stochastic mixing model with power law decay of variance a stochastic mixing model based on the law of large numbers is presented that describes the decay of the variance of a conserved scalar in decaying turbulence as a power law sigma2 c proportional t alpha a general lagrangian mixing process is modeled by a stochastic difference equation where the mixing frequency and the ambient concentration are random processes the mixing parameter lambda is introduced as a coefficient in the mixing frequency in order to account for initial length scale ratio of the velocity and scalar field and other physical dependencies we derive a nonlinear integral equation for the probability density function pdf of a conserved scalar that describes the relaxation of an arbitrary initial distribution to a delta function numerical studies of this equation are conducted and it is shown that lambda has a distinct influence on the decay rate of the scalar results obtained from the model for the evolution of the pdf are in a good agreement with direct numerical simulation dns data

saddlepoint expansions in finite population problems summary saddlepoint approximations are derived for the sample mean in the case where samples are obtained randomly without replacement from a finite population asymptotic properties are investigated the great accuracy is demonstrated by numerical examples extensions to more general statistics are discussed

relative invariants of 2 simple prehomogeneous vector spaces of type i abstract in the present paper by using equivariant maps we construct explicitly relative invariants of the 2 simple prehomogeneous vector spaces of type i which are classified in t kimura s kasai m inuzuka o yasukura a classification of 2 simple prehomogeneous vector spaces of type i j algebra 114 1988 369 400 this is a continuation of our previous work t kogiso g miyabe m kobayashi t kimura non regular 2 simple prehomogeneous vector spaces of type i and their relative invariants j algebra 251 2002 27 69 t kogiso g miyabe m kobayashi t kimura relative invariants of some 2 simple prehomogeneous vector spaces math comp 72 2003 865 889

boundary behavior of diffusion approximations to markov jump processes we show that diffusion approximations including modified diffusion approximations can be problematic since the proper choice of local boundary conditions if any exist is not obvious for a class of markov processes in one dimension we show that to leading order it is proper to use a diffusion fokker planck approximation to compute mean exit times with a simple absorbing boundary condition however this is only true for the leading term in the asymptotic expansion of the mean exit time higher order correction terms do not in general satisfy simple absorbing boundary conditions in addition the diffusion approximation for the calculation of mean exit times is shown to break down as the initial point approaches the boundary and leads to an increasing relative error by introducing a boundary layer we show how to correct the diffusion approximation to obtain a uniform approximation of the mean exit time we illustrate these considerations with a number of examples including a jump process which leads to kramers diffusion model this example represents an extension to a multivariate process

prehomogeneous spaces associated with complex nilpotent orbits abstract we compute the irreducible components and relative invariants of certain prehomogeneous spaces arising in the theory of nilpotent orbits in complex reductive lie algebras

singular perturbation analysis of boundary value problems for differential difference equations vi small shifts with rapid oscillations this paper continues the authors study of boundary value problems for singularly perturbed linear second order differential difference equations with small shifts this study was initiated in the companion paper siam j appl math 54 1994 pp 249 272 in this paper the study is extended to problems that have solutions that exhibit rapid oscillations restrictions on the sizes of the shifts in terms of the small parameter are found such that generally the shifted terms cannot be replaced with truncated taylor series in particular it is shown that even when the shifts are small relative to the width of an oscillation they can affect the solution to leading order the conclusion is that oscillatory solutions are more sensitive to small delays than are layer solutions it is shown that a suitably modified version of the standard wkb method can be used to obtain leading order oscillatory solutions of these differential difference equations these preliminary studies of differential difference eq

asymptotic theory of large deviations for markov chains a formal asymptotic expansion is constructed for the joint probability density function pdf of a stationary ergodic markov chain xn and its averages yn since the pair xn yn is markovian the joint pdf satisfies a forward kolmogorov equation whose solution is expanded asymptotically for large n an algorithm is proposed for the calculation of the full asymptotic series but only the three leading terms are found explicitly it is found that for small values of the average the asymptotic expansion coincides with the appropriate version of the central limit theorem the ideas and methods are generalized to a large class of averages and to vector valued markov chains

