@leslietownes btw did you miss my joke from earlier?

Well, I tried at least. But it was a low hanging fruit

Interesting that showing continuity by arguing 'if the source moves small then the target moves small' is quite used in a certain field.

@onepotatotwopotato Arguing how?

for example Hatcher's proof of Brouwer fixed point theorem in dim 2

@Jakobian i missed it then but see it now. +1

it's near noon here but I'm so sleepy.

I went to a library, and they don't have a single book about category theory. Dang.

well, category theory is about everything, so in a way they did. all books are about category theory.

i feel like i know this, but to check: Suppose I have a solid region $V\in \mathbb{R}^3$, and I want to define its projection onto the $z=0$ plane. is that just the set of points $(x,y)$ such that $(x,y,z)\in V$ for some $z$?

(mostly trying to figure out if there's a better way to do something in mathematica, which...not hopeful atm)

@leslietownes that's quite a sad reality

@Semiclassical Of course.

@onepotatotwopotato I don’t know Hstcher’s proof. The proof I routinely taught proves smoothness quite easily.

the setup is a continuous map $h:\Bbb D^2\to\Bbb D^2$.

This is routine to make rigorous, but Hatcher avoided the tiny bit of vector algebra.

As I said, one often wants smoothness.

@TedShifrin unfortunate. (tho that's really just a matter of "sucks that i don't know how to do this any way but terribly slowly in mathematica")

If you have some sort of convexity, you can look for apparent contours of the boundary.

yeah, it's a nice convex set

Oh, so find the points on the boundary where the tangent plane is vertical. The projection of that curve gives you the boundary of the image.

hmm

(specifically it's supposed to be a particular 2D projection from the 6D set of 4-by-4 positive matrices)

Oh, I was in 3D.

Think that through and generalize.

yeah. i also do have a description of the set if i only project down to 3D initially, so in principle i can just apply that method to that 3D projection instead

You want points where the projection drops rank.

hmm, yeah

This is a crucial notion for some of the integral geometry in my second paper.

the 3D description i have is kinda gross, tho. the set of $(x,y,z)\in [-1,1]^3$ such that $|(x-y)z|\leq (\sqrt{1-x^2}+\sqrt{1-z^2})\sqrt{1-y^2}$

but that's a representation i'm stealing from elsewhere, so there's probably a nicer way to write it

(something something Schur complement blah blah blah)

mostly i'm just trying to formalize a particular side-view of my 3D solid

right now it's just "what i get when i plot orthographically from this angle in mathematica", which is just...not great

Over my head.

fair. mostly it's just frustrating b/c it's not a polyhedra

and thus i can't just cheat on this

Just for fun, I tried tweaking the definition of continuity. Instead of open sets having open preimages, let F-sigma sets have F-sigma preimages. But what should I play with this notion, in spite of naming it?

First see what functions have the property and why it might be useful.

Don't do it. Topology is already generalized all the way to categories

Topology is already quite comprehensive and complex

without tweakers

The motivation was, I wanted to seek an intermediacy between continuity and measurability,

Continuity sends open sets back to open preimages, and measurability sends Borel sets back to Borel preimages. F-sigma sets are an intermediacy.

So what does this get you?

What’s an example?

One particular kind of example that is not continuous but fits the new notion: Discontinuous functions whose all discontinuities are jump discontinuities.

Is that iff?

Removable discontinuities also

And a non-example: Dirichlet's function.

sin(1/x) is an interesting example, I think?

I’m not thinking about it :)

:(

@AlessandroCodenotti may have some thoughts…

I know that ted is allergic to measure theory

This is more point-set.

actually I'm allergic to both of them

Who knows? Analysis and Algebra already has an offspring (Functional Analysis), so do Algebra and Topology (Algebraic Topology, Topological group/ring theory), so lemme have an offspring from Topology and Analysis.

