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

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13
votes
0answers
367 views

My first proof that uses the well-ordering principle (very simple number theory). Please mark/grade.

What do you think about my first proof that uses the well-ordering principle? Please mark/grade. Theorem The sum of the cubes of three consecutive natural numbers is a multiple of 9. Proof ...
10
votes
0answers
260 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 ...
8
votes
0answers
238 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
119 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
159 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 $x_{0}$ ...
7
votes
0answers
609 views

bisectors of an angle in a triangle intersect at a single point - proof verification

Let´s consider a general triangle ABC. Let´s draw two angle bisectors from vertices A and B. It is obvious that these two angle bisectors intersect at a single point X. Since X lies on the angle ...
6
votes
0answers
143 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
56 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
129 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
128 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
108 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 ...
6
votes
0answers
749 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 ...
6
votes
0answers
136 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
230 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 ...
5
votes
0answers
38 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
49 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
78 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
71 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
77 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
64 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
102 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
989 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} ...
5
votes
0answers
108 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
175 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
126 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
590 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
84 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. ...
4
votes
0answers
18 views

Iterated circumcenters - proving collinearity and establishing distance ratios

Let $P_0, P_1, P_2$ be three points on the circumference of a circle with radius $1$, where $P_1P_2 = t < 2$. For each $i \ge 3$, define $P_i$ to be the centre of the circumcircle of $\triangle ...
4
votes
0answers
57 views

Finding the expected value and variance of ${X^3}$

For a random variable $X$, $(X^3-1)$ is uniformly distributed in the interval $[0,7]$ I need to find the expected value and variance of $\color{blue}{X^3}$ and I know that: cumulative ...
4
votes
0answers
39 views

Too Many Members in a Finitely Generated Module are Linearly Dependent

I am new to module theory and as of now am not very comfortable with the subject. So can somebody please check whether my claim and its proof is okay? Consider the following statement: Let $M$ be ...
4
votes
0answers
71 views

How to Prove It, Exercise 6.5.9

Suppose $R$ is a relation on $A$ and $S$ is the transitive closure of $R$. If $(a, b) \in S$, then there is some positive integer $n$ such that $(a, b) \in R^n$, and therefore by the well-ordering ...
4
votes
0answers
33 views

subset of a compact set in $\mathbb{R}$ with nonempty interior has positive outer measure

Let $A\subset I=[a,b] \subset \mathbb{R}$, $a < b$ such that Int$(A) \neq \emptyset$. Show that $A$ has positive outer measure. What I have so far: Since Int$(A) \subseteq A$, by the ...
4
votes
0answers
97 views

Did I correctly derive the scheme for this PDE using the Crank Nicolson Method?

I'm taking an Applied Numerical Methods course this semester, and I was given the following homework problem: Basically, before I begin writing any sort of code, I would like to ensure that I have ...
4
votes
0answers
40 views

Show $\frac{\partial^2}{\partial ^2 x}+ \frac{\partial^2}{\partial ^2 y}= 4 \frac{\partial^2}{\partial z \partial{ \overline{z}}} $

I want to solve the following exercise: Show that: $$\frac{\partial^{2}}{\partial ^{2}x}+ \frac{\partial^{2}}{\partial ^{2}y}= 4 \frac{\partial^{2}}{\partial z \partial{ \overline{z}}} $$ My ...
4
votes
0answers
122 views

$\zeta(2)=\frac{\pi^2}{6}$ proof improvement.

Recently in one of my calculus exercise I have made out a (quite novel to me) proof for $\zeta(2)=\frac{\pi^2}{6}$ via the famous infinite product below: ...
4
votes
0answers
69 views

Prove that for all integers, if $a$ is even and $b$ is odd then $a^{2}+3b$ is odd.

Theorem: For all integers, if $a$ is even and $b$ is odd then $a^{2}+3b$ is odd. So far my proof is as follows: Let $a$ be any even integer Let $b$ be any odd integer By the definition of even ...
4
votes
0answers
179 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. ...
4
votes
0answers
50 views

Limit of continuous function

Prove or provide a counterexample: 1) $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous. If $(a_{n}) = f(n)$ converges to $L$, then $\lim_{x \rightarrow \infty} f(x) = L$. Counterexample: I ...
4
votes
0answers
52 views

My proof of: Every convergent real sequence is a Cauchy sequence.

