Related papers: Non-Archimedean Geometry and Physics on Adelic Spa…
A formalism of arithmetic partial differential equations (PDEs) is being developed in which one considers several arithmetic differentiations at one fixed prime. In this theory solutions can be defined in algebraically closed p-adic fields.…
Proposed is a program for what we call Geometric Arithmetic, based on our works on non-abelian zeta functions and non-abelian class field theory. Key words are stability and adelic intersection-cohomology theory.
Partial Differential Equations (PDEs) have long been recognized as powerful tools for image processing and analysis, providing a framework to model and exploit structural and geometric properties inherent in visual data. Over the years,…
The article deals with plastic and non-plastic sub-spaces $A$ of the real line ${\mathbb{R}}$ with the usual Euclidean metric $d$. It investigates non-expansive bijections, proves properties of such maps and demonstrates their relevance by…
Metrics on Grassmannians have a wide array of applications: machine learning, wireless communication, computer vision, etc. But the available distances between subspaces of distinct dimensions present problems, and the dimensional asymmetry…
The purpose of this book is to build up the fundament of an Arakelov theory over adelic curves in order to provide a unified framework for the researches of arithmetic geometry in several directions.
In order to compute with $l$--adic sheaves or crystals on a line over $\mathbb{F} _q$ a low-technology alternative to the traditional computation with the Hecke operators on the automorphic side could be helpful. A program which has evolved…
We consider the quasihyperbolic metric, and its generalizations in both the $n$-dimensional Euclidean space $R^n$, and in Banach spaces. Historical background, applications, and our recent work on convexity properties of these metrics are…
Double coset spaces of adelic points on linear algebraic groups arise in the study of global fields; e.g., concerning local-global principles and torsors. A different type of double coset space plays a role in the study of semi-global…
The development of algorithmic fractal dimensions in this century has had many fruitful interactions with geometric measure theory, especially fractal geometry in Euclidean spaces. We survey these developments, with emphasis on connections…
We show the existence of group-theoretic sections of the "etale-by-geometrically abelian" quotient of the arithmetic fundamental group of hyperbolic curves over $p$-adic local fields relative to a proper and flat model which are…
In the present paper we study stability of recurrence equations (which in particular case contain a dynamics of rational functions) generated by contractive functions defined on an arbitrary non-Archimedean algebra. Moreover,…
In this paper we consider applications of methods from wavelet analysis to nonlinear dynamical problems related to accelerator physics. In our approach we take into account underlying algebraical, geometrical and topological structures of…
We review basic notions and methods of noncommutative geometry and their applications to analysis and geometry on foliated manifolds.
This is a chapter in an incoming book on aperiodic order. We review results about the topology, the dynamics, and the combinatorics of aperiodically ordered tilings obtained with the tools of noncommutative geometry.
These lecture notes are based on [arXiv: math/0702714, 0907.4469, 0907.4470]. We introduce and study basic aspects of non-Euclidean geometries from a coordinate-free viewpoint.
We construct a family of quantum scalar fields over a $p-$adic spacetime which satisfy $p-$adic analogues of the G\aa rding--Wightman axioms. Most of the axioms can be formulated the same way in both, the Archimedean and non-Archimedean…
We lay foundations of the subject in the title, on which we build in another paper devoted to isometries in spaces of K\"ahler metrics.
Approximate symmetries of geodesic equations on 2-spheres are studied. These are the symmetries of the perturbed geodesic equations which represent approximate path of a particle rather than exact path. After giving the exact symmetries of…
Over the past six years, a detailed framework has been constructed to unravel the quantum nature of the Riemannian geometry of physical space. A review of these developments is presented at a level which should be accessible to graduate…