backward perturbation analysis of certain characteristic subspaces this paper gives optimal backward perturbation bounds and the accuracy of approximate solutions for subspaces associated with certain eigenvalue problems such as the eigenvalue problemax x the generalized eigenvalue problem ax bx and the singular value decomposition of a matrixa this paper also gives residual bounds for certain eigenvalues generalized eigenvalues and singular values

a note on the backward errors for hermite eigenvalue problems we discuss the backward error analysis of the hermite eigenproblems three kinds of problems are included generalized eigenproblem matrix polynomial eigenproblem and multiparameter eigenproblem in the case of the approximate eigenvalue being real these problems have already been solved by higham and higham siam j matrix anal 20 1998 493 tisseur lin alg appl 309 2002 339 and hachstenbach and plestenjak lin alg appl 375 2003 63 this note as a complement of their results considers the complex case

inverse eigenproblem for centrosymmetric and centroskew matrices and their approximation in this paper we first give the solvability condition for the following inverse eigenproblem iep given a set of vectors xi i 1m in cn and a set of complex numbers i i 1m find a centrosymmetric or centroskew matrix c in rn n such that xi i 1m and i i 1m are the eigenvectors and eigenvalues of c respectively we then consider the best approximation problem for the ieps that are solvable more precisely given an arbitrary matrix b in rn n we find the matrix c which is the solution to the iep and is closest to b in the frobenius norm we show that the best approximation is unique and derive an expression for it

on the generalized bi skew symmetric solutions of a linear matrix equation and its procrust problems in this paper the solvability conditions and the explicit expressions of the generalized bisymmetric and bi skew symmetric solutions of the matrix equation ax b are respectively established by applying two methods then the maximal and minimal ranks of the solutions are derived if the solvability conditions are not satisfied the generalized bisymmetric and bi skew symmetric least squares solutions of the matrix equation are considered and the generalized bisymmetric and bi skew symmetric least squares solutions with the minimum norm are also obtained in addition two algorithms are provided to compute the generalized bi skew symmetric least squares solution and some examples are given to illustrate that the algorithms are feasible

reducing stochasticity in the north atlantic oscillation index with coupled langevin equations we present a critical investigation of the functional relationship between the two pressure time series routinely used to define the index characterizing the north atlantic oscillation nao well known to regulate global climate variability and change first by a standard markov analysis we show that the standard nao index based on the pressure difference is not optimal in the sense of producing sufficiently reliable forecasts because it contains a dominating stochastic term in the corresponding langevin equation then we introduce a variationally optimized markov analysis involving two coupled langevin equations tailored to produce a nao quasi index having the desired minimum possible stochasticity the variationally optimized markov analysis is very general and can be applied in other physical situations involving two or more time series

the inverse eigenproblem with a submatrix constraint and the associated approximation problem for r s symmetric matrices abstract let r r n n and s r n n be nontrivial involutions i e r r 1 i and s s 1 i a matrix a r n n is called r s symmetric if r a s a this paper presents a r s symmetric matrix solution to the inverse eigenproblem with a leading principal submatrix constraint the solvability condition of the constrained inverse eigenproblem is also derived the existence the uniqueness and the expression of the r s symmetric matrix solution to the best approximation problem of the constrained inverse eigenproblem are achieved respectively an algorithm is presented to compute the r s symmetric matrix solution to the best approximation problem two numerical examples are given to illustrate the effectiveness of our results

minimal norm static feedbacks using dissipative hamiltonian matrices abstract in this paper we characterize the set of static state feedbacks that stabilize a given continuous linear time invariant system pair using dissipative hamiltonian matrices this characterization results in a parametrization of feedbacks in terms of skew symmetric and symmetric positive semidefinite matrices and leads to a semidefinite program that computes a static state stabilizing feedback this characterization also allows us to propose an algorithm that computes minimal norm static feedbacks the theoretical results extend to the static output feedback sof problem and we also propose an algorithm to compute the minimal norm sof we illustrate the effectiveness of our algorithm compared to state of the art methods for the sof problem on numerous numerical examples from the compleib library

crises sudden changes in chaotic attractors and transient chaos abstract the occurrence of sudden qualitative changes of chaotic dynamics as a parameter is varied is discussed and illustrated it is shown that such changes may result from the collision of an unstable periodic orbit and a coexisting chaotic attractor we call such collisions crises phenomena associated with crises include sudden changes in the size of chaotic attractors sudden appearances of chaotic attractors a possible route to chaos and sudden destructions of chaotic attractors and their basins this paper presents examples illustrating that crisis events are prevalent in many circumstances and systems and that just past a crisis certain characteristic statistical behavior whose type depends on the type of crisis occurs in particular the phenomenon of chaotic transients is investigated the examples discussed illustrate crises in progressively higher dimension and include the one dimensional quadratic map the two dimensional henon map systems of ordinary differential equations in three dimensions and a three dimensional map in the case of our study of the three dimensional map a new route to chaos is proposed which is possible only in invertible maps or flows of dimension at least three or four respectively based on the examples presented the following conjecture is proposed almost all sudden changes in the size of chaotic attractors and almost all sudden destruction or creations of chaotic attractors and their basins are due to crises

universal enveloping algebras of poisson hopf algebras abstract for a poisson algebra a by exploring its relation with lie rinehart algebras we prove a poincare birkhoff witt theorem for its universal enveloping algebra a e some general properties of the universal enveloping algebras of poisson hopf algebras are studied given a poisson hopf algebra b we give the necessary and sufficient conditions for a poisson polynomial algebra b x p to be a poisson hopf algebra we also prove a structure theorem for b e when b is a pointed poisson hopf algebra namely b e is isomorphic to b h b the crossed product of b and h b where h b is the quotient hopf algebra b e b e b

