# All Questions

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### Inequality in matrix norm

Let $\|\cdot\|$ be matrix norm on $M_n$.Why does $\|A\|_2 \le \|A\|^{\frac{1}{2}} \|A^*\|^{\frac{1}{2}}$? ($\|A\|_2 = \displaystyle\max_{\|x\|_2 = 1} \|Ax\|_2$)
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### John b.Conway chapter 2 section 2 exercise 4 [on hold]

Show that an idempotent is compact if and only if it has finite rank.
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### Who first used the notation $\mathcal{O}_K$ for ring of integers?

I think this is a standard notation since almost every author uses it, but who came up with the notation? After all, what does $\mathcal{O}$ in $\mathcal{O}_K$ stand for? Thanks in advance.
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### Closed subscheme of an affine scheme.

This is a portion of the exercise 2.3.11 (b) of Hartshorne. Suppose $Y$ is a closed subscheme of $X=\operatorname{Spec}A$. I would like to show that there are $f_{i}\in A$ so $D(f_{i})$ cover $X$ and ...
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### For $n > 2, n \in \mathbb{Z}$, show the sum of integers coprime to $n$ in the range $[1,n-1]$ is equal to $\frac{1}{2}n \phi(n)$

For $n > 2, n \in \mathbb{Z}$, show the sum of integers coprime to $n$ in the range $[1,n-1]$ is equal to $\frac{1}{2}n \phi(n)$ Firstly $\phi(n)$ is Euler's totient function, the number of ...
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### Subgroups of a finite $p$-group

Let $G$ be a group such that $|G|=p^n$ for some $p$ prime and $n\in\mathbb{N}$. I want to prove that if $k\le n$ then $G$ has a normal subgroup of order $p^k$. I want to use induction on $k$. If ...
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### Maximal p-subgroup of inertia group.

We know from the theory that if $\mathbb{L}$ is a finite Galois extension of the local field $\mathbb{K}$ then the ramification group $G_1$ is a $p$-group where $p$ is the characteristic of the ...
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### Infinite sum of reciprocals of pentagonal numbers

How do i find this sum: $$\sum_{n=1}^\infty \frac{1}{p(n)}$$ where $p(n)=\dfrac{n(3n-1)}{2}$ is the $n$th pentagonal number? Thank you for your help.
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### Manufacturing problem, exponential distribution

A manufacturing process produces $92%$ good chips (G) and $8%$ bad chips (B). The lifetime, in seconds, of chips is exponentially distributed $E(\lambda)$.For good chips, ...
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### Questions about soundness and completeness

I am doing a module about introduction to symbolic logic and I seem to understand most of it, apart from problems of the following kind involving that involve soundness and completeness: The ...
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### Residues and poles proof

Let the degree of the polynomials $P(z)=a_0+a_1z+a_2z^2+\cdots+a_nz^n$ $a_n\neq0$ and $Q(z)=b_0+b_1z+b_2z^2+\cdots+b_mz^m$ $b_m\neq 0$ be such that $m\geq n+2$. Show that if all the zeros of ...
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### If $f:\mathbb{R}\to\mathbb{R}^2$ is of class $C^1$, show that $f$ does not carry $\mathbb{R}$ onto $\mathbb{R}^2$

If $f:\mathbb{R}\to\mathbb{R}^2$ is of class $C^1$, show that $f$ does not carry $\mathbb{R}$ onto $\mathbb{R}^2$. In few words I have to show that $f(\mathbb{R})$ contains no open set of ...
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### Calculate the expectation of psi(x) with a gamma density x∼Gamma(α,β )

Suppose I have a Gamma distributed random Variable x∼Gamma(α,β). Now I want to calculate the following expectation values (integrals): E[psi(x)] with psi(x) being the digamma function Many thanks ...
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### Showing that a recursively defined sequence is decreasing.

A colleague of mine is interested in finding out how to show the following: Prove that the sequence $(a_n)$ defined by ...
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### What are the summands in $\mathbb{Z}^n$?
I am interested in knowing what all the rank $k$ submodules of the $\mathbb{Z}$ module $\mathbb{Z}^n$ are that are also summands. I know that $\mathbb{Z}^k$ sitting in $\mathbb{Z}^n$ in the standard ...