quadratic pencils and least squares piecewise polynomial approximation for a partition o o m denotes the space of piecewise polynomials of order k and of smoothness m 1 this space can be represented as the graph of the appropriate linear operator between two finite dimensional hilbert spaces it gives an approach to the c de boor problem 1972 on uniform boundedness with respect to in the l norm of the orthogonal projections onto stm and we give the detailed analysis of a quadratic pencil matrix valued polynomial of the second degree which appears in the case of a geometric mesh if 2m k the explicit calculations and estimates of zeros of the characteristic polynomial show that in the case sbx x the geometric mesh with the parameter x 0 m 0 and any partition ax n of the unit interval 0 1 01 0 1 0 1 n n i 1 we define the piecewise polynomial subspace 0 2 s t km n c m1 0 11 where p fel 01vfia k i pk e f ej l il f ia is a polnom1ial caxi of order k 0 s a s n each interior breakpoint a 1 a n generates m continuity conditions 0 3 ia c af ia ya i 0 in to make a functionf e pkt be an element of scm let q qs be the orthogonal projection onto s in l2 0 1 i e with respect to the inner product f g f x g x dx we are interested in q as a map in c 0 1 or l 0 1 and we would like to get the estimates of its norm iiqiioo sup iiqfli lflfloo f where ilf 11k0 esssupo x if x i f e l o 1 received september 29 1980 1980 mathematics subject classification primary 41a1 5 secondary 47a68 this paper was partly supported by nsf grant 7906079 1983 american mathematical society 0025 571 8 82 0000 0750 05 25

kernels regularization and differential equations many common machine learning methods such as support vector machines or gaussian process inference make use of positive definite kernels reproducing kernel hilbert spaces gaussian processes and regularization operators in this work these objects are presented in a general unifying framework and interrelations are highlighted with this in mind we then show how linear stochastic differential equation models can be incorporated naturally into the kernel framework and vice versa many kernel machines can be interpreted in terms of differential equations we focus especially on ordinary differential equations also known as dynamical systems and it is shown that standard kernel inference algorithms are equivalent to kalman filter methods based on such models in order not to cloud qualitative insights with heavy mathematical machinery we restrict ourselves to finite domains implying that differential equations are treated via their corresponding finite difference equations

on the finite section method for computing exponentials of doubly infinite skew hermitian matrices abstract computing the exponential of large scale skew hermitian matrices or parts thereof is frequently required in applications in this work we consider the task of extracting finite diagonal blocks from a doubly infinite skew hermitian matrix these matrices usually have unbounded entries which impede the application of many classical techniques from approximation theory we analyze the decay property of matrix exponentials for several classes of banded skew hermitian matrices then finite section methods based on the decay property are presented we use several examples to demonstrate the effectiveness of these methods

analytical inversion of symmetric tridiagonal matrices in this paper we present an analytical formula for the inversion of symmetrical tridiagonal matrices the result is of relevance to the solution of a variety of problems in mathematics and physics as an example the formula is used to derive an exact analytical solution for the one dimensional discrete poisson equation with dirichlet boundary conditions

decay rates for inverses of band matrices spectral theory and classical approximation theory are used to give a new proof of the exponential decay of the entries of the inverse of band matrices the rate of decay of a 1 can be bounded in terms of the essential spectrum of aa for general a and in terms of the essential spectrum of a for positive definite a in the positive definite case the bound can be attained these results are used to establish the exponential decay for a class of generalized eigenvalue problems and to establish exponential decay for certain sparse but nonbanded matrices we also establish decay rates for certain generalized inverses

a strongly polynomial algorithm for controlled queues we consider the problem of computing optimal policies of finite state finite action markov decision processes mdps a reduction to a continuum of constrained mdps cmdps is presented such that the optimal policies for these cmdps constitute a path in a graph defined over the deterministic policies this path contains in particular an optimal policy of the original mdp we present an algorithm based on this new approach that finds this path and thus an optimal policy in the general case this path might be exponentially long in the number of states and actions we prove that the length of this path is polynomial if the mdp satisfies a coupling property thus we obtain a strongly polynomial algorithm for mdp s that satisfies the coupling property we prove that discrete time versions of controlled m m 1 queues induce mdp s that satisfy the coupling property the only previously known polynomial algorithm for controlled m m 1 queues in the expected average cost model is based on linear programming and is not known to be strongly polynomial our algorithm works both for the discounted and expected average cost models and the running time does not depend on the discount factor

