For questions concerning a specific proof, asking for verification, identifying errors, suggestions for improvement, etc.

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12
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0answers
300 views

Show that in any group of order $23 \cdot 24$, the $23$-Sylow subgroup is normal.

Show that in any group of order $23 \cdot 24$, the $23$-Sylow subgroup is normal. Let $P_k$ denote the $k$-Sylow subgroup and let $n_3$ denote the number of conjugates of $P_k$. $n_2 \equiv 1 ...
10
votes
0answers
79 views

Function defined as a limit

Q. If $f(x)=\lim\limits_{n\to\infty}\dfrac{\log(2+x)-x^{2n}\sin(x)}{1+x^{2n}}$, then explain why the function does not vanish anywhere in the interval $[0,\pi/2]$, although $f(0)$ and $f(\pi/2)$ ...
10
votes
0answers
84 views

The order of subgroup generated by two distinct elements of order 2

$G$ is a group of order $2$6. If $x$ and $y$ are two distinct elements of order $2$, what could the order of $\langle x,y\rangle$ be? By lagranges theorem, $\langle x\rangle$ and $\langle ...
9
votes
0answers
64 views

Prove that $f_n(x)=\frac{x}{n}$, $n=1,2,\ldots$ does not converge uniformly on $\mathbb{R}$

Prove that $f_n(x)=\frac{x}{n}$, $n=1,2,\ldots$ does not converge uniformly on $\mathbb{R}$. It's clear this function converges pointwise to $0$ function. We have to show that there is ...
8
votes
0answers
90 views

Finding irreducible components of Spec$(R/I^n)$

Let $R= k[x,y,z]/(xy,yz,zx)$. Let $I=(x)$. What are the irreducible components of $\mathrm{Spec}(R/I^n)$ where $n \geq 2$ and $k$ is a field? For solving this problem I'm trying to use following ...
8
votes
0answers
272 views

A question on odd perfect numbers

Let $\sigma(x)$ be the sum of the divisors of the positive integer $x$. If $\sigma(M) = 2M$, then $M$ is said to be perfect. Currently, there are $48$ known examples of even perfect numbers -- on ...
7
votes
0answers
87 views

Fast convergence in $L^1$ implies convergence almost everywhere

This is a proof-verification request. Claim: Let $(X,\mathscr M,\mu)$ be a measure space. Let $f_n$ ($n\in\mathbb N$) and $f$ be measurable, integrable, real-valued functions such that ...
7
votes
0answers
266 views

Can fundamental theorem of algebra for real polynomials be proven without using complex numbers?

For polynomials with real coefficients, I am trying to prove the following version of fundamental theorem of algebra, which avoids using complex numbers in the proof. Existence of complex roots will ...
7
votes
0answers
136 views

An inequality for $|\zeta (s,a)|$, a detailed proof

In page 272 of [1], Apostol leaves as a reader's assigment to complete a proof of a related statement with Hurwitz zeta function, defined initially for $\sigma >1$ by the series ...
7
votes
0answers
173 views

Classify groups of order 171

This is a problem from Stanford Algebra Qualifying Exam, Fall 1998. I know the standard way is to use Sylow theorems and semidirect product. $171 = 9\cdot 19$. By Sylow theorems, $n_3|19$ and ...
7
votes
0answers
2k views

Proof of separability of Lp spaces

The following is a proof in Brezis book. It shows the separability of $L^{p}$ spaces: I have a few questions regarding the proof. Questions: It says 'it is easy to construct a function $f_{2} ...
7
votes
0answers
236 views

Higher Order Derivative Proof .

