Sarastro
Reputation
1,280
Next privilege 2,000 Rep.
 Mar14 comment Prove that if $N(z)$ is irreducible in $\Bbb{Z}$ then $z$ is irreducible in $\Bbb{Z}[\sqrt{\alpha}]$. In an integral domain $R$, $r,s\in R$ are associates iff $r=su$ for some unit $u\in R$. If $r$ divides $z$, $rs=z$ for some $s$. By my definition, if $z$ is irreducible, either $r$ or $s$ is a unit. So $r|z$ implies $r$ is a unit or $r$ is an associate of $z$. Mar14 comment Prove that if $N(z)$ is irreducible in $\Bbb{Z}$ then $z$ is irreducible in $\Bbb{Z}[\sqrt{\alpha}]$. For a ring $R$, $r\in R$ is a unit if $\exists s\in R$ such that $rs=sr=1$. An element in a ring is irreducible if it is not the product of two nonunits. Mar10 comment If h is a holomorphic non-vanishing function in the complex plane then h is the exponential of another function. It should be $\exp(g(0))=h(0)$. Feb11 comment The function given by the Rao-Blackwell theorem is a statistic No, $\mathbb E_\theta[V_n|T_n=t]=\sum_{\lbrace x:T_n(x)=t\rbrace}\mathbb P_\theta(X=x|T_n=t)V_n(x)$ follows by definition of expectation. We sum over all possibilities of $x$ given $T_n=t$. It is not true in general that $\mathbb E_\theta[V_n|T_n=t]=\sum_{\lbrace x:T_n(x)=t\rbrace}\mathbb P_\theta(V_n=V_n(x)|T_n=t)V_n(x)$ Jan22 comment Why is every sequential map uniformly continuous Any $\delta$ less than 1 works. Jan12 comment $f:[a,b]\to [c,d]$ is Riemann integrable bijective function imply $f^{-1}$ also Riemann integrable math.stackexchange.com/questions/19329/… Nov16 comment Eigenvalues, polynomials and minimal polynomials If $\lambda=0$, $Bv$ could be $0$, in which case your solution your solution to (a) wouldn't work (hence the stated hint). However, note that $det AB = det BA$, so... Nov1 comment $A$ is dense in $[0,1]$ and $f(x)=0$ for all $x$ in $A$ then the integral is zero Are you using Riemann or Lebesgue integrals? The Dirichlet function is not Riemann integrable. Nov1 comment How to find a non-Gaussian function f(x) that satisfies the following condition: Do you mean $\int_0^\infty f(x)^2 dx > 2 (\int_0^\infty f(x)dx)^2$? If so, the problem is much easier, but this function still works. Oct21 comment Last 3 digits of $7^{12341}$ Note also that $\varphi(125)=100$, so you really just need $7^{41} (\bmod 125)$. Sep8 comment Simple groups in group theory Simple groups are non-trivial: en.wikipedia.org/wiki/Simple_group Aug26 comment What if objective function $Z$ is also in the constraints? The constraint should instead be $\sum_j y_j = 1$ (there's also a constraint that $y\ge0$). Incidentally, this proves the Minimax Theorem for two-player zero-sum games. Jun2 comment If $y=x^{x^{x^{x^{x^{.^{.^{.}}}}}}}$ then how $y=x^y$? I haven't proven that the tower converges. I proved that $(b_n)$ converges, since the OP asked me to in the comments. May31 comment Variant on divergence theorem @custodia Yes, in particular we can choose $\vec{k}$ to be each of the basis vectors. May31 comment Variant on divergence theorem One way to show the identity is to apply the divergence theorem on $\vec{k}f$, where $\vec{k}$ is a constant vector, and noting that $\nabla\cdot\vec{k}f=\vec{k}\cdot \nabla f$, since $\vec{k}$ is constant. Oct16 comment Is it true that $\lvert \sin z \rvert \leq 1$ for all $z\in \mathbb{C}$? The second equation should read $4i=e^{iz}-e^{-iz}$. The error carries on from there. Sep21 comment On the roots of $t^4-6\sqrt3t^3+8t^2+2\sqrt3t-1=0$ Excellent, thanks. For the last sentence, you mean $n=0,1,2,4$. Mar30 comment If i randomly select $n+1$ integers smaller that $2n$ , must there include an integer which divisible with one of the others? I suppose negative integers are not permitted? Feb19 comment Conditional probability. What is the meaning of this explanation? In other words, the solution is $\frac{1}{6}\cdot0.4$. Feb12 comment Probability of an event I suspect what the question really means is that the test returns positive for .95 of the people with diabetes for .02 of the people without.