Wikipedia:Reference desk/Archives/Mathematics/2011 July 16

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July 16

are numbers sets?

In either naive or axiomatic set theory, are numbers themselves considered sets? And if so, what elements do they contain? Aacehm (talk) 17:24, 16 July 2011 (UTC)

Natural numbers are often defined as sets of all natural numbers less than them. Zero is the empty set ($\emptyset$), 1 is the set containing 0 ($\{0\}=\{\emptyset \}$, 2 is the set containing zero and 1 ($\{0,1\}=\{\emptyset ,\{\emptyset \}\}$), etc.. --Tango (talk) 18:05, 16 July 2011 (UTC)

I can almost remember from 25 years ago what purpose this definition served. Does anyone else? 157.22.42.3 (talk) 00:09, 17 July 2011 (UTC)

I believe the definition comes from Principia Mathematica (or perhaps Frege); the aim was to construct mathematics from the smallest possible set of basic concepts and axioms. Looie496 (talk) 00:47, 17 July 2011 (UTC)

The natural numbers are defined using the Peano axioms (or variations upon them). Once you have a set of axioms, it can be helpful to find a model for them. It has become common to try and define all mathematics in terms of ZF set theory, and the model I described is a natural way to model the Peano axioms in ZF set theory. One thing finding such a model does is prove that the axioms are consistent within the theory - if they weren't, then you would have a big problem. --Tango (talk) 02:37, 17 July 2011 (UTC)

So It's possible to write the number 3 as something like this: {{},{{}},{{},{{}}}}? Aacehm (talk) 15:36, 17 July 2011 (UTC)

That's 2, (the empty set is 0), but yes, the idea is right. Looie496 (talk) 16:59, 17 July 2011 (UTC)

No 3 in that interpretation,{} is the empty set and corresponds to 0, and {{}} which is the set containing the empty set corresponds to 1. Dmcq (talk) 18:03, 17 July 2011 (UTC)

Indeed, it's 3. --Tango (talk) 22:15, 18 July 2011 (UTC)

I don't know if this answers the question, or if it's a mathematical or philosophical answer, but I've heard a definition where any integer is a set of all sets with that cardinality.. e.g. 3 is 3 oranges, 3 cars, 3 planets, etc... Duga3 (talk) 01:23, 20 July 2011 (UTC)

I'm afraid mathematicians tend to start looking for cover if you start talking about the set of all sets. The usual thing to do is define a universal set in advance, but there's no easy way to do that in this case. Dmcq (talk) 07:42, 20 July 2011 (UTC)

This was essentially Frege's definition, I believe. In modern set theory it doesn't work, but can be rescued with Scott's trick. This is the usual way of defining cardinal numbers in the absence of choice. Algebraist 09:49, 20 July 2011 (UTC)

clothing with clothoids

I've lately become interested in Cornu spirals. Am I right in supposing that they can be determined by four points? How about replacing one or more with a tangent (not osculating) line or circle? (I don't mind if a solution is not unique, so long as the set of solutions is finite.) Where might I look for procedures? —Tamfang (talk) 00:12, 16 July 2011 (UTC)

Do you mean at most four points in general position? A straight line and a circle are both Euler spirals. Any two distinct points determine a unique straight line, while three non-linear points uniquely determine a circle. — Fly by Night (talk) 01:02, 16 July 2011 (UTC)

I don't know anything about these curves, but I imagine what Tamfang is saying is that given any four distinct points there is exactly one Cornu spiral that passes through all four. (Whether or not this is true, I don't know.) Sure, a straight line is determined by two distinct points, as long as you also have the information that it's a straight line; but if I just give you two distinct points on a Cornu spiral, you don't know whether the spiral is a straight line between them or a different shape of curve. Presumably, if I'm correctly interpreting what Tamfang is saying, you'd need to be given four collinear points to be sure that the curve is a straight line and not something else. Now, that actually seems to be false to me: I can pick out four collinear points on the curve in the illustration in the Euler spiral article. —Bkell (talk) 04:17, 16 July 2011 (UTC)

