Comment by Gro-Tsen on What's the earliest result (outside of logic) that...
The very first postulate in Euclid's Elements is “to draw a straight line from any point to any point” (“ᾘτήσθωἀπὸπαντὸςσημείουἐπὶπᾶνσημεῖονεὐθεῖανγραμμὴνἀγαγεῖν”). The fact that any two points in the...
View ArticleComment by Gro-Tsen on Motivation and physical interpretation of the Laplace...
This may be too trivial to qualify as an answer, but a very simple physical interpretation of the Laplace transform is if you have a radioactive decay process which starts as a mixture of components...
View ArticleComment by Gro-Tsen on Does a function from $\mathbb R^2$ to $\mathbb R$...
Where might one find a proof of the result concerning squares of any size parallel to the axes?
View ArticleComment by Gro-Tsen on Let $a_1, \dots, a_n$ be a finite set of positive...
By $X$ you mean the $\mathbb{Q}$-linear span of $a_1,\ldots,a_n$, right? The notation suggests the subring they generate.
View ArticleComment by Gro-Tsen on On the complexity of global fields isomorphism
I don't know about the complexity, but I can at least point out that finding an isomorphism from $k[y]/(Q)$ to $k[x]/(P)$ is the same as finding a Tschirnhaus transformation from $P$ to $Q$....
View ArticleComment by Gro-Tsen on Is the group of translations of an affine plane always...
Theorem 6.13(3) in Casse's book Projective Geometry: An Introduction (Oxford 2006) looks like it might be relevant (it says that in a projective plane the set of “elations” with a given line $\ell$ as...
View ArticleComment by Gro-Tsen on Consequences of Kirti Joshi's new preprint about...
OK, but now he really does say it: arxiv.org/abs/2403.10430
View ArticleComment by Gro-Tsen on Recommendations to learn about the use of toposes in...
@Z.A.K. Thanks. Fixed.
View ArticleComment by Gro-Tsen on How can I (semi-formally) convince myself that...
Your question seems related to the “problem of space”, also known as the Riemann-Helmholtz (or Helmholtz-Lie) problem which approximately speaking consists of identifying (hopefully “visually obvious”)...
View ArticleComment by Gro-Tsen on meaning of "the singletons of $\Gamma$ are a basis for...
I think this is supposed to mean that every nonempty set in class $\Gamma$ contains ($\supseteq$) a singleton in class $\Gamma$. But it could also conceivably mean that every nonempty set in class...
View ArticleComment by Gro-Tsen on Quasi polynomial algorithm for NP complete problem
I think this is a valid question for this forum (although maybe more suited to CS-SE or CSTheory-SE), but the wording (“huge time to solve”, “maybe in years”) is clumsy at best. Interpreted as “what...
View ArticleComment by Gro-Tsen on Randomly removing length 1 intervals in an interval (a...
@AlekseiKulikov Yes, I'd like to know at least the linear term $ct$, and perhaps something more precise (ideally up to $o(1)$, but already $o(t)$ would be nice).
View ArticleAnswer by Gro-Tsen for Examples of common false beliefs in mathematics
I just learned incidentally in the comments of another question here that it is not true that every proper subgroup is contained in a maximal (proper) subgroup. A counterexample is easy to find: the...
View ArticleHow exactly are realizability and the Curry-Howard correspondence related?
Consider, on the one hand:the Curry-Howard correspondence between, on the one hand, types and terms (programs) in various flavors of typed $\lambda$-calculus, and on the other, propositions and proofs...
View ArticleAnswer by Gro-Tsen for Proof of remark in algebraic number theory
Consider the distinct embeddings $\sigma_\ell\colon K \to \mathbb{C}$ of the number field $K$ in the complex numbers (up to equality on $K$, not just up to their image): by standard facts in Galois...
View ArticleIntuitionistic proofs of propositional formulae versus natural...
The setup: Given a formula $\varphi$ of intuitionistic propositional logic (i.e., made from the connectors $\Rightarrow$, $\land$, $\lor$, $\top$ and $\bot$ from propositional variables...
View ArticleAnswer by Gro-Tsen for When is an upper bound on the longest irreducible...
I can't answer in the case of Python or any such “natural” programming language, but let me show that whenever $\#\Sigma \geq 2$ there is an encoding such that the function you define is...
View ArticleConstructively valid reference for the soberness of discrete spaces and...
I am looking for constructively valid references for the following two related facts:discrete topological spaces are sober,the points of a locale coproduct are the disjoint union of the points of the...
View ArticleComputing the Heyting operation on the frame of nuclei
(The following definitions are meant to be standard and are reproduced for completeness of the question.) A frame is a partially ordered set in which every finite subset has a greatest lower bound...
View ArticleWhat is known about sublocales defined by regular nuclei?
(For basic terminology, which is supposed to be standard anyway, see this other question, which inspired this one.)I am interested in nuclei $j\colon L\to L$ on a frame $L$ which are regular elements...
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