on a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with nonglobally monotone coefficients we develop a perturbation theory for stochastic differential equations sdes by which we mean both stochastic ordinary differential equations sodes and stochastic partial differential equations spdes in particular we estimate the l p distance between the solution process of an sde and an arbitrary ito process which we view as a perturbation of the solution process of the sde by the l q distances of the differences of the local characteristics for suitable p q 0 as one application of the developed perturbation theory we establish strong convergence rates for numerical approximations of a class of sodes with nonglobally monotone coefficients as another application of the developed perturbation theory we prove strong convergence rates for spatial spectral galerkin approximations of solutions of semilinear spdes with nonglobally monotone nonlinearities including cahn hilliard cook type equations and stochastic burgers equations further applications of the developed perturbation theory include regularity analyses of solutions of sdes with respect to their initial values as well as small noise analyses for ordinary and partial differential equations

fixed set systems of equivariant infinite loop spaces we develop machinery enabling us to show that suitable g spaces including the equivariant version of bf are equivariant infinite loop spaces this involves a recognition principle for systems of spaces which behave for mally like the system of fixed sets of a g space that is we give a necessary and sufficient condition for such a system to be equivalent to the fixed set system of an equivariant infinite loop space the advantage of using the language of fixed set systems is that one can frequently replace the system of fixed sets of an ac tual g space by an equivalent formal system which is considerably simpler and which admits the requisite geometry necessary for delooping we also apply this machinery to construct equivariant eilenberg mac lane spaces corresponding to mackey functors

absolute continuity and numerical approximation of stochastic cahn hilliard equation with unbounded noise diffusion abstract in this article we consider the absolute continuity and numerical approximation of the solution of the stochastic cahn hilliard equation with unbounded noise diffusion we first obtain the holder continuity and malliavin differentiability of the solution of the stochastic cahn hilliard equation by using the strong convergence of the spectral gakerkin approximation then we prove the existence and strict positivity of the density function of the law of the exact solution for the stochastic cahn hilliard equation with sublinear growth diffusion coefficient which fills a gap for the existed result when the diffusion coefficient satisfies a growth condition of order 1 3 1 to approximate the density function of the exact solution we propose a full discretization based on the spatial spectral galerkin approximation and the temporal drift implicit euler scheme furthermore a general framework for deriving the strong convergence rate of the full discretization is developed based on the variation approach and the factorization method consequently we obtain the sharp mean square convergence rates in both time and space via sobolev interpolation inequalities and semigroup theories to the best of our knowledge this is the first result on the convergence rate of full discretizations for the considered equation

integral deligne cohomology for real varieties we develop an integral version of deligne cohomology for smooth proper real varieties for this purpose the role played by singular cohomology in the complex case has to be replaced by the ordinary bigraded gal mathbb c mathbb r equivariant cohomology of lewis et al bull am math soc n s 4 2 208 212 1981 the equivariant counterpart of singular cohomology the theory is aimed at giving more precise information about the 2 primary components of regulators we establish basic properties and give a geometric interpretation for the groups in dimension 2 in weights 1 and 2

the structure of some equivariant thom spectra we show that the equivariant thom spectra moz2 and moz2 do not split as wedges of equivariant eilenberg mac lane spectra as they do nonequivariantly this is done by finding two stage postnikov towers giving these spectra and determining the nontrivial k invariants we also consider the question in what sense is the spectrum moz2 representing unoriented bordism unique

s1 degree and global hopf bifurcation theory of functional differential equations abstract the recently developed s 1 degree and bifrucation theory are applied to provide a purely topological argument of a global hopf bifurcation theory for functional differential equations of mixed type in the special case where the equation is of retarded type the established result represents an analog of alexander and yorke s global hopf bifurcation theorem which has been obtained by chow fiedler mallet paret and nussbaum using different approaches

malliavin calculus for the stochastic cahn hilliard allen cahn equation with unbounded noise diffusion abstract the stochastic partial differential equation analyzed in this work is motivated by a simplified mesoscopic physical model for phase separation it describes pattern formation due to adsorption and desorption mechanisms involved in surface processes in the presence of a stochastic driving force this equation is a combination of cahn hilliard and allen cahn type operators with a multiplicative white space time noise of unbounded diffusion we apply malliavin calculus in order to investigate the existence of a density for the stochastic solution u in dimension one according to the regularity result in 5 u admits continuous paths a s using this property and inspired by a method proposed in 8 we construct a modified approximating sequence for u which properly treats the new second order allen cahn operator under a localization argument we prove that the malliavin derivative of u exists locally and that the law of u is absolutely continuous establishing thus that a density exists

