q}[q_{0},{\bf v}]]={\bf q}[q_{0},{\bf v}]$
and
(5.3)
$$\displaystyle{\bf u}[{\bf Q}[t,{\bf q}[q_{0},{\bf v}]]](C)$$
$$\displaystyle={\rm Re}\left(({\bf u}[q_{0}]+{\bf v})(C)\right)+\sqrt{-1}e^{-2t%
}{\rm Im}\left({\bf u}[q_{0}]+{\bf v})(C)\right)$$
for $t\geq 0$
and
$C\in H_{1}(\tilde{M}_{q_{0}},\tilde{\Sigma}_{ub}(q_{0}),\mathbb{R})^{-}$
when the lift $\widetilde{{\bf v}}$
of ${\bf v}\in{\rm Hom}(q_{0})$ is sufficiently short.
5.6. The hypercohomology group
Following Hubbard-Masur [17],
we recall the description of the holomorphic
tangent space $T_{q}\mathcal{Q}_{g,m}$
at $q\in\mathcal{Q}_{g,m}$
as the first hypercohomolgy group $\mathbb{H}^{1}(L^{\bullet})$
a complex of sheaves
(cf. [14] or [15]).
We will need the Kodaira-Spencer identification of the
tangent space of Teichmüller space
with the first cohomology group of the sheaf of holomorphic vector fields
(for instance, see [25]. See also [19] and [20]).
Let $X$ and $q$
be a holomorphic vector field
and a holomorphic quadratic differential
on an open set of a Riemann surface $M$.
Denote by $L_{X}q$ the Lie derivative of $q$ along $X$.
Let $\Theta_{M}$ and $\Omega_{M}^{\otimes 2}$ be the sheaves of germs of
holomorphic vector fields with zeroes at marked points
and meromorphic quadratic differentials on $M$ with (at most)
first order poles at marked points,
respectively.
Let $q_{0}\in\mathcal{Q}_{g,m}$
and $x_{0}=(M_{0},f_{0})\in\mathcal{T}_{g,m}$ with $q_{0}\in\mathcal{Q}_{x_{0}}$
($q_{0}$ need not to be generic).
The tangent space $T_{q_{0}}\mathcal{Q}_{g,m}$
is identified with the
first hypercohomology group of the complex of sheaves
$$\begin{CD}L^{\bullet}\colon\quad 0@>{}>{}>\Theta_{M_{0}}@>{L_{\cdot}q_{0}}>{}>%
\Omega_{M_{0}}^{\otimes 2}@>{}>{}>0.\end{CD}$$
The first cochain group is the direct sum
$C^{0}(M_{0},\Omega_{M_{0}}^{\otimes 2})\oplus C^{1}(M_{0},\Theta_{M_{0}})$.
Consider an appropriate covering $\mathcal{U}=\{U_{i}\}_{i}$ on $M_{0}$
such that $\mathbb{H}^{1}(L^{\bullet})\cong\mathbb{H}^{1}(\mathcal{U},L^{\bullet})$
(see the proof of [17, Proposition 4.5]).
A cochain $(\{\phi_{i}\}_{i},\{X_{ij}\}_{i,j})$
in $C^{0}(\mathcal{U},\Omega_{M_{0}}^{\otimes 2})\oplus C^{1}(\mathcal{U},\Theta_{%
M_{0}})$
is said to be a cocycle if
it satisfies
(5.4)
$$\delta\{X_{ij}\}_{i,j}=X_{ij}+X_{jk}+X_{ki}=0,\quad\delta\{\phi_{i}\}_{i}=\phi%
_{i}-\phi_{j}=L_{X_{ij}}(q_{0}).$$
A coboundary is a cochain $(\{\phi_{i}\}_{i},\{X_{ij}\}_{i,j})$
of the form
(5.5)
$$X_{ij}=Z_{i}-Z_{j}=\delta\{Z_{i}\}_{i},\quad\phi_{i}=L_{Z_{i}}(q_{0})$$
for some $0$-cochain $\{Z_{i}\}_{i}\in C^{0}(\mathcal{U},\Theta_{M_{0}})$
(cf. Figure 3).
For the hypercomology class $[(\{\phi_{i}\}_{i},\{X_{ij}\}_{i,j})]\in\mathbb{H}^{1}(L^{\bullet})$,
when the Kodaira-Spencer class of the
$1$-cochain $\{X_{ij}\}_{i,j}$ is trivial in $H^{1}(M_{0},\Theta_{M_{0}})$,
the hypercomology class $[(\{\phi_{i}\}_{i},\{X_{ij}\}_{i,j})]$
is associated to a holomorphic quadratic differential on $M_{0}$.
Indeed,
from (5.4) and (5.5),
$$\phi_{i}-\phi_{j}=L_{X_{ij}}(q_{0})=L_{Z_{i}}(q_{0})-L_{Z_{j}}(q_{0})$$
and $\{\phi_{i}-L_{Z_{i}}(q_{0})\}_{i}$ defines a holomorphic quadratic differential
on $M_{0}$.
5.7. Homomorphisms and hypercohomology classes
From Proposition 4.1,
we have a canonical inclusion
(5.6)
$${\rm Hom}(q_{0})\cong T_{q_{0}}\mathcal{Q}(\boldsymbol{\pi}(q))\hookrightarrow
T%
_{q_{0}}\mathcal{Q}_{g,m}\cong\mathbb{H}^{1}(L^{\bullet}).$$
Let ${\bf v}\in{\rm Hom}(q_{0})$
and $[(\{\phi_{i}\}_{i},\{X_{ij}\}_{i,j})]\in\mathbb{H}^{1}(L^{\bullet})$
the corresponding hypercohomology class via (5.6).
Take a $0$-cochain
$\{X_{i}\}_{i}$ of the sheaf
of $C^{\infty}$-vector fields such that
$X_{i}-X_{j}=X_{ij}$ on $U_{i}\cap U_{j}$,
and
each $X_{i}$ vanishes at any marked point of $M_{0}$.
The $1$-cochain $\{X_{ij}\}_{i,j}$
defines a holomorphic tangent vector at $x_{0}$
associated to the infinitesimal Beltrami differential
$-(X_{i})_{\overline{z}}$ on $M_{0}$
(cf. [33, (3.6)]).
The minus sign comes from our “$i,j$-convention”
in the definition of the hypercohomology
(compare with Equation $(7.27)$ in [19, §7.2.4]).
The holomorphic tangent vector from the $1$-cochain $\{X_{ij}\}_{i,j}$
coincides with the image of ${\bf v}\in{\rm Hom}(q_{0})$
($\hookrightarrow T_{q_{0}}\mathcal{Q}_{g,m}$)
via the differential of the projection $\mathcal{Q}_{g,m}\to\mathcal{T}_{g,m}$.
After choosing the covering $\mathcal{U}=\{U_{i}\}_{i}$ appropriately,
the right and left sides of the inclusion (5.6) is
related to the following formula:
(5.7)
$${\bf v}(C)=\int_{C}\Omega[q_{0},{\bf v}]$$
for $C\in H_{1}(\tilde{M}_{q_{0}},\tilde{\Sigma}_{ub}(q_{0}),\mathbb{R})^{-}$,
where $\Omega[q_{0},{\bf v}]$ is a $C^{\infty}$-closed $1$-form
on $\tilde{M}_{q_{0}}$ defined by
(5.8)
$$\boldsymbol{\Omega}[q_{0},{\bf v}]=\left(\frac{\tilde{\phi}_{i}}{2\omega_{q_{0%
}}}-\omega_{q_{0}}^{\prime}\tilde{X}_{i}-\omega_{q_{0}}(\tilde{X}_{i})_{z}%
\right)dz-\omega_{q_{0}}(\tilde{X}_{i})_{\overline{z}}d\overline{z}$$
on $U_{i}$,
and tildes in (5.7)
mean objects (differentials or vector fields etc.)
on the covering space $\tilde{M}_{q_{0}}$
which obtained as lifts of objects on $M_{0}$.
For a proof, see e.g. [33, Lemma 3.1].
Actually,
in [33, Lemma 3.1],
we discuss only in the case where $q_{0}$ is generic.
However, we can deduce (5.7)
the same argument
since we consider deformations of holomorphic
quadratic differentials along strata
(see also the discussion in [9, Lemma 5.6]).
Remark 3.
We notice the following:
(1)
Fix $\epsilon>0$ sufficiently small.
Then,
$\phi_{i}$ is the $\lambda$-derivative
of the infinitesimal deformation
of a holomorphic mapping
$\{|\lambda|<\epsilon\}\ni\lambda\mapsto{\bf q}[q_{0},\lambda{\bf v}]\in%
\mathcal{Q}(\boldsymbol{\pi}(q_{0}))$
on $U_{i}$ at $\lambda=0$
(e.g. [33, §3.3]).
Since
${\bf q}[q_{0},\lambda{\bf v}]$ varies in $\mathcal{Q}(\boldsymbol{\pi}(q_{0}))$,
we can see that $\boldsymbol{o}_{p}(\phi_{i})\geq\boldsymbol{o}_{p}(q_{0})-1$
for $p\in\Sigma_{s\setminus m}(q_{0})\cap U_{i}$
and
$\boldsymbol{o}_{p}(\phi_{i})\geq\boldsymbol{o}_{p}(q_{0})$
for $p\in\Sigma_{sm}(q_{0})\cap U_{i}$
(cf. [9, Lemma 5.2].
Hence the first term of the coefficient of $dz$ of
the differential $\boldsymbol{\Omega}[q_{0},{\bf v}]$
in (5.8)
is holomorphic around $\tilde{\Sigma}(q_{0})\cap U_{i}$.
(2)
For ${\bf v}\in{\rm Hom}(q_{0})$,
we define the complex conjugate
$\overline{{\bf v}}\in{\rm Hom}(q_{0})$
of ${\bf v}$
by
$$\overline{{\bf v}}(C)=\overline{{\bf v}(C)}$$
for $C\in H_{1}(\tilde{M}_{q_{0}},\tilde{\Sigma}_{ub}(q_{0}),\mathbb{R})^{-}$.
We can easily deduce from (5.7) that
(5.9)
$$\int_{C}\boldsymbol{\Omega}[q_{0},\overline{{\bf v}}]=\int_{C}\overline{%
\boldsymbol{\Omega}[q_{0},{\bf v}]}$$
for $C\in H_{1}(\tilde{M}_{q_{0}},\tilde{\Sigma}_{ub}(q_{0}),\mathbb{R})^{-}$.
Notice that the complex conjugate of ${\bf v}$ here
is thought of as a tangent vector
in the $(1,0)$-part in the complexification of the real tangent vector
space at $q_{0}$.
Compare [24, Proposition 1.5 in Chapter IX].∎
We claim
the following
(cf. [8] and [9, Lemma 5.6]).
Proposition 5.1.
Let $q_{0}\in\mathcal{Q}_{g,m}$.
Let ${\bf v}\in{\rm Hom}(q_{0})$
and $[\{\phi_{i}\}_{i},\{X_{ij}\}_{i,j})]$ the corresponding hypercohomology class.
When the Kodaira-Spencer class of $\{X_{ij}\}_{i,j}$ is trivial,
$$\boldsymbol{\Omega}[q_{0},{\bf v}]=\frac{\pi_{q_{0}}^{*}(\psi)}{\omega_{q_{0}}}$$
for some $\psi\in\mathcal{Q}_{x_{0}}^{T}(q_{0})$
Proof.
The assumption implies that there is a $0$-cochain
$\{Z_{i}\}_{i}\in C^{0}(\mathcal{U},\Theta_{M_{0}})$ such that
$Z_{i}-Z_{j}=X_{ij}$.
From (5.8)
we have
$$\boldsymbol{\Omega}[q_{0},{\bf v}]=\left(\frac{\tilde{\phi}_{i}}{2\omega_{q_{0%
}}}-\omega_{q_{0}}^{\prime}\tilde{Z}_{i}-\omega_{q_{0}}\tilde{Z}^{\prime}_{i}%
\right)dz=\frac{\tilde{\psi}_{i}}{2\omega_{q_{0}}}dz$$
where
(5.10)
$$\psi_{i}=\phi_{i}-L_{Z_{i}}(q_{0})=\phi_{i}-(q_{0}^{\prime}Z_{i}+2q_{0}Z_{i}^{%
\prime}).$$
As discussed in the last paragraph of §5.6,
$\{\psi_{i}\}_{i}$ defines a holomorphic quadratic differential $\psi$ on $M_{0}$.
We can check from (1) in Remark 3
and (5.10) that $\psi\in\mathcal{Q}_{x_{0}}^{T}(q_{0})$.
∎
Proposition 5.2 (Hodge-Kodaira decomposition).
Let $x_{0}=(M_{0},f_{0})\in\mathcal{T}_{g,m}$.
Suppose $q_{0}\in\mathcal{Q}_{x_{0}}$ is generic.
For ${\bf v}\in{\rm Hom}(q_{0})\cong T_{q_{0}}\mathcal{Q}_{g,m}$,
let ${\tt v}\left({\bf v},q_{0}\right)\in T_{x_{0}}\mathcal{T}_{g,m}$ be the
image of ${\bf v}$
of the differential of the projection $\mathcal{Q}_{g,m}\to\mathcal{T}_{g,m}$
at $q_{0}$.
Then,
$${\bf v}(C)=\int_{C}\frac{\pi_{q_{0}}^{*}(\eta_{{\tt v}\left(\overline{{\bf v}}%
,q_{0}\right)})}{\omega_{q_{0}}}+\int_{C}\overline{\left(\frac{\pi_{q_{0}}^{*}%
(\eta_{{\tt v}\left({\bf v},q_{0}\right)})}{\omega_{q_{0}}}\right)}$$
for $C\in H_{1}(\tilde{M}_{q_{0}},\mathbb{R})^{-}=H_{1}(\tilde{M}_{q_{0}},\tilde{%
\Sigma}_{ub}(q_{0}),\mathbb{R})^{-}$.
Proof.
From the definition of the $q_{0}$-realizations,
for $\phi\in\mathcal{Q}_{x_{0}}$,
(5.11)
$$\displaystyle\int_{M_{0}}\mu\phi$$
$$\displaystyle=\int_{M_{0}}\frac{\overline{\eta_{{\tt v}\left({\bf v},q_{0}%
\right)}}}{|q_{0}|}\phi=\frac{1}{2}\int_{\tilde{M}_{q_{0}}}\frac{\overline{\pi%
_{q_{0}}^{*}\left(\eta_{{\tt v}\left({\bf v},q_{0}\right)}\right)}}{|\omega_{q%
_{0}}|^{2}}\pi_{q_{0}}^{*}(\phi)$$
$$\displaystyle=-\frac{\sqrt{-1}}{4}\int_{\tilde{M}_{q_{0}}}\overline{\left(%
\frac{\pi_{q_{0}}^{*}\left(\eta_{{\tt v}\left({\bf v},q_{0}\right)}\right)}{%
\omega_{q_{0}}}\right)}\wedge\frac{\pi_{q_{0}}^{*}(\phi)}{\omega_{q_{0}}}.$$
Let $\boldsymbol{\Omega}[q_{0},{\bf v}]^{(0,1)}$ is the $(0,1)$-part of
$\boldsymbol{\Omega}[q_{0},{\bf v}]$.
From (5.8),
a Beltrami differential
$\boldsymbol{\Omega}[q_{0},{\bf v}]^{(0,1)}/\omega_{q_{0}}$
on $\tilde{M}_{q_{0}}$
is the lift of
the infinitesimal Beltrami differential $\mu$ on $M_{0}$
associated to ${\tt v}\left({\bf v},q_{0}\right)$.
Let $\boldsymbol{\Omega}^{h}+\overline{\boldsymbol{\Omega}^{ah}}$
be the harmonic form in the de Rham cohomology class of
$\boldsymbol{\Omega}[q_{0},{\bf v}]$,
where $\boldsymbol{\Omega}^{h}$ and $\boldsymbol{\Omega}^{ah}$ are holomorphic $1$-forms on
$\tilde{M}_{q_{0}}$.
Then,
(5.12)
$$\displaystyle\int_{M_{0}}\mu\phi$$
$$\displaystyle=\frac{1}{2}\int_{\tilde{M}_{q_{0}}}\frac{\boldsymbol{\Omega}[q_{%
0},{\bf v}]^{(0,1)}}{\omega_{q_{0}}}\pi_{q_{0}}^{*}(\phi)=-\frac{\sqrt{-1}}{4}%
\int_{\tilde{M}_{q_{0}}}\boldsymbol{\Omega}[q_{0},{\bf v}]\wedge\frac{\pi_{q_{%
0}}^{*}(\phi)}{\omega_{q_{0}}}$$
$$\displaystyle=-\frac{\sqrt{-1}}{4}\int_{\tilde{M}_{q_{0}}}\overline{%
\boldsymbol{\Omega}^{ah}}\wedge\frac{\pi_{q_{0}}^{*}(\phi)}{\omega_{q_{0}}}.$$
We can easily check that every holomorphic $1$-form in the $(-1)$-eigenspace of
the space of holomorphic $1$-forms
is presented as $\pi_{q_{0}}^{*}(\phi)/\omega_{q_{0}}$
for some $\phi\in\mathcal{Q}_{x_{0}}$.
From (5.11) and (5.12),
we have $\boldsymbol{\Omega}^{ah}=\pi_{q_{0}}^{*}\left(\eta_{{\tt v}\left({\bf v},q_{0}%
\right)}\right)/\omega_{q_{0}}$.
Since the harmonic differential in the de Rham cohomology class is unique,
from (5.9),
we deduce that
$\boldsymbol{\Omega}^{h}=\pi_{q_{0}}^{*}\left(\eta_{{\tt v}\left(\overline{{\bf
v%
}},q_{0}\right)}\right)/\omega_{q_{0}}$.
∎
For a generic differential $q_{0}\in\mathcal{Q}_{g,m}$,
we define
$${\rm Hom}_{0}(q_{0})=\{{\bf v}\in{\rm Hom}(q_{0})\mid{\tt v}\left({\bf v},q_{0%
}\right)=0\}.$$
From Proposition 5.2,
we have
Corollary 5.1.
Let $x_{0}\in\mathcal{T}_{g,m}$ and $q_{0}\in\mathcal{Q}_{x_{0}}$
a generic differential.
Then,
the mapping
$${\rm Hom}_{0}(q_{0})\ni{\bf v}\mapsto\eta_{{\tt v}\left(\overline{{\bf v}},q_{%
0}\right)}\in\mathcal{Q}_{x_{0}}$$
is a complex linear isomorphism.
Example 4.
For generic $q\in\mathcal{Q}_{g,m}$,
$$\displaystyle\int_{C}\frac{\pi_{q}^{*}(q)}{\omega_{q}}=\int_{C}\omega_{q}$$
$$\displaystyle={\bf u}[q](C)=\int_{C}\frac{\pi_{q}^{*}(\eta_{{\tt v}\left(%
\overline{{\bf u}[q]},q\right)})}{\omega_{q}}+\overline{\left(\frac{\pi_{q}^{*%
}(\eta_{{\tt v}\left({\bf u}[q],q\right)})}{\omega_{q}}\right)}$$
for $C\in H_{1}(\tilde{M}_{q},\mathbb{R})^{-}$.
Therefore,
$\eta_{{\tt v}\left(\overline{{\bf u}[q]},q\right)}=q$
and
$\eta_{{\tt v}\left({\bf u}[q],q\right)}=0$.
6. Stratification of Teichmüller space
6.1. Stratifications on $\mathcal{Q}_{x_{0}}$
Let $x_{0}\in\mathcal{T}_{g,m}$.
Dumas [9]
defined a stratification of $\mathcal{Q}_{x_{0}}$ by symbols
applying the Whitney stratification
(cf. [40] and [45]).
Indeed,
the stratification on $\mathcal{Q}_{g,m}$
provides a stratification on $\mathcal{Q}_{x_{0}}$
by complex-analytic sets.
This stratification can be refined as a complex-analytic stratrification
$\{Z_{i}\}_{i\in I}=\{Z_{i,x_{0}}\}_{i\in I}$ satisfying the following conditions:
(1)
Each $Z_{i}$ is a complex submanifold of $\mathcal{Q}_{x_{0}}-\{0\}$
invariant under the action of $\mathbb{C}^{*}$;
(2)
the symbol is constant on each stratum $Z_{i}$;
(3)
$\mathcal{Q}_{x_{0}}-\{0\}=\sqcup_{i\in I}Z_{i}$;
(4)
$Z_{i}\cap Z_{j}=\emptyset$ if $i\neq j$;
(5)
the closure $Z_{j}$ is a complex-analytic set,
and if $Z_{i}\cap\overline{Z_{j}}\neq\emptyset$ for $i,j\in I$,
then $Z_{i}\subset\overline{Z_{j}}$.
The refinement refers to changing the stratification
in such a way that each new stratum is entirely contained
in one of the old strata.
Under the situation in the above (5),
$\dim_{\mathbb{C}}Z_{i}<\dim_{\mathbb{C}}Z_{j}$
if $Z_{i}\neq Z_{j}$.
There is a unique stratum $Z_{\infty}$ consisting of all generic differentials
(we assume the index set $I$ contains a symbol “$\infty$”).
Since the stratification is locally finite,
we have
Lemma 6.1.
For any $q_{0}\in Z_{i}-\{0\}$,
there is a neighborhood $U$ in $\mathcal{Q}_{x_{0}}-\{0\}$ of $q_{0}$ such that
$I_{U}=\{i\in I\mid Z_{i}\cap U\neq\emptyset\}$ is a finite set;
and if $Z_{j}\cap U\neq\emptyset$,
then $Z_{i}\cap\overline{Z_{j}}\neq\emptyset$.
We extend an observation by Dumas
as follows
(cf. Lemma 5.2 in [9]).
Proposition 6.1 (Tangent space to the strata in fibers).
Let $\{Z_{i}\}_{i\in I}$ is the stratification of $\mathcal{Q}_{y_{0}}$
defined in §6.1.
Let $q_{0}\in Z_{i}$.
If we identify the tangent space $T_{q_{0}}Z_{i}$ as a subspace of $\mathcal{Q}_{x_{0}}$,
we have $T_{q_{0}}Z_{i}\subset\mathcal{Q}_{x_{0}}^{T}(q_{0})$.
Proof.
Let $p_{0}\in\Sigma_{s}(q_{0})$
and $k_{0}=\boldsymbol{o}_{p_{0}}(q_{0})$.
For simplicity,
$q_{0}$ is assumed to be represented as $q_{0}=z^{k_{0}}dz^{2}$
around $p_{0}$ with the coordinate $z$ with $z(p_{0})=0$.
From the universal deformation of the singularities,
the deformation of $q_{0}$ around $p_{0}$
is described as the Lie derivative
$$L_{X}(z^{k_{0}}dz^{2})=(k_{0}z^{k_{0}-1}X(z)+2z^{k_{0}}X^{\prime}(z))dz^{2}$$
along a holomorphic vector field $X=X(z)(\partial/\partial z)$ around $p_{0}$,
where $X(0)=0$ if $p_{0}\in\Sigma_{m}(q_{0})$
(cf. [17, Proposition 3.1].
See also Lemma 5.2 in [9]).
One can see that
the infinitesimal deformation
$\dot{q}$
satisfies
$$\boldsymbol{o}_{p_{0}}(\dot{q})\geq\begin{cases}k_{0}-1&(\mbox{if $p_{0}\in%
\Sigma_{s\setminus m}(q_{0})$})\\
k_{0}&(\mbox{if $p_{0}\in\Sigma_{sm}(q_{0})$})\end{cases}$$
and is contained in
$\mathcal{Q}_{x_{0}}^{T}(q_{0})$.
∎
6.2. Stratification of Teichmüller space
Let $x_{0}\in\mathcal{T}_{g,m}$.
Let $\{Z_{i}\}_{i\in I}$ be the stratification
of $\mathcal{Q}_{x_{0}}-\{0\}$ defined in §6.1.
Let $\mathcal{UQ}_{x_{0}}$ be the unit ball in $\mathcal{Q}_{x_{0}}$
with respect to the $L^{1}$-norm and set
$\Xi_{x_{0}}\colon\mathcal{UQ}_{x_{0}}\to\mathcal{T}_{g,m}$
be the Teichmüller homeomorphism discussed in §2.2.2.
For $i\in I$,
we define
$\mathcal{T}_{i}=\Xi_{x_{0}}(Z_{i}\cap\mathcal{UQ}_{x_{0}})$.
The purpose of this section is to show the following.
Theorem 5 (Stratification).
The collection $\{\mathcal{T}_{i}\}_{i\in I}$ is a stratification of
real-analytic submanifolds
in $\mathcal{T}_{g,m}-\{x_{0}\}$.
Since $\Xi_{x_{0}}$ is a homeomorphism,
$\{\mathcal{T}_{i}\}_{i\in I}$ is a stratification of topological manifolds in
$\mathcal{T}_{g,m}-\{x_{0}\}$.
Namely,
each $\mathcal{T}_{i}$ is a locally closed topological submanifold of $\mathcal{T}_{g,m}-\{0\}$,
the collection $\{\mathcal{T}_{i}\}_{i\in I}$ is a locally finite
and satisfies
(3’)
$\mathcal{T}_{g,m}-\{0\}=\sqcup_{i\in I}\mathcal{T}_{i}$;
(4’)
$\mathcal{T}_{i}\cap\mathcal{T}_{j}=\emptyset$ if $i\neq j$;
and
(5’)
if $\mathcal{T}_{i}\cap\overline{\mathcal{T}_{j}}\neq\emptyset$ for $i,j\in I$,
then $\mathcal{T}_{i}\subset\overline{\mathcal{T}_{j}}$.
(The numbers correspond to those in the properties of complex-analytic statifications given in §6.1.)
We will show that
the restriction of $\Xi_{x_{0}}$ to each $Z_{i}\cap\mathcal{UQ}_{x_{0}}$
is a real-analytic immersion.
The author does not know if the closure $\overline{\mathcal{T}_{i}}$ is an real-analytic subset of $\mathcal{T}_{g,m}-\{0\}$ for each $i\in I$.
Notice that Theorem 5
is recognized as a kind of refinements of
Masur’s result [30, Proposition 2.2].
Proof of Theorem 5.
Let $i\in I$.
Notice from the definition
that $Z_{i}$ is a complex submanifold of $\mathcal{Q}_{x_{0}}$.
From (2.2) and
§5.4,
the Teichmüller homeomorphism $\Xi_{x_{0}}$ on $Z_{i}$ is described as
$$\Xi_{x_{0}}(q)=\Pi({\bf Q}[\tanh^{-1}(\|q\|),q])$$
for $q\in Z_{i}\cap\mathcal{UQ}_{x_{0}}$.
From
Proposition 4.1
and Riemann’s formula,
the norm $Z_{i}\ni q\mapsto\|q\|$ varies real-analytically
(cf. [26, §1]
and [11, Chapter III]).
Hence the mapping
$$Z_{i}\ni q\mapsto{\bf Q}[\tanh^{-1}(\|q\|),q]\in\mathcal{Q}(\boldsymbol{\pi}(q%
_{0}))$$
is real-analytic.
Therefore,
$\Xi_{x_{0}}$ is also real-analytic on $Z_{i}$
since $\Pi$ is holomorphic.
Hence,
to complete the proof,
it suffices to show that the (real) differential of the restriction of $\Xi_{x_{0}}$ to $Z_{i}$
is non-singular.
Let $q_{0}\in Z_{i}\cap\mathcal{UQ}_{x_{0}}$ and
${\bf v}\in{\rm Hom}(q_{0})$
($\hookrightarrow T_{q_{0}}\mathcal{Q}(\boldsymbol{\pi}(q_{0}))$)
which tangent to $Z_{i}$.
For simplicity,
set $Q_{0}={\bf Q}[\tanh^{-1}(\|q_{0}\|),q_{0}]$
and $x_{1}=(M_{1},f_{1})=x_{\tanh^{-1}(\|q_{0}\|),q_{0}}\in\mathcal{T}_{g,m}$
(cf. §5.4).
From Proposition 6.1,
there is $\psi\in\mathcal{Q}^{T}_{x_{0}}(q_{0})$ such that
(6.1)
$${\bf v}(C)=\int_{C}\frac{\pi_{q_{0}}^{*}(\psi)}{\omega_{q_{0}}}$$
for $C\in H_{1}(\tilde{M}_{q_{0}},\tilde{\Sigma}_{ub}(q_{0}),\mathbb{R})^{-}$.
Let $f\colon\{|\lambda|<\epsilon\}\to Z_{i}$ be a holomorphic disk
with $q(0)=q_{0}$ and $q^{\prime}(0)={\bf v}$.
For simplicity,
set $K(t)=\exp(2\tanh^{-1}(\|q(t)\|)=(1+\|q(t)\|)/(1-\|q(t)\|)$ for $t\in\mathbb{R}$ with $|t|<\epsilon$.
From
(5.3)
and (6.1),
(6.2)
$$\displaystyle\left.\frac{d}{dt}\right|_{t=0}{\bf u}[{\bf Q}[\tanh^{-1}(\|q(t)%
\|),q(t)]](C)$$
$$\displaystyle={\rm Re}\int_{C}\frac{\pi_{q_{0}}^{*}(\psi)}{\omega_{q_{0}}}+%
\sqrt{-1}{\rm Im}\left(K(0)^{-1}\int_{C}\frac{\pi_{q_{0}}^{*}(\psi)}{\omega_{q%
_{0}}}+(K^{-1})^{\prime}|_{t=0}\int_{C}\omega_{q_{0}}\right).$$
for $C\in H_{1}(\tilde{M}_{q_{0}},\tilde{\Sigma}_{ub}(q_{0}),\mathbb{R})^{-}$.
Recall that the holomorphic tangent space is the $(1,0)$-part
of the complexification of the real tangent vector space
(cf. [24, Chapter IX]).
In general,
for a complex manifold with a local chart $z=(z_{1},\cdots,z_{n})$,
a holomorphic tangent vector $\sum_{j=1}^{n}a_{j}(\partial/\partial z_{j})$
is the $(1,0)$-part of a real tangent vector
$\sum_{j=1}^{n}(a_{j}(\partial/\partial z_{j})+\overline{a_{j}}(\partial/%
\partial\overline{z}_{j}))$ of the underlying differential structure.
The variation (6.2) stands for
the $(1,0)$-part of the image of the corresponding real tangent vector to ${\bf v}$
under the (real) differential of the map $Z_{i}\ni q\mapsto{\bf Q}[\tanh^{-1}(\|q\|),q]\in\mathcal{Q}(\boldsymbol{\pi}(q%
_{0}))$
around $q_{0}$.
We denote by ${\bf w}(C)$ the right-hand side of (6.2).
Then,
${\bf w}$ stands for a homomorphism
in ${\rm Hom}(Q_{0})$
($\subset T_{Q_{0}}\mathcal{Q}(\boldsymbol{\pi}(q_{0}))$)
via the isomorphism $H_{1}(\tilde{M}_{q_{0}},\tilde{\Sigma}_{ub}(q_{0}),\mathbb{R})^{-}\cong H_{1}(%
\tilde{M}_{Q_{0}},\tilde{\Sigma}_{ub}(Q_{0}),\mathbb{R})^{-}$
induced by the Teichmüller mapping
from $x_{0}$ to $x_{1}$.
Suppose that the derivative
$$\{|t|<\epsilon\mid t\in\mathbb{R}\}\ni t\mapsto\Pi\left({\bf Q}[\tanh^{-1}(\|q%
(t)\|),q(t)]\right)$$
at $t=0$ vanishes.
We will conclude ${\bf v}=0$.
Since $\Pi$ is holomorphic,
the differential of $\Pi$
sends the $(1,0)$-part $T_{Q_{0}}\mathcal{Q}_{g,m}$
of the complexification of the real tangent space at $Q_{0}$
to that at $x_{1}$.
From the assumption,
we deduce ${\tt v}\left({\bf w},Q_{0}\right)=0$ in $T_{x_{1}}\mathcal{T}_{g,m}$
(cf. [24, Proposition 2.9, Chapter IX]).
From Proposition 5.1,
there is $\phi\in\mathcal{Q}^{T}_{x_{1}}(Q_{0})$ such that
(6.3)
$${\bf w}(C)=\int_{C}\frac{\pi_{Q_{0}}^{*}(\phi)}{\omega_{Q_{0}}}$$
for $C\in H_{1}(\tilde{M}_{q_{0}},\tilde{\Sigma}_{ub}(q_{0}),\mathbb{R})^{-}\cong H%
_{1}(\tilde{M}_{Q_{0}},\tilde{\Sigma}_{ub}(Q_{0}),\mathbb{R})^{-}$.
Let $\psi^{\prime}\in\mathcal{Q}_{x_{0}}$ and $\phi^{\prime}\in\mathcal{Q}_{x_{1}}$
be the holomorphic quadratic differentials
defined by descending the squares $(\pi_{q_{0}}^{*}(\psi)/\omega_{q_{0}})^{2}$
and $(\pi_{Q_{0}}^{*}(\phi)/\omega_{Q_{0}})^{2}$
respectively.
Comparing the real parts of (6.2) and (6.3)
we have $v(\phi^{\prime})=v(\psi^{\prime})$ in $\mathcal{MF}$
(cf. [17, Lemma 4.3]).
Since $K(0)=e^{2d_{T}(x_{0},x_{1})}$,
(6.4)
$$K(0)^{-1}\|\psi^{\prime}\|=e^{-2d_{T}(x_{0},x_{1})}{\rm Ext}_{x_{0}}(v(\psi^{%
\prime}))\leq{\rm Ext}_{x_{1}}(v(\phi^{\prime}))=\|\phi^{\prime}\|$$
from the Kerckhoff formula
(see also [13, Lemma 4.1]).
By Riemann’s formula, (3.3) and (6.2),
(6.5)
$$\displaystyle\|\phi^{\prime}\|$$
$$\displaystyle=\frac{1}{2}\int_{\tilde{M}_{Q_{0}}}{\rm Re}\left(\frac{\pi_{Q_{0%
}}^{*}(\phi)}{\omega_{Q_{0}}}\right)\wedge{\rm Im}\left(\frac{\pi_{Q_{0}}^{*}(%
\phi)}{\omega_{Q_{0}}}\right)$$
(6.6)
$$\displaystyle=K(0)^{-1}\frac{1}{2}\int_{\tilde{M}_{q_{0}}}{\rm Re}\left(\frac{%
\pi_{q_{0}}^{*}(\psi)}{\omega_{q_{0}}}\right)\wedge{\rm Im}\left(\frac{\pi_{q_%
{0}}^{*}(\psi)}{\omega_{q_{0}}}\right)$$
$$\displaystyle\qquad+(K^{-1})^{\prime}|_{t=0}\frac{1}{2}\int_{\tilde{M}_{q_{0}}%
}{\rm Re}\left(\frac{\pi_{q_{0}}^{*}(\psi)}{\omega_{q_{0}}}\right)\wedge{\rm Im%
}(\omega_{q_{0}})$$
$$\displaystyle=K(0)^{-1}\|\psi^{\prime}\|+(K^{-1})^{\prime}(0)\,{\rm Re}\left\{%
\frac{\sqrt{-1}}{4}\int_{\tilde{M}_{q_{0}}}\frac{\pi_{q_{0}}^{*}(\psi)}{\omega%
_{q_{0}}}\wedge\overline{\omega_{q_{0}}}\right\}.$$
Since
(6.7)
$$\left.\frac{d\|q(t)\|}{dt}\right|_{t=0}={\rm Re}\left\{\frac{\sqrt{-1}}{2}\int%
_{\tilde{M}_{q_{0}}}\frac{\pi_{q_{0}}^{*}(\psi)}{\omega_{q_{0}}}\wedge%
\overline{\omega_{q_{0}}}\right\},$$
from (6.4)
and (6.5),
we obtain
$$\displaystyle 0$$
$$\displaystyle\leq\|\phi^{\prime}\|-K(0)^{-1}\|\psi^{\prime}\|=(K^{-1})^{\prime%
}(0)\,{\rm Re}\left\{\frac{\sqrt{-1}}{4}\int_{\tilde{M}_{q_{0}}}\frac{\pi_{q_{%
0}}^{*}(\psi)}{\omega_{q_{0}}}\wedge\overline{\omega_{q_{0}}}\right\}$$
$$\displaystyle=-\frac{1}{(1+\|q_{0}\|)^{2}}\left(\left.\frac{d\|q(t)\|}{dt}%
\right|_{t=0}\right)^{2},$$
and $(K^{-1})^{\prime}=0$ at $t=0$.
Therefore,
$\|\phi^{\prime}\|=K(0)^{-1}\|\psi^{\prime}\|$
from (6.5) again.
Hence,
we have $v(\psi^{\prime})=sv(q_{0})$ and $\psi^{\prime}=s^{2}q_{0}$
for some $s\geq 0$
from the uniqueness of the extremal problem for the Kerckhoff formula
(or the Teichmüller uniqueness theorem. See [19, Theorem 5.9]
or [1, §(3.5)]).
From (6.7),
we obtain
$$0=\left.\frac{d\|q(t)\|}{dt}\right|_{t=0}=s^{2}\|q_{0}\|$$
and $s=0$.
Therefore,
$\psi^{\prime}=0$ and
${\bf v}=0$ from (6.1).
∎
From (2.2),
we conclude the following.
Corollary 6.1.
The Teichmüller distance function $\mathcal{T}_{g,m}-\{x_{0}\}\ni x\mapsto d_{T}(x_{0},x)$
is real-analytic on each stratum of $\{\mathcal{T}_{i}\}_{i\in I}$.
Recall that the top stratum $Z_{\infty}$ of the stratification of $\mathcal{Q}_{x_{0}}$
is an open set which consists of generic differentials.
From Theorem 5,
the restriction of the Teichmüller homeomorphism $\Xi_{x_{0}}\colon Z_{\infty}\to\mathcal{T}_{\infty}$ is a real-analytic diffeomorphism.
Hence,
Corollary 6.1 is thought of as an extension of an observation by Rees
in [38, §2.3].
6.3. Non-generic strata are removable
Let us prove Theorem
2.
We use the following removable singuality theorem due to Blanchet
(see also [5]).
Proposition 6.2 (Blanchet [2]).
Let $\Omega$ be a domain in $\mathbb{C}^{N}$ and $V\subset\Omega$ be a $C^{1}$-real submanifold with positive real codimension.
Let $u$ be a function of class $C^{1}$ on $\Omega$.
Then, $u$ is plurisubharmonic on $\Omega$ if so is $u$ on $\Omega-V$.
We return to our setting.
Let $\{\mathcal{T}_{i}\}_{i\in I}$ be the stratification
in Theorem 5.
Let $u$ be a function of class $C^{1}$ on $\mathcal{T}_{g,m}-\{x_{0}\}$
which is bounded above around $x_{0}$.
Suppose that $u$ is plurisubharmonic on the top stratum $\mathcal{T}_{\infty}$.
Let $x_{1}\in\mathcal{T}_{g,m}-\{x_{0}\}$
and $\mathcal{T}_{i}$ the stratum containing $x_{1}$.
Suppose that $u$ is extended as a plurisubharmonic function
on $\mathcal{T}_{j}$ for all $j\in I$ with $\dim\mathcal{T}_{j}>\dim\mathcal{T}_{i}$.
From the locally finiteness of the stratification,
there is a small neighborhood $U$ of $x_{1}$ such that
$I(U)=\{j\in I\mid\mathcal{T}_{j}\cap U\neq\emptyset\}$ is a finite set
and $\mathcal{T}_{i}\cap\overline{\mathcal{T}_{j}}\neq\emptyset$
for $j\in I(U)$
from Lemma 6.1.
From the assumption,
$u$ is plurisubharmonic on $U-\mathcal{T}_{i}$.
Since $\mathcal{T}_{i}$ is a real-analytic submanifold of $\mathcal{T}_{g,m}$
with positive codimension,
by Blanchet’s extension theorem (Proposition 6.2),
$u$ is plurisubharmonic on $U$.
This inductive procedure guarantees that $u$ is plurisubharmonic
on $\mathcal{T}_{g,m}-\{x_{0}\}$.
Since $u$ is bounded above around $x_{0}$,
$u$ is extended as a pluriharmonic function on $\mathcal{T}_{g,m}$
(cf. [23, Theorem 2.9.22]).
∎
7. Pluricomplex Green function on the Teichmüller space
In this section,
we will show the following theorem
which implies Theorem 1,
since the Teichmüller distance is the Kobayashi distance on $\mathcal{T}_{g,m}$
(cf. [39] and §7.1).
Theorem 6 (Plurisubharmonicity).
Let $x_{0}\in\mathcal{T}_{g,m}$.
The log-tanh of the Teichmüller distance function
$$\mathcal{T}_{g,m}\ni x\mapsto u_{x_{0}}(x):=\log\tanh d_{T}(x_{0},x)$$
is plurisubharmonic on $\mathcal{T}_{g,m}$.
Earle [10]
showed that $u_{x_{0}}$ is of class $C^{1}$ on $\mathcal{T}_{g,m}-\{x_{0}\}$.
Since $u_{x_{0}}(x)\to-\infty$ as $x\to x_{0}$,
from Theorem 2,
it suffices to show that $u_{x_{0}}$ is plurisubharmonic on
the top stratum $\mathcal{T}_{\infty}$.
7.1. Complex analysis
Let $X$ be a complex manifold.
Let $p\in X$ and $z=(z_{1},\cdots,z_{n})$ be a holomorphic local chart
around $p$.
Let $u$ be a $C^{2}$ function around $p$ on $X$.
For $v=\sum_{i=1}^{n}v_{i}(\partial/\partial z_{i})\in T_{p}X$,
we define the Levi form of $u$
by
$$\mathcal{L}(u)[v,\overline{v}]=\sum_{i,j=1}^{n}\frac{\partial^{2}u}{\partial z%
_{i}\partial\overline{z}_{j}}(z(p))v_{i}\overline{v_{j}}.$$
Let $g\colon\{\lambda\in\mathbb{C}\mid|\lambda|<\epsilon\}\to X$
be a holomorphic mapping with $g(0)=p$ and $g_{*}(\partial/\partial\lambda)=v$.
Then, we see
(7.1)
$$\mathcal{L}(u)[v,\overline{v}]=\frac{\partial^{2}(u\circ g)}{\partial\lambda%
\partial\overline{\lambda}}(0).$$
A $C^{2}$-function $u$ on $X$ is called plurisubharmonic
if $\mathcal{L}(u)[v,\overline{v}]\geq 0$ for $v\in T_{p}X$ and $p\in X$.
In general,
a function $u$ on a domain $\Omega$ on $\mathbb{C}^{N}$ is called plurisubharmonic if for any $a\in\Omega$ and $b\in\mathbb{C}^{N}$,
$\lambda\mapsto u(a+\lambda b)$ is subharmonic or
identically $-\infty$ on every component of $\{\lambda\in\mathbb{C}\mid a+\lambda b\in\Omega\}$.
A bounded domain $\Omega$ in $\mathbb{C}^{N}$ is said to be hyperconvex
if it admits a negative continuous plurisubharmonic exhaustion
(cf. [41]).
Krushkal [27] showed that
Teichmüller space is hyperconvex (see also [33]).
Demailly [6] observed that
for any bounded hyperconvex domain $\Omega$ in $\mathbb{C}^{n}$
and $w\in\Omega$,
there is a unique plurisubharmonic function $g_{\Omega,w}\colon\Omega\to[-\infty,0)$
such that
(1)
$(dd^{c}g_{\Omega,w})^{n}=(2\pi)^{n}\delta_{w}$,
where $\delta_{w}$ is the Dirac measure with support at $w$;
and
(2)
$g_{\Omega,w}(z)=\sup_{v}\{v(z)\}$
where the supremum runs over all non-positive plurisubharmonic function $v$
on $\Omega$ with $v(z)\leq\log\|z-w\|+O(1)$ around $z=w$.
(cf. [6, Théorème 4.3]).
The function $g_{\Omega}(w,z)=g_{\Omega,w}(z)$ is called
the pluricomplex Green function on $\Omega$.
Klimek showed that
(7.2)
$$\log\tanh{\rm Car}_{\Omega}(z,w)\leq g_{\Omega}(z,w)\leq\log\tanh{\rm Kob}_{%
\Omega}(z,w)$$
for $z,w\in\Omega$
and the equality in the second inequality in (7.2)
holds if the third term of (7.2)
is plurisubharmonic,
where ${\rm Car}_{\Omega}$ and ${\rm Kob}_{\Omega}$ are
the Carathéodory distance and the Kobayashi distance on $\Omega$,
respectively
(cf. [22, Corollaries 1.2 and 1.4]).
7.2. Setting
Let $q_{0}\in Z_{\infty}\cap\mathcal{Q}_{x_{0}}$
and $x_{1}=\Xi_{x_{0}}(q_{0})$.
Let $v\in T_{x_{1}}\mathcal{T}_{g,m}$ and $g\colon\{|\lambda|<\epsilon\}\to\mathcal{T}_{\infty}$ a holomorphic mapping with $g(0)=0$ and
$g_{*}(\left.\partial/\partial\lambda\right|_{\lambda=0})=v$.
For the simplicity,
let
$$\displaystyle q_{\lambda}$$
$$\displaystyle=\Xi_{x_{0}}^{-1}(g(\lambda))\in Z_{\infty}\subset\mathcal{Q}_{x_%
{0}},$$
$$\displaystyle Q_{\lambda}$$
$$\displaystyle={\bf Q}[\tanh^{-1}(\|q_{\lambda}\|),q_{\lambda}]\in\mathcal{Q}_{%
g(\lambda)},$$
$$\displaystyle d(\lambda)$$
$$\displaystyle=d_{T}(x_{0},g(\lambda)),\ \mbox{and}\ d_{0}=d(0)=d_{T}(x_{0},x_{%
1})$$
for $\lambda\in\{|\lambda|<\epsilon\}$.
Notice again that each $Q_{\lambda}$ is generic
since the Teichmüller mapping preserves the order of singular points.
For calculations later,
we notice from the definition that
(7.3)
$$\|q_{\lambda}\|=\tanh(d(\lambda))\ \mbox{and}\ \|Q_{\lambda}\|=e^{-2d(\lambda)%
}\|q_{\lambda}\|.$$
7.2.1.
We define ${\bf v}_{1}$, ${\bf v}_{2}\in{\rm Hom}(q_{0})$
($\cong T_{q_{0}}\mathcal{Q}_{g,m}$)
by
(7.4)
$${\bf u}[q_{\lambda}]={\bf u}[q_{0}]+\lambda{\bf v}_{1}+\overline{\lambda}{\bf v%
}_{2}+o(|\lambda|)$$
as $\lambda\to 0$
on $H_{1}(\tilde{M}_{q_{0}},\mathbb{R})^{-}\cong H_{1}(\tilde{M}_{q_{0}},\tilde{%
\Sigma}_{ub}(q_{0}),\mathbb{R})^{-}$.
Since $q_{\lambda}\in\mathcal{Q}_{x_{0}}$ for all $\lambda$,
we deduce ${\tt v}\left({\bf v}_{i},q_{0}\right)=0$ for $i=1,2$
and ${\bf v}_{1}$,
${\bf v}_{2}\in{\rm Hom}_{0}(q_{0})$
($\cong\mathcal{Q}_{x_{0}}\subset T_{q_{0}}\mathcal{Q}_{g,m}$).
We will use the notation
(7.5)
$${\tt D}_{x_{1}}(v)={\bf v}_{1},\ \mbox{and}\ \overline{{\tt D}}_{x_{1}}(v)={%
\bf v}_{2}$$
after calculating the first derivative and the Levi form of the Teichmüller distance
(cf. §7.5).
However,
in the following calculation,
we will use the notation ${\bf v}_{1}$
and ${\bf v}_{2}$ for the simplicity.
From (5.2)
and (7.4),
$$\displaystyle{\bf u}[Q_{\lambda}]$$
$$\displaystyle={\rm Re}({\bf u}[q_{\lambda}])+\sqrt{-1}e^{-2d(\lambda)}{\rm Im}%
({\bf u}[q_{\lambda}])$$
$$\displaystyle={\bf u}[q_{0}]+\lambda{\bf w}_{1}+\overline{\lambda}{\bf w}_{2}+%
o(|\lambda|)$$
on $H_{1}(\tilde{M}_{q_{0}},\mathbb{R})^{-}$
as $\lambda\to 0$,
where
${\bf w}_{1}$ and ${\bf w}_{2}$
are in ${\rm Hom}(q_{0})$ defined by
(7.6)
$$\displaystyle\begin{cases}{\bf w}_{1}&={\displaystyle\frac{1+e^{-2d_{0}}}{2}{%
\bf v}_{1}+\frac{1-e^{-2d_{0}}}{2}\overline{{\bf v}_{2}}-d_{\lambda}\,e^{-2d_{%
0}}({\bf u}[q_{0}]-\overline{{\bf u}[q_{0}]})}\\
{\bf w}_{2}&={\displaystyle\frac{1-e^{-2d_{0}}}{2}\overline{{\bf v}_{1}}+\frac%
{1+e^{-2d_{0}}}{2}{\bf v}_{2}-\overline{d_{\lambda}}\,e^{-2d_{0}}({\bf u}[q_{0%
}]-\overline{{\bf u}[q_{0}]}),}\end{cases}$$
where $d_{\lambda}$ is the $\lambda$-derivative of the Teichmüller distance function $d(\lambda)$
at $\lambda=0$.
In
(7.6),
we canonically identify ${\rm Hom}(Q_{0})$ with ${\rm Hom}(q_{0})$,
and ${\bf w}_{1}$ and ${\bf w}_{2}$ stands for
tangent vectors in $T_{Q_{0}}\mathcal{Q}_{g,m}\cong{\rm Hom}(Q_{0})$,
while the right-hand sides of
(7.6)
are tangent vectors in ${\rm Hom}(q_{0})\cong T_{q_{0}}\mathcal{Q}_{g,m}$.
See the discussion in the proof of Theorem 5
and (2) of Remark 3.
Since $\Pi({\bf u}[Q_{\lambda}])=g(\lambda)$,
we have
(7.7)
$${\tt v}\left({\bf w}_{1},Q_{0}\right)=v\quad\mbox{and}\quad{\tt v}\left({\bf w%
}_{2},Q_{0}\right)=0$$
and ${\bf w}_{2}\in{\rm Hom}_{0}(Q_{0})$,
since $\Pi$ is holomorphic.
7.2.2.
Since $H_{1}(\tilde{M}_{q_{0}},\mathbb{R})^{-}\cong H_{1}(\tilde{M}_{q_{0}},\tilde{%
\Sigma}_{ub}(q_{0}),\mathbb{R})^{-}$,
${\rm Hom}(q_{0})$ is canonically isomorphic to the cohomology group
$H^{1}(\tilde{M}_{q_{0}},\mathbb{R})^{-}$.
We define the wedge product $\wedge$ on ${\rm Hom}(q_{0})$
by
$${\bf x}\wedge{\bf y}=\int_{\tilde{M}_{q_{0}}}\frac{\pi_{q_{0}}^{*}(\eta_{{\tt v%
}\left(\overline{{\bf x}},q_{0}\right)})}{\omega_{q_{0}}}\wedge\overline{\left%
(\frac{\pi_{q_{0}}^{*}(\eta_{{\tt v}\left({\bf y},q_{0}\right)})}{\omega_{q_{0%
}}}\right)}-\frac{\pi_{q_{0}}^{*}(\eta_{{\tt v}\left(\overline{{\bf y}},q_{0}%
\right)})}{\omega_{q_{0}}}\wedge\overline{\left(\frac{\pi_{q_{0}}^{*}(\eta_{{%
\tt v}\left({\bf x},q_{0}\right)})}{\omega_{q_{0}}}\right)}$$
for ${\bf x}$, ${\bf y}\in{\rm Hom}(q_{0})$.
For generic $q\in\mathcal{Q}_{g,m}$,
(7.8)
$$\displaystyle\sqrt{-1}{\bf u}[q]\wedge\overline{{\bf u}[q]}$$
$$\displaystyle=4\|q\|$$
(cf. (2.2) and Example 4).
Example 7 (Teichmüller disk).
The Teichmüller disk associated to $q_{0}$ is defined as an isometric holomorphic disk
in $\mathcal{T}_{g,m}$
defined by the holomorphic family of Beltrami differentials
$$\mathbb{D}\ni\lambda\mapsto\left(\frac{\lambda+\tanh(d_{0})}{1+\tanh(d_{0})%
\lambda}\right)\frac{\overline{q_{0}}}{|q_{0}|}$$
on $M_{0}$.
With the Teichmüller homeomorphism (2.1),
the Teichmüller disk is described as
(7.9)
$$\mathbb{D}\ni\lambda\mapsto\Xi_{x_{0}}\left(\left(\frac{\overline{\lambda}+%
\tanh(d_{0})}{1+\tanh(d_{0})\overline{\lambda}}\right)\frac{q_{0}}{\|q_{0}\|}%
\right).$$
For $\lambda\in\mathbb{D}$.
let ${\bf D}_{q_{0}}(\lambda)$ be the right-hand side of (7.9).
By definition,
${\bf D}_{q_{0}}(0)=x_{1}$ and ${\bf D}_{q_{0}}(-\tanh(d_{0}))=x_{0}$.
Then
(7.10)
$$\displaystyle{\bf u}[\Xi_{x_{0}}^{-1}({\bf D}_{q_{0}}(\lambda))]$$
$$\displaystyle=\frac{1}{\|q_{0}\|^{1/2}}\left(\frac{\overline{\lambda}+\tanh(d_%
{0})}{1+\tanh(d_{0})\overline{\lambda}}\right)^{1/2}{\bf u}[q_{0}]$$
$$\displaystyle={\bf u}[q_{0}]+\frac{\overline{\lambda}}{\sinh(2d_{0})}{\bf u}[q%
_{0}]+o(|\lambda|),$$
where the branch of the square root taken to be $1^{1/2}=1$.
7.3. The first variation of the Teichmüller distance
We give the first variational formula of the Teichmüller distance function
in our setting.
From Riemann’s formula,
(7.3) and (7.8),
we deduce
$$\displaystyle d_{\lambda}$$
$$\displaystyle=\left.(\tanh^{-1}(\|q_{\lambda}\|))_{\lambda}\right|_{\lambda=0}%
=\frac{1}{1-\|q_{0}\|^{2}}\frac{\sqrt{-1}}{4}({\bf v}_{1}\wedge\overline{u[q_{%
0}]}+u[q_{0}]\wedge\overline{{\bf v}_{2}})$$
(7.11)
$$\displaystyle=\frac{\sqrt{-1}\cosh^{2}(d_{0})}{4}({\bf v}_{1}\wedge\overline{u%
[q_{0}]}+u[q_{0}]\wedge\overline{{\bf v}_{2}}).$$
On the other hand,
Earle [10] gave the first variational formula
$$d_{\lambda}=\frac{1}{2\|Q_{0}\|}\int_{M_{1}}\mu Q_{0}$$
where $\mu$ is the infinitesimal Beltrami differential on $M_{1}$ representing $v$.
From (7.3),
(7.7)
and (7.8),
$$\displaystyle d_{\lambda}=\frac{1}{2\|Q_{0}\|}\int_{M_{1}}\mu Q_{0}=\frac{1}{2%
e^{-2d_{0}}\tanh(d_{0})}\int_{M_{1}}\frac{\overline{\eta_{v}}}{|Q_{0}|}Q_{0}$$
$$\displaystyle=\frac{e^{2d_{0}}}{2e\tanh(d_{0})}\frac{-\sqrt{-1}}{4}\int_{%
\tilde{M}_{Q_{0}}}\overline{\left(\frac{\pi_{Q_{0}}^{*}(\eta_{{\tt v}\left({%
\bf w}_{1},Q_{0}\right)})}{\omega_{Q_{0}}}\right)}\wedge\omega_{Q_{0}}=\frac{-%
\sqrt{-1}e^{2d_{0}}}{8\tanh(d_{0})}{\bf w}_{1}\wedge{\bf u}[Q_{0}]$$
$$\displaystyle=\frac{-\sqrt{-1}e^{2d_{0}}}{8\tanh(d_{0})}\left(\frac{1+e^{-2d_{%
0}}}{2}{\bf v}_{1}+\frac{1-e^{-2d_{0}}}{2}\overline{{\bf v}_{2}}-d_{\lambda}\,%
e^{-2d_{0}}({\bf u}[q_{0}]-\overline{{\bf u}[q_{0}]})\right)$$
$$\displaystyle\qquad\qquad\qquad\qquad\wedge\left(\frac{1+e^{-2d_{0}}}{2}{\bf u%
}[q_{0}]+\frac{1-e^{-2d_{0}}}{2}\overline{{\bf u}[q_{0}]}\right)$$
$$\displaystyle=\frac{-\sqrt{-1}e^{2d_{0}}}{8\tanh(d_{0})}\left(\frac{1-e^{-4d_{%
0}}}{4}\overline{{\bf v}_{2}}\wedge{\bf u}[q_{0}]+d_{\lambda}e^{-2d_{0}}\frac{%
1+e^{-2d_{0}}}{2}\overline{{\bf u}[q_{0}]}\wedge{\bf u}[q_{0}]\right.$$
$$\displaystyle\quad\qquad\qquad\qquad\qquad+\left.\frac{1-e^{-4d_{0}}}{4}{\bf v%
}_{1}\wedge\overline{{\bf u}[q_{0}]}-d_{\lambda}e^{-2d_{0}}\frac{1-e^{-2d_{0}}%
}{2}{\bf u}[q_{0}]\wedge\overline{{\bf u}[q_{0}]}\right)$$
$$\displaystyle=-\frac{\sqrt{-1}\cosh^{2}(d_{0})}{8}\left({\bf v}_{1}\wedge%
\overline{{\bf u}[q_{0}]}-{\bf u}[q_{0}]\wedge\overline{{\bf v}_{2}}\right)+%
\frac{\sqrt{-1}d_{\lambda}}{8\tanh(d_{0})}{\bf u}[q_{0}]\wedge\overline{{\bf u%
}[q_{0}]}$$
$$\displaystyle=-\frac{\sqrt{-1}\cosh^{2}(d_{0})}{8}\left({\bf v}_{1}\wedge%
\overline{{\bf u}[q_{0}]}-{\bf u}[q_{0}]\wedge\overline{{\bf v}_{2}}\right)+%
\frac{d_{\lambda}}{2}.$$
Therefore,
we obtain
(7.12)
$$d_{\lambda}=\frac{\sqrt{-1}\cosh^{2}(d_{0})}{4}\left(-{\bf v}_{1}\wedge%
\overline{{\bf u}[q_{0}]}+{\bf u}[q_{0}]\wedge\overline{{\bf v}_{2}}\right).$$
Thus,
from (7.11) and (7.12)
we conclude the following.
Lemma 7.1 (First variational formula).
Under the notations in §7.2,
we have
$$d_{\lambda}=\frac{\sqrt{-1}\cosh^{2}(d_{0})}{4}{\bf u}[q_{0}]\wedge\overline{{%
\bf v}_{2}}$$
and ${\bf v}_{1}\wedge\overline{{\bf u}[q_{0}]}=0$.
7.4. Levi form of $d_{T}$
Let $u[q_{\lambda}]_{\lambda}\in{\rm Hom}(q_{0})$ be the $\lambda$-derivative of
the family $\{{\bf u}[q_{\lambda}]\}_{|\lambda|<\epsilon}$
of the representation.
Namely,
$u[q_{\lambda}]_{\lambda}(C)=(u[q_{\lambda}](C))_{\lambda}$
for $C\in H_{1}(\tilde{M}_{q_{0}},\mathbb{R})^{-}$.
Notice from the notation in §7.2
that $u[q_{\lambda}]_{\lambda}={\bf v}_{1}$
at $\lambda=0$.
We also define $u[q_{\lambda}]_{\overline{\lambda}}$
and $u[q_{\lambda}]_{\lambda\overline{\lambda}}$ in the same manner.
From
Lemma 7.1,
the $\lambda$-derivative of $d(\lambda)=d_{T}(x_{0},g(\lambda))$
on a disk $\{|\lambda|<\epsilon\}$ is
rewritten as
$$d_{\lambda}(\lambda)=\frac{\sqrt{-1}\cosh^{2}(d(\lambda))}{4}{\bf u}[q_{%
\lambda}]\wedge\overline{{\bf u}[q_{\lambda}]_{\overline{\lambda}}}.$$
Therefore,
$$\displaystyle d_{\lambda\overline{\lambda}}(0)$$
$$\displaystyle=\frac{\sqrt{-1}}{4}\cdot 2\sinh(d_{0})\cosh(d_{0})d_{\overline{%
\lambda}}\cdot{\bf u}[q_{0}]\wedge\overline{{\bf v}_{2}}$$
$$\displaystyle\quad+\frac{\sqrt{-1}\cosh^{2}(d_{0})}{4}{\bf v}_{2}\wedge%
\overline{{\bf v}_{2}}+\frac{\sqrt{-1}\cosh^{2}(d_{0})}{4}{\bf u}[q_{0}]\wedge%
\overline{{\bf u}[q_{0}]_{\lambda\overline{\lambda}}\mid_{\lambda=0}}.$$
From Lemma 7.1 again,
$$\displaystyle 0$$
$$\displaystyle=\left({\bf u}[q_{\lambda}]\wedge\overline{{\bf u}[q_{\lambda}]_{%
\lambda}}\right)_{\lambda}={\bf u}[q_{\lambda}]_{\lambda}\wedge\overline{{\bf u%
}[q_{\lambda}]_{\lambda}}+{\bf u}[q_{\lambda}]\wedge\overline{{\bf u}[q_{%
\lambda}]_{\lambda\overline{\lambda}}}$$
Thus,
the Laplacian $d_{\lambda\overline{\lambda}}(0)$ of the distance function
$d(\lambda)$ at $\lambda=0$ is
(7.13)
$$\frac{\cosh^{3}(d_{0})\sinh(d_{0})}{8}|{\bf u}[q_{0}]\wedge\overline{{\bf v}_{%
2}}|^{2}+\frac{\sqrt{-1}\cosh^{2}(d_{0})}{4}({\bf v}_{2}\wedge\overline{{\bf v%
}_{2}}-{\bf v}_{1}\wedge\overline{{\bf v}_{1}}).$$
7.5. Complex tangent spaces of the spheres
We use the notation (7.5).
Notice that
$$\displaystyle T_{x_{1}}\mathcal{T}_{g,m}\ni v\mapsto{\tt D}_{x_{1}}(v)\in{\rm
Hom%
}_{0}(q_{0})$$
$$\displaystyle T_{x_{1}}\mathcal{T}_{g,m}\ni v\mapsto\overline{{\tt D}}_{x_{1}}%
(v)\in{\rm Hom}_{0}(q_{0})$$
are complex and anti-complex linear respectively.
From (7.1), Lemma 7.1 and
(7.13),
the first derivative and the Levi form of the Teichmüller distance
function $\mathcal{T}_{g,m}\ni x\mapsto d_{T}(x_{0},x)$
at $x_{1}\in\mathcal{T}_{g,m}-\{x_{0}\}$
are rewritten as
$$\displaystyle\partial\,d_{T}(x_{0},\,\cdot\,)[v]$$
$$\displaystyle=\frac{\sqrt{-1}\cosh^{2}(d_{0})}{4}{\bf u}[q_{0}]\wedge\overline%
{\overline{{\tt D}}_{x_{1}}(v)}$$
(7.14)
$$\displaystyle\mathcal{L}(d_{T}(x_{0},\,\cdot\,))[v,\overline{v}]$$
$$\displaystyle=\frac{\cosh^{3}(d_{0})\sinh(d_{0})}{8}|{\bf u}[q_{0}]\wedge%
\overline{\overline{{\tt D}}_{x_{1}}(v)}|^{2}$$
$$\displaystyle+\frac{\sqrt{-1}\cosh^{2}(d_{0})}{4}(\overline{{\tt D}}_{x_{1}}(v%
)\wedge\overline{\overline{{\tt D}}_{x_{1}}(v)}-{\tt D}_{x_{1}}(v)\wedge%
\overline{{\tt D}_{x_{1}}(v)})$$
for $v\in T_{x_{1}}\mathcal{T}_{g,m}$.
For $v_{1},v_{2}\in T_{x_{1}}\mathcal{T}_{g,m}$,
the Hermitian form of the Levi form
is represented as
(7.15)
$$\displaystyle\mathcal{L}(d_{T}(x_{0},\,\cdot\,))[v_{1},\overline{v_{2}}]$$
$$\displaystyle=\frac{\cosh^{3}(d_{0})\sinh(d_{0})}{8}{\bf u}[q_{0}]\wedge%
\overline{\overline{{\tt D}}_{x_{1}}(v_{1})}\cdot\overline{{\bf u}[q_{0}]}%
\wedge\overline{{\tt D}}_{x_{1}}(v_{2})$$
$$\displaystyle+\frac{\sqrt{-1}\cosh^{2}(d_{0})}{4}(\overline{{\tt D}}_{x_{1}}(v%
_{2})\wedge\overline{\overline{{\tt D}}_{x_{1}}(v_{1})}-{\tt D}_{x_{1}}(v_{1})%
\wedge\overline{{\tt D}_{x_{1}}(v_{2})})$$
from (7.14).
For $r>0$,
we consider the sphere $S(x_{0},r)=\{x\in\mathcal{T}_{g,m}\mid d_{T}(x_{0},x)=r\}$
of the Teichmüller distance.
For $x_{1}\in S(x_{0},r)$,
we define
$$\displaystyle H^{1,0}_{x_{1}}=H^{1,0}_{x_{1}}(S(x_{0},r))$$
$$\displaystyle=\{v\in T_{x_{1}}\mathcal{T}_{g,m}\mid\partial\,d_{T}(x_{0},\,%
\cdot\,)[v]=0\}$$
$$\displaystyle=\{v\in T_{x_{1}}\mathcal{T}_{g,m}\mid{\bf u}[q_{0}]\wedge%
\overline{\overline{{\tt D}}_{x_{1}}(v)}=0\}.$$
The subspace $H^{1,0}_{x_{1}}$ is
the $(1,0)$-part of the complex tangent space of the sphere $S(x_{0},r)$
(cf. [3, §7]).
Let $\boldsymbol{\nu}(x_{1})\in T_{x_{1}}\mathcal{T}_{g,m}$ be the tangent vector
associated to the infinitesimal Beltrami differential $\overline{Q_{0}}/|Q_{0}|$
where $Q_{0}$ is the terminal differential of the Teichmüller mapping
from $x_{0}$ to $x_{1}$.
The vector $\boldsymbol{\nu}(x_{1})\in T_{x_{1}}\mathcal{T}_{g,m}$ is tangent
to the Teichmüller disk passing $x_{0}$ and $x_{1}$.
From (7.10),
(7.16)
$$\displaystyle{\tt D}_{x_{1}}(\boldsymbol{\nu}(x_{1}))=0,\quad\overline{{\tt D}%
}_{x_{1}}(\boldsymbol{\nu}(x_{1}))=\frac{1}{\sinh(2d_{0})}{\bf u}[q_{0}].$$
Proposition 7.1 (CR tangent space).
The complex tangent space $H^{1,0}_{x_{1}}$ is perpendicular to
$\boldsymbol{\nu}(x_{1})$ with respect to the Levi form of
the Teichmüller distance.
Proof.
From (7.15) and (7.16),
$\mathcal{L}(d_{T}(x_{0},\,\cdot\,))[v,\overline{\boldsymbol{\nu}(x_{1})}]=0$
for $v\in H^{1,0}_{x_{1}}$.
∎
Lemma 7.2 (Non-negativity on $H^{1,0}$).
For $v\in H^{1,0}_{x_{1}}$,
$$\sqrt{-1}(\overline{{\tt D}}_{x_{1}}(v)\wedge\overline{\overline{{\tt D}}_{x_{%
1}}(v)}-{\tt D}_{x_{1}}(v)\wedge\overline{{\tt D}_{x_{1}}(v)})\geq 0$$
Proof.
From [33, Corollary 1.2],
the Teichmüller distance function $\mathcal{T}_{g,m}\ni x\mapsto d_{T}(x_{0},x)$ is plurisubharmonic
(see also [28, Corollary 3]).
Hence the Levi-form of the distance function is non-negative
on $H^{1,0}_{x_{1}}$.
The assertion follows from (7.14).
∎
7.6. Log-tanh of $d_{T}$ is plurisubharmonic on $\mathcal{T}_{\infty}$
We set
$$u_{x_{0}}(x)=\log\tanh(d_{T}(x_{0},x))$$
for $x\in\mathcal{T}_{g,m}$.
From (2.2),
$$u_{x_{0}}(\Xi_{x_{0}}(q))=\log\|q\|$$
for $q\in\mathcal{UQ}_{x_{0}}$.
Recall that $\mathcal{T}_{\infty}$
is the top stratum of the stratification of $\mathcal{T}_{g,m}-\{x_{0}\}$
which is obtained in Theorem 5.
Theorem 8 (Plurisubharmonicity).
$u_{x_{0}}$ is plurisubharmonic on $\mathcal{T}_{\infty}$.
Proof.
Let $x_{1}\in\mathcal{T}_{\infty}$ and $q_{0}\in\mathcal{T}_{x_{0}}$ with $x_{1}=\Xi_{x_{0}}(q_{0})$ as above.
Since $u_{x_{0}}(g(\lambda))=\log\|q_{\lambda}\|$
for $\lambda\in\{|\lambda|<\epsilon\}$,
from the direct calculation,
the Levi form of $u_{x_{0}}$
is given as
(7.17)
$$\displaystyle\mathcal{L}(u_{x_{0}})[v,\overline{v}]$$
$$\displaystyle=\frac{\sqrt{-1}}{4\tanh(d_{0})}\left(\overline{{\tt D}}_{x_{1}}(%
v)\wedge\overline{\overline{{\tt D}}_{x_{1}}(v)}-{\tt D}_{x_{1}}(v)\wedge%
\overline{{\tt D}_{x_{1}}(v)}\right)$$
$$\displaystyle\qquad\qquad-\frac{1}{16\tanh^{2}(d_{0})}|{\bf u}[q_{0}]\wedge%
\overline{\overline{{\tt D}}_{x_{1}}(v)}|^{2}$$
for $v\in T_{x_{1}}\mathcal{T}_{g,m}$.
From
(7.3) and (7.8),
(7.14)
and (7.16),
$$\displaystyle\mathcal{L}(u_{x_{0}})[\boldsymbol{\nu}(x_{1}),\overline{%
\boldsymbol{\nu}(x_{1})}]$$
$$\displaystyle=\frac{\sqrt{-1}}{4\tanh(d_{0})}\frac{{\bf u}[q_{0}]}{\sinh(2d_{0%
})}\wedge\overline{\frac{{\bf u}[q_{0}]}{\sinh(2d_{0})}}-\frac{1}{16\tanh^{2}(%
d_{0})}\left|{\bf u}[q_{0}]\wedge\overline{\frac{{\bf u}[q_{0}]}{\sinh(2d_{0})%
}}\right|^{2}=0.$$
From Lemma 7.2,
the Levi form of $u_{x_{0}}$ is non-negative on $H^{1,0}_{x_{1}}$.
Applying the calculation in Proposition 7.1
to (7.17),
we also deduce that
the normal vector $\boldsymbol{\nu}(x_{1})$ is perpendicular
to $H^{1,0}_{x_{1}}$ with respect
to the Levi form of $u_{x_{0}}$.
Therefore,
the Levi form of $u_{x_{0}}$ is non-negative
on the whole $T_{x_{1}}\mathcal{T}_{g,m}$.
∎
7.7. Topological description of the Levi form
The space $\mathcal{MF}$ carries a natural symplectic structure
with the Thurston symplectic form $\omega_{Th}$ (cf. [36]).
Dumas [9, Theorem 5.3] introduced a Kähler (symplectic)
structure on each stratum of $\mathcal{Q}_{x_{0}}$
discussed in §6.1
which defined from the Levi-form of the $L^{1}$-norm on $\mathcal{Q}_{x_{0}}$,
and observed that the Hubbard-Masur homeomorphism
(2.3)
is a real-analytic symplectomorphism on each stratum of $\mathcal{Q}_{x_{0}}$
(cf. [9, Theorem 5.8]).
In fact,
when $q_{0}\in\mathcal{Q}_{x_{0}}$ is generic,
Dumas showed that
the Hubbard-Masur homeomorphism
(2.3) is a diffeomorphism around $q_{0}$ and satisfies
(7.18)
$$\displaystyle\omega_{Th}(d\mathcal{V}_{x_{0}}(\psi_{1}),d\mathcal{V}_{x_{0}}(%
\psi_{2}))$$
$$\displaystyle={\rm Im}\int_{M_{0}}\frac{\psi_{1}\overline{\psi_{2}}}{4|q_{0}|}$$
$$\displaystyle=\frac{1}{8}\int_{\tilde{M}_{q_{0}}}{\rm Re}\left(\frac{\pi_{q_{0%
}}^{*}(\psi_{1})}{\omega_{q_{0}}}\right)\wedge{\rm Re}\left(\frac{\pi_{q_{0}}^%
{*}(\psi_{2})}{\omega_{q_{0}}}\right)$$
for $\psi_{1},\psi_{2}\in T_{q_{0}}\mathcal{Q}_{x_{0}}=\mathcal{Q}_{x_{0}}$
(cf. (3.3) and [9, §5.2]).
Remark 9.
Dumas [9] discussed the Hubbard-Masur homeomorphism
(2.3) by assigning the horizontal foliations to quadratic differentials
in accordance with Hubbard and Masur’s original discussion.
The original Hubbard-Masur homeomorphism
$\mathcal{H}_{x_{0}}\colon\mathcal{Q}_{x_{0}}\to\mathcal{MF}$
satisfies $\mathcal{V}_{x_{0}}(q)=\mathcal{H}_{x_{0}}(-q)$.
Hence,
$d\mathcal{V}_{x_{0}}(\psi)=-d\mathcal{H}_{x_{0}}(\psi)$
for $\psi\in\mathcal{Q}_{x_{0}}=T_{q_{0}}\mathcal{Q}_{x_{0}}$.
Thus,
the formula (7.18) also holds in our case.
Let us go back to the notion in §7.2.
Notice that
(7.19)
$$\displaystyle{\bf v}_{2}\wedge\overline{{\bf v}_{2}}-{\bf v}_{1}\wedge%
\overline{{\bf v}_{1}}$$
$$\displaystyle=({\bf v}_{2}+\overline{{\bf v}}_{1})\wedge(\overline{{\bf v}}_{2%
}+{\bf v}_{1})=\left({\rm Re}({\bf u}[q_{\lambda}])\right)_{\overline{\lambda}%
}\wedge\overline{\left({\rm Re}({\bf u}[q_{\lambda}])\right)_{\overline{%
\lambda}}}$$
$$\displaystyle=-\frac{\sqrt{-1}}{2}{\rm Re}\left({\bf u}[q_{\lambda}]_{\xi_{1}}%
\right)\wedge{\rm Re}\left({\bf u}[q_{\lambda}]_{\xi_{2}}\right)$$
at $\lambda=0$,
where $\lambda=\xi_{1}+\xi_{2}\sqrt{-1}$.
Therefore,
when
$$q_{\lambda}=\Xi_{x_{0}}(g(\lambda))=q_{0}+\xi_{1}\psi_{1}+\xi_{2}\psi_{2}+o(|%
\lambda|)$$
as $\lambda=\xi_{1}+\xi_{2}\sqrt{-1}\to 0$,
from (7.3),
(7.8),
(7.18) and (7.19),
$$\mathcal{L}(u_{x_{0}})[v,\overline{v}]=\frac{1}{\|q_{0}\|}\omega_{Th}(d%
\mathcal{V}_{x_{0}}(\psi_{1}),d\mathcal{V}_{x_{0}}(\psi_{2}))-\frac{|{\bf u}[q%
_{0}]\wedge\overline{\overline{{\tt D}}_{x_{1}}(v)}|^{2}}{16\|q_{0}\|^{2}},$$
and
$$\displaystyle{\bf u}[q_{0}]\wedge\overline{\overline{{\tt D}}_{x_{1}}(v)}$$
$$\displaystyle=8\,\omega_{Th}(d\mathcal{V}_{x_{0}}(q_{0}),d\mathcal{V}_{x_{0}}(%
\psi_{2}-\sqrt{-1}\psi_{1}))$$
$$\displaystyle\qquad+\sqrt{-1}\,8\,\omega_{Th}(d\mathcal{V}_{x_{0}}(q_{0}),d%
\mathcal{V}_{x_{0}}(\psi_{1}+\sqrt{-1}\psi_{2})).$$
In particular,
when $v\in H^{1,0}_{x_{1}}$ for $x_{1}=g(0)$,
we conclude
$$\mathcal{L}(u_{x_{0}})[v,\overline{v}]=\frac{1}{\|q_{0}\|}\omega_{Th}(d%
\mathcal{V}_{x_{0}}(\psi_{1}),d\mathcal{V}_{x_{0}}(\psi_{2})).$$
As a corollary,
we deduce
(7.20)
$$\omega_{Th}(d\mathcal{V}_{x_{0}}(\psi_{1}),d\mathcal{V}_{x_{0}}(\psi_{2}))\geq%
\frac{|{\bf u}[q_{0}]\wedge\overline{\overline{{\tt D}}_{x_{1}}(v)}|^{2}}{16\|%
q_{0}\|}\geq 0.$$
From the definition,
$d\mathcal{V}_{x_{0}}(\psi_{i})\in T_{v(q_{0})}\mathcal{MF}$
is the infinitesimal transverse cocycle (in the sense of Bonahon [4])
of the initial differentials $\{q_{\lambda}\}_{\lambda}$
associated along the $\xi_{i}$-direction at $\lambda=0$.
