Moonshine 196,883
What object can exist in 196,883 dimension ?
Simon Norton: "I can explain to you what Moonshine (Monster Group) is in one sentence.
"It is the voice of God."
While all the 5 mathematicians have been fully exhausted after 15 years of effort to categorize all Simple Groups in the Universe, only Simon Norton remains lonely in the search of the largest Monster Group (Moonshine).
Conway, who migrated from Cambridge to Princeton, does not want to touch anything on Group now, said, "Simon is the only person on earth who knows Moonshine".
Monster Group M, order |M|=
$latex 2^{46}. 3^{20}. 5^{9}. 11^{2}. 13^{3}. 17.19.23.29.31.41.47.59.71$
Moonshine Monster Group dimensions (dj) & relationship with Fourier expansion of coefficients (cj) in Modular Function:
$latex x^{-1} + 744+196,884x + 21,493,760 x^{2} + 864,229,970x^{3} +\dots $
$latex c_n= c_1+c_2+...c_{n-1} + d_{n}$
where
$latex d_1 = 196,883$
$latex d_2 = 21,296,876$
$latex d_3 = 842,609,326$
and
$latex c_1 = 1+ d_1 = 196,884$
$latex c_2 = c_1+d_2 = 21,493,760$
$latex c_3 = c_1 + c_2 + d_3 = 864,229,970$
What a coincidence! no wonder Conway said this discovery was the most exciting event in his life.
Tuesday, 30 April 2013
Monday, 29 April 2013
French Curve
The French method of drawing curves is very systematic:
"Pratique de l'etude d'une fonction"
Let f be the function represented by the curve C
Steps:
1. Simplify f(x). Determine the Domain of definition (D) of f;
2. Determine the sub-domain E of D, taking into account of the periodicity (eg. cos, sin, etc) and symmetry of f;
3. Study the Continuity of f;
4. Study the derivative of f and determine f'(x);
5. Find the limits of f within the boundary of the intervals in E;
6. Construct the Table of Variation;
7. Study the infinite branches;
8. Study the remarkable points: point of inflection, intersection points with the X and Y axes;
9. Draw the representative curve C.
Example:
$latex \displaystyle\text{f: } x \mapsto \frac{2x^{3}+27}{2x^2}$
Step 1: Determine the Domain of Definition D
D = R* = R - {0}
Step 2: There is no Periodicity and Symmetry of f
E = D = R*
[See Note below for Periodic and Symmetric example]
Step 3: Continuity of f
The function f is the quotient of 2 polynomial functions, therefore f is differentiable
=> f is continuous in $latex ]-\infty,0[ \cup ]0,+\infty[ $
[See previous post CID Relation]
Step 4: Determine f'
$latex \displaystyle\forall x \in R^{\star}, f'(x) = \frac{6x^{2}.2x^{2} - 4x (2x^{3}+27)}{4x^{4}} = \frac{4x^{4}-4.27x}{4x^{4}} = \frac{4x(x^{3}-27)}{4x^{4}}$
$latex \forall x \in R^{\star}, (x^{3} - 27 >0) \iff (x>3)$
