Posts for Nickolas

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Is there an option to display the forums and/or site with something approximating their old appearance, similar to what Reddit does with old.reddit.com? I vastly prefer the previous appearance and layout.
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FractalFusion wrote:
  • The edge shared by the purple squares perpendicularly bisects a chord of the circle, and is thus a diameter.
  • Draw the perpendicular bisector of the northwest green edge, which is also a chord of the circle. This perpendicular bisector is also a diameter.
  • Note that the green and purple diameters meet at a 45° angle at the center of the circle.
  • There are any number of ways to proceed from here, but you might, for example, imagine translating the top purple square left until its bottom-right corner coincides with the center of the circle.
  • It is then easy to see that Green-East trisects its purple edge, and the desired result quickly follows.
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OmnipotentEntity wrote:
I'm looking into some PPL (probabilistic programming language) stuff, and as part of that I'm trying to come up with approximations of the results of various functions applied to distributions. For instance, let's say I want to apply the sigmoid function 1/(1+e-x) to a normally distributed random variable. We can solve for the cdf of this distribution by feeding the inverse of the sigmoid through the cdf of a normal distribution. From this we wind up with a reasonably complicated distribution on (0, 1). I'd like to estimate this distribution with a beta distribution, but I have no idea how to go about it. Although I have an expression of the pdf of the resulting distribution, I cannot even calculate the expectation value of it. Outside of SVI or Monte Carlo methods is there a way to calculate this?
If I'm not mistaken, what you're describing is a logit-normal distribution, which unfortunately has no closed-form solutions for any of its moments. StackExchange has further practical discussion here: https://math.stackexchange.com/questions/207861/expected-value-of-applying-the-sigmoid-function-to-a-normal-distribution. For arbitrary functions of random variables, the Delta Method can be helpful for estimating moments and describing their asymptotic distributions.
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Tub's suggestion of git is all-encompassing, you can even run git locally if you have the necessary knowledge and experience. There are also lots of open-source standalone diffing and merging tools you could use to make your own general solution. If you're less technically inclined, take a look at WinMerge, from what I can tell it has a GUI that lets you diff entire directories recursively, make patches, and apply patches through the use of an extension. (Read the documentation for more information.)
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Warp wrote:
I must admit that rather than clarifying, this discussion only made me more confused about whether my original assertion is "correct" or not.
It was correct as far as General Relativity is concerned. Free-falling objects are not accelerating, they are following a geodesic in a curved spacetime. On the other hand, anything coupling with a Standard Model gauge field is accelerating. (That is, anything being acted upon by electromagnetism, the weak interaction, or the strong interaction.)
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Tub wrote:
On the other hand, worrying about 8 GB of disk space? What year is this?
2019
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OmnipotentEntity wrote:
(In actual reality, the force you feel due to acceleration is parallel to the direction of acceleration, whereas the force due to a massive body is radial to the body, so you will experience tidal forces which can be detected (if the body is under you, then your feet experience slightly lower acceleration than your head, so you are stretched by this imbalance of forces, and the force on your left arm is down and to the right, whereas the force on your right arm is down and to the left, squeezing you slightly left to right.) However, if you were to construct an infinitely large flat massive surface then gravity and acceleration would be indistinguishable, even in principal.)
If I'm not mistaken, Lorentz contraction means there's an acceleration gradient along the length of an accelerating object. (See this blurb on Rindler Observers.) That is, both gravitational and non-gravitational acceleration result in tidal forces, making them indistinguishable in this regard.
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Warp wrote:
"The second most irrational number" would refer to the "most irrational" number that's not equal to those. I suppose one could come up with infinite patterns of 1's and 2's in the continued fraction representation, a simple example would be alternating between them. I suppose it could be argued to be "more irrational" than sqrt(2), because the continued fraction approaches that value slower than the one for sqrt(2). Out of curiosity I tried to solve what 1+1/(2+1/(1+1/(2+... equals to, and I got (1±sqrt(3))/2.
Not quite, what's being said is that concept of a second most irrational number (or family of numbers, if you prefer) isn't well-defined. There's no such number in the same way that there isn't a smallest positive real number. Root 2 is [2, 2, 2, ...]. What's a more irrational number by the criterion you proposed at the start of this discussion? Well, there's an infinite amount such numbers, but for example, [1, 2, 2, 2, ...] is more irrational. Is that the second most irrational number, then? Well, no, [1, 1, 2, 2, 2, ...] is more irrational, etc. This sort of reasoning also suffices if you wish to only consider equivalence classes of numbers that have the same convergence characteristics in their continued fraction representation, as p4wn3r mentioned. Is [2, 2, 2, ...] the second most irrational class? No, [(1, 2), (1, 2), (1, 2), (...)] is more irrational. Is that the second most irrational class? Well, no, [(1, 1, 2), (1, 1, 2), (1, 1, 2), (...)] is more irrational. Is that the second most irrational class? Well, no, double the number of 1s between each 2 again, etc. You could likely extend the idea of continued fractions with something similar to a surreal infinitesimal to obtain, by definition, a number less rational than any non-phi number. I suppose that could reasonably be referred to as the second most irrational number, though it wouldn't actually represent a number any more than surreal ε does. (But what is a number anyway?) As to whether that would lead to interesting or useful mathematics ... I have my doubts. Assuming the "surreal continued fractions" can be well-defined, their structure is probably equivalent to something more natural.
