Trigonometric Cases Research Articles

Consistency of the Bäcklund Transformation for the Spin Calogero–Moser System

We prove the consistency of the Bäcklund transformation (BT) for the spin Calogero–Moser (sCM) system in the rational, trigonometric, and hyperbolic cases. The BT for the sCM system consists of an overdetermined system of ordinary differential equations; to establish our result, we construct and analyze certain functions that measure the departure of this overdetermined system from consistency. We show that these functions are identically zero and that this allows for a unique solution to the initial value problem for the overdetermined system.

Isogeometric discretizations with generalized B-splines: Symbol-based spectral analysis

We perform a spectral analysis of matrices arising from isogeometric discretizations based on hyperbolic and trigonometric generalized B-splines. Second-order differential problems with variable coefficients are considered and discretized by means of sequences of both nested and non-nested generalized spline spaces. We prove that an asymptotic spectral distribution always exists when the matrix-size tends to infinity and is compactly described by a so-called symbol, just as in the polynomial B-spline case. We observe a strong resemblance between the symbol expressions in the hyperbolic, trigonometric and polynomial cases, which results in similar spectral features of the corresponding matrices. The theoretical symbol analysis is illustrated with numerical examples, and we show how the symbol can be used to make an analytical prediction of spectral discretization errors.

The Am(1) Q-system

We propose a Q-system for the [Formula: see text] quantum integrable spin chain. We also find compact determinant expressions for all the Q-functions, both for the rational and trigonometric cases.

Gauge theory and boundary integrability. Part II. Elliptic and trigonometric cases

We consider the mixed topological-holomorphic Chern-Simons theory introduced by Costello, Yamazaki & Witten on a ℤ2 orbifold. We use this to construct semi- classical solutions of the boundary Yang-Baxter equation in the elliptic and trigonometric cases. A novel feature of the trigonometric case is that the ℤ2 action lifts to the gauge bundle in a z-dependent way. We construct several examples of K -matrices, and check that they agree with cases appearing in the literature.

Source Identities and Kernel Functions for the Deformed Koornwinder–van Diejen Models

We consider generalizations of the $BC$-type relativistic Calogero-Moser-Sutherland models, comprising of the rational, trigonometric, hyperbolic, and elliptic cases, due to Koornwinder and van Diejen, and construct an explicit eigenfunction for these generalizations. In special cases, we find the various kernel function identities, and also a Chalykh-Feigin-Sergeev-Veselov type deformation of these operators and their corresponding kernel functions, which generalize the known kernel functions for the Koornwinder-van Diejen models.

Spectral analysis of matrices in Galerkin methods based on generalized B-splines with high smoothness

We present a first step towards the spectral analysis of matrices arising from IgA Galerkin methods based on hyperbolic and trigonometric GB-splines. Second order differential problems with constant coefficients are considered and discretized by means of sequences of both nested and non-nested spline spaces. We prove that there always exists an asymptotic eigenvalue distribution which can be compactly described by a symbol, just like in the polynomial case. There is a complete similarity between the symbol expressions in the hyperbolic, trigonometric and polynomial cases. This results in similar spectral features of the corresponding matrices. We also analyze the IgA discretization based on trigonometric GB-splines for the eigenvalue problem related to the univariate Laplace operator. We prove that, for non-nested spaces, the phase parameter can be exploited to improve the spectral approximation with respect to the polynomial case. As part of the analysis, we derive several Fourier properties of cardinal GB-splines.

Source Identities and Kernel Functions for Deformed (Quantum) Ruijsenaars Models

We consider the relativistic generalization of the quantum A N-1 Calogero–Sutherland models due to Ruijsenaars, comprising the rational, hyperbolic, trigonometric and elliptic cases. For each of these cases, we find an exact common eigenfunction for a generalization of Ruijsenaars analytic difference operators that gives, as special cases, many different kernel functions; in particular, we find kernel functions for Chalykh–Feigin–Veselov–Sergeev-type deformations of such difference operators which generalize known kernel functions for the Ruijsenaars models. We also discuss possible applications of our results.

Kernel Functions for Difference Operators of Ruijsenaars Type and Their Applications

A unified approach is given to kernel functions which intertwine Ruijsenaars difference operators of type A and of type BC. As an application of the trigonometric cases, new explicit formulas for Koornwinder polynomials attached to single columns and single rows are derived.

BAKER–AKHIEZER FUNCTION AS ITERATED RESIDUE AND SELBERG-TYPE INTEGRAL

AbstractA simple integral formula as an iterated residue is presented for the Baker–Akhiezer function related toAn-type root system in both the rational and trigonometric cases. We present also a formula for the Baker–Akhiezer function as a Selberg-type integral and generalise it to the deformedAn,1-case. These formulas can be interpreted as new cases of explicit evaluation of Selberg-type integrals.

SU(3) Richardson–Gaudin models: three-level systems

We present the exact solution of the Richardson–Gaudin models associated with the SU(3) Lie algebra. The derivation is based on a Gaudin algebra valid for any simple Lie algebra in the rational, trigonometric and hyperbolic cases. For the rational case additional cubic integrals of motion are obtained, whose number is added to that of the quadratic ones to match, as required from the integrability condition, the number of quantum degrees of freedom of the model. We discuss different SU(3) physical representations and elucidate the meaning of the parameters entering in the formalism. By considering a bosonic mapping limit of one of the SU(3) copies, we derive new integrable models for three level systems interacting with two bosons. These models include a generalized Tavis–Cummings model for three level atoms interacting with two modes of the quantized electric field.

