Wikipedia:Reference desk/Archives/Mathematics/2010 August 2

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

August 2

The Generalised Solution to All Problems.

Is there a generalised solution to all mathematical problems? (Or non-mathematical problems?)--220.253.172.214 (talk) 09:12, 2 August 2010 (UTC)

The set of all (mathematical) problems is, like the set of all sets, a moving target. You question is itself a (mathematical) problem. Bo Jacoby (talk) 09:40, 2 August 2010 (UTC).

Well, to even attempt to answer this, you would need to be more precise about what qualifies as a "generalised solution" and what exactly a "mathematical probem" is. German mathematician David Hilbert asked a similar (but more precisely formulated) question, to which the answer was eventually found to be "no" - see Entscheidungsproblem. Gandalf61 (talk) 10:04, 2 August 2010 (UTC)

To get down to earth, it is perfectly possible and in fact likely that some very simple problems like Goldbach's conjecture are true but cannot be proved. Dmcq (talk) 13:15, 2 August 2010 (UTC)

Yes, but then you get into a discussion about the meaning of truth, and that invariably results in me getting a headache... --Tango (talk) 14:12, 2 August 2010 (UTC)

I moved my discussion to a new section, in order not to obscure the OP's question anymore than I already have. --COVIZAPIBETEFOKY (talk) 14:37, 3 August 2010 (UTC)

• There is a generalized solution for all mathematical problems, but, in order to keep our jobs, every mathematician is sworn not to talk about it to the uninitiated.
• As for all philosophical problems, yes, there is an general answer, we don't know it yet, but [huge computer is already working on it] and will be ready in seven and a half million years.
• The case about physical problems is similar.
• Biologists don't work on the general answer for all biological problems because of the dangers this could pose, of which the slow demise of humanity (see eg. "Lymphater's Formula" by Stanislaw Lem) is one of the less scary. (This applies to all the other general answers, but the other scientists don't feel responsable for possibilities opened by their research.)
• Some churches believe they have a final and general solution for all theological problems.

– b_jonas 20:28, 5 August 2010 (UTC)

Amazing! OMG!

Zhang Yi (strategist): I'm trying to work it to DYK, and guess what? It was 260 characters originally, and 5x would be 1300... exactly 200 under the 1500 minimum! What are the chances of that? (And, although this is a silly selfref, please make the probability theoretical, not empirical. :P) Kayau Voting IS evil 12:33, 2 August 2010 (UTC)

Empirical is the best we could possibly do. We would need to know the distribution of lengths of stubs, which we can only get empirically. Incidentally, are you aware that the DYK rules say it needs to be 5x expanded and at least 1500? Also, whenever you ask "what are the chance of that?" you need to be very clear (to yourself, more than anything) what "that" is. What other results would have surprised your equally? Would you be surprised by any multiple of 100? 50? 10? Any number greater than 200? Any number less than 200? Or is there something special about 200 itself that only that number would surprise you? Often, once you consider the whole set of outcomes that would be equally surprising, you find that the chance of it being one of them isn't that unlikely at all. For example, someone in your town wins the lottery and you think "Wow! What are the chances?" and you probably think it is millions to one (which are the odds one one specific ticket winning), but once you realise that you would be equally surprised by anyone in your town winning and by them winning in any week, you find that actually its quite likely and shouldn't be at all surprising. --Tango (talk) 12:43, 2 August 2010 (UTC)

Um, yes, I am aware that DYK needs to be both 1500 and 5x. I've just done that. The article is now at an (un)impressive 2196 characters, which exceeds both requirements. I'm just pretty surprised that I got such a neat number - any number ending with 00 surprises me. Since empirical is the only choice and we've been having so many DYKs over the years, I won't bother the Wikipedian mathematicians and withdraw this question. Kayau Voting IS evil 14:24, 2 August 2010 (UTC)

If you are interested in the probability of it being any multiple of 100 then we can probably say we quite reasonable accuracy that it is about 1 in 100. We're talking about the difference between two numbers of similar magnitude, so the idea behind Benford's law shouldn't be an issue, and I doubt people count words other than to get over the 1500 threshold, so there is little chance of human influence resulting in certain lengths being more likely that others. It won't be exactly 1 in 100, but it will be pretty close. --Tango (talk) 14:35, 2 August 2010 (UTC)

1500 was chosen deliberately, and deliberately chosen numbers of that magnitude are extremely likely to end with 00. 1300 was found as 5 times some other number, and assuming a sufficiently smooth distribution of the base number, the multiple has about 5% chance of ending with 00. So I'd say that the chance of the difference ending with 00 is about 5%. -- Meni Rosenfeld (talk) 17:38, 2 August 2010 (UTC)

mathematics

Is logtanx integrable? —Preceding unsigned comment added by Bapi2912 (talk • contribs) 14:28, 2 August 2010 (UTC)

Are you looking for a primitive function? There appears not to be one expressable in elementary functions. What you get is this. --Andreas Rejbrand (talk) 15:18, 2 August 2010 (UTC)

There's a good criterion to see if a function α is integrable (in terms of exps and logs, which allow to express trigonometric and hyperbolic functions, etc).

You need α to have the form

$\alpha =c_{1}{\frac {u_{1}'}{u_{1}}}+c_{2}{\frac {u_{2}'}{u_{2}}}+\cdots +c_{n}{\frac {u_{n}'}{u_{n}}}+v'$

where the $c_{i}$ are constants, and the $u_{i}$ and $v$ are functions you admit as elementary (usually rational functions involving logs and exps).

