Multiparameter spectral theory in Hilbert space

The object of this paper is to present a unified approach to multiparameter spectral theory of linear operators in Hilbert space. The theory is applicable to both bounded and unbounded operators and has application in the study of multiparameter spectral problems of ordinary differential operators. The main results include a Parseval equality and an eigenfunction expansion theorem.     

To solving spectral problems for multiparameter polynomial matrices

An approach to solving spectral problems for multiparameter polynomial matrices based on passing to accompanying pencils of matrices is described. Also reduction of spectral problems for multiparameter pencils of complex matrices to the corresponding real problems is considered. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 323, 2006, pp. 212–231.     

Additive comparisons of stopping values and supremum values for finite stage multiparameter stochastic processes

This paper concerns the optimal stopping problem for discrete time multiparameter stochastic processes with the index set Nd. In the classical optimal stopping problems, the comparisons between the expected reward of a player with complete foresight and the expected reward of a player using nonanticipating stop rules, known as prophet inequalities, have been studied by many authors. Ratio comparisons between these values in the case of multiparameter optimal stopping problems are studied by Krengel and Sucheston (1981) [9] and Tanaka (2007, 2006) [14] and [15]. In this paper an additive comparison in the case of finite stage multiparameter optimal stopping problems is given.     

An approach to solving multiparameter algebraic problems

An approach to solving the following multiparameter algebraic problems is suggested: (1) spectral problems for singular matrices polynomially dependent on q≥2 spectral parameters, namely: the separation of the regular and singular parts of the spectrum, the computation of the discrete spectrum, and the construction of a basis that is free of a finite regular spectrum of the null-space of polynomial solutions of a multiparameter polynomial matrix; (2) the execution of certain operations over scalar and matrix multiparameter polynomials, including the computation of the GCD of a sequence of polynomials, the division of polynomials by their common divisor, and the computation of relative factorizations of polynomials; (3) the solution of systems of linear algebraic equations with multiparameter polynomial matrices and the construction of inverse and pseudoinverse matrices. This approach is based on the so-called ΔW-q factorizations of polynomial q-parameter matrices and extends the method for solving problems for one- and two-parameter polynomial matrices considered in [1–3] to an arbitrary q≥2. Bibliography: 12 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 229, 1995, pp. 191–246. Translated by V. N. Kublanovskaya.     

Root Vectors for Geometrically Simple Multiparameter Eigenvalues

A class of multiparameter eigenvalue problems involving (generally) non self-adjoint and unbounded operators is studied. Bases for lower order root subspaces, at geometrically simple eigenvalues of Fredholm type of arbitrary finite index, are computed in terms of the underlying multiparameter system.     

Second Root Vectors for Multiparameter Eigenvalue Problems of Fredholm Type

A class of multiparameter eigenvalue problems involving (generally) non self-adjoint and unbounded operators is studied. A basis for the second root subspace, at eigenvalues of Fredholm type, is computed in terms of the underlying multiparameter system. A self-adjoint version of this result is given under a weak definiteness condition, and Sturm-Liouville and finite-dimensional examples are considered.

     

Mathematical modeling of heat conduction processes for structural elements of nuclear power plants by the method of <Emphasis Type="Italic">R</Emphasis>-functions

Mathematical models of heat exchange in the motion of an incompressible viscous liquid through channels with a helical type of symmetry, including polyzonal finning of fuel elements, in curvilinear nonorthogonal coordinates are proposed. For a laminar flow in the domain of thermal stabilization, we reduce the three-dimensional problem to the two-dimensional problem and investigate the influence of a twist parameter on the distribution of a temperature field. Numerical solutions of model multiparameter problems of calculation of stationary temperature fields by the R-function method (RFM) are considered, which enables us to choose appropriate structural means of the RFM for the subsequent solution of real problems.     

Discretization of multiparameter eigenvalue problems

Summary Although multiparameter eigenvalue problems, as for example Mathieu's differential equation, have been known for a long time, so far no work has been done on the numerical treatment of these problems. So in this paper we extend the spectral theory for one parameter (cf. [7, II, VII]) to multiparameter eigenvalue problmes, formulate in the framework of discrete approximation a convergent numerical treatment, establish algebraic bifurcation equations for the intersection points of the eigenvalue curves and illustrate this with some numerical examples.     

Newton’s method as a tool for finding the eigenvalues of certain two-parameter (multiparameter) spectral problems

An iterative algorithm is examined for finding the eigenvalues of the two-parameter (multiparameter) algebraic eigenvalue problem. This algorithm uses Newton’s method and an efficient numerical procedure for differentiating determinants. Some numerical examples are given.     

Some properties of multiparameter random fields

We study some properties of multiparameter random fields, namely, the problems of absolute continuity of measures and averaging in the multiparameter case. For a special stochastic system, we present inequalities of large deviations.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 12, pp. 1609–1621, December, 1995.     

Estimating common vector parameters in interlaboratory studies

The primary goal of this work is to extend two methods of random effects models to multiparameter situation. These methods comprise the DerSimonian-Laird estimator, stemming from meta-analysis, and the Mandel-Paule algorithm widely used in interlaboratory studies. The maximum likelihood estimators are also discussed. Two methods of assessing the uncertainty of these estimators are given.     

