a second eigenvalue bound for the dirichlet schrodinger operator let i v be the ith eigenvalue of the schrodinger operator with dirichlet boundary conditions on a bounded domain omega subset mathbb r n and with the positive potential v following the spirit of the payne polya weinberger conjecture and under some convexity assumptions on the spherically rearranged potential v we prove that 2 v 2 s1 v here s1 denotes the ball centered at the origin that satisfies the condition 1 v 1 s1 v

eigenvalues of the laplacian in two dimensions the eigenvalue problem for the laplace operator in two dimensions is classical in mathematics and physics nevertheless computational methods for estimating the eigenvalues are still of much current interest particularly in applications to acoustic and electromagnetic waveguides although our primary interest is with the computational methods there are a number of theoretical results on the behavior of the eigenvalues and eigenfunctions that are useful for understanding the methods and in addition are of interest in themselves these results are discussed first and then the various computational methods that have been used to estimate the eigenvalues are reviewed with particular emphasis on methods that give error bounds some of the more powerful techniques available are illustrated by applying them to a model problem

fisher information and stochastic complexity by taking into account the fisher information and removing an inherent redundancy in earlier two part codes a sharper code length as the stochastic complexity and the associated universal process are derived for a class of parametric processes the main condition required is that the maximum likelihood estimates satisfy the central limit theorem the same code length is also obtained from the so called maximum likelihood code

a numerical study of the spectral gap we present a numerical study of the spectral gap of the dirichlet laplacian k 2 k 1 k of a planar convex region k besides providing supporting numerical evidence for the long standing gap conjecture that k 3 2 d2 k where d k denotes the diameter of k our study suggests new types of bounds and several conjectures regarding the dependence of the gap not only on the diameter but also on the perimeter and the area one of these conjectures is a stronger version of the gap conjecture mentioned above a similar study is carried out for the quotient of the first two dirichlet eigenvalues

eigenvalue gaps for the cauchy process and a poincare inequality abstract a connection between the semigroup of the cauchy process killed upon exiting a domain d and a mixed boundary value problem for the laplacian in one dimension higher known as the mixed steklov problem was established in r banuelos t kulczycki the cauchy process and the steklov problem j funct anal 211 2004 355 423 from this a variational characterization for the eigenvalues n n 1 of the cauchy process in d was obtained in this paper we obtain a variational characterization of the difference between n and 1 we study bounded convex domains which are symmetric with respect to one of the coordinate axis and obtain lower bound estimates for 1 where is the eigenvalue corresponding to the first antisymmetric eigenfunction for d the proof is based on a variational characterization of 1 and on a weighted poincare type inequality the poincare inequality is valid for all symmetric stable processes 0 2 and any other process obtained from brownian motion by subordination we also prove upper bound estimates for the spectral gap 2 1 in bounded convex domains

the bezoutian state space realizations and fisher s information matrix of an arma process in this paper we derive some properties of the bezout matrix and relate the fisher information matrix for a stationary arma process to the bezoutian some properties are explained via realizations in state space form of the derivatives of the white noise process with respect to the parameters a factorization of the fisher information matrix as a product in factors which involve the bezout matrix of the associated ar and ma polynomials is derived from this factorization we can characterize singularity of the fisher information matrix

can one hear the shape of a drum revisited in a landmark paper mark kac in 1966 amer math monthly 73 pp 1 23 showed that geometric properties of regions in r 2 can be obtained by studying the asymptotic properties of the spectrum of the laplacian he also conjectured that the shape of a region might be completely determined by the spectrum we describe recent developments in this field and discuss the status of the kac conjecture as well as similar conjectures for operators other than the laplacian

inequalities for eigenvalues of fourth order elliptic operators in divergence form on riemannian manifolds abstract in this paper we study eigenvalue of linear fourth order elliptic operators in divergence form with dirichlet boundary condition on a bounded domain in a compact riemannian manifolds with boundary possibly empty and provide a general inequality for them as an application by using this inequality we study eigenvalues of this operator on compact domains of complete submanifolds in a euclidean space

