character correspondences and irreducible induction and restriction let s and g be linitc groups such that s acts on g and is ici 1 and let c denote the fixed point subgroup c c s it is well known in this situation that there exists a natural bijection from the set irr g of s invariant irreducible characters of g onto irr c the full set of irreducible characters of c when s is solvable this bijection was constructed by g glauberman 2 and in the remaining case where g is solvable of odd order a totally different construction 3 gives the desired character correspondence it was proved by t r wolf g that when both constructions apply when s is solvable and ig is odd the two constructions yield exactly the same map if 31 e irr g we write x e irr c to denote the uniquely determined corresponding character similarly if he g is s invariant and 0 e irr h we write f3 e irr hn c to denote the image of 8 under the character bijection induced by the action of s on h our main result is the following

mesh refinement processes based on the generalized bisection of simplices mesh refinement processes based on the generalized bisection of simplices are discussed and characterized in the context of the general theory of rheinboldt siam j numer anal 17 1980 pp 766 778 then it is proved that such processes allow the construction of sequences of naturally smooth conforming and nested nonuniform meshes in fact it is shown that there exists a generalized mesh refinement operator that for any conforming triangulation delta and for any v subset delta produces a nested smooth conforming triangulation delta containing successors of all elements of v and such that the minimum cell size of delta is bounded from below by half the minimum cell size of delta two conforming mesh refinement algorithms that allow the selective refinement of computational triangulations are explicitly given

topology optimization using polytopes abstract meshing complex engineering domains is a challenging task arbitrary polyhedral meshes can provide the much needed flexibility in automated discretization of such domains the geometric property of polyhedral meshes such as its unstructured nature and the connectivity of faces between elements makes them specially attractive for topology optimization applications numerical anomalies in designs such as the single node connections and checkerboard pattern can be naturally circumvented with polyhedrons in the current work we solve the governing three dimensional elasticity state equation using the virtual element method vem approach the main characteristic difference between vem and standard finite element methods fem is that in vem the canonical basis functions are not constructed explicitly rather the stiffness matrix is computed directly utilizing a projection map which extracts the linear component of the deformation such a construction guarantees the satisfaction of the patch test used by engineers as an indicator of optimal convergence of numerical solutions under mesh refinement finally the computations reduce to the evaluation of matrices which contain purely geometric surface facet quantities the present work focuses on the first order vem in which the degrees of freedom are associated with the vertices the features of the current optimization approach are demonstrated using numerical examples for compliance minimization and compliant mechanism problems

a superconvergent ldg hybridizable galerkin method for second order elliptic problems we identify and study an ldg hybridizable galerkin method which is not an ldg method for second order elliptic problems in several space dimensions with remarkable convergence properties unlike all other known discontinuous galerkin methods using polynomials of degree k 0 for both the potential as well as the flux the order of convergence in l 2 of both unknowns is k 1 moreover both the approximate potential as well as its numerical trace superconverge in l 2 like norms to suitably chosen projections of the potential with order k 2 this allows the application of element by element postprocessing of the approximate solution which provides an approximation of the potential converging with order k 2 in l 2 the method can be thought to be in between the hybridized version of the raviart thomas and that of the brezzi douglas marini mixed methods

unified hybridization of discontinuous galerkin mixed and continuous galerkin methods for second order elliptic problems we introduce a unifying framework for hybridization of finite element methods for second order elliptic problems the methods fitting in the framework are a general class of mixed dual finite element methods including hybridized mixed continuous galerkin nonconforming and a new wide class of hybridizable discontinuous galerkin methods the distinctive feature of the methods in this framework is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements since the associated matrix is sparse symmetric and positive definite these methods can be efficiently implemented moreover the framework allows in a single implementation the use of different methods in different elements or subdomains of the computational domain which are then automatically coupled finally the framework brings about a new point of view thanks to which it is possible to see how to devise novel methods displaying very localized and simple mortaring techniques as well as methods permitting an even further reduction of the number of globally coupled degrees of freedom

