escape rates over potential barriers variational principles and the hamilton jacobi equation abstract we describe a rigorous formalism to study some extrema statistics problems like maximum probability events or escape rate processes by taking into account that the hamilton jacobi equation completes in a natural way the required set of boundary conditions of the euler lagrange equation for this kind of variational problem we apply this approach to a one dimensional stochastic process driven by colored noise for a double parabola potential where we have one stable and one unstable steady states

on the first integral of the euler lagrange equation boundary conditions and escape rates for stochastic processes abstract following the path integral formalism for stochastic processes we study formal properties of the extremal path the euler lagrange e l equation and a first integral of it a special interest is focused in the connection between this first integral and the boundary conditions we are interested in the maximum probability for the variable to attain a given value or escape rate process so we analyze the variational problem of a mobile end point of the extremal path this is a transversality condition which in the present context means a stationary condition and gives a zero value for the first integral of the e l equation under a certain initial condition we can handle this differential equation as an initial value problem this more rigorous approach to the variational problem is applied to some specific potentials both for white and colored noise in order to see clearly the qualitative difference introduced by the noise correlation time the dependence of the extremal action exhibits features that result from the nonlinearity of the integral of motion even in the case of the quadratic potential this behavior of s has been first reported recently by the author for the bistable potential 1 the present results can provide a better insight in the study of first passage time problems

generalized point interactions in one dimensional quantum mechanics there is a four parameter family of point interactions in one dimensional quantum mechanics they represent all possible self adjoint extensions of the kinetic energy operator if time reversal invariance is imposed the number of parameters is reduced to three one of these point interactions is the familiar function potential but the other generalized ones do not seem to be widely known we present a pedestrian approach to this subject and comment on a recent controversy in the literature concerning the so called interaction we emphasize that there is little resemblance between the interaction and what its name suggests

energy dependent point interactions in one dimension we consider a new type of point interaction in one dimensional quantum mechanics it is characterized by a boundary condition at the origin that involves the second and or higher order derivatives of the wavefunction the interaction is effectively energy dependent it leads to a unitary s matrix for the transmission reflection problem the energy dependence of the interaction can be chosen such that any given unitary s matrix or the transmission and reflection coefficients can be reproduced at all energies generalization of the results to coupled channel cases is discussed

illumination recovery from image with cast shadows via sparse representation in this paper we propose using sparse representation for recovering the illumination of a scene from a single image with cast shadows given the geometry of the scene the images with cast shadows can be quite complex and therefore cannot be well approximated by low dimensional linear subspaces however it can be shown that the set of images produced by a lambertian scene with cast shadows can be efficiently represented by a sparse set of images generated by directional light sources we first model an image with cast shadows composed of a diffusive part without cast shadows and a residual part that captures cast shadows then we express the problem in an l1 regularized least squares formulation with nonnegativity constraints as light has to be non negative at any point in space this sparse representation enjoys an effective and fast solution thanks to recent advances in compressive sensing in experiments on synthetic and real data our approach performs favorably in comparison with several previously proposed methods

point interactions in one dimensional quantum mechanics with coupled channels we explore generalized point interactions in one dimensional quantum mechanics with two coupled channels they represent possible self adjoint extensions of the nonrelativistic kinetic energy operator assuming time reversal invariance we find a family of self adjoint extensions with seven parameters this can be compared with the one channel case in which the corresponding number of parameters is three

many body system with a four parameter family of point interactions in one dimension we consider a four parameter family of point interactions in one dimension this family is a generalization of the usual function potential we examine a system consisting of many particles of equal masses that are interacting pairwise through such a generalized point interaction we follow mcguire who obtained exact solutions for the system when the interaction is the function potential we find exact bound states with the four parameter family for the scattering problem however we have not been so successful this is because as we point out the condition of no diffraction that is crucial in mcguire s method is not satisfied except when the four parameter family is essentially reduced to the function potential

the mechanism of stochastic resonance it is shown that a dynamical system subject to both periodic forcing and random perturbation may show a resonance peak in the power spectrum which is absent when either the forcing or the perturbation is absent

