|
|
| Line 1: |
Line 1: |
| [[File:CantorFunction.svg|right|thumb|400px| The graph of the Cantor function on the [[unit interval]] ]]
| | Planning to build your own home is a brave decision. Brick by brick, block by block you are not just going to lay the foundation to a home, but also to a sense of belongingness, love and peace with your close ones.<br><br>When it comes to flooring of this dream home of yours, you will obviously want it to be absolutely speckles, seamless, and professional looking. You would not want the tiles to come off after a few months after you have moved in your new house. This is why there is need for special attention to be paid towards the sort of adhesives that you use while making these floors.<br><br>Epoxy resins are known to be the best variety of adhesives for this purpose. This is because it is one of the strongest forms of adhesive in its class and has been used for construction purposes since a long time. The trick is to use epoxy resin adhesives well, so that they create such a bond between the floor and the tiles that it gives a professional look and feel and helps the flooring to look new as ever even after years.<br><br>The most essential step while using this type of adhesive is to take extreme care when mixing the two parts of it well. It is usually available in two parts, a powder and a hardener. These are both essential to form that unbreakable bond which gives a good finish to the floor. When you loved this post and you wish to receive much more information with regards to [http://richardy1.soup.io/post/431589896/Implementaciones-Vers-tiles-Con-La-Resina-Epoxi resinas epoxi] kindly visit our web page. Mixing should be done in small batches in order to avoid any wastage. Especially when one is going to use a lot of adhesive, it is ideal to mix small batches at a time and use them up, because it tends to harden quickly, and cannot be left unattended for a long time. Usually one would be required to use one part of hardener to 5 parts of resins. This would form an ideal paste that can be used in different purposes of home making.<br><br>Epoxy resins have long been used in many different sectors and industries and it is because it is hassle free and safe way to form glue for a variety of purposes.<br><br>One can find competitive price offers for epoxy resin adhesives and it is wise to select one of these offers, when using the glue in large quantities. It can be fun to make your own home, but you need to find the best raw material for the same first. |
| | |
| In [[mathematics]], the '''Cantor function''', named after [[Georg Cantor]], is an example of a [[function (mathematics)|function]] that is [[continuous function|continuous]], but not [[absolute continuity|absolutely continuous]]. It is also referred to as the '''Devil's staircase'''.
| |
| | |
| ==Definition==
| |
| [[File:Cantor function.gif|300px|right]]
| |
| | |
| See figure. To formally define the Cantor function ''c'' : [0,1] → [0,1], let ''x'' be in [0,1] and obtain ''c''(''x'') by the following steps:
| |
| | |
| #Express ''x'' in base 3.
| |
| #If ''x'' contains a 1, replace every digit after the first 1 by 0.
| |
| #Replace all 2s with 1s.
| |
| #Interpret the result as a binary number. The result is ''c''(''x'').
| |
| | |
| For example:
| |
| * 1/4 becomes 0.02020202... base 3; there are no 1s so the next stage is still 0.02020202...; this is rewritten as 0.01010101...; when read in base 2, this is 1/3 so ''c''(1/4) = 1/3.
| |
| * 1/5 becomes 0.01210121... base 3; the digits after the first 1 are replaced by 0s to produce 0.01000000...; this is not rewritten since there are no 2s; when read in base 2, this is 1/4 so ''c''(1/5) = 1/4.
| |
| * 200/243 becomes 0.21102 (or 0.211012222...) base 3; the digits after the first 1 are replaced by 0s to produce 0.21; this is rewritten as 0.11; when read in base 2, this is 3/4 so ''c''(200/243) = 3/4.
