measurable kac cohomology for bicrossed products we study the kac cohomology for matched pairs of locally compact groups this cohomology theory arises from the extension theory of locally compact quantum groups we prove a measurable version of the kac exact sequence and provide methods to compute the cohomology we give explicit calculations in several examples using results of moore and wigner

remarks on competitive coexistence the asymptotic behavior of two predators competing exploitatively for the same prey in a constant environment is analyzed in this paper assuming that the prey have fast dynamics it is proved through a singular perturbation argument that the two predators can coexist if the prey and predators parameters lie within specified ranges it is also shown that coexistence occurs through a simple periodic behavior a poor season characterized by an almost endemic presence of the prey alternates with a rich season during which prey are abundant and predators regenerate this property follows from the geometric nature of the equilibrium manifold and is therefore generic the results are in agreement with some conjectures supported by computer simulation and with the most recent theoretical findings on competitive coexistence

non semi regular quantum groups coming from number theory in this paper we study c algebraic quantum groups obtained through the bicrossed product construction examples using groups of adeles are given and they provide the first examples of locally compact quantum groups which are not semi regular the crossed product of the quantum group acting on itself by translations does not contain any compact operator we describe all corepresentations of these quantum groups and the associated universal c algebras on the way we provide several remarks on c algebraic properties of quantum groups and their actions

exponential energy decay estimate for the solutions of internally damped wave equation in a bounded domain abstract here we are concerned about the stability of the solution of internally damped wave equation y y y with small damping constant 0 in a bounded domain in r n under mixed undamped boundary conditions a uniform exponential energy decay rate e t me t e 0 where m 1 and 0 0 being a quadratic function of is proved for the solution of this type of boundary value problem earlier authors have considered the undamped wave equation with certain forms of damped boundary conditions proving similar and faster energy decay rates

some uniqueness and exact multiplicity results for a predator prey model in this paper we consider positive solutions of a predator prey model with diffusion and under homogeneous dirichlet boundary conditions it turns out that a certain parameter m in this model plays a very important role a good understanding of the existence stability and number of positive solutions is gained when m is large in particular we obtain various results on the exact number of positive solutions our results for large m reveal interesting contrast with that for the well studied casem 0 i e the classical lotka volterra predator prey model

positive solutions of a prey predator model with predator saturation and competition abstract in this paper we study the existence multiplicity bifurcation and stability of positive solutions to a prey predator model with predator saturation and competition u u a u b v f u v x v v c v d u f u v x u v 0 x where f u v 1 1 u 1 v and parameters are all positive constants and u and v are the densities of the prey and predator respectively

backstepping boundary control of a wave pde with spatially distributed time invariant unknown disturbances an observer based boundary control scheme is suggested for an infinite dimensional system modeled as a wave partial differential equation pde with in domain and boundary disturbances we consider a collocated case where the sensors and the actuators are located at the same boundary the aim of the controller is to regulate the rate of change of the opposite boundary around a given reference the unknown in domain disturbance is spatially distributed but it is constant in time same as the unknown boundary disturbance the problem is reformulated as a stabilization of an linear time invariant lti system with simultaneous delays in input and output channels the exponential stability of the equilibrium of the closed loop system is proven the performance of the controller is shown using numerical simulations

free products of compact quantum groups we construct and study compact quantum groups from free products ofc algebras in this connection we discover two mysterious classes of natural compact quantum groups a u m anda o m thea u m s respectivelya o m s are non isomorphic to each other for differentm s and are not obtainable by the ordinary quantization method we also clarify some basic concepts in the theory of compact quantum groups

inclusions irreductibles de facteurs et unitaires multiplicatifs ii resume from an irreducible depth 2 inclusion m 0 m 1 of factors with a faithful semifinite normal operator valued weight verifying a regularity condition we construct the canonical tower m i i n we prove then that both relative commutants m 2 m 0 and m 3 m 1 bear woronowicz algebra structures dual to each other as defined by masuda and nakagami moreover there exists an outer action of m 3 m 1 on m 1 such that m 0 is the fixed point algebra and m 2 is the crossed product the tower appears then as the tower of successive crossed products

