I think it should go like this. Suppose you have a laser light shining through glass onto a glass-air-interface. If the laser is normal to the surface, you'll get no refraction.
If you now start to sweep the angle of the laser light, you'll find that the beam both strikes the interface farther along and you get refraction of the transmitted ray to a larger angle.
Eventually, that ray will become parallel to the surface and you'll get TIR.
I'm going to see the classic Riemann-Hilbert correspondence in part two of a for-fun lecture series, part 1 was today and covered connections, although in DG land
Well maybe I can come back and say something about it. The end of the talk was super rushed and introduced holonomy and monodromy and I didn't get to take the notes
Inner product on each tangent space, yeah. $g_p(\bullet, \bullet)$ is an inner product on $T_p M$. being smooth means $g(X, Y)$ is a smooth function on $M$ for smooth vector fields $X, Y$
@AkivaWeinberger If $f$ is a smooth function on $M$, $D_Z f$ is another smooth function on $M$ defined by $(D_Z f)(p) = D_{Z(p)} f$. Directional derivative of $f$ in the direction of $Z$
If I have a function $u(x,t)$ can I make partial Laplace wrt one variable like this $\mathcal{L}_x u_{xx}(s,t) = s^2 \mathcal{L}_x u(s,t)$ or do I need to include initial terms?
@Waiting Nothing. I need to get well first before doing any mathematics, though that might sound strange to you. But I will probably try to learn some languages this year, just for fun.
@AkivaWeinberger OK, so we have a connection on Riemannian manifold $(M, g)$ now which is compatible with the Riemannian metric. There's one more condition, which brings us back to the question you asked
Theorem: There is a unique connection $\nabla$ on a Riemannian manifold $(M, g)$ which is metric ($\nabla_Z g(X, Y) = g(\nabla_Z X, Y) + g(X, \nabla_Z Y)$) and torsionless ($\nabla_X Y - \nabla_Y X = \mathcal{L}_Y X$).
That means any geometry that arises from such a connection is intrinsic to the Riemannian geometry
Considering that tobias will not be interested in pondering about semirings and other weak structures for the moment being, and that Mathien and co. will keep the algebra going, this transition is expected to be smooth
But then, I have no idea what the next era will be, since we already have an integral era back in 2014
@Akiva If X and Y are two vector fields on the 2-sphere in R^3, the connection is directional derivative of Y in the direction of X, projected to the tangent plane of the sphere
@Waiting hmm, that will be good cause semi is good at special functions, and I have been recently investigating limit and epsilon delta proofs at the 1st principle level
but we still have a lot of variables, for we have an even distribution of regulars for all fields of maths right now
@Akiva In particular, definition: If $\gamma$ is a curve in $M$, it's said to be a geodesic if $\nabla_{\gamma'} \gamma' = 0$. $\gamma'$ means the tangent vector field to $\gamma$ (extended to all of $M$ by a bump function)
More or less, it means a particle falls off along $\gamma$ with no acceleration
The geodesics in the sphere turns out to be exactly the (arcs of the) great circles with the connection I wrote down
It's a wonder that this notion of geodesic agrees with what you'd think ("locally length minimizing curves" - that definition involves NO connections. You can define length using the Riemannian metric only: $\int \sqrt{g(\gamma', \gamma')}$)
I am still best known here in math chat for all sort of weird maths from division by zero algebras, to a real counterpart of a cauchy integral like formula, to tensor visualisation, infinite sets investigations and so on
Mike Miller is much harder, fo it will take me at least a decade to have enough knowledge on algbraic topology
Lemma: Non weird people are often just tolerating me, instead of being genuinely interested
5
Proof: In a thousand words
I have a full list of people who are non weird, and whenever they make the chat room non weird, shit postings will skyrocket
This is rare nowadays as ever since we got most of the topology bunch to become weird, it is very unlikely for the weirdness level of the chat t drop below critical levels
Balarka used to be a non-weird and together with Slereah frequently flipped into a neutrino, the two combined with 0celo to form the Phase Lock Trio, which result in huge drop in the weirdness level in h bar, thus result in a lot of shitposting floods
Ever since the chemistry incident, Balarka becomes weird, and for unknown reason, Slereah becomes more unlikely to become a neutrino, thus the Trio rarely Phase Lock now. In addition
Balarka helped me to grasp differential geometry, thus making it more comprehensible and hence more potential to generate weirdness, this is why h bar does not have much shitposting lately
We are not sure what exactly is the chemistry incident, but one guess might be Balarka start taking chemistry
So far all my integral attempts are based on trying to approach them from first principle, with the most recent attempt being trying to turn epsilon delta into an algorithm that can give me value of a given limit
I am more the former, because I like to turn things into algorithms and I prefer general cases
The limit stuff is still in progress, for so far I failed to use it to show that the harmonic series is divergent and the p series are convergent, so something's amiss
I guess the problem is that $x \mapsto \sum_{x=1}^{\infty}$ as a linear functional, is too nonlocal for its value to be determined by epsilon deltas
The map: $(x > M) \to (a < x :\sum_{k=a}^x () ? \sum_{k=a}^M ())$ is notrivial and requires knowing whether the stuff in $f()$ is nonegative or nonpositive in the interval $(a,M)$
anyway, it's time to sleep, 3:11 might explain why I am sounding a bit rambling
I have a question, im making a program in C++ and i need some math advice. The problem i have is the following: i have a function that reads 512 blocks of bytes from a file and stores them in memory. Assuming that i dont know how much bytes i need to read (lets call it X), i do know that X > 512 and that ceil(X/512) - floor(X/512) != 0. (ceil(...) will round things up, for example 2.78 will be 3, and floor will round down, for example 4.8 would be 4).
I got this idea to make a cicle that would read 512 bytes at a time from the file, floor(X/5120) times, and in the end i would do (X/512) - floor(X/512) , so i would get the 0,.. part and i would multiply that to 512 so i would get the missing bytes : 512 * 0,... . The problem is that i dont know if the 0,... would be finite or infinite, for example it could be 0,234 or 0,333333(3).
The problem with this is that if it was for example 0,333333(3), the computer would assume it was 0,33333333334 , and it could change the result of the multiplication by 512. Anny ideas on a better way to do this?
When someone start talking about maths in the next hour, the chat should go back to normal, which is much better than to stay fully weird and cut off visitors
Claim: If $R$ is a finite boolean ring with identity, then $R \simeq \Bbb{Z}_2 \times ... \times \Bbb{Z}_2$......I'm trying to figure out how to use the Chinese Remainder theorem, but I'm not having much luck. I could use a hint.
Let G be a finite simple group with |G| ≥ 3. Suppose that G acts on the set X = {1, 2, . . . , n}. Show that every element of G corresponds to an even permutation of X.
