# All Questions

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### Natural probability on integers

This is a follow-up to this classical question asked recently here: we now (e.g. using the second Borel-Cantelli Lemma) that no probability measure on $\mathbb{Z}$ has the property that $n\mathbb{Z}$ ...
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### divisible by all standard prime numbers

This question is about prime numbers in nonstandard models of Peano Arithmetic. Every such model looks like N+AxZ, where A is a dense linear order without end points. There are many nonstandard ...
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### References request: are there some references about simple modules of group algebras?

Are there some references about constructing the simples, determining the dimensions of simple modules and describing decompositions of tensor products of simple modules of group algebras? Thank you ...
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### Which function is $f(x)=x$ when $x>=0$ and $f(x)=0$ when $x=-1$ [on hold]

This is a very simple question that will not be a problem for any mathematician. With enough time I could come up with a non-optimal solution myself, but I know some of you may just provide the ...
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### What are all the factors of n=p^2q^3r? [on hold]

n=p^2q^3r It has a total of 24 factors. I need to show all of the factors.
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### $l$-weights and $l$-character of finite-dimensional highest $l$-weight representation of $L\mathfrak{g}$

I am trying to solve the following problem, which is related to relatively recent results, but I am not sure how to do it. Problem In this problem, $\mathfrak{g}=\mathfrak{sl}_{2}$. We study ...
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### Transitive closure of balanced bounded mass transport

Given two $\sigma$-finite measures $\mu$ and $\nu$ on $\mathbb{R}^n$, write $\mu \sim \nu$ iff there exist countable decompositions $\mu = \mu_1 + \mu_2 + \cdots$ and $\nu = \nu_1 + \nu_2 + \cdots$ ...
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### Upper bound involving random orthogonal projection

Let $R$ be an $n\times N$ random matrix with i.i.d. standard Gaussian entries, $n<N$, and let $M:=(RR^T)^{-1/2}R$. Let $u,v\in \mathbb{R}^N$ non-random and s.t. $u^Tv=0$ and $\|u\|>\|v\|$. I ...
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### Writing down gerbes explicitly over the projective line

Let $X = [\mathbb P^1/(\mathbb Z/2\mathbb Z)]$, where we take the trivial action of $\mathbb Z/2\mathbb Z$ on $\mathbb P^1$. Is this DM stack over $\mathbb C$ a gerbe over $\mathbb P^1$? Is it the ...
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### Rationality of intersection of algebraic groups

Suppose that $G$ (defined over $\mathbb{Q}$) and $H$ (defined over $\mathbb{R}$) are two algebraic subgroups of a larger algebraic group defined over $\mathbb{Q}$. Assume that $G(\mathbb{R})$ and ...
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### Historical (personal) examples of teaching-based research

The phrase "teaching-based research" brings to mind research about teaching, though important, it is not what I mean. Unfortunately, I couldn't come up with a better phrase, thus please bear with me ...
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### G-delta sets and Lebesgue measure [on hold]

The set S of all subsets in R^n which are of the form G\N, where G is a G-delta set and N a null-set (=outer Lebesgue measure zero) coincides with the set of all Lebesgue measurable sets. How could ...
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### Question on homogeneous measures

Let $\mu$ be a strictly positive measure ($m(a)=0$ iff $a=0$) on a Boolean algebra $B$. $\mu$ is called homogeneous if it have the same Maharam type on every $b\in B$. By additive measure algebra I ...
Let $x_0$ be a point contained inside a compact, convex set $C\subset\mathbb{R}^d$, which is of the form $C=\{x:f(x)\leq0\}$ for some explicit convex function $f$. Is there a computationally ...