Two results in classical measure theory

Some Special Results of Measure Theory

Revisiting Two Classical Results on Graph Spectra

Let (G) and min (G) be the largest and smallest eigenvalues of the adjacency matrix of a graph G. Our main results are: (i) If H is a proper subgraph of a connected graph G of order n and diameter D; then (G) (H) > 1 2D (G)n : (ii) If G is a connected nonbipartite graph of order n and diameter D, then (G) + min (G) > 2 2D (G)n : These bounds have the correct order of magnitude for large and D. ...

Two Results on Resource-Bounded Measure

We construct an oracle relative to which NP has p-measure 0 but D p has measure 1 in EXP. This gives a strong relativized negative answer to a question posed by Lutz Lut96]. Secondly, we give strong evidence that BPP is small. We show that BPP has p-measure 0 unless EXP = MA and the polynomial-time hierarchy collapses. This contrasts the work of Regan et al. RSC95], where it is shown that P=pol...

More Results on Dynamic Cumulative Inaccuracy Measure

In this paper, borrowing the intuition in Rao et al. (2004), we introduce a cumulative version of the inaccuracy measure (CIM). Also we obtain interesting and applicable properties of CIM for different cases based on the residual, past and interval lifetime random variables. Relying on various applications of stochastic classes in reliability and information theory fields, we stu...

Two Results in Combinatorial Set Theory

In this note we sketch the proofs of two results in combinatorial set theory. The common theme of the results is singular cardinal combinatorics: they involve the interaction between forcing, large cardinals, PCF theory and versions of Jensen’s weak square principle. 1. Changing cofinality Our first result involves an old question about changes of cofinality. Suppose that V,W are inner models o...