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C λ , A = n = 0 α n λ n where α n A n

with the following constraints on A :

a 0 = 0 , a k = 1 - λ , a i + 1 - a i > λ for 0 i k - 1 .

As in the case of the mid- α Cantor sets C λ , homogeneous Cantor sets exhibit self-similarity in the following sense:

C λ , A = j = 0 k λ · C λ , A + a j .

An example of this self-similarity can be seen in [link] .

The first, second, and third stages in the construction of a homogeneous Cantor set with λ = 0 . 2 and A = 0 , 0 . 3 , 0 . 8 .


Another relevant topological structure is the Cantorval . In loose terms, one could consider Cantorvals as "Cantor sets that contain intervals." To be more precise about this definition, we need to first define a gap of a set to be a bounded connected component of the complement. For example, in the Cantor ternary set T , the interval 1 3 , 2 3 is the largest gap of T .

Now we can formally define a Cantorval. We say that a compact, perfect set C R is an M -Cantorval if every gap of C is accumulated on each side by other gaps and intervals of C . An example of an M -Cantorval is given by Anisca and Chlebovec in [link] ; see [link] .

Similarly, we say that C is an L -Cantorval (or an R -Cantorval ) if every gap of C is accumulated on the left (or right) by gaps and intervals of C , and if each gap of C has an interval adjacent to its right (or left). See [link] .

An M -Cantorval constructed in a manner similar to the construction of the Cantor ternary set T , except one only removes intervals at the odd stages; no intervals get removed at even stages.
An example of an L -Cantorval. Note (or, given the limited resolution, imagine) that every gap has an interval on its right and is accumulated on the left by points, intervals, and gaps.

Sums of mid- α Cantor sets

The problem tackled in this study revolves around characterizing the topological properties of the sum of two mid- α Cantor sets C λ and C γ , given by

C λ + C γ = x + y x C λ , y C γ

in terms of λ and γ .

It is known that this sum can be an interval, as in the case of C 1 3 + C 1 3 = 0 , 2 . However, such a sum can result in another Cantor set, as with C 1 5 + C 1 5 . The proofs of these facts are in "Known Results" .

When studying this sum, it is more convenient to characterize it in terms of λ and λ θ = γ with θ 1 as opposed to simply just λ and γ . This is due to a result from [link] discussed below.

Hausdorff dimension

A useful way of characterizing these types of sum sets in terms of Hausdorff dimension . To define Hausdorff dimension, as done in [link] , we need first to define the Hausdorff α -measure . (Note that the α here is different from the α used to define the mid- α Cantor sets.)

Given a set K R and a finite covering U = U i i I of K by open intervals, we define i to be the length of U i , and then diam U to be the maximum of the i . Then, the Hausdorff α -measure m α K of K is

m α K = lim ε 0 inf U covers K diam U < ε i I i α .

Then, there is a unique number H D K such that for α < H D K , m α K = , and for α > H D K , m α K = 0 . We call this number the Hausdorff dimension of K .

From this definition, it is clear that any set with Hausdorff dimension less than 1 must have zero Lebesgue measure. Note that it is possible to have a Cantor set that has both Hausdorff dimension equal to 1 and Lebesgue measure zero. There is also another class of Cantor sets that have Hausdorff dimension 1 and positive Lebesgue measure.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
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Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
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