Tag Archives: Math

Hokkaido University Photo Gallery and Trip Guide

These photos are from a trip to Hokkaido University in August 2017. Hokkaido University is just walking distance away from Sapporo station, and is worth spending an afternoon there. Admission is free. Even in the hottest summer, Hokkaido has cool … Continue reading

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8 Facts About Infinity That Will Blow Your Mind

Nice article on infinity. Also little known is the fact that the symbol of infinity was introduced by clergyman and mathematician John Wallis, hundreds of years ago in 1655. Although not well-known, John Wallis was a talented individual as can … Continue reading

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Free Math Public Lecture at National Library

Interested students may want to attend. It is at the National Library, on 14th December 2017. The Liar Game: Truths & Proofs from Euclid to Turing Ng Kong Beng Public Lecture Series Mark Wildon, Royal Holloway, University of London, UK … Continue reading

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Union-closed sets conjecture

Just read about this conjecture: Union-closed sets conjecture. The conjecture states that for any finite union-closed family of finite sets, other than the family consisting only of the empty set, there exists an element that belongs to at least half of the … Continue reading

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He who loves learning is better than he who knows how to learn (Confucius)

From Baidu Baike: 知之者不如好之者,好之者不如乐之者: 对于学习,了解怎么学习的人,不如喜爱学习的人;喜爱学习的人,又不如以学习为乐的人。比喻学习知识或本领,知道它的人不如爱好它的人接受得快,爱好它的人不如以此为乐的人接受得更快。 Translation: He who knows how to learn, is not as good as he who likes learning. He who likes learning, is not as good as he who loves learning. (Confucius) I guess this applies to … Continue reading

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It’s mathematically impossible to beat aging, scientists say

According to Math, no one can live forever. So far, the only counterexample that I know of is Turritopsis dohrnii, also known as the “immortal jellyfish”. The article doesn’t seem to address this counterexample though. Source: Science Daily Aging is a … Continue reading

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Leibniz was a universal genius, but why is Isaac Newton more known? Does it have to do with Newton being British and Leibniz being German?

Answer by Albert Heisenberg, Science Historian M.A. Brown University Leibniz’s formulation of differential and integral calculus was more refined, elegant, and ‘generalizable’ than Newton’s ‘fluxions.’ Leibniz, the natural genius that he was, became interested in mathematics much later in his life … Continue reading

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Note on p-divisibility in Bockstein Spectral Sequence

If generates a copy of , then . Write , where , . Note that since , hence does not generate a copy of . The only way that is possible is when , i.e .

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Terence Tao Numberphile

Among current mathematicians, many people regard Professor Tao as the world’s finest… Opinions on such things vary, of course. Professor Tao kindly fielded some of our questions, including many submitted by Numberphile viewers. EXTRA FOOTAGE: https://youtu.be/48Hr3CT5Tpk (and more extras to … Continue reading

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“Actual” GEP Questions 2017 (from Forum)

Since the actual GEP papers are never released, the next best source is from those who have actually taken it and post on forums like Kiasuparents. Some Maths questions my girl remembers. “ In a fishing competition, five kids caught … Continue reading

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Why do people get so anxious about math? – Orly Rubinsten

View full lesson: http://ed.ted.com/lessons/why-do-peop… Have you ever sat down to take a math test and immediately felt your heart beat faster and your palms start to sweat? This is called math anxiety, and if it happens to you, you’re not … Continue reading

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Free Math Games For Everyone

Free Math Games For Everyone Multiplication of big numbers, complex mathematical problems – there are so many issues that people can face not only at school but also in their everyday lives! People need math everywhere and always! That is … Continue reading

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Recommended Books for Spectral Sequences

Best Spectral Sequence Book So far the most comprehensive book looks like McCleary’s book: A User’s Guide to Spectral Sequences. It is also suitable for those interested in the algebraic viewpoint. W.S. Massey wrote a very positive review to this … Continue reading

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Interesting Facts about Green’s Theorem

Firstly, Green’s Theorem is named after the mathematician George Green (14 July 1793 – 31 May 1841). Something remarkable about George Green is that he is almost entirely self-taught. He only went to school for one year (when he was 8 years … Continue reading

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Pierre-Simon de Laplace; French Newton

To increase your interest in mathematics, let me introduce the French mathematician Pierre-Simon de Laplace, also known as the “French Newton” or “Newton of France”. He helped to calculate projectile motion for Napoleon’s artillery. Laplace was also the examiner for Napoleon when … Continue reading

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5 Skills Students Need To Cope With School Pressures

According to an article published by the American Psychological Association (APA), many teenagers in the USA say they experience stress in patterns comparable to what adults go through. Teenagers also report higher stress levels than adults during the school year. … Continue reading

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Best Online Resources to Improve Your Math Skills

Best Online Resources to Improve Your Math Skills Although Maths is a compulsory subject in all the educational institutions all over the world, many students consider it as a complete waste of time and skip this issue, claiming that they … Continue reading

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An ancient Babylonian tablet known as Plimpton 322

Source: NY Times One of my favorite YouTube Math Professors, Norman Wildberger, has made a historical math discovery: that the ancient Babylonian tablet known as Plimpton 322 is actually a trigonometric table. “It’s a trigonometric table, which is 3,000 years … Continue reading

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Education and the Blockchain – Should We be Teaching Blockchain in Schools?

