Mathematics - 2023-04-11 (page 2 of 2)

Xander Henderson

19:00

1 is a unit, so, by definition 1 is not prime.

Again, there is no reason why we can't define the word "prime" so that 1 is prime, but we usually don't. It is generally not useful to regard 1 as a prime number.

AMDG

@XanderHenderson We have $ab = 1$, and $1b = 1$ or $1a = 1$ are permissible, so I don't follow you.

Xander Henderson

3 mins ago, by Xander Henderson

The more grown up definition is that a non-unit $p$ is prime if, whenever $ab = p$, either $a$ or $b$ is a unit.

Note: a non-unit $p$.

AMDG

Oh wait I missed the part non-unit

Xander Henderson

By definition, a prime is not a unit.

AMDG

Sorry my bad

Ted Shifrin

19:03

@AMDG Because then integers no longer have unique prime factorizations. End of debate.

Xander Henderson

@TedShifrin I am sure that @AMDG would argue that you just add the phrase "non-unit prime" to all existing theorems about prime numbers.

AMDG

Why would integers no longer have unique prime factorizations? What is a unique prime factorization?

Xander Henderson

@AMDG Are you familiar with the fundamental theorem of arithmetic?

Ted Shifrin

Oh good grief.

This room needs an hour time out.

AMDG

@XanderHenderson I am not. I'm reading up on it on Wolfram MathWorld

He put in parentheses "except the number 1". This is sad.

If I then go to the definition of "exactly one", we get "one and only one".

So now tell me: are you all equating essences with their representations directly, or not? Because $(1^x) = 1$ no differently than any $1 = \prod 1$, so no matter how you slice it, you would get that any representation identical with the essence of one is in fact the unique factorization of 1, not a unique factorization of 1, and the same can be said for any integer. I mean why should it matter if I replace an integer with a limiting sum in a prime factorization?

Koro

19:13

@XanderHenderson I think this is definition of irreducible element.

AMDG

Perhaps we should go on to say that there is no unique representation of the integers in any base because of infinitely many implicit trailing and leading zeros.

Koro

irreducible and prime are not generally the same thing.

shintuku

this is what happens when you try to make philosophy engage with actual mathematics. it doesn't, it can't, unless you suppose magical entities inside mathematical constructs

Koro

@AMDG unique representation in this context has a well defined meaning.

It often includes 'upto rearrangement and multiplication by associates'

AMDG

associates?

Koro

19:17

For example: 5= 1.5=5.1 = (-1)(-5) etc.

AMDG

And, again, representation supposes some reality which it signifies, in this case, numbers.

Koro

@AMDG two elements a and b such that a=ub, where u is a unit.

Xander Henderson

@Koro Yes. Primes are the irreducible elements in $\mathbb{Z}$.

AMDG

An ideological reality existing as ideas, but a reality (in that sense) and no less.

@Koro I assume the dots mean multiply

Xander Henderson

@AMDG What is "an essence"?

shintuku

19:18

magical skeleton

Koro

@AMDG yes, the usual multiplication in this case.

Xander Henderson

@AMDG This has nothing to do with what the Fundamental Theorem of Arithmetic says.

Koro

@AMDG the equalities in the example are considered as 'one' when we say unique representation.

AMDG

@XanderHenderson That which makes a thing to be what it is. The essence of 1 is what makes 1 to be the number 1 and not any other number.

shintuku

@XanderHenderson i tried asking this, received nonanswers for 4 hours, and it ended with God. i'm warning you

Koro

19:20

I'm not sure what your question is though.

AMDG

Eh, not really. I said this definition up front and, well, I'm not going to mention how it ended. Like I said, obliged in conscience against.

Koro

@shintuku how is game theory?

I'm thinking of taking it next sem.

I don't want to take Sieve theory, modular forms etc as I have no idea what these are.

