where is the nearest non regular pencil abstract this is a first step toward the goal of finding a way to calculate a smallest norm deregularizing perturbation of a given square matrix pencil minimal de regularizing perturbations have geometric characterizations that include a variable projection linear least squares problem and a minimax characterization reminiscent of the courant fischer theorem the characterizations lead to new computationally attractive upper and lower bounds we give a brief survey and illustrate strengths and weaknesses of several upper and lower bounds some of which are well known and some of which are new the ultimate goal remains elusive

an infinite dimensional morse theory with applications in this paper we construct an infinite dimensional extraordinary cohomology theory and a morse theory corresponding to it these theories have some special properties which make them useful in the study of critical points of strongly indefinite functionals by strongly indefinite we mean a functional unbounded from below and from above on any subspace of finite codimension several applications are given to hamiltonian systems the onedimensional wave equation of vibrating string type and systems of elliptic partial differential equations 0 introduction let e be a real hilbert space with an inner product and let be a twice continuously differentiable functional denote the frechet derivative and the gradient of at x by x and x respectively where as usual x y x y y e recall that a point x0 e is said to be critical if x0 0 or equivalently if x0 0 the level c r will be called regular if 1 c contains no critical points and critical if x0 0 for some x0 1 c let a b a b be two regular levels of denote m 1 a b and consider the restriction of to m in morse theory one is interested in the local topological structure of the level sets of m near a critical point and in the relation between this local structure and the topological structure of the set m to be more specific suppose that x0 m is an isolated critical point of then one defines a sequence of critical groups of at x0 by setting cq x0 hq c u u x0 q 0 1 2 0 1 where c x0 c x e x c hq is the q th singular homology group with coefficients in some field f and u is a neighbourhood of x0 define the morse index of x0 to be the maximal dimension of a subspace of e on which the quadratic form x0 y y is negative definite one shows that if x0 is a nondegenerate critical point i e if x0 e e is invertible then cq x0 f received by the editors march 20 1995 1991 mathematics subject classification primary 58e05 secondary 34c25 35j65 35l05 55n20 58f05

multiple periodic solutions for discrete hamiltonian systems abstract several existence results of multiple periodic solutions for discrete hamiltonian system are established where the nonlinearity is either asymptotically linear or superlinear at infinity the approach used here is based on morse theory

applications of equivariant degree for gradient maps to symmetric newtonian systems abstract we consider g s 1 with being a finite group for which the complete euler ring structure in u g is described the multiplication tables for d 6 s 4 and a 5 are provided in the appendix the equivariant degree for g orthogonal maps is constructed using the primary equivariant degree with one free parameter we show that the g orthogonal degree extends the degree for g gradient maps in the case of g s 1 introduced by g ba in k g ba w krawcewicz j wu an equivariant degree with applications to symmetric bifurcation problems i construction of the degree bull london math soc 69 1994 377 398 the computational results obtained are applied to a symmetric autonomous newtonian system for which we study the existence of 2 periodic solutions for some concrete cases we present the symmetric classification of the solution set for the systems considered

solutions of systems of elliptic differential equations on circular domains abstract we study global bifurcation of weak solutions of systems of elliptic differential equations considered on so 2 invariant domains we formulate sufficient conditions for the existence of unbounded continua of nontrivial solutions branching from the trivial ones as the main tool we use the degree for so 2 equivariant gradient maps defined by the second author in rybicki nonlinear anal tma 23 1 1994 83

bifurcation from higher eigenvalues in nonlinear elliptic equations continua that meet infinity on considere des problemes aux valeurs propres non lineaires pour des equations aux derivees partielles elliptiques i j 1 a ij x u xd i x j q x u f x u sur u 0 sur r n borne

unbounded sets of solutions of non cooperative elliptic systems on spheres abstract the aim of this paper is to show that any continuum of nontrivial solutions of a non cooperative system of elliptic equations on the sphere s n 1 bifurcating from the set of trivial solutions is unbounded moreover we characterize bifurcation points of this system at which the global symmetry breaking phenomenon occurs as the main tool we use the degree for so 2 invariant strongly indefinite functionals defined in 13

