yshklarov

I'd love to see the teaching video!

You could be right, but the way I read it is that those are things that everyone already knows, so there is no point in spelling them out.

In the words of Robert Morris, 40 years ago:

"The three golden rules to ensure computer security are: do not own a computer; do not power it on; and do not use it."

In some cases these are not available. For instance, archive.org has many books that don't seem to be available anywhere else.

Oops, my mistake: that is not the definition of curvature! The outer derivative must be taken with respect to arclength, rather than with respect to t.

Do you know whether it's usual nowadays in industrial design to make curves that are actually smooth (I mean, derivatives of all orders are continuous) and not just C2? Or is it technically challenging for some reason, or not useful?

That is roughly correct. For a curve with smooth parametrization f(t) = (x(t), y(t)), with f'(t) ≠ 0 for all t, the curvature at the point f(t) is defined to be κ := |(f'(t)/|f'(t)|)'| -- the magnitude of the first derivative of the unit tangent vector. This is a nonnegative quantity, and is well defined, that is, independent of the specific parametrization of the curve (as legosexmagic said, you need to do something like this to get a quantity independent of the parametrization, but it is essentially a "normalized" second derivative).

The curvature comb consists (it seems to me) of a normal line segment at each point, whose length is directly proportional to the curvature there.

> We do not write code for our computers, but rather we write it for humans to read and use.

Prime example of a "deepity".

The author says: (https://twitter.com/shr_id/status/1602691898162184192)

  In the beginning I didn't think it was possible without cheating something but finally found a way to "build everything perfectly"
  Due to the inifinite recursion, each level is aperiodic for both space and time, and you'll never see the same pattern no matter how much you scroll

That information is available in the last section of the paper.

We're bipedal animals by nature, but not every individual is able to walk. Likewise, although we're social animal by nature, not every individual can easily form close social ties.

There are many reasons a person might struggle to make friends. For instance, some people were poorly socialized, or have some kind of early-childhood developmental trauma that makes it difficult to feel the appropriate emotions when talking to people. Others naturally have trouble with empathy and so are unable to smoothly navigate social interactions. And some just naturally aren't interested in learning about other people's lives, or in telling others about themselves. So they have to force themselves somehow.

I agree with you: these artificial systems are pathological and unpleasant. But sometimes it can seem like the only option.

My impression is that you've never struggled with any of these very basic aspects of life. Consider that not everybody's experience is the same.

Apparently, you're welcome to ask him directly: https://communitywiki.org/wiki/LionKimbro

> In statistics, the mean and median are both averages, but are technically quite different.

But numerically, they are not too wildly different! The difference between the mean and the median is at most one standard deviation. If the distribution is unimodal, then the difference is at most sqrt(3/5) standard deviations.

Actually, you can if you bend the coin!

Actually, the "fluff" in this case is outstanding. The paper itself doesn't provide much context, but the Quanta piece does a great job at explaining why the result is important.

But no! The very first sentence clearly states: "In this post, we'll look at the natural problem of how to communicate an estimate of a real value in [0,1], using just 1 bit." The second paragraph sketches the motivation for the problem.

How much more exposition do you want from a casual blog post? There's a link to the academic paper if you want full details (you can follow up on the numerous references to other papers, which explain applications.)

Indeed. "Many widely-accepted theories ultimately get rejected. Perhaps this one will be, too." What use is an article like this to anyone?

A fair point, but don't you think it sidesteps the matter at hand? Never mind that other systems have flaws, too. What are the specific reasons that Node.js itself is hated?

I see lots of comments arguing with the post, but seeing it in its original context is important. The full article addresses many of the criticisms:

The roots of creativity and genius http://super-memory.com/articles/genius.htm

Institute for Applied Computational Science (IACS) Seminar

PRESENTER: David G. Stork, Rambus Lab

TALK ABSTRACT: The central insight underlying the field of computational sensing and imaging is that the joint design of optics and signal processing to yield a final digital image or estimate of some property of the scene can relax the traditional constraints on optical elements need to make an optical image that "looks good." In our lensless imagers, binary diffraction gratings with special mathematical properties yield blurry, blob-like optical images that nevertheless contain sufficient information that a digital image of the scene can be computed.

The article purports to teach calculus, so it should get the basics right. The Fundamental Theorem of Calculus is called fundamental for a reason: it's the most important concept in the entire subject. It's not just a technicality. The article defines things in a non-standard way so as to reduce FTC to a tautology, and nowhere does it correctly explain the connection between differentiation and integration (which can -- and should -- be understood visually and intuitively).

In short, I'm not convinced that the author knows what he's talking about.

I'm not sure what this has to do with the content of the article. Could you elaborate?

This is fantastic, I love it! I've been looking for something like this.

Your definition of "uniform" doesn't look quite right: It's not enough for the symmetry group to be vertex transitive, you also have to require that the faces be regular polygons (see the wikipedia page on isogonal solids for examples of vertex-transitive solids which don't have regular faces). Also, "vertex-transitive" means that for any pair of vertices, there is a rotation or reflection symmetry of the solid which sends the first vertex to the second -- this is not equivalent to the definition you give, for example, the pentagonal cupola is not vertex-transitive but it does seem to satisfy your definition. The standard definition of "uniform" for higher dimensions is recursive: A convex polytope is called uniform if its facets are uniform and its symmetry group (including reflections) is vertex-transitive; a 2-dimensional polytope is called uniform if it is regular (i.e., is a regular polygon). So, in three dimensions, a convex polyhedron is called uniform if it has regular faces and its symmetry group is vertex-transitive.

