a rank one reduction formula and its applications to matrix factorizations let a in r m times n denote an arbitrary matrix if x in r n and y in r m are vectors such that omega y t ax ne 0 then the matrix b a omega 1 axy t a a has rank exactly one less than the rank of a this wedderburn rank one reduction formula is easy to prove yet the idea is so powerful that perhaps all matrix factorizations can be derived from it the formula also appears in places such as the positive definite secant updates bfgs and dfp as well as the abs methods by repeatedly applying the formula to reduce ranks a biconjugation process analogous to the gram schmidt process with oblique projections can be developed this process provides a mechanism for constructing factorizations such as text ldm t qr and svd under a common framework of a general biconjugate decomposition v t au omega that is diagonal and nonsingular two characterizations of biconjugation provide new insight into the lanczos method and its breakdown one characterization shows that

the ubiquitous kronecker product abstract the kronecker product has a rich and very pleasing algebra that supports a wide range of fast elegant and practical algorithms several trends in scientific computing suggest that this important matrix operation will have an increasingly greater role to play in the future first the application areas where kronecker products abound are all thriving these include signal processing image processing semidefinite programming and quantum computing second sparse factorizations and kronecker products are proving to be a very effective way to look at fast linear transforms researchers have taken the kronecker methodology as developed for the fast fourier transform and used it to build exciting alternatives third as computers get more powerful researchers are more willing to entertain problems of high dimension and this leads to kronecker products whenever low dimension techniques are tensored together

localization and coherence in nonintegrable systems we study the irreversible dynamics of nonlinear nonintegrable hamiltonian oscillator chains approaching their statistical asymptotic states in systems constrained by more than one conserved quantity the partitioning of the conserved quantities leads naturally to localized and coherent structures if the phase space is compact the final equilibrium state is governed by entropy maximization and the coherent structures are stable lumps in systems where the phase space is not compact the coherent structures can be collapsed represented in phase space by a heteroclinic connection of some unstable saddle to infinity

generalized linear array models with applications to multidimensional smoothing summary data with an array structure are common in statistics and the design or regression matrix for analysis of such data can often be written as a kronecker product factorial designs contingency tables and smoothing of data on multidimensional grids are three such general classes of data and models in such a setting we develop an arithmetic of arrays which allows us to define the expectation of the data array as a sequence of nested matrix operations on a coefficient array we show how this arithmetic leads to low storage high speed computation in the scoring algorithm of the generalized linear model we refer to a generalized linear array model and apply the methodology to the smoothing of multidimensional arrays we illustrate our procedure with the analysis of three data sets mortality data indexed by age at death and year of death spatially varying microarray background data and disease incidence data indexed by age at death year of death and month of death

periodic nonlinear schrodinger equation and invariant measures in this paper we continue some investigations on the periodic nlseiu u iu xx u u p 2 p 6 started in lrs we prove that the equation is globally wellposed for a set of data of full normalized gibbs measrue e beta h phi hd phi x h phi tfrac 1 2 int left phi right 2 tfrac 1 p int left phi right p after suitablel 2 truncation the set and the measure are invariant under the flow the proof of a similar result for the kdv and modified kdv equations is outlined the main ingredients used are some estimates from b1 on periodic nls and kdv type equations

homoclinic jumping in the perturbed nonlinear schrodinger equation we study the damped periodically forced focusing nls equation with even periodic boundary conditions we prove the existence of complicated solutions that repeatedly leave and come back to the vicinity of a quasi periodic plane wave with two time scales for pure forcing we prove the existence of a complicated self similar family of homoclinic bifurcations for mode independent damping we construct jumping transients for mode dependent damping we find generalized silnikov type solutions that connect a periodic plane wave to itself through repeated jumps we also study the breakdown of the unstable manifold of plane waves through repeated jumping our results give a direct explanation for the numerical observations of bishop et al 1999 john wiley sons inc

eigencurves for two parameter sturm liouville equations this paper concerns two parameter sturm liouville problems of the form p x y q x y lambda r x mu y quad a leqslant x leqslant b with self adjoint boundary conditions at a and b the set of lambda mu in bf r 2 for which there exists a nontrivial y satisfying the differential equation and the boundary conditions turns out to be a countable union of graphs of analytic functions our focus is on these graphs which are termed eigencurves in the literature although eigencurves have been used in a variety of ways for about a century they seem comparatively underdeveloped in their own right our plan is to give motivation for the topic elementary properties of eigencurves illustrations on a simple example first studied by richardson in 1918 and since then by several authors and some natural questions which may whet the reader s appetite some of these questions lead to new types of inverse sturm liouville problems

