global solutions for a semilinear two dimensional klein gordon equation with exponential type nonlinearity we prove the existence and uniqueness of global solutions for a cauchy problem associated to a semilinear klein gordon equation in two space dimensions our result is based on an interpolation estimate with a sharp constant obtained by a standard variational method 2006 wiley periodicals inc

volterra method for the radial schrodinger equation a new volterra type integral equation method for the numerical solution of the radial schrodinger equation is investigated the method carried out in configuration space is based on the conversion of differential equations into a system of volterra type integral equations together with the application of a spectral type clenshaw curtis quadrature through numerical examples the volterra type integral equation method is shown to be more efficient and more accurate than the integral equation method based on the fredholm formulation j comput phys 134 1997 134 and better than finite difference methods accompanying c code for the volterra type method is available upon request

accurate computation of stokes flow driven by an open immersed interface we present numerical methods for computing two dimensional stokes flow driven by forces singularly supported along an open immersed interface two second order accurate methods are developed one for accurately evaluating boundary integral solutions at a point and another for computing stokes solution values on a rectangular mesh we first describe a method for computing singular or nearly singular integrals such as a double layer potential due to sources on a curve in the plane evaluated at a point on or near the curve to improve accuracy of the numerical quadrature we add corrections for the errors arising from discretization which are found by asymptotic analysis when used to solve the stokes equations with sources on an open immersed interface the method generates second order approximations for both the pressure and the velocity and preserves the jumps in the solutions and their derivatives across the boundary we then combine the method with a mesh based solver to yield a hybrid method for computing stokes solutions at n 2 grid points on a rectangular grid numerical results are presented which exhibit second order accuracy to demonstrate the applicability of the method we use the method to simulate fluid dynamics induced by the beating motion of a cilium the method preserves the sharp jumps in the stokes solution and their derivatives across the immersed boundary model results illustrate the distinct hydrodynamic effects generated by the effective stroke and by the recovery stroke of the ciliary beat cycle

inversion error condition number and approximate inverses of uncertain matrices abstract the classical condition number is a very rough measure of the effect of perturbations on the inverse of a square matrix first it assumes that the perturbation is infinitesimally small second it does not take into account the perturbation structure e g vandermonde similarly the classical notion of the inverse of a matrix neglects the possibility of large structured perturbations we define a new quantity the structured maximal inversion error that takes into account both structure and non necessarily small perturbation size when the perturbation is infinitesimal we obtain a structured condition number we introduce the notion of approximate inverse as a matrix that best approximates the inverse of a matrix with structured perturbations when the perturbation varies in a given range for a wide class of perturbation structures we show how to use convex semidefinite programming to compute bounds on the structured maximal inversion error and structured condition number and compute an approximate inverse the results are exact when the perturbation is unstructured we then obtain an analytic expression for the approximate inverse when the perturbation is unstructured and additive we recover the classical condition number the approximate inverse is the operator related to the total least squares orthogonal regression problem

on the numerical solution of two point boundary value problems abstract this paper presents a new numerical method for the solution of linear two point boundary value problems of ordinary differential equations after reducing the differential equation to a second kind integral equation the latter are discretized via a high order nystrom scheme a somewhat involved analytical apparatus is then constructed which allows for the solution of the discrete system using o n dot p2 operations where n is the number of nodes on the interval and p is the desired order of convergence thus the advantages of the integral equation formulation small condition number insensitivity to boundary layers insensitivity to end point singularities etc are retained while achieving a computational efficiency previously available only to finite difference or finite element methods keywords two point boundary value problems integral equations chebyshev polynomials approximation theory

orthogonality preservers in c algebras jb algebras and jb triples abstract we study orthogonality preserving operators between c algebras jb algebras and jb triples let t a e be an orthogonality preserving bounded linear operator from a c algebra to a jb triple satisfying that t 1 d is a von neumann regular element then t a e 2 r d every element in t a and d operator commute in the jb algebra e 2 r d and there exists a triple homomorphism s a e 2 r d such that t l d r d s where r d denotes the range tripotent of d in e an analogous result for a being a jb algebra is also obtained when t a b is an operator between two c algebras we show that denoting h t 1 then t orthogonality preserving if and only if there exists a triple homomorphism s a b satisfying h s z s z h h s z s z h and t z l h r h s z 1 2 h r h s z s z r h h this allows us to prove that a bounded linear operator between two c algebras is orthogonality preserving if and only if it preserves zero triple products

integral and integrable algorithms for a nonlinear shallow water wave equation an asymptotic higher order model of wave dynamics in shallow water is examined in a combined analytical and numerical study with the aim of establishing robust and efficient numerical solution methods based on the hamiltonian structure of the nonlinear equation an algorithm corresponding to a completely integrable particle lattice is implemented first each particle in the particle method travels along a characteristic curve the resulting system of nonlinear ordinary differential equations can have solutions that blow up in finite time we isolate the conditions for global existence and prove l1 norm convergence of the method in the limit of zero spatial step size and infinite particles the numerical results show that this method captures the essence of the solution without using an overly large number of particles a fast summation algorithm is introduced to evaluate the integrals of the particle method so that the computational cost is reduced from o n2 to o n where n is the number of particles the method possesses some analogies with point vortex methods for 2d euler equations in particular near singular solutions exist and singularities are prevented from occurring in finite time by mechanisms akin to those in the evolution of vortex patches the second method is based on integro differential formulations of the equation two different algorithms are proposed based on different ways of extracting the time derivative of the dependent variable by an appropriately defined inverse operator the integro differential formulations reduce the order of spatial derivatives thereby relaxing the stability constraint and allowing large time steps in an explicit numerical scheme in addition to the cauchy problem on the infinite line we include results on the study of the nonlinear equation posed in the quarter space time plane we discuss the minimum number of boundary conditions required for solution uniqueness and illustrate this with numerical examples

