information theoretic lengths of jacobi polynomials the information theoretic lengths of the jacobi polynomials p n x which are information theoretic measures renyi shannon and fisher of their associated rakhmanov probability density are investigated they quantify the spreading of the polynomials along the orthogonality interval 1 1 in a complementary but different way as the root mean square or standard deviation because contrary to this measure they do not refer to any specific point of the interval the explicit expressions of the fisher length are given the renyi lengths are found by the use of the combinatorial multivariable bell polynomials in terms of the polynomial degree n and the parameters the shannon length which cannot be exactly calculated because of its logarithmic functional form is bounded from below by using sharp upper bounds to general densities on 1 1 given in terms of various expectation values moreover its asymptotics is also pointed out finally several computational issues relative to these three quantities are carefully analyzed

lie symmetries for integrable evolution equations on the lattice in this work we show how to construct symmetries for the differential difference equations associated with the discrete schrodinger spectral problem we find the whole set of symmetries which in the continuous limit go into the lie point symmetries of the corresponding partial differential equation i e the korteweg de vries equation among these of particular relevance is the non autonomous symmetry which in the continuous limit goes into the dilation symmetry for the corresponding equation unlike the continuous case this symmetry turns out to be a master symmetry thus belonging to the infinite dimensional group of generalized symmetries

surface consistent deconvolution in the log fourier domain in the surface consistent hypothesis a seismic trace is the convolution of a source operator a receiver operator a reflectivity operator representing the subsurface structure and an offset related operator in the log fourier domain convolutions become sums and the log of signal amplitude at a given frequency is the sum of source receiver structural and offset related terms recovering the amplitude of the reflectivity for a given frequency is then a linear problem very similar to a surface consistent static correction problem however this linear system is underconstrained thus among the infinite number of possible solutions a particular one must be selected studies with real data support the choice of a spatially band limited solution the surface consistent operators can then be calculated very efficiently using an inverse hessian method applications to real seismic data show improvement compared with previous techniques surface consistent deconvolution is robust and fast in the log f

properties of the zeros of generalized basic hypergeometric polynomials we define the generalized basic hypergeometric polynomial of degree n in terms of the generalized basic hypergeometric function by choosing one of its parameters to allow the termination of the series after a finite number of summands in this paper we obtain a set of nonlinear algebraic equations satisfied by the n zeros of the polynomial moreover we obtain an n n matrix m defined in terms of the zeros of the polynomial which in turn depend on the parameters of the polynomial the eigenvalues of this remarkable matrix m are given by neat expressions that depend only on some of the parameters of the polynomial that is the matrix m is isospectral moreover in case the parameters that appear in the expressions for the eigenvalues of m are rational the matrix m has rational eigenvalues a diophantine property

full likelihood inference for abundance from continuous time capture recapture data capture recapture experiments are widely used cost effective sampling techniques for estimating population sizes or abundances in biology ecology demography epidemiology and reliability studies for continuous time capture recapture data existing estimation methods are based on conditional likelihoods and an inverse weighting estimating equation the corresponding wald type confidence intervals for the abundance may have severe undercoverage and their lower limits can be below the number of individuals captured we propose a full likelihood inference approach by combining a parametric or partial likelihood with the empirical likelihood under both parametric and semiparametric intensity models we demonstrate that the maximum likelihood estimator attains the semiparametric efficiency lower bound and that the full likelihood ratio statistic for the abundance is asymptotically 2 with 1 degree of freedom simulations indicate that compared with conditional likelihood based methods the maximum full likelihood estimator has a smaller mean square error and the likelihood ratio confidence intervals often have remarkable gains in coverage probability we illustrate the advantages of the proposed approach by analysing illegal immigrant data for the netherlands and prinia flaviventris data from hong kong

a discrete schrodinger spectral problem and associated evolution equations a recently proposed discrete version of the schrodinger spectral problem is considered the whole hierarchy of differential difference nonlinear evolution equations associated with this spectral problem is derived it is shown that a discrete version of the kdv sine gordon and liouville equations is included and that the so called inverse class in the hierarchy is local the whole class of related darboux and backlund transformations is also exhibited