generalized linear systems geometric and structural approaches abstract this paper presents a new way of introducing invariant subspaces for generalized systems this comes from somewhat known geometric algorithms but in most cases with initial conditions different from those usually seen in this context these definitions are shown to be consistent with other ones directly deduced from classical approaches like matrix pencil tools or discrete algebraic formalism the key starting point for their introduction is the extension of structural descriptions for these generalized systems such as have been available for proper systems many results from the proper case can thus be extended for instance the geometric definition for controllability indices and something similar to the famous morse canonical decomposition of the strictly proper case some links with the inversion algorithm are also sketched

thermodynamics of the general diffusion process time reversibility and entropy production we introduce an axiomatic thermodynamic theory for the general diffusion process and prove a theorem concerning entropy and irreversibility the equivalence among time reversibility zero entropy production symmetricity of the stationary diffusion process and a potential condition

an arithmetic for matrix pencils theory and new algorithms this paper introduces arithmetic like operations on matrix pencils the pencil arithmetic operations extend elementary formulas for sums and products of rational numbers and include the algebra of linear transformations as a special case these operations give an unusual perspective on a variety of pencil related computations we derive generalizations of monodromy matrices and the matrix exponential a new algorithm for computing a pencil arithmetic generalization of the matrix sign function does not use matrix inverses and gives an empirically forward numerically stable algorithm for extracting deflating subspaces

periodic solutions of autonomous newtonian systems partially asymptotically linear at infinity abstract the goal of this article is to study the existence continuation and global bifurcation of autonomous newtonian systems partially asymptotically linear at infinity the main idea is to prove a kind of splitting lemma and apply the degree for s 1 equivariant gradient maps defined by rybicki 1994 in 16

interpolation of stochastic and deterministic reduced dynamics abstract there are roughly speaking the two extreme types of dynamical density functional theories one is deterministic without noise and another is fully stochastic with noise we attempt to interpolate these extreme limiting theories for general reduced description of time evolution by making use of a series of projection operators which generalize the so called kawasaki gunton operator this helps to clarify though in a formal way the controversies or confusions surrounding the current dynamical density functional theories

parallel algorithms for algebraic riccati equations the matrix sign function is the basis of a parallel algorithm for solving the generalized algebraic riccati equation three forms of the algorithm were implemented and tested on a distributed memory hypercube multiprocessor performance results indicate that the method is an excellent means of solving large scale problems on a parallel computer

global bifurcations of solutions of emden fowler type equation δu x λf u x on an annulus in rn n 3 abstract in this article we study global and symmetry breaking bifurcations of solutions of emden fowler type elliptic differential equations as the main tool we use a topological invariant which is suitable for study of global bifurcations and continuations of critical points of equivariant functionals namely we will apply degree for s 1 equivariant orthogonal maps defined in rybicki 1994 nonlinear anal tma 23 83 102

an inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblems we discuss an inverse free highly parallel spectral divide and conquer algorithm it can compute either an invariant subspace of a nonsymmetric matrix a or a pair of left and right deflating subspaces of a regular matrix pencil a lambda b this algorithm is based on earlier ones of bulgakov godunov and malyshev but improves on them in several ways this algorithm only uses easily parallelizable linear algebra building blocks matrix multiplication and qr decomposition but not matrix inversion similar parallel algorithms for the nonsymmetric eigenproblem use the matrix sign function which requires matrix inversion and is faster but can be less stable than the new algorithm

the effect of interference of coloured additive and multiplicative white noises on escape rate in this paper we have calculated the escape rate from a meta stable state for coloured and correlated noise driven open systems based on the fokker planck description of the stochastic process we consider the effect of two correlation times due to the additive coloured noise and the correlation between additive coloured and multiplicative white noises the effect of the noise correlation strength on the rate has also been investigated

