Wikipedia:Reference desk/Archives/Mathematics/2012 September 23

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September 23

Seeking function

Does anyone have any feeling whether any function f exists to satisfy the following:

$\int _{0}^{1}{\frac {x}{z^{2}}}f(x)f\left(x\left({\frac {1}{z}}-1\right)\right)\,dx=1$

for all 1/2 < z < 1

Of course, actual solution(s) would be even better, but failing that an idea of whether there are likely to be no solutions, one solution, or many solutions would be of interest. 81.159.105.97 (talk) 00:59, 23 September 2012 (UTC)

Intuitively I think there should be such a function. It should be easy (though computationally intensive) to find for every $\epsilon >0$ a function such that $\left|\int _{0}^{1}{\frac {x}{z^{2}}}f(x)f\left(x\left({\frac {1}{z}}-1\right)\right)\,dx-1\right|<\epsilon$, but in my experimentations the functions didn't seem to converge as $\epsilon \to 0$. -- Meni Rosenfeld (talk) 10:03, 24 September 2012 (UTC)

Thanks, could you outline the method you used to find that series of functions?

I tried writing f as a power series, and looking for coefficients such that, on evaluation of the integral, the z's from the f(x(1/z-1)) component cancelled out 1/z^2, based on the idea that z^2 = 1 - 2(1/z-1) + 3(1/z-1)^2 - 4(1/z-1)^3 + ... However, I could not get anything that looked solvable, or even looked as if it would converge as a power series. 86.128.6.76 (talk) 20:24, 24 September 2012 (UTC)

I wrote f as a polynomial of some degree, took a few evenly spaced values of z, and used a loss function based on the sum of squared difference between the actual and desired value of the integral on those points. I found the coefficients which minimize this loss. This requires the use of a CAS.

In fact I worked with $u=1/z$ rather than z. -- Meni Rosenfeld (talk) 09:46, 25 September 2012 (UTC)

Would you be able to post an example of a good-fit polynomial that you found?

I'd have thought that since an integral is involved and there is unlikely to be a nice power series function satisfying it that using a Fourier series or some orthogonal polynomial series would be best for converging to something if it exists. Dmcq (talk) 09:10, 26 September 2012 (UTC)

86.129.16.19 (talk) 19:18, 25 September 2012 (UTC)

The substitution $u={1 \over z}-1,0<u<1$ seems better, yielding the two equations to be satisfied:

$(1+u)^{2}\int _{0}^{1}xf(x)f(ux)dx=1$

${d \over du}\left((1+u)^{2}\int _{0}^{1}xf(x)f(ux)dx\right)=0.$

Moving the differentiation inside the definite integral is tricky, because of the interaction with the second integration limit (I think). Once this is done, it should provide an equation with a definite integral involving quantities like $f(x)$ and $f'(u)$, which could lead to a differential equation in $f(u)$. We could get rid of the $x$ inside the integral if need be using the substition $y=x^{2},g(x^{2})=f(x)$, so $\int _{0}^{1}xf(x)f(ux)dx={\tfrac {1}{2}}\int _{0}^{1}g(y)g(u^{2}y)dy$. Anyone game for the differentiation step? — Quondum 04:45, 26 September 2012 (UTC)

Differentiation under the integral sign indicates that the derivative and fixed-limit integral commute. If I haven't slipped up, this gives us the identity

$\int _{0}^{1}g(y)g'(u^{2}y)dy=-{2 \over u(1+u)^{3}}.$

For $u=0$ this gives us that $\lim _{y\to 0}g'(y)=-\infty$ (sign not proven) and for $u=1$ that $g(0)^{2}-g(1)^{2}=f(0)^{2}-f(1)^{2}={\tfrac {1}{2}}$. Maybe this hints at the graph of $f(x)$? It feels to me that $f(x)$ is a decreasing function not far from 1 over the interval [0,1]. — Quondum 09:03, 26 September 2012 (UTC)

If you do the substitution $x=e^{t},u=e^{-v},f(x)=h(t)$ then the integral becomes.. I've fixed this below, which is a nasty autocorrelation relation on $h$ except only integrating from minus infinity to zero instead over the whole real line. Dmcq (talk) 09:44, 26 September 2012 (UTC)

Well in fact the upper limit can be plus infinity simply by having $h(t)$ zero when $t$ is positive. Dmcq(talk) 09:49, 26 September 2012 (UTC)

