pfedak

That surprised me too. I found that Mathematica could solve it if you give it the denominator as 1-u^2 for the eventual substitution. Perhaps it just doesn't try very hard to look for substitutions? I could believe it not giving a lot of "coverage" per time spent compared to more advanced techniques.

Are you perhaps thinking of https://thetech.com/ ? I'm not sure in what sense the Technology Review is a student newspaper - to my knowledge (and checking a few bylines), the journalists are not students.

The article you link about repeatedly heated oils, specifically reusing oil when deep frying. That doesn't seem relevant to the use described above.

I kind of get the sense Baez is being polite and humble there, not wanting to say something is nonsense if others seem to find value in it.

That said, I think I agree with Baez about the actual mathematical content of the paper. That is, the proof is fine, but the way they try to formalize "model" seems so weak as to be useless. The definition used in the proof is that R is a model of S if the state of S determines the state of R. There's nothing about preserving the fidelity of those states.

edit: Maybe the clearest way I can put this confusion is that a regulator based on measurement of the system it's regulating is a "model" per the paper. The authors choose a formalism which sounds like it was more common at the time that describes the regulation as a function of "disturbances"/inputs only. Through this lens their definition of model makes more sense to me, but only in those cases where there is no measurement or feedback.

The issue I'm having with the claims is that their use of the word "model" seems to connote exactly of the opposite of what someone reading the paper without seeing their definition would suggest, and indeed what other commenters here are taking it to mean.

The colloquial restatement of the theorem suggests a regulator "is a model of [a] system" if the regulator's action is a function of the system, and nothing else. As in, a constant regulator that has only one state still meets the criteria for being a "model". The possibility that is excluded is that the regulator has additional details that don't correspond to states of the system.

In your thermostat example, the theorem doesn't even begin to say that, even though that's what the title suggests, and what everyone wants to believe. In fact, the theorem merely says a thermostat that only depends on the temperature of the room is simpler than one that also cares if the walls are blue.

Explored rather amusingly in The Pale King

https://books.google.com/books?id=gW0K6X-ThtAC&lpg=PA12&dq=p...

I'm disappointed, I was looking forward to seeing a new fractal. The onion curve in 2d is just a spiral, as mentioned, but the word "spiral" doesn't appear at all. I think this paper is mostly without merit, unfortunately.

Egregiously, the table comparing the (reasonably defined, I think) clustering metric of the spiral with the Hilbert curve is only for rectangular queries which ask for almost the entire space, with the query side length differing from the overall space's side length by a constant.

An obvious advantage of the fractal nature of the Hilbert curve is that its clustering properties will be scale invariant, but the spiral will quickly degrade to a "space filling curve" that just goes row by row for small queries. Perhaps later the paper gives some reason to hope for the onion curve, but I'm disinclined to look.

I'm having trouble following the proof of lemma 1, and I can't tell if it's the notation being different from what I'm used to.

Is the expression for gamma prime equal to negative the square of the distance between x and y, that is, -<x-y,x-y>? I'm curious, because I would have expected a stronger property to be necessary for the parallelogram structure.

As pointed out in that aside, the general vector-bivector multiplication isn't needed to evaluate the reflection case, but I agree it's a little unsatisfying to sweep that under the rug.

You can work everything out by breaking it down to the unit basis vectors, for which the products are explicit. One way of writing it, which feels more comfortable to me possibly at the expense of being the "wrong" sort of intuition, is

u(v^w) = (u.v)w - v(u.w) + u^v^w

(using . for inner product and ^ for outer product). I guess you can also think of the first two terms as the rule for taking the inner product of a vector and a wedge product, though I haven't thought through this completely.

Because of the repeated vector in the reflection formula, that last trivector vanishes, so the result is just a vector, as it will be whenever the vector is coplanar with the bivector.

I was hoping to read some code analysis that supported your claim of off-by-one being more common in 0-based languages, but all I see in your [3] is a few different speculative reasons - am I missing something?