non symmetric perturbations of self adjoint operators abstract we investigate the effect of non symmetric relatively bounded perturbations on the spectrum of self adjoint operators in particular we establish stability theorems for one or infinitely many spectral gaps along with corresponding resolvent estimates these results extend and improve classical perturbation results by kato and by gohberg kre n further we study essential spectral gaps and perturbations exhibiting additional structure with respect to the unperturbed operator in the latter case we can even allow for perturbations with relative bound 1 the generality of our results is illustrated by several applications massive and massless dirac operators point coupled periodic systems and two channel hamiltonians with dissipation

fast adaptive 2d vortex methods we present a new approach to vortex methods for the 2d euler equations we obtain long time high order accuracy at almost optimal cost by using three tools fast adaptive quadrature rules a free lagrangian formulation and a useful new analysis of the consistency error our error analysis halves the order of differentiability required of the flow and suggests an efficient new balance of smoothing parameters which works well with fast summation schemes numerical experiments with our methods confirm our theoretical predictions and display excellent long time accuracy

scattering for a critical nonlinear wave equation in two space dimensions we show that the solutions to the cauchy problem for a wave equation with critical exponential nonlinearity in 2 space dimensions scatter for arbitrary smooth compactly supported initial data

quantum time of arrival goursat problem the construction of quantum time of arrival operator conjugate to the system hamiltonian leads to a particular linear homogeneous goursat problem in this work we demonstrate how to approximate the solution of the mentioned differential equation both semi analytically and numerically with the goal of calculating the largest eigenvalue of the associated confined time of arrival operator in the analytical approximation we used the partial sum expansion of the solution and showed that the resulting largest eigenvalue converges as the number of terms increases the result shows that for the parameters considered in this paper the approximation is sufficient up to the fourth order correction term in the numerical approximation we develop a non iterative formula to obtain the numerical solution of the goursat problem the performance of the non iterative method is compared with the known numerical techniques in literature numerical results show that the non iterative algorithm is more accurate and faster

graph sampling for matrix completion using recurrent gershgorin disc shift matrix completion algorithms fill missing entries in a large matrix given a subset of observed samples the problem of how to pre select a subset of entries for sampling to maximize the reconstructed matrix fidelity is largely unaddressed in this paper we propose two sampling algorithms to tackle this problem i a fast base sampling algorithm on general single graphs and ii an extended sampling algorithm from the base algorithm for active matrix completion assuming the double graph smoothness prior where rows columns of the matrix signal are smooth with respect to row column factor graphs we first show that a matrix signal can be recovered from partial observations by solving a system of linear equations mathbf q mathbf x mathbf y where the sought matrix is vectorized as mathbf x and interpreted as a signal residing on a single graph on this single graph to promote small reconstruction error and stability of the linear system we maximize the smallest eigenvalue lambda min of mathbf q by greedily selecting samples corresponding to the largest magnitude entries in the first eigenvectors of mathbf q based on a corollary of the gershgorin circle theorem extending to the active matrix completion case we alternately choose entries in the rows and columns corresponding to the largest magnitude entries in the first eigenvectors of the row column coefficient matrices mathbf q r and mathbf q c our algorithm benefits computationally from warm start as the first eigenvectors are computed recurrently for more samples extensive experiments on both synthetic and real world datasets show that our extended graph sampling algorithm outperforms existing sampling schemes for matrix completion when combined with a range of modern matrix completion algorithms

on deriving a locally corrected nystrom scheme from a quadrature sampled moment method a novel high order method of moment procedure with quadrature point based discretization is presented the scheme is equivalent to a moment method employing smooth basis and testing functions applying a fixed point numerical quadrature approximation for the outer integral mapping the current to the quadrature points then leads to a formulation that is equivalent to that derived via the locally corrected nystrom method the convergence properties of the current density and rcs for smooth and singular geometries in two and three dimensions are also studied

structured condition number for multiple right hand side linear systems with parameterized quasiseparable coefficient matrix abstract in this paper we consider the structured perturbation analysis for multiple right hand side linear systems with parameterized coefficient matrix especially we present the explicit expressions for structured condition numbers for multiple right hand sides linear systems with 1 1 quasiseparable coefficient matrix in the quasiseparable and the givens vector representations in addition the comparisons of these two condition numbers between themselves and with respect to unstructured condition number are investigated moreover the effective structured condition number for multiple right hand sides linear systems with 1 1 quasiseparable coefficient matrix is proposed the relationships between the effective structured condition number and structured condition numbers with respect to the quasiseparable and the givens vector representations are also studied numerical experiments show that there are situations in which the effective structured condition number can be much smaller than the unstructured ones

gauss jacobi quadratures for weakly strongly hyper and nearly singular integrals in boundary integral equation methods for domains with sharp edges and corners abstract we present gauss jacobi quadrature rules in terms of hypergeometric functions for the discretization of weakly singular strongly singular hypersingular and nearly singular integrals that arise in integral equation formulations of potential problems for domains with sharp edges and corners the rules are tailored to weight functions with algebraic endpoint singularities of a fairly general form thus allowing one to easily incorporate a wide class of domains into the analysis numerical examples illustrate the accuracy and stability of the proposed algorithms it is shown that the same level of high accuracy can be achieved for any choice of the external variable the usefulness of the method is exemplified by application to the solution of a singular integral equation that arises in time harmonic electromagnetic scattering by either closed or open perfectly conducting cylindrical objects with edges and corners such as polygon cylinders and bent strips some practical aspects concerning the role of nearby singularities in achieving a highly accurate solution of singular integral equations are also discussed

