We define two classes of rings calling them weakly clean rings and weakly exchange rings both equipped with the strong property. Although the classes of weakly clean rings and weakly exchange rings are different, their two proper subclasses above do coincide. This extends results due to W. Chen (Commun. Algebra, 2006) and Chin-Qua (Acta Math. Hungar., 2011). We also completely characterize stro...

Let A be a Banach algebra. We study those closed ideals I of A for which the first cohomology group of A with coefficients in I is trivial; i.e. H(A, I) = {0}. We investigate such closed ideals when A is weakly amenable or biflat. Also we give some hereditary properties of ideal amenability.

We describe a very general technique for generating families of combinatorial objects without isomorphs. It applies to almost any class of objects for which an inductive construction process exists. In one form of our technique, no explicit isomorphism testing is required. In the other form, isomorph testing is restricted to within small subsets of the entire set of objects. A variety of differ...

A hereditary property P is a class of k-graphs closed under isomorphism and taking induced sub-hypergraphs. Let P (k) n denote those k-graphs of P (k) on vertex set {1, . . . , n}. We prove an asymptotic formula for log2 |P (k) n | in terms of a Turán-type function concerning forbidden induced sub-hypergraphs. This result complements several existing theorems for hereditary and monotone graph a...

No convenient internal characterization of spaces that are productively Lindelöf is known. Perhaps the best general result known is Alster’s internal characterization, under the Continuum Hypothesis, of productively Lindelöf spaces which have a basis of cardinality at most א1. It turns out that topological spaces having Alster’s property are also productively weakly Lindelöf. The weakly Lindelö...

An acyclic coloring of a graph is a proper vertex coloring without bichromatic cycles. We show that the acyclic colorings of any weakly chordal graph G correspond to the proper colorings of triangulations of G. As a consequence, we obtain polynomial-time algorithms for the acyclic coloring problem and the perfect phylogeny problem on the class of weakly chordal graphs. Our results also imply li...

The study and recognition of graph families (or graph properties) is an essential part of combinatorics. Graph colouring is another fundamental concept of graph theory that can be looked at, in large part, as the recognition of a family of graphs that are colourable according to certain rules. In this thesis, we study additive induced-hereditary families, and some generalisations, from a colour...

We describe a general method of constructing nonlocal recursion operators for symmetries PDEs. As an example, the cotangent equation to 3D rdDym $$u_{yt} = u_xu_{xy} - u_yu_{xx}$$ is considered which two mutually inverse are found. The exposition includes rigorous criterion check hereditary property operators.

An additive hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphism. Let P and Q be additive hereditary properties of graphs. A (P ,Q)-total coloring ∗Research supported in part by Slovak VEGA Grant 2/0194/10. 210 M. Borowiecki, A. Kemnitz, M. Marangio and P. Mihók of a simple graphG is a coloring of the vertices V (G) and edgesE(G) of G...

We consider the approximation of weighted maximum subgraph problems by partitioning the input graph into easier subproblems. In particular, we obtain efficient approximations of the weighted independent set problem with performance ratios of O(n(log log n/ log n)) and (∆ + 2)/3, with the latter improving on a ∆/2 ratio of Hochbaum for ∆ ≥ 5. We also obtain a O(n/ log n) performance ratio for va...