total positivity and its applications preface i spline functions total positivity of the spline kernel and its applications b bojanov total positivity properties of lb splines t lyche l l schumaker total positivity and splines k morken totally positive systems and interpolation by subspaces of generalized splines m sommer h strauss ii matrix theory preservation of total nonnegativity under the hadamard product and related topics j garloff d g wagner vertex implications for totally nonnegative under the hadamard product and related topics j garloff on factorizations of totally positive matrices j garloff on factorizations of totally positive matrices m gasca j m pena iii geometric modeling total positivity and optimal bases j m carnicer j m pena total positivity and the shape of curves t n t goodman tchebycheff curves m l mazure h pottmann iv probability and mathematical biology analytical and probabilistic methods for totally positive matrices g s goodman some inequalities of total positivity in pure and applied mathematics s karlin z ziegler new directions and problems of multivariate total positivity motivated by genetic recombination probabilities s karlin v approximation theory tchebycheff spaces and total positivity j m carnicer j m pena cebysev and weak cebysev systems r a zalik vi complex analysis zero sets of entire fourier transforms of multiply positive density functions o m katkova vii statistics total nonnegativity of moment matrices and its application to e optimal designs in polynomial regression b heiligers analysis of regression dependence in 2 xn bivariate distributions and some applications in contingency tables s kasala m b rao viii real analysis total positivity and nonlinear analysis r q jia solutions of certain functional equations s karlin alternance and snakes in the nonlinear finite moment problem k tatalian ix combinatorics the applications of total positivity to combinatorics and conversely f brenti x integral equations spectral properties of totally positive kernels and matrices a pinkus index

rank preserving diagonal completions of a matrix abstract suppose a is an n by n matrix over a field f we prove that it is possible to complete the diagonal entries of a so that the resulting rank of a is as small as possible when n 3r where r is the off diagonal rank of a and n r 3 1

structure ranks of matrices abstract the structure rank of a matrix i e the maximum order of a nonsingular submatrix all of whose entries are located in a given structure a subset of m n where m n are the row and column index sets respectively were previously studied for the off diagonal part here results on more general ranks e g off block diagonal strictly upper triangular and strictly upper block triangular are presented in particular it is shown that for a nonsingular matrix a in each of the last mentioned three cases the ranks of a and a 1 coincide

perturbation of spectral inverses applied to a boundary layer phenomenon arising in chemical networks abstract a linear model showing behavior of the boundary layer type is developed the governing equations have the form t b t x t 0 t 0 where the coefficient matrix b depends continuously on two nonnegative parameters and has a discontinuity in rank at the origin of the parameter space the analysis depends on a regularization technique for linear systems and on a perturbation result for spectral group generalized inverses

a local convergence proof for the iterative aggregation method the iterative aggregation method for the solution of a system of linear algebraic equations x ax b where a 0 b 0 s 0 and s a s is proved to be locally convergent it is shown that the method can be considered a consistent nonstationary iterative method where the iteration matrix depends on the current iterate and that some norm of the iteration matrix is less than one in the vicinity of the solution

more on structure ranks of matrices abstract the structure rank of a matrix is the largest rank of a submatrix which lies within a specified structure we characterize those structures avoiding the main diagonal for which a matrix and its inverse have the same structure rank thereby generalizing some results of m fiedler

completing a matrix when certain entries of its inverse are specified abstract suppose a a 11 a 12 a 21 a 22 is an n n matrix where a 11 has order p q if a is nonsingular let a 1 have the transposed partitioning b 11 b 12 b 21 b 22 first we show that the nullities of a 11 and b 22 are equal the remainder of our paper deals with the following completion problem suppose m 1 m 2 n 1 n 2 are nonnegative integers such that m 1 m 2 n 1 n 2 n 0

probabilistic diffusion tractography with multiple fibre orientations what can we gain we present a direct extension of probabilistic diffusion tractography to the case of multiple fibre orientations using automatic relevance determination we are able to perform online selection of the number of fibre orientations supported by the data at each voxel simplifying the problem of tracking in a multi orientation field we then apply the identical probabilistic algorithm to tractography in the multi and single fibre cases in a number of example systems which have previously been tracked successfully or unsuccessfully with single fibre tractography we show that multi fibre tractography offers significant advantages in sensitivity when tracking non dominant fibre populations but does not dramatically change tractography results for the dominant pathways

