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 ...
Search found 375 matches
- Thu Dec 28, 2017 8:09 pm
- Forum: Algebraic Structures
- Topic: True or false statements
- Replies: 1
- Views: 3335
- Wed Dec 27, 2017 5:08 pm
- Forum: Linear Algebra
- Topic: Dimension of intersection of subspaces
- Replies: 1
- Views: 3954
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 ...
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 ...
- Tue Nov 07, 2017 12:12 pm
- Forum: Functional Analysis
- Topic: Hilbert space
- Replies: 0
- Views: 2782
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 ...
\(\displaystyle{\ell^2(H):=\left\{x:\mathbb{N}\to H\,,\sum_{n=1}^{\infty}||x_{n}||^2<\infty\right\}}\).
and
\(\displaystyle{\langle{x,y\rangle}:=\sum_{n=1}^{\infty}\langle{x_n,y_n\rangle}\,,\forall\,x\,,y\in \ell^2 ...
- Sat Oct 07, 2017 10:17 pm
- Forum: Algebra
- Topic: Locally free but no globally
- Replies: 1
- Views: 3910
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 ...
\(\displaystyle{x^2+y^2-1=x\cdot x+(y+1)\cdot (y-1)\in\langle{x,y-1\rangle}}\), 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 ...
- Sat Oct 07, 2017 2:53 pm
- Forum: Real Analysis
- Topic: Open subset
- Replies: 3
- Views: 4034
Re: Open subset
Thank you.
- Thu Oct 05, 2017 12:02 pm
- Forum: Real Analysis
- Topic: Open subset
- Replies: 3
- Views: 4034
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 \(\displaystyle{G=\mathbb{R}}\).
such that \(\displaystyle{x\pm y\in G\,,\forall\,x\,,y\in G}\). Prove that \(\displaystyle{G=\mathbb{R}}\).
- Mon Aug 14, 2017 11:47 am
- Forum: General Mathematics
- Topic: Inequality
- Replies: 2
- Views: 4232
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 ...
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 ...
- Sat Jul 15, 2017 8:59 pm
- Forum: General Topology
- Topic: On the mole metric
- Replies: 1
- Views: 6958
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 ...
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 ...
- Fri Jul 07, 2017 4:26 pm
- Forum: General Topology
- Topic: On a Cauchy sequence
- Replies: 1
- Views: 6457
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 ...
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 ...
- Sat Jun 10, 2017 11:12 pm
- Forum: Calculus
- Topic: \(\int_{0}^{+\infty}\frac{\log{x}}{(x+\alpha)(x+\beta)}\,dx\)
- Replies: 2
- Views: 3504
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 ...
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 ...