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Summary: Shapes, radius functions and persistent homology

This is a summary of a talk by Professor Herbert Edelsbrunner, IST Austria. The PDF slides can be found here: persistent homology slides.

Biogeometry (2:51 in video)

We can think of proteins as a geometric object by replacing every atom by a sphere (possibly different radii). Protein is viewed as union of balls in \mathbb{R}^3.

Decompose into Voronoi domains V(x), and take the nerve (Delaunay complex).

Inclusion-Exclusion Theorem: \displaystyle Vol(\bigcup B)=\sum_{Q\in D_r(x)}(-1)^{\dim Q}Vol(\bigcap Q).
Volume of protein (\bigcup B) is alternating sum over all simplices Q in Delaunay complex.

Nerve Theorem: Union of sets have same homotopy type as nerve (stronger than having isomorphic homology groups).

Wrap (14:04 in video)

Collapses: 01 collapse means 0 dimensional and 1 dimensional simplices disappear (something like deformation retract).

Interval: Simplices that are removed in a collapse (always a skeleton of a cube in appropriate dimension)

Generalised Discrete Morse Function (Forman 1998): Generalised discrete vector field = partition into intervals (for acyclic case only)

Critical simplex: The only simplex in an interval (when a critical simplex is added, the homotopy type changes)

Lower set of critical simplex: all the nodes that lead up to the critical simplex.

Wrap complex is the union of lower sets.

Persistence (38:00 in video)

Betti numbers in \mathbb{R}^3: \beta_0: \# components, \beta_1:\# loops, \beta_2: \# voids.

Incremental Algorithm to compute Betti numbers (40:50 in video). [Deffimado, E., 1995]. Every time a simplex is added, either a Betti number goes up (birth) or goes down (death).

\alpha is born when it is not in image of previous homology group.

Stability of persistence: small change in position of points leads to similar persistence diagram.

Bottleneck distance between two diagrams is length of longest edge in minimizing matching. Theorem: \displaystyle W_\infty(Dgm(f),Dgm(g))\leq\|f-g\|_\infty. [Cohen-Steiner, E., Hares 2007]. One of the most important theorems in persistent homology.

Expectation (51:30 in video)

Poisson point process: Like uniform distribution but over entire space. Number of points in region is proportional to size of region. Proportionality constant is density \rho>0.

Paper: Expectations in \mathbb{R}^n. [E., Nikitenko, Reitones, 2016]

Reduces to question (Three points in circle): Given three points in a circle, what is the probability that the triangle (with the 3 points as vertices) contains the center of the circle? Ans: 1/4 [Wendel 1963].

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Brain has 11 dimensions

One of the possible applications of algebraic topology is in studying the brain, which is known to be very complicated.


If you can call understanding the dynamics of a virtual rat brain a real-world problem. In a multimillion-dollar supercomputer in a building on the same campus where Hess has spent 25 years stretching and shrinking geometric objects in her mind, lives one of the most detailed digital reconstructions of brain tissue ever built. Representing 55 distinct types of neurons and 36 million synapses all firing in a space the size of pinhead, the simulation is the brainchild of Henry Markram.

Markram and Hess met through a mutual researcher friend 12 years ago, right around the time Markram was launching Blue Brain—the Swiss institute’s ambitious bid to build a complete, simulated brain, starting with the rat. Over the next decade, as Markram began feeding terabytes of data into an IBM supercomputer and reconstructing a collection of neurons in the sensory cortex, he and Hess continued to meet and discuss how they might use her specialized knowledge to understand what he was creating. “It became clearer and clearer algebraic topology could help you see things you just can’t see with flat mathematics,” says Markram. But Hess didn’t officially join the project until 2015, when it met (and some would say failed) its first big public test.

In October of that year, Markram led an international team of neuroscientists in unveiling the first Blue Brain results: a simulation of 31,000 connected rat neurons that responded with waves of coordinated electricity in response to an artificial stimulus. The long awaited, 36-page paper published in Cell was not greeted as the unequivocal success Markram expected. Instead, it further polarized a research community already divided by the audacity of his prophesizing and the insane amount of money behind the project.

Two years before, the European Union had awarded Markram $1.3 billion to spend the next decade building a computerized human brain. But not long after, hundreds of EU scientists revolted against that initiative, the Human Brain Project. In the summer of 2015, they penned an open letter questioning the scientific value of the project and threatening to boycott unless it was reformed. Two independent reviews agreed with the critics, and the Human Brain Project downgraded Markram’s involvement. It was into this turbulent atmosphere that Blue Brain announced its modest progress on its bit of simulated rat cortex.

