transition phenomena in cellular automata rule space abstract we define several qualitative classes of cellular automata ca behavior based on various statistical measures and describe how the space of all cellular automata is organized as a cellular automaton is changed by varying entries in its rule table abrupt changes in qualitative behavior may occur these abrupt changes have the character of bifurcations in smooth dynamical systems or of phase transitions in statistical mechanical systems the most complex ca rules exhibit many of the characteristics of second order transitions suggesting an association between computation complexity and critical phenomena

analysis of multiple scattering iterations for high frequency scattering problems i the two dimensional case we present an analysis of a recently proposed integral equation method for the solution of high frequency electromagnetic and acoustic scattering problems that delivers error controllable solutions in frequency independent computational times within single scattering configurations the method is based on the use of an appropriate ansatz for the unknown surface densities and on suitable extensions of the method of stationary phase the extension to multiple scattering configurations in turn is attained through consideration of an iterative neumann series that successively accounts for further geometrical wave reflections as we show for a collection of two dimensional cylindrical convex obstacles this series can be rearranged into a sum of periodic orbits of increasing period each corresponding to contributions arising from waves that reflect off a fixed subset of scatterers when these are transversed sequentially in a periodic manner here we analyze the properties of these periodic orbits in the high frequency regime by deriving precise asymptotic expansions for the currents i e the normal derivative of the fields that they induce on the surface of the obstacles as we demonstrate these expansions can be used to provide accurate estimates of the rate at which their magnitude decreases with increasing number of reflections which defines the overall rate of convergence of the multiple scattering series moreover we show that the detailed asymptotic knowledge of these currents can be used to accelerate this convergence and thus to reduce the number of iterations necessary to attain a prescribed accuracy

center manifolds and contractions on a scale of banach spaces we give a new proof for the existence of a ck center manifold at a nonhyperbolic equilibrium point of a finite dimensional vector field of class ck the problem is reduced to a fixed point problem on a scale of banach spaces these banach spaces consist of mappings with a certain maximal exponential growth at infinity we give conditions under which there is a unique fixed point depending differentiably on the parameters the main difficulty is that the mappings under consideration become only differentiable after composition with appropriate embeddings on the scale of banach spaces

center manifolds for invariant sets abstract we derive a general center manifolds theory for a class of compact invariant sets of flows generated by a smooth vector field in r n by applying the hadamard graph transform technique it is shown that associated to a natural dynamical characteristic of the linearized flow along the invariant set there exists an invariant manifold called a center manifold of the invariant set which contains every locally bounded solution in particular contains the invariant set and is persistent under small perturbations

computation theory of cellular automata self organizing behaviour in cellular automata is discussed as a computational process formal language theory is used to extend dynamical systems theory descriptions of cellular automata the sets of configurations generated after a finite number of time steps of cellular automaton evolution are shown to form regular languages many examples are given the sizes of the minimal grammars for these languages provide measures of the complexities of the sets this complexity is usually found to be non decreasing with time the limit sets generated by some classes of cellular automata correspond to regular languages for other classes of cellular automata they appear to correspond to more complicated languages many properties of these sets are then formally non computable it is suggested that such undecidability is common in these and other dynamical systems

self organized criticality in the game of life the question of saturation of self organized criticality in the game of life is analyzed through finite size scaling we have carried out connection machine simulations for lattices up to 1024 ifmmode times else texttimes fi 1024 in that range we find no evidence of saturation we do however find finite size exponents that seem inconsistent with a single length scale picture at criticality

fredholm operators semigroups and the asymptotic and boundary behavior of solutions of pdes abstract every semigroup t on a banach space x can be used to define elements u x of exponential type relative to t by requiring that u t s v for some s 0 and v x let then x and y be banach spaces in which the exponential type is characterized by the semigroups t and s respectively and let l l x y be fredholm it is shown that if l satisfies some compatibility conditions with respect to t and s and if f y has exponential type then every solution u x of lu f has exponential type as well when l is a differential operator it is often compatible in this sense and in suitable spaces with semigroups that embody an asymptotic or boundary behavior this yields a way to study such a behavior in solutions of pdes which is technically simple very general and delivers rather sharp results furthermore this approach is easily generalized to the nonlinear setting

a stable runge kutta methods for semilinear evolution equations we consider semilinear evolution equations for which the linear part generates a strongly continuous semigroup and the nonlinear part is sufficiently smooth on a scale of hilbert spaces in this setting we prove the existence of solutions which are temporally smooth in the norm of the lowest rung of the scale for an open set of initial data on the highest rung of the scale under the same assumptions we prove that a class of implicit a stable runge kutta semidiscretizations in time of such equations are smooth as maps from open subsets of the highest rung into the lowest rung of the scale under the additional assumption that the linear part of the evolution equation is normal or sectorial we prove full order convergence of the semidiscretization in time for initial data on open sets our results apply in particular to the semilinear wave equation and to the nonlinear schrodinger equation 2012 elsevier inc

brief paper preservation of exponential stability for linear non autonomous functional differential systems we consider preservation of exponential stability for a system of linear equations with a distributed delay under the addition of new terms and a delay perturbation as particular cases the system includes models with concentrated delays and systems of integrodifferential equations our method is based on bohl perron type theorems

