In the literature it is known that the decomposable numerical range W∧ k (A) of A ∈ Cn×n is not necessarily convex. But it is not known whether W∧ k (A) is star-shaped. We construct a symmetric unitary matrix A ∈ Cn×n such that the decomposable numerical range W∧ k (A) is not star-shaped and hence not simply connected. We then consider a real analog R∧ k (A) and show that R∧ k (A) is star-shape...

An additive hereditary graph property is a class of simple graphs which is closed under unions, subgraphs and isomorphisms. If P1, . . . ,Pn are graph properties, then a (P1, . . . ,Pn)-decomposition of a graph G is a partition E1, . . . , En of E(G) such that G[Ei], the subgraph of G induced by Ei, is in Pi, for i = 1, . . . , n. The sum of the properties P1, . . . ,Pn is the property P1 ⊕ · ·...

In this paper, we present an efficient algorithm for performing stepwise selection in the class of decomposable models. We focus on the forward selection procedure, but we also discuss how backward selection and the combination of the two can be performed efficiently. The main contributions of this paper are (1) a simple characterization for the edges that can be added to a decomposable model w...

We use some general results regarding positive maps to exhibit examples of non-decomposable maps and 2 ×2N , N ≥ 2, bound entangled states, e.g. non distillable bipartite states of N+N qubits.

A graph G of order n is called arbitrarily vertex decomposable if for each sequence (n1, . . . , nk) of positive integers such that ∑k i=1 ni = n, there exists a partition (V1, . . . , Vk) of vertex set of G such that for every i ∈ {1, . . . , k} the set Vi induces a connected subgraph of G on ni vertices. We consider arbitrarily vertex decomposable unicyclic graphs with dominating cycle. We al...

We give a holomorphic extension result from non generic CR submanifold of C of positive CR dimension. We consider N a non generic CR submanifold given by N = {N, h(N)} where N is a generic submanifold of some C and h is a CR map from N into C. We prove that if N is a hypersurface then any CR distribution on N extends holomorphically to a complex transversal wedge, we then generalize this result...

Using a technique based on "state splitting" it is shown that every n-state Markov system is decomposable into a whirl interconnection of n 1 two-state Markov systems.

A graph G of order n is said to be arbitrarily vertex decomposable if for each sequence (n1, ..., nk) of positive integers such that n1 + ... + nk = n there exists a partition (V1, ..., Vk) of the vertex set of G such that for each i ∈ {1, ..., k}, Vi induces a connected subgraph of G on ni vertices. In this paper we show that if G is a two-connected graph on n vertices with the independence nu...

In this paper we prove that the wrapped Butterry graph WBF(d;n) of degree d and dimension n is decomposable into Hamilton cycles. This answers a conjecture of D. Barth and A. Raspaud who solved the case d = 2.

We study effective categoricity of computable abelian groups of the form ⊕ i∈ω H, where H is a subgroup of (Q,+). Such groups are called homogeneous completely decomposable. It is well-known that a homogeneous completely decomposable group is computably categorical if and only if its rank is finite. We study ∆n-categoricity in this class of groups, for n > 1. We introduce a new algebraic concep...