Read more at the link above.

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How to explain this math magic trick?

Quite impressive math magic trick, that even impressed the very strict judge Simon Cowell. I am not sure how he did it, other than possible prearranged volunteers. Another possibility is that the calculator is modified.


London had the volunteers give their best guesses to different questions, including how many No. 1-selling artists Cowell has had on his record label, how many millions of records judge Mel B. sold worldwide with the Spice Girls and what year judge Heidi Klum started modeling.

Meanwhile, London asked host Tyra Banks multiply the three answers together using the calculator on her own phone. He then instructed Banks to close her eyes and add a random eight-digit number to the previous calculation. She revealed that the grand total came out to 73,928,547.

Watch the clip to see why that number left the judges and audience members stunned!

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Is There a Multi-dimensional Mathematical World Hidden in the Brain’s Computation?

Math Online Tom Circle

Algebraic Topology” can detect the Multi-dimensional neural network in our brain – by studying the Homology (同调) and co-Homology (上同调) with the help of Linear Algebra (multi-dim Matrix) & Computers.

Homology = compute the number of “holes” in multi-dim space.

Neurons formed in the brain can be modeled in Math (Topology) by Simplex 单纯 (plural : Simplices), billions of them interconnected into a complex -“Simplicial Complex” (单纯复体)。

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What is math? 

Math Online Tom Circle

What is Math ?

  • Mathematics= “that which is learned“ –(Pythagoras)

Math is not about calculation, it is understanding the nature, the universe, the philosophy (logic, intelligence – both “human” and “artificial”)…

What is Axiom, Lemma, Proposition ? Why rigorous Calculus was needed hundred years afterNewton & Leibnizhad invented it – “Epsilon-Delta” Analysis.

Difference between Riemann Integral & Lebesgue Integral ?

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Yitang Zhang’s Santa Barbara Beach Walk

Professor Yitang Zhang is a famous Math professor who made important progress in number theory (Twin Prime Conjecture). Most strikingly, he made this progress in his fifties, which is kind of rare in the mathematical world.

Source: Quanta Magazine

Yitang Zhang on the beach adjoining the University of California, Santa Barbara, after scratching a function in the sand related to his current work on the Landau-Siegel zeros problem.

As an adolescent during the Cultural Revolution in China, Yitang Zhang wasn’t allowed to attend high school. Later, in his 30s, he worked odd jobs in the United States and sometimes slept in his car. But Zhang always believed he would solve a great math problem someday. Still, despite becoming one of China’s top math students and completing his doctorate at Purdue University in Indiana, for seven years Zhang could not find work as a mathematician. At one point, he worked at a friend’s Subway sandwich restaurant to pay the bills.

“I was not lucky,” Zhang, who is both incredibly reserved and self-confident, told Quanta in a 2015 interview.

At 44, after finally being hired to teach math at the University of New Hampshire, he turned his attention to number theory, a subject he had loved since childhood. He analyzed problems in his head during long walks near his home and the university. In his 50s, well past what many mathematicians consider their prime years (indeed, the Fields Medal is awarded to mathematicians under the age of 40), he began trying to prove the twin primes conjecture, which predicts an infinite number of prime number pairs that have a difference of two, such as 5 and 7, 29 and 31, and 191 and 193. No one had been able to prove this in over 150 years, and top number theorists could not even prove the existence of a bounded prime gap of any finite size.

In 2013, at 58, Zhang published his proof of a bounded prime gap below 70 million in one of the world’s most prestigious journals, the Annals of Mathematics. The paper’s referees wrote that Zhang, who had been unknown to established mathematicians, had proved “a landmark theorem in the distribution of prime numbers.”

Read more at: Quanta

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Finally received WordAds payout by WordPress

Thanks to all readers of, I have received WordAds payout by WordPress. first joined WordAds in 01-2014, where the payout was quite miserable (less than $1 for 5000 views). The payout has since increased considerably, thanks to higher quality adverts localized in Singapore.

The advertisements shown are quite relevant (usually tuition by top tutors/schools, travel and other deals in Singapore), so do turn off ad-blockers when visiting Thanks once again!

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Singapore Math for  Primary School Kids

Math Online Tom Circle

Singapore scores the 2017 PISA Test World’s First in Math.

Below is the “Singapore Math” shown by an USA School adopting the pedagogy.

I had the opportunity to meet the inventor of the “Singapore Math” Prof Lee PengYee (李彝义) in a Math seminar at NUS few years ago, who said he took the inspiration of the ancient Chinese math 算术 (literary means “Counting Technique” sans Algebra), combined with the PolyaProblem Solving Methodology, with a visual tool called “The Model Diagram”.

