Mathtuition88

爱因斯坦数学差吗？告诉你什么是罗氏几何和黎曼几何

欧氏几何: 第5公理 : 直线上过一点, 只有 n=1 条平行线

非欧几何:

罗氏: n >1 (空间)
黎曼: n = 0 (球体)

The Inventors of the 10 Computer languages

Math Online Tom Circle

Python (Dutch Guido van Rossum, 1956)
Java (Canadian James Gosling 1955)
Javascript (USA Brendan Eich, 1961)
C (USA Dennis Ritchie, 1941 – 2011 )
C++ (Denmark Bjarne Stroustrup, 1950)
Ruby (JAPAN Yukihiro “Matz” Matsumoto, 1965)
Perl (USA Larry Wall, 1954)
Pascal (Switzerland Niklaus Wirth, 1934)
Lisp (USA John McCarthy, 1927 – 2011)
PHP (Denmark Rasmus Lerdorf, 1968)

https://www.technotification.com/2018/04/programming-languages-creators.html

Below the 3 hotest Functional Programming language influenced by Lisp:

11. Kotlin(Russia Andrey Breslav)

12. Scala (USA Martin Odersky)

13. Haskell (USA)

14. Clojure (USA Rich Hickey)

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Group Theory Game

Math Online Tom Circle

https://www.cut-the-knot.org/proofs/PegsAndGroups.shtml

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School System Video (Do not make a fish climb trees)

Singapore is being mentioned around 4:54. Very nice video. The truth is that the classroom of today is still nearly the same as the classroom of 150 years ago. There needs to be a “Educational Revolution” parallel to that of the Industrial Revolution. Many children cannot fit into the single classroom model, leading to growth in diagnosis of behavioral “problems” such as ADHD in developed nations.

Americans who are tired of “Common Core” may want to check out Singapore Math for their kids, which is highly acclaimed in the educational realm.

11-year old math and chess prodigy in Singapore

Source: Channel News Asia

Aarushi Maheshwari solved the famous “Cheryl’s Birthday Problem” when she was only 9. She is also a chess champion and can play blindfold chess.

Watch the video below to learn more!

Also read our previous post on The Most Accomplished 10-Year-Old (Gifted pupil).

For those who want to learn more about Olympiad Math and International Chess, check out the previous two links. Math and Chess are two of the most intellectually challenging activities that can develop the intelligence of kids.

Basics of Partial Differential Equations Summary

PDE: Separation of Variables

1) Let $u(x,y)=X(x)Y(y)$ .
2) Note that $u_x=X'Y$ , $u_y=XY'$ , $u_{xx}=X''Y$ , $u_{yy}=XY''$ , $u_{xy}=u_{yx}=X'Y'$ .
3) Rearrange the equation such that LHS is a function of $x$ only, RHS is a function of $y$ only.
4) Thus, LHS=RHS=some constant $k$ .
5) Solve the two separate ODEs.

Wave Equation
$\displaystyle c^2y_{xx}=y_{tt},$ where $y(t,0)=y(t,\pi)=0$ , $y(0,x)=f(x)$ , $y_t(0,x)=0$ .

Solution of Wave Equation (with Fourier sine coefficients)
$\displaystyle y(t,x)=\sum_{n=1}^\infty b_n\sin(nx)\cos(nct)$ where $\displaystyle b_n=\frac{2}{\pi}\int_0^\pi f(x)\sin(nx)\,dx.$

d’Alembert’s solution of Wave Equation
$\displaystyle y(t,x)=\frac{1}{2}[f(x+ct)+f(x-ct)].$

Heat Equation
$\displaystyle u_t=c^2u_{xx},$
$u(0,t)=u(L,t)=0$ , $u(x,0)=f(x)$ .

Solution of Heat Equation
$\displaystyle u(x,t)=\sum_{n=1}^\infty b_n\sin\left(\frac{n\pi x}{L}\right)\exp\left(-\frac{\pi^2n^2c^2}{L^2}t\right),$ where $\displaystyle b_n=\frac{2}{\pi}\int_0^\pi f(x)\sin(nx)\,dx$ are Fourier sine coefficients of $f(x)$ .

5 Good Habits to Develop for College

As soon as you start your freshman year in the university, you’ll be faced with countless challenges. Getting accepted into college is an accomplishment, but it is only one of the many steps you’ll need to take in order to succeed.

The good news is: there are habits you can develop and continue to practice in preparation for the challenges of college life.

The value of pre-college preps

But first: Before you can develop good habits that will serve you during your college years, it is essential that you establish a strong foundation for success in advance.

The best way to achieve your goals in college is to prepare for college way before you enter the university. While in high school, you can get help from the experts who can help you make smarter decisions about your higher education plans.

By working with college application consultants, you can get expert help every step of the way, including:

Assessing your current situation and future academic goals.
Establishing an organized system to identify and plan the schools that suit your skills and preferences.
Preparing and submitting winning applications on time.
Acing your college admissions interview.
Finding the best college options, from the institution to the course to study, that matches your goals and capabilities.

Once you find the right college for you, college goals such as developing good habits become easier to achieve.

Habits to achieve your college #goals

It’s important for you to develop good habits early on to have a successful college experience. As an incoming freshman, here are five good habits you should develop.

1. Establish your priorities

When you’re so accustomed to high school life, it may become hard for you to really visualize what college life will be like. The independence needed to cope with different class requirements and strictly following your personal study schedule are some of the things you need to consider.

College is a process of preparing for your future career. You should make it a habit to set priorities, and create and follow a schedule to manage your school workload. Make a list of your priorities from the most urgent projects to those that are due later.

This way, you can tackle them one at a time and reduce stress arising from the pressure of wanting to finish them all at once.

Plan ahead. The brain can’t handle things all at once. Overloading your mind with things to remember will only trigger stress. And before you know it, your stress may become unmanageable so you end up making mistakes, and missing assignments and projects.
Have a planner. Have a checklist of things you should bring to college. Create or buy an academic planner to help you stay organized. List down due dates for projects, assignments and many others. Make sure to stick to your planner and finish your work before the deadline. While you’re at it, you can also create backup alarms using your smartphone calendar app.
Break down big tasks into smaller ones. Writing a 12-page research paper and preparing for a huge exam at the same time can be difficult. Planning ahead means you have time to break down these huge tasks into smaller, manageable chunks so that they seem less daunting.

