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On April 28 2010 00:53 space_yes wrote:
All elementary functions are 'proportional to their derivatives'
This is not true. If it were true, then every elementary function would be a solution to some differential equation df/dx = kf, for k in R, which of course is not true. The exponential functions up to equivalence by scalar multiplication, i.e any e^kx for k in R, are clearly in bijection with R. Moreover, you could have proven this to yourself in two seconds by differentiating any other elementary function, or by reading a definition of proportionality which is obviously something you also don't understand.
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Yeah ignore everything I said about the Taylor series expansions, it adds unnecessary confusion and doesn't support my argument.
For anyone who is following this the Taylor series centered at a = 0 of degree 2 is given by:
f(a) + f'(a)(x - a) <-- notationally a is very common but not necessarily standard here, different texts use different variables
So e^x:
1 + x
and e^-x is:
1 - x
And for degree of 1 you have 1 and -1 respectively.
credit: Cascade
I pulled an all nighter doing homework so there were some errors in my posts.
So everyone is clear before you PM: I have consistently stated the exponential model is unnecessary and my original op was critical of its use. I don't know why everyone is hating on the OP's modeling function be^(ax) so hard. Its not that big of a deal. The OP probably used excel for an exponential fit and obviously didn't derive 5.4/x.
Regardless of how you decide to model gas collection it should generally confirm the underlying behavior of the system assuming your error bounds on your model are reasonable and your step size isn't do big (if its a linearization). Because the period for returning scv is inversely proportional to the number of SCVs an exponential fit is inappropriate given that nothing is being doubled or halved during a fixed interval of time.
After re-reading my comments from last night and considering some user PMs I want to emphasize it wasn't my intent to exclude anyone from the discussion by unnecessarily elevating the discourse so my apologies if you felt that this occurred or that I derailed the thread...I get excited talking about math and I was nerdraging about being called nonsensical (even if someone my points weren't correct or hard to understand).
Additionally, due to a specific TL user request I promise the community I will not post my mathematical analysis while high.
EDIT: for clarity, pm suggestions
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lol, space_yes got trolled.
His math is sound enough despite the dubious utility for analyzing sc2 gas mining. His vicious argument style and monumental nerdrage interferes with helping lesser nerds understand.
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very good read, people have really been going all out on the technical aspect of the game. however i dont think my zerg playstyle would allow for less than 6 drones on 2 gas from the start.
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science. stats. graphs
i find it hard to argue <3
very nice
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interesting to read this once. but i think this wont affect the playstyle anyway. i mean everybody should know the 3(2) > 3(1) thing since you see that the 3rd probe is always idel for a sec.
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On April 28 2010 03:23 QuantumPenguin wrote:Show nested quote +On April 28 2010 00:53 space_yes wrote:
All elementary functions are 'proportional to their derivatives'
This is not true. If it were true, then every elementary function would be a solution to some differential equation df/dx = kf, for k in R, which of course is not true. The exponential functions up to equivalence by scalar multiplication, i.e any e^kx for k in R, are clearly in bijection with R. Moreover, you could have proven this to yourself in two seconds by differentating any other elementary function, or by reading a definition of proportionality which is obviously something you also don't understand.
Yes, proportional is not correct and actually this was not what I was trying to express. See my edit above. PM me if you want to continue this discussion. I worked it out so you know I understand 
f(x) = e^x
f'(x) = e^x
f'(x) = kf -> e^x = ke^x where k = 1
f(x) = sin(x)
f'(x) = cos(x)
f'(x) = kf -> cos(x) = k * sin(x) where there is no constant k that makes this equation true for k in R
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On April 27 2010 18:00 space_yes wrote:Show nested quote +On April 27 2010 17:40 DarkChrono wrote:On April 27 2010 16:24 space_yes wrote:On April 26 2010 21:52 shoop wrote:Hm. I have some objections. On April 26 2010 17:27 Arrian wrote:
Now, a very compelling pattern emerges here, one that looks like an exponential function.
