Wikipedia:Reference desk/Archives/Mathematics/2010 February 1

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February 1

welcome to the palyndrome day 01022010 --pma 01:33, 1 February 2010 (UTC)

What's the standard of proof to tell if an operation on a vector yields a vector?

I know that just returning a triplet of components doesn't mean an operation has yielded a vector. So given some operation @ where A @ B yields a triplet of numbers C_x,C_y,C_z, what's a property these three numbers will have only if C is a vector?71.161.63.23 (talk) 02:29, 1 February 2010 (UTC)

I don't understand. What is the difference between three components and a vector in three-dimensional space? You can usually treat them as equivalent. —Bkell (talk) 04:16, 1 February 2010 (UTC)

It seems the OP is using the definition of "vector" described here. --Tango (talk) 11:38, 1 February 2010 (UTC)

The question seems somewhat unclear, but here's a guess as to what is meant: a triple of numbers represents a vector relative to some basis (or "coordinate system"). Suppose the input is one or more triples of scalars, and the output is a triple of scalars. Suppose you change the coordinate system (or the "basis") and then put in the same input but in the new coordinate system. Look at the output. Does it or doesn't it represent the same vector that you got before, but in the new coordinate system?

One could ask this about cross-products, for example. For those, the answer would be a bit of algebraic definition chasing. Before going into that in detail, maybe we should await further clarification of the question. Michael Hardy (talk) 04:22, 1 February 2010 (UTC)

My best-guess interpretation of the question agrees with Michael's. Another way to look at it is to say that the result of an operation is a vector if and only if the operator commutes with the linear transformations L(A_x,A_y,A_z) that represent changes of basis when they act on the co-ordinates on vectors. In other words

$L(A_{x},A_{y},A_{z})\otimes L(B_{x},B_{y},B_{z})=L((A_{x},A_{y},A_{z})\otimes (B_{x},B_{y},B_{z}))$

So the result of

$(A_{x},A_{y},A_{z})\otimes (B_{x},B_{y},B_{z})=(A_{x}+B_{x},A_{y}+B_{y},A_{z}+B_{z})$

produces the vector A+B, but the result of

$(A_{x},A_{y},A_{z})\otimes (B_{x},B_{y},B_{z})=(A_{x}+B_{x},0,0)$

is not a vector because it depends on the choice of coordinate system. The 3D cross-product is a special (and somewhat confusing) case because it commutes with linear transformations that have positive determinant but anticommutes with transformations that have negative determinant. The result of the 3D cross-product is called a pseudo-vector. Gandalf61 (talk) 10:03, 1 February 2010 (UTC)

This is the OP. I was asking because I was reading chapter one of Richard Feynman's book Six Not-So-Easy Pieces where he says, "Suppose we multiply a vector by a number α, what does this mean? We define it to mean a new vector whose components are αa_x,αa_y, and αa_z. We leave it as a problem for the student to prove that it is a vector." Well, in order to prove something you need to know how to prove it. —Preceding unsigned comment added by 20.137.18.50 (talk) 13:26, 1 February 2010 (UTC)

Indeed, and Feynman's point here was to get his students to think about just what it means for something to be a physically meaningful vector. The key quality is that a physical vector should not depend on your arbitrary choice of co-ordinate system. So if an object C is defined as the result of applying operation A to vector B, then if we change our co-ordinate system from P to Q, apply operation A in co-ordinate system Q, then change co-ordinate system back from Q to P, we should be the same result. Re-arranging this gives:

(Change co-ordinate system from P to Q)(Apply operation A) = (Apply operation A)(Change co-ordinate system)

In other words, operation A and the change of co-ordinate system commute. Muitiplying a vector's co-ordinates by α satisfies this rule, so it is a physically meaningful vector operation. Squaring a vector's co-ordinates does not satisfy this rule, so this is not a physically meaningful vector operation. Gandalf61 (talk) 13:47, 1 February 2010 (UTC)

Right. This is another example of the great caution with which one has to read physics texts.