fredholm integral equation method for the integro differential schrodinger equation a new method based on the clenshaw curtis quadrature for the numerical solution of the integro differential schrodinger equation is investigated the method shows that it converges quickly and the truncation errors decrease faster than any power of the inverse number of the chebyshev support points discretization technique is presented in detail accompanying c code for the fredholm type method is available upon request

accurate numerical methods for micromagnetics simulations with general geometries in current fft based algorithms for micromagnetics simulations the boundary is typically replaced by a staircase approximation along the grid lines either eliminating the incomplete cells or replacing them by complete cells sometimes the magnetizations at the boundary cells are weighted by the volume of the sample in the corresponding cell we show that this leads to large errors in the computed exchange and stray fields one consequence of this is that the predicted switching mechanism depends sensitively on the orientation of the numerical grid we present a boundary corrected algorithm to efficiently and accurately handle the incomplete cells at the boundary we show that this boundary corrected algorithm greatly improves the accuracy in micromagnetics simulations we demonstrate by using a arrott s example of a hexagonal element that the switching mechanism is predicted independently of the grid orientation

integral equation method for coupled schrodinger equations a new integral equation method for the numerical solution of the radial schrodinger equation in one dimension developed by the authors 1997 j comput phys 134 134 is extended to systems of coupled schrodinger equations with both positive and negative channel energies the method carried out in configuration space is based on the conversion of differential equations into a system of integral equations together with the application of a spectral type clenshaw curtis quadrature an accompanying general multichannel fortran code is available upon request

norm estimates of complex symmetric operators applied to quantum systems this paper communicates recent results in the theory of complex symmetric operators and shows through two non trivial examples their potential usefulness in the study of schrodinger operators in particular we propose a formula for computing the norm of a compact complex symmetric operator this observation is applied to two concrete problems related to quantum mechanical systems first we give sharp estimates on the exponential decay of the resolvent and the single particle density matrix for schrodinger operators with spectral gaps second we provide new ways of evaluating the resolvent norm for schrodinger operators appearing in the complex scaling theory of resonances

fast inversion algorithms for a class of structured operator matrices we study linear complexity inversion algorithms for diagonal plus semiseparable operator matrices applications to integral and differential equations and block matrices are obtained comparison of different algorithms is performed complexity choice of parameters and results of numerical experiments are analyzed

fast solution methods for fredholm integral equations of the second kind the main purpose of this paper is to describe a fast solution method for one dimensional fredholm integral equations of the second kind with a smooth kernel and a non smooth right hand side function let the integral equation be defined on the interval 1 1 we discretize by a nystrom method with nodes cos j n j 0 n this yields a linear system of algebraic equations with an n 1 n 1 matrixa generallyn has to be chosen fairly large in order to obtain an accurate approximate solution of the integral equation we show by fourier analysis thata can be approximated well by mathop a limits sim a low rank modification of the identity matrix replacinga by mathop a limits sim in the linear system of algebraic equations yields a new linear system of equations whose elements and whose solution tilde x can be computed ino n logn arithmetic operations if the kernel has two more derivatives than the right hand side function then tilde x is shown to converge optimally to the solution of the integral equation asn increases we also consider iterative solution of the linear system of algebraic equations the iterative schemes use botha anda they yield the solution ino n 2 arithmetic operations under mild restrictions on the kernel and the right hand side function finally we discuss discretization by the chebyshev galerkin method the techniques developed for the nystrom method carry over to this discretization method and we develop solution schemes that are faster than those previously presented in the literature the schemes presented carry over in a straightforward manner to fredholm integral equations of the second kind defined on a hypercube

nystrom type method in three dimensional electromagnetic diffraction by a finite pec rotationally symmetric surface time harmonic electromagnetic wave diffraction by a perfectly electrically conducting pec finite rotationally symmetric surface located in free space is investigated the problem is split to independent azimuth orders and reduced to the sets of coupled hypersingular and singular integral equations ies for the surface current components these ies are discretized by the nystrom type method of discrete singularities using the interpolation type quadrature formulas from the solutions of corresponding matrix equations the near and the far field patterns are obtained the presented method has guaranteed convergence for arbitrary not axially symmetric primary field

spectral integration and two point boundary value problems a numerical method for two point boundary value problems with constant coefficients is developed which is based on integral equations and the spectral integration matrix for chebyshev nodes the method is stable achieves superalgebraic convergence and requires o n log n operations where n is the number of nodes in the discretization although stable spectral methods have been constructed in the past they have generally been based on reformulating the recurrence relations obtained through spectral differentiation in an attempt to avoid the ill conditioning introduced by that process

on m complex symmetric weighted shift operators on cn abstract in this paper we study m complex symmetric weighted shift operators on c n let t be the backward weighted shift on c n for some n 2 we consider when t and t a the matrix of entries the moduli of those of t are both m complex symmetric with the same standard conjugation c give as well some unitary operators useful in the study and generalize to upper triangular matrices also we show that if t is 2k complex symmetric with the standard conjugation c for some k n with k n then t is 2 k 1 complex symmetric with the conjugation c

a min max theorem for complex symmetric matrices abstract we optimize the form re x t tx to obtain the singular values of a complex symmetric matrix t we prove that for 0 k n 2 min codim v k max x v x 1 re x t tx 2 k 1 where t is an n n complex symmetric matrix having singular values 1 n we also show that the singular values missed in this theorem i e 2 4 are obtained by a similar optimization over real subspaces