regularized solutions for abstract volterra equations abstract we introduce a two kernel dependent family of strong continuous operators defined in a banach space which allows us to consider in an unified treatment the notions of among others c 0 semigroups of operators cosine families n times integrated semigroups resolvent families and k generalized solutions the results are applied to the study of existence and uniqueness of solutions for the volterra equation of convolution type u f a au in the case a is not necessarily densely defined examples for equations defined in l p spaces are also given

existence results for fractional order semilinear functional differential equations with nondense domain abstract in this paper we establish sufficient conditions for existence and uniqueness of solutions for some nondensely defined semilinear functional differential equations involving the riemann liouville derivative our approach is based on integrated semigroup theory the banach contraction principle and the nonlinear alternative of leray schauder type

on a solution to fractional order integrodifferential equations with analytic semigroups abstract in this paper we study a class of fractional order integrodifferential equations considered in an arbitrary banach space using the theory of analytic semigroups we establish the existence uniqueness and regularity of a mild solution to these fractional order integrodifferential equations

coexistence states of a holling type ii predator prey system abstract this paper characterize the existence of coexistence states to a reaction diffusion predator prey model with holling type ii functional response subject to dirichlet boundary conditions we find the necessary and sufficient conditions for existence of coexistence states by fixed point index theory and bifurcation theory

coexistence of a diffusive predator prey model with holling type ii functional response and density dependent mortality in this paper we consider a two competitor one prey diffusive model in which both competitors exhibit holling type ii functional response and one of the competitors exhibits density dependent mortality rate first we study the local and global existence of strong solution by using the c0 analytic semigroup then we consider the local and global stability of the positive constant equilibrium by using the linearization method and laypunov functional method respectively furthermore we derive some results for the existence and non existence of non constant stationary solutions when the diffusion rate of a certain species is small or large the existence of non constant stationary solutions implies the possibility of pattern formation in this system

boundary stabilization of nonlinear vibrations of a flexible structure in a bounded domain in rn abstract we study a problem of boundary stabilization of the vibrations of elastic structure governed by the nonlinear integro differential equation u a 2 b u 2 d x u f in a bounded domain in r n with a smooth boundary under mixed boundary conditions to stabilize this system we apply a velocity feedback control only on a part of the boundary we prove that the solution of such system is stable subject to some restriction on the uncertain disturbing force f we also estimate the total energy of the system over any time interval 0 t with a tolerance level of the disturbances finally we establish the uniform decay of solution by a direct method with an explicit form of exponential energy decay estimate when this disturbing force f is insignificant

hopf von neumann algebra bicrossproducts kac algebra bicrossproducts and the classical yang baxter equations two groups g1 g2 are said to be a matched pair if each acts on the space of the other and these actions say obey a certain compatibility condition for every matched pair of locally compact groups g1 g2 we construct an associated coinvolutive hopf von neumann algebra m g1 l g2 by simultaneous cross product and cross coproduct for non trivial these bicrossproduct hopf von neumann algebras are non commutative and non cocommutative if the modules for the actions are also matched then these bicrossproducts are kac algebras in this case we show that the dual kac algebra is of the same form with the roles of g1 g2 and of interchanged examples exist with g1 a simply connected lie group and choices of g2 determined by suitable solutions of the classical yang baxter equations on the complexification of the lie algebra of g1

zip bifurcation in a competition model on considere le systeme s s 1 s k m 1 x 1 s a 1 s m 2 x 2 s a 2 s x 1 m 1 x 1 s a 1 s d 1 w 1 x 2 m 2 x 2 s a 2 s d 2 x 2 soit i a i d i m i d i on etudie le cas 1 2

qualitative analysis for a diffusive predator prey model with a transmissible disease in the prey population in this paper we study a diffusive predator prey model with a transmissible disease in the prey population we offer a complete discussion of the dynamical properties under the homogeneous neumann boundary condition we analyze local and global stabilities of nonnegative constant steady states and long time behaviors of the positive solutions moreover we study the existence nonexistence and bifurcation of nonconstant positive stationary solutions finally some numerical simulations are presented to support and strengthen our analytical analysis

