Mathematics - 2022-07-19 (page 1 of 2)

00:34

I have a good question about formal proofs

Can I paste a 14 line snippet of python code which looks surprisingly like real math (using syntax sugars)

Ted Shifrin

A. I’m not going to read this. My days of grading homework are long gone. B. I hope it won’t be searchable. Plenty of people (no longer me, obviously) are assigning graded homework out of the book. Having solutions easily available to be copied is not something that pleases me … any more than the illegal electronic versions available. @polite

Mathematical Emergency

from abc import *
from set import Set

def magma(A:str, op:str="*"):
A = Set(A)
A.closure = Axiom(([a,b] in A) >> (eval(f'a{op}b') in A))
return A

A = magma("A", "*")

with Proof((a in A) >> ((b in A) >> (a*b in A))) as p:
p.let(a in A)
p.QED # throws exception
# here p is not a valid proof (yet)

So any way how do I complete the proof at the bottom?

In other words if we know that $A$ is automatically closed under * by definition, then how do we prove that (a in A) => ((b in A) => (a*b in A))

Calvin Khor

@MathematicalEmergency

Mathematical Emergency

It's a currying thing, but I'm lost as to how we usually prove this as maths people. My library is going to be ZFC based but indirectly (I'm just implementing things that are interesting objects such as the integers. No one likes a formal proof that's 100 lines to prove that 2 < 3

So whatever proof I present to users will be at a high level

The library's datastructures reflect that so far

polite proofs

@TedShifrin Is there any way I can bribe you for part A? And of course, it won't be searchable unless you say otherwise.

Calvin Khor

00:40

@MathematicalEmergency that sounds precisely like the definition of being closed under *? So I would guess you need to feed it the axiom as the proof somehow

Mathematical Emergency

I guess you gave me an ideA

draw a proof tree

Calvin Khor

@politeproofs i think you should not push this any further

Mathematical Emergency

It should be simple. You say let(a in A). But how do you "introduce" the next implication?

It's kind of in between formal and informal proofs

so that's why I'm confused as heck

Calvin Khor

@MathematicalEmergency but that's just understanding how your program works right? where is the informal bit

Mathematical Emergency

The informal bit comes from the fact that the API has "informal English keywords"

So they resemble and only resemble English-math statements. I don't want even API users to see "how things work under the hood". Because how that works, I'm straying from standard forms of implementation

I'm just coding whatever is needed and no more

Ted Shifrin

00:45

@politeproofs Not unless you want to pay my hourly rate, which isn’t cheap. I can, as always, answer the occasional question in here.

Calvin Khor

in the natural number game, you would have been able to call A.closure with two inputs

which would then return the statement a*b in A

Mathematical Emergency

I might do stuff like automatically commute two to n propositions P1 & ... & Pn across &'s for you so you don't have to "enter in hypotheses exactly"

Ted Shifrin

The chatroom is malfunctioning, again.

Mathematical Emergency

@CalvinKhor that's interesting, what's that

Calvin Khor

what's what?

Mathematical Emergency

00:46

natural number game

Calvin Khor

it amazes me that you're looking into this and you don't know that :) ill get a link

polite proofs

I understand. In that case, what's wrong with 1.2.22? Let $\overrightarrow{AB} = x$ and $\overrightarrow{AC} = y$. Clearly $x$ and $y$ are not parallel, so by 1.2.20. (b), $bx + ay$ bisects the angle between $x$ and $y$. Since $\overrightarrow{AD}$ also bisects the angle between $x$ and $y$, it follows that $\overrightarrow{AD} = t(bx + ay)$ for some $t \in \mathbb{R}$.
Then $\overrightarrow{AD} = \overrightarrow{AP} + \overrightarrow{PD}$, but notice that $\triangle PBD \sim \triangle ABC$, so $\overrightarrow{AD} = s(x + y)$ for some $s \in \mathbb{R}$. Then by 1.1.10. (b), we must have t…

Calvin Khor

ma.imperial.ac.uk/~buzzard/xena/natural_number_game

polite proofs

But clearly, this goes against the illustration provided in the book.

