We show that, under reasonably mild hypotheses, the solution of the forced–dissipative rotating primitive equations of the ocean loses most of its fast, inertia–gravity, component in the small Rossby number limit as t → ∞. At leading order, the solution approaches what is known as “geostrophic balance” even under ageostrophic, slowly time-dependent forcing. Higherorder results can be obtained i...

An r-edge coloring of a graph G is a mapping h : E(G) → [r], where h(e) is the color assigned to edge e ∈ E(G). An exact r-edge coloring is an r-edge coloring h such that there exists an e ∈ E(G) with h(e) = i for all i ∈ [r]. Let h be an edge coloring of G. We say G is rainbow if no two edges in G are assigned the same color by h. The anti-Ramsey number, AR(G,n), is the smallest integer r such...

A k-matching in a hypergraph is a set of k edges such that no two of these edges intersect. The anti-Ramsey number of a k-matching in a complete s-uniform hypergraph H on n vertices, denoted by ar(n, s, k), is the smallest integer c such that in any coloring of the edges of H with exactly c colors, there is a k-matching whose edges have distinct colors. The Turán number, denoted by ex(n, s, k),...

Both the Continuum Hypothesis and Martin's Axiom allow induc-tive constructions to continue in circumstances where the inductive hypothesis might otherwise fail. The search for useful related axioms procedes naturally in two directions: towards "Super Martin's Axioms ," which extend MA to broader classes of orders; and towards "Anti-Martin's Axioms" (AMA's) which are strictly weaker than CH, bu...

We investigate the effect of adding a single real (for various forcing notions adding reals) on cardinal invariants associated with the continuum. We show: (1) adding an eventually different or a localization real adjoins a Luzin set of size continuum and a mad family of size ω1; (2) Laver and Mathias forcing collapse the dominating number to ω1, and thus two Laver or Mathias reals added iterat...