lambertian reflectance and linear subspaces we prove that the set of all lambertian reflectance functions the mapping from surface normals to intensities obtained with arbitrary distant light sources lies close to a 9d linear subspace this implies that in general the set of images of a convex lambertian object obtained under a wide variety of lighting conditions can be approximated accurately by a low dimensional linear subspace explaining prior empirical results we also provide a simple analytic characterization of this linear space we obtain these results by representing lighting using spherical harmonics and describing the effects of lambertian materials as the analog of a convolution these results allow us to construct algorithms for object recognition based on linear methods as well as algorithms that use convex optimization to enforce nonnegative lighting functions we also show a simple way to enforce nonnegative lighting when the images of an object lie near a 4d linear space we apply these algorithms to perform face recognition by finding the 3d model that best matches a 2d query image

what is the set of images of an object under all possible illumination conditions the appearance of an object depends on both the viewpoint from which it is observed and the light sources by which it is illuminated if the appearance of two objects is never identical for any pose or lighting conditions then in theory the objects can always be distinguished or recognized the question arises what is the set of images of an object under all lighting conditions and pose in this paper we consider only the set of images of an object under variable illumination including multiple extended light sources and shadows we prove that the set of n pixel images of a convex object with a lambertian reflectance function illuminated by an arbitrary number of point light sources at infinity forms a convex polyhedral cone in r n and that the dimension of this illumination cone equals the number of distinct surface normals furthermore the illumination cone can be constructed from as few as three images in addition the set of n pixel images of an object of any shape and with a more general reflectance function seen under all possible illumination conditions still forms a convex cone in r n extensions of these results to color images are presented these results immediately suggest certain approaches to object recognition throughout we present results demonstrating the illumination cone representation

exchange energy splitting for ground state h2 from fluctuation theory the quantum mechanical energy splitting for a ground state hydrogen molecular ion has been computed as the complex valued mean first passage time across the potential barrier between two wells by separating the schrodinger equation using spheroidal coordinates and applying a subsequent transformation so as to obtain three one dimensional hermitian operator equations one of these operators describes a particle oscillating between two wells separated by a potential barrier the splitting is then calculated as the energy difference between the symmetry adapted wavefunctions and a reference state whose probability density is concentrated on one side of the barrier the eigenfunctions are calculated as the solutions of a volterra integral equation where the energy difference is the eigenvalue

mixture of spherical distributions for single view relighting we present a method for simultaneously estimating the illumination of a scene and the reflectance property of an object from single view images a single image or a small number of images taken from the same viewpoint we assume that the illumination consists of multiple point light sources and the shape of the object is known first we represent the illumination on the surface of a unit sphere as a finite mixture of von mises fisher distributions based on a novel spherical specular reflection model that well approximates the torrance sparrow reflection model next we estimate the parameters of this mixture model including the number of its component distributions and the standard deviation of them which correspond to the number of light sources and the surface roughness respectively finally using these results as the initial estimates we iteratively refine the estimates based on the original torrance sparrow reflection model the final estimates can be used to relight single view images such as altering the intensities and directions of the individual light sources the proposed method provides a unified framework based on directional statistics for simultaneously estimating the intensities and directions of an unknown number of light sources as well as the specular reflection parameter of the object in the scene

the fermi pseudo potential in one dimension wu and yu recently examined point interactions in one dimension in the form of the fermi pseudo potential on the other hand there are point interactions in the form of self adjoint extensions saes of the kinetic energy operator we examine the relationship between the point interactions in these two forms in the one channel and two channel cases in the one channel case the pseudo potential leads to the standard three parameter family of saes in the two channel case the pseudo potential furnishes a ten parameter family of saes

properties of diffusive systems near a saddle point application to a quartic double well this paper aims at the analysis of diffusive properties of unidimensional mechanical systems in the environment of maxima and minima of the potential it begins with a study of the properties of the singular solutions of the hamilton jacobi yasue equation in the above mentioned environment in both strong or very small frictional forces for the quartic symmetrical double well potential approximate solutions are found for local validity and the diffusion operator is then calculated in the limits of deep wells and small temperature the regime being supposed to be aperiodic with high or moderate values of frictional coefficient this equation is proved to be nonunique this operator is then reduced to second order by imposing suitable boundary conditions thus an appropriate eigenvalue equation is obtained to describe stationary states in the environment of extremal points of the potential energy function the main interest of this work relies upon the fact that transition times between wells mainly depend upon fluctuations near the saddle point

