Beal's Conjecture

Polymath Project

Polymath Project initiated by Terrence Tao inviting mathematicians worldwide to improve on Yitang Zhang's 70 million bounded gap between primes.

Online Seminar by Terrence Tao on Zhang's paper:

http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/

Go to this site to get regular update on progress.

http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes

Derivative Meaning

The derivative of a function can be thought of as:

(1) Infinitesimal: the ratio of the infinitesimal change in the value of a function to the infinitesimal change in a function.

(2) Symbolic: The derivative of
$Latex x^{n} = nx^{n-1} $
the derivative of sin(x) is cos(x),
the derivative of f°g is f'°g*g',
etc.

(3) Logical:
$Latex \boxed{\text{f'(x) = d}} $
$Latex \Updownarrow $
$latex \forall \varepsilon, \exists \delta, \text{ such that }$
$latex \boxed{
0 < |\Delta x| < \delta,
\implies
\Bigr|\frac{f(x+\Delta x)-f(x)}{\Delta x} - d \Bigr| < \varepsilon
}$

(4) Geometric: the derivative is the slope of a line tangent to the graph of the function, if the graph has a tangent.

(5) Rate: the instantaneous speed of f(t), when t is time.

(6) Approximation: The derivative of a function is the best linear approximation to the function near a point.

(7) Microscopic: The derivative of a function is the limit of what you get by looking at it under a microscope of higher and higher power.

(8) The derivative of a real-valued function f in a domain D is the Lagrangian section of the cotangent bundle T*(D) that gives the connection form for the unique flat connection on the trivial R-bundle ßxR for which the graph of f is parallel.

[Source]: Extract from "On Proof and Progess in Mathematics" by William Thurston.

Sequence Limit

Definition: $latex \text{Sequence } (a_n) $
has limit a

$latex \boxed{\forall \varepsilon >0, \exists N, \forall n \geq N \text { such that } |(a_n) -a| < \varepsilon}$

$latex \Updownarrow $

$latex \displaystyle \boxed{ \lim_{n\to\infty} (a_n) = a }$

What if we reverse the order of the definition like this:

∃ N such that ∀ε > 0, ∀n ≥ N,
$latex |(a_n) -a| < \varepsilon$

This means:

$latex \boxed {\forall n \geq N, (a_n) = a }$

Example:

$latex \displaystyle (a_n) = \frac{3n^{2} + 2n +1}{n^{2}-n-3}$

$latex \displaystyle\text{Prove: } (a_n) \text { convergent? If so, what is the limit ?}$

Proof:
$latex \displaystyle (a_n) = 3 + \frac{5n +10}{n^{2}-n-3}$

$latex n \to \infty, (a_n) \to 3$

Let's prove it.

$latex \text {Let } \varepsilon >0$
$latex \text{Choose N such that } \forall n \geq N, $
$latex \displaystyle |(a_n) -3| = \Bigr|\frac{5n +10}{n^{2}-n-3}\Bigr| < \varepsilon$

$latex \text{Simplify: } \displaystyle \Bigr|\frac{5n +10}{n^{2}-n-3}\Bigr|$
$latex \text{Let } n > 10 $
$latex \displaystyle \Bigr|\frac{5n +10}{n^{2}-n-3}\Bigr| < \frac{6n}{\frac{1}{2}n^{2}}= \frac{12}{n} < \varepsilon$

$Latex \text{Choose } N = \max (10, \frac{12}{\varepsilon})$
$Latex \displaystyle\forall n \geq N,
|(a_n) -3 | < \frac{12}{n} < \varepsilon$

Therefore,
$latex \displaystyle \boxed{ \lim_{n\to\infty} (a_n) = 3 }$ [QED]

[Source]: Excellent Introduction in Modern Math:
"A Concise Introduction to Pure Mathematics (3rd Edition)"
by Martin Liebeck
CRC Press @ 2011

Abel Prize 2013

Algebraic Geometry

Belgian mathematician Pierre Deligne is a 'perfect' mathematician: he won all the coveted 'Oscar' Math Prizes:
Fields Medal, Wolf Prize and in 2013 $1m Abel Prize.