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Warp wrote:
On that note, both argue that phi, ie. the golden ratio, is "the most irrational number". The latter video tangentially, and perhaps serendipitously, also gives an argument for what could be considered "the second most irrational number", which would be sqrt(2). (It's all about their continued fractions, and how fast they approximate the actual value.) Indeed, it appears that if you need a "very irrational number" (that has the properties that their continued fractions have), sqrt(2) seems like an easy and good choice.
The whole parts of the denominators in phi's continued fraction are [1, 1, 1, 1, ...], which is the minimal such sequence, so it is indeed the most irrational number in that sense. While root 2 is also fairly irrational in that sense, it's not "the second most irrational number", nor is that concept well-defined. Consider that for any number but phi, you can obtain a more irrational number by, say, prepending a 1 to the list of denominators in its continued fraction representation.
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Neat concept! I'd like this to be (at least) vaulted so I can easily find it when I'm searching for runs of Aladdin.
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Warp wrote:
I saw this on a t-shirt: Is that really so?
Yes, but it looks quite peculiar. It's off in a way that makes you think it was composed by someone who haphazardly combined pieces but didn't understand what they were looking at, and they were lucky it ended up working. Like, why isn't 3π/9 reduced to π/3? That's elementary school arithmetic, it's just strange and silly when you're not doing something esoteric like trying to use every digit. Why is the cosine positioned so ambiguously? It looks weird and makes you wonder if it's intended to be part of the integral or not. Yeah, it's to the right of the dz, so in the most formal/technical sense it's not part of the integral, but any sane person would move the cosine in front of the integral, or otherwise visually separate them somehow. (It doesn't actually matter either way in this case, but it looks weird.) It would also be completely standard to omit all of the parentheses as shown in the original equation, and failing to do so kind of makes the whole thing resemble typing in ASCII or something, even though it's clearly typeset. Lastly, there should be space between z^2 and dz. A lot of this stuff might seem nitpicky, but math is in many ways the art of communicating consistently and unambiguously, which the shirt you sighted fails to do. Someone that knew what they were doing would probably write the equation like this, perhaps with a different choice of multiplication symbol:
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  • I think this TAS competently answers interesting questions and should be published.
  • I think the PAL port is noticeably deficient in a number of ways that many people will find off-putting (music wrong tempo, sound effects sound wrong, lakitu behavior looks very strange), so it doesn't make sense for a PAL run to obsolete the flagship NTSC run.
  • I think that for a game of this stature, it's reasonable to consider how best to serve the "average fan". When an average fan searches for an SMB TAS on this site, what do we expect them to want to see? I bet it's going to be the best NTSC any% TAS in the overwhelming majority of cases. Obsoleting the NTSC run -- in effect, hiding it -- would defy expectations and make the site more confusing and less user friendly.
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OmnipotentEntity wrote:
According to OEIS, the asymptotic behavior of ak(n) seems to be nk/kk. I do not know why this would be the case.
I happen to be the author of those OEIS entries. I mainly didn't link them in the OP because I didn't want to predispose anyone to thinking that information was gospel. I never had anything other than Bobo's statistical argument for the asymptotic behavior of ak(n).
It should be noted that the behavior of a5(n) seems to exhibit non-monotonicity quite a bit.
As k gets larger, it takes longer and longer for the function to settle down. It also takes longer for the function to "get going" and start being interesting, so really there are two thresholds worth considering: not just the last index at which ak(n+1) < ak(n), but also the first! So far I've had no success in elucidating the boundaries of the "interesting region". I'm really intrigued by Bobo's idea of the point of maximal interestingness, though, and your animated graphs are really cool. (Hmm, are there an n and k that have the most interesting "most interesting point"?) Another thing I tried to prove was that if you observed monotonicity for "long enough" (once you were within the interesting region, of course), you could be assured the function was fully monotonic at that point, but I didn't really get beyond a statistical argument with that, either. (That was partially motivated by trying to make calculating A129651 easier.)
