HN Theater @HNTheaterMonth

⬐ graycat

Gee, I can like that!

For the set of real numbers R, he starts with a function

f: R --> R

such that f(x) is zero for x <= 0, strictly positive otherwise, and infinitely differentiable. There is such a function, the same or similar, in an exercise in Rudin, Principles of Mathematical Analysis.

At one time I used that function in Rudin to show that for positive integer n and closed subset C of R^n there exists function

f: R^n --> R

so that f(x) = 0 for all x in C, strictly positive otherwise, and infinitely differentiable. That result is comparable with the classic Whitney extension theorem -- Whitney assumed a little more and got a little more.

I discovered this result for and used it in some work in the constraint qualifications of the Kuhn-Tucker conditions: I constructed a counterexample that showed that the Zangwill and Kuhn-Tucker constraint qualifications are independent.

So, back to Milnor's lecture!

I want to see if he drags out the inverse and implicit function theorems (essentially the local nonlinear version of what is standard from Gauss elimination in systems of linear equations!).

⬐ analognoise

Isn't f(x) = 0 in C a trivial solution?

⬐ graycat

No. The domain of f is all of R^n so that f(x) has to be defined for all x in R^n.

Then there is the requirement that f be strictly positive otherwise, that is, in the open set outside C.

The result is a bit amazing: It says that any closed set can be the level set for an infinitely differentiable function.

Can add some interest when consider how bizarre some closed set are:

So, in the plane can take C as a sample path of Brownian motion. So, there's an infinitely differentiable function zero on that sample path and positive otherwise.

Next, the Mandelbrot set is closed. So, there's an infinitely differentiable function positive everywhere but zero on the Mandelbrot set.

Next consider the Cantor set or Cantor sets of positive measure. Same story.

So, a really smooth, infinitely differentiable, function can have for a level set anything at all considering that any level set of a continuous function is closed.

There's a famous paper in mathematical economics by Arrow, Hurwicz, and Uzawa with a question with no answer, and my work answers the question. Yes, Arrow and Hurwicz won prizes in economics -- IIRC so far poor Uzawa has not won such a prize!

For something intuitive, consider an infinitely differentiable mountain range, and pour in some water to make a lake or many lakes (assume a porous mountain) all with the same altitude. Then, the lake boundary is a level set of an infinitely differentiable function. As we know from Mandelbrot, commonly, roughly lake boundaries are fractals.

So, we're fine; everything holds together: My result shows that for Mandelbrot's fractal lakes, the ground can be smooth, infinitely differentiable. Did Mandelbrot know that?

For rational p/q for positive whole numbers p and q with p/q in lowest terms, consider in R^2 a ray from the origin of length 1/q at angle p radians. Do this for all such countably infinitely many rationals. Then the set of all these rays is closed but is a spiny urchin, really bizarre. That's the closed set I used for my counterexample.

Can also apply this to zero day detection of anomalies in server farms and networks: So, suppose collect 20 times a second numerical data on 10 variables. Then that vector of 10 variables has a probability distribution. Suppose it also has a continuous density. Then pour in water to make lakes -- all the same altitude. Get a point in a lake, then declare an anomaly. The volume of land under the lakes is essentially the false alarm rate. Pour in more water and get a higher false alarm rate but also a higher detection rate. Here the set where we declare an anomaly has the highest possible area for the given false alarm rate, and there's another sense in which the detector has optimal combinations for false alarm rate and detection rate.

How 'bout that!

⬐ analognoise

See, this is why I am on HN. I'm interested but not as advanced. You included good examples; now I have to figure out wtf you are talking about. :)

⬐ graycat

A function f is differentiable if it has a derivative just as in calculus or, as in high school, a tangent line. If the derivative, another function, is differentiable in the same sense, then the function f is twice differentiable. For any positive integer n, in the same way, can say what it means for function f to be n-times differentiable. Finally, if for each positive integer n, function f is n-times differentiable, then function f is infinitely differentiable.

