Wikipedia:Reference desk/Archives/Mathematics/December 2005

December 1

Burning Energy in Transportation

Is it true that you burn roughly the same amount of calories traversing a mile long track with a bike, walking or running?

No. From our article on bicycle:

In both biological and mechanical terms, the bicycle is extraordinarily efficient. In terms of the amount of energy a person must expend to travel a given distance, investigators have calculated it to be the most efficient self-powered means of transportation. From a mechanical viewpoint, up to 99% of the energy delivered by the rider into the pedals is transmitted to the wheels, although the use of gearing mechanisms may reduce this by 10-15%. In terms of the ratio of cargo weight a bicycle can carry to total weight, it is also a most efficient means of cargo transportation.

On firm, flat, ground, a 70 kg man requires about 100 watts to walk at 5 km/h. That same man on a bicycle, on the same ground, with the same power output, can average 25 km/h, so energy expenditure in terms of kcal/kg/km is roughly one-fifth as much. Generally used figures are

• 1.62 kJ/(km∙kg) or 0.28 kcal/(mile∙lb) for cycling,
• 3.78 kJ/(km∙kg) or 0.653 kcal/(mile∙lb) for walking/running,
• 16.96 kJ/(km∙kg) or 2.93 kcal/(mile∙lb) for swimming.

Hope that helps. - Fredrik | tc 04:57, 1 December 2005 (UTC)

I hope the answer is appreciated; but this question is not a mathematics question. It would be better to ask it at the Science reference desk. Butchers and plumbers are both professionals, but you wouldn't want to ask one to do the other's job. --KSmrq^T 06:45, 1 December 2005 (UTC)

The energy needed to move an object under abstract condition (no friction, no speed propotional friction, etc) is the same for all speeds, while the power (energy per unit of time) is different. To accelerate a car (without friction, etc) from 0 to 100 km/h in 10 or 5 seconds you need the same energy (about 92 Cal for a 1000 kg car) but double the power (eg 100 horsepower instead of 50). Of course, in reality there are many different factors that make this untrue. --DelftUser 16:37, 9 December 2005 (UTC)

Twin prime number

Can someone tell mathematically why the middle number between twin prime number is always divisible by 6? Eg: between the twin prime number pair 17 and 19, 18 is divisible by 6.Similarly between the twin prime number pair 29 and 31, 30 is divisible by 6. Can someone give a mathematical proof of why this occurs?

Regards, Vijai12:13, 1 December 2005 (UTC)

If your first prime number is p, then p cannot be divisible by 2 or 3. Therefore it must be either 1 or 5 modulo 6. (See modular arithmetic). Similarly, the second prime number p + 2, since it cannot be divisible by 2 or 3, must be either 1 or 5 modulo 6. Putting this together, the only possiblity left over is that p is 5 mod 6, and p+2 is 1 mod 6. Therefore the number in the middle, p + 1, is 0 mod 6; that is, divisible by six. Dmharvey 12:45, 1 December 2005 (UTC)

Or, equivalently: Both primes are odd. Therefore, the number in between is even. One of every consecutive n integers is divisible by n and neither of the primes are divisible by 3 so the number in between is. If it's divisible by three and even, it's divisible by 6. Superm401 | Talk 20:42, 1 December 2005 (UTC)

Seems to me that 3 and 5 are considered twin primes. Both proofs must be amended to exclude this case. --KSmrq^T 21:53, 1 December 2005 (UTC)

Fair enough. It's not so much that the proofs need to be amended, but that the premise needs to be changed. The integer between two twin primes that are both greater than 3 is always divisible by 6. Superm401 | Talk 02:57, 3 December 2005 (UTC)

December 2

(no questions today)

December 3

proof that evaluation at a point is continuous

Time to try out the math reference desk for one of my own questions.

So a textbook I'm reading says that evaluation of maps between topological spaces is only a continuous operation on the space of maps (carrying the compact-open topology) if the domain is locally compact and Hausdorff (our page on compact-open topology suggests that this can be weakened to regularity). It's stated without proof, so I suppose it should be easy. I've tried my hand at the proof, but my proof doesn't seem to rely on Hausdorff, so it's either wrong, or I've got some unstated assumption.

Here it is:

---

I'm looking to show that the evaluation map e: X×Top(X,Y) → Y given by (x,f) ↦ f(x) is continuous if X is a locally compact Hausdorff space (this can apparently be weakened to regularity) and Top(X,Y) (the hom-set between X and Y, that is, the continuous maps between X and Y) carries the compact open topology.

Let V be an open set in Y, and (x₀,f₀) be an element of e⁻¹(V). We want so show that e⁻¹(V) is open, i.e. that every pair (x₀,f₀) is contained in a neighborhood in e⁻¹(V) open in Top(X,Y).

Since X is locally compact, every open neighborhood of x₀ contains a compact subset. f₀(x₀) ∈ V, therefore U=f₀⁻¹(V) contains x₀. It is open since f₀ is continuous. Let K be a compact neighborhood of x₀ in U, and M an open neighborhood of x₀ in K. K⊆U, so f₀(K) ⊆ V, and so f₀∈W(K,V), the set of continuous maps φ∈Top(X,Y) such that φ(K)⊆V, which is open in the compact-open topology by definition.

Thus (x₀,f₀)∈M×W(K,V), and it remains to show that M×W(K,V) is contained in e⁻¹(V). For any (ξ,φ)∈ M×W(K,V), since ξ∈K, then we have e(ξ,φ)=φ(ξ)∈V since φ(K)⊆V.

Therefore every point in M×W(K,V) is also in e⁻¹(V), i.e. M×W(K,V)⊆e⁻¹(V). Since every point (x₀,f₀) of e⁻¹(V) has an open neighborhood contained in e⁻¹(V), e⁻¹(V) is open, and e is continuous.

Somehow, I don't seem to have used that X is Hausdorff anywhere in this proof. So either my proof is wrong, or I've used it somewhere implicitly without knowing it.

---

So can anyone offer any comments? Thanks. -lethe ^talk 00:11, 3 December 2005 (UTC)

The only place I can see anything possibly going wrong is when you assert that x₀ is contained in a compact set K which itself contains an open neighbourhood of x₀. If X is Hausdorff then certainly this is what locally compact means. But if X is not Hausdorff, I'm not so sure. The article locally compact suggests that there are several definitions of "locally compact" in use for non-Hausdorff spaces. Dmharvey 01:25, 3 December 2005 (UTC)

Oh, I see. In my proof, I used the third definition of local compactness (every point has a local base of compact neighborhoods) listed at the bottom of that article. If I had taken the first definition (every point has a compact neighborhood) then my proof wouldn't have gone through, since I would have no guarantee that K⊆U. Then requiring that the space be Hausdorff is just to ensure that I'm allowed to use the stronger version of local compactness with impunity. That's a satisfactory answer to me. Thanks Dmharvey. -lethe ^talk 02:26, 3 December 2005 (UTC)

Right. In fact, I think (assuming I haven't screwed up) I can give you a specific example of the result breaking down when you use that weaker notion of locally compact. Take a nice metrisable but not locally compact space (e.g. any infinite dimensional hilbert space will do), call it H. Take X = H union {P} where P is an extra point, and take the topology of X to be all the open sets of H plus one extra open set -- the whole of X. This is non-Hausdorff. It is however locally compact according to the weak definition, since X itself is compact (in a stupid sort of way). Now take S = {0,1}, with the weird topology where {1} is open but {0} isn't. Then I believe that the evaluation map on X \times Hom(X, S) is not continuous. In particular, if you take the inverse image of {1}, then this set is not open. To show this, you could take a point (Q,f), where Q is a point of H and f is a characteristic function of some small open set U containing Q, and then check that (Q,f) cannot be contained in any rectangular basic open set contained in X \times Hom(X, S). Exercise: check this :-) I think it comes down to the fact that in H, every compact set is the intersection of all open sets that contain it. (Warning: this entire paragraph may be garbage.) Dmharvey 04:00, 3 December 2005 (UTC)

OK, let's see. The continuous maps here are the characteristic maps of open sets in X. It seems like the only compact neighborhood in X is X itself (if H has no compact neighborhoods, that is.) The singleton containing the characteristic function of X (which is just a constant function) is open in Hom(X,S). Thus it seems to me thatU×1 is an open neighborhood of Q contained in e⁻¹, and therefore evaluation is continuous for this space, unless I'm mistaken. Hmm... -lethe ^talk 17:23, 4 December 2005 (UTC)

No, U \times 1 does not contain (Q, f), since f is not equal to the constant 1 function on X. Dmharvey 20:30, 4 December 2005 (UTC)

Finding B*B

Since there is a Math reference desk, I thought I'd throw out a problem I tried to answer, but gave up on. I don't think it is quickly solvable. I was working on a project in college and I thought I was doing great by reducing a formula to C*C-A = B. B had to be a square value of an integer. Since I could just assume A was a square of something, I thought this could be the classic AA+BB=CC sort of equation. C is not a single number. It is a formula itself that produces a set of integers. Given C and A, it appears to me to impossible to quickly come up with a B that is the square of an integer (or even find a B that is an integer itself). Was I missing something? --Kainaw ^(talk) 05:05, 3 December 2005 (UTC)

I'm really confused. You're trying to find a possible value of C that makes C² − A the square of an integer? What is the formula for C? Is A an integer? You really need to give more explicit information. —Keenan Pepper 06:55, 3 December 2005 (UTC)

How can I say this diplomatically…? If this is the best you can do in describing the question, it is no surprise you couldn't answer it; nor can we. It sounds like perhaps there is a Diophantine equation in there somewhere, but you give far too little information. Such equations can be easy or hopeless, depending on the details. (Fermat's Last Theorem is such an integer equation.) Care to try again? --KSmrq^T 07:54, 3 December 2005 (UTC)

C, in the program, is not a single integer. It is an infinite set of integers (the program can go up to 256 digits). Given A, which is also an integer because that is all the program can produce, each C returns an integer B. However, most of them are not true - they are truncated because the program is unable to represent anything but an integer. So, the goal was to quickly find at least one C that would return a true integer B. Looking back at my notes, I rewrote the problem as: Given an integer, find two squares of integers that are that distance apart. IE: given 1328, find B and C such that BB-CC=1328. I can graph the answers in the real number set - which does hit an integer now and then. But quickly finding only integers is, in my opinion, impossible. --Kainaw ^(talk) 18:12, 3 December 2005 (UTC)

Ah, now I understand the problem. It depends on the factorization of A. B^2 − C^2 factors as (B − C)(B + C). B − C and B + C are either both odd or both even (just try all the possibilities). So, A must be either the product of two odd factors (so it's odd) or the product of two even factors (so it's a multiple of 4). If A is odd, then B = (A + 1)/2 and C = (A − 1)/2 is a solution. If A is a multiple of 4, then B = A/4 + 1 and C = A/4 − 1 is a solution. If A is even but not a multiple of 4, there is no solution.