entropy constrained vector quantization an iterative descent algorithm based on a lagrangian formulation for designing vector quantizers having minimum distortion subject to an entropy constraint is discussed these entropy constrained vector quantizers ecvqs can be used in tandem with variable rate noiseless coding systems to provide locally optimal variable rate block source coding with respect to a fidelity criterion experiments on sampled speech and on synthetic sources with memory indicate that for waveform coding at low rates about 1 bit sample under the squared error distortion measure about 1 6 db improvement in the signal to noise ratio can be expected over the best scalar and lattice quantizers when block entropy coded with block length 4 even greater gains are made over other forms of entropy coded vector quantizers for pattern recognition it is shown that the ecvq algorithm is a generalization of the k means and related algorithms for estimating cluster means in that the ecvq algorithm estimates the prior cluster probabilities as well experiments on multivariate gaussian distributions show that for clustering problems involving classes with widely different priors the ecvq outperforms the k means algorithm in both likelihood and probability of error

isoperimetric bounds for higher eigenvalue ratios for the n dimensional fixed membrane problem we give several results which extend our recent proof of the payne polya weinberger conjecture to ratios of higher eigenvalues in particular we show that for a bounded domain n the eigenvalues of its dirichlet laplacian obey where m denotes the m th eigenvalue and j p k denotes the k th positive zero of the bessel function j p x certain extensions of this result are given the most general being the bound where k 2 and l m denotes the number of nodal domains of an m th eigenfunction our results imply certain further conjectures of payne polya and weinberger concerning 3 2 and 4 3 in addition we find a resonably good bound on 4 1 we also briefly discuss extensions to schrodinger operators and other elliptic eigenvalue problems

asymptotic normality of the maximum likelihood estimator for general hidden markov models hidden markov models hmms have during the last decade become a widespread tool for modeling sequences of dependent random variables inference for such models is usually based on the maximum likelihood estimator mle and consistency of the mle for general hmms was recently proved by leroux in this paper we show that under mild conditions the mle is also asymptotically normal and prove that the observed information matrix is a consistent estimator of the fisher information

on trace identities and universal eigenvalue estimates for some partial differential operators in this article we prove and exploit a trace identity for the spectra of schrodinger operators and similar operators this identity leads to universal bounds on the spectra which apply to low lying eigenvalues eigenvalue asymptotics and to partition functions traces of heat operators in many cases they are sharp in the sense that there are specific examples for which the inequalities are saturated special cases corresponding to known inequalities include those of hile and protter and of yang introduction in this article we prove and exploit an identity for the spectra of self adjoint operators h modeled on the dirichlet laplacian or more generally on schrodinger operators of the form p a x 2 v x 1 where p 1i is the usual momentum operator in convenient units and a x is the magnetic vector potential we recover and extend several known inequalities involving sums differences and ratios of eigenvalues let j j 1 denote the ordered eigenvalues of the dirichlet laplacian on a bounded d dimensional domain with zero dirichlet boundary conditions and recall that hile and protter hipr80 proved that d 4 1 n n j 1 j n 1 j 2 thereby extending an earlier inequality of payne polya and weinberger papowe56 in the last few years it has become clear that these and many similar relationships can be realized as special cases of abstract variational bounds involving the interplay among commutators of 2 a cartesian coordinate xj and the corresponding derivative xj for this analysis and various extensions of 2 see ha88 ho90 ha93 hami95 while inequalities of this type have been fairly sharp for low lying eigenvalues they have been mostly disappointing for higher eigenvalues yang ya91 however received by the editors september 28 1995 1991 mathematics subject classification primary 35j10 35j25 58g25

