Let $R$ be a commutative ring with identity‎. ‎A proper ideal $P$ of $R$ is a $(n-1,n)$-$Phi_m$-prime ($(n-1,n)$-weakly prime) ideal if $a_1,ldots,a_nin R$‎, ‎$a_1cdots a_nin Pbackslash P^m$ ($a_1cdots a_nin Pbackslash {0}$) implies $a_1cdots a_{i-1}a_{i+1}cdots a_nin P$‎, ‎for some $iin{1,ldots,n}$; ($m,ngeq 2$)‎. ‎In this paper several results concerning $(n-1,n)$-$Phi_m$-prime and $(n-1,n)$-...

In this paper, an iterative method is proposed for solving matrix equation $sum_{j=1}^s A_jX_jB_j = E$. This method is based on the global least squares (GL-LSQR) method for solving the linear system of equations with the multiple right hand sides. For applying the GL-LSQR algorithm to solve the above matrix equation, a new linear operator, its adjoint and a new inner product are dened. It is p...

Let $x$ be a complex number which has positive real part, and $w_1,\ldots,w_N$ rational numbers. We show that $w^s \zeta_N (s, x \ |\ w_1,\ldots, w_N)$ can expressed as finite linear combination of the Hurwitz zeta functions over $\mathbf Q(x)$, where $\zeta_N (s,x is Barnes function $w$ explicitly determined by $w_1,\ldots, w_N$. Furthermore, we give generalizations Kummer's formula on gamma K...

Let $a_1,\ldots,a_n$ and $b_1,\ldots,b_m$ be fixed positive integers, let ${\mathcal S}$ denote the set of all nonnegative integer solutions equation $x_1a_1+\cdots +x_na_n=y_1b_1+\cdots...

Abstract It is well known that the diffeomorphism type of Milnor fibration a (Newton) nondegenerate polynomial function f uniquely determined by Newton boundary . In present paper, we generalize this result to certain degenerate functions, namely show (possibly degenerate) form $f=f^1\cdots f^{k_0}$ boundaries $f^1,\ldots , if $\{f^{k_1}=\cdots =f^{k_m}=0\}$ complete intersection variety for an...

let $r$ be a commutative ring with identity‎. ‎a proper ideal $p$ of $r$ is a $(n-1,n)$-$phi_m$-prime ($(n-1,n)$-weakly prime) ideal if $a_1,ldots,a_nin r$‎, ‎$a_1cdots a_nin pbackslash p^m$ ($a_1cdots a_nin pbackslash {0}$) implies $a_1cdots a_{i-1}a_{i+1}cdots a_nin p$‎, ‎for some $iin{1,ldots,n}$; ($m,ngeq 2$)‎. ‎in this paper several results concerning $(n-1,n)$-$phi_m$-prime and $(n-1,n)$-...

In this note, we study the polynomial representation of quantum Olshanetsky-Perelomov system for a finite reflection group $W$ type $B_n$. We endow ring ${\mathbb C} [x_1,\ldots\\\ldots, x_n]$ with structure module over Weyl algebra associated [x_1,\ldots,x_n]^{W}$ invariant polynomials under reflections Then differential operators $W$. use theory namely higher Specht and establish decompositio...

We show that a smooth embedded paper Moebius band must have aspect ratio at least $$\begin{aligned}\lambda _1= \frac{2 \sqrt{4-2 \sqrt{3}}+4}{\root 4 \of {3} \sqrt{2}+2 \sqrt{2 \sqrt{3}-3}} =1.69497\ldots \end{aligned}$$ This bound comes more than 3/4 of the way from old known $$\pi /2=1.5708\ldots $$ to conjectured $$\sqrt{3} = 1.732\ldots