In the case A = 1328, 1328 = 4*332 so B = 333 and C = 331 works. 333^2 − 331^2 = 1328. —Keenan Pepper 19:05, 3 December 2005 (UTC)

Is that, in fact, what you were asking? —Keenan Pepper 19:06, 3 December 2005 (UTC)

• Well, a very simple case is A = 2C - 1. I'll leave the proof to the reader. --jpgordon∇∆∇∆ 18:45, 3 December 2005 (UTC)
  • And an equally simple case is A = 4C - 4; and, if you wish to continue, A = 6C - 9. Oh, what the heck, for any integer N, A = 2NC - N^2. That's probably enough hints. --jpgordon∇∆∇∆ 18:55, 3 December 2005 (UTC)
    • (slight correction: for any integer N < C.) --jpgordon∇∆∇∆ 20:27, 3 December 2005 (UTC)

Thanks. That was pretty much where I ended. From memory, I think I even ended up twisting it into the quadratic formula, but with the 4AC part being the incoming series of integers. In the end, stepping through all possibilities of one formula was no better than stepping through all possibilities of another one. It still ran too slow and I quit and went to a different project. This little formula just stuck in my head because I wasn't sure that it didn't have a discrete solution. --Kainaw ^(talk) 23:20, 3 December 2005 (UTC)

Banded operators

My turn! I'm looking for references and reading suggestions, since I suspect the problem is hard or at least general.

Never mind, the question was worded poorly.

Suppose one has a bi-infinite sequence $\{a_{n}\}=...,a_{-1},a_{0},a_{1},a_{2},...$ of real or complex numbers. Define an operator M with matrix elements $M_{mn}=a_{n-m}$ so that each row of the matrix is just the $\{a_{n}\}$ shifted over by one. I want to find the eigenvalues and eigenvectors for this M. I believe this is more or less solvable when only a finite number of the $a_{n}$ aren't zero; e.g. the discrete Laplace operator is a famous special case. I'm interested in the general case, where the $\{a_{n}\}$ may be or may contain convergent subsequences; where the subsequences converge to zero, or not, or have convergents that are dense in an interval/disk. The simplest non-trivial case is probably the one with the $a_{n}\to 0$ as $n\to \pm \infty$. Theorems, special cases, equivalent problems, etc. are invited. linas 19:30, 3 December 2005 (UTC)

Could you give a little more context? I'm not a functional analyst and "banded operator" gets a few hits, but few enough that I wonder if it might be a typo for "bounded operator". In the context of vector spaces in general, you've described a vector space and a class of linear operators on it, and certainly it has some nontrivial eigenvectors. For example the sequence {...,4,2,1,1/2,1/4,...} is an eigenvector of eigenvalue 2, for the operator that shifts everything 1 space to the right.

If you want a normed linear space or something more (Banach space, Hilbert space) then maybe you should tell us what the extra structure is. There's no obvious "canonical" norm to put on this thing. --Trovatore 20:39, 3 December 2005 (UTC)

Oops, sorry, I misread your question. --Trovatore 20:42, 3 December 2005 (UTC)

Thanks, but I'm asking the wrong question. Yes, $x^{n}$ is a formal eigenvector. I'm trying to figure out what I want ask, and this wasn't quite it. linas 23:20, 3 December 2005 (UTC)

The operators you seemed to be asking about are so-called Laurent operators (or discrete convolution operators). They are typically studied on $l^{p}(Z)$ (little l) spaces. Their study is especially nice if p=2. In this case one can identify $l^{2}(Z)$ with $L^{2}(T)$ via Fourier transform (T denotes the unit circle). This identification maps Laurent operators on $l^{2}(Z)$ to multiplication operators on $L^{2}(T)$. The spectrum of the latter is the range of the function they are multiplying with. Literature: B"ottcher/Silbermann "Introduction to large truncated Toeplitz matrices" 134.225.1.162 14:37, 28 March 2006 (UTC)

December 4

help?

0 3 1
2 2 −1
1 −3 2

I have to expand the given terminate along the third colum, and I've no idea how 2 do that. Any chance that someone could explain it to me withou using rocket science? —Preceding unsigned comment added by 129.108.96.87 (talk • contribs)

See the article, Determinant. This section illustrates how to expand a determinant along a row or column. —Keenan Pepper 00:31, 4 December 2005 (UTC)

Requesting math logic statements

The soon-to-be-featured article Intentionally blank page discusses the usage of "This page is intentionally left blank", which is a self-refuting meta-reference, in that it falsifies itself by its very existence on the page in question. Can someone construct equivalent mathematical statements to be used in this article? Thanks! — BRIAN0918 • 2005-12-4 15:21

$P\to \neg P$  ?

No, that's equivalent to $\neg P$. I think a good answer to this would have to use Gödel numbering in order to make statements about other statements.

This is very similar to the kind of statement Gödel used to prove his incompleteness theorem. Actually, in a way, it's the opposite. The blank page notice says, "there exists no statement on this page", but it itself is a statement on the page so it must be false. Gödel's undecidable proposition says, "no statement of a certain type can be proven within this formal system", but it itself is a statement of that type, so it cannot be proven, although it is true. (The alternative, that it could be proven although it were false, would mean the system were inconsistent.) —Keenan Pepper 16:31, 4 December 2005 (UTC)

So is it possible for a mathematical logic statement to be constructed that is equivalent to what I'm asking for? — BRIAN0918 • 2005-12-4 16:38

To quote from Foundations of Higher Mathematics: "Some sentences that are not propositions are ... and "This sentence is false". Image the last sentence as a true-false question. The question would be unfair exactly because the sentence is not a proposition." E.g. paradoxes cannot be captured in propositional logic. Also note that the proposition "This page is intentionally left blank." does not make it untrue. It is the act of printing that proposition on the page that makes it false. In this sense the proposition is not a paradox at all, it is simply false. —R. Koot 21:21, 4 December 2005 (UTC)

But it sounds like Gödel numbering could work in this case. I don't know much about number theory, but it seems like the phrase could be represented by a relation that defines a set as being empty while being a part of that set. -- BRIAN0918  21:28, 4 December 2005 (UTC)

The "self-referential propositional calculus" of Yiannis N. Moschovakis is expressive enough to capture the liar. (Note that Gödel sentences do not capture the liar; they assert their own unprovability, not falsehood.) Moschovakis gives SRP a semantics using least-fixed-point recursion. The liar comes out neither true nor false using that semantics. --Trovatore 21:28, 4 December 2005 (UTC)

What kind of series is this?

Let $P(n)$ be a polynomial in n. What is the series

$\sum _{n=1}^{\infty }{\frac {z^{n}}{P(n)}}$

If $P(n)$ is a monomial, then this is the polylogarithm. If all of the roots of the polynomial are degenerate, then this is the Lerch transcendant. But what about the general case? Can it be solved or re-expressed in terms of known special functions? linas 16:00, 4 December 2005 (UTC)

Perhaps a Hypergeometric function? I'm not sure if its completely general. PAR 16:49, 4 December 2005 (UTC)

Yes, its a special case $\,_{p}F_{p-1}$ w/ degenerate args. I guess I'm fishing for insights from number theory or something. linas 19:40, 4 December 2005 (UTC)

December 5

Compact Operators

Let $X$, $Y$, and $Z$ be Banach spaces. Assume that $K$ is a compact operator from $X$ to $Y$, and $T$ is a linear operator bounded operator from $Z$ to $Y$. I am looking for a proof (or a counterexample) that if the image of $T$ is in the image of $K$ ($\operatorname {im} \,T\subseteq \operatorname {im} \,K$) then $T$ is also compact. Thanks! (Igny 03:55, 5 December 2005 (UTC))

Assume $Y$ is not a compact space; why would you expect $T$ to be a compact operator? --KSmrq^T 19:04, 5 December 2005 (UTC)

Precisely because its image is in the image of another compact operator, $\operatorname {im} \,K\subseteq Y$. Also, it may be important, that $T$ is assumed to be a bounded operator. (Igny 21:40, 5 December 2005 (UTC))

Sorry, that's a non-answer; you merely stated the conjecture again. Why do you think $K$ is relevant? --KSmrq^T 03:14, 6 December 2005 (UTC)

I am looking at a sufficient as well as easy to formulate and check criterion that a bounded operator from one Banach space to another is compact. This conjecture interests me now. A friend of mine claims that he proved this conjecture many years ago in college. He has forgotten the proof completely, and now challenges me to find a counterexample. (Igny 05:05, 6 December 2005 (UTC))

Igny, the only restriction on T is that it is linear and bounded, and that its image is contained in the image of K. Clearly, just being linear and bounded is not enough to conclude that T is compact, so you need to look at the assumption $\operatorname {im} \,T\subseteq \operatorname {im} \,K$. How much of a restriction is that on T? What do you know about the image of a compact operator? -- Jitse Niesen (talk) 09:39, 6 December 2005 (UTC)

This is a cute question. It feels like the statement is false, but I've been struggling to come up with a counterexample. I would be interested if anyone comes up with a solid answer either way. Dmharvey 03:37, 7 December 2005 (UTC)

Jitse, I am going to put properties about compact operators (everything I know) into talk:compact operator, and may be some of them into compact operator itself later tonight.(Igny 15:37, 7 December 2005 (UTC))

I am working on a proof (if it is indeed provable) based on one of the properties of compact operators (see talk:compact operator). It is known that there exist a neighbourhood of 0 in $X$, (denoted by $U$) and compact set $V\subset Y$, such that $K(U)\subset V$. I am now trying to construct a neighbourhood of 0 in $Z$ (denoted by $W$) such that $T(W)\subset V$. That would be sufficient to prove that T is compact.

Continuity of Derivatives

Hello! I need a little help with this question. I tried searching for it but was unable to find it on the web. Hypothesis: Let F(x) be a real valued differentiable function on the interval [a,b]. Let f(x) be its derivative. Conclusion f(x) is continuous on the same interval.

I wanted to know how to go about proving or disproving it.

Thank You,

Regards, Abhinav Mehta.

This is false. Try to find an example of a function whose derivative exists but is not continuous. Hint: One way this can happen is if the function oscillates faster and faster as it approaches a point, but it's contained within an envelope such that its derivative exists at that point, but is not continuous there. —Keenan Pepper 17:33, 5 December 2005 (UTC)

Correction: if F(x) is differentiable (that is its derivative, f(x), exists) at every point of an open interval (a,b), then f(x) is continuous on (a,b). Example: F(x)=xsin(1/x) is continuous but not differentiable at x=0, F'(0) does not exist, and f(x) is not continuous at x=0. A rather loose proof is based on definition of the derivative,

$\lim _{x\to x_{0}}f(x)-f(x_{0})=\lim _{x\to x_{0}}\left(\lim _{h\to 0}{\frac {F(x+h)-F(x)}{h}}-\lim _{h\to 0}{\frac {F(x_{0}+h)-F(x_{0})}{h}}\right)=0$

You can switch the limits, since all limits exist (F(x) is continuous and differentiable). (Igny 18:55, 5 December 2005 (UTC))

I disagree with Igny. Take $F(x)=x^{2}\sin(1/x)$ (extended to zero by F(0) = 0). Then F'(x) exists for all x, and you can check that F'(0) = 0, even though F'(x) is very discontinuous at x = 0. Dmharvey 19:20, 5 December 2005 (UTC)

Yes you are right. Indeed the convergence in lim_h is not uniform over x, so you can not easily switch limits.(Igny 21:29, 5 December 2005 (UTC))

December 6

Precision and Tolerance

PLEASE HELP ME!!

