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 Forum: Algebraic Structures   Topic: Find the number of homomorphism

Posted: Thu Dec 06, 2018 7:43 pm 

Replies: 1
Views: 27


Hi Ram_1729. Could you explain what $Q_8$ and $K_4$ stand for?

 Forum: General Topology   Topic: $\mathbb{R}^2 \setminus \mathbb{Q} \times \mathbb{Q}$

Posted: Thu Dec 06, 2018 7:43 pm 

Replies: 2
Views: 264


Thank you Ram_1729. Exactly! It was an exam's question!

 Forum: General Mathematics   Topic: An inequality

 Post subject: An inequality
Posted: Thu Nov 22, 2018 9:26 pm 

Replies: 0
Views: 29


Let $x_1, x_2, \dots, x_n$ be $n \geq 2$ positive numbers other than $1$ such that $x_1^2+x_2^2+\cdots +x_n^2=n^3$. Prove that:

$$\frac{\log_{x_1}^4 x_2}{x_1+x_2}+ \frac{\log_{x_2}^4 x_3}{x_2+x_3}+ \cdots + \frac{\log_{x_n}^4 x_1}{x_n+x_1} \geq \frac{1}{2}$$

 Forum: Algebraic Structures   Topic: Symetry group of Tetrahedron

Posted: Tue Nov 06, 2018 4:48 pm 

Replies: 2
Views: 112


⋅ The tetrahedron is a regular solid with $4$ vertices and $4$ triangular faces. The symmetry group is the alternating group $\mathcal{A}_4$. ⋅ The symmetry group of a cube is isomorphic to $\mathcal{S}_4$ , the permutation group on 4 elements. If we number the vertices of the c...

 Forum: Linear Algebra   Topic: Computation of determinant

 Post subject: Computation of determinant
Posted: Tue Oct 16, 2018 10:26 am 

Replies: 0
Views: 77


Let $A, B \in \mathcal{M}_{2 \times 2}$ be matrices with integer entries such that $AB = BA$ , $\det \left( A + B \right) =2$ and $\det \left( A^3 + B^3 \right) = 2^3$. Evaluate the determinant


$$\mathcal{D} = \det \left( A^2 + B^2 \right)$$

 Forum: Calculus   Topic: Series with general harmonic number

Posted: Sun Aug 12, 2018 8:27 pm 

Replies: 0
Views: 97


Let $\mathcal{H}_n$ denote the $n$ - th harmonic number. It holds that $$\sum\limits_{n=1}^{\infty}\mathcal{H}_{pn}x^n = -\frac{1}{p}\sum\limits_{k=0}^{p-1} \frac{\ln \varphi_k}{\varphi_k}$$ where $p \in \mathbb{N}$ and $\displaystyle \varphi_k = \varphi_k(x) = 1 - \sqrt[p]{x}\exp\left(\frac{-2\pi i...

 Forum: Analysis   Topic: Multiplicity of root

 Post subject: Re: Multiplicity of root
Posted: Wed Aug 01, 2018 9:40 am 

Replies: 1
Views: 551


Given the function $f(x)=e^x-x-1$ prove that $0$ is a zero of $f$ of multiplicity $2$. It suffices to prove that the limit $\displaystyle \lim \limits_{x \rightarrow 0} \frac{f(x)}{x^2}$ is finite. However, \begin{align*} \lim_{x\rightarrow 0} \frac{f(x)}{x^2} &= \lim_{x\rightarrow 0} \frac{e^x...

 Forum: General Mathematics   Topic: A sum!

 Post subject: Re: A sum!
Posted: Wed Aug 01, 2018 9:35 am 

Replies: 1
Views: 465


Tolaso J Kos wrote:
Evaluate the following sum:

$$\sum_{{\rm d}\mid 10!}\frac{1}{{\rm d}+\sqrt{10!}}$$

Source
Matha's notes.



$$\sum_{{\rm d}\mid 10!}\frac{1}{{\rm d}+\sqrt{10!}} = \frac{3\sqrt{7}}{112} $$

Full solution tomorrow morning !

 Forum: Number theory   Topic: Irrational number

 Post subject: Re: Irrational number
Posted: Wed Aug 01, 2018 9:33 am 

Replies: 1
Views: 529


Let $p \leq N$ be the last prime. If we prove that between $p$ and $N$ does not exist a number that has $p$ as a factor we are done. So, we need to prove that $2p>N$. But this is exactly what Bertrand's postulate says.

 Forum: Calculus   Topic: A definite Integral

 Post subject: Re: A definite Integral
Posted: Tue Jun 19, 2018 7:20 pm 

Replies: 2
Views: 224


Are you sure about the upper limit? Should not it be $\frac{\pi}{2}$ ?
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