Related papers: Discrete Morse functions for graph configuration s…
We prove several combinatorial results on path algebras over discrete structures related to directed graphs. These results are motivated by Morse theory on a manifold with boundary and, more generally, by Floer theory on a configuration…
Statistical properties of binary complex networks are well understood and recently many attempts have been made to extend this knowledge to weighted ones. There is, however, a subtle difference between networks where weights are continuos…
Multiparticle systems on complicated metric graphs might have many applications in physics, biology and social life. But the corresponding science still does not exist. Here we start it with simplest examples where there is quadratic…
In part III is realized the consistent development of Heisenberg--Dyson's two-layer matrix approximation to the graph formalism for postulating discrete physical objects (DPO) introduced in parts I-II in the form of discrete sets of…
Numerous models for supervised and reinforcement learning benefit from combinations of discrete and continuous model components. End-to-end learnable discrete-continuous models are compositional, tend to generalize better, and are more…
We formulate the statistics of the discrete multicomponent fragmentation event using a methodology borrowed from statistical mechanics. We generate the ensemble of all feasible distributions that can be formed when a single integer…
In this paper we investigate composition operators on discrete spaces. We establish the classification of underlying graphs of such operators. For one class of such graphs, namely graphs with one cycle, we obtain a characterization of…
We introduce a version of discrete Morse theory specific for manifolds with boundary. The idea is to consider Morse functions for which all boundary cells are critical. We obtain "Relative Morse Inequalities" relating the homology of the…
Without recourse to the sophisticated machinery of twisted group algebras, projective character tables and explicit values of 2-cocycles, we here present a simple algorithm to study the gauge theory data of D-brane probes on a generic…
The theory of sparse structures usually uses tree like structures as building blocks. In the context of sparse/dense dichotomy this role is played by graphs with bounded tree depth. In this paper we survey results related to this concept…
We define a distance metric between partitions of a graph using machinery from optimal transport. Our metric is built from a linear assignment problem that matches partition components, with assignment cost proportional to transport…
Many problems of interest in computer science and information theory can be phrased in terms of a probability distribution over discrete variables associated to the vertices of a large (but finite) sparse graph. In recent years,…
We examine discrete vortex dynamics in two-dimensional flow through a network-theoretic approach. The interaction of the vortices is represented with a graph, which allows the use of network-theoretic approaches to identify key…
INTRODUCTION This papers deals with partial differential equations of second order, linear, with constant and not constant coefficients, in two variables, which admit real characteristics. I face the study of PDEs with the mentality of the…
We propose a discretization of classical confocal coordinates. It is based on a novel characterization thereof as factorizable orthogonal coordinate systems. Our geometric discretization leads to factorizable discrete nets with a novel…
In some previous works, the analytic structure of the spectrum of a quantum graph operator as a function of the vertex conditions and other parameters of the graph was established. However, a specific local coordinate chart on the…
In the context of discrete Morse theory, we introduce Morse frames, which are maps that associate a set of critical simplexes to all simplexes. The main example of Morse frames are the Morse references. In particular, these Morse references…
We generate random functions locally via a novel generalization of Dyson Brownian motion, such that the functions are in a desired differentiability class, while ensuring that the Hessian is a member of the Gaussian orthogonal ensemble…
The focus of this article is to develop computationally efficient mathematical morphology operators on hypergraphs. To this aim we consider lattice structures on hypergraphs on which we build morphological operators. We develop a pair of…
Large graphs abound in machine learning, data mining, and several related areas. A useful step towards analyzing such graphs is that of obtaining certain summary statistics - e.g., or the expected length of a shortest path between two…