stochastic heart rate model can reveal pathologic cardiac dynamics a simple one dimensional langevin type stochastic difference equation can simulate the heart rate fluctuations in a time scale from minutes to hours the model consists of a deterministic nonlinear part and a stochastic part typical of gaussian noise and both parts can be directly determined from measured heart rate data data from healthy subjects typically exhibit the deterministic part with two or more stable fixed points studies of 15 congestive heart failure subjects reveal that the deterministic part of pathologic heart dynamics has no clear stable fixed points direct simulations of the stochastic model for normal and pathologic cases can produce statistical parameters similar to those of real subjects results directly indicate that pathologic situations simplify the heart rate control system

semidefinite inverse eigenvalue problems with prescribed entries and partial eigendata in this paper we study the semidefinite inverse eigenvalue problem of reconstructing a real n by n matrix c such that it is nearest to the original pre estimated real n by n matrix c o in the frobenius norm and satisfies the measured partial eigendata where the required matrix c should preserve the symmetry positive semidefiniteness and the prescribed entries of the pre estimated matrix c o we propose the alternating direction method of multipliers for solving the semidefinite inverse eigenvalue problem where three related iterative algorithms are presented we also extend our method to the case of lower bounds numerical experiments are reported to illustrate the efficiency of the proposed method for solving semidefinite inverse eigenvalue problems

on the kahan parlett jiang theorem a globally optimal backward perturbation error for two sided invariant subspaces abstract a crucial question in large non hermitian eigenproblems is whether or not any residual bounds can be very useful in practical situations for this problem the famous kahan parlett jiang theorem kahan et al 1982 15 seems to be the best compromise between generality and sharpness in essence it presents an a posteriori error bound on approximate solutions to eigenproblems which provides a powerful tool to evaluate quality of the computed two sided invariant subspaces of large scale non hermitian matrices however the perturbation error determined by this theorem is only locally optimal rather than globally optimal in this work we revisit this problem and derive a globally optimal backward perturbation error for given two sided approximate invariant subspaces our theorem enhances the kahan parlett jiang theorem and numerical experiments demonstrate the effectiveness of our theoretical results

langevin equations from time series we discuss the link between the approach to obtain the drift and diffusion of one dimensional langevin equations from time series and pope and ching s relationship for stationary signals the two approaches are based on different interpretations of conditional averages of the time derivatives of the time series at given levels the analysis provides a useful indication for the correct application of pope and ching s relationship to obtain stochastic differential equations from time series and shows its validity in a generalized sense for nondifferentiable processes originating from langevin equations

symmetric matrix pencils abstract a significant number of matrix eigenvalue problems quadratic or linear are best reformulated as pencils a m in which both a and m are real and symmetric some examples are given and then the canonical forms are re examined to explain the role of the sign characteristic attached to real eigenvalues in addition we examine the limitations on the use of the rayleigh quotient functional x ax x mx in describing the eigenvalues this sheds new light on the class of definite pencils and the stability of their eigenvalues under perturbations the reduction of indefinite pencils to useful sparse forms is mentioned

structured mapping problems for linearly structured matrices abstract given an appropriate class of structured matrices s we characterize matrices x and b for which there exists a matrix a s such that a x b and determine all matrices in s mapping x to b we also determine all matrices in s mapping x to b and having the smallest norm we use these results to investigate structured backward errors of approximate eigenpairs and approximate invariant subspaces and structured pseudospectra of structured matrices

prime regular hopf algebras of gk dimension one this paper constitutes the first part of a program to classify all affine prime regular hopf algebras h of gelfand kirillov dimension one over an algebraically closed field of characteristic zero we prove a number of properties of such an algebra list some classes of examples and then prove that when the pi degree of h is prime our list contains all such algebras

neighborhoods of periodic orbits and the stationary distribution of a noisy chaotic system the finest state space resolution that can be achieved in a physical dynamical system is limited by the presence of noise in the weak noise approximation the stochastic neighborhoods of deterministic periodic orbits can be computed from distributions stationary under the action of a local fokker planck operator and its adjoint we derive explicit formulas for widths of these distributions in the case of chaotic dynamics when the periodic orbits are hyperbolic the resulting neighborhoods form a basis for functions on the attractor the global stationary distribution needed for calculation of long time expectation values of observables can be expressed in this basis

on the structure and cohomology ring of connected hopf algebras abstract let p be a prime and k be a field of characteristic p we investigate the algebra structure and the structure of the cohomology ring for the connected hopf algebras of dimension p 3 which appear in the classification obtained in 20 the list consists of 23 algebras together with two infinite families we identify the morita type of the algebra and in almost all cases this is sufficient to clarify the structure of the cohomology ring

on a general link between anomalous diffusion and nonextensivity stochastic processes of systems with composable entropy measures are studied the processes are described by fokker planck equations which are nonlinear with respect to their probability densities and whose diffusion coefficients are derived from the entropy measures in questions in nonextensive extensive systems anomalous normal diffusion is found

primitive cohomology of hopf algebras abstract primitive cohomology of a hopf algebra is defined by using a modification of the cobar construction of the underlying coalgebra among many of its applications two classifications are presented firstly we classify all non locally pi pointed hopf algebra domains of gelfand kirillov dimension two and secondly we classify all pointed hopf algebras of rank one the first classification extends some results of brown goodearl and others in an ongoing project to understand all hopf algebras of low gelfand kirillov dimension the second generalizes results of krop radford and wang you chen which classified hopf algebras of rank one under extra hypothesis properties and algebraic structures of the primitive cohomology are discussed

