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 ...

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24
votes
0answers
667 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 ...
20
votes
0answers
850 views

How can I calculate $\int \frac{\sec x\tan x}{3x+5}\,\mathrm dx$

How can I calculate $$ \int{\sec\left(x\right)\tan\left(x\right) \over 3x + 5}\,{\rm d}x $$ My Try:: $\displaystyle \int \frac{1}{3x+5}\left(\sec x\tan x \right)\,\mathrm dx$ Now Using Integration ...
12
votes
0answers
363 views

Rotman's exercise 2.8 “$S_n$ cannot be imbedded in $A_{n+1}$”

This question is about the (in)famous Rotman's exercise 2.8 in "An Introduction to the Theory of Groups." I've searched and found similar questions here and in MO, but none of them contains a valid ...
11
votes
0answers
204 views

Polynomials with rational zeros

Find all polynomials $F(x)={a_n}{x^n}+\cdots+{a_1}x+a_0$ satisfying $a_n \neq0$; $(a_0, a_1, a_2, \ldots ,a_n)$ is a permutation of $(0, 1, 2 ... n)$; all zeros of $F(x)$ are rational.
10
votes
0answers
243 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 ...
9
votes
0answers
329 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 ...
9
votes
0answers
209 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. ...
8
votes
0answers
293 views

Proof that $1729$ is the smallest taxicab number

For homework I have to produce the proof (algebraic or otherwise) to show that $1729$ HAS to be the smallest taxi cab number. A taxicab number means that it is the sum of two different cubes and can ...
8
votes
0answers
434 views

Complicated exercise on ODE

I have this exercise extracted from a examination of qualitative theory of ODE (in which we study the existence and uniqueness of solutions, and stability using the function of Lyapunov) I don't know ...
7
votes
0answers
108 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 ...
7
votes
0answers
105 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 ...
7
votes
0answers
447 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\ ...
7
votes
0answers
168 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 ...
6
votes
0answers
109 views

Prove $\ell_1$ is complete

I try to prove that $\ell_1$, the space of absolutely convergent sequences in $\mathbb{C}$ with norm $\| x \| = \sum_{k=1}^{\infty} |x_k|$, is complete. I already proved that, if $\{ x_n \}$ is a ...
6
votes
0answers
195 views

Ramanujan-Nagell Theorem Proof Question

I'm currently working through Stewart and Tall's Algebraic Number Theory. In particular, section 4.9 of this book provides a proof of the Ramanujan-Nagell Theorem, which states that the only integer ...
6
votes
0answers
84 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) ...
6
votes
0answers
247 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 ...
6
votes
0answers
157 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
257 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
248 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
210 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
58 views

Minimising an expression - involving polynomial

I found this one on a forum but it has been unanswered from long there. I am curious to know if there is a solution to this problem. Here it is: Let n be a positive integer. Determine the smallest ...
5
votes
0answers
87 views

Lagrange Interpolation

So the problem is this; For Lagrange Interpolation on the nodes $x_0 < x_1 < ... < x_{n-1} < x_n$ of the data ${(x_i, f(x_i))}^{n}_{i=0}$ the interpolating polynomial is $\sum ...
5
votes
0answers
90 views

$f$ continuous nowhere implies $f$ not Riemann integrable

I am trying to prove the statement in the title, without using the theorem which says that the set of discontinuities of a Riemann integrable function must have measure $0$. I am aware that similar ...
5
votes
0answers
220 views

Span and Dimension: A subspace

If $A$ is finite set of linearly independent vectors then the dimension of the subspace spanned by $A$ is equal to the number of vectors in $A$. This is obviously true. Since $A$ is a finite set of ...
5
votes
0answers
79 views

Calculus 2: Trigonometric Intergrals

I attempted to solve this problem by taking the two out of the integral. Then, I changed $\sec^3(x)$ to $\frac{1}{\cos^2(x)\cos(x)}$ and attempted to replace $\cos^2(x)$ with $1 - \sin^2(x)$. ...
5
votes
0answers
101 views

Uniqueness solutions of $dx/dt = f^2(x) + e^{-t}$.

Someone can help me in the following problem? Is a question of Zhang. Let $f(x)$ be continuous for $x \in \mathbb{R}$, show that $dx/dt = f^2(x) + e^{-t}$ has the property of uniqueness of ...
5
votes
0answers
153 views

Stability for Nonlinear System

I am trying to assess the (Liapunov) stability of the equilibrium at $(0,0)$ of the system \begin{align*} x_1' &= -4x_2 + x_1^2 \\ x_2' &= 4x_1 + x_2^2. \end{align*} I plotted the phase ...
5
votes
0answers
558 views

Proof that a sequence of continuous functions $(f_n)$ cannot converge pointwise to $1_\mathbb{Q}$ on $[0,1]$

As a homework question, we got asked the following: Construct a function $f:[0,1] \rightarrow \mathbb{R}$ which is not the pointwise limit of any sequence of continuous functions Thinking about ...
5
votes
0answers
138 views

Integral mean value theorem- relate a point in the derivative to an integral

Question: Let $f:[a,b] \to \Bbb R$ be a continuously differentiable function s.t $f(a)=f(b)=0$ Prove that exists a point $c \in (a,b)$ such that $$ |f'(c)| \ge \frac 4{(b-a)^2} \int ^b_a f(x) dx $$ ...
5
votes
0answers
297 views

Example of a ring satisfying this variant definition of “symmetric” on nilpotent elements

I want an example to show that if $a,b$ are nilpotent elements of a ring $R$ with 1 and if $c$ is any element of $R$, then $abc=0\Rightarrow acb=0$ but $cab=0$ does not imply $acb=0$. This is unlike ...
5
votes
0answers
192 views

How to prove following integral equality?