asymptotic behavior near transition fronts for equations of generalized cahn hilliard form we consider the asymptotic behavior of perturbations of standing wave solutions arising in evolutionary pde of generalized cahn hilliard form in one space dimension such equations are well known to arise in the study of spinodal decomposition a phenomenon in which the rapid cooling of a homogeneously mixed binary alloy causes separation to occur resolving the mixture into its two components with their concentrations separated by sharp transition layers motivated by work of bricmont kupiainen and taskinen 5 we regard the study of standing waves as an interesting step toward understanding the dynamics of these transitions a critical feature of the cahn hilliard equation is that the linear operator that arises upon linearization of the equation about a standing wave solution has essential spectrum extending onto the imaginary axis a feature that is known to complicate the step from spectral to nonlinear stability under the assumption of spectral stability described in terms of an appropriate evans function we develop detailed asymptotics for perturbations from standing wave solutions establishing phase asymptotic orbital stability for initial perturbations decaying with appropriate algebraic rate

pin 2 equivariant seiberg witten floer homology and the triangulation conjecture we define pin 2 equivariant seiberg witten floer homology for rational homology 3 spheres equipped with a spin structure the analogue of froyshov s correction term in this setting is an integer valued invariant of homology cobordism whose mod 2 reduction is the rokhlin invariant as an application we show that there are no homology 3 spheres y of rokhlin invariant one such that y y bounds an acyclic smooth 4 manifold by previous work of galewski stern and matumoto this implies the existence of non triangulable high dimensional manifolds

a second order splitting method for the cahn hilliard equation a semi discrete finite element method requiring only continuous element is presented for the approximation of the solution of the evolutionary fourth order in space cahn hilliard equation optimal order error bounds are derived in various norms for an implementation which uses mass lumping the continuous problem has an energy based lyapunov functional it is proved that this property holds for the discrete problem

homotopy invariants of time reversible periodic orbits ii towards applications abstract for a reversible periodic orbit we apply the sequence of homotopy invariants deg n n 1 2 3 defined in fiedler heinze 1996 we use this sequence to prove a global bifurcation result for reversible periodic orbits with prescribed minimal period this result will be applied to second order systems with neumann boundary conditions a discussion and remarks on the sequence of degrees concludes the paper

on the dimension and the index of the solution set of nonlinear equations we study the covering dimension and the index of the solution set to multiparameter nonlinear and semilinear operator equations involving fredholm maps of positive index the classes of maps under consideration are pseudo a proper and either approximation essential or equivariant approximation essential applications are given to semilinear elliptic bvp s

finite element approximation of the cahn hilliard cook equation we study the nonlinear stochastic cahn hilliard equation perturbed by additive colored noise we show almost sure existence and regularity of solutions we introduce spatial approximation by a standard finite element method and prove error estimates of optimal order on sets of probability arbitrarily close to 1 we also prove strong convergence without known rate

equivariant vector fields on spheres we address the following question if g is a compact lie group and s m is the unit sphere of an r g module m then how many orthonormal g invariant vector fields can be found on s m we call this number the g field number of m under reasonable hypotheses on m we reduce this question to determining when the difference of two g vector bundles vanishes in a certain subquotient of the kog theory of a real projective space in this general setting we solve the problem for 2 groups for odd order groups and for abelian groups if m also has enough orbit types for example all of them then we solve the problem for arbitrary finite groups we also show that under mild hypotheses on m the g field number depends only on the dimensions of the fixed point sets of m acknowledgements i wish to take this opportunity to express my heartfelt thanks to peter may for his constant guidance and encouragement and for his careful reading of this paper to jim mcclure for many helpful ideas and enlightening conversations and to mel rothenberg for being a one man audience and for all his suggestions and improvements introduction in 1940 hopf 14 and later eckmann 9 observed that a theorem of hurwitz and radon could be used to construct orthonormal vector fields on spheres they showed that if n 2a 1 2c 4d where 0 c s 3 then p n 1 orthonormal vector fields can be constructed on sn where p n 8d 2c is the nth hurwitz radon number twenty years later adams 1 showed that this was best possible there do not exist p n orthonormal vector fields on sn the purpose of this paper is to address the following question if g is a compact lie group and s m is the unit sphere of a real g module m then how many orthonormal g invariant vector fields or g fields can be found on s m we shall call this number the g field number of m previous work has been done on this problem by becker 6 he deals with the case in which g is finite and acts freely on s m he shows that under mild hypotheses the g field number of m depends only on g and the real dimension of m he also shows that it coincides with the e field number of m where 2 is any 2 sylow subgroup of g received by the editors march 22 1982 1980 mathematics subject classification primary 57r25 57s25 secondary 55r25 55r50 the author was tragically killed in an automobile accident on december 23 1981 the editors thank peter may and jim becker for their help in bringing this paper to publication requests for reprints should be addressed to peter may university of chicago 1983 american mathematical society 0002 9947 82 00001355 10 25