spectral analysis of coupled linear complementarity problems abstract this note deals with the so called cone constrained bivariate eigenvalue problem the equilibrium model under consideration is a system of linear complementarity problems p x ax by x p q y cx dy y q involving two closed convex cones and their corresponding duals we study the set of pairs r 2 for which this system has a nontrivial solution x y r n m we discuss also the link between the cone constrained version and the unconstrained one

geometric integrators for the nonlinear schrodinger equation abstract recently an interesting new class of pde integrators multisymplectic schemes has been introduced for solving systems possessing a certain multisymplectic structure some of the characteristic features of the method are its local nature independent of boundary conditions and an equal treatment of spatial and temporal variables the nonlinear schrodinger equation nls has a multisymplectic formulation and in this paper we discuss the performance of both symplectic and multisymplectic integrators for the nls in the numerical experiments the multisymplectic concatenated midpoint scheme a centered cell discretization is shown to preserve the local conservation laws extremely well over long times and to preserve global invariants such as the norm and momentum within roundoff on the other hand an integrable hamiltonian semi discretization of nls from ablowitz and ladik al possesses a full set of global conservation laws and a noncanonical symplectic structure we generalize the generating function technique to develop symplectic integrators of arbitrary order for a general class of noncanonical systems carrying a symplectic structure of the al type another approach examined in the paper is the introduction of transformations to reduce the al system to either 1 separable form or 2 canonical form and then apply standard schemes in the new coordinates all of the discretizations are tested numerically using initial data for spatially periodic multiphase solutions the performance of the schemes as well as interrelations among various geometric features are discussed

numerical chaos symplectic integrators and exponentially small splitting distances two discrete versions of duffing s equation are derived as reductions of discretizations of the nonlinear schrodinger nls equation inheriting the properties of the nsl discretizations one is shown to be integrable and using a mel nikov type analysis the other discretization is shown to be nonintegrable arguments are given why it is possible for the nonintegrable scheme to restore the homoclinic orbit at an exponentially fast rate which is faster than the order of the approximation as h 0 this suggests why there can be fundamental differences between symplectic and nonsymplectic discretizations of certain continuous hamiltonian systems

the kronecker product and stochastic automata networks this paper can be thought of as a companion paper to van loan s the ubiquitous kronecker product paper j comput appl math 123 2000 85 we collect and catalog the most useful properties of the kronecker product and present them in one place we prove several new properties that we discovered in our search for a stochastic automata network preconditioner we conclude by describing one application of the kronecker product omitted from van loan s list of applications namely stochastic automata networks

rogue waves dissipation and downshifting abstract we investigate the effects of dissipation on the development of rogue waves and downshifting by adding nonlinear and linear damping terms to the one dimensional dysthe equation significantly rogue waves do not develop after the downshifting becomes permanent thus in our experiments permanent downshifting serves as an indicator that damping is sufficient to prevent the further development of rogue waves using the inverse spectral theory of the nls equation simulations of the damped dysthe equation for sea states characterized by jonswap spectrum consistently show that rogue wave events are well predicted by proximity to homoclinic data as measured by the spectral splitting distance the cut off distance c u t o f f decreases as the strength of the damping increases indicating that for stronger damping the jonswap initial data must be closer to homoclinic data for rogue waves to occur

guarantees for the kronecker fast johnson lindenstrauss transform using a coherence and sampling argument abstract in the recent paper jin kolda ward arxiv 1909 04801 it is proved that the kronecker fast johnson lindenstrauss transform kfjlt is in fact a johnson lindenstrauss transform which had previously only been conjectured in this paper we provide an alternative proof of this for when the kfjlt is applied to kronecker vectors using a coherence and sampling argument our proof yields a different bound on the embedding dimension which can be combined with the bound in the paper by jin et al to get a better bound overall as a stepping stone to proving our result we also show that the kfjlt is a subspace embedding for matrices with columns that have kronecker product structure lastly we compare the kfjlt to four other sketch techniques in numerical experiments on both synthetic and real world data

computational chaos in the nonlinear schrodinger equation without homoclinic crossings abstract a hamiltonian difference scheme associated with the integrable nonlinear schrodinger equation with periodic boundary values is used as a prototype to demonstrate that perturbations due to truncation effects can result in a novel type of chaotic evolution the chaotic solution is characterized by random bifurcations across standing wave states into left and right going traveling waves in this class of problems where the solutions are not subject to even constraints the traditional mechanism of crossings of the unperturbed homoclinic orbits manifolds is not observed

a filter active set algorithm for ball sphere constrained optimization problem in this paper we propose a filter active set algorithm for the minimization problem over a product of multiple ball sphere constraints by making effective use of the special structure of the ball sphere constraints a new limited memory bfgs l bfgs scheme is presented the new l bfgs implementation takes advantage of the sparse structure of the jacobian of the constraints and generates curvature information of the minimization problem at each iteration only two or three reduced linear systems are required to solve for the search direction the filter technique combined with the backtracking line search strategy ensures the global convergence and the local superlinear convergence can also be established under mild conditions the algorithm is applied to two specific applications the nearest correlation matrix with factor structure and the maximal correlation problem our numerical experiments indicate that the proposed algorithm is competitive with some recently custom designed methods for each ind

