French Flawed Elite Higher Education: Classes Préparatoires (CP)

3 top Classes Préparatoires (CP) (Baccalaureate + 2 years, equivalent to Bachelor of Math & Science) supply more than 50% of the elite Ecole Polytechnique (X) & Ecole Normale Supérieure (ENS) (Masters degree in Engineering or Business) students each year:
1. Lycée Louis-Le-Grande (LLG) – Paris
2. Lycée Privé Sainte-Geneviève – Paris (Versailles)
3. Lycée Henri IV – Paris

These students come from families in the rich Parisian regions, whose parents and grand-parents are also the alumni of X or ENS — this education phenomenon is called “In-Breeding”, or in Mathematical parlence “Closure”. The flaw of this “Social Immobility” is that “The rich gets richer, the poor gets poorer“, because poor provincial French families have no resources to send their children to these top “CPs” to better prepare for the elite Grandes Ecoles. Although the “Concours” (Entrance Exams 科举) is meritocratic and fair, the preparation for “Concours” in the 2-year excellent top “CP” is not a level-playing field.

This Concours which was inherited from ancient China had already manifested for thousand years in the flawed elitism of Chinese “Mandarin” bureaucracy (官僚制度) — the top scholars were from the same closed circles of elite families, either mandarin parents or rich merchants. China had few major revolutions initiated by failed “Concours” ( 科举) scholars of poor provincial sons, notably “黄巢”之乱 (Hwang Chao Revolution) in 9 CE which brought down the glorious Tang (唐) Dynasty, the 洪秀全 (太平天国 Taiping Revolution, 19 CE) also preluded 40 years before the demise of Qing (清) Dynasty. In certain resemblance, the poor farmer’s non-university-graduate son Mao ZeDong launched the “Cultural Revolution” to purge the “Smelly Elite Scholars” (臭老九) inherited from the “Old China”.

It is no surprise for France, after the glorious 30-year “5th Republique” (1950 – 1980), to see so many social unrests – “non-elite” university graduate high unemployment, Arabic-African Muslim immigrant descendents’ riots, ISIS terrorism …, while the mostly-white Grandes Ecoles elites monopolise the top social echelons in government, civil service, big state and private enterprises. The previous French President faced strong resistance from the elites to introduce “diversity” (ie reserve certain student quota for poor family children and minority races) in these top Grandes Ecoles.

Circle in Different Representations

1-Dimensional Objects:
Affine Line: {\mathbb {A}^1}
Circle: {\mathbb {S}^1}

Six Representations of a Circle: {\mathbb {S}^1}
1) Euclidean Geometry (O-level Math)
Unit Circle : x^2 + y^2 = 1

2) Curve: (A-level Math)
Transcendental Parameterization :
\boxed { e(\theta) = (\cos \theta, \sin  \theta) \qquad  0 \leq \theta \leq 2\pi }

Rational Parameterisation :
\boxed { e(h) = \left(\frac {1-h^2} {1+h^2}  \: , \: \frac {2h} {1+h^2}\right) \quad \text { h any number or } \infty }


3) Affine Plane (French Baccalaureate – equivalent A-level – Math) {\mathbb {A}^2}
1-Dim Sub-spaces = Projective Lines thru’ Origin

4) Polygonal Representation (Undergraduate Math)

5) Identifying Intervals: (closed loop) (Undergraduate Math)

6) \text {Translation } (\tau, {\tau}^{-1}) \text { on a Line } {\mathbb {A}^1}
(Honors Year Undergraduate / Graduate Math)

[Using Quotient Group Notation]:
\boxed {  {\mathbb {S}^1} \simeq {\mathbb {A}^1 } \Big/ { \langle \tau , {\tau}^{-1} \rangle} }
{\mathbb {S}^1} = \text { Space of all orbits}


Are Circle and Line the same 1-dimensional object, i.e. are they Homeomorphic (同胚) in Topology ?

Answer: To be continued in the next blog “Homeomorphism

Socratica: Abstract Algebra

Abstract Algebra is scary but not with this pretty lecturer!

1. Group
◇ Definition
◇ Symmetric Group
◇ Cayley Table

2. Linear Groups:
◇ GLn (R)
◇ SLn(R) — Determinant = 1

3.  Relationship between Group Structures:
◇ Homomorphism,
◇ Isomorphism, 
◇ Kernel

4. Ring Definition

5. Field Definition

Simplicial Complex (单纯 復形)

Simplex : 单纯 (plural Simplices)
0-dim (Point) \triangle_0
1-dim (Line) \triangle_1
2-dim (Triangle) \triangle_2
3-dim (Tetrahedron) \triangle_3

Simplicial Complex: 单纯復形 built by various Simplices under some rules.

Definitions of Simplex : S (v_0, v_1, ..., v_n)
Boundary (\delta )
\displaystyle \boxed {  \delta(S) = \sum_{i=0}^{n} (-1)^i (v_0 ...\hat v_i ...v_n)}


Theorem: \boxed { \delta ^2 (S) = 0} SO SIMPLE !!!

Follow the entire Algebraic Topology from University of New South Wales (3rd / 4th Year Math) :

Algebraic Topology: a beginner’s course – N J Wildberger: