August 2015 – Mathtuition88

Geometric n-simplex is convex

Given the definition of a geometric n-simplex:

$\displaystyle\sigma^n=\{x=\sum_{i=0}^{n}t_i a^i \mid t_i\geq 0\ \text{and }\sum_{i=0}^{n}=1\}\subseteq\mathbb{R}^n$

where $\{a^0,\dots, a^n\}$ are geometrically independent, we can show that the n-simplex is convex (i.e. given any two points, the line connecting them lies in the simplex).

Write $x=\sum_{i=0}^n t_i a^i$ , $y=\sum_{i=0}^n s_i a^i$ .

Consider the line from x to y: $\{ty+(1-t)x\mid 0\leq t\leq 1\}$ .

$\begin{aligned} ty+(1-t)x&=t\sum_{i=0}^n s_i a^i+(1-t)\sum_{i=0}^n t_i a^i\\ &=\sum_{i=0}^n (s_i t+t_i-tt_i)a_i\\ s_it+t_i-tt_i&=s_i t+t_i (1-t)\\ &\geq 0(0)+(0)(1-1)\\ &=0\\ \sum_{i=0}^n s_i t+t_i-tt_i &=t\sum_{i=0}^n s_i+\sum_{i=0}^n t_i -t\sum_{i=0}^n t_i\\ &=t(1)+(1)-t(1)\\ &=1 \end{aligned}$

Thus the line lies inside the simplex, and thus the simplex is convex.

Recommended Books for Math Majors

Life Algebra

Math Online Tom Circle

How to solve this ‘Life’ Algebra ?

The simultatneous inequality equation with 3 unknowns (t, e, m).

It has no solution but we can get the BEST approximation :
Retire after 55 before 60, then you get optimized {e, t, m} — still have good energy (e) with plenty of time (t) and sufficient pension money (m) in CPF & investment saving.

Beyond 60 if continuing to work, the solution of {e, t, m} -> {0, 0, 0}.

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Tiger Mom Amy Chua Sets Up Tuition Center in Singapore

Source: http://www.cnbc.com/2015/08/20/queen-of-the-tiger-moms-takes-on-singapore.html

“Tiger Mom” Amy Chua has started a Tuition Center in Singapore. Amy Chua is the famous author of Battle Hymn of the Tiger Mother, which is an interesting book which has both supporters and critics.

The Tuition Centre is called Keys Academy, located in North Bridge Road, Singapore.

Do check it out, and more importantly do read the book Battle Hymn of the Tiger Mother to see if you agree with the author! Amy Chua does have some good points to be made, as many of the top students in Western countries are Asians. Do check out her book to read about her method.

Some other books written by Amy Chua are:

The Triple Package: How Three Unlikely Traits Explain the Rise and Fall of Cultural Groups in America

Day of Empire: How Hyperpowers Rise to Global Dominance–and Why They Fall

World on Fire: How Exporting Free Market Democracy Breeds Ethnic Hatred and Global Instability

Cycle Decomposition of Permutations is Unique

Cycle decomposition of Permutations into disjoint cycles is unique (up to reordering of cycles).

A proof can be found here.

The condition of disjoint is crucial. For example, the permutation (1 3 2) can be factored into (2 3)(1 2), where the two cycles are not disjoint. (1 3 2)=(1 2)(1 3) is also another decomposition, the two cycles are also not disjoint.

Wolframalpha can calculate permutations, a useful tool to replace manual calculations. Take note though that Wolframalpha’s convention is multiplying permutations from left to right, while most books follow the convention of multiplying right to left.

