dynamical system for computing the eigenvectors associated with the largest eigenvalue of a positive definite matrix a dynamical system for computing the largest eigenpair of a given positive matrix is introduced certain qualitative properties of the proposed system are analyzed in detail the authors results suggest that the weight bounding term in pca algorithms for extracting the first principal component could assume a class of forms

qualitative behavior of non abelian toda like flows in this paper we analyze and compare the long term behavior of certain non abelian generalizations of the integrable toda lattice flow in particular we show that these flows exhibit generalized sorting behavior analogous to the sorting behavior of the original tridiagonal toda lattice we show that one of these generalizations the block double bracket equation is gradient with respect to a normal metric and exactly solvable this provides a new solvable case of the double bracket system

the quartic oscillator a non perturbative study by continuous unitary transformations the quantum quartic oscillator is investigated in order to test the many body technique of the continuous unitary transformations the quartic oscillator is sufficiently simple to allow a detailed study and comparison of various approximation schemes due to its simplicity it can be used as a pedagogical introduction to the unitary transformations both the spectrum and the spectral weights are discussed

differential equations for roaming pseudospectra paths to extremal points and boundary tracking when studying the varepsilon pseudospectrum of a matrix one is often interested in computing the extremal points having maximum real part or modulus this is a crucial step for example when computing the distance to instability of a stable system using the property that the pseudospectrum is determined via perturbations by rank 1 matrices we derive differential equations on the manifold of normalized rank 1 matrices whose solutions tend to the critical rank 1 perturbations associated with the extremal points of locally maximum real part and modulus this approach also allows us to track the boundary contour of the pseudospectrum in a neighborhood of the extremal points the technique we propose is related to an idea recently developed by guglielmi and overton who derived discrete dynamical systems instead of the continuous ones we present the method turns out to be fast in comparison with those previously proposed in the literature and appears to be promising in dealing with large sparse problems

optimal synthesis of circularly symmetric shaped beams the problem of the optimal power pattern synthesis of circularly symmetric shaped beams is addressed and solved in particular an existence criterion is provided to test a priori i e without solving the overall synthesis problem if a non superdirective source of given size can radiate or not a circularly symmetric power pattern lying in a given arbitrary mask in the affirmative case the proposed approach guarantees the achievement of the optimal solution i e that of maximum directivity or minimum size by using convex optimization techniques in general a number of equivalent source distributions i e radiating the same power pattern do exist which can be explicitly determined in a deterministic way numerical examples concerning applications of actual interest support the given theory and confirm the effectiveness of the developed synthesis procedure

orthogonal eigensubspace estimation using neural networks we present a neural network nn approach for simultaneously estimating all or some of the orthogonal eigenvectors of a symmetric nonindefinite matrix corresponding to its repeated minimum in magnitude eigenvalue this problem has its origin in the constrained minimization framework and has extensive applications in signal processing we recast this problem into the nn framework by constructing an appropriate energy function which the nn minimizes the nn is of feedback type with the neurons having sigmoidal activation function the proposed approach is analyzed to characterize the nature of the minimizers of the energy function the main result is that the matrix w is a minimizer of the energy function if and only if the columns of w are the orthogonal eigenvectors with a given norm corresponding to the smallest eigenvalue of the given matrix further all minimizers are global minimizers bounds on the integration time step that is required to numerically solve the system of differential equations which define the dynamics of the nn have also been derived results of computer simulations are presented to support our analysis

a coupled momentum method for modeling blood flow in three dimensional deformable arteries abstract blood velocity and pressure fields in large arteries are greatly influenced by the deformability of the vessel moreover wave propagation phenomena in the cardiovascular system can only be described considering wall deformability since blood is usually described as an incompressible fluid however computational methods for simulating blood flow in three dimensional models of arteries have either considered a rigid wall assumption for the vessel or significantly simplified or reduced geometries computing blood flow in deformable domains using standard techniques like the ale method remains a formidable problem for large realistic anatomic and physiologic models of the cardiovascular system we have developed a new method to simulate blood flow in three dimensional deformable models of arteries the method couples the equations of the deformation of the vessel wall at the variational level as a boundary condition for the fluid domain we consider a strong coupling of the degrees of freedom of the fluid and the solid domains and a linear membrane model enhanced with transverse shear for the vessel wall the effect of the vessel wall boundary is therefore added in a monolithic way to the fluid equations resulting in a remarkably robust scheme we present here the mathematical formulation of the method and discuss issues related to the fluid solid coupling membrane formulation time integration method and boundary and initial conditions implementation is discussed and results with simple geometries as well as large subject specific models are presented

