1,553 research outputs found
Nonlinear Integral-Equation Formulation of Orthogonal Polynomials
The nonlinear integral equation P(x)=\int_alpha^beta dy w(y) P(y) P(x+y) is
investigated. It is shown that for a given function w(x) the equation admits an
infinite set of polynomial solutions P(x). For polynomial solutions, this
nonlinear integral equation reduces to a finite set of coupled linear algebraic
equations for the coefficients of the polynomials. Interestingly, the set of
polynomial solutions is orthogonal with respect to the measure x w(x). The
nonlinear integral equation can be used to specify all orthogonal polynomials
in a simple and compact way. This integral equation provides a natural vehicle
for extending the theory of orthogonal polynomials into the complex domain.
Generalizations of the integral equation are discussed.Comment: 7 pages, result generalized to include integration in the complex
domai
Exact conserved quantities on the cylinder II: off-critical case
With the aim of exploring a massive model corresponding to the perturbation
of the conformal model [hep-th/0211094] the nonlinear integral equation for a
quantum system consisting of left and right KdV equations coupled on the
cylinder is derived from an integrable lattice field theory. The eigenvalues of
the energy and of the transfer matrix (and of all the other local integrals of
motion) are expressed in terms of the corresponding solutions of the nonlinear
integral equation. The analytic and asymptotic behaviours of the transfer
matrix are studied and given.Comment: enlarged version before sending to jurnal, second part of
hep-th/021109
Polynomial solutions of nonlinear integral equations
We analyze the polynomial solutions of a nonlinear integral equation,
generalizing the work of C. Bender and E. Ben-Naim. We show that, in some
cases, an orthogonal solution exists and we give its general form in terms of
kernel polynomials.Comment: 10 page
Nonparametric instrumental variables estimation of a quantile regression model
We consider nonparametric estimation of a regression function that is identified by requiring a specified quantile of the regression "error" conditional on an instrumental variable to be zero. The resulting estimating equation is a nonlinear integral equation of the first kind, which generates an ill-posed-inverse problem. The integral operator and distribution of the instrumental variable are unknown and must be estimated nonparametrically. We show that the estimator is mean-square consistent, derive its rate of convergence in probability, and give conditions under which this rate is optimal in a minimax sense. The results of Monte Carlo experiments show that the estimator behaves well in finite samples.Statistical inverse, endogenous variable, instrumental variable, optimal rate, nonlinear integral equation, nonparametric regression
Finite volume spectrum of N=1 superminimal models perturbed by
We describe an extension of the nonlinear integral equation (NLIE) tehnique
to N=1 superminimal models perturbed by . Along the way, we also
complete our previous studies of the finite volume spectrum of the N=1
supersymmetric sine-Gordon model by considering the attractive regime and more
specifically, breather states
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