Wikipedia:Reference desk/Archives/Mathematics/2009 April 6

= April 6 =

What is this called by definition
What is it called when two ratios are set equal to each other, used to compare 2 different things;make sure to write the words first.


 * I'm not really sure what you are looking for. Perhaps the answer is in one of the Proportionality (mathematics), Cross-multiplication, or Correlation articles.  If not, more details or an example might help. -- Tcncv (talk) 04:15, 6 April 2009 (UTC)

Maximal subgroups of a group
Hi - I'm trying to prove that if a finite group has only one maximal subgroup then its order must be the power of a prime. Unfortunately, I've been stuck on it for sometime. I've tried all sorts of arguments - first I noted that if a group has only one maximal subgroup, that maximal subgroup must be normal. Therefore, the quotient of the group by this maximal subgroup must have prime order p. I'm pretty sure the order of the group is a power of this particular p but I can't prove it. I thought maybe that I could show that there is only one maximal subgroup of the maximal subgroup of the group (and I'm sure that this is true) but it didn't work. The converse seems easy but this is hard. I also tried an inductive argument - assume that this is true for all groups having order less than n and prove it for any group having order n. By taking quotients I could have possibly exploited the hypothesis but even that didn't work. Could someone please help me? —Preceding unsigned comment added by 58.161.138.117 (talk) 06:30, 6 April 2009 (UTC)


 * By "one maximal subgroup", do you mean that there exists a proper subgroup H that contains all proper subgroups, or do you mean that there exists a unique proper subgroup H with maximal order? If the latter, try finding a counterexample.  If the former, use Sylow subgroups.  Eric.  131.215.159.99 (talk) 07:15, 6 April 2009 (UTC)
 * Thanks Eric but this problem is actually from chapter 2 of my textbook and Sylow theory comes at chapter 5. What is covered in chapter 2 is normarlizers, products of subgroups, normal subgroups, subgroups, centralizers of subgroups and cyclic groups. I am sure that the problem can be solved using only this theory. P.S I mean the former. —Preceding unsigned comment added by 58.161.138.117 (talk) 07:23, 6 April 2009 (UTC)
 * Ok, I found a way without using Sylow subgroups. Let G be our group and H the unique maximal proper subgroup.  Pick any element x outside of H.  What do we know about x?  We learn something essential about G.  Using this fact about G, and that G has a unique maximal proper subgroup, the result follows directly.  (In this approach we don't use any of the facts that you have determined above about H.  Normal subgroups, normalizers, centralizers, and quotients groups are all irrelevant.)  Eric.  131.215.159.99 (talk) 09:04, 6 April 2009 (UTC)
 * Thankyou very much! I worked out that if x is outside H, x must generate G. This is because (I presume), every proper subgroup of G is contained in H and therefore so is the cyclic group generated by x unless of course this group is not a proper one of G. Therefore, G is cyclic, and the result is trivial for cyclic groups having one maximal subgroup (in fact, I got the intuition that this was true from the example of cyclic groups!). Thankyou so much - I've been stuck on this for ages. —Preceding unsigned comment added by 58.161.138.117 (talk) 10:10, 6 April 2009 (UTC)

Divergence and Infinity
I have always felt (and i hope its true) that sequences and limits are so very similar: you have a similar definition, logic, theorems, everything. My doubt is this. Say there exists a sequence of integers 2^n,where n is a whole number. We say this sequence diverges, for large values of n.Now consider the function 2^x. Now, as x tends to infinity, should we say the limit of this function exists and is infinity, or should we say it diverges, or does not exist? My maths teacher told me it exists but is finite, but it doesn't make sense if you call the function going to infinity has a limit but a sequence going to infinity as divergent. Either you should call both convergent, or both divergent. I don't know if my teacher is right on this point, so i am asking the question here. —Preceding unsigned comment added by Rkr1991 (talk • contribs) 13:45, 6 April 2009 (UTC)
 * I suppose you are investigating the relationship that exists between the "limit of a function" and the "limit of a sequence". Well in fact there is a relationship. If the limit as x -> a of a function f(x) is p, then for any sequence xn converging to a, the limit as n -> posinf of f(xn) is p. With regards to the validity of the assertion made by your teacher, it is totally incorrect. The limit of the quoted function as x -> posinf does not exist and therefore cannot be finite. One can only speak of divergence with regards to sequences (and not functions (unless of course you are pedantic about it)). If you want to lurk deeper into this theory, you must consider first countable spaces which is a concept in topology. -- PS T  14:24, 6 April 2009 (UTC)


