Related papers: Linear and Sublinear Diversities
How can one meaningfully make a measurement, if the meter does not conform to any standard and its scale expands or shrinks depending on what is measured? In the present work it is argued that current evaluation practices for…
There are two very important subjects in physics: Symmetry of dynamical models and nonlinearity. All really fundamental models are invariant under some particular symmetry groups. There is also no true physics, no our Universe and life at…
In earlier papers we changed the concept of the inner product to a more general one, to the so-called Minkowski product. This product changes on the tangent space hence we could investigate a more general structure than a Riemannian…
In the paper, multivariate probability distributions are considered that are representable as scale mixtures of multivariate elliptically contoured stable distributions. It is demonstrated that these distributions form a special subclass of…
The aim of this paper is to study symmetries of linearly singular differential equations, namely, equations that can not be written in normal form because the derivatives are multiplied by a singular linear operator. The concept of…
We give a closed formula for the dimension of all linear systems in $\mathbb{P}^n$ with assigned multiplicity at arbitrary collections of points lying on a rational normal curve of degree $n$. In particular we give a purely geometric…
A proof using the theory of completely positive maps is given to the fact that if $A \in M_2$, or $A \in M_3$ has a reducing eigenvalue, then every bounded linear operator $B$ with $W(B) \subseteq W(A)$ has a dilation of the form $I \otimes…
To a finite group $G$, one can associate several notions of dimensions (or degrees). In this survey, we attempt to bring together some of the notions of dimensions or degrees defined using representations of the group in General Linear…
There are two basic ways of weakening the definition of the well-known metric regularity property by fixing one of the points involved in the definition. The first resulting property is called metric subregularity and has attracted a lot of…
Minkowski functionals constitute a family of order parameters which discriminate spatial patterns according to size, shape and connectivity. Here we point out, that these scalar descriptors can be complemented by vector-valued curvature…
Hindman's celebrated Finite Sums Theorem, and its high-dimensional version due to Milliken and Taylor, are extended from covers of countable sets to covers of arbitrary topological spaces with Menger's classic covering property. The methods…
We generalize some results of Borwein, Burke, Lewis, and Wang to mappings with values in metric (resp. ordered normed linear) spaces. We define two classes of monotone mappings between an ordered linear space and a metric space (resp.…
The purpose of this paper is to provide a new account of multiplicity for finite morphisms between smooth projective varieties. Traditionally, this has been defined using commutative algebra in terms of the length of integral ring…
A system of singular integral equations with monotone and concave nonlinearity in the subcritical case is investigated. The specified system and its scalar analog have direct applications in various areas of physics and biology. In…
This survey is dedicated to a new direction in the theory of dynamical systems: the dynamics of metrics in measure spaces and new (catalytic) invariants of transformations with invariant measure. A space equipped with a measure and a metric…
For a function defined on a convex set in a Euclidean space, midpoint convexity is the property requiring that the value of the function at the midpoint of any line segment is not greater than the average of its values at the endpoints of…
This paper presents a systematic study of the notion of surplus invariance, which plays a natural and important role in the theory of risk measures and capital requirements. So far, this notion has been investigated in the setting of some…
We address the problem of merging graph and feature-space information while learning a metric from structured data. Existing algorithms tackle the problem in an asymmetric way, by either extracting vectorized summaries of the graph…
We introduce the notion of difference equation defined on a structured set. The symmetry group of the structure determines the set of difference operators. All main notions in the theory of difference equations are introduced as invariants…
We formalize the general principle of significance with respect to binary relations which is a universal tool for description and analysis of various situations in and apart from mathematics. We derive the basic properties and focus on a…