> Reflection symmetry -> suffices to test odd and even functions separately.
Maybe reflection means flipping a function? Or when it goes above or below an axis?
> Translation invariance -> suffices to test individual plane waves (i.e., to inspect the Fourier multiplier symbol).
Fourier transform has something to do with frequencies. A plane wave sounds like part of a DnD spell.
> Dilation invariance [if unitary] -> suffices to test homogeneous functions.
If this didn't have "functions" in it I would think it's biology.
> Rotation invariance -> suffices to test the case of spherical harmonic behavior in angular variable (separation of variables).
Good thing the spherical harmonic behaviour sufficies, I do not know I would test the cubic dissonant one.
> Invariance under linear combinations / convex combinations / algebraic operations -> suffices to check basis elements / extreme elements / generators
Combinations has something to do with choosing things? Generators.. no, I give up.
> Invariance under tensor products -> suffices to check one-dimensional/irreducible case
Aha! Tensors! Like TensorFlow! I know one-dimensional too! This is the closest I've gotten to having a smart thought.
> Invariance under limits -> suffices to check a dense subclass
"Dense subclass" is exactly how I'm feeling right now.
> Multiplicative structure (in analytic number theory) -> suffices to check prime powers
¯\_(ツ)_/¯
> [Principle of induction] Preserved by successor -> suffices to test base case and/or limiting cases
I do know this one, but I actually wouldn't have figured it out if he hadn't pointed out this is induction.
I'm tired of showing how dumb I am for now, but you get me. I am as far as I think.
I think you're getting at the point with this though:
> I actually wouldn't have figured it out if he hadn't pointed out this is induction.
A lot of these other ones are not that complicated either, it's just hard to parse and sometimes requires a small bit of math knowledge, which I think is within reach if you're familiar with induction and would only take e.g. a minute for you to understand if someone explained it (ideally with a blackboard than in text).
Some of those are definitely nonsense to me. But, to put my money where my mouth is and to give some examples (as you spent the time to type that out), 'invariance under linear combinations' just means that, if something holds for f(x), then it holds for 2f(x), f(x) + 10, etc. (these are all linear combinations). So then saying 'suffices to check basis elements' means 'just check f(x), and the others all fall out for free'.
As another example, convex combinations: Google 'convex hull' for images, but basically this one is just saying just check the 'extreme elements', i.e. the 'corners' at the boundary, and everything in the middle falls out for free because we have 'invariance under convex combinations'. A convex combination here is just a some point in the middle of these extremes.
While these may not be immediately obvious when reading them, the pictures or ideas are actually sometimes quite simple.
> 'invariance under linear combinations' just means that, if something holds for f(x), then it holds for 2f(x), f(x) + 10, etc. (these are all linear combinations).
I'm not sure how Terence meant it, but for some people, it actually means that some property that holds for f(x) will also hold for f(ax + b).
> So then saying 'suffices to check basis elements' means 'just check f(x), and the others all fall out for free'.
Yes, but your statement confuses me more than Terence's :-)
A given vector space has basis elements (e.g. x, y and z unit vectors for 3-D Cartesian space). It means that if you can show the property is true for the basis elements, you've now shown it's true for any vector in that space. One needs to show linearity holds to assume this.
> As another example, convex combinations: Google 'convex hull' for images, but basically this one is just saying just check the 'extreme elements', i.e. the 'corners' at the boundary, and everything in the middle falls out for free because we have 'invariance under convex combinations'. A convex combination here is just a some point in the middle of these extremes.
You're right there. Or at least you definitely have a point.
I would question whether being able to see this pinhole perspective of the maths he's talking about really means I "understand" it, or if that's just a semantic game. I don't feel like I am any more able to do something with the information than a second ago, even though I "understand" it more.
Glad it made a bit of sense! But that's a fair point about whether it really constitutes 'understanding'.
I suppose it's equivalent here to whether knowing the meaning of a particular sentence in the alien language qualifies as understanding it, vs being able to meaningfully speak about it in this alien language, and use the sentence in a conversation. Seems like just semantics to me.
> Invariance under linear combinations / convex combinations / algebraic operations -> suffices to check basis elements / extreme elements / generators
Linear combinations of a basis (b1, b2, ..., bn) are sums of the form a1*b1 + a1*b2 + ... an*bn using some coefficients (a1, a2, ..., an). Convex combinations add the requirement that a1 + a2 + ... an = 1. Algebraic operations adds other things besides adding and multiplying by coefficients. Anyway, he's saying that if you have a function for which f(a1*b1 + a2*b2) = a1*f(b1) + a2*f(b2), then you only need to know what f does to b1 and b2 and the other basis elements in order to know what it does to anything.
> Multiplicative structure (in analytic number theory) -> suffices to check prime powers
Here he's talking about how (for example) some functions f(ab) = f(a)f(b) (sometimes with the extra condition that `a` and `b` have no factors in common).
For such functions with "multiplicative structure", if you want to know their value at any point, you only need to values at prime powers.
No, you're closer than it feels. Here are what they mean in more every day terms.)
Reflection symmetry - the laws of physics should be the same for an observer who is a mirror image of ourselves. (In classical mechanics this is so. In quantum mechanics you also have to reverse the sign of all charges.)
Translation symmetry - The laws of physics should be the same for an observer in a different position in space. (This one is true.)
Rotational symmetry - the laws of physics should be the same for an observer who is rotated from ourselves. (Again true.)
Combinations - just refers to combining things by some rule. A linear combination of vectors is just adding scalars times vectors. So the set of linear combinations of 2 vectors is the plane spanned by those vectors. A convex combination of points is any average of them. So the convex combinations of 3 points is a triangle. And if you allow some set of algebraic operations, from a fixed set of values you can generate a whole system. Those fixed values are generators for that system. So, for example, (1, 0) and (0, 1) with the algebraic operations of addition and subtraction generate the whole grid of points (n, m) with n, m integers. In all of these cases you can often work with just the few things from which the plane/triangle/grid/whatever was created, without having to look at the rest.
Dense subclass - for any point you choose, for any distance you choose, the dense subclass includes something at least that close. For example the rational numbers are dense within the reals. Pi is not a rational number, but we have no trouble finding rational numbers within 1 billionth of pi. So you can sometimes prove something for all rational numbers, and then find you've proven it for all real ones. (For example, with powers, roots, and division we can define 2^x for all rational numbers x and prove properties about it. From that we can actually define 2^x for all real numbers.)
I'm tired of showing how dumb I am for now, but you get me. I am as far as I think.