bicrossproducts of multiplier hopf algebras abstract in this paper we generalize majid s bicrossproduct construction we start with a pair a b of two regular multiplier hopf algebras we assume that b is a right a module algebra and that a is a left b comodule coalgebra the right action of a on b gives rise to the smash product a b the left coaction of b on a gives a possible coproduct on a b we discuss in detail the necessary compatibility conditions between the action and the coaction for to be a proper coproduct on a b the result is again a regular multiplier hopf algebra majid s construction is obtained when we have hopf algebras we also look at the dual case constructed from a pair c d of regular multiplier hopf algebras where now c is a left d module algebra while d is a right c comodule coalgebra we show that indeed these two constructions are dual to each other in the sense that a natural pairing of a with c and of b with d yields a duality between a b and the smash product c d we show that the bicrossproduct of an algebraic quantum group is again an algebraic quantum group i e a regular multiplier hopf algebra with integrals the algebra case is also considered some special cases are treated and they are related with other constructions available in the literature the basic example coming from a not necessarily finite group g with two subgroups h and k such that g k h and h k e where e is the identity of g is used to illustrate our theory more examples will be considered in forthcoming papers on the subject

quantum group gauge theory on quantum spaces we construct quantum group valued canonical connections on quantum homogeneous spaces including aq deformed dirac monopole on the quantum sphere of podles with quantum differential structure coming from the 3d calculus of woronowicz onsu q 2 the construction is presented within the setting of a general theory of quantum principal bundles with quantum group hopf algebra fibre associated quantum vector bundles and connection one forms both the base space spacetime and the total space are non commutative algebras quantum spaces

existence of asymptotically almost periodic solutions for damped wave equations abstract in this paper a class of nonlinear damped wave equations of the form u t u t a u t a u t f t u t t 0 satisfying with prescribed initial conditions are studied some sufficient conditions are established for the existence and uniqueness of an asymptotically almost periodic solution these results have significance in the study of vibrations of flexible structures possessing internal material damping finally an example is presented to illustrate the feasibility and effectiveness of the results

s shaped global bifurcation curve and hopf bifurcation of positive solutions to a predator prey model where d is a bounded domain in r n 1 with smooth boundary d and a b c d m are positive constants problem 1 0 is known as the predator prey model with holling tanner type interactions where u and v represent the densities of the prey and predator respectively hence we are only interested in positive solutions by which we mean that both u and v are positive in d it is easy to see that a 1 is necessary for the existence of positive solutions to 1 0 and we shall also assume that d 1 the case d 1 is rather different and will not be discussed here we are mainly interested in studying the number of positive solutions of 1 0 and the stability of these solutions in particular we shall show that when b d c and m fall into certain range the solution set u v a of 1 0 forms a s shaped smooth curve and that hopf bifurcation occurs along this curve this not only confirms rigorously similar numerical observations on 1 0 made in 4 but also shows that the corresponding parabolic system

existence of the mild solution for fractional semilinear initial value problems abstract in this paper we will study the existence and uniqueness of mild solution for the semilinear initial value problem of non integer order u t a u t f t u t g u t s u t where 0 1 and f t u t g u t s u t is a given function

bicrossproduct approach to the connes moscovici hopf algebra abstract we give a rigorous proof that the codimension one connes moscovici hopf algebra h cm is isomorphic to a bicrossproduct hopf algebra linked to a group factorisation of the diffeomorphism group diff r we construct a second bicrossproduct u cm equipped with a nondegenerate dual pairing with h cm we give a natural quotient hopf algebra k heis of h cm and hopf subalgebra u heis of u cm which again are in duality all these hopf algebras arise as deformations of commutative or cocommutative hopf algebras that we describe in each case finally we develop the noncommutative differential geometry of k heis by studying first order differential calculi of small dimension

bounded input bounded output stability of a damped nonlinear string in this paper we prove that the displacement of a damped nonlinear string under bounded distributed forces is bounded the proof is achieved by showing that an energy like lyapunov function corresponding to the string is bounded

