a simplification of a result by zellini on the maximal rank of symmetric three way arrays zellini 1979 theorem 3 1 has shown how to decompose an arbitrary symmetric matrix of ordern n as a linear combination of 1 2n n 1 fixed rank one matrices thus constructing an explicit tensor basis for the set of symmetricn n matrices zellini s decomposition is based on properties of persymmetric matrices in the present paper a simplified tensor basis is given by showing that a symmetric matrix can also be decomposed in terms of 1 2n n 1 fixed binary matrices of rank one the decomposition implies that ann n p array consisting ofp symmetricn n slabs has maximal rank 1 2n n 1 likewise an unconstrained indscal symmetric candecomp parafac decomposition of such an array will yield a perfect fit in 1 2n n 1 dimensions when the fitting only pertains to the off diagonal elements of the symmetric matrices as is the case in a version of parafac where communalities are involved the maximal number of dimensions can be further reduced to 1 2n n 1 however when the saliences in indscal are constrained to be nonnegative the tensor basis result does not apply in fact it is shown that in this case the number of dimensions needed can be as large asp the number of matrices analyzed

continuous discontinuous finite element method for convection diffusion problems with characteristic layers we study convergence properties of a numerical method for convection diffusion problems with characteristic layers on a layer adapted mesh the method couples standard galerkin with an h version of the nonsymmetric discontinuous galerkin finite element method with bilinear elements in an associated norm we derive the error estimate as well as the supercloseness result that are uniform in the perturbation parameter applying a post processing operator for the discontinuous galerkin method we construct a new numerical solution with enhanced convergence properties

typical rank and indscal dimensionality for symmetric three way arrays of order i 2 2 or i 3 3 a peculiar property of three way arrays is that the rank they typically have does not necessarily coincide with the maximum possible rank given their order typical tensorial rank has much been studied over algebraically closed fields however very few results have been found pertaining to the typical rank of three way arrays over the real field these results refer to arrays sampled randomly from continuous distributions arrays that consist of symmetric slices do not fit into this sampling scheme the present paper offers typical rank results over the real field for arrays containing symmetric slices of order 2 2a nd 3 3 symmetric arrays often appear to have lower typical ranks than their asymmetric counterparts this paper also examines whether or not the rank of a symmetric array coincides with the smallest number of dimensions that allow a perfect fit of indscal for all cases considered this is indeed true thus a full indscal solution may require fewer dimensions for a symmetric array than a full cp decomposition applied to an asymmetric array of the same size the reverse situation does not seem to arise next we examine in which cases cp solutions inevitably are indscal solutions finally the rank reducing impact of double standardizing the slices is discussed

the typical rank of tall three way arrays the rank of a three way array refers to the smallest number of rank one arrays outer products of three vectors that generate the array as their sum it is also the number of components required for a full decomposition of a three way array by candecomp parafac the typical rank of a three way array refers to the rank a three way array has almost surely the present paper deals with typical rank and generalizes existing results on the typical rank ofi j k arrays withk 2 to a particular class of arrays withk 2 it is shown that the typical rank isi when the array is tall in the sense thatjk j i jk in addition typical rank results are given for the case wherei equalsjk j

on mean values of mollifiers and l functions associated to primitive cusp forms we study the second moment of the l function associated to a holomorphic primitive cusp form of even weight perturbed by a new family of mollifiers this family is a natural extension of the mollifers considered by conrey and by bui conrey and young as an application we improve the current lower bound on critical zeros of holomorphic primitive cusp forms

interior penalty discontinuous approximations of convection diffusion problems with parabolic layers a nonsymmetric discontinuous galerkin finite element method with interior penalties is considered for two dimensional convection diffusion problems with regular and parabolic layers on an anisotropic shishkin type mesh with bilinear elements we prove error estimates uniformly in the perturbation parameter in an integral norm associated with this method on different types of interelement edges we derive the values of discontinuity penalization parameters numerical experiments complement the theoretical results

