Nonlinear matrix algebra and engineering applications Part 2: Polynomial form matrices

A matrix vector formalism is developed for systematizing the manipulation of sets of nonlinear algebraic equations. In this formalism all manipulations are performed by multiplication with specially constructed transformation matrices. For many important classes of nonlinearities, algorithms based on this formalism are presented for rearranging a set of equations so that their solution may be obtained by numerically searching along a single variable. Theory developed proves that all solutions are obtained.     

Effective matrix formalism for singularity analysis of differential equations and new intergrable system in nonlinear elasticity

We introduce a simple matrix formalism for Taylor series and generalized Laurent series that can be used for implementing the Taylor method for nonlinear ODEs and singularity analysis of differential equations. Advantages of this approach over conventional techniques are shown on model examples. Surprisingly, the same formalism can be used for proving C-integrability of a 3D model in nonlinear elasticity. An alternative proof is obtained by using similarity between the model in nonlinear elasticity and the classic Pohlmeier-Lund-Regge model from high energy physics. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Quantum inverse scattering method and algebraized matrix Bethe-Ansatz

Within the framework of the quantum inverse scattering method, an algebraic formalism is proposed for finding the eigenvectors and eigenvalues of the trace of the monodromy matrices of systems with internal degrees of freedom, i.e., a matrix Bethe-Ansatz. The results obtained are a generalization of the Gaudin-Yang method for multicomponent systems. The paper gives applications of the formalism developed, in particular, to the theory of exactly solvable models of quantum field theory with asymptotic freedom in two-dimensional space-time.     

The Hamilton formalism with fractional derivatives

Recently the traditional calculus of variations has been extended to be applicable for systems containing fractional derivatives. In this paper the passage from the Lagrangian containing fractional derivatives to the Hamiltonian is achieved. The Hamilton's equations of motion are obtained in a similar manner to the usual mechanics. In addition, the classical fields with fractional derivatives are investigated using Hamiltonian formalism. Two discrete problems and one continuous are considered to demonstrate the application of the formalism, the results are obtained to be in exact agreement with Agrawal's formalism.     

On rigged immersions

A self-contained account is given in an efficient formalism of rigged immersions of one manifold-with-connection in another, leading to the analogues of the Gauss, Codazzi and Ricci equations discovered by Schouten. The equations expressing their interdependence are then derived and it is shown that in general one of the two sets of “Codazzi” equations is a consequence of the other set and the Gauss and Ricci equations. The formalism is specialised to the Riemannian case, where it is shown that, for large codimension (specific limits being given), all butn components of the Codazzi equations are determined by the other equations. A local theorem on the existence of rigged immersions is proved.     

Analyzing games by Boolean matrix iteration

Finite win-draw-lose games with perfect information are studied, using a Boolean formulation with the intention of computational realization. The interdependence of the sets of winning strategies is expressed by means of Boolean matrix equations. Their solutions which describe the winning positions can be obtained by matrix iteration. In the case of last-player-winning games this method shows the existence of two kernels of a bipartite graph which are distinguished in the sense that they bound all other possible kernels. For some chess endings with three and four men all positions are completely analyzed.     

Variational formulations of differential equations and asymmetric fractional embedding

Variational formulations for classical dissipative equations, namely friction and diffusion equations, are given by means of fractional derivatives. In this way, the solutions of those equations are exactly the extremal of some fractional Lagrangian actions. The formalism used is a generalization of the fractional embedding developed by Cresson [Fractional embedding of differential operators and Lagrangian systems, J. Math. Phys. 48 (2007) 033504], where the functional space has been split in two in order to take into account the asymmetry between left and right fractional derivatives. Moreover, this asymmetric fractional embedding is compatible with the least action principle and respects the physical causality principle.     

A note on the iterative solutions of general coupled matrix equation

Recently, Ding and Chen [F. Ding, T. Chen, On iterative solutions of general coupled matrix equations, SIAM J. Control Optim. 44 (2006) 2269-2284] developed a gradient-based iterative method for solving a class of coupled Sylvester matrix equations. The basic idea is to regard the unknown matrices to be solved as parameters of a system to be identified, so that the iterative solutions are obtained by applying hierarchical identification principle. In this note, by considering the coupled Sylvester matrix equation as a linear operator equation we give a natural way to derive this algorithm. We also propose some faster algorithms and present some numerical results.     

Separating a unital divisor from a rectangular polynomial matrix

It is required to separate a unital divisor with a preassigned characteristic polynomial from a rectangular polynomial matrix over a field. Necessary and sufficient conditions of existence, under certain restrictions, are obtained for such a divisor, as well as a method for constructing it. By the approach used in this paper it is possible to completely solve this problem for rectangular polynomial matrices, all of whose elementary divisiors are pair-wise relatively prime. The results obtained are illustrated by solving matrix equations.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 8, pp. 1089–1094, August, 1990.     

Chebyshev rational matrix approximation with a priori error bounds for linear and Riccati matrix equations

This paper deals with initial value problems for Lipschitz continuous coefficient matrix Riccati equations. Using Chebyshev polynomial matrix approximations the coefficients of the Riccati equation are approximated by matrix polynomials in a constructive way. Then using the Fröbenius method developed in [1], given an admissible error ϵ > 0 and the previously guaranteed existence domain, a rational matrix polynomial approximation is constructed so that the error is less than ϵ in all the existence domain. The approach is also considered for the construction of matrix polynomial approximations of nonhomogeneous linear differential systems avoiding the integration of the transition matrix of the associated homogeneous problem.     