the empirical distribution of rate constrained source codes let x x sub 1 be a stationary ergodic finite alphabet source x sup n denote its first n symbols and y sup n be the codeword assigned to x sup n by a lossy source code the empirical kth order joint distribution q spl circ sup k x sup n y sup n spl rceil x sup k y sup k is defined as the frequency of appearances of pairs of k strings x sup k y sup k along the pair x sup n y sup n our main interest is in the sample behavior of this random distribution letting i q sup k denote the mutual information i x sup k y sup k when x sup k y sup k spl sim q sup k we show that for any sequence of lossy source code s of rate spl les r lim sup sub n spl rarr spl infin 1 k i q spl circ sup k x sup n y sup n spl rfloor spl les r 1 k h x sub 1 sup k h x a s where h x denotes the entropy rate of x this is shown to imply for a large class of sources including all independent and identically distributed i i d sources and all sources satisfying the shannon lower bound with equality that for any sequence of codes which is good in the sense of asymptotically attaining a point on the rate distortion curve q spl circ sup k x sup n y sup n spl rfloor spl rarr sup d p x sup k y sup k a s whenever p sub x sup k sub y sup k is the unique distribution attaining the minimum in the definition of the kth order rate distortion function consequences of these results include a new proof of kieffer s sample converse to lossy source coding as well as performance bounds for compression based denoisers

estimates on eigenvalues of laplacian in this paper we study eigenvalues of laplacian on either a bounded connected domain in an n dimensional unit sphere sn 1 or a compact homogeneous riemannian manifold or an n dimensional compact minimal submanifold in an n dimensional unit sphere sn 1 we estimate the k 1 th eigenvalue by the first k eigenvalues as a corollary we obtain an estimate of difference between consecutive eigenvlaues our results are sharper than ones of p c yang and yau 25 leung 19 li 20 and harrel ii and stubbe 12 respectively from weyl s asymptotical formula we know that our estimates are optimal in the sense of the order of k for eigenvalues of laplacian on a bounded connected domain in an n dimensional unit sphere sn 1

symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes we consider the following eigenvalue optimization problem given a bounded domain and numbers 0 a 0 find a subset d of area a for which the first dirichlet eigenvalue of the operator d is as small as possible

an isoperimetric problem for leaky loops and related mean chord inequalities we consider a class of hamiltonians in l2 r2 with attractive interaction supported by piecewise c2 smooth loops of a fixed length l formally given by x with 0 it is shown that the ground state of this operator is locally maximized by a circular we also conjecture that this property holds globally and show that the problem is related to an interesting family of geometric inequalities concerning mean values of chords of

a short note on the ratio between sign real and sign complex spectral radius of a real square matrix abstract for a real n n matrix a the sign real and the sign complex spectral radius invented by rump are respectively defined as r a max a x x r x r n 0 c a max a x x c x c n 0 for n 2 we prove r a n c a where the constant n 1 cos n 1 cos n is best possible

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

joint universal lossy coding and identification of stationary mixing sources with general alphabets in this paper we consider the problem of joint universal variable rate lossy coding and identification for parametric classes of stationary beta mixing sources with general polish alphabets compression performance is measured in terms of lagrangians while identification performance is measured by the variational distance between the true source and the estimated source provided that the sources are mixing at a sufficiently fast rate and satisfy certain smoothness and vapnik chervonenkis vc learnability conditions it is shown that for bounded metric distortions there exist universal schemes for joint lossy compression and identification whose lagrangian redundancies converge to zero as radic vn logn n as the block length n tends to infinity where vn is the vc dimension of a certain class of decision regions defined by the n dimensional marginal distributions of the sources furthermore for each n the decoder can identify n dimensional marginal of the active source up to a ball of radius o radic vnlogn n in variational distance eventually with probability one the results are supplemented by several examples of parametric sources satisfying the regularity conditions

