vector quantization of speech and speech like waveforms an algorithm for the design of vector quantizers that are locally optimum in the sense of minimizing an average quantitative distortion measure is used to design 1 and 2 bit sample vector quantizers for both real sampled speech and a simulated speech like auto regressive random process both weighted and unweighted squared error distortion measures are considered several comparisons are made and discussed based on the average distortions of the vector quantization schemes the results for the simulated speech are compared to mathematical performance bounds from information theory to provide an indication of how nearly globally optimal vector quantization is for such highly correlated sources a comparison of the results for the real speech and the simulated speech provides a quantitative measure of the accuracy of such models and hence of the applicability of information theory bounds and code designs based on probabilistic models the signal to quantization noise ratios of vector quantizers designed to minimize squared error distortion are compared to those of several popular speech waveform coding systems of similar rates

observation of symmetry breaking state selection and sensitivity in a noisy electronic system abstract measurements of the time evolution of the statistical density as a control parameter is swept through a bifurcation point in the presence of additive noise are presented we have used these densities to study branch selectivity which results from a small additive interaction a universal time scale for all such processes is established the results are in good agreement with a recent theory

nonlinear scattering theory at low energy abstract we study the scattering theory of a conservative nonlinear one parameter group of operators on a hilbert space x relative to a group of linear unitary operators under certain hypotheses the scattering operator carries a neighborhood of 0 in x into x the theory is designed to apply to the semilinear schrodinger and klein gordon equations

a pyramid vector quantizer the geometric properties of a memoryless laplacian source are presented and used to establish a source coding theorem motivated by this geometric structure a pyramid vector quantizer pvq is developed for arbitrary vector dimension the pvq is based on the cubic lattice points that lie on the surface of an l dimensional pyramid and has simple encoding and decoding algorithms a product code version of the pvq is developed and generalized to apply to a variety of sources analytical expressions are derived for the pvq mean square error mse and simulation results are presented for pvq encoding of several memoryless sources for large rate and dimension pvq encoding of memoryless laplacian gamma and gaussian sources provides rose improvements of 5 64 8 40 and 2 39 db respectively over the corresponding optimum scalar quantizer although suboptimum in a rate distortion sense because the pvq can encode large dimensional vectors it offers significant reduction in rose distortion compared with the optimum lloyd max scalar quantizer and provides an attractive alternative to currently available vector quantizers

well posedness and existence of bound states for a coupled schrodinger gkdv system abstract we derive some new results concerning the cauchy problem and the existence of bound states for a class of coupled nonlinear schrodinger gkdv systems in particular we obtain the existence of strong global solutions for initial data in the energy space h 1 r h 1 r generalizing previous results obtained in tsutsumi 1993 11 corcho and linares 2007 13 and dias et al submitted for publication 14 for the nonlinear schrodinger kdv system

on the construction of a special class of time inhomogeneous diffusion processes a special class of time inhomogeneous diffusion processes generated starting from gauss markov processes conditioned on the same initial state is considered this class includes many interesting diffusion processes with time dependent infinitesimal drift and variance for which the transition probability density function is explicitly determined moreover closed form results for the first passage time density through suitable time varying boundaries are obtained special cases generated starting from wiener and ornstein uhlenbeck processes are considered and widely discussed

decay of unstable states examination of the scaling theory abstract the behavior of the solution for an exactly solvable bistable model is analyzed the decay processes of unstable states are discussed the results of the scaling theory are tested it is concluded that the scaling theory is not valid for the model even when the potential barrier is very high

a generalized langevin equation for dealing with nonadditive fluctuations a suitable extension of the mori memory function formalism to the non hermitian case allows a multiplicative process to be described by a langevin equation of non markoffian nature this generalized langevin equation is then shown to provide for the variable of interest the same autocorrelation function as the well known theoretical approach developed by kubo the stochastic liouville equation sle theory it is shown furthermore that the present approach does not disregard the influence of the variable of interest on the time evolution of its thermal bath the stochastic process under study can also be described by a fokker planck like equation which results in a gaussian equilibrium distribution for the variable of interest the main flaw of the sle theory that resulting in an uncorrect equilibrium distribution is therefore completely eliminated

