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### convergence of a numerical series using information about an entire series [migrated]

I'm on a problem that seems simple but turns out to be a bit twisted. Let be $\sum_{n\epsilon N }^{}{u_nz^n}$ a power series with radius of convergence ρ = 1. Which of the following statements are ...
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### An example of module which is square-free, CS, NOT C3, and NOT nonsingular

Let $M$ be a right $R$-module ($R$ has unity). Recall that $M$ is called square-free if $M$ does not contain two nonzero isomorphic submodules with zero intersection. $M$ is called CS if every ...
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### Expected norm of a product of Gaussian matrices

Suppose $C_n$ is a product of $n$ $d\times d$ matrices with IID entries coming from standard normal. The following appears to be true. Is there an elementary proof? $$E[\|C_n\|_F^2]=d^{n+1}$$ This ...
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### Do the nearby cycle and Beilinson's vanishing cycle functors commute?

Let $X$ be a complex algebraic variety with a pair of regular functions $f_1,f_2$. To these functions we can associate various functors: the nearby cycles functor $\Psi_{f_i}$, the vanishing cycles ...
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### How exactly are realizability and the Curry-Howard correspondence related?

Consider, on the one hand: the Curry-Howard correspondence between, on the one hand, types and terms (programs) in various flavors of typed $\lambda$-calculus, and on the other, propositions and ...
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### Pfaffian elements and anomalies

If $X$ is a compact even dimensional spin manifold, then we have a family of chiral Dirac operators parametrized by $Met(X)$, the (infinite dimensional) manifold of Riemannian metrics on $X$. This is ...
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### Measuring to exact decimal places with ruler and compass exclusively [closed]

Do you know any way to construct a segment given its length in decimals, using only a ruler and compass, in an exact way? For example: a) 0.54896753 b) 12 decimals of acos(20°)
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### Orthogonal representation of free products of two groups

Suppose $A$ and $B$ are two countable, discrete, amenable groups. One definition of amenability tells us that there is a sequence of finitely supported, positive definite functions that converges to 1 ...
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### Research directions related to the Hilbert-Smith conjecture

The Hilbert-Smith Conjecture (HSC) is a famous open problem in geometric group theory stating "for every prime $p$ there are no faithful continuous action of the $p$-adic group of integers $A_p$ ...
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### When are the chirp signals orthogonal?

Assume that we have two bounded-time chirp signals, \begin{align} x(t)&=\exp\Big(j\pi(\alpha t^2+\beta t+\gamma)\Big),\quad 0\leq t\leq T,\\ y(t)&=\exp\Big(j\pi(\alpha' t^2+\beta' t+\gamma')\...
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### When does Morita equivalence between two Hopf-von Neumann algebras imply also equivalence of their categories of comodules?

Let $A$ and $B$ be two Hopf-von Neumann (bi)algebras. Furthermore, let us assume that we know that they are Morita equivalent as von Neumann algebras (i.e. their categories of appropriate ...
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### Exponential of Liouville Numbers

By Mahler classification of Transcendental real numbers (into the sets of $S$-, $T$- and $U$-numbers), we know that Any Liouville number is a $U$-number. $\log \alpha$ is either an $S$- or a $T$-...
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### Complemented C* Algebras

let $A$ and $B$ be unital separable commutative $C^*$ algebras, with $A\subset B$. Is it true that $A$ is complemented in $B$?
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### How to define the Sobolev quotient space $H^s(Γ)/{\mathbb R}$

Let $\Gamma$ be the boundary of a Lipschitz domain $\Omega\subset \mathbb R^3$. Denote by $H^s(\Gamma)$ the usual scalar Sobolev space for $s\in\mathbb R$. I want to know the definition of the ...
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### Large sets of nearly orthogonal integer vectors

This question is motivated by the Question 5 from the 2017 Asia Pacific Mathematical Olympiad. To paraphrase, the question asks what is the largest cardinality of a set $S \subset \mathbb{Z}^n$ such ...
Got stuck on this one for months. Given a sequence of non decreasing positive integers $a_1, .., a_n$, let there be $a_i$ balls labeled the number i, for each $1 \leq i \leq n$. Suppose there is a k ...