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

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2
votes
1answer
34 views

Very basic question about set theory: unions and intersection

Let $\{ E_n \}_{n=1}^{\infty} $be a collection of countable sets and let $$ F_k = E_k \setminus ( \bigcup_{j=1}^{k-1} E_j ) $$ Then $F_k$ are pairwise disjoint and $\bigcup^{\infty} F_k = ...
7
votes
3answers
83 views

valid proof of series $\sum \limits_{v=1}^n v$

$$\sum \limits_{v=1}^n v=\frac{n^2+n}{2}$$ please don't downvote if this proof is stupid, it is my first proof, and i am only in grade 5, so i haven't a teacher for any of this 'big sums' proof: if ...
0
votes
1answer
26 views

Find the cummulative distribution function and the density function of the random variable: $Y={1\over 1+U}$

Let $U$ have a uniform distribution on $[0,1]$. Find the cummulative distribution function and the density function of the random variable: $Y={1\over 1+U}$ My attempt: $F_Y(x)=P[Y\le x]=P[{1\over ...
0
votes
2answers
73 views

Fixing the closed form of $\sum_{k=1}^nk\sin^2(kx).$

I've been working on finding the closed form of this:$$\sum_{k=1}^nk\sin^2(kx).$$ Using the fact that:$$\sum_{k=1}^nku^k={u\over (1-u)^2}\bigg[nu^{n+1}-(n+1)u^n+1\bigg]\forall u\ge 1\quad (1)$$ I ...
3
votes
0answers
40 views

Proving continuity on Sobolev space with weak topology

Hi I am interested in proving that an operator $$\eta : W^{1,p}(\Omega) \times L^{p}(\Omega) \rightarrow L^{p'}(\Omega)$$ is (weak $\times$ norm, norm) continuous. I want to know if it is viable to ...
1
vote
1answer
33 views

Proof if $f$ continuous in $x_0$ then there is a neighbourhood of $x_0$ so f bounded.

I have this question : Proof if $f$ continuous in $x_0$ then there is a neighbourhood of $x_0$ so $f$ bounded. I want to know if my proof is valid : If continuous in $x_0$ then : $$\lim_{x \to ...
1
vote
1answer
57 views

Minimum number of edges to ensure connectedness

Question: Consider a simple graph G with n vertices. What is the minimum number of edges that G must have in order to ensure that it is connected? Justify your answer. My attempt: Let G = $(V, E)$. ...
2
votes
1answer
39 views

Showing a Group $G$ is not Simple [duplicate]

Let $G$ be a finite group of order $pq$, where $p,q$ are distinct prime numbers. Show that $G$ is not simple. Here is my attempt: $|G|=pq$. If $G$ is not simple, then it has non-trivial subgroups, ...
0
votes
1answer
30 views

How many peas one can win

$A$ and $B$ plays the following game. In a table there are $n>1$ plates which are empty at the beginning. In the beginning of every round, $A$ moves some plates to the right hand side of the board, ...
1
vote
0answers
31 views

Orthonormal set is a Hilbert basis $\iff$ Parseval's identity is true

Let $H$ be a Hilbert space and $\{e_k:k\in \mathbb{Z}\}$ an orthonormal set. Prove that the set is a Hilbert basis if and only if Parseval's identity is true. The direct theorem is almost ...
0
votes
2answers
19 views

Justify each step in the proof sequence

$[A \rightarrow ( B \lor C) ] \land B' \land C' \rightarrow A'$ I know how to read the proof sequence, but I don't know what it means to "justify" each step? Does this mean to just state what each ...
0
votes
1answer
34 views

Rubik's Slide Proof's and Symmetries in a Rubik's Slide

$\quad$In the February edition of The Mathematical Association of America Monthly there is a article called "$\mathit{Rubik's\ on\ the\ Torus}$". Where they are dealing with solving problems involving ...
28
votes
1answer
1k views

Checking a possible proof of Fermat's Last Theorem

Theorem 1.2 of Bennett and Skinner (Canad. J. Math., 2004) asserts that the Diophantine equation $x^{p} - 4y^{p} = z^{2}$ is unsolvable for every prime $p \geq 7.$ The following is a possible proof ...
1
vote
0answers
41 views

Working with the Mobius transformatios and linear algebra.