on bounded pseudodifferential operators in wiener spaces we aim at extending the definition of the weyl calculus to an infinite dimensional setting by replacing the phase space r2n by b2 where i h b is an abstract wiener space a first approach is to generalize the integral definition using the wigner function the symbol is then a function defined on b2 and belonging to a l1 space for a gaussian measure the weyl operator is defined as a quadratic form on a dense subspace of l2 b for example the symbol can be the stochastic extension on b2 in the sense of l gross of a function f which is continuous and bounded on h2 in the second approach this function f defined on h2 satisfies differentiability conditions analogous to the finite dimensional ones one needs to introduce hybrid operators acting as weyl operators on the variables of finite dimensional subset of h and as anti wick operators on the rest of the variables the final weyl operator is then defined as a limit and it is continuous on a l2 space under rather weak conditions it is an extension of the operator defined by the first approach we give examples of monomial symbols linking this construction to the classical pseudodifferential operators theory and other examples related to other fields or previous works on this subject

the game of life at finite temperature abstract in this paper we propose a cellular array model whose transition function is based on a continuously valued expression that includes a temperature parameter t and two shift parameters e 0 and x 0 this transition function coincides with the transition function of the game of life cellular automaton model when the temperature approaches the limit t 0 states in this model are continuously valued for t 0 and cell state transitions though deterministic are affected by thermal fluctuation spectral analysis of the cell state s time series shows a 1 f spectrum in the low temperature region as in the game of life and a lorentzian spectrum in the high temperature region analysis of the average entropy shows a division along approximately the same critical point into a low temperature region indicative of the game of life and a high temperature region with different behavior we show that the model is robust to deviations in the values of parameters e 0 and x 0 but that the robustness declines with an increase in temperature up to a critical temperature at which all robustness is lost

proof of a mckean conjecture on the rate of convergence of boltzmann equation solutions the present work provides a definitive answer to the problem of quantifying relaxation to equilibrium of the solution to the spatially homogeneous boltzmann equation for maxwellian molecules under really mild conditions on the initial datum and a weak physically consistent angular cutoff hypothesis our main result theorem 1 contains the first precise statement that the total variation distance between the solution and the limiting maxwellian distribution admits an upper bound of the form c e varlambda b t varlambda b being the least negative eigenvalue of the linearized collision operator and c a constant depending only on the initial datum the validity of this quantification was conjectured about fifty years ago by henry p mckean as to the proof of our results we have taken as point of reference an analogy between the problem of convergence to equilibrium and the central limit theorem of probability theory highlighted by mckean

semilinear pseudodifferential equations in spaces of tempered ultradistributions abstract we study a class of semilinear elliptic equations on spaces of tempered ultradistributions of beurling and roumieu type assuming that the linear part of the equation is a pseudodifferential operator of infinite order satisfying a suitable ellipticity condition we prove a regularity result in the functional setting above for weak sobolev type solutions

universality and complexity in cellular automata abstract cellular automata are discrete dynamical systems with simple construction but complex self organizing behaviour evidence is presented that all one dimensional cellular automata fall into four distinct universality classes characterizations of the structures generated in these classes are discussed three classes exhibit behaviour analogous to limit points limit cycles and chaotic attractors the fourth class is probably capable of universal computation so that properties of its infinite time behaviour are undecidable

a grobman hartman theorem for nonuniformly hyperbolic dynamics abstract we establish a version of the grobman hartman theorem in banach spaces for nonuniformly hyperbolic dynamics we also consider the case of sequences of maps which corresponds to a nonautonomous dynamics with discrete time more precisely we consider sequences of lipschitz maps a m f m such that the linear parts a m admit a nonuniform exponential dichotomy and we establish the existence of a unique sequence of topological conjugacies between the maps a m f m and a m furthermore we show that the conjugacies are holder continuous with holder exponent determined by the ratios of lyapunov exponents with the same sign to the best of our knowledge this statement appeared nowhere before in the published literature even in the particular case of uniform exponential dichotomies although some experts claim that it is well known in this case we are also interested in the dependence of the conjugacies on the perturbations f m we show that it is holder continuous with the same holder exponent as the one for the conjugacies we emphasize that the additional work required to consider the case of nonuniform exponential dichotomies is substantial in particular we need to consider several additional lyapunov norms

off spectral analysis of bergman kernels the asymptotic analysis of bergman kernels with respect to exponentially varying measures near emergent interfaces has attracted recent attention such interfaces typically occur when the associated limiting bergman density function vanishes on a portion of the plane the off spectral region this type of behavior is observed when the metric is negatively curved somewhere or when we study partial bergman kernels in the context of positively curved metrics in this work we cover these two situations in a unified way for exponentially varying weights on the complex plane we obtain a uniform asymptotic expansion of the coherent state of depthn rooted at an off spectral point which we also refer to as the root function at the point in question the expansion is valid in the entire off spectral component containing the root point and protrudes into the spectrum as well this allows us to obtain error function transition behavior of the density of states along the smooth interface previous work on asymptotic expansions of bergman kernels is typically local and valid only in the bulk region of the spectrum which contrasts with our non local expansions

