adaptive hp finite element viscoelastic flow calculations accuracy and stability remain key issues in viscoelastic flow simulation classical low order finite element techniques fail to converge when the elasticity of the fluid is increased in this paper an adaptive hp finite element method is used to solve differential viscoelastic flow problems an a posteriori error estimator based on some recent rigorous results of oden and wu for the navier stokes equations 1 is also used starting from an initial mesh local refinements h adaptivity or enrichments p adaptivity are applied in the spirit of the strategy proposed in 2 the approximation error is reduced to a given level of accuracy with a minimal set of additional degrees of freedom numerical results on two 2d model problems illustrate both the validity of the error estimator presented and the efficiency of the adaptive procedure

discriminant subspace analysis a fukunaga koontz approach the fisher linear discriminant fld is commonly used in pattern recognition it finds a linear subspace that maximally separates class patterns according to the fisher criterion several methods of computing the fld have been proposed in the literature most of which require the calculation of the so called scatter matrices in this paper we bring a fresh perspective to fld via the fukunaga koontz transform fkt we do this by decomposing the whole data space into four subspaces with different discriminabilities as measured by eigenvalue ratios by connecting the eigenvalue ratio with the generalized eigenvalue we show where the fisher criterion is maximally satisfied we prove the relationship between fld and fkt analytically and propose a unified framework to understanding some existing work furthermore we extend our theory to the multiple discriminant analysis mda this is done by transforming the data into intraclass and extraclass spaces followed by maximizing the bhattacharyya distance based on our fkt analysis we identify the discriminant subspaces of mda fkt and propose an efficient algorithm which works even when the scatter matrices are singular or too large to be formed our method is general and may be applied to different pattern recognition problems we validate our method by experimenting on synthetic and real data

a posteriori error estimators for the stokes and oseen equations the problem of obtaining posteriori estimates of the discretization error when one uses finite element methods to approximate problems with an incompressibility constraint is discussed a general approach to the treatment of the constraint condition and to the possible non self adjointness of the associated momentum equations is presented a posteriori error estimates are derived for adaptive h p and h p type finite element schemes key features are that the local error residual problems are not subject to an incompressibility constraint thereby avoiding the need for special finite element schemes and that the analysis is valid for essentially any discretization scheme including continuous and discontinuous pressure spaces the estimator bounds the actual error measured in an energy like norm

towards a generalized singular value decomposition we suggest a form for and give a constructive derivation of the generalized singular value decomposition of any two matrices having the same number of columns we outline its desirable characteristics and compare it to an earlier suggestion by van loan siam j numer anal 13 1976 pp 76 83 the present form largely follows from the work of van loan but is slightly more general and computationally more amenable than that in the paper cited we also prove a useful extension of a theorem of stewart siam rev 19 1977 pp 634 662 on unitary decompositions of submatrices of a unitary matrix

wiretap channel with causal state information and secure rate limited feedback in this paper we consider the secrecy capacity of a wiretap channel in the presence of causal state information and secure rate limited feedback in this scenario the causal state information from the channel is available to both the legitimate transmitter and the legitimate receiver in addition the legitimate receiver can send secure feedback to the transmitter at a limited rate r f we derive upper and lower bounds on the secrecy capacity and show that when the channel to the eavesdropper is degraded the bounds are tight and the secrecy capacity is completely characterized the capacity achieving scheme is based on wyner csiszar and korner wiretap coding and two steps of shared key generation one from the state information and one via the noiseless feedback the upper bound is more involved and requires a nontrivial recursive lemma extending previous results in the literature to include both state and feedback we conclude the paper by showing that a few interesting known results can be seen as special cases of the above as well as discussing the case where the source of local randomness at the encoder is limited

a dual mixed approximation method for a three field model of a nonlinear generalized stokes problem in this work a dual mixed approximation of a nonlinear generalized stokes problem is studied the problem is analyzed in sobolev spaces which arise naturally in the problem formulation existence and uniqueness results are given and error estimates are derived it is shown that both lowest order and higher order mixed finite elements are suitable for the approximation method numerical experiments that support the theoretical results are presented

history and generality of the cs decomposition abstract it is almost a quarter of a century since chandler davis and william kahan brought together the key ideas of what stewart later completed and defined to be the cs decomposition csd of a partitioned unitary matrix this paper outlines some germane points in the history of the csd pointing out the contributions of jordan of davis and kahan and of stewart and the relationship of the csd to the direct rotation of davis and kato the paper provides an easy to memorize constructive proof of the csd reviews one of its important uses and suggests a motivation for the csd which emphasizes how generally useful it is it shows the relation between the csd and generalized singular value decompositions and points out some useful nullity properties one form of the csd trivially reveals finally it shows how via the qr factorization the csd can be used to obtain interesting results for partitioned nonsingular matrices we suggest the csd be taught in its most general form with no restrictions on the two by two partition and initially with no mention of angles between subspaces

analysis and finite element approximation of an optimal control problem for the oseen viscoelastic fluid flow in this article we study a boundary control problem for an oseen type model of viscoelastic fluid flow the existence of a unique optimal solution is proved and an optimality system is derived by the first order necessary condition we investigate finite element approximations to a solution of the optimality system and a solution algorithm for the system based on the gradient method