Mathematical Emergency

That uses Lean which is based on CIC, I know that

but I'm doing my own thing as Lean is too esoteric for new users

I want 0 learning curve for new math users who know python already

So Lean is Type-theoretically founded. I'm just founding this upon "anything". I currently don't even care if there are paradoxes popping up

Any axioms

But the way I work with them is "all standard methods used by humans"

Calvin Khor

00:50

well the game makes it seem p easy to learn :P but since you're making this up on the fly, you just need to figure out how to make A.closure eat two proofs, one proof that a in A and another that b in A, and spit out a*b in A

or is that not satisfactory somehow and you want a different "idea"?

Mathematical Emergency

No, that's pretty much it

But there's a lot to be done internally

more than checking if expressions match up to variable sub, and recording the mapping

and mapping back to the user's chosen variables, etc

I have to store each step in the proof, for one

I could store it as the .py file, but I also want "objects" in memory

Ted Shifrin

Polite…. huh? Looks like garbage. What is $P$ and where did it come from?

Mathematical Emergency

I also want to convert (easily; unlike with Lean4) the proof steps to fluid English-math

+ KaTeX support

So you can pass along what you think appropriate LaTeX is for your symbols

polite proofs

@TedShifrin $P$ is a point on $\overrightarrow{AB}$

Mathematical Emergency

Or stuff like $\text{Hom}_C(A,B)$ will be generated for you because $\text{Hom}$ is standard notation

Calvin Khor

00:53

@MathematicalEmergency have you seen mizar.org/fm ?

Ted Shifrin

What for? It certainly has led you to absolute garbage.

polite proofs

I'm just describing $\overrightarrow{AD}$ as a linear combination of $x$ and $y$

Ted Shifrin

You are not using the important thing and have created garbage. What do you need to use about $D$ that you have not yet used?

Mathematical Emergency

@CalvinKhor heard of it before. I could spend 5 years trying to understand a single paper at that level. I prefer to just code until I understand the underlying issues

The underlying problems emerge naturally, everybody including my project needs some type of variable substitution awareness

e.g

polite proofs

I'm not sure... I'm not using $D$ being on $\overrightarrow{BC}$ too much

Ted Shifrin

00:57

Not at all. Clearly that is the way to go.

Calvin Khor

@MathematicalEmergency well, i am not really asking you to read the fine print. but the code is already human readable

Ted Shifrin

Depends on which human!

Mathematical Emergency

After all, Thierry Coquand had to do a lot of inventive coding I think to test out ideas. Since the end result was a useable PA.

polite proofs

$\overrightarrow{AD} = x + \overrightarrow{BD}$ and $\overrightarrow{AD} = y + \overrightarrow{CD}$

Mathematical Emergency

@CalvinKhor I'm not sure what you're trying to tell me :)

Ted Shifrin

01:00

Go think, polite.

Mathematical Emergency

:D

Calvin Khor

sciendo.com/article/10.1515/forma-2015-0010 if I understand correctly, the paper is compilable code as-is

Mathematical Emergency

brb

shintuku

01:20

did anyone uncover the secret to spending absurd amounts of time sitting studying

other than drugs

Mathematical Emergency

01:41

ample breaks and exercise are key

sitting is kind of like smoking

leslie townes

shin: i think your answer is 'no'

shintuku

@MathematicalEmergency sound advice

i will take that as an excuse to watch a movie

leslie townes

morbius

Ted Shifrin

Send Munchkin to lead calisthenics!

leslie townes

my wife is going to a conference and munchkin is 'helping' her pack, it's not going well

she's gonna show up and realize half her stuff has been replaced with stuffed animals

Ted Shifrin

01:47

ROFL

Send Munchkin to infiltrate and sabotage MaraLago.

leslie townes

i proposed this and for some reason the wife was not a fan

Ted Shifrin

You proposed which?

leslie townes

basically that i not be left alone with munchkin for 4 days

Ted Shifrin

Your wife underestimates your deleterious effscts.

