Wikipedia:Reference desk/Archives/Mathematics/2008 January 26

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[edit] January 26

[edit] tiling a hypersaddle

The smallest figure that tiles the hyperbolic plane (H2) is the "237" triangle, whose angles are π/2, π/3, π/7. What is the smallest tile in H3? —Tamfang (talk) 01:14, 26 January 2008 (UTC)

Does this answer your question: List of regular polytopes#Tessellations of Hyperbolic 3-space SpinningSpark 12:35, 27 January 2008 (UTC)
Thanks but no. The fundamental region of such a tiling is not the icosahedron, cube or dodecahedron mentioned, but an irregular tetrahedron (1/48 of the cube, 1/120 of the icosahedron or dodecahedron), and that's what I'm looking for. There are other tilable simplices that give rise to uniform but not regular tilings (see Talk:Polychoron#H3). I don't know how to measure the volume of any of them. —Tamfang (talk) 23:35, 28 January 2008 (UTC)

[edit] Most beautiful/interesting/elegant mathematical expression/function/other

Just curious about interesting maths. Those of you who are most passionate about math should be able to put forth your favorite(s). What is, in your opinion, the most beautiful, interesting, or elegant mathematical thing that you have found? HYENASTE 03:18, 26 January 2008 (UTC)

Euler's identity, eπi + 1 = 0 is oftenly mentioned as the most beautiful mathematical expression (check the article for more information.)
But, personally, I also like the series expansion of cosine: \cos x = \sum_{n=0}^\infty \frac{(-1)^nx^{2n}}{(2n)!}. We do have an article on mathematical beauty too, in case you're interested. — Kieff | Talk 03:31, 26 January 2008 (UTC)
I rather like f(x) = {x \over{x^2+1}}. It's the simplest (that is, having the smallest positive integral coeffecients possible) odd function that satisfies
\lim_{x \rightarrow +\infty} f(x) = \lim_{x \rightarrow -\infty} f(x) = \lim_{x \rightarrow 0} f(x) = 0 and
f^\prime(0) = 1 - SigmaEpsilon → ΣΕ 04:34, 26 January 2008 (UTC)

Mine is this "Genetic complementary learning autonomously generates fuzzy rule." I am still learning on Wikipedia how it all comes together. My first step is understanding Stochastic matrix. I just want to understand Stochastic matrix as fast as possible, so I can grasp some cool formulas published regarding GCL. :) --Obsolete.fax (talk) 07:44, 26 January 2008 (UTC)

Mine is Immerman-Szelepcsényi theorem. It is a genuinely suprising result and the proof is in some sense an elaborate mathematical joke. I also like Gödel's incompleteness theorems for similar reasons. Third mention is the proof that Hex (board game) is won by the first player to move. What I want to say - mathematical beauty is not related much to the importance underlying problems or to the simplicity of the result. It works more like jokes. You hear a story and in the end you get an additional surprising information, that completely changes the way you view the story. Take a look at the last link, the "Strategy" paragraph - anybody can understand it. Thorbadil (talk) 11:52, 26 January 2008 (UTC)

Nice challenge! My favorite is the formula

\int_{x=a}^\infty \frac{x^n dx}{e^x n!}=\sum_{i=0}^n\frac{a^i}{e^a i!}

It is not quite trivial, it has a some symmetry, and it is useful for converting between gamma distribution and poisson distribution. It is applicable when computing the probability that the best soccer team won, given only information on the outcome of a particular match. The beauty is entirely in the eyes of the beholder. Bo Jacoby (talk) 12:11, 26 January 2008 (UTC).


