super resolution on the sphere using convex optimization this paper considers the problem of recovering an ensemble of diracs on a sphere from its low resolution measurements the diracs can be located at any location on the sphere not necessarily on a grid we show that under a separation condition one can recover the ensemble with high precision by a three stage algorithm which consists of solving a semi definite program root finding and least square fitting the algorithm s computation time depends solely on the number of measurements and not on the required solution accuracy we also show that in the special case of non negative ensembles a sparsity condition is sufficient for recovery furthermore in the discrete setting we estimate the recovery error in the presence of noise as a function of the noise level and the super resolution factor

lowner based blind signal separation of rational functions with applications a new blind signal separation bss technique is proposed enabling a deterministic separation of signals into rational functions rational functions can take on a wide range of forms such as the well known pole like shape the approach is a possible alternative for the well known independent component analysis when the theoretical sources are not independent such as for frequency spectra or when only a small number of samples is available the technique uses a low rank decomposition on the tensorized version of the observed data matrix the deterministic tensorization with lowner matrices is comprehensively analyzed in this paper uniqueness properties are investigated and a connection with the separation into exponential polynomials is made finally the technique is illustrated for fetal electrocardiogram extraction and with an application in the domain of fluorescence spectroscopy enabling the identification of chemical analytes using only a single excitation emission matrix

moment maps scalar curvature and quantization of kahler manifolds building on donaldson s work on constant scalar curvature metrics we study the space of regular kahler metrics e i e those for which deformation quantization has been defined by cahen gutt and rawnsley after giving in sects 2 and 3 a review of donaldson s moment map approach we study the essential uniqueness of balanced basis i e of coherent states in a more general setting theorem 2 5 we then study the space e in sect 4 and we show in sect 5 how all the tools needed can be defined also in the case of non compact manifolds

diffusion in a potential with a time dependent discontinuity we investigate the one dimensional diffusion of a particle in a v shaped potential combined with a time dependent jump at the tip employing the matching conditions we calculate the exact green function of the corresponding smoluchowski equation we then specialize the analysis to a harmonically oscillating height of the potential discontinuity we calculate the particle s mean position as a function of time and study its nonlinear features our analysis reveals a new type of stochastic resonance namely the time asymptotic amplitude of the mean position oscillations exhibits a maximum at an optimal value of the v potential slope

scalar curvature and projective embeddings i we prove that a metric of constant scalar curvature on a polarised kahler manifold is the limit of metrics induced from a specific sequence of projective embeddings satisfying a condition introduced by h luo this gives as a corollary the uniqueness of constant scalar curvature kahler metrics in a given rational cohomology class the proof uses results in the literature on the asymptotics of the bergman kernel the arguments are presented in a general framework involving moment maps for two different group actions

on the partial realization problem abstract in this paper we take a unified approach to the partial realization problem in which we seek to incorporate ideas from numerical linear algebra most of which were originally developed in other contexts we approach the partial realization problem from several different angles and explore the connections to such topics as factoriza tion of hankel matrices block tridiagonalization generalizations of the lanczos process for biorthogonalization the euclidean algorithm and the principal part con tinued fractions of arne magnus the pade table and the berlekamp masseyalgo rithm in this way we are able to clarify some previous results by rissanen kalman and others and place them in a broader context this leads to several results and concepts which we think are new our analysis is restricted to the scalar case but some definitions and formulations have been rigged to facilitate an extension to the matrix case

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

factorization of finite rank hankel and toeplitz matrices the main purpose of this paper is to prove factorization results for finite hankel and toeplitz matrices that generalize known results for the positive semidefinite case generalized vandermonde matrices appear in the factorizations factorization theorems are proved for infinite hankel and toeplitz matrices of finite rank and the results for finite matrices are deduced via a technique of rank preserving extension

generalization of a theorem of caratheodory caratheodory stated the conditions to be obeyed by n complex numbers c1 cn in order that they can be written uniquely in the form cp mj 1 jjp with p 1 n js being different unimodular complex numbers and js strictly positive numbers we give the conditions to be obeyed for the former property to hold true if js are simply required to be real and different from zero the number of the possible choices of the signs of js are at most equal to the number of the distinct eigenvalues of the hermitian toeplitz matrix whose i j th entry is cj i where c p by definition is the complex conjugate of cp and c0 0 this generalization is relevant to neutron scattering its proof is made possible by a lemma stating the necessary and sufficient conditions to be obeyed by the coefficients of a polynomial equation for all the roots to lie on the unit circle this lemma is an interesting side result of our analysis

the mechanism of stochastic resonance it is shown that a dynamical system subject to both periodic forcing and random perturbation may show a resonance peak in the power spectrum which is absent when either the forcing or the perturbation is absent

one dimensional diffusion through single and double square barriers exact solutions are obtained for the diffusion through single and double square potentials by using the laplace transform method we compare transmission of a classical particle through one barrier of given length and through two barriers with a well between them of the same overall length the influence of the barriers heights and widths as well as of the distances to the reflecting walls is studied in detail the results obtained are used to estimate the limits of applicability of the kramers rate theory

homogeneous matrix equations and multivariate linear models abstract the growth curve model potthoff and roy 1964 and an extension von rosen 1989 are considered the mean structures for the models are given by abc and i a i b i c i respectively where the a and c matrices are known and the b matrices unknown the purpose is to discuss homogeneous matrix equations dbe 0 d i be i 0 i 1 2 s and d 1 b 1 e 1 d 2 b 2 e 2 0 regarded as restrictions on the parameter space in statistical models among other results we will see what kind of restrictions have to be imposed on the d and e matrices in order to obtain interpretable maximum likelihood estimators

a multivariate generalization of prony s method abstract prony s method is a prototypical eigenvalue analysis based method for the reconstruction of a finitely supported complex measure on the unit circle from its moments up to a certain degree in this note we give a generalization of this method to the multivariate case and prove simple conditions under which the problem admits a unique solution provided the order of the moments is bounded from below by the number of points on which the measure is supported as well as by a small constant divided by the separation distance of these points stable reconstruction is guaranteed in its simplest form the reconstruction method consists of setting up a certain multilevel toeplitz matrix of the moments compute a basis of its kernel and compute by some method of choice the set of common roots of the multivariate polynomials whose coefficients are given in the second step all theoretical results are illustrated by numerical experiments

randomized interpolation and approximationof sparse polynomials we present a randomized algorithm that interpolates a sparse polynomial in polynomial time in the bit complexity model the algorithm can be also applied to approximate polynomials that can be approximated by sparse polynomials the approximation is in the l 2 norm

a new aspect of hankel matrices via krylov matrix abstract an investigation is made of the krylov matrices associated to the companion matrix c of a monic polynomial p x of degree n in this paper we propose a new approach to the study of hankel matrices as krylov matrices of c t firstly we focus on algebraic structures of krylov matrices of c over a field f we then discuss an equivalent condition for the hankel matrix to be a krylov matrix of c t by the notion of compatibility as a result we derive new determinant formulas for such hankel matrices in terms of eigenvalues and eigenvectors of c respectively

balanced metrics on cartan and cartan hartogs domains this paper consists of two results dealing with balanced metrics in donaldson terminology on noncompact complex manifolds in the first one we describe all balanced metrics on cartan domains in the second one we show that the only cartan hartogs domain which admits a balanced metric is the complex hyperbolic space by combining these results with those obtained in loi and zedda mathematische annalen 2011 to appear we also provide the first example of complete kahler einstein and projectively induced metric g such that g is not balanced for all 0

music for multidimensional spectral estimation stability and super resolution this paper presents a performance analysis of the multiple signal classification music algorithm applied on d dimensional single snapshot spectral estimation while s true frequencies are located on the continuum of a bounded domain inspired by the matrix pencil form we construct a d fold hankel matrix from the measurements and exploit its vandermonde decomposition in the noiseless case music amounts to identifying a noise subspace evaluating a noise space correlation function and localizing frequencies by searching the s smallest local minima of the noise space correlation function in the noiseless case 2s d measurements guarantee an exact reconstruction by music as the noise space correlation function vanishes exactly at true frequencies when noise exists we provide an explicit estimate on the perturbation of the noise space correlation function in terms of noise level dimension d the minimum separation among frequencies the maximum and minimum amplitudes while frequencies are separated by 2 rayleigh length rl at each direction as a by product the maximum and minimum non zero singular values of the multidimensional vandermonde matrix whose nodes are on the unit sphere are estimated under a gap condition of the nodes under the 2 rl separation condition if noise is i i d gaussian we show that perturbation of the noise space correlation function decays like sqrt log bf n bf n as the sample size bf n increases when the separation among frequencies drops below 2 rl our numerical experiments show that the noise tolerance of music obeys a power law with the minimum separation of frequencies

