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Second year course in Mathematics with Fun

We are supposed to CREATE a new course in pure mathematics at the second year undergraduate level. It need NOT be analytic, logical, NOR rigorous. It could be descriptive without any proof. However, it must be intuitive, motivational toward current (or futuristic) research. The students should have FUN to work with this course. In addition to the topics listed below by my colleagues (I do not know what they are), be grateful if you could propose MORE with SPECIFIC POINTERS for us to follow up and to organise our course work. Thank you in advance.

• Can you hear the shape of a drum?
  
• Conservation principles and symmetry : Noether's Theorem
  
• Knotty problems
  
• A mathematical approach to weaving

Hope that this would be useful to many other colleges as well. Twma 01:35, 27 July 2006 (UTC)

How about calculations for an interstellar ship capable of flying humans to Proxima Centauri ? The students could do various calcs given engines capable of a given thrust, using a given amount of fuel, etc. The mass would be determined by fuel, food, water, and air stores, in addition to the fixed mass of the equipment and passengers. The mass of water and air would be influenced by the recycle rate, with far more needed at a 50% recycle rate than at a 99% rate, for example. Students could be given many different combos of ship engines, air recycling equipment, water recycling equipment, etc., and be asked to determine the size and mass of each sip and cost and length of each trip. (Note that the technology for this equipment does not currently exist, so fictional equipment parameters would be provided.) Come to think of it, this could make for a nice educational video game. :-) StuRat 03:48, 27 July 2006 (UTC)

With theory of relativity it'd be even more fun to calculate how old they would be when they finish their trip.(Igny 03:52, 27 July 2006 (UTC))

Thank you very much for the nice suggestion but it may not be able to satisfy the requirement to motivate the students towards current research. Personally, I believe that every mathematical model has its limitation, especially the mathematical description of relativity. When the inflation is high as described by mathematical data, the Government steps in and the model has to be replaced by another one.Twma 02:27, 28 July 2006 (UTC)

Aside from specific topics, here are some thoughts. Try for active and interactive participation involving multiple senses: sight, sound, touch, and maybe even taste and smell. Students at that age retain a lively curiosity, but are also highly motivated by anything connected with sex/social contacts and music. For specific topics, you might want to browse the Mathematical Atlas maintained by Dave Rusin. --KSmrq^T 15:55, 28 July 2006 (UTC)

I fully agree with you on participation involving multiple senses, sex/social contacts, music and video. I am looking forward to see our outcome. Thank you again.Twma 19:15, 30 July 2006 (UTC)

I strongly recommend that this item to be deleted because it will not get anywhere and because deletion could offer more room for other usage of this valuable resource.Twma 03:37, 2 August 2006 (UTC)

We don't usually delete pages from the reference desk, if that's what you're suggesting. They don't take up all that much space, and on principle, it's really easier to just leave them as they lie. If you really feel strongly about removing your own question, though, I guess we won't stand in your way. Black Carrot 03:50, 3 August 2006 (UTC)

Thanks.Twma 05:39, 3 August 2006 (UTC)

Just one thing, though: although I can't see any reason to stop you deleting this question in its entirety (it does seem like a strange thing to want to do, though) I do object to you removing pieces of things. It changes the flow of a discussion and removes valuable context. Black Carrot 06:44, 3 August 2006 (UTC)

Please accept my apology. To the best of my memory, I only removed one short simple line of yours and a few lines of mine. I thought both are IRRELEVANT to the title. Promise NOT to do it again. By the way, when our COMMITTEE makes a conclusion, this paragraph is likely to be beyond the display period of about 10 days. Twma 23:46, 3 August 2006 (UTC)

I'm having some trouble interpreting your writing style. Do the capitalized words mean you're getting snippy with me, or are they some kind of emphasis? And to what committee are you referring? Black Carrot 03:09, 4 August 2006 (UTC)

It appears that the more I write, the more misunderstanding I am creating and wasting more resources. The TITLE of this question is (what should be included in a proposed new) Second year course in Mathematics with Fun. CAPITAL letters are to emphasize something. Committe means our local organizing committee. It appears that we should shift ground from FUN course into a course ADVERTISING what an undergraduate in pure mathematics SHOULD KNOW. I propose PREVIEWS in measure theory, Fourier analysis on the unit circle group and on the real line, generalized functions. In addition, I also proposed topological properties such as Borsuk-Ulam theorem and fixed point theorem proved by combinatorial method without homotopy or homology groups. Finally, I appreciate the help from this site and have no intention to get snippy with anyone. To the best of memory, I deleted only one simple short line (Say What?) of yours. Hope that I am making friends. I shall say no more on issues that are NOT related to the title: (what should be included in a proposed new) Second year course in Mathematics with Fun.Twma 19:28, 5 August 2006 (UTC)

a quicker version of euclids algorithm

need some codes for finding a quicker version of euclids algorithm

Try binary GCD algorithm? I don't think there's anything faster than O(n²). —Keenan Pepper 04:41, 27 July 2006 (UTC)