Using this, you can for example prove that the error function is not elementary. Similarly, we have that

$\log(\tan(x))=\log \left(i{\frac {e^{ix}-e^{-ix}}{e^{ix}+e^{-ix}}}\right)={\frac {i\pi }{2}}+\log \left({\frac {e^{ix}-e^{-ix}}{e^{ix}+e^{-ix}}}\right)$

So working in the (differential) field

$\mathbb {C} \left(x,e^{ix},\log \left({\frac {e^{ix}-e^{-ix}}{e^{ix}+e^{-ix}}}\right)\right)$

Writing f for $\log \left({\frac {e^{ix}-e^{-ix}}{e^{ix}+e^{-ix}}}\right)$ we would have that if α has an elementary integral y, then

$y=v(x,e^{ix},f)+\sum _{i=1}^{n}c_{i}\log(u_{i}(x,e^{ix},f))$

Now we can use some of the really nice properties of the integral (I'm not going to take care of branches). It has no poles (or logarithmic singularities) whatsoever, so v has to be a polynomial function, and then we must be able to write

$\sum _{i=1}^{n}c_{i}\log(u_{i}(x,e^{ix},f))=\log(h(x,e^{ix},f))$ where h is a nonvanishing function with no singularities, i.e. it is a nonvanishing polynomial in x, e^ix and f. Hence we see that h must in fact be a polynomial in e^ix (by the fundamental theorem of algebra), and so $\log(h)$ is a polynomial (as we can't allow log terms which have singularities).

Now we have shown that y, if elementary, has to be polynomial in x, e^ix and f. We can't allow f because of singularities, and we can see that we can't have e^ix terms asymptotically. So it has to be a polynomial in x, but this is obviously not possible: for instance, as a real function, it is periodic.

This shows that $\log(\tan(x))$ has no elementary antiderivative. (I missed out some of the details, like dealing with the branches properly, maybe someone can fill that in!) -Sam^Talk 17:52, 2 August 2010 (UTC)

Graph of (-2) to the power of n

I take it that this is discontinuous - because if n =, for example, 1/2 or any rational number between 0 and 1 then there will be several answers - some of them complex.

What if n is an irrational number? _ i can't begin to get a handle as to what the answer could be

For example 2 to the power of an irrational number, say 'r' - I suppose the answer lies in between the well defined rational exponents which lie either side of r —Preceding unsigned comment added by 78.147.240.244 (talk) 19:41, 2 August 2010 (UTC)

(−2)ⁿ = 2ⁿ (−1)ⁿ = 2ⁿ cos(n π) + i 2ⁿ sin(n π) is a continuous, complex-valued function of n. You may prefer to write (−2)ⁿ = e^{n (log(2) + i π(1+2k))} for k an integer. This is a multivalued function. Choosing k=0 gives the former result. See exponentiation. Bo Jacoby (talk) 20:02, 2 August 2010 (UTC).

A negative number to an irrational power is not a well-defined expression. And if you want to deal with rational powers, you've got "multiple-valued" complex square roots, etc. So instead, just deal with complex-valued functions directly. Michael Hardy (talk) 16:46, 3 August 2010 (UTC)

Betting on the ponies

I've posted a multiple question over here [1], which might properly belong here. If you're into wagering on the sport of kings, would you mind taking a crack at the question anyway?

Many thanks, DaHorsesMouth (talk) 21:13, 2 August 2010 (UTC)

Constrained optimization problem

I have to find critical points of the constrained optimization problem

$f(x_{1},x_{2},x_{3})=2x_{1}^{2}+x_{2}^{2}+x_{3}^{2}$

subject to

$g(x_{1},x_{2},x_{3})=2x_{1}+x_{2}+x_{3}=4$

Using Lagrange multipliers I only come up with a critical point at (1,1,1). Solving for grad L I can't see how there can be any other critical points, but the question seems to indicate that there's more than one. —Preceding unsigned comment added by 118.208.156.45 (talk) 22:55, 2 August 2010 (UTC)

To be honest I don't remember how to do Lagrange multipliers, but thinking about the geometry of the example, there should only be one critical point of f on that plane (which is a minimum). Sometimes books phrase what you should be looking for in terms of the most general case so as not to give anything away about the answer. Rckrone (talk) 03:45, 3 August 2010 (UTC)

Roughly, the idea is to find points of the form x = (x₁, x₂, x₃) such that the gradient of f at x is parallel to the gradient of g at x ... (This should be made slightly more precise.) PST 04:16, 3 August 2010 (UTC)

Define z = 2x₁² + x₂² + x₃² and w = 2x₁ + x₂ + x₃. The challenge is to find extremum for z subject to the condition w = 4. Extremum for z means that dz = 0. Constant w means that dw = 0. For any Lagrange multiplier λ you have dz+λdw = 0. Now dz = 4x₁dx₁ + 2x₂dx₂ + 2x₃dx₃ and dw = 2dx₁ + dx₂ + dx₃. In order to eliminate dx₃ choose λ=−2x₃. Then dz+λdw = 4x₁dx₁ + 2x₂dx₂ + 2x₃dx₃ − 2x₃(2dx₁ + dx₂ + dx₃) = 4(x₁ − x₃)dx₁ + 2(x₂ − x₃)dx₂ = 0 for all values of dx₁ and dx₂, so x₁ − x₃ = 0 and x₂ − x₃ = 0. The condition w = 4 means that 2x₁ + x₂ + x₃ = 2x₁ + x₁ + x₁ = 4x₁ = 4, so x₁ = x₂ = x₃ = 1 is the only solution. Bo Jacoby (talk) 07:04, 3 August 2010 (UTC).

Another way of seeing this is the following: function f is strictly convex. But then, f is also strictly quasi-convex. A strictly quasi-convex function has one unique minimizer over a planar (convex) set [2]. 190.177.92.221 (talk) 20:19, 8 August 2010 (UTC)