random walk graph laplacian based smoothness prior for soft decoding of jpeg images given the prevalence of joint photographic experts group jpeg compressed images optimizing image reconstruction from the compressed format remains an important problem instead of simply reconstructing a pixel block from the centers of indexed discrete cosine transform dct coefficient quantization bins hard decoding soft decoding reconstructs a block by selecting appropriate coefficient values within the indexed bins with the help of signal priors the challenge thus lies in how to define suitable priors and apply them effectively in this paper we combine three image priors laplacian prior for dct coefficients sparsity prior and graph signal smoothness prior for image patches to construct an efficient jpeg soft decoding algorithm specifically we first use the laplacian prior to compute a minimum mean square error initial solution for each code block next we show that while the sparsity prior can reduce block artifacts limiting the size of the overcomplete dictionary to lower computation would lead to poor recovery of high dct frequencies to alleviate this problem we design a new graph signal smoothness prior desired signal has mainly low graph frequencies based on the left eigenvectors of the random walk graph laplacian matrix lerag compared with the previous graph signal smoothness priors lerag has desirable image filtering properties with low computation overhead we demonstrate how lerag can facilitate recovery of high dct frequencies of a piecewise smooth signal via an interpretation of low graph frequency components as relaxed solutions to normalized cut in spectral clustering finally we construct a soft decoding algorithm using the three signal priors with appropriate prior weights experimental results show that our proposal outperforms the state of the art soft decoding algorithms in both objective and subjective evaluations noticeably

scattering for the two dimensional energy critical wave equation we investigate existence and asymptotic completeness of the wave operators for the nonlinear klein gordon equation with a defocusing exponential nonlinearity in two space dimensions a certain threshold is defined based on the value of the conserved hamiltonian below which the exponential potential energy is dominated by the kinetic energy via a trudinger moser type inequality we prove that if the energy is below or equal to the critical value then the solution approaches a free klein gordon solution at the time infinity an interesting feature in the critical case is that the strichartz estimate together with sobolev type inequalities cannot control the nonlinear term uniformly on each time interval it crucially depends on how much the energy is concentrated thus we have to trace concentration of the energy along time in order to set up favorable nonlinear estimates and only after that we can apply bourgain s induction argument or any other similar one

on the boundary conditions characterizing j selfadjoint extensions of j symmetric operators the study of boundary value problems involving linear differentiai equations with real valued coefficients is by now a well established area of analysis on the other hand much less is known about the solvability of such problems when the coefficients or boundary conditions are known to be complex examples of this latter type arise naturally for example in nuclear physics e g the so called optical mode1 for low energy scattering 1 p ijo electromagnetic field theory dielectric waveguides with heat loss c f 19 i or the propagation of radio waves through inhomogeneous media s and elsewhere in cases like these the relevant differential expressions are no longer formally symmetric and hence the powerful methods associated with the spectral theory of selfadjoint operators are not available to facilitate the study of such problems glazman introduced in 4 the concept of a j symmetric operator in a complex hilbert space r let j be a given conjugation operator on 2 i e j is a conjugate linear involution with jx 3 u x for all x and y in 2 a closed densely defined linear operator t in 3 is said to be j symmetric if

fast accurate and small scale direct trajectory optimization using a gegenbauer transcription method this paper reports a novel direct gegenbauer ultraspherical transcription method gtm for solving continuous time optimal control oc problems ctocps with linear nonlinear dynamics and path constraints in elgindy et al 2012 1 we presented a gtm for solving nonlinear ctocps directly for the state and the control variables and the method was tailored to find the best path for an unmanned aerial vehicle mobilizing in a stationary risk environment this article extends the gtm to deal further with problems including higher order time derivatives of the states by solving the ctocp directly for the control u t and the highest order time derivative x n t n z the state vector and its derivatives up to the n 1 th order derivative can then be stably recovered by successive integration moreover we present our solution method for solving linear quadratic regulator lqr problems as we aim to cover a wider collection of ctocps with the concrete aim of comparing the efficiency of the current work with other classical discretization methods in the literature the proposed numerical scheme fully parameterizes the state and the control variables using gegenbauer expansion series for problems with various order time derivatives of the state variables arising in the cost function dynamical system or path terminal constraints the gtm seeks to fully parameterize the control variables and the highest order time derivatives of the state variables the time horizon is mapped onto the closed interval 0 1 the dynamical system characterized by differential equations is transformed into its integral formulation through direct integration the resulting problem on the finite interval is then transcribed into a nonlinear programming nlp problem through collocation at the gegenbauer gauss gg points the integral operations are approximated by optimal gegenbauer quadratures in a certain optimality sense the reduced nlp problem is solved in the gegenbauer spectral space and the state and the control variables are approximated on the entire finite horizon the proposed method achieves discrete solutions exhibiting exponential convergence using relatively small scale number of collocation points the advantages of the proposed direct gtm over other traditional discretization methods are shown through four well studied oc test examples the present work is a major breakthrough in the area of computational oc theory as it delivers significantly more accurate solutions using considerably smaller numbers of collocation points states and controls expansion terms moreover the gtm produces very small scale nlp problems which can be solved very quickly using the modern nlp software

numerical methods based on rational variable substitution for wiener hopf equations of the second kind this paper considers numerical methods for wiener hopf equations of the second kind y t 0 k t s y s ds g t 0 t by applying rational variable substitution to integrals on the semi infinite interval 0 and using the well known clenshaw curtis quadrature to the resulted integral we get a clenshaw curtis rational ccr quadrature rule we then apply the ccr quadrature to wiener hopf equations the reduction of singularities in the transformed equation is considered numerical examples are given to illustrate the efficiency of the numerical methods proposed in this paper

