perturbation of doubly periodic solution branches with applications to the cahn hilliard equation abstract in this paper we prove the existence of doubly periodic solutions of certain nonlinear elliptic problems on r 2 and study the geometry of their nodal domains in particular we will show that if we perturb a nonlinear elliptic equation exhibiting a small amplitude doubly periodic solution whose nodal domains form a checkerboard pattern then the perturbed equation will have a unique nearby solution which is still doubly periodic but for which the nodal line structure breaks up moreover we indicate what can happen if we start with a large amplitude doubly periodic solution whose nodal domains form a checkerboard pattern and we relate these solutions to the cahn hilliard equation and spinodal decomposition

spinodal decomposition for the cahn hilliard equation in higher dimensions nonlinear dynamics this paper addresses the phenomenon of spinodal decomposition for the cahn hilliard equation where n n 1 2 3 is a bounded domain with sufficiently smooth boundary and f is cubic like for example f u u u 3 based on the results of 26 the nonlinear cahn hilliard equation will be discussed this equation generates a nonlinear semiflow in certain affine subspaces of h 2 in a neighborhood u e with size proportional to e n around the constant solution where lies in the spinodal region we observe the following behavior within a local inertial manifold containing there exists a finite dimensional invariant manifold which dominates the behavior of all solutions starting with initial conditions from a small ball around with probability almost 1 the dimension of is proportional to e n and the elements of exhibit a common geometric quantity which is strongly related to a characteristic wavelength proportional to e

pattern formation in a nonlinear model for animal coats abstract several models have been proposed for describing the formation of animal coat patterns we consider reaction diffusion models due to murray which rely on a turing instability for the pattern selection in this paper we describe the early stages of the pattern formation process for large domain sizes this includes the selection mechanism and the geometry of the patterns generated by the nonlinear system on one two and three dimensional base domains these results are obtained by an adaptation of results explaining the occurrence of spinodal decomposition in materials science as modeled by the cahn hilliard equation we use techniques of dynamical systems viewing solutions of the reaction diffusion model in terms of nonlinear semiflows our results are applicable to any parabolic system exhibiting a turing instability

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

finite groups of symplectic automorphisms of k3 surfaces in positive characteristic we show that mukai s classication of nite groups which may act symplectically on a complex k3 surface extends to positive characteristic p under assumptions that i the order of the group is coprime to p and ii either the surface or its quotient is not birationally isomorphic to a supersingular k3 surface with artin invariant 1 in the case without assumption ii we classify all possible new groups which may appear we prove that assumption i on the order of the group is always satised if p 11 for p 2 3 5 11 we give examples of k3 surfaces with nite symplectic automorphism groups of order divisible by p which are not contained in mukai s list

synchronization of linear dynamical networks on time scales this paper studies the synchronization problem of complex dynamical networks cdns on time scales a pinning impulsive control scheme that takes into account of time delay effects is designed to achieve synchronization of cdns on time scales with the state of an isolated node based on the theory of time scales and the direct lyapunov method a synchronization criterion is established for linear cdns on general time scales our result shows that by impulsive control a small portion of nodes the consensus of cdns on time scales can be achieved according to our pinning impulsive control scheme different numbers of nodes will be selected at each impulsive instant and time delay is considered in the pinning impulses the modeling framework developed in this paper is a unification and generalization of many existing continuous time and discrete time cdn models while the pinning impulsive control scheme is an extension of the existing control scheme for synchronization of continuous time cdns moreover the idea of studying dynamical systems on time scales provide a unified approach to investigate continuous time system and its discrete time counterpart simultaneously numerical simulations are given to illustrate the effectiveness of the theoretical analysis

