Let $R$ be a commutative ring and let $M$ be an $R$-module. We define the small intersection graph $G(M)$ of $M$ with all non-small proper submodules of $M$ as vertices and two distinct vertices $N, K$ are adjacent if and only if $Ncap K$ is a non-small submodule of $M$. In this article, we investigate the interplay between the graph-theoretic properties of $G(M)$ and algebraic properties of $M...

Hyper-minimization is a state reduction technique that allows a finite change in the semantics. The theory for hyper-minimization of deterministic weighted tree automata is provided. The presence of weights slightly complicates the situation in comparison to the unweighted case. In addition, the first hyper-minimization algorithm for deterministic weighted tree automata, weighted over commutati...

in this note we consider the notion of intuitionistic fuzzy (weak) dual hyper k-ideals and obtain related results. then we classify this notion according to level sets. after that we determine the relationships between intuitionistic fuzzy (weak) dual hyper k-ideals and intuitionistic fuzzy (weak) hyper k-ideals. finally, we define the notion of the product of two intuitionistic fuzzy (weak) du...

In this paper, we study norms of circulant matrices F = Circ(F 0 , F (k) 1 , . . . ,F (k) n−1) , L = Circ(L 0 , L (k) 1 , . . . ,L (k) n−1) and r -circulant matrices Fr = Circr(F (k) 0 , F (k) 1 , . . . ,F (k) n−1) , Lr = Circr(L 0 , L (k) 1 , . . . ,L (k) n−1) , where F (k) n and L (k) n denote the hyper-Fibonacci and hyper-Lucas numbers, respectively. Mathematics subject classification (2010)...

We show that the validity of the non-commutative Khintchine inequality for some q with 1 < q < 2 implies its validity (with another constant) for all 1 ≤ p < q. We prove this for the inequality involving the Rademacher functions, but also for more general " lacunary " sequences, or even non-commutative analogues of the Rademacher functions. For instance, we may apply it to the " Z(2)-sequences ...

A sequence (xn)n=1? on the torus T exhibits Poissonian pair correlation if for all s>0,limN???1N#{1?m?n?N:|xm?xn|?sN}=2s. It is known that this condition implies equidistribution of (xn). We generalize result to four-fold differences: s>0 we havelimN???1N2#{1?m,n,k,l?N{m,n}?{k,l}:|xm+xn?xk?xl|?sN2}=2s then equidistributed. This notion generalizes higher orders, and any k show a exhibiting 2k-fo...

Let k be a commutative ring and let R be a commutative k−algebra. Given a positive integer n and a R−algebra A one can consider three functors of points from the category CR of commutative R−algebras to the small category of sets. All these functors are representable, namely • RepA represents the functor induced by B → homR(A,Mn(B)), where Mn(B) are the n× n matrices over B, for all B ∈ CR. • t...

In this note, we study the lattice structure on the class of all weak hyper K-ideals of a hyper K-algebra. We first introduce the notion of (left,right) scalar in a hyper K-algebra which help us to characterize the weak hyper K-ideals generated by a subset. In the sequel, using the notion of a closure operator, we study the lattice of all weak hyper K-ideals of ahyper K-algebra, and we prove a ...

Let n 1 be an integer. The hypercube Qn is the graph whose vertex set isf0;1gn, where two n-tuples are adjacent if they differ in precisely one coordinate. This graph has many applications in Computer sciences and other area of sciences. Inthe graph Qn, the layer Lk is the set of vertices with exactly k 1’s, namely, vertices ofweight k, 1 k n. The hyper-star graph B(n;k) is...

In this paper we prove that every n-Jordan homomorphis varphi:mathcal {A} longrightarrowmathcal {B} from unital Banach algebras mathcal {A} into varphi -commutative Banach algebra mathcal {B} satisfiying the condition varphi (x^2)=0 Longrightarrow varphi (x)=0, xin mathcal {A}, is an n-homomorphism. In this paper we prove that every n-Jordan homomorphism varphi:mathcal {A} longrightarrowmathcal...