# All Questions

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### Multisection of a Power Series Proof

Suppose that $$H_{N,k}(x)=\frac{x^ke^{\frac{-x}{N}}}{N^{k-1}k!\sum_{n=0}^{N-1}{w_N^{-nk}e^{\frac{w_N^nx}{N}}}}=\sum_{n=0}^\infty{A_n\frac{x^n}{n!}}$$ where $k\lt N, w_N=e^{\frac{2i\pi}{N}}$, and ...
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### Quotient is isomorphic exercise

Suppose $G$ is solvable, $N \vartriangleleft G$. Let $f \in Hom(G,H)$. We have a normal series $\{e\}=G_0 \vartriangleleft G_1 \vartriangleleft ... \vartriangleleft G_n = G$ with $G_{i+1}/G_i$ ...
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### Finding probability of getting into a university based on 3 factors

How would I go about calculating probability of getting into a college (CU Boulder) using a data set containing GPA (0.00-4.00), ACT Composite score (0-36), Class rank percentile (0-100), and whether ...
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### Order statistics and transformations

Assume random variables X$_1$, ... , X$_n$ and Y$_1$, ..., Y$_n$ are U(0,a)-distributed. Show that Z$_n$ = n*$log\frac{max(Y_{(n)},X_{(n)})}{min(Y_{(n)},X_{(n)})}$ has an Exp(1) Distribution. I've ...
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### In how many different ways can you order in line the letters of the word AAAAABBB?

In how many different ways can you order in line the letters of the word AAAAABBB?I was thinking - I have $8!$ for the whole letters including repeats, and then because each word repeats $5!$ for the ...
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### Proof for a periodic function

I have to solve the following exercise: The function $f : \mathbb{R} \rightarrow \mathbb{R}$ is a periodic function with $P = 2\pi$ so that $f(x) = f(x + 2\pi)$ is true for all $x \in \mathbb{R}$. ...
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### Bias of two estimators

I hope someone can help me. I have some trouble calculating the bias of two estimators.Unluckily it is really urgent because I hold a presentation next week. The topic is nonparametric local ...
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### integral vs. residue at infinity

I have an issue with residues at infinity. I am computing the integral $\displaystyle{\int_{C_3^+(0)} \dfrac{e^{3z}}{z^2(z^2+2z+2)} dz}$ Since all three poles ($0$ of order 2, $1\pm i$ of order 1) ...
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### Magnitude and Angle of Discrete Fourier Transform

I can't figure out how to get the magnitudes for periodic discrete Fourier transforms. For example if $x[n] = cos(\frac{\pi}{4}n + \frac{\pi}{2})$, I need to find and plot the magnitude $|X(e^{jw})|$...
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### Convergence result of a UI martingale

This Lemma is from Ethier and Kurtz's Markov Processes on page 205. My question is about the equation (5.47). I can prove it is a uniformly integrable martingale (bounded by 1), so converges to a ...
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Can every monad give rise to a monad transformer? In the paper Calculating monad transformers with category theory by Oleksandr Manzyuk, one finds a construction of monad transformers as ...
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### Constant Gauss curvature from bipolar projections.

Please help finding z-coordinate for constant positive and negative Gauss curvatures in Mongé form : $$x= \sqrt{R^2 + T^2} + R \cos u ,\, y= R \sin u ,\, z= f(x,y).$$ $R,T$ are constants. (...
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### How to solve this integral to find the exact length of an equation in the polar plane?

I hope it is only because it's late and I've been studying for a calculus exam for several hours, but I cannot see how to solve this integral. The problem states: Find the exact length of the ...
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### What determines the number of families to $1-4x-4(1-x^2)z = w^2$?

This is related to this post. First, we have, Theorem: "If $w_0, z_0$ is a solution to, $$1-4x-4(1-x^2)z = w^2\tag1$$ then, $$w = w_0+2(x^2-1)n$$ $$z = z_0+w_0\,n+(x^2-1)n^2$$ is also a ...
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### Collatz Conjecture: Literature on Convergence

Does anyone know of a paper showing that if all n converge, they must converge to unity for n>0. Else, any literature related to convergence properties would be appreciated. Thanks, Jordan
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### What process does this SDE weakly converge to?

So my question is motivated by the following: Note that the ODE $$dy_t = 2sgn(y_t)\sqrt{|y_t|}$$ $$y_0 = 0$$$$has no unique solution. However, consider the SDE as follows:$$ dy_t = 2sgn(y_t)\sqrt{...
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### A planar graph $G$ is the dual of its dual if and only if $G$ is connected.

That is, prove that a planar graph is the dual of it's dual iff it is connected. I know that in order for this to be true, G must be isomorphic to it's dual (G'), but I'm not sure how connectedness ...
### Young diagram for $S_5$
I am trying to draw the Young diagram for $S_5$. I know the following pieces of information about $S_5$. The order of the group is $120$. The number of conjugacy classes and so partitions is $7$. ...