Activity for celtschk
| Type | On... | Excerpt | Status | Date |
|---|---|---|---|---|
| Comment | Post #292671 |
That is indeed a very important information. With thousands of dimensions, reject sampling probably would not be a good strategy. Another important question is the number of vectors. there are of course at least (dimension+1) of them (or else you'd not have a finite volume), but it should make a huge... (more) |
— | 3 months ago |
| Comment | Post #292671 |
My guess would be that the most efficient way to sample would be to random sample from a well-chosen parallelogram-shaped bounding box, and then discard the point and repeat if the point lies outside of the shape. While you have the additional effort of generating more points than you ultimately end ... (more) |
— | 3 months ago |
| Comment | Post #292225 |
Are you sure you didn't miss a complex conjugation in the inner product formula? (more) |
— | 5 months ago |
| Comment | Post #292215 |
If you know in advance that the thing you search is in one of those four smaller rectangles (rather than anywhere in the 9 rectangles that make up the three squares), that would be exactly the kind of information which would change the probability. Except that if you knew it before even searching the... (more) |
— | 5 months ago |
| Comment | Post #292207 |
Of course it does. If the top right square now has a probability of $2/3$, then this means it is twice as likely in the upper right square than not. Which is a very strong hint that this square is the one which should be looked at next. Indeed, the probability distribution is exactly that: The inform... (more) |
— | 5 months ago |
| Comment | Post #292207 |
Well, I guess you could find a piece of paper with additional information in the visited square. *Somewhere* the extra information must come from, after all. (more) |
— | 5 months ago |
| Comment | Post #292189 |
Thank you for the answer and the reference. A small note: When you write $\lg$, you obviously mean the base 2 logarithm (the paper uses just comparisons with powers of 2). You should be more explicit about that, either by writing $\log_2$, or by using one of the standard notations for the base 2 loga... (more) |
— | 5 months ago |
| Edit | Post #292178 | Initial revision | — | 5 months ago |
| Question | — |
Is there a closed formula for multiplication of imaginary units in the direct limit of the Cayley-Dickson construction? The Cayley-Dickson construction is a way to systematically construct, starting from the real numbers, a sequence of ever higher-dimensional real algebras $Ak$ which starts with complex numbers and quaternions. The rules are as follows: Each algebra comes with an operation called conjugation,... (more) |
— | 5 months ago |
| Edit | Post #292099 |
Post edited: Fixed a typo |
— | 5 months ago |
| Edit | Post #292099 | Initial revision | — | 5 months ago |
| Answer | — |
A: What does it mean by saying that $C([0,1])$ is a subset of $L^\infty([0,1])$? This is actually an instance of a broader concept called identification. To understand the concept of identification, it is possibly better to first look at a simpler example, and only then see how this applies to $L^\infty([0,1])$ versus $C([0,1])$. What is identification? Example: $\mathbb N\su... (more) |
— | 5 months ago |
| Edit | Post #291882 |
Post edited: fixed a typo |
— | 6 months ago |
| Edit | Post #291882 |
Post edited: |
— | 6 months ago |
| Edit | Post #291882 | Initial revision | — | 6 months ago |
| Question | — |
For which spaces are all continuous functions either constant or the identity? The Sierpinski space has a particular property: All continuous functions to itself are either the identity or constant. Obviously the empty space and the singleton space share this property. My question is now: Are there other such spaces, and how would you find/construct them? I already found ... (more) |
— | 6 months ago |
| Comment | Post #290765 |
Correct me if I'm wrong, but AFAIK one of the assumptions of the Gödel incompleteness theorem is that you have a *finite* number of axioms or axiom schemes. Which makes sense given the fact that any axiom system we can actually handle will be finite in this way. However it doesn't preclude that there... (more) |
— | 7 months ago |
| Comment | Post #291687 |
Thank you. Together with the answer of the only-of part below, that completely answers my question.
(more) |
— | 7 months ago |
| Comment | Post #291691 |
Thank you. That's a quite elegant proof (I actually never considered that the odd numbers form a group modulo a power of two, though it's easy to check; that's a nice fact already by itself). (more) |
— | 7 months ago |
| Edit | Post #291687 |
Post edited: Fixed some errors |
— | 7 months ago |
| Comment | Post #291687 |
You're right on all accounts. I'll edit immediately. Sorry for the confusion. (more) |
— | 7 months ago |
| Comment | Post #287887 |
Thank you (and sorry for the late reaction; I've not been on the site for a long time). Your explanation why my construction fails to do what I intended was very helpful. (more) |
— | 7 months ago |
| Edit | Post #291687 | Initial revision | — | 7 months ago |
| Question | — |
All numbers are triangular modulo $N$ iff $N$ is a power of $2$? When thinking about binary representations of triangular numbers, I noticed an interesting property: In the cases I've tested, for the numbers from $0$ to $2^n-1$, each combination of the last $n$ bits occurs exactly once, that is, $k\mapsto k(k+1)/2 \bmod 2^n$ is a bijection on the set $\{0,\ldot... (more) |
— | 7 months ago |
| Comment | Post #288097 |
Thank you (and sorry for late reply, I wasn't on the site for quite some time).
