Mathematics - 2024-10-22

leslie townes

00:47

my daughter just learned about inequalities in her homework this week, and she isn't having it. "LESS THAN? i need to do MORE of LESS THAN?"

nice try, kid, but wordplay will get you nowhere

sku

01:34

I have possibly a naïve question on this answer from MSE. https://math.stackexchange.com/questions/898643/prove-that-if-an-infinite-series-converges-then-the-associative-property-holds

Where exactly in this proof shows the regrouping? The fact that we have put parentheses is not helping me see it… For example, I have a sequence (a1, a2, a3, … ), the sequence of partial sums is (a1, a1+a2, a1+a2+a3, a1+a2+a3+a4…). In this I don’t see a subsequence of the form (a4+a5+a6) assuming I want to group in 3s.

copper.hat

@sku if we let $s_n =a+1=...a_n$ and $s_n \to L$ as a sequence then we say that $\sum_k a_k = L$. The regrouping is a bit strange, but is straightforward.

to see what they mean by regrouping, pick some sequence $n_1\ge 1, n_2 > n_1,....$. Then let $b_1 = a_1+...+a_{n_1}$, $b_2 = a_{n_1+1}+...+ a_{n_2}$, etc.

Then show that $\sum_k b_k = L$.

sku

Algebraically I follow it. I want to understand from an analysis perspective. What is the original sequence of partial sums and what is the subsequence of partial sums? In above example, b1, b2 etc form a subsequence of which sequence?

think_meaning_builds

02:55

@leslietownes be sure to let her know that there is word-play built into inequalities in that the symbols: ≤ and ≥ are called "inequalities."

leslie townes

@sku the "regrouping" operation of this question is not the same as taking a subsequence. it is very possible to obtain one sequence from another by "regrouping," such that the sequence obtained by "regrouping" is not a subsequence of the original one. the're just different operations. copper has explained what "regrouping" is trying to get at. i would not confuse it with passing to a subsequence.

what's confusing in the background is that both of these two operations, passing to subsequences, and "regrouping," will, under a wide set of circumstances, not change the limit of the result.

but it's still apples and oranges

Zac U.

Anyone wanna hear my latest toy math puzzle?

leslie townes

zac: think of how depressing it would be if the answer were "no" :)

Zac U.

Indeed. This puzzle is one where you have a sequence of numbers and each progressive row stacks upwards. Each row contains the respective number of squares. The structure must be balanced. The first row should have no isolated squares, by the second row's completion the structure should be connected and if a row coming directly after a previous has a higher number of squares to be placed, then leaning over is allowed. You can clip the squares into rectangles

By leaning over I mean we would assume that having a 1x2 on top of only one square, would balance

By clip I mean the squares can become one with other squares for balancing purposes and if adjacent are considered connected. Tho that rule could be modified I suppose?

sku

03:36

@leslietownes I think I may have overthought it. The associative property works because we are always working on convergence of a finite sequence and then having this imply that the infinite series converges (by definition).

leslie townes

03:48

sku: well yes it is fairly straightforward with sums of nonnegative terms. it does get subtle with series involving terms that may not have the same sign, because grouping can strongly affect any "cancellation" that occurs or does not occur in computing partial sums. riemann's rearrangement theorem is the textbook example of this

or centuries earlier it was some monk who noticed perplexities inherent in where you put ( ) in 1 - 1 + 1 - 1 + 1 - ...

nickbros123

04:27

wanted to ask a question but it got solved halfway through typing lol

they should allow typing things in exams

leslie townes

just as a way of charging the memory banks? a keyboard not connected to anything?

nickbros123

a keyboard connected to a text editor with latex would be nice

leslie townes

oh yeah, i could see that

that reminds me of how a number of classmates made fun of me for "typing my homework" when i was in college, but it was like, hey, if i fuck up an argument and need to rewrite a few things i can just go in and do that and i'm only printing once at the very end before i turn it in, instead of this shambolic and borderline comical erasing and crossing out shit

i guess these days there isn't even a printing step? or is there

when i was last teaching all the assignments and stuff was on a course management system but there was no facility for turning in homework online because you couldn't count on people to have access to a scanner or phone with a good camera

so you'd still get this badly handwritten stuff like something my daughter would do on pieces of paper that were lined if you were lucky

nickbros123

04:58

@leslietownes there isnt. most of it for me atleast, is in google classroom. So there is no reason not to tex homework. Its good for compilation purposes as well. I also take notes in tex, so I have not only theorems done in class but also misc theorems and so on, along with assignment problems, so its like, when u are in a long bus ride u can just open github and revise some stuff :)

leslie townes

yeah that was the other thing i mentioned to my peers at the time, i actually have an archive of what i did that i can refer to and go back to (and, as needed, cut and paste from in other classes)

which you don't get with handwritten hogwash

nickbros123

yeah. with this setup one can revise stuff reasonably effectively cuz everything is in one github folder.

