Taking into account ties in the two-sample Wilcoxon test

One gives a probability proof of the well-know formula for the computation of the variance of the two-sample Wilcoxon test statistic under the presence of ties.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 119, pp. 195–197, 1982.     

Taking the particle interaction into account in a new concept of nucleation theory

Taking the interaction of particles into account in a new theory of nucleation leads to the occurrence of a phase transition in the Van der Waals theory. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 2, pp. 303–304, August, 2008.     

A new distribution generalizing the Bose-Einstein distribution

We present a distribution that generalizes the Bose-Einstein distribution.     

Taking into account the interaction between particles in the new nucleation theory, quasiparticles, quantization of vortices, and the two-particle distribution function

In the present paper, we consider in greater detail the problems that were discussed in the previous short communication [1].     

Taking the detail balance relations into account in the construction of approximations of kinetic equations

We propose an approach to the construction of approximations of kinetic equations of a lattice gas in the context of cluster approaches, taking account of the detail balance relations. This approach is applied to construct approximations for a one-component lattice gas on a homogeneous one-dimensional lattice, in which it is shown that in the absence of migration the additional requirement of unique solutions of the approximate equations leads to the well-known Kirkwood approximations in the context of the quasi-chemical approximations. We give the equations for a one-component lattice gas on a homogeneous rectangular lattice, describing a family of approximations satisfying these requirements and conditions on the additional relations needed to close the system. Bibliography: 5 titles. Translated fromProblemy Matematicheskoi Fiziki, 1998, pp. 96–112.     

A correction to the Maxwell distribution and the Bose-Einstein-type distribution in classical physics

We find new distributions in classical statistical physics that explain some old and new experiments. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 1, pp. 149–153, October, 2008.     

Records in the classical and quantum standard map

Record statistics is the study of how new highs or lows are created and sustained in any dynamical process. The study of the highest or lowest records constitute the study of extreme values. This paper represents an exploration of record statistics for certain aspects of the classical and quantum standard map. For instance the momentum square or energy records is shown to behave like that of records in random walks when the classical standard map is in a regime of hard chaos. However different power laws is observed for the mixed phase space regimes. The presence of accelerator modes are well-known to create anomalous diffusion and we notice here that the record statistics is very sensitive to their presence. We also discuss records in random vectors and use it to analyze the quantum standard map via records in their eigenfunction intensities, reviewing some recent results along the way.     

Convergence to equilibrium for classical and quantum spin systems

Summary The paper is devoted to stochastic equations describing the evolution of classical and quantum unbounded spin systems on discrete lattices and on Euclidean spaces. Existence and asymptotic properties of the corresponding transition semigroups are studied in a unified way using the theory of dissipative operators on weighted Hilbert and Banach spaces. This paper is an enlarged and rewritten version of the paper [7].Partially supported by the Italian National Project MURST Problemi nonlinearinell' Analisi... and by DRET under contract 901636/A000/DRET/DSISR.Partially sponsored by the KBN grant 2 2003 91 02 and by the KBN grant 2PO3A 082 08     

On the hidden parameter in quantum and classical physics

Mathematical Notes -     

On simulating the quantum and classical branching programs

The complexity classes defined on the basis of branching programs are considered. Some basic relations are established between the complexity classes defined by the probabilistic and quantum branching programs (measure-once, as well as measure-many), computing with bounded or unbounded error. To prove these relations, we developed a method of “linear simulation” of a quantum branching program and a method of “quantum simulation” of a probabilistic branching program.     

Non-standard power-law Lagrangians in classical and quantum dynamics

Recently, non-standard Lagrangians (NSL) have gained an increasing significance due to their wide implications in the theory of non-linear differential equations, complex dynamical systems and theoretical physics. In this work, we choose the power-law non-standard Lagrangian and we discuss some of its implications in classical and quantum theories.     

Long-time self-similarity of classical solutions to the bipolar quantum hydrodynamic models

In this paper, a bipolar transient quantum hydrodynamic model (BQHD) for charge density, current density and electric field is considered on the one-dimensional real line. This model takes the form of the classical Euler-Poisson system with additional dispersion caused by the quantum (Bohn) potential. We investigate the long-time behavior of the BQHD model and show the asymptotical self-similarity property of the global smooth solution. Namely, both of the charge densities tend to a nonlinear diffusion wave in large time, which is not a solution to the BQHD equation, but to the combined quasi-neutral, relaxation and semiclassical limiting model. Next, as a by-product, we can compare the large-time behavior of the bipolar quantum hydrodynamic models and of the corresponding classical bipolar hydrodynamic models. As far as we know, the nonlinear diffusion phenomena about the 1D BQHD is new.     

Projective modules in classical and quantum functional analysis

Along with the classical version, there are two “quantized” versions of the theory of operator algebras. In these lectures, the fundamental homological notion of a projective module is described in the framework of these three theories. Our initial definitions of projectivity do not go far from their prototypes in abstract algebra; however, the principal results concern essentially functional-analytic objects and, as a rule, have no purely algebraic analogs. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 7, pp. 7–84, 2007.     

Comparing quantum and classical approaches in statistical physics

We discuss hypotheses pertaining to the behavior of the mean energy of an equilibrium system in the quantum and classical theories. We justify the hypotheses for the cases of the harmonic oscillator and the potential well. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 3, pp. 516–520, June, 2000.     

Remarks on classical solutions to steady quantum Navier-Stokes equations

The paper of Dong [Dong, J. Classical solutions to one-dimensional stationary quantum Navier–Stokes equations, J. Math Pure Appl. 2011] which proved the existence of classical solutions to one-dimensional steady quantum Navier–Stokes equations, when the nonzero boundary value u ₀ satisfies some conditions. In this paper, we obtain a different version of existence theorem without restriction to u ₀. As a byproduct, we get the existence result of classical solutions to the stationary quantum Navier–Stokes equations.