iterative aggregation disaggregation techniques for nearly uncoupled markov chains iterative aggregation disaggregation methods provide an efficient approach for computing the stationary probability vector of nearly uncoupled also known as nearly completely decomposable markov chains three such methods that have appeared in the literature recently are considered and their similarities and differences are outlined specifically it is shown that the method of takahashi corresponds to a modified block gauss seidel step and aggregation whereas that of vantilborgh corresponds to a modified block jacobi step and aggregation the third method that of koury et al is equivalent to a standard block gauss seidel step and iteration for each of these methods a lemma is established which shows that the unique fixed point of the iterative scheme is the left eigenvector corresponding to the dominant unit eigenvalue of the stochastic transition probability matrix in addition conditions are established for the convergence of the first two of these methods convergence conditions for the third having already been established by stewart et al all three methods are shown to have the same asymptotic rate of convergence

quasi birth and death processes level geometric distributions an aggregation disaggregation approach a special class of homogeneous continuous time quasi birth and death qbd markov chains mcs which possess level geometric lg stationary distribution are considered a functional analytic approach is applied which provides not only a clear analytic interpretation of the concepts introduced elsewhere but also represents a suitable basis for computations an iterative aggregation disaggregation method is proposed as tool for numerical computations

generalized oscillatory matrices we define a new class of generalized oscillatory matrices shortly go matrices over a noncommutative ring with identity and a positive part similarly to the classical case this class consists of square generalized totally nonnegative matrices gtn matrices of which some power is generalized totally positive using the previously defined ordering of invertible gtn matrices we study in particular so called basic go matrices which form in a sense atoms of this ordering

a factorization of totally nonsingular matrices over a ring with identity abstract we say that a rectangular matrix over a ring with identity is totally nonsingular tns if for all k all its relevant submatrices either having k consecutive rows and the first k columns or k consecutive columns and the first k rows are invertible we prove that a matrix is tns if and only if it admits a certain factorization with bidiagonal type factors and certain invertible entries this approach generalizes the loewner neville factorization usually applied to totally positive matrices

a class of oscillatory matrices with exponent n 1 abstract a matrix a is called an oscillatory matrix if it is totally nonnegative and there exists a positive integer k such that a k is totally positive in this paper we describe the class of n n oscillatory matrices which have maximal exponent equal to n 1

hybrid ikebe newton s iteration for inverting general nonsingular hessenberg matrices after a concise survey the expanded ikebe algorithm for inverting the lower half plus the superdiagonal of an n n unreduced upper hessenberg matrix h is extended to general nonsingular upper hessenberg matrices by computing in the reduced case a block diagonal form of the factor matrix hl in the inverse factorization h 1 h l u 1 this factorization enables us to propose hybrid and accurate nongaussian procedures for computing h 1 thus hl is computed directly in the aim to be used as a fine initial guess for newton s iteration which converges to h 1 in a suitable number of iterations

two results on basic oscillatory matrices this paper continues the research of the authors on totally nonnegative and oscillatory matrices first the result of the second author stating that the hadamard product of oscillatory tridiagonal matrices of the same order is again an oscillatory tridiagonal matrix is extended to the class of basic oscillatory matrices introduced recently then it is shown that every oscillatory matrix contains a basic oscillatory matrix as a factor this explains the role of the class of basic oscillatory matrices within the class of oscillatory matrices

generalized hessenberg matrices we define and study generalized hessenberg matrices i e square matrices which have subdiagonal rank one here subdiagonal rank means the maximum order of a nonsingular submatrix all of whose entries are in the subdiagonal part we prove that the property of being generalized hessenberg matrix is preserved by post and premultiplication by a nonsingular upper triangular matrix by inversion for invertible matrices etc we also study a special kind of generalized hessenberg matrices

partition based mass clustering of tractography streamlines we describe a novel scalable clustering framework for streamlines obtained from diffusion tractography clustering is an attractive means of segmenting a large set of streamlines into anatomically relevant bundles for most existing methods however the large datasets produced in high resolution or multiple subject studies are problematical to achieve good scalability our method repeatedly divides the data into subsets which are then partitioned using hierarchical clustering a final partition is obtained by recombining the subsets in addition the recombination scheme provides a consistency measure for cluster assignment of individual streamlines which is used to clean up the final result the clusters have good anatomical plausibility and we show that three clusters corresponding to the three known segments of the arcuate fasciculus show excellent agreement with literature a major advantage of the method is the fact that it can find clusters in datasets of essentially arbitrary size this fact is exploited to find consistent clusters in concatenated tractography data from multiple subjects we expect the identification of bundles across subjects to be an important application of the method