I would appreciate if someone could check over my proof for this question and advise me if it is correct. My attempt so far; Now as $f$ is k times differentiable , it taylor series about $x_{0}$ ...
6
votes
0answers
42 views

Uniform limit of one-to-one analytic functions is either constant or one-to-one

Let $U$ be a complex domain, and $(f_n)_{n\in \mathbb{N}}$ be a sequence on one-to-one analytic functions defined on $U$. Suppose that $f_n$ converges to $f$ uniformly on every compact subset of $U$. ...
6
votes
0answers
120 views

From $\prod_{d\mid n}d=n^{\sigma_0(n)/2}$ to $n!=\operatorname{lcm}(1,\ldots,n)^{e(n)}$, where $\sigma_0(n)$ is the number of divisors

We know that $$\prod_{d\mid n}d=n^{\sigma_{0}(n)/2}$$ for every integer $n\geq 1$, where $\sigma_{0}(n)$ is the number of positive divisors of $n$, see for example [1] (exercise 10, page 47). And for ...
6
votes
0answers
155 views

Olympic number theory problem: is this solution fine and sufficiently well written?

Determine all the positive integers $m$ such that both the ratios $$ \frac{2(5^m+5)}{3^m+1}, \frac{9^m+1}{5^m+5}$$ are integers. Attempt to a solution: If the ratios are both integers, than their ...
6
votes
0answers
60 views

$E \to S$ surjective in degrees $\geq 1$ implies $\widetilde{E} \to \widetilde{S}$ surjective

In the proof of Theorem II.8.13 in Hartshorne (which is the Euler sequence), the author writes: Let $S = A[x_0, \ldots, x_n]$. [...] The exact sequence $$0 \to M \to E \to S$$ of graded ...
6
votes
0answers
139 views

Finitely additive function on an infinite set, s.t., $m(A)=0$ for any finite set and $m(X)=1$ (constructive approach)

Other exercise which I found in Dudley's Analysis book: Show that there is a measure on a infinite set $X$, defined on $2^X$ s.t. is finitely additive, $m(A)=0$ for any finite set and $m(X)=1$. ...
6
votes
0answers
135 views

Proposed proof of analysis result

Hi please advise on my proof of the following result: Assume that $I \subset \mathbb{R}^{n}$ is convex, bounded open set with Lipschitz boundary and let $u_{m},u$ be such that $$u_{m} ...
6
votes
0answers
184 views

Order of the Normalizer of a Sylow $p$-subgroup in $S_{p}$

Question: Let $p$ be a prime and $P \leq S_{p}$ with $|P| = p$. Prove that $|N_{S_{p}}(P)| = p(p-1)$. I have already solved this problem, but I have come across a proof that is much more elegant than ...
6
votes
0answers
143 views

Proof of inverse function theorem by approximation property

In proving the inverse function theorem using the approximation characterization of the derivative, we are given $F:\mathbb{R}^n \to \mathbb{R}^n$ such that  $$F(p_0 + h) - F(p_0) = DF_{p_0}(h) + ...
6
votes
0answers
248 views

Can one trisect $\arccos(6/7)$?

Is this proof correct? Proof: Here $\theta = \arccos(6/7)$. Now to show we can't trisect $\theta$, we show that $\theta/3$ is not constructible by finding the irreducible polynomial in $\mathbb ...
6
votes
0answers
371 views

Are these sets in $\mathbb{R}$ open and/or closed: $\{\frac{1}{n} : n \in \mathbb{N}\}$, $\{0\}\cup \{\frac{1}{n} : n \in \mathbb{N}\}$ and $[0,1)$.

In $\mathbb{R}$, are these sets open? Are they closed? $A = \{\frac{1}{n} : n \in \mathbb{N}\}$ $B = A \cup \{0\} $ $[0, 1)$ My thoughts: $A$ is not open as if we have an open ball with $r > ...
5
votes
0answers
36 views

Find the inverse of the following piecewise defined function

Find the inverse of $f$ if $f(x)=$ $$ \begin{cases} \sqrt{2-x}, &\text{for $x<0$}\\ 1-x^2, &\text{for $x \ge 0$} \\ \end{cases} $$ My effort For $y=\sqrt{2-x}$ ,we find ...
5
votes
0answers
67 views

How do I evaluate this integral :$ \int_0^\frac{\pi}{2} [\text{chi}(\cot^2x)+\text{shi}(\cot^2x)]\csc^2(x)e^{-\csc^2(x)}dx$?

I have tried to evaluate this integral :$$ \int_0^\frac{\pi}{2} [\text{chi}(\cot^2x)+\text{shi}(\cot^2 x)]\csc^2(x)e^{-\csc^2(x)}dx $$ where, $\text{shi}(x)=\int_{0}^{x}\frac{\sinh t}{t}dt$ , ...
5
votes
0answers
33 views

Am I using a correct method to derive these orders?