To expand on the previous example: let V be any 2 dimensional vector space and let a and b be fixed, independent vectors in V, which form an ordered basis. Consider the map f from V to V that proceeds by writing a vector in coordinates according to this ordered basis, squaring each of those coordinates, and then finding the vector in V that has those new coordinates. Certainly f is a well defined map from V to V, for each vector x in V, the result f(x) is (trivially) another vector in V. Also, it makes no difference "in which coordinate system we apply f" - because the coordinates of a vector in the basis {a,b} are the same no matter what other coordinate system one might consider. — Carl (CBM · talk) 14:22, 1 February 2010 (UTC)

square root of x equals negative one

$\scriptstyle {\sqrt {x}}=-1$

Solve for x? —Preceding unsigned comment added by 220.253.218.157 (talk) 03:25, 1 February 2010 (UTC)

By definition, ${\sqrt {x}}=-1$ is equivalent to $(-1)^{2}=x$ so that $x=1$. Now, can you solve for x in ${\sqrt {-1}}=x$? --PST 03:42, 1 February 2010 (UTC)

Well, by definition, the radical sign refers to the principal square root, which is never negative, so ${\sqrt {x}}=-1$ has no solution. On the other hand, it is true that −1 is a square root of 1, and if −1 is going to be the square root of anything that thing had better be 1; but −1 is not the square root of 1 denoted by ${\sqrt {\ }}$. So the equation ${\sqrt {1}}=-1$ is not true. —Bkell (talk) 04:13, 1 February 2010 (UTC)

What is the rate of data input on a wiki article over time?

What I am interested in knowing is, for a given wiki article [so let us say, on average], is there a pattern for data input, and if so what is it? By data input, I mean the content of a wiki article, such as alterations, additions, corrections, deletions etc...

I would suggest not including pictures, as they require data that is out of proportion to text data.

I am assuming that there would be an oscillating pattern relating to the amount of data available on the subject that hasn't yet been put up in the article. So something like lots of data input initially, then tapering off as new data on the subject becomes scarce, only to repeat whenever there is any substantial new amount of data made available on a subject.

I assume also that this pattern would vary due to controversies within a wiki article, such that so long as the controversy is 'hot' the rate of data input would be increased, and so too decrease as the controversy 'cools'.

For context, I am working on a project utilizing open source organization for the creation of specific projects and would like to have an idea as to any patterns in data input that might give a clue as to when a given 'open source project' might be either finishing up, or more likely, ending a cycle. In short, some point when one would be able to say that the project is basically done for now.

I don't necessarily need the detailed mathematics behind the pattern [though that would be nice I suppose], so much as an understanding on if there is a pattern, and if so, what is the pattern and how might it be used to determine when and if there is a point when one could say something like "this piece is done for now" or at least "nothing much new is going to be added to this piece in the near future".

Any help or info would be appreciated, thanks bunches EAshe (talk) 04:14, 1 February 2010 (UTC)

I don't know if this will help. the problem as you've laid it out has some serious difficulties, because it depends on editing style and article type issues that are difficult to quantify. for instance, on wikipedia I could say you need to distinguish between mainspace and talkspace changes (in some cases mainspace changes can be predicted from talk space volume, in other cases talk space volume follows brief flurries of changes in mainspace). further, you'd need to identify the (fairly minor but ongoing) process of link updates, citation fixes, bot entries and cleanup efforts from actual substantive content changes. I'd just scratch the effort to analyze it directly, and take a month's worth of raw date (e.g., pull every edit made for an entire month straight off the wikipedia servers) and analyze it statistically for determinable patterns. you may not be able to determine the causation of such patterns, but you can probably generalize that the pattern itself will translate across similar constructs. --Ludwigs2 08:04, 1 February 2010 (UTC)

The idea of data mining may be useful here, really more of a computing question than a math question though.--RDBury (talk) 09:34, 1 February 2010 (UTC)

It varies by article. You can download the complete history of any article through the m:API. You can download the complete edit history of most of the Wikipedias from m:dumps, but unfortunately no history dumps of the English Wikipedia have been released in the past couple of years, supposedly due to its size. There are some older enwiki history dumps floating around on the internet, and you can get more recent ones for most of the non-English wikis. There are various people who study the stuff you are asking about, but I don't know of anything published. Some qualitative discussion is in Ray Rosenzweig's well-known article about Wikipedia's history-related content.[1] Some other materials from that same site may also be of interest. 66.127.55.192 (talk) 16:19, 1 February 2010 (UTC)

WP:Statistics is a good place to start seeing what other people have done that way with Wikipedia. Dmcq (talk) 16:24, 1 February 2010 (UTC)

