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May 15

shape of solution space

The solution space of x+y=a is a straight line through (0,a) and (a,0). The solution space for $x^{2}+y^{2}=a$ is a circle center at 0,0 with radius ${\sqrt {a}}$ . What about $x^{3}+y^{3}=a$ ? What about higher powers?

The general problem I'm interested in is positive integer solutions for systems of n equations with n unknowns: ${\begin{cases}x_{1}+x_{2}+\cdots +x_{n}=a_{1}\\x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}=a_{2}\\x_{1}^{3}+x_{2}^{3}+\cdots +x_{n}^{3}=a_{3}\\\cdots \\x_{1}^{n}+x_{2}^{n}+\cdots +x_{n}^{n}=a_{n}\\\end{cases}}$

Is there any method of solving systems like that? I do know various constraints on the variables, if that helps. Thanks 67.122.209.126 (talk) 08:08, 15 May 2009 (UTC)[reply]

From the sums of powers of the unknowns,

a_{k}=\sum _{i=1}^{n}x_{i}^{k}

for k=1,2,...,n, you first compute the coefficients of the n 'th degree polynomial

p(x)=\prod _{i=1}^{n}(x-x_{i})

in which the unknowns

x_{i}

are roots. See Power sum symmetric polynomial. From the coefficients the roots are then computed numerically by the Durand-Kerner method. The roots are usually not positive integers but might even be nonreal complex numbers. Bo Jacoby (talk) 08:22, 15 May 2009 (UTC).[reply]

Thanks. In this particular problem (given where the data came from), the solutions are definitely integers. Finding them numerically seems reasonable. I think what I'm looking at is a bunch of intersecting shells in the first "2ⁿ-tant" (quadrant, octant, 16-tant, etc) that start out as a hyperplane, then part of a sphere, and get more and more cube-like as the exponent gets higher, and I need to find the points of intersection. 67.122.209.126 (talk) 01:58, 18 May 2009 (UTC)[reply]

logarithm

hey, does anyone how to solve this queston???

log_aa - log_aa² + log_aa³ - log_aa⁴ + .......

(please explain it in the simplest way possible, i'm new to this thing, and if possible, tell me any identities of logarithms (relevant or irrelevant to this question, i don't care)

thanx —Preceding unsigned comment added by 122.50.131.217 (talk) 09:42, 15 May 2009 (UTC)[reply]

sory, i have to make a change to this question.

the series ends at 99 terms, i.e. it ends at +log_aa⁹⁹ —Preceding unsigned comment added by 122.50.131.217 (talk) 09:59, 15 May 2009 (UTC)[reply]

Note that

log_{a}(a^{n})

equals n so your series becomes:

1 - 2 + 3 - 4 + 5 + ....... + 99

Famously performed by Gauss, we may note the following:

1 + 99 = 100

3 + 97 = 100

5 + 95 = 100

In this manner, we add all odd numbers with another term in the series (which you can verify exists) such that the sum is 100. Adding in this way, we obtain (25 x 100) since there are 25 odd numbers between 1 and 49 (inclusive) - note that adding odd numbers greater than 50 in this manner will repeat sums. Now, we have taken care of all odd numbers. Similarly, for the even numbers:

- 2 - 98 = - 100

- 4 - 96 = - 100

- 6 - 94 = - 100

In this manner, we add all even numbers with another term in the series (which you can verify exists) such that the sum is - 100. Again, by similar reasoning, since there are 24 even numbers between 2 and 48 inclusive, and - 50 has no other number associated to it, the sum is (24 x (-100)) - 50 = - 2400 - 50 = - 2450. The other sum we calculated was 2500. Since, 2500 - 2450 = 50, the sum is 50.

Let me note that the method I have provided is delibrately long. There is a much shorter way to obtain the answer to this problem (by pairing numbers as I have done to get the answer). See if you can find it (if it is obvious (which it should be), do not worry about it). Note however, this method will not work for infinite sums, contrary as it seems. In general, different rearrangements of an infinite series may yield different sums, unless certain conditions are imposed. See Convergent_series#Conditional_and_absolute_convergence for more details.

Regarding logarithms, I recommend looking at the article Logarithm which provides many of the basic identities that you will probably use, should you continue such problems. Hope this helps. --PS T 10:25, 15 May 2009 (UTC)[reply]

We may also look at our 1 − 2 + 3 − 4 + · · · article, section Divergence for partial sums sequence of this series. --CiaPan (talk) 11:09, 15 May 2009 (UTC)[reply]

Fatou's Lemma and liminfs

I am working on a problem that uses Fatou's Lemma. I have all but the very last detail finished. I have two sequences f_n and f_n' and they are equal almost everywhere. I need the equality of $\int \varliminf f_{n}'\,dx=\int \varliminf f_{n}\,dx$ and I think it is true. I know that if two functions are equal almost everywhere, then their integrals are equal. I assume that if I have two sequences of functions where for each n, the functions are equal almost everywhere, then their liminfs will be equal almost everywhere also, but I'm not exactly seeing this. I could see something weird happening where f_1 = f_1' except on some set of measure 0, f_2 = f_2' except on some other set of measure 0, and so on. Can any one help me out a bit? Thanks StatisticsMan (talk) 15:19, 15 May 2009 (UTC)[reply]

A countable union of null sets is a null set. Therefore there is a null set Z of the domain X such that for all x in X\Z and for all n in N you have f_n(x)=f '_n(x)(so here Z is just the union of the Z_n:={x: f_n(x)≠f '_n(x)} )--pma (talk) 17:03, 15 May 2009 (UTC)[reply]

Good point. Thanks. StatisticsMan (talk) 17:11, 15 May 2009 (UTC)[reply]

Irrationality of Pi

I pretty much get the proofs that pi is irrational, in that I can see that they're valid proofs, but I don't really understand why they work, and I can't see how anyone could have come up with them. Why is pi irrational? For comparison, the proof that e is irrational seems pretty reasonable to me. Black Carrot (talk) 21:07, 15 May 2009 (UTC)[reply]

You might consider the Hermite-Lindemann transcendence theorem to be a more direct approach. 207.241.239.70 (talk) 22:37, 15 May 2009 (UTC)[reply]

It was in the 18th century that π was first proved to be irrational, and I think the way it was done was that someone found a continued fraction expansion of the arctangent function, and consequently of particular values of the arctangent function, such as π/4. Some types of continued fractions (such as those in which every numerator is 1) terminate if the number they represent is rational, and this one did not terminate. Michael Hardy (talk) 22:56, 15 May 2009 (UTC)[reply]