dynamical bifurcation for the kuramoto sivashinsky equation abstract in this paper by using the center manifold reduction method together with the eigenvalue analysis we made bifurcation analysis for the kuramoto sivashinsky equation and proved that the kuramoto sivashinsky equation with constraint condition bifurcates an attractor a as crossed the first critical value 0 1 under the two cases our analysis was based on a new and mature attractor bifurcation theory developed by ma and wang 2005 17 18

galerkin finite element approximations for stochastic space time fractional wave equations the traditional wave equation models wave propagation in an ideal medium to characterize wave propagation in inhomogeneous media with frequency dependent power law attenuation the space time fractional wave equation is needed further incorporating the additive white gaussian noise external noise coming from many natural sources leads to the stochastic space time fractional wave equation this paper discusses the galerkin finite element approximations for the stochastic space time fractional wave equation forced by an additive space time white noise we first discretize the space time additive noise which introduces a modeling error and results in a regularized stochastic space time fractional wave equation then the regularity of the regularized equation is analyzed for the discretization in space the finite element approximation is used and the definition of the discrete fractional laplacian is introduced we derive the mean squared l 2 norm a priori estimates for the modeling error and for the

a second order accurate linearized difference scheme for the two dimensional cahn hilliard equation the cahn hilliard equation is a nonlinear evolutionary equation that is of fourth order in space in this paper a linearized finite difference scheme is derived by the method of reduction of order it is proved that the scheme is uniquely solvable and convergent with the convergence rate of order two in a discrete l2 norm the coefficient matrix of the difference system is symmetric and positive definite so many well known iterative methods e g gauss seidel sor can be used to solve the system

nonlinear aspects of the cahn hilliard equation abstract this paper treats phase separation within the context of the phenomenological cahn hilliard equation c t m c f c k 2 c where c is the concentration m c is the mobility and f c is homogeneou s free energy which is assumed here to be a fourth degree polynomial natural boundary conditions are introduced the full set of equilibrium solutions is specified a comparison theorem for stability criteria which was postulated by langer is proved here within the framework of the natural boundary conditions energy methods are used to define and estimate the limit of monotonic global stability it is pointed out that within the parameter region where the uniform homogeneous state is the only equilibrium solution there may still exist some internal excitable region in which the homogeneous solution possesses growing fluctuations furthermore a periodic instability is shown to exist in the metastable region in addition to the well known nucleation instability

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

strong and weak convergence rates of a spatial approximation for stochastic partial differential equation with one sided lipschitz coefficient strong and weak approximation errors of a spatial finite element method are analyzed for the stochastic partial differential equations spdes with one sided lipschitz coefficients including the s

strong convergence rate of finite difference approximations for stochastic cubic schrodinger equations abstract in this paper we derive a strong convergence rate of spatial finite difference approximations for both focusing and defocusing stochastic cubic schrodinger equations driven by a multiplicative q wiener process beyond the uniform boundedness of moments for high order derivatives of the exact solution the key requirement of our approach is the exponential integrability of both the exact and numerical solutions by constructing and analyzing a lyapunov functional and its discrete correspondence we derive the uniform boundedness of moments for high order derivatives of the exact solution and the first order derivative of the numerical solution which immediately yields the well posedness of both the continuous and discrete problems the latter exponential integrability is obtained through a variant of a criterion given by cox hutzenthaler and jentzen arxiv 1309 5595 as a by product of this exponential integrability we prove that the exact and numerical solutions depend continuously on the initial data and obtain a large deviation type result on the dependence of the noise with first order strong convergence rate

