Relationships for moments of order statistics from the right-truncated generalized half logistic distribution

In this paper, we establish several recurrence relations satisfied by the single and the product moments for order statistics from the right-truncated generalized half logistic distribution. These relationships may be used in a simple recursive manner in order to compute the single and the product moments of all order statistics for all sample sizes and for any choice of the truncation parameter P. These generalize the corresponding results for the generalized half logistic distribution derived recently by Balakrishnan and Sandhu (1995, J. Statist. Comput. Simulation, 52, 385–398).Earlier went by the name R. A. Sandhu.     

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.     

An Exact Procedure for the Resource-Constrained Weighted Earliness–Tardiness Project Scheduling Problem

In this paper we study the resource-constrained project scheduling problem with weighted earliness–tardinesss penalty costs. Project activities are assumed to have a known deterministic due date, a unit earliness as well as a unit tardiness penalty cost and constant renewable resource requirements. The objective is to schedule the activities in order to minimize the total weighted earliness–tardinesss penalty cost of the project subject to the finish–start precedence constraints and the constant renewable resource availability constraints. With these features the problem becomes highly attractive in just-in-time environments.We introduce a depth-first branch-and-bound algorithm which makes use of extra precedence relations to resolve resource conflicts and relies on a fast recursive search algorithm for the unconstrained weighted earliness–tardinesss problem to compute lower bounds. The procedure has been coded in Visual C++, version 4.0 under Windows NT. Both the recursive search algorithm and the branch-and-bound procedure have been validated on a randomly generated problem set.     

A linear filtering problem in complete correlated actions

In the last paper, the geometry of the Sz.-Nagy-Foia model for contraction operators on Hilbert spaces was used to advantage in several problems of multivariate analysis. The lifting of intertwining operators, one of the basic results from the Sz.-Nagy-Foia theory, is now recognized as the most adequate operatorial form of the deep classical results of the extrapolation theory. The labeling of the exact intertwining dilations given by [1]Acta Sci. Math. (Szeged) 40 9–32] and the recursive methods used there open a broad perspective for using the Sz.-Nagy-Foia model in multivariate filtering theory. In this paper, using the notion of correlated action (see [5 and 6] Rev. Roumaine Math. Pures Appl. 23, No. 9 1393–1423]) as a time domain, a linear filtering problem is formulated and its solution in terms of the coefficients of the analytic function which factorizes the spectral distribution of the known data and the coefficients of an analytic function which describes the cross correlations is given. In some special cases it is shown that the filter coefficients can be determined using recursive methods from the intertwining dilation theory, of the autocorrelation function of the known data and an intertwining operator, interpreted as the initial estimator given by the prior statistics.     

A higher-order moment method of the lattice Boltzmann model for the conservation law equation

In this paper, we proposed a higher-order moment method in the lattice Boltzmann model for the conservation law equation. In contrast to the lattice Bhatnagar–Gross–Krook (BGK) model, the higher-order moment method has a wide flexibility to select equilibrium distribution function. This method is based on so-called a series of partial differential equations obtained by using multi-scale technique and Chapman–Enskog expansion. According to Hirt’s heuristic stability theory, the stability of the scheme can be controlled by modulating some special moments to design the third-order dispersion term and the fourth-order dissipation term. As results, the conservation law equation is recovered with higher-order truncation error. The numerical examples show the higher-order moment method can be used to raise the accuracy of the truncation error of the lattice Boltzmann scheme for the conservation law equation.     

Extension of partial recursive functions and functions with a recursive graph

It is proved that every degree of complexity of mass problems, containing the decision problem of a recursive enumerable set, contains also the problem of extension of a partial recursive function, the graph of which is recursive. Some properties of functions with a recursive graph are considered.Translated from Matematicheskie Zametki, Vol. 5, No. 2, pp. 261–267, February, 1969.The author is deeply indebted to V. A. Uspenskii for discussion of the results.     

First-Quantized Fermions in Compact Dimensions

We discuss a path integral representation for fermionic particles and strings in the spirit of V. Ya. Fainberg and the author (Nucl. Phys. B, 306, 659–676, (1998); Phys. Lett. B, 211, 81–85, (1988)). We concentrate on the problems arising when some target-space dimensions are compact. We consider the partition function for a fermionic particle at a finite temperature or in compact time in detail as an example. We demonstrate that a self-consistent definition of the path integral generally requires introducing nonvanishing background Wilson loops and that modulo some common problems for real fermions in the Grassmannian formulation, these loops can be interpreted as condensates of world-line fermions. Properties of the corresponding string-theory path integrals are also discussed.     

