The derivative of "(1/2)*x*sqrt(4x2 + 1) + (1/4)*sinh-1(2x)" is "sqrt(4x2 + 1)".
The derivative is repeated exactly in the original function. Does that mean you could replace it with anything in the original function, and get it as the derivative of that function, or does it have to be precisely "sqrt(4x2 + 1)" for it to work like that?
In other words is the derivative of "(1/2)*x*f(x) + (1/4)*sinh-1(2x)" "f(x)"?
I only need the values in the range x=[-1, 1] (which of course means that y is in the range [0, 1]). Since the curve is symmetric, the range x=[0, 1] would be enough, of course, but since the polynomial is also symmetric I think it doesn't make much of a difference.
An approximation would be enough for the purpose. (Although in this particular case the approximation probably needs to be accurate to a quite numerous amount of decimal places. At least 10-12 decimals. Not that it's absolutely critical, but would be good.)
It's actually for a hobby project of mine. I'm trying to move along an x2 curve at constant speed (or, more precisely, I need points along the curve that are at equal distances from each other measured by the curve's length), but for the life of me I can't figure out a function for this.
I suppose the only way is to approximate by recursively subdividing and measuring distances along straight lines. That's a bit surprising. One wouldn't think of x2 to be that complicated.
I suppose that any sequence can be made "slower" by making a variation of it with more terms (such as, for example, and I suppose, in the Leibniz formula rather than using odd numbers directly, make each term a sequence that converges to said odd number, or something), but I suppose that the spirit of my question was not really "can you take an existing formula and create a variant of it that has more terms". Rather, if there's a completely different unrelated formula that's slower. (Although, I suppose, "unrelated" may be too fuzzy of a requirement because many sequences approaching a given value can be converted into each other by applying some transformations.)
It's hard to make abstract ideas unambiguous, when dealing with mathematics.
The Leibniz formula for pi is
4 * (1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + 1/13 + ...) = pi
One curious thing about this series is how incredibly slowly it approaches pi. For example, to get the 10 first decimal places of pi correct, you need about five billion terms.
This is in contrast with other methods, such as the continued fraction for pi, which approaches pi very rapidly. You only need a few terms to get a substantial amount of correct decimals.
I was wondering if there's a series that approaches pi even slower than the Leibniz formula.
The three-day minimum time before accepting a submission for publication was introduced so that the public would have time to raise any objections that may affect whether the run does qualify for publication or not (and also to an extent whether it should be published to moons or vault).
(The suggestion of this minimum period was put forward when some judges were accepting some submissions for publication within hours of them being submitted, which was quite controversial in at least one case, as there was possibly bias. But that's in the past now.)
Given that there exist some minimum requirements for publication (eg. no cheating, no abusing emulator bugs that do not exist in the original hardware, etc.) I don't think it would be a good idea to just automatically accept any submission that anybody sends, no matter which "tier" they would be put into. Those requirements need to be checked (and the more people check them, the better. After all, sometimes the problematic cases can be hard to spot.)
I was curious about what the derivative of exp(exp(x)) is (where exp(x) is ex). It has been so long since I last solved any derivatives that I'm really, really rusty, but I came up with the answer exp(exp(x)+x).
In order to somewhat verify that the answer is correct, I calculated what the tangent line of exp(exp(x)) at x=0 would be assuming that solution is correct, and I used a graphing calculator to draw it. The tangent line would be exp(exp(0))+exp(exp(0)+0) = e+x*e. And it indeed results in a line that's tangential to the exp(exp(x)) curve at x=0.
Out of curiosity I also graphed the curve exp(x) and its tangent at x=0, which would be exp(0)+x*exp(0), ie. 1+x. The result looked correct.
I noticed that both tangent lines, ie. e+x*e and 1+x, intersect at x=-1.
This of course brings up the conjecture that the same is true for all curves of the form exp(exp(...(exp(x))...)). However, this hypothesis is incorrect because the tangent for exp(exp(exp(x))) does not intersect the x axis at x=-1.
So I was wondering what exactly is the pattern? If the tangent lines at x=0 of both exp(x) and exp(exp(x)) intersect at x=-1, what other curves of that form have the same property? What's the pattern? Or is that just a huge coincidence that's true only for those two curves?
Set dy/dx = 0: 1-lnx = 0, x = e
Therefore y = e1/e
Surprise e!
Is there a reason why precisely e appears in this case? I mean, is there a simple correlation with the definition of e, and the curve x1/x?
(I find it curious how e was first discovered when mathematicians wondered what happens when you calculate compound interest continuously over a period of time. Rather than resulting in infinite interest, as one might hastily think, the maximum amount of gains is finite. And e was the factor that popped up as the limit.
