# The International Conference on Computational Mathematics ICCM-2004

## Abstracts

Statistic modeling and Monte Carlo methods

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

### Antipov M.V.

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

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