I've just had another look and the main problem with this is that we can't easily get a value for the integral when $v$ is negative, unfortunately this can't be a straight extension of the formula above. If one could do that then the Wiener–Khinchin theorem would give something very close to an answer overall. Since $h(t)$ is zero for positive $t$ the formula for negative $v$ is

$\int _{-\infty }^{v}h(t)h(t-v)dt.$

which is missing the bit from $v$ to zero otherwise we'd have been on a roll. Though there is the interesting bit that the power spectrum should give zero for positive values. Dmcq (talk) 12:05, 26 September 2012 (UTC)

Ulp silly me, one should simply use the same function with $v$ negated. In fact I think that should lead to some sort of cleanish solution if one plugs the business into the Mathematica online integrator. AN exercise for the originator or perhaps there's an error somewhere - anywayI better go. Dmcq (talk) 13:00, 26 September 2012 (UTC)

Thank you very much for your help and interest. Unfortunately, I am rather lost. I am aware of the online tool that does integration, but where have we reduced the problem to simply one of integration? 86.129.16.11 (talk) 21:06, 26 September 2012 (UTC)

No sorry it seems I haven't come anywhere near a solution and I'm not sure there is one though this sort of thing is not something I know very much about. Having the autocorrelation integral always about 1/4 when $v$ is large is a problem, but it looks otherwise fairly sensible - so I think I need to have a much better look at it. Dmcq (talk) 21:57, 26 September 2012 (UTC)

And to fix my math though it isn't any good for a solution, I should have done the substitution $x=e^{t},u=e^{-v},e^{t}f(e^{t})=h(t)$ we get

$\int _{-\infty }^{0}h(t)h(t-v)dt={e^{-v} \over {(1+e^{-v})^{2}}}.$

With $h(t)=0$ for $t>0$ the limits can be change to minus and plus infinity which is the autocorrelation I was talking about, and much better as the right hand side goes to zero as $v$ goes to infinity with the maximum at $v=0$ as things like this ought to be. Still found the version with $u$ better for estimating a function though. Dmcq (talk) 19:51, 29 September 2012 (UTC)

Since there hasn't been a function posted yet as far as I can see: f(x) = (1.3+x)/((x+0.6)²(x+1)²) shows integrals at x=1 for different values of z between 0.98 to 1.035, at least according to KmPlot I'm using. Just some values I got with trial and error, first time I use it, so I don't know how to let it search for optimised values. No idea what parameters to tweak in order to have the integral curves cross at the same point (and at x=1). Ssscienccce (talk) 15:15, 27 September 2012 (UTC)

In general one needs a formula for f where one can plug in parameters, e.g. the coefficients of a polynomial, and a function that evaluates the difference between what the integral gives with the trial function and what it should be which is always one here. Integrating the sum of the squares of this difference for z between a half and one then gives what Meni Rosenfeld called a loss function above, though you could just use the absolute maxium difference or fourth powers - just anything which gives something to optimize. You then need an optimization program that can adjust the parameters to minimize this loss function as you would normally be doing one or more of these steps numerically rather than symbolically - or it might just be easier anyway. A popular way is use a package implementing for instance a variant of the BFGS method which is essentially a multi-dimensional version of Newton's method in optimization, the number of dimensions is the number of coefficients. Dmcq (talk) 19:51, 29 September 2012 (UTC)

Topology Norm on a space over a complex field

Is there any "recognized" metric [norm] which satisfies the Pythagorean Theorem even when the lengths of [vectors representing] sides in a triangle are not real [complex] numbers? (e.g. the hypotenuse will be 0 in [by] that metric norm if the lenghs of [vectors representing] the other sides are 1 and i; In other words: the length of the vector (1,i) will be zero, because 1² + i² = 0²). Additionally, can such a metric [norm] be regarded as Euclidean (or pseudo-Euclidean) in some sense? Or rather, can it even be regarded as the simple Euclidean metric [norm] on a space over the complex field? Is it a legitimate "recognized" metric [norm]? 77.126.50.25 (talk) 08:58, 23 September 2012 (UTC)

Would the pseudo-Euclidean Lorentz metric (Minkowski space) qualify? This still uses real numbers, but modifies signs in the Pythagorean theorem. The question is essentially circular, inasmuch as what is considered to be orthogonal (a "right angle") is defined by the metric (usually a symmetric bilinear form), including in any number of dimensions. Pure imaginary numbers can be used to produce the same effect. — Quondum 10:11, 23 September 2012 (UTC)

The circularity you've pointed at can be avoided by defining the space over the complex field (rather than over the real field), so that the two vectors (x,0) (0,y) be orthogonal - because their inner product is zero, yet x may equal 1 and y may equal i, in which case - the metric I'm looking for - determines the length of their sum (1,i) to be zero (because 1² + i² = 0²).

