Radix sort stands out for having a better worse case theoretical bound than any comparison-based sort, for fixed length integers. Despite the fact that radix sort can be implemented either in-place or in parallel, there exists no parallel in-place implementation for radix sort that guarantees a sub-linear worst case span. The challenge arises due to read-write races when reading from and writing to the same array. In this thesis, I introduce Regions sort and use it to implement a parallel work-efficient in-place radix sorting algorithm. To sort integers from a range , and a parameter , the algorithm requires only (log log) auxiliary memory, (log) work and ((/ + log) log) span. Regions sort uses a divide-and-conquer paradigm. It first divides the array into sub-arrays, each of which is sorted independently in parallel. This decreases the irregularity in the input and reorders it into a set of regions, each of which has similar properties. Next, it builds a data structure, called the regions graph to represent the needed data movements to completely sort the array. Finally, Regions sort iteratively plans swaps that satisfies the required data movements. The algorithm then has to recurse on records with so-far equal keys to break ties. I compare two variants of Regions sort with the state-of-the-art sorting integer sort and comparison sort algorithms. Experiments show that the best variant of Regions sort usually outperforms other sorting algorithms. In addition, I perform different experiments showing that Regions sort maintains its superiority regardless input distribution and range. More importantly, the single-threaded implementation of Regions sort outperforms the highly optimized Ska Sort implementation for serial in-place radix sort.

Abstract. Consider an n-dimensional smooth Riemannian manifold (M n , g) with a given smooth measure m on it. We call such a triple (M n , g, m) a Riemannian measure space. Perelman introduced a variant of scalar curvature in his recent work on solving Poincaré's conjectureg , where dm = φ dvol(g) and R is the scalar curvature of (M n , g). In this note, we study the topological obstruction for the φ-stable minimal submanifold with positive P -scalar curvature in dimension three under the setting of manifolds with density. Manifold with densityBy a Riemannian measure space we mean a triple (M n , g, m), where (M n , g) is an n-dimensional smooth oriented Riemannian manifold and m is a smooth measure defined on M n . Given a Riemannian manifold, there is a natural measure associated with it, i.e. the Riemannian volume measure dvol(g). By the Radon-Nikodým theorem, there exists a smooth function φ > 0 on M n such that dm = φ dvol(g).Here φ > 0 is called the density of the manifold. The triple (M n , g, φ) is called the Riemannian manifold with density φ. Clearly, the study of Riemannian measure space is equivalent to the study of manifolds with density.The study of manifolds with density traces back to the work of Bakry-Émery [1] in the early 1980s, in which they introduced the Bakry-Émery-Ricci tensor in the study of the diffusion process:

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Abstract. In this note, we remark on a 2001 paper of S. Goyal and V. Goyal. The main result of this work is that they used some elementary method to find a class of functions K(x) = K(x 1 , x 2 , x 3 ) for which the solutions toon S 2 can be obtained. We observe that this class of functions that they studied is actually the trivial one, i.e. the class of positive constant functions.