simplicity transformations for three way arrays with symmetric slices and applications to tucker 3 models with sparse core arrays abstract tucker three way pca and candecomp parafac are two well known methods of generalizing principal component analysis to three way data candecomp parafac yields component matrices a e g for subjects or objects b e g for variables and c e g for occasions that are typically unique up to jointly permuting and rescaling columns tucker 3 analysis on the other hand has full transformational freedom that is the fit does not change when a b and c are postmultiplied by nonsingular transformation matrices provided that the inverse transformations are applied to the so called core array g this freedom of transformation can be used to create a simple structure in a b c and or in g this paper deals with the latter possibility exclusively it revolves around the question of how a core array or in fact any three way array can be transformed to have a maximum number of zero elements direct applications are in tucker 3 analysis where simplicity of the core may facilitate the interpretation of a tucker 3 solution and in constrained tucker 3 analysis where hypotheses involving sparse cores are taken into account in the latter cases it is important to know what degree of sparseness can be attained as a tautology by using the transformational freedom in addition simplicity transformations have proven useful as a mathematical tool to examine rank and generic or typical rank of three way arrays so far a number of simplicity results have been attained pertaining to arrays sampled randomly from continuous distributions these results do not apply to three way arrays with symmetric slices in one direction the present paper offers a number of simplicity results for arrays with symmetric slices of order 2 2 3 3 and 4 4 some generalizations to higher orders are also discussed as a mathematical application the problem of determining the typical rank of 4 3 3 and 5 3 3 arrays with symmetric slices will be revisited using a sparse form with only 8 out of 36 elements nonzero for the former case and 10 out of 45 elements nonzero for the latter one that can be attained almost surely for such arrays the issue of maximal simplicity of the targets to be presented will be addressed either by formal proofs or by relying on simulation results

on the finite element method for singularly perturbed reaction diffusion problems in two and one dimensions second order elliptic boundary value problems which are allowed to degenerate into zero order equations are considered the behavior of the ordinary galerkin finite element method without special arrangements to treat singularities is studied as the problem ranges from true second order to singularly perturbed

a uniformly convergent continuous discontinuous galerkin method for singularly perturbed problems of convection diffusion type abstract in this paper we introduce a coupled approach of local discontinuous galerkin and standard finite element method for solving singularly perturbed convection diffusion problems on shishkin mesh with linear elements a rate o n 1 ln n in an associated norm is established where n is the number of elements numerical experiments complement the theoretical results moreover a rate o n 2 ln 2 n in a discrete l norm and o n 2 in l2 norm are observed numerically on the shishkin mesh

layer adapted meshes for convection diffusion problems abstract in the present paper a classification and a survey are given of layer adapted meshes for stationary convection dominated convection diffusion problems for a number of standard numerical schemes theoretical results are presented that demonstrate that the use of properly layer adapted meshes yields robust methods i e methods that perform equally well no matter how dominant the convection we review a number of techniques used in the convergence analysis of these methods

finite element superconvergence on shishkin mesh for 2 d convection diffusion problems in this work the bilinear finite element method on a shishkin mesh for convection diffusion problems is analyzed in the two dimensional setting a superconvergence rate o n 2 ln2 n en 1 5lnn in a discrete e weighted energy norm is established under certain regularity assumptions this convergence rate is uniformly valid with respect to the singular perturbation parameter e numerical tests indicate that the rate o n 2ln2 n is sharp for the boundary layer terms as a by product an e uniform convergence of the same order is obtained for the l2 norm furthermore under the same regularity assumption an e uniform convergence of order n 3 2ln5 2n en 1ln1 2n in the l norm is proved for some mesh points in the boundary layer region

the carroll and chang conjecture of equal indscal components when candecomp parafac gives perfect fit abstract the candecomp parafac algorithm approximates a set of matrices x 1 x i by products of the form ac i b with c i diagonal i 1 i carroll and chang have conjectured that when the matrices are symmetric the resulting a and b will be column wise proportional for cases of perfect fit ten berge et al have shown that the conjecture holds true in a variety of cases but may fail when there is no unique solution in such cases obtaining proportionality by changing part of the solution seems possible the present paper extends and further clarifies their results in particular where ten berge et al solved all i 2 2 cases now all i 3 3 cases and also the i 4 4 cases for i 2 8 and 9 are clarified in a number of cases a and b necessarily have column wise proportionality when candecomp parafac is run to convergence in other cases proportionality can be obtained by using specific methods no cases were found that seem to resist proportionality