generalized foulkes modules and maximal and minimal constituents of plethysms of schur functions this paper proves a combinatorial rule giving all maximal and minimal partitions lambda such that the schur function s lambda appears in a plethysm of two arbitrary schur functions determining the decomposition of these plethysms has been identified by stanley as a key open problem in algebraic combinatorics as corollaries we prove three conjectures of agaoka on the partitions labeling the lexicographically greatest and least schur functions appearing in an arbitrary plethysm we also show that the multiplicity of the schur function labelled by the lexicographically least constituent may be arbitrarily large the proof is carried out in the symmetric group and gives an explicit non zero homomorphism corresponding to each maximal or minimal partition

bounds for entries of the inverse matrix of a tridiagonal matrix inequalities for the moduli of entries of the inverse matrix of a tridiagonal matrix are obtained these results are applicable to a relatively large family of matrices which occur in many branches of numerical analysis applied mathematics and engineering

highest weight vectors in plethysms we realize the mathrm gl n mathbb c modules s k s m mathbb c n and lambda k s m mathbb c n as spaces of polynomial functions on n times k matrices in the case k 3 we describe explicitly all the mathrm gl n mathbb c highest weight vectors which occur in s 3 s m mathbb c n and in lambda 3 s m mathbb c n respectively in particular we obtain alternative formulas for the multiplicities in these modules

the caldeira leggett quantum master equation in wigner phase space continued fraction solution and application to brownian motion in periodic potentials the continued fraction method of solving classical fokker planck equations has been adapted to tackle quantum master equations of the caldeira leggett type this was done taking advantage of the phase space wigner representation of the quantum density matrix the approach differs from those in which some continued fraction expression is found for a certain quantity in that the full solution of the master equation is obtained by continued fraction methods this allows us to study in detail the effects of the environment fluctuations and dissipation on several classes of nonlinear quantum systems we apply the method to the canonical problem of quantum brownian motion in periodic potentials both for cosine and ratchet potentials lacking inversion symmetry

inversion of tridiagonal matrices this paper presents a simple algorithm for inverting nonsymmetric tridiagonal matrices that leads immediately to closed forms when they exist ukita s theorem is extended to characterize the class of matrices that have tridiagonal inverses

decay properties for functions of matrices over c algebras abstract we extend previous results on the exponential off diagonal decay of the entries of analytic functions of banded and sparse matrices to the case where the matrix entries are elements of a c algebra

decay rates of inverses of banded m matrices that are near to toeplitz matrices the decaying behavior of inverses of positive definite band matrices is analysed for m matrices that are in some sense close to toeplitz matrices estimates based on the factorization are derived that are better than existing ones in particular for nonsymmetric matrices some examples are given

monotone convex sequences and cholesky decomposition of symmetric toeplitz matrices this paper studies off diagonal decay in symmetric toeplitz matrices it is shown that if the generating sequence of the matrix is monotone positive and convex then the monotonicity and positivity are maintained through triangular decomposition the work is motivated by recent results on explicit bounds for inverses of triangular matrices

analytical inversion of general periodic tridiagonal matrices in this paper we present an analytical forms for the inversion of general periodic tridiagonal matrices and provide some very simple analytical forms which immediately lead to closed formulae for some special cases such as symmetric or perturbed toeplitz for both periodic and non periodic tridiagonal matrices an efficient computational algorithm for finding the inverse of any general periodic tridiagonal matrices from the analytical form is given it is suited for implementation using computer algebra systems such as maple matlab macsyma and mathematica an example is also given to illustrate the algorithm

on computing jacobi matrices associated with recurrent and mobius iterated function systems a new technique for the computation of jacobi matrices associated with measures possessing scale invariance properties is described invariance is imposed via linear transformations and by mobius transformations in psl 2 r this technique is proven to be numerically stable and apt to compute jacobi matrices of large orders

non symbolic algorithms for the inversion of tridiagonal matrices a representation for the entries of the inverse of general tridiagonal matrices is based on the determinants of their principal submatrices it enables us to introduce through the linear recurrence relations satisfied by such determinants a simple algorithm for the entries of the inverse of any tridiagonal nonsingular matrix reduced as well as unreduced the numerical approach is preserved here without invoking the symbolic computation for tridiagonal diagonally dominant matrices a scaling transformation on the recurrences allows us to give another algorithm to avoid overflow and underflow

