The main goal of this paper is to present an alternative, real variable proof of the T(1)-Theorem for the Cauchy Integral. We then prove that the estimate from below of analytic capacity in terms of total Menger curvature is a direct consequence of the T(1)-Theorem. An example shows that the L 1-BMO estimate for the Cauchy Integral does not follow from L 2 boundedness when the underlying measur...

In this paper, the connection between Menger probabilistic norms and H"{o}hle probabilistic norms is discussed. In addition, the correspondence between probabilistic norms and Wu-Fang fuzzy (semi-) norms is established. It is shown that a probabilistic norm (with triangular norm $min$) can generate a Wu-Fang fuzzy semi-norm and conversely, a Wu-Fang fuzzy norm can generate a probabilistic norm.

In this paper propositions will be established concerning the collection of all the cut points of a given planef continuum. It will be shown that there does not exist an uncountable collection of mutually exclusive subcontinua of a given continuum M each of which contains at least one cut point of M. With the aid of this fundamental theorem it is shown, among other things, that the set of all t...

We establish a surprising connection between Menger’s classical covering property and Blass’s modern combinatorial notion of groupwise density. This connection implies a short proof of the groupwise density bound on the additivity number for Menger’s property.

E3 c(x, y, z)dH(x)dH(y)dH(z) where H1 is the 1-dimensional Hausdorff measure in Rn, c(x, y, z) is the inverse of the radius of the circumcircle of the triangle (x, y, z), that is, following the terminology of [6], the Menger curvature of the triple (x, y, z). A Borel set E ⊂ Rn is said to be “purely unrectifiable” if for any Lipschitz function γ : R → Rn, H1(E ∩ γ(R)) = 0 whereas it is said to ...

We give short and simple proofs of the following two theorems by Galil and Yu [3]. Let s and t be two vertices in an n-node graph G. (1) There exist k edge-disjoint s-t paths of total length O(n √ k). (2) If we additionally assume that the minimum degree of G is at least k, then there exist k edge-disjoint s-t paths, each of length O(n/k). Let G = (V , E) be an undirected n-node graph, with no ...

A Menger algebra of 2-place functions over a set Δ is (cf. [5]) a set 6 of functions mapping Δ x Δ into Δ, which is closed with respect to substitution or composition, i.e., has the property that, for any three functions F, G, H belonging to 6, the composite function F{G,H) belongs to 6. Here, F{G,H) is the function assuming the value F(G(x,y), H(x9y)) for each (x,y) in Δ x Δ. The purpose of th...

With the publication of Murray N. Rothbard’s monumental work An Austrian Perspective on the History of Economic Thought in 1995, contemporary Austrian economists have become aware that they are the modern heirs to a grand intellectual tradition. The first fruit of further research in the Austrian tradition was 15 Great Austrian Economists, a small volume edited by Randall G. Holcombe in 1999. I...

Erd˝ os conjectured that, given an infinite graph G and vertex sets A, B ⊆ V (G), there exist a set P of disjoint A–B paths in G and an A–B separator X 'on' P, in the sense that X consists of a choice of one vertex from each path in P. We prove, for countable graphs G, the extension of this conjecture in which A, B and X are allowed to contain ends as well as vertices, and where the closure of ...