You can also follow these valuable tips on how to adjust to college life.

2. Participate, get involved

Don’t show up in class just to get your attendance checked. Involve yourself by participating in class discussions.

When you go to college, large lecture halls may appear intimidating to you. Practice the habit of engaging yourself during lectures. This will help you get a better grasp of learning. If there’s a vacant seat in front, occupy it.

Grab the opportunity to sit close to the professor for you to feel more present. You shouldn’t take part just because you want to dominate discussions or to help you get a better grade. Choose to speak out for the sake of engaging yourself.

Foster relationships with professors. A close mentoring relationship with a professor is advantageous as he or she can provide you some guidance as you go about your college education.

3. Challenge yourself

When you get into college, be on the lookout for opportunities. Take some challenging classes to expose yourself to new fields. Be open to new things. The world is too big for you to confine yourself to the things you already know.

Branch out and try everything you can. The more you explore your interests, the earlier you’ll discover your college major. Develop the habit of constantly challenging yourself; it will teach you about the value of hard work and discipline.

4. Build your portfolio

In connection to the third tip, pick activities that’ll advance your knowledge and experience. Make it a habit to save great pieces or project works you’ve done during high school. These will work to your advantage the moment you apply for college.

Your portfolio will serve as a proof of how adept you are in the course you’re applying for. This will also increase your chances of getting in a university that you want.

5. Understand how you learn

Each student has different learning abilities. While some are visual learners, others may be auditory learners. It’s important to consider what type of learner you are.

Kinesthetic Learner – gains knowledge through feelings or experience
Visual Learner – gains knowledge by seeing
Auditory Learner – gains knowledge by hearing

Knowing how you learn will be beneficial to you once you start college. There will be no more spoon-feeding in college so you have to be quicker in learning new things. If you know that you are an auditory learner, grab the chance to sit in front to hear each lecture clearly.

To sum up

Every student is unique, so there really is no one-size-fits-all formula for achieving college success. But by getting expert help in finding the right college and excelling in your chosen institution by building good habits, college can be one of the best years of your life.

AUTHOR BIO

Brian Giroux is an experienced college admissions advisor and co-founder of Capital College Consulting. Brian is a Professional Member of Independent Educational Consulting Association (IECA). Brian has worked with students from over 30 countries to help provide guidance through the US admissions process.

Brian’s experience includes 18+ years in education serving multiple roles as educator, athletic director, and college admissions consultant.

The Most Accomplished 10-Year-Old (Gifted pupil)

Pan Annan is probably the most accomplished 10 Year-old student in Singapore, or perhaps even in the world.

Her list of accomplishments:

International rhythmic gymnastics champion
Youngest member of the Singapore National Youth Chinese Orchestra (SNYCO), where she plays the Pipa
Gifted Education Programme (GEP)
Math Olympiad trainee
Raffles Girls’ Primary pupil

Any one of the above accomplishments is enough to stand out among 10 year-olds, and she has all of them! The most amazing is how she manages her time. I am familiar with Chinese Orchestra trainings, that alone is enough to account for quite a significant amount of time after school, since there is group practicing, sectional practicing, not to mention practicing alone. Possibly Chinese Orchestra alone adds up to a minimum of 5-10 hours per week.

Also, the workload from RGPS GEP is very demanding. Her schedule and timetable can only be achieved with 100% efficiency and focus. (She even does her homework in the car to maximize efficiency and save time.)

Parents may want to read my previous blog post on Book by Truly Gifted Kid (GEP Book), where a similarly prodigious child genius Moshe Kai Cavalin outlines his secret, with input from his mom on parenting. Also, as you can see, the standard for GEP students nowadays is very high, you may read Recommended Books for GEP Selection Test and How to Get Into GEP for some tips on how to do some foundational preparation for the GEP.

Sincerely all the best to Pan Annan for achieving her dreams of being a gymnastic champion.

Population Differential Equations and Laplace Transform

Malthus Model
$\displaystyle \frac{dN}{dt}=BN-DN=kN$

$N$ : Total population

$B$ : Birth-rate per capita

$D$ : Death-rate per capita

$k=B-D$

Solution to D.E.:
$\displaystyle \boxed{N(t)=\widehat{N}e^{kt}},$

where $\widehat{N}=N(0)$ .

Logistic Equation
$\begin{aligned} D&=sN\\ \frac{dN}{dt}&=BN-sN^2\\ \widehat{N}&=N(0)\\ N_\infty&=B/s \end{aligned}$

Logistic Case 1: Increasing population ( $\widehat{N}<N_\infty$ )
$\begin{aligned} N(t)&=\frac{B}{s+(\frac{B}{\widehat{N}}-s)e^{-Bt}}\\ &=\frac{N_\infty}{1+(\frac{N_\infty}{\widehat{N}}-1)e^{-Bt}} \end{aligned}$

The second expression can be derived from the first: divide by $s$ in both the numerator and denominator.

Logistic Case 2: Decreasing population ( $\widehat{N}>N_\infty$ )
$\begin{aligned} N(t)&=\frac{B}{s-(s-\frac{B}{\widehat{N}})e^{-Bt}}\\ &=\frac{N_\infty}{1-(1-\frac{N_\infty}{\widehat{N}})e^{-Bt}} \end{aligned}$

Logistic Case 3: Constant population ( $\widehat{N}=N_\infty$ )
$\displaystyle N(t)=N_\infty$

Harvesting
Basic Harvesting Model: $\displaystyle \boxed{\frac{dN}{dt}=(B-sN)N-E}.$

$E$ : Harvest rate (Amount harvested per unit time)

Maximum harvest rate without causing extinction: $\boxed{\dfrac{B^2}{4s}}$ .

$\displaystyle \boxed{\beta_1,\beta_2=\frac{B\mp\sqrt{B^2-4Es}}{2s}}.$

$\beta_1$ : Unstable equilibrium population

$\beta_2$ : Stable equilibrium population

Extinction Time: $\displaystyle \boxed{T=\int_{\widehat{N}}^0\frac{dN}{N(B-sN)-E}}.$

Laplace transform of $f$
$\displaystyle F(s)=L(f)=\int_0^\infty e^{-st}f(t)\,dt$

Tip: Use this equation when the questions contains the words “show from the definition”.