 * Within the real numbers, the limit does not exist in either case. For convenience, for fun, and to streamline arguments, the numbers infinity and negative infinity are introduced to make the extended real numbers, where the limit is infinity in both cases. Limits in terms of sequences and limits of continuous functions are usually very similar. Black Carrot (talk) 14:14, 6 April 2009 (UTC)


 * Also, "diverges" and "limit does not exist" are the same thing as far as I know. Black Carrot (talk) 14:15, 6 April 2009 (UTC)
 * This does depend on the space you consider. In the extended real numbers (as you mentioned), the sequence defined by xn = 1 if n is even and -1 otherwise, for all natural numbers n, diverges. However, the sequence defined by xn = n for all natural numbers n converges (because we are considering the extended real number system). Therefore, there is a difference depending on context. On a different note, every sequence is in fact a function with domain the set of all natural numbers. -- PS T  14:24, 6 April 2009 (UTC)


 * Was it really necessary to insert your comments both before and after mine? Would it have hurt that much to just follow chronological order? Black Carrot (talk) 14:49, 6 April 2009 (UTC)
 * A point-set topological neighbourhood. --pma (talk) 19:59, 6 April 2009 (UTC)
 * I have never really understood why Wikipedia has to follow the lower limit topology. -- PS T  12:41, 7 April 2009 (UTC)
 * it this to because way more --pma (talk) 09:15, 10 April 2009 (UTC) easier write read is ordered and People way


 * I use "diverge" and "limit does not exist" as synonyms. Otherwise, we would be left with the sticky problem of needing a verb for limits that neither converge nor diverge to do.  Eric.  131.215.159.99 (talk) 20:23, 6 April 2009 (UTC)
 * As a general rule, at least in the context of infinite series (rather than sequences), the term divergent means "does not converge to a finite limit". So even if you're allowing infinity as a possible value, and even if the partial sums converge to infinity, we still say the series diverges.  This is a good usage, I think; otherwise you would have to check every time whether infinity is an allowed value, and it would just be confusing.
 * For infinite products things get even subtler -- an infinite product whose partial products are all positive, but converge to zero, is said to diverge to zero.
 * Some authors use a baroque set of rules to define convergence to zero, versus divergence to zero, in the case that factors of the infinite product are allowed to be zero themselves. It's something like this:  If there are only finitely many zero factors, then delete them.  If that infinite product converges to a finite, nonzero limit, then the original infinite product converges to zero.  However, if there are infinitely many zero factors, or if the partial products of the nonzero factors diverge, or converge to zero or infinity, then the infinite product diverges to zero.
 * On first reading the above paragraph, it probably doesn't make a lot of sense, but there are actually some OK reasons for this set of conventions. However as long as you're working with the reals, as opposed to the complex numbers, a far easier solution is just to ban zero factors from infinite products &mdash; they're not good for much anyway. --Trovatore (talk) 09:32, 10 April 2009 (UTC)

Just a quick answer check - volume integral
Hi there guys - was just wondering if someone could tell me whether my answer for this is wrong or not:

For the ellipsoid D, $$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2} \leq 1$$, evaluate $$\int x^2dxdydz$$ over D.

My answer is $$\frac{4a^3bc\pi}{15}$$ - is this correct?

Thanks, Spamalert101 (talk) 17:03, 6 April 2009 (UTC)
 * Yes. Algebraist 17:50, 6 April 2009 (UTC)

Thanks! Spamalert101 (talk) 19:40, 6 April 2009 (UTC)

Testing hypotheses about ratings
Let us say there are several sections of the same class which were taught by several different professors in a particular university in a particular semester. In each section (indexed k) $$n_k$$ students gave the rating $$r_k$$ (with std $$s_k$$) to the corresponding professor who taught the course. Let us say, I also taught this course this semester and got rating $$r_0$$ with std $$s_0$$ from $$n_0$$ students. How would you set up and test the hypothesis that I am better than the other professors? (Igny (talk) 21:39, 6 April 2009 (UTC))


 * Sounds like homework. Sum of normally distributed random variables might help. 75.62.6.87 (talk) 07:10, 7 April 2009 (UTC)