Though, I have heard that the instructor does not give more than 45 marks in the subject out of 100.

shintuku

@Koro have just done it up to nash equilibrium, haven't needed more yet

Koro

Does it have lot of maths in it?

shintuku

"that which makes a thing be what it is", i.e., a magic skeleton

@Koro the book i have looks like it can have a lot of math

but the hardcore math was the nash equilibrium proof

schn

19:23

Are there series such that $\sum_{j=1}^\infty\sum_{k=1}^\infty a_{jk}=L$ (for some finite $L$), but $\sum_{k=1}^\infty\sum_{j=1}^\infty a_{jk}=\infty$, i.e. can the row series converge while the column series diverges? Or do they converge or diverge together?

shintuku

i haven't taken a class yet

Koro

Ok. So do you think that it's a one semester course?

shintuku

only have had a section from microeconomics, but on game theory

Xander Henderson

@AMDG This isn't math.

shintuku

@Koro let me give you a reference you can check through, according to SE it is up there with rigorously mathematical approaches

2 sec

Koro

19:24

@schn I think there is a theorem related to this in Rudin's. You may look that up.

schn

ok, do you know which one?

Koro

@shintuku you get 1.78 s only. :P

shintuku

@Koro Rubinstein and Osborne - A course in game theory

Koro

@schn see chapter 8 or 9 of Rudin's PMA.

schn

ok, thanks

AMDG

19:26

@XanderHenderson Ted mentioned unique factorization. If we rephrase in consideration of essences, then we can define factorization in terms of the numbers themselves. Then, what that ultimately means is that we can have a product of integers considered in terms of what its representation simplifies to as being identical with its factorization rather than arbitrary products of units being considered unique factorizations.

Koro

@shintuku OK. I don't know what this subject will be about. And I have no expectations from 'teachers' here.

Xander Henderson

@AMDG The FTA can be summarized as "every integer has a unique prime factorization", which has nothing to do with representations of numbers by numerals with leading or trailing zeros, or repeating 9s, or whatever.

It has nothing to do with the base in which a number is represented. That simply isn't what the theorem says.

shintuku

@Koro can you get a syllabus?

schn

@Koro chapter 3 is on series though :) you are sure it's 8 or 9?

Koro

Noncooperative Games Games in normal form. Nash equilibrium and standard concepts, Applications of static games to economics, Games in extensive form. A refinement of Nash equilibrium: subgame perfection.Applications of extensive games. Other refinements. Applications to economic situations with incomplete information. Repeated games
and applications.
2. Cooperative Games Games in characteristic function form. Various solution concepts: core, bargaining set, Shapley value, etc. Application to economics.

@schn yes. Chapter 3 does not talk about series of functions

shintuku

19:30

yeah we didn't cover that much, and only did noncooperative games

AMDG

@XanderHenderson Then you aren't understanding what I'm saying. I'm not talking about those, and it's mostly just trying to address, "Then integers no longer have unique prime factorizations." itself because it implies that permitting one to be prime makes the factorizations non-unique in some way.

Xander Henderson

@AMDG It does.

Ted Shifrin

@schn Start fiddling around with conditionally convergent things. For example, image each row has an equal number of $+1$'s and $-1$'s and $0$'s elsewhere. Can you conclude that column sums make sense?

AMDG

@XanderHenderson How so?

Ted Shifrin

If it's in Rudin, it's in the multivariable or measure theory chapters. The relevant thing is Fubini's Theorem.

Koro

19:32

AMDG: I think it will help if you study UFD (unique factorization domains).

:-)

Xander Henderson

@AMDG The power of the FTA is that every positive integer has a unique representation as the product of primes, i.e. $18 = 2^1 \cdot 3^2$. If you call 1 prime, then this representation is no longer unique, as $18 = 1^m \cdot 2^1 \cdot 3^2$, where $m$ can be any natural number.

So there is a distinct factorization corresponding to every choice of $m$.

shintuku

oops, supposing magical essences leads to an incorrect conclusion about prime factorization. no wonder: as a historical fact this sort of philosophy has literally lead to 'proving' geocentrism

Xander Henderson

But, again, we could define prime differently, then state the FTA as "Every positive integer has a factorization as a product of primes, where that factorization is unique up to multiplication by some number of factors of 1".

AMDG

Yeah man, because a tried and true over novel and new philosophy that has existed since Aristotle isn't trustworthy, but I digress.

shintuku

tried and true, sure if you can call geocentrism that, hehe

Xander Henderson

19:37

This is more of a mouthful, and kind of obscures the underlying idea... but one could define things this way. We just usually don't, because it is a pain to do so.

AMDG

@XanderHenderson So here, ultimately what I'm seeing is the definition of factor here is contingent on the representation of a number via an algebraic expression because the simple reality is that $1^m = 1$ no matter what positive integer choice of $m$. I argue instead that it is more fitting to define factors as numbers in themselves, in which case we get that every such possible expression $1^m$ is in fact one unique factorization.

