a tensor singular values and its symmetric embedding eigenvalues national natural science foundation of china 11201092 11261012 scientific research starting foundation for doctors of guizhou normal university china program of young scientists of jiangxi province 20122bcb23003

time discretization of an integro differential equation of parabolic type the stability and convergence properties of two time discretizations of an integro differential equation of parabolic type are studied the methods reduce to the backward euler and crank nicolson methods if the integral term is absent the integral term is approximated in each case by a quadrature rule with relatively high order truncation error so that a relatively large time step can be used for the quadrature in order to reduce the memory and computational requirements of the method

singular values of a real rectangular tensor real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics in this paper we systematically study properties of singular values of a real rectangular tensor and give an algorithm to find the largest singular value of a nonnegative rectangular tensor numerical results show that the algorithm is efficient

comments on explicit criterion for the positive definiteness of a general quartic form the purpose of this note is to point out that the result in the above paper w h ku ieee trans autom control vol ac 10 no 3 pp 372 373 jul 1965 is incomplete and to give a complete and improved result

the degree of the e characteristic polynomial of an even order tensor the e characteristic polynomial of an even order supersymmetric tensor is a useful tool in determining the positive definiteness of an even degree multivariate form in this paper for an even order tensor we first establish the formula of its e characteristic polynomial by using the classical macaulay formula of resultants then give an upper bound for the degree of that e characteristic polynomial examples illustrate that this bound is attainable in some low order and dimensional cases

z eigenvalue methods for a global polynomial optimization problem as a global polynomial optimization problem the best rank one approximation to higher order tensors has extensive engineering and statistical applications different from traditional optimization solution methods in this paper we propose some z eigenvalue methods for solving this problem we first propose a direct z eigenvalue method for this problem when the dimension is two in multidimensional case by a conventional descent optimization method we may find a local minimizer of this problem then by using orthogonal transformations we convert the underlying supersymmetric tensor to a pseudo canonical form which has the same e eigenvalues and some zero entries based upon these we propose a direct orthogonal transformation z eigenvalue method for this problem in the case of order three and dimension three in the case of order three and higher dimension we propose a heuristic orthogonal transformation z eigenvalue method by improving the local minimum with the lower dimensional z eigenvalue methods and a heuristic cross hill z eigenvalue method by using the two dimensional z eigenvalue method to find more local minimizers numerical experiments show that our methods are efficient and promising

on approximate diagonalization of third order symmetric tensors by orthogonal transformations abstract in this paper we study the approximate orthogonal diagonalization problem of third order symmetric tensors we define several classes of approximately diagonal tensors including the ones corresponding to the stationary points of this problem we study the relationships between these classes and other well known objects such as tensor z eigenvalue and z eigenvector we also prove results on convergence of the cyclic jacobi or jacobi com2 algorithm

discontinuous galerkin methods for ordinary differential equations a class of galerkin methods derived from discontinuous piecewise polynomial spaces is analyzed for polynomials of degree k these methods lead to a family of one step schemes generating approximations up to order 2k 2 for the solution of an ordinary differential equation

the form of the spectral functions associated with a class of sturm liouville equations with integrable coefficient we consider the second order linear differential equation in the case where roughly q l 1 0 we devise a representation for the spectral function t associated with which is valid for t sufficiently large our results are the best possible

time stepping discontinuous galerkin methods for fractional diffusion problems time stepping hp hp versions discontinuous galerkin dg methods for the numerical solution of fractional subdiffusion problems of order alpha with 1 alpha 0 1 0 will be proposed and analyzed generic hp hp version error estimates are derived after proving the stability of the approximate solution for h h version dg approximations on appropriate graded meshes near t 0 t 0 we prove that the error is of order o k max 2 p frac alpha 2 o kmax 2 p 2 where k k is the maximum time step size and p ge 1 p 1 is the uniform degree of the dg solution for hp hp version dg approximations by employing geometrically refined time steps and linearly increasing approximation orders exponential rates of convergence in the number of temporal degrees of freedom are shown finally some numerical tests are given

the asymptotic form of the spectral function in sturm liouville problems with a large potential like xc c 2 a new formula is given for the spectral function in singular sturm liouville problems where the potential is like const x c 0

