moment based evidence for simple rational valued hilbert schmidt generic 2 2 separability probabilities employing the hilbert schmidt measure we explicitly compute and analyze a number of determinantal product bivariate moments k pt n k n 0 1 2 3 with pt denoting the partial transpose for both generic 9 dimensional two rebit and generic 15 dimensional two qubit 1 density matrices the results are then incorporated into a general formula parameterized by k n and with the case 2 presumptively corresponding to generic 27 dimensional quaternionic systems holding the dyson index like parameter fixed the induced univariate moments pt n and pt n are inputted into a legendre polynomial based least squares probability distribution reconstruction algorithm of provost 2005 mathematica j 9 727 yielding specific separability probability estimates since as the number of inputted moments grows estimates based on the variable pt strongly decrease while ones employing pt strongly increase and converge faster the gaps between upper and lower estimates diminish yielding sharper and sharper bounds remarkably for 2 with the use of 2325 moments a separability probability lower bound 0 999 999 987 as large as is found for 1 based on 2415 moments a lower bound results that is 0 999 997 066 times as large as a simpler still fractional value that had previously been conjectured slater 2007 j phys a math theor 40 14279 furthermore for employing 3310 moments the lower bound is 0 999 955 times as large as a rational value previously considered slater 2010 j phys a math theor 43 195302

products of rectangular random matrices singular values and progressive scattering we discuss the product of m rectangular random matrices with independent gaussian entries which have several applications including wireless telecommunication and econophysics for complex matrices an explicit expression for the joint probability density function is obtained using the harish chandra itzykson zuber integration formula explicit expressions for all correlation functions and moments for finite matrix sizes are obtained using a two matrix model and the method of biorthogonal polynomials this generalizes the classical result for the so called wishart laguerre gaussian unitary ensemble or chiral unitary ensemble at m 1 and previous results for the product of square matrices the correlation functions are given by a determinantal point process where the kernel can be expressed in terms of meijer g functions we compare the results with numerical simulations and known results for the macroscopic level density in the limit of large matrices the location of the end points of support for the latter are analyzed in detail for general m finally we consider the so called ergodic mutual information which gives an upper bound for the spectral efficiency of a mimo communication channel with multifold scattering

statistical properties of random density matrices statistical properties of ensembles of random density matrices are investigated we compute traces and von neumann entropies averaged over ensembles of random density matrices distributed according to the bures measure the eigenvalues of the random density matrices are analysed we derive the eigenvalue distribution for the bures ensemble which is shown to be broader then the quarter circle distribution characteristic of the hilbert schmidt ensemble for measures induced by partial tracing over the environment we compute exactly the two point eigenvalue correlation function

probability measure generated by the superfidelity we study the probability measure on the set of density matrices induced by the metric defined by using superfidelity we give the formula for the probability density of eigenvalues we also study some statistical properties of the set of density matrices equipped with the introduced measure and provide a method for generating density matrices according to the introduced measure

universal microscopic correlation functions for products of truncated unitary matrices we investigate the spectral properties of the product of m complex non hermitian random matrices that are obtained by removing l rows and columns of larger unitary random matrices uniformly distributed on the group u n l such matrices are called truncated unitary matrices or random contractions we first derive the joint probability distribution for the complex eigenvalues of the product matrix for fixed n l and m given by a standard determinantal point process in the complex plane the weight however is non standard and can be expressed in terms of the meijer g function the explicit knowledge of all eigenvalue correlation functions and the corresponding kernel allows us to take various large n and l limits at fixed m at strong non unitarity with l n finite the eigenvalues condense on a domain inside the unit circle at the edge and in the bulk we find the same universal microscopic kernel as for a single complex non hermitian matrix from the ginibre ensemble at the origin we find the same new universality classes labeled by m as for the product of m matrices from the ginibre ensemble keeping a fixed size of truncation l when n goes to infinity leads to weak non unitarity with most eigenvalues on the unit circle as for unitary matrices here we find a new microscopic edge kernel that generalizes the known results for m 1 we briefly comment on the case when each product matrix results from a truncation of different size lj

multivariate approximation theory and applications this introductory paper describes the main topics of this special issue dedicated to leonardo traversoni known at the international level as the promoter of the conference series multivariate approximation theory and applications to celebrate his 60th birthday

