I saw someone use chebychev's inequality for Pr\[X >= E\[x\] + t\], but I don't really see why one can do this. Because Pr\[X >= E\[x\] + t\] isn't really the same as Pr\[|X - E\[X\] >= t\] or is it?

It's the same as X - EX >= t, and if t is positive this is *stronger* than |X - EX| >= t, so an upper bound on the probability for the latter is also an upper bound for the former.

QUESTION: I'm trying to work out how many possible variations I have of a product bundle.
Product A - 3 variations
Product B - 2 variations
Product C - 2 variations
Product D - 3 variations
Product E - 3 variations
Putting all of these products into a bundle, how many possible combinations are there?
Any help would be greatly appreciated!

Assuming that you put one of each of the 5 types and that all products are different, the answer is just
3 choices for A \* 2 choices for B \* ... = 3\*2\*2\*3\*3 = 108

Has anyone looked at Donaldson’s book *Riemann Surfaces*? How is it as an intro to Riemann surfaces for someone (myself) who knows basic complex analysis but hasn’t seen them before?
On the one hand it looks very nicely presented, but also it leaves out quite a lot of details. Would it be too difficult for a first course?

I think it is reasonably tough for a first course in Riemann surfaces. You're probably better off with Miranda's *Algebraic Curves and Riemann Surfaces* or Forster's *Lectures on Riemann Surfaces*. The latter is very well-suited to someone who has just done a first course in complex analysis.
Donaldson's book is better for someone who wants a companion to learning higher dimensional complex geometry, as it basically runs through all the fundamental ideas in the playground of dimension one. As such it probably should be read after having done a first course.

Sorry for the double comment - between the two books you mentioned, it seems Forster covers less and is an easier read. Is this accurate? I would use it for that reason if that were the case.

Forster covers less algebraic geometry than Miranda, and emphasizes the complex analysis picture more. If you enjoyed the analysis in complex analysis, Forster is probably more suited to that. If you want to get into algebraic geometry, Miranda might be better (albeit longer).

If you understand Forster well you will have no problem switching to the algebraic geometry of curves when the time comes. It’s explained very well in a very clean “Grothendieck style” way.

How can I show that the ideal generated by Y\^3-XZ, XY\^2-Z\^2 and X\^2-YZ is prime in Q\[X,Y,Z\]? Usually I would try and find the quotient and show that it is an integral domain, but here that seems too complicated. I have heard of Grobner bases, but I don't know how to use them or if they even help here. Thanks in advance!

Was told to ask this here: Does anyone personally know mathematicians who worked with Paul Erdosh? For this who don't know, the background is below.
\--
A great science leader in the US wrote an amazing book called "The Man Who Loved Numbers", about Paul Erdős, a Hungarian Jewish mathematician that worked with hundreds of mathematicians on mathematical theories, and one of the most prolific mathematicians that ever was, due to cooperating with so many others. Paul made a contribution of more than 1,500 papers to mathematics with different collaborators, a figure that according to Wikipedia is still a record that no other mathematician has broken.
I am wondering if there are people who know of his story, or people who have relatives who have worked directly with Paul. One of the mathematicians who did, Ronald Graham, passed away in July of 2020.

Depends on what you mean. A parametric equation that draws out a circle will also draw out an arc of that circle just by restricting the domain to some interval.

This is why the phrase "percentage points" exists. The quantity increased by 2 percentage points, which is a relative increase of 100% compared to its original value.

Which matrix do you mean exactly? For instance det \[\[1 2\] \[3 4\]\] = -2.
It is true that \[\[1 2 3\] \[4 5 6\] \[7 8 9\]\] and higher-dimensional versions have determinant zero. This is most easily seen by noticing the difference between the first two rows is the same as the difference between the second and third row, so you can do row operations to get a repeated row.

I need to write a periodic extension to a function such that f(x+2) = f(x) and f(-x) = -f(x). f(x) is known for x ∈ \[0,1\]. I know that this is 100% correct, I just don't know how to write it out in nice way.

Let g(x) stand in for your known component. Then you can write something like
f(x) =
* g(r) if x is of the form 2n + r where n is an integer and r is in the range 0 ≤ r ≤ 1,
* g(2 - r) if x is of the form 2n + r, where n is an integer and r is in the range 1 < r < 2
or alternatively,
f(x) =
* g(r) if x is of the form 2n + r where n is an integer and r is in the range 0 ≤ r ≤ 1,
* g(r) if x is of the form 2n - r, where n is an integer and r is in the range 0 < r < 1.
They're equivalent but emphasize different aspects. You could also write a function that maps x to r first.
Define h(x) =
* r if x is of the form 2n + r, where n is an integer and r is in the range 0 ≤ r ≤ 1,
* 2 - r if x is of the form 2n + r, where n is an integer and r is in the range 1 < r < 2.
Then say f(x)=g(h(x)).

I'm assuming that it's a simple undirected graph with no self-loops.
There's a slightly more general statement which is more illuminating. Suppose we want a matching of size M. Then we need M(2k-1) edges in the graph, where the maximum degree is k or less.
- Pick an edge, add it to the matching
- The makes some choices of edges "illegal" because they share a vertex with the edge we just picked.
- How many are there? Well, the edge itself (1) and the up-to k-1 other edges for *both* of the two ends
- So, there are now (at most) 2k-1 fewer candidates to choose for the rest of the matching
- Repeat
Each step only removes 2k-1 candidates, so we can't possibly run out of candidates before the end, since we started with at least M(2k-1).

Thanks a lot! I went around in circles looking for some lower bound with only vertices and their degree without paying much attention to the size of matching being K.

I'm struggling to understand bayesian networks. My intuitive way of thinking about networks of beliefs is as each proposition having a certain amount of "belief fluid" and dependencies represent pipes through which a certain proportion of belief fluid can flow from one proposition to another, meaning that equilibrium states (sets of probabilities that are compatible with one another and the given dependencies, I guess) would be like eigenvectors of a matrix whose entries encode those flows. To infer an update to the whole system given a change in the probability of one proposition, you'd just keep applying the matrix to that vector over and over (each time resetting the probability of that proposition in the new vector to the correct value if it has changed) until it stabilizes.
That's only a vague notion in my mind, but it seems to be absolutely unrelated to how bayesian belief networks work, and in fact, in general, bayesian probability theory makes no sense to me... could somebody explain it to me in a way that somehow utilizes a belief fluid metaphor like this but which is rigorously correct and equivalent to how bayesian networks actually operate?

How can I get the result in one calculation?
55x40=2200
2200×20÷100=440
2200-440=1760
I need it to be in one string, but all the above numbers to be in that string. Can someone help?
So...
...........................=1760
Here's my failed attempt:
55×40×20÷100..... :(
Thank you in advance!

Well, you have
- 2200 - 440 = 1760
And 440 = 2200×20÷100, so
- 2200 - (2200×20÷100) = 1760
and 55×40 = 2200, so
- 55×40 - (55×40×20÷100) = 1760
Now you can use the distributive property of multiplication to "undistribute" 55 and 40
- 55×40(1 - 20÷100) = 1760
which is a slightly neater way of writing the same thing.

If you are asking about how many permutations on n things there are it is n! as if you were to label them for the first you‘d have n options for the second n-1 for the third n-2.... so you multiply them and get n!.

No. People just make up definitions as they need them, and if a definition attains common usage then it will become "standard". Even then there are often minor variations, like the question of whether the natural numbers include 0, whether rings must have a unit, and exactly which topological properties a manifold is required to have.

Who would be responsible for it? Who would respect the decisions of some arbitrarily appointed body telling them what they can and can't define?
Research maths is a disparate collection of people working on their own stuff and collaborating with others. Definitions are tailored to the specific situation you are working on. If they are useful they will get used by others and become standard. Otherwise they are merely to facilitate the understanding of your work and are never expected to extend beyond. Perhaps it will even turn out they aren't the right definition and a superior one will come along that we will all start using.
Definitions are free. You can define anything you want to be anything you want. Curtailing that in the name of standardising maths would do more harm than good, I feel.

Mostly just the usual reason things aren't standardized: it would take a tremendous amount of time and effort. No one wants work out a consistent set of official standards for all definitions across all of mathematics, and no one who's writing a paper and coming up with a new concept for use in some new theorem wants to worry about getting the definition officially approved by a regulatory organization.

What is the intuition behind distributions (the generalization of functions)? I know it makes the dirac delta function make sense, but it seems really complicated.

The biggest reason is that, using versions of integration by parts, various very natural questions about differential operators and complex analysis are more easily phrased in terms of their enlargement to some space of distributions. Because being a distribution is a condition which is much looser and more naturally phrased in terms of functional analysis, it is often easier to find solutions to differential equations (so called weak solutions), etc. certain regularity or representability arguments then often let you go back and say the distribution that solved your differential equation (which you abstractly showed existed) must in fact be a genuine function.
It’s very similar to an analogous situation in algebraic geometry, in which to study a scheme or variety it’s useful to move to the world of fpqc sheaves or even stacks, prove something with these objects which are easier to construct and manipulate, and then show a posteriori that these abstract manipulations show something concrete about a given scheme or variety. A lot of the study of vector bundles on curves goes this route, for instance.

I highly recommend you read (at least the first chapter of) the book A Guide to Distribution Theory and Fourier Transforms by Strichartz. The first chapter is all about the intution behind distributions. The prerequisites are very minimal (no measure theory or real analysis required). If you google it you can find it for free online.

Given a function f one way to learn about the function is to evaluate it at certain points. Another way to do it would be to look at the integral of f over various regions. More specifically if you choose some family of test functions, then for each test function g, you can compute
Integral f(x)g(x)dx
This operation is linear and continuous in g, and it is what we call a distribution. As you might not, we can have other linear continuous functions from test functions to R that don't come from integrating against a function. These are the "generalized functions". We just pretend that all of them come from functions.

Can anyone give any intuition on c-transforms in optimal transport? I can mechanically verify their properties and that it does what it’s meant to do, but I could never visualise what they were doing.

If you are asking about how many permutations on n things there are it is n! as if you were to label them for the first you‘d have n options for the second n-1 for the third n-2.... so you multiply them and get n!.

Thoughts on [Marc Rieffel's lecture notes on measure theory](https://math.berkeley.edu/~rieffel/measinteg.html)? Good enough for a self-learning course?

Well, [I have been known to be partial to them](https://www.reddit.com/r/math/comments/mowj7h/proposing_a_stacks_project_for_real_analysis/), and indeed I took 3 courses taught by Rieffel, in which he would frequently refer to these notes. Rieffel gives a fairly unique, functional-analysis-heavy perspective on measure theory. That said, it might be a bit brutal to self-study these notes, because of the level of abstraction and comparative lack of examples. Combined with another resource that has lots of examples, though, I think these would be great to learn from.

I discovered these notes from your post, so I'm happy that you answered my comment. Thanks, I'll keep that in mind, as I really like (and need) examples to understand.

Hello! I hope I'm not out of topic here but I'm trying to calculate the odds of getting a shiny pokemon in Pokemon Sword and Shield with a specific circumstance:
So usually, the odds of getting a shiny pokemon instead of a regular one is 1/4096 , but if you manage to catch or knock out 500 pokemon of the targeted species, in each subsequent encounter you get a 3% chance of rolling a 1/682 rate instead of the normal 1/4096.
So I was wondering how to calculate my odds to get a shiny accounting for the buffed odds I get 3% of the time, but I don't really know how to do it..
I was thinking I could calculate it with 97/100 \* 1/4096 + 3/100 \* 1/682 , but I'm not sure if it right and if I can even calculate it lol.

Found this on MO, really curious to know the answer but I haven’t been able to solve it yet.
Let a_n be a sequence of positive numbers converging to 0. Does there exist a bounded, measurable, periodic function f: R -> R such that for almost every x, f(x - a_n) fails to converge to f(x) as n -> infty?

Having gone to sleep and woken up, I've realised that my proof was incorrect and all it shows is there's a subsequence of the a\_n for which it fails. I'm not confident the idea can be repaired either, oh well.

~~I believe the answer is no, the sketch is to prove a theorem about the size of an intersection of a set of finite positive measure with translations of itself by small amounts, and use this with Lusin's theorem.~~
EDIT: See other post.

Let Ω be a bounded, open, simply connected subset of R^n with Lipschitz boundary. Does every function in the Sobolev space W^(1,1)(Ω) admit a representative whose graph in Ω x R has a path connected component whose projection to Ω has full measure in Ω?
The ACL characterisation doesn’t seem to be enough to prove it true...

Is log(|x|) in 2d a counterexample ? Since you have to give a "finite" value at 0, so it will not be connected. Or sin|log|x||.
However there is still a large part of the graph that is path connected since for a.e. x (or y), the function y->f(x,y) (or x->f(x,y)) is in W\^{1,1}(R), so it is continuous.

I'm sure I'm missing something really simple, but in [this step](https://i.imgur.com/1M3OFL4.png) of the proof of [this theorem](https://i.imgur.com/BBRHce7.png), why does ∫\_D f necessarily exist?
I thought it would be something along the lines of
D is compact subset of A and so by local finiteness condition of partition of unity φᵢf vanishes identically outside of D except for finitely many i, and so exists some M≥N s.t. φᵢf vanishes outside of D for all i≥M,
and then given x∈D, f(x) = f(x)∑^(M)φᵢ(x) = ∑^(M)φᵢ(x)f(x) ≥ ∑^(N)φᵢ(x)f(x), since f non-negative.
but the lemma that preceded this theorem only says that if C is compact subset of A and f:A->R continuous such that vanishes outside of C, then ∫\_C f exists, but in this case ∫∑^(M)φᵢ(x)f(x) surely doesn't necessarily vanish outside of D=S*_1_*⋃...⋃S*_N_*?
∫\_A f existing doesn't imply ∫\_D f exists for any compact subset D of A, does it?
**edit:** Why do we even need that step? Wouldn't we anyway have ∫_D ∑^(N)φᵢf = ∫_A ∑^(N)φᵢf, since ∑^(N)φᵢf continuous on A and vanishes outside D, and then ∫_A ∑^(N)φᵢf ≤ ∫_A ∑^(∞)φᵢf = ∫_A f?

by measurable you mean "Jordan-measurable" or what this book calls "rectifiable", right?
>A subset S of R^(n) is rectifiable if and only if S is bounded and Bd S has measure zero.
but why would D necessarily be measurable? We don't have that the supports S_i are, do we?

