application of the karhunen loeve procedure for the characterization of human faces the use of natural symmetries mirror images in a well defined family of patterns human faces is discussed within the framework of the karhunen loeve expansion this results in an extension of the data and imposes even and odd symmetry on the eigenfunctions of the covariance matrix without increasing the complexity of the calculation the resulting approximation of faces projected from outside of the data set onto this optimal basis is improved on average

the fokker planck equation with absorbing boundary conditions we study the initial boundary value problem for the fokker planck equation in an interval with absorbing boundary conditions we develop a theory of well posedness of classical solutions for the problem we also prove that the resulting solutions decay exponentially for long times to prove these results we obtain several crucial estimates which include hypoellipticity away from the singular set for the fokker planck equation with absorbing boundary conditions as well as the holder continuity of the solutions up to the singular set

some examples of kinetic schemes whose diffusion limit is il in s exponential fitting this paper is concerned with diffusive approximations of some numerical schemes for several linear or weakly nonlinear kinetic models which are motivated by wide range applications including radiative transfer or neutron transport run and tumble models of chemotaxis dynamics and vlasov fokker planck plasma modeling the well balanced method applied to such kinetic equations leads to time marching schemes involving a scattering s matrix itself derived from a normal modes decomposition of the stationary solution one common feature these models share is the type of diffusive approximation their macroscopic densities solve drift diffusion systems for which a distinguished numerical scheme is il in scharfetter gummel s exponential fitting discretization we prove that these well balanced schemes relax within a parabolic rescaling towards such type of discretization by means of an appropriate decomposition of the s matrix hence are asymptotic preserving

mean exit times for particles driven by weakly colored noise consider a particle whose velocity is an ornstein uhlenbeck process with correlation time varepsilon 2 frac dx dt frac sigma z varepsilon qquad frac dz dt frac z varepsilon 2 frac sqrt 2 xi t varepsilon where xi t is gaussian white noise this represents brownian motion with memory i e a non markovian brownian motion singular perturbation techniques reduce the mean exit time problem to a boundary layer problem which turns out to be the half range expansion problem for the fokker planck operator this is solved via complex variable theory it is found that the spatial density has a milne extrapolation length varepsilon sigma zeta tfrac 1 2 varepsilon sigma 1 46035 cdots where zeta is the riemann zeta function it is also found that the average time at which x t first escapes an interval a 0 b ven that it starts at x 0 0 and z 0 z 0 is e t text ex frac 1 2 sigma 2 left a varepsilo

asymptotic solution of the wang uhlenbeck recurrence time problem a langevin particle is initiated at the origin with positive velocity its trajectory is terminated when it returns to the origin in 1945 wang and uhlenbeck posed the problem of finding the joint probability density function pdf of the recurrence time and velocity naming it the recurrence time problem we show that the short time asymptotics of the recurrence pdf is similar to that of the integrated brownian motion solved in 1963 by mckean we recover the long time t ensuremath 3 2 decay of the first arrival pdf of diffusion by solving asymptotically an appropriate variant of mckean s integral equation

multispace kl for pattern representation and classification this work introduces the multispace karhunen loeve mkl as a new approach to unsupervised dimensionality reduction for pattern representation and classification the training set is automatically partitioned into disjoint subsets according to an optimality criterion each subset then determines a different kl subspace which is specialized in representing a particular group of patterns the extension of the classical kl operators and the definition of ad hoc distances allow mkl to be effectively used where kl is commonly employed the limits of the standard kl transform are pointed out in particular mkl is shown to outperform kl when the data distribution is far from a multidimensional gaussian and to better cope with large sets of patterns which could cause a severe performance drop in kl

nonlinear modeling of scattered multivariate data and its application to shape change we are given a set of points in a space of high dimension for instance this set may represent many visual appearances of an object a face or a hand we address the problem of approximating this set by a manifold in order to have a compact representation of the object appearance when the scattering of this set is approximately an ellipsoid then the problem has a well known solution given by principal components analysis pca however in some situations like object displacement learning or face learning this linear technique may be ill adapted and nonlinear approximation has to be introduced the method we propose can be seen as a nonlinear pca nlpca the main difficulty being that the data are not ordered we propose an index which favors the choice of axes preserving the closest point neighborhoods these axes determine an order for visiting all the points when smoothing finally a new criterion called generalization error is introduced to determine the smoothing rate that is the knot number for the spline fitting experimental results conclude this paper the method is tested on artificial data and on two databases used in visual learning