recursive fast algorithm and the role of the tensor product the use of the tensor product for modeling and designing fft algorithms is addressed the benefit of the tensor product approach lies in the strong connection between certain tensor product constructs and important computer architectures the scope of the tensor product approach is generalized to include a much larger class of fast recursive algorithms this greatly enhances the versatility of the tensor product technique and brings many different algorithms to the level of understanding and flexibility enjoyed by the fft

nonlinear power like iteration by polar decomposition and its application to tensor approximation low rank tensor approximation is an important subject with a wide range of applications most prevailing techniques for computing the low rank approximation in the tucker format often first assemble relevant factors into matrices and then update by turns one factor matrix at a time in order to improve two factor matrices simultaneously a special system of nonlinear matrix equations over a certain product stiefel manifold must be resolved at every update the solution to the system consists of orbit varieties invariant under the orthogonal group action which thus imposes challenges on its analysis this paper proposes a scheme similar to the power method for subspace iterations except that the polar decomposition is used as the normalization process and that the iteration can be applied to both the orbits and the cross sections the notion of quotient manifold is employed to factor out the effect of orbital solutions the dynamics of the iteration is completely characterized an isometric isomorphism between the tangent spaces of two properly identified riemannian manifolds is established to lend a hand to the proof of convergence

chaotic and homoclinic behavior for numerical discretizations of the nonlinear schro dinger equation abstract certain conservative discretizations of the nls can produce irregular behavior we consider the diagonal discretization as a conservative perturbation of the integrable discretization and study the homoclinic crossings in its nonlinear spectrum we find that irregularity sets in when two homoclinic structures are present and in this case many and continual homoclinic crossings occur throughout the irregular time series we indicate a melnikov analysis to study the consequences of this homoclinic behavior

fully discrete energy stable high order finite difference methods for hyperbolic problems in deforming domains a time dependent coordinate transformation of a constant coefficient hyperbolic system of equations which results in a variable coefficient system of equations is considered by applying the energy method well posed boundary conditions for the continuous problem are derived summation by parts sbp operators for the space and time discretization together with a weak imposition of boundary and initial conditions using simultaneously approximation terms sats lead to a provable fully discrete energy stable conservative finite difference scheme we show how to construct a time dependent sat formulation that automatically imposes boundary conditions when and where they are required we also prove that a uniform flow field is preserved i e the numerical geometric conservation law ngcl holds automatically by using sbp sat in time and space the developed technique is illustrated by considering an application using the linearized euler equations the sound generated by moving boundaries numerical calculations corroborate the stability and accuracy of the new fully discrete approximations

numerical simulation of nonlinear schro dinger systems a new conservative scheme we present a new numerical scheme for nonlinear schrodinger type equations the scheme conserves the energy and charge of the systems and it is linearly implicit numerical experiments for several physical problems show that our scheme is stable accurate and efficient

discretizations integrable systems and computation discretizations and associated numerical computation of solutions of certain integrable systems such as the nonlinear schrodinger equation nls and the sine gordon sg equations with periodic boundary values can lead to instabilities chaotic and spurious results the chaos can be due to truncation errors or even roundoff errors and can be traced to the fact that these integrable systems are strongly unstable when the initial values are in the neighbourhood of homoclinic manifolds by using the associated nonlinear spectral transform of the nls equation and tracking the evolution of relevant eigenvalues one can observe and relate crossing of homoclinic manifolds to the temporal chaos in the waveforms when the initial data is even for general initial values even though there is no crossing of the unperturbed homoclinic manifolds the waveforms still exhibit chaotic phenomena which can be related to the evolution of the spectrum this paper reviews the current understanding of this intriguing phenomena and also compares the implementation of certain symplectic integrators and runge kutta algorithms for the nls and sg equations in regions of phase space proximate to the homoclinic manifolds

transition to chaos in discrete nonlinear schrodinger equation with long range interaction discrete nonlinear schrodinger dnls equation describes a chain of oscillators with nearest neighbor interactions and a specific nonlinear term we consider its modification with long range interaction through a potential proportional to 1 l1 with fractional 2 and l as a distance between oscillators this model is called dnls it exhibits competition between the nonlinearity and a level of correlation between interacting far distanced oscillators that is defined by the value of we consider transition to chaos in this system as a function of and nonlinearity it is shown that decreasing of with respect to nonlinearity stabilize the system connection of the model to the fractional generalization of the nls called fnls in the long wave approximation is also discussed and some of the results obtained for dnls can be correspondingly extended to the fnls

