# All Questions

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### SVD and base changes matrices

I'm not hugely comfortable with linear algebra, so wanted to double check that the following reasoning was correct. Does it hold that, given two matrices R and B U R B Ut=U R Ut U B Ut= U R Ut D I ...
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### I have calculate an integral. Is it correct?

Could you please help me to check this integral? Is it correct? \begin{align} I&=\int \limits_{\gamma_0}^{\infty}\frac{exp(-x D_6)}{(x A_3 +1)(x A_4+1)^{m}}dx \\ &=\int ...
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### Divisibility of group exponents when the subogroup has finite index.

Let $G$ be a group (not necessary finite) and $H$ a subgroup of $G$ of index $n$. Show that $exp(G)|exp(H)\cdot n$. Remarks. -The case when $H$ is normal is a consequence of the group structure on ...
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### Double-check my answer in calculating gravitational potential energy

"A satellite with a mass of 200kg maintains its orbit at an altitude of 300km above the Earth's surface. The Earth has a radius of $6.38 * 10^6m$ and a mass of $5.97 * 10^24$" I was told this was the ...
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### Prove the Basis of Column Vectors

Let $V$ be a vector space over field $\mathbb{F}$. Let $B=\{u_1,...,u_n\}$ be an ordered basis of $V$. Show that $\{[u_1]_c,...[u_n]_c\}$ is a basis of $M_{n,1}(\mathbb{F})$ for every ordered basis ...
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### The cardinal of the set of all measures on $\mathbb{R}$

It is a very simple question that I don't know how to do: Let $M = \{\mu \colon \mathcal{B}(\mathbb{R})\to \mathbb{R} \colon \mu \text{ is a measure}\}$ $$|M| = \ ?$$ Any help will be appreciated.
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### Existence of nonstandard elementary extensions of $PA$?

My question follows from the 1958 result of MacDowell–Specker (located originally in Modelle der Arithmetik, J. Symbolic Logic Volume 38, Issue 4 (1973), 651-652) of the proof of the following ...
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### How to compute the volume of such a triangular prism shaped object

like this : where $z_i$ are not equal.
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### Where did I go wrong with this definite integration?

I'm trying to solve the definite integral $\int_0^n\pi^{ex}dx$ Wolfram says that the answer is $\frac{\pi^{en}-1}{e \ln(\pi)}$, but I got $\frac{\pi^{en}-1}{\ln(\pi)}$. Can anyone help me figure out ...
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### Error analysis of kernel density estimation

Let $X$ be a random variable with true density $f$, $Y = \{y_i\}_{i=1}^n$ be a realization of a random variable in $d$-dimensional space $R^d$, and $\hat{f}$ is the density estimator of $Y$ using ...
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### Find optimal matrix to maximize the expected expression

I am interested of the following function with $Q$. $$f(Q)=h^TQh-\frac{1}{(h^TQh+1)^2}-g^TQg+\frac{1}{(1+g^TQg)^2}$$ where $h$ and $g$ are both given $N\times 1$ vectors. And $Q$ is $N \times N$ ...
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### Why is $\hat{I}$ contained in the Jacobson radical $J(\hat{R})$?

Suppose $I$ is an ideal of a commutative ring $R$, and $\hat{R}$ is the $I$-adic completion. I don't follow why $\hat{I}$ is in $J(\hat{R})$. I know $\hat{R}$ is complete wrt the $\hat{I}$-adic ...
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### Nice approximations of sums by integrals.

Let $f(x):\Bbb Z^+\rightarrow \Bbb R^+$ be a non-monotone function. If for every $m\in\Bbb N$, $$S(m) =\sum_{n=1}^N\frac{1}{(1+f(n))^m}$$ be sum of interest, then is there a way to study this ...
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### Height and coheight of an ideal

Given an ideal $\mathfrak{a}$, Matsumura defined the height of $\mathfrak{a}$ as: $$\text{ht}(\mathfrak{a})=\inf_{\mathfrak{p}\in V(\mathfrak{a})}\text{ht}(\mathfrak{p})$$ He states that: ...
### tesnsor product of two copies of $\Bbb{R}$ over $\Bbb{R}$
I would like to know what would be tensor product of set of reals over reals would be? That is, $\Bbb{R} \otimes_\Bbb{R} \Bbb{R}$ I think it should be $\Bbb{R}^{2}$ as tensor product combines two ...