Yeah, that's true. — Fly by Night (talk) 04:53, 16 July 2011 (UTC)

Ah but can you pick out four arbitrarily spaced collinear points? Anyway, as I said, I'll be satisfied with a finite number of solutions. —Tamfang (talk) 06:24, 16 July 2011 (UTC)

It seems that Euler spirals have five degrees of freedom. According to Kimia, B. B.; Frankel, I.; Popescu, A. M. (2003). "Euler Spiral for Shape Completion" (PDF). International Journal of Computer Vision. 54 (1/2/3): 159–182. you need only specify a point p (two numbers), the angle of the tangent at p (one number), the curvature at p (one number), and the rate of change of the curvature at p (one number). Take a look at Proposition 1 at the bottom right of page 164, and the top left of page 165. For example, you could specify two points, say p and q, (giving four numbers) and then specify the direction of the tangent line at either p or q (one number). If you want an Euler spiral that passes through four fixed points then you are imposing eight conditions. That means that if you had a three parameter family of a quadruple of points, then you'd expect only one quadruple in the family to have an Euler spiral passing through each of the points. — Fly by Night (talk) 18:41, 16 July 2011 (UTC)

Every Euler/Cornu spiral has the same shape, differing from the "unit" spiral by rotation (one number), scale (one number) and translation (two numbers). The fifth apparent degree of freedom is consumed in positioning p along the curve. With four points, four of the eight degrees of freedom are thus consumed. Otherwise you wouldn't always be able to draw a line (two DoF) through two points (four numbers), or a circle (three DoF) through three points (six numbers).

"same shape" — oops, I forgot chirality. —Tamfang (talk) 05:47, 19 July 2011 (UTC)

Incidentally, today I sat down to compute the integral of squared derivative of curvature in Knuth's cubic, which is misstated in the article (p.162, eqn 1); but the expression soon became big enough that I felt more likely than not to make a blunder in the algebra .... someday I'll buy Mathematica. —Tamfang (talk) 09:46, 17 July 2011 (UTC)

I see what you mean. There is some quotient involved. For example, you can specify a unique line by a base point and an angle, but that same line is given by any base point on the line and the same angle. The space of unoriented lines in the plane is parametrised by the projective plane minus a point, i.e. is homeomorphic to the Möbius band; which is locally two dimensional. The paper says that you need five numbers to give an Euler spiral, $(x,y,\theta ,\kappa ,\kappa ').$ But there is some kind of quotient. For example, if we reverse the orientation then $(x,y,\theta ,\kappa ,\kappa ')\sim (x,y,\theta ,-\kappa ,-\kappa ').$ This seems like a non-trivial problem. Consider the space all initial conditions, which will be $\mathbb {R} ^{5}$, but then identify all points that give the same spiral (as a point-set). Like I said all the points $(x,y,\theta ,\kappa ,\kappa ')\sim (x,y,\theta ,-\kappa ,-\kappa ').$ But then we can "slide" $(x,y)$ along the curve, which changes $(\theta ,\kappa ,\kappa ')$, and all such points need to be identified. I recommend the book Olver, P. J. (2009). Equivalence, Invariants and Symmetry. Cambridge University Press. ISBN 0521101042., and especially the chapter about Jet and Contact transformations. — Fly by Night (talk) 20:32, 17 July 2011 (UTC)

P.S. What is the correct statement of Knuth's cubic, and how does the paper misstate it? — Fly by Night (talk) 21:41, 17 July 2011 (UTC)

$-r_{2}e^{i\theta _{2}}-s^{2}(1-s)$ should be $-r_{2}e^{-i\theta _{2}}s^{2}(1-s)$ : a 'minus' was moved from within an exponent to immediately after it, splitting the last term into two. —Tamfang (talk) 04:45, 18 July 2011 (UTC)