technical communique decay estimates for applications of razumikhin type theorems decay estimates are established for a family of razumikhin type retarded functional differential equations rfdes from the point of applications of razumikhin type theorems the work presented in this paper is an extension of the ones reported so far in the related literature since many contributions are based on razumikhin type theorems our results may provide effective methods for quantitative stability analysis of delayed systems encountered in engineering

a constructive approach to regularity of lagrangian trajectories for incompressible euler flow in a bounded domain the 3d incompressible euler equations are an important research topic in the mathematical study of fluid dynamics not only is the global regularity for smooth initial data an open issue but the behaviour may also depend on the presence or absence of boundaries for a good understanding it is crucial to carry out besides mathematical studies high accuracy and well resolved numerical exploration such studies can be very demanding in computational resources but recently it has been shown that very substantial gains can be achieved first by using cauchy s lagrangian formulation of the euler equations and second by taking advantage of analyticity results of the lagrangian trajectories for flows whose initial vorticity is holder continuous the latter has been known for about 20 years serfati in j math pures appl 74 95 104 1995 but the combination of the two which makes use of recursion relations among time taylor coefficients to obtain constructively the time taylor series of the lagrangian map has been achieved only recently frisch and zheligovsky in commun math phys 326 499 505 2014 podvigina et al in j comput phys 306 320 342 2016 and references therein here we extend this methodology to incompressible euler flow in an impermeable bounded domain whose boundary may be either analytic or have a regularity between indefinite differentiability and analyticity non constructive regularity results for these cases have already been obtained by glass et al ann sci ec norm sup 45 1 51 2012 using the invariance of the boundary under the lagrangian flow we establish novel recursion relations that include contributions from the boundary this leads to a constructive proof of time analyticity of the lagrangian trajectories with analytic boundaries which can then be used subsequently for the design of a very high order cauchy lagrangian method

initial value problems for creeping flow of maxwell fluids abstract we consider the flow of nonlinear maxwell fluids in the unsteady quasistatic case where the effect of inertia is neglected we study the well posedness of the resulting pde initial boundary value problem locally in time this well posedness depends on the unique solvability of an elliptic boundary value problem we first present results for the 3d case with sufficiently small initial data and for a simple shear flow problem with arbitrary initial data after that we extend our results to some 3d flow problems with large initial data we solve our problem using an iteration between linear subproblems the limit of the iteration provides the solution of our original problem

stability analysis for neural dynamics with time varying delays by using the usual additive neural network model a delay independent stability criterion for neural dynamics with perturbations of time varying delays is derived we extend previously known results obtained by gopalsamy and he 1994 to the time varying delay case and present decay estimates of solutions of neural networks the asymptotic stability is global in the state space of neuronal activations from the techniques used in this paper it is shown that our criterion ensures stability of neural dynamics even when the delay functions vary violently with time our approach provides an effective method for the stability analysis of neural dynamics with delays

partial linearization for planar nonautonomous differential equations abstract we prove a partial linearization theorem for planar nonautonomous differential equations with one center like direction and one hyperbolic direction gap conditions are formulated in terms of the dichotomy spectral intervals the result is applied to the duffing van der pol oscillator under a nonautonomous parametric perturbation

stability of x t ax t bx t tau a stability criterion for linear time delay systems described by a differential difference equation of the form dx t ax t bx t tau is proposed the result obtained includes information on the size of the delay and therefore can be a delay dependent stability condition its relation to existing delay independent stability criteria is also discussed

solutions blowing up on any given compact set for the energy subcritical wave equation abstract we consider the focusing energy subcritical nonlinear wave equation t t u u u p 1 u in r n n 1 given any compact set k r n we construct finite energy solutions which blow up at t 0 exactly on k the construction is based on an appropriate ansatz the initial ansatz is simply u 0 t x t a x 2 p 1 where a 0 vanishes exactly on k which is a solution of the ode h h p we refine this first ansatz inductively using only ode techniques and taking advantage of the fact that for suitably chosen a space derivatives are negligible with respect to time derivatives we complete the proof by an energy argument and a compactness method

global well posedness for the generalized large scale semigeostrophic equations we prove existence and uniqueness of global classical solutions to the generalized large scale semigeostrophic equations with periodic boundary conditions this family of hamiltonian balance models for rapidly rotating shallow water includes the l1 model derived by r salmon in 1985 and its 2006 generalization by the second author the results are under the physical restriction that the initial potential vorticity is positive as strong as those available for the euler equations of ideal fluid flow in two dimensions moreover we identify a special case in which the velocity field is two derivatives smoother in sobolev space as compared to the general case our results are based on careful estimates which show that although the potential vorticity inversion is nonlinear bounds on the potential vorticity inversion operator remain linear in derivatives of the potential vorticity this permits the adaptation of an argument based on elliptic lp theory proposed by yudovich in 1963 for proving existence and uniqueness of weak solutions for the two dimensional euler equations to our particular nonlinear situation

transformations and hardy krause variation using a multivariable faa di bruno formula we give conditions on transformations tau 0 1 m to cal x where cal x is a closed and bounded subset of mathbb r d such that f circ tau is of bounded variation in the sense of hardy and krause for all f in c m cal x we give similar conditions for f circ tau to be smooth enough for scrambled net sampling to attain o n 3 2 epsilon accuracy some popular symmetric transformations to the simplex and sphere are shown to satisfy neither condition some other transformations due to fang and wang monogr statist appl probab 51 crc press boca raton fl 1993 satisfy the first but not the second condition we provide transformations for the simplex that makes f circ tau smooth enough to fully benefit from scrambled net sampling for all f in a class of generalized polynomials we also find sufficient conditions for conditional inversion in mathbb r 2 and for importance sampling to be of bounded variation in the sense of ha