a second order accurate numerical method for a fractional wave equation we study a generalized crank nicolson scheme for the time discretization of a fractional wave equation in combination with a space discretization by linear finite elements the scheme uses a non uniform grid in time to compensate for the singular behaviour of the exact solution at t 0 with appropriate assumptions on the data and assuming that the spatial domain is convex or smooth we show that the error is of order k2 h2 where k and h are the parameters for the time and space meshes respectively

a procedure for the positive definiteness of forms of even order a positivity method for forms of even order is reported it is shown that testing for positivity of such forms can be reduced to a test for positive definiteness of parametric matrices special emphasis is placed on quartic and sixth order forms examples are given to illustrate the proposed criterion

discontinuous galerkin method for an evolution equation with a memory term of positive type we consider an initial value problem for a class of evolution equations incorporating a memory term with a weakly singular kernel bounded by c t s 1 where 0 1 for the time discretization we apply the discontinuous galerkin method using piecewise polynomials of degree at most q 1 for q 1 or 2 for the space discretization we use continuous piecewise linear finite elements the discrete solution satisfies an error bound of order k 9 h 2 l k where k and h are the mesh sizes in time and space respectively and l k max 1 log k 1 in the case q 2 we prove a higher convergence rate of order k 3 h 2 l k at the nodes of the time mesh typically the partial derivatives of the exact solution are singular at t 0 necessitating the use of non uniform time steps we compare our theoretical error bounds with the results of numerical computations

on fully discrete galerkin approximations for partial integro differential equations of parabolic type the subject of this work is the application of fully discrete galerkin finite element methods to initial boundary value problems for linear partial integro differential equations of parabolic type we investigate numerical schemes based on the pade discretization with respect to time and associated with certain quadrature formulas to approximate the integral term a preliminary error estimate is established which contains a term related to the quadrature rule to be specified in particular we consider quadrature rules with sparse quadrature points so as to limit the storage requirements without sacrificing the order of overall convergence for the backward euler scheme the crank nicolson scheme and a third order pade type scheme the specific quadrature rules analyzed are based on the rectangular the trapezoidal and simpson s rule for all the schemes studied optimal order error estimates are obtained in the case that the solution of the problem is smooth enough since this is important for our error analysis we also discuss the regularity of the exact solutions of our equations high order regularity results with respect to both space and time are given for the solution of problems with smooth enough data 17 refs

convergence and asymptotic stability of galerkin methods for linear parabolic equations with delays this paper is concerned with the convergence and asymptotic stability of semidiscrete and full discrete schemes for linear parabolic equations with delay these full discrete numerical processes include forward euler backward euler and crank nicolson schemes the optimal convergence orders are consistent with those of the original parabolic equation it is proved that the semidiscrete scheme backward euler and crank nicolson full discrete schemes can unconditionally preserve the delay independent asymptotic stability but some additional restrictions on time and spatial stepsizes of the forward euler full discrete scheme is needed to preserve the delay independent asymptotic stability numerical experiments illustrate the theoretical results

the asymptotic form of the titchmarsh weyl m function associated with a second order differential equation with locally integrable coefficient on deduit un developpement asymptotique pour la m fonction de titchmarsh weyl associee a l equation differentielle lineaire d ordre 2 y qy y sur 0 dans le cas ou la seule restriction sur la fonction a valeur reelle q est q l 1 0 e pour e 0

numerical solution of parabolic integro differential equations by the discontinuous galerkin method the numerical solution of a parabolic equation with memory is considered the equation is first discretized in time by means of the discontinuous galerkin method with piecewise constant or piecewise linear approximating functions the analysis presented allows variable time steps which as will be shown call then efficiently be selected to match singularities in the solution induced by singularities in the kernel of the memory term or by nonsmooth initial data the combination with finite element discretizat in space is also studied

discretization with variable time steps of an evolution equation with a positive type memory term we consider the linearized form of an evolution equation used to model viscoelasticity and heat conduction in materials with memory the equation is discretized in time using several schemes based on the backward euler and crank nicolson methods and incorporating appropriate quadratures for the memory term by permitting variable time steps we remove a restriction present in earlier related work by the first two authors

comments on a procedure for the positive definiteness of forms of even order hasan and hasan ieee trans automat control vol 41 p 615 17 1996 give a necessary and sufficient condition for the positive definiteness of a quadratic programming problem the purpose of this paper is to point out that the main result in the above mentioned paper by is erroneous