Recommended Math Books from Amazon

“偷得浮生半日闲”诗句分析—出现于中三课文《乌敏岛》

Chinese Tuition Singapore

在中三下华文课本中，有一篇题目为《乌敏岛》的课文。课文介绍了乌敏岛的自然风光和纯朴的人文环境。在文章的最后，作者写道“如果能’偷得浮生半日闲’，何不暂时摆脱现实的束缚，和三五好友到乌敏岛游玩呢？”。

“偷得浮生半日闲”出自于唐代诗人李涉的七言绝句《题鹤林寺僧舍》。全文如下：

终日昏昏醉梦间，忽闻春尽强登山。

因过竹院逢僧话，偷得浮生半日闲。

大意是：作者整日昏昏沉沉处于醉梦之中，消磨人生。忽然有一天才意识到春天就要过去了，于是勉强去爬山。在游览寺院的时候碰到一位高僧，便与其闲聊，难得在这纷纷扰扰的世事中获得片刻的清闲。

这首诗的创作背景是李涉官途不顺，被皇帝贬官后又流放到南方，所以其情绪消极终日昏昏沉沉。而在一次偶然机会，登山之时偶遇高僧，闲聊之中，不料解开了苦闷的心结，化解了世俗的烦扰，使得自己心情得以放松。

再回到课文《乌敏岛》，从课文的开始，作者就强调“踏上乌敏岛，映入眼帘的是一幅和繁忙市区截然不同的景象”。市区的人们熙熙攘攘，为生活而忙于奔走，有很多世事的烦扰。而乌敏岛却是一个别样的世界，这里没有喧嚣，人们的生活简单平静而又质朴。来到这里，看看美不胜收的风景，体验淳朴宁静的生活，相信你也会暂时忘记现实生活的烦恼和忧愁。

“偷得浮生半日闲”，人生漂浮不定，难得半日的清闲。

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Weierstrass M-test Proof and Special Case of Abel’s Theorem

First, let us recap what is Weierstrass M-test:

Weierstrass M-test:

Let $\{f_n\}$ be a sequence of real (or complex)-valued functions defined on a set A, and let $\{M_n\}$ be a sequence satisfying $\forall n\in\mathbb{N}, \forall x\in A$

$|f_n (x)|\leq M_n$ , and also $\sum_{n=1}^\infty M_n=M<\infty$ .

Then, $\sum_{n=1}^\infty f_n(x)$ converges uniformly on A (to a function f).

Proof:

Let $\epsilon >0$ . $\exists N\in\mathbb{N}$ such that $m\geq N$ implies $|M-\sum_{n=1}^m M_n|<\epsilon$ .

For $m\geq N, \forall x\in A$ ,

$\begin{aligned} |f(x)-\sum_{n=1}^m f_n(x)|&=|\sum_{n=m+1}^\infty f_n (x)|\\ &\leq\sum_{n=m+1}^\infty |f_n (x)|\\ &\leq \sum_{n=m+1}^\infty M_n\\ &=|M-\sum_{n=1}^m M_n|\\ &<\epsilon \end{aligned}$

Thus, $\sum_{n=1}^\infty f_n (x)$ converges uniformly.

Application to prove Abel’s Theorem (Special Case):

Consider the special case of Abel’s Theorem where all the coefficients $a_i$ are of the same sign (e.g. all positive or all negative).

Then, for $x\in [0,1]$ ,

$|a_n x^n|\leq |a_n|:=M_n$

Then by Weierstrass M-test, $\sum_{n=1}^\infty a_n x^n$ converges uniformly on [0,1] and thus $\lim_{x\to 1^-} \sum_{n=1}^\infty a_n x^n=\sum_{n=1}^\infty a_n$ .

Check out some books suitable for Math Majors here!

If Ratio Test Limit exists, then Root Test Limit exists, and both are equal

The limit for ratio test is $\lim_{n\to\infty} \frac{|a_{n+1}|}{|a_n|}$ , while the limit for root test is $\lim_{n\to\infty}|a_n|^{1/n}$ . Something special about these two limits is that if the former exists, the latter also exists and they are equal!

Proof:

Let $\lim_{n\to\infty}|\frac{a_{n+1}}{a_n}|=L$ . There exists $N\in\mathbb{N}$ such that $n\geq N \implies ||\frac{a_{n+1}}{a_n}|-L|<\epsilon$ .

i.e. $L-\epsilon<|\frac{a_{n+1}}{a_n}|<L+\epsilon$

For $n>N$ ,

$|a_n|=\frac{|a_n|}{|a_{n-1}|}\cdot \frac{|a_{n-1}|}{|a_{n-2}|}\cdots \frac{|a_{N+1}|}{|a_N|}\cdot |a_N| < (L+\epsilon)^{n-N}\cdot |a_N|$ .

