Related papers: Diversities, hyperconvexity and fixed points
We introduce an alternative approach for constrained mathematical programming problems. It rests on two main aspects: an efficient way to compute optimal solutions for unconstrained problems, and multipliers regarded as variables for a…
A new notion of displacement convexity on a matrix level is developed for density flows arising from mean-field games, compressible Euler equations, entropic interpolation, and semi-classical limits of non-linear Schr\"odinger equations.…
We show the existence of Lebesgue-equivalent conservative and ergodic $\sigma$-finite invariant measures for a wide class of one-dimensional random maps consisting of piecewise convex maps. We also estimate the size of invariant measures…
A diversity $\delta$ in $M$ is a function defined over every finite set of points of $M$ mapped onto $[0,\infty)$, with the properties that $\delta(X)=0$ if and only if $|X|\leq 1$ and $\delta(X\cup Y)\leq\delta(X\cup Z)+\delta(Z\cup Y)$,…
Some examples of three-dimensional metrics of constant curvature defined by solutions of nonlinear integrable differential equations and their generalizations are constructed. The properties of Riemann extensions of the metrics of constant…
The concept of illumination bodies studied in convex geometry is used to amend the halfspace depth for multivariate data. The proposed notion of illumination enables finer resolution of the sample points, naturally breaks ties in the…
Hyperuniformity refers to the suppression of density fluctuations at large scales. Typical for ordered systems, this property also emerges in several disordered physical and biological systems, where it is particularly relevant to…
Let $C$ be a convex subset of a locally convex space. We provide optimal approximate fixed point results for sequentially continuous maps $f\colon C\to\bar{C}$. First we prove that if $f(C)$ is totally bounded, then it has an approximate…
We introduce a new type of mappings in metric space which are three-point analogue of the well-known Chatterjea type mappings, and call them generalized Chatterjea type mappings. It is shown that such mappings can be discontinuous as is the…
The embedding of finite metrics in $\ell_1$ has become a fundamental tool for both combinatorial optimization and large-scale data analysis. One important application is to network flow problems in which there is close relation between…
The notion of a (metric) modular on an arbitrary set and the corresponding modular space, more general than a metric space, were introduced and studied recently by the author [V. V. Chistyakov, Metric modulars and their application, Dokl.…
We prove that a nonempty closed and geodesically convex subset of the $l_{\infty}$ plane $\mathbb{R}^2_{\infty}$ is hyperconvex and we characterize the tight spans of arbitrary subsets of $\mathbb{R}^2_{\infty}$ via this property: Given any…
The growing complexity of dynamical systems and advances in data collection necessitates robust data-driven control strategies without explicit system identification and robust synthesis. Data-driven stability has been explored in linear…
In this note, we highlight some properties of the metric projection onto a closed convex in a Hilbert space. In particular, we use some recent results on fixed points of nonexpansive potential operators.
The goal of this paper is to promote the use of fixed point strategies in data science by showing that they provide a simplifying and unifying framework to model, analyze, and solve a great variety of problems. They are seen to constitute a…
We consider a new type of mappings in metric spaces which can be characterized as mappings contracting perimeters of triangles. It is shown that such mappings are continuous. The fixed-point theorem for such mappings is proved and the…
The smallest hyperconvex metric space containing a given metric space X is called the tight span of X. It is known that tight spans have many nice geometric and topological properties, and they are gradually becoming a target of research of…
We investigate how to glue hyperconvex (or injective) metric spaces such that the resulting space remains hyperconvex. We give two new criteria, saying that on the one hand gluing along strongly convex subsets and on the other hand gluing…
Hypercontractive inequalities are a useful tool in dealing with extremal questions in the geometry of high-dimensional discrete and continuous spaces. In this survey we trace a few connections between different manifestations of…
In this research article, we discuss two topics. Firstly, we introduce SCC-Map and $\phi$-contraction type $T$-coupling. By using these two definitions, we generalize $\phi$-contraction type coupling given by H. Aydi et al. [3] to…