Homework Statement
Let \mathcal{C} be a category such that, for each object c \in Ob(\mathcal{C}), the slice category \mathcal{C}\,/c is equivalent to a small category, even though \mathcal{C} may not be small. Show that the functor category [ \mathcal{C}^{\text{ op}}, \bf{Set}] is an...
Homework Statement
I am trying to work out a solution to the following problem, where we are working in a field K complete with respect to a discrete valuation, with valuation ring \mathcal{O} and residue field k.
Q: Let f(X) be a monic irreducible polynomial in K[X]. Show that if f(0) \in...
Homework Statement
I've been given the following problem:
"Suppose that U is a finite-dimensional subspace of a Fréchet space (V,\tau). Show that the subspace topology on U is the usual topology (given for example by a Euclidean norm) and that U is a closed linear subspace of V."
I feel a bit...
Ah yes of course, I was being stupid :smile:
So then as you suggested we get out a quartic in the required variables, I'm just not sure what a "plane" quartic is? Thanks!
I'm a little confused about why you're doing this: rather than "rewriting" the old equation in terms of the images of X, Y, Z, wouldn't we take the map \varphi to "substitute in" YZ, XZ, XY for each instance of X, Y, Z respectively? For instance the first term of the equation would become...
Oh, of course, I'm sorry! I thought I wrote it down, obviously not :)
\varphi: \mathbb{P}^2 - \to \mathbb{P}^2 is the rational map given by (X_0:X_1:X_2) \to (X_1 X_2: X_0 X_2: X_0 X_1) = (1/X_0: 1/X_1 : 1/X_2): I have shown already that this map is not regular at the points (1:0:0), (0:1:0)...
You've been such a brilliant help Micromass, thank you so much :)
I do have one final question which has come up, if you don't mind. The final part of a problem I've been doing says: "let V \subset \mathbb{P}^2 be the plane curve given by the vanishing of the polynomial X_0^2X_1^3 + X_1^2 X_2^3...
Wonderful: that last post is fixed, I don't remember having to use itex the last time I used PF but perhaps that's just how long I've been away for! I concur with your blog, Hartshorne is boring and unintelligible, I don't like it at all but sadly it seems to be a bit of an industry standard, if...
Hi Micromass! Thanks for the quick response! I have primarily been using any online lecture notes I could find, but Hartshorne has come in handy once or twice too.
I have seen the notion of singularity, yes: so essentially the tangent space has dimension 2 at the origin whereas the curve...
Homework Statement
How would I go about showing that if X = \{(x,\,y) \in \mathbb{C}^2 | x^2 = y^3\} then X is birational but not isomorphic to the affine space \mathbb{A}^1[/tex]? I have found the obvious birational map, sending [itex](x,\,y) \to \frac{x}{y}, so I have shown the spaces...
Homework Statement
Suppose that S is a countably infinite subset of \ell_2 with the property that the linear span of S′ is dense in \ell_2 whenever S\S′ is finite. Show that there is some S′ whose linear span is dense in \ell_2 and for which S\S′ is infinite.
The Attempt at a Solution
I...
My definition of a Hilbert space is standard, i.e. a real or complex inner product space which is complete under the norm defined by the inner product. The spectrum of an operator T is, for me, the set of points 'p' in the complex plane for which T-pI is not invertible (I the identity map). I...
Homework Statement
Much as the title says, I need to construct an example of a linear operator on Hilbert space with empty spectrum. I can very easily construct an example with empty point-spectrum (e.g. the right-shift operator on l_2), but this has very far from empty spectrum.
If I...
I apologise if it sounded like I was trying to avoid doing the work or learning from the problems myself but instead trying to find a book from which to copy them - indeed, precisely the reason why I didn't ask any specific questions on the problem sheets is because I want to do them myself in...
Hello everyone - I'm a third year student at Cambridge university, and I've recently started taking a course on Riemann surfaces along with a number of other pure courses this year.
The problem is, the lecturer of the course is of a rather sub-par standard - whilst I don't doubt he's probably...
Ah of course, it makes perfect sense when you put it like that :) The argument is fairly simple once you spot it, I was definitely overcomplicating things - thankyou for being so patient!
I'm still not completely sure sorry, I'm obviously not getting this :( When you're looking at L over L∩K, the span of your elements will have to be smaller than or equal to the span of those elements over K, but I can't really see how to formulate this idea properly - sorry to keep asking! I'm...
I've just noticed also this looks a lot like the second isomorphism theorem - perhaps another way to go about the problem?
Anyway, i can see why L is a spanning set of vectors for KL over K, and I guess you could use AoC to pick a basis by selecting an element of L at random, then another not...
"Product Field"
Homework Statement
Let K and L be subfields of a field M such that M/K (the field extension M of K) is finite. Denote by KL the set of all finite sums ∑xiyi with xi ∈ K and yi ∈ L. Show that KL is a subfield of M, and that
[KL : K] ≤ [L : K ∩ L].
Homework Equations...
Homework Statement
Two long thin concentric perfectly conducting cylindrical shells of radii a and b (a<b) are connected together at one end by a resistor of resistance R, and at the other by a battery that establishes a potential difference V. Thus, a current I=V/R flows in opposite...
Hey there all!