Sorry, but this is nonsense. If the miners would not hold each other up, then the mean time between gas returns would obviously be inversely proportional to the number of miners; to be precise Yes, and that is what the OP 2nd's graph clearly shows. Also see this to convince yourself. I think you may be confused. e^x is given by: As you can see, the exponential function's value is proportional to its previous values. Considering e^-x it is inversely proportional to its previous values. It is this property of prior dependence that makes it particularly good for modeling growth and decay. lol What you've written here is nonsensical. There's no "previous values" to a real function, and the only function that's inversely proportional to it's "previous values" is 1 (given a reasonable definition of what this even means, e.g. f(x) = a/f(x-c), c>0, for all x). If a miner does a trip in 5 seconds, then two miners do two trips in 5 seconds, and k miners do k trips in 5 seconds, so miners do k/5 trips a second. Sticking with k miners, Let's call this rate R. If k miners do R trips a second, then it takes 1/R seconds for a trip to be done. Notice how we took the inverse? This shows that they are inversely proportional. (We've just discovered the obvious concept that period is the inverse of frequency.) The relevant function here was 1/x, not e^-x. I am trying to use easy to understand terms. Something inversely proportional is given by 1/x.The OP function is of the form be^(ax) + c = e^-x = 1/e^x. Look at the definition of the exponential function. You must not understand something. With respect to "previous values" I'm referring to the last value for x i.e. a Maclaurin series polynomial of degree 5 (or whatever we need for that interval of workers and accuracy). The exponential function is used to model over a specific domain so how is that not a real valued function? Here is the Taylor series expansion for e^x with a = 0: Look very hard at that series before you post claiming nonsense and wikipedia exponential function until you understand.
Don't get me wrong, it's great that you're taking an interest in math, but the ability to be honestly critical of your own (and others) arguments is an essential skill. It's ok to be wrong, even though it's the internet.
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On April 28 2010 04:59 space_yes wrote:
Yes, proportional is not correct and actually this was not what I was trying to express.
I guess you are meaning that all elementary functions have derivatives that can be expressed in terms of the original function, which is not strictly true. However there is something similar I have just found:
http://en.wikipedia.org/wiki/Pfaffian_function
Quite cool.
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Slightly off topic, but I don't think it serves much purpose to go into the whole discussion again.
Am I the only one who gets the feeling that the psychology involved in discussing math topics on an open forum is much more interesting than the actual "maths" itself? Reading through the thread just raises so many questions: what on earth drives people to write some of these responses? Why do we get so worked up over issues that are so unimportant in the grand scheme of things? It's also amazing how responses to a single post can vary so much in their assessment. For your enjoyment, I've collected some excerpts - decide for yourself who you agree with.
Brilliant
prOxi.swAMi: SCIENCE. Seriously, very interesting. Kudos for going so in-depth into it.
Clearout: I love it when people does this kind of thorough reasearch, and then writes it up good and befitting standards of a scientific magazine.
v3chr0: Good read very interesting... quite a work of art you have there
mfZOR: Shit mate, Awesome read. Very technical
Archerofaiur: seriously is there any game on the face of the earth that recieves this kind of dedicated research.
LiquiDLegend: Very in-depth analysis as others said, its really good.
RonNation: it seems this is mostly a lesson in common sense, but a good lesson nonetheless
Reborn8u: Your definitely going to be one of the professors at starcraft university
Korpze: My mind has been blown.
stork4ever: whoa, this is why i joined this forum
bay: Wow- really fantastic article. This is the kind of stuff I have been looking for! Thanks so much
So so
BladeRunner: reasonably good info but not very in-depth and way too verbose.
Zalan: Yeah, as most said: interesting info, but a bit redundant...
Toran7: Articles like this make me feel inadequate when it comes to math :/
crate: I definitely like the effort and SC2 needs more analysis on resource gathering especially with its implications on mapmaking.
