26 research outputs found
The Non-Archimedean Theory of Discrete Systems
In the paper, we study behavior of discrete dynamical systems (automata)
w.r.t. transitivity; that is, speaking loosely, we consider how diverse may be
behavior of the system w.r.t. variety of word transformations performed by the
system: We call a system completely transitive if, given arbitrary pair
of finite words that have equal lengths, the system , while
evolution during (discrete) time, at a certain moment transforms into .
To every system , we put into a correspondence a family of continuous maps of a suitable non-Archimedean metric space
and show that the system is completely transitive if and only if the family
is ergodic w.r.t. the Haar measure; then we find
easy-to-verify conditions the system must satisfy to be completely transitive.
The theory can be applied to analyze behavior of straight-line computer
programs (in particular, pseudo-random number generators that are used in
cryptography and simulations) since basic CPU instructions (both numerical and
logical) can be considered as continuous maps of a (non-Archimedean) metric
space of 2-adic integers.Comment: The extended version of the talk given at MACIS-201
T-functions revisited: New criteria for bijectivity/transitivity
The paper presents new criteria for bijectivity/transitivity of T-functions
and fast knapsack-like algorithm of evaluation of a T-function. Our approach is
based on non-Archimedean ergodic theory: Both the criteria and algorithm use
van der Put series to represent 1-Lipschitz -adic functions and to study
measure-preservation/ergodicity of these
T-Funtions of several variables: New Criteria for Transitivity
The paper presents new criteria for transitivity of T-functions of several
variables. Our approach is based on non-Archimedean ergodic theory.
The criteria: for any 1-lipschitz ergodic map there are 1-lipschitz ergodic
map and two bijection ,
that Comment: arXiv admin note: text overlap with arXiv:1112.5089 by other author
Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory
The goal of this present manuscript is to introduce the reader to the
nonstandard method and to provide an overview of its most prominent
applications in Ramsey theory and combinatorial number theory.Comment: 126 pages. Comments welcom