spectral asymptotics for the third order operator with periodic coefficients abstract we consider the self adjoint third order operator with 1 periodic coefficients on the real line the spectrum of the operator is absolutely continuous and covers the real line we determine the high energy asymptotics of the periodic antiperiodic eigenvalues and of the branch points of the lyapunov function furthermore in the case of small coefficients we show that either whole spectrum has multiplicity one or the spectrum has multiplicity one except for a small spectral nonempty interval with multiplicity three in the last case the asymptotics of this small interval is determined

a new matrix method for the casimir operators of the lie algebras w mathfrak s mathfrak p n hbox mathbb r and i mathfrak s mathfrak p 2n hbox mathbb r a method is given to determine the casimir operators of the perfect lie algebras and the inhomogeneous lie algebras in terms of polynomials associated with a parametrized 2n 1 2n 1 matrix for the inhomogeneous symplectic algebras this matrix is shown to be associated to a faithful representation we further analyse the invariants for the extended schrodinger algebra in n 1 dimensions which arises naturally as a subalgebra of the method is extended to other classes of lie algebras and some applications to the missing label problem are given

strong universal source coding subject to a rate distortion constraint the following problem in universal source coding is explored some members of a class of sources have various constraints on the maximum rate at which they may be encoded and thc remainder have various constraints on the maximum distortion that may result from encoding it is desired to find a universal code that will adapt its performance so that whatever source in the class is encoded the resulting performance meets the constraint and is optimal in the rate distortion function sense for that particular source it is shown that such codes exist when the class is totally bounded and the constraints are uniformly continuous in an appropriate sense a key result shows that the value of any uniformly continuous function on a totally bounded class can be uniformly well estimated from observations of the output of any source in

on a class of multilevel universal source coding schemes this work seeks to reduce the computational complexity and the coding rate associated with a class of universal source coding schemes for encoding fixed length m ary sequences into variable length sequences the lynch davisson 1966 minimax universal source coding scheme the tanaka ando leon garcia tal weighted universal source coding scheme 1985 and cover s general enumerative source coding scheme 1973 are modified the essence of the proposed modification is to rename the m symbols of the alphabet so that the most frequent symbol of the given input data block will be denoted 0 the second most frequent symbol will be denoted 1 and so on until the least frequent symbol will be denoted m 1 it is proven that out of all the m permutations of the m symbols the proposed one minimizes the computational complexity of the above mentioned schemes and preserves their unique decodability property moreover for the modified tal scheme it is proven that for block lengths n m further data compression can be achieved

minimax renyi redundancy the redundancy for universal lossless compression of discrete memoryless sources in campbell s setting is characterized as a minimax renyi divergence which is shown to be equal to the maximal alpha mutual information via a generalized redundancy capacity theorem special attention is placed on the analysis of the asymptotics of minimax renyi divergence which is determined up to a term vanishing in blocklength

robust source coding of weakly compact classes a fixed rate block code is said to be strongly robust for a class of sources lambda if its maximum distortion over lambda h is no larger than the maximum of the distortion rate functions of the sources in the class at the rate of the code it is shown that such codes exist at all positive rates whenever the class is compact with resect to the topology of weak convergence and satisfies certain additional but not very strong constraints on the alphabet and distortion measure examples of classes that satisfy these conditions are given in addition classes are exhibited for which there are no strongly robust codes these help to demarcate the boundary between classes for which strongly robust codes do and do not exist more specifically they show that strongly robust codes exist more widely than strongly universal codes but ess widely than weakly universal codes

invariant geometric evolutions of surfaces and volumetric smoothing the study of geometric flows for smoothing multiscale representation and analysis of two and three dimensional objects has received much attention in the past few years in this paper we first survey the geometric smoothing of curves and surfaces via geometric heat type flows which are invariant under the groups of euclidean and affine motions second using the general theory of differential invariants we determine the general formula for a geometric hypersurface evolution which is invariant under a prescribed symmetry group as an application we present the simplest affine invariant flow for convex surfaces in three dimensional space which like the affine invariant curve shortening flow will be of fundamental importance in the processing of three dimensional images

the asymptotic redundancy of bayes rules for markov chains we derive the asymptotics of the redundancy of bayes rules for markov chains of fixed order over a finite alphabet extending the work of barron and clarke 1990 on independent and identically distributed i i d sources the asymptotics are derived when the actual source is the class of spl phi mixing sources which strictly includes markov chains these results can be used to derive minimax asymptotic rates of convergence for universal codes when a markov chain of fixed order is used as a model

