Schatten properties,nuclearity and minimality of phase shift invariant spaces

We extend Feichtinger's minimality property on the smallest non-trivial time-frequency shift invariant Banach space, to the quasi-Banach case. Analogous properties are deduced for certain matrix spaces.We use these results to prove that the pseudo-differential operator $\mathrm{Op}(a)$ is a Schatten-q operator from $M^{\infty }$ to $M^p$ and r-nuclear operator from $M^{\infty }$ to $M^r$ when $a\in M^r$ for suitable p, q and r in $(0,\infty ]$.     

Regularity of optimal sets for some functional involving eigenvalues of an operator in divergence form

In this paper we consider minimizers of the functional$\min \{{\lambda }_1(\mathrm{\Omega })+\cdots +{\lambda }_k(\mathrm{\Omega })+\mathrm{\Lambda }|\mathrm{\Omega }|,:\mathrm{\Omega }\subset D\text{ open}\}$ where $D\subset {\mathbb{R}}^d$ is a bounded open set and where $0<{\lambda }_1(\mathrm{\Omega })\le \cdots \le {\lambda }_k(\mathrm{\Omega })$ are the first k eigenvalues on Ω of an operator in divergence form with Dirichlet boundary condition and with Hölder continuous coefficients. We prove that the optimal sets ${\mathrm{\Omega }}^{⁎}$ have finite perimeter and that their free boundary $\partial {\mathrm{\Omega }}^{⁎}\cap D$ is composed of a regular part, which is locally the graph of a $C^{1,\alpha }$-regular function, and a singular part, which is empty if $d<d^{⁎}$, discrete if $d=d^{⁎}$ and of Hausdorff dimension at most $d-d^{⁎}$ if $d>d^{⁎}$, for some $d^{⁎}\in \{5,6,7\}$.     

Maximum modulus principle for “holomorphic functions” on the quantum matrix ball

We describe the Shilov boundary ideal for a q-analog of the algebra of holomorphic functions on the unit ball in the space of $n\times n$ matrices and show that its $C^{?}$-envelope is isomorphic to the $C^{?}$-algebra of continuous functions on the quantum unitary group $U_q(n)$.     

On a problem of Specker about Euclidean representations of finite graphs

Say that a graph $G$ is representable in ${\mathbb{R}}^n$ if there is a map $f$ from its vertex set into the Euclidean space ${\mathbb{R}}^n$ such that $6f\left(x\right)?f\left(x^{\prime }\right)6=6f\left(y\right)?f\left(y^{\prime }\right)6$ iff $\{x,x^{\prime }\}$ and$\{y,y^{\prime }\}$ are both edges or both non-edges in $G$. The purpose of this note is to present the proof of the following result, due to Einhorn and Schoenberg in Einhorn and Schoenberg (1966): if $G$ finite is neither complete nor independent, then it is representable in ${\mathbb{R}}^{\left|G\right|?2}$. A similar result also holds in the case of finite complete edge-colored graphs.     

Vector-valued nonstationary Gabor frames

As an extension of Gabor frames, nonstationary Gabor (NSG) frames were recently introduced in adaptive signal analysis. They allow for efficient reconstruction with flexible sampling and varying window functions. In this paper we generalize the notion of NSG frames from $L^2(\mathbb{R})$ to the vector-valued Hilbert space $L^2(\mathbb{R},{\mathbb{C}}^L)$, and investigate the resulting vector-valued NSG frames. We derive a Walnut's representation of the mixed frame operator, and provide some necessary/sufficient conditions for a vector-valued NSG system to be a frame for $L^2(\mathbb{R},{\mathbb{C}}^L)$. Furthermore, we show the existence of painless vector-valued NSG frames, and of vector-valued NSG frames with fast decaying window functions.     

Complete permutation polynomials with the form (xpm−x+δ)s+axpm+bx over Fpn

In this paper, several classes of complete permutation polynomials with the form ${(x^{p^m}-x+\delta )}^s+a{x}^{p^m}+bx$ over ${\mathbb{F}}_{p^n}$ are proposed by the AGW criterion and determining the number of solutions of some equations. Our results also enrich constructions of known permutation polynomials.     