One unique factorization having infinitely many possible expressions.

But I mean... that can be said of anything in mathematics, can't it?

$2 = \frac{2f(x)}{f(x)}$

Koro

@XanderHenderson 1 = 5^0 3^0

AMDG

@shintuku Both geocentrism and heliocentrism are possible, just FYI

Mathematically, it should be clear why (unironically, considering how the astronomical bodies have been studied in the first place...)

For all we know, they both rotate about a central axis equidistant to the sun and the earth

@Koro 0 isn't a natural number. ~~It's unnatural.~~

Xander Henderson

@Koro 0 isn't a natural number. :P

AMDG

Huh

D.C. the III

19:51

@shintuku @Koro I was about to recommend Osbourne. It is the most rigourous Game Theory text at the undergrad level. Got it here on my bookshelf.

Xander Henderson

(or, if you define the natural numbers to include $0$, take my previous statement, and say "where $m$ can be any nonzero natural number", instead.)

@AMDG No. That kind of defeats the whole idea of a unique factorization, as described in the FTA.

Koro

@D.C.theIII thanks

D.C. the III

Is the "mass" of a region the same as the "volume of a region? In terms of doing integration? i.e $mass(\Omega) = vol(\Omega)$? Trying to understand the idea of the density function

Koro

I’m not sure though if I should take it next sem or not

AMDG

Again, we're defining factors, and factorization by extension, as a matter of the numbers themselves which multiply to give some integer, so assuming "simplifying an expression" is already rigorously defined, then the essence of a number is given by that expression which cannot be further simplified and is constituted entirely of constants.

Koro

19:55

I’ll have diff. geometry next sem.

D.C. the III

you would do fine in it, since you have the mathematical training. When I took it I was out of my league because my mathematical maturity was missing. Now when I do read through the text it is a a lot more understandable.

Koro

What is it about? How would you describe it?

Like ‘who’ are the stakeholders in the subject? In Real analysis for example, the stakeholders are derivatives, integrals, limits, functions, etc.

D.C. the III

A mathematically grounded way of guessing an opponent's next moves?

AMDG

I am then saying that this such expression directly represents the unique factorization of the number, so in the case of your $18$, we have $1^m \cdot 2 \cdot 3^2 = 1\cdot 2\cdot 3^2$. The latter cannot be simplified further, so this would be the identity of this definition of $18$'s factors.

Koro

Ah, so related to probability?

Or combinatorics?

D.C. the III

19:57

That's how I would explain it to a person who isn't familiar with math

more so probability

but combinatorics does play a role in probability anyways

Koro

I think. I’ll study Osbourne’s and then decide.

It was offered during my undergraduation but I hadn’t taken it.

That’s why probably Osborne sounded familiar to me.

schn

@TedShifrin that's a nice example, so the sum of all the column sums can diverge while the sum of all the row sums converges. Suppose though our terms $a_{jk}$ can only be positive or equal to zero. Is it then true that the sum of all column sums and that of all row sums converge or diverge together?

D.C. the III

If your into things like games of skill such as chess or poker, etc where you have to take into account all of the opponent's possibilities you will enjoy it

Xander Henderson

@AMDG Everything is definitions.

Koro

@D.C.theIII sounds interesting :-).

I’ll then try to finish it before next sem.

Because you know how ‘classes’ are here.

D.C. the III

20:05

A simple problem that you could read on to see what Game THeory is about is the "Prisoner's Dilemma".....it's a famous question has a wiki entry

Koro

Thanks. I’ll check that out.

AMDG

Game Theory?

D.C. the III

yes

AMDG

What is it exactly?

Ted Shifrin

@schn Yes, with everyone nonnegative it’s Fubini’s Theorem.

schn

20:09

Ok, thanks.

AMDG

Alrighty, time to press some buttons to cause a bunch of lights to flicker in a satisfying manner for a while

D.C. the III

20:29

@TedShifrin so is density $\delta(x)$ to be thought of as the "instantaneous mass/volume" of a function at the point $x$?

over a given region $\Omega$.

Ted Shifrin

Yes. There’s actually a discussion in the book.

D.C. the III

Yes. I've been reading it.. read it a few times over.