| |
| | |
| ==Properties==
| |
| The Cantor function challenges naive intuitions about [[continuous function|continuity]] and [[measure (mathematics)|measure]]; though it is continuous everywhere and has zero derivative [[almost everywhere]], ''c'' goes from 0 to 1 as ''x'' goes from 0 to 1, and takes on every value in between. The Cantor function is the most frequently cited example of a real function that is [[uniformly continuous]] (precisely, it is [[Hölder continuous]] of exponent ''α = log2/log3'') but not [[absolute continuity|absolutely continuous]]. It is constant on intervals of the form (0.''x''<sub>1</sub>''x''<sub>2</sub>''x''<sub>3</sub>...''x''<sub>n</sub>022222..., 0.''x''<sub>1</sub>''x''<sub>2</sub>''x''<sub>3</sub>...''x''<sub>n</sub>200000...), and every point not in the Cantor set is in one of these intervals, so its derivative is 0 outside of the Cantor set. On the other hand, it has no [[derivative]] at any point in an [[uncountable]] subset of the [[Cantor set]] containing the interval endpoints described above.
| |
| | |
| Extended to the left with value 0 and to the right with value 1, it is the [[cumulative distribution function|cumulative probability distribution function]] of a random variable that is uniformly distributed on the Cantor set. This distribution, called the [[Cantor distribution]], has no discrete part. That is, the corresponding measure is [[Atom (measure theory)|atomless]]. This is why there are no jump discontinuities in the function; any such jump would correspond to an atom in the measure.
| |
| | |
| However, no non-constant part of the Cantor function can be represented as an integral of a [[probability density function]]; integrating any putative [[probability density function]] that is not [[almost everywhere]] zero over any interval will give positive probability to some interval to which this distribution assigns probability zero.
| |
| | |
| The Cantor function is the standard example of a [[singular function]].
| |
| | |
| The Cantor function is non-decreasing, and so in particular its graph defines a [[rectifiable curve]]. The arc length of the graph is 2.<ref>http://math.stackexchange.com/questions/27859/arc-length-of-the-cantor-function</ref> | |
| | |
| == Alternative definitions ==
| |
| | |
| === Iterative construction ===
| |
| [[File:Cantor function sequence.png|250px|right]]
| |
| | |
| Below we define a sequence {''ƒ''<sub>n</sub>} of functions on the unit interval that converges to the Cantor function.
| |
| | |
| Let ''ƒ''<sub>0</sub>(''x'') = ''x''.
| |
| | |
| Then, for every integer {{nowrap|''n'' ≥ 0}}, the next function ''ƒ''<sub>''n''+1</sub>(''x'') will be defined in terms of ''ƒ''<sub>''n''</sub>(''x'') as follows:
| |
| | |
| Let ''ƒ''<sub>''n''+1</sub>(''x'') = {{nowrap|0.5 × ''ƒ''<sub>''n''</sub>(3''x'')}}, when {{nowrap|0 ≤ ''x'' ≤ 1/3 }};
| |
| | |
| Let ''ƒ''<sub>''n''+1</sub>(''x'') = 0.5, when {{nowrap|1/3 ≤ ''x'' ≤ 2/3 }};
| |
| | |
| Let ''ƒ''<sub>''n''+1</sub>(''x'') = {{nowrap|0.5 + 0.5 × ''ƒ''<sub>''n''</sub>(3 ''x'' − 2)}}, when {{nowrap|2/3 ≤ ''x'' ≤ 1}}.
| |
| | |
| The three definitions are compatible at the end-points 1/3 and 2/3, because ''ƒ''<sub>''n''</sub>(0) = 0 and ''ƒ''<sub>''n''</sub>(1) = 1 for every ''n'', by induction. One may check that ''ƒ''<sub>''n''</sub> converges pointwise to the Cantor function defined above. Furthermore, the convergence is uniform. Indeed, separating into three cases, according to the definition of ''ƒ''<sub>''n''+1</sub>, one sees that | |
| | |
| :<math>\max_{x \in [0, 1]} |f_{n+1}(x) - f_n(x)| \le \frac 1 2 \, \max_{x \in [0, 1]} |f_{n}(x) - f_{n-1}(x)|, \quad n \ge 1.</math>
| |
| | |
| If ''ƒ'' denotes the limit function, it follows that, for every ''n'' ≥ 0,
| |
| | |
| :<math>\max_{x \in [0, 1]} |f(x) - f_n(x)| \le 2^{-n+1} \, \max_{x \in [0, 1]} |f_1(x) - f_0(x)|.</math>
| |
| | |
| Also notice that the choice of starting function does not really matter, provided ''ƒ''<sub>0</sub>(0) = 0, ''ƒ''<sub>0</sub>(1) = 1 and ''ƒ''<sub>0</sub> is [[Bounded function|bounded]].