a monte carlo analysis of the two dimensional planar and step models using the metropolis monte carlo scheme the authors have analysed and compared the behaviour of the two dimensional classical planar and step models in particular they analyse the energy specific heat correlation function and susceptibility on a range of finite lattices while both models show similar numerical difficulties it is argued that in the case of the step model a phase transition is unlikely

robustness and approximation of escape times and large deviations estimates for systems with small noise effects for the purposes of estimating escape time from a given set or other statistical properties of systems with small noise effects it is generally assumed in applications that the system noise is white gaussian the gaussian assumption greatly simplifies the computation but is not adequate for many important classes of applications in control and communication theory for example when the noise is small the mean escape time from a set can be quite sensitive to the underlying statistics even though in the study of the effects of the noise over any fixed finite time interval the gaussian approximation might be a good one this paper is concerned with the sensitivity of these statistical quantities to the underlying statistical structure when the noise effects are small and also with the question of when the gaussian assumption makes sense consider a sequence of systems with small noise effects whose statistics converge in some sense to those of a limit system the techniques developed involve approx

new exact exponents for two dimensional self avoiding walks the author establishes using standard coulomb gas methods the values of a new set of geometrical exponents that have been previously conjectured these exponents give for any p the number of configurations of p two dimensional self avoiding walks of the same length l which are attached by their ends

asymptotic behavior of stochastic approximation and large deviations the theory of large deviations is applied to the study of the asymptotic properties of the stochastic approximation algorithms

partially reflected diffusion the radiation reactive or robin boundary condition for the diffusion equation is widely used in chemical and biological applications to express reactive boundaries the underlying trajectories of the diffusing particles are believed to be partially absorbed and partially reflected at the reactive boundary however the relation between the reaction constant in the robin boundary condition and the reflection probability is not well defined in this paper we define the partially reflected process as a limit of the markovian jump process generated by the euler scheme for the underlying ito dynamics with partial boundary reflection trajectories that cross the boundary are terminated with probability p sqrt delta t and otherwise are reflected in a normal or oblique direction we use boundary layer analysis of the corresponding master equation to resolve the nonuniform convergence of the probability density function of the numerical scheme to the solution of the fokker planck equation in a half space wi

a viscosity solution approach to the asymptotic analysis of queueing systems we consider a system of several interconnected queues with a single class of customers and model the state xt of the system as a jump markov process the problem of interest is to estimate the large deviations behavior of the rescaled system xt ext s corresponding to large time and large excursions of the original unscaled system the techniques employed are those of the theory of viscosity solutions to hamilton jacobi equations from the point of view of large deviation theory the interesting new problem here is the treatment of the process when one or more of the queues are nearly empty since an abrupt change in the jump measure occurs from the point of view of viscosity solutions the discontinuity of the jump measure leads to nonlinear boundary conditions on domains with corners for the associated partial differential equations much of the paper is devoted to proving uniqueness of viscosity solutions for these equations and these sections are of independent interest while our use of test functions in proving the uniqueness is an adaptation of the usual technique the construction of the test functions themselves via the legendre transform is new we obtain a representation for the solution of the equation in terms of a nonstandard optimal control problem which suggests the correct integrand in the large deviation rate functional since it is the treatment of the effects due to the boundaries that is novel we devote the majority of the paper to the detailed development of a simple two dimensional system that exhibits all the essential new features however the arguments may be applied to nueueing systems that are considerably more general and we attempt to indicate this generality as well

an asymptotic theory of large deviations for markov jump processes we present new asymptotic methods for the analysis of markov jump processes the methods based on the wkb and other singular perturbation techniques are applied directly to the kolmogorov equations and not to approximate equations that come e g from diffusion approximations for time homogeneous processes we construct approximations to the stationary density function and the mean first passage time from a given domain examples involving a random walk and a problem in queueing theory are presented to illustrate our methods for a class of time inhomogeneous processes we construct long time approximations to the transition probability density function and the probability of large deviations from a stable state the law of large numbers is obtained as a special case

asymptotic expansions for sums of weakly dependent random vectors it is shown that formal edgeworth expansions are valid for sums of weakly dependent random vectors the error of approximation has ordero n s 2 2 if the strong mixing coefficients in iii are decreasing at an exponential rate the above conditions can easily be checked and are often satisfied when the sequence of random vectors is a gaussian or a markov or an autoregressive process explicit formulas are given for the distribution of finite fourier transforms of a strictly stationary time series