We can so attest.

leslie townes

if she were reported to the authorities for abandoning her child to me, i could understand it.

i'm not saying do that

i'm just saying.

Ted Shifrin

01:54

You wsnt to switch legal expertise snd defend yourself ?

leslie townes

haha, our cat just attacked our daughter.

Ted Shifrin

Better to ship Munchkin to MaraLago.

Calvin Khor

02:52

@leslietownes cat's just looking out for you

geocalc33

03:30

-1

Q: nontrivial foliation of the cubed unit interval

Does $M=(0,1)^3$ admit a co-dimension 1 foliation whose leaves have the property that they admit closed loops and s.t. the leaves converge to two points (i.e. they converge to $011$ or $100$)? I can prove the trivial cases for $M$ in which the foliation is simply by planes, and the slightly more ...

Anyone know how to solve the last case?

Koro

05:29

Is $2^{\phi(k)}-1$ divisible by $k$?

where $\phi$ is Euler phi function.

where k is odd.

graffe

05:46

if I have two coins with probs p1 and p2 respectively of giving a head, what is the expected number of coin tosses until they are both heads?

is it just 1/p1+1/p2?

Calvin Khor

@Koro

Euler's theorem

In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that, if n and a are coprime positive integers, and φ ( n ) {\displaystyle \varphi (n)} is Euler's totient function, then a raised to the power φ ( n ) {\displaystyle \varphi (n)} is congruent to 1 modulo n; that is a φ ( n ) ≡...

@graffe assumin you flip coin 1 then coin 2, write down the outcome, then repeat. probability of HH in one flip is p1p2. The number of coin tosses until both heads is a geometric distribution with success parameter p1p2. So the expected number of tosses until both heads is 1/(p1p2) (if you count the last toss i.e. the HH toss; otherwise subtract 1)

S.M.T

06:04

Hii everyone. I have been going through a lot of Stress from past few months. I think what is the pont of studying science , maths or eating or travelling or like living if we die anyway. I study so hard for my maths but knowing i have to die 1 day. I just feel there is no point of working so hard on my maths. All that i worked hard , i will forget it.

Calvin Khor

@S.M.T what's your conclusion then, not to work at all? because nothing you said is specific to maths

S.M.T

No , its not specific to math problem. I just wanted help in my fear

Calvin Khor

if you're stressed try to take a break

if you're in a uni they should have counsellors

S.M.T

I have but i cant get this thought out of my head.

@CalvinKhor Ill be in uni in 3-4 months

Calvin Khor

the way i see it, you're going to die whether you do maths or not. doing maths is strictly better than not, and you don't have much time, so get going asap ;)

Not Tfue

06:29

Is this the same as for all x and y that is natural number such that x is composite or y is prime or (x>=y)?

Or is it just saying there doesn't exist x and y such that x is prime and y is composite and x<y?

I am confused about this.

I have heard negation of there exists for all but the professor was saying there doesn't exist.

Koro

@CalvinKhor I know the function. But I wondered if the statement I mentioned above is true or not.

Calvin Khor

@Koro the wikipedia page I linked is precisely the result you asked for in the case $a=2$

its not a link to the wiki page on Euler's Totient function, but rather the "Euler Totient Theorem"

@Koro or rather I should say, from the wikipedia page, if you put a=2, then the theorem stated there is precisely the result you asked for. can't edit anymore

@graffe slight slip-up in my earlier comment; what I called a "toss" is a pair of tosses. so multiply by 2...

Not Tfue

06:45

Ignore my question. I know the answer.

Koro

@CalvinKhor basically, that’s generalised version of Fermat’s theorem.

Calvin Khor

fermat's little. yeah

Koro

:-)

I don’t know how I missed that. Thanks a lot @Calvin.

Calvin Khor

yw!

i cant understand the proof on wikipedia cuz im too rusty, so feeling pretty lucky i managed to help lol

Mad Spaces

07:21

why?

Mad Spaces

07:33

i solved it by applying l hospital on the limit

nvm not

one potato two potato

09:40

In practice, how do people usually compute cohomology group of a manifold? Definition itself is quite complicated to me so I usually use universal coefficient theorem to compute it indirectly using homology.