a note on the 2d generalized zakharov kuznetsov equation local global and scattering results abstract we consider the generalized two dimensional zakharov kuznetsov equation u t x u x u k 1 0 where k 3 is an integer number for k 8 we prove local well posedness in the l 2 based sobolev spaces h s r 2 where s is greater than the critical scaling index s k 1 2 k for k 3 we also establish a sharp criteria to obtain global h 1 r 2 solutions a nonlinear scattering result in h 1 r 2 is also established assuming the initial data is small and belongs to a suitable lebesgue space

construction of solutions to the subcritical gkdv equations with a given asymptotical behavior we consider the subcritical generalized korteweg de vries equation ut uxx u4 x 0 t x r let rj t x qcj x xj cjt j 1 n be n soliton solutions to this equation denote u t the kdv linear group and let v be in an adequate weighted sobolev space we construct a solution u t to the generalized korteweg de vries equation such that limt u t u t v j 1nrj t h1 0

on the relaxation of fluctuations in the steady state of the stratonovich model a detailed investigation has been carried out of the relaxation of fluctuations in the steady state of the stratonovich model also known as the random growing rate model the autocorrelation time of this system driven by parametric white noise in the linear term has been measured for an electronic circuit model computed by digital simulation and calculated by use of a continued fraction expansion method the results are all consistent with each other and with those obtained from the matrix continued fraction method of jung and risken 1985 provided that explicit account is taken of the ways in which any macroscopic real physical system is bound to differ from the idealisation represented by the original form of the stratonovich model in particular it is necessary to recognise the existence of weak additive noise and a small additive constant the physical origins of some earlier seemingly discrepant calculations and experimental data are discussed

diffusion in a class of exactly solvable non harmonic potentials intrinsic effects induced by non linearities abstract this paper deals with the problem of a particle that diffuses in a potential with a reflecting barrier and has a point of stable equilibrium and a point of unstable equilibrium based on the exact solutions obtained earlier for the fokker planck equation of a class of these models we analyze the behavior of the probability density the mean path and the onset time which determines the transition from unimodal to bimodal probability densities the study is made over different initial positions two of them very close to the unstable point which permits a clear comparison among the subsequent evolutions and the observation of some intrinsic effects induced by non linearities

exact propagator and eigenfunctions for multistable models with arbitrarily prescribed n lowest eigenvalues the author presents a method to construct potentials for schrodinger equations with some prescribed features for which all eigenfunctions and the time dependent propagator can be explicitly calculated the prescribed features can be formulated by choosing arbitrarily the n lowest eigenvalues alternatively one can prescribe some qualitative behaviour for the potential like the number and relative depths of wells and barriers the results can also be applied to the construction of fokker planck models with prescribed properties and explicitly calculable transition probability density the method is based on ideas of supersymmetric quantum mechanics and the theory of solitons that can be traced back to the work of darboux 1894 and crum 1955

a vector quantizer for the laplace source the low complexity nearly optimal vector quantizer vq is a generalization of t r fischer s 1986 pyramid vq and is similar in structure to the unrestricted polar quantizers previously presented for the independent gaussian source an analysis of performance is presented with results for both the product code pyramid vq and the unrestricted version this analysis although asymptotic in nature helps to demonstrate the performance advantages of the vq implementation issues of the vq are discussed nonasymptotic results are considered in particular the author presents an approximate design algorithm for finite bit rate and demonstrates the usefulness of this vq through several example designs with monte carlo simulations of performance for the restricted form the pyramid vq the author provides further implementational information and low dimension analytical results

nonlinear small data scattering for the generalized korteweg de vries equation abstract we study the longtime stability of small solutions to the ivp for the generalized korteweg de vries equation we obtain a lower bound for the degrees of nonlinearity of the perturbation which guarantees that the small solutions of the nonlinear problem behave asymptotically like the solutions of the associated linear problem this behavior allows us to establish a nonlinear scattering result for small perturbations with these degrees for a class of small data we improve the value of this lower bound the new crucial ingrediants in our proofs are the l p decay estimates of the half derivative of the airy kernel