Thus,
the non-negativity derived from (7.20)
of the Thurston symplectic pairing
between the infinitesimal transverse cocycles $d\mathcal{V}_{x_{0}}(\psi_{1})$
and $d\mathcal{V}_{x_{0}}(\psi_{2})$ is a necessary condition
to describe complex-analytic deformations of Teichmüller
mappings
from the topological aspect in Teichmüller theory.
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Pilot Length Optimization for Spatially Correlated Multi-User MIMO Channel Estimation
Beatrice Tomasi, Maxime Guillaud
Mathematical and Algorithmic Sciences Laboratory, Huawei Technologies Co. Ltd.
France Research Center, 20 quai du Point du Jour, 92100 Boulogne Billancourt, France
email: {beatrice.tomasi,maxime.guillaud}@huawei.com
Abstract
We address the design of pilot sequences for channel estimation in the context of multiple-user Massive MIMO; considering the presence of channel correlation, and assuming that the statistics are known, we seek to exploit the spatial correlation of the channels to minimize the length of the pilot sequences, and specifically the fact that the users can be separated either through their spatial signature (low-rank channel covariance matrices), or through the use of different training sequences.
We introduce an algorithm to design short training sequences for a given set of user covariance matrices. The obtained pilot sequences are in general non-orthogonal, however they ensure that the channel estimation error variance is uniformly upper-bounded by a chosen constant over all channel dimensions.
We show through simulations using a realistic scenario based on the one-ring channel model that the proposed technique can yield pilot sequences of length significantly smaller than the number of users in the system.
Index Terms:
Pilot design, channel estimation, massive MIMO
I Introduction
Channel state information (CSI) acquisition represents an important problem in the multi-user Massive MIMO (Multiple-Input Multiple-Output) scenario [1]. Accurate downlink CSI is required in order to obtain the large multiplexing gain expected in massive MIMO system and achieve the rates shown e.g. in [2].
It is well known that, in the presence of i.i.d. channels, it is necessary to make the length of the pilot sequences at least as large as the total number of transmit antennas, in order to avoid the effect known as pilot contamination [3]; depending on the coherence time of the channel, the transmission of long training sequences instead of data-bearing symbols can represent a significant loss in spectral efficiency.
In this context, classical uplink CSI acquisition based on orthogonal pilot sequences across the users may not be efficient, since maintaining pilot orthogonality across many users requires the use of longer training sequences than required.
In the context of Massive MIMO however, the channels exhibit a large degree of correlation [4].
In fact, a denser antenna array improves the spatial resolution, and makes the received signal more spatially correlated, to the point of resulting in a rank deficient spatial correlation matrix. This correlation can potentially help reduce the required training overhead, since the requirement to maintain pilot orthogonality across many users can be relaxed.
In the literature, the problem of pilot design for MIMO correlated channels has been widely studied, for example in [5], [6], and [7]. However, in these works the pilot optimization is done for a MIMO system where all the transmitting antennas share the same spatial correlation subspace.
This assumption was lifted in [8], where it is proposed to schedule uplink CSI acquisition across the users such that the terminals can be separated in space; pilot orthogonality in time is therefore not required, yielding shorter training sequences. However, it is not clear how close to perfect separation in space a practical system can operate, considering realistic propagation conditions, and a finite number of users to choose from at the scheduling stage.
Another aspect of the problem, related to mitigating pilot contamination in the context of multi-cell Massive MIMO, has been considered in e.g. [9]; in this work, the same (orthogonal) pilot sequences are reused across the cells, and the problem of assigning each user to one of the pilot sequences is considered.
The object of the present article is to explore the options available for the design of pilot sequences when the users’ spatial covariances are not strictly orthogonal. We focus on the single-cell case, however we do not require orthogonality in time between the pilot sequences assigned to different users. In the following, we target this problem in the context of uplink CSI acquisition, when the channel covariances of the individual users are assumed to be known and arbitrary (i.e. they can be either mutually orthogonal, or have partly or fully overlapping spans). In Section III, we consider the noiseless case, and establish bounds on the length of the training sequences required to simultaneously learn all the users’ channels, by deriving necessary and sufficient conditions for channel identifiability. In Section IV, we introduce an algorithm to design pilot sequences with the objective of minimizing their length across all the users, while simultaneously ensuring that the channel estimation error variance is uniformly upper-bounded by a chosen constant over all channel dimensions. Finally, in Section V, we show through simulations using a realistic scenario based on the one-ring channel model that the proposed technique can yield pilot sequences of length significantly smaller than the number of users in the system.
II System Model
Let us consider a massive MIMO system with $K$ single-antenna user terminals (UT) and $M$ antennas at the BS.
The column vector of channel coefficients between the $k$-th single-antenna terminal and the $M$ antennas at the BS, $\bm{h}_{k}\in\mathbb{C}^{M}$ can be expressed as the product between the spatial correlation matrix $\mathbf{R}_{k}^{\frac{1}{2}\scriptscriptstyle}\in\mathbb{C}^{M\times r_{k}}$ where $r_{k}\leq M$ is the rank of the spatial correlation, and a vector of complex Gaussian i.i.d. random variables, $\bm{\eta}_{k}\in\mathbb{C}^{r_{k}}$ that represents the fast fading process, i.e.
$$\bm{h}_{k}=\mathbf{R}_{k}^{\frac{1}{2}\scriptscriptstyle}\bm{\eta}_{k},\quad\quad\forall k=1\ldots K.$$
(1)
We assume that the $\bm{\eta}_{k}\sim\mathcal{CN}(0,\mathbf{I}_{r_{k}})$ are independent across the users, and that the spatial correlation matrices $\mathbf{R}_{k}=\mathbb{E}[\bm{h}_{k}\bm{h}_{k}^{H}]$ are constant and known at the BS. Note that the single-antenna assumption is made here as a matter of notational simplicity, and can be trivially relaxed by treating the antennas of a multi-antenna user as multiple virtual users having the same channel covariance.
We assume that pilot sequences are sent simultaneously from all terminals to the BS in order to estimate the channel coefficients. Let $\bm{p}_{k}\in\mathbb{C}^{L}$ denote the sequence of length $L$ symbols transmitted by terminal $k$.
The signal received at the BS, $\mathbf{Y}=[\bm{y}(1),\dots,\bm{y}(L)]\in\mathbb{C}^{M\times L}$, is obtained as
$$\mathbf{Y}=\mathbf{H}\mathbf{P}^{T\scriptscriptstyle}+\mathbf{N},$$
(2)
where $\mathbf{H}=[\bm{h}_{1},\dots,\bm{h}_{K}]$ is the column concatenation of the channel vectors from the $K$ terminals to the $M$ antennas at the BS, $\mathbf{P}=[\bm{p}_{1},\dots,\bm{p}_{K}]\in\mathbb{C}^{L\times K}$ is the matrix containing the training sequences sent by the UTs, and $\mathbf{N}\in\mathbb{C}^{M\times L}$ represents additive noise.
By vectorizing the received signal in (2), it can be expressed as
$$\mathrm{vec}(\mathbf{Y})=\tilde{\mathbf{P}}\mathrm{vec}(\mathbf{H})+\mathrm{vec}(\mathbf{N}),$$
(3)
where $\tilde{\mathbf{P}}=(\mathbf{P}\otimes\mathbf{I}_{M})$.
Combining (1) with (3) yields
$$\bm{y}=\mathrm{vec}(\mathbf{Y})=\tilde{\mathbf{P}}\tilde{\mathbf{R}}^{\frac{1}{2}\scriptscriptstyle}\bm{\eta}+\mathrm{vec}(\mathbf{N}),$$
(4)
where
$$\tilde{\mathbf{R}}^{\frac{1}{2}\scriptscriptstyle}=\left(\begin{array}[]{ccc}\mathbf{R}_{1}^{\frac{1}{2}\scriptscriptstyle}&\dots&0\\
0&\mathbf{R}_{k}^{\frac{1}{2}\scriptscriptstyle}&0\\
0&\dots&\mathbf{R}_{K}^{\frac{1}{2}\scriptscriptstyle}\end{array}\right),$$
and $\bm{\eta}\in\mathbb{C}^{r}$ is the vector concatenation of the fast fading process coefficients, i.e. $\bm{\eta}^{T}=[\bm{\eta}_{1}^{T},\ldots,\bm{\eta}_{K}^{T}]$ with $r=\sum_{i=1}^{K}r_{i}$. The objective of the receiver is to jointly estimate $\bm{h}_{1}\ldots\bm{h}_{K}$ from the received signal $\mathbf{Y}$.
III Channel Identifiability
In this section, we establish bounds on the length of the probing sequence that is required to be able to identify the channel; to this aim, we focus on the noiseless case ($\mathbf{N}=\bf 0$), and establish for which pilot length $L$ the knowledge of $\mathbf{Y}$ is sufficient to uniquely identify $\mathbf{H}$. Note that because of (1) and since $\mathbf{R}_{k}^{\frac{1}{2}\scriptscriptstyle}$ is assumed full column rank, it is equivalent to identify $\bm{h}_{k}$ and $\bm{\eta}_{k}$. Furthermore, since in the noise-free case, (4) yields the trivial system of linear equations $\bm{y}=\tilde{\mathbf{P}}\tilde{\mathbf{R}}^{\frac{1}{2}\scriptscriptstyle}\bm{\eta}$, all the channel vectors $\bm{h}_{1},\ldots,\bm{h}_{K}$ can be estimated from $\mathbf{Y}$ iff $\tilde{\mathbf{P}}\tilde{\mathbf{R}}^{\frac{1}{2}\scriptscriptstyle}$ has full column rank. Therefore, we will focus on the identifiability condition
$$\mathrm{rank}\left(\tilde{\mathbf{P}}\tilde{\mathbf{R}}^{\frac{1}{2}\scriptscriptstyle}\right)=r.$$
(5)
III-A General Case
We first consider the general case where no particlar assumption is made on the relative position of the subspace spanned by the correlation matrices $\mathbf{R}_{1}^{\frac{1}{2}\scriptscriptstyle},\ldots,\mathbf{R}_{K}^{\frac{1}{2}\scriptscriptstyle}$.
Note that
$$\displaystyle\mathrm{rank}(\tilde{\mathbf{P}}\tilde{\mathbf{R}}^{\frac{1}{2}\scriptscriptstyle})$$
$$\displaystyle\leq$$
$$\displaystyle\min\left(\mathrm{rank}(\tilde{\mathbf{P}}),\mathrm{rank}(\tilde{\mathbf{R}}^{\frac{1}{2}\scriptscriptstyle})\right)$$
(6)
$$\displaystyle\leq$$
$$\displaystyle\min\left(M\cdot\mathrm{rank}(\mathbf{P}),r\right).$$
(7)
The identifiability criterion (5) imposes to fulfill (7) with equality, which requires $M\cdot\mathrm{rank}(\mathbf{P})\geq r$.
Finally, since $\mathbf{P}$ has dimension ${L\times K}$, we have $L\geq\mathrm{rank}(\mathbf{P})$, which yields the necessary condition for identifiability
$$L\geq\frac{r}{M}.$$
(8)
We now establish a sufficient condition. Using Sylvester’s rank inequality, we obtain
$$\displaystyle\mathrm{rank}(\tilde{\mathbf{P}}\tilde{\mathbf{R}}^{\frac{1}{2}\scriptscriptstyle})$$
$$\displaystyle\geq$$
$$\displaystyle\mathrm{rank}(\tilde{\mathbf{P}})+\mathrm{rank}(\tilde{\mathbf{R}}^{\frac{1}{2}\scriptscriptstyle})-KM$$
(9)
$$\displaystyle\geq$$
$$\displaystyle M\cdot\mathrm{rank}(\mathbf{P})+r-KM.$$
(10)
Note that taking
$$\mathrm{rank}(\mathbf{P})=K$$
(11)
in (10) yields $\mathrm{rank}(\tilde{\mathbf{P}}\tilde{\mathbf{R}}^{\frac{1}{2}\scriptscriptstyle})\geq r$, i.e. it guarantees identifiability. Since $\mathrm{rank}(\mathbf{P})\leq\min(L,K)$, the sufficient condition (11) also imposes $L\geq K$, which is consistent with the fact that we do not exploit the spatial properties of the channel to reduce the training length, and therefore $L$ must be at least equal to the number of users $K$ to ensure that they can be distinguished.
Note that the bounds on the training length $L$ obtained through the necessary condition (8) and the inequality $L\geq K$ associated to the sufficient condition (11) are not equal.
III-B Mutually Orthogonal Channel Subspaces
Let us consider the case of mutually orthogonal channel covariance matrices, i.e. $\mathrm{tr}\left(\mathbf{R}_{i}\mathbf{R}_{j}\right)=0\ \forall i\neq j$.
In that case, the necessary condition (8) can be shown to be sufficient as well. First, let us consider the case $r\leq M$. (8) indicates that a training sequence of length at least $L=1$ is required; we show that this is indeed sufficient. For this, consider the training sequence of user $k$, $\bm{p}_{k}=[p_{k}(1)]$ which is reduced to length 1, and assume that $p_{k}(1)\neq 0\ \forall k$.
It is trivial to see that
$$\tilde{\mathbf{P}}\tilde{\mathbf{R}}^{\frac{1}{2}\scriptscriptstyle}=\left(p_{k}(1)\mathbf{R}_{1}^{\frac{1}{2}\scriptscriptstyle}\ \ldots\ p_{k}(K)\mathbf{R}_{K}^{\frac{1}{2}\scriptscriptstyle}\right)$$
(12)
is full column rank thanks to the orthogonality of the column subspaces of $\mathbf{R}_{1}^{\frac{1}{2}\scriptscriptstyle}\ \ldots\ \mathbf{R}_{K}^{\frac{1}{2}\scriptscriptstyle}$, and therefore the identifiability condition (5) is fulfilled.
A similar result can be obtained for the case $r>M$, where it can be shown that a training sequence of length $L=\lceil\frac{r}{M}\rceil$ is sufficient to ensure identifiability.
We conclude that in the case of mutually orthogonal subspaces, the bound (8) is tight.
Note that this case is of particular interest in Massive MIMO, since it shows that when $\frac{r}{M}$ is small, $L$ can be made small, and in particular channel identifiability can be achieved using pilot sequences of length $L<K$.
III-C Identical Channel Subspaces
Let us now focus on the case where the channels of all the users live in the same linear subspace; this can be represented without loss of generality by taking $\mathbf{R}_{1}^{\frac{1}{2}\scriptscriptstyle}=\ldots=\mathbf{R}_{K}^{\frac{1}{2}\scriptscriptstyle}=\mathbf{R}^{\frac{1}{2}\scriptscriptstyle}$ for some full column rank matrix $\mathbf{R}^{\frac{1}{2}\scriptscriptstyle}\in\mathbb{C}^{M\times d}$.
In that case, we have $\tilde{\mathbf{P}}\tilde{\mathbf{R}}^{\frac{1}{2}\scriptscriptstyle}=\mathbf{P}^{T\scriptscriptstyle}\otimes\mathbf{R}^{\frac{1}{2}\scriptscriptstyle}$, and
$$\mathrm{rank}(\tilde{\mathbf{P}}\tilde{\mathbf{R}}^{\frac{1}{2}\scriptscriptstyle})=\mathrm{rank}(\mathbf{P})\cdot d.$$
(13)
Since $r=Kd$, combining (5) and (13) yields the identifiability condition $\mathrm{rank}(\mathbf{P})=K$.
Note that this is the same condition as (11), however in the case of identical subspaces across the users, it is both necessary and sufficient, while in the general case it is not necessary, as shown in Section III-B.
IV Minimum length pilot sequence under estimation error constraint
Let us now consider the more general setting where noise is present, and the covariance matrices of the users can be arbitrary; Linear Minimum Mean Square Error (LMMSE) estimation of the channels is assumed. In this context, we tackle the design of short pilot sequences under an estimation error constraint.
IV-A Error covariance matrix
The LMMSE estimator of the fast fading coefficients between all users and the BS array, $\hat{\bm{\eta}}$, is
$$\hat{\bm{\eta}}=\mathbf{C}_{\bm{\eta}\bm{y}}\mathbf{C}_{\bm{y}}^{-1}\bm{y},$$
(14)
where $\mathbf{C}_{\bm{\eta}\bm{y}}=\tilde{\mathbf{R}}^{\frac{H}{2}\scriptscriptstyle}\tilde{\mathbf{P}}^{H\scriptscriptstyle}$, and $\mathbf{C}_{\bm{y}}=\tilde{\mathbf{P}}\tilde{\mathbf{R}}\tilde{\mathbf{P}}^{H\scriptscriptstyle}+\sigma^{2}\mathbf{I}_{LM}$.
The covariance matrix of the estimation error for $\bm{\eta}$ is given by [10]
$$\displaystyle\mathbf{C}_{\bf e,\bm{\eta}}$$
$$\displaystyle=$$
$$\displaystyle\mathbb{E}[(\hat{\bm{\eta}}-\bm{\eta})(\hat{\bm{\eta}}-\bm{\eta})^{H\scriptscriptstyle}]$$
(15)
$$\displaystyle=$$
$$\displaystyle\mathbf{I}_{r}-\tilde{\mathbf{R}}^{\frac{H}{2}\scriptscriptstyle}\tilde{\mathbf{P}}^{H\scriptscriptstyle}(\tilde{\mathbf{P}}\tilde{\mathbf{R}}\tilde{\mathbf{P}}^{H\scriptscriptstyle}+\sigma^{2}\mathbf{I}_{LM})^{-1}\tilde{\mathbf{P}}\tilde{\mathbf{R}}^{\frac{1}{2}\scriptscriptstyle}.$$
(16)
Considering (1), we define $\hat{\bm{h}}=\tilde{\mathbf{R}}^{\frac{1}{2}\scriptscriptstyle}\hat{\bm{\eta}}$. The covariance matrix of the estimation error on $\bm{h}$ is therefore
$$\displaystyle\mathbf{C}_{\bf e}$$
$$\displaystyle=$$
$$\displaystyle\mathbb{E}[(\hat{\bm{h}}-\bm{h})(\hat{\bm{h}}-\bm{h})^{H\scriptscriptstyle}]$$
(17)
$$\displaystyle=$$
$$\displaystyle\tilde{\mathbf{R}}^{\frac{1}{2}\scriptscriptstyle}\mathbf{C}_{\bf e,\bm{\eta}}\tilde{\mathbf{R}}^{\frac{H}{2}\scriptscriptstyle}$$
(18)
$$\displaystyle=$$
$$\displaystyle\tilde{\mathbf{R}}-\tilde{\mathbf{R}}\tilde{\mathbf{P}}^{H\scriptscriptstyle}(\tilde{\mathbf{P}}\tilde{\mathbf{R}}\tilde{\mathbf{P}}^{H\scriptscriptstyle}+\sigma^{2}\mathbf{I}_{LM})^{-1}\tilde{\mathbf{P}}\tilde{\mathbf{R}}.$$
(19)
In order to control the accuracy of the channel estimation process across all the users, we assume that we wish to uniformly bound the estimation error on all dimensions of $\bm{h}$ by a given constant $\epsilon>0$. This can be done by requiring that all the eigenvalues of $\mathbf{C}_{\bf e}$ are lower or equal to $\epsilon$, which we denote111For two positive semidefinite matrices $\mathbf{A}$ and $\mathbf{B}$, $\mathbf{A}\preceq\mathbf{B}$ is a shorthand notation for the condition that $\mathbf{B}-\mathbf{A}$ is positive semidefinite. $\mathbf{C}_{\bf e}\preceq\epsilon\mathbf{I}$.
IV-B Pilot length minimization algorithm
We now formulate the pilot length minimization as a rank minimization problem over a convex set. Recall that we seek the minimum $L$ for which there exists a $L\times K$ matrix $\mathbf{P}$ that satisfies $\mathbf{C}_{\bf e}~{}\preceq~{}\epsilon\mathbf{I}$. Thus we seek to solve the optimization problem
$$\displaystyle\underset{\mathbf{P}\in\mathbb{C}^{L\times K}}{\min}$$
$$\displaystyle L$$
$$\displaystyle\mathrm{s.t.}$$
$$\displaystyle\mathbf{C}_{\bf e}\preceq\epsilon\mathbf{I}.$$
Letting $\mathbf{X}=\mathbf{P}^{H\scriptscriptstyle}\mathbf{P}\in\mathbb{C}^{K\times K}$, minimizing $L$ is equivalent to minimizing ${\mathrm{rank}}(\mathbf{X})$, i.e. (IV-B) becomes:
$$\displaystyle\underset{\mathbf{X}\succeq{\bf 0}}{\min}$$
$$\displaystyle{\mathrm{rank}}(\mathbf{X})$$
$$\displaystyle\mathrm{s.t.}$$
$$\displaystyle\mathbf{C}_{\bf e}\preceq\epsilon\mathbf{I}.$$
This problem can be solved efficiently, but approximately, through heuristics. Following [11], we consider a regularized smooth surrogate of the rank function, namely $\log\det(\mathbf{X}+\delta\mathbf{I})$ for some small $\delta$, which yields:
$$\displaystyle\underset{\mathbf{X}\succeq{\bf 0}}{\min}$$
$$\displaystyle\log\det(\mathbf{X}+\delta\mathbf{I})$$
$$\displaystyle\mathrm{s.t.}$$
$$\displaystyle\mathbf{C}_{\bf e}\preceq\epsilon\mathbf{I}.$$
The objective function in (IV-B) is concave, however it is smooth on the positive definite cone; a possible way to approximately solve this problem is to iteratively minimize a locally linearized version of the objective function, i.e. solve
$$\displaystyle\mathbf{X}_{t+1}=\arg$$
$$\displaystyle\underset{\mathbf{X}\succeq{\bf 0}}{\min}$$
$$\displaystyle\mathrm{Tr}(\mathbf{X}_{t}+\delta\mathbf{I})^{-1}\mathbf{X}$$
$$\displaystyle\mathrm{s.t.}$$
$$\displaystyle\mathbf{C}_{\bf e}\preceq\epsilon\mathbf{I}.$$
until convergence to some $\bar{\mathbf{X}}$. We suggest to initialize the algorithm by choosing $\mathbf{X}_{0}$ as the rank-1, all-ones matrix $\mathbf{1}_{K\times K}$.
Let us now focus on the constraint $\mathbf{C}_{\bf e}\preceq\epsilon\mathbf{I}$. Decomposing $\tilde{\mathbf{R}}^{\frac{1}{2}\scriptscriptstyle}=\mathbf{U}\mathbf{\Lambda}^{1/2}$, where $\mathbf{\Lambda}$ and $\mathbf{U}$ are respectively the diagonal matrix containing the eigenvalues of $\tilde{\mathbf{R}}$ and the associated eigenvectors, and using (19), we obtain
$$\mathbf{C}_{\bf e}\preceq\epsilon\mathbf{I}\ \Leftrightarrow\ \tilde{\mathbf{R}}^{\frac{H}{2}\scriptscriptstyle}\left(\mathbf{X}\otimes\mathbf{I}_{M}\right)\tilde{\mathbf{R}}^{\frac{1}{2}\scriptscriptstyle}\succeq\left(\epsilon^{-1}\mathbf{\Lambda}-\mathbf{I}_{r}\right)\sigma^{2}.$$
(24)
Note that since this constraint is convex, (IV-B) is a convex optimization problem that can be efficiently solved numerically.
Due to the various approximations involved in transforming (IV-B) into (IV-B), $\bar{\mathbf{X}}$ might not be strictly rank-deficient, but it can have some very small eigenvalues instead. It is therefore necessary to apply some thresholding on these eigenvalues to recover a stricly rank-deficient solution. Let us denote by $\bm{e}_{k}$ the eigenvector associated to the $k$-th eigenvalue $v_{k}$ of $\bar{\mathbf{X}}$, $k=1\ldots K$, with $v_{1}\geq\ldots\geq v_{K}\geq 0$. We then let
$$\displaystyle L$$
$$\displaystyle=$$
$$\displaystyle\max_{i=1\ldots K}i$$
$$\displaystyle\mathrm{s.t.}\quad v_{i}\geq\epsilon_{s}$$
for a suitably chosen (small) $\epsilon_{s}$, and obtain the matrix of optimized training sequences as $\bar{\mathbf{P}}=[\bm{e}_{1},\dots,\bm{e}_{L}]^{T}$.
The proposed algorithm is shown in Algorithm 1.
V Numerical results
Algorithm 1 has been bechmarked numerically. For each realization of the covariance matrices, we applied the proposed algorithm to compute $\bar{\mathbf{P}}$ (the solution to (IV-B) was obtained via the numerical solver CVX [12], and parameter $\delta$ was set to $10^{-4}$). We note that although we do not provide any convergence proof, the proposed method has demonstrated reliable convergence in the simulations. Note also that we can not claim any global optimality for the obtained solution $\bar{\mathbf{X}}$, and indeed simulations have shown that the convergence point depends on the initialization, with the rank-1 initialization $\mathbf{X}_{0}=\mathbf{1}_{K\times K}$ giving the best results.
The scenario considered in this section is that of a uniform circular array (UCA) of diameter $2$m, consisting of $M\in\{16,30\}$ antenna elements (AE), which serves $K\in\{5,10\}$ UTs randomly distributed around the BS at a distance between $250$ and $750$m. We assume that $200$ scatterers distributed randomly on a disc of radius $50$m centered on each terminal are causing fast fading (see Fig. 1).
The covariance matrices are generated by a ray-tracing procedure based on the one-ring channel model [13], with a central frequency of $1.8$ GHz, and are normalized such that $\mathrm{trace}(\mathbf{R}_{k})=...=\mathrm{trace}(\mathbf{R}_{K})=1$ (this can be achieved in reality via power control). According to this model, the support of the angle of arrivals associated to a given UT is limited, which yields covariance matrices with few large eigenvalues. We have applied a threshold to these eigenvalues to obtain the ranks $r_{k}$ that ensure that at least 99% of the energy of the full-rank matrix is captured by the rank-deficient model. The noise variance at each BS antenna element is chosen as $\sigma^{2}=10^{-4}$.
Fig. 2 shows the average length of the pilot sequences obtained by the proposed algorithm over $200$ realizations as a function of the error threshold $\epsilon$. Clearly, $L$ decreases when the constraint on the maximum estimation error is relaxed (for large $\epsilon$). In all scenarios, the proposed algorithm yields pilot sequences having an average $L$ significantly lower than $K$. Note also that this gain appears to be more pronounced for larger antenna arrays.
Fig. 3 depicts the histogram (over $200$ realizations) of $L$ in the scenario with $M=30$ AE and $K=10$ UTs, when the constraint on the maximum error is set to $\epsilon=10^{-4}$. In most of the realizations, the length of the pilot sequence is significantly smaller than the number of users.
Note that because of the thresholding procedure described in (IV-B), $\mathbf{X}=\bar{\mathbf{P}}^{H\scriptscriptstyle}\bar{\mathbf{P}}$ does not necessarily satisfy (24) exactly, although $\bar{\mathbf{X}}$ does. In order to verify that the maximum estimation error constraint is still approximately satisfied, we compute the maximum eigenvalue of $\mathbf{C}_{\bf e}$ for the length-$L$ pilot sequence obtained after thresholding. The results, in Fig. 4 for $(M,K)=(30,10)$, show that this error is not significantly greater than the target $\epsilon$.
VI Conclusions
We have analyzed how channel correlation and the knowledge of per-user covariance matrices can significantly reduce the duration of channel estimation in Massive MIMO systems. We have established bounds on the duration of estimation based on the extreme cases of identical and orthogonal channel covariance matrices. For the general case, we introduced an algorithm to design short training sequences for a given set of user covariance matrices. The obtained pilot sequences are in general non-orthogonal, however they ensure that the channel estimation error variance is uniformly upper-bounded by a chosen constant over all channel dimensions.
Simulations have shown that the proposed technique can yield pilot sequences of length significantly smaller than the number of users in the system.
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Searching for Cross-Correlation Between Stochastic Gravitational Wave Background and Galaxy Number Counts
Kate Z.Yang, ${}^{1}$ Vuk Mandic${}^{1}$, Claudia Scarlata${}^{1}$ and Sharan Banagiri${}^{1}$
${}^{1}$School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA
E-mail:yang5991@umn.edu
Abstract
Advanced LIGO and Advanced Virgo have recently published the upper limit measurement of persistent directional stochastic gravitational wave background (SGWB) based on data from their first and second observing runs (O1 and O2) (Abbott
et al., 2019c). In this paper we investigate whether a correlation exists between this maximal likelihood SGWB map and the electromagnetic tracers of matter structure in the universe, such as galaxy number counts. The method we develop will improve the sensitivity of future searches for anisotropy in the SGWB and expand the use of SGWB anisotropy to probe the formation of structure in the universe. In order to compute the cross-correlation, we used the spherical harmonic decomposition of SGWB in multiple frequency bands and converted them into pixel-based sky maps in HEALPix (Gorski et al., 1999) basis. For the electromagnetic (EM) part, we use the Sloan Digital Sky Survey (SDSS) galaxy catalog and form HEALPix sky maps of galaxy number counts at the same angular resolution as the SGWB maps. We compute the pixel-based coherence between these SGWB and galaxy count maps. After evaluating our results in different SGWB frequency bands and in different galaxy redshift bins, we conclude that the coherence between the SGWB and galaxy number count maps is dominated by the null measurement noise in the SGWB maps, and therefore not statistically significant. We expect the results of this analysis to be significantly improved by using the more sensitive upcoming SGWB measurements based on the third observing run (O3) of Advanced LIGO and Advanced Virgo.
keywords:
gravitational waves, galaxy, methods: statistical
††pubyear: 2020
1 Introduction
Recent detections of gravitational waves (GWs) from the mergers of binary black hole (BBH) (Abbott
et al., 2016d, c, 2017b, 2017h, 2017c, 2019a, 2019d) and binary neutron star (BNS) (Abbott
et al., 2017g) systems have initiated the field of multi-messenger astrophysics. These discoveries have triggered a broad range of studies including novel tests of General Relativity (Abbott
et al., 2016b, 2017e), constraints on the neutron star equation of state (Abbott
et al., 2017g), estimates of the BBH and BNS rates and studies of their progenitors (Abbott
et al., 2016d, e, 2017g), a new measurement of the Hubble constant $H_{0}$ (Abbott
et al., 2017d), kilonova interpretation of the observed BNS merger (Abbott
et al., 2017f), and others.
The observed rates of resolved BBH and BNS mergers also imply a relatively strong stochastic gravitational-wave background (SGWB) arising from the signal superposition of such mergers throughout the universe (Abbott
et al., 2016a, 2018b, 2019b).
As the ground-based Advanced LIGO (aLIGO) (Aasi
et al., 2015) and Advanced Virgo (aVirgo) (Acernese
et al., 2015) GW detectors improve their strain sensitivities, one of their primary targets will be the detection of this SGWB (Abbott
et al., 2017a, 2019b, 2019b). The projected strain sensitivity improvements (Abbott
et al., 2018a) combined with the recently proposed Bayesian search technique (Smith &
Thrane, 2018) have made the SGWB detection with advanced detectors a real possibility. It is worth noting that SGWB can also be generated through a variety of stochastic processes in the early Universe (Romano &
Cornish, 2017), including models of inflation, phase transitions, cosmic strings, and others.
Due to this discovery potential, there has been a recent surge in the literature studying the possible anisotropy of the SGWB due to BBH and BNS mergers (Contaldi, 2017; Jenkins et al., 2018, 2019a; Jenkins &
Sakellariadou, 2019; Jenkins
et al., 2019b; Bertacca
et al., 2020; Cusin
et al., 2017, 2018c, 2018b; Cusin et al., 2018a; Cusin
et al., 2019; Pitrou
et al., 2020; Cañas-Herrera et al., 2019; Stiskalek et al., 2020; Cavaglia &
Modi, 2020; Payne et al., 2020), as well as in cosmological SGWB models due to phase transitions (Geller
et al., 2018) and cosmic strings (Jenkins &
Sakellariadou, 2018). Several contributions have also investigated the possibility of correlating the SGWB anisotropy with the anisotropy observed in EM tracers of the large scale structure, such as galaxy counts and weak lensing (Cusin
et al., 2017, 2018c, 2018b, 2019; Cañas-Herrera et al., 2019; Scelfo et al., 2018; Oguri, 2016, 2016; Mukherjee
et al., 2020; Mukherjee &
Wandelt, 2018), or the Cosmic Microwave Background (CMB, Geller
et al., 2018). The first theoretical predictions of the cross-correlation power spectrum between the SGWB and the galaxy number counts have been made, as well as between the SGWB and the weak lensing convergence, including the dependence on SGWB frequency and the galaxy redshift distribution (Cusin
et al., 2018b).
Correlating the SGWB anisotropy with anisotropy in EM tracers offers multiple lines of inquiry. First, the GW-EM correlation method is likely to be more sensitive when trying to detect the SGWB anisotropy than the traditional techniques that rely on GW data alone. Second, the GW-EM correlation can be placed in a parameter estimation framework so as to measure the cosmological and astrophysical parameters of the model that gives rise to the SGWB anisotropy (Cusin
et al., 2017, 2018b, 2019; Cañas-Herrera et al., 2019; Mukherjee &
Silk, 2020)–for example, to constrain the formation and evolution of structure in the universe. Third, correlations with different EM tracers (e.g. galaxy counts vs CMB) may enable separating different SGWB contributions (e.g. due to binary mergers vs cosmological, respectively).
In this paper, we present the first analysis of the GW-EM anisotropy correlations, using the data from the second observation run of Advanced LIGO and Advanced Virgo correlated with the galaxy count survey from the Sloan Digital Sky Survey (SDSS). We observe no significant correlations in the data and hence place upper limits on the correlation parameter. The rest of this paper is organized as follows. In Section 2 we review the method for measuring the SGWB anisotropy and we apply it to different frequency bands of Advanced LIGO data to compute SGWB sky maps at different angular resolutions. In Section 3 we review the SDSS survey and compute the maps of the galaxy count distribution across the sky in several redshift slices. In Section 4 we compute the correlations between the SGWB and SDSS maps and establish the first upper limits on the correlation coefficients. In Section 5 we offer our concluding remarks and discuss the numerous ways of extending our study in the future.
2 SGWB Anisotropy Upper Limit
Stochastic gravitational-wave background arises as a superposition of waves from many incoherent GW sources. The SGWB therefore does not have a deterministic waveform, and is instead characterized by its energy density spectrum. In particular, we define the frequency and angular GW energy density spectrum $\Omega_{\rm{GW}}(f,\Theta)$ as:
$$\displaystyle\Omega_{\rm{GW}}(f,\Theta)=\frac{f}{\rho_{c}}\frac{d^{3}\rho_{\rm%
{GW}}}{dfd^{2}\Theta},$$
(1)
where $\rho_{\rm{GW}}$ is the GW energy density, $f$ is frequency, $\Theta$ represents a direction on the sky, and $\rho_{c}$ is the critical energy density needed to close the universe. Past searches for the SGWB anisotropy have assumed that this spectrum can be factorized into frequency and sky-direction parts (Thrane et al., 2009; Abbott
et al., 2019c, 2017a):
$$\displaystyle\Omega_{\rm{GW}}(f,\Theta)=\frac{2\pi^{2}}{3H_{0}^{2}}f^{3}H(f){%
\mathcal{P}}(\Theta),$$
(2)
where $H_{0}$ is the Hubble constant, ${\mathcal{P}}(\Theta)$ captures the angular dependence on the sky, and $H(f)$ describes the frequency dependence of the spectrum, typically assumed to take a power law form: $H(f)=(f/f_{\rm ref})^{\alpha-3}$, with some reference frequency $f_{\rm{ref}}$ and spectral index $\alpha$. For the SGWB due to BBH and BNS mergers $\alpha=2/3$ (Abbott
et al., 2016a, 2018b), but different values of the spectral index are appropriate for other models. In this paper we adopt $f_{\rm ref}=100$ Hz.
The spatial dependence can be decomposed into any set of basis functions on a sphere–we will use the spherical harmonics basis:
$$\displaystyle P(\Theta)=\sum_{lm}P_{lm}Y_{lm}(\Theta)$$
(3)
The objective of the SGWB anisotropy upper limit analysis is therefore to estimate the parameters $P_{lm}$. We adopt the approach developed in (Thrane et al., 2009) and used in past anisotropic SGWB searches (Abbott
et al., 2019c, 2017a), which starts with the cross-correlation between the strain time series data of GW detectors (LIGO Hanford (H1) and LIGO Livingston (L1) in our case):
$$\displaystyle C(f,t)=s_{1}^{*}(f,t)\ s_{2}(f,t),$$
(4)
where $t$ denotes a time segment, and $s_{1}$ and $s_{2}$ are Fourier transforms of the strain time series of H1 and L1 in this time segment. We then define the dirty map $X_{\nu}$:
$$\displaystyle X_{\nu}=\sum_{f,t}\gamma_{\nu}^{*}(f,t)\frac{H(f)}{P_{1}(f,t)P_{%
2}(f,t)}C(f,t),$$
(5)
where the sum is over all frequency bins $f$ and all time segments $t$. The index $\nu$ runs over the spherical harmonic components (i.e. $\nu\equiv(l,m)$), $P_{1,2}(f,t)$ are strain power spectral densities for the two detectors, and $\gamma_{\nu}(f,t)$ is a geometric factor that is a function of the separation and relative orientation of the LIGO detectors H1 and L1 (Christensen, 1992; Thrane et al., 2009).
The dirty map $X_{\nu}$ represents an estimate of the GW energy density sky distribution convolved with the response of the detectors’ antenna patterns. The corresponding uncertainty is described by the covariance matrix, also known as the Fisher matrix:
$$\displaystyle\Gamma_{\mu\nu}=\sum_{f,t}\gamma_{\mu}^{*}(f,t)\frac{H^{2}(f)}{P_%
{1}(f,t)P_{2}(f,t)}\gamma_{\nu}(f,t).$$
(6)
Estimators of the spherical harmonic coefficients $P_{lm}$, also known as the the clean map, are then given by (Thrane et al., 2009):
$$\displaystyle\hat{P}_{\mu}=\sum_{\nu}\big{(}\Gamma_{R}^{-1}\big{)}_{\mu\nu}X_{%
\nu}.$$
(7)
The covariance matrix corresponding to the $P_{lm}$’s is the inverse of the Fisher matrix, $\Gamma_{R}^{-1}$. In general, the Fisher matrix may be singular, reflecting the fact that the GW detector network may be insensitive to some directions on the sky. The inversion of this matrix therefore requires regularization, which is accomplished by diagonalizing the Fisher matrix and removing the eigenvalues that are close to zero (i.e. typically setting about 1/3 of the lowest eigenvalues to infinity) (Thrane et al., 2009). The subscript $R$ in $\Gamma_{R}^{-1}$ denotes that the Fisher matrix has been regularized. The regularization does not induce a bias on the clean map $\hat{P}_{\mu}$.
The angular resolution of this technique is set by a diffraction-like limit (Thrane et al., 2009):
$$\displaystyle\theta=\frac{c}{2df}\approx\frac{50\rm{Hz}}{f_{\alpha}},$$
(8)
where $\theta$ is in radians, $d$ is the distance between H1 and L1 (3000 km), and $f_{\alpha}$ is typically taken to be the most sensitive frequency in the detector band for a power law SGWB with spectral index $\alpha$, and for the given detector noise power spectra (Abbott
et al., 2017a; Romano &
Cornish, 2017). For the BBH/BNS background, $\alpha=2/3$ and the most sensitive frequency in the past searches was found to be $50-60$ Hz, implying a coarse angular resolution of order $\theta\approx\pi/3$ and therefore spherical harmonic decomposition up to $l_{\rm max}=\pi/\theta\approx 3-4$ (Abbott
et al., 2017a, 2019c).
In an attempt to probe finer angular scales, we will conduct the above analysis in several narrower frequency bands: 50-100 Hz, 100-150 Hz, 150-200 Hz, 200-250 Hz. The higher frequency bands will result in better angular resolution, specifically in $l_{max}=$ 4, 8, 12, and 16, respectively.
For the highest $l_{max}$=16 the corresponding angular resolution is $\theta\approx$ 7.3 deg. We note, however, that the sensitivity of the search is reduced at higher frequencies, both because of the poorer strain sensitivity of the GW detectors above $\sim 100$ Hz (Aasi
et al., 2015; Acernese
et al., 2015) and because of the $f^{3}$ term in Eq. 2.
We apply the above analysis procedure to the GW data from the second observing run (O2) of Advanced LIGOâs detectors H1 and L1. The O2 data are collected from 16:00:00 UTC on 30 November, 2016 to 22:00:00 UTC on 25 August, 2017 (Abbott
et al., 2019c). We follow closely the data processing procedure described in (Abbott
et al., 2017a, 2019c). The time-series data are divided into 50% overlapping segments of 192 seconds, passing through a cascading high-pass filter. The data segments are then Fourier transformed into the frequency domain, and the H1-L1 cross-correlation is computed for each 192 second long segment. The results from these overlapping time segments are optimally combined to produce the final cross-correlation estimate. We use the same data selection criteria described in (Abbott
et al., 2019c). Finally we compute the clean map estimates following Eq. 7, for each of the four frequency bands. The resulting clean maps, sigma maps and signal-to-noise (SNR) maps for the four frequency bands are shown in Figure 1.
3 Galaxy Count Anisotropy
As an example of an EM tracer of matter structure, we will use the distribution of galaxy counts across the sky. The most complete and largest area galaxy survey currently comes from the Sloan Digital Sky Survey (SDSS), whose Data Release 16 (DR16) contains observations through August 2018 (Ahumada
et al., 2020). The SDSS imaging data contain observations covering almost 1.5$\times 10^{4}$ deg${}^{2}$ or roughly 1/3 of the sky. The photometric catalog includes approximately 2$\times 10^{8}$ galaxies with r-band magnitude brighter than $m_{r}\approx 22.2$.
In addition to the imaging observations, SDSS acquired spectra for $\approx 1.8\times 10^{6}$ galaxies brighter than $m_{r}\approx 17.7$. For galaxies fainter than this limit, SDSS provides an estimate of the galaxy photometric redshift based on the analysis of the five photometric bands (hereafter, photo-$z$). Although the resulting redshifts are substantially more uncertain than those derived from spectroscopic observations, the use of photo-$z$ allows us to increase the sample size considerably.
From the SDSS archive, we select all galaxies with magnitudes in the 17$<m_{r}\leq 21$ range. To identify only galaxies, we use the SDSS type parameter ($type=3$). The magnitude range was chosen to ensure a survey completeness level of 90% or better, and to minimize the contamination to the galaxy sample by misclassified stars (see Wang
et al., 2013, for a discussion). We also constrain the analysis to include data in a fully contiguous area mostly in the northern Galactic hemisphere.
The final photometric galaxy catalog includes 23 million objects with median photometric redshift of 0.33.
For a subsample of 1.4 million galaxies, spectroscopic redshifts are available, with a median spectroscopic redshift of 0.39.
A number of systematic effects can potentially affect the spatial distribution of galaxies on the large scales relevant for the cross-correlation with the GW maps. Here we consider only the effects of atmospheric seeing variation and Milky Way extinction, as they impact the observed galaxy number counts on degree scales and above (Reid et al., 2016; Ross et al., 2017). We follow (Wang
et al., 2013) and we consider in the analysis only areas of good seeing and minimal Galactic extinction. We quantify the seeing using the average Full Width Half Maximum (FWHM) of the point spread function (PSF) during the observations and exclude from the analysis those sky regions with average FWHM $\geq 1.5^{\prime\prime}$. This cut is found to exclude $\sim$ 12% of the total area. Galactic extinction is characterized via the color excess, E(B-V), and we exclude areas with $E(B-V)>0.13$, or 15% of the total area.
The seeing and galactic extinction cuts can have significant effects on the average number of galaxies in some areas of the sky. In order to obtain unbiased galaxy count maps, we apply the following procedure. In the HEALPix basis, the full sky is divided into pixels of the same angular size (Gorski et al., 1999), a convenient choice for the computations of cross correlations with the GW sky maps. The number of pixels in the HEALPix basis is chosen to match the value of $l_{\rm max}$ for each frequency band:
$$\displaystyle{\rm{\#pixels}}\approx\frac{4\pi}{\theta^{2}}=\frac{4}{\pi}\cdot l%
_{max}^{2}.$$
(9)
While we ultimately need a galaxy count map of resolution corresponding to $l_{\rm max}=16$, corresponding to an angular scale of $\sim 7^{\circ}$, we start by producing the HEALPix map for the SDSS photometric catalog with a higher resolution (small pixels). The small pixels have an angular scale of 2.4${}^{\circ}$. This angular scale is small enough that the seeing and galactic extinction do not vary too much for the galaxies within the pixels, but large enough to ensure a large number of galaxies (on average $>10^{2}$ galaxies). Using all galaxies in each small pixel, we compute the average r-band seeing and extinction for that pixel. We reject all galaxies within a pixel whose average seeing is greater than 1.5 arcsecond or the galactic extinction is $>$ 0.13.
To correct for the missing pixels we then replace the counts in that rejected pixel with the average counts of the other small pixels inside a larger HEALPix pixel corresponding to $l_{\rm max}=16$. Since all objects in the SDSS spectroscopic catalog are also in the SDSS photometric catalog, we apply the same procedure to the spectroscopic catalog as well. The results for both catalogs are shown in Figure 2.
Since the SGWB due to BBH and BNS mergers at different angular scales is expected to be dominated by binaries at different redshifts (Cusin
et al., 2019), we will conduct our analysis in several redshift bins, i.e. compute the correlation between SGWB sky maps and the galaxy number sky maps in each redshift bin, respectively. We choose to divide both catalogs into redshift bins of width 0.1 (i.e. 0.0-0.1, 0.1-0.2,…). For the photometric catalog, we extend the analysis up to redshift 0.6, which includes 97% of all the galaxies. For the spectroscopic catalog, we go up to 0.7 and the redshift slicing includes 98% galaxies.
While the photometric and spectroscopic catalog maps including all redshifts do not appear to be correlated (as shown in Figure 2), we have confirmed that the photometric and spectroscopic maps in each redshift bin are highly correlated.
4 SGWB-EM Correlations
Having produced the sky maps for the SGWB in each of the four frequency bands and for the galaxy counts in each of the redshift bins, we next compute the correlations between these maps. Denoting the SGWB energy density in a pixel $i$ as $M_{{\rm GW},i}$ and the galaxy number count in the same pixel as $M_{{\rm GC},i}$, we define the corresponding fluctuations:
$$\displaystyle\delta M_{{\rm GW},i}=M_{{\rm GW},i}-\langle M_{\rm GW}\rangle,\ %
\ \delta M_{{\rm GC},i}=M_{{\rm GC},i}-\langle M_{\rm GC}\rangle.$$
(10)
Then we define the coherence between these fluctuations as:
$$\displaystyle\Gamma=\frac{\langle\delta M_{\rm GW}\cdot\delta M_{\rm GC}%
\rangle^{2}}{\langle\delta M_{\rm GW}^{2}\rangle\langle\delta M_{\rm GC}^{2}%
\rangle}.$$
(11)
The averages are computed over all pixels in the maps.
To assess the significance of the measured coherence, we use simulations. In particular, we generate 10,000 simulated SGWB noise maps assuming zero-mean multivariate Gaussian distribution for $P_{lm}$’s described by the regularized inverse Fisher matrix obtained from LIGO data (see Section 2). We then compute $\Gamma$ for these simulated maps and the galaxy count sky maps in different redshift bins respectively. We then compute the false alarm rate (FAR) as a function of coherence:
$$\displaystyle{\rm{FAR}}(\Gamma)=\frac{{\rm{\#\ of\ events}}>\Gamma}{{\rm{total%
\ \#\ of\ events}}}.$$
(12)
Figure 3 shows an example of the false alarm rate calculation for the specific case of the 50-100 Hz band SGWB map and the full photometric SDSS galaxy catalog. The blue curve is derived from the 10,000 simulations and the red dot denotes the actual measured coherence using the O2 LIGO data. The FAR value of the red dot then gives the p-value significance of the measured coherence.
We repeat this procedure for all four frequency bands of the GW data and all redshift bins of the SDSS data, and for both the photometric and spectroscopic SDSS catalogs. The resulting p-values are shown in Figure 4.
The p-values for the photometric catalog have wider spread than the spectroscopic catalog, which can be explained by the fact that the photometric catalog is more uncertain in redshift and therefore more noisy as galaxy count maps in redshift slices.
Above all it is evident that all p-values are at or above $10^{-1}$ - $10^{-2}$ for both the photometric and spectroscopic galaxy count maps,
indicating low statistical significance. We therefore observe no correlation between SGWB and galaxy count sky maps.
We note that this analysis can be extended to perform model selection and/or parameter estimation. For this paper, we consider a simple empirical model where we assume that the SGWB energy density fluctuations are proportional to the normalized galaxy density fluctuations:
$$\displaystyle\delta M_{{\rm GW},i}^{\rm model}=\lambda\cdot\frac{\delta M_{{%
\rm GC},i}}{\langle M_{{\rm GC},i}\rangle}+\delta M_{\rm GW}^{noise},$$
(13)
where the index $i$ =1,2 represents galaxy count maps of the photometric or spectroscopic catalogs in the full redshift range. The factor $\lambda$ can therefore be interpreted as the GW strain power
per normalized fluctuation in the galaxy number count. We can use the observed value of $\Gamma$ to constrain the model parameter $\lambda$. To do so, we scan the values of the scaling parameter $\lambda$; for each value of $\lambda$ we generate 1000 realizations of the SGWB noise map $\delta M_{\rm GW}^{noise}$ similarly to above, and for each realization we compute the coherence $\Gamma$ between the corresponding model $\delta M_{{\rm GW},i}^{\rm model}$ and the galaxy count map $\delta M_{{\rm GC},i}$.
Figure 5 shows an example of $\Gamma$ as a function of $\lambda$, computed using the SGWB map in the 50-100 Hz frequency band and the full photometric SDSS catalog. We define $\lambda_{95}$ to be the 95% confidence upper limit on the scaling factor, i.e. the value of $\lambda$ that yields coherence $\Gamma$ larger than the observed coherence in 95% of the simulations. For the example shown in Figure 5, we find that $\lambda_{95}=2.7\times 10^{-49}$ st${}^{-1}$.
The calculation is repeated for all frequency bands for both photometric and spectroscopic catalogs, and the corresponding $\lambda_{95}$ values are summarized in Table 1.
5 Discussion and Conclusions
Studying the cross correlations between the SGWB and EM tracers of matter structure offers both the possibility of detecting the SGWB anisotropy sooner and the possibility to probe cosmological and astrophysical parameters driving the formation of structure. In this paper we have laid out a formalism to measure such SGWB-EM correlations. We have used the LIGO data from the second observing run and the galaxy catalog data from the SDSS to study the correlations of different GW frequency bands and different redshift slices in galaxy catalogs. We found no evidence for correlations between the SGWB and galaxy catalogs in these data.
We emphasize that while this may be the first measurement of its kind, there are many possible directions that should be explored in future works. We outline some of the possibilities here:
•
Our work has used only galaxy counts to track the matter structure. This can be expanded to use weak lensing survey data, or the cosmic microwave and infrared background data (e.g. from Planck (Planck
Collaboration et al., 2019; Ade et al., 2014)), or the X-ray data measured by Chandra X-ray Observatory (Schwartz, 2004). Different EM tracers will potentially correlate with different components of the SGWB: for example, galaxy counts or weak lensing may correlate with BBH/BNS SGWB, while the CMB anisotropy may correlate with cosmological SGWB models. Hence, spatial correlations with different EM tracers may help distinguish different SGWB contributions.
•
Vast amounts of new data are expected in the coming decade, on both GW and EM fronts. Advanced LIGO and Advanced Virgo are soon to complete the third observation run, with a sequence of detector upgrades and additional observation runs being planned. The Euclid (Paykari
et al., 2020) and SPHEREx (Korngut
et al., 2018) missions are expected to produce unprecedented galaxy surveys. For example, Euclid will identify 3$\times 10^{7}$ emission line galaxies, and use them as tracers of the large scale structure at $1<z<2.5$, which will significantly expand upon the existing SDSS catalogs.
•
The recently proposed Bayesian approach to measuring the BBH SGWB (Smith &
Thrane, 2018; Ashton
et al., 2019) promises to be significantly more sensitive to this type of background than the traditional stochastic search techniques (used also in this paper). This approach can produce the Bayesian posterior distribution of the BBH sky positions (for the entire BBH population), which can then be used to study correlations with the EM tracers such as galaxy counts or weak lensing surveys. The Bayesian approach also offers the possibility of extracting the redshift distribution of the BBH population, giving rise to the possibility of studying 3D correlations (sky position plus redshift) between the BBH SGWB and the galaxy count catalogs. (Banagiri et al., 2020)
•
Our analysis included estimation of a scaling parameter in a simple empirical model of the correlation between SGWB and galaxy counts. This can be expanded to include more sophisticated models of the BBH/BNS SGWB that properly take into account the cosmological and astrophysical evolution (Cusin
et al., 2017, 2018c, 2018b, 2019; Pitrou
et al., 2020).
We conclude by noting that studying the SGWB-EM correlations is a good example of how multi-messenger data can be used to generate new probes of astrophysics and cosmology. Upcoming data sets from both GW and EM detectors and telescopes, combined with improvements in data analysis techniques, promise novel ways of probing the evolution of structure in the universe, and perhaps also models of the early universe.
Acknowledgements
The authors thank Giulia Cusin for numerous discussions and insights regarding this manuscript. This work was supported by the NSF grant PHY-1806630.
The authors are thankful for the computing resources provided by LIGO Laboratory and supported by the National Science Foundation grants PHYâ0757058 and PHYâ0823459.
The code for the analysis in this paper is available upon request.
This paper is assigned the LIGO document control number LIGO-P2000220.
Data Availability
The data that support the findings of this study for the LIGO side are openly available in "O2 Data Release" at https://www.gw-openscience.org/data/. For the SDSS data, the photometric catalog ("photoObj") is available at https://www.sdss.org/dr16/imaging/catalogs/, the spectroscopic catalogs are available at https://www.sdss.org/dr16/spectro/.
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Entanglement-assisted Enhanced Information Transmission Over a Quantum Channel with Correlated
Noise; A General Expression
A. Fahmi
fahmi@theory.ipm.ac.ir
Institute for Studies in Theoretical Physics and
Mathematics (IPM) P. O. Box 19395-5531, Tehran, Iran
Abstract
Entanglement and entanglement-assisted are useful resources to
enhance the mutual information of the Pauli channels, when the
noise on consecutive uses of the channel has some partial
correlations. In this paper, We study quantum communication
channels with correlated noise and derive a general expression for
the mutual information of quantum channel, for the product,
maximally entangled state coding and entanglement-assisted systems
with correlated noise in the Pauli quantum channels. Hence, we
suggest more efficient coding in the entanglement-assisted systems
for the transmission of classical information and derive a general
expression for the entanglement-assisted classical capacity. Our
results show that in the presence of memory, a higher amount of
classical information is transmitted by two or four consecutive
uses of entanglement-assisted systems.
pacs: 03.67.Hk, 05.40.Ca
I INTRODUCTION
One of the remarkable byproducts of the development of quantum
mechanics in recent years is quantum information and quantum
computation theories. Classical and quantum information theories
have some basic differences. Some of these differences are
superposition principle, uncertainty principle and non-local
effects. The non-locality associated with entanglement in quantum
mechanics is one of the most subtle and intriguing phenomena in
nature nil . Its potential usefulness has been demonstrated
in a variety of applications, such as quantum teleportation,
quantum cryptography, and quantum dense coding. On the other hand,
quantum entanglement is a fragile feature, which can be destroyed
by interaction with the environment. This effect, which is due to
decoherence Shor ,is the main obstacle for practical
implementation of quantum computing and quantum communication.
Several strategies have been devised against decoherence. Quantum
error correction codes, fault-tolerant quantum computation
nil and dechorence free subspaces Knill are among
them. One of the main problems in the quantum communication is the
decoherence effects in the quantum channels.
Recently, the study of quantum channels has attracted a lot of
attention. Early works in this direction were devoted, mainly, to
memoryless channels for which consecutive signal transmissions
through the channel are not correlated. The capacities of some of
these channels were determined Sch ; Caves and it was proven
that in most cases their capacities are additive for single uses
of the channel. For Gaussian channels under Gaussian inputs, the
multiplicativity of output purities was proven in Ser and
the additivity of the energy-constrained capacity, even in the
presence of classical noise and thermal noise, was proven in
Hir , under the assumption that successive uses of the
channel are represented by the tensor product of the operators
representing one single use of the channel, i.e., the channel is
memoryless.
In a recent letter, Bartlett et al. Bar showed that
it is possible to communicate with perfect fidelity, and without a
shared reference frame, at a rate that asymptotically approaches
one encoded qubit per transmitted qubit. They proposed a method to
encode a qubit, using photons in a decoherence-free subspace of
the collective noise model. Boileau et al. considered
collective-noise channel effects in the quantum key distribution
Boi1 and they gave a realistic roboust scheme for quantum
communication, with polarized entangled photon pairs Boi2 .
In the last few years much attention has been given to bosonic
quantum channels Gio .
Recently Macchiavello et al. Mac ; Bowen , considered a
different class of channels, in which correlated noise acts on
consecutive uses of channels. They showed that higher mutual
information can be achieved above a certain memory threshold, by
entangling two consecutive uses of the channel. This type of
channels and its extension to the bosonic case, has attracted a
lot of attention in the recent years Gio2 .
K. Banaszek et. al Ball implemented Macchiavello
et al. suggestion experimentally. They shown how
entanglement can be used to enhance classical communication over a
noisy channel. In their setting, the introduction of entanglement
between two photons is required in order to maximize the amount of
information that can be encoded in their joint polarization degree
of freedom, and they obtained experimental classical capacity with
entangled states and showed that it is more than $2.5$ times the
theoretical upper limit, when no quantum correlations are allowed.
Hence, recently some people show that provided the sender and
receiver share prior entanglement, a higher amount of classical
information is transmitted over Pauli channels in the presence of
memory, as compared to product and entangled state coding
AT .
In this Paper, we show that if parties use a semi-quantum approach
to entanglement-assisted coding, a higher amount of classical
information is transmitted. We derive a general expression for the
entanglement-assisted classical capacity and compare our results
with the product Bell states and entanglement-assisted coding for
various types of Pauli channels.
This Paper is organized as follows: In Sec. II we briefly review
some properties of quantum memory channels and derive the general
expression of classical capacity of quantum channel for product
and maximally entangled state coding. In Sec. III, we derive the
general expression of entanglement-assisted classical capacity of
quantum channels with correlated noise that was calculated
previously for Pauli channels by Arshed and Toor (AT). In Sec IV,
we show that AT model is not an optimal coding for transmission of
classical information, and we suggest another sets of states and
derive the general expression of entanglement-assisted classical
capacity. Our results show that a higher amount of classical
capacity can indeed be achieved for all values of memory, by two
or four consecutive uses of the entanglement-assisted systems for
depolarizing, flip, two Pauli and phase damping channels.
II Entanglement-enhanced information transmission over a quantum channel with correlated noise
Encoding classical information into quantum states of physical
systems gives a physical implementation of the constructs of
information theory. The majority of research into quantum
communication channels has focused on the memoryless case,
although there have been a number of important results obtained
for quantum channels with correlated noise operators or more
general quantum channels Mac ; Ha .
The action of transmission channels is described by Kraus
operators $A_{i}$ Kraus , which satisfy the
$\sum_{i}A^{\dagger}_{i}A_{i}\leq 1$, the equality holds when the
map is trace-preserving. Thus, if we send a qubit in a state
described by the density operator $\rho$, through the channel,
then the corresponding out put state is given by the map.
$$\displaystyle\rho\longrightarrow\varepsilon(\rho)=\sum_{i}A_{i}\rho A^{\dagger%
}_{i}$$
An interesting class of Kraus operators acting on individual
qubits can be expressed in terms of the Pauli operators
$\sigma_{x,y,z}$
$$\displaystyle A_{i}=\sqrt{p_{i}}\sigma_{i}\ ,$$
with $\sum_{i}p_{i}=1$ , $i=0,x,y,z$ and $\sigma_{0}=I$. A noise
model for these actions is, for instance, the application of a
random rotation by angle $\pi$ around the axis $\hat{\bf{x}},\hat{\bf{y}},\hat{\bf{z}}$ with the probabilities
$p_{x},p_{y},p_{z}$ respectively, and the identity with probability
$p_{0}$.
In the simplest scenario, the transmitter can send one qubit at a
time along the channel. In this case the codewords will be
restricted to the tensor products of the states of the individual
qubits. Quantum mechanics, however allows also the possibility to
entangle multiple uses of the channel.
Recently, a model for quantum channels with memory has been
proposed that can consistently define quantum channels with
Markovian correlated noise Mac . The model is also extended
to describe channels that act on transmitted states in such a way
that there is no requirement for interactions with an environment
within the model. A Markovian correlated noise channel of length
$n$, is of the form:
$$\displaystyle\varepsilon(\rho)$$
$$\displaystyle=$$
$$\displaystyle\sum_{i_{1}\cdots i_{n}}p_{i_{1}}p_{i_{2}|i_{1}}\dots p_{i_{n}|i_%
{n-1}}$$
$$\displaystyle\times(A_{i_{n}}\otimes\cdots\otimes A_{i_{1}})\rho(A^{\dagger}_{%
i_{1}}\otimes\cdots\otimes A^{\dagger}_{i_{n}})$$
where the $A_{i_{k}}$ are Kraus operators for single uses of the
channel on the state $k$ and $p_{i_{k}|i_{k-1}}$ can be interpreted
as the conditional probability that a $\pi$ rotation around the
axis $i_{k}$ is applied to the $k$-th qubit, given that a $\pi$
rotation around axis $i_{k-1}$ was applied on the $k-1$-th qubit.
They considered Mac the case of two consecutive uses of a
channel with a partial memory, i.e. $p_{i_{k}|i_{k-1}}=(1-\mu)p_{i_{k}}+\mu\delta_{i_{k}|i_{k-1}}$. This means that with the
probability $\mu$, the same rotation is applied to both qubits,
while with probability $1-\mu$ the two rotations are
uncorrelated. Then, they concentrated their attention on the
depolarizing channel, for which $p_{0}=1-p$ and $p_{i}=p/3,i=x,y,z$,
and showed that for the specific case of a quantum depolarizing
channel with collective noise, the transmission of classical
information can be enhanced by employing maximally entangled
states as carriers of information, rather than product states.
In this section, we would like derive a general expression for the
classical capacity of Pauli quantum channels with correlated noise
for Bell and product states. The maximum mutual information
$I(\varepsilon(\rho))$ of a general quantum channel $\varepsilon$
is given by the Holevo-Schumacher-Westmoreland bound Sch
$$\displaystyle C(\varepsilon)=Max_{[\pi_{i},\rho_{i}]}S(\varepsilon(\sum_{i}\pi%
_{i}\rho_{i}))-\sum_{i}\pi_{i}S(\varepsilon(\rho_{i}))$$
(2)
where $S(\omega)=-Tr(\omega\log_{2}\omega)$ is the von Neumann
entropy of the density operator $\omega$ and the maximization is
performed over all input ensembles $\pi_{i}$ and $\rho_{i}$. Note
that this bound incorporates maximization over all POVM (positive
operator value measures) measurements at the receiver, including
the collective ones over multiple uses of the channel.
In what follows, we shall derive $I(\varepsilon(\rho))$ for
maximally Bell and product states. For the Bell states which are
defined as:
$$\displaystyle|\Psi^{\pm}\rangle_{AB}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{\sqrt{2}}(|00\rangle\pm|11\rangle),$$
$$\displaystyle|\Phi^{\pm}\rangle_{AB}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{\sqrt{2}}(|01\rangle\pm|10\rangle)$$
(3)
we know that the maximally entangled input states can be derived
from each other by unitary transformations (for example
$\rho_{0}=|\Psi^{+}\rangle\langle\Psi^{+}|$), and with respect to
the von Neumann entropy, $S(\omega)$ is invariant under any
unitary transformation of a quantum state $\omega$. The second
term on the right hand side of equation (2) becomes:
$$\displaystyle\sum_{i}\pi_{i}S(\varepsilon(\rho_{i}))=S(\varepsilon(\rho_{0}))=%
-\sum_{i}\lambda_{i}\log_{2}\lambda_{i}$$
where $\lambda_{i}$ are eigenvalues of the transformed states. On
the other hand, with respect to the relation
$Tr_{k}\rho_{i}=\frac{1}{2}\textbf{I}_{l}$ (with $k=A,B$ and
$l=B,A$), we show that the Holevo limit can be attained by setting
$\pi_{i}=\frac{1}{2^{2}}$ (with $i=0,...,3$). The quantum state
$\varepsilon(\rho_{i})$, in the first term, can be writhen as:
$$\displaystyle\varepsilon(\sum_{i}\pi_{i}\rho_{i})=\frac{1}{2^{2}}\sum_{i}%
\varepsilon(\rho_{i})=\varepsilon(\frac{1}{2}\textbf{I}_{1}\otimes\frac{1}{2}%
\textbf{I}_{2})$$
To get the final result, we take suitable bases for the density
matrixes representation. Then, the mutual information
$I(\varepsilon(\rho))$ of quantum channel is given by:
$$\displaystyle I(\varepsilon(\rho))=2+\sum_{i}\lambda_{i}\log_{2}\lambda_{i}$$
(4)
which for Bell states have the eigenvalues:
$$\displaystyle\lambda_{1}$$
$$\displaystyle=$$
$$\displaystyle(1-\mu)[(p_{0}^{2}+p_{x}^{2}+p_{y}^{2}+p_{z}^{2})+\mu$$
(5)
$$\displaystyle\lambda_{2}$$
$$\displaystyle=$$
$$\displaystyle(1-\mu)2(p_{0}p_{z}+p_{x}p_{y})$$
$$\displaystyle\lambda_{3}$$
$$\displaystyle=$$
$$\displaystyle(1-\mu)2(p_{0}p_{x}+p_{y}p_{z})$$
$$\displaystyle\lambda_{4}$$
$$\displaystyle=$$
$$\displaystyle(1-\mu)2(p_{0}p_{y}+p_{x}p_{z})$$
With a similar approach, it is straightforward to verify that
eigenvalues of product states in the $z$ basis ${|jk\rangle_{z}}$
(with $j,k=0,1$) are give by:
$$\displaystyle\lambda^{z}_{1}$$
$$\displaystyle=$$
$$\displaystyle(1-\mu)(p_{0}+p_{z})^{2}+\mu(p_{0}+p_{z})$$
(6)
$$\displaystyle\lambda^{z}_{2}$$
$$\displaystyle=$$
$$\displaystyle(1-\mu)(p_{x}+p_{y})^{2}+\mu(p_{x}+p_{y})$$
$$\displaystyle\lambda^{z}_{3,4}$$
$$\displaystyle=$$
$$\displaystyle(1-\mu)(p_{0}+p_{z})(p_{x}+p_{y})$$
By inserting these eigenvalues in the eq. (4), the mutual
information is give for the Bell and product states.
It is not complicated to verify that if we use product states in
$x$ basis ${|jk\rangle_{x}}$ (with $j,k=0,1$ and
$|l\rangle_{x}=(|0\rangle+(-1)^{l}|\rangle)_{z}$, $l=j,k$), the
eigenvalues in this basis are identical to the above eigenvalues
with $p_{x}$ and $p_{z}$ being interchanged. Hence, if we use the
product states in the $y$ basis ${|jk\rangle_{y}}$ (with $j,k=0,1$
and $|l\rangle_{y}=(|0\rangle+i(-1)^{l}|\rangle)_{z}$, $l=j,k$),
the eigenvalues in this basis are identical to above eigenvalues
with, $p_{y}$ and $p_{z}$ being interchanged.
These eigenvalues reduce to previous amounts in the especial case
of depolarizing channels Mac . In the final section, we
compare these mutual information to the entanglement-assisted
classical capacity.
III Entanglement-assisted Enhanced classical capacity of quantum channels with correlated noise
Recently, Arshed and Toor AT considered an interesting
extension of classical capacity, and calculated the
entanglement-assisted classical capacity of Pauli channels for two
consecutive uses of the channels. They assumed that the sender
(Alice) and the receiver (Bob) share two (same or different)
maximally entangled Bell states $|\psi^{\pm}\rangle,$
$|\phi^{\pm}\rangle$ (as defined by the eq. (II)). The
first qubits of Bell states belongs to Alice while the second
qubits belongs to Bob. Alice sends her qubits through the channel.
They calculated eigenvalues of the pure density matrix,
transformed under the action of the depolarizing channel, for
probabilities $p_{0}=1-p$, $p_{x}=p_{y}=p_{z}=p/3$, and after some
simple calculations, they derived the entanglement-assisted
classical capacity for the Pauli channels in the presence of
partial memory.
As was suggested by AT, in the presence of partial memory, the
action of Pauli channels is described by the Kraus operators:
$$\displaystyle A_{i,j}(\mu)=\sqrt{p_{i}[(1-\mu)p_{j}+\mu\delta_{i,j}]}(\sigma_{%
i}^{A}\otimes I^{B})\otimes(\sigma_{j}^{A}\otimes I^{B})$$
The mutual information of quantum dense coding system, where Alice
and Bob shared quantum state $\rho^{AB}$ (which is statistical
mixture of the Bell states), is given by:
$$\displaystyle I^{AT}(\varepsilon(\rho))=4+\sum_{i}\lambda_{i}\log_{2}\lambda_{i}$$
(7)
where $\lambda_{i}$ are the eigenvalues of the transformed states.
For states that were considered by AT, eigenvalues in the general
case are give by:
$$\displaystyle\lambda_{1}$$
$$\displaystyle=$$
$$\displaystyle(1-\mu)p_{0}^{2}+\mu p_{0}$$
(8)
$$\displaystyle\lambda_{2}$$
$$\displaystyle=$$
$$\displaystyle(1-\mu)p_{x}^{2}+\mu p_{x}$$
$$\displaystyle\lambda_{3}$$
$$\displaystyle=$$
$$\displaystyle(1-\mu)p_{y}^{2}+\mu p_{y}$$
$$\displaystyle\lambda_{4}$$
$$\displaystyle=$$
$$\displaystyle(1-\mu)p_{z}^{2}+\mu p_{z}$$
$$\displaystyle\lambda_{5,6}$$
$$\displaystyle=$$
$$\displaystyle(1-\mu)p_{0}p_{x}$$
$$\displaystyle\lambda_{7,8}$$
$$\displaystyle=$$
$$\displaystyle(1-\mu)p_{0}p_{y}$$
$$\displaystyle\lambda_{9,10}$$
$$\displaystyle=$$
$$\displaystyle(1-\mu)p_{0}p_{z}$$
$$\displaystyle\lambda_{11,12}$$
$$\displaystyle=$$
$$\displaystyle(1-\mu)p_{x}p_{y}$$
$$\displaystyle\lambda_{13,14}$$
$$\displaystyle=$$
$$\displaystyle(1-\mu)p_{x}p_{z}$$
$$\displaystyle\lambda_{15,16}$$
$$\displaystyle=$$
$$\displaystyle(1-\mu)p_{y}p_{z}$$
These eigenvalues reduce those of AT approach in the especial
cases of depolarizing, flip, two-Pauli and phase damping channels
AT .
IV Semi-Entanglement-assisted Enhanced Information transmission over quantum channels with correlated noise
In the preceding section, we derived the classical information
capacity $C(\varepsilon)$ of a quantum dense coding system.
Although, the use of entanglement-assisted enhances classical
capacity of channels with correlated noise, in comparison with
both product and entangled state coding for all values of $\mu$,
here exists other quantum dense coding systems in which those
states have a higher amount of classical information transmission
in the channels with correlated noise.
In this section, we derive a general expression for the
entanglement-assisted classical capacity for two kinds of quantum
dense coding systems which we call semi-quantum dense coding. Our
results show that for channels with partial memory and with the
use of appropriate choice of entanglement-assisted states, higher
mutual information can indeed be achieved for all values of $\mu$,
by two or four consecutive uses of the channel. First, we consider
a simple modification of AT approach. As we saw, AT use two
maximally entangled Bell states, the first qubit belongs to the
Alice and the second qubit belongs to the Bob, we replace AT
states by following states:
$$\displaystyle|\Phi_{1,2}\rangle=|\psi^{+}\rangle_{A}|\psi^{+}\rangle_{B}\pm|%
\psi^{-}\rangle_{A}|\psi^{-}\rangle_{B}$$
(9)
$$\displaystyle|\Phi_{3,4}\rangle=|\psi^{-}\rangle_{A}|\psi^{+}\rangle_{B}\pm|%
\psi^{+}\rangle_{A}|\psi^{-}\rangle_{B}$$
$$\displaystyle|\Phi_{5,6}\rangle=|\varphi^{+}\rangle_{A}|\psi^{+}\rangle_{B}\pm%
|\varphi^{-}\rangle_{A}|\psi^{-}\rangle_{B}$$
$$\displaystyle|\Phi_{7,8}\rangle=|\varphi^{-}\rangle_{A}|\psi^{+}\rangle_{B}\pm%
|\varphi^{+}\rangle_{A}|\psi^{-}\rangle_{B}$$
$$\displaystyle|\Phi_{9,10}\rangle=|\psi^{+}\rangle_{A}|\varphi^{+}\rangle_{B}%
\pm|\psi^{-}\rangle_{A}|\varphi^{-}\rangle_{B}$$
$$\displaystyle|\Phi_{11,12}\rangle=|\psi^{-}\rangle_{A}|\varphi^{+}\rangle_{B}%
\pm|\psi^{+}\rangle_{A}|\varphi^{-}\rangle_{B}$$
$$\displaystyle|\Phi_{13,14}\rangle=|\varphi^{+}\rangle_{A}|\varphi^{+}\rangle_{%
B}\pm|\varphi^{-}\rangle_{A}|\varphi^{-}\rangle_{B}$$
$$\displaystyle|\Phi_{15,16}\rangle=|\varphi^{-}\rangle_{A}|\varphi^{+}\rangle_{%
B}\pm|\varphi^{+}\rangle_{A}|\varphi^{-}\rangle_{B}$$
In these states, the first Bell states belongs to Alice and the
second Bell states belongs to Bob, which are represented by
subscriptions in the Bell states respectively. These states are
equivalent to the Bell state which is shared by Alice and Bob. For
example, parties can construct
$|\Phi_{1}\rangle=|00\rangle_{A}|00\rangle_{B}+|11\rangle_{A}|11\rangle_{B}$
by performing cnot gates $C_{A1}$ and $C_{B2}$ on the
state
$(|00\rangle_{AB}+|11\rangle_{AB})|0\rangle_{1}|0\rangle_{2}$ ($A$
and $B$ are the controller and $1$ and $2$ are the targets).
Hence, Alice can transform $|\Phi_{i}\rangle$ (with $i=1,...,8$)
states to each other by a local operation on the his qubits. The
same transformation exists for $|\Phi_{i}\rangle$ (with
$i=9,...,16$). These two groups of states cannot transform to each
other by Alice’s local operation.
On the other hand, as we know, if Alice and Bob previously shared
entangled states between themselves, then, for noiseless quantum
channels, the amount of classical information transmitted through
a quantum channel is doubled in comparison with the unshared
states $C_{E}=2C$ BW . In other words, every previously
shared Bell state is equivalent with two quantum channels (or
twice the using quantum channel).
In our approach, Alice and Bob shared only one of the above states
(for example, $|\Phi_{1}\rangle$) and they had previously
compromised that if Alice states were received by Bob in the time
interval $\delta t_{0}=t_{1}-t_{0}$ ($\delta t_{1}=t_{2}-t_{1}$),
Bob would operate $I^{B}\otimes I^{B}$ ($I^{B}\otimes\sigma_{x}^{B}$) on the his qubits (the Bell states that were
represented by subscript $2$ in the (9)) Ano1 .