I suspect Minkowski space won't be sufficient, because of two reasons: First, it's not defined over the complex field - as opposed to what's expected from the space having the metric I'm looking for; Second, even if one realy tried to use Minkowski space over the complex field, its metric still doesn't fully obey the Pythagorean Theorem - but rather modifies signs - as you have already pointed out.

As long as I think about this, it seems that the metric I'm looking for is the simple Euclidean metric on a space over the complex field, isnt it? 77.126.50.25 (talk) 11:32, 23 September 2012 (UTC)

You are describing a space of two real dimmensions with a metric. If you choose to separately consider it as the complex field, this is irrelevant, since the complex multiplication is not used here except as you choose within the metric, and you are choosing to dismember it into it real and imaginary parts (i.e. two one-dimensional spaces over the real numbers) before defining the metric in terms of the pieces. You are separately defining (x,0) and (0,y) be orthogonal; you can't do this, it is determined by the metric (i.e. you have to find whether they are orthogonal under the metric you've defined). The metric you have described is exactly the Lorentz metric in two dimensions, where you have simply labelled the units on one axis with an "i". The two axes do turn out to be orthogonal, but vectors away from the axes behave a little counterintuitively, with those at 45° being self-orthogonal. This matches the behaviour of split-complex numbers rather than the normal complex numbers. If you were to try to define a "simple Euclidean metric" over the complex field in two dimensions, each "axis" would allow values from the whole field (you'd be dealing with 4 real dimensions). This construction works perfectly well, but I doubt that it is what you're looking for. — Quondum 13:00, 23 September 2012 (UTC)

Thanks to your comment, I've just fixed some mistakes in the old version of my original question, in order to make it clear that what I've been trying to describe - was not "a space of two real dimensions" - nor "two one-dimensional spaces over the real numbers", but rather one space of two complex dimensions. Btw, I didn't understand why I couldn't state that (x,0) and (0,y) are orthogonal; Doesn't this follow from the very definition of the orthogonality and the inner product? As to the name of the section, I've just changed it. 77.126.50.25 (talk) 14:14, 23 September 2012 (UTC)

The complex metric is "unrecognised" not because it is impossible to construct. It is "unrecognised" because fails to satisfy the triangle inequality, if not to mention that almost certainly evokes a lot of null vectors. And the existence of non-zero null vectors is a hint to impossibility of the Pythagorean Theorem in complex-valued "metric spaces" (which are actually not so by their core properties). BTW, I still do not realize why this section is named "Topology". Incnis Mrsi (talk) 13:28, 23 September 2012 (UTC)

As Quondum has indicated, physicists typically may not require the triangle inequality of a "metric" (although mathematicians typically do). 77.126.50.25 (talk) 14:38, 23 September 2012 (UTC)

For this I implicity assumed a definition of "metric" as quadratic form. Mathematicians typically require the triangle inequality of a "metric", whereas physicists don't. And in physics, the Lorentz "metric" is most certainly recognized and is used extensively. and yes, the section is misnamed, care to suggest another? — Quondum 14:08, 23 September 2012 (UTC)

See my response to your previous comment. 77.126.50.25 (talk) 14:15, 23 September 2012 (UTC)

Let's take a step back, and try to determine what you are looking for.

• We could define a quadratic form on an n-dimensional complex vector space. This will assign a complex value to each vector in this space. It could look mathematically very similar to the classical sum-of-squares in geometry, only with complex values (or the dot product on complexified vectors). It is full of null vectors, but that never bothered relativists. Regarded as a real vector space, it has a signature of (n,n).
• Alternatively, we can define a sesquilinear form (or else its associated "quadratic" form) on the same space. This will assign a real number to each vector. The end result is rather like (actually indistinguishable from) a 2n-dimensional (real) Euclidean space, with a signature of (2n,0).

