A graph G has a representation modulo n if there exists an injective map f : V (G) → {0, 1, . . . , n} such that vertices u and v are adjacent if and only if |f(u) − f(v)| is relatively prime to n. The representation number rep(G) is the smallest n such that G has a representation modulo n. We present new results involving representation numbers for stars, split graphs, complements of split gra...

We investigate how the matrix representation of SU(N) algebra approaches that of the Poisson algebra in the large N limit. In the adjoint representation, the (N2 − 1)× (N2 − 1) matrices of the SU(N) generators go to those of the Poisson algebra in the large N limit. However, it is not the case for the N × N matrices in the fundamental representation.

As with most computer science problems, representation of the data is the key to ecient and eective solutions. Piecewise linear representation has been used for the representation of the data. This representation has been used by various researchers to support clustering, classication, indexing and association rule mining of time series data. A variety of algorithms have been proposed to obtain...

Let S be a principally embedded ???????? 2 -subalgebra in n for n?3. A special case of results the third author and Gregg Zuckerman implies that there exists positive integer b(n) such any finite-dimensional irreducible -representation, V, an S-representation embedding V with dimension at most b(n). In 2017 paper (joint Hassan Lhou), they prove b(n)=n is sharpest possible bound, also address em...

This is also a SO(n) representation, the fundamental representation on vectors. Replacing v in this formula by an arbitrary element of C(n) we get a representation of Spin(n) (and also SO(n)) on C(n) which can be identified with its representation on Λ∗(Rn). This representation is reducible, the decomposition into irreducibles is just the decomposition of Λ∗(Rn) into the various Λ(R) for k = 0,...

We show that each Eulerian representation of n is the restriction of a representation of n+1. We describe the new representations, giving character formulae, and identify the one which restricts to the rst Eulerian representation as the tree representation.

Representation of a granule, relation and operation between two granules are mainly researched in granular computing. Hyperbox granular computing classification algorithms (HBGrC) are proposed based on interval analysis. Firstly, a granule is represented as the hyperbox which is the Cartesian product of $N$ intervals for classification in the $N$-dimensional space. Secondly, the relation betwee...

In this study, we introduce a new type of surface curves called $D$-type curve. This curve is defined by the property that the unit Darboux vector $vec{W}_{0} $ of a surface curve $vec{r}(s)$ and unit surface normal $vec{n} $ along the curve $vec{r}(s)$ satisfy the condition $leftlangle vec{n} ,vec{W}_{0} rightrangle =text{constant}$. We point out that a $D$-type curve is a geodesic curve or an...

‎for a set of non-negative integers~$l$‎, ‎the $l$-intersection number of a graph is the smallest number~$l$ for which there is an assignment of subsets $a_v subseteq {1,dots‎, ‎l}$ to vertices $v$‎, ‎such that every two vertices $u,v$ are adjacent if and only if $|a_u cap a_v|in l$‎. ‎the bipartite $l$-intersection number is defined similarly when the conditions are considered only for the ver...

in this paper we study a two-phase free boundary problem for a semilinear elliptic equation on a bounded domain $dsubset mathbb{r}^{n}$ with smooth boundary‎. ‎we give some results on the growth of solutions and characterize the free boundary points in terms of homogeneous harmonic polynomials using a fundamental result of caffarelli and friedman regarding the representation of functions whose ...