a parallel domain decomposition based implicit method for the cahn hilliard cook phase field equation in 3d we present a numerical algorithm for simulating the spinodal decomposition described by the three dimensional cahn hilliard cook chc equation which is a fourth order stochastic partial differential equation with a noise term the equation is discretized in space and time based on a fully implicit cell centered finite difference scheme with an adaptive time stepping strategy designed to accelerate the progress to equilibrium at each time step a parallel newton krylov schwarz algorithm is used to solve the nonlinear system we discuss various numerical and computational challenges associated with the method the numerical scheme is validated by a comparison with an explicit scheme of high accuracy and unreasonably high cost we present steady state solutions of the chc equation in two and three dimensions the effect of the thermal fluctuation on the spinodal decomposition process is studied we show that the existence of the thermal fluctuation accelerates the spinodal decomposition process and that the final steady morphology is sensitive to the stochastic noise we also show the evolution of the energies and statistical moments in terms of the parallel performance it is found that the implicit domain decomposition approach scales well on supercomputers with a large number of processors

equivariant cohomology and sheaves abstract ro g graded equivariant cohomology was originally defined by using equivariant spectra in this paper we construct a sheaf version of ro g graded equivariant cohomology theory by introducing a complex of presheaves we obtain an isomorphism theorem of two different cohomology theories we further apply the results to the study of real algebraic varieties

computation of the equivariant 1 stem abstract let g be a compact lie group in this paper combining a short exact sequence obtained by balanov and krawcewicz with some additional topological techniques we complete the computation of the first equivariant stem 1 g st using the exact sequence and a property of nonabelian connected compact lie groups whose proof was suggested to us by r oliver we show that this group is finite if and only if g is finite

the fourier collocation method for the cahn hilliard equation abstract in this paper a fourier collocation method for numerically solving cahn hilliard equations with periodic boundary conditions is developed we establish their semidiscrete and fully discrete schemes that inherit the energy dissipation property and mass conservation property from the associated continuous problem we prove existence and uniqueness of the numerical solution and derive the optimal error bounds we perform some numerical experiments which confirm our results

vector bundles over classifying spaces of compact lie groups the completion theorem of atiyah and segal as says that the complex k theory group k bg of the classifying space of any compact lie group g is isomorphic to r g the representation ring completed with respect to its augmentation ideal however the group k bg bg z bu does not directly contain information about vector bundles over the infinite dimensional complex bg itself the purpose of this paper is to compare the grothendieck group of vector bundles over bg which we denote k bg with both k bg and r g the main result is an algebraic description of k bg in terms of the representation rings of certain subgroups of g as one consequence we show that of the natural maps

spectra of manifolds less a small domain soit m une variete de riemann c connexe compacte et l operateur de laplace beltrami associee on etudie le spectre pour une sous variete m compacte e 0 et b e le voisinage tubulaire de m de rayon e d operateur e

global hopf bifurcation of differential equations with threshold type state dependent delay abstract we develop global hopf bifurcation theory of differential equations with state dependent delay using the s 1 equivariant degree and investigate a two degree of freedom mechanical model of turning processes for the model of turning processes we show that the extreme points of each vibration component of the non constant periodic solutions can be embedded into a manifold with explicit algebraic expression this observation enables us to establish analytical upper and lower bounds of the amplitudes of the periodic solutions in terms of the system parameters and to exclude certain periods using the achieved global bifurcation theory we reveal that if the relative frequency between the natural frequency and the turning frequency varies in a certain interval then generically every bifurcated continuum of periodic solutions must terminate at a bifurcation point this termination means that the underlying system with parameters in the stability region near the vertical asymptotes of the stability lobes is less subject to chatter in the process several sufficient conditions for the non existence of non constant periodic solutions are also obtained

a fourth order parabolic equation with nonlinear principal part in this paper we study a fourth order parabolic equation with nonlinear principal part modeling epitaxial thin film growth in two space dimensions on the basis of the schauder type estimates and campanato spaces we prove the global existence of classical solutions

area versus capacity and independence in the crushed ice model we study the heat flow from an infinite union b of disjoint closed balls in euclidean space rm kept at constant temperature 1 while the surrounding medium is at temperature 0 at time 0 we obtain conditions on the geometry of b such that the heat flow for small time is governed by the area of the boundary of b

heat content and hardy inequality for complete riemannian manifolds abstract upper bounds are obtained for the heat content of an open set d in a complete riemannian manifold provided the dirichlet laplace beltrami operator satisfies a strong hardy inequality and the distance function on d satisfies an integrability condition

