Author: tomcircle

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…

“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…

1. Python (Dutch Guido van Rossum, 1956)
2. Java (Canadian James Gosling 1955)
3. Javascript (USA Brendan Eich, 1961)
4. C (USA Dennis Ritchie, 1941 – 2011 )
5. C++ (Denmark Bjarne Stroustrup, 1950)
6. Ruby (JAPAN Yukihiro “Matz” Matsumoto, 1965)
7. Perl (USA Larry Wall, 1954)
8. Pascal (Switzerland Niklaus Wirth, 1934)
9. Lisp (USA John McCarthy, 1927 – 2011)
10. PHP (Denmark Rasmus Lerdorf, 1968)

https://www.technotification.com/2018/04/programming-languages-creators.html

Below the 3 hotest Functional Programming language influenced by Lisp:

11. Kotlin(Russia Andrey Breslav)

12. Scala (USA Martin Odersky)

13. Haskell (USA)

14. Clojure (USA Rich Hickey)

https://www.cut-the-knot.org/proofs/PegsAndGroups.shtml

http://time.com/5224618/bill-gates-hans-rosling-factfulness/

Prime Numbers since Euclid 2300 years ago remains a mystery today…

Riemann’s Hypothesis is the Top Conjecture Unsolved in 21st Century.

https://theconversation.com/why-prime-numbers-still-fascinate-mathematicians-2-300-years-later-92484

Key Words: 250 years anniversary

• Yesterdays: Fourier discovered Heat is a wave , Fourier Series, Fourier Transformation, Signal processing…
• Today: IT imaging JPEG compression, Wavelets, 3G/4G Telecommunications, Gravitational waves …
• Friends / bosses: Napoleon, Monge… Egypt Expedition with Napoleon Army.
• Taught at the newly established Military Engineering University “Ecole Polytechnique”.
• Scientific Research: Short period but intense.
• Before Fourier died (wrapped himself in thick carpet in hot summer), he was reviewing another young Math genius Evariste Galois’s paper on “Group Theory”.

https://news.cnrs.fr/articles/joseph-fourier-is-still-transforming-science

Lycée Louis-Le-Grand is the best high school (lycée) for Math in France – if not in the world – it produced many world-class mathematicians, among them “the Father of Modern Math” in 19th century the genius Evariste Galois. (See also: Unknown Math Teacher produced two World’s Math Grand Master Students ), Molière, Victor Hugo, 3 French Presidents, etc.

Its Baccalaureat (A-level) result 100% passed with 75% scoring distinctions. Each year 1/4 of Ecole Polytechnique (France Top Engineering Grande Ecole) students come from here.

More surprisingly, the “Seconde” (Secondary 4) students learn Chinese Math since 6ème (PSLE Primary 6).

Note: Louis Le Grand (= Louis 14th). He sent the Jesuits (天主教的一支: 耶稣会传教士) as the “French King’s Mathematicians”(eg. Bouvet 白晋) to the 16-year-old Chinese Emperor (康熙) KangXi’s Court.

Senior Wrangler is the First position in the Math Tripos in Cambridge. Singapore Prime Minister Lee Hsien Loong was the Senior Wrangler in 1973, the first Singaporean student with such great honors, among other senior wranglers like Arthur Cayley (Group Theory), J.J. Sylvester (Inventor of Matrix, private tuitor of the “inventor of Nursing” Florence Nightingale), J.E. Littlewood (partnered in a twin research team with G.H. Hardy), Frank Ramsey (Ramsey’s Theorem), Stokes, Pell, etc.

Some great mathematicians like Bertrand Russell (Logician, Nobel Litterature Prize) , G.H. Hardy (20th century greatest Pure Mathematician, mentored 2 geniuses: Indian Ramanujian and Chinese Hua Luogeng 华罗庚*) were not Senior Wrangler. Prof Hardy hated Math Tripos syllabus (revealed in his autobiography: “A Mathematician’s Apology“).

1914 Brian Charles Molony
1923 Frank Ramsey
1928 Donald Coxeter
1930 Jacob Bronowski
1939 James Wilkinson
1940 Hermann Bondi
1952 John Polkinghorne
1953 Crispin Nash-Williams
1959 Jayant Narlikar
1970 Derek…

Three Key Ideas:

1. Cloud of Points
2. Filter function: f (x,y,z) –> x
3. Cluster: Overlapping bins
4. Draw network:

• Vertices: clusters
• Edges: connecting lines between clusters.

Key Point:

Haskell & any FP compiler don’t check the Category Theory proof if your codes (eg. fmap) follow Functor’s Laws (eg. Preserve structure, identity) or Monad’s Laws !

https://www.quantamagazine.org/gunter-ziegler-and-martin-aigner-seek-gods-perfect-math-proofs-20180319/

http://www.thehindu.com/sci-tech/science/canadian-mathematician-langlands-wins-abel-prize/article23304119.ece/amp/

See also: Langland’s Program

Michael Bishop (Nobel Medicine 1989)
School Science subjects should be taught in this sequence (unless in single ‘Combined Science’):
1st Physics
2nd Chemistry
3rd Biology

Key Points:

1. The Foundation of Science has changed since Newtonian Era, except Biology / Medicine and Psychology which don’t keep up:

Math => Fractal Math

Newton Physics (Matter)=> Quantum Physics (Energy)

(Organic) Chemistry => Electro-Chemistry

Energy = Field = 气 Chi / Qi

3. Environment -> Mind -> Perception -> Genetic Control

4. Consciousness (5%), Subconscious-ness (95%)

Examples:

• Driving car: inexperienced driver (consciousness), experienced driver (subconscious).
• Dating (conscious behavior)

Note: The successful DeeplearningAI is using the concept of Perception called “Perceptrons” by analysing the Environmental (BIG DATA) pattern with the help of Mathematics (Calculus : Gradient Descent, Statistics : Bayesian Probability, Algebraic Topology, etc).

Most popular AI Deeplearning Book by MIT :

[English]

http://www.deeplearningbook.org/

[中文版]

https://exacity.github.io/deeplearningbook-chinese/

…
https://dev.to/drminnaar/choosing-a-programming-language-493h

…

As software becomes more complicated for high-speed trains, driverless cars, missile weapons. .. and AI deeplearning algorithm, we can’t depend our life safety on the programmers who don’t understand the advanced maths behind these algorithms.

The Advanced Math is the Category Theory – the most advanced math foundation above “Set Theory” since WW2. Functional Programming is based on Category Theory with mathematical functions – always output correctly with no “side-effects”.

https://www.extremetech.com/computing/259977-software-increasingly-complex-thats-dangerous

https://blog.hackerrank.com/which-country-would-win-in-the-programming-olympics/

The result is surprising … China, Russia, Poland are the top 3 countries based on the tests, not the usual expected countries such as USA, India, UK.

Reason: Mathematics ! China, Russia and Poland are strong in Advanced Math education in university. Functional Programming requires the Advanced Math eg. Category Theory.

1. Twin Primes Conjecture : there are infinitely many pairs of Twin Primes among the infinitely many prime numbers.

Note: 2013 Zhang YiTang proved the gap between the twin primes is no more than 70,000,000 [Terence Tao’s Polymath Project using Zhang’s method to further narrow the gap to 246 ]

2. Goldbach’s Conjecture

Chen’s “1+2” Theorem

3. Palindromic Primes (eg.11, 101, 16561)

4. Riemann Hypothesis

https://www.sciencealert.com/prime-number-theorems-conjectures-explained