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January 14

Nets

What does the net of a sphere look like? Sciencefordummies 03:20, 14 January 2007 (UTC)[reply]

What notion of net are you referring to? If you mean a net of a polyhedron, then a sphere is not a polyhedron, so the concept does not apply. (Although the sphere is homeomorphic to a polyhedron, this notion of net is not topological.) If you mean a net in a topological space, it might make sense to consider nets in a sphere, but then there are many nets and you would have to be more specific which net you mean, and perhaps also explain what you mean in this context by "look like". Finally, if you mean an ε-net, we need to know what probability distribution(s) on the sphere you have in mind. Or is this still some other notion of net? --Lambiam ^Talk 07:17, 14 January 2007 (UTC)[reply]

Thanks for the help (I meant a net in a topological space) Sciencefordummies 19:06, 14 January 2007 (UTC)[reply]

Like the latitude and longitude lines for the Earth ? StuRat 07:38, 15 January 2007 (UTC)[reply]

That's one of them, yeah. Basically, if you peel an orange (and manage to keep the skin intact in some manner all the way through), that topmost layer of skin is, in some degree, a net. (Not a polyhedral net, of course, because of the absence of any falt surfaces.) --JB Adder | Talk 02:55, 16 January 2007 (UTC)[reply]

Behaviour of parametric functions

Hello,

In reference to Area inside various curves, how does one know, with just the equation of a function, the behaviour of a function ? For example, how would one know there is a loop ? I suppose a crunode would be a clue to a loop but isn't sufficient. I looked at the parametric equation page, but it didn't give much detail. Of course, plotting a curve of the function would tell where to look (but in a non rigorous manner), but how would it be done without such a graph ? I suppose horizontal and vertical tangents are another clue, but well, aren't there some mathematical entities that could help ? I know of the winding number but I can't think of a way to use it... Furthermore, is there any way the the answer to the previous questions would also help to be able to invent equations that would describe certain curves (for example, knowing what must be present in the equation for there to be a loop (much like knowing that there must be a horizontal tangent for there to be a double root (and vice-versa), for functions of the type y=f(x)).

Thanks. --Xedi 14:38, 14 January 2007 (UTC)[reply]

Unless we restrict the kind of functions used for parameterization, this will be tough. When a curve crosses itself the same point occurs two (or more!) times with different tangents. For algebraic curves, the implicit form helps. Otherwise, "continuous function" without further restriction includes some extremely wild behaviors. --KSmrq^T 23:52, 14 January 2007 (UTC)[reply]

Well, would the assumptions of contitinuity and derivability help much ? Is there a simple way of calculating the turning number ? I mean, what are the methods in this domain ? How would the mathematicians that didn't have access to computer visualisation do ? (Of course the intuition of great mathematicians served as a "replacement"). Aren't there special things to look for in an equation that woud say if there is a loop ? For example, derivative of the x part that changes sign twice is needed for a loop... But anything more precise ? I don't know what assumptions are needed for the problem to be easier... just what would be a general method of knowing what a parametric function looks like without a graph of it ?

Thanks ! --Xedi 16:44, 15 January 2007 (UTC)[reply]

If you want full automation and guaranteed non-incorrectness, Lipschitz continuity of the derivatives of the functions may help. Without bounds on the wildness, you can only hope and pray. Since the winding number is an integer, if you have a family of parametrically represented curves indexed by a parameter, (like the family {<x = f(a, t), y = g(a, t)> for t in [0, 1]}_a indexed by a), the winding number as a function of the indexing parameter is discontinuous; therefore any putative algorithm for determining it must be non-terminating, or incorrect, or else incomplete (sometimes "giving up" instead of returning a result). Because of that I used "non-incorrectness" instead of "correctness". If you are willing to accept an occasional "I give up", you can construct a representation of a path by sampling for sufficiently small increments of the path parameter, building it up from finite segments. Using the Lipschitz condition you can determine bounds on the change of direction to check for a guarantee that the curve does not sneakily curve back on itself within one path segment; if the check fails, decrease the increment and try again. The condition also gives you a corridor within which the actual path must remain, which you can use to establish crossings or to check for a guarantee of non-crossing. Working this out correctly will require quite some effort and determination, and the question is if your application is worth it. If correctness is not vitally important, you can just take small steps and hope the curve is sufficiently tame. If it is truly vitally important, you may also want to guard against the effect of inexact floating-point arithmetic, for instance by using interval arithmetic. --Lambiam ^Talk 22:17, 15 January 2007 (UTC)[reply]

Sorry, but I wasn't talking about finding an algorithm, I was more thinking of how to be able to find out how a parametric function behaves without visualization, for example with only a pen and paper. I don't know what conditions are to be fulfilled for this process to be reasonable simple (and would like to know) but I would like not to take into account just complexity of the formulas involved (I mean, if the only limitation to finding out is to have to calculate a (very) hard integral, I don't mind). I understand, as you said, that such a process is very dependent on what the function itself is and a generalization is near impossible, but maybe some conditions may suffice to make the process only limited by complexity of calculations (an, albeit very simple, example, would be of the curve at crunode : the crunode itself indicates self intersection, with the continuity and the vertical tangent line at x=1, it implies a loop (well, sort of)). As such, if I stumble upon a parametric equation (or implicit cartesian equation too), with only pen and paper (and maybe ability to calculate about all integrals), how can I "guess" the behaviour of the function (surely some assumptions are to be made on the nature of the function...) ? (Another example would be the curve y²(1+x) = x²(3-x), there is a loop, how would I prove it ?)