Check out Prime Numbers - A Computational Perspective by Crandall and Pomerance (ISBN 0-387-25282-7); it contains pseudocode implementations of several algorithms that are quicker (and much more complicated). The best complexity they give is O(n (ln n)² (ln ln n)). (The complexity is actually O(M(n) ln n) where M(n) is the time needed to multiply two n-digit numbers. The best known multiplication algorithm is in O(n (ln n) (ln ln n))). Fredrik Johansson 12:04, 27 July 2006 (UTC)

For big integers, there's a trick to do multiple steps on the first word together and then you have to do fewer long multiplications. The advantage of this optimization is that – unlike with the binary Euclidean algorithm – you can get the modular inverses as well. You can read about this in Knuth vol 2.

If you want a fast implementation, not just theory, get the GNU Multi-Precision Library (GMP) package which is a free multiple-precision numeric library optimized for speed, and which is used in GnuPG too. – b_jonas 13:29, 27 July 2006 (UTC)

Horizontal asymptode

this is not homework. i was giving a practice test yesterday on blue.utb.edu and a question on functions came up which I couldnt figure out. could you please answer it?

     Find the horizontal asymptode of 
           f(x)=(6x^2-8x-4)/(8x^2-5x+3)

Hint: when x takes large values (either negative or positive) the value of one term in the numerator dominates all of the other numerator terms. Similarly, one term in the denominator dominates the rest of the denominator. As x gets larger, f(x) tends towards a limit which is the ratio of these two dominating terms - which in this case is a constant value, c. The horizontal asymptote is the line f(x)=c. Gandalf61 12:12, 27 July 2006 (UTC)

(The correct spelling has no "d", it's "asymptote".) It may be suggestive to evaluate at a large value, say x = ±1000. If you had the usual background to handle this properly, perhaps using limits, the problem would present little challenge. But here's a little trick you might try. Let x = 1/y and simplify. Now when y is zero, x is infinite. (Just be warned: this trick does not always suffice.) --KSmrq^T 16:52, 27 July 2006 (UTC)

Now that a day has passed, I'll add a tip to the trick: To convert the polynomials in x to ones in 1/y, merely reverse the coefficients. Formally, a₀+a₁x+⋯+a_nxⁿ becomes a₀+a₁(1⁄y)+⋯+a_n(1⁄y)ⁿ, which is then multiplied by yⁿ to give a₀yⁿ+a₁yⁿ⁻¹+⋯+a_n. This will suffice if the numerator and denominator have the same degree.

To amplify on the caution: Suppose, for example, we are given (8x³−1)/(1−x²). Because the numerator has larger degree it dominates at infinity; but the ratio goes to −∞ as x goes to +∞, and +∞ as x goes to −∞. If we try the substitution trick we get (8−y³)/(y³−y), and evaluation at y = 0 cannot tell us about the sign behavior. In algebraic geometry, the projective line merges positive and negative infinity, so there is no conflict. Exercise: Can a ratio of polynomials have different finite asymptotes at positive and negative infinity? --KSmrq^T 15:29, 28 July 2006 (UTC)

definition

Hi! I cannot search "ichimoku cloud" in Wikipedia. I understand that it is one of method that people use for stock/forex technical analysis, but I would like to know more in details. I wish to have this definition in Wikipedia sometime soon. Thank you. Best, Nan

Perhaps you can be bold and start a new article. --Proficient 15:49, 27 July 2006 (UTC)

The first hit on a Google search for "ichimoku cloud" is [1], which looks quite informative. The other hits look good too. Black Carrot 18:35, 27 July 2006 (UTC)

Thank You Very Much

Generating Function

I just started reading about generating functions, and I'm really, really confused. In what way is 1/(1-x), for instance, equal to x + x² + x³ ... ? There's not a whole lot of explanation given, here or elsewhere. Black Carrot 22:35, 27 July 2006 (UTC)

It is easy to see that

$(1-x)(1+x+x^{2}+\ldots +x^{n})=(1-x^{n+1})\,\!$

Then divide by (1-x) both sides and let n go to infinity under assumption that $|x|<1$ (Igny 23:18, 27 July 2006 (UTC))

These are formal power series, with no requirement that they converge in any sense. Anyway, Black Carrot asks for an equality that happens to be false, because it is not a correct paraphrase of the article. We are given a geometric progression beginning with 1 (not x), with ratio x, and the sum of this infinite sequence is discussed in the appropriate article (as a limit). In a typical formal education this fact would be as trivially familiar as 1+1 = 2 long before one encountered generating functions, which is why it makes a good example. The explicitly worked Fibonacci example is better still. Best of all is to read the cited book by Wilf, available online. (Why settle for a Wikipedia article when a top expert has provided a free text?!) --KSmrq^T 00:36, 28 July 2006 (UTC)

You're right, I should have followed more of the links. It makes sense now. Black Carrot 03:03, 28 July 2006 (UTC)