http://www.nature.com/news/mathematician-wins-award-for-shaping-algebra-1.12644

Turn Sphere Inside Out

Watch this amazing video, mathematically you can turn a sphere inside out, but not a circle:

http://www.snotr.com/video/3107/How_to_turn_a_sphere_inside_out

Prime Gap by Unheralded Mathematician

http://simonsfoundation.org/features/science-news/unheralded-mathematician-bridges-the-prime-gap/

On April 17, 2013, a paper arrived in the inbox of Annals of Mathematics, one of the discipline’s preeminent journals. Written by a mathematician virtually unknown to the experts in his field — a 50-something lecturer at the University of New Hampshire named Yitang Zhang — the paper claimed to have taken a huge step forward in understanding one of mathematics’ oldest problems, the twin primes conjecture.

Editors of prominent mathematics journals are used to fielding grandiose claims from obscure authors, but this paper was different. Written with crystalline clarity and a total command of the topic’s current state of the art, it was evidently a serious piece of work, and the Annals editors decided to put it on the fast track.


Yitang Zhang (Photo: University of New Hampshire)

Just three weeks later — a blink of an eye compared to the usual pace of mathematics journals — Zhang received the referee report on his paper.

“The main results are of the first rank,” one of the referees wrote. The author had proved “a landmark theorem in the distribution of prime numbers.”

Rumors swept through the mathematics community that a great advance had been made by a researcher no one seemed to know — someone whose talents had been so overlooked after he earned his doctorate in 1991 that he had found it difficult to get an academic job, working for several years as an accountant and even in a Subway sandwich shop.

“Basically, no one knows him,” said Andrew Granville, a number theorist at the Université de Montréal. “Now, suddenly, he has proved one of the great results in the history of number theory.”

Mathematicians at Harvard University hastily arranged for Zhang to present his work to a packed audience there on May 13. As details of his work have emerged, it has become clear that Zhang achieved his result not via a radically new approach to the problem, but by applying existing methods with great perseverance.

“The big experts in the field had already tried to make this approach work,” Granville said. “He’s not a known expert, but he succeeded where all the experts had failed.”



“There are a lot of chances in your career, but the important thing is to keep thinking,” Zhang said.

Relative Entropy & WinZip

Data compression:
Low entropy: a string of 1000 repeated 0 has little information content, can be compressed as "1000 x 0".

High Entropy: a string of 1000 random '0' and '1' cannot be compressed.

Relative Entropy: the best optimized compression.

Example:
Morse code let 1 dot '.' represents the most commonly used alphabet 'e', and less commonly used alphabet 'q' is '--.-'

Applications: (use WinZip tool)
1. Analyze two articles, if they are written by same author or two different authors: the later case has higher relative entropy, requires more disk space for the file. If the compressed file is smaller, likelihood they are from the same author.

2. Analyzing 52 European languages: French and Italian have low Relative Entropy, they belong to same language family (Latin); Swedish and Croatian have high Relative Entropy, they are from different family.

3. WinZip can tell if your article after compressed is only 1/3 of the original size, most likely 2/3 of its content are redundant.

4. WinZip could be used to analyze information from data string of DNA sequences or Stock market movements.

Gödel's Proof: God's Existence

Kurt Gödel's Mathematical Proof of God's Existence

Axiom 1: (Dichotomy) A property is positive if and only if its negation is negative.

Axiom 2: (Closure) A property is positive if it necessarily contains a positive property.

Theorem 1. A positive is logically consistent (i.e., possibly it has some instance).

Definition. Something is God-like if and only if it possesses all positive properties.

Axiom 3. Being God-like is a positive property.

Axiom 4. Being a positive property is (logical, hence) necessary.

Definition. A Property P is the essence of x if and and only if x has P and is necessarily minimal.

Theorem 2. If x is God-like, then being God-like is the essence of x.

Definition. NE(x): x necessarily exists if it has an essential property.

Axiom 5. Being NE is God-like.

Theorem 3. Necessarily there is some x such that x is God-like.

Source: Wang Hao (1987) Reflections on Kurt Goedel. MIT Press: Cambridge, Mass. (Page 195).