asymptotically efficient adaptive choice of control laws incontrolled markov chains we consider a controlled markov chain on a general state space whose transition probabilities are parameterized by an unknown parameter belonging to a compact metric space there is a one step reward associated with each pair of control and the following state of the process given a finite set of stationary control laws under each of which the markov chain is uniformly recurrent an optimal control law is this set is one that maximizes the long run average reward in ignorance of the parameter value we construct an adaptive control rule which uses the optimal control law s at a relative frequency of 1 o n 1 log n and show that this relative frequency gives an asymptotically optimal balance between the control objective and the amount of information needed to learn about the unknown parameter the basic idea underlying this construction is to introduce suitable uncertainty adjustments via sequential testing theory into the certainty equivalence rule thus resolving the apparent dilemma between control and information

surface critical exponents for models of polymer collapse and adsorption the universality of the theta and theta points the surface critical exponents are estimated both in ordinary and in special regimes from exact series expansions of up to 28 terms for a model of the theta point they are found to be in agreement with those derived for the theta point confirming the conjecture that the theta and theta points lie in the same universality class some essential features of the phase diagram for a self interacting self avoiding polymer in solution are also calculated to a higher degree of accuracy than previously obtained

controlled stochastic model of a communication system with multiple sources a stochastic model of buffering in a data communication system is considered with source and sink transmission parameters depending on the number of active sources for models in this control setting the authors analyze an effective numerical method for evaluating the equilibrium distribution of buffer content the theoretical basis of the method is established first then it is shown that the method has the same complexity in terms of the total number of sources as known analytical methods for the model with constant parameters asymptotics for tail probabilities at high buffer levels and under heavy load are also derived and the complexity of their computation is compared with that of evaluating explicit formulas in comparison to earlier results this approach reduces the complexity of computing the probability of overflow and its asymptotic estimates the speed up stems from the application of interpolation schemes

saddlepoint expansions for sums of markov dependent variables on a continuous state space based on the conjugate kernel studied in iscoe et al 1985 we derive saddlepoint expansions for either the density or distribution function of a sumf x1 f x n where thex i s constitute a markov chain the chain is assumed to satisfy a strong recurrence condition which makes the results here very similar to the classical results for i i d variables in particular we establish also conditions under which the expansions hold uniformly over the range of the saddlepoint expansions are also derived for sums of the formf x1 x0 f x2 x1 f x n xn 1 although the uniformity result just mentioned does not generalize

invariants nilpotent orbits and prehomogeneous vector spaces provisionally in the introduction we call such a triple g o ad g i i 0 a prehomogeneous vector space of vinberg type abbreviated pv of v type the purpose of this paper is to construct relative invariants of pv of v type in a unified way let us explain the way of construction of invariants we may assume that the original graded lie algebra g is zh graded i e g ie 2l g i we may also assume that i 2 without loss of generality cf 1 2 4 let p v be a representation of g and

asymptotic analysis of a state dependent m g 1 queueing system we present new asymptotic methods for the analysis of queueing systems these methods are applied to a state dependent m g 1 queue we formulate problems for and compute approximations to i the stationary density of the unfinished work ii the mean length of time until the end of a busy period iii the mean length of a busy period and iv the mean time until the unfinished work reaches or exceeds a specified capacity the methods are applied to the full kolmogorov equations scaled so that the arrival rate is rapid and the mean service is small thus we do not truncate equations as in diffusion approximations for state independent m g 1 queues our results are shown to agree with the known exact solutions we include comparisons both analytic and numerical between our results and those obtained from diffusion approximations

on the performance of state dependent single server queues new asymptotic methods for the analysis of queueing systems are introduced and applied to state dependent m g 1 queues the methods are used to compute approximations to the stationary density of the queue length the mean length of a busy period the mean number of customers served during a busy period as well as other quantities of interest we obtain results that are superior to those obtained from diffusion approximations in that they are uniformly valid for all values of the traffic intensity while diffusion approximations are adequate only when this quantity is close to one when specialized to state independent queues our approximations are shown to agree with the asymptotic expansions of known exact results finally we show that the behavior of the state dependent systems is markedly different from that of the corresponding state independent systems