poisson homology in degree 0 for some rings of symplectic invariants abstract let g be a finite dimensional semi simple lie algebra h a cartan subalgebra of g and w its weyl group the group w acts diagonally on v h h as well as on c v the purpose of this article is to study the poisson homology of the algebra of invariants c v w endowed with the standard symplectic bracket to begin with we give general results about the poisson homology space in degree 0 denoted by hp 0 c v w in the case where g is of type b n c n or d n results which support alev s conjecture then we are focusing the interest on the particular cases of ranks 2 and 3 by computing the poisson homology space in degree 0 in the cases where g is of type b 2 so 5 d 2 so 4 then b 3 so 7 and d 3 a 3 so 6 sl 4 in order to do this we make use of a functional equation introduced by y berest p etingof and v ginzburg we recover by a different method the result established by j alev and l foissy according to which the dimension of hp 0 c v w equals 2 for b 2 then we calculate the dimension of this space and we show that it is equal to 1 for d 2 we also calculate it for the rank 3 cases we show that it is equal to 3 for b 3 c 3 and 1 for d 3 a 3

the analytic solutions of some boundary layer problems in the theory of brownian motion the authors give analytic solutions of the generalised albedo and milne problems for the uhlenbeck ornstein process and discuss the relation between these and the original wang uhlenbeck 1945 problem of determining the distribution of first passage times

mixture densities maximum likelihood and the em algorithm the problem of estimating the parameters which determine a mixture density has been the subject of a large diverse body of literature spanning nearly ninety years during the last two decades the

the adaptive hough transform we introduce the adaptive hough transform aht as an efficient way of implementing the hough transform ht method for the detection of 2 d shapes the aht uses a small accumulator array and the idea of a flexible iterative coarse to fine accumulation and search strategy to identify significant peaks in the hough parameter spaces the method is substantially superior to the standard ht implementation in both storage and computational requirements in this correspondence we illustrate the ideas of the aht by tackling the problem of identifying linear and circular segments in images by searching for clusters of evidence in 2 d parameter spaces we show that the method is robust to the addition of extraneous noise and can be used to analyze complex images containing more than one shape

spectral clustering on multiple manifolds spectral clustering sc is a large family of grouping methods that partition data using eigenvectors of an affinity matrix derived from the data though sc methods have been successfully applied to a large number of challenging clustering scenarios it is noteworthy that they will fail when there are significant intersections among different clusters in this paper based on the analysis that sc methods are able to work well when the affinity values of the points belonging to different clusters are relatively low we propose a new method called spectral multi manifold clustering smmc which is able to handle intersections in our model the data are assumed to lie on or close to multiple smooth low dimensional manifolds where some data manifolds are separated but some are intersecting then local geometric information of the sampled data is incorporated to construct a suitable affinity matrix finally spectral method is applied to this affinity matrix to group the data extensive experiments on synthetic as well as real datasets demonstrate the promising performance of smmc

colored noise and a characteristic level crossing problem in this paper we discuss the activation of a dynamical system of one spatial variable forced by an external gaussian stationary colored noise the model is an analog of the smoluchowski equation when external fluctuations are present we assume that the underlying deterministic dynamics are derived from a potential forming a well we enlarge the state space by including the noise variable and then use half range expansion techniques singular perturbations and matched asymptotics to solve the boundary value problem for the steady state probability density function in the asymptotic limit of small correlation time and over the entire regime of noise strength we find a uniformly valid expression for the activation rate to the top of the potential barrier a characteristic point for the unforced dynamical system we show the important effects of the boundary behavior of the process on its activation rate if the intensity of the noise is small compared to the autocorrelation of the noise then the effective activation energy is modified by the noise autocorrelation we consider an example of the activation process in a double well potential and compare our uniformly valid results with results reported in the literature

principal components null space analysis for image and video classification we present a new classification algorithm principal component null space analysis pcnsa which is designed for classification problems like object recognition where different classes have unequal and nonwhite noise covariance matrices pcnsa first obtains a principal components subspace pca space for the entire data in this pca space it finds for each class i an m sub i dimensional subspace along which the class intraclass variance is the smallest we call this subspace an approximate null space ans since the lowest variance is usually much smaller than the highest a query is classified into class i if its distance from the class mean in the class ans is a minimum we derive upper bounds on classification error probability of pcnsa and use these expressions to compare classification performance of pcnsa with that of subspace linear discriminant analysis slda we propose a practical modification of pcnsa called progressive pcnsa that also detects new untrained classes finally we provide an experimental comparison of pcnsa and progressive pcnsa with slda and pca and also with other classification algorithms linear svms kernel pca kernel discriminant analysis and kernel slda for object recognition and face recognition under large pose expression variation we also show applications of pcnsa to two classification problems in video an action retrieval problem and abnormal activity detection