proof of certain diophantine conjectures and identification of remarkable classes of orthogonal polynomials certain diophantine conjectures are proven and to do so certain remarkable classes of orthogonal polynomials are identified yielding additional diophantine findings

recursion operator and bi hamiltonian structure for integrable multidimensional lattices the recursion operator and the bi hamiltonian structure for an integrable two dimensional version of the toda chain are algorithmically derived from the corresponding linear problem the intimate relation between two dimensional theories and non abelian one dimensional theories is emphasized the analogies and differences between discrete and continuous integrable two dimensional systems are pointed out and discussed

the hamiltonian structure of the nonabelian toda hierarchy we show that a subset of the whole class of nonlinear differential difference equations associated with the discrete analog of the matrix schrodinger operator is endowed with a hamiltonian structure and exhibits an infinite number of integrals of motion in involution we also establish the relation between these integrals of motion and the transmission coefficient of the underlying linear problem and show that such a relation implies that for the whole class previously introduced there exists an infinite number of conserved quantities

a complex deformation of the classical gravitational many body problem that features many completely periodic motions a complex deformation of the newtonian equations of motion of the classical gravitational many body problem is introduced namely a many body problem that features a parameter and that reduces when this parameter vanishes to the standard equations of motion of newtonian gravitation for an arbitrary number of pointlike bodies with arbitrary masses and it is shown that when this parameter is instead positive 0 there is an open set of complex initial data such that all the complex motions originating from it are completely periodic with period t 2 and that the infinite measure of this set is a finite nonvanishing fraction of the measure of the entire set of initial data

tridiagonal matrices orthogonal polynomials and diophantine relations i it is well known that the eigenvalues of tridiagonal matrices can be identified with the zeros of polynomials satisfying three term recursion relations and being therefore members of an orthogonal set a class of such polynomials is identified some of which feature zeros given by simple formulae involving integer numbers in the process certain neat formulae are also obtained which perhaps deserve to be included in standard compilations since they involve classical polynomials such as the jacobi polynomials and other named polynomials

a class of integrable hamiltonian systems whose solutions are perhaps all completely periodic we show that the dynamical system characterized by the complex equations of motion qj i qj k 1 k jnqjqkf qj qk j 1 n with f x x x is hamiltonian and integrable and we conjecture that all its solutions qj t j 1 n are completely periodic with a period that is a finite integral multiple of t 2 here n is an arbitrary positive integer is an arbitrary nonvanishing real constant y y is the weierstrass function with arbitrary semiperiods and are two arbitrary constants special cases are f x 2 coth x 1 r2 sinh2 x f x 2 coth x f x 2 sinh x f x 2 x 1 2x2 and of course f x 2 x these findings as well as the conjecture which is shown to be true in some of these special cases are based on the possibility to recast these equations of motion in the modified lax form l i l l m with l and m appropriate n n matrix functions of the n dynamical variables qj and of their time derivatives qj

lie algebra contractions and symmetries of the toda hierarchy the lie algebra l of generalized and point symmetries of the equations in the toda hierarchy is shown to be a semidirect sum of two infinite dimensional lie algebras one perfect the other abelian in the continuous limit the structure of the lie algebra changes a contraction occurs with the lattice spacing as the contraction parameter in particular for the toda equation itself a set of five elements involving both point symmetries and generalized ones contracts to the point symmetry algebra of the potential kdv equation

integrable hierarchies of nonlinear difference difference equations and symmetries we construct the hierarchy of nonlinear difference difference equations associated with the discrete schrodinger spectral problem as examples of equations contained in this hierarchy we obtain the discrete time toda and volterra lattice equations in the case of the time discrete toda lattice we construct its lie point and generalized symmetries finally we present its backlund transformations and relate it to the already constructed symmetries