Closed subspace

I wish to show that the range of the operator T:ℓ^∞-->ℓ^∞ where ℓ^∞ has the norm ||x||=sup|x_n|, T((x_n))=(x_n/n) is not a closed set. I tried taking a limit point of the range and a sequence converging to it and thereby tried to show that the limit point has an inverse image but nothing came out of it. What would be the correct approach. Thanks-Shahab (talk) 07:01, 1 February 2010 (UTC)

Consider the element y ∈ ℓ^∞ such that y_n:=1/√n. Prove that y is not in the image of T although is in its closure. --pma 07:51, 1 February 2010 (UTC)

Here is a hint for a more abstract proof. The range of T certainly contains the space c_c of all sequences with compact support, and certainly is contained in the space c₀ of all sequences vanishing at infinity. Note that the former is dense in the latter. So, were the range of T closed, it would be c₀. But then we would have a linear continuous bijection T:ℓ^∞ → c₀, hence invertible by the open mapping theorem, which is impossible, because ℓ^∞ and c₀ are not even homeomorphic (the latter is separable, whereas the former is not).

The second proof, or other similar indirect arguments may be convenient or even necessary for more difficult cases; however note that for the present problem it would be considered somehow out of place. We feel it mathematically impolite using indirect arguments and general principles (recall that there is the axiom of choice behind the open mapping theorem) in order to prove the existence of an object that could be easily exhibited. On the other hand, as soon as you are a bit acquainted with the basic of functional analysis, the second proof is what should naturally come to your mind (as it first came to mine) -there are no computations in it, but it's just a simple organization of known facts.

Moral: abstract functional analysis, as well as category theory and other general theories, is not a remedy for solving all concrete problems in mathematics; it is rather a guidance that tells you what should be true and why, and which dirction you should take. --pma 08:55, 1 February 2010 (UTC)

Your moral raises an interesting point; namely whether mathematics is about "pure problem solving" (that is, the formulation and solution of a given problem), or whether it is something deeper than that. Some mathematicians with whom I have collaborated have occassionally stated that they believe "problem solving" to be the whole story behind mathematics, but I feel otherwise. As you pointed out, mathematics seems more to be about developing intuition about the connections between different concepts; a good mathematician should have a feel for how certain "principles", for instance, are connected, and should be able to use this feel to "do mathematics" (in the realm of functional analysis, one could note, in some basic sense, that the theory of Von Neumann algebras is about the connection between the algebraic and topological structure of a *-algebra). Thus I believe that mathematics does not really undermine the procedure of "taking a problem, breaking it into simpler problems, and solving the simpler problems" (not in full generality but this possibly may work in concrete cases). I am probably delving into a controversial topic here, but I do agree with your moral, if I have interpreted some aspects of it correctly. PST 11:39, 1 February 2010 (UTC)

Bill Thurston wrote a well-known essay on that topic,[2] plus there are books like "The Mathematical Experience" (which I haven't read). 66.127.55.192 (talk) 16:24, 1 February 2010 (UTC)

Thank you all.-Shahab (talk) 03:55, 2 February 2010 (UTC)

are there inaccessible integers?

I'm trying to make sense of Edward Nelson's concept of predicative arithmetic. My question:

Is there a sentence T of the form $\exists x.\phi (x)$ where $\phi (x)$ is an arithmetic predicate, where T is a theorem of Peano arithmetic, but there is no PA theorem of the form $\phi (x)\land x=SSS\ldots 0$? This basically says a certain integer x exists but it's impossible to count up to it and know when you've gotten there. (And just to be sure: I think there is obviously no such sentence if $\phi (x)$ is required to be recursive, but am I mistaken?) Thanks. 66.127.55.192 (talk) 18:22, 1 February 2010 (UTC)

There are many such sentences. Just take $\phi (x)=((x=0\land \psi )\lor (x=1\land \neg \psi ))$, where $\psi$ is any sentence undecidable in PA. By a more complicated argument, you can also arrange that PA does not even prove $\phi (0)\lor \phi (1)\lor \dots \lor \phi ({\overline {n}})$ for any n. The property that a counterexample you want does not exist is called the numerical existence property; while the argument above shows that no reasonable classical arithmetic can have it, intuitionistic theories like Heyting arithmetic usually do have it. — Emil J. 18:42, 1 February 2010 (UTC)

Oh, and $\phi$ can't be recursive, as you say. — Emil J. 18:45, 1 February 2010 (UTC)

Hmm, thanks, I guess my question didn't capture what I was trying to get at, which is whether there are numbers (like the enormous ones that appear in Ramsey theory), that are finite according to PA, but that are too large to count to. I'll see if I can figure out a more accurate way to formalize this notion, without making it imply that PA is omega-inconsistent (although, hmm, maybe it really does imply exactly that). 66.127.55.192 (talk) 19:48, 1 February 2010 (UTC)

I don't know if you've completely grasped Nelson's point. When he says you can't write such a natural number as $SSS\ldots 0$ or whatever, he means you literally can't write it. You don't have enough time, enough chalk, enough space.