poisson co homology of polynomial poisson algebras in dimension four sklyanin s case abstract in this paper we compute the poisson co homology of a polynomial poisson structure given by two casimir polynomial functions which define a complete intersection with an isolated singularity

the density profile for the klein kramers equation near an absorbing wall we derive asymptotic series for the expansion coefficients of a function in terms of the pagani functions which occur in the boundary layer solutions of the klein kramers equation the results enable us to determine the density profile in the stationary solution of this equation near an absorbing wall from the numerically determined velocity distribution at the wall with an accuracy of about 2 we also obtain information about the analytic behavior of the density profile this profile increases near the wall with the square root of the distance to the wall finally the asymptotic analysis leads to an understanding of the slow convergence of variational approximations to the solution of the absorbing wall problem and of the exponents that occur when one studies the variational approximations to various quantities of interest as functions of the number of terms in the variational ansatz this is used to obtain a better variational estimate for the density at the wall

on the estimation of a covariance matrix in designing parzen classifiers the design of the parzen classifiers requires careful attention to the window width as well as kernel covariance matrices although a considerable amount of effort has been devoted to the selection of the window width the problem of estimating kernel covariance matrices has received little attention in the past in this paper we discuss the kernel covariance estimators for the design of the parzen classifiers we compare the performance of the parzen classifiers based on several kernel covariance estimators such as the toeplitz ness s and orthogonal expansion estimators on three artificial data sets from experimental results we recommend the use of the toeplitz estimator particularly in high dimensional spaces

use of fourier and karhunen loeve decomposition for fast pattern matching with a large set of templates we present a fast pattern matching algorithm with a large set of templates the algorithm is based on the typical template matching speeded up by the dual decomposition the fourier transform and the karhunen loeve transform the proposed algorithm is appropriate for the search of an object with unknown distortion within a short period patterns with different distortion differ slightly from each other and are highly correlated the image vector subspace required for effective representation can be defined by a small number of eigenvectors derived by the karhunen loeve transform a vector subspace spanned by the eigenvectors is generated and any image vector in the subspace is considered as a pattern to be recognized the pattern matching of objects with unknown distortion is formulated as the process to extract the portion of the input image find the pattern most similar to the extracted portion in the subspace compute normalized correlation between them at each location in the input image and find the location with the best score searching for objects with unknown distortion requires vast computation the formulation above makes it possible to decompose highly correlated reference images into eigenvectors as well as to decompose images in frequency domain and to speed up the process significantly

the analytic structure of the stationary kinetic boundary layer for brownian particles near an absorbing wall a method for solving kinetic boundary layer problems for the 1d klein kramers equation the kinetic equation for brownian particles was proposed a few years ago by marshall and watson 1987 for the special case of particles moving under the influence of a constant or zero external force in this paper the authors apply this method to a number of classical stationary boundary layer problems for completely or partially absorbing walls these are the milne problem in which a current flows towards the wall from infinity and the albedo problem in which particles are injected into the system at the wall with a prescribed velocity distribution the solutions are known to be non analytic at the wall for zero normal velocity the authors pay particular attention to the nature of this singularity for several special cases the results are compared with results obtained by approximate methods

the convex hull for a random acceleration process in two dimensions we compute exactly the mean perimeter l t and the mean area a t of the convex hull of a random acceleration process of duration t in two dimensions we use an exact mapping that relates via cauchy s formulae the computation of the perimeter and the area of the convex hull of an arbitrary two dimensional stochastic process x t y t to the computation of the extreme value statistics of the associated one dimensional component process x t the latter can be computed exactly for the one dimensional random acceleration process even though the process is non markovian physically our results are relevant in describing the average shape of a semi flexible ideal polymer chain in two dimensions

exact solution of the reaction diffusion problem for a particle generating band on a surface by riemann hilbert matching the existence of a class of exact solutions for diffusion reaction problems is demonstrated for the example of a band the method may be applied to any particle source which can be analytically mapped to a band or a disc situated on an infinite plane a simple asymptotic expression for the experimentally measurable collector efficiency current current response function has been found

