Related papers: Schmidt's game, Badly Approximable Linear Forms an…

We investigate the structure of $k$-positivity and Schmidt numbers for classes of linear maps and bipartite quantum states exhibiting symplectic group symmetries. Specifically, we consider (1) linear maps on $M_d(\mathbb{C})$ which are…

The famous theorem of R.Aumann and M.Maschler states that the sequence of values of an N-stage zero-sum game G_N with incomplete information on one side converges as N tends to infinity, and the error term is bounded by a constant divided…

Positive maps which are not completely positive are used in quantum information theory as witnesses for convex sets of states, in particular as entanglement witnesses and more generally as witnesses for states having Schmidt number not…

Given a measurable dynamical system $(X,\mathcal{X},\mu,T)$, where $X$ is a compact metric space, $\mathcal{X}$ is the Borel $\sigma$-algebra on $X$, $\mu$ is a $T$-invariant Borel probability measure and $T$ is a homeomorphism acting on…

Let $K/k$ be an extension of number fields, and let $P(t)$ be a quadratic polynomial over $k$. Let $X$ be the affine variety defined by $P(t) = N_{K/k}(\mathbf{z})$. We study the Hasse principle and weak approximation for $X$ in three…

We show that the map $\operatorname{Br} T \to (\operatorname{Br} T_{\bar k})^{\Gamma_k}$ is surjective for a torus $T$ defined over a field $k$ of characteristic $0$ when $k$ is a local or global field or $T$ is quasi-trivial.

In this paper, we address a natural question at the intersection of combinatorial game theory and computational complexity: "Can a sum of simple tepid games in canonical form be intractable?" To resolve this fundamental question, we…

In this paper, we show how one may (efficiently) construct two types of extremal combinatorial objects whose existence was previously conjectural. (*) Panchromatic Graphs: For fixed integer k, a k-panchromatic graph is, roughly speaking, a…

Let $K\subset\mathbb R^d$ be a compact subset equipped with a $\delta$-Ahlfors regular measure $\mu$. For any $\tau>1/d$ and any ``inhomogeneous'' vector $\boldsymbol{\theta}\in\mathbb R^d$, let $W_d(\psi_\tau,\boldsymbol{\theta})$ denote…

We say a natural number $n$ is matchable if there is a bijection from the set of $\tau(n)$ divisors of $n$ to the set $\{1,2,\dots,\tau(n)\}$, where corresponding numbers are relatively prime. We show that the set of matchable numbers has…

We first show that a continuous function f is nonnegative on a closed set $K\subseteq R^n$ if and only if (countably many) moment matrices of some signed measure $d\nu =fd\mu$ with support equal to K, are all positive semidefinite (if $K$…

A subset $R$ of integers is a set of Bohr recurrence if every rotation on $\mathbb{T}^d$ returns arbitrarily close to zero under some non-zero multiple of $R$. We show that the set $\{k!\, 2^m3^n\colon k,m,n\in \mathbb{N}\}$ is a set of…

For an integer $k\ge2$, a tuple of $k$ positive integers $(M_i)_{i=1}^{k}$ is called an amicable $k$-tuple if the equation \[ \sigma(M_1)=\cdots=\sigma(M_k)=M_1+\cdots+M_k \] holds. This is a generalization of amicable pairs. An amicable…

The problem of identifying the set of Dirichlet-to-Neumann (DtN) maps arising from conductivities on a smooth domain, among operators acting on functions on the boundary, is a challenging issue in the mathematical analysis of the Calder\'on…

We consider dynamical systems $(X,T,\mu)$ which have exponential decay of correlations for either H\"older continuous functions or functions of bounded variation. Given a sequence of balls $(B_n)_{n=1}^\infty$, we give sufficient conditions…

Given a suitably nested family $Z = \langle Z(m,k,\gamma) \rangle_{m,k \in \mathbb N, \gamma >0}$ of Borel subsets of matrices, and associated Borel measures and rate function, $\mu$, an entropy, $\chi^{\mu}(Z)$, is introduced which…

This paper concerns the compactness and separability properties of the normed Boolean algebras (N.B.A.) with respect to topology generated by a distance equal to the square root of a measure of symmetric difference between two elements. The…

The Rolling Ball Theorem asserts that given a convex body K in Euclidean space and having a smooth surface bd(K) with all principal curvatures not exceeding c>0 at all boundary points, K necessarily has the property that to each boundary…

A classical result of Kaufman states that, for each $\tau>1,$ the set of well approximable numbers \[ E(\tau)=\{x\in\mathbb{R}: \|qx\| < |q|^{-\tau} \text{ for infinitely many integers q}\} \] is a Salem set with Hausdorff dimension…

We compute the algebraic K-theory of the non-commutative ring k<x_1,...,x_n>/(m^a) when k is a perfect field of positive characteristic and m=(x_1,...,x_n). We express the answer in terms of the truncation poset Witt vectors developed in…