Koro

It seemed to me as if there were two moons in the sky.

D.C. the III

I'm just wrapping my head around the idea "properly" instead of how I had understood it previously

Koro

But one was reflection in a mirror far away. 😅

Ted Shifrin

20:35

When you learn measure theory, it’s a Radon-Nikodym derivative.

D.C. the III

So what is the difference between mass and volume?

Ted Shifrin

I think it really is how you understood it earlier.

Mass leads to weight in the presence of gravity. Volume does not.

D.C. the III

Hmm...I like that notion. Since I have zero physics background a lot of the physics notions are foreign to me. But this does make sense.

Need to get an undergrad physics course or textbook under my belt

Hash Nuke

21:36

Does continuous open map maps closed set to closed set?

D.C. the III

Do you mean if a continuous map, maps a closed set to a closed set?

Xander Henderson

21:52

I read the question to be "Do continuous open maps send closed sets to closed sets?", i.e. if $C$ is closed and $f$ is continuous and open (open sets go to open sets), is $f(C)$ closed. But neither of you seems to have written an easily parsable question. :/

Ted Shifrin

What are some well-known open maps?

That are not homeomorphisms, of course.

Xander Henderson

@TedShifrin No idea. My recollection is that it was only a concept that I saw in functional analysis(?), related to the open mapping theorem and the closed graph theorem.

(like, uh... the open mapping theorem is about open maps)

Ted Shifrin

Think submersions.

D.C. the III

Well I never knew of the notion of an "open map" per se, not yet at least....

Xander Henderson

@TedShifrin Yeah, I struggled through that class. :/

I have no intuition for submersions and immersions and what not. :\

Ted Shifrin

21:59

Bah.

Xander Henderson

@TedShifrin Submersions are the ones that are surjective on the tangent space?

Ted Shifrin

Yes. Think of the simplest nontrivial example.

Xander Henderson

Embeddings are submersions, right?

No... wait.

That goes the wrong way.

Ted Shifrin

Right, wrong.

Xander Henderson

Projections. Take a high dimensional space, and shove it into a low dimensional space.

Ted Shifrin

22:02

Right, right.

Xander Henderson

Why am I answering @HashNuke's question? :D

Ted Shifrin

I didn’t tell you to

Xander Henderson

Yeah, I know.

You just led me down the garden path. :D

In any event, I am supposed to be paying attention to a meeting.

Hash Nuke

Sorry, I was elsewhere

Ted Shifrin

But we’re more interesting.

Read the above and figure the rest out for yourself.

Hash Nuke

22:05

Yes @XanderHenderson thats the map I am talking about.

Ted Shifrin

So?

eigenvalue

22:37

@TedShifrin I am glad to see you here! Maybe you can help?

If given a semi-Riemannian meric $g=g_{jk}(x^{1},...,x^{n-1})dx^{i}dx^{k}, j,k\in\{1,...,n-1\}$ defined on some compact set, then the metric (or rather the components of the metric) has an absolute minimum at some G. So we can define $g'=G\delta_{ik}dx^{i}dx^{k},\:j,k\in\{1,...,n-1\}$.

Question:

I am wondering how the minimum of a matrix (here the matrix representation of the metric g) is usually defined. I have seen that one method is the Max-Min value method, but that's not what has been applied here.

Ted Shifrin

22:52

Some entries might be negative, you know.

You can always diagonalize, although there might be some difficulties with smoothness in some cases.

There is no such thing as minimum of a matrix.

eigenvalue

@TedShifrin yes, but they could be all positive, too. And I don't see why we can diagonalize globally. Do you understand what has been done with respect to g in first portion of my text?

Ted Shifrin

I suppose this could mean $A-A_0$ is always positive definite, but these are tensors on different tangent spaces.

eigenvalue

How would you approach the situation where the metric components are defined on some compact domain ( and therefore attain a minimum somewhere)?

Ted Shifrin

What is the point of all this? What is the goal?

eigenvalue

This is needed for some proof and by using compactness the proof becomes manageable. But there is this one step as I mentioned above where the metric g gets replaced by a simplified diagonal metric with the components G. And I don't see how this step is possible

Ted Shifrin

23:01

Well, they’re making a (local) change of chart, but I don’t see any rationale for the minimum. They must give more explanation than you have.

eigenvalue

This is the additional information:

Next, consider the function

$\tilde{G}:C_{0}\times\mathbb{S}^{n-2}\longrightarrow\mathbb{R}$

$\:\:(t,x^{1},...,x^{n-1},v^{1},...,v^{n-1})\mapsto g_{jk}(t,x^{1},...,x^{n-1})v^{j}v^{k}$.