What we usually say is that "in principle" the number could be written down, if we don't have to pay for chalk or space and are given enough time. But Nelson challenges you to figure out what this "in principle" actually means. What does it mean? If you're a formalist like Nelson, and don't accept (or at least don't rely on) the existence of ideal objects apart from our formalized reasoning about them, it's very hard to give a defensible account of what "in principle" means here. --Trovatore (talk) 19:55, 1 February 2010 (UTC)

I think he goes further. He doesn't like the induction scheme of PRA because the induction step is φ(n)→φ(n+1) for formulas φ that range over all the integers including the ones not yet shown to be numerals (i.e. PRA is an impredicative theory). He has been trying to prove PA is actually inconsistent (why he hopes to find an inconsistency even if PA is false, I'm not sure). He does say that multiplication is a legitimate operation (though exponentiation is not), so numbers like 1000*1000*1000*1000*1000*1000*1000 exist, even though there is not enough chalk in the world to write down that number in unary. I.e. he allows proofs "in principle", it's just a weaker principle than PRA. But, I'm having a hard time coming up with an example of a PA integer that he would say doesn't exist. 66.127.55.192 (talk) 20:27, 1 February 2010 (UTC)

I don't think he's literally accepting "in principle" in that case. Rather, he can see concretely enough that if you had a proof of a contradiction by allowing multiplication, you could violate intuitions he can actually check about accessible physical objects, that he's convinced multiplication is OK.

As to a specific example, I think he explicitly says that there is no justifiable way to get to $2^{2^{65536}}$. But it has been quite a long time since I looked at his stuff, so you may be more up-to-date on that. --Trovatore (talk) 21:13, 1 February 2010 (UTC)

I believe predicative arithmetic is something like PRA but with a weaker induction schema, so you can't use it on arbitrary formulas, you can only use it on formulas with a certain syntactic characteristic that fits Nelson's concept of predicativity, and it turns out from this that multiplication is total. The crucial difference between multiplication and exponentiation is that multiplication is associative. It is pretty interesting stuff. I haven't tried to read his book but have read some of his expository papers from his site. This one is just 9 pages: [3]. 66.127.55.192 (talk) 05:47, 2 February 2010 (UTC)

Derivation of volume of a pyramid

I've seen a derivation which embedded three pyramids of demonstrably equal volume into a prism, demonstrating that the volume of each is one third that of the prism? I looked for the derivation online, and couldn't find it. Does anyone know if there's a Wikipedia article or other website that illustrates this derivation?

Thanks, --129.116.47.49 (talk) 18:48, 1 February 2010 (UTC)

You can draw one yourself. Label A, B, and C the corners of the triangular base of a prism, and A', B', and C' the corresponding corners of the opposite triangular face. Then mark out three pyramids: The one linking A, B, C, and A', the one linking A', B, C, and B', and the one linking A', B', C', and C. You can show these are all the same volume if you assume that skewing a solid shape doesn't change its volume. The pyramid A'B'C'C can be made into a reflection of ABCA' by sliding the corner C to the point A, so they have the same volume. The pyramid A'B'CB you can slide the corner C to the point C', then the corner B to the point A and the pyramid is congruent to A'B'C'A, which is the same volume as ABCA'. Black Carrot (talk) 19:17, 1 February 2010 (UTC)

Cut a cube into three square based pyramids having a common summit. Bo Jacoby (talk) 22:44, 1 February 2010 (UTC).

What are the formulas for a Mercator Projection?

The article didn't have formulas for these situations. Perhaps they should be added.

Let's say Theta is the distance in miles (or kilometers) between two points on the same line of latitude. Let's also say Theta sub zero (in inches or centimeters) is the distance Theta on a mercator map at the equator. For latitude Phi, what is the distance in inches or centimeters for Theta miles (or kilometers) compared to Theta sub zero?

A related question: what is the distance Phi in inches (centimeters) representing the distance between two lines of latitude Theta sub one and Theta sub two, both either above or below the equator?

