connected graph representations of the quantum propagator and semiclassical expansions the time evolution operator u t s of a spinless nonrelativistic n body quantum system in euclidean space with real analytic time dependent scalar interaction v x t is studied a complete formal asymptotic expansion involving simple connected graphs is derived for the full coordinate and mixed coordinate momentum representation propagators the derivation is based on dyson s series for u t s and the combinatorics involved in the cluster expansion of the classical grand partition function these results provide an efficient means of generating nonperturbative propagator expansions in the physical variables mass m planck s constant h time displacement t s the structural bridge between the wkb and wigner kirkwood expansions is sketched for mixed representations of u t s in the heat equation context the graphical expansions are found for mixed representations of the density operator e beta h finally an explicit differential formula is obtained for wigner s distribution function f x p beta

heat equation asymptotics of nonminimal operators on differential forms let d d d e be a natural operator on differential forms for example dv j k kvj c1 j kvk c2rjkvk has this form it will be shown that when the lower order term e is absent the integrated heat kernel coefficients an d can be simply expressed in terms of the familiar coefficients an j for the laplacian operators on forms the first two coefficients when a zeroth order term is present are evaluated

on growth rates of subadditive functions for semiflows abstract let x t x be a semiflow on a compact metric space x a function f x t x is subadditive with respect to if f x t s f x t f x t nbsp s we define the maximal growth rate of f to be sup x x lim sup t 1 t f x t this growth rate is shown to equal the maximal growth rate of the subadditive function restricted to the minimal center of attraction of the semiflow applications to birkhoff sums characteristic exponents of linear skew product semiflows on banach bundles and average lyapunov functions are developed in particular a relationship between the dynamical spectrum and the measurable spectrum of a linear skew product flow established by r a johnson k j palmer and g r sell siam j math anal 18 1987 1 33 is extended to semiflows in an infinite dimensional setting

the einstein equation for signature type changing spacetimes we give a local existence and uniqueness theorem for solutions of einstein s equations with dust energy momentum tensor in the class of m dimensional analytic transverse signature type changing spacetimes where the initial condi tions are given on the hypersurface of signature type change we also prove a similar theorem in the case that the energy momentum tensor represents a scalar field

long time behavior of solutions to a class of stochastic parabolic equations with homogeneous white noise stratonovitch s case in this article we prove new results concerning the long time behavior of random fields that are solutions in some sense to a class of semilinear parabolic equations subjected to a homogeneous and multiplicative white noise our main results state that these random fields eventually homogeneize with respect to the spatial variable and finally converge to a non random global attractor which consists of two spatially and temporally homogeneous asymptotic states more precisely we prove that the random fields either stabilize exponentially rapidly with probability one around one of the asymptotic states or that they set out to oscillate between them in the first case we can also determine exactly the corresponding lyapunov exponents in the second case we prove that the random fields are in fact recurrent in that they can reach every point between the two asymptotic states in a finite time with probability one in both cases we also interpret our results in terms of stability properties of the global attractor and we provide estimates for the average time that the random fields spend in small neighborhoods of the asymptotic states our methods of proof rest upon the use of a suitable regularization of the brownian motion along with a related wong zakai approximation procedure

global attractivity in monotone and subhomogeneous almost periodic systems abstract a global attractivity theorem is first proved for a class of skew product semiflows then this result is applied to monotone and subhomogeneous almost periodic reaction diffusion equations ordinary differential systems and delay differential equations for their global dynamics

singularities of maximal surfaces we show that the singularities of spacelike maximal surfaces in lorentz minkowski 3 space generically consist of cuspidal edges swallowtails and cuspidal cross caps the same result holds for spacelike mean curvature one surfaces in de sitter 3 space to prove these we shall give a simple criterion for a given singular point on a surface to be a cuspidal cross cap

h cross smooth quantum solutions a semiclassical method the h cross to 0 semiclassical expansion of wigner 1932 and kirkwood 1933 is obtained for quantum systems of finitely many particles in a d dimensional euclidean space rd without boundaries quantum systems defined by a hamiltonian h given as the sum of the negative laplacian perturbed by a potential nu x are considered the semiclassical behaviour of the kernels of the semigroup family of operators e zh rez or 0 is determined in terms of an asymptotic expansion in the variable h cross the order of the expansion is proportional to the number of bounded derivatives nu x supports the expansion is uniform in rd rd and accompanied by explicit bounds for the error term the results are obtained for potentials nu x that are fourier images of complex bounded measures

