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...

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...

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...

The family of regular closed subsets of a topological space is used to introduce two concepts concerning a function f from a space X to a space Y . The first of them is the notion of f being rc-continuous. One of the established results states that a space Y is extremally disconnected if and only if each continuous function from a space X to Y is rc-continuous. The second concept studied is the...

We complement the literature by proving that for a fixed-point free map f : X → X the statements (1) f admits a finite functionally closed cover A with f [A]∩A = ∅ for all A ∈ A (i.e., a coloring) and (2) βf is fixed-point free are equivalent. When functionally closed is weakened to closed, we show that normality is sufficient to prove equivalence, and give an example to show it cannot be omitt...

Let f, g : X → G/K be maps from a closed connected orientable manifold X to an orientable coset space M = G/K where G is a compact connected Lie group, K a closed subgroup and dimX = dimM . In this paper, we show that if L(f, g) = 0 then N(f, g) = 0; if L(f, g) 6= 0 then N(f, g) = R(f, g) where L(f, g), N(f, g), and R(f, g) denote the Lefschetz, Nielsen, and Reidemeister coincidence numbers of ...

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...