perturbation analysis for the generalized singular value problem this paper discusses perturbation bounds for generalized singular values and for subspaces associated with the generalized singular value decomposition of two matrices having the same number of columns by a limiting procedure from our results we can derive those for the ordinary singular value problem which were proved by l mirsky quart j math oxford 11 1960 pp 50 59 and p a wedin bit 12 1972 pp 99 111

singular value and generalized singular value decompositions and the solution of linear matrix equations abstract the solution of the linear matrix equations i axb cyd e and ii axb fxg e h are considered new necessary and sufficient conditions for the consistency of the equations are derived some using the generalized singular value decomposition special cases iii ax yd e and iv axb e are treated using the singular value decomposition numerical algorithms for the solutions are also suggested

regularized discriminant analysis abstract linear and quadratic discriminant analysis are considered in the small sample high dimensional setting alternatives to the usual maximum likelihood plug in estimates for the covariance matrices are proposed these alternatives are characterized by two parameters the values of which are customized to individual situations by jointly minimizing a sample based estimate of future misclassification risk computationally fast implementations are presented and the efficacy of the approach is examined through simulation studies and application to data these studies indicate that in many circumstances dramatic gains in classification accuracy can be achieved

the truncated k moment problem for closure of open sets we solve the truncated k moment problem when k subseteq r n is the closure of a not necessarily bounded open set which includes the important cases k r n and k r n that is we completely characterize the interior of the convex cone of finite sequences that have a representing measure on k subseteq r n it is in fact the domain of the legendre fenchel transform associated with a certain convex function and so in this context detecting whether a sequence is in the interior of this cone reduces to solving a finite dimensional convex optimization problem this latter problem is related to maximum entropy methods for approximating an unknown density from knowing only finitely many of its moments interestingly the proposed approach is essentially geometric and of independent interest as it also addresses the abstract problem of characterizing the interior of a convex cone c which is the conical hull of a set continuously parametrized by a compact set m subset r n or m subseteq s n 1 where m is the closure of an open subset of r n resp s n 1 as a by product we also obtain a barrier function for the cone c

numerical approximation of a quasi newtonian stokes flow problem with defective boundary conditions in this article we study the numerical approximation of a quasi newtonian stokes flow problem where only the flow rates are specified at the inflow and outflow boundaries a variational formulation of the problem using lagrange multipliers to enforce the stated flow rates is given the existence and the uniqueness to the continuous and discrete variational formulations of the solution are shown an error analysis for the numerical approximation is also given numerical computations are included which demonstrate the approximation scheme studied

extrapolation techniques for ill conditioned linear systems in this paper the regularized solutions of an ill conditioned system of linear equations are computed for several values of the regularization parameter lambda then these solutions are extrapolated at lambda 0 by various vector rational extrapolations techniques built for that purpose these techniques are justified by an analysis of the regularized solutions based on the singular value decomposition and the generalized singular value decomposition numerical results illustrate the effectiveness of the procedures

a general representation theorem for partially ordered commutative rings an extension of the kadison dubois representation theorem is proved this extends both the classical version 3 and the preordering version given by jacobi in 5 it is then shown how this can be used to sharpen the results on representations of strictly positive polynomials given by jacobi and prestel in 6

a posteriori estimators for mixed finite element approximations of a fluid obeying the power law we study a posteriori error estimation in the approximation by the finite element method of a fluid which obeys the power law 2 d u r 2d u p f u 0 1 r 2 abstract and calculable a posteriori estimators are given in two fields and in a three fields version of this problem furthermore we give some examples of fe spaces satisfying the inf sup condition which relates the discrete space of tensors and the discrete space of velocities needed for this formulation

subspace model identification part 2 analysis of the elementary output error state space model identification algorithm the elementary moesp algorithm presented in the first part of this series of papers is analysed in this paper this is done in three different ways first we study the asymptotic properties of the estimated state space model when only considering zero mean white noise perturbations on the output sequence it is shown that in this case the moespl implementation yields asymptotically unbiased estimates an important constraint to this result is that the underlying system must have a finite impulse response and subsequently the size of the hankel matrices constructed from the input and output data at the beginning of the computations depends on the number of non zero markov parameters this analysis however leads to a second implementation of the elementary moesp scheme namely moesp2 the latter implementation has the same asymptotic properties without the finite impulse response constraint secondly we compare the moesp2 algorithm with a classical state space model identification scheme the latter

an extension of analytic solutions of a class of constrained matrix minimization problems abstract in this paper we consider an extension of analytic solutions of a class of constrained matrix minimization problems and propose analytic solutions of the following matrix minimization problem min s k s k h i n l k l k h i m det c i m k 1 s a k s k b k l k min s k s k h i n l k l k h i m t r c i m k 1 s a k s k b k l k where a k c m n b k c n m c is a complex number i m denotes m order identity matrix and det t r denote matrix determinant and trace function the proposed results improve the corresponding existing ones in xu et al 2019 numerical examples are provided to illustrate the efficiency of the proposed theoretical results