tree graphs and the solution to the hamilton jacobi equation a combinatorial method is used to construct solutions of the hamilton jacobi equation an exact expression for hamilton s principal function s is obtained for classical systems of finitely many particles interacting via a certain class of time dependent potentials if x p and t are the position momentum and time variables for n point particles of mass m it is shown that hamiltonians of the form h x p t 1 2m p2 v x t have complete integrals s that are analytic functions of the inverse mass parameter m 1 in a punctured disk about the origin if v x t is bounded c in the x variable and has controlled x derivative growth then the coefficients of the laurent expansion of s about m 1 0 may be expressed in terms of gradient structures associated with tree graphs this series expansion for s x t y t0 converges absolutely and uniformly for all x y for time displacements t t0 t 2k 1 m eu 1 2 where k and u are bounds associated with the space derivatives of the potential for t t0 t the classi

a phase space technique for the perturbation expansion of schrodinger propagators a perturbation theory for schrodinger and heat equations that is based on phase space variables is developed the dyson series representing the evolution kernel is described in terms of two basic classical quantities the free classical motion along flat space geodesics and the green function for the jacobi operator in phase space further for problems with abelian interactions it is demonstrated that the perturbation theory may be summed to all orders yielding an exponentiated connected graph description for the evolution kernel connected graph representations provide an efficient method of constructing various semiclassical approximations wherein expansion coefficients are directly determined by explicit cluster integrals this type of application is discussed for the case of schrodinger and heat equations with external electromagnetic fields detailed expressions for coefficients are obtained for both the gauge invariant large mass expansion as well as the short time schwinger dewitt expansion finally it is shown how to apply this phase space method so that it incorporates a recently proposed covariant perturbation theory

an asymptotic analysis of quantum evolution with electromagnetic fields a singular perturbation expansion of solutions to the schrodinger initial value problem is constructed using an approximate propagator for a nonrelativistic quantum system interacting with time dependent external electromagnetic fields this approximate propagator defines a gauge invariant semiclassical expansion that is realized by large mass scaling the asymptotic nature of this approximation is established by constructing error estimates that bound the hilbert space norm difference between the exact and approximate evolved states the maximum order of the approximation is determined explicitly as a function of the number of derivatives supported by the scalar and vector potentials the asymptotic expansion is obtained when the configuration space rd and also for problems where is a proper subset of rd and the self adjoint hamiltonian is defined using a supplementary boundary condition typically dirichlet or periodic

robust permanence for interacting structured populations abstract the dynamics of interacting structured populations can be modeled by d x i d t a i x x i where x i r n i x x 1 x k and a i x are matrices with non negative off diagonal entries these models are permanent if there exists a positive global attractor and are robustly permanent if they remain permanent following perturbations of a i x necessary and sufficient conditions for robust permanence are derived using dominant lyapunov exponents i of the a i x with respect to invariant measures the necessary condition requires max i i 0 for all ergodic measures with support in the boundary of the non negative cone the sufficient condition requires that the boundary admits a morse decomposition such that max i i 0 for all invariant measures supported by a component of the morse decomposition when the morse components are axiom a uniquely ergodic or support all but one population the necessary and sufficient conditions are equivalent applications to spatial ecology epidemiology and gene networks are given

a new algebraic approach for calculating the heat kernel in quantum gravity it is shown that the heat kernel operator for the laplace operator on any covariantly constant curved background i e in symmetric spaces may be presented in the form of an averaging over the lie group of isometries with some nontrivial measure using this representation the heat kernel diagonal i e the heat kernel in coinciding points is obtained related topics concerning the structure of symmetric spaces and the calculation of the effective action are discussed

ellipsoidal billiards in pseudo euclidean spaces and relativistic quadrics abstract we study the geometry of confocal quadrics in pseudo euclidean spaces of arbitrary dimension d and any signature and related billiard dynamics the goal is to give a complete description of periodic billiard trajectories within ellipsoids the novelty of our approach is the introduction of a new discrete combinatorial geometric structure associated with a confocal pencil of quadrics a colouring in d colours this is used to decompose quadrics of d 1 geometric types of a pencil into new relativistic quadrics of d relativistic types a study of what we term discriminant sets of tropical lines and and their singularities provides insight into the related geometry and combinatorics this yields an analytic criterion describing all periodic billiard trajectories including light like trajectories as a case of special interest

isometric immersions of mathbb r 2 into mathbb r 4 and perturbation of hopf tori we produce a new general family of flat tori in mathbb r 4 the first one since bianchi s classical works in the nineteenth century to construct these flat tori obtained via small perturbation of certain hopf tori in mathbb s 3 we first present a global description of all isometric immersions of mathbb r 2 into mathbb r 4 with flat normal bundle

transverse type changing pseudo riemannian metrics and the extendability of geodesics we study the geodesics and pre geodesics of a smooth manifold with smooth pseudo riemannian metric which changes bilinear type i e the signature changes on a hypersurface we classify all geodesics and pre geodesics that cross the hypersurface of type change transversely we then apply these results to the eikonal partial differential equation to find geometric conditions for the local existence or non existence of smooth transverse solutions

