Mathematics - 2020-09-29

4:59 AM

Is there a proof of the fundamental theorem of arithmetic that is at a high school level of mathematics? If no one knows of a good reference here I'll ask this question on the main site after checking for duplicates, and eh, providing motivation and all those other things.

In my previous message I called it the prime number theorem, but I had the name wrong. So I removed it while the google search for the real name loaded (was close to the 60 second mark).

robjohn

8:32 AM

@AMDG if you would like, you can use the Online LaTeX Equation Editor.

That is what many of us used before ChatJax was around.

You can copy the LaTeX it generates, but images are not as widely accepted on this site any more.

Stupid question inc

9:39 AM

chat jax doesn't work 😕

robjohn

10:00 AM

@Stupidquestioninc It works for most people here. What OS are you using?

. o O ( Chrome? )

$$\int_{-\infty}^\infty e^{-\pi x^2}\,\mathrm{d}x$$

Here is what I see

AMDG

11:37 AM

Yeah I've used that before.

Also ChatJax doesn't work for me either. Opera GX can't add the links as bookmarks.

Needs to be a Chromium plugin honestly

Lovely. It doesn't exist.

I'm not making a browser plugin, I'll tell you that much :P

Browsers are garbage... APIs are garbage... OSes are garbage... all is garbage D:

Hm, the resolution of that screenshot makes me question if that is from a retina display on a mac or just a really high-end monitor.

robjohn

11:53 AM

@AMDG It's a Retina display

AMDG

I was right!

robjohn

The fact that it was twice the normal resolution was probably a big hint

AMDG

Yeah. yuge pixels.

robjohn

Some programs think the display is 1440x900 and some 2880x1800. This causes some weird rendering problems. The biggest I had to overcome was Mathematica, which had some serious trouble with the Retina display.

AMDG

I tell you man... these APIs...

I don't even blame the developers... if anyone is to blame, it's professors, CEOs, poor management, and poor design. Oh and developers.

robjohn

11:59 AM

Progress dictates that standards need to be changed and supporting changing standards is what is extremely hard for developers.

platform divergence

AMDG

Progress dictates the need for a constant standard that never changes. It's the complete opposite of what people think today.

I write code. Once. Not twice.

Ever notice we all live in the same universe that's been the same since the beginning? Constant standard that never changes. Our ability to create depends on a reality that is unchanging.

robjohn

I'd have a hard time posting articles on the 320x200 display that has been standard since the PET and Apple II days.

AMDG

You don't need to hardcode the width and height of your software is what I'm saying...

Your software should have a standard so solid that it'll work a century from now without changing. Code to the principles, and abstract hardware, while also understanding the hardware that you use, and optimize where possible. Also VMs don't make sense.

robjohn

Just wait until the people with 4 color receptors start complaining and we need to update monitors to produce more colors.

AMDG

Software should have no need to change... the hardware should ideally be the only thing that changes over time, and ideally, nothing changes over time.

robjohn

12:05 PM

Sometimes, the things that need to be flexible are not quite obvious ahead of time.

AMDG

The only reason something has to change is to improve it or to fix a defect.

Anything else is vanity.

robjohn

That is pretty much what essential progress consists of: improvements and defect fixes.

AMDG

Thing is, the hardware shouldn't need to conform to my eyes. It should have been a universal standard in the first place and tuned to emitting all colors through a single universal emitter particle.

robjohn

@AMDG I'll order a universal emitter particle right away!

AMDG

Particle as in fundamental unit, not subatomic particle.

Our eyes will never evolve to the point where they change in substance, and three-color receptors are already optimal, so we won't have four-color receptors ever.

robjohn

12:12 PM

some people do and birds and some insects use more colors to great advantage.

AMDG

The pixel is the fundamental particle of the screen, so a universal pixel would emit all colors without being optimized for human eyes, but for emitting all wavelengths.

Alright, I stand corrected, but the substance will never change. They will always be eyes that see a narrow set of wavelengths.

But insects and birds have no need of seeing monitors, so... :)

robjohn

We would have to figure a way of recording such data (continuous spectrum at an arbitrary resolution) and find a way to scale it back for available hardware (which will only be able to handle a discrete subclass at any given time).

@AMDG not yet...

AMDG

Well obviously you don't want harmful wavelengths of radiation to be emitted... I don't think I've ever needed to emit gamma rays before...

You'd still want something that just emits all wavelengths of visible light but not all wavelengths of radiation.

But if you had to record all that data anyways, then just use multiple sensors on a very small scale the same way the eye does.

robjohn

@AMDG sure, but still a continuum of data is a nightmare to deal with, even if it is a limited range. $[0,1]$ has as many points as $(-\infty,\infty)$

AMDG

We already deal with continuums of data. Anything that is a contiguous structure is also a virtual continuum because it is a contiinuity of two discrete things. Something that is indefinite as a continuum would be perceptibly indifferent as it can converge to a single discrete point by changing the perspective without changing the indefinite nature of the thing we are handling.