Note that this pedagogy is based on concrete visualisation, while good for majority of young students to learn computation and problem solving, it doesn’t train the opposite skill in “abstractness” which is required in university Advanced Math, where the French, Russian and USA university students excel more than the Asian cohorts.

The Ideal Math Education for the next 22ndCentury:

Singapore Math (Computation for…

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Guide to Starting Javaplex (With Matlab)

Guide to Starting Javaplex (With Matlab)

Step 1)

Visit and download the Persistent Homology and Topological Data Analysis Library


Download the tutorial at and jump to section 1.3. Installation for Matlab.


In Matlab, change Matlab’s “Current Folder” to the directory matlab examples that you just extracted from the zip file.

(See to change current folder)

Type this in Matlab: cd /…/matlab_examples

Where … depends on where you put the folder

4) In the tutorial (from the link given in step 2), proceed to follow the instructions starting from “In Matlab, change Matlab’s “Current Folder” to the directory matlab examples that you just extracted from the zip file. In the Matlab command window, run the load javaplex.m file.”.

5) Test: Run example 3.2 (House example) by typing in the code (following the tutorial)

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Subtle Error in Wikipedia: Dedekind’s number

On Wikipedia (, it is stated that the Dedekind’s number M(n) is the the number of abstract simplicial complexes with n elements.

This is incorrect, at least based on the Wikipedia definition of abstract simplicial complex, which does not allow the empty set as a face.

The correct definition is found in another Wikpedia site:

The number of abstract simplicial complexes on up to n elements is one less than the nth Dedekind number.

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Habitica Mage Highest Intelligence Gear Guide

Habitica is a productivity app, something like a to-do list in a gamified form. Users gain experience and gold from completing tasks, and can join in quests with a party to defeat monsters.

The mage class has one of the highest damage in the game (after warrior), and its base stat is intelligence. The best mage gear with highest intelligence (as of June 2017) is as follows:

Armor: Jean Chalard’s Noble Tunic (+25 INT, +25 CON)

Headgear: Nameless Helm (+25 INT, +25 STR)

Shield-Hand Item: Diamond Stave (+16 INT)

Weapon: Trident of Crashing Tides (+15 INT)

For stats, put all stats into intelligence.

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Donald Trump Math: 17×6=?

What is 17×6? (without using calculator)

Practicing basic mental math without calculator is good. Even for PSLE, where calculator is allowed, it is a good idea not to rely on it too much.

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Pi hiding in prime regularities

Math Online Tom Circle

Three mysterious Math Objects:

  • Pi,
  • Complex Numbers,
  • Prime Numbers

are hiding in circle.

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5 Math Tips That Will Speed Up Your Calculations

5 Math Tips That Will Speed Up Your Calculations

For most students, regardless of their age, studying math seems like a nightmare. Taking tuition for maths helps when you or your child struggle with calculations.

Check out these 5 simple tricks for making mental math faster and more accurate:

  1. 11 Times Trick – It’s easy to multiply single digit numbers with 11: just repeat the number, but what about double digits? This trick is almost shockingly simple: just add a space between the two digits, and insert their sum in the middle. E.g., 32 x 11 = 3(3+2)2, or 352.

If the total is more than 9, add 1 to the first digit and insert the second number, e.g. 78 x 11 = 7(7+8)8, or 7(15)8. Move the 1, so 7+1(5)8 = 858.

  1. Large Sums Trick – To add large numbers quickly in your head, convert them into multiples of 10. E.g. 762 + 816 can be rounded off to 760 + 820, so 1580. Add up the remaining numbers taken aside while rounding off the two, i.e. +2 and -4, so -2. Then add these to the previous total, so 1580 – 2 = 1578.
  2. Binary & Bisect Trick – To multiply two numbers, one of which is even, here’s what to do. Divide the even number by 2 and multiply the other by two, and continue doing this till you reach numbers that are easy to calculate. E.g. 12 x 37 = 6 x 74 = 3 x 148 = 444, and 20 x 43 = 10 x 86 = 860.
  3. Multiplication Trick – To multiply numbers quickly, follow these rules:
  • Multiplying by 4: Multiply by 2 and then again by 2, e.g. 42 x 4 = 84 x 2 = 168.
  • Multiplying by 5: Multiply by 10 and then divide by 2, e.g. 190 x 5 = 1900/2 = 950.
  • Multiplying by 9: Multiply by 10 and then subtract the original number from the result, e.g. 26 x 9 = 260 – 26 = 234
  • Multiplying by 99: Multiply by 100 and then subtract the original number from the result, e.g. 51 x 99 = 5100 – 51 = 5049.
  1. Percentages Trick – “Percent” literally means per 100, so break down a number into 100s to find a certain percentage. E.g. 8% of 400 = 8 per 4 hundreds, so 8 x 4 = 32. If the number is under 100, move the decimal point. E.g. 8% of 50 = 8 x 5 or 40, and with the decimal point moved, 4.

