Related papers: Combinatorial Morse flows are hard to find
We review possible measures of complexity which might in particular be applicable to situations where the complexity seems to arise spontaneously. We point out that not all of them correspond to the intuitive (or "naive") notion, and that…
The Collatz conjecture is explored using polynomials based on a binary numeral system. It is shown that the degree of the polynomials, on average, decreases after a finite number of steps of the Collatz operation, which provides a weak…
We consider the set of polygons all of whose sides are vertical or horizontal with fixed combinatorics (for example all the figure "L"s). We show that there is a dense G $\delta$ subset of such polygons such that for each polygon in this G…
Let $X_1,\ldots, X_{d+2}$ be random points in $\mathbb R^d$. The classical Sylvester problem asks to determine the probability that the convex hull of these points, denoted by $P:= [X_1,\ldots, X_{d+2}]$, is a simplex. In the present paper,…
Strongly interacting binary mixtures of superparamagnetic colloidal particles confined to a two-dimensional water-air interface are examined by theory, computer simulation and experiment. The mixture exhibits a partial clustering in…
Simplicial complexes are a popular tool used to model higher-order interactions between elements of complex social and biological systems. In this paper, we study some combinatorial aspects of a class of simplicial complexes created by a…
In this article, we use concepts and methods from the theory of simplicial sets to study discrete Morse theory. We focus on the discrete flow category introduced by Vidit Nanda, and investigate its properties in the case where it is defined…
The paper investigates the problem of finding communities in complex network systems, the detection of which allows a better understanding of the laws of their functioning. To solve this problem, two approaches are proposed based on the use…
The problem of enumerating meanders -- pairs of simple plane curves with transverse intersections -- was formulated about forty years ago and is still far from solved. Recently, it was discovered that meanders admit a factorization into…
In 1946, M. Morse proposed a conjecture that an analytic topologically transitive systems is metrically transitive. We prove this Morse conjecture for flows on a closed orientable surface of negative Euler characteristic. As a consequence,…
It was recently shown that almost all solutions in the symmetric binary perceptron are isolated, even at low constraint densities, suggesting that finding typical solutions is hard. In contrast, some algorithms have been shown empirically…
When performing causal discovery, assumptions have to be made on how the true causal mechanism corresponds to the underlying joint probability distribution. These assumptions are labeled as causal razors in this work. We review numerous…
We study the structure of the asymptotic expansion of the probability that a combinatorial object is connected. We show that the coefficients appearing in those asymptotics are integers and can be interpreted as the counting sequences of…
Many random combinatorial objects have a component structure whose joint distribution is equal to that of a process of mutually independent random variables, conditioned on the value of a weighted sum of the variables. It is interesting to…
We show the existence of rigid combinatorial objects which previously were not known to exist. Specifically, for a wide range of the underlying parameters, we show the existence of non-trivial orthogonal arrays, $t$-designs, and $t$-wise…
We present a family of complete acyclic Morse matchings on the face lattice of a hypersimplex. Since a hypersimplex is a convex polytope, there is a natural way to form a CW complex from its faces. In a future paper we will utilize these…
In this paper, we propose an extension of quantum searches on graphs driven by quantum walks to simplicial complexes. To this end, we newly define a quantum walk on simplicial complex which is an alternative of preceding studies by authors.…
We show that the gradient flows associated with a recently found family of Morse functions converge exponentially to the roots of the symmetric continuous Hahn polynomials. By symmetry reduction the rate of the exponential convergence can…
The aim of this paper is twofold. First, we demonstrate how Riordan matrices can be employed to connect well-known concepts in geometric combinatorics, such as $f$-vectors, $h$-vectors $\gamma$-vectors, in a similar fashion to the McMullen…
Conley indices and Morse decompositions of flows can be found by using algorithms which rigorously analyze discrete dynamical systems. This usually involves integrating a time discretization of the flow using interval arithmetic. We compare…