a fast and reliable numerical solver for general bordered k tridiagonal matrix linear equations to improve on the shortcomings observed in symbolic algorithms introduced recently for related matrices a reliable numerical solver is proposed for computing the solution of the matrix linear equation a x b the n n matrix coefficient a is a nonsingular bordered k tridiagonal matrix the particular structure of a is exploited through an incomplete or full givens reduction depending on the singularity of its associated k tridiagonal matrix then adapted back substitution and sherman morrison s formula can be applied specially the inverse of the matrix a is computed moreover for a wide range of matrices a the solution of the vector linear equation a x b can be computed in o n time numerical comparisons illustrate the results

a priori sparsity patterns for parallel sparse approximate inverse preconditioners parallel algorithms for computing sparse approximations to the inverse of a sparse matrix either use a prescribed sparsity pattern for the approximate inverse or attempt to generate a good pattern as part of the algorithm this paper demonstrates that for pde problems the patterns of powers of sparsified matrices psms can be used a priori as effective approximate inverse patterns and that the additional effort of adaptive sparsity pattern calculations may not be required psm patterns are related to various other approximate inverse sparsity patterns through matrix graph theory and heuristics concerning the pde s green s function a parallel implementation shows that psm patterned approximate inverses are significantly faster to construct than approximate inverses constructed adaptively while often giving preconditioners of comparable quality

inversion of jacobi s tridiagonal matrix abstract this paper gives a simple algorithm for finding the explicit inverse of a general jacobi tridiagonal matrix a few illustrations are also provided to demonstrate the applicability of our algorithm

video deblurring algorithm using accurate blur kernel estimation and residual deconvolution based on a blurred unblurred frame pair blurred frames may happen sparsely in a video sequence acquired by consumer devices such as digital camcorders and digital cameras in order to avoid visually annoying artifacts due to those blurred frames this paper presents a novel motion deblurring algorithm in which a blurred frame can be reconstructed utilizing the high resolution information of adjacent unblurred frames first a motion compensated predictor for the blurred frame is derived from its neighboring unblurred frame via specific motion estimation then an accurate blur kernel which is difficult to directly obtain from the blurred frame itself is computed using both the predictor and the blurred frame next a residual deconvolution is applied to both of those frames in order to reduce the ringing artifacts inherently caused by conventional deconvolution the blur kernel estimation and deconvolution processes are iteratively performed for the deblurred frame simulation results show that the proposed algorithm provides superior deblurring results over conventional deblurring algorithms while preserving details and reducing ringing artifacts

generalized foulkes conjecture and tableaux construction foulkes conjectured that for n ab and a a we prove the generalized conjecture for a 2 and a 3 by explicitly constructing the corresponding tableaux we also prove the multiplicity constraint for certain cases for these proofs we develop a theory of construction conditions for tableaux giving rise to sym b wrsym a modules and in doing so completely classify all such tableaux for a 2 and a 3

a reliable givens lu approach for solving opposite bordered tridiagonal linear systems abstract a fast and reliable numerical solver is proposed to handle determined opposite bordered tridiagonal linear systems a x b a precise sequence of givens rotations on the nonsingular matrix coefficient a allows us to work with a simpler linear system this equivalent system which admits always a trivial lu factorization can be solved in linear time for large order matrices of use in applications the main advantage of this solver with respect to other special algorithms proposed is its stability and risk free performance as it is checked with comparative examples its extension to manage opposite bordered tridiagonal matrix equations is straightforward

asymptotically optimal truncated multivariate gaussian hypothesis testing with application to consensus algorithms in the interest of complexity reduction or to facilitate efficient distributed computation using consensus truncated versions of the optimal hypothesis test are considered for a canonical multivariate gaussian problem with l observations the truncated tests employ correlation terms involving any given observation the focus is on cases with a large l such that significant efficiency results with a truncation rule k as a function of l which increases very slowly with l a key result provides sufficient conditions on truncation rules and sequences of hypothesis testing problems which provide no loss in deflection performance as l approaches infinity when compared to the optimal detector the set of asymptotically optimal truncation rules satisfying these sufficient conditions varies with the scaling behavior of the difficulty of the hypothesis testing problem with l several popular classes of system and process models including observations from wide sense stationary limiting processes as l after the mean is subtracted are used as illustrative classes of examples to demonstrate the sufficient conditions are not overly restrictive in these examples significant truncation can be employed even when the difficulty of the hypothesis testing problem scales in the least favorable manner putting the most stringent conditions on the truncation rule in all the cases considered numerical results imply the fixed false alarm rate detection probability of the truncated detector converges to the detection probability of the optimal detector for our asymptotically optimal truncation in terms of deflection

