on the efficiency of subsumption algorithms the costs of subsumption algorithms are analyzed by an estimation of the maximal number of unification attempts worst case unification complexity made for deciding whether a clause c subsumes a clause d for this purpose the clauses c and d are characterized by the following parameters number of variables in c number of literals in c number of literals in d and maximal length of the literals the worst case unification complexity immediately yields a lower bound for the worst case time complexity first two well known algorithms chang lee stillman are investigated both algorithms are shown to have a very high worst case time complexity then a new subsumption algorithm is defined which is based on an analysis of the connection between variables and predicates in c an upper bound for the worst case unification complexity of this algorithm which is much lower than the lower bounds for the two other algorithms is derived examples in which exponential costs are reduced to polynomial costs are discussed finally the asymptotic growth of the worst case complexity for all discussed algorithms is shown in a table for several combinations of the parameters

representations of the schrodinger group and matrix orthogonal polynomials the representations of the schrodinger group in one space dimension are explicitly constructed in the basis of the harmonic oscillator states these representations are seen to involve matrix orthogonal polynomials in a discrete variable that have charlier and meixner polynomials as building blocks the underlying lie theoretic framework allows for a systematic derivation of the structural formulas recurrence relations difference equations rodrigues formula etc that these matrix orthogonal polynomials satisfy

a note on the maximization of matrix valued hankel determinants with applications in this note we consider the problem of maximizing the determinant of moment matrices of matrix measures the maximizing matrix measure can be characterized explicitly by having equal matrix valued weights at the zeros of classical one dimensional orthogonal polynomials the results generalize classical work of schoenberg indag math 62 1959 282 to the case of matrix measures as a statistical application we consider several optimal design problems in linear models which generalize the classical weighing design problems

spectral decomposition and matrix valued orthogonal polynomials the relation between the spectral decomposition of a self adjoint operator which is realizable as a higher order recurrence operator and matrix valued orthogonal polynomials is investigated a general construction of such operators from scalar valued orthogonal polynomials is presented two examples of matrix valued orthogonal poly nomials with explicit orthogonality relations and three term recurrence relation are pre sented which both can be considered as 2 2 matrix valued analogues of subfamilies of askey wilson polynomials

on polynomials orthogonal with respect to an inner product involving differences abstract in this paper we consider the inner product r x g x d x c g c where c r and is a distribution function with infinite spectrum such that has no points of increase in the interval c c 1 furthermore 0 and g are functions on r and c c 1 c let qn x be the sequence of monic orthogonal polynomials with respect to this inner product and pn x pnc x the sequences of monic standard orthogonal polynomials 0 with respect to d x and x c x c 1 d x respectively we derive an explicit representation for qn x in terms of pn x and pnc x and we present some results on the distribution of the zeros of qn x in relation to the zeros of pn x finally we treat the special case where pn x are charlier polynomials and c 0

on a class of matrix orthogonal polynomials on the real line abstract it is shown how to construct matrix orthogonal polynomials on the real line provided we have a sequence of scalar orthogonal polynomials over an algebraic harmonic curve we consider the moment matrix associated with a distribution on and related to a basis of the linear space of polynomials obtained through the polynomial which defines the curve this particular moment matrix can be regarded as a hankel block matrix and thus matrix orthogonal polynomials appear in a natural way

gaussian quadrature formulae for matrix weights we study gaussian quadrature formulae for a matrix weight we firstly show how to generate gaussian quadrature formulae such that the sum of the rank of the quadrature matrix coefficients is a given nonnegative integer l the nodes are the eigenvalues of certain perturbation of the truncated n jacobi matrix of size l or equivalently the zeros of certain matrix combination of two consecutive orthonormal polynomials and the quadrature coefficients are the coefficients in the partial fraction decomposition of the ratio between the inverse of this orthonormal matrix polynomial and the associated polynomial of the second kind we secondly prove that any gaussian quadrature formula for a matrix weight must be of this form

a survey on orthogonal matrix polynomials satisfying second order differential equations the subject of orthogonal polynomials cuts across a large piece of mathematics and its applications two notable examples are mathematical physics in the 19th and 20th centuries as well as the theory of spherical functions for symmetric spaces it is also clear that many areas of mathematics grew out of the consideration of problems like the moment problem that are intimately associated to the study of scalar valued orthogonal polynomials matrix orthogonality on the real line has been sporadically studied during the last half century since krein devoted some papers to the subject in 1949 see ams translations series 2 vol 97 providence rhode island 1971 pp 75 143 dokl akad nauk sssr 69 2 1949 125 in the last decade this study has been made more systematic with the consequence that many basic results of scalar orthogonality have been extended to the matrix case the most recent of these results is the discovery of important examples of orthogonal matrix polynomials many families of orthogonal matrix polynomials have been found that as the classical families of hermite laguerre and jacobi in the scalar case satisfy second order differential equations with coefficients independent of n the aim of this paper is to give an overview of the techniques that have led to these examples a small sample of the examples themselves and a small step in the challenging direction of finding applications of these new examples