a visibility graph averaging aggregation operator abstract the problem of aggregation is of considerable importance in many disciplines in this paper a new type of operator called visibility graph averaging vga aggregation operator is proposed this proposed operator is based on the visibility graph which can convert a time series into a graph the weights are obtained according to the importance of the data in the visibility graph finally the vga operator is used in the analysis of the taiex database to illustrate that it is practical and compared with the classic aggregation operators it shows its advantage that it not only implements the aggregation of the data purely but also conserves the time information meanwhile the determination of the weights is more reasonable

direct approach to the band completion problem we study the problem of completion of a matrix with a specified band in such a way that the inverse be a band matrix with the same width for the general block matrix case a new approach to solve this and related problems is proposed the relations with the class of quasiseparable matrices play an important role

iterative and semi iterative methods for computing stationary probability vectors of markov operators iterative and semi iterative methods for computing stationary prob ability vectors of markov type operators are proposed and their convergence properties are analyzed the methods studied apply to certain classes of prob lems in infinite dimensional spaces as well as to classical n x n stochastic ma trices

generalized totally nonnegative matrices we define a new class of generalized totally nonnegative matrices shortly gtn matrices over a noncommutative ring with identity and a positive part the class of gtn matrices is closed under multiplication every submatrix of a gtn matrix is a gtn matrix as well we also define a standard form of a gtn matrix we then study the invertible gtn matrices of a fixed order and classify them the classes form a partially ordered set can be noncommutatively multiplied and multiplication preserves ordering

a qz method based on semiseparable matrices this manuscript focusses on an alternative method for computing the eigenvalues of a pencil of two matrices based on semiseparable matrices an effective reduction of a matrix pair to lower semiseparable upper triangular form will be presented as well as a qz iteration for this matrix pair important to remark is that this reduction procedure also inherits a kind of nested subspace iteration as was the case when solving the standard eigenvalue problem with semiseparable matrices it will also be shown that the qz iteration for a semiseparable triangular matrix pair is closely related to the qz iteration for a hessenberg triangular matrix pair

on the structure rank of a class of structure matrices abstract we introduce a class of structure matrices called transpose complementary structure matrices we give a characterization of the n extendable transpose complementary structure matrices with trace k where k 0 2 we also prove that a nonsingular matrix and its inverse have the same s rank for this class of structure matrices s

a note on companion matrices abstract we show that the usual companion matrix of a polynomial of degree n can be factored into a product of n matrices n 1 of them being the identity matrix in which a 2 2 identity submatrix in two consecutive rows and columns is replaced by an appropriate 2 2 matrix the remaining being the identity matrix with the last entry replaced by possibly different entry by a certain similarity transformation we obtain a simple new companion matrix in a pentadiagonal form some generalizations are also possible

a cholesky lr algorithm for the positive definite symmetric diagonal plus semiseparable eigenproblem abstract we present a cholesky lr algorithm with laguerre s shift for computing the eigenvalues of a positive definite symmetric diagonal plus semiseparable matrix by exploiting the semiseparable structure each step of the method can be performed in linear time

a model based method for retrospective correction of geometric distortions in diffusion weighted epi the self diffusion tensor may be calculated from several echo planar image acquisitions preceded by different diffusion gradients unfortunately these diffusion gradients cause geometric distortion that must be corrected before estimation of the tensor in the present paper we suggest and implement a method for retrospective correction of these distortions firmly based on a physical model for the diffusion weighted images this method simultaneously estimates subject movement and distortion parameters by finding the set of parameters that minimizes residual error when fitting data to the diffusion tensor model we show how this notion can be formalized as a quadratic form thereby facilitating the implementation of a rapid algorithm in addition we suggest models for how distortions vary with slice position and gradient direction that allow us to substantially reduce the dimensionality of the parameter space our results indicate that we are able to estimate both eddy current induced distortion and subject movement directly from the data without need of any additional measurements

automatic clustering and population analysis of white matter tracts using maximum density paths we introduce a framework for population analysis of white matter tracts based on diffusion weighted images of the brain the framework enables extraction of fibers from high angular resolution diffusion images hardi clustering of the fibers based partly on prior knowledge from an atlas representation of the fiber bundles compactly using a path following points of highest density maximum density path mdp and registration of these paths together using geodesic curve matching to find local correspondences across a population we demonstrate our method on 4 tesla hardi scans from 565 young adults to compute localized statistics across 50 white matter tracts based on fractional anisotropy fa experimental results show increased sensitivity in the determination of genetic influences on principal fiber tracts compared to the tract based spatial statistics tbss method our results show that the mdp representation reveals important parts of the white matter structure and considerably reduces the dimensionality over comparable fiber matching approaches

