String diagrams (1 – 5)

2. **Natural Transformation:**

3. **Adjunction**

String diagrams (1 – 5)

2. **Natural Transformation:**

3. **Adjunction**

__Old Hardware__

- iMac 8,1 (2008), Intel Core 2 Duo, 2.4 GHz,
- Memory (2GB DDR2 SDRM 800 MHZ), upgraded to 6 GB (2017)
- OS with CD : Leopard (10.5.4), upgraded via purchased CD “Snow Leopard” (10.6.3), subsequent few more upgrades (via App Store) till (lastly) El Capitan 10.11.6 (last crashed here)

**Useful Keyboard Commands (hold all keys together while start-up /powering on, then release after seeing Apple Log****o**)

- <
**OPTION**> (aka <**Alt**>) – “**Start-Up Manager**“: choose the selected Start-up drive to boot. (Macintosh HD is the default, if it is corrupted, then likely the Recovery HD too. ) - <
**SHIFT**>:**Safe mode**, do some simple repairs; if ok, exit manually by restart. - <
**COMMAND**><**R**> /<**COMMAND**><**R**><**OPTION**>:**Recovery**Mode (<**Option>**is via**Internet**, screen right-hand-top corner “Wi-Fi” must set on). It will show Mac OS Installation Menu: (1…, 2.Reinstall OS…4. Disk Utility).

When Mac screen frozen in blank/grey / white / spinning rainbow Beachball / spinning wheel, the equivalent of “Control-C” in Microsoft windows is:

- <
**COMMAND**><**Esc**><**OPTION**>: This is Forced “Kill” the culprit app when screen hangs. The “Finder” will pop up, then “Force Quit” it. To kill the**foremost**culprit app, press additional <**SHIFT**> together with these 3 keys.

**How do you know the Start Up HD fails:**

Trying the above key commands in vain;

Try the various hardware Reset tricks in vain:

- Reset SMC
- Reset PRAM (after replace Hard disk)
- Reset NVRM
- etc

Watch this very well explained video series (Part 1 & 2) which cover ** ALL** tricks to fix common problems : (If still fail, go to next section ”

__Life Saver Tools :__

1. Bootable Mac OS USB Drive (with Mac OS Installer = OS Recovery Utilities + Mac OS)

2. External Back Up Drive (Time Machine)

** Notes**:

1. Mac OS Generations:

Leopard [CD] (10.5.4 – 10.5.8)

Snow Leopard [CD] (10.6.3 – 10.6.8) : first time introduced “*App Store”, **built-in at Mac Apple menu (drop down).*

Lion (10.7) – upgrade via App Store

Mountain Lion (10.8) – upgrade via App Store

Below free Mac OS since:

Maverick (10.9)

Yosemite (10.10)

**El Capitan (10.11 – 10.11.6)**

Sierra (10.12)

High Sierra (10.13)

Mojave (10.14)

Picture Order (From Left) :1. 2. 3. 4.

**1. ****Caucher Birkar**** ****(UK / Kurdish – Iran, 40**)

https://www.bbc.com/news/science-environment-45032422

**2. Alessio Figalli (Italy, 34**)

https://www.quantamagazine.org/alessio-figalli-a-mathematician-on-the-move-wins-fields-medal-20180801/

**3. Akshay Venkatesh (Australia / India, 36**)

Studies number theory and representation theory.

https://www.quantamagazine.org/fields-medalist-akshay-venkatesh-bridges-math-and-time-20180801/

**4. Peter Scholze (Germany, 30**)

Intersection between number theory and geometry

Keywords:

- Dijkstra, Edge Wyber (born 1930 Rotterdam)
- Goto is harmful
- Structures: sequence, selection, iteration

**Three Programming Paradigms**:

1. Structured Programming (1968 Dijkstra)

- Impose discipline on
**direct**transfer of control aka “*Goto*“. - “If/ then /else, do/while” control structures are structured.
- Language: Pascal, etc

2. Object-Oriented ‘OO’ (1966 Ole Johan Dahl & Kristen Nygaard)

- Impose discipline on
**Indirect**transfer of control (,*Polymorphism*,*Inheritance*‘**Encapsulation:**‘ function of class, it’s local variables =*constructor*variables).*instance* - Language: C++/ C# / Objective-C, Java / JavaScript, Go, Python, etc.

OO = Data + Function

=>

Class = Object + Method

3. Functional Programming ‘FP’ (1958 John McCarthy’s LISP language, based on Math “Lambda Calculus” from Alonzo Church 1936).

