long term prediction of non linear time series using multiresolution wavelet models the long term prediction of non linear dynamical time series based on identified multiresolution wavelet models from historically observed data sets is investigated and a new direct prediction approach is introduced prediction results based on the new direct scheme are compared with those from iterative methods and it is shown that improved predictions can be obtained using the new approach

structural properties of dendrite groups let g be the homeomorphism group of a dendrite we study the normal subgroups of g for instance there are uncountably many non isomorphic such groups g that are simple groups moreover these groups can be chosen so that any isometric g action on any metric space has a bounded orbit in particular they have the fixed point property fh

microstructures with finite surface energy the two well problem we study solutions of the two well problem i e maps which satisfy ueso n a so n b a c in n where a and b are n n matrices with positive determinants this problem arises in the study of microstructure in solid solid phase transitions under the additional hypothesis that the set e where the gradient lies in so n a has finite perimeter we show that u is locally only a function of one variable and that the boundary of e consists of subsets of hyperplanes which extend to and which do not intersect in this may not be the case if the assumption on e is dropped we also discuss applications of this result to magnetostrictive materials

convergence of a finite element based space time discretization in elastodynamics we study a finite element based space time discretization of the elastodynamics equation u varepsilon tt rm div sigma nabla u varepsilon varepsilon delta u t varepsilon 0 for varepsilon geq 0 where sigma d phi and phi is a nonconvex function the convergence for regularization parameters varepsilon 0 of iterates to weak solutions and for the limiting problem varepsilon 0 of iterates towards generalized solutions is shown in a general setting of data computational experiments are included to motivate formation and propagation of two dimensional microstructures for decreasing values of the regularization parameter

quasilinear hyperbolic parabolic equations of one dimensional viscoelasticity abstract we study the global existence of solutions of initial boundary value problems for a quasilinear hyperbolic parabolic equation describing the longitudinal motion of a one dimensional viscoelastic rod we treat a variety of nonhomogeneous boundary conditions requiring separate analyses because they lead to distinctive physical effects we employ a constitutive equation giving the stress as a general nonlinear function of the strain and the strain rate all global analyses of this and related problems except that of dafermos j differential equations 1969 71 86 have employed a stress that is merely affine in the strain rate dafermos s assumptions are far more appropriate for shearing motions than for longitudinal motions our constitutive equation satisfies the physically natural requirement that an infinite amount of compressive stress is needed to produce a total compression at any point of the rod this requirement is the source of a severe singularity in the governing partial differential equations which is particularly acute when time dependent dirichlet data are prescribed the further novel yet physically reasonable restrictions we impose on the constitutive function yield estimates that preclude a total compression anywhere at any finite time the resulting estimates are crucial for the global existence theory we obtain

a prediction error and stepwise regression estimation algorithm for non linear systems the identification of non linear systems based on a narmax non linear autoregressive moving average model with exogenous inputs model representation is considered and a combined stepwise regression prediction error estimation algorithm is derived the stepwise regression routine determines the model structure by detecting significant terms in the model while the prediction error algorithm provides optimal estimates of the final model parameters implementation of the algorithms is discussed in detail and several simulated examples and industrial applications are included to illustrate that parsimonious models of non linear systems can be identified using the algorithm

simplicity of some automorphism groups abstract let m be a countably infinite first order relational structure which is homogeneous in the sense of fraisse we show under the assumption that the class of finite substructures of m has the free amalgamation property along with the assumption that aut m is transitive on m but not equal to sym m that aut m is a simple group this generalises results of truss rubin and others the proof uses the polish group structure of the automorphism group and generalises to certain other homogeneous structures with prospects for further application

parametric and nonparametric identification of linear systems in the presence of nonlinear distortions a frequency domain approach this paper studies the asymptotic behavior of nonparametric and parametric frequency domain identification methods to model linear dynamic systems in the presence of nonlinear distortions under some general conditions for random multisine excitations in the first part a related linear dynamic system rlds approximation to the nonlinear system nls is defined and it is shown that the differences between the nls and the rlds can be modeled as stochastic variables with known properties in the second part a parametric model for the rlds is identified convergence in probability of this model to the rlds is proven a function of dependency is defined to detect and separate the presence of unmodeled dynamics and nonlinear distortions and to bound the bias error on the transfer function estimate

