pick conditions on a uniform algebra and von neumann inequalities let a be a uniform algebra and let m1 mn be points in the maximal ideal space of a we describe the set d in cn of all points w1 wn such that given e 0 there exists f a with f m w 1 n and f 1 e this problem was solved for the disk algebra by g pick in 1916 we generalize pick s condition and we also study the related problem of generalizations of von neumann s inequality for contraction operators on hilbert space in our work a certain gometric notion defined in the introduction definition 1 of a hyperconvex set in cn plays a key role

nonpolynomial spline collocation for volterra equations with weakly singular kernels volterra integral equations of the second kind with weakly singular kernels possess in general solutions which are not smooth near the left endpoint of the interval of integration since ordinary polynomial spline collocation cannot lead to high order convergence we introduce special nonpolynomial spline spaces which are modelled after the structure of these solutions near the point of nonsmooth behavior collocation in these spaces will once more lead to high order methods analogous results are derived for volterra integro differential equations with weakly singular kernels

two dimensional commutative banach algebras and von neumann inequality we show the following let b be a two dimensional commutative banach algebra with identity if b satisfies t b t 1 f t 1 whenever f is a polynomial satisfying f z 1 z 1 then b is isometric to a subalgebra of the algebra b h of all bounded linear operators on some hilbert space h and b satisfies tk b tk 1 k 1 n f t1 tn 1 whenever f is a polynomial in n variables satisfying f z1 zn 1 zk 1 k 1 n for all n

variational regularized 2 d nonnegative matrix factorization a novel approach for adaptive regularization of 2 d nonnegative matrix factorization is presented the proposed matrix factorization is developed under the framework of maximum a posteriori probability and is adaptively fine tuned using the variational approach the method enables 1 a generalized criterion for variable sparseness to be imposed onto the solution and 2 prior information to be explicitly incorporated into the basis features the method is computationally efficient and has been demonstrated on two applications that is extracting features from image and separating single channel source mixture in addition it is shown that the basis features of an information bearing matrix can be extracted more efficiently using the proposed regularized priors experimental tests have been rigorously conducted to verify the efficacy of the proposed method

the numerical treatment of volterra integro differential equations with unbounded delay abstract initial value problems for ordinary and partial volterra integro differential equations with infinite and finite delay arise in the mathematical modelling of various physical and biological phenomena in this paper we give a survey of some recent developments in the numerical treatment of such functional equations with the focus being on equations with infinite delay in addition we will discuss a number of numerical aspects that are not yet understood well

the piecewise polynomial collocation method for nonlinear weakly singular volterra equations second kind volterra integral equations with weakly singular kernels typically have solutions which are nonsmooth near the initial point of the interval of integration using an adaptation of the analysis originally developed for nonlinear weakly singular fredholm integral equations we present a complete discussion of the optimal global and local order of convergence of piecewise polynomial collocation methods on graded grids for nonlinear volterra integral equations with algebraic or logarithmic singularities in their kernels

norm preserving extensions of bounded holomorphic functions let v be an analytic subvariety of a domain omega in mathbb c n when does v have the property that every bounded holomorphic function f on v has an extension to a bounded holomorphic function on omega with the same norm an obvious sufficient condition is if v is a holomorphic retract of omega we shall discuss for what domains omega this is also necessary this is joint work with ukasz kosinski

piecewise homotopy perturbation method for solving linear and nonlinear weakly singular vie of second kind the purpose of this paper is to obtain the approximation solution of linear and strong nonlinear weakly singular volterra integral equation of the second kind especially for such a situation that the equation is of nonsmooth solution and the situation that the problem is a strong nonlinear problem for this purpose we firstly make a transform to the equation such that the solution of the new equation is as smooth as we like through modifying homotopy perturbation method an algorithm is successfully established to solve the linear and nonlinear weakly singular volterra integral equation of the second kind and the convergence of the algorithm is proved strictly comparisons are made between our method and other methods and the results reveal that the modified homotopy perturbation is effective

a block by block method for volterra integro differential equations with weakly singular kernel the theory of a block by block method for solving volterra integro differential equations with continuous kernels see makroglou 14 5d is adapted to volterra integrodifferential equations with weakly singular kernels and a rate of convergence is given

stability analysis of product t methods for abel integral equations of the second kind stability analysis of product methods for abel integral equations of the second kind based on the test equation y t 1 lambda int limits 0 1 frac y s t s alpha ds t geqq 0 0 0 and we investigate whether this property is inherited by numerical approximations toy

convolutive blind source separation of speech signals based on amplitude modulation decorrelation the problem of blind separation of a convolutive mixture of speech signals is considered signal separation is performed in the frequency domain the amplitude modulation decorrelation algorithm for convolutive blind source separation of speech is described the first ideas of which were presented by anemuller at the joint asa eaa daga meeting berlin 1999 the algorithm is based on the fact that the frequency specific envelopes of speech signals exhibit correlation across different frequencies this feature can be used to solve the permutation problem of frequency domain based blind source separation algorithms furthermore it leads to separation of good quality since it results in a high number of constraints which must be fulfilled by the unmixed signals results for the separation of speech signals are presented for different mixing scenarios including real room reverberant mixing

improving the accuracy of collocation solutions of volterra integral equations of the first kind by local interpolation we study the numerical solution of volterra integral equations of the first kind by collocation methods using piecewise polynomials of degreep we show that when superconvergence occurs the error in the collocation solution at the node points alternates in sign independent from the kernel or the exact solution evaluation at certain points of interpolating polynomials of degreep 2 annihilates the leading term in the asymptotic expansion of the error thus yielding convergence of orderp 3 the derivation of the asymptotic expansion of the error for the general equation is reduced to the special case of numerical differentiation by the use of an asymptotic continuity result for the projectors associated with the collocation method under compact perturbations we present some numerical results on the improved order of convergence forp 2 and 3

existence theorems for an integral equation of the chandrasekhar h equation with perturbation abstract we prove existence of positive solutions in thaces c 0 1 and c c 0 1 of an integral equation of the chandrasekhar h equation with perturbation

variable transformations in the numerical solution of second kind volterra integral equations with continuous and weakly singular kernels extensions to fredholm integral equations abstract a number of techniques that use variable transformations in numerical integration have been developed recently cf sidi numerical integration iv h brass g hammerlin eds birkhauser basel 1993 pp 359 373 laurie j comput appl math 66 1996 337 344 the use of these transformations resulted in increasing the order of convergence of the trapezoidal and the midpoint quadrature rule in this paper the application of variable transformation techniques of sidi and laurie type to the numerical solution of second kind volterra integral equations with continuous and weakly singular kernels is considered since the transformations are such that the end points of integration need not be used as mesh points the methods introduced can be used for vie with both continuous and weakly singular kernel in a uniform way the methods have also the advantages of simplicity of application and of achieving high order of convergence the application of the idea to fredholm integral equations with continuous and weakly singular equations is also considered numerical results are included and they verify the expected increased order of convergence they were obtained by using the trapezoidal formula for the evaluation of the transformed integrals

high order methods for the numerical solution of volterra integro differential equations abstract if a first order volterra integro differential equation is solved by collocation in the space of continuous polynomial splines of degree m 1 with collocation occuring at the gauss legendre points then the resulting approximation u converges at its knots like o h 2m while its derivative u exhibits only o h m convergence this paper deals with the question of how to choose the collocation points so that both u and u converge like o h q with q maximal

mathematical programming methods in the numerical solution of volterra integral and integro differential equations with weakly singular kernel abstract nonlinear volterra integral and integro differential equations with weakly singular kernel are considered and solved numerically using nonlinear mathematical programming methods based on minimax approximations in both cases polynomial and multiquadric approximation are used