distribution theory for schrodinger s integral equation much of the literature on point interactions in quantum mechanics has focused on the differential form of schrodinger s equation this paper in contrast investigates the integral form of schrodinger s equation while both forms are known to be equivalent for smooth potentials this is not true for distributional potentials here we assume that the potential is given by a distribution defined on the space of discontinuous test functions first by using schrodinger s integral equation we confirm a seminal result by kurasov which was originally obtained in the context of schrodinger s differential equation this hints at a possible deeper connection between both forms of the equation we also sketch a generalisation of kurasov s j math anal appl 201 1 297 323 1996 result to hypersurfaces second we derive a new closed form solution to schrodinger s integral equation with a delta prime potential this potential has attracted considerable attention including some controversy interestingly the derived propagator satisfies boundary conditions that were previously derived using schrodinger s differential equation third we derive boundary conditions for super singular potentials given by higher order derivatives of the delta potential these boundary conditions cannot be incorporated into the normal framework of self adjoint extensions we show that the boundary conditions depend on the energy of the solution and that probability is conserved this paper thereby confirms several seminal results and derives some new ones in sum it shows that schrodinger s integral equation is a viable tool for studying singular interactions in quantum mechanics

generalized fokker planck equation for non markovian processes this paper deals with a system of linear non markovian langevin equations with memory functions that are not constant in time and a nonzero initial instant of time a set of statistical means based on the application of a generalized furutsu novikov formula was used to derive a generalized fokker planck equation corresponding to this system and holding for both long and short instants of time considered as an example is the brownian motion of a particle in a viscoelastic fluid with a particular relaxation time

effective fokker planck equation path integral formalism we derive an effective fokker planck equation for a nonlinear non markovian stochastic process using pathintegral formalism the effective fokker planck equation of our approximation scheme is local in state space and local in time and its diffusion constant remains positive over the entire state space for all noise correlation time the mean first passage time mfpt and the stationary probability density function spdf for the case of a bistable potential driven by ornstein uhlenbeck noise are computed the mfpt is compared with that of earlier theories and with the numerical simulation results the spdf is compared with that of the results obtained through the matrix continued fraction method interesting connections between the path integral method and the functional calculus approach to the colored noise problem are brought out

boundary conditions at the derivative of a delta function the author derives the joining conditions on psi at a point where the potential is the nth derivative of a dirac delta function

asymptotic evaluation of the keesom integral a simple physically transparent formula is obtained in the asymptotic evaluation of the keesom integral k a for large values of the parameter a its derivation is described in some detail and the asymptotic formula is compared with the results obtained from direct three dimensional numerical integration

on photometric issues in 3d visual recognition from asingle 2d image we describe the problem of recognition under changing illumination conditions and changing viewing positions from a computational and human vision perspective on the computational side we focus on the mathematical problems of creating an equivalence class for images of the same 3d object undergoing certain groups of transformations mostly those due to changing illumination and briefly discuss those due to changing viewing positions the computational treatment culminates in proposing a simple scheme for recognizing via alignment an image of a familiar object taken from a novel viewing position and a novel illumination condition on the human vision aspect the paper is motivated by empirical evidence inspired by mooney images of faces that suggest a relatively high level of visual processing is involved in compensating for photometric sources of variability and furthermore that certain limitations on the admissible representations of image information may exist the psychophysical observations and the computational results that follow agree in several important respects such as the same apparent limitations on image representations

non markovian effects on the brownian motion of a free particle abstract non markovian effects on the brownian movement of a free particle in the presence as well as in the absence of inertial force are investigated under the framework of generalized fokker planck equations rayleigh and smoluchowski equations more specifically it is predicted that non markovian features can diminish the values of both the root mean square displacement and the root mean square momentum thereby assuring the mathematical property of analyticity of such physically observable quantities for all times t 0 accordingly the physical concept of non markovian brownian trajectory turns out to be mathematically well defined by differentiable functions for all t 0 another consequence of the non markovicity property is that the langevin stochastic equations underlying the fokker planck equations should be interpreted as genuine differential equations and not as integral equations according to a determined interpretation rule doob s rule for instance