perturbation analysis of the canonical subspaces abstract this paper presents a perturbation analysis of the canonical subspaces of a matrix pair a b it is shown that if the perturbations in the matrices a b do not change the ranks of each matrix then the perturbation bounds of the canonical subspaces depend upon the perturbations and the scaled condition numbers of the matrix pair as well as upon the gap of the corresponding canonical correlations

perturbation theory for the rank deficient equality constrained least squares problem this paper presents perturbation theory for the equality constrained least squares problem min f in s kf g 2 with s f lf h 2 min y in c n ly h 2 lse in which l and begin array 20 c l k end array may not be of full rank the analysis generalizes the results of elden 4 perturbation theorems for the lse problem are proved numerical experiments are also given to verify the error bounds

perturbation analysis for block downdating of a cholesky decomposition let the cholesky decomposition of x rm t x be x rm t x r rm t r where r is upper triangular the cholesky block downdating problem is to determine tilde r such that tilde r rm t tilde r r rm t r z rm t z where z is a block of rows from the data matrix x we analyze the sensitivity of this block downdating problem of the cholesky factorization a measure of conditioning for the cholesky block downdating is presented and compared to the single row downdating case

coupled generalized nonlinear stokes flow with flow through a porous medium in this article we analyze the flow of a fluid through a coupled stokes darcy domain the fluid in each domain is non newtonian modeled by the generalized nonlinear stokes equation in the free flow region and the generalized nonlinear darcy equation in the porous medium a flow rate is specified along the inflow portion of the free flow boundary we show existence and uniqueness of a variational solution to the problem we propose and analyze an approximation algorithm and establish a priori error estimates for the approximation

equivalent formulae for the supremum and stability of weighted pseudoinverses during recent decades there have been a great number of research articles studying interior point methods for solving problems in mathematical programming and constrained optimization stewart and o leary obtained an upper bound for scaled pseudoinverses sup w1 2x w1 2 2 of a matrix x w p where p is a set of diagonal positive definite matrices we improved their results to obtain the supremum of scaled pseudoinverses and derived the stability property of scaled pseudoinverses forsgren further generalized these results to derive the supremum of weighted pseudoinverses sup w1 2x w1 2 2 where w p p is a set of diagonally dominant positive semidefinite matrices by using a signature decomposition of weighting matrices w and by applying the binet cauchy formula and cramer s rule for determinants the results are also extended to equality constrained linear least squares problems in this paper we extend forsgren s results to a general complex matrix x to establish several equivalent formulae for sup w1 2x w1 2 2 where p is a w p set of diagonally dominant positive semidefinite matrices or a set of weighting matrices arising from solving equality constrained least squares problems we also discuss the stability property of these weighted pseudoinverses

a simultaneous decomposition for seven matrices with applications abstract let h m n be the set of all m n matrices over the quaternion algebra h in this paper we construct a simultaneous decomposition for seven matrices with compatible sizes a h m n b h m p 1 c h m p 2 d h m p 3 e h q 1 n f h q 2 n and g h q 3 n as applications of the simultaneous matrix decomposition we give some solvability conditions general solutions as well as the range of ranks of the general solutions to the following two generalized sylvester matrix equations b x e c y f d z g a and b x w e c y f d z g a where a b c d e f and g are given quaternion matrices moreover we provide some numerical examples to illustrate the results of this paper

secure transmission with multiple antennas part ii the mimome wiretap channel the capacity of the gaussian wiretap channel model is analyzed when there are multiple antennas at the sender intended receiver and eavesdropper the associated channel matrices are fixed and known to all the terminals a computable characterization of the secrecy capacity is established as the saddle point solution to a minimax problem the converse is based on a sato type argument used in other broadcast settings and the coding theorem is based on gaussian wiretap codebooks at high signal to noise ratio snr the secrecy capacity is shown to be attained by simultaneously diagonalizing the channel matrices via the generalized singular value decomposition and independently coding across the resulting parallel channels the associated capacity is expressed in terms of the corresponding generalized singular values it is shown that a semi blind masked multi input multi output mimo transmission strategy that sends information along directions in which there is gain to the intended receiver and synthetic noise along directions in which there is not can be arbitrarily far from capacity in this regime necessary and sufficient conditions for the secrecy capacity to be zero are provided which simplify in the limit of many antennas when the entries of the channel matrices are independent and identically distributed the resulting scaling laws establish that to prevent secure communication the eavesdropper needs three times as many antennas as the sender and intended receiver have jointly and that the optimum division of antennas between sender and intended receiver is in the ratio of 2 1

compatible spectral approximations for the velocity pressure stress formulation of the stokes problem the numerical approximation of the mixed velocity pressure stress formulation of the stokes problem using spectral methods is considered in addition to the compatibility condition between the discrete velocity and pressure spaces a second condition between the discrete velocity and stress spaces must also be satisfied in order to have a well posed problem the theory is developed by considering a doubly constrained minimization problem in which the viscous stress tensor is minimized subject to the constraint that the viscous forces are irrotational the discrete problem is analyzed and error estimates are derived a comparison between the mixed approach and the standard velocity pressure formulation is made in terms of the condition number of the resulting systems

