By Niva B Maslova

This quantity includes papers from major specialists in smooth continuum thought of composite fabrics. The papers divulge intimately principles, aproaches, effects and views during this extensively interdisciplinary box. The emphasis is at the mathematical modelling and version research of the mechanical behaviour and energy of composites, together with tools of predicting potent macroscopic homes (dielectric, elastic, nonlinear, inelastic, plastic and thermoplastic) from identified microstructures

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**Example text**

The following remarkable Carleman theorem [14] allows to describe solutions of the equation J ( f , f ) = 0: Carleman's theorem. 7). 9). All these facts were observed by Boltzmann in his first publications. The next step was made by Carleman. e. for the hard spheres model. Below we discuss some properties of the collision operators, generalizing Carleman's results. Since the collision integrals are linear with respect to the kernel B , it is sufficient to consider the function B satisfying with z = I(v,a)llvl-l, 7 E [Or11, bl > 0 to describe the operators with Grad's potentials.

4. 11) with 0 = R3. Notice that the well-posedness of problem 2 is not clear a priori even formally. Variants of the conditions at infinity are discussed in Chapter 5. We shall also discuss some problems with a prescribed symmetry with respect to spatial variables x, while full three dimensional velocity dependence is allowed. A special role of these problems stems from their applications in the boundary layers problems for complicated flows. One of the classical problem consists in finding a solution of the Cauchy problem, independent of X.

The next step was made by Carleman. e. for the hard spheres model. Below we discuss some properties of the collision operators, generalizing Carleman's results. Since the collision integrals are linear with respect to the kernel B , it is sufficient to consider the function B satisfying with z = I(v,a)llvl-l, 7 E [Or11, bl > 0 to describe the operators with Grad's potentials. 10) is fulfilled. An important property of the collision frequency is its unboundedness for large IvI: ~ ( f= ) O((v17) as lvl + oo.