also perhaps long term this is better; if say I want to revise RA-1 after a few months, it would be a nightmare going through all those piles of scattered notes, assignment sheets, handouts etc

Ryder Rude

09:31

@XanderHenderson thanks. i will check these out..

I've seen the The Thing

@ModularMindset yes

the spinors matrices/generators have half integer eigenvalues $\frac{k}{2}$ and when u exponentiate $e^{i4\pi \frac{k}{2}}$, u get the identity transformation. so we need a 4pi rotation

for the electron, the eigenvalue is 1/2 cuz it is a spin-1/2 particle. i think the only known spinors in nature are spin-1/2 ones. Don't think spin 3/2, 5/2... have been found

more specifically, the eigenvalues of $A$ range from -k/2 to k/2 with jumps of +1

so $\theta = 4\pi$ is the first point where $e^{iA\theta}$ is identity. then u have 8pi, 12pi ,etc

Jakobian

10:00

@RyderRude are you teaching yourself any math?

Ryder Rude

@Jakobian not right now..

i was learning formal logic the last time, but I've stopped for now

Ive been looking into measurement problem of quantum mechanics and philosophies of consciousness/subjective experience

Jakobian

@RyderRude if that's math

no disrespect for logicians, but I don't think it is

Ryder Rude

@Jakobian i would say it has axioms and theorems, so it is math

but it doesnt deal with specific axiomatic systems

Jakobian

10:15

@RyderRude that's a very shallow way to think of it

Ryder Rude

@Jakobian what is ur view on what should be called math

Jakobian

rather than my own view I'd be more interested to see claims from the community

historically the two are certainly different

13

A: In simple terms, what is the difference between logic in mathematics and philosophy?

The definitions of 'logic' and 'mathematics' are themselves subject to dispute. In particular, the word 'logic' is used in different senses. At its narrowest, it is concerned with the relationship of consequence between propositions or sentences. In a wider sense it is sometimes used as a synonym...

3

A: Is mathematics based on formal logic, or vice versa?

Historically, mathematics and logic evolved independently, though mathematicians have always used forms of logical inference. Euclid, for example, proved things by reductio ad contradictionem which is a kind of rule still used in modern logic. But the two main systems of ancient logic: Aristotle'...

we need to distinguish between the chicken and the egg

Ryder Rude

10:32

yes, all of mathematics is somewhat dependent on logic, as logic is the language of mathematics

in this sense, we can't say that logic itself is mathematics

but also, mathematical logic involves two languages : observer language and object language. this means that we are using the language of logic to study logic

the observer language is something we r not studying. and the object language is something we r studying like we would study any other mathematical structure like groups, topology, etc

Jakobian

I'd rather we use precise language or not discuss it, preferably the latter

either way saying that formal logic is mathematics rises an eyebrow and I'd need good evidence for such claim

It's not enough that you use some concepts the meaning I don't understand

Ryder Rude

i haven't studied it in detail myself. but the example books give is : suppose u r using English to study German. then German is the object language and English is the observer language

we study a language using another language. and we prove things about the language we are studying

so this aspect of mathematical logic can be called mathematics. it's just proving things about a system we r studying, no different from group theory or set theory

but, in general, mathematical logic cannot be called mathematics, i think

Jakobian

@RyderRude I don't think that I can agree that mathematics is a language the same way formal logic is a language

Ryder Rude

@Jakobian i agree. mathematics is specific axiomatic systems written using the formal logic language

Jakobian

So we must agree that mathematics and formal logic are different in significant ways then

and neither can be called the other

Ryder Rude

10:47

yes

formal logic just means the act of doing logic formally using symbols, such that there is no ambiguity

this is also what the philosophy SE answer says

this idea is distinct from mathematics

Jakobian

sure. At best we can say the two work closely with each other

Ryder Rude

yes. it is somewhat a two way relationship

mathematics has to use precise logic, as proofs should be unambiguous. this means mathematics is a consumer of logic