a tree of generalizations of the ordinary singular value decomposition abstract it is shown how to generalize the ordinary singular value decomposition of a matrix into a combined factorization of any number of matrices we propose to call these factorizations generalized singular value decompositions for two matrices this reduces to the product and quotient singular value decompositions one of the factorizations for three matrices is the restricted singular value decomposition these generalizations form a tree of factorizations where at level k for k matrices there are 2 k factorizations not all of which are independent the different levels are related to each other in a recursive fashion any generalized singular value decomposition for k matrices can be constructed from a decomposition for k 1 matrices this results in an inductive proof which uses only the ordinary singular value decomposition several examples are analysed in detail

powers of totally positive subgroups of fields abstract for a formally real field f a totally positive co torsion subgroup s of f and m 1 we prove that s m s m this extends a result of becker we prove several refinements of this fact related to higher pythagoras numbers real holomorphy rings and sums with mixed powers

generalized epsilon pseudospectra normal matrices have complete sets of orthonormal eigenvectors and therefore the spectral decomposition is useful in studying the properties of normal operators in contrast the eigenvectors of nonnormal matrices can be nearly linearly dependent and the eigenvalue problem may by highly ill conditioned thus additional concepts and analysis techniques are useful in examining nonnormal operators the concepts of epsilon pseudospectra introduced by l n trefethen has proven to be a powerful tool in the analysis of nonnormal operators the author generalizes epsilon pseudospectra and the associated computational algorithms to the generalized eigenvalue problem rank 1 perturbations are used to determine the epsilon pseudospectra

on the symmetric solutions of linear matrix equations abstract the symmetric solutions of linear matrix equations ax c and axb c are considered the necessary and sufficient conditions for the consistency of the equations with a symmetric condition on solutions are derived using the singular value decomposition and the generalized singular value decomposition numerical algorithms for solving the equations are suggested

polynomial programming lp relaxations also converge we consider the global minimization of a multivariate polynomial on a compact semialgebraic set using a result of krivine j analyse math 12 1964 pp 307 326 c r acad sci paris 258 1964 pp 3417 3418 it is shown that lp relaxations based on products of the original constraints in the spirit of the reformulation linearization technique procedure of sherali and adams converge to the global optimum this extends to compact semialgebraic sets previous results known for polytopes and 0 1 programs the asymptotic behavior of the lp relaxations is also analyzed and it is shown that in important cases finite convergence cannot occur

the matrix equations ax b xc d with px sxp constraint abstract the matrix equations ax b xc d with the constraint px sxp are considered where p is a given hermitian matrix satisfying p 2 i and s 1 using an eigenvalue decomposition of p the constrained problem can be equivalently transformed to two independent unconstrained problems of matrix equations in similar forms and hence the constrained problem can be solved in terms of the eigenvectors of p in this paper a simple and eigenvector free formula of the general solutions to the constrained problem is presented using moore penrose generalized inverses of the coefficient matrices a and c a similar problem of the matrix equations with generalized constraint is discussed too we also consider the least norm solution of the constrained problems

the symmetric solution of the matrix equations ax ya c axat bybt c and atxa btxb c d abstract the necessary and sufficient conditions for the existence of and the expressions for the symmetric solutions of matrix equations i ax ya c ii axa t byb t c and iii a t xa b t xb c d are derived in addition the minimum 2 norm least squares symmetric solution of equation i the minimum 2 norm symmetric solution of equation ii and the least squares solution of equation iii are obtained

the general linear model and the generalized singular value decomposition abstract a new derivation is given for the generalized singular value decomposition of two matrices x and f having the same number of rows it is shown how this decomposition reveals the structure of the general gauss markov linear model y x 2 ff and exhibits the structure and solution of the generalized linear least squares problem used to provide the best linear unbiased estimator for the model the decomposition is used to prove optimality of the estimator and to reveal the structure of the covariance matrix of the error of the estimator

goal oriented error estimation and adaptivity for the finite element method abstract in this paper we study a new approach in a posteriori error estimation in which the numerical error of finite element approximations is estimated in terms of quantities of interest rather than the classical energy norm these so called quantities of interest are characterized by linear functionals on the space of functions to where the solution belongs we present here the theory with respect to a class of elliptic boundary value problems and in particular show how to obtain accurate estimates as well as upper and lower bounds on the error we also study the new concept of goal oriented adaptivity which embodies mesh adaptation procedures designed to control error in specific quantities numerical experiments confirm that such procedures greatly accelerate the attainment of local features of the solution to preset accuracies as compared to traditional adaptive schemes based on energy norm error estimates

linear unbiased approximators of the disturbances in the standard linear model abstract the least squares residuals from the standard linear model have a variance matrix which is a function of the n q matrix of observations on the regressors we examine two classes of residuals which do not suffer from this defect our first class of residuals luz residuals has a variance matrix which is a scalar multiple of an n n idempotent matrix of rank n q specified by the user and our second class of residuals lus residuals has a variance matrix which is a scalar multiple of the n q n q identity matrix

