on the spectral norms of cauchy toeplitz and cauchy hankel matrices in this study we have found upper and lower bounds for the spectral norm of cauchy toeplitz and cauchy hankel matrices in the forms t n 1 a i j b n i j 1 h n 1 a i j b n i j 1

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

m tensors and nonsingular m tensors abstract the m matrix is an important concept in matrix theory and has many applications recently this concept has been extended to higher order tensors 18 in this paper we establish some important properties of m tensors and nonsingular m tensors an m tensor is a z tensor we show that a z tensor is a nonsingular m tensor if and only if it is semi positive thus a nonsingular m tensor has all positive diagonal entries an m tensor regarding as the limit of a sequence of nonsingular m tensors has all nonnegative diagonal entries we introduce even order monotone tensors and present their spectral properties in matrix theory a z matrix is a nonsingular m matrix if and only if it is monotone this is no longer true in the case of higher order tensors we show that an even order monotone z tensor is an even order nonsingular m tensor but not vice versa an example of an even order nontrivial monotone z tensor is also given

coupling of fem and bem in shape optimization in the present paper we consider the numerical solution of shape optimization problems which arise from shape functionals of integral type over a compact region of the unknown shape especially l 2 tracking type functionals the underlying state equation is assumed to satisfy a poisson equation with dirichlet boundary conditions we proof that the shape hessian is not strictly h 1 2 coercive at the optimal domain which implies ill posedness of the optimization problem under consideration since the adjoint state depends directly on the state we propose a coupling of finite element methods fem and boundary element methods bem to realize an efficient first order shape optimization algorithm fem is applied in the compact region while the rest is treated by bem the coupling of fem and bem essentially retains all the structural and computational advantages of treating the free boundary by boundary integral equations

symmetric coupling of boundary elements and raviart thomas type mixed finite elements in elastostatics both mixed finite element methods and boundary integral methods are important tools in computational mechanics according to a good stress approximation recently even low order mixed methods of raviart thomas type became available for problems in elasticity since either methods are robust for critical poisson ratios it appears natural to couple the two methods as proposed in this paper the symmetric coupling changes the elliptic part of the bilinear form only hence the convergence analysis of mixed finite element methods is applicable to the coupled problem as well specifically we couple boundary elements with a family of mixed elements analyzed by stenberg the locking free implementation is performed via lagrange multipliers numerical examples are included

coupling regularization and adaptive hp bem for the solution of a delamination problem in this paper we couple regularization techniques with the adaptive hp version of the boundary element method hp bem for the efficient numerical solution of linear elastic problems with nonmonotone contact boundary conditions as a model example we treat the delamination of composite structures with a contaminated interface layer this problem has a weak formulation in terms of a nonsmooth variational inequality the resulting hemivariational inequality is first regularized and then discretized by an adaptive hp bem we give conditions for the uniqueness of the solution and provide an a priori error estimate furthermore we prove the very first a posteriori error estimate for the nonsmooth variational problem utilizing a novel mixed regularized formulation thus enabling hp adaptivity various numerical experiments illustrate the behavior strengths and limitations of the proposed high order approximation scheme

spectra of uniform hypergraphs abstract we present a spectral theory of uniform hypergraphs that closely parallels spectral graph theory a number of recent developments building upon classical work has led to a rich understanding of symmetric hyperdeterminants of hypermatrices a k a multidimensional arrays symmetric hyperdeterminants share many properties with determinants but the context of multilinear algebra is substantially more complicated than the linear algebra required to address spectral graph theory i e ordinary matrices nonetheless it is possible to define eigenvalues of a hypermatrix via its characteristic polynomial as well as variationally we apply this notion to the adjacency hypermatrix of a uniform hypergraph and prove a number of natural analogs of basic results in spectral graph theory open problems abound and we present a number of directions for further study

