let $g$ be a non-abelian group of order $p^n$‎, ‎where $nleq 5$ in which $g$ is not extra special of order $p^5$‎. ‎in this paper we determine the maximal size of subsets $x$ of $g$‎ ‎with the property that $xyneq yx$ for any $x,y$ in $x$ with‎ ‎$xneq y$‎.

‎three infinite families of finite abelian groups will be‎ ‎described such that each member of these families has‎ ‎the r'edei $k$-property for many non-trivial values of $k$‎.

China is a country that is arousing a large amount of interest as a new mega-market, as it opens itself to a market-orientated economy and expands at a remarkable rate. However, the influx of international investment, new business opportunities and advertising from overseas has seen Chinese consumers being exposed to potentially offensive advertising products and images. While some products and...

the purpose of the present study is to introduce the sequence space[{l_p}(e,delta) = left{ x = (x_n)_{n = 1}^infty : sum_{n = 1}^infty left| sum_{j in {e_n}} x_j - sum_{j in e_{n + 1}} x_jright| ^p < infty right},]where $e=(e_n)$ is a partition of finite subsets of the positive integers and $pge 1$. the topological properties and inclusion relations of this space are studied. moreover, the prob...

In a visual cryptography scheme, a secret image is encoded into n shares, in the form of transparencies. The shares are then distributed to n participants. Qualified subsets of participants can recover the secret image by superimposing their transparencies, but non-qualified subsets of participants have no information about the secret image. Pixel expansion, which represents the number of subpi...

Let f (m, n) denote the number of relatively prime subsets of {m+ 1, m + 2,. .. , n}, and let Φ(m, n) denote the number of subsets A of {m + 1, m + 2,. .. , n} such that gcd(A) is relatively prime to n. Let f k (m, n) and Φ k (m, n) be the analogous counting functions restricted to sets of cardinality k. Simple explicit formulas and asymptotic estimates are obtained for these four functions. A ...

Kernel-based clustering generally maps the observed data to a high dimensional feature space and can usually achieve preferable classification by enlarging the difference among samples. Competitive kernel clustering creates a competitive environment by means of hierarchical method in which clusters compete for samples based on cardinalities in kernel space. Collaborative clustering implementing...

All graphs in this article are finite, undirected, without loops or multiple edges. Let G be a graph with vertices vl5 v2,..., vn. The complement in G of a subgraph H is the subgraph of G obtained by deleting all edges in H. The join GxvG2 of two graphs GY and G2 is obtained by adding an edge from each vertex in Gl to each vertex in G2. Let Kn be the complete graph and Pn the path on n vertices...