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This is a survey article in algebraic logic, where we take a magical tour from old concepts due to Henkin, Monk and Tarski like neat embeddings, to modern views and perspectives, culminating in the use of Erdos graphs in settling important…
By using a combination of algebraic, geometric, and dynamical techniques, together with input from higher dimensional Diophantine approximation, we give a complete characterization of all linearly repetitive cut and project sets with…
We survey some algebraic geometric aspects of mirror symmetry and duality in string theory. Some applications of computer algebra to algebraic geometry and string theory are shortly reviewed.
Solving polynomial equations is a subtask of polynomial optimization. This article introduces systems of such equations and the main approaches for solving them. We discuss critical point equations, algebraic varieties, and solution counts.…
An expository approach is given on the relationship between algebraic and geometric approaches to properties of isometries in the plane and the 2-sphere.
We prove time-dependent versions of Kingman's subadditive ergodic theorem, which can be used to study stochastic processes as well as propagation of solutions to PDE in time-dependent environments.
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 prove the constructive version of Birkhoff's ergodic theorem following Vyugin but trying to separate and state explicitly the combinatorial statement on which this proof is based. We pose some questions related to this statement (and the…
The purpose of this paper is to give, on one hand, a mathematical exposition of the main topological and geometrical properties of geometric transitions, on the other hand, a quick outline of their principal applications, both in…
The interplay rich between algebraic geometry and string and gauge theories has recently been immensely aided by advances in computational algebra. However, these symbolic (Gr\"{o}bner) methods are severely limited by algorithmic issues…
This paper studies ergodic properties of certain measures arising in the dynamics of holomorphic correspondences. These measures, in general, are not invariant in the classical sense of ergodic theory. We define a notion of ergodicity, and…
In this paper, for a given finitely generated algebra (an algebraic structure with arbitrary operations and no predicates) A we study finitely generated limit algebras of A, approaching them via model theory and algebraic geometry. Along…
The theory of matroids has been generalized to oriented matroids and, recently, to arithmetic matroids. We want to give a definition of "oriented arithmetic matroid" and prove some properties like the "uniqueness of orientation".
Geometric mechanics is usually studied in applied mathematics and most introductory texts are hence aimed at a mathematically minded audience. The present note tries to provide the intuition of geometric mechanics and to show the relevance…
Geodesics deviation equation (GDE) is itroduced. In "adiabatic" approximation exact solution of the GDE if found. Perturbation theory in general case is formulated. Geometrical criterion of local instability which may lead to chaos is…
We prove a version of pointwise Ergodic Theorem for non-stationary random dynamical systems. Also, we discuss two specific examples where the result is applicable: non-stationary iterated function systems and non-stationary random matrix…
Excitations of a relativistic geometry are used to represent the theory of quantum electrodynamics. The connection excitations and the frame excitations reduce, respectively, to the electromagnetic field operator and electron field…
We survey the development of Clifford's geometric algebra and some of its engineering applications during the last 15 years. Several recently developed applications and their merits are discussed in some detail. We thus hope to clearly…
We introduce the notion of stationary actions in the context of C*-algebras. We develop the basics of the theory, and provide applications to several ergodic theoretical and operator algebraic rigidity problems.
There are versions of "calculus" in many settings, with various mixtures of algebra and analysis. In these informal notes we consider a few examples that suggest a lot of interesting questions.