Related papers: An interesting spectral gap problem, from Jim Fill

The problem of determining the existence of a spectral gap in a lattice quantum spin system was previously shown to be undecidable for one [J. Bausch et al., "Undecidability of the spectral gap in one dimension", Physical Review X 10…

In a 1989 paper titled "shortest paths without a map", Papadimitriou and Yannakakis introduced an online model of searching in a weighted layered graph for a target node, while attempting to minimize the total length of the path traversed…

The purpose of this erratum is to fill a gap in the proof of the `Composite Braid Theorem' in the manuscript "Studying Links Via Closed Braids IV: Composite Links and Split Links (SLVCB-IV)", Inventiones Math, \{bf 102\} Fasc. 1 (1990),…

This note provides a very short proof of a spectral gap independent property of the simultaneous iterations algorithm for finding the top singular space of a matrix. See Rokhlin-Szlam-Tygert-2009, Halko-Martinsson-Tropp-2011 and…

A fundamental result in extremal graph theory is attributed to Mantel's theorem, which states that every graph on $n$ vertices with more than $\lfloor n^2/4 \rfloor$ edges must contain a triangle. Lov\'{a}sz and Simonovits (1975) provided a…

In this paper we study the spectral gap of the path graph and illustrate an interesting effect which has been described recently in the continuous setting. More explicitly, in the large-volume limit and in the presence of a certain external…

Let $\Lambda$ be a subgroup of an arithmetic lattice in SO(n+1,1). The quotient $\mathbb{H}^{n+1} / \Lambda$ has a natural family of congruence covers corresponding to primes in some ring of integers. We establish a super-strong…

Let $\emph{spex}_{\mathcal{OP}}(n,F)$ and $\emph{spex}_{\mathcal{P}}(n,F)$ be the maximum spectral radius over all $n$-vertex $F$-free outerplanar graphs and planar graphs, respectively. Define $tC_l$ as $t$ vertex-disjoint $l$-cycles,…

This paper is a natural continuation of paper "On rectifiable spaces and its algebraical equivalents, topological algebraic systems and Mal'cev algebras" published in arxiv:1309.4572. Thus we justify the need to present the entire material…

As researchers use computational methods to study complex social behaviors at scale, the validity of this computational social science depends on the integrity of the data. On July 2, 2015, Jason Baumgartner published a dataset advertised…

Mark Kac asked a famous question in 1966 entitled Can one hear the shape of a drum?, a spectral geometry problem that has intrigued mathematicians for the last six decades and is important to many other fields, such as architectural…

We provide a first systematic treatment of so-called rectangular multispectral perturbation theory. With their paper from 2003, Hochstenbach and Plestenjak ["Backward Error, Condition Numbers, and Pseudospectra for the Multiparameter…

The spectral theory of graphs provides a bridge between classical signal processing and the nascent field of graph signal processing. In this paper, a spectral graph analogy to Heisenberg's celebrated uncertainty principle is developed.…

Matrix completion is a fundamental problem that comes up in a variety of applications like the Netflix problem, collaborative filtering, computer vision, and crowdsourcing. The goal of the problem is to recover a k-by-n unknown matrix from…

There is a gap in Theorem 2.2 of the paper of Du (\cite{D_2010}). In this paper, we shall state the gap and repair it.

$\texttt{Randomstrasse101}$ is a blog dedicated to Open Problems in Mathematics, with a focus on Probability Theory, Computation, Combinatorics, Statistics, and related topics. This manuscript serves as a stable record of the Open Problems…

In the late 90's, Tom Wolff introduced the circle tangency counting problem in his expository article on the Kakeya conjecture. For collections of well-spaced circles, we break the $N^{3/2}$-barrier, proving that a set of $N$ well-spaced…

Dynamic Complexity was introduced by Immerman and Patnaik \cite{PatnaikImmerman97} (see also \cite{DongST95}). It has seen a resurgence of interest in the recent past, see…

A framework is developed to describe the Zariski topologies on the prime and primitive spectra of a quantum algebra $A$ in terms of the (known) topologies on strata of these spaces and maps between the collections of closed sets of…

Cecotti and Vafa proposed in 1993 a beautiful idea how to associate spectral numbers $\alpha_1,...,\alpha_n\in{\mathbb R}$ to real upper triangular $n\times n$ matrices $S$ with 1's on the diagonal and eigenvalues of $S^{-1}S^t$ in the unit…