Wikipedia:Reference desk/Archives/Mathematics/2009 December 31
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December 31
[edit]Ryan's hypo-whatsisis?
[edit]I've read milenium problems and wow! The math's mindboggling! My math teacher's a math degree and he sais he couldn't solve Ryan's hypo-whatsisis (I don't know what a hypothises is?). He's tried many calculators to solve it failed (he sais he'll solve it though since he's a math degree, but Wiki sais its to hard to solve). Now I know why math's there after 12th grade. Many thanx! Though there's another milenium problem Hodge conjecture? My math teacher sais it's fake math because there isn't numbers and formulas. Is Hodge real math? Please explain Hodge conjecture though! I know many of youy think I'm not serious but I'm serious. It's my English (i'm not a native speaker), so may be it sounds I'm not serious. But I now understand math's more than calculators, that's math is more than 12th grade (my high school teacher's wrong then). Sorry and many thanx. —Preceding unsigned comment added by 122.109.239.199 (talk) 04:09, 31 December 2009 (UTC)
Based on previous questions from the same IP address, this is not a good-faith, serious question. Expand to see off-topic discussion (which really belongs on WT:RD)
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Thanx but what's Hodge conjecture about? My math's teacher's a math degree and he sais it's fake math since there's no number/formula but there's weird cross symbol. He sais capital letters can't be crossed since math's about x's & y's crossed he's probably wrong since you all have dozen math degress (meni rose&field sais) and sais he's wrong. Please explain Hodge conjecture to me though?? What's it about? Many thanx. —Preceding unsigned comment added by 122.109.239.199 (talk) 06:47, 31 December 2009 (UTC)
Thanx but please explain Hodge though??? I want to understand Hodge because my math teacher doesn't sais it's fake math so I can be better than my math teacher (who's a math degree and's a math nerd). Please explain. I've solved Godbach. is it a math problem? You sais that math isn't about calculating? 4=2+2 6=3+3 8=5+3 10=7+3 ... does that solve goldbach math problem? Many thanx. Math hurted my brain but I wanted to challenge myself like it. Now I like it thanx! I wnat to understand Hodge though. —Preceding unsigned comment added by 122.109.239.199 (talk) 07:05, 31 December 2009 (UTC)
I can't speak english good sorry. English isn't my native language. I write my question quickly I make many mistakes. Please don't insult me though. —Preceding unsigned comment added by 122.109.239.199 (talk) 07:26, 31 December 2009 (UTC)
My friend help's me write my question. He sais he researches algebra and that algebra isn't calculators I didn't beleive him so he sais ask Wiki (he sais Wiki is very reliable). He helps me wrote my first question. Sorry for grammer and spelling error. He didn't help me anymore with questions. I written my questions quickly. Please don't insult me though. Thanx shahab. Why does meni rose&field insult me though??? —Preceding unsigned comment added by 122.109.239.199 (talk) 07:24, 31 December 2009 (UTC) Nobody has insulted you. Your explanations for certain strange things about your posts are unconvincing.Julzes (talk) 07:42, 31 December 2009 (UTC)
He or she is aching to be banned from wikipedia. I'd certainly treat it as an attack on the site. I don't know what the procedure is for dealing with that, of course, but someone here does.Julzes (talk) 07:55, 31 December 2009 (UTC)
Personally, I'm a little more trigger happy than you. But I'm also a little bit of a novice, so someone else will have to report it if that turns out to be more clearly necessary. I'll look into these incident procedures for future reference.Julzes (talk) 08:34, 31 December 2009 (UTC)
I cannot actually comprehend how one could mistake "Rosenfeld" as "Rose&field" (or how one could mistake "Riemann" as "Ryan", if you were referring to "Riemann's hypothesis"); it appears too awkward to be genuine. Please make an attempt to respect us, and we will make a similar attempt to respect your questions. Your (inconsistent) grammar and spelling are not really the problem were it not for your copious references to your "math teacher", how he possesses a "math degree" and how this excuses him from calling true mathematics "fake math" (I doubt your mathematics teacher actually exists, to be honest; more likely is that he (and your "algebraist friend") are hypothetical). If you actually ask us genuine questions (rather than seek a predictable disagreement with your "omniscient mathematics teacher's" words, and use this as a motivation to claim that he possesses a mathematics degree), I am sure that your posts will be treated with greater seriousness. However, at present, I would not recommend posting furthur questions; if you are not serious, it is likely that someone will report you (and even in the unlikely case that you are serious, we are quite convinced otherwise and may report you nonetheless). Consider posting furthur questions when you have thoroughly reflected on the comments given to you, and the time spent in answering your questions. --PST 09:19, 31 December 2009 (UTC)
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Gauss–Jordan elimination = reduced row echelon form?
[edit]Is Gauss–Jordan elimination the same as Reduced Row Echelon Form? --33rogers (talk) 07:46, 31 December 2009 (UTC)
- Gauss Jordan elimination involves taking the coefficient matrix and reducing it to reduced row echelon form, with corresponding changes on the right hand side. Then the system is solved by back substitution. This whole procedure is called Gauss Jordan elimination.[1] Reduced row echelon form is the type which the coefficent matrix ultimately becomes.-Shahab (talk) 07:51, 31 December 2009 (UTC)
Some basic topology of the torus
[edit]I want to show that the torus, described as "the surface of revolution obtained by rotating a circle about an axis in its plane and disjoint from it", is homeomorphic to (part of Problem 4 here). I suppose this is meant to be easy but my analytic geometry is sufficiently weak that I'm having a devil of a time turning this description into something concrete enough to write down an explicit map and formally verify all the conditions. Here's my attempt so far:
It's not too hard to see that . Define by so that the image of f is the circle of radius 1 in the xz-plane centred at (2,0,0). Let R(t) be rotation around the z-axis by the angle 2πt. Finally define by . It is clear that h is injective, and it's continuous because each component of is a linear polynomial in sines and cosines of s and t (as R(t) is just a rotation matrix). Since the domain of h is compact, this implies that h is a homeomorphism onto its image.