persistence and extinction of disease in non autonomous sirs epidemic models with disease induced mortality abstract in this paper non autonomous sirs epidemic models with bilinear incidence and disease induced mortality are studied under the quite weak assumptions the sufficient and necessary conditions on the permanence and strong persistence of the disease and the sufficient condition on the extinction of the disease are established some new threshold values of the integral form r 0 r 1 and r 2 are obtained we prove that the disease is permanent if and only if r 0 0 and if r 1 0 or r 2 0 then the disease is extinct as applications of the main results we discuss the periodic and almost periodic models the corresponding basic reproductive numbers r 0 are obtained we show that if r 0 1 then the disease is permanent and if r 0 1 then the disease is extinct

a quantum wkb approximation without classical trajectories the time evolution of a coherent state is studied in the limit 0 for a spinless nonrelativistic quantum system in euclidean space with smooth bounded time dependent scalar interaction v x t the lowest order wkb approximation is derived rigorously complete with an explicit bound on its l2 norm error over a specified finite time interval the method is based on a constructive series representation of the classical action thereby avoiding the need to analyze the behavior of classical trajectories the amplitude of the initial states considered is neither assumed to be of rapid decrease nor to depend on

limit cycles for competitor competitor mutualist lotka volterra systems abstract it is known that a limit cycle or periodic coexistence can occur in a competitor competitor mutualist lotka volterra system x 1 x 1 r 1 a 11 x 1 a 12 x 2 a 13 x 3 x 2 x 2 r 2 a 21 x 1 a 22 x 2 a 23 x 3 x 3 x 3 r 3 a 31 x 1 a 32 x 2 a 33 x 3 where r i a i j are positive real constants x liang j jiang the dynamical behavior of type k competitive kolmogorov systems and its applications to 3 dimensional type k competitive lotka volterra systems nonlinearity 16 2003 785 801 in this paper we shall construct an example with at least two limit cycles and furthermore we will show that the number of periodic orbits and hence a fortiori of limit cycles is finite it is also shown that contrary to three dimensional competitive lotka volterra systems nontrivial periodic coexistence does occur even if none of the three species can resist invasion from either of the others in this case new amenable conditions are given on the coefficients under which the system has no nontrivial periodic orbits these conditions imply that the positive equilibrium if it exists is globally asymptotically stable

on the maximal number of exceptional values of gauss maps for various classes of surfaces the main goal of this paper is to reveal the geometric meaning of the maximal number of exceptional values of gauss maps for several classes of immersed surfaces in space forms for example complete minimal surfaces in the euclidean three space weakly complete improper affine spheres in the affine three space and weakly complete flat surfaces in the hyperbolic three space for this purpose we give an effective curvature bound for a specified conformal metric on an open riemann surface

symplectic runge kutta and related methods recent results symplectic algorithms are numerical integrators for hamiltonian systems that preserve the symplectic structure in phase space in long time integrations these algorithms tend to perform better than their nonsymplectic counterparts some symplectic algorithms are derived by explicitly finding a generating function other symplectic algorithms are members of standard families of methods such as runge kutta methods that just turn out to preserve the symplectic structure here we survey what is known about the second type of symplectic algorithms

coexistence for species sharing a predator a class of equations describing the dynamics of two prey sharing a common predator are considered even though the boundary and internal dynamics can exhibit oscillatory behavior it is shown these equations are permanent if only if they admit a positive equilibrium going beyond permanence a subclass of equations are constructed that are almost surely permanent but not permanent there exists an attractor in the positive orthant that attracts lebesgue almost every but not every initial condition

propagators for relativistic systems with non abelian interactions the relativistic evolution of a system of particles in the proper time schwinger dewitt formalism is investigated for a class of interactions that can be represented as fourier transforms of bounded complex matrix valued measures a dyson series representation of the propagator is obtained this class of interactions is non abelian and includes both external electromagnetic and yang mills fields the study of the relativistic problem is facilitated by embedding the original quantum evolution into a larger class of evolution problems that result if one makes an analytic continuation of the metric tensor g this continuation is chosen so that the extended propagator shares for all signatures of g the gaussian decay properties typical of heat kernels estimates of the nth order dyson iterate kernels are found that ensure the absolute convergence of the perturbation series in this fashion a number of analytic and smoothness properties of the propagator are determined in particular it is demonstrate

a random fixed point theorem and the random graph transformation in the first part of this article we formulate a random fixed point theorem for random dynamical systems where a contraction condition is formulated as an expectation of a particular expression in the following we are going to formulate a random graph transformation this transformation defines a random dynamical system under certain conditions this random dynamical system has a random fixed point which can be found by the mentioned random fixed point theorem in one application we show that random dynamical systems given by particular differential equations have a random unstable invariant manifold

small ħ asymptotics for quantum partition functions associated to particles in external yang mills potentials to a gauge field on a principalg bundlep m is associated a sequence of quantum mechanical hamiltonians as planck s constant 0 and a sequence of representations n ofg is taken this paper studies the associated quantum partition functions trace exp th n and produces a complete asymptotic expansion as 0 1 n of which the principal term proportional to the classical partition function is the familiar classical limit