Jakobian

because when you consume it, it drains all your mental power

and you can't think logically anymore

Ryder Rude

xD

@Jakobian what r u learning these days

Jakobian

Wallman-Frink compactifications and related concepts

for example, recently I was checking that for a locally compact Hausdorff space $X$, its one-point compactification is a Wallman-Frink compactification

Ryder Rude

11:00

Oh

I'm not familiar with these ideas

Jakobian

From what I know, most compactifications that anyone cares about are Wallman-Frink compactifications

Ryder Rude

oh

Jakobian

not sure if all of them are, it was an open problem in the past, maybe it still is

Ryder Rude

I've been looking into philosophies of subjective experience

and the measurement problem

i cam across a new interpretation of QM. It's called transactional interpretation

it says there r two kinds of time. the one we have here. and another time in the spatiotemporal space. in that space, things from the future interact with things from the past

this is a paper about it arxiv.org/abs/2103.11245

this is an overview of the ideas arxiv.org/abs/1608.00660 . the previous paper discusses how our spacetime emerges from this idea

Jakobian

physics, as you might tell, is not particularly interesting to me

some physics maybe is, but philosophy of physics is particularly not interesting

Ryder Rude

11:10

oh. but this is pure physics, not philosophy. it is a fundamental theory idea, which is why it appears close to philosophy. it hasn't been tested

u were interested in spacetime and big bang stuff. this idea is about how our spacetime emerges from the other one

i seriously doubt this idea. it is just one of the proposals to solve the measurement problem

Jakobian

just because I want to have basic understanding, doesn't mean I want to study it

or that I am particularly interested in it - I am not

Ryder Rude

oh

i understand

u didn't mention physics in ur main interests

Jakobian

its not even my side interest, to be honest

Ryder Rude

are you interested in philosophies of consciousness?

Jakobian

no

Ryder Rude

11:17

Goff vs Kastrup debate 2020, Part 1

►

@Jakobian oh..

Claudio

12:48

I wonder if maybe I'm being overly cautious: when I check for the existence of partial derivatives, continuity, or so on, I often encounter expression like $$ \lim_{t \to 0} \frac{t^b}{(t^2)^{a}}, a \in \mathbb{R}$$ and usually in the solutions to the exercises $(t^2)^{a} = t^{2a}$ but I feel like this might be wrong in fact for $a = 1/2$ we have $t^{2a} = t \ne |t| = (t^2)^{1/2}$

maybe I can add the ulterior assumption $t >0$ but then the limit would be different

@SineoftheTime anyways, one of the answers to the question u linked yesterday did exactly what was done in the solution, namely if u have $1/(f+g), f,g \ge 0$ then u can find an upper bound $1/f[ 1/g] \text{ if } f \ge g [\hspace{0.2cm} g \ge f]$ so u search for two upper bounds in the two cases $|x|^c \ge |y|^d [\text{ or viceversa }]$

I wonder why they don't teach the most general case hahaha

Sine of the Time

13:15

@Claudio $(a^b)^c=a^{bc}$ is true if $b,c \in \Bbb R$ and $a$ is a positive real

Claudio

yeah I know, thats my point hahah

but $t$ isnt necessarily positive is it?

unlike $\rho$ for polar coords

Sine of the Time

It seems to me that you have to consider the left and right limits, so no we can't say that it's positive

@Claudio it would be too easy to get the right result in th exam :D

Claudio

13:31

yeah what I'm saying the solutions don't seem to care about this

which is kind of weird so maybe I'm overlooking something

Sine of the Time

14:02

@Claudio maybe in this case, since $2-2a=0$ holds for $a=1$, then $(h^2)^a=h^{2a}$

ModularMindset

14:37

I wonder what particle(s) a $2\pi*k$ rotation for $k$ a natural number correspond to, where the orientation of the vector never returns to its original orientation. This in contrast to fermionic particles.

