On the thermodynamic foundations of the theory of local-gradient mechanicothermodiffusion systems

In the context of the continuous-thermodynamic approach we generalize the Gibbs equation and obtain the initial relations of local-gradient mechanicothermodiffusion. We state the relation between the thermodynamic flows and forces in the form of functionals. We find influence functions that cause expansion of the phase space that determines the thermodynamic potentials by the gradients of the intensive parameters of the equilibrium state of the system. It is shown that such influence functions are connected with the undamped memory of the body of the action at the initial time. Translated fromMatematychni Metody ta Fizyko-Mekhanichni, Polya, Vol. 41, No. 1, 1998, pp. 62–72.     

Equivalence classes in the Weyl groups of type <Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

We consider two families of equivalence classes in the Weyl groups of type B _n which are suggested by the study of left cells in unequal parameter Iwahori-Hecke algebras. Both families are indexed by a non-negative integer r. It has been shown that the first family coincides with left cells corresponding to the equal parameter Iwahori-Hecke algebra when r=0; the equivalence classes in the second family agree with left cells corresponding to a special class of choices of unequal parameters when r is sufficiently large. Our main result shows that the two families of equivalence classes coincide, suggesting the structure of left cells for remaining choices of the Iwahori-Hecke algebra parameters.      

The equations of the thermodynamic model of magnetoelasticity of dielectric ferromagnetic bodies

We develop a thermodynamic approach to the mathematical modeling of magnetoelastic processes in dielectric ferromagnetic bodies that are subject to the action of force loading, heating, and an external electromagnetic field. The basis for the construction of the physical relations is the principle of local thermodynamic state. In the computations of the momentum balance equation of the magnetization process account is taken of its tensor character.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 36, 1992, pp. 30–34.     

Asymptotic theory for estimating the parameters of a Lévy process

Summary We consider consistency and asymptotic normality of maximum likelihood estimators (MLE) for parameters of a Lévy process of the discontinuous type. The MLE are based on a single realization of the process on a given interval [0,t]. Depending on properties of the Lévy measure we either consider the MLE corresponding to jumps of size greater than ε and, keepingt fixed, we let ε tend to 0, or we consider the MLE corresponding to the complete information of the realization of the process on [0,t] and lett tend to ∞. The results of this paper improve in both generality and rigor previous asymptotic estimation results for such processes.     

Construction of Separation Variables for Finite-Dimensional Integrable Systems

We consider a class of integrable systems such that solutions of the corresponding Hamilton–Jacobi equation depend on n+m arbitrary parameters and are represented as products of flat curves. The first n parameters are identified with the values of the integrals of motion. The remaining parameters enter the definition of the integrals of motion as arbitrary constants (charges) and can be used to find separation variables. We show that on the coadjoint orbits of Lie groups, the Casimir operators not only generate a family of integrals but also allow constructing separation variables.     

Thermodynamics from the differential geometry standpoint

We study the differential-geometric structure of the space of thermodynamic states in equilibrium thermodynamics. We demonstrate that this space is a foliation of codimension two and find variables in which the foliation fibers are flat. We show that we can introduce a symplectic structure on this space: the external derivative of the 1-form of the heat source, which has the form of the skew-symmetric product dT 蝃 dS in the found variables. The entropy S then plays the role of the Lagrange function (or Hamiltonian) in mechanics, completely determining the thermodynamic system. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 1, pp. 141–148, October, 2008.     

Integration of the equations of gas dynamics for 2.5-dimensional solutions

We integrate the equations of gas dynamics in finite form for the solutions in which the thermodynamic parameters depend only on one spatial variable. The corresponding motion of gas represents the nonlinear superposition of the one-dimensional gas motion corresponding to the invariant system and the two-dimensional motion determined by noninvariant functions. These motions are called 2.5-dimensional. We reduce the invariant system to a first-order implicit ordinary differential equation. We study various solutions of the latter. We construct some continuous and discontinuous solutions to the equations of gas dynamics and give their physical interpretation.     

Mathematical modeling of the temperature fields during the solidification of metal in molds of complicated profile

We present a computational model of the process of solidification of metal in molds of complicated configuration. The model is based on the theory of R-functions, which makes it possible to construct the normal equations of drawings and introduce additional parameters into these equations, by means of which one can assure the required behavior of the solution corresponding to the technological implementation of the process. Computational experiments were carried out using thePOLE system. Translated fromMatematichni Metodi i Fiziko-mekhanichni Polya, Vol. 40, No. 1, 1997, pp. 59–65.     