Well, that's nice. Just a curiosity? One number among others, that just happens to be the answer to this very particular question? Except that e turned out to be one of the most important and ubiquitous numbers in mathematics, as it tends to pop up very often in seemingly unrelated problems.)
Final Fantasy XV...
It looks good, and the setting is somewhat interesting. It's mostly a wide open sandbox, which is great.
I'm not very fond of the combat system. At first it's confusing and hard to really understand. But of course, as usual, when you play it enough, you start getting it, and getting more fluent at it. However, even after that I wasn't very fond of it. Dare I say that I even liked the combat system of FF XIII more (not that it was the best I have ever played, but I think it was better than the one in XV.)
(I really, really, really detest the idea that Square Enix is going to use the same (or a very similar) fighting mechanic in their upcoming remake of FF7. It's going to ruin the game. F*** them. Honestly.)
While the open world mechanic is quite nice and enjoyable, by the about 20 or 30 hours of gameplay it starts wearing out and feel repetitive (don't get me wrong, something like 30 hours of somewhat interesting gameplay is nothing to scoff at; it's just that I have played wide open sandbox games that have lasted a lot longer than that before getting boring). At some point it mostly consists of repetitive fetch quests, in addition to the main quests.
The story is a bit meh, in my opinion.
By the end parts of the game it becomes much more linear. (For example there's a really infamous chapter or two that everybody hates for being so linear, boring and devoid of anything interesting, and I can't blame them.) Somehow, for some reason, the difficulty also spikes enormously.
So much so, in fact, that I stopped playing the game.
Pinkie Pie is the kind of character that could very easily be highly annoying, but for the most part the writers have successfully avoided it. However, in the fourth episode of this season she was really annoying (to the viewers, or at least to me, in addition to being so in-universe). At least in the first half of the episode.
But cool thing that we learn Maud's full name.
(Btw, wouldn't she be a petrologist?)
Do you know what I hate?
- People having a quasi-religious attitude towards VR.
- Moderators in some online forums being complete a-holes (tasvideos is fortunately not one of those forums.)
- Some developer gaining too much influence in a collaborative open source project, so much so that he essentially has sole ownership of it, and absolutely nothing is implemented without his personal approval, and this goes to his head, and that, alongside some weird and stubborn principles that are detrimental to the quality of the project, causes it to go to s**t.
- The increasing attacks on free speech by western governments and many organizations.
- The old media trying to attack YouTube.
- The economic model of mobile games (ie. "free" to play). The fact that good old-fashioned fair way of selling products doesn't sell on those platforms.
- The fact that Square Enix hates the old-style (ie. pre-10) Final Fantasy game mechanics and has been avoiding it like the plague.
- Daylight saving time.
- The fact that console games do not support mouse&keyboard, even though the console itself does.
- The fact that you have to pay a yearly fee on consoles in order to play games online.
- The fact that most people don't know the difference between Godwin's Law and Reductio ad Hitlerum.
- Half-Life 2 speedruns.
- The de facto monopoly of Steam.
- Nvidia's greed with gsync.
Linux tends to indeed suffer significantly less of growing larger and larger over time, although, possibly, also Windows has become better on that front as of late.
Back when I had a Windows XP machine, with also Linux installed in it, I couldn't help but notice how the free space in the Windows partition was getting smaller and smaller over the months and years, even though I didn't keep adding anything to it (I just used it for games, and I uninstall all games once I'm done with them). Even after a lengthy cleanup process, the free space never got back to what it was at the very beginning. Windows just grew over time. On the other hand, Linux simply doesn't do that (at most, depending on the distro, it might keep some older versions of the kernel when it upgrades it, but usually it will do so only for a couple of them and clean up the older ones.)
However, Windows might have got a bit better on that front. I don't remember the problem being so bad with Windows7, nor Windows10. There might still be some of it, but Windows10 has kept itself quite nicely under that 80G limit that it has been since the it first installed itself here. I suppose it has managed to not grow.
On the other hand, Windows is probably quite a memory hog (once again something I don't think Linux is so bad at, although probably depends a bit on which window manager you are using, but probably not even then). Win10 might hog more RAM than Win7 and especially WinXP did. Windows is definitely not notorious for optimizing memory usage, and will quite happily hog hundreds of megabytes of it for the absolutely silliest of purposes. (Although, to be fair, Firefox seems to have taken a page from Microsoft's book, because it's absolutely horrendous in terms of RAM usage. I only have this page open at this moment, and it's taking a whopping 452 megabytes.)
It's also a fantastic motivator for shaking up the status quo.
I suppose that kudos to the show for daring to change the status quo from time to time, rather than hitting the reset button at the end of every episode, like most other shows do. But in some cases such a change is not necessarily for the better.