a subdomain residual method for a posteriori error estimates of finite element solutions of viscoelastic fluid flows abstract in this paper we present a subdomain residual method which gives a posteriori estimates of the errors in finite element solutions of viscoelastic fluid flows we employ a coupled approach in which the estimated errors in both velocity and extra stress are computed simultaneously by solving a series of local problems the subdomains on which the local problems are posed consist of patches of elements a given parameter which is independent of the element size is employed to determine the size of the subdomains to ensure that the error estimate improves as the element size is reduced the error estimator is applied to finite element solutions of newtonian and non newtonian fluid flows past a cylindrical tube in a duct the contour plots of the components of the estimated errors in velocity and extra stress are presented these errors are compared with reference errors

a low order mixed finite element method for a class of quasi newtonian stokes flows part i a priori error analysis abstract this is the second part of a work dealing with a low order mixed finite element method for a class of nonlinear stokes models arising in quasi newtonian fluids in the first part we showed that the resulting variational formulation is given by a twofold saddle point operator equation and that the corresponding galerkin scheme becomes well posed with piecewise constant functions and raviart thomas spaces of lowest order as the associated finite element subspaces in this paper we develop a bank weiser type a posteriori error analysis yielding a reliable estimate and propose the corresponding adaptive algorithm to compute the mixed finite element solutions several numerical results illustrating the efficiency of the method are also provided

residual a posteriori error estimator for a three field model of a non linear generalized stokes problem in this article we propose and analyze an a posteriori error estimator for a three field model of a generalized stokes problem the components of the a posteriori error estimator are defined via a non linear projection of the residues of the variational equations both upper and lower bounds for the approximation error are derived in terms of the components of the a posteriori error estimator the non linear projections do not need to be explicitly computed to construct the a posteriori error estimates

two sided coupled generalized sylvester matrix equations solving using a simultaneous decomposition for fifteen matrices abstract in this paper we investigate and analyze in detail the structure and properties of a simultaneous decomposition for fifteen matrices a i c p i t i b i c s i q i c i c p i t i 1 d i c s i 1 q i and e i c p i q i i 1 2 3 we show that from this simultaneous decomposition we can derive some necessary and sufficient conditions for the existence of a solution to the system of two sided coupled generalized sylvester matrix equations with four unknowns a i x i b i c i x i 1 d i e i i 1 2 3 apart from proving an expression for the general solutions to this system we derive the range of ranks of these solutions using the ranks of the given matrices a i b i c i d i and e i we provide some numerical examples to illustrate our results moreover we present a similar approach to consider the simultaneous decomposition for 5k matrices and the system of k two sided coupled generalized sylvester matrix equations with k 1 unknowns a i x i b i c i x i 1 d i e i i 1 k k 4 the main results are also valid over the real number field and the real quaternion algebra

the matrix nearness problem for symmetric matrices associated with the matrix equation atxa btxb c d abstract a direct method based on the projection theorem in inner products spaces the generalized singular value decomposition and the canonical correlation decomposition is presented for finding the optimal approximate solution x in the set s e to a given matrix x where s e denotes the least squares symmetric solution set of the matrix equation atxa btxb c d the analytical expression of the optimal approximate solution x is obtained and an algorithm for finding this solution is also suggested

optimal control for quasi newtonian flows with defective boundary conditions we consider a generalized newtonian fluid with defective boundary conditions where only flow rates or mean pressures are prescribed on parts of boundary the defect boundary condition problem is formulated as an optimal control problem in which a neumann or dirichlet boundary control is used for matching given flow rates or mean pressures for the constrained optimization problem an optimality system is derived from which a solution of the problem is obtained computational algorithms are discussed and numerical results are also presented

multiple input multiple output gaussian broadcast channels with confidential messages this paper considers the problem of secret communication over a two receiver multiple input multiple output mimo gaussian broadcast channel the transmitter has two independent messages each of which is intended for one of the receivers but needs to be kept asymptotically perfectly secret from the other it is shown that surprisingly under a matrix power constraint both messages can be simultaneously transmitted at their respective maximal secrecy rates to prove this result the mimo gaussian wiretap channel is revisited and a new characterization of its secrecy capacity is provided via a new coding scheme that uses artificial noise an additive prefix channel and random binning

the moment problem on compact semi algebraic sets in this chapter we begin the study of the multidimensional moment problem the passage to dimensions d 2 brings new difficulties and unexpected phenomena in sect 3 2 we derived solvability criteria of the moment problem on intervals in terms of positivity conditions it seems to be natural to look for similar characterizations in higher dimensions as well this leads us immediately into the realm of real algebraic geometry and to descriptions of positive polynomials on semi algebraic sets in this chapter we treat this approach for basic closed compact semi algebraic subsets of mathbb r d it turns out that for such sets there is a close interaction between the moment problem and positivstellensatze for strictly positive polynomials

