Wikipedia:Reference desk/Archives/Mathematics/2010 October 15

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October 15

Division by zero

1×0=0

       Therefore, 1=0÷0

2×0=0

       Therefore, 2=0÷0

Therefore, 1=2=0÷0

Similarly any number is equal to any number Please tell me the mistake in this. Jijo (talk) 05:55, 15 October 2010 (UTC)

Your error is assuming that division by zero is a legitimate operation. —Anonymous Dissident^Talk 06:03, 15 October 2010 (UTC)

Or, in greater generality, assuming that division by zero follows the same rules as division by other numbers. -- Meni Rosenfeld (talk) 06:14, 15 October 2010 (UTC)

You can also visit the Mathematical fallacy article for similar problems.--Email4mobile (talk) 06:15, 15 October 2010 (UTC)

I am not dividing it to get a result. I have simply used a division symbol and left it alone.Jijo (talk) 19:02, 15 October 2010 (UTC) —Preceding unsigned comment added by Jijo925 (talk • contribs) 19:01, 15 October 2010 (UTC)

Asserting that 1x0=0 implies 1=0/0 is in some senses a correct statement, this is because 0/0 is an indeterminant form, the statement itself as given is technically nonsense, as you can't perform division by 0, so it is nonsense to write it that way. What you alluding to is the fact that 0/0 is an indeterminant form. A better way to write this would be $\lim _{x\to 0^{+}}{\frac {x}{x}}=1$ and $\lim _{x\to 0^{+}}{\frac {2x}{x}}=2$. A math-wiki (talk) 01:03, 16 October 2010 (UTC)

I don't think the OP is necessarily alluding to indeterminant forms. I think they're following the rules of algebra more or less like a machine that was never told division by 0 doesn't follow those rules. In general, certainly, a*b = c means a = c/b, whenever b is not 0. In fact, "c/b" is merely a convenient symbol meaning "c times the multiplicative inverse of b", where that inverse must exist for "c/b" to be meaningful. 67.158.43.41 (talk) 06:40, 16 October 2010 (UTC)

You are dividing by 0. You're not evaluating the expression, but that doesn't matter.--203.97.79.114 (talk) 08:30, 16 October 2010 (UTC)

Partially correct. He isn't evaluating the zero division on the right, but he *is* evaluating it on the left. Just moving the number and changing the operation is a shortcut - what you're really doing is dividing both sides by the same number. That is, if you start with x×y=z, you pass through (x×y)/y=z/y in order to get to x=z/y. You have to evaluate the (x×y)/y in order to simplify it to x. That simplification works with all numbers except zero, as (x×y)/0 is undefined, for any value of x and y. You can't simplify (x×0)/0 to x anymore than you can go from the general simplification of "w/w is 1" to the specific statement "0/0 is 1". -- 174.24.199.14 (talk) 15:47, 16 October 2010 (UTC)

Variance

Hi! Say $L=\sum _{i=1}^{N}X_{i}$ where $N$ ~ Poisson$(\lambda )$ and $X_{i}=(Y|Y>K)$ and Y ~ Lognormal$(\mu ,\sigma ^{2})$. The $X_{i}$'s are all iid (and independent of N) and K is some positive number. I am trying to find the variance of L in terms of the parameters...

Using the law of total variance on L and N, I get

${\begin{aligned}\operatorname {Var} (L)&=\operatorname {E} [N]\operatorname {Var} [X]+\operatorname {Var} [N]\operatorname {E} [X]^{2}\\&=\lambda \operatorname {E} [X^{2}]\\\end{aligned}}$

Then using an answer to an earlier question I asked here (thanks for that), I get

${\begin{aligned}\operatorname {E} (X^{2})&=e^{2(\mu +\sigma ^{2})}{\frac {\Phi ({\frac {\mu +2\sigma ^{2}-\log {K}}{\sigma }})}{\Phi ({\frac {\mu -\log {K}}{\sigma }})}}\\\end{aligned}}$

My question is: is this correct? I don't think it is because a simulation exercise gives me a much smaller variance... Thanks for any help. --Mudupie (talk) 09:26, 15 October 2010 (UTC)

This looks correct to me. I've also done my own simulation and the values seem to match. What parameters did you use, and what result did you get? -- Meni Rosenfeld (talk) 11:18, 15 October 2010 (UTC)

Thank you for the confirmation - I guess there must be something wrong with my simulation then. I'm using the following parameters:

$\lambda =716.666...$

$\mu =7.06363497100939$

$\sigma ^{2}=7.90978323231687$

$K=1000$

The analytic std deviation then comes to 117,938,227 but my simulated std deviation is only 78,057,536. I used 500,000 trials... --Mudupie (talk) 11:33, 15 October 2010 (UTC)