iterative aggregation disaggregation method for computing stationary probability vectors of markov type operators a global convergence of a class of iterative aggregation disaggregation methods is presented and some applications as well as some conclusions of numerical experiments performed are shown

local convergence of the exact and inexact iterative aggregation method for linear systems and markov operators the iterative aggregation method for the solution of linear systems is extended in several directions to operators on banach spaces to the method with inexact correction i e to methods where the inner linear system is in turn solved iteratively and to the problem of finding stationary distributions of markov operators local convergence is shown in all cases convergence results apply to the particular case of stochastic matrices moreover an argument is given which suggests why the iterative aggregation method works so well for nearly uncoupled markov chains as well as for markov chains with other zero nonzero structures

consecutive column and row properties of matrices and the loewner neville factorization abstract we say that a square matrix over a ring with identity has the consecutive column property if for all k all its relevant submatrices having k consecutive rows and the first k columns are invertible similarly the consecutive row cr is defined we show that analogously to the commutative case for totally positive matrices a matrix has both cc and cr properties if and only if it admits a certain loewner neville type factorization with invertible entries this factorization is unique since the result is proved for matrices in such generality it holds also for block matrices over a field with all blocks square explicit both ways formulae are found between two sets of parameters the loewner neville coefficients in the factorization and the schur complements of relevant submatrices in relevant submatrices larger by one we also show that for lower triangular matrices the cc property is preserved by inversion

complementary basic matrices abstract we show that an n n matrix which has both subdiagonal and superdiagonal rank at most one even if we distribute the diagonal positions except the first and last completely between the subdiagonal and superdiagonal part then this matrix can be factorized into a product of n 1 matrices each consisting of a 2 2 principal submatrix in two consecutive rows and columns in all possible of the n 1 positions and completed by ones along the diagonal the converse is also true it is shown that the spectrum does not depend on the order of the factors

suborthogonality and orthocentricity of matrices we present some results on submatrices of orthogonal and unitary matrices and their relation to so called orthocentric matrices these are then completely characterized

intrinsic products and factorizations of matrices we say that the product of a row vector and a column vector is intrinsic if there is at most one nonzero product of corresponding coordinates analogously we speak about intrinsic product of two or more matrices as well as about intrinsic factorizations of matrices since all entries of the intrinsic product are products of entries of the multiplied matrices there is no addition we present several examples together with important applications these applications include companion matrices and sign nonsingular matrices

improved optimization for the robust and accurate linear registration and motion correction of brain images linear registration and motion correction are important components of structural and functional brain image analysis most modern methods optimize some intensity based cost function to determine the best registration to date little attention has been focused on the optimization method itself even though the success of most registration methods hinges on the quality of this optimization this paper examines the optimization process in detail and demonstrates that the commonly used multiresolution local optimization methods can and do get trapped in local minima to address this problem two approaches are taken 1 to apodize the cost function and 2 to employ a novel hybrid global local optimization method this new optimization method is specifically designed for registering whole brain images it substantially reduces the likelihood of producing misregistrations due to being trapped by local minima the increased robustness of the method compared to other commonly used methods is demonstrated by a consistency test in addition the accuracy of the registration is demonstrated by a series of experiments with motion correction these motion correction experiments also investigate how the results are affected by different cost functions and interpolation methods

a note on generalized hessenberg matrices two different generalizations of the concept of hessenberg matrices have appeared recently in the literature in this note we study their connections in more detail and give a general structure theorem for these classes of matrices

a nonlinear model involving m operators an amplification effect measured in the cascade of vision an amplification effect measured in the cascade of vision forms a basis to formulate an abstract nonlinear model offering a generalization of the cascade problem this model is analyzed and some conditions concerning the existence and or possible uniqueness and multiplicity of solutions are presented an essential feature of the model is that the appropriate operators governing the problem are not surjective for the case of unique solvability a numerical procedure is proposed and its convergence is proved

