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We study vector spaces associated to a family of generalized Euler integrals. Their dimension is given by the Euler characteristic of a very affine variety. Motivated by Feynman integrals from particle physics, this has been investigated…
This paper explores the connection between dynamical system properties and statistical physics of ensembles of such systems. Simple models are used to give novel phase transitions; particularly for finite N particle systems with many…
Euler systems are certain compatible families of cohomology classes, which play a key role in studying the arithmetic of Galois representations. We briefly survey the known Euler systems, and recall a standard conjecture of Perrin-Riou…
In this paper, we derive some interesting symmetric properties for the geenralized Euler numbers and polynomials.
A numerical method is devised for study of stochastic partial differential equations describing directed percolation, the contact process, and other models with a continuous transition to an absorbing state. Owing to the heightened…
We consider Smale spaces, a particular class of hyperbolic topological dynamical systems, which include the basic sets for Smale's Axiom A systems. We present a homology theory for such systems which is based on the dimension group in the…
This is an expository article. We survey some fundamental trends in representation theory of symmetric groups and related objects which became apparent in the last fifteen years. The emphasis is on connections with Lie theory via…
A categorical framework for modeling and analyzing systems in a broad sense is proposed. These systems should be thought of as `machines' with inputs and outputs, carrying some sort of signal that occurs through some notion of time. Special…
We introduce an axiomatic theory of spherical diagrams as a tool to study certain combinatorial properties of polyhedra in $\mathbb R^3$, which are of central interest in the context of Art Gallery problems for polyhedra and other…
We investigate the most general phase space of configurations, consisting of all possible ways of assigning elementary attributes, ``energies'', to elementary positions, ``cells''. We discuss how this space possesses structures that can be…
We introduce a fundamental theory for the kinetics of systems of classical particles. The theory represents a unification of kinetic theory, Brownian motion and field theory. It is self-consistent and is the dynamic generalization of the…
We develop a global cohomology theory for number fields by offering topological cohomology groups, an arithmetical duality, a Riemann-Roch type theorem, and two types of vanishing theorem. As applications, we study moduli spaces of…
We revisit static, spherically symmetric perfect-fluid stellar models in General Relativity within the framework of the $1+1+2$ semi-tetrad formalism. For locally rotationally symmetric static spacetimes, the Tolman-Oppenheimer-Volkoff…
This is a survey highlighting several recent results concerning well/ill posedness of the Euler system of gas dynamics. Solutions of the system are identified as limits of consistent approximations generated either by physically more…
The generalization of the n-dimensional cube, an n-dimensional chain, the exterior derivative and the integral of a differential n-form on it are introduced and investigated. The analogue of Stokes theorem for the differential space is…
We propose a high dimensional generalisation of the standard Klein bottle, going beyond those considered previously. We address the problem of generating continuous scalar fields (distributions) and dynamical systems (flows) on such state…
This work reflects on mechanics as an epistemological framework on the state of a physical system to regard dynamics as the distribution of mechanical properties over spacetime coordinates. The resulting distribution is taken to be the…
We survey an area of recent development, relating dynamics to theoretical computer science. We discuss the theoretical limits of simulation and computation of interesting quantities in dynamical systems. We will focus on central objects of…
Since physical theories employ mathematical models to describe and predict physical phenomena, our knowledge depends on the models available to that end. To increase their scope we present a particular type of simplified models, serial…
A generalization of metric space is presented which is shown to admit a theory strongly related to that of ordinary metric spaces. To avoid the topological effects related to dropping any of the axioms of metric space, first a new, and…