ince s limits for confluent and double confluent heun equations we find pairs of solutions to a differential equation which is obtained as a special limit of a generalized spheroidal wave equation this is also known as confluent heun equation one solution in each pair is given by a series of hypergeometric functions and converges for any finite value of the independent variable z while the other is given by a series of modified bessel functions and converges for z z0 where z0 denotes a regular singularity for short the preceding limit is called ince s limit after ince who have used the same procedure to get the mathieu equations from the whittaker hill ones we find as well that when z0 tends to zero the ince limit of the generalized spheroidal wave equation turns out to be the ince limit of a double confluent heun equation for which solutions are provided finally we show that the schrodinger equation for inverse fourth and sixth power potentials reduces to peculiar cases of the double confluent heun equation and its ince s limit respectively

comment on dirac equation in the background of the nutku helicoid metric j math phys 48 092301 2007 the dirac equation written on the boundary of the nutku helicoid space consists of a system of ordinary differential equations we tried to analyze this system and we found that it has a higher singularity than those of the heun equations which give the solutions of the dirac equation in the bulk we also lose an independent integral of motion on the boundary this facts explain why we could not find the solution of the system on the boundary in terms of known functions we make the stability analysis of the helicoid and catenoid cases and end up with an appendix which gives a new example wherein one encounters a form of the heun equation

relationship between the energy eigenstates of calogero sutherland models with oscillator and coulomb like potentials we establish a simple algebraic relationship between the energy eigenstates of the rational calogero sutherland model with harmonic oscillator and coulomb like potentials we show that there is an underlying su 1 1 algebra in both of these models which plays a crucial role in such an identification furthermore we show that our analysis is in fact valid for any many particle system in arbitrary dimensions whose potential term apart from the oscillator or the coulomb like potential is a homogeneous function of coordinates of degree 2 the explicit coordinate transformation which maps the coulomb like problem to the oscillator one has also been determined in some specific cases

a number of quasiexactly solvable n body problems we present several examples of quasiexactly solvable n body problems in one two and higher dimensions we study various aspects of these problems in some detail in particular we show that in some of these examples the corresponding polynomials form an orthogonal set and many of their properties are similar to those of the bender dunne polynomials we also discuss qes problems where the polynomials do not form an orthogonal set

the heun equation and the calogero moser sutherland system i the bethe ansatz method we propose and develop the bethe ansatz method for the heun equation as an application holomorphy of the perturbation for the bc1 inozemtsev model from the trigonometric model is proved

the heun equation and the calogero moser sutherland system iv the hermite krichever ansatz we develop a theory for the hermite krichever ansatz on the heun equation as a byproduct we find formulae which reduce hyperelliptic integrals to elliptic ones

on solvable potentials related to so 2 2 ii natanzon potentials general natanzon potentials related to the so 2 2 group are studied the s matrices for systems under consideration are related to intertwining operators between weyl equivalent most degenerate principal series representations of so 2 2

complex periodic potentials with a finite number of band gaps we obtain several new results for the complex generalized associated lame potential v x a a 1 msn2 y m b b 1 msn2 y k m m f f 1 msn2 y k m ik m m g g 1 msn2 y ik m m where y x k m 2 ik m 2 sn y m is the jacobi elliptic function with modulus parameter m and there are four real parameters a b f g first we derive two new duality relations which when coupled with a previously obtained duality relation permit us to relate the band edge eigenstates of the 24 potentials obtained by permutations of the parameters a b f g second we pose and answer the question how many independent potentials are there with a finite number a of band gaps when a b f g are integers and a b f g 0 for these potentials we clarify the nature of the band edge eigenfunctions we also obtain several analytic results when at least one of the four parameters is a half integer as a by product we also obtain new solutions of heun s differential equation

source identity and kernel functions for inozemtsev type systems the inozemtsev hamiltonian is an elliptic generalization of the differential operator defining the bcn trigonometric quantum calogero sutherland model and its eigenvalue equation is a natural many variable generalization of the heun differential equation we present kernel functions for inozemtsev hamiltonians and chalykh feigin veselov sergeev type deformations thereof our main result is a solution of a heat type equation for a generalized inozemtsev hamiltonian which is the source of all these kernel functions applications are given including a derivation of simple exact eigenfunctions and eigenvalues of the inozemtsev hamiltonian

