Wikipedia:Reference desk/Archives/Mathematics/2010 December 13

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December 13

Ambiguous problem(apparently)

How many numbers greater than 1000 but not greater than 4000 can be formed with digits 0,1,2,3,4 if repetition is allowed and not allowed. I got the answer for the first one easily as 3*5*5*5. But I calculated the second one as 2*3*4*5 and the text book says I am wrong. I actually began with the units place where I can fill it with 5 alternatives (0,1,2,3,4) and as we move to tens place etc... it becomes 4,3and finally 2 i.e I got 2*3*4*5. But the answer says 3*4*3*2! I am confused.. —Preceding unsigned comment added by 1.23.35.7 (talk) 01:06, 13 December 2010 (UTC)

Strictly speaking, it's 3*5*5*5-1, since you said it had to be greater than 1000. If you mean greater than or equal to 1000 then it is 3*5*5*5. The flaw with your second answer is that if you start with the units you can't be sure you'll have a 1, 2 or 3 left by the time you get to the thousands digit. You should start with the thousands and you'll find you get the correct answer. --Tango (talk) 01:19, 13 December 2010 (UTC)

Nope, it's 3*5*5*5. You can't have 1000, but you can have 4000. Taemyr (talk) 02:01, 13 December 2010 (UTC)

For the second question, it goes like this:

• There are 3 possible choices for the thousands place (1,2,3)
• For each of these, there are 4 possibilities for the hundreds place.
• For each of these, there are 3 possibilities for the tens place.
• For each of these, there are 2 possibilities for the ones place.

This gives 3*4*3*2, which is the answer given in the book. -- Bk314159 (Talk to me and find out what I've done) 03:56, 15 December 2010 (UTC)

Coding qualitative variables as dummy variables

From Elements of Statistical Learning, 2nd ed., page 10: "Here a K-level qualitative variable is represented by a vector of K binary variables or bits, only one of which is 'on' at a time. Although more compact coding schemes are possible, dummy variables are symmetric in the levels of the factor."

So, you could take this column of K states ("blue", "orange", "pink", etc.) and mix them in a different order and it makes no difference. Why is that an advantage vs. a "more compact coding scheme"? Or what is the advantage? 198.161.238.19 (talk) 16:26, 13 December 2010 (UTC)

Side-note, the 1st 3rd and 4th. editions of Elements Of Statistical Learning are available for apparantly legitimate download here: http://www-stat.stanford.edu/~tibs/ElemStatLearn/download.html Maybe later editions give better explainations. Not being a mathematician, and I may be telling you what you are already aware of or getting it completely wrong, but if this is something like multiple regression then you would have for example three columns for blue orange and pink respectively. If the item was say orange than it would be encoded as 0 1 0. If blue then 1 0 0, and so on. So the order of the columns would not make any difference. Perhaps - I'm guessing - if blue orange and pink were coded as 1 2 and 3 respectively in just one column, then their order would make a diference to the results. And you do not know beforehand what the right order should be. 92.28.245.105 (talk) 22:29, 13 December 2010 (UTC)

For "column" in the above, read "variable". 92.29.123.139 (talk) 23:47, 14 December 2010 (UTC)

The only obvious "more compact coding scheme" that comes to mind is to use K-1 vectors to code K states. In your example, "blue" could be (1,0), "orange" be (0,1), and "pink" be (0,0). You wouldn't want to code these categories with a single scalar (like "blue"=1, "orange"=2, and "pink"=3), as this imposes an order on the states. (I recall this can be done with ordinal data, but not purely categorical data.) I doubt, however, there is truly much of an advantage to using a symmetrical vs. a more compact coding scheme, particularly because any statistical inference shouldn't depend on the choice of (arbitrary) coding schemes. The only conceivable advantage that comes to mind with using a more compact coding scheme would be to save storage space; depending on the circumstances, that may or may not be a big deal. Nm420 (talk) 23:19, 15 December 2010 (UTC)

I'm still not clear why, exactly, being "symmetric in the levels of the factor" is an advantage. I get that it doesn't impose an order, but, again, why is not having an imposed order better? 198.161.238.19 (talk) 16:47, 16 December 2010 (UTC)

For all intents and purposes, there is no difference between using the symmetric coding scheme vs. the more compact one. If we were to think of this problem as a one-way ANOVA, comparing the means of K different treatments, you could set up the design matrix to have either K or K-1 columns. Using K columns, you would have the "symmetry" alluded to above, but the matrix would be of non-full rank; using the "compact" coding scheme with K-1 columns, the matrix would have full rank then. Estimation of the various parameters of interest goes by different methods in the two cases, though you will still draw the same inferences. The only conceivable advantage to the "symmetric" model, that I can think of anyhow, is the ease of interpretation of the results. However, a practiced statistician should have absolutely no problems with interpretation in either case, so this alleged advantage is really a misnomer.

