on the solution of stein s equation and fisher s information matrix of an armax process the main goal of this paper consists in expressing the solution of a stein equation in terms of the fisher information matrix fim of a scalar armax process a condition for expressing the fim in terms of a solution to a stein equation is also set forth such interconnections can be derived when a companion matrix with eigenvalues equal to the roots of an appropriate polynomial associated with the armax process is inserted in the stein equation the case of algebraic multiplicity greater than or equal to one is studied the fim and the corresponding solution to stein s equation are presented as solutions to systems of linear equations the interconnections are obtained by using the common particular solution of these systems the kernels of the structured coefficient matrices are described as well as some right inverses this enables us to find a solution to the newly obtained linear system of equations

conformal geometry and its applications on 3d shape matching recognition and stitching three dimensional shape matching is a fundamental issue in computer vision with many applications such as shape registration 3d object recognition and classification however shape matching with noise occlusion and clutter is a challenging problem in this paper we analyze a family of quasi conformal maps including harmonic maps conformal maps and least squares conformal maps with regards to 3d shape matching as a result we propose a novel and computationally efficient shape matching framework by using least squares conformal maps according to conformal geometry theory each 3d surface with disk topology can be mapped to a 2d domain through a global optimization and the resulting map is a diffeomorphism i e one to one and onto this allows us to simplify the 3d shape matching problem to a 2d image matching problem by comparing the resulting 2d parametric maps which are stable insensitive to resolution changes and robust to occlusion and noise therefore highly accurate and efficient 3d shape matching algorithms can be achieved by using the above three parametric maps finally the robustness of least squares conformal maps is evaluated and analyzed comprehensively in 3d shape matching with occlusion noise and resolution variation in order to further demonstrate the performance of our proposed method we also conduct a series of experiments on two computer vision applications i e 3d face recognition and 3d nonrigid surface alignment and stitching

on stein s equation vandermonde matrices and fisher s information matrix of time series processes part i the autoregressive moving average process abstract this paper introduces several forms of relationships between fisher s information matrix of an autoregressive moving average or arma process and the solution of a corresponding stein equation fisher s information matrix consists of blocks associated with the autoregressive and moving average parameters an interconnection with a solution of stein s equation is set forth for the block case as well as for fisher s information matrix as a global matrix involving all parameter blocks both cases have their importance for the interpretation of the estimated parameters the cases of distinct and multiple eigenvalues are addressed the obtained links involve equations with left and right inverses these can be expressed in terms of the inverse of appropriate vandermonde matrices a condition is set forth for establishing an equality between fisher s information matrix and a solution to stein s equation two examples are presented for illustrating some of the results obtained the global and off diagonal block case with distinct and multiple roots respectively are considered

existence of minimal nonsquare j symmetric factorizations for self adjoint rational matrix functions in this paper we show that any rational matrix function having hermitian values on the imaginary axis and with constant signature and constant pole signature admits a minimal symmetric factorization with possibly nonsquare factors our proof is based on a construction which shows that any such function can be extended preserving its mcmillan degree to a function that admits j symmetric factorization with square factors also we consider other properties of the factors in j symmetric factorizations particular attention is given to the study of the common invariant zero structure of these factors

the gaussian ceo problem for scalar sources with arbitrary memory in this paper we consider the achievable sum rate distortion tradeoff for the gaussian central estimation officer ceo problem with a scalar source having arbitrary memory we describe how the arbitrary memory problem can be fully characterized by using known results for the vector ceo problem and then we formulate the variational problem of minimizing the sum rate subject to a distortion constraint to solve the problem we extend the conventional lagrange method and show that if the solution exists it should consist of a zero part and a non zero part where the non zero part is determined by solving a set of euler equations by calculating the second variation of the min sum rate problem a sufficient condition is also found that can be used to determine if the necessary solution results in the minimal sum rate the special case of two terminals is examined in detail and it is shown that an analytical solution is possible in this case analysis and discussion with examples are provided to illustrate the theoretical results the general solution obtained in this paper is shown to be compatible with the previous results for cases such as the problem of rate evaluation for sources without memory

statistical resilience of random populations to random perturbations we consider populations represented by random collections of real valued points and explore their statistical resilience to random perturbations seeking populations whose statistics remain qualitatively unchanged by the action of arbitrary random perturbations of a certain type studying a general physical perturbation scheme we obtain an explicit characterization of statistically resilient populations show that these objects are fractal and comprehensively analyze their topological and statistical structures an application of statistical resilience attained is an alternative explanation of the ubiquity of power law statistics

on a class of rational matrix differential equations arising in stochastic control we prove a monotonicity and a comparison theorem for the solutions of a rational matrix differential equation appearing in stochastic control and derive existence and convergence results for the solutions of this differential equation moreover in the time invariant case we present conditions ensuring that the corresponding algebraic matrix equation has a stabilizing solution

the theory of v bezoutians and its applications abstract some propositions of the theory of v bezoutians are presented as an application the following problem is considered to construct an appropriate pontryagin space in which a given matrix boundary value problem containing the eigenvalue parameter in the boundary conditions is self adjoint the solution of this problem is given in three special cases with the help of v bezoutians of matrix polynomials entering the boundary conditions

constraints on deformable models recovering 3d shape and nongrid motion abstract inferring the 3d structures of nonrigidly moving objects from images is a difficult yet basic problem in computational vision our approach makes use of dynamic elastically deformable object models that offer the geometric flexibility to satisfy a diversity of real world visual constraints we specialize these models to include intrinsic forces inducing a preference for axisymmetry image based constraints are applied as extrinsic forces that mold the symmetry seeking model into shapes consistent with image data we describe an extrinsic force that applies constraints derived from profiles of monocularly viewed objects we generalize this constraint force to incorporate profile information from multiple views and use it to exploit binocular image data for time varying images the force becomes dynamic and the model is able to infer not only depth but nonrigid motion as well we demonstrate the recovery of 3d shape and nonrigid motion from natural imagery