global convergence of oja s pca learning algorithm with a non zero approaching adaptive learning rate a non zero approaching adaptive learning rate is proposed to guarantee the global convergence of oja s principal component analysis pca learning algorithm most of the existing adaptive learning rates for oja s pca learning algorithm are required to approach zero as the learning step increases however this is not practical in many applications due to the computational round off limitations and tracking requirements the proposed adaptive learning rate overcomes this shortcoming the learning rate converges to a positive constant thus it increases the evolution rate as the learning step increases this is different from learning rates which approach zero which slow the convergence considerably and increasingly with time rigorous mathematical proofs for global convergence of oja s algorithm with the proposed learning rate are given in detail via studying the convergence of an equivalent deterministic discrete time ddt system extensive simulations are carried out to illustrate and verify the theory derived simulation results show that this adaptive learning rate is more suitable for oja s pca algorithm to be used in an online learning situation

invariant blocks under coprime action let a and g be finite groups with a g 1 assume that a acts on g via automorphisms let b be an a invariant block of g if each irreducible character in b is a invariant and b has cyclic defect groups then a centralizes a defect group of b

convergence analysis of the high order mimetic finite difference method we prove second order convergence of the conservative variable and its flux in the high order mfd method the convergence results are proved for unstructured polyhedral meshes and full tensor diffusion coefficients for the case of non constant coefficients we also develop a new family of high order mfd methods theoretical result are confirmed through numerical experiments

a locking free weak galerkin finite element method for elasticity problems in the primal formulation this paper presents an arbitrary order locking free numerical scheme for linear elasticity on general polygonal polyhedral partitions by using weak galerkin wg finite element methods like other wg methods the key idea for the linear elasticity is to introduce discrete weak strain and stress tensors which are defined and computed by solving inexpensive local problems on each element such local problems are derived from weak formulations of the corresponding differential operators through integration by parts locking free error estimates of optimal order are derived in a discrete h 1 norm and the usual l 2 norm for the approximate displacement when the exact solution is smooth numerical results are presented to demonstrate the efficiency accuracy and the locking free property of the weak galerkin finite element method

bi iterative least square method for subspace tracking subspace tracking is an adaptive signal processing technique useful for a variety of applications in this paper we introduce a simple bi iterative least square bi ls method which is in contrast to the bi iterative singular value decomposition bi svd method we show that for subspace tracking the bi ls method is easier to simplify than the bi svd method the linear complexity algorithms based on bi ls are computationally more efficient than the existing linear complexity algorithms based on bi svd although both have the same performance for subspace tracking a number of other existing subspace tracking algorithms of similar complexity are also compared with the bi ls algorithms

a dual purpose principal and minor subspace gradient flow the dual purpose principal and minor subspace gradient flow can be used to track principal subspace ps and if altered simply by the sign it can also serve as a minor subspace ms trackor this is of practical significance in the implementations of algorithms in this paper a unified information criterion is proposed and a dual purpose principal and minor subspace gradient flow is derived based on the information criterion in this dual purpose gradient flow the weight matrix length is self stabilizing i e moving towards unit length in each learning step the energy function associated with the dual purpose gradient flow for tracking ps and ms is given and it exhibits a unique global minimum attained if and only if its state matrices span the ps or ms of the autocorrelation matrix of a vector data stream the other stationary points of its energy function are unstable saddle points the proposed dual purpose gradient flow can efficiently track an orthonormal basis of the ps or ms which is illustrated through simulation experiments

recurrent neural network approach based on the integral representation of the drazin inverse in this letter we present the dynamical equation and corresponding artificial recurrent neural network for computing the drazin inverse for arbitrary square real matrix without any restriction on its eigenvalues conditions that ensure the stability of the defined recurrent neural network as well as its convergence toward the drazin inverse are considered several illustrative examples present the results of computer simulations