Have you contacted the authors? It might be a good idea to ask them. Maybe they used a different representation of the curves that looks like an error. It's been peer reviewed, and will have undergone many revisions before and after original submission. Have you thought any more about the quotients that I mentioned? I think that this is a really interesting, non-trivial problem. What is the topology of the space of Euler spirals? Like I said, the space of unoriented lines in the plane is the Möbius band. A line is given by $ax+by+c=0$. Thus a line seems to be given by the point $(a,b,c)\in \mathbb {R} ^{3}$. However, the same line is given by $\lambda (ax+by+c)=(\lambda a)x+(\lambda b)y+(\lambda c)=0$, for all $\lambda \neq 0$. Thus, the points $(\lambda a,\lambda b,\lambda c)\in \mathbb {R} ^{3}$, for $\lambda \neq 0$ all give the same line. This is exactly the same equivalence relation as the projective plane, i.e. ${\mathbf {x} }\sim \lambda {\mathbf {x} }$ for all $\lambda \neq 0$. It's tempting to think that the space of lines is given by $(x:y:z)\in \mathbb {RP} ^{2}$, however, the point $(0:0:1)\in \mathbb {RP} ^{2}$ does not correspond to a line; the equation 0 = 1 does not give a line. Thus, the set of unoriented lines in the plane is parametrised by the projective plane, minus $(0:0:1)$, which is the Möbius band. Have you any ideas about the equivalence relations involved in parametrising the space of Euler spirals? I mentioned one earlier, i.e. that $(x,y,\theta ,\kappa ,\kappa ')\sim (x,y,\theta ,-\kappa ,-\kappa ')$ I think that this implies some unorientablity of the space; just like the space of unoriented lines in the plane. Like I said, it seems to be non-trivial, and it would be a very nice result if you could find the answer. — Fly by Night (talk) 22:14, 18 July 2011 (UTC)

On second thought I'm unsure about the minus sign on $i\theta _{2}$; the sign of $\theta _{2}$ is a choice of convention, unstated (so far as I can see) and unimportant for now. But with the extra minus operator after that exponent (where there ought to be multiplication), there's no longer a term parallel to $r_{0}e^{i\theta _{0}}s(1-s)^{2}$. Note the structure of the whole expression: $3s^{2}-2s^{3}$ has derivative zero at s=0 and s=1; the r₀ term provides the derivative at s=0, and the r₂ term provides the derivative at s=1, both of these terms being zero in each place. It would take more than inserting a single operator to express the curve in another equivalent form. It's a typo.

The shape of the space of clothoids is over my head, sorry. —Tamfang (talk) 06:06, 19 July 2011 (UTC)

1 radian as a pure number

The article Radian per second says that 2π rad/s = 1 Hz. Since the SI unit Hz is equal to (1/s), could we not divide each side by (1/s) and get 2π rad = 1 $\Rightarrow$ 1 rad = ${\frac {1}{2\pi }}$ which is a pure number not carrying any units? Widener (talk) 00:54, 16 July 2011 (UTC)

A radian is a pure number already, just like the 1 in the numerator of hz. — Carl (CBM · talk) 00:56, 16 July 2011 (UTC)

But is that pure number that the radian represents indeed the number ${\frac {1}{2\pi }}$?Widener (talk) 01:29, 16 July 2011 (UTC)

Yes, if you take the "1" in the numerator to mean "1 cycle". The identity you started with (2π rad/sec = 1 hz) is only sane when you read the "1 hz" to mean "1 cycle hz", that is, "1 cycle per second". You can do all sorts of crazy things with dimensionless units if you don't take care, for example (using the fact that radians are dimensionless):

2π rad/s = 1 hz = (1 rad) hz = 1 rad/s

This is what the article on radians per second is warning about when it talks about the importance of distinguishing angular frequency and ordinary frequency, both of which are dimensionally measured in hertz but which measure different things. Someone else may have a different way of explaining why it's nonsense. — Carl (CBM · talk) 01:52, 16 July 2011 (UTC)