singular value decomposition via gradient and self equivalent flows the task of finding the singular value decomposition svd of a finite dimensional complex linear operator is here addressed via gradient flows evolving on groups of complex unitary matrices and associated self equivalent flows the work constitutes a generalization of that of brockett on the diagonalization of real symmetric matrices via gradient flows on orthogonal matrices and associated isospectral flows it complements results of symes chu and other authors on continuous analogs of the classical qr algorithm as well as earlier work by the authors on svd via gradient flows on positive definite matrices

recurrent neural networks for computing pseudoinverses of rank deficient matrices three recurrent neural networks are presented for computing the pseudoinverses of rank deficient matrices the first recurrent neural network has the dynamical equation similar to the one proposed earlier for matrix inversion and is capable of moore penrose inversion under the condition of zero initial states the second recurrent neural network consists of an array of neurons corresponding to a pseudoinverse matrix with decaying self connections and constant connections in each row or column the third recurrent neural network consists of two layers of neuron arrays corresponding respectively to a pseudoinverse matrix and a lagrangian matrix with constant connections all three recurrent neural networks are also composed of a number of independent subnetworks corresponding to the rows or columns of a pseudoinverse the proposed recurrent neural networks are shown to be capable of computing the pseudoinverses of rank deficient matrices

a trace inequality with a subtracted term abstract for fixed real or complex matrices a and b the well known von neumann trace inequality identifies the maximum of tr uavb z sfnc as u and v range over the unitary group the maximum being a bilinear expression in the singular values of a and b this paper establishes the analogue of this inequality for real matrices a and b when u and v range over the proper real orthogonal group the maximum is again a bilinear expression in the singular values but there is a subtracted term when a and b have determinants of opposite sign

the large n behaviour of the lipkin model and exceptional points the ubiquitous lipkin model is investigated for an interaction parameter beyond the traditional critical point it is argued that a phase transition occurs higher up in the spectrum for such larger interaction where using appropriate scaling of the energies the position of the phase transition becomes independent of the particle number the phase transition is related to near singularities in the complex interaction plane the exceptional points consideration of finite temperature yields the well known physical features associated with phase transitions

nonholonomic double bracket equations and the gauss thermostat in this rapid communication we consider certain equations that arise from imposing a constant kinetic energy constraint on a one dimensional set of oscillators this is a nonlinear nonholonomic constraint on these oscillators and the dynamics are consistent with gauss s law of least constraint dynamics of this sort are of interest in nonequilibrium molecular dynamics we show that under certain choices of external potential these equations give rise to a generalization of the so called double bracket equations which are of interest in studying gradient flows and integrable systems such as the toda lattice in the case of harmonic potentials the flow is described by a symmetric bracket and periodic solutions are obtained

dynamical methods for polar decomposition and inversion of matrices we show how one may obtain polar decomposition as well as inversion of fixed and time varying matrices using a class of nonlinear continuous time dynamical systems first we construct a dynamic system that causes an initial approximation of the inverse of a time varying matrix to flow exponentially toward the true time varying inverse using a time parameterized homotopy from the identity matrix to a fixed matrix with unknown inverse and applying our result on the inversion of time varying matrices we show how any positive definite fixed matrix may be dynamically inverted by a prescribed time without an initial guess at the inverse we then construct a dynamical system that solves for the polar decomposition factors of a time varying matrix given an initial approximation for the inverse of the positive definite symmetric part of the polar decomposition as a byproduct this method gives another method of inverting time varying matrices finally using homotopy again we show how dynamic polar decomposition may be applied to fixed matrices with the added benefit that this allows us to dynamically invert any fixed matrix by a prescribed time

hessian riemannian gradient flows in convex programming in view of solving theoretically constrained minimization problems we investigate the properties of the gradient flows with respect to hessian riemannian metrics induced by legendre functions the first result characterizes hessian riemannian structures on convex sets as metrics that have a specific integration property with respect to variational inequalities giving a new motivation for the introduction of bregman type distances then the general evolution problem is introduced and global convergence is established under quasi convexity conditions with interesting refinements in the case of convex minimization some explicit examples of these gradient flows are discussed dual trajectories are identified and sufficient conditions for dual convergence are examined for a convex program with positivity and equality constraints some convergence rate results are established in the case of a linear objective function several optimality characterizations of the orbits are given optimal path of viscosi