Let $M=\left( \begin{array}{ccc} a & b \\ c & d \\ \end{array} \right) \in GL_{2}(\mathbb{C})$ and we recall that the Möbius transformation attached to $M$ is the map: $z \to ...
1
vote
1answer
23 views

Looking for a way to improve my inductive proof of a statement derived by Rolle's Theorem

The following problem is 'absolutely' clear: Problem: Let $f$ be continuous on the interval $[a,b]$ and $n$-times differentiable on $(a,b)$ and $f$ vanishes on $n+1$ points $x_0< x_1 < \dots ...
0
votes
0answers
21 views

Isomorphism class of groups

$G$={1,9,16,22,29,53,74,79,81} which is a subgroup of U(91). The question is to find the isomorphism classes of G. I have figured out that U(91) is isomorphic to U(7)+U(13), where plus is the direct ...
0
votes
1answer
35 views

Counting sets and adding an element

Let $A$ be a set with $n$ elements, where $n \in \mathbb{\omega}$. Suppose $s \notin A$, prove that $A \cup \{s\}$ has $n+1$ elements. Here is what I have done so far: By induction, let $P(n):$ if ...
3
votes
3answers
120 views

$p:E\to B$ is fibration then $p_*:map(X,E)\to map(X,B)$ is fibration as well.

$p:E\to B$ is fibration then for $X$ being compactly generated weakly Hausdorff space $p_*:map(X,E)\to map(X,B)$ is fibration as well. We'd like to show that for any $Y$ and continuous $f$ and ...
2
votes
2answers
29 views

When there is a proposition $(P\rightarrow Q)$, which row in the truth table of $\rightarrow $ should I use?

I solved one question in a book of analysis, and although I used an informal method to check it, I'd like to know more about what should be done. The question was the following: $A\subset X$ ...
2
votes
0answers
16 views

Suppose $H:= \{\sigma \in G| \sigma(1) = 1\}$, if for any $j \in \{1,2,…,n\}$ $t_j\in G$ such that $t_j(1) = j$. Show that $|G| = n|H|.$

Let G be a subgroup of the symmetric group $S_n$ in n letters. Consider the following subset of G: $$H:= \{\sigma \in G| \sigma(1) = 1\}$$ Suppose that G acts on the set $\{1,2,...,n\}$ transitively ...
0
votes
2answers
33 views

elementary algebra question On the generator of a group

(Def): Let $G$ be a group and $X \subset G$. Let $\{ H_{\alpha} \}_{\alpha \in \Gamma} $ be a collection of all subgroups of $G$ which contain $X$. Then $\bigcap_{\alpha \in \Gamma} H_{\alpha} $ is ...
1
vote
1answer
24 views

Poincare Inequality for 1-Dimensional Problem.

I am referring to the book Introduction to Functional Analysis to Boundary Value Problems and Finite Element by Daya Reddy (page ...
2
votes
1answer
21 views

Is $T:= \{g \in A_4|g^2 =(1)\}$ a subgroup of $A_4$?

Consider the subset $$T:= \{g \in A_4|g^2 =(1)\}$$ of the alternating group $A_4$ in four letters. Is T a subgroup of $A_4$? My Proof: Yes. If I am not wrong T is the Klein 4-group since only ...
0
votes
0answers
22 views

Proving the limit comparison test

I have the next attempt: Because $0<L< \infty$, we can find two positive and finite numbers, $m$ and $M$, such that $m<L<M$. Now, because $L = lim_{n\to \infty} \frac{a_{n}}{b_{n}}$ we ...
1
vote
0answers
43 views

Is my proof right of this result?

Suppose that: $\sum_{n=1}^{\infty}a_{n}$ converges absolutely and $\{b_n\}$ is bounded.Prove that $\sum_{n=1}^{\infty}a_{n}b_n$ converges absolutely. My attempt: Let $M$ be the upper bound of ...
2
votes
2answers
58 views

Verifying the convergence of a series.

I need to prove that the series $$\sum_{n=0}^{\infty}3^{-n}$$ converges and to find the limit. My attempt: We can express our series as: ...
2
votes
1answer
29 views

The space $BPV(0,1)$ is a separable metric space under certain metric.

