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.

Math Chants

Math Chants make learning Math formulas or Math properties fun and easy for memory . Some of them we learned in secondary school stay in the brain for whole life, even after leaving schools for decades.

Math chant is particularly easy in Chinese language because of its single syllable sound with 4 musical tones (like do-rei-mi-fa) - which may explain why Chinese students are good in Math, as shown in the International Math Olympiad championships frequently won by China and Singapore school students.

1. A crude example is the quadratic formula which people may remember as a little chant:
"ex equals minus bee plus or minus the square root of bee squared minus four ay see all over two ay."

$latex \boxed{
x = \frac{-b \pm \sqrt{b^{2}-4ac}}
{2a}
}$

2. $latex \mathbb{NZQRC}$
Nine Zulu Queens Rule China

3. $latex \boxed {\cos 3A = 4\cos^{3} A - 3\cos A }$

cos = $ 1 (kö, Singapore Chinese Fujian福建話 / Taiwan 台湾闽南语dialect) [一元]

cos 3= $1.3
4cos^3 = $4.3
3cos = $3

$1.3 = $4.3 - $3

4. $latex \boxed{\pi = 3.14159\dots}$
Chant in Chinese Mandarin (Beijing):
山顶一寺一壶酒
Chinese sound means: On the mountain (3) top (.) there is one (1) temple (4) with a (1) bottle (5) of wine (9)

Note : Below is the 22 decimal memory chant:
山顶一寺一壶酒，尔乐, 我三壶把酒吃，酒杀尔，杀不死，乐而乐。

$latex \boxed{
\pi = 3.14159 \:26 \:535897 \:932 \:384 \:626 \dots
}$


5. Group Properties:
$latex \boxed{\text{CAN I ?} }$
C: Closure
A: Associative
N: Neutral (e)
I: Inverse

If only 50%, it is Semi-Group(C, A) 半群
If No Inverse, it is MoNoId (C,A,N) 幺群
(MoNoId : No I)

6. The eccentric choice of letters (a,h,g,b,f,c, skipping iand e) for the coefficients in Conic equation: $latex \boxed
{ ax^{2} + 2hxy + 2gx + by^{2}+ 2fy + c = 0
}$
Math Chant for the coefficients:
{a h g b f}
"all hairy guys big feet"

c: constant term (understood)

Proof & Progress in Math

"On Proof and Progress in Mathematics" by the late William Thurston

Great article in Mathematics Education, enjoy reading !

http://www.ams.org/journals/bull/1994-30-02/S0273-0979-1994-00502-6/S0273-0979-1994-00502-6.pdf

成语数学

中国的小学离校考试 (PSLE) 「神题」: 猜成语
1) 20 除 3
2）1 除100
3）9寸+1寸=1尺
4）12345609
5）1,3,5,7,9

答案：:
1) 20/3= 6.666 六六大顺
2）百中挑一
3）得寸進尺
4）七零八落
5）举世无双

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).

More on 666

$latex 666 = 1^{6} - 2^{6} + 3^{6}$

$latex 666 = 6 +6 +6 +6^{3}+ 6^{3}+6^{3}$

$latex 666 = 2^{2}+ 3^{2}+ 5^{2}+ 7^{2}+ 11^{2}+ 13^{2}+ 17^{2}$

$latex \phi(666) = 666 $
where
$latex \phi(n) = \text{number of integers less than n and co-prime with n }$

Bible Code

Appeal to my Blog's international readers (currently at 41 countries):
As of today I have verified Bible Code works for 6 languages : English, Chinese, French, Irish, Japanese and Spanish.

Please help to comment below if the "Bible Code" works in your native language (Russian, Spanish, Italian, Korean, Indonesian...) It works best with Bible in King-James version.

Bible Code:

Step 1: From Genesis 1:1 chose any 1 word from the 10 words. (Eg. 'the')

Step 2: Count the length (n1) of the selected word ('the' = n1 = 3)

Step 3: Jump n1 (3) words to the next word (eg. 'created')

Step 4: count the length (n2) of the selected word ('created' = n2 = 7)

Step 5: Jump n2 (7) words to the next word.

...
Repeat till the selected word first appears in Genesis 1:3 verse.

You will always end with the word "God".

Note1: It also works for Chinese Bible with strokes笔划:
( )神 A blank to respect God's name ：can be counted as 1 or 2 words(上帝).

