On two theorems of Thompson

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A generalization of a formal system is considered, in which the axioms of the formal system can be withdrawn or supplemented, as mechanical experimentation proceeds through time and the consequences of various combinations of assumptions are realized.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 Theorems on Galois Cohomology1

Notice that we have dropped the hypothesis that both k and K be Galois over the rationals. To see how Theorem 1 generalizes Yokoi's result, remember that if G has prime order p, then multiplication by p annihilates all the cohomology groups. Thus in this case the cohomology groups are determined up to isomorphism by their order. The technique used to prove Theorem 1 can be used to prove other r...

Two theorems on perfect codes

TWO theorems nre proved on perfect codes. The first cne states tk.at Lloyd's theorem is true without tne assumption that the number of symbols in the alphabet is a prime power. The second thevrem asser?s the impossibility of perfect group codes over non-prime-pcwer-alphabets. IA V be a finite set, I VI = q > 2, and let 1 <_ e 2 ye be Ia:ional integers. We pu?iV= (1, 2,. .. . n). Forv=(Vi)~=l E ...

Two theorems on lattice expansions

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...