a new weak galerkin finite element method for elliptic interface problems a new weak galerkin wg finite element method is introduced and analyzed in this paper for solving second order elliptic equations with discontinuous coefficients and interfaces comparing with the existing wg algorithm for solving the same type problems the present wg method has a simpler variational formulation and fewer unknowns moreover the new wg algorithm allows the use of finite element partitions consisting of general polytopal meshes and can be easily generalized to high orders optimal order error estimates in both h 1 and l 2 norms are established for the present wg finite element solutions extensive numerical experiments have been conducted to examine the accuracy flexibility and robustness of the proposed wg interface approach in solving regular elliptic interface problems high order convergences are numerically confirmed by using piecewise polynomial basis functions of high degrees moreover the wg method is shown to be able to accommodate very complicated interfaces due to its flexibility in choosing finite element partitions finally in dealing with challenging problems with low regularities the piecewise linear wg method is capable of delivering a second order of accuracy in l norm for both c 1 and h 2 continuous solutions

discontinuous galerkin finite element methods for interface problems a priori and a posteriori error estimations discontinuous galerkin dg finite element methods were studied by many researchers for second order elliptic partial differential equations and a priori error estimates were established when the solution of the underlying problem is piecewise h 3 2 epsilon smooth with epsilon 0 however elliptic interface problems with intersecting interfaces do not possess such a smoothness in this paper we establish a quasi optimal a priori error estimate for interface problems whose solutions are only h 1 alpha smooth with alpha in 0 1 and hence fill a theoretical gap of the dg method for elliptic problems with low regularity the second part of the paper deals with the design and analysis of robust residual and recovery based a posteriori error estimators theoretically we show that the residual and recovery estimators studied in this paper are robust with respect to the dg norm i e their reliability and efficiency bounds do not depend on the jump provided that the distribution of coefficients is locally quasi monotone

residual based error estimate and quasi interpolation on polygonal meshes for high order bem based fem abstract only a few numerical methods can treat boundary value problems on polygonal and polyhedral meshes the bem based finite element method is one of the new discretization strategies which make use of and benefits from the flexibility of these general meshes that incorporate hanging nodes naturally the article in hand addresses quasi interpolation operators for the approximation space over polygonal meshes to prove interpolation estimates the poincare constant is bounded uniformly for patches of star shaped elements these results give rise to the residual based error estimate for high order bem based fem and its reliability as well as its efficiency are proven such a posteriori error estimates can be used to gauge the approximation quality and to implement adaptive fem strategies numerical experiments show optimal rates of convergence for meshes with non convex elements on uniformly as well as on adaptively refined meshes

a posteriori error estimates stopping criteria and adaptivity for multiphase compositional darcy flows in porous media in this paper we derive a posteriori error estimates for the compositional model of multiphase darcy flow in porous media consisting of a system of strongly coupled nonlinear unsteady partial differential and algebraic equations we show how to control the dual norm of the residual augmented by a nonconformity evaluation term by fully computable estimators we then decompose the estimators into the space time linearization and algebraic error components this allows to formulate criteria for stopping the iterative algebraic solver and the iterative linearization solver when the corresponding error components do not affect significantly the overall error moreover the spatial and temporal error components can be balanced by time step and space mesh adaptation our analysis applies to a broad class of standard numerical methods and is independent of the linearization and of the iterative algebraic solvers employed we exemplify it for the two point finite volume method with fully implicit euler time stepping the newton linearization and the gmres algebraic solver numerical results on two real life reservoir engineering examples confirm that significant computational gains can be achieved thanks to our adaptive stopping criteria already on fixed meshes without any noticeable loss of precision

fast principal component extraction by a weighted information criterion principal component analysis pca is an essential technique in data compression and feature extraction and there has been much interest in developing fast pica algorithms on the basis of the concepts of both weighted subspace and information maximization this paper proposes a weighted information criterion winc for searching the optimal solution of a linear neural network we analytically show that the optimum weights globally asymptotically converge to the principal eigenvectors of a stationary vector stochastic process we establish a dependent relation of choosing the weighting matrix on statistics of the input process through the analysis of stability of the equilibrium of the proposed criterion therefore we are able to reveal the constraint on the choice of a weighting matrix we develop two adaptive algorithms based on the winc for extracting in parallel multiple principal components both algorithms are able to provide adaptive step size which leads to a significant improvement in the learning performance furthermore the recursive least squares rls version of winc algorithms has a low computational complexity o np where n is the input vector dimension and p is the number of desired principal components in fact the winc algorithm corresponds to a three layer linear neural network model capable of performing in parallel the extraction of multiple principal components the winc algorithm also generalizes some well known pca psa algorithms just by adjusting the corresponding parameters since the weighting matrix does not require an accurate value it facilitates the system design of the winc algorithm for practical applications the accuracy and speed advantages of the winc algorithm are verified through simulations

