In this paper, we discuss generating theorems of polyhedral quadrangulations closed surfaces. We prove that the set eight reductional operations {R1, …, R8} defined for is finitizable any surface F2, is, there exist finitely many minimal F2 using such R1, R7 and R8. Furthermore, show proper subset not torus.

Real-world applications now deal with a massive amount of data, and information about the world is inaccurate, incomplete, or uncertain. Therefore, we present in our paper proposed model for solving problems. This based on class locally generalized closed sets, namely, simply* alpha closed* sets closed** (briefly, -sets -sets), open set. We also introduce various concepts their properties relat...

We consider union-closed set systems with infinite breadth, focusing on three particular configurations ${\mathcal T}_{\rm max}(E)$, min}(E)$ and ort}(E)$. show that these are not isolated examples; in any given system of at least one will occur as a subprojection. This characterizes those which have is the first general structural result for such systems.

‎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 $f$ 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 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.