Picard's Theorem - any entire function must take every complex value infinitely often with at most ONE exception which is taken finitely. -mattbuck 00:14, 27 January 2008 (UTC)
You have conflated the two theorems called "Picard's theorem". The first, "Little Picard", says that any entire function takes on every complex value at least once, with at most one exception which is not taken on. The second, "Big Picard", says that a holomorphic function with an essential singularity takes on every complex value infinitely often, with at most one exception. The additional assumption in the latter theorem is seen to be necessary by the example of the identity function. Tesseran (talk) 05:44, 30 January 2008 (UTC)

My favorite would a result in probability theory regarding supermartingales. Which basically says that if you are in an unfavorable game (where you will loose money on the average) then there is no betting strategy that will make the game fair or favorable. There are strategies to make the game "less unfair" but it will still be unfair. Furthermore, the optimum betting strategy, in any game, is actually the bold strategy in which you bet all or nothing. These results might sound obvious, but what is interesting is that they have been mathematically proven. A Real Kaiser (talk) 05:35, 27 January 2008 (UTC)

As my favourite is Eulers formula mentioned above, I will give you me second favourite. It is the proof that there are only five regular polyhedra in 3-space. Not only is the proof elegantly simple, but it leads to a stunningly surprising result. The infinite number of regular polygons in 2-space would suggest the opposite conclusion before working through the maths. Possibly my third favourite is the same in 4-space, the number of polytopes now surprisingly increases again (but still a little way short of infinity) to six and includes one that has no analogue in any other number of dimensions. SpinningSpark 13:00, 27 January 2008 (UTC)
Cantor's twin daughters, the Ordinals and Cardinals. The first-ever consistent, convincing study of the absolute infinite is simple enough (at the beginning) that you could teach it to a kid along with arithmetic, and they probably wouldn't know what the big deal was about. A runner up would be fractals - not just the pretty colors, but the math behind them. Black Carrot (talk) 20:51, 27 January 2008 (UTC)

[edit] real numbers set

as amatter of fact ,i asked this before but i got no answer.i just want here to say it could be something,it could be important after study it.you can also go to my talk and see the pdf file that contains this discussion. the purpose of this theory is an attempt to show that real numbers can be generated or counted randomly and intensively .

Consider we express the tow positive real numbers ,A&B as,

A=Σam[(10)^(n-m)] B=Σbm[(10)^(n-m)] Where,( n,m=0,1.2,……) am,bm,positive integer Now if, am+bm=pm+10,pm<10 pm,positive integer Then we define the relationship R, ARB={pm*(10)^(n)}+{(pm+1)*(10)^(n-1)+..... Obviously, R; looks like adding backwards.e.g, 341R283=525 =(3+2)=5,(4+8)=12,(1+1+3)=5 Lets now pick up arbitrarily the infinite sequence

S0=Σn\(10)^(n),+Σn\(10)^(n+1) + Σn\(10)^(n+2)+.....

Where n=1to9 ,10to99,100to999 ,...etc.respectively

i.e, S0=0.123456789101112131415161718192021222324.... n ,is apositive integer. In order to generate or count* the real numbers within the interval,e.g. (0,1),

We define ,F;

F:N→IR

Where ,

F(n)=SnR0.1

Where, Sn, the set of sequences

S1=s0 R 0.1

S2=s1 R 0.1

etc.

hypothesis

There are an infinite sequences,s1,s2 that we can make S1RS2 Close enough to any real number.

now in order to solve the equation xR0.1 =1,x got to approach 1.9999....but how to solve xRx =1?obviousely,R,is an equevalence relation on positive real numbers set,natural numbers set and positive rational number set.

