# All Questions

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### A short proof for $\dim(R[T])=\dim(R)+1$?

If $R$ is a commutative ring, it is easy to prove $\dim(R[T]) \geq \dim(R)+1$. For noetherian $R$, we have equality. Every proof I'm aware of uses quite a bit of commutative algebra and non-trivial ...
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### The Ring Game on $K[x,y,z]$

I recently read about the Ring Game on MathOverflow, and have been trying to determine winning strategies for each player on various rings. The game has two players and begins with a commutative ...
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### Does there exist a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
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### Generalizing $\int_{0}^{1} \frac{\arctan\sqrt{x^{2} + 2}}{\sqrt{x^{2} + 2}} \, \frac{\operatorname dx}{x^{2}+1} = \frac{5\pi^{2}}{96}$

The following integral \begin{align*} \int_{0}^{1} \frac{\arctan\sqrt{x^{2} + 2}}{\sqrt{x^{2} + 2}} \, \frac{dx}{x^{2}+1} = \frac{5\pi^{2}}{96} \tag{1} \end{align*} is called the Ahmed's integral ...
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### Is there a homology theory that counts connected components of a space?

It is well-known that the generators of the zeroth singular homology group $H_0(X)$ of a space $X$ correspond to the path components of $X$. I have recently learned that for Čech homology the ...
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### Geometric interpretation of the Riemann-Roch for curves

Let $X$ be a smooth projective curve of genus $g\geq2$ over an algebraically closed field $k$ and denote by $K$ a canonical divisor. I have some clues about the geometrical interpretation of the ...
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### Is there a categorical definition of submetry?

(Updated to include effective epimorphism.) This question is prompted by the recent discussion of why analysts don't use category theory. It demonstrates what happens when an analyst tries to use ...
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### Solving Special Function Equations Using Lie Symmetries

The Lie group and representation theory approach to special functions, and how they solve the ODEs arising in physics is absolutely amazing. I've given an example of its power below on Bessel's ...
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### Does every ring of integers sit inside a ring of integers that has a power basis?

Given a finite extension of the rationals, K, we know that $K=\mathbb{Q}[\alpha]$ by the primitive element theorem, so every $x \in K$ has the form $$x = a_0 + a_1 \alpha + \cdots + a_n \alpha^n$$ ...
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### When are nonintersecting finite degree field extensions linearly disjoint?

Let $F$ be a field, and let $K,L$ be finite degree field extensions of $F$ inside a common algebraic closure. Consider the following two properties: (i) $K$ and $L$ are linearly disjoint over $F$: ...
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Question : For every even $k\ge 4$, is the following $(\star)$ true? \begin{align}\text{If \beta=0.{a_1}^{k}{a_2}^{k}{a_3}^{k}\cdots\in\mathbb Q, then \alpha=0.a_1a_2a_3\cdots\in\mathbb Q.}\... 0answers 623 views ### Subgroups as isotropy subgroups and regular orbits on tuples Is there some natural or character-theoretic description of the minimum value of d such that G has a regular orbit on Ωd, where G is a finite group acting faithfully on a set Ω? Motivation: In some ... 0answers 334 views ### Irreducibility of ~\frac{x^{6k+2}-x+1}{x^2-x+1}~ over \mathbb Q[x] The Artin—Schreier polynomial ~x^n-x+1~ is always irreducible over \mathbb Q[x], unless n=6k+2, in which case it seems to have only two factors, one of which is always x^2-x+1. The ... 0answers 369 views ### Ring structure on the Galois group of a finite field Let F be a finite field. There is an isomorphism of topological groups (\mathrm{Gal}(\overline{F}/F),\circ) \cong (\widehat{\mathbb{Z}},+). It follows that the Galois group carries the structure ... 0answers 453 views ### Are the sums \sum_{n=1}^{\infty} \frac{1}{(n!)^k} transcendental? This question is inspired by my answer to the question "How to compute \prod_{n=1}^\infty\left(1+\frac{1}{n!}\right)?". The sums f(k) = \sum_{n=1}^{\infty} \frac{1}{(n!)^k} (for positive integer ... 0answers 167 views ### Painting the plane red and blue: Is it possible for each unit circumference to contain exactly n blue points? I recently stumbled upon the following problem: Consider the plane: You may color each point either red or blue. Is there a way to color it such that each unit circumference (centred anywhere) ... 0answers 605 views ### Formulae of the Year 2016 Decode the following limits to welcome the new year! This is my love limits (Created by me). I hope you Love it. LetA_{n}=\dfrac{n}{n^2+1}+\dfrac{n}{n^2+2^2}+\cdots+\dfrac{n}{n^2+n^2} show that ...
If $X$ is a normed vector space and if for each bounded operator $T \in B(X)$ with $\| T\| < 1$, the operator ${\rm id} - T$ is boundedly invertible, does it follow that $X$ is complete? Context: ...