Wikipedia:Reference desk/Archives/Mathematics/2019 October 2

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October 2

Cartoon asymptotic

On Sandy Cheeks' clipboard in "Sandy's Rocket" is the following expression:

$f(x)=_{x\to \infty +}{\sqrt {\frac {(x^{2}+x)^{3}}{\pi }}}$

This looks like an asymptotic expansion to me (it's remarkable that a cartoon expression turns out to make sense). Is there a well-known transcendental function with this property?--Jasper Deng (talk) 00:06, 2 October 2019 (UTC)

It doesn't make sense to me: I think this is just a cartoonist's idea of a mathematical formula. First, if f(x) is a function of x, it should equate to something that has a specific value depending on the value of x, but an expression that is to be evaluated as x goes to infinity is not that. In addition, the square-root expression obviously increases without bound as x increases, so nothing like a limit or asymptote seems to apply. Finally, I'm not aware of any notation where ${x\to \infty +}$ can be written "by itself" in that position without something to indicate what's being computed as x goes to infinity, e.g. "lim" to indicate a limit. Am I missing something? --76.69.116.4 (talk) 06:58, 2 October 2019 (UTC)

You very much are missing something. The precise meaning of this expression is $\lim _{x\to +\infty }f(x)/{\sqrt {\frac {(x^{2}+x)^{3}}{\pi }}}=1$ or $f(x)\sim {\sqrt {\frac {(x^{2}+x)^{3}}{\pi }}}({\text{as }}x\to +\infty )$. An asymptote need not be a bounded function or curve. Curvilinear asymptotes are a common thing.--Jasper Deng (talk) 07:06, 2 October 2019 (UTC)

Thanks for the interpretation. --76.69.116.4 (talk) 19:05, 2 October 2019 (UTC)

You are correct that the most reasonable way to interpret this equation is $\lim _{x\to +\infty }f(x)\left/{\sqrt {\frac {(x^{2}+x)^{3}}{\pi }}}\right.=1$. However:

1. That's not a standard way to write that; no mathematician would write it like that; and we can only guess that this is what was intended, it's far from obvious.
2. This is not an asymptotic expansion. It's just a simple asymptote - to be an expansion there have to be several terms. Actual asymptotic expansions are a fascinating and mysterious beast - they are similar to a Taylor series, except they do not converge, which sometimes allows them to converge faster than convergent series.

Anyway, it's easy to artificially construct a transcendental function with this asymptote. For example, $f(x)=(1+\exp(-x)){\sqrt {\frac {(x^{2}+x)^{3}}{\pi }}}$. -- Meni Rosenfeld (talk) 10:50, 2 October 2019 (UTC)

Looking at the expression again, I realize that in fact ${\sqrt {\frac {(x^{2}+x)^{3}}{\pi }}}$ is asymptotic to ${\frac {x^{3}}{\pi }}$. So, in addition to everything else, the original equation is unnecessarily cumbersome, since we could just as easily have written that f is asymptotic to ${\frac {x^{3}}{\pi }}$. -- Meni Rosenfeld (talk) 10:55, 2 October 2019 (UTC)

Yeah, I realized shortly after posting that this is not an expansion. I don't trust physicists to not abuse notation like that (Sandy after all seems to be much more a physicist than mathematician); nothing against them of course, but at my institution, the math department insisted on teaching undergraduate calculus after watching a physics professor teach "$\int _{0}^{\infty }f(x)dx$ converges if and only if $\lim _{x\to \infty }f(x)=0$". I was more hoping for something less contrived, something like a Bessel function, something that might be used in the context of rocket science. The simpler asymptote is $x^{3}/{\sqrt {\pi }}$ - you missed the square root--Jasper Deng (talk) 19:02, 2 October 2019 (UTC)

Sum of two squares; can it be factored sometimes??