understanding blind deconvolution algorithms blind deconvolution is the recovery of a sharp version of a blurred image when the blur kernel is unknown recent algorithms have afforded dramatic progress yet many aspects of the problem remain challenging and hard to understand the goal of this paper is to analyze and evaluate recent blind deconvolution algorithms both theoretically and experimentally we explain the previously reported failure of the naive map approach by demonstrating that it mostly favors no blur explanations we show that using reasonable image priors a naive simulations map estimation of both latent image and blur kernel is guaranteed to fail even with infinitely large images sampled from the prior on the other hand we show that since the kernel size is often smaller than the image size a map estimation of the kernel alone is well constrained and is guaranteed to succeed to recover the true blur the plethora of recent deconvolution techniques makes an experimental evaluation on ground truth data important as a first step toward this experimental evaluation we have collected blur data with ground truth and compared recent algorithms under equal settings additionally our data demonstrate that the shift invariant blur assumption made by most algorithms is often violated

temporal coherence based deblurring using non uniform motion optimization non uniform motion blur due to object movement or camera jitter is a common phenomenon in videos however the state of the art video deblurring methods used to deal with this problem can introduce artifacts and may sometimes fail to handle motion blur due to the movements of the object or the camera in this paper we propose a non uniform motion model to deblur video frames the proposed method is based on superpixel matching in the video sequence to reconstruct sharp frames from blurry ones to identify a suitable sharp superpixel to replace a blurry one we enrich the search space with a non uniform motion blur kernel and use a generalized patchmatch algorithm to handle rotation scale and blur differences in the matching step instead of using pixel based or regular patch based representation we adopt a superpixel based representation and use color and motion to gather similar pixels our non uniform motion blur kernels are estimated from the motion field of these superpixels and our spatially varying motion model considers spatial and temporal coherence to find sharp superpixels experimental results showed that the proposed method can reconstruct sharp video frames from blurred frames caused by complex object and camera movements and performs better than the state of the art methods

the inverse of a tridiagonal matrix in this paper explicit formulae for the elements of the inverse of a general tridiagonal matrix are presented by first extending results on the explicit solution of a second order linear homogeneous difference equation with variable coefficients to the nonhomogeneous case and then applying these extended results to a boundary value problem a formula for the characteristic polynomial is obtained in the process we also establish a connection between the matrix inverse and orthogonal polynomials in addition the case of a cyclic tridiagonal system is discussed

kernel optimum nearly analytical discretization kond algorithm applied to parabolic and hyperbolic equations abstract two applications of the kernel optimum nearly analytical discretization kond algorithm to the parabolic and the hyperbolic type equations are presented in detail to lead to novel numerical schemes with very high numerical accuracy it is demonstrated numerically that the two dimensional kond p scheme for the parabolic type yields much less numerical error by over 2 3 orders measured quantitatively by the root mean square deviation from analytical solutions and reduces the cpu time to about 1 5 for a common numerical accuracy compared with the conventional explicit scheme of reference it is also demonstrated numerically that the kond h scheme for the hyperbolic type yields much less diffusive error and has fairly high stability for both of the linear and the nonlinear wave propagations compared with other conventional schemes origins of numerical errors in data processing in general numerical schemes are discussed

semiparametric detection of significant activation for brain fmri functional magnetic resonance imaging fmri aims to locate activated regions in human brains when specific tasks are performed the conventional tool for analyzing fmri data applies some variant of the linear model which is restrictive in modeling assumptions to yield more accurate prediction of the time course behavior of neuronal responses the semiparametric inference for the underlying hemodynamic response function is developed to identify significantly activated voxels under mild regularity conditions we demonstrate that a class of the proposed semiparametric test statistics based on the local linear estimation technique follow x 2 distributions under null hypotheses for a number of useful hypotheses furthermore the asymptotic power functions of the constructed tests are derived under the fixed and contiguous alternatives simulation evaluations and real fmri data application suggest that the semiparametric inference procedure provides more efficient detection of activated brain areas than the popular imaging analysis tools afni and fsl

supremum norm posterior contraction and credible sets for nonparametric multivariate regression in the setting of nonparametric multivariate regression with unknown error variance we study asymptotic properties of a bayesian method for estimating a regression function f and its mixed partial derivatives we use a random series of tensor product of b splines with normal basis coefficients as a prior for f and the error variance is either estimated using the empirical bayes approach or is endowed with a suitable prior in a hierarchical bayes approach we establish pointwise l2 and supremum norm posterior contraction rates for f and its mixed partial derivatives and show that they coincide with the minimax rates our results cover even the anisotropic situation where the true regression function may have different smoothness in different directions using the convergence bounds we show that pointwise l2 and supremum norm credible sets for f and its mixed partial derivatives have guaranteed frequentist coverage with optimal size new results on tensor products of b splines are also obtained in the course

on the inverses of general tridiagonal matrices in this work the sign distribution for all inverse elements of general tridiagonal h matrices is presented in addition some computable upper and lower bounds for the entries of the inverses of diagonally dominant tridiagonal matrices are obtained based on the sign distribution these bounds greatly improve some well known results due to ostrowski 1952 23 shivakumar and ji 1996 26 nabben 1999 21 22 and recently given by peluso and politi 2001 24 peluso and popolizio 2008 25 and so forth it is also stated that the inverse of a general tridiagonal matrix may be described by 2n 2 parameters k k 2n and k k 1n 1 instead of 2n 2 ones as given by el mikkawy 2004 3 el mikkawy and karawia 2006 4 and huang and mccoll 1997 10 according to these results a new symbolic algorithm for finding the inverse of a tridiagonal matrix without imposing any restrictive conditions is presented which improves some recent results finally several applications to the preconditioning technology the numerical solution of differential equations and the birth death processes together with numerical tests are given

a fast and well conditioned spectral method a spectral method is developed for the direct solution of linear ordinary differential equations with variable coefficients and general boundary conditions the method leads to matrices that are al

preconditioned iterative methods for the nine point approximation to the convection diffusion equation iterative methods preconditioned by incomplete factorizations and sparse approximate inverses are considered for solving linear systems arising from fourth order finite difference schemes for convection diffusion problems simple recurrences for implementing the ilu 0 factorization of the nine point scheme are derived different sparsity patterns are considered for computing approximate inverses for the coefficient matrix and the quality of the preconditioner is studied in terms of plots of the field of values of the preconditioned matrices in terms of algebraic properties of the preconditioned matrices our experimental results show that incomplete factorizations give a preconditioner of better quality than approximate inverses comparison of the convergence rates of gmres applied to the preconditioned linear systems is done with respect to the field of values ritz and harmonic ritz values of the preconditioned matrices numerical results show that the gmres residual norm decreases rapidly when the difference between the ritz and harmonic ritz values becomes small we also describe the results of experiments when some well known krylov subspace methods are used to solve the linear system arising from the compact fourth order discretizations

spectra of sparse regular graphs with loops we derive exact equations that determine the spectra of undirected and directed sparsely connected regular graphs containing loops of arbitrary lengths the implications of our results for the structural and dynamical properties of network models are discussed by showing how loops influence the size of the spectral gap and the propensity for synchronization analytical formulas for the spectrum are obtained for specific lengths of the loops

analytical inversion of general tridiagonal matrices in this paper we give a complete analysis for general tridiagonal matrix inversion for both non block and block cases and provide some very simple analytical formulae which immediately lead to closed forms for some special cases such as symmetric or toeplitz tridiagonal matrices

efficient tridiagonal solvers on multicomputers three parallel algorithms namely the parallel partition lu ppt algorithm the parallel partition hybrid pph algorithm and the parallel diagonal dominant pdd algorithm are proposed for solving tridiagonal linear systems on multicomputers these algorithms are based on the divide and conquer parallel computation model the ppt and pph algorithms support both pivoting and nonpivoting the ppt algorithm is good when the number of processors is small otherwise the pph algorithm is better when the system is diagonal dominant the pdd algorithm is highly parallel and provides an approximate solution which equals the exact solution within machine accuracy computation and communication complexities of the three algorithms are presented all three methods have been implemented on a 64 node ncube 1 multicomputer the analytic results closely match the results measured from the ncube 1 machine

