In this paper we study a stochastic model of a neural network of Purkinje cells proposed by Axelrad et al. 2], 5]. In this model only the inhibitory interaction between the cells is considered. We analyze the stability properties of networks whose graph is complete N-partite. It is shown that if the parameters of the model are below some critical values, then the network converges to unique equ...

For a given r-uniform hypergraph F we study the largest blow-up of F which can be guaranteed in every large r-uniform hypergraph with many copies of F . For graphs this problem was addressed by Nikiforov, who proved that every n-vertex graph that contains Ω(n`) copies of the complete graph K` must contain a complete `-partite graph with Ω(logn) vertices in each class. We give another proof of N...

For any given integer $$r\geqslant 3$$ , let $$k=k(n)$$ be an with $$r\leqslant k\leqslant n$$ . A hypergraph is r-uniform if each edge a set of r vertices, and said to linear two edges intersect in at most one vertex. Let $$A_1,\ldots ,A_k$$ k-partition [n] $$|A_i|=n_i\geqslant 1$$ An H called k-partite e satisfies $$|e\cap A_i|\leqslant for $$1\leqslant i\leqslant k$$ In this paper, the numbe...

In [8], it is shown that the complete multipartite graph Kn(2t) having n partite sets of size 2t, where n ≥ 6 and t ≥ 1, has a decomposition into gregarious 6-cycles if n ≡ 0, 1, 3 or 4 (mod 6). Here, a cycle is called gregarious if it has at most one vertex from any particular partite set. In this paper, when n ≡ 0 or 3 (mod 6), another method using difference set is presented. Furthermore, wh...

We propose concurrence classes for general pure multipartite states based on an orthogonal complement of a positive operator valued measure on quantum phase. In particular, we construct W m class, GHZ, and GHZ class concurrences for general pure m-partite states. We give explicit expressions for W 3 and GHZ class concurrences for general pure three-partite states and for W , GHZ, and GHZ class ...

The set S of distinct scores (outdegrees) of the vertices of a k-partite tournament T(X1, X2, · · · , X k) is called its score set. In this paper, we prove that every set of n non-negative integers, except {0} and {0, 1}, is a score set of some 3-partite tournament. We also prove that every set of n non-negative integers is a score set of some k-partite tournament for every n ≥ k ≥ 2.

Combinatorial batch codes provide a tool for distributed data storage, with the feature of keeping privacy during information retrieval. Recently, Balachandran and Bhattacharya observed that the problem of constructing such uniform codes in an economic way can be formulated as a Turán-type question on hypergraphs. Here we establish general lower and upper bounds for this extremal problem, and a...

For integers k > 0 and r > 0, a conditional (k, r)-coloring of a graph G is a proper k-coloring of the vertices of G such that every vertex v of degree d(v) in G is adjacent to vertices with at least min{r, d(v)} different colors. The smallest integer k for which a graph G has a conditional (k, r)-coloring is called the rth order conditional chromatic number, denoted by χr(G). For different val...