block jacobi matrices and zeros of multivariate orthogonal polynomials a commuting family of symmetric matrices are called the block jacobi matrices if they are block tridiagonal they are related to multivariate orthogonal polynomials we study their eigenvalues and joint eigenvectors the joint eigenvalues of the truncated block jacobi matrices correspond to the common zeros of the multivariate orthogonal polynomials

relating h sub 2 and h sub spl infin norm bounds for sampled data systems we relate the h sub spl infin and h sub 2 norms for multi input multi output sampled data feedback control systems where a continuous time plant is controlled by a digital compensator with hold and sampler upper bounds on both h sub 2 and h sub spl infin norms are obtained based on fundamental relations derived by two different approaches namely the hybrid state space approach and the fast sampling and lifting approach

a new family of extended gauss quadratures with an interior interval constraint starting from two sequences ga c n and gd b n of ordinary gauss quadrature formulae with an orthogonality measure d on the open intervals a c and d b respectively we construct a new sequence ga b e n of extended gaussian quadrature formulae for d on a b which is based on some preassigned points the nodes of ga c n gd b n and the e n zeros contained in c d of a nonclassical orthogonal polynomial on a b with respect to a linear functional the principal result gives explicit formulae relating these polynomials and shows how their recurrence coefficients in the three term recurrence formulae are related thus a new class of gaussian quadratures having some nodes contained in a given interior interval can be computed directly by standard software for ordinary gauss quadrature formulae

multiple travelling wave solutions of nonlinear evolution equations using a unified algebraic method a new direct and unified algebraic method for constructing multiple travelling wave solutions of general nonlinear evolution equations is presented and implemented in a computer algebraic system compared with most of the existing tanh methods the jacobi elliptic function method or other sophisticated methods the proposed method not only gives new and more general solutions but also provides a guideline to classify the various types of the travelling wave solutions according to the values of some parameters the solutions obtained in this paper include a kink shaped and bell shaped soliton solutions b rational solutions c triangular periodic solutions and d jacobi and weierstrass doubly periodic wave solutions among them the jacobi elliptic periodic wave solutions exactly degenerate to the soliton solutions at a certain limit condition the efficiency of the method can be demonstrated on a large variety of nonlinear evolution equations such as those considered in this paper kdv mkdv ito s fifth mkdv hirota nizhnik novikov veselov broer kaup generalized coupled hirota satsuma coupled schrodinger kdv 2 1 dimensional dispersive long wave 2 1 dimensional davey stewartson equations in addition as an illustrative sample the properties of the soliton solutions and jacobi doubly periodic solutions for the hirota equation are shown by some figures the links among our proposed method the tanh method extended tanh method and the jacobi elliptic function method are clarified generally

commuting family of block jacobi matrices abstract the block jacobi matrices considered in this paper are a family of block tridiagonal matrices which are natural extensions of a singular jacobi matrix in the sense that they are associated with orthogonal polynomials in several variables we present the basic properties of these matrices

zero sets of bivariate hermite polynomials abstract we establish various properties for the zero sets of three families of bivariate hermite polynomials special emphasis is given to those bivariate orthogonal polynomials introduced by hermite by means of a rodrigues type formula related to a general positive definite quadratic form for this family we prove that the zero set of the polynomial of total degree n m consists of exactly n m disjoint branches and possesses n m asymptotes a natural extension of the notion of interlacing is introduced and it is proved that the zero sets of the family under discussion obey this property the results show that the properties of the zero sets considered as affine algebraic curves in r 2 are completely different for the three families analyzed

comparison of peak and rms gains for discrete time systems abstract a convolution system can have a frequency response which is small for all frequencies yet still greatly amplify the peaks of signals passing though it for finite dimensional systems however we establish the simple bound h 1 2 n 1 h h h where h 1 is the peak gain of the system h h is the maximum frequency response of the system and n its dimension the same result for continuous time systems is due to gohberg and doyle and it is mentioned in 6 the bound implies that h optimal controllers which minimize the maximum of some disturbance to error transfer function cannot have very large peak gains from the disturbance to error

weighted hankel norm approximation calculation of bounds abstract frequency domain bounds are computed for the approximation error involved in the optimum hankel norm approximation procedure the bounds differ from those arising in the unweighted optimal hankel norm approximation procedure by a multiplicative factor depending solely on the weighting function this factor becomes unity when the weighting function is constant