incomplete block matrix factorization preconditioning methods the ultimate answer abstract a considerable interest has been devoted to block matrix incomplete factorization preconditioning methods in recent time see 12 2 3 4 6 7 8 10 and 21 the purpose of this paper is to survey some of these results and to present new results showing how much more efficient and robust such methods can be as compared to other admittingly also efficient methods such as pointwise incomplete factorization preconditioning methods and multilevel methods with hierarchical basis functions both on scalar as well as on parallel or vectorizable computers

the analytic inversion of any finite symmetric tridiagonal matrix we use the theory of orthogonal polynomials to write down explicit expressions for the polynomials of the first and second kind associated with a given infinite symmetric tridagonal matrix h the green s function is the inverse of the infinite symmetric tridiagonal matrix h zi by calculating the inverse of the finite symmetric tridiagonal matrix we can find the analytical form of the inverse of the finite symmetric tridiagonal matrix

on the closed representation for the inverses of hessenberg matrices the general representation for the elements of the inverse of any hessenberg matrix of finite order is here extended to the reduced case with a new proof those entries are given with proper hessenbergians from the original matrix it justifies both the use of linear recurrences of unbounded order for such computations on matrices of intermediate order and some elementary properties of the inverse these results are applied on the resolvent matrix associated to a finite hessenberg matrix in standard form two examples on the unit disk are given

inversion of k tridiagonal matrices with toeplitz structure in this paper we consider an inverse problem with the k tridiagonal toeplitz matrices a theoretical result is obtained that under certain assumptions the explicit inverse of a k tridiagonal toeplitz matrix can be derived immediately two numerical examples are given to demonstrate the validity of our results

on a conjecture of foulkes abstract suppose that 1 2 ab for some non negative integers a and b denote by p a b the set of unordered partitions of into a parts of cardinality b in this paper we study the decomposition of the permutation module c p a b where c is the field of complex numbers in particular we show that c p 3 b is isomorphic to a submodule of c p b 3 for b 3 this settles the next unproven case of a conjecture of foulkes

efficient computation of tridiagonal matrices largest eigenvalue abstract this paper proposes a method for a fast estimation of the largest eigenvalue of an asymmetric tridiagonal matrix the proposed method is based on the power method and the computation of the square of the original matrix the matrix square is computed through a proposed fast algorithm designed specifically for tridiagonal matrices implementations for compressed column ccs and compressed row storage crs formats are provided discussed and compared to a standard scientific library we investigate the roundoff numerical errors showing that the proposed method provides errors no greater than the usual power method we provide numerical results with simulations in c c implementation in order to demonstrate the effectiveness of the proposed method

removing motion blur with space time processing although spatial deblurring is relatively well understood by assuming that the blur kernel is shift invariant motion blur is not so when we attempt to deconvolve on a frame by frame basis this is because in general videos include complex multilayer transitions indeed we face an exceedingly difficult problem in motion deblurring of a single frame when the scene contains motion occlusions instead of deblurring video frames individually a fully 3 d deblurring method is proposed in this paper to reduce motion blur from a single motion blurred video to produce a high resolution video in both space and time unlike other existing approaches the proposed deblurring kernel is free from knowledge of the local motions most importantly due to its inherent locally adaptive nature the 3 d deblurring is capable of automatically deblurring the portions of the sequence which are motion blurred without segmentation and without adversely affecting the rest of the spatiotemporal domain where such blur is not present our method is a two step approach first we upscale the input video in space and time without explicit estimates of local motions and then perform 3 d deblurring to obtain the restored sequence

two sided bounds on the inverses of diagonally dominant tridiagonal matrices abstract we establish upper and lower bounds for the entries of the inverses of diagonally dominant tridiagonal matrices these bounds improve the bounds recently given by shivakumar and ji moreover we show how to improve our bounds iteratively for an n n m matrix this iterative refinement yields the exact inverse after n 1 steps

inverse problem for multivariate time series using dynamical latent variables abstract factor analysis is a well known statistical method to describe the variability among observed variables in terms of a smaller number of unobserved latent variables called factors while dealing with multivariate time series the temporal correlation structure of data may be modeled by including correlations in latent factors but a crucial choice is the covariance function to be implemented we show that analyzing multivariate time series in terms of latent gaussian processes which are mutually independent but with each of them being characterized by exponentially decaying temporal correlations leads to an efficient implementation of the expectation maximization algorithm for the maximum likelihood estimation of parameters due to the properties of block tridiagonal matrices the proposed approach solves an ambiguity known as the identifiability problem which renders the solution of factor analysis determined only up to an orthogonal transformation samples with just two temporal points are sufficient for the parameter estimation hence the proposed approach may be applied even in the absence of prior information about the correlation structure of latent variables by fitting the model to pairs of points with varying time delay our modeling allows one to make predictions of the future values of time series and we illustrate our method by applying it to an analysis of published gene expression data from cell culture hela