improper affine maps we study a class of improper affine spheres with singularities called improper affine maps new examples of genus 0 and 1 are described and the classification of complete improper affine maps with at most two embedded ends will be given

linking legendrian linking and causality the set n of all null geodesics of a globally hyperbolic d 1 dimensional spacetime m g is naturally a smooth 2d 1 dimensional contact manifold the sky of an event x in m is the subset x of n consisting of all null geodesics through x and is an embedded legendrian submanifold of n diffeomorphic to s d 1 it was conjectured by low that for d 2 two events x and y are causally related if and only if x and y are linked in an appropriate sense we use the contact structure and knot polynomial calculations to prove this conjecture in certain particular cases and suggest that for d 3 smooth linking should be replaced with legendrian linking

periodic ellipsoidal billiard trajectories and extremal polynomials a comprehensive study of periodic trajectories of billiards within ellipsoids in d dimensional euclidean space is presented the novelty of the approach is based on a relationship established between periodic billiard trajectories and extremal polynomials on the systems of d intervals on the real line by leveraging deep but yet not widely known results of the krein levin nudelman theory of generalized chebyshev polynomials fundamental properties of billiard dynamics are proven for any d viz that the sequences of winding numbers are monotonic by employing the potential theory we prove the injectivity of the frequency map as a byproduct for d 2 a new proof of the monotonicity of the rotation number is obtained the case study of trajectories of small periods t d le t le 2d is given in particular it is proven that all d periodic trajectories are contained in a coordinate hyperplane and that for a given ellipsoid there is a unique set of caustics which generates d 1 periodic trajectories a complete catalog of billiard trajectories with small periods is provided for d 3

global stability and predator dynamics in a model of prey dispersal in a patchy environment on considere un systeme d equations differentielles autonomes non lineaires on determine l equilibre de ce systeme

on growth rates of sub additive functions for semi flows determined and random cases abstract let r m m be a measurable random dynamical systems on the compact metric space m over f p t t r with time r let m p and e p denote the set of all invariant measures on m and the set of all ergodic invariant measures whose marginal on coincide with p respectively a function f r m r is sub additive with respect to if f t s x f t x f s t t x we define the maximal growth rate of f to be lim sup t 1 t sup x m f t x for p a e it is shown that it is equal to max m p where lim t 1 t m f t x d and there exists e p such that max m p the result may have some applications in the study of the dynamical spectrum of infinite dimension random dynamical systems and robust permanence for differential equations

uniform persistence in processes with application to nonautonomous competitive models abstract an acyclicity theorem on uniform persistence is proved for continuous processes this result is applied to two species nonautonomous competitive lotka volterra models

pseudo riemannian geodesics and billiards abstract in pseudo riemannian geometry the spaces of space like and time like geodesics on a pseudo riemannian manifold have natural symplectic structures just like in the riemannian case while the space of light like geodesics has a natural contact structure furthermore the space of all geodesics has a structure of a jacobi manifold we describe the geometry of these structures and their generalizations we also introduce and study pseudo euclidean billiards emphasizing their distinction from euclidean ones we present a pseudo euclidean version of the clairaut theorem on geodesics on surfaces of revolution we prove pseudo euclidean analogs of the jacobi chasles theorems and show the integrability of the billiard in the ellipsoid and the geodesic flow on the ellipsoid in a pseudo euclidean space

on the validity of the wkb approximation we consider the wkb approximation to the non relativistic schrodinger equation for an n particle system with a possibly time dependent potential a recent rigorous and explicit error estimate given by molzahn 1988 is investigated by means of examples to this end we extend the class of potentials considered by molzahn and find that his estimates are almost optimal however these examples also reveal an inherent drawback of the wkb method

on the wkb approximation to the propagator for arbitrary hamiltonians this paper presents a general expression for the wkb approximation to the propagator corresponding to an arbitrary hamiltonian operator h for example if the correspondence rule used to pass from the classical hamiltonian hc to h is such that it associates apiq j 1 a q jpi to piq j then the formula gives kwkb kvvexp 1 2 a ft 2hc qi pi qc t pc t t dt where kvv 2 ih n 2 det m 1 2exp isc h is van vleck s well known formula sc being the action functional evaluated at the classical path qc pc and mij 2sc qa i qbj more generally the formula presented here applies to any system with n degrees of freedom described by a function f x t whose time evolution is given by h x k x t k t f x t 0 regardless of the form of h the schrodinger equation of quantum mechanics and the fokker planck equation of diffusion are obvious examples many examples are discussed this generalizes results obtained in a previous publication j math phys 18 786 90 1977