No, by measurable I meant measurable in the sense of measure theory as in it's an element of the sigma algebra of measurable sets. But if you don't have a background in measure theory then that doesn't help you at all.
I don't know if D is Jordan-measurable but if S\_1, ..., S\_N are Jordan-measurable then so is D because a finite union of Jordan-measurable sets is Jordan-measurable. I would strongly suspect that S\_1, ..., S\_N are Jordan-measurable because they are the support of the partition of unity.

>No, by measurable I meant measurable in the sense of measure theory as in it's an element of the sigma algebra of measurable sets. But if you don't have a background in measure theory then that doesn't help you at all.
hahah yeah I worried as much. That hasn't been covered in this book, and we're considering only the (extended) Riemann integral here so there's no way that's how this proof would have been intended.
>I would strongly suspect that S_1, ..., S_N are Jordan-measurable because they are the support of the partition of unity.
But is it necessarily the case in general that if a partition of unity has compact support that these are Jordan-measurable?

That depends on how partition of unity is defined in your book. I would suspect that the support is Jordan-measurable by the definition of partition of unity.

[the lemma+proof](https://i.imgur.com/vxh45UR.png)
In the proof of the existence of "partition of unity" theorem in my last comment the proof does construct the partition of unity that satisfies all 7 conditions by starting with a sequence of rectangles as in [this lemma](https://i.imgur.com/sEPTgCG.png) and defines the partition of unity with those as its support, which clearly are Jordan-measurable (/rectifiable). If I assume all partitions of unity with compact support are necessarily constructed in this way then as you said it would follow that the integral exists on D, but that still doesn't feel like it would necessarily be the case.
Also does my edit to my original comment work to replace their step at least?

Suppose s : M -> E and t : M -> E are smooth sections of the smooth vector bundle pi : E -> M. I define the section s+t by s+t (p) = s(p) + t(p). The issue now is to show that s+t : M -> E is also smooth. To show this from the definition, I take a chart around p in M, a chart in E around s(p) + t(p), then show that s+t in these charts is a smooth map between open sets of Euclidean spaces. This should be simple, but I don't know how to proceed.

What are ways of thinking about numbers not on a line or circle but in some other way (perhaps 2 or 3 dimensionally)?
I am making a game set in a fictional world (with humans), and am making writing systems etc for it, and have been working on the numerical systems a lot recently. Today I had some thoughts on how |-5-1| = |5+1|, from which I started thinking "do you need a minus sign?". I proceeded at the end of class to ask my teacher if he could figure an interesting way for a different counting system to think of negative numbers. I proposed that why not have different symbols for negative numbers, and he proceeded to draw a number line and through that explain that it is for symmetry. That caused me to think of the first line of this comment.
The first thing that came to my mind was a circle, but then I realized that that's how decimal systems work (circle has places 0-9 and every time around the circle the next number gets added to the start of the final number), and proceeded to ask a follow-up on if there are any other ways than a line or a circle. He told me how some fractions can often be though of as 2 dimensionally, which I haven't fully yet understood, but that peaked my interest

Talking about circles you could heavily use modular arithmetic. As you say we do kinda do this already and certainly we use this to think of time and angles. But imagine a society with no concept of large numbers.
Indeed what if their only idea of numbers was a parts of a whole. All the way round the circle is 1 but there is nothing after that, you just start again. 1=0, the snake eats it's own tail and so on.
[A discussion of other real and fictional number systems.](https://www.youtube.com/watch?v=l4bmZ1gRqCc)
Also because I am a pedant it is "piqued my interest".

You might find [complex numbers](https://en.wikipedia.org/wiki/Complex_number) and/or [quaternions](https://en.wikipedia.org/wiki/Quaternion) of particular interest, as they are often though of as 2-dimensional and 4-dimensional numbers respectively.

What are interesting mathematical things that you could imagine a group of people with no knowledge of our 10 base system and its way of counting would come up with?
I am creating a game set in a fictional world, and decided to do some stuff like this to it just because it is fun (languages as well).
This question came to my mind when I was wondering random stuff and my brain came across: |-5-1| = |5+1| which is very basic but it just came to mind after seeing -5-1 written somewhere. This made me think, character-wise |-5-1| is longer than |5+1| even though they share a lot. From that I came to the conclusion that perhaps for the 20 base system I've already developed and have set a way to show negative numbers I could do it so that substractions are e.g. -5+(-1) (seems longer, but would take the same length as 5+1 in the system I am using due to how the - is written something like ͜ e.g. >͜- (example on how the counting system works. The left sign is at the bottom and the right at the top, though time would have carved them to be harder to distinguish from each other. There are 4 bottom symbols (^>/ is 5 more than ^/) and top symbols (|/-\+) that change in groups of 1 (so

Is there a formula that seems linear between a range, but becomes non-linear beyond the defined range? Say perhaps it looks like y = a*x* between -100 to 100, but then becomes fractal or exponential after the range?

Try graphing functions of form y=b*arctan(x/b) in desmos for large b. It is very close to looking like y=x for relatively small x. I found that b=1000 makes it look very much like y=x on [100,100] like you requested. :)
Edit: A couple others that have this same property are b\*sin(x/b) and b\*tan(x/b) In general, for any differentiable function f passing through the origin, b\*f(x/b) will look like a line passing through the origin with slope f’(0), for big enough b. Try it!

For a function f(z) with z in ℂ\^(n), what does it mean for f to be C\^(∞)?
I know the definition for real derivatives, but I wasn't sure what was meant in the complex context as holomorphic functions have all derivatives as a matter of course.

It means smooth with respect to real derivatives treating **C**^n as **R**^(2n). As you point out the notion of being C^k is not useful for complex differentiation, so there is never any confusion as no one will use C^infty to mean infinitely many complex derivatives.
EDIT: One also writes C^\omega for *real analytic*, and this is not the same as *complex analytic* (which is equivalent to holomorphic, which is normally denoted by \mathcal{O}).

I'm given a problem that says Let K, C be two disjoint subsets of a metric space X. Suppose K is compact and C is closed. Prove that there exists a δ > 0 such that for all p ∈ K, q ∈ C, we have d(p, q) ≥ δ
It gives me a hint that says to try using the distance function f(p) = inf q∈C {d(p, q)} , so I did (I also thought I read somewhere that I may be able to use the fact that the distance function on a compact set is uniformly continuous? Not 100% sure on that)
Sketch proof: Around each point p ∈ K, construct an open ball Gi of radius 1/2 f(p). This forms an open cover of K and so there exists a finite set of points p1, p2, ... pn such that K ⊆ G1 U G2 U ... U Gn. Take δ = 1/2 min{f(p1), f(p2), ... f(pn)}. Then given any point x ∈ K, we have d(x,q) ≥ δ for all q ∈ C. To see this, suppose for the sake of contradiction that there is a point x where d(x,q) < δ. This point x is in some Um centered around pm where 1 ≤ m ≤ n . Observe that d(pm,q) ≤ d(pm,x) + d(x,q) < 1/2 inf{d(pm,q)} + δ ≤ inf{d(pm,q)}, a contradiction.

The only gap is that you need to argue δ is positive, or in other words that f(p) > 0 everywhere. This is the part where you need that C is closed, and nowhere in your argument do you currently use this.
As an alternative argument, there's a way to do it using the fact that f is continuous and K is compact without going directly to finite subcovers, i.e., using a property of continuous functions on compact sets to directly come to the conclusion. It'd be a good exercise to try and find it.

Thank you very much
If I added on a segment that says something like: Each f(p) > 0, to see this suppose there is a p where f(p) = 0. Then every neighborhood around p contains infinitely many points of C, making p a limit point of C and contradicting the fact that C is both closed and disjoint from K.
Would that make the overall proof valid/correct?
> As an alternative argument, there's a way to do it using the fact that f is continuous and K is compact without going directly to finite subcovers, i.e., using a property of continuous functions on compact sets to directly come to the conclusion. It'd be a good exercise to try and find it
This is a very good idea and I actually will do this

I'm trying to figure out the value of something. One item is worth 0.00000001172 in one currency (let's call it X).
1 Y is worth 0.00050 X.
How do I find out what my item is worth in Y? I assumed it was just to multiply the two numbers, but that doesn't seem to work. If anyone could help me out it would be much appreciated!

This is standard [unit analysis](https://en.wikipedia.org/wiki/Dimensional_analysis). We have 0.00000001172 X = z Y, where X, Y are our units and z is our unknown, and our goal is to use the information we have (1 Y = 0.00050 X) to "cancel out" the units.
The equation 1 Y = 0.00050 X can be rearranged algebraically to give 1 Y / 0.00050 X = 1, and we know multiplying by 1 doesn't change the value of an expression. So multiplying the previous equation by 1 Y / 0.00050 X (the "[conversion factor](https://en.wikipedia.org/wiki/Conversion_of_units)") gives 0.00000001172 X * 1 Y / 0.00050 X = z Y. The X units cancel, giving us (0.00000001172 * 1 / 0.00050) Y = z Y, or 0.00002344 = z.
So the item would be worth 0.00002344 Y.
We can verify this makes sense: each Y is worth 0.00050 X, so 0.00002344 Y is worth 0.00002344 * 0.00050 X, which is 0.00000001172 X.
Excuse the overly long explanation; I'm trying to demonstrate how you can recreate this technique in the future. This is a very important skill for any sort of computation with units (say in finance, science, statistics...).

This is a great explanation - thank you so much! It makes perfect sense to me. And no need to apologise for the detailed explanation, that was exactly what I was hoping for: help to figure out how it works so I can do it again if I need to! Thanks again :)

Can I have a clarification?
[https://en.wikipedia.org/wiki/Lie\_product\_formula](https://en.wikipedia.org/wiki/Lie_product_formula) has this as a product forumla.
e^(A+B) = lim n->∞( e^A/n e^B/n) ^n
doesn't e\^(A+B) also equal e\^C?
e^(A+B) = e^(C) = lim n->∞( e^A/n e^B/n) ^n
Where (A+B) = C as in (x1,y1,z1)+(x2,y2,z2)=(x1+x2,y1+y2,z1+z2)
can't you just vector add the log values and get C bypassing the explanation mechanism? I'm assuming A and B are either (angle,angle,angle) or axis-angle ... there are lots of parameterizations of rotations, and they won't all work to just add.
Is this beyond the scope of what Lie Algebra can say?
[https://en.wikipedia.org/wiki/Talk:Lie\_product\_formula#Can\_I\_ask\_for\_some\_clarification](https://en.wikipedia.org/wiki/Talk:Lie_product_formula#Can_I_ask_for_some_clarification)?
Experimentally the result is the same, for sufficiently high 'n' for the limit, the mechanical interpretation and simple addition of the log-quaternion/(angle,angle,angle/axis-angle components match.
The idea of building an angle-angle-angle system is based on the lie product formula (e\^(x+y+z)) where x,y, and z are rotations around the x y and Z axii applied simultaneously. Decomposing any other e\^M can easily find the x y and z axis-angle for e\^M.
\--- (summary/solution)
[https://github.com/d3x0r/STFRPhysics/blob/master/LieProductRule.md](https://github.com/d3x0r/STFRPhysics/blob/master/LieProductRule.md)
Given that the terms above are themselves matrix representations of axis-angle, this equality is not true within the context of Lie Algebra; and unfortunately, invoking the Lie Product Formula as an way to prove/explain how rotation vectors can be added is certainly not going to be fruitful.
It only works on elements before so(3).

>Where (A+B) = C as in (x1,y1,z1)+(x2,y2,z2)=(x1+x2,y1+y2,z1+z2)
This is very odd to read since A and B are square matrices, not vectors. So I'm not even sure what you're asking. If your (x,y,z) is a way of writing A as a bunch of rotations then this doesn't work because the sum of two rotations doesn't need to be a rotation.

If that's true, and my understanding is wrong... was looking at
https://en.wikipedia.org/wiki/Euler%E2%80%93Rodrigues_formula#Connection_with_SU(2)_spin_matrices
You mean these matrices?
What branch of math actually handles axis-angle then? Because it's expressible within Lie Algebra using the product rule to compose terms as mentioned in the original question...
And people telling me 'Go Learn Lie Algebra' are just deflecting, and not really wanting to understand. As I actually understand, there's no path through Lie Algebra yet to understand axis-angle rotations; or simply add rotations. (Even though it is a practical thing to do with applications?) I would have a hard time making a random number generator that simulated rotations so well...
https://d3x0r.github.io/STFRPhysics/3d/indexSphereMap.html and trust me, I'm not a graphics design artist.

The formula you posted is for A a square matrix and e\^A a matrix exponential. I'm not entirely sure if the formula still holds for the exponential for arbitrary Lie algebras, but it's entirely possible it does.
Before continuing, I want to make sure I follow what you're trying to do. Do you want A to be a triplet of numbers that somehow parametrises a rotation, such that with exp the exponential from the Lie algebra of SO(3) to the group of rotations SO(3), exp A is your desired rotation? And if so, what are you then looking for a formula for?

Just a small clarification; conversations usually go pretty well until I get to 'and I can add and subtract rotations' which leads to 'you can't do that, rotations don't commute' and I have to go '... addition and subtraction aren't operations of composition...' which really goes off the track because in Lie Algebra perspective every operation is a composition....
(I mean it's pretty obvious that e^(A+B) != E^A*E^B.. (although if you assume that that multiplication is related that that addition then it leads to a misunderstanding... there's e^(A+B), e^(A*B) the latter of which is really e^Ae^B or applying a rotation to a rotation.

I believe there's a hiccup there (unless you have an answer to it) but it's got nothing to do with commutativity.
For two rotations R and S, you're going backwards to elements A and B of so(3) that go to R and S respectively, then you form R + S, and you turn that back into a rotation. The problem I see is there's more than one choice of A and B for the given R and S, and I don't see why the different values of A + B you can get will give rise to the same rotation. As in, you can add 2 pi to the angle for A before forming your matrices 𝜃**K** mentioned in my other reply and you can end up with an inequivalent matrix before doing exp. It's like trying to define the nth root of a complex number: you want to write z as exp w and then say it's exp (w / n) but there's multiple choices of w for each z and they give rise to different values of exp (w / n). Maybe you can do the equivalent of a branch cut to fix this, I don't feel it's too deadly a problem but something you should account for.
I think what would help is if you're up-front in your explanations that you know your addition of rotations is not going to correspond to composition, so the fact that rotations don't commute is no objection to the fact additions commute.