By using this protocol, Alice and Bob get access to all Hilbert
space states in the states (9). This protocol (using
one shared Bell state between parties and two types of qubits
transmission), we call semi-quantum dense coding which has the
same cost as that of the AT approach (which uses two shared Bell
states between parties and one type of qubits transmission in the
quantum channel).
We consider the shortest variational time of the channel (fiber)
under thermal and mechanical fluctuations as $\tau_{fluc}$. If the
time lapse ($\tau_{lap}$ ) between the two used channel is small
compared to $\tau_{fluc}$, the effects of the channel on various
qubits can be considered as a correlated noise. For example, in
the experiment of K. Banaszek et. al Ball
$\tau_{lap}\approx 6ns$, which is much smaller than the
mechanical fluctuations of the fiber.
Similar to the previous cases, the action of transmission channels
is described by Karus operators $A_{i,j}$ , satisfying
$\sum_{i,j}A_{i,j}A_{i,j}^{\dagger}=1$, Where in the presence of
partial memory, the action of Pauli channels on $\rho$ is
described by
$$\displaystyle A_{i,j}(\mu)=\sqrt{p_{i}[(1-\mu)p_{j}+\mu\delta_{i,j}]}(\sigma_{%
i}\otimes\sigma_{j})\otimes(I\otimes I)$$
If Alice sends the density matrix $\rho$ through the quantum
channel, the corresponding output state is given by the mapping
$$\displaystyle\rho\longrightarrow\varepsilon(\rho)=\sum_{i,j=0}^{3}A_{i,j}(\mu)%
\rho A_{i,j}^{\dagger}(\mu)$$
(10)
That can be written for channels with partial memory as:
$$\displaystyle\varepsilon(\rho)$$
$$\displaystyle=$$
$$\displaystyle(1-\mu)\sum_{i,j=0}^{3}p_{i}p_{j}(\sigma_{i}\otimes\sigma_{j}%
\otimes I\otimes I)\rho(\sigma_{i}\otimes\sigma_{j}\otimes I\otimes I)+\mu\sum%
_{i=0}^{3}p_{i}(\sigma_{i}\otimes\sigma_{i}\otimes I\otimes I)\rho(\sigma_{i}%
\otimes\sigma_{i}\otimes I\otimes I)$$
Here, we would like to compute the entanglement-assisted classical
capacity of quantum channels by using quantum superdense coding
system, where Alice and Bob share a completely entangled state
$|\Phi^{AB}_{1}\rangle$, which belongs to an $(2^{2}\times 2^{2})$-dimensional (for $2^{2}$-dimensional Bell states) Hilbert
space $\emph{H}\otimes\emph{H}$, and the encoded system is sent
through a quantum channel, described by an arbitrary
trace-preserving completely positive map $\varepsilon$.
In a simple modification of Ban et al. Ban approach
to the calculation of classical capacity of quantum dense coding,
we assume Alice and Bob to share a statistical mixture of the
entangled states $|\Phi_{i}\rangle$ (with $i=1,...,16$), where the
shared quantum state $\rho^{AB}$ is given by:
$$\displaystyle\rho^{AB}=\sum_{j=0}^{16}\lambda_{j}|\Phi^{AB}_{j}\rangle\langle%
\Phi^{AB}_{j}|$$
(11)
with $\lambda_{j}\geq 0$ and $\sum_{j=0}^{16}\lambda_{j}=1$. To
encode $2\log_{2}2^{2}$ bits of classical information, Alice
applies one of $8\times 2$ operation, $8$ unitary operators $V_{k}$
$(k=0,...,8)$ to her part of the quantum state $\rho^{AB}$ and the
two types of state transmission. She sends the encoded system to
Bob through an arbitrary quantum channel $\varepsilon$. Then, Bob
obtains the quantum state:
$$\displaystyle\rho^{\prime AB}_{jk}=(\varepsilon^{A}\otimes I^{B})[(V^{A}_{j}%
\otimes v^{B}_{k})\rho^{AB}(V^{A\dagger}_{j}\otimes v^{B\dagger}_{k})]$$
(12)
In the above relation, $v^{B}_{k}$ depends on the type of qubits
sent by Alice, which operate by Bob on the his qubits. If the
prior probability of the classical information corresponding to
$V^{A}_{j}$ and $v^{B}_{k}$ (Alice and Bob operation on the shared
state) is $\pi_{jk}$, where $\pi_{jk}\geq 0$ and
$\sum_{j=0}^{8}\sum_{k=0}^{1}\pi_{jk}=1$, the maximum amount of
mutual information of the quantum dense coding system would be
given by BSST :
$$\displaystyle C^{E}(\varepsilon,{\pi_{jk}})$$
$$\displaystyle=$$
$$\displaystyle Max_{[\rho,{\pi_{jk}}]}I^{AB}(\varepsilon,{\pi_{jk}})$$
$$\displaystyle=$$
$$\displaystyle Max_{[\rho,{\pi_{jk}}]}S(\rho^{\prime AB})-\sum_{j,k}\pi_{jk}S(%
\varepsilon(\rho_{jk}))$$
where $\rho^{\prime AB}=\sum_{j}\sum_{k}\pi_{jk}\varepsilon(\rho^{AB}_{jk})$. In the above relation the maximization is
performed over all input ensembles $\pi_{jk}$ and $\rho_{ik}$.
Note that this bound incorporates maximization over all POVM
measurements (positive operator value measures) at the receiver,
including the collective ones over multiple uses of the channel.
In what follows, we shall derive $I(\varepsilon(\rho_{jk}))$ for
entangled states which were suggested in the eq.(9).
We know that the entangled input states $|\Phi_{i}\rangle$ can be
derived from $|\Phi_{1}\rangle$ by unitary transformations (Pauli
matrixes), and with respect to the von Neumann entropy,
$S(\omega)$ is invariant under any unitary transformation of a
quantum state $\omega$. The second term on the right hand side of
equation (IV) becomes:
$$\displaystyle\sum_{j,k}\pi_{jk}S(\varepsilon(\rho_{jk}))=S(\varepsilon^{A}%
\otimes\emph{I}^{B}(\rho^{AB}))$$
On the other hand, we show that the maximum amount of mutual
information can be attained by setting $\pi_{jk}=\frac{1}{2^{4}}$
(with $j=0,...,8$ and $k=0,1$). The quantum state $\varepsilon(\rho^{AB}_{jk})$, in the first term, can be writhen as:
$$\displaystyle\sum_{jk}\pi_{jk}\varepsilon(\rho^{AB}_{jk})=\frac{1}{2^{4}}\sum_%
{jk}\varepsilon(\rho^{AB}_{jk})=\varepsilon(\frac{1}{2}\textbf{I}_{1}\otimes..%
.\otimes\frac{1}{2}\textbf{I}_{4})$$
To get the final result, we take suitable bases for density
matrixes representation. Mutual information
$I_{1}^{AB,e-a}(\varepsilon,{\pi_{jk}})$ can be calculated for
the quantum channel with entangled states (9) in the
general case. Therefore, for symmetric and asymmetric Pauli
channels, we have:
$$\displaystyle I_{1}^{AB,e-a}(\varepsilon,{\pi_{jk}})=4-S(\varepsilon^{A}%
\otimes\emph{I}^{B}(\rho^{AB}))$$
(14)
The von Neumann entropy of the $\rho^{AB}$ state, transformed
under the action of Pauli channels, is given by:
$$\displaystyle S(\varepsilon^{A}\otimes\emph{I}^{B}(\rho^{AB}))=-\sum_{i}%
\lambda_{i}\log_{2}\lambda_{i}$$
(15)
where $\lambda_{i}$ are the eigenvalues of the transformed
$\rho^{AB}$ state, which have the explicit forms:
$$\displaystyle\lambda_{1}$$
$$\displaystyle=$$
$$\displaystyle(1-\mu)(p_{0}^{2}+p_{z}^{2})+\mu(p_{0}+p_{z})$$
(16)
$$\displaystyle\lambda_{2}$$
$$\displaystyle=$$
$$\displaystyle(1-\mu)(p_{x}^{2}+p_{y}^{2})+\mu(p_{x}+p_{y})$$
$$\displaystyle\lambda_{3}$$
$$\displaystyle=$$
$$\displaystyle(1-\mu)2p_{0}p_{z}$$
$$\displaystyle\lambda_{4}$$
$$\displaystyle=$$
$$\displaystyle(1-\mu)2p_{x}p_{y}$$
$$\displaystyle\lambda_{5,6}$$
$$\displaystyle=$$
$$\displaystyle(1-\mu)(p_{0}p_{x}+p_{y}p_{z})$$
$$\displaystyle\lambda_{7,8}$$
$$\displaystyle=$$
$$\displaystyle(1-\mu)(p_{0}p_{y}+p_{x}p_{z})$$
Hence, there exist two groups of eigenvalues which are equal to
each other. These states don’t mix with each other through
interaction with environment.
Although the above states are more efficient than the AT states
(as we shall see at the end of this section) for the transmission
of classical information, there exist other entangled states which
enhance the classical capacity of quantum channels through the use
of the entanglement-assisted. Similar to the previous case, we
consider semi-quantum approach, which uses one entangled state
augmented by quantum channels and show that in the channels with
partial memory, the use of four particle entanglement-assisted
enhances the amount of mutual information can indeed be achieved
for all values of $\mu$, by four consecutive uses of the channel
(at more of cases, for example, in the depolarizing, flip and two
Pauli channels). This approach has $8^{4}$ dimensional Hilbert
space, that in following we represent sixty four (one quarter of
total Hilbert space dimension) of them:
$$\displaystyle|\Psi_{1,i}\rangle=|\psi^{+}\rangle_{1}|\psi^{+}\rangle_{2}|%
\omega^{i}\rangle_{3}|\psi^{+}\rangle_{4}+|\psi^{-}\rangle_{1}|\psi^{-}\rangle%
_{2}|\omega^{i+1}\rangle_{3}|\psi^{-}\rangle_{4}+|\phi^{+}\rangle_{1}|\phi^{+}%
\rangle_{2}|\omega^{i+2}\rangle_{3}|\phi^{+}\rangle_{4}+|\phi^{-}\rangle_{1}|%
\phi^{-}\rangle_{2}|\omega^{i+3}\rangle_{3}|\phi^{-}\rangle_{4}$$
(17)
$$\displaystyle|\Psi_{2,i}\rangle=|\psi^{+}\rangle_{1}|\psi^{+}\rangle_{2}|%
\omega^{i}\rangle_{3}|\psi^{+}\rangle_{4}+|\psi^{-}\rangle_{1}|\psi^{-}\rangle%
_{2}|\omega^{i+1}\rangle_{3}|\psi^{-}\rangle_{4}-|\phi^{+}\rangle_{1}|\phi^{+}%
\rangle_{2}|\omega^{i+2}\rangle_{3}|\phi^{+}\rangle_{4}-|\phi^{-}\rangle_{1}|%
\phi^{-}\rangle_{2}|\omega^{i+3}\rangle_{3}|\phi^{-}\rangle_{4}$$
$$\displaystyle|\Psi_{3,i}\rangle=|\psi^{+}\rangle_{1}|\psi^{+}\rangle_{2}|%
\omega^{i}\rangle_{3}|\psi^{+}\rangle_{4}-|\psi^{-}\rangle_{1}|\psi^{-}\rangle%
_{2}|\omega^{i+1}\rangle_{3}|\psi^{-}\rangle_{4}-|\phi^{+}\rangle_{1}|\phi^{+}%
\rangle_{2}|\omega^{i+2}\rangle_{3}|\phi^{+}\rangle_{4}+|\phi^{-}\rangle_{1}|%
\phi^{-}\rangle_{2}|\omega^{i+3}\rangle_{3}|\phi^{-}\rangle_{4}$$
$$\displaystyle|\Psi_{4,i}\rangle=|\psi^{+}\rangle_{1}|\psi^{+}\rangle_{2}|%
\omega^{i}\rangle_{3}|\psi^{+}\rangle_{4}-|\psi^{-}\rangle_{1}|\psi^{-}\rangle%
_{2}|\omega^{i+1}\rangle_{3}|\psi^{-}\rangle_{4}+|\phi^{+}\rangle_{1}|\phi^{+}%
\rangle_{2}|\omega^{i+2}\rangle_{3}|\phi^{+}\rangle_{4}-|\phi^{-}\rangle_{1}|%
\phi^{-}\rangle_{2}|\omega^{i+3}\rangle_{3}|\phi^{-}\rangle_{4}$$
$$\displaystyle|\Psi_{5,i}\rangle=|\psi^{+}\rangle_{1}|\psi^{-}\rangle_{2}|%
\omega^{i}\rangle_{3}|\psi^{+}\rangle_{4}+|\psi^{-}\rangle_{1}|\psi^{+}\rangle%
_{2}|\omega^{i+1}\rangle_{3}|\psi^{-}\rangle_{4}+|\phi^{+}\rangle_{1}|\phi^{-}%
\rangle_{2}|\omega^{i+2}\rangle_{3}|\phi^{+}\rangle_{4}+|\phi^{-}\rangle_{1}|%
\phi^{+}\rangle_{2}|\omega^{i+3}\rangle_{3}|\phi^{-}\rangle_{4}$$
$$\displaystyle|\Psi_{6,i}\rangle=|\psi^{+}\rangle_{1}|\psi^{-}\rangle_{2}|%
\omega^{i}\rangle_{3}|\psi^{+}\rangle_{4}+|\psi^{-}\rangle_{1}|\psi^{+}\rangle%
_{2}|\omega^{i+1}\rangle_{3}|\psi^{-}\rangle_{4}-|\phi^{+}\rangle_{1}|\phi^{-}%
\rangle_{2}|\omega^{i+2}\rangle_{3}|\phi^{+}\rangle_{4}-|\phi^{-}\rangle_{1}|%
\phi^{+}\rangle_{2}|\omega^{i+3}\rangle_{3}|\phi^{-}\rangle_{4}$$
$$\displaystyle|\Psi_{7,i}\rangle=|\psi^{+}\rangle_{1}|\psi^{-}\rangle_{2}|%
\omega^{i}\rangle_{3}|\psi^{+}\rangle_{4}-|\psi^{-}\rangle_{1}|\psi^{+}\rangle%
_{2}|\omega^{i+1}\rangle_{3}|\psi^{-}\rangle_{4}-|\phi^{+}\rangle_{1}|\phi^{-}%
\rangle_{2}|\omega^{i+2}\rangle_{3}|\phi^{+}\rangle_{4}+|\phi^{-}\rangle_{1}|%
\phi^{+}\rangle_{2}|\omega^{i+3}\rangle_{3}|\phi^{-}\rangle_{4}$$
$$\displaystyle|\Psi_{8,i}\rangle=|\psi^{+}\rangle_{1}|\psi^{-}\rangle_{2}|%
\omega^{i}\rangle_{3}|\psi^{+}\rangle_{4}-|\psi^{-}\rangle_{1}|\psi^{+}\rangle%
_{2}|\omega^{i+1}\rangle_{3}|\psi^{-}\rangle_{4}+|\phi^{+}\rangle_{1}|\phi^{-}%
\rangle_{2}|\omega^{i+2}\rangle_{3}|\phi^{+}\rangle_{4}-|\phi^{-}\rangle_{1}|%
\phi^{+}\rangle_{2}|\omega^{i+3}\rangle_{3}|\phi^{-}\rangle_{4}$$
$$\displaystyle|\Psi_{9,i}\rangle=|\psi^{+}\rangle_{1}|\phi^{+}\rangle_{2}|%
\omega^{i}\rangle_{3}|\psi^{+}\rangle_{4}+|\psi^{-}\rangle_{1}|\phi^{-}\rangle%
_{2}|\omega^{i+1}\rangle_{3}|\psi^{-}\rangle_{4}+|\phi^{+}\rangle_{1}|\psi^{+}%
\rangle_{2}|\omega^{i+2}\rangle_{3}|\phi^{+}\rangle_{4}+|\phi^{-}\rangle_{1}|%
\psi^{-}\rangle_{2}|\omega^{i+3}\rangle_{3}|\phi^{-}\rangle_{4}$$
$$\displaystyle|\Psi_{10,i}\rangle=|\psi^{+}\rangle_{1}|\phi^{+}\rangle_{2}|%
\omega^{i}\rangle_{3}|\psi^{+}\rangle_{4}+|\psi^{-}\rangle_{1}|\phi^{-}\rangle%
_{2}|\omega^{i+1}\rangle_{3}|\psi^{-}\rangle_{4}-|\phi^{+}\rangle_{1}|\psi^{+}%
\rangle_{2}|\omega^{i+2}\rangle_{3}|\phi^{+}\rangle_{4}-|\phi^{-}\rangle_{1}|%
\psi^{-}\rangle_{2}|\omega^{i+3}\rangle_{3}|\phi^{-}\rangle_{4}$$
$$\displaystyle|\Psi_{11,i}\rangle=|\psi^{+}\rangle_{1}|\phi^{+}\rangle_{2}|%
\omega^{i}\rangle_{3}|\psi^{+}\rangle_{4}-|\psi^{-}\rangle_{1}|\phi^{-}\rangle%
_{2}|\omega^{i+1}\rangle_{3}|\psi^{-}\rangle_{4}-|\phi^{+}\rangle_{1}|\psi^{+}%
\rangle_{2}|\omega^{i+2}\rangle_{3}|\phi^{+}\rangle_{4}+|\phi^{-}\rangle_{1}|%
\psi^{-}\rangle_{2}|\omega^{i+3}\rangle_{3}|\phi^{-}\rangle_{4}$$
$$\displaystyle|\Psi_{12,i}\rangle=|\psi^{+}\rangle_{1}|\phi^{+}\rangle_{2}|%
\omega^{i}\rangle_{3}|\psi^{+}\rangle_{4}-|\psi^{-}\rangle_{1}|\phi^{-}\rangle%
_{2}|\omega^{i+1}\rangle_{3}|\psi^{-}\rangle_{4}+|\phi^{+}\rangle_{1}|\psi^{+}%
\rangle_{2}|\omega^{i+2}\rangle_{3}|\phi^{+}\rangle_{4}-|\phi^{-}\rangle_{1}|%
\psi^{-}\rangle_{2}|\omega^{i+3}\rangle_{3}|\phi^{-}\rangle_{4}$$
$$\displaystyle|\Psi_{13,i}\rangle=|\psi^{+}\rangle_{1}|\phi^{-}\rangle_{2}|%
\omega^{i}\rangle_{3}|\psi^{+}\rangle_{4}+|\psi^{-}\rangle_{1}|\phi^{+}\rangle%
_{2}|\omega^{i+1}\rangle_{3}|\psi^{-}\rangle_{4}+|\phi^{+}\rangle_{1}|\psi^{-}%
\rangle_{2}|\omega^{i+2}\rangle_{3}|\phi^{+}\rangle_{4}+|\phi^{-}\rangle_{1}|%
\psi^{+}\rangle_{2}|\omega^{i+3}\rangle_{3}|\phi^{-}\rangle_{4}$$
$$\displaystyle|\Psi_{14,i}\rangle=|\psi^{+}\rangle_{1}|\phi^{-}\rangle_{2}|%
\omega^{i}\rangle_{3}|\psi^{+}\rangle_{4}+|\psi^{-}\rangle_{1}|\phi^{+}\rangle%
_{2}|\omega^{i+1}\rangle_{3}|\psi^{-}\rangle_{4}-|\phi^{+}\rangle_{1}|\psi^{-}%
\rangle_{2}|\omega^{i+2}\rangle_{3}|\phi^{+}\rangle_{4}-|\phi^{-}\rangle_{1}|%
\psi^{+}\rangle_{2}|\omega^{i+3}\rangle_{3}|\phi^{-}\rangle_{4}$$
$$\displaystyle|\Psi_{15,i}\rangle=|\psi^{+}\rangle_{1}|\phi^{-}\rangle_{2}|%
\omega^{i}\rangle_{3}|\psi^{+}\rangle_{4}-|\psi^{-}\rangle_{1}|\phi^{+}\rangle%
_{2}|\omega^{i+1}\rangle_{3}|\psi^{-}\rangle_{4}-|\phi^{+}\rangle_{1}|\psi^{-}%
\rangle_{2}|\omega^{i+2}\rangle_{3}|\phi^{+}\rangle_{4}+|\phi^{-}\rangle_{1}|%
\psi^{+}\rangle_{2}|\omega^{i+3}\rangle_{3}|\phi^{-}\rangle_{4}$$
$$\displaystyle|\Psi_{16,i}\rangle=|\psi^{+}\rangle_{1}|\phi^{-}\rangle_{2}|%
\omega^{i}\rangle_{3}|\psi^{+}\rangle_{4}-|\psi^{-}\rangle_{1}|\phi^{+}\rangle%
_{2}|\omega^{i+1}\rangle_{3}|\psi^{-}\rangle_{4}+|\phi^{+}\rangle_{1}|\psi^{-}%
\rangle_{2}|\omega^{i+2}\rangle_{3}|\phi^{+}\rangle_{4}-|\phi^{-}\rangle_{1}|%
\psi^{+}\rangle_{2}|\omega^{i+3}\rangle_{3}|\phi^{-}\rangle_{4}$$
In the above states, $i=0,...,3$ and $|\omega^{i}\rangle$ are
define as:
$$\displaystyle|\omega^{0,1}\rangle=|\Psi^{\pm}\rangle\hskip 14.226378pt|\omega^%
{2,3}\rangle=|\Phi^{\pm}\rangle$$
the subscript $i$ is calculated in the $mod\hskip 5.690551pt4$. In
the above states, the first two Bell states of $1,2$ belongs to
Alice and the second two Bell states of $3,4$ belongs to Bob which
are represented by subscriptions in the Bell states respectively.
Similar to the previous case, Alice and Bob share only one of the
above states (for example, $|\Psi_{1,0}\rangle$) and Alice can
transform this state to one of the above states (one quarter of
the total Hilbert space dimension) by local operation on the her
qubits.
They had previously compromised that if Alice states received by
Bob were in the time intervals $\delta t_{0}=t_{1}-t_{0}$ or
$\delta t_{1}=t_{2}-t_{1}$ or $\delta t_{2}=t_{2}-t_{3}$ or
$\delta t_{3}=t_{4}-t_{3}$, Bob would operate $I^{B}\otimes I^{B}$
or $\sigma_{z}^{B}\otimes I^{B}$ or $I^{B}\otimes\sigma_{x}^{B}$
or $\sigma_{z}^{B}\otimes\sigma_{x}^{B}$ on his qubits
respectively Ano2 . By using this protocol, Alice and Bob
get access to all of the Hilbert space states.
Similar to the previous cases, Karus operators $A_{i,j,k,l}$ ,
satisfy $\sum_{i,j,k,l}A_{i,j,k,l}A_{i,j,k,l}^{\dagger}=1$, and in
the presence of partial memory, the action of Pauli channels on
the $\rho$ is described by:
$$\displaystyle A_{i,j,k,l}(\mu)=\sqrt{p_{i}[(1-\mu)p_{j}p_{k}p_{l}+\mu\delta_{i%
,j}\delta_{j,k}\delta_{k,l}]}\sigma_{i}\otimes\sigma_{j}\otimes\sigma_{k}%
\otimes\sigma_{l}\otimes I^{\otimes 4}$$
For simplicity, we consider only two types of sending particles:
i) The two Bell states in the Alice’s hand (subscribed by $1,2$ in
the states (17)) are sent at the same time
($\tau_{lap}\ll\tau_{fluc}$ for each pair). ii) The two Bell
states are sent with a time delay ($\tau_{fluc}\ll\tau_{lap}$ for
each pair). Although, we can consider the general case of the
transmission of quantum states, but its explicit form is very
complicated and doesn’t clarify any physical properties. With a
similar approach to the one we described in the previous case, the
mutual information for the entanglement-assisted systems can be
calculated. If Alice sends the density matrix $\rho$ through a
quantum channel, the corresponding output state is given by the
map:
$$\displaystyle\rho\longrightarrow\varepsilon(\rho)=\sum_{i,j,k,l=0}^{3}A_{i,j,k%
,l}(\mu)\rho A_{i,j,k,l}^{\dagger}(\mu)$$
For channels with partial memory output, the density matrix can be
written as following:
$$\displaystyle\varepsilon(\rho)$$
$$\displaystyle=$$
$$\displaystyle(1-\mu)\sum_{i,j,k,l=0}^{3}p_{i}p_{j}p_{k}p_{l}(\sigma_{i}\otimes%
\sigma_{j}\otimes\sigma_{k}\otimes\sigma_{l}\otimes I^{\otimes 4})\rho(\sigma_%
{i}\otimes\sigma_{j}\otimes\sigma_{k}\otimes\sigma_{l}\otimes I^{\otimes 4})$$
$$\displaystyle+\mu\sum_{i=0}^{3}p_{i}(\sigma_{i}\otimes\sigma_{i}\otimes\sigma_%
{i}\otimes\sigma_{i}\otimes I^{\otimes 4})\rho(\sigma_{i}\otimes\sigma_{i}%
\otimes\sigma_{i}\otimes\sigma_{i}\otimes I^{\otimes 4})$$
After a similar calculation to the previous case, we derive the
mutual information of the quantum channel, with correlated noise,
given by:
$$\displaystyle I_{2}^{AB,e-a}(\varepsilon,\pi_{i})=8-S(\varepsilon^{A}\otimes%
\emph{I}^{B}(\rho^{AB}))$$
(18)
The von Neumann entropy of the $\rho^{AB}$ state, transformed
under the action of Pauli channels, is given by:
$$\displaystyle S(\varepsilon^{A}\otimes\emph{I}^{B}(\rho^{AB}))=-\sum_{i}%
\lambda_{i}\log_{2}\lambda_{i}$$
(19)
where $\lambda_{i}$ are the eigenvalues of the transformed
$\rho^{AB}$ state with the explicit forms:
$$\displaystyle\lambda_{1}$$
$$\displaystyle=$$
$$\displaystyle(1-\mu)(p_{0}^{4}+p_{z}^{4}+p_{x}^{4}+p_{y}^{4})+\mu$$
$$\displaystyle\lambda_{2,3,4}$$
$$\displaystyle=$$
$$\displaystyle(1-\mu)[2(p_{0}p_{z})^{2}+2(p_{x}p_{y})^{2}]$$
$$\displaystyle\lambda_{4,6,7}$$
$$\displaystyle=$$
$$\displaystyle(1-\mu)[2(p_{0}p_{x})^{2}+2(p_{y}p_{z})^{2}]$$
$$\displaystyle\lambda_{8,9,10}$$
$$\displaystyle=$$
$$\displaystyle(1-\mu)[2(p_{0}p_{y})^{2}+2(p_{x}p_{z})^{2}]$$
$$\displaystyle\lambda_{11,...,14}$$
$$\displaystyle=$$
$$\displaystyle(1-\mu)[p_{0}p_{z}(p_{0}^{2}+p_{z}^{2})+p_{x}p_{y}(p_{x}^{2}+p_{y%
}^{2})]$$
$$\displaystyle\lambda_{15,...,26}$$
$$\displaystyle=$$
$$\displaystyle(1-\mu)[p_{0}p_{z}(p_{x}^{2}+p_{y}^{2})+p_{x}p_{y}(p_{0}^{2}+p_{z%
}^{2})]$$
$$\displaystyle\lambda_{27,...,30}$$
$$\displaystyle=$$
$$\displaystyle(1-\mu)[p_{0}p_{x}(p_{0}^{2}+p_{x}^{2})+p_{y}p_{z}(p_{y}^{2}+p_{z%
}^{2})]$$
$$\displaystyle\lambda_{31,...,38}$$
$$\displaystyle=$$
$$\displaystyle(1-\mu)[p_{0}p_{x}(p_{y}^{2}+p_{z}^{2})+p_{y}p_{z}(p_{0}^{2}+p_{x%
}^{2})]$$
$$\displaystyle\lambda_{39,...,42}$$
$$\displaystyle=$$
$$\displaystyle(1-\mu)[p_{0}p_{y}(p_{x}^{2}+p_{z}^{2})+p_{x}p_{z}(p_{0}^{2}+p_{y%
}^{2})]$$
$$\displaystyle\lambda_{43,...,46}$$
$$\displaystyle=$$
$$\displaystyle(1-\mu)[p_{0}p_{y}(p_{0}^{2}+p_{y}^{2})+p_{x}p_{z}(p_{x}^{2}+p_{z%
}^{2})]$$
$$\displaystyle\lambda_{47,...,52}$$
$$\displaystyle=$$
$$\displaystyle(1-\mu)[4p_{0}p_{x}p_{y}p_{z}]$$
$$\displaystyle\lambda_{53,...,60}$$
$$\displaystyle=$$
$$\displaystyle(1-\mu)[(p_{0}p_{z}+p_{x}p_{y})(p_{0}p_{x}+p_{y}p_{z})]$$
$$\displaystyle\lambda_{61,...,64}$$
$$\displaystyle=$$
$$\displaystyle(1-\mu)[(p_{0}p_{z}+p_{x}p_{y})(p_{0}p_{y}+p_{x}p_{z})]$$
Here we have four sets of equal eigenvalues (corresponding to each
four sets of states, in which every set has $64$ elements). These
states don’t mix with each other through the interaction with
environment.
In the following, we would like to consider some examples of Pauli
channels, both symmetric and asymmetric, and to work out their
entanglement-assisted classical capacity. For depolarizing
channels which are symmetric types of Pauli channel, for which
$p_{0}=1-p$ and $p_{i}=p/3,i=x,y,z$. In Fig.(1), we plot
the mutual information of quantum channel
$I^{AB}(\varepsilon(\rho))$ and the mutual information
corresponding to entanglement-assisted dense coding system $I^{AB,e-a}(\varepsilon(\rho))$ of the depolarization channel versus its
memory coefficient $\mu$. Fig.(1) shows that the use of
semi-quantum approach enhances mutual information of the quantum
channel, if one prior entanglement is shared by Alice and Bob.
That has better capacity over both products, entangled states and
entanglement-assisted coding for all values of $\mu$. Similar to
earlier works Mac , there exists another memory threshold
for two semi-quantum approaches. Explicit form of the memory
threshold is a very complicated relation, but numerical
calculation is shown in the Fig.(2) for $0<p_{i}<1/3$.
In Fig.(2) we see that as expected in the high error
channels, $p_{0}=p_{i}=p/3,i=x,y,z$, the memory threshold is equal
to zero, i.e. $\mu_{t}=0$. In other words, in the channels with
high errors, any output density matrix can be transformed to the
following form:
$$\displaystyle\varepsilon(\rho)=(1-\mu)\frac{1}{a^{b}}I^{\otimes b}+\mu\sigma$$
(20)
With $a=b=4$ for states (17) and $a=4,b=2$ for states
(9). Hence, $Tr\sigma=1,Tr\varepsilon(\rho)=1$.
Optimal mutual information is obtained by minimizing the output
entropy, and, for this, we must have a pure state at the output
channel. The optimization of the the mutual information can be
achieved by going to an appropriate bases that diagonaliyez
$\sigma$. If we assume that $\sigma$ has $k$ none-zero diagonal
elements, then, the entropy is given by:
$$\displaystyle S(\varepsilon(\rho))$$
$$\displaystyle=$$
$$\displaystyle k\{(1-\mu)a^{-b}+\mu\frac{1}{k}\}\log_{2}\{(1-\mu)a^{-b}+\mu%
\frac{1}{k}\}$$
$$\displaystyle+(a^{b}-k)\{(1-\mu)a^{-b}\}\log_{2}\{1-\mu)a^{-b}\}$$
Minimum value of the above relation can be obtained for $k=1$. In
the other words, $\sigma$ must be a pure state, and this happens
for the input states of (17).
On other hand, the maximum of memory threshold is give by $\mu_{t}^{Max}=0.185$. Thus, for channels with $\mu\geq\mu_{t}^{Max}$,
the use of (Se-Qu2) approach is more efficient.
In the following, we consider some examples of asymmetric Pauli
channels nil . The noise introduced by them is of three
types: namely, bit flip, phase flip and bit-phase flip.
Probability distribution for flip channels is given by $i,j=0,f$,
with probabilities $p_{0}=1-p$ and $p_{f}=p$. Here $f=x,y$ and
$z$ for bit flip, phase flip and bit-phase flip channels,
respectively. In Fig.(3), we plot the mutual information
of $I^{AB}(\varepsilon(\rho)$ and the mutual information
corresponding to entanglement-assisted dense coding system $I^{AB,e-a}(\varepsilon(\rho)$ of the bit flip channel versus its memory
coefficient $\mu$. Fig.(3) shows that the use of four
particles, semi-quantum dense coding, approach enhances the
capacity of quantum channel and the usual entanglement-assisted
coding and two particle semi-quantum dense coding have the same
plot for all values of $\mu$. This shows that in the asymmetric
Pauli channels, optimality in the unshared entangled states
strictly depends on the noise of the channel. For example, in the
bit flip channels, it is more appropriate to use $x$ basis.
At another stage, we consider the two-Pauli channels nil .
The probability distribution for the two-Pauli channels is given
by $i,j=0,x,y$, with probabilities $p_{0}=(1-p)$ and
$p_{x}=p_{y}=p/2$. In Fig.(4), we plot the mutual
information of $I^{AB}(\varepsilon(\rho)$ and the mutual
information corresponding to entanglement-assisted dense coding
system $I^{AB,e-a}(\varepsilon(\rho)$ of the two-Pauli channels,
versus its memory coefficient $\mu$. This shows that the use of
four particles semi-quantum dense coding approach enhances the
capacity of quantum channel for all values of $\mu$. Furthermore,
at the usual stage (unshared state) Mac , use of product
states in the $x$ or $y$
basis would be more efficient than in the $z$ basis and there exists a memory threshold where a higher amount of
classical information is transmitted with entangled states.
Explicit form of memory threshold is very complicated, the numerical value of memory threshold
for the
above error model is $\mu_{t}=0.409$.
Finally, we consider another type of asymmetric Pauli channels,
the so-called phase damping channel nil . The probability
distribution for phase damping channels are given by $i,j=0,z$
with probabilities $p_{0}=(1-p/2)$ and $p_{z}=p/2$.
Fig.(5) shows that for phase damping channels, the use
of two particles semi-quantum dense coding is more appropriate,
compared with the four particle approach to information
transmission, for all of values of $\mu$. This shows that in the
usual stage (previously unshared state) Mac , the use of
product states in the $z$
basis would be more efficient than in the $x$ or $y$ basis.
On the other hand, we show that $I_{n}({\cal E})$ is superadditive
in the presence of entanglement-assisted, i.e. we have
$I^{E}_{n+m}>I^{E}_{n}+I^{E}_{m}$ and therefore $C^{E}_{n}>C^{E}_{1}$. At this stage, the classical capacity $C^{E}$ of the
channel is defined by:
$$\displaystyle C^{E}=\lim_{n\to\infty}C^{E}_{n}\;.$$
It has been shown (similar to what we plotted in the figures) that
the amount of reliable information which can be transmitted per
use of the channel, is given by Sch :
$$\displaystyle C^{E}_{n}=\frac{1}{n}{\mbox{sup}}_{\cal E}I^{E}_{n}(\cal E)\ \ ,$$
One of the main applications of the above extension of memorial
channels is the extension of the standard quantum cryptography
BB84 Benn to protocols where the key is carried by quantum
states in a space of higher dimension, using two (or $d+1$)
mutually unbiased bases, which for high memorial channels have
very low error rates. This procedure ensures that any attempt by
any eavesdropper Eve to gain information on sender’s state induces
errors in the transmission, which can be detected by the
legitimate parties Cerf . On the other hand, if we are
interested in other quantum key distribution protocols (such as
the EPR protocol Eke ), we must encode a qubit in a
decoherence-free subspace of the collective noise for key
distribution Boi1 . If we are interested in the total
dimension of Hilbert space, we must revise the EPR protocol for
this new approach. Another application of the above extension can
be quantum coding and quantum superdence coding at higher
dimensions. The errors in the memorial channels can be considered
as a subset of collective noise which are considered in the
decoherence-free subspace approach. Some experimental results
Ball show that in some special cases the use of these
states are appropriate, because in the memorial channels we make
use of all maximally entangled states.
In conclusion, we have calculated a general expression for the
classical capacity of a quantum channel for product states,
maximally entangled states and entanglement-assisted coding in the
presence of memory. Hence, we have suggested another approach for
the transmission of information by using semi-quantum dense
coding. In this approach, we use an entangled pair that was
previously shared between the parties, and they compromised for a
transmission of quantum states through quantum channels in a
specific manner. Our results show that if noise in the consecutive
uses of the channels is assumed to be Markov-correlated quantum
noise, then, the use of semi-quantum approach to quantum dense
coding enhances classical capacity of quantum channels in various
types of Pauli channels.
We would like to thank M. Golshani for useful comments and
critical reading of the manuscript and S. Fallahi for his
helps.(This work was supported under the project: Entezar).
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A Population of Massive Globular Clusters in NGC 5128
Paul Martini
Harvard-Smithsonian Center for Astrophysics;
60 Garden Street, MS20; Cambridge, MA 02138; pmartini@cfa.harvard.edu
Luis C. Ho
Observatories of the Carnegie Institution of Washington;
813 Santa Barbara Street; Pasadena, CA 91101; lho@ociw.edu
Abstract
We present velocity dispersion measurements of 14 globular clusters in
NGC 5128 (Centarus A) obtained with the MIKE echelle spectrograph
on the 6.5m Magellan Clay telescope. These clusters are among the
most luminous globular clusters in NGC 5128 and have velocity dispersions
comparable to the most massive clusters known in the Local Group, ranging
from 10 – 30 km s${}^{-1}$.
We describe in detail our cross-correlation measurements, as well as
simulations to quantify the uncertainties.
These 14 globular clusters are the brightest NGC 5128 globular clusters with
surface photometry and structural parameters measured from the Hubble
Space Telescope.
We have used these measurements to derive masses and mass-to-light ratios for
all of these clusters and establish that the fundamental plane relations for
globular clusters extend to an order of magnitude higher mass than in the
Local Group.
The mean mass-to-light ratio for the NGC 5128 clusters is $\sim 3\pm 1$, higher
than measurements for all but the most massive Local Group clusters.
These massive clusters begin to bridge the mass gap between the most massive
star clusters and the lowest-mass galaxies.
We find that the properties of NGC 5128 globular clusters overlap quite well
with the central properties of nucleated dwarf galaxies and
ultracompact dwarf galaxies.
As six of these clusters also show evidence for extratidal light, we
hypothesize that at least some of these massive clusters are the nuclei of
tidally stripped dwarfs.
Subject headings:galaxies: individual (NGC 5128) — galaxies: star clusters
— globular clusters: general
††slugcomment: ApJ accepted [30 March 2004]
1. Introduction
Globular clusters provide valuable snapshots of the formation history of
galaxies and their large sizes and luminosities make them the most readily
observable sub-galactic constituents.
In addition, globular clusters exhibit surprisingly uniform properties
that suggests a common formation mechanism.
They are well-fit by single-mass, isotropic King models (King, 1966),
which describe clusters in terms of scale radii, central surface
brightness, and core velocity dispersion.
Detailed studies of globular clusters in our Galaxy have shown that in
fact they only inhabit a narrow range of the parameter space available
to King models (Djorgovski, 1995; McLaughlin, 2000) and other globular
cluster systems in the Local Group also appear to approximately follow the
same relations (Djorgovski et al., 1997; Dubath & Grillmair, 1997; Dubath et al., 1997; Barmby et al., 2002; Larsen et al., 2002).
These parameter correlations trace a globular cluster fundamental plane
that is analogous to, but distinct from, the fundamental plane for
elliptical galaxies
(Dressler et al., 1987; Djorgovski & Davis, 1987; Bender et al., 1992; Burstein et al., 1997).
Globular cluster studies that include internal kinematics have been
confined to the Local Group due to the faint apparent magnitudes of
more distant extragalactic globular clusters. These studies have therefore
only included the globular cluster systems of spiral and dwarf galaxies
and not the globular cluster systems of large ellipticals.
Yet the globular cluster systems of ellipticals are a particularly
interesting regime as they probe both a new morphological type and one likely
to have exhibited a different and more complex formation history.
The globular cluster systems of elliptical galaxies, as well as many spirals
such as our own, have bimodal color distributions suggestive of multiple
episodes of formation (e.g. Kundu & Whitmore, 2001; Larsen et al., 2001).
Models for the formation of these globular cluster systems posit that one of
these populations may be the intrinsic population of the galaxy and
subsequent mergers resulted in the second population
either as the result of a new episode of globular
cluster formation (Schweizer, 1987; Ashman & Zepf, 1992; Forbes, Brodie, & Grillmair, 1997) or accretion of
globular cluster systems from other galaxies, including accretion of
globular clusters by tidal stripping from other members of a cluster of
galaxies (Côté, Marzke, &
West, 1998).
NGC 5128 (Centarus A), as the nearest large elliptical
galaxy, is arguably the best source for extending detailed globular cluster
studies outside of the Local Group.
While NGC 5128 is the central galaxy of a large group, rather
than a giant elliptical in a cluster, it likely had a similar formation
history to its larger cousins. The most relevant similarity is the strong
evidence for a recent, gas-rich merger
(for a recent review of NGC 5128 see Israel, 1998).
Estimates of the size of the NGC 5128 globular cluster population suggest
that it has a total of $\sim 2000$ clusters, approximately a factor of
3 more than the entire Local Group (Harris et al., 1984).
Simple scaling arguments suggest that NGC 5128 should possess a number of
extremely massive globular clusters
and therefore is not only a good target for study of the globular cluster
system of an elliptical galaxy, but also for study of how well the
fundamental plane relations established locally apply to more massive
globular clusters.
A recent photometric and spectroscopic study of NGC 5128 by
Peng, Ford, & Freeman (2004a, b) concluded that the metal-rich globular clusters
may have a mean age of $5^{+3}_{-2}$ Gyr, while an analysis of their
photometric data yields a metallicity range of
$-2.0$ through $+0.3$ (Yi et al., 2004)
The most massive globular clusters can also be used to explore connections
between the formation processes for star clusters and galaxies.
While fundamental plane studies (e.g. Burstein et al., 1997) clearly illustrate
a significant mass gap between the most massive Galactic globular clusters and
the least massive dwarf galaxies, there have been encroachments into this gap
from both sides. For many years, studies have speculated that at least some
globular clusters may be the remains of tidally-stripped dwarf galaxy nuclei
(Zinnecker et al., 1988; Freeman, 1993; Bassino et al., 1994).
Two of the most massive globular clusters in the Local Group, $\omega$Cen in
our Galaxy and G1 in M31, have been interpreted as the nuclei of tidally
stripped dwarfs (Meylan et al., 2001; Gnedin et al., 2002; Bekki & Freeman, 2003).
From the galaxy side, recent studies of nucleated dwarf galaxies in the Virgo
cluster (Geha, Guhathakurta, & van der
Marel, 2002) and ultracompact dwarf galaxies in the Fornax cluster
(Drinkwater et al., 2003) show some similarities between these least-massive
galaxies and the most massive globular clusters. This mass gap may thus
reflect the scarcity of the most massive globular clusters and the difficulty
of kinematic measurements for the least massive, lowest surface-brightness
dwarf galaxies, rather than a physical separation.
In this paper we present velocity dispersion measurements for 14
globular clusters in NGC 5128. These globular clusters were selected from
the Hubble Space Telescope (HST) study of Harris et al. (2002) and
therefore have well-measured
structural parameters. These data are combined to estimate masses for
these clusters, masses that are among the largest known for any star clusters
and comparable to the nuclei of the lowest-mass galaxies. In the
next sections we describe the observations, data processing, and velocity
dispersion measurements. Analysis of these measurements is described
in §5 and the potential link between star clusters and galaxies is
explored in §6. Our results are summarized in the final section.
Throughout this paper we adopt the distance of $3.84\pm 0.35$ Mpc for
NGC 5128 determined by Rejkuba (2004) from the brightness of the tip of the
red giant branch and the Mira period-luminosity relation.
2. Observations
Spectra of 14 globular clusters in NGC 5128 were obtained in the
course of seven nights in March 2003 with the MIKE echelle spectrograph
(Bernstein et al., 2003) and the 6.5m Magellan Clay telescope at Las
Campanas Observatory (see Table 1).
MIKE is a double echelle spectrograph capable of resolution $R=19,000$ on
the red side when used with the $1^{\prime\prime}$ slit selected for these observations.
This corresponds to a velocity resolution FWHM of 15.8 km s${}^{-1}$ or
$\sigma=6.2$ km s${}^{-1}$.
MIKE was located at the East Nasmyth port but not directly attached to the
instrument rotator. The advantage of this configuration is that the
instrument is gravity invariant and therefore potential calibration
difficulties due to instrument flexure are completely avoided. The
disadvantage of this configuration is that targets could not be observed at
the parallactic angle and that they rotate with respect to the spectrograph
slit. In any event, this configuration was the only one available at the time.
To minimize lost light due to atmospheric dispersion, the position angle of
the spectrograph was set to correspond to the parallactic angle for an object
at airmass 1.3; below airmass $\sim 1.4$ the atmospheric dispersion correction
is effectively negligible and most of our observations took place in this
airmass range. Any lost light due to atmospheric dispersion is unlikely to
significantly impact our velocity dispersion measurements as the velocity
dispersion gradient of the clusters is not large. The
rotation of the slit also has a negligible impact as all of these globular
clusters are fairly round (Harris et al., 2002).
While MIKE was used to obtain both red and blue data, sufficient
signal-to-noise ratio (SNR) observations of these (intrinsically red) clusters
were only obtained on the red side.
To enhance the SNR, the observations were binned $2\times 2$, leading to
$0.286^{\prime\prime}$ pixel${}^{-1}$ and a velocity scale of 4.2 km s${}^{-1}$pixel${}^{-1}$.
Table 1 lists all 14 clusters according to their
identification in Harris et al. (2002), along with their apparent $V$ magnitude
from Peng et al. (2004a), the total integration time, and the average
SNR per pixel over the wavelength range 5000 – 6800 Å when binned to
0.2 Å pixel${}^{-1}$. The total integration time
listed is the sum of an integer number of 1800 s exposures. This exposure time
was chosen to facilitate cosmic ray removal, yet also avoid a significant
contribution from detector noise.
Spectra of 16 velocity dispersion templates were also obtained, ranging
in spectral type from G through M stars and luminosity classes from IV
through I.
These templates were observed in a
similar manner to the globular clusters. Typically three integrations of a
few seconds each were obtained for each template. Arc lamps for wavelength
calibration were obtained before or after the observations of each target.
Several sets of flatfield exposures using a quartz lamp were also obtained
over the course of the seven-night observing run.
3. Data Processing
The data were processed with a combination of tasks from the
IRAF111IRAF is distributed by the National Optical Astronomy
Observatories, which are operated by the Association of Universities for
Research in Astronomy, Inc., under cooperative agreement with the National
Science Foundation. echelle package and the sky-subtraction software
described by Kelson (2003). All images were overscan-subtracted and
a simple cosmic-ray cleaning algorithm was applied to the individual cluster
exposures. The images were also rotated such that the dispersion direction was
approximately parallel to the rows of the detector. Inspection of the several
sets of flatfield exposures obtained over the course of the run showed no
measurable variation and all of the flatfield exposures were therefore summed
to produce one master flatfield. This flat was then normalized with low-order
fits to each order and divided into all of the object exposures prior to
sky subtraction.
Sky subtraction of MIKE data is complicated by the nonlinear transformation
between the orthogonal (dispersion, spatial) and “natural” detector
(row, column) coordinate systems and the slight undersampling of the data.
The marginally undersampled sky lines are tilted with respect to the
detector coordinate system and this tilt angle varies as a function of
spatial position. As discussed in detail by Kelson (2003), the classical
approach of rectifying and rebinning the image prior to sky subtraction can
result in significant residuals in the case where the sky lines are
critically- or undersampled. Superior sky subtraction can be obtained if a sky
model is calculated and subtracted prior to any rectification and rebinning
of the data.
The key to this technique is to calculate the transformation between
the array coordinates $(x,y)$ and the orthogonal dispersion and spatial
coordinate system $(x_{r},y_{t})$. By collapsing the spectrum along the spatial
coordinate $y_{t}$, the profile of each sky line can then be measured as a
function of the coordinate $x_{r}$ in which the sky line is well-sampled (due
to the tilt of the observed spectrum). A two-dimensional sky model
can then be constructed with the same pixelization as the original data,
thus avoiding the residuals that result from rebinning marginally or
undersampled data. To employ Kelson’s method, we first calculated
the transformation $y_{t}=Y(x,y)$ with his getrect command, using a
flatfield frame to trace the boundaries of each order. These boundaries
were defined with findslits (treating them as individual slits
in a multislit mask). We then calculated $x_{r}=X(x,y_{t})$ with
getrect applied to each order. Accurate calculation of this
transformation requires several lines per order. As there are few
airglow lines in the higher orders (shorter wavelengths), we created a
composite image of spectroscopic lines by combining a very high SNR sky frame
(calculated by summing a large number of our exposures of the fainter
clusters) with ThAr and NeAr lamp spectra. As many ThAr lines saturated in the
13 reddest orders, these orders were masked out before construction of the
composite line image for getrect. A sky model image for each globular
cluster exposure was then calculated with skyrect and subtracted from
the flatfielded data. Observations of the template stars were sufficiently
short that no sky subtraction was deemed necessary.
The dispersion solution was calculated with the ecidentify task in the
IRAF echelle package and ThAr lamp spectra. Although most of the reddest
orders were saturated by bright lines as described above, a combination of
multiple lines in the bluer orders and several unsaturated lines in the
redder orders led to a good solution (rms $\sim$ 0.005 – 0.007 Å) with a
fourth-order Legendre polynomial in $x$ and $y$.
All of the object spectra were then extracted with apall. Because
most of the individual cluster spectra were too faint for a good trace,
particularly in the bluest orders, we used a reference trace from a brighter
object. This trace was recentered based on the centroid of the cluster trace
in the reddest, highest SNR orders. Each individual exposure was extracted in
this manner as several-pixel shifts were observed between a small number of
the multiple exposures on individual clusters. The width of the spectral
extraction window was 9 pixels or 2.6${}^{\prime\prime}$. The arc lamp was extracted for
each object with the same trace and the dispersion solution was calculated
with the ecreidentify task. The spectra were then placed on a linear
wavelength scale with the dispcor task for measurement of their
line-of-sight velocity dispersion $\sigma$. Template stars and the flux
standard HR 1544 were extracted in a similar manner. Orders 68 – 39 of C23,
the cluster with the highest SNR, are shown in
Figure 1.
The calcium triplet region (order 40) is shown for all 14 clusters
in Figure 2. No correction for telluric absorption was applied to
these data.
4. Velocity Dispersion Measurement
The two most commonly employed techniques for velocity dispersion measurement
are direct-fitting in pixel space (e.g. Rix & White, 1992; Kelson et al., 2000; Barth et al., 2002) and
cross-correlation techniques in the Fourier domain
(e.g. Illingworth, 1976).
After experiments with both techniques we chose the cross-correlation method
developed in detail by Tonry & Davis (1979) as implemented by the rvsao
package (Kurtz & Mink, 1998) in IRAF. The main advantage of the cross-correlation
function (hereafter CCF) is that it is less sensitive to the relative line
strengths in the targets and templates. This is likely to be the case for the
present study due to metallicity differences between the globular clusters
and the Galactic template stars observed. Trial implementations of
direct-fitting in pixel space confirmed that most of the stellar templates
are not good matches in detail to these globular clusters. Direct
pixel-fitting did show that the observed stellar templates with spectral types
between G8 and K3 were the best matches to the globular cluster spectral
energy distributions. The stellar templates used for this study are:
HD 80499 (G8III), HD 95272 (K0III), HD 88284 (K0III), HD 43827 (K1III),
HD 92588 (K1IV), HD 44951 (K3III), and HD 46184 (K3III).
The first step in implementation of the CCF technique is the calculation of
the relation between the CCF width and the observed line-of-sight velocity
dispersion $\sigma$. To measure this relationship,
we convolved a range of the observed stellar templates by Gaussian functions
with $\sigma=$ [5,7,10,14,20,28,40,56] km s${}^{-1}$and used a simple spline fit to
calculate the relation between CCF width and input $\sigma$.
We calculated this relationship for each of the spectroscopic orders and
determined that the uncertainty in $\sigma$ measurements in orders with
significant telluric absorption (50, 48, 47, 45, 42, and 41) was considerably
higher than those without. Given the large number of orders available, these
orders were excluded from subsequent analysis.
Based on measurements from the remaining orders we determined
that all of the globular clusters have $\sigma$ in the approximate range
of 10 – 30 km s${}^{-1}$. The velocity dispersions of these clusters are thus
well-resolved by this instrument configuration.
While the CCF technique is known to be relatively insensitive to template
mismatch, we tested the importance of this potential source of systematic
error by convolving each spectral type by the range of velocity dispersions
given above
and using other templates of similar but not identical temperature to measure
$\sigma$ (e.g. a K0III template to measure an artificially broadened K3III
template). This analysis showed that template mismatch produced at most an rms
uncertainty of 1 km s${}^{-1}$ when $\sigma$ was calculated for a single order.
Given the large number of orders available and our estimate of the
range in $\sigma$ of these clusters, we then experimented with combining
multiple orders together as well as binning the data to improve SNR.
A range of tests showed we obtain excellent and very stable results
over the wavelength range 5000 – 6800 Å and the spectra rebinned to
a scale of 0.2 Å pixel${}^{-1}$. The rms variation due to template mismatch
over this range was less than 0.5 km s${}^{-1}$. CCFs for all 14 clusters are
shown in Figure 3 and the measured $\sigma$ and estimated
uncertainties are listed in Table 1.
The 5000 – 6800 Å spectral range does not include the higher SNR CaT
region. While the CaT lines are quite strong for all of the clusters
(see Figure 2), these lines are in fact so strong that they
exhibit significant damping wings. These damping wings broaden the line
profiles of the template stars significantly and they are not a good
representation of the instrumental line profile. The CaT lines in the template
stars have intrinsic widths comparable to or larger than the measured
velocity dispersions of the globular clusters over the 5000 – 6800 Å
region. We therefore did not use these lines to measure the cluster
velocity dispersions. The neighboring, red orders have comparable SNR to the
CaT region, although these orders are contaminated by significant telluric
absorption. Although the telluric absorption is to some extent correctable,
these orders were also discarded for the velocity dispersion measurement
because the SNR in the 5000 — 6800 Å region is sufficiently high that SNR
is at most a modest contributor to the total error budget (see below).
While there are still reasonably strong lines in the 5000 – 6800 Å region,
notably Mg$b$ and H$\alpha$, these lines only make a minor contribution
to the velocity dispersion measurement.
Given the range of SNR and $\sigma$ of these data, we performed simulations
with our template stars to determine the quality of our measurements
as a function of both SNR and $\sigma$. These tests were performed by first
convolving the best-fitting template (HD 88284) with the range of velocity
dispersions
described above. We then added noise to these convolved spectra to
produce a set of output spectra with average SNR of [2,4,8,16,32,64,128] for
each velocity dispersion. As the SNR per pixel is quite variable due to both
the blaze
function and the red color of these clusters, we used an input SNR spectrum
from one of the cluster observations
(C23, shown in Figure 1)
to reproduce the pixel-to-pixel variation in the SNR, while
forcing the mean SNR in the spectra to equal the specified values.
The widths of these input spectra were then calculated in a similar fashion
to the globular cluster measurements described above. The results of
this analysis for $\sigma=10-28$ km s${}^{-1}$(the range most appropriate to
this study) are shown in Figure 4. This figure displays the
difference between the input $\sigma$ and the measured $\sigma$ at a given
SNR, normalized by the input $\sigma$.
As expected, the deviation between the input $\sigma$ and the measured $\sigma$
increases toward lower SNR. In particular, these simulations show that
at lower SNR the measurements will tend to overestimate $\sigma$. This
may be because the addition of noise systematically erases narrower features
more readily than broader features. The simulations also show that at a given
SNR the velocity dispersion measurement will be more accurate
if the intrinsic velocity dispersion is lower. This is likely a reflection of
the net SNR over the number of pixels that comprise a given feature. The fewer
pixels that span, e.g. the absorption lines employed here, the higher the net
SNR of the feature relative to a spectrum of the same mean SNR per pixel with
a larger $\sigma$. This systematic effect on the $\sigma$ measurement impacts
our observations at most at the 5% level. Three of the clusters with
velocity dispersions of on order 10 km s${}^{-1}$(C25, C41, C44) have a mean SNR less
than 10 over then 5000 – 6800 Å range in our rebinned spectra, while three
(C2, C31, C32) in the 14 km s${}^{-1}$range have a mean
SNR $\sim 10$ (see Table 1).
The measured velocity dispersions for these low-SNR clusters have been
adjusted to account for the expected overestimate.
The remaining clusters do not suffer from systematic errors due to low SNR.
The resolution of our rebinned spectra could also impact the measured $\sigma$
as the velocity dispersion resolution now varies from 12 km s${}^{-1}$at
5000 Å to 8.8 km s${}^{-1}$at 6800 Å. However, measurements of $\sigma$ with
a dispersion of 0.1 Å pixel${}^{-1}$ yield consistent results with
the values in Table 1, although with somewhat larger scatter
due to the lower SNR. The relationship between measured $\sigma$ and
SNR at each $\sigma$ shown in Figure 4 also demonstrates that
the measured $\sigma$ converges to the input value as the SNR increases.
The measured values quoted in Table 1 also agree well with
measurements obtained from single, unbinned orders and direct-pixel fitting,
although the latter approaches have larger scatter.
We note that preliminary velocity dispersions measurements from high-resolution
spectroscopy of four NGC 5128 globular clusters were obtained by
Dubath and reported in Harris et al. (2002).
These measurements are marginally consistent with our own.
The $1^{\prime\prime}\times 2.6^{\prime\prime}$ spectral extraction window we employed corresponds to
a physical area of $18.6\times 48.4$ pc, which is larger than the half-light
radius for all of these clusters (e.g. see Table 2).
In King models for globular clusters, the
velocity dispersion is a function of radius. Our measurements, because they
include most of the cluster, should correspond well to the global,
one-dimensional velocity dispersion $\sigma_{\infty}$. In contrast, the
core one-dimensional velocity dispersion $\sigma_{0}$ is approximately the
quantity measured for Galactic globular clusters and the most commonly quoted
quantity in the literature. The core value $\sigma_{0}$ is larger than
$\sigma_{\infty}$ in the King
models and the degree of difference depends on cluster concentration.
For the clusters studied here, this correction is on order 5 – 10% for the
range of cluster concentrations in this sample (see e.g. Djorgovski et al., 1997).
The errorbars quoted in Table 1 include the uncertainties in this
correction, which are caused by uncertainties in the measured structural
parameters reported by Harris et al. (2002) and the exact placement of the
slit, as well as the uncertainties due to template mismatch and SNR.
5. Analysis
5.1. Observed Cluster Properties
These 14 globular clusters are the brightest clusters with King model
structural parameters measured with HST by Harris et al. (2002).
These models are
described by the parameters $W_{0}$ (central potential), $r_{c}$ (core radius)
and $c$ (cluster concentration, where $c={\rm log}\,r_{t}/r_{c}$ and $r_{t}$ is
the tidal radius). Harris et al. (2002) also fit for ellipticity.
Both $r_{c}$ and $r_{h}$ (the model half-mass radius)
from their fits are provided in Table 2, converted to
parsecs, and their measurement of $c$ is also given.
The projected half-light radius is
$R_{h}\approx 0.73\,r_{h}$. As these values span the approximate
range of $1.4<c<2.1$, none are core-collapse clusters.
All 14 of these clusters were also included in the recent study by
Peng et al. (2004a), who have obtained $UBVRI$ photometry and radial velocities
for over 200 NGC 5128 globular clusters. As most of the Harris et al. (2002)
photometry is from unfiltered STIS images, we have listed instead
the Peng et al. (2004a) $V_{0}$, $(B-V)_{0}$, and $(V-I)_{0}$ values in the table, where
these
quantities have been dereddened based on their quoted $E(B-V)$ and assuming
$R_{V}=3.1$. We also include the central surface brightness $\mu_{V}^{0}$ from
Harris et al. (2002), but recalibrated by the difference between their quoted
$V$ magnitudes and those in Peng et al. (2004a), and have calculated the
mean, reddening-corrected surface brightness within the projected half-light
radius: $\langle\mu_{V}\rangle_{h}=V_{0}+0.75-2.5\,\log\,\pi R_{h}^{2}$. All of these
dereddened values only account for Galactic dust and do not include any dust
intrinsic to NGC 5128. While most of these clusters are not near NGC 5128’s
prominent dust lane, dust may affect the colors of some of these clusters.
The color distribution of NGC 5128 globular clusters is bimodal
(Held et al., 1997; Peng et al., 2004a), similar to the globular cluster systems of many
galaxies (e.g. Larsen et al., 2001; Kundu & Whitmore, 2001).
A bimodal color distribution is most commonly ascribed to a bimodal
distribution in metallicity as color is a reasonable proxy for metallicity
in old single stellar populations.
In the case of NGC 5128, this galaxy appears to
have undergone a major merger in the recent past. Based on the strength of
H$\beta$, Peng et al. (2004a) conclude that the more metal-rich globular clusters
formed $5^{+3}_{-2}$ Gyr ago, while Kaviraj et al. (2004) derive an age estimate
of 1 – 2 Gyr for the metal-rich population in their study of the metallicity
distribution function (derived from the Peng et al. (2004a) $U$ and $B$
photometry).
Figure 5 shows the $(B-V)_{0}$ and $(V-I)_{0}$ color distributions
for all of the globular clusters studied by Peng et al. (2004a). The clusters
in the present study are represented by the hatched histogram and span the full
range in color of the majority of the NGC 5128 globular cluster system.
Figure 6 demonstrates that these clusters do not show
any strong correlations between color and velocity dispersion, central
concentration, mass, or mass-to-light ratio (see next section).
5.2. Cluster Masses and Mass-to-Light Ratios
One of the most valuable and readily calculated properties of a globular
cluster is its total mass. There are multiple ways to estimate the mass
of a globular cluster, although the most straightforward approach is to use
the virial theorem:
$$M_{vir}=2.5\,\frac{\langle v^{2}\rangle\,r_{h}}{G},$$
(1)
where $\langle v^{2}\rangle\approx 3\sigma^{2}$ is the mean-square speed
of the cluster stars and $r_{h}$ is the half-mass radius (Binney & Tremaine, 1987).
The virial masses for these clusters (in units of $10^{6}$ M${}_{\odot}$) are listed in
Table 3.
An alternative approach is to use the King parameters to calculate the cluster
mass. From Illingworth (1976), the mass is
$$M_{King}=167\,r_{c}\,\mu\,\sigma_{0}^{2},$$
(2)
where $\mu$ is the dimensionless King model mass (King, 1966) and
$\sigma_{0}$ is the core velocity dispersion (see also Richstone & Tremaine, 1986).
As discussed previously, we have estimated $\sigma_{0}$ from the cluster
concentration parameter (Djorgovski et al., 1997).
The dimensionless mass $\mu$ is also a function of concentration.
The King model masses for these clusters are $\sim 50$% less than
the virial mass estimates. It is not unusual for these two estimates to
disagree (e.g. see Meylan et al., 2001, for $\omega$Cen and G1).
The uncertainties (not shown) in the King masses are larger than the virial
mass estimates due to uncertainties in the measurement of $c$ and $r_{c}$, in
addition to the uncertainties in $\sigma$. We therefore use the virial mass
in the subsequent discussion.
Once the mass has been estimated, the $V-$band mass-to-light ratio
in solar units $\Upsilon_{V,\odot}$ is readily calculated. The virial
mass is divided by the $V-$band luminosity $L_{V,\odot}$, which is computed
from the reddening-corrected magnitude $V_{0}$ (assuming $M_{V,\odot}=4.83$).
The mass-to-light ratios for these clusters are listed in column 5 of
Table 3.
These values fall in the range 1.4 – 5.7 and have an
average value of $\langle\Upsilon_{V,\odot}\rangle\sim 3$. This average is
larger than the mean core mass-to-light ratio of $1.45\pm 0.1$ for globular
clusters in the Galaxy (McLaughlin, 2000) and the comparable core and
global values of $1.53\pm 0.18$ from four globular clusters in
M33 (Larsen et al., 2002).
If we had adopted the King mass estimates to compute the
mass-to-light ratios, rather than the virial mass estimates, the mass-to-light
ratios would be $\sim 50$% less.
The difference between the NGC 5128 globular clusters and those in the Local
Group may be due to either intrinsic differences or due to measurement
uncertainties.
If the latter is the case, then the masses are
overestimated, the luminosities are underestimated, or both. An overestimate
in the mass could be explained by either an overestimate of $\sigma$ by
$\sim$ 50% or an overestimate of $r_{h}$ by a factor of 2, although both
of these possibilities vastly exceed the estimated uncertainties in these
quantities. The mass-to-light ratios between the different globular cluster
systems could also be brought into agreement if the luminosities are
underestimated by a factor of 2. This exceeds the estimated uncertainty in
the distance to NGC 5128 (Rejkuba, 2004), although intrinsic reddening could
also contribute and NGC 5128 is well-known for its prominent, large dust lanes.
The luminosities could also be underestimated if there is a
significant contribution to the surface brightness at large radii, where the
SNR is poor. While a conspiracy of all of these potential sources of
error could bring the mean mass-to-light ratios of these globular cluster
systems into agreement, we conclude that the differences are likely real.
Further support of these large mass-to-light ratios is provided by some of
the most massive
globular clusters in the Local Group, including the Galactic globular
cluster $\omega$Cen, which has $\Upsilon_{V,\odot}=2.4-3.5$
and G1 in M31 with $\Upsilon=3.6-7.5$ (Meylan et al., 1995, 2001).
The lower-right panel of Figure 6 shows the mass-to-light
ratios for the NGC 5128 globular clusters as a function of $(V-I)_{0}$. No obvious
trend with color is apparent, as might be expected if the generally redder,
more metal-rich clusters identified by Peng, Ford, & Freeman (2004b) formed more recently.
The massive, Local Group clusters $\omega$Cen and G1 are also shown for
comparison. For these clusters the average of the virial and King mass
estimates from Meylan et al. (2001) were used for the mass-to-light ratio,
while the integrated colors are either from the Harris (1996) compilation
($\omega$Cen) or from Heasley et al. (1988).
5.3. Fundamental Plane
NGC 5128 has a significantly larger population of the most massive and luminous
globular clusters than any galaxy in the Local Group. Our cluster
sample is dominated by these clusters due to observational constraints,
yet this population is arguably the most interesting as they provide an
opportunity to verify and extend relationships for local clusters to more
extreme examples of the population. The clusters in our sample are
comparable in mass to $\omega$Cen, the most massive Galactic globular cluster
at $M=5\times 10^{6}$ M${}_{\odot}$(Meylan et al., 1995) and M31’s G1, which with
$M=(7-17)\times 10^{6}$ M${}_{\odot}$(Meylan et al., 2001) is the most massive cluster
known in the Local Group.
The main relationship of interest is the globular cluster fundamental plane,
the approximately two-dimensional structure occupied by clusters in the
three-dimensional space defined by, e.g. central velocity dispersion,
surface brightness, and core radius. Djorgovski (1995) demonstrated that
the plane occupied by globular clusters is consistent with the expectations
for virialized cluster cores,
$$r_{c}\sim\sigma_{0}^{2}\,I^{-1}_{0}\,\Upsilon^{-1},$$
(3)
and a constant mass-to-light ratio. This appears to be the case for both
the core properties as well as properties derived at the half-light radius,
with the differences between these two regimes due to the degree of
central concentration.
In the top panels of Figure 7 we plot two projections of the
core fundamental plane from Djorgovski (1995) for the NGC 5128 globular
clusters, along with clusters in the Milky Way, M31, M33, and the
Magellanic Clouds. These projections show that the NGC 5128 clusters are
relatively large and have higher surface brightnesses than Galactic clusters,
although are more similar to globular clusters studied in other Local Group
galaxies. Half-light projections (bottom panels) show all of these
globular cluster systems follow similar trends to the Milky Way system,
although extragalactic globular clusters appear to have systematically
lower mean surface brightness within the half-light radius for a given
$\sigma$.
This may reflect a bias toward selection of bright objects that nevertheless
appear marginally resolved in ground-based images or a relative
overestimate of $R_{h}$ for more distant clusters.
Numerous literature sources were employed to obtain data for globular
clusters in the Local Group. Data for the Milky Way were obtained from
Pryor & Meylan (1993) and Harris (1996). Velocity dispersion data for M31 were
obtained from Dubath & Grillmair (1997) and Djorgovski et al. (1997), while structural
parameters were obtained from Barmby et al. (2002) or earlier work
(Battistini et al., 1982; Crampton et al., 1985; Bendinelli et al., 1993; Fusi Pecci et al., 1994; Rich et al., 1996; Grillmair et al., 1996).
Data for the Magellanic Clouds were obtained from Dubath et al. (1997) and
data for M33 were obtained from Larsen et al. (2002), while the photometry
for the latter was dereddened with the $E(V-I)$ values derived by
Sarajedini et al. (1998).
Data for the three Fornax dwarf galaxy’s globular clusters with velocity
dispersion
measurements and structural parameters were obtained from Dubath, Meylan, & Mayor (1992)
and Mackey & Gilmore (2003).
McLaughlin (2000) has recently recast discussion of the fundamental
plane for globular clusters in terms of mass-to-light ratio, luminosity,
binding energy, and Galactocentric radius. In the context of the four
independent parameters that characterize single-mass, isotropic King models,
he found that Galactic globular clusters could be described with a constant
mass-to-light ratio and a binding energy regulated only by luminosity and
Galactocentric radius: $E_{b}=7.2\times 10^{39}(L/L_{\odot})^{2.05}(r_{gc}/8\,{\rm kpc})^{-0.4}$ erg. Figure 8 shows that NGC 5128 clusters
follow approximately the same relation between binding energy and luminosity
as Milky Way and other globular clusters in the Local Group, although
the NGC 5128 and other clusters with higher estimated $M/L$ fall above the
relation traced by Milky Way clusters.
We did not explore a dependence of binding energy on projected galactocentric
radius due to distance uncertainties.
6. Connection to other Spheroidal Systems
Many efforts over the years have investigated the connection between globular
clusters and other spheroidal systems. One valuable approach is
the $\kappa-$space formalism developed by Bender et al. (1992). This space is
comprised of three orthogonal axes that are proportional to mass ($\kappa_{1}$),
mass-to-light ratio ($\kappa_{2}$), and a third perpendicular axis that scales
as surface brightness cubed times mass-to-light ratio ($\kappa_{3}$).
Burstein et al. (1997) compiled $\kappa$–space coordinates for a large number of
spheroidal systems, with masses from globular cluster scales to clusters of
galaxies. For globular clusters, the $\kappa-$space coordinates are derived as
$$\kappa_{1}=(\log\sigma_{e}^{2}+\log r_{e})/\sqrt{2}+0.11\\
$$
(4)
$$\kappa_{2}=(\log\sigma_{e}^{2}+2\,\log I_{e}-\log r_{e})/\sqrt{6}+0.06\\
$$
(5)
and
$$\kappa_{3}=(\log\sigma_{e}^{2}-\log I_{e}-\log r_{e})/\sqrt{3}+0.09,\\
$$
(6)
where $\sigma_{e}\approx\sigma_{\infty}$, $r_{e}\approx r_{h}$, and
$I_{e}$ is the mean surface brightness within $r_{e}$ in units of $B-$band
solar luminosities per square parsec.
In Figure 9 we plot the three $\kappa-$space projections of the
fundamental plane for NGC 5128 clusters (open circles), along with
Galactic globular clusters (crosses), those in M31 (stars), and
galaxies (diamonds and open triangles,
data from Burstein et al., 1997).
This figure indicates that the NGC 5128 clusters are quite similar, although
systematically more massive, than the Galactic clusters.
In the $\kappa_{1}$ coordinate, globular clusters are clearly separated
from galaxies. This does not
necessarily rule out the existence of objects in this gap, as a
simple extrapolation of the mass functions for both globular cluster systems
and dwarf galaxies suggest that such objects should be extremely rare and,
in the case of dwarf galaxies, quite difficult to detect. Simple scaling
arguments suggest that globular cluster systems in massive ellipticals such as
NGC 5128 are arguably the best place to identify objects in this region,
provided there is not some other physical mechanism that sets a maximum
globular cluster mass.
The globular clusters in the present study are a step toward eliminating the
mass gap.
There is also a separation in the $\kappa_{3}$ coordinate ($M/L$) between
globular clusters and galaxies, although not as stark as $\kappa_{1}$
(see Figure 9, left panel). The mean mass-to-light
ratio of Galactic globulars is 1.45 (McLaughlin, 2000), and
Larsen et al. (2002) found a comparable number for M33. While the higher
mass-to-light ratio for NGC 5128 clusters is somewhat surprising, this may be
a common feature of more massive clusters. For example, the mass-to-light
ratios of $\omega$Cen and G1 are in the same range as those in NGC 5128, as
are the mass-to-light ratios (or $\kappa_{3}$ values) of other massive
clusters in the Galaxy and M31.
The $\kappa_{3}$ values for dwarf and regular elliptical galaxies are higher
than for most globular clusters, reflecting their larger mass-to-light ratios,
although they overlap with the most massive globular clusters.
The fundamental plane relationship for spheroidal galaxies requires
$\Upsilon_{V}\propto L_{V}^{0.2}$ (van der Marel, 1991; Magorrian et al., 1998).
A relationship between mass (or luminosity) and mass-to-light ratio for
the most massive globular clusters is suggested by their apparently higher
$\kappa_{3}$ values.
The $\kappa_{2}$ ($I_{e}^{3}\times M/L$) coordinate is essentially an indicator of
surface brightness at the effective radius as its dependence on $M/L$ is
much weaker than $I_{e}$.
$\kappa_{2}$ exhibits a great deal of overlap in the properties of all
spheroidal systems. In fact, the galaxies overlap much more closely with
Galactic globular clusters
than those in NGC 5128. This is because $\kappa_{2}$ is dominated by the mean
surface brightness within the effective radius and the NGC 5128 clusters
are on average lower in mean surface brightness within this radius
(e.g. see Figure 7).
While the NGC 5128 globular clusters overlap more with other globular
clusters than galaxies, they are an even better match to the nuclei of the
dwarf ellipticals studied by Geha et al. (2002), who
measured both integrated and nuclear velocity dispersions and surface
brightnesses for a sample of Virgo dwarf ellipticals.
While the integrated
properties of nucleated dwarf galaxies (large, open squares) fall within
the region of $\kappa-$space inhabited by other dwarf ellipticals and
spheroids (diamonds), their nuclei (squares with crosses) overlap
very well with the NGC 5128 clusters.
The $\kappa_{1}-\kappa_{2}$ relation for giant elliptical galaxies and bulges
exhibits an anticorrelation between mass and surface brightness that sets them
nearly orthogonal to the dwarf galaxies, which are correlated in $\kappa_{1}-\kappa_{2}$. This correlation reflects the correlation between luminosity
and larger cores of higher surface brightness found by Kormendy (1985).
While this correlation suggests mass loss due to winds, Bender et al. (1992)
found that mass loss would produce too steep a relation between
$\kappa_{1}$ and $\kappa_{2}$, instead suggesting the observed dwarfs
are the remnants of a larger, unobserved population. The massive globular
clusters appear to exhibit a similar $\kappa_{1}-\kappa_{2}$ correlation,
although steeper than the dwarf galaxies.
7. Discussion
The strong dynamical and photometric similarities of these massive globular
clusters to the nuclei of dwarf galaxies has interesting implications for
models which posit stripped dwarf galaxy nuclei as the origin of
the most massive globular clusters. The most massive Local Group clusters,
such as $\omega$Cen in our own Galaxy and G1 in M31, have both been
discussed and modeled as stripped dwarf nuclei (Freeman, 1993; Meylan et al., 2001; Gnedin et al., 2002; Bekki & Freeman, 2003). The position of $\omega$Cen and G1 in Figures 7
and 9 show that the NGC 5128 clusters occupy a similar region
of the fundamental plane and $\kappa-$space.
All of these massive clusters have comparable masses,
mass-to-light ratios, and central surface brightnesses.
Support of the interpretation of these clusters as tidally-stripped
galaxy nuclei is provided by the tentative detection of extratidal light
by Harris et al. (2002) for six of the globular clusters (marked with an $x$ in
Table 2 and 3, partly filled circles in the
figures).
In the context of the stripped-dwarf model, it is tempting to view this
extratidal light as the last vestiges of the extended dwarf envelope around
these nuclei, although extratidal light may also be present due to the
evaporation of cluster stars by two-body relaxation.
Deeper observations of these and other massive clusters
would be extremely valuable to confirm and quantify this extratidal light.
Such observations would also better quantify the ellipticities of these
globular clusters, which are comparable to those of $\omega$Cen and G1 but
greater than those of dwarf nuclei.