global smooth solutions for the quasilinear wave equation with boundary dissipation abstract we consider the existence of global solutions of the quasilinear wave equation with a boundary dissipation structure of an input output in high dimensions when initial data and boundary inputs are near a given equilibrium of the system our main tool is the geometrical analysis the main interest is to study the effect of the boundary dissipation structure on solutions of the quasilinear system we show that the existence of global solutions depends not only on this dissipation structure but also on a riemannian metric given by the coefficients and the equilibrium of the system some geometrical conditions on this riemannian metric are presented to guarantee the existence of global solutions in particular we prove that the norm of the state of the system decays exponentially if the input stops after a finite time which implies the exponential stabilization of the system by boundary feedback

fractional cauchy problems with almost sectorial operators this paper concerns the abstract cauchy problem of fractional evolution equations with almost sectorial operators the suitable mild solutions of evolution equations with riemann liouville derivative and caputo derivative are introduced respectively the existence theorems of mild solutions of cauchy problems are established the results obtained here improve and generalize some known results finally an example is given for demonstration

on recent developments in the theory of abstract differential equations with fractional derivatives abstract this note is motivated from some recent papers treating the problem of the existence of a solution for abstract differential equations with fractional derivatives we show that the existence results in agarwal et al 2009 1 belmekki and benchohra 2010 2 darwish et al 2009 3 hu et al 2009 4 mophou and n guerekata 2009 6 7 mophou 2010 8 9 muslim 2009 10 pandey et al 2009 11 rashid and el qaderi 2009 12 and tai and wang 2009 13 are incorrect since the considered variation of constant formulas is not appropriate in this note we also consider a different approach to treat a general class of abstract fractional differential equations

embeddable quantum homogeneous spaces abstract we discuss various notions generalizing the concept of a homogeneous space to the setting of locally compact quantum groups on the von neumann algebra level we recall an interesting duality for such objects studied earlier by m izumi r longo s popa for compact kac algebras and by m enock in the general case of locally compact quantum groups a definition of a quantum homogeneous space is proposed along the lines of the pioneering work of vaes on induction and imprimitivity for locally compact quantum groups the concept of an embeddable quantum homogeneous space is selected and discussed in detail as it seems to be the natural candidate for the quantum analog of classical homogeneous spaces among various examples we single out the quantum analog of the quotient of the cartesian product of a quantum group with itself by the diagonal subgroup analogs of quotients by compact subgroups as well as quantum analogs of trivial principal bundles the former turns out to be an interesting application of the duality mentioned above

a note on the boundary stabilization of the wave equation we study the energy decay rates of the wave equation in a domain where boundary damping is present we generalize the geometrical conditions obtained earlier in j math purer appl 58 1979 pp 249 273 by using some more general multipliers of strauss comm pure appl math 28 1975 pp 265 278 the interaction between distributed damping and boundary damping is discussed a regulator problem is also formally discussed by the synthesis method

exact controllability stabilization and perturbations for distributed systems exact controllability is studied for distributed systems of hyperbolic type or for petrowsky systems like plate equations the control is a boundary control or a local distributed control exact controllability consists in trying to drive the system to rest in a given finite time the solution of the problems depends on the function spaces where the initial data are taken and also depends on the function space where the control can be chosen a systematic method named hum for hilbert uniqueness method is introduced as the terminology indicates it is based on uniqueness results classical or new and on hilbert spaces constructed in infinitely many ways by using uniqueness a number of applications are indicated having a general method for exact controllability implies having a general method for stabilization this leads to new and complicated nonlinear riccati s type pdes to be compared with direct methods when available in the last part of the paper we consider how all this behaves fo

periodic two predator one prey interactions and the time sharing of a resource niche a competition model involving two competing predator species and a single renewable resource prey species is studied under the assumption that the system parameters are periodic in time it is shown by means of global bifurcation techniques that a continuum of positive periodic solutions exists as a function of a selected averaged parameter and that the stability of these solutions at least locally near bifurcation depends on the direction of bifurcation in the special autonomous case of constant parameters the bifurcation is vertical and the spectrum of the continuum is a discrete point this autonomous case supports the principle of competitive exclusion in that coexistence of the predators on the single resource prey is possible in the sense that the system equilibrates only on a parameter set of measure zero in the more general case of periodic coefficients however it is shown that the spectrum can be an interval of positive length provided the predator parameter oscillations are out of phas