the darboux transformation and solvable double well potential models for schrodinger equations the darboux transformation a method used to transform a schrodinger type equation to a schrodinger equation with a new potential is discussed an exactly solvable double well potential model for the one dimensional schrodinger equation is obtained

on fluctuations and relaxation in systems described by a one dimensional fokker planck equation with a time dependent potential abstract we study the effect of a time variation of a bistable potential on kramers relaxation and on suzuki s fluctuation enhancement for a system described by a one variable fokker planck equation in the limit of a small constant diffusion coefficient we show that the two processes must be described with two different approximations 1 i kramers relaxation can be treated in a quasi adiabatic scheme we then show that for a large range of potential modulation rates the local populations in each well obey adiabatic balance equations the solution of which we discuss in various limits 2 ii suzuki s description of the fluctuation enhancement can be extended to the dynamical case we reformulate the technique set up by ahlers et al to describe the crossing of the bifurcation from a mono to a bistable potential we show that the extended suzuki approximation is valid provided that the crossing velocity is larger than a limit which we estimate

logarithmic pyramid vector quantization design and theoretical analysis vector quantization is an integral part of modern speech and audio codecs this study proposes the logarithmic pyramid vector quantizer lpvq which is a gain shape vector quantizer specifically designed for the quantization of laplace distributed memoryless sources the objective of the study is a theoretical analysis of the lpvq we determine the distortion of the shape quantizer with respect to rate and vector dimension for the quantization of an i i d laplace source furthermore we derive formulas for the quantization signal to noise ratio snr of the shape quantizer and the quantization snr of the lpvq giving reliable results for an effective bit rate per vector coordinate of 2bits and higher we study the snr behavior of the lpvq for infinite dimensions and compare it with the maximal achievable snr according to the rate distortion bound after having proposed a strategy for the allocation of the bit rate for the gain quantization and the quantization of the shape vector respectively we verify the derived formulas by simulations

non linear dynamics of systems coupled with external noise some exact results abstract we solve exactly a class of non linear stochastic differential equations describing a variety of physical and chemical relevant systems coupled to external sources of noise the static and dynamic solutions are valid in the whole space of parameters including the critical points of non equilibrium instabilities critical slowing down is found and an unexpected long time tail evolution is predicted near and at the critical point this last effect is completely due to the coupling with the external noise

exact results for the diffusion in a class of asymmetric bistable potentials we solve the fokker planck equations with drifts deriving from a class of asymmetric nonharmonic potentials which include bistable cases an analytical expression for the probability current over the potential barrier is obtained finally we compare our exact results with those obtained by kramers approximation

decision making in noisy bistable systems with time dependent asymmetry our work draws special attention to the importance of the effects of time dependent parameters on decision making in bistable systems here we extend previous studies of the mechanism known as speed dependent cellular decision making in genetic circuits by performing an analytical treatment of the canonical supercritical pitchfork bifurcation problem with an additional time dependent asymmetry and control parameter this model has an analogous behavior to the genetic switch in the presence of transient asymmetries and fluctuations slow passage through the critical region in both systems increases substantially the probability of specific decision outcomes we also study the relevance for attractor selection of reaching maximum values for the external asymmetry before and after the critical region overall maximum asymmetries should be reached at an instant where the position of the critical point allows for compensation of the detrimental effects of noise in retaining memory of the transient asymmetries

multiple local optima in vector quantizers corresp two results are presented on vector quantizers meeting necessary conditions for optimality first a simple generalization of well known centroid and moment properties of the squared error distortion measure to a weighted quadratic distortion measure with an input dependent weighting is presented the second result is an application of the squared error special case of the first result to a simulation study of the design of 1 bit per sample two and three dimensional quantizers for a memoryless gaussian source using the generalized lloyd technique the existence of multiple distinct local optima is demonstrated thereby showing that sufficient conditions for unique local optima do not exist for this simple common case it is also shown that at least three dimensions are required for a vector quantizer to outperform a scalar quantizer for this source

dispersion of small amplitude solutions of the generalized korteweg de vries equation abstract we study the long time behavior of small solutions of the initial value problem for the generalized korteweg de vries equation t u x 3 u x f u 0 gkdv u x 0 g x for the case where f w w s with s 1 4 23 57 3 8625 our results imply that if g l11 g l22 is sufficiently small then sup r 1 t 1 3 u t l in particular the solution tends to zero in the supremum norm the proofs make use of duhamel s formula and dispersion estimates for the linear propagator as well as chain and leibniz rules for fractional derivatives of compositions d f u lp and products d fg lp 0

