Strictly positive definite functions by integral transforms

We study the continuity and strict positive definiteness of positive definite functions on quasi-metric spaces given by integral transforms. We apply some of our findings to positive definite functions on the Euclidean space $R^m$ which are given by cosine transforms ($m=1$) and Fourier–Bessel transforms ($m>1$). We also apply the results to positive definite functions on a general quasi-metric space realized as extensions of certain real Laplace transforms defined by conditionally negative definite functions on the quasi-metric space itself.     

Dilations of semigroups on von Neumann algebras and noncommutative Lp-spaces

We prove that any weak* continuous semigroup ${(T_t)}_{t?0}$ of factorizable Markov maps acting on a von Neumann algebra M equipped with a normal faithful state can be dilated by a group of Markov ?-automorphisms analogous to the case of a single factorizable Markov operator, which is an optimal result. We also give a version of this result for strongly continuous semigroups of operators acting on noncommutative ${\mathrm{L}}^p$-spaces and examples of semigroups to which the results of this paper can be applied. Our results imply the boundedness of the McIntosh's ${\mathrm{H}}^{\infty }$ functional calculus of the generators of these semigroups on the associated noncommutative ${\mathrm{L}}^p$-spaces generalising some previous work from Junge, Le Merdy and Xu. Finally, we also give concrete dilations for Poisson semigroups which are even new in the case of ${\mathbb{R}}^n$.     

Representation and factorization theorems for almost-Lp-spaces

We extend the notions of $p$-convexity and $p$-concavity for Banach ideals of measurable functions following an asymptotic procedure. We prove a representation theorem for the spaces satisfying both properties as the one that works for the classical case: each almost $p$-convex and almost $p$-concave space is order isomorphic to an almost-$L^p$-space. The class of almost-$L^p$-spaces contains, in particular, direct sums of (infinitely many) $L^p$-spaces with different norms, that are not in general $p$-convex – nor $p$-concave –. We also analyze in this context the extension of the Maurey–Rosenthal factorization theorem that works for $p$-concave operators acting in $p$-convex spaces. In this way we provide factorization results that allow to deal with more general factorization spaces than $L^p$-spaces.     

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 $U_q(n)$.     

Separable discrete functions: Recognition and sufficient conditions

A discrete function of $n$ variables is a mapping $g:X_1\times \dots \times X_n\to A$, where $X_1,\dots ,X_n$, and $A$ are arbitrary finite sets. Function $g$ is called separable if there exist $n$ functions $g_i:X_i\to A$ for $i=1,\dots ,n$, such that for every input $x_1,\dots ,x_n$ the function $g(x_1,\dots ,x_n)$ takes one of the values $g_1\left(x_1\right),\dots ,g_n\left(x_n\right)$. Given a discrete function $g$, it is an interesting problem to ask whether $g$ is separable or not. Although this seems to be a very basic problem concerning discrete functions, the complexity of recognition of separable discrete functions of $n$ variables is known only for $n=2$. In this paper we will show that a slightly more general recognition problem, when $g$ is not fully but only partially defined, is NP-complete for $n\ge 3$. We will then use this result to show that the recognition of fully defined separable discrete functions is NP-complete for $n\ge 4$.The general recognition problem contains the above mentioned special case for $n=2$. This case is well-studied in the context of game theory, where (separable) discrete functions of $n$ variables are referred to as (assignable) $n$-person game forms. There is a known sufficient condition for assignability (separability) of two-person game forms (discrete functions of two variables) called (weak) total tightness of a game form. This property can be tested in polynomial time, and can be easily generalized both to higher dimension and to partially defined functions. We will prove in this paper that weak total tightness implies separability for (partially defined) discrete functions of $n$ variables for any $n$, thus generalizing the above result known for $n=2$. Our proof is constructive. Using a graph-based discrete algorithm we show how for a given weakly totally tight (partially defined) discrete function $g$ of $n$ variables one can construct separating functions $g_1,\dots ,g_n$ in polynomial time with respect to the size of the input function.     