Balarka Sen

@onepotatotwopotato cellular cohomology, Poincare duality, ...

clever use to Mayer-Vietoris

one potato two potato

Ok poincare duality and MV sequence. But people rarely compute cohomology from the very definition right? Or do they?

Balarka Sen

They dont

one potato two potato

alright thanks

Thorgott

10:07

usually computing cohomology is no harder than computing homology

not even in light of UCT, but most techniques for computing homology work just as well to compute cohomology

the only thing you ever compute by definition is the homology of a point

the rest can be thought of as abstract nonsense (not that that's a viewpoint I'd advise)

Balarka Sen

a computer can write down (co)homology given a cell structure, sure. the tools i mentioned are just to do it faster than computers in certain occasions

i dont need to write down the cellular chain complex to know the cohomology of a 3-manifold

one potato two potato

Is there a computer program that writes down (co)homology of a given cell structure?

Thorgott

10:25

@BalarkaSen computer would need simplicial, not cellular, right?

Not Tfue

10:35

Shouldn't the statement "everybody doesn't love someone" be equivalent to "nobody loves someone" instead of "nobody loves everybody"?

0

Q: Predicate Logic Expression: "Nobody loves anybody."

Express the following in predicate logic: "Nobody loves anybody." $$P(x): \text{x is a person.}$$ $$L(x,y): \text{x loves y.}$$ My attempt was: $$\neg[\exists x(P(x) \land \forall y P(y) \longrightarrow L(x,y))]$$ Although my instructor wrote it as: $$\neg[\exists x(P(x) \land ( \forall y P(...

discrete-mathematics logic predicate-logic

Balarka Sen

@Thorgott why would it need simplicial

Thorgott

cause I don't believe a computer can compute degrees

for reasonable functions, it can, but not for unreasonable functions

Balarka Sen

it depends on how you input attaching maps

Thorgott

fair, I don't even know how to input a map into a computer

or a space, for that matter

simplicial is straightforward cause that just needs discrete data

Balarka Sen

suppose theres an intermediate category, where you have a CW complex with cellular attaching maps

which should be amenable to coding

or something.

just do some brain printing

Thorgott

10:49

yeah, idk, but im also no willing to think hard about anything

everything in sight is triangulable

2

user4539917

13:11

Feeling any better today? @S.M.T

one potato two potato

13:44

Why the last statement: $H^p(X;R)\cong Hom_R(H_p(X;R),R)$ true?

Thorgott

13:59

try proving it by hand

the punchline should be that all sequences of vector spaces split, more precisely you can find retractions $C_p(X;R)\rightarrow Z_p(X;R)$ and $C_p(X;R)\rightarrow B_p(X;R)$ on cycles and boundaries respectively

the former will help establish surjectivity and the latter will help establish injectivity

and, actually, the former will be possible too if $R$ is just a PID, so in that case the map is still surjective, its kernel is computed by the UCT

one potato two potato

14:34

I was thinking of the proof by proving $H^p(X;R)\otimes_R H_p(X;R)\to R$ is isomorphism if $R$ is a field and taking adjoint proves the statement. You're talking about this or directly showing the statement?

Balarka Sen

How do you plan to prove that's an isomorphism

Hi @AminIdelhaj

Long time

Amin Idelhaj

What's up? It's been a while!

Thorgott

the map $H^p(X;R)\otimes_RH_p(X;R)\rightarrow R$ is usually not an isomorphism

thats a misconception at the level of linear algebra

$R^2\otimes R^2\rightarrow R,\,(x,y)\otimes(u,v)\mapsto xu+yv$ is not an isomorphism, but its adjoints are isomorphisms

anyway, the proofs shouldnt look too different regardless of whether you formulate them in terms of the pairing or the adjoint

Balarka Sen

@AminIdelhaj not too bad, what have you been up to

Amin Idelhaj

15:04

Hey soryr

So I've been kinda just trying mentally to get back in the game somewaht

I was home for a year during covid and I feel like since then something broke in me and I haven't been able to get work done anymore

Idk if it's burnout or motivation's gone or what

But I'm trying to get it back

@BalarkaSen How about you man?

