For a positive integer k, a k-relation on a set Ω is a non-empty subset ∆ of the k-fold Cartesian product Ω; ∆ is called a k-relation for a permutation group H on Ω if H leaves ∆ invariant setwise. The k-closure H of H, in the sense of Wielandt, is the largest permutation group K on Ω such that the set of k-relations for K is equal to the set of k-relations for H. We study k-relations for finit...

The dynamical system (f,R) is introduced and some of its properties are investigated. It is proven that there is an invariant set Λ on which the periodic points of f are dense.

We construct, for the Weil representation associated with any discriminant form, an explicit basis in which action of involves algebraic integers over its field definition. The a general element $${\text {SL}}_{2}(\mathbb {Z})$$ on many parts these bases is simple and explicit, fact that we use determining dimension space invariants some families forms.

in this paper, we prove some coupled coincidence point theorems for mappings satisfying generalized contractive conditions under a new invariant set in ordered cone metric spaces. in fact, we obtain sufficient conditions for existence of coupled coincidence points in the setting of cone metric spaces. some examples are provided to verify the effectiveness and applicability of our results.

For a separable rearrangement invariant space X on [0, 1] of fundamental type we identify the set all $$p\in [1,\infty ]$$ such that $$\ell ^p$$ is finitely represented in way unit basis vectors ( $$c_0$$ if $$p=\infty $$ ) correspond to pairwise disjoint and equimeasurable functions. This can be treated as follow up paper by first-named author related spaces $$(0,\infty )$$ .

We consider a linear problem over a finite set of integer vectors. We assume that there is a verification oracle, which is an algorithm being able to verify whether a given vector optimizes a given linear function over the feasible set. Given an initial solution, the algorithm proposed in this paper finds an optimal solution of the problem together with a path, in the 1-skeleton of the convex h...