In this case the human written statement of the theorem is small. Can there be bugs? Absolutely! But it's essentially a few lines of code worth of thinking.
The lean proof checker then checks to make sure the proof actually proves the statement.
In this case an AI is generating the proof, but if it "compiles" it's correct. The only thing humans need to check is the statement to be proven.
(I don't know anything about this project but I've played around with lean and used other proof checkers more sesrisously).
This is also an issue with non-AI and non-Lean proofs. Andrew Wiles' initial Fermat's Last Theorem proof initially had an error in it. That was spotted by peer review, fixed, and an updated proof was submitted.
Fun fact: cosine similarity's first use in recommendation systems to recommend usenet groups.
(https://dl.acm.org/doi/epdf/10.1145/192844.192905 although they don't call it cosine similarity; they do compute a "correlation coefficient" between two people by adding together the products of scores each gave to a post)
The Pearson correlation coefficient is covariance normalised to the range [-1, 1] by dividing with the standard deviations (https://en.wikipedia.org/wiki/Pearson_correlation_coefficien...). So not quite same as the normalised scalar product, even though the formulas look related.
Pearson correlation = cosine of the angle between centered random variables. Finite-variance centered random variables form a Hilbert space so it’s not a coincedence. Standard deviation is the length of the random variable as a vector in that space.
That makes sense; I don't actually know much about this.
That being said, weirdly, the normalization by standard deviation happens outside the call to `cov` in the paper (page 181, column 1, equations (unnumbered) 1 and 2). And in equation 2 they've expanded `cov` to be the sum of pointwise multiplication of the (scores - average score) people have given to posts.
Again, not my area of expertise, just looking at the math here.
The dot product is computed between two vectors.
For these use cases that dot product is equal to the cosine of the angle between these angles.
(Strictly speaking we have that the angle is actually defined in terms of the dot/inner product in more abstract spaces like function spaces or L^p/l^p)
It's grounded in basic trigonometry, i.e. it calculates the angle `theta` between two entities/vectors, `a` and `b`. If `theta` is close to 180 degrees, cos(theta) is -1, and cosine similarity dictates these are opposite concepts, i.e. unrelated.
Not sure why you're being downvoted. This is exactly what it is. Now mail-in ballots that previously had to be dropped in the box by election day have to be dropped off days in advance.
The postmark isn't the stamp (piece of paper). It's the ink seal that's stamped over the stamp with the date on it. It doesn't matter how you pay for the piece of mail; it now potentially get postmarked at a later date than previously.
Request a Manual Postmark: Customers may present a mail piece at a retail counter and request a "manual (local) postmark". This postmark is applied at the time of acceptance, so the date aligns with the date the USPS took possession.
I should have said manual postmark but it’s what I implied. They’re stamping or postmarking it with a date manually.
Every voting cycle for at least the last 4 you have seen a certain class of citizen not understand why the early numbers come in from small, rural precincts who can finish early and then the densest areas come in later and vote for other people than their guy. I’m sure that this is being fed by people whose livelihood depends on misunderstanding the problem, but it’s something people should be taught in school and the civics curriculum went away sometime around No Child Left Behind. So this has been going on at least since GW Bush, and some of this bullshit tracks all the way back to Reagan.
What's the consideration in the written offer? Promises aren't enforceable in court. For a contract to be enforceable, it has to be an exchange of something, not a one sided offer.
A while back the Met in New York had an exhibit of painted reconstructed statues where they let artists make reasonable guesses about what the statues would have looked like. It was pretty fantastic.
I've found that good tab AI-based tab completion is the sweet spot for me. I am still writing code, but I don't have to type all of it if it's obvious.
This has been my approach, as well. I've got a neovim setup where I can 1) open up a new buffer, ask a question, and then copy/paste from it and 2) prompt the remainder of the line, function, or class. (the latter two are commands I run, rather than keybinds).
Hypocrisy: You're just claiming a different community of native speakers are wrong.
For some of the samples on that site, it'd question whether they even have majority-support as "correct" when brought to people's conscious attention, as opposed to simply being a popular mistake they don't object-to. (Do any polls exist? The nature of the content evades easy search-terms.)
I don't think that's what they meant. It's the case you can use literal strings of bits to encode a (2^n)-tree node, so you use actual bitstring comparisons and operations to manipulate them. Rightshift gives you the parent and things like that.
I don't think this is something the article cares about, though.
The lean proof checker then checks to make sure the proof actually proves the statement.
In this case an AI is generating the proof, but if it "compiles" it's correct. The only thing humans need to check is the statement to be proven.
(I don't know anything about this project but I've played around with lean and used other proof checkers more sesrisously).
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