Math Blog

The Big Internet Math-Off, Round 1 – James Tanton v Nira Chamberlain | The Aperiodical

http://aperiodical.com/2018/07/the-big-internet-math-off-round-1-james-tanton-v-nira-chamberlain/

https://www.forbes.com/sites/kionasmith/2018/06/30/sundays-google-doodle-celebrates-mathematician-gottfried-wilhelm-leibniz/

My favorite mathematician is German Leibniz, who co-invented Calculus with Newton.

Today we thank Leibniz for his elegant Calculus symbols:

$latex boxed {frac{dy}{dx}}&s=3&fg=aa0000$

$latex boxed{int_{0}^{infty}x^{n}e^{-x}dx}&s=3&fg=aa0000$

Leibniz also invented 01 binary algebra, which he later found it was already in the 3,000-year-old Chinese “Yin-Yang” (阴阳 八卦), so impressed that he recommended to the most powerful western (French) king Louis XIV (14th) to use Chinese as the Universal Language of the world.

The rich Newton sued Leibniz for plagiarism of Calculus, until Leibniz died poor in bankruptcy, buried in a common unknown grave.

The war between Newton & Leibniz extended & lasted 100 years between UK Math Community and Continental Europe Math Community. As a result UK lost its math leadership after Newton, France (Lagrange, Fourier, Cauchy, Galois… ) followed by Germany (Felix Klein, Gauss, Hilbert, Riemann …) took over as the world…

Keywords:

1. Fermat’s Last Theorem (FLT):$latex boxed {x^{n} +y^{n} = z^{n} ; ; forall n >2 }&s=3&fg=aa0000$
2. Pierre de Fermat (France 17CE): FLT Conjecture or Prank ?
3. Euler (n= 3)
4. Taniyama-Shimura-Weil Conjecture: Modular Form = Elliptic Curve
5. Galois Group Symmetry
6. Andrew Wiles (UK Cambridge 1994): (Modular Form = Elliptic Curve) = FLT (q.e.d.)

Notes: Do not confuse Prof Andrew Wiles (proved FLT) with (French/American) Prof André Weil (Founder of Bourbaki School of Modern Math in POST-WW2 universities worldwide) who proved (4) laying the ground work for FLT.

Science Friday: How Abstract Math Can Analyze Social Injustice.

Dr. Eugenia Cheng

https://www.sciencefriday.com/segments/how-abstract-math-can-analyze-social-injustice/

[Sound]:

https://m.soundcloud.com/scifri/how-abstract-math-can-analyze-social-injustice

Interview: Yoneda Lemma

Worldcup 2018

Surprising Result:

World No.1 (Germany) Lost to World No.59 (Korea) – why ?

Reason: Math !

For weak Korea to win, the best strategy is to keep the goals number as low as possible with strongest defence.

Note : 2017 International Math Olympiad (IMO) World Champion Team was Korea.

The actual Game proved the Korean “Math” strategy was right: (2:0)

The “war” of OOP vs FP is akin to Applied Math vs Pure Math.

The formers (OOP & Applied Math) are not “rigourous” but practical, compared to the laters (FP & Pure Math) which are elegant but too abstract for popular usage.

OOP: SmallTalk and its followers – C++, C#, Objective-C, Java…

FP: LISP and its followers – Haskell, Clojure, …

The “hybrid” (OOP&FP): Scala, Kotlin (Google: Java ‘cousin’), Swift (Apple: Objective -C ‘cousin’), F# (MicroSoft)

The “cons” of OOP, which are bad for concurrency / parallel computing in multi-cored CPU:

1. State changed
2. Side-effect
3. Mutability of data
4. Polymorphism

https://blog.cleancoder.com/uncle-bob/2014/11/24/FPvsOO.html

Landau’s book “Symmetry” explains it as follow:

Automorphism = Congruence= 叠合 has
1). Proper 真叠合 (symmetry: left= left, right = right)
2). Improper 非真叠合 (non-symmetry: reflection: left changed to right, vice-versa).
Congruence => preserve size / length
=> Movement 运动 (translation 平移, rotation about O )
= Proper congruence (Symmetry)

In Space S, the Automorphism that preserves the structure of S forms a Group Aut(G).
=> Group Aut(G) describes the Symmetry of Space S.

Hence Group is the language to describe Symmetry.

 

Symmetry (hence Group) explains :
1. Conservation of Energy;
2. Conservation of Angular Momentum;
3. Periodic Table;
4. Laws of Thermodynamic.