Btw, I liked the second episode more than some others seemingly have. Not that it was one of the best episodes, but not completely bad either. Didn't like much the song part (not that the song itself was bad; it just came completely out of the blue and didn't fit at all). Trixie felt a bit shallow of a character (not as in a shallow personality, but as shallow scriptwriting. It lacked proper depth.)
Money is a fantastic motivator for retconning things in a merchandise driven franchise. It's a little surprising anyone expected "artistic integrity" from this show.
I suppose. But did they really have to destroy Twilight's cozy tree library in order to sell more "Princess of Friendship" castles?
(I haven't actually checked if there are such toys. I'm assuming there are.)
Although that's not really a retcon per se.
First there were just two alicorns. Then, out of nowhere, there was a third one, in another kingdom. Hmm, ok, I suppose. Then they kind of threw everything implied out of the window and just made the main protagonist convert into an alicorn, out of the blue, because reasons. Then they once again throw everything out of the window by having that third alicorn have a baby alicorn; because they could, I suppose. And now they are hinting that Starlight Glimmer might turn into an alicorn by the end of the season, maybe.
At this pace I wouldn't be surprised if they discover a colony of thousands of alicorns, all of whom are "princesses". Because why not. They make no sense anyway.
Everything else I would forgive (alicorn is just a fourth, much rarer kind of pony), but I still just don't buy Twilight becoming one. Especially not in that manner. She is not an alicorn. She's a unicorn. Those wings don't belong. They are just a bad hallucination created by Discord, or something.
I finally got around to watching the first episode of the season.
It felt really weird. During most of the episode, especially the first 5 minutes, it almost felt like I was watching some other series. It somehow felt so different. The characters were the same, their voices were the same, and they were referring to the last finale... but it still felt like a completely different show. It's hard to describe why.
There were also so many what felt like out-of-character moments. Perhaps Celestia being the most prominent example. And much of the writing felt really lazy (like Discord's role and lines in the whole episode).
Even the kind of build up that the episode was going towards (Starlight going away) didn't actually realize itself. Nothing actually changed.
This felt like a second episode of a season, not a season opener.
And I swear that the second that Starlight turns into a "princess", I think I'm done with the series. I'm glad they didn't go that route in the last season finale. I hope they don't go that route this season either.
I haven't made a guitar video in a long, long time, but finally I borrowed a camera from a friend, so I decided to make a couple new ones.
Link to videoLink to video
I think that what the original poster meant was that in both unassisted and assisted playing the goal is to complete the game as fast as is possible, using realistic expectations of what is possible unassisted or tool-assisted. The goal of a TAS is not to beat the unassisted run (in which case it would certainly be much easier), but to beat the game as fast as possible using all the tricks available, which can indeed make creating the run much more difficult, as many of those tricks are not even expected to be used in an unassisted run (eg. luck manipulation giving you 100% success rate).
I was one day watching Torje's stream on twitch, where he was speedrunning LoZ:OoT, and the conversation veered towards the OoT TAS and why nobody seems willing to do one, using modern routes and tricks. He mentioned something that might be the perfect example of what the original post (and the thread title) is asking.
Not that OoT any% is easy to speedrun, of course (getting even close to the time of the current any% record requires an enormous amount of practice and work), but it's of course perfectly possible. According to him, however, an optimal TAS would require such an astonishing amount of work to get certain tricks working that it's unyielding.
He specifically mentioned the floor clip trick inside the Deku tree. It's possible to do even in real time, as he demonstrated in that very stream, but requires a very specific setup that requires quite some time. A TAS would obviously need to do the trick without any setup at all, instead going right to it. And, apparently, it requires such an accurate positioning / running direction / timing or something, that it would be an almost insurmountable amount of work, even with TAS tools.
(I'm not sure I fully understood the reason, but he knows infinitely more about the internals of the game than I do, so I'm not questioning his expertise and knowledge.)
In which case we have the extra constraints that 1 ≤ a,c < 10 and that b and d are integers
Obviously they can't be integers because else you wouldn't be able to interpolate between them. (Well, I suppose you could, but you would be rounding it to the nearest whole number, which just wouldn't work very well as an interpolation.)
Thinking about a bit more, I realized that my approach to the problem was a bit off. Well, I was kind of there, but not quite. I realized that what I was really attempting was interpolating in logarithmic coordinates rather than linear ones.
If I have two numbers A and B, and I want to interpolate between them in the manner I described (ie. logarithmically), I should calculate the x and y in A=10x and B=10y, then interpolate between x and y linearly, ie. exp=x*(1-factor)+y*factor, and then number I'm looking for is 10exp.
By definition x=log10(A), and y=log10(B).
So, I suppose, the answer to my question is:
result = 10log10(A)*(1-factor)+log10(B)*factor
This ought to give me the answer to my original question.