I would assume from your description that the first is what you mean. There's nothing particularly recognized or not about it; it would be known as "an n-dimensional complex vector space with a nondegenerate quadratic form", or, if you're referring to the "metric" itself, "a nondegenerate quadratic form on an n-dimensional complex vector space". It trivially satisfies the Pythagorean identity for the addition of orthogonal vectors. — Quondum 15:15, 23 September 2012 (UTC)

Thank you very much, both for the information you presented - and for your patience; Yes, it seems like the first option is what I'm looking for (since the "magnitude"-function I'm looking for is supposed to assign a complex number, which - of course - may not be real). However, something still bothers me: I'm looking for a "magnitude"-function - being identical to the well celebrated Euclidean formula for calculating the length of the hypotenuse: this formula is supposed to look like a square root of sum of squares, but that formula does not have a "quadratic form", but rather a form of "a square root of a quadratic form". So are you still sure the first option may point at the correct direction for satisfying my requirement? 77.126.50.25 (talk) 18:50, 23 September 2012 (UTC)

It unfortunately does not easily lend itself to square roots, but best stays in the "squares form" a² + b² = c². You could take the square root of a complex number, at the cost of getting a two-valued complex "norm". We have for complex vectors a and b that a⊥b (by definition) when a⋅b=0. The "magnitudes squared" are a²=a⋅a and b²=b⋅b. We have then that (a+b)²=(a+b)⋅(a+b)=a²+b²+2a⋅b=a²+b². All in complex numbers. Then you can take the two-valued square root if you wish, still complex. It seems to fit, unless you wish to restrict this last to positive reals. You can get this last restriction with the sesquilinear form, but this involves all sorts of conjugates, so no true squaring in the Pythagorean expression. With more context on what is needed it could be easier to determine what you want. — Quondum 21:30, 23 September 2012 (UTC)

Thanks. Two points to be clarified:

1. You wrote in your previous response: "an n-dimensional complex vector space with a nondegenerate quadratic form...It trivially satisfies the Pythagorean identity for the addition of orthogonal vectors". Does every (nondegenerate) quadratic form (on a complex vector space) really satisfy the Pythagorean identity (for the addition of orthogonal vectors)? Aren't there (nondegenerate) quadratic forms (on a complex vector space) other than the form a² + b² = c²? I'm asking about that, because I'd like to be sure that - the verbal expression you used - really refers uniquley to what I'm looking for, that is: a legitimate/standard/known (verbal) expression/term/name, for all complex vector spaces - which have a "magnitude"-function satisfying the Pythagorean identity.

2. Why didn't mathematicians define the Euclidean space also over the complex field? Note that if they did, then the original Euclidean definition of the norm (on a real vector space) $\|\mathbf {x} \|={\sqrt {\mathbf {x} \cdot \mathbf {x} }}$ could still hold - even when one replaces the (original) real vector space by a complex vector space, except that the mathematicians wouldn't use the name "norm" (since the Euclidean definition of the norm doesn't satisfy the triangle inequality when the field is complex) but rather a similar name, e.g. "Pseudo-norm", or "length", or "magnitude", and likewise. Additionally, phisicists could apparently use the term "Euclidean space over a complex field" - without changing anything (even not the name "norm"), because they don't really care about the triangle inequality (as you've already pointed out), correct? 77.126.50.25 (talk) 12:04, 24 September 2012 (UTC)

a² + b² = c² is the addition formula relating the squares of orthogonal vectors being added; the quadratic form is $q(\mathbf {x} )=\sum _{i=1}^{n}\sum _{j=1}^{n}a_{ij}x_{i}x_{j},\mathbf {x} =(x_{1},...,x_{n})$, which is what assigns a complex number to each vector with the given complex components. Every non-degenerate quadratic form over complex vectors is equivalent: you can always diagonalize it to the form $q(\mathbf {x} )=\sum _{i=1}^{n}{x_{i}}^{2}$ by suitably choosing your basis (essentially the axes and their scaling). So, yes: for any given dimensionality, there is really only one "complex vector space with a non-degenerate quadratic form".

