Far from Being Exhausted

Mitochondria from Exhausted Rats

On Snarks that are far from being 3-Edge Colorable

In this note we construct two infinite snark families which have high oddness and low circumference compared to the number of vertices. Using this construction, we also give a counterexample to a suggested strengthening of Fulkerson’s conjecture by showing that the Petersen graph is not the only cyclically 4-edge connected cubic graph which require at least five perfect matchings to cover its e...

How Far Is an Element from Being Prime?

Let D be an integral domain. We investigate two invariants ω(D, x) and ω(D) which measure how far an x ∈ D is from being prime and how far an atomic integral domain D is from being a UFD, respectively. We give a new characterization of number fields with class number two. We also study asymptotic versions of these two invariants.

On Being Far from Far and on Dual Problems in Property Testing

This work studies a new type of problems in property testing, called dual problems. For a set Π in a metric space and δ > 0, denote by Fδ(Π) the set of elements that are δ-far from Π. Then, in property testing, a δ-tester for Π is required to accept inputs from Π and reject inputs from Fδ(Π). A natural dual problem is the problem of δ-testing the set of “no” instances, that is Fδ(Π): A δ-tester...

Setting up Exhausted Values

This paper argues that exhaustification is empirically and theoretically important as a tool in the semantic description of various constructions in natural language. Discussion focuses on the comparative construction, and presents a view of exhaustification as a ‘control element’ that influences the interpretation of a more basic element: the existential quantifier.