I mean that's basically the complex plane in a nutshell.

robjohn

12:24 PM

As far as I know, continuous data needs to be recorded analog or discretized (with a given resolution)

each has difficulties

AMDG

brb I need some coffee

Man, I didn't sleep very well. I think I'll have to leave this conversation for now, but thanks for the conversation @robjohn !

robjohn

@AMDG get some sleep. I should do the same.

AMDG

Eh, well... it's a matter of correcting my circadian rhythm.

I got into a fast which threw it off.

user193319

1:20 PM

Let $S$ be a ring and $F$ a free $S$-module with basis $e_1,...,e_k,...$ (infinite basis). I am trying to show that $Hom_{S}(F,F) is a rank $2$ $Hom_{S}(F,F)$-module. This a dumb question, but what is the module action of $Hom_{S}(F,F)$ on itself? Is function composition or pointwise function multiplication?

Thorgott

1:54 PM

Is the pointwise product of two ring homs a ring hom?

user193319

Hmm, now that you mention it I don't think so.

Thorgott

ye

Edward Evans

2:14 PM

Waddup

Thorgott

2:27 PM

$(\infty,2)$-groupoids

Edward Evans

no

Thorgott

agreed

AMDG

2:47 PM

What's the implicit equation of a cylinder? I have an idea.

Define some constant scalar a for rotation and movement. Let an indefinite horizontal cylinder centered on the x axis rotate by a in one direction, and move along the x axis by a in one direction. If you stuck a pin needle at the point of rotation for any given time t, do you not get sine or cosine, or are my hopes and dreams crushed? :P

Because it would seem to me that it is just a diagonal line on a plane that is then wrapped around a cylinder.

AMDG

3:17 PM

indefinite *right horizontal cylinder

Ted Shifrin

@AMDG You're making this awfully complicated. A right circular cylinder centered on the $x$-axis is given by $y^2+z^2=a^2$, where $a$ is the radius. You can parametrize it by $(x,a\cos\theta,a\sin\theta)$ as $x$ and $\theta$ vary.

AMDG

What exactly am I making complicated?

Also, thanks

Ted Shifrin

Just three lines of words for saying that we have circles of the same radius moving with their centers along the $x$-axis. (Maybe that's what you're saying.)

AMDG

Ah, I see. Well that isn't how I was thinking about it.

It's a rotating cylinder. I think that's pretty clear.

It isn't like I'm trying to find a real-valued closed form of sine or cosine or anything...

(/s)

So now I wonder if I can get something with square sine and cosine on a steinmetz solid.

Ted Shifrin

Well, now I have no idea what you're talking about.

AMDG

3:29 PM

Tell me what parts and I'll explain.

skullpatrol

in The h Bar, 4 hours ago, by Danu

I was actually here to bring some much more positive news: Ryan Unger has published his first paper, in collaboration with none other than Shing-Tung Yau! Congratulations to him! https://arxiv.org/abs/2009.12618

Thorgott

:O

Ted Shifrin

Wow. Well, that's impressive.

AMDG

I have a square sine and cosine. They're periodic functions of a square that map unique values to the perimeter of a square with side length of 2.

One steinmetz solid is the intersection of two cylinders whose projection from one perspective is a square, and a circle from another. I'm sure you can see where I'm going with this...

Oh, it's called a bicylinder.

Ted Shifrin

A bicylinder is the intersection of two (solid) circular cylinders, typically.

AMDG

3:35 PM

Right, so if I have square sine and cosine, I can pick a square from this perspective and obtain two points on a circle inset on a square of the same... I don't know what it's called. The diameter equals the length of one side of the square.

Steinmetz solid

In geometry, a Steinmetz solid is the solid body obtained as the intersection of two or three cylinders of equal radius at right angles. Each of the curves of the intersection of two cylinders is an ellipse. The intersection of two cylinders is called a bicylinder. Topologically, it is equivalent to a square hosohedron. The intersection of three cylinders is called a tricylinder. A bisected bicylinder is called a vault, and a cloister vault in architecture has this shape. Steinmetz solids are named after mathematician Charles Proteus Steinmetz, who solved the problem of determining the volume of...

Ted Shifrin

I never knew the name, but I know the shape well. And the tricylinder, as well.

Sayan Chattopadhyay

@skullpatrol Oh damn, that's epic. I was wondering what happened to Ryan, turns out he used the lockdown in a much better way than quite a lot of us.

AMDG

I don't know the names of a lot of the things that I have ideas for :D

Ted Shifrin

See p. 4 and following of this.

skullpatrol

@SayanChattopadhyay yeah, he went to Princeton before the lockdown started.

