Asymptotic Fourier Coefficients for a C ∞ Bell (Smoothed-“Top-Hat”) & the Fourier Extension Problem

In constructing local Fourier bases and in solving differential equations with nonperiodic solutions through Fourier spectral algorithms, it is necessary to solve the Fourier Extension Problem. This is the task of extending a nonperiodic function, defined on an interval , to a function which is periodic on the larger interval . We derive the asymptotic Fourier coefficients for an infinitely differentiable function which is one on an interval , identically zero for , and varies smoothly in between. Such smoothed “top-hat” functions are “bells” in wavelet theory. Our bell is (for x ≥ 0) where where . By applying steepest descents to approximate the coefficient integrals in the limit of large degree j , we show that when the width L is fixed, the Fourier cosine coefficients a j of on are proportional to where Λ( j ) is an oscillatory factor of degree given in the text. We also show that to minimize error in a Fourier series truncated after the N th term, the width should be chosen to increase with N as . We derive similar asymptotics for the function f ( x )= x as extended by a more sophisticated scheme with overlapping bells; this gives an even faster rate of Fourier convergencePeer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/43417/1/10915_2005_Article_9010.pd

A Spectral Embedding Method for the Incompressible Navier-Stokes Equations

In order to solve the incompressible Navier-Stokes equations in geometries of complex shape with a spectral type method, one uses an embedding approach based on Fourier expansions and boundary integrals equations. By using appropriate formulations of these equations, we propose algorithms that simply require efficient solvers of scalar elliptic equations. The capabilities of the "spectral embedding method" method are pointed out by considering the classical 2D driven cavity problem with comparisons to spectral Chebyshev results

Structure, thermal and mechanical properties of poly (ε-caprolactone)/organomodified clay bionanocomposites prepared in open air by in situ polymerization

The first example of the usefulness of titanium (IV) butoxide, as initiators for open air in-situ intercalative polymerization of ε-caprolactone (ε-CL) in the presence of organoclay is herein reported. The bionanocomposites based on poly(ε-caprolactone) (PCL) were prepared by in-situ Ring Opening Polymerization (ROP) of ε-caprolactone in the presence of different organomodified montmorillonite clay (ODA-MMT) loading (1, 3 and wt%). Structural, thermal and mechanical characterizations of the resulting bionanocomposites were investigated. The presence of the nanoclay increased PCL crystallinity, melting temperature and thermal stability, whereas some decrease in T was observed. TEM analyses confirmed the good dispersion of ODA-MMT with 1 and 3 wt% content into the PCL polymer as already asserted by XRD diffraction. Finally, the Young's modulus of the PCL nanocomposites was higher compared to the neat PCL, while a decrease of stress and strain at break for materials with different filler content was observed.Financial support for cooperative research came from the Hassan II Academy of Science and Technology of Morocco/CSIC-Spain (Project AH11STC-nano 2011 – 2012 and 2010MA003), the MESRSFC and CNRST of Morocco (PPR program), and the bilateral scientific committee programs of collaboration between Morocco and Tunisia (n 17/TM/20), and the Spanish Ministry of Science and Innovation (MICINN) through the project MAT2016-81138-R

A Comparative Theoretical and Computational Study on Robust Counterpart Optimization: II. Probabilistic Guarantees on Constraint Satisfaction

Probabilistic guarantees on constraint satisfaction for robust counterpart optimization are studied in this paper. The robust counterpart optimization formulations studied are derived from box, ellipsoidal, polyhedral, “interval+ellipsoidal”, and “interval+polyhedral” uncertainty sets (Li, Z.; Ding, R.; Floudas, C.A.A Comparative Theoretical and Computational Study on Robust Counterpart Optimization: I. Robust Linear and Robust Mixed Integer Linear Optimization. Ind. Eng. Chem. Res. 2011, 50, 10567). For those robust counterpart optimization formulations, their corresponding probability bounds on constraint satisfaction are derived for different types of uncertainty characteristic (i.e., bounded or unbounded uncertainty, with or without detailed probability distribution information). The findings of this work extend the results in the literature and provide greater flexibility for robust optimization practitioners in choosing tighter probability bounds so as to find less conservative robust solutions. Extensive numerical studies are performed to compare the tightness of the different probability bounds and the conservatism of different robust counterpart optimization formulations. Guiding rules for the selection of robust counterpart optimization models and for the determination of the size of the uncertainty set are discussed. Applications in production planning and process scheduling problems are presented