on a fem bem formulation for an exterior quasilinear problem in the plane we use a version of the fem bem method introduced by costabel boundary elements ix vol 1 c a brebbia et al eds springer verlag 1987 and han j comput math 8 1990 pp 223 232 to discretize an exterior quasilinear problem we provide error estimates for the galerkin method and propose a fully discrete scheme based on simple quadrature formulas furthermore we show that these numerical integration schemes preserve the optimal rates of convergence finally we present results of numerical experiments involving our discretization method

semismoothness of the maximum eigenvalue function of a symmetric tensor and its application abstract in this paper we examine the maximum eigenvalue function of an even order real symmetric tensor by using the variational analysis techniques we first show that the maximum eigenvalue function is a continuous and convex function on the symmetric tensor space in particular we obtain the convex subdifferential formula for the maximum eigenvalue function next for an mth order n dimensional symmetric tensor a we show that the maximum eigenvalue function is always th order semismooth at a for some rational number 0 in the special case when the geometric multiplicity is one we show that can be set as 1 2 m 1 n sufficient condition ensuring the strong semismoothness of the maximum eigenvalue function is also provided as an application we propose a generalized newton method to solve the space tensor conic linear programming problem which arises in medical imaging area local convergence rate of this method is established by using the semismooth property of the maximum eigenvalue function

biorthogonal wavelet approximation for the coupling of fem bem we apply multiscale methods to the coupling of finite and boundary element methods to solve an exterior dirichlet boundary value problem for the two dimensional poisson equation adopting biorthogonal wavelet matrix compression to the boundary terms with n degrees of freedom we show that the resulting compression strategy fits the optimal convergence rate of the coupling galerkin methods while the number of nonzero entries in the corresponding stiffness matrices is considerably smaller than n 2

weak solvability of interior transmission problems via mixed finite elements and dirichlet to neumann mappings we study the weak solvability of an interior linear nonlinear transmission problem arising in steady heat transfer and potential theory for the variational formulation we use a dirichlet to neumann mapping on the interface which is obtained from the application of the boundary integral method to the linear domain and we utilize a mixed finite element method in the nonlinear region existence and uniqueness of solution for the continuous formulation are provided and general approximation results for a fully discrete galerkin method are derived in particular a compatibility condition between the mesh sizes involved is deduced in order to conclude the solvability and stability of this galerkin scheme

a wavelets method for plane elasticity problem a neumann boundary value problem of plane elasticity problem in the exterior circular domain is reduced into an equivalent natural boundary integral equation and a poisson integral formula with the dtn method using the trigonometric wavelets and galerkin method we obtain a fast numerical method for the natural boundary integral equation which has an unique solution in the quotient space we decompose the stiffness matrix in our numerical method into four circulant and symmetrical or antisymmetrical submatrices and hence the solution of the associated linear algebraic system can be solved with the fast fourier transform fft and the inverse fast fourier transform ifft instead of the inverse matrix examples are given for demonstrating our method has good accuracy of our method even though the exact solution is almost singular

tensor decompositions and applications this survey provides an overview of higher order tensor decompositions their applications and available software a tensor is a multidimensional or n way array decompositions of higher order tensors i e n way arrays with n geq 3 have applications in psycho metrics chemometrics signal processing numerical linear algebra computer vision numerical analysis data mining neuroscience graph analysis and elsewhere two particular tensor decompositions can be considered to be higher order extensions of the matrix singular value decomposition candecomp parafac cp decomposes a tensor as a sum of rank one tensors and the tucker decomposition is a higher order form of principal component analysis there are many other tensor decompositions including indscal parafac2 candelinc dedicom and paratuck2 as well as nonnegative variants of all of the above the n way toolbox tensor toolbox and multilinear engine are examples of software packages for working with tensors

an eigenvalue method for testing positive definiteness of a multivariate form in this paper we present an eigenvalue method for testing positive definiteness of a multivariate form this problem plays an important role in the stability study of nonlinear autonomous systems via lyapunov s direct method in automatic control at first we apply the d andrea dickenstein version of the classical macaulay formulas of the resultant to compute the symmetric hyperdeterminant of an even order supersymmetric tensor by using the supersymmetry property we give detailed computation procedures for the bezoutians and specified ordering of monomials in this approach we then use these formulas to calculate the characteristic polynomial of a fourth order three dimensional supersymmetric tensor and give an eigenvalue method for testing positive definiteness of a quartic form of three variables some numerical results of this method are reported

