Let $R$ be a commutative ring with identity and let $M$ be an $R$-module. A proper submodule $P$ of $M$ is called strongly prime submodule if $(P + Rx : M)ysubseteq P$ for $x, yin M$, implies that $xin P$ or $yin P$. In this paper, we study more properties of strongly prime submodules. It is shown that a finitely generated $R$-module $M$ is Artinian if and only if $M$ is Noetherian and every st...

l k=0 (−1) m−k l k m − k n 2k k − 2l + m = l k=0 l k 2k n n − l m + n − 3k − l. On the basis of this identity, for d ∈ {0, 1, 2,. .. } and ε ∈ {0, ±1} we construct explicit f ε (d) and g ε (d) such that for any prime p > d we have p−1 k=1 k ε C k+d ≡ f ε (d) if p ≡ 1 (mod 3), g ε (d) if p ≡ 2 (mod 3), where C n denotes the Catalan number 1 n+1 2n n ; for example, if p 5 is a prime then 0<k<p−4 ...

The author considers a class of nonlocal convolution-type ordinary differential equations the form − A ∗ ( g ∘ u ) 1 ′ t = λ f , 0 < $$\begin{equation*} \quad -A{\left({\left(a*(g\,\circ\, u)\right)}(1)\right)}u^{\prime \prime }(t)=\lambda f{\left(t,u(t)\right)}, 1,\nonumber \end{equation*}$$ where $g$ is continuous function satisfying p q $(p,q)$ growth — that is, there exist constants c $c_1$...

Using a procedure based on interpolation via continued fractions supplemented by statistical sampling, we analyse proton magnetic form factor data obtained electron+proton scattering $Q^2 \in [0.027,0.55]\,$GeV$^2$ with the goal of determining radius. The approach avoids assumptions about function used for and ensuing extrapolation onto $Q^2\simeq 0$ extraction slope. In this way, find $r_M = 0...

Given a connected graph G = ( V , E ) $G=(V,E)$ and length function ℓ : → R $\ell :E\to {\mathbb{R}}$ we let d v w ${d}_{v,w}$ denote the shortest distance between vertex $v$ $w$ . A t $t$ -spanner is subset ′ ⊆ $E^{\prime} \subseteq E$ such that if ${d}_{v,w}^{^{\prime} }$ denotes distances in subgraph $G^{\prime} =(V,E^{\prime} )$ then ≤ }\le t{d}_{v,w}$ for all ∈ $v,w\in V$ We study size of ...

‎Let $G$ be a group with identity $e.$ Let $R$ be a $G$-graded‎ ‎commutative ring and $M$ a graded $R$-module‎. ‎In this paper‎, ‎we‎ ‎introduce several results concerning graded classical prime‎ ‎submodules‎. ‎For example‎, ‎we give a characterization of graded‎ ‎classical prime submodules‎. ‎Also‎, ‎the relations between graded‎ ‎classical prime and graded prime submodules of $M$ are studied‎.‎

We introduce the notion of ideal, prime ideal, lter, fuzzy ideal, fuzzy prime ideal, fuzzy lter of ordered $Gamma$-semiring and study their properties and relations between them. We characterize the prime ideals and lters of ordered $Gamma$-semiring with respect to fuzzy ideals and fuzzy l- ters respectively.

The mass relation $${M_{0^{+}}+3M_{1^{+\prime }}+5M_{2^{+}}= 9M_{1^{+}}}$$ miraculously holds for the P-wave charmonium $$(c\bar{c})$$ and bottomonium $$(b\bar{b})$$ systems with soaring precision. origin of such can be addressed from Quark Models, have been confirmed experimentally in a limited number cases. In this connection, we propose $$M_{0^{+}}+5M_{2^{+}}=3(M_{1^{+\prime }}+M_{1^{+}})$$ ...

A polynomial P (x) with integer coefficients is said to be transitive modulo m, if for every x, y ∈ Z there exists k ≥ 0 such that P (x) = y (mod m). In this paper, we construct new examples of transitive polynomials modulo prime powers and partially describe cubic and quartic transitive polynomials. We also study the orbit structure of affine maps modulo prime powers.

Abstract This paper is concerned with the n -dimensional strongly coupled parabolic systems triangular form in cylinder Ω × ( 0 , T stretchy="false">] \Omega \times (0,T] . We investigate <m:...