I'm allergic to discrete math, especially graph theory, btw.

i prefer indiscreete mathematics

but I prefer discrete set

seems i like extra es as well

@DannyuNDos In reasonable spaces, let's say Polish to be safe but separable metric is probably enough, this should be equivalent to being a Baire class 1 function

I knew Alessandro would know :)

Baire class $\xi$ functions are a natural notion in between "preimage of open is open" and "preimage of borel is borel", they are defined by "preimage of open is $\mathbf{\Sigma}^0_{\xi+1}$", so in particular Baire class 1 is defined by "preimage of open is $F_\sigma$"

In reasonable spaces open sets are $F_\sigma$, so your notion implies being Baire class 1, but maybe I was too quick in claiming an equivalence

@TedShifrin By coincidence I've been working a lot with Baire class 1 functions lately, so it immediately looked at least somewhat related to me!

why do we care Baire class 1 function?

In reasonable spaces being Baire class 1 is the same as being a pointwise limit of continuous functions, and this comes up in some places

I care about them because of dynamical systems reasons. Specifically there is a property of dynamical systems (tameness) that, in the metric case, is characterized by saying that every element of the Ellis semigroup of the system is Baire class 1

why 1?

What do you mean? Why 1 rather than 2 or $\omega^3$?

I mean why Baire class "1"? I questioned the meaning of 1 here.

Because they are the first class of functions in an infinite hierarchy. Baire class 1 functions are defined by "preimage of open sets is $\mathbf{\Sigma}^0_2$" and Baire class $\xi$ functions in general are defined by "preimage of open is $\mathbf{\Sigma}^0_{\xi+1}$"

I guess one could start at "preimage of open is $\mathbf{\Sigma}^0_1$", but those already have a name: continuous functions!

Thx a lot!

@AlessandroCodenotti what is the definition of tameness in dynamical system?

There are many equivalent definitions, the shortest to state and easiest to work with for metric systems is the one I gave above (in my opinion), namely that all elements of the Ellis semigroup are Baire class 1 (for non metric systems one has to talk about fragmentedness, which is a weird notion equivalent to Baire class 1 in the metric case but not in general)

oh

It's different from tameness I heard before. Thank you for the responds.

What's the one you've heard about? There are a billion equivalent definitions

I heard that word in some seminar talking about hyperbolic 3-manifolds.

That's probably a different notion then. "tame" is somewhat of a overused name

Even there, there're several notions of tame

I gotta mention that "separable" was such an unfortunate choice of terminology.

I want to calculate $\ \iint_D f(x,y) $ where $0\leq x\leq y$ and $x^2+y^2\leq1$ . Is this first quadrant disk. I

Can i say x is between 0 and 1? which is the biggest value of y ?

no thats wrong i wan the part of the first qudrant disk that is above the line y=x

so the angle is pi/4 to pi/2 r form 0 to 1 and polar coordinates

@TedShifrin see? Trolling can be fun

Originally we used Baire functions instead of the Borel hierarchy

You can probably find out more in a book about descriptive set theory

Though they might not include Baire functions

This is Moschovakis

Hello Everyone...

@Jakobian and this becomes problematic when we have $|x-a|<\delta$, because then, given $\epsilon=\frac12$, there may be a $\delta>0$ such that $|x-0|=|x|<\delta$ holds, but $|f(x)-0|=|f(x)|<\epsilon$ may not hold for those $x$ (in this case, for $x=0$; we'll always have $|f(x)|=1>\epsilon$). So the function you specified above would not have a limit using $|x-a|<\delta$.

I'm still baffled by the fact that apparently some books use $|x-a|<\delta$ in the limit definition of a function.

Engineering math classes feel like a total breeze compared to courses I take as a mathematics major (some classes overlap).

with the exception of materials science classes, I don’t like those

@sunny As has been pointed out before, some authors are lazy or sloppy, or may be making other implicit assumptions which are described earlier in the text. Books are not gospel, are written by humans, and can contain errors. I would stop obsessing over this.

You know what the correct definition is.

@XanderHenderson or just never thought about this

@Jakobian I would categorize this under "sloppy".