Why do we care about function f being infinitely differentiable? Because that's necessary for function f to have a Taylor series expansion. Why care? It says we have a start on a ready made way to approximate f as accurately as we wish with, also some good, first-cut information on the function from if only from the first few terms. With more, as Milnor mentioned, the function f might be analytic which means that it is equal to its Taylor series expansion.

In R^n, a set is open if it contains a little ball around each of its points. So, in R the interval (0,1) is open. A set C a subset of R^n is closed if and only if R^n - C is open. Open sets have some nice properties; so do closed sets.

E.g., we know from calculus from uses of deltas and epsilons what a continuous function is. Well we can generalize to topoloty, Milnor's subject: Quite generally, function f is continuous if the inverse image of each open set in the range set of f is open in the domain set of f -- and this definition works with astounding generality. Topology is heavily interested in continuity.

In R^n, a each bounded, closed set is compact which is really a generalization of it being finite and finger lick'n good: E.g., in calculus, each f: R --> R continuous on a compact set has a Riemann integral on that set. Biggie stuff.

Can take open and closed off to, say, Banach space (complete normed vector space) and the famous open mapping and closed graph theorems.

For the result I discussed, open and closed were central concepts.

The Kuhn-Tucker conditions are in optimization. So, imagine you are in a dark cave with walls and a floor that is not flat. If you put down a ball and it rolls, then you are not at the bottom. So, a necessary condition for being at the bottom is that the ball not roll. For that it may be that the floor would let the ball roll but the walls stop it. So, when do the walls do this? Roughly the normal vectors of the walls in contact with the marble block the motion for the marble to roll lower. But for the simple Kuhn-Tucker equations, in terms of gradients, work, the walls must be sufficiently nice, that is, satisfy some constraint qualifications. There are lots of constraint qualifications, some easier to verify than others. In nearly all routine engineering problems, the constraints do satisfy some, typically several, constraint qualifications. But if want to dig into lots of details, look at pathological cases, etc. then can wonder about constraint qualifications. Kuhn and Tucker have a statement of some constraint qualifications (CQ), and so does Zangwill. Are the two CQs equivalent? One direction was no. I showed that the other direction was also no and, thus, that the two CQs are independent, with neither implying the other. My proof was by counterexample, from my result, and was bizarre and tricky, which is likely why it was new. For my counterexample, I used my result about infinite differentiability, but that was much more than I needed. I showed infinite differentiability only out of curiosity -- just wondered if it was true.

The result is curious because intuitively an infinitely differentiable function is supposed to be very smooth (Milnor gave a precise definition but I was just being intuitive) but some of the level sets can be wildly irregular, e.g., the Mandelbrot set and Brownian motion. E.g., Brownian motion never has a tangent line, not even a first derivative, but it is the level set of an infinitely differentiable function. So, the juxtaposition is curious.

Yet again a math guy stands in awe of reality that math discovers one little theorem at a time.

So, now you know more about what the heck we are talking about!

⬐ tgb

I think I know such a function for the Mandelbrot set: http://iquilezles.org/www/articles/distancefractals/distance... (Well, the function that this one converges to.) Not sure about it's differentiability, though.

⬐ SamReidHughes

Neat. Did you do it by making such a function for the open ball, and then, given an open set defined as a countable union of open balls, define a sequence of such functions scaled appropriately so that their derivatives all converge uniformly?

Edit: Looks like you can make that work, according to the answer at https://math.stackexchange.com/questions/791248/every-closed...

⬐ graycat

I took a countable dense set in R^n - C, x_j, j = 1, 2, ..., and for each j took the closed circle with center x_j and radius d(x_j, C) (distance from x_j to C which is well defined due to compactness). Then I defined a function g_j: R^n --> R 0 outside the circle and on the circle with values on the radii much as in the function Milnor, Rudin, and I mentioned. So, g_j is infinitely differentiable, strictly positive on the interior of the circle at x_j, and 0 otherwise. Then I added the functions g_j in a convergent way, again using the exponential trick. The resulting function was as desired, 0 on C, strictly positive otherwise, and infinitely differentiable.