Osmoses: I agree that most of this article was unnecessary fluff, but there was one very important discovery that I actually think I will incorporate into my play
Rubbish
QuantumPenguin: As others have said, there is no science or maths here, just some annotated arithmetic. Furthermore it makes absolutely no sense to regress a function on a domain for which it is undefined. Quite a lot of pretentious nonsense in this article.
shoop: Sorry, but this is nonsense. If the miners would not hold each other up, then the mean time between gas returns would obviously be inversely proportional to the number of miners...
Markwerf: This article is so unneccesary long it's unbelievable
DarkChrono: I'm laughing so hard at gasexpfunction.png (2nd graph)
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On April 28 2010 06:20 DarkChrono wrote:
Don't get me wrong, it's great that you're taking an interest in math, but the ability to be honestly critical of your own (and others) arguments is an essential skill. It's ok to be wrong, even though it's the internet.
You have to be wrong to learn ^_^.
I guess you are meaning that all elementary functions have derivatives that can be expressed in terms of the original function, which is not strictly true. However there is something similar I have just found: http://en.wikipedia.org/wiki/Pfaffian_functionQuite cool.
Yes and that Pfaffian function you came across is pretty cool.
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Very very nice article, but i am wondering about one thing
* 1(1) | 0.74 gas/second
* 2(1) | 1.50 gas/second
* 2(2) | 1.48 gas/second
* 3(1) | 2.00 gas/second
* 3(2) | 2.23 gas/second
* 4(2) | 3.05 gas/second
* 5(2) | 3.33 gas/second
* 6(2) | 3.94 gas/second
it should be true in my opinion that
5(2) = 2(1)+3(1)
but result is different.
2(1)+3(1) = 1.5 g/s + 2.0 g/s = 3.5 g/s
while 5(2) = 3.33 g/s
Is this because of fact that one geyser is closer than second one ? Or is this just small deviation in data ?
Btw, i am curious how it is with minerals, especially how long should i wait in seconds to get minerals spend on worker back.
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* 4(2) | 3.05 gas/second
* 5(2) | 3.33 gas/second
* 6(2) | 3.94 gas/second
If you put a third worker on one geyser you get +0.28gas/second, but if you put a third worker on the second geyser you get another + 0.61gas/second, which doesn't make sense to me.
Anyone has an explanation for this?
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On April 28 2010 08:21 shoop wrote:
Rubbish
QuantumPenguin: As others have said, there is no science or maths here, just some annotated arithmetic.Furthermore it makes absolutely no sense to regress a function on a domain for which it is undefined.
Everything I said here was true. Even if one were to pretend that the graph has some meaning for a real number of workers (which it doesn't), the interpolation is still wrong, as every natural number of workers n for n > 6 has gas return interval equal to the gas return interval of n=6.
You do also realise that only one of the data points is even remotely close to its graphical position, right? That picture is just meaningless.
Edit: Oh, I thought you were classifying my post as rubbish, until I saw you'd self-quoted there My apologies.
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On April 28 2010 18:01 QuantumPenguin wrote:
Edit: Oh, I thought you were classifying my post as rubbish
I think you're quite right, actually But given how this thread developed I thought it would be better not to go into right and wrong too much anymore. After all, I already said what I think. I'm still thinking if I can come up with a good psychological theory to explain the way threads such as this often develop. It's crazy.
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On April 27 2010 00:00 Markwerf wrote:
This article is so unneccesary long it's unbelievable...
First of all the method and data is nice and all but really this could be summarized much easier. It is pretty safe to assume beforehand to that rate of gas income is constant while the gas geyser is running as we all know this from playing experience. Thus just putting 1, 2 and 3 workers on a geyser and just timing for a minute each and calculating how much you gas income you would get would suffice, doing the 2nd gas as well is completely unneccesary as it's obviously the same as the first.
So 4(2) is really the same as 2x 2(1)??, everyone could think of that beforehand..