center manifolds for infinite dimensional nonautonomous differential equations abstract we study a nonlinear integral equation for a center manifold of a semilinear nonautonomous differential equation having mild solutions we assume that the linear part of the equation admits in a very general sense a decomposition into center and hyperbolic parts the center manifold is obtained directly as the graph of a fixed point for a lyapunov perron type integral operator we prove that this integral operator can be factorized as a composition of a nonlinear substitution operator and a linear integral operator the operator is formed by the green s function for the hyperbolic part and composition operators induced by a flow on the center part we formulate the usual gap condition in spectral terms and show that this condition is in fact a condition of boundedness of on corresponding spaces of differentiable functions this gives a direct proof of the existence of a smooth global center manifold

asymptotic expansion of the dulac map and time for unfoldings of hyperbolic saddles local setting abstract in this paper we study unfoldings of planar vector fields in a neighbourhood of a hyperbolic resonant saddle we give a structure theorem for the asymptotic expansion of the local dulac time as well as the local dulac map with the remainder uniformly flat with respect to the unfolding parameters here local means close enough to the saddle in order that the normalizing coordinates provided by a suitable normal form can be used the principal part of the asymptotic expansion is given in a monomial scale containing a deformation of the logarithm the so called roussarie ecalle compensator especial attention is paid to the remainder s properties concerning the derivation with respect to the unfolding parameters

center manifolds for difference equations smooth parameter dependence abstract for sufficiently small c 1 perturbations of nonautonomous linear difference equations with a nonuniform exponential trichotomy we establish the existence of center manifolds with the optimal c 1 regularity we also consider the case of parameter dependent perturbations and we obtain the c 1 dependence of the center manifolds on the parameter in addition we consider arbitrary growth rates with the usual exponential estimates of the form e c t in the notion of exponential trichotomy replaced by e c t where is now an arbitrary function the proof of the regularity both of the center manifold and of its dependence on the parameter is based on the fiber contraction principle the most technical part of the argument concerns the continuity of the fiber contraction that essentially needs a direct argument

exterior powers of the adjoint representation and the weyl ring of e8 abstract i derive explicitly all polynomial relations in the character ring of e 8 of the form k e 8 p k 1 8 0 where k e 8 is an arbitrary exterior power of the adjoint representation and i is the ith fundamental character this has simultaneous implications for the theory of relativistic integrable systems seiberg witten theory quantum topology orbifold gromov witten theory and the arithmetic of elliptic curves the solution is obtained by reducing the problem to a large but finite dimensional linear problem which is amenable to an efficient solution via distributed computation

regularity of the geodesic flow of the incompressible euler equations on a manifold in this paper we investigate the regularity in time of the volume preserving geodesic flow which is associated with the incompressible euler equations on a compact d dimensional riemannian manifold with boundary our result which completes the local in time well posedness theory of ebin and marsden ann math 92 102 163 1970 states roughly that the time smoothness of geodesic curves is only limited by the smoothness of the manifold such regularity is measured in a broad class of ultradifferentiable functions which includes the real analytic and gevrey classes a by product of this simple and constructive proof is new ideas to design high order semi lagrangian methods for integrating the incompressible euler equations on a manifold

the existence of almost periodic solutions of some delay differential equations abstract a set of sufficient conditions is derived for the existence of almost periodic solutions of some delay differential equations by using exponential dichotomy and fixed point theorem this criterion provides a method for the existence analysis of almost periodic solution of time delay systems

sobolev regularity of the equation on the hartogs triangle the regularity of the overline partial problem on the domain left z 1 right left z 2 right 1 in mathbb c 2 is studied using l 2 methods estimates are obtained for the canonical solution in weighted l 2 sobolev spaces with a weight that is singular at the point 0 0 in particular the singularity of the bergman projection for the hartogs triangle is contained at the singular point and it does not propagate

dynamical and spatial aspects of sandpile cellular automata the bak tang and wiesenfeld cellular automaton is simulated in 1 2 3 4 and 5 dimensions we define a new set of scaling exponents by introducing the concept of conditional expectation values scaling relations are derived and checked numerically and the critical dimension is discussed we address the problem of the mass dimension of the avalanches and find that the avalanches are noncompact for dimensions larger than 2 the scaling of the power spectrum derives from the assumption that the instantaneous dissipation rate of the individual avalanches obeys a simple scaling relation primarily the results of our work show that the flow of sand down the slope does not have a 1 f power spectrum in any dimension although the model does show clear critical behavior with scaling exponents depending on the dimension the power spectrum behaves as 1 f 2 in all the dimensions considered

delay independent exponential stability criteria for time varying discrete delay systems derives delay independent exponential stability conditions for linear nonlinear time varying discrete delay systems since these conditions are of delay independence and easily verifiable they may provide handy tools for the stability analysis