@冥王Hades nothing new. Those classes aren't for mathematicians

probably not all of them are a breeze though, I'd rather take a theoretical class than an applied one with full of calculations and implementations

even if it's conceptually much easier

@sunny I think the definition from the book is fine

you can always talk about limits after restricting the domain of functions, in the case of the function you are talking about, it really only has a one sided limit

i dont see why it should have a limit as x -> 0 in the sense of it being defined in a whole open interval containing 0

like, you can make it have a limit using the books definition by defining the function only in (-infinity,0)

then 0 is a limit point, where the function isnt defined, and the limit makes sense

if you decide to make the function something defined on (-infinity,0] then yes, it no longer has a limit at 0

that being said, this isnt something ive thought about for a long time, and im more familiar with the deleted neighbourhood definition

if you have to bend 180 degrees to make one definition right, then it's not really fine

i mean, i think the definition invites more consideration of the manner in which a limit is taken tbh

so i view it as acceptable, like everything can be made to work with that definition, and it may make it more convenient to define things like one sided limits

that causes unnecessary confusion

it could in a first class in analysis yeah

by fine I meant legitimate

there is no logical pitfall to it

also, it makes some proofs more streamlined, like limits of compositions of functions - as the answerer points out in the post linked

If you're going to take the domain $\mathbb{R}\setminus \{a\}$ every time you want to take $\lim_{x\to a}$ then you might as well use the punctured neighbourhood definition

not only is it unnecessary, it's also weird

i think its a little more nuanced than that

if it makes even one more proof easier i would say there is something to it, its also been published in more than one book

that being said, i reiterate i dont mind either definition and im less familiar with this one

It's less intuitive

@porridgemathematics as already mentioned that could amount to laziness or sloppiness

a very common thread among mathematicians

so being published in more than one book doesn't mean it's a very conscious decision

"If all of your friends jumped off a cliff, would you jump, too?"

in fact I think the people that did choose it fully aware of it, might have been influenced by the sloppiness of others

fair point

Uhh don't push!

it is a lot less intuitive

@porridgemathematics No, it's wrong. :/

you become who you surround yourself with, so yes if my friends were consistently jumping off cliffs over time there would be a higher probability that I would in fact jump off a cliff as well

@XanderHenderson I mean there's nothing wrong with taking a dive if it's safe enough, sure

I'd jump off

@Jakobian Yes, but the whole point of that motherly expression is that it is not safe. :P

Just because a bunch of people are doing something stupid, doesn't mean you should do the same stupid thing.

@geocalc33 I'm not entirely sure of that

I would not jump off the cliff, but I would be influenced by environmental and social forces that would increase the probability

imo it is not wrong, i could even argue why its more natural. take a singleton space, and try to evaluate the limit of a function defined on the singleton space, to me the answer is obviously that there is a limit, and its the functions value at the point, with the deleted neighbourhood thing, you would get that it could be any limit

this is completely a matter of taste

its whether you believe limits should be meaningful in singleton spaces

that doesn't show it's more natural

it just shows it translates better in some pathological cases

im playing devils advocate, hence the I could argue, i didnt say this is a convincing argument for replacing the usual definition with this one

im just arguing why its a bit strong to call it, wrong

I think it's okay to call it wrong

I wouldn't say that myself, but it's definitely a definition you shouldn't use

ok, thats just semantics then, i internalised wrong as irrevocably, i agree its for all intents and purposes, or pedagogically, wrong

but still less black and white than that persons definition of even

The definition is there for subintervals of $\mathbb{R}$ or their unions, and we use it in real analysis. And it works just fine for them

indeed

in fact as argued before, it works more intuitively, etc. etc. I don't want to repeat myself

@porridgemathematics No, it really isn't.

The limit is defined by taking points in punctured neighborhoods.

$S=\Bbb Z - (C \times \lbrace 0 \rbrace)$ can one help me with this notation? Where $C$ happens to be that cantor set.

The thing you get without puncturing the neighborhood is a definition of continuity.

@geocalc33 Is it written like this?

@geocalc33 What are you trying to express?

yeah, but then you need to explain that the reason for that is behaviour everywhere other than at the point you're taking the limit

which you ought to in a class, but it still requires explanation

@porridgemathematics I don't understand what you are trying to say here.

