fractional laplacian phase transitions and boundary reactions a geometric inequality and a symmetry result we deal with symmetry properties for solutions of nonlocal equations of the type sv f v in rn where s 0 1 and the operator s is the so called fractional laplacian the study of this nonlocal equation is made via a careful analysis of the following degenerate elliptic equation div x u 0on rn 0 x ux f u on rn 0 where 1 1 y rn x 0 and u u y x this equation is related to the fractional laplacian since the dirichlet to neumann operator u r n 1 x ux r n 1 is 1 2 more generally we study the so called boundary reaction equations given by div x u g x u 0on rn 0 x ux f u on rn 0 under some natural assumptions on the diffusion coefficient and on the nonlinearities f and g we prove a geometric formula of poincare type for stable solutions from which we derive a symmetry result in the spirit of a conjecture of de giorgi

systematic encoding via grobner bases for a class of algebraic geometric goppa codes any linear code with a nontrivial automorphism has the structure of a module over a polynomial ring the theory of grobner bases for modules gives a compact description and implementation of a systematic encoder we present examples of algebraic geometric goppa codes that can be encoded by these methods including the one point hermitian codes

traveling wave solutions in a diffusive system with two preys and one predator abstract this paper is concerned with the traveling wave solutions in a diffusive system with two preys and one predator by constructing upper and lower solutions the existence of nontrivial traveling wave solutions is established the asymptotic behavior of traveling wave solutions is also confirmed by combining the asymptotic spreading with the contracting rectangles applying the theory of asymptotic spreading the nonexistence of traveling wave solutions is proved

asymptotic behavior of stochastic fitzhugh nagumo systems on unbounded thin domains abstract this article is concerned with the asymptotic behavior of solutions of a class of non autonomous stochastic fitzhugh nagumo systems driven by multiplicative white noise on unbounded thin domains the existence and uniqueness of tempered pullback random attractors are proved for the systems defined on n 1 dimensional unbounded narrow domains we also establish the upper semicontinuity of these attractors when a family of n 1 dimensional unbounded thin domains collapses onto an n dimensional unbounded domain a cut off method and a decomposition technique are employed to derive the pullback asymptotic compactness of solutions in order to overcome the difficulties caused by the non compactness of sobolev embeddings on unbounded domains as well as the lack of regularity of one component of the solutions

on the automorphism group of various goppa codes we determine the automorphism group of various goppa codes spl cscr sub l d g associated with certain function fields f f sub q of genus g 0 it is well known that for deg d n 2g 2 the automorphism group aut sub d g f f sub q can be embedded into aut spl cscr sub l d g as a subgroup we show that under certain conditions on the divisors d and g aut sub d g f f sub q is actually isomorphic to aut spl cscr sub l d g

spreading speeds of a lotka volterra predator prey system the role of the predator abstract this paper is concerned with the spreading speeds of a modified lotka volterra predator prey system the intention is to describe the propagation mode whereby the two species coinvade a new habitat under some assumptions it is proved that the spreading speed of the prey is slower than in the case where the predator vanishes and the density of the prey on the coexistence domain is also smaller compared with the case where interspecies actions disappear therefore the predator has a negative effect on the evolution of the prey by slowing the spreading speed and decreasing the population density of the prey

attractors for random dynamical systems a criterion for existence of global random attractors for rds is established existence of invariant markov measures supported by the random attractor is proved for spde this yields invariant measures for the associated markov semigroup the results are applied to reation diffusion equations with additive white noise and to navier stokes equations with multiplicative and with additive white noise

well posedness of the cauchy problem for the fractional power dissipative equations abstract this paper studies the cauchy problem for the nonlinear fractional power dissipative equation u t u f u for initial data in the lebesgue space l r r n with either r r d n b 2 d or the homogeneous besov space b p r n with 2 d b n p and 1 p where 0 f u f u or q d f u with q d being a homogeneous pseudo differential operator of order d 0 2 and f u is a function of u which behaves like u b u with b 0