global convergence analysis for the nic flow previously a family of fast subspace tracking algorithms based on a novel information criterion nic has been proposed and investigated it is known that these new algorithms are associated with a new kind of flow which is called the nic flow in this paper as in the case of the conventional oja subspace algorithms with the oja flow in this paper some fundamental questions about this new nic flow such as its solution existence and convergence are investigated in addition the convergence domain is characterized some important results on these issues are obtained via manifold theory

a priori and a posterior error estimate of new weak galerkin finite element methods for second order elliptic interface problems on polygonal meshes abstract in this paper we present a posteriori error estimate of a new weak galerkin wg finite element method for the second order elliptic interface problems instead of introducing the local weak derivatives on each element the bilinear form is computed through the inner product of traditional defined derivatives the proposed numerical scheme is symmetric and positive definite and can be applied to polygonal meshes in the a posteriori error estimate we construct a error indicator that can be applied to polygonal meshes or meshes with hanging nodes the reliability and efficiency of the designed error estimator have been proved in this work extensive numerical tests are performed to validate our algorithm these results demonstrate the effectiveness of the adaptive mesh refinement guided by the proposed error estimator

a dual purpose principal and minor component flow abstract principal component flows are flows converging to the eigenvectors associated with the largest eigenvalues of a given symmetric matrix similarly minor component flows converge to the eigenvectors associated with the smallest eigenvalues traditional flows required the matrix to be positive definite and moreover finding well behaved minor component flows appeared to be harder and unrelated to the principal component case this paper derives a flow which can be used to extract either the principal or the minor components and which does not require the matrix to be positive definite the flow is shown to be a generalisation of the oja brockett flow

iterative kernel principal component analysis for image modeling in recent years kernel principal component analysis kpca has been suggested for various image processing tasks requiring an image model such as e g denoising or compression the original form of kpca however can be only applied to strongly restricted image classes due to the limited number of training examples that can be processed we therefore propose a new iterative method for performing kpca the kernel hebbian algorithm which iteratively estimates the kernel principal components with only linear order memory complexity in our experiments we compute models for complex image classes such as faces and natural images which require a large number of training examples the resulting image models are tested in single frame super resolution and denoising applications the kpca model is not specifically tailored to these tasks in fact the same model can be used in super resolution with variable input resolution or denoising with unknown noise characteristics in spite of this both super resolution and denoising performance are comparable to existing methods

a weak galerkin finite element scheme for solving the stationary stokes equations a weak galerkin wg finite element method for solving the stationary stokes equations in two or three dimensional spaces by using discontinuous piecewise polynomials is developed and analyzed the variational form we considered is based on two gradient operators which is different from the usual gradient divergence operators the wg method is highly flexible by allowing the use of discontinuous functions on arbitrary polygons or polyhedra with certain shape regularity optimal order error estimates are established for the corresponding wg finite element solutions in various norms numerical results are presented to illustrate the theoretical analysis of the new wg finite element scheme for stokes problems

the nonconforming virtual element method for the stokes equations we present the nonconforming virtual element method vem for the numerical approximation of velocity and pressure in the steady stokes problem the pressure is approximated using discontinuous piecewise polynomials while each component of the velocity is approximated using the nonconforming virtual element space on each mesh element the local virtual space contains the space of polynomials of up to a given degree plus suitable nonpolynomial functions the virtual element functions are implicitly defined as the solution of local poisson problems with polynomial neumann boundary conditions as typical in vem approaches the explicit evaluation of the non polynomial functions is not required this approach makes it possible to construct nonconforming virtual spaces for any polynomial degree regardless of the parity for two and three dimensional problems and for meshes with very general polygonal and polyhedral elements we show that the nonconforming vem is inf sup stable and establish optimal a pri