nonlinear equations invariant under the poincare similitude and conformal groups in two dimensional space time all realizations of the lie algebras p 1 1 sim 1 1 and conf 1 1 are classified under the action of the group of local diffeomorphisms of r3 the result is used to obtain all second order scalar differential equations invariant under the corresponding poincare similitude and conformal groups the invariant equations are in general nonlinear and the requirement of linearity turns out to be very restrictive group invariant solutions of some of the conformally invariant equations are obtained either by quadratures or by a linearizing transformation

on calculating the capacity of an infinite input finite infinite output channel a version of the arimoto blahut algorithm for continuous channels involves evaluating integrals over an entire input space and thus is not tractable two generalized discrete versions of the arimoto blahut algorithm are presented for this purpose instead of calculating integrals both algorithms require only the computation of a sequence of finite sums this significantly reduces numerical computational complexity

matrix elements for the symplectic sp 4 lie algebra the closed gelfand tsetlin type formulae are constructed in this paper for the matrix elements of the symplectic sp 4 lie algebra in an arbitrary irreducible representation using the reduction chain sp 4 sp 2 sp 2 all resulting formulae for the matrix elements were obtained by directly solving the cartan weyl commutation relations of the sp 4 algebra

a strong version of the redundancy capacity theorem of universal coding the capacity of the channel induced by a given class of sources is well known to be an attainable lower bound on the redundancy of universal codes with respect to this class both in the minimax sense and in the bayesian maximin sense we show that this capacity is essentially a lower bound also in a stronger sense that is for most sources in the class this result extends rissanen s 1984 1986 lower bound for parametric families we demonstrate the applicability of this result in several examples e g parametric families with growing dimensionality piecewise fixed sources arbitrarily varying sources and noisy samples of learnable functions finally we discuss implications of our results to statistical inference

predictive analog to digital conversion for resistance to data outliers abstract we consider predictive source encoding for analog to digital conversion that is resistant to data outliers we focus on a gaussian stationary source and we develop a sequence of outlier resistant source oncoders the studied performance criteria include mean squared distortion and output entropy at the nominal gaussian source when data outliers are absent as well as outlier resistance measures such as breakdown point and influence function the proposed sequence of predictive encoders attains high resistance to data outliers at the expense of some increase in mean squared distortion and differential entropy at the nominal gaussian source

joint fixed rate universal lossy coding and identification of continuous alphabetmemoryless sources the problem of joint universal source coding and density estimation is considered in the setting of fixed rate lossy coding of continuous alphabet memoryless sources for a wide class of bounded distortion measures it is shown that any compactly parametrized family of rd valued independent and identically distributed i i d sources with absolutely continuous distributions satisfying appropriate smoothness and vapnik chervonenkis vc learnability conditions admits a joint scheme for universal lossy block coding and parameter estimation such that when the block length n tends to infinity the overhead per letter rate and the distortion redundancies converge to zero as o n 1 log n and o radicn 1 log n respectively moreover the active source can be determined at the decoder up to a ball of radius o radicn 1 log n in variational distance asymptotically almost surely the system has finite memory length equal to the block length and can be thought of as blockwise application of a time invariant nonlinear filter with initial conditions determined from the previous block comparisons are presented with several existing schemes for universal vector quantization which do not include parameter estimation explicitly and an extension to unbounded distortion measures is outlined finally finite mixture classes and exponential families are given as explicit examples of parametric sources admitting joint universal compression and modeling schemes of the kind studied here

nodal solutions of a nonlinear eigenvalue problem of the euler bernoulli equation abstract we are concerned with determining values of r for which there exist nodal solutions of the boundary value problems of the euler bernoulli equation p x u q x u r f x u u 0 x 0 l u 0 cos p u 0 sin 0 u 0 0 u l cos p u l sin 0 u l 0 where 0 2 p c 2 0 l and q c 1 0 l the proof of our main result is based upon bifurcation techniques