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Consider a partitioning function Pk(n), which lists all the ways you can write a positive integer n as the sum of k other positive integers, where order doesn't matter. For example:   P2(5) = { (1,4) , (2,3) }   P3(5) = { (1,1,3) , (1,2,2) }   P2(6) = { (1,5) , (2,4) , (3,3) } For any given list, you can find the LCM of each individual partition, for example:   LCM(1,4) = 4   LCM(2,3) = 6 Those are all of the 2-partitions of 5, and the biggest LCM is 6. That is,   Max( LCM( P2(5) ) ) = 6 For the sake of brevity, let's condense all of that into a single function:   ak(n) = Max( LCM( Pk(n) ) ) In general, you would expect ak(n) to get bigger as n gets bigger, and much faster than the lower bound of n-k+1 guaranteed by the partition (1, 1, ..., n-k+1). The bigger n is, the more ways there are to write it as the sum of smaller positive integers, and thus the more "chances" there are to have a big LCM. But it's not always the case that ak(n) grows. For example:   a2(5) = 6 ≥ 5 = a2(6)   a3(10) = 30 ≥ 21 = a3(11) So my questions are twofold:
  • What is the asymptotic behavior of ak(n)? Is there a nice closed form expression?
  • How often is it true that ak(n) ≥ ak(n+1)? If it only happens finitely many times, when is the last one?
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Unknown394 wrote:
OK, so I practiced the wall jump thing and sort of got the hang of it (that means failing nine times and getting it once), but the flagpole glitch is absolutely impossible. It requires a very exact landing spot, Xspeed and Yspeed, and also requires different controls on each frame (usually left+right, empty, right, Left+right+A, empty, right) which is impossible to be done in a real time speedrun.
Not only are you obviously and demonstrably wrong (watch the top SMB1 runners on twitch, the best have about a 33% success rate on the flagpole glitch), but your attitude is both harmful to you as a person and, more importantly, harmful to society. The inability of most people to understand — or even give the slightest consideration to! — the fact that there's an astonishingly vast world outside of their own limited experience is why the world sucks. The incurious masses seek refuge in echo chambers, desperate to validate their beliefs. Don't be one of them.
There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy.
Anyway, you don't understand how anyone could perform sequences of frame perfect inputs with any reasonable degree of accuracy. My first suggestion is that you should come to terms with your own lack of expertise, and accept that you're poorly suited to judge the fundamental limits of elite humans. No doubt you believe you're an elite gamer, but consider the evidence: lots of people can consistently do things you consider to be impossible. If you were actually an expert, shouldn't you be able to do the same things that other experts can do? How can you become an expert? The same way you get to Carnegie Hall: practice, practice, practice. That doesn't mean that you screw around until you manage to do something once, and then declare it random and impossible. That means you keep trying until you succeed 500, 1000, or 10000 times. Study how other people do what they do, and systematically explore a multitude of approaches toward replicating or improving upon their success. Then and only then might you be sufficiently experienced to form a reasonable estimate as to humanity's practical limits in that narrow endeavor. With respect to your specific example, you might try to identify visual and audio cues to enhance your accuracy. You might try practicing in slow motion, steadily increasing your speed over an extended period of time as your success rate increases. You might try practicing with a metronome, or a LUA script that scrolls the correct inputs down the screen DDR style. You might experiment with different controller grips, or different controllers entirely. You might do exercises to increase the speed and strength of the flexors and extensors in your hands. And if you do all of those things in good faith and still suck after tons and tons of practice? Well, not everyone can be Roger Federer -- but it would be extremely foolish to claim that what he does is impossible.
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It's pretty unlikely there's anything unusual about the water in the pool you were swimming in. All of the symptoms you describe are adequately explained by having insufficient cardiovascular fitness and exercising at altitude for the first time. I would venture a guess that even if you look fit and regularly work out, you don't often stress your heart and lungs. Maybe you run a couple miles a few times a week at a moderate pace, but you don't push yourself to improve your times, and you don't do sprints. Swimming is also particularly prone to under-performance at altitude because you can't just "run slower and breathe harder". You have to hold your breath while exerting and slowing down your rhythm actually slows down the pace at which you can breathe. You say you averaged 7 minutes per lap in an Olympic pool, so your speed was 0.53mph (0.86kph), assuming a 50m pool length, or half that speed again assuming a 25m pool length. (It's common for "short course" lap pools to be mislabeled as "Olympic".) That's a fairly slow pace. Your average, moderately fit, middle-aged swimmer can maintain 1.5-2mph (2.4-3.2kph) for extended periods of time without difficulty. Someone like Katie Ledecky can sustain a little over 3mph (4.8kph) essentially indefinitely, while the best males are a tad faster still.