Balarka Sen

yeah thats happened to quite a lot of people i know

its been a rough 2ish years

@AminIdelhaj im alright. trying to learn more topology, as always. reading novels.

Amin Idelhaj

Nice, any particular topics you've been looking at lately?

Balarka Sen

currently im thinking about lefschetz fibrations and kirby calculus

Amin Idelhaj

Good stuff! Right now I should start reading a paper on the "Kuznecov trace formula"

Balarka Sen

is this about automorphic forms

Amin Idelhaj

15:10

It's in that vein yeah. I haven't started reading this paper so I know nothing of it yet

Balarka Sen

gotcha

Amin Idelhaj

But the OG version is the Selberg trace formula. My quasi-joke is that this is all a very souped up version of saying that sum of the eigenvalues is the sum of the diagonal :P

Balarka Sen

aha

Amin Idelhaj

It's good stuff for sure, the general phrasing for topological groups specializes to the typical version for hyperbolic surfaces (or more generally symmetric spaces), as well as to Poisson summation and to Frobenius reciprocity

Balarka Sen

i dont know anything here but have heard something to the effect of length spectra of hyperbolic surfaces and if it determines the geometry or not

S.M.T

15:15

@user4539917 Nah

Amin Idelhaj

Yea, so for hyperbolic surfaces the statement is basically, write your surface as $\Gamma\setminus \mathbb{H}$ where $\Gamma$ has only the identity and hyperbolic elements

(Let's say compact for now)

one potato two potato

@Thorgott Right I misinterpreted adjoint isomoprhism between Hom and tensor. It seems the statement follows from UCT. Thanks anyway

Amin Idelhaj

Let $r_i$ be the spectral parameters, so the eigenvalues of the Laplacian are $\frac{1}{4} + r_i^2$, and let $G(S)$ be the set of closed oriented geodesics on $S$.

Balarka Sen

@onepotatotwopotato yes it follows from UCT but you can and should prove it by hand

@AminIdelhaj arright

Amin Idelhaj

$$\sum_{i=0}^{\infty} h(r_i) = \frac{\text{Area}(S)}{2\pi}\int_0^{\infty} rh(r)\tanh(\pi r) dr + \sum_{\gamma \in G(S)} \frac{\ell(\gamma_0)}{e^{\frac{1}{2}\ell(\gamma)} - e^{-\frac{1}{2}\ell(\gamma)} } g(\ell(\gamma))$$

Balarka Sen

15:21

h is?

also whats g

Amin Idelhaj

Here $g:\mathbb{R}\to \mathbb{R}$ is even, smooth, and compactly supported, $h$ is its Fourier transform, $\ell(\gamma)$ is the length of $\gamma$, and $\gamma_0$ is the unique oriented prime geodesic satisfying $\gamma_0^m = \gamma$

Balarka Sen

ah, neat!

Amin Idelhaj

So now depending on your choice of $g$ and $h$ you can use this to do different things. For instance, if you want to prove the Weyl law, you want to choose $g$ so that its support lies in $[-R,R]$ where $R$ is less than the length of any geodesic on $S$. You'd hope to then be able to $h$ to be characteristic of $[-X,X]$, but obv $h$ must have Paley-Wiener type, so you need to massage it a fair bit

Balarka Sen

in some precise sense (above formula), the length spectrum and the spectrum of the Laplacian are dual.

Amin Idelhaj

Yup!

Balarka Sen

15:24

@AminIdelhaj Gotcha, so can count geodesics with this

Thorgott

wtf is this analysis im seeing

Amin Idelhaj

This is number theory Thorgott

Balarka Sen

analytic number theory on riemann surfaces, Thorgott

Amin Idelhaj

It's also analysis and it's also hyperbolic geometry

Thorgott

number theory hasn't existed for centuries

it's just a dogwhistle for analysis or for algebra, depending on whom you talk to

Amin Idelhaj

15:26

@BalarkaSen Yeah, I don't know the details but there's basically the "prime geodesic theorem" that's proven using this business

Balarka Sen

cool stuff!