time discretization via laplace transformation of an integro differential equation of parabolic type we consider the discretization in time of an inhomogeneous parabolic integro differential equation with a memory term of convolution type in a banach space setting the method is based on representing the solution as an integral along a smooth curve in the complex plane which is evaluated to high accuracy by quadrature using the approach in recent work of lopez fernandez and palencia this reduces the problem to a finite set of elliptic equations with complex coefficients which may be solved in parallel the method is combined with finite element discretization in the spatial variables to yield a fully discrete method the paper is a further development of earlier work by the authors which on the one hand treated purely parabolic equations and on the other an evolution equation with a positive type memory term

a posteriori error bounds and global error control for approximation of ordinary differential equations the author analyzes a finite element method for the integration of initial value problems in ordinary differential equations general and contractive problems are treated and quasi optimal a priori and a posteriori error bounds obtained in each case in particular good results are obtained for a class of stiff dissipative problems these results are used to construct a rigorous and robust theory of global error control the author also derives an asymptotic error estimate that is used in a discussion of the behavior of the error in conclusion the properties of the error control are exhibited in a series of numerical experiments

an inverse problem involving the titchmarsh weyl m function let m denote the titchmarsh weyl m function associated with the differential equation with neumann boundary condition at 0 in the case q c m 0 e for some integer m and e 0 we prove the following result theorem 2 1 if for some integer 3 we know that im l 2 m 0 0 on the ray e i then q v 0 0 for all v in the set 0 1 m for which 2 mod

positivity and nonnegativity conditions of a quartic equation and related problems in this paper explicit conditions for positivity no real roots nonnegativity on positive real axis no positive real roots with odd multiplicity and stability aperiodicity all roots are real and negative and simple of a quartic or biquadratic equation are given the derived conditions from the known solution of the quartic equation are not only complete but simpler than those derived from sturm extended hurwitz inners and hankel methods because of abel s theorem no explicit solution in terms of the roots of an equation higher than quartic exists similar simplification for higher degree polynomial equations may not be possible furthermore explicit conditions for positivity and nonnegativity of equations of higher degree than four are extremely difficult to obtain and may not be possible the results of the paper will hopefully shed some light on a century old problem and thus enhance the engineering application of the derived condition to higher order systems

on the location of the weyl circles the paper deals with explicit estimates concerning certain circles in the complex plane which were associated with sturm liouville problems by h weyl by the use of riccati equations instead of linear integral equations improvements are obtained for results of everitt and halvorsen concerning the behaviour of the titchmarsh weyl m coefficient

minimal gersgorin tensor eigenvalue inclusion set and its approximation for a tensor its minimal gersgorin tensor eigenvalue inclusion set is presented by establishing a sufficient and necessary condition for the elements belonging to this set we give a sequence of approximation sets and prove that the limit of this sequence is the minimal gersgorin tensor eigenvalue inclusion set for an irreducible tensor

tensor principal component analysis via convex optimization this paper is concerned with the computation of the principal components for a general tensor known as the tensor principal component analysis pca problem we show that the general tensor pca problem is reducible to its special case where the tensor in question is super symmetric with an even degree in that case the tensor can be embedded into a symmetric matrix we prove that if the tensor is rank one then the embedded matrix must be rank one too and vice versa the tensor pca problem can thus be solved by means of matrix optimization under a rank one constraint for which we propose two solution methods 1 imposing a nuclear norm penalty in the objective to enforce a low rank solution 2 relaxing the rank one constraint by semidefinite programming interestingly our experiments show that both methods can yield a rank one solution for almost all the randomly generated instances in which case solving the original tensor pca problem to optimality to further cope with the size of the resulting convex optimization models we propose to use the alternating direction method of multipliers which reduces significantly the computational efforts various extensions of the model are considered as well

finite element methods for parabolic and hyperbolic partial integro differential equations on formule plusieurs methodes de galerkin a temps discret pour des equations integrodifferentielles paraboliques

maximum block improvement and polynomial optimization in this paper we propose an efficient method for solving the spherically constrained homogeneous polynomial optimization problem the new approach has the following three main ingredients first we establish a block coordinate descent type search method for nonlinear optimization with the novelty being that we accept only a block update that achieves the maximum improvement hence the name of our new search method maximum block improvement mbi convergence of the sequence produced by the mbi method to a stationary point is proved second we establish that maximizing a homogeneous polynomial over a sphere is equivalent to its tensor relaxation problem thus we can maximize a homogeneous polynomial function over a sphere by its tensor relaxation via the mbi approach third we propose a scheme to reach a kkt point of the polynomial optimization provided that a stationary solution for the relaxed tensor problem is available numerical experiments have shown that our new method works very efficiently for a majority of the test instances that we have experimented with the method finds the global optimal solution at a low computational cost