on the mdl principle for i i d sources with large alphabets average case universal compression of independent and identically distributed i i d sources is investigated where the source alphabet is large and may be sublinear in size or even larger than the compressed data sequence length n in particular the well known results including rissanen s strongest sense lower bound for fixed size alphabets are extended to the case where the alphabet size k is allowed to grow with n it is shown that as long as k o n instead of the coding cost in the fixed size alphabet case of 0 5logn extra code bits for each one of the k 1 unknown probability parameters the cost is now 0 5log n k code bits for each unknown parameter this result is shown to be the lower bound in the minimax and maximin senses as well as for almost every source in the class achievability of this bound is demonstrated with two part codes based on quantization of the maximum likelihood ml probability parameters as well as by using the well known krichevsky trofimov kt low complexity sequential probability estimates for very large alphabets kgtn it is shown that an average minimax and maximin bound on the redundancy is essentially to first order log k n bits per symbol this bound is shown to be achievable both with two part codes and with a sequential modification of the kt estimates for k theta n the redundancy is theta 1 bits per symbol finally sequential codes are designed for coding sequences in which only m min k n alphabet symbols occur

new trace formulae for a quadratic pencil of the schrodinger operator this work deals with the eigenvalue problem for a quadratic pencil of the schrodinger operator on a finite closed interval with the two point boundary conditions we will obtain new regularized trace formulas for this class of differential pencil

stochastic quantization for performance stability we consider stochastic stationary analog signals for such signals we consider quantizers whose performance remains stable under perturbations in the statistical description of the signals performance is defined as the induced average distortion the output entropy and the asymptotic consistency in the preservation of important signal characteristics we found that stochastic quantizers can in general realize performance stability while deterministic quantizers do not we consider block quantization for lower average distortion under fixed output rate and we propose predictive stochastic quantization for highly correlated signals the proposed predictive quantizers compress automatically to the output entropy we also propose nonpredictive block stochastic quantization for weakly correlated or uncorrelated signals we describe an elias type noiseless compressor for channel transmission and a matching decoder

outlier resistant filtering and smoothing abstract we consider a stationary gaussian information process transmitted through an additive noise channel we assume that the noise and information processes are mutually independent and we model the noise process as nominally gaussian with additive outliers for the above system model we first develop a theory for outlier resistant filtering and smoothing operations we then design specific such nonlinear operations and we study their performance the performance criteria are the asymptotic mean squared error at the gaussian nominal model the breakdown point and the influence function we find that the proposed operations combine excellent performance at the nominal model with strong resistance to outliers

density estimation by stochastic complexity the results by p hall and e j hannan 1988 on optimization of histogram density estimators with equal bin widths by minimization of the stochastic complexity are extended and sharpened in two separate ways as the first contribution two generalized histogram estimators are constructed the first has unequal bin widths which together with the number of the bins are determined by minimization of the stochastic complexity using dynamic programming the other estimator consists of a mixture of equal bin width estimators each of which is defined by the associated stochastic complexity as the main contribution in the present work two theorems are proved which together extend the universal coding theorems to a large class of data generating densities the first gives an asymptotic upper bound for the code redundancy in the order of magnitude achieved with a special predictive type of histogram estimator which sharpens a related bound the second theorem states that this bound cannot be improved upon by any code whatsoever

universal prediction this paper consists of an overview on universal prediction from an information theoretic perspective special attention is given to the notion of probability assignment under the self information loss function which is directly related to the theory of universal data compression both the probabilistic setting and the deterministic setting of the universal prediction problem are described with emphasis on the analogy and the differences between results in the two settings

nonlinear filters for linear models a robust approach we consider the altering problem for linear models where the driving noises may be quite general nonwhite and non gaussian and where the observation noise may only be known to belong to a finite family of possible disturbances using diffusion approximation methods we show that a certain nonlinear filter minimizes the asymptotic filter variance this nonlinear filter is obtained by choosing at each moment on the basis of the observations one of a finite number of kalman type filters driven by a suitable nonlinear transformation of the innovations as a byproduct we obtain also the asymptotic identification of the a priori unknown observation noise disturbance by yielding an asymptotically efficient filter in face of an unknown observation noise our approach may also be viewed as a robust approach to filtering for linear models

the rotation number for almost periodic potentials we define and analyze the rotation number for the almost periodic schrodinger operatorl d2 dx2 q x we use the rotation number to discuss i the spectrum ofl ii its relation to the korteweg de vries equation

universal coding information prediction and estimation a connection between universal codes and the problems of prediction and statistical estimation is established a known lower bound for the mean length of universal codes is sharpened and generalized and optimum universal codes constructed the bound is defined to give the information in strings relative to the considered class of processes the earlier derived minimum description length criterion for estimation of parameters including their number is given a fundamental information theoretic justification by showing that its estimators achieve the information in the strings it is also shown that one cannot do prediction in gaussian autoregressive moving average arma processes below a bound which is determined by the information in the data