estimates for n widths of multiplier operators of multiple walsh series abstract estimates for kolmogorov and gelfand n widths of multiplier operators of multiple walsh series are obtained upper and lower bounds are established for n widths of general multiplier operators these results are applied to get upper and lower bounds for n widths of specific multiplier operators which generate sets of finitely and infinitely differentiable functions in the dyadic sense it is shown that these estimates have order sharp in various important cases

classification of joint numerical ranges of three hermitian matrices of size three abstract the joint numerical range w f of three hermitian 3 by 3 matrices f f 1 f 2 f 3 is a convex and compact subset in r 3 generically we find that w f is a three dimensional oval assuming dim w f 3 every one or two dimensional face of w f is a segment or a filled ellipse we prove that only ten configurations of these segments and ellipses are possible we identify a triple f for each class and illustrate w f using random matrices and dual varieties

all teleportation and dense coding schemes we establish a one to one correspondence between 1 quantum teleportation schemes 2 dense coding schemes 3 orthonormal bases of maximally entangled vectors 4 orthonormal bases of unitary operators with respect to the hilbert schmidt scalar product and 5 depolarizing operations whose kraus operators can be chosen to be unitary the teleportation and dense coding schemes are assumed to be tight in the sense that all hilbert spaces involved have the same finite dimension d and the classical channel involved distinguishes d 2 signals a general construction procedure for orthonormal bases of unitaries involving latin squares and complex hadamard matrices is also presented

hilbert schmidt volume of the set of mixed quantum states we compute the volume of the convex n2 1 dimensional set n of density matrices of size n with respect to the hilbert schmidt measure the hyper area of the boundary of this set is also found and its ratio to the volume provides information about the structure of n similar investigations are also performed for the smaller set of all real density matrices as an intermediate step we analyse volumes of the unitary and orthogonal groups and of the flag manifolds

induced measures in the space of mixed quantum states we analyse several product measures in the space of mixed quantum states in particular we study measures induced by the operation of partial tracing the natural rotationally invariant measure on the set of all pure states of a n k composite system induces a unique measure in the space of n n mixed states or in the space of k k mixed states if the reduction takes place with respect to the first subsystem for k n the induced measure is equal to the hilbert schmidt measure which is shown to coincide with the measure induced by singular values of non hermitian random gaussian matrices pertaining to the ginibre ensemble we compute several averages with respect to this measure and show that the mean entanglement of n n pure states behaves as lnn 1 2

optimal and nearly optimal algorithms for approximating polynomial zeros abstract we substantially improve the known algorithms for approximating all the complex zeros of an n th degree polynomial p x our new algorithms save both boolean and arithmetic sequential time versus the previous best algorithms of schonhage 1 pan 2 and neff and reif 3 in parallel nc implementation we dramatically decrease the number of processors versus the parallel algorithm of neff 4 which was the only nc algorithm known for this problem so far specifically under the simple normalization assumption that the variable x has been scaled so as to confine the zeros of p x to the unit disc x x 1 our algorithms which promise to be practically effective approximate all the zeros of p x within the absolute error bound 2 b by using order of n arithmetic operations and order of b n n 2 boolean bitwise operations in both cases up to within polylogarithmic factors the algorithms allow their optimal work preserving nc parallelization so that they can be implemented by using polylogarithmic time and the orders of n arithmetic processors or b n n 2 boolean processors all the cited bounds on the computational complexity are within polylogarithmic factors from the optimum in terms of n and b under both arithmetic and boolean models of computation in the boolean case under the additional realistic assumption that n o b

free self decomposability and unimodality of the fuss catalan distributions we study properties of the fuss catalan distributions mu p r p ge 1 0 1 there exists r 0 p with p 1 r 0 p p such that the fuss catalan distribution mu p r is unimodal if and only if either r p or 0 r le r 0 p we prove rigorously some partial results

efficient approximation of quantum channel capacities we propose an iterative method for approximating the capacity of classical quantum channels with a discrete input alphabet and a finite dimensional output under additional constraints on the input distribution based on duality of convex programming we derive explicit upper and lower bounds for the capacity to provide an additive varepsilon close estimate to the capacity the presented algorithm requires o n vee m m 3 log n 1 2 varepsilon 1 steps where n denotes the input alphabet size and m denotes the output dimension we then generalize the method to the task of approximating the capacity of classical quantum channels with a bounded continuous input alphabet and a finite dimensional output this using the idea of a universal encoder allows us to approximate the holevo capacity for channels with a finite dimensional quantum mechanical input and output in particular we show that the problem of approximating the holevo capacity can be reduced to a multi dimensional integration problem for certain families of quantum channels we prove that the complexity to derive an additive varepsilon close solution to the holevo capacity is subexponential or even polynomial in the problem size we provide several examples to illustrate the performance of the approximation scheme in practice