exponential stability criteria of uncertain systems with multiple time delays the exponential stability with convergence rate of uncertain linear systems with multiple time delays is studied in this paper using the characteristic function of linear time delay system stability criteria are derived to guarantee stability sufficient conditions are also obtained for exponential stability of uncertain parametric systems with multiple time delays for two dimensional time invariant system with multiple time delays the proposed stability criteria are shown to be less conservative than those in the literature numerical examples are given to illustrate the validity of our new stability criteria

smooth invariant manifolds in banach spaces with nonuniform exponential dichotomy we establish the existence of smooth stable manifolds for semiflows defined by ordinary differential equations v a t v f t v in banach spaces assuming that the linear equation v a t v admits a nonuniform exponential dichotomy our proof of the ck smoothness of the manifolds uses a single fixed point problem in the unit ball of the space of ck functions with holder continuous kth derivative this is a closed subset of the space of continuous functions with the supremum norm by an apparently not so well known lemma of henry see proposition 3 the estimates showing that the functions maintain the original bounds when transformed under the fixed point operator are obtained through a careful application of the faa di bruno formula for the higher derivatives of the compositions see 31 and 35 as a consequence we obtain in a direct manner not only the exponential decay of solutions along the stable manifolds but also of their derivatives up to order k when the vector field is of class ck

robust stability criteria for dynamical systems including delayed perturbations considers the problem of robust stability of uncertain time delay dynamical systems a new robust stability criteria for linear dynamical systems subject to delayed time varying and nonlinear perturbations is derived the results obtained in this note are less conservative than the ones reported so far in the literature some analytical methods are employed to investigate the bound on the perturbations so that the systems are stable a numerical example is given to demonstrate the utilization of the authors results

1 f fluctuation in the game of life abstract the fourier power spectra are examined for the game of life and its variants the results show that only the game of life exhibits fluctuation by means of observing patterns and measuring entropy and difference pattern spreading rate we conclude that the game of life only belongs to wolfram s class iv cellular automata which are thought to be capable of supporting universal computation this predicts a connection between fluctuation and computational universality

ultradistributional boundary values of harmonic functions on the sphere abstract we present a theory of ultradistributional boundary values for harmonic functions defined on the euclidean unit ball we also give a characterization of ultradifferentiable functions and ultradistributions on the sphere in terms of their spherical harmonic expansions to this end we obtain explicit estimates for partial derivatives of spherical harmonics which are of independent interest and refine earlier estimates by calderon and zygmund we apply our results to characterize the support of ultradistributions on the sphere via abel summability of their spherical harmonic expansions

the fbi transforms and their use in microlocal analysis abstract using a more general class of fbi transforms introduced by s berhanu and j hounie in 12 we completely characterize regularity and microregularity in denjoy carleman non quasi analytic classes which includes the gevrey classes and m christ version of the fbi transform defined in 22 as examples we also exhibit a result on microlocal regularity for solutions of first order partial differential equations in these classes that do not seem possible to prove using the classical fbi transform

smooth center manifolds for nonuniformly partially hyperbolic trajectories abstract we establish the existence of unique smooth center manifolds for ordinary differential equations v a t v f t v in banach spaces assuming that v a t v admits a nonuniform exponential trichotomy this allows us to show the existence of unique smooth center manifolds for the nonuniformly partially hyperbolic trajectories in addition we prove that the center manifolds are as regular as the vector field our proof of the c k smoothness of the manifolds uses a single fixed point problem in an appropriate complete metric space to the best of our knowledge we establish in this paper the first smooth center manifold theorem in the nonuniform setting

a note on the critical nonlinear high order schrodinger equation abstract well posedness issues for some critical high order schrodinger equation with polynomial nonlinearity are investigated

concavity of the mutual information rate for input restricted memoryless channels at high snr we consider a memoryless channel with an input markov process supported on a mixing finite type constraint we continue the development of asymptotics for the entropy rate of the output hidden markov chain and deduce that at high signal to noise ratio the mutual information rate of such a channel is concave with respect to almost all input markov chains of a given order

asymptotics of input constrained erasure channel capacity in this paper we examine an input constrained erasure channel and we characterize the asymptotics of its capacity when the erasure rate is low more specifically for a general memoryless erasure channel with its input supported on an irreducible finite type constraint we derive partial asymptotics of its capacity using some series expansion type formula of its mutual information rate and for a binary erasure channel with its first order markovian input supported on the 1 infty rll constraint based on the concavity of its mutual information rate with respect to some parameterization of the input we numerically evaluate its first order markov capacity and further derive its full asymptotics the asymptotics obtained in this paper when compared with the recently derived feedback capacity for a binary erasure channel with the same input constraint enable us to draw the conclusion that feedback may increase the capacity of an input constrained channel even if the channel is memoryless

center manifolds for impulsive equations under nonuniform hyperbolicity abstract we establish the existence of smooth center manifolds under sufficiently small perturbations of an impulsive linear equation in particular we obtain the c 1 smoothness of the manifolds outside the jumping times we emphasize that we consider the general case of nonautonomous equations for which the linear part has a nonuniform exponential trichotomy

