kmill

38 ms·

301.
by kmill 7y ago
I got an iPad just to use LiquidText, back when I had to study for my PhD qualifying exam a few years ago. It has so many great UI affordances -- the way it shows search results, highlighted text, connections between notes and the text, the
302.
by kmill 7y ago
The Dedekind cut formulation of the real numbers is that a real number is defined to be a nonempty "downwards closed" subset of rational numbers. Then the min operation corresponds to taking intersections, and taking intersections
303.
by kmill 7y ago
The array is length N, and it says it contains the integers 1 through N inclusive, as in, all of them. If there were any repeats, then you'd have an array of length greater than N.
304.
by kmill 7y ago
Function composition is a monoid: (f ∘ g) ∘ h = f ∘ (g ∘ h). For example, (sin ∘ cos) ∘ tan = sin ∘ (cos ∘ tan). It's not commutative in general; sin ∘ cos != cos ∘ sin (in fact, sin(cos(x)) != cos(sin(x)) no matter the value of x).
305.
by kmill 7y ago
I thought I'd add: what's the "mono" in "monoid"? It's that the ports come in only one compatibility type. For example, devices that have exactly two ports, one male and one female, for some type of conn
306.
by kmill 7y ago
The word "order" is unfortunately overloaded. The first kind, like what you're talking about, is whether you use the left port or the right port of the first object when joining it to the second object. That certainly can ma
307.
by kmill 7y ago
A monoid is a set of objects each having a port on the left and on the right, and given two such objects you can join the left port of one object to the right port of the other, constructing a new object. There is also an "empty"
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by kmill 7y ago
I know you've been in the music business for a few hundred years, Josquin, but there's no need to be so harsh about this undergrad poster! From what I could figure, it's something like a final project for a class on Mathemat
309.
by kmill 7y ago
It looks like it's done here in the print routine: https://github.com/JeffBezanson/femtolisp/blob/master/print.... It adds 48 to each byte being printed, which is ASCII '0'.
310.
by kmill 7y ago
Though PageRank has the innovation of weighting the connections based on how important the information is. (Just for fun, here's one linear-algebra-free way to understand classic PageRank: imagine starting from a random page on the int
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by kmill 7y ago
Gauss, too, was excited by the way decimals formed cycles as a kid, and he went on to invent modular arithmetic to completely explain the phenomenon!
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Privacy implications of the Apple U1 chip and ultra-wideband(freedom-to-tinker.com)
2 points by kmill 7y ago|0 comments
313.
by kmill 7y ago
I considered mentioning this, but the way it works is editors ask specific scientists to review articles for them (for free). The editor sends the article (the "product") for them to review, in advance of possible publication. Som
314.
by kmill 7y ago
This isn't the best comparison, since journals publish the reviewed articles, not the reviews themselves. It's more like whether Amazon should pay the people who decide what should be in the Amazon Basics line, which I suspect the
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by kmill 7y ago
It lets you bias the maze to make interesting designs at least: https://jsbin.com/betediceyu/edit?html,output (you might have to click "Run with JS" to make it work properly)
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by kmill 7y ago
A similar question is why are some Turing machines universal and some aren't? Or, more generally, why can you prove some Turing machines halt on all inputs and for some you cannot? I think what you mean by "encoding" is what
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by kmill 7y ago
I find computer scientists have a stronger grasp on encodings than mathematicians, who tend to get bogged down in these details due to lack of familiarity. With what I wrote up, the point I failed to make is that it comes down to some clev
318.
by kmill 7y ago
That's the core idea of the proof inasmuch as parsing is the core idea of writing a compiler. Once you get that part out of the way, the incompleteness theorem is a clever diagonalization argument (and is a consequence of the Lawvere
319.
by kmill 7y ago
They're expecting you to know the spectral theorem for Hermitian matrices. Since M_j is obtained by striking out the jth row and jth column, it too is Hermitian, so it has has n-1 real eigenvalues (possibly with multiplicity). The no
320.
by kmill 7y ago
They're sneaking the Euclidean metric tensor in everywhere, which means there's no distinction between upper and lower indices. I agree it's confusing, and I'd rather they kept tensor orientations intact. The dot product
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by kmill 7y ago
I see I was rather oblique, but I brought up the fun fact because that equation is how Penrose proved it (to account for signs). By the way, the TWF article is citing the Bar-Natan paper.
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by kmill 7y ago
The paper doesn't really explain everything you need to be able to read the notation, but, being a tensor diagram aficionado, I have to say I found the way they encoded divergence and curl very exciting: it is all consistent and helpfu
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by kmill 7y ago
The conclusions section has a nice short history of some graphical notations, noting "graphical notations of tensor algebra have a history spanning over a century." Equation 18 in their paper is in Penrose's original paper (&
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by kmill 7y ago
Yeah, the "type of types" is called a kind in Haskell https://wiki.haskell.org/Kind https://en.wikipedia.org/wiki/Kind_(type_theory) Caveat: there seems to be some reason not to think of ki
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by kmill 7y ago
a1369209993 was answering my "I don't think [natural transformations from the identity functor] always exist" with "no, they don't." The natural transformation would be a function "a -> Const q a"
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by kmill 7y ago
Typo: a "closed manifold" is compact without boundary. The Poincaré conjecture states that closed simply connected 3-manifolds are all diffeomorphic to the 3-sphere. Even stronger, every closed 3-manifold whose fundamental gr
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by kmill 7y ago
I put this in another comment, but I think you're thinking 'a' is a category, but it's an object in a category. The type for a functor (in Haskell) is Hask -> Hask, which is represented by the star notation Sniffnoy
328.
by kmill 7y ago
> is the natural transformation from identity functor Sorry I confused you here. I meant, if you had a function of that type (a -> f a), you would want a naturality relation to hold for it, like how 'return' for monads is m
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by kmill 7y ago
Are you familiar with how the word "vector" changes meaning depending on which vector space your talking about? The same goes for "morphism," which depends on the category. In your example of "tmap : a -> f a&qu
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by kmill 7y ago
Functors don't include tmap. Yes, a functor maps objects from one category to objects of the second, but it does so on the sets of objects. There might not be a corresponding morphism between "a" and "f a". The

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