Ryder Rude

15:19

@ModularMindset when $k$ is a natural number,.the particle is a bosons. boson states return to original state after 2pi rotation

some bosons found in nature are Higgs Boson, photon, W and Z boson, Gluons

Higgs is spin-0, photon is 1. and idk about the others. they're probably 1

the spin statistics theorem says that half integer spin particles must be fermions and integer spins must be Bosons en.wikipedia.org/wiki/Spin%E2%80%93statistics_theorem

Claudio

17:02

@SineoftheTime nah the reasoning is for $a \in \mathbb{R}$. I think they just didnt put much effort in writing the solutions this time (they are always running late) :P

Sine of the Time

18:02

@Claudio 💀

Gian's Pizzeria

@SineoftheTime what is the integration theorem for exact shapes of class $C^0$? What is the sufficient condition for the exactness of shapes $C^1$ in a stellated open space? pls

Sine of the Time

@Gian'sPizzeria by shapes do you mean forms right?

Hi @pizza

When is your exam?

Pizza

Hi

in 2 days

Gian's Pizzeria

@SineoftheTime yes

Pizza

I hope it goes well

Sine of the Time

18:15

did you study the theory of differential forms? I don't know which theorems did you see

@Pizza you studied a lot, you have to manage well the time and don't panic

Gian's Pizzeria

Yes

Sine of the Time

@Gian'sPizzeria On star domains, closed is equivalent to exact if the form is C^1

Gian's Pizzeria

👍

Sine of the Time

Oct 16 at 17:30, by Sine of the Time

Did you see a theorem like this: $\omega$ is exact iff for all $x,y\in A$, for all $\gamma, \phi$ $\int_{\gamma} \omega=\int_{\phi}\omega$

read the messages after this one

Pizza

@SineoftheTime I'm just a little afraid that I'll forget things during the theory questions

Sine of the Time

18:18

I would focus first on the exercises

Pizza

yes indeed

Even the professor said it

Sine of the Time

do first the exercises you're sure you're able to solve

Gian's Pizzeria

@SineoftheTime thanks

Sine of the Time

like critical points, double integrals, diff equations

@Gian'sPizzeria np

Pizza

@SineoftheTime I have to be careful not to make careless mistakes

Sine of the Time

18:22

yeah

Pizza

What do you advise me to do tomorrow? It's the last day

Should I review the exercises or do more theory?

Sine of the Time

that's up to you

Gian's Pizzeria

Anyway

Sine of the Time

you can read the text of the exercises of the previous exams and see (in your head) if you can solve them

If you know how to solve them, don't do all the computation

Pizza

Oh okay

Sine of the Time

18:26

@Gian'sPizzeria are you preparing an exam?

Gian's Pizzeria

$|z^2-1|^2=1$ on which geometric plane are the roots of the equation placed?

Sine of the Time

what is $z$?

Gian's Pizzeria

If I solve it I'll find out how man got to the moon

@SineoftheTime Complex number

Sine of the Time

what do you mean by "on which geometric plane" ?

Gian's Pizzeria

@SineoftheTime It would be finding solutions

Sine of the Time

18:31

do you have an idea on how to solve these kind of exercises?

It does not seem hard

Gian's Pizzeria

@SineoftheTime No

Sine of the Time

write for example $z$ in the cartesian form

Gian's Pizzeria

Ok

Sine of the Time

19:54

@AlessandroCodenotti How are you? The situation is critic in Bologna, hope you did not have problems in the last days

Alessandro Codenotti

I was out of down during the weekend luckily. The street I live in got flooded, but not catastrophically so, it was mostly an issues for cellars and some ground floor shops and we were without electricty for a few hours, nothing compared to the seriously affected areas

And my flat is pretty safe being on the 5th floor

Sine of the Time

One of my friends was out of town and did not manage to get back to home because the street was flooded

Alessandro Codenotti

Yeah it would have been challenging for me to get back home on Sunday as well, but I returned yesterday and it was not a problem

Sine of the Time

glad to hear you're fine :)

think_meaning_builds

21:17

Internet Archive and Wayback Machine are down again. Yes, again. However, Archive-It and IA's blog are still up.

Claudio

$$ |\sqrt[3]{x^5}| = \sqrt[3]{|x|^5} $$ this can be written safely without the restriction $x>0$, correct?

@think_meaning_builds hacked again?

think_meaning_builds

Yup, fourth time this month.

Claudio

DANG

think_meaning_builds

:(

Why doesn't StackOverflow step in and help guard them?

