Hamid Enki
  • Member for 6 years, 6 months
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1 answers
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60 views
If $F$ is a field then $F[\sqrt{-d}]$ is always a field?
Accepted answer
2 votes

Yes, it is always a field. Because you are just adjoining an algebraic (in your case) or transcendental element to F. $\mathbf{Z}[\sqrt{-2}]$ Is not a field because $\mathbf{Z}$ is not a field.

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1 answers
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38 views
entire function Res(1/f)
2 votes

Pick a large circle around the origin containing all the zero's. Then your sum equals to $\frac{1}{2\pi i} \int_{|z|=R} \frac{1}{f}$ by residue theorem. Now apply ML inequality to see that the ...

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3 answers
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76 views
Prove that to every $A \in L(R^n,R^1)$ corresponds a unique $y \in R^n$ such that Ax=xy. Prove that $||A||=|y|$
1 votes

Hint: Consider a base of $R^n$ like $\{e_i\}$ and apply $A$ to each $e_i$, assuming such $y$ exists. What should be this $Ae_i$?

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2 answers
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178 views
Help with finding the limit of a recursive sequence.
Accepted answer
1 votes

Hint: Based on your proof of convergence for $a_1\in(1-\sqrt a, 1+\sqrt a)$ break the problem into few steps: $a_1 \geq 1+\sqrt a$; show that for all $n$, $a_n\geq1+\sqrt a$ as well. Then show that $...

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5 answers
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187 views
Express in the form of $a+ib$.
1 votes

Use polar form for $1-i$ which is $\sqrt2e^{-i \pi/4}$.

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1 answers
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51 views
Reducible/Irreducible 2x2 matrices
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How about $A=\begin{pmatrix} 1 & 0\\ 1 & 1 \end{pmatrix}$? $A^n=\begin{pmatrix} 1 & 0\\ n & 1 \end{pmatrix}$.

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2 answers
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65 views
Ceiling problem - Determining n0 for the convergence of 1/n
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As you pointed out we cannot. I would put $n_0=\lfloor\frac{1}{\epsilon}\rfloor+1$.

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3 answers
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42 views
What is the better condition to check for convergence of a complex series?
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$\log(n) $ stands as counter example to both.

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2 answers
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261 views
Let $\{a_n\}$, $\{b_n\}$ be sequences bounded above. prove $\limsup (a_n + b_n)\le \limsup(a_n) +\limsup(b_n)$
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If they represent $\gamma, \alpha$ and $\beta$ correspondingly and if $\gamma \gt \alpha+\beta$, then there should be m such that $\alpha+\beta\lt a_m +b_m \le\gamma$ which means $(\alpha-a_m)+(\beta-...

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2 answers
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138 views
Findinf the remaining eigenvalues of a $3\times 3$ matrix
-1 votes

I think the $\lambda_1 \lambda_2 n$ should be equal to $(-1)^n (-1)^n \det(M)$. Now putting them in a quadratic equation may help a little bit.

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