Every commutative quasirational language is regular

Every (λ, Κ)-regular Ultrafilter Is (λ, Κ)-regular

We prove the following: Theorem A. If D is a (λ+, κ)-regular ultrafilter, then either (a) D is (λ, κ)-regular, or (b) the cofinality of the linear order ∏ D〈λ, <〉 is cf κ, and D is (λ, κ′)-regular for all κ′ < κ. Corollary B. Suppose that κ is singular, κ > λ and either λ is regular, or cf κ < cf λ. Then every (λ+n, κ)-regular ultrafilter is (λ, κ)-regular. We also discuss some consequences and...

Learning Commutative Regular Languages

When every $P$-flat ideal is flat

In this paper‎, ‎we study the class of rings in which every $P$-flat‎ ‎ideal is flat and which will be called $PFF$-rings‎. ‎In particular‎, ‎Von Neumann regular rings‎, ‎hereditary rings‎, ‎semi-hereditary ring‎, ‎PID and arithmetical rings are examples of $PFF$-rings‎. ‎In the context domain‎, ‎this notion coincide with‎ ‎Pr"{u}fer domain‎. ‎We provide necessary and sufficient conditions for‎...

Every 4-regular graph is acyclically edge-6-colorable

An acyclic edge coloring of a graph G is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index a(G) of G is the smallest integer k such that G has an acyclic edge coloring using k colors. Fiamčik (1978) and later Alon, Sudakov and Zaks (2001) conjectured that a(G) ≤ ∆ + 2 for any simple graph G with maximum degree ∆. Basavaraju and Chandran (2009) show...

Almost Every Real Quadratic Map Is Either Regular or Stochastic

Stony Brook IMS Preprint #1997/8 July, 1997 Abstract. We prove uniform hyperbolicity of the renormalization operator for all possible real combinatorial types. We derive from it that the set of infinitely renormalizable parameter values in the real quadratic family Pc : x 7→ x 2 + c has zero measure. This yields the statement in the title (where “ regular” means to have an attracting cycle and ...