So I have been meaning to learn a little algebraic geometry for some time now, but have never gotten around to it. Since classes are just now winding down for the year, I figured that it was an ideal time to self-study a bit.
Now for a bit about my background: I know that commutative algebra...
Homework Statement
Simplify the following expression:
\sum_{ \alpha_1 + \cdots + \alpha_n = k} e^{i(\alpha_1 \theta_1 + \cdots + \alpha_n \theta_n)}
Homework Equations
\alpha_n = k - \sum_{j=1}^{n-1} \alpha_j
0 = \sum_{j=1}^{n} \theta_j
The Attempt at a Solution
I am trying to simplify...
As part of a larger problem involving classifying intertwining operators of two group representations, I came across the following question: If X is an n \times n diagonal matrix with n distinct non-zero eigenvalues, then exactly which n \times n matrices A satisfy the following equality...
Homework Statement
Fix an inertial reference frame and consider a particle moving with velocity \mathbf{u} in this frame. Let \mathbf{u'} denote the velocity of the particle as measured in an inertial frame moving at velocity \mathbf{v} with respect to the original frame. Show the following...
Here is the problem: If M is a manifold with boundary, then find a retraction r:U→∂M where U is a neighborhood of ∂M.
I realize the Collar Neighborhood Theorem essentially provides the desired map, but I am actually using this result to prove the aforementioned theorem. My thought on how to...
Recently I went through a bunch of my old solution sets and realized that I've been quite sloppy with my use of punctuation; in particular, examining the quality of my writing in solution sets over a period time, it appears that the lack of proper punctuation is getting more and more pronounced...
How to compute \chi(\mathbb{C}\mathrm{P}^2)?
This problem is from a class on differential topology, so we have defined the Euler characteristic as the sum of the indices of isolated zeros on a non-vanishing vector field. Off the top of my head, I cannot think of any theorems which really help...
I am trying to prove the following result: Fix a,b \in \mathbb{R} with a \neq 0. Let L = \{(x,y) \in \mathbb{R}^2:ax+by = 0\} and let \pi:\mathbb{R}^2 \rightarrow \mathbb{T}^2 be the canonical projection map. If \frac{b}{a} \notin \mathbb{Q}, then \pi(L) (with the subspace topology) is not a...
Homework Statement
Prove that the identity map \mathrm{id}_{S^{2k+1}} and the antipodal map -\mathrm{id}_{S^{2k+1}} are smoothly homotopic.
Homework Equations
N/A
The Attempt at a Solution
My attempt:
Fix k \in \mathbb{Z}_{\geq 0} and let \{e_i\}_{i=1}^{2k+2} be the standard basis for...
Recently I have been working through a text on Differential Topology and have come across the notion of smooth homotopy. Now the textbook (along with every other source I can find on the matter) defines a smooth homotopy of maps f,g:M \rightarrow N as a smooth map h:M \times [0,1] \rightarrow N...
In Dummit and Foote, a short exact sequence of R-modules 0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0 (\psi:A \rightarrow B and \phi:B \rightarrow C) is said to split if there is an R-module complement to \psi(A) in B. The authors are not really clear on what the phrase "an R-module...
Homework Statement
Let [a,b] be a closed, bounded interval of real numbers and consider L^{\infty}[a,b]. Let X be the subspace of L^{\infty}[a,b] comprising those equivalence classes that contain a continuous function. Show that such an equivalence class contains exactly one continuous...
Homework Statement
Let (X,||\cdot||) be a normed vector space and suppose that Y is a closed vector subspace of X. Show that the map ||x||_1=\inf_{y \in Y}||x-y|| defines a pseudonorm on X. Let (X/Y,||\cdot||_1) denote the normed vector space induced by ||\cdot||_1 and prove that the...
Let \varphi_e denote the p.c. function computed by the Turing Machine with code number e. Given M=\{x : \neg(y<x)[\varphi_y=\varphi_x]\} prove that M is infinite and contains no infinite c.e. subset.
I have already proved that M is infinite. A necessary and sufficient condition to prove that M...
Homework Statement
Find eight elements r \in \mathbb{Q}[x]/(x^4-16) such that r^2=r.