positivstellensatze for real function algebras we look for algebraic certificates of positivity for functions which are not necessarily polynomial functions similar questions were examined earlier by lasserre and putinar positivity and optimization for semi algebraic functions to appear proposition 1 and by putinar a striktpositivestellensatz for measurable functions corrected version to appear theorem 2 1 we explain how these results can be understood as results on hidden positivity the required positivity of the functions implies their positivity when considered as polynomials on the real variety of the respective algebra of functions this variety is however not directly visible in general we show how algebras and quadratic modules with this hidden positivity property can be constructed we can then use known results for example jacobi s representation theorem jacobi in math z 237 259 273 2001 theorem 4 or the krivine stengle positivstellensatz marshall in positive polynomials and sums of squares mathematical surveys and monographs 146 2008 page 25 to obtain certificates of positivity relative to a quadratic module of an algebra of real valued functions our results go beyond the results of lasserre and putinar for example when dealing with non continuous functions the conditions are also easier to check we explain the application of our result to various sorts of real finitely generated algebras of semialgebraic functions the emphasis is on the case where the quadratic module is also finitely generated our results also have application to optimization of real valued functions using the semidefinite programming relaxation methods pioneered by lasserre siam j optim 11 3 796 817 2001 lasserre in moments positive polynomials and their applications imperial college press london 2009 lasserre and putinar in positivity and optimization for semi algebraic functions to appear marshall in positive polynomials and sums of squares mathematical surveys and monographs 146 2008 page 25

dimension reduction in text classification with support vector machines support vector machines svms have been recognized as one of the most successful classification methods for many applications including text classification even though the learning ability and computational complexity of training in support vector machines may be independent of the dimension of the feature space reducing computational complexity is an essential issue to efficiently handle a large number of terms in practical applications of text classification in this paper we adopt novel dimension reduction methods to reduce the dimension of the document vectors dramatically we also introduce decision functions for the centroid based classification algorithm and support vector classifiers to handle the classification problem where a document may belong to multiple classes our substantial experimental results show that with several dimension reduction methods that are designed particularly for clustered data higher efficiency for both training and testing can be achieved without sacrificing prediction accuracy of text classification even when the dimension of the input space is significantly reduced

the expansion problem of anti symmetric matrix under a linear constraint and the optimal approximation this paper mainly discusses the following two problems problem i given a rn m b rm m x0 asrq q the set of q q anti symmetric matrices find x asrn n such that atxa b x0 x 1 q where x 1 q is the q q leading principal submatrix of matrix x problem ii given x rn n find x se such that x x minx se x x where is the frobenius norm and se is the solution set of problem i the necessary and sufficient conditions for the existence of and the expressions for the general solutions of problem i are given moreover the optimal approximation solution an algorithm and a numerical example of problem ii are provided

an optimization criterion for generalized discriminant analysis on undersampled problems an optimization criterion is presented for discriminant analysis the criterion extends the optimization criteria of the classical linear discriminant analysis lda through the use of the pseudoinverse when the scatter matrices are singular it is applicable regardless of the relative sizes of the data dimension and sample size overcoming a limitation of classical lda the optimization problem can be solved analytically by applying the generalized singular value decomposition gsvd technique the pseudoinverse has been suggested and used for undersampled problems in the past where the data dimension exceeds the number of data points the criterion proposed in this paper provides a theoretical justification for this procedure an approximation algorithm for the gsvd based approach is also presented it reduces the computational complexity by finding subclusters of each cluster and uses their centroids to capture the structure of each cluster this reduced problem yields much smaller matrices to which the gsvd can be applied efficiently experiments on text data with up to 7 000 dimensions show that the approximation algorithm produces results that are close to those produced by the exact algorithm

some new properties of the equality constrained and weighted least squares problem abstract in this paper some new properties of the equality constrained and weighted least squares problem wlse min w 1 2 kx g 2 subject to lx h are obtained we derive a perturbation bound based on an unconstrained least squares problem and deduce some equivalent formulae for the projectors of this unconstrained ls problem we also present a new way to compute the minimum norm solution x wlse of the wlse problem by using the qr decomposition of the corresponding matrices and propose an algorithm to compute x wlse using the qr factorizations some numerical examples are provided to compare different methods for solving the wlse problem

archimedean operator theoretic positivstellensatze abstract we prove a general archimedean positivstellensatz for hermitian operator valued polynomials and show that it implies the multivariate fejer riesz theorem of dritschel rovnyak and positivstellensatze of ambrozie vasilescu and scherer hol we also obtain several generalizations of these and related results the proof of the main result depends on an extension of the abstract archimedean positivstellensatz for algebras that is interesting in its own right

linear matrix equations from an inverse problem of vibration theory abstract the symmetric positive semidefinite and positive definite real solutions of matrix equations a t xa d and a t xa xa y ad d 0 are considered necessary and sufficient conditions for the existence of such solutions and their general forms are derived using the singular value decomposition the theory is motivated and illustrated with a problem of vibration theory

novel fisher discriminant classifiers at the present several applications need to classify high dimensional points belonging to highly unbalanced classes unfortunately when the training set cardinality is small compared to the data dimensionality small sample size problem the classification performance of several well known classifiers strongly decreases similarly the classification accuracy of several discriminative methods decreases when non linearly separable and unbalanced classes are treated in this paper we firstly survey state of the art methods that employ improved versions of linear discriminant analysis lda to deal with the above mentioned problems secondly we propose a family of classifiers based on the fisher subspace estimation which efficiently deal with the small sample size problem non linearly separable classes and unbalanced classes the promising results obtained by the proposed techniques on benchmark datasets and the comparison with state of the art predictors show the efficacy of the proposed techniques

