The existence of paths of low degree sum of their vertices in planar graphs is investigated. The main results of the paper are: 1. Every 3-connected simple planar graph G that contains a k-path, a path on k vertices, also contains a k-path P such that for its weight (the sum of degrees of its vertices) in G it holds

A set A ⊆ N is (k, `)-sum-free, for k, ` ∈ N, k > `, if it contains no solutions to the equation x1 + · · ·+xk = y1 + · · ·+y`. Let ρ = ρ(k− `) be the smallest natural number not dividing k − `, and let r = rn, 0 ≤ r < ρ, be such that r ≡ n (mod ρ). The main result of this note says that if (k − `)/` is small in terms of ρ, then the number of (k, `)-sum-free subsets of [1, n] is equal to (φ(ρ) ...

For integers k and n with k ≤ n a vector x ∈ R is said to be weakly k-majorized by a vector q ∈ R if the sum of the r largest components of x does not exceed the sum of the r largest components of q, for r = 1, . . . , k. For a given q the set of vectors weakly k-majorized by q defines a polyhedron P (q; k). We determine the vertices of both P (q; k) and its integer hull Q(q; k). Furthermore a ...

Let $s(n):= \sum _{d\,|\, n,\,d \lt n} d$ denote the sum of proper divisors $n$. It is natural to conjecture that for each integer $k\ge 2$, equivalence \[ \text {$n$ $k$th powerfree} \iff {$s(n)$ \] holds almost al

In this paper, we obtain asymptotic formula on the "hyperbolic" summation \begin{equation*} \underset{mn\leq x}{\sum }D_{k}\left( \gcd \left( m,n\right) \right) \text{ \ }\left( k\in \mathbb{Z}_{\geq 2}\right), \end{equation*} such that $D_{k}\left( n\right) = \dfrac{\tau _{k}\left( }{\tau_{k}^{\ast }$, where $\tau =\!\!\sum\limits_{n_{1}n_{2}\ldots n_{k}=n}\!\!1$ denotes Piltz divisor function...

This paper determines the sum-capacity of degraded Gaussian K-user interference channels (IFCs). To this end, the paper first develops a family of genie-MAC (multiple access channel) outer bounds for Gaussian K-user interference channels. This family is based on existing genie-aided bounding mechanisms, but differs from current approaches in its optimization problem formulation and application....

International Journal of Pharmaceutical Sciences Review and Research Available online at www.globalresearchonline.net © Copyright protected. Unauthorised republication, reproduction, distribution, dissemination and copying of this document in whole or in part is strictly prohibited. 153 Aniruddh Dash1*, Jitendra Narayan Senapati2, Binod Chandra Raulo1, Prasanta Kumar Brahma3, Mahesh Chandra Sah...

Let A be a finite nonempty set of integers. An asymptotic estimate of several dilates sum size was obtained by Bukh. The unique known exact bound concerns the sum |A + k ·A|, where k is a prime and |A| is large. In its full generality, this bound is due to Cilleruelo, Serra and the first author. Let k be an odd prime and assume that |A| > 8kk. A corollary to our main result states that |2 · A +...

Let $M_R$ be a module with $S=End(M_R)$. We call a submodule $K$ of $M_R$ annihilator-small if $K+T=M$, $T$ a submodule of $M_R$, implies that $ell_S(T)=0$, where $ell_S$ indicates the left annihilator of $T$ over $S$. The sum $A_R(M)$ of all such submodules of $M_R$ contains the Jacobson radical $Rad(M)$ and the left singular submodule $Z_S(M)$. If $M_R$ is cyclic, then $A_R(M)$ is the unique ...