approximation design of optimal controllers for nonlinear systems with sinusoidal disturbances abstract this paper considers an infinite time optimal damping control problem for a class of nonlinear systems with sinusoidal disturbances a successive approximation approach saa is applied to design feedforward and feedback optimal controllers by using the saa the original optimal control problem is transformed into a sequence of nonhomogeneous linear two point boundary value tpbv problems the existence and uniqueness of the optimal control law are proved the optimal control law is derived from a riccati equation matrix equations and an adjoint vector sequence which consists of accurate linear feedforward and feedback terms and a nonlinear compensation term and the nonlinear compensation term is the limit of the adjoint vector sequence by using a finite term of the adjoint vector sequence we can get an approximate optimal control law a numerical example shows that the algorithm is effective and robust with respect to sinusoidal disturbances

on the bezoutian for polynomial matrices abstract the bezoutian b of two polynomial matrices can be described as a solution of a linear matrix equation this fact yields a new proof of the barnett factorization of b

optimal tracking control with zero steady state error for time delay systems with sinusoidal disturbances the problem of optimal tracking control with zero steady state error for linear time delay systems with sinusoidal disturbances is considered based on the internal model principle a disturbance compensator is constructed such that the system with external sinusoidal disturbances is transformed into an augmented system without disturbances by introducing a sensitivity parameter and expanding power series around it the optimal tracking control problem can be simplified into the problem of solving an infinite sum of linear optimal control series without time delay and disturbance the obtained optimal tracking control law with zero steady state error consists of accurate linear state feedback terms and a time delay compensating term which is an infinite sum of an adjoint vector series the accurate linear terms can be obtained by solving a riccati matrix equation and a sylvester equation respectively the compensation term can be approximately obtained through a recursive algorithm a numerical simulation shows that the algorithm is effective and easily implemented and the designed tracking controller is robust with respect to the sinusoidal disturbances

on the eigenvalue estimation for solution to lyapunov equation abstract this paper deals with the problems of eigenvalue estimation for the solution to the perturbed matrix lyapunov equation we obtain some eigenvalue inequalities on condition that x is a positive semidefinite solution to the equation atxa x q which can be used in control theory and linear system stability

the maximal process of nonlinear shot noise abstract in the nonlinear shot noise system model shots statistics are governed by general poisson processes and shots decay dynamics are governed by general nonlinear differential equations in this research we consider a nonlinear shot noise system and explore the process tracking along time the system s maximal shot magnitude this maximal process is a stationary markov process following a decay surge evolution it is highly robust and it is capable of displaying both a wide spectrum of statistical behaviors and a rich variety of random decay surge sample path trajectories a comprehensive analysis of the maximal process is conducted including its markovian structure its decay surge structure and its correlation structure all results are obtained analytically and in closed form

on shape of plane elastic curves we study shapes of planar arcs and closed contours modeled on elastic curves obtained by bending stretching or compressing line segments non uniformly along their extensions shapes are represented as elements of a quotient space of curves obtained by identifying those that differ by shape preserving transformations the elastic properties of the curves are encoded in riemannian metrics on these spaces geodesics in shape spaces are used to quantify shape divergence and to develop morphing techniques the shape spaces and metrics constructed are novel and offer an environment for the study of shape statistics elasticity leads to shape correspondences and deformations that are more natural and intuitive than those obtained in several existing models applications of shape geodesics to the definition and calculation of mean shapes and to the development of shape clustering techniques are also investigated

nasal similarity measure of 3d faces based on curve shape space abstract we propose a novel method for measuring the nasal similarity among 3d faces firstly we construct a representation for the nose shape which is composed of a set of geodesic curves each crosses the bridge of the nose next using these geodesic curves we formulate a similarity measure to compare among noses in the curve shape space under the riemannian framework the shape space is a quotient space for which the scaling translation and rotation are removed since the nose similarity measure is based on the shape comparison the proposed method has the following advantages 1 the similarity measure is robust to facial expressions since the nose is not affected by facial expressions 2 the geometric features of the nose shape match well with the human perception 3 the similarity measure is independent of the mesh grid because the chosen nose curves are not sensitive to the triangular mesh model we construct a nasal hierarchical structure for noses organization which is based on nose similarity measure results in our experiments we evaluate the performance of the proposed method and compare it with competing methods on three public face databases namely frgc2 0 texas3d and bosphorusdb the results show superiority of the proposed method in terms of both the speed and the accuracy when the nasal measurements are processed in the nasal hierarchical structure and the nasal samples with low sampling rate 5 25 of original point cloud

computational methods for linear matrix equations given the square matrices a b d e and the matrix c of conforming dimensions we consider the linear matrix equation a mathbf x e d mathbf x b c in the unknown matrix mathbf x our aim is to provide an overview of the major algorithmic developments that have taken place over the past few decades in the numerical solution of this and related problems which are producing reliable numerical tools in the formulation and solution of advanced mathematical models in engineering and scientific computing

on a class of entire matrix function equations abstract this paper treats a class of linear entire matrix function equations which appear naturally in the study of krei n orthogonal matrix functions necessary and sufficient conditions for solvability are given in terms of root functions of the coefficients

how to count zeroes of polynomials on quadrature domains using the bezout matrix abstract classically the bezout matrix or simply bezoutian of two polynomials is used to locate the roots of the polynomial and in particular test for stability in this paper we develop the theory of bezoutians on real riemann surfaces of dividing type the main result connects the signature of the bezoutian of two real meromorphic functions to the topological data of their quotient which can be seen as the generalization of the classical cauchy index as an application we propose a method to count the number of zeroes of a polynomial in a quadrature domain using the inertia of the bezoutian we provide examples of our method in the case of simply connected quadrature domains

asymmetric inclusion process we introduce and explore the asymmetric inclusion process asip an exactly solvable bosonic counterpart of the fermionic asymmetric exclusion process asep in both processes random events cause particles to propagate unidirectionally along a one dimensional lattice of n sites in the asep particles are subject to exclusion interactions whereas in the asip particles are subject to inclusion interactions that coalesce them into inseparable clusters we study the dynamics of the asip derive evolution equations for the mean and probability generating function pgf of the sites occupancy vector obtain explicit results for the above mean at steady state and describe an iterative scheme for the computation of the pgf at steady state we further obtain explicit results for the load distribution in steady state with the load being the total number of particles present in all lattice sites finally we address the problem of load optimization and solve it under various criteria the asip model establishes bridges between statistical physics and queueing theory as it represents a tandem array of queueing systems with unlimited batch service and a tandem array of growth collapse processes