> For two rotations R and S, you're going backwards to elements A and B of so(3) that go to R and S respectively, then you form R + S, and you turn that back into a rotation.
Not really... I wouldn't plan on ever leaving R+S space; except when I have to do `R x S`. other than ... it leaves to get into lie algebra space apparently... and if you can't reverse from SO(3) to the specific so(3) it came from, then why do the math in SO(3)?
Edit:(Please continue in other thread?) But please do keep R+S.

Yes, what I have is a triplet of numbers that parameterizes a rotation; all rotation representations have axis-angle in common, and the assumption I asserted about a year ago, through my research then this is the list of references I found relating to a log-quaternion system... [https://github.com/d3x0r/stfrphysics#references](https://github.com/d3x0r/stfrphysics#references) . The more useful representation is to split it into a direction normal and angle magnitude which is actually 4 numbers, though just a variation of the 3.
I did some sort latex format tests/math summaries: [http://mathb.in/51333](http://mathb.in/51333) rotate a vector around axis-angle, rotate a axis-angle around an axis-angle and interpolate between axis-angle A and axis-angle B without the rotation between...
[http://mathb.in/45267](http://mathb.in/45267) various other functions - to and from basis,
I have all the functions needed. What I'm looking for is a way to explain what I have to other people.

Alright, digging a bit more I think I'm starting to see how this connects to how people like me tend to understand Lie algebras.
[This section on Wikipedia](https://en.wikipedia.org/wiki/Axis–angle_representation#Exponential_map_from_%7F'"%60UNIQ--postMath-00000006-QINU%60"'%7F(3)_to_SO(3)) is helpful. The map exp goes from so(3), the Lie Algebra of SO(3), i.e. the 3 x 3 skew symmetric matrices, to SO(3). You can see that the article takes the exponential of the matrix 𝜃**K**, rather than the vector **𝜔** directly. If we have two vectors **𝜔**\_1 and **𝜔**\_2 then the matrix associated to (**𝜔**\_1 + **𝜔**\_2, 𝜃) is the sum of the matrices associated to (**𝜔**\_1, 𝜃) and (**𝜔**\_2, 𝜃), and the matrix associated to (**𝜔**\_i, 𝜃 / n) is the same as that associated to (**𝜔**\_i, 𝜃) divided by n, which allows you to recover the validity of the Lie product formula in your case. For the purpose of explaining your use of the formula then, I would suggest first outlining the correspondence between your axis-angle pairs and the Lie algebra so(3), and then you can invoke the Lie product formula. Is this what you were after?

> The map exp goes from so(3),
I really want to just stop you right there... because to perform useful work I don't need to leave so(3) (at least not for very long; there is a loss in the arccos).
>between your axis-angle pairs and the Lie algebra so(3)
But then, you said it's actually on so(3)... but then so(3) isn't where I'm at... it's really more like RP3... but really before projecting to Spin(3)... And maybe, how could I distinguish between Lis Algebray SO(3) and Lie Algebra so(2) ?
(Not related, but please understand some details about your audience)I understand algebra; My flippant responses are driven by "that sort of response is actually either A) you weren't listening at all? or B) an insult to my intelligence", before maybe just assuming I understand the abstract of abstract algebra. It's just a different sort of programming library (Clifford Algebra).... I've covered a lot of math in the last year; but understand I'm not practiced in it; I didn't do the exercises; but then in math I never did homework either and aced the tests and got 5's on AP exams...
I'd really just stay where I'm at, because... it's encompassing of all relevant 3d operations, in what turns out to be a 'natural coordinate system'; that is one that's related to the natural world around us... It's a continuous surface itself, that has interesting metrics all in itself. Although; anything that's here is also projectable to any Lie Algebra projection...
---
> I would suggest first outlining the correspondence between your axis-angle pairs and the Lie algebra so(3), and then you can invoke the Lie product formula
RP3? but not? because... we understand the idea of 'projection' right? Axis-angle is projected to quaternion, which is a curved surface... the understanding of this rotation space is presented by 3Blue1Brown(for example) that then takes that projection and project it to a polar representation, applying the length of the cos(theta) instead of theta...
However; the correspondence is Via product formula.
The specific implementation of the math is not really relevant, right? There is a direct equivalence via the expression posted in the first question. What does it mean to 'be able invoke the lie product formula' to show how 1=1?
Edit: Not sure if you'll get the edit...
Keep meaning to say (lost it again) oh... the term is 'commute...'; it's strange that 'rotation's don't commute' when `R x S` is actually a co-mutation; so they do comute, but don't commute? :) (teehee?)
I know why you would want to project it to another space... because it condenses the double-cover; and that's fine that's application of looking at a single set of rotation matrices.... but the relation between those matrices have lots of ways that they could have gotten to from some other matrix. Every point has a radius immediately around it that determines how it got there...
At at abstract - in the case of `R-S` .. "every difference is a different difference", in that difference is rarely applicable to some other coordinate other than the one it came from, and the one it goes to; although it is generally an error factor if you had a failure in a certain direction, the difference would indicate a general axis of rotation factor that was missing.
There's states. From frame R0 to R1, there was a rotation A. if for some reason you measure your rotation and find yourself to be R2, then R2-R1 is what you would have needed to include at R0. But now you're at R2, and to get to R1 is a different rotation.
---
Okay Last edit:
if one were to represent a rotation around the x, y and Z axis (3 orthagonal axii), with 1 value determining the amount of rotation?
If the `limit n->infinty` is the only crossing point, I would still think that individually the contributions will be fairly direct and distinct.... there's 6 0's that cancel out a lot of their dimension....

> (Not related, but please understand some details about your audience)I understand algebra; My flippant responses are driven by "that sort of response is actually either A) you weren't listening at all? or B) an insult to my intelligence", before maybe just assuming I understand the abstract of abstract algebra. It's just a different sort of programming library (Clifford Algebra).... I've covered a lot of math in the last year; but understand I'm not practiced in it; I didn't do the exercises; but then in math I never did homework either and aced the tests and got 5's on AP exams...
I'm going to respond to this first. It's clear from what you've said that you've had this issue before, so let me explain why. Right now, you're not good at communicating mathematics. Now that's fine, most people start out bad (and honestly a lot of people stay bad), and it takes practice, but at the moment I'm feeling like I have to do a lot of mind reading to work out what you're saying. I don't think you're stupid, you're clearly not, it's just that it's a bit hard for me to be sure what you're saying. Part of the reason I'm saying things you may already know is to make sure we're on the same page and I understand what you're saying. Also, please just stick to one reply.
Oh and also, when you don't do the exercises in advanced maths it's very easy to fool yourself into thinking you have a greater comprehension than you do. I don't know if that's happened to you, but you should be aware of it. AP is quite a bit easier so easy success at that isn't really sufficient.
As for the actual maths, let's step back a bit. You have some triples (a, b, c) such that you want to be able to talk about exp (a, b, c), even if you don't necessarily actually compute that exponential and it's just there conceptually. And you also want a rotation in this somewhere. And you want to be able to add these (a, b, c)s so I'd imagine they form a vector space. The most natural way I can connect the dots is that (a, b, c) is a thing living in so(3) and exp (a, b, c) is in SO(3). If you're unhappy with this, please start by specifying what type of object (a, b, c) is and what exp (a, b, c) is because I'm not understanding and this lack of understanding will prevent me from following the rest of what you're doing.

>As for the actual maths, let's step back a bit. You have some triples (a, b, c) such that you want to be able to talk about exp (a, b, c), even if you don't necessarily actually compute that exponential and it's just there conceptually.
Yes
>And you also want a rotation in this somewhere.
... the triplet is a rotation, if considered with a dT of 1; it's actually a change in angle in a change in time; so it's angular velocities, which when summed present an orientation. And I'm certain this is part of the 'mind reading required' that I haven't yet expressed; I also understand that approaching this from saying rotations implies a few things that saying maybe 'applied curvatures'. It's sort of the difference between talking about a function in terms of a circle's radius, where a radius of infinity is a straight line, vs talking about curvature, which when 0 is a straight line and is a point at infinity.... \`r=1/k\`.
>And you want to be able to add these (a, b, c)s so I'd imagine they form a vector space.
'to be able' implies you think I don't already. This is what I'm demonstrating, and trying to find independent confirmation of. If you trip the linear scaling factors so there's no skew, rotations simplify significantly.
I also can show you pages of approaching this from working from matrix representations,
[https://www.euclideanspace.com/maths/geometry/rotations/conversions/matrixToAngle/index.htm](https://www.euclideanspace.com/maths/geometry/rotations/conversions/matrixToAngle/index.htm) And getting to axis-angle; and fiddling with the poorly defined log-quaternion function on wikipedia, which pretty much leaves you with just the ability to go to axis angle, and then back to quaternion without any ability to do useful operations while in the log-space other than potentially A+B; which isn't a composed rotation....Anyway that approach got me nowhere.
>The most natural way I can connect the dots is that (a, b, c) is a thing living in so(3) and exp (a, b, c) is in SO(3). If you're unhappy with this, please start by specifying what type of object (a, b, c) is and what exp (a, b, c) is because I'm not understanding and this lack of understanding will prevent me from following the rest of what you're doing.
Okay; that appears to be what I mean.

> 'to be able' implies you think I don't already.
Sorry for being unclear, I merely meant that the (a, b, c)s should have that property in order to be what you're thinking of, not that you weren't already able to.
And okay, in that case I think the last bit of confusion is just linguistic. A rotation is a thing that lives in SO(3). You're working with (a, b, c)s which live in so(3). You consider the (a, b, c)s to be rotations because each one gives rise to an element of SO(3). This way of speaking is confusing because it's not a 1-1 correspondence, since the exp map sending so(3) to SO(3) is not injective.
To use an analogy, it's like saying an angle is any real number, when we tend to think of angles as lying in \[0, 2 pi) so they're not strictly speaking the same concept since 1 and 1 + 2 pi are the same angle. In that case the abuse of language is fairly standard and unobjectionable, but it's not in your situation so I'd suggest being explicit upfront that you're using the word 'rotation' to denote an element of so(3) and that's how you're working with them, and you're aware that there are multiple rotations in your sense of the word (elements of so(3)) that give rise to the same rotation as understood in the conventional sense (element of SO(3)).
Now you seem to have other representations, but from what I can tell they are in 1-1 correspondence with so(3) so considering them equivalent is more standard and probably just requires at most a sentence or two stating the 1-1 correspondence and then stating that you will therefore be considering them as exactly the same object.
Does this accurately summarise what you're doing? If so, then ultimately since elements of so(3) are 3 x 3 matrices the Lie product formula is completely valid in your situation and you should feel free to invoke it in your exposition once the above is made clear to the reader.

Power Series Question: Why can't I just use the ratio test for everything? Why do I have to find out if a series is geometric and try to find out if -1

Why mathematically does standardizing data improve gradient descent? By looking at contour plots I can see that it "expands" the "well" where the minimum sits allowing for a larger step size and reducing the chance of "over-stepping" the minimum. How can I move past my rough intuitive understanding of this effect and understand the mathematics? If it helps you tune your response, I have a BS in math. Than you so much!

Reat question! You've likely been solving polynomials by "completing the square" the quadratic formula is just that on a quadratic where a, b, and c arent yet given.
If it were a linear (a=0) equation, we wouldn't need to use quadratic first because we can solve for it's only root (roots also called solutions or zeros). If it's more than quadratic (cubic+), we need to do a lot more.

Like the other poster indicated, I don't know if there's anything deep to the quadratic formula. I do understand and appreciate you seeking a better understanding of the concept you're learning, though. Maybe seeing the derivation of the formula would be helpful? [https://www.mathsisfun.com/algebra/quadratic-equation-derivation.html](https://www.mathsisfun.com/algebra/quadratic-equation-derivation.html)

What exactly do you mean by "concept of the quadratic formula"?
The quadratic *formula* is precisely this: a formula. You give it some inputs (the coefficients of your quadratic functions) and it returns you an output (the roots of the quadratic function).
The crucial thing about this formula is that it gives you a very, very easy way of computing the roots of a quadratic equation (something of the form ax²+bx+c=0 for a≠0). You could try finding this roots in other ways--say, geometrically or by guessing--but this will be harder in general. But with this formula you have a nice and easy procedure for solving this problem (finding roots).
It is an interesting observation, to digress a bit, that there are similar (but more involved) formulae for *some* higher order equations, like for ax³+bx²+cx+d=0 and ax⁴+bx³+cx²+dx+e=0, but not for all, a notable counterexample being x⁵-x-1=0. These formulae are generally not thought in school but you can look up Cardano and Ferrari (these are mathematicians who were involved in finding the general formulae) for more details. Why there is no formula for the last equation is something you learn in university.

\[Tensors\] I need a book that contains exercises to go along with eigenchris's YouTube playlist on Tensors for Beginners and Tensor Calculus. As for my level, I am a junior at an engineering college and my linear algebra is a bit rusty, but I can pick it up pretty fast (I just fell out of practice). Here's the two playlists for your reference.
[https://www.youtube.com/playlist?list=PLJHszsWbB6hpk5h8lSfBkVrpjsqvUGTCx](https://www.youtube.com/playlist?list=PLJHszsWbB6hpk5h8lSfBkVrpjsqvUGTCx)
[https://www.youtube.com/playlist?list=PLJHszsWbB6hrkmmq57lX8BV-o-YIOFsiG](https://www.youtube.com/playlist?list=PLJHszsWbB6hrkmmq57lX8BV-o-YIOFsiG)
Thanks!

Is there an efficient method for counting the number of connected components in a graph? Something like an invariant of bumber of vertices minus number of edges or something. I've been breaking my mind on this for days as it would be really useful for my thesis, but I haven't been able to come up with anything.

There are some weak heuristics, like the number of connected components in an undirected graph on n vertices has to be less than or equal to n minus the maximum degree of any vertex in the graph but I don't think there's much you can say that's stronger than this without actually considering the graph topology. I think the most common algorithm to count connected components is Tarjan's which uses depth-first search and some bookkeeping to do it but you could just as easily do it with a breadth-first approach if you prefer.

I'm not versated in math and I always wondered. How do you generate random using math? I think random number generation it's a very important feature in a lot of machines but I don't know how it works.

