All Questions

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Euler Transform elementary Proof

In this webpage Computing the Digits in π there is a proof of the Euler Transform (page 22). The proof there relies on measure theory and Lebesgue integration, I haven't studied that yet. In page ...
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2 smooth algebraic varieties, isomorphism $\mathcal{D}_{X \times Y} \to p_1^* \mathcal{D}_X \otimes p_2^*\mathcal{D}_Y$ as sheaves of rings?

Let $X$ and $Y$ be two smooth algebraic varieties. How do I see that I have an isomorphism$$\mathcal{D}_{X \times Y} \overset{\sim}{\to} p_1^* \mathcal{D}_X \otimes p_2^*\mathcal{D}_Y$$as sheaves of ...
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The second integral of the Killing form

Let $G$ be a lie group. Assume that $B$ is the Killing form of its Lie algebra $T_{e}G$. So $B$ is counted as a symmetric $2$-form on $G$ by translation. Is there a smooth function $f$ on $G$ ...
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Colliding Bullets

I saw this problem yesterday on reddit and I can't come up with a reasonable way to work it out. Once per second, a bullet is fired starting from $x=0$ with a uniformly random speed in $[0,1]$. If ...
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Given moments find extrema of expectation of a function

Consider random variables $Y$ with finite support: $\mathbb{P}(\vert Y\vert \le c) = 1$ and given moments: $\mathbb{E}Y = 0, \text{Var}Y = \sigma^2$. How to find such Y that maximizes functional ...
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Number of permutations for card decks in “Skat” and “Doppelkopf” games

In the card game Skat, you play with a deck of $32$ cards and each of the three players gets $10$ cards. This means that there are $32! \approx 2.63 × 10^{35}$ possible permutations of the card deck. ...
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Prove that a real-valued function in $n$ dimensions is integrable.`

Prove that a continuous, real-valued function on a closed interval in $E^n$ is integrable. Is it possible to deduct the above statement only using the following: A real valued function function $f$ ...
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A subset having volume implies the interior of that subset is same as volume of subset.

Define volume as $\text{vol}(A) = \int _A 1$, and A has volume if this exists. A couple questions similar to this has been asked, however, I (and everyone else) was unsure of their definition of ...
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The uniqueness of the cohomology of CW complexes.

Can anyone refer me to a proof of uniqueness of the cohomology of CW complexes, as follows? There is a natural isomorphism$$H^*(X, A; \pi) \cong H^*(C^*(X, A;\pi))$$under which the natural ...
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Is $\prod_{n=1}^\infty P_{2n-1}$ regularizable?

Assume that $P_n$ denotes the $n$'th prime for this entire question. Inspriation: I was dumbfounded by the fact that: $$\hat\prod_\limits{n=1}^\infty P_{n}=4\pi^2$$ After further investigation, I ...
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The $n$-cells of a modular lattice are antichains

The terminology "$n$-cell" is made up by me, and I'd love to hear if this has an official name. Given a poset $(X,\le)$, define the set $A_n$ of $n$-cells of $X$ recursively as follows: An ...
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Can't EF game theory be applied to finite languages WITH function symbols?

Let $\mathcal{M}$ and $\mathcal{N}$ be two structures in a language $\mathcal{L}$. We define the finite determined game $G_n(\mathcal{M},\mathcal{N})$ as a game with $n$ rounds where in each round ...
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Singular points of one-to-one mapping on rectifiable set

Let $E$ be an $s$-rectifiable set in $\mathbb{R}^n$ of positive $s$-dimensional Hausdorff measure $H^s(E)>0$. The original question I have is: Can there exist a one-to-one Lipschitz function ...
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Existence of Closed Curves around Bounded Components

I am stuck on part of a complex analysis proof that I think needs more justification than given. It's pretty purely a topological statement, but it may be that complex-analytic techniques would be ...
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Comparison of constrained optimization methods

I am trying to solve a constrained optimization problem using filter methods and came across two papers on the topic that I am having some problems with. The original filter method paper is the ...
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Finding zeta function of an elliptic curve

Let p=3 (mod 4) be a prime, and $E/F_{p^r}$ be the elliptic curve given by $y^2 = x^3 − x$ Find the zeta-function of $E/F_p$ and use it to determine $|E(F_{p^r} )|$ for all r>0.
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Using differential equation mixing problem for measuring combustion gasses

I'm trying to measure the concentration of carbon monoxide as the result of combustion as a factor of time in an enclosed space, with a relief valve (e.g., without incoming flow, no gas will flow ...
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This topic suggested me the following question: If $R$ is a commutative graded ring and $F$ a graded $R$-module which is free, then can we conclude that $F$ has a homogeneous basis (that is, a ...
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Complicated sum with binomial coefficients

I know how to prove, that $\frac{1}{2^{n}}\cdot\sum\limits_{k=0}^nC_n^k \cdot \sqrt{1+2^{2n}v^{2k}(1-v)^{2(n-k)}}$ tends to 2 if n tends to infinity for $v\in (0,\, 1),\ v\neq 1/2$. This can be proved ...
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On the change $u=x^{1+\frac{1}{p_n}}$ in $\log \zeta(s)=s\int_0^\infty\frac{\pi(x)}{x(x^s-1)}dx$, where $p_n$ is the nth prime number

In [1] Wikipedia say that for $\Re s>1$ the Riemann zeta function satisfies $$\log \zeta(s)=s\int_0^\infty\frac{\pi(x)}{x(x^s-1)}dx,$$ where $\pi(x)$ is the prime counting function, and say too ...