robust uniform persistence in discrete and continuous nonautonomous systems abstract this is an extension of the work in salceanu 2011 14 to nonautonomous systems of difference and differential equations on the positive cone of r m that exhibit a positively invariant boundary hyperplane x it is shown that when a compact subset of x which attracts all orbits in x is a robust uniform weak repeller robust uniform persistence for the complementary dynamics i e the dynamics in r m x is obtained additional assumptions are made to deal with the nonautonomous nature of the systems some particular cases that often occur in applications are discussed and then sufficient conditions for the robust uniform persistence of the disease in two epidemic models from the literature are given

criteria for cr robust permanence abstract let xi xifi x i 1 n be a cr vector field that generates a dissipative flow on the positive cone of rn is called permanent if the boundary of the positive cone is repelling is called crrobustly permanent if remains permanent for sufficiently small cr perturbations of the vector field a necessary condition and a sufficient condition for cr robust permanence involving the average per capita growth rates fi d with respect to invariant measures are derived the necessary condition requires that inf maxi fi d 0 where the infimum is taken over ergodic measures with compact support in the boundary of the positive cone the sufficient condition requires that the boundary flow admit a morse decomposition m1 mk such that every mj satisfies min maxi fi d 0 where the minimum is taken over invariant measures with support in mj as applications we provide a sufficient condition for cr robust permanence of lotka volterra models and a topological characterization of cr robust permanence for food chain models

constructive representations of propagators for quantum systems with electromagnetic fields the quantum evolution of an n body system of particles that mutually interact through scalar fields and couple to an arbitrary external electromagnetic field is rigorously described both operator and kernel valued solutions to the evolution problem are found based upon a particular realization of the dyson expansion a convergent series representation of the propagator the kernel of the schrodinger time evolution operator is obtained the basic approach is to embed the quantum evolution problem in the larger class of evolution problems that result if mass is allowed to be complex quantum evolution with real mass is considered to be the boundary value of the complex mass evolution problem the constructive representation of the propagator is determined for the class of analytic scalar and vector fields that are given as fourier transforms of time dependent scalar and vector valued measures

on thec1stability conjecture for flows we give a proof of thec1stability conjecture of palis and smale for flows which has reduced to proving thatc1structural stability implies axiom a the proof is based on the fundamental work of liao and mane and on the recent powerfulc1connecting lemma of hayashi

population dynamics under selection and mutation long time behavior for differential equations in measure spaces abstract we study the long time behavior of solutions to a measure valued selection mutation model that we formulated in 14 we establish permanence results for the full model and we study the limiting behavior even when there is more than one strategy of a given fitness a case that arises in applications we show that for the pure selection case the solution of the dynamical system converges to a dirac measure centered at the fittest strategy class provided that the support of the initial measure contains a fittest strategy thus we term this dirac measure an asymptotically stable strategy we also show that when the strategy space is discrete the selection mutation model with small mutation has a locally asymptotically stable equilibrium that attracts all initial conditions that are positive at the fittest strategy

path integrals in n 2 supersymmetric quantum mechanics by use of path integrals in n 2 supersymmetric quantum mechanics a formula for the kernel of the supersymmetric time evolution operator is found structurally it is similar to the feynman kac formula used in the path integrals of quantum mechanics crucial to the development of this formula is a generalization of the wiener measure based upon the free supercharges rather than the free hamiltonian

a dyson like expansion for solutions to the quantum liouville equation given a hamiltonian of the form h h v the convergence of a dyson like expansion in is constructed and shown for the wigner distribution function that solves the quantum liouville equation that corresponds to h here h is a quadratic polynomial in p q its coefficients may depend continuously on time the potential v is a function of p and t as well as q roughly speaking it is the fourier transform of a time dependent measure

asymptotic behaviors of the heat kernel in covariant perturbation theory the trace of the heat kernel is expanded in a basis of nonlocal curvature invariants of nth order the coefficients of this expansion the nonlocal form factors are calculated to third order in the curvature inclusive the early time and late time asymptotic behaviors of the trace of the heat kernel are presented with this accuracy the late time behavior gives the criterion of analyticity of the effective action in quantum field theory the latter point is exemplified by deriving the effective action in two dimensions

time decay and spectral kernel asymptotics for quantum systems in r3 defined by a hamiltonian h given as the sum of the negative laplacian perturbed by a real valued potential v x the large time behavior of the fundamental solution of the time dependent schrodinger equation is investigated for a suitably restricted class of potentials that have algebraic decay as x the continuous spectrum portion of the fundamental solution is characterized by an asymptotic expansion as t which is uniform in compact subsets of r3 r3 these results are then applied to derive the large energy asymptotic expansions of the spectral kernel associated with h

uniform persistence and chain recurrence soit 7b e d un espace metrique localement compact et e un sous ensemble clos de 7b e soit un systeme dynamique sur e on reconsidere la notion de persistance uniforme dans le cadre de la theorie de conley des ensembles invariants

