# All Questions

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### Unique minimal permuted inner products

Fix $n\in\Bbb N$. Denote $P$ to be $2n+c$ smallest consecutive primes all bigger than $n^2$ for some constant $c>2$. Pick $p_1,\dots,p_n$ and $q_1,\dots,q_n$ from $S$. What is the probability ...
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### embeds in $L(c_{0},\ell_{1})$

Let $c_{o}:=\lbrace x:\mathbb{N}\rightarrow \rightarrow\mathbb{R} :lim_{j\rightarrow\infty} x_{j}=0,sup_{j}\vert x_{j}\vert <\infty \rbrace$ denote the usual Banach sequence spaces. Given Banach ...
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### Can there be a numerical system in which logarithms can be expressed in terms of exponents in closed form?

The invention of complex numbers allowed to express trigonometric functions through hyperbolic ones in closed form. Is there possible an extension of real/complex numbers in which logarithms and ...
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### what's the detail to get the probability $p=1-(1-1/N)^{Tma_i}$ of extracting at least one ball in the urn(polya urn problem)

I am reading supplementary information of the paper Activity driven modeling of dynamic networks. It analogys the number of out degree of a activity node by Polya urns problem: it will equal to ...
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### Banach-Mazur distance from finite-dimensional subspaces of $\ell_p$ to the Hilbert space

I am reading a paper http://www.math.tamu.edu/~johnson/TF3.4.pdf by Bill Johnson and Andrzej Szankowski and having trouble grasping why $d_n(Z_m) \leq d_n(\ell_{p_{m+1}} ) = n^{|p_{m+1}-2|}$ in the ...
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### Definable curves in definable sets

Suppose that I have an unbounded subset $X \subset \mathbb{R}^n$, definable in the $o$-minimal structure $\mathbb{R}_{an, exp}$. Is it possible to find an unbounded, analytic and definable curve (i.e. ...
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### Ergodic, non-atomic measure on the circle which are $\times 2$ and $\times \frac12$ invariant

There any many ergodic, $T$-invariant, non-atomic measures on the space $X = [0,1)$, where $Tx = 2x \pmod 1$ is the doubling map. My question is: are any such measures also $T^{-1}$-invariant? BYO ...
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### Generating-bijective groups

We may say that two finitely generated groups $G$ and $H$ are generating-bijective when there exist homomorphisms $\phi:G\rightarrow H$ and $\psi:H\rightarrow G$ such that, for each ordered generating ...
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### How to write $\mathbb{C}[G/U_-]=\oplus_{\lambda} V_{\lambda}$ explicitly?

Let $G=GL_n$ and $U_-$ the set of all lower unipotent triangular matrices. Then by Gauss Decomposition, we have $G = U_-B$, where $B$ is the set of all upper triangular matrices. The group $U_-$ acts ...
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### Free cocompact action of discrete group gives a covering map [migrated]

I'd like to find a short proof of the following seemingly basic fact encountered on the second page of Atiyah's paper "Elliptic operators, discrete groups, and von Neumann algebras." ...
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### Chern-Simons form and Rarita-Schwinger operator

The Rarita-Schwinger (RS) operator naturally generalizes the Dirac operator and in Physics it describes particles with spin-3/2. I was wondering if there exists any reference concerning the ...
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### Example of a Schur-nontrivial group with no abelian subgroup of the form $H\times H$?

A group $G$ is Schur-nontrivial if the Schur multipler $H^2(G,U(1))$ is not the trivial group. I am trying to find an example of a Schur-nontrivial group which does not contain a subgroup of the form ...
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### Sequences of random variables converging in probability to the same limit a.s [migrated]

Let $(X_n)_{n \geq 1}$ and $(Y_n)_{n \geq 1}$ be two sequences of random variables s.t. $X_n$ converges to X and $Y_n$ to $Y$ both in probability. Furthemore, $X$ = $Y$ a.s. How can I prove that, for ...