This is exercise 2.42 from Leoni's book A First Course in Sobolev Spaces. The BPV is defined as the space contain the function $u$ such that $$ \text{Var}[u]:=\sup\left\{ ...
1
vote
1answer
36 views

Stability of $a$ implies $\lim _{t\to \infty} x(t)= a$

I have the differential equation $x'=f(x),x\in\mathbb{R}^n$. Let $a$ be a stable point of the differential equation, I want to prove that if $x(t)$ is a solution such that $\forall ...
2
votes
2answers
22 views

Markov Chain: Moving on a circle

A particle moves on 12 points situated on a circle. At each step it is equally likely to move one step in the clockwise or in the counterclockwise direction. Find the mean number of steps for ...
2
votes
3answers
134 views

Prove that $ 2^n \not \equiv 1 \pmod{n} $ for any $n > 1$.

I have proved this in following way. Assume that $ 2^n \equiv 1 \pmod{n} $. that means $n\mid(2^n -1)$. but by proof by contradiction, for $n=3$ this does not hold and we can say $n \nmid (2^n-1) ...
1
vote
1answer
39 views

Prove that any subfield of $\Bbb R$ contains $\Bbb Q$

Prove that any subfield of $\Bbb R$ must contain $\Bbb Q$. Now for any subfield $F$ of $\Bbb R$, $1\in F$ so, $\Bbb Z \subset F \Rightarrow \Bbb Q \subseteq F$. Have I done it correctly?
0
votes
2answers
51 views

Let M and N be normal subgroups of a group G such that $G = MN.$ Prove that $G/(M \cap N) \simeq G/M \times G/N$.

Let M and N be normal subgroups of a group G such that $G = MN.$ Prove that $G/(M \cap N) \simeq G/M \times G/N$. Claim 1: $M\cap N$ is a normal subgroup of G: Proof: $1_G \in M$ and $N$ since M ...
0
votes
0answers
42 views

Proving a function constant

The problem is to prove f is a constant function given that: $$f:R \to Q $$ is a continuous function. I was hoping to show two cases given a is rational and b is irrational and the equate the two. ...
0
votes
0answers
26 views

Eigenvalue and Eigenvector for the linear transformation in $ \mathbb{Z}_2^4$

I'm trying to find the Eigenvalue and Eigenvector for the Linear transformation: $T:\mathbb{Z}_2^4 \to \mathbb{Z}_2^4: (x_1,x_2,x_3,x_4)=(x_1+x_3,-2x_1-x_3,x_2+x_4,x_2-x_4)$ My problem is with ...
2
votes
2answers
45 views

Proving and Finding a limit

I need to find the following limit and prove using the definition of limits. $$\lim_{x\to1} {x \over x+1} = \frac 1 2$$. Following the definition: $$\forall \epsilon \exists \delta : \lvert x - c ...
3
votes
1answer
44 views

Show that a group is abelian.

Let G be a group and m be a positive integer. Suppose that for all $\alpha, \beta \in G$, $$(\alpha \beta)^m = \alpha^m \beta^m,$$ $$(\alpha \beta)^{m+1} = \alpha^{m+1}\beta^{m+1},$$ and $$(\alpha ...
0
votes
1answer
39 views

Prove that every connected undirected graph with n vertices has at least n-1 edges.

I would appreciate it if anyone can verify my proof. It is a proof by induction, but I attempt to reason things out rather than using a purely mathematical approach, in a similar vein to many other ...
0
votes
1answer
5 views

Show that the entries of the square of diagonal matrix are equal to the square of the entries of the diagonal matrix.

The question seems trivial which is why I have some trouble coming up with a proof that is mathematically correct. BTW I cannot yet use eigenvalues as we have not yet covered them in class. If ...
0
votes
0answers
9 views

Let p be a prime number and let n ≥ 1. Show that $\mathbb{F}_p$ contains an element of order n iff p ≡ 1(mod n).

