The Cantor Set
نویسنده
چکیده
Analysis is the science of measure and optimization. As a collection of mathematical fields, it contains real and complex analysis, functional analysis, harmonic analysis and calculus of variations. Analysis has relations to calculus, geometry, topology, probability theory and dynamics. We will focus mostly on ”the geometry of fractals” today. Examples are Julia sets which belong to the subfield of ”complex analysis” of ”dynamical systems”. ”Calculus of variations” is illustrated by the Kakeya needle set in ”geometric measure theory”, a glimpse of ”Fourier analysis” is seen by looking at functions which have fractal graphs, ”spectral theory” as part of functional analysis is represented by the ”Hofstadter butterfly”. As we take a tabloid approach and describe the topic with gossip about some ”pop icons” in each field, consider this page the center fold page of the ”Analytical Enquirer”.
منابع مشابه
Completeness in Probabilistic Metric Spaces
The idea of probabilistic metric space was introduced by Menger and he showed that probabilistic metric spaces are generalizations of metric spaces. Thus, in this paper, we prove some of the important features and theorems and conclusions that are found in metric spaces. At the beginning of this paper, the distance distribution functions are proposed. These functions are essential in defining p...
متن کاملPopulation dynamic of Long-spined tripodfish (Pseudotriacanthus strigilifer( Cantor, 1849 in the Persian Gulf (Hormozgan Province)
In the present study, we investigated biological characteristics and population dynamics of long-spined tripodfish, Pseudotriacanthus strigilifer, in coastal waters of Hormozgan province. A total of 391 fish samples were collected from September 2014 until September 2015 by the set net of Moshta. Length frequency data were analyzed using FiSAT software and ELEFAN 1 method. The longest and the s...
متن کاملON THE HAUSDORFF h-MEASURE OF CANTOR SETS
We estimate the Hausdorff measure and dimension of Cantor sets in terms of a sequence given by the lengths of the bounded complementary intervals. The results provide the relation between the decay rate of this sequence and the dimension of the associated Cantor set. It is well-known that not every Cantor set on the line is an s-set for some 0 ≤ s ≤ 1. However, if the sequence associated to the...
متن کاملN ov 2 00 7 Orbit equivalence for Cantor minimal Z 2 - systems
We show that every minimal, free action of the group Z2 on the Cantor set is orbit equivalent to an AF-relation. As a consequence, this extends the classification of minimal systems on the Cantor set up to orbit equivalence to include AF-relations, Z-actions and Z2-actions.
متن کاملCantor Sets, Binary Trees and Lipschitz Circle Homeomorphisms
We define the notion of rotations on infinite binary trees, and construct an irrational tree rotation with bounded distortion. This lifts naturally to a Lipschitz circle homeomorphism having the middle-thirds Cantor set as its minimal set. This degree of smoothness is best possible, since it is known that no C1 circle diffeomorphism can have a linearly self-similar Cantor set as its minimal set.
متن کاملFolding of Cantor String
In recent years, folding of various objects have been generated using different approaches. The classical Cantor set is an interesting mathematical construction with links to several areas of analysis and topology. The purpose of this paper is to represent the folding of Cantor string (compliment of Cantor set) using direct folding and folding by cut methods. Moreover, the results governing the...
متن کامل