anatomicuts hierarchical clustering of tractography streamlines based on anatomical similarity diffusion mri tractography produces massive sets of streamlines that contain a wealth of information on brain connections the size of these datasets creates a need for automated clustering methods to group the streamlines into meaningful bundles conventional clustering techniques group streamlines based on their spatial coordinates neuroanatomists however define white matter bundles based on the anatomical structures that they go through or next to rather than their spatial coordinates thus we propose a similarity measure for clustering streamlines based on their position relative to cortical and subcortical brain regions we incorporate this measure into a hierarchical clustering algorithm and compare it to a measure that relies on euclidean distance using data from the human connectome project we show that the anatomical similarity measure leads to a 20 improvement in the overlap of clusters with manually labeled tracts importantly this is achieved without introducing any prior information from a tract atlas into the clustering algorithm therefore without imposing the existence of any named tracts

a remark on oscillatory matrices in this note we revisit a classical criterion obtained by gantmacher and krein for determining when a totally nonnegative matrix is actually oscillatory a new proof of this criterion is presented by incorporating bidiagonal factorizations of invertible totally nonnegative matrices and utilizing certain associated weighted planar diagrams

structured eigenvalue condition numbers for parameterized quasiseparable matrices the development of fast algorithms for performing computations with n times n n n low rank structured matrices has been a very active area of research during the last two decades as a consequence of the numerous applications where these matrices arise the key ideas behind these fast algorithms are that low rank structured matrices can be described in terms of o n parameters and that these algorithms operate on the parameters instead on the matrix entries therefore the sensitivity of any computed quantity should be measured with respect to the possible variations that the parameters defining these matrices may suffer since this determines the maximum accuracy of a given fast computation in other words it is necessary to develop condition numbers with respect to parameters for different magnitudes and classes of low rank structured matrices but as far as we know this has not yet been accomplished in any case in this paper we derive structured relative eigenvalue condition numbers for the important class of low rank structured matrices known as 1 1 1 1 quasiseparable matrices with respect to relative perturbations of the parameters in the quasiseparable and in the givens vector representations of these matrices and we provide fast algorithms for computing them comparisons among the new structured condition numbers and the unstructured one are also presented as well as numerical experiments showing that the structured condition numbers can be small in situations where the unstructured one is huge in addition the approach presented in this paper is general and may be extended to other problems and classes of low rank structured matrices

a note on the nullity theorem in this paper we take a closer look at the nullity theorem as formulated by markham and fiedler in 1986 the theorem is a valuable tool in the computations with structured rank matrices it connects ranks of subblocks of an invertible matrix a with ranks of other subblocks in his inverse a 1 a little earlier barrett and feinsilver 1981 proved a theorem very close to the nullity theorem but restricted to semiseparable and tridiagonal matrices which are each others inverses we will adapt the ideas of barrett and feinsilver to come to a new alternative proof of the nullity theorem based on determinantal formulas in the second part of the paper we extend the nullity theorem to make it suitable for two types of decompositions namely the lu and the qr decomposition these theorems relate the ranks of subblocks of the factors l u and q to the ranks of subblocks of the factored matrix it is shown that a combination of the nullity theorem and his extended versions is suitable to predict in an easy manner the structure of decompositions and or of inverses of structured rank matrices e g higher order band higher order semiseparable hessenberg and many other types of matrices as examples to show the power of the nullity theorem and the related theorems we apply them to semiseparable and related matrices

generalized totally positive matrices abstract we say that a rectangular matrix over a in general noncommutative ring with identity having a positive part is generalized totally positive gtp if in all nested sequences of so called relevant submatrices the schur complements are positive here a relevant submatrix is such either having k consecutive rows and the first k columns or k consecutive columns and the first k rows this notion generalizes the usual totally positive matrices we prove e g that a square matrix is gtp if and only if it admits a certain factorization with bidiagonal type factors and certain invertible entries also the product of square gtp matrices of the same order is again a gtp matrix and its inverse has the checkerboard sign property

automatic fiber bundle segmentation in massive tractography datasets using a multi subject bundle atlas this paper presents a method for automatic segmentation of white matter fiber bundles from massive dmri tractography datasets the method is based on a multi subject bundle atlas derived from a two level intra subject and inter subject clustering strategy this atlas is a model of the brain white matter organization computed for a group of subjects made up of a set of generic fiber bundles that can be detected in most of the population each atlas bundle corresponds to several inter subject clusters manually labeled to account for subdivisions of the underlying pathways often presenting large variability across subjects an atlas bundle is represented by the multi subject list of the centroids of all intra subject clusters in order to get a good sampling of the shape and localization variability the atlas composed of 36 known deep white matter bundles and 47 superficial white matter bundles in each hemisphere was inferred from a first database of 12 brains it was successfully used to segment the deep white matter bundles in a second database of 20 brains and most of the superficial white matter bundles in 10 subjects of the same database