spectrum separation and inertia for operator polynomials abstract problems on spectrum separation and inertia of operator polynomials are solved using the concept of bezoutian of operator polynomials the main results extend those previously known for matrix polynomials applications of the main results include differential equations with constant coefficients and some problems concerning orthogonal operator polynomials

optimal output tracking control for nonlinear systems via successive approximation approach abstract this paper deals with the optimal output tracking control ootc problem of a class of nonlinear systems whose reference input to be tracked is produced by a generalized linear exosystem to solve the nonlinear ootc problem the nonlinear two point boundary value tpbv problem derived from the necessary conditions of the ootc problem is transformed into two decoupled iterative sequences of linear differential equations the ootc law obtained consists of accurate feedforward and feedback terms and a nonlinear compensation term the former can be found by solving a sylvester equation and a riccati equation and the latter can be approximated using a successive approximation approach saa a reduced order reference input observer is constructed to make the feedforward control law physically realizable a simulation example is employed to illustrate the validity of the results

on the active periods of nonlinear shot noise abstract we study the temporal on and off structure of the recently introduced nonlinear shot noise model i eliazar j klafter pnas 102 2005 13779 i eliazar j klafter nonlinear shot noise levy noah joseph physica a 2006 to appear in which i shots of random magnitudes bombard a system stochastically in time ii the magnitudes of incoming shots decay to zero nonlinearly and iii the overall effect of the shots on the system is additive when the shot inflow and shot decay satisfy certain conditions the resulting shot noise alternates randomly between active periods in which the noise is on and silent periods in which noise is off the statistical properties of the active and silent periods are analyzed and the resulting on and off shot noise structure is explored explicit formulae for means variances laplace transforms and other statistics are derived in closed form based on this analysis shot noise systems are categorized into three classes transient null recurrent and recurrent and these classes are characterized both analytically and probabilistically within the null recurrent class systems whose active periods are governed by heavy tailed probability distributions are further characterized

on the laplace beltrami operator and brain surface flattening in this paper using certain conformal mappings from uniformization theory the authors give an explicit method for flattening the brain surface in a way which preserves angles from a triangulated surface representation of the cortex the authors indicate how the procedure may be implemented using finite elements further they show how the geometry of the brain surface may be studied using this approach

piecewise linear manifolds einstein metrics and ricci flows this article provides an attempt to extend concepts from the theory of riemannian manifolds to piecewise linear p l spaces in particular we propose an analogue of the ricci tensor which we give the name of an einstein vector field on a given set of p l spaces we define and discuss normalized einstein flows p l einstein metrics are defined and examples are provided criteria for flows to approach einstein metrics are formulated second variations of the total scalar curvature at a specific einstein space are calculated

on a class of matrix polynomial equations abstract this paper deals with a class of matrix polynomial equations which appears in a natural way in the study of szeg krei n orthogonal matrix polynomials necessary and sufficient conditions for solvability are given in terms of left and right root functions of the coefficients the main result is a discrete version of the main theorem in 3 which concerns entire matrix functions

teichmuller shape descriptor and its application to alzheimer s disease study we propose a novel method to apply teichmuller space theory to study the signature of a family of nonintersecting closed 3d curves on a general genus zero closed surface our algorithm provides an efficient method to encode both global surface and local contour shape information the signature teichmuller shape descriptor is computed by surface ricci flow method which is equivalent to solving an elliptic partial differential equation on surfaces and is numerically stable we propose to apply the new signature to analyze abnormalities in brain cortical morphometry experimental results with 3d mri data from alzheimer s disease neuroimaging initiative adni dataset 152 healthy control subjects versus 169 alzheimer s disease ad patients demonstrate the effectiveness of our method and illustrate its potential as a novel surface based cortical morphometry measurement in ad research

shot noise displaying simultaneously the noah and joseph effects shot noise processes displaying the noah effect are of infinite variance hence the measurement of the correlation structure of such processes indicating whether the joseph effect is displayed as well cannot be conducted via their autocovariance functions which are undefined to circumvent this problem a poisson based analysis applicable to shot noise processes with infinite variance as well as to shot noise processes with divergent noise levels is developed a poissonian autocovariance function which characterizes the process distribution of general shot noise processes is introduced in particular this function governs and quantifies the processes stationary structure amplitudal behavior and correlation structure temporal behavior the poissonian methodology developed enables a precise quantitative analysis of shot noise processes displaying simultaneously the noah and joseph effects

three dimensional face recognition an expression invariant 3d face recognition approach is presented our basic assumption is that facial expressions can be modelled as isometries of the facial surface this allows to construct expression invariant representations of faces using the bending invariant canonical forms approach the result is an efficient and accurate face recognition algorithm robust to facial expressions that can distinguish between identical twins the first two authors we demonstrate a prototype system based on the proposed algorithm and compare its performance to classical face recognition methods the numerical methods employed by our approach do not require the facial surface explicitly the surface gradients field or the surface metric are sufficient for constructing the expression invariant representation of any given face it allows us to perform the 3d face recognition task while avoiding the surface reconstruction stage

optimal tracking control for large scale interconnected systems with time delays this paper considers an infinite horizon optimal tracking control problem for a class of large scale interconnected systems with state time delays by using the successive approximation approach two iteration sequences of vector differential equations are constructed meanwhile the large scale interconnected system is decomposed into finite decoupled subsystems the existence and uniqueness of the optimal solution is proved as well as the convergence of the solution sequence by finite iterations of the solution sequence a suboptimal tracking control law is obtained a reduced order reference input observer is designed to make the feedforward term of the optimal tracking control law physically realizable a numerical example shows that the presented algorithm is effective and easy to implement

brain surface conformal parameterization with the ricci flow in brain mapping research parameterized 3 d surface models are of great interest for statistical comparisons of anatomy surface based registration and signal processing here we introduce the theories of continuous and discrete surface ricci flow which can create riemannian metrics on surfaces with arbitrary topologies with user defined gaussian curvatures the resulting conformal parameterizations have no singularities and they are intrinsic and stable first we convert a cortical surface model into a multiple boundary surface by cutting along selected anatomical landmark curves secondly we conformally parameterize each cortical surface to a parameter domain with a user designed gaussian curvature arrangement in the parameter domain a shape index based on conformal invariants is computed and inter subject cortical surface matching is performed by solving a constrained harmonic map we illustrate various target curvature arrangements and demonstrate the stability of the method using longitudinal data to map statistical differences in cortical morphometry we studied brain asymmetry in 14 healthy control subjects we used a manifold version of hotelling s t2 test applied to the jacobian matrices of the surface parameterizations a permutation test along with the cumulative distribution of p values were used to estimate the overall statistical significance of differences the results show our algorithm s power to detect subtle group differences in cortical surfaces