symmetry breaking at non positive solutions of semilinear elliptic equations we consider symmetry breaking bifurcations at non positive radially symmetric solutions of semilinear elliptic equations on a ball with dirichlet boundary conditions for nonlinearities which are asymptotically affine linear we find solutions at which the symmetry breaks the kernel of the linearized equation at these solutions is an absolutely irreducible representation of the group o n for this kind of equation a transversality condition is satisfied if the perturbation of the affine linear problem is small enough thus we obtain by the equivariant branching lemma a large variety of isotropy subgroups of o n which occur as symmetries of the bifurcating solution branches

unexpectedly linear behavior for the cahn hilliard equation this paper gives theoretical results on spinodal decomposition for the cahn hillard equation we prove a mechanism which explains why most solutions for the cahn hilliard equation starting near a homogeneous equilibrium within the spinodal interval exhibit phase separation with a characteristic wavelength when exiting a ball of radius r namely most solutions are driven into a region of phase space in which linear behavior dominates for much longer than expected the cahn hilliard equation depends on a small parameter epsilon modeling the atomic scale interaction length we quantify the behavior of solutions as epsilon rightarrow 0 specifically we show that most solutions starting close to the homogeneous equilibrium remain close to the corresponding solution of the linearized equation with relative distance o epsilon 2 n 2 up to a ball of radius r in the h 2 omega norm where r is proportional to epsilon 1 rho n 4 as epsilon to 0 here n in 1 2 3 denotes the

slow dynamics for the cahn hilliard equation in higher space dimensions the motion of bubbles it is known that the van der waals cahn hilliard w c h dynamics can be approximated by a quasi static stefan problem with surface tension it turns out that the stefan problem has a manifold of equilibria equal in dimension to that of the domain any sphere of fixed radius with interface contained in the domain is an equilibrium indistinguishable from the point of view of the perimeter functional we resolve this degeneracy by showing that at the w c h level this manifold is replaced by a quasi invariant stable manifold on which the typical solution preserves its bubble like shape until it reaches the boundary moreover we show that the bubble moves superslowly we also obtain an equation that determines those special spheres that correspond to equilibria at the w c h level our work establishes the phenomenon of superslow motion in higher space dimensions in the class of single interface solutions

the automorphism groups of kummer surfaces associated with the product of two elliptic curves we calculate the automorphism groups of several kummer surfaces associated with the product of two elliptic curves we give their generators explicitly

synchronization control for nonlinear stochastic dynamical networks pinning impulsive strategy in this paper a new control strategy is proposed for the synchronization of stochastic dynamical networks with nonlinear coupling pinning state feedback controllers have been proved to be effective for synchronization control of state coupled dynamical networks we will show that pinning impulsive controllers are also effective for synchronization control of the above mentioned dynamical networks some generic mean square stability criteria are derived in terms of algebraic conditions which guarantee that the whole state coupled dynamical network can be forced to some desired trajectory by placing impulsive controllers on a small fraction of nodes an effective method is given to select the nodes which should be controlled at each impulsive constants the proportion of the controlled nodes guaranteeing the stability is explicitly obtained and the synchronization region is also derived and clearly plotted numerical simulations are exploited to demonstrate the effectiveness of the pinning impulsive strategy proposed in this paper

p elementary subgroups of the cremona group abstract we classify up to conjugacy the subgroups of the cremona group isomorphic to z p r where p is prime and r is maximal

convergence of the cahn hilliard equation to the hele shaw model we prove that level surfaces of solutions to the cahn hilliard equation tend to solutions of the hele shaw problem under the assumption that classical solutions of the latter exist the method is based on a new matched asymptotic expansion for solutions a spectral analysis for linearizd operators and an estimate for the difference between the true solutions and certain approximate ones

maximum norms of random sums and transient pattern formation many interesting and complicated patterns in the applied sciences are formed through transient pattern formation processes in this paper we concentrate on the phenomenon of spinodal decomposition in metal alloys as described by the cahn hilliard equation this model depends on a small parameter and one is generally interested in establishing sharp lower bounds on the amplitudes of the patterns as the parameter approaches zero recent results on spinodal decomposition have produced such lower bounds unfortunately for higher dimensional base domains these bounds are orders of magnitude smaller than what one would expect from simulations and experiments the bounds exhibit a dependence on the dimension of the domain which from a theoretical point of view seemed unavoidable but which could not be observed in practice in this paper we resolve this apparent paradox by employing probabilistic methods we can improve the lower bounds for certain domains and remove the dimension dependence we thereby obtain optimal results which close the gap between analytical methods and numerical observations and provide more insight into the nature of the decomposition process we also indicate how our results can be adapted to other situations