on the predictability of discrete dynamical systems iii abstract given an integer n 2 a metrizable compact topological n manifold x with boundary and a finite positive borel measure on x we prove that most homeomorphisms f x x are non sensitive almost everywhere on x moreover we also prove that for most homeomorphisms f x x the non wandering set f has measure zero

generic stationary measures and actions let g be a countably infinite group and let be a generating probability measure on g we study the space of stationary borel probability measures on a topological g space and in particular on z g where z is any perfect polish space we also study the space of stationary measurable g actions on a standard nonatomic probability space equip the space of stationary measures with the weak topology when has finite entropy we show that a generic measure is an essentially free extension of the poisson boundary of g when z is compact this implies that the simplex of stationary measures on z g is a poulsen simplex we show that this is also the case for the simplex of stationary measures on 0 1 g we furthermore show that if the action of g on its poisson boundary is essentially free then a generic measure is isomorphic to the poisson boundary next we consider the space of stationary actions equipped with a standard topology known as the weak topology here we show that when g has property t the ergodic actions are meager we also construct a group g without property t such that the ergodic actions are not dense for some finally for a weaker topology on the set of actions which we call the very weak topology we show that a dynamical property e g ergodicity is topologically generic if and only if it is generic in the space of measures there we also show a glasner king type 0 1 law stating that every dynamical property is either meager or residual

metastable patterns for the cahn hilliard equation part ii layer dynamics and slow invariant manifold abstract in this paper we study the dynamics of the 1 dimensional cahn hilliard equation u t 2 u xx w u xx in a finite interval in a neighborhood of an equilibrium with n 1 transition layers where is a small parameter and w is a double well energy density function with equal minima the lower bound of the layer motion speed is given explictly and the layer motion directions are described precisely if a solution of the cahn hilliard equation starts outside a neighborhood of the equilibrium of size o ln 1 it is proved that there is an n dimensional unstable invariant manifold which is a smooth graph over the approximate manifold constructed in j differential equations 111 1994 421 457 with its global lipschitz constant exponentially small and this unstable invariant manifold attracts solutions exponentially fast uniformly in

the relaxation of non quasiconvex variational integrals we show that the steepest descent algorithm in connection with wiggly energies yields minimizing sequences that converge to a global minimum of the associated non quasiconvex variational integrals we introduce a multi level infinite dimensional variant of the steepest descent algorithm designed to compute complex microstructures by forming non smooth minimizers from the smooth initial guesses we apply this multi level method to the minimization of the variational problems associated with martensitic branching

the dynamics of nucleation for the cahn hilliard equation when a constant metastable solution of the cahn hilliard equation is subjected to a spatially localized large amplitude perturbation a transition process may be triggered leading to a globally stable stationary solution in one space dimension the existence and instability of a third stationary solution with the same mass is proved a spike like solution called a canonical nucleus within the class of solutions which are even with respect to the center of the spike it has a one dimensional unstable manifold in addition the process of nucleation by formal arguments using two space scales and two timescales is described the last stage in the process can be approximated by a nonlinear stefan free boundary problem

smoothing data with local instabilities for the identification of chaotic systems chaotic systems are characterized by the local divergence of nearby orbits in state space this provokes a sensitive dependence on initial conditions and in turn drastically limits the accuracy of long term predictions this has important implications in the filtering of data generated by such systems in the present paper the use of global smoothers for chaotic data is investigated the ultimate objective is to be able to identify dynamically valid models from smoothed data when the identification from the original noisy data has completely failed the objective is to produce identified models that faithfully reproduce the dynamical invariants of the original system such as the geometry of the attractors in state space the largest lyapunov exponent fractal dimensions and poincare sections numerical examples are included which illustrate the main points of the paper

