Abstract: We provide a detailed explanation of the physical meaning of some concepts used in the new statistics corresponding to thermodynamics, including the notions of locally ideal gas, number of collective degrees of freedom, and jamming factor. The equation of state is treated as a surface in three-dimensional space, and the spinodal is viewed as an Arnold catastrophe for a quasistatic process. We show that the parameters derived according to the new statistics completely coincide with the parameters of the van der Waals gas and also make a comparison with nitrogen. Directions of research are outlined for the construction of statistics in mesoscopic physics. PubDate: 2015-01-01

Abstract: We discuss a mathematical link between the Quantum Statistical Mechanics and the logistic growth and decay processes. It is based on an observation that a certain nonlinear operator evolution equation, which we refer to as the Logistic Operator Equation (LOE), provides an extension of the standard model of noninteracting bosons. We discuss formal solutions (asymptotic formulas) for a special calibration of the LOE, which sets it in the number-theoretic framework. This trick, in the tradition of Julia and Bost-Connes, makes it possible for us to tap into the vast resources of classical mathematics and, in particular, to construct explicit solutions of the LOE via the Dirichlet series. The LOE is applicable to a range of modeling and simulation tasks, from characterization of interacting boson systems to simulation of some complex man-made networks. The theoretical results enable numerical simulations, which, in turn, shed light at the unique complexities of the rich and multifaceted models resulting from the LOE. PubDate: 2015-01-01

Abstract: We give a review of some of our works which generalize stochastic analysis tools for semigroups where there is no stochastic process. PubDate: 2015-01-01

Abstract: The heat kernel trace in a (D+1)-dimensional Euclidean spacetime is used to derive free energy in the finite temperature field theory. The spacetime presents a D-dimensional compact space (domain) with a (D-1)-dimensional boundary, and one closed dimension, whose volume is proportional to Planck’s inverse temperature. The thermal sum arises due to the topology of the closed Euclidean time. The free energy thus obtained is a functional of the Planck’s inverse temperature and the geometry of the system. Its ‘high temperature’ asymptotic expressions, given for (3+1) and (2+1) dimensions, contain two contributions defined by the volume of the domain and by the volume of boundary of the domain. No universal asymptotic of free energy exists while approaching the absolute zero temperature, which is forbidden topologically. PubDate: 2015-01-01

Abstract: We continue our study of gauge equivariant K-theory. We thus study the analysis of complexes endowed with the action of a family of compact Lie groups and their index in gauge equivariant K-theory. We introduce various index functions, including an axiomatic one, and show that all index functions coincide. As an application, we prove a topological index theorem for a family D = (D
b
)
b∈B
of gauge-invariant elliptic operators on a G-bundle X → B, where G → B is a locally trivial bundle of compact groups, with typical fiber G. More precisely, one of our main results states that a-ind(D) = t-ind(D) ∈ K
G
0
(X), that is, the equality of the analytic index and of the topological index of the family D in the gauge-equivariant K-theory groups of X. The analytic index ind
a
(D) is defined using analytic properties of the family D and is essentially the difference of the kernel and cokernel K
G
-classes of D. The topological index is defined purely in terms of the principal symbol of D. PubDate: 2015-01-01

Abstract: In the paper, we obtain order bounds for the Kolmogorov, linear, and Gelfand widths of weighted Sobolev classes W
p,g
r
(Ω) on a domain with John condition in the space L
q,v
(Ω). The weights v are functions of the distance to some h-set contained in the boundary of the domain. We consider here some limiting values of parameters describing the weight v. PubDate: 2015-01-01

Abstract: In this paper, we introduce and investigate a fractional integral operator which contains Fox’s H-function in its kernel. We find solutions to some fractional differential equations by using this operator. The results derived in this paper generalize the results obtained in earlier works by Kilbas et al. [7] and Srivastava and Tomovski [23]. A number of corollaries and consequences of the main results are also considered. Using some of these corollaries, graphical illustrations are presented and it is found that the graphs given here are quite comparable to the physical phenomena of decay processes. PubDate: 2015-01-01