Recurrence relations among moments of order statistics from two related sets of independent and non-identically distributed random variables

Some recurrence relations among moments of order statistics from two related sets of variables are quite well-known in the i.i.d. case and are due to Govindarajulu (1963a, Technometrics, 5, 514–518 and 1966, J. Amer. Statist. Assoc., 61, 248–258). In this paper, we generalize these results to the case when the order statistics arise from two related sets of independent and non-identically distributed random variables. These relations can be employed to simplify the evaluation of the moments of order statistics in an outlier model for symmetrically distributed random variables.     

Solution of the 2D ising model on a triangular lattice by the method of auxiliaryq-deformed Grassmann fields

A representation in the form of a functional integral is obtained for the partition function of the inhomogeneous 2D Ising model on a triangular lattice where the coupling parameters are arbitrary functions of coordinates. The method for transforming the partition function into an integral uses an auxiliary six-component Grassmann field in which the Grassmann fields corresponding to one of the components commute with the others. Thus, one pair of components realizes a representation of the q-deformed group SL_q(2, R) with q=–1 and the other two pairs correspond to the usual Grassmann spinors (q=1). An explicit expression in terms of the modified Pfaffian is found for the Gaussian integral over these fields and its relation to the ordinary Grassmann functional integral is established.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 109, No. 3, pp. 441–463, December, 1996.     

Numerical solutions of the first boundary-value problem for the telegraph equation on open contours

The first plane initial—boundary-value problem for the telegraph equation is reduced by a Chebyshev—Laguerre temporal integral transform to a sequence of stationary boundary-value problems for elliptic equations. Their solutions are sought in integral form. This leads to a recursive sequence of integral equations of the first kind that are solved by the collocation method with isolation of singularities. The sought function is determined by the inverse transform.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 72, pp. 57–62, 1990.     

On the linearization of systems of conservation laws for fluids at a material contact discontinuity

In order to investigate the linearized stability or instability of compressible flows, as it occurs for instance in Rayleigh–Taylor or Kelvin–Helmholtz instabilities, we consider the linearization at a material discontinuity of a flow modeled by a multidimensional nonlinear hyperbolic system of conservation laws. Restricting ourselves to the plane-symmetric case, the basic solution is thus a one-dimensional contact discontinuity and the normal modes of pertubations are solutions of the resulting linearized hyperbolic system with discontinuous nonconstant coefficients and source terms. While in Eulerian coordinates, the linearized Cauchy problem has no solution in the class of functions, we prove that for a large class of systems of conservation laws written in Lagrangian coordinates and including the Euler and the ideal M.H.D. systems, there exists a unique function solution of the problem that we construct by the method of characteristics.     

Equivalent formulations and necessary optimality conditions for the Lennard–Jones problem

The minimization of molecular potential energy functions is one of the most challenging, unsolved nonconvex global optimization problems and plays an important role in the determination of stable states of certain classes of molecular clusters and proteins. In this paper, some equivalent formulations and necessary optimality conditions for the minimization of the Lennard–Jones potential energy function are presented. A new strategy, the code partition algorithm, which is based on a bilevel optimization formulation, is proposed for searching for an extremal Lennard–Jones code. The convergence of the code partition algorithm is proved and some computational results are reported.     

A multivariate empirical characteristic function test of independence with normal marginals

This paper proposes a semi-parametric test of independence (or serial independence) between marginal vectors each of which is normally distributed but without assuming the joint normality of these marginal vectors. The test statistic is a Cramér–von Mises functional of a process defined from the empirical characteristic function. This process is defined similarly as the process of Ghoudi et al. [J. Multivariate Anal. 79 (2001) 191] built from the empirical distribution function and used to test for independence between univariate marginal variables. The test statistic can be represented as a V-statistic. It is consistent to detect any form of dependence. The weak convergence of the process is derived. The asymptotic distribution of the Cramér–von Mises functionals is approximated by the Cornish–Fisher expansion using a recursive formula for cumulants and inversion of the characteristic function with numerical evaluation of the eigenvalues. The test statistic is finally compared with Wilks statistic for testing the parametric hypothesis of independence in the one-way MANOVA model with random effects.     

Completeness of the Description of an Equilibrium Canonical Ensemble by a Two-Particle Partition Function

We show that in the equilibrium classical canonical ensemble of particles with pair interaction, the full Gibbs partition function can be uniquely expressed in terms of the two-particle partition function. This implies that for a fixed number N of particles in the equilibrium system and a fixed volume V and temperature T, the two-particle partition function fully describes the Gibbs partition as well as the N-particle system in question. The Gibbs partition can be represented as a power series in the two-particle partition function. As an example, we give the linear term of this expansion. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 1, pp. 123–132, October, 2005.     

On the Generalized Rogers–Ramanujan Continued Fraction

On page 26 in his lost notebook, Ramanujan states an asymptotic formula for the generalized Rogers–Ramanujan continued fraction. This formula is proved and made slightly more precise. A second primary goal is to prove another continued fraction representation for the Rogers–Ramanujan continued fraction conjectured by R. Blecksmith and J. Brillhart. Two further entries in the lost notebook are examined. One of them is an identity bearing a superficial resemblance to the generating function for the generalized Rogers–Ramanujan continued fraction. Thus, our third main goal is to establish, with the help of an idea of F. Franklin, a partition bijection to prove this identity.     

