The Kakutani's precompactness lemma

Lipschitz Precompactness for Closed Negatively Curved Manifolds

We prove that, given a integer n ≥ 3 and a group π, the class of closed Riemannian n-manifolds of uniformly bounded negative sectional curvatures and with fundamental groups isomorphic to π is precompact in the Lipschitz topology. In particular, the class breaks into finitely many diffeo-

Lecture 19 : The Vanishing Lemma and Lemma 8 . 8

Semi-precompactness in Šostak’s L-fuzzy topological spaces

In this paper, we introduce a good definition of semiprecompactness in L-fuzzy topological spaces in Šostak’s sense, where L is a fuzzy lattice. We define this concept on arbitrary L-fuzzy sets, obtain a different characterization and study some of its properties. 2010 AMS Classification: 54A40, 54D30, 53A05

The Generalized XOR Lemma

The XOR Lemma states that a mapping is regular or balanced if and only if all the linear combinations of the component functions of the mapping are balanced Boolean functions. The main contribution of this paper is to extend the XOR Lemma to more general cases where a mapping may not be necessarily regular. The extended XOR Lemma has applications in the design of substitution boxes or S-boxes u...

3 the Switching Lemma

Today we show that PARITY is not in AC0. AC0 is a family of circuits with constant depth, polynomial size, and unbounded fan-in for the AND and OR gates. We establish this result through an application of the Switching Lemma. This result is the first use of randomization in its full power in complexity. Circuits were defined in previous lectures. In this lecture, we always assume that the circu...