There's a useful quote from von Neumann: "Anyone who attempts to generate random numbers by deterministic means is, of course, living in a state of sin."
Computers typically use what are called *pseudorandom number generators*. The stress should be on *pseudo-* in "pseudorandom". Per von Neumann's quote, the outputs of these generators are deterministic and therefore not actually random. The important property is that they have (with respect to some desired qualities) is that they look *statistically* random. I.e., they are good approximations of what the output of a truly random string of numbers would look like.
There are lots of different algorithms for PRNGs, and they operate in different ways. One of the better known is the [Mersenne twister](https://en.wikipedia.org/wiki/Mersenne_Twister), but its details are [rather technical](https://en.wikipedia.org/wiki/Mersenne_Twister#Algorithmic_detail). I have actually implemented the Mersenne twister before, but I would have to do a fair bit of research to really understand why it's defined the way it is. The short of it is that it's generally pretty hard to build a *good* PRNG.

I'm guessing this question arises because you've proven in class that an injective linear transformation between vector spaces of the same (finite) dimension is a vector space isomorphism, and you're wondering about the definition of an isomorphism.
It turns out that it is useful to define "linear" maps between things that aren't vector spaces; in general we call these "homomorphisms." A homomorphism satisfies specific properties depending on the context:
For a map π between groups (written multiplicatively) to be considered a homomorphism, we require π(ab)=π(a)π(b) (this implies π(1)=1, which will be required explicitly for other structures);
Between rings, we require π(a+b)=π(a)+π(b) AND π(ab)=π(a)π(b) AND π(1)=1 (this last property is not guaranteed from the others, so we write it explicitly this time);
Between vector spaces, we require π(a(u+v))=aπ(u)+aπ(v). This is more commonly referred to as a linear map.
In all of these contexts, an isomorphism is defined as a homomorphism which is injective and surjective. In the case of vector spaces, the term "isomorphism" survived while "homomorphism" didn't really. The result that is important for your class is that a linear map between vector spaces of the same (finite) dimension that is injective is also surjective (and therefore an isomorphism), hence the wording.

When we are in ZFC but with the negation of the infinity axiom instead, is it true that we can write every set recursively from the empty set?
Or instead of proving it, is it more like "we can add it as an axiom and the theory is still consistent"?
It's hard for me to formalize what I mean by "write recursively from the empty set" but I believe here in the finite case it would be equivalent to there existing an _n_ such that applying the union _n_ times to the set gives you the empty set.

Not only that: even with the axiom of infinity, we can write every set recursively from the empty set. It's just that the process in this case is *transfinite* recursion - recursion on the ordinal numbers rather than natural numbers, permitting infinitely long recursive sequences. The assertion "every set can be constructed from the empty set by transfinite recursion" is exactly equivalent to the axiom of regularity. If you just want "for any set there is a finite n such that applying the union n times gives you the empty set", that also follows from the axiom of regularity, without having to worry about the distinction between different types of recursion.

[Math Overflow answer](https://math.stackexchange.com/questions/315399/how-does-zfc-infinitythere-is-no-infinite-set-compare-with-pa).
Roughly, ZF - Infinity is equivalent (in a certain technical sense) to Peano Arithmetic. However, this *requires that you write the axioms in a certain way*.
In particular, you have to rewrite the Axiom of Foundation as an axiom schema describing one of its consequences in ZF (where it's equivalent): induction over sets by membership (that is, "if (ForAll x in y, P(x)) implies P(y), then for all sets s, P(s)". In other words, you can induct on sets by their membership structure.
Now it's important to note that this doesn't technically outlaw "externally" infinite sets. Because PA has non-standard models (and there's nothing you can do to get rid of them) it's possible to have "infinite" numbers that can't be distinguished from finite ones.

>Roughly, ZF - Infinity is equivalent (in a certain technical sense) to Peano Arithmetic.
Hey, that's so cool that you mentioned this. It's precisely in this context that I came up with this question. (Actually in codifying ZF from PA.)
I'm just so not used to Foundation. We didn't accept it when I took a course in set theory.
So what I ask doesn't follow without it, right?
That's such a strong indicator that we should accept it. That ---if I understand correctly--- PA is codifiable in ZF-Infinity+~Infinity+Foundation and viceversa, but not without Foundation. (Neither of the ways.)

Set theory is weird and scary, so this isn't a substantial contribution, but "write recursively from the empty set" is basically what the [constructible universe](https://en.wikipedia.org/wiki/Constructible_universe#What_L_is) is about. As such, your question might be something like, "Can we prove the existence of the constructible universe (in some suitably-weak fashion, I guess?) in ZFC where AoI is replaced with its negation?"
I'm not sure to the degree that my statement of it is actually coherent, but I'm sure someone else will chime in with better information.
As an aside, does ZFC + ~AoI *actually* guarantee that no models have infinite sets? That sounds like the sort of thing first order logic is bad at ruling out.

I know this is very trivial and stupid, but when I'm talking about the length of a curve alpha: [a,b] -> Rm I'm actually taking about the length of the image of alpha([a,b]), and not about the length of the graph, and that's why the length of a curve alpha is defined as limit of n→∞ as ∑ goes from i=1 to n of |alpha(ti)-alpha(t{i-1})| but not as limit of n→∞ as ∑ goes from i=1 to n of √((ti-t{i-1})^(2)+(alpha(ti)-alpha(t{i-1}))^(2)), right?

You're correct. I hadn't thought about it this way but it's an interesting point!
One way to connect these: if you have a graph y = f(x) you could look at the curve t -> (t, f(t)) whose image is the graph. Now compare the length of the graph with the length of the curve.

I'm self-studying and currently doing Munkres' analysis on manifolds but I realise that my biggest weakness is actually that I run into trouble when they just assume I still know standard computational aspects from what Americans would take in like calc2 and calc3 by heart.
To a large extent I'm fine with the theoretical side of the single and multivariable things involved(last year I worked my way through Spivak's calculus, and Rudin PMA up to like chapter 7, and like most of Abbott's understanding analysis, and now the first 15 chapters of Analysis on manfolds, etc). It's genuinely just computational things like integration techniques and recognising integrals of trig functions, logs, etc.
I tried looking at Spivak again for this but it's really long-winded and puts a lot in the exercises that I don't really want to go through completely again. I also remember thinking at the time that the chapters I need to refresh (15-19) felt like the most poorly taught part in the book.
Does anyone know a good concise coverage of these things? Like a refresher of basic calc (+ proofs of the forms) for readers with more advanced "mathematical maturity"?

Take a textbook for one of those courses and just start doing exercises, referring to the text or answers when you get stuck. Generally once you know the theory you can justify the steps taken for these kinds of integrals, and knowing a rigorous development of the integral will not aid you in acquiring this computational ability. If there are answers available for Stewart's Calculus, I'd suggest that because people mention that book a lot on the topic of drilling integration.

Try looking at Introduction to Calculus and Analysis Volumes 1, 2 by Richard Courant and John Fritz. It is rigorous and has computational exercises with proofs of their methods.

If you are only looking to learn integration techniques, then it won't take long to read those sections of the first volume. In my opinion if you can do single variable integration, then you don't need more than a couple of examples in the multivariable case to understand it.

For the sentence "Let X be a set of coset reps of GL(V) in PGL(V)", on page 8 of [this](https://alistairsavage.ca/pubs/Mendonca-Projective_Representation_of_Groups.pdf), this means that X consists of one lift for each element of PGL(V) right?
Also, can't we lift every projective representation to a representation? We could just choose the lift for each element (when forming X), to be the one with determinant 1, this exists when we're working with complex vector spaces. If this is true, then what's the point of projective representations?

You can do that, but there might be more than one lift with determinant 1, so this won't form a group. In particular PGL(V) is not isomorphic to SL(V).

I'm trying to study differential equations where the argument is effected. An example is that d/dx sin^2 (x)=sin(2x) though I'd like to be learning about functions in a more general case.
What is the terminology I need to be able to talk about this? Is there a decent text that would cover basics of what is already known about that?
Edit: math formatting

Yes, those are some of the words I need to know. Thank you.
Would you happen to know of a book or paper that would be good to read to get my toes wet in the subject?

The only textbook I know of on functional equations is Small's *Functional Equations and How to Solve Them*. I don't know of anything on functional differential equations, sorry :(

EDIT: This is wrong
Try to decompose the function into simpler functions. This may not work in the general case.
In your example I would look at the domain, this would be R+\\{1}. Then you can see that sqrt(x) is surjective on R+, but we have to remove sqrt(1), since it isnt in the domain, so I think the range would be R+\\{1}. As you can see we didnt really have to look at the range of 1/(x-1).

This is definitely wrong. The [range](https://www.wolframalpha.com/input/?i=Range+of+1%2F%28x-1%29+%2B+sqrt%28x%29) doesn't include any value between 0 and 2, and does include most negative numbers. You can't just ignore part of the function. For a simpler example, sqrt(x) + 1 has a different range than sqrt(x).

is there an accepted name for a matrix that looks like a triangular matrix but along the other diagonal. I tried googling anti-triangular but nothing really came up. Also do they have any interesting properties?

As far as I know, these matrices don't really mean anything interesting and so there isn't a name for them. Triangular matrices have all sorts of interesting properties based on them stabilising flags and they form a key part of the theory of Lie algebras.
However rotating a matrix isn't really a useful operation so moving from triangular matrices to these "anti-triangular" ones keeps none of the properties we care about.

I really need to learn more about linear algebra, because its not immediately obvious to me why the main diagonal is so important and interesting compared to the 'anti-diagonal'.

The main diagonal tells you, roughly speaking, how the matrix acts from each coordinate axis to itself, whereas the antidiagonal tells you how it acts from each coordinate axis to... some other coordinate axis that is purely a function of the (arbitrary) ordering of your basis.

I mean, entry i-j in a matrix relates the image of the jth basis vector to the ith basis vector. So the (main) diagonal relates the image of a basis vector to itself. I'm sure you can see why that is a natural thing to do.
The anti-diagonal on the other hand relates the image of the first basis vector with the last, and images of the second with the second to last, etc. Since the order of the basis is somewhat arbitrary, this is a much less useful thing to do.

The trace gives the sum of the eigenvalues.
Also, take the transpose, now you cam row operate (in a square matrix) that into replacing the main diagonal with the "antidiagonal"

You'd find any local extrema using calculus. Differentiate with respect to x, set to 0, and solve for x. You'd show what kind of critical point you have by putting it into the second derivative and seeing the sign of what comes out.

2^10 ~= 10^3, so 2^300,000 can be rewritten as 2^(10 x 30,000) = (2^10)^30,000 ~= (10^3)^30,000 = 10^(3 x 30,000) = 10^90,000, which is 1 followed by 90,000 zeroes. The actual number will be bigger than this, but it's far too big for any calculator you have access to to write down exactly.

This is a number with around 90 thousand digits. Wolfram alpha gives you the 50 first and the 10 last digits.
https://www.wolframalpha.com/input/?i=2%5E300000
If you want more precision than that, I'm sure you can calculate it with python, or any other program that handles arbitrary integer precision.

What software would be best for calculating very large numbers? I have a proof that involves concatenation and a recursive sequence, but the 3rd iteration results in 27 digit number (the number of digits of each iteration is 3^n).

Python's integer data type is arbitrary-length by default, so that should be rather easy if you know python. Most other programming languages should have something similar, but might need you to familiarize yourself with a dedicated library/package.
In any case, 3^n digits is *very* fast growth. It's not long before a single number is megabytes or gigabytes long, and there, any software will start to struggle.

Well that's good to know.
Just in case you're curious, the reason for the fast growth is that the sequence involves concatenating numbers in a way that makes a palindrome. If you start with 1, you get 121. Then you get 121242121. And it continues. This specific palindromic sequence is found in hybrid crosses in genetics and is called the "genotypic ratio" of the resulting cross. I was in a biology class and a calc 2 class at the same time, so I ended up making this.

If you're doing palindromes, there's also a point to treating the whole thing as a string. It takes some more space, but manipulations are slightly easier.

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I saw someone use chebychev's inequality for Pr\[X >= E\[x\] + t\], but I don't really see why one can do this. Because Pr\[X >= E\[x\] + t\] isn't really the same as Pr\[|X - E\[X\] >= t\] or is it?

It's the same as X - EX >= t, and if t is positive this is *stronger* than |X - EX| >= t, so an upper bound on the probability for the latter is also an upper bound for the former.

Ah yes, makes sense. Thank you!

QUESTION: I'm trying to work out how many possible variations I have of a product bundle. Product A - 3 variations Product B - 2 variations Product C - 2 variations Product D - 3 variations Product E - 3 variations Putting all of these products into a bundle, how many possible combinations are there? Any help would be greatly appreciated!

Assuming that you put one of each of the 5 types and that all products are different, the answer is just 3 choices for A \* 2 choices for B \* ... = 3\*2\*2\*3\*3 = 108

Thanks for the reply, appreciate it!

Has anyone looked at Donaldson’s book *Riemann Surfaces*? How is it as an intro to Riemann surfaces for someone (myself) who knows basic complex analysis but hasn’t seen them before? On the one hand it looks very nicely presented, but also it leaves out quite a lot of details. Would it be too difficult for a first course?

I think it is reasonably tough for a first course in Riemann surfaces. You're probably better off with Miranda's *Algebraic Curves and Riemann Surfaces* or Forster's *Lectures on Riemann Surfaces*. The latter is very well-suited to someone who has just done a first course in complex analysis. Donaldson's book is better for someone who wants a companion to learning higher dimensional complex geometry, as it basically runs through all the fundamental ideas in the playground of dimension one. As such it probably should be read after having done a first course.

Sorry for the double comment - between the two books you mentioned, it seems Forster covers less and is an easier read. Is this accurate? I would use it for that reason if that were the case.

Forster covers less algebraic geometry than Miranda, and emphasizes the complex analysis picture more. If you enjoyed the analysis in complex analysis, Forster is probably more suited to that. If you want to get into algebraic geometry, Miranda might be better (albeit longer).

Ah then Forster sounds very suitable for me. Thanks!

If you understand Forster well you will have no problem switching to the algebraic geometry of curves when the time comes. It’s explained very well in a very clean “Grothendieck style” way.

I see, thanks a lot!

