Related papers: On a discrete John-type theorem

In this paper, we prove the existence of fixed points of mappings satisfying the condition (Da), a kind of generalized nonexpansive mappings, on a weakly compact convex subset in a Banach space satisfying Opial's condition. And we use…

Asymptotic expansions are derived for Gegenbauer (ultraspherical) polynomials for large order $n$ that are uniformly valid for unbounded complex values of the argument $z$, including the real interval $0 \leq z \leq 1$ in which the zeros in…

Let $\tau(n)$ be Ramanujan's tau function, defined by the discriminant modular form \[ \Delta(z) = q\prod_{j=1}^{\infty}(1-q^{j})^{24}\ =\ \sum_{n=1}^{\infty}\tau(n) q^n \,,q=e^{2\pi i z} \] (this is the unique holomorphic normalized…

We show that any $n\times m$ matrix $A$ can be approximated in operator norm by a submatrix with a number of columns of order the stable rank of $A$. This improves on existing results by removing an extra logarithmic factor in the size of…

We extend Polyak's theorem on the convexity of joint numerical range from three to any number of quadratic forms on condition that they can be generated by three quadratic forms with a positive definite linear combination. Our new result…

In this article, we consider a dominant rational self-map $f:X \dashrightarrow X$ of a normal projective variety defined over a number field. We study the arithmetic degree $\alpha_k(f)$ for $f$ and $\alpha_k(f,V)$ of a subvariety $V$,…

We establish conditions for the discrete versions of logarithmic concavity and convexity of the higher order regularized basic hypergeometric function with respect simultaneous shift of all its parameters. For a particular case of Heine's…

We generalize the dual notions of "expansion" and "collapse" so they can be applied to arbitrary metric spaces. We also expand the theory to allow for infinitely many such moves. Those tools are then employed to prove a variety of…

It is explained how a locally convex (lc) topology $\tau$ on a real vector space $V$ extends to a locally multiplicatively convex (lmc) topology $\overline{\tau}$ on the symmetric algebra $S(V)$. This allows the application of the results…

The general volume of a star body, a notion that includes the usual volume, the $q$th dual volumes, and many previous types of dual mixed volumes, is introduced. A corresponding new general dual Orlicz curvature measure is defined that…

I will discuss results of three different types in geometry and topology. (1) General vanishing and rigidity theorems of elliptic genera proved by using modular forms, Kac-Moody algebras and vertex operator algebras. (2) The computations of…

According to Suk's breakthrough result on the Erdos-Szekeres problem, any point set in general position in the plane, which has no $n$ elements that form the vertex set of a convex $n$-gon, has at most $2^{n+O\left({n^{2/3}\log n}\right)}$…

Let $\tau$ denote the divisor function, and $f$ be any multiplicative function that satisfies some mild hypotheses. We establish the asymptotic formula or non-trivial upper bound for the shifted convolution sum $\sum_{n \leq…

We construct (generalized) logarithmic derivatives for general n-dimensional local fields K of mixed characteristics (0,p) in which p is not necessarily a prime element with residue field k such that [k:k^p]=p^{n-1}. For the construction of…

Coverings of convex bodies have emerged as a central component in the design of efficient solutions to approximation problems involving convex bodies. Intuitively, given a convex body $K$ and $\epsilon> 0$, a covering is a collection of…

Differential calculus on discrete sets is developed in the spirit of noncommutative geometry. Any differential algebra on a discrete set can be regarded as a `reduction' of the `universal differential algebra' and this allows a systematic…

We generalize parts of the theory of associative geometries developed by Kinyon and the author in the framework of universal algebra: we prove that certain associoid structures, such as pregroupoids and principal equivalence relations, have…

Motivated by classical Euler's $Tonnetz$, we introduce and study the combinatorics and topology of more general simplicial complexes $Tonn^{n,k}(L)$ of "Tonnetz type". Out main result is that for a sufficiently generic choice of parameters…

In this paper we introduce generalised Markov numbers and extend the classical Markov theory for the discrete Markov spectrum to the case of generalised Markov numbers. In particular we show recursive properties for these numbers and find…

We extend the construction of [19] by introducing spaces of generalized tensor fields on smooth manifolds that possess optimal embedding and consistency properties with spaces of tensor distributions in the sense of L. Schwartz. We thereby…