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About the spoiler; I didn't cover that in my course, I just checked my notes. But, it is fairly 'obvious' when you consider vector spaces, so I suppose I was just supposed to deduce it. And I see the difference now. Thanks for the help, I'll have another go at the problem.

I can't see a difference between our definitions, but yours is certainly clearer. I don't know what that would be. I'd hazard a guess at matrices, but surely that relies on M being free as well?

Take P a free module over an arbitrary ring. Show P is projective. Definition of Projective: If f: M-->N and g: P-->N are module homomorphisms with f surjective, then if P is projective there exists homomorphism h such that h: P-->M with f(h(x))=g(x) for all x. Obviously f has a right...

I've formalised everything and I'm very happy with the answers now - I appreciate all the help

To some_dude, thanks for that clarification. And I haven't had the time to learn any Latex or Tex, but it is high on my to-do list. Thanks for the help

Thanks for the quick reply, and sorry, you're absolutely right the first part of (b) doesn't make sense as there's a typo. I'll make it clearer: (b)X is a compact metric space. F is a closed subset of X, and p is any point of X. Show there is a point q in F such that d(p,q)=infimum(d(p,q'))...

The question I'm doing is as follows: (a) Show that every compact subset of a Hausdorff space is closed. I've done this. (b) F is a compact metric space. F is a closed subset of X, and p is any point of X. Show there is a point q in F such that d(p,q)=infimum(d(p,q')) such that q' is in...

If that's correct, then the thing you posted is distributed as an F distribution, which is what I need? And would swapping beta.hat for beta.hat-beta make any difference to this?

Use the MLE sigma2.hat=(1/n)*||Y-Xbeta.hat||^2 ? This is distributed as a chi-squared n-1 variable if I remember correctly...

Take the linear model Y=X*beta+e, where e~Nn(0, sigma^2 * I), and it has MLE beta.hat First, find the distribution of (beta.hat-beta)' * X'*X * (beta.hat-beta), where t' is t transpose. I think I've done this. I think it's a sigma^2 chi-squared (n-p) distribution. Next, Hence find a...

I have Xi-Xbar ~ N(0, (1+1/n)sigma2) ?

We have X1,...,Xn~N(mu, sigma2) The crux of my problem is finding out the distribution of, say, X1-Xbar (where Xbar is the mean of the n RVs). This is going to end up proving the independence of Xbar and Sxx, btw. I know Xbar~N(mu, sigma2/n), but I don't know how to find the distribution...

Ah! I forgot about that double zero. Very much appreciated

At first glance, your g(z) has an odd number of roots within the disc, whereas f(z) would have an even number (I'm fairly sure of this, since f(z)+1 is even and the first peak's height is greater than 1 - look at a plot of it on wolframalpha or something if you're not convinced). So, I don't...

I believe I have it completely sorted now. Thanks very much, you've been a great help. Sorry if I came across as dim :P

I don't understand your first sentence. Are you saying you think there's a typo in the question I'm doing? And (I haven't tried it yet, but) I'm fairly certain that integrating along the lines parallel to the imaginary axis won't yield the integral -along- the imaginary axis. So I still see...

My bad, I worked the denominator out wrong. After multiplying by that, we have the integrand as: z(a-cos(z)-isin(z))/(a^2+1-2acos(z)) Which is nearly of the required form. But how is the numerator sorted out? We can't just take imaginary parts, because z has both real and imaginary...

That gives you a denominator somewhat resembling what is required, but you still have a |z| in there that I can't get rid of. (I got the denominator to be a^2+1-2|z|*a*cos(theta) ). And even forgetting the issue with that |z| it still leaves the question over what to do with the numerator...

The disc in question is {z: |z|<(n+1/2)pi}. I can't figure out how to apply Rouche to this. Any help would be appreciated. (This is in the context of showing all roots of zsin(z)=1 are real. I counted the zeros of zsin(z)-1 on the real axis and got 2n+2, and now I hope to get the same answer...

Hi I find my notes for how to calculate complex integrals woefully inadequate, and I'm hoping someone can explain to me how to do them. One that the notes particularly fail for is: 'Integrate z/(a-exp(-iz)) along the rectangle with vertices at pi, -pi, pi+iR, -pi+iR Hence integrate...

How would I show that every unit in Z[sqrt(2)] is of the form +/- (1 +/- sqrt(2) )^n ? I can show these are all units, but I can't show every unit is one of these. From some research, I'm aware this is a special case of Dirichlet's Unit Theorem, but that is far above the level I'm working at...

Hi, F is a finite field. The problem is set up as follows: Let V be a 2-dimensional vector space over F. Let Omega=set of all 1-dimensional subspaces of V. I've constructed PGL(2,F) by taking the quotient of GL(2,F) and the kernel of the action of GL(2,F) on Omega. Similarly for PSL(2,F)...

I'm probably coming across as quite dim here, but I still don't see how this answers my question. You have a number N. You try dividing it by every number less than sqrt(N) until you get a hit. If no hits, then it's prime. If you get a hit, redefine N as N/hit, and go again. This way, you...

What do you mean by 'the run time is determined by'? Do you mean that the factor it would take the longest to find would be the second largest? And if so, why exactly does this imply having only two factors would be quickest? I don't intuitively see why counting up to 17 5000 times will be...

What's the worst case for the factorisation of n using trial division? Worst case in terms of arithmetic operations that is. Many places tell me that it's n=pq with p and q prime and close to each other (and hence close to root(n) ), but I can't prove it. Help would be appreciated.

I know of linear programming, but I have no idea how to go about solving that using it unfortunately. And nor do I think I'm expected to know, this is a linear algebra question, which is the annoying thing... And I'd never heard of or encountered quadratic programming before.

Question: Given a1+a2+a3+...+an=0 and a1^2+a2^2+...+an^2=1, (all real numbers) find the maximal value of a1*a2+a2*a3+...+an*a1 Thoughts so far: I've treated the expression as a combination of n variables and differentiated - when it came to putting the constraints in it got to be a...

D'oh. I was right, it was very simple. Thanks for that!

Hi How would I show that the centre of mass of two point particles, one of mass m1 and the other m2, when released from rest from infinite simultaneously, stays in the same place? I feel this is a very simple question, I just can't seem to get it right. Everything I've tried still involves...

Thanks a lot, that's really very useful. James

Hi. The question is: Given X, Y and Z are all continuous, independant random variables uniformly distributed on (0,1), prove that (XY)^Z is also uniformly distributed on (0,1). I worked out the pdf of XY=W. I think it's -ln(w). I have no idea at all how to show that W^Z is U(0,1). What...

Thanks for that - everything now works.

Homework Statement div(J)=0 in volume V, and J.n=0 on surface S enclosing V, where n is the normal vector to the surface. Show that the integral over V of J dV is zero. Homework Equations The Attempt at a Solution I can't get anywhere with it! The divergence theorem doesn't...