Tight wavelet frame sets in finite vector spaces

Let $q\ge 2$ be an integer, and ${\mathbb{F}}_q^d$, $d\ge 1$, be the vector space over the cyclic space ${\mathbb{F}}_q$. The purpose of this paper is two-fold. First, we obtain sufficient conditions on $E?{\mathbb{F}}_q^d$ such that the inverse Fourier transform of $1_E$ generates a tight wavelet frame in $L^2({\mathbb{F}}_q^d)$. We call these sets (tight) wavelet frame sets. The conditions are given in terms of multiplicative and translational tilings, which is analogous with Theorem 1.1 ([20]) by Wang in the setting of finite fields. In the second part of the paper, we exhibit a constructive method for obtaining tight wavelet frame sets in ${\mathbb{F}}_q^d$, $d\ge 2$, q an odd prime and $q\equiv 3$ (mod 4).     

Distance matrices of a tree: Two more invariants,and in a unified framework

A classical result of Graham and Pollak (1971) states that the determinant of the distance matrix $D_T$ of any tree $T$ depends only on the number of edges of $T$. This and several other variants of $D_T$ have since been studied – including a $q$-version, a multiplicative version, and directed versions of these – and in all cases, $det\left(D_T\right)$ depends only on the edge-data.In this paper, we introduce a more general framework for bi-directed weighted trees that has not been studied to date; our work is significant for three reasons. First, our setting strictly generalizes – and unifies – all variants of $D_T$ studied to date (with coefficients in an arbitrary unital commutative ring) – including in Graham and Pollak (1971) above, as well as Graham and Lovász (1978), Yan and Yeh (2006), Yan and Yeh (2007), Sivasubramanian (2010), and others.Second, our results strictly improve on state-of-the-art for every variant of the distance matrix studied to date, even in the classical Graham–Pollak case. Here are three results for trees: (1) We compute the minors obtained by deleting arbitrary equinumerous sets of pendant nodes (in fact, more general sub-forests) from the rows and columns of $D_T$, and show these minors depend only on the edge-data and not the tree-structure. (2) We compute a second function of the distance matrix $D_T$: the sum of all its cofactors, termed $cof\left(D_T\right)$. We do so in our general setting and in stronger form, after deleting equinumerous pendant nodes (and more generally) as above – and show these quantities also depend only on the edge-data. (3) We compute in closed form the inverse of $D_T$, extending a result of Graham and Lovász (1978) and answering an open question of Bapat et al. (2006) in greater generality.Third, a new technique is to crucially use commutative algebra arguments – specifically, Zariski density – which to our knowledge are hitherto unused for such matrices/invariants, but are richly rewarding. We also explain why our setting is “most general”, in that for more general edgeweights, $det\left(D_T\right),cof\left(D_T\right)$ depend on the tree structure. In a sense, this completes the study of the invariants $det\left(D_T\right),cof\left(D_T\right)$ for distance matrices of trees $T$ with edge-data in a commutative ring.Our proofs use novel results for arbitrary bi-directed strongly connected graphs $G$: we prove a multiplicative analogue of an additive result by Graham et al. (1977), as well as a novel $q$-version thereof. In particular, we provide closed-form expressions for $det\left(D_G\right)$, $cof\left(D_G\right)$, and $D_G^{-1}$ in terms of their strong blocks. We then show how this subsumes the classical 1977 result, and provide sample applications to adding pendant trees and to cycle-clique graphs (including cactus/polycyclic graphs and hypertrees), subsuming variants in the literature. The final section introduces and computes a third – and novel – invariant for trees, as well as a parallel Graham–Hoffman–Hosoya type result for our “most general” distance matrix $D_T$.     