eigenvalue ratios for sturm liouville operators in this paper we prove various optimal bounds for eigenvalue ratios for the sturm liouville equation p x y q x y w x y and certain specializations our results primarily concern the regular case with dirichlet boundary conditions though various extensions and generalizations to other situations are possible our results here extend the result m 1 m2 obtained in a previous paper for the one dimensional schrodinger equation y q x y y on a finite interval with dirichlet boundary conditions and nonnegative potential q 0 in particular we obtain m 1 km2 k where the constants k k satisfy 0 k p x w x k for all x if q 0 lower bounds can also be obtained our methods involve a slight modification of the prufer variable techniques employed in the schrodinger case we also examine the consequences of our recent proof of the payne polya weinberger conjecture in the one dimensional sturm liouville setting finally we compare our general bounds to the detailed analyses of keller and of mahar and willner for the special case of the inhomogeneous stretched string

rearrangement inequalities and applications to isoperimetric problems for eigenvalues let be a bounded c 2 domain in r n where n is any positive integer and let be the euclidean ball centered at 0 and having the same lebesgue measure as consider the operator l div ar vr v on with dirichlet boundary condition where the symmetric matrix

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

strongly consistent code based identification and order estimation for constrained finite state model classes observations are made of data generated by a stationary ergodic finite alphabet information source according to an unknown statistical model two modeling problems the identification problem and the order estimation problem are considered it is required that the given model class in the identification problem and each given model class in the order estimation problem be a constrained finite state model class which is a type that includes many model classes of information theoretic interest strongly consistent decision rules are exhibited in both the identification problem and the order estimation problem the decision rules are code based in that a model class is chosen based on how well a certain code for that class encodes the observed data the code used for a model class is based on the maximum likelihood code for that class and asymptotic code performance is gauged by means of a key property of divergence rate distance

bounds for ratios of the membrane eigenvalues abstract for a bounded planar region in r 2 we obtain the ratios of lower order eigenvalues of laplace operator combining our results with the recursive formula in cheng and yang 2007 11 we can obtain better upper bound of the k 1 th k 3 membrane eigenvalues

asymptotic properties of the maximum likelihood estimator in autoregressive models with markov regime an autoregressive process with markov regime is an autoregressive process for which the regression function at each time point is given by a nonobservable markov chain in this paper we consider the asymptotic properties of the maximum likelihood estimator in a possibly nonstationary process of this kind for which the hidden state space is compact but not necessarily finite consistency and asymptotic normality are shown to follow from uniform exponential forgetting of the initial distribution for the hidden markov chain conditional on the observations

the method of fundamental solutions applied to some inverse eigenproblems in this work we address the application of the method of fundamental solution mfs as a forward solver in some shape optimization problems in two and three dimensional domains it is well known kac s problem that a set of eigenvalues does not determine uniquely the shape of the domain moreover even the existence problem is not well defined due to the ashbaugh benguria inequality although these results constitute counterexamples in the general problem of shape determination from the eigenfrequencies we can address simpler questions in shape determination using the mfs for instance we apply the mfs to build domains that include a specific finite set of eigenvalues or that have an eigenmode that verifies some prescribed conditions as a particular case an eigenmode that defines a certain nodal line

hidden markov processes an overview of statistical and information theoretic aspects of hidden markov processes hmps is presented an hmp is a discrete time finite state homogeneous markov chain observed through a discrete time memoryless invariant channel in recent years the work of baum and petrie 1966 on finite state finite alphabet hmps was expanded to hmps with finite as well as continuous state spaces and a general alphabet in particular statistical properties and ergodic theorems for relative entropy densities of hmps were developed consistency and asymptotic normality of the maximum likelihood ml parameter estimator were proved under some mild conditions similar results were established for switching autoregressive processes these processes generalize hmps new algorithms were developed for estimating the state parameter and order of an hmp for universal coding and classification of hmps and for universal decoding of hidden markov channels these and other related topics are reviewed

