Activity for clemens
| Type | On... | Excerpt | Status | Date |
|---|---|---|---|---|
| Comment | Post #296264 |
@#53890 Interesting; so both HTML and Markdown formatting works, it seems, but not putting the two on the same line?
Then why does this work, I wonder: <i>test HTML italics</i> *test Markdown italics*? (more) |
— | about 2 months ago |
| Comment | Post #296273 |
The usage in analysis, just from a cursory look, appears the same as that in topology to me. E.g. the term "[weak topology](https://en.wikipedia.org/wiki/Weak_topology)" is usually used for the coarsest topology that satisfies certain conditions. (more) |
— | 3 months ago |
| Comment | Post #296272 |
The parenthesized form (using \pmod in $\LaTeX$) is the one I'm most familiar with as well. As for Latin *modulo* I think it's ablative absolute; but oddly, [Disquisitiones Arithmeticæ](https://la.wikisource.org/wiki/Disquisitiones_arithmeticae/Sectio_secunda#art._13) doesn't use *modulo* at all ... (more) |
— | 3 months ago |
| Edit | Post #296262 |
Post edited: Put line breaks before and after the "align" environment to allow it to be formatted properly |
— | 3 months ago |
| Comment | Post #296264 |
It looks as though it works if you set it off by line breaks, as in my suggested edit. (more) |
— | 3 months ago |
| Suggested Edit | Post #296262 |
Suggested edit: Put line breaks before and after the "align" environment to allow it to be formatted properly (more) |
helpful | 3 months ago |
| Edit | Post #296264 |
Post edited: |
— | 3 months ago |
| Edit | Post #296264 | Initial revision | — | 3 months ago |
| Answer | — |
A: "align" in whatever software this site uses instead of MathJax `align` works for me: \begin{align} R^2 &= \frac{\sum(\text{fitted value minus sample mean value})^2}{\sum(\text{observed value minus sample mean value})^2} \\ & = \frac{\text{explained variability}}{\text{total variability}}. \end{align} I'm pretty sure Codidact uses MathJax. We have a ... (more) |
— | 3 months ago |
| Comment | Post #296065 |
@#118664 30 was the figure given in my statistics textbook (though I don't remember the title exactly) for when one could reasonably safely use the Z-test instead of the T-test. But that textbook was written before computers made the T-test practical.
(more) |
— | 3 months ago |
| Comment | Post #296123 |
Thanks.
By the way can you elaborate on the issue with \Pi and \mathcal? Is it simply that the \mathcal{CB} is not in the right size, shape, etc. when in the subscript of \Pi instead of \prod? (more) |
— | 3 months ago |
| Edit | Post #296088 |
Post edited: edited to remove dangling footnote |
— | 4 months ago |
| Edit | Post #296192 |
Post edited: |
— | 4 months ago |
| Edit | Post #296192 |
Post edited: |
— | 4 months ago |
| Edit | Post #296192 | Initial revision | — | 4 months ago |
| Answer | — |
A: For which spaces are all continuous functions either constant or the identity? All but one of the spaces under consideration are $T1$ This is not a complete answer, but note that one can straightforwardly prove that the space is either the Sierpiński space or a $T1$ space! (We know it is $T0$ by the argumentation in the OP---if it weren't $T0$, then swapping two topolo... (more) |
— | 4 months ago |
| Comment | Post #296181 |
Interesting; I'm still wondering what precise quantification you're doing over the $x$ and the $\delta x$ in calculus, though. Also, what is a sidebar deduction? (more) |
— | 4 months ago |
| Edit | Post #296181 |
Post edited: |
— | 4 months ago |
| Edit | Post #296181 |
Post edited: |
— | 4 months ago |
| Edit | Post #296181 | Initial revision | — | 4 months ago |
| Answer | — |
A: How to find dualized quantifier pairs in substitutionally quantified analysis? I confess I don't fully understand this question (I await further clarification), but in general dualization is not that complicated. If $P$ is the dual of $Q$ it just means that it acts on $\bot$ the same way that $Q$ acts on $\top$, and vice versa. So whenever you have $P$ (say, a quantifier), ... (more) |
— | 4 months ago |
| Comment | Post #296180 |
You say that $\forall$ and $\exists$ "don't seem to be the quantifiers that Newton had in mind", but why? Is this related to e.g. Cauchy's confusion between uniform continuity and continuity? (more) |
— | 4 months ago |
| Edit | Post #296168 |
Post edited: clarified and expanded last question |
— | 4 months ago |
| Edit | Post #296168 |
Post edited: clarified and expanded last question |
— | 4 months ago |
| Edit | Post #296167 |
Post edited: typofixes |
— | 4 months ago |
| Edit | Post #296168 |
Post edited: |
— | 4 months ago |
| Edit | Post #296168 | Initial revision | — | 4 months ago |
| Answer | — |
A: Is there an equational law satisfied by "linear kei" but not free kei? (This is about $\frac23$ of an answer.) Firstly, to answer this question it may be helpful to make a little diagram of the different classes of quandles we're dealing with here: Inclusion relations amongst linear kei, hyperplanar kei, subkei of conjugation kei, subquandles of conjugation q... (more) |
— | 4 months ago |
| Suggested Edit | Post #296167 |
Suggested edit: typofixes (more) |
helpful | 4 months ago |
| Comment | Post #294111 |
Not an answer but a couple of notes on this question: 1. A graph folding as defined in the OP is simply a graph quotienting/epimorphism that preserves labels. The use of "fundamental group" in this context I found a bit distracting. 2. I don't currently see the relevance of the normality of the c... (more) |
— | 4 months ago |
| Comment | Post #296096 |
@alx: I don't think so; at any rate, my impression is it has to decrease at least as fast as $\frac1n$. (more) |
— | 4 months ago |
| Comment | Post #296096 |
@alx Yes; the required optimization would have to prevent $p$ from being inversely proportional to $n$ (in the complementary, max-independent-set problem). (more) |
— | 4 months ago |
| Comment | Post #296096 |
@#53937 "Theorem 3" of that preprint says that running time for random graphs where $p = k/n$ will be at least proportional to $e^{\frac{n}e}$. But my hunch is that optimizations (like yours) can reduce that exponent quite a bit. (more) |
— | 4 months ago |
| Comment | Post #296096 |
@alx: The 2015 arXiv preprint https://arxiv.org/pdf/1505.04969 (dealing with the complementary maximum-independent-set problem) has as a theorem that, for random graphs for which $p = 1/n$, $p$ converges to 0 in such a way that running time is exponential; but I haven't looked at it yet.