What about 8% of 350? Add up the 8s for each 100, and half of an 8 for the remaining 50, so (8 x 3) + 4 = 28. The same for 8 x 35, but moving the decimal point, so 2.8. Percentages can also be flipped, so 32% of 5 is the same as 5% of 32.

Math doesn’t have to be scary, and an online math tutor can help you deal with advanced problems without getting overwhelmed.

About Author: Making education simple and easy to comprehend is Dana Jandhayala’s forte. She’s had a long career as an educator where she has taught in several different schools and institutes in multiple countries. Today, she helps students with personalized online tutorials by SchoolPage that help make concepts easy to understand, making learning fast and fun. She writes to help students study better, and to coach parents so they can facilitate the success of their children.


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AI “AlphaGo” beats World’s Best Go Player Ke Jie in 1st Match

Math Online Tom Circle

After the defeat in 1st match, Ke Jie said “AlphaGo is approaching God!”.

This version of AlphaGo is 10x more powerful than last year’s version which has beateb the South-korean Go player Lee.

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Google officially supports Kotlin from May 2017

Math Online Tom Circle

Kotlin is the “New Java” officially supported by Google from May 2017! It is less verbose (罗唆) than Java which is clumsy with boilerplates (样板), interoperates with Java on JVM, with modern functional programming features, and most importantly,it is Multi-Platform : Java, Android, Javascript, and future versions run as native codes on iOS, MacOS and Linux (Microsoft – work in progress). This eliminates the current headache of having to re-write the same applications for different platforms in different languages.

Google makes Kotlin a first-class language for writing Android apps

Kotlin Tutorials:


Two ways to program in Kotlin:

1) Google way: Download Android Studio 3.0 (with Kotlin and Java 8 Support):

2) (A better way): use Jetbrains “Intelli IDEA for kotlin” (bundled with Kotlin)


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Math Tricks found in Chess

Just read this very nice article on Quora, on the relationship between Math and Chess:

Also interesting is this YouTube documentary “My Brilliant Brain” featuring Susan Polgar.

Tom Boshoff
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French youngest President Emmanuel Macron and his Education 

Math Online Tom Circle

Emmanuel Macron is the youngest French President (39) since Napoleon Bonaparte (40).

A brilliant student since young, he impressed his secondary school Drama teacher 24 years older, finally married her.

Like any genius (Einstein, Galois, Edison, …) who doesn’t adapt well in the traditional education system, Macron entered the prestigious and highly competitiveClasse Préparatoire (Art Stream) Lycée Henri IVin Paris to prepare for the “Concours” (法国抄袭自中国的)”科举” Entrance Examsin France’s top Ecole Normale Supérieure (ENS). Like the 19CE Math geniusEvariste Galoiswho failed the Ecole Polytechnique Concours twice in 2 consecutive years, Macron also failed ENS “Concours” in 2 consecutive years.

He revealed recently, ”The truth was I didn’t play the game. I was too much in love (with my former teacher) for seriously preparing the Concours …”

Note: French traditional name for the elitist tertiary education (first 2 or 3 years if repeat…

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LaTeX Thesis Template

This site ( has a guide on how to create your own \LaTeX template for a thesis. Quite nice and simple, and easily customizable.

The actual Tex source code is found here:

I like it as the source code is neat and clean, you can easily edit it if you know some basic Latex. Some other templates out there are quite complex and convoluted, it is hard to customize it.

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Math Olympiad Tuition

Maths Olympiad Tuition

Tutor: Mr Wu (Raffles Alumni, NUS Maths Grad)

SMS/Whatsapp: 98348087


Syllabus: Primary / Secondary Maths Olympiad. Includes Number Theory, Geometry, Combinatorics, Sequences, Series, and more. Flexible curriculum tailored to student’s needs. I can provide material, or teach from any preferred material that the student has.

Target audience: For students with strong interest in Maths. Suitable for those preparing for Olympiad competitions, DSA, GEP, or just learning for personal interest.

Location: West / Central Singapore at student’s home

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Nanyang Girls NYGH starts school later at 8.15am

Very good idea by NYGH. Sleep is important for students.