I think they're trying to say the definition should be explained in class, but I see it as kind of irrelevant

that both definitions would work for all of mathematics, as long as people are making the necessary adjustments in either case, like accepting deleted neighbourhoods as part of the definition, or explaining that while the point at which you take the limit matters with the other definition, it really doesn't in the grand scheme of things

@porridgemathematics This doesn't make sense.

alright, im not explaining myself very well then, agree to disagree

"For every $\varepsilon > 0$, there exists some $\delta > 0$ such that $0 < |x-a| < \delta$ implies that $|f(x) - L| < \varepsilon$" is the definition of a limit.

i know.

$S = \mathbb R^2 - (C \times \{0\})$ @Jakobian @XanderHenderson (sorry, i made a clerical error).

"For every $\varepsilon > 0$, there exists some $\delta > 0$ such that $|x-a| < \delta$ implies that $|f(x) - L| < \varepsilon$" is the definition of a continuity..

Those are two different things.

no continuity involves f(a)?

@geocalc33 that looks like the difference of sets

it involves that the function is defined at the point

@porridgemathematics Exactly.

the L thing doesnt appear in the definition of continuity

is that what you mean?

i gather you are saying the author just garbled the definition of continuity with the definition of limit, which is possible

If $|f(x) - L| < \varepsilon$ for all $|x-a| < \delta$, this implies that it also holds for $f(a)$.

the definition of continuity concerns |f(x) - f(a)|, not |f(x) - L|

And, yes, if you make the additional assumption that $f(a)$ is not defined, fine. But then say so.

@porridgemathematics It might be good to mention why we use $0 < |x-a| < \delta$ instead of $|x-a| < \delta$, but maybe not strictly necessary

@porridgemathematics If the inequalities hold throughout, then $f(a) = L$.

@Jakobian yes but the $(C \times \lbrace 0 \rbrace)$ is not entirely clear to me. Is this just the Cantor set cartesianed with the set containing $0$? So it's just the cantor set subtracted from $\Bbb R^2$

@Jakobian sure

@XanderHenderson im unsure what you are saying now

@geocalc33 yeah, the copy of the Cantor set on the x-axis

the only difference in the definitions is deleted or not deleted neighbourhoods, and continuity wasnt being mentioned as far as I can see

if the immediate answer is one is wrong then thats fine , but usually definitions have motivations, and it just so happens that both can be motivated in this case, but one has a much stronger motivation because it captures what limits ought to measure (the deleted neighbourhood one) better than the other

@porridgemathematics If $f(a)$ is defined, and for all $\varepsilon > 0$, there exists some $\delta$ such that $|x-a|<\delta$ implies that $|f(x) - L| < \varepsilon$, then it is necessarily true that $f(a) = L$.

ah, yes I already conceded this earlier

i said you need to work in a domain without a for this definition to capture what the usual one captures

I mentioned working in (-infinity,0) earlier for that reason

@porridgemathematics Yes.

Which is the point of taking $0 < |x-a| < \delta$.

If you don't puncture the domain, you are defining continuity.

@XanderHenderson yes im aware

not literally but I get your point

such a heated debate

yeah sorry i wasnt trying to ruffle any feathers

i was just being a contrarian

no, I'm just surprised we argued about it this much

@porridgemathematics Sure, but when there are students in the room that are actively being confused by the sloppiness of what is written in some books, it is not helpful to be contrarian like this.

ah okay granted, i did forget the guy who brought this up was likely a student

@sunny sorry - to be clear, what you linked to is indeed NOT the definition

Let $g_t(r)$ be a 1-parameter family of 1d Riemannian metrics for quantized mathematical time $t \in \Bbb N.$ What is the mathematical significance/and or interpretation of the following expression: $\sum_{t \in \Bbb N} g_t(r)?$

pierogi with groat are a good combination

I suppose the natural thing to do would be to determine whether or not that sum is a metric or not.

and clearly it is

whether it preserves anything useful in terms of metric geometry is another question.

I would collect a certain class of riemannian metrics on a smooth manifold and study how big it is

it for sure satisfies completeness

@onepotatotwopotato could you elaborate on that please? thank you

@Jakobian "Groat"? Like kasha? or...?

which class would you collect. how would you study how big it is?