As $\tilde{G}$ is a smooth function defined on the compact domain $C_{0}\times\mathbb{S}^{n-2}$, it has an absolute minimum at some $G_{0}$. Hence, on $(U,\varphi)$ we can uniquely define $\tilde{g}_{0}=-t(dt)^{2}+G_{0}\delta_{ik}dx^{i}dx^{k},\:j,k\in\{1,...,n-1\}$.

Ted Shifrin

I detest saying the minimum value is AT … it is a value, not a point of the domain.

eigenvalue

agreed :-)

Ted Shifrin

Anyhow, they can define a new semidefinite metric. Note dimensions are off. What is $C_0$?

eigenvalue

We choose smooth coordinates with $t_{0}>0$, and $\xi_{0}>0$ such that $C_{0}:=[0,t_{0}]\times B_{\xi_{0}}^{n-1}=[0,t_{0}]\times\{x\in\mathbb{R}^{n-1}\mid{\displaystyle \sum_{k=1}^{n-1}(x^{k})^{2}\leq\xi_{0}^{2}\}\subset}\mathbb{R}^{n}$ is contained in the domain of the coordinate chart.

Ted Shifrin

23:13

So they’re putting the diagonal in the space coords but $-t$ in the time. Where did that come from?

The minimization is using only the space coordinates.

eigenvalue

Because there exists a foliation is the said coordinate neighborhood. I left out the time coordinate in my question as the minimum is only considered in the space coordinates. But the metric we actually consider is $\tilde{g}=-t(dt)^{2}+g_{jk}(t,x^{1},...,x^{n-1})dx^{i}dx^{k}$

Ted Shifrin

Oh, so that was there to start with.

eigenvalue

but maybe it clarifies the issue if you know the actual metric

Ted Shifrin

OK, so they can define the new thing. What happens next? They’re not minimizing the entries, by the way. That’s important.

They’re minimizing the quadratic form on the unit sphere. That’s like finding the minimum eigenvalue.

user10478

Are basic questions about causal modeling on-topic-ish enough to be here?

eigenvalue

23:20

how can I envision this? so then the metric can take the diagonal form?

Ted Shifrin

They’re not diagonalizing the metric. They’re defining a new one.

@user10478 Speaking for me, I have no idea what that means.

eigenvalue

"minimizing the quadratic form on the unit sphere". I probably have to refresh my memory/ or look into this...in order to fully understand.

Ted Shifrin

The point is that this diagonal metric gives a lower bound on the actual one (plug in tangent vectors).

eigenvalue

"They’re not minimizing the entries" is already helping me to leave the wrong thinking path! Glad you mentioned this!

Ted Shifrin

In my multivariable course I proved the spectral theorem by doing exact

Exactly this approach with maximizing on the unit sphere.

eigenvalue

23:24

nice. Do you have a good source I could read to understand this (method) better?

it sounds familiar, though. I probably forgot the details already

user10478

@TedShifrin Ahhh, it's basically the manipulation of directed acyclic graphs or associated probability algebra to determine when correlation is causative. Used to simulate randomized control trials from observational studies, and stuff in that ballpark.

Ted Shifrin

It’s standard in applied linear algebra, finding eigenvalues by max/min. But you can find my video lecture for this.

@user10478 Zero clue.

This sounds much more stat than math.

eigenvalue

@TedShifrin I just found it. It's embarrassing not to remember these methods. Studying math seems like a never ending Sisyphus activity

Ted Shifrin

It helps to teach/tutor stuff. Then you actually absorb it better.

user10478

kk

It's listed under philosophy of science on wikipedia but I have a feeling that's misleading.

eigenvalue

23:32

@TedShifrin I am really happy this chat exists ... and super grateful for your help. So I can get back on track again. Going to refresh my memory about the spectral theorem now!

Ted Shifrin

Yup, it makes perfect sense to me now that we unscrambled it, @eigenvalue.

eigenvalue

@TedShifrin made my day! haha

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$Mathematics$ Mathematics