flow equations and the strong coupling expansion for the hubbard model applying the method of continuous unitary transformations to a class of hubbard models we reexamine the derivation of thet u expansion for the strong coupling case the flow equations for the coupling parameters of the higher order effective interactions can be solved exactly resulting in a systematic expansion of the hamiltonian in powers oft u valid for any lattice in arbitrary dimension and for general band filling the expansion ensures a correct treatment of the operator products generated by the transformation and only involves the explicit recursive calculation of numerical coefficients this scheme provides a unifying framework to study the strong coupling expansion for the hubbard model which clarifies and circumvents several difficulties inherent to earlier approaches our results are compared with those of other methods and it is shown that the freedom in the choice of the unitary transformation that eliminates interactions between different hubbard bands can affect the effective hamiltonian only at ordert3 u2 or higher

non perturbative flow equations from continuous unitary transformations we use a novel parametrization of the flowing hamiltonian to show that the flow equations based on continuous unitary transformations as proposed by wegner can be implemented through a nonlinear partial differential equation involving one flow parameter and two system specific auxiliary variables the implementation is non perturbative as the partial differential equation involves a systematic expansion in fluctuations controlled by the size of the system rather than the coupling constant the method is applied to the lipkin model to construct a mapping which maps the non interacting spectrum onto the interacting spectrum to a very high accuracy this function is universal in the sense that the full spectrum for any large number of particles can be obtained from it in a similar way expectation values for a large class of operators can be obtained which also makes it possible to probe the structure of the eigenstates

an effective deterministic procedure for the synthesis of shaped beams by means of uniform amplitude linear sparse arrays a rigorous deterministic approach is proposed for the mask constrained power pattern synthesis of shaped beams through one dimensional sparse arrays having uniform amplitude excitations and variable phases the approach includes the well known density taper procedure as a special case the corresponding technique which exploits an analytical formulation of the problem and convex programming routines results in fast computation and is able to achieve optimal solutions without recurring to any global optimization numerical results assessing the capabilities of the proposed design method are provided opening the way to the development of an effective approach for the synthesis of circularly symmetric shaped beams via uniformly excited circular ring arrays

spectral analysis of fokker planck and related operators arising from linear stochastic differential equations we study spectral properties of certain families of linear second order differential operators arising from linear stochastic differential equations we construct a basis in the hilbert space of square integrable functions using modified hermite polynomials and obtain a representation for these operators from which their eigenvalues and eigenfunctions can be computed in particular we completely describe the spectrum of the fokker planck operator on an appropriate invariant subspace of rapidly decaying functions the eigenvalues of the fokker planck operator provide information about the speed of convergence of the corresponding probability distribution to steady state which is important for stochastic estimation and control applications we show that the operator families under consideration can be realized as solutions of differential equations in the double bracket form on an operator lie algebra which leads to a simple expression for the flow of their eigenfunctions

the influence of geometric imperfections in cardiovascular fsi simulations abstract we present a study of the fluid structure interaction in an idealized end to end anastomosis of a vascular bypass graft and an artery special attention is paid to the impact of geometric imperfections in the artery and the flow path of the upstream vessel segment on the hemodynamics a partitioned solution approach is applied and developed further to solve the coupled problem in an implicit manner to stabilize and accelerate the convergence of the staggered coupling iterations an interface quasi newton least squares method is applied while the finite volume method is used for the fluid mechanics subproblem high order finite elements serve to discretize the structural subproblem a convergence study shows the efficiency of the high order elements in the context of nearly incompressible anisotropic materials used to model circular and irregular shaped segments of an artery the fluid structure interaction simulations reveal a dominant influence of the upstream vessel s curvature which however decays rapidly in straight sections where the influence of geometric imperfections is dominant based on the proposed simulation approach hemodynamic parameters such as the oscillating shear index can be directly linked to the shape and the intensity of the imperfections

numerical solution of isospectral flows in this paper we are concerned with the problem of solving numerically isospectral flows these flows are characterized by the differential equation l b l l l 0 l 0 where l 0 is a d x d symmetric matrix b l is a skew symmetric matrix function of l and b l is the lie bracket operator we show that standard runge kutta schemes fail in recovering the main qualitative feature of these flows that is isospectrality since they cannot recover arbitrary cubic conservation laws this failure motivates us to introduce an alternative approach and establish a framework for generation of isospectral methods of arbitrarily high order

dynamical systems that sort lists diagonalize matrices and solve linear programming problems abstract we establish a number of properties associated with the dynamical system h h h n where h and n are symmetric n by n matrices and a b ab ba the most important of these come from the fact that this equation is equivalent to a certain gradient flow on the space of orthogonal matrices we are especially interested in the role of this equation as an analog computer for example we show how to map the data associated with a linear programming problem into h 0 and n in such a way as to have h h h n evolve to a solution of the linear programming problem this result can be applied to find systems which solve a variety of genetic combinatorial optimization problems and it even provides an algorithm for diagonalizing symmetric matrices

