It is a well-known fact that algebraicK-theory is homotopy invariant as a functor on regular schemes; if X is a regular scheme, then the natural map Kn(X) → Kn(X×A) is an isomorphism for all n ∈ Z. This is false in general for nonregular schemes and rings. To express this failure, Bass introduced the terminology that, for any contravariant functor P defined on schemes, a scheme X is called P-re...

In this paper we introduce R-right (left), L-left (right) cancellative and weakly R(L)-cancellative semigroups and will give some equivalent conditions for completely simple semigroups, (completely) regular right (left) cancellative semigroups, right (left) groups, rectangular groups, rectangular bands, groups and right (left) zero semigroups according to R-right (left), L-left (right) and weak...

We study certain interpolation and extension properties of the space of regular operators between two Banach lattices. Let R p be the space of all the regular (or equivalently order bounded) operators on L p equipped with the regular norm. We prove the isometric identity R p = (R ∞ , R 1) θ if θ = 1/p, which shows that the spaces (R p) form an interpolation scale relative to Calderón's interpol...

Let R be a ring, CR and ′CR be the set of regular and left regular elements of R (CR ⊆ ′CR). Goldie’s Theorem is a semisimplicity criterion for the classical left quotient ring Ql,cl(R) := C −1 R R. Semisimplicity criteria are given for the classical left regular left quotient ring ′Ql,cl(R) := ′C −1 R R. As a corollary, two new semisimplicity criteria for Ql,cl(R) are obtained (in the spirit o...

In 1960, Dirac put forward the conjecture that r-connected 4critical graphs exist for every r ≥ 3. In 1989, Erdős conjectured that for every r ≥ 3 there exist r-regular 4-critical graphs. A method for finding r-regular 4-critical graphs and the numbers of such graphs for r ≤ 10 have been reported in [6, 7]. Results of a computer search for graphs of degree r = 12, 14, 16 are presented. All the ...

This paper addresses the following general problem of tree regular model-checking: decide whether R * (L) ∩ L p = ∅ where R * is the reflexive and transitive closure of a successor relation induced by a term rewriting system R, and L and L p are both regular tree languages. We develop an automatic approximation-based technique to handle this – undecidable in general – problem in most practical ...

Distance-regular graphs have been a key concept in Algebraic Combinatorics and have given place to several generalizations, such as association schemes. Motivated by spectral and other algebraic characterizations of distance-regular graphs, we study ‘almost distance-regular graphs’. We use this name informally for graphs that share some regularity properties that are related to distance in the ...

Distributions over strings and trees can be represented by probabilistic regular languages, which characterise many models in natural language processing. Recently, several datasets have become available which represent natural language phenomena as graphs, so it is natural to ask whether there is an equivalent of probabilistic regular languages for graphs. This paper presents regular graph lan...