@s.harp Say $W$ is the curve (W for Warsaw circle, also a name for the thing). Let $f : S^n \to W$ be a map; if image of $f$ is disjoint from the bit of the y-axis in $W$ on which nearby things accumulate - aka the bit of $W$ where $W$ is not locally path connected - then $f$ is nullhomotopic, because complement of the y-axis in $W$ is an arc $(-1, 1)$.
If $p, q$ are points on $S^n$ in a neighborhood of $0$ such that $f(p)$ is on "the left" of $w$ and $f(q)$ is on "the right of $w$, then $f$ can't possibly be continuous at $0$; choose a path from $p$ to $q$ which goes through $0$ and you have a path $[0, 1] " =[p, q]" \to S^n$ which lies inside a neighborhood of the bad y-axis, and is on both "half" of it. That just breaks path-connectedness of the topologists's sine curve.
So $f$ has to miss some small topologists' sine curve subspace $T \subset W$, containing the $y$-axis. But $W - T$ is just... homeomorphic to $[0, 1)$. Contract. $f$ is nullhomotopic.
Modulo details this is how the proof should be like.
@Secret I think things I havent understood are super abstract, like algebraic QFT or anything where people do QFT in a mathematically sensible framework suddenly involves all knowledge about algebraic geometry, d-modules, infinite dimensional super-lie-algebras etc that has ever been compiled
@Balarka its not path connected so it cant be contractible
there is a book "Towards the Mathematics of QFT" by Paugam, there is a chapter called "Functorial Analysis" and in the introduction to n-categories he says "the student interested in learning regular category theory may simply set $n=1$ in the following"
Better a somewhat sloppy thing which I can understand enough to test than a 'mathematically respectable' formulation which is so abstract as to be inaccessible.
I understand that doing physics in the way physicists do is real science in a way mathematics is not, but my personal perspective is that doing it that way is horribly unfufilling
I do slightly better on concepts that looks like linear algebra, and more recently abstract algebra as well.
Interestingly, things I don't understand well, such as real analysis, is actually less abstract than abstract algebra, because well, how often you can draw an algebraic structure on paper...?
Hi everyone. Graph theory question. I know there is a paper out there that discusses metric equivalence between temporal and non-temporal graphs and states explicitly there is no equivalence between the reachability of non-temporal graphs and the reachability of their temporal counterparts (i.e. non-temporal reachabiltiy isn't a viable proxy / estimation of temporal reachability). But I can't find the paper any more and Google is not helpful. Does anyone happen to know/remember it?
One can make a defense of topos theory in music, to the effect of: "While it's hardly the obvious thing to do, maybe one really needs these mathematical tools to appreciate a structure as complex as Mozart's music."
But the comparison to the statistical example is so absurd as to undermine the very argument.
I'd rather say: because of the abstraction of topos theory it is not an unnatural position that using it can give you non-trivial results not accessible before
My point of contrast, though, is that statistics is (to some extent) quite literally the subject of counting large populations and coming to conclusions about those populations.
Not just "how many voters were in favor of X" but how representative the sample is of the general population, and how likely voters in various demographics are to actually vote.
the company would conveniently manipulate stats to make the numbers they want to improve better and numbers they didnt want to improve worse etc constantly i guess you could say it let a really bad taste in my mouth about statistical data
the election will ask everybody in some set $X$ for their choice, the distribution $P$ from which you have sampled can be very far away from an approximator to what people in $X$ say
If a finite state machine is formally defined with a set of states $\{q_0, q_1, q_2\}$, where $q_0$ is the start state, is it a valid machine if there is no path from $q_0$ to $q_1$ and no path from $q_2$ to $q_1$? In other words, there is no way to reach $q_1$ at all, but it is still considered ...
There's no way to give every ordinal smaller than it a name such that a) it's decidable whether a given word is the name of an ordinal and b) it's decidable whether one word represents a smaller ordinal than another word.
Last night dream: I was at a desktop computer in the middle of the night running some kind of 3D/4D modelling. Akiva was nearby as a looped shape with a spherical envelop is displayed on the screen. He mention how he observed there are 3 looped geometries within. Meanwhile as I take a closer look I realised that that's no loop, those are (simple) knots. The geometry shape is also indexed by the ordinal $\omega^{\omega^{\omega^{\omega^{\omega}}}}$, visible faintly rotating with the geometric obje
In set theory, an ordinal number α is an admissible ordinal if Lα is an admissible set (that is, a transitive model of Kripke–Platek set theory); in other words, α is admissible when α is a limit ordinal and Lα⊧Σ0-collection.
The first two admissible ordinals are ω and
ω
1
C
K
{\displaystyle \omega _{1}^{\mathrm {CK} }}
(the least non-recursive ordinal, also called the Church–Kleene ordinal). Any regular uncountable cardinal...
I suppose "$S_\infty$" is ambiguous (it could mean two things). It could mean the set of all permutations on those, or it could mean only the set of finite permutations (the ones made of finitely many cycles of finite size).
Well, of course. It's just a nice example where both properties are incredibly trivial to see.
@AkivaWeinberger Ummm, well... For $[\varphi] \in \operatorname{Out}(G)$, the isogredience number of $[\varphi]$ is the number of equivalence classes given by the relation $\forall \varphi_1,\varphi_2 \in [\varphi]: \varphi_1 \sim \varphi_2 \iff \exists \iota \in \operatorname{Inn}(G): \iota \circ \varphi_1 = \varphi_2 \circ \iota$
Hi everyone. Anyone know is there a common name for this identity? $\sum_{k=0}^n C(n,k) =2^n$ (I know it comes from the binomial theorem, but I wonder if there's a specific name for it).
@Semiclassical Yes. In fact, the class notes I'm writing develop that identity with another method: summing up all the possible ways of flipping $n$ coins and comparing it to the number of $n$-character binary strings.
@AkivaWeinberger In covering theory and fixed-point theory. It connects self-maps and lifts of self-maps on spaces with the induced automorphisms on the fundamental group.