Let L be subset of {3,4,…} and let Xn,M(L) the number cycles belonging to unicyclic components whose length is in random graph G(n,M). We find limiting distribution subcritical regime M=cn with c<1/2 critical M=n2(1+μn−1/3) μ=O(1). Depending on a condition involving series ∑ℓ∈Lzℓ/(2ℓ), we obtain limit either Poisson or normal as n→∞.

The symmetric group S , is a metric space with distance d(a , b ) = IE(a-'b)( where E ( c ) is the set of points moved by c E S,. Let L be a given subset of {I. 2, . . . , n}, a permutation clique A = A(L, n ) is any subset A c S , with d(a, b ) E L whenever a, b E A , a # b. We give a framework of new and known information o n some special A = A ( L , n): maximal, largest, largest subgroups of...

We prove that for any convex polytope $\Omega \subset \mathbb{R}^d$ which is centrally symmetric and whose faces of all dimensions are also symmetric, there exists a Riesz basis exponential functions in the space $L^2(\Omega)$. The result new $d$ greater than one.

Let L = L1 × · · · × Ln be the product of n lattices, each of which has a bounded width. Given a subset A ⊆ L, we show that the problem of extending a given partial list of maximal independent elements of A in L can be solved in quasi-polynomial time.

In this note we provide a Henneberg-type constructive characterization theorem of [k, l]-sparse graphs, that is, the graphs for which the number of induced edges in any subset X of nodes is at most k|X| − l. We consider the case 0 ≤ l ≤ k.

The mechanism by which the salivary gland lysate (SGL) of Lutzomyia longipalpis enables Leishmania infection remains under investigation. One possibility is that saliva promotes cellular recruitment leading to development of skin lesions. In this study, we investigated leukocyte recruitment induced by L. major, L. major + SGL, or SGL alone into the peritoneal cavity of BALB/c mice. The administ...

In this paper we consider the diffusive search for a bounded target $\Omega \in \R^d$ with its boundary $\partial \Omega$ totally absorbing. We assume that is surrounded by semipermeable interface given closed surface \calM$ \subset \calM\subset \R^d$. That is, surrounds and thus partially screens process. also position of diffusing particle (searcher) randomly resets to initial $\x_0$ accordin...

The famous theorem of Cobham says that, for multiplicatively independent integers k and l, any subset of N, which is both kand l-recognizable, is recognizable. Many of its proofs are based on so called Hansel’s lemma stating that such a kand l-recognizable set is syndetic. We consider these proofs and point out that some of them are inadequate.

A graph G with a list of colors L(v) and weight w(v) for each vertex v is (L,w)-colorable if one can choose a subset of w(v) colors from L(v) for each vertex v, such that adjacent vertices receive disjoint color sets. In this paper, we give necessary and sufficient conditions for a weighted path to be (L,w)-colorable for some list assignments L. Furthermore, we solve the problem of the free-cho...

let g be a group. a subset x of g is a set of pairwise noncommuting elements if xy ̸= yx for any two distinct elements x and y in x. if |x| ≥ |y | for any other set of pairwise non-commuting elements y in g, then x is said to be a maximal subset of pairwise non-commuting elements. in this paper we determine the cardinality of a maximal subset of pairwise non-commuting elements in any non-abelian...