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This paper investigates stability conditions of continuous-time Hopfield and firing-rate neural networks by leveraging contraction theory. First, we present a number of useful general algebraic results on matrix polytopes and products of…
We determine the smooth concordance order of the 3-stranded pretzel knots P(p,q,r) with p,q,r odd. We show that each one of finite order is, in fact, ribbon, thereby proving the slice-ribbon conjecture for this family of knots. As…
Network clustering reveals the organization of a network or corresponding complex system with elements represented as vertices and interactions as edges in a (directed, weighted) graph. Although the notion of clustering can be somewhat…
In this paper, we consider the problem of determining in polynomial time whether a given planar point set $P$ of $n$ points admits 4-connected triangulation. We propose a necessary and sufficient condition for recognizing $P$, and present…
We define invariants of braids rather than invariants of conjugacy classes of braids. For any pure three-braid we give effective upper and lower bounds for these invariants. This is done in terms of a natural syllable decomposition of the…
A procedure for the construction and the classification of multilattices in arbitrary dimension is proposed. The algorithm allows to determine explicitly the location of the points of a multilattice given its space group, and to determine…
Network compression is crucial to making the deep networks to be more efficient, faster, and generalizable to low-end hardware. Current network compression methods have two open problems: first, there lacks a theoretical framework to…
Clustering is an essential data mining tool for analyzing and grouping similar objects. In big data applications, however, many clustering algorithms are infeasible due to their high memory requirements and/or unfavorable runtime…
We show that approximating the trace norm contraction coefficient of a quantum channel within a constant factor is NP-hard. Equivalently, this shows that determining the optimal success probability for encoding a bit in a quantum system…
The cycling operation is a special kind of conjugation that can be applied to elements in Artin's braid groups, in order to reduce their length. It is a key ingredient of the usual solutions to the conjugacy problem in braid groups. In…
We generalize the Moishezon Teicher algorithm that was suggested for the computation of the braid monodromy of an almost real curve. The new algorithm suits a larger family of curves, and enables the computation of braid monodromy not only…
The Garside group, as a generalization of braid groups and Artin groups of finite types, is defined as the group of fractions of a Garside monoid. We show that the semidirect product of Garside monoids is a Garside monoid. We use the…
Unstructured pruning reduces the memory footprint in deep neural networks (DNNs). Recently, researchers proposed different types of structural pruning intending to reduce also the computation complexity. In this work, we first suggest a new…
We construct cobordisms of small genus between torus knots and use them to determine the cobordism distance between torus knots of small braid index. In fact, the cobordisms we construct arise as the intersection of a smooth algebraic curve…
Tensor network contractions are widely used in statistical physics, quantum computing, and computer science. We introduce a method to efficiently approximate tensor network contractions using low-rank approximations, where each intermediate…
Given a set of $n$ points on a plane, in the Minimum Weight Triangulation problem, we wish to find a triangulation that minimizes the sum of Euclidean length of its edges. This incredibly challenging problem has been studied for more than…
We study the complexity of determining the edge connectivity of a simple graph with cut queries. We show that (i) there is a bounded-error randomized algorithm that computes edge connectivity with $O(n)$ cut queries, and (ii) there is a…
We give a simple deterministic constant-round algorithm in the congested clique model for reducing the number of edges in a graph to $n^{1+\varepsilon}$ while preserving the minimum spanning forest, where $\varepsilon > 0$ is any constant.…
We study algorithms for solving three problems on strings. The first one is the Most Frequently String Search Problem. The problem is the following. Assume that we have a sequence of $n$ strings of length $k$. The problem is finding the…
We establish an explicit criterion for determining whether a $4 \times 4$ upper-triangular matrix is a contraction with respect to the Euclidean operator norm.