Wikipedia:Reference desk/Archives/Mathematics/2012 January 31

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January 31

Calculating the spinning of the earth at the 39th parallel

Hello, I have an assignment in precalculus class I need help with.

Assignment: Observer is in car at a point on the 39th parallel. how fast would the car have to be going to keep the sun on the horizon?

I am not asking for an answer, but how would one go about achieving this answer? Thanks!--Prowress (talk) 01:25, 31 January 2012 (UTC)

What is the total length of the 39th parallel? This is the distance the driver would need to travel in one day. Sławomir Biały (talk) 01:27, 31 January 2012 (UTC)

Here's a rather low-tech method: Measure the distance around a globe at the 39th parallel using a string, then apply the scale of the globe to get the distance in miles or km. This is the speed you would need to go per day to keep the Sun where it is. Divide by 24 to get the MPH or km/hr. Obviously this isn't the method your pre-calc teacher had in mind, but you could use this method to check your answer. StuRat (talk) 01:30, 31 January 2012 (UTC)

At any angle of latitude, the radius of the circle defined by that parallel is the radius of the earth multiplied by the cosine of the angle. Multiply by pi to get the circumference, divide by 24 to get the required per-hour speed. The earth's radius isn't constant, but I'd assume that you're meant to take it as such.←86.174.199.35 (talk) 11:02, 31 January 2012 (UTC)

Should be "Multiply by 2pi". hydnjo (talk) 01:04, 1 February 2012 (UTC)

Of course.←86.174.199.35 (talk) 17:20, 1 February 2012 (UTC)

Galois extensions and inert prime ideals

Hello, I'm trying to solve the following problem:

Let $L/K$ be a Galois extension of number fields. Suppose $Gal(L/K)$ is not cyclic. Show that no primes of $K$ are inert in $L$ (where a prime ${\mathfrak {p}}$ of $K$ is inert in $L$ if ${\mathfrak {p}}{\mathcal {O}}_{L}$ is prime).

However, I have absolutely no idea where to start here: I have plenty of results which may be relevant from a lecture course I took on local fields but I think I have so many because the course was so broad, that I'm not sure what to use. This is an exercise but not for homework; there's no rush to hand anything in but I would very much appreciate any thorough explanation you can provide since I do want to understand the material better, and I find all this confluence of number fields/galois theory/ramifications very confusing. I may well be familiar with the results you think are required to solve the problem, if they aren't too complex; it was given as an exercise in the lecture course I took. Thank you for any help in advance :) Spamalert101 (talk) 03:14, 31 January 2012 (UTC)

I don't really remember this stuff, but poking around through some books I have, I think the argument is roughly the following sans the details: Suppose ${\mathfrak {p}}$ is inert. Since ${\mathfrak {p}}{\mathcal {O}}_{L}$ is prime and prime ideals are maximal in a Dedekind domain, the quotient ring ${\mathcal {O}}_{L}/{\mathfrak {p}}$ is a field, and in particular a field extension of ${\mathcal {O}}_{K}/{\mathfrak {p}}$. Both of these are finite fields. Somehow you show that the Galois group of L over K is isomorphic to the Galois group of ${\mathcal {O}}_{L}/{\mathfrak {p}}$ over ${\mathcal {O}}_{K}/{\mathfrak {p}}$. The Galois groups of finite fields are always cyclic (I don't remember how to prove this), so then $Gal(L/K)$ is cyclic. Rckrone (talk) 05:35, 3 February 2012 (UTC)

Simple probability questions

I know that tossing a coin five times in a row would only rarely result in five heads, but I am completely lost when it comes to actually calculating the probability. Each toss has a 1/2 chance but I have no idea how the odds combine in multiple tosses. A similar problem is calculating the chance of having any rain at all if the forecast says there is a 30% chance on each of the following five days. Roger (talk) 11:58, 31 January 2012 (UTC)

If two events (call them A and B) are independent then the probability of A and B both happening is the two individual probabilities multiplied together. Formally, $P(A{\mbox{ and }}B)=P(A)P(B)$.