a note on stewart s theorem for definite matrix pairs abstract let a and b be n n hermitian matrices the matrix pair a b is called definite pair and the corresponding eigenvalue problem ax bx is definite if c a b inf x 1 h a ib x 0 in this note we develop a uniform upper bound for differences of corresponding eigenvalues of two definite pairs and so improve a result which is obtained by g w stewart 2 moreover we prove that this upper bound is a projective metric in the set of n n definite pairs

statistically relaxing to generating partitions for observed time series data we introduce a relaxation algorithm to estimate approximations to generating partitions for observed dynamical time series generating partitions preserve dynamical information of a deterministic map in the symbolic representation our method optimizes an essential property of a generating partition avoiding topological degeneracies we construct an energylike functional and use a nonequilibrium stochastic minimization algorithm to search through configuration space for the best assignment of symbols to observed data as each observed point may be assigned a symbol the partitions are not constrained to an arbitrary parametrization we further show how to select particular generating partition solutions which also code low order unstable periodic orbits in a given way hence being able to enumerate through a number of potential generating partition solutions

noetherian hopf algebra domains of gelfand kirillov dimension two abstract we classify all noetherian hopf algebras h over an algebraically closed field k of characteristic zero which are integral domains of gelfand kirillov dimension two and satisfy the condition ext h 1 k k 0 the latter condition is conjecturally redundant as no examples are known among noetherian hopf algebra domains of gk dimension two where it fails

qualitative and quantitative analysis of stochastic processes based on measured data i theory and applications to synthetic data analysis of stochastic processes governed by the langevin equation is discussed the analysis is based on a general method for non parametric estimation of deterministic and random terms of the langevin equation directly from given data separate estimation of the terms corresponds to decomposition of process dynamics into deterministic and random components such decomposition provides a basis for qualitative and quantitative analysis of process dynamics in part i the following analysis possibilities are described and illustrated using various synthetic datasets 1 qualitative inspection of the estimated terms presented as fields 2 reconstruction of the deterministic and stochastic evolution of the process and 3 approximation of the deterministic term by an analytical function and quantitative treatment of the equations obtained in part ii these analysis possibilities are applied to experimental datasets from metal cutting and laser beam welding

a note on the approximation of two sided rayleigh quotient to eigenvalues abstract we consider the two sided rayleigh quotient introduced by ostrowski as convergence acceleration tool for eigenvalue solvers an interesting property of the two sided rayleigh quotient is established its approximation to eigenvalues depends on the eigenvalue condition number this fact is valid not only for matrix but also for matrix polynomial multiparameter system and periodic matrix pairs

eigenvalues of rayleigh quotient matrices support thata is ann n hermitian matrix andx 1 y 1 aren 1 matrices satisfyingx 1 h x 1 y 1 h y 1 i 1 l n such that the column space x 1 is an eigenspace ofa and y 1 is an approximation of x 1 let a 1 x 1 h ax 1 h 1 y 1 h ay 1 r ay 1 y 1 h 1 and theta arccos x 1 h y 1 y 1 h x 1 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 geqslant 0 the matricesa 1 andh 1 are called the rayleigh quotient matrices ofa with respect tox 1 andy 1 respectively this paper derives new upper bounds of the difference between h 1 and a 1 the set of the eigenvaluesh 1 anda 1 in terms of the quantities sin and or r in a simple way here denotes a unitarily invariant norm

backward errors for eigenvalue and singular value decompositions we present bounds on the backward errors for the symmetric eigenvalue decomposition and the singular value decomposition in the two norm and in the frobenius norm through different orthogonal decompositions of the computed eigenvectors we can define different symmetric backward errors for the eigenvalue decomposition when the computed eigenvectors have a small residual and are close to orthonormal then all backward errors tend to be small consequently it does not matter how exactly a backward error is defined and how exactly residual and deviation from orthogonality are measured analogous results hold for the singular vectors we indicate the effect of our error bounds on implementations for eigenvector and singular vector computation in a more general context we prove that the distance of an appropriately scaled matrix to its orthogonal qr factor is not much larger than its distance to the closest orthogonal matrix

on the alternating direction method of multipliers for nonnegative inverse eigenvalue problems with partial eigendata we consider the nonnegative inverse eigenvalue problem with partial eigendata which aims to find a nonnegative matrix such that it is nearest to a pre estimated nonnegative matrix and satisfies the prescribed eigendata in this paper we propose several iterative schemes based on the alternating direction method of multipliers for solving the nonnegative inverse problem we also extend our schemes to the symmetric case and the cases of prescribed lower bounds and of prescribed entries numerical tests including a practical engineering application in vibrations show the efficiency of the proposed iterative schemes

classification of connected hopf algebras of dimension p 3 i abstract let p be a prime and k be an algebraically closed field of characteristic p in this paper we provide the classification of connected hopf algebras of dimension p 3 except for the case when the primitive space of the hopf algebra is a two dimensional abelian restricted lie algebra each isomorphism class is presented by generators x y z with relations and hopf algebra structures let be the multiplicative group of p 2 p 1 th roots of unity when the primitive space is one dimensional and p is odd there is an infinite family of isomorphism classes which is naturally parameterized by a k 1

lyapunov statistics and mixing rates for intermittent systems we consider here a recent conjecture stating that correlation functions and tail probabilities of finite time lyapunov exponents would have the same power law decay in weakly chaotic systems we demonstrate that this conjecture fails for a generic class of maps of the pomeau manneville type we show further that typically the decay properties of such tail probabilities do not provide significant information on key aspects of weakly chaotic dynamics such as ergodicity and instability regimes our approaches are firmly based on rigorous results particularly the aaronson darling kac theorem and are also confirmed by exhaustive numerical simulations

