Let p and q be two positive primes, let \(\ell\) an odd prime F a quadratic number field. K extension of degree such that is dihedral \({\mathbb {Q}}\), or else abelian \(\ell\)-extension unramified over whenever divides the class F. In this paper, we provide complete characterization division quaternion algebras \(H_{K}(p, q)\) K.

This paper investigates the class of rings in which every nn-absorbing ideal is a prime ideal, called nn-AB ring, where nn positive integer. We give characterization an ring. Next, for ring RR, we study concept Ω(R)={ωR(I);I proper R},Ω(R)={ωR(I);I R}, ωR(I)=min{n;I n-absorbing R}ωR(I)=min{n;I R}. show that if RR Artinian or Prüfer domain, then Ω(R)∩NΩ(R)∩N does not have any gaps (i.e., wheneve...

A ring R is said to be semi-commutative if whenever a, b ∈ such that ab = 0, then aRb 0. In this article, we introduce the concepts of g−semi-commutative rings and g−N−semi-commutative several results concerning these two concepts. Let a G-graded g supp(R, G). Then with aRgb Also, − N−semi-commutative for any N(R) ⋂ Ann(a), bRg ⊆ Ann(a). We an example which N-semi-commutative some G) but itself...

Abstract We consider functionals given by the sum of perimeter and double integral some kernel g : ℝ N × → + {g:\mathbb{R}^{N}\times\mathbb{R}^{N}\to\mathbb{R}^{+}} , multiplied a “mass parameter” ε. show...

Let $G$ be a group with identity $e$. $R$ commutative $G$-graded ring non-zero identity, $S\subseteq h(R)$ multiplicatively closed subset of and $M$ graded $R$-module. In this article, we introduce study the concept $S$-1-absorbing prime submodules. A submodule $N$ $(N:_{R}M)\cap S=\emptyset$ is said to prime, if there exists an $s_{g}\in S$ such that whenever $a_{h}b_{h'}m_{k}\in N$, then eith...

An almost self-centered graph is a connected of order $n$ with exactly $n-2$ central vertices, and an peripheral $n-1$ vertices. We determine (1) the maximum girth $n$; (2) independence number radius $r$; (3) minimum $k$-regular graph; (4) size (5) possible degrees (6) vertices degree in $n-4$ which second largest possible. Whenever extremal graphs have neat form, we also describe them.

Let (bn) = (b1, b2, . . . ) be a sequence of integers. A primitive prime divisor of a term bk is a prime which divides bk but does not divide any of the previous terms of the sequence. A zero orbit of a polynomial φ(z) is a sequence of integers (cn) where the n-th term is the n-th iterate of φ at 0. We consider primitive prime divisors of zero orbits of polynomials. In this note, we show that f...

A frequency square is a matrix in which each row and column permutation of the same multiset symbols. type $$(n;\lambda )$$ if it contains $$n/\lambda $$ symbols, occurs $$\lambda times per column. In case when =n/2$$ we refer to as binary. set k-MOFS k squares such that any two are superimposed, possible ordered pair equally often. k-maxMOFS not contained $$(k+1)$$ -MOFS . For even n, let $$\m...

For a graph H, letc∞(H)=limn→∞⁡max⁡|E(G)|n, where the maximum is taken over all graphs G on n vertices not containing H as minor. Thus c∞(H) asymptotic density of Employing structural lemma due to Eppstein, we prove new upper bounds for disconnected H. In particular, determine whenever union cycles. Finally, investigate behaviour c∞(sKr) fixed r, sKr denotes s disjoint copies complete r vertice...