on a discrete model of phase separation dynamics a spatially discrete model for phase separation with conserved order parameter is proposed this one dimensional model is obtained as the deterministic limit of an anisotropic lattice gas a particular choice is made for the jump rates which still fulfill detailed balance conditions so that the resulting model is mathematically tractable it exhibits a phase transition of first order type whose nonlinear dynamics is investigated using both analytical and numerical methods all the stationary solutions with zero current are found and parametrized in terms of jacobian elliptic functions showing a striking similarity with the nonlinear continuous cahn hilliard equation in the limit of infinite wavelength particular solutions are found which descrive isolated domains of arbitrary size embedded in an homogeneous infinite medium of the opposite phase new results are also presented on the structure of the set of solutions time dependent profiles are studied in the spinodal regime and the stability of bounded stationary solutions is also investigated in this context a description of time dependent profiles is proposed which considers only interactions between neighboring domains and makes use of isolated domain solutions this approach results in an analytic expression for the exponents characteristic of the instability of stationary solutions and is validated by comparison to numerical values qualitative results are also discussed and the relation to the cahn hilliard equation is emphasized

symmetric functional differential equations and neural networks with memory we establish an analytic local hopf bifurcation theorem and a topological global hopf bifurcation theorem to detect the existence and to describe the spatial temporal pattern the asymptotic form and the global continuation of bifurcations of periodic wave solutions for functional differential equations in the presence of symmetry we apply these general results to obtain the coexistence of multiple large amplitude wave solutions for the delayed hopfield cohen grossberg model of neural networks with a symmetric circulant connection matrix

strong convergence of a fully discrete finite element approximation of the stochastic cahn hilliard equation we consider the stochastic cahn hilliard equation driven by additive gaussian noise in a convex domain with polygonal boundary in dimension d le 3 we discretize the equation using a standard fi

the equivariant hurewicz map let g be a compact lie group y be a based g space and v be a g representation if v g y is the equivariant homotopy group of y in dimension v and h v g y fis the equivariant ordinary homology group of y with burnside ring coefficients in dimension v then there is an equivariant hurewicz map h v g y h v g y one should not expect this map to be an isomorphism since h v g y must be a module over the burnside ring but v g y need not be however here it is shown that under the obvious connectivity conditions on y this map induces an isomorphism between h v g y and an algebraically defined modification of v g y

second order schemes and time step adaptivity for allen cahn and cahn hilliard models abstract in this paper we focus on efficient second order in time approximations of the allen cahn and cahn hilliard equations first of all we present the equations generic second order schemes based on a mid point approximation of the diffusion term and some schemes already introduced in the literature then we propose new ways of deriving second order in time approximations of the potential term starting from the main schemes introduced in guillen gonzalez and tierra 2013 yielding to new second order schemes for these schemes and other second order schemes previously introduced in the literature we study the constraints on the physical and discrete parameters that can appear to assure the energy stability unique solvability and in the case of nonlinear schemes the convergence of newton s method to the nonlinear schemes moreover in order to save computational cost we have developed a new adaptive time stepping algorithm based on the numerical dissipation introduced in the discrete energy law in each time step finally we compare the behaviour of the schemes and the effectiveness of the adaptive time stepping algorithm through several computational experiments

strong convergence rate of splitting schemes for stochastic nonlinear schrodinger equations abstract in this paper we show that solutions of stochastic nonlinear schrodinger nls equations can be approximated by solutions of coupled splitting systems based on these systems we propose a new kind of fully discrete splitting schemes which possess algebraic strong convergence rates for stochastic nls equations key ingredients of our approach are using the exponential integrability and stability of the corresponding splitting systems and numerical approximations in particular under very mild conditions we derive the optimal strong convergence rate o n 2 1 2 of the spectral splitting crank nicolson scheme where n and denote the dimension of the approximate space and the time step size respectively

hecke algebras and cohomotopical mackey functors in this paper we define the concept of the cohomotopical mackey functor which is more general than the usual cohomological mackey functor and show that hecke algebra techniques are applicable to cohomotopical mackey functors our theory is valid for any possibly infinite discrete group some applications to topology are also given

on spectral asymptotics for domains with fractal boundaries we discuss the spectral properties of the laplacian for domains with fractal boundaries the main goal of the article is to find the second term of spectral asymptotics of the counting functionn or its integral transformations function function for domains with smooth boundaries the order of the second term ofn under billiard condition is one half of the dimension of the boundary in the case of fractal boundaries the well known weyl berry hypothesis identifies it with one half of the hausdorff dimension of and the modified weyl berry conjecture with one half of the minkowski dimension of we find the spectral asymptotics for three natural broad classes of fractal boundaries cabbage type bubble type and web type and show that the minkowski dimension gives the proper answer for cabbage type of boundaries due to one dimensional structure of the cabbage type fractals but the answers are principally different in the two other cases

