the dynamics of traveling waves for a nonlinear belousov zhabotinskii system abstract in this paper we consider the existence of traveling wave fronts in a belousov zhabotinskii system with delay by traveling wave transformation and time scale transformation we change the belousov zhabotinskii system with delay into a singularly perturbed differential system by applying geometric singular perturbation theory we construct a locally invariant manifold for the associated traveling wave equation and obtain the traveling wave fronts for the equation by using the fredholm orthogonality finally we discuss the asymptotic behaviors of traveling wave solutions by applying the asymptotic theory

wavefront solutions of degenerate quasilinear reaction diffusion systems with mixed quasi monotonicity abstract this paper is concerned with the existence of wavefront solutions to a system of degenerate quasilinear reaction diffusion equations of mixed quasi monotone properties in the form u i t d i u i u i f i u x t 0 for i 1 n the important features of this system are that some of the diffusion coefficients d i u i are density dependent and may vanish at certain value of u i and that each function f i is quasi monotone increasing for some components of u u 1 u n and decreasing for other components of u such systems model reaction diffusion processes with density driven diffusion mechanism under certain general conditions we prove the existence of a traveling wave solution that is between a pair of coupled upper and lower solutions a predator prey model with nonlinear diffusion is used as an illustration of application the presence of wavefront solutions flowing toward the coexistence states is established by constructing appropriate upper and lower solutions

evolution semigroups and sums of commuting operators a new approach to the admissibility theory of function spaces this paper is concerned with conditions for the admissibility of a translation invariant function space m with respect to a well posed linear evolution equation du dt au f t t r we propose a new approach to this problem by considering the sum of two commuting operators d dt dm and the operator of multiplication by a on m on the one hand the closure of this operator is the infinitesimal generator of the so called evolution semigroup associated with on the other hand the generator g of this semigroup relates a mild solution u of to the forcing term f by the rule gu f consequently various spectral criteria of the type dm a for the admissibility of the function space m with respect to can be proved in an elegant manner moreover they can be naturally extended to general classes of differential equations including higher order and abstract functional differential equations applications and examples are provided to illustrate the obtained results

monotone traveling wavefronts of the kpp fisher delayed equation abstract in the early 2000 s gourley 2000 wu et al 2001 ashwin et al 2002 initiated the study of the positive wavefronts in the delayed kolmogorov petrovskii piskunov fisher equation u t t x u t x u t x 1 u t h x u 0 x r m since then this model has become one of the most popular objects in the studies of traveling waves for the monostable delayed reaction diffusion equations in this paper we give a complete solution to the problem of existence and uniqueness of monotone waves in eq we show that each monotone traveling wave can be found via an iteration procedure the proposed approach is based on the use of special monotone integral operators which are different from the usual wu zou operator and appropriate upper and lower solutions associated to them the analysis of the asymptotic expansions of the eventual traveling fronts at infinity is another key ingredient of our approach

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

reflection groups in algebraic geometry after a brief exposition of the theory of discrete reflection groups in spherical euclidean and hyperbolic geometry as well as their analogs in complex spaces we present a survey of appearances of these groups in various areas of algebraic geometry

leech roots and vinberg groups this chapter investigates the remarkable properties of the leech roots which are the fundamental roots for the even unimodular lattice in lorentzian space r 25 1 and correspond one for one with the points of the leech lattice we give an extensive table of the leech roots in both euclidean and hyperbolic coordinates we also provide the first of what promise to be many applications by showing that the leech roots simplify and explain the remarkable results of vinberg kaplinskaja and meyer on the reflection groups of unimodular lorentzian lattices in dimensions below 20 they also enable us to make some progress on the study of these groups in the next few dimensions

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

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

on the existence of traveling waves for delayed reaction diffusion equations abstract we study the existence of traveling wave solutions for reaction diffusion equations with nonlocal delay where reaction terms are not necessarily monotone the existence of traveling wave solutions for reaction diffusion equations with nonlocal delays is obtained by combining upper and lower solutions for associated integral equations and the schauder fixed point theorem the smoothness of upper and lower solutions is not required in this paper

quadratic forms over c t1 t2 abstract we show that dimension 4 quadratic forms over c t1 t2 ti transcendental do not satisfy what might be naturally taken as a strong hasse principle though a weak hasse principle exists we reduced solvability to a question of determining the quadratic reciprocity group for the discriminant d the classes modulo d of elements x such that d is a quadratic residue modulo x we determine this group in terms of the class group of a function field we prove that the weak hasse principle does not hold for forms over c t1 d therefore witt and brauer groups of 2 dimensional function fields over c do not restrict monomorphically to their localizations

rational curves on the supersingular k3 surface with artin invariant 1 in characteristic 3 abstract we show the existence of 112 non singular rational curves on the supersingular k3 surface x with artin invariant 1 in characteristic 3 by several ways these non singular rational curves have the minimum degree with respect to a very ample divisor on x using these rational curves we have a 16 10 configuration and a 280 4 112 10 configuration on the k3 surface moreover we study the picard lattice by using the theory of the leech lattice the 112 non singular rational curves correspond to 112 leech roots

asymptotic behavior and traveling wave solutions for parabolic functional differential equations on presente une generalisation de la theorie des equations kpp et de diffusion bistables non lineaires on montre que des solutions onde de propagation existent pour des equations fonctionnelles paraboliques non lineaires

