Activity for clemens
| Type | On... | Excerpt | Status | Date |
|---|---|---|---|---|
| Edit | Post #295555 | Initial revision | — | 7 months ago |
| Answer | — |
A: Why are there no recent well-received questions? There are in fact multiple recent well-received questions, e.g. here, here, and here that received lengthy and useful answers. That those questions had not been asked at the time of the OP is I think more a coincidence than a sign of low community vitality. And there is an abundance of interes... (more) |
— | 7 months ago |
| Edit | Post #295549 |
Post edited: reformatted |
— | 7 months ago |
| Comment | Post #295545 |
If we don't have choice my understanding is that forcing gives us a great deal of freedom; e.g. we no longer have the restriction that the continuum needs to be a regular cardinal. So I would certainly be very surprised if the title question did not have a positive answer. But I probably know les... (more) |
— | 7 months ago |
| Edit | Post #295549 |
Post edited: |
— | 7 months ago |
| Edit | Post #295549 | Initial revision | — | 7 months ago |
| Answer | — |
A: Are there other topologies on $\mathbb R$ that make it a topological field? Here's an incomplete answer that I hope will be useful. The basic question of whether there are any non-standard ways to give a topological field structure to $\mathbb{R}$ is of course answerable in the positive, but I tried to answer the deeper question of characterizing the various ways in whic... (more) |
— | 7 months ago |
| Comment | Post #295427 |
It looks to me as though kuromajutsushi knows much more than I do about measure theory and has answered about as much of your question as he found decipherable. I should frankly suggest to find some different topics to ask about on Codidact.
For Codidact is not a discussion forum like Reddit ... (more) |
— | 7 months ago |
| Comment | Post #295536 |
See my comment: https://math.codidact.com/comments/thread/11565#comment-28620 (more) |
— | 7 months ago |
| Comment | Post #295536 |
Never mind---wrt. my first question, the projection of an ultrafilter has to be maximal because otherwise we could nontrivially extend the projection and thus nontrivially extend the original ultrafilter (which contradicts the definition of an ultrafilter); and wrt. my second question, it is easy... (more) |
— | 7 months ago |
| Comment | Post #295427 |
The user in Reddit makes reasonable points. And I must say I did not read carefully your entire post; I expected you to do that and check whether my conjectures made sense in its context.
I have certainly found the discussion enjoyable, but I shall have to say that, not only in our own persona... (more) |
— | 7 months ago |
| Comment | Post #295536 |
This is very helpful, thanks! But how is it that, when we take projections of an ultrafilter, we always get an ultrafilter instead of getting the whole space?
Also, I was somewhat surprised to learn that we're taking an ultrafilter on *all* the subsets of a topology which surprises me somewha... (more) |
— | 7 months ago |
| Comment | Post #295427 |
Interesting. But perhaps this is overkill.
Already if we decide that the marginals follow a Cauchy distribution (again, the marginals have to follow some distribution, and it can't be uniform if it's unbounded) your point (1) works: almost all sequences (hence also almost all functions) have n... (more) |
— | 7 months ago |
| Comment | Post #295427 |
Well, don't hold your breath. I do still intend to read the paper on prevalence. In the meantime, I suggest some further clarification. The current clarifications are very helpful but it remains not at all clear what kind of probability distribution on $\mathcal{P}(\mathbb{R})$ is being requested... (more) |
— | 7 months ago |
| Edit | Post #295527 |
Post edited: |
— | 7 months ago |
| Edit | Post #295527 |
Post edited: |
— | 7 months ago |
| Edit | Post #295527 |
Post edited: corrected an error |
— | 7 months ago |
| Edit | Post #295527 |
Post edited: corrected an error |
— | 7 months ago |
| Comment | Post #295527 |
Hello and welcome to Codidact! I would advise you to remove the "Works for me" endorsement, as in fact I realize that the Grothendieck group approach does not work properly. (Since adding $\aleph_n$ has to be invertible, everything ends up equalling everything else.)
Sobociński's approach can ... (more) |
— | 7 months ago |
| Edit | Post #295527 | Initial revision | — | 7 months ago |
| Answer | — |
A: Is there a stable/consistent extension of wheel theory to alephs? (Update: this answer is not complete; I made a misstep, as you can see in footnote 2. Sorry!) I shall use a slightly different formalization of wheel theory I read from Sobociński's excellent Graphical Linear Algebra1, where the numbers on the wheel are considered as linear relations between i... (more) |
— | 7 months ago |
| Comment | Post #295427 |
This weighting of domains does not work. For then by symmetry a random partial function on $\mathbb{R}$ has domain $\subseteq \mathbb{R}^+$ with probability ½ and domain $\subseteq \mathbb{R}^-$ with probability ½. Since those two are disjoint, almost all partial functions on $\mathbb{R}$ would t... (more) |
— | 7 months ago |
| Comment | Post #295427 |
OK, that's quite helpful.
You cannot have a uniform marginal probability distribution over the unbounded real numbers, though, any more than you can have a uniform probability distribution over the positive integers.
I'll try to look into the paper on prevalence you linked. Is it your unde... (more) |
— | 7 months ago |
| Comment | Post #295427 |
Hello again!
