I am following the book A First Course in Dynamics: With a Panorama of Recent Developments (Anatole Katok and Boris Hasselblatt).
Definition 4.1.3 in the book states:
A homeomorphism $f: X \rightarrow X$ is said to be topologically transitive if there exists a point $x \in X$ such that its orbit...
Can I have a counterexample for the following: If L/K is a field extension and V is a vector subspace of L that contains K, then V is a subfield of L (assuming 1 goes back into V)
nick: something like {a + b sqrt(2) + c sqrt(3): a, b, c in Q} is a Q-vector subspace of Q(sqrt(2),sqrt(3)) (or any field that contains this one, like R) that contains Q but is not closed under multiplication
maybe some work to do there but sqrt(2) sqrt(3) isn't in that set
for example
or {a + b 2^(1/3): a, b in Q} a similar example of smaller dimension (exercise, 2^(2/3) isn't in there)
Interesting. I'm not really familiar with the area they are in, but it looks like they are relating spectral data to geometric data on Riemannian manifolds. I might need to look at that a bit more later.
Hausdorff dimension is an infimum, so a complex valued Hausdorff dimension doesn't make sense to me. Which is not to say it doesn't exist, just that I didn't know what the definition would be.
@onepotatotwopotato In the context I know, the complex dimensions of a space describe a "geometric oscillation".
the paper I uploaded above is measuring the Hausdorff dimension of a Jordan curve in $S^2$ (as far as I know) and that Jordan curve has a fractal nature. It makes sense to talk about the complex fractal dimension of that Jordan curve.
@BalarkaSen no worries! Still doing some kind of geometry? Are you doing a PhD now? I'm doing well, I'm also still doing maths, I'm a postdoc in Bologna now
@PM2Ring I was playing around with GoL, and trying to stabilise too-large spaceships. This is what I came up with: 4b3o$3b5o$3b3ob2o$6b2o3$2bo3bo$o7bo$9bo$o8bo$b9o4$b9o$o8bo$9bo$o7bo$2b o3bo3$6b2o$3b3ob2o$3b5o$4b3o! But searching for this pattern on ConwayLife search(https://conwaylife.com/ref/mniemiec/lifxsrch.htm) yields no results. Is this pattern too useless to be important, or are there any better search methods?
Let $S$ be an oriented surface with a conformal structure $c$. If $S$ is compact, by the uniformization theorem, it is possible to find a unique hyperbolic metric $g$ whose conformal class is $c$. But it seems if $S$ is not compact, for example $S$ has an infinitely many genus, then one might not be able to find a hyperbolic metric $g$ whose conformal class is $c$.
@SohamSaha It's not useless, but that strategy was investigated intensively, years ago. It's called a flotilla. conwaylife.com/wiki/Spaceship_flotilla Another approach is to build a "track" from still lifes or small period oscillators to stabilise the ships.
@SohamSaha A very interesting & versatile pattern is the Herschel. conwaylife.com/wiki/Herschel But maybe get some more experience playing with gliders & spaceships first. :)
If you're seriously interested in GoL, take a look at the Life Lexicon. conwaylife.com/wiki/Life_Lexicon Randomly browsing the Lexicon is a great way to become familiar with a lot of the standard patterns. Of course, the Wiki on the Conwaylife site is great, too. But the Lexicon is a bit smaller (at least, it was last time I used it), and its linear organisation makes it a little easier to study. IMHO.
Using Herschel tracks, it's now possible to construct oscillators (and glider guns) of any desired period, above a minimum value. So some of those old oscillators aren't used much these days. But they're still of historical interest.
Another fun class of pattern is the rake conwaylife.com/wiki/Rake which is basically a mobile glider gun.
@PM2Ring I just found out about GoL a few years back, and have been trying to do something interesting with it. But I can’t even get started. So I just play around with random patterns in the LifeViewer hoping to find something interesting
@SohamSaha It can be a bit frustrating. Especially when you don't know the jargon. The Lexicon helps with that. And then the posts on the forum will be much more readable. :)
A lot of random searches were done manually. These days, people tend to use scripts to speed up the process. But it's still a good idea to do manual searches when you're learning, to get a feel for what works.
FWIW, Conway was a bit annoyed that GoL got so much publicity, when he did so much other stuff. But Life made him famous to a much larger circle than any of his other stuff could have done. ;)
@SohamSaha There are "programmable" constructors that read their instructions from a tape. And in fact, you can program such a constructor to build a replica of itself. Of course, it's a little more efficient to make a self-replicator that's not universal.
Yes, anything can be built from gliders. But what's really elegant is to construct stuff from glider salvos, that can "fly in" to the construction site. Rather than having to manually place the gliders into configurations that they cannot just glide into.
@SohamSaha Definitely.
But they aren't fast. Eg, Gemini takes 33,699,586 generations to replicate.
@SohamSaha I have no idea. I haven't touched Windows for years. And I haven't done much Life stuff since 2010, so I'm not up to date on the latest search software.