an efficient and versatile algorithm for computing the covariance function of an arma process computation of an arma covariance function is a common ingredient in analysis and synthesis of various problems in stochastic control estimation and signal processing several approaches can be used for this purpose in this paper we present an algorithm based on simple polynomial calculations compared with alternative strategies it has small computational load shows good numerical robustness and can be extended to handle multivariable arma processes even with complex valued coefficients

image processing tools for better incorporation of 4d seismic data into reservoir models reservoir engineers have to predict the behavior of a hydrocarbon reservoir by building a simulation model which can best reproduce the data collected in the field these data fall into two types static data which are invariable in time and dynamic data which evolve according to fluid motions in the reservoir in this paper we focus on the integration of dynamic data related to four dimensional 4d inverted seismic data such seismic data constitute an invaluable source of information on fluid displacement and geology over extensive areas of the reservoir however incorporating them in the reservoir model through a matching process is a challenging task classical formulations of the objective function which computes the misfit between observed data and responses computed by the reservoir model are not adapted to 4d inverted seismic data for example a least square based mismatch is not representative of the visual difference between two seismic images in this paper we define a new formulation of the objective function based on simplification of seismic data in order to extract relevant information this simplification involves filtering and segmentation techniques as well as image comparison methods rooted in image analysis more precisely we focus on the non local means algorithm for filtering on the level set framework for segmentation and on the local modified hausdorff distance for image comparison we investigate the efficiency of such techniques in the context of seismic data and illustrate their potential on a synthetic history matching reservoir example

integration with respect to the haar measure on unitary orthogonal and symplectic group we revisit the work of the first named author and using simpler algebraic arguments we calculate integrals of polynomial functions with respect to the haar measure on the unitary group u d the previous result provided exact formulas only for 2d bigger than the degree of the integrated polynomial and we show that these formulas remain valid for all values of d also we consider the integrals of polynomial functions on the orthogonal group o d and the symplectic group sp d we obtain an exact character expansion and the asymptotic behavior for large d thus we can show the asymptotic freeness of haar distributed orthogonal and symplectic random matrices as well as the convergence of integrals of the itzykson zuber type

finitely correlated states on quantum spin chains we study a construction that yields a class of translation invariant states on quantum spin chains characterized by the property that the correlations across any bond can be modeled on a finite dimensional vector space these states can be considered as generalized valence bond states and they are dense in the set of all translation invariant states we develop a complete theory of the ergodic decomposition of such states including the decomposition into periodic neel ordered states the ergodic components have exponential decay of correlations all states considered can be obtained as local functions of states of a special kind so called purely generated states which are shown to be ground states for suitably chosen finite range vbs interactions we show that all these generalized vbs models have a spectral gap our theory does not require symmetry of the state with respect to a local gauge group in particular we illustrate our results with a one parameter family of examples which are not isotropic except for one special case this isotropic model coincides with the one dimensional antiferromagnet recently studied by affleck kennedy lieb and tasaki

deterministic improvement of complex polynomial factorization based on the properties of the associated resultant abstract we show an improvement of the known best deterministic approach to factorization of a univariate polynomial p x over the complex field previously such an improvement was achieved by means of a random perturbation of the coefficients of the input polynomial we now apply some new techniques which lead to a similar result without any randomization the approach is unusual in exploiting some properties of the resultant of p x and p x which is completely distinct from the techniques usually applied in order to compute complex polynomial factorization

random bures mixed states and the distribution of their purity ensembles of random density matrices determined by various probability measures are analysed a simple and efficient algorithm to generate at random density matrices distributed according to the bures measure is proposed this procedure may serve as an initial step in performing the bayesian approach to quantum state estimation based on the bures prior we study the distribution of purity of random mixed states the moments of the distribution of purity are determined for quantum states generated with respect to the bures measure this calculation serves as an exemplary application of the deform and study approach in the theory of integrable systems leading to one of painleves transcendents