periodic solutions of symmetric autonomous newtonian systems abstract in this paper we show the existence and bifurcation of t periodic solutions of a special form for an autonomous newtonian system with symmetry if the phase space r 2 n is equipped with the structure of an orthogonal representation w w and the potential v w r is invariant then for every such a solution the set of indices of nonvanishing fourier coefficients is finite and depends on w only if the potential v depends on the squares of complex coordinates then for every such a solution t is the minimal period

the gap distance to the set of singular matrix pencils abstract we study matrix pencils s e a using the associated linear subspace ker a e the distance between subspaces is measured in terms of the gap metric in particular we investigate the gap distance of a regular matrix pencil to the set of singular pencils and provide upper and lower bounds for it a relation to the distance to singularity in the frobenius norm is provided

global bifurcation of planar and spatial periodic solutions from the polygonal relative equilibria for the n body problem abstract given n point masses turning in a plane at a constant speed this paper deals with the global bifurcation of periodic solutions for the masses in that plane and in space as a special case one has a complete study of n identical masses on a regular polygon and a central mass the symmetries of the problem are used in order to find the irreducible representations the linearization and with the help of the orthogonal degree theory all the symmetries of the bifurcating branches

perturbation analysis of singular subspaces and deflating subspaces perturbation expansions for singular subspaces of a matrix and for deflating subspaces of a regular matrix pair are derived by using a technique previously described by the author the perturbation expansions are then used to derive fr echet derivatives condition numbers and r th order perturbation bounds for the subspaces vaccaro s result on second order perturbation expansions for a special class of singular subspaces can be obtained from a general result of this paper besides new perturbation bounds for singular subspaces and deflating subspaces are derived by applying a general theorem on solution of a system of nonlinear equations the results of this paper reveal an important fact each singular subspace and each deflating subspace have individual perturbation bounds and individual condition numbers

ab algorithm and its modifications for the spectral problems of linear pencils of matrices this paper describes and algorithm and its modifications for solving spectral problems for linear pencils of matrices both regular as well as singular

the quasi weierstras form for regular matrix pencils regular linear matrix pencils a e k n n where k q r or c and the associated differential algebraic equation dae e u x ax are studied the wong sequences of subspaces are investigate and invoked to decompose the k n into v w where any bases of the linear spaces v and wtransform the matrix pencil into the quasi weierstras form the quasi weierstras form of the matrix pencil decouples the original dae into the underlying ode and the pure dae or in other words decouples the set of initial values into the set of consistent initial values vand pure inconsistent initial values w 0 furthermore vand ware spanned by the general ized eigenvectors at the finite and infinite eigenvalues resp the quasi weierstras form is used to show how chains of generalized eigenvectors at finite and infinite eigenvalues of a e lead to the well known weierstras form so the latter can be viewed as a generalized jordan form finally it is shown how eigenvector chains constitute a basis for the solution space of e u x ax

the generalized newton iteration for the matrix sign function in this paper we present modified algorithms for computing deflating subspaces of matrix pairs using the matrix sign function our new algorithms achieve a considerable reduction of the computational cost of the generalized newton iteration for the matrix sign function and improve the accuracy of the computed deflating subspaces the matrix sign function is thus revealed as an effective technique for applications in which bases for the deflating subspaces are required when partial or complete information about the eigenspectrum is desired the matrix sign function can be used as an initial divide and conquer technique the basic kernels involved in this iteration are especially appropriate for current high performance architectures

degenerate branching points of autonomous hamiltonian systems abstract this paper deals with non constant 2 periodic solutions of x t j h x t where 0 and h c 2 r n r n r with 2 h x 0 i 0 0 2 2 h x 0 for x 0 h 1 0 sufficient conditions for the existence of connected branches of such solutions bifurcating from x 0 0 have been formulated the corresponding theorem concerning connected branches of arbitrary periodic nonstationary trajectories of the hamiltonian system x t j h x t emanating from x 0 has been proved minimal periods of trajectories near x 0 have been described