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

the maximum order of finite groups of automorphisms of k3 surfaces let g be a finite group of automorphisms of a k 3 surface x we shall show that g 3840 and if g 3840 then the pair x g is unique up to isomorphisms

classification of k3 surfaces with involution and maximal symplectic symmetry k3 surfaces with antisymplectic involution and compatible symplectic actions of finite groups are considered in this situation actions of large finite groups of symplectic transformations are shown to arise via double covers of del pezzo surfaces and a complete classification of k3 surfaces with maximal symplectic symmetry is obtained

convergence of the cahn hilliard equation to the mullins sekerka problem in spherical symmetry abstract we show that as e 0 the solution of the cahn hilliard equation formula converges to a solution of the mullins sekerka problem formula where denotes a normal v the normal velocity and k the sum of principal curvatures of the interface provided the solutions are radially symmetric we use energy type estimates to show that the solution of the cahn hilliard equation can be approximated by the well known stationary wave solution that corresponds to the potential w

monte carlo simulations for spinodal decomposition this paper addresses the phenomenon of spinodal decomposition for the cahn hilliard equation namely we are interested in why most solutions to the cahn hilliard equation which start near a homogeneous equilibrium u 0 in the spinodal interval exhibit phase separation with a characteristic wavelength when exiting a ball of radius r in a hilbert space centered at u 0 there are two mathematical explanations for spinodal decomposition due to grant and to maier paape and wanner in this paper we numerically compare these two mathematical approaches in fact we are able to synthesize the understanding we gain from our numerics with the approach of maier paape and wanner leading to a better understanding of the underlying mechanism for this behavior with this new approach we can explain spinodal decomposition for a longer time and larger radius than either of the previous two approaches a rigorous mathematical explanation is contained in a separate paper our approach is to use monte carlo simulations to examine the dependence of r the radius to which spinodal decomposition occurs as a function of the parameter e of the governing equation we give a description of the dominating regions on the surface of the ball by estimating certain densities of the distributions of the exit points we observe and can show rigorously that the behavior of most solutions originating near the equilibrium is determined completely by the linearization for an unexpectedly long time we explain the mechanism for this unexpectedly linear behavior and show that for some exceptional solutions this cannot be observed we also describe the dynamics of these exceptional solutions

much ado about mathieu abstract eguchi ooguri and tachikawa observed that the coefficients of the elliptic genus of type ii string theory on k3 surfaces appear to be dimensions of representations of the largest mathieu group subsequent work by several people established a candidate for the elliptic genus twisted by each element of m 24 in this paper we prove that the resulting sequence of class functions are true characters of m 24 proving the eguchi ooguri tachikawa mathieu moonshine conjecture the integrality of multiplicities is proved using a small generalisation of sturm s theorem while positivity involves a modification of a method of hooley for finding an effective bound on a family of selberg kloosterman zeta functions at s 3 4 we also prove the evenness property of the multiplicities as conjectured by several authors and use that to investigate the proposal of gaberdiel hohenegger volpato that mathieu moonshine lifts to the conway groups co 0 and co 1 we identify the role group cohomology plays in both mathieu moonshine and monstrous moonshine in particular this gives a cohomological interpretation for the non fricke elements in norton s generalised monstrous moonshine conjecture and gives a lower bound for h 3 b c o 1 c