generic well posedness of nonconvex optimal control problems the tonelli existence theorem in the calculus of variations and its subsequent modifications were established for integrands f which satisfy convexity and growth conditions in our previous work a generic existence and uniqueness result with respect to variations of the integrand of the integral functional without the convexity condition was established for a class of optimal control problems satisfying the cesari growth condition in this paper we extend this generic existence and uniqueness result to a class of optimal control problems in which the right hand side of differential equations is also subject to variations

multiset canonical correlations using globality preserving projections with applications to feature extraction and recognition multiset features extracted from the same patterns always represent different characteristics of data thus it is very valuable to perform the extraction on multiple feature sets this paper addresses the issue of multiset correlation feature extraction mcfe in multiple feature representations a novel method is proposed to carry out the mcfe for classification called multiset canonical correlations using globality preserving projections mcc gps which can perform joint dimensionality reduction for high dimensional data mcc gp integrates correlational characteristics of feature pairs and global geometric information of data in the transformed low dimensional space this makes mcc gps have better discriminant ability than a previous method proposed by the authors called multiset integrated canonical correlation analysis micca which only considers correlations for recognition tasks furthermore mcc gp can subsume two popular feature extraction methods into its framework under some constraints this also provides a new insight for these two methods the proposed method is applied to pattern recognition and examined using the coil 100 and eth 80 object databases and ar cmu pie and yale face databases extensive experimental results show that mcc gp outperforms micca and multiset canonical correlation analysis in terms of classification accuracy and efficiency

numerical simulation of quasi periodic solutions of the sine gordon equation abstract analytic solutions of the sine gordon equation corresponding to periodic boundary conditions can be complicated especially if initial values are chosen in the vicinity of homoclinic orbits for such initial values it has been demonstrated that numerical solutions develop instabilities and may become chaotic because of the analytical complexities one cannot easily calculate the accuracy of the numerical solutions and therefore compare different numerical schemes in a straightforward manner in this note we evaluate numerical methods in terms of the nonlinear spectrum in particular it provides one with a means of comparing symplectic and nonsymplectic integrators for integrable infinite dimensional hamiltonian systems

on the preservation of phase space structure under multisymplectic discretization in this paper we explore the local and global properties of multisymplectic discretizations based on finite differences and fourier spectral approximations multisymplectic ms schemes are developed for two benchmark nonlinear wave equations the sine gordon and nonlinear schrodinger equations we examine the implications of preserving the ms structure under discretization on the numerical scheme s ability to preserve phase space structure as measured by the nonlinear spectrum of the governing equation we find that the benefits of multisymplectic integrators include improved resolution of the local conservation laws dynamical invariants and complicated phase space structures

generic well posedness of optimal control problems without convexity assumptions the tonelli existence theorem in the calculus of variations and its subsequent modifications were established for integrands f which satisfy convexity and growth conditions in a j zaslavski nonlinear anal to appear a generic existence and uniqueness result with respect to variations of the integrand of the integral functional without the convexity condition was established for a class of optimal control problems satisfying the cesari growth condition in this paper we extend the generic existence and uniqueness result in a j zaslavski nonlinear anal to appear to a class of optimal control problems in which constraint maps are also subject to variations the main result of the paper is obtained as a realization of a variational principle extending the variational principle introduced in a d ioffe and a j zaslavski siam j control optim 38 2000 pp 566 581

generic well posedness of variational problems without convexity assumptions the tonelli existence theorem in the calculus of variations and its subsequent modifications were established for integrands f which satisfy convexity and growth conditions in our previous work a generic well posedness result with respect to variations of the integrand of the integral functional without the convexity condition was established for a class of optimal control problems satisfying the cesari growth condition in this paper we extend this generic well posedness result to two classes of variational problems in which the values at the end points are also subject to variations the main results of the paper are obtained as realizations of a general variational principle

vectorial phase retrieval of 1 d signals reconstruction of signals from measurements of their spectral intensities also known as the phase retrieval problem is of fundamental importance in many scientific fields in this paper we present a novel framework denoted as vectorial phase retrieval for reconstruction of pairs of signals from spectral intensity measurements of the two signals and of their interference we show that this new framework can alleviate some of the theoretical and computational challenges associated with classical phase retrieval from a single signal first we prove that for compactly supported signals in the absence of measurement noise this new setup admits a unique solution next we present a statistical analysis of vectorial phase retrieval and derive a computationally efficient algorithm to solve it finally we illustrate via simulations that our algorithm can accurately reconstruct signals even at considerable noise levels