turbulence amalgamation and generic automorphisms of homogeneous structures we study topological properties of conjugacy classes in polish groups with emphasis on automorphism groups of homogeneous countable structures we first consider the existence of dense conjugacy classes the topological rokhlin property we then characterize when an automorphism group admits a comeager conjugacy class answering a question of truss and apply this to show that the homeomorphism group of the cantor space has a comeager conjugacy class answering a question of akin hurley and kennedy finally we study polish groups that admit comeager conjugacy classes in any dimension in which case the groups are said to admit ample generics we show that polish groups with ample generics have the small index property generalizing results of hodges hodkinson lascar and shelah and arbitrary homomorphisms from such groups into separable groups are automatically continuous moreover in the case of oligomorphic permutation groups they have uncountable cofinality and the bergman property these results in particular apply to automorphism groups of many stable 0 categorical structures and of the random graph in this connection we also show that the infinite symmetric group s has a unique non trivial separable group topology for several interesting groups we also establish serre s properties fh and fa

model selection through a statistical analysis of the global minimum of a weighted nonlinear least squares cost function this paper presents a model selection algorithm for the identification of parametric models that are linear in the measurements it is based on the mean and variance expressions of the global minimum of a weighted nonlinear least squares cost function the method requires the knowledge of the noise covariance matrix but does not assume that the true model belongs to the model set unlike the traditional order estimation methods available in literature the presented technique allows to detect undermodeling the theory is illustrated by simulations on signal modeling and system identification problems and by one real measurement example

model validity tests for non linear signal processing applications abstract time series model validity tests based on general correlations are presented in this paper it is shown that the tests xi xi xi xi 2 and xi xi xi 1 t2 can only detect a subset of any possible unmodelled terms in the residuals whereas xi 2 xi 2 detects all possible terms these basic results are then extended to include functions of process or residual terms as entries in the correlation simulation studies are included to demonstrate the effectiveness of the tests when applied to estimated models of both simulated and real data sequences

independence in topological and c dynamics we develop a systematic approach to the study of independence in topological dynamics with an emphasis on combinatorial methods one of our principal aims is to combinatorialize the local analysis of topological entropy and related mixing properties we also reframe our theory of dynamical independence in terms of tensor products and thereby expand its scope to c dynamics

improved structure selection for nonlinear models based on term clustering in this paper the concepts of term clusters and cluster coefficients are defined and used in the context of system identification it is argued that if a certain type of term in a nonlinear model is spurious the respective cluster coefficient is small compared with the coefficients of the other clusters represented in the model once the spurious clusters have been detected the corresponding terms can be deleted from the set of candidate terms the consequences of doing this are i a drastic reduction in the size of the set of candidate terms and consequently a substantial gain in computation time is achieved ii the final estimated model is more likely to reproduce the dynamics of the original system and iii the final model is more robust to overparametrization numerical examples are included to illustrate the new procedure

metastable patterns for the cahn hilliard equation part i abstract in this paper we study the dynamics of the one dimensional cahn hilliard equation in a neighborhood of an equilibrium having n 1 transition layers an approximation for an n dimensional invariant manifold and slow channel are defined and it is proved that ff a solution of the cahn hilliard equation starts outside but close to the slow channel around the approximate manifold then it will approach the channel at an exponentially large speed after it enters the slow channel it will follow the approximate manifold with speed o e c and stay in the channel for exponentially large time

approximate and real trajectories for generic dynamical systems abstract in this paper we show that generic dynamical systems have a weak shadowing property namely that for a generic dynamical system in the space of discrete continuous dynamical systems on a compact manifold m with the usual c 0 topology given 0 there exists a 0 such that trajectories are close to real trajectories there are no dimensional restrictions on this result we also give some results on the inverse problem of tracing real trajectories by trajectories these problems are motivated by questions regarding the validity of numerical solutions of differential equations

stable patterns in a viscous diffusion equation we consider a pseudoparabolic regularization of a forward backward nonlinear diffusion equation ut a f u vut motivated by the problem of phase separation in a viscous binary mixture the function f is nonmonotone so there are discontinuous steady state solutions corresponding to arbitrary arrangements of phases we find that any bounded measurable steady state solution u x satisfying f u constant f u x 0 a e is dynamically stable to perturbations in the sense of convergence in measure in particular smooth solutions may achieve discontinuous asymptotic states furthermore stable states need not correspond to absolute minimizers of free energy thus violating gibbs principle of stability for phase mixtures