As an exercise I am calculating some orders of some elements but I don't know if my method is correct. Please can you tell me if my solutions (not the results but the steps to deduce the ...
5
votes
0answers
166 views

Alternative Proof of $\sqrt{2}$ is irrational

Can anyone check if this proof is correct. Thank you. Proof that $\sqrt{2}$ is irrational. Let x = $\sqrt{2}$ then $x^2=2$ and $x^2-2=0$ By the Rational Root Theorem, we have: the number $1$ ...
5
votes
0answers
65 views

Ornstein-Uhlenbeck SDE solution

I'm following this solution of $$dX_t=\kappa(\theta-X_t)\,dt+\sigma\,dW_t \tag1 $$ And the question is whether its solution $$X_t=\theta+e^{-\kappa(t-s)}(X_s-\theta)+\sigma\int_s^t ...
5
votes
0answers
64 views

Is this limit correct??

$$\lim_{h \rightarrow 0}h^2\cos \left ( \frac{1}{h} \right )=0$$ since $$-h^2 \leq h^2\cos \left ( \frac{1}{h} \right ) \leq h^2$$ and both the limits of the upper bound and lower bound are zero as ...
5
votes
0answers
123 views

Me vs. Wikipedia (Lacunary function)?

I was recently reading this wikipedia page: https://en.wikipedia.org/wiki/Lacunary_function and found atleast the example they are giving must be wrong because I have kind of managed to analytically ...
5
votes
0answers
44 views

For finitely generated free abelian groups $A,B$ if there is an onto homomorphism $A \to B$, then $\operatorname{rank}(A) \geq \operatorname{rank}(B)$

$\newcommand{\rank}{\operatorname{rank}}$For two finitely generated, free abelian groups $A,B$ prove that if there is an onto homomorphism $A \rightarrow B$, then $\rank(A) \geq \rank(B)$ Assume that ...
5
votes
0answers
186 views

A natural number has exactly 10 divisors including 1 and itself.How many distinct prime factors can this natural number have?

A natural number has exactly 10 divisors including 1 and itself.How many distinct prime factors can this natural number have? options given: (a) either $1$ or $2$ (b)$1$ or $3$ (c)either $2$ or ...
5
votes
0answers
82 views

Showing that only $(n+1)^{n-1}$ of all the possible $n^n$ choices assure a full car park

This exercise is taken from the site of Queen Mary University of London: A car park has $n$ spaces, numbered from $1$ to $n$, arranged in a row. $n$ drivers each independently choose a favourite ...
5
votes
0answers
83 views

Evaluate $\int_{-\infty}^{\infty}\frac{\sqrt{a+ix}}{a^2+x^2}\,dx$ using residue calculus

I'm asked to evaluate $$\int_{-\infty}^{\infty}\frac{\sqrt{a+ix}}{a^2+x^2}\,dx$$ $\mathbb{R}\ni a>0$, using residue calculus (where $\sqrt{\cdot}$ is the PV $\sqrt{}$). My approach is as follows: ...
5
votes
0answers
82 views

Prove $(\overline{A \cap B}) \subseteq \overline{A} \cap \overline{B}$.

Let $\overline{A}$ be the closure of $A$. My attempt: Since $A \subseteq \overline{A}$ and $B \subseteq \overline{B}$, we have $$A \cap B \subseteq \overline{A} \cap \overline{B}.$$ Since ...
5
votes
0answers
596 views

Every tree has two leaves. Is my proof ok?

A tree is a connected acyclic graph. A leaf is a vertex of degree one. The distance $d(u,v)$ between two vertices $u$ and $v$ of a graph is the length of the shortest path from $u$ to $v$. Theorem. ...
5
votes
0answers
82 views

Check proof about range of bounded linear operator.