CAN SOMEONE PLEASE DO THE FIRST ONE I REALLY DONT UNDERSTAND!! CAN YOU SHOW ME HOW IT IS DONT PLEASE!!What is the precision, tolerance, accuracy for the following:

4.7 m

5.21 ft

6.03 s

39 2/3 in.

It is impossible to know the accuracy without knowing the true value. Other than that, see Accuracy and precision and do your own homework. —Keenan Pepper 00:13, 6 December 2005 (UTC)

Thank you so much, but this isn't my homework!! but I thank you for all the help but if i may ask one more question: What do you mean by the true value? I really dont understand im sorry!

If these are exact values, then their accuracy is perfect (0). If instead they are measurements of physical things, the accuracy is the difference between the real value and the measured value. Say the 4.7m figure is the length of a rope. If the rope is actually 4.85m, then the measurement is accurate to +/- 0.15m, because that's how much it's off by. If you only have the measurement, but not the true value, then you have no way of knowing how accurate it is. It could be dead on (high accuracy), or it could be a really bad measurement (low accuracy). —Keenan Pepper 02:17, 6 December 2005 (UTC)

HELP ME

WHAT IS THE PRECISION ACCURACY AND TOLERANCE FOR 4.3 M AND EXPLAIN PLEASE I DONT UNDERSTAND THIS AT ALL. CAN YOU GIVE DETAILED ANSWER AS TO WHY? —Preceding unsigned comment added by 69.154.55.231 (talk • contribs) 01:22, 2005 December 6

Where did you get "4.3 M" from? The answer is going to depend on where that measurement came from. What type of instrument was used to measure it? A ruler? If so, how was the ruler marked--just with marks every meter, every 10 cm, every cm, every mm? Is 4.3m a single measurement, or is it the average of several measurements? The answer to your questions depends on where your number came from. Chuck 02:56, 6 December 2005 (UTC)

Dear anonymous IP requesting help: You are making several mistakes that greatly reduce your chances of getting a positive response.

1. You are writing in all caps, LIKE THIS, which is not only hard to read, but the equivalent of shouting at us.
2. You have posted the same question twice, which is rude and self-centered.
3. You have asked an elementary question, gotten a link to an explanatory page you could have found yourself, and ignored the answer.
4. You ask a question that looks exactly like a homework exercise, despite your denial.
5. You do not provide background information you are told is necessary to give a meaningful answer.
6. You ask a question more appropriate to a science forum than here.

By proceeding in this fashion, you are wasting your time and ours. Before you ask another question anywhere, do yourself a favor and read How To Ask Questions The Smart Way. --KSmrq^T 03:47, 6 December 2005 (UTC)

Source for Hilbert's Hotel

Repeating a question I've asked on Talk:Hilbert's paradox of the Grand Hotel:

No source is given for the claim that this story is due to Hilbert. AFAIK the earliest source for this problem is:

• George Gamow, 1947. One, two, three... Infinity. New York: Dover.

where on p17 he claims it is an example Hilbert gave in a lecture, claiming in turn to have got this story from an R. Courant who was working on a then unpublished book to be called "The Complete Collection of Hilbert Stories". I'm guessing this has to be Richard Courant, who would be a good source for this attribution, but as it stands I see only an attribuition of an attribution... Does anyone have a better source, or know anymore about the manuscript Gamow talked about?

I'm guessing this page has a wider readership, hence the repost. --- Charles Stewart 17:56, 6 December 2005 (UTC)

December 7

Hadamard matrices

Hi there! I am having trouble proving this very simple assertion: Given that hadamard matrices contain only +1 or -1, and that A^TA = nI, how do I show that |det A| = n^n/2 ? Also, could someone provide the significance of the equality? Thanks very much! --HappyCamper 02:51, 7 December 2005 (UTC)

The assertion has nothing to do with Hadamard matrices. If A is any matrix satisfying $A^{T}A=nI_{n}$ (where $I_{n}$ is the n by n identity matrix), then

$\det(A^{T}A)=\det(A^{T})\det A=\det A\cdot \det A=(\det A)^{2}$.

The determinant of $nI_{n}$ is nⁿ. Voila. Dmharvey 03:36, 7 December 2005 (UTC)

Ah, of course, silly me :D - Why didn't I see that before? The step I missed was that det(nI) = nⁿ. I made the mistake that det(nI) = n - thank you very much! --HappyCamper 03:40, 7 December 2005 (UTC)

Statistics Question

If I know that length of a particular caterpillar species at 20 days of age is 60mm with a standard deviation of 5mm. I want to test a new kind of caterpillar food for growth improvement.

After testing on 300 caterpillars (again for 20 days), I measured them and get the result of 70mm with standard deviation of 8mm. What are the probability that the new caterpillar food has no growth improvement from the normal caterpillar food. That is to say that the result occurs by pure random variations?

The reason I asked this question is:

Chowser writes to tell us the AZStarNet is running an article stating that North Carolina scientists claim to have identified a gene that affects IQ in males. The difference is apparently quite striking, with the average IQ difference between those that had the gene and those that didn't being approximately 20 points.

300 10-year-olds from neighboring areas? Any variation in a sample that size is just signal noise. The genetic->IQ link has always been a contentious subject... This is only fuel for the fire. Ohanian 23:03, 7 December 2005 (UTC)

I'm not a statistician but I'll see what I can say. (Someone else please feel free to shoot me down.) First thing is that you probably need to rephrase your question as something like: "IF there is no growth improvement from the super-food, THEN what is the probability of obtaining a result as extreme as mean = 70 and std. dev = 8?" Phrasing it this way allows you to treat the question as a problem in hypothesis testing. Secondly, I don't know how to incorporate the stddev = 8 piece of information. Maybe someone knows how to do this, but I don't. So now the question becomes "If there no growth improvement from the super-food, then what is the probability of obtaining a result as extreme as mean = 70?" Here is how you do this. Let's assume the original distribution (which corresponds to our null hypothesis) is a normal distribution with mean 60 and standard deviation 5. Then if you make 300 independent measurements of this, the mean of your measurements should again be a normal distribution with mean 60 and standard deviation ${\frac {5}{\sqrt {300}}}\approx 0.289$ (since the variance is 300 times the original variance which is 25). Therefore your measurement of 70 is about 34.6 standard deviations above the mean, which is of course, rather unlikely. Dmharvey 01:52, 8 December 2005 (UTC)

Someone had better shoot you down. I think m=60 s=5 means that you can make a zillion measurements and you'll still get m=60 s=5 and not s=5/sqrt(zillion). The square-root only applies to uncertain measurements of a value that has no intrinsic variation, however, caterpillars have intrinsic differences in length. linas 16:20, 8 December 2005 (UTC)

What he's saying is "Suppose I have an infinite population of caterpillars, with m=60, s=5. Suppose I choose N=300 out of this population. What is the likelyhood that m=70 for my sample? Given that I, in fact, did manage to draw an m=70 sample, no matter how unlikely this seems, what would be the expected s be for my sample?" linas 16:27, 8 December 2005 (UTC)

So I should be clearer about what I said. I interpret the statement "m = 60, s = 5" to mean that if you draw a caterpillar at random out of the infinitely large population, then the length X has the distribution

$X\sim N(60,5^{2}).$

Then, if you draw 300 independently, let's call these lengths X_k, then each one has the distribution

$X_{k}\sim N(60,5^{2}).$

Therefore

$\sum _{k=1}^{300}X_{k}\sim N(300\cdot 60,300\cdot 5^{2}).$

So if you take the mean of your measurements (which was what I assume s/he meant by "I measured them and get the result of 70mm"), then you find that

${\frac {1}{300}}\sum _{k=1}^{300}X_{k}\sim N(60,5^{2}/300)$

and then I computed the probability that a random selection from such a normal distribution is at least 70, which is the same as asking what is the probability that a normal random variable is 34.6 standard deviations above the mean. Dmharvey 18:45, 8 December 2005 (UTC)

I hope it's not necessary to point this out, but correlation does not imply causation. With other words, even though your caterpillar results is very unlikely under the null hypothesis (and the same goes for the IQ result), it does not follow that the new food causes the caterpillars to grow better; there for example might be a joint effect. -- Jitse Niesen (talk) 11:36, 8 December 2005 (UTC)

Before getting into details, you might want to consider using the logarithm of the length rather than the length itself. DO NOT just take logarithms of the mean and SD; take logarithms of all the lengths separately and go from there. Do a rankit plot of the logarithms and one of the original lengths and see if either better approximates a sample from the normal distribution than the other one does. One thing that suggests this to me is that the length is necessarily positive and that your SD got bigger. Possible you'd see no significant change is SD if you work with logarithms. You say you know the mean and SD of the hypothetical population. Do you also know whether it's approximately normally distributed, or whether its logarithm is? More later, maybe... Michael Hardy 19:09, 8 December 2005 (UTC)

Thanks to everyone. I have learned a lot from this. Ohanian 22:12, 8 December 2005 (UTC)

December 8

Digits division

OK, I've been thinking about this for a while now, and now I can finally query about it. :) Consider the series 1,4,7,10,13... or:

T_n = 3n - 2

(sorry for not being able to do formula notation)

Remember that series. Now let me proceed to the question.Take any number, say 321, and add up all the digits. You get 6. Now divide the original number by this sum (321 / 6). This gives 53.5. Now add up all the digits in this answer (including decimal digits). In this case we get 5+3+5 = 13. Note that 13 is a term in the series mentioned earlier. My theory is that this answer always adds up to a number which is a term in that series.

This always seems to work, bar a few exceptions:

• Dividing by prime numbers that produce non-terminating decimal digits
• Dividing by any multiple of the above-mentioned prime numbers
• Dividing by 9 or any multiple of 9, because all multiples of 9 will add up to 9 anyway

Otherwise, this always seems to work. I can provide as many examples as you need. 53 / 8 = 6.625 (6+6+2+5 = 19). 681 / 15 = 45.4 (4+5+4 = 13). Also note you can add up the digits of the divisor here, making 681 / 6 = 113.5 (1+1+3+5 = 10). 10, 13 and 19 are all terms in the series.