Inverse transform of $F(s)$
$\displaystyle f(t)=L^{-1}(F(s))$

Linearity
$\begin{aligned} L(af(t)+bg(t))&=aL(f)+bL(g)\\ L^{-1}(aF(s)+bG(s))&=aL^{-1}(F)+bL^{-1}(g) \end{aligned}$

List of common Laplace Transforms

$\begin{aligned} L(e^{at})&=\frac{1}{s-a}\\ L(1)&=\frac{1}{s}\\ L(\cos wt)&=\frac{s}{s^2+w^2}\\ L(\sin wt)&=\frac{w}{s^2+w^2}\\ L(t^n)&=\frac{n!}{s^{n+1}}\\ L(f')&=sL(f)-f(0)\\ L(f'')&=s^2L(f)-sf(0)-f'(0)\\ L(f^{(n)})&=s^nL(f)-s^{n-1}f(0)\\ &\quad -s^{n-2}f'(0)-\dots-f^{(n-1)}(0)\\ L\left(\int_0^t f(\tau)\,d\tau\right)&=\frac{1}{s}L(f) \end{aligned}$

$s$ -shifting
If $L(f)=F(s)$ , $s>a$ , then $\displaystyle \boxed{L(e^{ct}f(t))=F(s-c)},$
$s-c>a$ .

Tip: Use this when doing Laplace Transform of a function with an exponential factor $e^{ct}$ . Note that the reverse direction can sometimes be used as well: $\displaystyle L^{-1}[F(s-c)]=e^{ct}f(t).$

$t$ -shifting
If $L(f(t))=F(s)$ , then $\displaystyle \boxed{L(f(t-a)u(t-a))=e^{-as}F(s)}.$

Tip: Frequently, we use the reverse direction $\displaystyle L^{-1}[e^{-as}F(s)]=f(t-a)u(t-a).$

Delta function
$\delta(t)$ : infinitely tall and narrow spike at $t=0$ .

$\delta(t-a)$ : infinitely tall and narrow spike at $t=a$ .

$\boxed{L[\delta(t-a)]=e^{-as}}$

Two properties of delta function
$\begin{aligned} \int_0^\infty\delta(t-a)\,dt&=1\\ \int_0^\infty \delta(t-a)g(t)\,dt&=g(a) \end{aligned}$
for $a\geq 0$ .

Tip: Use delta function when the keywords “suddenly”, “burst”, etc. appear.

Unit step function
$\displaystyle u(t-a)=\begin{cases} 0, &t<a\\ 1, &t>a. \end{cases}$

For $0<a<b$ , $\displaystyle u(t-a)-u(t-b)=\begin{cases} 0, &t<a\\ 1, &a<t<b\\ 0, &t>b. \end{cases}$

Tip: Use unit step function for questions that require a force to “switch on / switch off” at certain times.

$\displaystyle \boxed{L(u(t-a))=\frac{e^{-as}}{s}}$

Free Money from PayLah (Quick! Before it is fully redeemed)

Last call to redeem the offer by PayLah! $5 just for downloading the app is pretty worth it.

Mathtuition88

Hey! You can get S$5 when you register for PayLah! with my Referral Code WILDOF087 (last 3 digits are numbers zero-eight-seven) by 31 December 2018. Download PayLah! from http://www.dbs.com.sg/paylah now and enter the above Code during sign-up before this offer is fully redeemed. T&Cs apply.–

PayLah! can be downloaded from Apple Store or Android store (Singapore). Getting your $5 should be instantaneous. Be sure to redeem it today since there is a cap of 40000 people who can redeem. (The offer is still available now, but who knows next week it may be fully redeemed.)

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NTUC Fairprice Receipt Number

For those participating in lucky draws, often it requires you to SMS the NTUC Fairprice Receipt Number. However, the latest version of NTUC receipts, upon closer inspection contains many numbers but none that are explicitly specified as “receipt number”.

In the old NTUC receipt, the receipt number is clearly stated.

What is NTUC Receipt Number?

It is extremely confusing to find the NTUC Receipt Number (out of the many numbers in the receipt). After some “research”, I come to the conclusion is that the receipt number is the same as the transaction number, labelled by Tr!

How I came to this conclusion: Look at the image above. The receipt number is 65539, the transaction number is Tr: 65539. It matches, right? Hence, it is logical that the NTUC Fairprice receipt number is precisely the Tr (transaction) number.

Similarly, I believe that this logic applies to Giant, Sheng Siong, Cold Storage receipt numbers too, it should be the transaction number, in the event that the receipt number is not clearly stated in the receipt. Same for NTUC Finest, NTUC Extra and all other types of NTUC Fairprice receipts.

I hope this helps those who are desperately searching for the receipt number!

Money Saving “Lobang” (Tips):

Get free $10 NTUC FairPrice Voucher by signing up for Trust Bank!

100 Changi Rewards points for you

Changi points are quite useful, can be used to redeem the following:

Attraction Passes
Exclusive Items
Exclusive Events
Travel and Airline Miles
Changi Rewards e-Vouchers
1-for 1 Deals
Hotel Stays
and other Changi Rewards’ Partners Deals

URL to sign up and get 100 points instantly: Changi Points

Hans Rosling’s Best TED Talk & Book “Factfulness”

Math Online Tom Circle

http://time.com/5224618/bill-gates-hans-rosling-factfulness/

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Why Prime Numbers Still A Mistery

Math Online Tom Circle

Prime Numbers since Euclid 2300 years ago remains a mystery today…

Riemann’s Hypothesis is the Top Conjecture Unsolved in 21st Century.

https://theconversation.com/why-prime-numbers-still-fascinate-mathematicians-2-300-years-later-92484

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Free Money from PayLah (Quick! Before it is fully redeemed)

110 Nanyang Girls’ High students fall sick during boarding school programme

Quite worrying. Based on my personal experience and online research, food in Singapore cannot be left in room temperature for more than 4 hours, due to our hot and humid temperature. To be safe, food needs to be put in the refrigerator as soon as possible if not consumed immediately.