penalized least squares estimation of volterra filters and higher order statistics volterra filters vfs and higher order statistics hos are important tools for nonlinear analysis processing and modeling despite their highly desirable properties the transfer of vfs and hos to real world signal processing problems has been hindered by the requirement of very large data records needed to obtain reliable estimates the identification of vfs and the estimation of hos both fall into the category of ill posed estimation problems we develop penalized least squares pls estimation methods for vfs and hos it is shown that pls is a very effective way to incorporate prior information of the problem at hand without directly constraining the estimation procedure hence pls produces much more reliable estimates the main contributions of this paper are the development of appropriate penalizing functionals and cross validation procedures for pls based vf identification and hos estimation

improved parallel qr method for large least squares problems involving kronecker products abstract a new algorithm is presented for the efficient solution of large least squares problems in which the coefficient matrix of the linear system is a kronecker product of two smaller dimension matrices the solution algorithm is based on qr factorizations of the smaller dimension matrices near perfect load balancing is achieved by exploiting a commutativity property of the kronecker product and communication requirements are minimized by employing a binary exchange algorithm for matrix transposition the parallel algorithm is presented and timing results are shown from test runs on an intel i 860 computer

tensor product basis approximations for volterra filters the paper studies approximations for a class of nonlinear filters known as volterra filters although the volterra filter provides a relatively simple and general representation for nonlinear filtering it is often highly overparameterized due to the large number of parameters the utility of the volterra filter is limited the overparameterization problem is addressed in the paper using a tensor product basis approximation tpba in many cases a volterra filter may be well approximated using the tpba with far fewer parameters hence the tpba offers considerable advantages over the original volterra filter in terms of both implementation and estimation complexity furthermore the tpba provides useful insight into the filter response the paper studies the crucial issue of choosing the approximation basis several methods for designing an appropriate approximation basis and error bounds on the resulting mean square output approximation error are derived certain methods are known to be nearly optimal

generic flows generated by continuous vector fields in banach spaces soit f un champ vectoriel defini sur un sous ensemble ouvert d un espace de banach reel on considere le probleme de cauchy x f t x x 0 y 7b e un sous ensemble non vide de points initiaux on etudie les f qui engendrent un flot pour y 7b e

restoring images degraded by spatially variant blur restoration of images that have been blurred by the effects of a gaussian blurring function is an ill posed but well studied problem any blur that is spatially invariant can be expressed as a convolution kernel in an integral equation fast and effective algorithms then exist for determining the original image by preconditioned iterative methods if the blurring function is spatially variant however then the problem is more difficult in this work we develop fast algorithms for forming the convolution and for recovering the original image when the convolution functions are spatially variant but have a small domain of support this assumption leads to a discrete problem involving a banded matrix we devise an effective preconditioner and prove that the preconditioned matrix differs from the identity by a matrix of small rank plus a matrix of small norm numerical examples are given related to the hubble space telescope hst wide field planetary camera the algorithms that we develop are applicable to other ill posed integral equations as well

on perturbation bounds of kronecker product linear systems and their level 2 condition numbers the classical perturbation theory of linear systems ax b is extended to kronecker product linear systems email protected b x d upper bounds are derived for the normwise and componentwise condition number the nearness to singularity and the sensitivity of the condition numbers are analyzed

mel nikov analysis of numerically induced chaos in the nonlinear schro dinger equation abstract certain hamiltonian discretizations of the periodic focusing nonlinear schrodinger equation nls have been shown to be responsible for the generation of numerical instabilities and chaos in this paper we undertake a dynamical systems type of approach to modeling the observed irregular behavior of a conservative discretization of the nls using heuristic mel nikov methods the existence of a pair of isolated homoclinic orbits is indicated for the perturbed system the structure of the persistent homoclinic orbits that are predicted by the mel nikov theory possesses the same features as the wave form observed numerically in the perturbed system after the onset of chaotic behavior and appears to be the main structurally stable feature of this type of chaos the mel nikov analysis implemented in the pde context appears to provide relevant qualitative information about the behavior of the pde in agreement with the numerical experiments in a neighborhood of the persistent homoclinic orbits the existence of a horseshoe is investigated and related with the onset of chaos in the numerical study

variational principles and well posedness in optimization and calculus of variations the concluding result of the paper states that variational problems are generically solvable and even well posed in a strong sense without the convexity and growth conditions always present in individual existence theorems this and some other generic well posedness theorems are obtained as realizations of a general variational principle extending the variational principle of deville godefroy zizler

global optimality of the successive maxbet algorithm the maxbet method is an alternative to the method of generalized canonical correlation analysis and of procrustes analysis contrary to these methods it does not maximize the inner products covariances between linear composites but also takes their sums of squares variances into account it is well known that the maxbet algorithm which has been proven to converge monotonically may converge to local maxima the present paper discusses an eigenvalue criterion which is sufficient but not necessary for global optimality however in two special cases the eigenvalue criterion is shown to be necessary and sufficient for global optimality the first case is when there are only two data sets involved the second case is when the inner products between all variables involved are positive regardless of the number of data sets