a game theoretic approach to robust filtering a game theoretic approach to the filtering or smoothing problem is presented a family of stationary information carrying processes and generalized models for the noise channel and the filter is considered sufficient conditions for the existence of saddle point type solutions are stated in addition the solution for a special case of noise channel a family of information carrying processes and a nonlinear filter are found

invariant polynomials and molien functions the molien function associated with a finite dimensional representation of a compact lie group is a useful tool in representation theory because it acts as a generating function for counting the number of invariant polynomials on the representation space the main purpose of this paper is to introduce a more general and apparently new generating function which in the same sense counts not only the number of invariant real polynomials in a real representation or the number of invariant complex polynomials in a complex representation as a function of their degree but encodes the number of invariant real polynomials in a complex representation as a function of their bidegree the first and second component of this bidegree being the number of variables in which the polynomial is holomorphic and antiholomorphic respectively this is obviously an additional and non trivial piece of information for representations which are truly complex i e not self conjugate or are pseudo real but it provides ad

kotani theory for one dimensional stochastic jacobi matrices we consider families of operators h on l2 given by h u n u n 1 u n 1 v n u n wherev is a stationary bounded ergodic sequence we prove analogs of kotani s results including that for a e ac h is the essential closure of the set ofe where e the lyaponov index vanishes and the result that ifv is non deterministic then ac is empty

the infinite period limit of the inverse formalism for periodic potentials soit l operateur de hill h p d 2 dx 2 v p x quand v p x est une fonction p periodique de x on montre que les formules pour la solution du probleme inverse ont une limite si on prend v p comme une repetition periodique d une fonction a support compact v x et que l on fait tendre p vers l infini

higher order trace formulas of the buslaev faddeev type for the half line schrodinger operator with long range potentials we deal with trace formulas for half line schrodinger operators with long range potentials we generalize the buslaev faddeev trace formulas to the case of square integrable potentials the exact relation between the number of the trace formulas and the number of integrable derivatives of the potential is also given the main results are optimal

the spectral theory of the vibrating periodic beam we study the spectral theory of the fourth order eigenvalue problem left a x u x right prime prime lambda rho x u x infty x infty where the functionsa and are periodic and strictly positive this equation models the transverse vibrations of a thin straight periodic beam whose physical characteristics are described bya and we examine the structure of the spectrum establishing the fact that the periodic and antiperiodic eigenvalues are the endpoints of the spectral bands we also introduce an entire function which we denote bye connected to the spectral theory whose zeros at least the ones of odd multiplicity are shown to lie on the negative real axis where they define a collection of pseudogaps next we prove some inverse results in the spirit of two old theorems of borg for the hill s equation we finish with a determinant formula i e a multiplicative trace formula and some comments on its role in the formulation of the general inverse problem

optimal error exponents in hidden markov models order estimation we consider the estimation of the number of hidden states the order of a discrete time finite alphabet hidden markov model hmm the estimators we investigate are related to code based order estimators penalized maximum likelihood ml estimators and penalized versions of the mixture estimator introduced by liu and narayan 1994 we prove strong consistency of those estimators without assuming any a priori upper bound on the order and smaller penalties than previous works we prove a version of stein s lemma for hmm order estimation and derive an upper bound on underestimation exponents then we prove that this upper bound can be achieved by the penalized ml estimator and by the penalized mixture estimator the proof of the latter result gets around the elusive nature of the ml in hmm by resorting to large deviation techniques for empirical processes finally we prove that for any consistent hmm order estimator for most hmm the overestimation exponent is null

minimax universal noiseless coding for unifilar and markov sources corresp constructive upper bounds are presented for minimax universal noiseless coding of unifilar sources without any ergodicity assumptions these bounds are obtained by quantizing the estimated probability distribution of source letters with respect to the relative entropy they apply both to fixed length to variable length fv and variable length to fixed length vf codes unifilar sources are a generalization of the usual definition of markov sources so these results apply to markov sources as well these upper bounds agree asymptotically with the lower bounds given by davisson for fv coding of stationary ergodic markov sources

branching rules for sp 2n algebra reduction on the chain sp 2n 2 sp 2 in this paper branching rules for the reduction sp 2n sp 2n 2 sp 2 are found and a new pattern for labeling vectors belonging to the base for unitary irreducible representations buir s is found