an a priori error analysis of the local discontinuous galerkin method for elliptic problems in this paper we present the first a priori error analysis for the local discontinuous galerkin ldg method for a model elliptic problem for arbitrary meshes with hanging nodes and elements of various shapes we show that for stabilization parameters of order one the l2 norm of the gradient and the l2 norm of the potential are of order k and k 1 2 respectively when polynomials of total degree at least k are used if stabilization parameters of order h 1 are taken the order of convergence of the potential increases to k 1 the optimality of these theoretical results is tested in a series of numerical experiments on two dimensional domains

superconvergence of the numerical traces of discontinuous galerkin and hybridized methods for convection diffusion problems in one space dimension in this paper we uncover and study a new superconvergence property of a large class of finite element methods for one dimensional convection diffusion problems this class includes discontinuous galerkin methods defined in terms of numerical traces discontinuous petrov galerkin methods and hybridized mixed methods we prove that the so called numerical traces of both variables superconverge at all the nodes of the mesh provided that the traces are conservative that is provided they are single valued in particular for is local discontinuous galerkin method we show that the superconvergence is order 2p 1 when polynomials of degree at most p are used extensive numerical results verifying our theoretical results are displayed

how accurate is the streamline diffusion finite element method we investigate the optimal accuracy of the streamline diffusion finite element method applied to convection dominated problems for linear bilinear elements the theoretical order of convergence given in the literature is either o h 3 2 for quasi uniform meshes or o h 2 for some uniform meshes the determination of the optimal order in general was an open problem by studying a special type of meshes it is shown that the streamline diffusion method may actually converge with any order within this range depending on the characterization of the meshes

asymptotically exact a posteriori ldg error estimates for one dimensional transient convection diffusion problems in this paper new a posteriori error estimates for the local discontinuous galerkin ldg formulation applied to transient convection diffusion problems in one space dimension are presented and analyzed these error estimates are computationally simple and are computed by solving a local steady problem with no boundary conditions on each element we first show that the leading error term on each element for the solution is proportional to a p 1 degree right radau polynomial while the leading error term for the solution s derivative is proportional to a p 1 degree left radau polynomial when polynomials of degree at most p are used these results are used to prove that for smooth solutions these error estimates at a fixed time converge to the true spatial errors in the l 2 norm under mesh refinement more precisely we prove that our ldg error estimates converge to the true spatial errors at o h p 5 4 rate finally we prove that the global effectivity indices in the l 2 norm converge to unity at o h 1 2 rate our computational results indicate that the observed numerical convergence rates are higher than the theoretical rates

simplicity and typical rank results for three way arrays matrices can be diagonalized by singular vectors or when they are symmetric by eigenvectors pairs of square matrices often admit simultaneous diagonalization and always admit block wise simultaneous diagonalization generalizing these possibilities to more than two non square matrices leads to methods of simplifying three way arrays by nonsingular transformations such transformations have direct applications in tucker pca for three way arrays where transforming the core array to simplicity is allowed without loss of fit simplifying arrays also facilitates the study of array rank the typical rank of a three way array is the smallest number of rank one arrays that have the array as their sum when the array is generated by random sampling from a continuous distribution in some applications the core array of tucker pca is constrained to have a vast majority of zero elements both simplicity and typical rank results can be applied to distinguish constrained tucker pca models from tautologies an update of typical rank results over the real number field is given in the form of two tables

a uniformly convergent galerkin method on a shishkin mesh for a convection diffusion problem abstract a galerkin finite element method that uses piecewise bilinears on a simple piecewise equidistant mesh is applied to a linear convection dominated convection diffusion problem in two dimensions the method is shown to be convergent uniformly in the perturbation parameter of ordern 1 ln nin a global energy norm and of ordern 1 2ln3 2 npointwise near the outflow boundary where the total number of mesh points iso n2