SINGAPORE: For almost a year now, Nanyang Girls High (NYGH) students have been starting school at 8.15am – a good 45 minutes later than most secondary schools.

And the results have been telling.

The school in Bukit Timah has been taking part in ground-breaking sleep studies conducted by Duke-NUS Medical School researchers – whose studies have shown that 80 per cent of teens here don’t get enough sleep, which affects their health, grades and cognitive abilities.

It was what the teachers of NYGH had been suspecting all along.

Mrs Ho-Sam Choon Juen, NYGH dean of student systems and info management, said: “For a long time, we’d known that our girls were not sleeping enough because of their academic and extra-curricular demands.

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Jurong East Maths Tuition

Maths Tuition

Tutor (Mr Wu):
– Raffles Alumni
– NUS 1st Class Honours in Mathematics

Experience: More than 10 years experience, has taught students from RJC, NJC, ACJC and many other JCs. Also has experience teaching Additional Math (O Level, IP).

Personality: Friendly, patient and good at explaining complicated concepts in a simple manner. Provides tips for how to check for careless mistakes, and tackle challenging problems.

SMS: 98348087

Areas teaching (West / Central Singapore, including Bukit Batok, Dover, Clementi, Jurong)

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BM Category Theory II 8: F-Algebra, Lambek’s Lemma , Catamorphism, Coalgebra, Anamorphism

Math Online Tom Circle

[Continued from previous BM Category Theory …]

$latex boxed {
text {type Algebra f a = f a} to text {a}

Intuition: [Artificial Intelligence] You teach the computer, like a Primary 6 kid, that Algebra is atype of expression (f) which, after evaluation, returns a value.

If a = i (initial)[or u (terminal)],
$latex boxed {
text {(f i} to text {i )} implies
text {f = Fix-point}

Intuition: Fix-point because, the Initial “i” after the evaluating the expression f, returns itself “i”.

Lambek’s Lemma
$latex boxed {
text {Endo-functor = Isomormphism}

Note: Endo-functor is a functor (equivalent tofunction in Set Theory) within the same Category (Endo = Self = 自)

Video 8.1F-Algebras & Lambek’s Lemma

Video 8.2Catamorphism & Anamorphism

foldr ~ catamorphism (浅层变质) of a Fix-point endo-functor on a List.

Examples: Fibinacci, Sum_List

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Bukit Batok Maths Tuition

Maths Tuition

Tutor (Mr Wu):
– Raffles Alumni
– NUS 1st Class Honours in Mathematics

Experience: More than 10 years experience, has taught students from RJC, NJC, ACJC and many other JCs. Also has experience teaching Additional Math (O Level, IP).

Personality: Friendly, patient and good at explaining complicated concepts in a simple manner. Provides tips for how to check for careless mistakes, and tackle challenging problems.

SMS: 98348087

Areas teaching (West / Central Singapore, including Bukit Batok, Dover, Clementi, Jurong)

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Renowned Chinese mathematician Wu Wenjun dies at 98


Wu Wenjun, distinguished mathematician, member of the Chinese Academy of Sciences (CAS), and winner of China’s Supreme Scientific and Technological Award winner, died at the age of 98 on Sunday in Beijing, according to the CAS.
Wu was born in Shanghai on May 12, 1919. In 1940, he graduated from Shanghai Jiao Tong University, and received a PhD from the University of Strasbourg, France in 1947.
In 1951, Wu returned to China and served as a math professor at Peking University. He made great contributions to the field of topology by introducing various principles now recognized internationally.
In the field of mathematics mechanization, Wu suggested a computerized method to prove geometrical theorems, known as Wu’s Method in the international community.
He was elected as a member of the CAS in 1957 and as a member of the Third World Academy of Sciences in 1990.
Wu Wenjun was given China’s Supreme Science and Technology Award by the then President Jiang Zemin in 2000, when this highest scientific and technological prize in China began to be awarded.

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Rise in WordAds earnings March 2017

WordPress and WordAds have been doing a good job, I must say. In this year (2017), the earnings have increased gradually for my site (view count is approximately constant).

Certainly a great improvement from 2016, where I got a measly $0.70 for 11824 ad impressions in one month.

Keep it up, WordPress!

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Ye Sons and Daughters (O Filii et Filiae)

Nice traditional hymn.

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What is a Module (模)? 

Math Online Tom Circle

Replace Field scalar as in a Vector Space to Ring scalar in a Module.

Module is more powerful than Vector Space – because Ring has an “Ideal” (理想) which can partition it to Quotient Ring, but Field (scalar in a Vector Space) can’t.