critical properties of the izergin korepin and solvable o n models and their related quantum spin chains the central charge of the izergin korepin model the corresponding quantum spin chain and the o n model is calculated analytically via the bethe ansatz the calculation extends a technique recently developed for the zamolodchikov fateev model in addition critical exponents and the central charge for these models are obtained from numerical solutions of bethe ansatz equations for finite systems as a physical application the authors find the exponents v 12 23 and gamma 53 46 for the theta transition of polymers in two dimensions

the dimensions of spaces of siegel cusp forms of general degree abstract in this paper we give a dimension formula for spaces of siegel cusp forms of general degree with respect to neat arithmetic subgroups the formula was conjectured before by several researchers cf 40 43 45 75 the dimensions are expressed by special values of shintani zeta functions for spaces of symmetric matrices at non positive integers this formula was given by shintani for only a small part of the geometric side of the trace formula see 64 to be precise it is the contribution of unipotent elements corresponding to the partitions 2 j 1 2 n 2 j where n denotes the degree and 0 j n hence our work is to show that all the other contributions vanish therefore one finds that shintani s formula means the dimension itself combining our formula and an explicit formula of the shintani zeta functions which was discovered by ibukiyama and saito cf 44 42 61 we can derive an explicit dimension formula for the principal congruence subgroups of level greater than two cf 43 45 in this explicit dimension formula the dimensions are described by degree n weight k level n and the bernoulli numbers b m

boundary critical behavior of d 2 self avoiding walks on correlated and uncorrelated vacancies in this paper we present exact results for the critical exponents of interacting self avoiding walks with ends at a linear boundary effective interactions are mediated by vacancies correlated and uncorrelated on the dual lattice by choosing different boundary conditions several ordinary and special regimes can be described in terms of clusters geometry and of critical and lowtemperature properties of the mathcal o n 1 model in particular the problem of boundary exponents at the point is fully solved and implications for point universality are discussed the surface crossover exponent at the special transition of noninteracting self avoiding walks is also interpreted in terms of percolation dimensions

series analysis of the two dimensional step model an n fit series analysis of the susceptibility of the two dimensional step model is carried out the work follows a similar analysis of ferer and velgakis of the planar and xy models due to the nature of the interaction function it is expected that a vortex inducted transition is not possible for the step model this is confirmed by the n fit analysis

markov additive processes i eigenvalue properties and limit theorems on etudie d un point de vue general les proprietes d existence et de regularite des valeurs propres et des fonctions propres de l operateur de feynman kac

singular perturbation analysis of boundary value problems for differential difference equations v small shifts with layer behavior in this paper an investigation is initiated of boundary value problems for singularly perturbed linear second order differential difference equations with small shifts i e where the second order derivative is multiplied by a small parameter and the shift depends on the small parameter similar boundary value problems are associated with expected first exit times of the membrane potential in models for neurons in particular this paper focuses on problems with solutions that exhibit layer behavior at one or both of the boundaries the analyses of the layer equations using laplace transforms lead to novel results it is shown that the layer behavior can change its character and even be destroyed as the shifts increase but remain small in the companion paper siam j appi math 54 1994 pp 273283 similar boundary value problems with solutions that exhibit rapid oscillations are studied

surface fractal dimension of two dimensional percolation the author determines the surface fractal dimension ds for two dimensional percolation from monte carlo data in finite systems the results agree with the prediction from conformal invariance ds 2 3

construction of irreducible relative invariant of the prehomogeneous vector space sl5 gl4 λ2 c5 c4 abstract we explicitly construct the irreducible relative invariant of the prehomogeneous vector space sl 5 gl 4 2 c 5 c 4 this prehomogeneous vector space has been known as the most difficult case in irreducible regular prehomogeneous vector spaces

diffusion approximations for the analysis of digital phase locked loops corresp recent results for getting diffusion limits of a sequence of suitably scaled stochastic processes are applied to the synchronization problem for a digital phase locked loop dpll for small filter gains and symbol intervals a diffusion gauss markov process approximation is rigorously obtained this approximation yields approximate error variances passage time distributions and correlation properties among other properties for the dpll the problem of tracking a drifting clock is also treated

