Mathematics - 2023-12-17

Jakobian

00:01

@TedShifrin I'm tired but, why is this stating uniqueness for every connected open neighbourhood $U\subseteq U_0$ of $x_0$?

I feel like this might be stronger than just claiming there exists unique such function on a connected neighbourhood of $x_0$, but I'm not sure

Ted Shifrin

I think it gives you the result on a maximal connected neighborhood, which your statement wouldn’t. I personally never state it this way, but it’s good.

Jakobian

00:16

It says that no matter how much you shrink your neighbourhood, as long as you keep it connected, you have uniqueness

I think they're avoiding a possible scenario of the function being defined on some way smaller neighbourhood, just not being able to be extended further

Ted Shifrin

00:29

I think the point is the reverse. Think of various branched coverings.

John Zimmerman

01:26

Anyone know of a differential equation with $n$ temporal variables and $n$ spatial variables?

Jakobian

@TedShifrin I still don't get it. Maybe it'll come to me later

John Zimmerman

in other words time and space are treated on equal footing

oscarmetal break

n temporal variables?

John Zimmerman

yeah like a mathematical time laplacian

you take a time laplacian and a classic laplacian and equate them maybe

oscarmetal break

wow, I can't imagine this.

Is it similar to say each spatial variable has dependence on $n$ temporal variables?

John Zimmerman

01:43

also you could take the $\nabla$ operator

$\nabla_t f =\nabla_x f$

Semiclassical

Closest I can think of would be working with correlation functions, e.g., you track some number of physical fields and measure each at some position/time

If you take an example where all the positions lie along a line, then that’s one spatial coordinate and one temporal coordinate per particle

Thing is I’m not sure that gives you differential operators which have that form

John Zimmerman

what are they nonlinear?

Semiclassical

No, that’s not the issue

John Zimmerman

oh correlation functions, meaning probability theory

Semiclassical

Well, more like statistical mechanics

Correlation functions are bread and butter for field theorists

Another idea is the multi-time formalism in the pilot wave interpretation but that’s probably way too involved

John Zimmerman

01:57

could vary the space and time variables as well. instead of $n$ space and $n$ time could have 15 space 9 time or anything else

Obliv

02:17

$\frac{\partial^2 y}{\partial x^2} = \frac{1}{v^2}\frac{\partial^2 y}{\partial t^2}$

so that has 2 spatial and 1 temporal?

derived from $y=f(x\pm vt)$ so just include another variable, say $t'$ and do the same process to $y = f(x \pm vt +t')$ maybe idk

oscarmetal break

It also need to be equal footing, isn't it? also need to satisfy $\nabla_t f=\nabla_x f$

John Zimmerman

@obliv no that equation is 1 space and 1 time

Obliv

i'm defining $t'$ to be another time variable tho

or is that not the problem

like $f(x,t,t')$

which might be 1 spatial 2 temporal actually since the function isn't necessarily mapped to $\Bbb{R}^2$

John Zimmerman

$\nabla_t f=\nabla_x f $ initially I was thinking about this equation where you expand the operators on both sides to $n$ dimensions each. But you could very well expand $\nabla_t$ to $k$ dimensions to give the equation a lot of flexibility

Ted Shifrin

02:51

@JohnZimmerman one-dimensional heat and wave equations

Obliv

what is a one-dimensional wave equation? pressure waves like sound?

hmm maybe my first equation counts then if $y$ is horizontal displacement

Obliv

03:07

do fourier transforms exist for spatial integration

Ted Shifrin

Guitar strings.

@Obliv Yes, and more generally than that.