Taking nth roots,

$|a_n|^{1/n}<(L+\epsilon)^\frac{n-N}{n}\cdot |a_N|^{\frac{1}{n}}$

Taking limits,

$\lim_{n\to\infty}|a_n|^{\frac{1}{n}}\leq (L+\epsilon)$

Since $\epsilon$ is arbitrary, $\lim_{n\to\infty}|a_n|^{\frac{1}{n}}\leq L$ .

Similarly, we can show $\lim_{n\to\infty}|a_n|^\frac{1}{n}\geq L$ .

Thus, $\lim_{n\to\infty}|a_n|^\frac{1}{n}=L$ .

This is considered a rather tricky (though not that difficult) proof, hope it helps whoever is searching for it!

Note that the converse is false, we can see that by considering the “rearranged” geometric series: 1/2,1, 1/8, 1/4, 1/32, … (source: https://www.maa.org/sites/default/files/0025570×33450.di021200.02p0190s.pdf)

where the ratio alternates from 2 to 1/8 and hence does not exist.

However, the root test limit of the first 2n terms is defined:

$\begin{aligned} |a_{2n}|&=\frac{|a_{2n}|}{a_{2n-1}}\cdot \frac{|a_{2n-1}|}{|a_{2n-2}|}\cdot \frac{|a_2|}{|a_1|}\cdot |a_1|\\ &=2\cdot \frac{1}{8}\cdot 2 \cdot \frac{1}{8} \cdots 2 \cdot \frac{1}{2}\\ &=2^n \cdot (\frac{1}{8})^{n-1}\cdot \frac{1}{2}\\ &=(\frac{1}{4})^{n-1} \end{aligned}$

Thus, $|a_{2n}|^\frac{1}{2n}=\frac{1}{4}^{\frac{n-1}{2n}}\to \frac{1}{2}$ .

To learn more about epsilon-delta proofs, check out one of the Recommended Analysis Books for Undergraduates.

Z[Sqrt(-2)] is a Principal Ideal Domain Proof

It turns out that to prove $\mathbb{Z}[\sqrt{-2}]$ is a Principal Ideal Domain, it is easier to prove that it is a Euclidean domain, and hence a PID.

(Any readers who have a direct proof that $\mathbb{Z}[\sqrt{-2}]$ is a PID, please comment below, as it would be very interesting to know such a proof. 🙂 )

Proof:

As mentioned above, we will prove that it is a Euclidean domain.

Let $a, b\in\mathbb{Z}[\sqrt{-2}], b\neq 0$ .

We need to show: $\exists q, r\in \mathbb{Z}[\sqrt{-2}]$ such that $a=bq+r$ , with $N(r)<N(b)$ .

Consider $\frac{a}{b}=c_1+c_2 \sqrt{-2} \in \mathbb{Q}[\sqrt{-2}]$ . Define $q=q_1+q_2 \sqrt{-2}$ where $q_1, q_2$ are the integers closest to $c_1, c_2$ respectively.

Then, $\frac{a}{b}=q+\alpha$ , where $\alpha=\alpha_1+\alpha_2 \sqrt{-2}$ .

$a=bq+b\alpha$ .

Take $r=b\alpha$ .

$\begin{aligned} N(r)&=N(b\alpha)\\ &=N(b)\cdot N(\alpha)\\ &=N(b)\cdot (\alpha_1^2+2\alpha_2^2)\\ &\leq N(b)\cdot ({\frac{1}{2}}^2+2(\frac{1}{2})^2)\\ &=N(b)\cdot (\frac{3}{4})\\ &<N(b) \end{aligned}$

Check out recommended Abstract Algebra books: Recommended Books for Math Undergraduates

Is Z[x] a Principal Ideal Domain?

In the previous post, we showed that a Euclidean domain is a Principal Ideal Domain (PID).

Consider the Polynomial Ring $\mathbb{Z}[x]$ . We can show that it is not a PID and hence also not a Euclidean domain.

Proof: Consider the ideal $<2,x>=\{ 2f(x)+xg(x)\vert f(x), g(x) \in \mathbb{Z} [x]\}$ .