The only interesting thing in here is how much less efficient the 3rd worker is then the first 2 ones using the geyser. The raw gas income is also slightly interesting.
The inverse graph is completely useless and add's nothing to the whole issue, that entire paragraph should be scrapped really.
Also the investment and income table at the end hardly makes sense as it counts the costs of workers which you would be having anyway. It neglects the oppurtunity cost of making the refinery as well which though different for each race can't just be neglected either.
The whole writing and setup of this piece is just not good. Needlessly trying to do complicated things which add nothing to the whole subject...
Congrats on proving that you're a moron. Do you want to confess to murdering JonBenét Ramsey now, too?
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On April 26 2010 17:27 Arrian wrote:
Granted this analysis is limited in scope. While I strongly suspect there are no differences in gas collection rates between the races, my analysis was limited only to Terran, on Lost Temple at 6 o'clock. Of course, I am assuming that these are typical geyser positions and there are no differences between the gas collection rates between the races, and there were other moderate sources of error, namely a non-uniform starting place for the gas collection, but this is normed by the large number of data points and the absolute nature of gas collection. Thus, by linear regression, an approximation of the rate of gas collection can be ascertained, but more importantly, its relation to other methods of gas collection.
There is a slight advantage to gas that will be placed on a "pure" horizontal or vertical pattern. When I play Z I try at times to take that little extra inch to get a kind of benefit overtime with an xtra hatch placed in this "optimal" way. After a few comparisons I won about 30-60 gas in two minutes with that hatch instead of letting the original hatch be the gathering point.
Shame I actually play T but I think that on some maps that has geysers placed diagonally some Z's could undergo a few bo's optimized for those type of maps, given if that xtra hatch would be part of a viable plan, that is. ^^
Excellent post, it just adds up with the little things I found myself!
Little edit what I pretty much did on my part (but too lazy to share) was to make up a list of gathering gas with a 1(1), 2(1) and 3(1) for every different timing of a building, unit, upgrade and research, then grouped up the stats.
I'm finally and slowly starting to think WHEN should I really take that gas, but what you just showed here simply gives me more to ponder and to readjust.
I did notice after writing down my "gas over time" that it wasn't possible to have a beginning estimate of over 20 seconds for example and then think "it will be the double harvested in 40 seconds" because, just as you said yourself, the chunky way the gatherers are bringing the resource to our main building; that's why I ended up doing an estimate for every specific timing that exists (only for Terran; I didn't have the courage to do the different ones for the two other races). Besides I didn't want a deeper understanding of "how gas works", I only wanted to plug in the amount gained over the given times to implement them into the relative timings and triggers. But you just showed us that deeper understanding would have important things to know :-)
If I ever feel like playing Z again (was Z on bw), I might give a try of some bo's with an xtra hatch but I'll just place it as close as possible to geysers that are originally put in a diagonal. I remain certain that it could be a considerable advantage over time against another race that won't be too fancy about making a CC/Nexus just for a geyser.
Last edit: I profusely apologize for making this so long but I just realised that the "triple" gas gained by doing 2(1st) and 1(2nd) can become a very very little difference if you play on a map where the geysers are positioned further away (LT is in his case of observations the shortest distance you can profit from) because the pause coming from a scv/probe/drone as the 3rd one being in one gas becomes nearly non existent.
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very informative, you actually went through the process of showing statistics with the graphs and all.
great work =]
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Great read to start the day off. ;-)
Thanks for the effort!
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Very informative! this shall come in handy lol
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EPT![[image loading]](http://upload.wikimedia.org/math/3/7/c/37ccb48bb92538c2f68a741ded35732a.png)
![[image loading]](http://upload.wikimedia.org/math/c/f/5/cf5d03795de766b013f91c4cb409bd9c.png)
![[image loading]](http://upload.wikimedia.org/math/6/4/f/64fba9599a29e87d1967f9c6410ddc5a.png)
