Uniform approximation and explicit estimates for the prolate spheroidal wave functions

For fixed $c,$ Prolate Spheroidal Wave Functions (PSWFs), denoted by $\psi_{n, c},$ form an orthogonal basis with remarkable properties for the space of band-limited functions with bandwith $c$. They have been largely studied and used after the seminal work of D. Slepian and his co-authors. In several applications, uniform estimates of the $\psi_{n,c}$ in $n$ and $c,$ are needed. To progress in this direction, we push forward the uniform approximation error bounds and give an explicit approximation of their values at $1$ in terms of the Legendre complete elliptic integral of the first kind. Also, we give an explicit formula for the accurate approximation the eigenvalues of the Sturm-Liouville operator associated with the PSWFs

Spectral Decay of Time and Frequency Limiting Operator

For fixed $c,$ the Prolate Spheroidal Wave Functions (PSWFs) $\psi_{n, c}$ form a basis with remarkable properties for the space of band-limited functions with bandwidth $c$. They have been largely studied and used after the seminal work of D. Slepian, H. Landau and H. Pollack. Many of the PSWFs applications rely heavily of the behavior and the decay rate of the eigenvalues $(\lambda_n(c))_{n\geq 0}$ of the time and frequency limiting operator, which we denote by $\mathcal Q_c.$ Hence, the issue of the accurate estimation of the spectrum of this operator has attracted a considerable interest, both in numerical and theoretical studies. In this work, we give an explicit integral approximation formula for these eigenvalues. This approximation holds true starting from the plunge region where the spectrum of $\mathcal Q_c$ starts to have a fast decay. As a consequence of our explicit approximation formula, we give a precise description of the super-exponential decay rate of the $\lambda_n(c).$ Also, we mention that the described approximation scheme provides us with fairly accurate approximations of the $\lambda_n(c)$ with low computational load, even for very large values of the parameters $c$ and $n.$ Finally, we provide the reader with some numerical examples that illustrate the different results of this work.Comment: arXiv admin note: substantial text overlap with arXiv:1012.388

Novel Analytical Model of Stress Concentration around Broken Fibers of Unidirectional Composite Plate

During the fiber fracture of unidirectional composite a distribution of stress around the neighbored fibers happens, this mechanism is called the local redistribution efforts. Referring to the "shear lag" method, the researches wanted to predict the stress concentration in the surrounding area of broken fibers as well as the longitudinal resistance of the unidirectional composite which presents a fiber breaking. The goal of this paper is to develop a new probabilistic model of unidirectional composite plate to calculate the stress concentration at the broken fibers and their neighboring fiber intact. The "shear lag" method has been generalized to see the broken fibers interference on the stress concentration factor variation of surrounding sound fibers

Agro-morphological and quality attributes of Moroccan carob

In Morocco, the carob tree is currently enjoying renewed interest with the introduction of a local development strategy based on the development of this resource. Sixty accessions of carob fruits from four regions in northern and southern Morocco are studied for phenotypic and physico-chemical criteria of pods. The phenotypic characterization of the fruits showed that the Chaouen region (north) is characterized by long (13.3 cm), wide (2.02 cm), thick (0.72 cm) and heavy (13.0 g) pods, a high pulp weight (10.8 g) and the lowest seed yield. The southern regions (Beni Mellal and Tafraout) are characterized by relatively short pods with high seed yield. The physico-chemical analysis of the pulp showed a significant variation between the studied regions, ranging from 35.3 to 51.5 % for total sugars, 5.3 to 11.3 % for polyphenols and 11.1 to 14.2 % for total fibers. The three minerals investigated (Fe, Ca, Mg) were detected in all carob samples, mainly in Tafraout, and their contents vary from one region to another.  Keywords: Morocco, Ceratonia siliqua L, pod, morphological characterization, chemical features, pul