Wikipedia:Reference desk/Archives/Mathematics/2008 June 2

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June 2

Quasiconvex subgroups of a finitely generated group

I'm trying to show that if G is a finitely generated group, and H is a quasiconvex subgroup of G (that is, there's some natural k such that for any a, b in G, any geodesic γ from a to b in the Cayley graph of G lies within the k-neighbourhood of H), then H is itself finitely generated. I'm having trouble because I don't really know how to show that a subgroup is finitely generated short of coming up with an actual generating set for it, but H seems too arbitrary for an explicit generating set formulation. Google has found me a couple of people asserting the result, but no proofs. Can anybody suggest any approaches to take? Thanks. Maelin (Talk | Contribs) 11:44, 2 June 2008 (UTC)

Aha! Figured it out with a friend. Thanks to anybody who spent some time thinking about it! Maelin (Talk | Contribs) 03:25, 3 June 2008 (UTC)

Complex cube root

Real and imaginary part of complex square roots can be expressed as radicals: We're looking for the real solutions for x and y for given reals a and b so that: $(x+i\cdot y)^{2}=a+i\cdot b$

Solution:
$x=\pm {\sqrt {\frac {a+{\sqrt {a^{2}+b^{2}}}}{2}}}$
$y={\frac {b}{2\;x}}$

Is there a similar representation for the cube root? I derived an equation of degree 12 which looks like it's maybe not impossible to transform it to a quartic equation of x³ or a cubic equation of x⁴:

$x^{12}+63\;x^{9}-3a\;x^{8}-45a\;x^{6}+3a^{2}\;x^{4}-(18a^{2}+27b)\;x^{3}-a^{3}=0$

Icek (talk) 15:13, 2 June 2008 (UTC)

It is certainly possible. The derivation is not nice though. Let x be the required root, and z the initial value. Then obviously x^3=z. This expression in polar form after applying demoivre's theorem yields rx^3(cos(3θx)+isin(3θx))=rz(cos(θz)+isin(θz)), where rx is the radius of x, θx is its angle, and respectively rz and θz are the radius and angle of z. Without loss of generality, we can make both rz and rx one. If |z| does not equal 1, then let z'=z/|z| and x'=x/|x|. |x| can easily be solved from |z| and x can be solved from x' and |x|. So, we can find x from x' and |z|. With this, the statement is simply (cos(3θx)+isin(3θx))=(cos(θz)+isin(θz)). Using trig identities, 4cos^3(θx)-3cos(θx)+3isin(θx)-4isin^3(θx)=cos(θz)+isin(θz). Seperating real and imaginary parts, this yields

4cos^3(θx)-3cos(θx)=cos(θz)'

3sin(θx)-4sin^3(θx)=sin(θz).

These two expressions are equivalent, so we choose the first one. Let a=cos(θx) and C=cos(θz). We get the expression 4a^3-3a=C, which is solveable through the cubic equation. By backtracking through our process with the found value of a, you can find x', and subsquently x. Indeed123 (talk) 16:24, 2 June 2008 (UTC)

Thank you very much - that seems to be a much easier way. Icek (talk) 19:53, 2 June 2008 (UTC)

An easier method would be to just convert the a+ib into $re^{i\theta }$ for $0\leq \theta <2\pi$ and $r\in \mathbb {R} _{>0}$. Then, $x+iy=se^{i\phi }$ where $s=r^{1/3}$ and $3\phi =\theta \mod 2\pi$. Polynomials are overrated. -mattbuck (Talk) 20:07, 2 June 2008 (UTC)

Well, that's beside the point, I knew that way. Icek (talk) 21:21, 2 June 2008 (UTC)

What is the point, by the way? Why not

$x+i\cdot y=(a+i\cdot b)^{1/3}$

Bo Jacoby (talk) 14:08, 3 June 2008 (UTC).

I think the OP is looking for the real and imaginary parts of the cube root, so just writing it like that doesn't help. --Tango (talk) 14:13, 3 June 2008 (UTC)

I think the answer is no. You may be able to reduce the question to a cubic equation, but that cubic equation will be such that it cannot be solved explicitly without taking the cube roots of a complext number. I've read that it is actually proven that you cannot solve all cubic equations explicitly with just radicals over real numbers and basic operations, you do need either radicals over complex numbers or trigonometric functions. Obviously you could get an approximation that converges fast or something like that if you don't want to do trig functions. – b_jonas 14:34, 3 June 2008 (UTC)

Yes, it seems so. Trying to solve Indeed123's cubic equation with the standard method you'll encounter a term sqrt(c² - 27), and c is a certain cosine. Is there a theorem saying that you cannot solve higher roots of complex numbers (just the basic roots, not whole polynomial equations) with radicals in real number (we need only consider prime exponents here)? Icek (talk) 01:11, 4 June 2008 (UTC)

This is a pre-computer problem. Before the time of the computer, people used tables of logarithms and trigonometric functions of real arguments, and so it was interesting to reduce algebraic problems to real root extraction and trigonometry. This was called an explicite solution. When a computer is available, there is no advantage in reducing the problem to extraction roots of real numbers. Even when it is possible, it is not a shortcut to the solution. Just solve the original equation z³=a using the Durand-Kerner method. Bo Jacoby (talk) 19:49, 3 June 2008 (UTC).

