A. V. Mikhalev

Linear Codes and Polylinear Recurrences over Finite Rings and Modules (A Survey)

We give a short survey of the results obtained in the last several decades that develop the theory of linear codes and polylinear recurrences over finite rings and modules following the well-known results on codes and polylinear recurrences over finite fields. The first direction contains the general results of theory of linear codes, including: the concepts of a reciprocal code and the MacWilliams identity; comparison of linear code properties over fields and over modules; study of weight functions on finite modules, that generalize in some natural way the Hamming weight on a finite field; the ways of representation of codes over fields by linear codes over modules. The second one develops the general theory of polylinear recurrences; describes the algebraic relations between the families of linear recurrent sequences and their periodic properties; studies the ways of obtaining “good” pseudorandom sequences from them. The interaction of these two directions leads to the results on the representation of linear codes by polylinear recurrences and to the constructions of recursive MDS-codes. The common algebraic foundation for the effective development of both directions is the Morita duality theory based on the concept of a quasi-Frobenius module.

Characterization of the space of Riemann integrable functions by means of cuts of the space of continuous functions. II

AbstractThe space RI of Riemann integrable functions and its relations (with respect to order cuts) with the space C of continuous bounded functions are considered. It is proved that the Riemann completion C ↣ RI/

Functional identities in?rings and their applications

nmathcal{N}n

Characterization of general Radon integrals

, where

Оптимальное использование вейвлет-компонент@@@Optimal use of wavelet components

nmathcal{N}n

Цикловые типы семейств полилинейных рекуррент и датчики псевдослучайных чисел

is the ideal of all sets of zero Jordan measure, is a more complicated analogue of the Dedekind completion ℚ ↣ ℝ when new additional structure on the spaces C and RI/