On your second point, you could define an ℝ-linear "norm" $\|\mathbf {x} \|={\sqrt {|q(\mathbf {x} )|}}={\sqrt {|\mathbf {x} \cdot \mathbf {x} |}}$, which might be useful in scaling, e.g. $\mathbf {x} /\|\mathbf {x} \|$ to get a "unit" vector. Alternatively, as I've said before, a ℂ-linear "norm" $\|\mathbf {x} \|={\sqrt {q(\mathbf {x} )}}=\pm {\sqrt {\mathbf {x} \cdot \mathbf {x} }}$ (with true ambiguity on the sign due to the square root of a complex number), for a similar purpose. The rest is simply a debate about naming, which is a problematic area. — Quondum 14:55, 24 September 2012 (UTC)

I feel I'm approaching (slowly) to my target, but unfortunately I haven't reached it yet. Let's put it clearer: My target is to reach a legitimate verbal expression denoting a (quadratic) space (over a complex field), with the ℂ-linear "norm" $\|\mathbf {x} \|$ satisfying: $\|\mathbf {x} \|^{2}=\mathbf {x} \cdot \mathbf {x}$. However, if I don't define the "norm" in advance, then how can I derive the ℂ-linear "norm" $\|\mathbf {x} \|^{2}=\mathbf {x} \cdot \mathbf {x}$ from the very fact that the space has a (nondegenerate) quadratic form $q(x)=a_{1}x_{1}^{2}+a_{2}x_{2}^{2}$??? Btw, if one really defines the "norm" as the ℂ-linear "norm" $\|\mathbf {x} \|$ satisfying: $\|\mathbf {x} \|^{2}=\mathbf {x} \cdot \mathbf {x}$, then one no longer needs the information telling us that the space has a quadratic form, does one? 77.126.50.25 (talk) 17:51, 24 September 2012 (UTC)

You need to define only one of the following, the other two follow naturally ("are induced") [ with derivations ]:

• A quadratic form q(x) [ = B(x,x) = ||x||² ]
• A symmetric bilinear form B(x,y) (also known as a scalar product, written x⋅y) [ = (q(x+y) − q(x) − q(y))/2 ]
• A ℂ-linear "norm", which we could (somewhat unconventionally) write ||x|| [ = ±(q(x))^1/2 ]

Most of this is in the linked articles. It is usually cleanest to start with one of the first two. The original wording I gave you suffices; anything further would amount to stating that or showing how the remaining aspects (uniqueness up to isomorphism, dot product, "norm") follow naturally from the description. — Quondum 19:17, 24 September 2012 (UTC)

I didn't quite answer "from the very fact that...". This is a little more involved, but not too complicated. It is essentially all in Quadratic form, though it is not all that clear that the equivalence is rooted in a suitable change of basis, and that complex vector spaces have exactly one case for non-degenerate quadratic forms, unlike real vector spaces, which have n+1 cases. — Quondum 19:43, 24 September 2012 (UTC)

1. Is it really natural/conventional/common/standard to regard the quadratic form as a magnitude/norm/length?

2. Why did you write "norm" with quotation marks? Was that just because this "norm" isn't really a norm - because it doesn't satisfy the triangle inequality when the field is complex?

3. You write: "You need to define only one of the following...The original wording I gave you suffices; anything further would amount to stating that...the remaining aspects...follow naturally...". Naturally, i.e. not logically/mathematically - unless you define (some of) them in advance, i.e. one could talk about a "complex vector space with a nondegenerate quadratic form", but with an (unnatural) "norm" $\|\mathbf {x} \|$ other than: $\mathbf {x} \cdot \mathbf {x}$, couldn't one?

4. Btw, how about the shorter expression "nondegenerately-quadratic space over a complex field"? 77.126.50.25 (talk) 20:54, 24 September 2012 (UTC)

1. I'm not trained in the area (especially terminology); terminology can be dizzyingly confusing. All of these would be unconventional.
2. I expect you to have problems finding a simple accepted term for the square-root-of-the-quadratic-form. Hence my quotation marks.
3. You can define any structure you like, as long as it's mathematically consistent. You could even (re)define your terminology. You could define twenty different simultaneous scalar products; it's just that when the structures are directly related (induced by each other), they are more natural, as when a particular construction leads to a related structure in a unique way (e.g. up to an isomorphism). If one needs to make an arbitrary choice, it's unnatural (or that's how I interpret this not-too-well-defined term). So "n-dimensional complex vector space with a nondegenerate quadratic form" is pretty unambiguous; if you then refer to the "naturally induced scalar product", and the "cannonical quadratic form", you'll probably be understood. I'd then perhaps use the suggestive term such as "complex magnitude" for its bivalued square root, introduced with clarification.
4. Maybe "nondegenerate quadratic complex space"? Depending on context, you could introduce the term with a more comprehensive definition to clarify exactly what you mean, possibly further shortening it (e.g. drop the "nondegenerate") if you wish. What you would want to call the length-substitute I have no idea. Judging by the length of this thread, you may get more attention from the real mathematicians by starting a new section simply for vetting a proposal on terminology. — Quondum 03:04, 25 September 2012 (UTC)

Ok, here is how I see it now, thanks to your comments: the expression "n-dimensional complex vector space with a non-degenerate quadratic form" - suffices to describe what I'm looking for, because such a space naturally induces a (complex) magnitude being the [square root of the] canonical quadratic form - what really makes the space satisfy the Pythagorean identity [gives the magnitude a form identical to the form of the Euclidean norm] (= what I need).

