We show that the statement CCFC = “the character of a maximal free filter F of closed sets in a T1 space (X, T ) is not countable” is equivalent to the Countable Multiple Choice Axiom CMC and, the axiom of choice AC is equivalent to the statement CFE0 = “closed filters in a T0 space (X, T ) extend to maximal closed filters”. We also show that AC is equivalent to each of the assertions: “every c...

Two countable lters on ! are incompatible if they have no common innnite pseudo-intersection. Letting a(P f) denote the minimal size of a maximal uncountable family of pairwise incompatible countable lters on !, we prove the consistency of t < a(P f).

Some properties of minimal open sets and maximal open sets are studied in [1, 2]. In this paper, we define dual concepts of them, namely, maximal closed set and minimal closed set. These four types of subsets appear in finite spaces, for example. More generally, minimal open sets and maximal closed sets appear in locally finite spaces such as the digital line. Minimal closed sets and maximal op...

let g be a group. a subset x of g is a set of pairwise noncommuting elements if xy ̸= yx for any two distinct elements x and y in x. if |x| ≥ |y | for any other set of pairwise non-commuting elements y in g, then x is said to be a maximal subset of pairwise non-commuting elements. in this paper we determine the cardinality of a maximal subset of pairwise non-commuting elements in any non-abelian...

a r t i c l e i n f o a b s t r a c t Keywords: Valuation independence Generalized series fields Fields of Puiseux series Restricted exponential function We investigate valued fields which admit a valuation basis. Given a countable ordered abelian group G and a real closed or algebraically closed field F with subfield K , we give a sufficient condition for a valued subfield of the field of gene...

In this paper we investigate the one-dimensional hyperbolic mean curvature flow for closed plane curves. More precisely, we consider a family of closed curves F : S × [0, T ) → R which satisfies the following evolution equation ∂F ∂t (u, t) = k(u, t) ~ N(u, t)− ▽ρ(u, t), ∀ (u, t) ∈ S1 × [0, T ) with the initial data F (u, 0) = F0(u) and ∂F ∂t (u, 0) = f(u) ~ N0, where k is the mean curvature an...

A classic result of Johnson and Lindenstrauss asserts that any set of n points in d-dimensional Euclidean space can be embedded into k-dimensional Euclidean space—where k is logarithmic in n and independent of d—so that all pairwise distances are maintained within an arbitrarily small factor. All known constructions of such embeddings involve projecting the n points onto a spherically random k-...

let $mathbb{f}$ be an algebraically closed field of prime characteristic $p>2$ and $(g, [p])$ a finite-dimensional restricted lie superalgebra over $mathbb{f}$. it is showed that anyfinite-dimensional indecomposable $g$-module is a module for a finite-dimensional quotient of the universal enveloping superalgebra of $g$. these quotient superalgebras are called the generalized reduced enveloping ...

to demonstrate more visibly the close relation between thecontinuity and integrability, a new proof for the banach-zareckitheorem is presented on the basis of the radon-nikodym theoremwhich emphasizes on measure-type properties of the lebesgueintegral. the banach-zarecki theorem says that a real-valuedfunction $f$ is absolutely continuous on a finite closed intervalif and only if it is continuo...

(0, 1,2}. The set of three-valued logical functions (i.e., f: E” --t E for n=l,2,... ) is denoted by P3. A subset F of P3 is said to be closed if it contains all superpositions (i.e., compositions or substitutions) of its members (cf. [l, 1 l-131). For closed sets F and H such that FC H (proper inclusion), F is an H-maxima/ set if FC G c H for no closed set G. A subset F of H is complete in H i...