Activity for Derek Elkinsâ€
| Type | On... | Excerpt | Status | Date |
|---|---|---|---|---|
| Comment | Post #293102 |
We're unioning sets of axioms, not combining models. Having two models that give different assignments to propositional variables (which is what I meant by "conflicting") doesn't mean that there's a contradiction. Even for the same set of axioms we can have multiple models, e.g. there are at least th... (more) |
— | 5 days ago |
| Comment | Post #293102 |
To answer the question in your first footnote (which seems to be missing some words, but I believe the intent is clear enough): The statement says what it says, which is that each finite subset has a model. There is no assumption that there is one model that will work for every finite subset, though ... (more) |
— | 6 days ago |
| Edit | Post #293102 |
Post edited: TeXify. Remove set-theory tag. Change footnotes into real footnotes, which may not be an improvement. Break the text into paragraphs, though it can probably be done better or more in line with the original source. |
— | 7 days ago |
| Suggested Edit | Post #293102 |
Suggested edit: TeXify. Remove set-theory tag. Change footnotes into real footnotes, which may not be an improvement. Break the text into paragraphs, though it can probably be done better or more in line with the original source. (more) |
helpful | 7 days ago |
| Comment | Post #293102 |
This looks like a compactness theorem, not a completeness theorem. (more) |
— | 7 days ago |
| Edit | Post #293013 | Initial revision | — | 26 days ago |
| Answer | — |
A: How to validate if the horizontal and vertical tangent lines exist for implicit functions? Answering your questions a bit out of order, I'll start with the "non-rigorousness" of talking about $dx$ by itself. While this seems to be commonly poorly explained, the derivative (of a function from and to reals) is an operation that takes functions to functions. Let's consider the typical high sc... (more) |
— | 26 days ago |
| Edit | Post #292410 |
Post edited: fix typo I introduced, cap -> cup, tweak title, remove symm tag |
— | 4 months ago |
| Suggested Edit | Post #292410 |
Suggested edit: fix typo I introduced, cap -> cup, tweak title, remove symm tag (more) |
helpful | 4 months ago |
| Edit | Post #292410 |
Post edited: Correct typo and formatting errors, also ^{\prime} -> ' |
— | 4 months ago |
| Edit | Post #292418 | Initial revision | — | 4 months ago |
| Answer | — |
A: Why $\gamma\cdot\operatorname{grad}u<0$ in the Theorem? (Nirenberg academic paper) As the proof of Theorem 2 suggests, this follows immediately from Theorem 2.1. While it's a bit ambiguously worded, to apply Theorem 2.1, we need $b1(x)=0$ in $\Delta u + b1(x)u{x1} + f(u) = 0$, $u > 0$ in $\Omega$, $u = 0$ on a part of $\partial\Omega$, and some basic continuity conditions. The assu... (more) |
— | 4 months ago |
| Suggested Edit | Post #292410 |
Suggested edit: Correct typo and formatting errors, also ^{\prime} -> ' (more) |
helpful | 4 months ago |
| Edit | Post #292320 |
Post edited: |
— | 4 months ago |
| Edit | Post #292320 | Initial revision | — | 4 months ago |
| Answer | — |
A: Reflection in the plane with polar coordinates Perhaps the simplest way to get the formula is to think geometrically. Let's say we wanted to reflect a point $x=(r\cos\theta,r\sin\theta)$ across the $x$-axis. In that case, we can simply negate $\theta$ giving $(r\cos(-\theta),r\sin(-\theta))=(r\cos\theta,-r\sin\theta)$ as expected. If we wan... (more) |
— | 4 months ago |
| Edit | Post #292225 |
Post edited: Formatting and minor typo fixes and grammar tweaks. Backslashes sometimes need to be escaped leading to things like \\\\ for \\. |
— | 4 months ago |
| Edit | Post #292231 | Initial revision | — | 4 months ago |
| Answer | — |
A: Complex functions and inner product $\langle \frac{\partial f}{\partial z} , g\rangle $ There are multiple issues with how you compute the exponent of $r$. The first issue is you compute $r^{j-1}r^k = r^{j+k}$ rather than $r^{j+k-1}$. You compound this error when you substitute $j = k+1$ into $r^{j+k}$ (which should be $r^{j+k-1}$) in the formula for $\left\langle\frac{\partial f}... (more) |
— | 4 months ago |
| Suggested Edit | Post #292225 |
Suggested edit: Formatting and minor typo fixes and grammar tweaks. Backslashes sometimes need to be escaped leading to things like \\\\ for \\. (more) |
helpful | 4 months ago |
| Edit | Post #292202 | Initial revision | — | 5 months ago |
| Answer | — |
A: Find the value of $\sum_{k=1}^\infty\frac{k^2}{k!}$ This answer is, in some ways, "just" a rephrasing of the power series answer by Snoopy, but the broader perspective and name-dropping the relevant tools may be useful. The relevant tool being generating functions. While not the best book on the topic, Herbert Wilf's generatingfunctionology is at l... (more) |
— | 5 months ago |
| Comment | Post #292189 |
$\lg$ (versus $\log$) is commonly used for the base 2 logarithm, though it is certainly isn't unambiguous. I've never seen $\operatorname{ld}$ or $\operatorname{lb}$ used for this, so it seems like a terrible choice for clarity. $\log_2$ is, of course, unambiguous. (more) |
— | 5 months ago |
| Comment | Post #291919 |
Answering for Peter Taylor, it's a rotation matrix. You could derive it, as you could any matrix, by considering a rotation operator and computing where it sends basis vectors. In this case, you could compute that the vector $(1,0)$ gets sent to $(\cos\alpha,\sin\alpha)$ by a (counter-clockwise) rota... (more) |
— | 6 months ago |
| Edit | Post #291823 | Initial revision | — | 6 months ago |
| Answer | — |
