Let $kge 1$ be an integer, and let $G$ be a finite and simple graph with vertex set $V(G)$.A weak signed Roman $k$-dominating function (WSRkDF) on a graph $G$ is a function$f:V(G)rightarrow{-1,1,2}$ satisfying the conditions that $sum_{xin N[v]}f(x)ge k$ for eachvertex $vin V(G)$, where $N[v]$ is the closed neighborhood of $v$. The weight of a WSRkDF $f$ is$w(f)=sum_{vin V(G)}f(v)$. The weak si...

Let $G$ be a graph with vertex set $V(G)$. For any integer $kge 1$, a signed (total) $k$-dominating functionis a function $f: V(G) rightarrow { -1, 1}$ satisfying $sum_{xin N[v]}f(x)ge k$ ($sum_{xin N(v)}f(x)ge k$)for every $vin V(G)$, where $N(v)$ is the neighborhood of $v$ and $N[v]=N(v)cup{v}$. The minimum of the values$sum_{vin V(G)}f(v)$, taken over all signed (total) $k$-dominating functi...

In this work, we study the k-means cost function. The (Euclidean) k-means problem can be described as follows: given a dataset X ⊆ R and a positive integer k, find a set of k centers C ⊆ R such that Φ(C,X) def = ∑ x∈X minc∈C ||x− c|| 2 is minimized. Let ∆k(X) def = minC⊆Rd Φ(C,X) denote the cost of the optimal k-means solution. It is simple to observe that for any dataset X, ∆k(X) decreases as ...

in this paper, some results of singh, gopalakrishna and kulkarni (1970s) have been extended to higher order derivatives. it has been shown that, if $sumlimits_{a}theta(a, f)=2$ holds for a meromorphic function $f(z)$ of finite order, then for any positive integer $k,$ $t(r, f)sim t(r, f^{(k)}), rrightarrowinfty$ if $theta(infty, f)=1$ and $t(r, f)sim (k+1)t(r, f^{(k)}), rrightarrowinfty$ if $th...

In 1969, Kannan [1] proved a new mapping which improved the Banach Contraction theorem. The purpose of this paper is to generalize the Kannan mapping and also to prove it, in space of fractals.

let $k$ be a field of characteristic$p>0$, $k[[x]]$, the ring of formal power series over $ k$,$k((x))$, the quotient field of $ k[[x]]$, and $ k(x)$ the fieldof rational functions over $k$. we shall give somecharacterizations of an algebraic function $fin k((x))$ over $k$.let $l$ be a field of characteristic zero. the power series $finl[[x]]$ is called differentially algebraic, if it satisfies...

Lemma A.1 (a) A real-valued convex function is also 0-convex and hence k-convex for all k ≥ 0. A k1-convex function is also a k2-convex function for k1 ≤ k2. (b) If f1(y) and f2(y) are k1-convex and k2-convex respectively, then for α, β ≥ 0, αf1(y) + βf2(y) is (αk1 + βk2)-convex. (c) If f(y) is k-convex and w is a random variable, then E{f(y − w)} is also k-convex, provided E{|f(y − w)|} < ∞ fo...

In this paper, some results of Singh, Gopalakrishna and Kulkarni (1970s) have been extended to higher order derivatives. It has been shown that, if $sumlimits_{a}Theta(a, f)=2$ holds for a meromorphic function $f(z)$ of finite order, then for any positive integer $k,$ $T(r, f)sim T(r, f^{(k)}), rrightarrowinfty$ if $Theta(infty, f)=1$ and $T(r, f)sim (k+1)T(r, f^{(k)}), rrightarrowinfty$ if $Th...

we consider the coupled system$f(x,y)=g(x,y)=0$,where$$f(x, y)=bs 0 {m_1}   a_k(y)x^{m_1-k}mbox{ and } g(x, y)=bs 0 {m_2} b_k(y)x^{m_2-k}$$with entire functions $a_k(y), b_k(y)$.we    derive a priory estimates  for the sums of the rootsof the considered system andfor the counting function of  roots.