And then what is the Pythagorean theorem? Start at latitude Phi sub one and end at Phi sub two (both above or below the equator), start at longitude Theta sub one and end at Theta sub two?

I promise I'm not in school and haven't been for twenty-five years. I saw a Mercator map on TV and just started wondering, and these formulas don't appear to be in the article.Vchimpanzee · talk · contributions · 19:02, 1 February 2010 (UTC)

I think you mean "for latitude Phi" rather than longitude in your first question, and I think the answer is just csc φ times θ₀. The second answer is also given by the difference between cosecants. I don't think the Pythagorean theorem applies. 66.127.55.192 (talk) 19:58, 1 February 2010 (UTC)

You are correct on the latitude. I fixed it. Thanks.Vchimpanzee · talk · contributions · 20:42, 1 February 2010 (UTC)

The scaling factor for distances measured along lines of constant latitude φ (horizontal lines on the map) is 1 / cos(φ), also known as sec(φ) - this gives a scaling factor that is 1 at the equator (φ=0) and approaches infinity as you approach the poles (φ = +/- 90 degrees). The vertical distance on the map between two points with the same longitude is more complex and depends on their respective latitudes - it is:

$\ln \left({\frac {\tan(\phi _{1})+\sec(\phi _{1})}{\tan(\phi _{2})+\sec(\phi _{2})}}\right)$

Gandalf61 (talk) 10:46, 2 February 2010 (UTC)

Can these formulas be added to the article?Vchimpanzee · talk · contributions · 15:53, 2 February 2010 (UTC)

They are already there, more or less - see Mercator Projection#Mathematics of the projection. Gandalf61 (talk) 16:01, 2 February 2010 (UTC)

Okay, the new formula may be. The ones above it aren't. I just noticed lambda was used for longitude. When I took a math class that dealt with related issues, we used Theta and Phi.Vchimpanzee · talk · contributions · 18:46, 2 February 2010 (UTC)

In spherical coordinate systems mathematicians traditionally use θ for elevation and φ for azimuth. In a geographic coordinate system, on the other hand, cartographers use φ for latitude and λ for longitude. Gandalf61 (talk) 09:41, 3 February 2010 (UTC)

Capacitance between 3 concentric spherical shell capacitors

Hi all,

I was just wondering if I could get a quick answer to this: if I have 3 spherical shells (concentric), at radii a, b and c (a < b < c), then how do I calculate the capacitance of the system? The shells are at potentials (respectively) 0, V, 0, and I've managed to obtain a general formula for both the potential and the electric field: I'm just not really sure what formula I use to calculate C - I know C=Q/V in 2-capacitor situations, but what do I do here? Do i treat a-b and b-c as 2 separate capacitor pairs and then add their capacitances after, for example, or what?

Many thanks - no great detail of explanation is needed, I just need to know how I should be calculating it so don't go out of your way with a long answer!

Otherlobby17 (talk) 22:16, 1 February 2010 (UTC)

Sounds like two capacitors in series, described by the usual formula. Maybe I'm missing something. 66.127.55.192 (talk) 00:23, 2 February 2010 (UTC)

Capacitance is always defined between precisely two points: you add +Q here and -Q there, measure the voltage between those two points (more precisely, its change when you added the charges), and divide. The two is very important; when we speak of the "capacitance of an object" (like a capacitor), we're implicitly talking about the capacitance between its two terminals. When we speak of the capacitance of "two capacitors in series", we mean the capacitance between the two terminals that aren't connected to the other (constituent) capacitor. When we speak of the capacitance of one electrically-connected object (like a sphere), we usually mean the capacitance between it and "infinity" (the limit of the capacitance between it and an enclosing sphere whose radius grows without bound). (The common components called capacitors do not involve a "capacitor pair"; the word you may be looking for for "half a capacitor" is "plate".)

In your case, my guess (based on the potentials you mentioned) would be that it's the capacitance between the two spheres and the middle sphere. Since we're treating them as one object, we're constraining them to have the same voltage in this thought experiment; that's equivalent to running a very thin wire between them (through a tiny hole in the middle sphere). So it's just Q/V again in your case, bearing in mind that the charges on the plates are not -Q/+Q/-Q, but are rather a/+Q/b where $a+b=-Q$. --Tardis (talk) 02:23, 2 February 2010 (UTC)

Thanks ever so much, that's been a great help :) Otherlobby17 (talk) 22:47, 3 February 2010 (UTC)