Fluctuation theory for level-dependent Lévy risk processes

A level-dependent Lévy process solves the stochastic differential equation $dU\left(t\right)=dX\left(t\right)??\left(U\left(t\right)\right)dt$, where $X$ is a spectrally negative Lévy process. A special case is a multi-refracted Lévy process with ${?}_k\left(x\right)={\sum }_{j=1}^k{\delta }_j{\mathbf{1}}_{\{x\ge b_j\}}$. A general rate function $?$ that is non-decreasing and locally Lipschitz continuous is also considered. We discuss solutions of the above stochastic differential equation and investigate the so-called scale functions, which are counterparts of the scale functions from the theory of Lévy processes. We show how fluctuation identities for $U$ can be expressed via these scale functions. We demonstrate that the derivatives of the scale functions are solutions of Volterra integral equations.     

On the power of standard information for tractability for L∞ approximation of periodic functions in the worst case setting

We study multivariate approximation of periodic functions in the worst case setting with the error measured in the $L_{\infty }$ norm. We consider algorithms that use standard information ${\mathrm{\Lambda }}^{\mathrm{std}}$ consisting of function values or general linear information ${\mathrm{\Lambda }}^{\mathrm{all}}$ consisting of arbitrary continuous linear functionals. We investigate equivalences of various notions of algebraic and exponential tractability for ${\mathrm{\Lambda }}^{\mathrm{std}}$ and ${\mathrm{\Lambda }}^{\mathrm{all}}$ under the absolute or normalized error criterion, and show that the power of ${\mathrm{\Lambda }}^{\mathrm{std}}$ is the same as the one of ${\mathrm{\Lambda }}^{\mathrm{all}}$ for various notions of algebraic and exponential tractability. Our results can be applied to weighted Korobov spaces and Korobov spaces with exponential weights. This gives a special solution to Open Problem 145 as posed by Novak and Woźniakowski (2012) [40].     

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 $(0,\infty ]$.     

Central Sets Theorem on noncommutative semigroups

Let $D$ be a directed set without maximal element, $S$ be an infinite semigroup and ${}^{D}S$ be the collection of all functions from $D$ into $S$. It is shown that for a commutative semigroup $S$, $A?S$ is a $C$-set with respect to ${}^{\mathbb{N}}S$ if and only if $A$ is a $C$-set with respect to ${}^{D}S$. We investigate the Central Sets Theorem for arbitrary semigroups. In fact the Central Sets Theorem is stated with respect to ${}^{S}S$ for arbitrary semigroups.     

The crooked property

Crooked permutations were introduced twenty years ago to construct interesting objects in graph theory. These functions, over ${\mathbb{F}}_{2^n}$ with odd n, are such that their derivatives have as image set a complement of a hyperplane. The field of applications was extended later, in particular to cryptography. However binary crooked functions are rare. It is still unknown if non quadratic crooked functions do exist. We extend the concept and propose to study the crooked property for any characteristic. A function F, from ${\mathbb{F}}_{p^n}$ to itself, satisfies this property if all its derivatives have as image set an affine subspace. We show that the partially-bent vectorial functions and the functions satisfying the crooked property are strongly related. We later focus on the components of these functions, establishing that the existence of linear structures is here decisive. We then propose a symbolic approach to identify the linear structures. We claim that this problem consists in solving a system of linear equations, and can often be seen as a combinatorial problem.     

Affine flag varieties and quantum symmetric pairs,II. Multiplication formula

We establish a multiplication formula for a tridiagonal standard basis element in the idempotent version, i.e., the Lusztig form, of the coideal subalgebras of quantum affine ${\mathfrak{gl}}_n$ arising from the geometry of affine partial flag varieties of type C. We apply this formula to obtain the stabilization algebras ${\dot{\mathbf{K}}}_n^{\mathfrak{c}}$, ${\dot{\mathbf{K}}}_{\mathfrak{n}}^{??}$, ${\dot{\mathbf{K}}}_{\mathfrak{n}}^{??}$ and ${\dot{\mathbf{K}}}_{\eta }^{??}$, which are idempotented coideal subalgebras of quantum affine ${\mathfrak{gl}}_n$. The symmetry in the formula leads to an isomorphism of the idempotented coideal subalgebras ${\dot{\mathbf{K}}}_{\mathfrak{n}}^{??}$ and ${\dot{\mathbf{K}}}_{\mathfrak{n}}^{??}$ with compatible monomial, standard and canonical bases.