Descartes 6th Meditation

Descartes 6th Meditation

Body is by nature divisible.

If so and if Mind and Body are one and the same,
then Mind is also divisible.

However, the Mind is entirely indivisible.
It follows that the Mind and Body are not the same.

French Taupe: 3/2 & 5/2

French elite Grandes Écoles (Engineering College), established since Napoleon with the first Military College (1794) École Polytechnique (nickname X because the College logo shows two crossed swords like X), entry only through very competitive 'Concours' Entrance Exams - to gauge its difficulty, Évariste Galois failed in two consecutive years.

Before taking Concours, there are two years of Prépas, or Classe Préparatoire (Preparatory class) housed in a Lycée (High school) to prepare the top Math / Science post-Baccalaureat students. These two undergraduate years are so torturous that French call these students Taupes (Moles) - they don't see sunlight because most of the time they are studying 24x7, minus sleeping and meal time.

Most students take 2 years to prepare (Year 1: Mathématiques Supérieures, Year 2: Mathématiques Spéciales) for the Concours in order to enter X. These students are nicknamed 3/2 (Trois-Demi), so called playfully by the integration of X:

$latex \displaystyle\int_{1}^{2} xdx= \frac{1}{2}x^{2}\Bigr|_{1}^{2}=\frac{3}{2}$

If by the end of second year some students fail the Concours, they can repeat the second year, then these repeat students are called 5/2 (Cinq-Demi) - integrating X from Year 2 to Year 3:

$latex \displaystyle\int_{2}^{3} x dx=\frac{1}{2}x^{2}\Bigr\vert_{2}^{3}=\frac{5}{2}$


Évariste Galois was 5/2 yet he still failed X, not because of his intelligence but the incompetent X Examiner at whom the angry Galois threw the chalk duster. (Well done !)

Another famous 5/2 is René Thom (Fields medal 1958) who discovered 'Chaos Theory'.

There are few rare cases of 7/2 (Sept-Demi):
$latex \displaystyle\int_{3}^{4} x dx=\frac{1}{2}x^{2}\Bigr\vert_{3}^{4}=\frac{7}{2}$
for those who insist on attempting 3 times to enter X or other elite Grandes Écoles. Equally good - if not better - is École Normale Supérieure (ENS) where Galois finally entered after having failed X twice. The tragic Galois was expelled by ENS for his involvement in the Revolution.

Note: Only 200 years later that ENS officially apologized in recent year, during the Évariste Galois Anniversary ceremony, for wrongfully expelled the greatest Math genius of France and mankind.




One of the top Classe Préparatoire "Lycée Pierre de Fermat" named after the 17th century great Mathematician of the "Last Theorem of Fermat", in his hometown Toulouse, Southern France.

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]

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."

Abstract Algebra compulsory

Prof S.S. Chern 陈省身 (the “Gauss No. 2" in Differential Geometry) retired from Berkeley University , he went back to his Chinese Alma mata Nankai University 南开大学 in Tianjing city. (Nankai produced the first prime minister Zhou Enlai 周恩来 and the recent Prime Minister Wen Jiabao 溫家宝).

Knowing the important role of Abstract Algebra in linking all branches of Math and all sciences, Prof Chern  made "Abstract Algebra" compulsory with effect 2001 for all students of Maths, Science, Engineering, IT, Finance:

19 weeks of 3 hrs per week = 57 hrs.

Syllabus:

Group, Ring, Field, Galois Theory.

Note: The French Classe Preparatoire for Grande Ecoles already implemented Abstract Algebra for more than 100 years.

Shimura Memoire on André Weil

Goro Shimura (志村 五郎 born 23 February 1930) is a Japanese mathematician, and currently a professor emeritus of mathematics (former Michael Henry Strater Chair) at Princeton University

Shimura is known to a wider public through the important Modularity Theorem (previously known as the Taniyama-Shimura conjecture before being proven in the 1990s); Kenneth Ribet has shown that the famous Fermat's Last Theorem (FLT) follows from a special case of this theorem. Shimura dryly commented that his first reaction on hearing of 1994 Andrew Wiles's proof of the semi-stable case of the FLT theorem was 'I told you so'.

Shimura's mémoire on the 20th century great French mathematician André Weil (Fields Medal, Founder of Bourbaki):


1. Weil advised us not to stick to a wrong idea too long. “At some point you must be able to tell whether your idea is right or wrong; then you must have the guts to throw away your wrong idea.”