Emmy Noether Theorem (1918): Conservation Laws owes to Symmetry :
1. In Linear motion
=> Conservation of Momentum

2. In Angular movement
=> Conservation of Angular Momentum

3. In Time
=> Conservation of Energy

https://www.sciencenews.org/article/emmy-noether-theorem-legacy-physics-math

It is not only owed to the Ecole Normale Superieure where the 11 Fields medalists were educated, but the prominent “Math Culture” in French society.

This is similar to the International Math Olympiad (IMO) “craze” in China since 1980s till today, where the parents send their primary school kids to drill in IMO boot- camps, because that is a “direct-entry” gateway to enter top university, bypassing the highly competitive “Gao-kao” 高考 (High-school College Entrance Exams for 500,000 places among 7 million students each year).

https://www.theguardian.com/world/2012/may/31/europa-french-maths-ecoles-normales-superieures

Quora: read how this middle-age (48) French recounts his “French Math” education since 6 to 15 years old:

The curriculum was designed in the late 60s in part by a group of real mathematicians, the Bourbaki. It was very abstract. I learned about basic set theory when I was 6. I did learn about the basic operators but not before I was able to perform…

中国剩余定理CRT (Chinese Remainder Theorem)
X ≡ 2 (mod 3)
X ≡ 3 (mod 5)
X ≡ 2 (mod 7)
Solve X?
明. 程大位 “算法统宗” (1593)
3人同行70稀
5树梅花21支
7子团圆半个月(15)
除百零五(105)便得知
Let remainders:
$Latex r_3=2, r_5=3, r_7=2$
$Latex r= r_3.70+ r_5.21 + r_7.15 (mod 3.5.7)$
r= 2.70 +3.21 +2.15 (mod 105)
r= 140 +63 +30 (mod 105)
r= 233 (mod 105)
$latex r= 23 = x_{min}$
or X= 23 +105Z (23 + multiples of 105)

——————————————-
CRT: Why 3:70, 5:21, 7:15
X ≡ 2 (mod 3)
X ≡ 3 (mod 5)
X ≡ 2 (mod 7)

1) Find A such that
A ≡ 1 (mod 3)
A ≡ 0 (mod 5)
A ≡ 0 (mod 7)

=> 5|A, 7|A => 35 |A

Chinese Remainder Theorem (CRT): 韩信点兵 (中国剩余定理)

南北朝 《孙子算经》(4 AD)

中国南北朝 : 《孙子算经》(4AD) 鸡兔同笼问题 (Chicken – Rabbits Problem)

Problem:
The Orchestra conductor has been asked to conduct a series of concert for 2 consecutive months.
The Orchestra has to play few pieces of classical music with the following conditions requested by the organising committee:

1) Play at least 1 piece of music a day;
2) Avoid playing exactly the same music in 2 consecutive days (although partially is allowed)
3) Due to resource constraint, it is impossible to play all pieces of music within a single day.

Question:
At least how many (N) pieces of music the conductor has to prepare in order to fulfill these conditions?

Solution:
Using Yijing （易经）notation:
* a piece of music being played in a day is denoted by
__ (solid line);
while not being played is denoted by
– – (broken line)
* 2 consecutive months in any year have maximum 62 (= 31+31) days.
* A bagua (8…

https://people.cs.pitt.edu/~kirk/cs1501/Pruhs/Spring2006/assignments/editdistance/Levenshtein%20Distance.htm

The wonderful iMac is a “long life” machine able to last more than 10 years, but it has an annoying weakness : the frequent spinning “beachball” which slows down or freezes the computer.

Below is a very simple trick to fix the problem.

Note: to get the “Force Quit Application” Screen: press 《Command + Option + Esc》3 keys simultaneously.

Key Points:

• 1900 Hilbert’s 17th Conjecture: Non-negative Polynomial <=> sum of 2 squares (Proved by Emile Artin in 1927)
• Computing Math : approximate by optimisation with “Linear Programs” which are faster to compute.
• Princeton Mathematicians applied it to self – driving cars.

https://www.wired.com/story/a-classical-math-problem-gets-pulled-into-self-driving-cars/amp

Explain:

Sum of 2 Squares <=> always non-negative ( ≥0)

$latex 13 = 4 + 9 = 2^{2} + 3^{2} $

$latex P (x) = 5x^2+16x+13 = (x+2)^{2} + (2x+3)^{2} geq 0 $

Self – driving Car: Trajectory = P (x)

P(x) < 0 where the car’s position in the trajectory;

Obstacles are positions where P (x) ≥ 0.

https://www.scientificamerican.com/article/maria-agnesi-the-greatest-female-mathematician-youve-never-heard-of/

https://www.smithsonianmag.com/science-nature/18th-century-lady-mathematician-who-changed-how-calculus-was-taught-180969078/

George Boole [2/11/ 1815 – 8/12/ 1864]: 《The Laws of Thought》: symbolic logic representation of thought.

