Let n = 2λm ≥ 526 with m ∈ {2, 3, 5, 7, 11}, and let S be a sequence of elements in Cn⊕Cn with |S| = n2 +2n−2. Let N 0 (S) denote the number of the subsequences with length n2(=|G|) and with sum zero. Among other results, we prove that either N 0 (S) = 1 or N |G| 0 (S) ≥ n2 + 1.

Finding the mean (average) of n arbitrary 3D rotations or orientations is conceptually challenging both in its algebra and its geometric interpretation. To my knowledge, no solutions have been found for the general case, although a number of approximate solutions have been proposed [3, 4, 5] to accurately and e ciently handle those limiting cases typical of engineering interest. This report pro...

We give natural constructions of number rigid determinantal point processes on the unit disc $${\mathbb{D}}$$ with sub-Bergman kernels form $${K_\Lambda}(z,w) = \sum\limits_{n \in \Lambda} {(n + 1){z^n}{{\bar w}^n}} ,\,\,\,\,\,z,w {\mathbb{D}},$$ Λ an infinite subset non-negative integers. Our are given in both deterministic and probabilistic methods. In method, our proofs involve classical Blo...

In this paper, we developed some new formulae for exclusive sum number of the complete bipartite graph , n m K . Also we show that for     , 2 2 , 6 n n K E n K n    2 4, 2 6, 3 n n n          and 2 4 n   .

Cameron and Erdős [6] raised the question of how many maximal sum-free sets there are in {1, . . . , n}, giving a lower bound of 2bn/4c. In this paper we prove that there are in fact at most 2(1/4+o(1))n maximal sum-free sets in {1, . . . , n}. Our proof makes use of container and removal lemmas of Green [8, 9] as well as a result of Deshouillers, Freiman, Sós and Temkin [7] on the structure of...

Dynamic resource allocation is an important means to increase the sum capacity of fading multiple-access channels (MACs). In this paper, we consider vector multiaccess channels (channels where each user has multiple degrees of freedom) and study the effect of power allocation as a function of the channel state on the sum capacity (or spectral efficiency) defined as the maximum sum of rates of u...

Given an integer r 0, let G r = (V r ; E) denote a graph consisting of a simple nite undirected graph G = (V; E) of order n and size m together with r isolated vertices K r. Then jV j = n, jV r j = n + r, and jEj = m. Let L : V r ! Z + denote a labelling of the vertices of G r with distinct positive integers. Then G r is said to be a sum graph if there exists a labelling L such that for every d...

A d-uniform hypergraph H is a sum hypergraph iff there is a finite S ⊆ IN such that H is isomorphic to the hypergraph H+d (S) = (V, E), where V = S and E = {{v1, . . . , vd} : (i 6= j ⇒ vi 6= vj)∧ ∑d i=1 vi ∈ S}. For an arbitrary d-uniform hypergraph H the sum number σ = σ(H) is defined to be the minimum number of isolated vertices w1, . . . , wσ 6∈ V such that H ∪ {w1, . . . , wσ} is a sum hyp...