an even order symmetric b tensor is positive definite it is easily checkable if a given tensor is a b tensor or a b0 tensor or not in this paper we show that a symmetric b tensor can always be decomposed to the sum of a strictly diagonally dominated symmetric m tensor and several positive multiples of partially all one tensors and a symmetric b0 tensor can always be decomposed to the sum of a diagonally dominated symmetric m tensor and several positive multiples of partially all one tensors when the order is even this implies that the corresponding b tensor is positive definite and the corresponding b0 tensor is positive semi definite this gives a checkable sufficient condition for positive definite and semi definite tensors this approach is different from the approach in the literature for proving a symmetric b matrix is positive definite as that matrix approach cannot be extended to the tensor case

lower bounds for the norms of cauchy toeplitz and cauchy hankel matrices in this study we have found lower bounds for the spectral norms and euclidean norms of cauchy toeplitz and cauchy hankel matrices in the forms t n 1 g i j h i j 1 n h n 1 g i j h i j 1 n

coupling of finite and boundary element methods for an elastoplastic interface problem a class of transmission problems is considered in which a nonlinear variational problem in one domain is coupled with a linear elliptic problem in a second domain a typical example is a problem from three dimensional elasticity theory where an elastoplastic material is embedded into a linear elastic material the nonlinear problem is given in variational form with a strictly convex functional the linear elliptic problem is described by boundary integral equations on the coupling boundary the typical saddle point structure of such problems is analyzed galerkin approximations are studied which consist of a finite element approximation in the first domain coupled with a boundary element method on the coupling boundary the convergence of the galerkin approximation is based on the saddle point structure which is shown to hold for the exact as well as the discretized problems

on the coupled bem and fem for a nonlinear exterior dirichlet problem in r2 in this paper we apply the coupling of boundary integral and finite element methods to solve a nonlinear exterior dirichlet problem in the plane specifically the boundary value problem consists of a nonlinear second order elliptic equation in divergence form in a bounded inner region and the laplace equation in the corresponding unbounded exterior region in addition to appropriate boundary and transmission conditions the main feature of the coupling method utilized here consists in the reduction of the nonlinear exterior boundary value problem to an equivalent monotone operator equation we provide sufficient conditions for the coefficients of the nonlinear elliptic equation from which existence uniqueness and approximation results are established then we consider the case where the corresponding operator is strongly monotone and lipschitz continuous and derive asymptotic error estimates for a boundary finite element solution we prove the unique solvability of the discrete operator equations and based on a strang type abstract error estimate we show the strong convergence of the approximated solutions moreover under additional regularity assumptions on the solution of the continous operator equation the asymptotic rate of convergenceo h is obtained

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

infinite and finite dimensional hilbert tensors abstract for an m order n dimensional hilbert tensor hypermatrix h n h i 1 i 2 i m h i 1 i 2 i m 1 i 1 i 2 i m m 1 i 1 i m 1 2 n its spectral radius is not larger than n m 1 sin n and an upper bound of its e spectral radius is n m 2 sin n moreover its spectral radius is strictly increasing and its e spectral radius is nondecreasing with respect to the dimension n when the order is even both infinite and finite dimensional hilbert tensors are positive definite we also show that the m order infinite dimensional hilbert tensor hypermatrix h h i 1 i 2 i m defines a bounded and positively m 1 homogeneous operator from l 1 into l p 1 p and the norm of corresponding positively homogeneous operator is smaller than or equal to 6