conditions for the existence of higher symmetries of evolutionary equations on the lattice in this paper we construct a set of five conditions necessary for the existence of generalized symmetries for a class of differential difference equations depending only on nearest neighboring interaction these conditions are applied to prove the existence of new integrable equations belonging to this class

the toda molecule equation and the 3 algorithm one of the well known convergence acceleration methods the e algorithm is investigated from the viewpoint of the toda molecule equation it is shown that the error caused by the algorithm is evaluated by means of solutions for the equation the acceleration algorithm based on the discrete toda molecule equation is also presented

on isochronous bruschi ragnisco ruijsenaars toda lattices equilibrium configurations behaviour in their neighbourhood diophantine relations and conjectures isochronous versions of the bruschi ragnisco ruijsenaars toda lattice and of some of its also integrable variants are introduced their equilibrium configurations are found when they exist and by investigating the motions of these systems near equilibrium some diophantine relations are obtained as well as some insight into the solution of those of these integrable models whose solutions are not yet known

a technique to identify solvable dynamical systems and a solvable generalization of the goldfish many body problem a simple approach is discussed which associates to solvable matrix equations solvable dynamical systems generally interpretable as interesting many body problems possibly involving auxiliary dependent variables in addition to those identifying the positions of the moving particles we then focus on cases in which the auxiliary variables can be altogether eliminated reobtaining thereby via this unified approach well known solvable many body problems and moreover a solvable extension of the goldfish model

variance estimation for systematic designs in spatial surveys summary in spatial surveys for estimating the density of objects in a survey region systematic designs will generally yield lower variance than random designs however estimating the systematic variance is well known to be a difficult problem existing methods tend to overestimate the variance so although the variance is genuinely reduced it is over reported and the gain from the more efficient design is lost the current approaches to estimating a systematic variance for spatial surveys are to approximate the systematic design by a random design or approximate it by a stratified design previous work has shown that approximation by a random design can perform very poorly while approximation by a stratified design is an improvement but can still be severely biased in some situations we develop a new estimator based on modeling the encounter process over space the new striplet estimator has negligible bias and excellent precision in a wide range of simulation scenarios including strip sampling distance sampling and quadrat sampling surveys and including populations that are highly trended or have strong aggregation of objects we apply the new estimator to survey data for the spotted hyena crocuta crocuta in the serengeti national park tanzania and find that the reported coefficient of variation for estimated density is 20 using approximation by a random design 17 using approximation by a stratified design and 11 using the new striplet estimator this large reduction in reported variance is verified by simulation

boundary conditions for integrable discrete chains a method of searching for boundary conditions consistent with the integrability property is suggested for discrete chains generating functions of conserved quantities of the corresponding finite dimensional reductions are given in terms of the l a pairs

open capture recapture models with heterogeneity ii jolly seber model summary estimation of abundance is important in both open and closed population capture recapture analysis but unmodeled heterogeneity of capture probability leads to negative bias in abundance estimates this article defines and develops a suite of open population capture recapture models using finite mixtures to model heterogeneity of capture and survival probabilities model comparisons and parameter estimation use likelihood based methods a real example is analyzed and simulations are used to check the main features of the heterogeneous models especially the quality of estimation of abundance survival recruitment and turnover the two major advances in this article are the provision of realistic abundance estimates that take account of heterogenetiy of capture and an appraisal of the amount of overestimation of survival arising from conditioning on the first capture when heterogeneity of survival is present

fully automated near surface analysis by surface consistent refraction method abstractindustry practices for near surface analysis indicate difficulties in coping with the increased number of channels in seismic acquisition systems and new approaches are needed to fully exploit the resolution embedded in modern seismic data sets to achieve this goal we have developed a novel surface consistent refraction analysis method for low relief geology to automatically derive near surface corrections for seismic data processing the method uses concepts from surface consistent analysis applied to refracted arrivals the key aspects of the method consist of the use of common midpoint cmp offset azimuth binning evaluation of mean traveltime and standard deviation for each bin rejection of anomalous first break fb picks derivation of cmp based traveltime offset functions conversion to velocity depth functions evaluation of long wavelength statics and calculation of surface consistent residual statics through waveform crosscorrelation residual time lags are evaluated in multiple cm