These globular clusters may also simply represent the upper end of the
globular cluster mass function in NGC 5128. The similarities between these
globular clusters and the most massive globular clusters in the Local
Group are more established than the interpretation of the better-studied
Local Group clusters as tidally-stripped dwarf nuclei.
In addition, there is evidence
for yet more massive young star clusters that demonstrate that there is
overlap between the masses of the most massive star clusters and the
least massive dwarf galaxies.
The most massive star cluster known is W3, a cluster with $\sigma=45$ km s${}^{-1}$
in the merger remnant NGC 7252 (Maraston et al., 2004). The inferred mass of this
cluster is nearly $10^{8}$ M${}_{\odot}$, even more massive than the
nuclei of the dwarf galaxies studied by Geha et al. (2002).
Even if we were to make the extreme suggestion that all of the NGC 5128 globular
clusters are in fact relic dwarf nuclei, the existence of W3 demonstrates
there is overlap between the masses of galaxies and star clusters.
On the other side of the mass gap, the ultra-compact dwarf galaxies in the
Fornax cluster (Drinkwater et al., 2003) are actually less massive than W3
and overlap with the most massive NGC 5128 clusters. These dwarfs have velocity
dispersions of $\sigma=24-37$ km s${}^{-1}$, effective radii of 10 – 30 pc,
masses in the range $10^{7-8}$ M${}_{\odot}$, and mass-to-light ratios of 2 – 4 in
solar units (Drinkwater et al., 2003). They are thus very comparable to the larger
and more massive of the NGC 5128 globular clusters. Numerical models for these
ultra-compact dwarfs have shown that they can form from nucleated dwarfs that
have been stripped of their envelopes by tidal forces in a cluster
(Bekki et al., 2003). This stripping process can explain the ultra-compact
dwarfs in Fornax, although it will operate over approximately a factor of
2 smaller radius in a smaller group like NGC 5128 due to decreased tidal shear.
The radius for the tidal stripping of NGC 5128 globular cluster progenitors is
likely to be on order 10 kpc (Bekki et al., 2003, see their Figure 7), comparable
to the projected distances of 5 – 23 kpc for some of the NGC 5128 globular
clusters in this study. It is therefore plausible that tidal stripping
could have transformed at least some nucleated dwarf galaxies into
these NGC 5128 globular clusters.
Another aspect of the tidally-stripped dwarf model is that their dark matter
halos cannot be too cuspy as otherwise the relatively concentrated
dark matter core will be too effective at retaining the stellar envelope.
In their study, Bekki et al. (2003) used the dark matter profile of
Salucci & Burkert (2000) (originally proposed by Burkert (1995) for
dwarf galaxies), rather than the more commonly adopted
profile of Navarro, Frenk, & White (1996), because it has a flatter core.
However, this requirement for a relatively flat dark matter core may remove
one possible explanation for the higher mass-to-light ratio of these massive
globular clusters. While a residual dark matter halo from their dwarf galaxy
past is a possible explanation for the larger mass-to-light ratios of these
NGC 5128 clusters, a dark matter profile with a flat, low-density core will
also be more efficiently stripped away (Bekki et al., 2003).
An alternative explanation for the high mass-to-light ratios of these
massive globular clusters is if they formed in a starburst with a
truncated stellar initial mass function (Charlot et al., 1993). A mass function with
relatively more low-mass stars would produce a higher mass-to-light
ratio.
The detailed shapes and kinematics of the nuclei of nucleated dwarfs and
the most massive globular clusters may provide one way to further investigate
the potential connection between these two populations. The most massive
globular clusters have significant ellipticities (Harris et al., 2002), while
this does not appear to be the case for the nuclei of nucleated dwarfs
(Geha et al., 2002).
However, the same tidal forces that strip a dwarf envelope may also induce
significant ellipticities.
The importance of rotational flattening and anisotropies
in the velocity distribution may also serve to distinguish between
dwarf nuclei and globular clusters.
The hypothesis that some of the most massive globular clusters are the nuclei
of galaxies offers an appealing explanation for recent evidence of an
intermediate-mass black hole in G1 (Gebhardt, Rich, & Ho, 2002). The mass of this
black hole falls on the same $M_{BH}-\sigma$ relationship for galaxies
and suggests that the formation mechanisms for black holes in star clusters
and galaxy spheroids are similar. If G1 is instead simply a tidally-stripped,
nucleated dwarf
galaxy, the problem is reduced in complexity and only one physical mechanism
for black hole growth is required to explain the $M_{BH}-\sigma$ relation.
An alternate interpretation of the similarity between the most massive
globular clusters and the nuclei of nucleated dwarfs is that the later are
simply star clusters that have migrated to or formed at the centers of these
dwarfs.
The properties of these objects would then be probes of massive star clusters
in different environments, rather than of actual overlap between the properties
of the most massive star clusters and the least massive galaxies.
This interpretation also explains their similar location in $\kappa-$space,
although stands in contrast to the simple explanation for the
intermediate-mass black hole in G1.
8. Summary
We have measured velocity dispersions for a sample of 14
globular clusters in the nearby, luminous elliptical galaxy NGC 5128,
the first such study of the globular cluster system of a luminous
elliptical galaxy and the first such study outside the Local Group.
These clusters have velocity dispersions in the range 10 – 30 km s${}^{-1}$,
comparable to the largest previously measured values for globular clusters.
We have used measured King model structural parameters for these clusters
from the literature to derive masses for all 14
clusters. These clusters are comparable in mass to the most massive
Galactic globular cluster $\omega$Cen and M31’s G1.
From these data we find the following:
1. The globular clusters in NGC 5128 approximately follow the same
fundamental plane relationships as Local Group globular clusters and extend
them to approximately an order of magnitude higher mass and luminosity.
2. The mean mass-to-light ratio of these clusters is larger than for
typical Local Group globular clusters, although comparable
to the more massive Local Group clusters.
3. These clusters begin to bridge the mass gap between the most massive
globular clusters and the least massive dwarf galaxies. In particular, there
is very good overlap in the photometric, structural, and kinematic
properties of these clusters and the properties of both nucleated dwarf
elliptical nuclei and ultra-compact dwarf galaxies.
4. The large masses of these clusters, combined with the possible detection
of extratidal light for some objects by Harris et al. (2002), suggest that
some of these globular clusters are in fact the nuclei of tidally stripped
dwarf galaxies.
The common properties of the most massive star clusters and the nuclei of
the least massive dwarfs suggest that both the formation mechanisms for
star cluster and galaxies can produce objects in the same region of the
fundamental plane or $\kappa-$space.
Alternately, the nuclei of nucleated dwarf galaxies may simply be star clusters
that happen to lie in the centers of galaxies, rather than true galaxy nuclei.
Future spectroscopic observations of additional massive
globular clusters could quantify the relative contribution of relic dwarf
nuclei to this population through kinematics and with stellar population
models, while deep, high-resolution images could provide better measurements
of structural parameters, particularly in the core, and search for and study
diffuse, low surface-brightness envelopes.
We would like to thank the staff of Las Campanas Observatory for their
excellent support, in particular for making the MIKE spectrograph
available after domestic security concerns delayed PANIC.
We acknowledge useful discussions with Dan Kelson and thank Francois
Schweizer, John Huchra, and the referee for helpful comments on the manuscript.
PM received support from a Carnegie Starr Fellowship and a Clay Fellowship.
The research of LCH is supported by the Carnegie Institution of Washington and
by NASA grants from the Space Telescope Science Institute (operated by AURA,
Inc., under NASA contract NAS5-26555).
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Interaction-driven transition between the Wigner crystal and the fractional Chern insulator in topological flat bands
Michał Kupczyński
michal.kupczynski@pwr.edu.pl
Department of Theoretical Physics,
Faculty of Fundamental Problems of Technology,
Wrocław University of Science and Technology,
PL-50370 Wrocław, Poland
Błażej Jaworowski
blazej@phys.au.dk
Max-Planck-Institut für Physik komplexer Systeme, D-01187 Dresden, Germany
Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark
Arkadiusz Wójs
arkadiusz.wojs@pwr.edu.pl
Department of Theoretical Physics,
Faculty of Fundamental Problems of Technology,
Wrocław University of Science and Technology,
PL-50370 Wrocław, Poland
Abstract
We investigate an interaction-driven transition between crystalline and liquid states of strongly correlated spinless fermions within topological flat bands at low density (with filling factors $\nu=1/5$, $1/7$, $1/9$). Using exact diagonalization for finite size systems with periodic boundary conditions, we distinguish different phases, whose stability depends on the interaction range, controlled by the screening parameter of the Coulomb interaction. The crystalline phases are identified by a crystallization strength, calculated from the Fourier transforms of pair correlation density, while the Fractional Chern insulator phases are characterized using momentum counting rules, entanglement spectrum, and overlaps with corresponding Fractional Quantum Hall states. The type of the phase depends on a particular single particle model and its topological properties. We show that for $\nu=1/7$ and $1/5$ it is possible to tune between the Wigner crystal and Fractional Chern insulator phase in the kagome lattice model with the band carrying the Chern number $C=1$. In contrast, in the $C=2$ models, the Wigner crystallization was absent at $\nu=1/5$, and appeared at $\nu=1/9$, suggesting that $C=2$ FCIs are more stable against the formation of crystalline order.
I Introduction
One of the most remarkable findings of the solid state physics in the last few decades, is the discovery of topological orders.
They have changed the paradigm of matter phase classification and provided a potential way to construct a fault-tolerant quantum computer, based on the non-Abelian fractional (anyonic) statistics [1, 2].
Among the most thoroughly studied examples of the topological orders are the fractional quantum Hall (FQH) states [3, 4, 5].
They were initially observed in the 2D electron gas in a strong magnetic field, where they appear at a fractional filling of the first or second Landau level [6].
Alternative realizations were proposed in lattice systems.
In this case, the role of a Landau level is played by a topological flat band, i.e. an energy band with a small dispersion and nonzero Chern number, and the FQH-like states arising in this setting are called fractional Chern insulators (FCIs) [7, 8, 9, 10].
They were experimentally realized in moiré lattices in graphene in a strong magnetic field [11], and there are strong indications that they can also be created without an external magnetic field [12].
Alternative realizations include the optical lattices [13, 14, 15, 16, 17, 18, 19, 20, 21] or arrays of optical cavities [22, 23, 24, 25, 26, 26, 27, 28], which can be easier to control than electronic systems.
The existence of FCIs in simple lattice models
of spinless fermions is now well established by many theoretical works [7, 8, 9, 10, 29, 30, 31, 32, 33, 34, 35, 36, 37].
The FCIs can exhibit several phenomena which are missing in the usual continuum FQH effect.
The most striking is the possibility of obtaining bands with arbitrary Chern number $C$. For $|C|=1$ the FCIs are the lattice analogs of the well-known FQH states (Laughlin, Moore-Read etc.) [7, 8, 9, 10, 29, 30].
However, in the case of $|C|>1$, one finds a new series of states [31, 32, 33, 34, 38, 39], which are a modified version of the multi-layer Halperin FQH states [40].
In order to design experiments, it is important to determine the stable regions of the desired phase. For both FQH and FCI, one of the factors determining this stability is the competition with other phases, e.g. the charge order. Being essentially flat bands, the Landau levels allow for the existence of a Wigner crystal (WC).
In the presence of the long range Coulomb interaction, WC becomes lower in energy than the FQH states as the filling factor decreases [41, 42, 43, 44, 45, 46, 47, 48, 49], which was confirmed experimentally [50, 51].
The competition of FCIs with charge density waves was studied for large filling factors [52, 53, 54, 55, 56, 57, 58, 59].
The existence of such charge-ordered states depends on the commensuration with the lattice.
On the other hand, in our earlier work [60], we have shown that analogously to FQH systems, Wigner crystals also emerge at a low filling factor of topological flat bands for certain types of long-range interaction.
In such cases, significant commensuration effects were absent.
In general, the existence of Wigner crystal depends on the filling factor and the type of interaction.
For example, in the Landau levels, one can use the Haldane pseudopotential formalism to construct a parent Hamiltonian for a Laughlin state at arbitrarily low filling [5], for which a Coulomb interaction would generate a Wigner crystal.
Thus, by changing the pseudopotential parameters, one can trigger a transition from FQH state to WC at a constant filling factor [48] (some control of these parameters in an experiment can be exerted e.g. by changing the width of a quantum well [61]).
Inspired by these findings, we investigate the stability of various liquid and crystal phases, and transitions between them, in different lattice models.
In this work, we study fractionally filled topological flat bands in the presence of a density-density interaction with a screened Coulomb (Yukawa) potential by utilizing the exact diagonalization (ED) method to compute the spectra and eigenstates for finite-size systems, with periodic boundary conditions. Our main results are following: (i) it is possible to trigger a transition between FCI and WC on bands with $C=1$ at fillings $\nu=1/7$ and $\nu=1/5$ by varying the range of the interaction, (ii) the FCIs at $C=2$ are more stable against WC formation than the FCIs at $C=1$, (iii) nevertheless it may be possible to observe a WC-FCI transiton in $C=2$ bands at filling $\nu=1/9$.
The article is organized as follows. In Section II we describe the $C=1$ and $C=2$ tight-binding models used by us, as well as the details of the exact diagonalization procedure. Next, in Section III, we show that for $\nu=1/7$ of a $C=1$ band, the long- and short-range interaction give rise to respectively WC and FCI phases, we introduce the WC and FCI characteristics and study them as a function of interaction range. Then, in Section IV, we compare the $C=1$ and $C=2$ systems at filling $\nu=1/5$, showing that the former display a WC-FCI transition, while in the latter we observe the FCI for all considered interaction ranges. Also, we study the $C=2$ bands at $\nu=1/9$, and observe both the FCI and WC phases, although their behaviour as a function of the interaction range strongly depends on the system size and chosen lattice model. The section V summarizes the results.
II Models and methods
II.1 Lattice models
We consider various tight-binding models which exhibit a non-zero Chern number of the lowest band. Within an energy band, the crystal momentum eigenstates are given by the Bloch wavefunctions,
$$\psi_{\mathbf{k}}(\mathbf{r})=e^{i\mathbf{k}\cdot\mathbf{r}}u_{\mathbf{k}}(\mathbf{r}),$$
(1)
where $u_{\mathbf{r}}(\mathbf{r})$ is lattice-periodic. The Chern number is defined as the integral of Berry curvature divided by $2\pi$, i.e.
$$C=\frac{i}{2\pi}\iint_{BZ}\left[\braket{\frac{\partial u_{\mathbf{k}}}{\partial k_{y}}}{\frac{\partial u_{\mathbf{k}}}{\partial k_{x}}}-\braket{\frac{\partial u_{\mathbf{k}}}{\partial k_{x}}}{\frac{\partial u_{\mathbf{k}}}{\partial k_{y}}}\right]\mathrm{d}k_{x}\mathrm{d}k_{y}$$
(2)
where BZ denotes the Brillouin zone.
The Chern number (2) is proportional to the Hall conductivity of the fully filled band.
As an example of a model with $C=1$ of the lowest band, we take the kagome lattice described by the Hamiltonian [62, 10]
$$H_{\mathrm{kag}}=-\sum_{\langle i,j\rangle}({t}_{1}\pm{\rm{i}}{\lambda}_{1}){c}_{i}^{\dagger}{c}_{j}-\sum_{\langle\langle i,j\rangle\rangle}({t}_{2}\pm{\rm{i}}{\lambda}_{2}){c}_{i}^{\dagger}{c}_{j},$$
(3)
where $c_{i}({c}_{i}^{\dagger})$ anihilates (creates) a particle on $i$-th lattice site, $\langle\rangle$, $\langle\langle\rangle\rangle$ denote nearest and next-nearest neighbours, respectively. The “$+$” corresponds to the hoppings along the arrows in Fig. 1 (a) and the “$-$” to the hoppings in the opposite direction. At $\lambda_{1}=\lambda_{2}=t_{2}=0$, the model is gapless, with one band being exactly flat.
Introducing nonzero $\lambda_{1}$ creates a pattern of effective magnetic flux, which is zero on the average, but breaks the time-reveresal symmetry, which is necessary for nonzero Chern numbers. At $\lambda_{2}=t_{2}=0$, the middle band has $C=0$, and the two other bands have opposite Chern numbers with $|C|=1$ (except from $\lambda_{1}=0$ and $\lambda_{1}=\pm\sqrt{3}t_{1}$ where the model is gapless). Inclusion of nonzero second-neighbour hoppings $t_{2}$, $\lambda_{2}$ allows to tune the band dispersion. Although the lowest band has $C=1$ in a wide range of parameters, it is nearly flat (resembling a Landau level) only in a certain part of this range. Moreover, even if we disregard the single-particle energies in the ED computation (see Sec. II.2), making the bands artificially flat, not all parameter values are favourable for FCIs, due to e.g. the fluctuations of the Berry curvature [10]. Considering systems with $\nu=1/5$ filling, we use the parameters $t_{1}=1$, $t_{2}=-0.3$, $\lambda_{1}=0.28$ and $\lambda_{2}=0.2$, corresponding to a nearly flat lowest band with $C=1$ [62], which shown to host a fermionic FCI phase at $\nu=1/3$ [10]. As we will show later, the $\nu=1/5$ FCI can also exist there. In the case of $\nu=1/7$, for which the FCI phase is much less stable than for higher fillings, we keep $t_{1}=1$ and $t_{2}=-0.3$, but we use $\lambda_{1}=0.5$ and $\lambda_{2}=0.2$, for which allow to increase the FCI stability (see Appendix A).
In addition to the kagome lattice, we also study two models with a $C=2$ lowest band. In general, the $|C|>1$ models [33, 63] can be created systematically by stacking several layers of $|C|=1$ models. However, the first model we use, the triangular lattice model, was found independently from this method [64]. It is defined by the Hamiltonian
$$H_{\mathrm{tri}}=\pm t\sum_{\langle i,j\rangle}\exp(i\phi_{ij})c^{\dagger}_{i}c_{j}\pm t^{\prime}\sum_{\langle\langle i,j\rangle\rangle}c^{\dagger}_{i}c_{j}\exp(i\phi_{ij}^{\prime}).$$
(4)
Here, “$+$” and “$-$” refer to the hopping denoted by solid and dashed lines in Fig. 1 (b), respectively. Each of the phases has three possible values, $\phi^{\prime}_{ij}\in\{-2\phi,0,2\phi\}$, $\phi_{ij}\in\{-\phi,0,\phi\}$, where the positive sign refers to the hopping along the arrows in Fig. 1 (b), negative sign to the hopping in the opposite direction, and 0 to the hoppings without an arrow. Following Ref. [64], we choose $t=1$, $t^{\prime}=1/4$, $\phi=\pi/3$, for which we obtain a nearly-flat lowest band with $C=2$, while the two other bands have $C=-1$ each. At these values of parameters, the lowest band can host bosonic FCIs at different fillings [64, 34], thus we expect that the same will happen for fermions.
The second $C=2$ model is a generalized Hofstadter model on the square lattice, i.e. a Hofstadter model with second-neighbor hoppings [65],
$$H_{\mathrm{Hof}}=-\sum_{n,m}\big{(}tc^{\dagger}_{n,m}c_{n+1,m}+\lambda^{\mathrm{od}}e^{2\pi i\phi(n+\frac{1}{2})}c^{\dagger}_{n,m}c_{n+1,m+1}+\\
+\lambda^{\mathrm{od}}e^{-2\pi i\phi(n+\frac{1}{2})}c^{\dagger}_{n,m}c_{n+1,m-1}+\lambda^{\mathrm{d}}e^{2\pi i\phi n}c^{\dagger}_{n,m}c_{n,m-1}\big{)}.$$
(5)
Here, instead of labeling sites with a single index $i$, we label them with two indices $n,m$ denoting their $x$ and $y$ positions in the lattice, respectively. The model is highly tunable. At $\lambda^{\mathrm{od}}=0$, $\lambda^{\mathrm{od}}=t$
it reduces to an ordinary Hofstadter model on square lattice [66]. At rational values of the flux, $\phi=a/b$, with $a\in\mathbb{Z}$, $b\in\mathbb{N}^{+}$ and $a,b$ coprime, the model has $b$ bands. At small flux, these bands are a lattice approximation of the continuum Landau levels, and thus have $|C|=1$. At higher flux, the Landau level structure is no longer visible, but the bands still are topologically nontrivial. The second-neighbour hopping $\lambda^{\mathrm{od}}$ can mix them, leading to topological phase transitions and changes in the Chern numbers. Ref. [65] shows that even in the relatively simple case of $\phi=1/3$ one can obtain a rich phase diagram, with Chern number up to $|C|=4$ in the middle band and up to $|C|=2$ in two other bands. In this work, we use $\phi=1/3$, which results in the unit cell containing three sites. The other parameters are fixed at $t=1$, $\lambda^{\mathrm{d}}=1$, $\lambda^{\mathrm{od}}=-1/2$, which leads to the presence of a nearly-flat $C=2$ lowest band, which was shown to host a $\nu=1/3$ bosonic FCI [67].
II.2 Interaction Hamiltonian
We study finite size systems with $N_{1}\times N_{2}$
unit cells along two real space lattice vectors and $N_{\mathrm{part}}$ particles. We impose periodic boundary conditions, so the total momentum is a good quantum number. We consider two-body interaction in a form of the screened Coulomb (Yukawa) potential
$$\hat{V}=\sum_{i,j}V(r_{ij})n_{i}n_{j}$$
(6)
where $n_{i}$, $n_{j}$ are the particle densities at $i$ and $j$ sites, $r_{ij}$ is the smallest distance between the sites $i$ and $j$ on the torus (i.e. with periodic boundary conditions taken into account).
The interaction potential is
$$V(r)=\exp\left(-\alpha\left(r-r_{NN}\right)\right)/r,$$
(7)
where $r_{NN}$ is the distance between the nearest-neighboring sites, and $\alpha$ is the screening parameter that will be varied to trigger the phase transition.
We note that in general, instead of considering only the closest periodic image of a given site, we could have calculated the sum of contributions of all its periodic images, however, since we consider strong screening, we expect that the differences between these two approaches will be small. Without the loss of generality, we set $r_{NN}=1$.
To focus only on the interaction effects and to reduce the computational complexity of our calculations, instead of the full Hamiltonian, we diagonalize
$$H_{\mathrm{int}}=P\hat{V}P,$$
(8)
where $P$ is the operator of projection to the lowest band. Thus, we first diagonalize the single-particle Hamiltonians given by Eqs. (3), (4), (5), and then construct the many-particle basis by distributing $N_{\mathrm{part}}$ particles over the momentum eigenstates in the lowest band. In order to focus on interaction effects, we use the flat band approximation by neglecting the band dispersion. For a given filling factor $\nu=\frac{N_{\mathrm{part}}}{N_{1}N_{2}}$, the resulting Hamiltonian matrix is diagonalized using the Implicitly Restarted Arnoldi Method implemented in the ARPACK package.
III Phase transition between crystal and liquid phases at filling $\nu=1/7$
To study the crystal and liquid phases, we need indicators which have large values when the given phase is stable and small (or vanishing) values when the phase is absent.
In subsections III.1 and III.2, these indicators are introduced on the example of $N_{1}\times N_{2}=5\times 7$ plaquette of the kagome lattice at filling factor $\nu=1/7$.
Finally, in subsection III.3 the phase transition between the liquids and crystal phases at filling $\nu=1/7$ is analysed.
III.1 The crystal phase for C=1 band at $\nu=1/7$
We begin with the long-range limit of screened Coulomb interaction, in which the Wigner crystallization is expected. The crystalline properties of many-body eigenstates $|\psi\rangle$ are determined using the pair correlation density (PCD), defined as
$$G(i,j)=\frac{\braket{\psi}{c^{\dagger}_{i}c^{\dagger}_{j}c_{j}c_{i}}{\psi}}{\braket{\psi}{c^{\dagger}_{i}c_{i}}{\psi}}.$$
(9)
For future analysis, we replace every site with a Gaussian function and make the PCD continuous (see Appendix B for details).
Fig. 2(a) shows the many-body spectrum for $\alpha=0.5$.
The energy differences are very small and decrease as the screening increases (which means that distant particles interact less strongly),
thus the energies are renormalized through division by the next-nearest neighbour interaction strength $V_{NNN}$ (in kagome lattice, $V_{NNN}=V(\sqrt{3})$).
Fig. 2(b) shows the PCD of the ground state. The PCD in the plaquette is shown together with its periodic repetitions.
The peaks of the PCD and the fixed particle indicated by red triangles form an almost-hexagonal lattice.
Within the plaquette, there are four peaks, which correspond to the four particles, plus one fixed particle giving $N_{\mathrm{part}}=5$.
The periodicity of the Wigner crystal can be characterized by the peaks in the Fourier transform of PCD either in Cartesian or polar coordinates, similarly as it has been done in our previous work [60]. The Cartesian transform is performed along two real space lattice vectors, yielding the quantity $F_{mn}$, where $m,n$ are integers describing the momenta (see Appendix B for definition).
For the comparison of different plaquettes, it is convenient to normalize $F_{mn}$ by the ${F}_{00}=N_{\mathrm{part}}-1$. We define $\tilde{F}_{mn}$ as $\tilde{F}_{mn}=F_{mn}/F_{00}$. To avoid the effects of the periodicity related to the periodic repetition of the considered plaquette, only the Fourier peaks $\tilde{F}_{mn}$ of $m$ and $n$ smaller than plaquettes sizes are taken into account.
In Fig. 2(c) we plot the magnitude of the normalized Cartesian Fourier coefficients $|\tilde{F}_{mn}|$.
A clear reciprocal lattice, with a unit cell smaller than in the reciprocal lattice of the underlying kagome lattice, is seen as brighter peaks around the peak at zero. This is a necessary condition for the presence of the Wigner crystals. The magnitude of these Fourier peaks decay as we move away from $m=0$, $n=0$, which is a consequence of particles having a finite spatial extent (see Ref. [60] for details).
We note that the Fourier transform of PCD is in fact much less anisotropic than it looks in Fig. 2(c) at the first glance. The apparent anisotropy comes from the fact that the Fourier transform is performed in the direction of the two primitive vectors of the kagome lattice, which are not orthogonal –- the angle between them is 60°. The $m$ and $n$ integers describe the Fourier components in these directions – in other words, they describe the coordinates of the points in reciprocal space along the reciprocal lattice vectors, the angle between which is 120°. Thus, if we plotted Fig. 2(c) in true reciprocal space, with 120° angle between the axes, and with $n$ and $m$ rescaled according to Eq. 14, the Fourier peaks would be arranged in a lattice much closer to hexagonal (see Fig. 1 in [60]). However, plotting $\tilde{F}_{mn}$ as in 2(c) makes it easier to determine $m$ and $n$ of the peaks.
At a given particle number $N_{\mathrm{part}}$, there is only a finite number $N_{W}$ of possible Wigner lattices, each characterized by two Fourier components at momenta $(m_{i},n_{i})$ and $(o_{i},p_{i})$, $i=1,2,\dots N_{W}$, corresponding to the two fundamental vectors of the reciprocal Wigner lattice.
To obtain periodicity in both directions, both of these components should be nonzero.
We define the crystallization strength as the square product of the magnitudes of these two components, normalized by the zeroth momentum component, maximized over all possible crystals
$$W=\max_{i\in[1,N_{W}]}\sqrt{|\tilde{F}_{m_{i}n_{i}}||\tilde{F}_{o_{i}p_{i}}|}.$$
(10)
In contrast to our previous work, Ref. [60], the square root is added to the definition of $W$ to obtain a magnitude of $W$ comparable to a single Fourier peak $\tilde{F}_{mn}$. Ideally, $W$ should be 0 for a perfectly flat PCD and 1 for an array of Dirac deltas (i.e. point-like particles). However, because of the existence of “exchange-correlation hole” around the fixed particle, the PCD is never perfectly flat even for liquids, and thus even liquids can have small nonzero $W$.
Nevertheless, the transition between a liquid and a crystal should be accompanied with by an increase of $W$.
The crystallization strength $W$ is shown as a color scale in Fig. 2(a) and one can notice that a set of the lowest energy states are of a crystalline character.
Alternatively, we can look at polar coordinates and obtain the transform $\tilde{F}_{\theta}(r,k_{\theta})$ in the angular direction only (see Appendix B for details). Here, $r$ is the distance to the fixed particle and $k_{\theta}$ is an integer describing momentum in the angular direction. To define angular crystallization strength, let us first define the peak strength at given $k_{\phi}$ as the normalized magnitude of the Fourier component maximized over all possible values of the radius
$$\tilde{F}_{\mathrm{peak}}(k_{\phi})=\max_{r<r_{\mathrm{max}}}|\tilde{F}_{\theta}(r,k_{\theta})|,$$
(11)
where $r_{\mathrm{max}}$ is defined in Appendix B. The angular crystallization strength is defined as $W_{\theta}=\max_{k_{\theta}=2,4,6}\tilde{F}_{\mathrm{peak}}(k_{\theta})$. The $W_{\phi}$ alone is not sufficient to determine the existence of the crystal, as $2-$fold symmetry is exhibited also e.g. by the stripe order. On the other hand, it probes not only the existence of WC but also its symmetry.
The angular Fourier transform is shown in Fig. 2(d). The range $r$ corresponds to the red circle indicated in Fig. 2(b). There is a maximum of angular density at $k_{\theta}=0$ (the zeroth Fourier component) around maximal $r$, which corresponds to the six peaks closest to the fixed particle. At this radius, we observe also a relatively strong $k_{\theta}=6$ Fourier component, showing that the PCD is approximately six-fold rotationally-symmetric, i.e. close to the hexagonal lattice. Note that we also have a nonzero Fourier component at $k_{\theta}=4$ (and, weaker, at $k_{\theta}=2$), which occurs because the Wigner crystal is not perfectly hexagonal, as the perfectly hexagonal Wigner lattice is not permitted by the boundary conditions for a $5\times 7$ system.
III.2 The liquid phase for C=1 band at $\nu=1/7$
In the limit of short-range interaction the FCI state is expected as the ground state. To study this state, we fix the parameter as $\alpha=6.0$.
The fractional Chern insulator phase is identified by looking at various signatures of topologically nontrivial liquid state [9, 68, 69, 70, 71, 72, 40].
For the Laughlin states at filling $\nu=1/q$, $q\in\mathbb{N}^{+}$, we expect $q$ quasi-degenerate states separated by a gap from the rest of the spectrum.
The momenta of these states are determined by the appropriate generalized Pauli principle [68, 9].
Fig. 3(a) shows the seven nearly degenerated states separated by the energy gap to excited states.
The momenta of the quasi-degenerate ground states agree with predictions from the generalized Pauli principle of $\nu=1/7$ Laughlin FCI [9, 68].
To avoid confusion, we note that throughout this work, in the cases where we observe quasi-degeneracy, we will use the phrase “quasi-degenerate ground states” to refer to the entire manifold and “absolute ground state” to refer to the lowest-energy one.
The ground state momentum counting rule is not a definite proof of FCI existence. It should be supplemented e.g. by the analysis of the particle entanglement spectrum [69, 70], which should reveal a nonzero gap $\Delta\zeta$ between the low energy sector with an appropriate number of states below the gap in each momentum sector that agree with the appropriate generalized Pauli principle.
While typically in the literature one constructs the density matrix as an equally-weighted superposition of pure state density matrices of all the quasi-degenerate ground states, we construct it from a single ground state (see Appendix C), which turns out to be sufficient to obtain the correct FCI entanglement energy level counting. The entanglement spectrum is obtained after tracing out all but $N_{A}=2$ particles (we use $N_{A}=2$ for all the systems investigated in this work).
In Fig. 3(b) $\Delta\zeta$ is denoted by the red dash line with a correct number of states below the gap confirming FCI. We note that there are also more gaps higher in the entanglement spectrum. They are absent for model FQH states, where the lowest gap is infinite. For FCIs, some of these gaps were connected with another generalized Pauli principles, which may reflect different types of correlations that can be generated by the Hamiltonian [10]. However, for the identification of the type of topological order in the given state, only the lowest gap is relevant.
We also calculate the entanglement entropy $S$, computed in the particle partition (the same as for the entanglement spectrum).
The numerical value for $S$ can be compared with exact bounds, $S_{\mathrm{max}}$ – the largest entropy permitted by the generalized Pauli principle, and $S_{\mathrm{min}}$ – the entropy of a single Slater determinant. For FQHE, $S$ was shown to be close to the former limit [71, 72], and we expect that the same will happen for FCI.
Although, as we will see later, this approach is less reliable than the entanglement spectrum, in some cases it does detect the transition between FCI and WC. The definitions of entanglement-related quantities can be found in Appendix C.
In Fig. 3(c), entanglement entropy is shown as the color scale on the energy spectrum. The quasi-degenerated ground state is characterized by the high entropy values.
For the investigated system, the lower and upper bounds for the entanglement entropy are $S_{\mathrm{min}}\approx 2.30$ and $S_{\mathrm{max}}\approx 5.95$ (see Appendix C for the definitions). The entropy for all states in the ground state manifold is close to the upper limit, as in the FQH systems [71, 72].
We also compare the overlaps $O=|\braket{\psi}{\psi_{FQH}}|^{2}$ between the state $\ket{\psi}$ and the ground state $\ket{\psi_{FQH}}$ of a model FQH Hamiltonian within the same momentum subspace. The gauge is fixed according to the prescription from Ref. [40]. The color scale in Fig. 3(a) shows corresponding overlaps which are $O>0.88$ for the seven quasi-degenerate ground states.
The comparison is performed only for the momentum subspaces, in which there is a model ground state. For all the others, we simply assign a zero overlap.
More details of these calculation can be found in Appendix D.
Fig. 3(d) shows the PCD of the absolute ground state at a limit of short range interactions for $\alpha=6$. A nearly uniform PCD, apart from the vicinity of the fixed particle, indicates it is a liquid state.
This is confirmed by the Cartesian Fourier spectrum shown in Fig. 3(e) and the polar Fourier spectrum shown in Fig. 3(f), which do not show any clear reciprocal Wigner lattice, thus the state is approximately rotationally and translationally invariant.
III.3 Phase transition between crystal and liquid phases for C=1 band at $\nu=1/7$
In the two previous subsections, we have shown a few signatures which allow for distinguishing liquid and crystal phases.
In the example system, the FCI state is a true ground state in the limit of the short-range interaction, and many low-energy states are Wigner crystals in the limit of the long-range interaction.
Thus, the phase transition between liquid and crystal is expected by tuning $\alpha$ parameter in the screened Coulomb interaction (6).
We begin from studies of the phase transition on the previously considered kagome plaquette $N_{1}\times N_{2}=5\times 7$ at the filling $\nu=1/7$.
Fig. 4 shows the evolution of the energy spectrum as a function of $\alpha$, measured with respect to the absolute ground state energy, with crystallization and liquid signatures indicated by color scales.
The color scale in Fig. 4(a) shows the Cartesian crystallization strength $W$ and the polar crystallization strength $W_{\theta}$ in Fig. 4(b).
For low $\alpha$, there is a single ground state with relatively large $W$ and $W_{\theta}$.
Low-lying excited states also display a crystalline order with even larger $W$ and $W_{\theta}$, compared to the ground state crystallization strength.
As the interaction range is decreased (larger $\alpha$), crystallization strength $W$ and $W_{\theta}$ decreases for all states.
Above $\alpha\approx 1.32$ the seven states with the lowest energy become separated from the rest of the spectrum by the gap indicated by a red dashed line.
The momenta of these states agree with predictions from the generalized Pauli principle of $\nu=1/7$ Laughlin FCI [9].
The energy split between that states is minimized at $\alpha\approx 1.65$, and leads to the level crossing at this point.
These seven states are characterized by different crystallization strengths $W$ and $W_{\theta}$.
After the crossing, for larger $\alpha$, the crystallization strength of low-energy states is small, but there are some states above the energy gap, which display relatively large $W$ and $W_{\theta}$ even above $\alpha=2$. However, for sufficiently large $\alpha$ the crystalline order disappears from all the states.
The quasi-degenerate states at large $\alpha$ are indeed the FCI states, which was confirmed by calculating the FCI signatures in Fig 3(c)-(e).
The overlap between the energy eigenstates and the model FQHE ground states is denoted in Fig. 4(c).
The seven states forming the ground state manifold have overlap with the model states $O>0.55$ in the entire range of $\alpha$, even below the gap closure at $\alpha\simeq 1.32$, but reach $O>0.88$ in the limit of large $\alpha$.
The overlap of the model ground state with the excited states is close to zero.
The evolution of the gap in the entanglement spectrum $\Delta\zeta$ is shown as a color scale in Fig. 4(d). It can be seen that this gap is open for the seven quasi-degenerate ground states at large $\alpha$, and decreases (eventually vanishing), as $\alpha$ decreases. In contrast, in the excited states the entanglement gap is much smaller or nonexistent.
The evolution of entanglement entropy of the states is shown as a color scale in Fig. 4(e).
For the investigated system, the lower and upper bounds for the entanglement entropy are $S_{\mathrm{min}}\approx 2.30$ an $S_{\mathrm{max}}\approx 5.95$ (see Appendix C for the definitions).
For $\alpha\approx 6$ the entropy for all states is $S\approx 5.8$, which is close to the upper limit, as in the FQH systems [71, 72].
As $\alpha$ is lowered and the system undergoes the transition to WC, the entropy decreases. The entropy can therefore be a good signature of the FCI. Nevertheless, even for crystalline states it remains well above the minimal value corresponding to a single Slater determinant. While some excited states have similar values of entropy as the ground states we note that the bound $S_{\mathrm{max}}$ is valid only for the ground states, so the comparison with this bound does not tell us anything about the nature of excited states. We also notice that the low-lying excited states (some of which exhibit crystalline order even when the FCI ground state manifold is fully formed) have significantly lower entanglement entropy than the quasi-degenerate ground states. Also, we observe that in the entire energy spectrum the entanglement entropy decreases with decreasing $\alpha$.
We can conclude that the crystalline phase is stable for the long-range interactions, and is replaced by the FCI phase in the limit of the short-range interaction on the considered $5\times 7$ kagome plaquette.
For the analysis of finite size effects, the signatures of both phases are plotted in Fig. 5 for three kagome plaquettes with sizes $4\times 7$, $5\times 7$, $6\times 7$ at filling $\nu=1/7$. We do not consider a $7\times 7$ system, as on $N_{1}=N_{2}$ plaquettes the degeneracy of the crystal may prevent its detection [60].
Because these characteristics behave differently for each state, here we plot them for a state selected in the following way: we choose the momentum subspace in which the absolute ground state is located at low $\alpha$ (e.g. $\mathbf{K}=[0,0]$ for the $5\times 7$ system), and then we select the lowest state from this subspace at each $\alpha$.
Because this subspace fulfills the FCI generalized Pauli principle, the state in question turns into one of the states from quasi-degenerate FCI manifold as $\alpha$ increases.
The full spectrum with the WC and FCI indicators is shown in the appendix E.1.
For every considered system, the crystallization strength, both Cartesian $W$ (Fig. 5(a)) and polar $W_{\theta}$ (Fig. 5(b)), decreases when $\alpha$ increases from values around $0.3-0.2$ in the crystal phase to, below $0.2$ in the liquid phase.
In the limit of the short-range interaction for all plaquettes seven quasi-degenerated states are separated by the energy gap from the rest of the spectrum.
Momenta of that ground states are in full agreement with the counting rules for the Laughlin state $\nu=1/7$.
Moreover, the increase of overlap $O$ with FQHE state (Fig. 5(c)), and normalized entanglement entropy (Fig. 5(e)) when crystallization strengths decrease, proves that the the crystal phase is replaced by the FCI phase.
The similar behavior is visible in the gap in the entanglement spectrum (Fig. 5(d)) for plaquettes $5\times 7$ and $6\times 7$.
The gap in the entanglement spectrum is not visible for the chosen state in the $4\times 7$ system, but it exists in a few other quasi-degenerate ground states, so that the value of the average gap over all FCI states is non-zero.
It indicates that the Fractional Chern Insulator in that system is not as stable in the smallest systems, as in the bigger ones.
It is important to notice here, that calculating entanglement spectrum for only one state instead of the superposition of all FCI states is not the standard approach, so the lack of the gap in the entanglement spectrum is not equivalent to the lack of the FCI state.
Previous subsections identify the crystal and the liquid phases in the limit of small $\alpha$ (long range interaction, a crystal limit) and large $\alpha$ (short range interaction, a liquid limit). In Figs. 4 and 5, one can notice that that the phase transition does not occur abruptly, it is rather continuous: the phase indicators change smoothly and can remain relatively large even when a given phase fully vanishes. However, this lack of sharp jumps may be a result of the small size of investigated systems.
Moreover, the fact overlap remain relatively high in the WC phase strongly suggest that FCI states have some crystal-like correlations built in.
Similar behaviour was reported for analogous phase transition in FQHE models [48].
Because the changes in most FCI/WC characteristics are gradual, it is hard to define a transition point. Definitions using some threshold on the values of these indicators would always be arbitrary. Another way to define the transition point is to look at characteristics which can take only the binary “yes” or “no” values, for example the existence of the energy gap above the seven states described by generalized Pauli principle. The level crossing leading to the closure of this gap occurs in all three systems, at $\alpha\approx 1.8$, $\alpha\approx 1.32$, $\alpha\approx 0.7$ for $4\times 7$, $5\times 7$ and $6\times 7$ systems, respectively (they are denoted by dashed vertical lines in Fig. 5). While from these results there seems to be a general trend of the transition point moving to the lower $\alpha$ with increasing system sizes, in such small systems, strong finite size effects prevent us from estimating the transition point in the thermodynamic limit.
We note that while our results are limited to the kagome lattice model and a certain set of parameter values, in our previous paper we have shown that the stability of the Wigner crystal does not strongly depend on the model [60]. Thus, the observed phase transition should be visible in any model for any set of parameter values, for which the FCI phase exists for short-range interaction. Moreover, these phenomena should not be limited to the considered screened Coulomb interaction, but should be visible in other similar types of density-density interaction, which allows manipulation of the interaction range.
IV Crystal-liquid phase transition on the $C=2$ models
In the previous section, the phase transition between Wigner crystal and Fractional Chern Insulator phases has been analysed at filling $\nu=1/7$ on the flat band with Chern number $C=1$.
In this section, the results are extended to the models with Chern number $C=2$ at $\nu=1/5$ and $\nu=1/9$ (in the former case, we also compare it with a $C=1$ band at the same filling factor).
The $C>1$ FCIs can be understood as multilayer FQH states with “color-entangling” boundary conditions, which mix the layers [31, 32, 40, 73].
For example, the $\nu=1/5$ and $\nu=1/9$ FCIs on $C=2$ bands, studied in this section, are modified Halperin states at filling $\nu=2/5$ and $\nu=2/9$, respectively (the difference in fillings is a consequence of different definitions of filling factor for FCI and FQHE).
The indicators of crystal and liquid phases at $C=2$ are the same as for $C=1$. Counting rules are more complicated comparing to the $C=1$ case [73], but instead of implementing them, we simply compare the momenta of the ground states (or entanglement energy levels) with results for a model FQH system [40, 73]. These model states can also be used to compute overlaps.
IV.1 Topological phase transition at $\nu=1/5$ for C=1 and C=2 bands
The FCIs in $C=2$ bands do not exist at $\nu=1/7$. Therefore, to compare the $C=1$ and $C=2$ cases, we study the $\nu=1/5$ filling. In the case of $C=2$ at $\nu=1/5$,
Fig. 6(a) shows a low-energy spectrum as a function of the $\alpha$ parameter on the $6\times 5$ triangular lattice plaquette with $N_{\mathrm{part}}=6$ particles. The energy gap between five low energy states and higher energy states is visible in the whole range of $\alpha$ parameter.
Momenta of these states agree with the momenta of the FQH states, which strongly suggests that the system is in the FCI phase. The color scale denotes strength of the Cartesian Fourier transform of the Wigner crystallization $W$.
The low values of $W<0.05$ mean that none of the quasi-degenerate ground states is a Wigner crystal. The lack of crystallization is confirmed by visual inspection of the PCDs.
To prove that the system is in the FCI phase, the overlap of the energy eigenstates with the FQHE ground states is shown in Fig. 6(c). It is high for the ground state on the whole $\alpha$ range. Its value is the highest in the limit of the long-range interactions, and its minimum value is equal to approximately $O\approx 0.75$ in the limit of short-range interaction.
In conclusion, in the considered system the WC phase does not exist, and the FCI state is more stable for the long-range interaction than in the short-range limit.
The results for Chern number $C=2$ at filling $\nu=1/5$ are compared with the system with the same filling but with Chern number $C=1$.
In Fig. 6 (a) the crystallization strength $W$ is presented on the low-energy spectrum of $6\times 5$ kagome plaquette with $N_{\mathrm{part}}=6$ particles, and Fig. 6 (c) shows the overlap with model FQHE states for the same system.
The Cartesian crystallization strength is about $W\approx 0.17$ in the absolute ground state at low $\alpha$ (which is doubly degenerate in this case) and even higher in some excited states, indicating the presence of crystalline order. The five quasi-degenerate ground states become separated from the rest of the spectrum at $\alpha\approx 1.80$. The momenta of these states fully agree with the counting rules. The ground states become FCI in the limit of the short-range interaction, confirmed by calculating the overlap with the FQHE states, which achieves $O\approx 0.9$.
Thus, the phase transition between FCI and WC phases, exist at $C=1$ at filling $\nu=1/5$, and is similar as the one occurring at $\nu=1/7$. This indicates that the lack of phase transition in the system with Chern number $C=2$ is not an effect of the filling $\nu=1/5$ only, and it suggests that it could be an effect of the Chern number value.
We study the phases at filling $\nu=1/5$ for different plaquettes and models (including the generalized Hofstadter model).
The results for a single, selected state are shown in Fig. 7.
The state is chosen in the same way as in Fig. 5. If there is exact degeneracy, the degenerate states display similar characteristics, so just one subspace is chosen.
The full spectrum for these other systems with the WC and FCI indicators is shown in the appendix E.2 for the case with $C=1$ and in the appendix E.3 in the case with $C=2$.
From Fig. 7 we can see that the phase transition occurs in systems with the Chern number $C=1$ and does not occur in considered systems with $C=2$. The crystallization strength for $C=1$ is, in general, smaller for $\nu=1/5$ than for $\nu=1/7$. This shows that the Wigner crystals at $\nu=1/5$ are more fragile than for $\nu=1/7$, which is in line with the results from [60], showing that the crystallization strength increases as the filling is lowered (although we note that here we use different single-particle parameters at $\nu=1/5$ and $\nu=1/7$, see Sec. II.1). The gap closing for $C=1$ occurs at $\alpha\approx 1.78$, $\alpha\approx 0.96$, $\alpha\approx 2.16$ for $6\times 5$, $7\times 5$ and $8\times 5$ systems, respectively (see the dashed vertical lines in Fig. 7). Contrary to the $\nu=1/7$ case, here we do not observe any clear trend in the location of the transition point as a function of the system size, which may be due to geometric effects.
The results presented in this subsection show, that the considered phase transition and the stability of the crystal strongly depend on the value of the Chern number.
IV.2 Topological phase transition at $\nu=1/9$ for $C=2$ band
The next available filling for a fermionic $C=2$ FCI is $\nu=1/9$.
For such a case, we study the following systems: $4\times 9$ and $6\times 6$, both with $N_{\mathrm{part}}=4$, for the generalized Hofstadter model, and $4\times 9$, $6\times 6$ and $5\times 9$ with $N_{\mathrm{part}}=4$, $N_{\mathrm{part}}=4$ and $N_{\mathrm{part}}=5$, for the triangular lattice model.
The $N_{\mathrm{part}}=5$ case is not considered for the Hofstadter model, as for rectangular Bravais lattice the WCs are degenerate on the classical level, which can prevent their detection using $W$ or $W_{\theta}$.
Fig. 8 shows the energy spectra color-coded with $W$ and overlap $O$ for these systems. Let us start by analyzing the overlaps. For all systems but one (Hofstadter $6\times 6$), we observe the presence of nine states with quite high overlap with model FQH states (we have $O>0.75$ in all these systems at some values of $\alpha$). For each of these systems, there is a range of $\alpha$ where these states are the lowest. Obviously, these states have also the same momenta as the model FQH ground states. Therefore, it seems that at these values of $\alpha$ the systems is in the FCI phase.
Further WC and FCI characteristics for selected states are shown in Fig. 9. The procedure of choosing the state is similar as in Fig. 5 and 7, but we have to adjust it for two reasons. First, the absolute ground state at low $\alpha$ does not always lie in a subspace consistent with FCI counting rules. Therefore, we focus only on the momenta corresponding to model FQH ground states. From these subspaces, we choose the ones where, at $\alpha=0.5$, the lowest state has the lowest energy. Typically, this energy level is exactly degenerate, as degeneracy is common in the $\nu=1/9$ case. Therefore, there are several such subspaces. Secondly, unlike the $\nu=1/5$ cases, the degenerate states can differ significantly in crystallization strengths (i.e. the data points in rows 1 and 3 of Fig. 8 can coincide with ones with higher or lower $W$). Therefore, among the selected subspaces, we choose the one in which the lowest state at low $\alpha$ has highest $W$. Then, we plot all the characteristics for the lowest state of this subspace for all $\alpha$.
From Fig. 9 (d) one can see that only the $5\times 9$ triangular lattice system displays an entanglement gap. Similarly, the entanglement entropy of the selected state is closest to $S_{\mathrm{max}}$ for this system (Fig. 9 (e)). In contrast, for the $4\times 9$ Hofstadter plaquette the entanglement entropy is far from the maximal value ($\tilde{S}=(S-S_{\mathrm{min}})/(S_{\mathrm{max}}-S_{\mathrm{min}})\approx 0.6$. This suggests that the FCI state is weaker and less stable than in the cases studied previously.
Moreover, the behaviour of the FCI phase differs qualitatively between systems, as can be seen in Fig. 8. In the $6\times 6$ and $5\times 9$ triangular lattice systems (Fig. 8 (i), (j)) the energy gap above the FCI ground-state manifold remains open in the entire investigated range of $\alpha$. In the $4\times 9$ systems of both lattices (Fig. 8 (c), (h)) we observe two $\alpha$ values where the gap closes – an upper and lower limit to the FCI phase (note that the upper limit was not observed for the systems investigated previously in this work). Moreover, in the $4\times 9$ Hofstadter system the gap is very small compared to the energy splitting of the ground-state manifold. In the $6\times 6$ Hofstadter system (Fig. 8 (d)), there is no FCI phase. Qualitative differences between systems can also be seen in Fig. 9, where the curves of FCI and WC characteristics can have significantly different shape for different plaquettes. These differences might be another signature of the fact that the FCI phase is weak and unstable, but may also have geometrical reasons – the shape of our systems varies strongly. We study the plaquettes with aspect ratio 1 or close to 1 ($6\times 6$ triangular, $4\times 9$ Hofstadter – remember that the Hofstadter unit cell has aspect ratio 3), as well as elongated ones ($6\times 6$ Hofstadter, $4\times 9$ and $5\times 9$ triangular).
In addition to the FCI, we also observe Wigner crystals. By investigating $W$ and $W_{\theta}$ (rows 1 and 3 of Fig. 8 and Fig. 9 (a), (b)), as well as inspecting the PCDs visually, we find that all the systems exhibit some form of crystalline order at low $\alpha$. This happens even for the $5\times 9$ triangular system, where $W$ is low for all states, and all values of $\alpha$. In this case the crystal is weak (i.e. the particles are not as localized as in Fig. 2 (b)) and deformed (i.e. the PCD maxima are displaced from their ideal periodic position), which may be a reason for low values of $W$. Nevertheless, there are four PCD maxima, signifying the localization of particles, and the formation of the crystal coincides with a slight increase in the $W$.
If we define the phase transition point as the FCI gap closing, then such transitions exist only in the $4\times 9$ systems, at $\alpha\approx 1.72$, $\alpha\approx 2.90$ (Hofstadter) and $\alpha\approx 1.33$, $\alpha\approx 4.90$ (triangular). Nevertheless, as noted above, we observe some form of crystalline order also for other triangular lattice systems. In the cases investigated before within this work, we observed that the crystalline order starts to develop already at $\alpha$ higher than the gap closing. This may also be the case here, i.e. the gap might close at $\alpha<0.5$. Another possibility is that the gap remains open due to the finite-size effects, and will close in the thermodynamic limit (provided that neither WC nor FCI disappears in infinite systems).
Another interesting case is the $4\times 9$ Hofstadter system. Figs. 8 (a) and 9 (a) show that $W$ remains high even when the system is in the FCI phase, while, as noted before, the entanglement entropy is far from $S_{\mathrm{max}}$ (Fig. 9 (e)). The visual inspection of the PCD shows that it displays crystalline order, i.e. the system simultaneously exhibits characteristics of WC and FCI. This is in line with the suggestion by Yang et al. [48], (discussed also in Sec. III.3), that the FQH states have some crystal-like correlations built in. Because of such effects, the gap closing is not necessarily a good definition of transition point for the $C=2$ $\nu=1/9$.
The differences between the systems studied in this subsections make any extrapolation to the thermodynamic limit even less reliable than for other cases considered in this work. The importance of the geometric effects can be seen when one compares the $4\times 9$ and $6\times 6$ plaquettes of the same lattice model - even though the number of sites is the same in both cases, and both have $N_{\mathrm{part}}=4$, the difference in aspect ratio leads to significantly different behaviuour of the two systems.
In summary, both Wigner crystal and FCI can exist at $\nu=1/9$ of $C=2$ bands (at least in finite-size systems), and the transition between them can be triggered by controlling the interaction range. However, there are strong, qualitative differences in the behaviour of these phases in systems of various size, shape and underlying lattice model. Moreover, by combining these findings with results from Sec. IV.1, we conclude that forming a Wigner crystal is harder in $C=2$ bands than in $C=1$ ones, in the sense that one has to consider lower filling factors. In other words, the $C=2$ FCIs seem to be more stable against WC formation than their $C=1$ counterparts. The difference between $C=2$ and $C=1$ is striking, compared to the small difference between the $C=1$ and $C=0$ cases reported in [60], although we note that the comparison in Ref. [60] was made for $\alpha$ too small for the FCI to be observed at $C=1$.
V Conclusions
In this work, we performed a finite-size exact-diagonalization study of transition between the FCI and Wigner crystal as a function of interaction range in the $C=1$ and $C=2$ flat-band lattice models.
First, we studied the example of the $C=1$ band of the kagome lattice at $\nu=1/7$. We analyzed five different characteristics of FCI and WC, all leading to the same conclusion: the FCI and WC emerge respectively for short- and long-range interaction, and hence it is possible to trigger a WC-FCI transition by controlling the interaction range. The results were qualitatively similar for three investigated systems.
Next, to see how the WC formation is affected by band topology, we compared the behaviour of $C=1$ and $C=2$ models at $\nu=1/5$. The former displayed an FCI-WC transition, although the WC was weaker than for $\nu=1/7$. In the latter, however, the WC was absent, which suggests that the $C=2$ FCIs are more stable against the crystal formation than their $C=1$ counterparts at the same filling.
Finally, we studied the $C=2$ models at $\nu=1/9$. In such a case, we observed both FCI and WCs, suggesting that one may be able to observe the FCI-WC transition for $C=2$ systems. However, the behaviour of these phases as a function of $\alpha$ exhibited significant, qualitative differences between the lattice models and system sizes.
We note that for the systems whose size is small enough for exact diagonalization, the geometry of the system and the number of particles plays an important role, e.g. by limiting the possible Wigner crystals consistent with the periodic boundary conditions. This may be an explanation for qualitative differences between the systems. Therefore, our results, strictly speaking, can be applied to finite-size systems only, and while we can speculate about the thermodynamic limit, we cannot make any definite conclusion about it. However, working on few-particle systems, with similar number of particles as discussed in this work, might be a feasible way of creating an FCI in optical lattices [74], and in such case one does not need to analyze the thermodynamic limit. While the periodic boundary conditions were chosen by us for computational convenience, we note that an optical-lattice realization of a fractional Chern insulator in a torus geometry was proposed [75]. However, typical schemes of creating an FCI in optical lattices consider short-range interaction[14, 15, 16, 17, 18], so creating a tunable long-range interaction remains an experimental challenge.
Further exploration of the transition for larger systems (perhaps also with open boundary conditions) may be performed using the DMRG method [76] or using model wavefunctions [40, 77], as these methods were successful in investgating the WC-to-FQH transition in Landau levels [42, 43, 49].
Acknowledgements.
We thank Pawel Potasz and Alina Wania Rodrigues for helpful comments and careful proofreading of the manuscript. B. J. thanks Zhao Liu for fruitful discussions regarding the implementation of the overlap computations.
M. K. was supported by the National Science Centre (NCN, Poland) under grant: 2019/33/N/ST3/03137.
B. J. was supported by Foundation for Polish Science (FNP) START fellowship no. 032.2019 and Independent Research Fund Denmark under Grant Number 8049-00074B.
Our calculations were performed at the Wrocław Center for Networking and Supercomputing.
Appendix A Choice of kagome lattice parameters
The stability of the FCI phase depends not only on the many-body interaction, but also on the lattice parameters.
In most cases, we have used well-known parameters from the literature [64, 65, 10, 62]. The only exception is the kagome lattice with filling $\nu=1/7$, for which the FCI phase was not stable enough for various considered system sizes and interaction parameters.
To determine more suitable values of parameters, we calculated signatures of the FCI phase as the function of the Hamiltonian (3) parameters $\lambda_{1}$ and $\lambda_{2}$ with fixed values $t_{1}=1$, $t_{2}=-0.3$.
We focused on the $4\times 7$ plaquette with $N=4$ particles with the screened Coulomb interaction in short range limit (screening parameter $\alpha=6.0$), see Fig. 10.
The simplest signature of the FCI state is the energy gap $\Delta E$ between the 7-fold quasi-degenerate FCI ground state at lattice momenta $\mathbf{K}\in\{(2,0),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)\}$ and the first excited state.
If one of the six lowest states has a momentum which does not belong to this set, i.e. the generalized Pauli principle is not fulfilled, we set $\Delta E=0$.
Additionally, we calculated average particle entanglement entropy $\langle S\rangle$ and average overlap with FQHE state $\langle O\rangle$.
The average is taken over seven states with the lowest energy.
By looking for that three signatures of the FCI state, we chose $\lambda_{1}=0.5$ and $\lambda_{2}=0.2$.
These parameters are marked by the $\times$ sign on the Fig. 10. We note here, that the question what is the influence of single-particle parameter on the stability of WC and properties of WC-FCI transition is still open.
Appendix B Signatures of the Wigner crystal – details
Here we provide a more detailed summary of the definitions of crystalization strength. For even more details, see [60].
The PCD is turned into a continuous quantity by replacing every site with a Gaussian,
$$G_{i}(\mathbf{r})=\sum_{j=1}^{N}G(i,j)\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{|\mathbf{r}-\mathbf{r_{j}}|^{2}}{2\sigma}\right),$$
(12)
where $\sigma$ is the width of the Gaussian, and $\mathbf{r}$ is the vector connecting site $i$ (where the fixed particle is located) and a given point in space. We use $\sigma=0.5$. Typically, the results do not differ significantly for different starting sites $i$, hence we can choose any site and drop this index, provided that we measure $\mathbf{r}$ with respect to that site.
To obtain the Cartesian Fourier transform, we first discretize this continuum PCD on a regular grid $N_{\mathrm{grid,1}}\times N_{\mathrm{grid,2}}$, obtaining a matrix
$$\tilde{G}_{mn}=G\left(\frac{mN_{1}}{N_{\mathrm{grid,1}}}\mathbf{a}_{1}+\frac{nN_{2}}{N_{\mathrm{grid,2}}}\mathbf{a}_{2}\right),$$
(13)
where $\mathbf{a}_{1}$, $\mathbf{a}_{2}$ are the lattice vectors of the tight-binding model. We perform a discrete Fourier transform of $\tilde{G}_{mn}$ using the FFT algorithm and obtain the Fourier coefficients $F_{mn}$, which we normalize by dividing by the magnitude of the zeroth component $\tilde{F}_{mn}=F_{mn}/|F_{00}|$. Thus, the pair correlation density in the momentum space is given, up to normalization, by
$$F_{c}(\mathbf{k})=\sum_{mn}\tilde{F}_{mn}\delta(\mathbf{k}-\frac{m}{N_{1}}\mathbf{b}_{1}-\frac{n}{N_{2}}\mathbf{b}_{2}),$$
(14)
where $\mathrm{c}$ denotes “Cartesian”, and $\mathbf{b}_{1}$, $\mathbf{b}_{2}$ are the reciprocal lattice vectors obtained from $\mathbf{a}_{1}$, $\mathbf{a}_{2}$. The Fourier peaks located at $m,n$ being multiples of $N_{1},N_{2}$, respectively corresponds to the reciprocal lattice of the tight-binding model. Any $m,n$ smaller than $N_{1},N_{2}$, respectively, are responsible for features varying on a scale larger than a single unit cell, i.e. a possible Wigner crystal. However, not every such pattern is a WC: it needs to be periodic in two directions, i.e. two Fourier components should be nonzero, and the number of maxima of the corresponding real-space pattern should match the number of particles. Thus, for every $N_{\mathrm{part}}$ there is a finite number $N_{W}$ of possible Wigner crystals, each labelled with two integer vectors $(m_{i},n_{i}),(o_{i},p_{i})$. We determine them by listing all possible combinations of these integers, and neglecting all these with incorrect number of maxima. Then, for every possible crystal, we calculate the corresponding crystalization strength by multiplying the two Fourier components described by these vectors. Next, we take the maximum value of this product over all crystals as the crystalization strength (Eq. (10)). We note that $\tilde{G}_{mn}$ does not have the exact periodicity of the reciprocal lattice of the Wigner crystal, as the “hole” at the position of the fixed particle breaks the periodicity of the original PCD (see the Appendix A.3 of [60] for details). This can generate nonzero $W$ for a non-crystalline PCD pattern, but, compared to $W$ for a Wigner crystal in the same system, it is generally smaller.
We can also perform the angular Fourier transform, which we do by discretizing $G(\mathbf{r})$ on a polar grid and performing FFT at each $r$ separately. This yields the $r$-dependent Fourier coefficients $F_{\theta}(r,k_{\theta})$. As noted in the main text, we look at $k_{\theta}=2,4,6$, related to 2,4,6-fold rotational symmetry. However, we should bear in mind that PCD is periodic with plaquette periodicity. That is, even if PCD is uniform far away from the fixed particle, the periodic images of the “hole” around its position will introduce an artificial angular periodicity. Thus, we need to introduce a cutoff radius $r_{\mathrm{max}}$. As a compromise between avoiding the “holes” and capturing as many particles as possible, we choose $r_{\mathrm{max}}$ equal 0.6 times the distance to nearest periodic image of the fixed particle. Having this in mind, we define the normalized angular Fourier transform as
$$\tilde{F}_{\theta}(r,k_{\theta})=\frac{F_{\theta}(r,k_{\theta})}{\max_{r<r_{\mathrm{max}}}|F_{\theta}(r,0)|}.$$
(15)
Then we proceed as described in Sec. III.1.
Appendix C Entanglement signatures of FCI
The existence of the FCI phase can be seen using entanglement methods. Here, we focus on the particle partition. Typically in the FCI literature [70, 9], one constructs a density matrix as an equal-weight superposition of the pure-state density matrices of all $q$ quasi-degenerate ground states,
$$\rho=\frac{1}{q}\sum_{i=1}^{q}\ket{\psi_{i}}\bra{\psi_{i}}.$$
(16)
Then, one divides the system into two subsystems $A$ and $B$, with $N_{A}$ and $N_{B}$ particles ($N_{A}+N_{B}=N_{\mathrm{part}}$), and performs a trace over the $B$ subsystem, $\rho_{A}=\mathrm{Tr}_{B}\rho$. From the eigenvalues $\lambda_{i}$ of the reduced density matrix $\rho_{A}$ one constructs the entanglement energies $\zeta_{i}=-\ln\lambda_{i}$.
In our work, we follow this approach, but instead of using $\rho$ defined by Eq. (16) we construct a pure-state density matrix of each energy eigenstate separately
$$\rho_{i}=\ket{\psi_{i}}\bra{\psi_{i}}.$$
(17)
This definition also works for $i>q$, i.e. the excited states.
As we noted in the main text, in most of the studied cases even such a single-state entanglement spectrum displays the gap and correct counting of states below it for FCI. We define the entanglement gap in the following way. Let $n_{\mathrm{P}}(K_{1},K_{2})$ be the number of entanglement energy levels consistent with the generalized Pauli principle [9, 68] in the $\mathbf{K}=[K_{1},K_{2}]$ subspace. We denote the $i$th entanglement energy level in the $\mathbf{K}$ subspace as $\zeta_{i}(K_{1},K_{2})$, (we assume that they are sorted in an increasing order, i.e. $\zeta_{i}(K_{1},K_{2})\leq\zeta_{j}(K_{1},K_{1})$ for $i<j$). We define two sets of entanglement energies, $\zeta_{\mathrm{below}}=\{\zeta_{i}(K_{1},K_{2}):i\leq n_{\mathrm{P}}(K_{1},K_{2}),K_{1}=0,\dots N_{1}-1,K_{2}=0,\dots N_{2}-1\}$ and
$\zeta_{\mathrm{above}}=\{\zeta_{i}(K_{1},K_{2}):i>n_{\mathrm{P}}(K_{1},K_{2}),K_{1}=0,\dots N_{1}-1,K_{2}=0,\dots N_{2}-1\}$. The entanglement gap is defined as
$$\Delta\zeta=\max\left\{0,\min\left\{\zeta_{\mathrm{above}}\right\}-\max\left\{\zeta_{\mathrm{below}}\right\}\right\}.$$
(18)
In the case of $C=2$ states, instead of implementing the generalized Pauli principle, we compare the entanglement spectra of the topological flat band systems, obtained using (17), to the entanglement spectra of the model continuum Halperin-like states, computed using (16). That is, $n_{\mathrm{P}}(K_{1},K_{2})$ used in the calculation of $\Delta\zeta$ for the investigated systems is the number of entanglement energy levels below the gap in the corresponding model state.
Instead of looking at the structure of the entanglement spectrum, we can use the entanglement entropy,
$$S=-\sum_{i}\lambda_{i}\ln\lambda_{i}.$$
(19)
In Ref. [71, 72], an exact upper bound for the entanglement entropy of Laughlin states was obtained,
$$S_{\mathrm{max}}=\ln\sum_{K_{1},K_{2}}n_{\mathrm{P}}(K_{1},K_{2}).$$
(20)
It is derived by assuming that all $\zeta_{i}$ below the gap are equal, and all the others are infinite, i.e. the corresponding $\lambda_{i}$ equal 0 and do not contribute to the entropy. The authors of Refs. [71, 72] found numerically that the entanglement entropy of the continuum Laughlin states is close to that bound. We expect that the same will happen for FCI, both at $C=1$ and $C=2$.
There is also a lower bound on the entanglement entropy, obtained by assuming that the state $\ket{\psi_{i}}$ is a single Slater determinant,
$$S_{\mathrm{min}}=\ln{{N_{\mathrm{part}}}\choose{N_{A}}}.$$
(21)
We expect that the entanglement entropy of the FCIs will be far larger than this minimum value.
Appendix D Overlap with model FQH states
To calculate the overlap with a model wavefunction, three problems need to be solved. First of all, the FQH and FCI states should carry the same quantum numbers. That is, in the single-particle bases $\ket{\phi_{\mathrm{FCI}}(\mathbf{k})}$, $\ket{\phi_{\mathrm{FQH}}(\mathbf{k})}$ for FQH and FCI, the same sets of values of $\mathbf{k}=[k_{1},k_{2}]$ momenta should be allowed. If we fully exploit the translational symmetry of the FCI fully, we have $k_{1}=0,\dots,N_{1}-1$ and $k_{2}=0,\dots,N_{2}-1$ as allowed momenta. However, in the Landau gauge for the FQH systems, the Brillouin zone is different – it is 1-dimensional, with $k=0,1,\dots N_{1}N_{2}$, Therefore, another basis for FQH systems should be used.
The second problem is that on a torus, we can add a phase $e^{i\gamma_{1}}$, $e^{i\gamma_{2}}$ at the boundary conditions in the directions $\mathbf{a}_{1}$, $\mathbf{a}_{2}$. After such a modification, the system stays in the FCI/FQH phase – indeed, these phases are varied during the calculation of FCI/FQH signatures, such as spectral flow or many-body Chern number (see e.g. [78, 8]). These phases control the Berry phase of a particle encircling the torus around its fundamental cycles (large Wilson loops), and to maximize the overlap, we should demand that the respective large Wilson loops are equal for FCI and FQH. This does not necessarily mean that the boundary condition phases are equal for FQH and FCI. Therefore, we fix $\gamma_{1}=\gamma_{2}=0$ for FCI and search for the optimal $\gamma_{1}$, $\gamma_{2}$ in FQH.
Thirdly, we should specify the mapping between FQH and FCI precisely. To compute the overlap $\braket{\psi}{\psi_{FQH}}$ between the ED result and the model wavefunction, we need to know $\braket{\phi_{\mathrm{FCI}}(\mathbf{k})}{\phi_{\mathrm{FQH}}(\mathbf{k}^{\prime})}$, i.e. the overlap between the single-particle basis functions for FCI and FQH systems. Since $\ket{\phi_{FCI}(\mathbf{k})}$, $\ket{\phi_{\mathrm{FQH}}(\mathbf{k})}$ describe different systems, it is up to us to define the relation between them by fixing the values of $\braket{\phi_{\mathrm{FCI}}(\mathbf{k})}{\phi_{\mathrm{FQH}}(\mathbf{k}^{\prime})}$. It is natural to identify the states with the same momenta, i.e. to set $\braket{\phi_{\mathrm{FCI}}(\mathbf{k})}{\phi_{\mathrm{FQH}}(\mathbf{k}^{\prime})}=0$ if $\mathbf{k}\neq\mathbf{k}^{\prime}$. However, this still leaves us with some ambiguity. Let us assume that the basis $\ket{\phi_{\mathrm{FCI}}(\mathbf{k})}$ are the lowest-band eigenfunctions resulting from the numerical diagonalization of the single-particle model. We can define a different basis for the FCI, by multiplying every basis vector by a momentum-dependent phase $\ket{\tilde{\phi}_{\mathrm{FCI}}(\mathbf{k})}=e^{i\theta_{\mathbf{k}}}\ket{\phi_{\mathrm{FCI}}(\mathbf{k})}$. We can require either that $\braket{\phi_{\mathrm{FCI}}(\mathbf{k})}{\phi_{\mathrm{FQH}}(\mathbf{k})}=1$ or $\braket{\tilde{\phi}_{\mathrm{FCI}}(\mathbf{k})}{\phi_{\mathrm{FQH}}(\mathbf{k})}=1$. These two options result in two different values of the overlap. Therefore, we have to find a mapping for which $\braket{\psi}{\psi_{FQH}}$ is maximal. That is, given $\ket{\phi_{\mathrm{FCI}}(\mathbf{k})}$, we have to find the phases $\theta_{\mathbf{k}}$ which transforms it into an another basis $\ket{\tilde{\phi}_{\mathrm{FCI}}(\mathbf{k})}$, which maximizes $\braket{\psi}{\psi_{FQH}}$ under the condition $\braket{\tilde{\phi}_{\mathrm{FCI}}(\mathbf{k})}{\phi_{\mathrm{FQH}}(\mathbf{k})}=1$. This is what we mean by “fixing the gauge”.
The solutions for all the three problems were given in Ref. [40]. The authors proposed a Bloch basis for FQH systems, indexed by a momentum in $N_{1}\times N_{2}$ Brillouin zone, and an algorithm which provides appropriate $\theta_{\mathbf{k}}$, $\gamma_{1}$ and $\gamma_{2}$. The algorithm computes the Berry connection and the large Wilson loops in the FCI case and compares with the result for FQH, adjusting the $\gamma_{1}$, $\gamma_{2}$ accordingly. The gauge $\theta_{\mathbf{k}}$ is found by imposing a discrete analog of Coulomb gauge condition for the FCI and solving a discretized Poisson equation with Berry curvature fluctuation as a source. The algorithm was implemented in the DiagHam software [79], and in our work we use a DiagHam-based code to optimize the overlap.
The basis $\ket{\tilde{\phi}_{\mathrm{FCI}}(\mathbf{k})}$ is then fed to the ED calculation, i.e. the Hamiltonian (8) is diagonalized in the many-particle basis constructed as Slater determinants of these wavefunctions. The model FQH states for the overlap are constructed by diagonalizing the appropriate pseudopotential Hamiltonian in the Bloch basis, taking into account the boundary condition phases $\gamma_{1}$,$\gamma_{2}$. This is true for both $C=1$ and $C=2$. In the latter case, a bilayer system is considered, with boundary conditions mixing the layers. The Bloch basis is constructed following Refs. [40, 73]. The model wavefunctions for $\nu=1/5$ are obtained with two first pseudopotentials $V_{0}=V_{1}=1$ and the rest equal to zero, while for $\nu=1/9$ four first pseudopotentials are equal to unity, and the rest is zero.
Appendix E System size analysis
In this appendix, we present the signatures of WC and FCI phases in the low-energy spectrum of the systems partially described in the main article.
E.1 $C=1$, $\nu=1/7$
In section III we have shown the WC and FCI signatures for the kagome lattice system at filling $\nu=1/7$, for the whole energy spectrum for the $5\times 7$ plaquette and a single selected state for $4\times 7$ and $6\times 7$ plaquettes.
The energy spectrum with the WC and FCI signatures for the last two systems is plotted in Fig. 11.
The Cartesian WC strength $W$ is shown in Fig. 11 (a) and (b), and the overlap with the model FQHE states is shown in Fig. 11 (c) and (d).
The behaviour of these systems is similar to the $5\times 7$ case described in Sec. III, in the sense that for large $\alpha$ we obtain an FCI with seven quasi-degenerate ground states, and as we lower $\alpha$ this ground-state manifold splits while the Wigner crystals emerge.
The spectra from Fig. 11 display also some differences with respect to the $5\times 7$ case.
For the $4\times 7$ system, the gap above the FCI ground state manifold closes temporarily between $\alpha\approx 3.1$ and $\alpha\approx 3.6$.
Nevertheless, in that region, these states still have a large overlap with model FQHE states ($O>0.83$).
As for the $6\times 7$ system, we can see that the gap closing occurs for much smaller $\alpha$ than for two other systems.
We also note that similarly to the $5\times 7$ case, the Wigner crystals arise in the excited states as well, sometimes with larger $W$ than in the ground state. Interestingly, in both the $4\times 7$ and the $6\times 7$ system, in some of the excited states we obtain a different Wigner lattice than in the ground states.
E.2 $C=1$, $\nu=1/5$
In section IV.1 we have shown the phase transition between WC and FCI phases on the kagome lattice system at filling $\nu=1/5$, for the whole energy spectrum for the $6\times 5$ plaquette and the one selected state for $7\times 5$ and $8\times 5$ plaquettes.
Fig. 12 (a) and (b) show the energy spectra as a function of $\alpha$ color-coded with Cartesian crystallization strength for the plaquette $7\times 5$ and $8\times 5$, respectively.
The overlap with the FQHE state for the $7\times 5$ system is shown in Fig. 12 (c).
The results for the plaquette $8\times 5$ are obtained only in the limited range of $\alpha$, because of the numerical complexity of the computation and problems with numerical convergence of the diagonalization problem, especially in the limit of high $\alpha$ values. For all systems at large $\alpha$, we observe five quasi-degenerate ground states, which momenta match FCI counting rules.
Also, all ground states have crystalline order in the long-range-interaction limit.
From Fig. 12 and the consideration in section IV.1, it can be seen that as we decrease $\alpha$, the gap above the FCI quasi-degenerate ground state manifold decreases and eventually closes. This process looks slightly different than in the case of $\nu=1/7$: not all ground states cross before the gap closing occurs (e.g. in Fig. 12 (a) the absolute ground state does not cross with any other state all through the transition). As $\alpha$ is lowered, the crystallization strength increases in one or two states which eventually become the absolute ground states. It is however interesting to note that in the $7\times 5$ system, the crystallization strength in the ground state has a maximum at $\alpha=0.91$. Upon further decrease of $\alpha$, the crystallization strength drops, and at $\alpha=0.5$ the Wigner crystal is nonexistent. This is an explicit example that a too-small screening can be detrimental for Wigner crystals.
Similarly to the $\nu=1/7$ case, in a few excited state, the crystalline order exists and is visible even when the FCI phase is well established.
E.3 $C=2$, $\nu=1/5$
In section IV.1 we have shown the phase transition between WC and FCI phases at filling $\nu=1/5$ on the lattice systems with Chern number $C=2$.