209.8.244.39 (talk) 12:31, 26 January 2008 (UTC)husseinshimaljasimdini

Your post is very hard to read. Please learn LaTeX (Help:Formula can help you with the Wikipedia version), write more clearly and format the post to be more readable (correct punctuation and capitalization will help). -- Meni Rosenfeld (talk) 14:07, 27 January 2008 (UTC)
Oh come on, it's not that hard. He's describing an adjustment of addition. Instead of carrying to the left, you carry to the right. So, 16(+)4=10.1, instead of 20. The decimal system remains the same, and is still intended to represent the real numbers. He describes a sequence of numbers in which each is the previous "plus" 0.1, and says that the "sums" of pairs of consecutive members of the sequence are dense in an interval. In the case of the number he started with, the unit interval. He says he's now trying to solve equations involving this operation, including x(+)0.1=1 and x(+)x=1. He doesn't seem to completely understand the words he's using, but he's still got an interesting question. Off the top of my head, 0.99999...(+)0.1=1, assuming normal rules of carrying to infinity, and there's no solution to the other question because the first digit will have to be 5 (0.5), and the second will have to be a digit n so that 2n+1=10m for some m, which is impossible. The article P-adics might be of interest. Black Carrot (talk) 20:42, 27 January 2008 (UTC)
Corrections. The first digit could be zero, so either x=0 or the same problem. Also, I meant 1.9999... for the other, which I see he has. Problem, though. If 1.99999....=2, and 2(+)0.1=2.1, and 1.999...(+)0.1=1, then 1=2.1, which is an odd system. May have to be careful with the infinite decimals. Black Carrot (talk) 20:46, 27 January 2008 (UTC)
I don't know what I was thinking. 50(+)50=1, of course. Black Carrot (talk) 20:57, 27 January 2008 (UTC).

Dear Meni Rosenfeld,Black carrot is right.that is exactly what i meant.Husseinshimaljasimdini (talk) 12:37, 28 January 2008 (UTC)husseinshimaljasimdini-baghdad-iraq

Okay. -- Meni Rosenfeld (talk) 13:19, 28 January 2008 (UTC)
Well done to Black Carrot for successfully interpreting the original post. Seems to me that the binary operation it describes is base-dependent - for example, in decimal, 2(+)2=4, but in binary 2(+)2 = 102 (+) 102 = 1. This means it is can have little or no serious mathematical use or interest, in my opinion. Gandalf61 (talk) 13:28, 28 January 2008 (UTC)

by reviewing the general mapping,R,where,ARB={pm*(10)^(n)}+{(pm+1)*(10)^(n-1)+..... it seems that binary operation is irrelevant.my main consern here is,can this relation with the set of sequences obove be applied to generate or count uniformly the irrational numbers set independently of considering cardinals concept or cauchy sequences?also can this relation be generalized to solve equations like xR1=10? OR xRx=(-1)?or xR1=(-1)? 210.5.236.35 (talk) 14:04, 28 January 2008 (UTC)husseinshimaljasimdini. also i mean here, can the mathemetical constructure of R,be fixed to be an equlevance relation involves negative numbers?210.5.236.35 (talk) 14:14, 28 January 2008 (UTC)Husseinshimaljasimdini

Negative numbers would be tricky. One way to define the negative of a number, say -a, is as the number such that aR(-a)=0. So, -1 would be the number x such that 1Rx=0, which would make it equal to 9.99999... . I still recommend reading about the P-adics. For xR1=10, x=19.999999... . For xRx=(-1)=9.9999..., x=(-50)=59.99999... . For xR1=(-1)=9.99999..., x=(-2)=8.99999... . I don't understand what you want to do with the sequences S1 and S2 etc. By "generate or count uniformly" do you mean generate randomly? Black Carrot (talk) 04:47, 30 January 2008 (UTC)

Thank you very much Black carrot for your advice.as amatter of fact the department of mathematics in my college has ran out of experts, specially after the war and when i asked them the were making fun of me because iam physicist,thats why i am annoying you guys here whith my questions.85.17.231.25 (talk) 09:06, 30 January 2008 (UTC)husseinshimaljasimdini