diffusion process with two reflecting barriers in a time dependent potential we consider a brownian particle which is driven by a harmonically oscillating force the motion of which is restricted by two reflecting boundaries we solve the fokker planck equation using the finite element method and focus on the dynamics of the mean position of the particle in the time asymptotic regime as a function of the strength of the external force the response of the system i e the amplitude of the mean position and the dynamical shift in the stationary limit shows a resonancelike behavior as a function of the diffusion coefficient for certain parameter regimes we explain these numerical results heuristically and give some qualitative estimates

sublinear time algorithms for compressive phase retrieval in the problem of compressed phase retrieval the goal is to reconstruct a sparse or approximately k sparse vector x in mathbb c n given access to y phi x where v denotes the vector obtained from taking the absolute value of v in mathbb c n coordinate wise in this paper we present sublinear time algorithms for a few for each variants of the compressive phase retrieval problem which are akin to the variants considered for the classical compressive sensing problem in theoretical computer science our algorithms use pure combinatorial techniques and near optimal number of measurements

polynomials and hankel matrices abstract compatibility of a hankel n n matrix h and a polynomial f of degree m m n is defined if m n compatibility means that hc t f c f h where c f is t companion matrix of f with a suitable generalization of c f this theorem is generalized to the case that m n

radial balanced metrics on the unit ball of the kepler manifold abstract we show that there is no radial balanced metric on the unit ball of the kepler manifold with not too wild boundary behavior additionally we identify explicitly the weights corresponding to radial metrics with such boundary behavior which satisfy the balanced condition as far as germs at the boundary are concerned related results for poincare metrics are also established

vandermonde decomposition of multilevel toeplitz matrices with application to multidimensional super resolution the vandermonde decomposition of toeplitz matrices discovered by caratheodory and fejer in the 1910s and rediscovered by pisarenko in the 1970s forms the basis of modern subspace methods for 1 d frequency estimation many related numerical tools have also been developed for multidimensional md especially 2 d frequency estimation however a fundamental question has remained unresolved as to whether an analog of the vandermonde decomposition holds for multilevel toeplitz matrices in the md case in this paper an affirmative answer to this question and a constructive method for finding the decomposition are provided when the matrix rank is lower than the dimension of each toeplitz block a numerical method for searching for a decomposition is also proposed when the matrix rank is higher the new results are applied to study the md frequency estimation within the recent super resolution framework a precise formulation of the atomic ell 0 norm is derived using the vandermonde decomposition practical algorithms for frequency estimation are proposed based on the relaxation techniques extensive numerical simulations are provided to demonstrate the effectiveness of these algorithms compared with the existing atomic norm and subspace methods

numerical solution of bivariate and polyanalytic polynomial systems finding the real solutions of a bivariate polynomial system is a central problem in robotics computer modeling and graphics computational geometry and numerical optimization we propose an efficient and numerically robust algorithm for solving bivariate and polyanalytic polynomial systems using a single generalized eigenvalue decomposition in contrast to existing eigen based solvers the proposed algorithm does not depend on grobner bases or normal sets nor does it require computing eigenvectors or solving additional eigenproblems to recover the solution the method transforms bivariate systems into polyanalytic systems and then uses resultants in a novel way to project the variables onto the real plane associated with the two variables solutions are returned counting multiplicity and their accuracy is maximized by means of numerical balancing and newton raphson refinement numerical experiments show that the proposed algorithm consistently recovers a higher percentage of solutions and is at the sa

vandermonde factorization and canonical representations of block hankel matrices abstract we study to which extent well known facts concerning vandermonde factorization or canonical representation of scalar hankel matrices transfer to block hankel matrices with p q blocks it is shown that nonsingular block hankel matrices can be factored like in the scalar case into nonconfluent vandermonde matrices and that the theorem on full rank factorization of arbitrary hankel matrices transfers in a weak version to the 2 2 block case but not to larger block sizes in general the minimal rank of a vandermonde factorization both with finite nodes and affine is described in terms of the hankel matrix the main tools are realization partial realization and moebius transformations

the lanczos algorithm and hankel matrix factorization abstract we explore the connections between the lanczos algorithm for matrix tridiagonalization and two fast algorithms for hankel matrix factorization we show how the asymmetric lanczos process is related to the berlekamp massey algorithm and how the symmetrized lanczos process is related to the phillips algorithm we also discuss conditions under which the analysis of phillips applies

