optimal binary index assignments for a class of equiprobable scalar and vector quantizers the problem of scalar and vector quantization in conjunction with a noisy binary symmetric channel is considered the issue is the assignment of the shortest possible distinct binary sequences to quantization levels or vectors so as to minimize the mean squared error caused by channel errors by formulating the assignment as a matrix or vector in the scalar case and showing that the mean squared error due to channel errors is determined by the projections of its columns onto the eigenspaces of the multidimensional channel transition matrix a class of source quantizer pairs is identified for which the optimal index assignment has a simple and natural form among other things this provides a simpler and more accessible proof of the result of crimmins et al 1969 that the natural binary code is an optimal index assignment for the uniform scalar quantizer and uniform source it also provides a potentially useful approach to further developments in source channel coding

deformed oscillator algebra approach of some quantum superintegrable lissajous systems on the sphere and of their rational extensions we extend the construction of 2d superintegrable hamiltonians with separation of variables in spherical coordinates using combinations of shift ladder and supercharge operators to models involving rational extensions of the two parameter lissajous systems on the sphere these new families of superintegrable systems with integrals of arbitrary order are connected with jacobi exceptional orthogonal polynomials of type i or ii and supersymmetric quantum mechanics moreover we present an algebraic derivation of the degenerate energy spectrum for the one and two parameter lissajous systems and the rationally extended models these results are based on finitely generated polynomial algebras casimir operators realizations as deformed oscillator algebras and finite dimensional unitary representations such results have only been established so far for 2d superintegrable systems separable in cartesian coordinates which are related to a class of polynomial algebras that display a simpler structure we also point out how the structure function of these deformed oscillator algebras is directly related with the generalized heisenberg algebras spanned by the nonpolynomial integrals

combined source channel vector quantization using deterministic annealing the authors present a new approach to combined source channel vector quantization the method derived within information theory and probability theory utilizes deterministic annealing to avoid some local minima that trap conventional descent algorithms such as the generalized lloyd algorithm the resulting vector quantizers satisfy the necessary conditions for local optimality for the noisy channel case they tested the method against several versions of the noisy channel generalized lloyd algorithm on stationary first order gauss markov sources with a binary symmetric channel the method outperformed other methods under all test conditions with the gains over noisy channel gla growing with the codebook size the quantizers designed using deterministic annealing are also shown to behave robustly under channel mismatch conditions as a comparison with a separate source channel system over a large range of test channel conditions the method outperformed a bandwidth equivalent system incorporating a hamming code also for severe channel conditions the method produces solutions with explicit error control coding

optimal quantizer design for noisy channels an approach to combined source channel coding we present an analysis of the zero memory quantization of memoryless sources when the quantizer output is to be encoded and transmitted across a noisy channel necessary conditions for the joint optimization of the quantizer and the encoder decoder pair are presented and an iterative algorithm for obtaining a locally optimum system is developed the performance of this locally optimal system obtained for the class of generalized gaussian distributions and the binary symmetric channel is compared against the optimum performance theoretically attainable using rate distortion theoretic arguments as well as against the performance of lloyd max quantizers encoded using the natural binary code and the folded binary code it is shown that this optimal design could result in substantial performance improvements the performance improvements are more noticeable at high bit rates and for broad tailed densities

the q version of a theorem of bochner askey and wilson 1985 found a family of orthogonal polynomials in the variable s k 1 2 k 1 k that satisfy a q difference equation of the form a k p n s qk p n s k b k p n s k q p n s k theta n p n s k n 0 1 we show here that this property characterizes the askey wilson polynomials the proof is based on an operator identity of independent interest this identity can be adapted to prove other characterization results indeed it was used in grunbaum and haine 1996 to give a new derivation of the result of bochner alluded to in the title of this paper we give the appropriate identity for the case of difference equations leading to the wilson polynomials but pursue the consequences only in the case of q difference equations leading to the askey wilson and big q jacobi polynomials this approach also works in the discrete case and should yield the results in leonard 1982

infinitely many shape invariant potentials and cubic identities of the laguerre and jacobi polynomials we provide analytic proofs for the shape invariance of the recently discovered odake and sasaki phys lett b 679 414 2009 two families of infinitely many exactly solvable one dimensional quantum mechanical potentials these potentials are obtained by deforming the well known radial oscillator potential or the darboux poschl teller potential by a degree l l 1 2 eigenpolynomial the shape invariance conditions are attributed to new polynomial identities of degree 3l involving cubic products of the laguerre or jacobi polynomials these identities are proved elementarily by combining simple identities

orthogonal polynomial solutions of linear ordinary differential equations abstract this paper surveys the latest known results concerning the classification of differential equations of the form k 1 n a k x y k x y x having a sequence of polynomial eigenfunctions p n x n 0 that are orthogonal with respect to some real bilinear form since the publishing of the erice report in 1990 several new significant results and generalizations have been discovered

the generalized quantum isotonic oscillator recently it has been proved that a nonlinear quantum oscillator generalization of the isotonic one is exactly solvable for certain values of its parameters here we show that the schrodinger equation for such an oscillator can be transformed into a confluent heun equation we give a very simple and efficient algorithm to solve it numerically no matter what the values of the parameters are algebraic quasi polynomial solutions for particular values of the parameters are found