(more) |
— | 4 months ago |
| Edit | Post #296096 |
Post edited: |
— | 4 months ago |
| Edit | Post #296096 |
Post edited: added solution |
— | 4 months ago |
| Comment | Post #296096 |
@alx: I'm not entirely sure right now. But of interest: it looks to me as though the most difficult random graphs for SAT have a $p$ of about $1-\frac{4.26}{2n}$. (And random SAT is still an exponentially hard problem for the best algorithms.) So heuristically speaking we expect connectivity to b... (more) |
— | 4 months ago |
| Comment | Post #296096 |
@alx: Very interesting. The difficulty remains the small size of the samples. [Random SAT instances](https://cs.stackexchange.com/questions/144382/how-hard-is-random-sat) seem to have exponentially increasing hardness—but at about a 1% increase per additional variable. This would correspond (via ... (more) |
— | 4 months ago |
| Comment | Post #296071 |
> Why do you emphasize "actual"?
Actual contrasts with possible.
> Can you give an example of this?
I already did but another prototypical example is Brownian motion. (more) |
— | 4 months ago |
| Comment | Post #296071 |
@Ooker That's correct; probability measures deal *either* with the set of all *actual* bodies (in which case you'd get a measure over a finite set, which is pretty trivial to work with from a mathematician's perspective) *or* with the set of all bodies in some *probability* distribution (say, a m... (more) |
— | 4 months ago |
| Edit | Post #296123 |
Post edited: added expression for the cube |
— | 4 months ago |
| Edit | Post #296123 | Initial revision | — | 4 months ago |
| Question | — |
Stone–Čech compactification and ultrafilters Munkres defines the Stone–Čech compactification $\beta X$ of a topological space $X$ as based on the embedding of completely regular spaces into the cube $[0,1]^{\mathcal{CB}(X)}$ where I use $\mathcal{CB}(X)$ to denote the space of bounded continuous functions on $X$. Specifically, any point $h$... (more) |
— | 4 months ago |
| Comment | Post #296103 |
"Raise a matrix to matrix power"—do you mean the $\log$ and $\exp$ functions are total over this algebra? (more) |
— | 4 months ago |
| Comment | Post #296071 |
Of course $\sigma(\text{height }>1.72{ m})$ makes sense—though, more precisely, we should write it $\sigma(\{\text{situation}: \text{height }>1.72\text{ m in situation}\})$. It computes the measure of an *event* i.e. a set of situations, in this case the set of situations-in-which-one's-height-ex... (more) |
— | 4 months ago |
| Comment | Post #296071 |
Now, there is a measure that we use to compute conditional probabilities over the height, weight, etc. (this is done via the Radon–Nikodym theorem). But in that case the height is the *domain* not the *codomain* of the measure.
Unless you're talking about computing the mean height, mean weight... (more) |
— | 4 months ago |
| Comment | Post #296071 |
@Ooker: No. The weight, height, age, etc. are not in the codomain of the measure. If they were in the codomain of the measure then what would be in the domain?
The measure has to take an event and return (usually) a number. E.g. say $\sigma(\text{height } < 72 \text{ in}) = 0.6$; then we know ... (more) |
— | 4 months ago |
| Edit | Post #296110 |
Post edited: Might it be better to use semicolons to distinguish the $x_i$ from the $y$? |
— | 4 months ago |
| Comment | Post #296095 |
@Ripheus24 The continuum in AD is not an aleph? Wow! Is it incommensurable with the alephs, then? (more) |
— | 4 months ago |
| Comment | Post #296095 |
Interesting! Wrt. the divisibility of space—and this, I'm thinking, could perhaps somehow be material for another Codidact post—it's occurred to me that what Aristotle was groping towards with his actual/potential infinity distinction has to do with *compactness*: a closed, bounded interval/regio... (more) |
— | 4 months ago |