p mathcal t symmetric non self adjoint operators diagonalizable and non diagonalizable with a real discrete spectrum consider in the operator family h0 a 1a1 a dad d 2 is the quantum harmonic oscillator with rational frequencies w is a symmetric bounded potential and g is a real coupling constant we show that if g being an explicitly determined constant the spectrum of h g is real and discrete moreover we show that the operator has a real discrete spectrum but is not diagonalizable

evaluation of single layer potentials over curved surfaces the evaluation of nearly singular single layer potentials encountered in boundary element methods is treated by a new approach the potential is expressed as a sum of a one dimensional integral and a correction term that vanishes for planar surfaces the small variance of the second term s integrand allows the use of a quasi monte carlo quadrature numerical results show that a significant reduction in computational time is obtained over algorithms employing domain subdivisions

high order corrected trapezoidal quadrature rules for singular functions a class of quadrature formulae is presented applicable to both nonsingular and singular functions generalizing the classical endpoint corrected trapezoidal quadrature rules while the latter rules are usually derived by means of the euler maclaurin formula their generalizations are obtained as solutions of certain systems of linear algebraic equations a procedure is developed for the construction of very high order quadrature rules applicable to functions with a priori specified singularities and relaxing the requirements on the distribution of nodes the scheme applies not only to nonsingular functions but also to a wide class of functions with monotonic singularities numerical experiments are presented demonstrating the practical usefulness of the new class of quadratures tables of quadrature weights are included for singularities of the form s x x lambda for a variety of values of lambda and s x log x

grothendieck s inequalities for real and complex jbw triples we prove that if m 4 1 2 3 and e 0 if v and w are complex jbw triples with preduals v andw respectively and if u is a separately weak continuous bilinear form on v w then there exist norm one functionals 1 2 v and 1 2 w satisfying u x y m u x 2 2 e 2 x 2 1 1 2 y 2 2 e 2 y 2 1 1 2 for all x y v w here for a norm one functional on a complex jb triple v stands for the prehilertian seminorm on v associated to in bf1 we arrive in this grothendieck s inequality through results of c h chu b iochum and g loupias cil and a corrected version of the little grothendieck s inequality for complex jb triples due to t barton and y friedman bf1 we also obtain extensions of these results to the setting of real jb triples 2000 mathematics subject classification 17c65 46k70 46l05 46l10 and 46l70

strichartz type estimates and the well posedness of an energy critical 2d wave equation in a bounded domain abstract we study the well posedness of the cauchy problem with dirichlet or neumann boundary conditions associated to an h 1 critical semilinear wave equation on a smooth bounded domain r 2 first we prove an appropriate strichartz type estimate using the l q spectral projector estimates of the laplace operator our proof follows burq lebeau and planchon 2008 4 then we show the global well posedness when the energy is below or at the threshold given by the sharp moser trudinger inequality finally in the supercritical case we prove an instability result using the finite speed of propagation and a quantitative study of the associated ode with oscillatory data

complex symmetric operators and applications ii a bounded linear operator t on a complex hilbert space h is called complex symmetric if t ct c where c is a conjugation an isometric antilinear involution of h we prove that t cj t where j is an auxiliary conjugation commuting with t t t we consider numerous examples including the poincare neumann singular integral bounded operator and the jordan model operator compressed shift the decomposition t cj t also extends to the class of unbounded c selfadjoint operators originally introduced by glazman in this context it provides a method for estimating the norms of the resolvents of certain unbounded operators

conditions for bound states in a periodic linear chain and the spectra of a class of toeplitz operators in terms of polylogarithm functions conditions for bound states for a periodic linear chain are given within the framework of an exactly solvable non relativistic quantum mechanical model in three dimensional space these conditions express the strength parameter in terms of the distance between two consecutive centres of the chain and of the range interaction parameter this expression can be formulated in terms of polylogarithm functions and in some particular cases in terms of the riemann zeta function an interesting mathematical result is that these expressions also correspond to the spectra of toeplitz complex symmetric operators the non trivial zeros of the riemann zeta function are interpreted as multiple points at the origin of the spectra of these toeplitz operators

j self adjointness of a class of dirac type operators in this note we prove that the maximally defined operator associated with the dirac type differential expression m q iddxim q q ddxim where q represents a symmetric m m matrix i e q x q x a e with entries in l1loc r is j self adjoint where j is the antilinear conjugation defined by j 1c 1 0imim0 and c a1 am b1 bm a1 am b1 bm the differential expression m q is of significance as it appears in the lax formulation of the non abelian matrix valued focusing nonlinear schrodinger hierarchy of evolution equations

fast graph sampling set selection using gershgorin disc alignment graph sampling set selection where a subset of nodes are chosen to collect samples to reconstruct a smooth graph signal is a fundamental problem in graph signal processing gsp previous works employ an unbiased least squares ls signal reconstruction scheme and select samples via expensive extreme eigenvector computation instead we assume a biased graph laplacian regularization glr based scheme that solves a system of linear equations for reconstruction we then choose samples to minimize the condition number of the coefficient matrix specifically maximize the smallest eigenvalue lambda min circumventing explicit eigenvalue computation we maximize instead the lower bound of lambda min designated by the smallest left end of all gershgorin discs of the matrix to achieve this efficiently we first convert the optimization to a dual problem where we minimize the number of samples needed to align all gershgorin disc left ends at a chosen lower bound target t algebraically the dual problem amounts to optimizing two disc operations i shifting of disc centers due to sampling and ii scaling of disc radii due to a similarity transformation of the matrix we further reinterpret the dual as an intuitive disc coverage problem bearing strong resemblance to the famous np hard set cover sc problem the reinterpretation enables us to derive a fast approximation scheme from a known sc error bounded approximation algorithm we find an appropriate target t efficiently via binary search extensive simulation experiments show that our disc based sampling algorithm runs substantially faster than existing sampling schemes and outperforms other eigen decomposition free sampling schemes in reconstruction error