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American Genius Series 1 1of8 Jobs vs Gates (2015)

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Structure Preserving

Math Online Tom Circle

Abstract Algebra studies all kinds ofmathematical structures(Group, Ring, Vector Space, Category, …) , and relationship between them if the structures are preserved after mapping.

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The Cry of the Poor by John Michael Talbot

Recently I discovered a classical guitar Christian composer on YouTube – John Michael Talbot. His music is quite amazing, and one of a kind.

  1. The Cry of the Poor

2. Be not afraid

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Five Loaves and Two Fishes – Corrinne May (Illustrated)

Corrinne May is a Singaporean singer.

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Mathematics: Beauty vs Utility – Numberphile

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2 minutes pour le théorème de Bézout

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Maths -Topologie

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How the Staircase Diagram changes when we pass to derived couple (Spectral Sequence)

Set A_{n,p}^1=H_n(X_p) and E_{n,p}^1=H_n(X_p,X_{p-1}). The diagram then has the following form:

When we pass to the derived couple, each group A_{n,p}^1 is replaced by a subgroup A_{n,p}^2=\text{Im}\,(i_1: A_{n,p-1}^1\to A_{n,p}^1). The differentials d_1=j_1k_1 go two units to the right, and we replace the term E_{n,p}^1 by the term E_{n,p}^2=\text{Ker}\, d_1/\text{Im}\,d_1, where the d_1‘s refer to the d_1‘s leaving and entering E_{n,p}^1 respectively.

The maps j_2 now go diagonally upward because of the formula j_2(i_1a)=[j_1a]. The maps i_2 and k_2 still go vertically and horizontally, i_2 being a restriction of i_1 and k_2 being induced by k_1.

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How do you overcome the fear of the future? Pope Francis provides the keys


Have you ever been gripped by a fear of the future that left you walking in circles or paralyzed from moving forward? Today, Pope Francis provided a deeper answer to this common human fear, urging Christians to remember that faith is the anchor that keeps our lives moored to the heart of God, amid every storm and difficulty.

Speaking to faithful and pilgrims at today’s general audience in St. Peter’s Square, the pope reminded them that, wherever we go, God’s love goes before us.

“There will never be a day in our lives when we will cease being a concern for the heart of God,” he said, as he continued his series on Christian hope.

“I am with you”

Reflecting on St. Matthew’s Gospel, Pope Francis observed that it begins with the birth of Jesus as Emmanuel — “God is with us” — and concludes with the Risen Lord’s promise to his disciples: “I am with you always, even to the end of the age” (Mt 28:20). He said:

The whole gospel is enclosed in these two quotations, words that communicate the mystery of a God whose name, whose identity is ‘to be with,’ in particular, to be with us, with the human creature. Ours is not an absent God … He is a God who is “passionately” in love with man, and so tender a lover that he is incapable of separating Himself from him. We humans are able to break bonds and bridges. He is not. If our heart cools, his remains incandescent. Our God always accompanies us, even if we unfortunately forgot about Him. […] Christians especially do not feel abandoned, because Jesus promises not to only wait for us at the end of our long journey, but to accompany us during each of our days.

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Relative Homology Groups

Given a space X and a subspace A\subset X, define C_n(X,A):=C_n(X)/C_n(A). Since the boundary map \partial: C_n(X)\to C_{n-1}(X) takes C_n(A) to C_{n-1}(A), it induces a quotient boundary map \partial: C_n(X,A)\to C_{n-1}(X,A).

We have a chain complex \displaystyle \dots\to C_{n+1}(X,A)\xrightarrow{\partial_{n+1}}C_n(X,A)\xrightarrow{\partial_n}C_{n-1}(X,A)\to\dots where \partial^2=0 holds. The relative homology groups H_n(X,A) are the homology groups \text{Ker}\,\partial_n/\text{Im}\,\partial_{n+1} of this chain complex.

Relative cycles
Elements of H_n(X,A) are represented by relative cycles: n– chains \alpha\in C_n(X) such that \partial\alpha\in C_{n-1}(A).

Relative boundary
A relative cycle \alpha is trivial in H_n(X,A) iff it is a relative boundary: \alpha=\partial\beta+\gamma for some \beta\in C_{n+1}(X) and \gamma\in C_n(A).