Top 10 Tough Math

These are the top 10 tough Mathematics:
1. Motivic cohomology or cohomology Theory 上同调理论
2. Langlands Functoriality Conjecture
3. Advanced Number Theory (eg. Fermat's Last Theorem) 高等数论
4. Quantum Group 量子群
5. Infinite Dimensional Banach Space 无穷维度巴拿哈空间
6. Local and Micro-local Analysis of Large Finite Group 大有限群之局部与微局分析
7. Large and Inaccessible Cardinals 大与不可达基数
8. Algebraic Topology 代数拓扑学
9. Super-String Theory 超弦论
10. Langlands Theory 非阿贝尔互反性，自守性表现和模数变化

Self-study Advanced Math

I came across this review at Amazon in 2007 on how to study Advanced Math on your own. Wonderful advice !

Give yourself 10-15 years, with passion, interest, dedicated commitment, disciplined, you could self-study Math to be a next Fermat, or Hua Luogen - both learned Math by themselves through self-learning from books.

http://www.amazon.com/gp/richpub/syltguides/fullview/R1GE1P236K3YSV/ref=cm_syt_dtpa_f_1_rdssss1/102-4263436-5550568?pf_rd_m=ATVPDKIKX0DER&pf_rd_s=sylt-center&pf_rd_r=00JK8KDA3S1T2JRNBCVC&pf_rd_t=201&pf_rd_p=253457301&pf_rd_i=0821839675

Google Linear Algebra

Google Search Engine & Linear Algebra:
1) Let M(nxn) matrix of size n (say 1 billion) web pages:

$latex
\begin{pmatrix}
m_{11} & m_{12} & \ldots & m_{1n}\\
m_{21} & m_{22} & \ldots & m_{2n}\\
\vdots & \vdots & m_{jk} & \vdots\\
m_{n1} & m_{n2} &\ldots & m_{nn}
\end{pmatrix}
$

$latex m_{jk} = \begin{cases} 1, & \text{if Page }j \text{ linked to Page k} \\
0, & \text{if } \text{ not}
\end{cases}
$

Note: This PageRank of 0 & 1 is over-simplied. The actual PageRank is a fuzzy number between 0 and 1, based on Larry Page's patented PageRank formula, taking into accounts of the importance of the pages linked from and to, plus many other factors.

2) Let v(n) eigenvector of n webpages' PageRank ak:
$latex \begin{pmatrix}
a_1 \\
a_2 \\
\vdots\\
a_k \\
\vdots\\
a_n
\end{pmatrix}
$ $latex \displaystyle \implies a_k= \sum_{j}m_{jk} $
(all Page j pageRanks)
The page k pointed to by all pages j.

3) Let λ eigenvalue: M.v =λ.v

4) Iterate n times: $latex \boxed{(M^{n}).v = \lambda{^n}.v}$

=> page k is ranked more important if many important pages j point to it;
& pages j themselves pointed by other important pages, ...(iterate n times).
=> highest ranked pages appear first in Search list.

Harvard Abstract Algebra Video

The excellent lecture videos of "Introduction to Abstract Algebra", taught by Prof Benedict Gross at Harvard, can be downloaded here:
http://www.extension.harvard.edu/openlearning/math222/

I met Prof Benedict Gross in Singapore last year at the NUS Public Lecture. I thanked him for these Abstract Algebra videos recorded many years earlier which helped me to understand and follow this advanced public lecture on "Elliptic Curve". He was thrilled that remotely he could influence an unknown student through his Internet lectures.

Analysis by Timothy Gowers

Why easy analysis problems are easy
by Timothy Gowers (UK, Fields Medal 1998)

Timothy Gowers is teaching in Cambridge, he wrote the thick volume of "Princeton Math Encyclopedia."

He is a very good mathematician, who likes to explain simple fundamental Math questions (like why 2+2=4, multiplication is commutative,...), in the process making abstract math simple to understand.

"If you have recently met epsilons and deltas for the first time, then you may find the problems you are asked to solve on examples sheets very hard. On the other hand, you will notice that your lecturers, supervisors etc. do not find them hard at all. Why is this? " Read on ...

https://www.dpmms.cam.ac.uk/~wtg10/autoanalysis.html

Below is my attempt to rewrite the Example 1 with Latex epsilon-delta notation for easy reading.