nonlinear langevin equations and the time dependent density functional method to study the time dependent density functional method tddfm two streaming velocity reversible terms are reformulated in the nonlinear langevin equation mori s prog theor phys 33 423 1965 projection operator method shows a variety of nonlinear langevin equations this is because the equations depend on the choice of phase space functions employed in the projection if phase space functions include particular functions however the streaming velocity term has an invariable form the form is independent of the choice of other phase space functions since the invariable streaming velocity term does not introduce the tddfm the second viewpoint is presented in this the linearization of the streaming velocity term agrees with the frequency term in the linear langevin equation since only the second streaming velocity term introduces the tddfm one needs to be cautious in the derivation of the tddfm

existence and continuation of solutions for a nonlinear neumann problem abstract in this article we study the existence continuation and bifurcation from infinity of nonconstant solutions for a nonlinear neumann problem we apply the leray schauder degree and the degree for s o 2 equivariant gradient operators defined by the second author in s rybicki s o 2 degree for orthogonal maps and its applications to bifurcation theory nonlinear anal tma 23 1 1994 83 102

levy flights in confining potentials we analyze confining mechanisms for levy flights when they evolve in suitable external potentials their variance may exist and show signatures of a superdiffusive transport two classes of stochastic jump type processes are considered those driven by langevin equation with levy noise and those named topological levy processes occurring in systems with topological complexity such as folded polymers or complex networks whose langevin representation is unknown and possibly nonexistent our major finding is that both above classes of processes stay in affinity and may share common stationary probability density even if their detailed dynamical behavior look different this near equilibrium observation seems to be generic to a broad class of levy noise driven processes such as e g superdiffusion on folded polymers geophysical flows and even climatic changes

entropy production in nonequilibrium statistical mechanics we consider systems of nonequilibrium statistical mechanics driven by nonconservative forces and in contact with an ideal thermostat these are smooth dynamical systems for which one can define natural stationary states srb in the simplest case and entropy production e minus the sum of the lyapunov exponents in the simplest case we give exact and explicit definitions of the entropy production e for the various situations of physical interest we prove that e 0 and indicate cases where e 0 the novelty of the approach is that we do not try to compute entropy production directly but make it depend on the identification of a natural stationary state for the system

principle of maximum entropy and reduced dynamics a new method to obtain a series of reduced dynamics at various stages of coarse graining is proposed this ranges from the most coarse grained one which agrees with the deterministic time evolution equation for averages of the relevant variables to the least coarse grained one which is the generalized fokker planck equation for the probability distribution function of the relevant variables the method is based on the extention of the kawasaki gunton operator with the help of the principle of maximum entropy

multiple periodic solutions for γ symmetric newtonian systems abstract the existence of periodic solutions in symmetric newtonian systems x f x can be effectively studied by means of the o 2 equivariant gradient degree with values in the euler ring u o 2 in this paper we show that in the case of being a finite group the euler ring u o 2 and the related basic degrees are effectively computable using euler ring homomorphisms the burnside ring a o 2 and the reduced o 2 degree with no free parameters we present several examples of newtonian systems with various symmetries for which we show existence of multiple periodic solutions we also provide exact value of the equivariant topological invariant for those problems

linear relations and the kronecker canonical form abstract we show that the kronecker canonical form which is a canonical decomposition for pairs of matrices is the representation of a linear relation in a finite dimensional space this provides a new geometric view upon the kronecker canonical form each of the four entries of the kronecker canonical form has a natural meaning for the linear relation which it represents these four entries represent the jordan chains at finite eigenvalues the jordan chains at infinity the so called singular chains and the multi shift part or to state it more concisely for linear relations the kronecker canonical form is the analogue of the jordan canonical form for matrices

ascent descent nullity defect and related notions for linear relations in linear spaces for a linear relation in a linear space the concepts of ascent descent nullity and defect are introduced and studied it is shown that the results of a e taylor and m a kaashoek concerning the relationship between ascent descent nullity and defect for the case of linear operators remain valid in the context of linear relations sometimes under the additional condition that the linear relation does not have any nontrivial singular chains in particular it is shown for a linear relation a with a trivial singular chain manifold whose ascent p is finite and whose nullity and defect are equal and finite that the linear space h is a direct sum of ker a p and ran a p furthermore it is shown that the various results which require the absence of singular chains are not valid when such chains are present c 2007 elsevier inc all rights reserved

