Variations on a Theme of Widder

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

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 of Horowitz

Horowitz [Hor1] showed that for every n ≥ 2, there exist elements w1, . . . , wn in F2 = free(a, b) which generate non-conjugate maximal cyclic subgroups of F2 and which have the property that trace(ρ(w1)) = · · · = trace(ρ(wn)) for all faithful representations ρ of F2 into SL2(C). Randol [Ran] used this result to show that the length spectrum of a hyperbolic surface has unbounded multiplicity....

Variations on a Theme of Borel