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- Fri Nov 06, 2020 5:35 am
- Forum: Algebraic Structures
- Topic: Not a Hopfian group
- Replies: 1
- Views: 994
Re: Not a Hopfian group
Well, we define $f:\mathcal{G} \rightarrow \mathcal{G}$ by $f(x)=x^2$ and $f(y)=y$ and extend it to $\mathcal{G}$ homomorphically. Since $\mathcal{G}$ is well defined then $f$ is a surjective because $$f\left ( y^{-1} xy x^{-1} \right ) = x$$ but not an isomorphism because if we take $z=y^{-1} x y$ ...
- Sat Dec 14, 2019 3:16 pm
- Forum: Calculus
- Topic: \(\sum_{n=2}^{\infty} \frac{(-1)^n}{n\log{n}}\)
- Replies: 1
- Views: 1925
Re: \(\sum_{n=2}^{\infty} \frac{(-1)^n}{n\log{n}}\)
Basically it equals to
$$\int_1^\infty \left( 1+\left(2^{1-s}-1\right)\zeta(s) \right) \, \mathrm{d}s$$
However, the $\zeta$ function does not behave well under integrals. So, I would not expect a closed form to exist ... !
$$\int_1^\infty \left( 1+\left(2^{1-s}-1\right)\zeta(s) \right) \, \mathrm{d}s$$
However, the $\zeta$ function does not behave well under integrals. So, I would not expect a closed form to exist ... !
- Wed Oct 23, 2019 7:58 pm
- Forum: General Mathematics
- Topic: An inequality
- Replies: 1
- Views: 2883
Re: An inequality
The Engels form of the Cauchy – Schwartz inequality gives us: \begin{align*} \sum \frac{\log_{x_1}^4 x_2}{x_1+x_2} & \geq \frac{\left (\sum \log_{x_1}^2 x_2 \right )^2}{\sum (x_1+x_2)} \\ &= \frac{\left ( \sum \log_{x_1}^2 x_2 \right )^2}{2\sum x_1} \\ &\!\!\!\!\!\!\overset{\text{AM-GM}}{\geq } \fra...
- Sun Oct 20, 2019 6:15 pm
- Forum: General Mathematics
- Topic: Arithmotheoretic limit
- Replies: 0
- Views: 2871
Arithmotheoretic limit
Evaluate the limit:
$$\ell= \lim_{n \rightarrow +\infty} \frac{1}{n^2} \sum_{m=1}^{n} n \pmod m$$
$$\ell= \lim_{n \rightarrow +\infty} \frac{1}{n^2} \sum_{m=1}^{n} n \pmod m$$
- Sat Oct 12, 2019 12:46 pm
- Forum: Blog Discussion
- Topic: A logarithmic Poisson integral
- Replies: 1
- Views: 1842
Re: A logarithmic Poisson integral
It is closely related to another famous integral; namely $$\int_{0}^{\pi}\frac{\mathrm{d} \theta}{1-2a\cos \theta+a^2}\quad , \quad |a|<1$$ Evaluation of the integral: For $|a|<1$ we have successively: \begin{align*} \int_{0}^{\pi} \frac{{\rm d}x}{1-2a \cos x+a^2} &= \frac{1}{2} \int_{-\pi}^{\pi} \f...
- Mon Sep 30, 2019 2:54 pm
- Forum: Linear Algebra
- Topic: Linear Projection
- Replies: 2
- Views: 2150
Re: Linear Projection
Hi ,
I'm sorry but I do not understand what exactly you wrote down! Could you please elaborate?
I'm sorry but I do not understand what exactly you wrote down! Could you please elaborate?
- Wed Sep 25, 2019 2:49 pm
- Forum: General Mathematics
- Topic: Inequality in a triangle
- Replies: 0
- Views: 1811
Inequality in a triangle
Let $ABC$ be a triangle and denote $a, b, c$ the lengths of the sides $BC , CA$ and $AB$ respectively. If $abc \geq 1$ then prove that
$$\sqrt{\frac{\sin A}{a^3+b^6+c^6}} + \sqrt{\frac{\sin B}{b^3+c^6+a^6}} + \sqrt{\frac{\sin C}{c^3 + a^6+b^6}} \leq \sqrt[4]{\frac{27}{4}}$$
$$\sqrt{\frac{\sin A}{a^3+b^6+c^6}} + \sqrt{\frac{\sin B}{b^3+c^6+a^6}} + \sqrt{\frac{\sin C}{c^3 + a^6+b^6}} \leq \sqrt[4]{\frac{27}{4}}$$
- Sun Sep 22, 2019 7:51 pm
- Forum: Meta
- Topic: Welcome to the new and improved mathimatikoi.org
- Replies: 7
- Views: 7192
Re: Welcome to the new and improved mathimatikoi.org
Oh, also how about adding an Equation Editor button next to the Preview Button so that it links to an equation editor to speed up typesetting?
- Sun Sep 22, 2019 5:36 pm
- Forum: Linear Algebra
- Topic: Linear Projection
- Replies: 2
- Views: 2150
Linear Projection
Let $\mathcal{V}$ be a linear space over $\mathbb{R}$ such that $\dim_{\mathbb{R}} \mathcal{V} < \infty$ and $f:\mathcal{V} \rightarrow \mathcal{V}$ be a linear projection such that any non zero vector of $\mathcal{V}$ is an eigenvector of $f$. Prove that there exists $\lambda \in \mathbb{R}$ such t...
- Sun Sep 22, 2019 5:31 pm
- Forum: Meta
- Topic: Welcome to the new and improved mathimatikoi.org
- Replies: 7
- Views: 7192
Re: Welcome to the new and improved mathimatikoi.org
I really like the add of the topic tags. Now every topic can be sorted into categories and be found easier. Hey , what about a live topic preview? That would be fantastic!