solitary wave and periodic wave solutions for the thermally forced gravity waves in atmosphere by introducing a new transformation a new direct and unified algebraic method for constructing multiple travelling wave solutions of general nonlinear evolution equations is presented and implemented in a computer algebraic system which extends fan s direct algebraic method to the case when r 4 the solutions of a first order nonlinear ordinary differential equation with a higher degree nonlinear term and fan s direct algebraic method of obtaining exact solutions to nonlinear partial differential equations are applied to the combined kdv mkdv gkdv equation which is derived from a simple incompressible non hydrostatic boussinesq equation with the influence of thermal forcing and is applied to investigate internal gravity waves in the atmosphere as a result by taking advantage of the new first order nonlinear ordinary differential equation with a fifth degree nonlinear term and an eighth degree nonlinear term periodic wave solutions associated with the jacobin elliptic function and the bell and kink profile solitary wave solutions are obtained under the effect of thermal forcing most importantly the mechanism of propagation and generation of the periodic waves and the solitary waves is analysed in detail according to the values of the heating parameter which show that the effect of heating in atmosphere helps to excite westerly or easterly propagating periodic internal gravity waves and internal solitary waves in atmosphere which are affected by the local excitation structures in atmosphere in addition as an illustrative sample the properties of the solitary wave solution and jacobin periodic solution are shown by some figures under the consideration of heating interaction

commuting extensions and cubature formulae based on a novel point of view on 1 dimensional gaussian quadrature we present a new approach to d dimensional cubature formulae it is well known that the nodes of 1 dimensional gaussian quadrature can be computed as eigenvalues of the so called jacobi matrix the d dimensional analog is that cubature nodes can be obtained from the eigenvalues of certain mutually commuting matrices these are obtained by extending adding rows and columns to certain noncommuting matrices a1 ad related to the coordinate operators x1 xd in rd we prove a correspondence between cubature formulae and commuting extensions of a1 ad satisfying a compatibility condition which in appropriate coordinates constrains certain blocks in the extended matrices to be zero thus the problem of finding cubature formulae can be transformed to the problem of computing and then simultaneously diagonalizing commuting extensions we give a general discussion of existence and of the expected size of commuting extensions and briefly describe our attempts at computing them

new relations between norms of system transfer functions it is well known that the h2 norm and h norm of a transfer function can differ arbitrarily since both norms reflect fundamentally different properties however if the pole structure of the transfer function is known it is possible to bound the h norm from above by a constant multiple of the h2 norm it is desirable to compute this constant as tightly as possible in this article we derive a tight bound for the h norm given knowledge of the h2 norm and the poles of a transfer function we compute the bound in closed form for multiple input multiple output transfer functions in continuous and discrete time furthermore we derive a general procedure to compute the bound given a weighted l2 norm

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

cubature formulae and orthogonal polynomials abstract the connection between orthogonal polynomials and cubature formulae for the approximation of multivariate integrals has been studied for about 100 yr the article j radon published about 50 yr ago j radon zur mechanischen kubatur monatsh math 52 1948 286 300 has been very influential in this text we describe some of the results that were obtained during the search for answers to questions raised by his article

reduction of dispersionless coupled korteweg de vries equations to the euler darboux equation a quasilinear hyperbolic system of two first order equations is introduced the system is linearized by means of the hodograph transformation combined with riemann s method of characteristics in the process of linearization the main step is to explicitly express the characteristic velocities in terms of the riemann invariants the procedure is shown to be performed by quadrature only for specific combinations of the parameters in the system we then apply the method developed here to the dispersionless versions of the typical coupled korteweg de vries ckdv equations including the broer kaup ito hirota satsuma and bogoyavlenskii equations and show that these equations are transformed into the classical euler darboux equation a more general quasilinear system of equations is also considered with application to the dispersionless ckdv equations for the jaulent miodek and nutku oguz equations

relating h 2 and h bounds for finite dimensional systems for a linear time invariant system the infinity norm of the transfer function can be used as a measure of the gain of the system this notion of system gain is ideally suited to the frequency domain design techniques such as h infinity optimal control another measure of the gain of a system is the h 2 norm which is often associated with the lqg optimal control problem the only known connection between these two norms is that for discrete time transfer functions the h 2 norm is bounded by the h infinity norm it is shown in this paper that given precise or certain partial knowledge of the poles of the transfer function it is possible to obtain an upper bound of the h infinity norm as a function of the h 2 norm both in the continuous and discrete time cases it is also shown that in continuous time the h 2 norm can be bounded by a function of the h infinity norm and the bandwidth of the system

a class of bivariate orthogonal polynomials and cubature formula the existence of gaussian cubature for a given measure depends on whether the corresponding multivariate orthogonal polynomials have enough common zeros we examine a class of orthogonal polynomials of two variables generated from that of one variable