Could someone please either show why this is true, or a counterexample (that isn't an exception I mentioned). -- Daverocks 04:35, 8 December 2005 (UTC)

Easy. The sum of a number's decimal digits is congruent to it modulo 9 (see modular arithmetic if you're unfamiliar with this). 321 is 6 mod 9, and the sum of its digits, 6, is also 6 mod 9. Assuming the decimal terminates (your first two exceptions) means if you add enough zeros to the number it will divide evenly. In this case, 3210 / 6 = 535. The numerator and denominator of this fraction are the same mod 9. Assume they are not 0 mod 9 (your third exception). If they are 1, 2, 4, 5, 7, or 8 mod 9, then you are allowed to do the division and the answer is 1 mod 9. If they are 3 or 6 mod 9, then you can divide both of them by 3 and the answer might not be 1 mod 9 but it will definitely be 1 mod 3. In this case, 3210 / 6 = 1070 / 2 = 2 / 2 = 1 (mod 3). The integers which are 1 mod 3 are 1, 4, 7, 10... exactly the sequence you defined at the beginning. —Keenan Pepper 07:19, 8 December 2005 (UTC)

So then, obviously, 535 is 1 mod 3 (it's in the sequence), and the sum of its digits is congruent to it mod 9 (so automatically by 3 also), so the final answer of 13 is 1 mod 3 (it's in the sequence). Sorry that wasn't a rigorous proof but it should give you a feeling for why it's true. —Keenan Pepper 07:24, 8 December 2005 (UTC)

Wow, thanks! I didn't think it would be that simple actually. I was trying to come up with ways of expressing the sum of the digits (such as a+b+c when the number is 100a+10b+c) but I forgot all about mod! You also conveniently proved an extra fact for me; when the sum of the digits of a number x (x mod 9) is not divisible by 3, the result has to be 1 mod 9, more specific than the 1 mod 3 series. Finally a maths proof I understand! Lol. -- Daverocks 10:30, 8 December 2005 (UTC)

What the hell does Sin mean?

Sin is the abbreviation for the Sine function. It's abbreviated by one letter so that it matches length with cosine (cos), tangent (tan), et al. — Lomn | Talk / RfC 22:27, 8 December 2005 (UTC)

Also, in many languages other than English, like Swedish, Danish and German, sine actually is called "sinus". That means the abbreviation makes a little bit more sense. :) TERdON 00:00, 9 December 2005 (UTC)

(OT) May I just congratulate the questioner on an excellent example of the art of punning. -- AJR | Talk 23:44, 8 December 2005 (UTC)

I'm not sure I can define Sin formally, but I know it when I see it.[1] It has something to do with undulating up and down…. ;-) —KSmrq^T 23:58, 8 December 2005 (UTC)

Not Original ... Charles Matthews 14:05, 9 December 2005 (UTC)

December 9

polygons

what are the names for 200-gons, 300-gons, 400-gons,500-gons,600,700,800,900 sided figures without using the numeral value 200-gon, etc?

• There is no standard mathematical name for such shapes; most mathematicians would prefer the names you list above. However, you could probably extrapolate a reasonable Greek name for them, like bihectagon perhaps for a 200-gon. See more information at Polygon. ESkog | ^Talk 01:05, 9 December 2005 (UTC)
  • Actually, there is a note on Talk:Hectagon which you might find useful. I would recommend verifying your source elsewhere, but it looks like your answer may be tricosagon for 300, etc. ESkog | ^Talk 01:08, 9 December 2005 (UTC)
    • That system (which would, for example, make a 475-gon an unwieldly tetracosaheptacontapentagon) seems reasonable, although I can't find any evidence of anyone actually using it. I say it seems reasonable since the greek numbers for 200-900 all have the root -κοσια, which could give -kosa- or -cosa- in English, and polygon naming uses greek. -- AJR | Talk 01:47, 9 December 2005 (UTC)
      • Having had four years of Ancient Greek at school, I agree. ナイトスタリオン ✉ 07:01, 9 December 2005 (UTC)

what is a quatrieon?

Did you mean quaternion? --- Charles Stewart 20:14, 9 December 2005 (UTC)

December 10

Text correlation

We all know about Bayes text classification, but rather than something that classifies text into discrete categories what about something that correlates text on a linear continuum? For example, something that could learn to predict SAT scores based on samples of writing. Does anyone know a way to do that? ---Nacional 02:55, 10 December 2005 (UTC)

Hmm...not sure. It's a hot topic of research for sure. Off the top of my head, perhaps some sort of artificial intelligence or something to do with correlative entropy...but realistically, I don't think the technology or theory is there to do this yet. It is exceptionally difficult to train a computer to "understand" something written. Linguistic semantics is exceptionally hard to quantify. Think of it this way: the day we can do this, the day we can have automatic robots write Wikipedia! --HappyCamper 02:59, 10 December 2005 (UTC)

Bayes's theorem on which Bayesian filtering is based, applies to continuous as well as discrete processes, I believe. linas 03:01, 10 December 2005 (UTC)

Yes, that is true. Also good for statistical signal processing. (That filtering article needs a cleanup!) --HappyCamper 03:16, 10 December 2005 (UTC)

I didn't mean anything as complex as NLP, just naive classification. A Bayesian classifier doesn't know of the grammar, it just counts up word frequencies. Maybe I can use a Bayesian method to do a correlation, I just don't know how technically you would set that up. ---Nacional 16:56, 10 December 2005 (UTC)

Well, here's one way: a probabilistic model of P(score|f(text)), where f is a mapping from text to a vector of features (discrete or not, like whether a particular word occurs in a text, or some coherence measure). Now just collect some data and use some machine learning technique to learn the model. Details vary. --Ornil 04:21, 20 December 2005 (UTC)

How to find the seventh root of a number?

Greetings,

I have a Texas Instruments TI-83 Plus, and am trying to calculate the following.

${\sqrt[{7}]{1.714}}$

I would like to know how to enter that into the calculator.

I thank in advance whoever will be of assistance,

Grumpy Troll ^(talk) 13:12, 10 December 2005 (UTC).

(1.714)^(1/7) - Fredrik | tc 13:14, 10 December 2005 (UTC)

You can also do it by writing "7", pressing the math-button and finding the sign that looks like a small x with a root sign on it, and then typing "1.714". Although, 1.714^(1/7) is probably easiest. 85.194.22.34 14:15, 11 December 2005 (UTC)

December 11

Matricies and exponentials

Under what conditions does the exponential of a matrix exist? I have always assumed that it exists, but certainly there are more rigorous definitions of the convergence of the power series on which it is defined? Supposing this is the case, does this mean that the sine and cosine of matricies are also well defined, and do these matrix functions satisfy any sort of periodicity? Thanks for your help! --HappyCamper 03:18, 10 December 2005 (UTC)

If you have any matrix of real numbers (or more generally complex numbers), then the exponential is well-defined. It is given by the usual power series. For this to make sense, you need to decide what it means for a sequence of matrices to be "approaching zero"; one decent definition is that all of its entries approach zero.

Another way to think about it is to first understand what happens with diagonalisable matrices. Say if $A=M^{-1}DM$ where D is diagonal, then the exponential turns out to be $e^{A}=M^{-1}e^{D}M$, where $e^{D}$ means the matrix obtained by replacing each diagonal entry with its exponential. (You can prove this by playing around with the series). If the matrix is not diagonalisable, you need to work a bit harder, because you need to find the exponential of each Jordan block.

You need to be careful with certain properties. For example, it is not always true that $e^{A+B}=e^{A}e^{B}$, but it is true if A and B commute with each other.

Yes, sine and cosine can be defined in a similar way. Dmharvey 03:41, 10 December 2005 (UTC)

Perhaps we can treat this more simply. We define exp(A) with the familiar power series

$\exp(A)=1+A+{\frac {A^{2}}{2!}}+{\frac {A^{3}}{3!}}+\cdots +{\frac {A^{k}}{k!}}+\cdots .$

If each matrix entry converges, then the matrix converges. But suppose A is an n×n matrix with largest (magnitude) entry a. Then the largest entry possible in A² is n a²; in A³, it's n² a³; in A⁴, it's n³ a⁴; and so on. If the real (or complex) power series converges, think you can prove the matrix does? --KSmrq^T 08:36, 10 December 2005 (UTC)

Seems no-one answered the question about periodicity. Let $I={\bigg (}{\begin{matrix}0&1\\-1&0\end{matrix}}{\bigg )}$ and let a and b be real numbers. Then (where 1 denotes the identity matrix)

$\sin a1+bI=\sin a1\cos bI+\cos a1\sin bI\,$ (only because a1 and bI commute!) which you can check gives $(\sin a\cosh b).1+(\cos a\sinh b).I\,$, just like taking sin of a complex number.

But sinh and cosh aren't periodic, so clearly there isn't any very simple periodicity statement that works in general. —Blotwell 11:26, 11 December 2005 (UTC)

The matrix sine and cosine are periodic, in the sense that

$\sin A=\sin(A+2\pi I)\quad {\mbox{and}}\quad \cos A=\cos(A+2\pi I),$

where I is the identity matrix. You can convince yourself of this either by learning holomorphic functional calculus, or via the diagonalization definition outlined by Dmharvey, or experimentation (Matlab and Maple have functions for computing the matrix sine). -- Jitse Niesen (talk) 14:20, 11 December 2005 (UTC)

Let me provide some background regarding this question. I am doing something related to coupled cluster theory in which the Baker-Campbell-Hausdorff formula is ubiquitous. In a certain implementation that I have, some specific and peculiar matrix elements simply refuse to converge (and others exhibit this odd periodicity), so I began to wonder whether there was something pathological with what I was doing. Is there really no more subtelty beyond that of the power series? I might take a closer look at things related to Matrix exponential and Holomorphic functional calculus. The literature I am familiar with usually does not present the quantum mechanical problem from this perspective, so it is definitely something I want to explore further. Thanks for the responses! --HappyCamper 17:17, 11 December 2005 (UTC)

Some remarks: BCH does not always converge. If you compute the matrix exponential numerically, you have to be careful and the power series is usually not the best approach. I assumed that your matrix is finite-dimensional and not something like <ψ|A|φ> in bra-ket notation; this might mess things up. Perhaps User:Linas can help; he recently wrote resolvent formalism which seems to be half-way between physics and Holomorphic functional calculus. -- Jitse Niesen (talk) 18:37, 11 December 2005 (UTC)

I think might do that - I'll follow up with this question in a little bit. Thanks so much for your replies! --HappyCamper 01:56, 13 December 2005 (UTC)

No, I can't help. I imagine that what is needed is a formal treatment of the theory of the "radius of convergence" for BCH. I've never seen such. There may well be tricks, but I can only guess: try to work in a basis so that the commutator is either "as small as possible", or "as close to the identity as possible". The theory of Lie groups just says "exponential of matrices exist", but I've not seen anything in Lie groups dealing with algorithms. However, do perform a search for "Lie Group" in journals for numerical algorithms, that might get what you want. linas 02:13, 14 December 2005 (UTC)

Topology, vector fields, and hurricanes

Just yesterday, I chanced upon this book by this author. I flipped through one page describing in colloquial terms this theorem. Does anyone know where the proof originated from? Our article does not seem to provide enough details for me to find the seminal paper (if there is one). Thank you for your help! --HappyCamper 17:24, 11 December 2005 (UTC)

Everything2 says that "The hairy ball theorem is usually attributed to Brouwer or Poincaré."[2] Couldn't find anything more specific, though. —Ilmari Karonen (talk) 01:22, 12 December 2005 (UTC)

I don't have original sources to check, but Milnor, in Topology from the Differentiable Viewpoint, ISBN 0813901812, says (in discussing the degree of a map), "As an application, following Brouwer, we show that Sⁿ admits a smooth field of nonzero tangent vectors if and only if n is odd." And the only Brouwer citation in the bibliography is

• Brouwer, L. E. J., "Uber Abbildung von Mannigfaltigkeiten", Math. Annalen 71 (1912), 97–115.

