We introduce the notion of a matron M = {Mn,m} whose submodules M∗,1 and M1,∗ are non-Σ operads. We construct a functor from PROP to matrons and its inverse, the universal enveloping functor. We define the free matron H∞, generated by a singleton in each bidegree (m, n) 6= (1, 1), and define an A∞-bialgebra as an algebra over H∞. We realize H∞ as the cellular chains of polytopes {KKn,m = KKm,n}...

We introduce a class of generalized vector quasivariational-like inequality problems in Banach spaces. We derive some new existence results by using KKM-Fan theorem and an equivalent fixed point theorem. As an application of our results, we have obtained as special cases the existence results for vector quasi-equilibrium problems, generalized vector quasivariational inequality and vector quasi-...

We consider the weakly efficient solution for a class of nonconvex and nonsmooth vector optimization problems in Banach spaces. We show the equivalence between the nonconvex and nonsmooth vector optimization problem and the vector variational-like inequality involving set-valued mappings. We prove some existence results concerned with the weakly efficient solution for the nonconvex and nonsmoot...

In this paper, some new W-GPH-KKM theorems are established in pseudo H-spaces without any linear and convex structure under much weaker assumptions, and next as their applications, some new coincidence theorems, maximal element theorems, and existence theorems of solutions to generalized equilibrium problems are proved in pseudo H-spaces. The results represented in this paper unify and extend s...

For an even integer m at least 4 and any positive integer t, it is shown that the complete equipartite graph Kkm(2t) can be decomposed into edge-disjoint gregarious m-cycles for any positive integer k under the condition satisfying (m−1)+3 4m < k. Here it will be called a gregarious cycle if the cycle has at most one vertex from each partite set.

This research is to examine the quality of school learning which can be identified by existence KKM, where determination Minimum Completeness Criteria (KKM) initial stage implementing assessment outcomes. The reality in field teachers setting KKM not based on analysis and does pay attention principles steps for determining it, therefore it necessary have activities at beginning year that provid...