In LISP:

Data = Function

- Impose discipline upon
**assignment**(,*side effect*of data,*immutability*__Referential Transparency *)____.__ - Category Theory = Program : Monad, etc
- Language:
**Pure**FP: {Lisp, Clojure, Haskell},**Hybrid**OO+FP: {Scala, Kotlin}, etc.

4. Any more ?

All Programs can be constructed from just 3 structures (Böhm and Jacopini, 1966):

__Sequence / Selection / Iteration.__

Dijkstra’s Math Proofs for:

1. **Sequence** – by simple enumeration.

- Math Technique: trace the inputs of the sequence to the outputs of the sequence.

2. **Selection** – by reapplication of enumeration.

- Each path thru the selection was enumerated. If both paths eventually produced appropriate Math results, then the proof was solid.

3. **Iteration** – by induction.

- Proved the case for 1 by enumeration.
- Assume if N case was correct.
- Proved N+1 case correct by enumeration.
- Also proved the starting and ending criteria of the iteration by enumeration.

**Note** (*): **Referential Transparency** means – a function (f) with a given parameter always returns the same result.

Eg. Trigonometric function (f) = sin 30 = 0.5 (always! )

In FP, a program is many layers of composition of functions of function, with each function guaranteed (math proven) always returning the same result for given parameters (aka arguments). This is software safety with no surprising unexpected result due to side effects (like database search / Web search / IO output errors).

** Reference**:

Clean Architecture – A Craftsman’s Guide to Software Structure and Design (by Robert C. Martin)

[Singapore National Library NLB #004.22]

“Docker” – the latest IT revolution of “Containers” in OS Virtualization was created by a young French IT graduate of the Parisian (private) Grande Ecole “Epitech” in 2010 – Solomon Hykes (born 1983).

**Why we built Docker?**

**2014 Google “Kubernetes” embraces “Docker” in its Container Orchestration Automation tool:**

https://www.sumologic.com/blog/devops/kubernetes-vs-docker/

**Epitech Interview (in French):**

**One afternoon 5 friends rented 3 bikes from 5 p.m. to 6:30 p.m. and took turns to ride on them. At any time, 3 of them cycled while the other 2 friends rested. **

**If each of them had the same amount of cycling time, how many minutes did each person ride on a bike?**

**Note**: PSLE (Primary Schools Leaving Exams) is the Singapore National Exams for all 12 year-old pupils at Primary 6 year end. The result of which will determine which secondary school the pupil is qualified to enter the following year. Math subject, besides Science, English and mother tongue (Chinese or Malay or Tamil) are tested in PSLE.

[**Answer**] Try before you scroll down below ….

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Answer = each student rides 18 mins per bike (= 90 mins /5 ).

The “3” bikes are tricky “smokes” not relevant, it could be any “n” (<6) bikes , as long as total 90 mins, and each student rides same duration.

From Mainframe/Mini Server-based Monolithic Application since 80s, to

Browser-based thin-client-thick-Server Application from mid-90s, to

MicroServices-based Applications in 2018’s…

Small is beautiful !!

“Dinausaur” Monolithic Applications give way to Microservice-based Application.

Microservice is a realisation of “mini”-SAAS (Software As A Service).

Component-based Software a la hardware components VLSI is becoming a reality.

https://dzone.com/articles/convert-legacy-applications-into-future-proof-appl

When more **specific behavior** needed, use “interface”.

eg. Interface Vehicle: drive()

-> car.drive ();

-> motorcycle.drive ();

If it is a common **general behavior**, use **extension**.

https://blog.oozou.com/interface-in-kotlin-and-when-to-use-it-5d32caab19be

Why Monad:

1. Functional Programming:

“pure” function = no side effects

2. All applications without side effects are useless!

3. Monad wraps side-effect as if it is “pure” for FP composition.

https://dzone.com/articles/scalafp-lets-find-the-reasons-behind-monads

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/

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

Today we thank Leibniz for his elegant Calculus symbols:

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 center of math. After WW2 many German mathematicians (mostly Jewish eg. Noether, Gödel, Artin, …) fled to the USA which is now the Kingdom of Advanced Math.

__World Cup 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 ****goal**** number as ****low**** as po**ssible with strongest **defence**.

**Conclusion**: Highest chance for Korea to beat Germany is either 1 or 2 goals !

Germany lost by not launching a strong attack to score as many goals as possible.