(more) |
— | 7 months ago |
| Comment | Post #291562 |
Actually for $x\ne 0$, $\left|x\right|$ is differentiable; it's derivative there agrees with the sign function. So as long as you exclude $0$ from the domain, there is no problem with taking the derivative. Of course that also means that the result also can only be used for $x\ne 0$, but then, it is ... (more) |
— | 7 months ago |
| Comment | Post #291588 |
Thank you for your answer.
Either I don't correctly understand your answer, or I see a gap in your argument.
Let's denote the product topology of $V\times V$ when $V$ is equipped with the topology $\mathcal T_k$, with $\mathcal T_k\otimes\mathcal T_k$. Then obviously, the fact that $+$ is conti... (more) |
— | 7 months ago |
| Edit | Post #287880 |
Post edited: Removed further application because I now think I got it wrong |
— | almost 2 years ago |
| Edit | Post #287880 | Initial revision | — | almost 2 years ago |
| Question | — |
Unification and generalization of limit and colimit The recent post by r reminded me of my own idea to unify and generalise limit and colimit. I also only occasionally dabble in category theory, and thus I wonder if the following is a sound construction, and if so, if it is a known concept. The idea is as follows: We have a category $\mat... (more) |
— | almost 2 years ago |
| Edit | Post #287848 |
Post edited: fixed some grammar errors |
— | almost 2 years ago |
| Edit | Post #287848 |
Post edited: fixed typo |
— | almost 2 years ago |
| Edit | Post #287848 | Initial revision | — | almost 2 years ago |
| Answer | — |
A: Criterion in terms of the bases for determining whether one topology is finer than another A counterexample would be the set $S=\{0,1,2\}$ with the topologies $\mathcal T = \{\emptyset, \{0\}, \{0,1\}, S\}$and $\mathcal T' = \{\emptyset,\{0\},S\}$. Clearly $\mathcal T'$ is not finer than $\mathcal T$. Now $\mathcal T$ is generated by the basis $\mathscr B = \{\{0\},\{0,1\},S\}$ and $\ma... (more) |
— | almost 2 years ago |
| Comment | Post #287846 |
Your (2') is never true, as $\emptyset\in\mathscr B$, but there certainly doesn't exist a nonempty $B'\subset\emptyset$. In (2) the empty set is excluded by the demand that $x\in\mathscr B$. You need to explicitly exclude $B$ from being the empty set in order for (2') to be possibly true. (more) |
— | almost 2 years ago |
| Comment | Post #287764 |
On reputation not limiting your actions: It is an intentional design. Codidact *does* limit your abilities, but it doesn't do so on reputation, but on measures that relate to the abilities. For example, you cannot gain edit ability by answering many questions, but by suggesting many edits that got ac... (more) |
— | almost 2 years ago |
| Comment | Post #287764 |
While I agree it would make sense to separate the two, note that having a high reputation on Q&A doesn't imply you're a good mathematician either. You may gather reputation by simply asking many good questions. Or you may gather upvotes by giving good pedagogical answers to simple mathematical questi... (more) |
— | almost 2 years ago |
| Edit | Post #287724 | Initial revision | — | almost 2 years ago |
| Answer | — |
A: Proving $|{\bf R}^{\bf R}|=|2^{\bf R}|$ using the Schroeder-Bernstein Theorem To apply the Schröder-Bernstein theorem, we need injections in two directions. Given that the Schröder-Bernstein theorem doesn't require the axiom of choice (AC), there's a value in avoiding anything that requires AC (such as the theorem that existence of a surjection from $A$ to $B$ implies the exis... (more) |
— | almost 2 years ago |
| Comment | Post #287518 |
I don't think it's technically possible to block display math on parts of a page (but I might be wrong on this). But maybe submitting a title with it can be blocked. (more) |
— | about 2 years ago |
| Edit | Post #287410 |
Post edited: Changed displaymath to inline math in title |
— | about 2 years ago |
| Suggested Edit | Post #287410 |
Suggested edit: Changed displaymath to inline math in title (more) |
helpful | about 2 years ago |
| Edit | Post #287481 | Initial revision | — | about 2 years ago |
| Question | — |
LaTeX formulas not rendered correctly in summary I've noticed that the question preview of several questions on the Q&A page contains “misplaced &” with yellow background, which turns out to be a MathJax error. However opening one of them shows that the post itself is just fine. However the formula that doesn't correctly show up in the preview c... (more) |
— | about 2 years ago |
| Comment | Post #287419 |
@#53407 Well, the “length” is defined in the title as the number of the points that come to lie on the perimeter of the convex hull. It's not what conventionally would be called “length” (yes, it's another case of confusing wording, but in this case the exact intended definition is contained in the q... (more) |
— | about 2 years ago |
| Comment | Post #287419 |
I see, so you should simply remove the word "grid" to make your question vastly more understandable. You have a square with randomly distributed points in it. No grid that is of any relevance. (more) |
— | about 2 years ago |
| Comment | Post #287410 |
The product as you've written it now is not well defined, and I think not what you wanted anyway. It is an infinite product. (more) |
— | about 2 years ago |
| Comment | Post #287419 |
So what is the point of the grid in the first place? (more) |
— | about 2 years ago |
| Comment | Post #287419 |
Err, actually even in the case of identical points we have $l=4$ for all configurations, as the identical points still need to be counted separately. But still a counterexample. (more) |
— | about 2 years ago |
| Comment | Post #287419 |
Actually thinking about it, it's a counterexample in the other case, too, as there are more configurations with two points identical than all points different. (more) |
— | about 2 years ago |