I prefer not to use the term "radian" at all. The angle you are talking about is simply "1" and a full circle is just "2π". Angles don't have units, they are just numbers - they are defined as the ratio of two lengths (radius and arc length), which is just a number. The only time you might need to use the word "radian" is to make it clear that you aren't working in degrees, but that's almost never the case in maths beyond school-level since degrees are a completely arbitrary and useless concept from a mathematical point of view. --Tango (talk) 18:10, 16 July 2011 (UTC)

I respectfully disagree with Tango. The radian and the turn are both perfectly natural units of measurement of angle. See Angle#Units. The turn per second has the name hertz, but the radian per second has got no such special name. The rpm is also a commonly used unit for frequency. Angular frequency and ordinary frequency is the same concept, but measured in different units. Wavelengths are usually per wave rather than per radian. The unit of the planck constant is Js, meaning joule per hertz rather than in joule per radian per second. We cannot close our eyes to the fact that the radian is not the only unit of angle around. Bo Jacoby (talk) 19:22, 16 July 2011 (UTC).

Ok, turns can be useful. I stand by my statement that degrees aren't, though. Radians are still more natural mathematically than turns - a lot of useful formulae need an extra 2π in them if you work in turns rather than radians (some, of course, allow you to get rid of a 2π, but far fewer as far I as am aware). Waves don't really involve angles, but rather things that are analogous to them, so whether we throw a 2π into the analogy or not is really a matter of choice (basically, a choice between using k or ω). As for the Planck constant, a lot of physicists prefer the "reduced Planck constant" (denoted ħ), which is in radians. --Tango (talk) 23:32, 16 July 2011 (UTC)

When the unit is omitted we pay the price of having several notations for the same concept, ν and ω for frequency, h and ħ for planck constant. E=hν and E=ħω. The "reduced Planck constant" is not reduced but merely measured in another unit. You don't drive slower by measuring your speed in miles per hour rather than kilometers per hour, and an oscillation is not faster by being measured in rad per second rather than in turn by second. So, avoid confusion by saying radian, if that is what you mean, or turn or period or cycle or wave or tau (2π), if that is what you mean. The degree is probably here to stay for yet a while, even if we don't like it very much. Bo Jacoby (talk) 05:29, 17 July 2011 (UTC).

Taylor convergence

I'm working on code that uses a Taylor series to approximate $\int _{0}^{x}e^{f(t)}dt$, where f is a polynomial. The series coefficients $a_{n}$ are generated as follows:

$a_{0}=0$

$g\leftarrow e^{f(0)}$ (a constant polynomial)

$n\leftarrow 1$

loop

$a_{n}=g(0)$

$n\leftarrow n+1$

$g\leftarrow (g^{\prime }+gf^{\prime })/n$

endloop

I know the code is correct (though I could have mistranslated it to math notation here!). I've been arbitrarily using 100 terms. To speed it up, I want to estimate the remainder:

$L=|t_{max}|$

$k=|f^{\prime }(0)|$

$M=kL^{1+m}$ where m is the degree of f′

loop

...

$L\leftarrow ML/n$

if k*L < tolerance: break

...

endloop

Am I on the right track? —Tamfang (talk) 15:01, 16 July 2011 (UTC)

I think this is only reasonable if x is small. Otherwise think about what will happen if $f^{\prime }(0)=0$. Looie496 (talk) 17:48, 16 July 2011 (UTC)

Oops, I think I meant k to be the highest-ranking coefficient of f', rather than its constant term. —Tamfang (talk) 17:56, 16 July 2011 (UTC)

... I used $M=f_{a}^{\prime }(t_{max})$, where $f_{a}$ is f with each coefficient replaced by its absolute value. Didn't work. :( —Tamfang (talk) 19:46, 17 July 2011 (UTC)

And then I saw my blunder and changed M to $t_{max}f_{a}^{\prime }(t_{max})$. Now it works beautifully. —(Forgot to sign this yesterday) Tamfang (talk) 21:09, 18 July 2011 (UTC)