ck centre unstable manifolds we consider the existence and smoothness of global centre unstable manifolds for finite and infinite dimensional flows or maps we show that every global centre unstable manifold can be expressed as a graph of a c k map provided that the nonlinearities are c k smooth the proofs are based on a lemma by d henry on a necessary and sufficient condition for a lipschitz map to be continuously differentiable

a new algorithm for computing the multivariate faa di bruno s formula abstract a new algorithm for computing the multivariate faa di bruno s formula is provided we use a symbolic approach based on the classical umbral calculus that turns the computation of the multivariate faa di bruno s formula into a suitable multinomial expansion we propose a maple procedure whose computational times are faster compared with the ones existing in the literature some illustrative applications are also provided

comparison of fast boundary element methods on parametric surfaces we compare fast black box boundary element methods on parametric surfaces in r3 these are the adaptive cross approximation the multipole method based on interpolation and the wavelet galerkin scheme the surface representation by a piecewise smooth parameterization is in contrast to the common approximation of surfaces by panels nonetheless parametric surface representations are easily accessible from computer aided design cad and are recently topic of the studies in isogeometric analysis especially we can apply two dimensional interpolation in the multipole method a main feature of this approach is that the cluster bases and the respective moment matrices are independent of the geometry this results in a superior compression of the far field compared to other cluster methods

explicit error estimates for spline approximation of arbitrary smoothness in isogeometric analysis in this paper we provide a priori error estimates with explicit constants for both the l 2 projection and the ritz projection onto spline spaces of arbitrary smoothness defined on arbitrary grids this extends the results recently obtained for spline spaces of maximal smoothness the presented error estimates are in agreement with the numerical evidence found in the literature that smoother spline spaces exhibit a better approximation behavior per degree of freedom even for low smoothness of the functions to be approximated first we introduce results for univariate spline spaces and then we address multivariate tensor product spline spaces and isogeometric spline spaces generated by means of a mapped geometry both in the single patch and in the multi patch case

divided differences of implicit functions under general conditions the equation g x y 0 implic itly denes y locally as a function of x in this article we express divided dierences of y in terms of bivariate divided dierences of g generalizing a recent result on divided dierences of inverse functions

approximate solutions and micro regularity in the denjoy carleman classes abstract we begin with a sequence m of positive real numbers and we consider the denjoy carleman class c m we show how to construct m approximate solutions for complex vector fields with c m coefficients we then use our construction to study micro local properties of boundary values of approximate solutions in general m involutive structures of codimension one where the approximate solution is defined in a wedge whose edge where the boundary value exists is a maximally real submanifold we also obtain a c m version of the edge of the wedge theorem

essential statistics and tools for complex random variables complex random signals play an increasingly important role in array communications and biomedical signal processing and related fields however the fundamental properties of complex valued signals and mathematical tools needed to process them are scattered in literature we provide a concise unified and rigorous treatment of essential properties and tools of complex random variables and apply these fundamentals to derive complex extensions of leibniz rule faa di bruno s formula and taylor s series the extensions allow establishing relationships among complex moments and cumulants and characterizing the circularity property we propose measures for testing and quantifying circularity and observe that non circularity may be more common in practical applications than previously thought all results are rigorously proved and supplemented with clarifying examples

robustness of nonuniform exponential trichotomies in banach spaces abstract we establish the robustness under sufficiently small linear perturbations of nonuniform exponential trichotomies defined by linear equations x a t x in banach spaces we also establish the continuous dependence on the perturbation of the constants in the notion of trichotomy we consider both trichotomies in semi infinite intervals and trichotomies in r

center manifolds for smooth invariant manifolds this is the published version also available here http dx doi org 10 1090 s0002 9947 00 02443 0 first published in trans amer math soc in 2000 published by the american mathematical society

smoothness of invariant manifolds for nonautonomous equations for semiflows generated by ordinary differential equations v a t v admitting a nonuniform exponential dichotomy we show that for any sufficiently small perturbation f there exist smooth stable and unstable manifolds for the perturbed equation v a t v f t v as an application we establish the existence of invariant manifolds for the nonuniformly hyperbolic trajectories of a semiflow in particular we obtain smooth invariant manifolds for a class of vector fields that need not be c1 for any 0 1 to the best of our knowledge no similar statement was obtained before in the nonuniformly hyperbolic setting we emphasize that we do not need to assume the existence of an exponential dichotomy but only the existence of a nonuniform exponential dichotomy with sufficiently small nonuniformity when compared to the lyapunov exponents of the original linear equation furthermore for example in the case of stable manifolds we only need to assume that there exist negative lyapunov exponents while we also allow zero exponents our proof of the smoothness of the invariant manifolds is based on the construction of an invariant family of cones

derivatives of entropy rate in special families of hidden markov chains consider a hidden markov chain obtained as the observation process of an ordinary markov chain corrupted by noise recently zuk et al showed how in principle one can explicitly compute the derivatives of the entropy rate of at extreme values of the noise namely they showed that the derivatives of standard upper approximations to the entropy rate actually stabilize at an explicit finite time we generalize this result to a natural class of hidden markov chains called black holes we also discuss in depth special cases of binary markov chains observed in binary symmetric noise and give an abstract formula for the first derivative in terms of a measure on the simplex due to blackwell