(A rough translation of the title would be "On maps of manifolds".) In case it's not obvious, for n = 2 Milnor is stating the hairy ball theorem. However, it is quite possible that Brouwer was merely demonstrating an old result with new machinery; I do not know. The reference to Poincaré may be because of the index theorem, another route to the result. If you do pin down the original source of the theorem, and also when the amusing name was first used, I'd like to hear about it. --KSmrq^T 07:54, 12 December 2005 (UTC)

By the way, this book is fantastic - all high school students considering university should be made to read it. Dmn 17:15, 12 December 2005 (UTC)

What's not to be found on the internet these days: See [3], middle of the page, Satz 2: Ein stetiges Vektorfeld auf einer Kugel gerader Dimenzionenzahl besitzt wenigstens einen singulären Punkt. Transl. Theorem 2: A continuous vectorfield on a sphere of even dimensions has at least one singular point. Please somebody confirm that this is viewable from a non-university computer. Stefán Ingi 15:05, 14 December 2005 (UTC)

I confirm it. -lethe ^talk 15:44, 14 December 2005 (UTC)

Yes, I was able to download the article as PDF. Well done! One fine point: I quoted Milnor's citation verbatim, but the correct title is clearly "Über …", with umlaut. --KSmrq^T 02:27, 15 December 2005 (UTC)

December 12

(No questions today)

December 13

Chi-squared test for national lottery

On page 3 of [4], can anyone explain how the author derived the chi-squared test statistic in that form? I've fiddled with it, and it seems that the author multiplied the value by 48/43 for no apparent reason. --Fangz 09:01, 13 December 2005 (UTC)

Well I see another problem, I think there is an X(i) missing! Chi-square test equation is $(X(i)-E)^{2}$ and not $X(i)^{2}-E^{2}$.--DelftUser 19:59, 13 December 2005 (UTC)

December 14

what is a parallelogram

what is a paraellelogram.

See Parallelogram. Halcatalyst 00:40, 14 December 2005 (UTC)

what does pqrs stand for

what does pqrs stand for.

Assuming that this has something to do with the previous question on parallelograms, PQRS is commonly chosen to label the four corners of a quadrilateral. So, P is one corner of the quadrilateral, Q is another, and so are R and S, and when put together, "PQRS" means the quadrilateral. Enochlau 00:53, 14 December 2005 (UTC)

Distribution Function

How do you take raw data and determine what type of distribution function best fits the data? i.e. a Normal Distribution, Gamma Distribution, Possion Distribution? etc....

Jim T.

--192.85.47.1 03:04, 14 December 2005 (UTC)

Make a histogram. —Keenan Pepper 05:24, 14 December 2005 (UTC)

Most typically, you determine the distribution from model analysis - i.e. you assume a distribution based on how you think the data should look, and then test it against the data you have. A normal distribution, for example, would be used if you believe the data you have is based on a large number of independent components summed together. If you don't base your choice of distribution on a model, then you can potentially get in all sorts of trouble. After all, you can generate an arbitary distribution function to match any data perfectly, and still have that distribution be totally meaningless and unhelpful. --Fangz 08:41, 14 December 2005 (UTC)

when is the boundary of a boundary equal to the boundary?

I wrote in the article boundary (topology) that the boundary of the boundary of a set is equal to the boundary of that set if the set is open or closed. It is a sufficient, but not necessary condition for this equality to hold, for example, a half-open interval satisfies. I wonder if anyone knows a necessary and sufficient condition? At first I thought maybe F_σ/G_δ, but the rationals are F_σ, and they don't satisfy. -lethe ^talk 13:47, 14 December 2005 (UTC)

In functional analysis, a boundary of U, subset of a metric space X, is the set of points which are neither internal points of U nor internal points of its complement ($\partial U=\{x:x\not \in \operatorname {int} \,U\cup \operatorname {int} \,(X\backslash U)\}$). Since $\operatorname {int} \,(\partial U)=\emptyset$, $\partial \partial U=\partial U$. (Igny 17:21, 14 December 2005 (UTC))

Well, it's not true in general that the interior of a boundary is empty, not even in a metric space (consider an interval of rationals in R). However, the comment is helpful, sincethe emptiness of the boundary is clearly a more general condition than that U is either open or closed. I wonder if it's a necessary condition. -lethe ^talk 17:49, 14 December 2005 (UTC)

I think emptiness of bd(S) is not a necessary condition. Consider an interval of rationals in some space that the reals are dense in, like say the hyperreals. The boundary has empty interior, yet I do not have bd(bd(S))=bd(S) here. I'm not too familiar with hyperreals, so I'm not sure if this is correct. -lethe ^talk 18:06, 14 December 2005 (UTC)

We have that bd(S)=cl(S)\int(S), thus bd(bd(S)) = cl(bd(S))\int(bd(S)) = {since bd(S) is closed} = bd(S)\int(bd(S)). Thus, "int(bd(S)) is empty" is necessary and sufficient condition for bd(bd(S))=bd(S). When I was reading about hyperreal numbers a sentence "Unlike the reals, the hyperreals do not form a standard metric space" caught my attention. What are open/closed sets in hyperreals?(Igny 19:36, 14 December 2005 (UTC))

That proof looks correct to me. There must be something wrong with my example of the hyperreals, I guess. I'm not sure why the hyperreals are not metrizable, but I know they do carry the order topology. -lethe ^talk 03:52, 15 December 2005 (UTC)

Probably the reals are not dense in the hyperreals. -lethe ^talk 04:02, 15 December 2005 (UTC)

I'm confused. What's wrong with the half-open interval? The boundary of [0,1) is {0,1}, and the boundary of that is again {0,1}. Dmharvey 17:54, 14 December 2005 (UTC)

I'm looking for a necessary condition that bd(bd(S)) = bd(S). The half-open interval satisfies that equation, as you note, but it is neither open nor closed. I conclude that openness or closedness is not a necessary condition for bd(bd(S))=bd(S). -lethe ^talk 18:06, 14 December 2005 (UTC)

Oops, you are right. If U is dense in X and has empty interior, then $\partial U=X$. I think that $\operatorname {int} \,\partial U=\emptyset$ is a necessary and sifficient condition then (duh). (Igny 18:04, 14 December 2005 (UTC))

Following the previous thought, if there is an open set V such that both U and X\U are dense in V, then $V\subset \operatorname {int} \,\partial U$ and $\partial \partial U\not =\partial U$.(Igny 18:11, 14 December 2005 (UTC))

However, while $\partial \partial U$ may not equal $\partial U$, this is as far as it goes: $\partial \partial \partial U=\partial \partial U$ for all U. I couldn't find the proof of this in the book I thought I saw it in, but it's pretty easy to sketch:

• The boundary of any set is closed.
• A closed set contains its boundary.
• The boundary of a closed set has no interior points (follows from above).

The boundary of a boundary, being the boundary of a closed set, is therefore equal to its own boundary, QED. —Ilmari Karonen (talk) 15:22, 15 December 2005 (UTC)

Yeah, we've got that already. Thanks though. -lethe ^talk 18:27, 15 December 2005 (UTC)

As for the original question, I'm not sure we can do any better than the trivial condition given by Igny above. For example, for any closed set $S\subset \mathbb {R} ^{n}$, we can construct a set $U=\partial S\cup (S\cap \mathbb {Q} ^{n})$ such that $S=\partial U$. The same trick works in any space that has a dense set with empty interior. —Ilmari Karonen (talk) 20:47, 15 December 2005 (UTC)

Another factoid: If If ∂U = ∂∂U and ∂V = ∂∂V, then ∂(U∪V) = ∂∂(U∪V) and ∂(U∩V) = ∂∂(U∩V). I've no idea if these rules are sufficient to construct all such sets from unions and intersections of closed and open sets, though, and it's too late in the evening for me to start looking for counterexamples. Time to get some sleep. —Ilmari Karonen (talk) 01:51, 16 December 2005 (UTC)

Topologic Maps

Continuous function (topology) perhaps? Or maybe topographic map? Dmharvey 19:16, 14 December 2005 (UTC)

What is exactly the question ? Google suggests : "Did you mean: Topological Maps " -- Harvestman 00:14, 18 December 2005 (UTC)

Here is a citation :

"Animated visualisation of dynamic sequences. To organise and better understand a data set, topological maps are often employed. This produces a still image where the similarity of datum instances is locally preserved. "

Ata Kaban’s projects page. My conclusion for today : try googling also. -- Harvestman 00:14, 18 December 2005 (UTC)

December 15

Two parabola questions

1. Given three points, how do you find the equation of the parabola that passes through all three?

2. Given the vertex and another point on the parabola, how do you find the equation of the parabola?

Thanks, anon.

Choose a coordinate system where the axis of the parabola is vertical, then its equation is y=Ax²+Bx+C. If your three points are (x₁,y₁), (x₂,y₂), and (x₃,y₃), then these three equations must be solved simultaneous for A, B, and C: y₁=Ax₁²+Bx₁+C, y₂=Ax₂²+Bx₂+C, and y₃=Ax₃²+Bx₃+C.

For the second, if the vertex is (x₀,y₀), then the parabola satisfies y − y₀ = A(x − x₀)². If the second point is (x₁,y₁), substitute that in to solve: A=(y₁ − y₀)/(x₁ − x₀)². -lethe ^talk 03:40, 15 December 2005 (UTC)

Please be aware that neither question has a definite answer unless an assumption like "vertical axis" is included. Also, any degree 2 curve like a parabola has both an implicit equation, f(x,y) = 0, and a parametric equation, (x,y) = (X(t),Y(t)). Neither of these is unique; the implicit equation can be scaled and the parametric equation can be reparameterized. Lastly, some arrangements of points (such as three points on a horizontal line) allow no solution; others (such as three coincident points) allow more than one solution. Numerical computations can have trouble near such arrangements. --KSmrq^T 04:04, 15 December 2005 (UTC)

Let me extend Lethe's answer by showing how to solve the equation system

$y_{1}=Ax_{1}^{2}+Bx_{1}+C;y_{2}=Ax_{2}^{2}+Bx_{2}+C;y_{3}=Ax_{3}^{2}+Bx_{3}+C$.

There are two easy ways to solve such an equation, but the simpler to understand method is this. First, let

$A_{1}:=1;B_{1}:=-x_{2}-x_{3};C_{1}:=x_{2}x_{3}$,

this way,

$0=A_{1}x_{2}^{2}+B_{1}x_{2}+C_{1}=A_{1}x_{3}^{2}+B_{1}x_{3}+C_{1}$.

Now let

$F_{1}:=A_{1}x_{1}^{2}+B_{1}x_{1}+C_{1}$.

Similarly, let $A_{2}:=1;B_{2}:=-x_{1}-x_{3};C_{2}:=x_{1}x_{3}$, so that $0=A_{2}x_{1}^{2}+B_{2}x_{1}+C_{2}=A_{2}x_{3}^{2}+B_{2}x_{3}+C_{2}$, and $F_{2}:=A_{2}x_{2}^{2}+B_{2}x_{2}+C_{2}$; and let $A_{3}:=1;B_{3}:=-x_{1}-x_{2};C_{3}:=x_{1}x_{2}$, so that $0=A_{3}x_{1}^{2}+B_{3}x_{1}+C_{3}=A_{3}x_{2}^{2}+B_{3}x_{2}+C_{3}$, and $F_{3}:=A_{3}x_{3}^{2}+B_{3}x_{3}+C_{3}$.

Now, the solution is a combination of these solutions:

$A:=y_{1}A_{1}/F_{1}+y_{2}A_{2}/F_{2}+y_{3}A_{3}/F_{3};$ $B:=y_{1}B_{1}/F_{1}+y_{2}B_{2}/F_{2}+y_{3}B_{3}/F_{3};$ $C:=y_{1}C_{1}/F_{1}+y_{2}C_{2}/F_{2}+y_{3}C_{3}/F_{3}$.

Indeed, this way for example

$Ax_{1}^{2}+Bx_{1}+C=$$(y_{1}A_{1}/F_{1}+y_{2}A_{2}/F_{2}+y_{3}A_{3}/F_{3})x_{1}^{2}+$$(y_{1}B_{1}/F_{1}+y_{2}B_{2}/F_{2}+y_{3}B_{3}/F_{3})x_{1}+(y_{1}C_{1}/F_{1}+y_{2}C_{2}/F_{2}+y_{3}C_{3}/F_{3})=$$(y_{1}/F_{1})(A_{1}x_{1}^{2}+B_{1}x_{1}+C_{1})+$$(y_{2}/F_{2})(A_{2}x_{1}^{2}+B_{2}x_{1}+C_{2})+(y_{3}/F_{3})(A_{3}x_{1}^{2}+B_{3}x_{1}+C_{3})=$$(y_{1}/F_{1})F_{2}+(y_{2}/F_{2})0+(y_{3}/F_{3})0=$$y_{1}$,

and the other equations stand for similar reasons.

– B jonas 09:32, 15 December 2005 (UTC)

Question about cones

What is the formula for the surface area of a cone that has an elliptical base. if possible please show how it was derived.

the surface area of any cone is 1/2 Bh, where B is the perimeter of the base, and h is the height of the cone. The perimeter of an ellipse can be given by an elliptic integral or approximately by π[3(a + b) − √(3a + b)(a + 3b)], where a and b are the semimajor and semiminor axes. The formula is simply from the formula for the area of a triangle. If you want to include the area of the base, it's πab. -lethe ^talk 23:00, 15 December 2005 (UTC)

Are you sure about the 1/2 Bh part? MathWorld suggests $\pi r{\sqrt {r^{2}+h^{2}}}$ for a cone with a circular base. If we use your formula, $B=2\pi r$, and thus $A=\pi rh$, which doesn't quite seem the same. Enochlau 23:09, 15 December 2005 (UTC)

Errr... whoops. I guess it should be 1/2 Bs, where s is the diagonal length of the cone. This is how we get a triangle. Only, in the elliptical case, that's not constant, so I'll have to admit I don't know the answer off-hand. Thanks for catching my error, and my apologies for the misinformation. -lethe ^talk 23:47, 15 December 2005 (UTC)

It would appear that mathworld actually has the exact answer for the elliptical cone: it's 2a√(b² + h²) times elliptic integral of some messy radical that I don't want to type in. -lethe ^talk 23:51, 15 December 2005 (UTC)

The formula for the elliptic cone at mathworld is

$2asE\left({\frac {eh}{s}}\right)\approx \pi as\left[1-\left({\frac {1}{2}}\right)^{2}\left({\frac {eh}{s}}\right)^{2}-{\frac {1}{3}}\left({\frac {1\cdot 3}{2\cdot 4}}\right)^{2}\left({\frac {eh}{s}}\right)^{4}-{\frac {1}{5}}\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right)^{2}\left({\frac {eh}{s}}\right)^{6}+\cdots \right]$

where s is the diagonal length to the semiminor axis s=√(b²+h²) and a > b, e is the eccentricity e = √(1-b²/a²) and E is the complete elliptic integral of second kind. -lethe ^talk 00:10, 16 December 2005 (UTC)

Perhaps we should now address the origin of such fancy formulae. The heart of the matter is that we need to know the length of the perimeter of the ellipse which forms the base. The surface area is then obtained by sweeping that vertically and scaling to a point. The latter integral is easy, since the integrand is a constant (the unscaled perimeter) times a linear function of height. In sharp contrast, there is no elementary closed formula for the perimeter length of an ellipse (excepting the special case of a circle). Since such calculations are of great practical interest (historically, for describing the orbits of planets), special functions were defined to give the required answer. We know a great deal about the mathematical properties of these so-called "elliptic integrals", but in practical applications we are still forced to work with numerical approximations. One thing we can say exactly, is that the "shape" of an ellipse is completely determined by one parameter, called the eccentricity. Intuitively, eccentricity tells us how much the ellipse differs from a circle. We can separate that from the "size" of the ellipse. Thus it is convenient to define an "ellipse length" function that depends only on eccentricity, assuming a fixed size; we then scale that by a factor dilating or contracting the fixed size to our actual size. (Pure mathematicians today are interested in elliptic integrals for more sophisticated reasons, so the definitions and discussions can seem baffling, and decidedly unphysical.) --KSmrq^T 08:08, 16 December 2005 (UTC)

December 16

Homotopy groups of spheres

Where can I find the homotopy groups of order n+k of the n-dimenstional sphere [i.e. π_n+k (Sⁿ) ] for k>20 and n>19 ? It's for the article Homotopy groups of spheres. Tompw 00:36, 16 December 2005 (UTC)

I was hoping there might be something more in the updated (online) version of Ravenel's Complex Cobordism and Stable Homotopy Groups of Spheres [5], but I didn't spot anything. Keep in mind, one reason people are interested in these things is because they are notoriously irregular and difficult to calculate. It's not an area in which I play, and I've got no special insight. I just didn't want the question to be ignored. By the way, it might be helpful to add a link to Ravenel, and also one to some discussion by John Baez [6]. Good luck! --KSmrq^T 11:01, 17 December 2005 (UTC)

Bach Mathematics

I'm planning to write an article about the mathematics in Bach's works. And try to collect some ideas here. I know he used the golden ratio in many works. Also his name and certain 'holy' numbers. Which works should I look at? helohe (talk) 12:22, 16 December 2005 (UTC)

Try to find a copy of Gödel, Escher, Bach, and follow the links in the Wikipedia article. --KSmrq^T 12:59, 16 December 2005 (UTC)

I have the book. helohe (talk) 13:21, 16 December 2005 (UTC)

One of the works that uses a lot of number symbolism is the third book of the Clavierübung, in which everything is done in threes; 27 (3x3x3) pieces, 9 (3x3) chorale preludes, three trios, prominent use of the major third etc. [7] David Sneek 20:08, 16 December 2005 (UTC)

Of course, the major third is 4 semitones, and it's about a 5/4 frequency ratio... —Keenan Pepper 20:51, 16 December 2005 (UTC)

December 17

Total Number of Mathematicians

Can someone give an estimate of the number people in U.S., E.U., or the world whose understanding of mathematics goes beyond abstract algebra, topology, et cetera?

--John Doe —Preceding unsigned comment added by 66.81.21.224 (talk • contribs)

It may be helpful to look for the numbers of PhDs in mathematics and related fields as a first approximation. Akriasas 21:40, 17 December 2005 (UTC)

You'd probably also want to add in people working on a Master's and/or PhD in math, and also those who research certain funkier fields of theoretical physics - cosmologists, string theorists, that sort of thing. And those few of us who didn't pursue it past undergrad but who still just can't get enough geometry. --George 21:49, 17 December 2005 (UTC)

• Can someone use the type of estimating that is used in many introductory physics textbooks to come up with an answer?

--66.81.200.216 01:11, 18 December 2005 (UTC)

I'm afraid your question is not terribly clear. What problem are you trying to solve, and what technique are you trying to use? --George 05:42, 18 December 2005 (UTC)

You can use anything you want to "come up with an answer", the question is whether the answer means anything. —Keenan Pepper 06:42, 18 December 2005 (UTC)

OK, something screwy is going on. Someone misinterpreted this as being an unrelated question and marked it "(no subject entered)" when really it was a follow-up to the "Total Number of Mathematicians" question. I'll try to fix it and avoid even more confusion. —Keenan Pepper 07:51, 18 December 2005 (UTC)

'twas I because of the differing IPs :-( --hydnjo talk 13:08, 18 December 2005 (UTC)

• Is it at least possible to find out the number of people who have taken and obtained a high score -- 90%+ correct -- on the GRE subject test in mathematics [8] in the United States, or similar exams in other countries, since the examination has been offered? To be even more specific, I wish to know if it is possible to estimate the number of people who can solve almost any problem from arithmetic to topology? A long time ago, I read how Enrico Fermi estimated the number of piano tuners in San Francisco -- I don't remember the details, though? --John Doe

See Math Quiz One to learn that three 'simple' problems (two arithmetic and one about differentials) are not solved since last century. -- Harvestman 08:38, 18 December 2005 (UTC)

That's because they're stupid. Task 3 is okay, but Tasks 1 and 2 aren't actually questions about math, they're questions about certain ill-defined representations of math, which I can't stand. —Keenan Pepper 08:56, 18 December 2005 (UTC)

Agreed. There's an infinite number of polynomial functions that can be used to fit the conditions in Q1 and Q2.--Fangz 14:03, 18 December 2005 (UTC)

If you know basic statistics, you should be able to figure out the answer to this question using the data found at this website [9]. That website has a lot of other statistical info on the GRE which you might find interesting. --George 21:20, 18 December 2005 (UTC)

I think the original question is oddly phrased. There is nothing "beyond" abstract algebra or topology, as both are active areas of research about which more is discovered daily. As for the second question, estimating "the number of people who can solve almost any problem from arithmetic to topology", I'd estimate that as approximately zero, "the ultimate round number" as L. Neil Smith once put it. --Trovatore 22:04, 18 December 2005 (UTC)

December 18

no questions today

pity ☢ Ҡieff⌇↯

December 19

math

Can you solve this question? (320-270)/50 what does the forward slash mean used in this manner?

I shall look forward to an answer.Thanks in advance.Patrick

It probably means "divided by". So you need to work out 320 minus 270, and then divide the answer by 50. Dmharvey 19:06, 19 December 2005 (UTC)

Or, if you're lazy, you could just Google for it. :-) —Ilmari Karonen (talk) 15:10, 21 December 2005 (UTC)

formula

looking for the formula to figure out the gallon capacity of a pit, or lagoon?