clifford theory for mackey algebras we develop a clifford theory for mackey algebras for simple mackey functors using their classification we prove mackey algebra versions of clifford s theorem and the clifford correspondence let r g be the mackey algebra of a finite group g over a commutative unital ring r a nd let 1 n be the unity of r n where n is a normal subgroup of g observing that 1n r g 1n is a crossed product of g n over r n a number of results concerning group graded algebras are extended to the context of mackey algebras including fong s theorem green s indecomposibility theorem and some reduction and extension techniques for indecomposable mackey functors

oja s algorithm for graph clustering markov spectral decomposition and risk sensitive control given a positive definite matrix m and an integer n m 1 oja s subspace algorithm will provide convergent estimates of the first n m eigenvalues of m along with the corresponding eigenvectors it is a common approach to principal component analysis this paper introduces a normalized stochastic approximation implementation of oja s subspace algorithm as well as new applications to the spectral decomposition of a reversible markov chain recall that this means that the stationary distribution satisfies the detailed balance equations meyn tweedie 2009 equivalently the statistics of the process in steady state do not change when time is reversed stability and convergence of oja s algorithm are established under conditions far milder than that assumed in previous work applications to graph clustering markov spectral decomposition and multiplicative ergodic theory are surveyed along with numerical results

invariant constituents and invariant blocks under coprime action abstract let a and g be finite groups with a g 1 we assume that a acts on g via automorphism let n be an a invariant normal subgroup of g let be an a invariant irreducible brauer character of n if a is of prime power order then the induced brauer character g contains an a invariant irreducible constituent if g n is p solvable then g contains an a invariant irreducible constituent let b be an a invariant block of g then under glauberman isaacs correspondence the set irr a b is a union of blocks of c g a say b 1 b 2 b s let q i be a defect group of b i then there is a defect group d of b such that q i d

a discontinuous skeletal method for advection diffusion reaction on general meshes we design and analyze an approximation method for advection diffusion reaction equa tions where the generalized degrees of freedom are polynomials of order k 0 at mesh faces the method hinges on local discrete reconstruction operators for the diffusive and advective derivatives and a weak enforcement of boundary conditions fairly general meshes with poly topal and nonmatching cells are supported arbitrary polynomial orders can be considered including the case k 0 which is closely related to mimetic finite difference mixed hybrid finite volume methods the error analysis covers the full range of peclet numbers including the delicate case of local degeneracy where diffusion vanishes on a strict subset of the domain computational costs remain moderate since the use of face unknowns leads to a compact stencil with reduced communications numerical results are presented

global convergence of oja s subspace algorithm for principal component extraction oja s principal subspace algorithm is a well known and powerful technique for learning and tracking principal information in time series a thorough investigation of the convergence property of oja s algorithm is undertaken in this paper the asymptotic convergence rates of the algorithm is discovered the dependence of the algorithm on its initial weight matrix and the singularity of the data covariance matrix is comprehensively addressed

a hybrid high order locking free method for linear elasticity on general meshes abstract we devise an arbitrary order locking free method for linear elasticity the method relies on a pure displacement primal formulation and leads to a symmetric positive definite system matrix with compact stencil the degrees of freedom are vector valued polynomials of arbitrary order k 1 on the mesh faces so that in three space dimensions the lowest order scheme only requires 9 degrees of freedom per mesh face the method can be deployed on general polyhedral meshes the key idea is to reconstruct the symmetric gradient and divergence inside each mesh cell in terms of the degrees of freedom by solving inexpensive local problems the discrete problem is assembled cell wise using these operators and a high order stabilization bilinear form locking free error estimates are derived for the energy norm and for the l 2 norm of the displacement with optimal convergence rates of order k 1 and k 2 respectively for smooth solutions on general meshes the theoretical results are confirmed numerically and the cpu cost is evaluated on both standard and polygonal meshes

a weak galerkin mixed finite element method for second order elliptic problems abstract a new weak galerkin wg method is introduced and analyzed for the second order elliptic equation formulated as a system of two first order linear equations this method called wg mfem is designed by using discontinuous piecewise polynomials on finite element partitions with arbitrary shape of polygons polyhedra the wg mfem is capable of providing very accurate numerical approximations for both the primary and flux variables allowing the use of discontinuous approximating functions on arbitrary shape of polygons polyhedra makes the method highly flexible in practical computation optimal order error estimates in both discrete h and l norms are established for the corresponding weak galerkin mixed finite element solutions