characterizing dynamics with covariant lyapunov vectors a general method to determine covariant lyapunov vectors in both discrete and continuous time dynamical systems is introduced this allows us to address fundamental questions such as the degree of hyperbolicity which can be quantified in terms of the transversality of these intrinsic vectors for spatially extended systems the covariant lyapunov vectors have localization properties and spatial fourier spectra qualitatively different from those composing the orthonormalized basis obtained in the standard procedure used to calculate the lyapunov exponents

the hermitian p k 1 anti reflexive solutions of a linear matrix equation in this paper we first establish the necessary and sufficient conditions for the existence and the explicit expressions of the hermitian p k 1 anti reflexive solutions of the matrix equation a x b and meanwhile the best approximation solution is considered then if the solvability conditions are not satisfied the least squares hermitian p k 1 anti reflexive solutions and the least squares hermitian p k 1 anti reflexive solutions with the minimum norm of the above matrix equation are respectively derived in addition two algorithms are shown to compute the least squares hermitian p k 1 anti reflexive solutions and the corresponding numerical examples are also given to illustrate the feasibility of the algorithms

chatter onset in non regenerative cutting a numerical study a non linear model of non regenerative cutting is analyzed when damped the model has a stable fixed point for small values of chip thickness as chip thickness is increased the fixed point becomes unstable via a sub critical hopf bifurcation this transition corresponds to the onset of chatter when dynamic noise is added to the system chatter onset can occur in the sub critical parameter region although chatter free cutting is predicted by the stability analysis of the fixed point a noisy bifurcation diagram is compared qualitatively to a diagram obtained experimentally in regenerative cutting similar features regarding the onset of chatter are observed in both diagrams

lifting of quantum linear spaces and pointed hopf algebras of orderp3 abstract we propose the following principle to study pointed hopf algebras or more generally hopf algebras whose coradical is a hopf subalgebra given such a hopf algebraa consider its coradical filtration and the associated graded coalgebra gr a then gr ais a graded hopf algebra since the coradicala0ofais a hopf subalgebra in addition there is a projection gr a a0 letrbe the algebra of coinvariants of then by a result of radford and majid ris a braided hopf algebra and gr ais the bosonization or biproduct ofranda0 gr a r a0 the principle we propose to studyais first to studyr then to transfer the information to gr avia bosonization and finally to lift toa in this article we apply this principle to the situation whenris the simplest braided hopf algebra a quantum linear space as consequences of our technique we obtain the classification of pointed hopf algebras of orderp3 pan odd prime over an algebraically closed field of characteristic zero with the same hypothesis the characterization of the pointed hopf algebras whose coradical is abelian and has indexporp2 and an infinite family of pointed nonisomorphic hopf algebras of the same dimension this last result gives a negative answer to a conjecture of i kaplansky

linearization in the large of nonlinear systems and koopman operator spectrum abstract according to the hartman grobman theorem a nonlinear system can be linearized in a neighborhood of a hyperbolic stationary point here we extend this linearization around stable unstable equilibria or periodic orbits to the whole basin of attraction for both discrete diffeomorphisms and flows we discuss the connection of the linearizing transformation to the spectrum of koopman operator

linear response of the lorenz system the present numerical study provides strong evidence that at standard parameters the response of the lorenz system to small perturbations of the control parameter r is linear this evidence is obtained not only directly by determining the response in the observable a x z but also indirectly by validating various implications of the assumption of a linear response like a quadratic response at twice the perturbation frequency a vanishing response in a x x the kramers kronig relations and relations between different response functions since the lorenz system is nonhyperbolic the present results indicate that in contrast to a recent speculation the large system limit thermodynamic limit need not be invoked to obtain a linear response for chaotic systems of this type

residual bounds of approximate solutions of the algebraic riccati equation let tilde x geq 0 approximate the unique hermitian positive semi definite solution x to the algebraic riccati equation are where f g geq 0 a f is stabilizable and a g is detectable let be the residual of the are with respect to tilde x and define the linear operator vec l by by applying a new forward perturbation bound to the optimal backward perturbation corresponding to the approximate solution tilde x we obtained the following result if a f tilde x is stable and if 4 vec l 1 vec l 1 hat r f 1 for any unitarily invariant norm then

statistics of the stochastically forced lorenz attractor by the fokker planck equation and cumulant expansions we investigate the fokker planck description of the equal time statistics of the three dimensional lorenz attractor with additive white noise the invariant measure is found by computing the zero or null mode of the linear fokker planck operator as a problem of sparse linear algebra two variants are studied a self adjoint construction of the linear operator and the replacement of diffusion with hyperdiffusion we also access the low order statistics of the system by a perturbative expansion in equal time cumulants a comparison is made to statistics obtained by the standard approach of accumulation via direct numerical simulation theoretical and computational aspects of the fokker planck and cumulant expansion methods are discussed

