Related papers: Strictly and non-strictly positive definite functi…
A spherical conical metric $g$ on a surface $\Sigma$ is a metric of constant curvature $1$ with finitely many isolated conical singularities. The uniformization problem for such metrics remains largely open when at least one of the cone…
We obtain formulae for the expected number and height distribution of critical points of smooth isotropic Gaussian random fields parameterized on Euclidean space or spheres of arbitrary dimension. The results hold in general in the sense…
Spin (spherical) random fields are very important in many physical applications, in particular they play a key role in Cosmology, especially in connection with the analysis of the Cosmic Microwave Background radiation. These objects can be…
We survey recent results regarding the study of dynamical properties of the space of positive definite functions and characters of higher rank lattices. These results have several applications to ergodic theory, topological dynamics,…
A fold map is a smooth map at each singular point of which it is represented as the product map of a Morse function and the identity map on an open ball. A special generic map is a fold map such that the Morse function can be taken as a…
The standard method to generate dynamical models with a finite extent is to apply a truncation in binding energy to the distribution function. This approach has the disadvantages that one cannot choose the density to start with, that the…
We construct and discuss new numerical homotopy invariants of topological spaces that are suitable for the study of functions on loop and sphere spaces. These invariants resemble the Lusternik-Schnirelmann category and provide lower bounds…
We construct a family of measures for random fields based on the iterated subdivision of simple geometric shapes (triangles, squares, tetrahedrons) into a finite number of similar shapes. The intent is to construct continuum limits of scale…
We present an explicit characterization for the real, continuous, isotropic and strictly positive definite kernels on a product of compact two-point homogeneous spaces, in the cases in which at least one of the spaces is a sphere of…
The Dagum family of isotropic covariance functions has two parameters that allow for decoupling of the fractal dimension and Hurst effect for Gaussian random fields that are stationary and isotropic over Euclidean spaces. Sufficient…
A key fact in the theory of Boolean functions $f : \{0,1\}^n \to \{0,1\}$ is that they often undergo sharp thresholds. For example: if the function $f : \{0,1\}^n \to \{0,1\}$ is monotone and symmetric under a transitive action with…
The fundamental functional summary statistics used for studying spatial point patterns are developed for marked homogeneous and inhomogeneous point processes on the surface of a sphere. These are extended to point processes on the surface…
In this article we continue the study of properties of squeezing functions and geometry of bounded domains. The limit of squeezing functions of a sequence of bounded domains is studied. We give comparisons of intrinsic positive forms and…
In general relativity, the Einstein equations provide spherically symmetric static spacetimes with dynamics defined as an evolution along the radial coordinate $r$. The geometrical sector becomes a one-dimensional mechanical system, with…
Characteristic functions that are radially symmetric have a dual interpretation, as they can be used as the isotropic correlation functions of spatial random fields. Extensions of isotropic correlation functions from balls into…
Let U denote the Urysohn sphere and consider U as a metric structure in the empty continuous signature. We prove that every definable function from U^n to U is either a projection function or else has relatively compact range. As a…
We present sufficient condition for a family of positive definite kernels on a compact two-point homogeneous space to be strictly positive definite based on their representation as a series of spherical harmonics. The family analyzed is a…
Special functions are often defined as a Fourier or Laplace transform of a positive measure, and the positivity of the measure manifests as positive definiteness of certain matrices. The purpose of this expository note is to give a sample…
The stability of rotating isotropic spherical stellar systems is investigated by using N-body simulations. Four spherical models with realistic density profiles are studied: one of them fits the luminosity profile of globular clusters,…
With a simple generic approach, we develop a classification that encodes and measures the strength of completeness (or compactness) properties in various types of spaces and ordered structures. The approach also allows us to encode notions…