mimo multiple access channel with an arbitrarily varying eavesdropper secrecy degrees of freedom a two transmitter gaussian multiple access wiretap channel with multiple antennas at each of the nodes is investigated the channel matrices of the legitimate users are fixed and revealed to all the terminals whereas the channel matrices of the eavesdropper are arbitrarily varying and only known to the eavesdropper the secrecy degrees of freedom s d o f region under a strong secrecy constraint is characterized a transmission scheme that orthogonalizes the transmit signals of the two users at the intended receiver and uses a single user wiretap code for each user is shown to achieve the s d o f region the converse involves establishing an upper bound on a weighted sum rate expression this is accomplished by using induction where at each step one combines the secrecy and multiple access constraints associated with an adversary eavesdropping a carefully selected group of sub channels

the m n symmetric procrustes problem abstract an p q matrix a is said to be m n symmetric if man man t for given m r n p n r q n in this paper the following m n symmetric procrustes problem is studied find the m n symmetric matrix a which minimizes the frobenius norm of ax b where x and b are given rectangular matrices we use project theorem the singular value decomposition and the generalized singular value decomposition of matrices to analysis the problem and to derive a stable method for its solution the related optimal approximation problem to a given matrix on the solution set is solved furthermore the algorithm to compute the optimal approximate solution and the numerical experiment are given

inverse problems for nonsymmetric matrices with a submatrix constraint abstract in this paper the following problems are considered problem i a given matrices x r n p with full column rank b r p p and a 0 r r r find a matrix a r n n such that x t ax b a 1 r a 0 where a 1 r is the r r leading principal submatrix of the matrix a problem i b given matrices x r n p b r p p and a 0 r r r find a matrix a r n n such that x t ax b min s t a 1 r a 0 problem ii given a matrix a r n n with a 1 r a 0 find a s e such that a a inf a s e a a where s e is the solution set of problem i a by applying the generalized singular value decomposition gsvd and the canonical correlation decomposition ccd of a matrix pair the solvability conditions for problem i a and the general solution of problem i are derived the expression of the solution of problem ii is presented a numerical algorithm for solving the problems is provided

a posteriori error estimates for spectral element solutions to viscoelastic flow problems abstract in this paper we develop an a posteriori error estimate for viscoelastic flow calculations using spectral elements the error estimate is compared with the exact error obtained by computing the difference between the numerical solution and a reference calculation for the benchmark problem of viscoelastic flow past a rigid sphere in a tube

numerical analysis of a three fields model for a quasi newtonian flow abstract we study a finite element approximation of a non linear three fields version of stokes system for incompressible fluids such a problem arises when one considers a white metzner type model for a viscoelastic fluid flow and sets the relaxation time to zero the non linear aspect comes from the fact that a fraction of the viscosity of the fluid depends on the second invariant of the rate of deformation tensor we give existence and unicity results for the continuous problem and its approximation and we prove the error bound

on the structure and geometry of the product singular value decomposition the product singular value decomposition is a factorization of two matrices which can be considered as a generalization of the ordinary singular value decomposition at the same level of generality as the quotient generalized singular value decomposition a constructive proof of the product singular value decomposition is provided which exploits the close relation with a symmetric eigenvalue problem several interesting properties are established the structure and the non uniqueness properties of the so called contragredient transformation which appears as one of the factors in the product singular value decomposition are investigated in detail finally a geometrical interpretation of the structure is provided in terms of principal angles between subspaces

a continuous jacobi like approach to the simultaneous reduction of real matrices abstract the problem of simultaneous reduction of real matrices by either orthogonal similarity or orthogonal equivalence transformations is considered based on the jacobi idea of minimizing the sum of squares of the complementary part of the desired form to which matrices are reduced the projected gradient method is used in this paper it is shown that the projected gradient of the objective function can be formulated explicitly this gives rise to a system of ordinary differential equations that can be readily solved by numerical software the advantages of this approach are that the desired form to which matrices are reduced can be almost arbitrary and that if a desired form is not attainable then the limit point of the corresponding differential equation gives a way of measuring the distance from the best reduced matrices to the nearest matrices that have the desired form the general procedure for deriving these differential equations is discussed the framework can be generalized to the complex valued case some applications are given

least squares solution with the minimum norm for the matrix equation axb gxh c d based on the projection theorem in hilbert space by making use of the generalized singular value decomposition and the canonical correlation decomposition an analytical expression of the least squares solution for the matrix equation axb gxh c d with the minimum norm is derived an algorithm for finding the minimum norm solution is described and some numerical results have been given

a representation theorem for certain partially ordered commutative rings let a be a commutative ring with 1 let p subset a be a preordering of higher level i e 0 1 in p 1 not in p p p subset p p cdot p subset p and a 2n subset p for some n in n and let m subset a be an archimedean p module i e 1 in m 1 not in m m m subset m p cdot m subset m and forall a in a exists n in n n a in m we endow x m varphi in rm hom a r mid varphi m subset r with the weak topology with respect to all mappings widehat a x m rightarrow r widehat a varphi varphi a and consider the representation phi m a rightarrow cal c x m r a mapsto widehat a we find that x m is a non empty compact hausdorff space further we prove that

