# All Questions

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### Random links and $3$-manifolds

In Jeffrey Weeks book "The Shape of Space" he explaines at the end of Chapter 18 (on page 255) the following about the geometrization conjecture: A non-trivial connected sum $M_1\# M_2$ admits a ...
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### “Dimension” of ideals in $F_q[x]/\langle x^n-1\rangle$?

I'm very much confused by algebra. Hoping to get a bit more comfortable I tried to compute different things and see what happens... Let $F_q$ be the finite field with $q$ elements and ...
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### Is a n-th derivative of elliptic modular function $j(z)$ modular form?

Let $j(z)$ be a elliptic modular function. Then $j(\gamma z) = j(z)$ for all $\gamma \in SL(2,\mathbb{Z})$. I would like to see whether $\frac{d^n}{dz^n}j(z)$ be a modular form for $n = 1,2,3$. ...
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### research statement for assistant professorship [migrated]

What should a research statement for an application for an assistant professorship contain (pure mathematics in Germany)? How long should it be?
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### A reference about Grassmannian over finite fields

Suppose $Gr_k(k,n)$ the Grassmannian which classifies all the dimension $k+1$ sub-spaces of a dimension $n+1$ linear space over the field $k$. For the case over a finite field $\mathbb F_{q}$, we can ...
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### Lifting of a Diffeomorphism to an Orientable Double Cover [on hold]

Let $M$ be a non-orientable smooth $d$-dimensional manifold. Let $\tilde{M}$ be the oriented double cover for $M$; and let $f\in Diff^{1+\beta}(M)$. Define $\tilde{f}:\tilde{M}\rightarrow \tilde{M}$ ...
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### Convergence of a triple sum involving the imaginary part of the Riemann zeta function's non trivial zeros

Let $N>0$ an integer, $k>0$ a real parameter and let $\rho = \beta +i \gamma$ a non trivial zero of the Riemann zeta function. For a work I need to find the best $k$ possible such that ...
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### Brownian motion - probability of striking a sphere in $\mathbb{R}^n$ (a clarification)

This is primarily in reference to this question on MO. Serguei Popov's answer gives an explicit formula for the probability of a Brownian particle starting at the origin in $\mathbb{R}^n$ hitting the ...
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### $L^1$ convergence to equilibrium of solutions of heat equation

Let $u$ and $v$ be the weak solutions of $$u_t - \Delta u = f$$ $$u(0)=u_0$$ and $$-\Delta v = f$$ $$|\Omega|^{-1}\int_\Omega v =0$$ on a bounded domain $\Omega$, where $u$ and $v$ satisfy homogeneous ...
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### A question on decreasing function [on hold]

Let $t\in (0,1)$ and ${a_n}{x^n} + .... + {a_1}{x^1} + f(t) = 0$ $f(t)$ is continuous decreasing function of $t$. $a_i\ge0$ for all $i$. $y(t)$ is positive real zero of the first equition. Can ...
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### Collection of dense subsets as a “fingerprint” for Hausdorff topologies?

Let $(X,\tau)$ be a Hausdorff space and let ${\cal D}$ denote the collection of dense subsets of $(X,\tau)$. Is it possible that there is another Hausdorff topology $\tau_1 \neq \tau$ on $X$ such that ...
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### Circle actions on graph C*-algebras

Do all graph C*-algebras admit actions of the circle? Suppose we have a graph C*-algebra which we know is the quotient of a graph algebra by a circle action. Is it possible to read off the original ...
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### splitting property of etale covering

Theorem (Global Splitting): Let $X$ be an integral separated normal scheme flat and of finite type over $\mathbb Z$. Let $\phi: Y\rightarrow X$ be a connected etale covering which splits completely ...
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### Differential geometry [migrated]

If we have integrable distribution D of rank k on a manifold, and we have k functions which are constants on the associated integral manifolds. can we glue together these functions to obtain global ...
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### Advice on dealing with the gap [on hold]

A young mathematician AA is writing a paper proving a property X for a certain model. There have been quite a few articles proving the property X for various models. One of the first ones, let's call ...
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### Representation theorem for modular lattices?

Birkhoff's representation theorem implies that every distributive lattice embeds into the lattice of subsets of a set. Is there also some representation theorem for modular lattices? For example, I ...
Is there an easy way to separate the points of $\Bbb C \Bbb P^n$ or $\Bbb R \Bbb P^n$ (viewed as real-analytic manifolds) with real-analytic functions? If two points lie in a coordinate patch where a ...