optimization via intermittency with a self organizing neural network one of the major obstacles in using neural networks to solve combinatorial optimization problems is the convergence toward one of the many local minima instead of the global minima in this letter we propose a technique that enables a self organizing neural network to escape from local minima by virtue of the intermittency phenomenon it gives rise to novel search dynamics that allow the system to visit multiple global minima as meta stable states numerical experiments performed suggest that the phenomenon is a combined effect of kohonen type competitive learning and the iterated softmax function operating near bifurcation the resultant intermittent search exhibits fractal characteristics when the optimization performance is at its peak in the form of 1 f signals in the time evolution of the cost as well as power law distributions in the meta stable solution states then queens problem is used as an example to illustrate the meta stable convergence process that sequentially generates in a single run 92 solutions to the 8 queens problem and 4024 solutions to the 17 queens problem

asymptotic expansion of multivariate conservative linear operators in this paper we study the asymptotic expansion of the partial derivatives of a sequence of linear conservative operators we find sufficient conditions on the sequence that guarantee that such an expansion can be derived from the asymptotic formula for the nondifferentiated sequence our results are valid even if the operators of the sequence present difficult conservative properties different form the classical preservation of the usual convexities we use our theorems to obtain the complete asymptotic expansions and voronovskaja formulae for the partial derivatives of multivariate versions of the meyer konig and zeller operators and the bleimann butzer and hahn operators

analyticity of lagrangian trajectories for well posed inviscid incompressible fluid models abstract we discuss general incompressible inviscid models including the euler equations the surface quasi geostrophic equation incompressible porous medium equation and boussinesq equations all these models have classical unique solutions at least for short time we show that they have real analytic lagrangian paths more precisely we show that as long as a solution of any of these equations is in a class of regularity that assures holder continuous gradients of velocity the corresponding lagrangian paths are real analytic functions of time the method of proof is conceptually straightforward and general and we address the combinatorial issues head on

interpolation properties of generalized plane waves this paper is aimed at developing new shape functions adapted to the scalar wave equation with smooth possibly vanishing coefficients and investigates the numerical analysis of their interpolation properties the interpolation is local but high order convergence is shown with respect to the size of the domain considered the new basis functions are then implemented in a numerical method to solve a scalar wave equation problem with a mixed boundary condition the main theoretical result states that any given order of approximation can be achieved by an appropriate choice of parameters for the design of the shape functions the convergence is studied with respect to the size of the domain which is referred to in the literature as h h convergence

analysis of the domain mapping method for elliptic diffusion problems on random domains in this article we provide a rigorous analysis of the solution to elliptic diffusion problems on random domains in particular based on the decay of the karhunen loeve expansion of the domain perturbation field we establish decay rates for the derivatives of the random solution that are independent of the stochastic dimension for the implementation of a related approximation scheme like quasi monte carlo quadrature stochastic collocation etc we propose parametric finite elements to compute the solution of the diffusion problem on each individual realization of the domain generated by the perturbation field this simplifies the implementation and yields a non intrusive approach having this machinery at hand we can easily transfer it to stochastic interface problems the theoretical findings are complemented by numerical examples for both stochastic interface problems and boundary value problems on random domains

a multivariate faa di bruno formula with applications a multivariate faa di bruno formula for computing arbitrary partial derivatives of a function composition is presented it is shown by way of a general identity how such derivatives can also be expressed in the form of an infinite series applications to stochastic processes and multivariate cumulants are then delineated

stability criterion for linear systems with nonlinear delayed perturbations a stability criterion for linear systems with nonlinear time varying delayed perturbations has been derived in light of razumikhin type methods as the proposed criterion is easily applicable and independent of the delay size it provides an effective method for the stability analysis of time delay systems

consistent results on life abstract the findings presented by bennett and bourzutschky that the cellular automaton game of life is not self organized critical which was earlier suggested by bak et al have been questioned most recently by alstrom and leao the reason for this is that the two groups used systems with different boundary conditions and measured different properties we measure the same properties as bak et al and find that the power law seen for smaller lattices disappears for larger lattices the conclusion is that the game of life is subcritical but with a rather long length scale

some extensions of faa di bruno s formula with divided differences the well known formula of faa di bruno s for higher derivatives of a composite function has played an important role in combinatorics in this paper we generalize the divided difference form of faa di bruno s formula and give an explicit formula for the n th divided difference of a multicomposite function more generally we establish the relationships of the bell polynomials with respect to multicomposite functions applying these to multicomposite functions we obtain some extensions of faa di bruno s formula

kaleidoscope of life a 24 neighbourhood outer totalistic cellular automaton abstract one of the challenges of cellular automaton research is finding models with a low complexity and at the same time a rich dynamics a measure of low complexity is the number of states in the model and the number of transition rules to switch between those states in this paper we propose a 2 dimensional 2 state cellular automaton that though governed by a single simple transition rule has a sufficiently rich dynamics to be computationally universal according to the transition rule a cell s state is determined by the sum of the states of the cells at orthogonal or diagonal distances one or two from the cell distance 2 moore neighbourhood but not by the previous state of the cell itself notwithstanding its simplicity this model is able to generate a great variety of patterns including several types of stable configurations oscillators and patterns that move over cellular space gliders we prove the computational universality of the model by constructing a universal set of logic gates not and and from these patterns a key element in this proof is the shifting of phases and positions of signals such that they meet the input requirements of the logic gates similarities of the model with classical spin systems are also discussed