Ben Steffan

🤔

think_meaning_builds

21:30

I'm thinking about asking on cybersecurity.SE or StackOverflow meta...

Ben Steffan

I think you have a funny idea of how cybersecurity works

What exactly is SE supposed to do here?

This isn't like a physical """battle""" where you can just pull up with more troops

Xander Henderson

@Claudio Assuming that $x$ is real, sure.

think_meaning_builds

Security advice?

Share their expertise.

Ben Steffan

About a system that has nothing to do with SE?

Claudio

@XanderHenderson thank you, that's reassuring

Ben Steffan

21:33

That's not a bad idea per se, but it's like pouring a bucket of water onto a housefire :)

besides, there's organizations & companies better suited to give that sort of advice

think_meaning_builds

Help rebuild it more securely.

Ben Steffan

Cybersecurity is not SE's area of expertise.

Xander Henderson

It's getting DDoS'd... there are some things that can be done to mitigate against such an attack, but not a lot...

Ben Steffan

SE is a Q&A network.

Xander Henderson

Q&A?

Ben Steffan

21:35

@XanderHenderson ...yes :)

"Questions & Expertise"

think_meaning_builds

They have been using volunteer help for so long and now would be a good time to help another volunteer organization, imo.

Ben Steffan

The point is they haven't really got much to offer on this front.

Xander Henderson

@think_meaning_builds My house is on fire, and the fire department can only send one truck. I should probably call my neighbor. He has a garden hose. It doesn't quite reach the house, but I clearly need more help.

think_meaning_builds

This begs the question, who can help?

Xander Henderson

@think_meaning_builds That is not what that phrase means.

And, again, this is a DDoS attack. There are some things that can be done about this kind of attack, but not a lot. What "help" are you expecting?

think_meaning_builds

21:44

I'm not using the phrase in the formal sense, but the everyday language sense, sir.

Sine of the Time

what is the internet archive?

Xander Henderson

@think_meaning_builds That usage is an abomination.

think_meaning_builds

A library.

Ben Steffan

@SineoftheTime it's where they store all the old websites that have gone away (literally)

among other things

it's pretty much what it says on the tin

think_meaning_builds

Internet Archive

The Internet Archive is an American nonprofit organization founded in 1996 by Brewster Kahle that runs a digital library website, archive.org. It provides free access to collections of digitized materials including websites, software applications, music, audiovisual, and print materials. The Archive also advocates a free and open Internet. As of September 5, 2024, the Internet Archive held more than 42.1 million print materials, 13 million videos, 1.2 million software programs, 14 million audio files, 5 million images, 272,660 concerts, and over 866 billion web pages in its Wayback Machine. Its...

Sine of the Time

21:48

ah ok

think_meaning_builds

If it gets destroyed, it will be a loss to all of us.

Ben Steffan

a huge loss

it's very important in all sorts of ways

think_meaning_builds

The library of Alexandria is burning.

And nobody cares.

QED

psie

In Introduction to Topology by Gamelin and Greene, the Cauchy-Schwarz inequality is stated without the absolute value on the left hand side (unlike on Wikipedia). It reads $$\int f\overline{g}\leq\sqrt{\int |f|^2}\sqrt{\int |g|^2}.$$Can this be right? Isn't the left hand side possibly a complex number, and we are comparing it to a real number, which only makes sense if the left hand side is also real.

s.harp

they are probably dealing with real hilbert spaces then

psie

22:04

s.harp

then there is an absolute value missing

psie

I believe so too. There was a previous exercise where one had to prove the inequality for real-valued functions, but this time it is for complex-valued functions.

Ben Steffan

22:19

An $\infty$-operad is a fibration of $\infty$-operads with codomain the commutative $\infty$-operad

how's that for a definition

I have no idea how I will fit even a single proof into this talk

hell, I have no idea how I will even fit all of the relevant definitions

Thorgott

22:44

@BenSteffan are we defining a fibration of $\infty$-operads before $\infty$-operads?

Ben Steffan

@Thorgott obviously :^)

Thorgott

lovely

Ben Steffan

and other tricks to avoid spelling out the definition of $\infty$-operad

Thorgott

i think ill have to learn what an $\infty$-topos is soon

Ben Steffan

oh, also fun

one of these days I will do so as well and go and finally read the ABG+ papers

Transcript for

$Mathematics$ Mathematics