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《一路北上》 December 19, 2010

Posted by choonyee in Uncategorized.
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越过新柔海峡

踏上回家路途

夜幕低垂

巴士在南北大道上奔驰

阴柔的月光中

看见迎面而来的路牌

那是熟悉的马来文字

云锁孤月

巴士在永平附近抛描了

着急的乘客

纷纷致电回家报平安

那是熟悉的福建口音

人造灯光

点缀着游子之都

想起在这打拚的朋友们

你们多久没回家了？

我在回家途中

晨曦载曜

抚摸着粽油园的幼苗

拉紧了高压塔的电缆

路旁的石碑上写着

Sempadan Negeri Pulau Pinang

日上三竿

142天的别离

650公里的车程

越过槟威大桥

我回到家了

An interesting function June 16, 2010

Posted by choonyee in Mathematics.
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This post discusses the function $f(x)=x^x$ and another similar function of this form (I’m unsure if there is a name associated with this family of functions). The first encounter with this function is probably from Calculus on the study of indeterminate form $0^0$ .

First of all, we can show that $f$ is well-defined for all $x>0$ by taking natural logarithm,

$\log f(x) = \log (x^x) = x\log x$ . (1)

Then, we can evaluate the indeterminate form $0^0$ by taking limit

$\displaystyle\lim_{x\to 0^+} x^x = \displaystyle\lim_{x\to 0^+} \exp(x \log x)$ .

Now we encounter another indeterminate form of $0\times \infty$ which can be dealt with by L’Hopital’s Rule, giving

$\displaystyle\lim_{x\to 0^+} \frac{\log x}{1/x} = \displaystyle\lim_{x\to 0^+} \frac{1/x}{-1/x^2} = 0$ .

Hence, we have the following result

$\displaystyle\lim_{x\to 0^+} x^x = e^0 = 1$ .

So far all the discussions above can be found from standard Calculus textbook. Now let us explore this function further by considering a few problems. A natural problem to look at is the derivative of $f$ . Is $f$ differentiable for $x>0$ ? Or in the first place, is $f$ continuous for $x>0$ ? The answers to both questions are the same — YES. We can of course invoke the rigorous $\varepsilon-\delta$ definition but in this case it is much easier to take advantage of equation (1). Note that for $x>0$ , $x^x = \exp(x\log x)$ is just a composition of continuous and differentiable functions.

It is important to emphasize that conventional differentiation rules for powers/exponents do not apply here since both the base and exponent are variable now. For instance,

$\frac{d}{dx} (x^x) \neq x(x^{x-1})$ .

Nonetheless, the derivative $f'$ can still be computed by implicitly differentiating equation (1),

$\frac{1}{f(x)}f'(x) = x(\frac{1}{x})+\log x$ ,

$f'(x) = (1+\log x)x^x$ .

After obtaining the derivative, we can find the stationary point of $f$ by letting $f'(x)=0$ . Since $f$ is never 0, we have $1+\log x = 0$ which gives $x=e^{-1}$ . Thus, $f$ has only one stationary point $(e^{-1},e^{-e^{-1}})$ . To determine whether this is a min or max point, we need to compute the second derivative

$f''(x) = [\frac{1}{x}+(1+\log x)^2]x^x.$

At $x=e^{-1}, f''(x)>0$ implies that it is a minimum point. Now we can combine all the information obtained to sketch the graph of $f$ for $x>0$ . It will look like a skewed U-shaped curve with a min point and goes to infinity as $x$ increases.

Next, let’s turn our attention to negative values of $x$ . In particular, we would like to verify whether the following claim (posted in a math forum) is true

For $x<0, x^x$ is real-valued if and only if $x$ is negative integer.

One of the direction is easy to verify, i.e. if $x$ is negative integer, then $x^x$ is real-valued. However, the other direction is rather tricky. For example,

$(-\frac{1}{2})^{-1/2} = -\sqrt{2} \, i$ ,

shows that $x^x$ can be complex-valued. Take another example says $x=-\frac{1}{3}$ , then we’ll encounter the n-th root of unity, namely $(-1)^{1/3}$ , which yields one real and two complex values. This suggests that $x^x$ becomes a multi-valued function and further discussion will wander far off Calculus and drift into the realm of Complex Analysis.

After experimenting with a few more examples show that $x^x$ is complex-valued for most of negative $x$ except at integer points. However, providing examples does not secure a mathematical proof and thus a solid proof is still sought after. My guess is we do need to use complex analysis to prove that $x^x$ is real-valued for negative $x$ only at integer points.

From the above discussion, we see that $x^x$ is rather wild at the negative side. Why don’t we deal away with the negative values by considering the following function

$g(x) = (x^2)^{x^2}$ ,

which is well-defined for all values of $x\in\mathbb{R}-\{0\}$ . At the origin, we encounter the indeterminate form $0^0$ again. But this time we can evaluate both one-sided limits which, after similar computations as above, give

$\displaystyle\lim_{x\to 0^-} g(x) = \displaystyle\lim_{x\to 0^+} g(x) = 1$ .