Given what? —Keenan Pepper 23:51, 19 December 2005 (UTC)

If it's a regular shape, use the appropriate volume formula. Otherwise, you'll need to use integration; most likely one of the approximations given to first-year calculus students, such as fitting rectangles under a curve, will be the best approach. — Lomn | Talk / RfC 14:05, 20 December 2005 (UTC)

December 20

(no questions today)

pity again ☢ Ҡieff⌇↯

December 21

Wiki Math/Science textbooks

Hello. OK, this is pie in the sky, but is anyone at wikipedia or wikimedia considering a project to enable the creation of free textbooks? General mathematics and science textbooks would be of tremendous help for those who simply want to learn new material but cannot afford $100+ college texts. The commercial 'teach-yourself' and 'for-dummies' books for math are utterly worthless. Hell, I recently checked out the local adult education center hoping to find some math courses and found not a single one. There's a real need for this... so - just a suggestion. Thanks!

http://en.wikibooks.org/ Dmharvey 00:16, 21 December 2005 (UTC)

I almost never buy textbooks, and certainly not $100+ ones. Instead I get them from libraries (including the university library). It's cheaper. – B jonas 11:44, 21 December 2005 (UTC)

determinant for magic square 4*4

I want to know ,did someone solve this determinant

See http://www.mathpages.com/home/kmath310.htm —Keenan Pepper 08:20, 21 December 2005 (UTC)

how do you read this number

658,000,000,000,000,000,000,000

6.58×10²³. —Ilmari Karonen (talk) 15:07, 21 December 2005 (UTC)

six hundred fifty-eight sextillion should do. —Joehaer 15:10, 21 December 2005 (UTC)

• The previous reading doesn't need a comma -- check an arithmetic textbook on reading numerals. Thus, I deleted the comma in six hundred, fifty-eight sextillion to make it six hundred fifty-eight sextillion.

--66.81.25.5 14:51, 22 December 2005 (UTC)

I believe that it would depend on what country you are in. If it is British English, the reading would be different than in American English. --Think Fast 21:20, 21 December 2005 (UTC)

Yeah, but the British are crazy. So, you know. --George 23:35, 21 December 2005 (UTC)

According to Names of large numbers, sextillion (American) =10²¹ whilst sextillion (British) =10³⁶. hydnjo talk 23:54, 21 December 2005 (UTC)

• 6.58×10²³ can be read as six and fifty-eight hundredths times ten to the twenty-third power.

--66.81.25.5 14:42, 22 December 2005 (UTC)

• • I would read that as six point five eight times ten to the power of twenty-three - doesn't really matter either way, minor differences... enochlau (talk) 08:44, 23 December 2005 (UTC)

• It's pretty close to a Avogadro's number... --Shanedidona 02:02, 24 December 2005 (UTC)
  • ... Isn't Avogadro's number approximately 6.02x10²³, nowhere near that? x42bn6 Talk 13:14, 24 December 2005 (UTC)

Well, it's about 1.09 moles. --84.64.190.1 12:22, 27 December 2005 (UTC)

December 22

Prerequisites for Topology

Are non-Euclidean geometries prerequisites for topology? In other words, is it necessary to have a clear understanding of spherical and hyperbolic geometries before studying topology?

--66.81.25.5 14:45, 22 December 2005 (UTC)

Not at all. Some familiarity with metric spaces is useful for point-set topology, but since this is a topic that is usually well-covered, not even that is necessary. At the level of point-set topology, most geometric questions are fairly irrelevant. On the other hand, in differential topology, one often studies topological invariants that are constructed geometrically, like the characteristic classes. Therefore, when studying differential topology, some knowledge of differential geometry is important. But even then, this isn't so much a question of knowing the difference between elliptic and hyperbolic geometry, but rather just knowing about arbitrary Riemannian geometries and connections on bundles. Could you specify what branch of topology you had in mind? -lethe ^talk 15:02, 22 December 2005 (UTC)

• I was referring to the topology that is needed to answer the questions on the GRE test in subject mathematics.

--John Doe

At that level, it's mostly basic point-set topology with a touch of real analysis. One book I'd suggest is Munkres' Topology. [10]. I think that's where I first learned all that stuff. Dmharvey 01:31, 23 December 2005 (UTC)

Study of non-Euclidean geometries would not hurt, but plays such a minor role in topology that it is by no means a prerequisite. Many examples and motivations in topology naturally come from other areas of mathematics, and familiarity with those topics can be helpful in better understanding topology. On the other hand, the basic definitions in (point-set) topology require only the simplest of set theory, and many of the examples require no more than understanding real numbers. Although its applications are much broader today, topology evolved from questions in the differential and integral calculus (analysis), and differential topology obviously requires some understanding of those topics. As for what specifically might be covered by a GRE subject test in mathematics, it's almost certain to be skewed toward the connections with analysis rather than geometry. To get a better score, study for the test, not the topic. (Caution: This is not the way to get a good education!) Many test preparation books are available, either for purchase or in a school library; and recent test takers (entering graduate students) may offer helpful guidance. Practice, sleep well, try to relax, and good luck! --KSmrq^T 01:40, 23 December 2005 (UTC)

December 23

How are values or worth of companies established

You could start here: Stock valuation. Dmharvey 01:33, 23 December 2005 (UTC)

December 24

How would one derive this?

I've forgotten a lot of my integration knowledge over this holiday season.  :( Anyway,

$\int {\sqrt {1-\cos t}}\,dt=-2{\sqrt {1-\cos t}}\,\cot {\left({\frac {t}{2}}\right)}+c$

using The Integrator (courtesy of Lists of integrals).

How do you derive this? Half-angles were never in my syllabus, and I have never understood them.  :( x42bn6 Talk 13:30, 24 December 2005 (UTC)

This sounds like a homework question, so you only get a hint. The answer the computer gave demonstrates why such tools are of limited usefulness in answering homework questions :-) You need to know the double-angle formulae, for example $\cos t=2\cos ^{2}(t/2)-1$. Using this, you can rewrite the integrand, and then it becomes a much simpler integral. (You should also check that the simpler answer you get this way is really the same as the more complicated one that the computer gave you.) Dmharvey 13:55, 24 December 2005 (UTC)

OK, thanks. However, it does prove that my teacher has set homework involving things outside my syllabus.  :( x42bn6 Talk 07:42, 25 December 2005 (UTC)

Well, if your concern is the double-angle formula, it's at a level somewhat below calculus, so your teacher might not even have thought of it. In any case, remember it—it's a useful trick in many, many cases. -- SCZenz 07:46, 25 December 2005 (UTC)

At A-Levels, not all topics are covered by certain examining boards. For example, I don't even need to do conics! Nevertheless, I'll keep this one in mind. x42bn6 Talk 01:46, 26 December 2005 (UTC)

December 25

Factoring Absolute Values

Given $f(x)=x+{\sqrt {|x|}}$. I want to find $\lim _{x\to -\infty }f(x)$.

Therefore, for $x<0,f(x)=x+{\sqrt {-x}}$. How do I factor ${\sqrt {-x}}$ out of$x+{\sqrt {-x}}$ in order to find the limit at negative infinity? --66.81.26.44 18:28, 25 December 2005 (UTC)

You should already have a good guess about what the limit will be. To prove that $f(x)\to -\infty$ as $x\to -\infty$, you need to prove that for any $M$ you can find an $x_{M}$ such that $f(x)<M$ when $x<x_{M}$. —Ilmari Karonen (talk) 18:46, 25 December 2005 (UTC)

For $x<x_{M}\Rightarrow \ x+{\sqrt {-x}}<M$, but I don't know how to prove anything since $\lim _{x\to -\infty }x+{\sqrt {-x}}=-\infty +\infty =$indeterminate. I can graph the function and say that the limit is $-\infty$, but how do I show it using algebraic techniques? --66.81.192.159 19:08, 25 December 2005 (UTC)

You do it just like I wrote above; that's how the limit of a function at infinity is defined: $\lim _{x\to -\infty }f(x)=-\infty \,\Leftrightarrow \,\forall M\,\exists x_{M}:(x<x_{M}\Rightarrow f(x)<M)$. —Ilmari Karonen (talk) 17:07, 26 December 2005 (UTC)

Another approach might be: if x is negative, then $x=-{\sqrt {-x}}{\sqrt {-x}}$. (Why?) Then you can probably factor it and do something. Dmharvey 19:40, 25 December 2005 (UTC)

$\lim _{x\to -\infty }x+{\sqrt {-x}}=\lim _{x\to -\infty }{\sqrt {-x}}(-{\sqrt {-x}}+1)=\infty \times -\infty =-\infty$. Thank you very much!

December 26

Interval subsequence-filling sequences

Is there a name for sequences which fill an interval densely? I.e. for any real number within the interval, there exists a subsequence which converges to the value. An example would be the sequence:

1/2, 1/3, 2/3, 1/4, 2/4, 3/4, 1/5... which goes through all the positive rational numbers.

Does the set of these sequences, or a partition of them, have any sort of group structure? (Just wondering if there is a way to construct the Reals by this method...)--81.79.139.17 23:19, 26 December 2005 (UTC)

How about dense sets? Not sure about any structure though.(Igny 05:46, 27 December 2005 (UTC))

It seems these are, surprisingly enough, called dense sequences. Unfortunately we don't appear to have an article about them, but Googling for "dense sequence" plus some disambiguation term like "interval" or "reals" turns up quite a few hits. A rather nice example of a dense sequence is the van der Corput sequence, which we also have no article on, but MathWorld does (see also [11]). —Ilmari Karonen (talk) 15:17, 27 December 2005 (UTC)

Update: I just started an article on the van der Corput sequence. It turns out we did already have a definition at Constructions of low-discrepancy sequences, though it wasn't particularly accessible to the general public. —Ilmari Karonen (talk) 16:17, 27 December 2005 (UTC)

Are Low-discrepancy sequences necessarily dense? My math at that level is weak, but it would seem to be no. In any case our article doesn't mention anything about it. The goal of Low-discrepancy sequences seems to be to cover the interval reasonably well within the first x terms, not necesarily to be dense in the limit. - Taxman ^Talk 20:56, 29 December 2005 (UTC)

I'm not really sure, but it would seem likely that any sequence with sufficiently low discrepancy ought to be dense. Quoting from the article, "Roughly speaking, the discrepancy of a sequence is low if the number of points falling into a set B is close to the number one would expect from the measure of B.". That implies that, if one continues the sequence far enough, any set of non-zero measure ought to contain a point in the sequence with probability approaching 1. Conversely, if a sequence is not dense, there must be some open set that contains no points in the sequence, which ought to affect the discrepancy of the sequence sooner or later. None of this is in any sense rigorous, of course. —Ilmari Karonen (talk) 22:13, 29 December 2005 (UTC)

December 27

t-test and z-test

Can you please explain the t-test and z-test and how both differ from each other for a high school student who has not studied statistics? Please use any analogy or other aids and examples to explain the t-test in very simple terms. I will appreciate any comments in advance. --66.81.18.155 01:47, 27 December 2005 (UTC)

Nothing?

--John Doe

I assume you mean the Student's t-distribution as the t-test and a Normal distribution for the Z-test (capital Z). The t-test is used for small values of n, the number of trials. This is because for small values of n, the approximation of the population variance is much less approximate for small values of n. Hence for hypothesis testing, small values of n use the t-test and larger values of n use the Z-test. Small usually means a value of 25 or less. Does that help? x42bn6 Talk 06:13, 29 December 2005 (UTC)

• This was a nice explanation. Can the Z-test be used to determinate whether the means of two groups are statistically different?

--John Doe 18:11, 29 December 2005 (UTC)

• I just read the Z-test article and the answer seems to be affirmative. --John Doe after 2 minutes.

... As for your two means question, I'm not sure. Though I do believe that both the Z- and t-tests can be used to statistically determine whether they are different. Again, it depends on the number of trials. I never learnt and probably won't learn about the two means part, though, so don't put too much weighting inside this answer. x42bn6 Talk 01:19, 30 December 2005 (UTC)

--Thank you.