on multiplicity and stability of positive solutions of a diffusive prey predator model abstract in the paper we consider the following diffusive prey predator model u u a u v 1 m u in u 0 on v v b m v m u in v 0 on we are mainly concerned with the positive solutions of the system in the case that the parameter m is large and obtain a complete understanding for the existence multiplicity and stability of positive solutions

existence and uniqueness of mild solutions to impulsive fractional differential equations this paper is concerned with the existence and uniqueness of a mild solution to impulsive fractional semilinear differential equations the results are obtained by means of fixed point methods we also give an example of such problems

qualitative analysis of a ratio dependent predator prey system with diffusion ratio dependent predator prey models are favoured by many animal ecologists recently as they better describe predator prey interactions where predation involves a searching process when densities of prey and predator are spatially homogeneous the so called michaelis menten ratio dependent predator prey system which is an ordinary differential system has been studied by many authors the present paper deals with the case where densities of prey and predator are spatially inhomogeneous in a bounded domain subject to the homogeneous neumann boundary condition its main purpose is to study qualitative properties of solutions to this reaction diffusion partial differential system in particular we will show that even though the unique positive constant steady state is globally asymptotically stable for the ordinary differential equation dynamics non constant positive steady states exist for the partial differential equation model this demonstrates that stationary patterns arise as a result of diffusion

couple assorti de systemes de kac et inclusions de facteurs de typeii1 resume soit k l une inclusion irreductible d indice fini de facteurs de type ii 1 on suppose que l est engendre par les sous facteurs intermediaires m et n respectivement isomorphes aux produits croises de k par les algebres de kac a et b on donne des conditions pour qu il existe une inversion t sur a et b telle que l soit isomorphe au produit croise de k par le t produit tensoriel de a et b l inversion t permet de definir une action a gauche b de b sur a et une action a droite a de a sur b m et n sont alors respectivement isomorphes au produit croise de k b par a b b et de k a par b a a reciproquement si le t produit tensoriel de deux algebres de kac a et b de dimension finie agit sur un facteur k de type ii 1 par une action exterieure l inclusion k k a t b remplit ces conditions et on obtient un carre bicommutatif de facteurs let k l be a irreducible inclusion of type ii 1 factors with finite indice we assume that l is generated by the intermediate subfactors m and n which are respectively isomorphic to the crossed product of k by the kac algebras a and b we give conditions so that there is an inversion t on a and b such that l is isomorphic to the crossed product of k by the t tensor product of a and b the inversion t defines a left action b of b on a and a right action a of a on b then m and n are respectively isomorphic to the crossed product of k b by a b b and of k a by b a a conversely if the t tensor product of two kac algebras a and b of finite dimension acts on a ii 1 factor k by an outer action the inclusion k k a t b fulfills these conditions and we get a bicommutative square of factors

bicrossproduct structure of the null plane quantum poincare algebra a nonlinear change of basis allows us to show that the non standard quantum deformation of the 3 1 poincare algebra has a bicrossproduct structure quantum universal r matrix pauli lubanski and mass operators are presented in the new basis

abstract fractional cauchy problems with almost sectorial operators abstract of concern are the cauchy problems for linear and semilinear time fractional evolution equations involving in the linear part a linear operator a whose resolvent satisfies the estimate of growth 1 0 in a sector of the complex plane which occurs when one considers for instance the partial differential operators in the limit domain of dumb bell with a thin handle or in the space of holder continuous functions by constructing a pair of families of operators in terms of the generalized mittag leffler type functions and the resolvent operators associated with a for the first time and a deep analysis on the properties for these families we obtain the existence and uniqueness of mild solutions and classical solutions to the cauchy problems moreover we present three examples to illustrate the feasibility of our results

non existence of non constant positive steady states of two holling type ii predator prey systems strong interaction case abstract we prove the non existence of non constant positive steady state solutions of two reaction diffusion predator prey models with holling type ii functional response when the interaction between the predator and the prey is strong the result implies that the global bifurcating branches of steady state solutions are bounded loops

the interaction of steady state and hopf bifurcations in a two predator one prey competition model the existence of a stable periodic solution in the positive octant of a two predator one prey competition model is shown to result from the interaction of a steady state bifurcation with a hopf bifurcation this result confirms the conjectures based on numerical simulations of several authors that coexistence in the form of periodic oscillations in all three population sizes takes place for some parameter values of the model