a return to the hardy littlewood integral inequality this paper is concerned with the extension and generalization of a classic integral inequality first considered by hardy and littlewood in 1932 the integral inequality is big int b a p f 2 q f 2 big 2 leqslant k int b a w f 2 int b a w w 1 pf qf 2 f in delta where k is a positive number the coefficients p q and w are real valued and w 0 on the interval a b infty a b leqslant infty and p 1 q w in l loc a b delta is the maximal linear manifold of functions f for which the two integrals on the right hand side are both finite the hardy littlewood inequality is the special case a 0 b infty p w 1 and q 0 on 0 infty here k 4 and all the cases of equality can be identified in this paper the results of everitt in 1971 connecting the above inequality with the titchmarsh weyl m coefficient of the associated differential equation py qy lambda wy on a b are restated in an improved form and are shown to hold under the above minimal conditions on the coefficients p q and w it is proved that the best possible value for the number k satisfies 1 k leqslant infty and that within these bounds all values of k can be realized in the course of establishing these results new information of independent interest is obtained about the behaviour of the m coefficient near the real axis examples are given to exhibit the wide range of possibilities for the integral inequality recent results on the asymptotic form of the m coefficient and on the transformation theory of the associated differential equation are shown to have applications for the general properties of the inequality all recent contributions to this general integral inequality are listed in the references and their content is surveyed during the course of the paper

properties of some blind equalization criteria in noisy multiuser environments blind equalization in noisy multiuser channels has met with increasing attention with the advent of multiaccess digital communication systems we examine blind equalizer performance in cases where perfect equalization proves unattainable due to noise and interference from concurrent users in particular we obtain a characterization of stationary points and extrema for a family of blind criteria in undermodeled cases which assimilates the influence of differing source statistics and background noise correlation properties relations to mean square equalization measures are then obtained as a byproduct by re examining a gradient search procedure we obtain domains of attraction of each extremum in a special sufficient order setting we also derive a global step size bound for undermodeled cases which ensures convergence of a gradient search procedure to an extremum of a blind cost function we likewise confirm that the super exponential algorithm results from an optimal choice of this step size parameter

a property of the asymptotic series for a class of titchmarsh weyl m functions in an earlier paper 6 we showed that if q c n 0 e for some e 0 then the titchmarsh weyl m function associated with the second order linear differential equation has the asymptotic expansion as a in a sector of the form 0 we show that if the real valued function q admits the expansion in a neighbourhood of 0 then

d eigenvalues of diffusion kurtosis tensors diffusion kurtosis imaging dki is a new model in medical engineering where a diffusion kurtosis dk tensor is involved a dk tensor is a fourth order three dimensional fully symmetric tensor in this paper we introduce d eigenvalues for a dk tensor the largest the smallest and the average d eigenvalues of a dk tensor correspond with the largest the smallest and the average apparent kurtosis coefficients akc of a water molecule in the space respectively we present their computational methods and discuss related anisotropy value of akc

mixed finite element approximations of parabolic integro differential equations with nonsmooth initial data we analyze the semidiscrete mixed finite element methods for parabolic integro differential equations that arise in the modeling of nonlocal reactive flows in porous media a priori l 2 error estimates for pressure and velocity are obtained with both smooth and nonsmooth initial data more precisely a mixed ritz volterra projection introduced earlier by ewing et al in siam j numer anal 40 2002 pp 1538 1560 is used to derive optimal l 2 error estimates for problems with initial data in h 2 cap h 0 1 in addition for homogeneous equations we derive optimal l 2 error estimates for initial data just in l 2 here we use an elementary energy technique and duality argument