Exactly solvable discrete BCS-type Hamiltonians and the six-vertex model

We propose the new family of the exactly solvable discrete state BCS-type Hamiltonians based on its relationship to the six-vertex model in the quasiclassical limit both in the rational and the trigonometric cases. We establish the relation of the BCS Hamiltonian and its eigenfunctions to the form of the monodromy matrix in the F-basis. Using the algebraic Bethe ansatz method for the standard BCS model with equal coupling the expression for the general scalar product and the determinant expressions for the physically interesting correlation functions for the finite number of sites which can be used in the numerical and analytical computations are obtained. We also compare the correlators with the results obtained by means of the variational method.

Notes on Inequalities with Doubling Weights

Various important weighted polynomial inequalities, such as Bernstein, Marcinkiewicz, Nikolskii, Schur, Remez, etc., inequalities, have been proved recently by Giuseppe Mastroianni and Vilmos Totik under minimal assumptions on the weights. In most of the cases this minimal assumption is the doubling condition. Sometimes, however, as in the weighted Nikolskii inequality, the slightly stronger A ∞condition is used. Throughout their paper the L p norm is studied under the assumption 1⩽ p<∞. In this note we show that their proofs can be modified so that many of their inequalities hold even if 0< p<1. The crucial tool is an estimate for quadrature sums for the pth power (0< p<∞ is arbitrary) of trigonometric polynomials established by Lubinsky, Máté, and Nevai. For technical reasons we discuss only the trigonometric cases.

On Uniqueness of Commutative Rings of Weyl Group Invariant Differential Operators

The uniqueness of commutative rings of classical Weyl group invariant differential operators is discussed. We show this uniqueness for trigonometric or elliptic potential cases under some order conditions. For rational potential cases, counter examples are constructed.

Approximation of Density Functions by Sequences of Exponential Families

1. Introduction. Consider the estimation of a probability density func- tion p(x) defined on a bounded interval. We approximate the logarithm of the density by a basis function expansion consisting of polynomials, splines or trigonometric series. The expansion yields a regular exponential family within which we estimate the density by the method of maximum likelihood. This method of density estimation arises by application of the principle of maxi- mum entropy or minimum relative entropy subject to empirical constraints. We show that if the logarithm of the density has r square-integrable deriva- tives, f IDr log p12 < ox, then the sequence of density estimators P converges to p in the sense of relative entropy (Kullback-Leibler fp log(p/ pn) at rate Opr(1/m2r + m/n) as m -X c and m2/n -* 0 in the spline and trigonometric cases and m3/n -O 0 in the polynomial case, where m is the dimension of the family and n is the sample size. Boundary conditions are assumed for the density in the trigonometric case. This convergence rate specializes to Op r(n-2r/(2r+ 1)) by setting m = nl/(2r+ 1) when the log-density is known to have degree of smoothness at least r. Analogous convergence results for the relative entropy are shown to hold in general, for any class of log-density functions and sequence of finite-dimensional linear spaces having L2 and L. approximation properties. The approximation of log-densities using polynomials has previously been considered by Neyman (1937) to define alternatives for goodness-of-fit tests, by Good (1963) as an application of the method of maximum entropy or minimum relative entropy, by Crain (1974, 1976a, b 1977) who demonstrates existence and consistency of the maximum likelihood estimator and by Mead and

On the almost everywhere divergence of Lagrange interpolation (complex and trigonometric cases)

On the almost everywhere divergence of Lagrange interpolation (complex and trigonometric cases)

On Multivariate Density Estimates Based on Orthogonal Expansions

The topics of orthogonality and Fourier series occupy a central position in analysis. Nevertheless, there is surprisingly little statistical literature, with the exception of that of time series and regression, which involves Fourier analysis. In the last decade however, several papers have appeared which deal with the estimation of orthogonal expansions of distribution densities and cumulatives. Cencov [1] and Van Ryzin [6] considered general properties of orthogonal expansion based density estimators and the latter applied these properties to obtain classification procedures. Schwartz [3] and the authors [2] and [4] investigated respectively the Hermite and Trigonometric special cases. The authors also obtained certain general results which apply not only to estimators of the population density but also to estimators of the population cumulative [2], [4] and [5]. In this paper several results derived for the univariate case are extended to the multivariate case. Also a new relationship is obtained which involves general Fourier expansions and estimators. Although there is some reason for calling the Gram-Charlier estimation of distribution densities a Fourier method, one fundamental aspect of Fourier methods is not shared by Gram-Charlier estimation. Gram-Charlier techniques make no use of Parseval's Formula or related error relationships of Fourier analysis. The ease with which the mean integrated square error (MISE) is evaluated, when Fourier methods are applied, accounts for most of the recent interest in this area. Section 1 of this paper deals with an investigation of two general MISE relationships for multivariate estimates of Fourier expansions. The relationship given in Theorem 2 is particularly simple and yet includes the four MISE's which are involved in the estimation problem. In Section 2 the choice of orthogonal functions is restricted to the trigonometric polynomials. It is shown that the MISE of multidimensional trigonometric polynomial estimators are related in a simple way to the Fourier coefficients of the distribution which is being estimated. This result is of considerable utility since it yields a rule for deciding which terms should be included in the estimate of the multivariate density.

Null Trigonometric Series in Differential Equations

In this paper it is shown how trigonometric series which are Cesàro-summable to zero may be used to solve differential equations. The explicit solution of the general ordinary linear equation with constant coefficients is found in terms of trigonometric series and special cases are dealt with.