We study in this paper polarized infinite exponent partition relations. We apply our results to constructing a model for the theory “ZF+DC+ω1 is the only regular, uncountable cardinal ≤ ωω1+1.” This gives a partial answer to a question of Woodin. In 1994, J. Steel proved what had long been suspected: that assuming AD, the regular cardinals of L(R) below Θ are all measurable, where Θ is the leas...

Countable Lawvere theories model computational effects such as exceptions, side-effects, interactive input/output, nondeterminism and probabilistic nondeterminism. The category of countable Lawvere theories has sums, tensors, and distributive tensors, modelling natural combinations of such effects. It is also closed under taking images. Enrichment in a category such as Cpo allows one to extend ...

We introduce a notion of L-Betti numbers for locally compact, second countable, unimodular groups. We study the relation to the standard notion of L-Betti numbers of countable discrete groups for lattices. In this way, several new computations are obtained for countable groups, including lattices in algebraic groups over local elds, and Kac-Moody lattices. We also extend the vanishing of reduce...

No countable C,-free graph contains every countable C,-free graph as a subgraph, for n 2 4. For n = 4, this was proved earlier by J. Pach.

The distribution behavior of ofloxacin enantiomers was examined in the aqueous and organic solvent of a two-phase system containing chiral selector L-tartarate. The effect of extraction equilibrium time, buffer pH, organic solvents and length of alkyl chain of L-tartarate on the partition coefficients and enantioselectivity of racemic ofloxacin were investigated, respectively. The L-tartara...

We present necessary and sufficient conditions for the existence of a countably additive measure on a Boolean σ-algebra. For instance, a Boolean σ-algebra B is a measure algebra if and only if B−{0} is the union of a chain of sets C1 ⊂ C2 ⊂ ... such that for every n, (i) every antichain in Cn has at mostK(n) elements (for some integerK(n)), (ii) if {an}n is a sequence with an / ∈ Cn for each n,...

For a graph $G = (V, E)$, a partition $pi = {V_1,$ $V_2,$ $ldots,$ $V_k}$ of the vertex set $V$ is an textit{upper domatic partition} if $V_i$ dominates $V_j$ or $V_j$ dominates $V_i$ or both for every $V_i, V_j in pi$, whenever $i neq j$. The textit{upper domatic number} $D(G)$ is the maximum order of an upper domatic partition. We study the properties of upper domatic number and propose an up...

This paper is devoted to settling the following problem on (infinite, partially) ordered sets: Is there always a partition (2-coloring) of an ordered set X so that all nontrivial maximal chains ofX meet both classes (receive both colors)? We show this is true for all countable ordered sets and provide counterexamples of cardinality H3. Variants of the problem are also considered and open proble...

In this paper, spaces with σ-locally countable weak-bases are characterized as the weakly open msss-images of metric spaces (or g-first countable spaces with σ-locally countable cs-networks). To find the internal characterizations of certain images of metric spaces is an interesting research topic on general topology. Recently, S. Xia introduced the concept of weakly open mappings, by using it,...

We study the relationship between the countable axiom of choice and the Tychonoff product theorem for countable families of topological spaces.