inexact and incremental bilinear lanczos components algorithms for high dimensionality reduction and image reconstruction recently matrix based methods have gained wide attentions in pattern recognition and machine learning communities the generalized low rank approximations of matrices glram and the bilinear lanczos components blc algorithm are two popular algorithms that treat data as the native two dimensional matrix patterns however these two algorithms often require heavy computation time and memory space in practice especially for large scale problems in this paper we propose inexact and incremental bilinear lanczos components algorithms for high dimensionality reduction and image reconstruction we first introduce the thick restarting strategy to the blc algorithm and present a thick restarted lanczos components trblc algorithm in this algorithm we use the ritz vectors as approximations to dominant eigenvectors instead of the lanczos vectors in our implementation the iterative matrices are neither formed nor stored explicitly thanks to the characteristic of the lanczos procedure then we explore the relationship between the reconstruction error and the accuracy of the ritz vectors so that the computational complexities of eigenpairs can be reduced significantly as a result we propose an inexact thick restarted lanczos components inex trblc algorithm moreover we investigate the problem of incremental generalized low rank approximations of matrices and propose an incremental and inexact trblc incr trblc algorithm numerical experiments illustrate the superiority of the new algorithms over the glram algorithm and its variations as well as the blc algorithm for some real world image reconstruction and face recognition problems highlightswe proposed an inexact algorithm for image reconstructionthis algorithm is efficient both in storage and running time we analyze its convergence and backward errors an incremental version of the algorithm is given to handle dynamic increased data set

continuity properties of the 1 inverse and perturbation bounds for the drazin inverse there are known results about the continuity of the important pseudoinverses we extend some well known results to the 1 inverse as for the drazin inverse which is not a 1 inverse we present new perturbation bounds

on the variation of the spectrum of a normal matrix abstract let a and a be two n n normal matrices with spectra and j then by the hoffman wielandt theorem there is a permutation of 1 2 n such that n j 1 j j 2 a a f where f denotes the frobenius norm however if a is normal but a nonnormal it may be asked how to relate the eigenvalues of a to those of a an answer is given in this paper there is a permutation of 1 2 n such that n j 1 j j 2 n a a f and the factor n is best possible as a corollary we have j j n a a 2 for the spectral norm 2 thus the known upper bound 2n 1 a a 2 is reduced by a factor of about two

liouvillian dynamics of the hopf bifurcation two dimensional vector fields undergoing a hopf bifurcation are studied in a liouville equation approach the liouville equation rules the time evolution of statistical ensembles of trajectories issued from random initial conditions but evolving under the deterministic dynamics the time evolution of the probability densities of such statistical ensembles can be decomposed in terms of the spectrum of the resonances i e the relaxation rates of the liouvillian operator or the related frobenius perron operator the spectral decomposition of the liouvillian operator is explicitly constructed before at and after the hopf bifurcation because of the emergence of time oscillations near the hopf bifurcation the resonance spectrum turns out to be complex and defined by both relaxation rates and oscillation frequencies the resonance spectrum is discrete far from the bifurcation and becomes continuous at the bifurcation this continuous spectrum is caused by the critical slowing down of the oscillations occurring at the hopf bifurcation and it leads to power law relaxation as 1 square root of t of the probability densities and statistical averages at long times t infinity moreover degeneracy in the resonance spectrum is shown to yield a jordan block structure in the spectral decomposition

a robust nonparametric framework for reconstruction of stochastic differential equation models abstract in this paper we employ a nonparametric framework to robustly estimate the functional forms of drift and diffusion terms from discrete stationary time series the proposed method significantly improves the accuracy of the parameter estimation in this framework drift and diffusion coefficients are modeled through orthogonal legendre polynomials we employ the least squares regression approach along with the euler maruyama approximation method to learn coefficients of stochastic model next a numerical discrete construction of mean squared prediction error mspe is established to calculate the order of legendre polynomials in drift and diffusion terms we show numerically that the new method is robust against the variation in sample size and sampling rate the performance of our method in comparison with the kernel based regression kbr method is demonstrated through simulation and real data in case of real dataset we test our method for discriminating healthy electroencephalogram eeg signals from epilepsy ones we also demonstrate the efficiency of the method through prediction in the financial data in both simulation and real data our algorithm outperforms the kbr method

hochschild cohomology and the coradical filtration of pointed coalgebras applications abstract in this paper we show that there is a close connection between the coradical filtration of a pointed coalgebra and the hochschild cohomology of that coalgebra with coefficients in some one dimensional bicomodules as an application for a given prime numberpand an algebraically closed fieldkof characteristic 0 we classify all pointed hopf algebras of dimensionp3overk

resonances in a chaotic attractor crisis of the lorenz flow local bifurcations of stationary points and limit cycles have successfully been characterized in terms of the critical exponents of these solutions lyapunov exponents and their associated covariant lyapunov vectors have been proposed as tools for supporting the understanding of critical transitions in chaotic dynamical systems however it is in general not clear how the statistical properties of dynamical systems change across a boundary crisis during which a chaotic attractor collides with a saddle this behavior is investigated here for a boundary crisis in the lorenz flow for which neither the lyapunov exponents nor the covariant lyapunov vectors provide a criterion for the crisis instead the convergence of the time evolution of probability densities to the invariant measure governed by the semigroup of transfer operators is expected to slow down at the approach of the crisis such convergence is described by the eigenvalues of the generator of this semigroup which can be divided into two families referred to as the stable and unstable ruelle pollicott resonances respectively the former describes the convergence of densities to the attractor or escape from a repeller and is estimated from many short time series sampling the state space the latter is responsible for the decay of correlations or mixing and can be estimated from a long times series invoking ergodicity it is found numerically for the lorenz flow that the stable resonances do approach the imaginary axis during the crisis as is indicative of the loss of global stability of the attractor on the other hand the unstable resonances and a fortiori the decay of correlations do not flag the proximity of the crisis thus questioning the usual design of early warning indicators of boundary crises of chaotic attractors and the applicability of response theory close to such crises