a sequential algorithm for the universal coding of finite memory sources the estimation and universal compression of discrete sources are considered and a sequential algorithm for the universal coding of finite memory sources attaining asymptotically minimum redundancy is presented the algorithm performs an online estimation of the source states and uses an arithmetic code

the floquet theory of the periodic euler bernoulli equation abstract we continue the study initiated in 17 of the spectral theory of the fourth order eigenvalue problem a x u x x u x where the functions a and are periodic and strictly positive this equation models the transverse vibrations of a thin straight periodic beam whose physical characteristics are described by a and the equality of the algebraic and geometric multiplicities of the periodic and antiperiodic eigenvalues is established also a spectrum like set that we called pseudospectrum or spectrum is introduced or rather discovered this spectrum is shown to lie on the negative real axis and have a band gap structure various open questions and conjectures are mentioned at the end of the paper

positive solutions of nonlinear fourth order boundary value problems with local and non local boundary conditions we establish new existence results for multiple positive solutions of fourth order nonlinear equations which model deflections of an elastic beam we consider the widely studied boundary conditions corresponding to clamped and hinged ends and many non local boundary conditions with a unified approach our method is to show that each boundary value problem can be written as the same type of perturbed integral equation in the space c 0 1 involving a linear functional u but although we seek positive solutions the functional is not assumed to be positive for all positive u the results are new even for the classic boundary conditions of clamped or hinged ends when u 0 because we obtain sharp results for the existence of one positive solution for multiple solutions we seek optimal values of some of the constants that occur in the theory which allows us to impose weaker assumptions on the nonlinear term than in previous works our non local boundary conditions contain multi point problems as special cases and for the first time in fourth order problems we allow coefficients of both signs

algebraic model for the evolution of the genetic code a search for symmetries based on the compact simple lie algebras is performed to verify to what extent the genetic code is a manifestation of some underlying symmetry an exact continuous symmetry group cannot be found to reproduce the present genetic code however a unique approximate symmetry group is compatible with codon assignment for the fundamental amino acids and the termination codon in order to obtain the actual genetic code the symmetry must be slightly broken

on testing trained vector quantizer codebooks this paper discusses a criterion for testing a vector quantizer vq codebook that is obtained by training when a vq codebook is designed by a clustering algorithm using a training set time average distortion which is called the training set distortion tsd is usually calculated in each iteration of the algorithm since the input probability function is unknown in general and cumbersome to deal with the algorithm stops when the tsd ceases to significantly decrease in order to test the resultant codebook validating set distortion vsd is calculated on a separate validating set vs codebooks that yield small difference between the tsd and the vsd are regarded as good ones however the difference vsd tsd is not necessarily a desirable criterion for testing a trained codebook unless certain conditions are satisfied a condition that is previously assumed to be important is that the vs has to be quite large to well approximate the source distribution this condition implies greater computational burden of testing a codebook in this paper we first discuss the condition under which the difference vsd tsd is a meaningful codebook testing criterion then convergence properties of the vsd a time average quantity are investigated finally we show that for large codebooks a vs size as small as the size of the codebook is sufficient to evaluate the vsd this paper consequently presents a simple method to test trained codebooks for vq s experimental results on synthetic data and real images supporting the analysis are also provided and discussed

the periodic euler bernoulli equation we continue the study of the floquet spectral theory of the beam equation namely the fourth order eigenvalue problem a x u x x u x x where the functions a and p are periodic and strictly positive this equation models the transverse vibrations of a thin straight periodic beam whose physical characteristics are described by a and p here we develop a theory analogous to the theory of the hill operator d dx 2 q x we first review some facts and notions from our previous works including the concept of the pseudospectrum or spectrum our new analysis begins with a detailed study of the zeros of the function f a k for any given quasimomentum k c where f k 0 is the floquet bloch variety of the beam equation the hill quantity corresponding to f k is 2 cos kb where a a is the discriminant and b the period of q we show that the multiplicity m of any zero of f a k can be one or two and m 2 for some k if and only if is also a zero of another entire function d a independent of k furthermore we show that d a has exactly one zero in each gap of the spectrum and two zeros counting multiplicities in each gap if is a double zero of f a k it may happen that there is only one floquet solution with quasimomentum k thus there are exceptional cases where the algebraic and geometric multiplicities do not agree next we show that if is an open gap of the pseudospectrum i e then the floquet matrix t a has a specific jordan anomaly at a a and a we then introduce a multipoint dirichlet type eigenvalue problem which is the analogue of the dirichlet problem for the hill equation we denote by n n z the eigenvalues of this multipoint problem and show that n n z is also characterized as the set of values of a for which there is a proper floquet solution f x a such that f 0 a 0 we also show theorem 7 that each gap of the l 2 r spectrum contains exactly one n and each gap of the pseudospectrum contains exactly two n s counting multiplicities here when we say gap or gap we also include the endpoints so that when two consecutive bands or bands touch the in between collapsed gap or gap is a point we believe that n n z can be used to formulate the associated inverse spectral problem as an application of theorem 7 we show that if is a collapsed closed gap then the floquet matrix t is diagonalizable some of the above results were conjectured in our previous works however our conjecture that if all the gaps are closed then the beam operator is the square of a second order hill type operator is still open

