‎let us consider the set of non-abelian finite simple groups which‎ ‎admit non-trivial irreducible projective representations of degree $le 7$ over‎ ‎an algebraically closed field $mathbb{f}$ of characteristic $pgeq 0$‎. ‎we survey some recent results which‎ ‎lead to the complete list of the groups in this set which are‎ ‎$(2,3,7)$-generated and of those which are $(2,3)$-generated‎.

Let f be a diffeomorphism of a closed C∞ three-dimensional manifold. In this paper, we introduce the notion of C1-stably weak shadowing for a closed f -invariant set, and prove that C1-generically, for an aperiodic chain component Cf of f isolated in the chain recurrent set, if f|Cf is C 1-stably weak shadowing, then there are a C1-neighborhood U(f) of f and an open and dense subset V of U(f) s...

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...

Let G be a (p, q) graph. Let k be an integer with 2 ≤ k ≤ p and f from V (G) to the set {1, 2, . . . , k} be a map. For each edge uv, assign the label |f(u) − f(v)|. The function f is called a k-difference cordial labeling of G if |νf (i) − vf (j)| ≤ 1 and |ef (0) − ef (1)| ≤ 1 where vf (x) denotes the number of vertices labelled with x (x ∈ {1, 2 . . . , k}), ef (1) and ef (0) respectively den...

In this paper, we introduce a new function called tri-ĝ continuous in tri-topological spaces. A map f : X → Y is if f-1(V) closed for each tri-closed set V Y. Also define, quasi tri-ĝ. perfectly tri-ĝ, contra totally and strongly maps are introduced some of their properties studied.

It is shown how many techniques of categorical domain theory can be expressed in the general context of topical categories (where \topical" means internal in the category Top of Grothendieck toposes with geometric morphisms). The underlying topos machinery is hidden by using a geometric form of constructive mathematics, which enables toposes as \generalized topological spaces" to be treated in ...

We study exponentiability of homomorphisms in varieties of universal algebras close to classical ones. After describing an “almost folklore” general result, we present a purely algebraic proof of “étale implies exponentiable”, alternative to the topologically motivated proof given in one of our previous papers, in a different context. We prove that only isomorphisms are exponentiable homomorphi...

let g be a (p, q) graph. let k be an integer with 2 ≤ k ≤ p and f from v (g) to the set {1, 2, . . . , k} be a map. for each edge uv, assign the label |f(u) − f(v)|. the function f is called a k-difference cordial labeling of g if |νf (i) − vf (j)| ≤ 1 and |ef (0) − ef (1)| ≤ 1 where vf (x) denotes the number of vertices labelled with x (x ∈ {1, 2 . . . , k}), ef (1) and ef (0) respectively den...

1.2. Proposition. — π : PnA → SpecA is a closed morphism. Proof. Suppose Z ↪→ PnA is a closed subset. We wish to show that π(Z) is closed. Suppose y / ∈ π(Z) is a closed point of SpecA. We’ll check that there is a distinguished open neighborhood D(f) of y in SpecA such that D(f) doesn’t meet π(Z). (If we could show this for all points of π(Z), we would be done. But I prefer to concentrate on cl...

A topological space X is said to be completely regular if whenever F is a nonempty closed set and x ∈ X \F , there is a continuous function f : X → [0, 1] such that f(x) = 0 and f(F ) = {1}. A completely regular space need not be Hausdorff. For example, ifX is any set with more than one point, then the trivial topology, in which the only closed sets are ∅ and X, is vacuously completely regular,...