a discontinuous galerkin method for stochastic cahn hilliard equations abstract in this paper a discontinuous galerkin method for the stochastic cahn hilliard equation with additive random noise which preserves the conservation of mass is investigated numerical analysis and error estimates are carried out for the linearized stochastic cahn hilliard equation the effects of the noises on the accuracy of our scheme are also presented numerical examples simulated by monte carlo method for both linear and nonlinear stochastic cahn hilliard equations are presented to illustrate the convergence rate and validate our conclusion

error estimates with smooth and nonsmooth data for a finite element method for the cahn hilliard equation a finite element method for the cahn hilliard equation a semilinear parabolic equation of fourth order is analyzed both in a spatially semidiscrete case and in a completely discrete case based on the backward euler method error bounds of optimal order over a finite time interval are obtained for solutions with smooth and nonsmooth initial data a detailed study of the regularity of the exact solution is included the analysis is based on local lipschitz conditions for the nonlinearity with respect to sobolev norms and the existence of a ljapunov functional for the exact and the discretized equations is essential a result concerning the convergence of the attractor of the corresponding approximate nonlinear semigroup upper semicontinuity with respect to the discretization parameters is obtained as a simple application of the nonsmooth data error estimate

second phase spinodal decomposition for the cahn hilliard cook equation we consider the dynamics of a nonlinear partial differential equation perturbed by additive noise assuming that the underlying deterministic equation has an unstable equilibrium we show that the nonlinear stochastic partial differential equation exhibits essentially linear dynamics far from equilibrium more precisely we show that most trajectories starting at the unstable equilibrium are driven away in two stages after passing through a cylindrical region most trajectories diverge from the deterministic equilibrium through a cone shaped region which is centered around a finite dimensional subspace corresponding to strongly unstable eigenfunctions of the linearized equation and on which the influence of the nonlinearity is surprisingly small this abstract result is then applied to explain spinodal decomposition in the stochastic cahn hilliard cook equation on a domain g this equation depends on a small interaction parameter e 0 and one is generally interested in asymptotic results as e 0 specifically we show that linear behavior dominates the dynamics up to distances from the deterministic equilibrium which can reach e 2 dim g 2 with respect to the h 2 g norm

hopf bifurcations of functional differential equations with dihedral symmetries abstract we discuss the joint impact of temporal delay and spatial dihedral symmetries on the occurence and multiplicity of hopf bifurcations for a system of fdes by applying the equivariant degree theory we establish a result on the existence of multiple branches of nonconstant periodic solutions and classify their symmetries general results are illustrated by a ring of identical oscillators with identical coupling between adjacent cells

erratum finite element approximation of the cahn hilliard cook equation we prove an additional result on the linearized cahn hilliard cook equation to fill a gap in the main argument in our paper that was published in siam j numer anal 49 2011 pp 2407 2429 the result is a pathwise error estimate which is proved by an application of the factorization argument for stochastic convolutions

local minimisers and singular perturbations we construct local minimisers to certain variational problems the method is quite general and relies on the theory of convergence the approach is demonstrated through the model problem it is shown that in certain nonconvex domains n and for e small there exist nonconstant local minimisers u e satisfying u e 1 except in a thin transition layer the location of the layer is determined through the requirement that in the limit u e u 0 the hypersurface separating the states u 0 1 and u 0 1 locally minimises surface area generalisations are discussed with for example vector valued u and anisotropic perturbations replacing u 2

a nonconforming finite element method for the two dimensional cahn hilliard equation the cahn hilliard equation is a nonlinear evolutionary equation that is fourth order in space in this paper a continuous in time finite element galerkin approximation is considered we use the nonconforming morley element and derive optimal order error bounds in l 2

analysis of a splitting scheme for damped stochastic nonlinear schrodinger equation with multiplicative noise in this paper we investigate the damped stochastic nonlinear schrodinger nls equation with multiplicative noise and its splitting based approximation when the damped effect is large enough we

structure and dimension of global branches of solutions to multiparameter nonlinear equations this paper is concerned with the topological dimension of global branches of solutions appearing in different problems of nonlinear analysis in particular multiparameter including infinite dimensional continuation and bifurca tion problems by considering an extension of the notion of essential maps defined on sets and using elementary point set topology we are able to unify and extend in a selfcontained fashion most of the recent results on such problems our theory applies whenever any generalized degree theory with the boundary dependence property may be used but with no need of a gebraic structures our applications to continuation and bifutcation follow from the nontriviality of a local invariant in the stable homotopy group of a sphere and give information on the local dimension and behavior of the sets of solutions of bifurcation points and of continuation points