vector field approximation using radial basis functions in this article we investigate the performance of rbf pde methods for approximating solenoidal fields it is well known that global rbf collocation methods present a trade off principle which means that smoothness implies high convergence order plus ill conditioning on the other hand local methods for solving this problem have recently appeared in the literature in this paper we perform a numerical investigation of the differences between rbf global and local methods in order to investigate the possible advantage of using local methods for the approximation of vector fields more precisely we compare the local hermite interpolation technique using inverse multiquadrics against the non symmetric collocation method of kansa

on a class of free levy laws related to a regression problem abstract the free meixner laws arise as the distributions of orthogonal polynomials with constant coefficient recursions we show that these are the laws of the free pairs of random variables which have linear regressions and quadratic conditional variances when conditioned with respect to their sum we apply this result to describe free levy processes with quadratic conditional variances and to prove a converse implication related to asymptotic freeness of random wishart matrices

composed ensembles of random unitary matrices composed ensembles of random unitary matrices are defined via products of matrices each pertaining to a given canonical circular ensemble of dyson we investigate statistical properties of spectra of some composed ensembles and demonstrate their physical relevance we also discuss the methods of generating random matrices distributed according to invariant haar measure on the orthogonal and unitary group

hall normalization constants for the bures volumes of the n state quantum systems we report the results of certain integrations of quantum theoretic interest relying in this regard upon recently developed parametrizations of boya et al 1998 preprint quant ph 9810084 of the n n density matrices in terms of squared components of the unit n 1 sphere and the n n unitary matrices firstly we express the normalized volume elements of the bures minimal monotone metric for n 2 and 3 thereby obtaining bures prior probability distributions over the two and three state systems then as a first step in extending these results to n 3 we determine that the hall normalization constant cn for the marginal bures prior probablity distribution over the n 1 dimensional simplex of the n eigenvalues of the n n density matrices is for n 4 equal to 71 680 2 since we also find that c3 35 it follows that c4 is simply equal to 211c3 c2 itself is known to equal 2 the constant c5 is also found it too is associated with a remarkably simple decomposition involving the product of the eight consecutive prime numbers from 3 to 23 we also preliminarily investigate several cases n 5 with the use of quasi monte carlo integration we hope that the various analyses reported will prove useful in deriving a general formula which evidence suggests will involve the bernoulli numbers for the hall normalization constant for arbitrary n this would have diverse applications including quantum inference and universal quantum coding

the detection and recovery of discontinuity curves from scattered data we present a numerical method for the detection of faults and gradient faults of two dimensional functions when uniformly scattered and noisy data are given the method recognizes the presence of a discontinuity curve by a suitable statistic moreover we discuss the recovery of such curves and provide a computational bound for the pointwise error and convergence properties

a new procedure for the construction of hierarchical high order hdiv and hcurl finite element spaces this paper considers a systematic procedure for the construction of a hierarchy of high order finite element approximation for hdiv and hcurl spaces based on triangular and quadrilateral partitions of bidimensional domains the principle is to choose an appropriate set of vectors based on the geometry of each element which are multiplied by an available set of h 1 hierarchical scalar basic functions this strategy produces vector basis functions with continuous normal or tangent components on the elements interfaces properties that characterise functions in hdiv or hcurl respectively we also present a numerical study to evaluate the correct balancedness of the resulting hdiv spaces of degree k and l 2 spaces of degree k 1 on the resolution of the mixed formulation for a steklov eigenvalue problem

subnormalized states and trace nonincreasing maps we investigate the set of completely positive trace nonincreasing linear maps acting on the set mn of mixed quantum states of size n extremal point of this set of maps are characterized and its volume with respect to the hilbert schmidt hs euclidean measure is computed explicitly for an arbitrary n the spectra of partially reduced rescaled dynamical matrices associated with trace nonincreasing completely positive maps belong to the n cube inscribed in the set of subnormalized states of size n as a by product we derive the measure in mn induced by partial trace of mixed quantum states distributed uniformly with respect to the hs measure in mn2

generating random density matrices we study various methods to generate ensembles of random density matrices of a fixed size n obtained by partial trace of pure states on composite systems structured ensembles of random pure states invariant with respect to local unitary transformations are introduced to analyze statistical properties of quantum entanglement in bi partite systems we analyze the distribution of schmidt coefficients of random pure states such a distribution is derived in the case of a superposition of k random maximally entangled states for another ensemble obtained by performing selective measurements in a maximally entangled basis on a multi partite system we show that this distribution is given by the fuss catalan law and find the average entanglement entropy a more general class of structured ensembles proposed containing also the case of bures forms an extension of the standard ensemble of structureless random pure states described asymptotically as n by the marchenko pastur distribution