the staircase method integrals for periodic reductions of integrable lattice equations we show in full generality that the staircase method papageorgiou et al 1990 phys lett a 147 106 14 quispel et al 1991 physica a 173 243 66 provides integrals for mappings and correspondences obtained as traveling wave reductions of systems of integrable partial difference equations we apply the staircase method to a variety of equations including the korteweg de vries equation the five point bruschi calogero droghei equation the quotient difference qd algorithm and the boussinesq system we show that in all these cases if the staircase method provides r integrals for an n dimensional mapping with then one can introduce q 2r variables which reduce the dimension of the mapping from n to q these dimension reducing variables are obtained as joint invariants of k symmetries of the mappings our results support the idea that often the staircase method provides sufficiently many integrals for the periodic reductions of integrable lattice equations to be completely integrable we also study reductions on other quad graphs than the regular lattice and we prove linear growth of the multi valuedness of iterates of high dimensional correspondences obtained as reductions of the qd algorithm

relation between backlund transformations and higher continuous symmetries of the toda equation in this paper we study one aspect of the continuous symmetries of the toda equation namely we establish a correspondence between backlund transformations and continuous symmetries of the toda equation a symmetry transformation acting on a solution of the toda equation can be seen as a superposition of backlund transformations conversely a backlund transformation can be written at least formally as a composition of infinitely many higher symmetry transformations this result reinforces the opinion that the presence of sufficiently many continuous symmetries for discrete equations is an indication of their integrability

algebraic structure of discrete zero curvature equations and master symmetries of discrete evolution equations an algebraic structure related to discrete zero curvature equations is established it is used to give an approach for generating master symmetries of the first degree for systems of discrete evolution equations and an answer to why there exist such master symmetries the key of the theory is to generate nonisospectral flows t l l 0 from the discrete spectral problem associated with a given system of discrete evolution equations three examples are given

capture recapture when time and behavioral response affect capture probabilities summary we consider a capture recapture model in which capture probabilities vary with time and with behavioral response two inference procedures are developed under the assumption that recapture probabilities bear a constant relationship to initial capture probabilities these two procedures are the maximum likelihood method both unconditional and conditional types are discussed and an approach based on optimal estimating functions the population size estimators derived from the two procedures are shown to be asymptotically equivalent when population size is large enough the performance and relative merits of various population size estimators for finite cases are discussed the bootstrap method is suggested for constructing a variance estimator and confidence interval an example of the deer mouse analyzed in otis et al 1978 wildlife monographs62 93 is given for illustration

continuous symmetries of difference equations lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations in this article we review the results of a much more recent program the use of lie groups to study difference equations we show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners one is to use generalized symmetries acting on solutions of difference equations but leaving the lattice invariant the other is to restrict them to point symmetries but to allow them to also transform the lattice

properties of the zeros of the polynomials belonging to the q askey scheme in this paper we provide properties which are to the best of our knowledge new of the zeros of the polynomials belonging to the q askey scheme these findings include diophantine relations satisfied by these zeros when the parameters characterizing these polynomials are appropriately restricted

maximum empirical likelihood estimation for abundance in a closed population from capture recapture data summarycapture recapture experiments are widely used to collect data needed for estimating the abundance of a closed population to account for heterogeneity in the capture probabilities huggins 1989 and alho 1990 proposed a semiparametric model in which the capture probabilities are modelled parametrically and the distribution of individual characteristics is left unspecified a conditional likelihood method was then proposed to obtain point estimates and wald type confidence intervals for the abundance empirical studies show that the small sample distribution of the maximum conditional likelihood estimator is strongly skewed to the right which may produce wald type confidence intervals with lower limits that are less than the number of captured individuals or even are negative in this paper we propose a full empirical likelihood approach based on huggins and alho s model we show that the null distribution of the empirical likelihood ratio for the abundance is asymptotically chi squared with one degree of freedom and that the maximum empirical likelihood estimator achieves semiparametric efficiency simulation studies show that the empirical likelihood based method is superior to the conditional likelihood based method its confidence interval has much better coverage and the maximum empirical likelihood estimator has a smaller mean square error we analyse three datasets to illustrate the advantages of our empirical likelihood approach