I think you may need a bigger sample. Try running 5 batches of 100,000 trials and find the variance of each batch. If the results vary significantly it will support this hypothesis. -- Meni Rosenfeld (talk) 12:17, 15 October 2010 (UTC)

Thanks. I think you are right. Surprisingly, though, the simulated std deviation does not vary significantly among the batches. I narrowed the problem down to the simulation of lognormal random numbers - the problem occurs when $\sigma ^{2}$ is large (more than 4ish) even when K is zero. I'm using Excel's LOGINV function. I simulated 6,000,000 numbers from Lognormal(7,8) distribution and the simulated variance was significantly lower than what it should have been (even though the simulated 90th, 91st, ... 99th, 99.1th, 99.2th, ... 99.9th and even 99.99th percentiles were accurate). I'll post a follow up question when I have the data. One more question for now: how does one determine what an appropriate number of simulations is for these types of problems? --Mudupie (talk) 20:56, 16 October 2010 (UTC)

I think what's happening is that the distribution gives enormous values once in a while, say, once in a few millions. They will have a large impact on the variance, but not on the 99.99th percentile. If you're running 500,000 trials it's very likely not to encounter any extreme value and end up with a low value. But if you did happen to encounter such an outlier then your sample variance will be much higher.

To estimate the required sample size, you can first take an initial sample and use it to roughly estimate the moments of the distribution up to the 4th. Use these to estimate the variance of the n-trial estimate of the variance of X, $\mathbb {V} \left[{\overline {X^{2}}}-{\overline {X}}^{2}\right]$. Then find the n required to get a sufficiently accurate estimate. Note that this may not work so well for misbehaving distributions, as this one seems to be.

Anyway, try to test this with much lower $\mu$ and $\sigma$ - this will have a much more benign behavior with which you can check the accuracy of the formulas. -- Meni Rosenfeld (talk) 10:30, 17 October 2010 (UTC)

Vector Fields and the Indices

Let M be a smooth (n–1)-dimensional manifold in Rⁿ. Let X be a smooth vector field over M with only isolated singularities. Let p be such an isolated singularity, i.e. X(p) = 0. How do I compute the index of X at p using connections? Does anyone know a nice formula? — Fly by Night (talk) 12:39, 15 October 2010 (UTC)

Coefficients in lengthy expansions

For an expression such as $(a+b+c)(a+b+d)(a+c+d)(b+c+d)$, is there an efficient way to determine how many times abcd, say, will appear in the expansion? My feeling is that the binomial coefficients could be used, but I can't put my finger on how. Thanks for the help. —Anonymous Dissident^Talk 13:01, 15 October 2010 (UTC)

After running a few of these "cyclic" products through Wolfram|Alpha and entering the sequence of coefficients into the OEIS, it's clear that the coefficient of the degree one term in n variables corresponds to the derangements of n objects (... 2, 9, 44, 265, ...). I don't really understand this result, though; can anyone explain? —Anonymous Dissident^Talk 13:30, 15 October 2010 (UTC)

Rearrange as

$(b+c+d)(a+c+d)(a+b+d)(a+b+c)$

then note that coefficient of abcd is number of ways of permuting abcd such that a occurs in position 2,3 or 4 but not position 1; b occurs in position 1, 3 or 4 but not 2 etc. i.e. coefficient of abcd is counting derangements of abcd, as you said. Gandalf61 (talk) 13:48, 15 October 2010 (UTC)

That makes good sense. Thanks. Unfortunately, counting derangements is about as time-intensive as just expanding the product, so I suppose a particularly convenient method does not exist. —Anonymous Dissident^Talk 00:59, 16 October 2010 (UTC)

The article on derangements includes closed form formulas for computing the n-th derangement. The most useful is that the number of derangements of $n$ objects is the integer nearest to ${\frac {n!}{e}}$. If you need to compute other coefficients, however, it seems more difficult. Eric. 82.139.80.27 (talk) 18:07, 16 October 2010 (UTC)

Here's a suggestion that may or may not help you: Let $s=a_{1}+a_{2}+\cdots +a_{n}$, and rewrite your product as

$(s-a_{1})(s-a_{2})\cdots (s-a_{n})=\sum _{i=0}^{n}(-1)^{i}s^{n-i}e_{i}(a_{1},...,a_{n})$

where $e_{i}$ are the elementary symmetric polynomials. Furthermore $s$ is simply $e_{1}$, so depending on what you want to do with the product, you may be able to save work by staying in the ring of symmetric functions as far as possible. –Henning Makholm (talk) 00:12, 17 October 2010 (UTC)

could you explain cayley graph

Hey, i am a 8th grade student interested in graph theory but i am having trouble to understand your articles. :-( is there any way you can explain "Cayley graph" to me in an easier way, thanks a million!!!! —Preceding unsigned comment added by 85.181.50.90 (talk) 21:41, 15 October 2010 (UTC)