characterization of compactness for resolvents and its applications abstract in this paper we derive some characterizations of compactness for resolvents as well as the subordination principle associated with the compactness as applications we obtain the existence of fractional evolution equations by using schauder s fixed point theorem a simple example about fractional heat equation is also given to illustrate our theory

diffusion self diffusion and cross diffusion u1 x 0 u 1 0 x u2 x 0 u 2 0 x i n 0 where 2 n 1 2 x 2 is the laplace operator 0 is a bounded smooth domain of r n with n1 0 and 0 are the boundary and the closure of 0 respectively 0t 0 0 t and 0t 0 0 t for some t 0 is the outward unit normal vector on 0 di ai bi ci i 1 2 are all positive constants while ij i j 1 2 denote non negative constants the initial values u1 0 and u2 0 are non negative smooth func tions which are not identically zero the mathematical model 1 1 was proposed by shigesada et al 27 in their study of spatial segregation of interacting species where u1 and u2 represent the densities of two competing species d1 and d2 are their diffu sion rates a1 and a2 denote the intrinsic growth rates b1 and c2 account for intra specific competitions b2 and c1 are the coefficients of inter specific competitions 11 and 22 are usually referred as self diffusion pressures and 12 and 21 are cross diffusion pressures an interesting feature of 1 1 is that the movements of these two competing species are affected by the population pressures due to the mutual interference between the individuals and it was hoped that the nonlinear dispersive force due to the

the steady states of a non cooperative model of nuclear reactors abstract this paper characterizes the existence of coexistence states in a reaction diffusion model arising in the theory of nuclear reactors from a mathematical point of view the importance of this model relies upon the fact that the associated variational systems are of non cooperative type and consequently the comparison techniques available for cooperative systems fail to work out although in higher spatial dimensions the dynamics of the model might be rather involved by the absence of limitations for the number of steady states we can prove the uniqueness of the steady state in the one dimensional prototype model our results complement and eventually sharpen the findings of arioli g arioli long term dynamics of a reaction diffusion system j differential equations 235 2007 298 307

extensions of locally compact quantum groups and the bicrossed product construction abstract in the framework of locally compact quantum groups we study cocycle actions we develop the cocycle bicrossed product construction starting from a matched pair of locally compact quantum groups we define exact sequences and establish a one to one correspondence between cocycle bicrossed products and cleft extensions in this way we obtain new examples of locally compact quantum groups

mapping ideals of quantum group multipliers abstract we study the dual relationship between quantum group convolution maps l 1 g l g and completely bounded multipliers of g for a large class of locally compact quantum groups g we completely isomorphically identify the mapping ideal of row hilbert space factorizable convolution maps with m c b l 1 g yielding a quantum gilbert representation for completely bounded multipliers we also identify the mapping ideals of completely integral and completely nuclear convolution maps the latter coinciding with l 1 b g where b g is the quantum bohr compactification of g for quantum groups whose dual has bounded degree we show that the completely compact convolution maps coincide with c b g our techniques comprise a mixture of operator space theory and abstract harmonic analysis including fubini tensor products the non commutative grothendieck inequality quantum eberlein compactifications and a suitable notion of inner co amenable quantum group that we introduce and for which we exhibit examples from the bicrossed product construction our main results are new even in the setting of group von neumann algebras v n g for quasi sin locally compact groups g

note on boundary stabilization of wave equations an energy decay rate is obtained for solutions of wave type equations in a bounded region in mathbb r n whose boundary consists partly of a nontrapping reflecting surface and partly of an energy absorbing surface unlike most previous results on this subject the results presented here are potentially valid for regions having connected boundaries

stabilization of quasi linear vibrations of an inhomogeneous beam in this note we study the stabilization of a vibrating inhomogeneous beam under distributed disturbing force we prove that the amplitude of such vibration is bounded subject to some restriction on the uncertain disturbing force we also estimate the total energy of the system over time interval 0 t for any time t 0 together with a tolerance level of the disturbance finally we establish the uniform decay of solution with an explicit form of exponential energy decay estimate when the uncertain disturbances are insignificant