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

travelling wave fronts in reaction diffusion systems with spatio temporal delays this paper deals with the existence of travelling wave fronts in reaction diffusion systems with spatio temporal delays our approach is to use monotone iterations and a nonstandard ordering for the set of profiles of the corresponding wave system new iterative techniques are established for a class of integral operators when the reaction term satisfies different monotonicity conditions following this the existence of travelling wave fronts for reaction diffusion systems with spatio temporal delays is established finally we apply the main results to a single species diffusive model with spatio temporal delay and obtain some existence criteria of travelling wave fronts by choosing different kernels

evolution semigroups and spectral criteria for almost periodic solutions of periodic evolution equations abstract we investigate spectral criteria for the existence of almost periodic solutions to linear 1 periodic evolution equations of the formdx dt a t x f t with in general unbounded a t and almost periodicf using the evolution semigroup associated with the evolutionary process generated by the equation under consideration we show that if the spectrum of the monodromy operator does not intersect the set e isp f then the above equation has an almost periodic mild solutionxfwhich is unique if one requiressp xf 2 k k z sp f we emphasize that our method allows us to treat the equations without assumption on the existence of floquet representation this improves recent results on the subject in addition we discuss some particular cases in which the spectrum of monodromy operator does not intersect the unit circle and apply the obtained results to study the asymptotic behavior of solutions finally an application to parabolic equations is considered

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

the automorphism group of the 26 dimensional even unimodular lorentzian lattice abstract the group mentioned in the title is shown to be a certain infinitely generated coxeter group extended by the negative of the identity operation and the group of all automorphisms of the notorious leech lattice

perron theorem in the monotone iteration method for traveling waves in delayed reaction diffusion equations abstract in this paper we revisit the existence of traveling waves for delayed reaction diffusion equations by the monotone iteration method we show that perron theorem on existence of bounded solution provides a rigorous and constructive framework to find traveling wave solutions of reaction diffusion systems with time delay the method is tried out on two classical examples with delay the predator prey and belousov zhabotinskii models

travelling waves for delayed reaction diffusion equations with global response we develop a new approach to obtain the existence of travelling wave solutions for reaction diffusion equations with delayed non local response the approach is based on an abstract formulation of the wave profile as a solution of an operational equation in a certain banach space coupled with an index formula of the associated fredholm operator and some careful estimation of the nonlinear perturbation the general result relates the existence of travelling wave solutions to the existence of heteroclinic connecting orbits of a corresponding functional differential equation and this result is illustrated by an application to a model describing the population growth when the species has two age classes and the diffusion of the individual during the maturation process leads to an interesting non local and delayed response for the matured population

on the decidability of diophantine problems in combinatorial geometry on montre que la decidabilite algorithmique de la resolubilite des equations diophantiennes dans le corps des nombres rationnels est equivalente a une conjecture de b grunbaum sur la coordination rationnelle en geometrie combinatoire

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

traveling wave solutions for reaction diffusion systems abstract this paper is concerned with traveling waves of reaction diffusion systems the definition of coupled quasi upper and quasi lower solutions is introduced for systems with mixed quasimonotone functions and the definition of ordered quasi upper and quasi lower solutions is also given for systems with quasimonotone nondecreasing functions by the monotone iteration method it is shown that if the system has a pair of coupled quasi upper and quasi lower solutions then there exists at least a traveling wave solution moreover if the system has a pair of ordered quasi upper and quasi lower solutions then there exists at least a traveling wavefront as an application we consider the delayed system of a mutualistic model

problems equivalent to rational diophantine solvability muthemutics research group alubamu state university montgomery alabama 36195 communicated by walter feit received august 5 1987 under the hypothesis that rational diophantine equations are unsolvable we classify a number of other problems as solvable or unsolvable such as gv c h w for g e g h e h algebraic groups v w subspaces existence of monomorphisms and epimorphisms of modules and algebras existence of submodules of various types