Obliv

okay, so like in my textbook for optics we derived the far-field formula for the E-field given at some point on a screen that passed through a single slit. It said if the field strength wasn't uniform across the slit, we could use the FT of the function that describes the field strength at various points within the aperture..

so when I looked up what a FT was it mentions converting a function that depends on time to one of frequency

and in the context of that section of my book i didn't see it immediately applicable. but yeah if it works for spatial stuff too then that makes sense

but then we'd have an integral with two functions inside..

like specifically in the derivation instead of just integrating $e^{iks \sin\theta}$ over the slit position $s$ we'd have another function that varies the amplitude by $s$ so $F(s)e^{iks \sin\theta}$ as the integrand i think

that reminds me of convolutions from ODE

Ted Shifrin

03:42

Both are integrals :)

Obliv

03:55

that got me nostalgic and i was hoping to see my old community college professor teaching ODE this spring so I could visit but turns out he isn't teaching anything :( i hope he didn't retire

people at my university were telling me they didn't cover a lot of the stuff i did in my ODE class in their own class so I'm guessing I had a special teacher

Ted Shifrin

Different places have different texts and syllabi.

Obliv

yeah, wish I kept all of my HWs and notes lol, could use them

but at least I still have the book

Ted Shifrin

And, yes, you may have had a particularly dedicated teacher. There are more of those in community colleges than in universities.

Obliv

what was your career journey like @TedShifrin

or where can I read about it, if you've already talked about it

I wonder if you ever had stints teaching at CCs

would be quite a ridiculous over qualification lol

Ted Shifrin

I kept all my notes and stuff until after I graduated, and most math stuff until I retired snd had to downsize to move cross-country. I got tid of modt of my own lecture notes, exams, etc., except for LaTex stuff on my computer.

Obliv

04:00

that's a shame, although inevitable. I wish I had all of that stored away

Ted Shifrin

No, I never taught in CC. I did get certification to do so in case I needed to teach my last year of grad school, but I was fortunate that my adviser supported me on his grant and had me teach a bunch of his diff geo classes.

I did grading for him, too, and ran the geometry seminar. A good deal for me ….

Obliv

Sounds like a fair deal, I think? Oh, the other day I was looking at some old treatise in physics and it occurred to me.. how did people take notes back then? Like, we take for granted the simplest things like pencils and paper..

I know making copies by hand was a pain, esp. limited by the tools of the time. But a lot of important stuff got pressed by Gutenberg's printing press or some variation thereof.

But notes for your own sake, I have no clue.

I guess you just had to be conservative with your writing

leslie townes

05:30

not sure how 'old' the treatise you were looking at was, or what 'back then' is, but gutenberg's movable type was mid 1400s, and cheap paper for students was available in a lot of places that formalized education was available by the mid-late 1800s. with most of what we might recognize as classroom education coming into existence between those times, during which many less advanced students were probably not expected to keep personal notes, and more advanced scholars had the resources to do so.

until very recently, anything resembling math or physics was mostly the province of monks and bored rich people.

if old novels are accurate, a lot of students in the 19th century would have written in chalk on slate in the classroom, with no taking that home and studying it later.

there was a cold war era joke that soviet math was often written in a very concise and elegant style, not for aesthetic reasons, but because of paper shortages induced by a planned economy. at least, i think it was a joke.

nickbros123

07:31

I haven't studied rigorously how these things are defined, I just have been going with the flow of what physics books do- can I consider a 1 dimensional sort of "up and down", ie, let's say $x(t)=sin(t)$, or some similar 1-d thing , from $t=0 \to t=2\pi$ or an equivalent period, is this considered a "closed loop"?

leslie townes

at the level of vibes, i see nothing preventing you from doing that. but also at the level of vibes, the precise choice of words would not matter. if there is something particular that a "closed loop" needs to be, however, that data might or might not count as one, depending on those particulars.

e.g. something happening in a space that is "only" a subset of R^1 might not be "interesting" enough to model whatever you would want to analyze via "closed loops."

in math, but maybe not all physics, people will sometimes call a function from an interval like [0, 2pi] into a space X a "curve" [or some other term like "curve"] in X, and call it a "closed" curve [or maybe even a "loop"] in X if the endpoints of the interval get sent to the same thing in X. i don't know what relation this usage would have to meanings given to loops in physics.