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

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

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**

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:

- State changed
- Side-effect
- Mutability of data
- Polymorphism

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

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 (University Entrance Exams for 500,000 places among 7 million students each year, only 7% successful chance !! vs Singapore 40%).

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 aboutbasic set theory when I was 6. I did learn about the basic operators but not before I was able to perform them in arbitrary bases from 2 to 10. I was taught the properties ofordering relationship, equivalence classes and partitions at age 11,vector calculus at age 13. Basic differential calculus was taught at age 15, complex numbers and integral calculus at 17. This was carried over to physics where we were happily solvingODEs for mechanics and electrical circuits at age 16. This was for almost everyone except the most dyed-in-the-wool literary person. I was taught elementary proof techniques at age 13. By age 15 I knew about formal logic, contradictions and recursive proof.

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)

My wonderful iMac 8,1 (2008) is a “long life” machine able to last more than 10 years, but it has an annoying weakness : the frequent spinning “Beachball” (*known as the infamous “**SBBOD**” : Spinning Beach Ball Of Death*) which slows down or freezes the computer.

Fret not ! 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.

Additional Info: If the above trick doesn’t fix the turning beachball problem, most likely you also face another related problem : slow booting at startup.

This can be fixed below: using command line tool “fsck” to repair the hard disk:

1. Restart iMac

2. Press and hold both <Command> <S> keys until see a black screen with white letters. Then release the keys.

3. Wait till the prompt “**root #**” at the end of white letters scrolling down the screen (take about few mins): type

**/sbin/fsck -fy**

If the repair sucessfully, the screen should be shown with last message:

** **The volume Macintosh HD was repaired successfully ****

then at <root #> prompt, type :

**reboot**

Hit RETURN key. The iMac should now boot up at normal speed. Congrats !

If the repair fails: (example below)

then **repeat “fsck” tool 1 or 2 ti**mes until repair successfully.

**Health-Check**: Disk Utility’s “First Aid”

http://osxdaily.com/2012/05/24/check-hard-drive-health-mac-disk-utility/

**Other Tip to kill Spinning Beachball: Update dyld (need sudo Admin password)**

https://www.lifewire.com/fix-spod-spinning-pinwheel-of-death-2260811

iMac (2008) still usable today (2018) :

**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 “
which are faster to compute.**Linear Programs”** - 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)

**Self-driving Car**: Trajectory = P (x)

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

Obstacles are positions where P (x) ≥ 0.

This is one of the many cases of Pure Math turned to be Applied Math in last few decades. Other examples:

- Encryption in Blockchain / Bitcoin using Modular Form / Elliptic Curve (with which Prof Andrew Wiles proved the 350-year-old Fermat’s Last Theorem in 1994);
- Linear Algebra (Matrix, eigenVector/eigenValue) used in Google PageRanking Search Algorithm.
- etc.

**Is Applied Math => Pure Math ?**

Yes!

- The ancient“Chinese Remainder Problem” (韩信点兵 ) since Han dynasty (200 BC) was re-discovered by 19 CE Gauss in Modulus Arithmetic (“Number Theory” in Pure Math).

Summary: Simplix is an imperative language (not Object-Oriented, nor Functional), it has the “minimalist” features with extensibility from Java VM libraries by intraoperability (aka embedding) Java/C/C++ codes:

- Only
**1 data type: List**. The rest (collection, map, structure, array…) are just made of trees of lists. **Functions****Services**: group of functions.

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: **

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

then, **Associativity** **Law:**

A sheep is a sheep,

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)

w = European

then **Distributive Law**:

If t = Chinese

then all non-Chinese men = {**x – t}**

If s = Singaporean,

then

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

**Example:**

In English, 3 most common words:

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

=> “** the**” is

Zipf’s Law : the frequency of the

nthranked word is proportional to1/n.

** Reference**:

Hadamard estimated that :

About

90%of mathematiciansthinkvisually,10%thinkformally.

Usually, they think in steps:

- Get the right idea, often think vaguely about structural issues, leading to some kind of strategic vision;
- Tactics to implement it;
- 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].

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.

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 :

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, other than the “technical” axioms & theorems.

If a number system (Calculator arithmetic) has 4 operations (+ – * ÷ ), then it is a “**Field**” (域) – Eg. Real, Complex, Z/pZ (Integer mod Prime)…

If a number system with +, – and * (but no ÷), then it is a “**Ring**” (环).

eg1. Clock arithmetic {1,2, 3,…,12} = Z/12 (note: 12 is non-prime). [Note: the clock shape is like a ring, hence the German called this **Clock number** a “Ring”.]

eg2: Matrix (can’t ÷ matrices)

eg3. Polynomial is a ring (can’t ÷ 0 which is also a polynomial).

