Wikipedia:Reference desk/Archives/Mathematics/2010 August 16

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

August 16

trying to compare homicides

I need some raw data to try some different things and comparisons...

How many murders have been committed in Minneapolis, Minnesota in year 2010?

If available the same data for Hennepin County and the state as a whole

How many murders have been committed in the Greater London Area (boroughs plus The Square Mile) in the year 2010?

How many in England & Wales?

I am really curious as to WHERE to find this data so I can directly do my own research in the future. My intent is to play around with the population and educational level variables and just experiment with this rather depressing data. Thanks I have a reference question (talk) 05:25, 16 August 2010 (UTC)

The miscellaneous reference desk would be better for a question like that. Also have you tried searching using Google or a suchlike search engine? Dmcq (talk) 08:37, 16 August 2010 (UTC)

Crime statistics for Greater London are available from the Metropolitan Police here - for example, there were 125 homicides (murder, manslaughter, corporate manslaughter and infanticide) in Greater London in the 12 months to July 2010. Crime statistics for England and Wales are available from the Home Office - their report on Crime in England and Wales 2009/2010 is available here. Gandalf61 (talk) 12:39, 16 August 2010 (UTC)

Laurent Series for ln(z) in annulus around z=0?

Hello, is there a laurent series expanded around z=0 for ln(z)? Thanks.Rich (talk) 11:54, 16 August 2010 (UTC)

ln(z) around 0 is multi valued or discontinuous, while laurent series are single valued and continuous. So no. Bo Jacoby (talk) 12:58, 16 August 2010 (UTC).

Hmm thanks that's interesting. Does that mean ln(z) has an essential singularity at 0?-Rich Peterson24.7.28.186 (talk) 13:36, 17 August 2010 (UTC)

No, because that would require the function to be holomorphic in a neighbourhood of 0 minus 0 itself, and as Bo explained, this condition fails for whatever branch of ln(z) you choose. If the function had an essential singularity in 0, it would also have a Laurent series around 0.—Emil J. 13:43, 17 August 2010 (UTC)

This type of singularity is called a "branch point singularity". Note that functions with such a singularity at some point can have well defined values at that point. The word "singularity" refers to non-analytic behavior. So, e.g. the function z^p for positive non-integer p has a branch point singularity at z = 0, even though the function is well defined at z = 0. Count Iblis (talk) 14:43, 17 August 2010 (UTC)

We have an article titled branch point (I haven't looked at it recently, so I can't promise anything about what it says. Michael Hardy (talk) 15:02, 17 August 2010 (UTC)

Thanks everyone.Rich (talk) 02:46, 19 August 2010 (UTC)

SO(n) vs Spin group

I think I understand this discussion of why SO(3) is not simply connected. I gather that Spin(3) is simply connected, but I'm curious what this exactly means. As I understand it, it should be possible, with any simply connected manifold, to continuously deform any path between two points on the manifold to any other path with the same end points. So, suppose I represent a rotation changing over time using Slerp. Suppose further that I use Slerp, or an extension like Squad to mediate a rotation of $2\pi$ about some axis. Is it possible to continuously "deform" this rotation until it represents no rotation over time whatsoever? Thanks.--Leon (talk) 13:51, 16 August 2010 (UTC)

Slerp does not explain why the circle S¹ is not simply connected but the spheres Sⁿ for n>1 are simply connected. Note that Spin(3) is the 3-sphere. The main point you should try to understand is the difference between the circle and the 2-sphere with regard to simple connectivity. Tkuvho (talk) 14:23, 16 August 2010 (UTC)

I'm not too clear on what you mean. And I'm not sure why the circle S¹ is relevant.--Leon (talk) 14:31, 16 August 2010 (UTC)

Thinking in terms of Slerp, what would be your guess concerning the circle: does Slerp suggest it is simply connected, or not? Tkuvho (talk) 14:39, 16 August 2010 (UTC)

Well I believe the answer is "no", but I'm still not entirely sure what you mean.

What I'm after is an intuitive geometrical understanding of the difference between the two groups when used to represent rotations in $\mathbb {R} ^{3}$, and for a theoretical engineering application. Also, does the statement

"Surprisingly, if you run through the path twice, i.e., from north pole down to south pole and back to the north pole so that φ runs from 0 to 4π, you get a closed loop which can be shrunk to a single point"

mean that if I were to construct a rotation of $4\pi$ about some axis, I could continuously deform such a rotation to no rotation at all?--Leon (talk) 14:48, 16 August 2010 (UTC)

Exactly. More precisely, you can deform such a family of rotations, to the trivial (i.e. constant) family. You are unlikely to be able to do that with something as canonical as the Slerp deformations, but perhaps you could. Tkuvho (talk) 15:01, 16 August 2010 (UTC)

To summarize what's involved: rotations of 3-space can be represented by unit quaternions q. Two "opposite" quaternions, q and -q, represent the same rotation of 3-space. Hence the space of rotations is the quotient of the unit quaternions by the antipodal map. Thus, while unit quaternions form a 3-sphere (simply-connected), the rotations form a manifold which is its antipodal quotient, and therefore non-simply connected. Tkuvho (talk) 15:05, 16 August 2010 (UTC)

Cool, thanks. But, with regards to the earlier bit, is there any "intuitive, geometrical" difference between the two groups when used to represent rotations? To me it seems that the significance is one can continuously deform a path described by elements of Spin(3) to any other such path, keeping the same end points the same, and this isn't true of SO(3). As a $2\pi$ rotation in Spin(3) is not an example of an element going to itself (quaternions: 1->i->-1 is a $2\pi$ rotation but it isn't an element travelling some path to end where it started) it couldn't be contracted to a point in any case. Or am I missing something...--Leon (talk) 15:13, 16 August 2010 (UTC)