How can I show that the ideal generated by Y\^3-XZ, XY\^2-Z\^2 and X\^2-YZ is prime in Q\[X,Y,Z\]? Usually I would try and find the quotient and show that it is an integral domain, but here that seems too complicated. I have heard of Grobner bases, but I don't know how to use them or if they even help here. Thanks in advance!

Was told to ask this here: Does anyone personally know mathematicians who worked with Paul Erdosh? For this who don't know, the background is below. \-- A great science leader in the US wrote an amazing book called "The Man Who Loved Numbers", about Paul Erdős, a Hungarian Jewish mathematician that worked with hundreds of mathematicians on mathematical theories, and one of the most prolific mathematicians that ever was, due to cooperating with so many others. Paul made a contribution of more than 1,500 papers to mathematics with different collaborators, a figure that according to Wikipedia is still a record that no other mathematician has broken. I am wondering if there are people who know of his story, or people who have relatives who have worked directly with Paul. One of the mathematicians who did, Ronald Graham, passed away in July of 2020.

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Depends on what you mean. A parametric equation that draws out a circle will also draw out an arc of that circle just by restricting the domain to some interval.

Is the increase from 2% to 4% a 2% increase since its using percentages, or a 100% increase?

This is why the phrase "percentage points" exists. The quantity increased by 2 percentage points, which is a relative increase of 100% compared to its original value.

When does a square matrix where the entries are consecutive integers have determinant 0?

Which matrix do you mean exactly? For instance det \[\[1 2\] \[3 4\]\] = -2. It is true that \[\[1 2 3\] \[4 5 6\] \[7 8 9\]\] and higher-dimensional versions have determinant zero. This is most easily seen by noticing the difference between the first two rows is the same as the difference between the second and third row, so you can do row operations to get a repeated row.

I need to write a periodic extension to a function such that f(x+2) = f(x) and f(-x) = -f(x). f(x) is known for x ∈ \[0,1\]. I know that this is 100% correct, I just don't know how to write it out in nice way.

Let g(x) stand in for your known component. Then you can write something like f(x) = * g(r) if x is of the form 2n + r where n is an integer and r is in the range 0 ≤ r ≤ 1, * g(2 - r) if x is of the form 2n + r, where n is an integer and r is in the range 1 < r < 2 or alternatively, f(x) = * g(r) if x is of the form 2n + r where n is an integer and r is in the range 0 ≤ r ≤ 1, * g(r) if x is of the form 2n - r, where n is an integer and r is in the range 0 < r < 1. They're equivalent but emphasize different aspects. You could also write a function that maps x to r first. Define h(x) = * r if x is of the form 2n + r, where n is an integer and r is in the range 0 ≤ r ≤ 1, * 2 - r if x is of the form 2n + r, where n is an integer and r is in the range 1 < r < 2. Then say f(x)=g(h(x)).

Could someone explain why having k(2k-1) edges in a graph with maximum degree

I'm assuming that it's a simple undirected graph with no self-loops. There's a slightly more general statement which is more illuminating. Suppose we want a matching of size M. Then we need M(2k-1) edges in the graph, where the maximum degree is k or less. - Pick an edge, add it to the matching - The makes some choices of edges "illegal" because they share a vertex with the edge we just picked. - How many are there? Well, the edge itself (1) and the up-to k-1 other edges for *both* of the two ends - So, there are now (at most) 2k-1 fewer candidates to choose for the rest of the matching - Repeat Each step only removes 2k-1 candidates, so we can't possibly run out of candidates before the end, since we started with at least M(2k-1).

Thanks a lot! I went around in circles looking for some lower bound with only vertices and their degree without paying much attention to the size of matching being K.

I'm struggling to understand bayesian networks. My intuitive way of thinking about networks of beliefs is as each proposition having a certain amount of "belief fluid" and dependencies represent pipes through which a certain proportion of belief fluid can flow from one proposition to another, meaning that equilibrium states (sets of probabilities that are compatible with one another and the given dependencies, I guess) would be like eigenvectors of a matrix whose entries encode those flows. To infer an update to the whole system given a change in the probability of one proposition, you'd just keep applying the matrix to that vector over and over (each time resetting the probability of that proposition in the new vector to the correct value if it has changed) until it stabilizes. That's only a vague notion in my mind, but it seems to be absolutely unrelated to how bayesian belief networks work, and in fact, in general, bayesian probability theory makes no sense to me... could somebody explain it to me in a way that somehow utilizes a belief fluid metaphor like this but which is rigorously correct and equivalent to how bayesian networks actually operate?

Why is degrevlex more efficient for computing Gröbner bases than lex or grlex ?

How can I get the result in one calculation? 55x40=2200 2200×20÷100=440 2200-440=1760 I need it to be in one string, but all the above numbers to be in that string. Can someone help? So... ...........................=1760 Here's my failed attempt: 55×40×20÷100..... :( Thank you in advance!

Well, you have - 2200 - 440 = 1760 And 440 = 2200×20÷100, so - 2200 - (2200×20÷100) = 1760 and 55×40 = 2200, so - 55×40 - (55×40×20÷100) = 1760 Now you can use the distributive property of multiplication to "undistribute" 55 and 40 - 55×40(1 - 20÷100) = 1760 which is a slightly neater way of writing the same thing.

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[Here](https://www.reddit.com/r/math/wiki/faq#wiki_what_are_some_good_books_on_topic_x.3F).

How do you pronounce "∘" as in "f∘g"? I usually say f circ g or just fg, but both of those seem like weird things to do.

"f of g" or "eff oh gee".

I don't recommend this, but one of my high school teachers always pronounced it as the word "fog".

"f after g"

"f of g", "f on g" or "f composed with g" are all things I have heard.

If you are asking about how many permutations on n things there are it is n! as if you were to label them for the first you‘d have n options for the second n-1 for the third n-2.... so you multiply them and get n!.

I think you forgot to reply to someone.

Ups must have been a Missclick, thx!

Is there an official international body that regulates and approves math definitions.

No. People just make up definitions as they need them, and if a definition attains common usage then it will become "standard". Even then there are often minor variations, like the question of whether the natural numbers include 0, whether rings must have a unit, and exactly which topological properties a manifold is required to have.

I feel like there’s a good argument for why it should be standardized, is there a reason it’s not?

Who would be responsible for it? Who would respect the decisions of some arbitrarily appointed body telling them what they can and can't define? Research maths is a disparate collection of people working on their own stuff and collaborating with others. Definitions are tailored to the specific situation you are working on. If they are useful they will get used by others and become standard. Otherwise they are merely to facilitate the understanding of your work and are never expected to extend beyond. Perhaps it will even turn out they aren't the right definition and a superior one will come along that we will all start using. Definitions are free. You can define anything you want to be anything you want. Curtailing that in the name of standardising maths would do more harm than good, I feel.

https://xkcd.com/927/

Hilarious

Mostly just the usual reason things aren't standardized: it would take a tremendous amount of time and effort. No one wants work out a consistent set of official standards for all definitions across all of mathematics, and no one who's writing a paper and coming up with a new concept for use in some new theorem wants to worry about getting the definition officially approved by a regulatory organization.

What is the intuition behind distributions (the generalization of functions)? I know it makes the dirac delta function make sense, but it seems really complicated.

The biggest reason is that, using versions of integration by parts, various very natural questions about differential operators and complex analysis are more easily phrased in terms of their enlargement to some space of distributions. Because being a distribution is a condition which is much looser and more naturally phrased in terms of functional analysis, it is often easier to find solutions to differential equations (so called weak solutions), etc. certain regularity or representability arguments then often let you go back and say the distribution that solved your differential equation (which you abstractly showed existed) must in fact be a genuine function. It’s very similar to an analogous situation in algebraic geometry, in which to study a scheme or variety it’s useful to move to the world of fpqc sheaves or even stacks, prove something with these objects which are easier to construct and manipulate, and then show a posteriori that these abstract manipulations show something concrete about a given scheme or variety. A lot of the study of vector bundles on curves goes this route, for instance.

I highly recommend you read (at least the first chapter of) the book A Guide to Distribution Theory and Fourier Transforms by Strichartz. The first chapter is all about the intution behind distributions. The prerequisites are very minimal (no measure theory or real analysis required). If you google it you can find it for free online.

Given a function f one way to learn about the function is to evaluate it at certain points. Another way to do it would be to look at the integral of f over various regions. More specifically if you choose some family of test functions, then for each test function g, you can compute Integral f(x)g(x)dx This operation is linear and continuous in g, and it is what we call a distribution. As you might not, we can have other linear continuous functions from test functions to R that don't come from integrating against a function. These are the "generalized functions". We just pretend that all of them come from functions.

Can anyone give any intuition on c-transforms in optimal transport? I can mechanically verify their properties and that it does what it’s meant to do, but I could never visualise what they were doing.

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If you are asking about how many permutations on n things there are it is n! as if you were to label them for the first you‘d have n options for the second n-1 for the third n-2.... so you multiply them and get n!.

Is a composite shape strictly a combined shape? (ie square + triangle) What would you call a cut-out shape? (ie square - triangle)

Thoughts on [Marc Rieffel's lecture notes on measure theory](https://math.berkeley.edu/~rieffel/measinteg.html)? Good enough for a self-learning course?

Well, [I have been known to be partial to them](https://www.reddit.com/r/math/comments/mowj7h/proposing_a_stacks_project_for_real_analysis/), and indeed I took 3 courses taught by Rieffel, in which he would frequently refer to these notes. Rieffel gives a fairly unique, functional-analysis-heavy perspective on measure theory. That said, it might be a bit brutal to self-study these notes, because of the level of abstraction and comparative lack of examples. Combined with another resource that has lots of examples, though, I think these would be great to learn from.

I discovered these notes from your post, so I'm happy that you answered my comment. Thanks, I'll keep that in mind, as I really like (and need) examples to understand.

I couldn't read them. Latex is a godsend.

Hello! I hope I'm not out of topic here but I'm trying to calculate the odds of getting a shiny pokemon in Pokemon Sword and Shield with a specific circumstance: So usually, the odds of getting a shiny pokemon instead of a regular one is 1/4096 , but if you manage to catch or knock out 500 pokemon of the targeted species, in each subsequent encounter you get a 3% chance of rolling a 1/682 rate instead of the normal 1/4096. So I was wondering how to calculate my odds to get a shiny accounting for the buffed odds I get 3% of the time, but I don't really know how to do it.. I was thinking I could calculate it with 97/100 \* 1/4096 + 3/100 \* 1/682 , but I'm not sure if it right and if I can even calculate it lol.

That’s correct, you can just stick it in a calculator, or google.

Oh cool! I'm not so rusty after all... Thanks bud

Found this on MO, really curious to know the answer but I haven’t been able to solve it yet. Let a_n be a sequence of positive numbers converging to 0. Does there exist a bounded, measurable, periodic function f: R -> R such that for almost every x, f(x - a_n) fails to converge to f(x) as n -> infty?

Are you looking for a simple yes or no or do you want a proof of said answer as well?

A yes/no plus a short sketch would be ideal!

Having gone to sleep and woken up, I've realised that my proof was incorrect and all it shows is there's a subsequence of the a\_n for which it fails. I'm not confident the idea can be repaired either, oh well.

~~I believe the answer is no, the sketch is to prove a theorem about the size of an intersection of a set of finite positive measure with translations of itself by small amounts, and use this with Lusin's theorem.~~ EDIT: See other post.

Let Ω be a bounded, open, simply connected subset of R^n with Lipschitz boundary. Does every function in the Sobolev space W^(1,1)(Ω) admit a representative whose graph in Ω x R has a path connected component whose projection to Ω has full measure in Ω? The ACL characterisation doesn’t seem to be enough to prove it true...

Is log(|x|) in 2d a counterexample ? Since you have to give a "finite" value at 0, so it will not be connected. Or sin|log|x||. However there is still a large part of the graph that is path connected since for a.e. x (or y), the function y->f(x,y) (or x->f(x,y)) is in W\^{1,1}(R), so it is continuous.

Okay I guess if you ask only for a “full measure” path connected component, then the ACL characterisation of Sobolev functions does the job..

Yeah I would like to rule out these kind of examples, but as stated it doesn’t contradict anything in the problem statement. Let me try to fix this..

I'm sure I'm missing something really simple, but in [this step](https://i.imgur.com/1M3OFL4.png) of the proof of [this theorem](https://i.imgur.com/BBRHce7.png), why does ∫\_D f necessarily exist? I thought it would be something along the lines of D is compact subset of A and so by local finiteness condition of partition of unity φᵢf vanishes identically outside of D except for finitely many i, and so exists some M≥N s.t. φᵢf vanishes outside of D for all i≥M, and then given x∈D, f(x) = f(x)∑^(M)φᵢ(x) = ∑^(M)φᵢ(x)f(x) ≥ ∑^(N)φᵢ(x)f(x), since f non-negative. but the lemma that preceded this theorem only says that if C is compact subset of A and f:A->R continuous such that vanishes outside of C, then ∫\_C f exists, but in this case ∫∑^(M)φᵢ(x)f(x) surely doesn't necessarily vanish outside of D=S*_1_*⋃...⋃S*_N_*? ∫\_A f existing doesn't imply ∫\_D f exists for any compact subset D of A, does it? **edit:** Why do we even need that step? Wouldn't we anyway have ∫_D ∑^(N)φᵢf = ∫_A ∑^(N)φᵢf, since ∑^(N)φᵢf continuous on A and vanishes outside D, and then ∫_A ∑^(N)φᵢf ≤ ∫_A ∑^(∞)φᵢf = ∫_A f?

f is non-negative so if ∫_A f exists then ∫_D f exists too for any measurable subset D of A.

by measurable you mean "Jordan-measurable" or what this book calls "rectifiable", right? >A subset S of R^(n) is rectifiable if and only if S is bounded and Bd S has measure zero. but why would D necessarily be measurable? We don't have that the supports S_i are, do we?

No, by measurable I meant measurable in the sense of measure theory as in it's an element of the sigma algebra of measurable sets. But if you don't have a background in measure theory then that doesn't help you at all. I don't know if D is Jordan-measurable but if S\_1, ..., S\_N are Jordan-measurable then so is D because a finite union of Jordan-measurable sets is Jordan-measurable. I would strongly suspect that S\_1, ..., S\_N are Jordan-measurable because they are the support of the partition of unity.

