In this paper, we suggest and analyze a new class of three-step projection iterative methods for solving the extended general variational inequalities, which are obtained using the updating technique of the solution in conjunction with projection technique. We also consider the convergence criteria of these new iterative methods under some mild conditions. Since the extended general variational inequalities include the general variational inequalities and other related optimization problems as special cases, results obtained in this paper continue to hold for these problems. Results obtained in this paper may be viewed as a refinement and improvement of the known results.

In this paper, a common fixed point theorem for $R$-weakly commuting maps in intuitionistic fuzzy metric spaces is proved.

Skew polynomial rings have invited attention of mathematicians and various properties of these rings have been discussed. The nature of ideals (in particular prime ideals, minimal prime ideals, associated prime ideals), primary decomposition and Krull dimension have been investigated in certain cases.

This article concerns transparent (decomposable) rings. Recall that a ring $\mathrm{R}$ is said to be a Transparent ring if in $\mathrm{R}$ there exist irreducible ideals $I_j,1\le j\le n$ such that ${\cap }_{j=1}^n{I}_j=0$ and each $R/I_j$ has a right Artinian quotient ring.

Now let $\mathrm{R}$ be a ring, which is an order in an Artinian ring $\mathrm{S}$. Let $\sigma$ and $\tau$ be automorphisms of $\mathrm{R}$ and $\delta$ be a $(\sigma ,\tau )$-derivation of $\mathrm{R}$; i.e. $\delta :R\to R$ is an additive mapping satisfying $\delta \left(ab\right)=\sigma \left(a\right)\delta \left(b\right)+\delta \left(a\right)\tau \left(b\right)$ for all $\text{a},b\in R$. We define an extension of $\mathrm{R}$, namely $R[x,\sigma ,\tau ,\delta ]=\{f=\sum _{i=0}^n{x}^i{a}_i,a_i\in R\}$, subject to the relation $ax=x\sigma \left(\tau \right(a\left)\right)+\delta \left(a\right)$ for all $a\in R$.

We show that if $\mathrm{R}$ is a commutative Noetherian $\mathrm{\mathbb{Q}}$-algebra, $\sigma$ and $\tau$ as usual, then there exists an integer $m\ge 1$ such that the extension ring $R[x,\alpha ,\beta ,\vartheta ]$ is a Transparent ring, where $\alpha ={\sigma }^m$, $\beta ={\tau }^m$ and $\vartheta$ is an $(\alpha ,\beta )$-derivation of $\mathrm{R}$ with $\alpha \left(\vartheta \right(a\left)\right)=\vartheta \left(\alpha \right(a\left)\right)$, and $\beta \left(\vartheta \right(a\left)\right)=\vartheta \left(\beta \right(a\left)\right)$, for all $a\in R$.

New expansions for some functions related to the Zeta function in terms of the Pochhammer polynomials are given (coefficients $b_k$, $d_k$ and ${\widehat{d}}_k$). In some formal limit our expansion $b_k$ obtained via the alternating series gives the regularized expansion of Maslanka for the Zeta function. The real and the imaginary part of the function on the critical line is obtained with a good accuracy up to .

Then, we give the expansion (coefficient ${\widehat{d}}_k$) for the derivative of $\ln \left[\right(s-1\left)\zeta \right(s\left)\right]$. The critical function of the derivative, whose bounded values for at large values of $k$ should ensure the truth of the Riemann Hypothesis (RH), is obtained either by means of the primes or by means of the zeros (trivial and non-trivial) of the Zeta function. In a numerical experiment performed up to high values of $k$ i.e. up to $k={10}^{14}$ we obtain a very good agreement between the two functions, with the emergence of fourteen oscillations with stable amplitude.

Genus two curves covering elliptic curves have been the object of study of many articles. For a fixed degree $n$ the subloci of the moduli space ${\mathcal{M}}_2$ of curves having a degree $n$ elliptic subcover has been computed for $n=3,5$ and discussed in detail for $n$ odd; see [17, 22, 3, 4]. When the degree of the cover is even the case in general has been treated in [16]. In this paper we compute the sublocus of ${\mathcal{M}}_2$ of curves having a degree $4$ elliptic subcover.