A: Interpreting $\text{Prop}$ in Set Not asserting an axiom only broadens the class of possible models. Unless you're adding an axiom that states that the type theory is definitively not proof irrelevant, any proof irrelevant model is still a model. Whether or not proof irrelevance is assumed would only matter if it excluded some model,... (more) |
— | 6 months ago |
| Edit | Post #291617 |
Post edited: Make more legible and grammatical. |
— | 7 months ago |
| Comment | Post #291588 |
Maybe I don't understand what you intend with the notation, but ${+}^{-1}(U)$ is a subset of $V \times V$ and so definitely isn't in $\mathcal T_k$. (more) |
— | 7 months ago |
| Comment | Post #291617 |
$\alpha$ and $\beta$ need to be homomorphisms of the entropic structure. Presumably, this means $\alpha(x \odot y) = \alpha(x) \odot \alpha(y)$. $x^2$ doesn't seem like a homomorphism for the entropic structure you defined. That is, $(x + 2y)^2 = x^2 + 4xy + 4y^2 \neq x^2 + 2y^2$ even mod $3$. (more) |
— | 7 months ago |
| Suggested Edit | Post #291617 |
Suggested edit: Make more legible and grammatical. (more) |
helpful | 7 months ago |
| Edit | Post #291652 | Initial revision | — | 7 months ago |
| Answer | — |
A: Why does the method of separating variables work? I don't think there's a satisfying answer to this question currently. The very first problem – which is probably surprising – is that there isn't a widely accepted, general definition of "separation of variables". An obvious approach to studying separation of variables would be to appl... (more) |
— | 7 months ago |
| Comment | Post #291500 |
I simply substituted in the value for $\beta$. (more) |
— | 7 months ago |
| Comment | Post #291500 |
$\mathbf s\cdot(\alpha\mathbf u -\beta\mathbf v)=\mathbf s\cdot\alpha\mathbf u -\mathbf s\cdot\beta\mathbf v$, but we know $\mathbf s\cdot\mathbf v=0$. While I didn't write it in the clearest possible way, this whole equation is $\mathbf s\cdot\mathbf s = 10$ and then we're expanding $\mathbf s$ twic... (more) |
— | 7 months ago |
| Edit | Post #291500 | Initial revision | — | 7 months ago |
| Answer | — |
A: How to find a point on a vector equation with another vector equation and perpendicular distance? What you've done seems fine; you just need to continue to use the information that you have. In particular, that $s^2=10$. This is definitely an example where generalizing makes things more clear and simple. Abstracting from the details and removing unnecessary information makes the computation and l... (more) |
— | 7 months ago |
| Edit | Post #291317 | Initial revision | — | 8 months ago |
| Answer | — |
A: Is there a $\Delta$-complex structure on the sphere with less than three $0$-simplices? If it wasn't for the right to left implication of condition 3, you could do this with a single map out of the 0-simplex. If there are 2 or fewer points in the image of a map from $\Delta^0$, then that means at least two vertices of the image of every map out of the 2-simplex must be the same. Sinc... (more) |
— | 8 months ago |
| Edit | Post #291255 |
Post edited: Had my adjunction backward which made the proof nonsense |
— | 9 months ago |
| Edit | Post #291255 |
Post edited: Nicer proof by being more categorically inspired |
— | 9 months ago |
| Edit | Post #291255 | Post undeleted | — | 9 months ago |
| Edit | Post #291255 |
Post edited: Correct mistake |
— | 9 months ago |
| Edit | Post #291255 | Post deleted | — | 9 months ago |
| Edit | Post #291255 | Initial revision | — | 9 months ago |
| Answer | — |
A: Prove that $\forall x\in\Bbb R:\lfloor x^2\rfloor-\lfloor rx\rfloor\ge-1\iff|r|\le2$. This is definitely more complicated than it needs to be. First, we can rewrite the inequality as $$\lfloor x^2 + 1\rfloor \geq \lfloor rx \rfloor$$ Proof 1 (This is the first proof I wrote, but the second one is nicer and takes a more categorical perspective.) $\lfloor x^2 + 1 \rfloor = 1... (more) |
— | 9 months ago |
| Comment | Post #290803 |
I don't understand many of your questions. They are answered in the article itself. For example, "What are the four axioms regarding equality [... a]nd the fourth?" The answer is the four it lists in the body of the article. The paragraph you quote is a summary of the article and is not meant to be s... (more) |
— | 10 months ago |
| Comment | Post #290727 |
I used "specification language" to emphasize that FOL is a complete system by itself and to emphasize the distinction between the language in which we are specifying something, i.e. FOL, and the thing being specified, in this case ZFC, or, more precisely, the $\in$ predicate. I call it a "specificati... (more) |
— | 11 months ago |
| Comment | Post #290727 |
If you drop the Axiom of Infinity, you're not working in ZFC any longer. The word "rejecting" is a bit ambiguous in this context. It can mean either "not accepting," or it can mean asserting the negation. Not accepting the Axiom of Infinity doesn't mean all sets are finite. It just means can't prove ... (more) |
— | 11 months ago |
| Comment | Post #290727 |
FOL is a specification language and quantifiers and logical connectives, including implication, are what you use to write the specification. You introduce predicate symbols, such as $\in$, to give a name for the things you are describing. Equality is usually taken as part of FOL, but you could also t... (more) |
— | 11 months ago |
| Comment | Post #290637 |
This question has more or less been asked here: https://math.codidact.com/posts/279044 and my answer there captures pretty much what I would say here. Nevertheless, to answer the precise question here, the textbooks are wrong, at least when taken out of context. They are simply doing linguistics but ... (more) |
— | 11 months ago |