Homework questions are welcome as long as they are asked honestly, explain the problem, and show sufficient effort. Please do not use this as the only tag for a question. For the answers on homework questions, helpful hints or instructions are preferred to a complete solution. Please do not add ...
27
votes
0answers
1k views
Simplicial homology of real projective space by Mayer-Vietoris
Consider the $n$-sphere $S^n$ and the real projective space $\mathbb{RP}^n$. There is a universal covering map $p: S^n \to \mathbb{RP}^n$, and it's clear that it's the coequaliser of $\mathrm{id}: S^n ...
20
votes
0answers
411 views
Computing the Chern-Simons invariant of SO(3)
I am an undergraduate learning about gauge theory and I have been tasked with working through the two examples given on pages 65 and 66 of "Characteristic forms and geometric invariants" by Chern and ...
16
votes
0answers
266 views
A homotopy sphere
My question is part of an exercise in Hatcher's 'Algebraic Topology'.
Consider a CW complex $X$, constructed from a circle and two 2-disks $e_2$ and $e_3$, attached to that circle by maps of degree 2 ...
11
votes
0answers
580 views
Spivak's “Differential Geometry” Volume 1, Chapter 1 ,Problem #20 part (b)
Problem 20 part (b) of Chapter 1 asks us to show that the infinite-holed torus is homeomorphic to the "infinite jail cell window." His hint helped me to get started (I think).
(I apologize for not ...
9
votes
0answers
439 views
Hatcher Problem 1.2.11: Cell decomposition of Mapping Torus $T_f$ of $S^1 \times S^1$
Suppose we have continuous function $f : X \to X$ that sends the basepoint of $X$ to itself, viz. $f(x_0) = x_0$ where $x_0$ is the basepoint of $X$. Recall the definition of the mapping torus $T_f$ ...
8
votes
0answers
91 views
Number of automorphisms of saturated models
I have the following assignment question: Let $M$ be an $L$-model of cardinality $\kappa$. Assume $M$ is saturated. How can you show that $|\text{Aut}(M)|=2^{|M|}$?
I see two possible ...
8
votes
0answers
147 views
Equality of integrals
this is q.2 of ahlfors p. 241: Show that $$\int_{-1}^{1}\frac{dt}{\sqrt {(1-t^2)(1-k^2t^2)}}=\int_{1}^{\frac{1}{k}}\frac{dt}{\sqrt {(t^2-1)(1-k^2t^2)}}$$ if and only if $k=(\sqrt{2}-1)^2$ . Thank you.
...
8
votes
0answers
116 views
Consider the sequence defined: $a_1=0, a_{n+1}=3+\sqrt{11+a_n}$, show that is bounded above and increasing using induction.
Consider the sequence defined: $a_1=0, a_{n+1}=3+\sqrt{11+a_n}$
a) Show, using induction, that this sequence is bounded above by 14; b) prove that the sequence is increasing; c) Why must it converge?; ...
8
votes
0answers
101 views
Variational field on $S^n$
This is my first post/question here, so I hope that I do everything right... I'm currently preparing for an exam and therefore trying to solve this exercise:
Let $p\in S^n$ and $v \in T_p S^n$. ...
7
votes
0answers
91 views
Class group of $k[x,y,z,w]/(xy-zw)$
I had a homework problem (II.6.5 in Hartshorne) to compute the (Weil divisor) class group of $X=\operatorname{Spec} k[x,y,z,w]/(xy-zw)$. I have accomplished this; however, I used some results I don't ...
7
votes
0answers
62 views
Way to Tietze's Transformation Theorem
during our knot-theory lecture we have talking about the following theorem:
Given two finite presentations of the same group, one can be obtained from the other by a finite sequence of Tietze ...
7
votes
0answers
143 views
Gödelian incompleteness; Smullyan's Puzzle
I am currently doing exercises on the Gödelian theorems; and we are confronted with the introductory puzzle of R. Smullyan's book, which is as follows:
Suppose we have a machine which prints strings ...