On Tzitzéica Vortex Streets and Their Reciprocals in Subsonic Gas Dynamics

Hydrodynamic vortex street solutions are generated based on an elliptic Tzitzéica model. A multiparameter class of reciprocal transformations is coupled with the action of a Bäcklund transformation to construct compressible vortex solutions of breather type valid in subsonic flow régimes of a generalized Kármán–Tsien gas.     

Variational approach to the solution of linear multiparameter eigenvalue problems

We associate a multiparameter spectral problem in a real Euclidean space with a variational problem of finding a minimum of a certain functional. We establish the equivalence of the spectralproblem and the variational problem. On the basis of the gradient procedure, we propose a numerical algorithm for the determination of its eigenvalues and eigenvectors. The local convergence of the algorithm is proved.     

The resultant approach to computing vector characteristics of multiparameter polynomial matrices

Known types of resultant matrices corresponding to one-parameter matrix polynomials are generalized to the multiparameter case. Based on the resultant approach suggested, methods for solving the following problems for multiparameter polynomial matrices are developed: computing a basis of the matrix range, computing a minimal basis of the right null-space, and constructing the Jordan chains and semilattices of vectors associated with a multiple spectrum point. In solving these problems, the original polynomial matrix is not transformed. Methods for solving other parametric problems of algebra can be developed on the basis of the method for computing a minimal basis of the null-space of a polynomial matrix. Issues concerning the optimality of computing the null-spaces of sparse resultant matrices and numerical precision are not considered. Bibliography: 19 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 323, 2005, pp. 182–214.     

含两组状态变量的等变分歧问题在左右等价群下的开折

基于奇点理论中光滑映射芽的左右等价关系, 讨论多参数等变分歧问题关于左右等价的开折．将这种等变分歧问题的状态变量分为两组,其中属于同一组的诸状态变量可以独立地变化,而属于另一组的诸状态变量在变化过程中依赖于前一组中的诸状态变量．应用光滑映射芽开折理论中的相关方法和技巧,得到了一个含两组状态变量的多参数等变分歧问题的开折是通用开折的充要条件．     

Regular multiparameter eigenvalue problems with several parameters in the boundary conditions

This paper extends the work of J. Walter on regular eigenvalue problems with eigenvalue parameter in the boundary condition to the multiparameter setting. The main results include several completeness and expansion theorems.     

A particle swarm optimization-based hybrid algorithm for minimum concave cost network flow problems

Traditionally, minimum cost transshipment problems have been simplified as linear cost problems, which are not practical in real applications. Some advanced local search algorithms have been developed to solve concave cost bipartite network problems. These have been found to be more effective than the traditional linear approximation methods and local search methods. Recently, a genetic algorithm and an ant colony system algorithm were employed to develop two global search algorithms for solving concave cost transshipment problems. These two global search algorithms were found to be more effective than the advanced local search algorithms for solving concave cost transshipment problems. Although the particle swarm optimization algorithm has been used to obtain good results in many applications, to the best of our knowledge, it has not yet been applied in minimum concave cost network flow problems. Thus, in this study, we employ an arc-based particle swarm optimization algorithm, coupled with some genetic algorithm and threshold accepting method techniques, as well as concave cost network heuristics, to develop a hybrid global search algorithm for efficiently solving minimum cost network flow problems with concave arc costs. The proposed algorithm is evaluated by solving several randomly generated network flow problems. The results indicate that the proposed algorithm is more effective than several other recently designed methods, such as local search algorithms, genetic algorithms and ant colony system algorithms, for solving minimum cost network flow problems with concave arc costs.     

带有多个分歧参数的等变分歧问题的万有开折

对于含一个分歧参数的分歧问题,已有万有开折定理阻［１,２］．本文考虑带有多个分歧参数的等变分歧问题,并且允许它的状态空间与靶空间可以不同,给出了等变万有开折定理的更一般的形式,［１,２］中相关结果其特殊情形．     

Application of an optimization problem in Max-Plus algebra to scheduling problems

The problem tackled in this paper deals with products of a finite number of triangular matrices in Max-Plus algebra, and more precisely with an optimization problem related to the product order. We propose a polynomial time optimization algorithm for 2×2 matrices products. We show that the problem under consideration generalizes numerous scheduling problems, like single machine problems or two-machine flow shop problems. Then, we show that for 3×3 matrices, the problem is NP-hard and we propose a branch-and-bound algorithm, lower bounds and upper bounds to solve it. We show that an important number of results in the literature can be obtained by solving the presented problem, which is a generalization of single machine problems, two- and three-machine flow shop scheduling problems. The branch-and-bound algorithm is tested in the general case and for a particular case and some computational experiments are presented and discussed.     

Koszul cohomology for finite families of comodule maps and applications

The Koszul cohomology for finite families of comodule endomorphisms is used to study finite families of polynomials with operator coefficients. In particular, we consider multiparameter eigenvalue problems.