Comment by Gro-Tsen on Teaching suggestions for Kleene fixed point theorem
I was about to make similar suggestions, but I realized that OP is talking about Kleene's fixed-point theorem for continuous functions on directed-complete partial order, whereas your answer is about...
View ArticleComment by Gro-Tsen on Does the first fundamental representation of...
Yes. In fact, if $V_k$ (for $1\leq k\leq n$) denotes the $k$-th fundamental representation in the order of the nodes of the Dynkin diagram, and $V_0$ the trivial representation, then $\bigwedge^k V_1 =...
View ArticleAnswer by Gro-Tsen for Does the first fundamental representation of...
(Copied from my own comments.) Yes. In fact, if $V_k$ (for $1\leq k\leq n$) denotes the $k$-th fundamental representation in the order of the nodes of the Dynkin diagram, and $V_0$ the trivial...
View ArticleComment by Gro-Tsen on Why does Weihrauch reducibility make use of...
While it doesn't exactly answer your question, I recommend Kihara's paper “Rethinking the notion of oracle”, which considers all sorts of kinds of reductions (oracles you can only use once vs, oracles...
View ArticleComment by Gro-Tsen on A homogeneous manifold that does not admit an...
There is no Riemannian metric on the circle invariant under all self-diffeomorphisms of the circle. Not even of we restrict ourselves to the (finite-dimensional, but non compact) subgroup $PGL_2$...
View ArticleComment by Gro-Tsen on What are the Nash equilibria of the “aim for the...
@MichaelGreinecker Right: sorry, I should have excluded that case.
View ArticleComment by Gro-Tsen on Etymology of “real numbers"
Meta: I don't understand the downvote(s) on this question. The origin of the term “real number” is a perfectly good and sensible question on the history of mathematics / mathematical terminology, and...
View ArticleComment by Gro-Tsen on Why is it not possible to define the necessity...
A formal answer to your question has already been given by Simon Henry. But let me make the following more informal remark: even just classically, the whole point of modal operators is that they give...
View ArticleComment by Gro-Tsen on The constructive Eudoxus reals
The terminology of constructive real numbers is a giant pile of mess because subtly different concepts sometimes go by the same name or vice versa, and not every author takes the trouble to recall...
View ArticleComment by Gro-Tsen on Examples of concrete games to apply Borel determinacy to
OK, that meets my stated conditions, but concerning (3) I was hoping for a game whose borelness would be much easier to see — if I understand what you say, it's not at all obvious, so it's hard to give...
View ArticleExamples of concrete games to apply Borel determinacy to
I'm teaching a course on various mathematical aspects of games, and I'd like to find some examples to illustrate Borel determinacy. Open or closed determinacy is easy to motivate because it proves the...
View ArticleComment by Gro-Tsen on Reference: If $X$ is metrizable, then $X$ is...
I'm not sure whether the difficult part is “if” or “only if”, but just in case, the “if” part (i.e., metrizable of non-measurable cardinality implies realcompact) is in Gillman & Jerison, Rings of...
View ArticleComment by Gro-Tsen on Sufficient condition for a continuous function to be...
Simpleminded counterexample to the question as stated: let $K'$ be the closed unit disk in $\mathbb{R}^2$ and $K = \partial K'$ be the unit circle (so also $\partial K = K$), and let $f\colon K\to K'$...
View ArticleAnswer by Gro-Tsen for Alternate descriptions of finite fields
Instead of having a single polynomial-quotient (aka “rupture”) step $\mathbb{F}_p[X]/(P)$ with $P \in \mathbb{F}_p[X]$ irreducible, you can also construct finite fields in several steps, i.e.,...
View ArticleComment by Gro-Tsen on Non-isomorphic $T_0$-spaces with order-isomorphic...
For the specialization order? You mean like the usual topology on $\mathbb{R}$ and the discrete topology on $\mathbb{R}$? 😉
View ArticleWhat are the possible symmetry groups of n-point constructions in the...
Let $k$ be an infinite field, perhaps take $k = \mathbb{C}$ if it simplifies matters.I will be asking a question about $\mathbb{P}^2$ for definiteness and to simplify definitions/notations, but feel...
View ArticleComment by Gro-Tsen on Can Gomoku(five in a row) draw on an infinite board?...
It's unclear what kind of answer you're expecting. You did a lengthy summary of the state of the art, from which it is obvious that it is not known for which $n$ optimal play results in a draw, and MO...
View ArticleComment by Gro-Tsen on Ordinal realizability vs the constructible universe
Strongly related question (except that I asked about $L_\alpha$ rather than $L$ itself), which I'm mainly linking because the “related” tab didn't discover it on its own.
View ArticleFormulas for the line joining two points in the projective plane over a...
Let $K$ be a[n associative] division algebra (= skew field). By the “projective plane”$\mathbb{P}^2(K)$ over $K$ I mean, as usual, the set of triples $(x,y,z)$ of elements of $K$, not all zero, up to...
View ArticleComment by Gro-Tsen on Convergence of sequences formed by orthocenters,...
This is reminiscent of a kind of “iterated mean” process (see this question on this subject), but in dimension 2 and invariant under Euclidean similitudes rather than in dimension 1. To prove...
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