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

on the continuous realization of iterative processes many important mathematical problems are solved by iterative methods many of these iterative schemes may be regarded as the discrete realization of certain continuous dynamical systems this paper summarizes some of the recent developments in the continuous realization of several popular basic iterative methods

on the thermodynamic limit of the lipkin model the thermodynamic limit of the lipkin model is investigated while the limit turns out to be rather elusive the analysis gives strong indications that the limit yields two analytically dissociated operators one for the normal and one for the deformed phase while the lipkin hamiltonian is hermitian and has a second order phase transition in finite dimensions finite particle number both properties seem to be destroyed in the thermodynamic limit

a theory for learning by weight flow on stiefel grassman manifold recently we introduced the concept of neural network learning on stiefel grassman manifold for multilayer perceptron like networks contributions of other authors have also appeared in the scientific literature about this topic this article presents a general theory for it and illustrates how existing theories may be explained within the general framework proposed here

on quadratic stability of systems with structured uncertainty this paper considers the problem of stability robustness with respect to a class of nonlinear time varying perturbations which are bounded in a component wise rather than aggregated manner a family of robustness bounds is parameterized in terms of a non singular symmetric matrix it is shown that the problem of computing the largest robustness bound over the set of non singular symmetric matrices can be approximated by a smooth minimization problem over a compact set a convergent algorithm for computing an optimal robustness bound is proposed in the form of a gradient flow

global convergence of nonlinear cascade flows with morse bott zero dynamics abstract we derive a sufficient condition that a flow captures the dynamics on an invariant submanifold this leads to a refinement of the lasalle invariance principle as a consequence we generalize a well known global asymptotic stability result of nonlinear cascade systems to show global convergence to a compact invariant set this includes the case where a globally asymptotically stable system is coupled to a morse bott flow

p fems for hyperelastic anisotropic nearly incompressible materials under finite deformations with applications to arteries simulation the displacement formulation of p fems is extended to nearly incompressible hyper elastic anisotropic materials under finite deformations in a three dimensional setting to demonstrate the efficiency and accuracy of the formulation we derive analytical solutions that serve for the verification of the p fe results the locking free properties at the limit of incompressibility the high convergence rates and the robustness with respect to large aspect ratios of the p fes are demonstrated by numerical experiments and compared in terms of degrees of freedom and cpu times to equivalent classical formulations using h fems p fems are then exploited to investigate artery like structures having complex constitutive models and particularly the influence of slight allowable compressibility of orders of several percents on the stress levels copyright 2011 john wiley sons ltd

the p version of the finite element method for three dimensional curved thin walled structures in this paper we present an implementation of a three dimensional p version for structural problems of solids with almost arbitrarily curved surfaces applying the blending function method complex structures can often be modelled by a few p elements being the basis for a higher order approximation numerical examples will demonstrate that the p version with anisotropic ansatz spaces allows to predict the structural behaviour of three dimensional plates and shells with approximately the same amount of degrees of freedom as in the two dimensional case yet significantly more accurate due to the three dimensional model furthermore it is advantageous to compute complex structures exclusively with three dimensional discretizations as no special elements are needed to model the transition from dimensionally reduced formulations like plates or shells to fully three dimensional solid elements using the p version with anisotropic ansatz spaces the whole structure can be efficiently discretized with solid elements even if the aspect ratio of the elements becomes very large copyright 2001 john wiley sons ltd

explicit magnus expansions for nonlinear equations in this paper we develop and analyse new explicit magnus expansions for the nonlinear equation y a t y y defined in a matrix lie group in particular integration methods up to order four are presented in terms of integrals which can be either evaluated exactly or replaced by conveniently adapted quadrature rules the structure of the algorithm allows us to change the step size and even the order during the integration process thus improving its efficiency several examples are considered including isospectral flows and highly oscillatory nonlinear differential equations

in memory of kenneth g wilson kenneth wilson had an enormous impact on the renormalization group and field theories in general i had the great pleasure to work in three fields to which he contributed essentially critical phenomena gauge invariance in duality and confinement and flow equations and similarity renormalization

analytical and computational assessment of locking in the hp finite element method locking is the phenomenon by which the numerical approximation of parameter dependent problems deteriorates for values of the parameter close to a limiting value in this paper we give a definition of locking and develop precise computable and analytic ways to quantify it using the example of nearly incompressible elasticity we show by means of computational and theoretical results the difference between the h version and php version in combatting locking our results establish the superiority of high order elements both h p and hp when the standard variational form is used we also discuss other issues such as curved elements mixed methods and locking phenomena for problems over anisotropic materials and over thin domains