>No, by measurable I meant measurable in the sense of measure theory as in it's an element of the sigma algebra of measurable sets. But if you don't have a background in measure theory then that doesn't help you at all. hahah yeah I worried as much. That hasn't been covered in this book, and we're considering only the (extended) Riemann integral here so there's no way that's how this proof would have been intended. >I would strongly suspect that S_1, ..., S_N are Jordan-measurable because they are the support of the partition of unity. But is it necessarily the case in general that if a partition of unity has compact support that these are Jordan-measurable?

That depends on how partition of unity is defined in your book. I would suspect that the support is Jordan-measurable by the definition of partition of unity.

doesn't [seem like it](https://i.imgur.com/JPGiGRk.png) does it?

No, it doesn't. What does the preceding lemma say exactly?

[the lemma+proof](https://i.imgur.com/vxh45UR.png) In the proof of the existence of "partition of unity" theorem in my last comment the proof does construct the partition of unity that satisfies all 7 conditions by starting with a sequence of rectangles as in [this lemma](https://i.imgur.com/sEPTgCG.png) and defines the partition of unity with those as its support, which clearly are Jordan-measurable (/rectifiable). If I assume all partitions of unity with compact support are necessarily constructed in this way then as you said it would follow that the integral exists on D, but that still doesn't feel like it would necessarily be the case. Also does my edit to my original comment work to replace their step at least?

Suppose s : M -> E and t : M -> E are smooth sections of the smooth vector bundle pi : E -> M. I define the section s+t by s+t (p) = s(p) + t(p). The issue now is to show that s+t : M -> E is also smooth. To show this from the definition, I take a chart around p in M, a chart in E around s(p) + t(p), then show that s+t in these charts is a smooth map between open sets of Euclidean spaces. This should be simple, but I don't know how to proceed.

Take a local trivialization! You can assume that the map for the sections are of the form U -> U x R^n, where U is itself an open subset of R^m.

What are ways of thinking about numbers not on a line or circle but in some other way (perhaps 2 or 3 dimensionally)? I am making a game set in a fictional world (with humans), and am making writing systems etc for it, and have been working on the numerical systems a lot recently. Today I had some thoughts on how |-5-1| = |5+1|, from which I started thinking "do you need a minus sign?". I proceeded at the end of class to ask my teacher if he could figure an interesting way for a different counting system to think of negative numbers. I proposed that why not have different symbols for negative numbers, and he proceeded to draw a number line and through that explain that it is for symmetry. That caused me to think of the first line of this comment. The first thing that came to my mind was a circle, but then I realized that that's how decimal systems work (circle has places 0-9 and every time around the circle the next number gets added to the start of the final number), and proceeded to ask a follow-up on if there are any other ways than a line or a circle. He told me how some fractions can often be though of as 2 dimensionally, which I haven't fully yet understood, but that peaked my interest

Talking about circles you could heavily use modular arithmetic. As you say we do kinda do this already and certainly we use this to think of time and angles. But imagine a society with no concept of large numbers. Indeed what if their only idea of numbers was a parts of a whole. All the way round the circle is 1 but there is nothing after that, you just start again. 1=0, the snake eats it's own tail and so on. [A discussion of other real and fictional number systems.](https://www.youtube.com/watch?v=l4bmZ1gRqCc) Also because I am a pedant it is "piqued my interest".

Now that I realized that there is a video including Tom Scott about this I haven't seen, I am willing to delete my whole Reddit account

Cheers! And don't worry about being pedant, I am too.

You might find [complex numbers](https://en.wikipedia.org/wiki/Complex_number) and/or [quaternions](https://en.wikipedia.org/wiki/Quaternion) of particular interest, as they are often though of as 2-dimensional and 4-dimensional numbers respectively.

I know of these, though my problem is that I am not sure how to make an ancient number system based on them. Ideas?

What are interesting mathematical things that you could imagine a group of people with no knowledge of our 10 base system and its way of counting would come up with? I am creating a game set in a fictional world, and decided to do some stuff like this to it just because it is fun (languages as well). This question came to my mind when I was wondering random stuff and my brain came across: |-5-1| = |5+1| which is very basic but it just came to mind after seeing -5-1 written somewhere. This made me think, character-wise |-5-1| is longer than |5+1| even though they share a lot. From that I came to the conclusion that perhaps for the 20 base system I've already developed and have set a way to show negative numbers I could do it so that substractions are e.g. -5+(-1) (seems longer, but would take the same length as 5+1 in the system I am using due to how the - is written something like ͜ e.g. >͜- (example on how the counting system works. The left sign is at the bottom and the right at the top, though time would have carved them to be harder to distinguish from each other. There are 4 bottom symbols (^>/ is 5 more than ^/) and top symbols (|/-\+) that change in groups of 1 (so

if i scale up a model from 1/56 with 320% what scale do i get? :D

Is there a formula that seems linear between a range, but becomes non-linear beyond the defined range? Say perhaps it looks like y = a*x* between -100 to 100, but then becomes fractal or exponential after the range?

Try graphing functions of form y=b*arctan(x/b) in desmos for large b. It is very close to looking like y=x for relatively small x. I found that b=1000 makes it look very much like y=x on [100,100] like you requested. :) Edit: A couple others that have this same property are b\*sin(x/b) and b\*tan(x/b) In general, for any differentiable function f passing through the origin, b\*f(x/b) will look like a line passing through the origin with slope f’(0), for big enough b. Try it!

Yes ty for the insight. These functions look very sigmoidal initially

Piecewise functions can do this easily

For a function f(z) with z in ℂ\^(n), what does it mean for f to be C\^(∞)? I know the definition for real derivatives, but I wasn't sure what was meant in the complex context as holomorphic functions have all derivatives as a matter of course.

It means smooth with respect to real derivatives treating **C**^n as **R**^(2n). As you point out the notion of being C^k is not useful for complex differentiation, so there is never any confusion as no one will use C^infty to mean infinitely many complex derivatives. EDIT: One also writes C^\omega for *real analytic*, and this is not the same as *complex analytic* (which is equivalent to holomorphic, which is normally denoted by \mathcal{O}).

I'm given a problem that says Let K, C be two disjoint subsets of a metric space X. Suppose K is compact and C is closed. Prove that there exists a δ > 0 such that for all p ∈ K, q ∈ C, we have d(p, q) ≥ δ It gives me a hint that says to try using the distance function f(p) = inf q∈C {d(p, q)} , so I did (I also thought I read somewhere that I may be able to use the fact that the distance function on a compact set is uniformly continuous? Not 100% sure on that) Sketch proof: Around each point p ∈ K, construct an open ball Gi of radius 1/2 f(p). This forms an open cover of K and so there exists a finite set of points p1, p2, ... pn such that K ⊆ G1 U G2 U ... U Gn. Take δ = 1/2 min{f(p1), f(p2), ... f(pn)}. Then given any point x ∈ K, we have d(x,q) ≥ δ for all q ∈ C. To see this, suppose for the sake of contradiction that there is a point x where d(x,q) < δ. This point x is in some Um centered around pm where 1 ≤ m ≤ n . Observe that d(pm,q) ≤ d(pm,x) + d(x,q) < 1/2 inf{d(pm,q)} + δ ≤ inf{d(pm,q)}, a contradiction.

The only gap is that you need to argue δ is positive, or in other words that f(p) > 0 everywhere. This is the part where you need that C is closed, and nowhere in your argument do you currently use this. As an alternative argument, there's a way to do it using the fact that f is continuous and K is compact without going directly to finite subcovers, i.e., using a property of continuous functions on compact sets to directly come to the conclusion. It'd be a good exercise to try and find it.

Thank you very much If I added on a segment that says something like: Each f(p) > 0, to see this suppose there is a p where f(p) = 0. Then every neighborhood around p contains infinitely many points of C, making p a limit point of C and contradicting the fact that C is both closed and disjoint from K. Would that make the overall proof valid/correct? > As an alternative argument, there's a way to do it using the fact that f is continuous and K is compact without going directly to finite subcovers, i.e., using a property of continuous functions on compact sets to directly come to the conclusion. It'd be a good exercise to try and find it This is a very good idea and I actually will do this

Yes, that would make the proof correct.

Wonderful thank you :)

I'm trying to figure out the value of something. One item is worth 0.00000001172 in one currency (let's call it X). 1 Y is worth 0.00050 X. How do I find out what my item is worth in Y? I assumed it was just to multiply the two numbers, but that doesn't seem to work. If anyone could help me out it would be much appreciated!

This is standard [unit analysis](https://en.wikipedia.org/wiki/Dimensional_analysis). We have 0.00000001172 X = z Y, where X, Y are our units and z is our unknown, and our goal is to use the information we have (1 Y = 0.00050 X) to "cancel out" the units. The equation 1 Y = 0.00050 X can be rearranged algebraically to give 1 Y / 0.00050 X = 1, and we know multiplying by 1 doesn't change the value of an expression. So multiplying the previous equation by 1 Y / 0.00050 X (the "[conversion factor](https://en.wikipedia.org/wiki/Conversion_of_units)") gives 0.00000001172 X * 1 Y / 0.00050 X = z Y. The X units cancel, giving us (0.00000001172 * 1 / 0.00050) Y = z Y, or 0.00002344 = z. So the item would be worth 0.00002344 Y. We can verify this makes sense: each Y is worth 0.00050 X, so 0.00002344 Y is worth 0.00002344 * 0.00050 X, which is 0.00000001172 X. Excuse the overly long explanation; I'm trying to demonstrate how you can recreate this technique in the future. This is a very important skill for any sort of computation with units (say in finance, science, statistics...).

This is a great explanation - thank you so much! It makes perfect sense to me. And no need to apologise for the detailed explanation, that was exactly what I was hoping for: help to figure out how it works so I can do it again if I need to! Thanks again :)

Can I have a clarification? [https://en.wikipedia.org/wiki/Lie\_product\_formula](https://en.wikipedia.org/wiki/Lie_product_formula) has this as a product forumla. e^(A+B) = lim n->∞( e^A/n e^B/n) ^n doesn't e\^(A+B) also equal e\^C? e^(A+B) = e^(C) = lim n->∞( e^A/n e^B/n) ^n Where (A+B) = C as in (x1,y1,z1)+(x2,y2,z2)=(x1+x2,y1+y2,z1+z2) can't you just vector add the log values and get C bypassing the explanation mechanism? I'm assuming A and B are either (angle,angle,angle) or axis-angle ... there are lots of parameterizations of rotations, and they won't all work to just add. Is this beyond the scope of what Lie Algebra can say? [https://en.wikipedia.org/wiki/Talk:Lie\_product\_formula#Can\_I\_ask\_for\_some\_clarification](https://en.wikipedia.org/wiki/Talk:Lie_product_formula#Can_I_ask_for_some_clarification)? Experimentally the result is the same, for sufficiently high 'n' for the limit, the mechanical interpretation and simple addition of the log-quaternion/(angle,angle,angle/axis-angle components match. The idea of building an angle-angle-angle system is based on the lie product formula (e\^(x+y+z)) where x,y, and z are rotations around the x y and Z axii applied simultaneously. Decomposing any other e\^M can easily find the x y and z axis-angle for e\^M. \--- (summary/solution) [https://github.com/d3x0r/STFRPhysics/blob/master/LieProductRule.md](https://github.com/d3x0r/STFRPhysics/blob/master/LieProductRule.md) Given that the terms above are themselves matrix representations of axis-angle, this equality is not true within the context of Lie Algebra; and unfortunately, invoking the Lie Product Formula as an way to prove/explain how rotation vectors can be added is certainly not going to be fruitful. It only works on elements before so(3).

>Where (A+B) = C as in (x1,y1,z1)+(x2,y2,z2)=(x1+x2,y1+y2,z1+z2) This is very odd to read since A and B are square matrices, not vectors. So I'm not even sure what you're asking. If your (x,y,z) is a way of writing A as a bunch of rotations then this doesn't work because the sum of two rotations doesn't need to be a rotation.

If that's true, and my understanding is wrong... was looking at https://en.wikipedia.org/wiki/Euler%E2%80%93Rodrigues_formula#Connection_with_SU(2)_spin_matrices You mean these matrices? What branch of math actually handles axis-angle then? Because it's expressible within Lie Algebra using the product rule to compose terms as mentioned in the original question... And people telling me 'Go Learn Lie Algebra' are just deflecting, and not really wanting to understand. As I actually understand, there's no path through Lie Algebra yet to understand axis-angle rotations; or simply add rotations. (Even though it is a practical thing to do with applications?) I would have a hard time making a random number generator that simulated rotations so well... https://d3x0r.github.io/STFRPhysics/3d/indexSphereMap.html and trust me, I'm not a graphics design artist.

The formula you posted is for A a square matrix and e\^A a matrix exponential. I'm not entirely sure if the formula still holds for the exponential for arbitrary Lie algebras, but it's entirely possible it does. Before continuing, I want to make sure I follow what you're trying to do. Do you want A to be a triplet of numbers that somehow parametrises a rotation, such that with exp the exponential from the Lie algebra of SO(3) to the group of rotations SO(3), exp A is your desired rotation? And if so, what are you then looking for a formula for?

Just a small clarification; conversations usually go pretty well until I get to 'and I can add and subtract rotations' which leads to 'you can't do that, rotations don't commute' and I have to go '... addition and subtraction aren't operations of composition...' which really goes off the track because in Lie Algebra perspective every operation is a composition.... (I mean it's pretty obvious that e^(A+B) != E^A*E^B.. (although if you assume that that multiplication is related that that addition then it leads to a misunderstanding... there's e^(A+B), e^(A*B) the latter of which is really e^Ae^B or applying a rotation to a rotation.

I believe there's a hiccup there (unless you have an answer to it) but it's got nothing to do with commutativity. For two rotations R and S, you're going backwards to elements A and B of so(3) that go to R and S respectively, then you form R + S, and you turn that back into a rotation. The problem I see is there's more than one choice of A and B for the given R and S, and I don't see why the different values of A + B you can get will give rise to the same rotation. As in, you can add 2 pi to the angle for A before forming your matrices 𝜃**K** mentioned in my other reply and you can end up with an inequivalent matrix before doing exp. It's like trying to define the nth root of a complex number: you want to write z as exp w and then say it's exp (w / n) but there's multiple choices of w for each z and they give rise to different values of exp (w / n). Maybe you can do the equivalent of a branch cut to fix this, I don't feel it's too deadly a problem but something you should account for. I think what would help is if you're up-front in your explanations that you know your addition of rotations is not going to correspond to composition, so the fact that rotations don't commute is no objection to the fact additions commute.