Suppose to the contrary $<2,x>=<p(x)>=\{ f(x)p(x)\vert f(x)\in \mathbb{Z}[x]\}$ .

Note that $2\in <2,x>$ , hence $2\in <p(x)>$ .

2=f(x)p(x)

p(x)=2 or -2.

<p(x)>=<2>

However, $x\in <2,x>$ but $x\notin <2>$ . (contradiction!)

Check out this page for Recommended Singapore Math books!

Proof that a Euclidean Domain is a PID (Principal Ideal Domain)

Previously, we defined what is a Euclidean Domain and what is a PID. Now, we will prove that in fact a Euclidean Domain is always a PID (Principal Ideal Domain). This proof will be elaborated, it can be shortened if necessary.

Proof:

Let R be a Euclidean domain.

Let I be a nonzero ideal of R. (If I is a zero ideal, then I=(0) )

Choose $b\in I, b\neq 0$ such that $d(b)=\min \{ d(i): i\in I\}$ , where d is the Euclidean function. By the well-ordering principle, every non-empty set of positive integers contains a least element, hence b exists.

Let $a\in I$ be any element in I. $\exists q,r \in R$ such that $a=bq+r$ , with either r=0, or d(r)<d(b). (This is the property of Euclidean domain.)

We can’t have d(r)<d(b) as that will contradict minimality of d(b). Thus, r=0, and a=bq. Hence every element in the ideal is a multiple of b, i.e. I=(b). Thus R is a PID (Principal Ideal Domain).

Check out other pages on our blog:

What is Singapore Math + Recommended Singapore Math Books

Definition of Euclidean Domain and Principal Ideal Domain (PID)

A Euclidean domain is an integral domain $R$ with a function $d:R\setminus \{0\}\to \mathbb{N}$ satisfying the following:

(1) $d(a)\leq d(ab)$ for all nonzero $a,b$ in $R$ .

(2) for all $a,b \in R$ , $b\neq 0$ , $\exists q, r, \in R$ such that $a=bq+r$ , with either $r=0$ or $d(r)<d(b)$ .

(d is known as the Euclidean function)

On the other hand, a Principal ideal domain (PID) is an integral domain in which every ideal is principal (can be generated by a single element).

Video on Computing Homology Groups

This video follows after the previous video on Simplicial Complexes.

If you are looking for quality Math textbooks to study from (including Linear Algebra and Calculus, the two most popular Math courses), check out my page on Recommended Math Books for students!

Egg Mathematics

Math Online Tom Circle

I highly recommend this Harvard Online Course “Science & Cooking” for food and Math lover:

http://online-learning.harvard.edu/course/science-and-cooking

Example of the Course :

How much boiled water you need to cook a perfect egg ?

By conservation of heat (energy), the heat (Q) of boiled water is transferred to the egg (assume no loss of heat to the environment: container, air, etc).

Secondary school Physics :

Q = m.C. (T’-T)
m = mass
C=Specific Heat
T’= Final Température
T= Initial Temperature

Chef’s tip: a perfect egg cooked at around 64 C.

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Chinese Tuition (West Side of Singapore)

If you live near the West side of Singapore (e.g. Buona Vista, Dover, Clementi, Jurong), and are looking for a patient and dedicated Chinese Tutor, do check out:

ChineseTuition88.com

Chinese Tuition Singapore

新加坡华文补习老师

Tutor: Ms Gao (高老师）

Ms Gao is a patient tutor, and also effectively bilingual in both Chinese and English.

A native speaker of Mandarin, she speaks clearly with perfect accent and pronunciation. She is also well-versed in Chinese history, idioms and proverbs.

Ms Gao is able to teach Chinese at the Primary and Secondary school level. She will teach in an exam-oriented style, but will also try her best to make the lesson interesting for the student.

Ms Gao graduated from Huaqiao University from Fujian, China.

Contact:

Email: chinesetuition88@gmail.com

(Preferably looking for students staying in the West side of Singapore)

Video on Simplices and Simplicial Complexes

Professor Wildberger is extremely kind to upload his videos which would be very useful to any Math student studying Topology. Simplices / Simplicial Complexes are usually the first chapter in a Algebraic Topology book.