I preferred to say "nondegenerately-quadratic space over a complex field", because if I said "nondegenerate quadratic complex space" then it would mean that the space itself is not degenerate, i.e. that it contains more than the zero-vector, whereas by "non-degenerately" I refer - not to the space itself - but rather to its being quadratic. Anyways, "nondegenerately quadratic complex space" is really shorter. 77.126.50.25 (talk) 09:23, 25 September 2012 (UTC)

I don't agree with "a (complex) magnitude being the canonical quadratic form - what really makes the space satisfy the Pythagorean identity". What I call the complex magnitude squares to the quadratic form, so it is not the same thing. And don't confuse the cannonical quadratic form with the Pythagorean identity (though it does make seeing its relation to scaled basis vectors simpler): it does not have to be in any particular form for this identity to hold: the sum of the squares of the complex magnitudes of any set of mutually orthogonal vectors equals the square of the complex magnitude of the sum of the vectors. — Quondum 10:29, 25 September 2012 (UTC)

You could say: "... such a space naturally induces a (complex) magnitude and orthogonality condition that satisfy the Pythagorean identity". (The orthogonality condition is, of course, x⋅y=0, the dot being the naturally induced scalar product) Simpler, no? — Quondum 10:59, 25 September 2012 (UTC)

Yes, I forgot the word: "squares". I also confused the canonical quadratic form with the Pythagorean identity. I've just fixed all of that in my previous response. I also accept your last section - which takes into account the concept of orthogonality (being a concept I'm trying to ignore), but please pay attention again - also to how I see all of that (while ignoring the concept of orthogonality): The expression "n-dimensional complex vector space with a non-degenerate quadratic form" - suffices to describe what I'm looking for, because such a space naturally induces a (complex) magnitude being the square root of the canonical quadratic form - what really gives the magnitude a form identical to the form of the Euclidean norm (= what I need). 77.126.50.25 (talk) 12:58, 25 September 2012 (UTC)

Sounding good, but implies a dependence of the magnitude on the quadratic form being in canonical form; this is not the case. I would say: "... a (complex) magnitude being the square root of the quadratic form - which in canonical form results in the magnitude being identical in form to the Euclidean norm". Or even more specific: "which in canonical form results in the magnitude being the complexification of the Euclidean norm". — Quondum 16:35, 25 September 2012 (UTC)

How does it "result in" the required magnitude? Probably by multiplying it by something, right? So the expression "n-dimensional complex vector space with a non-degenerate quadratic form" still needs some additions in order to "result in a magnitude being identical in form to the Euclidean norm", unless we see that multiplication - as "natural", so we no longer need the "canonical" part, and we can say the following:

• The expression "n-dimensional complex vector space with a non-degenerate quadratic form" - suffices to describe what I'm looking for, because such a space naturally induces a (complex) magnitude being identical in form to the Euclidean norm (= what I need).

Or even simpler:

• The magnitude function I'm looking for is the complexification of the Euclidean norm. i.e. the space I'm looking for is the complex vector space whose magnitude function is the complexification of the Euclidean norm.

Would it be correct to say that? 77.126.50.25 (talk) 18:23, 26 September 2012 (UTC)

My bad; I was thinking of the Euclidean norm for an orthonormal basis; quite unnecessary. But yes, I can't quibble with your phrasing now. — Quondum 19:46, 26 September 2012 (UTC)

Ok, so I prefer your last suggestion about the "magnitude function being the complexification of the Euclidean norm", because it points exactly at what I need - without having to rely on a "natural inducement", as opposed to your first suggestion - involving the quadratic form. 77.126.50.25 (talk) 22:17, 26 September 2012 (UTC)