automorphisms of enriques surfaces which act trivially on the cohomology groups let s be a smooth surface over the complex number field r an a u t o m o r p h i s m q of s is cohomologically trivial resp numerically trivial if the induced a u t o m o r p h i s m q of the cohomology ring h s 7 resp h s i1 is trivial we denote by a o s resp a s the quot ient o f the group o fcohomolog ica l ly resp numerically trivial au tomorph i sms of s by its connected component i t is known tha t a s is a finite g roup i f s is kiihler 4 3 a s is even trivial for example in the case of ra t ional surfaces abelian surfaces and k3 surfaces 9 5 in this article we shall s tudy a s for an enriques surface s in this case it is no more true that ao s is always trivial so far two examples are known

the leech lattice new proofs of several known results about the leech lattice are given in particular i prove its existence and uniqueness and prove that its covering radius is the square root of 2 i also give a uniform proof that the 23 holy constructions of the leech lattice all work

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

traveling wave solutions in a delayed diffusive competition system abstract in this paper we first investigate the existence of traveling wave fronts in a delayed diffusive competition system by constructing a pair of upper and lower solutions then we consider the asymptotic behavior of traveling wave solutions at the minus plus infinity by means of the bilateral laplace transform finally the monotonicity and uniqueness up to the translation of traveling wave solutions are also obtained by the strong comparison principle and the sliding method

projective models of the supersingular k3 surface with artin invariant 1 in characteristic 5 abstract let x be a supersingular k3 surface in characteristic 5 with artin invariant 1 then x has a polarization that realizes x as the fermat sextic double plane we present a list of polarizations of x with degree 2 whose intersection number with this fermat sextic polarization is less than or equal to 5 and give the defining equations of the corresponding projective models we also present a method to describe birational morphisms between these projective models explicitly as a by product a non projective automorphism of the fermat sextic double plane is obtained

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

on traveling wave solutions in general reaction diffusion systems with time delays abstract we study the existence of traveling wave solutions in a general class of mixed quasi monotone reaction diffusion systems with time delays first by applying the schauder fixed point theorem we prove the existence of a traveling wave solution between classically defined upper and lower solutions for better applications of the upper lower solution method on various real life models the existence result is further extended under weak form or piecewise smooth upper lower solutions in several reaction diffusion systems with time delays single species logistic growth n species competition and ratio dependent predator prey with gompertz growth we apply our main result to establish the existence of traveling wave solutions flowing towards the positive or coexistent states under reasonable conditions on ecological parameters

traveling waves and competitive exclusion in models of resource competition and mating interference abstract in this article we study reaction diffusion models for two closed related biological species u and v under resource competition and mating interference from v species to u species the effect of one sided sexual competition makes the trivial state and u dominance state both unstable while the v dominance state is shown to be asymptotically stable with attraction regions and convergent rates depending on the biological parameters using the upper lower solution method we further prove that for a family of wave speeds there exist traveling wave solutions connecting the u dominance state and the v dominance state at this confirms an earlier conjecture that unbalanced mating interference will lead to competitive exclusion these results can also be obtained on an extended model with instantaneous effects of resource competition and temporal delay on mating interference through a transformation into three equation system we prove that the temporal delay does not affect the stability of the steady states and the existence of the traveling waves but causes changes on the attraction regions and convergence rates finally numerical simulations are also presented to illustrate the theoretical results

reflection groups on the octave hyperbolic plane for two different integral formskof the exceptional jordan algebra we show that aut kis generated by octave reflections these provide geometric examples of discrete reflection groups acting with finite covolume on the octave or cayley hyperbolic plane oh2 the exceptional rank one symmetric space the isometry group of the plane is the exceptional lie groupf4 20 our groups are defined in terms of coxeter s discrete subring k of the nonassociative division algebra o and we interpret them as the symmetry groups of lorentzian lattices over k we also show that the reflection group of the hyperbolic cell over k is the rotation subgroup of a particularrealreflection group acting onh8 oh1 part of our approach is the treatment of the jordan algebra of matrices that are hermitian with respect to any real symmetric matrix

traveling waves for a model of the belousov zhabotinsky reaction abstract following j d murray we consider a system of two differential equations that models traveling fronts in the noyes field theory of the belousov zhabotinsky bz chemical reaction we are also interested in the situation when the system incorporates a delay h 0 as we show the bz system has a dual character it is monostable when its key parameter r 0 1 and it is bistable when r 1 for h 0 r 1 and for each admissible wave speed we prove the uniqueness of monotone wavefronts next a concept of regular super solutions is introduced as a main tool for generating new comparison solutions for the bz system this allows to improve all previously known upper estimations for the minimal speed of propagation in the bz system independently whether it is monostable bistable delayed or not special attention is given to the critical case r 1 which to some extent resembles to the zeldovich equation