I think perhaps you're trying to take a bit too much from the quoted line in the text. This quote is still only in the introductory chapter of the book, where the authors are just trying to give an indication of the many different ways to look at various statistical problems. In any type of problem, observing some type of symmetry will generally aid one in getting a quicker and fuller comprehension of the problem. From what I can tell, this is the reason the quoted line exists. Nm420 (talk) 20:33, 16 December 2010 (UTC)

I took to heart Feynman's injunction to understand every line. I know this line is inconsequential in the context of the book. That line asserted the value of symmetry over compactness and I was curious. Thanks for the reply. 198.161.238.19 (talk) 22:11, 16 December 2010 (UTC)

Theory of constant-Jacobian coordinate changes

Is there any general theory behind the set of coordinate transforms from $\mathbb {R} ^{n}\mapsto \mathbb {R} ^{n}$ whose Jacobian determinant has a constant value? By "general" theory, I mean, are such coordinate transforms classified in some way, do they belong to some lie group? If so, can someone direct me to the relevant literature? Thanks.--Leon (talk) 18:26, 13 December 2010 (UTC)

You may want to look at the theory of volume preserving maps? Ray^Talk 20:28, 15 December 2010 (UTC)

Integral from Biot-Savart law

Trying to find the general form of the magnetic field generated by a circular current of radius r in terms of x and y:

$B={\frac {\mu _{0}I}{4\pi }}\int _{0}^{2\pi }{\frac {-rsin\theta \ (y-rcos\theta )-rcos\theta (x+rsin\theta )}{((x+rsin\theta )^{2}+(y-rcos\theta )^{2})^{\frac {3}{2}}}}d\theta$

Is this correct, and does this integral evaluate to a closed form expression? I also tried putting x and y in terms of $r_{2}$ and $\theta _{2}$, but I wasn't able to evaluate that either. 149.169.148.62 (talk) 22:28, 13 December 2010 (UTC)

If you don't get a reply in a day or two then I'd recommend making a post on the science reference desk, there should be some physicist there to help. Most contributors on this page seem to deal with pure maths and statistics. I could be wrong thought. — Fly by Night (talk) 23:12, 13 December 2010 (UTC)

History: irrationality of powers of pi

Hey all. I am trying to figure out who (and when) first proved positive integer powers of pi are irrational. I have found that $\pi$ was proved irrational in 1761 by Lambert (Ref: Hardy and Wright). I found a proof specific to $\pi ^{2}$ as well (in Hardy and Wright), but no reference as to who (or when) came up with it. And, I have not seen proofs of any other powers specifically. It seems to me that most of the powers of $\pi$ were proved irrational when "Lindemann proved in 1882 that $e^{\alpha }$ is transcendental for every non-zero algebraic number $\alpha$, thereby establishing that $\pi$ is transcendental (see below)." according to this article Lindemann–Weierstrass theorem. Because, if $\pi ^{n}$ were rational, then $\pi ^{n}={\frac {a}{b}}$ for some integers $a,b$ which means $\pi$ is a root of $ax^{n}-b$. This contradicts that $\pi$ is transcendental so $\pi ^{n}$ is irrational for all positive integers n. So, is there any previous proof that the powers of $\pi$ are irrational? Or was this the first one? Thanks. StatisticsMan (talk) 22:33, 13 December 2010 (UTC)

Isn't that true for all transcendental numbers? It's transcendental if it's not algebraic, i.e. if it's not the root of a polynomial with rational coefficients. Assume that t is a transcendental number such that tⁿ is rational, i.e. there exist integers a and (non-zero) b such that tⁿ = a/b. This means that btⁿ – a = 0, i.e. t is algebraic. That contradicts our assumption that t was transcendental. Meaning there does not exists a rational number a/b such that tⁿ = a/b, i.e. all integer powers of transcendental numbers are (at least) irrational. The Pi article says that in 1882, Ferdinand von Lindemann proved that π is transcendental. — Fly by Night (talk) 23:26, 13 December 2010 (UTC)

Whoosh!

