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 Steward
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Mar
14
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 $z$.
Mar
14
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.
Mar
10
comment If h is a holomorphic non-vanishing function in the complex plane then h is the exponential of another function.
It should be $\exp(g(0))=h(0)$.
Feb
11
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 $\mathbb E_\theta[V_n|T_n=t]=\sum_{\lbrace x:T_n(x)=t\rbrace}\mathbb P_\theta(V_n=V_n(x)|T_n=t)V_n(x)$
Jan
22
comment Why is every sequential map uniformly continuous
Any $\delta$ less than 1 works.
Jan
12
comment $f:[a,b]\to [c,d]$ is Riemann integrable bijective function imply $f^{-1}$ also Riemann integrable
math.stackexchange.com/questions/19329/…
Nov
16
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...
Nov
1
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.
Nov
1
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.
Oct
21
comment Last 3 digits of $7^{12341}$
Note also that $\varphi(125)=100$, so you really just need $7^{41} (\bmod 125)$.
Sep
8
comment Simple groups in group theory
Simple groups are non-trivial: en.wikipedia.org/wiki/Simple_group
Aug
26
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.
Jun
2
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.
May
31
comment Variant on divergence theorem
@custodia Yes, in particular we can choose $\vec{k}$ to be each of the basis vectors.
May
31
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.
Oct
16
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.
Sep
21
comment On the roots of $t^4-6\sqrt3t^3+8t^2+2\sqrt3t-1=0$
Excellent, thanks. For the last sentence, you mean $n=0,1,2,4$.
Mar
30
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?
Feb
19
comment Conditional probability. What is the meaning of this explanation?
In other words, the solution is $\frac{1}{6}\cdot0.4$.
Feb
12
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.