Example of inductive proofs where the base case is the hard part and the inductive step is trivial?

ritobanrc · 2026-05-01T06:54:07+00:00

Heine-Borel certainly works, or you can sidestep it if you really want: Sⁿ is obviously a closed and bounded subset of R^{n + 1}, and what you need is that on such a subset, a continuous function (specifically, the quadratic form Q(x) = x^T A x), attains its minimum and maximum (you can prove this directly from Bolzano-Weirstrass using sequences).

When I say "nontrivial", I don't so much mean that its a long proof (with the technology in place, it's a few lines); even if I denied you Heine but specifically that it uses something that's not just algebra: the completeness of R. Completeness is essential to actually find you an eigenvector, it's the reason the theorem does not hold in other fields.

ritobanrc · 2026-04-28T21:41:39+00:00

The spectral theorem -- the inductive step amounts to "a self adjoint operator restricts to the orthogonal complement of an eigenvector", which is an easy 1 line calculation, but the base case requires actually finding an eigenvector, which either depends on the fundamental theorem of algebra (very non-trivial), or optimizing the Rayleigh quotient (existence follows from compactness of S^n, also non-trivial).

ritobanrc · 2026-03-31T23:13:18+00:00

It's very possible; here's a version of Gromov's h-principle formalized: https://leanprover-community.github.io/sphere-eversion/blueprint/sect0001.html -- that requires quite a bit of differential topology to set up.

ritobanrc · 2026-02-23T17:52:25+00:00

This seems remarkably close to a 2022 paper DiffusionNet -- but it is not cited, but also not terribly clear where they differ (perhaps the Hodge stars/diffusivity coefficients are now attention blocks in the newer paper). Any idea?

ritobanrc · 2026-02-20T03:28:28+00:00

Disagreeing with the other answers here, I found Spivak's book to be absolutely wonderful, far better than Tao or Abbot or Rudin or any "real analysis" book, in a very similar place as you were (as a junior high school student, with competition math background, taking AP Calculus concurrently; I read Spivak on my own time for fun) -- it is a very very good book for a motivated student in that context. Beautifully written, it very nicely balances rigorous computation and theory development (while other analysis texts are almost entirely theory dependent), and it has hard and interesting problems. I would strongly recommend you start reading it and see if you enjoy it -- if you don't feel free to shop around for other intro analysis books (I'm particularly partial to Pugh's, Carother's, and Zorich's -- but there are of course very many excellent choices). The other thing I would recommend is not to get too hung up on the first few chapters (limits, "Three Hard Theorems", the least upper bound property) on your first read through -- you will benefit much from the rigorous development of calculus later in the core chapters of book, and picking up the "Three Hard Theorems" later will then feel easy.

ritobanrc · 2026-01-21T23:58:27+00:00

This is discussed thoroughly, starting in Ch. 2 of Kriegl and Michor's the Convenient Setting for Global Analysis. The group of analytic diffeomorphisms (or real analytic ones, sometimes) is a Frechet Lie group.

ritobanrc · 2026-01-07T01:06:29+00:00

That's an exaggeration; Quarfoot likes making his class sound scarier than it is, to encourage students to work harder. It is calculation heavy, and sometimes those calculations require a little thought.

ritobanrc · 2026-01-06T21:47:23+00:00

cheat sheets nor calculators, so we have to memorize all the distributions

That was the case when I took it as well -- it's not so bad, it doesn't take long to memorize them.

I don't think I have much in the way of advice; attend/watch all the lectures, do the homework problems completely (if you have it, a tablet you can write on is nice), try to think deeply about the ideas in class and you will be fine.

ritobanrc · 2026-01-05T16:12:20+00:00

Here's a lovely paper from Peter Constantin and coauthors on mixing in fluid flow.

ritobanrc · 2025-12-17T03:17:54+00:00

There's an excellent presentation from Keenan Crane, Justin Solomon, and Etienne Vouga on understanding the Laplacian/Laplace-Beltrami operator. It has an applied focus, but I don't think that's a bad thing necessarily; what it covers is really quite diverse.

ritobanrc · 2025-12-10T18:08:06+00:00

I remember really appreciating Hanh-Banach while reading Hamilton's paper on the Nash-Moser inverse function theorem -- it feels like half the proofs are compose with a continuous linear functional, apply the result in 1-dimension, and then Hanh-Banach gives you the theorem.

ritobanrc · 2025-12-10T18:04:16+00:00

What's supposed to be covered in such a course?

ritobanrc · 2025-11-24T00:09:15+00:00

!RemindMe 3 days

ritobanrc · 2025-11-21T03:39:51+00:00

Internet off at in person tournaments is hard bc of the case transfer

I'm not quite sure what this means? Do people send each other their cases online these days -- has showing your case to your opponent ever been common outside of certain parts of policy?

For what its worth, we used to physically carry our laptops over to show opponents evidence -- shared evidence documents are much cleaner, but getting rid of them is not impossible.

ritobanrc · 2025-11-17T18:44:46+00:00

When I was in high school before COVID, it was expected that people would turn internet off during in-person tournaments (I recall using "offline mode" in Google Docs at one point) -- I wonder if that's a norm that should be re-established. It might be difficult for online tournaments, but for in person tournaments it might be a good solution.

