Four color theorem, using concept of free-planar graphs, is discussed.

After first determining criteria for wild ramification of L/K (which can only happen at primes above 2), the above result is obtained for n = 2 (e ≥ 3) by computing TL/K(OL) explicitly, and is then extended to the general case. This approach does not rely on Leopoldt’s Theorem, in contrast to the techniques used in [4]. The explicit nature of the calculations used to compute I(L/K) leads to the...

Let G be a plane graph and T an even subset of its vertices. It has been conjectured that if all T -cuts of G have the same parity and the size of every T -cut is at least k, then G contains k edge-disjoint T -joins. The case k = 3 is equivalent to the Four Color Theorem, and the cases k = 4, which was conjectured by Seymour, and k = 5 were proved by Guenin. We settle the next open case k = 6.

[This corrects the article on p. 267 in vol. 42, PMID: 26015880.].

In this note the Stone-Cech compactification is used to produce short proofs of two theorems on the structure of free topological groups. The first is: The free topological group on any Tychonoff space X contains, as a closed subspace, a homeomorphic copy of the product space X". This is a generalization of a result of B. V. S. Thomas. The second theorem proved is C. Joiner's, Fundamental Lemma.

Strengths-based solution-focused approaches are gaining ground in statutory child protection work, but few studies have asked front line practitioners how they navigate the complex worker-client relationships such approaches require. This paper describes one component of a mixed-methods study in a large Canadian statutory child protection agency in which 225 workers described how they applied t...

This paper shows that the general hypersurface of degree ≥ 6 in projective four space cannot support an indecomposable rank two vector bundle which is Arithmetically CohenMacaulay and four generated. Equivalently, the equation of the hypersurface is not the Pfaffian of a four by four minimal skewsymmetric matrix.