a posteriori error estimates for a discontinuous galerkin approximation of second order elliptic problems several a posteriori error estimators are introduced and analyzed for a discontinuous galerkin formulation of a model second order elliptic problem in addition to residual type estimators we introduce some estimators that are couched in the ideas and techniques of domain decomposition results of numerical experiments are presented

global analysis of oja s flow for neural networks a detailed study of oja s learning equation in neural networks is undertaken in this paper not only are such fundamental issues as existence uniqueness and representation of solutions completely resolved but also the convergence issue is resolved it is shown that the solution of oja s equation is exponentially convergent to an equilibrium from any initial value moreover the necessary and sufficient conditions are given on the initial value for the solution to converge to a dominant eigenspace of the associated autocorrelation matrix as a by product this result confirms one of oja s conjectures that the solution converges to the principal eigenspace from almost all initial values some other characteristics of the limiting solution are also revealed these facilitate the determination of the limiting solution in advance using only the initial information two examples are analyzed demonstrating the explicit dependence of the limiting solution on the initial value in another respect it is found that oja s equation is the gradient flow of generalized rayleigh quotients on a stiefel manifold

convergence analysis of a deterministic discrete time system of oja s pca learning algorithm the convergence of oja s principal component analysis pca learning algorithms is a difficult topic for direct study and analysis traditionally the convergence of these algorithms is indirectly analyzed via certain deterministic continuous time dct systems such a method will require the learning rate to converge to zero which is not a reasonable requirement to impose in many practical applications recently deterministic discrete time ddt systems have been proposed instead to indirectly interpret the dynamics of the learning algorithms unlike dct systems ddt systems allow learning rates to be constant which can be a nonzero this paper will provide some important results relating to the convergence of a ddt system of oja s pca learning algorithm it has the following contributions 1 a number of invariant sets are obtained based on which we can show that any trajectory starting from a point in the invariant set will remain in the set forever thus the nondivergence of the trajectories is guaranteed 2 the convergence of the ddt system is analyzed rigorously it is proven in the paper that almost all trajectories of the system starting from points in an invariant set will converge exponentially to the unit eigenvector associated with the largest eigenvalue of the correlation matrix in addition exponential convergence rate are obtained providing useful guidelines for the selection of fast convergence learning rate 3 since the trajectories may diverge the careful choice of initial vectors is an important issue this paper suggests to use the domain of unit hyper sphere as initial vectors to guarantee convergence 4 simulation results will be furnished to illustrate the theoretical results achieved

the glauberman correspondence and stable multiplicity free subgroups abstract for an action by automorphisms of a finite group a on a group g of relatively prime order the equivalence of the permutation actions of a on the set of complex irreducible characters of g and on the set of conjugacy classes of g was one of the first applications of the glauberman isaacs correspondence for such a and g with a solvable and h an a stable multiplicity free subgroup of g we show similarly using the glauberman correspondence and an idea of navarro the equivalence of the actions of a on the set of h classes of g and on the set of pairs of irreducible characters of g and of h with over

a weak galerkin finite element method for second order elliptic problems this paper introduces a finite element method by using a weakly defined gradient operator over generalized functions the use of weak gradients and their approximations results in a new concept called discrete weak gradients which is expected to play an important role in numerical methods for partial differential equations this article intends to provide a general framework for managing differential operators on generalized functions as a demonstrative example the discrete weak gradient operator is employed as a building block in the design of numerical schemes for a second order elliptic problem in which the classical gradient operator is replaced by the discrete weak gradient the resulting numerical scheme is called a weak galerkin wg finite element method it can be seen that the weak galerkin method allows the use of totally discontinuous functions in the finite element procedure for the second order elliptic problem an optimal order error estimate in both a discrete h 1 and l 2 norms are established for the corresponding weak galerkin finite element solutions a superconvergence is also observed for the weak galerkin approximation