approximation of smooth functions on compact two point homogeneous spaces estimates of kolmogorov n widths dn bpr lq and linear n widths n bpr lq 1 q of sobolev s classes bpr r 0 1 p on compact two point homogeneous spaces ctphs are established for part of p q 1 1 sharp orders of dn bpr lq or n bpr lq were obtained by bordin et al j funct anal 202 2 2003 307 in this paper we obtain the sharp orders of dn bpr lq and n bpr lq for all the remaining p q our proof is based on positive cubature formulas and marcinkiewicz zygmund type inequalities on ctphs

optimal cubature formulas on compact homogeneous manifolds abstract we find lower bounds for the rate of convergence of optimal cubature formulas on sets of differentiable functions on compact homogeneous manifolds of rank i or two point homogeneous spaces it is shown that these lower bounds are sharp in the power scale in the case of s 2 the unit sphere in r 3

hermite interpolation by rational gk motions of low degree interpolation by rational spline motions is an important issue in robotics and related fields in this paper a new approach to rational spline motion design is described by using techniques of geometric interpolation this enables us to reduce the discrepancy in the number of degrees of freedom of the trajectory of the origin and of the rotational part of the motion a general approach to geometric interpolation by rational spline motions is presented and two particularly important cases are analyzed i e geometrically continuous quartic rational motions and second order geometrically continuous rational spline motions of degree six in both cases sufficient conditions on the given hermite data are found which guarantee the uniqueness of the solution if the given data do not fulfill the solvability conditions a method to perturb them slightly is described numerical examples are presented which confirm the theoretical results and provide evidence that the obtained motions have nice shapes

non uniform interpolatory subdivision via splines we present a framework for deriving non uniform interpolatory subdivision algorithms closely related to non uniform spline interpolants families of symmetric non uniform interpolatory 2n point schemes of smoothness c n 1 are presented for n 2 3 4 and even higher order as well as a variety of non uniform 6 point schemes with c 3 continuity

novel fast computation algorithm of the second order statistics for autoregressive moving average processes very often random signals are modeled as autoregressive moving average arma processes in engineering and scientific applications the acquisition of the second order statistics or the calculation of the autocorrelation function for an arma process is very important in those applications in this paper we discuss two kinds of arma autocorrelation computation approaches namely the recursive approach and the direct approach to overcome the problems of memory usage and computational burden we design a new arma autocorrelation calculation algorithm which belongs to the direct approach our novel algorithm originates from the partial fractional decomposition for the rational trigonometric functions and the associated definite integrals for comparison we also provide the theoretical analysis of computational complexity for our new scheme and the conventional recursion based algorithm from our studies it can be shown that our new arma autocorrelation calculation method is more efficient than the conventional algorithm when the underlying autocorrelation function is long or the corresponding power spectrum possesses the narrowband characteristics on the other hand the conventional algorithm is more efficient than our new algorithm when the model orders of the arma process are large and the underlying autocorrelation function is short the corresponding power spectrum is wideband since our proposed new scheme involves the root finding of a polynomial the effect of the error tolerance in solving polynomials on the induced complexity is also discussed in this paper

geometries of quantum states the quantum analog of the fisher information metric of a probability simplex is searched and several riemannian metrics on the set of positive definite density matrices are studied some of them appeared in the literature in connection with cramer rao type inequalities or the generalization of the berry phase to mixed states they are shown to be stochastically monotone here all stochastically monotone riemannian metrics are characterized by means of operator monotone functions and it is proven that there exist a maximal and a minimal among them a class of metrics can be extended to pure states and a constant multiple of the fubini study metric appears in the extension

bures volume of the set of mixed quantum states we compute the volume of the n2 1 dimensional set n of density matrices of size n with respect to the bures measure and show that it is equal to that of an n2 1 dimensional hyper hemisphere of radius 1 2 for n 2 we obtain the volume of the uhlmann hemisphere s3 4 we find also the area of the boundary of the set n and obtain analogous results for the smaller set of all real density matrices an explicit formula for the bures hall normalization constants is derived for an arbitrary n