further results on cauchy tensors and hankel tensors in this article we present various new results on cauchy tensors and hankel tensors we first introduce the concept of generalized cauchy tensors which extends cauchy tensors in the current literature and provide several conditions characterizing positive semi definiteness of generalized cauchy tensors with nonzero entries furthermore we prove that all even order generalized cauchy tensors with positive entries are completely positive tensors which means every such that generalized cauchy tensor can be decomposed as the sum of nonnegative rank 1 tensors we also establish that all the h eigenvalues of nonnegative cauchy tensors are nonnegative secondly we present new mathematical properties of hankel tensors we prove that an even order hankel tensor is vandermonde positive semi definite if and only if its associated plane tensor is positive semi definite we also show that if the vandermonde rank of a hankel tensor a is less than the dimension of the underlying space then positive semi definiteness of a is equivalent to the fact that a is a complete hankel tensor and so is further equivalent to the sos property of a thirdly we introduce a new class of structured tensors called cauchy hankel tensors which is a special case of cauchy tensors and hankel tensors simultaneously sufficient and necessary conditions are established for an even order cauchy hankel tensor to be positive definite

cross hill the heuristic cross hill method proposed by qi et al 2009 14 was recently extended from finding the z eigenvalues of tensors to quantum separation problem by han and qi 2013 5 in this paper we show that it can be extended to solve general global optimization problems the heuristic cross hill method is a combination of a local optimization method and a global optimization method with lower dimension at each iteration it first uses the local optimization method to find a local solution then using this point and an arbitrary orthogonal vector it solves a two dimensional optimization problem to find a better solution than that the local approach was able to find preliminary experimental results are very encouraging

time discretization of parabolic problems by the hp version of the discontinuous galerkin finite element method the discontinuous galerkin finite element method dgfem for the time discretization of parabolic problems is analyzed in the context of the hp version of the galerkin method error bounds which are explicit in the time steps as well as in the approximation orders are derived and it is shown that the hp dgfem gives spectral convergence in problems with smooth time dependence in conjunction with geometric time partitions it is proved that the hp dgfem results in exponential rates of convergence for piecewise analytic solutions exhibiting singularities induced by incompatible initial data or piecewise analytic forcing terms for the h version dgfem algebraically graded time partitions are determined that give the optimal algebraic convergence rates a fully discrete hp scheme is discussed exemplarily for the heat equation the use of certain mesh design principles for the spatial discretizations yields exponential rates of convergence in time and space numerical examples confirm the theoretical results

convergence of an algorithm for the largest singular value of a nonnegative rectangular tensor abstract in this paper we present an iterative algorithm for computing the largest singular value of a nonnegative rectangular tensor we establish the convergence of this algorithm for any irreducible nonnegative rectangular tensor

parallel in time algorithm with spectral subdomain enhancement for volterra integral equations nsf of china 11201077 11071203 91130002 nsf of fujian province 2012j01007 fuzhou university 0460022456 hong kong research grant council gif grants hong kong baptist university frg grants

post processing discontinuous galerkin solutions to volterra integro differential equations analysis and simulations this paper presents a superconvergence extraction technique for volterra integro differential equations with smooth and non smooth kernels specifically extracting superconvergence is done via a post processed discontinuous galerkin dg method obtained from interpolating the dg solution using lagrange polynomials at the nodal points a global superconvergence error bound in the l norm is established for a non smooth kernel a family of non uniform time meshes is used to compensate for the singular behaviour of the exact solution near t 0 the derived theoretical results are numerically validated in a sample of test problems demonstrating higher than expected convergence rates

hp discontinuous galerkin time stepping for volterra integrodifferential equations we present an hp error analysis of the discontinuous galerkin time stepping method for volterra integrodifferential equations with weakly singular kernels we derive new error bounds that are explicit in the time steps the degrees of the approximating polynomials and the regularity properties of the exact solution it is then shown that start up singularities can be resolved at exponential rates of convergence by using geometrically graded time steps our theoretical results are confirmed in a series of numerical tests

eigenvalues and invariants of tensors abstract a tensor is represented by a supermatrix under a co ordinate system in this paper we define e eigenvalues and e eigenvectors for tensors and supermatrices by the resultant theory we define the e characteristic polynomial of a tensor an e eigenvalue of a tensor is a root of the e characteristic polynomial in the regular case a complex number is an e eigenvalue if and only if it is a root of the e characteristic polynomial we convert the e characteristic polynomial of a tensor to a monic polynomial and show that the coefficients of that monic polynomial are invariants of that tensor i e they are invariant under co ordinate system changes we call them principal invariants of that tensor the maximum number of principal invariants of mth order n dimensional tensors is a function of m and n we denote it by d m n and show that d 1 n 1 d 2 n n d m 2 m for m 3 and d m n m n 1 m for m n 3 we also define the rank of a tensor all real eigenvectors associated with nonzero e eigenvalues are in a subspace with dimension equal to its rank

