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## Search found 171 matches

- Fri Nov 06, 2020 5:35 am
- Forum: Algebraic Structures
- Topic: Not a Hopfian group
- Replies:
**1** - Views:
**1224**

### 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:
**2195**

### 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:
**3286**

### 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:
**3158**

### 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:
**2212**

### 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:
**2520**

### 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:
**2024**

### 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:
**8406**

### 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:
**2520**

### 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:
**8406**

### 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!