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  1. J

    Algebraic Geometry Textbook Recommendations

    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...
  2. J

    Simplifying an expression involving complex exponentials

    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...
  3. J

    Similar Diagonal Matrices

    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...
  4. J

    Relativist Addition of Velocities

    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...
  5. J

    Neighborhood Retract of Boundary

    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...
  6. J

    Punctuation in mathematical writing

    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...
  7. J

    Euler characteristic of complex projective plane

    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...
  8. J

    Irrational Winding of the Torus

    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...
  9. J

    Homotopy between identity and antipodal map

    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...
  10. J

    Smooth homotopy

    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...
  11. J

    Short Exact Sequences: Splitting

    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...
  12. J

    Prove [itex]C[a,b][/itex] a closed linear subspace of [itex]L^{\infty}[a,b][/itex]

    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...
  13. J

    Open mappings

    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...
  14. J

    Recursion Theorem and c.e. sets

    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...
  15. J

    Polynomial Rings

    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...
  16. J

    Units in a matrix ring

    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...
  17. J

    Numbering Turing Programs

    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...
  18. J

    Godel Numbers

    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...
  19. J

    Independence of Vector Space Axioms

    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...
  20. J

    Well-ordering in a first-order language

    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,<)...
  21. J

    Order isomorphism

    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...
  22. J

    Ord well-ordered by ∈

    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,<)...
  23. J

    Translating a Formal Language

    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...
  24. J

    Collection of Well-Formed Formulas with no Equivalent, Independent Subset

    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...
  25. J

    Classification of p-groups

    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...
  26. J

    Coulomb Integral for Diatomic Hydrogen Ion

    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...
  27. J

    Potential Energy Surface

    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...
  28. J

    Automorphisms of Algebraic Numbers

    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...
  29. J

    Almost All Graphs are Non-Planar

    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...
  30. J

    Graphs: Expected Number of Triangles and Variance

    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...
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