起10-＞ 空8->神9->1:3 神
初7-＞神...是9 -> 神9 ->1:3 ( )神
神9-＞混12-＞水4->1:3( )神
创6-＞空8-＞神9-＞1:3( )神
造10 ->面9->水4 ->1:3( )神
天4-＞空8-＞神9-＞1:3( )神
地6-＞沌7-＞灵7-＞1:3( )神

Note 2: it works for French Bible too!
1.1 Au 2 ->Dieu 4 -> la2 ->Or 2 ->terre5 -> les 3 ->l'abîme 6 ->sur 3
->1.3 Dieu

Note 3: It works for Japanese Bible too!
1:1 さ3->地6 ->な4->み3->の2-> も3
->あ4->の2->が5->て2->お4->い2
->1:3神

Note 4: It also works for Irish Bible !
1:1 báire 5 -> talamh 6 ->an 2-> agus 4 -> aghaidh 7-> ag 2 -> os 2 ->na 2 ->
1:3 (Dúirt : said) Dia (noun God)

Note 5: It works for Spanish too !
(à 1 mot près : +/- 1 word,
God said = Said God
Dios dijo = dijo Dios)

Mozart Pieces

Mozart Piece P

$latex \boxed{P= 0.027465 + 0.157692K + 0.000159446K^{2}}$
where
K = Köchel number (sequenced by time)

Note: This formula is 85% accurate, error not exceeding 2.

Example:
Mozart No. 40 (G Minor) Symphony
is K.550

Background: Wolfgang Amadeus Mozart (1756-1791) was one of the most prolific composers of all time. In 1862, the German musicologist Ludwig von Köchel made a chronological list of Mozart's musical work. This list is the source of the Köchel numbers, or "K numbers", that now accompany the titles of Mozart's pieces (e.g., Sinfonia Concertante in E-flat major, K. 364). The table below gives the Köchel numbers and composition dates of 17 of Mozart's works.

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

The Eye of God

This NASA picture is "The Eye of God" with a majestic look over the Earth and the humans He created. He speaks to us in "The Language of God" - Mathematics!

"The great book of nature," said Galileo, "can be read only by those who know the language in which it was written. And this language is mathematics."

Note: There are three universal things (3M) which transcend races, cultures and languages, that God never changed after the Tower of Babel:
M: Music (formed by 7 musical notes)
M: Money (based on Gold)
M: Mathematics (derived from basic structures $latex \mathbb{NZQRC}$)

It was also observed by Leibniz that Music and Math are connected: "Human mind appreciates music through calculation without knowing."

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

Origamics

Origami + Mathematics = Origamics

The Origamics was invented and coined by its inventor a biologist Prof Kazuo Haga (Japan) in 1994.

Haga's First Theorem

Fold a square paper of size 1 unit x 1 unit: join the right-bottom vertex to the mid-point of the top edge.

This one fold creates respective edge points which mark out various ratios:

$latex \frac{1}{2}, \frac{1}{6}, \frac{1}{8}, \frac{3}{8}, \frac{\sqrt{5}}{2}, \frac{5}{24}, \dots $


Without ruler, this is the most accurate way to obtain length of $latex \frac{1}{3}, \sqrt{5} \dots $
As summarized in the diagram:

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.

Cut a cake 1/5

Visually cut a cake 1/5 portions of equal size:

1) divide into half:




2) divide 1/5 of the right half:




3) divide half, obtain 1/5 = right of (3)

$latex \frac{1}{5}= \frac{1}{2} (\frac{1}{2}(1- \frac{1}{5}))= \frac{1}{2} (\frac{1}{2} (\frac{4}{5}))=\frac{1}{2}(\frac{2}{5})$





4) By symmetry another 1/5 at (2)=(4)




5) divide left into 3 portions, each 1/5

$latex \frac{1}{5}= \frac{1}{3}(\frac{1}{2}+ \frac{1}{2}.\frac{1}{5}) = \frac{1}{3}.\frac{6}{10}$

Mathew Effect of e^x

"For unto every one that hath shall be given, and he shall have abundance: but from him that hath not shall be taken even that which he hath."

—Matthew 25:29, King James Version

Or, "the rich gets richer, the poor gets poorer."

Mathematically, this is $latex e^{x} \text { increases much faster than x increases} $

$latex \frac{d}{dx} e^{x} = e^{x}$

Pigeonhole Principle

$latex \pi = 3.14159265358979323846264 $
$latex \text{Let } a_1, a_2,\dots a_{24} \text{ represent the first 24 digits of } \pi$
Prove:
$latex (a_1 - a_2)(a_3 - a_4) \dots (a_{23} - a_{24}) \text{ is even}$

Proof:

13 Odd digits = {3.14159265358979323846264 }

11 Even digits

$latex \text {12 brackets :}(a_1 - a_2)(a_3 - a_4) \dots (a_{23} - a_{24})$

Put 13 odds into 12 brackets, by Pigeonhole Principle, there is certainly one bracket where
$latex (a_j - a_k) \text{ is a difference of 2 odds, which is an even = 2n}$

2n multiplies with any number will always give even.
The product of 2n with the other 11 brackets will always be even.

Therefore
$latex (a_1 - a_2)(a_3 - a_4) \dots (a_{23} - a_{24}) \text { is even}$

Love + Hate Math

Book on Loving and Hating Mathematics:

ISBN: 978-0-691-142470
Reuben Hershey and Vera John-Steiner
Princeton Press

Φ and 666

We still don't understand these mysterious numbers appearing everywhere in the Nature.
Mathematicians call them 'Transcendental number' (超函数) - basically they are irrational numbers (like $latex \sqrt {2}$ ), in additional, they are NOT solution of any polynomial equations (otherwise, they are called Algebraic number).

Top 3 mysterious numbers are:
$latex \pi, e, \phi$
connected by this formula:
$latex \boxed{e^{i.\pi} + 2\phi = \sqrt{5}}$
$latex \pi \text { involves anything in circle}$

$latex e \text{ anything with exponential growth:}$ $latex \text{ e.g. epidemic, bank interest...}$

$latex \phi \text{ the golden ratio involves symmetry and beauty. }$


Golden Ratio Φ

$latex \boxed{ \phi = 1.61803 \dots = -2 \sin 666 ^\circ}$

Recognize the Satanic number 666?

No wonder we praise in Chinese a gorgeous pretty lady having 魔鬼身材 ('satanic' body shape). It has mathematical truth with the golden ratio of beauty Φ and 666. :)

See also my next blog "More on 666 "

Note:
$latex \boxed{\frac{6}{5} \phi^{2}=\pi }$

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

Mind, Brain And Education

Mind, Brain And Education
1. Spaced Repetition
2. Retrieval Practice

Tool: Test
Not to assess what students know, but to reinforce it.
Memory is like a storage tank, a test as a kind of dipstick that measure how much information we've put in there.

But that's not how the brain works.

Every time we pull up a memory, we make it stronger and more lasting, so that testing doesn't just measure, it changes learning.

Simply reading over materials to be learnt, or even taking notes and making outlines, as many homework assignments require, doesn't have this effect.

Language learner: 80% retained.
Science: 50% retained.

Self-quizzing (focus less on input of knowledge by passive reading, focus more on output by calling out that same information from brain.)

Cognitive disfluency:
Tough topic, recall better.
Interleaved assignment: mix up different kinds of problems instead of grouping by type.

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)

Beautiful Trigonometry

Beautiful Trigonometry
tan α +tan β +tan γ = tan α.tan β.tan γ

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.

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.

IMO Technique

(a+b)³ = a³ + 3a²b+ 3ab² + b³

Different equivalent forms:
(1)：(a+b)³ = a³ + b³+3ab(a+b)
(2)：a³ + b³ = (a+b)³ - 3ab(a+b)
(3): a³ + b³ = (a+b)(a² -ab + b²)
(4)：(a+b)³ - ( a³ + b³ ) = 3ab(a+b)

1997 USAMO Q5:
Prove:
$latex \frac{1}{a^{3}+b^{3}+abc} +
\frac{1}{b^{3}+c^{3}+abc} +
\frac{1}{c^{3}+a^{3}+abc} \leq
\frac{1}{abc}$

Proof:
Apply (3):
a³ + b³ = (a+b)(a² -ab + b²) ≥ (a+b)ab

Note:
a² -ab + b²= (a-b)² + ab ≥ ab
since (a-b)² ≥ 0

$latex \frac{abc}{a^{3}+b^{3}+abc}
\leq \frac{abc}{(a+b)ab + abc}
= \frac{c}{a+b+c}$


Symmetrically,
$latex \frac{abc}{b^{3}+c^{3}+abc}
\leq \frac{a}{a+b+c}$

$latex \frac{abc}{c^{3}+a^{3}+abc}
\leq \frac{b}{a+b+c}$

Add 3 RHS:
$latex \frac{a+b+c}{a+b+c} = 1$

$latex \frac{abc}{a^{3}+b^{3}+abc} +
\frac{abc}{b^{3}+c^{3}+abc} +
\frac{abc}{c^{3}+a^{3}+abc} \leq 1$

$latex \frac{1}{a^{3}+b^{3}+abc} +
\frac{1}{b^{3}+c^{3}+abc} +
\frac{1}{c^{3}+a^{3}+abc} \leq
\frac{1}{abc}$

[QED]

Multi-variable Limit

Analysis is the study of Functions, using the main tool "Limit".