| |
| | |
| === Fractal volume ===
| |
| The Cantor function is closely related to the [[Cantor set]]. The Cantor set ''C'' can be defined as the set of those numbers in the interval [0, 1] that do not contain the digit 1 in their [[base (exponentiation)|base-3 (triadic) expansion]], except if the 1 is followed by zeros only (in which case the tail 1000<math>\ldots</math> can be replaced by 0222<math>\ldots</math> to get rid of any 1). It turns out that the Cantor set is a [[fractal]] with (uncountably) infinitely many points (zero-dimensional volume), but zero length (one-dimensional volume). Only the D-dimensional volume <math> H_D </math> (in the sense of a [[Hausdorff dimension|Hausdorff-measure]]) takes a finite value, where <math> D = \log(2)/\log(3) </math> is the fractal dimension of ''C''. We may define the Cantor function alternatively as the D-dimensional volume of sections of the Cantor set
| |
| | |
| : <math>
| |
| f(x)=H_D(C \cap (0,x)).
| |
| </math>
| |
| | |
| There is <ref>{{cite book |ref=harv |last= X. -J.|first= Yang. |title= Advanced Local Fractional Calculus and Its Applications |publisher= World Science Publisher |year=2012|isbn= 978-1-938576-01-0}}</ref> <br/>
| |
| : <math>
| |
| H_D \left( {C\cap \left( {a,b} \right)} \right)+H_D \left( {C\cap \left(
| |
| {b,c} \right)} \right)=H_D \left( {C\cap \left( {a,c} \right)} \right), </math>
| |
| | |
| where the set <math> C </math> is a Cantor set.
| |
| | |
| If the set <math> C </math> is a Cantor set, then we have a Lebesgue-cantor staircase
| |
| function,<ref>{{cite book |ref=harv |last= X. -J.|first= Yang. |title= Local Fractional Functional Analysis and Its Applications |publisher= Asian Academic Publisher |year=2011|isbn= 978-988-19132-1-0}}</ref><ref>{{cite journal |journal= Advances in Mechanical Engineering and its Applications |year=2012 |volume=1 |issue=3 |pages=47–53 |title= Heat transfer in discontinuous media |author= Yang, X. -J. }}</ref> namely, <br/>
| |
| : <math>
| |
| H_D \left( {C\cap \left( {a,b} \right)} \right)=\Gamma \left( {1+D}
| |
| \right)_a I_b^{\left( D \right)} 1,</math>
| |
| | |
| where <br/>
| |
| : <math>
| |
| _a I_b^{\left( D \right)} 1=\frac{1}{\Gamma \left( {1+\alpha }
| |
| \right)}\int_a^b {\left( {dt} \right)^D} </math>
| |
| | |
| denotes local fractional integral operator.<ref>{{cite journal |journal= Thermal Science |year=2013 |volume=17 |issue=2 |pages=625–628 |title= Fractal heat conduction problem solved by local fractional variation iteration method |author= Yang, X. -J., Baleanu. D.|doi= 10.2298/TSCI121124216Y}}</ref>
| |
| | |
| As a direct result, we have <br/>
| |
| : <math>
| |
| H_D \left( {C\cap \left( {a,b} \right)} \right)=(b-a)^D. </math>
| |
| | |
| such that<br/>
| |
| : <math>
| |
| f\left( x \right)=H_D \left( {F\cap \left( {0,x} \right)} \right)=x^D,</math>
| |
| | |
| : <math>
| |
| f\left( x \right)=H_D \left( {F\cap \left( {a,x} \right)} \right)=(x-a)^D.