random dynamics of fractional stochastic reaction diffusion equations on rn without uniqueness this paper is concerned with the asymptotic behavior of the solutions of the nonautonomous fractional stochastic reaction diffusion equations on rn with continuous but not necessarily differentiable nonlinear drift terms which leads to the nonuniqueness of solutions and hence a multivalued random dynamical system with the solution operators of the equation we first show the existence and uniqueness of random attractors for such a dynamical system and then establish the upper semicontinuity of these attractors as the intensity of noise approaches zero the measurability of the random attractors is proved by the method of weak upper semicontinuity of multivalued functions and the pullback asymptotic compactness of the system is derived by the idea of uniform estimates on the tails of the solutions this paper is concerned with the asymptotic behavior of the solutions of the nonautonomous fractional stochastic reaction diffusion equations on rn with continuous but not necessarily differentiable nonlinear drift terms which leads to the nonuniqueness of solutions and hence a multivalued random dynamical system with the solution operators of the equation we first show the existence and uniqueness of random attractors for such a dynamical system and then establish the upper semicontinuity of these attractors as the intensity of noise approaches zero the measurability of the random attractors is proved by the method of weak upper semicontinuity of multivalued functions and the pullback asymptotic compactness of the system is derived by the idea of uniform estimates on the tails of the solutions

asymptotic behavior of non autonomous fractional stochastic reaction diffusion equations abstract we first apply the galerkin method to prove the existence and uniqueness of solutions for a class of non autonomous fractional reaction diffusion equations driven by multiplicative noise we then establish the existence and uniqueness of tempered pullback random attractors for the equations in an appropriate hilbert space the upper semicontinuity of random attractors is also obtained when the intensity of noise approaches zero

asymptotic spreading in a lotka volterra predator prey system abstract this paper is concerned with the asymptotic spreading of a lotka volterra predator prey system with self limit effect in which the intrinsic growth rates of both species are positive the spreading speeds of both species are estimated which implies the role of nonlinearities in asymptotic spreading biologically our results confirm that the prey may contribute to the asymptotic spreading of the predator

traveling waves in diffusive predator prey equations periodic orbits and point to periodic heteroclinic orbits we demonstrate the existence of periodic traveling wave train and traveling front solutions for a diffusive predator prey system the analysis is in the three dimensional phase space of the nonlinear ordinary differential equation system given by the diffusive predator prey system in the traveling wave variable the analysis shows the existence of periodic orbits heteroclinic orbits and a heteroclinic connection of a point and a periodic orbit the proof uses shooting techniques invariant manifold theory and the qualitative theory of ordinary differential equations

well posedness and dynamics of a fractional stochastic integro differential equation abstract in this paper we investigate the well posedness and dynamics of a fractional stochastic integro differential equation describing a reaction process depending on the temperature itself existence and uniqueness of solutions of the integro differential equation is proved by the lumer phillips theorem besides under appropriate assumptions on the memory kernel and on the magnitude of the nonlinearity the existence of random attractor is achieved by obtaining first some a priori estimates moreover the random attractor is shown to have finite hausdorff dimension

automorphism groups of one point codes from the curves y sup q y x sup qr 1 the automorphism groups are determined for the one point codes c sub m on the curve over f sub q 2r defined by y sup q y x sup qr 1 where r is an odd number this generalizes xing s theorem see ibid vol 41 p 1629 35 nov 1995 and extends a result of wesemeyer see ibid vol 44 p 630 43 march 1998 to the case of the above curve

invasion traveling wave solutions of a predator prey system abstract this paper is concerned with the traveling wave solutions of a predator prey system in which intra specific conflictions are involved by using the comparison principle and the asymptotic speed of spreading the existence nonexistence and minimal wave speed of non negative traveling wave solutions are established such a traveling wave solution can model the spatial temporal process where the predator invades the territory of the prey and they eventually coexist