semiflexible polymer in the half plane and statistics of the integral of a brownian curve a continuum model of a polymer with non zero bending energy fluctuating without overhangs in the half plane is considered the exact partition function is obtained from the marshall watson solution of the klein kramers equation for brownian motion in the half space the partition function contains information on probabilities associated with the integral of a brownian curve and reproduces sinai s t 5 4 result for the asymptotic first passage time density the t 5 2 dependence of a different passage probability implies a first order polymer adsorption transition for short range pinning potentials

algebras associated to elliptic curves this paper completes the classification of artin schelter regular algebras of global dimension three for algebras generated by elements of degree one this has been achieved by artin schelter tate and van den bergh we are therefore concerned with algebras which are not generated in degree one we show that there exist some exceptional algebras each of which has geometric data consisting of an elliptic curve together with an automorphism just as in the case where the algebras are assumed to be generated in degree one in particular we study the elliptic algebras a a and a a where a p2 which were first defined in an earlier paper we omit a set s p2 consisting of 11 specified points where the algebras a a become too degenerate to be regular theorem let a represent a a or a a where a p2 s then a is an artin schelter regular algebra of global dimension three moreover a is a noetherian domain with the same hilbert series as the appropriately graded commutative polynomial ring in three variables this combined with our earlier results completes the classification

the growth of small droplets from a gas mixture results and estimates for very small radii abstract to test and to complement the moment method for solving kinetic equations in the space outside of a sphere that absorbs one component out of a gas mixture we develop a solution method especially suitable for radii smaller or not much larger than a mean free path the method is used to calculate the stationary current towards an absorbing sphere suspended in a gas mixture that is in equilibrium far from the sphere for the simplest kinetic equation a variant of the linear bgk equation and for radii below two mean free paths this procedure gives the current towards the sphere to within an accuracy of at most a few tenths of a percent for radii around two mean free paths the new results agree with those of a variant of the moment method for lower radii discrepancies of up to a few percent are found our new method does not work for the klein kramers equation a kinetic equation for brownian particles for the latter equation we obtain an estimate for the current at very low radii from the free space solution by uhlenbeck and ornstein its numerical evaluation strongly indicates an analytic behavior for small radii different from the one found in the bgk case

ray solution of a singularly perturbed elliptic pde with applications to communications networks we analyze a second order linear elliptic pde with mixed boundary conditions this problem arose as a limiting case of a markov modulated queueing model for data handling switches in communications networks we use singular perturbation methods to analyze the problem in particular we use the ray method to solve the pde in the limit where convection dominates diffusion we show that there are both interior and boundary caustics as well as a cusp point where two caustics meet an internal layer boundary layers and a corner layer our analysis leads to approximate formulas for the queue length or buffer content distribution at the switch

the kinetic boundary layer for the linearized boltzmann equation around an absorbing sphere we construct approximate solutions of the linearized boltzmann equation for a gas outside of a completely absorbing sphere a simple model for a liquid droplet growing in a supersaturated vapor the solutions are linear combinations of two chapman enskog type solutions which carry heat and particle currents and boundary layer eigenfunctions that decay with increasing distance from the sphere on a distance of the order of a mean free path to construct the boundary layer eigenfunctions and the linear combination that satisfies the boundary condition at the sphere we expand the solution in burnett functions and truncate the resulting system of equations for the expansion coefficients for one particular truncation prescription which generalizes grad s 13 moment scheme good initial convergence with increasing order of truncation is obtained for both moderately small and large radii of the sphere the results for small radii extrapolate smoothly toward the known limit of zero radius we present results for the reaction rate the particle current arriving at the sphere divided by the density at infinity and for the density and temperature profiles in the boundary layer the explicit calculations are carried out for maxwell molecules but the method appears to be suitable for more general intermolecular potentials

half range solutions of indefinite sturm liouville problems abstract for diffusion equations of indefinite sturm liouville type we develop two equivalent methods of constructing explicit representations for the solutions the first method is based on an eigenfunction expansion whereas the second uses a wiener hopf type integral equation and factorization some illustrative examples are worked out

first passage times for the uhlenbeck ornstein process a numerical solution is obtained for the laplace transformed backward kramers equation from which the mean first passage time may be obtained the main difficulties are associated with a the parabolic nature of the time development operator and b the existence of a double structure in the solution near the absorbing barrier both of these difficulties are resolved by computational methods derived from boundary layer theory the reliability of the method is assessed by comparing its results with an earlier analytic solution for the case of a uniform force field the authors also present the results for a harmonic force field for which no analytic solution is yet known

