一道几乎无人会做的平面几何难题

givdn any Triangle ABC, trisect each angle A,B,C, the 6 trisecting lines (2 each at A,B,C) meet at points M, N,Q inside.

Prove: Triangle MNQ is a regular triangle ?

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The Modular Form

Synopsis 概要:
A Modular Form (模型式) is a type of function studied in a field of mathematics called complex Analysis.

The study of complex analysis reveals that Modular Forms have something called ‘q-expansion,’ like a generalized polynomial. The coefficients of these expansions come in patterns (Monster Group). There is a relationship between Partition Theory and Modular Form. The number theorists regard Modern Form as a basic part of their toolkit in important applications eg. Proof of the 350-year-old Fermat’s Last Theorem by Prof Andrew Wiles in 1994

Form” : Function with special properties – eg.

  • Space Forms: manifolds with certain shape.
  • Quadratic Forms (of weight 2): x^2+3xy+7z^2
  • Cubic Forms (of weight 3): x^3+{x^2}y + y^3
  • Automorphic Forms (particular case: Modular Forms): auto (self), morphic (shape).

1. Non-Euclidean Geometry

1.1 Hyperbolic Plane : is the Upper-Half in Complex plane H (positive imaginary part) where :

  • Through point p there are 2 lines L1 & L2 (called “geodesic“) parallel to line L.
  • Distance between p & q in H: \boxed {\int_{L} \frac {ds}{y}}
    where L the “line” segment (the arc of the semicircle or the vertical segment) and ds^2 = dx^2+dy^2

1.2 Group of Non-Euclidean Motions:
f: H \rightarrow H

  1. Translation: z \rightarrow {z + b} \quad \forall b \in \mathbb {R}
  2. Dilation: z \rightarrow {az } \quad \forall a \in \mathbb {R^{+}}
  3. Inversion: z \rightarrow - \frac {1} {z} \quad \forall z \in H \implies z \neq 0
  4. Flip about axis (or line): z \rightarrow - \bar{z}

Note:
z = x + iy
\bar{z} = x - iy
-\bar{z} = -x + iy

Let = Group of the above 1 & 2 & 3 motions (exclude 4 since Flip is NOT complex-differentiable function of z)

\boxed {G^{0} = \{\gamma (z) = \frac {az+b}{cz+d} \quad \text {;} \quad ad - bc > 0\}}

Fractional Linear Transformation:

z \rightarrow \gamma(z)

\gamma = \begin{bmatrix} a & b \\ c & d \end{bmatrix}

z \rightarrow \begin{bmatrix} a & b \\ c & d \end{bmatrix} (z)

2. Group of Matrix M_2 (A)

Revision: Group = “CAN I

Matrix (K) with entries (a, b, c, d) from Set A (eg. Z, R, C…):

K = \begin{bmatrix} a & b \\ c & d \end{bmatrix}

Determinant = det (K) = ad – bc

Provided det (K) \neq 0
\displaystyle { \begin{bmatrix} a & b \\ c & d \end{bmatrix}}^{-1}= {\frac {1}{ad - bc}} {\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}}

2.1 General Linear Group GL_2(A)

\boxed {GL_2(\mathbb {R}) = \{ K \in M_2 (\mathbb {R}) \: | \: det (K) \neq 0\}}

\boxed {GL{_2}^{+}(\mathbb {R}) = \{ K \in M_2 (\mathbb {R}) \: | \: det (K) > 0\}}

\boxed {GL_2(\mathbb {C}) = \{ K \in M_2 (\mathbb {C}) \: | \: det (K) \neq 0\}}

\boxed {GL_2(\mathbb {Z}) = \{ K \in M_2 (\mathbb {Z}) \: | \: det (K) = \pm 1\}}

2.2 Special Linear Group SL_2(\mathbb{Z}) \subset GL_2(\mathbb{Z})

\boxed {SL_2(\mathbb {Z}) = \{ K \in M_2 (\mathbb {Z}) \: | \: det (K) = +1\}}

The Group SL_{2}(\mathbb {Z})= \{S, T\} acts on the upper half-plane H

T = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} \quad \boxed {T (z) = z+1}

S = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \quad \boxed {S (z) = -\frac {1}{z}}

Notes:

S^2 = -I \implies S^{4} = I

T^{k}= \begin{bmatrix} 1 & k \\ 0 & 1 \end{bmatrix} \quad \forall k \in {\mathbb {Z}}

3. Modular Form : M_{k} is an Analytic Function of weight k (k being a nonnegative Even Integer) f : H \rightarrow C with 2 properties:

(1) Transformation property
\boxed {f(\gamma (z)) = (cz+d)^{k}f (z)}

(2) Growth property: possess a “q-expansion” of the form:
\boxed {f(z) = a_0 + a_{1}q +a_{2}q^{2}+... }
where all aj are constants, and
q=e^{2\pi{iz}}

Cusp Form of weight k : S_{k}
\boxed{f(z) = a_{1}q +a_{2}q^{2}+... }

Note: S for Spitze (German: Cusp) – “尖点” (A pointed end where 2 curves meet.)

Note: q(z+1) = q(z) [hint:] e^{2i\pi} = 1
More generally, with an automorphy factor \phi (X)
g(X+1) = \phi {(X)}g(X)
eg. g(X) = e^{X} \implies g(X+1) = e^{X+1}=e.e^X = e.{g(X)} \text { ;} \quad \phi (X)=e

(Complex) Vector Spaces (V) = \{S_{k} \subset M_{k}\}
fulfilling:
(V1) V is nonempty.
(V2) For any function v in V, and any complex number c, the function cv is also in V.
(V3) For any function v and w in V, the function v + w is also in V.