periodic solutions of asymptotically linear autonomous newtonian systems with resonance abstract in this paper we study the existence continuation and bifurcation from infinity of 2 periodic solutions of autonomous newtonian systems we underline that the resonant case is considered to prove the results we apply the degree for s 1 equivariant gradient maps defined by rybicki 1994 in 15 and the angle condition introduced by bartsch and li 1997 in 16

langevin equation for the density of a system of interacting langevin processes we present a simple derivation of the stochastic equation obeyed by the density function for a system of langevin processes interacting via a pairwise potential the resulting equation is considerably different from the phenomenological equations usually used to describe the dynamics of non conserved model a and conserved model b particle systems the major feature is that the spatial white noise for this system appears not additively but multiplicatively this simply expresses the fact that the density cannot fluctuate in regions devoid of particles the steady state for the density function may however still be recovered formally as a functional integral over the coursed grained free energy of the system as in models a and b

global bifurcations of solutions of elliptic differential equations abstract the aim of this article is to prove global bifurcation theorems for s 1 equivariant potential operators of the form compact perturbation of identity as an application we prove that components of the set of nontrivial solutions of system formula which bifurcate from the set of trivial solutions are unbounded in suitably chosen space the important point to note here is that our results apply only to systems with an even number of equations as the main tool we use degree theory for s 1 equivariant potential operators constructed by the author in 16 and developed in 17

stochastic operators information and entropy for a stochastic operatoru on andl1 space i eu is linear positive and norm preserving on the positive cone ofl1 it is shown thatu decreases relative information between two nonnegativel1 functions furthermore it is shown that the following properties ofu are closely related u is energy decreasing energy preserving u ish decreasing whereh is boltzmann sh functional and the maxwell distributions are fixed points ofu

global bifurcations of critical orbits of g invariant strongly indefinite functionals abstract let h be a separable hilbert space which is an orthogonal representation of a compact lie group g and let h r be a g invariant strongly indefinite functional of the class c 1 to study critical orbits of we have defined the degree for g invariant strongly indefinite functionals which is an element of the euler ring u g using this degree we have formulated the rabinowitz alternative for g invariant strongly indefinite functionals the abstract results are applied to the study of global bifurcations of weak solutions of non cooperative systems of elliptic differential equations

on the trend to equilibrium for some dissipative systems with slowly increasing a priori bounds we prove convergence to equilibrium with explicit rates for various kinetic equations with relatively bad control of the distribution tails in particular boltzmann type equations with smoothed soft potentials we compensate the lack of uniform in time estimates by the use of precise logarithmic sobolev type inequalities and the assumption that the initial datum decays rapidly at large velocities our method not only gives explicit results on the times of convergence but is also able to cover situations in which compactness arguments apparently do not apply even mere convergence to equilibrium was an open problem for soft potentials

existence and continuation of periodic solutions of autonomous newtonian systems abstract in this article we study the existence and the continuation of periodic solutions of autonomous newtonian systems to prove the results we apply the infinite dimensional version of the degree for so 2 equivariant gradient operators defined by the third author in nonlinear anal theory methods appl 23 1 1994 83 102 and developed in topol meth nonlinear anal 9 2 1997 383 417 using the results due to rabier symmetries topological degree and a theorem of z q wang j math 24 3 1994 1087 1115 and wang symmetries and calculation of the degree chinese ann math 10 1989 520 536 we show that the leray schauder degree is not applicable in the proofs of our theorems because it vanishes

periodic solutions of asymptotically linear autonomous hamiltonian systems abstract in this paper we study the existence and the continuation of periodic solutions of autonomous hamiltonian systems we apply the degree for g invariant strongly indefinite functionals defined in gol biewska and rybicki 2011 2 we consider resonant as well as nonresonant case at stationary solutions

existence of periodic solutions for semilinear reaction diffusion systems abstract in this paper we consider the existence of periodic solutions of reaction diffusion systems by using s 1 degree theory due to dylawerski et al see jodel et al ann pol math 41 1991 243

a qr type reduction for computing the svd of a general matrix product quotient in this paper a qr type reduction technique is developed for the computation of the svd of a general matrix product quotient a a 1 s 1 a 2 s 2 cdots a m s m with a i in bf r n times n hbox and s i 1 or s i 1 first the matrix a is reduced by at most m qr factorizations to the form q 1 11 q 1 21 1 where q 1 11 q 1 21 in bf r n times n and q 1 11 tq 1 11 q 1 21 tq 1 21 i then the svd of a is obtained by computing the csd cosine sine decomposition of q 1 11 and q 1 21 using the matlab command gsvd the performance of the proposed method is verified by some numerical examples

perturbation analysis of the matrix sign function perturbation expansions and new perturbation bounds for the matrix sign function are derived by using a simple technique the perturbation expansions are then used to derive frechet derivatives th

equivariant bifurcation index abstract we consider a bifurcation index bif g k 0 1 u g defined in terms of the degree for g equivariant gradient maps see g ba 1997 21 rybicki 1994 22 rybicki 2005 23 where g is a real compact connected lie group and u g is the euler ring of g see tom dieck 1977 29 tom dieck 1987 30 the main result of this article is the following bif g k 0 1 u g iff bif t k 0 1 u t where t g is a maximal torus of g it is also shown that all the bifurcation points of weak solutions of the following problem u f u in b n u 0 on s n 1 are global bifurcation points additionally the global symmetry breaking bifurcation points are characterised

global continua of periodic solutions of singular first order hamiltonian systems of n vortex type the paper deals with singular first order hamiltonian systems of the form begin aligned gamma k dot z k t j nabla z k h z t quad z k t in omega subset mathbb r 2 k 1 dots n end aligned where j in mathbb r 2 times 2 defines the standard symplectic structure in mathbb r 2 gamma 1 dots gamma n in mathbb r setminus 0 are given and the hamiltonian h is of n vortex type begin aligned h z 1 dots z n frac 1 2 pi sum genfrac 0 0pt j k 1 j ne k n gamma j gamma k log left z j z k right f z end aligned this is defined on the configuration space z 1 ldots z n in omega n z j ne z k text for j ne k of n different points in the domain omega subset mathbb r 2 the function f omega n rightarrow mathbb r may have additional singularities near the boundary of omega n we prove the existence of a global continuum of periodic solutions z t z 1 t dots z n t in omega n that emanates after introducing a suitable singular limit scaling from a relative equilibrium z t in mathbb r 2n of the n vortex problem in the whole plane where f 0 examples for z include thomson s vortex configurations or equilateral triangle solutions the domain omega need not be simply connected a special feature is that the associated action integral is not defined on an open subset of the space of 2 pi periodic h 1 2 functions the natural form domain for first order hamiltonian systems this is a consequence of the singular character of the hamiltonian our main tool in the proof is a degree for s 1 equivariant gradient maps that we adapt to this class of potential operators

parallel algorithm for solving some spectral problems of linear algebra abstract an algorithm is described for computing the deflating subspaces of a regular linear matrix pencil b a more precisely the algorithm is intended to compute the projection matrices p and i p onto the deflating subspaces of matrix pencils corresponding to the eigenvalues inside and outside the unit circle this algorithm can be considered as an orthogonal version of the matrix sign function method it possesses a quadratic convergence rate and simultaneously with the projection operators operators computes a condition number of the problem we point out the extent to which this algorithm can be parallelized applications are made to solving the riccati equation

