We investigate two density questions for Sobolev, Besov and Triebel--Lizorkin spaces on rough sets. Our main results, stated in the simplest Sobolev space setting, are that: (i) an open set $\Omega\subset\mathbb R^n$, $\mathcal{D}(\Omega)$ is dense $\{u\in H^s(\mathbb R^n):{\rm supp}\, u\subset \overline{\Omega}\}$ whenever $\partial\Omega$ has zero Lebesgue measure $\Omega$ "thick" (in sense o...

The misorientation of 515 grain boundaries has been determined using electron backscatter diffraction data from an 18 μm thick copper foil with columnar structure and a preferential {110} surface orientation. energy the was dihedral angles in vicinity boundary thermal grooves. experimental vs. angle shows deep minima for low-angle small corresponding to Σ3 Σ9 boundaries. Only fraction coinciden...

in this article, thermo-elastic-behavior of a thick-walled cylinder made from a polystyrene nanocomposite reinforced with functionally graded (fg) single-walled carbon nanotubes (swcnts) was carried out in radial direction while subjected to a steady state thermal field. the swcnts were assumed aligned, straight with infinite length and a uniform layout. two types of variations in the volume fr...

We estimate the abundance of Compton-thick (CT) active galactic nuclei (AGN) based on our joint model of X-ray and infrared backgrounds. At Lrest2−10keV > 10 42 erg/s, the CT AGN density predicted by our model is a few ×10 Mpc from z=0 up to z=3. CT AGN with higher luminosity cuts (> 10, 10 & 10 erg/s) peak at higher z and show a rapid increase in the number density from z=0 to z∼2-3. The CT to...

This paper studies a generalization of the Schumpeterian models with endogenous market structure that allows the overall production structure to be more than linear in the growthdriving factor and yet generates endogenous growth, defined as steady-state, constant, exponential growth of income per capita. This version of modern growth theory, therefore, is robust in the sense that its key result...

M. Beiglböck, V. Bergelson, and A. Fish proved that if G is a countable amenable group and A and B are subsets of G with positive Banach density, then the product set AB is piecewise syndetic. This means that there is a finite subset E of G such that EAB is thick, that is, EAB contains translates of any finite subset of G. When G = Z, this was first proven by R. Jin. We prove a quantitative ver...

we introduce center-like subsets z*(r,f), z**(r,f) and z1(r,f), where r is a ring and f is a map from r to r. for f a derivation or a non-identity epimorphism and r a suitably-chosen prime or semiprime ring, we prove that these sets coincide with the center of r.

let $r$ be a commutative ring with identity and $m$ be an$r$-module. let $fspec(m)$ denotes the collection of all prime fuzzysubmodules of $m$. in this regards some basic properties of zariskitopology on $fspec(m)$ are investigated. in particular, we provesome equivalent conditions for irreducible subsets of thistopological space and it is shown under certain conditions$fspec(m)$ is a $t_0-$spa...

Thick shells have attracted much attention in recent years as intelligent and functional graded materials because of their unique properties. In this review paper, some critical issues and problems in the development of thick shells made from Functionally graded piezoelectric material (FGPM) are discussed. This review has been conducted on various types of methods which are available for thick ...

From high resolution spectra of 61 cool dwarfs and giants, Mg isotopic abundance ratios Mg:Mg:Mg are derived from spectral synthesis of the MgH A-X lines near 5140Å. Our sample spans the range −2.5 ≤ [Fe/H] ≤ 0.1, including the first measurements of Mg isotope ratios in stars with metallicities below [Fe/H]=−2.0. We confirm the decrease in Mg/Mg with decreasing [Fe/H] as predicted by recent mod...