Thorgott

i like geodesics, but the rest looks dubious

Balarka Sen

prime geodesics are like prime numbers, thorgott

so you can count how many are there upto some length and ask for a PNT

nOmBeR ThEorY

Amin Idelhaj

FWIW, the LHS of this expression is called the "spectral side" of the trace formula, and the RHS is called the geometric side

And my quasi-joke except the more time goes on the more I wonder how much of a joke this

Thorgott

what makes them prime

Amin Idelhaj

15:29

Is that "sum of eigenvalues = sum of diagonals" is similar. LHS = spectral side, RHS = geometric side :P

Balarka Sen

@AminIdelhaj makes a lot of sense

@Thorgott closed geodesic which is not a power of some geodesic.

Amin Idelhaj

Prime geodesics are basically those that "don't wind around"

Yup

Balarka Sen

can you do this with hyperbolic graphs

Thorgott

ah ok

Balarka Sen

count prime cycles instead

Amin Idelhaj

15:32

But yeah stuff like Weyl Law, prime geodesic theorem, and bounds you get in arithmetic quantum chaos (here you want to consider Hecke operators as well, and you get an "amplified trace formula") are all applications where you're sorta choosing g and h in a clever way. Some applications (big in Langlands) are where you vary the group. These I don't understand in the least lol

Hmm, I've seen something similar in Lubotzky-Phillips-Sarnak

Where they construct Ramanujan graphs

Balarka Sen

i know the result but not details

Amin Idelhaj

The proof of the trace formula basically boils down to, these two expressions are both different ways of writing the trace of a certain integral operator

Balarka Sen

i guess that must be the right perspective though, not to think about hyperbolic graphs (which are like H^2, has very little closed geodesics) but their finite quotients. Ramanujan graphs are exactly like large quotients of p-regular trees

@AminIdelhaj hm ok cool

Amin Idelhaj

I'm not quite familiar with the term "hyperbolic graph" as such, in general I actually tend to think of k-regular graphs as being analogous to hyperbolic surfaces

Because the (p+1)-regular tree is basically "p-adic hyperbolic plane" in a way

Balarka Sen

oh, i was just thinking of Cayley graphs of delta-hyperbolic groups. They're more like universal covers

@AminIdelhaj right

its exactly $\mathrm{SL}_2(\Bbb Q_p)/\mathrm{SL}_2(\Bbb Z_p)$, the real version being $T^1 X(1)$

Amin Idelhaj

15:38

And yeah there's some weird connections in this way. I went to this conference about a month and a half ago about Laplacians on random graphs and surfaces

It was a bit packed (6 hours a day of talks) and I was fairly fatigued so I missed a good bit of stuff and am a bit salty as a result

But a lot of what I did go to was p sick

Balarka Sen

if you read about p-adics too much you'd get p-sick

Amin Idelhaj

Hahahahaha

But yeah it started off talking about why you can speak of a "random surface", first talk just kinda stated that volume of moduli space is finite

Second did go into some detail on Mirzakhani's work on integrating on moduli space

Balarka Sen

@Thorgott: random surfaces are everywhere see

Amin Idelhaj

And this set the stage for some cool stuff. One common theme for hyperbolic surfaces is the Selberg eigenvalue conjecture, that for Gamma a congruence subgroup of SL(2,Z), the smallest non-zero eigenvalue of the Laplacian on Gamma\H is at least 1/4

Selberg was able to prove that it's at least 3/16

Balarka Sen

interesting

Thorgott

15:42

my eyes and ears are shut

"p-adic hyperbolic plane"

do you also believe in p-adic Lie groups?