4. Congruence Groups (of Level N)

\boxed {\Gamma (N) = \{ \gamma \in SL_{2}(\mathbb {Z}) | \gamma \equiv I \: (mod \: N)\}}

\boxed {\Gamma_{0} (N) = \{ \gamma \in SL_{2}(\mathbb {Z}) | c \equiv 0 \: (mod \: N)\}}

\boxed {\Gamma_{1} (N) = \{ \gamma \in SL_{2}(\mathbb {Z}) | c \equiv {a - 1} \equiv {d - 1} \equiv 0 \: (mod \: N)\}}

Note: It is one of the mysteries, or facts, of the theory that the above 3 are the main Congruence Subgroups needed to do most of the work that number theorists demand from Modular Form.

5. Applications

5.1 L-Function: when 2 different objects have the same L-function, this can mean that there is a very profound and often very useful tight connection between them.

5.2 Elliptic Curve

y^2 = x^3 + ax^2 + bx + c

5.3 Galois Representation

\rho : G_{Q} \rightarrow GL_{n}(K) \, | \, \rho ({\sigma}{\tau})= \rho(\sigma) \rho (\tau)

5.4 Monstrous “Monshine” – largest Simple Group

j(z) = q^{-1} +744 + 196884q + 21493760q^{2} + ...

The connection between j-function and the Monster Group was discovered by Simon Norton and John Conway, fully explained by Richard Borcherds in 1992 for which (partly) he was awarded the Fields Medal.

5.5 Fermat’s Last Theorem

5.6 Sato-Tate Conjecture

Note: “Operator” is synonymous to “Function of functions” (eg. Hecke Operator), just like “Form” is synonym for “Function”

Reference: [National Library NLB # 512.7]

Amazon Review: https://www.amazon.com/gp/aw/cr/0691170193/ref=mw_dp_cr

Other related links:

1. Shimura-Taniyama Weil Theorem

2.费马大定理 Fermat’s Last Theorem

School System Video (Do not make a fish climb trees)

Singapore Maths Tuition

Singapore is being mentioned around 4:54. Very nice video. The truth is that the classroom of today is still nearly the same as the classroom of 150 years ago. There needs to be a “Educational Revolution” parallel to that of the Industrial Revolution. Many children cannot fit into the single classroom model, leading to growth in diagnosis of behavioral “problems” such as ADHD in developed nations.

Americans who are tired of “Common Core” may want to check out Singapore Math for their kids, which is highly acclaimed in the educational realm.

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Population Differential Equations and Laplace Transform

Singapore Maths Tuition

Malthus Model
$latex displaystyle frac{dN}{dt}=BN-DN=kN$

$latex N$: Total population

$latex B$: Birth-rate per capita

$latex D$: Death-rate per capita

$latex k=B-D$

Solution to D.E.:
$latex displaystyle boxed{N(t)=widehat{N}e^{kt}},$

where $latex widehat{N}=N(0)$.

Logistic Equation
$latex begin{aligned}
D&=sN
frac{dN}{dt}&=BN-sN^2
widehat{N}&=N(0)
N_infty&=B/s
end{aligned}$

Logistic Case 1: Increasing population ($latex widehat{N}<N_infty$)
$latex begin{aligned}
N(t)&=frac{B}{s+(frac{B}{widehat{N}}-s)e^{-Bt}}
&=frac{N_infty}{1+(frac{N_infty}{widehat{N}}-1)e^{-Bt}}
end{aligned}$

The second expression can be derived from the first: divide by $latex s$ in both the numerator and denominator.

Logistic Case 2: Decreasing population ($latex widehat{N}>N_infty$)
$latex begin{aligned}
N(t)&=frac{B}{s-(s-frac{B}{widehat{N}})e^{-Bt}}
&=frac{N_infty}{1-(1-frac{N_infty}{widehat{N}})e^{-Bt}}
end{aligned}$

Logistic Case 3: Constant population ($latex widehat{N}=N_infty$)
$latex displaystyle N(t)=N_infty$

Harvesting
Basic Harvesting Model: $latex displaystyle boxed{frac{dN}{dt}=(B-sN)N-E}.$

$latex E$: Harvest rate (Amount harvested per unit time)

Maximum harvest rate without causing extinction: $latex boxed{dfrac{B^2}{4s}}$.

$latex displaystyle boxed{beta_1,beta_2=frac{Bmpsqrt{B^2-4Es}}{2s}}.$

$latex beta_1$: Unstable equilibrium population

$latex beta_2$: Stable equilibrium population

Extinction Time: $latex displaystyle boxed{T=int_{widehat{N}}^0frac{dN}{N(B-sN)-E}}.$

Laplace transform of $latex f$
$latex displaystyle F(s)=L(f)=int_0^infty e^{-st}f(t),dt$

Tip: Use this equation when the questions…

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11-year old math and chess prodigy in Singapore

Singapore Maths Tuition

Source: Channel News Asia

Aarushi Maheshwari solved the famous “Cheryl’s Birthday Problem” when she was only 9. She is also a chess champion and can play blindfold chess.

Watch the video below to learn more!

Also read our previous post on The Most Accomplished 10-Year-Old (Gifted pupil).

For those who want to learn more about Olympiad Math and International Chess, check out the previous two links. Math and Chess are two of the most intellectually challenging activities that can develop the intelligence of kids.

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