on li yorke pairs the li yorke definition of chaos proved its value for interval maps in this paper it is considered in the setting of general topological dynamics we adopt two opposite points of view on the one hand sufficient conditions for li yorke chaos in a topological dynamical system are given we solve a long standing open question by proving that positive entropy implies li yorke chaos on the other hand properties of dynamical systems without li yorke pairs are investigated in addition to having entropy 0 they are minimal when transitive and the property is stable under factor maps arbitrary products and inverse limits finally it is proved that minimal systems without li yorke pairs are disjoint from scattering systems

limit sets of typical continuous functions abstract given a metrizable compact topological n manifold x with boundary and a finite positive borel measure on x we prove that for the typical continuous function f x x it is true that for every point x in a full measure subset of x the limit set f x is a cantor set of hausdorff dimension zero f maps f x homeomorphically onto itself each point of f x has a dense orbit in f x and f is non sensitive at each point of f x moreover the function x f x is continuous almost everywhere

dynamical entropy in banach spaces we introduce a version of voiculescu brown approximation entropy for isometric automorphisms of banach spaces and develop within this framework the connection between dynamics and the local theory of banach spaces as discovered by glasner and weiss our fundamental result concerning this contractive approximation entropy or ca entropy characterizes the occurrence of positive values both geometrically and topologically this leads to various applications for example we obtain a geometric description of the topological pinsker factor and show that a c algebra is type i if and only if every multiplier inner automorphism has zero ca entropy we also examine the behaviour of ca entropy under various product constructions and determine its value in many examples including isometric automorphisms of l p for 1 p and noncommutative tensor product shifts

unique global solvability for initial boundary value problems in one dimensional nonlinear thermoviscoelasticity the balance laws of mass momentum and energy are considered for a broad class of one dimensional nonlinear thermoviscoelastic materials for the initial boundary value problem corresponding to pinned endpoints held at constant temperature we establish existence and uniqueness of temporally global classical solutions for initial data of unrestricted size our approach also applies to all boundary conditions involving pinned or stress free endpoints which are either held at constant temperature or insulated an additional and novel feature of the theory is that solid like and gaseous materials are treated in a unified way

modelling and analysis of non linear time series the modelling of non linear time series is reviewed and new results are introduced by employing some ideas from the identification of non linear control systems both global and local conditions for stationary and invertibility are established for the general non linear time series model and it is shown how these results provide a framework for time series estimators methods of computing multistep ahead predictions are studied and the usefulness of polynomial models is discussed

graph theoretic structure of maps of the cantor space abstract in this paper we develop unifying graph theoretic techniques to study the dynamics and the structure of spaces h 0 1 n and c 0 1 n the space of homeomorphisms and the space of self maps of the cantor space respectively using our methods we give characterizations which determine when two homeomorphisms of the cantor space are conjugate to each other we also give a new characterization of the comeager conjugacy class of the space h 0 1 n the existence of this class was established by kechris and rosendal and a specific element of this class was described concretely by akin glasner and weiss our characterization readily implies many old and new dynamical properties of elements of this class for example we show that no element of this class has a li yorke pair implying the well known glasner weiss result that there is a comeager subset of h 0 1 n each element of which has topological entropy zero our analogous investigation in c 0 1 n yields a surprising result there is a comeager subset of c 0 1 n such that any two elements of this set are conjugate to each other by an element of h 0 1 n our description of this class also yields many old and new results concerning dynamics of a comeager subset of c 0 1 n

weakly almost periodic functions model theoretic stability and minimality of topological groups we investigate the automorphism groups of aleph 0 categorical structures and prove that they are exactly the roelcke precompact polish groups we show that the theory of a structure is stable if and only if every roelcke uniformly continuous function on the automorphism group is weakly almost periodic analysing the semigroup structure on the weakly almost periodic compactification we show that continuous surjective homomorphisms from automorphism groups of stable aleph 0 categorical structures to hausdorff topological groups are open we also produce some new wap trivial groups and calculate the wap compactification in a number of examples

identification of models for chaotic systems from noisy data implications for performance and nonlinear filtering abstract this paper investigates the identification of global models from chaotic data corrupted by additive noise it is verified that noise has a strong influence on the identification of chaotic systems in particular there seems to be a critical noise level beyond which the accurate estimation of polynomial models from chaotic data becomes very difficult similarities with the estimation of the largest lyapunov exponent from noisy data suggest that part of the problem might be related to the limited ability of predicting the data records when these are chaotic a nonlinear filtering scheme is suggested in order to reduce the noise in the data and thereby enable the estimation of good models this prediction based filtering incorporates a resetting mechanism which enables the filtering of chaotic data and which is also applicable to non chaotic data

good measures on cantor space while there is up to homeomorphism only one cantor space i e one zero dimensional perfect compact nonempty metric space there are many measures on cantor space which are not topologically equivalent the clopen values set for a full nonatomic measure is the countable dense subset u u is clopen of the unit interval it is a topological invariant for the measure for the class of good measures it is a complete invariant a full nonatomic measure is good if whenever u v are clopen sets with u v there exists w a clopen subset of v such that w u these measures have interesting dynamical properties they a re exactly the measures which arise from uniquely ergodic minimal systems on cantor space for some of them there is a unique generic measure preserving homeomorphism that is within the polish group of such homeomorphisms there is a dense g conjugacy class

generically there is but one self homeomorphism of the cantor set we describe a self homeomorphism r of the cantor set x and then show that its conjugacy class in the polish group h x of all homeomorphisms of x forms a dense g subset of h x we also provide an example of a locally compact second countable topological group which has a dense conjugacy class

homogeneous probability measures on the cantor set abstract we show that every homeomorphism between closed measure zero subsets extends to a measure preserving auto homeomorphism whenever the cantor set is endowed with a suitable probability measure this is valid both for the standard product measure as well as for the universal homogeneous rational measure

weak shadowing for discrete dynamical systems on nonsmooth manifolds abstract in j math anal appl 189 1995 409 423 corless and pilyugin proved that weak shadowing is a c 0 generic property in the space of discrete dynamical systems on a compact smooth manifold m in our paper we give another proof of this theorem which does not assume that m has a differential structure moreover our method also works for systems on some compact metric spaces that are not manifolds such as a hilbert cube or generally a countably infinite cartesian product of manifolds with boundary and a cantor set

ordered structures and large conjugacy classes abstract this article is a contribution to the following problem does there exist a polish non archimedean group equivalently automorphism group of a fraisse limit that is extremely amenable and has ample generics as fraisse limits whose automorphism groups are extremely amenable must be ordered i e equipped with a linear ordering we focus on ordered fraisse limits we prove that automorphism groups of the universal ordered boron tree and the universal ordered poset have a comeager conjugacy class but no comeager 2 dimensional diagonal conjugacy class we formulate general conditions implying that there is no comeager conjugacy class comeager 2 dimensional diagonal conjugacy class or non meager 2 dimensional topological similarity class in the automorphism group of an ordered fraisse limit we also provide a number of applications of these results

metastable patterns in solutions of ut ϵ2uxx f u we consider the equation in question on the interval 0 x 1 having neumann boundary conditions with f u f u where f is a double well energy density with equal minima at u 1 the only stable states of the system are patternless constant solutions but given two phase initial data a pattern of interfacial layers typically forms far out of equilibrium the ensuing nonlinear relaxation process is extremely slow patterns persist for exponentially long times proportional to exp a l where a f 1 and l is the minimum distance between layers physically a tiny potential jump across a layer drives its motion we prove the existence and persistence of these metastable patterns and characterise accurately the equations governing their motion the point of view is reminiscent of center manifold theory a manifold parametrising slowly evolving states is introduced a neighbourhood is shown to be normally attracting and the parallel flow is characterised to high relative accuracy proofs involve a detailed study of the dirichlet problem spectral gap analysis and energy estimates