Aftermath of the Chernobyl Catastrophe from the Point of View of the Security Concept

The paper deals with uncertainty relations for time and energy operators, and the aftermath of the Chernobyl catastrophe is considered as an example. The mathematical approach developed by Holevo is analyzed, which allows us to assign the corresponding observables to non-self-adjoint operators and to establish uncertainty relations for nonstandard canonical conjugate pairs.

Relations for calculating the minimal time interval in which the energy jump can be discovered are given. Based on the intensity parameter introduced by the author, which is related to a special statistics called Gentile statistics and to the polylogarithm function, properties of stable chemical elements, such as time fluctuations and the jump of specific energy in the transition from the Bose—Einstein distribution to the Fermi—Dirac distribution, are mathematically described with regard to experimental data. The obtained data are arranged in a table for 255 stable chemical elements.

The mathematical approach developed by the author of the present paper allows one to describe the “antipode” (in a certain sense) of the standard thermodynamics, i.e., the thermodynamics of nuclear matter. This field of nuclear physics is very important for the study of properties of radioactive elements and, accordingly, from the standpoint of ensuring nuclear safety.

     

q-Deformed grassmann fields and the two-dimensional Ising model

We construct an exact representation of the Ising partition function in the form of the SL_q(2, R)-invariant functional integral for the lattice-free q-fermion field theory (q=–1). It is shown that the q-fermionization allows one to rewrite the partition function of the eight-vertex model in an external field through a functional integral with four-fermion interaction. To construct these representations, we define a lattice (l, q, s)-deformed Grassmann bispinor field and extend the Berezin integration rules to this field. At q=–1, l=s=1, we obtain the lattice q-fermion field which allows us to fermionize the two-dimensional Ising model. We show that the Gaussian integral over (q, s)-Grassmann variables is expressed through the (q, s)-deformed Pfaffian which is equal to square root of the determinant of some matrix at q=±1, s=±1.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 103, No. 3, pp. 388–412. June, 1995.     

Phase coexistence in Ising, Potts and percolation models

We study phase coexistence (separation) phenomena in Ising, Potts and random cluster models in dimensions d3 below the critical temperature. The simultaneous occurrence of several phases is typical for systems with appropriately arranged (mixed) boundary conditions or for systems satisfying certain physically natural constraints (canonical ensembles). The various phases emerging in these models define a partition, called the empirical phase partition, of the space. Our main results are large deviations principles for (the shape of) the empirical phase partition. More specifically, we establish a general large deviation principle for the partition induced by large (macroscopic) clusters in the Fortuin–Kasteleyn model and transfer it to the Ising–Potts model where we obtain a large deviation principle for the empirical phase partition induced by the various phases. The rate function turns out to be the total surface free energy (associated with the surface tension of the model and with boundary conditions) which can be naturally assigned to each reasonable partition. These LDP-s imply a weak law of large numbers: asymptotically, the law of the phase partition is determined by an appropriate variational problem. More precisely, the empirical phase partition will be close to some partition which is compatible with the constraints imposed on the system and which minimizes the total surface free energy. A general compactness argument guarantees the existence of at least one such minimizing partition. Our results are valid for temperatures T below a limit of slab-thresholds conjectured to agree with the critical point T_c. Moreover, T should be such that there exists only one translation invariant infinite volume state in the corresponding Fortuin–Kasteleyn model; a property which can fail for at most countably many values and which is conjectured to be true for every T≠T_c.     

Decomposition of Green polynomials of type A and Springer modules for hooks and rectangles

This paper deals with graded representations of the symmetric group on the cohomology ring of flags fixed by a unipotent matrix. We consider a combinatorial property, called the “coincidence of dimension” of the graded representations, and give an interpretation in terms of representation theory of the symmetric group in the case where the corresponding partition of the unipotent matrix is a hook or a rectangle. The interpretation is equivalent to a recursive formula of Green polynomials at roots of unity.     

Matrix Models: Geometry of Moduli Spaces and Exact Solutions

We study the connection between characteristics of moduli spaces of Riemann surfaces with marked points and matrix models. The Kontsevich matrix model describes intersection indices on continuous moduli spaces, and the Kontsevich–Penner matrix model describes intersection indices on discretized moduli spaces. Analyzing the constraint algebras satisfied by various generalized Kontsevich matrix models, we derive time transformations that establish exact relations between different models appearing in mathematical physics. We solve the Hermitian one-matrix model using the moment technique in the genus expansion and construct a recursive procedure for solving this model in the double scaling limit.