qualitative robustness in time series abstract we consider robust operations in time series we present a definition and subsequent qualitative analysis of robustness we also present meaningful definitions of performance criteria such as the breakdown point and the sensitivity of robust operations we present some specific classes of robust operations and we discuss and analyze their properties finally we fully analyze a particular class of robust predictors and interpolators for a linearly contaminated class of stationary stochastic processes

efficient universal noiseless source codes although the existence of universal noiseless variable rate codes for the class of discrete stationary ergodic sources has previously been established very few practical universal encoding methods are available efficient implementable universal source coding techniques are discussed in this paper results are presented on source codes for which a small value of the maximum redundancy is achieved with a relatively short block length a constructive proof of the existence of universal noiseless codes for discrete stationary sources is first presented the proof is shown to provide a method for obtaining efficient universal noiseless variable rate codes for various classes of sources for memoryless sources upper and lower bounds are obtained for the minimax redundancy as a function of the block length of the code several techniques for constructing universal noiseless source codes for memoryless sources are presented and their redundancies are compared with the bounds consideration is given to possible applications to data compression for certain nonstationary sources

sliding block encoders that are rho bar continuous functions of their input corresp sliding block encoders operating within a statistically contaminated source environment are considered stationary analog sources are studied using the rho bar distance as the measure of contamination sufficient constructive conditions are developed for the design of sliding block encoders that guarantee rho bar stability of their output when operating within a statistically contaminated source environment the relationship of this stability to the robustness of the encoding scheme is discussed where robustness is a local stability property applied on the output entropy and the average distortion

robust recursive estimation in the presence of heavy tailed observation noise a first order approximation is derived for the conditional prior distribution of the state of a discrete time stochastic linear dynamic system in the presence of contaminated normal observation noise this distribution is then used to derive a first order approximation of the conditional mean minimum variance estimator if the observation noise distribution can be represented as a member of the e contaminated normal neighborhood then the conditional prior is also to first order an analogous perturbation from a normal distribution whose first two moments are given by the kalman filter moreover the perturbation is itself of a special form combining distributions whose parameters are given by banks of parallel kalman filters and optimal smoothers

universal lossless compression with unknown alphabets the average case universal compression of patterns of sequences generated by independent and identically distributed i i d sources with unknown possibly large alphabets is investigated a pattern is a sequence of indices that contains all consecutive indices in increasing order of first occurrence if the alphabet of a source that generated a sequence is unknown the inevitable cost of coding the unknown alphabet symbols can be exploited to create the pattern of the sequence this pattern can in turn be compressed by itself it is shown that if the alphabet size k is essentially small then the average minimax and maximin redundancies as well as the redundancy of every code for almost every source when compressing a pattern consist of at least 0 5log n k3 bits per each unknown probability parameter and if all alphabet letters are likely to occur there exist codes whose redundancy is at most 0 5log n k2 bits per each unknown probability parameter where n is the length of the data sequences otherwise if the alphabet is large these redundancies are essentially at least theta n 2 3 bits per symbol and there exist codes that achieve redundancy of o n 1 2 bits per symbol two suboptimal low complexity sequential algorithms for compression of patterns are presented and their description lengths analyzed also pointing out that the pattern average universal description length can decrease below the underlying i i d entropy for large enough alphabets

local spectral properties of reflectionless jacobi cmv and schrodinger operators we prove that jacobi cmv and schrodinger operators which are reflectionless on a homogeneous set e in the sense of carleson under the assumption of a blaschke type condition on their discrete spectra accumulating at e have purely absolutely continuous spectrum on e