coordinate systems for conformal cerebellar flat maps methods quasi conformal flat maps of the human cerebellum 1 4 were created from a high resolution tl weighted mri volume 5 a topologically correct surface was produced from a cerebellar volume defined by a plane parallel to the posterior commisure obex line and orthogonal to a plane passing through the vermal midline our quasi conformal flattening procedure 2 3 was applied to this surface to produce flat maps the cortical surface was parcellated according to 6 and activations produced by a target interception task were imposed eleven readily identifiable anatomical landmarks e g the apex of the fourth ventricle and the rostral most tip of the lingula were defined for the hyperbolic coordinate system one landmark was used as the map center and a second to specify the map orientation we also used a polar coordinate system on the hyperbolic flat map with equiangular grid lines radiating from the map focus and equidistant circles surrounding the focus for the spherical coordinate system one landmark was used as the north pole one as the south pole and a third as a distinguished equatorial point we used the usual latitude and longitude coordinates similar to 7 results and conclusions a major benefit of quasi conformal flat maps is that they can be used to generate canonical surface based coordinate systems for a cortical surface since our fiat maps to the hyperbolic plane and the sphere are l l and onto the coordinate systems produced by these flat maps can be used to quantitatively localize structure and function furthermore these flat maps are mathematically unique different choices of landmarks and coordinate systems will increase the utility of flat maps for studying functional imaging dam we believe that the use of cerebellar flat maps and canonical coordinate systems based on reproducible anatomical landmarks will improve our ability to localize functional activity on the cerebellar cortex and to quantify anatomical and functional differences between individual subjects and groups of subjects

singular sturmian comparison theorems for linear hamiltonian systems abstract in this paper we prove singular comparison theorems on unbounded intervals for two nonoscillatory linear hamiltonian systems satisfying the sturmian majorant condition and the legendre condition at the same time we do not impose any controllability condition the results are phrased in terms of the comparative index and the numbers of proper focal points of the minimal principal solutions of these systems at both endpoints of the considered interval the main idea is based on an application of new transformation theorems for principal and antiprincipal solutions at infinity and on new limit properties of the comparative index involving these solutions this work generalizes the recently obtained sturmian separation theorems on unbounded intervals for one system by the authors 2019 as well as the sturmian comparison theorems and transformation theorems on compact intervals by j elyseeva 2016 and 2018 we note that all the results are new even in the completely controllable case

combinatorial ricci flows on surfaces we show that the analogue of hamilton s ricci flow in the combinatorial setting produces solutions which converge exponentially fast to thurston s circle packing on surfaces as a consequence a new proof of thurston s existence of circle packing theorem is obtained as another consequence ricci flow suggests a new algorithm to find circle packings

one hundred years of the design of experiments on and off the pages of biometrika the earliest important papers on planned experiments in biometrika appeared within a year of each other in 1917 and 1918 roughly speaking one was concerned with design optimality and industrial experiments the other with agricultural trials and blocking we find this approximate division of the subject into two parts helpful in describing its development and growth as a result of the hostility between fisher and karl pearson much of the development of designs for agriculture in the 1920s and 1930s occurred off the pages of biometrika despite this we are able to trace a coherent history of the development of the subject from field trials and response surface methods to clinical trials and bayesian versions of design optimality

factored forms for solutions of ax xb c and x axb c in companion matrices abstract the main concern of this paper is linear matrix equations with block companion matrix coefficients it is shown that general matrix equations ax xb c and x axb c can be transformed to equations whose coefficients are block companion matrices c l x xc m diag i 0 0 and x c l xc m diag i 0 0 respectively where l and cm stand for the first and second block companion matrices of some monic r r matrix polynomials l si s 1j 0 jlj and m ti t7minus 1j 0 jmj the solution of the equat with block companion coefficients is reduced to solving vector equations sx where the matrix s is r2l r2l l max s t and enjoys some symmetry properties

mertens equimodular matrices of redheffer type abstract asymptotic behavior of the mertens function m n k 0 n k which is equal to the determinant of the n n redheffer matrix is known to be closely related to the riemann hypothesis an infinite matrix whose nth leading principal minor is equal to m n for all n 1 is called a mertens equimodular matrix we use riordan matrices to find a large class of mertens equimodular matrices each element of which is called by us a riordan redheffer matrix briefly an r r matrix we also give the generating function for the characteristic polynomials of r r matrices as a result we introduce several examples of r r matrices that reveal interesting spectral properties further we pose two conjectures on the eigenvalues of those r r matrices finally we find a sufficient condition for the riemann hypothesis using the smallest singular value of a r r matrix

birkhoff james e orthogonality sets of vectors and vector valued polynomials abstract consider a complex normed linear space x and let x with 0 motivated by recent works on rectangular matrices and on normed linear spaces we study the birkhoff james e orthogonality set of with respect to give an alternative definition for this set and explore its rich structure we also introduce the birkhoff james e orthogonality set of polynomials in one complex variable whose coefficients are members of x and survey and record extensions of results on matrix polynomials to these vector valued polynomials

optimal mass transport for shape matching and comparison surface based 3d shape analysis plays a fundamental role in computer vision and medical imaging this work proposes to use optimal mass transport map for shape matching and comparison focusing on two important applications including surface registration and shape space the computation of the optimal mass transport map is based on monge brenier theory in comparison to the conventional method based on monge kantorovich theory this method significantly improves the efficiency by reducing computational complexity from o n 2 to o n for surface registration problem one commonly used approach is to use conformal map to convert the shapes into some canonical space although conformal mappings have small angle distortions they may introduce large area distortions which are likely to cause numerical instability thus resulting failures of shape analysis this work proposes to compose the conformal map with the optimal mass transport map to get the unique area preserving map which is intrinsic to the riemannian metric unique and diffeomorphic for shape space study this work introduces a novel riemannian framework conformal wasserstein shape space by combing conformal geometry and optimal mass transport theory in our work all metric surfaces with the disk topology are mapped to the unit planar disk by a conformal mapping which pushes the area element on the surface to a probability measure on the disk the optimal mass transport provides a map from the shape space of all topological disks with metrics to the wasserstein space of the disk and the pullback wasserstein metric equips the shape space with a riemannian metric we validate our work by numerous experiments and comparisons with prior approaches and the experimental results demonstrate the efficiency and efficacy of our proposed approach

