Four variations on a fractal theme

Variations on a theme.

A ring satisfying x = x is necessarily commutative. We consider a variety of weaker forms of this condition and show that many but not all of them imply commutativity. We also present a variety of elementary proofs of the fact that x = x implies commutativity.

Variations on a Theme

Characterizing Non-Deterministic Circuit Size (Four variations on one theme)

Consider the following simple communication problem. Fix a universe U and a family Ω of subsets of U . Players I and II receive, respectively, an element a ∈ U and a subset A ∈ Ω. Their task is to find a subset B of U such that |A∩B| is even and a ∈ B. With every Boolean function f we associate a collection Ωf of subsets of U = f−1(0), and prove that its (one round) communication complexity com...

Variations on a Montagovian theme

What are the objects of knowledge, belief, probability, apriority or analyticity? For at least some of these properties, it seems plausible that the objects are sentences, or sentence-like entities. However, results from mathematical logic indicate that sentential properties are subject to severe formal limitations. After surveying these results, I argue that they are more problematic than ofte...

Variations on a Theme by James Stirling