improved bounds for the inverses of diagonally dominant tridiagonal matrices abstract we obtain new bounds for the entries of the inverse of a diagonally dominant tridiagonal matrix which improve the best previous ones due to h b li et al we apply our bounds to the tridiagonal matrices arising in the second order finite difference discretization of certain boundary value problems of parabolic type establishing asymptotically optimal bounds

vectorizable preconditioners for elliptic difference equations in three space dimensions abstract until now research in vectorizable preconditioners has mainly focused on problems in two space dimensions employing all degrees of freedom we here present preconditioners for 3d matrices based on point line and plane factorizations both in vectorizable and for reasons of comparisons in recursive versions by a limit analysis and accompanying numerical tests of elliptic difference equations with anisotropy we find that methods based on line block factorizations are to be preferred in general among the vectorizable methods a line method where euler expansion of the factors of the pivot blocks is used is the most robust with respect to the problem parameters furthermore this method turns out to approach the best of the recursive methods in efficiency in terms of operation counts it has the added bonus of almost maximal vectorlength

a new recursive algorithm for inverting general tridiagonal and anti tridiagonal matrices in the current article the authors present a new recursive symbolic computational algorithm for inverting general tridiagonal and anti tridiagonal matrices an illustrative example is given

symbolic algorithms for the inverses of general k tridiagonal matrices two symbolic algorithms for inverting k tridiagonal matrices have been recently found by el mikkawy and atlan 2014 2015 these two algorithms are mainly based on the doolittle lu factorization of the k tridiagonal matrix in the current paper we present a new explicit analytic expression for the inverses of general tridiagonal matrices at first by using a block diagonalization technique we then relate k tridiagonal matrix inversion to tridiagonal matrix inversion meanwhile an efficient algorithm is derived for computing the inverses of nonsingular k tridiagonal matrices with the help of any algorithm for computing the inverses of tridiagonal matrices three examples are given in order to illustrate the performance and efficiency of the proposed algorithms

on plethysm conjectures of stanley and foulkes abstract independently siemons and wagner i j siemons o wagner private communication 1986 and stanley r p stanley positivity problems and conjectures in algebraic combinatorics in mathematics frontiers and perspectives amer math soc 2000 pp 295 319 conjecture that if a partition dominates its conjugate partition then an fs n homomorphism 1 n s n s s n 1 n s n s s n is injective we demonstrate a method of adding columns to the partition that preserves the injectivity of

complex jacobi matrices complex jacobi matrices play an important role in the study of asymptotics and zero distribution of formal orthogonal polynomials fops the latter are essential tools in several fields of numerical analysis for instance in the context of iterative methods for solving large systems of linear equations or in the study of pade approximation and jacobi continued fractions in this paper we present some known and some new results on fops in terms of spectral properties of the underlying infinite jacobi matrix with a special emphasis to unbounded recurrence coefficients here we recover several classical results for real jacobi matrices the inverse problem of characterizing properties of the jacobi operator in terms of fops and other solutions of a given three term recurrence is also investigated this enables us to give results on the approximation of the resolvent by inverses of finite sections with applications to the convergence of pade approximants

a numerical solver for general bordered tridiagonal matrix equations abstract to overcome several limitations of symbolic algorithms introduced recently for matrices of large order a fast numerical solver is proposed for the matrix linear equation a x b where the n n coefficient matrix a is a general nonsingular bordered tridiagonal matrix its sparse structure is preserved through partial givens reduction in particular the matrix inverse of a can be computed for a wide range of bordered tridiagonal linear systems a x b the solution is computed in linear time using back substitution and sherman morrison s formula numerical comparisons illustrate the results

on the h 1 h1 stability of the l_2 l2 projection onto finite element spaces we study the stability in the h 1 h 1 seminorm of the l 2 l 2 projection onto finite element spaces in the case of nonuniform but shape regular meshes in two and three dimensions and prove in particular stability for conforming triangular elements up to order twelve and conforming tetrahedral elements up to order seven