final state problem for korteweg de vries type equations we study the final state problem for the korteweg de vries type equations ut 1 x 1ux u2ux t x r r u t fs t l2 0 as t where r the function fs t we call a final state defined by the final data u we show that there does not exist a nontrivial solution of this equation in the case of fs t u t u where u t is the free evolution group of this equation we construct the modified wave operator for the korteweg de vries type equations under the conditions that the final data u arc real valued functions and the fourier transform u vanishes at the origin

solvable models of the fokker planck equation an approach based on the gel fand levitan method from a given solvable fokker planck equation one can construct a number of other solvable models for diffusion in a stable or bistable potential fields using the gel fand levitan method of the inverse scattering theory the simplest way of achieving this is to change the lowest eigenvalue and or the normalization of the lowest eigenfunction of the ordinary differential equation obtained by separating the time dependent part for these cases it is shown that the new probability distribution is expressible in terms of integrals involving the original probability distribution and the gel fand levitan kernel the possibility of changing the lowest eigenvalue enables one to find bistable potential fields which would correspond to a well defined long time relaxation rate for the probability current

a vector distribution model and an effective nearest neighbor search method for image vector quantization in this correspondence a modified version of hunt s 1980 image model is used to interpret the distribution of image data vectors the model suggests that the diagonal line of the coordinates system is a good approximation of the principal axis of the image data vector set the validity of the model is supported by experiments following this suggestion an effective nearest neighbor search method for vector quantization of image data is developed the method is based on partitioning the vector space using hyperplanes which are perpendicular to the diagonal direction of the coordinate system the validity of the method is assessed by analyzing its complexity and comparing its performance to those of existing algorithms on a number of images

diffusion in a bistable potential at intermediate and high friction we study the motion of a brownian particle in a bistable potential for intermediate and high friction following ideas of titulaer we perform a high friction expansion of the distribution functionp v x t in velocity and space we show for arbitrary potential that the expansion coefficients obey simple recursion relations which allow them to be calculated easily when terms of order 5 are neglected the resulting differential equations can be transformed into hermitian schrodinger type equations using the wkb technique we solve these equations analytically for the case of the bistable potential and discuss the various time regimes involved in the system in particular we show that the final approach to equilibrium is governed by the kramers rate our results become exact in the limit of low temperatures

the effect of noise on pitchfork and hopf bifurcations we present the results of an experimental and numerical investigation into the effects of noise on pitchfork and hopf bifurcations good quantitative agreement is found between calculations and experiment in the case of the pitchfork we find that natural imperfections override the effects of the noise however novel noise amplification effects have been uncovered in the study of the hopf bifurcation these destroy the critical event found in the noise free case and could be of considerable practical importance in systems containing dynamic bifurcations

rate distortion speech coding with a minimum discrimination information distortion measure an information theory approach to the theory and practice of linear predictive coded lpc speech compression systems is developed it is shown that a traditional lpc system can be viewed as a minimum distortion or nearest neighbor system where the distortion measure is a minimum discrimination information between a speech process model and an observed frame of actual speech this distortion measure is used in an algorithm for computer aided design of block source codes subject to a fidelity criterion to obtain a 750 bits s speech compression system that resembles an lpc system but has a much lower rate a larger memory requirement and requires no on line lpc analysis quantitative and informal subjective comparisons are made among our system and lpc systems

exact solution for the diffusion in bistable potentials we solve analytically the fokker planck equation for a one parameter family of symmetric attractive nonharmonic potentials which include double well situations the exact knowledge of the eigenfunctions and eigenvalues allows us to fully discuss the transient behavior of the probability density in particular for the bistable potentials we can give analytical expressions for the probability current over the working barrier and for the onset time which characterizes the transition from uni to bimodal probability densities