some properties of solutions of yule walker type equations abstract the distribution of spectra of scalar and matrix polynomials generated by solutions of yule walker type equations with respect to the real line and the unit circle is investigated a description of the spectral distribution is given in terms of the inertia of the corresponding hermitian block hankel or block toeplitz matrix these results can be viewed as matrix analogues of m g krein s theorems on polynomials orthogonal on the unit circle

stochastic ornstein uhlenbeck capacitors we introduce and study a class of random capacitor systems which are both charged and discharged stochastically a capacitor is fed by a random inflow with stationary and independent increments discharging occurs according to a markovian rate which is linear in the capacitor s level the resulting capacitor dynamics are markovian stochastically cyclic and regenerative we coin these systems levy charged ornstein uhlenbeck capacitors various random quantities associated with these systems are analyzed including the time to discharge the duration of the charging cycle the trajectory and the peak height of the capacitor level during a charging cycle and the capacitor s stationary equilibrium level furthermore we show that there are sharp distinctions between these capacitor systems and corresponding standard levy driven ornstein uhlenbeck systems

shape analysis of planar multiply connected objects using conformal welding shape analysis is a central problem in the field of computer vision in 2d shape analysis classification and recognition of objects from their observed silhouettes are extremely crucial but difficult it usually involves an efficient representation of 2d shape space with a metric so that its mathematical structure can be used for further analysis although the study of 2d simply connected shapes has been subject to a corpus of literatures the analysis of multiply connected shapes is comparatively less studied in this work we propose a representation for general 2d multiply connected domains with arbitrary topologies using conformal welding a metric can be defined on the proposed representation space which gives a metric to measure dissimilarities between objects the main idea is to map the exterior and interior of the domain conformally to unit disks and circle domains unit disk with several inner disks removed using holomorphic 1 forms a set of diffeomorphisms of the unit circle mathbb s 1 can be obtained which together with the conformal modules are used to define the shape signature a shape distance between shape signatures can be defined to measure dissimilarities between shapes we prove theoretically that the proposed shape signature uniquely determines the multiply connected objects under suitable normalization we also introduce a reconstruction algorithm to obtain shapes from their signatures this completes our framework and allows us to move back and forth between shapes and signatures with that a morphing algorithm between shapes can be developed through the interpolation of the beltrami coefficients associated with the signatures experiments have been carried out on shapes extracted from real images results demonstrate the efficacy of our proposed algorithm as a stable shape representation scheme

the ricci flow on the 2 sphere the classical uniformization theorem interpreted differential geomet rically states that any riemannian metric on a 2 dimensional surface ispointwise conformal to a constant curvature metric thus one can con sider the question of whether there is a natural evolution equation whichconformally deforms any metric on a surface to a constant curvature met ric the primary interest in this question is not so much to give a newproof of the uniformization theorem but rather to understand nonlinearparabolic equations better especially those arising in differential geome try a sufficiently deep understanding of parabolic equations should yieldimportant new results in riemannian geometry the question in the preceding paragraph has been studied by richardhamilton 3 and brad osgood ralph phillips and peter sarnak 6 in 3 hamilton studied the following equation which we refer to as

bezoutians of rational matrix functions matrix equations and factorizations abstract the notion of bezoutian for a quadruple of rational matrix functions introduced by the authors in previous works is shown to be an adequate connecting link between certain factorization problems for rational matrix functions and quadratic and linear matrix equations particular attention is given to various types of coprime factorizations a characterization of bezoutians as angle operators between certain pairs of subspaces is also given

optimizing equiareality of nurbs surfaces using composite mobius transformations the equiareality of nurbs surfaces greatly affects the results of visualization and tessellation applications especially when dealing with extruding and intruding shapes to obtain more equiareal parameterizations of nurbs surfaces than the mobius based method an optimization algorithm is presented in this paper based on the more flexible composite reparameterization functions for fixed knots the optimal composite reparameterization can be obtained using the levenberg marquardt method for a given tolerance a uniform subdivision scheme is interleaved with the optimization procedure and this process finishes until the change of the equiareal deviation energy is less than the given tolerance examples are given to show the performance of our algorithm for visualization and tessellation applications

generalized gradients priors on minimization flows this paper tackles an important aspect of the variational problem underlying active contours optimization by gradient flows classically the definition of a gradient depends directly on the choice of an inner product structure this consideration is largely absent from the active contours literature most authors explicitely or implicitely assume that the space of admissible deformations is ruled by the canonical l 2 inner product the classical gradient flows reported in the literature are relative to this particular choice here we investigate the relevance of using i other inner products yielding other gradient descents and ii other minimizing flows not deriving from any inner product in particular we show how to induce different degrees of spatial consistency into the minimizing flow in order to decrease the probability of getting trapped into irrelevant local minima we report numerical experiments indicating that the sensitivity of the active contours method to initial conditions which seriously limits its applicability and efficiency is alleviated by our application specific spatially coherent minimizing flows we show that the choice of the inner product can be seen as a prior on the deformation fields and we present an extension of the definition of the gradient toward more general priors

restarted global fom and gmres algorithms for the stein like matrix equation x m x c abstract in this paper we propose the restarted global full orthogonalization method gl fom and global generalized minimum residual gl gmres method to solve the stein like matrix equation x m x c with m x a x b m x a x b m x a x b or m x a x h b respectively where x is an unknown matrix to be solved first by using a real inner product in complex matrix spaces a generalized global arnoldi process is given then we demonstrate how to employ the restarted gl fom and gl gmres algorithms for solving the stein like matrix equation the proposed algorithms take advantage of the shifted structure of the matrix equation and are implemented by the original coefficient matrices finally some numerical examples are given to illustrate the effectiveness with comparison to some existing methods