a kam theorem for finitely differentiable hamiltonian systems abstract given l 2 2 d 4 we prove the persistence of a cantor family of kam tori of measure o e 1 2 l for any non degenerate nearly integrable hamiltonian system of class c l d t d where 1 is the diophantine power of the frequencies of the persistent kam tori and d r d is a bounded domain provided that the size e of the perturbation is sufficiently small this extends a result by d salamon in 25 according to which we do have the persistence of a single kam torus in the same framework moreover as for the persistence of a single torus the regularity assumption is essentially optimal

asynchronous game of life abstract this paper proposes the 8 state asynchronous equivalence of the well known game of life gl our cellular automaton called asynchronous game of life simulates exactly the behavior of the gl such as universal computation and self organization no matter whether the update of cells is simultaneous or independent according to some updating scheme like a step driven or time driven method see biosystems 51 1999 123 we employ the updating scheme of blok and bergersen phys rev e 59 1999 3876 such that at every time step each cell has a certain probability to be updated and investigate the statistical properties of our model through power spectral analyses

delay independent stability conditions and decay estimates for time varying functional differential equations this paper presents sufficient delay independent conditions that guarantee stability of nonlinear time varying functional differential equations fde s estimates on the rate of decay of solutions are also obtained

simple stability criteria for single and composite linear systems with time delays easy ways to test the stability of systems involving time delays have been sought in this paper several stability conditions with an extremely simple form are provided first some criteria for single linear systems with time delays are presented then these results are extended to the composite linear systems with time delays which exist in each subsystem and in the interconnections between subsystems among these criteria those which are expressed by scalar inequalities also permit us to assess the transient behaviour of systems

quantitative measures of robustness for systems including delayed perturbations asymptotically stable linear systems subject to delayed time varying and nonlinear perturbations are considered razumikhin type theorems are used to obtain easy to compute bounds on the perturbations so that the systems remain stable results indicate that if delayed perturbations are included then the bound is reduced as compared to the one for nondelayed perturbations however in certain cases previously obtained bounds for the nondelayed perturbations guarantee stability even when delayed perturbations are in effect

stability theory and lyapunov regularity abstract we establish the stability under perturbations of the dynamics defined by a sequence of linear maps that may exhibit both nonuniform exponential contraction and expansion this means that the constants determining the exponential behavior may increase exponentially as time approaches infinity in particular we establish the stability under perturbations of a nonuniform exponential contraction under appropriate conditions that are much more general than uniform asymptotic stability the conditions are expressed in terms of the so called regularity coefficient which is an essential element of the theory of lyapunov regularity developed by lyapunov himself we also obtain sharp lower and upper bounds for the regularity coefficient thus allowing the application of our results to many concrete dynamics it turns out that using the theory of lyapunov regularity we can show that the nonuniform exponential behavior is ubiquitous contrarily to what happens with the uniform exponential behavior that although robust is much less common we also consider the case of infinite dimensional systems

quantification of spatiotemporal phenomena by means of cellular automata techniques abstract quantification methods for spatiotemporal patterns are introduced which are based on nearest neighbor considerations inspired by cellular automata as well as by more complex spatiotemporal dynamics in particular spatial and temporal structures which can be found in aggregation and clustering phenomena are quantified by introducing the concept of cellular automata ca measures for homogeneity and for the amount of fluctuations contributing to the observed dynamics the aim of this paper is to define such measures and then test their performance on theoretically generated data sets these tests with exemplary spatiotemporal patterns show the applicability of these methods for data analysis for example with the help of the ca fluctuation number we are able to extract the amount of noise present in spatiotemporal data the observables constructed with the help of such ca techniques have straightforward applications in life sciences where the quantitative analysis of spatiotemporal patterns still suffers from the lack of a standardized set of analysis techniques

center manifolds for infinite delay we obtain a c1 center manifold theorem for perturbations of delay difference equations in banach spaces with infinite delay our results extend in several directions the existing center manifold theorems besides considering infinite delay equations we consider perturbations of nonuniform exponential trichotomies and generalized trichotomies that may exhibit stable unstable and central behaviors with respect to arbitrary asymptotic rates ec n for some diverging sequence n this includes as a very special case the usual exponential behavior with n n

the cauchy lagrangian method for numerical analysis of euler flow a novel semi lagrangian method is introduced to solve numerically the euler equation for ideal incompressible flow in arbitrary space dimension it exploits the time analyticity of fluid particle trajectories and requires in principle only limited spatial smoothness of the initial data efficient generation of high order time taylor coefficients is made possible by a recurrence relation that follows from the cauchy invariants formulation of the euler equation zheligovsky and frisch 2014 44 truncated time taylor series of very high order allow the use of time steps vastly exceeding the courant friedrichs lewy limit without compromising the accuracy of the solution tests performed on the two dimensional euler equation indicate that the cauchy lagrangian method is more and occasionally much more efficient and less prone to instability than eulerian runge kutta methods and less prone to rapid growth of rounding errors than the high order eulerian time taylor algorithm we also develop tools of analysis adapted to the cauchy lagrangian method such as the monitoring of the radius of convergence of the time taylor series certain other fluid equations can be handled similarly