a dual dual formulation for the coupling of mixed fem and bem in hyperelasticity we combine a dual mixed finite element method with the boundary integral equation method to study the solvability and galerkin approximations of a linear nonlinear transmission problem arising in plane elastostatics our approach is based on the introduction of the strain tensor as an auxiliary unknown in the finite element domain and it leads to what we call a dual dual mixed formulation since the corresponding operator equation becomes of a twofold saddle point structure we derive existence and uniqueness of solution for the continuous and discrete formulations and provide the associated error analysis in particular we extend the finite element space for plane elasticity given by the peers element and define an explicit finite element boundary element subspace satisfying all the required compatibility conditions most of our analysis makes use of an extension of the classical babuska brezzi theory to a class of nonlinear saddle point problems

fem bem coupling methods for tokamak plasma axisymmetric free boundary equilibrium computations in unbounded domains abstract incorporating boundary conditions at infinity into simulations on bounded computational domains is a repeatedly occurring problem in scientific computing the combination of finite element methods fem and boundary element methods bem is the obvious instrument and we adapt here for the first time the two standard fem bem coupling approaches to the free boundary equilibrium problem the johnson nedelec coupling and the bielak maccamy coupling we recall also the classical approach for fusion applications dubbed according to its first appearance von hagenow lackner coupling and present the less used alternative introduced by albanese blum and de barbieri in 2 we show that the von hagenow lackner coupling suffers from undesirable non optimal convergence properties that suggest that other coupling schemes in particular johnson nedelec or albanese blum de barbieri are more appropriate for non linear equilibrium problems moreover we show that any of such coupling methods requires newton like iteration schemes for solving the corresponding non linear discrete algebraic systems

on the adaptive coupling of fem and bem in 2 d elasticity this paper concerns the combination of the finite element method fem and the boundary element method bem using the symmetric coupling as a model problem in two dimensions we consider the hencky material a certain nonlinear elastic material in a bounded domain with navier lame differential equation in the unbounded complementary domain using some boundary integral operators the problem is rewritten such that the galerkin procedure leads to a fem bem coupling and quasi optimally convergent discrete solutions beside this a priori information we derive an a posteriori error estimate which allows up to a constant factor the error control in the energy norm since information about the singularities of the solution is not available a priori in many situation and having in mind the goal of an automatic mesh refinement we state adaptive algorithms for the h version of the fem bem coupling illustrating numerical results are included

least squares methods for the coupling of fem and bem in the present paper we propose least squares formulations for the numerical solution of exterior boundary value problems the partial differential equation is a first order system in a bounded subdomain and the unbounded subdomain is treated by means of boundary integral equations the first order system is derived from a strongly elliptic second order system the analysis of the present least squares formulations is reduced to the analysis of the galerkin method for the coupling of finite element and boundary element methods fem and bem of the second order problem the least squares approach requires no stability condition however it requires the computation of negative as well as of half integer sobolev norms the arising linear systems can be preconditioned to have condition numbers sim 1 the present methods benefit strongly from the use of biorthogonal wavelets on the coupling boundary and the computation of corresponding equivalent norms in sobolev spaces in particular the application of green s formula leads to an efficient discretization of least squares formulations

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

a fem bem coupling method for a nonlinear transmission problem modelling coulomb friction contact a nonlinear transmission problem modelling elastoplastic interface problems with contact and coulomb friction is reduced to a boundary domain variational inequality we present a corresponding fem bem coupling procedure and derive for its galerkin solution an a priori error estimate furthermore we reformulate the problem as an equivalent saddle point problem whose discretization can be solved by the uzawa algorithm convergence of the fem bem coupling method is proved and numerical results are given

a fully discrete bem fem method for an exterior elasticity system in the plane abstract we present a modified version of the usual bem fem coupling for the exterior elasticity problem in the plane cf costabel and stephan siam j numer anal 27 1990 1212 1226 this new formulation allows us to take advantage of techniques from hsiao et al computing 25 1980 557 566 to compute the boundary integral terms using simple quadrature formulas we provide error estimates for the galerkin method and prove that the corresponding fully discrete scheme preserves the optimal rates of convergence

