Tag Archives: Math

(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

Posted in math | Tagged , | Leave a comment

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

Posted in math | Tagged , | Leave a comment

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

Posted in math | Tagged , , | Leave a comment

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

Posted in math | Tagged , | Leave a comment

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

Posted in math | Tagged , , | Leave a comment

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

Posted in math | Tagged | Leave a comment

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.

Posted in math | Tagged , , , | Leave a comment

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

Posted in math | Tagged , | Leave a comment

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

Posted in math | Tagged , | Leave a comment

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

Posted in math | Tagged , | Leave a comment

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

Posted in math | Tagged , | Leave a comment

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

Posted in math | Tagged | Leave a comment

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

Posted in math | Tagged , | Leave a comment

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

Posted in math | Tagged , | Leave a comment

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

Posted in math | Tagged , , | Leave a comment

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

Posted in math | Tagged , , | 2 Comments

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

Posted in math | Tagged , | Leave a comment

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

Posted in math | Tagged , | Leave a comment

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

Posted in math | Tagged , | Leave a comment

Real World Applications of Algebra, Geometry and Topology

Quite a nice video here:

Posted in math | Tagged | 2 Comments

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

Posted in math | Tagged , | Leave a comment

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

Posted in math | Tagged , | Leave a comment

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

Posted in math | Tagged , , | Leave a comment

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

Posted in math | Tagged | 3 Comments

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

Posted in math | Tagged , | Leave a comment

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

Posted in math | Tagged | Leave a comment

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

Posted in math | Tagged , | Leave a comment

Recent Interview of Shing-Tung Yau (in Chinese)

Excellent interview of S.T. Yau, Fields Medalist. One mischievous student tried to ask a trick question that is a variant of the Missing Dollar Problem. The interviewer is Sa Beining, who is a famous celebrity in China. Not much mathematical content … Continue reading

Posted in math | Tagged , | Leave a comment

The Reason Why Singaporean Students are Top in Maths (PISA)

Quite interesting analysis on how and why Singapore topped the ranking for PISA in Math/Science. One possible reason is the difficulty of PSLE trains students to solve tricky and difficult (for that level) math questions. It is well known that … Continue reading

Posted in math | Tagged , | 2 Comments

Normal Extension

An algebraic field extension is said to be normal if is the splitting field of a family of polynomials in . Equivalent Properties The normality of is equivalent to either of the following properties. Let be an algebraic closure of … Continue reading

Posted in math | Tagged , | Leave a comment

Some Linear Algebra Theorems

Linear Algebra Diagonalizable & Minimal Polynomial: A matrix or linear map is diagonalizable over the field if and only if its minimal polynomial is a product of distinct linear factors over . Characteristic Polynomial: Let be an matrix. The characteristic … Continue reading

Posted in math | Tagged , | Leave a comment

Even physicists are ‘afraid’ of mathematics

Interesting news, since it is widely known that physicists are the most mathematically literate out of all the sciences. Perhaps what the research really shows is that huge chunks of equations may obscure the meaning of the research and thus is … Continue reading

Posted in math | Tagged | 1 Comment

Sufficient condition for “Weak Convergence”

This is a sufficient condition for something that resembles “Weak convergence”: for all Suppose that a.e.\ and that , . If , we have for all , . Note that the result is false if . Proof: (Case: , where … Continue reading

Posted in math | Tagged , | Leave a comment

Square root x is not Lipschitz on [0,1]

is not Lipschitz on : Suppose there exists such that for all , , By Mean Value Theorem, this means that for some between and . However, is unbounded on , a contradiction. Note however, that is absolutely continuous on … Continue reading

Posted in math | Tagged , | Leave a comment

Young’s Convolution Theorem

Let and , and let . If and , then and Amazing Theorem! If , then .

Posted in math | Tagged , | Leave a comment

Relationship between L^p convergence and a.e. convergence

It turns out that convergence in Lp implies that the norms converge. Conversely, a.e. convergence and the fact that norms converge implies Lp convergence. Amazing! Relationship between convergence and a.e. convergence: Let , . If , then . Conversely, if … Continue reading

Posted in math | Tagged , | Leave a comment

98-Year-Old NASA Mathematician Katherine Johnson: ‘If You Like What You’re Doing, You Will Do Well’

Source: http://people.com/human-interest/nasa-katherine-johnson-mathematician-advice-interview/ Despite her age, Johnson isn’t slowing down anytime soon. “I like to learn,” she says. “That’s an art and a science. I’m always interested in learning something new.” As a young girl she’d stop by the library on her … Continue reading

Posted in math | Tagged , , , | Leave a comment

Donald Trump’s Answer to Math Question: 2+2=?