2. According to him, one of the best way to learn French or any foreign language was to see the same movie in that language again and again, staying in the same seat in the same movie theatre.

3. A French gentleman’s ideal is to have three concurrent loves: the first one, whom he cares about at present; the second, a potential one, whom he has his eye on with the hope that she will eventually be his principal love; the third, the past one, with whom he hasn’t completely cut off his relations. Then he observed: “It’s a good idea for a mathematician to have three mathematical loves in the same sense.”

4. As to Fields medals, he said: “It’s a kind of lottery. There are so many eligible candidates, and the whole selection process is a matter of chance. Therefore the prize could be given to any of them as in a lottery.”

5. He used to say that a good mathematician must have two good ideas. “It is possible for someone to have a really good idea, but it may be just a fluke. Once the person has a second good idea, then there is a good chance for him to develop into a better mathematician.”

6. In the summer of 1970 after the Nice Congress, I was talking with him somewhere in the Institute about French mathematicians. He observed that there were three young mathematicians in Paris who started brilliantly, and so there were high expectations for them. He mentioned three well-known names and said, “What happened to them? They utterly failed to produce anything great.” After around 1975 he expressed, more than once, his pessimistic view that French mathematics had been declining for some time.

{Note: This recorded memoire of Shimura made the French very unhappy, for which Shimura refused to delete it from the book }

7. Weil told me several anecdotes about Hardy. “Hardy’s opinion that mathematics is a young man’s game is nonsense,” Weil said.

8. When I prodded the guests to tell their ambitions in their next lives... “I want to be a Chinese scholar studying Chinese poems,” said Weil. After visiting China twice, he had been reading English translations of Chinese standard literature like ＂The Dream of the Red Chamber＂(红楼梦).

{Note: Weil met Hua LuoGeng 华罗庚 and remarked if every Chinese is like Hua, very soon in future the Westerners will have to learn Math in Chinese}

9. Weil said, "I would like to see the Riemann hypothesis settled before I die, but that is unlikely.” Weil died at 90+ in 1996.

Why French excel in math ?

Mathematics and quantitative finance, France

Since 1990, there have been 22 winners of the Fields Medal, widely regarded as the Nobel Prize of mathematics. Thirteen came from just two countries, Russia and France. Russia has more winners (seven), but more than twice the population, so the honours go to France, with six winners.

Cédric Villani, the 2010 Fields Medallist, cited national character. “Maths is an abstract way of looking at the world, which fits well with the French mentality. We apply algebra to everything.” Elite institutions help too. France’s brightest school leavers progress to the grandes écoles, which traditionally educate top scientists, administrators and presidents. For maths, you want Monsieur Villani’s alma mater, the École Normale Supérieure (ENS). All 10 French Fields Medallists learnt there. At ENS, no teacher can stay longer than 10 years. Instead of ancient dons, students get tutors at the forefront of mathematics. Many try, but only 40 mathematicians a year enter the ENS.

The French have applied their maths genius to the money markets too. The Financial Times business schools rankings suggest France leads the world in producing “financial engineering” experts, with six institutions in the top 10 masters courses in finance. France can thus claim to dominate quantitative finance, the highly mathematical specialism involved in about half of all financial trades.

They should thank Michel Crouhy. In 1986, at the École des Hautes Études Commerciales (EHESS) in Paris, he devised the world’s first masters course in financial engineering. “The business school students didn’t have good enough maths, so I said ‘Let’s take only maths graduates, engineers. I won’t have to spend forever explaining the equations.’ It worked; the EHESS still offers the world’s best finance masters course, according to the Financial Times.

“Americans told me they wanted to start a course like ours but they weren’t allowed,” says Crouhy. “Because US MBA programmes were so strong, the universities worried a finance masters would compete with their MBA and destroy the MBA’s franchise.” America’s hesitation seems to have cost them.
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Baccalaureat Origin

1. Arabic Origin: Latin translation of Arabic word 'Bi-haqq al-ruwayeh'. Arabs learnt from China.

2. Chinese Origin: Chinese ‘秀才' (xiu-cai) scholar system (equivalent to Cambridge GCE A level).

3. First appeared as 'Baccalareus' in the University of Paris (1232 AD) by Pope Gregory IX, means 'The right (license) to teach on the authority of another'.

'Zero' (Love) in Tennis Game

Why they say 'Love' instead of ‘Zero’?
'Love' is actually French's l'oeuf (egg), sound alike.