@geocalc33 for example collect all possible metrics with certain curvature you can assign on your smooth manifold. And find some bijective correspondence with some known space

@XanderHenderson kasza, yeah

I think those are my favourite kind of pierogi

and see how the deformation of metrics corresponds to the bijective space.

I remember I had some pierogi with groat and some meat once, rabbit maybe? It was pretty good

for hyperbolic metric, it's called teichmuller space

No sorry, it was duck

@Jakobian Ah, the Polish spelling.

arguably it's less confusing than using ch, sh and zh for West Slavic people

because we use ch as h often

I can read Cyrillic too if you prefer

@onepotatotwopotato thanks for the suggestion. Is this trivial for a class of metrics with 0 curvature?

actually don't tell me, I want to figure it out myself

Splitting fields are pretty important if we want algebraic closures and stuff

It reminds me of one of the stupid questions I asked: 'isn't it the existence of the splitting field trivial? one can just add the roots of the given polynomial'

I think that question is in the top 3 of mine.

alright never mind, the existence of algebraic closure only uses Zorn's lemma and stuff

together with size of any algebraic extension being bounded

@Semiclassical Here's a plot of the surface, at least.

sagecell.sagemath.org/…

@PM2Ring nice. one cute instance of this is the y=0 slice:

That looks blank.

it's almost just [-1,1]^2 but the corners are rounded off

i think it has the following geometric characterization. suppose we arbitrarily pick two different pairs of orthogonal vectors $(a_1,a_2)$, and $(b_1,b_2)$, and compute $(a_1\cdot b_1,a_2\cdot b_2)$. then the above is the set of all possible such pairs of dot products

@Jakobian other than electromagnetism and fluid dynamics which I actually found to be challenging (and fun) the rest were not that hard

And even then the mathematics involved in those two classes was easier to grasp compared to the actual concepts.

(i do have reasons for thinking that's true, but as i work that out mentally that seems implausible...hrm)

Thermodynamics for example, which most people find difficult, was easy for me because it’s literally just differential equations, some cool diagrams and some obvious stuff

@冥王Hades the main thing with thermo is that it's where the usual physics way of teaching calculus sorta runs into issues

so that it's hard to make sense of all the partial derivative identities etc

@PM2Ring are those "holes" in the surface a function of the software?

@geocalc33 what's being plotted is the case of equality for the $\leq$ inequality I referenced

in addition, though, one is only interested in points inside [-1,1]^3

so one should really take those portions of the surface and "fill in" the gaps that lie along the faces of the cube

that happens automatically when you plot 3D regions in mathematica, but Sage doesn't have 3D region plots afaik so the best you can do is plot those parts of the surface

yeah, my conjectured interpretation doesn't work. what am i doing wrong...

@冥王Hades Any physics is hard to grasp to me because I don't understand the concepts involved. People like to teach physics intuitively but it's not how I think

in mathematics on the other hand, there's no experiments, everything is from the ground up

there's no modeling

things are clearly defined and unambiguous

if I don't understand some concept in mathematics, it's only because I never read its definition

if I don't understand some concept in physics, well I don't know

except for the frustrating cases where different definitions exist

tho those are usually more historical

(insert my usual reference to the French definition of limit)

@Semiclassical we were talking about this today, and why the French definition is wrong

wrong as in, pedagogically for instance

hah

the benefit apparently is that it makes compositions easier to deal with

@Semiclassical You can also ask it to plot several contours, to get a view of the internal structure. The 2nd image on doc.sagemath.org/html/en/reference/plot3d/sage/plot/plot3d/… has a simple example involving a sphere.

@PM2Ring oh blah, i figured it out. i had a typo in what i reported. should be $|(x-z)y|$ not $|(x-y)z|$

@Semiclassical Ah. That looks more symmetrical. :)

yeah

@Jakobian That seems related to a comment that I got earlier today.

I remember we agonized over that French definition 8 years ago or so. Yes, let's just say all functions are continuous.

So now we can have "Freedom Limits"?

as with most questions of sticky definitions, there's usually not a wrong definition but there are definitions which are easier to apply / harder to misuse

but as long as you stick to them---and recognize that you've made the choice to do so!---there's no risk

For example, in my multivariable math book, I proved only that the composition of continuous functions is continuous, not anything about limits of compositions. But why obscure what's going on by defining limits only for continuous functions? Surely that gets troublesome in more advanced mathematics.