Limit problems appear in:
1. Continuity
2. Derivative
3. Integral
4. Sequence

Multi-variable Functions have different approach of Limit compared to Single-value functions.
Eg. L'Hôpital Rule is not applicable to Multi-variable Functions.

Case 1: Find the Limit (L) of
$latex \displaystyle f(x,y)= \frac{xy}{x^{2}+y^{2}} \text{ at point (0,0)} $

Solution:
Consider the point P(x,y) on f(x,y)
$latex \displaystyle \lim_{x\to 0} f(x,y)=f(0,y)= 0 \text{...(1)}$
$latex \displaystyle\lim_{y\to 0} f(x,y)=f(x,0)=0 \text{...(2)}$

but when P moves along y=x straight line approaching (0,0),
ie. x->0, y=x->0,
$latex \displaystyle\lim_{x\to 0} \frac{xy}{x^{2}+y^{2}}= \frac{x^2}{2.x^2} = \frac{1}{2} \text{...(3)}$

From (1),(2),(3) there are 3 limits {0, 0, 1/2}, hence the Limit L does not exist.

Case 2:
Find limit L of
$latex \displaystyle f(x,y)= \frac{x^{2}+y^{2}}{x^{4}+y^{4}}
\text { at point} (\infty, \infty)$

Solution:
Let y= kx
$latex \displaystyle f(x,y)= \frac{x^{2}+k^{2}y^{2}}{x^{4}+k^{4}y^{4}} = \frac{1+k^{2}}{(1+k^{4})x^2}$
When x -> $latex \infty$,
f(x,y) independent of k => possible limit of 0

Prove: $latex \displaystyle\lim_{(x,y)\to \infty} f(x,y)= L = 0$

$latex \displaystyle \left| \frac{x^{2}+y^{2}}{x^{4}+y^{4}} - 0 \right| = \frac{x^{2}+y^{2}}{x^{4}+y^{4}}
\leq \frac{x^{2}+y^{2}}{2x^{2}y^{2}} = \frac{1}{2x^{2}} + \frac{1}{2y^{2}}$

Note:
$latex (x^{2} - y^{2})^2 \geq 0$
$latex x^{4}-2x^{2}y^{2}+y^{4}\geq 0 $
$latex x^{4}+y^{4} \geq 2x^{2}y^{2}$

$latex \displaystyle\forall \epsilon > 0 \text{, take } \delta \geq \frac{1}{\sqrt{\epsilon}} \text{ such that}$
$latex |x|>\delta, |y|>\delta$
$latex \implies \displaystyle \left| \frac{x^{2}+y^{2}}{x^{4}+y^{4}} - 0 \right| \leq \frac{1}{2x^{2}} + \frac{1}{2y^{2}}
< \frac{1}{2{\delta}^{2}}+\frac{1}{2{\delta}^{2}}$

$latex \implies \displaystyle \left| \frac{x^{2}+y^{2}}{x^{4}+y^{4}} - 0 \right| \leq \frac{1}{2}\left({\frac{1}{\sqrt{\epsilon}}}\right)^{-2}+\frac{1}{2}\left({\frac{1}{\sqrt{\epsilon}}}\right)^{-2} = \frac{\epsilon}{2}+ \frac{\epsilon} {2}$
$latex \implies \displaystyle \left| \frac{x^{2}+y^{2}}{x^{4}+y^{4}} - 0 \right| \leq \epsilon$
$latex \implies \displaystyle\lim_{(x,y)\to \infty} f(x,y)= L =0 $
[QED]

Source: Prof Zhang ShiZao 章士藻(1940-) "Collected Works of Mathematics Education" 数学教育文集
for Ecole Normale Supérieure in China 高等师范学院

Generalized Analytic Geometry

Generalized Analytic Geometry

Find the equation of the circle which cuts the tangent 2x-y=0 at M(1,4), passing thru point A(4,-1).

Solution:

1st generalization:
Let the point circle be:
(x-1)² + (y-4)² =0

2nd generalization:
It cuts the tangent 2x-y=0
(x-1)² + (y-4)² +k(2x-y) =0 ...(C)

Pass thru A(4,-1)
x=4, y= -1
=> k= -2
(C): (x-3)² + (y-1)² = 0
[QED]

Galois Theory Simplified

Galois discovered Quintic Equation has no radical (expressed with +,-,*,/, square root) solutions, but his new Math "Group Theory" also explains:
$latex x^{5} - 1 = 0 \text { has radical solution}$
but
$latex x^{5} -x -1 = 0 \text{ has no radical solution}$

Why ?