the potts model on the bethe lattice abstract the approach developed by thompson for the ising model is used to calculate the free energy of the potts model on the bethe lattice this is accomplished by introducing a simplification in obtaining the recursion relationship for the local magnetization and a generalization of the hyperbolic functions

the hermite krichever ansatz for fuchsian equations with applications to the sixth painleve equation and to finite gap potentials several results including integral representation of solutions and hermite krichever ansatz on heun s equation are generalized to a certain class of fuchsian differential equations and they are applied to equations which are related with physics we investigate linear differential equations that produce painleve equation by monodromy preserving deformation and obtain solutions of the sixth painleve equation which include hitchin s solution the relationship with finite gap potential is also discussed we find new finite gap potentials namely we show that the potential which is written as the sum of the treibich verdier potential and additional apparent singularities of exponents 1 and 2 is finite gap which extends the result obtained previously by treibich we also investigate the eigenfunctions and their monodromy of the schrodinger operator on our potential

properties of predictors for autoregressive time series abstract the prediction of the n s th observation of the pth order autoregressive process is investigated the mean square of the predictor error through terms of order n 1 conditional on yn y n 1 y n p 1 is obtained for the stationary normal process the mean squared error expression is similar to the usual regression formula for the variance of the predictor error the usual regression formula for the estimated variance of a predictor error and its generalization to s period prediction is shown to provide a consistent estimator of the mean squared error of the least squares predictor for both stationary and non stationary processes

banded regularization of autocovariance matrices in application to parameter estimation and forecasting of time series summary the paper addresses a large p small n problem in a time series framework and considers properties of banded regularization of an empirical autocovariance matrix of a time series process utilizing the banded autocovariance matrix enables us to fit a much longer auto regressive ar p model to the observed data than typically suggested by the akaike information criterion while controlling how many parameters are to be estimated precisely and the level of accuracy we present results on asymptotic consistency of banded autocovariance matrices under the frobenius norm and provide a theoretical justification on optimal band selection by using cross validation remarkably the cross validation loss function for banded prediction is related to the conditional mean square prediction error and thus may be viewed as an alternative model selection criterion the procedure proposed is illustrated by simulations and application to predicting the sea surface temperature index in the nino 3 4 region

partially solvable quantum many body problems in d dimensional space d 1 2 3 a simple technique employed almost three decades ago to manufacture partially solvable quantum many body problems is revisited a quantum problem is partially solvable if only some of its eigenvalues and eigenfunctions can be exhibited the models thereby generated are characterized by hamiltonians of normal form i e standard kinetic plus momentum independent potential energy in most cases the latter features three body in addition to two body and one body interactions the setting refers to d dimensional space the examples focus on d 1 d 2 and d 2 and include generalizations of and additional results on cases recently discussed in the literature as well as new models

inverse moment bounds for sample autocovariance matrices based on detrended time series and their applications abstract in this paper we assume that observations are generated by a linear regression model with short or long memory dependent errors we establish inverse moment bounds for k n dimensional sample autocovariance matrices based on the least squares residuals also known as the detrended time series where k n n k n and n is the sample size these results are then used to derive the mean square error bounds for the finite predictor coefficients of the underlying error process based on the detrended time series we further estimate the inverse of the n dimensional autocovariance matrix r n 1 of the error process using the banded cholesky factorization by making use of the aforementioned inverse moment bounds we obtain the convergence of moments of the difference between the proposed estimator and r n 1 under spectral norm

quantum mass correction for the twisted kink we present an analytic result for the 1 loop quantum mass correction in semiclassical quantization for the twisted 4 kink on s1 without explicit knowledge of the fluctuation spectrum for this purpose we use the contour integral representation of the spectral zeta function by solving the bethe ansatz equations for the n 2 lam equation we obtain an analytic expression for the corresponding spectral discriminant we discuss the renormalization issues of this model an energetically preferred size for the compact space is finally obtained

