# All Questions

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### What is common notation for “disjoint union of copies of $\mathbb{R}$”?

I'm looking at a question out of Lee's Smooth Manifolds: Show that a disjoint union of uncountably many copies of $\Bbb{R}$ is locally Euclidian and Hausdorff but not second countable. My ...
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### Watches on a table

From Peter Winkler's 'Mathematical puzzles', taken from an All USSR Mathematical Competition, 1976: 50 accurate watches lie on a table. Prove that there exists a moment in time when the sum of the ...
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### Is this what universal covering spaces are used for?

From the perspective of real analysis, we have: $$\int_{-1}^{1}\frac{1}{1+x^2} = \mathrm{tan}^{-1}(1)-\mathrm{tan}^{-1}(-1) = 2\mathrm{tan}^{-1}(1) = 2 \cdot \frac{\pi}{4} = \frac{\pi}{2}$$ Something ...
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### Volume of solid lies under $z=x^2+y^2$

Find the volume of solid lies under $z=x^2+y^2$ above $x$-$y$ plane and inside the cylinder $x^2+y^2=2x$. I know, for volume we have to us $V=\iiint { \mathrm dx\mathrm dy\mathrm dz}$ but i was not ...
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### order of multiple quantifiers

Problem: For a, b, c, d restricted to the universe of positive integers, explain why ∀a ∃b ∀c ∃d a/b = c/d is true, but ∀a ∃d ∀c ∃b a/b = c/d is false. I understand that the order of ...
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### How to integrate $\frac{dx}{(x-p)\sqrt {(x-p)(x-q)}}$?

How to integrate $\frac{dx}{(x-p)\sqrt {(x-p)(x-q)}}$ ? I tried substituting $x=1/t$ but that's making it more complicated.Any suggestions?
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### Reduce Number of digit

I have 24 digit number like 281564785148270616103860.I want to reduce this number with any optimization technique to 12 digit number . So i can use this to decode with the same technique.Any answer ...
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### Is there any approach to computer vision that doesn't make use of geometry?

I've long been interested in applying my background in functional analysis (especially wavelets) and other related areas to actually create something with "real world" value (not that I don't enjoy ...
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### trigonometry to calculate height

Can not solve this problem I can help Edit : From the top of a hill a person finds losangulos Depression three consecutive stones and indicator of kilometers of a straight road level are α, β and γ....
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### Conditions for Non-negativity

Let's consider $A$ to be a square symmetric matrix whose entries are non-negative real numbers that sums to one. Even more, we shall consider its diagonal elements to be equal to zero. The question is:...
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### Existence of a basis $B$ such that $M(\phi,B)=E$

Considering the matrix $$A=\begin{bmatrix}{2}&{1}&{0}\\{1}&{0}&{-1}\\{0}&{-1}&{-2}\end{bmatrix}\in M_3(R).$$ And $\phi:R^3 \times R^3\longrightarrow R$ the bilinear ...
### How to represent $GF(q)$ element in Matlab
Everyone, I am trying to simulate in Matlab the Galois field $GF(q)$ where $q$, for example, is $8$. I know the elements of $GF(8)$ are $0,1,\alpha,\alpha^2,\alpha^3,\ldots,\alpha^6$. Also, we can ...
$\mathrm{sgn}(\pi\circ\sigma)=\mathrm{sgn}(\pi)\cdot \mathrm{sgn}(\sigma)$ I am familiar with how to count inversions and any insight for why this formula holds true would be very helpful.