properties of the zeros of generalized hypergeometric polynomials abstract we define the generalized hypergeometric polynomial of degree n as follows p n 1 p 1 q z m 0 n n m 1 m p m z n m m 1 m q m z n f q p 1 n 1 p 1 q 1 z here n is an arbitrary positive integer p and q are arbitrary nonnegative integers the p q parameters j and k are arbitrary generic possibly complex numbers m is the pochhammer symbol and f q p 1 0 1 p 1 q z is the generalized hypergeometric function in this paper we obtain a set of n nonlinear algebraic equations satisfied by the n zeros n of this polynomial we moreover manufacture an n n matrix l in terms of the 1 p q parameters n j k characterizing this polynomial and of its n zeros n and we show that it features the n eigenvalues m m k 1 q k 1 m m 1 n these n eigenvalues depend only on the q parameters k implying that the n n matrix l is isospectral for variations of the p parameters j and they clearly are integer or rational numbers if the q parameters k are themselves integer or rational numbers a nontrivial diophantine property

inference on population size in binomial detectability models many models for biological populations including simple mark recapture models and distance sampling models involve a binomially distributed number n of observations x 1 x n on members of a population of size n two popular estimators of n where is a vector parameter are the maximum likelihood estimator and the conditional maximum likelihood estimator c c based on the conditional distribution of x 1 x n given n we derive the large n asymptotic distributions of and c c and give formulae for the biases of log and log c we show that the difference c is remarkably of order 1 and we give a simple formula for the leading part of this difference simulations indicate that in many cases this formula is very accurate and that confidence intervals based on the asymptotic distribution have excellent coverage an extension to product binomial models is given

finite dimensional representations of difference operators and the identification of remarkable matrices two square matrices of arbitrary order n are introduced they are defined in terms of n arbitrary numbers zn and of an arbitrary additional parameter a respectively q and provide finite dimensional representations of the two operators acting on a function f z as follows f z a f z a respectively f qz f z q 1 z these representations are exact in a sense explained in the paper when the function f z is a polynomial in z of degree less than n this formalism allows to transform difference equations valid in the space of polynomials of degree less than n into corresponding matrix vector equations as an application of this technique several remarkable square matrices of order n are identified which feature explicitly n arbitrary numbers zn or the n zeros of polynomials belonging to the askey and q askey schemes several of these findings have a diophantine character

initial value problems for lattice equations we describe how to pose straight band initial value problems for lattice equations defined on arbitrary stencils in finitely many directions we arrive at discrete goursat problems and in the remaining directions we find cauchy problems next we consider s1 s2 periodic initial value problems in the goursat directions the periodic solutions are generated by correspondences in the cauchy directions assuming the equation to be multi linear the periodic solution can be obtained uniquely by iteration of a particularly simple mapping whose dimension is a piecewise linear function of s1 s2

nonlinear harmonic oscillators the existence is noted of assemblies of an arbitrary number of complex oscillators or equivalently of an arbitrary even number of real oscillators characterized by newtonian equations of motion acceleration equal force with one body velocity dependent linear forces and many body velocity independent cubic forces all the nonsingular solutions of which are isochronous completely periodic with the same period as for the singular solutions as usual they emerge in the context of the initial value problem from a closed domain in phase space having lower dimensionality