analysis of reflected diffusions via an exponential time based transformation let x t be a time homogeneous diffusion process with state space 0 infty where 0 is a reflecting or entrance endpoint and let z denote a random variable that describes the process x t evaluated at an exponentially distributed random time we propose a method to obtain closed form expressions for the conditional density and the mean of a new diffusion process y t with the same state space and with the same infinitesimal variance whose drift depends on the infinitesimal moments of x t and on the hazard rate function of z this method also allows us to obtain the laplace transform of the first passage time density of y t through a lower constant boundary we then discuss the relation between y t and the process x t subject to catastrophes as well as the interpretation of y t as a diffusion in a decreasing potential we study in detail some special cases concerning diffusion processes obtained when x t is the wiener ornstein uhlenbeck bessel and rayleigh process

decay and scattering of small solutions of a generalized boussinesq equation abstract we study the long time behavior of small solutions of the initial value problem for a generalized boussinesq equation we obtain a lower bound for the degrees of nonlinearity which allows us to establish a nonlinear scattering result for small perturbations that is the small solutions of the nonlinear problem behave asymptotically like the solution of the associated linear problem under certain hypotheses we can construct a scattering operator for the boussinesq equation which carries a neighborhood of 0 in the energy space x into x

brownian motion in the bistable potential at intermediate and high friction relaxation from the instability point abstract we consider a brownian particle in a double well potential at intermediate and high friction and study the evolution of an initial distribution which is located at the top of the barrier the evolution equation is obtained from the fokker planck klein kramers equation by an inverse friction expansion and is transformed to a time dependent schrodinger equation which we solve approximately using the wkb method the result becomes exact for temperatures low compared with the barrier height we calculate explicitly the characteristic time in which the distribution changes from a single peak to a double peak structure and show that the decay process is mainly described by a scaling function which reduces to suzuki s scaling solution in the limit of high friction

product code vector quantizers for waveform and voice coding memory and computation requirements imply fundamental limitations on the quality that can be achieved in vector quantization systems used for speech waveform coding and linear predictive voice coding lpc one approach to reducing storage and computation requirements is to organize the set of reproduction vectors as the cartesian product of a vector codebook describing the shape of each reproduction vector and a scalar codebook describing the gain or energy such shape gain vector quantizers can be applied both to waveform coding using a quadratic error distortion measure and to voice coding using an itakura saito distortion measure in each case the minimum distortion reproduction vector can be found by first selecting a shape code word and then based on that choice selecting a gain codeword several algorithms are presented for the design of shape gain vector quantizers based on a traning sequence of data or a probabilistic model the algorithms are used to design shape gain vector quantizers for both the waveform coding and voice coding application the quantizers are simulated and their performance is compared to that of previously reported vector quantization systems

uniform decay estimates and the lorentz invariance of the classical wave equation on etudie le comportement asymptotique des solutions de u 0 ou t 2 1 2 n 2 pour des conditions initiales u 0 u t g x en t 0 avec g reguliere a support compact dans r n

taylor series expansion for solutions of the korteweg devries equation with respect to their initial values abstract the initial value problem ivp for the korteweg devries kdv equation t u u x u 3 x u 0 u x 0 x for x r t r establishes a nonlinear map k from the space hs r to the space c t t hs r it has been known for many years that this map k is continuous bona and smith 1975 kato 1983 and it was proved recently that this map is lipschitz continuous kenig et al 1993 it is shown in this paper that the nonlinear map k is infinitely many times frechet differentiable from hs r to c t t hs r furthermore it is proved that k has a taylor series expansion at any given hs r i e formula where kn the nth derivative of k at is an n linear map from the n fold space of hs r to c t t hs r and the series converges in the space c t t hs r uniformly for h for some 0 each term yn kn hn in the series solves a linearized kdv equation thus any solutions of the ivp can be obtained by solving a series of linear problems by contrast the corresponding map kp established by the initial value problem for the periodic kdv equation t u u x u 3 x u 0 u x 0 x for x s t r where s is the unit length circle in the plane is known to be continuous only from hs s to c t t hs s this is due to the lack of smoothing effects for solutions of the periodic kdv equation in this paper it is shown that kp is lipschitz continuous from hs 1 s to c t t hs s and is n times frechet differentiable from hs n 1 s to c t t hs s for any n 1 the method developed in this paper applies to other nonlinear dispersive wave equations