a class of n body problems with nearest and next to nearest neighbour interactions we obtain the exact ground state and part of the excitation spectrum in one dimension on a line and the exact ground state on a circle in the case where the n particles are interacting via nearest and next to nearest neighbour interactions furthermore using the exact ground state we establish a mapping between these n body problems and the short range dyson models introduced recently to model intermediate spectral statistics using this mapping we compute the one and two point functions of a related many body theory in the thermodynamic limit and show the absence of long range order however quite remarkably we prove the existence of an off diagonal long range order in the symmetrized version of the related many body theory generalization of the models to other root systems is also considered besides we also generalize the model on the full line to higher dimensions finally we consider a model in two dimensions in which all the states exhibit novel correlations

renormalisation groups with periodic and aperiodic orbits motivated by recent results for some frustrated hierarchical models the authors analyse the consequences of renormalisation group transformations with periodic and aperiodic orbits they find that the free energy has in general an infinite number of singularities this occurs already when the transformation has a stable periodic orbit they present an idealised example and a new class of hierarchical potts models to which the theory applies

a new perspective on single and multi variate differential equations we elaborate upon a new method of solving linear differential equations of arbitrary order which is applicable to a wide class of single and multi variate equations our procedure separates the operator part of the equation under study in to a part containing a function of the euler operator and constants and another one retaining the rest the solution of the equation is then obtained from the monomials or the monomial symmetric functions for the multi variate case which are the eigenfunctions of the euler operator novel exponential forms of the solutions of the differential equations enable one to analyze the underlying symmetries of the equations and explore the algebraic structures of the solution spaces in a straightforward manner the procedure allows one to derive various properties of the orthogonal polynomials and functions in a unified manner after showing how the generating functions and rodriguez formulae emerge naturally in this method we briefly outline the generalization of the present approach to the multi variate case

examples of heun and mathieu functions as solutions of wave equations in curved spaces we give examples of where the heun function exists as solutions of wave equations encountered in general relativity as a new example we find that while the dirac equation written in the background of nutku helicoid metric yields mathieu functions as its solutions in four spacetime dimensions the trivial generalization to five dimensions results in the double confluent heun function we reduce this solution to the mathieu function with some transformations

novel correlations in two dimensions two body problem we discuss a many body hamiltonian with two and three body interactions in two dimensions introduced recently by murthy bhaduri and sen apart from an analysis of some exact solutions in the many body system we analyse in detail the two body problem which is completely solvable we show that the solution of the two body problem reduces to solving a known differential equation due to heun we show that the two body spectrum becomes remarkably simple for large interaction strengths and the level structure resembles that of the landau levels we also clarify the ultraviolet regularization which is needed to define an inverse square potential properly and discuss its implications for our model

model averaging by jackknife criterion in models with dependent data abstract the past decade witnessed a literature on model averaging by frequentist methods for the most part the asymptotic optimality of various existing frequentist model averaging estimators has been established under i i d errors recently hansen and racine hansen b e racine j 2012 jackknife model averaging journal of econometrics 167 38 46 developed a jackknife model averaging jma estimator which has an important advantage over its competitors in that it achieves the lowest possible asymptotic squared error under heteroscedastic errors in this paper we broaden hansen and racine s scope of analysis to encompass models with i a non diagonal error covariance structure and ii lagged dependent variables thus allowing for dependent data we show that under these set ups the jma estimator is asymptotically optimal by a criterion equivalent to that used by hansen and racine a monte carlo study demonstrates the finite sample performance of the jma estimator in a variety of model settings

uniform moment bounds of fisher s information with applications to time series in this paper a uniform over some parameter space moment bound for the inverse of fisher s information matrix is established this result is then applied to develop moment bounds for the normalized least squares estimate in nonlinear stochastic regression models the usefulness of these results is illustrated using time series models in particular an asymptotic expression for the mean squared prediction error of the least squares predictor in autoregressive moving average models is obtained this asymptotic expression provides a solid theoretical foundation for some model selection criteria