Therefore f' has the same sign as $latex x \mapsto x(x-3)$
$latex \begin{cases} \forall x \in ]-\infty,0[ \cup ]3,+\infty[, & f'(x)>0 \\
\forall x \in ]0,3[ , & f'(x)<0
\end{cases}$
Step 5a: Limit at x=0
$latex \displaystyle\lim_{x\to 0}(2x^{3}+27) = 27$
$latex \displaystyle\lim_{x\to 0} 2x^{2} = 0 , (\forall x \in R^{\star}, x^{2} >0)$
Therefore, $latex \displaystyle\lim_{x\to 0}f(x) = + \infty$
Step 5b: Limit at $latex x= + \infty$
$latex \displaystyle\lim_{x\to +\infty} f(x) =\lim_{x\to +\infty} \frac{2x^{3}+27}{2x^{2}}=\lim_{x\to +\infty} \frac{2x^{3}}{2x^{2}} = \lim_{x\to +\infty} x = +\infty$
Step 5c: Limit at $latex x= - \infty$
Similarly,
$latex \displaystyle\lim_{x\to -\infty} f(x) = \lim_{x\to -\infty} x = -\infty$
Step 6: Construct the Table of Variation
$latex \begin{array}{|l|l|l|}
\hline
x & - \infty \rightarrow \: \: \: \: 0 & 0 \:\:\:\:\: \rightarrow \:\:3 \rightarrow \:\:\: +\infty \\
\hline
f'(x) & \:\: \: \: \:\: \: + & \:\:\:\: - \:\:\:\:\:\:\:\:\: 0 \:\:\:\:\:\:\: + \\
\hline
f(x) & -\infty \nearrow +\infty & +\infty \searrow \: \frac{9}{2} \nearrow +\infty\\
\hline
\end{array}$
Step 7: Study the infinite branches
7a) $latex \displaystyle\lim_{x\to 0}f(x) = + \infty$
=> y-axis is the asymptote
7b) $latex \displaystyle\forall x \in R^{\star}, f(x) = \frac{2x^{3}+27}{2x^{2}}= x+\frac{27}{2x^{2}}$
$latex \displaystyle\lim_{x\to +\infty}\frac{27}{2x^{2}} = 0$ , $latex \displaystyle\lim_{x\to -\infty}\frac{27}{2x^{2}} = 0$
=>
$latex \displaystyle\lim_{x\to +\infty}f(x) = x$ , $latex \displaystyle\lim_{x\to -\infty}f(x) = x$
=> y= x is another asymptote
$latex \forall x \in R^{\star}, \frac{27}{2x^{2}} >0$
=> The curve C is above the asymptote y=x
Step 8: Study the remarkable points: intersection points with x-axis
$latex \forall x \in R^{\star},(2x^{3}+27 =0)
\iff (x^{3}=-\frac{27}{2})
\iff (x=-\frac{3}{\sqrt[3]{2}}) = -2.38$
Step 9: Draw the representative curve C of f.
[caption id="attachment_2564" align="alignnone" width="500"]
french curve[/caption]
Note:
$latex \displaystyle\text{Let g: } x \mapsto \frac{sin x}{2- cos^{2}x}$
D = R
g(x) is periodic of 2π => E = [0 , 2π]
$latex \displaystyle\forall x \in R, g(-x)= \frac{sin (-x)}{2-cos^{2}(-x)}=-\frac{sin x}{2-cos^{2}x}=-g(x) $
=> g(x) is symmetric with respect to the origin point O
We can restrict our study of g(x) in E = [0,π]
$latex \displaystyle\forall x \in R, g(\pi-x)= \frac{sin (\pi-x)}{2-cos^{2}(\pi-x)}=\frac{sin x}{2-cos^{2}x}=g(x) $
=> g(x) is symmetric w.r.t. to the equation x= π/2
Finally, we can further restrict our study of g(x) in E = [0, π/2]
"Pratique de l'etude d'une fonction"
Let f be the function represented by the curve C
Steps:
1. Simplify f(x). Determine the Domain of definition (D) of f;
2. Determine the sub-domain E of D, taking into account of the periodicity (eg. cos, sin, etc) and symmetry of f;
3. Study the Continuity of f;
4. Study the derivative of f and determine f'(x);
5. Find the limits of f within the boundary of the intervals in E;
6. Construct the Table of Variation;
7. Study the infinite branches;
8. Study the remarkable points: point of inflection, intersection points with the X and Y axes;
9. Draw the representative curve C.
Example:
$latex \displaystyle\text{f: } x \mapsto \frac{2x^{3}+27}{2x^2}$
Step 1: Determine the Domain of Definition D
D = R* = R - {0}
Step 2: There is no Periodicity and Symmetry of f
E = D = R*
[See Note below for Periodic and Symmetric example]
Step 3: Continuity of f
The function f is the quotient of 2 polynomial functions, therefore f is differentiable
=> f is continuous in $latex ]-\infty,0[ \cup ]0,+\infty[ $
[See previous post CID Relation]
Step 4: Determine f'
$latex \displaystyle\forall x \in R^{\star}, f'(x) = \frac{6x^{2}.2x^{2} - 4x (2x^{3}+27)}{4x^{4}} = \frac{4x^{4}-4.27x}{4x^{4}} = \frac{4x(x^{3}-27)}{4x^{4}}$
$latex \forall x \in R^{\star}, (x^{3} - 27 >0) \iff (x>3)$
Therefore f' has the same sign as $latex x \mapsto x(x-3)$
$latex \begin{cases} \forall x \in ]-\infty,0[ \cup ]3,+\infty[, & f'(x)>0 \\
\forall x \in ]0,3[ , & f'(x)<0
\end{cases}$
Step 5a: Limit at x=0
$latex \displaystyle\lim_{x\to 0}(2x^{3}+27) = 27$
$latex \displaystyle\lim_{x\to 0} 2x^{2} = 0 , (\forall x \in R^{\star}, x^{2} >0)$
Therefore, $latex \displaystyle\lim_{x\to 0}f(x) = + \infty$
Step 5b: Limit at $latex x= + \infty$
$latex \displaystyle\lim_{x\to +\infty} f(x) =\lim_{x\to +\infty} \frac{2x^{3}+27}{2x^{2}}=\lim_{x\to +\infty} \frac{2x^{3}}{2x^{2}} = \lim_{x\to +\infty} x = +\infty$
Step 5c: Limit at $latex x= - \infty$
Similarly,
$latex \displaystyle\lim_{x\to -\infty} f(x) = \lim_{x\to -\infty} x = -\infty$
Step 6: Construct the Table of Variation
$latex \begin{array}{|l|l|l|}
\hline
x & - \infty \rightarrow \: \: \: \: 0 & 0 \:\:\:\:\: \rightarrow \:\:3 \rightarrow \:\:\: +\infty \\
\hline
f'(x) & \:\: \: \: \:\: \: + & \:\:\:\: - \:\:\:\:\:\:\:\:\: 0 \:\:\:\:\:\:\: + \\
\hline
f(x) & -\infty \nearrow +\infty & +\infty \searrow \: \frac{9}{2} \nearrow +\infty\\
\hline
\end{array}$
Step 7: Study the infinite branches
7a) $latex \displaystyle\lim_{x\to 0}f(x) = + \infty$
=> y-axis is the asymptote
7b) $latex \displaystyle\forall x \in R^{\star}, f(x) = \frac{2x^{3}+27}{2x^{2}}= x+\frac{27}{2x^{2}}$
$latex \displaystyle\lim_{x\to +\infty}\frac{27}{2x^{2}} = 0$ , $latex \displaystyle\lim_{x\to -\infty}\frac{27}{2x^{2}} = 0$
=>
$latex \displaystyle\lim_{x\to +\infty}f(x) = x$ , $latex \displaystyle\lim_{x\to -\infty}f(x) = x$
=> y= x is another asymptote
$latex \forall x \in R^{\star}, \frac{27}{2x^{2}} >0$
=> The curve C is above the asymptote y=x
Step 8: Study the remarkable points: intersection points with x-axis
$latex \forall x \in R^{\star},(2x^{3}+27 =0)
\iff (x^{3}=-\frac{27}{2})
\iff (x=-\frac{3}{\sqrt[3]{2}}) = -2.38$
Step 9: Draw the representative curve C of f.
[caption id="attachment_2564" align="alignnone" width="500"]
french curve[/caption]Note:
$latex \displaystyle\text{Let g: } x \mapsto \frac{sin x}{2- cos^{2}x}$
D = R
g(x) is periodic of 2π => E = [0 , 2π]
$latex \displaystyle\forall x \in R, g(-x)= \frac{sin (-x)}{2-cos^{2}(-x)}=-\frac{sin x}{2-cos^{2}x}=-g(x) $
=> g(x) is symmetric with respect to the origin point O
We can restrict our study of g(x) in E = [0,π]
$latex \displaystyle\forall x \in R, g(\pi-x)= \frac{sin (\pi-x)}{2-cos^{2}(\pi-x)}=\frac{sin x}{2-cos^{2}x}=g(x) $
=> g(x) is symmetric w.r.t. to the equation x= π/2
Finally, we can further restrict our study of g(x) in E = [0, π/2]
Saturday, 27 April 2013
TeX Math Editor
1989 Stanford Professor Donald Knuth published the first version of TeX mathematical software.
Subsequent versions follow π:
3.14,
3.141,
3.1415,
....
3.1415926 (current version)
Subsequent versions follow π:
3.14,
3.141,
3.1415,
....
3.1415926 (current version)
La Ligne Directe du Dieu
Cédric Villani (Médaille Fields 2010) "Théorème Vivant":
"La fameuse ligne directe, quand vous recevez un coup de fil du dieu de la mathématique, et qu'une voix résonne dans votre tête. C'est très rare, il faut l'avouer!"
"The famous direct line, when you receive a 'telephone call' from the God of the Mathematic, and that a voice resonates in your head. It is very rare, one has to admit."
"La fameuse ligne directe, quand vous recevez un coup de fil du dieu de la mathématique, et qu'une voix résonne dans votre tête. C'est très rare, il faut l'avouer!"