point cloud denoising via feature graph laplacian regularization point cloud is a collection of 3d coordinates that are discrete geometric samples of an object s 2d surfaces imperfection in the acquisition process means that point clouds are often corrupted with noise building on recent advances in graph signal processing we design local algorithms for 3d point cloud denoising specifically we design a signal dependent feature graph laplacian regularizer sdfglr that assumes surface normals computed from point coordinates are piecewise smooth with respect to a signal dependent graph laplacian matrix using sdfglr as a signal prior we formulate an optimization problem with a general ell p norm fidelity term that can explicitly remove only two types of additive noise small but non sparse noise like gaussian using ell 2 fidelity term and large but sparser noise like laplacian using ell 1 fidelity term to establish a linear relationship between normals and 3d point coordinates we first perform bipartite graph approximation to divide the point cloud into two disjoint node sets red and blue we then optimize the red and blue nodes coordinates alternately for ell 2 norm fidelity term we iteratively solve an unconstrained quadratic programming qp problem efficiently computed using conjugate gradient with a bounded condition number to ensure numerical stability for ell 1 norm fidelity term we iteratively minimize an ell 1 ell 2 cost function using accelerated proximal gradient apg where a good step size is chosen via lipschitz continuity analysis finally we propose simple mean and median filters for flat patches of a given point cloud to estimate the noise variance given the noise type which in turn is used to compute a weight parameter trading off the fidelity term and signal prior in the problem formulation extensive experiments show state of the art denoising performance among local methods using our proposed algorithms

some inertia theorems in euclidean jordan algebras abstract this paper deals with some inertia theorems in euclidean jordan algebras first based on the continuity of eigenvalues we give an alternate proof of kaneyuki s generalization of sylvester s law of inertia in simple euclidean jordan algebras as a consequence we show that the cone spectrum of any quadratic representation with respect to a symmetric cone is finite second we present ostrowski schneider type inertia results in euclidean jordan algebras in particular we relate the inertias of objects a and x in a euclidean jordan algebra when l a x 0 or s a x 0 where l a and s a denote lyapunov and stein transformations respectively

perturbation of spectral subspaces and solution of linear operator equations abstract let a and b be normal operators on a hilbert space let k a and k b be subsets of the complex plane at distance at least from each other let e be the spectral projector for a belonging to k a and let f be the spectral projector for b belonging to k b our main results are estimates of the form ef c e a b f in some special situations the constant c is as low as 1 as an application we prove for an absolute constant d that if the space is finite dimensional and if a and b are normal with a b d then the spectrum of b can be obtained from that of a multiplicities counted by moving each eigenvalue by at most our main results have equivalent formulations as statements about the operator equation aq qb s let a and b be normal operators on perhaps different hilbert spaces assume a k a and b k b where k a k b and are as before then we give estimates of the forms q c aq qb

quadrature rule for indefinite integral of algebraic logarithmic singular integrands an automatic quadrature method is presented for approximating the indefinite integral of functions having algebraic logarithmic singularities q x y c f email protected x yf t t c alog t c dt 1 1 within a finite range 1 1 for some smooth function f t that is approximated by a finite sum of chebyshev polynomials we expand the given indefinite integral in terms of chebyshev polynomials by using auxiliary algebraic logarithmic functions present scheme approximates the indefinite integral q x y c f uniformly namely bounds the approximation error independently of the value c as well x and y this fact enables us to evaluate the integral transform q x y c f with varied values of x y and c efficiently some numerical examples illustrate the performance of the present quadrature scheme

bipartite approximation for graph wavelet signal decomposition to compactly represent a graph signal in the frequency domain critically sampled biorthogonal wavelet filterbanks have been proposed to decompose signals on bipartite graphs however in practice graph signals often reside on general graph structures that are not bipartite thus an original nonbipartite graph must be decomposed into a sequence of bipartite graph approximations so that the filterbanks can be applied successively for signal decomposition in this paper unlike previous proposals that are heuristic in nature we design new bipartite approximation strategies for model based and empirically derived probability distributions in the first case a signal prior assumes a gaussian markov random field gmrf model parametrized by the original graph in the second case beyond the gmrf signal prior empirical signal observations are available to compute a posterior probability distribution in both cases we optimize for energy compaction in the bipartite subgraphs with two criteria the kullback leibler divergence metric which encourages preserving the spectral characteristics of the original graph and multiplicity of eigenvalue at frequency 1 for graph laplacian which is the frequency with minimal energy discrimination the relative importance between the two criteria is determined by a proposed measure that evaluates the degree of mismatch between the signal prior and posterior given these criteria we first design a global numerical optimization and then propose a local heuristic approach for fast implementation with simplifications leading to local metric computation experimental results show that our proposed bipartite subgraph decomposition outperforms competing proposals in terms of energy compaction

the integral equation method for a steady kinematic dynamo problem with only a few exceptions the numerical simulation of cosmic and laboratory hydromagnetic dynamos has been carried out in the framework of the differential equation method however the integral equation method is known to provide robust and accurate tools for the numerical solution of many problems in other fields of physics the paper is intended to facilitate the use of integral equation solvers in dynamo theory in concrete the integral equation method is employed to solve the eigenvalue problem for a hydromagnetic dynamo model with an isotropic helical turbulence parameter for the case of spherical geometry three examples of the function r with steady and oscillatory solutions are considered a convergence rate proportional to the inverse squared of the number of grid points is achieved based on this method a convergence accelerating strategy is developed and the convergence rate is improved remarkably typically quite accurate results can be obtained with a few tens of grid points in order to demonstrate its suitability for the treatment of dynamos in other than spherical domains the method is also applied to x2 dynamos in rectangular boxes the magnetic fields and the electric potentials for the first eigenvalues are visualized