adaptive minor component extraction with modular structure an information criterion for adaptively estimating multiple minor eigencomponents of a covariance matrix is proposed it is proved that the proposed criterion has a unique global minimum at the minor subspace and that all other equilibrium points are saddle points based on the gradient search approach of the proposed information criterion an adaptive algorithm called adaptive minor component extraction amex is developed the proposed algorithm automatically performs the multiple minor component extraction in parallel without the inflation procedure similar to the adaptive lattice filter structure the amex algorithm also has the flexibility wherein increasing the number of the desired minor component does not affect the previously extracted minor components the amex algorithm has a highly modular structure and the various modules operate completely in parallel without any delay simulation results are given to demonstrate the effectiveness of the amex algorithm for both the minor component analysis mca and the minor subspace analysis msa

a posteriori error estimates for the virtual element method an posteriori error analysis for the virtual element method vem applied to general elliptic problems is presented the resulting error estimator is of residual type and applies on very general polygonal polyhedral meshes the estimator is fully computable as it relies only on quantities available from the vem solution namely its degrees of freedom and element wise polynomial projection upper and lower bounds of the error estimator with respect to the vem approximation error are proven the error estimator is used to drive adaptive mesh refinement in a number of test problems mesh adaptation is particularly simple to implement since elements with consecutive co planar edges faces are allowed and therefore locally adapted meshes do not require any local mesh post processing

on stochastic approximation of the eigenvectors and eigenvalues of the expectation of a random matrix abstract in applications of signal processing and pattern recognition eigenvectors and eigenvalues of the statistical mean of a random matrix sequence are needed iterative methods are suggested and analyzed in which no sample moments are used convergence is shown by stochastic approximation theory

socles and radicals of mackey functors we study the socle and the radical of a mackey functor m for a finite group g over a field k usually of characteristic p 0 for a subgroup h of g we construct bijections between some classes of the simple subfunctors of m and some classes of the simple kn g h submodules of m h we relate the multiplicity of a simple mackey functor sh vg in the socle of m to the multiplicity of v in the socle of a certain kn g h submodule of m h we also obtain similar results for the maximal subfunctors of m we then apply these general results to some special mackey functors for g including the functors obtained by inducing or restricting a simple mackey functor mackey functors for a p group the fixed point functor and the burnside functor bkg for instance we find the first four top factors of the radical series of bkg for a p group g and assuming further that g is an abelian p group we find the radical series of bkg

pinning consensus in networks of multiagents via a single impulsive controller in this paper we discuss pinning consensus in networks of multiagents via impulsive controllers in particular we consider the case of using only one impulsive controller we provide a sufficient condition to pin the network to a prescribed value it is rigorously proven that in case the underlying graph of the network has spanning trees the network can reach consensus on the prescribed value when the impulsive controller is imposed on the root with appropriate impulsive strength and impulse intervals interestingly we find that the permissible range of the impulsive strength completely depends on the left eigenvector of the graph laplacian corresponding to the zero eigenvalue and the pinning node we choose the impulses can be very sparse with the impulsive intervals being lower bounded examples with numerical simulations are also provided to illustrate the theoretical results

asymptotic distributions associated to oja s learning equation for neural networks we perform a complete asymptotic performance analysis of the stochastic approximation algorithm denoted subspace network learning algorithm derived from oja s learning equation in the case where the learning rate is constant and a large number of patterns is available this algorithm drives the connection weight matrix w to an orthonormal basis of a dominant invariant subspace of a covariance matrix our approach consists in associating to this algorithm a second stochastic approximation algorithm that governs the evolution of ww sup t to the projection matrix onto this dominant invariant subspace then using a general result of gaussian approximation theory we derive the asymptotic distribution of the estimated projection matrix closed form expressions of the asymptotic covariance of the projection matrix estimated by the snl algorithm and by the smoothed snl algorithm that we introduce are given in the case of independent or correlated learning patterns and are further analyzed it is found that the structures of these asymptotic covariance matrices are similar to those describing batch estimation techniques the accuracy or our asymptotic analysis is checked by numerical simulations and it is found to be valid not only for a small learning rate but in a very large domain finally improvements brought by our smoothed snl algorithm are shown such as the learning speed misadjustment tradeoff and the deviation from orthonormality