a growth collapse model levy inflow geometric crashes and generalized ornstein uhlenbeck dynamics we introduce and study a stochastic growth collapse model the growth process is a steady random inflow with stationary independent and non negative increments crashes occur according to an arbitrary renewal process they are geometric and their magnitudes are random and are governed by an arbitrary distribution on the unit interval if the system s pre crash level is x 0 and the crash magnitude is 0 c 1 then an avalanche of size cx takes place after which the system collapses to the post crash level 1 c x this results in a stochastic growth collapse growth collapse system evolution governed by generalized non markovian ornstein uhlenbeck dynamics we explore the behavior of this growth collapse model and compute various system statistics including mean variance auto correlation laplace transform and in the case of heavy tailed inflows probability tails special emphasis is devoted to scale free systems where the crash magnitudes are governed by a power law distribution we show that i there is a hidden regular ornstein uhlenbeck structure underlying all scale free systems ii when crash rates tend to infinity these systems yield regular ornstein uhlenbeck limits and iii when the inflow is selfsimilar these systems yield linnik distributed equilibria

on the history of the bezoutian and the resultant matrix abstract it is shown how the bezoutian and the resultant matrix evolved from euler s work in elimination theory

hyperbolic harmonic mapping for surface registration automatic computation of surface correspondence via harmonic map is an active research field in computer vision computer graphics and computational geometry it may help document and understand physical and biological phenomena and also has broad applications in biometrics medical imaging and motion capture industries although numerous studies have been devoted to harmonic map research limited progress has been made to compute a diffeomorphic harmonic map on general topology surfaces with landmark constraints this work conquers this problem by changing the riemannian metric on the target surface to a hyperbolic metric so that the harmonic mapping is guaranteed to be a diffeomorphism under landmark constraints the computational algorithms are based on ricci flow and nonlinear heat diffusion methods the approach is general and robust we employ our algorithm to study the constrained surface registration problem which applies to both computer vision and medical imaging applications experimental results demonstrate that by changing the riemannian metric the registrations are always diffeomorphic and achieve relatively high performance when evaluated with some popular surface registration evaluation standards

remarks on the identifiability of thurstonian paired comparison models under multiple judgment it is well known that the representations of the thurstonian models for difference judgment data are not unique it has been shown that equivalence classes can be formed to provide a more meaningful partition of the covariance structures of the thurstonian ranking models in this paper we examine the equivalence relations between thurstonian covariance structure models for paired comparison data obtained under multiple judgment and discuss their implications on the general identification constraints and methods to check for parameter identifiability in restricted models

metric optimization for surface analysis in the laplace beltrami embedding space in this paper we present a novel approach for the intrinsic mapping of anatomical surfaces and its application in brain mapping research using the laplace beltrami eigen system we represent each surface with an isometry invariant embedding in a high dimensional space the key idea in our system is that we realize surface deformation in the embedding space via the iterative optimization of a conformal metric without explicitly perturbing the surface or its embedding by minimizing a distance measure in the embedding space with metric optimization our method generates a conformal map directly between surfaces with highly uniform metric distortion and the ability of aligning salient geometric features besides pairwise surface maps we also extend the metric optimization approach for group wise atlas construction and multi atlas cortical label fusion in experimental results we demonstrate the robustness and generality of our method by applying it to map both cortical and hippocampal surfaces in population studies for cortical labeling our method achieves excellent performance in a cross validation experiment with 40 manually labeled surfaces and successfully models localized brain development in a pediatric study of 80 subjects for hippocampal mapping our method produces much more significant results than two popular tools on a multiple sclerosis study of 109 subjects

bezout and hankel matrices associated with row reduced matrix polynomials barnett type formulas abstract based on the approach introduced by b d o anderson and e i jury in 1976 the definition of finite hankel and bezout matrices corresponding to matrix polynomials is extended to the case where the denominator of the corresponding rational matrix function is not necessarily monic but is row reduced the matrices introduced keep most of the well known properties that hold in the monic case in particular we derive extensions of formulas giving a connection with polynomials in the companion matrix usually called barnett formulas of the inversion theorem and of formulas concerning alternating products of hankel and bezout matrices

generalized bezoutian and matrix equations abstract a natural generalization of the classical bezout matrix of two polynomials is introduced for a family of several matrix polynomials the main aim of the paper is to show that this generalized bezoutian serves as an adequate connecting link between the class of equations in matrix polynomials m y z l r and the class of linear matrix equations ax xb c each equation in one of these classes is coupled with a certain equation in the other class so that for each couple the generalized bezoutian corresponding to a solution y z of the equation in matrix polynomials is a solution of the matrix equation and conversely any solution x of the matrix equation is a generalized bezoutian corresponding to a certain solution of the equation in matrix polynomials in particular either both equations are solvable or both have no solutions explicit formulas connecting the solutions of the two equations are given also various representation formulas for the generalized bezoutian are derived and its relation to the resultant matrix and the greatest common divisor of several matrix polynomials is discussed

common eigenvalues divisors and multiples of matrix polynomials a review abstract the main objective of this paper is to review some results obtained in the last decade using methods of spectral theory and concerning common multiples common divisors and common eigenvalues for matrix polynomials this includes discussion of resultant and bezoutian matrices for matrix polynomials and descriptions of their kernels in terms of spectral data some extensions to operator polynomials whose coefficients are bounded linear operators on a hilbert space are also included

computing quasiconformal maps using an auxiliary metric and discrete curvature flow surface mapping plays an important role in geometric processing which induces both area and angular distortions if the angular distortion is bounded the mapping is called a quasiconformal mapping qc mapping many surface mappings in our physical world are quasiconformal the angular distortion of a qc mapping can be represented by the beltrami differentials according to qc teichmuller theory there is a one to one correspondence between the set of beltrami differentials and the set of qc surface mappings under normalization conditions therefore every qc surface mapping can be fully determined by the beltrami differential and reconstructed by solving the so called beltrami equation in this work we propose an effective method to solve the beltrami equation on general riemann surfaces the solution is a qc mapping associated with the prescribed beltrami differential the main strategy is to define an auxiliary metric am on the domain surface such that the original qc mapping becomes conformal under the auxiliary metric the desired qc mapping can then be obtained by using the conventional conformal mapping method in this paper we first formulate a discrete analogue of qc mappings on triangular meshes then we propose an algorithm to compute discrete qc mappings using the discrete yamabe flow method to the best of our knowledge it is the first work to compute the discrete qc mappings for general riemann surfaces especially with different topologies numerically the discrete qc mapping converges to the continuous solution as the mesh grid size approaches to 0 we tested our algorithm on surfaces scanned from real life with different topologies experimental results demonstrate the generality and accuracy of our auxiliary metric method