on the many dirichlet laplacians on a non convex polygon and their approximations by point interactions abstract by birman and skvortsov it is known that if is a planar curvilinear polygon with n non convex corners then the laplace operator with domain h 2 h 0 1 is a closed symmetric operator with deficiency indices n n here we provide a kre n type resolvent formula for any self adjoint extensions of such an operator i e for the set of self adjoint non friedrichs dirichlet laplacians on and show that any element in this set is the norm resolvent limit of a suitable sequence of friedrichs dirichlet laplacians with n point interactions

a vector quantization approach to universal noiseless coding and quantization a two stage code is a block code in which each block of data is coded in two stages the first stage codes the identity of a block code among a collection of codes and the second stage codes the data using the identified code the collection of codes may be noiseless codes fixed rate quantizers or variable rate quantizers we take a vector quantization approach to two stage coding in which the first stage code can be regarded as a vector quantizer that quantizes the input data of length n to one of a fixed collection of block codes we apply the generalized lloyd algorithm to the first stage quantizer using induced measures of rate and distortion to design locally optimal two stage codes on a source of medical images two stage variable rate vector quantizers designed in this way outperform standard one stage fixed rate vector quantizers by over 9 db the tail of the operational distortion rate function of the first stage quantizer determines the optimal rate of convergence of the redundancy of a universal sequence of two stage codes we show that there exist two stage universal noiseless codes fixed rate quantizers and variable rate quantizers whose per letter rate and distortion redundancies converge to zero as k 2 n sup 1 log n when the universe of sources has finite dimension k this extends the achievability part of rissanen s theorem from universal noiseless codes to universal quantizers further we show that the redundancies converge as o n sup 1 when the universe of sources is countable and as o n sup 1 spl epsiv when the universe of sources is infinite dimensional under appropriate conditions

rates of convergence of minimum distance estimators and kolmogorov s entropy on construit des estimateurs robustes et on montre que le taux de convergence resultant depend naturellement d une fonction entropie

extremal eigenvalue gaps for the schrodinger operator with dirichlet boundary conditions we consider the problems of minimizing and maximizing the gap between the two lowest dirichlet eigenvalues of the schrodinger operator v x on a bounded domain in rn when the potential v is subjected to a p norm constraint we give characterization theorems for extremizing potentials we prove in particular that a second eigenvalue corresponding to a minimizing potential is always single

bounds on eigenvalues of dirichlet laplacian in this paper we investigate an eigenvalue problem of dirichlet laplacian on a bounded domain in an n dimensional euclidean space rn if k 1 is the k 1 th eigenvalue of dirichlet laplacian on then we prove that for n 41 and k geq 41 lambda k 1 leq k frac2n lambda 1 and for any n and k lambda k 1 leq c 0 n k k frac2n lambda 1 with c 0 n k leq j 2 n 2 1 j 2 n 2 1 1 where jp k denotes the k th positive zero of the standard bessel function jp x of the first kind of order p from the asymptotic formula of weyl and the partial solution of the conjecture of polya we know that our estimates are optimal in the sense of order of k

universal bounds for eigenvalues of the biharmonic operator abstract in this paper we study the eigenvalues of the clamped plate problem 2 u u in d u d u d 0 where d is a bounded connected domain in an n dimensional complete minimal submanifold of a unit m sphere s m 1 or of an m dimensional euclidean space r m let 0 1 2 k be the eigenvalues of the above problem we obtain universal bounds on k 1 in terms the first k eigenvalues independent of the domains for example when d is contained in an n dimensional complete minimal submanifold of s m 1 we show that k 1 1 k i 1 k i 1 k n i 1 k k 1 i 1 2 2 n 4 i 1 2 n 2 1 2 i 1 k k 1 i 1 2 4 i 1 2 n 2 1 2 from which one can obtain a more explicit upper bound on k 1 in terms of 1 k see corollary 1 when d is contained in a complete n dimensional minimal submanifold of r m we prove the inequality k 1 1 k i 1 k k 8 n 2 n 2 1 2 1 k i 1 k i k 1 i 1 2 which generalizes the main theorem in cheng and yang 2006 10 that states that the same estimate holds when d is a connected and bounded domain in r n

rates of convergence for empirical processes of stationary mixing sequences classical empirical process theory for vapnik cervonenkis classes deals mainly with sequences of independent variables this paper extends the theory to stationary sequences of dependent variables it establishes rates of convergence for mixing and mixing empirical processes indexed by classes of functions the method of proof depends on a coupling of the dependent sequence with sequences of independent blocks to which the classical theory can be applied a uniform o ns 1 s rate of convergence over v c classes is established for sequences whose mixing coefficients decay slightly faster than o n s

on the optimization of the principal eigenvalue for single centre point interaction operators in a bounded region we investigate relations between spectral properties of a single centre point interaction hamiltonian describing a particle confined to a bounded domain with a dirichlet boundary and the geometry of for this class of operators krein s formula yields an explicit representation of the resolvent in terms of the integral kernel of the unperturbed one d z 1 we use a moving plane analysis to characterize the behaviour of the ground state energy of the hamiltonian with respect to the point interaction position and the shape of in particular we establish some conditions showing how to place the interaction to optimize the principal eigenvalue

optimization of spectral functions of dirichlet laplacian eigenvalues we consider the shape optimization of spectral functions of dirichlet laplacian eigenvalues over the set of star shaped symmetric bounded planar regions with smooth boundary the regions are represented using fourier cosine coefficients and the optimization problem is solved numerically using a quasi newton method the method is applied to maximizing two particular nonsmooth spectral functions the ratio of the nth to first eigenvalues and the ratio of the nth eigenvalue gap to first eigenvalue both of which are generalizations of the payne polya weinberger ratio the optimal values and attaining regions for n 13 are presented and interpreted as a study of the range of the dirichlet laplacian eigenvalues for both spectral functions and each n the optimal attaining region has multiplicity two nth eigenvalue

optimal lower bounds for eigenvalues of linear and nonlinear neumann problems in this paper we prove a sharp lower bound for the first non trivial neumann eigenvalue 1 for the p laplace operator p 1 in a lipschitz bounded domain in n our estimate does not require any convexity assumption on and it involves the best isoperimetric constant relative to in a suitable class of convex planar domains our bound turns out to be better than the one provided by the payne weinberger inequality

a sharp bound for the ratio of the first two eigenvalues of dirichlet laplacians and extensions the first author s work was partially supported by fondecyt chile project 0132 88 and by a summer research fellowship provided by the research council of the university of missouri columbia he would like to thank the physics department and others at the universidad de chile for their hospitality during his visit in april 1990 when much of this research was completed the second author was supported in part by fondecyt chile projects 0132 88 and 1238 90 both authors also thank fritz gesztesy for general comments and encouragement

spectral geometry in a rotating frame properties of the ground state abstract we investigate spectral properties of the operator describing a quantum particle confined to a planar domain rotating around a fixed point with an angular velocity and demonstrate several properties of its principal eigenvalue 1 we show that as a function of rotation center position it attains a unique maximum and has no other extrema provided the said position is unrestricted furthermore we show that as a function the eigenvalue attains a maximum at 0 unique unless has a full rotational symmetry finally we present an upper bound to the difference 1 1 b where the last named eigenvalue corresponds to a disk of the same area as

symmetric decreasing rearrangement can be discontinuous suppose f xl x2 0 is a continuously differentiable function supported in the unit disk in the plane its symmetric decreasing rearrangement is the rotationally invariant function f xl x2 whose level sets are circles enclosing the same area as the level sets of f such rearrangement preserves lp norms but decreases convex gradient integrals e g p p 1 p 0 j 1 2 3 is a sequence of infinitely differentiable functions also supported in the unit disk which converge uniformly together with first derivatives to f the symmetzed functions also converge uniformly the real question is about convergence of the derivatives of the symmetrized functions we announce that the derivatives of the symmetrized functions need not converge strongly e g it can happen that fj f p 0 for every p we further characterize exactly those f s for which convergence is assured and for which it can fail

