Search results for "Polynomials"
showing 10 items of 144 documents
The discretized harmonic oscillator: Mathieu functions and a new class of generalized Hermite polynomials
2003
We present a general, asymptotical solution for the discretised harmonic oscillator. The corresponding Schr\"odinger equation is canonically conjugate to the Mathieu differential equation, the Schr\"odinger equation of the quantum pendulum. Thus, in addition to giving an explicit solution for the Hamiltonian of an isolated Josephon junction or a superconducting single-electron transistor (SSET), we obtain an asymptotical representation of Mathieu functions. We solve the discretised harmonic oscillator by transforming the infinite-dimensional matrix-eigenvalue problem into an infinite set of algebraic equations which are later shown to be satisfied by the obtained solution. The proposed ansa…
Indefinite integrals for some orthogonal polynomials obtained using integrating factors
2020
A method has been presented recently for deriving integrals of special functions using two kinds of integrating factor for the homogeneous second-order linear differential equations which many spec...
Error analysis of the orthogonal series solution of linear time-invariant systems
1989
Similarities in the error analysis of the polynomial series solution of linear time-invariant systems are pointed out.
Biorthogonal Multiwavelets Originated from Hermite Splines
2015
This chapter presents multiwavelet transforms that manipulate discrete-time signals. The transforms are implemented in two phases: 1. Pre (post)-processing, which transforms a scalar signal into a vector signal (and back). 2. Wavelet transforms of the vector signal. Both phases are performed in a lifting way. The cubic interpolating Hermite splines are used as a predicting aggregate in the vector wavelet transform. Pre(post)-processing algorithms which do not degrade the approximation accuracy of the vector wavelet transforms are presented. A scheme of vector wavelet transforms and three pre(post)-processing algorithms are described. As a result, we get fast biorthogonal algorithms to trans…
Real symplectic formulation of local special geometry
2006
We consider a formulation of local special geometry in terms of Darboux special coordinates $P^I=(p^i,q_i)$, $I=1,...,2n$. A general formula for the metric is obtained which is manifestly $\mathbf{Sp}(2n,\mathbb{R})$ covariant. Unlike the rigid case the metric is not given by the Hessian of the real function $S(P)$ which is the Legendre transform of the imaginary part of the holomorphic prepotential. Rather it is given by an expression that contains $S$, its Hessian and the conjugate momenta $S_I=\frac{\partial S}{\partial P^I}$. Only in the one-dimensional case ($n=1$) is the real (two-dimensional) metric proportional to the Hessian with an appropriate conformal factor.
Rolle's Theorem for Polynomials of Degree Four in a Hilbert Space
2002
AbstractIn an infinite-dimensional real Hilbert space, we introduce a class of fourth-degree polynomials which do not satisfy Rolle's Theorem in the unit ball. Extending what happens in the finite-dimensional case, we show that every fourth-degree polynomial defined by a compact operator satisfies Rolle's Theorem.
Hybrid approximation for solutions of high-order integro-differential equations including variable delay
2020
Abstract In this study, a numerical technique with hybrid approximation is developed for solving high-order linear integro-differential equations including variable delay under the initial conditions. These type of problems are of applications in mathematical physics, mechanics, natural sciences, electronics and computer science. The aim of this work is to investigate an approximation with the matrix forms of Taylor and Laguerre polynomials along with standard collocation points. By the reduction of the solution of this problem with regard to the matrix relations, the solution of a system of algebraic equations has been obtained. The usefulness of this algorithm has been demonstrated by num…
On an Inequality for Legendre Polynomials
2020
This paper is concerned with the orthogonal polynomials. Upper and lower bounds of Legendre polynomials are obtained. Furthermore, entropies associated with discrete probability distributions is a topic considered in this paper. Bounds of the entropies which improve some previously known results are obtained in terms of inequalities. In order to illustrate the results obtained in this paper and to compare them with other results from the literature some graphs are provided.
Optical Quality Comparison between Spherical and Aspheric Toric Intraocular Lenses
2014
Purpose To measure and compare the optical quality of spherical and aspheric toric intraocular lenses (IOLs). Methods Wavefront aberrations of AcrySof Toric and IQ Toric IOLs (Alcon Laboratories) for different powers were measured at 3- and 5-mm pupils by Nimo TR0805 instrument. The Zernike coefficients of trefoil, coma, tetrafoil, secondary astigmatism, and spherical aberration were evaluated. The point spread functions (PSFs) of each IOL evaluated were calculated from the wavefront aberrations. The PSF images also were calculated from the IOL wavefront aberrations, adding the cornea's aberrations to simulate the optical quality after their implantation. Results Spherical toric IOLs showed…
Novel Stability Criteria for T--S Fuzzy Systems
2014
In this paper, novel stability conditions for Takagi-Sugeno (T-S) fuzzy systems are presented. The so-called nonquadratic membership-dependent Lyapunov function is first proposed, which is formulated in a higher order form of both the system states and the normalized membership functions than existing techniques in the literature. Then, new membership-dependent stability conditions are developed by the new Lyapunov function approach. It is shown that the conservativeness of the obtained criteria can be further reduced as the degree of the Lyapunov function increases. Two numerical examples are given to demonstrate the effectiveness and less conservativeness of the obtained theoretical resul…