The signatures of both phases have been plotted on the low-energy spectrum for the $6\times 5$ Hofstadter lattice in the Fig. 6 and for the one selected state in the Fig. 7 for the following plaquettes: $3\times 10$, $5\times 7$ Hofstadter model and $7\times 5$ triangular lattice.
In the Fig. 13 is shown the full spectrum of the last mentioned plaquette with the color-coded Cartesian crystallization strength $W$ (in Figs. 13 (a), (b), (c)) and the overlap with the FQH states ( in Figs. 13 (d), (e), (f)).
The behaviour of each system is similar: in the entire studied range of $\alpha$ a five-fold quasi-degenerate ground state manifold is separated by a gap from the rest of the spectrum (see the dashed red horizontal line in Fig. 13).
The momenta of these states are the same as the momenta of the model FQH system, and the overlap with them remains large (the lower row of Fig. 13).
Also, the crystallization strength of the ground state (the upper row of Fig. 13) remains small for all $\alpha$ values for all states, but a few excited states are characterized by a larger value of the WC strength $W$ in the small values of $\alpha$.
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Prospects of the local Hubble parameter measurement using gravitational waves from double neutron stars
Naoki Seto${}^{1}$ and Koutarou Kyutoku${}^{2,3,4,5}$
${}^{1}$Department of Physics, Kyoto University, Kyoto 606-8502, Japan
${}^{2}$Theory Center, Institute of Particle and Nuclear Studies, KEK,
Tsukuba 305-0801, Japan
${}^{3}$Department of Particle and Nuclear Physics, the Graduate University
for Advanced Studies (Sokendai), Tsukuba 305-0801, Japan
${}^{4}$Interdisciplinary Theoretical Science (iTHES) Research Group, RIKEN,
Wako, Saitama 351-0198, Japan
${}^{5}$Center for Gravitational Physics, Yukawa Institute for Theoretical
Physics, Kyoto University, Kyoto 606-8502, Japan
Abstract
Following the detection of the GW170817 signal and its associated
electromagnetic emissions, we discuss the prospects of the local Hubble
parameter measurement using double neutron stars (DNSs). The kilonova
emissions of GW170817 are genuinely unique in
terms of the rapid evolution of color and magnitude and we
expect that, for a good fraction $\ga 50\%$ of the DNS events within
$\sim 200$Mpc, we could identify their host galaxies, using their
kilonovae. At present, the estimated DNS merger rate
$(1.5^{+3.2}_{-1.2})\times 10^{-6}{\rm Mpc^{-3}yr^{-1}}$ has a large
uncertainty. But, if it is at the high end, we could measure the local
Hubble parameter $H_{L}$ with the level of $\Delta H_{L}/H_{L}\sim 0.042$
($1\sigma$ level), after the third observational run (O3). This
accuracy is four times better than that obtained from GW170817 alone,
and we will be able to examine the Hubble tension at $2.1\sigma$
level.
keywords:
1 introduction
The gravitational wave (GW) signal GW170817 was detected at the signal-to-noise ratio (SNR) of 32.4 that is the largest value among the GW signals detected so far (Abbott et al. 2017a). From the estimated masses, the signal is considered to be generated by a DNS inspiral. After the GW detection, the associated electromagnetic (EM) emissions were discovered worldwide at various wavelengths (Abbott et al. 2017b). These sequential events brought profound impacts broadly on astronomical and physical communities. Here, in the face of the current torrent of research papers, we do not mention general aspects of GW170817, but rather concentrate on our main topic, observational cosmology.
It has been long known that, using GWs from binary inspirals (often called the standard sirens), we can estimate the distance to the source, solely based on the first principle of physics (Schutz 1986, Krolak & Schutz 1987). This shows a remarkable contrast to the traditional distance ladder that relies heavily on various empirical relations.
Meanwhile, because of the simple scaling property of general relativity, it is not straightforward to estimate the redshift of the binary only from GWs (see also Chernoff & Finn 1993, MacLeod & Hogan 2008, Messenger & Read 2012). Therefore, in order to utilize the standard sirens efficiently, it would be crucially advantageous, if we can identify the transient EM signals associated with the GW events (see e.g. Holz & Hughes 2005, Nissanke et al. 2010). But, we had been far from confident whether such multi-messenger observations actually work.
Now, this concern is largely untangled by the followup observations of
GW170817 and the resulting identification of its host galaxy NGC4993 at
$z=0.010$ (after the peculiar velocity correction, Abbott et al. 2017c, see also Hjorth et al. 2017). In fact, its kilonova (also called the macronova) emission turned out to be genuinely unique in
terms of the rapid evolution of color and magnitude, also showing a characteristic time
profile. It
is true that we only have the single DNS event and additional ones are
essential to understand the basic properties of the EM
counterparts. But, now, we can expect long-term development of
observational cosmology, by using DNSs as a powerful probe.
In the near future, around the entrance of this new avenue, our primary target would be the Hubble parameter, as already discussed in the pioneering work by Schutz (1986) more than 30 years ago. Indeed, using the distance $\sim 40$Mpc estimated from the GW170817 signal and the redshift of its host galaxy, the LIGO-Virgo team reported the Hubble parameter $H_{0}=70^{+12}_{-8}~{}{\rm km~{}sec^{-1}Mpc^{-1}}$ (Abbott et al. 2017c). Here the error bar represents 68.3% probability range.
The Hubble parameter is one of the most fundamental cosmological
parameters, since the discovery of the cosmic expansion in 1929. But
this parameter has attracted much attention quite recently. We have a
9% mismatch between the locally estimated value $73.24\pm 1.74~{}{\rm km~{}sec^{-1}Mpc^{-1}}$ and that determined from the cosmic microwave background $66.93\pm 0.62~{}{\rm km~{}sec^{-1}Mpc^{-1}}$ (Riess et al. 2016, Planck Collaboration 2016). This tension might be caused by an unidentified systematic error in the two types of measurements or might, in fact, imply a challenge to the standard cosmological model. In any case, the newly established method based on the DNSs could make a notable contribution to the Hubble tension.
In this paper, we discuss the prospects of gravitational-wave
observational cosmology with the forthcoming third observation run (O3)
of the LIGO-Virgo collaboration (Abbott et al. 2016) and its follow-on operations (see Zhao & Wen 2017 for the third generation detectors).
Our results would be also useful to discuss related topics such as the efforts to suppress the amplitude calibration error of the GW measurement (see e.g. Vitale et al. 2012, Tuyenbayev et al. 2017, Cahillane et al. 2017) or the observational strategy for the EM counterpart search (Cowperthwaite et al. 2017).
This paper is organized as follows. In §2, we discuss the kilonova signals and contaminations at the host galaxy identification, taking into account the actual observational results of GW170817. In §3, we derive analytical expressions to evaluate the expected number of DNS detections and the averaged distance error at the GW data analysis. Then, in §4, we discuss the prospects of the Hubble parameter measurement in the near future. §5 is a brief summary of this paper.
Following the standard convention, we assume the DNS masses at $1.4M_{\odot}+1.4M_{\odot}$ whose chirp mass is only $\sim 2\%$ different from that of GW170817.
2 kilonova signals for the host galaxy identification
The loudness of the optical sky always stands in the way to identify
kilonovae in followup observations (Cowperthwaite & Berger 2015, Tanaka
2016, Cowperthwaite et al. 2017). Here, we argue that a
good fraction $\gtrsim 50\%$ of kilonovae for DNS events within $\sim$200\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$$ can be identified by followup observations in optical
(and hopefully near-infrared) bands incorporating insights obtained from
observations of the kilonova associated with GW170817. Once an
electromagnetic counterpart such as the kilonova is successfully
identified, the host galaxy will be determined relatively easily,
because the expected cosmological redshift will at most be
0.05–0.1. Real-time identification is not necessary for our purpose,
i.e., determining the host galaxy, and accordingly we do not worry about
the lack of template images (Cowperthwaite & Berger 2015).
The key findings from the kilonova associated with GW170817 are
summarized as follows (see, e.g., McCully et al. 2017, Shappee et
al. 2017, Siebert et al. 2017, Utsumi et al. 2017). The emission becomes
bright right after the merger , say $\sim$1\text{\,}\mathrm{d}\mathrm{a}\mathrm{y}$$, peaking in blue
optical bands (Shappee et al. 2017). While the magnitude in optical
bands such as the g-band drops very rapidly (Siebert et
al. 2017), near-infrared bands sustain bright emission for a few to
several days, where longer emission is found in longer wavelengths
(Utsumi et al. 2017). Even in the relatively red z-band, the
emission becomes dim by $2.5\text{\,}\mathrm{m}\mathrm{a}\mathrm{g}$ in only 6 days (Utsumi et
al. 2017). The corresponding
change in the peak wavelength makes the color evolution exceedingly
rapid (McCully et al. 2017). The rapid decline of magnitudes and the
rapid reddening are both distinctive features of the kilonova (see also
Cowperthwaite & Berger 2015, Cowperthwaite et al. 2017). Furthermore,
the spectrum is mostly featureless and becomes red as early as a few
days after the merger (McCully et al. 2017, Siebert et al. 2017). Such
spectra are not observed for other known transients, and thus make the
kilonova very unique (Shappee et al. 2017). It should be worth noting
that GW170817 appears to be observed from relatively polar directions
(Abbott et al. 2017a).
Turning now to identification of future kilonovae. Fast optical
transients as significant contaminants are summarized comprehensively in
Cowperthwaite & Berger (2015). Among the fast transients, the type Ia
supernova outnumbers the kilonova at given brightness. Fortunately, they
will be easily distinguished due to their significantly slow time
evolution compared to that of kilonovae (see also Scolnic et
al. 2017). Elimination of type Ia supernovae could further benefit from
the line structures in the spectrum if it is taken and the presumably
high redshift. On the other extreme, stellar flares last less than a day
and will be eliminated by requiring detections in multiple
nights. Quiescent emission of the underlying star could be detected
later for further secure elimination.
Taking the estimated rate of fast transients, we expect that the type
.Ia supernova and so-called Pan-STARRS fast transients can serve as
significant contaminants (Cowperthwaite & Berger 2015). A remarkable
point is that the peak brightness of the kilonova associated with
GW170817 was found to be brighter by $\gtrsim$1.5\text{\,}\mathrm{m}\mathrm{a}\mathrm{g}$$ (Utsumi et al. 2017), or
equivalently by a factor of $\gtrsim 4$ than the model adopted in
Cowperthwaite & Berger (2015). This means that the number of fast transients that can
compete with kilonovae will be reduced by a factor of 8. Thus, the
number of type .Ia supernovae and Pan-STARRS fast transients will only
be $\approx 0.2$ and $\lesssim 1$, respectively, even for
$100\text{\,}\mathrm{d}\mathrm{e}\mathrm{g}^{2}$ sky localization. Furthermore, the rapid decline of the
kilonova associated with GW170817 is hardly reproduced by Pan-STARRS
fast transients. Thus, a large fraction of Pan-STARRS fast transients,
say $\gtrsim 80\%$, would be removed by requiring a moderate decline
rate that does not significantly remove blue kilonovae. Overall, we
expect that more than $\gtrsim 50\%$ of blue kilonovae can be identified
based on the brightness and decline rates.
The rapid reddening and/or red color, once detected, will serve as a
powerful tool to distinguish kilonovae from other transients as
previously thought (Cowperthwaite & Berger 2015, Tanaka 2016). Note
that the blueness of the kilonova associated with GW170817 does not
exclude existence of red kilonovae. Particularly, we still expect to
observe kilonovae without blue components for edge-on binaries, for
which lanthanide-enriched dynamical ejecta should be obscuring any blue
emission (Kasen, Fernández & Metzger 2015). A substantial fraction
of red kilonovae can be securely identified as electromagnetic
counterparts to DNS events further aided by rapid magnitude
evolution. Quantitatively, requiring $i-z\gtrsim$0.4\text{\,}\mathrm{m}\mathrm{a}\mathrm{g}$$ will
remove most of the contaminants, say $\gtrsim 90\%$, without discarding
red kilonovae significantly (Cowperthwaite & Berger 2015). The
requirement may be loosened to $\gtrsim$0\text{\,}\mathrm{m}\mathrm{a}\mathrm{g}$$ when we
additionally require the rise or decline time to be $\lesssim$4\text{\,}\mathrm{d}\mathrm{a}\mathrm{y}$$, but about a half of kilonovae may be lost for the cut
based on the decline (Cowperthwaite & Berger 2015).
To summarize, we expect that more than 50% of kilonovae may be
identified successfully by first seeking rapid blue components, and next
red components. In any case, properties of kilonovae discussed here rely
heavily on a single event GW170817 combined with theoretical knowledge,
and future detections of a variety of kilonovae in followup observations
to DNS events are crucially important to refine the selection
criteria. The fraction of identifiable kilonovae may be increased in the
near future by understanding their characteristics with actual
observations.
3 signal detection and analysis
In this section, we discuss analytical expressions for the detectable volume of DNSs and the appropriately averaged their distance estimation errors. We basically follow the formulation in Cutler & Flanagan (1994). In addition, using the method developed in Seto (2015), we perturbatively include the geometrical information of the detector network. We also count the differences in sensitivities among detectors, and derive expressions convenient for statistical study of the local Hubble parameter measurement.
3.1 detectable volume
The SNR $\rho_{i}$ of a detector $i$ can be written as
$$\rho_{i}^{2}=10\left(\frac{d_{h,i}}{D}\right)^{2}[F_{+,i}(\mbox{\boldmath${n}$%
},\psi)^{2}\frac{(1+v^{2})}{4}+F_{\times,i}(\mbox{\boldmath${n}$},\psi)^{2}v^{%
2}],$$
(1)
where $F_{+,i}$ and $F_{\times,i}$ are the beam pattern functions that depend on the source direction ${n}$ and the polarization angle $\psi$ (Thorne 1987). The quantity $d_{h,i}$ is the horizon distance for $\rho_{i}=10$.
We have the relation $d_{h,i}=2.26d_{r,i}$ with the detection range
$d_{r,i}$ for the same SNR (see e.g. Chen et al. 2017). In
eq.(1), $D$ is the (luminosity) distance to the binary and
$v$ is the cosine of its inclination angle with $|v|=1$ for face-on and $v=0$ for edge-on.
If the detector noises are uncorrelated, the total SNR $\rho$ of a detector network is given by
$$\rho^{2}=\sum_{i}\rho_{i}^{2},$$
(2)
Following Cutler & Flanagan (1994), we express $\rho^{2}$ in the form
$$\rho^{2}=\frac{\sigma(\mbox{\boldmath${n}$})}{D^{2}}[c_{0}(v)+\epsilon(\mbox{%
\boldmath${n}$})c_{1}(v)\cos 4\psi]$$
(3)
with $c_{0}(v)\equiv(1+v^{2})^{2}/4+v^{2}$ and $c_{1}(v)\equiv(1+v^{2})^{2}/4-v^{2}$.
Here the function $\sigma(\mbox{\boldmath${n}$})$ shows the total sensitivity of the network to GWs coming from the direction ${n}$. Meanwhile, the function $\epsilon(\mbox{\boldmath${n}$})$ represents the asymmetry of the sensitivities to two (appropriately decomposed) orthogonal polarization modes. We generally have $0\leq\epsilon(\mbox{\boldmath${n}$})\leq 1$. If the network is blind to one of the modes, we have $\epsilon(\mbox{\boldmath${n}$})=1$.
From eq.(3), for a given SNR threshold $\rho_{T}$, the maximum detectable distance $D_{max}$ is given by
$$D_{max}=\frac{1}{\rho_{T}}\sigma(\mbox{\boldmath${n}$})^{1/2}[c_{0}(v)+%
\epsilon(\mbox{\boldmath${n}$})c_{1}(v)\cos 4\psi]^{1/2},$$
(4)
and can be regarded as a function of the four angular parameters $(\mbox{\boldmath${n}$},\psi,v)$.
To simplify expressions below, we introduced the averaging operation with the angular parameters;
$$\int dA[\cdots]\equiv\frac{1}{4\pi}\int_{4\pi}d\mbox{\boldmath${n}$}~{}\frac{1%
}{2\pi}\int_{0}^{2\pi}d\psi~{}\frac{1}{2}\int_{-1}^{1}dv[\cdots].$$
(5)
Then the effective volume $V$ for the detection threshold $\rho_{T}$ is given as
$$V=\int dA\int_{0}^{D_{max}}dD4\pi D^{2}=\rho_{T}^{-3}U,$$
(6)
where we defined
$$U\equiv\frac{4\pi}{3}\rho_{T}^{3}\int dAD_{max}^{3}.$$
(7)
Using eq.(4), we formally have
$$U=\frac{4\pi}{3}\int dA\sigma(\mbox{\boldmath${n}$})^{3/2}[c_{0}(v)+\epsilon(%
\mbox{\boldmath${n}$})c_{1}(v)\cos 4\psi]^{3/2}.$$
(8)
We now evaluate this expression.
Note that, because of the power 3/2, the four-dimensional integrals $dA$ cannot be performed separately (Seto 2015). But we can overcome this difficulty by perturbatively expanding the term proportional to $\epsilon(\mbox{\boldmath${n}$})$ as follows
$$\displaystyle U$$
$$\displaystyle=$$
$$\displaystyle\frac{4\pi}{3}\int dA\sigma(\mbox{\boldmath${n}$})^{3/2}c_{0}(v)^%
{3/2}\Big{[}1+\frac{3}{2}\frac{\epsilon(\mbox{\boldmath${n}$})c_{1}(v)\cos 4%
\psi}{c_{0}(v)}$$
(9)
$$\displaystyle+\frac{3}{8}\left(\frac{\epsilon(\mbox{\boldmath${n}$})c_{1}(v)%
\cos 4\psi}{c_{0}(v)}\right)^{2}+\cdots\Big{]}.$$
After performing integrals separately, we obtain
$$\displaystyle U$$
$$\displaystyle=$$
$$\displaystyle\frac{4\pi}{3}g\times 0.821(1+0.01s_{2}+2.1\times 10^{-4}s_{4}+%
\cdots).$$
(10)
Here we used numerical results such as
$$\frac{1}{2}\int_{-1}^{1}dvc_{0}(v)^{3/2}=0.821,$$
(11)
and also defined
$$\displaystyle g$$
$$\displaystyle\equiv$$
$$\displaystyle\frac{1}{4\pi}\int_{4\pi}d\mbox{\boldmath${n}$}\sigma(\mbox{%
\boldmath${n}$})^{3/2},$$
(12)
$$\displaystyle s_{m}$$
$$\displaystyle\equiv$$
$$\displaystyle\frac{1}{4\pi g}\int_{4\pi}d\mbox{\boldmath${n}$}\sigma(\mbox{%
\boldmath${n}$})^{3/2}\epsilon(\mbox{\boldmath${n}$})^{m}$$
(13)
for even $m$.
For the quantity $U$, all the geometrical information of the network are contained in $g$ and $s_{m}$.
Since $0\leq\epsilon(\mbox{\boldmath${n}$})\leq 1$, we have the following inequalities for the integrals $s_{m}$
$$0\leq\cdots\leq s_{4}\leq s_{2}\leq 1.$$
(14)
Therefore, after dropping the corrections $\propto s_{m}$ in eq.(10), we get a good approximation
$$U\simeq 3.44g$$
(15)
with relative error less than 1% (Seto 2015, see also Schutz 2011).
For a network with a single detector $i$, we identically have $\epsilon(\mbox{\boldmath${n}$})=1$ and $s_{m}=1$. We also have
$$V=\frac{4\pi}{3}\left(\frac{10}{\rho_{i}}\right)^{3}d_{r,i}^{3}$$
(16)
because of the definition of the detection range $d_{r,i}$.
3.2 distance error
We assume that the source direction ${n}$ is accurately determined by
the sky position of the EM counterparts such as the kilonova. Then from the information related to the extrinsic properties of GWs, we need to simultaneously estimate just the four extrinsic parameters, $D$, $\psi$, $v$ and the initial phase of the wave.
From the Fisher matrix of these parameters (Cutler & Flanagan 1994), the variance of the relative distance error for a binary is given by
$$\left\langle\left(\frac{\Delta D}{D}\right)^{2}\right\rangle={4D^{2}}\frac{(1+%
v^{2})-\epsilon(\mbox{\boldmath${n}$})(1-v^{2})\cos 4\psi}{\sigma(\mbox{%
\boldmath${n}$})(1-\epsilon(\mbox{\boldmath${n}$})^{2})(1-v^{2})^{2}}.$$
(17)
This expression depends on the four angular parameters $(\mbox{\boldmath${n}$},\psi,v)$ as well as the distance $D$. Due to the singularity of the Fisher matrix, this expression diverges at $|v|\to 1$ (face-on) and overestimates the variance, compared with a more elaborate nonlinear estimation (see e.g. Nissanke et al. 2013, Rodriguez et al. 2014). In contrast, for the edge-on binaries, eq.(19) would be an underestimation, especially for low SNRs.
Assuming a homogeneous and isotropic binary distribution, we can derive
an averaged error in the relative distance $\sigma_{\rm lnD}$ for
binaries with $\rho>\rho_{T}$ as
$$\sigma_{\rm lnD}^{2}=\left[\frac{\int dA\int_{0}^{D_{max}}dD\left\langle\left(%
\frac{\Delta D}{D}\right)^{2}\right\rangle^{-1}4\pi D^{2}}{\int dA\int_{0}^{D_%
{max}}dD4\pi D^{2}}\right]^{-1}.$$
(18)
After the $dD$ integral, we have
$$\sigma_{\rm lnD}^{2}=\rho_{T}^{-2}\frac{U}{X}$$
(19)
where we defined
$$\displaystyle X$$
$$\displaystyle\equiv$$
$$\displaystyle\pi\int dA\sigma(\mbox{\boldmath${n}$})(1-\epsilon^{2})(1-v^{2})^%
{2}$$
(21)
$$\displaystyle\times[(1+v^{2})-\epsilon(\mbox{\boldmath${n}$})(1-v^{2})\cos 4%
\psi]^{-1}\int_{0}^{D_{max}}dD$$
$$\displaystyle=$$
$$\displaystyle\pi\int dA\sigma(\mbox{\boldmath${n}$})^{3/2}(1-v^{2})^{2}[c_{0}(%
v)+\epsilon(\mbox{\boldmath${n}$})c_{1}(v)\cos 4\psi]^{1/2}$$
$$\displaystyle\times(1-\epsilon^{2})[(1+v^{2})-\epsilon(\mbox{\boldmath${n}$})(%
1-v^{2})\cos 4\psi]^{-1}.$$
As shown in the $dD$ integral in eq.(21), $X$ is not dominated by
small $D(\ll D_{max})$. Therefore, for a sufficiently large number of
DNS events, the statistical fluctuations of our estimation $\sigma_{\rm lnD}$ would be small.
In the same manner as $U$ in the previous subsection, after expanding the relevant factors in $X$ and averaging with the four angular parameters, we get
$$X=g(0.966-0.574s_{2}-0.158s_{4}-0.068s_{6}-0.037s_{8}-\cdots).$$
(22)
This expression is our new result and would be useful for statistical discussion on the local Hubble parameter measurement.
4 Prospects of O3 and beyond
Based on the expressions derived in the previous section, we now discuss the prospects of the forthcoming observational runs.
4.1 observation plan
In Table 1, we summarize the actual results of the past two runs, O1 and O2, and the planned parameters for the future runs O3 and O4. Here we denote the 2020+run (in Abbott et al. 2016) simply by O4. In Table 1, the duration $T_{d}$ for O1 and O2 are the total time for the simultaneous operation of the two LIGO detectors (based on Abbott et al. 2017a). For O3 and O4, the observational time relevant for the Hubble parameter estimation should be
$$T_{obs}=f_{d}T_{d}$$
(23)
with the time fraction $f_{d}$ for the simultaneous operation of all the three detectors. The duration $T_{d}$ of O4 is not explicitly presented in Abbott et al. (2016).
In Table 1, the detection ranges $d_{r,i}$ are given for the threshold $\rho_{T}=10$ (in contrast to the conventional $\rho_{T}=8$). In the 6th column, we present the four-dimensional volume $VT_{d}$ using eq.(6) for $\rho_{T}=10$.
We also present $s_{2}$, $s_{4}$, and $\sqrt{U/X}$ required for the estimation of the relative distance error $\sigma_{\rm lnD}$.
At the stage 2024+ (Abbott et al .2016), KAGRA is planned to join the detector network with $d_{r,i}=112$Mpc, in addition of two LIGOs ($d_{r,i}=152$Mpc) and Virgo ($d_{r,i}=100$Mpc). For these four detectors, we have $V=68\times 10^{6}{\rm Mpc^{3}}$ and $\sqrt{U/X}=2.5$.
The DNS merger rate $R$ estimated after the GW170817 detection is (Abbott et al. 2017a)
$$R=(1.5^{+3.2}_{-1.2})\times 10^{-6}{\rm Mpc^{-3}yr^{-1}}$$
(24)
(90% probability range).
We hereafter denote $R=f_{R}R_{0}$ with the median value $R_{0}=1.5\times 10^{-6}{\rm Mpc^{-3}yr^{-1}}$ and the scaling parameter $f_{R}$.
Then the expected DNS events is given by
$$N=RV(f_{d}T_{d})=(f_{R}f_{d})R_{0}T_{d}U\rho_{T}^{-3}.$$
(25)
In reality, we will be able to identify the host galaxies for not all of the $N$ events. Therefore, we introduce the probability $f_{E}$ for the successful host galaxy identification, and
use the total DNS events $N_{E}=f_{E}N$ for estimation of the local Hubble parameter.
Here, for simplicity, we neglect the dependence of $f_{E}$ on the distance $D$ and the inclination $v$.
Using Table 1, we explicitly have
$$N_{E}=A(f_{R}f_{d}f_{E})\left(\frac{T_{d}}{\rm 1yr}\right)\left(\frac{\rho_{T}%
}{10}\right)^{-3}$$
(26)
with the numerical coefficients $A=32$ for O3 and 69 for O4.
In Table 2, we summarize the parameters that appear in this expression and are also useful for discussions below.
4.2 local Hubble parameter measurement
Next, we discuss the error in estimation of the local Hubble parameter
$H_{L}$. For each DNS (label $j=1,\cdots,N_{E}$) with identified host galaxies,
we can estimate the Hubble parameter
$$H_{j}=\frac{cz_{j}}{D_{j}},$$
(27)
using the measured redshift $z_{j}$ of the host galaxy and the distance $D_{j}$ from GW data analysis. But both of them contain errors $\delta z_{j}$ and $\delta D_{j}$. The former would be dominated by the local peculiar velocity $v_{j}$ as
$$\delta z_{j}\sim v_{j}/c,$$
(28)
and the latter $\delta D_{j}$ would be the parameter estimation error at GW data analysis.
Then we have
$$\frac{\delta H_{j}}{H_{L}}\simeq\frac{v_{j}}{cz_{j}}+\frac{\delta D_{j}}{D_{j}}.$$
(29)
From eq.(19) and Table 1, the magnitude of the relative distance error is roughly estimated as
$$\frac{\delta D_{j}}{D}\sim\frac{1}{\rho_{T}}\sqrt{\frac{U}{X}}\sim 0.3\left(%
\frac{\rho_{T}}{10}\right)^{-1}.$$
(30)
Meanwhile, given the typical one-dimensional velocity of galaxies $\sim 400\,{\rm km\,sec^{-1}}$ (Strauss & Willick 1995), we have ${v_{j}}/{cz_{j}}\sim 0.05$ for DNS distance $D_{j}\sim 100$ Mpc ($cz_{j}\sim 7000{\rm km~{}sec^{-1}}$).
Therefore, for individual DNS events, the error for the Hubble parameter is approximately given by
$$\frac{\delta H_{j}}{H_{L}}\simeq\frac{\delta D_{j}}{D_{j}}.$$
(31)
Statistically using totally $N_{E}$ DNSs, we have the estimation error for the local Hubble parameter
$$\frac{\Delta H_{L}}{H_{L}}\sim f_{F}\frac{\sigma_{\rm lnD}}{\sqrt{N}_{E}}%
\simeq\sqrt{\frac{U}{X}}\frac{f_{F}\rho_{T}^{1/2}}{(f_{d}f_{R}f_{E}R_{0}T_{d})%
^{1/2}}.$$
(32)
As mentioned earlier, the original expression (19) based on the Fisher matrix could be both over- and underestimate the variance, compared with a more elaborate nonlinear analysis. To include these mismatches, we introduced the correction factor $f_{F}$ in eq.(32). We should have $f_{F}\to 1$ in the limit $\rho_{T}\to\infty$.
4.3 case study for O3 and beyond
Using expressions in the previous subsections and assuming the planned
detector sensitivity in Table 1, we have eq. (26) for the
expected DNS events with identified host galaxies, and
$${\Delta H_{L}}/H_{L}\sim Bf_{F}(f_{R}f_{d}f_{E})^{-1/2}(\rho_{T}/10)^{1/2}(T_{%
d}/1{\rm yr})^{-1/2}$$
(33)
for the error of the local Hubble parameter. Here we have the numerical factors $(A,B)=(32,0.049)$ for O3 and (69,0.035) for O4.
These expressions still depend on the six parameters $(f_{R},f_{d},f_{E},f_{F},T_{d},\rho_{T})$.
Therefore, we evaluate eqs. (26) and (33) for the three cases below.
We commonly assume $T_{d}=1$yr for O3 and 2yr for O4, and also fix the SNR threshold at $\rho_{T}=10$. In reality, at the stage of O4, we can use the results of O3. By combining O3 and O4, the number $N_{E}$ would be 23% larger and the error ${\Delta H_{L}}/H_{L}$ would be $\sim 11\%$ smaller. But, for simplicity, we discuss the results O3 and O4 separately.
4.3.1 optimistic case.
We assume the higher end of the DNS merger rate $f_{R}=3.1$ and the high efficiencies $f_{E}=f_{d}=0.8$ with the correction factor $f_{F}=1.2$.
At the end of O3, we have $\Delta H_{L}/H_{L}\sim 0.042$ with $N_{E}\sim 63$.
Therefore, if the current mismatch $\sim 9\%$ between the local and global
Hubble parameters might be examined at $2.1\sigma$ level. Then, after O4, we will have $\Delta H_{L}/H_{L}=0.021$ with $N_{E}=274$.
The cosmic variance due to the coherence of the peculiar velocity field is estimated to be $\sim 1\%$ at the survey depth of $D\sim 200$Mpc (Shi & Turner 1998) and could become a potential concern at O4. Even for O3, there would be a strong motivation to suppress the amplitude calibration error much lower than $5\%$ (Tuyenbayev et al. 2017).
4.3.2 standard case
We assume the median value $f_{R}=1.0$ for the merger rate, and the efficiencies $f_{E}=f_{d}=0.6$ with the factor $f_{E}=1.3$. As easily understood from Eqs. (26) and (33), the factor $f_{R}$ is the major cause of the difference from the optimistic case above.
With O3, we have $\Delta H_{L}/H_{L}\sim 0.11$ with $N_{E}\sim 12$. Among the expected twelve events, the maximum SNR is roughly estimated to be $10\times 12^{1/3}\sim 23$, and smaller than that of GW170817. The error $\Delta H_{L}/H_{L}\sim 0.11$ is not so different form $0.15$ obtained from GW170817.
After O4, we have $\Delta H_{L}/H_{L}\sim 0.053$ with $N_{E}=50$, and can examine the tension at $2\sigma$ level.
4.3.3 pessimistic case
We assume the lower end of the rate $f_{R}=0.2$, and the efficiencies
$f_{E}=f_{d}=0.5$ with $f_{F}=1.4$. We have $N_{E}\sim 2.3$ for O3 and 10 for
O4. Even with O4, the error $\Delta H_{L}/H_{L}$ remains 0.15 and is
comparable to that from GW170817. If the time fraction $f_{d}$ is less
0.5, the error becomes even larger. We expect that GW observation is
not likely to play a critical role to examine the Hubble tension in
the next five years for this case.
5 summary
The GW170817 event clearly demonstrated that the DNSs could become ideal standard sirens accompanied by characteristic EM signals for host galaxy identification (Abbott et al. 2017a). The kilonova of GW170817 is genuinely unique in
terms of the rapid evolution of color and magnitude and we expect that, for a good fraction $\ga 50\%$ of the DNS events within $\sim 200$Mpc, we can identify their host galaxies, using their kilonovae. Therefore, the DNSs will become a powerful and reliable tool for observational cosmology.
Our immediate target would be the locally measured Hubble parameter that
currently has a 9% tension with the value obtained form the cosmic microwave background (Riess et al. 2016, Planck Collaboration 2016). Considering these circumstances, we discussed the prospects of the Hubble parameter measurement using DNSs observed during the forthcoming LIGO-Virgo observational runs (Abbott et al. 2016).
In order to evaluate the measurement error of the Hubble parameter estimated form multiple DNSs, we derived convenient expressions, and applied them for the three observational scenarios.
If the DNS merger rate is at the high end of the current constraint $\sim 3.7\times 10^{-6}{\rm Mpc^{-3}yr^{-1}}$ and the planed sensitivities are realized for LIGO and Virgo, we could attain the accuracy $\Delta H_{L}/H_{L}\sim 0.042$ with O3. Then, the Hubble tension might be verified at $2.1\sigma$ or we might indicate a potential systematic error for the traditional cosmoligical distance probes.
Also, this precision would give a strong motivation to suppress the amplitude calibration errors of the ground-based detectors. On the other hand, if the DNS merger rate is at the low end $\sim 0.2\times 10^{-6}{\rm Mpc^{-3}yr^{-1}}$, even with the 2020+ observation, it would be unlikely to go significantly beyond the level $\Delta H_{L}/H_{L}\sim 0.15$ already obtained by GW170817 whose $SNR=32.4$ is contrastingly at the high end tail.
In any case, additional DNS events with O3 would be indispensable to further constrain the DNS rate and also better understand the EM counterparts, especially the anisotropies of kilonova emissions.
Acknowledgments
We would like to than Masaomi Tanaka for helpful comments.
This work is supported by JSPS Kakenhi Grant-in-Aid for Scientific Research
(Nos. 15K65075, 16H06342, 17H01131, 17H06358).
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An Optimized and Safety-aware Maintenance Framework: A Case Study on Aircraft Engine
Muhammad Ziyad${}^{1}$, Kenrick Tjandra${}^{2}$, Zulvah${}^{3}$, Mushonnifun Faiz Sugihartanto${}^{4}$, Mansur Arief${}^{2}$
${}^{1}$Muhammad Ziyad is with Sustainable Production Development, KTH Royal Institute of Technology, Södertälje, Sweden${}^{2}$ Kenrick Tjandra and Mansur M. Arief are with the Department of Mechanical Engineering, Carnegie Mellon University, USA${}^{3}$Zulvah is with Aquaculture Industry, Tangerang, Indonesia${}^{4}$Mushonnifun Faiz Sugihartanto is with the Department of Business Management, Institut Teknologi Sepuluh Nopember, Surabaya, IndonesiaThe code for this paper is made available at https://github.com/mansurarief/optimized-safety-aware-aircraft-maintenance
Abstract
The COVID-19 pandemic has recently exacerbated the fierce competition in the transportation businesses. The airline industry took one of the biggest hits as the closure of international borders forced aircraft operators to suspend their international routes, keeping aircraft on the ground without generating revenues while at the same time still requiring adequate maintenance. To maintain their operational sustainability, finding a good balance between cost reductions measure and safety standards fulfillment, including its maintenance procedure, becomes critical. This paper proposes an AI-assisted predictive maintenance scheme that synthesizes prognostics modeling and simulation-based optimization to help airlines decide their optimal engine maintenance approach. The proposed method enables airlines to utilize their diagnostics measurements and operational settings to design a more customized maintenance strategy that takes engine operations conditions into account. Our numerical experiments on the proposed approach resulted in significant cost savings without compromising the safety standards. The experiments also show that maintenance strategies tailored to the failure mode and operational settings (that our framework enables) yield 13% more cost savings than generic optimal maintenance strategies. The generality of our proposed framework allows the extension to other intelligent, safety-critical transportation systems.
1 Introduction
Intelligent transportation systems have undeniably shown rapid development in recent years[1]. Its existence has benefited humans in multiple aspects of our daily lives, e.g. through the public bus tracking system, assisted parking technology, or highway collision avoidance system [2]. In the coming years, implementing more advanced intelligent transportation systems will put heavier reliance on these systems [3] and less intervention or dependency on human aspects, requiring each system to maintain its reliability, often through routine inspection and maintenance procedures. The heavily regulated air transportation industry serves as an excellent example of designing a safety-critical deployment for an intelligent transportation system [4].
Over the last few decades, airline industries have seen rapid growth [5]. Statistics show that pre-COVID-19 pandemic, the number of passengers traveling by airplanes has increased significantly, from around 1.95 billion people in 2004 to almost 4.6 billion in 2019 [6]. This huge market potential has attracted numerous airline companies, resulting in unprecedented fierce competition and forcing them to operate as efficiently as possible to gain competitiveness while maintaining compliance with safety standards and regulations and delivering quality services. Since 2020, COVID-19 has exacerbated the situation, causing the airline revenues to plummet to only 40% of the previous year’s value [7]. The significantly heavier pressure forced companies to find means to reduce their operational costs to stay viable in the market, including adopting artificial intelligence (AI) in assisting decision-making process data [8].
One of the components of operational costs that are often put under scrutiny for potential expenses reduction is maintenance [9]. The total aircraft maintenance costs contribute on average 10%-15% of the total operational costs [10], with a value reaching half of billion dollars. For instance, in 2019, the maintenance costs of full-service airlines such as Qatar Airways, Garuda Indonesia, KLM amounted to 10%, 12%, 14% of the total operating costs, respectively [11, 12, 13]. For low-cost airlines such as AirAsia Berhad, the percentage of maintenance cost is around 11% in 2019 [14].
If optimized, maintenance strategy could deliver appealing cost reductions [15]. However, too much inclination on cost reductions could increase the safety risk of passengers and crews, especially if the maintenance standard is compromised to the bottom margin of the minimum safety standard. Therefore, the decision-makers should be equipped with decision-making tools to weigh safety considerations properly.
Despite being among the easiest to implement, preventive maintenance strategy is prone to underestimation and overestimation without systematic assessment. Several drawbacks, for example, are (potentially unnecessarily) high maintenance cost, human error or excessive operational usage, which accelerate engine(s) deterioration, and longer maintenance time in total [16, 17]. To get ahead of these barriers, some manufacturers are moving toward more advanced prognosis and health management of the aircraft engine to predict the failure. The advent of sensor measurements provides relatively cheap and fast means to collect signals on the current airworthiness of the engines. The availability of these data enables the prognostics approach and has led to the advancement of more accurate prognostics based on data analytics, machine learning, and AI [18, 19, 20, 8].
In this paper, we leverage on data-driven nature of prognostics to make the first leap in building an AI-assisted framework for an optimized maintenance decision. Our goal is to make safe decisions and use sensor measurements to estimate the hazard score of aircraft engines over time, which serves as a proxy for the safety level of the engine. We then use this score to determine the airworthiness of the engines. Specifically, we use the Cox Proportional-Hazards regression model for our framework and employ simulations to account for the uncertainty of optimal threshold selection in minimizing the total maintenance costs (performance restoration costs for preventive approach or LLP replacement costs for reactive approach). This threshold can then be used as a criterion to assess the operability of an aircraft, striking a balance between preventive and reactive maintenance measures. Since the problem can be identified in advance, maintenance cost, lead time, and duration can be minimized, and assets utilization can be maximized.
Our contribution is threefold. First, we advance the use of prognostics for optimizing engine maintenance decisions under uncertainty. By synthesizing the power of proportional hazard model and versatility of simulation-based optimization method, using NASA Turbofan public dataset [21] we demonstrate an optimized maintenance strategy in the numerical experiments that attains 37%-50% cost reduction compared to a baseline model that could schedule maintenance either overly conservatively or extremely dangerously. Second, our framework allows the decision-makers to construct strategies based on their priority on safety and cost, an important feature to make cognizant maintenance decisions [22]. Finally, the data-driven nature of our approach enables a more directed maintenance program based on the characteristics of the engines, rather than a simple and generic one that often be used when lacking proper prognostics. We believe that the use of more advanced prognostics techniques that AI promises in the near future will further enhance the robustness of our framework and scale up its real-world applications, including connected and autonomous vehicles.
2 Related Works
In this section, we briefly review the concept of aircraft engine maintenance, data-driven prognostics approaches, and optimal maintenance models.
2.1 Aircraft Engine Maintenance
The maintenance approach for turbofan engines can be classified into two major categories (or the combinations thereof): scheduled (or preventive) maintenance and unscheduled (reactive) maintenance. Most scheduled maintenance approaches are based on the modules, and the LLP installed on the engines [23]. The scheduled maintenance for turbofan engine is further divided into engine modules performance restoration, Life Limited Parts (LLP) replacement [23], and engine re-delivery program [24]. Moreover, engine maintenance accounts for up to 80% of the total aircraft maintenance costs [23].
One popular LLP-based approach is the engine removal forecast. This method uses some parameters such as flight cycles or flight hours and advocates that the turbofan engine should be maintained or replaced in the workshop when the LLP reaches its limit. The scheduling often gets highly complicated as each engine consists of many different types of LLPs. Engine re-delivery program is another type of scheduled maintenance program. This approach aims to restore the turbofan engine condition to its best condition by repairing or replacing modules and LLP before being delivered back to the engine lessor. This type of maintenance occurs at the end of the engine operation/usage phase.
In contrast, reactive maintenance is unforeseen in nature. It could occur due to various reasons such as hard landing, ground damage, tail strikes, lightning strikes, bird strikes, or high engine temperature [25]. Due to the uncertainty of these events, it is difficult to predict their occurrence, the number of resources needed, and the costs to allocate. Frequently, the maintenance events are viewed as a combination of both scheduled and unscheduled maintenance.
The engine maintenance event can be defined based on the bathtub curve [23, 17]. Based on the best practices, the proportion of maintenance is subject to what stage the engine is in its life cycle. In the preliminary stage, the scheduled maintenance event is around 30%, and the unscheduled maintenance event is around 70% from the total event. After the engine reaches the mature stage, scheduled maintenance takes a higher proportion than the unscheduled maintenance event (around 70% for scheduled maintenance and 30% unscheduled) as its failure rate is lower in the mature stage. In the wear-out stage, the failure rate of the engine tends to increase as the engine age increases. The unscheduled maintenance occurs more frequently at this stage. Thus, the proportion of scheduled maintenance is around 30%, and the unscheduled maintenance is around 70% [23, 17].
By optimizing the maintenance event, the airlines could minimize their maintenance cost and maintain their customers’ safety level.
2.2 Data-driven Prognostics Approach and Survival Analysis
Prognostics approaches have led to the advancement of predictive maintenance strategies using data analytics and machine learning techniques prevalent success in minimizing the risk of failure and improve efficiencies in many areas including healthcare and engineering [26, 27]. At a high level, prognostics is an effort to predict the remaining lifetime or the failure event using the signals obtained from the measurements. At its core, prognostics uses historical data to find the interrelation between parameters, extract the essential pattern and predict the future state of the system under study [28, 29, 30, 31, 32, 33]. Recent work uses a more complex AI system to improve the prediction accuracy [8]. While many of these works have been able to predict the remaining useful life of the engines with sufficient accuracy, using the predictions to formulate an optimal maintenance decision remains a challenge, especially when we are interested in accounting for the prediction error.
A more straightforward prognostics approach is to estimate the airworthiness of the engine based on sensor measurements to model the failure events. A useful technique is called survival analysis. Survival analysis deals with problems with a significant emphasis on the time until the occurrence of a particular event (classical examples include death, breakdowns, failures, etc., hence the term “survival”) [34].
2.3 Optimal Maintenance Models
Maintenance decisions have been among the classical optimization techniques applications with tremendous success history. The literature on the use of mathematical modelling to analyze, plan, and optimize maintenance operations in various industrial contexts is abundant [35, 36, 37]. The approaches are mainly classified into two: fixed-time and condition-based models, with a good review provided in [38, 39]. We are mainly interested in the later models. Many approach employs linear programming formulations to identify the optimal solutions [40, 41]. The strength of these workers includes the characterization of the solutions, which often offers valuable insights on how to formulate even higher-level strategies for the companies. However, these approaches are designed based on a prior understanding of the nature of the failures, which could be expensive or even impractical to obtain in real life. Hence, we start with a simulation-based approach to optimize our hazard threshold to be as generic as possible. This choice also allows us to quantify and output the uncertainty of our approach, which can be useful to account for by the decision-makers.
3 Metholodogy
Our framework mainly consists of two components: engine hazard estimation using Cox Proportional-Hazards model and hazard threshold simulation
3.1 Hazard Estimation via Cox-Proportional Hazards Model
In this work, we adopt the Cox Proportional-Hazards model, which is a semi-parametric approach to determine the effect of covariates $x_{t}\in\mathbb{R}^{d}$ on the survival at time $t$. The hazard function for this model is given by
$$h(t|x_{t})=h_{0}(t)\hat{h}(x_{t}),$$
(1)
which consists of time-varying baseline hazard rate $h_{0}(t)$ and partial hazard $\hat{h}(x_{t})$ defined as
$$\hat{h}(x_{t})=\exp\left(\beta^{\intercal}(x_{t}-\bar{x}_{t})\right),$$
(2)
that signifies the effect of the covariates $x_{t}$. Here $\bar{x}_{t}$ denotes the mean of $x_{t}$ (computed during training). The parameters $\beta\in\mathbb{R}^{d}$ and the baseline hazard $h_{0}(t)$ is fitted using $\mathtt{lifelines}$ package [42].
3.2 Hazard Threshold Optimization via Simulation
The survival analysis allows us to estimate the current hazard level $\log\hat{h}(x_{t})$ if we can collect sensor measurements $x_{t}$ at time $t$. With this hazard level estimate as a proxy, we can set a threshold as a decision criterion, say $\lambda$, to determine the airworthiness of the aircraft engines at time $t$. A straightforward binary decision $y_{t}(\lambda)\in\{0,1\}$ at time $t$ (where 0 means no maintenance required and 1 means otherwise) could therefore be
$$y_{t}(\lambda)=\begin{cases}1,&\text{if }\log\hat{h}(x_{t})<\lambda,\\
0,&\text{otherwise}.\end{cases}$$
(3)
This means we allow the aircraft to fly at time $t$ if the preflight hazard level $\log\hat{h}(x_{t})$ is below our threshold $\lambda$. Otherwise, we cancel the flight and perform maintenance.
We note, however, that our estimate can be very noisy thus needs post-processing. We use a set of training data $\mathcal{D}_{\text{train}}=\{x_{[1:t_{i}]},t_{i}\}_{i=1}^{n}$ that contains $n$ engines to optimize $\lambda$. For engine $i$, we assume to have $t_{i}$ measurements $x_{[1:t_{i}]}=[x_{1},x_{2},\cdots,x_{t_{i}}]$, i.e. the sensor measurements from time $t=1$ to $t=t_{i}$ after which the engine fails. We use this dataset to simulate and compute the optimal threshold $\lambda^{*}$ that balances safety and cost.
For simplicity, suppose that from each flight, the airline incurs $C_{1}$ engine performance restoration cost if performing preventive maintenance and $C_{2}>C_{1}$ engine LLP replacement cost if the engine fails (all assumed a fixed constant). Then, for a threshold $\lambda$, the total maintenance cost can be written as
$$\displaystyle\text{TotalCost}(\lambda)$$
$$\displaystyle=\sum_{i=1}^{n}C_{1}\left(\max_{t=1,\cdots,t_{i}}\{\log\hat{h}(x_{t})\}\geq\lambda\right)$$
$$\displaystyle+\sum_{i=1}^{n}C_{2}\left(\max_{t=1,\cdots,t_{i}}\{\log\hat{h}(x_{t})\}<\lambda\right).$$
(4)
The first term in the RHS of (4) shows that restoration cost is incurred if the hazard score is above the threshold. The second term highlights that replacement cost is applied if no maintenance is done because the engine eventually fails at time $t_{i}$. The optimal threshold $\lambda^{*}$ is obtained by minimizing the total maintenance cost
$$\lambda^{*}=\arg\min_{\lambda}\text{TotalCost}(\lambda).$$
(5)
We then use the optimized $\lambda^{*}$ and evaluate the resulting maintenance strategies using data in the test set $\mathcal{D}_{\text{test}}$.
In addition to the total maintenance cost, we also output the estimated probability of engine failures given $\lambda$:
$$\text{FailureProb}(\lambda)=\frac{1}{n}\sum_{i=1}^{n}\mathbb{I}\left(\max_{t=1,\cdots,t_{i}}\{\log\hat{h}(x_{t})\}<\lambda\right),$$
(6)
where $\mathbb{I}(\cdot)$ is an indicator function that outputs 1 if the supplied argument is true and outputs 0 otherwise. Since this probability describes the notion of safety, we need to account for the uncertainty in our estimation.
To do this, we employ a simulation-based optimization approach. Instead of using the whole $n$ samples, we sample with replacement $n_{0}<n$ engines for $k$ replications and compute the means and standard deviations of the total cost (4) and failure probability (6). The approach helps provide additional robustness to the framework, especially when the estimated hazard scores are very noisy. Finally, we note that while we use constant values for $C_{1}$ and $C_{2}$ across time and engines, the approach can directly be extended for when these values are variable.
4 Numerical Experiments
We use NASA Turbofan engine datasets in our experiment. The datasets are provided by NASA Ames Research Center. These datasets are simulation results generated using simulation software known as Commercial Modular Aero-Propulsion System Simulation (C-MAPSS). C-MAPSS simulates turbofan engines model with the engine specification of 90,000 lb thrust class, operating altitudes between sea level to 40,000 ft, Mach number ranging from 0 to 0.90, and sea-level temperature ranging from -60${}^{\circ}$ to 103${}^{\circ}$ F. Besides, the power management systems can be operated over a wide range of thrust levels across a full range of flight [21].
The datasets are divided into the training set and the test set. There are six combinations of operating conditions that consider the altitude, TRA, and Mach number. The fault modes considered include single fault mode (HPC fault) and multiple fault modes (HPC and fan fault). In total, there are 708 data points (combined), which are then partitioned into four subsets (FD001, FD002, FD003, and FD004).
The turbofan engine datasets were used to construct a model that can minimize the maintenance cost while maintaining the company’s safety level . In the experiment, we use the following parameters: performance restoration cost $C_{1}=3.5\times 10^{6}$, LLP replacement cost $C_{2}=4\times 10^{6}$, number of sampled engines $n_{0}=30$, and simulation replications $k=10$. We enumerate a range of $\lambda$ values and simulate the system using the training set to obtain $\lambda^{*}$.
We first implement the proposed method to obtain the optimized threshold for all engines (combined dataset), giving us the generic maintenance strategy for all operating conditions. The costs for various $\lambda$ values are summarized in Fig. 1 while the probabilities are shown in Fig. 2. The colored lines in these figures represent the means for the costs and failure probability, while the shadows represent their one-sigma confidence intervals, i.e. the interval $[\text{mean}-\text{std},\text{mean}+\text{std}]$. Our simulations show that this choice of $\lambda^{*}$ gives about $0.28$ probability of engine failures, with simulated hazard trajectories shown in Fig. 3.
Finally, we implement the proposed approach to all four partitions of the datasets (FD001, FD002, FD003, and FD004). We then compare the costs and safety (in terms of probability of engine failures) for the optimized threshold. The results are summarized in Fig. 4 and 5.
5 Discussion
In this section, we discuss our findings. First, Fig. 1 shows that the optimal cost for engine maintenance for the combined turbofan engine dataset is achieved with $\lambda^{*}=9.0$. When the hazard score (log-partial hazard) exceeds this threshold, the engines are likely to fail due to HPC or fan faults. If it happens, the failure event would then accrue more maintenance costs to the airlines, leading to higher LLP replacement costs. The optimal maintenance cost for the combined turbofan dataset estimated at around US$ 76 Million. In this case, we can reduce the maintenance costs 37% under the optimized maintenance strategy.
Next, we highlight in Fig. 2 the failure probability on different failure threshold. While $\lambda^{*}=9.0$ indeed minimize the cost, the decision-makers could adjust this threshold to account for his safety priority. For instance, if the current engine failure probability ($\text{FailureProbability}(\lambda^{*})=0.28$) is deemed too high, thus too risky, then a lower $\lambda$ values could be chosen. From Fig. 3, one shall see that doing so will increase maintenance frequency, thus more engines will be restored, and less will be left broken and replaced since the threshold for the hazard will be lower. This setting, in turn, will increase safety (but comes with higher maintenance costs accordingly). This sort of analysis is one of our approach’s advantages: it empowers decision-makers with quantitative measures to weigh safety and efficiency considerations carefully.
In Fig. 4, we show the optimal cost for engine maintenance for FD001, FD002, FD003, and FD004 turbofan engine dataset. The optimal thresholds w.r.t. minimum maintenance costs are attained at 17.5, 10, 22, and 9 for FD001, FD002, FD003, and FD004, respectively. From the figure, we see that in the case when $\lambda<\lambda^{*}$, performance restoration costs dominate the maintenance cost. In this case, maintenance is performed too excessively. On the contrary, when $\lambda>\lambda^{*}$, the replacement costs dominate, suggesting that maintenance is too rarely scheduled, causing far too many engines to be broken and have to be replaced. The optimized threshold $\lambda^{*}$ balances the two costs. In these cases, we see that the maintenance costs can be reduced by 40%-50% under the optimized maintenance strategy.
In Fig. 5, we show the probability of engine failures under various $\lambda$ values for FD001, FD002, FD003, and FD004 turbofan engine dataset. The optimal thresholds yield “profit-optimized” failure probability of 0.28, 0.15, 0.22, and 0.55 for FD001, FD002, FD003, and FD004 respectively. One might argue that these failure probabilities are perceivably high, especially for FD004, in which the simulations suggest that the estimated hazard scores are very noisy, suggesting risky airline operations. This situation poses significantly more challenges in determining the optimal threshold (partially captured by the wider confidence intervals for the total cost of FD004 in Fig. 4), leading to more than half of the engines failing and thus being replaced. Hence, one could adopt a more safety-aware decision, shifting $\lambda^{*}$ slightly lower. Although such adaptation leads to higher costs, it provides an additional buffer against noisy estimations and safer maintenance decisions.
Finally, we note that the cost reductions of using a more directed strategy (using clustered datasets, like FD001, FD002, FD003, or FD004) provide higher cost reductions (about 13% more) compared to a generic strategy (i.e. using a combined dataset). The data-driven approach allows a more targeted strategy depending on the situations described by the data. However, we note that in the real world scenario, it might be difficult for an airline to classify the engine based on the failure mode and operating conditions a priori. Therefore, a generic strategy that can cover all engines could be initially implemented while the operators look for ways to identify the engine classifications or clusters based on their failure mode or operational settings.
6 Conclusion
In this work, we propose an optimized maintenance strategy developed by combining prognostics and simulation-based optimization. The proposed framework uses the Cox Proportional-Hazards regression model to estimate the hazard score of an aircraft engine and simulation to optimize the hazard threshold, minimizing the total maintenance cost while maintaining a certain safety level. Compared to the generic optimal maintenance method, our proposed work yields 13% cost reductions. The method also benefits decision-makers, enabling them to design a more directed maintenance strategy if the datasets are clustered based on some characteristics and conditions of the engines. More advanced machine learning methods and AI applications will extend the applicability of the proposed framework to deal with more complex maintenance strategies.
Acknowledgment
The authors would like to thank the GMF AeroAsia team for providing valuable discussions and insights about maintenance approaches in airline industries. The first author is extremely grateful for the funding provided by the Swedish Institute to study at KTH Royal Institute of Technology.
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A Polyhedral Proof of the Matrix Tree Theorem
Aaron Dall
and
Julian Pfeifle
Departament de Matemàtica Aplicada II,
Universitat Politècnica de Catalunya,
Jordi Girona 1-3,
E-08034 Barcelona
[aaron.dall,julian.pfeifle]@upc.edu
(Date:: 15 April 2014)
Abstract.
The classical matrix tree theorem relates the number of spanning trees of a connected graph with the product of the nonzero eigenvalues of its Laplacian matrix.
The class of regular matroids generalizes that of graphical matroids, and a generalization of the matrix tree theorem holds for this wider class.
We give a new, geometric proof of this fact by showing via a dissect-and-rearrange argument that two combinatorially distinct zonotopes associated to a regular matroid have the same volume.
Along the way we prove that for a regular oriented matroid represented by a unimodular matrix, the lattice spanned by its cocircuits coincides with the lattice spanned by the rows of the representation matrix.
Key words and phrases:Regular matroid, unimodular matrix, cocircuit lattice, zonotope
2010 Mathematics Subject Classification: Primary 52B40; Secondary 52C40.
Both authors were partially supported by the project MINECO MTM2012-30951/FEDER. The first author received additional support from the MCINN grant BES-2010-030080. The second author received additional support from grants EUI-EURC-2011-4306, MTM 2011-24097 and 2009-SGR-1040.
Introduction
The matrix tree theorem is a classical result in algebraic graph theory that relates the number of spanning trees of a connected graph $G$ with the product of the nonzero eigenvalues of the Laplacian matrix of $G$.
Theorem 1 (Kirchoff [5]).
Let $G$ be a connected graph on $n$ vertices with $s$ spanning trees and whose Laplacian L has nonzero eigenvalues $\lambda_{1},\dots,\lambda_{n-1}$. Then
$$\prod_{i\in[n-1]}\lambda_{i}=ns.$$
A classical proof of this theorem proceeds as follows (see [1], [4]).
First one shows that every principal minor of L is equal to the sum of the squares of the maximal minors of the signed vertex-edge incidence matrix of $G$.
Then one computes that such a minor is equal to $\pm 1$ if the edges of $G$ corresponding to the columns of the submatrix span a tree.
Finally, one verifies that the maximal principal minor is precisely $1/n$ times the product of the nonzero eigenvalues of L using the characteristic polynomial.
Many generalizations of the theorem exist including for weighted graphs, simplicial complexes, and regular matroids.
In the last case, the regular matroid matrix tree theorem is the following particular case of Theorem 3 in [6].
Theorem 2.
Let $\mathcal{M}$ be a rank $d$ regular matroid represented by a $d\times n$ unimodular matrix M of full rank, and let ${\textsf{L}}={\textsf{M}}{\textsf{M}}^{\top}$.
Then the number of bases of $\mathcal{M}$ is $\lambda_{1}\cdots\lambda_{d}$, where $\lambda_{1},\dots,\lambda_{d}$ are the eigenvalues of L .
In this paper we recast this result into the domain of polyhedral geometry by considering the zonotopes generated by the columns of the matrices M and L and proving that the volumes of these two zonotopes are the same although their combinatorial structures are in general vastly different.
Theorem 3.
Let M be a unimodular matrix of full rank, and ${\textsf{L}}={\textsf{M}}{\textsf{M}}^{\top}$.
Then the volume of the zonotope ${\textsf{Z}}({\textsf{M}})$, the volume of the zonotope ${\textsf{Z}}({\textsf{L}})$, and the product of the eigenvalues of L are all equal.
When M has full row rank, then so does L and so the zonotope ${\textsf{Z}}({\textsf{L}})$ is a $d$-dimensional parallelepiped and L has $d$ real nonzero eigenvalues.
It follows that the volume of ${\textsf{Z}}({\textsf{L}})$ is exactly the determinant of L, which in turn is the product of the eigenvalues of L.
This shows that the last two quantities in Theorem 3 are equal, and so the crucial part of the proof is to show that the zonotopes have the same volume.
After some preliminary results are presented in Section 1, we will prove Theorem 3 in Section 2 via a novel dissect-and-rearrange argument. Finally, in Section 3 we give a new polyhedral proof of the classical Matrix Tree Theorem that, while similar to the general proof for full rank matrices in the previous section, copes with the fact that the defining matrices M and L do not have full rank. The classical proof of the matrix tree theorem involves matrix calculations that rely on the total unimodularity of the signed vertex-edge incidence matrix of a graph $G$, i.e., that one has a totally unimodular representation of the matroid $\mathcal{M}(G)$. Our polyhedral approach works even when the representation of $\mathcal{M}(G)$ is only unimodular.
1. Preliminaries
1.1. Matrices
Let $M$ be an $m\times n$ matrix.
For $i\in[m]$ and $j\in[n]$ we write $m_{ij}$ for the entry in the $i^{\text{th}}$ row and $j^{\text{th}}$ column of $M$ and $M_{j}$ for the $j^{\text{th}}$ column of $M$.
Throughout this paper M will denote a $d\times n$ integer matrix of rank $r$.
The matrix M is unimodular if all of its maximal minors are in $\{-1,0,1\}$ and is totally unimodular if all of its minors are in $\{-1,0,1\}$.
We denote by ${\textsf{L}}={\textsf{L}}({\textsf{M}})$ the $d\times d$ symmetric matrix defined by ${\textsf{L}}:={\textsf{M}}{\textsf{M}}^{\top}$ (where ${\textsf{M}}^{\top}$ denotes the transpose of M).
As L is symmetric, its eigenvalues $\lambda_{1},\dots,\lambda_{d}$ are all real.
For any ring $R$ we denote the set of all $R$-combinations of the columns of $M$ by ${{}_{R}}{\left\langle M\right\rangle}$.
Note that when $R=\mathbb{Z}$ and M is a unimodular matrix we have ${{}_{\mathbb{Z}}}{\left\langle{\textsf{M}}\right\rangle}=\mathbb{Z}^{m}\cap%
\operatorname{im}{\textsf{M}}$.
We note here that in this paper we reserve the term lattice for a free discrete subgroup of a vector space.
1.2. Matroids
We now give the pertinent definitions and facts about matroids and oriented matroids, essentially following [7] and [3], respectively.
A matroid $\mathcal{M}=(E,\mathcal{I})$ is an ordered pair consisting of a ground set $E$ and a collection $\mathcal{I}$ of subsets of $E$ that satisfy the following independent set axioms:
I1
$\emptyset\in\mathcal{I}$;
I2
$\mathcal{I}$ is closed with respect to taking subsets; and
I3
if $I_{1},I_{2}\in\mathcal{I}$ with $|I_{1}|\leq|I_{2}|$, then there is some $e\in I_{2}\setminus I_{1}$ such that $I_{1}\cup\{e\}\in\mathcal{I}$.
The bases $\mathcal{B}$ of a matroid $\mathcal{M}$ is the subset of $\mathcal{I}$ consisting of independent sets of maximal size.
Clearly, the sets $\mathcal{B}$ and $\mathcal{I}$ of a matroid determine each other.
Let $M$ be an $m\times n$ matrix with entries in a field $\mathbb{F}$.
Then the archetypal example of an independent set matroid is $\mathcal{M}=([n],\mathcal{I}(M))$ where $[n]:=\{1,\dots,n\}$ is an indexing set for the columns of $M$ and $\mathcal{I}(M)$ consists of all subsets of (indices of) columns of $M$ that are linearly independent in the $m$-dimensional vector space $V(m,\mathbb{F})$ over $\mathbb{F}$.
In this case we write $\mathcal{M}=\mathcal{M}(M)$.
A matroid $\mathcal{M}$ is called $\mathbb{F}$-representable if there exists a matrix $M$ with entries in $\mathbb{F}$ such that $\mathcal{M}=\mathcal{M}(M)$.
It is immediate that $B\in\mathcal{B}$ is a basis of an $\mathbb{F}$-representable matroid $\mathcal{M}(M)$ if and only if the collection $\{M_{i}\nonscript\;{\left.\kern-1.2pt\vphantom{}\middle|\right.\kern-1.2pt}%
\nonscript\;i\in B\}$ is a basis for $V(m,\mathbb{F})$.
Let $\mathcal{M}=([n],\mathcal{I})$ be an arbitrary matroid.
A subset $C\subset[n]$ is a circuit of $\mathcal{M}$ if $C$ is a minimal dependent set.
The set of circuits of $\mathcal{M}$ is denoted $\mathcal{C}=\mathcal{C}(\mathcal{M})$.
A matroid $\mathcal{M}$ is connected if for every pair of elements $e\neq f$ in $E$ there is a circuit $C\in\mathcal{C}$ containing both.
Given a matroid $\mathcal{M}=([n],\mathcal{I})$ with bases $\mathcal{B}$, let $\mathcal{B}^{*}=\{E-B\nonscript\;{\left.\kern-1.2pt\vphantom{}\middle|\right.%
\kern-1.2pt}\nonscript\;B\in\mathcal{B}\}$ and let $\mathcal{I}^{*}$ be the collection of all subsets of elements of $\mathcal{B}^{*}$.
Then the matroid $\mathcal{M}^{*}:=([n],\mathcal{I}^{*})$ is the dual matroid of $\mathcal{M}$.
The sets $\mathcal{B}^{*},\mathcal{I}^{*},\mathcal{C}^{*}$ of bases, independent sets, and circuits of the dual $\mathcal{M}^{*}$ are called, respectively, the cobases, coindependent sets and cocircuits of $\mathcal{M}$.
Let $\mathcal{C}=\{C_{1},\dots,C_{m}\}$ be the circuits of $\mathcal{M}$.
Then the circuit incidence matrix of $\mathcal{M}$ is the $m\times n$ matrix $A(\mathcal{C})$ with entry $a_{ij}$ equal to 1 if $j$ is in $C_{i}$ and equal to 0 otherwise.
The cocircuit incidence matrix $A(\mathcal{C}^{*})$ is defined analogously.
The matroid $\mathcal{M}$ is orientable if one replace some of the nonzero entries of $A^{\prime}(\mathcal{C})$ and $A^{\prime}(\mathcal{C}^{*})$ by $-1$ such that, if a circuit $C$ and cocircuit $C^{*}$ have nonempty intersection, then both of the sets $\{i\in[n]\nonscript\;{\left.\kern-1.2pt\vphantom{}\middle|\right.\kern-1.2pt}%
\nonscript\;\mathbf{a}_{C}(i)=\mathbf{a}_{C^{*}}(i)\neq 0\}$ and $\{i\in[n]\nonscript\;{\left.\kern-1.2pt\vphantom{}\middle|\right.\kern-1.2pt}%
\nonscript\;\mathbf{a}_{C}(i)=-\mathbf{a}_{C^{*}}(i)\neq 0\}$ are nonempty.
In this paper we will work with the class of regular matroids characterized in the following theorem. (See Lemma 2.2.21, Theorem 6.6.3, and Corollary 13.4.6 in [7].)
Theorem 4.
For a matroid $\mathcal{M}$ the following are equivalent:
(1)
$\mathcal{M}$ is regular;
(2)
$\mathcal{M}$ is $\mathbb{F}_{2}$-representable and orientable;
(3)
$\mathcal{M}$ is representable over $\mathbb{R}$ by a unimodular matrix;
(4)
$\mathcal{M}$ is representable over $\mathbb{R}$ by a totally unimodular matrix; and
(5)
the dual of $\mathcal{M}$ is regular.
Moreover, if $\mathcal{M}$ is regular, M is a totally unimodular matrix that represents $\mathcal{M}$ over $\mathbb{R}$ and $\mathbb{F}$ is any other field, then M is an $\mathbb{F}$-representation of $\mathcal{M}$ when viewed as a matrix over $\mathbb{F}$.
We now turn briefly to oriented matroids that are representable over $\mathbb{R}$.
Let $\mathcal{A}=\{\mathbf{a}_{1},\dots,\mathbf{a}_{n}\}\subset\mathbb{R}^{d}$ be a vector configuration that spans the vector space $\mathbb{R}^{d}$.
Then the covectors of the oriented matroid $\mathcal{M}(\mathcal{A})$ are the elements of the set
$$\displaystyle\mathcal{V}^{*}:$$
$$\displaystyle=\{(\operatorname{sign}f(\mathbf{a}_{1}),\dots,\operatorname{sign%
}f(\mathbf{a}_{n}))\nonscript\;{\left.\kern-1.2pt\vphantom{}\middle|\right.%
\kern-1.2pt}\nonscript\;f:\mathbb{R}^{d}\to\mathbb{R}\text{ linear functional}\}$$
$$\displaystyle\subseteq\{-1,0,1\}^{n}.$$
The cocircuits of the oriented matroid $\mathcal{M}(\mathcal{A})$ are the minimal elements of the poset $(\mathcal{V}^{*},\prec)$, where the relation $\prec$ is defined by extending $0\prec\pm 1$ component-wise.
In the next subsection, we recall how this poset is related to the face poset of the zonotope generated by the matrix $M$ whose columns are the $\mathbf{a}_{i}$.
Here we conclude by recalling how to retrieve the covectors of the oriented matroid $\mathcal{M}(\mathcal{A})$ from a certain subspace arrangement in $\mathbb{R}^{n}$ and how to obtain the underlying unoriented matroid $\underline{\mathcal{M}}(\mathcal{A})$.
Let $M$ be the matrix whose columns are $\mathbf{a}_{1},\dots,\mathbf{a}_{n}$.
Then the covectors of $\mathcal{M}(\mathcal{A})$ can be read off from the hyperplane arrangement induced by the coordinate hyperplanes of $\mathbb{R}^{n}$ in the rowspace of $M$.
To see this first consider the $n$ columns of $M$ as elements of the dual vector space $(\mathbb{R}^{d})^{*}$.
Then each $\mathbf{a}_{i}$ defines a hyperplane in $\mathbb{R}^{d}$ given by $\mathcal{H}_{i}:=\{x\in\mathbb{R}^{d}\nonscript\;{\left.\kern-1.2pt\vphantom{}%
\middle|\right.\kern-1.2pt}\nonscript\;\left\langle\mathbf{a}_{i},x\right%
\rangle=0\}$ for $i\in[n]$.
Defining $\mathcal{H}_{i}^{+}:=\{x\in\mathbb{R}^{d}\nonscript\;{\left.\kern-1.2pt%
\vphantom{}\middle|\right.\kern-1.2pt}\nonscript\;\left\langle\mathbf{a}_{i},x%
\right\rangle>0\}$ and $\mathcal{H}_{i}^{-}=\mathbb{R}^{d}\setminus(\mathcal{H}_{i}\cup\mathcal{H}_{i}%
^{+})$, assign to each $x\in\mathbb{R}^{d}$ a sign vector $\sigma(x)\in\{-1,0,1\}^{n}$ whose $i^{\text{th}}$ coordinate is $1$ (respectively, $0,-1$) if $x$ is in $\mathcal{H}_{i}^{+}$ (respectively, $\mathcal{H}_{i},\mathcal{H}_{i}^{-}$).
The set of all points in $\mathbb{R}^{d}$ that receive the same sign vector $\sigma$ form a relatively open topological cell (which we label with $\sigma$) and the union of all such cells is $\mathbb{R}^{d}$.
The sign vectors that occur are precisely the covectors of the oriented matroid $\mathcal{M}(\mathcal{A})$, and the sign vectors that label $1$-dimensional cells are the cocircuits.
Now consider the subspace arrangement in the rowspace of $M$ induced by the coordinate hyperplane arrangement in $\mathbb{R}^{n}$ (oriented in the natural way), which we denote by $\mathscr{H}(M)$.
A point $y$ in the rowspace of $M$ satisfies $y_{i}=0$ (respectively, $y_{i}>0,y_{i}<0$) if and only if any point $x\in\mathbb{R}^{d}$ with $y=M^{\top}\!x$ lies on the hyperplane $\mathcal{H}_{i}$ (respectively, in $\mathcal{H}_{i}^{+},\mathcal{H}_{i}^{-}$) as defined in the previous paragraph.
So the oriented matroid coming from the hyperplane arrangement in the rowspace of $M$ induced by the coordinate hyperplanes in $\mathbb{R}^{n}$ is exactly $\mathcal{M}(\mathcal{A})$.
The preceding discussion tells us that the rowspace of $M$ intersects exactly those cells of the coordinate hyperplane arrangement labeled by the covectors of $\mathcal{M}(M)$.
It should be noted, however, that in general the covectors themselves do not lie in the rowspace of $M$ even when $M$ is a totally unimodular matrix (see the upcoming Remark 11).
We show in Theorem 10 that, when M is a unimodular matrix, every cocircuit of the oriented matroid $\mathcal{M}({\textsf{M}})$ does lie in the rowspace of M, and in fact, the set of cocircuits is a spanning set for the lattice ${{}_{\mathbb{Z}}}\!\left\langle{\textsf{M}}^{\top}\right\rangle$.
Finally, to each oriented matroid $\mathcal{M}$ one associates the underlying unoriented matroid $\underline{\mathcal{M}}$ whose cocircuits are obtained from the cocircuits of $\mathcal{M}$ by forgetting signs, i.e., if $C^{*}$ is a cocircuit of $\mathcal{M}$, then $\underline{C}^{*}$ is a cocircuit of $\underline{\mathcal{M}}(M)$ where $(\underline{C}^{*})_{i}=|(C^{*})_{i}|$.
An oriented matroid is regular if its underlying unoriented matroid is.
Many statistics of an orientable matroid (e.g., the number of bases or the number of independent sets) remain invariant after orientation, and so when discussing these properties with respect to a given matroid $\mathcal{M}(M)$, we often disregard the difference between the oriented matroid and the underlying unoriented matroid when no confusion can arise.
1.3. Zonotopes
A zonotope is a Minkowski sum of a finite number of line segments.
For an $m\times n$ matrix $M$ the zonotope generated by $M$, denoted ${\textsf{Z}}(M)$, is the Minkowski sum of the line segments $\operatorname{conv}\{\boldsymbol{0},M_{i}\}$.
If $M$ has rank $d$ then the zonotope ${\textsf{Z}}(M)$ is a $d$-dimensional convex polytope that is centrally symmetric about its barycenter.
We write ${\textsf{Z}}_{0}(M)$ for the translated copy of ${\textsf{Z}}(M)$ whose barycenter is at the origin, i.e., the Minkowski sum $\sum_{i\in[n]}S_{i}$ where $S_{i}=\operatorname{conv}\{-\frac{1}{2}{\textsf{M}}_{i},\frac{1}{2}{\textsf{M}%
}_{i}\}$.
A parallelepiped is half open if it is it the Minkowski sum of half-open line segments. The next result, due to Stanley, gives a decomposition of a zonotope into half-open parallelepipeds of various dimensions.
Theorem 5 ([10] Lemma 2.1).
Let $M$ be a rank $d$ matrix and $\mathcal{I}$ be the independent sets of the matroid $\mathcal{M}(M)$.
Then the zonotope ${\textsf{Z}}(M)$ is the disjoint union of half-open parallelepipeds
$$\Pi_{I}:=\left\{\sum_{i\in I}\alpha_{i}\widetilde{M_{i}}\nonscript\;{\left.%
\kern-1.2pt\vphantom{}\middle|\right.\kern-1.2pt}\nonscript\;\alpha_{i}\in[0,1%
)\right\},$$
where $\widetilde{M_{i}}$ is either $M_{i}$ or $-M_{i}$.
The parallelepipeds of maximal dimension in the above theorem are generated by maximal independent subsets of the columns of $M$.
As their union covers ${\textsf{Z}}(M)$ up to a set of measure zero, it follows that the volume of the zonotope is the sum of the volumes of the parallelepipeds generated by the bases of $\mathcal{M}(M)$.
For a fixed basis $B\in\mathcal{B}(M)$, the volume of $\Pi_{B}$ is simply the absolute value of the determinant of (the matrix whose columns are elements of) $B$.
When $M$ is unimodular each of these determinants is
$\pm 1$ and so we have the following corollary:
Corollary 6.
The volume of a zonotope generated by a unimodular matrix $M$ is equal to the number of bases in the regular matroid $\mathcal{M}(M)$, i.e., $\operatorname{vol}({\textsf{Z}}(M))=|\mathcal{B}(M)|$.
A zonotope ${\textsf{Z}}(M)\subset\mathbb{R}^{d}$ generated by a representation $M$ of a regular matroid $\mathcal{M}$ can be used to tile its affine span.
More precisely, a polytope $P\subset\mathbb{R}^{d}$ is said to tile its affine span $S$ if there is a polyhedral subdivision of $S$ whose maximal cells are translates of $P$.
The next theorem, due to Shepard [9], tells us that a zonotope tiles its affine span exactly when the underlying matroid is regular.
Theorem 7.
A zonotope ${\textsf{Z}}(M)$ tiles its affine span if and only if the matroid $\mathcal{M}(M)$ is regular.
Note that in the above theorem $M$ is not required to be unimodular but only a representation over $\mathbb{R}$ of a regular matroid.
This distinction will become important later on when we discuss the space-tiling properties of the zonotope generated by the Laplacian of a connected graph which, though not itself a unimodular matrix, is nevertheless a representation of the regular matroid $\mathcal{M}({\textsf{N}}^{\top})$, where N is the signed vertex-edge incidence matrix of the graph.
To conclude this subsection we fulfill our promise from the previous one and give the relationship between covectors of an oriented matroid $\mathcal{M}(M)$ (for an arbitrary matrix $M$) and the faces of the zonotope ${\textsf{Z}}_{0}(M)$.