linear operators preserving the sign real spectral radius abstract define the sign real spectral radius of a real n n matrix a as 0 s a max s s 0 sa where 0 a max a real eigenvalue of a is the real spectral radius of a and s denotes the set of signature matrices i e s s s i the absolute value of matrices being meant entrywise in this paper we show that linear invertible operators f on the space of n n real matrices m n r preserving the sign real spectral radius are exactly the operators of the form f a p t d 1 sa t dp with p a permutation matrix d a diagonal matrix and s a signature matrix

on the redundancy of lossy source coding with abstract alphabets the redundancy problem of lossy source coding with abstract source and reproduction alphabets is considered for coding at a fixed rate level it is shown that for any fixed rate r 0 and any memoryless abstract alphabet source p satisfying some mild conditions there exists a sequence c sub n sub n 1 sup spl infin of block codes at the rate r such that the distortion redundancy of c sub n defined as the difference between the performance of c sub n and the distortion rate function d p r of p is upper bounded by spl part d p r spl part r ln n 2n o ln n n for coding at a fixed distortion level it is demonstrated that for any d 0 and any memoryless abstract alphabet source p satisfying some mild conditions there exists a sequence c sub n sub n 1 sup spl infin of block codes at the fixed distortion d such that the rate redundancy of c sub n defined as the difference between the performance of c sub n and the rate distortion function r p d of p is upper bounded by 7ln n 6n o ln n n these results strengthen the traditional berger s 1968 1971 abstract alphabet source coding theorem and extend the positive redundancy results of zhang yang and wei see ibid vol 43 no 1 p 71 91 1997 and ibid vol 42 p 803 21 1996 on lossy source coding with finite alphabets and the redundancy result of wyner see ibid vol 43 p 1452 64 1997 on block coding of memoryless gaussian sources

universal inequalities for eigenvalues of the buckling problem on spherical domains in this paper we study the eigenvalues of the buckling problem on domains in a unit sphere we obtain universal bounds on the k 1 th eigenvalue in terms of the first k eigenvalues independent of the domains

a second proof of the payne polya weinberger conjecture without using product representations or elaborate comparisons of zeros we prove the two key properties of the bessel function ratiojp 1jp 1 1x jpjp 1x that we used to prove the payne polya weinberger conjecture in these new proofs we use only differential equations and the rayleigh ritz method for estimating lowest eigenvalues the new proofs admit generalization to other related problems where our previous proofs fail

the minimum description length principle in coding and modeling we review the principles of minimum description length and stochastic complexity as used in data compression and statistical modeling stochastic complexity is formulated as the solution to optimum universal coding problems extending shannon s basic source coding theorem the normalized maximized likelihood mixture and predictive codings are each shown to achieve the stochastic complexity to within asymptotically vanishing terms we assess the performance of the minimum description length criterion both from the vantage point of quality of data compression and accuracy of statistical inference context tree modeling density estimation and model selection in gaussian linear regression serve as examples

the ppw conjecture in curved spaces abstract in euclidean 1 and hyperbolic 5 space and the round hemisphere 2 geodesic balls maximize the gap 2 1 of dirichlet eigenvalues among domains with fixed 1 we prove an upper bound on 2 1 for domains in manifolds with certain curvature bounds the inequality is sharp on geodesic balls in spaceforms

a universally acceptable smoothing factor for kernel density estimates we define a minimum distance estimate of the smoothing factor for kernel density estimates based on a methodology first developed by yatracos it is shown that if f nh denotes the kernel density estimate on r d for an i i d sample of size n drawn from an unknown density f where h is the smoothing factor and if f n is the kernel estimate with the same kernel and with the proposed new data based smoothing factor then under a regularity condition on the kernel k formula math this is the first published smoothing factor that can be proven to have this property

best possible upper and lower bounds for the zeros of the bessel function _ let j k denote the k th positive zero of the bessel function j x in this paper we prove that for 0 and k 1 2 3 ak 21 3 j k ak 21 3 3 20 ak 21 3 1 3 these bounds coincide with the first few terms of the well known asymptotic expansion j k ak 21 3 3 20 ak 21 3 1 3 as k being fixed where ak is the k th negative zero of the airy function ai x and so are best possible

nonasymptotic universal smoothing factors kernel complexity and yatracos classes we introduce a method to select a smoothing factor for kernel density estimation such that for all densities in all dimensions the l 1 error of the corresponding kernel estimate is not larger than three times the error of the estimate with the optimal smoothing factor plus a constant times log n n where n is the sample size and the constant depends only on the complexity of the kernel used in the estimate the result is nonasymptotic that is the bound is valid for each n the estimate uses ideas from the minimum distance estimation work of yatracos as the inequality is uniform with respect to all densities the estimate is asymptotically minimax optimal modulo a constant over many function classes

sums of laplace eigenvalues rotationally symmetric maximizers in the plane abstract the sum of the first n 1 eigenvalues of the laplacian is shown to be maximal among triangles for the equilateral triangle maximal among parallelograms for the square and maximal among ellipses for the disk provided the ratio area 3 moment of inertia for the domain is fixed this result holds for both dirichlet and neumann eigenvalues and similar conclusions are derived for robin boundary conditions and schrodinger eigenvalues of potentials that grow at infinity a key ingredient in the method is the tight frame property of the roots of unity for general convex plane domains the disk is conjectured to maximize sums of neumann eigenvalues

on low eigenvalues of the laplacian with mixed boundary conditions abstract by means of the so called symmetrization we study the eigenvalue problem for the laplace operator with mixed boundary conditions we obtain various bounds for combinations of the low eigenvalues and some sharp comparison results for the first eigenfunction in terms of bessel functions

perron frobenius theory for complex matrices abstract the purpose of this paper is to present a unified perron frobenius theory for nonnegative for real not necessarily nonnegative and for general complex matrices the sign real spectral radius was introduced for general real matrices this quantity was shown to share certain properties with the perron root of nonnegative matrices in this paper we introduce the sign complex spectral radius again this quantity extends many properties of the perron root of nonnegative matrices to general complex matrices various characterizations will be given and many open problems remain

universal bounds for eigenvalues of the biharmonic operator on riemannian manifolds in this paper we consider eigenvalues of the dirichlet biharmonic operator on compact riemannian manifolds with boundary possibly empty and prove a general inequality for them by using this inequality we study eigenvalues of the dirichlet biharmonic operator on compact domains in a euclidean space or a minimal submanifold of it and a unit sphere we obtain universal bounds on the k 1 th eigenvalue on such objects in terms of the first k eigenvalues independent of the domains the estimate for the k 1 th eigenvalue of bounded domains in a euclidean space improves an important inequality obtained recently by cheng and yang

brownian cuboid and its geometric characteristics abstract a class of random cuboid model which is defined brownian cuboid is constructed in this paper based on the structure characteristics of brownian cuboid and the properties of brownian motion geometric characteristics of brownian cuboid are discussed by which the probability estimates of the length of the space diagonal surface area and volume are obtained and moreover the small deviation estimates and some relevant results for the geometric characteristics of brownian cuboid are given additionally we directly derive the upper and lower bounds for the smallest positive zero of the bessel functions of the first kind

information theoretic asymptotics of bayes methods in the absence of knowledge of the true density function bayesian models take the joint density function for a sequence of n random variables to be an average of densities with respect to a prior the authors examine the relative entropy distance d sub n between the true density and the bayesian density and show that the asymptotic distance is d 2 log n c where d is the dimension of the parameter vector therefore the relative entropy rate d sub n n converges to zero at rate log n n the constant c which the authors explicitly identify depends only on the prior density function and the fisher information matrix evaluated at the true parameter value consequences are given for density estimation universal data compression composite hypothesis testing and stock market portfolio selection