Off-by-one can show up in a bunch of ways that are agnostic to the underlying indexing, so it isn't obvious to me that there would be a substantial difference at all. Also, given what's said below about the ease of writing code which fails to consider custom indices, I'm not sure I believe just switching to 0-based or anything else in Julia is as painless as you suggested earlier - it could introduce its own off-by-one errors!

The algorithm won't assign them that way in that case. A more thorough explanation is elsewhere in the comments, but essentially if Alice got placed at B despite preferring A, they had to have gotten bumped from A, then bumped someone (or found an open space) at B. But Bob can't have been the person that bumped them: if there was an open space at B, Bob would never even have considered A; if Alice bumped someone at school B, then in your situation it must have been Bob, and so for Bob to end up at A when Alice couldn't there had to be someone else with priority between Alice and Bob for A, exactly the person getting screwed over by the swap.

The article seems deliberately deceptive, entirely ignoring the second half of what it claims net neutrality means except to criticize Google for not treating all the bits it sends the same, and then encouraging us to think net neutrality creates a "bevy of regulations" for Google's competitors. What new regulations apply to non ISPs? How does net neutrality change that other companies have to make deals with service providers?

There's no argument here, just a complaint that Google is huge which, while probably valid, seems irrelevant.

Debugging tools in n64 emulators (such as nemu64) are sufficient to make tracking down particular game logic more in the neighborhood of a fun puzzle than a slog through (roughly) a megabyte of assembly. Super Mario 64 in particular has had an active ROM hacking community for over a decade, so there is a large body of knowledge about the game's internals, though it resembles folklore more than a systematic approach. The hard parts to discover through disassembly are the 3d graphics api and memory-mapped IO, but that was already worked out quite a while ago for emulators (based largely on patent documents).

In fact, there are level editors for SM64! Kaze, the creator of this hack, occasionally streams development on twitch; it's a lot of cross-referencing different pieces of accumulated documentation.

Edit: An example of some of the documentation, a list of ROM addresses of various textures: http://wiki.origami64.net/super_mario_64/textures

Galois generators of the sort in the article can be described naturally using the finite field Z_2[x]/p(x), that is, polynomials with coefficients taken mod two, where we consider polynomial p(x) is equivalent to zero. If p has degree n, this field has 2^n elements - the 2^n polynomials with degree less than n are distinct, but x^n is equivalent to x^n-p(x), which has degree less than n, so everything with higher degree is already accounted for. This is, however, only actually a field if p is irreducible.

Now we can describe the generator as multiplication by x, where the leftmost bit is the lowest power, since every bit moves right except for the rightmost, which corresponds to turning x^n into x^n-p(x) as above. The cycle length is the smallest k such that x^k is equivalent to 1. Now, an interesting property of finite fields is that there is always a "primitive" element, the powers of which generate every other element (you can prove this inductively by counting elements z such that z^d = 1 for different d, and noting that z^d-1, as a degree d polynomial, has at most d roots in the field). If x is primitive, it will cycle through all other values before reaching 1. In the specific case of n=17, however every element of the field is primitive (this is easy group theory, the nonzero elements form a group under multiplication, and that group has a prime number of elements).

This means any degree-17 polynomial which is irreducible in Z_2[x] will give rise to a full-cycle-length generator. Luckily, finding an irreducible polynomial isn't too hard - of the 2^16 options (ignoring the ones without a constant term, which are obviously divisible by x), 7710 are irreducible by my quick Mathematica computation.

To add to this, the "walking in place" animation happens when Mario's position + velocity would end up out of bounds, which the game doesn't generally permit. When the player turns and allows the speed to drop, the motion will take Mario to a valid position, and he will zoom until he gets stuck again.

I'm having trouble with your "therefore" conclusion. The projection of the perimeter of the inner box could easily be shorter than the sum of the lengths of the projections of each edge. It seems like you need a less obvious (to me, anyway) statement about the projection of a collection of lines.

The Museum of Math in NYC has a wall with several intricate examples.

http://momath.org/wp-content/gallery/main-gallery/img_2757.j...