Locality estimation of parallel algorithm for distributed memory computers

Locality is an algorithm characteristic describing a usage level of fast access memory. For example, in case of distributed memory computers we focus on memory of each computational node. To achieve the high performance of algorithm implementation one should choose the best possible locality option. Studying the parallel algorithm locality is to estimate the number and volume of data communications. In this work, we formulate and prove the statements for computers with distributed memory that allow us to estimate the asymptotic volume of data communication operations. These estimation results are useful while comparing alternative versions of parallel algorithms during data communication cost analysis.

On the asymptotic behavior of the dimension of spaces of harmonic functions with polynomial growth

Suppose {(M^{n},g)} is a Riemannian manifold with nonnegative Ricci curvature, and let {h_{d}(M)} be the dimension of the space of harmonic functions with polynomial growth of growth order at most d. Colding and Minicozzi proved that {h_{d}(M)} is finite. Later on, there are many researches which give better estimates of {h_{d}(M)}. In this paper, we study the behavior of {h_{d}(M)} when d is large. More precisely, suppose {(M^{n},g)} has maximal volume growth and has a unique tangent cone at infinity. Then when d is sufficiently large, we obtain some estimates of {h_{d}(M)} in terms of the growth order d, the dimension n and the asymptotic volume ratio {\alpha=\lim_{R\rightarrow\infty}\frac{\mathrm{Vol}(B_{p}(R))}{R^{n}}}. When {\alpha=\omega_{n}}, i.e., {(M^{n},g)} is isometric to the Euclidean space, the asymptotic behavior obtained in this paper recovers a well-known asymptotic property of {h_{d}(\mathbb{R}^{n})}.

A necessary and sufficient condition for some steady Ricci solitons to have positive asymptotic volume ratio

In this paper, we firstly establish a useful ODE relationship between R1(c) and V1(c) on the steady Ricci soliton. Based on this, we obtain a necessary and sufficient condition for some complete noncompact steady gradient Ricci solitons to have positive asymptotic volume ratio.