The difference of two squares (a^2 - b^2) can always be factored as (a-b)(a+b). For example:

a^4 - 4b^4 = (a^2-2b^2)(a^2+2b^2)

Do you see how this can be factored as the difference of two squares??

In general, the sum of two squares cannot be factored unless it's the square of an odd power greater than 1, such as a^100 + b^100; it can be factored and its first factor is (a^4 + b^4).

However, sometimes independent of being an odd power greater than 1 it is possible to factor the sum of two squares, for example:

a^4 + 4b^4 = (a^2-2ab+2b^2)(a^2+2ab+2b^2)

Any more general polynomial that can be factored (avoiding complex numbers) despite being the sum of two squares?? My example was a^4 plus 4b^4; I want to know if there's anything more general than this. Georgia guy (talk) 01:21, 2 October 2019 (UTC)

Sure. $a^{6}+b^{6}$ is also a sum of cubes and is factored thus: $(a^{2}+b^{2})(a^{4}-b^{2}a^{2}+b^{4})$. $a^{8}+b^{8}$ is a special case of what you gave. Not sure what the question is.--Jasper Deng (talk) 01:35, 2 October 2019 (UTC)

I'm talking about a polynomial that can be factored despite being the sum of two squares but not the sum of two like odd powers greater than 1. Your example doesn't qualify because it's a sum of two cubes and can be factored as such. Georgia guy (talk) 01:38, 2 October 2019 (UTC)

Can't any power of 2 greater than 2 be factored by reduction to your example? The like powers case reduces the question to factoring sums of odd prime powers so that will always be factorable. You need to introduce something more general to get something that's irreducible.--Jasper Deng (talk) 01:42, 2 October 2019 (UTC)

I'm not sure what your question is. Every real polynomial can be factored into real quadratic and linear terms. It doesn't need to be "square of an odd power" or anything like that.

Your own example $a^{100}+b^{100}$ doesn't have squares of an odd power. 50 isn't odd.

The most basic example, $a^{4}+b^{4}$, can be factored as $(a^{2}+{\sqrt {2}}ab+b^{2})(a^{2}-{\sqrt {2}}ab+b^{2})$.

Perhaps you've meant to ask about factoring into terms with only integer coefficients? Then it's no longer always possible.

Specifically for $a^{n}+b^{n}$, I believe it can be factored over the integers whenever n is not a power of 2 (which might be what you incorrectly called "square of an odd power"). -- Meni Rosenfeld (talk) 10:30, 2 October 2019 (UTC)

But $a^{4}+4b^{4}$ can be factored even though the largest possible exponent is a power of 2 (2 to be specific.) Georgia guy (talk) 10:52, 2 October 2019 (UTC)

@Georgia guy: I think then you need to specify your base field, for that polynomial is irreducible over $\mathbb {Q}$ but not over $\mathbb {Q} ({\sqrt {2}})$.--Jasper Deng (talk) 18:54, 2 October 2019 (UTC)

I'm using Q. The polynomial I'm talking about is $a^{4}+4b^{4}$ . Georgia guy (talk) 21:30, 2 October 2019 (UTC)

@Georgia guy: A general factorization of a sum of squares is $x^{2}+y^{2}=(x+{\sqrt {2xy}}+y)(x-{\sqrt {2xy}}+y)$ which will be over $\mathbb {Q}$ if and only if ${\sqrt {2xy}}$ is. Thus, at least one of $x,y$ must be divisible by 2 (i.e. have divide coefficient when considering degree-one terms in $a,b$) and the product of x and y must be a perfect square. With $x=a^{2},y=2b^{2}$ we have that satisfied, but we do not have that satisfied with $x=a^{2},y=b^{2}$. Thus, the power of a and b must also be even in this factorization. If both are odd, the procedure for odd powers applies. This is only a sufficient condition still, as other factorizations can't be ruled out. The remaining case is the sum of two odd powers with unequal coefficients, which is addressed by extending the case of an unweighted sum of odd powers. If the factorization fails with all of the factors of the degree (not just prime factors) other than one, the polynomial is irreducible. The special case $x^{n}+a=0$ is simply whether, for any factor k > 1 of n, the kth root of -a is rational or not. I'm sure the generalization to two variables is precisely what I said before.--Jasper Deng (talk) 07:08, 6 October 2019 (UTC)

