xkcd

Qaanol

Why is there a ghost behind Cueball?

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Thanks for the suggestion! I'm not sure how to get around the fact that x and y are in vector spaces of different dimension, though. I know that invertible transformations do exist, but I couldn't even begin to figure out how to construct T(t) which is invertible for t>0, but approaches something l...

Qaanol

Here’s something to try:

Form a continuous family of transformations T(t) such that T(0) = C, and T is invertible when t≠0. For example, I think T(t) = t·I + (1−t)·C should work. Find the general solution in terms of T when t≠0, then take the limit as t→0.

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SonofRojBlake wrote:Wait... so Cueball has been naked this whole time?

Mind. Blown.

Going back to reread all the previous comics with that in mind.....

All xkcd characters are naked except when specifically shown wearing clothes.

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I found a typo in the Who’s On First section. When it gets to the pitcher and Costello asks “You don’t want to tell me today?” it has Abbott respond “I’m tell you, man.”

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If you beat Advent.exe you get an option to save the image. I'm not sure of any way to link directly to the result, so I'm going to imgur this. The URL from which the image downloaded for me is https://xkcd.com/1975/v6xso1_right_click_save.png . Not sure if that is a temporary address or permanent ...

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83.1 ~ 83.33... = 5/6 Take it up to 4 and see what happens? Guessing what's going on from one data point is going to be pretty impossible. I’ll try that when I have time to update the code. I’m trying to find an analytical solution. With n IID normal variates, their position in ℝ n is spherically-s...

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I had a silly typo in my simulation code.

The *actual* results are consistently close to 83.1% for H = J, and 6.4% for H = L.

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Here’s the procedure: • We choose n numbers from a standard normal distribution, and sort them so x 1 ≤ x 2 ≤ ⋯ ≤ x n . • Then we find the midpoint of each consecutive pair, m i = (x i + x i+1 ) / 2. • These midpoints partition the real line into intervals, one of which, call it J, contains 0. • If ...

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Thanks!

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Here’s the scenario: • You have a book, and some of the words are underlined. • I hand you another book. • You have to figure out if it is possible to start with your book, erase some of the non-underlined words, and end up with my book (or at least, the same words in the same order as my book). The...

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>-) wrote:I'm under the impression if lim n -> inf of f(n) = infinity, then it means for every w, there exists n such that f(n) > w.

Not quite: that limit means for every w, eventually f(x) will never drop below w again. In other words, there exists n such that f(x) > w for *every* x>n.

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>-) wrote:For every real number w, there is an integer n, such that there exists an integer k >= n such that S_k/sqrt(k) > w, with probability 1.

Not quite: limsup means it keeps happening out to infinity. In other words, for *every* integer n there is a k>n with f(k) > w.

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https://imgs.xkcd.com/comics/memorable_quotes.png Mouseover: “"Since there's no ending quote mark, everything after this is part of my quote. —Randall Munroe” • • • Most xkcd comics have all text majuscule, but there is a lowercase letter in this one! Also, I’m probably going to use a few of t...

Qaanol

Okay, try this on for size: Can you prove that, for every positive integer n there exists a positive real number a n such that, if x is within distance a n of 0 then your function’s magnitude is less than the magnitude of x n ? In other words, by choosing x “close enough” to zero, can you show that ...

Qaanol

At x=0, can you prove that your function…
• is continuous?
• is differentiable?
• is twice-differentiable?

What do you imagine a proof that it is infinitely-differentiable would look like?

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This sounds like homework.

Can you tell us what “k-algebra” means?

Can you tell us what “End(V)” means?

Can you tell us what “Hom(V, V)” means?

Can you tell us what “V” means?

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Xeio wrote:Todays Google doodle is nice.

They appear to measure “shortest solution” in terms of fewest instructions, not least movement.

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chrisjwmartin wrote:
Heimhenge wrote:Why stop with averaging just Celsius and Fahrenheit? Throw Rankine and Kelvin into the mix too. Call it RKCF. I leave the formula as an exercise for the reader.

I hope you're not going to be so timid as to use the arithmetic mean for RKCF.

Obviously it should use the arithmetic-geometric mean.

Qaanol

↶ One degree is approximately 1.75 percent In what way? It's not 1.75 percent of a circle, or even of a quarter-arc. (Vaguely related, 1px in CSS is about 1.25 arcminutes - the CSS length units are technically angle units, since they scale by viewing distance to subtend the same fraction of your vi...

