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Thu Dec 28, 2017 7:09 pm
Forum: Algebraic Structures
Topic: True or false statements
Replies: 1
Views: 1011

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... Wed Dec 27, 2017 4:08 pm Forum: Linear Algebra Topic: Dimension of intersection of subspaces Replies: 1 Views: 1248 ### 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)... Tue Nov 07, 2017 11:12 am Forum: Functional Analysis Topic: Hilbert space Replies: 0 Views: 969 ### 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)... Sat Oct 07, 2017 8:17 pm Forum: Algebra Topic: Locally free but no globally Replies: 1 Views: 1406 ### 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...
Sat Oct 07, 2017 12:53 pm
Forum: Real Analysis
Topic: Open subset
Replies: 3
Views: 1414

### Re: Open subset

Thank you.
Thu Oct 05, 2017 10:02 am
Forum: Real Analysis
Topic: Open subset
Replies: 3
Views: 1414

### 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 $\displaystyle{G=\mathbb{R}}$.
Mon Aug 14, 2017 9:47 am
Forum: General Mathematics
Topic: Inequality
Replies: 2
Views: 1419

Replies: 2
Views: 1194

### 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...