operators without eigenvalues in finite dimensional vector spaces abstract we introduce the concept of a canonical subspace of c d z and among other results prove the following statements an operator in a finite dimensional vector space has no eigenvalues if and only if it is similar to the operator of multiplication by the independent variable on a canonical subspace of c d z an operator in a finite dimensional pontryagin space is symmetric and has no eigenvalues if and only if it is isomorphic to the operator of multiplication by the independent variable in a canonical subspace of c d z with an inner product determined by a full matrix polynomial nevanlinna kernel

constant coefficient differential algebraic operators and the kronecker form abstract we consider constant coefficient differential algebraic equations from an operator theoretic point of view we show that the kronecker form allows to determine the nullspace and range of the corresponding differential algebraic operators this yields simple matrix theoretic characterizations of features like closed range and fredholmness

bifurcation and symmetry breaking of solutions of systems of elliptic differential equations in this article we study global bifurcations of weak solutions of the following variational system of elliptic differential equations u au u u in u 0on we prove sufficient conditions for the existence of unbounded continua of nontrivial solutions emanating from the trivial ones and necessary conditions for the existence of bounded continua moreover we describe a symmetry breaking phenomenon that occurs on that continua

symmetric liapunov center theorem for minimal orbit abstract using the techniques of equivariant bifurcation theory we prove the existence of non stationary periodic solutions of symmetric systems q t u q t in any neighborhood of an isolated orbit of minima q 0 of the potential u we show the strength of our result by proving the existence of new families of periodic orbits in the lennard jones two and three body problems and in the schwarzschild three body problem

noise and conditional entropy evolution abstract we study the convergence properties of the conditional kullback leibler entropy in stochastic systems we have proved general results showing that asymptotic stability is a necessary and sufficient condition for the monotone convergence of the conditional entropy to its maximal value of zero additionally we have made specific calculations of the rate of convergence of this entropy to zero in a one dimensional situation illustrated by ornstein uhlenbeck and rayleigh processes higher dimensional situations and a two dimensional ornstein uhlenbeck process with a stochastically perturbed harmonic oscillator and colored noise as examples we also apply our general results to the problem of conditional entropy convergence in the presence of dichotomous noise in both the one dimensional and multidimensional cases we show that the convergence of the conditional entropy to zero is monotone and at least exponential in the specific cases of the ornstein uhlenbeck and rayleigh processes as well as the stochastically perturbed harmonic oscillator and colored noise examples we obtain exact formulae for the temporal evolution of the conditional entropy starting from a concrete initial distribution the rather surprising result in this case is that the rate of convergence of the entropy to zero is independent of the noise amplitude

fluctuations and non equilibrium thermodynamics abstract a closed macroscopic system with fluctuating local properties is treated as a thermodynamic system with internal degrees of freedom gibb s entropy postulate is used to define the systems entropy as a functional of the probability density in internal coordinate space it is shown that application of the scheme of thermodynamics of irreversible processes then leads directly to the theory of fluctuations as markov processes described by a multivariate fokker planck equation in this perspective fluctuation theory may be said to have become integrated into non equilibrium thermodynamics

a new class of discretization methods for the solution of linear differential algebraic equations with variable coefficients we discuss new discretization methods for linear differential algebraic equations daes with variable coefficients we introduce numerical methods to compute the local invariants of such daes that were introduced by the authors in a previous paper p kunkel and v mehrmann j comput appl math 56 1994 pp 225 251 using these quantities we are able to determine numerically global invariances like the strangeness index which generalizes the differentiation index for daes that in particular include undetermined solution components based on these methods we then obtain regularization schemes which allow us to employ general solution methods the new methods are tested on a number of numerical examples

the structure of linear relations in euclidean spaces the structure of a linear relation multivalued operator in a euclidean space is completely determined a linear relation can be written as a direct sum of three relations of different classes which are jordan relations completely singular relations and multishifts all three classes of relations are characterized in terms of the spectrum and their chain structure which leads to a generalization of the classical jordan canonical form

dynamical density functional theory for interacting brownian particles stochastic or deterministic we aim to clarify confusions in the literature as to whether or not dynamical density functional theories for the one body density of a classical brownian fluid should contain a stochastic noise term we point out that a stochastic as well as a deterministic equation of motion for the density distribution can be justified depending on how the fluid one body density is defined i e whether it is an ensemble averaged density distribution or a spatially and or temporally coarse grained density distribution