Balarka Sen

wrong person to ask

Amin Idelhaj

And one of the results presented in this conference was that for $\varepsilon > 0$, the probability that a random surface in $\mathcal{M}_g$ has $\lambda_1 < \frac{3}{16} - \varepsilon$ goes to $0$ as $g\to\infty$

Balarka Sen

hes all about that

@AminIdelhaj are there examples which are lower than 3/16 lol

im confused, didnt you say selberg proved $\lambda_1 \geq 3/16$ always

Amin Idelhaj

ONly for arithmetic surfaces

This is for a general surface

Balarka Sen

ah

Amin Idelhaj

15:47

As for examples, I don't know any myself and I'm not even sure what's in general known. iirc we don't know an infinite family of arithmetic surfaces whose lambda_1 is bounded below by 1/4

It seems like this kinda thing is hard to compute

But yeah I tend to think of Ramanujan graphs as being analogues of what we hope arithmetic surfaces are, namely that in each case the spectrum of the Laplacian is controlled by that of the universal cover

Balarka Sen

gotcha

Amin Idelhaj

There's sorta two directions at the moment regarding my possible research area. I originally was gonna be looking at bounding spherical functions but that just wasn't doing it for me. Not sure if it's because I'm just not into the problem or because of burnout holdover or what

But one possibility would be estimating the volume of a surface whose lambda_1 is at least some T

By doing "linear programming on h in the trace formula"

Balarka Sen

that sounds cool.

Amin Idelhaj

Another possibility would be expander graphs/complexes

Balarka Sen

i like explicit examples

Amin Idelhaj

15:59

Anyway I gotta get going for now but yeah it was good catching up, I should ask you more about the stuff you've been doing as well! :)

Also Thorgott careful there's a random surface behind you

Balarka Sen

sure, take care!

@AminIdelhaj lol

Joe Shmo

is it true that $L^1$ functions satisfy $\int_X |f| d \mu < \epsilon$ for $\mu(X) < \delta$?

Thorgott

@AminIdelhaj almost surely?

@JoeShmo what's $\varepsilon$, what's $\delta$? that's hardly a complete statement

Joe Shmo

small

$X$ a meager enough set

if $f \in L^p$ for $p > 1$, then it is true

oh, lets say on a probability space

Thorgott

that does not answer my question

Joe Shmo

16:07

it's self explanatory: for $\varepsilon > 0$, does there exist $\delta > 0$ so that $\mu(X) < \delta \implies \int_X |f| d\mu < \varepsilon$?

if it's true, then the proof should be obvious

you know what? take $X = \{|f| > M\}$ such that it satisfies $\mu(X) < \delta$, where $\delta$ is small enough to your satisfaction

the latter is possible, since $f \in L^1$

does there exist $M$ large enough so that $\int_X |f| < \varepsilon$?

Balarka Sen

just use dominated convergence theorem

Joe Shmo

yes

Balarka Sen

@JoeShmo this is obvious by partitioning the whole space into $|f| < M$ and $|f| > M$

Joe Shmo

yeah yeah

ty

Thorgott

your quantification is messed up

were you asking this for all $X$ s.t. $\mu(X)<\delta$ or are you asking for the existence of an $X$ you can freely choose

cause if it's the latter, you might as well be lazy and choose $X=\emptyset$

Joe Shmo

16:13

right, which is why I restricted to $X$ being a superlevel set

user2103480

what a mathematician's question @Thorgott

Thorgott

a frenchman's question

so what's the actual question now, with proper quantification of all appearing variables?

Joe Shmo

the last paragraph where $X$ is a superlevel set is unambiguous

user2103480

@Thorgott for all epsilon is there a delta such that for all measurable sets X, if mu(X) < delta, then int_X |f| < eps

Balarka Sen

yes there is a delta

$\delta$

there

delta male mindset

2

Thorgott

16:18

ambiguity isn't a concern cause no proper statement has been formulated yet

Joe Shmo

thats false

Thorgott

or rather, you have formulated a proper statement before, but it's apparently not the one you actually meant to ask about

user2103480

@JoeShmo what is false

Joe Shmo

@Thorgott's second to last comment

Balarka Sen

"what is false"?

Joe Shmo

16:19

bye

user2103480

byw

Thorgott

it was indeed false, which is why I corrected it right after

on the other hand, you still have to produce the statement you actually mean to ask about

unless you meant to ask about the original one and everything about superlevel sets was meaningless

Joe Shmo

Thor, let's take a moment to appreciate that this is an incredibly pointless discussion about our feelings.