Is my proof correct? Let $(x_n)_{ n \in \mathbb{N} }$ be a real sequence. $\textbf{Definition 1.}$ $(x_n)$ is $\textit{convergent}$ iff there is an $x \in \mathbb{R}$ such that, for every ...
4
votes
0answers
85 views

Prove that: $ \left( \sum_{i\neq j}a_{i}b_{j} \right)^2 \geq \left( \sum_{i\neq j}a_{i}a_{j} \right) \left( \sum_{i\neq j}b_{i}b_{j} \right)$

Let $a_{1}, \cdots, a_{n}, b_{1}, \cdots, b_{n}$ be positive real numbers. Prove that: $$ \left( \sum_{i\neq j}a_{i}b_{j} \right)^2 \geq \left( \sum_{i\neq j}a_{i}a_{j} \right) \left( \sum_{i\neq ...
4
votes
0answers
128 views

Show that a finite group G generated by two elements of order 2 is isomorphic to a dihedral group $D_{2n}$ for some n. (Proof Verification)

Show that a finite group G generated by two elements of order 2 is isomorphic to a dihedral group $D_{2n}$ for some n. (Proof Verification) Proof: Let G be generated by c, b, where $c^2 = b^2 = 1$. ...
4
votes
0answers
170 views

6-digit password - a special decoding method

Consider the situation of decoding a 6-digit password that consists of the symbols A to Z and 0 to 9, where all possible combinations are tried randomly and uniformly. Consider the ...
4
votes
0answers
67 views

$\mu$-clubs and stationary sets consisting of elements with cofinality $\mu$

Let $\mu < \kappa$ be infinite cardinals. A set $C$ is called a $\mu$-club in $\kappa$, if it is unbounded in $\kappa$ and contains all its limit points of cofinality $\mu$. Now let $T \subset S ...
4
votes
0answers
108 views

Compact family of Lip functions under the sup norm metric, proof verification.

Hi everyone I'd like to know if the following is correct, I'd appreciate your opinion and also any suggestion to improve my argument. Thanks in advance for your time. If $(K,d)$ is a compact ...
4
votes
0answers
86 views

$\aleph_\alpha^{\aleph_1} = \aleph_\alpha^{\aleph_0}\cdot 2^{\aleph_1}$ for all $\omega \le \alpha < \omega_1$

I'd like someone to check my proof that $\aleph_\alpha^{\aleph_1} = \aleph_\alpha^{\aleph_0} \cdot 2^{\aleph_1}$ for all $\omega \le \alpha < \omega_1$. First, let's prove that ...
4
votes
0answers
73 views

Two question about how to compute this integral limit

Let $f: (-\pi,\pi]\to \mathbb R$ be continuous and let $p_x (u) = {(f(u+x) - f(x)) \cos ({u \over 2}) \over \sin ({u \over 2}) }$. I want to show that $$ \int_{-\pi}^\pi p_x(u) \sin (Nu) du \to 0$$ ...
4
votes
0answers
112 views

If $f_n(t):=f(t^n)$ converges uniformly to continuous function then $f$ constant

Let $f \colon [0,1] \to \mathbb R$ be continuous and let $f_n$ be defined by $$ f_n(x):=f(x^n), \qquad x \in [0,1], \,\, n \in \mathbb N. $$ Suppose there exists a continuous function $g$ on ...
4
votes
0answers
114 views

Countable transitive $T \vDash ZFC-P$ with $\approx$ not absolute: why do we need $H(\aleph_3)$?

I am trying to solve Exercise IV.3.31 from Kunen's Foundations of Mathematics. I think I have a solution but I am confused by one of the hints. By request, here is the text of the exercise. ...
4
votes
0answers
113 views

Proof “correctness” : Cycle structure of conjugate permutations

My Algebra lecturer is a very strict about proofs(w.r.t Completeness , correctness and format ) more so than I have encountered in the past or any of my lecturers of the courses I am take concurrent. ...
4
votes
0answers
75 views

Prove that $\lim\limits_{t \rightarrow 0} \int\limits_{0}^{\infty} e^{-tx} f(x) dx = 1$ for $t>0$.

Suppose $f \in \mathcal{R}$ on $[0,A]$ for all $A < \infty$, and $f(x) \rightarrow 1$ as $x \rightarrow + \infty$. Prove that $\lim\limits_{t \rightarrow 0} \int\limits_{0}^{\infty} e^{-tx} f(x) ...