This probably is just a coincidence, but I've always found it interesting and I wanted to put some feelers out there to see if maybe there really is something to it after all.
There are these collections of powers, and in one case a sum of powers in two different ways, that have similar digits t...
I think one thing I will try to ensure it is not really a statistical fluke is to see if similar phenomenon arises in other number bases, and if there is, grab an arbitrary base $b$ to see how digits in $b$ changes in these expressions
There are 20 calculations with singular hessians. Taking account of the molecule structures, it seems there are way too many potential undulation minimas than reasonable, thus I am suspecting instead some numerical error stuff going and is investigating it with the technical support
Guys Im having trouble rembering something. It had to do with definite integrals of sine and cosine being 0 for all values up until 15 then the pattern stops and the integral is posivtive. Anyone herd of something like that.
Is there a Fourier type basis for the functions space $L^2(\mathbb{R}^2)$? I know the fourier series is a basis for functions restricted to the unit circle in R^2, but what about the entire space itself?
We knew that gears and cogs can fit together under a variety of orientations, types and sizes. But what about the topology of a configuration of cogs. For example, it is known that an odd number of cogs will frustrate and thus no rotation can occur while an even number of cogs can rotate. Is there some kind of mathematical notion that governs this property of cogs?
@TedShifrin a bit of progress. I thought about the example I had before (in "quaternionic notation"), $\ell=\langle (I,0)+i(J,0)\rangle$, $D=\ell\oplus \langle (0,1)+i(0,K) \rangle=\ell\oplus q$. Following the suggestion of anon & Mike, I starting thinking: What about we try to change $\ell$ (not q). A tiny bit of octonionic algebra seems to indicate that if we are looking for$\ell=e_m+ie_n$, the only options are $m=1,n=2$ or $m=6,n=5$ (so $\ell=\langle (0,J)+i(0,I)\rangle$ is the second option)
Now I want to say that I can in fact pick $\ell$ to be any ray in the complex 2-plane $\langle (I,0)+i(J,0)\rangle\oplus \langle (0,J),+i(0,I)\rangle$, sweeping out a copy of $\Bbb CP^1$, and giving me the fiber over the line $q$, which we regard as an element of the quadric $Q_5$
This might still be nonsense :P If you want a more detailed story please let me know.
Also, Group Theory in the Bedroom (forgot the author) might count, and there's The Colossal Book of Mathematics by Martin Gardner, which is essentially a big compilation of his Scientific American articles about recreational mathematics.
I'm a Patreon supporter (he's actually the only one I support on Patreon at the moment) so I can say that the next video will probably be about quantum mechanics, neural networks, or probability. I have no idea which though.
No problem. I've learned something new and interesting from every 3B1B video, even those in subjects I knew (or rather, "knew"). Which was most of them.
Maybe it's only good for a math enthusiast like me, we the masses don't know anything about high level academic math, so we tend to stand in awe of these kind of things, please forgive my ignorance @BalarkaSen
@FuzzyPixelz Not really. I have found myself getting thoroughly excited by a lot of elementary mathematics, or popular explanations of mathematics that I already know. If a person likes thinking, various perspectives about known facts are always bound to interest them.
I wasn't throwing a snide comment on popular mathematics and that channel. It was more of an inside joke on how I procrastinate in the worst possible way by watching memes on internet than watching at least a pop-math talk.
[Waiting's harmonic series] Hmm, it seems there might be somethign nice here, but, that does not look quite in the shape of a recursion relation, but close
$H_{k,2}$ is the 2nd order kth generalised harmonic number, which is basically that sum of $\frac{1}{n^2}$ given a name
One thing to be checked is that in the infinite series limit, $\lim_{k\to \infty} H_{k,2}$ should tend to $\zeta (2)$
The remaining term can be unravelled, which give reciprocals of the form $\frac{1}{n(n-u)}$ where $u \in [0,n)$. However at present I am not sure how to sum those yet
Even if such a closed form can be obtained, I am not sure whether you will call a sum of n special functions an elegant closed form
since the harmonic number is basically that sum given a name
Meanwhile, the generalised case $$\sum_{n=1}^{k}\left(\frac{H_n}{n}\right)^{\ell}$$ may have a closed form as a long sum of special functions by doing similar expansion and then binomial expand the powers
Thanks to the help of Semiclassic, simpleart and Leaky, I am starting to have some idea on how to handle infinite series. For now, I will restrict to investigation using elementary manipulation (i.e. +, - , *, / and use of relevant special functions and their identities, thus no integral or differential representations, nor representations and approximation in terms of other series) and try to learn as much of the dynamics of series as possible
Currently, the technique I learnt are generating functions, which is very useful in shifting the terms within a series in a linear fashion. More methods and intuition are to be learnt soon
Suppose I have a coil of wire and a bar magnet. I've got an ammeter attached to the coil of wire, so it's a complete circuit and I'll be able to detect any current running through it
I still get electricity running on the coil in the appropriate orientation (which I'm too lazy to figure out), don't I? Because the there is a relative motion between the coil and the bar magnet
the somewhat tricky part is that, if you draw the field lines emanating from the north pole, then what matters aren't the components which point along the axis of the bar magnet i.e. perpendicular to the plane of the wire
that's the same direction I'm dragging the electrons, so the cross product of that velocity with this part of the magnetic field is zero
a picture makes this more apparent but ugh don't wanna draw it
anyways. If I instead hold the coil fixed and move the magnet, then this explanation seems not to hold anymore since I'm not moving the conduction elecrtons.
