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General Discussion / Re: Maths thread.
« on: June 01, 2019, 06:42:02 pm »Thanks! This is a quality explanation.
Your ln(0)s have to be ln(1)s, right?
yep, fixed
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yep, fixed
77
General Discussion / Re: Maths thread.
« on: June 01, 2019, 04:39:14 pm »I'm somewhat confused about complex numbers right now. Or rather about how WolframAlpha deals with them, which is presumably the correct way.
If I ask for solutions to the equation x^5 = 1, it gives me all the complex numbers with angles n * 72°, n = 0,1,2,3,4. (Left.) If I ask for solutions to the equation x^{5/2} = 1, it gives me the ones with n = 0, 2, 3. (Right)
So what exactly is the property that the points with angles 72° and 288° don't have?
It’s because the function which raises a complex number to the fifth power is a real function, whereas the function which raises a complex number to the 5/2 power is not, it’s a doubly valued multifunction. In all 5 of these cases, one of the two values of the multifunction is 1, but only in the three given solutions is 1 the *principal* value of the multifunction.
When you say double valued multifuction, you mean a function p : \C -> \powerset(\C) defined by p(c) = {[first square root of c^5], [second square root of c^5]}?
So what's the definition / consensus on what is the principal value?
Yeah that's one way to think about it (though the more proper and technical setting for this involves Riemann surfaces).
To define principal values, we need to talk about arguments, logarithms, and power functions.
The argument of a complex number is its angle in the complex plane measured in radians from the positive real axis. Of course, this is not a single number, but a set of numbers, denoted arg z, since, e.g., 0 and 2pi refer to the same angle. So we single out one of these angles to be called the principal argument, which (more or less arbitrarily) is the angle A in arg z satisfying -pi < A <= pi.
Then we note that we can write a complex number z in exponential notation, as z = r*e^(i*a), where r=|z| is the modulus and a is one of the arguments of z. Here we can choose any of the arguments of z, since the complex exponential function is periodic with period 2*pi*i (for the same reason that adding 2*pi to an angle gives the same angle). This means in particular that the complex exponential function is no longer one-to-one as the real exponential function is, and hence the inverse function (the logarithm) is not a real function.
But we decide that we want to talk about logarithms anyway, so we define it as a multifunction--the log of a nonzero complex number z is now the set of complex numbers which exponentiate to z. It turns out that log z = log (re^ia) = ln|z| + i*arg z, where ln denotes the real logarithm of the positive real number |z| and arg z is again the full set of arguments of z. This is an infinite set, since arg z is an infinite set of angles differing by multiples of 2pi. If we insist on defining at a single-valued version of the log function, we by convention usually use the principle argument A, which gives the principal branch of the log function, denoted Log. That is, Log z = ln|z| + i*A.
This now lets us define power functions. Given z and another complex number c, we can define z^c by z^c = e^(c*logz) = e^(c*ln|z| + i*c*arg z). Since log is multi-valued, the power function typically is as well, but we can single one of these out by using the principal branch of the log function, e^(c*Log z) = e^(c*ln|z| + i*c*A). It happens to be the case that the power function is infinitely-valued unless c is rational, in which case it has as many values as the denominator of c written in lowest terms. When c is an integer, the denominator is 1, so power functions with integer exponents are in fact all single-valued.
So for example, let's apply this to some of your numbers. The number z_1 at angle 72 degrees has principal argument of 2pi/5 radians, and a modulus of 1, so the principal value of z_1^(5/2) is
z_1^(5/2) = e^(5/2*ln(1) + i*(5/2)*(2pi/5)) = e^(i*pi)=-1.
The number z_3 at angle 3*72 = 216 degrees has argument of 6pi/5 radians, but its principal argument is -4pi/5. Its modulus is also 1, so the principal value of z_3^(5/2) is
z_3^(5/2) = e^(5/2*ln(1) + i*(5/2)*(-4pi/5)) = e^(i*(-2pi))=1.
