Search found 375 matches

by Papapetros Vaggelis
Fri Jan 06, 2017 12:43 pm
Forum: ODE
Topic: Function
Replies: 2
Views: 4002

Function

Find the differentiable function \(\displaystyle{f:\mathbb{R}\to \mathbb{R}}\) which satisfies

\(\displaystyle{f(0)=1}\) and \(\displaystyle{f^\prime(x)=f^2(x)\,f(-x)\,,\forall\,x\in\mathbb{R}}\).
by Papapetros Vaggelis
Tue Nov 22, 2016 8:46 pm
Forum: Complex Analysis
Topic: Uniform convergence
Replies: 1
Views: 3705

Uniform convergence

Examine if there exists a sequence \(\displaystyle{\left(p_n(z)\right)_{n\in\mathbb{N}}}\) of

complex polynomials such that \(\displaystyle{p_n(z)\to \dfrac{1}{z}}\) uniformly to

\(\displaystyle{C_{r}=\left\{z\in\mathbb{C}: |z|=r\right\}}\) (\(\displaystyle{r>0}\)).
by Papapetros Vaggelis
Thu Nov 10, 2016 4:36 pm
Forum: General Mathematics
Topic: Double inequality
Replies: 0
Views: 2745

Double inequality

If \(\displaystyle{n\in\mathbb{N}\,,n\geq 2}\), then prove that

\(\displaystyle{n\,\left(\sqrt[n]{n+1}-1\right)<\sum_{k=1}^{n}\dfrac{1}{k}<n\,\left(1-\dfrac{1}{\sqrt[n]{n+1}}+\dfrac{1}{n+1}\right)}\).
by Papapetros Vaggelis
Fri Oct 28, 2016 11:46 am
Forum: Real Analysis
Topic: Bounded sequence
Replies: 6
Views: 8471

Re: Bounded sequence

Here is a solution. If \(\displaystyle{x=0}\) or \(\displaystyle{x=2\,\pi}\), then \(\displaystyle{s_n=0\,,n\in\mathbb{N}}\) and we are done. It is sufficient to prove the argument if \(\displaystyle{x\in\left(0,2\,\pi\right)}\) since the \(\displaystyle{\sin}\) - function is a \(\displaystyle{2\,\p...
by Papapetros Vaggelis
Wed Oct 26, 2016 6:51 pm
Forum: Real Analysis
Topic: Bounded sequence
Replies: 6
Views: 8471

Bounded sequence

Let \(\displaystyle{x\in\mathbb{R}}\). Prove that the sequence

\(\displaystyle{s_n=\sin\,x+...+\sin\,(n\,x)\,,n\in\mathbb{N}}\) is bounded.
by Papapetros Vaggelis
Mon Oct 24, 2016 8:40 pm
Forum: Linear Algebra
Topic: Vector space
Replies: 0
Views: 2273

Vector space

Let \(\displaystyle{A}\) be a non-empty set and \(\displaystyle{\mathcal{F}}\) be a non-empty collection of \(\displaystyle{1-1}\) and onto functions \(\displaystyle{f:A\to \mathbb{R}^n}\) such that : if \(\displaystyle{f\,,g\in\mathcal{F}}\) then \(\displaystyle{f\circ g^{-1}:\mathbb{R}^n\to \mathb...
by Papapetros Vaggelis
Sun Oct 16, 2016 11:55 am
Forum: Real Analysis
Topic: A simple integral
Replies: 2
Views: 2916

Re: A simple integral

Hi Riemann. Hi Grigorios. Here is another solution. \(\displaystyle{\begin{aligned} \int_{0}^{1}\dfrac{\sqrt{1-x^2}}{(1+x)^2}\,\mathrm{d}x&\stackrel{x=\sin\,u}{=}\int_{0}^{\pi/2}\dfrac{\sqrt{1-\sin^2\,u}\,\cos\,u}{(1+\sin\,u)^2}\,\mathrm{d}u\\&=\int_{0}^{\pi/2}\dfrac{\cos^2\,u}{(1+\sin\,u)^2...
by Papapetros Vaggelis
Thu Oct 13, 2016 3:07 pm
Forum: Real Analysis
Topic: Integral of an unknown $f$
Replies: 2
Views: 2840

Re: Integral of an unknown $f$

A function which satisfies all the conditions of the above exercise is \(\displaystyle{f(x)=1-x\,,x\in\mathbb{R}}\).
by Papapetros Vaggelis
Thu Oct 13, 2016 3:05 pm
Forum: Real Analysis
Topic: Integral of an unknown $f$
Replies: 2
Views: 2840

Re: Integral of an unknown $f$

Hi Riemann. We have that \(\displaystyle{f(f(0))=0\implies f(1)=0}\) and since \(\displaystyle{f}\) is continuous and strictly decreasing at \(\displaystyle{\left[0,1\right]}\) we get \(\displaystyle{f(\left[0,1\right])=\left[f(1),f(0)\right]=\left[0,1\right]}\). Also, according to the relation \(\d...
by Papapetros Vaggelis
Tue Oct 11, 2016 2:13 pm
Forum: Algebraic Structures
Topic: Even permutation
Replies: 1
Views: 2777

Re: Even permutation

Let \(\displaystyle{\sigma:S_{n}\to \mathbb{Z}_{2}}\) be the sign map, which is ring epimorphism. Then, \(\displaystyle{\begin{aligned} \sigma(a^{-1}\,\beta^{-1}\,a\,\beta)&=-\sigma(a)-\sigma(b)+\sigma(a)+\sigma(b)\\&=0\end{aligned}}\) so, the permutation \(\displaystyle{a^{-1}\,\beta\,a\,\b...