I'm a little confused by the concept of Givens rotations and was hoping someone could help elaborate a little bit with the following problem for me - if i could get some help understanding how to approach the problem it would be a really great help to me.
'Let A be an n x n...
Homework Statement
Hi all,
I've just calculated the first three nonzero terms of the Laurent series of 1/(cos(z)-1) in the region |z|<2pi, and now I've been asked to 'find the three non-zero central terms of the Laurent expansion valid for 2pi<|z|<4pi' - firstly, what does it mean by...
Homework Statement
Hi all, I'm having some trouble seeing why this question isn't trivial, maybe someone can help explain what I actually need to show - shouldn't take you long! :)
Suppose h:\mathbb{C} \to \mathbb{C}-\{0\} is analytic with no zeros. Show there is an analytic function...
Thanks so much, i didn't appreciate that the cross ratio is invariable under rotation so you can just place the points in a more convenient position and work it out that way. Have a good day!
Homework Statement
Hi all,
I wasn't sure whether to ask this here or not, but I've tried to make progress and so far had no such luck. I've not managed to make any headway on this so far, I don't think I can quite get my head around the geometry that's in play.
If u,v\in \mathbb{C} correspond...
Homework Statement
Hi all - I've been battering away at this for an hour or so, and was hoping someone else could lend a hand!
Q: Show that any Mobius transformation T not equal to 1 on \mathbb{C}_{\infinity} has 1 or 2 fixed points. (Done) Show that the Mobius transformation corresponding...
Hi all,
I'm trying to get to grips with the Frechet derivative, and whilst I think I've got all the fundamental concepts down, I'm having trouble evaluating some of the trickier limits I've come up against.
The two I'm struggling with currently are the further derivatives of the functions...
Thanks ever so much, how stupid of me not to spot that! Is it true of every order then, that there exists some matrix which isn't the n'th power of any matrix? (I couldn't come up with an example this late at night for the cube root, but I may be being slow ;-))
Homework Statement
Hi again all,
I've just managed to prove the existence (non-constructively) of a 'square root function' f on some open epsilon-ball about the identity matrix 'I' such that [f(A)]^2=A\qquad \forall\, A \,\text{ s.t.}\, \|I-A\|<\epsilon within Mn, the space of n*n matrices...
Hi all, I'm looking at the following problem:
Suppose that f:\mathbb{R}^2\to\mathbb{R} is such that \frac{\partial{f}}{\partial{x}} is continuous in some open ball around (a,b) and \frac{\partial{f}}{\partial{y}} exists at (a,b): show f is differentiable at (a,b).
Now I know that if both...
I figured yf(y^{-1}\textbf{x})=yf(\frac{x}{y},1), so set f_x=\frac{\partial}{\partial{x}}f(\frac{x}{y},1), f_{yy}=\frac{\partial^2}{\partial{y^2}}f(\frac{x}{y},1) and so on:
then we get
\frac{\partial{}}{\partial{x}}(yf(\frac{x}{y},1))=yf_x \Rightarrow...
Homework Statement
How do I show that if f\in C^2 \text{(}\mathbb{R}\text{)} is convex then the function yf(y^{-1}\textbf{x}) is convex on (x,y):y>0?
Homework Equations
I know the standard definitions and whatnot about convexity, but I tried chugging through the algebra and didn't have any...
Homework Statement
Show that any two subspaces of the same dimension in a finite-dimensional vector space have a common complementary subspace. [You may wish to consider first the case where the subspaces have dimension 1 less than the space.]
The Attempt at a Solution
I've managed to sort...
Homework Statement
Hi all - I've done all but the last part of this problem, for which I have the solution, but I was hoping someone could explain why the solution is what it is :)
q:The Hamiltonian of a quantum system suddenly changes by a finite amount. Show that the wavefunction must...
Hi there all,
I've been doing some work to try and keep myself sharp over the holidays but I've reached a point where I'm doing maths I have no notes for, and one particular topic has me completely stuffed - a few more difficult derivatives, primarily between matrices...
Homework Statement
Is the following statement true: for every compact metric space X there is a constant N S.T. every subcover of X by balls of radius one has a subcover with at most N balls?
Homework Equations
The Attempt at a Solution
I know you're meant to post your working but I...
Homework Statement
If R and S are two equivalence relations on the same set A, we deﬁne R ◦ S =
{(x, z ) ∈ A × A : there exists y ∈ A such that (x, y) ∈ R and (y, z ) ∈ S }.
Show that the following conditions are equivalent:
(i) R ◦ S is a symmetric relation on A ;
(ii) R ◦ S is a...
Sorry, there were 2 parts to the question phrased in a very unrelated way but obviously they were related, implying that yes, A is singular. Thanks very much for the help :)
Apologies, the title messed up - was meant to be 'existence of a C: AC=CA=0 for 2x2 matrices'.
Homework Statement
How would one show 'nicely' that for any 2x2 non-zero matrix A, there exists some 2x2 non-zero matrix C such that AC=CA=0? I can see how to show it by showing that the...
Homework Statement
A horizontal table oscillates with a displacement A sin ωt , where A = (Ax , 0, Az ) is the amplitude vector and ω the angular frequency in an inertial frame of reference with the z axis vertically upwards, normal to the table. A block sitting on the table has mass m and...