Best way to show that the only constant curvature curves on the Euclidean plane are circles and lines

What is the easiest way to show this result? The assumption should only require twice continuous differentiability but my result ended up assuming at least three times. I began by positing that the derivative be always a unit vector, where the parametrization is the arc length (I am quite certain this is permissible WLOG), and also positing the curvature to be unity (again, I believe this can be done without loss of generality, since zero curvature is easily shown to be a line, while dilating the plane should allow any constant-curvature curve's nonzero curvature to be set to 1). After some tedious manipulations beginning with the Cartesian formula for curvature in the plane, I ultimately showed that both components of the derivative satisfy $(u')^{2}=a^{2}(1-u^{2})$ whose solution is $\sin(as+C)$ (where s is the arc length parameter). However, this method seems deficient because it relies on third derivatives of position, and also requiring special treatment for the points at which $u=1$. Perhaps more geometrically, it can be shown from these assumptions that the acceleration is both normal and of constant magnitude, but I'm not sure how that rigorously goes to showing the curve to be a circular arc, other than being used to derive the above ODE.--Jasper Deng (talk) 08:39, 2 October 2019 (UTC)

This all seems unnecessary to me, since curvature in 2D is defined as 1/R, therefore constant curvature means constant radius, which, by definition, is a circle/circular arc. And the special/degenerate case of zero curvature/infinite radius is a line/ray/line segment. SinisterLefty (talk) 09:18, 2 October 2019 (UTC)

@SinisterLefty: This does not straightforwardly follow from the definition of curvature as $|{\frac {d\mathbf {T} }{ds}}|$, and in any case is circular as the definition of turning radius for general curves relies on this definition of curvature. "Radius" is not meaningful without a circle to refer to and the definition of curvature constructs one for that purpose.--Jasper Deng (talk) 09:55, 2 October 2019 (UTC)

How about the following:

We know that the unit tangent is always $(\cos \theta ,\sin \theta )$ for some $\theta$. Let $\theta (t)$ be the appropriate $\theta$ for every t. We have $|dT/ds|=|dT/dt|=|\theta '(t)|$. The curvature is constant (wlog equal to 1), so $|\theta '(t)|$ is constant, and by continuity $\theta '(t)$ is constant (rather than flipping signs all of a sudden). So $\theta (t)=a+t$ and $T(t)=(\cos(a+t),\sin(a+t))$. Since the curve is parameterized by length, $u'(t)=T(t)=(\cos(a+t),\sin(a+t))$, so $u(t)=(x_{0},y_{0})+(\sin(a+t),-\cos(a+t))$, a circle.

(There's some subtlety in that the choice of $\theta$ is multivalued, but we can find a continuous $\theta (t)$ which matches the curve). -- Meni Rosenfeld (talk) 10:15, 2 October 2019 (UTC)

The angle θ is the tangential angle and you can define curvature as dθ/ds. This simplifies things a bit: if dθ/ds is constant then θ = ks+c, from which dx/ds = cos(ks+c), dy/ds = sin(ks+c), and integrating gives x = -1/k sin(ks+c)+d, y = 1/k cos(ks+c)+e. This parameterizes a circle of radius 1/k and center (d, e). If k=0 you get dx/ds = cos(c), dy/ds = sin(c), x=cos(c)s+d, y=sin(c)s+e, which is a line. Tangential angle is a useful tool that used to be covered in any standard calculus text, not so much now though. For example it can be used to easily derive the equation of the catenary. --RDBury (talk) 12:01, 2 October 2019 (UTC)