In our earlier publication we have shown how to compute by iteration a rational number ${u_{2,k}}$ in the two-term Machin-like formula for pi of kind $$\frac{\pi}{4}=2^{k-1}\arctan\left(\frac{1}{u_{1,k}}\right)+\arctan\left(\frac{1}{u_{2,k}}\right),\qquad k\in \mathbb{Z},\quad k\ge 1,$$ where ${u_{1,k}}$ can be chosen as an integer ${u_{1,k}} = \left\lfloor{{a_k}/\sqrt{2-a_{k-1}}}\right\rfloor$...

Let $p(z)$ be a polynomial of degree $n$. Then the polar derivative with respect to real or complex number $\alpha$ is defined by $D_\alpha p(z)=n p(z)+(\alpha-z) p^{\prime}(z)$. Govil and Mctume [14] proved that if $n$ having all its zeros in $|z| \leq k, k \geq 1$, then for $|\alpha| 1+k+k^n$,$$\begin{aligned}\max _{|z|=1}\left|D_\alpha p(z)\right| & n\left(\frac{|\alpha|-k}{1+k^n}\right)...

Let $G$ be a $(p,q)$ graph. Let $f:V(G)to{1,2, ldots, k}$ be a map where $k in mathbb{N}$ and $k>1$. For each edge $uv$, assign the label $gcd(f(u),f(v))$. $f$ is called $k$-Total prime cordial labeling of $G$ if $left|t_{f}(i)-t_{f}(j)right|leq 1$, $i,j in {1,2, cdots,k}$ where $t_{f}(x)$ denotes the total number of vertices and the edges labelled with $x$. A graph with a $k$-total prime cordi...

Let $$S=\left\langle s_1,\ldots ,s_n\right\rangle $$ be a numerical semigroup generated by the relatively prime positive integers $$s_1,\ldots ,s_n$$ . $$k\geqslant 2$$ an integer. In this paper, we consider following k-power variant of Frobenius number S defined as $$\begin{aligned} {}^{k\!}r\!\left( S\right) := \text { largest } k {-power integer not belonging to S. \end{aligned}$$ investigat...

Recently, for single series, the necessary and sufficient conditions $\left\vert C,0\right\vert\Rightarrow \left\vert A_{f}\right\vert_{k}$ vise versa, A_{f}\right\vert \Rightarrow C,0\right\vert_{k}$ versa have been established, where $1 < k \infty $ $A$ is a factorable matrix. The present study extends these results to double summability, also provides some new results.

We propose some refinements for the second inequality in $$ \frac{1}{2}\left\Vert A\right\Vert \le w(A)\le \left\Vert A\right\Vert, where A ∈ B(H). In particular, if is hyponormal, then, by refining Young with Kantorovich constant K K(⋅, ⋅), we show that \frac{1}{2{\operatorname{inf}}_{\left\Vert x\right\Vert =1}\zeta (x)}\left|\left\Vert A\right.\right|++\left|\left.{A}^{\ast}\right\Vert \righ...

Given a minimal model of an elliptic curve, $E/K$, over finite extension, $K$, ${\mathbb Q}_{p}$ for any rational prime, $p$, and point $P \in E(K)$ infinite order, we determine precisely $\min \left( v \phi_{n}(P) \right), \psi_{n}^{2}(P) \right) \right)$, where $v$ is normalised valuation on $K$ $\phi_{n}(P)$ $\psi_{n}(P)$ are polynomials arising from multiplication by $n$ this the curve.