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A long time ago, back when the speedrun mode of Within a Deep Forest was still enabled, a number of you did speedruns. If any videos of those runs exist, there are some as-yet empty run categories on speedrun.com/wadf just waiting to be filled in:
  • Glitchless
  • Any% Hard
  • 100% Hard
Additionally, the Glitchless Hard run is over 10 years old and pretty beatable!
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Warp wrote:
In this Numberphile video Matt Parker proves that the ratio between consecutive Fibonacci numbers approaches the golden ratio. He does this by noting that the definition of the Fibonacci numbers is that each number is the sum of the two previous numbers, and then assuming that as we go along the sequence, the ratio between two consecutive pairs of numbers approaches the same value, ie, that Xn/Xn-1 = Xn-1/Xn-2, and then solves what that ratio must be so that the equality holds (which, of course, is the golden ratio). But isn't he just assuming that the ratios of consecutive Fibonacci numbers converges to a particular value? He is also assuming that the ratios of two consecutive pairs approach each other as we advance in the sequence. On which basis can these assumptions be made? Couldn't it be perfectly possible that the ratios diverge, or oscillate, as we go along the series? Can the convergence simply be assumed? Shouldn't it be proved?
No, not really. You may as well ask, "Shouldn't he start with the Peano postulates and go from there? Isn't he just assuming you can add and multiply numbers? Shouldn't it be proved?" As in everything, context is important. The context of the video is one in which you only need algebra to derive interesting and useful math, such as the relationship between Fibonacci-like sequences and the golden ratio. Formally defining the idea of series convergence and then formally proving that the ratio of successive terms converges is beyond the scope of the video. (And while it is technically possible to use nothing but algebra to show that the square of the difference of successive ratios is always decreasing -- which algebraically hints at the idea that the ratio of successive terms is approaching a fixed value -- why bother? It's a 5-minute algebra video, not a textbook.) Furthermore, the way a lot of real math is done by real mathematicians is to just sort of assume something that seems like it might be interesting or true, and to see where it leads. Afterwards you can always go back and build a framework that formally justifies your assumptions. Or not, maybe you were wrong. Furtherfurthermore, one could also argue that the point of videos like this is to inspire curiosity. If the video makes you asks questions, then perhaps you should try to answer them. (Mission accomplished!)
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Cool! I picked up SMM a few days ago and I'm having a hell of a time. Here's a level I've been iterating on: Weird Science (v4) - 23A5-0000-017A-F486. It's one of those one-screen puzzles, and the main concept behind it is that it's literally impossible* to beat. No, not Panga-hard, actually just impossible. It also has a cool, optional, automated sub-level that's fun to watch at least once. I figure if I'm going to make an impossible level, I might as well put on a show! You can additionally check out previous versions of the level in my profile, but they aren't nearly as polished, and also can be cheesed to varying extents in order to clear them. * 99.999999999999999999999999875664384% guaranteed! (Not a joke.)
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jlun2 wrote:
Wouldn't the any% be more like this this though: [1205] NES The Legend of Zelda "2nd quest" by Baxter & Morrison in 24:59.83 Which may make it somewhat redundant?
I must admit some confusion. I watched RAT926's glitch route in 05:47 video and it's qualitatively very different than the existing any% video. I think there's value in having both. Where do you perceive redundancy?
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jlun2 wrote:
That got me thinking. Since there's already a 2nd quest category, a swordless and also a 100%ish, would it be fair that a warped route obsolete the any%? Mind you, the any% route also has the "heavy glitch abuse" tag too.
I think there's an interesting difference between this new warp route and the existing any% route, which more or less resembles the way humans play the game, only with inhuman precision and luck. I like the idea of having both runs listed. Just my 2¢.
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scrimpeh wrote:
Can someone recognize where the music in the first 20 seconds of this VOD from twitch is from? I swear for my life that I've heard it before, but I can't find it anywhere. Any help would be greatly appreciated.
Just ask the streamer. Send him a Twitch PM or ask in chat while he's streaming.
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Warp wrote:
The only thing that article tells me is that "yes, using the term in this context is a misnomer".
Then the only question you asked was adequately answered, I believe.
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Warp wrote:
It's common to use the term "non-euclidean geometry" in video games where the level geometry does not follow logical physics. For example you are seemingly inside a close room with a column at its center, but if you walk around the column you end up in a completely different place (typically there's some kind of "portal" surface at one side of the column that connects the level to another completely different level.) For example the game Antichamber is based on this. Is "non-euclidean geometry" a misnomer in this case? It doesn't sound like a case of hyperbolic or elliptic geometry.
http://en.wikipedia.org/wiki/Non-Euclidean_geometry