differential coefficients of orthogonal matrix polynomials we find explicit formulas for raising and lowering first order differential operators for orthogonal matrix polynomials we derive recurrence relations for the coefficients in the raising and lowering operators some examples are given

exceptional laguerre and jacobi polynomials and the corresponding potentials through darboux crum transformations a simple derivation is presented of the four families of infinitely many shapeinvariant hamiltonians corresponding to the exceptional laguerre and jacobi polynomials the darboux crum transformations are applied to connect the well known shape invariant hamiltonians of the radial oscillator and the darboux p oschl teller potential to the shape invariant potentials of odake sasaki dutta and roy derived the two lowest members of the exceptional laguerre polynomials by this method the method is expanded to its full generality and many other ramifications including the aspects of the generalized bochner problem and the bispectral property of the exceptional orthogonal polynomials are discussed

matrix valued orthogonal polynomials on the real line some extensions of the classical theory in the work presented below the classical subject of orthogonal polynomials on the real line is discussed in the matrix setting an analogue of the determinant definition of orthogonal polynomials is presented the classical properties such as the recurrence relation the kernel polynomials and the christoffel darboux formula are discussed a function for the system of matrix valued orthogonal polynomials on the real line is presented some properties of the functions are investigated

a class of matrix orthogonal polynomials on the unit circle abstract the study of matrix orthogonal polynomials on the unit circle is connected with problems of applied mathematics such as filtering theory prediction theory and scattering theory in this paper some families of matrix orthogonal polynomials on the unit circle are given explicitly the decomposition a z z a z z h 1 a z where is the linear space of complex algebraic polynomials is the main idea for this approach

matrix measures moment spaces and favard s theorem for the interval 0 1 and 0 abstract in this paper we study the moment spaces corresponding to matrix measures on compact intervals and on the nonnegative line 0 a representation for nonnegative definite matrix polynomials is obtained which is used to characterize moment points by properties of generalized hankel matrices we also derive an explicit representation of the orthogonal polynomials with respect to a given matrix measure which generalizes the classical determinant representation of the one dimensional case moreover the coefficients in the recurrence relations can be expressed explicitly in terms of the moments of the matrix measure these results are finally used to prove a refinement of the well known favard theorem for matrix measures which characterizes the domain of the underlying measure of orthogonality by properties of the coefficients in the recurrence relationships

unified bundling and registration of brain white matter fibers magnetic resonance diffusion tensor imaging is being widely used to reconstruct brain white matter fiber tracts to characterize structural properties of the tracts reconstructed fibers are often grouped into bundles that correspond to coherent anatomic structures for further group analysis of fiber bundles it is desirable that corresponding bundles from different studies are coregistered to address these needs simultaneously a unified fiber bundling and registration ufibre framework is proposed in this work the framework is based on maximizing a posteriori bayesian probabilities using an expectation maximization algorithm given a set of segmented template bundles and a whole brain target fiber set the ufibre algorithm optimally bundles the target fibers and registers them with the template the bundling component in the ufibre algorithm simplifies fiber based registration into bundle to bundle registration and the registration component in turn guides the bundling process to find bundles consistent with the template experiments with in vivo data demonstrate that the estimated bundles have an 80 consistency with ground truth and the root mean square error between their bundle medial axes is less than one voxel the proposed algorithm is highly efficient offering potential routine use for group analysis of white matter fibers

trace formulas and borg type theorems for matrix valued jacobi and dirac finite difference operators abstract borg type uniqueness theorems for matrix valued jacobi operators h and supersymmetric dirac difference operators d are proved more precisely assuming reflectionless matrix coefficients a b in the self adjoint jacobi operator h as a s b with s the right left shift operators on the lattice z and the spectrum of h to be a compact interval e e e e we prove that a and b are certain multiples of the identity matrix an analogous result which however displays a certain novel nonuniqueness feature is proved for supersymmetric self adjoint dirac difference operators d with spectrum given by e 1 2 e 1 2 e 1 2 e 1 2 0 e e our approach is based on trace formulas and matrix valued exponential herglotz representation theorems as a by product of our techniques we obtain the extension of flaschka s borg type result for periodic scalar jacobi operators to the class of reflectionless matrix valued jacobi operators

cyclic tridiagonal pairs higher order onsager algebras and orthogonal polynomials abstract the concept of cyclic tridiagonal pairs is introduced and explicit examples are given for a fairly general class of cyclic tridiagonal pairs with cyclicity n we associate a pair of divided polynomials the properties of this pair generalize the ones of tridiagonal pairs of racah type the algebra generated by the pair of divided polynomials is identified as a higher order generalization of the onsager algebra it can be viewed as a subalgebra of the q onsager algebra for a proper specialization at q the primitive 2 n th root of unity orthogonal polynomials beyond the leonard duality are revisited in light of this framework in particular certain second order dunkl shift operators provide a realization of the divided polynomials at n 2 or q i