polynomial roots from companion matrix eigenvalues in classical linear algebra the eigenvalues of a matrix are sometimes defined as the roots of the characteristic polynomial an algorithm to compute the roots of a polynomial by computing the eigenvalues of the corresponding companion matrix turns the tables on the usual definition we derive a first order error analysis of this algorithm that sheds light on both the underlying geometry of the problem as well as the numerical error of the algorithm our error analysis expands on work by van dooren and dewilde in that it states that the algorithm is backward normwise stable in a sense that must be defined carefully regarding the stronger concept of a small componentwise backward error our analysis predicts a small such error on a test suite of eight random polynomials suggested by toh and trefethen however we construct examples for which a small componentwise relative backward error is neither predicted nor obtained in practice we extend our results to polynomial matrices where the result is essentially the same but the geometry becomes more complicated

isospectral flows on a class of finite dimensional jacobi matrices abstract we present a new matrix valued isospectral ordinary differential equation that asymptotically block diagonalizes n n zero diagonal jacobi matrices employed as its initial condition this o d e features a right hand side with a nested commutator of matrices and structurally resembles the double bracket o d e studied by r w brockett in 1991 we prove that its solutions converge asymptotically that the limit is block diagonal and above all that the limit matrix is defined uniquely as follows for n even a block diagonal matrix containing 2 2 blocks such that the super diagonal entries are sorted by strictly increasing absolute value furthermore the off diagonal entries in these 2 2 blocks have the same sign as the respective entries in the matrix employed as the initial condition for n odd there is one additional 1 1 block containing a zero that is the top left entry of the limit matrix the results presented here extend some early work of kac and van moerbeke

computer modeling of cardiovascular fluid structure interactions with the deforming spatial domain stabilized space time formulation hemodynamic factors such as the wall shear stress are believed to affect a number of cardiovascular diseases including atherosclerosis and aneurysm since resolving phenomena in a living human body is currently beyond the capabilities of in vivo measurement techniques computer modeling is expected to play an important role in gaining a better understanding of the relationship between the cardiovascular diseases and the hemodynamic factors we have developed a computer modeling technique for cardiovascular hemodynamic simulations with this modeling technique patient specific 3d geometry of an artery can be analyzed we take into account some of the important factors in human body for the purpose of demonstrating in vivo situations in a virtual world the interaction between the blood flow and the deformation of the arterial walls is a factor that we are specifically focusing on for such fluid structure interactions we have developed a computer modeling tool based on the deforming spatial domain stabilized space time dsd sst formulation this simulation tool is applied to a patient specific model under pulsatile blood flow conditions the simulations show that the flow behavior with compliant arterial walls is different from what we see with rigid arterial walls consequently the distribution of the wall shear stress on the compliant arterial walls is significantly different from that on the rigid arterial walls we deduce that the compliance of the arterial walls needs to be taken into account in cardiovascular hemodynamic simulations and the computer modeling tool we have developed can be effective in investigation of cardiovascular diseases

eigenvalue distributions from a star product approach we use the well known isomorphism between operator algebras and function spaces equipped with a star product to study the asymptotic properties of certain matrix sequences in which the matrix dimension d tends to infinity our approach is based on the su 2 coherent states which allow for a systematic 1 d expansion of the star product this produces a trace formula for functions of the matrix sequence elements in the large d limit which includes higher order finite d corrections from this a variety of analytic results pertaining to the asymptotic properties of the density of states eigenstates and expectation values associated with the matrix sequence follows it is shown how new and existing results in the settings of collective spin systems and orthogonal polynomial sequences can be readily obtained as special cases in particular this approach allows for the calculation of higher order corrections to the zero distributions of a large class of orthogonal polynomials

double bracket equations and geodesic flows on symmetric spaces in this paper we consider the geometry of hamiltonian flows on the cotangent bundle of coadjoint orbits of compact lie groups and on symmetric spaces a key idea here is the use of the normal metric to define the kinetic energy this leads to hamiltonian flows of the double bracket type we analyze the integrability of geodesic flows according to the method of thimm we obtain via the double bracket formalism a quite explicit form of the relevant commuting flows and a correspondingly transparent proof of involutivity we demonstrate for example integrability of the geodesic flow on the real and complex grassmannians we also consider right invariant systems and the generalized rigid body equations in this setting

moyal implementation of flow equations a non perturbative approach to quantum many body systems we show how wegner s flow equations can be reformulated as ordinary differential equations through the use of the moyal bracket in finite dimensional hilbert spaces the introduction of the moyal bracket leads naturally to the identification of a small expansion parameter namely the inverse of the dimensionality of the space this expansion corresponds to a non perturbative treatment of the coupling constant in the case of infinite dimensional spaces plays the role of the small parameter and the moyal formulation then allows for a semi classical treatment of the flow equation we demonstrate these statements for the lipkin and dicke models as well as for the symmetric x4 and double well potentials