maximal lyapunov exponent at crises we study the variation of lyapunov exponents of simple dynamical systems near attractor widening and attractor merging crises the largest lyapunov exponent has universal behavior showing abrupt variation as a function of the control parameter as the system passes through the crisis point either in the value itself in the case of an attractor widening crisis or in the slope for an attractor merging crisis the distribution of local lyapunov exponents is very different for the two cases the fluctuations remain constant through a merging crisis but there is a dramatic increase in the fluctuations at a widening crisis

diffusion effusion and chaotic scattering an exactly solvable liouvillian dynamics we study diffusion and chaotic scattering in a chain of baker maps coupled together which forms an area preserving mapping of an infinitely extended strip onto itself this exactly solvable mapping sustains chaotic behaviors and diffusion processes the relationship between the diffusion coefficient the lyapunov exponent and the entropy per unit time is derived the long lived classical resonances of the liouville evolution operator are proved to converge toward the eigenvalues of the phenomenological diffusion equation in this sense there is a quasi isomorphism between the resonance spectrum of the liouville evolution and the eigenvalue spectrum of the phenomenological diffusion equation furthermore we show that a fractal repeller is associated to each non equilibrium state in the isolated and finite multibaker chain the nonequilibrium states are all unstable with respect to the equilibrium validating a weak form of the second principle of thermodynamics for the present dynamical system consequences of nonequilibrium fractals on classical measurements are discussed we then describe the open multibaker chain as a scattering system fractal properties of chaotic scattering are here shown to be related to diffusion in the chain

linear response susceptibility and resonances in chaotic toy models abstract we consider simple examples illustrating some new features of the linear response theory developed by ruelle for dissipative and chaotic systems d ruelle smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics j stat phys 95 1999 393 468 in this theory the concepts of linear response susceptibility and resonance which are familiar to physicists have been revisited due to the dynamical contraction of the whole phase space onto attractors in particular the standard framework of the fluctuation dissipation theorem breaks down and new resonances can show up outside the power spectrum in previous papers we proposed and used new numerical methods to demonstrate the presence of the new resonances predicted by ruelle in a model of chaotic neural networks in this article we deal with simpler models which can be worked out analytically in order to gain more insights into the genesis of the stable resonances and their consequences on the linear response of the system we consider a class of two dimensional time discrete maps describing simple rotator models with a contracting radial dynamics onto the unit circle and a chaotic angular dynamics t 1 2 t mod 2 a generalisation of this system to a network of interconnected rotators is also analysed and related with our previous studies b cessac j a sepulchre stable resonances and signal propagation in a chaotic network of coupled units phys rev e 70 2004 056111 b cessac j a sepulchre transmitting a signal by amplitude modulation in a chaotic network chaos 16 2006 013104 1 13104 12 these models permit us to classify the different types of resonances in the susceptibility and to discuss in particular the relation between the relaxation time of the system to equilibrium with the mixing time given by the decay of the correlation functions also it enables one to propose some general mechanisms responsible for the creation of stable resonances with arbitrary frequencies widths and dependence on the pair of perturbed observed variables

nonnegative inverse eigenvalue problems with partial eigendata in this paper we consider the inverse problem of constructing an n n real nonnegative matrix a from the prescribed partial eigendata we first give the solvability conditions for the inverse problem without the nonnegative constraint and then discuss the associated best approximation problem to find a nonnegative solution we reformulate the inverse problem as a monotone complementarity problem and propose a nonsmooth newton type method for solving its equivalent nonsmooth equation under some mild assumptions the global and quadratic convergence of our method is established we also apply our method to the symmetric nonnegative inverse problem and to the cases of prescribed lower bounds and of prescribed entries numerical tests demonstrate the efficiency of the proposed method and support our theoretical findings

stochastic time series with strong correlated measurement noise markov analysis in n dimensions an extension and generalization of a recently presented approach for the analysis of langevin type stochastic processes in the presence of strong measurement noise is presented for a stochastic process in n dimensions which is superimposed with strong exponentially correlated gaussian distributed measurement noise it is possible to extract the strength and the correlation functions of the noise as well as polynomial approximations of the drift and diffusion functions of the underlying process

relative eigenvalues of a definite matrix pair abstract let a and b be hermitian matrices and let c a b min x 2 1 xh a ib x the matrix pair a b is called a definite pair and the corresponding eigenvalue problem ax bx is definite if c a b 0 the relationship between the eigenvalues of a b and those of a definite pair of the form xh1ax1 xh1bx1 is studied in this paper the eigenvalues of xh1ax1 xh1bx1 are said to be the relative eigenvalues of a b with respect to the subspace r x1 from the main result of this paper one can deduce a corresponding conclusion on hermitian matrices

eigenvector free solutions to ax b with px xp and xh sx constraints abstract the matrix equation ax b with px xp and xh sx constraints is considered where p is a given hermitian involutory matrix and s 1 by an eigenvalue decomposition of p we equivalently transform the constrained problem to two well known constrained problems and represent the solutions in terms of the eigenvectors of p using moore penrose generalized inverses of the products generated by matrices a b and p the involved eigenvectors can be released and eigenvector free formulas of the general solutions are presented similar strategy is applied to the equations ax b xc d with the same constraints

stochastic model for heart rate fluctuations a normal human heart rate shows complex fluctuations in time which is natural because the heart rate is controlled by a large number of different feedback control loops these unpredictable fluctuations have been shown to display fractal dynamics long term correlations and 1 f noise these characterizations are statistical and they have been widely studied and used but much less is known about the detailed time evolution dynamics of the heart rate control mechanism here we show that a simple one dimensional langevin type stochastic difference equation can accurately model the heart rate fluctuations in a time scale from minutes to hours the model consists of a deterministic nonlinear part and a stochastic part typical to gaussian noise and both parts can be directly determined from the measured heart rate data studies of 27 healthy subjects reveal that in most cases the deterministic part has a form typically seen in bistable systems there are two stable fixed points and one unstable one