But remember that this whole principal business is rather arbitrary. Using the argument 12pi/5 for z_1 gives
z_1^(5/2) = e^(5/2*ln(1) + i*(5/2)*(12pi/5)) = e^(i*6pi)=1,
and using the argument 6pi/5 for z_3 gives
z_3^(5/2) = e^(5/2*ln(1) + i*(5/2)*(6pi/5)) = e^(i*3pi)=-1.
78
General Discussion / Re: Maths thread.
« on: June 01, 2019, 12:14:36 pm »I'm somewhat confused about complex numbers right now. Or rather about how WolframAlpha deals with them, which is presumably the correct way.
If I ask for solutions to the equation x^5 = 1, it gives me all the complex numbers with angles n * 72°, n = 0,1,2,3,4. (Left.) If I ask for solutions to the equation x^{5/2} = 1, it gives me the ones with n = 0, 2, 3. (Right)
So what exactly is the property that the points with angles 72° and 288° don't have?
It’s because the function which raises a complex number to the fifth power is a real function, whereas the function which raises a complex number to the 5/2 power is not, it’s a doubly valued multifunction. In all 5 of these cases, one of the two values of the multifunction is 1, but only in the three given solutions is 1 the *principal* value of the multifunction.
79
General Discussion / Re: Random Stuff Part IV
« on: May 15, 2019, 10:17:19 pm »I am researching how to apply for a US visa right now... The example visa application that the US government provides is both cute and somewhat embarrassing:
There's a typo. Should be Spouse/Cousin.
Spouse/Aunt
80
General Discussion / Re: Maths thread.
« on: May 14, 2019, 10:36:20 am »2,3,5,7,11,13,17,19,23,29,31,33,36,40,44,47,51,57,59,62,65,69,73,83,86,89,100,101,104,107,110,116,117,113,115,117,114,119,124,124,125,126,128,132
Can you guess where I stopped?
Man this thing sucks at computing powers of 2.
81
General Discussion / Re: Brag Board
« on: May 14, 2019, 10:29:32 am »My wife is pregnant with our first child! Due in December. Got to tell our families on mother's day, which was fun
Congrats! It is a wild ride which will reprogram your life and brain in ways you can’t possibly conceive! But it’s good times.
82
General Discussion / Re: Maths thread.
« on: May 04, 2019, 02:57:56 pm »Oh I didn't see the 3 was outside. Well you can use the trig rules for cos(x+y) to get that cos(3x) = 4cos³(x) - 3cos(x), and then solve for cos(x).I was working on a problem, and had to calculate cos(atan(45*sqrt(3)/28)/3), which wolfram alpha helpfully tells me is equal to 4/sqrt(19). Does anybody know how it figured that out?tan x = a/b in some rectangular triangle, and cos x = b/c. So atan(a/b) = acos(b/c). Also since everything is rectangular, c = sqrt(a² + b²).
thus atan(a/b) = acos(b/sqrt(a² + b²)) = acos(sqrt(b²/(a² + b²)) = acos(sqrt(1/(a²/b² + 1))). This can be written as atan(x) = acos(sqrt(1/(x² + 1))).
I think the rest should follow easily.
I already knew that, but what about the divided by 3?
And solve a cubic, which is what I was trying to do in the first place?
It's a depressed cubic, and so at this point you can use e.g. Cardano's method to solve it explicitly.
83
General Discussion / Re: Brag Board
« on: April 27, 2019, 11:33:17 am »I have a baby!
Congrats and good luck and you can do this!!!
Thanks!
How's everything going? Baby and parents being awesome?
Going well so far. Sometimes it seems a little easier than I expected and sometimes it seems about as hard as I expected, but it hasn’t really been any harder than I expected so far. I’m sure that will change eventually, though. I also have extremely generous paternity leave which is going to help a ton.
The real question is whether or not you have any family nearby. We made the fatal mistake of moving hundreds of miles from any family before settling down and popping out kids. It's been nightmare mode ever since. When family visits or we visit family we get glimpses of what life could be like and it appears to be lullaby dream mode in comparison.
Yeah both of our families are ~1000 miles away so that has indeed been a challenge.
