16,077 research outputs found
Dimensions of Copeland-Erdos Sequences
The base- {\em Copeland-Erd\"os sequence} given by an infinite set of
positive integers is the infinite sequence \CE_k(A) formed by concatenating
the base- representations of the elements of in numerical order. This
paper concerns the following four quantities.
The {\em finite-state dimension} \dimfs (\CE_k(A)), a finite-state version
of classical Hausdorff dimension introduced in 2001.
The {\em finite-state strong dimension} \Dimfs(\CE_k(A)), a finite-state
version of classical packing dimension introduced in 2004. This is a dual of
\dimfs(\CE_k(A)) satisfying \Dimfs(\CE_k(A)) \geq \dimfs(\CE_k(A)).
The {\em zeta-dimension} \Dimzeta(A), a kind of discrete fractal dimension
discovered many times over the past few decades.
The {\em lower zeta-dimension} \dimzeta(A), a dual of \Dimzeta(A)
satisfying \dimzeta(A)\leq \Dimzeta(A).
We prove the following.
\dimfs(\CE_k(A))\geq \dimzeta(A). This extends the 1946 proof by Copeland
and Erd\"os that the sequence \CE_k(\mathrm{PRIMES}) is Borel normal.
\Dimfs(\CE_k(A))\geq \Dimzeta(A).
These bounds are tight in the strong sense that these four quantities can
have (simultaneously) any four values in satisfying the four
above-mentioned inequalities.Comment: 19 page
Dimension of Scrambled Sets and The Dynamics of Tridiagonal Competitive-Cooperative System
One of the central problems in dynamical systems and differential equations is the analysis of the structures of invariant sets. The structures of the invariant sets of a dynamical system or differential equation reflect the complexity of the system or the equation. For example, any omega-limit set of a finite dimensional differential equation is a singleton implies that each bounded solution of the equation eventually stabilizes at some equilibrium state. In general, a dynamical system or differential equation can have very complicated invariant sets or so called chaotic sets. It is of great importance to classify those systems whose minimal invariant sets have certain simple structures and to characterize the complexity of chaotic type sets in general dynamical systems. In this thesis, we focus on the following two important problems: estimates for the dimension of chaotic sets and stable sets in a finite positive entropy system, and characterizations of minimal sets of nonautonomous tridiagonal competitive-cooperative systems
Metric characterizations of spherical, and Euclidean buildings
A building is a simplicial complex with a covering by Coxeter complexes
(called apartments) satisfying certain combinatorial conditions. A building
whose apartments are spherical (respectively Euclidean) Coxeter complexes has a
natural piecewise spherical (respectively Euclidean) metric with nice geometric
properties. We show that spherical and Euclidean buildings are completely
characterized by some simple, geometric properties.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol5/paper17.abs.htm
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