connected hopf algebras of dimension p2 abstract let h be a finite dimensional connected hopf algebra over an algebraically closed field k of characteristic p 0 we provide the algebra structure of the associated graded hopf algebra gr h then we study the case when h is generated by a hopf subalgebra k and another element and the case when h is cocommutative when h is a restricted universal enveloping algebra we give a specific basis for the second term of the hochschild cohomology of the coalgebra h with coefficients in the trivial h bicomodule k finally we classify all connected hopf algebras of dimension p 2 over k

structured backward errors for kkt systems karush kuhn tucker kkt systems are linear systems with coefficient matrices of the form graphics where h is symmetric a normwise structured backward error for kkt systems is defined and a com

pointed hopf algebras of dimensionp3 abstract we give a structure theorem for pointed hopf algebras of dimension p 3 having coradical kc p where k is an algebraically closed field of characteristic zero combining this with previous results we obtain the complete classification of all pointed hopf algebras of dimension p 3

perturbation theorems for the generalized eigenvalue problem abstract given a matrix pair z a b the perturbation of its eigenvalues is studied considering two pairs z w as points of the grassman manifold g n 2 n and its eigenvalues as points in g 1 2 the projective complex plane the distance of the spectra measured in the chordal metric in g 1 2 is bounded by some distance of the matrix pairs in g n 2 n analogs of the bauer fike theorem henrici s theorem and the hoffman weilandt theorem are obtained from which the classical results can be derived

spectral signature of the pitchfork bifurcation liouville equation approach the time evolution of probability densities of one dimensional nonlinear vector fields is studied using a liouville equation approach it is shown that the liouville operator admits a discrete spectrum of eigenvalues of decaying type if the vector field is far from bifurcation the associated right and left eigenvectors are explicitly constructed for simple models and shown to be distributions rather than regular functions on the other hand the spectrum of the liouville operator may become continuous at the bifurcation point a phenomenon illustrated explicitly in the paper in the case of the pitchfork bifurcation the relationship between the spectral decompositions of the liouville and of the fokker planck equations is discussed in particular the spectral decompositions constructed here for the liouville equation are obtained as the noiseless limit of the well known spectral decompositions of the fokker planck equation of the associated stochastic process

on estimating the separation of two regular matrix pairs let a b and c d be two n times n n n and m times m m m regular matrix pairs respectively the main contribution of this paper is to derive lower and upper bounds of the separation between a b and c d in terms of their generalized schur decompositions and weierstrass canonical forms the results are illustrated by numerical examples

lyapunov measure for almost everywhere stability this paper is concerned with the analysis and computational methods for verifying global stability of an attractor set of a nonlinear dynamical system based upon a stochastic representation of deterministic dynamics a lyapunov measure is proposed for these purposes this measure is shown to be a stochastic counterpart of stability transience just as an invariant measure is a counterpart of the attractor recurrence it is a dual of the lyapunov function and is useful for the study of more general weaker and set wise notions of stability in addition to the theoretical framework constructive methods for computing approximations to the lyapunov measures are presented these methods are based upon set oriented numerical approaches several equivalent descriptions including a series formula and a system of linear inequalities are provided for computational purposes these descriptions allow one to carry over the intuition from the linear case with stable equilibrium to nonlinear systems with globally stable attractor sets finally in certain cases the exact relationship between lyapunov functions and lyapunov measures is also given

on hopf algebras of dimension 4p abstract in this paper we prove that a non semisimple hopf algebra h of dimension 4 p with p an odd prime over an algebraically closed field of characteristic zero is pointed provided h contains more than two group like elements in particular we prove that non semisimple hopf algebras of dimensions 20 28 and 44 are pointed or their duals are pointed and this completes the classification of hopf algebras in these dimensions

on a class of alternating coefficient matrices quadratic eigenvalue problem in this paper we consider a class of alternating coefficient matrices quadratic eigenvalue problem aqep appearing in a wide range of applications at first the aqep is reformulated as a generalized eigenvalue problem gep by applying tisseur s linearization technique then we point out a property about skew symmetric tridiagonal matrix eigenvalue problem consequently we give that the corresponding gep can be solved by applying the cholesky ql algorithm or lanczos algorithm without complex arithmetic the structured backward error for qep would be discussed because of the special structure of the polynomial eigenvalue problems we develop structured condition number the upper bound and the lower bound are given some computational experiments and some concluding remarks are provided

decay of correlations dedicated to micheline ishay i would like to thank p boyland l chierchia v donnay g de martino c gole j l lebowitz m lyubich m rychlik i g schwarz s vaienti and especially g gallavotti for helpful and enlightening discussions particularly warm thanks go to n chernov for carefully reading and finding a mistake in an early version the present paper benefits from several improvements due to his sharp criticism in addition i am indebted to p collet and most of all m wojtkowski for introducing me to the magical world of cones finally i thank j milnor director of the institute for mathematical sciences at stony brook university where i was visiting during part of this work and the italian c n r gnfm for providing travel funds