a fast solver for the stokes equations with distributed forces in complex geometries we present a new method for the solution of the stokes equations the main features of our method are 1 it can be applied to arbitrary geometries in a black box fashion 2 it is second order accurate and 3 it has optimal algorithmic complexity our approach to which we refer as the embedded boundary integral method ebi is based on anita mayo s work for the poisson s equation the fast solution of poisson s and the biharmonic equations on irregular regions siam journal on numerical analysis 21 1984 285 299 we embed the domain in a rectangular domain for which fast solvers are available and we impose the boundary conditions as interface jump conditions on the velocities and tractions we use an indirect boundary integral formulation for the homogeneous stokes equations to compute the jumps the resulting equations are discretized by nystrom s method the rectangular domain problem is discretized by finite elements for a velocity pressure formulation with equal order interpolation bilinear elements q1 q1 stabilization is used to circumvent the inf sup condition for the pressure space for the integral equations fast matrix vector multiplications are achieved via an n logn algorithm based on a block representation of the discrete integral operator combined with kernel independent singular value decomposition to sparsify low rank blocks the regular grid solver is a krylov method conjugate residuals combined with an optimal two level schwartz preconditioner for the integral equation we use gmres we have tested our algorithm on several numerical examples and we have observed optimal convergence rates

volterra type integral equation method for the radial schrodinger equation single channel case a new volterra type integral equation method for the numerical solution of the single channel radial schrodinger equation is investigated the method carried out in configuration space is based on the conversion of differential equations into a system of volterra type integral equations together with the application of a spectral type clenshaw curtis quadrature through numerical examples the volterra type integral equation method is shown to be more efficient and more accurate than the integral equation method based on the fredholm formulation 1 and better than finite difference methods accompanying c code for the volterra type method is available upon request

integral equations a tool to solve the schrodinger equation from the integral equation equivalent to the one dimensional schrodinger equation with the boundary conditions appropriate for a bound state problem an exact three point integration rule is derived the approximate evaluation of this rule by means of the euler mclaurin sum rule gives rise to various o h2 and o h4 integration methods associated with the different ways of splitting the starting equation into a free part and an interaction term the method is particularly useful to deal with the singularities of the potential

multilevel fast multipole acceleration in the nystrom discretization of surface electromagnetic integral equations for composite objects the multilevel fast multipole algorithm mlfma based on the nystrom discretization of surface integral equations sies is developed for solving electromagnetic em scattering by large composite objects traditionally the mlfma is based on the method of moments mom discretization for the sies and it usually works well when the robust rao wilton glisson rwg basis function is enough to represent unknown currents however the rwg basis function may not represent both the electric and magnetic current in solving the electric field integral equation efie and magnetic field integral equation mfie for penetrable objects and how one represents another current could be a problem in the mom in this work we use the nystrom method as a tool to discretize the sies and incorporate the mlfma to accelerate the solutions for electrically large problems the advantages of the nystrom discretization include the simple mechanism of implementation lower requirements on mesh quality and no use of basis and testing functions these benefits are particularly desired in the mlfma because the solved problems are very large and complex in general numerical examples are presented to demonstrate the effectiveness of the proposed scheme

image denoising by sparse 3 d transform domain collaborative filtering we propose a novel image denoising strategy based on an enhanced sparse representation in transform domain the enhancement of the sparsity is achieved by grouping similar 2d image fragments e g blocks into 3d data arrays which we call groups collaborative altering is a special procedure developed to deal with these 3d groups we realize it using the three successive steps 3d transformation of a group shrinkage of the transform spectrum and inverse 3d transformation the result is a 3d estimate that consists of the jointly filtered grouped image blocks by attenuating the noise the collaborative filtering reveals even the finest details shared by grouped blocks and at the same time it preserves the essential unique features of each individual block the filtered blocks are then returned to their original positions because these blocks are overlapping for each pixel we obtain many different estimates which need to be combined aggregation is a particular averaging procedure which is exploited to take advantage of this redundancy a significant improvement is obtained by a specially developed collaborative wiener filtering an algorithm based on this novel denoising strategy and its efficient implementation are presented in full detail an extension to color image denoising is also developed the experimental results demonstrate that this computationally scalable algorithm achieves state of the art denoising performance in terms of both peak signal to noise ratio and subjective visual quality

locally corrected multidimensional quadrature rules for singular functions accurate numerical integration of singular functions usually requires either adaptivity or product integration both interfere with fast summation techniques and thus hamper large scale computations this paper presents a method for computing highly accurate quadrature formulas for singular functions which combine well with fast summation methods given the singularity and the n nodes we first construct weights which integrate smooth functions with order k accuracy then we locally correct a small number of weights near the singularity to achieve order k accuracy on singular functions as well the method is highly efficient and runs in o nk 2d n log 2 n time and o k 2d n space we derive precise error bounds and time estimates and confirm them with numerical results which demonstrate the accuracy and efficiency of the method in large scale computations as part of our implementation we also construct a new adaptive multidimensional product gauss quadrature routine with an effective error

embedded methods for the numerical solution of the schrodinger equation abstract new methods for the approximate numerical integration of the one dimensional schrodinger equation are developed in this paper complete phase lag analysis of the new methods is included these new methods are called embedded methods because of a simple natural error control mechanism numerical results obtained for a one dimensional schrodinger equation show the validity of the developed theory

