maximal violation of bell s inequalities for continuous variable systems we generalize bell s inequalities to biparty systems with continuous quantum variables this is achieved by introducing the bell operator in perfect analogy to the usual spin 1 2 systems it is then demonstrated that two mode squeezed vacuum states display quantum nonlocality by using the generalized bell operator in particular the original einstein podolsky rosen states which are the limiting case of the two mode squeezed vacuum states can maximally violate bell s inequality due to clauser horne shimony and holt the experimental aspect of our scheme is briefly considered

nonlocal dispersal equations in time periodic media principal spectral theory limiting properties and long time dynamics abstract the present paper is devoted to the investigation of the following nonlocal dispersal equation u t t x d m j x y u t y d y u t x f t x u t x t 0 x where r n is a bounded and connected domain with smooth boundary m 0 2 is the cost parameter d 0 is the dispersal rate 0 characterizes the dispersal range j 1 n j is the scaled dispersal kernel and f is a time periodic nonlinear function of generalized kpp type this paper is a continuation of the works of berestycki et al 3 4 where f was assumed to be time independent we first study the principal spectral theory of the linear operator associated to the linearization of the equation at u 0 we establish an easily verifiable general and sharp sufficient condition for the existence of the principal eigenvalue as well as important sup inf characterizations of the principal eigenvalue next we study the influences of the principal spectrum point on the global dynamics and confirm that the principal spectrum point being zero is critical it is followed by the investigation of the effects of the dispersal rate d and the dispersal range characterized by on the principal spectrum point and the positive time periodic solution in particular we prove various limiting properties of the principal spectrum point and the positive time periodic solution as d 0 or to achieve these we develop new techniques to overcome fundamental difficulties caused by the lack of the usual l 2 variational formula for the principal eigenvalue the lack of the regularizing effects of the semigroup generated by the nonlocal dispersal operator and the presence of the time dependence of the nonlinearity f finally we establish the maximum principle for time periodic nonlocal dispersal operators

vortex methods and vortex motion vortex phenomena in fluid flows and the experimental theoretical and numerical methods used to characterize them are discussed in reviews by leading experts chapters are devoted to an overview of vortex methods the convergence of vortex methods graphic displays from numerical simulations physical vortex visualizations the four principles of vortex motion the visualization and computation of hovering mode vortex dynamics turbulence and vortices in superfluid he and statistical mechanics approaches to vortices and turbulence extensive photographs and sample computer graphics are provided

the periodic logistic equation with spatial and temporal degeneracies in this article we study the degenerate periodic logistic equation with homogeneous neumann boundary conditions tu u au b x t u in 0 u 0 on 0 u x 0 u0 x 6 0 in where r n 2 is a bounded domain with smooth boundary a and p 1 are constants the function b c r 0 1 is t periodic in t nonnegative and vanishes i e has a degeneracy in some subdomain of r we examine the effects of various natural spatial and temporal degeneracies of b x t on the long time dynamical behavior of the positive solutions our analysis leads to a new eigenvalue problem for periodic parabolic operators over a varying cylinder and certain parabolic boundary blow up problems not known before the investigation in this paper shows that the temporal degeneracy causes a fundamental change of the dynamical behavior of the equation only when spatial degeneracy also exists but in sharp contrast whether or not temporal degeneracy appears in the equation the spatial degeneracy always induces fundamental changes of the behavior of the equation though such changes differ significantly according to whether or not there is temporal degeneracy