Hope all affected students get well soon.

Source: 110 Nanyang Girls’ High students fall sick during boarding school programme
Read more at https://www.channelnewsasia.com/news/singapore/110-nanyang-girls-high-students-sick-boarding-school-programme-10097196

Second Order Linear ODE Summary

Second Order Linear D.E. Summary

Homogenous D.E.
$y''+ay'+by=0$ .

Solve the Characteristic Equation: $\lambda^2+a\lambda+b=0$ .
Case 1) Two real roots $\lambda_1,\lambda_2$ : $\implies \boxed{y=c_1e^{\lambda_1x}+c_2e^{\lambda_2x}}$

Case 2) Real double root $\lambda$ : $\implies \boxed{y=c_1e^{\lambda x}+c_2xe^{\lambda x}}$

Case 3) Complex Conjugate root $\lambda_1,\lambda_2=-\frac{a}{2}\pm iw$ , where $w=\sqrt{b-\frac{a^2}{4}}$ : $\implies \boxed{y=e^{-\frac{a}{2}x}(c_1\cos wx+c_2\sin wx)}$

Non-homogenous D.E.
General solution of non-homogenous D.E.: $\displaystyle y=y_h+y_p,$ where $y_h$ is the general solution of the homogenous equation, and $y_p$ is the particular solution (with no arbitrary constants).

Method of Undetermined Coefficients (Guess and try method)
$y''+p(x)y'+q(x)y=r(x)$ .

Only works if $r(x)$ is polynomial, exponential, sine or cosine (or sum/product of these).

Polynomial: Try $y$ =Polynomial (e.g. $y=Ax^2+Bx+C$ or $y=Bx+C$ .)

Exponential ( $e^{kx}$ ): Try $y=ue^{kx}$ , where $u$ is a function of $x$ .

Trigonometric ( $\sin kx$ or $\cos kx$ ): Convert to complex differential equation by replacing $y$ with $z$ , replace $\sin kx$ / $\cos kx$ by $e^{ikx}$ .

Try $z=ue^{ikx}$ , where $u$ is a function of $x$ . After solving for $z$ , take real/imaginary part of $z$ for cosine/sine respectively.

Method of variation of parameters
$y''+p(x)y'+q(x)y=r(x)$ .

[Step 1)] Solve the homogenous D.E. $y''+p(x)y'+q(x)y=0$ .

Get solution of the form $y_h=c_1y_1+c_2y_2$ .

[Step 2)]
Let $\displaystyle u=-\int\frac{y_2r}{W}\,dx$ and $\displaystyle v=\int\frac{y_1r}{W}\,dx$ where $W$ is the Wronskian $\displaystyle W=y_1y_2'-y_1'y_2.$

Particular solution: $y_p=uy_1+vy_2$ .

General solution: $y=y_h+y_p$ .

Forced Oscillations
Let $F_0$ be the amplitude of the driving (external) force. If $F_0=0$ , by Newton’s Second Law, $m\ddot{x}=-kx$ , hence $\displaystyle \boxed{\ddot{x}=-\omega^2 x},$
where $\omega=\sqrt{k/m}$ . The value $\omega$ is called the natural frequency.

If $F_0\neq 0$ , then $\displaystyle \boxed{m\ddot{x}+kx=F_0\cos\alpha t},$
where $\alpha$ is the driving (external) frequency.

At resonance (when $\alpha=\omega$ ), $\displaystyle \boxed{x=\frac{F_0t}{2m\omega}\sin(\omega t)}.$

Joseph Fourier is Still Transforming Science

Math Online Tom Circle

Key Words: 250 years anniversary

Yesterdays: Fourier discovered Heat is a wave , Fourier Series, Fourier Transformation, Signal processing…
Today: IT imaging JPEG compression, Wavelets, 3G/4G Telecommunications, Gravitational waves …
Friends / bosses: Napoleon, Monge… Egypt Expedition with Napoleon Army.
Taught at the newly established Military Engineering University “Ecole Polytechnique”.
Scientific Research: Short period but intense.
Before Fourier died (wrapped himself in thick carpet in hot summer), he was reviewing another young Math genius Evariste Galois’s paper on “Group Theory”.

https://news.cnrs.fr/articles/joseph-fourier-is-still-transforming-science

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10 Things You Don’t Know About Albert Einstein

Source: Thoughtco

Some of the 10 facts are:

He Loved to Sail
Einstein’s Brain
Einstein and the Violin
Presidency of Israel

For those interested in the theory of relativity (special and general), which is the main core of Einstein’s research, do check out this book Relativity: The Special and the General Theory, 100th Anniversary Edition. Highly rated on Amazon, it is one of the few books that tells the story of Einstein and also includes real scientific content.

Drawing Chemistry Diagrams in LaTeX

The web application “mol2chemfig” is very amazing. (http://py-chemist.com/mol_2_chemfig).

Especially the search function, where one can just type the name of the chemical and out comes the chemfig code where it can be pasted to LaTeX. The mol2chemfig package needs to be imported in LaTeX before the code can be compiled. Alternatively, one can just save the PDF image (vector graphics) generated by the web app.

Below is a screenshot of what is happening:

Linear First Order ODE, Bernoulli Equations and Applications

Linear First Order ODE
DE of the form: $y'+P(x)y=Q(x)$ .

Integrating factor: $R(x)=e^{\int P(x)\,dx}$ .
$\begin{aligned} R'&=RP\\ Ry'+RPy&=RQ\\ (Ry)'&=RQ\\ Ry&=\int RQ\,dx \end{aligned}$
$\displaystyle \boxed{y=\frac{\int RQ\,dx}{R}}$
(Remember to have a constant $C$ when integrating the numerator $\int RQ\,dx$ .)

Integration by parts
$\displaystyle \boxed{\int uv'\,dx=uv-\int u'v\,dx}$

Acronym: LIATE (Log, Inverse Trig., Algebraic, Trig., Exponential), where L is the best choice for $u$ . (This is only a rough guideline.)

Bernoulli Equations
DE of the form: $y'+p(x)y=q(x)y^n$ .

$y^{-n}y'+y^{1-n}p(x)=q(x)$

Set $\boxed{y^{1-n}=z}$ .

