Welcome to mathimatikoi.org forum; Enjoy your visit here.

Search found 375 matches

by Papapetros Vaggelis
Thu Dec 28, 2017 7:09 pm
Forum: Algebraic Structures
Topic: True or false statements
Replies: 1
Views: 783

Re: True or false statements

i. True statement. If \(\displaystyle{n\in\mathbb{N}\,,n\geq 2}\), then ,we define \(\displaystyle{\mathcal{x}:S_n\to \mathbb{C}^{\star}}\) by \(\displaystyle{\mathcal{x}(\sigma)=1}\) if \(\displaystyle{\sigma}\) is an even permutation and \(\displaystyle{\mathcal{x}(\sigma)=-1}\) if \(\displaystyl...
by Papapetros Vaggelis
Wed Dec 27, 2017 4:08 pm
Forum: Linear Algebra
Topic: Dimension of intersection of subspaces
Replies: 1
Views: 912

Re: Dimension of intersection of subspaces

Hi Riemann. Let \(\displaystyle{(x,y,z)\in W_1\cap W_2\cap W_3}\). Then, \(\displaystyle{(x,y,z)\in W_1\implies x+y-z=0\,\,(I)}\) \(\displaystyle{(x,y,z)\in W_2\implies 3\,x+y-2\,z=0\,\,(II)}\) \(\displaystyle{(x,y,z)\in W_3\implies x-7\,y+3\,z=0\,\,(III)}\). The relations \(\displaystyle{(I)\,,(II)...
by Papapetros Vaggelis
Tue Nov 07, 2017 11:12 am
Forum: Functional Analysis
Topic: Hilbert space
Replies: 0
Views: 781

Hilbert space

Let \(\displaystyle{\left(H,\langle{,\rangle}\right)}\) be a Hilbert space. We set \(\displaystyle{\ell^2(H):=\left\{x:\mathbb{N}\to H\,,\sum_{n=1}^{\infty}||x_{n}||^2<\infty\right\}}\). and \(\displaystyle{\langle{x,y\rangle}:=\sum_{n=1}^{\infty}\langle{x_n,y_n\rangle}\,,\forall\,x\,,y\in \ell^2(H)...
by Papapetros Vaggelis
Sat Oct 07, 2017 8:17 pm
Forum: Algebra
Topic: Locally free but no globally
Replies: 1
Views: 1056

Re: Locally free but no globally

hI PJPu17. The ring \(\displaystyle{R}\) is commutative. We observe that \(\displaystyle{x^2+y^2-1=x\cdot x+(y+1)\cdot (y-1)\in\langle{x,y-1\rangle}}\), so, \(\displaystyle{\langle{x^2+y^2-1\rangle}\leq \langle{x,y-1\rangle}\leq \mathbb{K}[x,y]}\) and according to the 3rd Ring Isomorphism Theorem, w...
by Papapetros Vaggelis
Sat Oct 07, 2017 12:53 pm
Forum: Real Analysis
Topic: Open subset
Replies: 3
Views: 1134

Re: Open subset

Thank you.
by Papapetros Vaggelis
Thu Oct 05, 2017 10:02 am
Forum: Real Analysis
Topic: Open subset
Replies: 3
Views: 1134

Open subset

Let \(\displaystyle{G}\) be a non-empty and open subset of \(\displaystyle{\left(\mathbb{R},|\cdot|\right)}\)

such that \(\displaystyle{x\pm y\in G\,,\forall\,x\,,y\in G}\). Prove that \(\displaystyle{G=\mathbb{R}}\).
by Papapetros Vaggelis
Mon Aug 14, 2017 9:47 am
Forum: General Mathematics
Topic: Inequality
Replies: 2
Views: 1033

Re: Inequality

Hi Riemann. It is sufficient to prove that \(\displaystyle{a+b+c\geq \sqrt{3\,a\,b\,c}}\), where \(\displaystyle{a\,,b\,,c>1}\) and \(\displaystyle{a\,b+b\,c+c\,a=a\,b\,c}\). So, \(\displaystyle{\begin{aligned}a+b+c\geq \sqrt{3\,a\,b\,c}&\iff (a+b+c)^2\geq 3\,a\,b\,c\\&\iff (a^2+b^2+c^2)+2\,(a\,b+b\...
by Papapetros Vaggelis
Sat Jul 15, 2017 6:59 pm
Forum: General Topology
Topic: On the mole metric
Replies: 1
Views: 2024

Re: On the mole metric

Hi Riemann. We observe that \(\displaystyle{0\leq f(x)\leq 1\,,\forall\,x\geq 0\,,0\leq f(x)\leq x\,,\forall\,x\geq 0}\) and that \(\displaystyle{f}\) is strictly increasing at \(\displaystyle{\left[0,+\infty\right)}\). Also, \(\displaystyle{f(x+y)\leq f(x)+f(y)\,,\forall\,x\,,y\geq 0}\) and \(\disp...
by Papapetros Vaggelis
Fri Jul 07, 2017 2:26 pm
Forum: General Topology
Topic: On a Cauchy sequence
Replies: 1
Views: 1800

Re: On a Cauchy sequence

It's obvious that \(\displaystyle{d}\) is a metric. i. Let \(\displaystyle{\epsilon>0}\). There exists \(\displaystyle{n_0\in\mathbb{N}}\) such that \(\displaystyle{\dfrac{2}{n_0}<\epsilon}\). Then, for every \(\displaystyle{n\,,m\in\mathbb{N}}\) such that \(\displaystyle{n\,,m\geq n_0}\) holds \(\d...
by Papapetros Vaggelis
Sat Jun 10, 2017 9:12 pm
Forum: Calculus
Topic: \(\int_{0}^{+\infty}\frac{\log{x}}{(x+\alpha)(x+\beta)}\,dx\)
Replies: 2
Views: 1004

Re: \(\int_{0}^{+\infty}\frac{\log{x}}{(x+\alpha)(x+\beta)}\,dx\)

Let \(\displaystyle{F(a,b)=\int_{0}^{\infty}\dfrac{\ln\,x}{(x+a)\,(x+b)}\,\mathrm{d}x\,\,,0<a<b}\) Using the substituton, we get \(\displaystyle{F(a,b)=-\int_{\infty}^{0}\dfrac{\ln(1/t)\,t^2}{(1+a\,t)\,(1+b\,t)}\,\dfrac{1}{t^2}\,\mathrm{d}t=\int_{0}^{\infty}\dfrac{-\ln\,x}{(a\,x+1)\,(b\,x+1)}\,\math...