The path you have described illustrates the point very nicely. As you mentioned, the path is not closed. However, under the antipodal quotient, it does descend to a closed path in the quotient space, namely SO(3)=RP³. This is because 1 and -1 define the same rotation, namely the trivial one. By general covering space theory, whenever a non-closed path descends to a loop, the resulting loop cannot be contracted to a point. Tkuvho (talk) 15:18, 16 August 2010 (UTC)

Really, I struck it out because I felt like an idiot for writing it! In any case, thanks for your help.--Leon (talk) 15:21, 16 August 2010 (UTC)

Some notions in topology

Hi, Does anybody know hwat is the meaning of the $\pi -weight$ and $\pi -base$ ? —Preceding unsigned comment added by 212.199.96.133 (talk) 13:56, 16 August 2010 (UTC) Topologia clalit (talk) 13:58, 16 August 2010 (UTC)

A π-base is a collection B of nonempty open sets such that every nonempty open set contains some member of B. I guess that the π-weight of a space is the cardinality of its smallest π-base.—Emil J. 14:01, 16 August 2010 (UTC)

Since it may not be immediately obvious what is the difference between a base and a π-base, here's some basic info: any base is also a π-base, but not vice versa. Consider the Sorgenfrey line S. Since every nonempty basic set in S contains a nonempty open set in the standard topology of the reals, the set {(p,q): p,q in Q, p < q} is a π-base of S, and in particular, S has countable π-weight. However, this set is not a base. In fact, if B is any base of S, then for each real a there is a set U_a in B such that $a\in U_{a}\subseteq [a,a+1)$. Since this makes a = min(U_a), all these sets have to be distinct, hence S has weight $2^{\aleph _{0}}$.—Emil J. 15:10, 16 August 2010 (UTC)

Thanks! Topologia clalit (talk) 07:19, 18 August 2010 (UTC)

correlation conundrum

Hi there - I hope someone can help me out - it's been a long time since college stats classes...

I have five business groups, for each one I have a performance metric (a percentage score). I am looking for factors which might explain that metric (I understand that correlation <> causation!)

The groups have offices (between 20 and 90, each group has a different number), they have revenue (varies with each group), and the offices are in different locations. Some of the offices are in locations that are on a list of 20 places that are problematic - we'll call this the bad location list. All the metrics are for the group as a whole, not the individual offices.

OK, so, number of offices correlates well with performance - r2 - .87. Percentage of offices that are on the list correlates well (but negatively) about -.8. So far so good - the more locations, the better, and the fewer of them that are on the bad list the better. What I'm worried about is that the larger the number of offices, the lower the percentage on the bad list CAN be - that is to say there are only 20 'bad' locations, so if you have more than 20 office locations you are automatically going to have a lower percentage on the bad list - how can I deal with this? I want to disagregate the effect of size per-se from the effect of bad locations.

Thanks so much! —Preceding unsigned comment added by 97.115.64.81 (talk) 14:54, 16 August 2010 (UTC)

Where did this list of 20 bad locations come from? You might be better off going back to the starting point and using whatever metric was used to compile the list rather than using the list itself. --Tango (talk) 08:18, 18 August 2010 (UTC)

5 data points is very few, so whatever results you'll find will probably be highly insignificant. When there's more data, one thing you can do is to look at all points where the total number of offices is some fixed value, and see how the performance varies with number of bad offices.

Perhaps you can use some form of ANOVA where you do multiple regression for various choices of variables and see which contribute the most to explaining the variance. If you use a baseline model using only the total number of offices, and a second model including both that and the number of bad offices, then a large difference in the RSS between them will tell you that the number of bad offices indeed correlates with performance.

This is of course assuming that a linear model is at all appropriate. -- Meni Rosenfeld (talk) 07:40, 19 August 2010 (UTC)

Identifying Multiples of low primes

It's easy to identify multiples of 2 (last digit even), 3 (digits sum to 3 or a multiple of 3), 5 (last digit 0 or 5) and 11 (alternate digits sum to 11). I seem to remember that there's also a trick to spotting multiples of 7, but I can't remember what it is. Can anyone help? Thanks Rojomoke (talk) 16:26, 16 August 2010 (UTC)

Divisibility rule#Divisibility by 7. Algebraist 16:38, 16 August 2010 (UTC)

The powers of 10 are 1, 10, 100, 1000 and so on. Modulo 7 these powers are 1, 3, 2, -1, -3, -2, 1, 3, 2, -1, -3, -2, and so on. So the rule is: if the ones plus thrice the tenths plus twice the hundreds minus the thousands minus thrice the tenthousands minus twice the hundredthousands and so on, is a multiple of seven, then so is the original number. Bo Jacoby (talk) 16:52, 16 August 2010 (UTC).

If you do the same thing as for 11 except with thousands you can make the checks for 7, 11 and 13 easier. For instance with 134,768,345,558 add 134+345 and 768+558 to give 479 and 1326. Subtract the smaller from the larger to give 847 and then just check this number 847 instead of the original. It is divisible by 7 but not by 11 or 13 so the original number is divisible by 7 but not 11 or 13. This is because 1001 = 7.11.13 Dmcq (talk) 19:12, 16 August 2010 (UTC)

I see there is an article about all this Divisibility rule Dmcq (talk) 19:18, 16 August 2010 (UTC)