Related papers: Constant Degree Direct Product Testers with Small …

A topological space $X$ is called almost discrete, if it has precisely one nonisolated point. In this paper, we get that for a countable product $X=\prod X_i$ of almost discrete spaces $X_i$ the space $C_p(X)$ of continuous real-valued…

In this work we show that high dimensional expansion implies locally testable code. Specifically, we define a notion that we call high-dimensional-expanding-system (HDE-system). This is a set system defined by incidence relations with…

We prove that the size of the product set of any finite arithmetic progression $\mathcal{A}\subset \mathbb{Z}$ satisfies \[|\mathcal A \cdot \mathcal A| \ge \frac{|\mathcal A|^2}{(\log |\mathcal A|)^{2\theta +o(1)} } ,\] where…

We establish a cubic lower bound on the Dehn function of a certain finitely presented subgroup of a direct product of 3 free groups.

We define a general product of two $n$-dimensional tensors $\mathbb {A}$ and $\mathbb {B}$ with orders $m\ge 2$ and $k\ge 1$, respectively. This product is a generalization of the usual matrix product, and satisfies the associative law.…

The algebraic properties of formal power series, whose coefficients show factorial growth and admit a certain well-behaved asymptotic expansion, are discussed. It is shown that these series form a subring of $\mathbb{R}[[x]]$. This subring…

This report summarizes progress made towards a new approach for enlarging the dynamic aperture of particle accelerators. Unlike prior methods which attempted to move the location of select resonances outward in phase space, our approach…

We show that, under suitably general formulations, covering properties, accumulation properties and filter convergence are all equivalent notions. This general correspondence is exemplified in the study of products. Let $X$ be a product of…

Let $\Omega\subset\mathbb{R}^n$ be an open, connected subset of $\mathbb{R}^n$, and let $F\colon\Omega-\Omega\to\mathbb{C}$, where $\Omega-\Omega=\{x-y\colon x,y\in\Omega\}$, be a continuous positive definite function. We give necessary and…

This article deals with two different problems in commutative algebra. In the first part, we give a proof of generalized forms of the Direct Summand Theorem (DST (or DCS)) for module-finite extension rings of mixed characteristic $R\subset…

We give a strong direct sum theorem for computing $xor \circ g$. Specifically, we show that for every function g and every $k\geq 2$, the randomized query complexity of computing the xor of k instances of g satisfies…

Let $U_n=U_n(P,Q)$ be a nondegenerate Lucas sequence with $Q=\pm 1$ and discriminant $\Delta=P^2+4Q>0$. We study Diophantine equations \[ A y^k=\prod_{i=1}^r U_{n_i}(P,Q), \qquad k\geq 2, \] where the indices $n_1,\ldots,n_r$ are pairwise…

We provide an explicit construction of finite 4-regular graphs $(\Gamma_k)_{k\in \mathbb N}$ with ${girth \Gamma_k\to\infty}$ as $k\to\infty$ and $\frac{diam \Gamma_k}{girth \Gamma_k}\leqslant D$ for some $D>0$ and all $k\in\mathbb{N}$. For…

High-dimensional expanders are a generalization of the notion of expander graphs to simplicial complexes and give rise to a variety of applications in computer science and other fields. We provide a general tool to construct families of…

This is an expository survey on recent sum-product results in finite fields. We present a number of sum-product or "expander" results that say that if $|A| > p^{2/3}$ then some set determined by sums and product of elements of $A$ is nearly…

This paper presents a reformulation of the Leibniz product rule as a finite sum that expresses the fractional derivative of the product of two differentiable functions. This paper then proves the cases for when the product consists of an…

Let $K_1$, $\dots$, $K_n$ be bounded, complete, separable metric spaces. Let $\lambda_i$ be a Borel probability measure on $K_i$ for each $i$. Let $f:\prod_i K_i \to \mathbb{R}$ be a bounded and continuous potential function, and let…

Direct verification of the existence of an infinite set of multicritical non-perturbative FPs (Fixed Points) for a single scalar field in two dimensions, is in practice well outside the capabilities of the present standard approximate…

This 1995 paper contains a sharp version of the classical Marcinkiewicz multiplier theorem for the class of homogeneous Fourier multipliers in two dimensions; here a one-dimensional Marcinkiewicz condition is sufficient. Examples are given…

A compact space $Y$ is called homeo-product-minimal if given any minimal system $(X,T)$, it admits a homeomorphism $S:Y\to Y$, such that the product system $(X\times Y,T\times S)$ is minimal. We show that a large class of cofrontiers is…