Tag Archives: Math

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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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

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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

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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

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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

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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

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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

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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

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Young’s Convolution Theorem

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

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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

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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

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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

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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

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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

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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

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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

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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

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Summation by parts / Abel’s Lemma

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

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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

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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

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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

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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

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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.

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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

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Laurent Series (Example)

The Laurent series is something like the Taylor series, but with terms with negative exponents, e.g. . The below Laurent Series formula may not be the most practical way to compute the coefficients, usually we will use known formulas, as … Continue reading

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Implicit Function Theorem

The implicit function theorem is a strong theorem that allows us to express a variable as a function of another variable. For instance, if , can we make the subject, i.e. write as a function of ? The implicit function … Continue reading

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Differentiable Manifold

Differentiable manifold An -dimensional (differentiable) manifold is a Hausdorff topological space with a countable (topological) basis, together with a maximal differentiable atlas. This atlas consists of a family of charts where the domains of the charts, , form an open … Continue reading

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Lie Groups

One of the best books on Lie Groups is said to be Representations of Compact Lie Groups (Graduate Texts in Mathematics). It is one of the rarer books from the geometric approach, as opposed to the algebraic approach. Lie group A … Continue reading

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lim sup & lim inf of Sets

The concept of lim sup and lim inf can be applied to sets too. Here is a nice characterisation of lim sup and lim inf of sets: For a sequence of sets , consists of those points that belong to … Continue reading

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What is a Degree in Math and Why is it Valuable?

Very interesting article on why you should consider a degree in math if you are interested in math. Source: http://www.snhu.edu/about-us/news-and-events/2016/08/what-is-a-degree-in-math-and-why-is-it-valuable Mathematics is the study of quantity, structure, space and change. As abstract as that may seem, math is, at its core, … Continue reading

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Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is one of the most amazing and important theorems in analysis. It is a non-trivial result that links the concept of area and gradient, two seemingly unrelated concepts. Fundamental Theorem of Calculus The first part … Continue reading

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