$latex x^{5} - 1 = 0 \text { has 5 solutions: } \displaystyle x = e^{\frac{ik\pi}{5}}$
$latex \text{where k } \in \{0,1,2,3,4\}$
which can be expressed in
$latex x= cos \frac{k\pi}{5} + i.sin \frac{k\pi}{5} $
hence in {+,-,*,/, √ }
ie
$latex x_0 = e^{\frac{i.0\pi}{5}}=1$
$latex x_1 = e^{\frac{i\pi}{5}}$
$latex x_2 = e^{\frac{2i\pi}{5}}$
$latex x_3 = e^{\frac{3i\pi}{5}}$
$latex x_4 = e^{\frac{4i\pi}{5}}$
$latex x_5 = e^{\frac{5i\pi}{5}}=1=x_0$

=>
$latex \text {Permutation of solutions }{x_j} \text { forms a Cyclic Group: }
\{x_0,x_1,x_2,x_3,x_4\} $

Theorem: All Cyclic Groups are Solvable
=>
$latex x^{5} -1 = 0 \text { has radical solutions.}$

However,
$latex x^{5} -x -1 = 0 \text{ has no radical solution } $
because the permutation of solutions is A5 (Alternating Group) which is Simple
ie no Normal Subgroup
=> no Symmetry!

Life Changing Book

The book which changed their life:
1. GH Hardy: by Carmille Jordan's Cours d'Analyse:
"I shall never forget the astonishment with which I read the remarkable work ... and I learnt for the first time as I read it what mathematics really meant."

2. Ramanujan : George Carr's
"A Synopsis of Elementary Results in Pure & Applied Mathematics"
(4,400 results without proofs)

3. Riemann : Legendre's book

4. Hardy/Littlewood:
Landau 2-volume "Handbuch der Lehre von Der Verteilung der Primzahlen"
(Handbook of the Theory of the Distribution of Prime Numbers)

5. Atle Selverg (Norway): Ramanujan's "Collected Papers"

Note: This blogger's mathematics 'fire' is rekindled by John Derbyshire's "Unknown Quantity"

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.

Proofs of Isomorphism

Two ways to prove f is Isomorphism:

1) By definition:
f is Homomorphism + f bijective (= surjective + injective)

2) f is homomorphism + f has inverse map $latex f^{-1}$

Note: The kernel of a map (homomorphism) is the Ideal of a ring.

Two ways to construct an Ideal:
1) use Kernel of the map
2) by the generators of the map.

Two ways to prove Injective:

1) By definition of Injective Map:
f(x) = f(y)
prove x= y

2) By Kernel of homomorphism:
If f is homomorphism
$latex f: A \mapsto B$
Prove Ker f = {0}

Note: Lemma:
$Latex \text{f is injective} \iff Ker f = {0}$

Proof Isomorphism 4 Steps:
1. Define function f：S -> T
Dom(f) = S
2. Show f is 1 to 1(injective)
3. Show f is onto (surjective)
4. Show f(a*b) = f(a). f(b)

Example:
Let T = even Z
Prove (Z,+) and (T,+) isomorphic
Proof:
1. Define f: Z -> T by
f(a) = 2a
2. f(a)=f(b)
2a=2b
a=b
=> f injective

3. Suppose b is any even Z
then a= b/2 ∈ Z and
f(a)=f(b/2)=2(b/2)=b
=> f onto

4. f(a+b) =2(a+b) =2a+2b =f(a)+f(b)
Hence (Z,+) ≌ (T,+)

Isomorphism (≌)
1. To prove G not isomorphic to H:
=> Prove |G| ≠ |H|

2. Isomorphism class = Equivalence Class
B’cos “≌” is an Equivalence relation.

3. Properties of Isomorphism (≌):
i) match e
$latex \phi(e) = e'$
ii) match inverse
$latex \phi (g^{-1} )= (\phi (g))^{-1}$

iii) match power
$latex \phi(g^{k})= (\phi(g))^{k}$

Visualize Isomorphism

This is the Chinese graphical way to visualize Abstract Algebra:

Third Theorem of Isomorphism





Second Theorem of Isomorphism

Group G with normal subgroup N,
H subgroup of G (H ≤ G ).
=>
1.(H ∩ N) normal subgroup of H;
2. HN/N ≌ H /(H ∩ N)