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By J. Litwiniszyn

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Extra resources for Stochastic Methods in Mechanics of Granular Bodies: Course held at the Department of General Mechanics October 1972

Example text

It therefore becomes necessary to impose additional conditions on operator F , and to select an auxiliary set of operators which fulfil those conditions. Let the additional condition be linearity of the operator, which is the simplest mathematical possibility. Departure from this assumption and the attempt to assume nonlinearity of the operator encounters difficulty in the choice of such a nonlinearity, because the number of dif ferent kinds of nonlinearity, that can be conceived is unlimited. Many problems can also be interpreted within a sufficiently narrow range by means of linear models and such models are therefore useful from a practical standpoint.

In this method, certain conditions is imposed on the class of function 't~k . The conditions concem the moments of the transition function. x3 the transition functions tp~k are 11 m concentrated 11 in point II, 11 that everywhere beyond point II the value of function 't~k tends towards zero, and that ~~k have finite moments of the first, second and third order. g. x 3 disappears completely lii II beyend the cylinder with its base in plane II and centre in point II, and height (x 3- x3) . 27 and its physical interpretation.

Alization of the so-called equation of Smoluchowski describing the class of Markow's continuous basic processes in the theory of diffusion and Brownian movements. 12 is the basic equation for interpretation of displacement of granular media from standpoint of the concept presented herein. 12 can be obtained by means of the integral transformation of Fourier, and another by constructing a system of differential equations similar to the method employed by Kolmogorow. 4. e. if these functions depend only on the difference of coordinates.

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