discrete singular convolution for the sine gordon equation abstract this paper explores the utility of a discrete singular convolution dsc algorithm for the integration of the sine gordon equation the initial values are chosen close to a homoclinic manifold for which previous methods have encountered significant numerical difficulties such as numerically induced spatial and temporal chaos a number of new initial values are considered including a case where the initial value is exactly on the homoclinic orbit the present algorithm performs extremely well in terms of accuracy efficiency simplicity stability and reliability

the spectra of multiplicative attribute graphs abstract a multiplicative attribute graph is a random graph in which vertices are represented by random words of length t in a finite alphabet and the probability of adjacency is a symmetric function t t 0 1 these graphs are a generalization of stochastic kronecker graphs and both classes have been shown to exhibit several useful real world properties we establish asymptotic bounds on the spectra of the adjacency matrix and the normalized laplacian matrix for these two families of graphs under certain mild conditions as an application we examine various properties of the stochastic kronecker graph and the multiplicative attribute graph including the diameter clustering coefficient chromatic number and bounds on low congestion routing

chaotic transport and integrable instabilities in a near integrable many particle hamiltonian lattice abstract this paper is a report of detailed numerical experiments on a large dimensional hamiltonian lattice which arises as a nearly integrable discretization of the periodic sine gordon equation through four different one parameter families of initial data we focus on the interplay between numerically observable near integrable behavior quantified by modern statistical diagnostics and the integrable pde solutions and phase space geometry associated with the initial data transitions are observed within each family from quasiperiodic behavior to chaotic motion which is sometimes accompanied by transport toward equipartition these transitions in the near integrable system are correlated with integrable features of the initial data in particular transitions from neutrally stable to exponentially unstable sine gordon solutions and associated homoclinic orbits geometrically these correspond to transitions from neutrally stable tori to whiskered tori our results indicate that 1 positive lyapunov exponents our messure of chaos in near integrable orbits coincide one to one with integrable pde instabilities and homoclinic orbits 2 the statistical transition to equipartition of energy on a long fixed timescale occurs only if but not necessarily if there are integrable instabilities and associated whiskered tori the rate of transport appears to depend on the amplitude of the homoclinic degrees of freedom 3 isolated cases of discreteness induced instabilities and action transport are observed these return to quasiperiodic flow at finer discretizations 4 a simple measure such as energy of the initial data is not an accurate predictor of chaotic or recurrent behavior one family of initial data exhibits as the energy increases alternating windows of quasiperiodic and chaotic orbits

kronecker products unitary matrices and signal processing applications discrete unitary transforms are extensively used in many signal processing applications and in the development of fast algorithms kronecker products have proved quite useful in this semitutorial paper we briefly review properties of kronecker products and direct sums of matrices which provide a compact notation in treating patterned matrices a generalized matrix product which inherits some useful algebraic properties from the standard kronecker product and allows a large class of discrete unitary transforms to be generated from a single recursion formula is then introduced the notation is intimately related to sparse matrix factorizations and examples are included illustrating the utility of the new notation in signal processing applications finally some novel characteristics of hadamard transforms and polyadic permutations are derived in the framework of kronecker products

the matrix equation xa bx r and its applications abstract we study the well known sylvester equation xa bx r in the case when a and b are given and r is known up to its first n 1 rows we prove new results on the existence and uniqueness of x our results essentially state that in case a is a nonderogatory matrix there always exists a solution to this equation a solution is uniquely determined by its first row x 1 and there is an interesting relationship between x 1 and the rows of r we also give a complete characterization of the nonsingularity of x in this case as applications of our results we develop direct methods for constructing symmetrizers and commuting matrices computing the characteristic polynomial of a matrix and finding the numbers of common eigenvalues between a and b some well known important results on symmetrizers bezoutians and inertia are recovered as special cases

kronecker products and shuffle algebra relates three classical concepts viz mixed radix number system kronecker product of matrices and perfect shuffle it presents an algebra which describes the hardware organization of the computation of a product mv where m is a matrix in kronecker product form and v is a vector the algebraic formalism describes both the blockwise structure of the computation and the various possible connection patterns

high order split step exponential methods for solving coupled nonlinear schrodinger equations numerical integration schemes for coupled time dependent nonlinear schrodinger equations are examined using exponential splitting step methods exponentiation of the nonlinear potential term is reduced to the exponential of a kinetic energy term which can be calculated by fast fourier transforms high order iteration schemes involving a minimum number of product operators are shown to yield highly accurate amplitude and phase these new splitting methods are shown to be highly efficient both with respect to accuracy and integration time