Let $P:=(\mathcal{V}^{*},\prec)$ be the poset of covectors of $\mathcal{M}(M)$ where $\prec$ is the component-wise extension of $0\prec\pm 1$, and let $\mathcal{F}$ be the poset whose elements are the faces of ${\textsf{Z}}_{0}(M)$ ordered by inclusion.
Then $P$ is anti-isomorphic to $\mathcal{F}$ as witnessed by the order-reversing bijection that sends a covector $\mathbf{v}=(v_{1},\dots,v_{n})$ to the face
$$F_{\mathbf{v}}=\sum_{i:v_{i}=1}\tfrac{1}{2}M_{i}-\sum_{i:v_{i}=-1}\tfrac{1}{2}%
M_{i}+\sum_{i:v_{i}=0}S_{i},$$
where $S_{i}=\operatorname{conv}\left\{-\frac{1}{2}M_{i},\frac{1}{2}M_{i}\right\}$.
Note that the facets of ${\textsf{Z}}_{0}(M)$ correspond to the cocircuits of the oriented matroid $\mathcal{M}(M)$.
Now consider the barycenters $\pm\beta_{1},\dots,\pm\beta_{r}$ of the facets $\pm F_{1},\dots,\pm F_{r}$ of ${\textsf{Z}}_{0}(M)$.
If $\mathcal{C}^{*}_{i}$ is the cocircuit corresponding to the facet $F_{i}$, then it is clear from the above expression that $\beta_{i}=\frac{1}{2}M\mathcal{C}^{*}_{i}$.
For the formulation of Corollary 12 below, it turns out to be more appropriate to work with the scaled barycenter matrix $B=B(M)$ whose columns are the $\beta_{i}$ scaled by a factor of $2$.
1.4. Lattices
Before turning to the proof of the main result in the next section, we review the necessary terminology and results from lattice theory.
Our notation follows [12] and all proofs can be found either there or in [11].
Let $\mathcal{L}\subset\mathbb{R}^{n}$ be a lattice, i.e., a free discrete subgroup of $\mathbb{R}^{n}$.
Then $\mathcal{L}$ gives rise to the oriented matroid $\mathcal{M}(\mathcal{L})$ whose covectors are $\mathcal{V^{*}}=\{\operatorname{sign}(\mathbf{v})\nonscript\;{\left.\kern-1.2%
pt\vphantom{}\middle|\right.\kern-1.2pt}\nonscript\;\mathbf{v}\in\mathcal{L}\}.$
The support of a vector $\mathbf{v}\in\mathcal{L}$ is the set $\mathrm{supp}\>(\mathbf{v})=\{i\in[n]\nonscript\;{\left.\kern-1.2pt\vphantom{}%
\middle|\right.\kern-1.2pt}\nonscript\;\mathbf{v}_{i}\neq 0\}$.
A nonzero vector $\mathbf{v}\in\mathcal{L}$ is elementary if its coordinates lie in $\{-1,0,1\}$ and it has minimal support in $\mathcal{L}\setminus{\boldsymbol{0}}$.
Two vectors in $\mathcal{L}$ are conformal if their component-wise product is in $\mathbb{R}^{n}_{\geq 0}$.
A zonotopal lattice is a pair $(\mathcal{L},\left\langle\cdot,\cdot\right\rangle)$ where $\mathcal{L}\subset\mathbb{Z}^{n}$ is a lattice, $\left\langle\cdot,\cdot\right\rangle$ is an inner product on $\mathbb{R}^{n}$ such that the canonical basis vectors are pairwise orthogonal, and for every $\mathbf{v}\in\mathcal{L}\setminus\{\boldsymbol{0}\}$ there is an elementary vector $\mathbf{u}\in\mathcal{L}$ such that $\mathrm{supp}\>(\mathbf{u})\subseteq\mathrm{supp}\>(\mathbf{v})$.
The next proposition (Lemma 3.2 in [12]) tells us that zonotopal lattices are generated by the cocircuits of their oriented matroids in an especially nice way.
Proposition 8.
The elementary vectors of a zonotopal lattice $\mathcal{L}$ are exactly the cocircuits of the oriented matroid $\mathcal{M}(\mathcal{L})$.
Moreover, every vector $\mathbf{v}\in\mathcal{L}$ is the sum of pairwise conformal elementary vectors, and if the support of $\mathbf{v}$ equals the support of some elementary vector $\mathbf{u}$, then $\mathbf{v}$ is a scalar multiple of $\mathbf{u}$.
As noted in Remark 4.2 of [12], the oriented matroid of a zonotopal lattice is regular.
Historically this was taken as the definition of a regular matroid (see Section 1.2 of [11]).
We reestablish this connection and give a modern proof for the fact that, for a regular oriented matroid $\mathcal{M}({\textsf{M}})$ with cocircuits $\mathcal{C}^{*}$ and M unimodular, the lattices generated by M and $\mathcal{C}^{*}$ coincide (see Theorem 10).
2. Proof of the Polyhedral Matroid Matrix Tree Theorem
Let $\mathcal{M}$ be a regular rank $d$ matroid.
If M is a unimodular representation of $\mathcal{M}$, then by Corollary 6 the volume of the zonotope generated by M is equal to the number of bases of $\mathcal{M}$, $\operatorname{vol}({\textsf{Z}}({\textsf{M}}))=|\mathcal{B}(\mathcal{M})|$.
When M has full-row rank then so does the square matrix L, and so the zonotope ${\textsf{Z}}({\textsf{L}})$ is a parallelepiped with volume $\det({\textsf{L}})=\lambda_{1}\cdots\lambda_{d}$, where the $\lambda_{i}$ are the eigenvalues of L.
Using row operations that preserve unimodularity and then deleting any rows of zeros, any unimodular represention M of $\mathcal{M}$ can be transformed into a full row-rank unimodular representation of $\mathcal{M}$, so without loss of generality we may assume M is a full rank unimodular representation of $\mathcal{M}$.
The proof of Theorem 3 will be complete once we show that the zonotopes ${\textsf{Z}}({\textsf{M}})$ and ${\textsf{Z}}({\textsf{L}})$ have the same volume.
Remark 9.
When M has nontrivial corank (as is the case, for example, when ${\textsf{M}}={\textsf{N}}(G)$ is the signed incidence matrix of a graph), the zonotope ${\textsf{Z}}({\textsf{L}})$ is no longer a parallelepiped.
This means that some care must be taken when showing that the volume of ${\textsf{Z}}({\textsf{L}})$ is the product of its nonzero eigenvalues.
We sweep this detail under the rug in this section for ease of exposition, dealing with it in detail in the next section where we use our techniques to prove the graphical matrix tree theorem.
Our first goal is to see that, when M is a unimodular representation of a regular matroid, the lattices generated by L and the scaled barycenter matrix $B$ coincide.
(Note that we do not require M to have full rank nor to be totally unimodular.)
This fact is an immediate corollary of the following theorem.
Theorem 10.
Let $\mathcal{M}$ be a regular oriented matroid on $n$ elements and M be a unimodular matrix representing $\mathcal{M}$ over $\mathbb{R}$. Then the lattices ${{}_{\mathbb{Z}}}\!\left\langle{\textsf{M}}^{\top}\right\rangle$ and ${{}_{\mathbb{Z}}}\!\left\langle\mathcal{C}^{*}\right\rangle$, generated by the columns of ${\textsf{M}}^{\top}$ and by the cocircuits of $\mathcal{M}$, respectively, coincide.
Proof.
Recall that the subspace arrangement $\mathscr{H}=\mathscr{H}({\textsf{M}})\subset\mathbb{R}^{n}$ is obtained by intersecting the rowspace of M with the coordinate hyperplane arrangement in $\mathbb{R}^{n}$.
Clearly, the closure of any cell of $\mathscr{H}$ is the positive hull of the rays of $\mathscr{H}$ it contains and the sign vector of a cell is conformal to each of the rays contained in its closure.
By the discussion in Section 1.2, the cocircuits of $\mathcal{M}({\textsf{M}})$ are the sign vectors that label the rays of this arrangement.
Let $\rho$ be such a ray, labeled with the sign vector $\sigma$.
We claim $\rho=\operatorname{posHull}(\sigma)$.
Consider the polytope $\rho\cap[-1,1]^{n}$.
The equations for the rowspace of M are given by the kernel of M and it follows from Theorem 4 that one can find a unimodular basis for $\ker{\textsf{M}}$ (see [2] Lemma 2.10 for details when M is a full-rank totally unimodular matrix).
Thus the line segment $\rho\cap[-1,1]^{n}$ is the intersection of hyperplanes and halfspaces whose normal vectors can be viewed as the rows of a unimodular matrix.
Moreover, the equations and inequalities of the segment all have integer (in fact $\{0,\pm 1\}$) right-hand sides.
Thus, by Theorem 19.2 in [8], we obtain that $\rho\cap[-1,1]^{n}=\operatorname{conv}\{\boldsymbol{0},\mathbf{v}\}$ is a lattice segment with $\mathbf{v}\in\{-1,0,1\}^{n}$ .
But then $\mathbf{v}=\operatorname{sign}(\mathbf{v})=\sigma$, and so $\rho=\operatorname{posHull}(\sigma)$.
In particular, every cocircuit $C^{*}$ of $\mathcal{M}({\textsf{M}})$ is in the rowspace of M, and hence ${{}_{\mathbb{Z}}}\!\left\langle\mathcal{C}^{*}\right\rangle\subseteq{{}_{%
\mathbb{Z}}}\!\left\langle{\textsf{M}}\right\rangle$.
For the opposite inclusion, let $\mathbf{w}\in{{}_{\mathbb{Z}}}\!\left\langle{\textsf{M}}\right\rangle$ with sign vector $\sigma_{\mathbf{w}}$.
Then, as the cell of $\mathscr{H}$ labelled with $\sigma_{\mathbf{w}}$ is the positive hull of the rays it contains and the labels on these rays have minimal support, for any such ray $\rho$ we have $\mathrm{supp}\>(\rho)\subseteq\mathrm{supp}\>(\sigma_{\mathbf{w}})$.
It follows immediately that ${{}_{\mathbb{Z}}}\!\left\langle{\textsf{M}}\right\rangle$ together with the standard inner product on $\mathbb{R}^{n}$ is a zonotopal lattice.
Moreover, the elementary vectors of ${{}_{\mathbb{Z}}}\!\left\langle{\textsf{M}}\right\rangle$ are those $\{-1,0,1\}$-vectors in the rowspace of M that have minimal support, i.e., lie on a ray of the arrangement induced by the coordinate hyperplane arrangement.
It follows that the elementary vectors of ${{}_{\mathbb{Z}}}\!\left\langle{\textsf{M}}\right\rangle$ are the cocircuits of $\mathcal{M}({\textsf{M}})$ and, since elementary vectors of a zonotopal lattice span the lattice by Proposition 8, the theorem follows.
∎
Remark 11.
As we already mentioned, the covectors of $\mathcal{M}$ do not always lie in the rowspace of M. Consider the totally unimodular matrix
$${\textsf{N}}_{K_{3}}\ =\ \begin{pmatrix}-1&-1&0\\
1&0&-1\\
0&1&1\end{pmatrix}.$$
By Theorem 10, the rowspace of ${\textsf{N}}_{K_{3}}$ has a basis of cocircuits, for example
$$\mathcal{C}^{*}\ =\ \begin{pmatrix}1&1&0\\
0&1&1\end{pmatrix}.$$
The lattice point $(1,2,1)$ lies in the rowspace of $\mathcal{C}^{*}$. However, taking signs yields the covector $(1,1,1)$ of $\mathcal{M}$, which does not lie in the rowspace of $\mathcal{C}^{*}$.∎
Recall that for an arbitrary matrix $M$, the columns of the scaled barycenter matrix $B=B(M)$ are the barycenters $\beta_{i}=\frac{1}{2}M\mathcal{C}^{*}_{i}$ of ${\textsf{Z}}_{0}(M)$, scaled by $2$.
Corollary 12.
Let $\mathcal{M}$ be a regular oriented matroid on $n$ elements and M be a unimodular matrix representing $\mathcal{M}$ over $\mathbb{R}$.
Then the lattices generated by the columns of L and the columns of $B$ are equal.
Proof.
Theorem 10 tells us that the lattices generated respectively by ${\textsf{M}}^{\top}$ and $\mathcal{C}^{*}$ coincide, and therefore so do their images ${\textsf{L}}={\textsf{M}}{\textsf{M}}^{\top}$ and $B={\textsf{M}}\mathcal{C}^{*}$ under M.
∎
We now use the fact that the columns of L are a basis for $\mathbb{R}^{d}$ to define a subdivision of ${\textsf{Z}}_{0}({\textsf{M}})$.
For each sign vector $\epsilon\in\{+,-\}^{d}$ we define the following objects:
•
the simplicial cone $\sigma_{\epsilon}:=\operatorname{posHull}\{\epsilon_{i}{\textsf{L}}_{i}%
\nonscript\;{\left.\kern-1.2pt\vphantom{}\middle|\right.\kern-1.2pt}\nonscript%
\;i\in[d]\}$ (see
0a);
•
the vector $v_{\epsilon^{-}}:=\sum_{i:\,\epsilon_{i}=-}{\textsf{L}}_{i}$;
•
the polytope $P_{\epsilon}:=\sigma_{\epsilon}\cap{\textsf{Z}}_{0}({\textsf{M}})$ (see Figure 0b);
•
the polytope $Q_{\epsilon}:=P_{\epsilon}+v_{\epsilon^{-}}$ (see Figure 0c).
Example.
Consider the path on three vertices with edges oriented so that $i\rightarrow j$ if $i<j$.
A full-rank representation for the independent set matroid $\mathcal{M}({\textsf{N}})$ of the signed incidence matrix of this graph is given by the matrix on the left below, while the corresponding Laplacian is the matrix on the right:
$${\textsf{M}}=\begin{pmatrix}-1&0\\
1&-1\end{pmatrix}\hskip 44.0pt{\textsf{L}}=\begin{pmatrix}1&-1\\
-1&2\end{pmatrix}.$$
Both of the zonotopes ${\textsf{Z}}({\textsf{M}})$ and ${\textsf{Z}}({\textsf{L}})$ are two dimensional parallelepipeds and Figure 1 illustrates the families $\sigma_{\epsilon},P_{\epsilon}$, and $Q_{\epsilon}$ as $\epsilon$ varies over all sign vectors for this example after a suitable coordinate transformation.
Note that the zonotope of the Laplacian is the parallelepiped in the positive quadrant shaded dark grey. ∎
Clearly the union of the $P_{\epsilon}$ over all sign vectors is the zonotope ${\textsf{Z}}_{0}({\textsf{M}})$ and the intersection of any two of them is a face of both.
We now prove Theorem 3 by showing that the union of the $Q_{\epsilon}$ is in fact ${\textsf{Z}}({\textsf{L}})$ and that any two $Q_{\epsilon}$ intersect in a set of measure zero.
Theorem 3.
Let $\mathcal{M}$ be a regular oriented matroid on $n$ elements, let M be a unimodular matrix representing $\mathcal{M}$ over $\mathbb{R}$, and put ${\textsf{L}}={\textsf{M}}{\textsf{M}}^{\top}$.
Then the volume of the zonotope ${\textsf{Z}}({\textsf{L}})$ equals the volume of the zonotope ${\textsf{Z}}({\textsf{M}})$.
Proof.
By Corollary 12, the line segment $\operatorname{conv}\{\boldsymbol{0},{\textsf{L}}_{i}\}$ intersects some proper face of ${\textsf{Z}}_{0}({\textsf{M}})$ and the point of intersection is the barycenter of both.
In particular, the distance between any two points of ${\textsf{Z}}_{0}({\textsf{M}})$ in the direction parallel to ${\textsf{L}}_{i}$ is less than or equal to $||{\textsf{L}}_{i}||$, with equality if and only if the points lie in opposite faces of ${\textsf{Z}}_{0}({\textsf{M}})$ intersected by the line ${}_{\mathbb{R}}\langle{\textsf{L}}_{i}\rangle$.
First we show that $\bigcup_{\epsilon}Q_{\epsilon}\subseteq{\textsf{Z}}({\textsf{L}})$.
Let $H_{1}=\langle{\textsf{L}}_{i}\nonscript\;{\left.\kern-1.2pt\vphantom{}\middle|%
\right.\kern-1.2pt}\nonscript\;i\in\{2,\dots,d\}\rangle_{\mathbb{R}}$ be the hyperplane spanned by all columns of L except for ${\textsf{L}}_{1}$ and let $H^{+}_{1}$ be the open halfspace bounded by $H_{1}$ and containing ${\textsf{L}}_{1}$.
For $p\in{\textsf{Z}}_{0}({\textsf{M}})$ define $\mathcal{L}_{1,p}:=p+\langle{\textsf{L}}_{1}\rangle$ to be the line through $p$ parallel to ${\textsf{L}}_{1}$ and let $q_{1}=\mathcal{L}_{1,p}\cap H_{1}$ (see Figure 2).
Since the width of ${\textsf{Z}}_{0}({\textsf{M}})$ parallel to ${\textsf{L}}_{i}$ is at most $||{\textsf{L}}_{i}||$, it follows that $p=\sum\alpha_{i}{\textsf{L}}_{i}$ for some unique set of $\alpha_{i}$ with $|\alpha_{i}|\leq 1$.
For example, when $i=1$ we have $||p-q_{1}||\leq||{\textsf{L}}_{1}||$ and,
as $p-q_{1}$ is parallel to ${\textsf{L}}_{1}$ by construction, it follows that $p-q_{1}=\alpha_{1}{\textsf{L}}_{1}$ where
$$\alpha_{1}:=\pm\frac{||p-q_{1}||}{||{\textsf{L}}_{1}||}$$
is positive (respectively negative, zero) if and only if $p\in H^{+}_{1}$ (respectively $p\in H_{1}^{-}$, $p\in H_{1}$).
Given $p=\sum\alpha_{i}{\textsf{L}}_{i}$, define the sign vector $\epsilon$ by
$$\epsilon_{i}=\begin{cases}\operatorname{sign}(\alpha_{i})&\text{if $\alpha_{i}%
\neq 0$}\\
+&\text{else}.\end{cases}$$
Then each $\delta_{k}$ in the expression
$$p+v_{\epsilon}=\sum_{i\in[d]}\alpha_{i}{\textsf{L}}_{i}+\sum_{j:\,\epsilon_{j}%
=-}{\textsf{L}}_{j}=\sum_{k}\delta_{k}{\textsf{L}}_{k}$$
is in $[0,1]$ and it follows that $Q_{\epsilon}\subseteq{\textsf{Z}}({\textsf{L}})$ (see Figure 3).
Now we prove ${\textsf{Z}}({\textsf{L}})\subseteq\bigcup_{\epsilon}Q_{\epsilon}$.
Let $q=\sum_{i\in[d]}\gamma_{i}{\textsf{L}}_{i}\in{\textsf{Z}}({\textsf{L}})$, so that $\gamma_{i}\in[0,1]$ for all $i$ by definition.
Since facet-to-facet shifts of ${\textsf{Z}}_{0}({\textsf{M}})$ tile the column space of M, the point $q$ lies in some translate of ${\textsf{Z}}_{0}({\textsf{M}})$.
Since to pass from one tile to a neighboring one through a facet is to add some vector $w$ in ${}_{\mathbb{Z}}\langle B\rangle={}_{\mathbb{Z}}\langle{\textsf{L}}\rangle$ (Corollary 12), we have $q\in{\textsf{Z}}_{0}({\textsf{M}})+\sum_{i\in[d]}a_{i}{\textsf{L}}_{i}$,
where the $a_{i}\in\mathbb{Z}$, so that
$$q=\sum_{i\in[d]}\alpha_{i}{\textsf{L}}_{i}+\sum_{i\in[d]}a_{i}{\textsf{L}}_{i}$$
with $\alpha_{i}\in[-1,1]$.
Moreover, all $a_{i}\geq 0$ because $q$ lies in the positive hull of the $L_{i}$’s.
Comparing coefficients in the two expressions for $q$ and using that the ${\textsf{L}}_{i}$ form a basis yields $\alpha_{i}+a_{i}=\gamma_{i}.$
Since $a_{i}$ is a nonnegative integer and $\gamma_{i}\in[0,1]$, we have $a_{i}\in\{0,1\}$ (notice that the degenerate case $\alpha_{i}=-1$ and $\gamma_{i}=1$, in which case $a_{i}$ would equal $2$, cannot occur), and
$$a_{i}=\begin{cases}1\text{ if }\alpha_{i}\in[-1,0)\text{ and }\\
0\text{ if }\alpha_{i}\in(0,1].\end{cases}$$
Let $\epsilon$ be the sign vector defined by $\epsilon_{i}=-$ (respectively $+$) if $a_{i}=1$ (respectively, $0$).
Then $q\in Q_{\epsilon}$ and hence ${\textsf{Z}}({\textsf{L}})\subseteq\bigcup_{\epsilon}Q_{\epsilon}$.
Finally, we show that for any two sign vectors $\epsilon,\epsilon^{\prime}$ the intersection of the relative interiors of $Q_{\epsilon}$ and $Q_{\epsilon^{\prime}}$ is empty.
Let $\phi:\bigcup(\operatorname{rel~{}int}P_{\epsilon})\to\bigcup(\operatorname{rel%
~{}int}Q_{\epsilon})$ be the map that sends $\operatorname{rel~{}int}P_{\epsilon}\to\operatorname{rel~{}int}Q_{\epsilon}$.
There are two points $p\neq p^{\prime}\in{\textsf{Z}}_{0}({\textsf{M}})$ with $\phi(p)=\phi(p^{\prime})=:q$ if and only if
$$q\in({\textsf{Z}}_{0}({\textsf{M}})+v_{\epsilon})\cap({\textsf{Z}}_{0}({%
\textsf{M}})+v_{\epsilon^{\prime}})$$
for two sign vectors $\epsilon$ and $\epsilon^{\prime}$.
So $q$ lies on the boundary of both translates of ${\textsf{Z}}_{0}({\textsf{M}})$.
But then $p$ and $p^{\prime}$ both lie on the boundary of ${\textsf{Z}}_{0}({\textsf{M}})$ which contradicts the fact that they were in the relative interior of their respective cells.
Thus $\phi$ is a bijective map onto
$${\textsf{Z}}({\textsf{L}})\setminus\left(\partial{\textsf{Z}}({\textsf{L}})%
\cup\bigcup_{\epsilon}\partial Q_{\epsilon}\right).$$
So we have produced a volume-preserving bijection between ${\textsf{Z}}({\textsf{L}})$ and ${\textsf{Z}}({\textsf{M}})$ (up to a set of measure zero), which completes the proof.
∎
3. The Graphical Case
Let $G=([n],E)$ be a connected graph on $n$ vertices with signed vertex-edge incidence matrix N and Laplacian L.
The rank of N (and hence of L) is equal to the maximal size of a linearly independent subset of the columns of N.
This is exactly the number of edges in a spanning tree of $G$, i.e., $\operatorname{rank}{\textsf{N}}=\operatorname{rank}{\textsf{L}}=n-1$.
It follows that $0$ is an eigenvalue of L of multiplicity $1$, and it is easy to check that the all-ones vector $\boldsymbol{1}_{n}$ is a corresponding eigenvector.
So the zonotope ${\textsf{Z}}({\textsf{L}})$ is no longer a parallelepiped and its volume is no longer obtained by computing the determinant of L, as was the case in the previous section.
Nonetheless, we now modify our techniques from the previous section to obtain a polyhedral proof of the classical matrix tree theorem.
Recall from the introduction that the original formulation for the matrix tree theorem states that, for $G$ and ${\textsf{L}}={\textsf{N}}{\textsf{N}}^{\top}$ as in the previous paragraph, $n$ times the number of spanning trees is equal to the product of the nonzero eigenvalues of L.
The classical proof of this version of the matrix tree theorem proceeds in three steps.
First one uses the fact that $0$ is an eigenvalue of L of multiplicity 1 with corresponding eigenvector $\boldsymbol{1}_{n}$ to show that all $n$ of the maximal principal minors of L are equal and that the coefficient $c_{1}$ on the linear term of the characteristic polynomial of L is equal to $n$ times any maximal principal minor.
Then one uses the Cauchy-Binet theorem and the total unimodularity of N to prove that each of these minors equals the number of spanning trees of $G$.
Finally the theorem follows from the observation that, since L is symmetric and $0$ is an eigenvalue of multiplicity $1$, the characteristic polynomial of L factors over $\mathbb{R}$ and hence the coefficient $c_{1}$ is the product of the nonzero eigenvalues of L.
Our polyhedral proof of the matrix tree theorem follows a similar tack.
First we show in Proposition 13 that the zonotope ${\textsf{Z}}({\textsf{L}})$ decomposes into $n$ parallelepipeds all having the same volume.
Then we explain how results from the previous sections show that the volume of one (and hence any) of these parallelepipeds is equal to the number of spanning trees of $G$.
Finally we show that the volume of ${\textsf{Z}}({\textsf{L}})$ is the product of the nonzero eigenvalues of L as follows: First we construct two full-dimensional zonotopes, one having $d$-dimensional volume equal to $n$ times the $(d-1)$-dimensional volume of ${\textsf{Z}}({\textsf{L}})$ and the other having volume equal to $n$ times the product of the nonzero eigenvalues of L.
Then we show that these two zonotopes have the same volume using a proof technique reminiscent of that used to prove Theorem 3.
Moreover, we prove these results in greater generality whenever possible.
Our first goal is to see how the factor of $n$ in the Matrix Tree Theorem manifests itself in the polyhedral set-up, the idea being that the zonotope of the Laplacian of $G$ is the union of $n$ zonotopes all having the same volume.
We formalize this in the following result which holds in the more general case that the matrix M is only unimodular, i.e., it has all maximal minors in $\{-1,0,1\}$.
Proposition 13.
Let M be a unimodular matrix and let ${\textsf{L}}={\textsf{M}}{\textsf{M}}^{\top}$.
Then the zonotope ${\textsf{Z}}({\textsf{L}})$ decomposes into $|\mathcal{B}({\textsf{M}}^{\top})|$ top dimensional parallelepipeds all having the same volume.
Proof.
As $\operatorname{im}({\textsf{M}}^{\top})$ is orthogonal to $\ker{\textsf{M}}$, an independent set in the matroid $\mathcal{M}({\textsf{M}}^{\top})$ remains independent after multiplication by M, i.e., $\mathcal{M}({\textsf{M}}^{\top})$ and $\mathcal{M}({\textsf{L}})$ are isomorphic matroids.
As M is unimodular, so is ${\textsf{M}}^{\top}$, and so any set of columns $B$ of ${\textsf{M}}^{\top}$ corresponding to a basis of its matroid is a $\mathbb{Z}$-basis for the lattice $\mathcal{L}=\mathbb{Z}^{n}\cap\operatorname{im}({\textsf{M}}^{\top})$, that is, ${\textsf{Z}}(B)$ is a fundamental parallelepiped of $\mathcal{L}$.
It follows that every top dimensional parallelepiped in a maximal cubical decomposition of ${\textsf{Z}}({\textsf{L}})$ is a fundamental parallelepiped for the lattice ${\textsf{M}}\mathcal{L}={{}_{\mathbb{Z}}}{\left\langle{\textsf{L}}\right\rangle}$, the image of $\mathcal{L}$ under M.
The result now follows from the fact that the volume of a fundamental parallelepiped of a lattice is a lattice invariant.
∎
Example.
Consider the complete graph $K_{4}$ on four vertices with edges oriented so that $i\rightarrow j$ if $i<j$. The signed vertex-edge incidence matrix N and the Laplacian L are
$${\textsf{N}}=\begin{pmatrix}-1&-1&-1&0&0&0\\
1&0&0&-1&-1&0\\
0&1&0&1&0&-1\\
0&0&1&0&1&1\end{pmatrix},\hskip 44.0pt{\textsf{L}}=\begin{pmatrix}3&-1&-1&-1\\
-1&3&-1&-1\\
-1&-1&3&-1\\
-1&-1&-1&3\end{pmatrix}.$$
The three dimensional zonotope ${\textsf{Z}}({\textsf{N}})\subset\mathbb{R}^{4}$ is a translate of the classical permutahedron obtained by taking the convex hull of all points obtained from $[1,2,3,4]$ by permuting coordinates.
The zonotope ${\textsf{Z}}({\textsf{L}})$ is the cubical zonotope (all of its facets are 2-cubes) displayed in Figure 3a. By Proposition 13 it is the union of four parallelepipeds of equal volume; see Figure 3b for the subdivision of ${\textsf{Z}}({\textsf{L}})$ into parallelepipeds and Figure 3c for an exploded view of the subdivision.
In the graphical case, Proposition 13 tells us that the zonotope ${\textsf{Z}}({\textsf{L}})$ decomposes into $n$ parallelepipeds in ${{}_{\mathbb{R}}}{\left\langle{\textsf{N}}\right\rangle}$ all having the same volume.
More explicitly the decomposition is ${\textsf{Z}}({\textsf{L}})=\bigcup_{i}\Pi_{i}$ where, for $i\in[n]$, the parallelepiped $\Pi_{i}$ is generated by all of the columns of L save for the $i^{\text{th}}$.
We now show that the volume of one (and hence, any) of these parallelepipeds is equal to the number of spanning trees of $G$.
To see this first note that Theorem 6 holds regardless of the corank of the unimodular matrix involved, so in our case the volume of ${\textsf{Z}}({\textsf{N}})$ equals the number of spanning trees of $G$ (recall here that volume is taken with respect to the affine hull of the columns of N.)
Also independent of the corank of the defining matrix is Lemma 12, in which we showed that the lattice generated by the columns of the Laplacian is equal to the lattice generated by the matrix $B$ whose columns are the barycenters of the facets of ${\textsf{Z}}({\textsf{N}})$ scaled by a factor of 2.
Since any $n-1$ columns of L form a lattice basis for
${{}_{\mathbb{Z}}}{\left\langle{\textsf{L}}\right\rangle}={{}_{\mathbb{Z}}}{%
\left\langle B\right\rangle}$, we only need to check that an appropriate modification of Theorem 3 still holds when we drop the full-rank condition.
Indeed, in the proof of the theorem the full-rank condition guaranteed us that the columns of L formed a basis for their $\mathbb{Z}$-span, whereas when the corank of L is greater than 0 the columns over-determine the $\mathbb{Z}$-span.
Nonetheless, the proof of Theorem 3 at the end of Section 2 goes through verbatim for the following theorem in which M is allowed to have arbitrary corank.
Theorem 14.
Let $\mathcal{M}$ be a regular matroid and M be a unimodular representation of $\mathcal{M}$ over $\mathbb{R}$.
Let ${\textsf{L}}={\textsf{M}}{\textsf{M}}^{\top}$ and let $\overline{{\textsf{L}}}$ be the matrix obtained by taking any basis for ${}_{\mathbb{Z}}{\left\langle{\textsf{L}}\right\rangle}$ from among the columns of L.
Then the volume of ${\textsf{Z}}({\textsf{M}})$ equals the volume of ${\textsf{Z}}\left(\overline{{\textsf{L}}}\right)$.
Proof.
∎
In the graphical case, taking Proposition 13 and Theorem 14 together shows that the volume of ${\textsf{Z}}({\textsf{L}})$ is $n$ times the number of spanning trees.
So all that remains is to show that the volume of ${\textsf{Z}}({\textsf{L}})$ is the product of nonzero eigenvalues of L.
We will achieve this by defining two new full-dimensional zonotopes ${\textsf{Z}}(\Lambda)$ and ${\textsf{Z}}(\Gamma)$ and then showing that
(i)
$\operatorname{vol}{\textsf{Z}}(\Lambda)=n\lambda_{1}\cdots\lambda_{n-1}$,
(ii)
$\operatorname{vol}{\textsf{Z}}(\Gamma)=n\operatorname{vol}{\textsf{Z}}({%
\textsf{L}})$, and
(iii)
$\operatorname{vol}{\textsf{Z}}(\Lambda)=\operatorname{vol}{\textsf{Z}}(\Gamma)$.
To construct these new zonotopes, define the matrices $\Lambda$ and $\Gamma$ by setting $\Lambda_{ij}={\textsf{L}}_{ij}+1$ and letting $\Gamma=[{\textsf{L}}|\boldsymbol{1}]$ be the matrix obtained from L by appending a column of ones.
To prove (i), observe that the columns of $\Lambda$ arise by summing the vector $\boldsymbol{1}$ to each column of the rank $(n-1)$ matrix L, and that $\boldsymbol{1}$ is orthogonal to each of these columns.
In consequence, the columns of the $n\times n$ matrix $\Lambda$ are linearly independent.
Thus, the zonotope ${\textsf{Z}}(\Lambda)$ is an $n$-dimensional parallelopiped with volume equal to the product of the eigenvalues of $\Lambda$.
If $\lambda\in\operatorname{Spec}({\textsf{L}})$ is a nonzero eigenvalue with eigenvector $v$, then the sum of the coordinates of $v$ is zero.
It follows that $\Lambda v={\textsf{L}}v=\lambda v$, and so $\lambda$ is also an eigenvalue of $\Lambda$.
Since $\boldsymbol{1}\in\ker L$, it follows that $\Lambda\boldsymbol{1}=n\boldsymbol{1}$, and so $\operatorname{Spec}\Lambda=(\operatorname{Spec}({\textsf{L}})\setminus\{0\})%
\cup\{n\}$, and $\operatorname{vol}{\textsf{Z}}(\Lambda)=n\,\lambda_{1}\cdots\lambda_{n-1}$.
For (ii), first observe that $\det({\textsf{N}}_{P_{n}}|\boldsymbol{1})=n$, where ${\textsf{N}}_{P_{n}}$ is the signed incidence matrix of the path on $n$ vertices. Thus, the volume of any zonotope that is a prism ${\textsf{Z}}(M|\boldsymbol{1})={\textsf{Z}}(M)\times\boldsymbol{1}$ over a unimodular cube ${\textsf{Z}}(M)$ is $n$.
Our claim $\operatorname{vol}{\textsf{Z}}(\Gamma)=n\operatorname{vol}{\textsf{Z}}({%
\textsf{L}})$ now follows from the following general fact:
Proposition 15.
Let $P\in\mathbb{R}^{n}$ be an $(n-1)$-dimensional lattice polytope with affine span $S$ and let $\mathcal{L}=S\cap\mathbb{Z}^{n}$ be the induced lattice.
For $v\in\mathbb{Z}^{n}\setminus S$ let $Q$ be the prism $P\times v$.
Then
$$\operatorname{vol}(Q)\ =\ h_{S}(v)\operatorname{vol}_{S}(P),$$
where $h_{S}(v)$ is the lattice height of $v$ from $S$ and $\operatorname{vol}_{S}$ is the induced volume form on $\operatorname{aff}S$.
Proof.
Without loss of generality we may assume that $\boldsymbol{0}\in S$ so that $S$ is a linear hyperplane with primitive normal vector $u\in\mathbb{Z}^{d}$, say.
For any $i\in\mathbb{Z}$ define $S_{i}$ to be the parallel translate of $S$ given by $\{x\in\mathbb{R}^{n}\nonscript\;{\left.\kern-1.2pt\vphantom{}\middle|\right.%
\kern-1.2pt}\nonscript\;\langle x,u\rangle=i\}$.
Then for every $v\in\mathbb{Z}^{n}$ there is an $i\in\mathbb{Z}$ such that $v\in S_{i}$, and this is precisely the lattice height of $v$ with respect to $S$, $h_{S}(v)=i$.
Suppose $v\in\mathbb{Z}^{n}$ satisfies $h_{S}(v)=1$.
In the $k$-th dilate of $Q$, the only lattice points of $\mathbb{Z}^{n}$ lie on the sections $Q\cap H_{i}$ where $H_{i}=\{x\in\mathbb{R}^{n}:h_{S}(x)=i\}$ for $0\leq i\leq k$.
Moreover, the distribution of lattice points is the same in each section $Q\cap H_{i}$.
Thus, the number of lattice points in the $k^{\text{th}}$ dilate of $Q=P\times v$ is exactly
$$\displaystyle\#\left(Q\cap\frac{1}{k}\mathbb{Z}^{n}\right)$$
$$\displaystyle=(k\,h_{S}(v)+1)\;\#\!\left(P\cap\frac{1}{k}\mathcal{L}\right)$$
$$\displaystyle=(k+1)\;\#\!\left(P\cap\frac{1}{k}\mathcal{L}\right).$$
So in this case we have
$$\displaystyle\operatorname{vol}(Q)$$
$$\displaystyle=\lim_{k\to\infty}\frac{1}{k^{n}}\#\left(Q\cap\frac{1}{k}\mathbb{%
Z}^{n}\right)$$
$$\displaystyle=\lim_{k\to\infty}\frac{k+1}{k}\frac{1}{k^{n-1}}\#\left(P\cap%
\frac{1}{k}\mathcal{L}\right)$$
$$\displaystyle=\lim_{k\to\infty}\frac{1}{k^{n-1}}\#\left(P\cap\frac{1}{k}%
\mathcal{L}\right)$$
$$\displaystyle=\operatorname{vol}_{S}(P).$$
Since $Q$ is a full-dimensional prism, its lattice volume and Euclidean volume coincide.
It follows that $\operatorname{vol}(P\times v)=\operatorname{vol}_{S}(P)$ for any $v\in\mathbb{R}^{n}$ with $h_{S}(v)=1$.
For an arbitrary $v\in\mathbb{Z}^{n}$ with $h_{S}(v)=i$, the prism $Q$ decomposes into $i$ (typically rational) polytopes which are slices of $Q$ sitting between the affine hyperplanes $S_{j-1}$ and $S_{j}$ where $j\in[i]$.
Each of these slices is a translated copy of a height-one prism over $P$ and hence has volume $\operatorname{vol}(P)$.
As there are $h_{S}(v)$ many of them, the result follows.
∎
The missing claim (iii), $\operatorname{vol}{\textsf{Z}}(\Lambda)=\operatorname{vol}{\textsf{Z}}(\Gamma)$, is true in much greater generality, and it is this generalization that we state in Theorem 16, the proof of which uses a technique analogous to the proof of Theorem 3.
Theorem 16.
For any set $B=\{b_{1},\dots,b_{n}\}$ of points that linearly span $\mathbb{R}^{n}$, let $\beta=\frac{1}{n}\sum_{i\in[n]}b_{i}$ be their barycenter and let $\Pi={\textsf{Z}}(B)$ be the zonotope they generate.
Let $P$ be the zonotope generated by $\beta$ together with the points $b_{i}-\beta$ for $i\in[n]$.
Then $\operatorname{vol}\Pi=\operatorname{vol}P$.
Note that we obtain claim (iii) as a special case by taking $B,\Pi,$ and $P$ to be the columns of $\Lambda$, the zonotope ${\textsf{Z}}(\Lambda)$, and the zonotope ${\textsf{Z}}(\Gamma)$, respectively.
Before proceeding with the proof in the general case, let us illustrate the techniques to be used:
Example.
For the complete graph $K_{3}$ on three vertices, the zonotope ${\textsf{Z}}(\Gamma)$ is the prism over the hexagon ${\textsf{Z}}({\textsf{L}})$ shown in blue in Figure 5 intersecting the red parallelepiped ${\textsf{Z}}(\Lambda)$.
For each sign vector $\epsilon\in\{+,-\}^{3}$, the simplicial cone spanned by $\epsilon{\textsf{L}}=\{\epsilon_{i}{\textsf{L}}_{i}\}$ intersects ${\textsf{Z}}(\Gamma)$ and these intersections are the $P_{\epsilon}$.
By construction, all of the $P_{\epsilon}$ are full-dimensional except for $P_{\{-,-,-\}}$ which consists only of the origin.
The seven full-dimensional pieces are illustrated center-left in Figure 6.
Six of the seven $P_{\epsilon}$ are visible in the figure, while the colored hexagon beneath the prism suggests the location of the invisible piece.
By translating each $P_{\epsilon}$ by the sum of all $\Lambda_{i}$ such that $\epsilon_{i}$ is negative, we obtain the union of the $Q_{\epsilon}$ as seen center-right in Figure 6.
This union is exactly the zonotope of $\Lambda$.
Proof of Theorem 16.
We prove that there is a decomposition of $P$ into full dimensional polytopal cells and a set of translations (one for each polytope in the decomposition) such that the union of the translated cells is exactly $\Pi$ and that if two shifted cells intersect, they do so only on their boundaries.
First we show that for every point $p\in P$ there is a sign vector $\epsilon=\epsilon(p)\in\{+,-\}^{n}$ such that $p\in{\textsf{Z}}(\epsilon B)$ where $\epsilon B:=\{\epsilon_{i}b_{i}\nonscript\;{\left.\kern-1.2pt\vphantom{}%
\middle|\right.\kern-1.2pt}\nonscript\;i\in[n]\}$.
As $P$ is a zonotope, given any $p\in P$ there is an $\alpha\in[0,1]^{n+1}$ such that
$$\displaystyle p$$
$$\displaystyle=\alpha_{n+1}\beta+\sum_{i\in[n]}\alpha_{i}(b_{i}-\beta)$$
$$\displaystyle=\sum_{i\in[n]}\frac{1}{n}\left(n\alpha_{i}+\alpha_{n+1}-\sum_{j%
\in[n]}\alpha_{j}\right)b_{i}$$
$$\displaystyle=\sum_{i\in[n]}\frac{1}{n}\left((n-1)\alpha_{i}+\alpha_{n+1}-\sum%
_{j\in[n]\setminus\{i\}}\alpha_{j}\right)b_{i}.$$
Let us abbreviate this last expression to $p=\sum_{i\in[n]}\gamma_{i}b_{i}$, where the $\gamma_{i}$ are unique because the $b_{i}$ form a basis of $\mathbb{R}^{n}$. Since each $\alpha_{j}$ is in $[0,1]$, it follows that $\gamma_{i}\in[-1,1]$ for all $i$. Therefore, setting $\epsilon_{i}=\operatorname{sign}\gamma_{i}$ if $\gamma_{i}\neq 0$ (and $\epsilon_{i}=\pm$ arbitrarily if $\gamma_{i}=0$) proves the claim.
For each $\epsilon\in\{+,-\}^{n}$, define $P_{\epsilon}:=P\cap{\textsf{Z}}(\epsilon B)$ and $v_{\epsilon}=\sum_{i:\epsilon_{i}=-}b_{i}$, see Figure 5.
By the previous paragraph we know that $P$ is the union of the $P_{\epsilon}$ and we now show that the union of the translated polytopes $P_{\epsilon}+v_{\epsilon}$ is $\Pi$.
To see this let $q=\sum_{i\in[n]}\alpha_{i}b_{i}\in{\textsf{Z}}(B)$.
If $q=0$ then $q\in P$ so we may assume there is a nonnegative integer $k$ such that $\sum_{i\in[n]}\alpha_{i}\in(k,k+1]$.
Moreover, we may assume (after permuting indices if necessary) that the $\alpha_{i}$ are decreasing, i.e., $\alpha_{1}\geq\alpha_{2}\geq\cdots\geq\alpha_{n}$.
Now we define $\epsilon$ to be the sign vector with $\epsilon_{i}=-$ if and only if $i\leq k$.
It follows that
$$\displaystyle q-v_{\epsilon}$$
$$\displaystyle=q-\sum_{i=1}^{k}b_{i}$$
$$\displaystyle=\sum_{i<k}(\alpha_{i}-1)(b_{i}-\beta+\beta)+(\alpha_{k}-1)b_{k}+%
\sum_{j>k}\alpha_{j}(b_{j}-\beta+\beta)$$
$$\displaystyle=\sum_{i<k}(\alpha_{i}-1)(b_{i}-\beta)+(\alpha_{k}-1)b_{k}+\sum_{%
j>k}\alpha_{j}(b_{j}-\beta)+\left(-(k-1)+\sum_{i\in[n]\setminus k}\alpha_{i}%
\right)\beta.$$
Since $b_{k}=n\beta-\sum_{i\neq k}b_{i}$, we can express the second summand as
$$\displaystyle(\alpha_{k}-1)b_{k}$$
$$\displaystyle=-(\alpha_{k}-1)\left(-n\beta+\sum_{i\neq k}b_{i}\right)$$
$$\displaystyle=-(\alpha_{k}-1)\left(-\beta+\sum_{i\neq k}(b_{i}-\beta)\right)$$
$$\displaystyle=\sum_{i<k}(1-\alpha_{k})(b_{i}-\beta)+(\alpha_{k}-1)\beta+\sum_{%
j>k}(1-\alpha_{k})(b_{j}-\beta),$$
so that
$$\displaystyle q-v_{\epsilon}$$
$$\displaystyle=\sum_{i<k}(\alpha_{i}-\alpha_{k})(b_{i}-\beta)+\sum_{j>k}(\alpha%
_{j}-\alpha_{k}+1)(b_{j}-\beta)+\left(-k+\sum_{i\in[n]}\alpha_{i}\right)\beta$$
and so $q-v_{\epsilon}\in P$ since $\alpha_{i}\geq\alpha_{k}$ if $i\leq k$ and $\alpha_{k}\geq\alpha_{i}$ otherwise. Moreover, our choice of $k$ guarantees that all coefficients in this linear combination lie in $[0,1]$.
Finally, in order to prove that our decomposition and rearrangement preserves volume, we must show that if two translated cells $P_{\epsilon}+v_{\epsilon}$ and $P_{\epsilon^{\prime}}+v_{\epsilon^{\prime}}$ intersect then they do so on a set of measure zero.
To see this let $p\in P_{\epsilon}$ and $p^{\prime}\in P_{\epsilon^{\prime}}$ be such that $\epsilon\neq\epsilon^{\prime}$ and $p+v_{\epsilon}=p^{\prime}+v_{\epsilon^{\prime}}\in{\textsf{Z}}(B)$, where $v_{\epsilon}=\sum_{i:\epsilon_{i}=-}b_{i}$ as before.
Then
$$\displaystyle 0$$
$$\displaystyle=p+v_{\epsilon}-p^{\prime}-v_{\epsilon^{\prime}}$$
$$\displaystyle=\sum\alpha_{i}b_{i}+v_{\epsilon}-\sum\beta_{i}b_{i}-v_{\epsilon^%
{\prime}}$$
$$\displaystyle=\sum_{i\in\epsilon^{-}\setminus\epsilon^{\prime-}}(\alpha_{i}-%
\beta_{i}+1)b_{i}+\sum_{j\in\epsilon^{\prime-}\setminus\epsilon^{-}}(\alpha_{j%
}-\beta_{j}-1)b_{j}+\sum_{k:\epsilon_{k}=\epsilon^{\prime}_{k}}(\alpha_{k}-%
\beta_{k})b_{k}.$$
Since $B$ is a basis it follows that the coefficient on any $b_{i}$ in the final expression equals zero.
Therefore $\alpha_{k}=\beta_{k}$ if $\epsilon_{k}=\epsilon^{\prime}_{k}$ and otherwise either $\alpha_{i}=0$ and $\beta_{i}=1$ or vice versa.
It follows from the definitions of $P_{\epsilon}$ and $P_{\epsilon^{\prime}}$ that $p\in\partial P_{\epsilon}$ and $p^{\prime}\in\partial P_{\epsilon^{\prime}}$, and the proof is complete.
∎
Taken together, Proposition 15 and Theorem 16 tell us that when M is a unimodular matrix with corank 1, we can recover the product of the nonzero eigenvalues of L by constructing a certain full-rank matrix $\Lambda$ associated to L and analyzing the zonotope it generates.
This construction essentially replaces the eigenvalue $0$ of L with the eigenvalue $n$ while fixing the other eigenvalues.
We suspect that this can be strengthened to allow for unimodular representations of regular matroids of arbitrary corank in the statement of Theorem 3.
Presently we have no proof for this fact, and so we leave it as a conjecture.
Conjecture 17.
Let $\mathcal{M}$ be a regular matroid and M a unimodular $m\times n$ representation of $\mathcal{M}$ with corank greater than 1.
Then there is a $m\times m$ matrix $\Lambda$ with full rank such that every nonzero eigenvalue of L is an eigenvalue of $\Lambda$ and every other eigenvalue of $\Lambda$ depends only on the ambient dimension $m$.
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Not All Samples Are Created Equal:
Deep Learning with Importance Sampling
Angelos Katharopoulos
François Fleuret
Abstract
Deep neural network training spends most of the computation on
examples that are properly handled, and could be ignored.
We propose to mitigate this phenomenon with a principled importance
sampling scheme that focuses computation on “informative” examples,
and reduces the variance of the stochastic gradients during
training. Our contribution is twofold: first, we derive a tractable
upper bound to the per-sample gradient norm, and second we derive an
estimator of the variance reduction achieved with importance sampling,
which enables us to switch it on when it will result in an actual
speedup.
The resulting scheme can be used by changing a few lines of code in a standard
SGD procedure, and we demonstrate experimentally, on image classification, CNN
fine-tuning, and RNN training, that for a fixed wall-clock time budget, it
provides a reduction of the train losses of up to an order of magnitude and a
relative improvement of test errors between 5% and 17%.
variance reduction, importance sampling, deep learning
1 Introduction
The dramatic increase in available training data has made the use of deep
neural networks feasible, which in turn has significantly improved the
state-of-the-art in many fields, in particular computer vision and natural
language processing. However, due to the complexity of the resulting
optimization problem, computational cost is now the core issue in training
these large architectures.
When training such models, it appears to any practitioner that not all samples
are equally important; many of them are properly handled after a few epochs of
training, and most could be ignored at that point without impacting the final
model. To this end, we propose a novel importance sampling scheme that
accelerates the training of any neural network architecture by focusing the
computation on the samples that will introduce the biggest change in the
parameters which reduces the variance of the gradient estimates.
For convex optimization problems, many works (Bordes et al., 2005; Zhao & Zhang, 2015; Needell et al., 2014; Canévet et al., 2016; Richtárik & Takáč, 2013) have taken advantage of the difference in importance
among the samples to improve the convergence speed of stochastic optimization
methods. On the other hand, for deep neural networks, sample
selection methods were mainly employed to generate hard negative samples for
embedding learning problems or to tackle the class imbalance problem
(Schroff et al., 2015; Wu et al., 2017; Simo-Serra et al., 2015).
Recently, researchers have shifted their focus on using importance sampling to
improve and accelerate the training of neural networks (Alain et al., 2015; Loshchilov & Hutter, 2015; Schaul et al., 2015). Those works, employ either the
gradient norm or the loss to compute each sample’s importance. However, the
former is prohibitively expensive to compute and the latter is not a
particularly good approximation of the gradient norm.
Compared to the aforementioned works, we derive an upper bound to the per
sample gradient norm that can be computed in a single forward pass. This
results in reduced computational requirements of more than an order of
magnitude compared to Alain et al. (2015). Furthermore, we quantify the
variance reduction achieved with the proposed importance sampling scheme and
associate it with the batch size increment required to achieve an equivalent
variance reduction. The benefits of this are twofold, firstly we provide an
intuitive metric to predict how useful importance sampling is going to be, thus
we are able to decide when to switch on importance sampling during training. Secondly, we also
provide theoretical guarantees for speedup, when variance reduction is above a
threshold. Based on our analysis, we propose a simple to use algorithm that can
be used to accelerate the training of any neural network architecture.
Our implementation is generic and can be employed by adding a single line of code in
a standard Keras model training. We validate it on three independent tasks:
image classification,
fine-tuning and sequence classification with recurrent neural networks.
Compared to existing batch selection schemes, we show that our method
consistently achieves lower training loss and test error for equalized
wall-clock time.
2 Related Work
Existing importance sampling methods can be roughly categorized in methods
applied to convex problems and methods designed for deep neural networks.
2.1 Importance Sampling for Convex Problems
Importance sampling for convex optimisation problems has been extensively
studied over the last years. Bordes et al. (2005) developed LASVM, which is
an online algorithm that uses importance sampling to train kernelized support
vector machines. Later, Richtárik & Takáč (2013) proposed a generalized
coordinate descent algorithm that samples coordinate sets in a way that
optimizes the algorithm’s convergence rate.
More recent works (Zhao & Zhang, 2015; Needell et al., 2014) make a
clear connection with the variance of the gradient estimates of stochastic
gradient descent and show that the optimal sampling distribution is
proportional to the per sample gradient norm. Due to the relatively simple
optimization problems that they deal with, the authors resort to sampling
proportionally to the norm of the inputs, which in simple linear
classification is proportional to the Lipschitz constant of the per sample loss function.
Such simple importance measures do not exist for Deep Learning and the direct
application of the aforementioned theory (Alain et al., 2015), requires
clusters of GPU workers just to compute the sampling distribution.
2.2 Importance Sampling for Deep Learning
Importance sampling has been used in Deep Learning mainly in the form of
manually tuned sampling schemes. Bengio et al. (2009) manually design a
sampling scheme inspired by the perceived way that human children learn; in
practice they provide the network with examples of increasing difficulty in an
arbitrary manner. Diametrically opposite, it is common for deep embedding
learning to sample hard examples because of the plethora of easy non
informative ones (Simo-Serra et al., 2015; Schroff et al., 2015).
More closely related to our work, Schaul et al. (2015) and
Loshchilov & Hutter (2015) use the loss to create the sampling distribution.
Both approaches keep a history of losses for previously seen samples, and sample
either proportionally to the loss or based on the loss ranking. One of the main limitations of history based sampling,
is the need for tuning a large number of hyperparameters that control the
effects of “stale” importance scores; i.e. since the model is constantly
updated, the importance of samples fluctuate and previous observations may
poorly reflect the current situation. In particular,
Schaul et al. (2015) use various forms of
smoothing for the losses and the importance sampling weights, while
Loshchilov & Hutter (2015) introduce a large number of hyperparameters that
control when the losses are computed, when they are sorted as well as how the
sampling distribution is computed based on the rank.
In comparison to all the above methods, our importance sampling scheme based on
an upper bound to the gradient norm has a solid theoretical basis with clear
objectives, very easy to choose hyperparameters, theoretically guaranteed
speedup and can be applied to any type of network and loss function.
2.3 Other Sample Selection Methods
Finally, we would like to mention two other related methods for speeding up the
training of neural networks. Wu et al. (2017), design a distribution
(suitable only for the distance based losses) that maximizes the diversity of
the losses in a single batch. Fan et al. (2017) use reinforcement
learning to train a neural network that selects samples for another neural
network in order to optimize the convergence speed. Although their preliminary
results are promising, the overhead of training two networks makes the
wall-clock speedup unlikely and their proposal not as appealing.
3 Variance Reduction for Deep Neural Networks
Importance sampling aims at increasing the convergence speed of Stochastic
Gradient Descent (SGD) by focusing computation on samples that actually
induce a change in the model parameters. This formally translates into a
reduced variance of the gradient estimates for a fixed computational cost. In
the following sections, we analyze how this works and present an efficient
algorithm that can be used to train any Deep Learning model.
3.1 Introduction to Importance Sampling
Let $x_{i}$, $y_{i}$ be the $i$-th input-output pair from the training set,
$\Psi(\cdot;\theta)$ be a Deep Learning model parameterized by the vector
$\theta$, and $\mathcal{L}(\cdot,\cdot)$ be the loss function to be minimized during
training.
The goal of training is to find
$$\theta^{*}=\operatorname*{arg\,min}_{\theta}\frac{1}{N}\sum_{i=1}^{N}\mathcal{%
L}(\Psi(x_{i};\theta),y_{i})$$
(1)
where $N$ corresponds to the number of examples in the training set.
We use an SGD procedure with learning rate $\eta$, where the update at iteration $t$ depends on the sampling distribution $p^{t}_{1},\dots,p^{t}_{N}$ and re-scaling coefficients $w^{t}_{1},\dots,w^{t}_{N}$. Let $I_{t}$ be the data point sampled at that step, we have $P(I_{t}=i)=p^{t}_{i}$ and
$$\theta_{t+1}=\theta_{t}-\eta w_{I_{t}}\nabla_{\theta_{t}}\mathcal{L}(\Psi(x_{I%
_{t}};\theta_{t}),y_{I_{t}})$$
(2)
Plain SGD with uniform sampling is achieved with $w^{t}_{i}=1$ and $p^{t}_{i}=\frac{1}{N}$ for all $t$ and $i$.
If we define the convergence speed $S$ of SGD as the reduction of the distance
of the parameter vector $\theta$ from the optimal parameter vector $\theta^{*}$
in two consecutive iterations $t$ and $t+1$
$$S=-\operatorname{\mathbb{E}}_{P_{t}}\!\left[\left\lVert\theta_{t+1}-\theta^{*}%
\right\rVert_{2}^{2}-\left\lVert\theta_{t}-\theta^{*}\right\rVert_{2}^{2}%
\right],$$
(3)
and if we have $w_{i}=\frac{1}{Np_{i}}$ such that
$$\operatorname{\mathbb{E}}_{P_{t}}\!\left[w_{I_{t}}\nabla_{\theta_{t}}\mathcal{%
L}(\Psi(x_{I_{t}};\theta_{t}),y_{I_{t}})\right]=\nabla_{\theta_{t}}\frac{1}{N}%
\sum_{i=1}^{N}\mathcal{L}(\Psi(x_{i};\theta_{t}),y_{i}),$$
(4)
and set $G_{i}=w_{i}\nabla_{\theta_{t}}\mathcal{L}(\Psi(x_{i};\theta_{t}),y_{i})$,
then we get (this is a different derivation of the result by Wang et al., 2016)
$$\displaystyle S$$
$$\displaystyle=-\operatorname{\mathbb{E}}_{P_{t}}\!\left[\left(\theta_{t+1}\!-%
\!\theta^{*}\right)^{T}\!\left(\theta_{t+1}\!-\!\theta^{*}\right)-\left(\theta%
_{t}\!-\!\theta^{*}\right)^{T}\!\left(\theta_{t}\!-\!\theta^{*}\right)\right]$$
(5)
$$\displaystyle=-\operatorname{\mathbb{E}}_{P_{t}}\!\left[\theta_{t+1}^{T}\!%
\theta_{t+1}\!-\!2\theta_{t+1}\theta^{*}-\theta_{t}^{T}\!\theta_{t}+2\theta_{t%
}\theta^{*}\right]$$
$$\displaystyle=-\operatorname{\mathbb{E}}_{P_{t}}\!\left[\left(\theta_{t}\!-\!%
\eta G_{I_{t}}\right)^{T}\!\left(\theta_{t}\!-\!\eta G_{I_{t}}\right)+2\eta G_%
{I_{t}}^{T}\!\theta^{*}\!-\!\theta_{t}^{T}\!\theta_{t}\right]$$
$$\displaystyle=-\operatorname{\mathbb{E}}_{P_{t}}\!\left[-2\eta\left(\theta_{t}%
\!-\!\theta^{*}\right)G_{I_{t}}+\eta^{2}G_{I_{t}}^{T}\!G_{I_{t}}\right]$$
$$\displaystyle=2\eta\left(\theta_{t}\!-\!\theta^{*}\right)\operatorname{\mathbb%
{E}}_{P_{t}}\!\left[G_{I_{t}}\right]-\eta^{2}\operatorname{\mathbb{E}}_{P_{t}}%
\!\left[G_{I_{t}}\right]^{T}\!\operatorname{\mathbb{E}}_{P_{t}}\!\left[G_{I_{t%
}}\right]-$$
$$\displaystyle\quad\,\eta^{2}\text{Tr}\left(\operatorname{\mathbb{V}}_{P_{t}}\!%
\left[G_{I_{t}}\right]\right)$$
Since the first two terms, in the last expression, are the speed of batch
gradient descent, we observe that it is possible to gain a speedup by sampling
from the distribution that minimizes $\text{Tr}\left(\operatorname{\mathbb{V}}_{P_{t}}\!\left[G_{I_{t}}\right]\right)$. Several works
(Needell et al., 2014; Zhao & Zhang, 2015; Alain et al., 2015) have shown
the optimal distribution to be proportional to the per-sample gradient norm.
However, computing this distribution is computationally prohibitive.
3.2 Beyond the Full Gradient Norm
Given an upper bound $\hat{G}_{i}\geq\left\lVert\nabla_{\theta_{t}}\mathcal{L}(\Psi(x_{i};\theta_{t}%
),y_{i})\right\rVert_{2}$ and due to
$$\displaystyle\operatorname*{arg\,min}_{P}\text{Tr}\left(\operatorname{\mathbb{%
V}}_{P_{t}}\!\left[G_{I_{t}}\right]\right)=\operatorname*{arg\,min}_{P}%
\operatorname{\mathbb{E}}_{P_{t}}\!\left[\left\lVert G_{I_{t}}\right\rVert_{2}%
^{2}\right],$$
(6)
we propose to relax the optimization problem in the following way
$$\displaystyle\min_{P}\operatorname{\mathbb{E}}_{P_{t}}\!\left[\left\lVert G_{I%
_{t}}\right\rVert_{2}^{2}\right]\leq\min_{P}\operatorname{\mathbb{E}}_{P_{t}}%
\!\left[w_{I_{t}}^{2}\hat{G}_{I_{t}}^{2}\right].$$
(7)
The minimizer of the second term of equation 7,
similar to the first term, is $p_{i}\propto\hat{G}_{i}$. All that remains, is to
find a proper expression for $\hat{G}_{i}$ which is significantly easier to
compute than the norm of the gradient for each sample.
In order to continue with the derivation of our upper bound $\hat{G}_{i}$, let us introduce some notation specific to a multi-layer perceptron. Let $\theta^{(l)}\in\mathbb{R}^{M_{l}\times M_{l-1}}$ be the weight matrix for layer $l$ and $\sigma^{(l)}(\cdot)$ be a Lipschitz continuous activation function. Then, let
$$\displaystyle x^{(0)}$$
$$\displaystyle=x$$
(8)
$$\displaystyle z^{(l)}$$
$$\displaystyle=\theta^{(l)}\,x^{(l-1)}$$
(9)
$$\displaystyle x^{(l)}$$
$$\displaystyle=\sigma^{(l)}(z^{(l)})$$
(10)
$$\displaystyle\Psi(x;\Theta)$$
$$\displaystyle=x^{(L)}$$
(11)
Although our notation describes
simple fully connected neural networks without bias, our analysis holds for any
affine operation followed by a slope-bounded non-linearity ($|\sigma^{\prime}(x)|\leq K$).
With
$$\displaystyle\Sigma_{l}^{\prime}(z)$$
$$\displaystyle=diag\left(\sigma^{\prime(l)}(z^{(l-1)}_{1}),\dots,\sigma^{\prime%
(l)}(z^{(l-1)}_{M_{l}})\right),$$
(12)
$$\displaystyle\Delta^{(l)}$$
$$\displaystyle=\Sigma_{l}^{\prime}(z_{l})\theta_{l+1}^{T}\dots\Sigma_{L-1}^{%
\prime}(z_{L-1})\theta_{L}^{T},$$
(13)
$$\displaystyle\nabla_{x_{i}^{(L)}}\mathcal{L}$$
$$\displaystyle=\nabla_{x_{i}^{(L)}}\mathcal{L}(\Psi(x_{i};\Theta),y_{i})$$
(14)
we get
$$\displaystyle\left\lVert\nabla_{\theta_{l}}\mathcal{L}(\Psi(x_{i};\Theta),y_{i%
})\right\rVert_{2}$$
(15)
$$\displaystyle=$$
$$\displaystyle\left\lVert\left(\Delta^{(l)}\Sigma_{L}^{\prime}(z_{L})\nabla_{x_%
{i}^{(L)}}\mathcal{L}\right)\left(x_{i}^{(l-1)}\right)^{T}\right\rVert_{2}$$
(16)
$$\displaystyle\leq$$
$$\displaystyle\left\lVert\Delta^{(l)}\right\rVert_{2}\left\lVert\Sigma_{L}^{%
\prime}(z_{L})\nabla_{x_{i}^{(L)}}\mathcal{L}\right\rVert_{2}\left\lVert x_{i}%
^{(l-1)}\right\rVert_{2}$$
(17)
$$\displaystyle\leq$$
$$\displaystyle\underbrace{\max_{l,i}\left(\left\lVert x_{i}^{(l-1)}\right\rVert%
_{2}\left\lVert\Delta^{(l)}\right\rVert_{2}\right)}_{\rho}\left\lVert\Sigma_{L%
}^{\prime}(z_{L})\nabla_{x_{i}^{(L)}}\mathcal{L}\right\rVert_{2}$$
(18)
Various weight initialization (Glorot & Bengio, 2010) and activation
normalization techniques
(Ioffe & Szegedy, 2015; Ba et al., 2016) uniformise the activations
across samples in most high-performance architectures.
As a result, the variation of the gradient norm is mostly
captured by the gradient of the loss function with respect to the
pre-activation outputs of the last layer of our neural
network. Consequently we can derive the following upper bound to the
gradient norm of all the parameters
$$\displaystyle\left\lVert\nabla_{\Theta}\mathcal{L}(\Psi(x_{i};\Theta),y_{i})%
\right\rVert_{2}\leq\underbrace{L\rho\left\lVert\Sigma_{L}^{\prime}(z_{L})%
\nabla_{x_{i}^{(L)}}\mathcal{L}\right\rVert_{2}}_{\hat{G}_{i}},$$
(19)
which can be computed in a closed form in terms of $z_{L}$
and is marginally more difficult to compute than the value of the loss.
However, our upper bound depends on the time step $t$, thus we cannot generate
a distribution once and sample from it during training. This is intuitive
because the importance of each sample changes as the model changes.
3.3 When is Variance Reduction Possible?
Computing the importance score from equation 19 is more than
an order of magnitude faster compared to computing the gradient norm for each
sample. Nevertheless, it still costs one forward pass through the network and
can be wasteful. For instance, during the first iterations of training, the
gradients with respect to every sample have approximately equal norm; thus we
would waste computational resources trying to sample from the uniform
distribution. In addition, computing the importance score for the whole dataset
is still prohibitive and would render the method unsuitable for online
learning.
In order to solve the problem of computing the importance for the whole
dataset, we pre-sample a large batch of data points, compute the
sampling distribution for that batch and re-sample a smaller batch with
replacement. The above procedure upper bounds both the speedup and variance
reduction. Given a large batch consisting of $B$ samples and a small one consisting
of $b$, we can achieve a maximum variance reduction of $\frac{1}{b^{2}}-\frac{1}{B^{2}}$ and a maximum speedup of $\frac{B+3b}{3B}$ assuming that the
backward pass requires twice the amount of time as the forward pass.
Due to the large cost of computing the importance per sample, we only perform
importance sampling when we know that the variance of the gradients can be
reduced. In the following equation, we show that the variance reduction is
proportional to the squared $L_{2}$ distance of the sampling distribution, $g$,
to the uniform distribution $u$. Let $g_{i}\propto\left\lVert\nabla_{\theta_{t}}\mathcal{L}(\Psi(x_{i};\theta_{t}),y%
_{i})\right\rVert_{2}$ and $u=\frac{1}{B}$ the uniform probability.
$$\displaystyle\text{Tr}\left(\operatorname{\mathbb{V}}_{u}\!\left[G_{i}\right]%
\right)-\text{Tr}\left(\operatorname{\mathbb{V}}_{g}\!\left[G_{i}\right]\right)$$
(20)
$$\displaystyle=\operatorname{\mathbb{E}}_{u}\!\left[\left\lVert G_{i}\right%
\rVert_{2}^{2}\right]-\operatorname{\mathbb{E}}_{g}\!\left[w_{i}^{2}\left%
\lVert G_{i}\right\rVert_{2}^{2}\right]$$
(21)
$$\displaystyle=\frac{1}{B}\sum_{i=1}^{B}\left\lVert G_{i}\right\rVert_{2}^{2}-%
\left(\frac{1}{B}\sum_{i=1}^{B}\left\lVert G_{i}\right\rVert_{2}\right)^{2}$$
(22)
$$\displaystyle=\frac{\left(\sum_{i=1}^{B}\left\lVert G_{i}\right\rVert_{2}%
\right)^{2}}{B^{3}}\sum_{i=1}^{B}\left(B^{2}\frac{\left\lVert G_{i}\right%
\rVert_{2}^{2}}{(\sum_{i=1}^{B}\left\lVert G_{i}\right\rVert_{2})^{2}}-1\right)$$
(23)
$$\displaystyle=\frac{\left(\sum_{i=1}^{B}\left\lVert G_{i}\right\rVert_{2}%
\right)^{2}}{B}\sum_{i=1}^{B}\left(g_{i}^{2}-u^{2}\right)$$
(24)
$$\displaystyle=\frac{\left(\sum_{i=1}^{B}\left\lVert G_{i}\right\rVert_{2}%
\right)^{2}}{B}\sum_{i=1}^{B}\left(g_{i}-u\right)^{2}$$
(25)
$$\displaystyle=\left(\frac{1}{B}\sum_{i=1}^{B}\left\lVert G_{i}\right\rVert_{2}%
\right)^{2}B\left\lVert g-u\right\rVert_{2}^{2}.$$
(26)
Equation 26 already provides us with a useful metric to decide
if the variance reduction is significant enough to justify using importance
sampling. However, choosing a suitable threshold for the $L_{2}$ distance squared
would be tedious and unintuitive. We can do much better by dividing the
variance reduction with the original variance to derive the increase in the
batch size that would achieve an equivalent variance reduction. Assuming that
we increase the batch size by $\tau$, we achieve variance reduction
$\frac{1}{\tau^{2}}$; thus we have
$$\displaystyle\frac{\left(\frac{1}{B}\sum_{i=1}^{B}\left\lVert G_{i}\right%
\rVert_{2}\right)^{2}B\left\lVert g-u\right\rVert_{2}^{2}}{\frac{1}{B}\sum_{i=%
1}^{B}\left\lVert G_{i}\right\rVert_{2}^{2}}=$$
(27)
$$\displaystyle\frac{1}{\sum_{i=1}^{B}g_{i}^{2}}\left\lVert g-u\right\rVert_{2}^%
{2}=1-\frac{1}{\tau^{2}}\iff$$
(28)
$$\displaystyle\frac{1}{\tau}=\sqrt{1-\frac{1}{\sum_{i=1}^{B}g_{i}^{2}}\left%
\lVert g-u\right\rVert_{2}^{2}}$$
(29)
Using equation 29, we have a hyperparameter that
is very easy to select and can now design our training procedure which is
described in pseudocode in algorithm 1. Computing $\tau$ from
equation 29 allows us to have guaranteed speedup when $B+3b<3\tau b$. However, as it is shown in the experiments, we can use
$\tau_{th}$ smaller than $\frac{B+3b}{3b}$ and still get a significant speedup.
The inputs to the algorithm are the pre-sampling size $B$, the batch size $b$,
the equivalent batch size increment after which we start importance sampling
$\tau_{th}$ and the exponential moving average parameter $a_{\tau}$ used to
compute a smooth estimate of $\tau$. $\theta_{0}$ denotes the initial parameters
of our deep network. We would like to point out that in line
\comment15 15 of the algorithm, we compute $g_{i}$ for
free since we have done the forward pass in the previous step.
The only parameter that has to be explicitly defined for our algorithm is the
pre-sampling size $B$ because $\tau_{th}$ can be set using
equation 29. We provide a small ablation study for $B$ in
the supplementary material.
4 Experiments
In this section, we analyse experimentally the performance of the proposed
importance sampling scheme based on our upper-bound of the gradient
norm. In the first subsection, we compare the variance reduction achieved with
our upper bound to the theoretically maximum achieved with the true gradient
norm. We also compare against sampling based on the loss, which is commonly
used in practice. Subsequently, we conduct experiments which demonstrate that
we are able to achieve non-negligible wall-clock speedup for a variety of tasks
using our importance sampling scheme.
In all the subsequent sections, we use uniform to refer to the usual
training algorithm that samples points from a uniform distribution, we use
loss to refer to algorithm 1 but instead of sampling
from a distribution proportional to our upper-bound to the gradient norm
$\hat{G}_{i}$ (equations 7 and 19), we sample
from a distribution proportional to the loss value and finally
upper-bound to refer to our proposed method. All the other baselines
from published methods are referred to using the names of the authors.
Experiments were conducted using Keras (Chollet et al., 2015) with TensorFlow
(Abadi et al., 2016), and the code to reproduce the experiments will be
provided under an open source license when the paper will be published. For all
the experiments, we use Nvidia K80 GPUs and the reported time is calculated by
subtracting the timestamps before starting one epoch and after finishing one;
thus it includes the time needed to transfer data between CPU and GPU memory.
Our implementation provides a wrapper around models that substitutes
the standard uniform sampling with our importance-sampling
method. This means that adding a single line of code to call
this wrapper before actually fitting the model is sufficient to switch
from the standard uniform sampling to our importance-sampling
scheme. And, as specified in § 3.3 and
Algorithm 1, our procedure reliably estimates at
every iteration if the importance sampling will provide a speed-up and
sticks to uniform sampling otherwise.
4.1 Ablation study
As already mentioned, several works (Loshchilov & Hutter, 2015; Schaul et al., 2015) use the loss value, directly or indirectly, to generate
sampling distributions. In this section, we present experiments that validate
the superiority of our method with respect to the loss in terms of variance
reduction. For completeness, in the supplementary material we include a
theoretical analysis that explains why sampling based on the loss also
achieves variance reduction during the late stages of training.
Our experimental setup is as follows: we train a wide residual network
(Zagoruyko & Komodakis, 2016) on the CIFAR100 dataset (Krizhevsky, 2009),
following closely the training procedure of Zagoruyko & Komodakis (2016) (the
details are presented in § 4.2). Subsequently, we sample $1,024$
images uniformly at random from the dataset. Using the weights of the trained
network, at intervals of $3,000$ updates, we resample $128$ images from the
large batch of $1,024$ images using uniform sampling or importance
sampling with probabilities proportional to the loss, our
upper-bound or the gradient-norm. The gradient-norm is
computed by running the backpropagation algorithm with a batch size of 1.
Figure 1 depicts the variance reduction achieved
with every sampling scheme in comparison to uniform. We measure this
directly as the distance between the mini-batch gradient and the batch gradient
of the $1,024$ samples. For robustness we perform the sampling $10$ times and
report the average. We observe that our upper bound and the gradient norm
result in very similar variance reduction, meaning that the bound is relatively
tight and that the produced probability distributions are highly correlated.
This can also be deduced
by observing figure 2, where the probabilities proportional
to the loss and the upper-bound are plotted against the optimal
ones (proportional to the gradient-norm). We observe that our upper
bound is almost perfectly correlated with the gradient norm, in stark contrast
to the loss which is only correlated at the regime of very small gradients.
Quantitatively the sum of squared error of $16,384$ points in
figure 2 is $0.017$ for the loss and $0.002$ for our
proposed upper bound.
Furthermore, we observe that sampling hard examples (with high loss), increases
the variance, especially in the beginning of training. Similar behaviour has
been observed in problems such as embedding learning where semi-hard sample
mining is preferred over sampling using the loss (Wu et al., 2017; Schroff et al., 2015).
4.2 Image classification
In this section, we use importance sampling to improve the training of a wide
residual network on CIFAR10 and CIFAR100. We follow the experimental setup of
Zagoruyko & Komodakis (2016), specifically we train a wide resnet 28-2 with SGD
with momentum. We use batch size $128$, weight decay $0.0005$, momentum $0.9$,
initial learning rate $0.1$ divided by $5$ after $20,000$ and $40,000$
parameter updates. Finally, we train for a total of $50,000$ iterations. In
order for our history based baselines to be compatible with the data
augmentation of the CIFAR images, we pre-augment both datasets to generate $1.5\times 10^{6}$ images for each one. It is important to mention, that our method
does not have this limitation since it can work on infinite datasets in a true
online fashion. To compare between methods, we use a learning rate schedule
based on wall-clock time and we also fix the total seconds available for
training. A faster method should have smaller training loss and test error
given a specific time during training.
For this experiment, we compare the proposed method to uniform,
loss, online batch selection by Loshchilov & Hutter (2015) and the
history based sampling of Schaul et al. (2015). For the method of
Schaul et al. (2015), we use their proportional sampling since the
rank based is very similar to Loshchilov & Hutter (2015) and we select the
best parameters from the grid $a=\{0.1,0.5,1.0\}$ and $\beta=\{0.5,1.0\}$. Similarly, for online batch selection, we use $s=\{1,10,10^{2}\}$ and
a recomputation of all the losses every $r=\{600,1200,3600\}$ updates.
For our method, we use a presampling size of $640$. One of the goals of this
experiment is to show that even a smaller reduction in variance can effectively
stabilize training and provide wall-clock time speedup; thus we set $\tau_{th}=1.5$. For robustness, we perform $3$ independent runs and report the average.