You are asking big questions! My answer is brief but I hope it helps a little.

By axiomatizing a definition we can begin to prove theorems rigorously. If we formally abstract a concept and deduce formal statements from it (which can themselves be quite unituitive), we can be confident that the statements apply to the concrete situation at hand. On the other hand, if we always worked only with concrete or physical examples, we would have to figure out everything from scratch every time. A good general theory is one which concisely explains many specific cases at once.

For example: Can every periodic continuous function be approximated uniformly to whatever degree of precision we like by a partial Fourier sum? If we abstract away what's important, and study the situation in the general context of Hilbert spaces, we can answer this question not only for this specific case but also for a broad class of families of approximants in one fell swoop. We save a lot of effort by doing this, and we also usually gain extra insight into the problem.

Still, you don't want to abstract too early. It's important to understand the concrete case first before jumping a level in abstraction, otherwise you end up understanding nothing. If you're having any trouble with metric spaces (I'm not sure if you are) then you might find it helpful to look at a few specific examples to see what they're used for. Examples: In coding theory, we often use what's called the Hamming metric, the distance between two words. In graph theory, there is a natural geodesic metric: the shortest distance between two vertices as a walk along the edges. If that's not concrete enough, consider the popular "6 degrees of separation" rule: people are vertices and relationships are edges. Of course there are the usual examples of Euclidian space R^n, unitary space C^n, and the other various normed spaces you're studying in functional analysis.

By working with the axioms of metric spaces we can prove theorems which apply to all such cases at once. Here's an example of a tricky theorem: Let x be a point in a metric space. Suppose a subsequence satisfies the condition that each subsequence has a subsequence which converges to x. Then the entire sequence converges to x. You wouldn't want to prove this theorem from scratch in every specific concrete case! It's more efficient to abstract out the essential features (the axioms) and then prove the theorem in the general setting.

As for your other question: Set theory is (in my opinion) not a fundamental feature of mathematics. It just so happens that mathematicians today like to axiomatize everything in terms of set theory. Still, sets will always be useful even if we don't choose to define everything in terms of sets. A set is (loosely speaking) nothing more than a collection of objects, and we'll probably always find it useful to deal with collections of objects.

As for symbols: What more is a symbol than a hook on which we havg an abstraction? How can we manipulate abstractions if we don't have symbols for them? I'm using the word "symbol" in the broad sense here, which includes diagrams (or pieces of diagrams) and words in a natural language. If you want to see abstractions manipulated via diagrams and words, take a look at classical sources (say, pre-1500's). I believe our modern notation is far more amenable to manipulation and understanding.

In short: In mathematics, abstraction and symbol manipulation are the name of the game. If we have no symbols then we have no abstraction and no mathematics.

I don't understand why so many people recommend baby Rudin (Principles of Mathematical Analysis). The presentation in Rudin is not merely terse, but also quite dry and unmotivated. I suggest you avoid it--regardless of how much talent or maturity you have. There are plenty of more interesting texts which will teach you just as much: Spivak and Pugh are nice, I also recommend the recent two-volume work by Zorich.

By the way, as you aquire experience you'll gain confidence and get over the urge to always check your answers. Here's a good exercise with a built-in answer key: When reading a text, every time you get to a result (claim, theorem, etc) try to prove it on your own before you continue. You probably should be doing that more often than not.

In any case, don't stick to just one text/source. Shop around, read a few pages here and there before you settle on something. There's no way a stranger on the internet can make a good recommendation: Find what works for you. The most important thing is that you're fully engrossed!

This is an absolute joke. Take a look at the data. https://sci-hub.hk/https://www.sciencedirect.com/science/art...

The sample size is far too low to draw any conclusions. Only a small handful of rats in each group got tumors. The authors must know this! They got a p-value <0.05 for just one of the numbers, the male rats at 50 V/m, 1.4% incidence of Schwannomas. The entire group (both female and male) was only at 1.2% and this was not considered statistically significant. By the way, the one-tenth-dose group (5 V/m) had fully 1.5% incidence of Schwannomas.

And yet the conclusion claims "These experimental studies provide sufficient evidence to call for the re-evaluation of IARC conclusions regarding the carcinogenic potential of RFR in humans." I have no words.

Taking this to the extreme, you could "store" 65535 1-bit values in one 16-bit value. You just have to keep track of how many of them are equal to 1.

I did not mean to imply that the volume of code was directly tied to its value. I understand that value stems from functionality and not from quantity of code. But functionality does stem from code, as jsprogrammer said. If your code does X, and X is valuable, then the code is valuable -- I don't see what's wrong with this line of reasoning.

This strikes me as some sort of Orwellian contortion. It really is a matter of your point of view. Insofar as code is correlated with functionality (and it is at least to some degree), code can be viewed as an asset. In addition, code can be sold for cash, and this makes it an asset from the strictly financial point of view. It may have maintenance costs associated with it -- and these certainly need to be taken into account -- but that is not the same as being a liability.

A correction: The chapter at

http://www.biostat.wisc.edu/~craven/hunter.pdf

is actually the first chapter of "Artificial Intelligence and Molecular Biology":

http://mitpress.mit.edu/books/artificial-intelligence-and-mo...