an h p version of the continuous petrov galerkin finite element method for volterra integro differential equations with smooth and nonsmooth kernels we present an h p version of the continuous petrov galerkin cpg finite element method for linear volterra integro differential equations with smooth and nonsmooth kernels we establish a priori error estimates in the l 2 h 1 and l infty norms that are completely explicit with respect to the local discretization and regularity parameters for singular solutions caused by the weakly singular kernels we prove that algebraic convergence rates can be achieved for the h p version of the cpg method with quasi uniform meshes moreover we show that exponential rates of convergence can be achieved for solutions with start up singularities by using geometric time partitions and linearly increasing polynomial degrees numerical experiments are provided to illustrate the theoretical results

the asymptotic form of the spectral functions associated with a class of sturm liouville equations on utilise une forme modifiee de la formule d inversion de stieltjes avec des resultats recents sur la forme asymptotique des m fonctions de titchmarch weyl pour obtenir une expression asymptotique pour les fonctions spectrales t associees a l equation differentielle y q x y y o x b

nonnegative polynomial optimization over unit spheres and convex programming relaxations we consider approximation algorithms for nonnegative polynomial optimization over unit spheres such optimization models have wide applications e g in signal and image processing high order statistics and computer vision since polynomial functions are nonconvex the problems under consideration are all np hard in this paper based on convex polynomial optimization relaxations we propose polynomial time approximation algorithms with new approximation bounds numerical results are reported to show the effectiveness of the proposed approximation algorithms

a superconvergent discontinuous galerkin method for volterra integro differential equations smooth and non smooth kernels abstract we study the numerical solution for volerra integro differential equations with smooth and non smooth kernels we use an h version discontinuous galerkin dg method and derive nodal error bounds that are explicit in the parameters of interest in the case of non smooth kernel it is justified that the start up singularities can be resolved at superconvergence rates by using non uniformly graded meshes our theoretical results are numerically validated in a sample of test problems

a new eigenvalue inclusion set for tensors and its applications abstract a new tensor eigenvalue inclusion set is given and proved to be tighter than those in l q qi 2005 18 and c q li y t li x kong 2014 12 in addition we study the eigenvalues lying on the boundary of the eigenvalue inclusion set provided by qi 2005 for weakly irreducible tensors as applications we give new bounds for the spectral radius of nonnegative tensors some sufficient conditions for a tensor to be a strong m tensor and some inequalities to identify the positive definiteness for an even order real supersymmetric tensor

an hp version discontinuous galerkin method for integro differential equations of parabolic type we study the numerical solution of a class of parabolic integro differential equations with weakly singular kernels we use an hp version discontinuous galerkin dg method for the discretization in time we derive optimal hp version error estimates and show that exponential rates of convergence can be achieved for solutions with singular temporal behavior near t 0 caused by the weakly singular kernel moreover we prove that by using nonuniformly refined time steps optimal algebraic convergence rates can be achieved for the h version dg method we then combine the dg time stepping method with a standard finite element discretization in space and present an optimal error analysis of the resulting fully discrete scheme our theoretical results are numerically validated in a series of test problems

an h 1 galerkin mixed finite element method for an evolution equation with a positive type memory term an h1 galerkin mixed finite element method is analyzed for a class of evolution equations with memory when a classical mixed method is applied to such problems it has not been possible to obtain any estimate for the flux however the proposed approach yields optimal order convergence without the lbb consistency condition and quasi uniformity of the finite element mesh compared to the results proved for one space variable the l2 estimate of the flux is not optimal for problems in two and three space dimensions therefore a modification of the method is proposed and analyzed a maximum norm estimate is also derived in one and two space variables a backward euler approximation of the modified method is also analyzed

criterions for the positive definiteness of real supersymmetric tensors the positive definiteness of an even degree homogeneous polynomial form f x plays an important role in the stability study of nonlinear autonomous systems via lyapunov s direct method in automatic control and the positive definiteness of f x is equivalent to that of an even order supersymmetric tensor which defines f x in this paper we provide some criterions for identifying the positive definiteness of an even order real supersymmetric tensor moreover an iterative algorithm for identifying the positive definiteness of an even order real supersymmetric tensor is obtained numerical examples are given to verify the corresponding results