The results are depicted in figure 3. We observe that in the
relatively easy CIFAR10 dataset, all methods can provide some speedup over
uniform sampling. However, the case is very different for the more complicated
CIFAR100 where only sampling with our proposed upper-bound to the
gradient norm reduces the variance of the gradients and provides faster
convergence. Examining the training evolution in detail, we observe that on
CIFAR10 our method is the only one that achieves a significant improvement in
the test error even in the first stages of training ($4,000$ to $8,000$
seconds). Quantitatively, on CIFAR10 we achieve more than an order of magnitude
lower training loss and $8\%$ lower test error from $0.087$ to $0.079$ while on
CIFAR100 approximately $3$ times lower training loss and $5\%$ lower test error
from $0.34$ to $0.32$ compared to uniform sampling.
At this point, we would also like to discuss the performance of the loss
compared to other methods that also select batches based on this metric. Our
experiments show, that using “fresh” values for the loss combined with a
warmup stage so that importance sampling is not started too early outperforms
all the other baselines on the CIFAR10 dataset.
4.3 Fine-tuning
Our second experiment shows the application of importance sampling to the
significant task of fine tuning a pre-trained large neural network on a new
dataset. This task is of particular importance because there exists an
abundance of powerful models pre-trained on large datasets such as ImageNet
(Deng et al., 2009).
The details of our experimental setup are the following, we fine-tune a
ResNet-50 (He et al., 2015) that is previously trained on the ImageNet ILSVRC 2012
dataset. We replace the last classification layer and then train the whole
network end-to-end to classify indoor images among 67 possible
categories (Quattoni & Torralba, 2009). We use SGD with learning rate
$10^{-3}$ and momentum $0.9$. We set the batch size to $16$ and for our
importance sampling algorithm we pre-sample $48$. The variance reduction
threshold is set to $2$ as designated by equation 29.
To assess the performance of both our algorithm and our gradient norm
approximation, we compare the convergence speed of our importance sampling
algorithm using our upper-bound and using the loss. Once again,
for robustness, we run $3$ independent runs and report the average.
The results of the experiment are depicted in figure 4. As
expected, importance sampling is very useful for the task of fine-tuning since
a lot of samples are handled correctly very early in the training process. Our
upper-bound, once again, greatly outperforms sampling proportionally to
the loss when the network is large and the problem is non trivial. Compared to
uniform sampling, in just half an hour importance sampling has converged close
to the best performance ($28.06\%$ test error) that can be expected on this
dataset without any data augmentation or multiple crops (Razavian et al., 2014),
while uniform achieves only $33.74\%$.
4.4 Pixel by Pixel MNIST
The goal of our final experiment is to show that our method is not limited to
convolutional networks with ReLU activations. To that end, we use our
importance sampling algorithm to accelerate the training of an LSTM in a
sequence classification problem. We use the pixel by pixel classification of
randomly permuted MNIST digits (LeCun et al., 2010), as defined by
Le et al. (2015). The problem may seem trivial at first, however as shown
by Le et al. (2015) it is particularly suited to benchmarking the training
of recurrent neural networks, due to the long range dependency problems
inherent in the dataset ($784$ time steps).
The details of our experimental setup are the following. We fix a permutation
matrix for all the pixels to generate a training set of $60,000$ samples with
$784$ time steps each. Subsequently, we train an LSTM
(Hochreiter & Schmidhuber, 1997) with $128$ dimensions in the hidden space,
$\text{tanh}(\cdot)$ as an activation function and $\text{sigmoid}(\cdot)$ as
the recurrent activation function. Finally, we use a linear classifier on top
of the LSTM to choose a digit based on the hidden representation.
To train the aforementioned architecture, we use the Adam optimizer
(Kingma & Ba, 2014) with a learning rate of $10^{-3}$ and a batch size of
$32$. We have also found gradient clipping to be necessary for the training
not to diverge; thus we clip the norm of all gradients to $1$.
The results of the experiment are depicted in figure 5. For
both the loss and our proposed upper-bound, importance sampling
starts at around $2,000$ seconds by setting $\tau_{th}=1.8$ and the presampling
size to $128$. We could set $\tau_{th}=2.33$
(equation 29) which would only result in our algorithm
being more conservative and starting importance sampling later. We clearly
observe that sampling proportionally to the loss hurts the convergence in this
case. On the other hand, our algorithm achieves $20\%$ lower training loss and
$7\%$ lower test error in the given time budget.
5 Conclusions
We have presented an efficient algorithm for accelerating the training of deep
Neural Networks using importance sampling. Our algorithm takes advantage of a
novel upper bound to the gradient norm of any Neural Network that can be
computed in a single forward pass. In addition, we show an equivalence of the
variance reduction achieved with importance sampling to increasing the batch
size; thus we are able to quantify both the variance reduction and the speedup
and intelligently decide when to stop sampling uniformly.
Our experiments show that our algorithm is effective in reducing the training
time for several tasks both on image and sequence data. More importantly, we
show that not all data points matter equally in the duration of training, which
can be exploited to gain a speedup or better quality gradients or both.
Our analysis opens several avenues of future research. The two most important
ones that were not investigated in this work are automatically tuning the
learning rate based on the variance of the gradients and decreasing the batch
size. It has been theorized that variance is advantageous for improving the
generalization performance of a Neural Network. We can keep the variance stable
and increase the convergence speed by automatically increasing the learning
rate proportionally to the batch increment. Secondly, instead of sampling more
data and re-sampling afterwards, we could instead start decreasing the batch
size thus keeping again the variance stable but reducing the time per update.
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Appendix
Appendix A Ablation study on $B$
The only hyperparameter that is somewhat hard to define in our algorithm is the
pre-sampling size $B$. As mentioned in the main paper, it controls the maximum
possible variance reduction and also how much wall-clock time one iteration
with importance sampling will require.
In figure 6 we depict the results of training with
importance sampling and different pre-sampling sizes on CIFAR10. We follow the
same experimental setup as in the paper.
We observe that larger presampling size results in lower training loss, which
follows from our theory since the maximum variance reduction is smaller with
small $B$. In this experiment we use the same $\tau_{th}$ for all the methods
and we observe that $B=384$ reaches first to $0.6$ training loss. This is
justified because computing the importance for $1,024$ samples in the beginning
of training is wasteful according to our analysis.
According to this preliminary ablation study for $B$, we conclude that choosing
$B=kb$ with $2<k<6$ is a good strategy for achieving a speedup. However,
regardless of the choice of $B$, pairing it with a threshold $\tau_{th}$
designated by the analysis in the paper guarantees that the algorithm will be
spending time on importance sampling only when the variance can be greatly
reduced.
Appendix B Importance Sampling with the Loss
In this section we will present a small analysis that provides intuition
regarding using the loss as an approximation or an upper bound to the per
sample gradient norm.
Let $\mathcal{L}(\psi,y):D\to\mathbb{R}$ be either the negative log likelihood through
a sigmoid or the squared error loss function defined respectively as
$$\displaystyle\mathcal{L}_{1}(\psi,y)$$
$$\displaystyle=-\log\left(\frac{\exp(y\psi)}{1+\exp(y\psi)}\right)$$
$$\displaystyle y\in\{-1,1\}\quad D=\mathbb{R}$$
(30)
$$\displaystyle\mathcal{L}_{2}(\psi,y)$$
$$\displaystyle=\left\lVert y-\psi\right\rVert_{2}^{2}$$
$$\displaystyle y\in\mathbb{R}^{d}\quad D=\mathbb{R}^{d}$$
Given our upper bound to the gradient norm, we can write
$$\displaystyle\left\lVert\nabla_{\theta_{t}}\mathcal{L}(\Psi(x_{i};\theta_{t}),%
y_{i})\right\rVert_{2}\leq L\rho\left\lVert\nabla_{\psi}\mathcal{L}(\Psi(x_{i}%
;\theta_{t}),y_{i})\right\rVert_{2}.$$
(31)
Moreover, for the losses that we are considering, when $\mathcal{L}(\psi,y)\to 0$
then $\left\lVert\nabla_{\psi}\mathcal{L}(\Psi(x_{i};\theta_{t}),y_{i})\right\rVert_%
{2}\to 0$. Using this
fact in combination to equation 31, we claim that so does the
per sample gradient norm thus small loss values imply small gradients. However,
large loss values are not well correlated with the gradient norm which can also
be observed in § 4.1 in the paper.
To summarize, we conjecture that due to the above facts, sampling
proportionally to the loss reduces the variance only when the majority of the
samples have losses close to 0. Our assumption is validated from our
experiments, where the loss struggles to achieve a speedup in the early
stages of training where most samples still have relatively large loss values. |
Perfect refiners for permutation group backtracking algorithms
Christopher Jefferson,
Rebecca Waldecker,
Wilf A. Wilson
Abstract
Backtrack search is a fundamental technique for computing with finite permutation groups, which has been formulated in terms of points, ordered partitions, and graphs. We provide a framework for discussing the most common forms of backtrack search in a generic way.
We introduce the concept of perfect refiners to better understand and compare the pruning power available in these different settings.
We also present a new formulation of backtrack search, which allows the use of graphs with additional vertices, and which is implemented in the software package Vole.
For each setting,
we classify the groups and cosets for which there exist perfect refiners.
Moreover, we describe perfect refiners for many naturally-occurring examples of stabilisers and transporter sets, including applications to normaliser
and subgroup conjugacy problems for 2-closed groups.
1 Introduction
A careful backtrack search through the elements of a symmetric group is the fastest general purpose technique, in practice, for solving many computational problems in finite permutation groups.
This includes finding normalisers and intersections of subgroups, and
stabilisers of sets and graphs.
At the heart of these backtracking algorithms are refiners.
Refiners are functions which are used to prune redundant parts of the search.
They provide the main facility for taking into account the structural aspects of the problem at hand, and making clever deductions. Better refiners are more likely to prune more efficiently, and can reduce search times by orders of magnitude.
Refiners, therefore, are an obvious and ongoing target of further research (see
for example [JPW19] for recent work in this area), and they are the focus of this article.
Roughly speaking, backtracking algorithms organise permutations as bijective maps
on sets of combinatorial structures. The first version was originally formulated by Sims,
organised around a base and strong generating set (see [Sim71]).
Two decades later, Leon reformulated this in terms of ordered partitions [Leo91, Leo97], and thereby obtained significantly improved performance in many cases.
This technique was enhanced with orbital graphs in [The97, JPW19].
More recently, this use of orbital graphs inspired a reformulation in [JPWW21]
around labelled digraphs.
The motivation for organising a search around more sophisticated structures is to allow some sets of permutations to be represented with greater fidelity,
so that refiners can prune more effectively.
However, when choosing the appropriate search infrastructure for a particular problem, we need to keep in mind that more complicated structures do not always offer better pruning, that they are more expensive to compute with, and that it is easy to find relatively small cases where all existing backtracking algorithms exhibit poor performance. This holds especially for subgroup conjugacy and normaliser problems.
It is therefore important to develop tools for understanding which problems are well suited to which kinds of backtrack search and refiners, and to understand the limitations of the existing techniques, with the aim of developing improvements.
This article has three main purposes.
Firstly, we introduce the concept of perfect refiners, which are those with maximal pruning power within a given search framework. This concept is also meant to initiate a discussion of the quality of refiners, both within and across various backtrack search techniques.
Secondly, we describe an extension to the graph backtracking technique that permits the use of graphs with additional vertices, and which enables many more perfect refiners than previous techniques. There already is an implementation available, in Vole [CJW21].
Finally, we partially classify, and give examples of, perfect refiners in each setting of backtrack search.
Our results show that graph backtracking and extended graph backtracking admit perfect refiners for many natural problems. This goes some way to explaining the experimental data in [JPWW21, Section 9], which demonstrated that many problems could be solved without actual backtracking happening during the search.
Furthermore, we find that extended backtracking enables improved refiners for normaliser and conjugacy problems.
In particular, we give refiners for normalisers and conjugacy
within the extended graph backtracking framework that are perfect in some cases.
This article is organised as follows.
In Section 2, we present some necessary background definitions and notation;
in particular, we briefly describe backtrack search in finite symmetric groups.
We then give definitions and results about arbitrary refiners in Section 3 and perfect refiners in Section 4.
In Section 5, we examine perfect refiners for stabiliser and transporter problems.
In Section 6, we introduce extended graph backtracking.
In Section 7,
we compare the various kinds of backtrack search and their potentials for perfect refinement.
In Section 8,
we give examples of perfect refiners for many natural problems.
We conclude with some final remarks in Section 9.
Acknowledgments
The results discussed here build on work in [JPWW21], and we include elements of an earlier draft of that article [JPWW19, Section 5.1].
We therefore thank the VolkswagenStiftung (Grant no. 93764)
and the Royal Society (Grant code URF\R\180015)
again for their financial support of this earlier work.
For financial support during the more recent advances,
we thank the DFG (Grant no. WA 3089/9-1) and again the Royal Society
(Grant codes RGF\EA\181005 and URF\R\180015).
2 Background and notation
In this section, we give some notation and definitions, and we introduce background concepts including
backtrack search in the symmetric group.
Our terminology and notation closely follows that of [JPWW21].
Sets and lists feature prominently in this article: a set is an unordered duplicate-free collection of objects, whereas a list has an ordering, and may contain duplicates.
If $L$ and $K$ are lists, then $|L|$ denotes the number of elements in $L$, $L\|K$ denotes the concatenation of $L$ and $K$,
and if $i\in\{1,\ldots,|L|\}$, then $L[i]$ denotes the element of $L$ in the $i$-th position.
Throughout this article, $\Omega$ is a nonempty finite
set, and $\operatorname{Sym}(\Omega)$ is the symmetric group on $\Omega$.
We denote any action of $\operatorname{Sym}(\Omega)$ by exponentiation.
Given an action of $\operatorname{Sym}(\Omega)$ on a set $\mathcal{O}$, we iteratively define an action of $\operatorname{Sym}(\Omega)$ induced on the set of all subsets of $\mathcal{O}$, and on the set of all finite lists with elements in $\mathcal{O}$.
In more detail, for all $x_{1},\ldots,x_{k}\in\mathcal{O}$ and $g\in\operatorname{Sym}(\Omega)$,
we define $\{x_{1},\ldots,x_{k}\}^{g}\coloneqq\{x_{1}^{g},\ldots,x_{k}^{g}\}$
and $[x_{1},\ldots,x_{k}]^{g}\coloneqq[x_{1}^{g},\ldots,x_{k}^{g}]$.
For example, if $\Omega\coloneqq\{1,2,3,4\}$, then the image of the set-of-lists
$\{[1,2],[2,3],[3,2]\}$ under the permutation $(1\,2)(3\,4)$ is
$$\{[1,2],[2,3],[3,2]\}^{(1\,2)(3\,4)}=\{[1,2]^{(1\,2)(3\,4)},[2,3]^{(1\,2)(3\,4)},[3,2]^{(1\,2)(3\,4)}\}=\{[2,1],[1,4],[4,1]\}.$$
Let $\mathcal{O}$ be a set on which $\operatorname{Sym}(\Omega)$ acts.
We define $\operatorname{Stacks}(\mathcal{O})$ to be the set of all finite lists with entries in $\mathcal{O}$
(we call these lists stacks), and $\operatorname{Stacks}(\mathcal{O})^{\operatorname{Stacks}(\mathcal{O})}$ to be the set of all functions from $\operatorname{Stacks}(\mathcal{O})$ to itself.
If $x,y\in\mathcal{O}$,
then the transporter set from $x$ to $y$ in $\operatorname{Sym}(\Omega)$ is denoted by $\operatorname{Transp}(x,\,y)\coloneqq\{g\in\operatorname{Sym}(\Omega):x^{g}=y\}$, and the stabiliser of $x$ in $\operatorname{Sym}(\Omega)$ is the subgroup $\operatorname{Stab}(x)\coloneqq\operatorname{Transp}(x,\,x)$ of all permutations in $\operatorname{Sym}(\Omega)$ that fix $x$ under the action.
Note that
for all $h\in\operatorname{Sym}(\Omega)$, $\operatorname{Transp}(x,\,y^{h})=\operatorname{Transp}(x,\,y)\cdot h$.
The setwise stabiliser of a collection $x_{1},\ldots,x_{n}$ of objects is the stabiliser of the set $\{x_{1},\ldots,x_{n}\}$,
whereas the pointwise stabiliser is $\bigcap_{i=1}^{n}\operatorname{Stab}(x_{i})$, which is equal to the stabiliser of any list of the objects, such as $[x_{1},\ldots,x_{n}]$.
The backtrack search techniques considered in this article are organised around lists of points, ordered partitions, and graphs.
An ordered partition of a set $V$ is a list of nonempty disjoint subsets of $V$
whose union is $V$.
A graph on a set $V$ is a pair $(V,E)$ of vertices $V$ and a set of $2$-subsets of $V$ called edges.
Similarly, a digraph on $V$ is a pair $(V,A)$ of vertices $V$ and a set of ordered pairs in $V$ called arcs.
The action of $\operatorname{Sym}(V)$ on the sets of all graphs and digraphs on $V$ is defined, respectively, by $(V,E)^{g}\coloneqq(V,E^{g})$ and $(V,A)^{g}\coloneqq(V,A^{g})$ for all $g\in\operatorname{Sym}(V)$, edge sets $E$, and arc sets $A$.
A labelled digraph is a digraph with an assignment of a label to each
vertex and arc. See [JPWW21, Section 2.1] for more information.
Whenever, in this article, we write that ‘$(V,E)$ is a graph’, then this means that $V$ is the set of vertices and $E$
is the set of edges as explained above. In the same way, we use the remaining notation introduced here without further explanation.
2.1 Classical backtracking, partition backtracking, and graph backtracking
We briefly summarise the concept of backtrack search in $\operatorname{Sym}(\Omega)$, and introduce some terminology.
Let $U_{1},\ldots,U_{k}$ be subsets of $\operatorname{Sym}(\Omega)$ for some $k\in\mathbb{N}$, and suppose that we wish to search for the intersection $U_{1}\cap\cdots\cap U_{k}$.
For this technique to be useful in practice, it should be computationally cheap, for each $i\in\{1,\ldots,k\}$,
to determine whether any given element of $\operatorname{Sym}(\Omega)$ is contained in $U_{i}$.
Many typical search problems can be formulated in this way.
For example, if $U_{1}$ is a subgroup of $\operatorname{Sym}(\Omega)$ given by generators and $U_{2}\coloneqq\operatorname{Transp}(\Gamma,\,\Delta)$ for some graphs $\Gamma$ and $\Delta$ with vertex set $\Omega$,
then searching for an element of $U_{1}\cap U_{2}$ solves the graph isomorphism problem for $\Gamma$ and $\Delta$ in $U_{1}$.
Let $\mathcal{O}$ be a set on which $\operatorname{Sym}(\Omega)$ acts (such as $\Omega$ itself, or the set of all ordered partitions of $\Omega$).
In this paper we will present all search techniques in a common framework. The fundamental idea of this framework is to organise a backtrack search for $U_{1}\cap\cdots\cap U_{k}$ around a pair of stacks of objects in $\mathcal{O}$.
At any point in the algorithm, when the pair of stacks is $(S,T)$ for some $S,T\in\operatorname{Stacks}(\mathcal{O})$, the set of permutations being searched is $\operatorname{Transp}(S,\,T)$.
The stacks are initially empty, which means that the search begins with the whole symmetric group.
In each step down into the recursion, new stacks are appended to the existing ones. This may be done by a refiner, in an attempt to remove redundant parts of the search space (Section 3), or by a splitter, in order to divide the search space into smaller parts that can be searched recursively (see [JPWW21, Section 6]).
For such a backtrack search to be practical, the set $\mathcal{O}$ should be easy to compute with, and in particular, it should be relatively cheap to compute stabilisers and transporter sets in $\operatorname{Sym}(\Omega)$ of elements in $\mathcal{O}$, or at least to obtain close overapproximations of them (see [JPWW21, Section 5]).
On the other hand, the set $\mathcal{O}$ should be sufficiently rich and varied that refiners can construct stacks in $\mathcal{O}$ that encode useful information about the problem at hand.
So far, backtrack search in finite symmetric groups has been formulated and implemented in several settings.
We use the term classical backtracking for the case that $\mathcal{O}=\Omega$; this is essentially the original backtrack search in $\operatorname{Sym}(\Omega)$ introduced in [Sim71], although presented differently.
Partition backtracking is backtrack search where the objects are ordered partitions of $\Omega$; this is essentially the technique introduced in [Leo91].
We use the term graph backtracking when the objects are labelled digraphs on $\Omega$ [JPWW21].
In Section 6, we introduce an advancement of this latter technique, which we call extended graph backtracking.
3 Refiners for backtrack search
Throughout this section and Sections 4 and 5, we let $\mathcal{O}$ be a set on which $\operatorname{Sym}(\Omega)$ acts, so that the setting is backtrack search organised around stacks in $\mathcal{O}$. In particular, all definitions and results are relative to the set $\mathcal{O}$, even if we do not explicitly repeat that every single time.
Definition 3.1.
A refiner for a set of permutations $U\subseteq\operatorname{Sym}(\Omega)$ is a pair $(f_{L},f_{R})$ of functions in $\operatorname{Stacks}(\mathcal{O})^{\operatorname{Stacks}(\mathcal{O})}$ such that:
$$\text{For all}\ S,T\in\operatorname{Stacks}(\mathcal{O}),\ U\cap\operatorname{Transp}(S,\,T)\subseteq\operatorname{Transp}(f_{L}(S),\,f_{R}(T)).$$
($$\ast$$)
In practice, we focus on refiners for subsets of $\operatorname{Sym}(\Omega)$ that are subgroups, cosets of subgroups, or empty.
We remark that any pair of functions in $\operatorname{Stacks}(\mathcal{O})^{\operatorname{Stacks}(\mathcal{O})}$ is a refiner for the empty set and that it is necessary to include the empty set
as a possibility because, for example, it is common to search for transporter sets, which may be empty.
Let $(f_{L},f_{R})$ be a refiner for a set $U\subseteq\operatorname{Sym}(\Omega)$,
let $S,T\in\operatorname{Stacks}(\mathcal{O})$ with $|S|=|T|$,
and suppose that we are part way through a search for an intersection of subsets of $\operatorname{Sym}(\Omega)$ that includes $U$,
with $\operatorname{Transp}(S,\,T)$ as the current search space.
In this situation, the aim of applying this refiner to the stacks $S$ and
$T$ is to prune elements of $\operatorname{Sym}(\Omega)\setminus U$ from the current search space, by moving to the transporter set corresponding to the lengthened stacks ${S}\mathop{\|}{f_{L}}(S)$ and ${T}\mathop{\|}{f_{R}}(T)$.
By ($\ast$ ‣ 3.1), $\operatorname{Transp}({S}\mathop{\|}{f_{L}}(S),\,{T}\mathop{\|}{f_{R}}(T))$ retains the elements of $\operatorname{Transp}(S,\,T)$ that are in $U$,
but on the other hand, $\operatorname{Transp}({S}\mathop{\|}{f_{L}}(S),\,{T}\mathop{\|}{f_{R}}(T))$ is contained in $\operatorname{Transp}(S,\,T)$, perhaps properly, and may therefore lack some elements of $\operatorname{Transp}(S,\,T)$ that are not in $U$.
In particular, if $\operatorname{Transp}({S}\mathop{\|}{f_{L}}(S),\,{T}\mathop{\|}{f_{R}}(T))$ is empty, then the search can backtrack.
Leon used the term $\mathcal{P}$-refinement in his work on partition backtracking [Leo91, Leo97] for a similar concept
that is essentially compatible with our notion of a refiner,
although he presents it in a significantly different fashion.
Definition 3.1 matches the notion of a refiner used in [JPWW21, Section 5].
We remark that the definition of a refiner guarantees no particular success at pruning.
For example, if $\iota$ is the identity map on $\operatorname{Stacks}(\mathcal{O})$, and if $\varepsilon$ is the constant map on $\operatorname{Stacks}(\mathcal{O})$ whose image is the empty stack in $\operatorname{Stacks}(\mathcal{O})$,
then $(\iota,\iota)$ and $(\varepsilon,\varepsilon)$ are refiners for every subset of $\operatorname{Sym}(\Omega)$, even though neither performs any pruning. Thus it is desirable to have a measure of the quality of a refiner.
In the following section we introduce perfect refiners, which are those that have maximal pruning power.
To simplify some forthcoming exposition, we introduce a way of applying permutations to functions in $\operatorname{Stacks}(\mathcal{O})^{\operatorname{Stacks}(\mathcal{O})}$.
It is straightforward to verify that this defines an action of $\operatorname{Sym}(\Omega)$.
Notation 3.2.
For any $f\in\operatorname{Stacks}(\mathcal{O})^{\operatorname{Stacks}(\mathcal{O})}$ and $x\in\operatorname{Sym}(\Omega)$, we define $f^{x}$ as follows:
$f^{x}(S)\coloneqq f(S^{x^{-1}})^{x}$ for all $S\in\operatorname{Stacks}(\mathcal{O})$.
Lemma 3.3.
For all $f\in\operatorname{Stacks}(\mathcal{O})^{\operatorname{Stacks}(\mathcal{O})}$ and $x\in\operatorname{Sym}(\Omega)$, the map $f^{x}$ explained above is in
$\operatorname{Stacks}(\mathcal{O})^{\operatorname{Stacks}(\mathcal{O})}$.
Moreover, $\operatorname{Sym}(\Omega)$ acts on $\operatorname{Stacks}(\mathcal{O})^{\operatorname{Stacks}(\mathcal{O})}$.
The following results show the close relationships between the two functions that comprise a refiner for a subgroup (Lemma 3.4) or more generally for a coset of a subgroup (Lemma 3.5).
We note that Lemma 3.5 is a reformulation of [JPWW21, Lemma 4.6] that uses Notation 3.2.
Lemma 3.4 (Lemma 4.4 in [JPWW21]; cf. [Leo91, Lemma 6] and [Leo97, Prop 2]).
If $(f_{L},f_{R})$ is a refiner for a subgroup of $\operatorname{Sym}(\Omega)$, then $f_{L}=f_{R}$.
Proof.
Suppose that $(f_{L},f_{R})$ is a refiner for a group $G$ and let $S\in\operatorname{Stacks}(\mathcal{O})$.
Since $1_{G}\in\operatorname{Transp}(S,\,S)$, it follows by ($\ast$ ‣ 3.1) that $1_{G}\in\operatorname{Transp}(f_{L}(S),\,f_{R}(S))$, which means that $f_{L}(S)=f_{L}(S)^{1_{G}}=f_{R}(S)$.
∎
Lemma 3.5.
Let $G\leq\operatorname{Sym}(\Omega)$, $x\in\operatorname{Sym}(\Omega)$, and $f_{L},f_{R}\in\operatorname{Stacks}(\mathcal{O})^{\operatorname{Stacks}(\mathcal{O})}$.
Then $(f_{L},f_{R})$ is a refiner for the right coset $Gx$ if and only if $(f_{L},f_{L})$ is a refiner for $G$ and $f_{R}={f_{L}}^{x}$.
By Lemmas 3.4 and 3.5, a refiner
for a group or coset is built from a single function from $\operatorname{Stacks}(\mathcal{O})$ to itself. The following lemma gives equivalent conditions for such a function to yield a refiner (always with respect to the set $\mathcal{O}$, as mentioned earlier):
Lemma 3.6.
Let $G\leq\operatorname{Sym}(\Omega)$ and $f\in\operatorname{Stacks}(\mathcal{O})^{\operatorname{Stacks}(\mathcal{O})}$.
The following are equivalent:
(i)
$(f,f)$ is a refiner for $G$.
(ii)
$f(S^{x})=f(S)^{x}$ for all $S\in\operatorname{Stacks}(\mathcal{O})$ and $x\in G$.
(iii)
$G\leq\operatorname{Stab}(f)$.
Proof.
Suppose that (i) holds and let $S\in\operatorname{Stacks}(\mathcal{O})$ and $x\in G$.
By Definition 3.1, $G\cap\operatorname{Transp}(S,\,S^{x})\subseteq\operatorname{Transp}(f(S),\,f(S^{x}))$.
Then (ii) holds, since $x\in G\cap\operatorname{Transp}(S,\,S^{x})$.
Conversely, suppose that (ii) holds and let $S,T\in\operatorname{Stacks}(\mathcal{O})$.
If $x\in G\cap\operatorname{Transp}(S,\,T)$, then $T=S^{x}$, and since $f(S)^{x}=f(S^{x})$ by assumption, it follows that $x\in\operatorname{Transp}(f(S),\,f(T))$. Therefore (i) holds by Definition 3.1.
The
equivalence of (ii) and (iii) is clear from Notation 3.2 and the definition of $\operatorname{Stab}(f)$.
∎
Next, we show a way for refiners to be combined to give a refiner for an intersection of sets.
Notation 3.7.
For all $f,g\in\operatorname{Stacks}(\mathcal{O})^{\operatorname{Stacks}(\mathcal{O})}$,
we define the function $f\mathop{\|}g\in\operatorname{Stacks}(\mathcal{O})^{\operatorname{Stacks}(\mathcal{O})}$ by
$(f\mathop{\|}g)(S)\coloneqq f(S)\mathop{\|}g(S)$ for all $S\in\operatorname{Stacks}(\mathcal{O})$.
Lemma 3.8.
Let $(f,\sigma)$ be a refiner for the set $U\subseteq\operatorname{Sym}(\Omega)$,
and let $(g,\tau)$ be a refiner for $V\subseteq\operatorname{Sym}(\Omega)$.
Then $(f\mathop{\|}g,\sigma\mathop{\|}\tau)$ is a refiner for $U\cap V$.
Proof.
We show that the set $U\cap V$ and the pair of functions $(f\mathop{\|}g,\sigma\mathop{\|}\tau)$ satisfy ($\ast$ ‣ 3.1).
Let $S,T\in\operatorname{Stacks}(\mathcal{O})$.
If $|f(S)|\neq|\sigma(T)|$, then $\operatorname{Transp}(f(S),\,\sigma(T))=\varnothing$, and so $U\cap\operatorname{Transp}(S,\,T)=\varnothing$, since $(f,\sigma)$ is a refiner for $U$.
In particular, $(U\cap V)\cap\operatorname{Transp}(S,\,T)=\varnothing$.
If instead $|f(S)|=|\sigma(T)|$, then
$$\displaystyle(U\cap V)\cap\operatorname{Transp}(S,\,T)$$
$$\displaystyle=(U\cap\operatorname{Transp}(S,\,T))\cap(V\cap\operatorname{Transp}(S,\,T))$$
$$\displaystyle\subseteq\operatorname{Transp}(f(S),\,\sigma(T)))\cap\operatorname{Transp}(g(S),\,\tau(T))$$
$$\displaystyle=\operatorname{Transp}(f(S)\|g(S),\,\sigma(T)\mathop{\|}\tau(T))$$
$$\displaystyle=\operatorname{Transp}((f\mathop{\|}g)(S),\,(\sigma\mathop{\|}\tau)(T)).\qed$$
We remark that the $\|$ operation is associative, and that we may repeatedly apply Lemma 3.8 to obtain a refiner for any finite intersection of sets, given a refiner for each set.
4 Perfect refiners
Lemma 4.1.
Let $U\subseteq\operatorname{Sym}(\Omega)$,
let $f_{L},f_{R}\in\operatorname{Stacks}(\mathcal{O})^{\operatorname{Stacks}(\mathcal{O})}$, and suppose that
$$U\cap\operatorname{Transp}(S,\,T)=\operatorname{Transp}({S}\mathop{\|}{f_{L}}(S),\,{T}\mathop{\|}{f_{R}}(T))$$
($$\circledast$$)
for all $S,T\in\operatorname{Stacks}(\mathcal{O})$ with $|S|=|T|$.
Then $(f_{L},f_{R})$ is a refiner for $U$.
Proof.
To show that ($\ast$ ‣ 3.1) holds, let $S,T\in\operatorname{Stacks}(\mathcal{O})$.
If $\operatorname{Transp}(S,\,T)=\varnothing$, then this is clear, so suppose otherwise; in particular, suppose that $|S|=|T|$. Then
$$\displaystyle U\cap\operatorname{Transp}(S,\,T)$$
$$\displaystyle=\operatorname{Transp}({S}\mathop{\|}{f_{L}}(S),\,{T}\mathop{\|}{f_{R}}(T))$$
$$\displaystyle=\operatorname{Transp}(S,\,T)\cap\operatorname{Transp}(f_{L}(S),\,f_{R}(T))$$
$$\displaystyle\subseteq\operatorname{Transp}(f_{L}(S),\,f_{R}(T)).\qed$$
Definition 4.2.
Refiners with the property ($\circledast$ ‣ 4.1) from Lemma 4.1 are called perfect refiners (with respect to $\mathcal{O}$, which we will usually omit).
We give several examples of perfect refiners in Section 8.
Note that a refiner for a set $U\subseteq\operatorname{Sym}(\Omega)$ is also a refiner for any proper subset of $U$, but never a perfect refiner.
The equation ‘$|S|=|T|$’ in ($\circledast$ ‣ 4.1) is included for
convenience:
in principle, two stacks of different lengths (whose transporter set is empty) could be extended to stacks with nonempty transporter set.
However, a refiner is only ever applied to pairs of stacks of equal lengths, so we remove the need to deal with this complication.
During a search, a perfect refiner for $U$ can be used to extend the stacks $S$ and $T$ (representing the current search space $\operatorname{Transp}(S,\,T)$) to obtain the potentially-smaller search space $\operatorname{Transp}(S,\,T)\cap U$.
Thus a perfect refiner for $U$ allows $U$ to be represented with full fidelity,
and no effort must be wasted considering elements of $\operatorname{Sym}(\Omega)$ that are not in $U$. In other words:
Perfect refiners are the refiners with maximal pruning power.
A perfect refiner therefore needs to be applied at most once in any branch of search, and should therefore be applied at the root node to avoid unnecessary repetition.
It is for this reason that perfect refiners often comprise constant functions in practice; see Lemma 4.3.
Note that if each set $U_{i}$ in a search for $U_{1}\cap\cdots\cap U_{n}$ is given with a perfect refiner, then the search can terminate at the root node without splitting or backtracking.
This is because the search begins with $S$ and $T$ being empty stacks, and so their transporter set is initially $\operatorname{Sym}(\Omega)$. Applying the perfect refiners in turn then gives stacks with transporter set $U_{1}\cap\cdots\cap U_{n}$, as required.
In the following lemmas, we see that constructing a perfect refiner for a set $U$ is equivalent to finding a pair of stacks in $\mathcal{O}$ with transporter set $U$.
It follows that the existence of a perfect refiner for any given subset of
$\operatorname{Sym}(\Omega)$, in backtrack search organised around stacks in $\mathcal{O}$, depends on the choice of $\mathcal{O}$. But as before, we will not always add “with respect to $\mathcal{O}$”.
Lemma 4.3.
Let $U\subseteq\operatorname{Sym}(\Omega)$, and suppose there exist $A,B\in\operatorname{Stacks}(\mathcal{O})$ such that $U\subseteq\operatorname{Transp}(A,\,B)$.
Let $f_{A}$ and $f_{B}$ be constant functions on $\operatorname{Stacks}(\mathcal{O})$ with images $A$ and $B$, respectively.
Then $(f_{A},f_{B})$ is a refiner for $U$.
Moreover, if $U=\operatorname{Transp}(A,\,B)$, then this refiner is perfect.
Proof.
The pair of functions satisfy the condition in Definition 3.1, since for all $S,T\in\operatorname{Stacks}(\mathcal{O})$:
$$\operatorname{Transp}(S,\,T)\cap U\subseteq U\subseteq\operatorname{Transp}(A,\,B)=\operatorname{Transp}(f_{A}(S),\,f_{B}(T)).$$
If $U=\operatorname{Transp}(A,\,B)$, then for all $S,T\in\operatorname{Stacks}(\mathcal{O})$ with $|S|=|T|$, condition ($\circledast$ ‣ 4.1) is satisfied, since
$$\displaystyle\operatorname{Transp}(S,\,T)\cap U$$
$$\displaystyle=\operatorname{Transp}(S,\,T)\cap\operatorname{Transp}(A,\,B)$$
$$\displaystyle=\operatorname{Transp}(S,\,T)\cap\operatorname{Transp}(f_{A}(S),\,f_{B}(T))$$
$$\displaystyle=\operatorname{Transp}({S}\mathop{\|}{f_{A}}(S),\,{T}\mathop{\|}{f_{B}}(T)).\qed$$
Corollary 4.4.
Let $G\leq\operatorname{Sym}(\Omega)$, and suppose there exists $A\in\operatorname{Stacks}(\mathcal{O})$ such that $G\leq\operatorname{Stab}(A)$.
Define $f_{A}$ to be the constant function on $\operatorname{Stacks}(\mathcal{O})$ with image $A$.
Then $(f_{A},f_{A})$ is a refiner for $G$.
Moreover, if $G=\operatorname{Stab}(A)$, then this refiner is perfect.
Lemma 4.5.
Let $U\subseteq\operatorname{Sym}(\Omega)$.
There exists a perfect refiner for $U$ if and only if $U=\operatorname{Transp}(S,\,T)$ for some $S,T\in\operatorname{Stacks}(\mathcal{O})$.
In particular, there exists a perfect refiner for a group $G\leq\operatorname{Sym}(\Omega)$ if and only if $G=\operatorname{Stab}(S)$ for some $S\in\operatorname{Stacks}(\mathcal{O})$.
Proof.
Let $(f_{L},f_{R})$ be a perfect refiner for $U$ and let $S$ be the empty stack in $\operatorname{Stacks}(\mathcal{O})$.
Note that $\operatorname{Transp}(S,\,S)=\operatorname{Sym}(\Omega)$. Then $U=\operatorname{Transp}(f_{L}(S),\,f_{R}(S))$ by Definition 4.2.
The converse implication follows from Lemma 4.3.
∎
Corollary 4.6.
If there exists a perfect refiner for a subset $U\subseteq\operatorname{Sym}(\Omega)$,
then either $U$ is empty, or $U$ is a subgroup, or a coset of a subgroup, of $\operatorname{Sym}(\Omega)$.
Lemma 4.7 (cf. Lemma 3.5).
Let $G\leq\operatorname{Sym}(\Omega)$, let $x\in\operatorname{Sym}(\Omega)$, and let $f_{L},f_{R}\in\operatorname{Stacks}(\mathcal{O})^{\operatorname{Stacks}(\mathcal{O})}$.
Then $(f_{L},f_{R})$ is a perfect refiner for $Gx$ if and only if $(f_{L},f_{L})$ is a perfect refiner for $G$ and $f_{R}=f_{L}^{x}$.
In particular, there exists a perfect refiner for a group $G\leq\operatorname{Sym}(\Omega)$ if and only if there exist perfect refiners for one, and hence all, cosets of $G$ in $\operatorname{Sym}(\Omega)$.
Proof.
We prove that for all $x,y\in\operatorname{Sym}(\Omega)$, $(f_{L},f_{L}^{x})$ is a perfect refiner for $Gx$ if and only if $(f_{L},f_{L}^{y})$ is a perfect refiner for $Gy$. The lemma follows from this by choosing $y\coloneqq 1_{G}$ and by Lemma 3.5.
Let $S,T\in\operatorname{Stacks}(\mathcal{O})$ with $|S|=|T|$. Suppose that $(f_{L},f_{L}^{x})$ is a perfect refiner for $Gx$.
Note that $Gy\cap\operatorname{Transp}(S,\,T)=(Gx\cap\operatorname{Transp}(S,\,T^{y^{-1}x}))\cdot x^{-1}y$.
Then by assumption:
$$\displaystyle Gx\cap\operatorname{Transp}(S,\,T^{y^{-1}x})$$
$$\displaystyle=\operatorname{Transp}({S}\mathop{\|}{f_{L}}(S),\,{T^{y^{-1}x}}\mathop{\|}{f_{L}^{x}}(T^{y^{-1}x}))$$
$$\displaystyle=\operatorname{Transp}({S}\mathop{\|}{f_{L}}(S),\,T^{y^{-1}x}\|{f_{L}^{y}}(T)^{y^{-1}x})$$
$$\displaystyle=\operatorname{Transp}({S}\mathop{\|}{f_{L}}(S),\,({T}\mathop{\|}{f_{L}^{y}}(T))^{y^{-1}x})$$
$$\displaystyle=\operatorname{Transp}({S}\mathop{\|}{f_{L}}(S),\,{T}\mathop{\|}{f_{L}^{y}}(T))\cdot y^{-1}x.$$
Hence $(f_{L},f_{L}^{y})$ is a perfect refiner for $Gy$ by ($\circledast$ ‣ 4.1).
The converse implication follows by symmetry.
∎
The following lemma shows that combining perfect refiners with the $\|$ operation of Notation 3.7 preserves perfectness.
This implies that the collection of groups and cosets which have perfect refiners is closed under intersection.
Lemma 4.8.
Let $(f,\sigma)$ and $(g,\tau)$ be perfect refiners for the sets $U,V\subseteq\operatorname{Sym}(\Omega)$, respectively.
Then $(f\|g,\sigma\|\tau)$ is a perfect refiner for the set $U\cap V$.
Proof.
We show that condition ($\circledast$ ‣ 4.1) from Lemma 4.1 holds,
for the set $U\cap V$ and the pair of functions $(f\|g,\sigma\|\tau)$.
Let $S,T\in\operatorname{Stacks}(\mathcal{O})$ with $|S|=|T|$.
By an argument similar to that used in the proof of Lemma 3.8, if $|f(S)|\neq|\sigma(T)|$, then the condition holds, so suppose otherwise. Then
$$\displaystyle(U\cap V)\cap\operatorname{Transp}(S,\,T)$$
$$\displaystyle=(U\cap\operatorname{Transp}(S,\,T))\cap(V\cap\operatorname{Transp}(S,\,T))$$
$$\displaystyle=\operatorname{Transp}(S\|f(S),\,T\|\sigma(T))\cap\operatorname{Transp}(S\|g(S),\,T\|\tau(T))$$
$$\displaystyle=\operatorname{Transp}(S,\,T)\cap\operatorname{Transp}(f(S),\,\sigma(T))\cap\operatorname{Transp}(g(S),\,\tau(T))$$
$$\displaystyle=\operatorname{Transp}(S\cap(f\|g)(S),\,T\cap(\sigma\|\tau)(T)).\qed$$
5 Perfect refiners for stabilisers and transporter sets
Many computations that are commonly performed with backtrack search can be formulated as stabiliser or transporter problems.
This includes computing normalisers, determining subgroup conjugacy, solving graph isomorphism, and finding sets that are phrased in such terms, like set stabilisers.
Therefore, in order to most successfully solve such problems with backtrack search organised around stacks in $\mathcal{O}$, we require a way of constructing refiners for the appropriate stabilisers and transporter sets.
These refiners should be cheap to compute, and ideally, they should be perfect if possible.
Lemma 4.3 constructively shows that this task can be reduced to translating the given stabiliser or transporter problem into one concerning stacks in $\mathcal{O}$.
For a problem that is already given in terms of $\operatorname{Stacks}(\mathcal{O})$, or at least in terms of $\mathcal{O}$, it follows that little to no translation is needed; this is stated explicitly in the following immediate corollary to Lemma 4.3.
Corollary 5.1.
For each $x\in\mathcal{O}$ and $S\in\operatorname{Stacks}(\mathcal{O})$, let $[x]$ be the stack with unique entry $x$, and let $f_{S}$ be the constant function in $\operatorname{Stacks}(\mathcal{O})^{\operatorname{Stacks}(\mathcal{O})}$ with image $S$.
Let $x,y\in\mathcal{O}$ and $S,T\in\operatorname{Stacks}(\mathcal{O})$.
Then $(f_{[x],}f_{[y]})$ is a perfect refiner for $\operatorname{Transp}(x,\,y)$, and
$(f_{S},f_{T})$ is a perfect refiner for $\operatorname{Transp}(S,\,T)$.
For other transporter problems, more work is required in order to apply Lemma 4.3, i.e. to construct stacks in $\mathcal{O}$ whose own transporter set closely contains (and ideally equals) the given one. The analogous statement holds for stabiliser problems.
While this translation can be done on an ad-hoc basis, it is desirable to instead have a systematic method that works for a whole class of objects that we wish to stabilise and to compute transporter sets of.
To that end, we present Theorem 5.3.
Definition 5.2.
Let $X$ and $Y$ be sets on which $\operatorname{Sym}(\Omega)$ acts, and let $f:X\rightarrow Y$ be a function. Then $f$ is called $\operatorname{Sym}(\Omega)$-invariant if $f(x^{g})={f(x)}^{g}$ for all $x\in X$ and $g\in G$.
Theorem 5.3.
Let $\Sigma$ and $\Theta$ be sets on which $\operatorname{Sym}(\Omega)$ acts,
let $\pi:\Sigma\rightarrow\Theta$ be a
$\operatorname{Sym}(\Omega)$-invariant function,
and let $x,y\in\Sigma$.
Then the following statements hold:
(i)
$\operatorname{Transp}(x,\,y)\subseteq\operatorname{Transp}(\pi(x),\,\pi(y))$.
In particular, $\operatorname{Stab}(x)\leq\operatorname{Stab}(\pi(x))$,
and any refiner for $\operatorname{Transp}(\pi(x),\,\pi(y))$ is a refiner for $\operatorname{Transp}(x,\,y)$.
(ii)
If $\pi$ is injective, then $\operatorname{Transp}(x,\,y)=\operatorname{Transp}(\pi(x),\,\pi(y))$;
in particular, $\operatorname{Stab}(x)=\operatorname{Stab}(\pi(x))$,
and any perfect refiner for $\operatorname{Transp}(\pi(x),\,\pi(y))$ is a perfect refiner for $\operatorname{Transp}(x,\,y)$.
Proof.
If $g\in\operatorname{Transp}(x,\,y)$, then ${\pi(x)}^{g}=\pi(x^{g})=\pi(y)$,
i.e. $g\in\operatorname{Transp}(\pi(x),\,\pi(y))$.
Therefore (i) holds.
Suppose that $\pi$ is injective.
If $g\in\operatorname{Transp}(\pi(x),\,\pi(y))$, then ${\pi(x)}^{g}=\pi(x^{g})$ and ${\pi(x)}^{g}=\pi(y)$, and so the injectivity of $\pi$ implies that $x^{g}=y$.
Therefore (ii) holds.
∎
Theorem 5.3 suggests a strategy for systematically constructing refiners for the stabilisers and transporter sets of objects in a set on which $\operatorname{Sym}(\Omega)$ acts, which is the foundation of our approach in Section 8.
More precisely, given such a set $\Sigma$, we identify another set $\Theta$ on which $\operatorname{Sym}(\Omega)$ acts, and for which we have identified refiners for all stabilisers and transporter sets.
We then define an injective $\operatorname{Sym}(\Omega)$-invariant function $\pi:\Sigma\rightarrow\Theta$.
The desired refiners are then inherited via $\pi$ as described in Theorem 5.3, with Corollary 5.1 providing the base of this recursive procedure.
Remark 5.4.
Let $\mathcal{O}$ and $\mathcal{O}^{\prime}$ be sets on which $\operatorname{Sym}(\Omega)$ acts, let $\pi:\operatorname{Stacks}(\mathcal{O})\rightarrow\operatorname{Stacks}(\mathcal{O}^{\prime})$ be an injective $\operatorname{Sym}(\Omega)$-invariant function, and let $U\subseteq\operatorname{Sym}(\Omega)$.
Then Lemmas 4.3 and 4.5 and Theorem 5.3(ii)
together constructively show that if $U$ has a perfect refiner in backtrack search organised around $\operatorname{Stacks}(\mathcal{O})$, then $U$ also has a perfect refiner in backtrack search organised around $\operatorname{Stacks}(\mathcal{O}^{\prime})$.
In Section 8, we also make frequent use of the following lemma, which shows that the $\|$ operation can be used to construct refiners for the stabilisers and transporter sets of lists.
This means that lists do need not to be considered separately from the objects that they contain.
Lemma 5.5.
Let $n\in\mathbb{N}$ with $n\geq 2$.
For each $i\in\{1,\ldots,n\}$, let $x_{i}$ and $y_{i}$ be objects from a set on which $\operatorname{Sym}(\Omega)$ acts, and let $(f_{L,i},f_{R,i})$ be a refiner for $\operatorname{Transp}(x_{i},\,y_{i})$.
Then $(f_{L,1}\|\cdots\|f_{L,n},\ f_{R,1}\|\cdots\|f_{R,n})$ is a refiner for $\operatorname{Transp}([x_{1},\ldots,x_{n}],\,[y_{1},\ldots,y_{n}])$.
Moreover, if each refiner $(f_{L,i},f_{R,i})$ is perfect, then the resulting refiner is perfect, too.
Proof.
Since $\operatorname{Transp}([x_{1},\ldots,x_{n}],\,[y_{1},\ldots,y_{n}])=\bigcap_{i=1}^{n}\operatorname{Transp}(x_{i},\,y_{i})$, the result follows from
Lemmas 3.8
and 4.8.
∎
Corollary 5.6.
Let $\Sigma$ be a set on which $\operatorname{Sym}(\Omega)$ acts.
Then there exist perfect refiners for all stabilisers and for all transporter sets of objects in $\Sigma$ if and only if there exist perfect refiners for all stabilisers and transporter sets of finite lists in $\Sigma$.
6 Extended graph backtracking, with extra vertices
With graph backtracking,
a search in $\operatorname{Sym}(\Omega)$ is organised around stacks of labelled digraphs
on the vertex set $\Omega$.
Although this technique gives significantly smaller search sizes in some
cases [JPWW21, Section 9],
and enables some important perfect refiners
(Section 8.2),
the requirement that the digraphs have vertex set $\Omega$ limits the
possibilities for perfect refiners.
This is shown explicitly in Corollary 7.3.
In this section, we introduce an extension to graph backtracking, which accommodates digraphs that are allowed to have additional vertices.
The technique requires only a small adjustment to the theory of graph
backtracking, and has already been implemented in Vole [CJW21].
In fact the necessary concepts from [JPWW21] extend naturally, and re-stating and re-proving the corresponding definitions and results do not give any new insights, which is why we do not include these technical details in this article.
Instead, we discuss the extended graphs themselves, and their stacks, in much detail, because this is necessary for formulating and examining refiners in this setting.
We require some additional definitions and notation for the rest of this article.
We fix $\Lambda$ as an infinite well-ordered set containing $\Omega$,
where $\alpha<\beta$ for all $\alpha\in\Omega$ and $\beta\in\Lambda\setminus\Omega$.
In practice, and in the forthcoming examples, we use $\Omega\coloneqq\{1,\ldots,n\}$ for some $n\in\mathbb{N}$, and $\Lambda\coloneqq\mathbb{N}$.
For any subset $V$ of $\Lambda$ that contains $\Omega$,
we regard $\operatorname{Sym}(\Omega)$ and $\operatorname{Sym}(V\setminus\Omega)$ as subgroups of $\operatorname{Sym}(V)$,
by identifying each permutation of $V$ that fixes $V\setminus\Omega$ pointwise with its restriction to $\Omega$,
and identifying each permutation of $V$ that fixes $\Omega$ pointwise with its restriction to $V\setminus\Omega$.
In this way, $\operatorname{Sym}(\Omega)$ and $\operatorname{Sym}(V\setminus\Omega)$ inherit from $\operatorname{Sym}(\Omega)$ an action on the set of all labelled digraphs on $V$.
Furthermore, since we regard $\operatorname{Sym}(\Omega)$ and $\operatorname{Sym}(V\setminus\Omega)$ as subgroups of
$\operatorname{Sym}(V)$, they
commute, and this means that
$\operatorname{Sym}(\Omega)$ permutes the set of orbits of $\operatorname{Sym}(V\setminus\Omega)$ on labelled digraphs on $V$.
For a labelled digraph $\Gamma$ on $V$, we use the notation $\overline{\Gamma}$ for the orbit of $\Gamma$ under $\operatorname{Sym}(V\setminus\Omega)$.
The action of $\operatorname{Sym}(\Omega)$ is then given by ${(\overline{\Gamma})}^{g}\coloneqq\overline{\Gamma^{g}}$ for all labelled digraphs $\Gamma$ on $V$ and all $g\in\operatorname{Sym}(\Omega)$.
We define an extended graph on $\Omega$
to be an orbit of $\operatorname{Sym}(V\setminus\Omega)$ on the set of labelled digraphs on $V$,
for some finite set $V$ with $\Omega\subseteq V\subseteq\Lambda$.
This will be illustrated in Example 6.1.
An extended graph stack on $\Omega$ is any list of extended graphs on $\Omega$.
We remark that different entries in an extended graph stack are allowed to be defined in relation to different subsets of $\Lambda$.
Extended graph backtracking in $\operatorname{Sym}(\Omega)$ is then backtrack search organised around extended graph stacks on $\Omega$.
The results of Sections 3–5 hold in the setting of extended graph backtracking in $\operatorname{Sym}(\Omega)$.
Example 6.1.
Let $\Omega\coloneqq\{1,\ldots,4\}$ and $V\coloneqq\{1,\ldots,6\}$.
Depicted in Figure 1 are labelled digraphs $\Gamma$ and $\Gamma^{(5\,6)}$ on $V$,
which comprise an orbit of $\operatorname{Sym}(V\setminus\Omega)$ on the set of labelled digraphs on $V$.
Therefore $\overline{\Gamma}=\{\Gamma,\Gamma^{(5\,6)}\}$ is an extended graph on $\Omega$.
Note that $\operatorname{Stab}(\overline{\Gamma})=\langle(1\,4),(1\,2)(3\,4)\rangle$,
which is the setwise stabiliser of $\{\{1,4\},\{2,3\}\}$ in $\operatorname{Sym}(\Omega)$.
In Section 8.3.1 we will discuss a similar construction.
Lemma 6.2.
Let $A$ and $B$ be extended graphs on $\Omega$, defined relative to subsets $U$ and $V$ of $\Lambda$, respectively, and let $\Gamma\in A$ and $\Delta\in B$ be arbitrary.
If $U\neq V$, then $\operatorname{Transp}(U,\,V)=\varnothing$. Otherwise,
$\operatorname{Transp}(A,\,B)$ is the restriction of
$T\coloneqq\{x\in\operatorname{Sym}(V):x\ \text{preserves}\ \Omega\ \text{setwise, and}\ \Gamma^{x}=\Delta\}$ to $\Omega$.
Proof.
Suppose that $U=V$ and let $g$ be the restriction to $\Omega$ of some $x\in T$.
Then since ${g}^{-1}x\in\operatorname{Sym}(V\setminus\Omega)$, it follows that
$A^{g}=\overline{\Gamma}^{g}=\overline{\Gamma^{g}}=\overline{\Gamma^{g(g^{-1}x)}}=\overline{\Gamma^{x}}=\overline{\Delta}=B$.
Conversely, if $g\in\operatorname{Transp}(A,\,B)=\operatorname{Transp}(\overline{\Gamma},\,\overline{\Delta})$, then
$\Gamma^{g}=\Delta^{h}$ for some $h\in\operatorname{Sym}(V\setminus\Omega)$.
Thus $g$ is the restriction of $gh^{-1}\in T$ to $\Omega$.
∎
Lemma 6.2 implies that we do not need to enumerate whole orbits of labelled digraphs in order to compute stabilisers and transporter sets of extended graphs.
In practice, we construct extended graphs that consist of labelled digraphs where the labels used for vertices in $\Omega$ are never used for vertices in $\Lambda\setminus\Omega$.
This guarantees that any permutation that maps one such labelled digraph to another necessarily preserves $\Omega$ as a set.
Therefore, in the notation of Lemma 6.2, the set $T$ is simply the transporter set in $\operatorname{Sym}(V)$ from $\Gamma$ to $\Delta$.
7 The groups and cosets with perfect refiners
By Lemmas 4.5 and 4.7, the task of finding a perfect refiner for a subgroup of $\operatorname{Sym}(\Omega)$, or for a coset of a subgroup,
is equivalent to finding a stack whose stabiliser in $\operatorname{Sym}(\Omega)$ is the corresponding group.
In this section, and for each setting of backtracking, we classify the groups (and correspondingly the cosets of subgroups) for which there exists a perfect refiner.
This information is summarised in Table 1.
It follows from these classifications that, in a sense, partition backtracking inherits the perfect refiners of classical backtracking, that graph backtracking inherits the perfect refiners of partition backtracking, and that extended graph backtracking inherits the perfect refiners of graph backtracking.
This can be shown constructively: it is straightforward to map lists of points to lists of ordered partitions, and lists of ordered partitions to lists of labelled digraphs, in injective $\operatorname{Sym}(\Omega)$-invariant ways.
As mentioned in Remark 5.4, perfect refiners can then be translated via these maps.
The classifications in Table 1 for classical backtracking (stabilisers in $\operatorname{Sym}(\Omega)$ of lists in $\Omega$) and partition backtracking (stabilisers in $\operatorname{Sym}(\Omega)$ of lists of ordered partitions of $\Omega$) are trivial to verify.
For graph backtracking, we require the notion of the 2-closure of a permutation group.
Definition 7.1 (2-closure, 2-closed; cf. [DM96, Section 3.2]).
The 2-closure of a group $G\leq\operatorname{Sym}(\Omega)$ is the largest subgroup of $\operatorname{Sym}(\Omega)$ with the same orbits on $\Omega\times\Omega$ as $G$; it is the pointwise stabiliser in $G$ of the set of all orbital graphs of $G$.
A 2-closed group is one that is equal to its 2-closure.
The 2-closure of any subgroup of $\operatorname{Sym}(\Omega)$ that acts at least 2-transitively on $\Omega$ is $\operatorname{Sym}(\Omega)$. Therefore no proper subgroup of $\operatorname{Sym}(\Omega)$ acts at least 2-transitively on $\Omega$ and is 2-closed.
Lemma 7.2.
Let $G$ be a subgroup of $\operatorname{Sym}(\Omega)$.
Then $G$ is the stabiliser of a labelled digraph stack on $\Omega$ if and only if $G$ is 2-closed.
Proof.
$(\Leftarrow)$ Labelling all vertices and arcs of a digraph with a fixed label preserves its stabiliser. Thus, if $S$ is a stack of labelled digraphs formed from all the orbital graphs of $G$ in this way, then $G=\operatorname{Stab}(S)$.
$(\Rightarrow)$
By [JPWW21, Lemma 3.6], we may assume that $G=\operatorname{Stab}(\Gamma)$ for a labelled digraph $\Gamma$ on $\Omega$.
Let $A_{1},\ldots,A_{k}\subseteq\Omega\times\Omega$ and $B_{1},\ldots,B_{l}\subseteq\Omega$ be the orbits of $G$ on the sets of arcs and vertices of $\Gamma$, respectively, for some $k\in\mathbb{N}\cup\{0\}$ and $l\in\mathbb{N}$.
For each $i\in\{1,\ldots,k\}$ and $j\in\{1,\ldots,l\}$, we define digraphs $\Delta_{i}\coloneqq(\Omega,A_{i})$ and $\Sigma_{j}\coloneqq(\Omega,\{(b,b):b\in B_{j}\})$.
Each such digraph is an orbital graph of $G$, and so its stabiliser contains $G$. Furthermore, if a permutation $g\in\operatorname{Sym}(\Omega)$ stabilises each of these digraphs
pointwise, then by construction, $g$ maps each vertex of $\Gamma$ to a vertex with the same label, and it maps each arc of $\Gamma$ to an arc with the same label, and so $g\in\operatorname{Stab}(\Gamma)$.
Thus we have proved that
$$G\leq\left(\bigcap_{i=1}^{k}\operatorname{Stab}(\Delta_{i})\right)\cap\left(\bigcap_{j=1}^{l}\operatorname{Stab}(\Sigma_{j})\right)\leq\operatorname{Stab}(\Gamma)=G.$$
Since $G$ is the pointwise stabiliser of a subset of its orbital graphs, it is 2-closed.
∎
Corollary 7.3.
In graph backtracking, there exists a perfect refiner for a group $G\leq\operatorname{Sym}(\Omega)$ if and only if $G$ is 2-closed.
For extended graph backtracking, we require the following result.
Proposition 7.4 ([Bou69, Kea]).
Let $G\leq\operatorname{Sym}(\Omega)$.
There exists a finite set $V$ containing $\Omega$
and a labelled digraph $\Gamma$ on $V$,
such that the stabiliser of $\Gamma$ in $\operatorname{Sym}(V)$ preserves $\Omega$ setwise,
and the restriction of this stabiliser to $\Omega$ is $G$.
Lemma 7.5.
Every subgroup of $\operatorname{Sym}(\Omega)$ has a perfect refiner in extended graph backtracking.
Proof.
Let $G\leq\operatorname{Sym}(\Omega)$, and let $\Gamma$ be a finite labelled digraph of the kind described in Proposition 7.4 for $G$.
By renaming the vertices that are not in $\Omega$, we may assume that the vertex set of $\Gamma$ is a subset of the infinite set $\Lambda$ that we use throughout this article.
By Lemma 6.2 and Proposition 7.4, the extended graph stack $[\overline{\Gamma}]$ has stabiliser $G$ in $\operatorname{Sym}(\Omega)$.
The result follows by Lemma 4.5.
∎
We primarily include Lemma 7.5 for its theoretical interest.
Given a group $G$, the number of additional vertices required to construct a witness for the result in Proposition 7.4 may be of similar size to $|G|$, which would normally be impractically large for use in a backtrack search. We are therefore also interested in how many extra vertices are required to represent different groups.
8 Examples of perfect refiners
In this section,
we describe some refiners for the stabilisers and transporter sets of some commonly-occurring objects on which $\operatorname{Sym}(\Omega)$ acts.
In most cases, we prove that the refiners are perfect, making frequent use of Corollary 5.1, Theorem 5.3, and Lemma 5.5.
The results for perfect refiners are summarised in Table 2.
As discussed in Section 7, a perfect refiner for a subset of $\operatorname{Sym}(\Omega)$ in classical backtracking can be easily converted into a perfect refiner for the same set in partition backtracking, and so on.
In this hierarchical sense, each class of objects in Table 2 appears by the ‘least’ kind of backtracking in which all of their stabilisers and transporter sets have perfect refiners.
A given kind of backtracking may not have perfect refiners for all stabilisers and transporter sets of some class of objects, but it may have perfect refiners for those of an important subclass of the objects.
In order to optimise implementations, it is important to understand these special cases well, but a thorough investigation of this topic is beyond the scope of this article.
Nevertheless, to demonstrate the principle, we consider sets of subsets of $\Omega$ with pairwise distinct sizes
in partition backtracking
(Section 8.1.2), and sets of nonempty pairwise disjoint subsets of $\Omega$
in graph backtracking
(Section 8.2.2), even though only extended graph backtracking has perfect refiners for all stabilisers and transporter sets of sets of subsets of $\Omega$ (Section 8.3.1).
8.1 Examples of perfect refiners in partition backtracking
To justify the part of Table 2 that relates to partition backtracking, we first show, for each kind of object, and with $|\Omega|\geq 5$, that not all stabilisers and transporter sets have perfect refiners in classical backtracking.
By Corollary 5.6 and Table 1, the following examples suffice:
If $n\in\mathbb{N}$ with $n\geq 5$ and $\Omega\coloneqq\{1,\ldots,n\}$, then the ordered partition $[\{1,2\},\{3,\ldots,n\}]$ of $\Omega$, the subset $\{1,2\}$ of $\Omega$, and the set of subsets $\left\{\{1,2\},\{3,\ldots,n\}\right\}$ of $\Omega$ have stabiliser $\langle(1\,2),(3\,4),(3\,\ldots\,n)\rangle\cong\mathcal{C}_{2}\times\mathcal{S}_{n-2}$ in $\operatorname{Sym}(\Omega)$.
8.1.1 Stabilisers and transporters of sets of points
We define a function $\pi$ from the set of all subsets of $\Omega$ to the set of all stacks of ordered partitions of $\Omega$.
We define $\pi(\varnothing)$ to be the empty stack on $\Omega$, and $\pi(\Omega)$ to be the stack consisting of the ordered partition $[\Omega]$, and for any nonempty proper subset $A$ of $\Omega$, we define $\pi(A)$ to be the stack consisting of the ordered partition $[A,\Omega\setminus A]$.
Since $\pi$ is an injective $\operatorname{Sym}(\Omega)$-invariant function, Theorem 5.3 and Corollary 5.1 give the desired perfect refiners.
8.1.2 Stabilisers and transporters of sets of subsets with pairwise distinct sizes
We define an injective $\operatorname{Sym}(\Omega)$-invariant function $\sigma$ that maps each set of subsets of $\Omega$ with pairwise distinct sizes to a list of subsets of $\Omega$ as follows.
Let $A$ be any such set.
There exists $m\in\mathbb{N}\cup\{0\}$ and subsets $A_{1},\ldots,A_{m}$ of $\Omega$, uniquely indexed by increasing size, such that $A=\{A_{1},\ldots,A_{m}\}$.
We define $\sigma(A)\coloneqq[A_{1},\ldots,A_{m}]$, and remark that a permutation of $\Omega$ stabilises the set of subsets $A$ if and only if it stabilises each of the subsets that it contains.
Perfect refiners for the stabilisers and transporters can thus be obtained from $\sigma$ via Section 8.1.1, Lemma 5.5, and Theorem 5.3.
8.2 Examples of perfect refiners in graph backtracking
For the kinds of objects listed by graph backtracking in Table 2, it is easy to find examples for all $|\Omega|\geq 4$ where the stabiliser in $\operatorname{Sym}(\Omega)$ is not a direct product of symmetric groups, which means that the stabiliser has no perfect refiner in partition backtracking (see Table 1).
It remains to show that there are perfect refiners in graph backtracking for the stabilisers and transporter sets of these objects.
8.2.1 Graph and digraph automorphisms and isomorphisms
We recall our standard notation for graphs from Section 2.
Let $\pi$ be the function that maps each graph $(\Omega,E)$ to the digraph $(\Omega,\{(\alpha,\beta),(\beta,\alpha):\{\alpha,\beta\}\in E\})$. This means that each edge $\{\alpha,\beta\}$ is replaced by the arcs $(\alpha,\beta)$ and $(\beta,\alpha)$.
In addition, we define $x$ to be some fixed label, and we let $\sigma$ be the function that maps each digraph on $\Omega$ to the labelled digraph on $\Omega$ formed by assigning the label $x$ to all of its vertices and arcs.
Then $\pi$ and $\sigma$ are injective $\operatorname{Sym}(\Omega)$-invariant functions, and so Theorem 5.3 and Corollary 5.1 give perfect refiners for all stabilisers and transporter sets in $\operatorname{Sym}(\Omega)$ of graphs and digraphs on $\Omega$ in graph backtracking.
8.2.2 Stabilisers and transporters of sets of nonempty pairwise disjoint sets
We define a function $\phi$ from the set of all subsets of $\Omega$ to the set of all digraphs on $\Omega$ as follows.
Let $A\coloneqq\{A_{1},\ldots,A_{k}\}$, for some $k\in\mathbb{N}\cup\{0\}$, be a set of subsets of $\Omega$,
and define $\phi(A)$ to be the digraph on $\Omega$ that consists of a clique (plus loops) on the vertices of each member of $A$, i.e.
$$\phi(A)\coloneqq\big{(}\Omega,\ \{(\alpha,\beta)\in\Omega\times\Omega:\{\alpha,\beta\}\subseteq A_{i}\ \text{for some}\ i\in\{1,\ldots,k\}\}\big{)}.$$
See Figure 2 for an example.
Then $\phi$ is $\operatorname{Sym}(\Omega)$-invariant, and so by Theorem 5.3 and Section 8.2.1, we obtain refiners in graph backtracking for the stabilisers and transporter sets of all sets of subsets of $\Omega$.
Furthermore, when restricted to sets of nonempty pairwise disjoint subsets of $\Omega$,
the function $\phi$ is injective and $\operatorname{Sym}(\Omega)$-invariant, and so the corresponding refiners are perfect.
8.2.3 Permutation centraliser and conjugacy, and subgroup centraliser
We consider $\operatorname{Sym}(\Omega)$ acting on itself by conjugation.
Let $\psi$ be the function from $\operatorname{Sym}(\Omega)$ to the set of all digraphs on $\Omega$ where for each $g\in\operatorname{Sym}(\Omega)$, $\psi(g)\coloneqq\left(\Omega,\{(\alpha,\beta)\in\Omega\times\Omega:\alpha^{g}=\beta\}\right)$. See Figure 3, where we explain an example for illustration.
Then $\psi$ is injective and $\operatorname{Sym}(\Omega)$-invariant, giving perfect refiners in graph backtracking for all centralisers in $\operatorname{Sym}(\Omega)$ of elements of $\operatorname{Sym}(\Omega)$, and for the transporter sets for permutation conjugacy (see Theorem 5.3 and Section 8.2.1).
By Lemma 5.5, this gives perfect refiners for all stabilisers of lists of permutations in $\operatorname{Sym}(\Omega)$ under conjugation.
The centraliser in $\operatorname{Sym}(\Omega)$ of a subgroup of $\operatorname{Sym}(\Omega)$ is the pointwise stabiliser of any of its generating sets, and thus we can obtain a perfect refiner in graph backtracking for the centraliser in $\operatorname{Sym}(\Omega)$ of any subgroup of $\operatorname{Sym}(\Omega)$ given by a generating set.
8.3 Examples of perfect refiners in extended graph backtracking
By Lemma 7.5, all stabilisers and transporter sets have perfect refiners in extended graph backtracking, but this knowledge is not necessarily immediately useful:
the extended graphs underpinning this lemma may require impractically many additional vertices, and anyway, the construction requires knowing the stabiliser or transporter set in advance.
In order to avoid too many additional vertices or circular arguments, we wish to construct refiners in extended graph backtracking using only facts about the relevant objects that can be computed cheaply.
In the forthcoming examples, we specify an extended graph
by giving one of the labelled digraphs that it contains.
Therefore, in each case, we must prove that this is well-defined.
The fixed set of vertices $\Lambda$ is totally ordered, and we always choose vertices from this set in ascending order.
Thus it suffices to show, in each case, that our choices lead to labelled digraphs that differ only by a permutation of their vertices in $\Lambda\setminus\Omega$.
8.3.1 Stabilisers and transporters of sets of sets
We define a function $\phi$ from the set of all sets of subsets of $\Omega$ to the set of all extended graphs on $\Omega$.
Let $A\coloneqq\{A_{1},\ldots,A_{k}\}$ be a set of subsets of $\Omega$,
where $k\in\mathbb{N}\cup\{0\}$, indexed arbitrarily,
and let $V_{k}\coloneqq\{\beta_{1},\ldots,\beta_{k}\}\subseteq\Lambda\setminus\Omega$ comprise the $k$ least elements of $\Lambda\setminus\Omega$.
We define $\Gamma_{A}$ to be the labelled digraph on $\Omega\cup V_{k}$ with arcs
$\bigcup_{i=1}^{k}\{(\alpha,\beta_{i}):\alpha\in A_{i}\}$,
where vertices in $\Omega$ are labelled white, and all other vertices and arcs are labelled black.
Thus there is a white vertex in $\Gamma_{A}$ for each member of $\Omega$, and a black vertex for each member of $A$; each vertex in $\Omega$ has arcs to the black vertices corresponding to the members of $A$ that contain it.
We define $\phi(A)\coloneqq\overline{\Gamma_{A}}$ to be the extended graph containing $\Gamma_{A}$.
The only choice involved in constructing a labelled digraph using the method described above is the indexing of the members of $A$. Therefore the labelled digraphs produced can differ only by a permutation of the names of the vertices outside of $\Omega$,
and $\phi$ is well-defined.
It is clear that $\phi$ is injective and $\operatorname{Sym}(\Omega)$-invariant, and so perfect refiners are given by Theorem 5.3 and Corollary 5.1.
Next, we discuss a situation where there is no perfect refiner in graph backtracking.
Let $H$ be the subgroup $\langle(1\,2\,3)(4\,5\,6),(1\,2)(3\,5)\rangle$ of $\operatorname{Sym}(\{1,\dots,6\})$.
It can be shown that $H$ is a proper subgroup of $\operatorname{Sym}(\{1,\dots,6\})$ that acts 2-transitively on $\Omega$, and which is therefore not 2-closed.
If we define $O$ to be the orbit of ${\{1,2,3\}}$ under $H$, then $H$ is exactly the stabiliser of $O$ in $\operatorname{Sym}(\{1,\ldots,6\})$.
Let $n\in\mathbb{N}$ with $n\geq 7$ and let $\Omega\coloneqq\{1,\ldots,n\}$.
It follows that the group $G\coloneqq H\times\operatorname{Sym}(\{7,\ldots,n\})$
(identifying this direct product with a subgroup of $\operatorname{Sym}(\Omega)$ as in Table 1) is the stabiliser in $\operatorname{Sym}(\Omega)$ of the set of subsets $O$, but it is not 2-closed, and therefore it has no perfect refiner in graph backtracking.
8.3.2 Stabilisers and transporters of sets of lists
We define a function $\pi$ that maps each set of lists in $\Omega$ to an extended graph on $\Omega$.
Let $A\coloneqq\{A_{1},\ldots,A_{k}\}$ be a set of nonempty lists in $\Omega$, for some $k\in\mathbb{N}\cup\{0\}$, indexed arbitrarily. Moreover,
let $r\coloneqq\prod_{i=1}^{k}|A_{i}|$ be the product of the lengths of these lists,
let $V_{r}$ be a set of the least $r$ elements of $\Lambda\setminus\Omega$,
and let $\rho$ be an arbitrary bijection from $\{(i,j):i\in\{1,\ldots,k\}\text{\ and\ }j\in\{1,\ldots,|A_{i}|\}\}$ to $V_{r}$.
We define $\pi(A)$ to be the extended graph that contains the labelled digraph on $\Omega\cup V_{r}$ with arcs
$$\left\{\big{(}\rho(i,j),\ A_{i}[j]\big{)}:1\leq i\leq k,\ 1\leq j\leq|A_{i}|\right\}\cup\left\{\big{(}\rho(i,j),\ \rho(i,j+1)\big{)}:1\leq i\leq k,\ 1\leq j<|A_{i}|\right\},$$
where the vertices in $\Omega$ are labelled white, and the vertices in $V_{r}$ and all arcs are labelled black.
Each vertex in $V_{r}$ corresponds via $\rho$ to a position in a list in $A$;
the arcs between vertices in $V_{r}$ encode the ordering of each list,
and the arcs towards vertices in $\Omega$ encode the entry at each position.
For a set of lists in $\Omega$ that includes the empty list, $[\ ]$, we proceed as above, except that the labelled digraph has an additional isolated vertex (the next least element of $\Lambda\setminus\Omega$) with label black.
It is straightforward to see that $\pi$ is well-defined (once a bijection $\rho$ is fixed), injective, and $\operatorname{Sym}(\Omega)$-invariant,
and so with this method, it is possible to use Theorem 5.3 and Corollary 5.1 to produce perfect refiners in extended graph backtracking for the stabilisers and transporter sets of any sets of lists in $\Omega$.
In combination with Lemma 7.2, the following lemma implies that for all sets $\Omega$ with $|\Omega|>3$, there exist sets of lists in $\Omega$ whose stabilisers and transporter sets do not have perfect refiners in graph backtracking.
This result also gives a different constructive proof of Lemma 7.5.
However, the method described in this section is impractical for constructing a perfect refiner for an arbitrary subgroup $G\leq\operatorname{Sym}(\Omega)$, since it would produce a labelled digraph with $(|G|+1)|\Omega|$ vertices.
Lemma 8.1.
Every subgroup of $\operatorname{Sym}(\Omega)$ is the stabiliser in $\operatorname{Sym}(\Omega)$ of a set of lists in $\Omega$.
Proof.
Let $G\leq\operatorname{Sym}(\Omega)$ and $A$ be an enumeration of $\Omega$.
Then $G$ is the stabiliser in $\operatorname{Sym}(\Omega)$ of $A^{G}$.
∎
8.3.3 Stabilisers and transporters of sets of graphs, digraphs, or labelled digraphs
We define a function $\psi$ from the set of all sets of labelled digraphs on $\Omega$ to the set of all extended graphs on $\Omega$ that is injective and $\operatorname{Sym}(\Omega)$-invariant.
Therefore $\psi$ can be combined with the functions of Section 8.2.1, via Theorem 5.3, to give perfect refiners in extended graph backtracking for the stabilisers and transporter sets of sets of graphs or digraphs on $\Omega$ that are not necessarily labelled.
First, we fix labels $\#$ and $\circledast$ that are not allowed to be used as labels in any labelled digraph on $\Omega$.
Let $A\coloneqq\{\Gamma_{1},\ldots,\Gamma_{k}\}$ be a set of labelled digraphs on $\Omega$, for some $k\in\mathbb{N}\cup\{0\}$, indexed arbitrarily.
Roughly speaking, we build a new labelled digraph from a disjoint union of copies of the $k$ members of $A$, retaining labels,
and adding arcs that anchor each copy, and arcs that maintain the correspondences between $\Omega$ and the vertices of the copies.