phase diagram of the one state potts model on the cayley tree the phase diagram of the one state potts model on the closed asymmetric cayley tree with branching ratior 2 is obtained from the bethe peierls map the route to chaos via the period doubling cascade is obtained by considering the antiferromagnetic coupling limit the connection of the potts model with the percolation problem is shown by calculating the order parameter its susceptibility the internal energy and the specific heat as well as their asymptotic behavior at the paramagnetic ferromagnetic critical point due to the type of the lattice and to the polynomial character of the map this is the simplest known example of a mckay berker kirkpatrick spin glass

potentials of the heun class we review different methods of generating potentials such that the one dimensional schrodinger equation odse can be transformed into the hypergeometric equation we compare our results with previous studies and complement the subject with new findings our main result is to derive new classes of potentials such that the odse can be transformed into the heun equation and its confluent cases the generalized heun equation is also considered

integrability versus separability for the multi centre metrics the multi centre metrics are a family of euclidean solutions of the empty space einstein equations with self dual curvature for this full class we determine which metrics do exhibit an extra conserved quantity quadratic in the momenta induced by a killing stackel tensor our systematic approach brings to light a subclass of metrics which correspond to new classically integrable dynamical systems within this subclass we analyze on the one hand the separation of coordinates in the hamilton jacobi equation and on the other hand the construction of some new killing yano tensors

bootstrap automata the dynamics of destruction abstract simulations for bootstrap cellular automata on an oriented square lattice are shown to be consistent with exact results on the critical threshold p c 1 and a finite size scaling bound regimes above below and close to p c for finite samples are detailed and characterized by a wide choice of updating and coding algorithms

the perturbation of the quantum calogero moser sutherland system and related results the hamiltonian of the trigonometric calogero sutherland model coincides with a certain limit of the hamiltonian of the elliptic calogero moser model in other words the elliptic hamiltonian is a perturbed operator of the trigonometric one

algebraic solution for the natanzon confluent potentials an algebraic method is used to solve the bound state problem for the most general natanzon potential for which the schrodinger equation can be reduced to the confluent hypergeometric form hence confluent potentials the solution is obtained straightforwardly a simple argument is given to sustain that the natanzon potential is the most general confluent potential

algebraic treatment of second poschl teller morse rosen and eckart equations the method of an earlier paper see ibid vol 20 p 4075 1987 is applied to the non compact case to solve a family of second poschl teller morse rosen and eckart equations with quantised coupling constants both discrete and continuous spectra bound state and scattering wavefunctions transmission coefficients are found from the matrix elements of group representations

limiting classification on linearized eigenvalue problems for 1 dimensional allen cahn equation i asymptotic formulas of eigenvalues abstract we are interested in classical linearized eigenvalue problems for one dimensional reaction diffusion equations with small diffusion coefficient previously authors have obtained the precise asymptotic formulas of all eigenvalues and eigenfunctions for f u sin u which suggest interesting structure of eigenpairs existence of two special eigenpairs and the limiting classification of the others by special eigenpairs in this paper we study the allen cahn case f u u u 3 and obtain the asymptotic formulas of eigenvalues which leads us to the limiting classification of eigenvalues

on algebro geometric simply periodic solutions of the kdv hierarchy in this paper we show that as tau rightarrow sqrt 1 infty any zero of the lame function converges to either infty or a finite point p satisfying text re p frac 1 2 and e 2 pi i p being an algebraic number our proof is based on studying a special family of simply periodic kdv potentials with period 1 i e algebro geometric simply periodic solutions of the kdv hierarchy we show that except the pole 0 all other poles of such kdv potentials locate on the line text re z frac 1 2 we also compute explicitly the eigenvalue set of the corresponding l 2 0 1 eigenvalue problem for such kdv potentials thus extends takemura s works commun math phys 235 467 494 2003 and electron j differ equ 2004 15 1 30 2004 our main idea is to apply the classification result for simply periodic kdv potentials by gesztesy et al trans am math soc 358 603 656 2006 and the darboux transformation