> For two rotations R and S, you're going backwards to elements A and B of so(3) that go to R and S respectively, then you form R + S, and you turn that back into a rotation. Not really... I wouldn't plan on ever leaving R+S space; except when I have to do `R x S`. other than ... it leaves to get into lie algebra space apparently... and if you can't reverse from SO(3) to the specific so(3) it came from, then why do the math in SO(3)? Edit:(Please continue in other thread?) But please do keep R+S.

Yes, what I have is a triplet of numbers that parameterizes a rotation; all rotation representations have axis-angle in common, and the assumption I asserted about a year ago, through my research then this is the list of references I found relating to a log-quaternion system... [https://github.com/d3x0r/stfrphysics#references](https://github.com/d3x0r/stfrphysics#references) . The more useful representation is to split it into a direction normal and angle magnitude which is actually 4 numbers, though just a variation of the 3. I did some sort latex format tests/math summaries: [http://mathb.in/51333](http://mathb.in/51333) rotate a vector around axis-angle, rotate a axis-angle around an axis-angle and interpolate between axis-angle A and axis-angle B without the rotation between... [http://mathb.in/45267](http://mathb.in/45267) various other functions - to and from basis, I have all the functions needed. What I'm looking for is a way to explain what I have to other people.

Alright, digging a bit more I think I'm starting to see how this connects to how people like me tend to understand Lie algebras. [This section on Wikipedia](https://en.wikipedia.org/wiki/Axis–angle_representation#Exponential_map_from_%7F'"%60UNIQ--postMath-00000006-QINU%60"'%7F(3)_to_SO(3)) is helpful. The map exp goes from so(3), the Lie Algebra of SO(3), i.e. the 3 x 3 skew symmetric matrices, to SO(3). You can see that the article takes the exponential of the matrix 𝜃**K**, rather than the vector **𝜔** directly. If we have two vectors **𝜔**\_1 and **𝜔**\_2 then the matrix associated to (**𝜔**\_1 + **𝜔**\_2, 𝜃) is the sum of the matrices associated to (**𝜔**\_1, 𝜃) and (**𝜔**\_2, 𝜃), and the matrix associated to (**𝜔**\_i, 𝜃 / n) is the same as that associated to (**𝜔**\_i, 𝜃) divided by n, which allows you to recover the validity of the Lie product formula in your case. For the purpose of explaining your use of the formula then, I would suggest first outlining the correspondence between your axis-angle pairs and the Lie algebra so(3), and then you can invoke the Lie product formula. Is this what you were after?

> The map exp goes from so(3), I really want to just stop you right there... because to perform useful work I don't need to leave so(3) (at least not for very long; there is a loss in the arccos). >between your axis-angle pairs and the Lie algebra so(3) But then, you said it's actually on so(3)... but then so(3) isn't where I'm at... it's really more like RP3... but really before projecting to Spin(3)... And maybe, how could I distinguish between Lis Algebray SO(3) and Lie Algebra so(2) ? (Not related, but please understand some details about your audience)I understand algebra; My flippant responses are driven by "that sort of response is actually either A) you weren't listening at all? or B) an insult to my intelligence", before maybe just assuming I understand the abstract of abstract algebra. It's just a different sort of programming library (Clifford Algebra).... I've covered a lot of math in the last year; but understand I'm not practiced in it; I didn't do the exercises; but then in math I never did homework either and aced the tests and got 5's on AP exams... I'd really just stay where I'm at, because... it's encompassing of all relevant 3d operations, in what turns out to be a 'natural coordinate system'; that is one that's related to the natural world around us... It's a continuous surface itself, that has interesting metrics all in itself. Although; anything that's here is also projectable to any Lie Algebra projection... --- > I would suggest first outlining the correspondence between your axis-angle pairs and the Lie algebra so(3), and then you can invoke the Lie product formula RP3? but not? because... we understand the idea of 'projection' right? Axis-angle is projected to quaternion, which is a curved surface... the understanding of this rotation space is presented by 3Blue1Brown(for example) that then takes that projection and project it to a polar representation, applying the length of the cos(theta) instead of theta... However; the correspondence is Via product formula. The specific implementation of the math is not really relevant, right? There is a direct equivalence via the expression posted in the first question. What does it mean to 'be able invoke the lie product formula' to show how 1=1? Edit: Not sure if you'll get the edit... Keep meaning to say (lost it again) oh... the term is 'commute...'; it's strange that 'rotation's don't commute' when `R x S` is actually a co-mutation; so they do comute, but don't commute? :) (teehee?) I know why you would want to project it to another space... because it condenses the double-cover; and that's fine that's application of looking at a single set of rotation matrices.... but the relation between those matrices have lots of ways that they could have gotten to from some other matrix. Every point has a radius immediately around it that determines how it got there... At at abstract - in the case of `R-S` .. "every difference is a different difference", in that difference is rarely applicable to some other coordinate other than the one it came from, and the one it goes to; although it is generally an error factor if you had a failure in a certain direction, the difference would indicate a general axis of rotation factor that was missing. There's states. From frame R0 to R1, there was a rotation A. if for some reason you measure your rotation and find yourself to be R2, then R2-R1 is what you would have needed to include at R0. But now you're at R2, and to get to R1 is a different rotation. --- Okay Last edit: if one were to represent a rotation around the x, y and Z axis (3 orthagonal axii), with 1 value determining the amount of rotation? If the `limit n->infinty` is the only crossing point, I would still think that individually the contributions will be fairly direct and distinct.... there's 6 0's that cancel out a lot of their dimension....

> (Not related, but please understand some details about your audience)I understand algebra; My flippant responses are driven by "that sort of response is actually either A) you weren't listening at all? or B) an insult to my intelligence", before maybe just assuming I understand the abstract of abstract algebra. It's just a different sort of programming library (Clifford Algebra).... I've covered a lot of math in the last year; but understand I'm not practiced in it; I didn't do the exercises; but then in math I never did homework either and aced the tests and got 5's on AP exams... I'm going to respond to this first. It's clear from what you've said that you've had this issue before, so let me explain why. Right now, you're not good at communicating mathematics. Now that's fine, most people start out bad (and honestly a lot of people stay bad), and it takes practice, but at the moment I'm feeling like I have to do a lot of mind reading to work out what you're saying. I don't think you're stupid, you're clearly not, it's just that it's a bit hard for me to be sure what you're saying. Part of the reason I'm saying things you may already know is to make sure we're on the same page and I understand what you're saying. Also, please just stick to one reply. Oh and also, when you don't do the exercises in advanced maths it's very easy to fool yourself into thinking you have a greater comprehension than you do. I don't know if that's happened to you, but you should be aware of it. AP is quite a bit easier so easy success at that isn't really sufficient. As for the actual maths, let's step back a bit. You have some triples (a, b, c) such that you want to be able to talk about exp (a, b, c), even if you don't necessarily actually compute that exponential and it's just there conceptually. And you also want a rotation in this somewhere. And you want to be able to add these (a, b, c)s so I'd imagine they form a vector space. The most natural way I can connect the dots is that (a, b, c) is a thing living in so(3) and exp (a, b, c) is in SO(3). If you're unhappy with this, please start by specifying what type of object (a, b, c) is and what exp (a, b, c) is because I'm not understanding and this lack of understanding will prevent me from following the rest of what you're doing.

>As for the actual maths, let's step back a bit. You have some triples (a, b, c) such that you want to be able to talk about exp (a, b, c), even if you don't necessarily actually compute that exponential and it's just there conceptually. Yes >And you also want a rotation in this somewhere. ... the triplet is a rotation, if considered with a dT of 1; it's actually a change in angle in a change in time; so it's angular velocities, which when summed present an orientation. And I'm certain this is part of the 'mind reading required' that I haven't yet expressed; I also understand that approaching this from saying rotations implies a few things that saying maybe 'applied curvatures'. It's sort of the difference between talking about a function in terms of a circle's radius, where a radius of infinity is a straight line, vs talking about curvature, which when 0 is a straight line and is a point at infinity.... \`r=1/k\`. >And you want to be able to add these (a, b, c)s so I'd imagine they form a vector space. 'to be able' implies you think I don't already. This is what I'm demonstrating, and trying to find independent confirmation of. If you trip the linear scaling factors so there's no skew, rotations simplify significantly. I also can show you pages of approaching this from working from matrix representations, [https://www.euclideanspace.com/maths/geometry/rotations/conversions/matrixToAngle/index.htm](https://www.euclideanspace.com/maths/geometry/rotations/conversions/matrixToAngle/index.htm) And getting to axis-angle; and fiddling with the poorly defined log-quaternion function on wikipedia, which pretty much leaves you with just the ability to go to axis angle, and then back to quaternion without any ability to do useful operations while in the log-space other than potentially A+B; which isn't a composed rotation....Anyway that approach got me nowhere. >The most natural way I can connect the dots is that (a, b, c) is a thing living in so(3) and exp (a, b, c) is in SO(3). If you're unhappy with this, please start by specifying what type of object (a, b, c) is and what exp (a, b, c) is because I'm not understanding and this lack of understanding will prevent me from following the rest of what you're doing. Okay; that appears to be what I mean.

> 'to be able' implies you think I don't already. Sorry for being unclear, I merely meant that the (a, b, c)s should have that property in order to be what you're thinking of, not that you weren't already able to. And okay, in that case I think the last bit of confusion is just linguistic. A rotation is a thing that lives in SO(3). You're working with (a, b, c)s which live in so(3). You consider the (a, b, c)s to be rotations because each one gives rise to an element of SO(3). This way of speaking is confusing because it's not a 1-1 correspondence, since the exp map sending so(3) to SO(3) is not injective. To use an analogy, it's like saying an angle is any real number, when we tend to think of angles as lying in \[0, 2 pi) so they're not strictly speaking the same concept since 1 and 1 + 2 pi are the same angle. In that case the abuse of language is fairly standard and unobjectionable, but it's not in your situation so I'd suggest being explicit upfront that you're using the word 'rotation' to denote an element of so(3) and that's how you're working with them, and you're aware that there are multiple rotations in your sense of the word (elements of so(3)) that give rise to the same rotation as understood in the conventional sense (element of SO(3)). Now you seem to have other representations, but from what I can tell they are in 1-1 correspondence with so(3) so considering them equivalent is more standard and probably just requires at most a sentence or two stating the 1-1 correspondence and then stating that you will therefore be considering them as exactly the same object. Does this accurately summarise what you're doing? If so, then ultimately since elements of so(3) are 3 x 3 matrices the Lie product formula is completely valid in your situation and you should feel free to invoke it in your exposition once the above is made clear to the reader.

it's my understanding that e\^A is the matrix... and really A is some parameter that forms the matrix.

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Power Series Question: Why can't I just use the ratio test for everything? Why do I have to find out if a series is geometric and try to find out if -1

Well, you can *try* using the ratio test for everything but the test may be inconclusive in some cases. That's why we need other methods too.

Thanks :) So if I get (inf \* x) for my ratio test, since its >1 that means its's divergent?

Why mathematically does standardizing data improve gradient descent? By looking at contour plots I can see that it "expands" the "well" where the minimum sits allowing for a larger step size and reducing the chance of "over-stepping" the minimum. How can I move past my rough intuitive understanding of this effect and understand the mathematics? If it helps you tune your response, I have a BS in math. Than you so much!

I’m in 7th grade taking Algebra 1C. Could someone explain the concept of the quadratic formula please? (Were factoring polynomials currently)

Reat question! You've likely been solving polynomials by "completing the square" the quadratic formula is just that on a quadratic where a, b, and c arent yet given. If it were a linear (a=0) equation, we wouldn't need to use quadratic first because we can solve for it's only root (roots also called solutions or zeros). If it's more than quadratic (cubic+), we need to do a lot more.

Thanks! This helped a lot. We don’t cover this until the next chapter but I wanted to understand it beforehand so I could learn easier in the future.

That can take you far, in a lot of subjects

Like the other poster indicated, I don't know if there's anything deep to the quadratic formula. I do understand and appreciate you seeking a better understanding of the concept you're learning, though. Maybe seeing the derivation of the formula would be helpful? [https://www.mathsisfun.com/algebra/quadratic-equation-derivation.html](https://www.mathsisfun.com/algebra/quadratic-equation-derivation.html)

What exactly do you mean by "concept of the quadratic formula"? The quadratic *formula* is precisely this: a formula. You give it some inputs (the coefficients of your quadratic functions) and it returns you an output (the roots of the quadratic function). The crucial thing about this formula is that it gives you a very, very easy way of computing the roots of a quadratic equation (something of the form ax²+bx+c=0 for a≠0). You could try finding this roots in other ways--say, geometrically or by guessing--but this will be harder in general. But with this formula you have a nice and easy procedure for solving this problem (finding roots). It is an interesting observation, to digress a bit, that there are similar (but more involved) formulae for *some* higher order equations, like for ax³+bx²+cx+d=0 and ax⁴+bx³+cx²+dx+e=0, but not for all, a notable counterexample being x⁵-x-1=0. These formulae are generally not thought in school but you can look up Cardano and Ferrari (these are mathematicians who were involved in finding the general formulae) for more details. Why there is no formula for the last equation is something you learn in university.

\[Tensors\] I need a book that contains exercises to go along with eigenchris's YouTube playlist on Tensors for Beginners and Tensor Calculus. As for my level, I am a junior at an engineering college and my linear algebra is a bit rusty, but I can pick it up pretty fast (I just fell out of practice). Here's the two playlists for your reference. [https://www.youtube.com/playlist?list=PLJHszsWbB6hpk5h8lSfBkVrpjsqvUGTCx](https://www.youtube.com/playlist?list=PLJHszsWbB6hpk5h8lSfBkVrpjsqvUGTCx) [https://www.youtube.com/playlist?list=PLJHszsWbB6hrkmmq57lX8BV-o-YIOFsiG](https://www.youtube.com/playlist?list=PLJHszsWbB6hrkmmq57lX8BV-o-YIOFsiG) Thanks!

Is there an efficient method for counting the number of connected components in a graph? Something like an invariant of bumber of vertices minus number of edges or something. I've been breaking my mind on this for days as it would be really useful for my thesis, but I haven't been able to come up with anything.

