Basic conjugacy theorems forG 2

The Conjugacy Theorems

We use the theory of differential forms, orientation, and mapping degree to prove that all maximal tori in a compact connected Lie group are conjugate. We also prove that all Lie groups are orientable, and that if G is a compact connected Lie group and T a maximal torus of G, then dim G/T is even.

Karlin’s Basic Composition Theorems and Stochastic Orderings

Suppose λ,x,ζ traverse the ordered sets Λ, X and Z, respectively and consider the functions f(λ,x,ζ) and g(λ,ζ) satisfying the following conditions, (a) f(λ,x,ζ) > 0 and f is TP2 in each pairs of variables when the third variable is held fixed and (b) g(λ,ζ) is TP2. Then the function h(λ,x) =∫Z f(λ,x,ζ)g(λ,ζ)dµ(ζ), defined on Λ×X is TP2 in (λ,x). The aim of this note is to use a new stochast...

Basic Theorems About Security

We build a mathematical structure in which we can ask questions about the methods for achieving security properties, such as confidentiality and integrity, and functionality properties, such as safety and liveness. The structure allows us to consider many different choices for the meaning of “confidentiality” and “integrity” and so on, and to compare and contrast security properties with functi...

Basic Elements: Theorems from Algebraic Avtheory

New Proofs of Basic Theorems in Calculus

In this note we present new proofs of three basic theorems in calculus. Although these theorems are well-known, in each proof we obtain something which seems to be unknown. We start with the Heine-Cantor theorem about uniform continuity and obtain explicitly the optimal delta for the given epsilon. We then proceed with the Weierstrass extreme value theorem and present two proofs of it: the “env...