Related papers: Gap Theorem for Separated Sequences without Pain

We establish a structure theorem for minimizing sequences for the isoperimetric problem on noncompact $\mathsf{RCD}(K,N)$ spaces $(X,\mathsf{d},\mathcal{H}^N)$. Under the sole (necessary) assumption that the measure of unit balls is…

A matrix is well separated if all its Gershgorin circles are away from the unit circle and they are separated from each other. In this article, the region of relative errors in the eigenvalues is obtained as a quadratic oval for non…

We address a two-dimensional nonlinear elliptic problem with a finite-amplitude periodic potential. For a class of separable symmetric potentials, we study the bifurcation of the first band gap in the spectrum of the linear Schr\"{o}dinger…

In this paper we extend the results of the research started by the first author, in which Karlin-McGregor diagonalization of certain reversible Markov chains over countably infinite general state spaces by orthogonal polynomials was used to…

We present a new proof of the existence of Morley sequences in simple theories. We avoid using the Erd\H{o}s-Rado theorem and instead use only Ramsey's theorem and compactness. The proof shows that the basic theory of forking in simple…

There are several ideal boundaries and completions in General Relativity sharing the topological property of being sequential, i.e., determined by the convergence of its sequences and, so, by some limit operator $L$. As emphasized in a…

We consider the problem of determining Weierstrass gaps and pure Weierstrass gaps at several points. Using the notion of relative maximality in generalized Weierstrass semigroups due to Delgado \cite{D}, we present a description of these…

The aim of this note is to provide a short and self-contained proof of H\"ormander's theorem about the smoothness of transition probabilities for a diffusion under H\"ormander's "brackets condition". While both the result and the technique…

Suppose $A\subset \mathbb{R}$ of size $k$ has distinct consecutive $r$--differences, that is for $1 \leq i \leq k -r$, the $r$--tuples $$(a_{i+1} - a_i , \ldots , a_{i+r} - a_{i + r -1})$$ are distinct. Then for any finite $B \subset…

Inspired by Goette-Semmelmann \cite{GSSU2002}, we derive an estimate for the scalar curvature without a nonnegativity assumption on curvature operator. As an application, we show that, on an even dimensional closed manifold with nonzero…

In this paper we characterize the Nazarov-Sklyanin hierarchy for the classical periodic Benjamin-Ono equation in two complementary degenerations: for the multi-phase initial data (the periodic multi-solitons) at fixed dispersion and for…

These informal notes, not intended for publication, provide an approach to the Borsuk--Ulam theorem via Stokes' theorem, in a similar spirit to Lima's proof of the Brouwer fixed point theorem. They are intended to be accessible to anyone…

Let $V \subset \mathbb{R}$ be a finite set with $|V| = n $ and suppose we are given each pairwise distance independently with probability $p$. We show that if $p = (1+\epsilon)/n$, for some fixed $\epsilon >0$, then we can reconstruct a…

The Ulam sequence, described by Stanislaw Ulam in the 1960s, starts $1,2$ and then iteratively adds the smallest integer that can be uniquely written as the sum of two distinct earlier terms: this gives $1,2,3,4,6,8,11,\dots$. Already in…

Let $p_{k}$ denote the $k$-th prime and $d(p_{k}) = p_{k} - p_{k - 1}$, the difference between consecutive primes. We denote by $N_{\epsilon}(x)$ the number of primes $\leq x$ which satisfy the inequality $d(p_{k}) \leq (\log p_{k})^{2 +…

We extend the criterion on the existence of telescopers for hypergeometric terms to the case of P-recursive sequences. This criterion is based on the concept of integral bases and the generalized Abramov-Petkovsek reduction for P-recursive…

We prove a splitter theorem for tight multimatroids, generalizing the corresponding result for matroids, obtained independently by Brylawski and Seymour. Further corollaries give splitter theorems for delta-matroids and ribbon graphs.

Let M be a family of sequences (a_1,...,a_p) where each a_k is a flat in a projective geometry of rank n (dimension n-1) and order q, and the sum of ranks, r(a_1) + ... + r(a_p), equals the rank of the join a_1 v ... v a_p. We prove upper…

A cap of spherical radius $\alpha$ on a unit $d$-sphere $S$ is the set of points within spherical distance $\alpha$ from a given point on the sphere. Let $\mathcal F$ be a finite set of caps lying on $S$. We prove that if no hyperplane…

This is an expository article on the recent marvellous theorem of Goldston, Pintz, and Yildirim on small gaps between prime numbers.