The trouble is that a simplified explanation will lack formal mathematical rigor, and these are complicated and subtle concepts. A Cayley graph is a representation of a discrete group. This is a concept that requires subtle understanding of continuity and discreteness (as well as formal definitions of graphs and groups). These are concepts to which you probably have not been introduced, even if you are an advanced eighth-grader. Don't be discouraged - graph theory is very complicated and takes a while to wrap your mind around. You might start with the Graph theory article. Then, read about formal definitions for a "group"; and finally, discrete group. A Cayley graph is just a graph for a particular discrete group. Nimur (talk) 21:55, 15 October 2010 (UTC)

I'm not dumb! Lady Ada invented binary language which all computers everywhere speak, so girl's can learn this things too. I just need someone to explain to me in a simple way, I am very smart, what is a discrete group? I read your graph theory article and understood it PERFECTLY 100% ALL OF IT. But your discrete group article is too hard. I don't want to give up, I know I can understand if I try!!! Please just explain discrete group to me very easy, I understand graph theory!!!! I'M SMART I JUST NEED SUM HELP —Preceding unsigned comment added by 85.181.50.90 (talk) 22:16, 15 October 2010 (UTC)

maybe there could be a better name for it. if you could call it anything, what would you call a discrete group if you didn't know what discrete group was? 85.181.50.90 (talk) 22:40, 15 October 2010 (UTC)

No one was implying that you were dumb, it's not a matter of intelligence, it's a matter of knowledge. I'm currently taking my first upper-division (300 level) mathematics courses myself and I couldn't make sense of the information in that article either as I simply don't know enough about the relevant material. I would recommend learning about topology, group theory, and graph theory a fair amount before returning to this particular object, perhaps then you'd have the necessary information to make sense of the article. Graph Theory is a very interesting subject I had the opportunity to study it some in high school Discrete Math and I very much enjoyed studying them. A math-wiki (talk) 01:12, 16 October 2010 (UTC)

To the original questioner: I think it's awesome that you're interested in this stuff. Here are my suggestions for you. First, you need to understand just a little bit of group theory. In particular, you need to understand what a group is, and you should study some examples of groups to get a feel for how they work and to build up a collection of examples for yourself to play with. You also need to understand the idea of a generating set of a group. Then a Cayley graph is sort of a "road map" of a group. There's one vertex that represents the identity element of the group, and for each generator of the group there is an edge leading to a vertex representing another element, and then edges leading from those vertices, and so on. If the group you're thinking about is finite (which is a good place to start), then these paths will eventually lead back around in loops and cycles and create a finite graph, with one vertex for every element in the group. If there are two or more ways to get from the identity element to some other element in this graph, then that means there are two or more ways to write that element in terms of the generators of the group. This is only a very brief introduction to this whole thing, so I haven't explained anything in very much detail. If you have other questions, please feel free to ask. How much of the things I have mentioned in this reply did you already know? If you already understand what a group is, and you can give me an example of a finite group that you're comfortable thinking about, then I can give you an explanation of its Cayley graph that I think you'll understand. —Bkell (talk) 05:55, 16 October 2010 (UTC)

Good response Bkell. Once the OP can get a specific finite group in mind, we can help her see how to draw the Cayley graph. OP: You might be interested in starting with 'clock groups', i.e. groups of integers that use Modular_arithmetic for the group operation. --SemanticMantis (talk) 14:39, 18 October 2010 (UTC)

In order to understand the concept Cayley graph, you need to understand the abstract definition of a graph, and the definition of a group. However, IMHO, it is not at all necessary to know about the difference between a discrete and a continuous group, to grasp the concepts.

Moreover, both the definitions you need are rather simple. However, they are also rather abstract.

The first question should be: Do you feel comfortable about using sets for defining things? Set theory is a very useful tool for more abstract definitions... JoergenB (talk) 21:22, 18 October 2010 (UTC)

Cayley graphs are a special type of graph that you can generate from a rather abstract object called a "Group". Part of why Cayley graphs are useful is because they can be used to visualize these abstract algebraic structures. If you want to learn about groups visually, I think you'd enjoy looking into symmetry groups, especially Frieze_groups. Feel free to ask separate followup questions in a new thread :) --SemanticMantis (talk) 22:58, 18 October 2010 (UTC)