Source: https://preply.com/ It goes without saying that tech progress is moving at a rapid pace. Futurists point to Moore’s law – the idea that tech capabilities double every two years – as evidence for tech’s expansion into nearly every facet of … Continue reading

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The scientist nuns: In pursuit of faith and reason

Source: Aleteia Making a career out of science, just like joining a religious order, requires dedication and discipline. Some tireless souls have managed to do both. In 1965, Mary Kenneth Keller became the first woman to obtain a PhD in … Continue reading

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Theorem of the Day

Just to recommend this excellent website: Theoremoftheday where they feature one mathematical theorem each day. The nice thing is that each theorem is a one-page summary, good for getting acquainted with the theorem, and subsequently you may read it up in more … Continue reading

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Topology application to Physics

Source: https://www.scientificamerican.com/article/the-strange-topology-that-is-reshaping-physics/?W The Strange Topology That Is Reshaping Physics Topological effects might be hiding inside perfectly ordinary materials, waiting to reveal bizarre new particles or bolster quantum computing Charles Kane never thought he would be cavorting with topologists. “I don’t think … Continue reading

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How to do Proof by Cases in LaTeX

If one searches online, one will find many different methods to do “proof by cases” in LaTeX. The most simple and convenient method in my opinion is to use the description environment. Something like this: \begin{proof} Proceed by cases. \begin{description} … Continue reading

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(Important Changes) PSLE Math: Arrow -> vs Equal=

Source: Facebook For those taking PSLE, please take note of this important update regarding the difference between arrow and equal sign. Forward this to your friends taking PSLE! Basically, I think MOE is trying to instill students to be mathematically … Continue reading

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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 … Continue reading

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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. Site: https://www.wired.com/story/the-mind-boggling-math-that-maybe-mapped-the-brain-in-11-dimensions/ If you can call understanding the dynamics of a virtual rat brain a real-world problem. In a multimillion-dollar supercomputer … Continue reading

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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. Site: http://www.usmagazine.com/entertainment/news/americas-got-talent-contestant-performs-crazy-magic-trick-w488672 London had the volunteers … Continue reading

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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 … Continue reading

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Subtle Error in Wikipedia: Dedekind’s number

On Wikipedia (https://en.wikipedia.org/wiki/Dedekind_number), 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 … Continue reading

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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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Math Tricks found in Chess

Just read this very nice article on Quora, on the relationship between Math and Chess: https://www.quora.com/What-math-tricks-are-hidden-in-chess Also interesting is this YouTube documentary “My Brilliant Brain” featuring Susan Polgar. Author: Tom Boshoff, Engineering student and math enthusiast Updated May 8 There’s lots … Continue reading

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Math Olympiad Tuition

Maths Olympiad Tuition Tutor: Mr Wu (Raffles Alumni, NUS Maths Grad) SMS/Whatsapp: 98348087 Email: mathtuition88@gmail.com 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 … Continue reading

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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, … Continue reading

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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, … Continue reading

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Renowned Chinese mathematician Wu Wenjun dies at 98

Source: https://news.cgtn.com/news/3d517a4e33637a4d/share_p.html 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 … Continue reading

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

Set and . The diagram then has the following form: When we pass to the derived couple, each group is replaced by a subgroup . The differentials go two units to the right, and we replace the term by the … Continue reading

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Relative Homology Groups

Given a space and a subspace , define . Since the boundary map takes to , it induces a quotient boundary map . We have a chain complex where holds. The relative homology groups  are the homology groups of this … Continue reading

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Exact sequence (Quotient space)

Exact sequence (Quotient space) If is a space and is a nonempty closed subspace that is a deformation retract of some neighborhood in , then there is an exact sequence where is the inclusion and is the quotient map . … Continue reading

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Reduced Homology

Define the reduced homology groups to be the homology groups of the augmented chain complex where . We require to be nonempty, to avoid having a nontrivial homology group in dimension -1. Relation between and Since , vanishes on and … Continue reading

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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! Source: https://math.stackexchange.com/questions/907176/klein-bottle-as-two-m%C3%B6bius-strips

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Mayer-Vietoris Sequence applied to Spheres

Mayer-Vietoris Sequence For a pair of subspaces such that , the exact MV sequence has the form Example: Let with and the northern and southern hemispheres, so that . Then in the reduced Mayer-Vietoris sequence the terms are zero. So … Continue reading

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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 … Continue reading

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Real World Applications of Algebra, Geometry and Topology

Quite a nice video here:

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SO(3) diffeomorphic to RP^3

} Proof: We consider as the group of all rotations about the origin of under the operation of composition. Every non-trivial rotation is determined by its axis of rotation (a line through the origin) and its angle of rotation. We … Continue reading

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SU(2) diffeomorphic to S^3 (3-sphere)

(diffeomorphic) Proof: We have that Since , we may view as Consider the map It is clear that is well-defined since if , then . If , it is clear that . So is injective. It is also clear that … Continue reading

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Persistent Homology Algorithm

Algorithm for Fields In this section we describe an algorithm for computing persistent homology over a field. We use the small filtration as an example and compute over , although the algorithm works for any field. A filtered simplicial complex … Continue reading

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To Live Your Best Life, Do Mathematics

This article is a very good read. 100% Recommended to anyone interested in math. The ancient Greeks argued that the best life was filled with beauty, truth, justice, play and love. The mathematician Francis Su knows just where to find … Continue reading

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viXra vs arXiv

viXra (http://vixra.org/) is the cousin of arXiv (http://arxiv.org/) which are electronic archives where researchers can submit their research before being published on a journal. The difference is that viXra allows anyone to submit their article, whereas arXiv requires an academic … Continue reading

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Free Math Notes by AMS

Just learnt from Professor Terence Tao’s blog that there is a new series of free math notes by the American Mathematical Society: http://www.ams.org/open-math-notes. Many of the notes there are of exceptionally high quality (check out “A singular mathematical promenade”, by Étienne Ghys). … Continue reading

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Proof of Equivalent Conditions for Split Exact Sequence

Attached is a proof of the equivalent conditions for a Split Exact Sequence, based on the nice proof in Hungerford using the Short Five Lemma. Very neat proof. Split Exact Sequence Proof

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