unitary colligations reproducing kernel hilbert spaces and nevanlinna pick interpolation in several variables abstract recently j agler studied the class s d of scalar valued analytic functions of d complex variables f for which f t 1 t d has norm at most 1 for any collection of d commuting contractions t 1 t d on a hilbert space h among other results he obtained a characterization of such functions in terms of a positivity property and in terms of a representation as the transfer function of a certain type of d variable linear system as well as a nevanlinna pick interpolation theorem for this class of functions in this note we examine the system theory aspects and uniqueness of the transfer function representation and give a simpler proof of the nevanlinna pick interpolation theorem for the class s d and obtain a d variable version of the toeplitz corona theorem by using ideas of arov and grossman introduced for 1 variable problems as a bonus we obtain a collection of linear fractional maps which parametrize the set of all s d solutions of an interpolation problem

solution of a class of volterra integral equations with singular and weakly singular kernels a class of volterra integral equations are solved in this paper where the involved kernel may be weakly singular singular and hypersingular for various cases existence and uniqueness of such volterra integral equations are established furthermore by reducing such equations to ordinary differential equations analytic solutions have been determined explicitly when restricting the solutions to certain special function classes the number of suitable solutions is given

product integration for volterra integral equations of the second kind with weakly singular kernels we introduce a new numerical approach for solving volterra integral equations of the second kind when the kernel contains a mild singularity we give a convergence result we also present numerical examples which show the performance and efficiency of our method

order of convergence of one step methods for volterra integral equations of the second kind nice proofs of convergence and asymptotic expansions are known for one step methods for ordinary differential equations it is shown that these proofs can be generalized in a natural way to extended one step methods for volterra integral equations of the second kind furthermore the convergence of mixed one step methods is investigated for both types general volterra runge kutta methods are considered as examples

two dimensional q algebras abstract two dimensional commutative banach algebra b with a unit has a simple form a bb for some fixed b in b and for a b in c when b is an operator algebra on a hilbert space we show that the norm on b is explicitly determined and then b is a q algebra moreover we describe completely two dimensional q algebras with their norms

a survey of recent advances in the numerical treatment of volterra integral and integro differential equations abstract the present survey paper samples recent advances in the numerical analysis of volterra integral equations of the first and second kind and of integro differential equations including equations with weakly singular kernels except for some important earlier references the discussion focuses on the development which has taken place during the last dozen years a fairly extensive bibliography selected to be representative rather than comprehensive complements the paper

stability results for collocation methods for volterra integral equations abstract stability behavior is investigated for polynomial spline collocation applied to volterra integral equations with special emphasis on a weakly singular test equation the characterization of the corresponding stability domain is related to stability results for solutions of finite recursion relations as they occur for methods applied to ordinary differential equations estimates of the stability domains are established and as a consequence conditions for a 2 stability are obtained exploiting these estimates collocation approximations with constant linear and continuous splines are considered in more depth and several numerical illustrations are presented

a stability analysis of the trapezoidal method for volterra integral equations with completely positive kernels abstract the solution of the volterra integral equation with completely positive kernel y t 0t b t s y s ds u0 0t b t s g s ds t 0 is nonnegative and nonincreasing provided that g is nonincreasing and 0 g t u0 for any t 0 we prove that under some additional hypotheses this property is inherited by the solution of the recurrence relation resulting from applying the trapezoidal method to this equation

numerical solution of non linear volterra integral equations this paper deals with non linear volterra integral equations of the type y x f x 0x h t x y t y x dt convergence criteria are given in the same sense of the maximum and ca norms for the numerical solution of this type of volterra integral equation several numerical methods are compared

product integration methods for second kind abel integral equations the construction and convergence of high order product integration methods for the second kind abel equation are discussed and the results of de hoog and weiss are generalised backward difference methods are introduced and numerical results are presented which verify the theoretical rates of convergence

stability of difference volterra equations direct liapunov method and numerical procedure abstract a procedure to construct liapunov functionals for discrete volterra equations is proposed using this procedure stability conditions are derived for general volterra difference equations some applications of the proposed procedure for obtaining stability conditions for linear multistep methods for volterra integro differential equations are presented