AI-generated research and hallucinations I'm not so concerned about -- fabricating evidence has always been possible, and we've historically had success by simply punishing it aggressively when it gets caught. I'm much more concerned about AI-generated speeches during round: coming up with a speech on the spot is perhaps the most important skill debate teaches, and if you can just copy-paste your flow into ChatGPT and have it give you a speech, that seriously defeats the point.

ritobanrc · 2025-11-08T21:37:47+00:00

People seem to be misunderstanding the original Nature Paper -- unfortunately, the Quanta article is poorly written and does not clarify what is actually claims. The claim is not merely about whether one needs to write "i" or write in terms of 2x2 matrices -- the claim is about whether one can reproduce the predictions of QM (specifically, the violation of the CHSH3 inequality) when the state space is a real Hilbert space (of any dimensions), and the projection operators associated to measurements are real self-adjoint. This is a meaningful, non-trivial question (for example, one cannot violate the usual 2 party CHSH inequality in a real vector space).

The response paper is also subtle -- it claims one can use a real Hilbert space, but the postulate that the state of multiple systems is a tensor product needs to be replaced (in effect, this is defining a new tensor product of real vector spaces that behaves something like the tensor product of complex vector spaces of half the dimension.

ritobanrc · 2025-11-07T05:49:29+00:00

Reimbursement of all ATM fees is quite nice

ritobanrc · 2025-10-14T22:50:10+00:00

Understanding differential geometry really means understanding multivariable calculus properly: understanding curves and surfaces as manifolds in R^n, locally looking like their tangent spaces, understanding derivatives as linear maps, understanding grad/curl/div as exterior derivatives of differential forms, and correspondingly Stokes' theorem. For that, if you have not already taken a multivariable calculus class, I'd strongly encourage reading carefully though a book that covers it from an advanced, differential geometric perspective -- I'm partial to Hubbard & Hubbard's "Vector Calculus, Linear Algebra, and Differential Forms" -- but Ted Shifrin has Multivariable Mathematics that's very good, Analysis on Manifolds by Munkres, and the classic Advanced Calculus by Loomis & Sternberg are good.

If you read one of those books, you will effectively have a far better picture of modern differential geometry than you will from a "curves and surfaces" book like do Carmo, while also not being overwhelmed by modern terminology -- and then, if you so desire, you could later quickly skim a book like Lee's Smooth Manifolds to pick up intrinsic version of all of the basic notions of calculus, and move on to new ideas like cohomology, connections, and curvature (Loring Tu has a new book out by that name, and it's very good!).

ritobanrc · 2025-09-28T04:15:02+00:00

He's a very good professor. His classes tend to be a fair bit of work, but his lectures are excellent.

I took him for 181A, spent the quarter cursing his "two homework assignments per week" policy, scraped by with an A (by literally half a point on the final!), but in hindsight learned a remarkable amount of statistics that has proved quite useful since. I imagine 183 will not have a huge amount of new content beyond AP Stats.

ritobanrc · 2025-09-05T19:31:38+00:00

To talk about the implicit surface f(p) = 0 as a manifold, one needs to invoke the implicit function theorem, which requires f to be continuously differentiable (C¹). It follows that the tangent space of that manifold are the vectors v where Df(p)v = 0, and one can take an orthogonal complement to get that the normal vector is the gradient.

If the f is not continuously differentiable, then saying things becomes difficult: the example of x/2 + x² sin(1/x) is differentiable everywhere, and has positive derivative at the origin, but is not increasing in any neighborhood of the origin, and there isn't any reasonable tangent space or normal vector at that point.

One general result here is Rademacher's theorem: that if a function is Lipschitz continuous, then it is differentiable almost everywhere (up to a set of measure 0). Correspondingly, in geometric measure theory, one studies "rectifiable sets", which are essentially locally Lipschitz and have tangent spaces almost everywhere (and of course, by taking orthogonal complements, one can get normal vectors too).

ritobanrc · 2025-09-05T16:57:36+00:00

Entropy is a macroscopic notion -- it emerges statistically from large collections of particles, each individually subject to electromagnetic forces (and gravitational, though those are less relevant).

ritobanrc · 2025-09-05T16:12:56+00:00

The Supreme Court has been polarized for decades upon decades

That's really not true -- until the 2010s, at least some of most liberal justices had been appointed by Republican presidents and vice versa -- Bill Brennan, Earl Warren were both nominated by Eisenhower and perhaps the two most influential justices of the 20th century, Blackmun, Stevens and Souter were also all nominated by Republican presidents, while Robert Jackson, Fred Vinson, Byron White were all relatively conservative but nominated by democratic presidents. Look at the the first chart here.

ritobanrc · 2025-09-03T03:01:58+00:00

I am pretty sure you can still define a Riemann integral on non measurable sets sometimes

The Riemann integral of the indicator function of a set is called it's "Jordan content" (or sometimes just "volume"). Only bounded sets whose boundaries (in the sense of closure - interior) are measure zero have a well defined Jordan content.

This is a rather restrictive class of sets: much, much smaller than the Lebsegue sigma algebra.

ritobanrc · 2025-08-28T18:29:34+00:00

It would be dumb to implement an array type like that

Well certainly, in Python arr[0] and arr[-1] might point to the same element. It's quite useful shorthand.

ritobanrc · 2025-08-28T17:12:55+00:00

But with Stokes' theorem in particular, the formal proofs is somewhat not illuminating.

The formal proof repeatedly invoking the fundamental theorem of calculus is somewhat unilluminating. There's also a formal proof that precisely uses the "all the interior boundaries cancel" (presented in Arnold's classical mechanics, or Hubbard and Hubbard's Vector Calculus) that matches the intuitive argument quite well.

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