your map would be a closed curve or loop in R^1 under that kind of definition.

nickbros123

08:28

Thanks for the info 👍

PM 2Ring

08:58

Slate (writing)

A slate is a thin piece of hard flat material, historically slate stone, which is used as a medium for writing. == Composition == The writing slate consisted of a piece of slate, typically either 4x6 inches or 7x10 inches, encased in a wooden frame.A slate pencil was used to write on the slate board. It was made from a softer and lighter coloured stone such as shale or chalk. Usually, a piece of cloth or slate sponge was used to clean it and this was sometimes attached with a string to the bottom of the writing slate. == History == The exact origins of the writing slate remain unclear....

> By the nineteenth century, writing slates were used around the world in nearly every school and were a central part of the slate industry. At the dawn of the twentieth century, writing slates were the primary tool in the classroom for students. In the 1930s (or later) writing slates began to be replaced by more modern methods.

@Obliv ^

Jakobian

11:56

Could someone help me find Lebesgue differentiation theorem for Bochner integrals?

Or failure thereof

@porridgemathematics maybe you know of any such results?

Jakobian

12:14

I found something for $f:\mathbb{R}\to E$ where $E$ is a Banach space, which is case that I'm interested in

Mats Granvik

15:31

Is the q-series associated with the modular form for the Weierstrass equation x^2+y^2 = z^2 in Fermats Last Theorem theorem, equal to the the characteristic sequence of 1? The characteristic sequence of 1 being the sequence 1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,...

Jakobian

Just 50 more zeros to be sure

2

Thomas Finley

9

A: Can Boolean ring without unit be embedded into a boolean ring?

Part of modern, abstract algebra is turning imagination into reality by fiat. This is a rather general technique: say you have some concrete object $A$ and you want an element with property $P$: then just adjoin a formal element to $A$ and quotient the result by the collection of all relations th...

What is $A[\epsilon]?

I never came accross this symbol until today

Jakobian

@ThomasFinley they mean ring of polynomials

Thomas Finley

15:46

@Jakobian So, it consists of all polynomials with coefficients in R and powers of x. The elements of (R[x]) have the form $a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0,$ where $a_i$ are elements of R and n is a non-negative integer, correct?

Jakobian

you've changed notation?

Thomas Finley

Again, what's $I$ supposed to mean in that context?

Jakobian

@ThomasFinley an ideal of $R[x]$

Thomas Finley

@Jakobian Ah, missed it while generalising the notation. I changed $A$ to $R$ and $\epsilon$ to $x$. I mean is that what is meant by $R[x]$ (or $A[\epsilon]$ )?

@Jakobian Oh, thanks!

John Zimmerman

16:07

$$ s^n \frac{\partial^{2n}}{\partial s^{2n}}\Omega^n_s(x)=n^{2n}x^n\bigg(\frac{\partial}{\partial x}\Omega_s(x) \bigg)^n $$

setting $n=1$ results in a degenerate 1d linear diffusion equation

Jakobian

17:01

@leslietownes Let $x_n$ be a sequence of units vectors with $\|y\|\leq (1+r_n)\|y+\lambda x_{n+1}\|$ for all $\lambda$ and $y\in \text{span}(x_1, ..., x_n)$. How to show that $x_n$ is a basic sequence?