quadrature rules for weakly singular strongly singular and hypersingular integrals in boundary integral equation methods we present n point gauss gegenbauer quadrature rules for weakly singular strongly singular and hypersingular integrals that arise in integral equation formulations of potential problems in domains with edges and corners the rules are tailored to weight functions with algebraic endpoint singularities related to the geometrical singularities of the domain each rule has two different expressions involving legendre functions and hypergeometric functions respectively numerical examples amply demonstrate the accuracy and stability of the proposed algorithms application to the solution of a singular integral equation is exemplified

nystrom clenshaw curtis quadrature for integral equations with discontinuous kernels a new highly accurate numerical approximation scheme based on a gauss type clenshaw curtis quadrature for fredholm integral equations of the second kind x t ab k t s x s ds y t whose kernel k t s is either discontinuous or not smooth along the main diagonal is presented this scheme is of spectral accuracy when k t s is infinitely differentiable away from the diagonal t s relation to the singular value decomposition is indicated application to integro differential schrodinger equations with nonlocal potentials is given

a fast higher order solver for scattering by penetrable bodies in three dimensions in this paper we introduce a new fast higher order solver for scattering by inhomogeneous media in three dimensions as in previously existing methods the low complexity of our integral equation method o nlogn operations for an n point discretization is obtained through extensive use of the fast fourier transform fft for the evaluation of convolutions however the present approach obtains significantly higher order accuracy than these previous approaches yielding at worst third order far field accuracy or substantially better for smooth scatterers even for discontinuous and complex refractive index distributions possibly containing severe geometric singularities such as corners and cusps the increased order of convergence of our method results from i a partition of unity decomposition of the green s function into a smooth part with unbounded support and a singular part with compact support and ii replacement of the possibly discontinuous scatterer by an appropriate fourier smoothed scatterer the resulting convolutions can then be computed with higher order accuracy by means of o nlogn ffts we present a parallel implementation of our approach and demonstrate the method s efficiency and accuracy through a variety of computational examples for a very large scatterer considered earlier in the literature with a volume of 3648 l 3 where l is the wavelength using the same number of points per wavelength and in computing times comparable to those required by the previous approach the present algorithm produces far field values whose errors are two orders of magnitude smaller than those reported previously

unitary equivalence to a complex symmetric matrix an algorithm abstract we present a necessary and sufficient condition for a 3 3 matrix to be unitarily equivalent to a symmetric matrix with complex entries and an algorithm whereby an arbitrary 3 3 matrix can be tested this test generalizes to a necessary and sufficient condition that applies to almost every n n matrix the test is constructive in that it explicitly exhibits the unitary equivalence to a complex symmetric matrix

the minimax principle and related topics in the jordan setting abstract we prove a minimax principle for weakly compact jb triples characterizing geometrically the singular values of an element among the consequences of this principle we present a weyl inequality on the perturbation of the singular values and a cauchy poincare interlacing theorem we also obtain a version of the ky fan maximum principle in the setting of weakly compact jb triples we study perturbations of the spectral resolutions showing that small perturbations of an element produces small perturbations of the corresponding spectral resolutions as a consequence we obtain that weakly compact jb triples satisfy the property that perturbations of a convex combination of elements in the closed unit ball coincide with a convex combination of perturbations of the elements also in the closed unit ball all these results hold true when particularized to weakly compact jb algebras

calculation of the two body scattering k matrix in configuration space by an adaptive spectral method a spectral integral method iem for solving the two body one variable lippmann schwinger equation for the wavefunction in configuration space is generalized to the case of the two variable scattering k matrix the main difficulty is that in this case the driving term of the integral equation is discontinuous it is found that the desirable features of the iem such as the economy of mesh points for a given required accuracy are carried over also to this case even though the result is also discontinuous the main innovation is a judicious choice of the partitions in coordinate space plus a new recursion relation forward and backward to the point of discontinuity for a simple exponential potential an accuracy of 7 significant figures is achieved for the k matrix with the number n of chebyshev support points in each partition equal to 17 for a potential with a large repulsive core such as the potential between two he atoms an accuracy of 7 significant figures requires that n is increased to 65 support points per partition

an analytic family of cal p cal t symmetric hamiltonians with real eigenvalues we study nonlinear perturbative expansions for symmetric local schrodinger operators the schrodinger operator is a sum of the harmonic oscillator hamiltonian and a local symmetric potential depending in general nonlinearly on the perturbation parameter a specific class of models having real spectrum for any value of the parameter is proposed

iterative solution of integral equations on a basis of positive energy sturmian functions an improvement of weinberg s quasiparticle method for solving general one dimensional integral equations is presented the method uses simple auxiliary sturmian functions for positive or negative energies and corrects iteratively for the truncation errors of the sturmian expansion of the solution numerical examples are given for the solution of the lippmann schwinger integral equation for the scattering of a particle from a potential with a repulsive core an accuracy of 1 10 6 is achieved after 14 iterations and 1 10 10 after 20 iterations the calculations are carried out in configuration space with an accuracy of 1 10 11 by using a spectral expansion method in terms of chebyshev polynomials the method can be extended to solving a schrodinger equation with coulomb and or nonlocal potentials

integral equation method for the continuous spectrum radial schrodinger equation a new approach to the numerical solution of boundary value problems for differential equations which originated in recent papers by greengard and rokhlin is improved and adapted to the numerical solution of the radial schrodinger equation the approach is based on the conversion of the differential equation into an integral equation together with the application of a spectral type clenshaw curtis quadrature method through numerical examples the integral equation method is shown to be superior to finite difference methods