Then $(1-n)y^{-n}y'=z'$ . The given DE becomes
$\displaystyle \boxed{z'+(1-n)p(x)z=(1-n)q(x)}.$

Fundamental Theorem of Calculus (FTC)
Part 1: $\displaystyle \frac{d}{dx}\int_a^x f(t)\,dt=f(x)$

Part 2: $\displaystyle \int_a^b F'(t)\,dt=F(b)-F(a)$

Hyperbolic Functions
$\begin{aligned} \sinh x&=\frac{e^x-e^{-x}}{2}\\ \cosh x&=\frac{e^x+e^{-x}}{2}\\ \cosh^2 x-\sinh^2 x&=1\\ \end{aligned}$
$\begin{aligned} \frac{d}{dx}\sinh x&=\cosh x\\ \frac{d}{dx}\cosh x&=\sinh x\\ \frac{d}{dx}\sinh^{-1}x&=\frac{1}{\sqrt{x^2+1}}\\ \frac{d}{dx}\cosh^{-1}x&=\frac{1}{\sqrt{x^2-1}} \end{aligned}$
$\displaystyle \int \tanh(ax)\,dx=\frac{1}{a}\ln(\cosh(ax))+C.$

Uranium-Thorium Dating
Starting Equations:
$\displaystyle \begin{cases} \frac{dU}{dt}=-k_U U\implies U=U_0e^{-k_Ut}\\ \frac{dT}{dt}=k_UU-k_TT. \end{cases}$

$\frac{dT}{dt}+k_T T=k_U U_0e^{-k_Ut}$

$R=e^{\int k_T}=e^{k_T t}$ .

$\displaystyle \boxed{T(t)=\frac{k_U}{k_T-k_U}U_0(e^{-k_Ut}-e^{-k_Tt})}$

$\displaystyle \boxed{\frac{T}{U}=\frac{k_U}{k_T-k_U}[1-e^{(k_U-k_T)t}]}$

Louis-Le-Grand, un lycée d’élite 法国(巴黎)精英学校: 路易大帝中学

Math Online Tom Circle

Lycée Louis-Le-Grand is the best high school (lycée) for Math in France – if not in the world – it produced many world-class mathematicians, among them “the Father of Modern Math” in 19th century the genius Evariste Galois. (See also: Unknown Math Teacher produced two World’s Math Grand Master Students ), Molière, Victor Hugo, 3 French Presidents, etc.

Its Baccalaureat (A-level) result 100% passed with 75% scoring distinctions. Each year 1/4 of Ecole Polytechnique (France Top Engineering Grande Ecole) students come from here.

More surprisingly, the “Seconde” (Secondary 4) students learn Chinese Math since 6ème (PSLE Primary 6).

Note: Louis Le Grand (= Louis 14th). He sent the Jesuits (天主教的一支: 耶稣会传教士) as the “French King’s Mathematicians”(eg. Bouvet 白晋) to the 16-year-old Chinese Emperor (康熙) KangXi’s Court.

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How to draw polygons with angles (LaTeX Tikz)

Just learnt some amazing angle-drawing techniques using LaTeX Tikz.

Although cumbersome to code, the benefit of using LaTeX to draw diagrams is that it matches the font of your document, and is easier to customize and edit. Also, it is “vector graphics” in the sense that even if you zoom in, it is in perfect resolution.

Important: In order to use Tikz to draw angles you need to load the following packages:

\usepackage{tikz}
\usetikzlibrary{calc,patterns,angles,quotes}

Sample Tikz code for drawing the above figure (polygon angles)

\begin{figure}[htbp]
\begin{center}
\begin{tikzpicture}
\coordinate (v0) at (-2.2,1.2);
\coordinate (v1) at (0,0);
\coordinate (v2) at (2.2,1.8);
\coordinate (v3) at (0,3);

\filldraw
(v1) circle (2pt) node[align=left, below] {$v_1$}
-- (v2) circle (2pt) node[align=center, below, right] {$v_2$}
-- (v3) circle (2pt) node[align=left, above] {$v_3$}
-- (v0) circle (2pt) node[align=left, above, left] {$v_0$}
-- (v1);
\pic [draw, -, "$\alpha_0$", angle eccentricity=1.5] {angle = v1--v0--v3};
\pic [draw, -, "$\alpha_1$", angle eccentricity=1.5] {angle = v2--v1--v0};
\pic [draw, -, "$\alpha_2$", angle eccentricity=1.5] {angle = v3--v2--v1};
\pic [draw, -, "$\alpha_3$", angle eccentricity=1.5] {angle = v0--v3--v2};
\end{tikzpicture}
\caption{}
\label{fig:quad}
\end{center}
\end{figure}

Clearly, the code can be modified for any polygon be it triangle, pentagon, etc. Also, if you are curious how to post source code in WordPress.com, it is by surrounding your source code with [ code ] [/ code]. It also supports a “lang=” option, see more at https://en.support.wordpress.com/code/posting-source-code/.

Bukit Batok / Hillview Maths Tuition

Please contact us for one-to-one Maths Tuition at Bukit Batok / Hillview area.

Nearby areas such as Clementi / Jurong East are also suitable.

Patient and dedicated Maths Tutor.

Senior Wrangler: Singapore Prime Minister

Math Online Tom Circle

Senior Wrangler is the First position in the Math Tripos in Cambridge. Singapore Prime Minister Lee Hsien Loong was the Senior Wrangler in 1973, the first Singaporean student with such great honors, among other senior wranglers like Arthur Cayley (Group Theory), J.J. Sylvester (Inventor of Matrix, private tuitor of the “inventor of Nursing” Florence Nightingale), J.E. Littlewood (partnered in a twin research team with G.H. Hardy), Frank Ramsey (Ramsey’s Theorem), Stokes, Pell, etc.

Some great mathematicians like Bertrand Russell (Logician, Nobel Litterature Prize) , G.H. Hardy (20th century greatest Pure Mathematician, mentored 2 geniuses: Indian Ramanujian and Chinese Hua Luogeng 华罗庚*) were not Senior Wrangler. Prof Hardy hated Math Tripos syllabus (revealed in his autobiography: “A Mathematician’s Apology“).