bivariate polynomial interpolation on the square at new nodal sets abstract as known the problem of choosing good nodes is a central one in polynomial interpolation while the problem is essentially solved in one dimension all good nodal sequences are asymptotically equidistributed with respect to the arc cosine metric in several variables it still represents a substantially open question in this work we consider new nodal sets for bivariate polynomial interpolation on the square first we consider fast leja points for tensor product interpolation on the other hand for interpolation in p n 2 on the square we experiment four families of points which are asymptotically equidistributed with respect to the dubiner metric which extends to higher dimension the arc cosine metric one of them nicknamed padua points gives numerically a lebesgue constant growing like log square of the degree

hodograph solutions of the dispersionless coupled kdv hierarchies critical points and the euler poisson darboux equation it is shown that the hodograph solutions of the dispersionless coupled kdv dckdv hierarchies describe critical and degenerate critical points of a scalar function which obeys the euler poisson darboux equation singular sectors of each dckdv hierarchy are found to be described by solutions of higher genus dckdv hierarchies concrete solutions exhibiting shock type singularities are presented

an analysis of polynomial chaos approximations for modeling single fluid phase flow in porous medium systems we examine a variety of polynomial chaos motivated approximations to a stochastic form of a steady state groundwater flow model we consider approaches for truncating the infinite dimensional problem and producing decoupled systems we discuss conditions under which such decoupling is possible and show that to generalize the known decoupling by numerical cubature it would be necessary to find new multivariate cubature rules finally we use the acceleration of monte carlo to compare the quality of polynomial models obtained for all approaches and find that in general the methods considered are more efficient than monte carlo for the relatively small domains considered in this work a curse of dimensionality in the series expansion of the log normal stochastic random field used to represent hydraulic conductivity provides a significant impediment to efficient approximations for large domains for all methods considered in this work other than the monte carlo method

a hierarchy of coupled korteweg de vries equations and the normalisation conditions of the hilbert riemann problem the authors discuss the normalisation conditions of the hilbert riemann problem which is of importance for the construction of the ladder of soliton solutions for a given spectral problem they provide the prescription for constructing the appropriate normalisation condition all this is applied to a new hierarchy of coupled korteweg de vries equations to find the soliton solution which displays some peculiar phenomena

recurrence formulas for multivariate orthogonal polynomials in this paper necessary and sufficient conditions are given so that multivariate orthogonal polynomials can be generated by a recurrence formula as a consequence orthogonal polynomials of total degree n in d variables that have dim i id common zeros can now be constructed recursively the result is important to the construction of gaussian cubature formulas

multivariate orthogonal polynomials quantum decomposition deficiency rank and support of measure abstract in this paper we investigate the multivariate orthogonal polynomials based on the theory of interacting fock spaces our framework is on the same stream line of the recent paper by accardi barhoumi and dhahri 1 the classical coordinate variables are decomposed into non commuting quantum operators called creation annihilation and preservation operators in the interacting fock spaces getting the commutation relations which follow from the commuting property of the coordinate variables between themselves we can develop the reconstruction theory of the measure namely the favard s theorem we then further develop some related problems including the marginal distributions and the rank theory of the jacobi operators we will see that the deficiency rank of the jacobi operator implies that the underlying measure is supported on some algebraic surface and vice versa we will provide with some examples

approximative model reconstruction of cascade systems this letter considers how to approximately reconstruct a cascade system from a given unstructured system estimate many system identification methods including subspace methods provide reliable but generally unstructured black box models the problem we consider is how to find cascade systems that are close to such black box models for this we use model matching techniques and optimal weighted hankel norm approximation to obtain accurate low order cascade systems we show that it is possible to bound the reconstruction error in terms of an error tolerance parameter and weighted hankel singular values the suggested methods are illustrated on both a numerical example and a real double tank system with experimental data

gaussian cubature and bivariate polynomial interpolation gaussian cubature is used to study bivariate polynomial interpolation based on the common zeros of quasi orthogonal polynomials

multi soliton and pfaffian solutions of a 2 1 dimensional nonlinear evolution equation via the jaulent miodek hierarchy abstract under investigation in this paper is a 2 1 dimensional nonlinear evolution equation generated by the jaulent miodek hierarchy via the hirota bilinear method multi soliton solutions of such equation are obtained soliton propagation and interactions are analyzed and both the elastic and inelastic interactions are seen with some parameters selected furthermore it is proved that the equation possesses the n soliton solutions expressed in terms of gramm type pfaffian and can be written in the form of a pfaffian identity