So $P({\mbox{5 heads}})=[P({\mbox{1 head}})]^{5}=\left({\frac {1}{2}}\right)^{5}={\frac {1}{32}}=3.125\%$

As for the rain question, to find the probability of some rain, it's easier to find the probability of no rain and subtract it from 1.

So $P({\mbox{some rain}})=1-P({\mbox{no rain}})$. Now, we can find the probability of there being no rain over the five days like we did in the coins question. We know the chance of rain on a single day is 30%, so the chance of it not raining is 70%, or 0.7 in decimal.

The probability of there being no rain over the five days is then ${0.7}^{5}=0.16807=16.807\%$.

So the probability of having some rain over the five days is $1-0.16807=0.83193=83.193\%$. Readro (talk) 12:50, 31 January 2012 (UTC)

Thanks! Roger (talk) 14:43, 31 January 2012 (UTC)

Note, however, that "a 30% chance of rain" is a rather vague statement. Before you could use that for any actual probability calculations, you'd need to answer some questions:

1) Is that for one specific point or over a large area ?

2) If for the large area, does that mean there is a 30% chance of rain occurring at just one point within the area, at each point within the area, or at all points within the area on the same day (sequentially or simultaneously) ?

3) What constitutes "rain" ? Is a single raindrop enough ? How about dew or sleet ? StuRat (talk) 19:30, 31 January 2012 (UTC)

... and, of course, in the rain example, if the days are consecutive or even close together, then the rain events on separate days are not independent so you can't just multiply, but many examination questions ignore the realities of meteorology! Dbfirs 17:07, 1 February 2012 (UTC)

Of course, in real life rain one day affects the probability of it raining on the following day, but for a question about introductory level probability I didn't feel it warranted a lecture about independence! Readro (talk) 17:18, 1 February 2012 (UTC)

I disagree. Statistics classes spend entirely too much time on various formulae, and yet never seem to cover the basics like how to tell dependent events from independent or how to take an unbiased sample. Thus, we get statistics with impeccable math based on absurdly biased surveys. StuRat (talk) 18:09, 1 February 2012 (UTC)

... and an innocent mother gets put in prison because an expert medical witness didn't understand the basic concept of independence in probabilities (he subsequently lost his job). Dbfirs 23:21, 1 February 2012 (UTC)

Now this is far too interesting to just leave like this so can we have a cite please. Roger (talk) 06:59, 3 February 2012 (UTC)

Sorry, yes, I remember using the case as an example of understanding independence in probability, but I'd forgotten the names of the people involved. Thisisn't a reliable source, but gives one view of the scandal. Dbfirs 09:39, 3 February 2012 (UTC)

I guess our article 'Sally Clark' isn't a reliable source either, but it's well referenced. Qwfp (talk) 13:11, 3 February 2012 (UTC)

Thanks. I forgot to look for that article. Dbfirs 12:45, 4 February 2012 (UTC)

The lesson is that the aftermath needn't have been so disastrous, if only the before-math had been better. -- Jack of Oz ^{[your turn]} 05:01, 5 February 2012 (UTC)

Resolved

1) Is that for one specific point or over a large area ?

2) If for the large area, does that mean there is a 30% chance of rain occurring at just one point within the area, at each point within the area, or at all points within the area on the same day (sequentially or simultaneously) ?

There is no difference between the two: A 30% chance of rain at any given spot within the defined area in 24 hours is the same as 30% of the defined area will have rain within the next 24 hours.

3) What constitutes "rain" ? Is a single raindrop enough ? How about dew or sleet ?

According to the South African Weather Service (their forecast for my location is what prompted my question) rain is precipitation of more than 0.1mm arriving on the ground as liquid - that exludes snow, frost and dew.

