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December 1, 2011
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The distribution functions of the solutions of the systems of linear algebraic equations (SLAE) , in general, have a cumbersome form; the order of these systems in some practical problems is large, therefore, the asymptotic behavior of the solutions should be studied in increasing order n of the system to infinity. A general form of the limit theorems of solutions of the systems of linear algebraic equations with independent random coefficients and components , i, j 1, . . . , n , are given in this survey. By the tradition of choosing the names of laws in probability theory (Arcsine law, Law of iterated logarithm, etc.) we call the unusual behavior of the solutions of (SLAERC) as Inverse Tangent Law .

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December 1, 2011
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The theorems of Baxter type were established for a class of random processes with K-increments. The obtained results can be used in the statistics of random processes, in particular for obtaining sufficient conditions of singularity of measures generated by random processes. This study was focused on the LevyBaxter limit theorems for random processes. A new class of random processes was created. This method of investigation can be used to establish the Baxter type theorems for random fields.

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December 1, 2011
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This four-part paper stems from previous work of certain of the authors, where the issue of inducing distributions on lower dimensional spaces arose as a natural outgrowth of the main goal: the estimation of conditional probabilities, given other partially specified conditional probabilities as a premise set in a probability logic framework. This paper is concerned with the following problem. Let 1 m < n be fixed positive integers, some open domain, and a function yielding a full partitioning of D into a family, denoted M ( h ), of lower-dimensional surfaces/manifolds via inverse mapping h 1 as D M ( h ), where M ( h ) d { h 1 ( t ) : t in range( h )}, noting each h 1 ( t ) can also be considered the solution set of all X in D of the simultaneous equations h ( X ) t . Let X be a random vector (rv) over D having a probability density function (pdf) . Then, if we add sufficient smoothness conditions concerning the behavior of h (continuous differentiability, full rank Jacobian matrix dh ( X )/ dX over D , etc.), can an explicit elementary approach be found for inducing from the full absolutely continuous distribution of X over D a necessarily singular distribution for X restricted to be over M ( h ) that satisfies a list of natural desirable properties? More generally, for fixed positive integer r , we can pose a similar question concerning rv ( X ), when is some bounded a.e. continuous function, not necessarily admitting a pdf.

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December 1, 2011
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We present a proof for the existence and uniqueness of weak solutions for a cut-off and non-cut-off model of non-linear diffusion equation in finite-dimensional space useful for modeling flows on porous medium with saturation, turbulent advection, etc. and subject to deterministic or stochastic (white noise) stirrings. In order to achieve such goal, we use the powerful results of compacity on functional L p spaces (the AubinLion Theorem). We use such results to write a path-integral solution for this problem. Additionally, we present the rigorous functional integral solutions for the linear diffusion and wave equations defined in infinite-dimensional spaces (separable Hilbert spaces). These further results are presented in order to be useful to understand Polymer cylindrical surfaces probability distributions and functionals on String theory.

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December 1, 2011
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We consider a mixed stochastic differential equation involving both standard Brownian motion and fractional Brownian motion with Hurst parameter H > 1/2. The mean-square rate of convergence of Euler approximations of solution to this equation is obtained.