generalized hermitian codes over hboxgf 2 r in this correspondence a generalization of hermitian function field proposed by garcia and stichtenoth is studied a weierstrass semigroup of the point at infinity for the case q 2 r ges 3 is calculated it turns out that unlike for the hermitian case there are already three generators for the semigroup this result then is applied to codes constructed on generalized hermitian gh function fields further results of kirfel and pellikaan are applied to estimating a feng rao designed distance for gh codes which improve on the goppa designed minimum distance next the question of codes dual to gh codes is studied it is shown that the duals are also gh codes and an explicit formula is given in particular this formula enables one to calculate the parameters of a dual code a new record giving 32 16 ges 12 code over gf 8 is presented as one of the examples

traveling wave solutions of diffusive lotka volterra equations a heteroclinic connection in we establish the existence of traveling wave solutions for a reaction dif fusion system based on the lotka volterra differential equation model of a predator and prey interaction the waves are of transition front type analogous to the solutions discussed by fisher and kolmogorov et al for a scalar reaction diffusion equation the waves discussed here are not necessarily monotone there is a speed c 0 such that for c c there is a traveling wave moving with speed c the proof uses a shooting argument based on the nonequivalence of a simply connected region and a nonsimply connected region together with a liapunov function to guarantee the existence of the traveling wave solution the traveling wave solution is equivalent to a heteroclinic orbit in 4 dimensional phase space

fractional ginzburg landau equation for fractal media abstract we derive the fractional generalization of the ginzburg landau equation from the variational euler lagrange equation for fractal media to describe fractal media we use the fractional integrals considered as approximations of integrals on fractals some simple solutions of the ginzburg landau equation for fractal media are considered and different forms of the fractional ginzburg landau equation or nonlinear schrodinger equation with fractional derivatives are presented the agrawal variational principle and its generalization have been applied

space time fractional schrodinger equation with time independent potentials abstract we develop a space time fractional schrodinger equation containing caputo fractional derivative and the quantum riesz fractional operator from a space fractional schrodinger equation in this paper by use of the new equation we study the time evolution behaviors of the space time fractional quantum system in the time independent potential fields and two cases that the order of the time fractional derivative is between zero and one and between one and two are discussed respectively the space time fractional schrodinger equation with time independent potentials is divided into a space equation and a time one a general solution which is composed of oscillatory terms and decay ones is obtained we investigate the time limits of the total probability and the energy levels of particles when time goes to infinity and find that the limit values not only depend on the order of the time derivative but also on the sign positive or negative of the eigenvalues of the space equation we also find that the limit value of the total probability can be greater or less than one which means the space time fractional schrodinger equation describes the quantum system where the probability is not conservative and particles may be extracted from or absorbed by the potentials additionally the non markovian time evolution laws of the space time fractional quantum system are discussed the formula of the time evolution of the mechanical quantities is derived and we prove that there is no conservative quantities in the space time fractional quantum system we also get a mittag leffler type of time evolution operator of wave functions and then establish a heisenberg equation containing fractional operators

regularity estimates for the solution and the free boundary of the obstacle problem for the fractional laplacian we use a characterization of the fractional laplacian as a dirichlet to neumann operator for an appropriate differential equation to study its obstacle problem we write an equivalent characterization as a thin obstacle problem in this way we are able to apply local type arguments to obtain sharp regularity estimates for the solution and study the regularity of the free boundary

asymptotic behavior of stochastic fractional power dissipative equations on rn abstract we study a reaction fractional diffusion equation with additive noise on the entire space r n with particular interest in the asymptotic behavior of solutions we first transform the equation into a random equation whose solutions generate a random dynamical system a priori estimates for solutions are derived when the nonlinearity satisfies certain growth conditions using estimates for far field values of solutions and a cut off technique asymptotic compactness is proved thus the existence of a random attractor in l 2 r n is established