 * What part of I am a professor didn't you understand? (Igny (talk) 13:44, 7 April 2009 (UTC))


 * The part where you state it as fact rather than as a hypothetical. —Tamfang (talk) 17:03, 7 April 2009 (UTC)
 * You're flying a plane from New York to San Francisco. The wind is west by south at 50 knots.  Altitude, 20000 feet; temperature, -10 C.  Total weight, 50 tonnes.  It's one day after the vernal equinox, and the Moon is in Saggitarius.  The tide at Pacifica is 6 ft with swells to 8 ft.
 * How old is the pilot? --Trovatore (talk) 23:04, 10 April 2009 (UTC)
 * Well, I wasn't born yesterday. —Tamfang (talk) 06:56, 11 April 2009 (UTC)


 * What on earth has the student rating got to do with being a good professor? I'd first ask what is the difference between a good professor and a bad or mediocre one and find a way of measuring that. For a teaching professor I'd suggest it would have something to do with say how well the students are doing in a couple of years time. One could then see if there is any correlation between that and student ratings. Dmcq (talk) 23:56, 7 April 2009 (UTC)
 * Could I suggest rating the students after 2 years as:
 * +2 does something notable (except a unabomber)
 * +1 does postgrad work
 * 0 Gets a job
 * -1 no job
 * -2 becomes a dropout, or switches to something else and denounces the field of study
 * Then one can rate the professors, teaching assistants etc by backward propagation as in a neural network. Dmcq (talk) 09:29, 8 April 2009 (UTC)


 * Well that is not a question of me being good or bad professor. It is a problem to prove to University's bureaucrats that I deserve a better pay. I did not invent the game, but I wonder if there is a way to win it conclusively, not just by a judgment call. At first I thought that the first suggestion of an IP to look at the sum of the normal variables was a mere guess or a trivial solution of a trivial problem. But now I see that he had a point. A rating of professor #i by a student #j could indeed modelled as a sum of two normal variables,
 * $$X_i+Y_j$$
 * where X (an ability of professor to influence students opinion positively or negatively) is from N(0,sx), and Y (student's opinion about an "average professor") is from N(0,sy).(Igny (talk) 23:05, 8 April 2009 (UTC))

If you use a sophisticated statistical test, the beancounters won't understand it. I suggest you try a graphical method. Compare a histogram of your scores against a histogram of the scores for the others. If you stand out as better, it will be obvious. If it isn't obvious, the beancounters won't believe your statistical test either (and they would be right since you might have designed the test after seeing the data). McKay (talk) 03:18, 9 April 2009 (UTC)


 * Right. I think it is obvious, but I wondered if it is also statistically significant. (Igny (talk) 17:13, 9 April 2009 (UTC))


 * To test statistical significance I think I'd use the Wilcoxon signed-rank test for this type data. I'm not sure what a statistician would do. I originally learned about this test in a nice text "Practical Statistics for non-mathematical people" by Russell Langley but I'm not sure where a really accessible description of the test would be nowadays. Dmcq (talk) 13:36, 12 April 2009 (UTC)

Converting quadratic Bézier curve into cubic Bézier curve
Hi all, I'm using TikZ for LaTeX to draw a quadratic curve. However, TikZ only has support to draw a cubic Bézier, so I was wondering what would be the easiest way of "converting" the quadratic into a cubic?

Although this in itself is impossible for a generic case, the quadratic curve I have is a parabola so it is symmetric, so the starting points are the same but the control points differ - but the cubic control points will also be symmetric about the turning point.

This thread is obviously useless without pictures, so:

I drew that with the  environment but I need to do some advanced formatting so I have to draw it in TikZ.

The control point for this parabola is (-1, -1).

Thanks in advance. x42bn6 Talk Mess 23:51, 6 April 2009 (UTC)
 * Actually, solved my problem. For anyone who finds this and wants answers, you want something called degree elevation: .  There's also something called degree reduction which allows the approximation for reducing orders of Bézier curves.  x42bn6 Talk Mess  01:33, 7 April 2009 (UTC)
 * We have it in the Wikipedia article as well, see the last point in Bézier curve. — Emil J. 10:23, 7 April 2009 (UTC)
 * Ah, thanks. x42bn6 Talk Mess  12:40, 7 April 2009 (UTC)