in this paper, we introduce a new generalization of weakly prime ideals called $i$-prime. suppose $r$ is a commutative ring with identity and $i$ a fixed ideal of $r$. a proper ideal $p$ of $r$ is $i$-prime if for $a, b in r$ with $ab in p-ip$ implies either $a in p$ or $b in p$. we give some characterizations of $i$-prime ideals and study some of its properties.  moreover, we give conditions  ...

Abstract We study the one-dimensional Kirchhoff-type equation − ( b + a ‖ u accent="false">′ 2 ) accent="true">″ x = <...

Abelian class group Cl(D) of imaginary quadratic order with odd squarefree discriminant D is used in public key cryptosystems, based on discrete logarithm problem in class group and in cryptosystems, based on isogenies of elliptic curves. Discrete logarithm problem in Cl(D) is hard if #Cl(D) is prime or has large prime divisor. But no algorithms for generating such D are known. We propose proba...

For nonnegative real numbers α, β, γ , A, B, and C such that B C > 0 and α β γ > 0, the difference equation xn 1 α βxn γxn−1 / A Bxn Cxn−1 , n 0, 1, 2, . . . has a unique positive equilibrium. A proof is given here for the following statements: 1 For every choice of positive parameters α, β, γ , A, B, and C, all solutions to the difference equation xn 1 α βxn γxn−1 / A Bxn Cxn−1 , n 0, 1, 2, . ...

We claim that the prime/irreducible elements of R are the associates of primes in Z different from p. Indeed, suppose that a/p, with a ∈ Z is an irreducible. Since powers of p are units, we may assume k = 0 and p a. Since −1 is also unit, we may also assume a > 0. If a is not prime, it factors in Z as a = bc, with b, c 6= ±1. Since p b, c [as p a and p is prime], they are not units of R and hen...

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 $$M_{0^{+}}+5M_{2^{+}}=3(M_{1^{+\prime }}+M_{1^{+}})$$ ...

In this work, by employing the Krasnosel'skii fixed point theorem, we study the existence of positive solutions of a three-point boundary value problem for the following fourth-order differential equation begin{eqnarray*} left { begin{array}{ll} u^{(4)}(t) -f(t,u(t),u^{prime prime }(t))=0 hspace{1cm} 0 leq t leq 1, & u(0) = u(1)=0, hspace{1cm} alpha u^{prime prime }(0) - beta u^{prime prime pri...

in this work, by employing the krasnosel'skii fixed point theorem, we study the existence of positive solutions of a three-point boundary value problem for the following fourth-order differential equation begin{eqnarray*} left { begin{array}{ll} u^{(4)}(t) -f(t,u(t),u^{prime prime }(t))=0 hspace{1cm} 0 leq t leq 1, &amp; u(0) = u(1)=0, hspace{1cm} alpha u^{prime prime }(0) - beta u^{prime prime...

This paper deals with the oscillation problems on nonlinear differential equation $(a(t)|x^{\prime }|^{p(t)-2}x^{\prime })^{\prime }+b(t)|x|^{\lambda -2}x=0$ involving $p(t)$-Laplacian. Sufficient conditions are given under which all proper solutions oscillatory. In addition, we give a-priori estimates for nonoscillatory and propose an open problem.