modelling and residual analysis of nonlinear autoregressive time series in exponential variables abstract an approach to modelling and residual analysis of nonlinear autoregressive time series in exponential variables is presented the approach is illustrated by analysis of a long series of wind velocity data which has first been detrended and then transformed into a stationary series with an exponential marginal distribution the stationary series is modelled with a newly developed type of second order autoregressive process with random coefficients called the near 2 model it has a second order autoregressive correlation structure but is nonlinear because its coefficients are random the exponential distributional assumptions involved in this model highlight a very broad four parameter structure which combines five exponential random variables into a sixth exponential random variable other applications of this structure are briefly considered dependency in the near 2 process not accounted for by standard autocorrelations is explored by developing a residual analysis for time series having autoregressive correlation structure this involves defining linear uncorrelated residuals which are dependent and then assessing this higher order dependence by standard time series computations application of this residual analysis to the wind velocity data illustrates both the utility and difficulty of nonlinear time series modelling

convergence of solutions to the equation of quasi static approximation of viscoelasticity with capillarity the long time behavior of solutions to the quasi steady approximation of the equation of viscoelasticity with capillarity is investigated we consider the equation with a number of boundary conditions including such that the set of equilibria is necessarily at least one dimensional we prove that any solution converges to an equilibrium point we admit some kind of initial data with infinite energy we use tools based on the assumption that the nonlinear term is real analytic

shifts of finite type as fundamental objects in the theory of shadowing shifts of finite type and the notion of shadowing or pseudo orbit tracing are powerful tools in the study of dynamical systems in this paper we prove that there is a deep and fundamental relationship between these two concepts let x be a compact totally disconnected space and f x rightarrow x a continuous map we demonstrate that f has shadowing if and only if the system f x is conjugate to the inverse limit of a directed system satisfying the mittag leffler condition and consisting of shifts of finite type in particular this implies that in the case that x is the cantor set f has shadowing if and only if f x is the inverse limit of a sequence satisfying the mittag leffler condition and consisting of shifts of finite type moreover in the general compact metric case where x is not necessarily totally disconnected we prove that f has shadowing if f x is a factor of the inverse limit of a sequence satisfying the mittag leffler condition and consisting of shifts of finite type by a quotient that almost lifts pseudo orbits

dynamic stability of non minimizing phase mixtures we study the dynamics of pattern formation in the one dimensional partial differential equation u tt w u x x u xxt u 0 proposed recently by ball holmes pego and swart as a mathematical cartoon for the dynamic formation of microstructures observed in various crystalline solids here w is a double well potential like 1 4 u x 2 1 2 this problem possesses infinitely many equilibrium solutions corresponding to different microstructures or phase mixtures all locally energetically unstable in the natural norms however many are locally dynamically stable theorem 1 1 and the paper proves this theorem and investigates other properties of these equilibria

restrictions on microstructure we consider the following question given a set of matrices with no rank one connections does it support a nontrivial young measure limit of gradients our main results are these a a young measure can be supported on four incompatible matrices b in two space dimensions a young measure cannot be supported on finitely many incompatible elastic wells c in three or more space dimensions a young measure can be supported on three incompatible elastic wells and d if supports a nontrivial young measure with mean value 0 then the linear span of must contain a matrix of rank one

on characters of inductive limits of symmetric groups abstract in the paper we completely describe characters central positive definite functions of simple locally finite groups that can be represented as inductive limits of products of symmetric groups under block diagonal embeddings each such group g defines an infinite graded graph that encodes the embedding scheme the group g acts on the space x of infinite paths of the associated graph by changing initial edges of paths assuming the finiteness of the set of ergodic measures for the system x g we establish that each non regular indecomposable character g c is uniquely determined by the formula g 1 fix g 1 k fix g k where 1 k are g ergodic measures fix g x x g x x and 1 k 0 1 we illustrate our results on the group of rational permutations of the unit interval