universal secure network coding via rank metric codes the problem of securing a network coding communication system against an eavesdropper is considered the network implements linear network coding to deliver n packets from source to each receiver and the adversary can eavesdrop on arbitrarily chosen links the objective is to provide reliable communication to all receivers while guaranteeing that the source information remains information theoretically secure from the adversary a coding scheme is proposed that can achieve the maximum possible rate of n packets the scheme which is based on rank metric codes has the distinctive property of being universal it can be applied on top of any communication network without requiring knowledge of or any modifications on the underlying linear network code the only requirement of the scheme is that the packet length be at least n which is shown to be strictly necessary for universal communication at the maximum rate a further scenario is considered where the adversary is allowed not only to eavesdrop but also to inject up to t erroneous packets into the network and the network may suffer from a rank deficiency of at most in this case the proposed scheme can be extended to achieve the rate of n 2t packets this rate is shown to be optimal under the assumption of zero error communication

mimo broadcast channel with an unknown eavesdropper secrecy degrees of freedom we study a multi antenna broadcast channel with two legitimate receivers and an external eavesdropper we assume that the channel matrix of the eavesdropper is unknown to the legitimate terminals but satisfies a maximum rank constraint as our main result we characterize the associated secrecy degrees of freedom for the broadcast channel with common and private messages we show that a direct extension of the single user wiretap codebook does not achieve the secrecy degrees of freedom our proposed optimal scheme involves decomposing the signal space into a common subspace which can be observed by both receivers and private subspaces which can be observed by only one of the receivers and carefully transmitting a subset of messages in each subspace we also consider the case when each user s private message must additionally remain confidential from the other legitimate receiver and characterize the s d o f region in this case

secrecy in mimo networks with no eavesdropper csit we consider two fundamental multi user channel models the multiple input multiple output mimo wiretap channel with one helper wth and the mimo multiple access wiretap mac wt channel in each case the eavesdropper has k antennas while the remaining terminals have n antennas each we consider a fast fading channel where the channel state information csi of the legitimate receiver is available at the transmitters but no csi at the transmitters csit is available for the eavesdropper s channel we determine the optimal sum secure degrees of freedom s d o f for each channel model for the regime k leq n and show that in this regime the mac wt channel reduces to the wth in the absence of eavesdropper csit for the regime n leq k leq 2n we obtain the optimal linear s d o f and show that the mac wt channel and the wth have the same optimal s d o f when restricted to linear encoding strategies in the absence of any such restrictions we provide an upper bound for the sum s d o f of the mac wt channel in the regime n leq k leq 2n our results show that unlike in the single input single output case there is loss of s d o f for even the wth due to lack of eavesdropper csit when k geq n

the secrecy capacity region of the gaussian mimo multi receiver wiretap channel in this paper we consider the gaussian multiple input multiple output mimo multi receiver wiretap channel in which a transmitter wants to have confidential communication with an arbitrary number of users in the presence of an external eavesdropper we derive the secrecy capacity region of this channel for the most general case we first show that even for the single input single output siso case existing converse techniques for the gaussian scalar broadcast channel cannot be extended to this secrecy context to emphasize the need for a new proof technique our new proof technique makes use of the relationships between the minimum mean square error and the mutual information and equivalently the relationships between the fisher information and the differential entropy using the intuition gained from the converse proof of the siso channel we first prove the secrecy capacity region of the degraded mimo channel in which all receivers have the same number of antennas and the noise covariance matrices can be arranged according to a positive semi definite order we then generalize this result to the aligned case in which all receivers have the same number of antennas however there is no order among the noise covariance matrices we accomplish this task by using the channel enhancement technique finally we find the secrecy capacity region of the general mimo channel by using some limiting arguments on the secrecy capacity region of the aligned mimo channel we show that the capacity achieving coding scheme is a variant of dirty paper coding with gaussian signals

perturbation theory for orthogonal projection methods with applications to least squares and total least squares the stabilized versions of the least squares ls and total least squares tls methods are two examples of orthogonal projection methods commonly used to solve the overdetermined system of linear equations ax b when a is nearly rank deficient in practice when this system represents the noisy version of an exact rank deficient zero residual problem tls usually yields a more accurate estimate of the exact solution however current perturbation theory does not justify the superiority of tls over ls in this paper we establish a model for orthogonal projection methods by reformulating the parameter estimation problem as an equivalent problem of nullspace determination when the method is based on the singular value decomposition of the matrix a b the model specializes to the well known tls method we derive new lower and upper perturbation bounds for orthogonal projection methods in terms of the subspace angle which shows how the perturbation of the approximate nullspace affects the accuracy of the solution in situations where tls is typically used such as in signal processing where the noise free compatible problem is exactly rank deficient our upper bounds suggest that the tls perturbation bound is usually smaller than the one for ls which means that tls is usually more robust than ls under perturbations of all the data also the bounds permit a comparison between the ls and tls solutions as well as for any two competing orthogonal projection methods we include numerical simulations to illustrate our conclusions