# All Questions

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### Distribution mod 1 of Factorial Multiples of Real Numbers.

Let $c$ be an irrational real number. Let $\{\cdot\}$ be the fractional part operator. I would like to get some sense of how in-the-dark we are about the distribution of values of $\{cn!\}$, for ...
619 views

### Diameter of a circle in an embedded Riemannian manifold

This question was inspired by an answer to the "Magic trick based on deep mathematics" question. I wanted to post it as a comment, but I ran out of characters! I'm sure there must be a collection of ...
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### How should I find a tutor for math-overflow level mathematics? [closed]

Searching for maths tutors online finds people willing to teach up to A-level. I'm looking for help at a more advanced level. At the moment I'm trying to teach myself category theory from downloaded ...
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### What is a proper stack?

I have seen the use of the word "proper Deligne-Mumford stack". Now, it is clear to me what it means for a morphism f of stacks to be proper: as usual it should be representable, and every morphism ...
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### Is there a limit of cos (n!)?

Hi, I encountered a problem today to prove that cos (n!) does not have a limit. I have no idea how to do it formally. Could someone help? The simpler the proof (by that i mean less complex theorems ...
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### Generate harmonic polynomials for a finite group

Let $G$ be a finite group and $H_G$ be the set of harmonic polynomials. In the case of the symmetric group these polynomials are spanned by the Vandermonde determinant and all its partial derivatives. ...
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### On the Frobenius coin problem [on hold]

Is there an $n_0\in\Bbb N$ such that for every $n\in\Bbb N_{>n_0}$ there are $a,b\in\Bbb N$ with $n<a,b<2n$ with $\mathsf{gcd}(a,b)=1$ such that 1. if $ax+by=rt$ for some $x,y>0$ with ...
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### Quantifier elimination - Existence of solution of a differential equation [on hold]

We consider the ring $\mathbb{C}[x]$ and the language $\{+, \frac{d}{dx}, 0, 1\}$. I want to eliminate the quantifier from the formula $\exists y \ Ly=f$. The elements of the ring are of the form ...
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### Mixed tensor index position significance

What is the significance of tensor index position? For example the fourth order Riemann curvature tensor \begin{align} R^m_{ijk} \end{align} or \begin{align} R^{\phantom{i}m}_{i\phantom{m}jk}. ...
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### Generalizing von Staudt's synthetic construction of the complex numbers

Starting from the real projective plane described synthetically/axiomatically, it is possible to construct the complex projective plane directly without passing through coordinates: one adjoins two ...
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### Confusion with practically implementing rational approximations

Writing a program visualizing Ford circles I've encountered a seemingly purely programmatic puzzle but then gradually realized there are some mathematical aspects of it which I don't understand. Let ...
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### Asymptotic enumeration of magic squares

An order-$n$ magic square is an $n \times n$ matrix over the numbers $\{1, ... ,n^2\}$, each appearing exactly once, whose row and column sums are all equal. Sometimes the sums of the diagonals are ...
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### does Gorenstein imply reduced? [on hold]

Let X be a projective scheme over a field, if X is Gorenstein then must X be reduced? The definition of Gorenstein I know is that all local rings have finite injective dimension as modules over ...
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### Reference request: category of sheaves of O-modules with coherent cohomology

Suppose $X$ is a smooth algebraic variety (say, in characteristic $0$). It's a folklore result that $D^b\text{Coh}(X)$ is equivalent to the derived category of complexes of sheaves of ...
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### Existence of nonlinear equation

How can we prove that equation (1) has solutions for every $p$. I mean, is there an analytic method that can be used to show that there exist solutions for every $p$ for this nonlinear equation: ...
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### All compact surfaces $S\subseteq \mathbb{R}^3$ are rigid?

Recently I've come across a lecture in differential geometry by Fernando Codá (in portuguese!) in which he stated that the following problem is (at least, up to 2014) open: Given ...
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### A weak topology generated by weakly $p$-summable sequences

Let $1\leq p<\infty$ and $X$ be a Banach space. $N_{p}(X)$ is to denote the subspace $\{x^{**}\in X^{**}:$ there exists a weakly $p$-summable sequence $(x_{n})_{n}$ in $X$ such that the sequence ...
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### Which commutative rings have irreducible (maximal) spectra?

Does there exist any term (or, maybe, a "description"?) for commutative unital noetherian rings such that their Jacobson ideals are prime (and so, their maximal spectra are irreducible)? What is the ...
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### Product-like structures on spheres

For $i=1,2$, let $j_i$ denote the inclusion of $S^n$ into the product $S^n \times S^n$ as the $i^{\text{th}}$ factor. I would very much like to know the answer to the following question, which seems ...
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### Systems with zero output for periodic inputs [on hold]

I need help with the following question: Suppose you have an LTI system which produces the 0 output in response to any periodic input with period T. Show that the impulse response h(t) of the system ...
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### Some examples of non trivial principal bundles

1.Is there a nontrivial pricipal bundle $P(M,G)$, with $G$ connected, such that the total space $P$ admit a foliation such that each leaf is diffeomorphic to $M$(Not necessarily via projection ...
Given a sequence of exchangeable random variables $X_1,\ldots,X_n$, and a measurable function $g: \mathbb{R}^n \to \mathbb{R}$. Let $S_n=g(X_1,\ldots,X_n)$. Then what is a natural construction of a ...