Let x = class of sheep’s

y = white

=> white sheep = xy = yx = sheep white

then Commutativity Law:

$latex boxed {xy = yx}&fg=aa0000&s=3 $

Let x= rivers, y = estuaries河口, z= navigable 通航

then, AssociativityLaw:

$latex boxed {(xy)z= x(yz)}&fg=aa0000&s=3 $

A sheep is a sheep,

$latex boxed {xx = x^{2} = x}&fg=aa0000&s=3 $

Note: x = 0 or 1 fulfills the above equation.

If x = class of men

y = class of women

z = class of adults (either men or women)

$latex boxed {z = x + y}&fg=aa0000&s=3 $

w = European

then Distributive Law:

$latex boxed {w(x+y) = wx + wy}&fg=aa0000&s=3 $

If t = Chinese

then all non-Chinese men = {x – t}

If s = Singaporean,

then

$latex boxed {s(x – t…

https://simple.m.wikipedia.org/wiki/Zipf%27s_law

Example:

In English, 3 most common words:

1. “the” : occurring 7% of the time;
2. “of” : 3.5% = 7/2
3. “and” : 2.8% ~ 7/3

=> “the” is 2x occurs more often than the 2nd “of“, 3x than the 3rd “and” …

Zipf’s Law : the frequency of the nth ranked word is proportional to 1/n.

Reference:

https://www.researchgate.net/publication/220469172_Mandelbrot’s_Model_for_Zipf’s_Law_Can_Mandelbrot’s_Model_Explain_Zipf’s_Law_for_Language

Hadamard estimated that :

About 90% of mathematicians thinkvisually, 10% think formally.

Usually, they think in steps:

1. Get the right idea, often think vaguely about structural issues, leading to some kind of strategic vision;
2. Tactics to implement it;
3. Rewrite everything in formal terms to present a clean, logical story. (Gauss’s removal of ‘scaffolding’ – middle working steps)

Source: [NLB #510.922]

“A mathematician is a machine for turning coffee into theorems.”

– Alfréd Rényi

It’s a pun in German, where the word Satz means both ‘theorem’ and ‘(coffee) grounds‘ [咖啡渣].

How likely is it that a mathematics student can’t solve IMO problems?

Is there a fear of embarrassment in being a math Ph.D. who can’t solve problems that high-school students can? by Cornelius Goh

https://www.quora.com/How-likely-is-it-that-a-mathematics-student-cant-solve-IMO-problems-Is-there-a-fear-of-embarrassment-in-being-a-math-Ph-D-who-cant-solve-problems-that-high-school-students-can/answer/Cornelius-Goh?share=311c5a88&srid=oZzP

Problem A3

A function f is defined on the positive integers by:

for all positive integers n,

$latex f(1) = 1 $
$latex f(3) = 3$
$latex f(2n) = f(n)$
$latex f(4n + 1) = 2f(2n + 1) – f(n) $
$latex f(4n + 3) = 3f(2n + 1) – 2f(n) $

Determine the number of positive integers n less than or equal to 1988 for which f(n) = n.

What is the explanation of the solution of problem 3 from IMO 1988? by Alon Amit

https://www.quora.com/What-is-the-explanation-of-the-solution-of-problem-3-from-IMO-1988/answer/Alon-Amit?share=7719956f&srid=oZzP

Encryption & Decryption: ECC (Elliptic Curve Cryptography):

Sending End: Encryption

1) SHA algorithm generates “Digital Signature” ;

2) Generate random “Private Key”.