Long Exact Sequence (Relative Homology)
There is a long exact sequence of homology groups:
\begin{aligned}  \dots\to H_n(A)\xrightarrow{i_*}H_n(X)\xrightarrow{j_*}H_n(X,A)\xrightarrow{\partial}H_{n-1}(A)&\xrightarrow{i_*}H_{n-1}(X)\to\dots\\  &\dots\to H_0(X,A)\to 0.  \end{aligned}

The boundary map \partial:H_n(X,A)\to H_{n-1}(A) is as follows: If a class [\alpha]\in H_n(X,A) is represented by a relative cycle \alpha, then \partial[\alpha] is the class of the cycle \partial\alpha in H_{n-1}(A).

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Prayer of St Thomas Aquinas for Students (English and Latin)


A Prayer Before Study

Ineffable Creator,
Who, from the treasures of Your wisdom,
have established three hierarchies of angels,
have arrayed them in marvelous order
above the fiery heavens,
and have marshaled the regions
of the universe with such artful skill,

You are proclaimed
the true font of light and wisdom,
and the primal origin
raised high beyond all things.

Pour forth a ray of Your brightness
into the darkened places of my mind;
disperse from my soul
the twofold darkness
into which I was born:
sin and ignorance.

You make eloquent the tongues of infants.
refine my speech
and pour forth upon my lips
The goodness of Your blessing.

Grant to me
keenness of mind,
capacity to remember,
skill in learning,
subtlety to interpret,
and eloquence in speech.

May You
guide the beginning of my work,
direct its progress,
and bring it to completion.

You Who are true God and true Man,
who live and reign, world without end.


Ante Studium

Creator ineffabilis,
qui de thesauris sapientiae tuae
tres Angelorum hierarchias designasti,
et eas super caelum empyreum
miro ordine collocasti,
atque universi partes elegantissime disposuisti,

tu inquam qui
verus fons
luminis et sapientiae diceris
ac supereminens principium

infundere digneris
super intellectus mei tenebras
tuae radium claritatis,
duplices in quibus natus sum
a me removens tenebras,
peccatum scilicet et ignorantiam.

Tu, qui linguas infantium facis disertas,
linguam meam erudias
atque in labiis meis gratiam
tuae benedictionis infundas.

Da mihi
intelligendi acumen,
retinendi capacitatem,
addiscendi modum et facilitatem,
interpretandi subtilitatem,
loquendi gratiam copiosam.

Ingressum instruas,
progressum dirigas,
egressum compleas.

Tu, qui es verus Deus et homo,
qui vivis et regnas in saecula saeculorum.


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Exact sequence (Quotient space)

Exact sequence (Quotient space)
If X is a space and A is a nonempty closed subspace that is a deformation retract of some neighborhood in X, then there is an exact sequence
\begin{aligned}  \dots\to\widetilde{H}_n(A)\xrightarrow{i_*}\widetilde{H}_n(X)\xrightarrow{j_*}\widetilde{H}_n(X/A)\xrightarrow{\partial}\widetilde{H}_{n-1}(A)&\xrightarrow{i_*}\widetilde{H}_{n-1}(X)\to\dots\\  &\dots\to\widetilde{H}_0(X/A)\to 0  \end{aligned}
where i is the inclusion A\to X and j is the quotient map X\to X/A.

Reduced homology of spheres (Proof)
\widetilde{H}_n(S^n)\cong\mathbb{Z} and \widetilde{H}_i(S^n)=0 for i\neq n.

For n>0 take (X,A)=(D^n,S^{n-1}) so that X/A=S^n. The terms \widetilde{H}_i(D^n) in the long exact sequence are zero since D^n is contractible.

Exactness of the sequence then implies that the maps \widetilde{H}_i(S^n)\xrightarrow{\partial}\widetilde{H}_{i-1}(S^{n-1}) are isomorphisms for i>0 and that \widetilde{H}_0(S^n)=0. Starting with \widetilde{H}_0(S^0)=\mathbb{Z}, \widetilde{H}_i(S^0)=0 for i\neq 0, the result follows by induction on n.

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Reduced Homology

Define the reduced homology groups \widetilde{H}_n(X) to be the homology groups of the augmented chain complex \displaystyle \dots\to C_2(X)\xrightarrow{\partial_2}C_1(X)\xrightarrow{\partial_1}C_0(X)\xrightarrow{\epsilon}\mathbb{Z}\to 0 where \epsilon(\sum_i n_i\sigma_i)=\sum_in_i. We require X to be nonempty, to avoid having a nontrivial homology group in dimension -1.