Balarka Sen

facts over feelings my friend

user2103480

I don't think it's pointless to talk about feelings

4

Joe Shmo

16:22

that said, I am satisfied with just restricting my original question to superlevel sets,

which is standard procedure in probability theory

Balarka Sen

lets just say, for the sake of argument, there is a question

2

lets stipulate that, ok?

Joe Shmo

and such that the choice of the original $X$ is unambiguous

Thorgott

so $X$ is a fixed superlevel set?

Joe Shmo

yeah, so you could quantify as follows: $X_M = \{|f| > M\}$

then $\mu(X_M) \xrightarrow{M \to \infty} 0$

Thorgott

interesting

Joe Shmo

16:25

which part

Thorgott

because that's a different statement from the one you get when you take $X$ to be fixed in the previous statement

Joe Shmo

again, this is all standard procedure in probability

no, I would argue that they next to identical

Thorgott

they are pretty different

Ted Shifrin

pops popcorn

Joe Shmo

you might take $X = \{|f| > M\}$ for a fixed $M$, such that $\mu(X) < \delta$, $\delta$ small enough to your satisfcation

I know when I'm being trolled

I know

Thorgott

16:28

what you're neglecting is the difference between $M$ being fixed beforehand and quantifying over all $M$ such that $\mu(X_M)<\delta$

Joe Shmo

^^that is IDENTICAL to the formulation with the $X_M$'s

agree to disagree

user2103480

How do you define identical here? If these statements all hold true for an L^1 function, then they're trivially equivalent

Joe Shmo

identical, as in it's a standard analysis formulation

user2103480

You mean: Assuming your statement about small sets yielding small integrals, and proving that the sets X_M you take become small, then the integrals over the sets X_M become small?

Thorgott

identical statements should not mean identical truth values

Balarka Sen

16:35

@user2103480 no, no, the quantification is $\exists f$

Thorgott

lmao

user2103480

@BalarkaSen what about the bimbofication

Thorgott

it's
$\forall[f]\in L^1\forall M\in\mathbb{R}\forall\varepsilon>0\exists\delta>0\colon\mu(X_M)<\delta\Rightarrow\int_{X_M}|f|d\mu<\varepsilon$
vs.
$\forall[f]\in L^1\forall\varepsilon>0\exists\delta>0\forall M\in\mathbb{R}\colon\mu(X_M)<\delta\Rightarrow\int_{X_M}|f|d\mu<\varepsilon$

important difference

user2103480

first off, you deserve to be banished from here for quantifying elements of L^1 as equivalence classes

alg*braist

Joe Shmo

I'm with you @user2103480

Balarka Sen

16:42

whoa, there, what happened to freedom of speech

Joe Shmo

freedom of speech was scrapped for snowflake culture

disagreement is fighting words. havent you heard

Mad Spaces

In physics, it is common to write infinite sums as integrals, without chicking their rigor (If riemann sum upper and lower actually converge to the same value)
Is there a quick way to check if the calculation done is rigoros, without falling back on the riemannian sums (or rather should i say, Valid)

Joe Shmo

but not all sums are integrals

wdym

infinite riemann sums?

Mad Spaces

Well, i can not answer this question. Physicists, books, lectures, tend to just write any sum that goes to infinity as an integral, and i am trying to find out, if there is a good way to check for the validity of this calculations, physicsts do not really care about rigor, but surely, the people who found out the theories and calculated these sums, probably checked for the validity. but most physicists do not follow those steps and just write sums as integrals as "understandable step"

leslie townes

16:58

there's no general method i know of for checking the validity of this stuff. physicists are fond of writing integrals that have no meaning, and integrals that have the exact meaning that you see as written. with no real way of telling the difference other than learning physics.

Mad Spaces

Is there some theorem that states if x and y in sum is that and this then sum = integral?

Leslie, i study physics.
You do not learn this stuff there....

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$Mathematics$ Mathematics