84
General Discussion / Re: Brag Board
« on: April 12, 2019, 12:53:54 pm »I have a baby!
Congrats and good luck and you can do this!!!
Thanks!
How's everything going? Baby and parents being awesome?
Going well so far. Sometimes it seems a little easier than I expected and sometimes it seems about as hard as I expected, but it hasn’t really been any harder than I expected so far. I’m sure that will change eventually, though. I also have extremely generous paternity leave which is going to help a ton.
85
General Discussion / Re: Brag Board
« on: April 11, 2019, 03:28:43 pm »I have a baby!
Congrats and good luck and you can do this!!!
Thanks!
87
Other Games / Re: Baba Is You
« on: March 30, 2019, 12:53:43 pm »
I just tried this at a friend's house and it's awesome.
88
General Discussion / Re: Random Stuff Part IV
« on: March 14, 2019, 04:09:46 pm »I've spent significantly more time on this forum than I have actually playing Dominion.
Is Dominion a new Forum Game?
Actually, Forum is a new(ish) card in Dominion.
Spending time here is strictly better than spending time on warehouse.dominionstrategy.com.
89
General Discussion / Re: Maths thread.
« on: January 20, 2019, 12:28:40 pm »The main reason I don't like the term is that it's confusing to beginners. It's usually the first time that something that is called multiplication doesn't behave like any other type of multiplication that they are used to. I know I at first was wondering why in the world it was defined in the way that it is. "Why not make it like matrix addition and use the component-wise product? That has all of the standard properties of multiplication." Of course we don't use the component-wise product because it's rarely useful, but back then it was radically different than any other kind of multiplication I had seen, and it seemed so arbitrary.
This kind of reminds me of tau vs. pi, or whether we should really call imaginary numbers "imaginary" numbers. It's probably too late to change things now, but it would make more sense, especially for beginners, if the terms were different.
This is all totally valid. Two things I would say:
1. Some terminology, while being confusing to beginners, is very intuitive and natural in a higher level context, or sometimes it's just so well established that you just have to deal with the fact that it's what everyone uses. In either case, though, we preumably want some of these beginners to become non-beginners some day, and we should think about the trade-offs between using counter-intuitive terminology in the introduction of a topic vs. making people re-learn proper terminology later on so that they can communicate with the rest of the community. (As an aside, this reminds me a little bit of learning how to ski. We teach young kids to make wedge-turns and then have to have them un-learn this and make proper turns later on).
2. I like your point about how "we don't use the component-wise product because it's rarely useful, but back then it was radically different than any other kind of multiplication I had seen, and it seemed so arbitrary." I think in a well-taught class there should be some serious time spent on getting students to understand why a component-wise product would be rarely useful, and why the proper definition is in fact not so arbitrary. The problem is that so many classes just introduce matrices and show how to multiply them with no context as to why we might want to do such a thing.
You can define anything you want, but the things worth studying in mathematics have a reason for the definition being what it is and not something else. I can define a binary operation on the set of all functions from the reals to the reals by
(f#g)(x) = f(x)g(x-2)+f(-2x)[g(x)]^2
Perfectly good binary operation. I could even prove theorems about it. But what's it good for? Not much as far as I know. Whereas the binary operation defined by
(f*g)(x) = int_{-infty}^infty f(y)g(x-y)dy (when convergent)
is good for lots of things.
Hmm; I'm not sure if I've ever posted in this thread. Anyway, Linear Algebra is hard to teach.
This is very true. It's also somewhat unique among introductory math courses in that there are two pretty radically different ways to approach the subject, which I generally think of as the "Begin with systems of linear equations" approach and the "Begin with the definition of an abstract vector space" approach. Courses titled "Linear Algebra" might be either one and they have a very different feel to them.
90
General Discussion / Re: Maths thread.
« on: January 19, 2019, 12:48:09 am »Barely related, but I feel like matrix multiplication isn't "multiplication". It's more like composition.
Eh, what is "multiplication" then? It's a binary operation and it distributes over addition, might as well call it that.
Do you call the composition of two functions a multiplication?