lowest nonzero eigenvalue of the diffusion equation for a brownian particle in a symmetric double well with low friction in this work a particle undergoing brownian motion in a symmetric double well is considered in the limit of low frictional forces and small temperature it is assumed that the particle fluctuations over equilibrium are governed by a second order diffusion equation in which the drift velocity solves a hjy equation while the diffusion coefficient is determined from the stationary condition the excited state is constructed by imposing suitable boundary conditions at the saddle point by means of an integral equation which is solved to a first approximation by iteration in the limit of high barrier the result is a complex eigenvalue for the relaxation of the fluctuation which is interpreted as a damped stochastic resonance the relation to kramer s rate is discussed

a computable evolution equation for the joint response excitation probability density function of stochastic dynamical systems by using functional integral methods we determine a computable evolution equation for the joint response excitation probability density function of a stochastic dynamical system driven by coloured noise this equation can be represented in terms of a superimposition of differential constraints i e partial differential equations involving unusual limit partial derivatives the first one of which was originally proposed by sapsis athanassoulis a connection with the classical response approach is established in the general case of random noise with arbitrary correlation time yielding a fully consistent new theory for non markovian systems we also address the question of computability of the joint response excitation probability density function as a solution to a boundary value problem involving only one differential constraint by means of a simple analytical example it is shown that in general such a problem is undetermined in the sense that it admits an infinite number of solutions this issue can be overcome by completing the system with additional relations yielding a closure problem which is similar to the one arising in the standard response theory numerical verification of the equations for the joint response excitation density is obtained for a tumour cell growth model under immune response

solving ill conditioned and singular linear systems a tutorial on regularization it is shown that the basic regularization procedures for finding meaningful approximate solutions of ill conditioned or singular linear systems can be phrased and analyzed in terms of classical linear algebra that can be taught in any numerical analysis course apart from rewriting many known results in a more elementary form we also derive a new two parameter family of merit functions for the determination of the regularization parameter the traditional merit functions from generalized cross validation gcv and generalized maximum likelihood gml are recovered as special cases

classifying the general family of 1d point interactions a scattering approach we propose a classification scheme for the complete family of 1d point interactions to do so we first review the solutions of the wavefunctions and green functions of the problem second we derive the exact time dependent propagators in such a way that we can write the expressions for the k s in a very compact form as they should such expressions do reduce to the known results in the literature according to the potential parameter values then we analyse in general terms how the different point interactions scatter off arbitrary initially localized wave packets finally we show that the physical features associated with the scattering process can be used to establish a classification procedure moreover these physical characteristics are directly related to the potential parameters leading to the many formulae for the k s as an application we present numerical calculations for gaussian wave packets

a fourier theory for cast shadows cast shadows can be significant in many computer vision applications such as lighting insensitive recognition and surface reconstruction nevertheless most algorithms neglect them primarily because they involve nonlocal interactions in nonconvex regions making formal analysis difficult however many real instances map closely to canonical configurations like a wall a v groove type structure or a pitted surface in particular we experiment with 3d textures like moss gravel and a kitchen sponge whose surfaces include canonical configurations like v grooves this paper takes a first step toward a formal analysis of cast shadows showing theoretically that many configurations can be mathematically analyzed using convolutions and fourier basis functions our analysis exposes the mathematical convolution structure of cast shadows and shows strong connections to recent signal processing frameworks for reflection and illumination

jwkb connection formula problem revisited again via borel summation silverstone phys rev lett 55 2523 2526 1985 has expressed the opinion that the traditional version of the jwkb connection formulas associated with a classical turning point is incorrect and

langer cherry derivation of the multi instanton expansion for the symmetric double well the multi instanton expansion for the eigenvalues of the symmetric double well is derived using a langer cherry uniform asymptotic expansion of the solution of the corresponding schrodinger equation the langer cherry expansion is anchored to either one of the minima of the potential and by construction has the correct asymptotic behavior at large distance while the quantization condition amounts to imposing the even or odd parity of the wave function this method leads to an efficient algorithm for the calculation to virtually any desired order of all the exponentially small series of the multi instanton expansion and with trivial modifications can also be used for nonsymmetric double wells

one dimensional schrodinger operators with interactions singular on a discrete set on etudie le caractere auto adjoint des operateurs de schrodinger sur y u j i x j y j ou on suppose y j x j 1