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Sergio Albeverio, Song Liang
December 1, 2004

### Abstract

Consider the lattice approximation of a -quantum field model with different lattice cutoffs a ′ and a in the free and interacting parts, respectively. In [1] it was shown that the corresponding continuum limit measure exists if lim a →0 a ′| log a | 5/4 < ∞ and it coincides with the original - field measure if lim a →0 a ′| log a | 2 < ∞. In this paper, a result is given indicating that the new continuum limit measure might be different from the original one if a ′ is too big compared with a .

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Saralees Nadarajah, Gokarna Aryal
December 1, 2004

### Abstract

A random variable X is said to have the skew-uniform distribution if its pdf is ƒ( x ) = 2 g ( x ) G ( λx ), where g (·) and G (·), respectively, denote the pdf and the cdf of the Uniform (− θ , θ ) distribution. This distribution – in spite of its simplicity – appears not to have been studied in detail. The only work that appears to give some details of this distribution is Gupta et al. [ Random Operators and Stochastic Equations , 10 , 2002, 133–140], where expressions for the pdf, moment generating function, expectation, variance, skewness and the kurtosis of X are given. Unfortunately, all of these expressions appear to contain some errors. In this paper, we provide a comprehensive description of the mathematical properties of X . The properties derived include the k th moment, the k th central moment, variance, skewness, kurtosis, moment generating function, characteristic function, hazard rate function, mean deviation about the mean, mean deviation about the median, Rényi entropy, Shannon entropy and the asymptotic distribution of the extreme order statistics. We also consider estimation and simulation issues.

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K.-H. Fichtner, W. Freudenberg, V. Liebscher
December 1, 2004

### Abstract

We develop from the exchange of particles in random point configurations a similar concept for states of boson systems. The central part is devoted to the study of the corresponding class of unitary operators, modelling an interaction between two Boson systems based on the exchange of particles (Bosons) only.

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Nassar H.S. Haidar, Adnan M. Hamzeh, Soumaya M. Hamzeh, Edward El-Nakat
December 1, 2004

### Abstract

This is an extension of a previous work of the first author [1], on monomial density functions, to study the approximation features over [ a, b ] of, the based on α x exponential density functions h α ( x ) when α > 0 is discrete and/or fractional. If α is discrete, a random variable of h n ( x ) is proved to form a martingale over a reversed filtration and is compared with a similar situation that happens to hold with respective monomial density functions. In the case of fractional α , we advance a new stochastic operator which generates, via a nonlinear technique, unique exponential and monomial spline approximants over [ a, b ] to functions ( x ) ∈ , a certain noncommutative inner product space.

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V. F. Kadankov, T. V. Kadankova
December 1, 2004

### Abstract

For a semicontinuous homogeneous process ξ(t) with independent increments the distribution of the its total duration of stay in an interval is obtained. In the case E ξ (1) = 0, E ξ (1) 2 < ∞, the limit theorem on a weak convergence of the time of duration of stay in an interval of the process to distribution of the time of duration of stay of Wiener process in the interval(0, 1) is obtained. For Wiener process the distribution of the total duration of stay in an interval is found.

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A. Yu. Pilipenko
December 1, 2004

### Abstract

The properties of stochastic flow generated by stochastic equation with reflection are studied. We investigate an instant of time and a place where particles carrying by such a flow can coalesce.

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Francesco Fidaleo
December 1, 2004

### Abstract

We continue the analysis of nontrivial examples of quantum Markov processes. This is done by applying the construction of entangled Markov chains obtained from classical Markov chains with infinite state–space. The formula giving the joint correlations arises from the corresponding classical formula by replacing the usual matrix multiplication by the Schur multiplication. In this way, we provide nontrivial examples of entangled Markov chains on , F being any infinite dimensional type I factor, J a finite interval of , and the bar the von Neumann tensor product between von Neumann algebras. We then have new nontrivial examples of quantum random walks which could play a rôle in quantum information theory. In view of applications to quantum statistical mechanics too, we see that the ergodic type of an entangled Markov chain is completely determined by the corresponding ergodic type of the underlying classical chain, provided that the latter admits an invariant probability distribution. This result parallels the corresponding one relative to the finite dimensional case. Finally, starting from random walks on discrete ICC groups, we exhibit examples of quantum Markov processes based on type II 1 von Neumann factors.