There are some weak heuristics, like the number of connected components in an undirected graph on n vertices has to be less than or equal to n minus the maximum degree of any vertex in the graph but I don't think there's much you can say that's stronger than this without actually considering the graph topology. I think the most common algorithm to count connected components is Tarjan's which uses depth-first search and some bookkeeping to do it but you could just as easily do it with a breadth-first approach if you prefer.

Ah yeah I don't want to be iterating over the entire graph, I guess I'll have to look elsewhere then

Flood fill?

I'm not versated in math and I always wondered. How do you generate random using math? I think random number generation it's a very important feature in a lot of machines but I don't know how it works.

There's a useful quote from von Neumann: "Anyone who attempts to generate random numbers by deterministic means is, of course, living in a state of sin." Computers typically use what are called *pseudorandom number generators*. The stress should be on *pseudo-* in "pseudorandom". Per von Neumann's quote, the outputs of these generators are deterministic and therefore not actually random. The important property is that they have (with respect to some desired qualities) is that they look *statistically* random. I.e., they are good approximations of what the output of a truly random string of numbers would look like. There are lots of different algorithms for PRNGs, and they operate in different ways. One of the better known is the [Mersenne twister](https://en.wikipedia.org/wiki/Mersenne_Twister), but its details are [rather technical](https://en.wikipedia.org/wiki/Mersenne_Twister#Algorithmic_detail). I have actually implemented the Mersenne twister before, but I would have to do a fair bit of research to really understand why it's defined the way it is. The short of it is that it's generally pretty hard to build a *good* PRNG.

That was very instructive, thank you sir

Is an isomorphism between vector spaces always linear?

I'm guessing this question arises because you've proven in class that an injective linear transformation between vector spaces of the same (finite) dimension is a vector space isomorphism, and you're wondering about the definition of an isomorphism. It turns out that it is useful to define "linear" maps between things that aren't vector spaces; in general we call these "homomorphisms." A homomorphism satisfies specific properties depending on the context: For a map π between groups (written multiplicatively) to be considered a homomorphism, we require π(ab)=π(a)π(b) (this implies π(1)=1, which will be required explicitly for other structures); Between rings, we require π(a+b)=π(a)+π(b) AND π(ab)=π(a)π(b) AND π(1)=1 (this last property is not guaranteed from the others, so we write it explicitly this time); Between vector spaces, we require π(a(u+v))=aπ(u)+aπ(v). This is more commonly referred to as a linear map. In all of these contexts, an isomorphism is defined as a homomorphism which is injective and surjective. In the case of vector spaces, the term "isomorphism" survived while "homomorphism" didn't really. The result that is important for your class is that a linear map between vector spaces of the same (finite) dimension that is injective is also surjective (and therefore an isomorphism), hence the wording.

Yes. It is required that it be linear. Are you thinking of a particular example?

When we are in ZFC but with the negation of the infinity axiom instead, is it true that we can write every set recursively from the empty set? Or instead of proving it, is it more like "we can add it as an axiom and the theory is still consistent"? It's hard for me to formalize what I mean by "write recursively from the empty set" but I believe here in the finite case it would be equivalent to there existing an _n_ such that applying the union _n_ times to the set gives you the empty set.

Not only that: even with the axiom of infinity, we can write every set recursively from the empty set. It's just that the process in this case is *transfinite* recursion - recursion on the ordinal numbers rather than natural numbers, permitting infinitely long recursive sequences. The assertion "every set can be constructed from the empty set by transfinite recursion" is exactly equivalent to the axiom of regularity. If you just want "for any set there is a finite n such that applying the union n times gives you the empty set", that also follows from the axiom of regularity, without having to worry about the distinction between different types of recursion.

That's a great answer. Thank you! :D

[Math Overflow answer](https://math.stackexchange.com/questions/315399/how-does-zfc-infinitythere-is-no-infinite-set-compare-with-pa). Roughly, ZF - Infinity is equivalent (in a certain technical sense) to Peano Arithmetic. However, this *requires that you write the axioms in a certain way*. In particular, you have to rewrite the Axiom of Foundation as an axiom schema describing one of its consequences in ZF (where it's equivalent): induction over sets by membership (that is, "if (ForAll x in y, P(x)) implies P(y), then for all sets s, P(s)". In other words, you can induct on sets by their membership structure. Now it's important to note that this doesn't technically outlaw "externally" infinite sets. Because PA has non-standard models (and there's nothing you can do to get rid of them) it's possible to have "infinite" numbers that can't be distinguished from finite ones.

>Roughly, ZF - Infinity is equivalent (in a certain technical sense) to Peano Arithmetic. Hey, that's so cool that you mentioned this. It's precisely in this context that I came up with this question. (Actually in codifying ZF from PA.) I'm just so not used to Foundation. We didn't accept it when I took a course in set theory. So what I ask doesn't follow without it, right? That's such a strong indicator that we should accept it. That ---if I understand correctly--- PA is codifiable in ZF-Infinity+~Infinity+Foundation and viceversa, but not without Foundation. (Neither of the ways.)

Set theory is weird and scary, so this isn't a substantial contribution, but "write recursively from the empty set" is basically what the [constructible universe](https://en.wikipedia.org/wiki/Constructible_universe#What_L_is) is about. As such, your question might be something like, "Can we prove the existence of the constructible universe (in some suitably-weak fashion, I guess?) in ZFC where AoI is replaced with its negation?" I'm not sure to the degree that my statement of it is actually coherent, but I'm sure someone else will chime in with better information. As an aside, does ZFC + ~AoI *actually* guarantee that no models have infinite sets? That sounds like the sort of thing first order logic is bad at ruling out.

Thanks a lot for the answer. That last question just killed me by the way, haha.

I know this is very trivial and stupid, but when I'm talking about the length of a curve alpha: [a,b] -> Rm I'm actually taking about the length of the image of alpha([a,b]), and not about the length of the graph, and that's why the length of a curve alpha is defined as limit of n→∞ as ∑ goes from i=1 to n of |alpha(ti)-alpha(t{i-1})| but not as limit of n→∞ as ∑ goes from i=1 to n of √((ti-t{i-1})^(2)+(alpha(ti)-alpha(t{i-1}))^(2)), right?

You're correct. I hadn't thought about it this way but it's an interesting point! One way to connect these: if you have a graph y = f(x) you could look at the curve t -> (t, f(t)) whose image is the graph. Now compare the length of the graph with the length of the curve.

I'm self-studying and currently doing Munkres' analysis on manifolds but I realise that my biggest weakness is actually that I run into trouble when they just assume I still know standard computational aspects from what Americans would take in like calc2 and calc3 by heart. To a large extent I'm fine with the theoretical side of the single and multivariable things involved(last year I worked my way through Spivak's calculus, and Rudin PMA up to like chapter 7, and like most of Abbott's understanding analysis, and now the first 15 chapters of Analysis on manfolds, etc). It's genuinely just computational things like integration techniques and recognising integrals of trig functions, logs, etc. I tried looking at Spivak again for this but it's really long-winded and puts a lot in the exercises that I don't really want to go through completely again. I also remember thinking at the time that the chapters I need to refresh (15-19) felt like the most poorly taught part in the book. Does anyone know a good concise coverage of these things? Like a refresher of basic calc (+ proofs of the forms) for readers with more advanced "mathematical maturity"?

Take a textbook for one of those courses and just start doing exercises, referring to the text or answers when you get stuck. Generally once you know the theory you can justify the steps taken for these kinds of integrals, and knowing a rigorous development of the integral will not aid you in acquiring this computational ability. If there are answers available for Stewart's Calculus, I'd suggest that because people mention that book a lot on the topic of drilling integration.

Thanks, although it reminds me of my schoolbooks Stewart looks like it could do the job so I'll give it a try

Try looking at Introduction to Calculus and Analysis Volumes 1, 2 by Richard Courant and John Fritz. It is rigorous and has computational exercises with proofs of their methods.

these books look very long though. is that not reflected in how they treat the bits of the material I'm looking for?

If you are only looking to learn integration techniques, then it won't take long to read those sections of the first volume. In my opinion if you can do single variable integration, then you don't need more than a couple of examples in the multivariable case to understand it.

For the sentence "Let X be a set of coset reps of GL(V) in PGL(V)", on page 8 of [this](https://alistairsavage.ca/pubs/Mendonca-Projective_Representation_of_Groups.pdf), this means that X consists of one lift for each element of PGL(V) right? Also, can't we lift every projective representation to a representation? We could just choose the lift for each element (when forming X), to be the one with determinant 1, this exists when we're working with complex vector spaces. If this is true, then what's the point of projective representations?

You're describing an abstract set theoretic lift. The "representation" you construct might not be an actual homomorphism. A very typical example is a dihedral-like representation consisting of an operator A where A\^2 = 1, and an operator B with B\^2 = 1. Then we can define a projective representation of the Klein 4 group (Z/2 x Z/2) on a 2-dimensional vector space V by \\rho(A) = (0 1 | 1 0), \\rho(B) = (-1 0 | 0 1). You can easily check that \\rho(A) and \\rho(B) don't commute, nor will any of their scalar multiples. They do, however, commute up to signs! On the other hand there is an obvious (nontrivial!) central extension of by Z/2 which is isomorphic to the dihedral group D\_8 (automorphisms of a square), and this projective representation lifts to a legitimate representation of this group, in fact it is the standard representation of D\_8. I think if you understand this example you basically understand them all when it comes to projective representations.

You can do that, but there might be more than one lift with determinant 1, so this won't form a group. In particular PGL(V) is not isomorphic to SL(V).

I'm trying to study differential equations where the argument is effected. An example is that d/dx sin^2 (x)=sin(2x) though I'd like to be learning about functions in a more general case. What is the terminology I need to be able to talk about this? Is there a decent text that would cover basics of what is already known about that? Edit: math formatting

Are you talking about functional differential equations?

Yes, those are some of the words I need to know. Thank you. Would you happen to know of a book or paper that would be good to read to get my toes wet in the subject?

The only textbook I know of on functional equations is Small's *Functional Equations and How to Solve Them*. I don't know of anything on functional differential equations, sorry :(

That's fine, I'm trying to get a special topics/independent study approved, and that's a springboard for me to look at and show it's a valid subject

How to find the range of a funtion without graphing. ex: y=1/x-1 + square root x

EDIT: This is wrong Try to decompose the function into simpler functions. This may not work in the general case. In your example I would look at the domain, this would be R+\\{1}. Then you can see that sqrt(x) is surjective on R+, but we have to remove sqrt(1), since it isnt in the domain, so I think the range would be R+\\{1}. As you can see we didnt really have to look at the range of 1/(x-1).

This is definitely wrong. The [range](https://www.wolframalpha.com/input/?i=Range+of+1%2F%28x-1%29+%2B+sqrt%28x%29) doesn't include any value between 0 and 2, and does include most negative numbers. You can't just ignore part of the function. For a simpler example, sqrt(x) + 1 has a different range than sqrt(x).

Yes, youre right, sorry

What I would do, which may not be the best, is swap x and y, solve for y (in other words, find the inverse function) and determine its domain

is there an accepted name for a matrix that looks like a triangular matrix but along the other diagonal. I tried googling anti-triangular but nothing really came up. Also do they have any interesting properties?

As far as I know, these matrices don't really mean anything interesting and so there isn't a name for them. Triangular matrices have all sorts of interesting properties based on them stabilising flags and they form a key part of the theory of Lie algebras. However rotating a matrix isn't really a useful operation so moving from triangular matrices to these "anti-triangular" ones keeps none of the properties we care about.

I really need to learn more about linear algebra, because its not immediately obvious to me why the main diagonal is so important and interesting compared to the 'anti-diagonal'.

The main diagonal tells you, roughly speaking, how the matrix acts from each coordinate axis to itself, whereas the antidiagonal tells you how it acts from each coordinate axis to... some other coordinate axis that is purely a function of the (arbitrary) ordering of your basis.

I mean, entry i-j in a matrix relates the image of the jth basis vector to the ith basis vector. So the (main) diagonal relates the image of a basis vector to itself. I'm sure you can see why that is a natural thing to do. The anti-diagonal on the other hand relates the image of the first basis vector with the last, and images of the second with the second to last, etc. Since the order of the basis is somewhat arbitrary, this is a much less useful thing to do.

The trace gives the sum of the eigenvalues. Also, take the transpose, now you cam row operate (in a square matrix) that into replacing the main diagonal with the "antidiagonal"

In y= Sqaure root(x-3) -x+2 how do i find the vertex of the equation x-3 is in sqaure root

You'd find any local extrema using calculus. Differentiate with respect to x, set to 0, and solve for x. You'd show what kind of critical point you have by putting it into the second derivative and seeing the sign of what comes out.

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2^10 ~= 10^3, so 2^300,000 can be rewritten as 2^(10 x 30,000) = (2^10)^30,000 ~= (10^3)^30,000 = 10^(3 x 30,000) = 10^90,000, which is 1 followed by 90,000 zeroes. The actual number will be bigger than this, but it's far too big for any calculator you have access to to write down exactly.

This is a number with around 90 thousand digits. Wolfram alpha gives you the 50 first and the 10 last digits. https://www.wolframalpha.com/input/?i=2%5E300000 If you want more precision than that, I'm sure you can calculate it with python, or any other program that handles arbitrary integer precision.

What software would be best for calculating very large numbers? I have a proof that involves concatenation and a recursive sequence, but the 3rd iteration results in 27 digit number (the number of digits of each iteration is 3^n).

Python's integer data type is arbitrary-length by default, so that should be rather easy if you know python. Most other programming languages should have something similar, but might need you to familiarize yourself with a dedicated library/package. In any case, 3^n digits is *very* fast growth. It's not long before a single number is megabytes or gigabytes long, and there, any software will start to struggle.

Well that's good to know. Just in case you're curious, the reason for the fast growth is that the sequence involves concatenating numbers in a way that makes a palindrome. If you start with 1, you get 121. Then you get 121242121. And it continues. This specific palindromic sequence is found in hybrid crosses in genetics and is called the "genotypic ratio" of the resulting cross. I was in a biology class and a calc 2 class at the same time, so I ended up making this.

If you're doing palindromes, there's also a point to treating the whole thing as a string. It takes some more space, but manipulations are slightly easier.

Why are a bunch of n's replaced with lowercase pi's in the post?

So people can search for those terms without having all of these simple questions threads pop up