Source: http://www.attn.com/stories/6407/george-takei-impersonates-donald-trump Question: What is 2+2? Answer: “I have to say a lot of people have been asking this question. No, really. A lot of people come up to me and they ask me. They say, ‘What’s 2+2’? And I tell … Continue reading

Posted in america, california, United States | Tagged , , , , , | Leave a comment

Wheeden Zygmund Measure and Integration Solutions

Here are some solutions to exercises in the book: Measure and Integral, An Introduction to Real Analysis by Richard L. Wheeden and Antoni Zygmund. Done by a graduate student, so there may be some errors. Chapter 1,2: analysis1 Chapter 3: analysis2 Chapter … Continue reading

Posted in math | Tagged , | Leave a comment

Absolute Continuity of Lebesgue Integral

The following is a wonderful property of the Lebesgue Integral, also known as absolute continuity of Lebesgue Integral. Basically, it means that whenever the domain of integration has small enough measure, then the integral will be arbitrarily small. Suppose is … Continue reading

Posted in math | Tagged , | Leave a comment

Why Math Education in the U.S. Doesn’t Add Up

The U.S. has some of the best universities in Math (think Harvard, Princeton, MIT), however the state of high school math is subpar and well below other developed nations. The main reason, according to this article, is the curriculum that … Continue reading

Posted in america, United States | Tagged , | 2 Comments

Support

Mathtuition88.com is a Math Education Blog that aims to provide useful Math / Education content that helps people worldwide. Over the years, Mathtuition88.com has grown to reach 500,000 total views, thanks to support of fans! If you find our website … Continue reading

Posted in math | Tagged , | Leave a comment

Fatou’s Lemma for Convergence in Measure

Suppose in measure on a measurable set such that for all , then . The proof is short but slightly tricky: Suppose to the contrary . Let be a subsequence such that (using the fact that for any sequence there … Continue reading

Posted in math | Tagged , | Leave a comment

Summation by parts / Abel’s Lemma

This is an amazing identity by Abel. Let and be two sequences. Then,  

Posted in math | Tagged , , | Leave a comment

Lebesgue’s Dominated Convergence Theorem for Convergence in Measure

Lebesgue’s Dominated Convergence Theorem for Convergence in Measure If satisfies on and , then and . Proof Let be any subsequence of . Then on . Thus there is a subsequence a.e.\ in . Clearly . By the usual Lebesgue’s … Continue reading

Posted in math | Tagged , | Leave a comment

Trigo Formulae

The following formulae will be useful when integrating Trigonometric functions. Taken from the MF15 formula sheet for JC. Addition Formulae Double Angle Formulae Remark: The second identity is useful for integrating and . Factor Formulae Remark: The factor formulae are … Continue reading

Posted in math | Tagged , | Leave a comment

Basel Problem using Fourier Series

A very famous mathematical problem known as the “Basel Problem” is solved by Euler in 1734. Basically, it asks for the exact value of . Three hundred years ago, this was considered a very hard problem and even famous mathematicians of … Continue reading

Posted in math | Tagged , , , | 3 Comments

A Limit that Converges to e

(L’Hopital’s Rule Proof) This limit is a useful and interesting result to know. Note especially that the method “” is incorrect. Proof: We will prove instead, and this implies First, we will find the limit . So . Exercise If … Continue reading

Posted in math | Tagged | Leave a comment

Laurent Series with WolframAlpha

WolframAlpha can compute (simple) Laurent series: https://www.wolframalpha.com/input/?i=series+sin(z%5E-1) Series[Sin[z^(-1)], {z, 0, 5}] 1/z-1/(6 z^3)+1/(120 z^5)+O((1/z)^6) (Laurent series) (converges everywhere away from origin) Unfortunately, more “complex” (pun intended) Laurent series are not possible for WolframAlpha.

Posted in math | Tagged , , | Leave a comment

Mathematicians Are Overselling the Idea That “Math Is Everywhere”

This article provides an alternative viewpoint on whether mathematics is useful to society. A good read if you are writing a GP (General Paper) essay on the usefulness of mathematics, to provide both sides of the argument. Source: http://blogs.scientificamerican.com/guest-blog/mathematicians-are-overselling-the-idea-that-math-is-everywhere/?WT.mc_id=SA_WR_20160817 Excerpt: Most people … Continue reading

Posted in math | Tagged , , , | 1 Comment