Author Archives: tomcircle

Originally posted on Math Online Tom Circle:
(Traditional Method learnt at O-Level) Integration by Parts: NEW Technique taught by this UC-Berkley Chinese Mathematician: “DI” Method https://youtu.be/2I-_SV8cwsw First Stop: “D column at 0 “ 2nd Stop: 3rd Stop:

Originally posted on Math Online Tom Circle:
https://youtu.be/EWdcOID8zxU

Originally posted on Math Online Tom Circle:
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…

Originally posted on Math Online Tom Circle:
https://youtu.be/_gTClIbHreI

Originally posted on Math Online Tom Circle:
https://youtu.be/A54_QPXdkU0

Originally posted on Math Online Tom Circle:
Fermat Little Theorem: For any prime integer p, any integer m When m = 2, Note: 九章算数 Fermat Little Theorem (m=2) Pascal Triangle (1653 AD France ）= (杨辉三角 1238 AD – 1298 AD)…

Originally posted on Math Online Tom Circle:
[Hint] : Assume 3 cases, then by Elimination using Mr Brown’s hint (somethingS right, somethingS wrong)

Originally posted on Math Online Tom Circle:
Fibonacci Number 1 1 2 3 5 8 13 21… https://m.youtube.com/watch?v=VK7XR-wlpAk

Originally posted on Math Online Tom Circle:
［中文］历史 History: https://youtu.be/NeoDdnSlRjk Sir Atiyah Proof on 2018 September : (5 Pages PDF download here) https://drive.google.com/file/d/17NBICP6OcUSucrXKNWvzLmrQpfUrEKuY/view https://youtu.be/jXugkzFW5qY Bigger Lecture Notes video (at 34 mins, inspired in Singapore) https://youtu.be/oe4Jo4i3w_w Analytic function: preserve angle after…

Originally posted on Math Online Tom Circle:
What is your favorite theorem ? I have 2 theorems which trigger my love of Math : Chinese Remainder Theorem: 韩信点兵, named after a 200 BCE Han dynasty genius general Han Xin （韩信）…

Originally posted on Math Online Tom Circle:
China Nanjing Aviation University set the WiFi password as the answer of this integral (first 6 digits). Can you solve it? Answer : Break the integral (I) into 2 parts: I = A(x)…

Originally posted on Math Online Tom Circle:
Why did the approach to mathematics change with Common Core, rather than providing more structure to how it was previously taught? by Peter Kruger https://www.quora.com/Why-did-the-approach-to-mathematics-change-with-Common-Core-rather-than-providing-more-structure-to-how-it-was-previously-taught/answer/Peter-Kruger?share=84fb2ef6&srid=oZzP

Originally posted on Math Online Tom Circle:
Solve: Since the exponents (3+x) on both sides are equal, => 2 = 3

Originally posted on Math Online Tom Circle:
Bible Math: 153 St. Peter Fish [John 21:3-11] 3  So they went out and got into the boat, but that night they caught nothing. 6 He said, “Throw your net on the right side of…

Originally posted on Math Online Tom Circle:
Below are two French Baccalaureate Math textbooks (Volume 1 & 2) for Grade 12th (or Junior College Year 2, GCE A Level, Lycée: Terminale C) students in the late 1970s. There was a…

Originally posted on Math Online Tom Circle:
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.…

Originally posted on Math Online Tom Circle:
Key Points: Bus Waiting Time Pattern Biology Chicken eyes Physics: Random matrix / Eigenvalues / Riemann Zeta-function for Prime Numbers Universal Pattern: Normal Distribution Curve https://www.quantamagazine.org/the-universal-pattern-popping-up-in-math-physics-and-biology-20180823/

Originally posted on Math Online Tom Circle:
Key Points: Normed Distance: Euclidean Expander Graph https://www.quantamagazine.org/universal-method-to-sort-complex-information-found-20180813/

Originally posted on Math Online Tom Circle:
The most interesting “Category Theory” （范畴论) for Programmers course III by Dr. Bartosz Milewski , a follow-up of last year’s course II. Prerequisites: Fundamental of Category Theory: Functor, Natural Transformation, etc. (Course II…

Originally posted on Math Online Tom Circle:
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…

Originally posted on Math Online Tom Circle:
https://youtu.be/BzyOoo4AOxs

Originally posted on Math Online Tom Circle:
Functional Programming (FP) Languages : Lisp, Haskell, Scala, Kotlin, etc. Other non-FP influenced by Lambda Calculus: Python, Ruby, Javascript, Java 8, C++ Inventor of Lambda Calculus : Alonzo Church (1903 – 1995), whose…

Originally posted on Math Online Tom Circle:
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://youtu.be/M7e-LByl_68

Originally posted on Math Online Tom Circle:
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: Leibniz also invented 01 binary algebra, which he later found it was…

Originally posted on Math Online Tom Circle:
https://youtu.be/AoecIZ64sIk Keywords: Fermat’s Last Theorem (FLT): Pierre de Fermat (France 17CE): FLT Conjecture or Prank ? Euler (n= 3) Taniyama-Shimura-Weil Conjecture: Modular Form = Elliptic Curve Galois Group Symmetry Andrew Wiles (UK Cambridge…

Originally posted on Math Online Tom Circle:
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

Originally posted on Math Online Tom Circle:
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…

Originally posted on Math Online Tom Circle:
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)…

Originally posted on Math Online Tom Circle:
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…

Originally posted on Math Online Tom Circle:
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…

Originally posted on Math Online Tom Circle:
https://www.sciencenews.org/article/emmy-noether-theorem-legacy-physics-math

Originally posted on Math Online Tom Circle:
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…

Originally posted on Math Online Tom Circle:
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: r= 2.70…

Originally posted on Math Online Tom Circle:
Chinese Remainder Theorem (CRT): 韩信点兵 (中国剩余定理) 南北朝 《孙子算经》(4 AD) https://youtu.be/Ope18waS1GU

Originally posted on Math Online Tom Circle:
中国南北朝 : 《孙子算经》(4AD) 鸡兔同笼问题 (Chicken – Rabbits Problem) https://youtu.be/tnKaBeDVFLU

Originally posted on Math Online Tom Circle:
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…

Originally posted on Math Online Tom Circle:
https://people.cs.pitt.edu/~kirk/cs1501/Pruhs/Spring2006/assignments/editdistance/Levenshtein%20Distance.htm

Originally posted on Math Online Tom Circle:
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.…

Originally posted on Math Online Tom Circle:
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.…

Originally posted on Math Online Tom Circle:
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/

Originally posted on Math Online Tom Circle:
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 =…

Originally posted on Math Online Tom Circle:
https://simple.m.wikipedia.org/wiki/Zipf%27s_law Example: In English, 3 most common words: “the” : occurring 7% of the time; “of” : 3.5% = 7/2 “and” : 2.8% ~ 7/3 => “the” is 2x occurs more often than…

Originally posted on Math Online Tom Circle:
Hadamard estimated that : About 90% of mathematicians think visually, 10% think formally. Usually, they think in steps: Get the right idea, often think vaguely about structural issues, leading to some kind of…

Originally posted on Math Online Tom Circle:
“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].

Originally posted on Math Online Tom Circle:
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…

Originally posted on Math Online Tom Circle:
Problem A3 A function f is defined on the positive integers by: for all positive integers n, Determine the number of positive integers n less than or equal to 1988 for which f(n)…

Originally posted on Math Online Tom Circle:
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…

Originally posted on Math Online Tom Circle:
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 https://youtu.be/XkVskCFA8r8

Originally posted on Math Online Tom Circle:
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…