Unanswered Questions

134
votes
1answer
4k views

Is there a 0-1 law for the theory of groups?

For each first order sentence $\phi$ in the language of groups, define : $$p_N(\phi)=\frac{\text{number of nonisomorphic groups $G$ of order} \le N\text{ such that } \phi \text{ is valid in } ...
72
votes
0answers
3k views

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 ...
61
votes
0answers
2k views

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 ...
57
votes
0answers
1k views

Identification of a curious function

During computation of some Shapley values (details below), I encountered the following function: $$ f\left(\sum_{k \geq 0} 2^{-p_k}\right) = \sum_{k \geq 0} \frac{1}{(p_k+1)\binom{p_k}{k}}, $$ where ...
9
votes
1answer
91 views

Integral of $\sqrt{x^3 + 8}$?

I have issues solving the following integral: $$\int\sqrt{x^3+8}~dx$$ I tried substitution and integration by parts, but with no use. I'm guessing I have to use some trigonometric substitution. ...
7
votes
0answers
70 views
+100

Why is there no theory of $G$-ic varieties, for linear algebraic groups $G$?

A toric variety is an algebraic variety $X$ with an embedding $T \hookrightarrow X$ of an algebraic torus $T$ as a dense open set, such that $T$ acts on $X$ and the embedding is equivariant. It ...
6
votes
0answers
100 views
+100

On distributivity of lattice of group topologies

Let $\frak L$ be the set of all topologies $\mathcal T$ on $\Bbb Q$ (the additive group of all rational numbers) such that $(\Bbb Q,\mathcal T)$ is a topological group. Then $(\frak L,\subseteq)$ is a ...
4
votes
0answers
74 views
+100

Compositions of filters on finite unions of Cartesian products

Let $\Gamma$ be the lattice of all finite unions of Cartesian products $A\times B$ of two arbitrary sets $A,B\subseteq U$ for some set $U$. See this note for other equivalent ways to describe the set ...
3
votes
0answers
18 views

Holder regularity for the heat potentials

First I apologize for my bad English and for any error: this is my first question. I need some regularity results for the single and double layer heat potentials. If $\Gamma(t,x)$ is the fundamental ...
3
votes
0answers
127 views
+50

What is the smallest possible angle of this polygon?

A convex polygon contains a square with side-length 1 and is contained in a parallel square with side-length 2 (which is its smallest containing square). What is the smallest possible angle of the ...
2
votes
0answers
15 views

Is an orthogonal projector bounded in $L_p$-spaces?

Let $P$ be an orthogonal projector on $L_2([0,1])$. In particular, $P$ is a projector from $C^\infty([0,1])$ to itself. For $0<p<\infty$, we define for $f \in C^\infty$ the norm (quasi-norm if ...
2
votes
0answers
22 views

Monoids, Semigroups, and a Reflective Subcategory.

The following is reflective in the category $\mathbf{Sem}$ of semigroups and their homomorphisms: we add a new neutral element to each semigroup, even monoids; then the reflections are identity ...
2
votes
0answers
74 views
+100

Existence of a bounded function satisfying a second order differential equation

This question is a variation version from here. Let $\phi:\mathbb{R}\mapsto\mathbb{R}$ be the standard normal density, $$\phi(x)=\frac1{\sqrt{2\pi}}e^{-\frac{x^2}{2}}, \forall x\in\mathbb{R}.$$ ...
2
votes
0answers
113 views
+50

Vector space basis change: is this “index-free” notation correct?

There are already quite a number of questions on basis change in a vector space. Nevertheless, to fully grasp the underlying idea I made up the following notation and I have some doubts on it (note: ...
2
votes
1answer
39 views

Maxima of this function

I am looking for extrema of the function $$g(a,b):=\frac{\pi}{2|b|}\left(\left(\text{C}\left[\frac{a-2 |b|}{\sqrt{2|b| \pi }}\right]-\text{C}\left[\frac{a+2 |b|}{\sqrt{2|b| \pi ...

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