(By the way, isn't there a theory related to all this ? Something I could read about ? I really can't find much...)

Still, thanks again for your help. Maybe after all it's too hard to generalize effectively. --Xedi 22:50, 15 January 2007 (UTC)[reply]

You could examine the turning points, i.e. points where the tangent is horizontal or vertical. Once you know the turning points you can aproximate the curve by a piecewise linear curve which becomes more tractable for some aplications. --Salix alba (talk)

I don't understand what you mean by being able to approximate the curve by a piecewise linear curve by knowing the turning points. I suppose knowing the turning points helps being able to decompose the function in functions of the type y=f(x) (or x=f(x)) and the curve definitely becomes more easy to approach, but why linear curve approximations ? --Xedi 22:55, 15 January 2007 (UTC)[reply]

If you can break the curve into segments so that the x and y coordinates are monotonically increasing or decreasing in each segment, that tells you something. It also gives you a set of boxes in which each segment is contained, i.e. a box where the two turning points are opposite corners. If none of the boxes intersect then the curve can't intersect. Your close to shifting the problem from a continuous to a discret graph theoretic one. --Salix alba (talk) 01:47, 16 January 2007 (UTC)[reply]

Degree might help. If the curve of a polynomial in both coordinates then you know the curve is not closed, but it dies tell you something about how many time the curve can self intersect, I think degree 2 curves can't intersect, and degree three curves can only have one intersection. Interesting side question is their a formula for the number of times a degree-n curve can self-intersect? --Salix alba (talk) 22:54, 15 January 2007 (UTC)[reply]

Sadly none of the degree articles talk about parametric/implicit functions... I suppose the intersection depends of the degree of the polynomial because the number of times the derivative can change sign depends on the degree of the polynomial. I would have thought that a degree 3 curve could only intersect once, as for a loop a coordinate needs to increase, decrease then increase. But then for higher degree, I don't know. Thanks. --Xedi 23:16, 15 January 2007 (UTC)[reply]

Degree of a continuous mapping talks about winding number, its actually a very deep concept with a lot of alegebraic topology behind it.

Another fun way of approaching the problem is through singularity theory. Consider the polynomial which just the highest degree terms, these will generally have a singularity at the origin, which may be one of crunode, acnode, cusp, Tacnode or many other higher degree forms (most of which are known). You can then look at the unfolding of the singularity to see the way it breaks up, you unfold a singularity by adding some of the lower degree terms. Quite a fun example is (t^4+a t^2,t^3+b t), which illustrates the dificulty of the problem, taking a=-1 and various values of b you can have 0, 1 or 3 intersections, and cases with, one cusp, one intersection and two cusps, a tacnode, other values of a and b can give a tacnode and an intersection, and probably quite a bit besides, a=0, b=0 gives a quite high level singularity whos names escapes me (possibly butterfly). --Salix alba (talk) 01:47, 16 January 2007 (UTC)[reply]

Integration in Maxima

I'm experimenting with open-source computer algebra systems, and am trying out Maxima. I've encountered a problem when doing symbolic integration. The value returned from integrate, although displayed as an algebraic expression, behaves differently from the same expression entered by hand. Here is an example of what is going on:

f(x):=45^(-x^2);

$\,f\left(x\right):={45}^{-{x}^{2}}$

F(x):=integrate(f(x),x);

$\,F(x):=\operatorname {integrate} (f(x),x)$

F(x);

${\frac {{\sqrt {\pi }}\,\operatorname {erf} \left({\sqrt {\log \left(5\right)+2\,\log \left(3\right)}}\,x\right)}{2\,{\sqrt {\log \left(5\right)+2\,\log \left(3\right)}}}}$

// Seems reasonable...

// Now I'd like to evaluate F(x) for a specific value of x

F(5);

Attempt to integrate wrt a number: 5#0: F(x=5)

-- an error. To debug this try debugmode(true);

// Hmm, why doesn't it simply substitute x=5?

// Ok, lets try defining G(x) = F(x) from scratch, and see what happens...

G(x):=sqrt(%pi)*erf(sqrt(log(5)+2*log(3))*x)/(2*sqrt(log(5)+2*log(3)));

${\frac {{\sqrt {\pi }}\,\operatorname {erf} \left({\sqrt {\log \left(5\right)+2\,\log \left(3\right)}}\,x\right)}{2\,{\sqrt {\log \left(5\right)+2\,\log \left(3\right)}}}}$

G(5);

${\frac {{\sqrt {\pi }}\,\operatorname {erf} \left(5\,{\sqrt {\log \left(5\right)+2\,\log \left(3\right)}}\right)}{2\,{\sqrt {\log \left(5\right)+2\,\log \left(3\right)}}}}$

float(G(5));

$\,0.45422679973746$

F(t)/G(t);

$\,1$

F(5)/G(5);

Attempt to integrate wrt a number: 5#0: F(x=5)

-- an error. To debug this try debugmode(true);

My question is this: How do I make F(x) behave as though it had been entered by hand. --NorwegianBlue^talk 19:19, 14 January 2007 (UTC)[reply]

When you wrote F(x):=integrate(f(x),x) you intended the right side to be evaluated before the function was defined, but that is not what you actually said. So what is happening is that F(5) means integrate(f(5),5) and the error is exactly what you would expect. Try the quote-quote input:

F(x):=''(integrate(f(x),x));

This forces the integral to be evaluated before the function is defined. --KSmrq^T 23:36, 14 January 2007 (UTC)[reply]

Yes, that fixed it. Thanks! --NorwegianBlue^talk 20:59, 15 January 2007 (UTC)[reply]