Hence, in parallel to conventions, it is wise to define $0^0 = 1$ so that the function $g(x)$ is now continuous and differentiable everywhere. The graph of $g$ will look like a W-shaped (with smooth corners) curve, with three turning points and symmetrical about y-axis since $g$ is an even function.

《再别南大》 May 5, 2010

Posted by choonyee in University.
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大学毕业在即，写了一首《再别南大》与同学们共勉之。

骊歌冉冉晕开时，岁月悠悠四夏驰。

大学之道未闻知，茫然背考为何值？

花香鸟语树成荫，夜宵琴声众成群。

若干年后喜相聚，当歌对酒叹光阴。

注释：

骊歌：离别之歌，毕业之歌。
冉冉：慢慢地。
晕开：散开。取自方文山作品《青花瓷》，“月色被打捞起，晕开了结局。”
四夏：新加坡常年如夏，故一夏一年，四夏四年。
大学之道：取自四书之一《大学》开篇第一句，“大学之道，在明明德，在亲民，在止于至善。
闻知：见闻，知道。
背考：背书考试。
众成群：众学生三五成群。含贬义，指学生按照国籍而分群。
若干年后：几年之后。
当歌对酒：取自曹操《短歌行》开篇第一句，“对酒当歌，人生几何？”

FYP May 5, 2010

Posted by choonyee in Mathematics.
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It has been a very long time since I write something on this blog. The main reason is that I have been occupied with my Final Year Project (FYP) for the whole semester. As the FYP is all over now, I shall share the fruit of my hard labor here.

My project is mainly a study of a relatively new numerical methods, the moving mesh methods, for solving time-dependent PDEs. I’ll share the abstract here and if the reader is interested, then the full report is available for download as well.

Abstract:

Moving mesh methods have gained substantial popularity over the past two decades as an adaptive strategy to solve time-dependent partial differential equations (PDEs). The main idea of moving mesh methods is to find an invertible mapping that maps localized structure in physical domain to a smooth one in computational domain. This can be done by introducing moving mesh PDEs (MMPDEs) that constitute the core of moving mesh methods. Two important characteristics of this method are the dynamic adaptation of mesh with the solution of physical PDEs and concentration of mesh points to region of large solution variations. Various numerical experiments are performed to illustrate the idea and highlight the advantages of moving mesh methods.

The full report is available here FYP Report. There are some numerical results that are converted into video clips but unfortunately, I haven’t figure out how to upload videos here.

What’s the next to go missing? December 24, 2009

Posted by choonyee in Politics.
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Prerequisite: Manglish+Malay+Hokkien

Ah Beng: Wah, u guys hear the news bo? One F5 fighter jet engine went missing from the Sungai Besi air force base le.

Akmal: Not one engine, but two le! Each one worth RM 500 million. Those thieves must be damn rich now.

Ahmad: Sure bo? Who can steal jet engine wo? Must be an inside job gua.

Ah Beng: What gua. Of coz la! And the engines were stolen two years ago le, now only the govn let us know nia. And you guess who was the Defense minister two years ago? Hehe…

Ahmad: Ooo….is him AGAIN ah?

Akmal: Aiyo, how we know le? He says the investigation will be open and promise no cover-up wo. But isn’t that delay of 2 years a cover up also?

Ah Beng: Ini panggil sendiri slap sendiri face. Haha.

Akmal: But his face so thick… nvm la.

Ahmad: This case quite serious le. If suddenly got people attack us then how?

Ah Beng: Aiyo, last time the jet fly fly sendiri can jatuh mia. Now the jet fly also cannot fly ah! Sia sui nia…

Ahmad: Eh u think the police can catch the thieves bo?

Akmal: If the thieves are some high ranking military personnel or got involve some govn people, u think the police dare to catch them meh. I think they’ll just find a few kelefe to eat the dead cat la.

Ah Beng: If got some blogger write something sensitive or some people take it to the street, they kena ISA and masuk Kamunting. If got some small fish suspected of rasuah they kena MACC investigation sampai ada orang mati di situ. Now so big case hor, u hear the govn speak of ISA or MACC?

Akmal: Ini panggil Satu Malaysia. Dua standard! Tau tak.

Ahmad: Argh, mereka pengkhianat negara. Mesti mati ditembak! :@

Ah Beng: Orang bawah ditembak. Orang atas enjoy life ~

Akmal: I bet this is tip of iceberg nia. If engine also can go missing, other military equipments sure can go missing too.

Ahmad: Yeah. Sooner or later the country also go missing!