degenerate bifurcation points of periodic solutions of autonomous hamiltonian systems abstract we study connected branches of nonconstant 2 periodic solutions of the hamilton equation x t j h x t where 0 h c 2 r n r n r and 2 h x 0 a 0 0 b for x0 h 1 0 the hessian 2h x0 can be singular we formulate sufficient conditions for the existence of such branches bifurcating from given x0 0 as a consequence we prove theorems concerning the existence of connected branches of arbitrary periodic nonstationary trajectories of the hamiltonian system x t j h x t emanating from x0 we describe also minimal periods of trajectories near x0

some geometric considerations about the kronecker normal form abstract in the first part of this paper we develop general algorithms from the kronecker normal form of any matrix pencil and give some of their properties in the second part these results are applied to the so called restricted matrix pencil associated with a given c a b system and introduced by jaffe and karcanias 1981 this allows us to make a link with well known algorithms of the geometric approach particularly the a b invariant subspace algorithm and the almost a b controllability subspace algorithm finally relationships between morse s invariants morse 1973 and the invariants of the restricted matrix pencil are exhibited

microscopic derivation of time dependent density functional methods time dependent density functional methods tddfm are studied from the microscopic viewpoint using projection operator methods in classical liquids a density field is defined without averaging so that a time evolution equation of the density field is derived with a random force the derived equation includes a free energy functional which is different from that defined in the tddfm the projection operator method provides the exact expression of the free energy functional another definition of the density field by an average leads to the equation of the tddfm in addition an equation describing fluctuations is also derived

right left characteristic sequences and column row minimal indices of a singular pencil for a general singular pencil sf g m n s the right left characteristic sequences r f g l f g are defined and they are shown to be of the piecewise arithmetic progression type the sets of column row minimal indices c f g r f g are defined by a singular points analysis of r f g l f g respectively thus c f g r f g emerge as numerical invariants of the ordered pair f g and they may be computed by rank tests on toeplitz matrices defined on f g

existence of limit cycles for coupled van der pol equations abstract in this paper we consider the existence of limit cycles of coupled van der pol equations by using s1 degree theory due to dylawerski et al see ann polon math 62 1991 243

solving the algebraic riccati equation with the matrix sign function abstract this paper presents some improvements to the matrix sign function algorithm for the algebraic riccati equation a simple reorganization changes nonsymmetric matrix inversions into symmetric matrix inversions scaling accelerates convergence of the basic iteration and yields a new quadratic formula for certain 2 by 2 algebraic riccati equations numerical experience suggests the algorithm be supplemented with a refinement strategy similar to iterative refinement for systems of linear equations refinement also produces an error estimate the resulting procedure is numerically stable it compares favorably with current schur vector based algorithms

temporal behavior of the conditional and gibbs entropies we study the temporal approach to equilibrium of the gibbs and conditional entropies for stochastic systems in the presence of white noise the conditional entropy will either remain constant or monotonically increase to its maximum of zero however the gibbs entropy may have a variety of patterns of approach to its final value ranging from a monotone increase or decrease to an oscillatory approach we have illustrated all of these behaviors using examples in which both entropy dynamics can be determined analytically

distance problems for dissipative hamiltonian systems and related matrix polynomials abstract we study the characterization of several distance problems for linear differential algebraic systems with dissipative hamiltonian structure since all models are only approximations of reality and data are always inaccurate it is an important question whether a given model is close to a bad model that could be considered as ill posed or singular this is usually done by computing a distance to the nearest model with such properties we will discuss the distance to singularity the distance to the nearest high index problem and the distance to instability for dissipative hamiltonian systems while for general unstructured differential algebraic systems the characterization of these distances are partially open problems we will show that for dissipative hamiltonian systems and related matrix polynomials there exist explicit characterizations that can be implemented numerically