6
votes
0answers
49 views
How can I calculate $\displaystyle \int \frac{\sec x\tan x}{3x+5}\,\mathrm dx$
How can I calculate $\displaystyle \int \frac{\sec x\tan x}{3x+5}\,\mathrm dx$
My Try:: $\displaystyle \int \frac{1}{3x+5}\left(\sec x\tan x \right)\,\mathrm dx$
Now Using Integration by Parts::
We ...
6
votes
0answers
70 views
An example of a compact multiplicatively unbounded ring
My teacher asked me to build an associative topological Hausdorff compact ring $R$ with $1$, which is multiplicatively unbounded. That means there is a neighborhood $U\ni 1$ such that $FU\not=R$ for ...
6
votes
0answers
273 views
Condition of an eigenvector problem
Please, somebody help me with this problem.
[Ciarlet 2.3-5] Let ${A}$ and ${B} = {A} + \delta{A}$ be two symmetric matrices with eigenvalues
$$\alpha_1\ \leq\ \alpha_2\ \leq\ \ldots\ \leq\ ...
6
votes
0answers
122 views
cohomological proof of Maschke's theorem
I have been working on the following problem..
I have spent plenty of time trying to solve it myself. I am, however, unable to prove one small step in the argument. Beneath you can find my attempt. ...
6
votes
0answers
315 views
Two parallel lines and distance between three points on them.
I'm solving one hard problem in my homework textbook (it is from the list of hardest problems in the end of book with stars). I reduced it to very simple question which I can prove by two, but very ...
6
votes
0answers
419 views
Galois closure of a $p$-extension is also a $p$-extension
I'm working on a problem in Dummit & Foote and I'm quite stumped. The problem reads:
Let $p$ be a prime and let $F$ be a field. Let $K$ be a Galois extension of $F$ whose
Galois group is a ...
6
votes
0answers
124 views
Virtually cyclic groups
Let $G$ be a group with finite generating set $A$, define the distance $$d(g,h)=\mathrm{min}\{n:gh^{-1}=a_1^{\varepsilon_1}\dots a_n^{\varepsilon_n}, a_i\in A,\varepsilon_i=\pm1\}.$$ Define the ball ...
6
votes
0answers
140 views
Fixed Points of a Reflection
This question is about problem 2.C.5 from Allen Hatcher's Topology. The statement of the problem is as follows:
Let $M$ be a closed orientable surface embedded in $\mathbb{R}^3$ in such a way that ...
6
votes
0answers
169 views
Using the compactness theorem to show a set of first-order formula is equivalent to a set of quantifier-free formula
I am going through some theorems in Hodges' ``A shorter model theory'' and I have realized that I do not understand a certain argument regarding compactness. My question has two forms, I am sure that ...
6
votes
0answers
162 views
Characterize the continuous functions with finite right-hand derivative for at least one point of $[0, 1]$
Let $(E,d_\infty)$ the metric space of continuous functions defined on $ [0,1] $, with $$ d_\infty(f,g)=\sup\{ |f(x) - g(x)| : x\in [0,1] \}. $$
For all $ n\in \mathbb{N} $ let
$$ F_n = \{ f: \exists ...
5
votes
0answers
41 views
Prove that if $Q^tQ = I$ and $A = QR$, then $\|Ax - b\| = \|Rx - Q^tb\|$
I have a linear algebra final tomorrow and was practicing a few proofs. I want to make sure this proof is correct.
Prove that: If $Q^tQ = I$ and $A = QR$, then $\|Ax - b\| = \|Rx - Q^tb\|$
...
5
votes
0answers
124 views
Operator completly continuous
For $\lambda>0$, let $v(t)=\lambda \phi(t)$. Consider the BVP
consisting of the equation $$u'''=\lambda[f(t,[u-v]^*+\gamma)+M(t)] ,t\in (0,1)$$
and (BC):$u(0)=u'(p)=\int_q^1 w(s)u''(s)=0 ...
5
votes
0answers
39 views
Is there another, better way to write the following product?
I have the following expression
$$ \prod_{k=0}^n k + \alpha(-1)^{k+1} $$
which is, for example, $(0-\alpha)(1+\alpha)(2-\alpha)$ for $n = 2$. Is there a way to write this using something like a ...