Ah Beng: If that GUY put our country on SALE, do u think our rich neighbor will be interested to buy? LOL…

Birthday primes (part 2) November 10, 2009

Posted by choonyee in Mathematics.
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We have seen in the previous post that anyone whose YOB is odd will only have either 1 or 0 birthday prime. It is natural to ask what about people whose YOB is even. It turns out that from the limited observation of the table in the previous post, anyone whose YOB is even will always have at least 1 birthday prime if they live long enough. This leads me to conjecture that

Given any positive even number, we can ALWAYS find a prime such that their sum is another prime.

(Remark: I’m unsure if this is already a known result. I just find out from Wikipedia about Polignac’s conjecture which looks similar to my question. But I’m not sure if my question here is equivalent to or just a special case of the Polignac’s conjecture.)

I find no counterexample so far for the first 1 million even numbers with the first 1000 primes. I understand that 1 million is nothing compared to the infinitude of primes, but it convinces me a little that this conjecture holds. I shall sketch out my idea here:

Let $\mathbb{P}$ be the set of all odd prime numbers and $\mathbb{P}_n$ be the set of odd prime numbers strictly less than $n$ . Let $E$ be the set of all positive even numbers and for an odd prime $n$ , let $E_n=\{n-p\ |\ p\in\mathbb{P}_n\}$ . For examples,

$E_{11}=\{4,6,8\}$ , $E_{19}=\{2,6,8,12,14,16\}$ .

The above conjecture is then equivalent to prove that

$E=\bigcup_{n\in\mathbb{P}}E_n.$

To be continued …

Birthday primes November 8, 2009

Posted by choonyee in Mathematics.
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We define birthday primes as a pair (year, age) where your age on that year are both prime numbers. I have written a Matlab program to list out all the year of birth (YOB) of people born in the 20th century who have some positive number (#) of birthday primes (BP). The only assumption made is life span of 80 years. The YOB that are not listed have 0 birthday prime.

YOB # of BP

1900           5
1902           6
1904           3
1905           1
1906           4
1908           6
1910           3
1911           1
1912           4
1914           6
1916           2
1918           4
1920           9
1922           3
1924           3
1926           9
1928           5
1929           1
1930           5
1931           1
1932           8
1934           3
1936           5
1938           7
1940           5
1942           4
1944           8
1946           6
1947           1
1948           3
1949           1
1950           9
1952           3
1954           3
1956          10
1958           5
1960           6
1962           6
1964           4
1966           6
1968           9
1970           7
1971           1
1972           3
1974           9
1976           6
1977           1
1978           2
1980          10
1982           6
1984           5
1985           1
1986           9
1988           5
1990           6
1991           1
1992           8
1994           5
1995           1
1996           6
1997           1
1998           7

For my own case (1986), my birthday primes are as follows:

Year Age

1993           7
1997          11
1999          13
2003          17
2017          31
2027          41
2029          43
2039          53
2053          67

Furthermore, people whose YOB is odd can either have only 1 birthday prime or 0 birthday prime, even assuming immortality, by a simple observation:

odd YOB +2 = odd (either prime or composite)

odd YOB + odd ‘prime age’ = even (composite)

‘Prime’ Birthday November 7, 2009

Posted by choonyee in Uncategorized.
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Today is 7th of Nov (11th month) and I’m 23. It just happens that 7, 11, 23 are all prime numbers! What a ‘prime’ birthday.

I am looking forward to year 2017 for a ‘all-prime’ birthday, where day, month, year and age (7,11,2017,31) are all primes! Wohoo~~

Metric function is continuous November 5, 2009

Posted by choonyee in Mathematics.
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A metric on a set X is a function $d:X\times X\to\mathbb{R}$ . For any $x,y,z\in X$ , the function $d$ satisfies the following conditions

(i) $d(x,y)\geq 0$ and $d(x,y)=0$ if and only if $x=y$ ,

(ii) $d(x,y)=d(y,x)$ ,

(iii) $d(x,z)\leq d(x,y)+d(y,z)$ .

The purpose of this post is to prove that $d$ is a continuous function. I’m unsure whether this proposition is so trivial that it is often omitted in standard books on analysis and topology. Anyway, here’s my proof.

Let $x=(x_1,x_2)$ and $y=(y_1,y_2)$ . We define a new metric $D$ on $X\times X$ by

$D(x,y)=d(x_1,y_1)+d(x_2,y_2).$

We can show that this definition satisfies all the three conditions above since $d$ itself is a metric. We also require the following inequality

$|d(p,q)-d(r,s)|\leq d(p,r)+d(q,s),$

which can be shown by using the triangle inequality

$d(p,q)\leq d(p,r)+d(r,s)+d(s,q).$

Since we are working on metric spaces, we shall use the classical definition for a function to be continuous. Given $\varepsilon>0$ , we set $\delta=\varepsilon>0$ . Then for $D(x,y)<\delta=\varepsilon$ , we have