let r be a commutative ring with identity and m be a unitary r-module. let : s(m) −! s(m) [ {;} be a function, where s(m) is the set of submodules ofm. suppose n  2 is a positive integer. a proper submodule p of m is called(n − 1, n) − -prime, if whenever a1, . . . , an−1 2 r and x 2 m and a1 . . . an−1x 2p(p), then there exists i 2 {1, . . . , n − 1} such that a1 . . . ai−1ai+1 . . . an−1x...

In this research, we develop some differential subordination results involving harmonic means of $f_{b}(z),f_{b}(z)+zf_{b}^{\prime}(z)$ and $f_{b}(z)+\frac{zf_{b}^{\prime}(z)}{f_{b}(z)},$ where $f_{b}(z)=\frac{z}{\left(1-z^{n}\right) ^{b}},$ $b\geq0;n\in\mathbb{N}=1,2,3...$ is an $n$-fold symmetric Koebe type functions defined in the unit disk with $f_{b}(0)=0,f_{b}^{\prime}(z)\neq0.$ By using ...

For $n\geq 1$, let $C_n$ denote a cyclic group of order $n$. Let $G=C_n\oplus C_{mn}$ with 2$ and $m\geq $k\in [0,n-1]$. It is known that any sequence $mn+n-1+k$ terms from $G$ must contain nontrivial zero-sum length at most $mn+n-1-k$. The associated inverse question to characterize those sequences maximal $mn+n-2+k$ fail subsequence $k\leq this the for Davenport Constant. $k=n-1$, $\eta(G)$ i...

Let $p$ be a prime, let $K$ discretely valued extension of $\mathbb{Q}_p$, and $A_{K}$ an abelian $K$-variety with semistable reduction. Extending work by Kim Marshall from the case where $p>2$ $K/\mathbb{Q}_p$ is unramified, we prove $l=p$ complement Galois cohomological formula Grothendieck for $l$-primary part N\'eron component group $A_{K}$. Our proof involves constructing, each $m\in \math...

Background • Legendre Sequence For a prime p > 2 let (s n) be the Legendre sequence defined as s n = 1, n p = −1, 0, otherwise, n ≥ 0, where. p denotes the Legendre symbol. • Sidelnikov Sequence Let q be an odd prime power, g a primitive element of F q , and let η denote the quadratic character of F Background • Legendre Sequence For a prime p > 2 let (s n) be the Legendre sequence defined as s...