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

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2
votes
3answers
116 views

How to show distributivity in a ring, and what is wrong with my algebra?

I am trying to show the following is a commutative ring with unity, however I am encountering a problem. First, addition and multiplication are defined as: $$a \oplus b=a+b-1$$$$a \odot ...
0
votes
1answer
15 views

Conditions for magic square.

So I've messing around with magic squares and something occured to me: Assume we have a nxn grid of numbers which respects the sum conditions of a magic square as in it has the appropriate column, ...
1
vote
2answers
47 views

Suppose G is a group, p is prime , Then the number of elements of G of order p is multiple of (p-1) [on hold]

I need Help . "Suppose $G$ is a group, $p$ is prime , Then the number of elements of G of order $p$ is multiple of $(p-1) $". Give me any advise or note
2
votes
2answers
88 views

There is no homomorphism from $\mathbb{Z_8} \times \mathbb{Z_2} \times \mathbb{Z_2}$ onto $\mathbb{Z_4} \times \mathbb{Z_4}$

If such a homomorphism $\phi$ existed, then the first isomorphism theorem says that $|\ker \phi| = 2$. Since $\mathbb{Z_8} \times \mathbb{Z_2} \times \mathbb{Z_2}$ is abelian, then every subgroup is ...
3
votes
1answer
49 views

If $n\mid m$ prove that the canonical surjection $\pi: \mathbb Z_m \rightarrow \mathbb Z_n$ is also surjective on units

Not sure if this is the right proof (i found it online): Since $n\mid m$, if we factor $m = p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}$, then $n = p_1^{\beta_1}p_2^{\beta_2}\cdots ...
0
votes
1answer
20 views

Lambert W function identity from differential equation

For constants $v,K$ and a function $C(t)$, can you prove that if : $$ \frac{dc}{dt} = - \frac{vc(t)}{K + c(t)},~\text{with } c(0) = c_0 $$ Then the solution: $$ \left[ K \ln c(t) + c(t) ...
0
votes
1answer
8 views

Lambert W function multiplication with scalar

Let $W$ be the Lambert W function, $Y$ be a real valued function and $x \in \mathbb{R} $. Given $ Ye^Y = x \iff Y = W(x) $ is it true that $Y = kW(\frac{1}{k}x)$ for non-zero $k \in \mathbb{R} $ ? ...
0
votes
2answers
38 views

If H and K are finite subgroups of G (another proof )

I have a question and it's solution , but I want another proof if there exist . Thanks
1
vote
1answer
21 views

Trivial mathematical analysis problem

Let $\mathbb{R} \supset E \neq \varnothing$. Put $\alpha = \sup E $. Then, for all $n \in \mathbb{N}$, $\alpha - \frac{1}{n} $ is not an upper bound of $E$, but $\alpha + \frac{1}{n}$ is an upper ...
1
vote
0answers
31 views

Proof of Bolzano-Weierstrass for functions over countable domains

Theorem A bounded sequence of functions defined over a countable domain has a convergent subsequence. The attempted proof uses Bolzano-Weierstrass for real sequences (nested bisections) proof and a ...
1
vote
0answers
18 views

Help with proof about merge two heaps to one heap…

We have two heaps: $H_1,H_2$ that have $n_1,n_2$ elements ($H_1$ have $n_1$ elements and $H_2$ have $n_2$ elements). We know that the smallest element at $H_1$ is bigger the root (the biggest element) ...
1
vote
1answer
26 views

An element $u$ is an upper bound of $E$ if and only if $t>u$ implies $t\notin E$

Let $S$ be an ordered field and $S \supset E\neq \varnothing$. Then, the following are equivalent: $u \in S$ is an upper bound of $E$. $t \in S$ and $t > u$ implies $t \notin E $. My Try: ...
0
votes
0answers
14 views

Surjective $\gamma \colon I \to M^1$, $\gamma (t_1)=\gamma (t_2)$ can be extended to a periodic parametrization of $M^1$