Well, I'm a dynamicist so kinda. By which I just mean that for me the notation f^2(x) generally means f(f(x)) and not [f(x)*f(x)] (exceptions being things like trig functions where the notation is standard but annoyingly inconsistent). But this is maybe confusing the issue...
To get at the heart of it, there are two things at play here:
(i) Terms in math mean whatever they are defined to mean and nothing else
(ii) Terms should ideally be chosen to be useful and consistent
Importantly, "multiplication" does not have a precise mathematical definition as a stand-alone term, so we can't appeal to (i). But there are certain binary operations on certain sets with certain properties that perhaps warrant using the same name for all of them, to highlight exactly those common properties.
On sets that already have a commutative binary operation which we've agreed to call "addition" and have agreed to denote with the symbol "+", we often have another binary operation (which we maybe denote by *) satisfying a*(b+c) = a*b+a*c and (a+b)*c = a*c+b*c. Seems useful and consistent to refer to such operations as "multiplication."
For general functions, composition definitely does not satisfy this property (sin(a+b) =/= sin a +sin b), so it seems less useful to call it "multiplication." Could we? Sure, but then what property is shared by all "multiplications?"
You can debate these things of course. I've avoided using the word "ring" but maybe you want to restrict "multiplication" to only refer to a ring operation. But multiplication in rings must be associative. Is that important to you? Maybe, but then you lose the ability to call the cross product of vectors in R^3 "multiplication." Do you care?
Now of course in the realm of matrices, we have such a binary operation which distributes over addition, so we do tend to call it "multiplication". It also happens to relate to linear transformations of vector spaces, and in a sense coincides with the notion of "composition" in that context (where you should now think about what a useful and consistent definition of "composition" would be).
At the end of the day I'd argue that it's better to talk about multiplying matrices as opposed to composing them. You'd not be insane to take the operation we call matrix multiplication and call it something else, as long as it is defined precisely, but you should think about whether that term is useful and consistent. In this case, you'd be clashing with established nomenclature, though, which is hard to overcome even when it's not useful or consistent (like sin^2(x) v. sin^(-1)(x)).
As a final quiz, how should we define multiplication of ordered pairs of positive integers? For a,b,c,d positive integers, which is more natural (consistent and useful)?
(i) (a,b)*(c,d) = (ac,bd)
(ii) (a,b)*(c,d) = (ac-bd, ad+bc)
91
General Discussion / Re: Maths thread.
« on: January 18, 2019, 11:29:27 pm »Barely related, but I feel like matrix multiplication isn't "multiplication". It's more like composition.
Eh, what is "multiplication" then? It's a binary operation and it distributes over addition, might as well call it that.
92
Dominion Articles / Re: The Top 20 Best Designed Cards
« on: January 14, 2019, 03:38:27 pm »For me it's mostly the fact that both things are so similar, really. It would be easier if the parts felt like they complimented each other, like on the other overpay cards, or were actually the same, like on Noble Brigand. I don't even know whether the card is good or bad, just that I never felt I could explain it in short words to my play group. That said, Guilds is still one of our favourite expansions, and sees a lot of use at our table. Baker's setup is perfect.Doctor was the first card I ever owned where I am not to date able to tell you what it does by heart. It's some trashing, top of deck, I think with matches? And then it does the same thing as overpay, or no, a very similar but not identical thing... Maybe it makes sense once you play a few games with it, but this never left our box.We had this experience during Renaissance testing. Doctor was in the game, and someone said, man I don't want to read that.
This is a similar issue I have with Graverobber. You're upgrading or pulling from the trash, there are some cost restrictions, something has to be an action, sometimes it goes on top of deck, but I cannot ever remember which goes with what.
93
General Discussion / Re: Movies: Any movie buffs?
« on: January 02, 2019, 12:48:26 pm »I was expecting to be annoyed by Spider-Ham. He was just in small enough doses that I was able to accept him.
John Mulaney is a national treasure.
This movie was fantastic, and gives further evidence that basically everything from Lord/Miller is worth watching. There was even a Clone High billboard easter egg in the background!