spherical contractions and interpolation problems on the unit ball in this note fractional representations of multipliers on vector valued functional hilbert spaces are used to give a proof of arveson s version of von neumann s inequality for n contractions on the unit ball we prove a commutant lifting theorem for operators on the classical hardy space over the unit ball in mathbb c n as applications we obtain interpolation results for functions in the schur class we deduce a toeplitz corona theorem on the unit ball and we give a simplified definition of arveson s curvature invariant for n contractions with finite dimensional defect space in the final part we describe a solution of the operator valued nevanlinna pick problem with uniform bounds on uniqueness sets in the unit ball

numerical solutions of volterra integral equations with a solution dependent delay on demontre l existence et l unicite des solutions pour l equation ainsi qu une limitation a priori de la solution on definit une solution numerique et on demontre que la solution numerique converge vers la solution de l equation integrale

nonnegative matrix factorization with the itakura saito divergence with application to music analysis this letter presents theoretical algorithmic and experimental results about nonnegative matrix factorization nmf with the itakura saito is divergence we describe how is nmf is underlaid by a well defined statistical model of superimposed gaussian components and is equivalent to maximum likelihood estimation of variance parameters this setting can accommodate regularization constraints on the factors through bayesian priors in particular inverse gamma and gamma markov chain priors are considered in this work estimation can be carried out using a space alternating generalized expectation maximization sage algorithm this leads to a novel type of nmf algorithm whose convergence to a stationary point of the is cost function is guaranteed we also discuss the links between the is divergence and other cost functions used in nmf in particular the euclidean distance and the generalized kullback leibler kl divergence as such we describe how is nmf can also be performed using a gradient multiplicative algorithm a standard algorithm structure in nmf whose convergence is observed in practice though not proven finally we report a furnished experimental comparative study of euclidean nmf kl nmf and is nmf algorithms applied to the power spectrogram of a short piano sequence recorded in real conditions with various initializations and model orders then we show how is nmf can successfully be employed for denoising and upmix mono to stereo conversion of an original piece of early jazz music these experiments indicate that is nmf correctly captures the semantics of audio and is better suited to the representation of music signals than nmf with the usual euclidean and kl costs

continuous collocation approximations to solutions of first kind volterra equations in this paper we give necessary and sufficient conditions for convergence of continuous collocation approximations of solutions of first kind volterra integral equations the results close some longstanding gaps in the theory of polynomial spline collocation methods for such equations the convergence analysis is based on a runge kutta or ode approach

operator algebras of idempotents abstract we introduce the concept of a matricial schur ideal which serves as a dual object for operator algebras generated by a finite set of idempotents using matricial schur ideals and some factorization theorems for tensor products of operator algebras we are able to obtain matrix valued interpolation results for general product domains these results include and generalize the recent matrix valued interpolation results on the bidisk

subtractive clustering for seeding non negative matrix factorizations non negative matrix factorization is a multivariate analysis method which is proven to be useful in many areas such as bio informatics molecular pattern discovery pattern recognition document clustering and so on it seeks a reduced representation of a multivariate data matrix into the product of basis and encoding matrices possessing only non negative elements in order to learn the so called part based representations of data all algorithms for computing non negative matrix factorization are iterative therefore particular emphasis must be placed on a proper initialization of nmf because of its local convergence the problem of selecting appropriate starting matrices becomes more complex when data possess special meaning as in document clustering in this paper we propose the adoption of the subtractive clustering algorithm as a scheme to generate initial matrices for non negative matrix factorization algorithms comparisons with other commonly adopted initializations of non negative matrix factorization algorithms have been performed and the proposed scheme reveals to be a good trade off between effectiveness and speed moreover the effectiveness of the proposed initialization to suggest a number of basis for nmf when data distances are estimated is illustrated when nmf is used for solving clustering problems where the number of groups in which the data are grouped is not known a priori the influence of a proper rank factor on the interpretability and the effectiveness of the results are also discussed

haar wavelet method for some nonlinear volterra integral equations of the first kind we present here a simple efficient haar wavelet method for numerical solution of a class of nonlinear volterra integral equations of the first kind the present method is based on converting nonlinear volterra integral equations of the first kind into linear volterra integral equations of the second kind numerical examples are given to illustrate efficiency and accuracy of the present method comparison with numerical methods in the past has also been done solving some nonlinear first kind volterra integral equations using haar wavelet haar wavelet method is simple and fast solved examples demonstrate the accuracy and efficiency of the present method present method gives better results than the numerical methods described in past necessary accuracy may be obtained by increasing the number of calculation points