lie algebraic aspects of the finite nonperiodic toda flows in this paper we review various aspects of nonperiodic full and tridiagonal toda flow theory in particular we consider gradient structures in the dynamics and geometry of these systems and we compare and contrast a number of formulations of the nonperiodic toda equations we describe the structure of these systems on general semisimple lie algebras in the case of the full kostant asymmetric toda flow we describe some of the background behind its integrability and explain the role of noncommutative integrability in its qualitative behavior we describe the relationship between the asymmetric toda flows and the symmetric and indefinite toda flows and show how one may conjugate from the full kostant toda flows to the full symmetric toda flows via a poisson map we describe briefly related toda systems such as nonabelian toda and the peakon flows

the structure of operators in effective particle conserving models for many particle systems defined on lattices we investigate the global structure of effective hamiltonians and observables obtained by means of a suitable basis transformation we study transformations which lead to effective hamiltonians conserving the number of excitations the same transformation must be used to obtain effective observables the analysis of the structure shows that effective operators give rise to a simple and intuitive perspective on the initial problem the systematic calculation of n particle irreducible quantities becomes possible constituting a significant progress details on how to implement the approach perturbatively for a large class of systems are presented

excitation retrieval of microwave linear arrays from phaseless far field data a methodology to recover the excitations of microwave linear arrays from the measurements of far field magnitude only is proposed the approach combines tools from convex optimization to solve the phase retrieval problem and a simple measurement procedure to mitigate the non uniqueness of the solution numerical simulations in various representative and realistic configurations of noise and measurement sampling are presented and discussed they show that it is possible to perfectly retrieve the array excitations from only the knowledge of far field magnitude by simply solving a convex optimization problem in addition the proposed approach that only calls for readily available routines is stable with respect to noise since the reconstruction performances degrades gracefully as the signal to noise ratio decreases

least squares matching problems abstract a large class of matching problems can be thought of in the following way one is given a group g which acts on a riemannian manifold and one is given two points x 1 and x 2 in the manifold the problem is to find g g such that d g x 1 x 2 is as small as possible if the group has finite cardinality this is a combinatorial optimization problem whereas if g is a lie group one can use the calculus to obtain necessary conditions our main results establish circumstances under which one can relax the combinatorial problem to an appropriate lie group problem and then solve it by the method of steepest descent an interesting class of applications involves embedding the permutation group on n letters in the orthogonal group o n with the latter acting on r n n 1 2 via a symmetricized tensor product the convergence properties of the descent equation which arises in this way are analyzed in some detail

statistical physics algorithms that converge in recent years there has been significant interest in adapting techniques from statistical physics in particular mean field theory to provide deterministic heuristic algorithms for obtaining app

asymptotic hamiltonian dynamics the toda lattice the three wave interaction and the non holonomic chaplygin sleigh abstract in this paper we discuss asymptotic stability in energy preserving systems which have an almost poisson structure in particular we consider a class of poisson systems which includes the toda lattice in standard hamiltonian systems one of course does not expect asymptotic stability the key here is the structure of the phase space of the poisson or almost poisson systems and the nature of the equilibria our systems are in some sense generalizations of the integrable toda lattice system but are not integrable in general as a particular example we point out an interesting connection between three mechanical two degree of freedom systems that exhibit asymptotic stability two of them are classical hamiltonian systems while the third is a non holonomic system non holonomic systems are generally energy preserving but not hamiltonian but the system analyzed here turns out to have a phase space which is the union of hamiltonian ones we also discuss various higher dimensional examples

universal computation and other capabilities of hybrid and continuous dynamical systems we explore the simulation and computational capabilities of hybrid and continuous dynamical systems the continuous dynamical systems considered are ordinary differential equations odes for hybrid systems we concentrate on models that combine odes and discrete dynamics e g finite automata we review and compare four such models from the literature notions of simulation of a discrete dynamical system by a continuous one are developed we show that hybrid systems whose equations can describe a precise binary timing pulse exact clock can simulate arbitrary reversible discrete dynamical systems defined on closed subsets of r n the simulations require continuous odes in ir2n with the exact clock as input all four hybrid systems models studied here can implement exact clocks we also prove that any discrete dynamical system in rn can be simulated by continuous odes in rt2n 1 we use this to show that smooth odes in ri3 can simulate arbitrary turing machines and hence possess the power of universal computation we use the famous asynchronous arbiter problem to distinguish between hybrid and continuous dynamical systems we prove that one cannot build an arbiter with devices described by a system of lipschitz odes on the other hand all four hybrid systems models considered can implement arbiters even if their odes are lipschitz

