相关论文: Value distribution theoretical properties of the G…

It is well known that the classical Gauss sum, normalized by the square-root number of terms, takes only finitely many values. If one restricts the range of summation to a subinterval, a much richer structure emerges. We prove a limit law…

We consider random geometric graphs on the plane characterized by a non-uniform density of vertices. In particular, we introduce a graph model where $n$ vertices are independently distributed in the unit disc with positions, in polar…

We study the asymptotic laws for the spatial distribution and the number of connected components of zero sets of smooth Gaussian random functions of several real variables. The primary examples are various Gaussian ensembles of real-valued…

Let $M$ be an $n$-dimensional smooth oriented complete embedded minimal hypersurface in $\mathbb{R}^{n+1}$ with Euclidean volume growth. We show that if the image under the Gauss map of $M$ avoids some neighborhood of a half-equator, then…

Random geometric graphs defined on Euclidean subspaces, also called Gilbert graphs, are widely used to model spatially embedded networks across various domains. In such graphs, nodes are located at random in Euclidean space, and any two…

We investigate the universality of singular value and eigenvalue distributions of matrix valued functions of independent random matrices and apply these general results in several examples. In particular we determine the limit distribution…

In a recent paper the author proved a theorem to the effect that the matrix of normalized Euclidean distances on the set of specially distributed random points in the $n$-dimensional Euclidean space $\mathbb R^{n}$ with independent…

We study probability distributions of eigenvalues of Hermitian and non-Hermitian Euclidean random matrices that are typically encountered in the problems of wave propagation in random media.

Gaussian double Markovian models consist of covariance matrices constrained by a pair of graphs specifying zeros simultaneously in the covariance matrix and its inverse. We study the semi-algebraic geometry of these models, in particular…

In this paper, we study general rotational surfaces in the 4- dimensional pseudo-Euclidean space E4-2 and obtain a characterization of flat general rotation surfaces with pointwise 1-type Gauss map in E4-2 and give an example of such…

A Lorentz surface in the four-dimensional pseudo-Euclidean space with neutral metric is called quasi-minimal if its mean curvature vector is lightlike at each point. In the present paper we obtain the complete classification of…

The curse of dimensionality is a common phenomenon which affects analysis of datasets characterized by large numbers of variables associated with each point. Problematic scenarios of this type frequently arise in classification algorithms…

To study spacelike surfaces of codimension two in the Lorentz-Minkowski space $\Bbb R^{n+1}_1,$ we construct a pair of maps whose values are in $HS_r:=H_+^n(\textbf v,1)\cap \{x_{n+1}=r\},$ called $\textbf n_r^{\pm}$-Gauss maps. It is…

Given an m-dimensional compact submanifold $\mathbf{M}$ of Euclidean space $\mathbf{R}^s$, the concept of mean location of a distribution, related to mean or expected vector, is generalized to more general $\mathbf{R}^s$-valued functionals…

We study spatial embeddings of random graphs in which nodes are randomly distributed in geographical space. We let the edge probability between any two nodes to be dependent on the spatial distance between them and demonstrate that this…

Let $A$ be an annular end of a complete minimal surface $S$ in $\mathbb R^m$ and let $V$ be a $k$-dimension projective subvariety of $\mathbb P^n(\mathbb C)\ (n=m-1)$. Let $g$ be the generalized Gauss map of $S$ into $V\subset\mathbb…

We demonstrate the parallel between the properties of Gaussian vectors and the Euclidean geometry. In particular we study the Markov property and give various equivalent Euclidean and probabilistic characterizations. We also give a simple…

Real-world road networks have an approximate scale-invariance property; can one devise mathematical models of random networks whose distributions are {\em exactly} invariant under Euclidean scaling? This requires working in the continuum…

We apply the invariant theory of surfaces in the four-dimensional Euclidean space to the class of general rotational surfaces with meridians lying in two-dimensional planes. We find all minimal super-conformal surfaces of this class.

It is shown that the topologies and nestings of the zero and nodal sets of random (Gaussian) band limited functions have universal laws of distribution. Qualitative features of the supports of these distributions are determined. In…