Maximality of Ciani curves over finite fields

In this paper, we will study Ciani curves in characteristic $p\ge 3$, in particular their standard forms $C:x^4+y^4+z^4+r{x}^2{y}^2+s{y}^2{z}^2+t{z}^2{x}^2=0$. It is well-known that any Ciani curve is a non-hyperelliptic curve of genus 3, and its Jacobian variety is isogenous to the product of three elliptic curves. As a main result, we will show that if C is superspecial, then $r,s,t$ belong to ${\mathbb{F}}_{p^2}$ and C is maximal or minimal over ${\mathbb{F}}_{p^2}$. Moreover, in this case we will provide a simple criterion in terms of $r,s,t,p$ that tells whether C is maximal (resp. minimal) over ${\mathbb{F}}_{p^2}$.     

On a Cheeger type inequality in Cayley graphs of finite groups

Let $G$ be a finite group. It was remarked in Breuillard et al. (2015) that if the Cayley graph $C(G,S)$ is an expander graph and is non-bipartite then the spectrum of the adjacency operator $T$ is bounded away from $-1$. In this article we are interested in explicit bounds for the spectrum of these graphs. Specifically, we show that the non-trivial spectrum of the adjacency operator lies in the interval $\left[-1+\frac{h{\left(\mathbb{G}\right)}^4}{\gamma },1-\frac{h{\left(\mathbb{G}\right)}^2}{2d^2}\right]$, where $h\left(\mathbb{G}\right)$ denotes the (vertex) Cheeger constant of the $d$ regular graph $C(G,S)$ with respect to a symmetric set $S$ of generators and $\gamma =2^9{d}^6{(d+1)}^2$.     

Regularity theory for Ln-viscosity solutions to fully nonlinear elliptic equations with asymptotical approximate convexity

We develop interior $W^{2,p,\mu }$ and $W^{2,\text{BMO}}$ regularity theories for $L^n$-viscosity solutions to fully nonlinear elliptic equations $T(D^{2}u,x)=f(x)$, where T is approximately convex at infinity. Particularly, $W^{2,\text{BMO}}$ regularity theory holds if operator T is locally semiconvex near infinity and all eigenvalues of $D^{2}T(M)$ are at least $?C{6M6}^{?(1+{\sigma }_0)}$ as $M\to \infty$. $W^{2,\text{BMO}}$ regularity for some Isaacs equations is given. We also show that the set of fully nonlinear operators of $W^{2,\text{BMO}}$ regularity theory is dense in the space of fully nonlinear uniformly elliptic operators.     

Free moment measures and laws

In [3], it was shown that convex, almost everywhere continuous functions coordinatize a broad class of probability measures on ${\mathbb{R}}^n$ by the map $U?{\left(\mathrm{?}U\right)}_{\mathrm{\#}}{e}^{?U}dx$. We consider whether there is a similar coordinatization of non-commutative probability spaces, with the Gibbs measure $e^{?U}dx$ replaced by the corresponding free Gibbs law. We call laws parameterized in this way free moment laws. We first consider the case of a single (and thus commutative) random variable and then the regime of n non-commutative random variables which are perturbations of freely independent semi-circular variables. We prove that free moment laws exist with little restriction for the one dimensional case, and for small even perturbations of free semi-circle laws in the general case.     

L-orthogonality in Daugavet centers and narrow operators

We study the presence of L-orthogonal elements in connection with Daugavet centers and narrow operators. We prove that if $\mathrm{dens}(Y)?{\omega }_1$ and $G:X?Y$ is a Daugavet center with separable range then, for every non-empty $w^{?}$-open subset W of $B_{X^{??}}$, it follows that $G^{??}(W)$ contains some L-orthogonal to Y. In the context of narrow operators, we show that if X is separable and $T:X?Y$ is a narrow operator, then given $y\in B_X$ and any non-empty $w^{?}$-open subset W of $B_{X^{??}}$ then W contains some L-orthogonal u so that $T^{??}(u)=T(y)$. In the particular case that $T^{?}(Y^{?})$ is separable, we extend the previous result to $\mathrm{dens}(X)={\omega }_1$. Finally, we prove that none of the previous results holds in larger density characters (in particular, a counterexample is shown for ${\omega }_2$ under the assumption $2^c={\omega }_2$).     

Additive G-codes over Fq and their dualities

We define additive G-codes over finite fields. We prove that if C is an additive G-code over ${\mathbb{F}}_q$ with duality M then its dual with respect to this duality $C^M$ is an additive G-code. We prove that if M and $M^{\prime }$ are two dualities, then $C^M$ and $C^{M^{\prime }}$ are equivalent codes. Finally, we study the existence of self-dual codes for a variety of dualities and relate them to formally self-dual and linear self-dual codes.