Here $\prod (1+r_n) \leq 1+r$, $r_n, r > 0$

I'm taking some $y\in \overline{\text{span}\{x_1, x_2, ...\}}$ but not sure where to go next

This is proof of Mazur's theorem

If $\sum_{n=1}^N a_n x_n = 0$, then $\|\sum_{n=1}^{N-1} a_n x_n\| = 0$ so that ... so $a_1 = a_2 = ... = a_N = 0$

If $z_n, z_m\in \text{span}\{x_1, x_2, ...\}$ then $\|z_n-z_m\|\geq \frac{1}{1+r}\|P_k(z_n-z_m)\|$

where $P_k(\sum_{n=1}^N a_n x_n) = \sum_{n=1}^{\min(N, k)} a_n x_n$

It follows that the coefficients $a_k(z_n)$ for fixed $k$ converge to some number $b_k$

(here $z_n\to y$)

$\|y-z_n\|\geq \frac{1}{1+r}\|P_k(z_n) - \sum_{n=1}^k b_kx_k\|$

$$\limsup_{k\to\infty} \|z_n - \sum_{n=1}^k b_nx_n\| \leq (1+r)\|y-z_n\|$$

Maybe like, $\|\sum_{n=1}^k b_nx_n - y\| \leq \|\sum_{n=1}^k b_nx_n - z_n\| + \|z_n-y\| \leq (2+r)\|y-z_n\|$

where $k$ is large enough

So now we take $\limsup_{k\to \infty}$ and then $n\to \infty$ to show $\sum b_nx_n$ is convergent and equals $y$

Jakobian

17:40

Okay, and now if $\sum a_n x_n = 0$, then $\|\sum_{n=1}^N a_nx_n\| \leq \varepsilon$ for big enough $N$, so $|a_1|\leq (1+r)\varepsilon$ which proves $a_1 = 0$. Then similarly $a_2 = 0$ and so on

Okay I see it

Jakobian

17:50

This is one of those proofs that I can follow details of, but I hardly get the idea behind

Jakobian

19:32

researchgate.net/profile/Sanket-Tikare/publication/…

I think this article is bullshit

they're proving that indefinite integral of a gauge integrable function is absolutely continuous which is ridiculous

leslie townes

20:10

jakobian, please don't undermine my otherwise rock-solid confidence in articles by people with gmail and yahoo email addresses in the bulletin of the kerala mathematics association

is the error at least pedagogically interesting, or is it something pedestrian

Jakobian

If what they said were true, then all gauge integrable functions would be Lebesgue integrable

which is a big error to make

I don't think any of the authors took a course in gauge integration (or even read about it)

I can't pin down the precise place where the error occurs because the proof is confusing

The error is that they seem to be trying to parition $[r_k, s_k]$ but the tags used in the partition of it are all equal to $\xi_k$

this makes it impossible for it to be $\delta$-fine in general (or a tagged partition, really)

the variation of Saks-Henstock lemma they're using, idk, seems false but I wasn't able to refute it (it won't matter if the Banach space is finite-dimensional so not a big error)

(this is actually what I was interested in)

John Zimmerman

20:39

Is anyone here?

Sine of the Time

no

Ted Shifrin

21:13

^^ Shades of Winnie the Pooh, Pooh asking at the entrance to Rabbit’s home, “Is anyone there?” Only to be told “No.”

I’m sure Munchkin has read this story.

leslie townes

21:36

ooh, i should introduce her to those stories. she was a little too young for them until recently.

Ted Shifrin

21:54

I misquoted. But the right gist. I still love those stories. Eeyore is my idol.

Jakobian

I'm giving up on trying to refute a.e. differentiability of integrals of gauge integrable functions taking values in infinite dimensional Banach space. It just sounds too hard

I found something like Pettis integral? But without differentiability anywhere. But seems like gauge integrable here would imply Bochner which is false

Sine of the Time

23:13

How to prove that the subspace $\{(x_n)_{n\in \Bbb N} \in \ell^2 | \, x_1=2x_2-x_3\}$ is closed?

Jakobian

@SineoftheTime do we agree that $x\mapsto -x_1+2x_2-x_3$ is continuous?

Sine of the Time

man, I did this trick dozens of times

How I did not see it

I should retire

Jakobian

It happens

Dannyu NDos

23:58

Is there a standard way of topologizing the complexity class $\mathsf{ALL}$? I tried doing it, and the result didn't seem to be Hausdorff.

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