on explicitly solvable gradient systems of moser karmarkar type abstract a generalization and a classification of explicitly solvable gradient systems which appear in linear programming and neurodynamics are studied it is shown that nonlinear dynamical systems of moser karmarkar type generically take both a lax representation and a gradient form a related optimization problem on a simplex is discussed an integration procedure and a complete description of the invariant manifolds and the phase portrait of the dynamical systems of moser karmarkar type are also investigated

flow equations for hamiltonians contrasting different approaches by using a numerically solvable model to contrast different generators for flow equations for hamiltonians and to discuss the dependence of physical quantities on unitarily equivalent but effectively different initial hamiltonians a numerically solvable model is considered which is structurally similar to impurity models by this we discuss the question of optimization for the first time a general truncation scheme is established that produces good results for the hamiltonian flow as well as for the operator flow nevertheless it is also pointed out that a systematic and feasible scheme for the operator flow on the operator level is missing for this an explicit analysis of the operator flow is given for the first time we observe that truncation of the series of the observable flow after the linear or bilinear terms does not yield satisfactory results for the entire parameter regime as especially close to resonances even high orders of the exact series expansion carry considerable weight

hamiltonian structure of dynamical systems which solve linear programming problems abstract a new class of completely integrable hamiltonian systems which solve linear programming problems is introduced the relations with reduced toda flows double bracket equations interior point methods are established an interpretation in terms of total actions on kahler manifolds is given a complete description of the phase portrait including the explicit description of action angle variables is provided

on solving an isospectral flow in this paper we expand the solution of the matrix ordinary differential system originally due to bloch and iserles of the form x n x 2 t 0 x 0 x 0 sym n n s o n where sym n denotes the space of real n n symmetric matrices and s o n denotes the lie algebra of real n n skew symmetric matrices the flow is solved using explicit magnus expansion which respects the isospectrality of the system we represent the terms of expansion as binary rooted trees and deduce an explicit formalism to construct the trees recursively

stabilization of nonholonomic systems using isospectral flows in this paper we derive and analyze a discontinuous stabilizing feedback for a lie algebraic generalization of a class of kinematic nonholonomic systems introduced by brockett new directions in applied mathematics p hilton and g young eds springer verlag new york 1982 pp 11 27 the algorithm involves discrete switching between isospectral and norm decreasing flows we include a rigorous analysis of the convergence

the p version of the finite element method in the p version of the finite element method the triangulation is fixed and the degree p of the piecewise polynomial approximation is progressively increased until some desired level of precision is reached in this paper we first establish the basic approximation properties of some spaces of piecewise polynomials defined on a finite element triangulation these properties lead to an a priori estimate of the asymptotic rate of convergence of the p version the estimate shows that the p version gives results which are not worse than those obtained by the conventional finite element method called the h version in which h represents the maximum diameter of the elements when quasi uniform triangulations are employed and the basis for comparison is the number of degrees of freedom furthermore in the case of a singularity problem we show under conditions which are usually satisfied in practice that the rate of convergence of the p version is twice that of the h version with quasi uniform mesh inve

mask constrained power synthesis of linear arrays with even excitations an existing effective approach for the synthesis of linear arrays providing power patterns lying in arbitrary masks is extended to the case where an even distribution is required on element excitations the approach consists of a linear programming optimization plus a polynomial factorization and it allows one to deal with a considerably reduced computational burden with respect to the previous approaches by so doing it is now possible to solve problems that cannot be successfully tackled by the previous techniques an extension of the overall strategy to the synthesis of fields guaranteeing optimal performances in terms of directivity or ripple in the main beam region is also given

analog computation with dynamical systems a b s t r a c t physical systems exhibit various levels of complexity their long term dynamics may converge to fixed points or exhibit complex chaotic behavior this paper presents a theory that enables to interpret natural processes as special purpose analog computers since physical systems are naturally described in continuous time a definition of computational complexity for continuous time systems is required in analogy with the classical discrete theory we develop fundamentals of computational complexity for dynamical systems discrete or continuous in time on the basis of an intrinsic time scale of the system dissipative dynamical systems are classified into the computational complexity classes pd co rpd npd and exp t corresponding to their standard counterparts according to the complexity of their long term behavior the complexity of chaotic attractors relative to regular ones leads to the conjecture pa fi npj continuous time flows have been proven useful in solving various practical problems our theory provides the tools for an algorithmic analysis of such flows as an example we analyze the continuous hopfield network 1998 elsevier science b v all rights reserved

optimal control and geodesic flows in this paper we analyze and generalize from the point of view of the maximum principle a class of nonlinear optimal control problems originally introduced in brockett 1994 the optimal control problems discussed here lie on the adjoint orbits of compact lie groups or on grassmann manifolds and the structure of the equations that arise depends on the metric used the so called normal metric we consider some special cases of the general problem and their meaning from the point of view of riemannian geometry and hamiltonian mechanics