regularized generalized canonical correlation analysis regularized generalized canonical correlation analysis rgcca is a generalization of regularized canonical correlation analysis to three or more sets of variables it constitutes a general framework for many multi block data analysis methods it combines the power of multi block data analysis methods maximization of well identified criteria and the flexibility of pls path modeling the researcher decides which blocks are connected and which are not searching for a fixed point of the stationary equations related to rgcca a new monotonically convergent algorithm very similar to the pls algorithm proposed by herman wold is obtained finally a practical example is discussed

graph regularized multiset canonical correlations with applications to joint feature extraction abstract multiset canonical correlation analysis mcca is a powerful technique for analyzing linear correlations among multiple representation data however it usually fails to discover the intrinsic geometrical and discriminating structure of multiple data spaces in real world applications in this paper we thus propose a novel algorithm called graph regularized multiset canonical correlations grmccs which explicitly considers both discriminative and intrinsic geometrical structure in multiple representation data grmcc not only maximizes between set cumulative correlations but also minimizes local intraclass scatter and simultaneously maximizes local interclass separability by using the nearest neighbor graphs on within set data thus it can leverage the power of both mcca and discriminative graph laplacian regularization extensive experimental results on the ar cmu pie yale b at t and eth 80 datasets show that grmcc has more discriminating power and can provide encouraging recognition results in contrast with the state of the art algorithms

information transmission using the nonlinear fourier transform part ii numerical methods in this paper numerical methods are suggested to compute the discrete and the continuous spectrum of a signal with respect to the zakharov shabat system a lax operator underlying numerous integrable communication channels including the nonlinear schrodinger channel modeling pulse propagation in optical fibers these methods are subsequently tested and their ability to estimate the spectrum are compared against each other these methods are used to compute the spectrum of various signals commonly used in the optical fiber communications it is found that the layer peeling and the spectral methods are suitable schemes to estimate the nonlinear spectra with good accuracy to illustrate the structure of the spectrum the locus of the eigenvalues is determined under amplitude and phase modulation in a number of examples it is observed that in some cases as signal parameters vary eigenvalues collide and change their course of motion the real axis is typically the place from which new eigenvalues originate or are absorbed into after traveling a trajectory in the complex plane

fast fourier transforms for nonequispaced data a group of algorithms is presented generalizing the fast fourier transform to the case of noninteger frequencies and nonequispaced nodes on the interval pi pi the schemes of this paper are based on a combination of certain analytical considerations with the classical fast fourier transform and generalize both the forward and backward ffts each of the algorithms requires o n cdot log n n cdot log 1 varepsilon arithmetic operations where varepsilon is the precision of computations and n is the number of nodes the efficiency of the approach is illustrated by several numerical examples

low rank estimation of higher order statistics low rank estimators for higher order statistics are considered in this paper the bias variance tradeoff is analyzed for low rank estimators of higher order statistics using a tensor product formulation for the moments and cumulants in general the low rank estimators have a larger bias and smaller variance than the corresponding full rank estimator and the mean squared error can be significantly smaller this makes the low rank estimators extremely useful for signal processing algorithms based on sample estimates of the higher order statistics the low rank estimators also offer considerable reductions in the computational complexity of such algorithms the design of subspaces to optimize the tradeoffs between bias variance and computation is discussed and a noisy input noisy output system identification problem is used to illustrate the results

a second order exponential time differencing scheme for non linear reaction diffusion systems with dimensional splitting abstract a second order l stable exponential time differencing etd method is developed by combining an etd scheme with approximating the matrix exponentials by rational functions having real distinct poles rdp together with a dimensional splitting integrating factor technique a variety of non linear reaction diffusion equations in two and three dimensions with either dirichlet neumann or periodic boundary conditions are solved with this scheme and shown to outperform a variety of other second order implicit explicit schemes an additional performance boost is gained through further use of basic parallelization techniques

on condition numbers for least squares with quadric inequality constraint abstract in this paper we will study normwise mixed and componentwise condition numbers for the linear mapping of the solution for general least squares with quadric inequality constraint glsqi and its standard form lsqi we will introduce the mappings from the data space to the interested data space and the frechet derivative of the introduced mapping can deduced through matrix differential techniques based on condition number theory we derive the explicit expressions of normwise mixed and componentwise condition numbers for the linear function of the solution for glsqi and lsqi also easier computable upper bounds for mixed and componentwise condition numbers are given numerical example shows that the mixed and componentwise condition numbers can tell us the true conditioning of the problem when its data is sparse or badly scaled compared with normwise condition numbers the mixed and componentwise condition number can give sharp perturbation bounds

inverse eigenvalue problems a collection of inverse eigenvalue problems are identified and classified according to their characteristics current developments in both the theoretic and the algorithmic aspects are summarized and reviewed in this paper this exposition also reveals many open questions that deserve further study an extensive bibliography of pertinent literature is attached

