High density piecewise syndeticity of product sets in amenable groups

M. Beiglb\"ock, 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 $G=\mathbb{Z}$, this was first proven by R. Jin. We prove a quantitative version of the aforementioned result by providing a lower bound on the density (with respect to a F\o lner sequence) of the set of witnesses to the thickness of $$. When $G=\mathbb{Z}^d$, this result was first proven by the current set of authors using completely different techniques.Comment: 7 pages; the proof of the main result has been simplified and sharpened by removing a technical assumption; also, a lower density version has been adde

Comments on "Basic properties of Rayleigh surface wave propagation along curved surfaces", by F. Jin, Z. Wang, K. Kishimoto, International Journal of Engineering Science 43 (2005) 250-261

In this ‘Letter to the Editor’, it is demonstrated that the main results obtained in the work ‘‘Basic properties of Rayleigh surface wave propagation along curved surfaces”, by F. Jin, Z. Wang, K. Kishimoto, International Journal of Engineering Science, 43 (2005) 250–261, have been established ten years earlier

MINIMAL PAIRS AND ITS INFLUENCE TOWARD STUDENTS PRONUNCIATION IN ENGLISH FRICATIVE SOUNDS

The aim of this paper is to describe the way to overcome students difficulties in pronouncing some English fricatives / f, v, , , , , s, z/ by using minimal pairs. Minimal pairs are two different forms are identical in every way expect for one sound segment that occurs in the same place in the string. Fricatives are formed by narrowing of the air passage at the some points so that, when air is expelled by pressure from lungs, it escapes with a kind of hissing sound. English fricatives sounds are given in a couple sounds and explained how to produce them in order to the students know the differences and easy to pronounce English fricatives

The Book of Poetry and Diplomacy : Who is the boss?

This episode uses two scenes of reciting odes to explore the struggle for hegemony. The first shows how a fugitive Jin Prince declare his ambition despite his precarious and dependent position. The second features a Jin leader pushing back against the hubristic self-aggrandizement of a Chu prince