umbral moonshine and k3 surfaces recently 23 cases of umbral moonshine relating mock modular forms and finite groups have been discovered in the context of the 23 even unimodular niemeier lattices one of the 23 cases in fact coincides with the so called mathieu moonshine discovered in the context of k3 non linear sigma models in this paper we establish a uniform relation between all 23 cases of umbral moonshine and k3 sigma models and thereby take a first step in placing umbral moonshine into a geometric and physical context this is achieved by relating the ade root systems of the niemeier lattices to the ade du val singularities that a k3 surface can develop and the configuration of smooth rational curves in their resolutions a geometric interpretation of our results is given in terms of the marking of k3 surfaces by niemeier lattices

pinning control of scale free dynamical networks abstract recently it has been demonstrated that many large complex networks display a scale free feature that is their connectivity distributions have the power law form in the present work control of a scale free dynamical network by applying local feedback injections to a fraction of network nodes is investigated the specifically and randomly pinning schemes are considered the specifically pinning of the most highly connected nodes is shown to require a significantly smaller number of local controllers as compared to the randomly pinning scheme the method is applied to an array of chua s oscillators as an example

the alternating group of degree 6 in the geometry of the leech lattice and k3 surfaces the alternating group of degree 6 is located at the junction of three series of simple non commutative groups simple sporadic groups alternating groups and simple groups of lie type it plays a very special role in the theory of finite groups we shall study its new roles both in a finite geometry of a certain pentagon in the leech lattice and also in the complex algebraic geometry of k3 surfaces

finiteness and quasi simplicity for symmetric k3 surfaces we compare the smooth and deformation equivalence of actions of finite groups on k3 surfaces by holomorphic and anti holomorphic transformations we prove that the number of deformation classes is finite and in a number of cases establish the expected coincidence of the two equivalence relations more precisely in these cases we show that an action is determined by the induced action in the homology on the other hand we construct two examples to show that first in general the homological type of an action does not even determine its topological type and second that k3 surfaces x and x with the same klein action do not need to be equivariantly deformation equivalent even if the induced action on h 2 0 x is real i e reduces to multiplication by 1 1 1 questions in this paper we study equivariant deformations of complexk3 surfaces with symmetry groups where by a symmetry we mean an either holomor phic or anti holomorphic transformation of the surface although the automor phism group of a particular k3 surface may be infinite we confine ourselves to finite group actions and address the following two questions see 1 4 1 6 for precise definitions finiteness whether the number of actions counted up to equivariant deforma

spinodal decomposition for the cahn hilliard equation in higher dimensions part i probability and wavelength estimate this paper is the first in a series of two papers addressing the phenomenon of spinodal decomposition for the cahn hilliard equation where is a bounded domain with sufficiently smooth boundary and f is cubic like for example f u u u3 we will present the main ideas of our approach and explain in what way our method differs from known results in one space dimension due to grant 26 furthermore we derive certain probability and wavelength estimates the probability estimate is needed to understand why in a neighborhood of a homogeneous equilibrium u0 of the cahn hilliard equation with mass in the spinodal region a strongly unstable manifold has dominating effects this is demonstrated for the linearized equation but will be essential for the nonlinear setting in the second paper 37 as well moreover we introduce the notion of a characteristic wavelength for the strongly unstable directions

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

ergodicity of stochastic cahn hilliard equations with logarithmic potentials driven by degenerate or nondegenerate noises abstract we study the asymptotic properties of the stochastic cahn hilliard equation with the logarithmic free energy by establishing different dimension free harnack inequalities according to various kinds of noises the main characteristics of this equation are the singularities of the logarithmic free energy at 1 and 1 and the conservation of the mass of the solution in its spatial variable both the space time colored noise and the space time white noise are considered for the highly degenerate space time colored noise the asymptotic log harnack inequality is established under the so called essentially elliptic conditions and the harnack inequality with power is established for non degenerate space time white noise