Check out also Professor Wildberger’s book on Rational Trigonometry, something that is quite novel and a new approach to the subject of Trigonometry. For instance, it can be used for rational parametrisation of a circle.

#SG50 Singapore’s Birthday (National Day Song)

This year’s national day song is being sung by JJ Lin, a very famous Singaporean songwriter and artiste who has become famous in China and Taiwan.

Hope you enjoy the song and music video! JJ Lin’s vocals are indeed very good, and for him to sing the national day song is really a good thing.

From an educational perspective, one thing special about Singapore is Singapore Math. I have written a very long article on what is Singapore Math, and the benefits of Singapore Math, and also some key books that exemplify the techniques of Singapore Math.

Hope that more and more people can appreciate and utilise Singapore Math for the benefit of their students and children! Singapore Math is the key secret that has led Singapore to progress up in the educational rankings, especially in early education stages. Students and parents all over the world, including the United States, have been using Singapore Math syllabus to great success.

For readers who are interested to learn more about Singapore Math, do check out some of my earlier blog posts on Singapore Math:

SG50 Happy Birthday + Qoo10 Best Offers

Wishing Singapore a very happy SG50 birthday this weekend!

Do check out some of the SG50 sales at Qoo10, many of the items are going at half price! Definitely cheaper than buying at retail stores.

[S$599.00][LG Electronics]2015 New LG Robot Vacuum Cleaner VR6470LVM VR6471LVM Ccordless/ Dual Eye Cleaner Hombot Support English Chinese

WWW.QOO10.SG

[S$499.00][1DAY Super Big Deal!]Canon EOS 100D 1855 Lens Kit Save $500! 50% SALE!!

WWW.QOO10.SG

Also check out my previous posts on Best Deals at Qoo10!

MacBook Air for Math Students

Tired of “blue screens of death” that are so common in Windows? Don’t want to wait 10 minutes for Windows to “start up”? Switch to Mac OS!

My old computer (ASUS) has lasted me 5 years, but has recently gotten to the point that it slows down to a crawl. Booting up Windows takes up to 15 minutes, and “blue screens of death” occurs extremely frequently. It hasn’t spoiled completely yet, I still use it for printing documents.

I have since switched to MacBook Air, and so far it has been a great experience.

Tips for Math Students using MacBook Air / Pro

For Math students, some apps that you may want to install are MacTeX. It is the LaTeX Mac version. The initial download is over 2 GB, so it might take a while. I downloaded the installer from the main website, it took around one hour.

Another app is Google Chrome, which works very well on Mac. WordPress.com runs better on Chrome, for instance the LaTeX expressions are rendered better on the Chrome browser versus the Safari browser.

MacBook Air is one of the lightest notebooks around. The downside is that it does not have a few features, for example Ethernet Adapter and Optical Drive. Not to worry, one can purchase add-ons to remedy the problem. (Listed below)

Thunderbolt to Gigabit Ethernet Adapter

Aluminum External USB DVD+RW,-RW Super Drive for Apple–MacBook Air, Pro, iMac, Mini

MacBook takes some time to get used to, hence it is good to play around with it to discover the hidden shortcuts. For instance, Copy on Mac is command-C instead of control-C.

If you have any tips for using the MacBook, do feel free to share it in the comments section below!

The Fundamental Group

Source: Topology (2nd Economy Edition)

If we pick a point $x_0$ of the space X to serve as a “base point” and consider only those paths that begin and end at $x_0$ , the set of these path-homotopy classes does form a group under *. It will be called the fundamental group of X.

The important thing about the fundamental group is that it is a topological invariant of the space X, and it will be crucial in studying homeomorphism problems.

Definition of fundamental group:
Let X be a space; let $x_0$ be a point of X. A path in X that begins and ends at $x_0$ is called a loop based at $x_0$ . The set of path homotopy classes of loops based at $x_0$ , with operation *, is defined as the fundamental group of X relative to the base point $x_0$ . It is denoted by $\pi_1 (X,x_0)$ .

Previously, we have shown that the operation * satisfies the axioms for a group. (See our earlier blog posts on the associativity properties and other groupoid properties of the operation.