I'm sure the OP is not interested in finding the root numerically in practice, but rather in the aesthetics of being able to express a quantity with a limited set of tools. The main driving force of mathematical advancement is its beauty, not any immediately obvious applications. -- Meni Rosenfeld (talk) 20:05, 3 June 2008 (UTC)

I agree. Yet the equation z³=a is beautiful and should be considered an answer rather than a question. Mathematicians often consider an equation like F(a,x) = 0 a problem, and an equation like x = f(a) a solution. My point is: stop doing that! The equation F(a,x) = 0 can be solved numerically if you want to do so, but until then, don't even try. Let it be the answer. Bo Jacoby (talk) 20:29, 3 June 2008 (UTC).

Bo Jacoby, are you actually implying that the Galois theory of the solution of equations by radicals is ugly and pointless, or am I misinterpreting you? Algebraist 09:46, 4 June 2008 (UTC)

No. The Galois theory itself teaches us that solving algebraic equations by radicals is not an attractive path in general, as most equations surprisingly cannot be solved that way. Since antiquity mathematicians had reduced quadratic equations to extraction of square roots, and since the renaissance mathematicians had reduced cubic equations to the extraction of square roots and cube roots. So it was generally conjectured that solving equations by radicals was the way to go. But square roots of negative numbers appeared in the process, so the result was not quite as nice as hoped. And the indispensable use of complex numbers made the reduction to radicals less useful, as tables of roots of complex numbers were inconvenient. (The square root of a complex number was computed by first transforming the complex number into polar coordinates, looking up the square root of the positive modulus, and halving the argument, and transforming back to rectangular coordiantes, using trigonometric tables). On the other hand the fundamental theorem of algebra replaces this ugliness with beauty: every polynomial can be factored into linear polynomials. The lesson is that the polynomial defines it roots, and using radicals has no advantage. Bo Jacoby (talk) 14:09, 4 June 2008 (UTC).

A series

Is there a closed formula for $\sum _{i=1}^{\infty }{\frac {1}{x^{2}+1}}$? Thanks, --Dr Zimbu (talk) 15:32, 2 June 2008 (UTC)

Yes, assuming you meant $\sum _{i=1}^{\infty }{\frac {1}{i^{2}+1}}$, it is equal to ${\frac {1}{2}}\left(-1+\pi {\frac {e^{\pi }+e^{-\pi }}{e^{\pi }-e^{-\pi }}}\right)$. -- Meni Rosenfeld (talk) 15:49, 2 June 2008 (UTC)

Three-dimensional geometry question

If a pyramid is built with its apex on top of the center of a unit square, what is the relationship between the height of the pyramid and the angle subtended by each side of the square at the apex? --Masatran (talk) 18:38, 2 June 2008 (UTC)

Call the length of the sides of the base b. Call the apex point A. From A drop a perpendicular to the base and call the intersection point B and the length of the segment h (I assume that's the height you are talking about). Now from B draw a perpendicular to one of the edges of the base and call the point of intersection C (it will be the midpoint of the edge since the base is a square). The distance BC is b/2. ABC forms a right triangle, and we are trying to find the hypotenuse AC. By the pythagorean theorem it is ${\sqrt {h^{2}+(b/2)^{2}}}$. --Prestidigitator (talk) 01:08, 3 June 2008 (UTC)

Diagrams are fun! Made this to assist. Chris M. (talk) 06:04, 3 June 2008 (UTC)

Prestidigitator has not given the final solution, but I was able to extend his argument to the final solution thus:

Let the line segment of which C is the mid-point, be DE. Note that $b$ equals $1$. Let $\theta =\angle {DAE}$ be the angle subtended by each side of the square at the apex.

$\tan({\tfrac {\theta }{2}})={\frac {1}{2{\sqrt {h^{2}+{\tfrac {1}{4}}}}}}$

It would be great if someone updated the image.

--Masatran (talk) 16:44, 3 June 2008 (UTC)

Oh yeah. I kinda dropped the ball half-way, huh? Completely lost track of what the final goal was. That's funny. Sorry about that. --Prestidigitator (talk) 22:53, 3 June 2008 (UTC)

Image, updated. I knew there was something missing :P. Chris M. (talk) 03:42, 4 June 2008 (UTC)

Error in the figure: Note that $\theta =\angle {DAE}\neq \angle {ACB}$ --Masatran (talk) 08:12, 4 June 2008 (UTC)