the existence of periodic travelling waves for singularly perturbed predator prey equations via the conley index this paper is concerned with the existence of periodic travelling wave solutions of reactiondiffusion equations which arise as models of predator prey interactions in mathematical ecology the relevant equations are somewhat similar to the fitz hugh nagumo equations and our results apply to this latter system as well in studying the fitz hugh nagumo equations conley 3 i initiated a new topological approach to this problem his method depends on the construction of an isolating neighborhood cf section 2a about a singular periodic solution associated with such a neighborhood is a topological invariant the conley index which when nontrivial implies the existence of a smooth solution near the singular one the new aspects of the present discussion are in the method used to construct the isolating neighborhood in ifi4 and in the computation of the index with regard to the isolating neighborhood our construction is global in that a parameter taken to be small in conley s equations is now allowed to assume large values in order to compute the index we continue our equations to those studied by conley actually we go one step further and continue conley s equations to the van der pol equations crossed with a pair of linear equations admitting a repelling critical point the index for this latter system is easily computed and turns out to be non trivial it follows from the invariance of the index under continuation 21 that the desired

asymptotic compactness and absorbing sets for 2d stochastic navier stokes equations on some unbounded domains we introduce a notion of an asymptotically compact ac random dynamical system rds we prove that for an ac rds the limit set b of any bounded set b is nonempty compact strictly invariant and attracts the set b we establish that the 2d navier stokes equations nses in a domain satisfying the poincare inequality perturbed by an additive irregular noise generate an ac rds in the energy space h as a consequence we deduce existence of an invariant measure for such nses our study generalizes on the one hand the earlier results by flandoli crauel 1994 and schmalfuss 1992 obtained in the case of bounded domains and regular noise and on the other hand the results by rosa 1998 for the deterministic nses

traveling wavefronts for delayed reaction diffusion systems via a fixed point theorem abstract by using schauder s fixed point theorem we prove some existence results for traveling wavefronts of reaction diffusion systems with quasimonotonicity reactions more precisely we reduce the existence of traveling wavefronts to the existence of an admissible pair of supersolution and subsolution which are easy to construct in practice finally to illustrate our main results we study the existence of traveling wavefronts for a delayed predator prey model with diffusion as well as the reaction diffusion system with the well known belousov zhabotinskii reaction and the obtained results improve the existing ones

asymptotic spreading fastened by inter specific coupled nonlinearities a cooperative system abstract this paper is concerned with the asymptotic spreading of a lotka volterra cooperative system by using the theory of asymptotic spreading of nonautonomous equations the asymptotic speeds of spreading of unknown functions formulated by a coupled system are estimated our results imply that the asymptotic spreading of one species can be significantly fastened by introducing a mutual species which indicates the role of cooperation described by the coupled nonlinearities

dynamics of the 3 d fractional complex ginzburg landau equation abstract we study the initial boundary value problem of the fractional complex ginzburg landau equation in three spatial dimensions with the dissipative effect given by a fractional laplacian a priori estimates are derived when the nonlinearity satisfies certain growth conditions using galerkin s method the existence of a global smooth solution is established uniqueness is also proved furthermore the existence of a global attractor is proved and estimates of the hausdorff and fractal dimensions for the global attractor are obtained

stability of travelling wave solutions of diffusive predator prey systems the stability of travelling wave solutions of singularly perturbed diffusive predator prey systems is proved by showing that the linearized operator about such a solution has no unstable spectrum and that the translation eigenvalue at a 0 is simple the proof illustrates the application of some recently developed geometric and topological methods for counting eigenvalues

asymptotic behavior for non autonomous fractional stochastic ginzburg landau equations on unbounded domains in this paper we first prove the existence and uniqueness of tempered pullback random attractors for a non autonomous stochastic fractional ginzburg landau equation driven by multiplicative noise with 0 1 in l2r3 then we obtain the upper semicontinuity of random attractors when the intensity of noise approaches zero due to the lack of the compactness of sobolev embeddings on unbounded domains we establish the pullback asymptotic compactness of solutions in l2 r3 by the tail estimates of solutions

asymptotic behavior of stochastic wave equations with critical exponents on mathbb r 3 the existence of a random attractor in h 1 3 x l 2 3 is proved for the damped semilinear stochastic wave equation defined on the entire space 3 the nonlinearity is allowed to have a cubic growth rate which is referred to as the critical exponent the uniform pullback estimates on the tails of solutions for large space variables are established the pullback asymptotic compactness of the random dynamical system is proved by using these tail estimates and the energy equation method