a faddeev sequence method for solving lyapunov and sylvester equations lyapunov and sylvester equations play an important role in linear systems theory this paper deals with a method of solving such equations of the form ap pb k and p apb k with a rm m b rn n and p k rm n by exploiting the matrix algebra structure of the problem no use is made of kronecker products and the largest matrices occurring in the algorithms are of sizes m m n n and m n the faddeev method for matrix inversion lies at the very heart of the algorithms presented it occurs on several levels of the problem for the matrices a and b and for the lyapunov and sylvester operators the resulting algorithms are capable of solving the equations in a finite number of recursion steps they are very much apt for symbolic calculation it is shown how a solution can be quickly obtained for an equation with an arbitrary right hand side k provided a solution is known for a right hand side xyt of rank 1 where a x and bt y are reachable pairs the concept of a faddeev reachability matrix introduced here turns out to be very useful it establishes a close connection between the controller canonical companion form of a reachable pair a b and the faddeev sequence of a if a is already on controller form then its faddeev sequence takes on an especially simple form also in the symmetric case where a bt many important simplifications arise for this case alternative algorithms that require less iterations are developed the paper concludes with some examples concerning the symbolic solution of the lyapunov equation ap pat bbt with a b on controller form showing the potential of the algorithms

bezoutian and schur cohn problem for operator polynomials abstract a generalized bezout operator bezoutian for a pair of operator polynomials is introduced and its kernel is described in terms of common spectral data of the underlying polynomials the location of the spectrum of an operator polynomial with compact spectrum with respect to the unit circle infinite dimensional version of the schur cohn problem is expressed via the inertia of a suitable bezoutian an application to the geometric dichotomy problem for difference equations with operator coefficients is given as well

non linear shot noise levy noah joseph abstract we introduce and study a generic non linear shot noise system model shots of random magnitudes arrive to the system stochastically following an arbitrary time homogeneous poisson point process after hitting the system the magnitude of an arriving shot decays to zero the decay is governed by an arbitrary differential equation dynamics shots are independent and their overall effect on the system is additive the system s noise level at time t equals the sum of the magnitudes at time t of all the shots arriving to the system prior to time t the resulting shot noise is i a levy process when the decay dynamics are degenerate ii a levy driven ornstein uhlenbeck process when the decay dynamics are linear and iii a stationary non markov process when the decay dynamics are non linear the resulting shot noise admits an underlying levy structure which we explicitly compute and can yield both the noah effect and the joseph effect closed form analytic formulae for various statistics are derived including the log laplace transform and cumulants of the stationary noise level the process auto covariance function and the process range of dependence

ricci flow embedding for rectifying non euclidean dissimilarity data pairwise dissimilarity representations are frequently used as an alternative to feature vectors in pattern recognition one of the problems encountered in the analysis of such data is that the dissimilarities are rarely euclidean while statistical learning algorithms often rely on euclidean dissimilarities such non euclidean dissimilarities are often corrected or a consistent euclidean geometry is imposed on them via embedding this paper commences by reviewing the available algorithms for analysing non euclidean dissimilarity data the novel contribution is to show how the ricci flow can be used to embed and rectify non euclidean dissimilarity data according to our representation the data is distributed over a manifold consisting of patches each patch has a locally uniform curvature and this curvature is iteratively modified by the ricci flow the raw dissimilarities are the geodesic distances on the manifold rectified euclidean dissimilarities are obtained using the ricci flow to flatten the curved manifold by modifying the individual patch curvatures we use two algorithmic components to implement this idea firstly we apply the ricci flow independently to a set of surface patches that cover the manifold second we use curvature regularisation to impose consistency on the curvatures of the arrangement of different surface patches we perform experiments on three real world datasets and use these to determine the importance of the different algorithmic components i e ricci flow and curvature regularisation we conclude that curvature regularisation is an essential step needed to control the stability of the piecewise arrangement of patches under the ricci flow

stability and inertia abstract the purpose of this paper is to present a brief overview of matrix stability and inertia theory a few applications of inertia and stability theorems and a nonspectral implicit matrix equation method for determining stability and inertia of a nonhermitian matrix are also presented inter relationships between different theorems are explicitly stated whenever appropriate the paper concludes with some problems for future research in this area

matrix representations of bezoutians abstract an approach is presented to get matrix representations for classical and more general r bezoutians with entries from a noncommutative ring formulas due to ammar and gader barnett gohberg and semencul gohberg and heinig kravitsky and russakovskii lerer and tismentsky ptak and wimmer as well as some generalizations and modifications are obtained in a unified way

focusr feature oriented correspondence using spectral regularization a method for precise surface matching existing methods for surface matching are limited by the tradeoff between precision and computational efficiency here we present an improved algorithm for dense vertex to vertex correspondence that uses direct matching of features defined on a surface and improves it by using spectral correspondence as a regularization this algorithm has the speed of both feature matching and spectral matching while exhibiting greatly improved precision distance errors of 1 4 percent the method focusr incorporates implicitly such additional features to calculate the correspondence and relies on the smoothness of the lowest frequency harmonics of a graph laplacian to spatially regularize the features in its simplest form focusr is an improved spectral correspondence method that nonrigidly deforms spectral embeddings we provide here a full realization of spectral correspondence where virtually any feature can be used as an additional information using weights on graph edges but also on graph nodes and as extra embedded coordinates as an example the full power of focusr is demonstrated in a real case scenario with the challenging task of brain surface matching across several individuals our results show that combining features and regularizing them in a spectral embedding greatly improves the matching precision to a submillimeter level while performing at much greater speed than existing methods

solution to generalized sylvester matrix equations an explicit solution to the generalized sylvester matrix equation ax by exf is established this solution is expressed in terms of the r controllability matrix of e a b a generalized symmetric operator matrix and an observability matrix moreover based on this solution solutions to some other matrix equations are also derived the results may provide great convenience for the analysis and synthesis problems related to these equations