1914 Brian Charles Molony
1923 Frank Ramsey
1928 Donald Coxeter
1930 Jacob Bronowski
1939 James Wilkinson
1940 Hermann Bondi
1952 John Polkinghorne
1953 Crispin Nash-Williams
1959 Jayant Narlikar
1970 Derek…

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Topological Data Analysis

Math Online Tom Circle

Three Key Ideas:

Cloud of Points
Filter function: f (x,y,z) –> x
Cluster: Overlapping bins
Draw network:

Vertices: clusters
Edges: connecting lines between clusters.

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Crash course in Category Theory

Math Online Tom Circle

Key Point:

Haskell & any FP compiler don’t check the Category Theory proof if your codes (eg. fmap) follow Functor’s Laws (eg. Preserve structure, identity) or Monad’s Laws !

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God’s Book of Perfect Proof

Math Online Tom Circle

https://www.quantamagazine.org/gunter-ziegler-and-martin-aigner-seek-gods-perfect-math-proofs-20180319/

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Canadian mathematician Langlands wins Abel Prize

Math Online Tom Circle

http://www.thehindu.com/sci-tech/science/canadian-mathematician-langlands-wins-abel-prize/article23304119.ece/amp/

Michael Bishop’s Science

Math Online Tom Circle

Michael Bishop (Nobel Medicine 1989)
School Science subjects should be taught in this sequence (unless in single ‘Combined Science’):
1st Physics
2nd Chemistry
3rd Biology

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Mind over Matter : Bruce Lipton

Math Online Tom Circle

Key Points:

1. The Foundation of Science has changed since Newtonian Era, except Biology / Medicine and Psychology which don’t keep up:

Math => Fractal Math

Newton Physics (Matter)=> Quantum Physics (Energy)

(Organic) Chemistry => Electro-Chemistry

Energy = Field = 气 Chi / Qi

3. Environment -> Mind -> Perception -> Genetic Control

4. Consciousness (5%), Subconscious-ness (95%)

Examples:

Driving car: inexperienced driver (consciousness), experienced driver (subconscious).
Dating (conscious behavior)

Note: The successful DeeplearningAI is using the concept of Perception called “Perceptrons” by analysing the Environmental (BIG DATA) pattern with the help of Mathematics (Calculus : Gradient Descent, Statistics : Bayesian Probability, Algebraic Topology, etc).

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“Deep Learning” Online Book

Math Online Tom Circle

Most popular AI Deeplearning Book by MIT :

[English]

http://www.deeplearningbook.org/

[中文版]

https://exacity.github.io/deeplearningbook-chinese/

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NS Overseas Notification SMS Number (NOT 36767)

The new number to SMS is no longer 36767.

SMS to 82411633 instead.

The format is still the same (separated by space):

MNC NRIC DestinationCountry DepartureDate (DD/MM/YYYY) ReturnDate YourHPNumber

Notification for NSmen Overseas Trip is only needed if you travel for more than 14 days.

Source: ns.sg

Disclaimer: The above is correct at the time of posting. Please double confirm with the official sources in case there are any updates.

Update 2019: NSmen no longer need to notify Govt of overseas travel under 6 months. Read more at: https://www.todayonline.com/singapore/nsmen-no-longer-need-notify-govt-overseas-travel-under-6-months

Creator of ‘Grand Unified Theory of Mathematics’ Wins Prestigious Math Prize

Professor Langlands is very famous in the field of Number Theory. Congratulations to Prof. Langlands for winning the very prestigious Abel Prize.

Source: LiveScience

A mathematician who developed what some consider the “grand unified theory of mathematics” has won one of the most prestigious prizes in mathematics.

Robert Langlands, an emeritus professor at the Institute for Advanced Study at Princeton University, has won the Abel Prize, a prestigious mathematics prize that honors a lifetime of groundbreaking work, organizers of the prize announced yesterday (March 20).

Langlands, 81, won the prize for work in which he found deep connections between two seemingly disparate areas of mathematics: number theory and harmonic analysis, according to a statement from the organizers of the prize.

How to help gifted children deal with hyper-sensitivity

This should be useful for parents of gifted kids.

Linear Algebra Online Calculator (Matrix Calculator)

Just came across this online tool (http://matrix.reshish.com/). Probably one of the nicest online matrix calculators that can do most stuff like calculating determinant, inverse, rank, Gauss-Jordan Elimination, etc.

Very intuitive and easy to use. Recommended to bookmark it for further use!

Cannot Login to SP Services

The SP Services website (https://services.spservices.sg/#/login) seems to be down as of 21 March 2018.

The error message received is:

Error: We’re currently facing issues communicating your request, please try again in a few minutes.

SP Services Utilities for Resale Flat

By the way, something I learnt from calling the hotline is that there is now no need for the old owner to cancel or to transfer the account. The new owner can simply proceed to open a new utilities account, and the old one will be cancelled automatically one day before the new account is opened. There will be no disruption to the water/electricity supply.

This is an improvement over the old method where the SP account has to be cancelled before opening a new SP services account.

Choosing a Programming Language

Math Online Tom Circle

…
https://dev.to/drminnaar/choosing-a-programming-language-493h

…

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Programmers need Advanced Math ?

Math Online Tom Circle

As software becomes more complicated for high-speed trains, driverless cars, missile weapons. .. and AI deeplearning algorithm, we can’t depend our life safety on the programmers who don’t understand the advanced maths behind these algorithms.

The Advanced Math is the Category Theory – the most advanced math foundation above “Set Theory” since WW2. Functional Programming is based on Category Theory with mathematical functions – always output correctly with no “side-effects”.

https://www.extremetech.com/computing/259977-software-increasingly-complex-thats-dangerous

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Which Countries Produce Best Programmers

Math Online Tom Circle

https://blog.hackerrank.com/which-country-would-win-in-the-programming-olympics/

The result is surprising … China, Russia, Poland are the top 3 countries based on the tests, not the usual expected countries such as USA, India, UK.

Reason: Mathematics ! China, Russia and Poland are strong in Advanced Math education in university. Functional Programming requires the Advanced Math eg. Category Theory.