recurrence relations of the multi indexed orthogonal polynomials v racah and q racah types in previous papers we discussed the recurrence relations of the multi indexed orthogonal polynomials of the laguerre jacobi wilson and askey wilson types in this paper we explore those of the racah and q racah types for the m indexed q racah polynomials we derive 3 2m term recurrence relations with variable dependent coefficients and 1 2l term l m 1 recurrence relations with constant coefficients based on the latter the generalized closure relations and the creation and annihilation operators of the quantum mechanical systems described by the multi indexed q racah polynomials are obtained in appendix b and appendix c we present a proof and some data of the recurrence relations with constant coefficients for the multi indexed wilson and askey wilson polynomials in previous papers we discussed the recurrence relations of the multi indexed orthogonal polynomials of the laguerre jacobi wilson and askey wilson types in this paper we explore those of the racah and q racah types for the m indexed q racah polynomials we derive 3 2m term recurrence relations with variable dependent coefficients and 1 2l term l m 1 recurrence relations with constant coefficients based on the latter the generalized closure relations and the creation and annihilation operators of the quantum mechanical systems described by the multi indexed q racah polynomials are obtained in appendix b and appendix c we present a proof and some data of the recurrence relations with constant coefficients for the multi indexed wilson and askey wilson polynomials

a variational principle for markov processes in this note we first present a result concerning a variational principle for general markov processes then we apply it to spin particle systems to obtain a full form of a variational principle characterizing the stationary markov laws of the systems a related extreme decomposition for any stationary distribution of such markov systems is also given

the generalized confluent supersymmetry algorithm representations and integral formulas we extend the confluent version of the higher order supersymmetry susy formalism to general linear differential equations of second order closed form representations of transformation functions their wronskians and of the general solutions to susy transformed equations are derived we use these results to construct formulas for resolving multiple integrals of special functions in terms of wronskians

large deviations for almost markovian processes large deviation theorems for the empirical distribution of almost markovian processes are proven relative entropy identifies the rate function and its definition depends only on the process

n fold supersymmetry and quasi solvability associated with x2 laguerre polynomials we construct a new family of quasi solvable and n fold supersymmetric quantum systems where each hamiltonian preserves an exceptional polynomial subspace of codimension 2 we show that the family includes as a particular case the recently reported rational radial oscillator potential whose eigenfunctions are expressed in terms of the x2 laguerre polynomials of the second kind in addition we find that the two kinds of the x2 laguerre polynomials are ingeniously connected with each other by the n fold supercharge

asymptotic and interlacing properties of zeros of exceptional jacobi and laguerre polynomials abstract in this paper we state and prove some properties of the zeros of exceptional jacobi and laguerre polynomials generically the zeros of exceptional polynomials fall into two classes the regular zeros which lie in the interval of orthogonality and the exceptional zeros which lie outside that interval we show that the regular zeros have two interlacing properties one is the natural interlacing between zeros of consecutive polynomials as a consequence of their sturm liouville character while the other one shows interlacing between the zeros of exceptional and classical polynomials a heine mehler type formula is provided for the exceptional polynomials which allows to derive the asymptotic behaviour of their regular zeros for large degree n and fixed codimension m we also describe the location and the asymptotic behaviour of the m exceptional zeros which converge for large n to fixed values

on channel constrained vector quantization and index assignment for discrete memoryless channels this correspondence introduces an approach to the design of channel robust vector quantizers design criteria are derived based on a new expression for the channel distortion as a function of the code vectors of the quantizer the introduced framework for design and analysis holds for arbitrary discrete memoryless channels simulations demonstrate good performance the correspondence also presents new results regarding the channel distortion as a function of the index assignment given to the code vectors of a vector quantizer

infinite families of non hermitian hamiltonians associated with exceptional xm jacobi polynomials using an appropriate change of variable the schr dinger equation is transformed into a second order differential equation satisfied by recently discovered jacobi type xm exceptional orthogonal polynomials this facilitates the derivation of infinite families of exactly solvable hermitian as well as non hermitian trigonometric scarf potentials and a finite number of hermitian and an infinite number of non hermitian symmetric hyperbolic scarf potentials the bound state solutions of all these potentials are associated with the aforesaid exceptional orthogonal polynomials these infinite families of potentials are shown to be extensions of the conventional trigonometric and hyperbolic scarf potentials by the addition of some rational terms characterized by the presence of classical jacobi polynomials all the members of a particular family of these rationally extended polynomial dependent potentials have the same energy spectrum and possess translational shape invariant symmetry the obtained non hermitian trigonometric scarf potentials are shown to be quasi hermitian in nature ensuring the reality of the associated energy spectra

exactly solvable potentials with finitely many discrete eigenvalues of arbitrary choice we address the problem of possible deformations of exactly solvable potentials having finitely many discrete eigenvalues of arbitrary choice as kay and moses showed in 1956 reflectionless potentials in one dimensional quantum mechanics are exactly solvable with an additional time dependence these potentials are identified as the soliton solutions of the korteweg de vries kdv hierarchy an n soliton potential has the time t and 2n positive parameters k1 kn and cj j 1 n corresponding to n discrete eigenvalues kj2 the eigenfunctions are elementary functions expressed by the ratio of determinants the darboux crum krein adler transformations or the abraham moses transformations based on eigenfunction deletions produce lower soliton number potentials with modified parameters cj we explore various identities satisfied by the eigenfunctions of the soliton potentials which reflect the uniqueness theorem of gel fand levitan marchenko equations for separable degenerate kernels