a stable numerical method for inverting shape from moments we derive a stable technique based upon matrix pencils for the reconstruction of or approximation by polygonal shapes from moments we point out that this problem can be considered the dual of 2 d numerical quadrature over polygonal domains an analysis of the sensitivity of the problem is presented along with some numerical examples illustrating the relevant points finally an application to the problem of gravimetry is explored where the shape of a gravitationally anomalous region is to be recovered from measurements of its exterior gravitational field

a fast method to diagonalize a hankel matrix abstract we consider a vandermonde factorization of a hankel matrix and propose a new approach to compute the full decomposition in o n 2 operations the method is based on the use of a variant of the lanczos method to compute a tridiagonal matrix whose eigenvalues are the modes generating the entries in the hankel matrix by adapting existing methods to solve for these eigenvalues and then for the coefficients one arrives at a method to compute the entire decomposition in o n 2 operations the method is illustrated with a simple numerical example

bergman kernels tyz expansions and hankel operators on the kepler manifold for a class of o n 1 r invariant measures on the kepler manifold possessing finite moments of all orders we describe the reproducing kernels of the associated bergman spaces discuss the corresponding asymptotic expansions of tian yau zelditch and study the relevant hankel operators with conjugate holomorphic symbols related reproducing kernels on the minimal ball are also discussed finally we observe that the kepler manifold either does not admit balanced metrics or such metrics are not unique

matrix pencil method for estimating parameters of exponentially damped undamped sinusoids in noise a study of a matrix pencil method for estimating frequencies and damping factors of exponentially damped and or undamped sinusoids in noise is presented comparison of this method to a polynomial method svd prony method shows that the matrix pencil method and the polynomial method are two special cases of a matrix prediction approach and that the pencil method is more efficient in computation and less restrictive about signal probes it is found through perturbation analysis and simulation that for signals with unknown damping factors the pencil method is less sensitive to noise than the polynomial method an expression of the cramer rao bound for the exponential signals is presented

stability results for scattered data interpolation by trigonometric polynomials a fast and reliable algorithm for the optimal interpolation of scattered data on the torus mathbb t d by multivariate trigonometric polynomials is presented the algorithm is based on a variant of the conjugate gradient method in combination with the fast fourier transforms for nonequispaced nodes the main result is that under mild assumptions the total complexity for solving the interpolation problem at m arbitrary nodes is of order cal o m log m this result is obtained by the use of localized trigonometric kernels where the localization is chosen in accordance with the spatial dimension d numerical examples show the efficiency of the new algorithm

strictly pseudoconvex domains in c n the theory which we have developed so far will be expanded in two ways 1 in this chapter we shall use the cauchy fantappie calculus to transform the bochner martinelli kernel into a kernel adapted to strictly pseudoconvex domains in n and prove a number of finiteness and vanishing theorems for these domains 2 in the next chapter we shall pass from n to arbitrary hermitian manifolds and establish the analogue of the bmk formula in that context obtaining along the way the hodge decomposition theorem on compact manifolds both lines of thought will merge in the subsequent chapters and lead to a solution of the levi problem and finiteness theorems for strictly pseudoconvex manifolds as well as to a solution of the overline partial neumann problem to be formulated later

factorizations of hermitian block hankel matrices abstract the basic results of n i achiezer and m g krein from the classical polynomial moment theory are concerned with certain representations of elements of a positive difinite hankel matrix these results are generalized to the case of an arbitrary scalar or block hermitian hankel matrix and are shown to be closely related to some factorizations of block hankel matrices matrix polynomials and indefinite scalar products

vandermonde factorization of toeplitz matrices and applications in filtering and warping by deriving a factorization of toeplitz matrices into the product of vandermonde matrices we demonstrate that the euclidean norm of a filtered signal is equivalent with the euclidean norm of the appropriately frequency warped and scaled signal in effect we obtain an equivalence between the energy of frequency warped and filtered signals while the result does not provide tools for warping per se it does show that the energy of the warped signal can be evaluated efficiently without explicit and complex computation of the warped transform the main result is closely related to the vandermonde factorization of hankel matrices

a simple hankel interpretation of the berlekamp massey algorithm abstract a simple interpretation of the berlekamp massey algorithm in the light of the hankel matrix is presented the salient result is that the jump of the linear feedback shift register lfsr length is derived almost trivially from the so called iohvidov index of the hankel matrix prior to making any reference to the berlekamp massey algorithm itself next the hankel system of equations that yields the updated connection polynomial is solved via the natural lu factorization of the hankel matrix which itself leads to the berlekamp massey algorithm in a simple and transparent manner