# All Questions

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### Triangles - sin, cos etc.

I know this is a quite simple question for most of you out there. However it has been a little troubling for me, and would like to get a little help if possible. I have a triangle $ABC$ where I know ...
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### integrate $\int \frac{x\cos x}{\sin^2x}dx$

$$\int \frac{x\cos x}{\sin^2x}dx$$ $$\int \frac{x\cos x}{\sin^2x}dx=\int \frac{x\cos x}{1-\cos^2x}dx=\int \frac{x\cos x}{(1-\cos x)(1+\cos x)}dx$$ How can I find the two fractions? if there are ...
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### Paradox on a linear system with semi-orthogonal matrix

Let $XA^{T} = B$ be a linear system on $X$ with $A \in \mathbb{R}^{m \times n}$ a semi-orthogonal matrix. Hence, $A^{T}A = I$, but $AA^{T} \neq I$ if $m > n$. Assume the case $m > n$. By ...
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### How could someone conclude $\check{H}^i (M, \mathbb{R}) = 0$ for arbitrary $M$?

sorry if this is a very stupid question and I'm missing something very trivial, though I could not solve it after thinking for a while. In page 18-19 of ...
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### Is there a closed form expression for the following sum?

Is there a closed form expression for the following sum? $$\sum_{0\le i_1<i_2<\cdots<i_k\le n}r^{i_1+i_2+\cdots+i_k}$$ I can understand that there are $\binom{n}{k}$ such terms and the ...
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### Is $\pi^k$ any closer to $[\pi^k]$ than expected?

Particular questions such as Why is $\pi$ so close to $3$? or Why is $\pi^2$ so close to $10$? may be regarded as the first two cases of the question sequence Why is $\pi^k$ so close to $[\pi^k]$? ...
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### difference between slopes of lines represented by an equation

The question is find the difference between slopes of lines give represented by equation of pair of lines which is $$x^2(\tan^2(\theta)+\cos^2(\theta))-2xy\tan(\theta)+y^2(\sin^2(\theta))=0$$ i have ...
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### UFDs with exactly two irreducibles

Let $p$ be a prime number in $\mathbb{Z}$. Let $R = R_p = \{x \in \mathbb{Q}\ |\ ord_p(x)\geq0\}$, which is a subring of $\mathbb{Q}$. Can you generalise this to construct UFDs with exactly ...
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### Several options using Black-Scholes equation(s)

Could someone provide me some information about the modelling of several options at the same time by using Black-Scholes (probably coupled) equations? Any reference to papers and/or books shall be ...
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### Convergent Bounded Linear Maps

I'm not sure how to show that the composition of two convergent bounded linear maps converges to the composition of their limits. I've shown that the composition of bounded linear maps is a bounded ...
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### A thief and a policeman [closed]

A policeman desperately tries to catch a thief that is $a$ meters away. The thief has the constant velocity $v$, and the policeman has the constant velocity $k\cdot v$, with $k > 1$. The policeman ...
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### Group of finite order where every element has infinite order

An often given example of a group of finite order where every element has infinite order is the group $\dfrac{\mathbb{Q, +}}{\mathbb{Z, +}}$. But I don't see why every element necessarily has finite ...
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