94
General Discussion / Re: Drinking
« on: December 21, 2018, 12:24:56 pm »Cuzz, are we still in the same hood? Cause I definitely want an invite to the next cocktail party.
Also, congrats on the upcoming little one!
Alas, no. I moved to Chicago about 3 years ago.
95
General Discussion / Re: Drinking
« on: December 21, 2018, 11:37:48 am »
Hey, joth is back!
I threw a cocktail party a few months ago and made huge batches of whiskey smashes (the signature cocktail from my wedding), mai tais, and a sort of paloma-type thing with an IPA float that came out really delicious. I tend to wildly swing between trying to make precise versions of classic cocktails and inventing my own with whatever I have on hand. Mixed results on both counts.
Since my wife got pregnant I've been leaning more into beer, though. My local bottle shop sells a beer advent calendar that I ended up with two of as gifts from people, so I've been working my way through those. A Metropolitan Dunkel Rye and a Half Acre IPL were standouts so far.
I threw a cocktail party a few months ago and made huge batches of whiskey smashes (the signature cocktail from my wedding), mai tais, and a sort of paloma-type thing with an IPA float that came out really delicious. I tend to wildly swing between trying to make precise versions of classic cocktails and inventing my own with whatever I have on hand. Mixed results on both counts.
Since my wife got pregnant I've been leaning more into beer, though. My local bottle shop sells a beer advent calendar that I ended up with two of as gifts from people, so I've been working my way through those. A Metropolitan Dunkel Rye and a Half Acre IPL were standouts so far.
96
Dominion General Discussion / Re: Taking Notes - Against the Rules?
« on: December 17, 2018, 02:38:55 pm »After consideration of my opponents inability to memorize certain aspects of the game (they cannot remember if they bought all of the cards for Museum), I prefer to make the game more 'playable' for them (if they prefer to take notes, then I can take notes myself, but I would rather want them to play the game and enjoy it. Just because I can remember things better should be my advantage. I want my ability to know how to play the kingdom at hand to be the deciding factor, not my ability that I have the advantage of having more VP then they do, and end the game on piles.
No problem playing that way if you want; so long as you recognize that you are playing a variant.
Yes to be safe you'll want everyone to fill out the Variant Contract in which all players affirm their understanding that an actual game of Dominion has not been played, and that they will under no circumstances claim that they have played a game of Dominion in speech, print or semaphore to any other entity living, dead, or fictional.
97
General Discussion / Re: The Legend of Zelda: Ocarina of Time turns 20
« on: December 12, 2018, 04:59:43 pm »Ha, I read that piece today and was wondering how you pitched it to them.
Yeah, I generally don't have to get my pitches pre-approved at my normal place of business.
Yeah, I don't know how these things work in media. But I don't think I had seen any other video game pieces there so I just thought it was cool that you were able to write something like that for them.
98
General Discussion / Re: The Legend of Zelda: Ocarina of Time turns 20
« on: December 11, 2018, 07:58:33 pm »
Ha, I read that piece today and was wondering how you pitched it to them.
I'm not much of a gamer at all but man do I love that game. Every time I start it up again I feel immediately transported to my parent's family room on Christmas morning in 1998. I haven't touched a console in months but now I feel like I need to play it this weekend.
I'm not much of a gamer at all but man do I love that game. Every time I start it up again I feel immediately transported to my parent's family room on Christmas morning in 1998. I haven't touched a console in months but now I feel like I need to play it this weekend.
99
Dominion: Renaissance Previews / Re: Renaissance rulebook is up
« on: November 07, 2018, 08:01:40 am »Patron gets added to the list of cards that clearly only exist to stick it to people like me who have responded to fan cards by saying "it's not a good idea to have a reaction that reacts to being revealed".
Why was this not considered to be a good idea?
100
Dominion: Renaissance Previews / Re: Fun or interesting combos with Capitalism
« on: November 07, 2018, 07:50:38 am »
Mine becomes a discount Expand for your new treasures.
Mint becomes a Disciple variant (buy it before you buy Capitalism).
Mint becomes a Disciple variant (buy it before you buy Capitalism).