scale invariance in three dimensional isotropic turbulence a paradox and its resolution if the reynolds number is large enough turbulence is expected to exhibit scale invariance in an intermediate inertial range of wave numbers as shown by power law behaviour of the energy spectrum and also by a constant rate of energy transfer through wave number however although it has long been known that the first of these is true there has been little recognition of the fact that if the second is to hold then there is a contradiction between the definition of the energy flux as the integral of the transfer spectrum and the observed behaviour of the transfer spectrum itself this is because the transfer spectrum t k is invariably found to have a zero crossing at a single point at k0 say implying that the corresponding energy flux cannot have an extended plateau but must instead have a maximum value at k k0 we outline the resulting paradox and note that it may be resolved by the observation that the symmetry of the triadic interactions means that t k is not the relevant transfer term in determining the energy flux instead the relevant term is a filtered partitioned version herein denoted by t k kc where k kc is the cut off wave number for low high pass filtering it is known from studies of spectral subgrid transfer that t k kc is zero over an extended range of wave numbers as this is the case for quite modest reynolds numbers it not only resolves the paradox but may also shed some light on the embarrassment of success of the kolmogorov theory

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

le nombre de solutions de certains problemes semi lineaires elliptiques abstract in this paper we compute the exact number of solutions for certain semi linear elliptic problems in the first part we derive some complementary results to a theorem of ambrosetti and prodi the global behaviour of bifurcating branches is then studied for some classes of problems in the context of a concave or a convex nonlinearity in the second part we consider the branch of positive solutions lastly in the third part in dimension one for an autonomous equation we obtain a precise description of all the solutions to these problems the proofs rely on simple eigenvalue comparison arguments

the local structure of turbulence in incompressible viscous fluid for very large reynolds numbers 1 we shall denote by u p u x 1 x 2 x 3 t 1 2 3 the components of velocity at the moment t at the point with rectangular cartesian coordinates x 1 x 2 x 3 in considering the turbulence it is natural to assume the components of the velocity u p at every point p x 1 x 2 x 3 t of the considered domain g of the four dimensional space x 1 x 2 x 3 t are random variables in the sense of the theory of probabilities cf for this approach to the problem millionshtchikov 1939 denoting by the mathematical expectation of the random variable a we suppose that 2 and d u d x 2 are finite and bounded in every bounded subdomain of the domain g

a path integral for turbulence in incompressible fluids in this paper a classical path integral is formulated for incompressible fluids that evolve according to the navier stokes equation the path integral propagates probability distributions deterministically on the space gvol of solenoidal velocity fields we construct a set of isp 2 charges associated with the geometry of gvol and its poisson structure and a pair of supersymmetry charges connected with the hamiltonian these charges generate exact symmetries of the classical path integral when the viscosity is set equal to zero when the effect of dissipation is included the charges associated with the poisson structure and the hamiltonian are no longer conserved charges that generate kolmogorov scaling and galilean transformations are also constructed the classical path integral is formulated in terms of vorticity as well

maximal violation of bell inequalities for mixed states we show that mixed states can produce maximal violations in the bell inequality due to clauser horne shimony and holt chsh this follows from the degeneracy of the operator which is naturally associated with the bell inequality here called the bell operator we have calculated the form of all the eigenvalues for the generic chsh bell operator finally we consider several examples which demonstrate the utility of studying the eigenvalues and eigenvectors of the bell operator

forced waves of the fisher kpp equation in a shifting environment abstract this paper concerns the equation 0 1 u t u x x f x c t u x r where c 0 is a forcing speed and f s u r r r is asymptotically of kpp type as s we are interested in the questions of whether such a forced moving kpp nonlinearity from behind can give rise to traveling waves with the same speed and how they attract solutions of initial value problems when they exist under a sublinearity condition on f s u we obtain the complete existence and multiplicity of forced traveling waves as well as their attractivity except for some critical cases in these cases we provide examples to show that there is no definite answer unless one imposes further conditions depending on the heterogeneity of f in s r