nevanlinna pick interpolation on distinguished varieties in the bidisk abstract this article treats nevanlinna pick interpolation in the setting of a special class of algebraic curves called distinguished varieties an interpolation theorem along with additional operator theoretic results is given using a family of reproducing kernels naturally associated to the variety the examples of the neil parabola and doubly connected domains are discussed

uniform error estimates of galerkin methods for monotone abel volterra integral equations on the half line we consider galerkin methods for monotone abel volterra integral equations of the second kind on the half line the l2 theory follows from kolodner s theory of monotone hammerstein equations we derive the loo theory from the l2 theory by relating the l2and loo spectra of operators of the form x b ax to one another here denotes convolution and b e l1 and a e loo as an extra condition we need b t o t a 1 with a 0 we also prove the discrete analogue in particular we verify that the galerkin matrix satisfies the discrete conditions

resolvents for weakly singular kernels and fractional differential equations the solution r t s of the resolvent equation 1 r t s b t s stb t u r u s du where b t s denotes a given weakly singular matrix is obtained by means of fixed point mappings the result is a series that begins with some singular terms after which the remainder of the terms defines a continuous function in particular the resolvent is calculated for the kernel b t s t s q 1 of the scalar abel integral equation of the second kind 2 x t a t 0t1 t s 1 qx s ds where 0 q 1 it is then used to derive closed form formulas for the resolvents corresponding to q 1 2 and 1 3 furthermore a closed form formula for the resolvent for b t s s t2 s2 q 1 is derived thereby demonstrating that the results of this paper apply not only to kernels of convolution type but also to those of non convolution type as well finally the resolvent for the kernel of 2 is used to find a general solution of a linear fractional differential equation of caputo type

numerical solution of a nonuniquely solvable volterra integral equation using extrapolation methods in this work the numerical solution of a volterra integral equation with a certain weakly singular kernel depending on a real parameter is considered although for certain values of this equation possesses an infinite set of solutions we have been able to prove that euler s method converges to a particular solution it is also shown that the error allows an asymptotic expansion in fractional powers of the stepsize so that general extrapolation algorithms like the e algorithm can be applied to improve the numerical results this is illustrated by means of some examples

a perspective on the numerical treatment of volterra equations abstract we discuss the properties and numerical treatment of various types of volterra and abel volterra integral and integro differential equations

underdetermined reverberant acoustic source separation using weighted full rank nonnegative tensor models in this paper a fusion of k models of full rank weighted nonnegative tensor factor two dimensional deconvolution k wntf2d is proposed to separate the acoustic sources that have been mixed in an underdetermined reverberant environment the model is adapted in an unsupervised manner under the hybrid framework of the generalized expectation maximization and multiplicative update algorithms the derivation of the algorithm and the development of proposed full rank k wntf2d will be shown the algorithm also encodes a set of variable sparsity parameters derived from gibbs distribution into the k wntf2d model this optimizes each sub model in k wntf2d with the required sparsity to model the time varying variances of the sources in the spectrogram in addition an initialization method is proposed to initialize the parameters in the k wntf2d experimental results on the underdetermined reverberant mixing environment have shown that the proposed algorithm is effective at separating the mixture with an average signal to distortion ratio of 3 db

numerical solutions of a perturbed chandrasekhar h equation abstract numerical solutions of perturbed chandrasekhar h type integral equations are analysed convergence criteria in the sense of the maximum norm and the c norm are studied several numerical methods are compared

fractional linear multistep methods for abel volterra integral equations of the second kind fractional powers of linear multistep methods are suggested for the numerical solution of weakly singular volterra integral equations the proposed methods are convergent of the order of the underlying multistep method also in the generic case of solutions which are not smooth at the origin the stability properties stability region a stability a a stability are closely related to those of the underlying linear multistep method