Activity for clemens
| Type | On... | Excerpt | Status | Date |
|---|---|---|---|---|
| Edit | Post #295465 |
Post edited: |
— | 7 months ago |
| Edit | Post #295465 |
Post edited: |
— | 7 months ago |
| Edit | Post #295465 |
Post edited: |
— | 7 months ago |
| Edit | Post #295465 | Initial revision | — | 7 months ago |
| Answer | — |
A: Compactness of the Propositional Calculus I think perhaps a visualization of this proof will be most useful. This is because the general theorem here can be understood as a straightforward consequence of Weak Kőnig's lemma: the lemma that every infinite binary tree has an infinite path. In proving completeness, we'll be working with ... (more) |
— | 7 months ago |
| Edit | Post #295461 |
Post edited: Added a formula at end to help users understand what I am doing |
— | 7 months ago |
| Edit | Post #295457 |
Post edited: |
— | 7 months ago |
| Edit | Post #295461 |
Post edited: changed first paragraph to better reflect the nature of the solution |
— | 7 months ago |
| Comment | Post #295434 |
Good question. As I'm very new here I do not know precisely what the customs are about modifying questions after they are answered. In StackExchange I think it is discouraged: see https://math.meta.stackexchange.com/questions/20103/is-that-ok-to-modify-a-question-significantly . But it would be g... (more) |
— | 7 months ago |
| Edit | Post #295461 | Initial revision | — | 7 months ago |
| Answer | — |
A: Multivariate urn/coupon collector problem without replacement I must admit to not yet understanding how we get the expected values in your partial answers (1) and (2). But here is a solution to the main question, giving us the number of ways to collect balls of at most $n1$ different colors in the first $n2$ draws without replacement, from a set of colored ... (more) |
— | 7 months ago |
| Edit | Post #295457 |
Post edited: I figured out how to explain better why measurability of $\mathbb{R}$ and $\mathbb{R}^2$ are equivalent |
— | 7 months ago |
| Comment | Post #295458 |
Given that $\frac{(c_i)_n}{(m)_n}$ is the probability of getting $n$ consecutive balls of the $i$th color from an urn with $m$ balls total, I wonder how one derives such a simple solution as $\frac{m+1}{c_i+1}$ in your first partial solution above? And, relatedly, I am frankly a bit unsure about ... (more) |
— | 7 months ago |
| Comment | Post #285086 |
I was interested in your mention of Project Euler, as I also have been very much enjoying it for some months now. I wonder, given this "intersection of niches", do you think it might be opportune to at some point raise awareness about Math.CD on the Project Euler chat (emphasizing the special fea... (more) |
— | 7 months ago |
| Edit | Post #295460 | Initial revision | — | 7 months ago |
| Question | — |
Going to "Account" tab incorrectly notifies me about earning "Participate" ability Steps to reproduce: If I go to my own profile page, then click the "Account" tab, I receive two notifications informing me (incorrectly) that I've earned the "Participate" and "Participate Everywhere" abilities on Mathematics Codidact. The expected behavior would be not receiving any notificat... (more) |
— | 7 months ago |
| Edit | Post #295457 |
Post edited: |
— | 7 months ago |
| Edit | Post #295457 | Initial revision | — | 7 months ago |
| Answer | — |
A: Defining a explicit function, without axiom of choice, that is not Lebesgue integrable on any interval? It is not provable in ZF alone that there is an function $f: \mathbb{R} → \mathbb{R}$ that is not Lebesgue-integrable on any interval. Here is why. ZF is consistent with the Axiom of Determinacy which shows (by a relatively straightforward but tedious topological game argument; see Mycielski a... (more) |
— | 7 months ago |
| Comment | Post #295434 |
I'm afraid I don't understand this question; what relation, for instance does your **Question 1** have to the title of the question?
Also, I was about to ask how an "example" of a function $f$ is supposed to "disprove the claim" that $f_{(c,d)}$ has undefined expected value. But after rereadi... (more) |
— | 7 months ago |
| Edit | Post #295456 | Initial revision | — | 7 months ago |
| Answer | — |
A: How can we grow this community? Hello, I'm new here (joined MSE 2 months ago, before hearing about Codidact). Hence please let me know if you'd prefer new users not to write on this Meta thread. One of the most attractive things about MSE, for me, is its gamification of teaching mathematics: one can write expositions of inte... (more) |
— | 7 months ago |
| Comment | Post #295007 |
Note that, if we start with an already existing topology on $\mathbb{Q}$, the topology on the algebraic real numbers $\mathbb{R} \cap \overline{\mathbb{Q}}$ is mostly fixed. This is because all polynomials are continuous functions. As a simple example, suppose we're extending the usual topology o... (more) |
— | 7 months ago |
| Comment | Post #295007 |
To complete your thought–it's very easy to get new topologies by pulling back the usual topology on $\mathbb{R}$ along any field endomorphism $\mathbb{R} → \mathbb{R}$. But, interestingly, this doesn't give us all the topologies on $\mathbb{R}$ whose restriction to $\mathbb{Q}$ is the usual topol... (more) |
— | 7 months ago |
| Comment | Post #293500 |
The most intuitive way to see that the topology in your last paragraph is not a topological group (or ring, or field) is that the group operations have to be not only continuous but, since they're invertible, \emph{homeomorphisms}. And clearly, since you're giving $\mathbb{Q}$ the usual topology ... (more) |
— | 7 months ago |
| Edit | Post #295451 |
Post edited: changed the summary to be somewhat clearer |
— | 7 months ago |
| Edit | Post #295451 | Initial revision | — | 7 months ago |
| Answer | — |
A: What complexity implications are there from *not* being in an abstract family of languages? Transform your statement \[\text{If a language is not an $X$, then it must have space/time complexity $\omega(f)$ (or $\Omega(f)$)}\] into its contrapositive: \[\text{If a language has complexity $O(g), g < f$, then it is an X}\] Now perhaps it is easier to see why this problem tends... (more) |
— | 7 months ago |