integrable and solvable many body problems in the plane via complexification a simple prescription allows us to transform by appropriate complexification any one dimensional many body problem describing motions on a line with newtonian acceleration equals forces equations of motion featuring forces which depend analytically on the positions and possibly on the velocities of the particles and which are moreover scaling or translation invariant into a two dimensional many body problem describing motions in the plane with rotation invariant equations of motion if the original one dimensional model is hamiltonian the two dimensional model is also hamiltonian in fact bi hamiltonian in this manner starting from known integrable or solvable one dimensional many body problems integrable or solvable many body problems in the plane generally featuring much richer dynamics are obtained several examples are exhibited finally another more direct prescription is outlined to transform by complexification almost any one dimensional many body problem into a rota

an isochronous variant of the ruijsenaars toda model equilibrium configurations behavior in their neighborhood diophantine relations an isochronous variant of the ruijsenaars toda integrable many body problem is introduced an equilibrium configuration of this dynamical system is identified and by investigating the motions in its neighborhood diophantine relations are obtained

likelihood and observed geometries a partir du point de vue de la conditionnalite on etablit un parallele observe a la theorie de chentsov efron amari etc en utilisant l information observee et un tenseur d asymetrie observee

recapture models under equality constraints for the conditional capture probabilities alessio farcomeni uniroma1 it summary we introduce a general class of capture recapture models in which capture probabilities depend on the previous capture history we discuss constrained versions of the saturated model based on equality constraints despite the seeming complexity inference can be performed through a simple estimating equation the approach is illustrated on a data set about the population of great copper butterflies in willamette valley of o regon it has long been recognized that in capture recapture exper iments the previous capture history may influence capture probabilities and failure to acknowl edge this fact may lead to biased estimates of the unknown population size one of the first model s used denoted by mb included behavioural response to capture by using different probabi lities before and after first capture otis et al 1978 pollock 1991 other examples are given by the markov chain models of yang chao 2005 and the persistence models detailed in ramsey usner 2003 and ramsey severns 2010 in this paper we propose a general class of models which are based on conditional capture probabilities we restrict the satura ted model through equality constraints arguing that the resulting class of models is rich enough to s ummarize a plethora of real data situations the restricted class can be also defined through partitions of the capture histories where capture probabilities conditional on different hist ories belonging to the same partition are equal surprisingly enough the approach based on maximizing the conditional likelihood leads to a straightforward inference based on a simple estimating equation an interesting feature of the proposed model class is that the previous capture histor y can be allowed to influence the capture probability in very flexible ways different models can be fitted and compared through information criteria or likelihood ratio tests

evolution equations associated with the discrete analog of the matrix schrodinger spectral problem solvable by the inverse spectral transform through the generalized wronskian technique we derive the whole class of nonlinear differential difference equations associated with the discrete analog of the matrix schrodinger spectral problem for such equations we briefly discuss soliton solutions continuum limit and backlund transformations

a technique to identify solvable dynamical systems and another solvable extension of the goldfish many body problem we take advantage of the simple approach recently discussed which associates to solvable matrix equations solvable dynamical systems interpretable as interesting many body problems possibly involving auxiliary dependent variables in addition to those identifying the positions of the moving particles starting from a solvable matrix evolution equation we obtain the corresponding many body model and note that in one case the auxiliary variables can be altogether eliminated obtaining thereby an also hamiltonian extension of the goldfish model the solvability of this novel model and of its isochronous variant is exhibited a related as well solvable model is also introduced as well as its isochronous variant finally the small oscillations of the isochronous models around their equilibrium configurations are investigated and from their isochronicity certain diophantine relations are evinced

relativistic toda systems we present and study poincare invariant generalizations of the galilei invariant toda systems the classical nonperiodic systems are solved by means of an explicit action angle transformation

goldfishing a new solvable many body problem a recent technique allows one to identify and investigate solvable dynamical systems naturally interpretable as classical many body problems being characterized by equations of motion of newtonian type generally in two dimensional space in this paper we tersely review results previously obtained in this manner and present novel findings of this kind mainly solvable variants of the goldfish many body model including models that feature isochronous classes of completely periodic solutions different formulations of these models are presented the behavior of one of these isochronous dynamical systems in the neighborhood of its equilibrium configuration is investigated and in this manner some remarkable diophantine findings are obtained