(that was the point going a mile over your head) --COVIZAPIBETEFOKY (talk) 23:59, 13 December 2010 (UTC)

"Fly by night", you missed the point. The original poster noted Lindemann's proof that π is transcendental in 1882, and asked whether there were earlier proofs of the weaker result that integer powers of π are irrational. Michael Hardy (talk) 05:02, 14 December 2010 (UTC)

Michael, I see. I read it as though he's posed and answered his own question without knowing it. I just wanted to emphasise that it's true for all transcendental number. My apologies to the Stat'sman. (It's the thought that counts.) COVIZAPIBETEFOKY, there's no need for that at all. I could easily have ridiculed one of your earlier answers; but that would have been rude. So come on, let's play nicely, shall we? — Fly by Night (talk) 16:03, 14 December 2010 (UTC)

Go ahead, I don't mind. I'd rather people point out when I'm being an idiot than ignore it. --COVIZAPIBETEFOKY (talk) 18:02, 14 December 2010 (UTC)

I think the point is that you can point out things to people without being a jerk about it. And, just because you are fine with being treated one way doesn't mean every other person in the world is fine with it as well. So, you should use tact instead and consider others. There is a saying that you catch more flies with honey than with vinegar. If you really want to correct people, doing it without being a jerk is actually going to work better. So, if that really is what you care about, then you will be more careful about how you do it. StatisticsMan (talk) 17:43, 15 December 2010 (UTC)

Check out the papter at vixra.org/pdf/1102.0058v1.pdf

Intersecting loops

Let's say I have n loops in the plane, labelled a_i (1 $\leq i\leq n$), and I can arrange them however I like. Given any n x n symmetric matrix, B = (b_ij), with entries 0 or 1, is it possible to arrange the loops so that if b_ij = 0 ($i\neq j$) then loops a_i and a_j are disjoint, and if b_ij = 1 ($i\neq j$), then a_i and a_j intersect at least twice? If not, is there a way to classify the situations where it isn't possible? I've tried to come up with a canonical way of drawing the loops so that they intersect in the desired way, I'm just worried that there will be a situation where the ith loop needs to intersect the jth, but that they're fenced off from each other by loops that loop i isn't allowed to intersect. Any situation like that which I've managed to construct has been remedied by drawing the loops in a different way...I'm just thinking lack of imagination might be causing me to miss a counterexample. Thanks, Icthyos (talk) 23:16, 13 December 2010 (UTC)

What you are describing is very similar to the idea of a string graph, but I don't know much more about these graphs than what is said in the article. —Bkell (talk) 03:08, 14 December 2010 (UTC)

Try this one:

${\begin{pmatrix}0&0&0&1&1&1\\0&0&0&1&1&1\\0&0&0&1&1&1\\1&1&1&0&0&0\\1&1&1&0&0&0\\1&1&1&0&0&0\\\end{pmatrix}}$

Gandalf61 (talk) 11:19, 14 December 2010 (UTC)

Unless I'm missing something: we want curves 1, 2, 3, 4, 5, and 6 so that 1-3 are disjoint, 4-6 are disjoint, and each of 1, 2 and 3 intersect 4, 5 and 6 at least twice. Without resorting to ASCII: draw 4, 5 and 6 as disjoint circles with centres on the x-axis, and draw 1, 2 and 3 as disjoint 'cigars' stretching across 4, 5 and 6.

I think I might have something, though - suppose I draw the curves in such a way that one, curve A, is 'caged off' (as in the original post) by a set of intersecting curves. If A needs to intersect some curve B outside this cage, there are two possibilities: if A is allowed to intersect at least one curve in the cage, then we simply let A pass through this curve, and hit B; if A is not allowed to intersect any member of the cage, then we redraw, with A not inside the cage! I believe this does away with all possible counterexamples - does anyone object? Icthyos (talk) 11:39, 14 December 2010 (UTC)

I suppose it's not quite that simple, because it might be a cage itself that is caged off, by a larger one. In cases like that, if need be, you'd have to move the entire (smaller) cage from inside to outside... Icthyos (talk) 11:56, 14 December 2010 (UTC)

Ah - I missed the point that intersecting loops intersect "at least twice" - I was assuming exactly twice. Hmmm ... Gandalf61 (talk) 12:18, 14 December 2010 (UTC)

Yessss...I started off trying to prove that actually, but then realised (luckily, as it wasn't true!) I didn't need 'exactly', just 'at least'. Icthyos (talk) 12:21, 14 December 2010 (UTC)