extending romanovski polynomials in quantum mechanics some extensions of the third class romanovski polynomials also called romanovski pseudo jacobi polynomials which appear in bound state wavefunctions of rationally extended scarf ii and rosen morse i potentials are considered for the former potentials the generalized polynomials satisfy a finite orthogonality relation while for the latter an infinite set of relations among polynomials with degree dependent parameters is obtained both types of relations are counterparts of those known for conventional polynomials in the absence of any direct information on the zeros of the romanovski polynomials present in denominators the regularity of the constructed potentials is checked by taking advantage of the disconjugacy properties of second order differential equations of schrodinger type it is also shown that on going from scarf i to scarf ii or from rosen morse ii to rosen morse i potentials the variety of rational extensions is narrowed down from types i ii and iii to type iii only

orthogonal polynomials from hermitian matrices a unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of hermitian matrices of finite or infinite dimensions it can be considered as a matrix version of exactly solvable schrodinger equations the hermitian matrices factorizable hamiltonians are real symmetric tridiagonal jacobi matrices corresponding to second order difference equations by solving the eigenvalue problem in two different ways the duality relation of the eigenpolynomials and their dual polynomials is explicitly established through the techniques of exact heisenberg operator solution and shape invariance various quantities the two types of eigenvalues the eigenvalues and the sinusoidal coordinates the coefficients of the three term recurrence the normalization measures and the normalisation constants etc are determined explicitly

quantization of the lienard ii equation and jacobi s last multiplier in this paper the role of jacobi s last multiplier in mechanical systems with a position dependent mass is unveiled in particular we map the lienard ii equation to a position dependent mass system the quantization of the lienard ii equation is then carried out using the point canonical transformation method together with the von roos ordering technique finally we show how their eigenfunctions and eigenspectrum can be obtained in terms of associated laguerre and exceptional laguerre functions

the design of joint source and channel trellis waveform coders the generalized lloyd algorithm is applied to the design of joint source and channel trellis waveform coders to encode discrete time continuous amplitude stationary and ergodic sources operating over discrete memoryless noisy channels experimental results are provided for independent and autoregressive gaussian sources binary symmetric channels and absolute error and squared error distortion measures performance of the joint codes is compared with the tandem combination of a trellis source code and a trellis channel code on the independent gaussian source using the squared error distortion measure operating over an additive white gaussian noise channel it is observed that the jointly optimized codes achieve performance close to or better than that of separately optimized tandem codes of the same constraint length performance improvement via a predictive joint source and channel trellis code is demonstrated for the autoregressive gaussian source using the squared error distortion measure

statistical mechanics of probabilistic cellular automata we investigate the behavior of discrete time probabilistic cellular automata pca which are markov processes on spin configurations on ad dimensional lattice from a rigorous statistical mechanics point of view in particular we exploit whenever possible the correspondence between stationary measures on the space time histories of pcas on d and translation invariant gibbs states for a related hamiltonian on d 1 this leads to a simple large deviation formula for the space time histories of the pca and a proof that in a high temperature regime the stationary states of the pca are gibbsian we also obtain results about entropy fluctuations and correlation inequalities and demonstrate uniqueness of the invariant state and exponential decay of correlations in a high noise regime we discuss phase transitions in the low noise or low temperature regime and review toom s proof of nonergodicity of a certain class of pcas

combined state adding and state deleting approaches to type iii multi step rationally extended potentials applications to ladder operators and superintegrability type iii multi step rationally extended harmonic oscillator and radial harmonic oscillator potentials characterized by a set of k integers m1 m2 mk such that m1 m2 mk with mi even resp odd for i odd resp even are considered the state adding and state deleting approaches to these potentials in a supersymmetric quantum mechanical framework are combined to construct new ladder operators the eigenstates of the hamiltonians are shown to separate into mk 1 infinite dimensional unitary irreducible representations of the corresponding polynomial heisenberg algebras these ladder operators are then used to build a higher order integral of motion for seven new infinite families of superintegrable two dimensional systems separable in cartesian coordinates the finite dimensional unitary irreducible representations of the polynomial algebras of such systems are directly determined from the ladder operator action on the constituent one dimensional hamiltonian eigenstates and provide an algebr

exceptional orthogonal polynomials and the darboux transformation we adapt the notion of the darboux transformation to the context of polynomial sturm liouville problems as an application we characterize the recently described xm laguerre polynomials in terms of an isospectral darboux transformation we also show that the shape invariance of these new polynomial families is a direct consequence of the permutability property of the darboux crum transformation