the distribution of exit times for weakly colored noise we analyze the exit time first passage time problem for the ornstein uhlenbeck model of brownian motion specifically consider the positionx t of a particle whose velocity is an ornstein uhlenbeck process with amplitude and correlation time e2 dx dt sigma z varepsilon dz dt z varepsilon 2 2 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 xi t varepsilon where t is gaussian white noise let the exit timetex be the first time the particle escapes an interval a t by directly solving the fokker planck equation in brief after taking a laplace transform we use singular perturbation methods to reduce the fokker planck equation to a boundary layer problem this boundary layer problem turns out to be a half range expansion problem which we solve via complex variable techniques this yields the laplace transform off t to within a transcendentally smallo e a e e b e error we then obtainf t by inverting the transform order by order in e in particular by lettingb we obtain the solution to wang and uhlenbeck s unsolved problem b througho e2 2 a1 this solution is f t erf left frac a varepsilon sigma alpha varepsilon sigma z 0 2 sigma t varepsilon 2 kappa 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 right for frac t varepsilon 2 1 andf 1 otherwise here 1 2 1 4603 where is the riemann zeta function and the constant is 0 22749

the kinetic boundary layer for the klein kramers equation a new numerical approach we explore a numerical technique for determining the structure of the kinetic boundary layer of the klein kramers equation for noninteracting brownian particles in a fluid near a wall that absorbs the brownian particles the equation is of interest in the theory of diffusion controlled reactions and of the coagulation of colloidal suspensions by numerical simulation of the langevin equation equivalent to the klein kramers equation we amass statistics of the velocities at the first return to the wall and of the return times for particles injected into the fluid at the wall with given velocities the data can be used to construct the solutions of the standard problems at an absorbing wall the milne and the albedo problem we confirm and extend earlier results by burschka and titulaer obtained by a variational method vexed by the slow convergence of the underlying eigenfunction expansion we briefly discuss some further boundary layer problems that can be attacked by exploiting the results reported here

half range completeness for the fokker planck equation half range completeness theorems are proved for eigenfunctions associated to the one dimensional fokker planck equation in a semi infinite medium existence and uniqueness results for perfectly absorbing partially absorbing and purely specularly reflecting boundary conditions are deduced for the stationary and time dependent problems similar results are obtained for a slab geometry

efficient calculation of primary images from a set of images a set of images is modeled as a stochastic process and karhunen loeve expansion is applied to extract the feature images although the size of the correlation matrix for such a stochastic process is very large we show the way to calculate the eigenvectors when the rank of the correlation matrix is not large we also propose an iterative algorithm to calculate the eigenvectors which save computation time andc omputer storage requirements this iterative algorithm gains its efficiency from the fact that only a significant set of eigenvectors are retained at any stage of iteration simulation results are also presented to verify these methods

half range completeness for the fokker planck equation with an external force this paper deals with a generalization of a classical result obtained by r beals and v protopopescu for the fokker planck equations to the case in which a constant external force is present

the kinetic boundary layer for the fokker planck equation with absorbing boundary the fokker planck equation for the distribution of position and velocity of a brownian particle is a particularly simple linear transport equation its normal solutions and an apparently complete set of stationary boundary layer solutions can be determined explicitly by a numerical algorithm we select linear combinations of them that approximately fulfill the boundary condition for a completely absorbing plane wall and that approach a linearly increasing position space density far from the wall various aspects of these approximate solutions are discussed in particular we find that the extrapolated asymptotic density reaches zero at a distance xm beyond the wall we find xm 1 46 in units of the velocity persistence length of the brownian particle this study was motivated by certain problems in the theory of diffusion controlled reactions and the results might be used to test approximate theories employed in that field

time at which the maximum of a random acceleration process is reached we study the random acceleration model which is perhaps one of the simplest yet nontrivial non markov stochastic processes and is key to many applications for this non markov process we present exact analytical results for the probability density p tm t of the time tm at which the process reaches its maximum within a fixed time interval 0 t we study two different boundary conditions which correspond to the process representing respectively i the integral of a brownian bridge and ii the integral of a free brownian motion our analytical results are also verified by numerical simulations