brown s spectral distribution measure for r diagonal elements in finite von neumann algebras abstract in 1983 l g brown introduced a spectral distribution measure for non normal elements in a finite von neumann algebra m with respect to a fixed normal faithful tracial state in this paper we compute brown s spectral distribution measure in case t has a polar decomposition t uh where u is a haar unitary and u and h are free when ker t 0 this is equivalent to that t t is an r diagonal pair in the sense of nica and speicher the measure t is expressed explicitly in terms of the s transform of the distribution t t of the positive operator t t in case t is a circular element i e t x1 ix2 2 where x1 x2 is a free semicircular system then sp t d the closed unit disk and t has constant density 1 on d

estimates of n widths of sobolev s classes on compact globally symmetric spaces of rank one estimates of kolmogorov s and linear n widths of sobolev s classes on compact globally symmetric spaces of rank 1 i e on sd pd r pd c pd h p16 cay are established it is shown that these estimates have sharp orders in different important cases new estimates for the p q norms of multiplier operators k k n are given we apply our results to get sharp orders of best polynomial approximation and n widths

product of ginibre matrices fuss catalan and raney distributions squared singular values of a product of s square random ginibre matrices are asymptotically characterized by probability distributions p s x such that their moments are equal to the fuss catalan numbers of order s we find a representation of the fuss catalan distributions p s x in terms of a combination of s hypergeometric functions of the type s f s 1 the explicit formula derived here is exact for an arbitrary positive integer s and for s 1 it reduces to the marchenko pastur distribution using similar techniques involving the mellin transform and the meijer g function we find exact expressions for the raney probability distributions the moments of which are given by a two parameter generalization of the fuss catalan numbers these distributions can also be considered as a two parameter generalization of the wigner semicircle law

quantum state filter with disturbance and noise a quantum state filter qsf is proposed in this paper to estimate a low rank quantum density matrix from informationally incomplete and contaminated measurements there exist sparse disturbances on the quantum density matrix and gaussian noise in the measurements a proximal jacobian variant of the alternating direction method of multipliers pj admm is proposed to design the qsf the closed form solutions to three resulting subproblems are given and the iterative qsf is developed the proposed qsf is proved to be convergent and its superiority is demonstrated in the numerical illustrations compared with different state of the art methods

a multiplier version of the bernstein inequality on the complex sphere we prove a multiplier version of the bernstein inequality on the complex sphere included in this is a new result relating a bivariate sum involving jacobi polynomials and gegenbauer polynomials which relates the sum of reproducing kernels on spaces of polynomials irreducibly invariant under the unitary group with the reproducing kernel of the sum of these spaces which is irreducibly invariant under the action of the unitary group

random graph states maximal flow and fuss catalan distributions for any graph consisting of k vertices and m edges we construct an ensemble of random pure quantum states which describe a system composed of 2m subsystems each edge of the graph represents a bipartite maximally entangled state each vertex represents a random unitary matrix generated according to the haar measure which describes the coupling between subsystems dividing all subsystems into two parts one may study entanglement with respect to this partition a general technique to derive an expression for the average entanglement entropy of random pure states associated with a given graph is presented our technique relies on weingarten calculus and flow problems we analyze the statistical properties of spectra of such random density matrices and show for which cases they are described by the free poissonian marchenko pastur distribution we derive a discrete family of generalized fuss catalan distributions and explicitly construct graphs which lead to ensembles of random states characterized by these novel distributions of eigenvalues

spectral density of generalized wishart matrices and free multiplicative convolution we investigate the level density for several ensembles of positive random matrices of a wishart like structure w xx where x stands for a non hermitian random matrix in particular making use of the cauchy transform we study the free multiplicative powers of the marchenko pastur mp distribution mp s which for an integer s yield fuss catalan distributions corresponding to a product of s independent square random matrices x x 1 x s new formulas for the level densities are derived for s 3 and s 1 3 moreover the level density corresponding to the generalized bures distribution given by the free convolution of arcsine and mp distributions is obtained we also explain the reason of such a curious convolution the technique proposed here allows for the derivation of the level densities for several other cases

eigenvalues and singular values of products of rectangular gaussian random matrices we derive exact analytic expressions for the distributions of eigenvalues and singular values for the product of an arbitrary number of independent rectangular gaussian random matrices in the limit of large matrix dimensions we show that they both have power law behavior at zero and determine the corresponding powers we also propose a heuristic form of finite size corrections to these expressions which very well approximates the distributions for matrices of finite dimensions

gradient field approximation application to registration in image processing we study a spline based approximation of vector fields in the conservative case the gradient vector field derives from a potential function we introduce a minimization problem on a hilbert space for which the existence and uniqueness of the solution is given we apply this approach to a registration process in image processing