Abstract Given a dynamical simplex K on Cantor space X , we consider the set $G_K^*$ of all homeomorphisms which preserve elements and have no non-trivial clopen invariant subset. Generalizing theorem Yingst, prove that for generic element g measures is equal to . We also investigate when there exists conjugacy class in this happens exactly has only one element, unique measure associated some o...

In 2006, Jonathan Sondow gave a nice geometric proof that e is irrational. Moreover, he said that a generalization of his construction may be used to prove the Cantor’s theorem. But, he didn’t do it in his paper, see [1]. So, this work will give a geometric proof to Cantor’s theorem using Sondow’s construction. After, it is given an irrationality measure to some Cantor series, for that, we gene...

This is a survey of topological, group theoretical and some graph theoretical aspects of ends. After discussing the notion of ends in topology, we consider ends of graphs and show that the metric end topology of connected graphs is metrizable. The “1–2–Cantor theorem” is proved for graphs whose ends are all limit ends, that is, ends which are accumulation points of an orbit of the group of auto...

The spectrum of one-dimensional discrete Schrödinger operators associated to strictly ergodic dynamical systems is shown to coincide with the set of zeros of the Lyapunov exponent if and only if the Lyapunov exponent exists uniformly. This is used to obtain Cantor spectrum of zero Lebesgue measure for all aperiodic subshifts with uniform positive weights. This covers, in particular, all aperiod...

We show that sample paths of Brownian motion (and other stable processes) intersect the same sets as certain random Cantor sets constructed by a branching process. With this approach, the classical result that two independent Brownian paths in four dimensions do not intersect reduces to the dying out of a critical branching process, and estimates due to Lawler (1982) for the long-range intersec...

In this paper we introduce projective geometry and one of its important theorems. We begin by defining projective space in terms of homogenous coordinates. Next, we define homgenous curves, and describe a few important properties they have. We then introduce Bezout’s Theorem, which asserts that the number of intersection points of two homogenous curves is less than or equal to the product of th...

We deal with Krull’s intersection theorem on the ideals of a commutative Noetherian ring in the fuzzy setting. We first characterise products of finitely generated fuzzy ideals in terms of fuzzy points. Then, we study the question of uniqueness and existence of primary decompositions of fuzzy ideals. Finally, we use such decompositions and a form of Nakayama’s lemma to prove the Krull intersect...

If each four spheres in a set of five unit spheres in R have nonempty intersection, then all five spheres have nonempty intersection. This result is proved using Grace’s theorem: the circumsphere of a tetrahedron encloses none of its escribed spheres. This paper provides self-contained proofs of these results; including Schläfli’s double six theorem and modified version of Lie’s line-sphere tra...

It is known from a previous paper [3] that Katona’s Intersection Theorem follows from the Complete Intersection Theorem by Ahlswede and Khachatrian via a Comparison Lemma. It also has been proved directly in [3] by the pushing–pulling method of that paper. Here we add a third proof via a new (k,k+1)-shifting technique, whose impact will be exploared elsewhere. The fourth and last of our proofs ...

Using a class sum and a collection of related Radon transforms, we present a proof G. James’s Kernel Intersection Theorem for the complex unipotent representations of the finite general linear groups. The approach is analogous to that used by F. Scarabotti for a proof of James’s Kernel Intersection Theorem for the symmetric group. In the process, we also show that a single class sum may be used...