large deviations theory and efficient simulation of excessive backlogs in a gi gi m queue the problem of using importance sampling to estimate the average time to buffer overflow in a stable gi gi m queue is considered using the notion of busy cycles estimation of the expected time to buffer overflow is reduced to the problem of estimating p sub n p buffer overflow during a cycle where n is the buffer size the probability p sub n is a large deviations probability p sub n vanishes exponentially fast as n to infinity a rigorous analysis of the method is presented it is demonstrated that the exponentially twisted distribution of s parekh and j walrand 1989 has the following strong asymptotic optimality property within the nonparametric class of all gi gi importance sampling simulation distributions as n to infinity the computational cost of the optimal twisted distribution of large deviations theory grows less than exponentially fast and conversely all other gi gi simulation distributions incur a computational cost that grows with strictly positive exponential rate

first order dynamics driven by rapid markovian jumps the effect of rapid random jumps on the behavior of deterministically stable dynamics is considered the random driving force noise is taken to be a state dependent markov jump process with either finite or infinite state space while the jump rate is high o 1 varepsilon the jump size is assumed to be o 1 for a particle that is deterministically confined to a potential well the stationary joint probability density function of the state and noise variables the probability distribution of the exit points and the mean first passage time from the well are computed in contrast to the case of white noise where these quantities depend on a barrier height determined solely by the potential here they depend on an effective barrier height determined by both the potential and the noise process the method used here is to introduce the small parameter varepsilon where 1 varepsilon is a measure of the rapid jump rate and to employ singular perturbation methods to solve the forward and

high temperature series analysis of an o 2 symmetric spin model with discretely valued interaction on 2d lattices an analysis is given of high temperature series expansions for the susceptibility of an o 2 symmetric spin model with discretely valued spin spin interaction on triangular and square lattices this analysis does not yield strong evidence for a phase transition in either case although it is weakly consistent with this possibility the authors also comment on the high temperature series for the free energy and specific heat

gradient stabilized and destabilized invasion percolation abstract the effects of gravity stabilization and destabilization on the slow displacement of a wetting fluid by a non wetting fluid in two dimensional and three dimensional porous media have been investigated experimentally the characteristic features of the resulting displacement patterns can be reproduced quite well by invasion percolation models with a spatial gradient added to the usual random threshold distribution in the case of destabilized displacement the patterns can be described in terms of a string of blobs of size that form a directed walk the internal structure of the blobs is like that of an invasion percolation cluster with trapping in the two dimensional case in the stabilized case the structure is like that of a fractal invasion percolation cluster on short length scales lengths and is uniform on longer lengths the correlation length also describes the maximum hole diameter the invasion front is a self similar fractal on length scales shorter than and flat on longer length scales in both the experiments and the simulations the correlation length is related to the bond number b0 the ratio between buoyancy and capillary forces by b0 1 where is the ordinary percolation correlation length exponent in accord with the theoretical arguments of wilkinson phys rev a 30 1984 520 34 1986 1380

metastability in stochastic dynamics of disordered mean field models we study a class of markov chains that describe reversible stochastic dynamics of a large class of disordered mean field models at low temperatures our main purpose is to give a precise relation between the metastable time scales in the problem to the properties of the rate functions of the corresponding gibbs measures we derive the analog of the wentzell freidlin theory in this case showing that any transition can be decomposed with probability exponentially close to one into a deterministic sequence of admissible transitions for these admissible transitions we give upper and lower bounds on the expected transition times that differ only by a constant factor the distributions of the rescaled transition times are shown to converge to the exponential distribution we exemplify our results in the context of the random field curie weiss model

precise asymptotics of small eigenvalues of reversible diffusions in the metastable regime we investigate the close connection between metastability of the reversible diffusion process x defined by the stochastic differential equation d xt f xt d t 2e d wt e 0 and the spectrum near zero of its generator l f where f d and w denotes brownian motion on d for generic f to each local minimum of f there corresponds a metastable state we prove that the distribution of its rescaled relaxation time converges to the exponential distribution as 0 with optimal and uniform error estimates each metastable state can be viewed as an eigenstate of l with eigenvalue which converges to zero exponentially fast in 1 modulo errors of exponentially small order in 1 this eigenvalue is given as the inverse of the expected metastable relaxation time the eigenstate is highly concentrated in the basin of attraction of the corresponding trap