Suppose that $\gamma \colon I \to M^1$ is a smooth surjective curve in a Riemannian connected 1-dimensional manifold. Furthermore, suppose that it is parametrized via arc lenght i.e.:$$||\dot ...
2
votes
0answers
23 views

On the greatest lower bound property

Proposition: Let $S$ be an ordered field and $S \supset E \neq \varnothing $. $E$ is bounded below. Then $ \inf E = - \sup ( - E ) $ Try: Write $- E = \{ -x : x \in E \} $ and let $l $ be a lower ...
0
votes
1answer
34 views

Use Resolution to proove a sentence in First Order Logic

I was just wondering if anyone could tell me if I've solved this problem right. If wrong, I would like to know what I did wrong. "Use resolution to prove Green(Linn) given the information below. You ...
0
votes
1answer
14 views

Another characterization of the supremum of a set

$u$ is an upper bound of a set $E \subset S$ if given any $\epsilon >0$, there is $\delta \in E $ such that $u - \epsilon < \delta$. PROBLEM: An upper bound $u$ of $E \subset S$ ($E \neq ...
0
votes
1answer
53 views

System with two quadratic equations

Respected All. I am unable to find out what's so wrong in the following. Please help me. It is given that $t$ is a common root of the following two equations given by \begin{align} &x^2-bx+d=0 ...
0
votes
2answers
37 views

Is this a valid proof of the Quotient rule?

In an emergency in high-school, I once derived the quotient rule from the chain and product rules. I now wonder whether this was actually a valid proof. I reconstructed it as well as I could remember: ...
1
vote
2answers
47 views

Prove there exists dense open set

Let $G$ be an open set in $X$ and $D$ be a dense open set in $G$.Show there exists a dense open subset $V$ of $X$ such that $V\cap G=D$. Since $D$ is open in $G$, there exists $V$ open in $X$ ...
1
vote
1answer
25 views

big-Oh prove or disprove 2^n is in big-Oh(3^n)

the definiton of Big-Oh says $\exists c\in$R+,$\exists B\in$ N,$\forall n\in$N, $n \geq B$$\implies$$2^n \leq c\times 3^n$. I believe $2^n \in O(3^n)$, but how to prove it? can anyone help. This this ...
0
votes
0answers
21 views

Do these statements prove this formula?

$$ \frac{d^n}{dx^n}[g(x)^{f(x)}] = g(x)^{f(x)} B_n(d_1,\cdots,d_n) $$ Calling $$ d_n = \frac{d^n}{dx^n}[ln(g(x))f(x)] $$ Since faa di bruno's formula states $$ \frac{d^n}{dx^n}[f(g(x))] = ...
1
vote
0answers
34 views

Proof Verification for $n2^{n-1} = \sum\limits_{k=1}^n k\binom{n}{k}$

$(1+x)^n = \sum\limits_{k=0}^n\binom{n}{k}x^k$ by binomial theorem $\frac{d}{dx}(1+x)^n =\frac{d}{dx}\sum\limits_{k=0}^n\binom{n}{k}x^k$ $n(1+x)^{n-1} = \sum\limits_{k=1}^n k\binom{n}{k}x^{k-1}$ ...
1
vote
1answer
19 views

Give an combinatorial argument

I need to find the possible value of $R_i$ and prove it by giving combinatorial argument, for following identity. I was able to give an argument like this. Consider double counting. Count ...
3
votes
1answer
31 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
votes
1answer
36 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
votes
2answers
63 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
vote
2answers
57 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
votes
1answer
37 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
votes
2answers
43 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
votes
5answers
246 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
votes
0answers
25 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 ...
2
votes
1answer
49 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 ...
0
votes
1answer
41 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
votes
1answer
33 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
71 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
24 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
49 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
37 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
vote
1answer
41 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
37 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
29 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
votes
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
votes
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
votes
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
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
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
18 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
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
votes
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 ...