semilinear fractional integro differential equations with compact semigroup abstract in this paper we study the local and global existence of mild solutions to a class of fractional integro differential equations in an arbitrary banach space associated with operators generating compact semigroup on the banach space

l p maximal regularity for second order cauchy problems we introduce the concept of l p maximal regularity for second order cauchy problems we prove l p maximal regularity for an abstract model problem and we apply the abstract results to prove existence uniqueness and regularity of solutions for nonlinear wave equations

bounded components of positive solutions of abstract fixed point equations mushrooms loops and isolas abstract in this work a general class of nonlinear abstract equations satisfying a generalized strong maximum principle is considered in order to study the behavior of the bounded components of positive solutions bifurcating from the curve of trivial states u 0 at a nonlinear eigenvalue 0 with geometric multiplicity one since the unilateral theorems of rabinowitz j funct anal 7 1971 487 theorems 1 27 and 1 40 are not true as originally stated cf the very recent counterexample of dancer bull london math soc 34 2002 533 in order to get our main results the unilateral theorem of lopez gomez spectral theory and nonlinear functional analysis research notes in mathematics vol 426 crc press boca raton fl 2001 theorem 6 4 3 is required our analysis fills some serious gaps existing is some published papers that were provoked by a direct use of rabinowitz s unilateral theory actually the abstract theory developed in this paper cannot be covered with the pioneering results of rabinowitz 1971 since in rabinowitz s context any component of positive solutions must be unbounded by a celebrated result attributable to dancer arch rational mech anal 52 1973 181

pattern formation of a coupled two cell brusselator model abstract in this paper we study the stationary problems for the coupled two cell brusselator model as follows d 1 u 1 b 1 u b u 2 v c w u in d 2 v u u 2 v in d 1 w 1 b 1 w b w 2 z c u w in d 2 z w w 2 z in u v w z 0 on we first study the stability of the unique positive constant solution u v w z 1 1 1 1 then we give a priori estimate positive upper and lower bounds to the positive solution at last we obtain the non existence and existence of positive non constant solutions as parameters d 1 d 2 and b varied

positive solutions of a diffusive predator prey model with modified leslie gower and holling type ii schemes abstract in this paper we investigate the existence multiplicity and stability of positive solutions to a prey predator model with modified leslie gower and holling type ii schemes p u u a 1 b u c 1 v u k 1 in v v a 2 c 2 v u k 2 in u 0 v 0 in u v 0 on where r n n 1 is a bounded domain with a smooth boundary the parameters a i b c i k i i 1 2 are positive numbers u and v are the respective populations of prey and predator here we say u v with u v 0 is a positive solution of problem p if u v is a solution of p and u v 0 in

c statistical quantum groups and weyl algebras c statistical quantum groups and algebras are obtained by a process of transmutation they are like super quantum groups and algebras but with statistics given now by a complex number among the examples it is shown that the weyl algebra of canonical quantization may be viewed equivalently as a c statistical plane related examples are the noncommutative torus and a double loop variable quantization of photons these results lead toward a reformulation of ordinary quantum mechanics as a classical theory with c statistics in this reformulation the role of the 1 factors of super statistics is played by the free particle time evolution

coexistence theorems of steady states for predator prey interacting systems on donne des conditions necessaires et suffisantes pour l existence des solutions positives des etats stationnaires pour des systemes predateur proie sous des conditions aux limites de dirichlet sur r n

physics for algebraists non commutative and non cocommutative hopf algebras by a bicrossproduct construction abstract the initial part of this paper presents physics for algebraists in the context of quantum mechanics combined with gravity such physical notions as the yang baxter equations position observables momentum space momentum and position quantization etc are described many readers may wish to just read this initial part of the paper the physics leads to the search for self dual algebraic structures and finally to non commutative and non cocommutative hopf algebras by a bicrossproduct construction the entire paper contains numerous examples the non commutative and non cocommutative hopf algebras are obtained as a simultaneous smash product and smash coproduct and denoted h1 h2 among the examples is one obtained by modifying the weyl algebra we also give the context in which the compatibility requirements on the structure maps reduce to the classical yang baxter equations and an example related to drinfel d s double hopf algebra d h