Homework Equations
N/A
The Attempt at a Solution
The elements 0+(x^4-16) and 1+(x^4-16) clearly satisfy the desired properties, but I still need six more elements. Can anyone help me figure out a...
Homework Statement
Let R be a commutative ring with 1. What are the units of M_n(R)?
Homework Equations
N/A
The Attempt at a Solution
If R is a field, then I know that we can characterize the units as those matrices with non-zero determinant (since those are the invertible...
Homework Statement
Assign a code number n(P) to every Turing program P.
Homework Equations
N/A
The Attempt at a Solution
Let pi denote the ith prime number. Let Q = {q0,q1,...} be internal states, let {B,1} denote tape symbols and let {L,R} denote direction symbols. Assign a code...
I am working through a computability theory textbook and right now the author is discussing assigning Godel numbers to each Turing Program. To do this, he suggests assigning each internal state, each of the elements of {1,B} and each of the elements of {L,R} a number. Then using these numbers...
Homework Statement
Determine whether the commutativity of (V,+) is independent from the remaining vector space axioms.
Homework Equations
N/A
The Attempt at a Solution
I am having a really hard time with this problem. Off the top of my head I could not think of any way to prove...
Is it possible to express well-ordering in a first-order language?
For example:
If X is a set and < is a binary relation on X such that (X,<) \vDash \forall x \; \neg(x<x) and (X,<) \vDash \forall x \forall y \forall z((x<y \wedge y<z) \rightarrow x<z), then (X,<) is a partial order.
If (X,<)...
I am trying to prove the following results: If α and β are ordinals, then the orderings (α,∈) and (β,∈) are isomorphic if and only if α = β.
So far, I have only proved that the class Ord is transitive and well-ordered by ∈. I can prove this result with the following lemma: If f:α→β is an...
Homework Statement
Let Ord denote the class of ordinals. Prove that Ord is is transitive and well-ordered by ∈.
Homework Equations
A set X is called transitive if x ∈ X and y ∈ x imply y ∈ X.
A set X is called ordinal if it is transitive and well-ordered by ∈.
A total order (X,<)...
Homework Statement
Translate \forall x (Nx \rightarrow Ix \rightarrow \neg \forall y (Ny \rightarrow Iy \rightarrow \neg x<y)) into English where N translates to "is a number", I translates to "is interesting" and "<" translates to is less than.
Homework Equations
N/A
The Attempt at...
Homework Statement
Find a collection of well-formed formulas Ʃ such that Ʃ has no independent equivalent subset.
Homework Equations
N/A
The Attempt at a Solution
So far I have been able to show that Ʃ must be infinite. However, after this, I get stuck. Could anyone give me a hint...
I am working on classifying all groups of order less than or equal to 100. For most orders, this is fairly straightforward, since we can just utilize Cauchy's Theorem/Sylow's Theorems to show that the group can be expressed as a semi-direct product and then find the desired automorphism...
Homework Statement
What does the non-negativity of the Coulomb Integral for H2+ suggest about the relative strengths of the attractive electron-proton force and the repulsive proton-proton forces?
Homework Equations
N/A
The Attempt at a Solution
I want to say that it means the...
Homework Statement
Estimate the ground-state potential energy surface for H2+ using the first-order perturbative change in the energy.
Homework Equations
N/A
The Attempt at a Solution
I can calculate the first-order correction to the energy using the fact that E^1_0 = \langle \mathrm{1s}_A...
Homework Statement
1) Find more than two automorphisms of A.
2) Do automorphisms of C fix A∩R?
Homework Equations
N/A
The Attempt at a Solution
I managed to figure out the second question since a map which preserves the additive structure of C will fix Q. And since the maps preserves...
Homework Statement
Prove that almost all graphs are not planar graphs.
Homework Equations
Kuratowski's Theorem
The Attempt at a Solution
Kuratowski's Theorem states that a graph is planar if and only if it does not contain a subgraph homeomorphic to K5 or K3,3. Therefore, if I...
Homework Statement
Let G be a random graph on n vertices:
1) What is the expected number of triangles in G?
2) What is the variance in the number of triangles?
Homework Equations
N/A
The Attempt at a Solution
I think I can do (1) by using indicator variables. In particular, let...