a connected graph expansion of the anharmonic oscillator propagator two series representations of the propagator associated with the quantum anharmonic oscillator are developed a closed form for the dyson series expansion of the propagator is obtained by using a phase space perturbation technique for abelian interactions the dyson series can be rearranged into an exponentiated connected graph series this representation is structurally similar to the cluster expansions of propagators associated with perturbations of the kinetic energy hamiltonian the connected graph series is particularly useful for the development of semiclassical expansions such as the wkb expansion the analytic structure in the physical parameters such as mass time and planck s constant is readily extracted from the connected graph series because of the explicit closed form formula for the series

reconstruction of singularities for solutions of schrodinger s equation we determine the behavior in time of singularities of solutions to some schrodinger equations onr n we assume the hamiltonians are of the formh 0 v where h 0 1 2 delta 1 2 sum limits k 1 n omega k 2 x k 2 and wherev is bounded and smooth with decaying derivatives when all k 0 the kernelk t x y of exp ith is smooth inx for every fixed t y when all 1 are equal but non zero the initial singularity reconstructs at times t frac m pi omega 1 and positionsx 1 m y just as ifv 0 k is otherwise regular in the general case the singular support is shown to be contained in the union of the hyperplanes x x js 1 l js y js when j t l j forj j 1 j r

covariant algebraic method for calculation of the low energy heat kernel using our recently proposed covariant algebraic approach the heat kernel for a laplace like differential operator in a low energy approximation is studied neglecting all the covariant derivatives of the gauge field strength yang mills curvature and the covariant derivatives of the potential term of third order and higher a closed formula for the heat kernel as well as its diagonal is obtained explicit formulas for the coefficients of the asymptotic expansion of the heat kernel diagonal in terms of the yang mills curvature the potential term and its first two covariant derivatives are obtained

uniform and strict persistence in monotone skew product semiflows with applications to non autonomous nicholson systems abstract we determine sufficient conditions for uniform and strict persistence in the case of skew product semiflows generated by solutions of non autonomous families of cooperative systems of odes or delay fdes in terms of the principal spectrums of some associated linear skew product semiflows which admit a continuous separation our conditions are also necessary in the linear case we apply our results to a noncooperative almost periodic nicholson system with a patch structure whose persistence turns out to be equivalent to the persistence of the linearized system along the null solution

the construction of spinor fields on manifolds with smooth degenerate metrics we examine some of the subtleties inherent in formulating a theory of spinors on a manifold with a smooth degenerate metric we concentrate on the case where the metric is singular on a hypersurface that partitions the manifold into lorentzian and euclidean domains we introduce the notion of a complex spinor fibration to make precise the meaning of continuity of a spinor field and give an express ion for the components of a local spinor connection that is valid in the absence of a frame of local orthonormal vectors these considerations enable one to construct a dirac equation for the discussion of the behavior of spinors in the vicinity of the metric degeneracy we conclude that the theory contains more freedom than the spacetime dirac theory and we discuss some of the implications of this for the continuity of conserved currents

kolmogorov vector fields with robustly permanent subsystems the following results are proven all subsystems of a dissipative kolmogorov vector field i xifi x are robustly permanent if and only if the external lyapunov exponents are positive for every ergodic probability measure with support in the boundary of the nonnegative orthant if the vector field is also totally competitive its carrying simplex is c1 applying these results to dissipative lotka volterra systems robust permanence of all subsystems is equivalent to every equilibrium x satisfying fi x 0 whenever xi 0 if in addition the lotka volterra system is totally competitive then its carrying simplex is c1

the stability under perturbations of repulsive sets on considere le systeme suivant sur r n x f x ou f f 1 f n x x 1 x n on demontre qu un theoreme de fonction de lyapunov inverse est verifie garantissant l existence d une fonction de lyapunov u et que l on peut choisir u de lipschitz

marginally trapped submanifolds in lorentzian space forms and in the lorentzian product of a space form by the real line we give local explicit representation formulas for n dimensional spacelike submanifolds which are marginally trapped in the minkowski space r1n 2 the de sitter space d x1d54a n 2 the anti de sitter space ad x1d54a n 2 and the lorentzian products x1d54a n 1 and n 1 of the sphere and the hyperbolic space by the real line

surfaces of constant curvature in r3 with isolated singularities abstract we prove that finite area isolated singularities of surfaces with constant positive curvature k 0 in r 3 are removable singularities branch points or immersed conical singularities we describe the space of immersed conical singularities of such surfaces in terms of the class of real analytic closed locally strictly convex curves in s 2 with admissible cusp singularities characterizing when the singularity is actually embedded in the global setting we describe the space of peaked spheres in r 3 i e compact convex surfaces of constant curvature k 0 with a finite number of singularities and give applications to harmonic maps and constant mean curvature surfaces

