Notes on number theory and dynamical systems (original) (raw)

On the Application of the Matrix Formalism for the Heat Kernel to Number Theory

Journal of Mathematical Sciences, 2019

Earlier, in the study of combinatorial properties of the heat kernel of the Laplace operator with covariant derivative, a diagram technique and matrix formalism were constructed. In particular, the obtained formalism allows one to control the coefficients of the heat kernel, which is useful for calculations. In this paper, we consider a simple case with an Abelian connection in the twodimensional space. This model allows us to give a mathematical description of the operators and find a relation between these operators and generating functions of numbers. Bibliography: 22 titles. Dedicated to M. A. Semenov-Tian-Shansky on the occasion of his 70th birthday

Some applications of homogeneous dynamics to number theory

2002

This survey paper is not a complete reference guide to number-theoretical applications of ergodic theory. Instead, the plan is to consider an approach to a class of problems involving Diophantine properties of n-tuples of real numbers, namely, describe a specific dynamical system which is naturally connected with these problems.

Dynamical Numbers: Interplay between Dynamical Systems and Number Theory

2010

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A new discrete dynamical system of signed integer partitions

European Journal of Combinatorics, 2016

In this paper we investigate a family of algebras endowed with a suitable non-degenerate bilinear form that can be used to define two different notions of dual for a given right ideal. We apply our results to the classification of the right ideals and their duals in the cyclic group algebra, in the Taft algebra and in another example of Hopf algebra arising as bosonization.

Recent Work on the Partition Function

2013

This expository article describes recent work by the authors on the partition function p(n). This includes a finite formula for p(n) as a "trace" of algebraic singular moduli, and an overarching � -adic structure which controls partition congruences modulo powers of primes � ≥ 5.