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Prime Number Theorems Conjectures Explained

Math Online Tom Circle

1. Twin Primes Conjecture : there are infinitely many pairs of Twin Primes among the infinitely many prime numbers.

Note: 2013 Zhang YiTang proved the gap between the twin primes is no more than 70,000,000 [Terence Tao’s Polymath Project using Zhang’s method to further narrow the gap to 246 ]

2. Goldbach’s Conjecture

Chen’s “1+2” Theorem

3. Palindromic Primes (eg.11, 101, 16561)

4. Riemann Hypothesis

https://www.sciencealert.com/prime-number-theorems-conjectures-explained

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Trigonometric Identities and Differential Equations

Trigonometric Identities
$\begin{aligned} \sin(A\pm B)&=\sin A\cos B\pm\cos A\sin B\\ \cos(A\pm B)&=\cos A\cos B\mp\sin A\sin B\\ \tan(A\pm B)&=\frac{\tan A\pm \tan B}{1\mp \tan A\tan B} \end{aligned}$
$\begin{aligned} \sin A+\sin B&=2\sin\frac{1}{2}(A+B)\cos\frac{1}{2}(A-B)\\ \sin A-\sin B&=2\cos\frac{1}{2}(A+B)\sin\frac{1}{2}(A-B)\\ \cos A+\cos B&=2\cos\frac{1}{2}(A+B)\cos\frac{1}{2}(A-B)\\ \cos A-\cos B&=-2\sin\frac{1}{2}(A+B)\sin\frac{1}{2}(A-B) \end{aligned}$

Reduction to Separable Form
Equations of the form
$y'=g(\frac{y}{x}).$
Set $\frac{y}{x}=v$ . Then $y=vx$ and $y'=v+xv'$ . Thus the original equation becomes $v+xv'=g(v)$ which is separable.

Linear Change of Variable
$y'=f(ax+by+c)$ can be solved by setting $u=ax+by+c$ .

Newton’s Law of Cooling
Rate of change of temperature of an object ( $\frac{dT}{dt}$ ) is proportional to the difference between its own temperature ( $T$ ) and the ambient temperature ( $T_\text{env}$ ). That is, $\frac{dT}{dt}=-k(T-T_\text{env})$ .

Funniest Math Video

This is probably one of the funniest Math video I have watched:

Another one that is very humorous is: Mr Bean Calculus and Trigonometry Math Exam.

Best “Do-nothing” Saving Accounts in Singapore (Better than Fixed Deposit)

Albert Einstein once said that “Compound interest is the eighth wonder of the world. He who understands it, earns it … he who doesn’t … pays it.”

In E Maths, students will learn that the formula for Compound Interest is $\text{Total amount}=P(1+r/100)^n$ . The exponential “to the power of n” explains the powerful effect of compound interest.

Choosing the right savings account can earn you hundreds of dollars or more per year (depending on how much savings you have).

There are many savings accounts that require a lot of “action” on your part, notably you need to credit your salary (with SAL code) to the bank, use their card, buy their investment/insurance/loan products, etc. These accounts include DBS Multiplier, OCBC 360, BOC Smart Saver, etc.

However, many people may not meet their criteria (or may not want to use so much of their products). For instance, self-employed people may not have SAL code salary.

Currently, I think the 2 best “Do-nothing” saving accounts in Singapore are:

Citi MaxiGain

From Citibank, the interest rate of MaxiGain is tied to the SIBOR rate. Starting interest will be 80% of SIBOR, stepping up 0.1% for 12 counters. Final rate is around 2% or more. This is frankly better than many fixed deposits. The only downside is that if you withdraw from the account, the counter will drop to 0 (or to 6 once you reach the checkpoint of 6 counters). But even with the counter dropping to zero, the interest is around 0.8% or more which is pretty decent.

Citibank is a very reliable bank from America, with bigger market capital than DBS. Tip: According to online sources, your account may be temporarily “frozen” for inactivity if you don’t have any transaction for too long a period. Hence, it is recommended to set up an automatic credit of $1 monthly to your Citibank Maxigain account.

CIMB FastSaver Account

Most people would not have heard of CIMB bank. It is a Malaysian bank. Not to worry though, they are listed under Singapore Deposit Insurance, that means you are insured up to $50,000 even if something bad happens to the bank. CIMB FastSaver is quite amazing in that it gives 1% interest, with no minimum balance or penalty for withdrawing. That means you can withdraw as and when you like. Their account is entirely online, you don’t have to step into their branch even for creating your account. It is a simple and effective savings account.

Happy Anniversary with WordPress.com! You registered on WordPress.com 5 years ago.

Just received this nice message from WordPress. 🙂

Happy Anniversary with WordPress.com!

You registered on WordPress.com 5 years ago.

Thanks for flying with us. Keep up the good blogging.

When religion (Pope Francis) met science (Stephen Hawking)

Also interesting is that the founder of the “Big Bang” theory was a Catholic priest!

A Catholic priest, Fr. Georges Lemaître, is “Father of the Big Bang Theory”

Source: https://aleteia.org/2016/12/03/when-religion-pope-francis-met-science-stephen-hawking/

The meeting was cordial, in spite of their diametrically opposed views regarding the existence of God.

[A surprising encounter took place between the pope, who holds that God’s mercy is infinite, and a scientist who rejects the existence of God due to the infinity of the Universe. On December 2, Stephen Hawking participated in a conference in Rome on the Big Bang, today considered the most widely accepted theory for the origin of the universe—a theory proposed by the Catholic priest Georges Lemaître (1894-1966) in 1927. ]

Pope Francis and Stephen Hawking: Faith and reason reach out to each other

British scientist Stephen Hawking and Pope Francis met face to face on November 28 in the Vatican, just days after that renowned astrophysicist had confidently declared that asking “what was before the Big Bang” is meaningless, since it “is like asking what lies south of the South Pole.”

The meeting between these two charismatic figures, world-renowned representatives of the Catholic Faith (Francis) and of reason based on scientific knowledge (Hawking), was rare but cordial, although they have completely opposite stances regarding the idea of God.

Pope Francis considers God to be an infinitely merciful Father, greater than the Universe He created — the same Universe that Hawking has been studying for decades and which he believes arose from nothing, without divine intervention.