distributed adaptive convex optimization on directed graphs via continuous time algorithms this note considers the distributed optimization problem on directed graphs with nonconvex local objective functions and the unknown network connectivity a new adaptive algorithm is proposed to minimize a differentiable global objective function by introducing dynamic coupling gains and updating the coupling gains using relative information of system states the nonconvexity of local objective functions unknown network connectivity and the uncertain dynamics caused by locally lipschitz gradients are tackled concurrently consequently the global asymptotic convergence is established when the global objective function is strongly convex and the gradients of local objective functions are only locally lipschitz when the communication graph is strongly connected and weight balanced the algorithm is independent of any global information then the algorithm is naturally extended to unbalanced directed graphs by using the left eigenvector of the laplacian matrix associated with the zero eigenvalue several numerical simulations are presented to verify the results

the coupling of boundary integral and finite element methods for the exterior steady oseen problem abstract in this article we present a new numerical method for solving the steady oseen equations in an unbounded plane domain the technique consists in coupling the boundary integral and the finite element methods an artificial smooth boundary is introduced separating an interior inhomogeneous region from an exterior homogeneous one the solution in exterior region is represented by an integral equation over the artificial boundary this integral equation is incorporated into a velocity pressure formulation for the interior region and a finite element method is used to approximate the resulting variational problem finally the optimal error estimates of the numerical solution are derived computer results will be discussed in a forthcoming paper

existence results for a nonlinear transmission problem abstract let o and i be open bounded regular subsets of r n such that the closure of i is contained in o let f o be a regular function on o and let f and g be continuous functions from i r to r by exploiting an argument based on potential theory and on the leray schauder principle we show that under suitable and completely explicit conditions on f and g there exists at least one pair of continuous functions u o u i such that u o 0 in o cl i u i 0 in i u o x f o x for all x o u o x f x u i x for all x i i u o x i u i x g x u i x for all x i where the last equality is attained in certain weak sense a simple example shows that such a pair of functions u o u i is in general neither unique nor locally unique if instead the fourth condition of the problem is obtained by a small nonlinear perturbation of a homogeneous linear condition then we prove the existence of at least one classical solution which is in addition locally unique

on eigenvalue problems of real symmetric tensors abstract we use variational methods to give a positive answer to a conjecture posed by liqun qi l qi eigenvalues of a real supersymmetric tensor j symbolic comput 40 2005 1302 1324 regarding the real eigenvalues of certain higher order tensors

some remarks on p p0 b and b0 tensors abstract recently song and qi extended the concept of p p 0 and b matrices to p p 0 b and b 0 tensors obtained some properties about these tensors and proposed many questions for further research in this paper we answer three questions mentioned as above and obtain further results about p p 0 b and b 0 tensors

wavelet least squares methods for boundary value problems this paper is concerned with least squares methods for the numerical solution of operator equations our primary focus is the discussion of the following conceptual issues the selection of appropriate least squares functionals their numerical evaluation in the special light of recent developments of wavelet methods and a natural way of preconditioning the resulting systems of linear equations we describe first a general format of variational problems that are well posed in a certain natural topology in order to illustrate the scope of these problems we identify several special cases such as second order elliptic boundary value problems their formulation as a first order system transmission problems the system of stokes equations or more general saddle point problems particular emphasis is placed on the separate treatment of essential nonhomogeneous boundary conditions we propose a unified treatment based on wavelet expansions in particular we exploit the fact that weighted sequence norms of wavelet coefficients are equivalent to relevant function norms arising in the least squares context this provides access to difficult norms efficient preconditioners and in the case of first order systems optimal l2 error estimates

notes on hilbert and cauchy matrices inspired by examples of small hilbert matrices the author proves a property of symmetric totally positive cauchy matrices called at property and consequences for the hilbert matrix

the h p boundary element method for solving 2 and 3 dimensional problems abstract in this survey paper we report on recent developments of the hp version of the boundary element method we consider weakly singular and hypersingular integral equations on polygons and polyhedral surfaces we show that the galerkin solutions computed with the hp version on geometric meshes converge exponentially fast towards the exact solutions of the integral equations the implementation of the hp version of the boundary element method is discussed and adaptive algorithms are given model problems for the coupling of finite elements and boundary elements are also considered numerical results are presented which underline the theoretical results