Wikipedia:Reference desk/Archives/Mathematics/2008 September 18

= September 18 =

Formula for planet temperature
I'm having trouble converting this formula from "FUNDAMENTALS OF ASTRONOMY" page 318 (QB43.3.B37 2006) by Dr.Cesare Barbieri, professor of Astronomy at the university of Pauda, Italy. It is not simplified for the Sun/Earth, it is the full formula with data for Gliese 581/581 c. Can someone help?

=(((((0.0000000567051)*(3840^4))/(4*PI*((0.0613*149597876600)^2))) * ((4*PI*((0.29*695500000)^2))/(4*PI*((11162)^2)))*((PI*((11162)^2)*(1-0.64))))/0.0000000567051)^0.25

$$T_p= \frac {(5.67051E-8)*(3840^4)} {(4*pi*(0.0613*1AU)^2)} *\ {{4*pi*(0.29*{R_\odot})^2} {(4*pi*(R)^2}} {pi*(R)^2*(1-0.296){} 0.0000000567051)^0.25 }$$

--GabrielVelasquez (talk) 04:09, 18 September 2008 (UTC)


 * I think you will have to provide a few more details here. You have written two different expressions - which one (if either) is correct ? And in what way are you trying to "convert" the formula - what are you converying from and to ? Gandalf61 (talk) 08:59, 18 September 2008 (UTC)


 * I found the reference on Google books. You appear to be trying to use the second equation on the page, which looks like this (rewriting quantities that refer to the Sun to refer to a general star e.g R⊙ becomes R∗ and marking those which refer to the planet explicitly, e.g. T becomes Tp):
 * $$\sigma\cdot T_{\mathrm{p}}^4 = \frac{\sigma\cdot T_{\ast}^4}{4\pi\cdot d^2}\cdot\frac{4\pi\cdot R_{\ast}^2}{4\pi\cdot R_{\mathrm{p}}^2}\cdot\pi\cdot R_{\mathrm{p}}^2(1-A_{\mathrm{p}})$$
 * However I don't see why you are trying to use this equation, as many of the terms in it cancel each other out, making it unnecessarily cumbersome to work with. In fact it is rearranged on the next line as follows:
 * $$T_{\mathrm{p}} = T_{\ast}\left[ \left(\frac{R_{\ast}}{d}\right)^2 \frac{1-A_{\mathrm{p}}}{4} \right]^{\frac{1}{4}}$$
 * This expression is far easier to work with (note for example that the radius of the planet cancels out) and gives exactly the same results! If you want to try it for yourself, I recommend Google calculator, which helpfully includes various units conversions and the constant r_sun, so for a planet orbiting, say, 0.1 AU from a 3500 K star with a radius 0.3 times that of the Sun and a bolometric albedo of 0.5, you could do a Google search for: 3500 K*((0.3*r_sun/0.1 AU)^2*(1-0.5)/4)^(1/4). Icalanise (talk) 12:53, 18 September 2008 (UTC)

Both of those formulas are in the book I mentioned, I just had trouble with the second divided terms in the code, thanks. I want the full version because I have seen other scientists use a different formula that looks wrong in comparison, but I know Selsis et al at Gliese 581 c simplifed it to factor out the Luminosity variables (L=4·pi·R^2·a·T^4)and it just looks wrong if you don't have the full version. GabrielVelasquez (talk) 21:16, 20 September 2008 (UTC)

Finding an average flood height overtime
dates of floods                  height of each flood 1965..........................   5.22 1972..........................    5.14 1993.........................     6.31 1997...........................   6.13

From this information iam required to find the average height of these floods for a 1 in 10 year event and a 1 in 100 year event

please help thankyou

if you need to email me it  the 0 is a zero —Preceding unsigned comment added by 124.170.38.135 (talk) 04:51, 18 September 2008 (UTC)


 * To find the average of values, just add them up, and divide by the number there are. For example, the average of 1, 2, 3, 4, and 5 is 3 because 1+2+3+4+5=15 and 15/5=3.  Ζρς ι'β' ¡hábleme! 21:53, 18 September 2008 (UTC)