"The famous direct line, when you receive a 'telephone call' from the God of the Mathematic, and that a voice resonates in your head. It is very rare, one has to admit."
Thursday, 25 April 2013
QuYuan 屈原 Symmetry
屈原 QuYuan (343–278 BCE) Symmetry:
http://en.wikipedia.org/wiki/Qu_Yuan
离騷《天问》
1. "九天之际, 安放安属,
隅隈多有, 谁知其数 ?"
=> 天 (Sky) 和 地 (Earth) must be 2 symmetric spheres.
If 地 (Earth) were flat, then there would be (隅隈) edges and angles at the 天 (Sky) & 地 (Earth) boundary (九天之际).
2. "东西南北, 其修孰多,
南北顺橢, 其衍几何。"
=> 南北顺橢 = The Earth is ellipse (橢), with north-south (南北) slightly flatten.
几何 = Geometry
3. How did QuYuan know this advanced astronomy & geometry in ~ 300 BCE?
[caption id="attachment_2393" align="alignnone" width="201"]
屈原[/caption]
http://en.wikipedia.org/wiki/Qu_Yuan
离騷《天问》
1. "九天之际, 安放安属,
隅隈多有, 谁知其数 ?"
=> 天 (Sky) 和 地 (Earth) must be 2 symmetric spheres.
If 地 (Earth) were flat, then there would be (隅隈) edges and angles at the 天 (Sky) & 地 (Earth) boundary (九天之际).
2. "东西南北, 其修孰多,
南北顺橢, 其衍几何。"
=> 南北顺橢 = The Earth is ellipse (橢), with north-south (南北) slightly flatten.
几何 = Geometry
3. How did QuYuan know this advanced astronomy & geometry in ~ 300 BCE?
[caption id="attachment_2393" align="alignnone" width="201"]
屈原[/caption]
墨子 Mozi & Force
[caption id="attachment_2380" align="alignnone" width="166"]
Mozi[/caption]
墨子 Mozi (468 BCE~ 376 BCE), 2000 years earlier than Newton
"墨子" : “力, 行之所以奋也。”
行: Moving
奋: Acceleration
力: Force is due to acceleration by the moving object.
F ∝ a
F = m.a
Mozi[/caption]墨子 Mozi (468 BCE~ 376 BCE), 2000 years earlier than Newton
"墨子" : “力, 行之所以奋也。”
行: Moving
奋: Acceleration
力: Force is due to acceleration by the moving object.
F ∝ a
F = m.a
εδ Confusion in Limit & Continuity
1. Basic:
|y|= 0 or > 0 for all y
2. Limit: $latex \displaystyle\lim_{x\to a}f(x) = L$ ; x≠a
|x-a|≠0 and always >0
hence
$latex \displaystyle\lim_{x\to a}f(x) = L$
$latex \iff $
For all ε >0, there exists δ >0 such that
$latex \boxed{0<|x-a|<\delta}$
$latex \implies |f(x)-L|< \epsilon$
3. Continuity: f(x) continuous at x=a
Case x=a: |x-a|=0
=> |f(a)-f(a)|= 0 <ε (automatically)
So by default we can remove (x=a) case.
Also from 1) it is understood: |x-a|>0
Hence suffice to write only:
$latex |x-a|<\delta$
f(x) is continuous at point x = a
$latex \iff $
For all ε >0, there exists δ >0 such that
$latex \boxed{|x-a|<\delta}$
$latex \implies |f(x)-f(a)|< \epsilon$
|y|= 0 or > 0 for all y
2. Limit: $latex \displaystyle\lim_{x\to a}f(x) = L$ ; x≠a
|x-a|≠0 and always >0
hence
$latex \displaystyle\lim_{x\to a}f(x) = L$
$latex \iff $
For all ε >0, there exists δ >0 such that
$latex \boxed{0<|x-a|<\delta}$
$latex \implies |f(x)-L|< \epsilon$
3. Continuity: f(x) continuous at x=a
Case x=a: |x-a|=0
=> |f(a)-f(a)|= 0 <ε (automatically)
So by default we can remove (x=a) case.
Also from 1) it is understood: |x-a|>0
Hence suffice to write only:
$latex |x-a|<\delta$
f(x) is continuous at point x = a
$latex \iff $
For all ε >0, there exists δ >0 such that
$latex \boxed{|x-a|<\delta}$
$latex \implies |f(x)-f(a)|< \epsilon$
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