# All Questions

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### Probability in ball coloring

You have exactly $n^2$ balls each one of which can be colored in one of $n^2$ ways. That is total colors is $n^2$ but I am not saying all the $n^{2}$ balls are distinctly colored. However assume each ...
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### Schwarz's lemma $\Rightarrow$ an analytic conformal map UHP$\to$UHP must be an FLT?

I read a solution to a conformal mapping problem that made the claim, "Schwarz's lemma implies that any analytic conformal map taking the upper half-plane to the upper half-plane must be a fractional ...
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### Why can moving averages of the prices indicate the trend of a stock?

In technical analysis of stock trading, we can use the moving averages of the historical prices of a stock to indicate whether it is currently in the uptrend or downtrend. Let me exemplify the idea ...
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### Definition clarification on orientation on a manifold.

I have been trying to self-learn differential geometry. I think I may have misunderstood/missed out on something along the way. It is said that for $X$ an $n$-form, $M$ a differentiable manifold, ...
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### Sigma-algebras on the Natural Numbers

I know that we can say something very general about all of the sigma-algebras on the natural numbers, but I don't quite know what it is. Can anybody please help? To elaborate further, I have read that ...
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### length of a walk in product graph

I was doing tensor product of graphs. We know that to find a walk between every two vertices x and y of any arbitrary length l in G, the graph must contain an odd cycle. I am stuck here. Is it ...
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### Clarification of a convex space

Just for be clear: if a convexity space is all set of lines that runs from all different points contained in such space that all can be connected with a line (beeline), does all the set of points in ...
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### A group $G$ with $G'$ abelian and every abelian normal subgroup finite

Let $G$ be a group such that $G'$ abelian and any abelian normal subgroup of $G$ is finite. Show that $G$ is finite.
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### Find the coordiantes of a rectangle using vectors

I have two coordinates which are vertices of a rectangle. They form the diagonal of the rectangle. Is it possible, using vectors, to calculate the other two coordinates? I have tried calculating ...
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### Inequality proofs

How do I prove that $$\ln(1+t)\geq t(1-t/2), \;\forall\; t\geq 0.$$ and that $$\ln2t\leq t\ln2, \;\forall\; t\geq 2.$$ How does the latter imply $$(\ln2t)^2\leq 2t\ln2, \;\forall\; t\geq 2.$$
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### Remainders of binomial coefficients.

Motivation: It is easy to notice that a polynomial map $f: \mathbb{Z} \to \mathbb{Z}$ does not need to have only integer coefficient. For example, $f(x) = \frac{x(x-1)}{2}$ does have rational ...
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### Rotate shape on a graph

I have some shape built out of points. The coordinates given for each point . (see pic below) I need to rotate this shape by a particular angle (for example 60° see pic below) Is there some formula ...
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### Variational formulation and change of coordinate

I was wondering if a change of coordinate (e.g cylindrical change) could affect the variational formulation with respect to the metric. I mean, does the metric $\mathbf{dx}$ which appears in the ...
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### Case of near equality in triangle inequality

Let $a_i$, $1 \leq i \leq n$ be complex numbers such that $\sum a_i=1$, and $\sum |a_i| \leq 1+\epsilon$, where $\epsilon>0$ is small. This corresponds to a case of "near equality" in the triangle ...
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### find rotation needed to minimize distance of pairs of vectors

Given two sets of vectors, how can I find the rotation needed to rotate one set onto the other? The sets are ordered and of same size, with vector n from set 1 corresponding to vector n from set 2. ...
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### Simplify$[(ABC)' + (B' + C)']'$

Need help on how to simplify $[(ABC)' + (B' + C)']'$. Here is my attempt: \begin{eqnarray} &&[(ABC)' + (B' + C)']'\\ &=&(ABC) + (B' + C)\\ &=&B'+C(AB + 1)\\ &=&B' + ...
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### proving isomorphism of two $k$-algebras

Let $k$ be a field. I would like to prove that $k[x,y]/(x^3-y^2) \cong k[t^2,t^3]$. Of course, intuitively, i can readily see that this must be the case. More formally, i define a homomorphism ...
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### Does the Lie Bracket automatically exist?

Let $g$ be a Matrix Lie Group. The Lie Algebra of $g := Lie(g)$ is defined as $Lie(g) = \{ \dot{\gamma}(0) | \gamma:(-\epsilon, \epsilon) \rightarrow g, \gamma \in C^1, \gamma(0) = \mathbb{I} \}$ ...
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### Why is this finite difference operator nonnegative?

If a mesh function is given by $\Psi_i=|\Phi_0|+\max_{0\leq j\leq N}|L^N_\epsilon\Phi_j|+\Phi_i$ for any mesh function $\Phi$. I know that $\Psi_0\geq 0$ but cannot figure out why ...
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### Simplicity and isolation of the first eigenvalue associated with some differential operators

Consider the operator $\Delta$ or more generally a second order differential operator $L$, in which the principal part is symmetric and positive definite. It can be proved (see here page 336) that the ...
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### What are the differences between the Lebesgue measure on the Hilbert cube $[0,1]^\mathbb{N}$ and the standard Lebesgue measure on $[0,1]^n$?

There are no Lebesgue measure on infinite dimensional Banach space. However, there is a Lebesgue measure on the Hilbert cube $[0,1]^\mathbb{N}$. What are the differences between this measure and the ...
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### Formal definition of a geometric plane explained

Do any one can give a formal definition of a geometric plane and explain it? I try to answer if any plane P in a space P is convex, but for that I need to know exactly what is a plane.
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### Solve following trigonometric equation

How to solve $$2\sin(x) / (1+2\cos(x)) = \sqrt{3}/2$$ where $0< x <180$. (Final answer may be in inverse form)
### Is there a closed form solution for partial sums of $1/(2^{2^0}) + 1/(2^{2^1}) + 1/(2^{2^2}) + \ldots$
Title says it all, this is such a classical looking series, $$\frac1{2^{2^0}} + \frac1{2^{2^1}} + \frac1{2^{2^2}} + \ldots.$$ So, I was just wondering, is there a closed form solution known for the ...