Related papers: Mean arc theorem for exploring domains with random…
Random walks are used for modeling various dynamics in, for example, physical, biological, and social contexts. Furthermore, their characteristics provide us with useful information on the phase transition and critical phenomena of even…
The capability of recurrent neural networks to approximate trajectories of a random dynamical system, with random inputs, on non-compact domains, and over an indefinite or infinite time horizon is considered. The main result states that…
Returning walks on a lattice are sequences of moves that start at a given lattice site and return to the same site after $n$ steps. Determining the total number of returning walks of a given length $n$ is a typical graph-theoretical problem…
In this survey, we will focus on the mean curvature flow theory with sphere theorems, and discuss the recent developments on the convergence theorems for the mean curvature flow of arbitrary codimension inspired by the Yau rigidity theory…
There are considered vector fields and quaternionic $\alpha$-hyperholomorphic functions in a domain of $R^2$ which generalize the notion of solenoidal and irrotational vector fields. There are established sufficient conditions for the…
Motivated by the Cauchy-Davenport theorem for sumsets, and its interpretation in terms of Cayley graphs, we prove the following main result : There is a universal constant e > 0 such that, if G is a connected, regular graph on n vertices,…
We consider the statistical analysis of trajectories on Riemannian manifolds that are observed under arbitrary temporal evolutions. Past methods rely on cross-sectional analysis, with the given temporal registration, and consequently may…
Many problems in physics, biology, and economics depend upon the duration of time required for a diffusing particle to cross a boundary. As such, calculations of the distribution of first passage time, and in particular the mean first…
In this paper we provide a local Cauchy theory both on the torus and in the whole space for general Vicsek dynamics at the kinetic level. We consider rather general interaction kernels, nonlinear viscosity and nonlinear friction.…
A fundamental question in rough path theory is whether the expected signature of a geometric rough path completely determines the law of signature. One sufficient condition is that the expected signature has infinite radius of convergence,…
Network connectivity is usually addressed for convex domains where a direct line of sight exists between any two transmitting/receiving nodes. Here, we develop a general theory for the network connectivity properties across a small opening,…
We consider convex hulls of random walks whose steps belong to the domain of attraction of a stable law in $\mathbb{R}^d$. We prove convergence of the convex hull in the space of all convex and compact subsets of $\mathbb{R}^d$, equipped…
This paper gives a partial confirmation of a conjecture of P. Agarwal, S. Har-Peled, M. Sharir, and K. Varadarajan that the total curvature of a shortest path on the boundary of a convex polyhedron in the 3-dimensional Euclidean space…
Interesting data often concentrate on low dimensional smooth manifolds inside a high dimensional ambient space. Random projections are a simple, powerful tool for dimensionality reduction of such data. Previous works have studied bounds on…
This paper is devoted to the study of unparameterized simple curves in the plane. We propose diverse canonical parameterizations of a 2D-curve. For instance, the arc-length parameterization is canonical, but we consider other natural…
In this paper we consider the prescribed mean curvature flow of a non-compact space-like Cauchy hypersurface of bounded geometry in a generalized Robertson-Walker space-time. We prove that the flow preserves the space-likeness condition and…
Rudin's version of the classical Julia-Wolff-Carath\'eodory theorem is a cornerstone of holomorphic function theory in the unit ball of $\mathbb{C}^d$. In this paper we obtain a complete generalization of Rudin's theorem for a holomorphic…
We analyze large-scale patterns in three-dimensional turbulent convection in a horizontally extended square convection cell by Lagrangian particle trajectories calculated in direct numerical simulations. A simulation run at a Prandtl number…
Convex records have an appealing purely geometric definition. In a sequence of $d$-dimensional data points, the $n$-th point is a convex record if it lies outside the convex hull of all preceding points. We specifically focus on the…
The Laplacian matrix and its pseudo-inverse for a strongly connected directed graph is fundamental in computing many properties of a directed graph. Examples include random-walk centrality and betweenness measures, average hitting and…