time and lag recursive computation of cumulants from a state space model time and lag recursive algorithms for the computation of the cumulants of the state vector and the output process of a multiple input multiple output time varying state space model are derived using a kronecker product representation for the cumulants of vector processes the noise processes are not assumed to be stationary symmetry relations for the cumulants of vector processes are discussed computational aspects are examined in detail it is conjectured that the recursive equations derived will be useful in developing state estimators and even optimal controllers based on higher order statistics

sampling of stochastic operators we develop sampling methodology aimed at identifying stochastic operators that satisfy a support size restriction on the autocorrelation of the operators stochastic spreading function the data that we use to reconstruct the operator or in some cases only the autocorrelation of the spreading function is based on the response of the unknown operator to a known deterministic test signal

affinity learning with diffusion on tensor product graph in many applications we are given a finite set of data points sampled from a data manifold and represented as a graph with edge weights determined by pairwise similarities of the samples often the pairwise similarities which are also called affinities are unreliable due to noise or due to intrinsic difficulties in estimating similarity values of the samples as observed in several recent approaches more reliable similarities can be obtained if the original similarities are diffused in the context of other data points where the context of each point is a set of points most similar to it compared to the existing methods our approach differs in two main aspects first instead of diffusing the similarity information on the original graph we propose to utilize the tensor product graph tpg obtained by the tensor product of the original graph with itself since tpg takes into account higher order information it is not a surprise that we obtain more reliable similarities however it comes at the price of higher order computational complexity and storage requirement the key contribution of the proposed approach is that the information propagation on tpg can be computed with the same computational complexity and the same amount of storage as the propagation on the original graph we prove that a graph diffusion process on tpg is equivalent to a novel iterative algorithm on the original graph which is guaranteed to converge after its convergence we obtain new edge weights that can be interpreted as new learned affinities we stress that the affinities are learned in an unsupervised setting we illustrate the benefits of the proposed approach for data manifolds composed of shapes images and image patches on two very different tasks of image retrieval and image segmentation with learned affinities we achieve the bull s eye retrieval score of 99 99 percent on the mpeg 7 shape dataset which is much higher than the state of the art algorithms when the data points are image patches the ncut with the learned affinities not only significantly outperforms the ncut with the original affinities but it also outperforms state of the art image segmentation methods

picture of all solutions of successive 2 block maxbet problems the maxbet method is a generalized principal components analysis of a data set where the group structure of the variables is taken into account similarly 3 block 12 13 partial maxdiff method is a generalization of covariance analysis where only the covariances between blocks 1 2 and 1 3 are taken into account the aim of this paper is to give the global maximum for the 2 block maxbet and 3 block 12 13 partial maxdiff problems by picking the best solution from the complete solution set for the multivariate eigenvalue problem involved to do this we generalize the characteristic polynomial of a matrix to a system of two characteristic polynomials and provide the complete solution set of the latter via sylvester resultants examples are provided

on a multivariate eigenvalue problem part i algebraic theory and a power method multivariate eigenvalue problems for symmetric and positive definite matrices arise from multivariate statistics theory where coefficients are to be determined so that the resulting linear combinations of sets of random variables are maximally correlated by using the method of lagrange multipliers such an optimization problem can be reduced to the multivariate eigenvalue problem for over 30 years an iterative method proposed by horst psychometrika 26 1961 pp 129 149j has been used for solving the multivariate eigenvalue problem yet the theory of convergence has never been complete the number of solutions to the multivariate eigenvalue problem also remains unknown this paper contains two new results by using the degree theory a closed form on the cardinality of solutions for the multivariate eigenvalue problem is first proved a convergence property of horst s method by forming it as a generalization of the so called power method is then proved the discussion leads to new formulations of nume

structured data sparse approximation to high order tensors arising from the deterministic boltzmann equation we develop efficient data sparse representations to a class of high order tensors via a block many fold kronecker product decomposition such a decomposition is based on low separation rank approximations of the corresponding multivariate generating function we combine the sinc interpolation and a quadrature based approximation with hierarchically organised block tensor product formats different matrix and tensor operations in the generalised kronecker tensor product format including the hadamard type product can be implemented with the low cost an application to the collision integral from the deterministic boltzmann equation leads to an asymptotical cost o n 4 log n o n 5 log n in the one dimensional problem size n depending on the model kernel function which noticeably improves the complexity o n 6 log n of the full matrix representation

structured mixed and componentwise condition numbers of some structured matrices structured matrices such as cauchy vandermonde toeplitz hankel and circulant matrices are considered in this paper we apply a kronecker product based technique to deduce the structured mixed and componentwise condition numbers for the matrix inversion and for the corresponding linear systems