‎the concept of the bipartite divisor graph for integer subsets has been considered in [‎m‎. ‎a‎. ‎iranmanesh and c‎. ‎e‎. ‎praeger‎, ‎bipartite divisor graphs for integer subsets‎, ‎{em graphs combin.}‎, ‎{bf 26} (2010) 95--105‎.]‎. ‎in this paper‎, ‎we will consider this graph for the set of character degrees of a finite group $g$ and obtain some properties of this graph‎. ‎we show that if $g...

A subalgebra B of a Leibniz algebra L is called weak c-ideal if there subideal C such that L=B+C and B∩C⊆BL where BL the largest ideal contained in B. This analogous to concept weakly c-normal subgroup, which has been studied by number authors. We obtain some properties c-ideals use them give characterizations solvable supersolvable algebras generalizing previous results for Lie algebras. note ...

We give a survey of new characterizations of finite solvable groups and the solvable radical of an arbitrary finite group which were obtained over the past decade. We also discuss generalizations of these results to some classes of infinite groups and their analogues for Lie algebras. Some open problems are discussed as well.

In this paper we classify all finite solvable groups satisfying the following property P5: their orders of representatives are set-wise relatively prime for any 5 distinct non-central conjugacy classes.

Let Ω ⊂ Rd, d ≥ 3, be a bounded Lipschitz domain. For Laplace’s equation ∆u = 0 in Ω, we study the Dirichlet and Neumann problems with boundary data in the weighted space L2(∂Ω, ωαdσ), where ωα(Q) = |Q−Q0|α, Q0 is a fixed point on ∂Ω, and dσ denotes the surface measure on ∂Ω. We prove that there exists ε = ε(Ω) ∈ (0, 2] such that the Dirichlet problem is uniquely solvable if 1 − d < α < d − 3 +...

This thesis1 investigates various aspects of weak system models for agreement problems in fault-tolerant distributed computing. In Part I we provide an introduction to the context of this work, discuss related literature and describe the basic system assumptions. In Part II of this thesis, we introduce the Asynchronous Bounded-Cycle (ABC) model, which is entirely time-free. In contrast to exist...

We consider the algebra $\dot\Sigma(\mathcal L)$ generated by inner-limit derivations over ${\rm GICAR}$ of a fermion gas populating an aperiodic Delone set $\mathcal L$. Under standard physical assumptions such as finite interaction range, Galilean invariance and continuity with respect to lattice, we demonstrate that image $\dot \Sigma(\mathcal through Fock representation can be completed gro...

In 2003, it was claimed that the following problem was solvable in polynomial time: do there exist k edge-disjoint paths of length exactly 3 between vertices s and t in a given graph? The proof was flawed, and we show that this problem is NP-hard even if we disallow multiple edges. We use a reduction from Partial Orientation, a problem recently shown by Pálvölgyi to be NP-hard. In [2], Bley dis...

We study list homomorphism problems L-HOM(H) for the class of reflexive digraphs H (digraphs in which each vertex has a loop). These problems have been intensively studied in the case of undirected graphs H, and appear to be more difficult for digraphs. However, it is known that each problem L-HOM(H) is NP-complete or polynomial time solvable. In this paper we focus on reflexive digraphs. We in...

for some l > 0 and for all x E 0 and C; E R" . We also assume a;j = aj;. For such operators, the Dirichlet problem is solvable in n if and only if it is solvable for the Laplace operator, according to a theorem of Littman, Stampacchia and Weinberger [I]. This means that if n ~ R" is a sufficiently nice bounded region (the unit ball, B, is an example) and J is a given continuous function on the ...