Several transitive properties and Devaney’s chaos

The relation among transitivity, indecomposability and Z-transitivity is discussed. It is shown that for a non-wandering system (each point is non-wandering), indecomposability is equivalent to transitivity, and for the dynamical systems without isolated points, Z-transitivity and transitivity are equivalent. Besides, a new transitive level as weak transitivity is introduced and some equivalent conditions of Devaney's chaos are given by weak transitivity. Moreover, it is proved that both d-shadowing property and d-shadowing property imply weak transitivity.     

Critically indecomposable graphs

An undirected graph G=(V,E) with a specific subset X⊂V is called X-critical if G and G(X), induced subgraph on X, are indecomposable but G(V−{w}) is decomposable for every w∈V−X. This is a generalization of critically indecomposable graphs studied by Schmerl and Trotter [J.H. Schmerl, W.T. Trotter, Critically indecomposable partially ordered sets, graphs, tournaments and other binary relational structures, Discrete Mathematics 113 (1993) 191-205] and Bonizzoni [P. Bonizzoni, Primitive 2-structures with the (n−2)-property, Theoretical Computer Science 132 (1994) 151-178], who deal with the case where X is empty.We present several structural results for this class of graphs and show that in every X-critical graph the vertices of V−X can be partitioned into pairs (a₁,b₁),(a₂,b₂),…,(a_m,b_m) such that G(V−{a_j1,b_j1,…,a_jk,b_jk}) is also an X-critical graph for arbitrary set of indices {j₁,…,j_k}. These vertex pairs are called commutative elimination sequence. If G is an arbitrary indecomposable graph with an indecomposable induced subgraph G(X), then the above result establishes the existence of an indecomposability preserving sequence of vertex pairs (x₁,y₁),…,(x_t,y_t) such that x_i,y_i∈V−X. As an application of the commutative elimination sequence of an X-critical graph we present algorithms to extend a 3-coloring (similarly, 1-factor) of G(X) to entire G.     

A note on semiorthogonal indecomposability of some Cohen-Macaulay varieties

In this short note, we observe that the criterion proven in [12] for semiorthogonal indecomposability of the derived category of smooth DM stacks based on the canonical bundle can be extended to the case of projective varieties with Cohen-Macaulay singularities. As a consequence, all projective curves of positive arithmetic genus have weakly indecomposable bounded derived categories and indecomposable categories of perfect complexes.     

On Decompositions of Compact Convex Sets

The present paper generalizes M. Edelstein's theorem on the indecomposability of compact convex sets in locally convex linear topological spaces to spherical and hyperbolic geometry. Moreover, the indecomposability of compact intervals in EU¹ w.r.t. homeomorphisms of EU¹ onto itself is shown.     

Further criteria for the indecomposability of finite pseudometric spaces

We continue the study of indecomposable finite (consisting of a finite number of points) pseudometric spaces (i.e., spaces whose only decomposition into a sum is the division of all distances in equal proportion). We prove that the indecomposability property is invariant under the following operation: connect two disjoint points by an additional simple chain, which is the inverted copy of the shortest path connecting these points. The indecomposability of the spaces presented by the graphsK _m,n (m ≥ 2,n ≥ 3) with edges of equal length is also proved. Translated fromMatematicheskie Zametki, Vol. 63, No. 3, pp. 421–424, March, 1998.     

Indecomposability and the number of links

Let (ℋ, ℳ) be a linear matrix problem induced from a finite dimensional algebra ∧. Then anṉ ×ṉ matrix M in R(ℋ, ℳ) is indecomposable if and only if the number of links in the canonical formM ^(∞) of M is equal to. ℳ-dimṉ − 1. On the other hand, the dimension of the endomorphism ring of M is equal to ℋ-dimṉ − σ(M).     

Stable indecomposability of loop spaces on symplectic groups

We prove that is stably indecomposable if or .