solutions of nonlinear planar elliptic problems with triangle symmetry abstract in this paper we continue the study of the nodal domain structure of doubly periodic solutions of certain nonlinear elliptic problems initiated in fifeet al physica d100 1997 257 278 more precisely we consider small amplitude solutions of u f u 0 in r 2whose nodal domains consist of equilateral triangles tiling the plane if this equation is suitably perturbed then for genericfwe prove the existence of unique nearby solutions with triangle symmetry and show how their nodal domain geometry breaks up furthermore we treat the non generic rectangular cases which had to be excluded in fifeet al physica d100 1997 257 278 as well as other nodal domain structures

synchronization of coupled reaction diffusion neural networks with time varying delays via pinning impulsive controller in this paper global exponential synchronization stability in an array of linearly diffusively coupled reaction diffusion neural networks with time varying delays is investigated by adding impulsive controller to a small fraction of nodes pinning impulsive controller in order to overcome the difficulty resulting from the fact that the impulsive controller affects only the dynamical behaviors of the controlled nodes a new analysis method is developed by using the developed method two known lemmas on stability of delayed functional differential equation with and without impulses and lyapunov stability theory several novel and easily verified synchronization criteria guaranteeing the whole network will be pinned to a homogenous solution are derived moreover the effects of the pinning impulsive controller and the dynamics of the uncontrolled nodes and the diffusive couplings on the synchronization process are explicitly expressed in the obtained criteria our results also show that we can always de

front migration in the nonlinear cahn hilliard equation the method of matched asymptotic expansions is used to describe solutions of the nonlinear cahn hilliard equation for phase separation in n 1 space dimensions the expansion is formally valid when the thickness of internal transition layers is small compared with the distance separating layers and with their radii of curvature on the dominant slowest timescale the interface velocity is determined by the mean curvature of the interface by a non local relation which is identical to that in a well known quasi static model of solidification which exhibits a shape instability discovered by mullins sekerka j appl phys 34 323 329 1963 on a faster timescale the cahn hilliard equation regularizes a classic two phase stefan problem similarity solutions of the two phase stefan problem should describe boundary layers existence and uniqueness of such similarity solutions which admit metastable states is proved rigorously in an appendix

birational automorphisms of quartic hessian surfaces we find generators of the group of birational automorphisms of the hessian surface of a general cubic surface its nonsingular minimal model is a k3 surface with the picard lattice of rank 16 which embeds naturally in the even unimodular lattice ii 1 25 of rank 26 and signature 1 25 the generators are related to reflections with respect to some leech roots a similar observation was made first in the case of quartic kummer surfaces in the work of kondo we shall explain how our generators are related to the generators of the group of birational automorphisms of a general quartic kummer surface which is birationally isomorphic to a special hessian surface

atlas of finite groups maximal subgroups and ordinary characters for simple groups this atlas covers groups from the families of the classification of finite simple groups recently updated incorporating corrections

periodicity of scalar dynamic equations and applications to population models easily verifiable sufficient criteria are established for the existence of periodic solutions of a class of nonautonomous scalar dynamic equations on time scales which incorporate as special cases many single species models governed by ordinary differential and difference equations when the time scale is the set of all real and all integer numbers respectively

a survey of finite groups of symplectic automorphisms of k3 surfaces we give a survey of the finite symplectic automorphisms of k3 surfaces in particular we mention the relation between them and the mathieu group m 24 due to mukai moreover we give its further developments and related works

automorphisms of finite order on rational surfaces with an appendix by i dolgachev abstract we classify minimal pairs x g for smooth rational projective surface x and finite group g of automorphisms on x we also determine the fixed locus x g and the quotient surface y x g as well as the fundamental group of the smooth part of y the realization of each pair is included mori s extremal ray theory and recent results of alexeev and also ambro on the existence of good anti canonical divisors are used