@robjohn Only if they come with Freedom Fries.

it's a question of power vs ease of use to some extent. if real analysis is where your math career tops out, then using the more restricted but simpler definition may be defensible. but the whole point of real analysis is largely that it's not your last math class

No, I disagree with that. Even in calculus, we cannot discuss $\lim_{x\to 0}\frac{\sin x}x$ unless we define the function at $0$ in the first place.

Before the discussion I had today I didn't think much of the French limits, as they're called. After it happened, I got pretty reinforced that they're wrong, and for multiple reasons.

again, i hesitate to call them wrong so much as...unprofitable?

I'll keep on calling them wrong

lol

The repaired function, with a simple radial colour map:

@TedShifrin Really? I would think that we would need to define the function at $0$ if we want it to be continuous (which $\operatorname{sinc}(x)$ is).

sagecell.sagemath.org/…

The inverse of the sinc function should be float.

>gurgle<

my guess would be that, done properly, what you get is not less expressive but that said expressiveness is distributed differently. e.g., the subtleties are carried not by the function's domain but by the limit's domain.

in the standard definition, of course, the notion of a limit itself having a domain isn't taken up

@robjohn That's what I said (I think).

I have enough issues with the US high school and college teachers insisting that $f(x)=1/x$ is not continuous at $0$. Good grief.

for instance, you can still write stuff like lim_{x →0; x ≠ 0} f(x) = 1 for $f(x)=\sin(x)/x$ in the French approach. but ehhhhhhhh

just because you can find a way to make it work, doesn't mean i like it

I think I remember arguing with all the French crowd here (back in the days there was a French crowd) about this ... Hippa, Astyx, et al.

same

the deleted neighbourhood definition is good enough for unions of intervals, and that's all you'll ever need at the level they are introduced

the only exception would be trying to define them on isolated points - but that's not a problem

@TedShifrin Calculus may not be common in high schools in your state in the near future... From scottaaronson.blog/?p=7425

I guess I should say, glorified limits of filters (i.e. $(x-r, x+r)\setminus \{x\}$)

> why did California just vote to approve the “California Math Framework,” which (though thankfully watered down from its original version) will discourage middle schools from offering algebra or any “advanced” math at all, on the argument that offering serious math leads to “inequitable outcomes”? [...] even as Jelani Nelson and other STEM experts testified about what a disaster the CMF would be, especially for the underprivileged and minority students who are its supposed beneficiaries?

This is the same thing we went through with Common Core all over 10 years ago. But, honestly, I think AP calculus has become an abomination. Taught by insufficiently qualified teachers to far too many students, training for the tests.

And the AP board in Princeton years ago decided that a score of 60% on the AP BC exam was sufficient to give a 5 score (which exempts students from at least two semesters of calculus everywhere). Well, one can get that score with basically no understanding of sequences and series material at all.

to me it's all of a kind with how underfunded education tends to be. that's where arguments about where the money should go arise

@PM2Ring They'll find any reason to fight for their imaginary ideals

talking up the ideologies being battled out seems silly if one doesn't foreground with the material constraints which are creating that battleground in the first place

Fighting for what's actually beneficial for the whole is replaced by what's politically correct

It reminds me environmental activists that do next to nothing to improve the world, but throw out their ideals without much thought about the complexity of the issue

people are living in the clouds

@Jakobian I thought we were talking about real analysis here

@PM2Ring here's something else that's fun. suppose you look at that inequality and decide that dealing with square roots is annoying. one way to get rid of that is to replace the variables with cosines ($x\mapsto \cos x$ etc): sagecell.sagemath.org/…

that surface looks a bit more familiar, eh? :)

(albeit it's not the regular version of such)

@TedShifrin That's sad. People who don't really need calculus shouldn't be rote-learning a butchered version of it. They should be learning mathematics that they can use. And not made to feel that the kids who do want calculus are ivory tower elitists.

i'm a bit more resigned to it. as idealistic as i'd like to be about why students take AP calculus, it's b/c they perceive it as a way to get a leg up on university...and the reason they go to university is b/c it's framed as "the way you get a career"

reading this convo I just imagined being forced to teach high school students, what a nightmare

yeah, no thanks

teaching intro physics is already pretty frustrating

having to do that in a context where everything is scrutinized to the point that it loses its meaning? no thanks

@Semiclassical Nice. :) I'm reminded of the superquadric that goes from a cube to an octahedron, via a sphere. Which I posted here a while ago (& recently re-linked).