computation of matrix valued formally orthogonal polynomials and applications we present a computational procedure for generating formally orthogonal polynomials associated with a given bilinear hankel form with rectangular matrix valued moments our approach covers the most general case of moments of any size and is not restricted to square moments moreover our algorithm has a built in deflation procedure to handle linearly dependent or almost linearly dependent columns and rows of the block hankel matrix associated with the bilinear form possible singular or close to singular leading principal submatrices of the deflated block hankel matrix are avoided by means of look ahead techniques applications of the computational procedure to eigenvalue computations reduced order modeling the solution of multiple linear systems and the fast solution of block hankel systems are also briefly described

on weyl titchmarsh theory for singular finite difference hamiltonian systems we develop the basic theory of matrix valued weyl titchmarsh m functions and the associated green s matrices for whole line and half line self adjoint hamiltonian finite difference systems with separated boundary conditions

polynomial interpolation and gaussian quadrature for matrix valued functions abstract the techniques for polynomial interpolation and gaussian quadrature are generalized to matrix valued functions it is shown how the zeros and rootvectors of matrix orthonormal polynomials can be used to get a quadrature formula with the highest degree of precision

on some n 1 symmetric linear functionals abstract many authors have dealt with problems related to the symmetrization of sequences of orthogonal polynomials on a real line or on a unit circle particular aspects are treated for quadratic or cubic decompositions in this paper we present a technique that unifies the treatment of these topics for an arbitrary positive order of decomposition n by considering a more general definition of orthogonality this technique is based on the decomposition with respect to the cyclic group of order n the main theorem is applied to the orthogonality on n 1 symmetric curves some particular cases are singled out these results may be useful in studying hermite pade approximations vector continued fractions and dynamical systems

matrix differential equations and scalar polynomials satisfying higher order recursions abstract we show that any scalar differential operator with a family of polynomials as its common eigenfunctions leads canonically to a matrix differential operator with the same property the construction of the corresponding family of matrix valued polynomials has been studied in a duran a generalization of favard s theorem for polynomials satisfying a recurrence relation j approx theory 74 1993 83 109 a duran on orthogonal polynomials with respect to a positive definite matrix of measures canad j math 47 1995 88 112 a duran w van assche orthogonal matrix polynomials and higher order recurrence relations linear algebra appl 219 1995 261 280 but the existence of a differential operator having them as common eigenfunctions had not been considered this correspondence goes only one way and most matrix valued situations do not arise in this fashion we illustrate this general construction with a few examples in the case of some families of scalar valued polynomials introduced in f a grunbaum l haine bispectral darboux transformations an extension of the krall polynomials int math res not 8 1997 359 392 we take a first look at the algebra of all matrix differential operators that share these common eigenfunctions and uncover a number of phenomena that are new to the matrix valued case

on a multivariable extension of jacobi matrix polynomials the classical jacobi matrix polynomials only for commutative matrices were first studied by defez et al e defez l jodar a law jacobi matrix differential equation polynomial solutions and their properties comput math appl 48 2004 789 803 the main aim of this paper is to construct a multivariable extension with the help of the classical jacobi matrix polynomials jmps generating matrix functions and recurrence relations satisfied by these multivariable matrix polynomials are derived furthermore general families of multilinear and multilateral generating matrix functions are obtained and their applications are presented

on a weyl titchmarsh theory for discrete symplectic systems on a half line recently bohner and sun introduced basic elements of a weyl titchmarsh theory into the study of discrete symplectic systems we extend this development through the introduction of weyl titchmarsh functions together with a preliminary study of their properties a limit point criterion is described and characterized green s function for the half line is introduced as a limit of such functions in the regular case and half line solutions obtained are seen to satisfy lambda dependent boundary conditions at infinity

a class of orthogonal polynomials on the radial rays in the complex plane abstract we introduce a class of polynomials orthogonal on some radial rays in the complex plane and investigate their existence and uniqueness a recurrence relation for these polynomials a representation and the connection with standard polynomials orthogonal on 0 1 are derived it is shown that their zeros are simple and distributed symmetrically on the radial rays with the possible exception of a multiple zero at the origin an analogue of the jacobi polynomials and the corresponding problem with the generalized laguerre polynomials are also treated

orthogonal matrix polynomials and higher order recurrence relations abstract it is well known that orthogonal polynomials on the real line satisfy a three term recurrence relation and conversely every system of polynomials satisfying a three term recurrence relation is orthogonal with respect to some positive borel measure on the real line we extend this result and show that every system of polynomials satisfying some 2 n 1 term recurrence relation can be expressed in terms of orthonormal matrix polynomials for which the coefficients are n n matrices we apply this result to polynomials orthogonal with respect to a discrete sobolev inner product and other inner products in the linear space of polynomials as an application we give a short proof of krein s characterization of orthogonal polynomials with a spectrum having a finite number of accumulation points