2008年6月20日 星期五

托運行李箱

1 號行李箱 : 12 x 21 x 30
2 號行李箱 : 10 x 18 x 26
3 號行李箱 : 11 x 20 x 30
4 號行李箱 : 9 x 15 x 23
5 號行李箱 : 12 x 19 x 29
6 號行李箱 : 11 x 16 x 25

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x, y, z 是第一個行李箱的尺寸，a, b, c 是第二個行李箱的尺寸。不妨假設 x+y+z >= a+b+c 。若只考慮體積，以上問題是

2008年6月19日 星期四

A "Football Theorem"

With the European Football Championship being a hot topic these days (this is especially true for me as I am now in a country that is crazy about the sport!), let me share with you a theorem about football. It goes like this:

"Place the ball in the center of the court, and start the game. The next time when the ball is being placed at the center of the court again, there exists one point on the ball which is at exactly the same position as it was in the first time."

The proof only involves a simple concept in elementary Linear Algebra, so do think before looking at the hint below!

Highlight the space in the right for the hint: Use the fact that every rotation has an eigenvalue 1.

2008年6月17日 星期二

"Generalize"的雜談（一）

1 + 2 + 3 + ... + 100
100+ 99 + 98 + ... + 1

2008年6月14日 星期六

TOPIC: The Secret of Accurate Transmission - Coding Theory

SPEAKER: Mr Andy Chan (University of Illinois at Chicago)

DATE: June 21, 2008 (Sat)

TIME: 5pm--6:15pm

VENUE: Room P4909 (Purple Zone, 4/F) in City University of Hong Kong

Anyone who are interested in Mathematics are welcomed to this seminar.

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The Secret of Accurate Transmission - Coding Theory

Data transfer is frequent nowadays. It plays an important role in our daily life. It can be as simple as filling forms manually, or as complicated as data transmission between computers via optical fibres. How can we reduce errors during the course of transmission with mathematics? The method is coding theory! In the seminar, we will discuss several error-detecting and error-correcting methods. The mathematical structure behind will be introduced as well.

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1. 編碼學的用途
2. 最基本的例子：重覆編碼
3. 最常見的例子：檢測碼
4. 測錯距離及除錯距離
5. Hamming 碼

2008年6月9日 星期一

百年一遇

Just read the following article from Mingpao. You'll know right away why I'm sharing it here. :p

【明報專訊】前日一場「百年一遇」大雨，令香港多處成澤國，不過，「百年一遇」大雨今年已過，不等於明年沒有，因為明年出現的機率與今年一樣；加上「重遇期」可適用於不同地方，故同年內不同地區都有可能發生「X年一遇」的大雨。

2008年6月1日 星期日

Investigating lengths and areas

Inspired by Kahoo, who wrote an article on some possible investigations that a primary school student can carry out of the Fibonacci sequence, let me also try posting some questions in geometry, so that some of you may try your hands on them. (Unfortunately I think the questions that follow will require knowledge from high school mathematics :p)

Suppose we are given a rope of length L.

1. What is the area of the largest rectangle that one can enclose using the given rope?

Do you have any intuition about this problem? Let's try some extreme cases. A rectangle of a given perimeter L is determined by its length, which can vary from 0 to L/2. Suppose we are in these extreme cases. What would the areas of the rectangles be? Suppose we gradually move from one extreme to another. What will happen to the area of the rectangle? Does that give you some intuition, and suggest how you could prove it?

Students who know one-variable calculus can solve this as a maximization problem; students who does not know calculus can solve it by some simple algebra (for instance, using the difference of squares formula, which is essentially the essence of the calculus method).

Graduate students in mathematics can also do this by flowing the rectangle into a square :p

2. What is the area of the largest triangle that one can enclose using the given rope?

A triangle with given perimeter is determined by 2 parameters (e.g. the length of two of its sides). So this would look like a two-dimensional maximization problem. Students who know multi-variable calculus can solve this by invoking Heron's formula. What if you don't know two-variable calculus? Is there a way of reducing the problem to one-variable?

The answer is yes, provided that you know the maximum exists in the first place: then look at the configuration for which the area of the triangle is a maximum, fixing one of the sides and vary the other two, one realizes (using one-variable calculus now, because the length of one of the variable sides determines the length of the other!) that the two variable sides have the same length. This would show indeed that the maximum can only be achieved when the lengths of all three sides of the triangle are the same, i.e. when it is an equilateral triangle.

Actually a simple geometric symmetry argument already leads to the above observation, and one can bypass calculus altogether!

3. What is the area of the largest shape that one can enclose using the given rope? (This is the famous isoperimetric inequality.)

First, what is the intuition?

Now that the shape of the enclosed area is arbitrary, we have an infinite degree of freedom when we maximize the area; indeed if one formulates the problem correctly, one can do a maximization involving only "countably many variables". For instance, check out the following two Fourier analytic proofs of the isoperimetric inequality in The everything seminar, that makes use of the Fourier series of nice periodic functions.

I'm also thinking that maybe one can also prove the isoperimetric inequality using calculus of variations (an infinitely many variable version of calculus). Any suggestions?

4. Can the problem be further generalized?

Sure, how about a higher dimensional version of the isoperimetric problem?

As it turns out, the isoperimetric problem is closely related to some other important inequalities in analysis, like the Sobolev inequalities (say for smooth functions with compact supports)

$\left(\int_{\mathbb{R}^n}|f(x)|^{\frac{n}{n-1}}dx\right)^{\frac{n-1}{n}} \leq C_n \int_{\mathbb{R}^n}|\nabla f(x)|dx$

(which allows one to control the size of a function from its derivatives).