# All Questions

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### Frobenius condition in infinite dimensions

My setting is this: $X$ a smooth manifold. $\phi_t:X\rightarrow X$ a flow of a vector field. $f\in \mathcal C^{\infty}(X;\mathbb R)$ an element of the space of smooth functions on $X$ and \begin{...
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### PDF of $X+Y$ for $X$ and $Y$ independent with given PDFs

Consider two independent random variables $X$ and $Y$. Let $f_X(x)=1-\frac{x}{2}$ if $0 \leq x \leq 2$ and $0$ otherwise. Let $f_Y(y)=2-2y$ if $0 \leq y \leq 1$ and $0$ otherwise. Find the ...
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### Is my understanding of conjugation in a group correct?

I'm reading some lecture notes and watching some videos, but the purpose of group conjugation was never really made explicit. After a good amount of time, this is what I've come up with, in the ...
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### Is the category of all finite dimensional modules of $g$ equivalent to the category of all finite dimensional modules of $U_q(g)$?

Let $g$ be a complex simple Lie algebra and $U_q(g)$ the corresponding quantum group. Is the category of all finite dimensional modules of $g$ equivalent to the category of all finite dimensional ...
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### probability in rock paper and scissors

This is actually a programming Question I had faced 2 times. Q) Mark was left behind on Mars by his crew. He found species there similar to humans.He taught them the game of Rock-Paper-Scissor (one ...
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### Notating an arbitrarily long series of series

Is there a better notation for the following sum? $$y = \sum\limits_{x_1}^{S_{x_0}} \sum\limits_{x_2}^{S_{x_1}} \dots \sum\limits_{x_{n}}^{S_{x_{n-1}}} f(x_n)$$ The fact ...
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### Linear Algebra - Field of 2 elements

I'm working on the following problem: $F = \mathbb{F}^2$ is the field of two elements. Let $U \subset F^4$ be the subspace generated by $(1,1,1,1),(1,1,0,0),(0,1,1,0)$ and $V \subset F^4$ be ...
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### Can a non-trivial factor of a strong Fermat-pseudoprime always be found efficiently?

Suppose, $N$ is a composite Fermat-pseudoprime to base $a$ : $$a^{N-1}\equiv 1\ (\ mod \ N)$$ If $N$ is NOT strong Fermat-pseudoprime to base $a$, a non-trivial factor of $N$ can be found ...
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### How to Find the Range and the set notation

For appropriate sets A and B , determine the range of a function f : A → B that assigns (a) to each integer the sum of that integer and its negative. (b) to each pair of ...
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### Area Between Polynomials as a Similarity Measure?

I'm trying to find a conceptually easy way to calculate a "similarity" number between two polynomials. The answer to this question compares the roots of each polynomial, but it isn't obvious to me how ...
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### Akra-Bazzi method - constructive proof

As I was familiarizing myself with different methods of computing complexities of recurrences, I stumbled upon the Akra-Bazzi method. Seeing such a beautiful result literally made my day. I was able ...
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### necessary and sufficient conditions for a function to “have a tangent vector at almost all points”

If I have a nice curve $z(t)$ and a reparametrization $t(s)$, what does it mean for $z(t(s))$ to have a tangent vector at all s, (or more specifically, at almost all s)
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### An apparent contradiction in the simple Lie algebra $E_8$

The following is the Dynkin diagram for simple Lie algebra $E_8$ My question is the following: It is clear that $e_i+e_j$ for $i \neq j$ is a positive root. Let $\alpha _8$ be the fundamental ...
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### Stochastic withdrawals from finitely-lived stock

Suppose an energy source has n quanta of energy in storage, all of which are available now (t=0) until t = T, at which time the energy source disappears (or is no longer available). Suppose there are ...
As this is homework I do not wan't a complete solution, just pointers where to look :) given there are $n$ persons at the bottom of a building, level 0. what is the probability that $k$ persons ...