 * Probably the best thing to do for real world problems is ask someone who deals with this in practice. Both the cases of flooding by the sea and of inland areas are quite complicated. You really don't have enough figures to come to any sort of even inaccurate conclusion without some model to plug them into. If you just want to use flood heights with nothing else you'd need many more figures from lower level floods so you can fit a probability distribution to them. Your figures above would be some way along one side and highly skewed and there's too few to see how quickly they go down. And anyway Global warming will make a mess of any figure just derived from historical data. Personally I'm rather surprised at the difference between 1972 and 1993 - I'd guess someone changed the environment of the area between the two. Dmcq (talk) 18:28, 20 September 2008 (UTC)
 * Even if the trend is towards higher floods, there is still going to be random fluctuations from year to year, it's not at all surprising that a later flood was a little lower than an earlier one. --Tango (talk) 19:51, 20 September 2008 (UTC)

Word for the "terms" in a union
Is there a word along the lines of "summand", "integrand", "radicand" for the individual "terms" in a union? For example, in $$A\cup B\cup C$$, what word describes the role of $$A$$, $$B$$, and $$C$$? Is "term" acceptable, or is there a niftier word like "unand" or "unitand"? While we're at it, what about intersections? —Bkell (talk) 05:01, 18 September 2008 (UTC)


 * I think terms in acceptable. As far as I know, there is no particular word associated to this. Thenub314 (talk) 07:31, 18 September 2008 (UTC)


 * I also have never heard such a specific term. If you wanted to invent one, then some people call union set summation, so you could call them summands. Or you could use the analogy with the logical OR operator and call them disjuncts. Algebraist 11:00, 18 September 2008 (UTC)

0 dimensional object in 4 dimensional world
In a tesseract, if the cube (3d) serves the function analogous to a 2d surface for a cube, and the plane serves the function analogous to the edge of a cube in 3d, what would be the status of a point(0 dimensional) in regard to four dimensions? It couldn't be the -1st dimension, could it?Leif edling (talk) 07:03, 18 September 2008 (UTC)


 * The analogy breaks down, or was never really more then an analogy. In the 4 dimensional cube you gain one more thing then you had before, you don't just translate all the things from a 3 dimensional cube. So in a 3 dimensional cube you have squares (2 dimensional cubes), line segments (1 dimensional cubes, this is a bit degenerate), and points (0 dimensional cubes, very degenerate).  In a 4 dimensional cube there are, 3 dimensional cubes, squares, line segments and points. Thenub314 (talk) 07:40, 18 September 2008 (UTC)


 * Yeah, because if you stepped into 4d land, you'd have almost no mass and no weight, because you are infinitesimal thin. See Paper mario that would be you. Sentriclecub (talk) 11:48, 18 September 2008 (UTC)


 * My second answer, is just like a point in 3d. A point is so thin, so small, that even though I can look on a map and see a point 37 degrees noth, 48 degrees east.  If I go to that actual location in the world, I still would not be able to even begin to see the "point" which exists, its smaller even the planck lengths.  So my answer is that its just the same as in a 3d world.  Remember, a point has no volume, so as soon as you give it a relative size, you instantly defined a cube (which is a 3d structure, which has volume). Sentriclecub (talk) 11:53, 18 September 2008 (UTC) My third answer is that it is a line, because given my second answer, that would say that a cube in 3d is not analogous to a tesseract (as you actually asked--I failed your original question) because a cube can exist in 4d just as a sheet of paper (to me at least) represents a 2D plane in my 3d world.

I read this "analogy" perspective:

Cells, Ridges, Edges: The upshot of all this is that in 4D, objects have a much richer structure than in 3D. In 3D, a polyhedron like the cube has vertices, edges, and faces, and fill a 3D volume. The cube is bounded by faces, which are 2D. Every pair of faces meet at an edge, which is 1D, and edges meet at vertices, which are 0D.

In 4D, objects like the hypercube not only has vertices, edges, and faces, but also cells. A 2D boundary is insufficient to bound a 4D object. Instead, 4D objects are bounded by 3D cells. Each pair of cells meet not at edges, but at 2D faces, also called ridges. The ridges themselves meet at edges, and edges meet at vertices.

The point here is that in 4D, 3D volumes play the role analogous to surfaces in 3D, and 2D ridges play the role analogous to edges. Because of this, it is important to visualize 4D objects by thinking in terms of bounding volumes, and not 2D surfaces. A 2D surface only covers the equivalent area of a thin string in 4D! When you see a 2D surface in the projection of a 4D image, you should understand that it is only a ridge, and not a bounding surface.

I got it here:

So, do the above answers signify that a tesseract does not have 2d planes analogous to the edges of 3d cube? (As is said in the last paragraph above)Leif edling (talk) 15:49, 18 September 2008 (UTC)

Elliptic Curves Isomorphism
What is an isomorphism when it comes to elliptic curves? I have an idea of what it is based on reading about it in two different books, but neither seems very clear to me.

Knapp's book "Elliptic Curves", which I don't have with me so I can't tell you exactly what it says, basically says two elliptic curves are isomorphic if they are related by an admissible change of variables. That's the definition of isomorphic. But, it never says anything about what that means. I assume it means the group structures are the same. But, is that all it means?

The other book I looked at is Dale Husemoller's "Elliptic Curves". Again, I don't have it with me, but I believe one theorem says two elliptic curves are isomorphic if and only if they have the same j-invariant. This is why I am not entirely sure on the isomorphism meaning only group structure. If this is what it means, then since there are an infinite number of j-invariants, there must be an infinite number of elliptic curve groups possible. The problem is Knapp's book doesn't seem very clear on this. It says only that elliptic curves for ranks up to 12 are known but a fact like there are an infinite number of such groups means clearly elliptic curves of much higher rank exist (higher than any given number).

Can any one help me understand this better? Thanks. StatisticsMan (talk) 13:46, 18 September 2008 (UTC)
 * You seem to be a bit confused about what base field (or whatever) you are working over. If one has a change of variables making two curves isomorphic, the group structure will be the same, sure. But if I understand your question correctly, when you say "the group structures are the same" this means that points over some field form groups which are isomorphic. But this is not enough to say that the curves are necessarily isomorphic.  If one looks at things over the complex numbers, all elliptic curve groups are isomorphic as abstract groups, or even as real Lie groups, being tori, the product of two circle groups.  (But they aren't isomorphic as complex Lie groups / Riemann surfaces unless they have the same j-invariant.)


 * When Knapp? says that elliptic curves with ranks up to 12 are known, he is talking about the Q-rational points on the curve as a finitely generated abelian group (cf Mordell–Weil theorem). Again, this fga group does not characterize the curve.  The infinitude of different groups doesn't follow from the infinitude of j-invariants, as many curves could have the same group of Q-rational points.  A theorem of Barry Mazur shows that there are only finitely many possible torsion subgroups of the Q-points, so  the possible infinitude of different groups must come from there being (possibly) infinitely many possible ranks, but as Knapp says, only ranks up to 12 are known.John Z (talk) 01:34, 19 September 2008 (UTC)


 * Well, you are correct that I meant over the rationals, so I apologize for not saying that. But, I still do not think I understand what an isomorphism for elliptic curves over Q really means.  You told me that it's not just the group.  Can you please tell me what it is since the books I have read don't bother to explain this?  My question was based on Mazur's theorem which says there are only 15 possible torsion subgroups of Q-points.  So, since I do not understand what an isomorphism means, I made a guess that it meant only the group.  If it means only the group, it follows there must be an infinite number of different ranks.  But, Knapp says only up to 12 are known.  So, it's either not just the group or it is and his statement does not say all it should. StatisticsMan (talk) 15:24, 19 September 2008 (UTC)


