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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 .

Cantorvals

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

what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
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Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
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Yasmin
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Cesar
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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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