symbolic dynamics of noisy chaos abstract one model of randomness observed in physical systems is that low dimensional deterministic chaotic attractors underly the observations a phenomenological theory of chaotic dynamics requires an accounting of the information flow from the observed system to the observer the amount of information available in observations and just how this information affects predictions of the system s future behavior in an effort to develop such a description we discuss the information theory of highly discretized observations of random behavior metric entropy and topological entropy are well defined invariant measures of such an attractor s level of chaos and are computable using symbolic dynamics real physical systems that display low dimensional dynamics are however inevitably coupled to high dimensional randomless e g thermal noise we investigate the effects of such fluctuations coupled to deterministic chaotic systems in particular the metric entropy s response to the fluctuations we find that the entropy increases with a power law in the noise level and that the convergence of the entropy and the effect of fluctuations can be cast as a scaling theory we also argue that in addition to the metric entropy there is a second scaling invariant quantity that characterizes a deterministic system with added fluctuations i0 the maximum average information obtainable about the initial condition that produces a particular sequence of measurements or symbols

structured backward errors for generalized saddle point systems abstract in this paper we investigate structured backward errors for three kinds of generalized saddle point systems where the matrix is not symmetric and its 2 2 block is not zero and has perturbations computable formula of backward errors are derived the expressions are useful for testing the stability of practical algorithms

isomorphism classes of finite dimensional connected hopf algebras in positive characteristic abstract we classify all finite dimensional connected hopf algebras with large abelian primitive spaces we show that they are hopf algebra extensions of restricted enveloping algebras of certain restricted lie algebras for any abelian matched pair associated with these extensions we construct a cohomology group which classifies all the extensions up to equivalence moreover we present a 1 1 correspondence between the isomorphism classes and a group quotient of the cohomology group deleting some exceptional points where the group respects the automorphisms of the abelian matched pair and the exceptional points represent those restricted lie algebra extensions

parametric estimation from approximate data non gaussian diffusions we study the problem of parameters estimation in indirect observability contexts where x t in r r is an unobservable stationary process parametrized by a vector of unknown parameters and all observable data are generated by an approximating process y varepsilon t which is close to x t in l 4 norm we construct consistent parameter estimators which are smooth functions of the sub sampled empirical mean and empirical lagged covariance matrices computed from the observable data we derive explicit optimal sub sampling schemes specifying the best paired choices of sub sampling time step and number of observations we show that these choices ensure that our parameter estimators reach optimized asymptotic l 2 convergence rates which are constant multiples of the l 4 norm y varepsilon t x t

on the approximation of complicated dynamical behavior we present efficient techniques for the numerical approximation of complicated dynamical behavior in particular we develop numerical methods which allow us to approximate sinai ruelle bowen srb measures as well as almost cyclic behavior of a dynamical system the methods are based on an appropriate discretization of the frobenius perron operator and two essentially different mathematical concepts are used our idea is to combine classical convergence results for finite dimensional approximations of compact operators with results from ergodic theory concerning the approximation of srb measures by invariant measures of stochastically perturbed systems the efficiency of the methods is illustrated by several numerical examples

minimization problems for r s symmetric and r s skew symmetric matrices let r cm m and s cn n be nontrivial involutions thus r r 1 i and s s 1 i we say that a cm n is r s symmetric r s skew symmetric if ras a ras a let s be the class of m n r s symmetric matrices or the class of m n r s skew symmetric matrices let z cn q and w cm q we study the following problems i give necessary and sufficient conditions for the existence of an a s such that az w and find all such matrices if the conditions are met ii find z w mina s az w and characterize the class s z w a s az w z w iii if b cm n is arbitrary find z w b mina s z w a b and find a s z w such that a b z w b we obtain explicit formulas for z w b z w and all the matrices in question

estimating long term behavior of flows without trajectory integration the infinitesimal generator approach the long term distributions of trajectories of a flow are described by invariant densities i e fixed points of an associated transfer operator in addition global slowly mixing structures such as almost invariant sets which partition phase space into regions that are almost dynamically disconnected can also be identified by certain eigenfunctions of this operator indeed these structures are often hard to obtain by brute force trajectory based analyses in a wide variety of applications transfer operators have proven to be very efficient tools for an analysis of the global behavior of a dynamical system the computationally most expensive step in the construction of an approximate transfer operator is the numerical integration of many short term trajectories in this paper we propose to directly work with the infinitesimal generator instead of the operator completely avoiding trajectory integration we propose two different discretization schemes a cell based discretization and a spectral coll

analysis of time series from stochastic processes analysis of time series from stochastic processes governed by a langevin equation is discussed several applications for the analysis are proposed based on estimates of drift and diffusion coefficients of the fokker planck equation the coefficients are estimated directly from a time series the applications are illustrated by examples employing various synthetic time series and experimental time series from metal cutting

fokker planck perspective on stochastic delay systems exact solutions and data analysis of biological systems stochastic delay systems with additive noise are examined from the perspective of fokker planck equations for a linear system the exact stationary probability density is derived by means of a delay fokker planck equation we show how to determine the delay equation of the linear system from experimental data and corroborate a fundamental result previously obtained by kuchler and mensch we also propose a method to derive delay equations of nonlinear systems from experimental data to this end the theory of multivariate fokker planck equations is used the applicability of this method is demonstrated for stochastic models describing tracking and pointing movements of humans

trace formula for noisy flows a trace formula is derived for the fokker planck equation associated with ito stochastic differential equations describing noisy time continuous nonlinear dynamical systems in the weak noise limit the trace formula provides estimations of the eigenvalues of the fokker planck operator on the basis of the pollicott ruelle resonances of the noiseless deterministic system which is assumed to be non bifurcating at first order in the noise amplitude the effect of noise on a periodic orbit is given in terms of the period and the derivative of the period with respect to the pseudo energy of the onsager machlup freidlin wentzell scheme