persistence versus extinction under a climate change in mixed environments abstract this paper is devoted to the study of the persistence versus extinction of species in the reaction diffusion equation u t u f t x 1 c t y u t 0 x where is of cylindrical type or partially periodic domain f is of fisher kpp type and the scalar c 0 is a given forced speed this type of equation originally comes from a model in population dynamics see 3 17 18 to study the impact of climate change on the persistence versus extinction of species from these works we know that the dynamics is governed by the traveling fronts u t x 1 y u x 1 c t y thus characterizing the set of traveling fronts plays a major role in this paper we first consider a more general model than the model of 3 in higher dimensional space where the environment is only assumed to be globally unfavorable with favorable pockets extending to infinity we consider in two frameworks the reaction term is time independent or time periodic dependent for the latter we study the concentration of the species when the environment outside becomes extremely unfavorable and further prove a symmetry breaking property of the fronts

generalizations of the pseudospin operator to test the bell inequality for the tmsv state chen et al 2002 phys rev lett 88 040406 proposed a method to measure the entanglement of a bipartite quantum system involving continuous variables the approach is based upon the construction of a pseudospin operator an important difference with the spin case is an infinite degeneracy of the eigenkets associated with the pseudospin in this paper we analyze the role played by such a degeneracy in the study of the locality of the two mode squeezed vacuum state we construct other representations for the pseudospin and study the corresponding bell inequalities

scale invariance and the inertial range spectrum in three dimensional stationary isotropic turbulence the assumption of scale invariance when taken in conjunction with the symmetries of the navier stokes equation is shown to lead to the energy spectrum e k e2 3tk 5 3 in the limit of infinite reynolds number here et is the flux of energy due to inertial transfer while the prefactor is determined by an integral over triple correlations of the phases of the system it is argued that this form of the prefactor provides an answer to the landau kraichnan criticism of the original kolmogorov 1941 theory k41

propagation dynamics of a nonlocal dispersal fisher kpp equation in a time periodic shifting habitat abstract this paper is devoted to the study of the propagation dynamics of a nonlocal dispersal fisher kpp equation in a time periodic shifting habitat we first show that this equation admits a periodic forced wave with the speed at which the habitat is shifting by using the monotone iteration method combined with a pair of generalized super and sub solutions then we establish the nonexistence uniqueness and global exponential stability of periodic forced waves by applying the sliding technique and the comparison argument finally we obtain the spreading properties for a large class of solutions

study of the hopf functional equation for turbulence duhamel principle and dynamical scaling we consider a formulation for the hopf functional differential equation which governs statistical solutions of the navier stokes equations by introducing an exponential operator with a functional derivative we recast the hopf equation as an integro differential functional equation by the duhamel principle on this basis we introduce a successive approximation to the hopf equation as an illustration we take the burgers equation and carry out the approximations to the leading order scale invariance of the statistical navier stokes equations in d dimensions is formulated and contrasted with that of the deterministic navier stokes equations for the statistical navier stokes equations critical scale invariance is achieved for the characteristic functional of the d th derivative of the vector potential in d dimensions the deterministic equations corresponding to this choice of the dependent variable acquire the linear fokker planck operator under dynamic scaling in three dimensions it is the vorticity gradient that behaves like a fundamental solution more precisely source type solution of deterministic navier stokes equations in the long time limit physical applications of these ideas include study of a self similar decaying profile of fluid flows moreover we reveal typical physical properties in the late stage evolution by combining statistical scale invariance and the source type solution this yields an asymptotic form of the hopf functional in the long time limit improving the well known hopf titt solution in particular we present analyses for the burgers equations to illustrate the main ideas and indicate a similar analysis for the navier stokes equations

existence and uniqueness of the solution of a space time periodic reaction diffusion equation this paper is concerned with the study of the periodic solutions and the entire solutions of the equation 1 tu a t x u q t x u f t x u where the diffusion matrix a the advection term q and the reaction term f are periodic in t and x we prove that the sign of the periodic principal eigenvalue associated with the linearized problem determines the existence and the uniqueness of the periodic solution introducing another eigenvalue we are able to state uniqueness conditions for the entire solution and to derive the asymptotic behavior of the solutions of the associated cauchy problem

generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains using three different notions of the generalized principal eigenvalue of linear second order elliptic operators in unbounded domains we derive necessary and sufficient conditions for the validity of the maximum principle as well as for the existence of positive eigenfunctions for the dirichlet problem relations between these principal eigenvalues their simplicity and several other properties are further discussed 2015 wiley periodicals inc

liouville type result and long time behavior for fisher kpp equation with sign changing and decaying potentials abstract this paper concerns the liouville type result for the general semilinear elliptic equation s a i j x i j u x k q i x i u x f x u x 0 a e in r n where f is of the kpp monostable nonlinearity as a continuation of the previous works of the second author 31 32 the novelty of this work is that we allow f s x 0 to be sign changing and to decay fast up to a hardy potential near infinity first we introduce a weighted generalized principal eigenvalue and use it to characterize the liouville type result for eq s that was proposed by h berestycki secondly if a i j is the identity matrix and q is a compactly supported divergence free vector field we find a condition that eq s admits no positive solution for k k where k is a certain positive threshold to achieve this we derive some new techniques thanks to the recent results on the principal spectral theory for elliptic operators 14 to overcome some fundamental difficulties arising from the lack of compactness in the domain this extends a nice result of berestycki hamel nadirashvili 5 on the limit of eigenvalues with large drift to the case without periodic condition lastly the well posedness and long time behavior of the evolution equation corresponding to s are further investigated the main tool of our work is based on the maximum principle for elliptic and parabolic equations however it is far from being obvious to see if the comparison principle for s holds or not

existence uniqueness and qualitative properties of positive solutions of quasilinear elliptic equations we study the following quasilinear elliptic equation p pu x a x up 1 b x g u 0in rn where p 1 a b l rn b g 0 b 0 and lloc rn infrn we provide a sharp criterion in term of generalized principal eigenvalues for existence nonexistence of positive solution of p in suitable classes of functions uniqueness result for p in those classes is also derived under additional conditions on we further show that i either for every 0 nonexistence phenomenon occurs ii or there exists a threshold value 0 in the sense that for every 0 existence and uniqueness phenomenon occurs and for every nonexistence phenomenon occurs in the latter case we study the limits as 0 and of the sequence of positive solutions of p our results are new even in the case p 2

some remarks on brs transformations anomalies and the cohomology of the lie algebra of the group of gauge transformations we show that ghosts in gauge theories can be interpreted as maurer cartan forms in the infinite dimensional group of gauge transformations we examine the cohomology of the lie algebra of and identify the coboundary operator with the brs operator we describe the anomalous terms encountered in the renormalization of gauge theories triangle anomalies as elements of these cohomology groups

spatial temporal dynamics of a lotka volterra competition model with nonlocal dispersal under shifting environment abstract we consider a competitive system with nonlocal dispersals in a 1 dimensional environment that is worsening with a constant speed reflected by two shifting growth functions by analyzing the spatial temporal dynamics of the model system we are able to identify certain ranges for the worsening speed c respectively for i extinction of both species ii extinction of one species but persistence of the other iii persistence of both species in the case of persistence of a species it is achieved through spreading to the direction of favorable environment with certain speed s and some estimates of these speeds are also obtained we also present some numeric simulation results which confirm our theoretical results and in the mean time motivate some challenging problems for future work

propagation and blocking in periodically hostile environments we study the persistence and propagation or blocking phenomena for a species in periodically hostile environments the problem is described by a reaction diffusion equation with the zero dirichlet boundary condition we first derive the existence of a minimal nonnegative nontrivial stationary solution and study the large time behavior of the solution of the initial boundary value problem in addition to the main goal we then study a sequence of approximated problems in the whole space with reaction terms with very negative growth rates which are outside the domain under investigation finally for a given unit vector by using the information of the minimal speeds of approximated problems we provide a simple geometric condition for the blocking of propagation and we derive the asymptotic behavior of the approximated pulsating travelling fronts moreover for the case of the constant diffusion matrix we provide two conditions for which the limit of approximated minimal speeds is positive