The soil nematode Caenorhabditis Elegans (C. elegans) is the most investigated of all multicellular organisms. Since the proposal to use it as a model organism, a series of research projects have been undertaken, investigating various aspects of this organism. As a result, the complete cell lineage, neural circuitry, and various genes and their functions have been identified. The complete C. el...

The main goal of this article is to present several connections between perfect codes in the Johnson scheme and designs, and provide new tools for proving Delsarte conjecture that there are no nontrivial perfect codes in the Johnson scheme. Three topics will be considered. The first is the configuration distribution which is akin to the weight distribution in the Hamming scheme. We prove that i...

Let $ (A,| cdot |) $ be a real Banach algebra, a complex algebra $ A_mathbb{C} $ be a complexification of $ A $ and $ | | cdot | | $ be an algebra norm on  $ A_mathbb{C}  $  satisfying a simple condition together with the norm $ | cdot | $ on $ A$.  In this paper we first show that $ A^* $ is a real Banach $ A^{**}$-module if and only if $ (A_mathbb{C})^* $ is a complex Banach $ (A_mathbb{C})^{...

A vertex 2-coloring of a graph is said to be perfect with parameters (aij) k i,j=1 if for every i, j ∈ {1, ..., k} every vertex of color i is adjacent with exactly i vertices of color j. We consider the perfect 2-colorings of the distance-2 graph of the 24-cube {0, 1}24 with parameters ((20+ c, 256− c)(c, 276− c)) (i.e., with eigenvalue 20). We prove that such colorings exist if c = 3, 6, 8, 9,...

In a Steiner triple system STS(v)=(V,B), for each pair {a, b} ⊂ V , the cycle graph Ga,b can be defined as follows. The vertices of Ga,b are V \{a, b, c} where {a, b, c} ∈ B. {x, y} is an edge if either {a, x, y} or {b, x, y} ∈ B. The Steiner triple system is said to be perfect if the cycle graph of every pair is a single (v − 3)-cycle. Perfect STS(v) are known only for v = 7, 9, 25 and 33. We ...

A vertex coloring of a graph is said to be perfect with parameters (aij) k i,j=1 if for every i, j ∈ {1, ..., k} every vertex of color i is adjacent with exactly aij vertices of color j. We consider the perfect 2-colorings of the distance-2 graph of the 24-cube {0, 1}24 with parameters ((20+ c, 256− c)(c, 276− c)) (i.e., with eigenvalue 20). We prove that such colorings exist for all c from 1 t...

the application of the [pd{c6h3(ch2ch2nh2)-4-ome-5-κ2-c,n}(μ-br)]2 complex of 2-methoxyphenethylamine in the heck coupling reaction was considered under both conventional and microwave irradiation conditions, and their results were compared. this complex is an efficient, stable and non-sensitive to air and moisture catalyst for the vinylation of substituted aryl halides with different electroni...

Given a one-factorization F of the complete bipartite graph Kn,n, let pf(F) denote the number of Hamiltonian cycles obtained by taking pairwise unions of perfect matchings in F . Let pf(n) be the maximum of pf(F) over all one-factorizations F of Kn,n. In this work we prove that pf(n) > n2/4, for all n > 2. 1 Perfect one-factorizations A one-factorization of a graph G is a partition of its set o...