last passage time for linear diffusions and application to the emptying time of a box we study the statistics of last passage time for linear diffusions first we present an elementary derivation of the laplace transform of the probability density of the last passage time thus recovering known results from the mathematical literature we then illustrate them on several explicit examples in a second step we study the spectral properties of the schrodinger operator associated to such diffusions in an even potential u x u x unveiling the role played by the so called weyl coefficient indeed in this case our approach allows us to relate the last passage times for dual diffusions i e diffusions driven by opposite force fields and to obtain new explicit formulae for the mean last passage time we further show that for such even potentials the small time t expansion of the mean last passage time on the interval 0 t involves the korteveg de vries invariants which are well known in the theory of schrodinger operators finally we apply these results to study the emptying time of a one dimensional box of size l containing n independent brownian particles subjected to a constant drift in the scaling limit where both n rightarrow infty and l rightarrow infty keeping the density rho n l fixed we show that the limiting density of the emptying time is given by a gumbel distribution our analysis provides a new example of the applications of extreme value statistics to out of equilibrium systems

analytical methods for recovering coefficients in differential equations from spectral data in this paper analytical methods for recovering coefficients from spectral data are discussed properties of these methods are given and mathematical models for which these methods might apply are presented

order estimation and sequential universal data compression of a hidden markov source by the method of mixtures we consider first the estimation of the order i e the number of states of a discrete time finite alphabet stationary ergodic hidden markov source hms our estimator uses a description of the observed data in terms of a uniquely decodable code with respect to a mixture distribution obtained by suitably mixing a parametric family of distributions on the observation space this procedure avoids maximum likelihood calculations the order estimator is shown to be strongly consistent with the probability of underestimation decaying exponentially fast in the number n of observations while the probability of overestimation does not exceed cn sup 3 where c is a constant next we present a sequential algorithm for the uniquely decodable universal data compression of the hms which performs an on line estimation of source order followed by arithmetic coding this code asymptotically attains optimum average redundancy

the trace formula for schrodinger operators on the line this paper discusses certain aspects of the spectral and inverse spectral problems for the schrodinger operator l q frac d 2 dx 2 q x for q x c the space of bounded continuous functions the trace formula of the title is the relation q 0 lambda 0 sum limits j 1 infty lambda 2j lambda 2j 1 2 mu j with appropriate choices of j j 0 and j j 1 which is a familiar relation in the theory of hill s equation we characterize the set subseteq c of potentials for which this holds further extensions of the theory of hill s equation are also obtained from the spectrum l q a torust q is constructed which is in general infinite dimensjional everyq x can be mapped to a continuous path ont q described by the auxiliary spectrum j j 1 under certain geometrical conditions on l q this path is the orbit of ac 1 vector field ont q and the mapping extends to one from the hull q overline q x xi xi in mathbb r to the closure of this orbit in particular q is compact these results have applications in the theory of schrodinger operators with ergodic potentials

robust linear filtering for multivariable stationary time series we consider the problem of asymptotic noncausal linear filtering for multivariable second order stationary time series under spectral uncertainty in both the signal and the noise processes the spectral uncertainty is modeled by e contaminated and p point classes for the case where both the signal and noise spectra are e contaminated we assume that the eigenvectors of the corresponding nominal spectral density matrices are identical the problem is formulated as a game whose saddle point solutions are found and analyzed

minimax redundancy for the class of memoryless sources let x sup n x sub 1 x sub n be a memoryless source with unknown distribution on a finite alphabet of size k we identify the asymptotic minimax coding redundancy for this class of sources and provide a sequence of asymptotically minimax codes equivalently we determine the limiting behavior of the minimax relative entropy min sub qxn max sub pxn d p sub xn spl par q sub xn where the maximum is over all independent and identically distributed i i d source distributions and the minimum is over all joint distributions we show in this paper that the minimax redundancy minus k 1 2 log n 2 spl pi e converges to log spl int spl radic det i spl theta d spl theta log spl gamma 1 2 sup k spl gamma k 2 where i spl theta is the fisher information and the integral is over the whole probability simplex the bayes strategy using jeffreys prior is shown to be asymptotically maximin but not asymptotically minimax in our setting the boundary risk using jeffreys prior is higher than that of interior points we provide a sequence of modifications of jeffreys prior that put some prior mass near the boundaries of the probability simplex to pull down that risk to the asymptotic minimax level in the limit