Let p be a prime number and let n ≥ 1. Show that $\mathbb{F}_p$ contains an element of order n iff p ≡ 1(mod n). For the reverse direction, assume $p \equiv 1(mod n). \text{Let } g \in \mathbb{F}_p ...
0
votes
1answer
28 views

Prove that if lim sup$_{n} n^{2}a_{n} = 0 \Rightarrow \sum_{k = 1}^{\infty} a_{k}$ converges

Prove that if lim sup$_{n} n^{2}a_{n} = 0 \Rightarrow \sum_{k = 1}^{\infty} a_{k}$ converges. Also assume that the sequence is positive. lim sup$_{n} n^{2}a_{n} = 0$ means that for every $\epsilon$, ...
1
vote
1answer
22 views

Prove that lim inf$_{n} na_{n} > 1 \Rightarrow \sum_{k = 1}^{\infty} a_{k}$ diverges

Prove that lim inf$_{n} na_{n} > 1 \Rightarrow \sum_{k = 1}^{\infty} a_{k}$ diverges The solution proof goes like: lim inf$_{n} na_{n} > 1 \Rightarrow$ there exists an $N \in \mathbb{N}$ such ...
1
vote
1answer
44 views

Induction proof for Fibonacci numbers

I am trying to get a hang on the induction method for proof, but I'm still dubious of many aspect of this proof regarding its application to sequences of integers, such as the Fibonacci sequence. ...
1
vote
0answers
19 views

Question from Stewart's Calculus regarding proof of independence of path and conservative vector fields.

Please look over this proof. In the proof, it says: "Notice that the first of these integrals does not depend on $x$, so..." How is that so? $C_1$ does depend on $x$. How/Why does it not ...
1
vote
0answers
25 views

Holomorphic funtions

Let $U$ be an open connected subset of $\mathbb{C}^n$, and $O(U)$ the ring of holomorphic functions on $U$. Prove that $O(U)$ is an integral domain. I have done If $fg\equiv0$ in $U$, then $f$ ...
0
votes
2answers
25 views

Transitivity relation in the set of Integers

Prove or disprove that R is transitive, where $R=\{ (a,a^2)| a \in \Bbb Z \}$ is a relation on $\Bbb Z$ By definition: $R$ is transitive $iff$ $$ (a,b)\in R \wedge (b,c) \in R\implies (a,c) ...
0
votes
1answer
18 views

For $f(\theta)= e^{\theta}$. Is it true that $\hat{f}(n)(1-in)=0$ for all $n\in \mathbb Z.$?

(This is motivated from the following question) Fact: If $f \in C^1(\mathbb{T})$, then the Fourier coefficients $\widehat{f'}(n)$ of the derivative $f′$ can be expressed in terms of the Fourier ...
1
vote
3answers
39 views

For $A,B,C\subset X$where $X$ is a metric space under some $d$, check if the triangle inequality holds for $d_m(A,B)=\min_{x\in A,y\in B}\{d(x,y)\} $

$$d_m(A,B)=\min_{x\in A,y\in B}\{d(x,y)\} $$ Is it the case that $$d_m(A,C)\leq d_m(A,B)+d_m(B,C)$$ based on the definition of $d_m$ and the fact that $d$ is already some arbitrary metric on $X$? I ...
0
votes
0answers
52 views

Proving symmetry of metric (single linkage between clusters using arbitrary dissimilarity measure)

I am told to assume that our dissimilarity measure $d$ satisfies the properties required of it, what seems to be the definition of a metric: $d(x,y) \geq0 $ and $d(x,y)=0 \Longleftrightarrow x=y$ ...
0
votes
0answers
40 views

For what values $q,r$ does the improper integral $\int_0^1 x^q (1-x^2)^r dx$ converge?

Question: For what values $q,r$ does the improper integral $\int_0^1 x^q (1-x^2)^r dx$ converge? I had some ideas from a previous thread but now I have a different attempt that I would like to ...
1
vote
1answer
25 views

Define $\phi:G/H \rightarrow G/K$ by $\phi(Ha)=Ka$. Prove: $\phi$ is a well defined function.

Let $H$ and $K$ be normal subgroups of a group $G$, with $H \subseteq K$. Define $\phi:G/H \rightarrow G/K$ by $\phi(Ha)=Ka$. Prove: $\phi$ is a well defined function. [That is, if $Ha=Hb$, ...