I'm glad to help. I think I've figured out what you're trying to say. You have a sequence of numbers dense in an interval. In other words, given a point in the interval, you can approximate that point more and more closely with a subsequence of the original sequence. You want to use the indexes of the subsequence to keep track of the given irrational number. Is that right? Black Carrot (talk) 17:19, 30 January 2008 (UTC)

yes black carrot.but the part of countablity conserns me.you know very well that ,G,is countable set if the sub sets that form ,G,are countable,now by reviwing the general concepts of countable set, we know that,K,is countable,if there exists an injective function F:K→N,either,K, is empty or there exists a surjective function,F:N→K.in the example that i gave above ,dont you think that ,F:N→(0,1),where,F=(SnR0.1),is asurjective?i also think F:(0,1)→N,is aninjective .(N,is the natural numbers set).if such function exists,does this make the positive irrational numbers set countable?because we know it is not.88.116.163.226 (talk) 12:31, 31 January 2008 (UTC)husseinshimaljasimdini

[edit] Calculus in Computer Science

I am a highschool senior, and I am looking at attending college with a CS major. I looked over the course requirements, and I saw that I will have to take Calculus I, II, and III. I was wondering, what on earth would you use calculus for while writing computer programs?
Thanks.
J.delanoygabsadds 23:50, 26 January 2008 (UTC)

Well, it could simply be analytical thinking that is required, or maybe just to show that you can work effectively without using actual numbers. -mattbuck 00:12, 27 January 2008 (UTC)
There are quite a few concepts covered in calculus that are very important in computer science and applied computer programming. Off the top of my head, there's optimization, asymptotic behavior of functions, limits and continuity of functions, vector operations, mathematical descriptions of shapes and surfaces, numerical methods for computing integrals and finding roots of polynomials, and a little bit of physics. —Bkell (talk) 03:55, 27 January 2008 (UTC)
Regardless of what's the use, please, don't be discouraged of taking CS because of calculus or mathematics. For one, mathematics is a great tool used frequently in any science, and you shouldn't dislike it. Secondly, picking an area of study based on "the less math the better" is a horrible way to make any important decisions in life.
I don't know if you have any of these thoughts, but I just wanted to point these out. Sadly, that sort of thinking happens a lot. :( — Kieff | Talk 05:45, 27 January 2008 (UTC)
Computer science is not about writing computer programs.  --Lambiam 10:20, 27 January 2008 (UTC)
I was going to say the same, but then realized that I couldn't coherently explain what CS is about - especially since the distinction can be blurry at times (a big part of CS is finding algorithms, which is only a step away from writing an actual program). Now that you've broken the ice, I'll say that CS is considered by many (most?) to be a branch of mathematics, so there should be no surprise in having to learn calculus for what is essentially a math major. Additionally, asymptotic behavior of functions is of key importance in definitions of computational complexity, and some definitions in, say, cryptography, use δ's and ε's in a way reminiscent of that found in limits. -- Meni Rosenfeld (talk) 12:22, 27 January 2008 (UTC)
And depending on what you intend on doing with that Computer Science, you may need to use calculus, or at least know a little bit about it, in some applications - for example, video game design would require physics (if you plan on making a game with a realistic physics engine), which in turn requires knowledge of the basics of calculus. That said, a friend of mine once specifically chose a university degree in "Design Computing" (which, for some reason, was housed in the Faculty of Architecture), over "Computer Science" or a similarly science-related stream, because it meant she didn't have to do any mathematics. Confusing Manifestation(Say hi!) 12:59, 27 January 2008 (UTC)
Also remember that “Computer Science” does not mean “Programming.” There is a lot more in the field than just programming. And while nearly all of the core courses in a typical computer science degree involve programming, probably less than a third to a half of them are about programming. GromXXVII (talk) 13:00, 27 January 2008 (UTC)
IMO it sounds that they focus far to much on calculus. While math is extremely important for CS, Discrete mathematics and Linear algebra is the topics that will matter. While computational complexity theory considers asymptotic behavior it's from a digital perspective and thus in a discreet model so calculus is not really applicable. Of course calculus is used to model an awful lot of things so when you start using what you learn in CS you will probably find that you are going to need it as well. Taemyr (talk) 10:50, 28 January 2008 (UTC)
Just wondering, is that a high school with two or three semesters per year? – b_jonas 07:28, 30 January 2008 (UTC)