stable maps into free g spaces in this paper we introduce a systematic method for calculating the group of stable equivariant maps x y g 3 18 into a g free space or spectrum y in fact the method applies without restriction on x whenever g is a p group and y is p complete and satisfies standard finiteness assumptions the method is an adams spectral sequence based on a new equivariant coho mology theory c x which we introduce in 1 this spectral sequence is quite calculable and provides a natural generalisation of the classical adams spectral sequence based on ordinary modp cohomology it also geometrically realises certain inverse limits of nonequivariant adams spectral sequences which have been useful in the study of the segal conjecture 19 5 21 9

sur une conjecture de m kac we consider the following heat conduction problem let k be a compact set in euclidean space 3 suppose that k is held at the temperature 1 while the surrounding medium is at the temperature 0 at time 0 following spitzer we investigate the asymptotic behaviour of the integral e k t which represents the total energy flow in time t from the set k to the surrounding medium 3 k an asymptotic expansion is given for e k t which refines a theorem due to spitzer this expansion also verifies and improves a formal calculation of kac similar results are proved in higher dimensions up to the constant m k the quantity e k t can be interpreted as the expected value of the volume of the wiener sausage associated with k and a d dimensional brownian motion this point of view both plays a major role in the proofs and leads to a probabilistic interpretation of the different terms of the expansion

strong convergence of euler type methods for nonlinear stochastic differential equations traditional finite time convergence theory for numerical methods applied to stochastic differential equations sdes requires a global lipschitz assumption on the drift and diffusion coefficients in practice many important sde models satisfy only a local lipschitz property and since brownian paths can make arbitrarily large excursions the global lipschitz based theory is not directly relevant in this work we prove strong convergence results under less restrictive conditions first we give a convergence result for euler maruyama requiring only that the sde is locally lipschitz and that the pth moments of the exact and numerical solution are bounded for some p 2 as an application of this general theory we show that an implicit variant of euler maruyama converges if the diffusion coefficient is globally lipschitz but the drift coefficient satisfies only a one sided lipschitz condition this is achieved by showing that the implicit method has bounded moments and may be viewed as an euler maruyama approximation to a perturbed sde of the same form second we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial

extremal geometry of a brownian porous medium the path w 0 t of a brownian motion on a d dimensional torus mathbb t d run for time t is a random compact subset of mathbb t d we study the geometric properties of the complement mathbb t d setminus w 0 t as t rightarrow infty for d ge 3 in particular we show that the largest regions in mathbb t d setminus w 0 t have a linear scale varphi d t d log t d 2 kappa d t 1 d 2 where kappa d is the capacity of the unit ball more specifically we identify the sets e for which mathbb t d setminus w 0 t contains a translate of varphi d t e and we count the number of disjoint such translates furthermore we derive large deviation principles for the largest inradius of mathbb t d setminus w 0 t as t rightarrow infty and the varepsilon cover time of mathbb t d as varepsilon downarrow 0 our results which generalise laws of large numbers proved by dembo et al electron j probab 8 15 1 14 2003 are based on a large deviation estimate for the shape of the component with largest capacity in mathbb t d setminus w rho t 0 t where w rho t 0 t is the wiener sausage of radius rho t with rho t chosen much smaller than varphi d t but not too small the idea behind this choice is that mathbb t d setminus w 0 t consists of lakes whose linear size is of order varphi d t connected by narrow channels we also derive large deviation principles for the principal dirichlet eigenvalue and for the maximal volume of the components of mathbb t d setminus w rho t 0 t as t rightarrow infty our results give a complete picture of the extremal geometry of mathbb t d setminus w 0 t and of the optimal strategy for w 0 t to realise extreme events

wiener sausage and self intersection local times abstract let b bt t 0 be a standard brownian motion in r 2 for every e 0 and every compact subset k of r 2 the wiener sausage of radius e associated with k is defined as the union of the sets bs ek s 0 1 the present paper gives full asymptotic expansions for the area of the wiener sausage when the radius e goes to 0 the kth term of the expansion is of order log e k and involves a random variable which measures the number of k multiple self intersections of the process such random variables are called renormalized self intersection local times and have been recently introduced and studied by e b dynkin a self contained construction of these local times is given together with a number of new approximations