the structure of the stationary kinetic boundary layer for the linear bgk equation recently very accurate numerical methods for solving kinetic boundary layer problems for linear kinetic equations have been developed to test such methods detailed information about exactly solvable models can be very helpful to this purpose the authors present results for the stationary 1d linear bgk equation obtained by the singular eigenfunction method that extend results already known in the literature both in scope and in precision special attention is paid to the nature of the singularity in the distribution function for small velocities near the boundary this singularity cannot be reproduced exactly by the numerical methods available explicit expressions are presented for the milne problem and for the albedo problem with input particles of a single velocity they compare the results with those of a recently developed variant of the two stream moment method they find excellent agreement except close to the singularity for many quantities of interest the accuracy obtainable by the novel two stream moment method cannot be reproduced with comparable numerical effort by evaluation of the exact solution

analytic solution for a boundary value problem in the theory of brownian motion a closed form formula for the solution of the first boundary value problem for the kolmogorov equation is found by constructing explicitly its green function

poisson co homology and isolated singularities abstract to each polynomial f x y z is associated a poisson structure on f 3 a surface and a poisson structure on this surface when is weight homogeneous with an isolated singularity we determine the poisson cohomology and homology of the two poisson varieties obtained

a singular perturbation approach to kramers diffusion problem we consider kramers diffusion problem which seeks to calculate the rate of escape of a particle from one potential well over a barrier to another presumably deeper and therefore more stable well though kramers introduced the problem as a model for chemical reactions it applies to numerous rate processes including atomic migration and ionic conductivity in crystals and transitions due to noise between stable states of dynamical systems with multi stable states to name but a few we propose a new approach not based on a fokker planck equation but rather on the solution of a singularly perturbed boundary value problem specifically we relate the rate of escape to the first passage time from the domain of attraction of the stable point corresponding to the first well the first passage time is then characterized via the ito calculus as a solution of an elliptic partial differential equation of singular perturbation type finally this equation is solved asymptotically by methods previously develop

homologie des invariants d une algebre de weyl sous l action d un groupe fini resume soit g un sous groupe fini de sp 2 n c operant par automorphismes dans l algebre de weyl a n c nous calculons les groupes d homologie et de cohomologie de hochschild de l algebre d invariants a n c g let g be a finite subgroup of sp 2 n c acting by automorphisms in the weyl algebra a n c we compute the hochschild homology and cohomology groups of the invariant algebra a n c g

full and partial range eigenfunction expansions for sturm liouville problems with indefinite weights this article is concerned with eigenvalue problems of the form au tu in a hilbert space h where ais a selfadjoint positive operator generated by a second order sturm liouville differential expression and t a selfadjoint indefinite multiplicative operator which is one to one emphasis is on the full range and partial range expansionproperties of the eigenfunctions

partial range completeness and existence of solutions to two way diffusion equations separating variables to solve a two way diffusion equation leads to a nonstandard eigenvalue problem in one variable it is shown that the eigenfunctions having negative eigenvalues are complete on the part of the domain where initial conditions are imposed while those with positive eigenvalues are complete where final conditions are imposed the corresponding exponentially growing and exponentially decaying solutions may be used to expand arbitrary solutions on semi infinite intervals in the time variable a natural iterative procedure for obtaining solutions on finite intervals is shown to converge in some cases a linearly growing solution must also be taken into account

transparent boundary conditions as dissipative subgrid closures for the spectral representation of scalar advection by shear flows we consider the evolution of a passive scalar in a shear flow in its representation as a system of lattice differential equations in wave number space when the velocity field has small support the interaction in wave number space is local and can be studied in terms of dispersive linear lattice waves we close the restriction of the system to a finite set of wave numbers by implementing transparent boundary conditions for lattice waves this closure is studied numerically in terms of energy dissipation rate and energy spectrum both for a time independent velocity field and for a time dependent synthetic velocity field whose fourier coefficients follow independent ornstein uhlenbeck stochastic processes

reactive boundary conditions as limits of interaction potentials for brownian and langevin dynamics a popular approach to modeling bimolecular reactions between diffusing molecules is through the use of reactive boundary conditions one common model is the smoluchowski partial adsorption condition which uses a robin boundary condition in the separation coordinate between two possible reactants this boundary condition can be interpreted as an idealization of a reactive interaction potential model in which a potential barrier must be surmounted before reactions can occur in this work we show how the reactive boundary condition arises as the limit of an interaction potential encoding a steep barrier within a shrinking region in the particle separation where molecules react instantly upon reaching the peak of the barrier the limiting boundary condition is derived by the method of matched asymptotic expansions and is shown to depend critically on the relative rate of increase of the barrier height as the width of the potential is decreased limiting boundary conditions for the same interaction potential in both the overdamped fokker planck equation brownian dynamics and the kramers equation langevin dynamics are investigated it is shown that different scalings are required in the two models to recover reactive boundary conditions that are consistent in the high friction limit where the kramers equation solution converges to the solution of the fokker planck equation