| |
| </math>
| |
| | |
| == Generalizations ==
| |
| Let
| |
| | |
| : <math>y=\sum_{k=1}^\infty b_k 2^{-k}</math>
| |
| | |
| be the [[dyadic rational|dyadic]] (binary) expansion of the real number 0 ≤ ''y'' ≤ 1 in terms of binary digits ''b''<sub>''k''</sub> = {0,1}. Then consider the function | |
| | |
| : <math>C_z(y)=\sum_{k=1}^\infty b_k z^{k}.</math>
| |
| | |
| For ''z'' = 1/3, the inverse of the function ''x'' = 2 ''C''<sub>1/3</sub>(''y'') is the Cantor function. That is, ''y'' = ''y''(''x'') is the Cantor function. In general, for any ''z'' < 1/2, ''C''<sub>''z''</sub>(''y'') looks like the Cantor function turned on its side, with the width of the steps getting wider as ''z'' approaches zero.
| |
| | |
| [[Minkowski's question mark function]] visually loosely resembles the Cantor function, having the general appearance of a "smoothed out" Cantor function, and can be constructed by passing from a continued fraction expansion to a binary expansion, just as the Cantor function can be constructed by passing from a ternary expansion to a binary expansion. The question mark function has the interesting property of having vanishing derivatives at all rational numbers.
| |
| | |
| ==References==
| |
| <references />
| |
| | |
| == External links ==
| |
| * [http://www.encyclopediaofmath.org/index.php/Cantor_ternary_function ''Cantor ternary function'' at Encyclopaedia of Mathematics]
| |
| * [http://demonstrations.wolfram.com/CantorFunction/ Cantor Function] by Douglas Rivers, the [[Wolfram Demonstrations Project]].
| |
| * {{MathWorld |title= Cantor Function |urlname= CantorFunction}}
| |
| | |
| [[Category:Fractals]]
| |
| [[Category:Measure theory]]
| |
| [[Category:Special functions]]
| |
Planning to build your own home is a brave decision. Brick by brick, block by block you are not just going to lay the foundation to a home, but also to a sense of belongingness, love and peace with your close ones.
When it comes to flooring of this dream home of yours, you will obviously want it to be absolutely speckles, seamless, and professional looking. You would not want the tiles to come off after a few months after you have moved in your new house. This is why there is need for special attention to be paid towards the sort of adhesives that you use while making these floors.
Epoxy resins are known to be the best variety of adhesives for this purpose. This is because it is one of the strongest forms of adhesive in its class and has been used for construction purposes since a long time. The trick is to use epoxy resin adhesives well, so that they create such a bond between the floor and the tiles that it gives a professional look and feel and helps the flooring to look new as ever even after years.
The most essential step while using this type of adhesive is to take extreme care when mixing the two parts of it well. It is usually available in two parts, a powder and a hardener. These are both essential to form that unbreakable bond which gives a good finish to the floor. When you loved this post and you wish to receive much more information with regards to resinas epoxi kindly visit our web page. Mixing should be done in small batches in order to avoid any wastage. Especially when one is going to use a lot of adhesive, it is ideal to mix small batches at a time and use them up, because it tends to harden quickly, and cannot be left unattended for a long time. Usually one would be required to use one part of hardener to 5 parts of resins. This would form an ideal paste that can be used in different purposes of home making.
Epoxy resins have long been used in many different sectors and industries and it is because it is hassle free and safe way to form glue for a variety of purposes.
One can find competitive price offers for epoxy resin adhesives and it is wise to select one of these offers, when using the glue in large quantities. It can be fun to make your own home, but you need to find the best raw material for the same first.