Related papers: Tropical mathematics, classical mechanics and geom…
This is the first paper in a series of eight where in the first three we develop a systematic approach to the geometric algebras of multivectors and extensors, followed by five papers where those algebraic concepts are used in a novel…
A brief review of some selected topics in p-adic mathematical physics is presented.
The chapter provides an introduction to the basic concepts of Algebraic Topology with an emphasis on motivation from applications in the physical sciences. It finishes with a brief review of computational work in algebraic topology,…
We present a study of geometric phases in classical wave and polarisation optics using the basic mathematical framework of quantum mechanics. Important physical situations taken from scalar wave optics, pure polarisation optics, and the…
We collect some classical results related to analysis on the Riemann surfaces. The notes may serve as an introduction to the field: we suppose that the reader is familiar only with the basic facts from topology and complex analysis. the…
In this paper we review a proposed geometrical formulation of quantum mechanics. We argue that this geometrization makes available mathematical methods from classical mechanics to the quantum frame work. We apply this formulation to the…
We discuss classical and quantum computations in terms of corresponding Hamiltonian dynamics. This allows us to introduce quantum computations which involve parallel processing of both: the data and programme instructions. Using mixed…
In these four lectures, aiming at senior undergraduate and junior graduate Physics and Mathematics students, basic elements of the theory of elliptic functions are presented. Simple applications in classical mechanics are discussed,…
The aim of this paper is twofold: First, we give a formal introduction to the basics of the mathematical framework of classical mechanics. Along the way, we prove a Hamiltonian and a Lagrangian version of Noether's Theorem, an important…
Mathematical aspects of contemporary classical and quantum gauge theory are sketched.
Symplectic and Poisson geometry emerged as a tool to understand the mathematical structure behind classical mechanics. However, due to its huge development over the past century, it has become an independent field of research in…
We survey the salient features and problems of conformal and superconformal mechanics and portray some of its developments over the past decade. Both classical and quantum issues of single- and multiparticle systems are covered.
Geometric aspects play an important role in the construction and analysis of structure-preserving numerical methods for a wide variety of ordinary and partial differential equations. Here we review the development and theory of symplectic…
The theory of tropical series, that we develop here, firstly appeared in the study of the growth of pluriharmonic functions. Motivated by waves in sandpile models we introduce a dynamic on the set of tropical series, and it is…
Algebraic systems called the local geodesic loops and their tangent Akivis algebras are considered. Their possible role in theory of gravity is considered. Quantum conditions for the infinitesimal quantum events are proposed.
We propose to study the tropical geometry specifically arising from convergent Hahn series in multiple indeterminates. One application is a new view on stable intersections of tropical hypersurfaces. Another one is perturbations of rank one…
This paper presents the basic ideas and properties of elliptic functions and elliptic integrals as an expository essay. It explores some of their numerous consequences and includes applications to some problems such as the simple pendulum,…
Some of the basic concepts of topology are explored through known physics problems. This helps us in two ways, one, in motivating the definitions and the concepts, and two, in showing that topological analysis leads to a clearer…
Our aim is to introduce the tropical tensor product and investigate its properties. In particular we show its use for solving tropical matrix equations.
We discuss the classical mechanics on the Grassmannian and the Disc modeled on the ideal L^(2,\infty)(H). We apply methods of geometric quantization to these systems. Their relation to a flat symplectic space is also discussed.