Let's have the equality $$ \int \limits_{-\infty}^{\infty} \left[ [\nabla_{\mathbf r'} \times \mathbf A (\mathbf r' )] \times \frac{\mathbf r' - \mathbf r}{|\mathbf r' - \mathbf r ...
5
votes
0answers
173 views

An entire function of strict order 2

Here is a problem from Stein and Shakarchi Complex Analysis, can somebody help me to solve it? I guess we can use Phragmen-Lindelof theorem but I don't know the exact way. Suppose $f(z)$ is an entire ...
5
votes
0answers
157 views

Simple problem of a differentiable function

Please, can somebody help me with this problem? I tried to use the Mean Value Theorem, but couldn't solve it. Let $g: [a,b]\rightarrow\mathbb{R}$ a differentiable function on $[a,b]$. If ...
5
votes
0answers
63 views

Homomorphism for a fixed graph NP-complete?

Let $G$ be the following Graph: We want to decide whether for an input structure $\mathcal{S}$ there exists a homomorphism $S \to G$. We will call this problem $HOM_G$. The task at hand is to show ...
5
votes
0answers
155 views

Evaluate integral $ I_s(x) \leq \frac{C}{(\pi(1-2 \alpha s))^{d/2}}\exp\left(\frac{\alpha}{1-2 \alpha s }|x|^2\right) $

For all $ x \in \mathbb{R}^n ,\hspace{5mm} 0 \leq s<t ,\hspace{5mm} t \in \mathbb{R}^+$ $$ I_s(x)=\int_{\mathbb{R}^n}\left|v\left(y \sqrt{2s}+x\right)\right|\exp(-|y|^2) \, \mathrm dy. $$ How we ...
5
votes
0answers
154 views

Each new member from the second divides the sum of all previous

All integers from $1$ to $13$ are recorded in a sequence such that each number (from the second onwards) divides the sum of all previous numbers. What numbers can be in the third place and why? I ...
5
votes
0answers
142 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
113 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
157 views

Non-Isomorphic Group Extensions

This is a question from a problem set on group cohomology, a subject I've just begun to learn. Let $B$ be a finite group and $A$ be abelian. I am looking for two groups $G_1$ and $G_2$ such that ...
5
votes
0answers
131 views

Prove $(\mathbb Z \times \mathbb Z, \Sigma)$ to be a partial order and tell if its subset $T'$ is a lattice

Let $T = (\mathbb Z\times\mathbb Z, \Sigma) $ be defined as follows: $$\begin{aligned} (a,b) \text{ } \Sigma \text { } (c,d) \Leftrightarrow (a,b) = (c,d) \text{ or } a^2b^2<c^2d^2\end{aligned}$$ ...
5
votes
0answers
351 views

Digital Numbers using all digits from 1-9

Call a number a digital number if it consists of all the digits from 1-9, each used exactly once. What is the probability that a digital number will be divisible by 7 ? What is the probability that a ...
5
votes
0answers
272 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
154 views

Problem 24 of section 2 of *Noncommutative Algebra* by Farb & Dennis

Problem 24 of section 2 of Noncommutative Algebra by Farb & Dennis states: Let $R$ be an artinian ring and let $G$ be a finite group. Show that $R[G]$ is semisimple if and only if $R$ is ...
5
votes
0answers
104 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 ...
5
votes
0answers
241 views

Abelian categories, axioms AB5 and AB5* and incompatability

This is a homework exercise, so please don't post full solutions to the question below. Grothendieck (I believe) introduced several axioms an abelian category A voluntarily could satisfy. In ...
5
votes
0answers
200 views

The category of presheaves on a possibly-large category

Suppose $\mathcal{C}$ is a category such that for every $c \in \mathrm{Ob}(\mathcal{C})$, the slice category $\mathcal{C}/c$ is equivalent to a small category. I need to show that the category of ...
4
votes
0answers
85 views

Line integrals and path independence

Consider $\textbf{F}(x,y)=\frac{-y}{x^2+y^2}\textbf{i}+\frac{x}{x^2+y^2}\textbf{j}$. Let $C_1$ be the upper half of the unit circle oriented counterclockwise, and let $C_2$ be the lower half of the ...
4
votes
0answers
21 views

Artin-Schreier Question from Corps Locaux

I have a question from Serre's book "Corps Locaux", namely question 5a in section 2 of chapter IV. It is as follows: "Let $e_K$ be the absolute ramification index of K, and let n be a positive ...
4
votes
0answers
67 views

Proof of a theorem about Baire categories

Problem: prove that the set of $C([0, 1])$ functions whose derivative is defined at every point (and it is either finite or infinite) is of the first Baire category. I have no idea how to approach ...