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Assuming the sampling rate is significantly higher than the frequencies of the signal, there is indeed an approach that uses a relatively small number of calls to the oracle.

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Is this for work?

If so, how much is a solution worth to your employer?

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CorruptUser wrote:I often go to portland, maine...

Huzzah!

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This is a cut-and-paste repost of a thread on this very forum from 2010: link.

Edit: So is the smoking one (albeit with the typo in the subject fixed)

Edit 2: Pretty sure the product/service one is as well: link

Edit 3: And the “intelligent information” one is from 2009: link

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I’ve been tinkering with the α·max + β·min approximation to the distance formula. As a quick refresher, if you know the x and y distances to some point and want to estimate the straight-line distance, you can take the dot product of ⟨x, y⟩ with an appropriately chosen vector ⟨α, β⟩. Since the dot pr...

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Cauchy wrote:Don't you do that determinant thing that gets you the cross product?

Thank you much!

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Given a basis for a hyperplane in ℝⁿ, what’s the best way to obtain a vector orthogonal to it? In particular, given n independent vectors in ℝⁿ, is there an efficient way to calculate a vector orthogonal to the hyperplane containing all of their differences ( v i − v 0 )? The general problem would b...

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As another minor thing, you can improve speed a little by looking at distance^2 rather than distance. The square root operation is one of the slowest basic operations a computer can do. Things often go quite a bit faster if you check whether distance^2<R^2 rather than doing the square root to calcu...

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I mean, submitting a middling score on someone necessarily means having less impact on that person's chances than giving a 0 or a 5, right? Voters tend to have their preferences decided and want to vote strategically to the advantage of the candidates they prefer and the disadvantage of alternative...

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Cueball knows what’s up, approval voting is *way* better than IRV, and Condorcet methods are generally better as well. There’s a new method being proposed in Oregon called “star voting” (aka. score-runoff) where you rate each candidate on a 0–5 scale (like website reviews) and the 2 highest-scoring ...

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In either order there is a double plosive from the final consonant of the first word and the initial consonant of the second. This makes is a fairly awkward phrase to say out loud, especially when speaking quickly, so perhaps people avoid it for that reason. I’m not convinced this explains it. Afte...

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I was reading about how adjectives in English have a certain order in which they appear (shown in this list ), and it occurred to me that “good big” is a pairing which obeys the ordering rule, yet nonetheless does not naturally appear in English (at least apart from compound-word situations like “bi...

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Depending on exactly what you are doing, and how the data are organized, and how many data points there are, the following approaches may be beneficial: • Split the universe into spatial bins (small boxes, like a 3D checkerboard) by coordinates, and keep track of which atoms are in which bins. This ...

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I’m a fan of Swift nowadays.

Qaanol

Are you also opposed to sort() returning a sorted array? Will you argue that arrays are an unsorted type, and it's valid for sort() to return an unsorted array, and it's the caller's job to handle that? Well if you’re working with an array of IEEE–754 floating-point values, some of which are NaN, a...

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But how can it be that assuming the n = k case proves the n = (k + 1) case? I'm more confused now than before, I'm afraid. Okay, I think we are making progress now. Proof by induction is not magic. The n=k case doesn’t *automatically* prove the n=(k+1) case. There are plenty of statements for which...

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I don't understand how we're not assuming both the k case and the (k + 1) case at the same time - isn't that what all of the algebraic manipulations are about? Can you explain what you're getting at using different words? I'm sorry; I'm just not understanding your answer or your question about (k +...

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I don't understand why we want "(k + 1)(k + 2)/2" on the right side. What am I missing? What does the statement “(1+2+3+…+n) = n(n+1)/2” say when n=(k+1)? I'm not sure what you're asking, my apologies. In particular, I don't know how we're proving the (k + 1) case by assuming both the (k)...

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That's another thing that I don't get - how is it that what I "know about k" is that (1 + 2 + 3 + ... + k) = k(k+1)/2? Am I not just assuming that, as opposed to knowing it? You know that about k because you assume it about k. You are proving something about *every* number k for which (1+...

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That's fair. What I'm having trouble making sense of is how the three layers interact, and why three layers are necessary. I only see 2 layers. We have a program “willHalt” which somebody claims can analyze programs to determine whether they halt with given input data. And we concoct a program to t...

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