an always convergent algorithm for the largest eigenvalue of an irreducible nonnegative tensor in this paper we propose an iterative method to calculate the largest eigenvalue of a nonnegative tensor we prove this method converges for any irreducible nonnegative tensor we also apply this method to study the positive definiteness of a multivariate form

deterministic approximation algorithms for sphere constrained homogeneous polynomial optimization problems due to their fundamental nature and numerous applications sphere constrained polynomial optimization problems have received a lot of attention lately in this paper we consider three such problems i maximizing a homogeneous polynomial over the sphere ii maximizing a multilinear form over a cartesian product of spheres and iii maximizing a multiquadratic form over a cartesian product of spheres since these problems are generally intractable our focus is on designing polynomial time approximation algorithms for them by reducing the above problems to that of determining the l 2 diameters of certain convex bodies we show that they can all be approximated to within a factor of log n n d 2 1 deterministically where n is the number of variables and d is the degree of the polynomial this improves upon the currently best known approximation bound of 1 n d 2 1 in the literature we believe that our approach will find further applications in the design of approximation algorithms for polynomial optimization problems with provable guarantees

nonsmooth data error estimates for approximations of an evolution equation with a positive type memory term we study the numerical approximation of an integro differential equation which is intermediate between the heat and wave equations the proposed discretization uses convolution quadrature based on the first and second order backward difference methods in time and piecewise linear finite elements in space optimal order error bounds in terms of the initial data and the inhomogeneity are shown for positive times without assumptions of spatial regularity of the data

error estimates and adaptive time step control for a class of one step methods for stiff ordinary differential equations we prove new optimal a priori error estimates for a class of implicit one step methods for stiff ordinary differential equations obtained by using the discontinuous galerkin method with piecewise polynomials of degree zero and one starting from these estimates we propose a new algorithm for automatic time step control and we discuss the relation between this algorithm and earlier algorithms implemented in packages for the numerical solution of stiff ordinary differential equations

iterative algorithms for computing us and u eigenpairs of complex tensors abstract this paper is devoted to the computation of us eigenpairs of complex symmetric tensors and u eigenpairs of complex tensors based on the takagi factorization of complex symmetric matrices we derive an iterative algorithm for computing us eigenpairs of complex symmetric tensors denoted as qrcst algorithm we also observe that multiple us eigenpairs can be found from a local permutation heuristic which is effectively a tensor similarity transformation resulting in the permuted version of qrcst we then generalize our techniques to general complex tensors finally we derive a higher order power type method for computing a us or a u eigenpair similar to the higher order power method for computing z eigenpairs of real symmetric tensors or a best rank one approximation of real tensors we illustrate our algorithms via numerical examples

a second order accurate numerical method with graded meshes for an evolution equation with a weakly singular kernel abstract a second order accurate numerical method with graded meshes is proposed and analyzed for an evolution equation with a weakly singular kernel the graded meshes are employed to compensate for the singular behavior of the exact solution at t 0 for the time discretization the product integration rule is used to approximate the riemann liouville fractional integral a generalized crank nicolson time stepping is considered and shown that the error is of order k 2 where k denotes the maximum time step a fully discrete difference scheme is constructed with space discretization by compact difference method numerical experiment is carried out to support the theoretical results the comparison between the method on uniform grids and graded grids shows the efficiency of our method

an hp version of the discontinuous galerkin time stepping method for nonlinear second order delay differential equations with vanishing delays abstract we present and analyze an h p version of the discontinuous galerkin dg time stepping method for nonlinear second order delay differential equations with vanishing delays we derive a priori error bounds in the l 2 and l norm that are fully explicit with respect to the local time steps the local approximation orders and the local regularity of the exact solution we further prove that the h p dg scheme based on geometrically refined time steps and on linearly increasing approximation orders attains exponential rates of convergence for analytic solutions with start up singularities moreover we also propose an h p dg scheme for the state dependent delay differential equations we illustrate the theoretical results with a series of numerical experiments

high order contrasts for independent component analysis this article considers high order measures of independence for the independent component analysis problem and discusses the class of jacobi algorithms for their optimization several implementations are discussed we compare the proposed approaches with gradient based techniques from the algorithmic point of view and also on a set of biomedical data