on the slow dynamics for the cahn hilliard equation in one space dimension we study the limiting behaviour of the solution of the cahn hilliard equation using energy type methods we assume that the initial data has a transition layer structure i e u epsilon approx pm 1 except near finitely many transition points we show that in the limit as epsilon rightarrow 0 the solution maintains its transition layer structure and the transition layers move slower than any power of epsilon

finite time stability and settling time estimation of nonlinear impulsive systems abstract this paper studies the problem of finite time stability fts for nonlinear impulsive systems based on impulsive control theory several lyapunov based fts theorems involving stabilizing impulses and destabilizing impulses are established respectively our proposed results provide sufficient conditions for estimating the settling time with respect to suitable classes of impulse time sequences it is shown that the settling time of nonlinear impulsive systems depends not only on the initial state but also on the impulse effect as compared with the case without using stabilizing impulses a smaller bound of setting time can be derived when a fts system is subject to stabilizing impulses conversely a larger bound of settling time can be derived when the fts system is subject to destabilizing impulses as compared with the case without using destabilizing impulses as a special case we extend the ideas to nonlinear impulsive delay systems and derive some delay independent fts results examples and their simulations are given to demonstrate the applicability of the proposed results

brief paper exponential stability of nonlinear time delay systems with delayed impulse effects the problem of exponential stability for nonlinear time delay systems with delayed impulses is addressed in this paper lyapunov based sufficient conditions for exponential stability are derived respectively for two kinds of delayed impulses i e destabilizing delayed impulses and stabilizing delayed impulses it is shown that if a nonlinear impulsive time delay system without impulse input delays is exponentially stable then under some conditions its stability is robust with respect to small impulse input delays moreover it is also shown that for a stable nonlinear impulsive time delay system if the magnitude of the delayed impulses is sufficiently small then under some conditions the delayed impulses do not destroy the stability irrespective of the sizes of the impulse input delays the efficiency of the proposed results is illustrated by three numerical examples

the 290 fixed point sublattices of the leech lattice abstract we determine the orbits of fixed point sublattices of the leech lattice with respect to the action of the conway group co 0 there are 290 such orbits detailed information about these lattices the corresponding coinvariant lattices and the stabilizing subgroups is tabulated in several tables

existence and uniqueness of solutions to singular cahn hilliard equations with nonlinear viscosity terms and dynamic boundary conditions abstract we prove global existence and uniqueness of solutions to a cahn hilliard system with nonlinear viscosity terms and nonlinear dynamic boundary conditions the problem is highly nonlinear characterized by four nonlinearities and two separate diffusive terms all acting in the interior of the domain or on its boundary through a suitable approximation of the problem based on abstract theory of doubly nonlinear evolution equations existence and uniqueness of solutions are proved using compactness and monotonicity arguments the asymptotic behaviour of the solutions as the diffusion operator on the boundary vanishes is also shown

the classification of purely non symplectic automorphisms of high order on k3 surfaces abstract an automorphism of order n of a k3 surface is called purely non symplectic if it multiplies the holomorphic symplectic form by a primitive n th root of unity we give the classification of purely non symplectic automorphisms with n 12 where denotes the euler totient function

towards a classification of symplectic automorphisms on manifolds of k3 n type the present paper is devoted to the classification of symplectic automorphisms of some hyperkahlermanifolds the result presented here is a proof that all finite groups of symplectic automorphisms of manifolds of k3 n typeare contained in conway s group co 1

higher order cahn hilliard equations with dynamic boundary conditions abstract our aim in this paper is to study the well posedness and the dissipativity of higher order cahn hilliard equations with dynamic boundary conditions more precisely we prove the existence and uniqueness of solutions and the existence of the global attractor

spinodal decomposition for the cahn hilliard cook equation this paper gives theoretical results on spinodal decomposition for the stochastic cahn hilliard cook equation which is a cahn hilliard equation perturbed by additive stochastic noise we prove that most realizations of the solution which start at a homogeneous state in the spinodal interval exhibit phase separation leading to the formation of complex patterns of a characteristic size