convergence of gaussian quasi likelihood random fields for ergodic levy driven sde observed at high frequency this paper investigates the gaussian quasi likelihood estimation of an exponentially ergodic multidimensional markov process which is expressed as a solution to a levy driven stochastic differential equations whose coefficients are supposed to be known except for the finite dimensional parameters to be estimated we suppose that the process is observed under the condition for the rapidly increasing experimental design by means of the polynomial type large deviation inequality the mighty convergence of the corresponding statistical random fields is derived which especially leads to the asymptotic normality at rate p nhn for all the target parameters and also to the convergence of their moments in our results the diffusion coefficient may be degenerate or even null although the resulting estimator is not asymptotically efficient in the presence of jumps we do not require any specific form of the driving levy measure rendering that the proposed estimation procedure is practical and somewhat robust to underlying model specification

exchange operator formalism for n body spin models with near neighbours interactions we present a detailed analysis of the spin models with near neighbours interactions constructed in our previous paper enciso et al 2005 phys lett b 605 214 by a suitable generalization of the exchange operator formalism we provide a complete description of a certain flag of finite dimensional spaces of spin functions preserved by the hamiltonian of each model by explicitly diagonalizing the hamiltonian in the latter spaces we compute several infinite families of eigenfunctions of the above models in closed form in terms of generalized laguerre and jacobi polynomials

a large deviation result for parameter estimators and its application to nonlinear regression analysis elaborating on the work of ibragimov and has minskii 1981 we prove a law of large deviations lld for m estimators i e those estimators which maximize a functional continuous in the parameter of the observations this lld is applied using the results of petrov 1975 to the problem of parametrical nonlinear regression in the situation of discrete time independent errors and regression functions which are continuous in the parameter this improves a result of prakasa rao 1984

strange attractor in the ising model with competing interactions on the cayley tree on formule le modele d ising avec interactions de competition sur un arbre de cayley comme une application non lineaire a 2 dimensions

real root property of the spectral polynomial of the treibich verdier potential and related problems abstract we study the spectral polynomial of the treibich verdier potential such spectral polynomial which is a generalization of the classical lame polynomial plays fundamental roles in both the finite gap theory and the ode theory of heun s equation in this paper we prove that all the roots of such spectral polynomial are real and distinct under some assumptions the proof uses the classical concept of sturm sequence and isomonodromic theories we also prove an analogous result for a polynomial associated with a generalized lame equation where we apply a new approach based on the viewpoint of the monodromy data

order selection for same realization predictions in autoregressive processes assume that observations are generated from an infinite order autoregressive ar process shibata ann statist 8 1980 147 164 considered the problem of choosing a finite order ar model allowing the order to become infinite as the number of observations does in order to obtain a better approximation he showed that for the purpose of predicting the future of an independent replicate akaike s information criterion aic and its variants are asymptotically efficient although shibata s concept of asymptotic efficiency has been widely accepted in the literature it is not a natural property for time series analysis this is because when new observations of a time series become available they are not independent of the previous data to overcome this difficulty in this paper we focus on order selection for forecasting the future of an observed time series referred to as same realization prediction we present the first theoretical verification that aic and its variants are still asymptotically elficient in the sense defined in section 4 for same realization predictions to obtain this result a technical condition easily met in common practice is introduced to simplify the complicated dependent structures among the selected orders estimated parameters and future observations in addition a simulation study is conducted to illustrate the practical implications of aic this study shows that aic also yields a satisfactory saute realization prediction in finite samples on the other hand a limitation of aic in same realization settings is pointed out it is interesting to note that this limitation of aic does not exist for corresponding independent cases

a new realisation of dynamical groups and factorisation method a new method of algebraisation of quantum mechanical eigenvalue equations is presented in this method the dynamical algebra is represented on the space of group matrix elements the ladder operators of the dynamical algebra are obtained from infeld hull miller factorisations the method is used to study the first poschl teller equation even in the non symmetric case the energy spectrum and the exact normalised solutions are obtained in agreement with the results of non algebraic methods

satellite potentials for hypergeometric natanzon potentials as a result of the so 2 1 treatment of the hypergeometric natanzon potentials vn a set of potentials related to a given one is determined these are the satellite potentials and all belong to the natanzon class the set arises as the result of the action of the so 2 1 generators on the carrier space of an irreducible representation the results are compared to those obtained from supersymmetric quantum mechanics susyqm for some natanzon potentials the chains of natanzon potentials constructed using the idea of a satellite potential are in most cases different from the susyqm chains

