New Refinements and Improvements of Some Trigonometric Inequalities Based on Padé Approximant

A multiple-point Padé approximant method is presented for approximating and bounding some trigonometric functions in this paper. We give new refinements and improvements of some trigonometric inequalities including Jordan’s inequality, Kober’s inequality, and Becker-Stark’s inequality. The analysis results show that our conclusions are better than the previous conclusions.

OPTICAL PROPERTIES OF MOLECULAR SYSTEM COUPLED TO THE SOLVENT

Analytic approximations for the absorption coefficient and refraction index have been obtained for a two-level molecular system in the presence of a four-wave mixing signal. Stochastic collisions among solute–solvent molecules produce a broadening of the upper level of energy. In this work, we have used four generalized Lorentzian approximants of the Voigt function as probability distribution, calculated by the two-point quasi-rational approximant method previously denoted as asymptotical Padé technique, for the evaluation of the average values of the Fourier components associated to the induced coherences. The innovation of this type of distributions allows us, by taking appropriate limits, to find similar profiles of optical properties as those using Gaussian and Lorentzian distribution functions.

On the commutative factorization of n × n matrix Wiener–Hopf kernels with distinct eigenvalues

In this article, we present a method for factorizing n × n matrix Wiener–Hopf kernels where n >2 and the factors commute. We are motivated by a method posed by Jones (Jones 1984 a Proc. R. Soc. A 393 , 185–192) to tackle a narrower class of matrix kernels; however, no matrix of Jones' form has yet been found to arise in physical Wiener–Hopf models. In contrast, the technique proposed herein should find broad application. To illustrate the approach, we consider a 3×3 matrix kernel arising in a problem from elastostatics. While this kernel is not of Jones' form, we shall show how it can be factorized commutatively. We discuss the essential difference between our method and that of Jones and explain why our method is a generalization. The majority of Wiener–Hopf kernels that occur in canonical diffraction problems are, however, strictly non-commutative. For 2×2 matrices, Abrahams has shown that one can overcome this difficulty using Padé approximants to rearrange a non-commutative kernel into a partial-commutative form; an approximate factorization can then be derived. By considering the dynamic analogue of Antipov's model, we show for the first time that Abrahams' Padé approximant method can also be employed within a 3×3 commutative matrix form.

Modeling of a constant Q: Methodology and algorithm for an efficient and optimally inexpensive viscoelastic technique

Linear anelastic phenomena in wave propagation problems can be well modeled through a viscoelastic mechanical model consisting of standard linear solids. In this paper we present a method for modeling of constant Q as a function of frequency based on an explicit closed formula for calculation of the parameter fields. Several standard linear solids connected in parallel can be tuned through a single parameter to yield an excellent constant Q approximation. The proposed method enables substantial savings in computations and memory requirements. Experiments show that the new method also yields higher accuracy in the modeling of Q than, e.g., the Padé approximant method.