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

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0answers
12 views

Give an combinatorial argument

I need to find what could be the value of $R_i$ and by giving the combinatorial argument prove it, for following identity. I was able to give an argument like this. Consider double counting. count ...
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0answers
17 views

Direct sum of simple modules and Schur's Lemma

Suppose $M,N$ are two non-isomorphic simple $R$-modules. For $m,n\geq1$, is it true that $$ \text{Hom}_R(M^{\oplus m},N^{\oplus n})\cong\hat{0}\,? $$ I think it's true by Schur's Lemma. ...
0
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1answer
23 views

King Arthur and knights at the round table puzzle

Can you help me with this math problem: Each of the K knights from the round table needs to choose a card which is marked with a number from 1 to N, N >= K. The cards all have different number. ...
2
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2answers
51 views

Show that a Series Diverges

Question: Let a sequence ($a_n$) have the property $\lim \limits_{n \to \infty} na_n = a > 0$ Show that the series $\sum_{n=1}^\infty a_n$ diverges Attempts: Basically, I firstly tried ...
1
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1answer
45 views

Is my proof regarding continuity at irrationals correct?

Consider the Thomae's function $$f(x)=\begin{cases} 0 \text{ ; when } x \text{ is irrational} \\\frac 1 q \text{ ; for } x=\frac p q \text{ irreducible fraction}\end{cases}$$ In the following proof ...
5
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1answer
35 views

Sequence of $f_n \in R[0, 1]$ that converges pointwise to $f \in R[0, 1]$ such that $\lim_{n \to \infty} \int_0^1 f_n dx \neq \int_0^1 f dx$.

Question: Find a sequence of functions $f_n \in R[0, 1]$ that converges pointwise to $f \in R[0, 1]$ such that $\lim_{n \to \infty} \int_0^1 f_n dx \neq \int_0^1 f dx$. ($R$ means ...
2
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2answers
39 views

Prove if $3$ does not divide $n$, then $n^2=1+3k$ for some integer $k$

I am proving by cases but am getting confused. I am not sure if this leads to a contradiction or not. Here's what I have so far: Direct Proof. Suppose $3$ does not divide $n$. Case 1: remainder ...
12
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5answers
198 views

How to prove $n < \left(1+\frac{1}{\sqrt{n}}\right)^n$

I want to know how to prove the following inequality. For $n = 1, 2, 3, \ldots $ $$ n < \left(1+\frac{1}{\sqrt{n}} \right)^n $$ I tried with math induction but I failed.
3
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0answers
21 views

Every $\sigma-$finite measure is semifinite. $(X, \mathcal{M}, \mu)$ is a measure space.

Definition 1: Say $X = \bigcup_{n=1}^{\infty} E_n $ where $E_n \in \mathcal{M}$ and $\mu( E_n ) < \infty $ for all $n$, we call $\mu$ $\sigma$-finite. More generally, if $E = \bigcup^{\infty} E_n ...
1
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1answer
38 views

Find the chance that $a^3 + b^3 \equiv 0 (\mod 3)$

We are given set of integer numbers $\{1,2, \dots N\}$. $N \ge 3$ Then perform a drawing with replacement of two elements $a$ and $b$. Problem is to find the probability of following statement holding ...
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1answer
36 views

If $a+b \geq x$ is known to be true does that mean $a+b\geq x-1$ contradicts it?

So I was proving something and I'm wondering if this line of argument is correct. Suppose that it is true that given conditions $M,N,O$; $a+b\geq x$. That is given those conditions the minimum value ...
2
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1answer
28 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 = ...
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3answers
64 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
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1answer
22 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
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2answers
46 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
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0answers
36 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
25 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
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1answer
38 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
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1answer
36 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
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1answer
28 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, ...
2
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0answers
26 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
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1answer
12 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
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0answers
17 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
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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 ...
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0answers
40 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
17 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
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0answers
16 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
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1answer
32 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 ...
2
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1answer
73 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 completely 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
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2answers
26 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
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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 ...
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2answers
29 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
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1answer
17 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
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0answers
20 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 ...
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0answers
42 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
57 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
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1answer
28 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\{ ...
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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
20 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
125 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
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1answer
38 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
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2answers
49 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
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0answers
40 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
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0answers
25 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
41 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
36 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 ...