5
votes
0answers
124 views
Probability, choose a box and then take exactly two white balls
There are $5$ boxes. There are $5$ white and $3$ black balls in two boxes, and $4$ white and $6$ black balls in the other three boxes. One box is randomly chosen. $3$ balls are randomly taken from ...
5
votes
0answers
53 views
how to prove this element is strictly positive?
Let $A$ be a $C^*\text{-algebra}$ and $A_+$ denote the positive elements. An element $a\in A_+$ is called strictly positive if $\overline{aAa}=A$. Want to prove: if $(e_n)$ is an approximate identity ...
5
votes
0answers
76 views
Non-isomorphic countable Boolean algebras
I'm trying to solve the next exercise:
Construct a sequence $\mathcal{B}_0,\mathcal{B}_1, \ldots$ of countable Boolean algebras such that for all $m \neq n$ then $\mathcal{B}_0 \ncong \mathcal{B}_1$.
...
5
votes
0answers
125 views
Trying to solve $\lambda^3 - 3.250\lambda^2 + \lambda - 0.063 = 0$ using Newton-Raphson method
This is what I've atempted so far in solving $\lambda^3 - 3.250\lambda^2 + \lambda - 0.063 = 0$. The following are the steps:
step 1: $f(\lambda) = \lambda^3 - 3.250\lambda^2 + \lambda - 0.063 $
...
5
votes
0answers
223 views
expansion for $1-|t|$
Let $f$ be a continuous function on $\mathbb{R}$ with compact support with exactly one maximum. Form the functions
$$
f_{m,k}(x)=f^m\left(x-\frac{k}{2^m}\right)
$$
I am wondering if one can expand ...
5
votes
0answers
173 views
Taxicab numbers.
I think most people know these numbers. Find $x,\ y,\ z,\ w$ such that $x^3 + y^3 = z^3 + w^3$ and $x,\ y,\ z,\ w$ are not equal to each other.
The first is $1729$.
I'm trying to figure out if ...
5
votes
0answers
97 views
Ramification of an integral closure of $\mathbb{C}\{z\}$
Let $\mathbb{C}\{z\}$ be the ring of convergent series in one
variable over $\mathbb{C}$, $K$ the fraction field of
$\mathbb{C}\{z\}$, $E$ a Galois extension of $K$ and
$\mathcal{O}_{E}$ the integral ...
4
votes
0answers
45 views
On an exercise from Hartshone's Algebraic Geometry, Ch I sect 4
My question is about the Ex. 4.9 page 31 in the book GTM52 by Robin Hartshone.
Let $X$ be a projective variety of dimension $r$ in $\mathbb{P}^n$, with $n\geq r+2$. Show that for suitable choice ...
4
votes
0answers
32 views
Group structure of $\mathbb{Q}_p ^* / \mathbb{Q}_p ^{*3}$
Let p be 1 mod 3 (separate question: work out 2 mod 3). What is the group structure of the abelian group $\mathbb{Q}_p ^* / \mathbb{Q}_p ^{*3}$?
$\mathbb{Q}_p ^*$ refers to the group of units in ...
4
votes
0answers
63 views
Determine the least prime $p$ for which $2^{p-1} \equiv 1 \pmod {p^2}$.
Determine the least prime $p$ for which $2^{p-1} \equiv 1 \pmod {p^{2}}$ .
4
votes
0answers
19 views
Oribt Space question
So I'm wondering how I would show that the orbit space $I^2/D_4$ is homeomorphic to $D^2$ and $\Delta XYZ$ where X is the origin, Y is a vertex of $I^2$ and Z is the midpoint of an edge of $I^2$. I ...
4
votes
0answers
64 views
C* algebra of bounded Borel functions
Let $T\in B(H)$ is normal, and $B(\sigma(T))$ denote the $C^*$ algebra of all bounded Borel functions on $\sigma(T)$. Then is it true that $B(\sigma(T))$ is a closed $C^*$ algebra under the sup. norm ...
4
votes
0answers
30 views
Conformal sets in $\Bbb C^n$
I'm trying to show whether these two sets are conformally equivalent.
