Let's discuss Mathematics

Atticus said:
You just mistyped the first line, it should be (nx-1)/n, on the second line it's correct. ;)

I didn't like topology that much. Functional analysis was my favourite course. Our course contained good deal of topology at the beginning, the best part of it, I'd say, Baire's category theorem and such. Maybe that's why the actual topology course felt boring. Plus I really never saw the idea of more general topologies than those induced by a metric. Sure, they are more general, and so on, but they felt very artificial. I think I've seen non-metric topology one or two times part from the classes.

EDIT: X-post. Adult content.
Click to expand...

The one-point compactification (for example the circle as compactification of the reals) is one example which can't be done in a metric space. It is quite useful.
 
Is that adding the point at infinity to the reals? Like the Riemann sphere in complex analysis?

Does infinity have a neighbourhood with that model? Would it be the union of (k, inf) and (-inf, -k) or something?
 
I might be missing something, but I don't think it's hard to get an exact % for a run of x in y trials, given the values of y & x. Coming up with a general formula might take a bit longer, I'll see what I can do tomorrow.
 
The right hand side of the distribution is easy to calculate.

Define the variable L(n,r) as the probability that the longest run in n tosses is r.

If r > n/2 there is only 1 possible run of this length, and it must be surrounded by tosses that aren't of the same kind, but apart from those "fixed" values the others toss results can be anything we like.

The case for r < n/2 is trickier. If r = n/2 that's not hard to calculate either.
 
I think you're overcomplicating things by looking at possible runs. The tosses will always be in order: a tails, b heads, c tails, d heads, etc. All you need to look at is the chance that one of a, b, c, d is equal to or greater than x. You know the average length of a, b, c, d, etc is 2 (actually a tiny bit below, it will only equal 2 when y is infinite.)

So the probability is that you will have y/2 runs.

The probability of a given run being x is 1/2^x.

The probability of a given run being less than x is 1/2 + 1/4 + 1/8 + ... + 1/2^(x-1)

Therefore the probability of a given run being x or greater is 1/2^(x-1)

So the probability of a given run being less than x is [2^(x-1)-1]/2^(x-1)

The probability that all runs are less than x is {[2^(x-1)-1]/2^(x-1)}^y/2

The probability of at least one run of x or greater is 1 - {[2^(x-1)-1]/2^(x-1)}^y/2


I don't think there's any problems in there, except for using y/2 as the approximation.

The actual run length won't be y/2, it will be y/(1/2 + 2/4 + 3/8 + 4/16 + ... + y/2^y) It's too late at night for me to try and get a formula for that sum. But come up with one, and you should have an exact formula for the probability of a run of at least x in y flips, for y>1
 
ParadigmShifter said:
Is that adding the point at infinity to the reals? Like the Riemann sphere in complex analysis?

Does infinity have a neighbourhood with that model? Would it be the union of (k, inf) and (-inf, -k) or something?
Click to expand...

Yeah, I think it's similar to the Riemann sphere
http://en.wikipedia.org/wiki/One-point_compactification

The stereographic projection of R is the most insightful example. Imagine the real line as the x-as in R^2 and the circle S^1 as a circle with radius 1/2 around (0,1/2). Given a point x in R, we draw a line from x to the north pole (0,1) of the circle. This line intersects the circle in another point y. When we identify x and y, we have found a continuous bijection between R and S^1\{(0,1)}. We then add (0,1) to the circle and identify it with +/- infinity, the resulting space is compact.

I Y is an open neighbourhood of infinity in this new space if the complement of Y is compact.
 
Why can't you add a metric to that space?

Can't you define d(x,inf) = inf for real x and d(inf,inf) = 0?

I think this still works with the triangle inequality

d(a,c) <= d(a,b) + d(b,c)

Or does d have to be S X S -> R?

EDIT: I guess divergent sequences wouldn't converge to inf in that metric, therefore the space wouldn't be compact.

EDIT2: Because there's really no way to express "getting closer to infinity" with my "metric".
 
I've never come across with anything requiring that, or at least don't remember, or if I have, it has been pretty trivial. If my grandfather could live without it, I need it neither.
 
Knots and classification of surfaces (manifolds)?

Topology is basically study of invariants innit.
 
No, I mean that specific one point compactification of R that Dutchfire mentioned.

Metric topology is of course essential. And there's nothing bad about general topology either, but I just found it boring. One reason for that boringness is that exotic topologies feel artificial to me. Also I'm more into analysis. Topology without metric is too algebra-like for my taste.
 
Well the complex variant (Riemann sphere) is very useful.
 
Do you really need some special topology with it? Isn't it just the ordinary C and infinity treated as special case?

Complex analysis is one of my embarrassing weak spots...
 
Another interesting topological exercise is proving that there is an infinite number of primes. You do it by defining the sets N_(a,b)=a+b Z as the base for your topology. These sets N_(a,b) are all clopen. It can be shown that all open sets in this topolgy are either infinite or empty. Now: Z\{1,-1}= union over primes p of N_(0,p). If there were a finite number of primes, this would be a finite union of closed sets, so Z\{1,-1} would be closed, {1,-1} would be open but that contradicts "all open sets in this topolgy are either infinite or empty".
 
Atticus said:
No, I mean that specific one point compactification of R that Dutchfire mentioned.

Metric topology is of course essential. And there's nothing bad about general topology either, but I just found it boring. One reason for that boringness is that exotic topologies feel artificial to me. Also I'm more into analysis. Topology without metric is too algebra-like for my taste.
Click to expand...

Well, defining lim f(x) -> infinity gets quite messy in metric sense, while it's quite clean topologically. That's true for many things done using topology. You don't need all those epsilons and deltas.
 
Don't ask me advice on dusting. I'm not bad at probability and statistics (if rather rusty).
 
For a company the following data is given.
Code: 
Quarter|Employees|Total costs 
----------------------------------
  1        342     $  4540
2,3,4      721     $ 37221

(so for the months april - december you have 721 people who work for you and you have costs of $ 37221)

How do you find the 'Average Cost per Employee per Year'?

My answer: 1/(1/4) * 1/342 * 4540 + 1/(3/4) * 1/721 * 37221 = 121.9
 
Yeah, it's a terrible table. To me the answer is (1/4 * 4540/342 + 3/4 * 37221/721)*4 = $168
 
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