Roger (talk) 07:15, 3 February 2012 (UTC)

To demonstrate my point, let's say there are only two places where rain is measured within the area in question. If there is a 30% chance of rain at each location, and if we consider them independent events (they aren't, of course, but could correlate either positively or negatively), that would give us a 9% chance of rain in both locations (0.3 × 0.3 = 0.09). So, would they report this as a 30% chance of rain or 9% ? Also note having it rain 30% of the time at a given location means much more accumulation than the 0.1 mm required if it only has to rain once at each location to qualify as "having rained". That 0.1 mm seems rather generous. I don't think most people would say "it rained today" if that was all they got (for one thing, they wouldn't have noticed unless they were outside at the time, as presumably it would immediately evaporate). StuRat (talk) 20:56, 3 February 2012 (UTC)

Classical mathematical texts

Are they still worth reading? Isaak Newton's work seems impenetrable, but the most recent Theory of Games and Economic Behavior of von Neumann appears, at a first glance, to be of some value. 212.170.181.95 (talk) 19:17, 31 January 2012 (UTC)

It depends on your reasons for reading. Is your goal to learn new material? Or is it to gain better perspective, depth, and historical understanding of something that you already have some familiarity with? One merit of old math texts is that their informational content and correctness usually isn't diminished with time. The truths described in Euclid's elements are as true today as when they were written. In contrast, I have a nice copy of "Plant-animals" (on google books here [1]) from 1912, which, though interesting from a history of science perspective, is quite useless to modern students, and "wrong" in many aspects. Another reason one might read old math texts is to get the style and voice of e.g. an inventor of the calculus, as opposed to a pedagogue living centuries later. Keep in mind though, that Newton was writing to communicate new research, and that a few hundred years of pedagogy and notational development can work wonders. For instance, learning e.g. PDE from a rigorous contemporary text is probably easier for many than slogging through older, more cumbersome notation used in the original research. SemanticMantis (talk) 19:53, 31 January 2012 (UTC)

One other thing to look out with old maths papers for is that notation will almost certainly have changed greatly. For example, a lot the geometry that people like Klein and Riemann came up with a century ago is still perfectly valid, and forms the basis for a big chunk of modern physics (general relativity and gauge theory in particular), but the papers they published (eg. Klein's Erlangen program papers) are almost unreadble these days because they only had a loose grasp on the concepts they were working with, which 150 years of research have subsequently polished. (While working my Masters project, I made the mistake of thinking I could bypass tracking down information on Cartan connections in textbooks by simply reading Cartan's papers directly. It turned out his definitions of fibre bundles are very different to modern ones, even though he's describing exactly the same objects, and I never managed to make his definitions compatible with the gauge theory-based definitions used in modern physics books). Smurrayinchester 22:07, 31 January 2012 (UTC)

If this is to satisfy your own quest for enlightenment -then go for what interests you most !. I learnt more from dating my lectures daughter, than I ever did from the the revered professor himself. --Aspro (talk) 22:13, 31 January 2012 (UTC)

I spend a lot of time reading older texts, thanks mostly to Google who has put a large number of them on the internet. Some other good sites to look at are U of M Hist. Math. Collection and arXiv.org at the Cornell Library. Notation and terminology has changed a great deal even from 100 years ago, so it takes a bit of patience to put it in modern language. Subject matter and methodology have changed significantly as well, for example it common into the late 1800's to use infinitesimals in mathematical proofs. Also, geometry was much more a part of the culture 100 years ago so you can expect a much more geometrical flavor to the material. To me, the best reason for going to the extra effort, especially if you're an amateur, is that much of the mathematics from 100 years ago has been forgotten. The math is just as elegant and interesting as it was then, but subjects tend to get played out after a while and when there is little hope of new results being published professionals tend to ignore those areas. An example is analytical statics which figured heavily in the mathematical curricula of the 1800's but you'd be hard pressed to find it even mentioned in a math classroom today.--RDBury (talk) 18:48, 1 February 2012 (UTC)

(edit conflict) They are very much worth reading. But I would never recommend using one to learn a topic. Topics, and the way they are understood, undergo changes over time. They often become more "refined", more "user friendly" over time. Modern texts also use modern language and notation that fits in better with mathematics as it stands today. On top of that, you'll be able to find the books in the bibliography! In short: reading classic books is a good thing to do if you're interested in the history of mathematics, but if you want to learn, and cross-reference that knowledge, the modern texts are the best. That doesn't mean that the most modern is the best; there is often a subject bible from 30 years ago, but I'm classing that as modern. — Fly by Night (talk) 19:03, 1 February 2012 (UTC)