dynamics as a mechanism preventing the formation of finer and finer microstructure we study the dynamics of pattern formation in the one dimensional partial differential equation u u w u x x u xxt u 0 text u u x t text x in 0 1 text t 0 proposed recently by ball holmes james pego swart bhjps as a mathematical cartoon for the dynamic formation of microstructures observed in various crystalline solids here w is a double well potential like 1 4 ux 2 1 2 what makes this equation interesting and unusual is that it possesses as a lyapunov function a free energy consisting of kinetic energy plus a nonconvex elastic energy but no interfacial energy contribution which does not attain a minimum but favours the formation of finer and finer phase mixtures e u u t int limits 0 1 frac u t 2 2 w u x frac u 2 2 dx our analysis of the dynamics confirms the following surprising and striking difference between statics and dynamics conjectured in bhjps on the basis of numerical simulations of swart holmes sh while minimizing the above energy predicts infinitely fine patterns mathematically weak but not strong convergence of all minimizing sequences unvn of e u v in the sobolev space w1p 0 1 l2 0 1 solutions to the evolution equation of ball et al typically develop patterns of small but finite length scale mathematically strong convergence in w1p 0 1 l2 0 1 of all solutions u t ut t with low initial energy as time t

the topological rohlin property and topological entropy for a compact metric space x let g h x denote the group of self homeomorphisms with the topology of uniform convergence the group g acts on itself by conjugation and we say that x satisfies the topological rohlin property if this action has dense orbits we show that the hilbert cube the cantor set and with a slight modification also even dimensional spheres satisfy this property we also show that zero entropy is generic for homeomorphisms of the cantor set whereas it is infinite entropy which is generic for homeomorphisms of cubes of dimension d 2 and the hilbert cube

generic automorphisms of homogeneous structures let g be a permutation group on the set usually we take n o we seek a notion of genericity for members of g the idea is that a member of g should be generic if it is in some sense typical we argue that the following is the correct definition suppose that g is endowed with a metric so that it becomes a complete metric space it follows that the baire category theorem holds and we may use the notions of meagre and comeagre sets

attainment results for nematic elastomers we consider a class of non quasiconvex frame indifferent energy densities which includes ogden type energy densities for nematic elastomers for the corresponding geomet rically linear problem we provide an explicit minimizer of the energy functional satisfying a nontrivial boundary condition other attainment results both for the nonlinear and the lin earized model are obtained by using the theory of convex integration introduced by muller and u sverak in the context of crystalline solids

chaos among self maps of the cantor space glasner and weiss have shown that a generic homeomorphism of the cantor space has zero topological entropy hochman has shown that a generic transitive homeomorphism of the cantor space has the property that it is topologically conjugate to the universal odometer and hence far from being chaotic in any sense we show that a generic self map of the cantor space has zero topological entropy moreover we show that a generic self map of the cantor space has no periodic points and hence is not devaney chaotic nor devaney chaotic on any subsystem however we exhibit a dense subset of the space of all self maps of the cantor space each element of which has infinite topological entropy and is devaney chaotic on some subsystem

a phase plane discussion of convergence to travelling fronts for nonlinear diffusion the paper is concerned with the asymptotic behavior as t of solutions u x t of the equation u t u xx f left u right 0 x in left infty infty right in the case f 0 f 1 0 with f u non positive for u 0 sufficiently close to zero and f u non negative for u 1 sufficiently close to 1 this guarantees the uniqueness but not the existence of a travelling front solution u u x ct u 0 u and it is shown in essence that solutions with monotonic initial data converge to a translate of this travelling front if it exists and to a stacked combination of travelling fronts if it does not the approach is to use the monotonicity to take u and t as independent variables and p ux as the dependent variable and to apply ideas of sub and super solutions to the diffusion equation for p

global existence for a quasi linear evolution equation with a non convex energy we establish the existence of global in time weak solutions to the initial boundary value problem related to the dynamics of coherent solid solid phase transitions in viscoelasticity the class of the stored energy functionals includes the double well potential and a general convolution damping term is considered

evolution equations of second order with nonconvex potential and linear damping existence via convergence of a full discretization global existence of solutions for a class of second order evolution equations with damping is shown by proving convergence of a full discretization the discretization combines a fully implicit time stepping with a galerkin scheme the operator acting on the zero order term is assumed to be a potential operator where the potential may be nonconvex a linear symmetric operator is assumed to be acting on the first order term applications arise in nonlinear viscoelasticity and elastodynamics

asymptotic behaviour and changes of phase in one dimensional nonlinear viscoelasticity in this paper we study the asymptotic behaviour as t co of solutions u x t to the nonlinear partial differential equation ut m4 a o o 1 1 with initial conditions u 0 q x u x u x o 0 1 3a