Two Eigenvector Theorems

Two Pebbling Theorems

Two conductance theorems, two canonical path theorems, and two walks on directed Cayley graphs

We show two Conductance-like theorems for mixing time of non-reversible non-lazy walks. These bounds involve a measure of expansion which expresses how well ergodic flow is distributed among vertices, which while conceptually similar to Blocking Conductance apply to non-lazy non-reversible Markov chains as well. As an application we derive two canonical path theorems for mixing time of non-reve...

Two common fixed Point theorems for compatible mappings

Recently, Zhang and Song [Q. Zhang, Y. Song, Fixed point theory forgeneralized $varphi$-weak contractions,Appl. Math. Lett. 22(2009) 75-78] proved a common fixed point theorem for two mapssatisfying generalized $varphi$-weak contractions. In this paper, we prove a common fixed point theorem fora family of compatible maps. In fact, a new generalization of Zhangand Song's theorem is given.

Two theorems on experimental logics

A generalization of a formal system is considered, in which the axioms of the formal system can be withdrawn or supplemented, as mechanical experimentation proceeds through time and the consequences of various combinations of assumptions are realized. The "theorems" of these experimental logics are taken to be those assertions possessing a proof which remains valid for all "sufficiently large t...

Two Rigidity Theorems for Gravity

In this paper we prove two theorems on initial data for general relativity. The results present “rigidity phenomena” for the extrinsic curvature, caused by the negativity of the scalar curvature. More precisely, Theorem 1 states that in the case of non-vacuum initial data if the metric has everywhere non-positive scalar curvature then the extrinsic curvature cannot be compactly supported. Theor...