on the automorphism group of generalized hermitian codes we determine the full automorphism group of the generalized hermitian curve denoted by gh which generalizes the hermitian curve the automorphism group of a code is important in coding theory and in this way we determine completely the automorphism group of the one point ag codes over the gh curve

well posedness of the cauchy problem for the fractional power dissipative equation in critical besov spaces abstract in this paper we study the cauchy problem for the semilinear fractional power dissipative equation u t u f u for the initial data u 0 in critical besov spaces b 2 r with n 2 2 d b where 0 f u p d u b 1 with p d being a homogeneous pseudo differential operator of order d 0 2 and b 0 being an integer making use of some estimates of the corresponding linear equation in the frame of mixed time space spaces the so called mono norm method which is different from the kato s double norm method fourier localization technique and littlewood paley theory we get the well posedness result in the case n 2

a stochastic pitchfork bifurcation in a reaction diffusion equation we study in some detail the structure of the random attractor for the chafee infante reaction diffusion equation perturbed by a multiplicative white noise d u u u u 3 d t u o d w t x d r m first we prove for m 5 a lower bound on the dimension of the random attractor which is of the same order in as the upper bound we derived in an earlier paper and is the same as that obtained in the deterministic case then we show for m 1 that as passes through 1 the first eigenvalue of the negative laplacian from below the system undergoes a stochastic bifurcation of pitchfork type we believe that this is the first example of such a stochastic bifurcation in an infinite dimensional setting central to our approach is the existence of a random unstable manifold

random attractors for stochastic reaction diffusion equations on unbounded domains the existence of a pullback attractor is established for a stochastic reaction diffusion equation on all n dimensional space the nonlinearity is dissipative for large values of the state and the stochastic nature of the equation appears as spatially distributed temporal white noise the reaction diffusion equation is recast as a random dynamical system and asymptotic compactness for this is demonstrated by using uniform a priori estimates for far field values of solutions

on automorphisms of geometric goppa codes the algebraic geometric codes which were introduced by v d goppa in 1977 7 8 we call them geometric goppa codes can be used to prove the existence of long linear codes which are better than the gilbert varshamov bound there are many good expositions of this important result ll 13 15 181 in that context one has to consider curves of high genus which admit many rational points on the other hand new interesting codes were found among the geometric goppa codes by looking at special curves e g elliptic curves hermite curves or artin schreier curves 3 5 8 13 16 171 in coding theory one is often interested in codes admitting many automorphisms e g cyclic codes goppa ls already observed that automorphisms of the underlying algebraic curve induce automorphisms of the associated geometric codes since goppa s paper is not quite correct we shall give a detailed exposition of this connexion in section 2 section 3 is the central part of our paper it is devoted to the case of geometric goppa codes associated with a curve of genus 0 so called rational goppa codes these codes are particularly interesting since many well known classes of codes e g bch codes classical goppa codes reed solomon codes cauchy codes 6 can be represented as rational goppa codes or subfield subcodes of them cl our main result is theorem 3 1 it states that in general all automorphisms of a rational goppa code are induced by automorphisms of the rational curve i e by projective transformations of the projective line

existence of the global smooth solution to the period boundary value problem of fractional nonlinear schrodinger equation in this paper we prove the existence and uniqueness of the global smooth solution to the period boundary value problem of fractional nonlinear schrodinger equation by using energy method

algebraic curves with a large non tame automorphism group fixing no point let k be an algebraically closed field of characteristic p 0 and let x be a curve over k of genus g 2 assume that the automorphism group aut x of x over k fixes no point of x the following result is proven if there is a point p on x whose stabilizer in aut x contains a p subgroup of order greater than gp p 1 then x is birationally equivalent over k to one of the irreducible plane curves ii iii iv v listed in the introduction