mixed type surfaces with bounded gaussian curvature in three dimensional lorentzian manifolds abstract a mixed type surface is a connected regular surface in a lorentzian 3 manifold with non empty spacelike and timelike point sets the induced metric of a mixed type surface is a signature changing metric and their lightlike points may be regarded as singular points of such metrics in this paper we investigate the behavior of gaussian curvature at a non degenerate lightlike point of a mixed type surface to characterize the boundedness of gaussian curvature at a non degenerate lightlike points we introduce several fundamental invariants along non degenerate lightlike points such as the lightlike singular curvature and the lightlike normal curvature moreover using the results by pelletier and steller we obtain the gauss bonnet type formula for mixed type surfaces with bounded gaussian curvature

smoothness of the carrying simplex for discrete time competitive dynamical systems a characterization of neat embedding abstract the paper is concerned with the question of smoothness of the carrying simplex s for a discrete time dissipative competitive dynamical system we give a necessary and sufficient criterion for s being a c 1 submanifold with corners neatly embedded in the nonnegative orthant formulated in terms of inequalities between lyapunov exponents for ergodic measures supported on the boundary of the orthant this completes one thread of investigation occasioned by a question posed by m w hirsch in 1988 besides amenable conditions are presented to guarantee the c r r 1 smoothness of s in the time periodic competitive kolmogorov systems of odes examples are also presented one in which s is of class c 1 but not neatly embedded the other in which s is not of class c 1

schrodinger semigroups for vector fields suppose h is the hamiltonian that generates time evolution in an n body spin dependent nonrelativistic quantum system if r is the total number of independent spin components and the particles move in three dimensions then the hamiltonian h is an r r matrix operator given by the sum of the negative laplacian x on the d 3n dimensional euclidean space rd plus a hermitian local matrix potential w x uniform higher order asymptotic expansions are derived for the time evolution kernel the heat kernel and the resolvent kernel these expansions are respectively for short times high temperatures and high energies explicit formulas for the matrix valued coefficient functions entering the asymptotic expansions are determined all the asymptotic expansions are accompanied by bounds for their respective error terms these results are obtained for the class of potentials defined as the fourier image of bounded complex valued matrix measures this class is suitable for the n body problem since interactio

exponential separation and principal lyapunov exponent spectrum for random nonautonomous parabolic equations abstract the purpose of the paper is to extend the principal eigenvalue and principal eigenfunction theory for time independent and periodic parabolic equations to random and general nonautonomous ones in the random case a notion of principal lyapunov exponent serving as an analog of principal eigenvalue is introduced it is shown that the principal lyapunov exponent is deterministic and of simple multiplicity it is also shown that there is a one dimensional invariant random subbundle corresponding to the solutions that are globally defined and of the same sign which serves as an analog of principal eigenfunction in addition monotonicity of the principal lyapunov exponent with respect to the zero order terms both in the equation and in the boundary condition is proved when the second and first order terms are deterministic it is proved that the principal lyapunov exponent is greater than or equal to the principal eigenvalue of the associated time averaged equation in the general nonautonomous case the concepts of principal spectrum which serves as an analog of principal eigenvalue and principal lyapunov exponents are introduced as is known the principal spectrum is a compact interval it is proved in the paper that the principal spectrum contains all the principal lyapunov exponents when the second and first order terms are time independent a lower estimate of the infimum of the principal spectrum is given in terms of an associated time averaged equation

complete linear weingarten surfaces of bryant type a plateau problem at infinity in this paper we study a large class of weingarten surfaces which includes the constant mean curvature one surfaces and flat surfaces in the hyperbolic 3 space we show that these surfaces can be parametrized by holomorphic data like minimal surfaces in the euclidean 3 space and we use it to study their completeness we also establish some existence and uniqueness theorems by studing the plateau problem at infinity when is a given curve on the ideal boundary the asymptotic boundary of a complete surface in our family and how many embedded solutions are there

error estimates for approximate solutions of the riccati equation with real or complex potentials a method is presented for obtaining rigorous error estimates for approximate solutions of the riccati equation with real or complex potentials our main tool is to derive invariant region estimates for complex solutions of the riccati equation we explain the general strategy for applying these estimates and illustrate the method in typical examples where the approximate solutions are obtained by gluing together wkb and airy solutions of corresponding one dimensional schrodinger equations our method is motivated by and has applications to the analysis of linear wave equations in the geometry of a rotating black hole

schrodinger spectral kernels high order asymptotic expansions the large energy behavior of the spectral kernel for the n body schrodinger hamiltonian is obtained in a setting of a d dimensional euclidean space without boundaries the schrodinger hamiltonian h is the sum of the negative laplacian plus a real valued local potential v x the class of potentials studied is the family of bounded and continuous functions that are formed from the fourier transforms of complex bounded measures these potentials are suitable for the n body problem since they do not necessarily decrease as x let e x y r be the family of spectral kernels generated by h in the limit explicit higher order asymptotic expansions are obtained for e x y and its associated riesz means the asymptotic expansion is uniform in x and y and is accompanied by estimates of the error term

on the fine structure of the global attractor of a uniformly persistent flow abstract we study the internal structure of the global attractor of a uniformly persistent flow we show that the restriction of the flow to the global attractor has duality properties which can be expressed in terms of certain attractor repeller decompositions we also study some natural morse decompositions of the flow and calculate their morse equations these equations provide necessary and sufficient conditions for the existence of attractors with the shape of s 1 or such that their suspension has spherical shape

the geometry of fronts we shall introduce the singular curvature function on cuspidal edges of surfaces which is related to the gauss bonnet formula and which characterizes the shape of cuspidal edges moreover it is closely related to the behavior of the gaussian curvature of a surface near cuspidal edges and swallowtails

learning by natural gradient on noncompact matrix type pseudo riemannian manifolds this paper deals with learning by natural gradient optimization on noncompact manifolds in a riemannian manifold the calculation of entities such as the closed form of geodesic curves over noncompact manifolds might be infeasible for this reason it is interesting to study the problem of learning by optimization over noncompact manifolds endowed with pseudo riemannian metrics which may give rise to tractable calculations a general theory for natural gradient based learning on noncompact manifolds as well as specific cases of interest of learning are discussed

exponential cluster solutions to quantum transport equations connected graph methods are used to solve the transport equation arising in a semiclassical representation of the schrodinger propagator k t s x y for a spinless quantum system whose hamiltonian is a smooth position and momentum dependent perturbation of the laplacian in rd exponential cluster expansions for k are obtained these formally exact solutions to the time dependent schrodinger equation employ explicit graphically determined derivatives of the hamiltonian s symbol integrated along geodesics for the special case of particles interacting with external electromagnetic fields the propagator s gauge invariant derivative expansion coefficients are determined in closed form in terms of the lorentz force a tree graph formula for hamilton s principal function is extracted from this result

uniform persistence for nonautonomous and random parabolic kolmogorov systems abstract the purpose of this paper is to investigate uniform persistence for nonautonomous and random parabolic kolmogorov systems via the skew product semiflows approach it is first shown that the uniform persistence of the skew product semiflow associated with a nonautonomous random parabolic kolmogorov system implies that of the system various sufficient conditions in terms of the so called unsaturatedness and or lyapunov exponents for uniform persistence of the skew product semiflows are then provided among others it is shown that if the associated skew product semiflow has a global attractor and its restriction to the boundary of the state space has a morse decomposition which is unsaturated or whose external lyapunov exponents are positive then it is uniformly persistent more specific conditions are discussed for uniform persistence in n species particularly 3 species random competitive systems

time evolution kernels uniform asymptotic expansions for a wide class of self adjoint schrodinger hamiltonians a detailed description of the time evolution kernel is obtained in a setting of a d dimensional euclidean space without boundaries the schrodinger hamiltonian h is the sum of the negative laplacian plus a real valued local potential v x the class of potentials studied is the family of bounded and continuous functions that are formed from the fourier transforms of complex bounded measures these potentials are suitable for the n body problem since they do not necessarily decrease as x an asymptotic expansion in the complex parameter z around z 0 is derived for the family of kernels uz x y corresponding to the analytic semigroup e zh re z 0 which is uniform in the coordinate variables x and y the asymptotic expansion has a simple semiclassical interpretation furthermore an explicit bound for the remainder term in the asymptotic expansion is found the expansion and the remainder term bound continue to the time axis boundary z it

persistence in dynamical systems soit un espace metrique localement compact de metrique d et soit e un sous ensemble clos de a frontiere e et interieur e on suppose que l on a un flot continu 7b f defini sur e tel que e est invariant par 7b f on considere la notion de persistance et on donne un critere de persistance

a borel like estimate for the rest of the heat kernel expansion it is shown that under some simple assumptions the rest of the truncated asymptotic expansion for the heat kernel has the form required for borel summability if n the number of terms in the truncation is sufficiently large rigorously speaking this result does not imply borel summability which requires a uniform bound in n but it could be useful as a starting point for further investigation on global estimates for the zeta function

uniform persistence and hopf bifurcations in r n we consider parameterized families of flows in locally compact metrizable spaces and give a characterization of those parameterized families of flows for which uniform persistence continues on the other hand we study the generalized poincare andronov hopf bifurcations of parameterized families of flows at boundary points of r n or more generally of an n dimensional manifold and show that this kind of bifurcations produce a whole family of attractors evolving from the bifurcation point and having interesting topological properties in particular in some cases the bifurcation transforms a system with extreme non permanence properties into a uniformly persistent one we study in the paper when this phenomenon happens and provide an example constructed by combining a holling type interaction with a pitchfork bifurcation