symplectic reflection algebras calogero moser space and deformed harish chandra homomorphism to any finite group sp v of automorphisms of a symplectic vector space v we associate a new multi parameter deformation h of the algebra v smash product of with the polynomial algebra on v the parameter runs over points of r where r number of conjugacy classes of symplectic reflections in the algebra h called a symplectic reflection algebra is related to the coordinate ring of a poisson deformation of the quotient singularity v this leads to a symplectic analogue of mckay correspondence which is most complete in case of wreath products if is the weyl group of a root system in a vector space and v then the algebras h are certain rational degenerations of the double affine hecke algebra introduced earlier by cherednik

algebraic index theorem we prove the atiyah singer index theorem where the algebra of pseudodifferential operators is replaced by an arbitrary deformation quantization of the algebra of functions on a symplectic manifold

an accurate two stream moment method for kinetic boundary layer problems of linear kinetic equations the authors develop a method for solving 1d kinetic boundary layer problems for linear equations that uses separate hermite moment expansions in the velocity variable for particles moving towards and away from a plane wall this so called two stream method is tested for two especially simple kinetic equations the linear bgk equation and the klein kramers equation the kinetic equation for brownian particles for both of these equations extensive exact information is available for two simple boundary layer problems the milne and the albedo problem a comparison of these exact results with those obtained with this version of the two stream moment method shows that the accuracy obtainable by this method exceeds that of earlier methods by several orders of magnitude in particular the method allows them to obtain accurate results not only for the moments of the distribution function but also for the distribution function itself

phase space diffusion in a system with a partially permeable wall a new boundary condition for a diffusive system with a partially permeable wall described by the klein kramers equation is proposed the green functions for such a system are given and a generalization to the system with a series of walls is briefly discussed it is shown that the well known boundary condition for the case of a partially absorbing wall which gives the so called radiation boundary condition for the diffusion equation cannot be used for the system under consideration

on the wang uhlenbeck problem in discrete velocity space the arguably simplest model for dynamics in phase space is the one where the velocity can jump between only two discrete values v with rate constant k for this model which is the continuous space version of a persistent random walk analytic expressions are found for the first passage time distributions to the origin since the evolution equation of this model can be regarded as the two state finite difference approximation in velocity space of the kramers klein equation this work constitutes a solution of the simplest version of the wang uhlenbeck problem formal solution in laplace space of generalizations where the velocity can assume an arbitrary number of discrete states that mimic the maxwell distribution is also provided

a drop of ink falls from my pen it comes to earth i know not when the authors obtain for a brownian particle in a uniform force field the mean and asymptotic first passage times as functions of the particle s initial position and velocity with the recurrence times given as a special case they discuss the region of phase space for which the diffusion model of brownian motion provides an adequate approximation and conclude that there is no possibility of obtaining the recurrence times within that model they find that the nature of the boundary value problem is profoundly altered when the motion is treated as a process in phase rather than configuration space because the time development operator is then parabolic rather than elliptic they argue that such a change in the treatment of brownian motion places it within the sphere of transport theory rather than diffusion theory and that consequently results such as ours have relevance to the study of phenomena such as radiative transfer and neutron transport

flux to a trap the flux of particles to a single trap is investigated for two systems 1 particles in 3d space which jump a fixed step lengthl the rayleigh flight and are adsorbed by a spherical surface and 2 particles on a lattice jumping to nearest neighbor sites with a single adsorbing site initially the particles are uniformly distributed outside the traps when the jump length goes to zero both processes go over to regular diffusion and the first case yields the diffusive flux to a sphere as solved by smoluchowski for nonzero step length the flux for large times is given by a modified form of smoluchowski s result with the effective radius replaced byr cl wherec 0 29795219 andcl is the milne extrapolation length for this problem for the second problem a similar expression for the flux is found with the effective trap radius a function of the lattice sc bcc fcc being considered

on the fokker planck equation with force term it has been suggested that the half range sturm liouville problem produced in the study of the one dimensional fokker planck equation with constant force term can be replaced by two definite sturm liouville problems the authors show that this appears not to be the case they also draw attention to some corrections to their earlier papers