But it's the same relative motion, so should have the same physical effect. What's the explanation then?
to elaborate a little: the electrons aren't put into motion by my moving the coil, so for them to start moving there must be an electric field present. since the only thing that's changing in this scenario is the magnetic field, it must be the case that a varying magnetic field induces an electric field.
moreover, from the symmetry of the problem one should be able to deduce that the resulting electric field lines are closed circles. so this effect is different from Coulomb's law, where all field lines have to start and terminate at charges. (in terms of math, the electric field so generated isn't curl-free.)
So from such considerations one actually can infer Faraday's law of magnetic induction.
for the punchline to this, consider the following paragraph:
"It is known that Maxwell's electrodynamics—as usually understood at the present time—when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena. Take, for example, the reciprocal electrodynamic action of a magnet and a conductor. The observable phenomenon here depends only on the relative motion of the conductor and the magnet, whereas the customary view draws a sharp distinction between the two cases in which either the one or the other of these bodies is in motion. For if the magnet is in motion and the conductor at rest,
@Semiclassical That's an interesting perspective. So you don't really need Faraday's law (that changing magnetic field induces an electric field) to explain why, if I move the coil keeping the bar magnet same, current flows through the coil. And by moving the bar magnet while keeping the coil fixed, we actually can conclude Faraday's law because the two situations produce the same physical scenario
They basically wrote up Singer's course. I have never studied it in detail. If I recall correctly, you'll find it has the principal bundle approach, full of differential forms.
I was upset that a UGA grad student who absconded with several thousand dollars of my books (free) then sold some of them off for a profit when he left pure math for applied.
Semiclassic, I think I posted my "homework" for the physics majors with exercises using forms and Hodge star to get div, grad, curl in different coordinate systems ... and there was a problem relating to the delta function, too.
category theory is pretty uncharted territory for me
I think I know how to do electrodynamics using forms
though that statement's a bit ambiguous, since one can do it either in 3 dimension or in 3+1 dimensions
3 dimensions is easier when you're doing div/grad/curl stuff. but you need 3+1 if you want to talk about the field strength tensor
Given how much physicists love stuff like point charges and line currents etc, I suspect that a full description of E&M would need to have k-currents in there rather than just k-forms
I actually put a little section on 3+1 and Hodge star and Maxwell's equations in my book, Semiclassic, but it was a throw-away. I've never lectured it. No time.
It is a differential operator, but I don't think of it that way most of the time. I mean, a vector is a derivative because it gives you a directional derivative at a point. That's all.
I think my point is just that, when you ask a physicist to write down an example of a vector field, they'd do something like $\vec{E}=(x\hat{x}+y\hat{y}+z\hat{z})/(x^2+y^2+z^2)^{3/2}$
It's good for understanding the difference between $C^k$ and smooth, @Danu. In fact, you can only use the differential operator approach on smooth manifolds. That's important, @Semiclassic. I should send you one of my exercises (which I stole from Spivak).
Yes, I would do that too in my multivariable math course. What's the big deal?
@Semiclassic: At any rate, unless you're in the smooth world, it's important to think of tangent vectors as equivalence classes of curves in the manifold. But meh ...
By the way, @Ted, do you have any idea how the field of sort of "general structure theory" for 3- 4- or maybe even higher-dimensional complex manifolds is doing? I caught a glimpse while writing about those uniqueness theorems in complex geometry and it seemed pretty nice (minimal model program?).
Let $A$ and $B$ be sets. Is it true that $A-B = B^c - A^c$, where $A^c = U - A$ and $U$ is the universe? I think I proved it but I would like verification.
Actually I now have the doubt that the proof I saw last year was the fixed point proof in disguise. I remember a lot of estimates that are basically the same one does to prove the Banach fixed point theorem
@Eric How interested are you in octonions...? (just kiddin, you don't wanna get into this :P)
Also, I was under the impression that Vanderweghe had to be Belgian for the longest time @Ted... Finally a successor for Henin-Ardenne & Clijsters, I thought.