Nudelman interpolation and the band method

In previous work the authors developed a new addition of the band method based on a Grassmannian approach for solving a completion/extension problem in a general, abstract framework. This addition allows one to obtain a linear fractional representation of all solutions of the abstract completion problem from special extensions which are not necessarily band extensions (for the positive case) or triangular extensions (for the contractive case). In this work we extend this framework to a somewhat more general setting and show how one can obtain formulas for the required special extensions from solutions of a system of linear equations. As an application we show how the formalism can be applied to the bitangential Nevanlinna-Pick interpolation problem, a case which, up to now, was not amenable to the band method.The first author was partially supported by National Science Foundation grant DMS-9500912.     

ON HERMITE MATRIX POLYNOMIALS AND HERMITE MATRIX FUNCTIONS

In this paper properties of Hermite matrix polynomials and Hermite matrix functions are studied. The concept ot total set with respect to a matrix functional is introduced and the total property of the Hermite matrix polynomials is proved. Asymptotic behaviour of Hermite matrix polynomials is studied and the relationship of Hermite matrix functions with certain matrix differential equations is developed. A new expression of the matrix exponential for a wide class of matrices in terms of Hermite matrix polynomials is proposed.     

About fractional quantization and fractional variational principles

In this paper, a new method of finding the fractional Euler–Lagrange equations within Caputo derivative is proposed by making use of the fractional generalization of the classical Faá di Bruno formula. The fractional Euler–Lagrange and the fractional Hamilton equations are obtained within the 1 + 1 field formalism. One illustrative example is analyzed.     

Asymptotic expansions for partial solutions of the sixth Painlevé equation

A formalism for an averaging method for the Painlevé equations, in particular, the sixth equation, is developed. The problem is to describe the asymptotic behavior of the sixth Painlevé transcendental in the case where the module of the independent variable tends to infinity. The corresponding expansions contain an elliptic function (ansatz) in the principal term. The parameters of this function depend on the variable because of the modulation equation. The elliptic ansatz and the modulation equation for the sixth Painlevé equation are obtained in their explicit form. A partial solution of the modulation equation leading to a previously unknown asymptotic expansion for the partial solution of the sixth Painlevé equation is obtained.     

On perturbations of matrix pencils with real spectra, a revisit

This paper continues earlier studies by Bhatia and Li on eigenvalue perturbation theory for diagonalizable matrix pencils having real spectra. A unifying framework for creating crucial perturbation equations is developed. With the help of a recent result on generalized commutators involving unitary matrices, new and much sharper bounds are obtained.

     

Finite iterative method for solving coupled Sylvester-transpose matrix equations

This paper is concerned with solutions to the so-called coupled Sylvester-transpose matrix equations, which include the generalized Sylvester matrix equation and Lyapunov matrix equation as special cases. By extending the idea of conjugate gradient method, an iterative algorithm is constructed to solve this kind of coupled matrix equations. When the considered matrix equations are consistent, for any initial matrix group, a solution group can be obtained within finite iteration steps in the absence of roundoff errors. The least Frobenius norm solution group of the coupled Sylvester-transpose matrix equations can be derived when a suitable initial matrix group is chosen. By applying the proposed algorithm, the optimal approximation solution group to a given matrix group can be obtained by finding the least Frobenius norm solution group of new general coupled matrix equations. Finally, a numerical example is given to illustrate that the algorithm is effective.     

Homogenization of very rough interfaces for the micropolar elasticity theory

In this paper, the homogenization of a very rough two-dimensional interface separating two dissimilar isotropic micropolar elastic solids is investigated. The interface is assumed to oscillate between two parallel straight lines. The main aim is to derive homogenized equations in explicit form. These equations are obtained by the homogenization method along with the matrix formalism of the theory of micropolar elasticity. Since obtained homogenized equations are totally explicit, they are a powerful tool for solving various practical problems. As an example, the reflection and transmission of a longitudinal displacement plane wave at a very rough interface of tooth-comb type is investigated. The closed-form formulas for the reflection and transmission coefficients have been derived. Based on these formulas, some numerical examples are carried out to show the dependence of the reflection and transmission coefficients on the incident angle and the geometry parameter of the interface.     

Painlevé analysis, Lie symmetries and exact solutions for (2+1)-dimensional variable coefficients Broer-Kaup equations

A (2+1) dimensional Broer-Kaup system which is obtained from the constraints of the KP equation is of importance in mathematical physics field. In this paper, the Painlevé analysis of (2+1)-variable coefficients Broer-Kaup (VCBK) equation is performed by the Weiss-Kruskal approach to check the Painlevé property. Similarity reductions of the VCBK equation to one-dimensional partial differential equations including Burger’s equation are investigated by the Lie classical method. The Lie group formalism is applied again on one of the investigated partial differential equation to derive symmetries, and the ordinary differential equations deduced from the optimal system of subalgebras are further studied and some exact solutions are obtained.     

Perturbation theory for the periodic Anderson model: II. Superconducting state

A new diagram technique, which has been developed for strongly correlated electron systems, is used to study the periodic Anderson model in the superconducting state. To treat both normal and anomalous Green's functions on an equal footing, we introduce an additional charge quantum number that distinguishes creation and annihilation operators. We derive the Dyson equations for the Green's functions of band and localized electrons in the presence of superconductivity. The equations obtained admit both singlet-type and triplet-type superconductivity. For singlet-type superconductivity, we establish the correspondence between these equations and the spinor Gor'kov-Nambu formalism. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 116, No. 3, pp. 456–473, September, 1998.