⬐ delish

Long ago I read a quote from a post about mathematicians who are good communicators. I'd love to come across it again.

Of John Milnor, this post proclaimed, "When he speaks, you understand."

I try to notice rare compliments, like that one. Feynman received one like that:

A female engineer says about him: "Yes, [Feynman's sexism] really annoys me," she said. "On the other hand, he is the only one who ever explained quantum mechanics to me as if I could understand it."

The only one!

http://longnow.org/essays/richard-feynman-connection-machine...

⬐ tgb

What causes the halos in this film? I notice whenever there is an unusually light or dark spot, it gets a halo around it of the opposite lightness. See his hand as he starts writing shortly after this time stop to see what I mean: https://youtu.be/1LwkljjLBns?list=PLelIK3uylPMFHC6Xny11XFXgw...

⬐ semi-extrinsic

Looks like someone has applied an unsharp mask.

Example: https://upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Us...

⬐ theoh

Could also be a high dynamic range operator: they do the same kind of thing to preserve contrast across large ranges of intensity. See Larson's paper http://gaia.lbl.gov/btech/papers/39882.pdf etc.

⬐ ddumas

It's not a filter, as others have suggested, but rather a standard artifact of vacuum-tube-based television cameras that were in use at the time. (I don't know why this lecture wasn't filmed, but the extensive dark halo effect makes it clear this was shot with a TV camera. This was the early era for magnetic video tape, but I assume that's how the recording was preserved.)

Anyway, the point is that these TV cameras are based on the fact that incoming light will dislodge electrons from a thin plate in a vaccum tube in an amount proportional to brightness. A very bright spot in the image produces a shower of electrons that is more powerful than the rest of the tube (the part that detects the electrons) can deal with. The net result of this "splash" of electrons is a mild desensitization of the detection apparatus around the bright spot. This makes the nearby stuff appear darker.

You mentioned that dark spots also seem to have a bright halo, but I don't see that in the video, and it isn't consistent with the usual artifacts of these cameras. Are you sure?

⬐ tgb

Very interesting, thanks.

There do seem to be subtle halos around dark spots, though. See here: https://youtu.be/1LwkljjLBns?list=PLelIK3uylPMFHC6Xny11XFXgw... It's not very noticeable in a still but is pretty clear when his hand shadow moves.

⬐ figure8

I really appreciate how he writes and draws the slides as he speaks. With math lectures, this always encourages me to think deeply about the details and implications of each line. I find it better than pre-written slides, now so easily created and presented using Power Point.

⬐ jordigh

I think this style of presentation is still very common in maths departments. Mathematicians tend to favour low-tech presentations. Blackboards are still in vogue!

⬐ mrcactu5

does anyone else dance to the music in the beginning?

⬐ cplease

It would have been nice to list the source of these lectures where they are freely available rather than a pirated Youtube link:

https://www.simonsfoundation.org/science_lives_video/profess...

Published sequel here: http://www.ams.org/notices/201106/rtx110600804p.pdf

⬐ nairboon

So how do you pirate a Youtube link?

⬐ cplease

"Link to an unauthorized copy on Youtube." Feel better?

⬐ cft

Unauthorized by whom? You should contact the copyright owner and alert them of the infringement then!

⬐ jordigh

It's not freely available if they are not free to copy... and I really wonder who could possibly be harmed by copying these videos around. I can't imagine that Milnor would object to having his work widely disseminated. Do you think if we copy something he made 50 years ago (which really should be in the public domain by now), he'll be less motivated to generate new lectures at his 84 years of age?

Your usage of the term "pirated" seems to indicate more indignation than is in order.

⬐ wodenokoto

Yes, it is called "free as in beer".

⬐ jordigh

It's not even free beer. If I go to an event and get a free beer, I'm not forbidden from giving my free beer to anyone else. It is ridiculous that we are forbidden from sharing a video that was created 50 years ago just because Disney wants to make sure Mickey Mouse never falls out of copyright.