This group is called the first homotopy group of X. There is also a second homotopy group, and even groups $\pi_n (X,x_0)$ for all $n\in \mathbb{Z}^+$ .

An example of a fundamental group:

$\pi_1 (\mathbb{R}^n,x_0)$ is the trivial group (the group consisting of just the identity). This is because if f is a loop in $\mathbb{R}^n$ based at $x_0$ , the straight line homotopy is a path homotopy between f and the constant path at $x_0$ .

An interesting question (discussed in the next upcoming blog posts) would be how the group depends on the base point $x_0$ .

Things to Make and Do in the Fourth Dimension: A Mathematician’s Journey through Narcissistic Numbers, Optimal Dating Algorithms, at Least Two Kinds of Infinity, and More, by Matt Parker

Check out the latest new Math book on the Fourth Dimension! The Fourth Dimension is the mysterious dimension which cannot be seen. Check out also our previous post on the Fourth Dimension Explained.

Things to Make and Do in the Fourth Dimension: A Mathematician’s Journey Through Narcissistic Numbers, Optimal Dating Algorithms, at Least Two Kinds of Infinity, and More

Mathematics popularizer Matt Parker, an Australian based in England, is a self-proclaimed “standup mathematician” perhaps best known for his numerous contributions to the Numberphile YouTube channel. He is also the Public Engagement in Mathematics Fellow at Queen Mary, University of London, and his new book, Things to Make and Do in the Fourth Dimension, is an ambitious and delightful addition to the current age’s plethora of high-quality volumes on recreational mathematics—even if most of the material he covers is focused on 2-D and 3-D. Like the extensive writings of legendaryScientific American columnist Martin Gardner this book seeks to make mathematics come alive for an intelligent and curious audience by engaging the reader in a lively informal style, and with irresistible invocations to roll up one’s sleeve and experiment. Parker also enlivens his chapters with numerous surprises.

Source: http://www.scientificamerican.com/article/how-to-get-to-the-fourth-dimension/?WT.mc_id=SA_WR_20150805

These 2 words help to get into Harvard or Stanford

Math Online Tom Circle

If applying to Harvard, use ‘Father and Mother’ to address your parents, to Stanford, use ‘Dad & Mom’…

http://m.fastcompany.com/3049289/most-creative-people/use-these-two-words-on-your-college-essay-to-get-into-harvarda

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New Book: 《Birth of a theorem : a mathematical adventure》

Math Online Tom Circle

By Cédric Villani, Fields Medalist (2010)

– Translated from the French by Malcolm DeBevoise – illustrations by Claude Gondard.” Villani, Cedric

NLB Call Number: English 510.92 VIL

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Anti-bacterial Vacuum

Just to share this site TTMLW (Tips to Make Life Wonderful). Hope it is useful to some of our readers!

GEP Test Dates 2015

Source: http://www.moe.gov.sg/education/programmes/gifted-education-programme/faq/gep-pupils/

Just a gentle reminder that the dates for 2015 GEP Test would be as follows:

Schedule for 2015

GEP Screening Test: 28 Aug 2015
GEP Selection Test: 20 and 21 Oct 2015
Invitation to join GEP: Early November 2015

(Do check the website above for updates)

Students interested to buy books relevant to GEP can check out one of my most popular blog posts on Recommended Books for GEP Screening / Selection Test. The truth is that at age 9 there is little difference between normal and gifted kids, i.e. normal kids with some training and excellent family support / learning environment can easily be on par with gifted students. Gifted students are nothing really special, they do have more training and good family learning background, but normal students with additional exposure and training can be as good as gifted students.

Singapore’s educational experts and professors have recently called for Singapore as a nation to read more books, posted prominently in Straits Times. The correct choice of books is critical, as reading books meant for entertainment like Harry Potter or Twilight is unlikely to benefit students a great deal. Worse still is reading FaceBook or Twitter, as they are often in broken English. Singaporeans are notoriously known for reading very few books, leading to bookstores like Borders and PageOne completely shutting down in Singapore. Students who wish to enter GEP would need to read even more books, as GEP would require a broad knowledge base, which is tested in the vocabulary and logic section of the screening tests.

