Tag Archives: Math

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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Gradient Theorem (Proof)

This amazing theorem is also called the Fundamental Theorem of Calculus for Line Integrals. It is quite a powerful theorem that sometimes allows fast computations of line integrals. Gradient Theorem (Fundamental Theorem of Calculus for Line Integrals) Let be a … Continue reading

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Liouville’s Theorem

Liouville’s Theorem Every bounded entire function must be constant. That is, every holomorphic function for which there exists such that for all is constant.

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Multivariable Version of Taylor’s Theorem

Multivariable calculus is an interesting topic that is often neglected in the curriculum. Furthermore it is hard to learn since the existing textbooks are either too basic/computational (e.g. Multivariable Calculus, 7th Edition by Stewart) or too advanced. Many analysis books skip multivariable calculus … Continue reading

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Pasting Lemma (Elaboration of Wikipedia’s proof)

The proof of the Pasting Lemma at Wikipedia is correct, but a bit unclear. In particular, it does not clearly show how the hypothesis that X, Y are both closed is being used. It actually has something to do with subspace … Continue reading

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One-Sided Limit that Does Not Exist

Offhand, it is hard to think of a function that does not have even a one-sided limit. This video shows one!

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Convolution Video

This is one of the more illuminating videos on Convolution. I believe after watching it, many will have a better understanding of what is the convolution.

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Algebra and Analysis Theorems

The following are two lists of useful algebra and analysis theorems that are covered during university. Algebra Theorems Mathtuition88 Analysis Theorems Mathtuition88

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Maths Tuition – What are the Benefits?

Maths Tuition – What are the Benefits? Maths tuition brings about many benefits that can be seen for the parent, the teacher and especially the student who is struggling with their mathematics subject in school. For starters, it will have … Continue reading

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Lusin’s Theorem and Egorov’s Theorem

Lusin’s Theorem and Egorov’s Theorem are the second and third of Littlewood’s famous Three Principles. There are many variations and generalisations, the most basic of which I think are found in Royden’s book. Lusin’s Theorem: Informally, “every measurable function is … Continue reading

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Why Singapore’s kids are so good at maths

Source: http://www.ft.com/cms/s/0/2e4c61f2-4ec8-11e6-8172-e39ecd3b86fc.html Sie Yu Chuah smiles when asked how his parents would react to a low test score. “My parents are not that strict but they have high expectations of me,” he says. “I have to do well. Excel at my … Continue reading

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The most Striking Theorem in Real Analysis

Lebesgue’s Theorem (see below) has been called one of the most striking theorems in real analysis. Indeed it is a very surprising result. Lebesgue’s Theorem (Monotone functions) If the function is monotone on the open interval , then it is … Continue reading

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There are two kinds of talented students.

Just read this interesting article. Will the new PSLE system reward students of the first kind or second kind? From my experience as student and tutor, Singapore has many talented students of the first kind, but very few talented students of … Continue reading

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小学生厌恶数学写诗:数学是死亡之源

Source: http://news.sina.com.cn/s/2015-02-08/031931495241.shtml 武汉的董女士前天在家帮女儿清理书包,从书包里搜出一张纸,上面赫然写着一首诗:数学是死亡之源,它像入地狱般痛苦。它让孩子想破脑汁,它让家长急得转圈。它让校园死气沉沉,它使生命慢慢离去。生命从数学中走去,一代代死得超快。那是生命的敌人,生命从数学中走去。珍惜宝贵的生命吧,一代代死得超快。数学是死亡之源。 读完这首诗,董女士惊呆了:“没想到她厌恶数学到了这般田地。” 董女士的女儿晶晶,今年10岁,读小学五年级,从进小学开始就特别不喜欢数学,尤其讨厌应用题,只要碰到追及问题和工程问题,晶晶那绝对是“一个脑袋两个大”。这次期末考试晶晶的数学考了70分,全班倒数第七。 在董女士的逼问下,晶晶终于交代,这首诗是和班上另外两个女生一起创作的。她们三个都对数学不感冒,联合创作了此诗,抒发忧伤。 网友们看到这首诗后,也表达了不同的观点。网友“@飞不动的咋咋鸟”说,“很有才的小学生,这搞不好以后是余秀华第二啊!”网友“@左边追寻”则说,“想用自己的血泪史告诉这位妹妹,学好数学很重要!”据《武汉晚报》 (原标题:小学生厌恶数学写诗:数学是死亡之源)

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