analytic continuation of eigenvalues of the lame operator abstract eigenvalues of the lame operator are studied as complex analytic functions in period of an elliptic function we investigate the branching of eigenvalues numerically and clarify the relationship between the branching of eigenvalues and the convergent radius of a perturbation series

new class of conditionally exactly solvable potentials in quantum mechanics motivated by an idea of dutra 1993 we obtain a new class of one dimensional conditionally exactly solvable potentials for which the entire spectra can be obtained in an algebraic manner provided one of the potential parameters is assigned a fixed negative value it is shown that using shape invariant potentials as input one may generate different classes of such potentials even in more than one dimension we also illustrate that wkb and supersymmetry inspired wkb methods provide very good approximations for these potentials with the latter doing comparatively better

integral representation of solutions to fuchsian system and heun s equation we obtain integral representations of solutions to special cases of the fuchsian system of differential equations and heun s differential equation in particular we calculate the monodromy of solutions to the fuchsian equation that corresponds to picard s solution of the sixth painleve equation and to heun s equation

nonparametric estimation of large covariance matrices of longitudinal data summary estimation of an unstructured covariance matrix is difficult because of its positivedefiniteness constraint this obstacle is removed by regressing each variable on its predecessors so that estimation of a covariance matrix is shown to be equivalent to that of estimating a sequence of varying coefficient and varying order regression models our framework is similar to the use of increasing order autoregressive models in approximating the covariance matrix or the spectrum of a stationary time series as an illustration we adopt fan zhang s 2000 two step estimation of functional linear models and propose nonparametric estimators of covariance matrices which are guaranteed to be positive definite for parsimony a suitable order for the sequence of auto regression models is found using penalised likelihood criteria like aic and bic some asymptotic results for the local polynomial estimators of components of a covariance matrix are established two longitudinal datasets are analysed to illustrate the methodology a simulation study reveals the advantage of the nonparametric covariance estimator over the sample covariance matrix for large covariance matrices

the 192 solutions of the heun equation a machine generated list of 192 local solutions of the heun equation is given they are analogous to kummer s 24 solutions of the gauss hypergeometric equation since the two equations are canonical fuchsian differential equations on the riemann sphere with four and three singular points respectively tabulation is facilitated by the identification of the automorphism group of the equation with n singular points as the coxeter group d n each of the 192 expressions is labeled by an element of d 4 of the 192 24 are equivalent expressions for the local heun function hl and it is shown that the resulting order 24 group of transformations of hl is isomorphic to the symmetric group s 4 the isomorphism encodes each transformation as a permutation of an abstract four element set not identical to the set of singular points

complex analytic dynamics on the riemann sphere dichotomie dynamique de fatou et julia points periodiques consequences du theoreme de montel l ensemble de julia est la fermeture de l ensemble des points periodiques repulseurs resultats classiques sur l ensemble de fatou classification de sullivan de l ensemble de fatou une condition pour le developpement sur l ensemble de julia la dynamique des polynomes l ensemble de mandelbrot et le travail de douady et hubbard le theoreme d application de riemann mesurable et la dynamique analytique

on reducing the heun equation to the hypergeometric equation abstract the reductions of the heun equation to the hypergeometric equation by polynomial transformations of its independent variable are enumerated and classified heun to hypergeometric reductions are similar to classical hypergeometric identities but the conditions for the existence of a reduction involve features of the heun equation that the hypergeometric equation does not possess namely its cross ratio and accessory parameters the reductions include quadratic and cubic transformations which may be performed only if the singular points of the heun equation form a harmonic or an equianharmonic quadruple respectively and several higher degree transformations this result corrects and extends a theorem in a previous paper which found only the quadratic transformations siam j math anal 10 3 1979 655

on polynomial solutions of the heun equation by making use of a recently developed method to solve linear differential equations of arbitrary order we find a wide class of polynomial solutions to the heun equation we construct the series solution to the heun equation before identifying the polynomial solutions the heun equation extended by the addition of a term x is also amenable for polynomial solutions