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Forum: Algebraic Structures Topic: Find the number of homomorphism 
Riemann 
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: $\mathbb{R}^2 \setminus \mathbb{Q} \times \mathbb{Q}$ 
Riemann 
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 
Riemann 
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 
Riemann 
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 
Riemann 
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 
Riemann 
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 $$\sum\limits_{n=1}^{\infty}\mathcal{H}_{pn}x^n = \frac{1}{p}\sum\limits_{k=0}^{p1} \frac{\ln \varphi_k}{\varphi_k}$$ 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 
Riemann 
Posted: Wed Aug 01, 2018 9:40 am


Replies: 1 Views: 551

Given the function $f(x)=e^xx1$ 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! 
Riemann 
Posted: Wed Aug 01, 2018 9:35 am


Replies: 1 Views: 465

Tolaso J Kos wrote: Evaluate the following sum: $$\sum_{{\rm d}\mid 10!}\frac{1}{{\rm d}+\sqrt{10!}}$$
$$\sum_{{\rm d}\mid 10!}\frac{1}{{\rm d}+\sqrt{10!}} = \frac{3\sqrt{7}}{112} $$
Full solution tomorrow morning ! 


Forum: Number theory Topic: Irrational number 
Riemann 
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 
Riemann 
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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