Cantor set and compactness

The article on the Cantor set says that every compact metric space is the continuous image of the Cantor set. If I do not misunderstand it means that for every compact space X there is a continuous onto function with domain the Cantor set and range X. But won't the cardinality of X be at most the continuum as if f is an onto function from A to B then |B|<=|A|? What if X has cardinality more then the continuum?-Shahab (talk) 15:42, 23 September 2012 (UTC)

We can show that there are no compact metric spaces with cardinality larger than the continuum. (This is off the top of my head, so I apologize if it is ugly:-)) First, we show that any second countable metric space has cardinality no larger than the continuum, then we show that every compact metric space is second countable. Let X be a second countable metric space and D a countable dense subset. Since X is hausdorff, x is in the closure of D iff there is a sequence in D limiting to x; thus, since the closure of D is X, there is an onto map from sequences in D that have a limit in X and points in X. There are at most continuum many such sequences, thus X cannot have cardinality larger than the continuum. Now, let Y be compact metric, we show Y is second countable. Let U(n) be the set of all open balls of radius 1/n; clearly, U(n) is a cover of Y, so a finite subcover F(n) exists. Pick a center for each ball in F(n) and call this set C(n), we'll show that the union C of all the C(n) is dense (it's obviously countable). Let y be any point in Y, since y is in some ball in F(n) there is a point y(n) in C(n) that is no further than 1/n from y; clearly, y(n) limits to y, thus, C is dense.Phoenixia1177 (talk) 19:57, 23 September 2012 (UTC)

Number of partitions

How does it come that math packages (for example, sagemath) can calculate the number of partitions in less than a sec? And that's apply even to numbers like 34349834 or like that. How is it so effective?Ptg93 (talk) 22:54, 23 September 2012 (UTC)

Modern computers can literally make billions of calculations per second, and sufficiently fast approximation formulas for the partition function are known, and indeed are discussed in the very article you linked to in your question, "partition (number theory)". —SeekingAnswers (reply) 05:58, 24 September 2012 (UTC)

Hard graft by programmers also counts. Even something like multiplication can involve involve implementing lots of complicated algorithms like the Schönhage–Strassen algorithm. Dmcq (talk) 08:35, 24 September 2012 (UTC)

I didn't find a satisfactory answer in the article. I don't know about Sage, but Mathematica can find the number of partitions of $10^{8}$ in 4 seconds, exactly apparently. The approximations are approximations, and the exact methods described seem to scale quadratically, meaning $10^{8}$ would take weeks even with billions of calculations per second.

So what algorithm is used? What is its runtime complexity? -- Meni Rosenfeld (talk) 09:59, 24 September 2012 (UTC)

There is a good description of what's done at Sage Days 35: Algorithms in Number Theory and FLINT - Fast Special Functions which describes the process with lots of detail. That is for Sage but Mathematica is basically the same on this. It isn't straightforward even with a fast machine to get it done so quickly. Dmcq (talk) 11:42, 24 September 2012 (UTC)

Hm, I was going to look at Mathematica's implementation notes after playing around a bit, but then it slipped my mind. For reference, it says "PartitionsP[n] uses Euler's pentagonal formula for small n and the non-recursive Hardy-Ramanujan-Rademacher method for larger n."

Fredrick Johansson, didn't he use to hang out around here? -- Meni Rosenfeld (talk) 12:49, 24 September 2012 (UTC)

Just for reference for those who would like more context/information:

• Euler's pentagonal formula is discussed in detail in the article "pentagonal number theorem", and I just added a note about it to the "partition (number theory)" article, which previously lacked any mention of the formula.
• The Hardy-Ramanujan-Rademacher method is discussed in the article "partition (number theory)".
• Fredrik Johansson (not Fredrick Johansson) is one of the programmers of Sage.

—SeekingAnswers (reply) 19:25, 24 September 2012 (UTC)

Probably User:Fredrik, but it seems they abandoned the place two years ago unfortunately. Dmcq (talk) 13:34, 24 September 2012 (UTC)

The code for Sage is at [1] and the original author is given as Jonathan Bober. Who would have thought the best way to compute it would involve pi, sqrt, sin, cos, cosh and sinh? Dmcq (talk) 14:41, 24 September 2012 (UTC)

No longer active here, though I make occasional edits without bothering to log in. If you want even more detail on computing the partition function, see http://arxiv.org/abs/1205.5991. I've thought about updating partition (number theory) with my results, but I don't particularly like self-promotion. Fredrik Johansson 04:57, 30 September 2012 (UTC)