$\Delta_n=\{(z_1,\dots,z_n):|z_i|<1,1\le i\le n|\}$ and $\Omega=\{(z_1,\dots,z_n):\text{Im}(z_1)>0\}$ where $n\ge 2$Thank ...
4
votes
0answers
51 views
Integration by parts of a normalized function - [copied from Physics.SE]
By using integration by parts, I need to show for $$A = \frac{\mathrm d}{\mathrm dx} + \tanh x, \qquad A^{\dagger} = - \frac{\mathrm d}{\mathrm dx} + \tanh x,$$ that
...
4
votes
0answers
77 views
Bijective map from two injective functions
A bijection $f \colon X \to Y$ is constructed from the injective functions $g\colon X \to Y$ and $h \colon Y \to X$. Suppose $X = Y = \mathbb N$. We let $g\colon X \to Y$ and $h \colon Y \to X$ be ...
4
votes
0answers
58 views
Show that degree of constant map is zero
Let $f \colon S^n \to S^n$ be a constant map, $n > 0$. I want to show that $\deg f = 0$. I will do it by definition. Let $\sum_k g_k \sigma^n_k$ be a singular chain, $g_k \in \mathbb{Z}$, ...
4
votes
0answers
48 views
Prove a function has $k$ continuous derivatives from its Fourier series
Here is the problem.
Let $k\in \mathbb{N}$. Suppose that there is a constant $C$ such that $|c_n|<\frac{C}{|n|^{k+1}}$ ($c_n$ here is the $n$th Fourier coefficient). Prove that ...
4
votes
0answers
64 views
find the torsion and the curvature of this curve… (it's horrible)
Let's consider the following curve:
$\varphi(t)=\begin{cases}
(t,0,e^{-\frac{1}{t^{2}}}) & t>0\\
(0,0,0) & t=0\\
(t,e^{-\frac{1}{t^{2}}},0) & t<0
\end{cases}
$
I have to compute ...
4
votes
0answers
119 views
Prove that the integral operator is bounded
Prove that the following operator is bounded on $L^{2}(0, \infty)$:
$Af(x)$ = $\frac{1}{\pi} \int_{0}^{\infty} \frac{f(y)}{x+y}dy$
with $||A|| \le 1$.
Attempt at Solution
It can be shown that:
...
4
votes
0answers
117 views
Disintegration of Measures
I was thinking about this exercise and I can't see how to end it. I'm sorry about the long post and thank you for the attention. Before asking the question, I need some background.
Let $(\Omega, ...
4
votes
0answers
55 views
A question on surfaces
If $S$ is the surface given by the function $z=y^2-x^2$, if I have the points $A=(1,0,-1)$, $B=(0,1,1)$, $C=(1,1,0)$, how can I use the Gaussian curvature to determine if there is an isometry of $S$ ...
4
votes
0answers
80 views
An exercise about the regular Borel measures
I want to prove following (from Folland, Ex. 3.26): If $\lambda$ and $\mu$ are positive, mutually singular Borel measures on $R^n$ and $\lambda + \mu$ is regular, then so are $\lambda$ and $\mu$.
For ...
4
votes
0answers
147 views
Semi-direct product isomorphic to direct product
I would like some help on the following problem from anyone who would like to help.
Let $f: H \to G$ be a group homomorphism. For $h \in H$, define $\rho(h) = \phi_{f(h)} \in Aut(G)$.
The situation ...
4
votes
0answers
67 views
Prove that (M,+,*) is a field
Prove that the multiplication $*:M \times M \to M$ defined by this table: * | 0 1 -------- 0 | 0 0 1 | 0 1 together with the commutative group (M,+), is a field (M,+,*).
Group axioms:
1) ...
4
votes
0answers
107 views
Find all permutations that commute with $\omega$=(1 9 7 10 12 2 5)(4 11)(3 6 8) in $S_{12}$
I'm asked in this exercise to find all permutations that commute with $\omega$=(1 9 7 10 12 2 5)(4 11)(3 6 8) in $S_{12}$.
What I have so far:
We could write $x$(1 9 7 10 12 2 5)(3 6 8)=(1 9 7 10 12 ...