overview of theoretical models for reaction rates there is enormous recent interest in the development of models for rate processes because rates are an almost universal characterization in the physical and biological sciences in this paper we provide an introduction to several of the problems to be discussed in greater depth by other speakers at a symposium held at the national institutes of health on may 6 8 1985 this review will focus on 1 the smoluchowski model for reaction rates together with its extension by onsager 2 first passage time formalism for discrete and continuous master equations and fokker planck equations 3 the kramers model and its extensions 4 diffusion in the presence of trapping centers

half range expansion analysis for langewin dynamics in the high friction limit with a singular absorbing boundary condition noncharacteristic case we consider the dynamics of a brownian particle given by the langevin equation in a strip under the effects of a deterministic force the trajectories of particles originate at a source whose spatial location in the phase space coincides with the location of adsorbing boundaries this leads to singular behavior of trajectories in the high friction limit we use the half range expansion technique and systematic asymptotics to solve a boundary value problem for the fokker planck operator and to calculate the steady state transition probability density the mean time to absorption and the distribution of exit points we do not make assumptions about other parameters in the problem except that they areo 1 relative to the friction coefficient we calculate explicitly the correct location of the milne type extrapolation for absorbing boundary conditions for the smoluchowski approximation to the langevin equation

a robust competitive clustering algorithm with applications in computer vision this paper addresses three major issues associated with conventional partitional clustering namely sensitivity to initialization difficulty in determining the number of clusters and sensitivity to noise and outliers the proposed robust competitive agglomeration rca algorithm starts with a large number of clusters to reduce the sensitivity to initialization and determines the actual number of clusters by a process of competitive agglomeration noise immunity is achieved by incorporating concepts from robust statistics into the algorithm rca assigns two different sets of weights for each data point the first set of constrained weights represents degrees of sharing and is used to create a competitive environment and to generate a fuzzy partition of the data set the second set corresponds to robust weights and is used to obtain robust estimates of the cluster prototypes by choosing an appropriate distance measure in the objective function rca can be used to find an unknown number of clusters of various shapes in noisy data sets as well as to fit an unknown number of parametric models simultaneously several examples such as clustering mixture decomposition line plane fitting segmentation of range images and estimation of motion parameters of multiple objects are shown

conjugate dynamical systems classical analogue of the quantum energy translation an aspect of quantum mechanics that has not been fully understood is the energy shift generated by the time operator in this study we introduce the use of the eigensurfaces of dynamical variables and commutators in classical mechanics to study the classical analogue of the quantum translation of energy we determine that there is a conjugate dynamical system that is conjugate to hamilton s equations of motion and then we generate the analogue of the time operator and use it in the translation of points along the energy direction i e the classical analogue of the pauli theorem the theory is illustrated with a nonlinear oscillator model

hydrodynamic singular regimes in 1 1 kinetic models and spectral numerical methods classical results from spectral theory of stationary linear kinetic equations are applied to efficiently approximate two physically relevant weakly nonlinear kinetic models a model of chemotaxis involving a biased velocity redistribution integral term and a vlasov fokker planck vfp system both are coupled to an attractive elliptic equation producing corresponding mean field potentials spectral de compositions of stationary kinetic distributions are recalled based on a variation of case s elementary solutions for the first model and on a sturm liouville eigenvalue problem for the second one well balanced godunov schemes with strong stability properties are deduced moreover in the stiff hydrodynamical scaling an hybridized algorithm is set up for which asymptotic preserving properties can be established under mild restrictions on the computational grid several numerical validations are displayed including the consistency of the vfp model with burgers hopf dynamics on the velocity field after blowup of the macroscopic density into dirac masses

a numerical approach to study the kramers equation for finite geometries boundary conditions and potential fields the kramers equation for the phase space function which models the dynamics of an underdamped brownian particle is the subject of our study numerical solutions of this equation for natural boundaries unconfined geometries have been well reported in the literature but not much has been done on the kramers equation for finite confining geometries which require a set of additional constraints imposed on the phase space function at physical boundaries in this paper we present numerical solutions for the kramers equation with a variety of potential fields namely constant linear harmonic and periodic in the presence of fully absorbing and fully reflecting boundary conditions bcs the choice of the numerical method and its implementation take into consideration the type of bcs in order to avoid the use of ghost points or artificial conditions we study and assess the conditions under which the numerical method converges various aspects of the solutions for the phase space function are presented with figures and discussed in detail