In my earlier post on Recommended GEP Books, I recommended some books to tackle the notoriously difficult GEP Screening Test, including the Vocabulary Section, Math Section, and Logic Section. Children with weaker English levels would definitely need to brush up on their vocabulary, as words like “gregarious“, “amicable“, “cantankerous” would pop up in GEP tests, leading to students being “flabbergasted“.

For the Math section, the harder GEP Screening Math questions are undoubtedly of a Math Olympiad style that would flounder all but those who are trained in the art of Math Olympiad. It is a truth that a P3 student scoring 100 marks in normal Math, most likely cannot solve a P3 Math Olympiad problem due to lack of training. However, once he/she is trained, Math Olympiad is just a trick and will be easily solved. Check out some Recommended Books for Math Olympiad Self-learning.

Finally, remember the Cheryl Birthday Puzzle that went viral? This is an example of a logic puzzle. Logic is not taught anywhere in the syllabus, and hence students would need to self learn to master the art of logic puzzles. This skill will be critical again for DSA / GAT / HAST, as they will be testing the similar logic puzzles again for P6 DSA.

To all students taking the GEP test, all the best. Keep calm and good luck!

Measure Theory: What does a.e. (almost everywhere) mean

Source: Elements of Integration by Professor Bartle

Students studying Mathematical Analysis, Advanced Calculus, or probability would sooner or later come across the term a.e. or “almost everywhere”.

In layman’s terms, it means that the proposition (in the given context) holds for all cases except for a certain subset which is very small. For instance, if f(x)=0 for all x, and g(x)=0 for all nonzero x, but g(0)=1, the function f and g would be equal almost everywhere.

For formally, a certain proposition holds $\mu$ -almost everywhere if there exists a subset $N\in \mathbf{X}$ with $\mu (N)=0$ such that the proposition holds on the complement of N. $\mu$ is a measure defined on the measure space $\mathbf{X}$ , which is discussed in a previous blog post: What is a Measure.

Two functions $f, g$ are said to be equal $\mu$ -almost everywhere when $f(x)=g(x)$ when $x\notin N$ , for some $N\in X$ with $\mu (N)=0$ . In this case we would often write $f=g$ , $\mu$ -a.e.

Similarly, this notation can be used in the case of convergence, for example $f=\lim f_n$ , $\mu$ -a.e.

The idea of “almost everywhere” is useful in the theory of integration, as there is a famous Theorem called “Lebesgue criterion for Riemann integrability”.

(From Wikipedia)

A function on a compact interval [a, b] is Riemann integrable if and only if it is bounded and continuous almost everywhere (the set of its points of discontinuity has measure zero, in the sense of Lebesgue measure). This is known as the Lebesgue’s integrability condition or Lebesgue’s criterion for Riemann integrability or the Riemann—Lebesgue theorem.^[4] The criterion has nothing to do with the Lebesgue integral. It is due to Lebesgue and uses his measure zero, but makes use of neither Lebesgue’s general measure or integral.

Reference book:

Post-Modern Algebra

Trigonometry in abstract algebra Group Theory… this is a new look of Elementary Math (E. Math) from a higher level (Abstract Algebra : Group Theory) — just as the Tang Poem said “欲穷千里目, 更上一层楼” (To see further distance away, just climb up to higher level).

Math Online Tom Circle

Modern Algebra: Based on the 1931 influential book “Modern Algebra” written by Van de Waerden (the student of E. Noether). Pioneered by the 20th century german Göttingen school of mathematicians, it deals with Mathematics in an abstract, axiomatic approach of mathematical structures such as Group, Ring, Vector Space, Module and Linear Algebra. It differs from the computational Algebra in 19th century dealing with Matrices and Polynomial equations.

This phase of Modern Algebra emphasises on the algebraization of Number Theory: {N, Z, Q, R, C}

Post-Modern Algebra: The axiomatic, abstract treatment of Algebra is viewed as boring and difficult. There is a renewed interest in explicite computation, reviving the 19th century invariant theory. Also the structural coverage (Group, Ring, Fields, etc) in Modern Algebra is too narrow. There is emphasis on other structures beyond Number Theory, such as Ordered Set, Monoid, Quasigroup, Category, etc.

Example:
The non-abelian Group S3 (order…

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