the sobolev orthogonality and spectral analysis of the laguerre polynomials l n k for positive integers k for k n we consider the analysis of the classical laguerre differential expression l k y x 1 x ke x x k 1e xy x rx ke xy x x 0 where r 0 is fixed in several nonisomorphic hilbert and hilbert sobolev spaces in one of these spaces specifically the hilbert space l2 0 x ke x it is well known that the glazman krein naimark theory produces a self adjoint operator a k generated by l k that is bounded below by rl where i is the identity operator on l2 0 x ke x consequently as a result of a general theory developed by littlejohn and wellman there is a continuum of left definite hilbert spaces h s k v s k s k s 0 and left definite self adjoint operators b s k s 0 associated with the pair l2 0 x ke x a k for a k and each of the operators b s k it is the case that the tail end sequence l n k n k of laguerre polynomials form a complete set of eigenfunctions in the corresponding hilbert spaces in 1995 kwon and littlejohn introduced a hilbert sobolev space w k 0 in which the entire sequence of laguerre polynomials is orthonormal in this paper we construct a self adjoint operator in this space generated by the second order laguerre differential expression l k having l n k n 0 as a complete set of eigenfunctions the key to this construction is in identifying a certain closed subspace of w k 0 with the kth left definite vector space v k k

exceptional laguerre and jacobi polynomials and the corresponding potentials through darboux crum transformations a simple derivation is presented of the four families of infinitely many shapeinvariant hamiltonians corresponding to the exceptional laguerre and jacobi polynomials the darboux crum transformations are applied to connect the well known shape invariant hamiltonians of the radial oscillator and the darboux p oschl teller potential to the shape invariant potentials of odake sasaki dutta and roy derived the two lowest members of the exceptional laguerre polynomials by this method the method is expanded to its full generality and many other ramifications including the aspects of the generalized bochner problem and the bispectral property of the exceptional orthogonal polynomials are discussed

disconjugacy regularity of multi indexed rationally extended potentials and laguerre exceptional polynomials the power of the disconjugacy properties of second order differential equations of schrodinger type to check the regularity of rationally extended quantum potentials connected with exceptional orthogonal polynomials is illustrated by re examining the extensions of the isotonic oscillator or radial oscillator potential derived in kth order supersymmetric quantum mechanics or multistep darboux backlund transformation method the function arising in the potential denominator is proved to be a polynomial with a nonvanishing constant term whose value is calculated by induction over k the sign of this term being the same as that of the already known highest degree term the potential denominator has the same sign at both extremities of the definition interval a property that is shared by the seed eigenfunction used in the potential construction by virtue of disconjugacy such a property implies the nodeless character of both the eigenfunction and the resulting potential

new ladder operators for a rational extension of the harmonic oscillator and superintegrability of some two dimensional systems new ladder operators are constructed for a rational extension of the harmonic oscillator associated with type iii hermite exceptional orthogonal polynomials and characterized by an even integer m the eigenstates of the hamiltonian separate into m 1 infinite dimensional unitary irreducible representations of the corresponding polynomial heisenberg algebra these ladder operators are used to construct a higher order integral of motion for two superintegrable two dimensional systems separable in cartesian coordinates the polynomial algebras of such systems provide for the first time an algebraic derivation of the whole spectrum through their finite dimensional unitary irreducible representations

large deviations for gibbs random fields a large deviation principle for gibbs random fields on zd is proven and a corresponding large deviations proof of the gibbs variational formula is given a generalization of the lanford theory of large deviations is also obtained

orthogonality of the jacobi polynomials with negative integer parameters it is well known that the jacobi polynomials pn x are orthogonal with respect to a quasi definite linear functional whenever and 1 are not negative integer numbers recently sobolev orthogonality for these polynomials has been obtained for a negative integer and not a negative integer and also for the case negative integer numbers in this paper we give a sobolev orthogonality for the jacobi polynomials in the remainder cases

the hadamard transform a tool for index assignment we show that the channel distortion for maximum entropy encoders due to noise on a binary symmetric channel is minimized if the vector quantizer can be expressed as a linear transform of a hypercube the index assignment problem is regarded as a problem of linearizing the vector quantizer we define classes of index assignments with related properties within which the best index assignment is found by sorting not searching two powerful algorithms for assigning indices to the codevectors of nonredundant coding systems are presented one algorithm finds the optimal solution in terms of linearity whereas the other finds a very good but suboptimal solution in a very short time

recurrence relations of the multi indexed orthogonal polynomials in paper ii we presented conjectures of the recurrence relations with constant coefficients for the multi indexed orthogonal polynomials of laguerre jacobi wilson and askey wilson types in this paper we present a proof for the laguerre and jacobi cases their bispectral properties are also discussed which gives a method to obtain the coefficients of the recurrence relations explicitly this paper extends to the laguerre and jacobi cases the bispectral techniques recently introduced by gomez ullate et al j approx theory 204 1 2016 e print arxiv 1506 03651 math ca to derive explicit expressions for the coefficients of the recurrence relations satisfied by exceptional polynomials of hermite type

asymptotic bounds on optimal noisy channel quantization via random coding asymptotically optimal zero delay vector quantization in the presence of channel noise is studied using random coding techniques first an upper bound is derived for the average rth power distortion of channel optimized k dimensional vector quantization at transmission rate r on a binary symmetric channel with bit error probability spl epsiv the upper bound asymptotically equals 2 sup rrg spl epsiv k r where k k r 1 log sub 2 l 2 spl radic spl epsiv 1 spl epsiv spl les g spl epsiv k r spl les 1 for all spl epsiv spl ges 0 lim sub spl epsiv spl rarr 0 g spl epsiv k r 1 and lim sub k spl rarr spl infin g spl epsiv k r 1 numerical computations of g spl epsiv k r are also given this result is analogous to zador s 1982 asymptotic distortion rate of 2 sup rr for quantization on noiseless channels next using a random coding argument on nonredundant index assignments a useful upper bound is derived in terms of point density functions on the minimum mean squared error of high resolution regular vector quantizers in the presence of channel noise the formula provides an accurate approximation to the distortion of a noisy channel quantizer whose codebook is arbitrarily ordered finally it is shown that the minimum mean squared distortion of a regular noisy channel vq with a randomized nonredundant index assignment is in probability asymptotically bounded away from zero

on the performance and complexity of channel optimized vector quantizers the performance and complexity of channel optimized vector quantizers are studied for the gauss markov source observations on the geometric structure of these quantizers are made which have an important implication on the encoding complexity for the squared error distortion measure it is shown that an operation equivalent to a euclidean distance measurement with respect to an appropriately defined set of points used to identify the encoding regions can be used to perform the encoding this implies that the encoding complexity is proportional to the number of encoding regions it is then demonstrated that for very noisy channels and a heavily correlated source when the codebook size is large the number of encoding regions is considerably smaller than the codebook size implying a reduction in encoding complexity

joint source and noisy channel trellis encoding corresp in a trellis encoding communication system the decoder is a time invariant nonlinear filter with finite memory sliding block code and the encoder is a trellis search algorithm matched to the decoder a coding theorem is established for a trellis encoding of a stationary and ergodic source over a discrete memoryless noisy channel which shows that such communication systems can perform arbitrarily close to the source distortion rate function evaluated at the channel capacity

the strong ergodic theorem for densities generalized shannon mcmillan breiman theorem let xi x2 be a stationary process with probability densities f x1 x2 xn with respect to lebesgue measure or with respect to a markov measure with a stationary transition measure it is shown that the sequence of relative entropy densities 1 n log f xi x2 x0 converges almost surely this long conjectured result extends the l convergence obtained by moy perez and kieffer and generalizes the shannon mcmillanbreiman theorem to nondiscrete processes the heart of the proof is a new martingale inequality which shows that logarithms of densities are l dominated

space time large deviations for interacting particle systems the asymptotic behavior of the space time empirical averages of several classes of interacting particle systems is studied here the authors prove that the empirical averages obey a process level large deviation principle whose rate function is given explicitly in the case of spin systems the authors establish some relevant properties of the rate function such as the correspondence between its minima and the invariant measures for the process finally for discrete time non ergodic systems the authors give some results on the critical behavior of the empirical averages

hamming coding of dct compressed images over noisy channels theoretical and simulation results of using hamming codes with the two dimensional discrete cosine transform 2d dct at a transmitted data rate of 1 bit pixel over a binary symmetric channel bsc are presented the design bit error rate ber of interest is 10 2 the 7 4 15 11 and 31 26 hamming codes are used to protect the most important bits in each 16 by 16 transformed block where the most important bits are determined by calculating the mean squared reconstruction error mse contributed by a channel error in each individual bit a theoretical expression is given which allows the number of protected bits to achieve minimum mse for each code rate to be computed by comparing these minima the best code and bit allocation can be found objective and subjective performance results indicate that using the 7 4 hamming code to protect the most important 2d dct coefficients can substantially improve reconstructed image quality at a ber of 10 2 furthermore the allocation of 33 out of the 256 bits per block to channel coding does not noticeably degrade reconstructed image quality in the absence of channel errors

image coding using robust quantization for noisy digital transmission a robust quantizer is developed for encoding memoryless sources and transmission over the binary symmetric channel bsc the system combines channel optimized scalar quantization cosq with all pass filtering the latter performed using a binary phase scrambling descrambling method applied to a broad class of sources the robust quantizer achieves the same performance as the gaussian cosq for the memoryless gaussian source this quantizer is used in image coding for transmission over a bsc the peak signal to noise ratio psnr performance degrades gracefully as the channel bit error rate increases

rational extensions of the quantum harmonic oscillator and exceptional hermite polynomials we prove that every rational extension of the quantum harmonic oscillator that is exactly solvable by polynomials is monodromy free and therefore can be obtained by applying a finite number of state deleting darboux transformations on the harmonic oscillator equivalently every exceptional orthogonal polynomial system of hermite type can be obtained by applying a darboux crum transformation to the classical hermite polynomials exceptional hermite polynomial systems only exist for even codimension 2m and they are indexed by the partitions of m we provide explicit expressions for their corresponding orthogonality weights and differential operators and a separate proof of their completeness exceptional hermite polynomials satisfy a 2l 3 recurrence relation where l is the length of the partition explicit expressions for such recurrence relations are given

large deviations and ergodicity for spin particle systems in this paper we investigate the large deviation principle ldp for spin particle systems with possibly vanishing flip rates the situation turns out to be much more complicated if the flip rates are allowed to be zero than the one considered by dai where the systems are assumed to have strictly positive flip rates the upper and lower large deviation bounds are studied respectively the two governing rate functions are compared and a variational principle is given we then apply the results to obtain some new large deviation estimates for the occupation times of attractive systems in particular we prove a strong form of exponential convergence for ergodic systems