a posteriori error estimation for discontinuous galerkin finite element approximation it is shown that the interelement discontinuities in a discontinuous galerkin finite element approximation are subordinate to the error measured in the broken h 1 seminorm one consequence is that the dg norm of the error is equivalent to the broken energy seminorm computable a posteriori error bounds are obtained for the error measured in both the dg norm and the broken energy seminorm and are shown to be efficient in the sense that they also provide lower bounds up to a constant and higher order data oscillation terms the estimators are completely free of unknown constants and provide guaranteed numerical bounds for the error

some basic formulations of the virtual element method vem for finite deformations abstract we present a general virtual element method vem framework for finite elasticity which emphasizes two issues element level volume change volume average of the determinant of the deformation gradient and stabilization to address the former issue we provide exact evaluation of the average volume change in both 2d and 3d on properly constructed local displacement spaces for the later issue we provide a new stabilization scheme that is based on the trace of the material tangent modulus tensor which captures highly heterogeneous and localized deformations two vem formulations are presented a two field mixed and an equivalent displacement based which is free of volumetric locking convergence and accuracy of the vem formulations are verified by means of numerical examples and engineering applications are demonstrated

regression on fixed rank positive semidefinite matrices a riemannian approach the paper addresses the problem of learning a regression model parameterized by a fixed rank positive semidefinite matrix the focus is on the nonlinear nature of the search space and on scalability to high dimensional problems the mathematical developments rely on the theory of gradient descent algorithms adapted to the riemannian geometry that underlies the set of fixed rank positive semidefinite matrices in contrast with previous contributions in the literature no restrictions are imposed on the range space of the learned matrix the resulting algorithms maintain a linear complexity in the problem size and enjoy important invariance properties we apply the proposed algorithms to the problem of learning a distance function parameterized by a positive semidefinite matrix good performance is observed on classical benchmarks

robust recursive least squares learning algorithm for principal component analysis a learning algorithm for the principal component analysis pca is developed based on the least square minimization the dual learning rate parameters are adjusted adaptively to make the proposed algorithm capable of fast convergence and high accuracy for extracting all principal components the proposed algorithm is robust to the error accumulation existing in the sequential pca algorithm we show that all information needed for pca can he completely represented by the unnormalized weight vector which is updated based only on the corresponding neuron input output product the updating of the normalized weight vector can be referred to as a leaky hebb s rule the convergence of the proposed algorithm is briefly analyzed we also establish the relation between oja s rule and the least squares learning rule finally the simulation results are given to illustrate the effectiveness of this algorithm for pca and tracking time varying directions of arrival

discontinuous hp finite element methods for advection diffusion reaction problems we consider the hp version of the discontinuous galerkin finite element method dgfem for second order partial differential equations with nonnegative characteristic form this class of equations includes second order elliptic and parabolic equations advection reaction equations as well as problems of mixed hyperbolic elliptic parabolic type our main concern is the error analysis of the method in the absence of streamline diffusion stabilization in the hyperbolic case an hp optimal error bound is derived here we consider only advection reaction problems which satisfy a certain standard positivity condition in the self adjoint elliptic case an error bound that is h optimal and p suboptimal by frac 1 2 a power of p is obtained these estimates are then combined to deduce an error bound in the general case for elementwise analytic solutions the method exhibits exponential rates of convergence under p refinement the theoretical results are illustrated by numerical experiments

fast subspace tracking and neural network learning by a novel information criterion we introduce a novel information criterion nic for searching for the optimum weights of a two layer linear neural network nn the nic exhibits a single global maximum attained if and only if the weights span the desired principal subspace of a covariance matrix the other stationary points of the nic are unstable saddle points we develop an adaptive algorithm based on the nic for estimating and tracking the principal subspace of a vector sequence the nic algorithm provides a fast on line learning of the optimum weights for the two layer linear nn we establish the connections between the nic algorithm and the conventional mean square error mse based algorithms such as oja s algorithm oja 1989 lmser past apex and gha the nic algorithm has several key advantages such as faster convergence which is illustrated through analysis and simulation