Gandalf's example is possible even if you require the curves to intersect exactly twice. Imagine two intersecting circles, like a typical two-class Venn diagram, and redraw each of the two circles as three disjoint concentric circles sufficiently close together. —Bkell (talk) 12:49, 14 December 2010 (UTC)

Icthyos, if your proposed proof that there is no counterexample holds water, then I think what you have proved is that every graph is a string graph. I don't know for sure that this is not the case, but I infer from the existence of the term "string graph" that there are graphs that are not string graphs—otherwise I would expect a theorem analogous to the circle packing theorem that applies to all graphs, showing that every graph is a string graph, and there would be no use for the term "string graph." I think what you should do is to try to find an example of a graph that is not a string graph, and see if that provides a counterexample for you. —Bkell (talk) 12:56, 14 December 2010 (UTC)

Subdivided K₅

Ah, if I read the string graph article less hastily, I see that it provides an example of a graph that is not a string graph—the subdivided K₅ shown to the right. This corresponds to the following matrix (apparently it's too big for LaTeX):

0 0 0 0 0  1 1 1 1  0 0 0  0 0  0
0 0 0 0 0  1 0 0 0  1 1 1  0 0  0
0 0 0 0 0  0 1 0 0  1 0 0  1 1  0
0 0 0 0 0  0 0 1 0  0 1 0  1 0  1
0 0 0 0 0  0 0 0 1  0 0 1  0 1  1

1 1 0 0 0  0 0 0 0  0 0 0  0 0  0
1 0 1 0 0  0 0 0 0  0 0 0  0 0  0
1 0 0 1 0  0 0 0 0  0 0 0  0 0  0
1 0 0 0 1  0 0 0 0  0 0 0  0 0  0

0 1 1 0 0  0 0 0 0  0 0 0  0 0  0
0 1 0 1 0  0 0 0 0  0 0 0  0 0  0
0 1 0 0 1  0 0 0 0  0 0 0  0 0  0

0 0 1 1 0  0 0 0 0  0 0 0  0 0  0
0 0 1 0 1  0 0 0 0  0 0 0  0 0  0

0 0 0 1 1  0 0 0 0  0 0 0  0 0  0

This should be the counterexample you're looking for. —Bkell (talk) 13:24, 14 December 2010 (UTC)

...and I shall have a great deal of fun verifying that, I'm sure! Hehe, thanks a lot. Icthyos (talk) 13:31, 14 December 2010 (UTC)

Well, the easy way to verify it is to understand why your idea is essentially the same as string graphs, and then trust that this subdivided K₅ is not a string graph. :-) Of course, the string graph article provides references if you want to see how that proof goes. This particular example has a lot of nice structure to it—there are five curves that are mutually disjoint, and for each pair of these five curves there is a curve that intersects precisely that pair—which may mean that direct verification is feasible. My guess is that you'll have to use the fact that K₅ is not planar, since that's the fundamental fact upon which this example is based. —Bkell (talk) 13:52, 14 December 2010 (UTC)

Hehe, yes - I picked up my pen to start doodling curves, before thinking better of it. I've already requested one of the books that is referenced in the article from my university library, so we shall see what comes of it. I wonder if there is a counter-example with fewer than 15 vertices...but we'd need some other argument, since K₅ is the smallest non-planar complete graph. Icthyos (talk) 15:31, 14 December 2010 (UTC)

(In fact, K₅ is the smallest non-planar graph, complete or not.) Some thoughts of mine toward a direct verification: Consider the five mutually disjoint curves corresponding to the original vertices of the K₅. We can't have three of them all nested inside each other, because otherwise there would be no way to have a curve that intersects the innermost and the outermost without intersecting the middle one. So these five curves are all “outside” each other—visualize them as small circles well separated from each other. (The case in which we have one big curve that encloses the other four is equivalent to this. There is no topological difference between one curve that encloses the other four, and five curves all “outside” each other—each curve divides the surface into two regions, one of which contains all four of the other curves.) Now, if there were some way to draw the remaining ten curves with the appropriate intersections, we'd essentially have a planar drawing of K₅, which we know to be impossible. —Bkell (talk) 15:57, 14 December 2010 (UTC)

I see. Thanks a lot - this has been a helpful (if slightly disappointing!) discussion. Icthyos (talk) 11:32, 15 December 2010 (UTC)