a bochner type characterization theorem for exceptional orthogonal polynomials abstract it was recently conjectured that every system of exceptional orthogonal polynomials is related to a classical orthogonal polynomial system by a sequence of darboux transformations in this paper we prove this conjecture which paves the road to a complete classification of all exceptional orthogonal polynomials in some sense this paper can be regarded as the extension of bochner s result for classical orthogonal polynomials to the exceptional class as a supplementary result we derive a canonical form for exceptional operators based on a bilinear formalism and prove that every exceptional operator has trivial monodromy at all primary poles

admissibility condition for exceptional laguerre polynomials abstract we prove a necessary and sufficient condition for the integrability of the weight associated to the exceptional laguerre polynomials this condition is very much related to the fact that the associated second order differential operator has no singularities in 0

time dependent potentials associated with exceptional orthogonal polynomials we introduce a method for generating rational extensions of time dependent potentials such that the associated schrodinger equation admits solutions in terms of exceptional orthogonal polynomials our method is applicable to position dependent schrodinger equations as well as to their conventional counterparts for constant mass

rational extension and jacobi type xm solutions of a quantum nonlinear oscillator we construct a rational extension of a recently studied nonlinear quantum oscillator model our extended model is shown to retain exact solvability admitting a discrete spectrum and corresponding closed form solutions that are expressed through jacobi type xm exceptional orthogonal polynomials

conditionally exactly solvable potentials and exceptional orthogonal polynomials it is shown that polynomials associated with solutions of certain conditionally exactly solvable potentials obtained via unbroken as well as broken supersymmetry belong to the category of exceptional orthogonal polynomials some properties of such polynomials e g recurrence relation ladder operators differential equations etc have been obtained

exceptional orthogonal polynomials and generalized schur polynomials we show that the exceptional orthogonal polynomials can be viewed as confluent limits of the generalized schur polynomials introduced by sergeev and veselov

a class of exactly solvable rationally extended non central potentials in two and three dimensions we start from a seven parameter six continuous and one discrete family of non central exactly solvable potentials in three dimensions and construct a wide class of ten parameters six continuous and four discrete family of rationally extended exactly solvable non central real as well as pt symmetric complex potentials the energy eigenvalues and the eigenfunctions of these extended non central potentials are obtained explicitly and it is shown that the wave eigenfunctions of these potentials are either associated with the exceptional orthogonal polynomials or some type of new polynomials which can be further re expressed in terms of the corresponding classical orthogonal polynomials similarly we also construct a wide class of rationally extended exactly solvable non central real as well as complex pt invariant potentials in two dimensions we start from a seven parameter six continuous and one discrete family of non central exactly solvable potentials in three dimensions and construct a wide class of ten parameters six continuous and four discrete family of rationally extended exactly solvable non central real as well as pt symmetric complex potentials the energy eigenvalues and the eigenfunctions of these extended non central potentials are obtained explicitly and it is shown that the wave eigenfunctions of these potentials are either associated with the exceptional orthogonal polynomials or some type of new polynomials which can be further re expressed in terms of the corresponding classical orthogonal polynomials similarly we also construct a wide class of rationally extended exactly solvable non central real as well as complex pt invariant potentials in two dimensions

detecting nonergodicity in continuous time spin systems we develop an empirical test aimed at detecting nonergodicity from a single sample of a spin system we show that the test is asymptotically correct and we give explicit asymptotics for the error probability the key tool consists in some new large deviation estimates

solvable rational extensions of the morse and kepler coulomb potentials we show that it is possible to generate an infinite set of solvable rational extensions from every exceptional first category translationally shape invariant potential this is made by using darboux backlund transformations based on unphysical regular riccati schrodinger functions which are obtained from specific symmetries associated with the considered family of potentials

classical quantum correspondence for shape invariant systems a quantization procedure which has recently been introduced for the analysis of painleve equations is applied to a general time independent potential of a newton equation this analysis shows that the quantization procedure preserves the exact solvability property for the class of shape invariant potentials when a general potential is considered the quantization procedure involves the solution of a gambier xxvii transcendental equation explicit examples involving classical and exceptional orthogonal laguerre and jacobi polynomials are discussed

multi indexed meixner and little q jacobi laguerre polynomials as the fourth stage of the project multi indexed orthogonal polynomials we present the multi indexed meixner and little q jacobi laguerre polynomials in the framework of discrete quantum mechanics with real shifts defined on the semi infinite lattice in one dimension they are obtained in a similar way to the multi indexed laguerre and jacobi polynomials reported earlier from the quantum mechanical systems corresponding to the original orthogonal polynomials by multiple application of the discrete analogue of the darboux transformations or the crum krein adler deletion of virtual state vectors the virtual state vectors are the solutions of the matrix schrodinger equation on all the lattice points having negative energies and infinite norm this is in good contrast to the q racah systems defined on a finite lattice in which the virtual state vectors satisfy the matrix schrodinger equation except for one of the two boundary points

a generalization of a theorem of bochner we describe all polynomial solutions to the general second order operator equation in the askey wilson operators thus extending a theorem of bochner from differential operators to askey wilson divided difference operators

on finite block length quantization distortion we investigate the upper and lower bounds on the quantization distortions for independent and identically distributed continuous sources in the finite block length regime we derive a lower bound on the quantization distortion as well as an upper bound on the quantization distortion based on random quantization codebooks moreover we apply the obtained bounds to the continuous gaussian source for the gaussian source we propose a computationally tractable method to numerically compute the upper and lower bounds for both bounded and unbounded quantization codebooks numerical results show that the gap between the upper and lower bounds is small for the block length of several hundreds and the upper and lower bounds are tighter than those proposed in the existing works