We give an example in Figure 6 and discuss the precise construction
in the following paragraph.
Let $\{V_{1},\ldots,V_{k}\}$ be an arbitrary partition of the least $k|\Omega|$ elements of $\Lambda\setminus\Omega$ into $k$ parts of size $|\Omega|$, and for each $i\in\{1,\ldots,k\}$, let $\tau_{i}$ be an arbitrary bijection from $\Omega$ to $V_{i}$.
In addition, let $X\coloneqq\{x_{1},\ldots,x_{k}\}$ be the least $k$ elements of $\Lambda\setminus(\Omega\cup V_{1}\cup\cdots\cup V_{k})$, indexed arbitrarily.
Then we define $\psi(A)$ to be the extended graph that contains the labelled digraph on $\Omega\cup V_{1}\cup\cdots\cup V_{k}\cup X$ with arcs
$$\bigcup_{i=1}^{k}\left(\left\{(\tau_{i}(\alpha),\alpha),\ (\tau_{i}(\alpha),x_{i}):\alpha\in\Omega\right\}\cup\left\{(\tau_{i}(\alpha),\tau_{i}(\beta)):(\alpha,\beta)\ \text{is an arc of}\ \Gamma_{i}\right\}\right),$$
where vertices in $\Omega$ and arcs ending in $\Omega$ are labelled $\#$,
vertices in $X$ and arcs ending in $X$ are labelled $\circledast$,
and for all $\alpha,\beta\in\Omega$ and $i\in\{1,\ldots,k\}$, the vertex $\tau_{i}(\alpha)$ has the label of $\alpha$ in $\Gamma_{i}$, and the arc $(\tau_{i}(\alpha),\tau_{i}(\beta))$ (if present) has the label of $(\alpha,\beta)$ in $\Gamma_{i}$.
Different choices in the construction lead to labelled digraphs that differ only by a permutation of the vertices in $\Lambda\setminus\Omega$, so $\psi$ is well-defined.
The injectivity of $\psi$ is guaranteed by the vertices in $X$ and their arcs; these could be omitted for sets of connected labelled digraphs.
It is clear that $\psi$ is $\operatorname{Sym}(\Omega)$-invariant.
Remark 8.2.
(a) The proof of Lemma 8.1 can be adapted to show that when $|\Omega|>3$,
there exist stabilisers and transporter sets of sets of labelled digraphs on $\Omega$ without perfect refiners in graph backtracking.
(b) The fundamental idea behind this technique can be adapted to build perfect refiners for the stabilisers and transporter sets of extended graphs in extended graph backtracking.
8.3.4 Stabilisers and transporters of sets of labelled digraph stacks
Let $\zeta$ be the injective $\operatorname{Sym}(\Omega)$-invariant function from the set of all labelled digraphs stacks on $\Omega$ to the set of all labelled digraphs on $\Omega$ that is given in [JPWW21, Section 3.1].
Moreover, let $k\in\mathbb{N}_{0}$.
Then the function that maps any set $\{S_{1},\ldots,S_{k}\}$ of labelled digraph stacks on $\Omega$ to the set of labelled digraphs $\{\zeta(S_{1}),\ldots,\zeta(S_{k})\}$ is injective and $\operatorname{Sym}(\Omega)$-invariant.
Thus we can use Theorem 5.3 in combination with Section 8.3.3 to construct perfect refiners in extended graph backtracking for the stabilisers and transporter sets of arbitrary sets of labelled digraphs stacks on $\Omega$.
This allows perfect refiners in graph backtracking for the stabilisers and transporter sets of one kind of object to be converted into perfect refiners in extended graph backtracking for the stabilisers and transporter sets of sets of those objects.
8.3.5 Normalisers and sets of conjugating elements of subgroups
Any subgroup of a group $K$ acts on the set of subgroups of $K$ by conjugation and the stabiliser of a subgroup $G$ under conjugation by a subgroup $H$ is the normaliser of $G$ in $H$, denoted $N_{H}(G)$.
Computing normalisers and deciding subgroup conjugacy are typical use cases for backtrack search, but existing techniques seem to find these problems particularly difficult.
Describing refiners for normalisers has been and continues to be an important area of research, see for example [The97, Cha21].
One intuitive explaination for the difficulty is that a normaliser may permute structures of a group (such as orbits) that would be fixed in the context of other search problems, which may reduce the potential for pruning.
In the earlier parts of Section 8.3 and in Table 2, we have seen constructions of perfect refiners in extended graph backtracking for the stabilisers and transporter sets in $\operatorname{Sym}(\Omega)$ of many kinds of sets of objects that do not have perfect refiners in existing backtracking settings.
This suggests that the extended graph backtracking technique may lend itself well to the development of better refiners for normalisers and for sets of conjugating elements of subgroups.
The strategy would be to identify structures that are permuted by the normaliser, and to then develop refiners for the stabilisers and transporter sets of sets of such structures.
As a first step, we examine orbital graphs.
Proposition 8.3.
Let $G,H\leq\operatorname{Sym}(\Omega)$ and $x\in\operatorname{Sym}(\Omega)$.
If $G^{x}=H$, then $x$ transports the set of orbital graphs of $G$ to the set of orbital graphs of $H$.
In particular, the normaliser of $G$ in $\operatorname{Sym}(\Omega)$ stabilises the set of orbital graphs of $G$.
Proof.
It is routine to verify that if $G^{x}=H$, then for all $\alpha,\beta\in\Omega$, $x$ transports the orbital graph of $G$ with base-pair $(\alpha,\beta)$ to the orbital graph of $H$ with base-pair $(\alpha^{x},\beta^{x})$.
∎
Proposition 8.3 implies that the function $\mu$ that maps a subgroup of $\operatorname{Sym}(\Omega)$ to its set of orbital graphs is $\operatorname{Sym}(\Omega)$-invariant.
In Section 8.3.3, we described perfect refiners in extended graph backtracking for the stabilisers and transporter sets of arbitrary sets of digraphs,
and so by Theorem 5.3, we can use $\mu$ to obtain refiners for the normalisers and sets of conjugating elements of subgroups.
However, if $|\Omega|\geq 4$, then $\mu$ is not injective (consider different 2-transitive subgroups of $\operatorname{Sym}(\Omega)$), so these refiners are not necessarily perfect.
Nevertheless, $\mu$ gives many instances of perfect refiners.
For example, it is clear that the set of 2-closed subgroups of $\operatorname{Sym}(\Omega)$ is closed under conjugation by $\operatorname{Sym}(\Omega)$, and that the restriction of $\mu$ to this set is injective. Therefore those refiners are perfect.
Furthermore, by Proposition 8.3, the refiner described above for the normaliser of a subgroup $G\leq\operatorname{Sym}(\Omega)$ is perfect if and only if $N_{\operatorname{Sym}(\Omega)}(G)$ is equal to the stabiliser in $\operatorname{Sym}(\Omega)$ of the set of orbital graphs of $G$.
By Lemma 8.4,
these are the subgroups whose normaliser in $\operatorname{Sym}(\Omega)$ coincides with the normaliser of its 2-closure.
Example 8.5 shows that this includes more than just the 2-closed subgroups of $\operatorname{Sym}(\Omega)$.
Lemma 8.4.
Let $G\leq\operatorname{Sym}(\Omega)$.
The stabiliser in $\operatorname{Sym}(\Omega)$ of the set of orbital graphs of $G$ is the normaliser in $\operatorname{Sym}(\Omega)$ of the 2-closure of $G$.
Proof.
Let $K$ denote the 2-closure of $G$.
The orbital graphs of $G$ and $K$ coincide by definition, and so $N_{\operatorname{Sym}(\Omega)}(G)$ is contained in the stabiliser by Proposition 8.3.
Conversely, let $x\in\operatorname{Sym}(\Omega)$ stabilise the set of orbital graphs of $G$ (and $K$) and let $g\in K$.
For all orbital graphs $\Gamma$ of $K$, $\Gamma^{x^{-1}}$ is an orbital graph of $K$ by assumption, and $\Gamma^{x^{-1}g}=\Gamma^{x^{-1}}$ since $g\in K$. Hence $\Gamma^{x^{-1}gx}=\Gamma$, and $x^{-1}gx\in K$.
∎
Example 8.5.
Let $\Omega\coloneqq\{1,\ldots,6\}$ and let $G\coloneqq\langle(1\,2\,3)(4\,5\,6),(1\,4)(2\,5)\rangle\leq\operatorname{Sym}(\Omega)$.
The orbital graphs of $G$ are depicted in Figure 7.
The normaliser of $G$ in $\operatorname{Sym}(\Omega)$ is the stabiliser in $\operatorname{Sym}(\Omega)$ of the set of orbital graphs of $G$, even though $G$ is not 2-closed. We note that $G<N_{\operatorname{Sym}(\Omega)}(G)<\operatorname{Sym}(\Omega)$.
9 Closing remarks
We have introduced the concept of perfect refiners, which gives a way of comparing the available pruning power in the various backtracking frameworks and which, within a framework, gives a way of comparing refiners for a given set.
This is naturally complemented by the introduction of extended graph backtracking.
We have also discussed the existence of perfect refiners in the different frameworks, and given concrete examples of perfect refiners that are implemented in Vole.
This work suggests several obvious questions and directions for further investigation.
For any given search problem, which backtracking framework is the most appropriate for solving it, and how should this decision be made? Having decided upon a framework, which refiners should be used? In which other ways can we compare and understand refiners?
One obvious line of inquiry is the development of further methods for comparing different refiners for the same set, and of measuring the ‘quality’ of a given refiner.
The notion of perfectness is binary, and it fails to capture any nuance in the way that a refiner may fail to be perfect.
It would also be useful to better understand the performance implications of using a given refiner in a backtrack search algorithm.
Roughly speaking, a perfect refiner for a given set is best possible in terms of search size.
On the other hand, like any refiner, a perfect refiner may be impractically expensive to compute with, and this might partially or fully override the search-size advantage.
For refiners in extended graph backtracking, we could begin to understand the interplay between these effects by
distinguishing refiners according to how many vertices and arcs their extended graphs require.
We have seen in Proposition 7.4 and Lemma 7.5 that every group and coset has a perfect refiner in extended graph backtracking, but that the digraphs in [Bou69, Kea] that underpin these results may be impractical for computation.
Extended graph backtracking would therefore benefit from a better understanding of the theoretical and practical possibilities for representing groups by digraphs of the kind in Proposition 7.4, because then we could work on finding perfect refiners, or decide that that is infeasible.
As mentioned in Section 8.3.5, extended graph backtracking appears to be well suited to the development of better refiners for use in normaliser and subgroup conjugacy computations. This is already an active area of research, motivated partly by the fact that good algorithms for normaliser search exist (e.g. in MAGMA, see [BCP97]).
Our work in this direction will be continued in the future.
Finally, we note that there is ongoing work to apply the ideas of the graph backtracking and extended graph backtracking frameworks to canonisation algorithms in finite symmetric groups.
References
[BCP97]
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PhD thesis, University of St Andrews, 2021.
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Mathematics.
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https://arxiv.org/abs/1911.04783.
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Christopher Jefferson, Markus Pfeiffer, Wilf A. Wilson, and Rebecca Waldecker.
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Journal of Algebra, 585:723–758, 11 2021.
[Kea]
Keith Kearnes.
Can every permutation group be realized as the automorphism group of
a graph (acting on a subset of the vertices)?
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Jeffrey S. Leon.
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[Leo97]
Jeffrey S. Leon.
Partitions, refinements, and permutation group computation.
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Charles C. Sims.
Computation with permutation groups.
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[The97]
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mit Anwendungen in der Konstruktion primitiver Gruppen.
Verlag der Augustinus-Buchh., 1997. |
Nilpotent Spacelike Jorden Osserman pseudo-Riemannian manifolds
P. Gilkey and S. Nikčević
PG: Mathematics Department, University of Oregon,
Eugene Or 97403 USA.
Email: gilkey@darkwing.uoregon.edu
SN: Mathematical Institute, Sanu,
Knez Mihailova 35, p.p. 367,
11001 Belgrade,
Yugoslavia
Email: stanan@mi.sanu.ac.yu
Abstract.
Pseudo-Riemannian manifolds of balanced signature which are both spacelike
and timelike Jordan Osserman nilpotent of order $2$ and of order $3$ have been
constructed previously. In this short note, we shall construct pseudo-Riemannian manifolds
of signature $(2s,s)$ for any $s\geq 2$ which are spacelike Jordan Osserman
nilpotent of order 3 but which are not timelike Jordan Osserman. Our example and techniques
are quite different from known previously both in that they are not in neutral signature and
that the manifolds constructed will be spacelike but not timelike Jordan
Osserman.
Key words and phrases:Jacobi operator, Osserman conjecture
2000 Mathematics Subject Classification. 53B20
1. Introduction
Let $(M,g)$ be a pseudo-Riemannian manifold of signature $(p,q)$. Let
$$S^{\pm}(M,g):=\{x\in TM:(x,x)=\pm 1\}$$
be the bundles of unit spacelike and unit timelike vectors, respectively. Let $R$ be the
associated Riemann curvature tensor. If $x\in T_{P}M$, then the Jacobi operator
$J(x)$ is the self-adjoint linear map of $T_{P}M$ which is characterized by the identity:
(1.a)
$$g(J(x)y,z)=R(y,x,x,z).$$
One says that $(M,g)$ is spacelike Osserman or timelike Osserman if the eigenvalues of $J$ are constant on $S^{+}(M,g)$ or on $S^{-}(M,g)$,
respectively. These are equivalent notions if $p\geq 1$ and $q\geq 1$ [12] so such
manifolds are simply said to be Osserman.
If $p=0$, and similarly if $q=0$, then one is in the Riemannian
setting. If $(M,g)$ is a rank $1$ symmetric space or if $(M,g)$ is flat, then the local
isometries of
$(M,g)$ act transitively on $S^{+}(M,g)$ so the eigenvalues of $J$ are constant on
$S^{+}(M,g)$. Osserman [19] wondered if the converse
held. Work of Chi [7] and of
Nikolayevsky [17] has shown this to be the case if the dimension is different from $8$ and $16$.
If $p=1$, and similarly if $q=1$, then one is in the
Lorentzian setting. Blažić, Bokan and Gilkey [1] and
García–Río, Kupeli and Vázquez-Abal [9] have shown that Lorentzian
Osserman manifolds have constant sectional curvature.
The situation is quite different in
the higher signature setting where $p\geq 2$ and $q\geq 2$. There exist Osserman
pseudo-Riemannian manifolds which are not symmetric spaces
[2, 3, 4, 5, 11]; we refer to [10] for an excellent
and quite comprehensive treatment of the subject.
In the higher signature setting, it is natural to impose a more restrictive hypothesis and
study the Jordan normal form of the Jacobi operator. We say that $(M,g)$ is spacelike
Jordan Osserman or is timelike Jordan Osserman if the Jordan normal form of $J(x)$
is constant on $S^{+}(M,g)$ or on $S^{-}(M,g)$, respectively. Relatively few examples of such
manifolds are known.
The eigenvalue $0$ is distinguished. One says that $(M,g)$ is nilpotent Osserman if
$J(x)^{p+q}=0$ or equivalently if $0$ is the only eigenvalue of
$J(x)$ for any $x\in TM$. The orders of nilpotency $n(x)$ and $n(M)$ are then defined by
the properties:
$$J(x)^{n(x)}=0,\quad J(x)^{n(x)-1}\neq 0,\quad\text{and}\quad n(M):=\sup_{x\in
TM%
}n(x)\,.$$
Fiedler and Gilkey [8] gave examples of $m$ dimensional
pseudo-Riemannian manifolds for any $m\geq 4$ where
$n(M)=m-2$; thus
$n(M)$ can be arbitrarily large. However for these examples,
$n(x)$ was constant neither on $S^{+}(M,g)$ or on $S^{-}(M,g)$ so these manifolds were neither
spacelike nor timelike Jordan Osserman.
Results of Gilkey and Ivanova [13] show that if $(M,g)$ is spacelike Jordan Osserman of signature $(p,q)$
where $p<q$, then the Jacobi operator is diagonalizable and hence $(M,g)$ can not be not nilpotent. Thus we suppose
$p\geq q$ henceforth. Examples of spacelike and timelike Jordan Osserman
manifolds of neutral signature $(s,s)$ which are nilpotent of order $2$ have been constructed
Gilkey, Ivanova, and Zhang [14] for any $s\geq 2$. Examples of spacelike
and timelike Jordan Osserman manifolds of signature $(2,2)$ which are nilpotent of order $3$
have been constructed by García-Rió, Vázquez-Abal and Vázquez-Lorenzo
[11]. This brief note is devoted to the proof of the following result:
Theorem 1.1.
If $s\geq 2$, then there exist pseudo-Riemannian manifolds of signature
$(2s,s)$ which are spacelike Jordan Osserman nilpotent of order $3$ and
which are not timelike Jordan Osserman.
Our examples is quite different in flavor from those described in
[11, 14] in several respects. The primary feature is that we are not in
the balanced setting where
$p=q$; the extra timelike directions play a central role in our construction.
Additionally, the examples of [11, 14] are also timelike Jordan Osserman;
this is not the case for our examples.
To prove Theorem 1.1, it is convenient to work first in a purely algebraic
context. In Section
2, we shall construct a family of algebraic curvature tensors $R$ on a vector space
$V$ of signature
$(2s,s)$ which are spacelike Jordan Osserman nilpotent of order $3$ and which are not
timelike Jordan Osserman. We complete the discussion in Section
3 by realizing this family geometrically. Our construction will show
that in fact there are many such examples; although we shall use quadratic polynomials to
define the metric in question, this is an inessential feature.
2. Algebraic curvature tensors
Let $V$ be a finite dimensional real vector space which is equipped with a non-degenerate
symmetric bilinear form
$g(\cdot,\cdot)$ of signature $(p,q)$. Let $R\in\otimes^{4}V^{*}$. We say that $R$ is an algebraic curvature tensor if $R$ satisfies the symmetries of the Riemann curvature tensor:
$$\displaystyle R(x,y,z,w)=-R(y,x,z,w),$$
(2.a)
$$\displaystyle R(x,y,z,w)=R(z,w,x,y),$$
$$\displaystyle R(x,y,z,w)+R(y,z,x,w)+R(z,x,y,w)=0\,.$$
The associated Jacobi operator is then defined using equation (1.a) and the notions
spacelike Jordan Osserman and so forth are defined analogously.
Let $s\geq 2$. Let $\mathcal{U}:=\{U_{1},...,U_{s}\}$,
$\mathcal{V}:=\{V_{1},....,V_{s}\}$, and $\mathcal{T}:=\{T_{1},...,T_{s}\}$ comprise a basis for
$\mathbb{R}^{3s}$. We let indices $a,b,c,d$ range from $1$ through $s$.
Lemma 2.1.
Let $g_{ab}=g_{ba}$ be an arbitrary symmetric matrix. Define a metric
$g$ of signature $(2s,s)$ on $\mathbb{R}^{3s}$ whose non-zero components are:
$$g(U_{a},U_{b})=g_{ab},\quad g(U_{a},V_{b})=g(V_{b},U_{a})=\delta_{ab},\quad g(%
T_{a},T_{b})=-\delta_{ab}\,.$$
Let $R^{(1)}$ and $R^{(2)}$ be algebraic curvature tensors on
$\operatorname{Span}\{U_{a}\}$. Define a $4$ tensor
$R=R(R^{(1)},R^{(2)})$ on
$\mathbb{R}^{3s}$ whose non-zero entries are
$$\displaystyle R(U_{a},U_{b},U_{c},U_{d})$$
$$\displaystyle:=$$
$$\displaystyle R^{(1)}(U_{a},U_{b},U_{c},U_{d}),$$
$$\displaystyle R(U_{a},U_{b},U_{c},T_{d})$$
$$\displaystyle=$$
$$\displaystyle R(U_{a},U_{b},T_{c},U_{d})=R(U_{a},T_{b},U_{c},U_{d})$$
$$\displaystyle=$$
$$\displaystyle R(T_{a},U_{b},U_{c},U_{d}):=R^{(2)}(U_{a},U_{b},U_{c},U_{d})\,.$$
(1)
$R$ is an algebraic curvature tensor on $\mathbb{R}^{3}$.
(2)
If $R^{(2)}(U_{a},U_{b},U_{c},U_{d}):=\delta_{ad}\delta_{bc}-\delta_{ac}\delta_{bd}$, then
$R$ is spacelike Jordan Osserman nilpotent of order $3$ and not timelike Jordan Osserman.
Proof.
The sum of algebraic curvature tensors is again an algebraic curvature tensor.
If $R^{(2)}=0$, then clearly $R$ is an algebraic curvature tensor since we may assume
$x,y,z,w\in\mathcal{U}$ in establishing the relations of display
(2.a). We may therefore set
$R^{(1)}=0$ and consider only the effect of
$R^{(2)}$ in proving assertion (1). In that case, exactly one of the vectors $x,y,z,w$ must be
taken from $\mathcal{T}$ and the remaining $3$ vectors must be taken from
$\mathcal{U}$. Suppose, for example, $x\in\mathcal{T}$ while
$y,z,w\in\mathcal{U}$. Then replacing $x$ by the corresponding element $\bar{x}\in\mathcal{U}$ replaces $R$ by $R^{(2)}$ and thus the relations of display (2.a)
follow for $R$ because from the corresponding relations for $R^{(2)}$. This proves assertion
(1).
The tensor $R^{(2)}$ of assertion (2) is the algebraic curvature tensor of constant sectional
curvature
$+1$ with respect to the standard metric $(U_{a},U_{b})=\delta_{ab}$. Consequently, it is
invariant under the action of the orthogonal group $O(s)$.
Expand a spacelike vector $X\in\mathbb{R}^{3s}$ in the form $X=u_{a}U_{a}+v_{a}V_{a}+t_{a}T_{a}$
where we adopt the Einstein convention and sum over repeated indices.
Then
$$g(X,X)=g_{ab}u_{a}u_{b}+2\delta_{ab}u_{a}v_{b}-\delta_{ab}t_{a}t_{b}.$$
If $\vec{u}=0$, then $g(X,X)\leq 0$. Consequently $\vec{u}\neq 0$. By making an orthogonal
rotation in the $U$ vectors and the same orthogonal rotation in the $V$ and in the $T$ vectors
and by rescaling
$X$, we may suppose without loss of generality that the bases $\mathcal{U}$, $\mathcal{V}$,
and $\mathcal{T}$ have been chosen so that the general form of $g$ and $R$ is the same, so
that $u_{1}=1$, and so that $u_{a}=0$ for $a>1$. For $1\leq a,b,c,d\leq s$, define
$R_{abcd}^{(2)}:=R^{(2)}(U_{a},U_{b},U_{c},U_{d})$. Then:
$$\begin{array}[]{lll}(J(X)U_{a},U_{b})=C_{ab},&(J(X)U_{a},T_{b})=R^{(2)}_{a11b}%
,&(J(X)U_{a},V_{b})=0,\\
(J(X)T_{a},U_{b})=R^{(2)}_{a11b},&(J(X)T_{a},T_{b})=0,&(J(X)T_{a},V_{b})=0,\\
(J(X)V_{a},U_{b})=0,&(J(X)V_{a},T_{b})=0,&(J(X)V_{a},V_{b})=0,\end{array}$$
where $C_{ab}=C_{ba}$ is an appropriately chosen matrix. We then have:
$$J(X)U_{a}=C_{ab}V_{b}-R^{(2)}_{a11b}T_{b},\ J(X)T_{a}=R^{(2)}_{a11b}V_{b},\ J(%
X)V_{a}=0\,.$$
It is now clear that $J(X)^{3}=0$. We have $J(X)X=0$ and $J(X)V_{a}=0$. Since $R_{a11b}^{(2)}=0$
if
$a=1$ or
$b=1$,
$J(X)T_{1}=0$. Set $R_{a11b}^{(2)}=\delta_{ab}$ for $a\geq 2$. Since $u_{1}=1$,
$\{X,U_{2},...,U_{s},T_{1},...,T_{s},V_{1},...,V_{s}\}$ is a basis for $V$. Consequently:
$$\displaystyle\operatorname{Range}(J(X))=\operatorname{Span}\{J(X)X,J(X)U_{2},.%
..,J(X)U_{s},J(X)T_{1},...,J(X)T_{s},$$
$$\displaystyle\phantom{\operatorname{Range}(J(X))=\operatorname{Span}\{}J(X)V_{%
1},...,J(X)V_{s}\}$$
$$\displaystyle\phantom{\operatorname{Range}(J(X))}=\operatorname{Span}\{J(X)U_{%
2},...,J(X)U_{s},J(X)T_{2},...,J(X)T_{s}\}$$
$$\displaystyle\phantom{\operatorname{Range}(J(X))}=\operatorname{Span}\{C_{2b}V%
_{b}-T_{2},...,C_{sb}V_{b}-T_{s},V_{2},...,V_{s}\}\,.$$
The set $\{C_{2b}V_{b}-T_{2},...,C_{sb}V_{b}-T_{s},V_{2},...,V_{s}\}$ is linearly independent.
Furthermore:
$$\displaystyle\operatorname{Range}(J(X))\cap\ker(J(X))=\operatorname{Span}\{V_{%
2},...,V_{s}\},$$
$$\displaystyle\operatorname{Range}(J(X)^{2})=\operatorname{Span}\{V_{2},...,V_{%
s}\}\,.$$
It is now clear that $R$ is spacelike Jordan Osserman nilpotent of order $3$.
Since $J(T_{1})=0$ while $J(U_{1}-V_{1})=J(U_{1})\neq 0$, $R$ is not timelike Jordan
Osserman.
∎
3. Geometric Realizations
We complete the proof of Theorem 1.1 by showing that the structures of Lemma
2.1 are geometrically realizable. The metrics we shall consider are similar those
described in different contexts in
[6, 15, 18]. We take coordinates of the form
$(u_{1},...,u_{s},v_{1},...,v_{s},t_{1},...,t_{s})$ on
$\mathbb{R}^{3s}$. Let
$$U_{a}:=\textstyle\frac{\partial}{\partial u_{a}},\quad V_{a}:=\textstyle\frac{%
\partial}{\partial v_{a}},\quad\text{and}\quad T_{a}:=\textstyle\frac{\partial%
}{\partial t_{a}}$$
be the associated coordinate frame for the tangent bundle. We let the index $r$ range from
$1$ to $3s$ and index the full coordinate frame
$$\{e_{1},...,e_{3s}\}:=\{U_{1},...,U_{s},V_{1},...,V_{s},T_{1},...,T_{s}\}\,.$$
Theorem 1.1 will follow from Lemma 2.1 and from the following Lemma:
Lemma 3.1.
Let $R^{(2)}$ be a fixed algebraic curvature tensor on
$\mathbb{R}^{s}$.
Define a metric $g$ of signature
$(2s,s)$ on $\mathbb{R}^{2s,s}$ whose non-zero inner products are given by:
$$\displaystyle g(U_{a},U_{b})=\psi_{abcd}u_{c}t_{d}\text{ where }\psi_{abcd}=%
\psi_{bacd}:=\textstyle-\frac{2}{3}(R_{acdb}^{(2)}+R_{adcb}^{(2)}),$$
$$\displaystyle g(U_{a},V_{b})=g(V_{b},U_{a})=\delta_{ab},\quad\text{and}\quad g%
(T_{a},T_{b})=-\delta_{ab}\,.$$
Let $R^{(1)}_{abcd}(u,t):=R(U_{a},U_{b},U_{c},U_{d})(u,t)$. Then
$R(u,t)=R(R^{(1)}(u,t),R^{(2)})$.
Proof.
At this point, we change our indexing convention slightly for the remainder of
the proof. We shall let indices
$a,b,c$ index elements of
$\mathcal{U}$, indices $\alpha,\beta,\gamma$ index elements of $\mathcal{V}$, and indices
$i,j,k$ index elements of $\mathcal{T}$. Indices $r_{\mu}$ will index the full coordinate
basis. By an abuse of notation, we shall set
$\Gamma_{abc}=g(\nabla_{U_{a}}U_{b},U_{c})$,
$\Gamma_{abi}=g(\nabla_{U_{a}}U_{b},T_{i})$, etc.
We replace an element of $\mathcal{T}$ by the corresponding element of $\mathcal{U}$ to define
$\tilde{\psi}_{abci}$,
$\tilde{R}^{(2)}_{abci}$,
$\tilde{R}^{(2)}_{abic}$, $\tilde{R}^{(2)}_{aibc}$, and $\tilde{R}^{(2)}_{iabc}$. The non-zero
Christoffel symbols of the metric are:
(3.a)
$$\begin{array}[]{l}\Gamma_{abc}=\textstyle\frac{1}{2}(\tilde{\psi}_{bcai}+%
\tilde{\psi}_{acbi}-\tilde{\psi}_{abci})t_{i},\\
\Gamma_{iab}=\Gamma_{aib}=-\Gamma_{abi}=\textstyle\frac{1}{2}\tilde{\psi}_{%
abci}u_{c}\,.\end{array}$$
We raise indices to see:
(3.b)
$$\Gamma_{r_{1}r_{2}}{}^{a}=0,\quad\Gamma_{r_{1}r_{2}}{}^{i}=-\Gamma_{r_{1}r_{2}%
i},\quad\text{and}\quad\Gamma_{r_{1}r_{2}}{}^{\alpha}=\Gamma_{r_{1}r_{2}a}\,.$$
The curvature tensor is given by:
$$R_{r_{1}r_{2}r_{3}r_{4}}=e_{r_{1}}\Gamma_{r_{2}r_{3}r_{4}}-e_{r_{2}}\Gamma_{r_%
{1}r_{3}r_{4}}+\Gamma_{r_{1}r_{5}r_{4}}\Gamma_{r_{2}r_{3}}{}^{r_{5}}-\Gamma_{r%
_{2}r_{5}r_{4}}\Gamma_{r_{1}r_{3}}{}^{r_{5}}\,.$$
If $r_{5}$ indexes an element of $\mathcal{V}$, then $\Gamma_{\star r_{5}\star}=0$ by equation
(3.a) while if $r_{5}$ indexes an element of $\mathcal{U}$, then
$\Gamma_{\star\star}{}^{r_{5}}=0$ by equation (3.b). Thus $r_{5}$ must index an element
of $\mathcal{T}$ and consequently, we may express:
(3.c)
$$R_{r_{1}r_{2}r_{3}r_{4}}=e_{r_{1}}\Gamma_{r_{2}r_{3}r_{4}}-e_{r_{2}}\Gamma_{r_%
{1}r_{3}r_{4}}+\Gamma_{r_{1}ir_{4}}\Gamma_{r_{2}r_{3}}{}^{i}-\Gamma_{r_{2}ir_{%
4}}\Gamma_{r_{1}r_{3}}{}^{i}\,.$$
Thus by equation (3.a), quadratic terms in $\Gamma$ can only appear in equation
(3.c) if
$r_{1}$, $r_{2}$, $r_{3}$, and $r_{4}$ all index elements of $\mathcal{U}$.
The only other non-zero curvatures occur when exactly one of $r_{\nu}$ indexes an
element of $\mathcal{T}$ and the remaining $r_{\nu}$ index elements of $\mathcal{U}$. We may
therefore compute the proof by computing:
$R(U_{a},U_{b},U_{c},T_{i})=U_{a}\Gamma_{bci}-U_{b}\Gamma_{aci}=\textstyle\frac%
{1}{2}(\tilde{\psi}_{acbi}-\tilde{\psi}_{bcai})$
$=-\textstyle\frac{1}{3}(\tilde{R}_{abic}^{(2)}+\tilde{R}_{aibc}^{(2)}-\tilde{R%
}_{baic}^{(2)}-\tilde{R}_{biac}^{(2)})$
$=-\textstyle\frac{1}{3}(2\tilde{R}^{(2)}_{abic}-\tilde{R}^{(2)}_{iabc}-\tilde{%
R}^{(2)}_{ibca})$
$=-\textstyle\frac{1}{3}(2\tilde{R}^{(2)}_{abic}+\tilde{R}^{(2)}_{icab})=\tilde%
{R}^{(2)}_{abci}$.
∎
Remark 3.2.
It is worth giving a very specific example. Define an inner product $g$ on
$\mathbb{R}^{6}$ whose non-zero components are, up to the usual $\mathbb{Z}_{2}$ symmetries given
by:
$$\displaystyle g(U_{1},U_{1})=-2u_{2}t_{2},\quad g(U_{2},U_{2})=-2u_{1}t_{1},%
\quad g(U_{1},U_{2})=u_{1}u_{2},$$
$$\displaystyle g(U_{1},V_{1})=g(U_{2},V_{2})=-g(T_{1},T_{1})=-g(T_{2},T_{2})=1.$$
This manifold has signature $(4,2)$. It is spacelike Jordan Osserman nilpotent of order $3$.
It is not timelike Jordan Osserman. Furthermore, it is curvature homogeneous up to order $0$ as defined by
Kowalski, Tricerri, and Vanhecke
[16].
Acknowledgments
Research of P. Gilkey partially supported by the NSF (USA) and
MPI (Leipzig). Research of S. Nikčević partially supported by the Dierks Von Zweck
Stiftung (Essen), DAAD (Germany) and MM 1646 (Srbija). It is a pleasant task to thank Professor E.
García–Río for helpful discussions on this matter.
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Inverse cascade in decaying 3D magnetohydrodynamic turbulence
Mattias Christensson[†]
Mark Hindmarsh[‡]
Centre for Theoretical Physics, University of Sussex, Brighton BN1 9QJ, U.K.
Axel Brandenburg[⋆]
Nordita, Blegdamsvej 17, DK-2100 Copenhagen, Denmark;
Department of Mathematics, University of Newcastle, Newcastle upon Tyne, NE1 7RU,
U.K.
Abstract
We perform direct numerical simulations
of three-dimensional freely decaying magnetohydrodynamic (MHD) turbulence.
For helical magnetic fields an inverse cascade effect is observed in
which power is transfered from smaller scales to larger scales.
The magnetic field reaches a scaling regime with self-similar evolution,
and power law behavior at high wavenumbers. We also find power law decay in the
magnetic and kinematic energies, and power law growth in the characteristic
length scale of the magnetic field.
pacs: PACS numbers: 95.30.Q, 47.65, 47.40, 98.80
SUSX-TH/00-018
NORDITA-2000-103 AP
[
]
I Introduction
Within cosmology, astrophysics or geophysics one often needs to deal with
electrically conducting plasmas
at high kinematic and magnetic Reynolds numbers where magnetic fields are
dynamically important. Indeed, much of the turbulence in the interstellar
medium is magnetohydrodynamic in nature.
Hydromagnetic turbulence has been explored extensively in connection with
the generation of large scale magnetic fields in astrophysical bodies
such as planets, stars, accretion discs and galaxies through dynamo theories.
Non-driven, freely decaying turbulence may also be of interest in connection
with both the physics of the interstellar medium and cosmology. Our interest
was inspired by the cosmology of primordial magnetic fields, which
is sometimes considered as a possible source for providing the
seed for the galactic dynamo [4].
There have been various related works on decaying MHD turbulence,
by authors interested in different contexts
[5, 6, 7, 8, 9, 10].
Most directly comparable to our work,
Biskamp and Müller [9] studied the energy decay in incompressible
3D magnetohydrodynamic turbulence in numerical simulations at relatively high
Reynolds number, and in a companion letter [10] studied the
scaling properties of the energy power spectrum.
We are here especially interested in the inverse cascade of magnetic helicity
whereby magnetic energy is transferred from small to large scale
fluctuations. This is important for a primordial magnetic field to reach a
large enough scale with sufficient amplitude to be relevant for seeding the
galactic dynamo [11].
It should be noted that due to the conformal invariance of MHD in the
radiation era the MHD equations in an expanding universe can be converted
into the relativistic MHD equations in flat spacetime by an appropriate scaling
of the variables and by using conformal time [12].
The equations of [12] differ slightly from the ordinary
non-relativistic MHD equations. However, in order to facilitate comparison
with earlier work, we use the non-relativistic equations.
We perform 3D simulations both with and without magnetic helicity,
starting from statistically homogeneous and isotropic random initial
conditions, with power spectra suggested by cosmological applications.
We find a strong inverse cascade in the helical case, with equivocal
evidence for a weak inverse cascade when only helicity fluctuations are present.
In the helical case we also find a
self-similar power spectrum with an approximately $k^{-2.5}$ behavior at
high $k$.
We present energy decay laws which are
comparable to those found in the incompressible case by Biskamp and
Müller [9], and in the compressible case by
Mac Low et al. [8].
II The model
We consider the equations for an isothermal compressible gas with a
magnetic field, which is governed by the momentum equation, the continuity
equation, and the induction equation, written here in the form
$$\displaystyle\frac{\partial{\mathbf{u}}}{\partial t}=-{\mathbf{u}}\cdot\mbox{%
\boldmath$\nabla$}{\mathbf{u}}-c_{s}^{2}\mbox{\boldmath$\nabla$}\ln\rho+\frac{%
{\mathbf{J}}\times{\mathbf{B}}}{\rho}$$
$$\displaystyle+\frac{\mu}{\rho}\left(\nabla^{2}{\mathbf{u}}+\frac{1}{3}\mbox{%
\boldmath$\nabla$}\mbox{\boldmath$\nabla$}\cdot{\mathbf{u}}\right),$$
(1)
$$\frac{\partial\ln\rho}{\partial t}=-{\mathbf{u}}\cdot\mbox{\boldmath$\nabla$}%
\ln\rho-\mbox{\boldmath$\nabla$}\cdot{\mathbf{u}},$$
(2)
$$\frac{\partial{\mathbf{A}}}{\partial t}={\mathbf{u}}\times{\mathbf{B}}+\eta%
\nabla^{2}{\mathbf{A}},$$
(3)
where ${\mathbf{B}}=\nabla\times{\mathbf{A}}$ is the magnetic field in terms of the magnetic
vector potential ${\mathbf{A}}$, ${\mathbf{u}}$ is the velocity, ${\mathbf{J}}$ is the current density,
$\rho$ is the density, $\mu$ is the dynamical viscosity, and $\eta$ is the
magnetic diffusivity.
The code for solving these equations [13]
uses a variable third order Runge-Kutta
timestep and sixth order explicit centered derivatives in space.
All our runs are performed on a $120^{3}$ grid, and
we use periodic boundary conditions,
which means that the average plasma density
$\langle\rho_{0}\rangle=\rho_{0}$ is conserved during runs.
Here $\rho_{0}$ is the value of the initially uniform density, and the
brackets denote volume average.
We adopt nondimensional quantities by measuring ${\mathbf{u}}$ in units of $c$,
where $c$ is the speed of light, ${\mathbf{k}}$ in units of $k_{1}$, where $k_{1}$
is the smallest wavenumber in the box, which has a size of
$L_{\rm BOX}=2\pi$, density in units of $\rho_{0}=1$, and ${\mathbf{B}}$ is measured
in units of $\sqrt{\mu_{0}\rho_{0}}c$, where $\mu_{0}$ is the magnetic
permeability. This is equivalent to putting
$c=k_{1}=\rho_{0}=\mu_{0}=1$.
In the following we will refer to the mean kinematic viscosity $\nu$ which
we define as $\nu\equiv\mu/\rho_{0}$. The sound speed $c_{s}$ takes
the value $c_{s}=1/\sqrt{3}$, as appropriate for a relativistic fluid.
With $c=1$, the unit of time is such that the light crossing time of the
box is $2\pi$.
Our equations are similar to those for the
relativistic gas in the early universe using scaled variables and
conformal time for non-relativistic bulk velocities [12].
We expect our results to change little using the true relativistic equations,
as our advection velocity is at most only mildly relativistic, and this only
at the beginning of the simulation.
III On the role of the inverse cascade
The magnetic helicity $H_{\rm M}$ is given by
$$H_{\rm M}=\int{\mathbf{A}}\cdot{\mathbf{B}}\,d^{3}x$$
(4)
and characterizes the linkage between magnetic field lines.
$H_{\rm M}$ is conserved in the absence of ohmic dissipation,
although it is still possible to
have local, small scale helicity fluctuations.
Helicity plays an important role in dynamo theory
[14, 15],
where turbulence is driven.
In many astrophysical and cosmological situations the magnetic Reynolds
number $Re_{\rm M}$ is very large. We define
the magnetic Reynolds number as $Re_{\rm M}=Lv/\eta$,
where $L$ and $v$ are the typical length scale and velocity of the
system under consideration and $\eta$ is the resistivity.
The magnetic Reynolds number is a measure of the relative
importance of flux freezing versus resistive diffusion.
In a cosmological context this number can be extraordinarily
large: causality imposes the weak limit $L\leq ct$ and relativity
demands $v<c$.
With conductivities relevant to the era when the electroweak phase transition
took place [16], one can in principle
obtain a magnetic Reynolds number of about $10^{16}$.
This is often taken to mean that the magnetic field is frozen into
the plasma, and the scale length of the field increases only with the
expansion of the Universe.
However, this simple picture does not necessarily give a full description
of the dynamics because
the MHD equations, especially at high Reynolds numbers where nonlinear terms
are important, exhibit turbulent behavior, which
can lead to a redistribution of magnetic energy over different
length scales [12].
Energy in a turbulent magnetic field can undergo an inverse cascade
and be transferred from high frequency modes to low frequency modes,
increasing the overall comoving correlation length [14].
This process is due to the
nonlinear terms giving rise to interactions between many different length
scales.
We will take the initial primordial power spectrum as given
and address the question of how such a primordial
spectrum evolves as a consequence of the nonlinear equations of motion.
IV Initial conditions
Since one of the aims of the present work is to investigate the role of magnetic
helicity in the inverse cascade we describe how the initial conditions
for our simulations were set up. We chose our initial condition by setting up
magnetic fluctuations with an initial power spectrum
$P_{\rm M}(k)\equiv\langle{\mathbf{B}}^{*}_{\mathbf{k}}\cdot{\mathbf{B}}_{%
\mathbf{k}}\rangle\approx k^{n}$ in Fourier
space (and averaged over shells of constant $k=|{\mathbf{k}}|$),
for low values of the wavenumber $k$, using a exponential cutoff $k_{c}$.
[The shell-averaged power spectrum, $P_{\rm M}(k)$, is not to be confused with the
shell-integrated energy spectrum, $E_{\rm M}=4\pi k^{2}\times{1\over 2}P_{\rm M}(k)$,
which is shown in the plots below.]
The magnetic field fluctuations are drawn from a Gaussian random field
distribution fully determined by its power spectrum in Fourier space according to
the following procedure. For each grid point we use the corresponding wavenumber
to select an amplitude from a Gaussian distribution centered on zero and with the
width
$$P_{\rm M}(k)=P_{\rm M,0}k^{n}\exp(-(k/k_{c})^{4})$$
(5)
where $k=|\bf k|$.
We then transform the field back into real space to obtain the
field at each grid point. This is done independently for each field component.
There is a requirement in cosmology that $n\geq 2$, which is set by causality
demanding that the
correlation function of the magnetic field vanishes at large distances, and
the fact that
the magnetic field is divergence-free [17].
In the simulations presented we chose the slope of the power spectrum
to be $n=2$. We also chose $k_{c}=30$, unless specified otherwise,
which gives a power spectrum peaked at a relatively large value of $k$.
Biskamp and Müller [9, 10] started with a spectrum
peaked at $k_{c}=4$, which
may account for the different slope in the late-time power spectrum which we
observe (see Section V.1).
Our velocity power spectrum was chosen in a similar way, but with
$n=0$ corresponding to white noise at large scales (there is no requirement for
incompressibility in the early Universe).
The initial magnetic energy was taken equal to the kinetic energy, and
had the value $5\times 10^{-3}$ in all runs, as the primordial field is
thought likely to be weak.
In order to introduce a non-zero average magnetic helicity into the system it is useful
to represent the vector potential in terms of its projection onto an orthogonal
basis formed by $\hat{{\mathbf{e}}}_{+}$, $\hat{{\mathbf{e}}}_{-}$ and $\hat{{\mathbf{k}}}$.
The two basis vectors $\hat{{\mathbf{e}}}_{+}$ and $\hat{{\mathbf{e}}}_{-}$ can be chosen
to be the unit vectors for circular polarization, right-handed and left-handed
respectively. That is
$\hat{{\mathbf{e}}}_{\pm}=\hat{{\mathbf{e}}}_{1}\pm i\hat{{\mathbf{e}}}_{2}$
where $\hat{{\mathbf{e}}}_{1}$ and $\hat{{\mathbf{e}}}_{2}$ are unit vectors orthogonal to
each other and to ${\mathbf{k}}$. They are given by
$\hat{{\mathbf{e}}}_{1}={\mathbf{k}}\times\hat{{\mathbf{z}}}/|{\mathbf{k}}%
\times({\mathbf{k}}\times\hat{{\mathbf{z}}})|$ and
$\hat{{\mathbf{e}}}_{2}={\mathbf{k}}\times({\mathbf{k}}\times\hat{{\mathbf{z}}}%
)/|{\mathbf{k}}\times\hat{{\mathbf{z}}}|$
respectively. $\hat{{\mathbf{z}}}$ is a reference direction.
Note that since
$$i\hat{{\mathbf{k}}}\times\hat{{\mathbf{e}}}_{s}=sk\hat{{\mathbf{e}}}_{s}$$
(6)
where $s=\pm 1$, this corresponds to an expansion of the magnetic
vector potential into helical modes.
Using these basis vectors it is easily seen that the magnetic energy
spectrum is
$$E_{\rm M}(k)=2\pi k^{2}\langle|{\mathbf{B}}_{{\mathbf{k}}}|^{2}\rangle$$
(7)
where the amplitude of the magnetic field is given by
$$|{\mathbf{B}}_{{\mathbf{k}}}|^{2}=(|A_{{\mathbf{k}}}^{+}|^{2}+|A_{{\mathbf{k}}%
}^{-}|^{2})|{\mathbf{k}}|^{2}$$
(8)
and the expression for the magnetic helicity spectrum $H_{\rm M}(k)$ is
$$H_{\rm M}(k)=4\pi k^{2}\langle{\mathbf{A}}_{{\mathbf{k}}}^{*}\cdot{\mathbf{B}}%
_{{\mathbf{k}}}\rangle$$
(9)
where
$${\mathbf{A}}_{{\mathbf{k}}}^{*}\cdot{\mathbf{B}}_{{\mathbf{k}}}=(|A_{{\mathbf{%
k}}}^{+}|^{2}-|A_{{\mathbf{k}}}^{-}|^{2})|{\mathbf{k}}|.$$
(10)
The function
$H_{\rm M}(k)$ is a sensitive measure of the correlation between the vector potential
and the magnetic field. $H_{\rm M}(k)$ may, of course, be positive in one part
of Fourier space and negative in another part. It is, however, bounded in
magnitude by the inequality
$$|H_{\rm M}(k)|\leq 2k^{-1}E_{\rm M}(k).$$
(11)
A field which saturates the above inequality is maximally helical.
The amplitudes $A_{{\mathbf{k}}}^{\pm}$ can be chosen independently, provided
$A^{*\pm}_{-{\mathbf{k}}}=A^{\pm}_{{\mathbf{k}}}$, which is just the condition that
the vector potential be real.
Therefore it is possible to adjust the amplitudes $|A_{{\mathbf{k}}}^{+}|$ and
$|A_{{\mathbf{k}}}^{-}|$ freely and in so doing obtaining a magnetic field with
arbitrary magnetic helicity. With our
method we are able to put statistically random but maximally helical fields
in our initial conditions. In our runs with initial helicity we take
$H_{\rm M}=H_{\rm max}$.
Because we evolve our dynamical fields on a discrete lattice we have to
be careful when using derivative operations in Fourier space.
In general, the wave vector, which is an eigenvalue of the
derivative operator, needs to be replaced by some function
$k_{\rm eff}(k)$, which is an eigenvalue of the discrete derivative
operator on the lattice. In our case we have, for the sixth order explicit
centered derivative
$$k_{\rm eff}(k)={\textstyle\frac{1}{30}}\left[\sin(3k)-9\sin(2k)+45\sin(k)%
\right].$$
(12)
In order to be consistent with the scheme used in the simulation, we use
$k_{\rm eff}(k)$ when calculating the initial condition in Fourier space.
V Results
In all runs the mean kinematic viscosity $\nu$ and the resistivity $\eta$
were chosen to be equal
with values between $\nu=\eta=5\times 10^{-4}-5\times 10^{-5}$.
In our simulations we typically obtain Reynolds numbers
of the order of $100-200$.
The Reynolds numbers in our simulations are
evaluated using the magnetic Taylor microscale which
we calculate here as the ratio of the rms magnetic field and the rms current
density, $L_{\rm T}=2\pi B_{\rm rms}/J_{\rm rms}$. The $2\pi$ factor is
here included so that $L_{\rm T}$ represents the typical wave length (and not
the inverse wavenumber) of structures in the current density.
V.1 Spectral evolution
The inverse magnetic cascade for decaying MHD turbulence is best
visualized in terms of magnetic energy spectra $E_{\rm M}(k)$ because
information on nonlinear interaction between different scales is
contained in $E_{\rm M}(k)$.
In Fig. 1 we show a run with initial magnetic helicity.
In Fig. 1 we see evidence for a dual energy transfer
both toward higher and lower wavenumbers.
The inverse cascade is characterized by the transfer of energy from small
scale structures in the magnetic field to larger ones.
In Fig. 1 this behavior is clearly seen as
indicated by the rise in the energy spectrum at small wavenumbers.
Some energy is also being transported to smaller scales where
the spectrum is decaying due to diffusive effects.
We also note that at wavenumbers above the peak $k_{p}(t)$ the spectrum
develops a power law shape. This power law has approximately a $k^{-2.5}$
slope.
This differs from the approximately $k^{-5/3}$ law found by Müller and Biskamp
[10]. We suggest that this is due to finite size effects, which
affect the spectrum if the initial scale separation between $k_{p}$ and the
smallest wavenumber in the box ($k=1$) is insufficient, and if the flow is
strongly helical so that its spectrum is governed by inverse cascading. In
order to check this we have performed
a run with larger initial length scale, $k_{c}=5$. In this
case the magnetic energy spectrum develop into an approximate $k^{-5/3}$ law at
late times. However, this occurs only after the peak of the spectrum has left
the simulation box,
i.e. after finite size effects have begun to play a role.
To check if the magnetic field evolution is self-similar one can make the
following ansatz for the energy spectrum
$$E_{\rm M}(k,t)=\xi(t)^{-q}g_{\rm M}(k\xi).$$
(13)
Here $\xi$ is the characteristic length scale of the magnetic field,
taken to be the magnetic Taylor microscale defined above, and $q$ is a parameter
whose value is some real number.
We call $g_{\rm M}(k\xi)$ the magnetic scaling function.
In Fig. 2 we have plotted $\xi(t)^{q}E_{\rm M}(k,t)$ versus the
scaled variable $k\xi(t)$.
The value of the parameter $q$ in this run is $q=0.7$.
It is seen that for each different value of time $t$, the data collapses
onto a single curve given by the scaling function $g_{\rm M}(k\xi)$, demonstrating the
self-similarity of the magnetic field evolution.
We also performed runs in which the magnetic helicity was zero, in the statistical
sense. Magnetic helicity was present due to fluctuations, but was of very small
amplitude. In these runs no significant inverse cascade was observed.
Fig. 3 shows the energy spectrum for such a run
with only small magnetic helicity fluctuations present in the initial conditions.
It is seen that only a weak inverse cascade
is present at the lowest wavenumbers, much smaller than in the helical case.
However, that it seem to be present at all is interesting as the effect could
become more pronounced for higher Reynolds numbers.
It is possible that this effect is due to the magnetic helicity fluctuations
even though they were small.
One simulation was performed with identically zero initial magnetic helicity
fluctuations. In this case random fluctuations develop rapidly
and no differences between the two cases, were observed.
V.2 Energy decay
In Fig. 4 we show the time evolution of the magnetic
energy $E_{\rm M}(t)$ and the kinetic energy $E_{\rm K}(t)$
for a run with initial helicity and a
$k^{4}$ initial energy spectrum slope.
It is seen that the asymptotic decay rate for $E_{\rm M}(t)$ is
approximately $t^{-0.7}$. The Reynolds number for
this run was around $Re\sim 200$ at late times. In another run with
$Re\sim 100$ the decay rate was seen to be $t^{-0.8}$, so
there seems to be a dependence of the decay rate of the magnetic field
on the Reynolds number and perhaps the resulting slope of the spectrum.
The kinetic energy also decays with a power law behavior at late times.
In the case of runs with initial helicity the kinetic energy
$E_{\rm K}(t)$ decays with a different, faster rate than $E_{\rm M}(t)$.
The asymptotic decay rate is close to $t^{-1.1}$.
In runs without initial helicity the decay rates of $E_{\rm M}(t)$ and
$E_{\rm K}(t)$ are approximately the same, close to $t^{-1.1}$.
In our runs with $E_{\rm K}=E_{\rm M}$ initially, the kinetic energy spectrum shows
no evidence of an inverse cascade at any scale.
However, when the initial velocity distribution
is zero the kinetic spectrum grows on all scales initially and in the low
wavenumber region the energy continues to grow even after the high wavenumber
modes start to decay.
V.3 Coherence length evolution
During the course of the simulations the initially small scale structures
gain in size. A convenient length scale is the magnetic Taylor microscale
$L_{\rm T}$, which was defined above. This length scale is mostly
characteristic of the small scales but even they grow during the course of
the simulations.
In Fig. 5 we show the evolution of $L_{\rm T}$
for a run with initial helicity.
The asymptotic behavior of the length scale is seen to grow
approximately as $L_{\rm T}\sim t^{0.5}$.
In runs with non-helical initial conditions the growth of the magnetic Taylor
microscale is slower than in the case of helical initial conditions.
In this case the magnetic Taylor microscale grows approximately as
$L_{\rm T}\sim t^{0.4}$.
The discussion so far has mainly been concerned with the evolution of causally
generated magnetic fields using an initial $k^{4}$ slope in the magnetic energy
spectrum.
Now we briefly comment on the other cases we have looked at.
For a white noise initial spectrum $E_{\rm M}(k)\sim k^{2}$, the evolution is
qualitatively and quantitatively similar to the causal case. For helical fields
we observe an inverse cascade, while for non-helical fields a much smaller
inverse cascade is present only for the lowest modes.
VI Discussion
Our simulations show the decay rate of magnetic energy for compressible turbulence
being sensitive to the initial helicity of the magnetic field configuration.
A similar result was found in [9] in the case of incompressible
turbulence.
The fact that magnetic helicity is conserved (except for resistive changes),
and the magnetic energy decays slower for helical fields, is connected with
the observed inverse cascade in which magnetic energy is transported toward
larger scales because of nonlinear dynamics.
The decay of kinetic energy does not seem to depend on the initial helicity and its
decay rate $E_{K}(t)\sim t^{-1.1}$ is consistent with the earlier work of
[8, 9]. Note that in the helical case we observe the
kinetic energy decaying more rapidly than the magnetic one;
this behavior was also found in [9].
While these results are not directly applicable to the
evolution of primordial magnetic fields in the early universe, they do suggest that
nonlinear magnetohydrodynamical effects may play an important role in this case.
In any case, it is interesting to compare our results with the work of other authors
interested in the decay properties of cosmological magnetic fields
[18, 19, 20, 21]. Ideal MHD has a scale invariance which
leads to the scaling law [18, 20]
$$E_{M}(t,k)=k^{-1-2h}\psi(k^{1-h}t),$$
(14)
where $\psi$ is an unknown function, related to $g_{\rm M}$.
Assuming it is peaked somewhere, and $h<0$,
the characteristic scale of the field goes as $L(t)\sim t^{1/(1-h)}$. It is
also often assumed that $\psi(0)$ exists and is non-zero: thus $h$ is
determined by the initial power spectrum. Hence for a magnetic power spectrum
of index $n$, $h=-(n+3)/2$ and
$$L(t)\sim t^{2/(n+5)}.$$
(15)
This law can also
be recovered by assuming that the characteristic time scale for the decay of
turbulence on a scale $l$ is the eddy turnover time $\tau_{l}=l/v_{l}$, where
$v_{l}\sim l^{-(n+3)}$ is the velocity averaged on a scale $l$ [19].
If the characteristic scale of the field is that scale which is just decaying,
then $\tau_{L}\sim t$, and we again find Eq. (15).
One should note that these arguments ignore helicity conservation.
We recall that our non-helical runs had $n=2$ for the magnetic power
spectrum and $n=0$ for the velocity power spectrum. The observed growth law
for the magnetic Taylor microscale, $t^{0.4}$, is not consistent with
the predicted power law for $n=2$, although it does square with the growth
law for $n=0$, and it is possible that the growth in the magnetic field
length scale is being controlled by the velocity field. Simulations at higher
Reynolds numbers seem required to resolve this issue.
One would expect on integrating the helicity power spectrum that $H_{M}\sim L_{I}E_{M}$,
where $L_{I}$ is the integral scale. We would expect that $L_{I}\sim L_{T}$, and hence,
if magnetic helicity is conserved,
$$E_{M}\sim L_{T}^{-1}.$$
(16)
However, magnetic helicity is not conserved exactly: we observe a decrease in
$H_{\rm M}$ by a factor of about 2 in a run with viscosity
$\nu=5\times 10^{-5}$.
Indeed, with $L_{T}\sim t^{0.5}$ we find a somewhat steeper relation:
$E_{M}\sim t^{-0.7}\sim L_{T}^{-1.4}$.
Finally, it is interesting to note that Son’s numerical simulations of
decaying turbulence [19], performed in the Eddy-Damped Quasi-Normal Markovian
(EDQNM) approximation, show some evidence of a power law developing at high $k$,
the slope being close to $k^{-2.5}$,
although there was no net helicity present, and no inverse cascade.
Furthermore, Field and Carroll [21], again using the EDQNM
approximation, found that there were self-similar solutions with
$E_{M}\sim t^{-2/3}\sim L_{T}^{-1}$.
VII Conclusions
We have shown that for an isothermal and compressible magnetized turbulent fluid,
when undergoing a process of free decay, a substantial inverse cascade
is present for helical magnetic field configurations,
which transfer energy from smaller scale magnetic fluctuation to larger scale ones.
For non-helical magnetic fields only a weak inverse cascade was
observed on the largest scales.
The energy spectrum of the magnetic field shows evidence for a self-similar evolution
with a development of a power law of roughly $k^{-2.5}$ beyond the peak.
Decay laws for both the kinematic and magnetic energy were found. The kinetic energy
decay was approximately $t^{-1.1}$ for both helical and non-helical magnetic fields.
The decay of the magnetic field energy was found to be strongly dependent on the the
initial helicity, decaying roughly as $t^{-0.7}$ and $t^{-1.1}$ for helical and
non-helical initial conditions respectively. For the helical case, the magnetic energy
decay rate showed a dependence on the Reynolds number, with a slower decay rate for
larger Reynolds numbers.
We also observed power law behavior in the characteristic length scale of the
magnetic field, defined as the Taylor microscale $L_{T}$. In the helical case
$L_{T}\sim t^{0.5}$, whereas for non-helical fields the growth was somewhat slower,
$L_{T}\sim t^{0.4}$, and we ascribe the faster growth rate to the presence of the
inverse cascade in the helical case.
Acknowledgements.This work was conducted on the Cray T3E and SGI Origin platforms using COSMOS
Consortium facilities, funded by HEFCE, PPARC and SGI. We also acknowledge
computing support from the Sussex High Performance Computing Initiative.
References
[†]
Electronic address:
m.christensson@sussex.ac.uk
[‡]
Electronic address:
m.b.hindmarsh@sussex.ac.uk
[⋆]
Electronic address:
brandenb@nordita.dk
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Phys. Rev. D62, 103008 (2000). |
Pulsar Wind Nebulae inside Supernova Remnants as Cosmic-Ray PeVatrons
Yutaka Ohira, Shota Kisaka, Ryo Yamazaki
Department of Physics and Mathematics, Aoyama Gakuin University, 5-10-1 Fuchinobe, Sagamihara 252-5258, Japan
Abstract
We propose that cosmic-ray PeVatrons are pulsar wind nebulae (PWNe) inside supernova remnants (SNRs).
The PWN initially expands into the free expanding stellar ejecta.
Then, the PWN catches up with the shocked region of the SNR, where particles can be slightly accelerated by the back and forth motion between the PWN and the SNR, and some particles diffuse into the PWN.
Afterwards the PWN is compressed by the SNR, where the particles in the PWN are accelerated by the adiabatic compression.
By using a Monte Carlo simulation, we show that particles accelerated by the SNR to $0.1~{}{\rm PeV}$ can be reaccelerated to $1~{}{\rm PeV}$ until the end of the PWN compression.
pacs: 52.35.Tc; 52.65.Pp; 96.50.Pw; 96.50.S-; 97.60.Gb; 98.38.Mz
Introduction.—
The origin of cosmic-ray (CR) PeVatrons is a long standing problem in the Astrophysics.
The CR spectrum has a spectral break at $\sim 10^{15}~{}{\rm eV}=1~{}{\rm PeV}$ (so called, the knee energy).
The diffusive shock acceleration (DSA) at supernova remnants (SNRs) is believed to be the acceleration mechanism of CRs up to the knee energy axford77 .
Although recent gamma-ray observations support the idea ohiraetal11 , there are still many problems.
One is the knee problem.
It was estimated that SNRs cannot accelerate CRs to the knee energy for a parallel shock without a magnetic field amplification lagage83 .
In order to accelerate CRs to the knee energy, magnetic fields must be amplified in the shock upstream region.
Several mechanisms of the magnetic field amplification in the shock upstream region have been proposed bell04 .
However, no simulations demonstrate that the upstream magnetic field is sufficiently amplified to accelerate CRs to the knee energy.
In contrast to the shock upstream region, magnetic fields are expected to be easily amplified to the equipartition level in the shock downstream region giacalone07 .
Super-Alfvénic turbulence amplifies the magnetic field by stretching the magnetic field line.
The downstream turbulence is generated by interactions between upstream density fluctuations and the shock front.
In addition, the downstream turbulence is generated by the Rayleigh-Taylor instability at the contact discontinuity.
As a result, the magnetic field in the shocked region is amplified by the turbulence.
If only the downstream magnetic field is amplified, the acceleration time scale of DSA is predominantly determined by the upstream residence time of accelerated particles, which depends on the shock velocity, $u_{\rm sh}$, and the diffusion coefficient, $D$, ohirayamazaki16 .
Then the acceleration time scale is given by
$$\displaystyle t_{\rm acc}$$
$$\displaystyle\approx$$
$$\displaystyle\frac{4D}{u_{\rm sh}^{2}}=\frac{4cE}{3eu_{\rm sh}^{2}B_{\rm up}}$$
(1)
$$\displaystyle\approx$$
$$\displaystyle 10^{4}~{}{\rm yr}\left(\frac{E}{1~{}{\rm PeV}}\right)\left(\frac%
{u_{\rm sh}}{3\times 10^{3}~{}{\rm km}~{}{\rm s}^{-1}}\right)^{-2}\left(\frac{%
B_{\rm up}}{3~{}{\rm\mu G}}\right)^{-1}~{}~{},$$
where we assume the shock compression ratio of 4, the Bohm diffusion coefficient, $D=cE/3eB_{\rm up}$, and $c,e,E$ and $B_{\rm up}$ are the speed of light, elementary charge, particle energy, and upstream magnetic field strength, respectively.
After the free expansion phase ($t>t_{\rm Sedov}$), the velocity of the forward shock decreases with time.
The Sedov time, $t_{\rm Sedov}$, is given by
$$t_{\rm Sedov}\approx 10^{3}~{}{\rm yr}\left(\frac{E_{\rm SN}}{10^{51}~{}{\rm
erg%
}}\right)^{-\frac{1}{2}}\left(\frac{M_{\rm ej}}{3~{}{\rm M}_{\odot}}\right)^{%
\frac{5}{6}}\left(\frac{n}{0.1~{}{\rm cm}^{-3}}\right)^{-\frac{1}{3}}~{}~{},$$
(2)
where $E_{\rm SN},M_{\rm ej}$ and $n$ are the explosion energy, ejecta mass, and the ambient number density, respectively.
From the condition, $t_{\rm acc}=t_{\rm Sedov}$, the maximum energy of particles accelerated at the forward shock is given by
$$\displaystyle E_{\rm max}\approx 0.1~{}{\rm PeV}\left(\frac{E_{\rm SN}}{10^{51%
}~{}{\rm erg}}\right)^{\frac{1}{2}}\left(\frac{M_{\rm ej}}{3~{}{\rm M}_{\odot}%
}\right)^{-\frac{1}{6}}$$
$$\displaystyle\left(\frac{n}{0.1~{}{\rm cm}^{-3}}\right)^{-\frac{1}{3}}\left(%
\frac{B_{\rm up}}{3~{}{\rm\mu G}}\right)~{}~{},$$
(3)
where $u_{\rm sh}=(2E_{\rm SN}/M_{\rm ej})^{1/2}$ is assumed.
This is about 10 times smaller than the knee energy.
The other possible solution for the knee problem is DSA at the perpendicular shocks jokipii87 .
Since accelerated particles cannot propagate to the far upstream region,
the acceleration time scale becomes small for the perpendicular shock.
Although the injection to DSA was thought to be difficult for the perpendicular shock,
it was showed by three-dimensional hybrid simulations that
particles are injected to DSA at the perpendicular shock in a partially ionized plasma, so that
particles are rapidly accelerated by the perpendicular shock ohira16 .
However, DSA at the perpendicular shocks has another problem.
For the case of the perpendicular shocks, the maximum energy is limited by the size of acceleration region, $R$.
The available potential drop is $\Delta\phi=RB_{\rm up}u_{\rm sh}/c$,
so that the maximum energy of accelerated protons is given by
$$\displaystyle E_{\rm max}$$
$$\displaystyle=$$
$$\displaystyle e\Delta\phi$$
(4)
$$\displaystyle\approx$$
$$\displaystyle 0.1~{}{\rm PeV}\left(\frac{R}{10~{}{\rm pc}}\right)\left(\frac{B%
_{\rm up}}{3~{}{\rm\mu G}}\right)\left(\frac{u_{\rm sh}}{3\times 10^{3}~{}{\rm
km%
}~{}{\rm s}^{-1}}\right)~{}~{},$$
which is again 10 times smaller than the knee energy.
In order to accelerate CRs to the knee energy at the perpendicular shock, we need an exceptional condition takamoto15 .
In this Letter, we propose a reacceleration mechanism from $0.1~{}{\rm PeV}$ to $1~{}{\rm PeV}$ by pulsar wind nebulae (PWNe) inside SNRs.
As mentioned above, the magnetic field in the shocked region of SNRs is large enough to scatter high-energy particles.
The magnetic field in young PWNe is also large compared with that in the interstellar medium,
which is about $B_{\rm PWN}=0.1-1~{}{\rm mG}$ tanaka10 .
The PWN initially expands into the freely expanding stellar ejecta toward the shocked region of the SNR.
Since this system can be interpreted as two walls approaching each other,
particles are accelerated, shuttling between the PWN and the shocked region of the SNR.
After the PWN reaches the reverse shock of the SNR, the PWN is compressed and particles inside the PWN are accelerated by the adiabatic compression blondin01 ; vanderswaluw01 .
In the next section, by using Monte Carlo simulation, we show that the PWN-SNR system actually accelerates particles
from $0.1~{}{\rm PeV}$ to $1~{}{\rm PeV}$.
Monte Carlo simulations.— In order to investigate the particle acceleration by the PWN-SNR system, we first provide evolution of an SNR and a PWN inside the SNR.
As a first step, we consider a spherically symmetric structure.
For constant ejecta and ambient density profiles, the approximate time evolution of the forward and reverse shock radii, $R_{\rm SNR,fs}$ and $R_{\rm SNR,rs}$, are given by mckee95 .
For a constant spindown luminosity of a pulsar and a constant ejecta density profile,
the analytical solution for the time evolution of the PWN radius, $R_{\rm PWN}$, is given by vanderswaluw01 .
Fig 1 shows the time evolutions of the forward and reverse shock radii of the SNR and the radius of PWN,
where we assume the constant ejecta profile with the ejecta mass of $M_{\rm ej}=3~{}{\rm M}_{\odot}$, the explosion energy of $E_{\rm SN}=10^{51}~{}{\rm erg}$, the constant ambient matter profile with the density of $n=0.1~{}{\rm cm}^{-3}$, and the constant pulsar spindown luminosity of $L_{\rm sd}=3\times 10^{38}~{}{\rm erg}~{}{\rm s}^{-1}$.
For these parameters, the PWN catches up with the reverse shock of the SNR at $t_{\rm c}\approx 2\times 10^{3}~{}{\rm yr}$.
Afterwards, the PWN is compressed by the shocked region of the SNR.
In this Letter, we simply assume that the velocity of the PWN during the compression is $v_{\rm PWN}=-v_{\rm PWN}(t_{\rm c})/2$ and the final size of the PWN is $R_{\rm PWN}=R_{\rm PWN}(t_{\rm c})/5$.
Then, the PWN size becomes $\approx 1.2~{}{\rm pc}$ at $t_{\rm end}\approx 5\times 10^{3}~{}{\rm yr}$.
These assumptions are reasonable to simulate evolution of a spherical PWN (e.g. gelfand09, ).
We next give the velocity structure in the PWN-SNR system.
The expansion velocity of the shocked ejecta of the SNR in the observer frame is given by
$$v_{\rm SNR,shocked}=\frac{R_{\rm SNR,rs}/t-v_{\rm SNR,rs}}{4}+v_{\rm SNR,rs}~{%
}~{},$$
(5)
where we assume the compression ratio at the SNR reverse shock is $4$, and $v_{\rm SNR,rs}=dR_{\rm SNR,rs}/dt$ is the propagation velocity of the reverse shock in the observer frame.
The expansion velocity of the PWN is given by $v_{\rm PWN}=dR_{\rm PWN}/dt$ vanderswaluw01 .
Until the PWN reaches the reverse shock of the SNR, the velocity structures in both the SNR and the PWN are assumed to be uniform in this Letter.
After the PWN interacts with the reverse shock of the SNR, the velocity structure in the PWN is assumed as
$${\vec{v}}_{\rm PWN,in}(t,{\vec{r}})=-\frac{v_{\rm PWN}(t_{\rm c})}{2}\frac{{%
\vec{r}}}{R_{\rm PWN}(t)}~{}~{}.$$
(6)
Since the shocked region of the SNR and the PWN region are expected to be highly turbulent porth14 ,
motion of high-energy particles could be approximated as the random walk.
Using the above hydrodynamical structure, we perform a test-particle Monte Carlo simulation.
Simulation particles are isotropically scattered in the local fluid frame.
The scattering time is assumed to be the Bohm scattering, $t_{\rm sc}=\Omega_{\rm c}^{-1}(E/m_{\rm p}c^{2})$,
where $\Omega_{\rm c}\approx 10^{-2}~{}{\rm s}^{-1}(B/1~{}{\rm\mu G})$ is the proton cyclotron frequency, $E$ is the particle energy, and $m_{\rm p}$ is the proton mass.
During the compression phase of the PWN ($t_{\rm c}<t<t_{\rm end}$),
we do not follow the particle motion once the particle escapes from the PWN.
In this Letter, we set the magnetic field to be $B=3\times 10^{2}~{}{\rm\mu G}$ both in the PWN and the shocked region of the SNR,
and $B=0$ in the free expanding ejecta.
So that, particles are not scattered in the free expanding ejecta.
At $t=10^{3}~{}{\rm yr}$, we impulsively inject simulation particles isotropically on the reverse shock sphere, $r=R_{\rm SNR,rs}$.
Since the SNR can accelerate particles to about $0.1~{}{\rm PeV}$ (see introduction), we set the initial energy to be $0.1~{}{\rm PeV}$.
Fig 2 shows the energy spectrum of the accelerated particles at the end of the PWN compression ($t=t_{\rm end}$).
The black histogram shows the energy spectrum of particles that were in the shocked region of the SNR when the PWN reached the reverse shock of the SNR ($t=t_{\rm c}$).
They are accelerated up to twice the initial energy by the back and forth motion between the PWN and the shocked region of the SNR.
However, they are not accelerated by the PWN compression.
The energy gain in each cycle is $\Delta E/E\sim\Delta v/c$, where $\Delta v=v_{\rm SNR,shocked}-v_{\rm PWN}$ is the relative velocity.
The time scale in each cycle is $\Delta t\sim\Delta R/c$, where $\Delta R=R_{\rm SNR,rs}-R_{\rm PWN}$ is the relative distance.
Then, the acceleration time scale for the reciprocation is given by $t_{\rm acc}=\Delta t(E/\Delta E)\sim\Delta R/\Delta v$ that is the same as the dynamical time scale in which the PWN catches up with the SNR reverse shock.
Since the acceleration time scale, $t_{\rm acc}$, does not depend on the particle energy, it takes $t_{\rm acc}$
to accelerate particles to twice the initial energy.
Therefore, the maximum energy during the approaching phase becomes twice the initial energy.
The red histogram shows the energy spectrum of particles confined in the PWN at $t=t_{\rm c}$.
They are further accelerated to $1~{}{\rm PeV}$ by the PWN compression.
The maximum energy gain during the compression is
$\Delta E/E\sim R_{\rm PWN}(t_{\rm end})/R_{\rm PWN}(t_{\rm c})=5$,
so that the particles are finally accelerated to ten times the initial energy.
Hence, the PWN-SNR system can accelerate particles to the knee energy.
Discussion.— We first discuss on the acceleration of heavy nuclei.
CRs are organized not only by protons but also by heavy nuclei whose origin is also a long standing problem ohiraetal16 .
Since the supernova ejecta is metal rich,
the reverse shock propagating into the supernova ejecta is thought to be the origin of heavy CR nuclei ptuskin13 .
However, the maximum energy of the accelerated particle at the reverse shock is not so large because
the magnetic field in the expanding supernova ejecta is expected to be very small.
Furthermore, particles accelerated by the reverse shock suffer the adiabatic cooling.
Our reacceleration model can boost the maximum energy of accelerated heavy nuclei,
so that the PWN-SNR system could be important for the production of heavy CR nuclei.
Next, we discuss the energy spectrum of accelerated particles.
As shown in Fig 2, most particles are boosted by only twice
and a few particles are accelerated to $1~{}{\rm PeV}$.
Therefore, the spectrum seems to be too steep to explain the CR spectrum observed at the Earth.
We considered only particle diffusion by the magnetic field fluctuation, but diffusion by turbulence could be significant.
If more particles diffuse deeper inside the PWN by the turbulent diffusion, more particles will be accelerated to $1~{}{\rm PeV}$.
Furthermore, since there is the potential difference of about $1~{}{\rm PV}$ in the PWN,
protons could be accelerated to $1~{}{\rm PeV}$ by drifting the toroidal magnetic field bell92 ,
which was not considered in this Letter.
Hence, further studies are needed to understand the energy spectrum of particles accelerated by the PWN-SNR system.
In addition, to understand the source spectrum of CRs that are injected to our Galaxy, we have to consider
the particle spectrum that have escaped from the SNR ohiraetal10 .
Therefore, when and how accelerated particles escape from the PWN-SNR system is an important issue.
This issue could be addressed by gamma-ray observations of the PWN-SNR system like SNR G327.1-1.1, W44 and so on.
As a first step, in this Letter we assumed spherical symmetry for the PWN-SNR system and
the Bohm scattering with constant magnetic field strength for the random walk.
In reality, a pulsar has a kick velocity, and the supernova ejecta, the ambient matter, the PWN have asymmetry.
In addition, the Rayleigh-Taylor instability amplifies the asymmetry vanderswaluw04 ,
so that the PWN-SNR system is actually more complicated.
In particular, the strong turbulence could play an important role, that amplifies the magnetic field,
affecting the particle motion porth16 , and accelerates particles by turbulent acceleration ohira13 .
In order to address above problems, we need a more realistic magnetohydrodynamical simulation.
This will be addressed in future work.
Summary.— We have proposed that PWNe inside SNRs are the CR PeVatron.
Firstly, the SNR shock accelerates protons to $\sim 0.1~{}{\rm PeV}$.
Then, the protons diffuse into the interior of the SNR and are reaccelerated to $\sim 0.2~{}{\rm PeV}$ by the back and forth motion between the SNR and the PWN.
Finally, the protons diffuse into the PWN and are accelerated to $\sim 1~{}{\rm PeV}$ by the adiabatic compression while the PWN is compressed by the SNR.
In addition, we have argued that the PWN-SNR system could be the origin of heavy CR nuclei.
Acknowledgements.
Numerical computations were carried out on the XC30
system at the Center for Computational Astrophysics (CfCA)
of the National Astronomical Observatory of Japan.
This work was supported in part by Grants-in-Aid for Scientific Research of the Japanese Ministry of Education, Culture, Sports, Science and Technology No.16K17702(YO), No.16J06773(SK), and No.15K05088(RY).
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