regularity of random attractors for fractional stochastic reaction diffusion equations on rn abstract we investigate the regularity of random attractors for the non autonomous non local fractional stochastic reaction diffusion equations in h s r n with s 0 1 we prove the existence and uniqueness of the tempered random attractor that is compact in h s r n and attracts all tempered random subsets of l 2 r n with respect to the norm of h s r n the main difficulty is to show the pullback asymptotic compactness of solutions in h s r n due to the noncompactness of sobolev embeddings on unbounded domains and the almost sure nondifferentiability of the sample paths of the wiener process we establish such compactness by the ideas of uniform tail estimates and the spectral decomposition of solutions in bounded domains

existence of travelling wave solutions of predator prey systems via the connection index we consider a reaction diffusion system which arises as a model of predator prey interactions in mathematical ecology the system admits four constant equilibria two of which are stable the main result is an existence theorem for travelling wave solutions which connect the two stable equilibria most previous efforts with regard to travelling waves for systems employ singular perturbations to locate the connecting orbit the new aspect of the present discussion lies in devising some novel methods for locating the connecting solution in phase space which do not require that the equations be singularly perturbed in certain cases the main result holds for an arbitrary choice of diffusion coefficients

existence and upper semicontinuity of random attractors for non autonomous fractional stochastic ginzburg landau equations this paper deals with the dynamical behavior of solutions for nonautonomous stochastic fractional ginzburg landau equations driven by multiplicative noise with 0 1 we first transform the stochastic equation into a random equation the solutions of which generate a random dynamical system consequently we establish the existence and uniqueness of tempered pullback random attractors for the equations in a bounded domain finally we obtain the upper semicontinuity of random attractors when the intensity of noise approaches zero

global weak solutions of the fractional landau lifshitz maxwell equation abstract the fractional landau lifshitz maxwell equation is considered in this paper we show the global existence of suitable weak solutions by galerkin method and the vanishing viscosity method the main difficulties in this study are due to the loss of compactness of this system and the fact that the nonlinear term is nonlocal and of the same order of the equation to overcome these difficulties we introduce the commutator estimates and some cancellation properties to this equation which prove to be proper tools in the study of landau lifshitz type equations

dynamical behavior of non autonomous fractional stochastic reaction diffusion equations abstract in this paper we investigate the asymptotic behavior of solutions for the non autonomous fractional stochastic reaction diffusion equations driven by multiplicative noise with s 0 1 we prove the existence and uniqueness of tempered pullback random attractors for the equations in a bounded domain u by the compactness of sobolev embedding h s u l 2 u which is different from the previous work 46

existence of traveling wave solutions for diffusive predator prey type systems abstract in this work we investigate the existence of traveling wave solutions for a class of diffusive predator prey type systems whose each nonlinear term can be separated as a product of suitable smooth functions satisfying some monotonic conditions the profile equations for the above system can be reduced as a four dimensional ode system and the traveling wave solutions which connect two different equilibria or the small amplitude traveling wave train solutions are equivalent to the heteroclinic orbits or small amplitude periodic solutions of the reduced system applying the methods of wazewski theorem lasalle s invariance principle and hopf bifurcation theory we obtain the existence results our results can apply to various kinds of ecological models

dynamics of fractional stochastic reaction diffusion equations on unbounded domains driven by nonlinear noise abstract this paper is concerned with the asymptotic behavior of the solutions of the fractional reaction diffusion equations with polynomial drift terms of arbitrary order driven by locally lipschitz nonlinear diffusion terms defined on r n we first prove the well posedness of the equation based on pathwise uniform estimates as well as uniform estimates on average we then define a mean random dynamical system via the solution operators and prove the existence and uniqueness of weak pullback mean random attractors we finally establish the existence of invariant measures when the diffusion terms are globally lipschitz continuous the uniform estimates on the tails of solutions are employed to prove the tightness of a family of probability distributions of solutions in order to overcome the non compactness of usual sobolev embeddings on unbounded domains

on automorphism groups of the hermitian codes we determine the exact automorphism groups of the goppa geometric codes from the hermitian curves over gf q sup 2