binary lattice vector quantization with linear block codes and affine index assignments we determine analytic expressions for the performance of some low complexity combined source channel coding systems the main tool used is the hadamard transform in particular we obtain formulas for the average distortion of binary lattice vector quantization with affine index assignments linear block channel coding and a binary symmetric channel the distortion formulas are specialized to nonredundant channel codes for a binary symmetric channel and then extended to affine index assignments on a binary asymmetric channel various structured index assignments are compared our analytic formulas provide a computationally efficient method for determining the performance of various coding schemes one interesting result shown is that for a uniform source and uniform quantizer the natural binary code is never optimal for a nonsymmetric channel even though it is known to be optimal for a symmetric channel

performance of quantizers on noisy channels using structured families of codes achievable distortion bounds are derived for the cascade of structured families of binary linear channel codes and binary lattice vector quantizers it is known that for the cascade of asymptotically good channel codes and asymptotically good vector quantizers the end to end distortion decays to zero exponentially fast as a function of the overall transmission rate and is achieved by choosing a channel code rate that is independent of the overall transmission rate we show that for certain families of practical channel codes and binary lattice vector quantizers the overall distortion can be made to decay to zero exponentially fast as a function of the square root of transmission rate this is achieved by carefully choosing a channel code rate that decays to zero as the transmission rate grows explicit channel code rate schedules are obtained for several well known families of channel codes

a study of vector quantization for noisy channels several issues related to vector quantization for noisy channels are discussed an algorithm based on simulated annealing is developed for assigning binary codewords to the vector quantizer code vectors it is shown that this algorithm could result in dramatic performance improvements as compared to randomly selected codewords a modification of the simulated annealing algorithm for binary codeword assignment is developed for the case where the bits in the codeword are subjected to unequal error probabilities resulting from unequal levels of error protection an algorithm for the design of an optimal vector quantizer for a noisy channel is briefly discussed and its robustness under channel mismatch conditions is studied numerical results for a stationary first order gauss markov source and a binary symmetric channel are provided it is concluded that the channel optimized vector quantizer design algorithm if used carefully can result in a fairly robust system with no additional delay the case in which the communication channel is nonstationary as in mobile radio channels is studied and some preliminary ideas for quantizer design are presented

pseudo gray coding a pseudo gray code is an assignment of n bit binary indexes to 2 points in a euclidean space so that the hamming distance between two points corresponds closely to the euclidean distance pseudo gray coding provides a redundancy free error protection scheme for vector quantization vq of analog signals when the binary indexes are used as channel symbols on a discrete memoryless channel and the points are signal codevectors binary indexes are assigned to codevectors in a way that reduces the average quantization distortion introduced in the reproduced source vectors when a transmitted index is corrupted by channel noise a globally optimal solution to this problem is generally intractable due to an inherently large computational complexity a locally optimal solution the binary switching algorithm is introduced based on the objective of minimizing a useful upper bound on the average system distortion the algorithm yields a significant reduction in average distortion and converges in reasonable running times the sue of pseudo gray coding is motivated by the increasing need for low bit rate vq based encoding systems that operate on noisy channels such as in mobile radio speech communications

robust vector quantization by a linear mapping of a block code in this paper we propose a novel technique for vector quantizer design where the reconstruction vectors are given by a linear mapping of a binary block code lmbc the lmbc framework provides a relation between the index bits and the reconstruction vectors through mapping properties we define a framework show its flexibility and give optimality conditions we consider source optimized vector quantization vq where the objective is to directly obtain a vq with inherent good channel robustness properties several instructive theoretical results and properties of the distortion experienced due to channel noise are demonstrated these results are used to guide the design process both optimization algorithms and a block code selection procedure are devised experimental results for gauss markov sources show that quantization performance close to an unconstrained vq is obtained with a short block code which implies a constrained vq the resulting vqs have better channel noise robustness than conventional vqs designed with the generalized lloyd algorithm gla and splitting initialization even when a post processing index assignment algorithm is applied to the gla based vq we have thus demonstrated a unique method for direct design resulting in an inherent good index assignment combined with small losses in quantization performance

invariance properties of wronskian type determinants of classical and classical discrete orthogonal polynomials abstract given a finite set f f 1 f k of nonnegative integers written in increasing order of magnitude and a classical discrete family p n n of orthogonal polynomials charlier meixner krawtchouk or hahn we consider the casorati determinant det p f i x j 1 i j 1 k in this paper we prove an invariance property of this kind of casorati determinants when the set f is substituted by the set i f 0 1 2 max f max f f f f our approach uses orthogonal polynomials that are eigenfunctions of higher order difference operators krall discrete polynomials these polynomials are orthogonal with respect to certain christoffel transforms of the classical discrete measures by passing to the limit this invariance property is extended to wronskian type determinants whose entries are hermite laguerre and jacobi polynomials