AMDG

3:41 PM

@TedShifrin Thanks, that's quite useful!

Sayan Chattopadhyay

Yeah I knew that. Didn't knew he would put out a paper. How long has he been there though?

skullpatrol

dunno

Ted Shifrin

This is first year.

Oh, now second year, starting this fall. I lose track of time.

Sayan Chattopadhyay

Wow isn't that like super early to have a publication?

skullpatrol

With a fields medalist

Ted Shifrin

3:44 PM

Yup. But two coauthors, one slightly famous, the other with lots of publications.

AMDG

Surely there's a way to use a bicylinder to map the distortion of travel along a square to travel along a circle to get linear rotation...

Oh right.

By distortion, I mean the distortion caused by putting a square on the surface of a sphere such that it's projection is a circle. The distortion is then what I presume is a non-linear map of travel along the same square to travel along the corresponding circle on the sphere.

skullpatrol

4:30 PM

@SayanChattopadhyay he's in the hbar right now

user153330

4:53 PM

hi yall! having a bit of trouble proving something from real analysis. so assume u have a sequence $x_n>0$ such that $\sum x_n$ converges. now i need to prove that there exists another sequence $y_n>0$ such that $\sum x_ny_n$ converges and $y_n$ goes to infinity. the best i could do is to reframe the problem as showing that must exist some $y_n>0$ such that $\sum x_ny_n$ is bounded and $y_n\to+\infty$, but I don't know how to attack it. i'd appreciate any hint

AMDG

6:49 PM

I'm having trouble understanding this notation here, mathworld.wolfram.com/LineElement.html with regards to mathworld.wolfram.com/Catenoid.html

rschwieb

7:59 PM

Hoping to learn something homological today. Suppose I want to conclude that a cyclic module $M$ is not a coherent $R$ module. If I use one homomorphism from $R^k$ onto $M$ and I can prove its kernel is not finitely generated, does that mean it holds for any such homomorphism from $R^k$ onto $M$?

In my particular case, $M$ was simple and I was convinced any homomorphism $R$ onto $M$ had a kernel which was not finitely generated.

I feel like there must be some basic homological thing that says "if it's finitely presented for one homomorphism, it is for the rest." but I am not knowledgeable enough about it :/

Sonofgreek

8:49 PM

Hello everyone! Suppose I have a set defined by $S= {s + t\sqrt{2} \, \, | s,t \in \mathbb{Q}}$ And $x \neq 0, x \in S$, how can I show that $\frac{1}{x} \in S$ as well? Any guideline or tips would be very much appreciated.

robjohn

@Sonofgreek $\frac1{s+t\sqrt2}=\frac{s-t\sqrt2}{s^2-2t^2}$

user153330

@Sonofgreek :multiply by the conjugate

Sonofgreek

I see! And then because of how S is defined it would $s-t\sqrt(2)$ part will show that it is in S?

user153330

@Sonofgreek yeah

robjohn

@Sonofgreek if $s,t\in\mathbb{Q}$, can you show that $\frac{s}{s^2-2t^2}\in\mathbb{Q}$ and $\frac{t}{s^2-2t^2}\in\mathbb{Q}$

Charlie

9:09 PM

nvm lol

Sonofgreek

I think I might just be stupid, would this mean that $s$ could have been $\frac{a}{a^2-2b^2}$ and $t$ could have been $\frac{b}{a^2-2b^2}$. And as long as $s,t$ were still rational numbers, they would be in $S$?

Sorry, if I am asking very obvious questions

Everstudent

Hello guys, I am still looking for the perfect Abstract Algebra book. I've tried a lot of books so far, but nothing clicks perfectly.
Dummit and Foote seems too long/verbose/wordy/dry.
Pinter is okay I guess, I haven't give up on it yet, but it's annoying how a lot of important material is introduced in the exercise without proper motivation/explanation as to why is it important(like for example - Normalizers, Conjugacy, etc.)
Allan Clark - I like this one, but it's annoying that there are no solutions, so I can't check mine

user153330

@Everstudent i've heard artin is the book but i didn't read it

pinter was very good for me, dummit&foote while wordy seems worth digesting even if it takes time

Everstudent

Besides Pinter, I can't seem to convince myself to invest time in any other book, yet I got stuck in it because many of the exercises are not motivated, so I do them mechanically and then forget them because I do not get their importance, and honestly a lot of the exercises are not new material, but some are

Artin is a lot about Linear Algebra and seems a bit quirky to me

I am stuck around Cauchy/Group Actions/Sylow.

