# Институт вычислительной математики

и математической геофизики

# The International Conference on Computational Mathematics

ICCM-2004

June, 21-25, 2004
Novosibirsk, Academgorodok

## Abstracts

*Statistic modeling and Monte Carlo methods*

## Theoretical and Practical Solution of Problem of Random Values Generation

**Institute of Computational Mathematics and Mathematical Geophysics SB RAS (Novosibirsk)**
documentstyle[12pt]{article}
itle{Theoretical and Practical Solution of Problem
of Random Values Generation}
author{M.V.,Antipov}
egin{document}
maketitle
egin{center}
{small
Institute of Computational Mathematics and Mathematical
Geophysics
Novosibirsk, Siberian Branch, Russian Academy of Sciences}
end{center}
vspace*{1mm}
In work [1,2] is proven, that the solution of a randomness problem
it is impossible outside of modernization of a mathematical views
on the basis of the concept of the restriction principle. It concerns
and modeling of random numbers to the full. Traditional way of randomness
imitation uses separate algorithms $ALG_i$,. They possess quite
evaluated but insuperable defects. For this reason even the physical
algorithms are pseudorandom [1,2] and have a finite measure
of pseudorandomness $mu_{ps} (ALG_i)$,.
At the same time as is shown in [3] the operator of congruent
summation
$$
z_{(m)}^{(n)} equiv left{,sumlimits_{i=1}^{m+1},y_i^{(n)},
ight}({
m mod},1),, quad y_i^{(n)} = left(,x_i^{(1)},,
x_i^{(2)},,...,,x_i^{(n)},
ight),, eqno{(1)}
$$
brings in sharp increase of a measure of pseudorandomness
$mu_{ps}{z_{(m)}^{(n)}} stackrel{m o infty}{Longrightarrow} infty$ for resulting algorithm $ALG(m)$
in parallel with all characteristics.
{f Lemma.} At increase of summation's parameter $m$ and
fulfilment some easily achievable conditions with decrease of
algorithmical dependence $ALG_i$,, the pseudorandomness
measure $mu_{ps} {ALG (m)}$ exceeds any given size.
{f Theorem.} The opportunity of numerical modeling $z_{(m)}^{(n)}$
of unlimitely increasing pseudorandomness measure $mu_{ps}$
permits the problem of randomness.
Thus, the imitation of randomness with the help of natural or any other
processes is impossible, and the physical generation so named <>
values has not theoretical and even practical prospects because of
inefficiency, vagueness and groundlessness. Mathematical algorithm (1)
at increase of a pseudorandomness measure $mu_{ps}$ surpasses any
natural realization with guarantee. The opportunity of imitation of
randomness and chaos as phenomena of pseudorandomness is provided
by algorithms of increasing complexity.
vspace*{3mm}
{small
[1]. Antipov M.V. Congruent Operator in Simulation of Continuous
Distributions, {it Journ. Numer. Math.}, Vol. 42, N 11, 2002, pp.1572 - 1580.
[2]. Antipov M.V. {it The Restricton Principle}, Novosibirsk: Ross.
Akad. Nauk, 1998, pp. 444.
[3]. Antipov M.V. Multiple Congruent Convolutions of Probability
Densities and Estimates in the $L^{infty}([0,1)^n)$ Space,
{it Journ. Numer. Math.}, Vol. 40, N 2, 2000, pp. 293 -- 303.
}
end{document}

*Note. Abstracts are published in author's edition*

© 1996-2000, Siberian Branch of Russian Academy of Sciences, Novosibirsk

Last update: 06-Jul-2012 (11:52:06)