global existence of small solutions to nonlinear evolution equations abstract we study the global existence and asymptotic behaviour of small solutions of a large class of nonlinear partial differential equations if the nonlinear terms are of high degree the solutions will be asymptotic to solutions of the linear equation

pseudospectral solution of the fokker planck equation with equilibrium bistable states the eigenvalue spectrum and the approach to equilibrium the fokker planck equation with a constant diffusion coefficient and a particular polynomial drift coefficient can exhibit a bistable equilibrium distribution such model systems have been used to study chemical reactions nucleation climate optical bistability and other phenomena in this paper we consider a particular choice for the drift coefficient of the form a x x 5 x 3 to exemplify the statistical behaviour of such systems the transformation of the fokker planck equation to a schrodinger equation leads to a potential that belongs to the class of potentials in supersymmetric susy quantum mechanics a pseudospectral method based on nonclassical polynomials is used to determine the spectrum of the fokker planck operator and of the schrodinger equation the converged numerical eigenvalues are compared with wkb and swkb approximations of the eigenvalues

well posedness of the initial value problem for the korteweg de vries equation 1 1 itu axu u1xu o x t e r u x 0 uo x the kdv equation which was first derived as a model for unidirectional propagation of nonlinear dispersive long waves 21 has been considered in different contexts namely in its relation with the inverse scattering method in plasma physics and in algebraic geometry see 24 and references therein our purpose is to study local and global well posedness of the ivp 1 1 in classical sobolev spaces hs r we shall say that the ivp 1 1 is locally resp globally well posed in the function space x if it induces a dynamical system on x by generating a continuous local resp global flow it was established in the works of bona and smith 3 bona and scott 2 saut and temam 30 and kato 1 5 that the ivp 1 1 is locally resp globally well posed in hs with s 3 2 resp s 2 roughly speaking global well posedness in hs depends on the available local theory and on the conservation laws satisfied by solutions of 1 1 namely

first exit times of harmonically trapped particles a didactic review we revise the classical problem of characterizing first exit times of a harmonically trapped particle whose motion is described by a one or multidimensional ornstein uhlenbeck process we start by recalling the main derivation steps of a propagator using langevin and fokker planck equations the mean exit time the moment generating function and the survival probability are then expressed through confluent hypergeometric functions and thoroughly analyzed we also present a rapidly converging series representation of confluent hypergeometric functions that is particularly well suited for numerical computation of eigenvalues and eigenfunctions of the governing fokker planck operator we discuss several applications of first exit times such as the detection of time intervals during which motor proteins exert a constant force onto a tracer in optical tweezers single particle tracking experiments adhesion bond dissociation under mechanical stress characterization of active periods of trend following and mean reverting strategies in algorithmic trading on stock markets relation to the distribution of first crossing times of a moving boundary by brownian motion some extensions are described including diffusion under quadratic double well potential and anomalous diffusion

large time asymptotics of solutions to the generalized korteweg de vries equation we study the asymptotic behavior for large time of solutions to the cauchy problem for the generalized korteweg de vries gkdv equationut u 1 u x 13uxxx 0 wherex t rwhen the initial data are small enough if the power of the nonlinearity is greater than 3 then the solution of the cauchy problem has a quasilinear asymptotic behavior for large time more precisely we show that the solutionu t satisfies the decay estimate u t l c 1 t 1 3 1 1 for 4 uux t l ct 2 3 1 t 1 3and using these estimates we prove the existence of the scattering stateu l2such that u t u t u l2 ct 3 3for any small initial data belonging to the weighted sobolev spaceh1 1 f l2 1 x 2 1 2 1 2x 1 2 f l2 whereu t is the airy free evolution group

vector quantizers and predictive quantizers for gauss markov sources low rate vector quantizers are designed and simulated for highly correlated gauss markov sources and the resulting performance is compared with arnstein s optimized predictive quantizer and with huang and schultheiss optimized transform coder two implementations of vector quantizers are considered full search vector quantizers which are optimal but require large codebook searches and tree searched vector quantizers which are suboptimal but require far less searching the various systems are compared on the basis of performance complexity and generality of design techniques