Rank-one convexity and polyconvexity of Hencky-type energies

We investigate a family of isotropic volumetric-isochorically decoupled strain energies based on the Hencky-logarithmic (true, natural) strain tensor log U. The main result of this note is that for n = 2 the considered energies are rank-one convex for suitable values of two material parameters. We also conjecture that there are values of the material parameters such that the corresponding energies are polyconvex. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Parameterized computational complexity of finding small-diameter subgraphs

Finding subgraphs of small diameter in undirected graphs has been seemingly unexplored from a parameterized complexity perspective. We perform the first parameterized complexity study on the corresponding NP-hard s-Club problem. We consider two parameters: the solution size and its dual.     

On the use of phase‐type distributions for inventory management with supply disruptions

Maintaining the continuity of operations becomes increasingly important for systems that are subject to disruptions due to various reasons. In this paper, we study an inventory system operating under a (q, r) policy, where the supply can become inaccessible for random durations. The availability of the supply is modeled by assuming a single supplier that goes through ON and OFF periods of stochastic duration, both of which are modeled by phase‐type distributions (PTD). We provide two alternative representations of the state transition probabilities of the system, one with integral and the other employing Kolmogorov differential equations. We then use an efficient formulation for the analytical model that gives the optimal policy parameters and the long‐run average cost. An extensive numerical study is conducted, which shows that OFF time characteristics have a bigger impact on optimal policy parameters. The ON time characteristics are also important for critical goods if disasters can happen. Copyright © 2011 John Wiley & Sons, Ltd.     

Vertex fusion under distance constraints

In this paper, we analyze parameter improvement under vertex fusion in a graph G. This is a setting in which a new graph G^′ is obtained after identifying a subset of vertices of G in a single vertex. We are interested in distance parameters, in particular diameter, radius and eccentricity of a vertex v. We show that the corresponding problem is NP-Complete for the three parameters. We also find graph classes in which the problem can be solved in polynomial time.     

New Relations in the Algebra of the Baxter Q-Operators

We consider irreducible cyclic representations of the algebra of monodromy matrices corresponding to the R-matrix of the six-vertex model. At roots of unity, the Baxter Q-operator can be represented as a trace of a tensor product of L-operators corresponding to one of these cyclic representations, and this operator satisfies the TQ equation. We find a new algebraic structure generated by these L-operators and consequently by the Q-operators.     

A central limit theorem for extremal characters of the infinite symmetric group

The asymptotic behavior of the lengths of the first rows and columns in the random Young diagrams corresponding to extremal characters of the infinite symmetric group is studied. We consider rows and columns with linear growth in n and prove a central limit theorem for their lengths in the case of distinct Thoma parameters. We also prove a more precise statement relating the growth of rows and columns of Young diagrams to a simple independent random sampling model.     

Supercritical fluid of particles with a Yukawa potential: A new approximation for the direct correlation function and the Widom line

We propose an approximation of a direct correlation function corresponding to the linearization with respect to ?β?(r) of a generalized mean spherical approximation for a hard-core multi-Yukawa system of particles. We use the results to study the behavior of maximums of thermodynamic response functions in the supercritical region of a fluid with a two-term Yukawa potential imitating the Lennard-Jones potential.     

Penalty Approach to the HJB Equation Arising in European Stock Option Pricing with Proportional Transaction Costs

We present a novel penalty approach to the Hamilton-Jacobi-Bellman (HJB) equation arising from the valuation of European options with proportional transaction costs. We first approximate the HJB equation by a quasilinear 2nd-order partial differential equation containing two linear penalty terms with penalty parameters λ ₁ and λ ₂ respectively. Then, we show that there exists a unique viscosity solution to the penalized equation. Finally, we prove that, when both λ ₁ and λ ₂ approach infinity, the viscosity solution to the penalized equation converges to that of the corresponding original HJB equation.     

Haldane–Wu Statistic and Rogers Dilogarithm

The Haldane–Wu exclusion statistic is considered from the generalized extensive statistics point of view and certain related mathematical aspects are investigated. A series representation for the corresponding generating function is obtained. Equivalence of two formulas for the central charge derived for the Haldane–Wu statistic via the thermodynamic Bethe ansatz is established. As a corollary, a series representation with a free parameter for the Rogers dilogarithm is found. It is shown that the generating function, entropy, and central charge for the Gentile statistic majorize those for the Haldane–Wu statistic (under an appropriate choice of parameters). This fact is applied in derivation of a dilogarithm inequality. Bibliography: 14 titles.     

Survival analysis in Johnson–Mehl Tessellation

The crystallization process is represented here by a generalized Boolean model, whose parameters are usually unknown. A better understanding of the model may be obtained if we estimate the corresponding parameters. In this paper, we provide non-parametric estimators for the parameters of the model. Among them, the degree of crystallinity at time t is the probability that an arbitrary point in the space has been captured by a crystal before time t. We estimate it following the Kaplan–Meier approach extended to the context of a Johnson–Mehl incomplete tessellation. Three estimators are defined, according to the kind of data we dispose. The results are also illustrated by simulations. We also provide estimators for the parameters describing geometrical aspects of the phenomenon.