@SohamSaha Well, it's got to do a lot of work. Golly is much faster, since it has the hashlife algorithm, which speeds up calculation of patterns containing a lot of repetition in time or space.
@SohamSaha It probably looks better in monochrome. :)
@PM2Ring one last thing: In the Marilyn code, the gliders are going around and getting reflected between two reflectors and somehow they are being ‘copied’ when they reach the output section right? And the gliders sequences getting reflected around are the ‘1’s and ‘0’s for a single row in the image?
In Rudin's PMA, when he proves the Stirling formula, i.e. $$\lim_{x\to\infty}\frac{\Gamma(x+1)}{(x/e)^x\sqrt{2\pi x}}=1,$$ he defines a continuous function $h(u)$ such that $h(u)\to\infty$ as $u\to -1$ and $h(u)\to 0$ as $u\to\infty$. Then he derives an integral expression for $\Gamma(x+1)$ in terms of the function $\psi_x(s)$ defined in the screenshot. He claims this function converges uniformly on $[-A,A]$ for every $A<\infty$.
I think I know which theorem in the book he uses to justify this claim, however, that theorem deals with sequences of functions, whereas here it seems we have a function that is indexed by a continuous variable $x\in (0,\infty)$. How can I deal with this?
It's causing me some confusion that we have $\psi_x$ and not $\psi_n$, where $n$ is a natural number.
I am asked to show that if $A \cap B$ be nonempty then $\sup A \cap B \leq \inf \{\sup A, \sup B \}$
When I go ahead and do that I don't really seem to use the hypothesis that $A \cap B$ nonempty at all...is it embedded in me asserting (tacitly) that $\sup A \cap B$ exists at all?
My proof: We have for all $x$ that $x \in A \cap B$ implies $x \in A$ and $x \in B$, so that $x \leq \sup A$ and $x \leq \sup B$. That is, $\sup A$ and $\sup B$ are upper bounds for $A \cap B$. Then by definition of the supremum we must have $\sup A \cap B \leq \sup A, \sup B$, so that $\sup A \cap B$ is a lower bound for $\{\sup A, \sup B\}$, whence, by definition of the infimum, we must have $\sup A \cap B \leq \inf \{\sup A, \sup B\}$.
@psie I've been stuck for hours now :( does anyone have a hunch, just a hunch, of what it means for $\psi_x(s)$ to converge uniformly to $\psi(s)$, where $x$ varies continuously, not just in the naturals?
I guess what I'm trying to ask is, in the context of this question, why do the authors specify $A \cap B$ nonempty? My thought was that we want to specify $A \cap B$ has a supremum at all but, as you point out THorgott, the nonemptiness of $A \cap B$ is nowhere near enough to guarantee this
key phrase is assuming the relevant suprema and infima exist
so it comes down to what I said above: either you're happy to either assume your poset has a minimal element $-\infty$ or to adjoin such an element to it and define $\sup \emptyset := -\infty$, or you leave it undefined, which should answer your question
Perhaps I am being obtuse (quite possible), but my question is the following: if $\sup A \cap B$ exists (as per "assume all relevant suprema/infima exist" then isn't $A \cap B$ being nonempty immaterial? i.e. the result then holds either way?
Immaterial? No, but if you work under the assumption that $A \cap B$ has a supremum and that $\sup \emptyset$ should be undefined then it follows from a suitable reading of "assuming the relevant suprema and infima exist"
splitting hairs, anyhow
the authors seem to feel a need to highlight this special case
which is kind of them, since the yoga of the empty set is usually left to the reader :)
There are some related questions, e.g. here (but with no answer). I'm concerned about the proof of Stirling's formula in Rudin's PMA. I've spent a good portion of the day trying to figure out with what tools presented in the book so far I can best understand this.
For a full overview of the part ...
Final means last, and I have it on good authority that putting a co- on something makes it go the other way, so cofinal must mean first. Similarly, initial means cofinal means first, and coinitial means cocofinal message final means last.
so you get gems like "What we call cofinal is called final, what we call coinitial is called cofinal,"
and I haven't even mentioned the option of calling them left/right cofinal yet
@Alessandro "Traditionally, final functors were called ‘cofinal functors’; but this use of ‘co’ is potentially misleading as it has nothing to do with dualization — it is derived from the Latin ‘cum’ rather than ‘contra’ — and so it is now generally omitted."
(unfortunately, this quote is from an older reference when the name 'final' was the prominent choice, but then HTT came out and reversed the conventions again)
speaking of awful conventions, @Ben it's killing me that Lurie denotes operads by $\mathcal{LM}$ or $\mathcal{BM}$ and then goes ahead and talks about the monoidal $\infty$-category $\mathcal{M}$
I just need an intuitive description of the inverse equivalence in the recognition principle for the handwaving "Discussion" section in my thesis, but the ideas are currently drowning in a sea of formalism