Hawking stated that “the discovery of the expansion of the Universe” — the “big bang” theory — has been one of the “most important intellectual discoveries” of recent times. However, “With James Hartle from the University of California, I worked out what physical conditions the early universe must have, if space-time had no boundaries in the past. Our model became known as the ‘No-Boundary Proposal,’” said Hawking, during the Science and Sustainability conference organized by the Pontifical Academy of Sciences at the Casina Pio IV in the Vatican.

Although the Big Bang is a well-accepted theory among experts, and it is fundamental for the formation of galaxies and planets, there is still no convincing answer to what happened before the explosion, what caused it, and consequently the reason for existence. Nevertheless, the theory fits into a logical explanation of the universe. “The message of science is that we are the children of logic and not of chaos,” explained Italian nuclear physicist Antonino Zichichi, a colleague of Hawking, in an interview with Vatican Radio.

Happy Pi Day!

Happy Pi Day to all readers of Mathtuition88.com!

Check out our previous posts on Pi:

Did you know, Pi day is also Einstein’s Birthday?

How to get Pi on Calculator – Without pressing the Pi Button

The Mystery of e^Pi-Pi (Very Mysterious Number)

Euler’s proof of Pi^2/6 (Basel Problem)

Category Theory: How to Make Pi

Pi hiding in prime regularities

Also, check out our cooking blog for the following Pie recipes!

Beef Pie

Strawberry Cheesecake Pie

Lemon Pie

Easy Banana Pie

Fudgy Fudge Pie

Zuji Promo Codes

Zuji is a famous flight and hotel booking service. By using the below Promo Codes, you can save some money from their cheap and good bookings.

(Please use the links provided below to enter the official Zuji website!)

Product: Flights
Period: 19 March 2018 – 1 April 2018
Coupon Code: ZUJI15DFLT
Coupon Value: S$15
Minimum Spend: S$500
Travel by: 30 September 2018

Product: Hotels
Period: 19 March 2018 – 1 April 2018
Coupon Code: ZUJI33DHTL
Coupon Value: S$33
Minimum Spend: S$500
Travel by: 30 September 2018

Product: Packages
Period: 19 March 2018 – 1 April 2018
Coupon Code: ZUJI22DPKG
Coupon Value: S$22
Minimum Spend: Nil
Travel by: 30 September 2018

Stephen Hawking dies aged 76

RIP Stephen Hawking.

Source: http://www.bbc.com/news/uk-43396008

The British physicist was known for his work with black holes and relativity, and wrote several popular science books including A Brief History of Time.

“We are deeply saddened that our beloved father passed away today,” a family statement said.

At the age of 22 Stephen Hawking was given only a few years to live after being diagnosed with a rare form of motor neurone disease.

Researchers studying the Flynn effect have found that scores on IQ tests have actually improved over time.

This is an interesting read for parents and students. Basically, IQ is a complex notion that is dependent on various factors, including environment and educational factors.

Source: ThoughtCo

You’ve probably heard someone lament the state of “kids today”: that current generations aren’t as smart as the ones that came before them. However, psychologists who study intelligence have found that there isn’t much support for this idea; instead, the opposite may actually be true. Researchers studying the Flynn effect have found that scores on IQ tests have actually improved over time. Below, we’ll review what the Flynn effect is, some possible explanations for it, and what it tells us about human intelligence.

What Is the Flynn effect?

The Flynn effect, first described in the 1980s by researcher James Flynn, refers to the finding that scores on IQ tests have increased in the past century. Researchers studying this effect have found wide support for this phenomenon. One research paper, published by psychologist Lisa Trahan and her colleagues, combined the results of other published studies (which included a total of over 14,000 participants) and found that IQ scores have indeed increased since the 1950s. Although researchers have documented some exceptions, IQ scores have generally increased over time. Trahan and her colleagues observed, “The existence of the Flynn effect is rarely disputed.”

Why Does the Flynn Effect Happen?

Researchers have put forward several theories to explain the Flynn effect. One explanation has to do with improvements in health and nutrition. For example, the past century has seen a decrease in smoking and alcohol use in pregnancy, discontinuation of the use of harmful lead paint, improvements in the prevention and treatment of infectious diseases, and improvements in nutrition.

As Scott Barry Kaufman writes for Psychology Today, “The Flynn effect serves as a reminder that when we give people more opportunities to prosper, more people do prosper.”

In other words, the Flynn effect could be partially due to the fact that, over the twentieth century, we’ve started addressing many of the public health issues that prevented people in earlier generations from reaching their full potential.

Another explanation for the Flynn effect has to do with societal changes that have occurred in the past century as a result of the Industrial Revolution. In a TED talk, Flynn explains that the world today is “a world where we’ve had to develop new mental habits, new habits of mind.” Flynn has found that IQ scores have increased the most rapidly on questions that ask us to find similaritiesbetween different things, and more abstract types of problem solving — both of which are things that we need to do more of in the modern world.

Several ideas have been put forward to explain why modern society might lead to higher scores on IQ tests. For example, today, many more of us have demanding, intellectually rigorous jobs. Schoolshave also changed: whereas a test at school in the early 1900s might have been more focused on memorization, a recent test might be more likely to focus on explaining the reasons for something. Additionally, more people today are likely to finish high school and go on to college. Family sizes tend to be smaller, and it has been suggested that this may allow children to pick up on new vocabulary words while interacting with their parents. It’s even been suggested that the entertainment we consume is more complex today.

Trying to understand and anticipate plot points in a favorite book or TV drama may actually be making us smarter.

The value of pre-college preps

Habits to achieve your college #goals

1. Establish your priorities

2. Participate, get involved

3. Challenge yourself

4. Build your portfolio

5. Understand how you learn

List of common Laplace Transforms

What is NTUC Receipt Number?

Money Saving “Lobang” (Tips):

Second Order Linear D.E. Summary

Sample Tikz code for drawing the above figure (polygon angles)

Why are many gifted children challenging and exhausting?

SP Services Utilities for Resale Flat

Citi MaxiGain

CIMB FastSaver Account

The meeting was cordial, in spite of their diametrically opposed views regarding the existence of God.

What Is the Flynn effect?

Why Does the Flynn Effect Happen?