absolute summability of the trace relation for certain schrodinger operators a recently established general trace formula for one dimensional schrodinger operators is systematically studied in the context of short range potentials potentials which approach different spatial asymptotes sufficiently fast and appropriate impurity defect interactions in one dimensional solids we prove the absolute summability of the trace formula and establish its connections with scattering quantities such as reflection coefficients in each case

trace formulae for the schrodinger equation with energy dependent potential we consider the eigenvalue problems for the schrodinger equation with energy dependent potential subject to the eigenparameter boundary conditions and quasiperiodic boundary conditions respectively two new formulae of the regularized trace for this class of schrodinger equation with boundary conditions are obtained in an explicit form

complexity of strings in the class of markov sources shannon s self information of a string is generalized to its complexity relative to the class of finite state machine fsm defined sources unlike an earlier generalization the new one is valid for both short and long strings the definition is justified in part by a theorem stating that asymptotically the mean complexity provides a tight lower bound for the mean length of all so called regular codes this also generalizes shannon s noiseless coding theorem for a large subclass of fsm sources a simple algorithm is described for computing the complexity

differential invariants for symplectic lie algebras realized by boson operators the study of algebraic symplectic models in high energy physics and even in molecular genetics has received much attention in the past decades in this paper we first survey one known bosonic realization with n types of bosons operators in a 2j 1 dimensional space with j being the semi integer then using the general theory of differential invariants we determine two sets of sp 2j 1 differential invariants in one set when n 2j 1 all independent invariants of any order are obtained in a simple closed form when n 2j 1 we could present only particular matrix forms in the latter case the differential invariants up to the second order are explicitly presented for the symplectic algebras sp 2 sp 4 and sp 6 realized by up to three types of bosons as applications we present the simplest sp 2 invariant lagrangians null lagrangians new solutions to the helmholtz inverse problem and evolution equations which will be of fundamental importance in the construction of dynamical systems invariant under the symplectic group

eigenvalue estimates and trace formulas for the matrix hill s equation abstract estimates for floquet multipliers and periodic eigenvalues are developed for the matrix hill s equation basic estimates and a description of the periodic eigenvalues as roots of an entire function are adequate for establishing trace formulas with a residue computation refined estimates and power sum formulas are available for restricted pairs of hill s operators

almost periodicity of some random potentials in this paper we study some random potentials with purely absolutely continuous spectrum in one dimension there are no examples of random potentials with absolutely continuous spectrum that are not almost periodic though it is known from kotani 3 that such potentials are deterministic hence one could conjecture that any random potential with purely absolutely continuous spectrum is almost periodic we show this result for random potentials with infinitely many bands in their spectrum such that the gaps behave well in the finite bands case concini and johnson l discussed the same problem a precise definition of the behavior of the gaps could be found in condition 2 3 we make use of the results of levitan 7 kotani 3 4 and moser 9 for doing this we also discuss the behavior of the lyapunov exponent asymptotically and its relation to the smoothness of the potentials

uniqueness theorems in inverse spectral theory for one dimensional schrodinger operators new unique characterization results for the potential v x in connection with schrodinger operators on r and on the half line 0 are proven in terms of appropriate krein spectral shift functions particular results obtained include a generalization of a well known uniqueness theorem of borg and marchenko for schrodinger operators on the half line with purely discrete spectra to arbitrary spectral types and a new uniqueness result for schrodinger operators with confining potentials on the entire real line

almost periodic schrodinger operators iii the absolutely continuous spectrum in one dimension we studyh d2 dx2 v x withv x limit periodic e g v x ancos x 2n with an we prove that for a genericv and for generican in the explicit example h is a cantor nowhere dense perfect set for a dense set the spectrum is both cantor and purely absolutely continuous and therefore purely recurrent absolutely continuous

on the measure of the spectrum for the almost mathieu operator we obtain partial results on the conjecture that for the almost mathieu operator at irrational frequency the measure of the spectrum s 4 2 for 2 we show that if n is rational and alpha n to alpha irrational then s alpha n lambda theta to 4 2 lambda

outlier resistant algorithms for detecting a change in a stochastic process outlier resistant algorithms that detect a change from a given nominal stationary process to another such process are given the nominal processes are assumed to be mutually independent and to satisfy some general regularity conditions the outlier sequences are assumed to be independently and identically distributed and independent of the nominal processes the proposed algorithms are sequential and consist of uniformly bounded steps the asymptotic performance of the algorithms is analyzed both in the absence and the presence of outliers breakdown points and influence functions are defined and analyzed the algorithms are studied in more detail for gaussian autoregressive nominal processes