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Let $a +_K b$ denote Knuth addition. It is easy to check that $a +_K b \equiv a + b$ mod $2$ (in fact mod $4$), so the odd numbers are Knuth-sum-free. On the other hand, note that if $a +_K b = a +_K …

I realized perhaps my question is equivalent to asking for a characterization of sets $S\subseteq [a,b]$ that can not support a singular continuous measure. Actually, here we may assume that $S$ is …

Yes, AC gives us a continuums-sized measure 0 set without the perfect set property. For the construction of just any continuums-sized subset of $\mathbb{R}$, what matters is that $\mathbb{R}$ has card …

There is also a quick abstract proof via representation theory: $S^2_0(\mathbb{R}^4)$ is a 9-dimensional representation of $\mathrm{SO}(4)/\{\pm I_4\}\simeq \mathrm{SO}(3)\times\mathrm{SO}(3)$ and, h …

I'm not sure of the precise answer to your question. I suspect a greedy algorithm that chooses the next number for $A$ or $\Omega\setminus A$ based on which has the lower variation at that stage will …

Here's an example that came up in practice. Theorem: (Cauchy-Pompeiu) Let $U \subset \mathbb{C}$ be a bounded open set with piecewise- $C^1$ boundary $\partial U$ oriented positively (see appendix B …

For the sake of trying to offer some sort of answer based off of Tim's first comment: Let $\mathcal{E}$ be a coherent sheaf on $M$ and $(((E_i, \nabla_i)\to U_i \subset M_i)_{i \in I}$ be a local reso …

It follows from the comment of Moisha Kohan above. Another way to see it: take the closed oriented geodesics $\gamma, \eta$ in $M$ realizing the monodromy of $g$ and $h$ respectively. Since $g$ and $h …

The assumption that $\Omega$ is bounded is in fact required. (So your attempt is the correct proof, once you fix the omission in the statement.) Counterexample: let $\Omega$ be the upper half plane. L …

I will keep the notation and terminology from that question. First we can compute the inverse of the Cartan matrix: The matrix $C_D^{-1}$ has entries $$ (C_D)^{-1}_{(i,j)}= \begin{cases} 1, & \text{if …

This is a very classical question, and concepts like having an $A_n$ structure were conceived with work on this problem in mind. In recent years there has been new interest and spectacular new result …

There is a counterexample when $C'$ won't be finitely generated. The example was suggested by Jason Starr in the comments above. Let $C$ be $\mathbb{C}[a, b]$ and, for every element $f(a, b)\in C$, le …

I don't completely understand the question, because I don't generally see a problem with locally small ordinary categories. My own experience has generally been that for some esoteric things category …

As phrased, this problem looks just as difficult as determining whether a given element of $\mathbb{Q}_p$ lies in $\mathbb{Q}$. There are real numbers which are unknown to be rational (e.g. $\pi + e$) …

I thought about this a bit when trying to understand $\lambda$-rings – the same example that Wraith offers – and I think the answer is basically what is remarked in the cited lecture notes: it is nece …