Example 1.

I wish to prove that the sequence (1,0,1,0,1,0,...) does not converge.

$latex \text{Let me set the sequence }
\{a_n\} \text{ to be:} $

$latex
\{a_n\}=
\begin{cases}
1, & \text{if }n \text{ is odd} \\
0, & \text{if }n\text{ is even}
\end{cases}
$

$latex \Large\text{ Then the statement that }
\{a_n\} \Large\text{ converges to } a \Large\text{ can be written: }$

$latex \exists a, \forall \varepsilon >0 ,\:\:\exists N ,\:\:\forall n > N , \:\:|a_n - a| < \varepsilon $

For divergence, we want to write the negation of the above as:

$latex \boxed{\forall a,\: \exists \varepsilon >0, \:\:\forall N, \:\:\exists n > N, \:\:|a_n-a| \geq \varepsilon}$

Take arbitrary a as below:

$latex a_n = 1 \text{ if n is odd, choose }a < 1/2$
$latex a_n = 0 \text{ if n is even, choose }a \geq 1/2$


$latex \text {Let } \varepsilon = \frac {1}{2}$
For either case whether n is even or odd,
$latex \forall N, \:\:\exists n > N, \:\: |a_n- a| \geq \frac{1}{2}$

$latex \iff \{a_n\} \:\: diverges$

Exercise:
Prove:
1-1+1-1+1...
=1, or
=0, or
= 1/2 (Leibniz said 50% -1 50% 0) ?

Sub-group Test

2-Step Test subgroups:
H subset of group G is subgroup if:
1. H is non-empty
(check: identity of G ∈ H)
2. $Latex a.b^{-1} \in H$

Prove Subset not a subgroup:

1. For infinite Group: sufficient to prove subset doesn't contain e (identity).

2. For finite group: sufficient to prove subset not closed.
H is subgroup of G
$latex \iff a*b^{-1} \in H, \forall a, b \in H$

Proof Set Technique

Proof Set Technique:
Let sets A, B
1. Prove A ⊆ B:
∀x ∈ A
(show x ∈ B )
=> A ⊆ B
2. Prove B ⊆ A:
∀x ∈ B
(show x ∈ A )
=> B ⊆ A
3. Prove A = B:
(A ⊆ B) & (B ⊆ A)
=> A = B

Group Problem

Let G be a group.
∀x,y ∈ G
Prove that xy and yx have the same order n ?
ie. $latex (xy)^{n}= (yx)^{n} = e$

Proof:
Let
$latex (xy)^{n}=e$
(xy)(xy).....(xy)(xy) = e
[(xy) n times]
By associativity,
x.(yx)(yx)...(yx).y=e
$latex (yx)(yx)...(yx).y = x^{-1}$
$latex (yx)^{n-1}= x^{-1}.y{^-1}$
$latex (yx)^{n-1}= (y.x)^{-1}$
$latex (yx) ^{n-1}.(yx) = e$
$latex (yx)^{n} = e$
ie yx has order n.
[QED]

4 Pillars of Mathematics

4 Pillars of Mathematics:

1. Fundamental Theorem of Arithmetics (Prime)
2. Fundamental Theorem of Algebra (Gauss)
3. Fundamental Theorem of Calculus (Leibniz)
4.Fundamental Theorem of Finite Group (Galois)

庖丁解牛数学方法

"庖丁解牛"数学方法
庄子讲庖丁(butcher)解牛有三个功夫階段：
1st Level: 看见一只全牛 (Whole Cow)

2nd Level: 三年后，不见全牛，只见牛的生理结構(Anatomy) ：骨骼，肌肉，筋腱。

3rd Level: 不以目视而是神视，＂与桑林之舞合拍，与经首之会同律。＂达到了"物我"两忘的境界。Intuition.

数学的方法也如此。

1st Level: Whole Math (Primary school to High School)

2nd Level: Component Structure - (Undergraduate Math)：
Macro- structure (Algebra : Group, Ring, Field, Vector Space... );
Micro-structure (Analysis : Calculus, Topology, etc)

3rd Level: 无处不数 Ubiquitous Math - (Graduate Math)
eg. Fermat's Last Theorem used all Math theories available today to prove.