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

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

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

the supremum of some canonical processes 1 introduction consider a random variable r v h such that eh2 l distributed like h to each t tk k 1 of 2 one can associate a r v xi ek l tkhk this defines a stochastic process xt t e2 by stochastic process we simply mean a collection of r v this process could be called the canonical process associated to h a long range and rather ambitious program would be to understand given h for which subsets t of 2 the process xt tct is bounded let us first discuss the case where h is centered gaussian the special importance of that case stems from the fact that for the question of boundedness discussed here gaussian processes of the type xt tct are the most general gaussian processes this fact is closely related to the rotational invariance of gaussian measures for a stochastic process xt tct where the set t is possibly uncountable we set

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

pathological outcomes of observational learning this paper explores how bayes rational individuals learn sequentially from the discrete actions of others unlike earlier informational herding papers we admit heterogeneous preferences not only may type specific herds eventually arise but a new robust possibility emerges confounded learning beliefs may converge to a limit point where history offers no decisive lessons for anyone and each type s actions forever nontrivially split between two actions to verify that our identified limit outcomes do arise we exploit the markov martingale character of beliefs learning dynamics are stochastically stable near a fixed point in many bayesian learning models like this one

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

regularity of gaussian processes on donne la validite de la conjecture de fernique 1974 sur l existence de mesures de majoration pouvant caracteriser la limitation des processus gaussiens

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

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

majorizing measures the generic chaining majorizing measures provide bounds for the supremum of stochastic processes they represent the most general possible form of the chaining argument going back to kolmogorov majorizing measures arose from the theory of gaussian processes but they now have applications far beyond this setting the fundamental question is the construction of these measures this paper focuses on the tools that have been developed for this purpose and in particular the use of geometric ideas applications are given to several natural problems where entropy methods are powerless

chaos in social learning with multiple true states abstract most existing social learning models assume that there is only one underlying true state in this work we consider a social learning model with multiple true states in which agents in different groups receive different signal sequences generated by their corresponding underlying true states each agent updates his belief by combining his rational self adjustment based on the external signals he received and the influence of his neighbors according to their communication we observe chaotic oscillation in the belief evolution which implies that neither true state could be learnt correctly by calculating the largest lyapunov exponents and hurst exponents

elliptically contoured distributions given two covariance matricesr ands for a given elliptically contoured distribution we show how simple inequalities between the matrix elements imply thate r f e s f e g whenx xi1 i2 in is a multiindex vector and f x mathop min limits i 1 mathop max limits i 2 mathop min limits i 3 max x i 1 i n orf x is the indicator function of sets such as mathop cap limits i 1 mathop cup limits i 2 mathop cap limits i 3 cup x i 1 i n mathop limits lambda i 1 i n of which the well known slepian s inequality n 1 is a special case

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

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

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

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

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