I'm not sure how satisfactory a solution to the original problem this is. Any advice appreciated! — merge 13:37, 31 December 2009 (UTC)
- This is not much of an answer, but can you intuitively feel why the result is true? Can you imagine the torus as a cartesian product in the manner described? In my eyes, the most important aspect of topology is attaining an intuitive feel for the topology of a space; although many textbooks encourage defining explicit homeomorphisms between topological spaces, it is not necessary as long as you can intuitively visualize the homeomorphism. However, when confronted with a first example, it is sometimes instructive to construct an explicit map. In this case, consider using your intuitive feel for the torus as a cartesian product when constructing a homeomorphism. Does this help? --PST 14:59, 31 December 2009 (UTC)
- Thanks for your response! Yes, the basic intuition is clear enough to me for all the spaces mentioned in the problem. What I'm specifically concerned about at this stage is grounding the intuition in correct proof. The above was exactly my attempt to do the explicit construction you suggested (modulo writing out all the rotation matrix details, which don't seem all that relevant). — merge 15:28, 31 December 2009 (UTC)
How about this:
Michael Hardy (talk) 01:48, 1 January 2010 (UTC)
- Thanks! I'm less concerned about the actual formula than whether the argument is sound, though (and whether there's a better way to do it). — merge 11:20, 1 January 2010 (UTC)
- What you did seems fine. Demonstrating a homeomorphism seems like a good way to go about proving that they're homeomorphic. 67.100.146.151 (talk) 05:01, 2 January 2010 (UTC)
- Yes, after looking at it for a while it seems all right to me. I think I've also managed to show the other spaces in the problem are equivalent. Thank you! — merge 12:21, 2 January 2010 (UTC)
- What you did seems fine. Demonstrating a homeomorphism seems like a good way to go about proving that they're homeomorphic. 67.100.146.151 (talk) 05:01, 2 January 2010 (UTC)
Looking for random "shopping" algorithm
[edit]I'm looking to see if an algorithm for a certain problem (described below) exists. I've run into this problem a number of times in different contexts, so suspect it might be solved, but don't know what it might be named. For the time being, I've coined the name "random shopping algorithm".
The problem begins with presented with a collection of "goods" with different costs and a given sum of money. The algorithm must then choose a random selection of goods to purchase whose total is equal to the given sum. It does not work to simply choose items at random until the sum is reached as that would select one high cost item more often than a lot of small cost items.
--jwandersTalk 22:25, 31 December 2009 (UTC)
- I'm not sure what your intended criteria for selection is but see subset sum problem. If you want all subsets with an equal sum to have the same probability then you have to know the total number of solutions and then pick one of the solutions. Or do you want each individual item which is both inside and outside at least one solution to have the same probability or as close to the same as possible? Then we would also have to define a closeness measure when there are items with different probabilities. PrimeHunter (talk) 23:33, 31 December 2009 (UTC)
- Please: "This criterion is..." or "These criteria are....". Michael Hardy (talk) 01:15, 1 January 2010 (UTC)
Hmm, no, I don't think the subset sum problem quite matches. As an example, say you have $20 and the items available are A for $10 or B for $1. The three possible outcomes here are (2As), (1A + 10Bs), or (20Bs). The algorithm I'm trying to find would chose one of these three with an equal probability. You right that I need some way of A) counting the number of possibilities N, randomly selecting n<N, and B) creating possibility n directly. As I said above, I'm hoping this is a solved problem that someone will recognise and know the name of. --jwandersTalk 00:16, 1 January 2010 (UTC)
- So your algorithm is:
- generate the N possibilities by the subset sum algorithm, and remember them in a list
- choose a random number between 1 and N, call it x
- output the x-th element of the list in step 1
- Doesn't that do it? Staecker (talk) 01:19, 1 January 2010 (UTC)
- In fact, if you have a method of enumerating all the possible combinations (perhaps using a recursive algorithm), then you don't even need to store them all in a list; the following algorithm will work to pick a random combination uniformly:
- Let k equal 1 and let X be the first combination.
- If there are no more combinations to generate, stop and return X.
- Otherwise, let Y be the next combination and increment k by 1.
- With probability 1/k, set X ← Y.
- Repeat from step 2.
- In fact, with a slight modification this algorithm can be used to uniformly pick n random combinations:
- Let k equal n and let X1, ..., Xn be the first n combinations.
- If there are no more combinations to generate, stop and return X1, ..., Xn.
- Otherwise, let Y be the next combination and increment k by 1.
- Generate a uniformly distributed random integer j between 1 and k inclusive.
- If j ≤ n, set Xj ← Y.
- Repeat from step 2.
- If you shuffle the first n combinations in step 1, the output should also be uniformly shuffled. However, it's just as easy to shuffle the output afterwards if you want. (Of course, there are cases where Staecker's method is better, especially if you expect to need more combinations later and if storing the list is cheaper than regenerating it.) −Ilmari Karonen (talk) 08:49, 4 January 2010 (UTC)
- In fact, if you have a method of enumerating all the possible combinations (perhaps using a recursive algorithm), then you don't even need to store them all in a list; the following algorithm will work to pick a random combination uniformly: