6 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
Ergodicity criteria for non-expanding transformations of 2-adic spheres
In the paper, we obtain necessary and sufficient conditions for ergodicity
(with respect to the normalized Haar measure) of discrete dynamical systems
on 2-adic spheres of radius
, , centered at some point from the ultrametric space of
2-adic integers . The map is
assumed to be non-expanding and measure-preserving; that is, satisfies a
Lipschitz condition with a constant 1 with respect to the 2-adic metric, and
preserves a natural probability measure on , the Haar measure
on which is normalized so that
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
DISKRETNOST IZAZIVA TALASE
In the paper, we show that matter waves can be derived from discreteness and causality. Namely we show that matter waves can naturally be ascribed to finite discrete causal systems, the Mealy automata having binary input/output which are bit sequences. If assign real numerical values (‘measured quantities’) to bit sequences, the waves arise as a correspondence between the numerical values of input sequences (‘impacts’) and output sequences (‘system-evoked responses’). We show that among all discrete causal systems with arbitrary (not necessarily binary) inputs/outputs, only the ones with binary input/output can be ascribed to matter waves ψ(x, t) = ei(kx−ωt).U ovom radu pokazujemo da talasi materije mogu biti izvedeni iz diskretnosti i kauzalnosti. Naime, pokazujemo da talasi materije mogu biti prirodno pripisani konačnim diskretnim kauzalnim sistemima, Mealy automatima kod kojih su ulaz/izlaz binarni nizovi bitova. Ako nizovi bitova imaju realne numeričke vrednosti (merljive veličine), tada talasi nastaju kao veza izmedju numeričkih vrednosti ulaznih nizova (izazova) i izlaznih nizova (odgovora sistema). Pokazujemo da od svih diskretnih kauzalnih sistema sa proizvoljnim (ne obavezno binarnim) ulazima/izlazima,samo onima sa binarnim ulazom/izlazom mogu se pripisati talasi materije ψ(x, t) = ei(kx−ωt)