第3-6步骤:

3) ECC encrypts the text with “Private Key”;

4) From the Private Key generates a “Public Key”;

5) Send out the “original message” and the “Public Key” with the “encrypted message” from 3);

Receiving End: Decryption

6) ECC with Public Key generates Digital Signature 1 (S1);

7) Use SHA algorithm on the original message generates Digital Signature 2 (S2);

8) If S1 = S2, then accept transaction, otherwise reject.

https://mp.weixin.qq.com/s/cLhycZBxkcl5oYNDsElUTg

Let N = 2n > 6

哥德巴赫猜想 Conjecture “1+1”: N = p1 + p2 (pj all primes)

陈景润 Chen Theorem “1+2”: N = p1+ p2.p3

The 7 Fields Medalists are:

2014 – Maryam Mirzakhani (1977-2017) – 1st lady Fields medalist

2010 – Cédric Villani (1973- )

2006 – Grigori Perelman (1966- ) – 1st declined the award

1998 – Andrew Wiles (1953- ) [silver plaque] – Fermat’s Last Theorem

1990 – Edward Witten (1951- ) – Physicist won Fields medal

1982 – Alain Connes (1947- ) – Quantum Theory

1966 – Alexander Grothendieck (1928-2014) – Hermit mathematician

https://www.newscientist.com/article/2166283-7-mathematicians-you-should-have-heard-of-but-probably-havent/

http://www.thecrimson.com/article/2018/4/27/bill-gates-event/

Bill Gates, a top Math student at Harvard entrance exams, recalled his first year Harvard “Math55” Course (Advanced Calculus & Linear Algebra) – the toughest at his time because 4 years of Math coursewares condensed into 1 year (2 semesters) !

Note: Harvard “Math55” is even tougher than the “notorious” French Classe Préparatoire, which is a 3-year Math undergraduate courseware squeezed in 2 years : 1st year (code-name “un-demi” or “1/2”) Mathématiques Supérieures; 2nd year (“trois-demi” or “3/2”) Mathématiques Spéciales.

Math55 Syllabus:
Though Math 55 bore the official title “Honors Advanced Calculus and Linear Algebra”, advanced topics in complex analysis, point set topology, group theory, and/or differential geometry could be covered in depth at the discretion of the instructor, in addition to single and multivariable real analysis and abstract linear algebra. In 1970, for example, students studied thedifferential geometryofBanach…

After 40 years of learning Abstract Algebra (aka Modern Math yet it is a 200-year-old Math since 19CE Galois invented Group Theory), through the axioms and theorems in math textbooks and lectures, then there is an Eureka “AHA!” revelation when one studies later the “Category Theory” (aka “Abstract Nonsense”) invented only in 1950s by 2 Harvard professors.

A good Abstract Math teacher is best to be a “non-mathematician” , who would be able to use ordinary common-sense concrete examples to explain the abstract concepts: …

Let me explain my points with the 4 Pillars of Abstract Algebra :

$latex boxed {text {(1) Field (2) Ring (3) Group (4) Vector Space}}&fg=aa0000&s=3$

Note: the above “1-2-3 & 4″ sequence is a natural intuitive learning sequence, but the didactical / pedagogical sequence is “3-2-1 & 4″, that explains why most students could not grasp the philosophical essence of Abstract Algebra…

Topology (2nd Edition)

This book is the best introductory book on Topology, an upper undergraduate/graduate course taken in university. I have written a short book review on it.

Excerpt:

Book Review: Topology
Book’s Author: James R. Munkres
Title: Topology
Prentice Hall, Second Edition, 2000

It is often said that one must not judge a book by its cover. The book with a plain cover, simply titled “Topology”, is truly a rare gem and in a class of its own among Topology books.

One striking aspect of the book is that it is almost entirely self-contained. As stated in the preface, there are no formal subject matter prerequisites for studying most of the book. The author begins with a chapter on Set Theory and Logic which covers necessary concepts like DeMorgan’s laws, Countable and Uncountable Sets, and the Axiom of Choice.

The first part of the book is on General…

“Form” : Function with special properties – eg.

• Space Forms: manifolds with certain shape.
• Quadratic Forms (of weight 2): $latex x^2+3xy+7z^2 $
• Cubic Forms (of weight 3): $latex x^3+{x^2}y + y^3 $
• AutomorphicForms (particular case: ModularForms): 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: $latex boxed {int_{L} frac {ds}{y}}&fg=aa0000&s=2$
  where L the “line” segment (the arc of the semicircle or the vertical segment) and $latex ds^2 = dx^2+dy^2$

1.2 Group of Non-Euclidean Motions:
$latex f: H rightarrow H$

1. Translation: $latex z rightarrow {z + b} quad forall b in mathbb {R}$
2. Dilation: $latex z rightarrow {az } quad forall a in mathbb {R^{+}}$
3. Inversion: $latex…