We dealt in a series of previous publications with some geometric aspects of the mappings by functions obtained as analytic continuations to the whole complex plane of general Dirichlet series. Pictures illustrating those aspects contain a lot of other information which has been waiting for a rigorous proof. Such a task is partially fulfilled in this paper, where we succeeded among other things...

sitive to prolan (the anterior pituitary-like hormone), than the immature female mouse, the rat has been tried as an experimental animal for pregnancy-hormone tests. Eberson and Silverberg (1931), who used immature rats, obtained results as early as thirty-six to fortyeight hours. Walker and Walker (1938) modified the above test and reduced the time to thirty hours. Later, Kelso (1940), and Fra...

2 The first line of conjecture 5 of [1] was proven by Dembo, Cover and Thomas in Theorem 23 of [2]. The proof is a consequence of the properties of the discrete Rényi entropies, defined as R p (x) = p 1 − p/2 ln(x p /x 2) p ≥ 1 (1) where x p is the Hölder p−norm

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...