On readability: I shall repeat my advice that, since you are looking for a single measure that satisfies three conditions, the conditions ought to be merged into a single question. Otherwise it adds unneeded difficulty for readers to go back and forth from one question to another,... (more) |
— | 7 months ago |
| Comment | Post #295461 |
Well, that's embarrassing! But it should be corrected now, and I've added an explanation for how it's derived from the IEP. (more) |
— | 7 months ago |
| Edit | Post #295461 |
Post edited: added further explanation for the formula (and corrected it, per Dylan Callaghan's comments) |
— | 7 months ago |
| Edit | Post #295461 |
Post edited: added further explanation for the formula (and corrected it, per Dylan Callaghan's comments) |
— | 7 months ago |
| Comment | Post #295425 |
What I have is not much of an answer, as already stated. (more) |
— | 7 months ago |
| Comment | Post #295425 |
Looking over it cursorily, it seems you want to define an "extended mean" of $f$ based on the limit of almost all sequences of bounded functions with pointwise limit $f$; is that correct? (more) |
— | 7 months ago |
| Edit | Post #295461 |
Post edited: corrected the formula and added explanation |
— | 7 months ago |
| Comment | Post #295461 |
You are correct, sorry.
I think the coefficient $-1^{n_1-|K|}$ just has to be adjusted to $-1^{n_1-|K|} {n \choose (n_1-|K|)}$.
(more) |
— | 7 months ago |
| Comment | Post #295425 |
Yes, that link works; thanks.
Given the application area of your work I would suggest you look into the techniques used in theoretical physics e.g. with Feynman path integrals and quantum chromodynamics. This kind of regularization of undefined sums/integrals to have finite and defined values ... (more) |
— | 7 months ago |
| Edit | Post #295482 | Initial revision | — | 7 months ago |
| Question | — |
How can one prove Tychonoff's theorem using ultrafilters? Tychonoff's theorem, as a reminder, says that the product of a possibly infinite number of compact spaces is compact (a compact space being a space every open cover of which has a finite subcover; this is sometimes called quasi-compactness, as in the Stacks Project). Interlude: the typical way... (more) |
— | 7 months ago |
| Comment | Post #295425 |
Sorry, I cannot read this link (https://acrobat.adobe.com/id/urn:aaid:sc:VA6C2:36a3a008-d76b-42a0-9e70-01efc0f26f01). Do you have the paper in a different format?
Anyway what relevance do "families" have to this question? I do not see them mentioned. (more) |
— | 7 months ago |
| Comment | Post #295425 |
Well, to answer any of these questions, I would want to know what standards of elegance this "measure" is supposed to meet, so to speak.
You write that you want the mean of a random function $f$ to be undefined a.a. Alright – and it's easy enough to do this (just take the Cauchy distribution (... (more) |
— | 7 months ago |
| Comment | Post #295425 |
OK, that makes sense.
Since the main question seems (to me) to be about defining the measure, I would suggest merging the three questions into one and putting the three statements in this question and in pts. 1 and 2 at the end. (more) |
— | 7 months ago |
| Comment | Post #295425 |
You linked, at the bottom of your post, to two other questions. What, more succinctly, is the difference between those two questions and your current question? They seem almost exactly the same.
(more) |
— | 7 months ago |
| Edit | Post #295470 |
Post edited: Added information about why the question as stated in the title is the same as the question as stated in the text |
— | 7 months ago |
| Edit | Post #295470 | Initial revision | — | 7 months ago |
| Question | — |
Why are regular languages closed under intersection and complementation? In Worrell's lecture notes I happened to find a very interesting remark: > Recall that a regular language is a language accepted by a nondeterministic finite automaton (NFA). Recall also that the class of regular languages is closed under intersection and complementation. (emphasis mine) T... (more) |
— | 7 months ago |
| Comment | Post #293102 |
@#84008 The linked lecture notes by Worrell are very interesting but what exactly do they have to do with the question of compactness in your OP? (more) |
— | 7 months ago |
| Comment | Post #295439 |
I am not sure, although the user seems to know what he's talking about. Maybe when I look at the referenced paper I can tell. I did not previously realize that there exist functions whose graphs have a Hausdorff dimension of 2; it sounds interesting.
(more) |
— | 7 months ago |
| Comment | Post #295439 |
Unless we're missing very important context, the statement of the user you quoted is patently incorrect "that an everywhere-surjective function on every subinterval must be wildly pathological and cannot be Lebesgue-integrable, so the integral defining the expectation fails to exist". Look up, e.... (more) |
— | 7 months ago |
| Edit | Post #295457 |
Post edited: Corrected answer to take into account unboundedness of functions |
— | 7 months ago |
| Edit | Post #295457 | Post undeleted | — | 7 months ago |
| Edit | Post #295457 | Post deleted | — | 7 months ago |
| Comment | Post #295469 |
Note that it is not correct that I myself was advised not to make major edits. It is simply general policy on StackExchange, and I would assume also here, that you can edit a question to clarify it, but not entirely change a question, especially after the original question has been answered and c... (more) |
— | 7 months ago |
| Edit | Post #295461 |
Post edited: added information on what variant of the question this answer answers |
— | 7 months ago |
| Edit | Post #295465 |
Post edited: |
— | 7 months ago |