in this paper we study finding the $(k,l)$-core problem on a tree which the vertices have positive or negative weights. let $t=(v,e)$ be a tree. the $(k,l)$-core of $t$ is a subtree with at most $k$ leaves and with a diameter of at most $l$ which the sum of the weighted distances from all vertices to this subtree is minimized. we show that, when the sum of the weights of vertices is negative, t...

Let G be a (p,q) graph. An injective map f : E(G) → {±1,±2,...,±q} is said to be an edge pair sum labeling if the induced vertex function f*: V (G) → Z - {0} defined by f*(v) = ΣP∈Ev f (e) is one-one where Ev denotes the set of edges in G that are incident with a vertex v and f*(V (G)) is either of the form {±k1,±k2,...,±kp/2} or {±k1,±k2,...,±k(p-1)/2} U {±k(p+1)/2} according as p is even or o...

In this article, we assign the D∗ s3(2860) to be a D-wave cs̄ meson, study the vertices D∗ s3(2860)DK and D ∗ s3(2860)D ∗K in details to select the pertinent tensor structures, calculate the hadronic coupling constants GD∗ s3(2860)DK and GD∗ s3(2860)DK with the three-point QCD sum rules and obtain the decay widths Γ (D∗ s3(2860) → D ∗K) and Γ (D∗ s3(2860) → DK). The predicted ratio R = Γ (D∗ s3(...

Let $mathcal {A} $ and $mathcal {B} $ be C$^*$-algebras. Assume that $mathcal {A}$ is of real rank zero and unital with unit $I$ and $k>0$ is a real number. It is shown that if $Phi:mathcal{A} tomathcal{B}$ is an additive map preserving $|cdot|^k$ for all normal elements; that is, $Phi(|A|^k)=|Phi(A)|^k $ for all normal elements $Ainmathcal A$, $Phi(I)$ is a projection, and there exists a posit...

We prove that some exact geometric pattern matching problems reduce in linear time to k -SUM when the has a fixed size k. This holds real RAM model for searching similar copy of set \(k\ge 3\) points within n plane, and an affine image d+2\) d-space. As corollaries, we obtain improved algorithms decision trees two problems. In particular, they can be solved by algebraic near-linear height.

Let $\mathscr{M}_{(2,1)}(N)$ be the infimum of largest sum-free subset any set $N$ positive integers. An old conjecture in additive combinatorics asserts that there is a constant $c=c(2,1)$ and function $\omega(N)\to\infty$ as $N\to\infty$, such $cN+\omega(N)<\mathscr{M}_{(2,1)}(N)<(c+o(1))N$. The $c(2,1)$ determined by Eberhard, Green, Manners, while existence $\omega(N)$ still wide open. In t...