I was able to understand the Isomorphism theorem, but I still can't get the deep usefulness of normalizers and quotient groups.(besides being kernels of homomorphisms)

I want to be able to do Algebraic Number Theory in time, but I thought that it would be best to get a solid Abstract Algebra foundation first.

user153330

@Everstudent books aren't perfect, if they don't tell you something look it up online here on stackexchange for example

Everstudent

9:22 PM

Well, my problem is that all the time I feel like I found the perfect book, but after a few chapters I am demotivated.

user153330

you can't sit there and expect to happen on a perfect book that has all the motivations and subtleties in it

@Everstudent motivation is a bitch, persistence is key

Everstudent

Like, some years ago, I tried Rudin's Real Analysis intro book, did some chapters but it was taking too much time, like 2-3 weeks per chapter or something so that demotivated me a lot.

With Pinter, the first chapters are a breeze, so that's really nice

But after Homomorphisms it got a lot more complex in it expecting that I remember all the exercises and their proofs

And as I've said, many of the exercises do not properly explain themselves so I just do them mechanically

So how many books should I pick and use?

I have access to a lot of digital books, while I can only get cheap ones on paper, like Pinter or Allan Clark.

user153330

@Everstudent i have that same problem, and it holds me back, i still don't know how to overcome it

Everstudent

It's taking me too much time to try too many books I suppose.

For me the problem is that it's distracting having to check so many books on the topic.

Also digital books distract me more than paper ones, because I can't skim as I do with paper ones. So I tend to start reading a lot of them before I figure if I like it or not.

By the way when studying a book, do you write a lot on paper? I tend to do so, but I need a lot of concentration for that. So a lot of time I just read the chapters first without writing anything.

robjohn

@Sonofgreek No, those are the new $s$ and $t$ (with a sign change)

41 mins ago, by robjohn

@Sonofgreek $\frac1{s+t\sqrt2}=\frac{s-t\sqrt2}{s^2-2t^2}$

nbro

10:00 PM

7

Q: How can AI researchers avoid "overfitting" to commonly-used benchmarks as a community?

In fields such as Machine Learning, we typically (somewhat informally) say that we are overfitting if improve our performance on a training set at the cost of reduced performance on a test set / the true population from which data is sampled. More generally, in AI research, we often end up testi...

robjohn

How does that make you feel?

Lucas Henrique

hey chat

I'm defining my personal theorem environments in $\LaTeX$ and I'm trying to get one environment to work with both section (from the article document class) and chapter (from reports) numberings. Any idea on that? Thanks

Mike Miller

11:26 PM

@TedShifrin A student wants to read about curves & surfaces with me. What sources do you recommend?

Ted Shifrin

11:43 PM

@MikeMiller How strong a student and what background? Obviously, in general, I'm partial to my own notes (particularly for variety of good exercises).

Mike Miller

Getting a feel for that. Analysis & algebra but not a ton of topology. I'd be glad to know any number of references though, I can talk to him about what seems appropriate.

Ted Shifrin

I prefer not to get buried in abstract smooth manifold stuff at the beginning. But baby doCarmo belabors that for pages and pages for surfaces. Student strong with differential forms? If so, look at Jeanne Clelland's book. If not, definitely have him look at mine.

Mike Miller

I'll ask about forms. I suspect not, but sometimes analysis does cover that here.

Ted Shifrin

Plus mine is free :)

Mike Miller

I did advise he avoid the abstract stuff (he had some bad experiences recently with very technical stuff that it's hard to love, and...)

Ted Shifrin

11:49 PM

There are more classic options (Struik, Milman & Parker), but you won't like those.

Mike Miller

But would a student?

I'm an old dog but young enough to learn new tricks.

Ted Shifrin

Have him look at mine for starters. I can send you my homeworks if you want guidance.

They're tensor anslysis flavor, full of indices.

Mike Miller

You win. Not a trick I want to learn.

Send it to me.

Ted Shifrin

You can download from link in profile.

If you finish through Gauss Bomnet fast, you can always do more advanced stuff from baby doCarmo.

Mike Miller

The homeworks?

I got the document.

Ted Shifrin

11:53 PM

Oh, gotcha. oK, will send.

Mike Miller

I think we're probably going to take our time and enjoy it.

@TedShifrin I wasn't clear. Thanks!

Oh, your document has exercises too.

Ted Shifrin

Yes, lots. Some challenging and interesting ones, plus routine ones and medium ones. My homeworks have three levels (flower = computation, pinecone = middle, pyramid = more challenging).

Mike Miller

Will I be able to do all of them? I usually do every problem I assign beforehand to make sure I'm not overdoing it, and want to make sure that won't overload me here. :) (oops, last week I gave a homework which was too much despite this...)

Ted Shifrin

I only assigned 5 each assignment, so it's up to him how much he chooses to do. You want exams too?

Way shorter assignments than I typically assign in almost any other course.

You can always ask me for more suggestions if you want.

Mike Miller

I think no exams.

Ted Shifrin

11:59 PM

OK. Sending.

Mike Miller

Wait, I mean I don't need them.