quantizers with uniform decoders and channel optimized encoders scalar quantizers with uniform decoders and channel optimized encoders are studied for a uniform source on 0 1 and binary symmetric channels two families of affine index assignments are considered the complemented natural code cnc introduced here and the natural binary code nbc it is shown that the nbc never induces empty cells in the quantizer encoder whereas the cnc can nevertheless we show that the asymptotic distributions of quantizer encoder cells for the nbc and the cnc are equal and are uniform over a proper subset of the source s support region empty cells act as a form of implicit channel coding an effective channel code rate associated with a quantizer designed for a noisy channel is defined and computed for the codes studied by explicitly showing that the mean squared error mse of the cnc can be strictly smaller than that of the nbc we also demonstrate that the nbc is suboptimal for a large range of transmission rates and bit error probabilities this contrasts with the known optimality of the nbc when either both the encoder and decoder are not channel optimized or when only the decoder is channel optimized

families of superintegrable hamiltonians constructed from exceptional polynomials we introduce a family of exactly solvable two dimensional hamiltonians whose wave functions are given in terms of laguerre and exceptional jacobi polynomials the hamiltonians contain purely quantum terms which vanish in the classical limit leaving only a previously known family of superintegrable systems additional higher order integrals of motion are constructed from ladder operators for the considered orthogonal polynomials proving the quantum system to be superintegrable

recurrence relations of the multi indexed orthogonal polynomials vi meixner pollaczek and continuous hahn types in previous papers we discussed the recurrence relations of the multi indexed orthogonal polynomials of the laguerre jacobi wilson askey wilson racah and q racah types in this paper we explore those of the meixner pollaczek and continuous hahn types for the m indexed meixner pollaczek and continuous hahn polynomials we present 3 2m term recurrence relations with variable dependent coefficients and 1 2l term l m 1 recurrence relations with constant coefficients based on the latter the generalized closure relations and the creation annihilation operators of the quantum mechanical systems described by the multi indexed meixner pollaczek and continuous hahn polynomials are obtained

quantizers with uniform encoders and channel optimized decoders scalar quantizers with uniform encoders and channel optimized decoders are studied for uniform sources and binary symmetric channels it is shown that the natural binary code nbc and folded binary code fbc induce point density functions that are uniform on proper subintervals of the source support whereas the gray code gc does not induce a point density function the mean squared errors mse for the nbc fbc gc and for randomly chosen index assignments are calculated and the nbc is shown to be mean squared optimal among all possible index assignments for all bit error rates and all quantizer transmission rates in contrast it is shown that almost all index assignments perform poorly and have degenerate codebooks

from pca s to equilibrium systems and back stationary measures for probabilistic cellular automata pca s ind dimensions give rise to space time histories whose statistics may naturally be described by gibbs states ind 1 dimensions for an interaction energy obtained from the pca in this note we study the converse question do all gibbs states for this correspond to statistical space time histories for the pca our main result states that the answer is yes at least for translation invariant or periodic gibbs states thus ergodicity questions for pca s can at least partially be formulated as questions of uniqueness of gibbs states

a gibbs variational principle in space time for infinite dimensional diffusions we show that the set of stationary weak solutions for a class of infinite dimensional stochastic differential equations coincides with the set of shift invariant space time gibbs fields for a certain potential the key step consists in proving the gibbs variational principle for space time gibbs fields

large deviations for empirical processes of interacting particle systems in this paper we study process level large deviations for spin flip particle systems and more general markov processes we prove that a modification of a well known donsker varadhan entropy function can be used to govern the large deviation lower bounds for any such process starting from a class of initial distributions related to an extremal invariant measure of the process in some specific cases we obtain the full large deviation principle

an extended class of orthogonal polynomials defined by a sturm liouville problem abstract we present two infinite sequences of polynomial eigenfunctions of a sturm liouville problem as opposed to the classical orthogonal polynomial systems these sequences start with a polynomial of degree one we denote these polynomials as x 1 jacobi and x 1 laguerre and we prove that they are orthogonal with respect to a positive definite inner product defined over the compact interval 1 1 or the half line 0 respectively and they are a basis of the corresponding l 2 hilbert spaces moreover we prove a converse statement similar to bochner s theorem for the classical orthogonal polynomial systems if a self adjoint second order operator has a complete set of polynomial eigenfunctions p i i 1 then it must be either the x 1 jacobi or the x 1 laguerre sturm liouville problem a rodrigues type formula can be derived for both of the x 1 polynomial sequences

whittaker hill equation and semifinite gap schrodinger operators a periodic one dimensional schrodinger operator is called semifinite gap if every second gap in its spectrum is eventually closed we construct explicit examples of semifinite gap schrodinger operators in trigonometric functions by applying darboux transformations to the whittaker hill equation we give a criterion for the regularity of the corresponding potentials and investigate the spectral properties of the new operators

heun s equation generalized hypergeometric function and exceptional jacobi polynomial we study heun s differential equation in the case that one of the singularities is apparent in particular we propose a conjecture that solutions of heun s equation in this case also satisfy a generalized hypergeometric equation which can be described in a more general form and establish it in some cases we demonstrate the application of our results to exceptional jacobi polynomials

exceptional orthogonal polynomials exactly solvable potentials and supersymmetry we construct two new exactly solvable potentials giving rise to bound state solutions to the schrodinger equation which can be written in terms of the recently introduced laguerre or jacobi type x1 exceptional orthogonal polynomials these potentials extending either the radial oscillator or the scarf i potential by the addition of some rational terms turn out to be translationally shape invariant as their standard counterparts and isospectral to them