 * I'm not familiar with this stuff at all, but it probably means they're isomorphic as algebraic varieties. Algebraist 22:03, 19 September 2008 (UTC)
 * Yes, that is all that it is, an invertible morphism of varieties. In StatisticsMan's words "related by an admissible change of variables" - we just need to change variables in an invertible way. An elliptic curve may be given by an equation like y^2=x^3+ax^2+bx+c. If we did a simple invertible change of variable, say replacing x by x+1, we get an equation of the same type that thus gives an isomorphic curve embedded trivially differently in the plane.  The j-invariant can be defined algebraically from the coefficients (a,b,c) of such an equation and is invariant under any such coordinate change. An isomorphism of the groups of rational points of two different curves with different j-invariants is not necessarily induced by an algebraic change of coordinates relating the two curves.  With elliptic curves it is practical to see modern algebraic geometry at work in a very concrete fashion. For this one might want to look at the lucid paper of John Tate, The Arithmetic of Elliptic Curves, Inventiones Math.(23) 1974, 179-206, which later expository works on elliptic curves owe much to.  Things like Mazur's theorem are much harder.John Z (talk) 06:10, 20 September 2008 (UTC)


 * So, just to be clear, we are saying that curves with isomorphic groups may have different j-invariants, so they are not isomorphic as curves, even thought their groups are isomorphic. So, although it is conjectured that there is no upper limit on the possible rank of a curve group, the fact that we can construct curves with an infinite number of different j-invariants does not prove this conjecture.
 * Incidentally, the search for curves whose groups have high rank has moved on a bit - this page says that the highest rank of an elliptic curve whose group's rank is known exactly is 18, and the highest known lower limit on a curve's group's rank is 28 (i.e. there is a curve whose group's rank is known to be at least 28, but it may be higher). Both examples were found by Noam Elkies in 2006. Gandalf61 (talk) 09:04, 20 September 2008 (UTC)


 * Alright, thanks a lot everyone. This has been very helpful. I appreciate that you took the time to answer my question.StatisticsMan (talk) 15:15, 20 September 2008 (UTC)

Is there any limitation to defining all x in Q as...
$$x = \frac{m}{n}$$, where $$n, m \in \mathbb{Z}$$ and $$n > 0 \,$$? I do not mean we need to show that Q is a field, but I have seen other "loose" definitions such as $$n, m \in \mathbb{Z}, n \not = 0$$ and I just thought that having n and m as negative was redundant, since we only need the numerator, m to be in Z.


 * Yes, a common way of doing this is defining $$\mathbb{Q}$$ as the set of equivalence classes of these formal fractions $$\frac{m}{n}$$, defining two fractions as equal if $$m_1 n_2 = m_2 n_1$$. When doing this, it does not matter at all if you require $$n$$ to be positive or just non-zero. The reason for choosing just non-zero is probably that then the same process will work for non-ordered domains. In other words: for constructing rationals from integers, your approach is just as good; but it's less adaptable. -- Jao (talk) 16:30, 18 September 2008 (UTC)
 * One use for the signs of the integers in computational work is to specify that $$(-a)/b$$ and $$a/b$$ (ie not $$a/(-b)$$ and $$(-a)/(-b)$$) mean that the fraction is not known to be in its lowest terms. It's a nice convenient flag that means "you do not need to try and reduce this fraction" Robinh (talk) 07:03, 19 September 2008 (UTC)

Algebra
What are some applications of groups, either to other mathematical structures or to the real world? Other than a little bit of Galois theory and a mention of homology, my textbooks haven't given much indication of what it's for. Black Carrot (talk) 22:49, 18 September 2008 (UTC)


 * Quantum chromodynamics, The Standard Model, and related areas of particle physics are littered with group theory. Dragons flight (talk) 22:52, 18 September 2008 (UTC)


 * Group theory is used extensively in chemistry, too. I don't know very much chemistry, but I believe groups are used in crystallography at least. Perhaps a chemist will come along and can provide more information. Another critically important application of group theory is the analysis of the Rubik's Cube. ;-) —Bkell (talk) 04:29, 19 September 2008 (UTC)


 * Correct about chemistry/crystallography. Spectroscopy as well.  The types of events that give rise to spectroscopic signals are determined by group theoretical properties of the molecule's symmetry.  See the book mentioned here for several such applications.  Baccyak4H (Yak!) 16:20, 19 September 2008 (UTC)


 * Groups have lots of applications in functional analysis, harmonic analysis and algebraic topology (although vector spaces are also used).

Topology Expert (talk) 05:51, 20 September 2008 (UTC)


 * Elliptic curve cryptography. Gandalf61 (talk) 08:51, 19 September 2008 (UTC)