I have to prove that the range $\mathcal{R}(T)$ of bounded linear operator $T:X\rightarrow Y$; $X,Y$ normed spaces need not be closed in $Y$. As a hint I'm given that I could consider ...
5
votes
0answers
136 views

In which of the three topologies is $f(t)=(t, 2t, 3t, 4t, \ldots)$ continuous? Here, $f$ is a function from $\mathbb{R}$ to $\mathbb{R}^\omega$.

Can someone please verify my proof or offer suggestions for improvement? Consider the product, uniform, and box topologies on $\mathbb{R}^\omega$. In which of the three topologies is $f(t)=(t, 2t, ...
5
votes
0answers
120 views

Jech, “Set theory” exercises 12.11 - Is my proof right?

I try to prove the Jech's "Set theory", exercises 12.11: 12.11. If $\kappa$ is an inaccessible cardinal, then $V_\kappa\models \text{there is a countable model of ZFC}$. My attempt. Since ...
5
votes
0answers
116 views

A power of the characteristic polynomial

Let $A$ be a square matrix with real or complex coefficients of size $n$. Define its characteristic polynomial by $\chi_A(X) = \det(A-XI_n)$ (or $\det(XI_n-A)$ if you prefer). The question is : Prove ...
5
votes
0answers
215 views

Complex Conjugation as an $F$-Automorphism of $K$?

I am struggling with the following problem: Let $f \in F[x]$ be an irreducible quintic polynomial with splitting field $K$, where $\mathbb{Q} \subseteq F$. Supposing that $f$ has three real roots ...
5
votes
0answers
137 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
777 views

The tangent bundle of a Lie group is trivial

I'm trying to recreate the proof given by Alex Youcis at http://math.stackexchange.com/a/308798/86801 and got everything except $\displaystyle (L_h)_\ast \left.\frac{\partial}{\partial x_i}\right|_e = ...
5
votes
0answers
1k views

Prove that every compact metric space $K$ has a countable base, and that $K$ is therefore separable.

Prove that every compact metric space $K$ has a countable base, and that $K$ is therefore separable. How does the following look? Proof: For each $n \in \mathbb{N}$, make an open cover of $K$ by ...
5
votes
0answers
88 views

representation theorem on the path space

I'm working on a project and have done some work. However, there are some point where I'm unsure if my thoughts are correct. It would be appreciated if someone could share their thoughts about it. ...
5
votes
0answers
167 views

Point Group of a pattern

I need to determine the point group of the following patterns: I think the one on the left is $D_1$ and the one on the right is $D_2$
5
votes
0answers
330 views

Prove the FHHF theorem using as much abstract non-sense as possible

This is my second attempt to solve exercise 1.6H from Vakil: Assume $F$ is a covariant right-exact functor and show that we get a map $HF^i \rightarrow H^iF$. Attempt to a solution: Apply $F$ to the ...
5
votes
0answers
171 views

Sequence of convex functions converges uniformly

I am working on the following problem. Let $f_{n}: [a, b] \rightarrow \mathbb{R}$ be a sequence of convex functions. Furthermore, for each fixed $x \in [a, b]$, suppose $f(x) = \lim_{n ...
4
votes
0answers
42 views

Is the following proof of $Z(S_n)=\{id\}$ correct?

I wanted to prove that the center of the symmetric group $S_n$, $n\geq 3$ is trivial. Is my argument correct? Suppose $\alpha\in Z(S_n)$, that is $\alpha\beta=\beta\alpha$ for all $\beta\in S_n$. We ...
4
votes
0answers
38 views

A is positive definite implies Ker(A) = 0?

If $A$ is a positive definite matrix can it be concluded that the kernel of $A$ is $\{0\}$? pf: R.T.P $\ker(A) = 0$, Suppose not i.e there exists some $x$ in $\ker(A)$ s.t $x \neq 0$, then $$Ax = 0 ...
4
votes
0answers
61 views

If $m(A)=0$ and $f:[0,1]\rightarrow [0,1]$ has bounded derivative then $m(f(A))=0$

Let $A\subset [0,1]$. Show that if $m(A)=0$ and $f:[0,1]\rightarrow [0,1]$ has bounded derivative then $m(f(A))=0$. ($m$ denotes Lebesgue measure on $[0,1]$) If $m(A)=0$ then for any ...