Ellipse Graphing

I'm being asked to graph an ellipse that's not in the usual format (example: $((x-3)^{2})/49+((y-5)^{2})/81=1$ but rather in a different format. The problem is $4x^{2}+y^{2}-8x+6y-3=0$. How do I convert it to the first format? Thanks, anon.

$(4x^{2}-8x\quad )+(y^{2}+6y\quad )=3\Rightarrow 4(x-1)^{2}+(y+3)^{2}=16$. Now divide by 16; so one has a vertical ellipse centered at (1,-3) with semiminor axis of length 2 and semimajor axis of length 4.

--66.81.26.100 19:34, 27 December 2005 (UTC)

That's called completing the square. —Keenan Pepper 02:45, 28 December 2005 (UTC)

December 28

Why is mathematics so difficult?

Yes, it's a 'funny' question - but looking at other subjects such as sociology, history, economics, marketing, psychology etc etc, they all 'seem' to be easier than mathematics. Can people give a reason for this? Mjm1964 09:47, 28 December 2005 (UTC)

Mathematics are quite easy. Look at Composition this one is quite hard... :-) helohe (talk) 14:15, 28 December 2005 (UTC)

I can't give a full answer. Many people fear from mathematics and think it is hard (maybe because of the way their teachers have thought them), or think that it doesn't have any connection to the real world. This is not the case for biology, history, economics etc, which many people just don't know but don't fear of it.

You may want to read Raymond Smullyan's funny books or Innumeracy: Mathematical Illiteracy and its Consequences from John Allen Paulos (Hungarian translation Számvakság). – b_jonas 15:05, 28 December 2005 (UTC)

Not all of mathematics is difficult, but there is a reason why mathematicians tend to be attracted to difficult problems: mathematicians tend to value as part of the interest of a problem how powerful and general the problem is (cf. Hardy's apology), which naturally is correlated with difficulty. --- Charles Stewart 15:08, 28 December 2005 (UTC)

My advisor had a rather controversial theory that minds of different people get wired differently, probably due to genetics and/or different upbringing. As a result, certain people can not understand even basics in mathematics, while others can not put two tunes together, and others can not draw an egg, not to mention a portrait. Yet each of these activities seem trivial for mathematicians/composers/artists.(Igny 17:11, 28 December 2005 (UTC))

• Richard Feynman was a great artist. You should see his drawings and sketches. Perhaps, he was genetically superior in many scholastic areas.

--66.81.26.84 21:55, 28 December 2005 (UTC)

You say economics. It is true that basic economics can be qualitative and easy. However, those who have chosen economics as a career and make a living from it use partial differential equations, linear algebra, and statistics. Look at the mathematics used by Nobel Prize winners in economics, the board member of the Federal Reserve in U.S., the Economic Advisors of the U.S. President in the White House, international corporations, and many other people and groups. Mathematics is exceedingly difficult because when you thoroughly understand and apply it, then you truly know what you are talking about and can effect palpable differences, for instance, in engineering. (No wonder that some people have suggested paying those individuals who pursue advanced mathematics.) --66.81.194.126 17:42, 28 December 2005 (UTC)

I don't know if it's even right to say that math is more difficult than other fields. I suspect that the number of people who can come up with a really ingenious economics experiment, a new approach to the philosophy of mind, or a great poem is probably just as small as the number who can come up with new insights in math.

That said, learning math (say in college) is probably harder than learning those other fields. First, most people find math boring and so won't study it with as much gusto as they might, say, their human sexuality course. Math also uses a sort of thinking that just doesn't come up much in day-to-day life, or indeed throughout most of your school career if you're not a math major, so becoming adept with it takes a lot of practice (which, again, most people won't do). Finally, as others have said, people tend to be afraid of it. If you believe you just can't do something, odds are you can't. --George 22:09, 28 December 2005 (UTC)

I've thought about this a lot in the context of teaching math and have come to the conclusion that it has a few factors that combining them all makes learning math more work for most people than other subjects. To be successful with math you have to successfully handle concepts, notation, and mechanics. The concepts can be simply difficult to understand and grasp either because they are complicated or as other people noted, just different than other things people normally have to interact with. The notation can be a sticking point because it's almost like learning another language and any of it you forget can trip you up. Finally the mechanics such as algabraic manipulation, substitutions and other tricks are similar to the situation with notation. Math builds on itself so you can't forget much of what you've learned and still be able to master new concepts. All three of these are present in math in higher quantities than most other subjects. I've often though that math classes were much more dense with the number of concepts presented than other subjects. I think though that if you're aware of these, the rewards for learning math such as making advanced study of other subjects more accessible. For example, in comparison to the rigorous math track in college, the economics classes were nearly trivial. - Taxman ^Talk 20:29, 29 December 2005 (UTC)

Oops! It's easier than psychology ;)! I somehow agree with the wiring theory. deeptrivia (talk) 04:09, 31 December 2005 (UTC)

Gödel

I read somwehere that Gödel thought that the universe revolved around each individual observer acording to the work he did on the mathematics of Einstein's relativity theory. is that true? if it is, can someone explain me how without using too much math and instead using more logic? please, thank you.

Well, Gödel never thought that The Universe™ (i.e. the universe that we live in) was rotating. He just showed that there exists a solution (the Gödel metric) to the Einstein field equations (which are the equations which tell you what shapes the universe can come in) which more or less looks like a whole rotating universe. The solution is fairly unphysical; its not isotropic (dosn't look the same from every angle), while our universe is highly isotropic. But far worse, it has closed timelike curves (CTCs). Those are paths which travel back in time to before they begin. Universes with CTCs let you do fun things like time travel, but also nasty things happen, like it's really hard (or impossible) to solve equations given initial data. Cause and effect gets called into question. The existence of this nasty solution caused Einstein to doubt his theory, but I don't think it's what caused Gödel to starve himself to death. I don't think anyone ever thought that it described our universe, just that it illustrated some potentially awkward solutions to Einstein's equations. -lethe ^talk 23:09, 28 December 2005 (UTC)

wow! did Gödel starve himself to death? I didn't know that :P and ... well thank you, there are a lot of people out there that believe he did describe our universe, there's a guy on the net, that has a page about it... anyway,do you think our universe is like that? I mean, that it has wormholes that go back to before they where made?

Our universe is definitely not a Gödel universe, which can be confirmed by a simple observation. -lethe ^talk 05:01, 29 December 2005 (UTC)

How can this be seen by simple observation?--Hypergeometric2F1[a,b,c,x] 09:24, 31 December 2005 (UTC)

well yes, but...isn't QM like really weird? so it might as well be weird as hell concerning things we don't see everyday.

It's true that quantum mechanics is unintuitive on a physical level - though mathematically it's quite nice, I think. That doesn't really have much to do with Gödel's solution of the GR equations, though. Anyway, as Lethe has pointed out, our universe is most assuredly not described by Gödel's solution. While there may be people out there who believe it is, those people are just plain wrong. There are plenty of quacks in the world - see our article on List of alternative, speculative and disputed theories - and fundamental physics attracts a lot of them because it's grand and complex and there are plenty of books that give the layperson a general overview but unfortunately not enough knowledge to see where they might be wrong.

Getting back to your original question, though: Gödel certainly didn't believe that; even he didn't believe that his solution applied to the real world. I suspect whoever wrote that was mixing Gödel's solution with something else commonly pointed out in popular treatments of astronomy: Namely, that the universe is expanding away from any given point within it and, therefore, one can say half-jokingly that each of us is the center of the expanding universe. --George 19:37, 29 December 2005 (UTC)

I seeeeeeeeeee!!! :D thank you :D how old are you?(I'm 20) I dislike new agey speculation so much! you have no idea! :( and there are a looot of people here in latin american colleges that believe those things, I feel like a lone skeptic...it's like people are so prone to believe something whitout evidence just because they'd like it to be so, without wanting to know the right thing. wow wow, I didn't know he was schizophrenic! (I'm a psychology student) how come crazy people are the smartest sometimes? I think we should re define mental health.

December 29

Conversion Rate Format

How could I find out what is the correct way to indicate rate of exchange for currencies? I have read that it is not the usual ratio notation with the unit at the bottom, but with its arithmetic inverse, like this 1USD/value expressed in foreign currency

Thank you for your assistance.

Having a look at Exchange rate should cover it for you. No explanation can keep it from being a little confusing and it takes some time to get used to. - Taxman ^Talk 20:12, 29 December 2005 (UTC)

"Connection" between two numbers?

In a GMAT test there are some questions about "connection" between two numbers. "Connection" is defined as a ratio between the LCM (least common multiple) and the product. Example: Connection between 6 and 9 is: LCM (6,9) = 18 Product 6x9 = 54 Connection = 18 / 54 = 1/3 or Connection (6,9) = 1/3.

My questions are: 1) Is this operation / definition in Wikipedia? (I couldn't find it) 2) What is this? 3) Where do we use it? (an example of the use of "connection")

Thanks a lot!

My Name is Adam Szymanski (email address deleted by Dmharvey) I'd appreciate a help from anybody.

Thanks!

This "connection" is actually something very simple and familiar. Here's a hint: the product of the GCD and LCM of X and Y is the product of X and Y. —Keenan Pepper 22:05, 29 December 2005 (UTC)

December 30

Source for Hilbert's Hotel

This is a reposting of a question archived here, and subsequently posted here --HappyCamper 02:26, 30 December 2005 (UTC)

Repeating a question I've asked on Talk:Hilbert's paradox of the Grand Hotel:

No source is given for the claim that this story is due to Hilbert. AFAIK the earliest source for this problem is:

• George Gamow, 1947. One, two, three... Infinity. New York: Dover.

where on p17 he claims it is an example Hilbert gave in a lecture, claiming in turn to have got this story from an R. Courant who was working on a then unpublished book to be called "The Complete Collection of Hilbert Stories". I'm guessing this has to be Richard Courant, who would be a good source for this attribution, but as it stands I see only an attribuition of an attribution... Does anyone have a better source, or know anymore about the manuscript Gamow talked about?

I'm guessing this page has a wider readership, hence the repost. --- Charles Stewart 17:56, 6 December 2005 (UTC)

If you've made that much of an effort to verify it and can't, remove it from the article. It ca always be put back in later if verified. - Taxman ^Talk 15:52, 31 December 2005 (UTC)

If Gamow says it's from Hilbert, I see no reason to assume otherwise. If it was a lecture example, we'll probably have to rely on secondary sources anyway; even if a primary source existed in the form of surviving lecture notes or such, those aren't likely to be easily available. Of course, the source of the claim should be mentioned in the article. —Ilmari Karonen (talk) 22:40, 31 December 2005 (UTC)

• I agree with Ilmari that Gamow is a good enough source for the article, but it looks likely that Courant has written something about this: it would be much nicer to cite him, since he is one of the main Hilbert authorities. I'll post this question to Historia Mathematica and see if they can help. --- Charles Stewart 23:31, 31 December 2005 (UTC)

December 31

Trajectory derivations

Help at Trajectory#Uphill/downhill in uniform gravity in a vacuum would be greatly appreciated! Samw 13:49, 31 December 2005 (UTC)