Abstract: A modified version of the ASEP model is interpreted as a two-dimensional model of ideal gas. Its properties are studied by simulating its behavior in different situations, using an animation program designed for that purpose. PubDate: 2015-01-01

Abstract: In this paper, we consider poly-Bernoulli and higher-order poly-Bernoulli polynomials and derive some new and interesting identities of those polynomials by using umbral calculus. PubDate: 2015-01-01

Abstract: By equating to zero all components of the Kröner incompatibility tensor of rank 2n − 4 or of the Riemann tensor dual to the Kröner tensor, n
2(n
2 − 1)/12 independent consistency equations for the stresses in an n-dimensional isotropic elastic medium are derived. The problem concerning the equivalence of the system of these equations to systems following from equating to zero either all n(n + 1)/2 components of the Ricci tensor or only one curvature invariant is investigated. It is shown that the answer to this question depends on the dimension of the space. Three cases are singled out: n = 2 (plane problem of elasticity theory), n = 3 (spatial problem of elasticity theory), and n ⩾ 4. PubDate: 2015-01-01

Abstract: The aim of this paper is to construct new generating functions for Hermite base Bernoulli type polynomials, which generalize not only the Milne-Thomson polynomials but also the two-variable Hermite polynomials. We also modify the Milne-Thomson polynomials, which are related to the Bernoulli polynomials and the Hermite polynomials. Moreover, by applying the umbral algebra to these generating functions, we derive new identities for the Bernoulli polynomials of higher order, the Hermite polynomials and numbers of higher order, and the Stirling numbers of the second kind. PubDate: 2015-01-01

Abstract: For slow-fast Hamiltonian systems with one fast degree of freedom, we describe the construction of the complete adiabatic invariant and the complete adiabatic term at once in all asymptotic orders by using the small parameter “dynamics” and parallel translations in the phase space. PubDate: 2015-01-01

Abstract: Let g be a finite-dimensional Lie algebra and L be a Lie algebra bundle (LAB). A given coupling Ξ between the LAB L and the tangent bundle TM of a manifold M generates the so-called Mackenzie obstruction Obs(Ξ) ∈ H
3 (M; ZL) to the existence of a transitive Lie algebroid (K. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, 2005, p. 279). We present two cases of evaluating the Mackenzie obstruction.
In the case of a commutative algebra g, the group Aut(g)
δ
is isomorphic to the group of all matrices GL(g) with the discrete topology. We show that the Mackenzie obstruction for coupling Obs(Ξ) vanishes.
The other case describes the Mackenzie obstruction for simply connected manifolds. We prove that, for simply connected manifolds, the Mackenzie obstruction is also trivial, i.e. Obs(Ξ) = 0 ∈ H
3(M; ZL; ∇
Z
). PubDate: 2014-10-01

Abstract: In this paper, we study some properties of several polynomials arising from umbral calculus. In particular, we investigate the properties of orthogonality type of the Frobeniustype Eulerian polynomials which are derived from umbral calculus. By using our properties, we can derive many interesting identities of special polynomials associated with Frobeniustype Eulerian polynomials. An application to normal ordering is presented. PubDate: 2014-10-01

Abstract: This paper is a report on the results of computer experiments with an algorithm that takes classical knots to what we call their “normal form” (and so can be used to identify the knot). The algorithm is implemented in a computer animation that shows the isotopy joining the given knot diagram to its normal form. We describe the algorithm, which is a kind of gradient descent along a functional that we define, present a table of normal forms of prime knots with 7 crossings or less, compare it to the knot table of normal forms of wire knots (obtained in [1] by mechanical experiments with real wire models) and (regretfully) present simple examples showing that normal forms obtained by our algorithm are not unique for a given knot type (sometimes isotopic knots can have different normal forms). PubDate: 2014-10-01

Abstract: The Cauchy problem for a parabolic equation with a small parameter multiplying the highest derivatives is considered. The dynamical system corresponding to the limit equation of the first order has an asymptotically stable equilibrium. A Lyapunov function known in a neighborhood of this equilibrium is used to construct a barrier in the Cauchy problem for the original parabolic equation. This result is applied to study the dynamical system with respect to random perturbations of the “white noise” type. PubDate: 2014-10-01