If a system (G) with 1 operation (○) and a set of elements {x y z w …} that is “closed” (kaki-lang 自己人, any 2 elements x ○ y = z still stay inside G ) , associative (ie bracket orderless) :(xy)z = x(yz), a neutral element (e) s.t. x+e = x = e+x, and inversible (, … still inside G), then G is a **Group**.

eg. {Integer, +}: 2’s inverse (-2), neutral 0, (2+3)+4=2+(3+4)

eg2. Triangle rotation 120 degree & flip about 3 inner axes.

If a non-empty system V ={v u w z …} that is “closed”if any of its 2 elements (called vectors v, u) v + u = w still in V,

AND if any vector multiply it by a scalar “λ” s.t. “λv” still in V, then V is a **Vector Space** (向量空间)。

eg1. Matrix (Vector) Space

eg2. Function (Vector) Space

eg3. Polynomial (Vector) Space

Summarise the above 4 or more systems into 1 Big System called “**Category**” (C) 范畴, then study relation (arrow or morphism) between f: C1 -> C2, this is “**Category** **Theory**“.

In any number system (aka algebraic structure), you can find the “Yin / Yang” (阴阳) duality : eg. “Algebra” **[#]** / “Co-Algebra”, Homology (同调) / Co-Homology (上同调)… if we find it difficult to solve a problem in the “Yang”-aspect. eg. In “Algebraic Topology”: “Homology” (ie “Holes”) with only “+” operation, then we could study its “Yin”-aspect Co-Homology in Ring structure, ie with the more powerful “*” multiplicative operation.

**Note [#]: **“Algebra” (an American invented structure) is a “Vector Space” plus multiplication between vectors. (Analogy in Physics : Cross Product of vectors).

Reference:https://www.youtube.com/playlist?list=PLi01XoE8jYoi3SgnnGorR_XOW3IcK-TP6

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 the differential geometry of Banach manifolds in the second semester of Math 55.^{[4]}

**Math55 Survivors :**

Of those students who could handle the workload, some became math or physics professors,^{[4]} including members of the Harvard Math Department such as Benedict Gross and Joe Harris; also, Harvard physics professor Lisa Randall ’84^{[15]} and Harvard economics professor Andrei Shleifer ’82.^{[16]}

In addition to these professors, past students of Math 55 include Bill Gates^{[18]} and Richard Stallman.^{[4]}

**The Professors teaching Math55**: include Siu Yum Tong 萧荫堂(China/HK).

How to keep state (counter, threshold, etc) in Functional Programming without BAD side effect ?

https://dzone.com/articles/functionalfun-states-in-functions-cache-lazy-and-c

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**

**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):
- Cubic Forms (of weight 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 “
“) parallel to line L.*geodesic* - Distance between p & q in H:

where L the “line” segment (the arc of the semicircle or the vertical segment) and

**1.2 Group of Non-Euclidean Motions:**

- Translation:
- Dilation:
- Inversion:
- Flip about axis (or line):

**Note**:

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

**Fractional Linear Transformation: **

2. __Group of Matrix__

Revision: Group = “CAN I“

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

**Determinant = det (K) = ad – bc**

Provided

2.1 General Linear Group

2.2 **Special Linear Group**

The Group acts on the upper half-plane H

**Notes**:

3. __ Modular Form : __ is an Analytic Function of weight

(1) Transformation property

(2) Growth property: possess a “q-expansion” of the form:

where all aj are constants, and

**Cusp Form of weight k **:

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

**Note**: q(z+1) = q(z) [hint:]

More generally, with an automorphy factor

eg.

**(Complex) Vector Spaces** (V) =

fulfilling:

(V1) V is nonempty.

(V2) For any function ** v** in V, and any complex number

(V3) For any function

4. **Congruence Groups (of Level 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**

5.3 **Galois Representation**

5.4 **Monstrous “Monshine”** – largest Simple Group

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

Amateur mathematician partially solves 60-year-old problem : “4-color” solution is **out!**

**芯片 (Chips) 的种类:**

- CPU 中央处理机
- Memory 记忆体储存

**材料: 硅** (\gui) Silicon 取自 沙 (sand)

**中兴 ZTE 禁令之芯片 为什么这么难做**？

步骤:

**Design 设计**(最难 !)**Manufacture**制作**Test & Packaging**测试&封装

2003 上海交大 微电子学院院长 陈进 的上亿骗局 “汉芯”一号 (其实是 Motorola Chip) 害中国退后十多年 !

