Tire track geometry: Variations on a theme

Tire track geometry: variations on a theme

We study closed smooth convex plane curves Γ enjoying the following property: a pair of points x, y can traverse Γ so that the distances between x and y along the curve and in the ambient plane do not change; such curves are called bicycle curves. Motivation for this study comes from the problem how to determine the direction of the bicycle motion by the tire tracks of the bicycle wheels; bicyc...

Geometry, Statistics, Probability: Variations on a Common Theme

ion Mathematicians like to talk about how things behave, rather than what they are. Nonmathematicians often prefer the reverse. "Anything that behaves this way is a vector (because I said so)" is not a natural way of talking to the nonmathematician. Mathematicians ought to be more sensitive to this than they typically are. Repeating examples in different words, emphasizing the equivalence of th...

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...