optimal sinusoidal disturbances damping for singularly perturbed systems with time delay abstract this paper considers an optimal damping vibration control problem for linear time delay singularly perturbed systems affected by external sinusoidal disturbances by using the slow fast decomposition theory of singularly perturbed systems the system is first decomposed into a fast subsystem without time delay and a slow subsystem with time delay affected by external sinusoidal disturbances then the successive approximation approach saa and the feedforward compensation techniques are proposed to solve the slow time scale time delay optimal control problem and damp the external sinusoidal disturbances respectively the conditions of existence and uniqueness of the finite and infinite horizon feedforward and feedback composite control ffcc laws are presented and the design approaches are given the ffcc laws consist of nondelay feedback terms disturbances compensation terms and a time delay compensation term which is the limit of the solution sequence of the adjoint vector equations simulation examples indicate that the saa is valid and the ffcc laws are easy to implement and more effective with respect to damping the external sinusoidal disturbances than that of the classical feedback composite control fcc laws

sylvester and lyapunov equations and some interpolation problems for rational matrix functions abstract the main theme of this paper is the interrelations between interpolation problems for rational matrix functions and the linear matrix equations of sylvester and lyapunov type with suitable controllability and observability conditions using these interrelations we are able to solve certain interpolation problems for rational matrix functions which include for example the problem of constructing a rational matrix function of minimal size with prescribed zeros poles and their multiplicities as to matrix equations we prove for example a result which is in some sense the inverse to the well known chen wimmer inertia theorem an application to system theory is also included

bezoutians of rational matrix functions abstract various concepts of a bezoutian of two rational matrix functions are introduced thereby extending this concept previously studied in the framework of matrix and operator polynomials and analystic functions beyond the class of analytic functions basic properties of the bezoutians are established the bezoutian based on realizations of the functions in question turns out to be most adequate and is studied in depth in particular its kernel is discribed in terms of the realizations the result serves as a crucial ingredient in the proof of the key theorem of the paper providing a characterization of the common zero data of two square size rational matrix functions in terms of the kernel of their bezoutian

ricci flow for 3d shape analysis ricci flow is a powerful curvature flow method which is invariant to rigid motion scaling isometric and conformal deformations we present the first application of surface ricci flow in computer vision previous methods based on conformal geometry which only handle 3d shapes with simple topology are subsumed by the ricci flow based method which handles surfaces with arbitrary topology we present a general framework for the computation of ricci flow which can design any riemannian metric by user defined curvature the solution to ricci flow is unique and robust to noise we provide implementation details for ricci flow on discrete surfaces of either euclidean or hyperbolic background geometry our ricci flow based method can convert all 3d problems into 2d domains and offers a general framework for 3d shape analysis we demonstrate the applicability of this intrinsic shape representation through standard shape analysis problems such as 3d shape matching and registration and shape indexing surfaces with large nonrigid anisotropic deformations can be registered using ricci flow with constraints of feature points and curves we show how conformal equivalence can be used to index shapes in a 3d surface shape space with the use of teichmuller space coordinates experimental results are shown on 3d face data sets with large expression deformations and on dynamic heart data

controllability observability and the solution of ax xb c a closed form finite series representation of the unique solution x of the matrix equation ax xb c is developed using this representation the image kernel and rank of x are related to the controllable and unobservable subspaces of the a c and c b pairs respectively bounds on the rank of x are obtainedin terms of the dimensions of these subspaces in the case that c has unitary rank an exact calculation of rank x is made the generic rank of x with a b fixed and c generic is evaluated

optimal hankel norm model reductions and weiner hopf factorization i the canonical case we consider the problem of approximating a given stable rational matrix function g s of mcmillan degree n by a function s f s where has mcmillan degree l n and f is antistable it is known that the minimum possible l norm of the error g f l or equivalently the minimum possible hankel norm of the stable part of the error g h is equal to the l 1 st hankel singular value l 1 g s of g we give an explicit linear fractional map parametrization for the class of all functions s f s as above which satisfy g f l l 1 g s the coefficients of the linear fractional map are completely determined by the matrices a b and c in a realization g s c si a 1b for g s and the unique solutions of a pair of lyapunov equations involving these matrices note that without loss of generality g 0 the basic idea is to use the approach of ball and helton to reduce the problem to a symmetric wiener hopf factorization problem which in turn can be solved by applying a result of kaashoek and ran the results obtained here are equivalent to the results of glover but our analysis gives an alternative more geometric approach

bezout equation vandermonde operators and common multiples for operator polynomials soit h un espace de hilbert complexe on considere des polynomes d operateurs de type p j 0 m i p j ou p j sont des operateurs agissant dans h et est un parametre complexe on considere des p pour lequel le spectre p est un ensemble compact

a variance minimization criterion to feature selection using laplacian regularization in many information processing tasks one is often confronted with very high dimensional data feature selection techniques are designed to find the meaningful feature subset of the original features which can facilitate clustering classification and retrieval in this paper we consider the feature selection problem in unsupervised learning scenarios which is particularly difficult due to the absence of class labels that would guide the search for relevant information based on laplacian regularized least squares which finds a smooth function on the data manifold and minimizes the empirical loss we propose two novel feature selection algorithms which aim to minimize the expected prediction error of the regularized regression model specifically we select those features such that the size of the parameter covariance matrix of the regularized regression model is minimized motivated from experimental design we use trace and determinant operators to measure the size of the covariance matrix efficient computational schemes are also introduced to solve the corresponding optimization problems extensive experimental results over various real life data sets have demonstrated the superiority of the proposed algorithms

genus zero surface conformal mapping and its application to brain surface mapping we developed a general method for global conformal parameterizations based on the structure of the cohomology group of holomorphic one forms for surfaces with or without boundaries gu and yau 2002 gu and yau 2003 for genus zero surfaces our algorithm can find a unique mapping between any two genus zero manifolds by minimizing the harmonic energy of the map in this paper we apply the algorithm to the cortical surface matching problem we use a mesh structure to represent the brain surface further constraints are added to ensure that the conformal map is unique empirical tests on magnetic resonance imaging mri data show that the mappings preserve angular relationships are stable in mris acquired at different times and are robust to differences in data triangulation and resolution compared with other brain surface conformal mapping algorithms our algorithm is more stable and has good extensibility

common zero structure of rational matrix functions abstract the paper is concerned with description of common zero data of two square size rational matrix functions the notion of a common null pair which encodes the data is introduced and studied from various points of view the main result of the paper is a characterization of the common null pair in terms of realization of the functions in question connections with common minimal divisors of rational matrix functions are elucidated the recently developed theory of null pole triples and sylvester data sets is extensively used in the proofs