Relation between H_n and \widetilde{H}_n
Since \epsilon\partial_1=0, \epsilon vanishes on \text{Im}\,\partial_1 and hence induces a map \tilde{\epsilon}:H_0(X)\to\mathbb{Z} with \ker\tilde{\epsilon}=\ker\epsilon/\text{Im}\,\partial_1=\widetilde{H}_0(X). So H_0(X)\cong\widetilde{H}_0(X)\oplus\mathbb{Z}. Clearly, H_n(X)\cong\widetilde{H}_n(X) for n>0.

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8 JCs to merge (i.e. 4 JCs to close down)

The latest education news in Singapore is that 4 pairs of JCs to merge as student numbers shrink; 14 primary and 6 secondary schools also affected.

The effect on the primary and secondary schools is not that significant, due to the large number of primary and secondary schools. However, there are only around 20 JCs in Singapore, the effect is quite big for JCs.

8 JCs merging is just a nice way of saying 4 JCs to be shut down permanently. RIP Serangoon, Tampines, Innova and Jurong JCs.

The most affected would be O level students in the next 5 years. Yes, there is declining birthrate but that is gradual. So for the next 5 years, there is approximately the same number of students competing for 4 less JCs.

So by “Demand and Supply” logic, we have:
– similar demand for JCs (approx. same number of students in the next 5 years)
– lower supply of JCs (due to the 4 axed JCs)

By Economic Theory: If supply decreases and demand is unchanged, then it leads to a higher equilibrium price.

Hence the logical conclusion is that the “price” will rise, that is, cutoff points for JCs may become lower. To add on to that, the 4 axed JCs cater mainly to the 13-20 pointers. So students falling in that L1R5 range will be especially affected.

Also check out: Which JC is good?

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Math Online Tom Circle


$latex (a + x)^{frac {1}{3}} + (a – x)^{frac {1}{3}} = 2(a)^{frac {1}{3}} + 2(x)^{frac {1}{3}} $

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The World’s Best Mathematician (*)

Math Online Tom Circle

Terence Tao:

  • Weakness in Math: Algebraic Topology
  • Collaborative & “Lone Wolf”
  • Not attempting the Riemann Hypothese: tools not there yet.

3 Phases of Math Training:

  1. Pre-rigourous: intuition
  2. Rigorous: formal prove
  3. Post-rigourous: 1 + 2

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The Legend of Question Six (IMO 1988)

Math Online Tom Circle

This question was submitted by West Germany to the IMO Committee, the examiners could not solve it in 6 hours.

In the IMO (1988) only 11 contestants solved it, one of them proved it elegantly. Terence Tao (Australia) only got 1 mark out of 7 in this question.


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The Shortest Ever Papers 

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Klein Bottle as Gluing of Two Mobius Bands

This is a nice picture on how the Klein bottle can be formed by gluing two Mobius bands together. Very neat and self-explanatory!


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Mayer-Vietoris Sequence applied to Spheres

Mayer-Vietoris Sequence
For a pair of subspaces A,B\subset X such that X=\text{int}(A)\cup\text{int}(B), the exact MV sequence has the form
\begin{aligned}  \dots&\to H_n(A\cap B)\xrightarrow{\Phi}H_n(A)\oplus H_n(B)\xrightarrow{\Psi}H_n(X)\xrightarrow{\partial}H_{n-1}(A\cap B)\\  &\to\dots\to H_0(X)\to 0.  \end{aligned}

Example: S^n
Let X=S^n with A and B the northern and southern hemispheres, so that A\cap B=S^{n-1}. Then in the reduced Mayer-Vietoris sequence the terms \tilde{H}_i(A)\oplus\tilde{H}_i(B) are zero. So from the reduced Mayer-Vietoris sequence \displaystyle \dots\to\tilde{H}_i(A)\oplus\tilde{H}_i(B)\to\tilde{H}_i(X)\to\tilde{H}_{i-1}(A\cap B)\to\tilde{H}_{i-1}(A)\oplus\tilde{H}_{i-1}(B)\to\dots we get the exact sequence \displaystyle 0\to\tilde{H}_i(S^n)\to\tilde{H}_{i-1}(S^{n-1})\to 0.
We obtain isomorphisms \tilde{H}_i(S^n)\cong\tilde{H}_{i-1}(S^{n-1}).

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Resucitó (Spanish Easter Song)

Wishing all readers a happy Easter!

This is a famous Spanish Easter Song: Resucitó, with translation.

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Spectral Sequence

Spectral Sequence is one of the advanced tools in Algebraic Topology. The following definition is from Hatcher’s 5th chapter on Spectral Sequences. The staircase diagram looks particularly impressive and intimidating at the same time.

Unfortunately, my LaTeX to WordPress Converter app can’t handle commutative diagrams well, so I will upload a printscreen instead.

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