# All Questions

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### Representations of $\text{SL}(n,\mathbb{F}_p)$ and $\text{Sp}(2n,\mathbb{F}_p)$ whose dimensions are $p^k$

I should preface this by saying that I am not a representation theorist, so I apologize if this can easily be found in standard sources (but sadly I cannot seem to extract it from any of the books I ...
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### Odds of 7 straight spins on a roulette wheel falling within the same group of 12 numbers [on hold]

What are the odds of 7 straight spins on a roulette wheel rendering a number within the same group of 12 numbers? (i.e. 7 numbers within 1-12, 7 within 13-24 or 7 within 25-36). This would be a ...
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### Construct the best piece-wise linear continuous function fitting given curve

How to construct the optimal piece-wise linear continuous function fitting given curve and given number of knots (optimal knots positions also must be determined by this method)?
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### A conjectural convergence condition for a weakened Elliott-Halberstam conjecture

For $a$ and $q$ positive integers such that $a\lt q$ and $(a,q)=1$, let $\pi(x;q,a)$ be the number of primes $p\equiv a\pmod q$ below $x$. One can show that $\pi(x;q,a)\sim \dfrac{\pi(x)}{\varphi(q)}$ ...
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### Eigenfunction and fourier transform [on hold]

Assume that $f_1$ is the first eigenfunction of the Dirichlet Laplacian $-\Delta f = \lambda f$, $U=\{z\in \mathbf{C}:|z|<1\}$ and $f|_{|z|=1}=0$. Assume also that $\lambda_1$ is the first ...
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### topologically finitely generated residually finite group

Suppose that $G$ is a topologically finitely generated profinite group and $H$ is a subgroup of countably infinite index. Can I say that $H$ must be topologically finitely generated with the subspace ...
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### How can one determine if a singularity is simple?

Let $f(z_1,z_2,\dots ,z_n)$ be an analytic function in $\mathbb{C}[[z_1,z_2,\dots ,z_n]]$ whose leading term defines an isolated singularity at the origin. If we have the following types of ...
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### Does a singularly perturbed cadlag process has sample paths in a Polish space?

In the theory of stochastic processes it is often said in the broader literature that Polish state spaces are the only important ones appearing in practice. Are there also examples of stochastic ...
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### Marten's proof of torelli theorem

I am trying to read the proof of torelli theorem by Henrik H.Martens "A new proof of torelli's theorem" Annals of mathematics vol78 no. 1 .The proof seems to me like using mysterious combination of 3 ...
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### Kodaira dimension of co-adjoint orbit

Let $G$ be a compact Lie group and $a\in\mathfrak{g}^*$ (dual of Lie algebra of Lie group $G$). Then let $\mathcal O_a$ be a coadjoint orbit. Then every co-adjoint orbit is Kähler manifold and also ...
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### Why is the constraint “ Rank (W) = 1” nonconvex? [on hold]

The SDR (semidefinite relaxation) is introduced to handle the SDP (semidefinite programming) problem with Rank (W) = 1, where W is a positive semidefinite matrix. I wonder why is the constraint " ...
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### Why $\gcd \{ {\rm ord}_p(q)\colon q\mid p-1 \}$ likes to be large?

For a prime $p$, let $F_p$ denote the greatest common divisor of the orders modulo $p$ of all prime divisors of $p-1$: $$F_p = \gcd \{ {\rm ord}_p(q)\colon q\mid p-1 \};$$ thus, for instance, ...
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### Are angles between points enough to decide the realizability?

Let n points in the plane be given whose coordinates we don't know. Assume, however, that for any triple of the points we know the angle. Question: Can we decide whether the n points are realizable ...
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### Concept of synchronizability

This thread is about the concept of synchronizability. It's a concept I tried to formalize in its most general sense but without success. The goal of this thread is therefore to try to formalize it in ...
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### countably-infinite-index subgroup of a finitely generated profinite group

Suppose that $G$ is a profinite group with the property that every open compact subgroup is topologically finitely generated and just infinite. Suppose that $H$ is a commensurated subgroup of $G$ with ...
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### Integrating new vectors of GL(n,F)

I'd be interested in the following: Let $\pi$ be an irreducible admissible generic representation of $GL(2n,F)$, $F$ a p-adic field. Assume that $\pi$ is ramified and let $W$ be a (non-trivial) new ...
Suppose $C_{1},C_{2},...,C_{n}$ are $0-1$ vectors of length $m$. Given $C_{i} \in \{0,1\}^{m}$ with $C_{i}=x_{i1}x_{i2}...x_{im}$ we say $C_{i}'=x_{i1}'x_{i2}'...x_{im}'$ is a subvector of $C_{i}'$ if ...