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

functional dissipation microarrays for classification in this article we describe a new method of extracting information from signals called functional dissipation that proves to be very effective for enhancing classification of high resolution texture rich data our algorithm bypasses to some extent the need to have very specialized feature extraction techniques and can potentially be used as an intermediate feature enhancement step in any classification scheme functional dissipation is based on signal transforms but uses the transforms recursively to uncover new features we generate a variety of masking functions and extract features with several generalized matching pursuit iterations in each iteration the recursive process modifies several coefficients of the transformed signal with the largest absolute values according to the specific masking function in this way the greedy pursuit is turned into a slow controlled dissipation of the structure of the signal that for some masking functions enhances separation among classes our case study in this paper is the classification of crystallization patterns of amino acids solutions affected by the addition of small quantities of proteins

on computation of the stability radius for nonlinearly structured perturbations this paper considers the problem of finding a perturbation matrix with the least spectral norm such that a matrix valued function becomes singular where the dependence of the function on the perturbation is allowed to be nonlinear it is proved that such a problem can be approximated by a smooth unconstrained minimization problem with compact sublevel sets a computational procedure proposed based on this result is demonstrated to be effective in both linear and nonlinear cases

reaching an optimal consensus dynamical systems that compute intersections of convex sets in this paper multi agent systems minimizing a sum of objective functions where each component is only known to a particular node is considered for continuous time dynamics with time varying interconnection topologies assuming that each node can observe a convex solution set of its optimization component and the intersection of all such sets is nonempty the considered optimization problem is converted to an intersection computation problem by a simple distributed control rule the considered multi agent system with continuous time dynamics achieves not only a consensus but also an optimal agreement within the optimal solution set of the overall optimization objective directed and bidirectional communications are studied respectively and connectivity conditions are given to ensure a global optimal consensus in this way the corresponding intersection computation problem is solved by the proposed decentralized continuous time algorithm we establish several important properties of the distance functions with respect to the global optimal solution set and a class of invariant sets with the help of convex and non smooth analysis

ordinary differential equations and the symmetric eigenvalue problem in this paper the authors develop a general framework for calculating the eigenvalues of a symmetric matrix using ordinary differential equations new algorithms are suggested and old algorithms including qr are interpreted

flow equations and normal ordering a survey first we give an introduction to the method of diagonalizing or block diagonalizing continuously a hamiltonian and explain how this procedure can be used to analyse the two dimensional hubbard model then we give a short survey on applications of this flow equation on other models finally we outline how symmetry breaking can be introduced by means of a symmetry breaking of the normal ordering not of the hamiltonian

experimental computation of real numbers by newtonian machines following a methodology we have proposed for analysing the nature of experimental computation we prove that there is a 3 dimensional newtonian machine which given any point x 2 0 1 can generate an infinite sequence pn qn for n 1 2 of rational number interval approximations that converges to x as n 1 the machine is a system for scattering and collecting particles the theorem implies that every point x 2 0 1 is computable by a simple newtonian kinematic system that is bounded in space and mass and for which the calculation of the n th approximation of x takes place in o n time with o n energy we describe a set of variants of the scatter machine which explain why our machine is non deterministic

locking effects in the finite element approximation of plate models we analyze the robustness of various standard finite element schemes for a hierarchy of plate models and obtain asymptotic convergence estimates that are uniform in terms of the thickness d we identify h version schemes that show locking i e for which the asymptotic convergence rate deteriorates as d 0 and also show that the p version is free of locking in order to isolate locking erects from boundary layer erects which also arise as d 0 our analysis is carried out for the periodic case which is free of boundary layers we analyze in detail the lowest model of the hierarchy the well known reissner mindlin model and show that the locking and robustness of finite element schemes for higher models of the hierarchy are essentially identical to the riessner mindlin case

numerical integration methods for the double bracket flow in this paper new methods up to order four based on the magnus expansion are proposed for the numerical integration of the double bracket equation the magnus series is constructed term by term by means of recurrences and a bound on the convergence domain is also provided the new integrators preserve the most salient qualitative features of the flow and are computationally more efficient than other standard lie group solvers such as the runge kutta munthe kaas class of algorithms