bigraded equivariant cohomology of real quadrics we give a complete description of the bigraded bredon co homology ring of smooth projective real quadrics with coefficients in the constant mackey functor z these invariants are closely related to the integral motivic cohomology ring which is not known for these va rieties some of the results and techniques introduced can be applied to other geometrically cellular real varieties

spatial approximation of stochastic convolutions we study linear stochastic evolution partial differential equations driven by additive noise we present a general and flexible framework for representing the infinite dimensional wiener process which drives the equation since the eigenfunctions and eigenvalues of the covariance operator of the process are usually not available for computations we propose an expansion in an arbitrary frame we show how to obtain error estimates when the truncated expansion is used in the equation for the stochastic heat and wave equations we combine the truncated expansion with a standard finite element method and derive a priori bounds for the mean square error specializing the frame to biorthogonal wavelets in one variable we show how the hierarchical structure support and cancelation properties of the primal and dual bases lead to near sparsity and can be used to simplify the simulation of the noise and its update when new terms are added to the expansion

a transfer homomorphism for compact lie group actions here a space is perfectly normal if it is normal and every open subspace is a countable union of closed subspaces transfer maps for actions of finite groups were defined by floyd 10 and used in particular for studying orbit spaces for actions on contractible spaces when g is a compact lie group acting freely on x then n x h x g is a fiber bundle with fiber g h and the transfer constructed by becker and gottlieb 1 applies in this case so our goal here is to generalize both of these constructions one obvious consequence of the above theorem is that

equivariant fixed point index and fixed point transfer in nonzero dimensions dold s fixed point index and fixed point transfer are generalized for certain coincidence situations namely maps which change the equivariant di mension those invariants change the dimension correspondingly a formula for the index of a situation over a space with trivial group action is exhibited for the transfer a generalization of dold s lefschetz hopf trace formula is proved

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

faceting and coarsening dynamics in the complex swift hohenberg equation the complex swift hohenberg equation models pattern formation arising from an oscillatory instability with a finite wave number at onset and finds applications in lasers optical parametric oscillators and photorefractive oscillators we show that with real coefficients this equation exhibits two classes of localized states localized in amplitude only or localized in both amplitude and phase the latter are associated with phase winding states in which the real and imaginary parts of the order parameter oscillate periodically but with a constant phase difference between them the localized states take the form of defects connecting phase winding states with equal and opposite phase lag and can be stable over a wide range of parameters the formation of these defects leads to faceting of states with initially spatially uniform phase depending on parameters these facets may either coarsen indefinitely as described by a cahn hilliard equation or the coarsening ceases leading to a frozen faceted structure

existence and regularity of solution for a stochastic cahn hilliard allen cahn equation with unbounded noise diffusion the cahn hilliard allen cahn equation with noise is a simplified mean field model of stochastic microscopic dynamics associated with adsorption and desorption spin flip mechanisms in the context of surface processes for such an equation we consider a multiplicative space time white noise with diffusion coefficient of sub linear growth using technics from semigroup theory we prove existence and path regularity of stochastic solution depending on that of the initial condition our results are also valid for the stochastic cahn hilliard equation with unbounded noise diffusion for which previous results were established only in the framework of a bounded diffusion coefficient we prove that the path regularity of stochastic solution depends on that of the initial condition and are identical to those proved for the stochastic cahn hilliard equation and a bounded noise diffusion coefficient if the initial condition vanishes they are strictly less than 2 d 2 in space and 1 2 d 8 in time as expected from the theory of parabolic operators in the sense of petrovski the bi laplacian operator seems to be dominant in the combined model

locking of periodic patterns in cahn hilliard models for langmuir blodgett transfer the influence of a periodic spatial forcing on the pattern formation in a generalized cahn hilliard model describing langmuir blodgett transfer is studied the occurring locking effects enable a control mechanism for the pattern formation process in the one dimensional case the parameter range in which patterns are created is increased and the patterns properties can be adjusted in a broader range in two dimensions one dimensional stripe patterns can be destabilized giving rise to a multitude of complex two dimensional structures including oblique and lattice patterns

on the nonexistence of elements of kervaire invariant one we show that the kervaire invariant one elements j 2j 1 2s exist only for j 6 by browder s theorem this means that smooth framed manifolds of kervaire invariant one exist only in dimensions 2 6 14 30 62 and possibly 126 except for dimension 126 this resolves a longstanding problem in algebraic topology