**Global Foundry: how a CPU is made ?**

Python is the best Language in 2018 but not yet in mobile apps. However 2 Python frameworks (still primitive) for mobile: Kivy and BeeWare (more native-look).

https://dbader.org/blog/python-mobile-development-kivy-vs-beeware

- Python (Dutch Guido van Rossum, 1956)
- Java (Canadian James Gosling 1955)
- Javascript (USA Brendan Eich, 1961)
- C (USA Dennis Ritchie,
**1941 – 2011**) - C++ (Denmark Bjarne Stroustrup, 1950)
- Ruby (JAPAN Yukihiro “Matz” Matsumoto, 1965)
- Perl (USA Larry Wall, 1954)
- Pascal (Switzerland Niklaus Wirth, 1934)
- Lisp (USA John McCarthy,
**1927 – 2011**) - 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)

JλVλ 8 = “λ – Calculus” Java functional programming.

__History of Java:__

https://dzone.com/articles/j%CE%BBv%CE%BB-8-a-comprehensive-look

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

The above author recommends 5 best AI languages:

**Python**- Java (compatible: Google
**Kotlin***) __Lisp__(#modern ‘clone’:**Clojure***)__Prolog__(#)**C++**

Note 1: I have reservation for # __3) & #4)__ which are the 1970s / 1980s obsolete languages due to the issues in performance and no “practical” platforms (in mobile phone age), besides lack of major SW/HW vendor support (Google, Oracle, Microsoft, etc), and the small user community unlike the other three languages.

Note 2: Functional Programming (FP) is the modern AI language MUST-HAVE feature – only Kotlin & Clojure are “FP”.

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 (he wrapped himself with thick blanket 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

Join John Baez’s Azimuth Math Forum 导读 (Study Tour Guide) in Applied Category Theory (CT):

John Baez (1961-) is the world’s expert in Category Theory. He gave a talk on CT in Hokkaido University last year.

The 导读 is using a book by 2 mathematicians Brenden Fong and David Spivak in “Applied Category Theory” – **Download the free book of this course here.**

- Brendan Fong and David Spivak,
*Seven Sketches in Compositionality: An Invitation to Applied Category Theory*.

**Note**: Both John Baez and his wife Lisa Raphals (Professor in Chinese) work now in National University of Singapore – Center of Quantum Technologies & Philosophy, respectively.

Lycée Louis-Le-Grand, founded since 1563, is the best high school (lycée, 高中 1~3) 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, Charles Hermite, the 20th CE PolyMath Henri Poincaré, (See also: Unknown Math Teacher produced two World’s Math Grand Master Students ), Molière, Romaine Rolland (罗曼.罗兰), Jean-Paul Satre, Victor Hugo, 3 French Presidents, etc.

Its Baccalauréat (A-level) result is outstanding – 100% passed with 77% scoring distinctions. Each year 1/4 of Ecole Polytechnique (*) (France Top Engineering Grande Ecole ) students come from here.

More surprisingly, the “Seconde” (Secondary 4 ~ 中国/法国 “高一”) students learn Chinese Math since 6ème (Primary 6).

**Note **: Below is the little girl Heloïse (on blackboard in Chinese Math Class) whose admission application letter to the high school :

Translation– I practise Chinese since 6ème (Primary 6), 5 hours a week. I know that your school teaches 1 hour in Chinese Math, which very much interests me because Chinese and Mathematics are actually the 2 subjects I like most.

Interviewerasked Heloïse :Q: Why do you learn Chinese?

A: It is to prepare (myself) for working in China in the future, to immerse now in the language of environment. Anyway, the Chinese mode of operation is so different from ours.

**Note **: Louis Le Grand (= Louis 14th). He sent in 1687 AD 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 teach the 26-year-old Chinese Emperor (康熙) KangXi in Euclidean Geometry, etc.

**Note (*):** 5 Singaporeans (out of 300+ French Scholarship students) had entered Ecole Polytechnique through Classes Préparatoires / Concours aux Grandes Ecoles in native French language since 1980 to 2011. It is possible one day some of these elite French boys and girls could enter China top universities via “Gaokao” (高考 ~ “Concours”) in native Chinese language.

If you lose the Java source codes, you still can decompile from Java bytecodes.