@Semiclassical I was thinking more about puberty and antisocial behaviour

@PM2Ring tbh this specific phenomenon has always weirded me out

@冥王Hades Those applied math courses are actually what I'm going to study. Is this also the right forum to ask questions about engineering math, although mine are machine learning math.

i kinda sorta get it: if you pick some 3D unit vectors and ask about their dot products, then you're secretly asking questions about their angles

so the lesson is evidently that constraints on those angles are linear, so they'll be nice even as the constraints on dot products aren't

and then when you move to dot products, that naturally ends up preserving the edges but inflating the faces outwards

won't say i know how to actually derive said linear constraints on angles from scratch tho

@s.harp I also hate interacting with immature people

Thankfully this disappears with age for most people. But it can also be long time before that happens, as a negative

I heard for most people, brain matures around the age of 25

Most people on university undergrad level will still be under 25 though

I think people would also benefit greatly from having mandatory philosophy classes in school

While myself I'm more of a free thinker, I think some guidance wouldn't hurt

people should at least read their marx and their lenin before age 20

I know this is a joke but it still hurts

you can make up for gaps in your students' education by including quotes in front of each problem in a problem set

I've put a shakespeare quote into an exercise sheet before, does that count?

yes

"Our schoolteacher should be raised to a standard he has never achieved, and cannot achieve, in bourgeois society. This is a truism and requires no proof. We must strive for this state of affairs by working steadily, methodically and persistently to raise the teacher to a higher cultural level, to train him thoroughly for his really high calling and—mainly, mainly and mainly—to improve his position materially." VI Lenin

quote this the next time you are negotiating a pay increase

I have to supervise school teachers in training for a curves and surfaces course next semester

maybe ill send them this quote

LOL

I came up with a problem, but it's equivalent to an open problem...great.

does open problem mean it has to be well known?

no

anyway, yeah, task failed successfully

reducing to an open problem isn't always a bad thing, mind. if there's already insights about said open problem, then that can translate into insights about the other version

that's true

theguardian.com/world/2023/jul/26/…

what?

sounds like same old stuff, claims about aliens based entirely on "trust me bro"

no, this is very recent

what's surprising is that they claim that under an oath

well: "Representative Alexandria Ocasio-Cortez asked the three witnesses, "If you were me, where would you look?" regarding answers to UAP questions and evidence to validate his claims. Grusch replied, "I'd be happy to give you that in a closed environment. I can tell you specifically."[1]"

so he evidently wasn't willing to say that under oath

@leslietownes Every time I taught differential topology, I included an Oscar Wilde quote.

@TedShifrin i did manage to follow up on what you were suggesting yesterday vis-a-vis finding the shadow by looking at the tangent planes

Oh?

without getting into the weeds, the following curve turns out to make an appearance: $t\to (\cos t,\cos 3t)$

Interesting.

which isn't entirely surprising to me, but it is neat

Glad I could help!

i had to do a lot of mathematica work to get it down to that, but the starting point was indeed just to look at the tangent planes

Are correlation matrices and cross-correlation matrices the same thing?

Also, from Wikipedia, "Cross-covariance is related to the more commonly used cross-correlation of the processes in question...In signal processing, the cross-covariance is often called cross-correlation." What?

I'm trying to mentally organize all the correlation, covariance, convolution, cross/auto, matrix, etc. terminology. I did find the (incomplete) table of links on Wikipedia but I'm getting more confused the more I click through. Do these terms mean anything sensible or is it just madness?

For example, I can't tell if the convolution matrix in, i.e., image processing is constructed the same as the cross-correlation matrix, except using convolution instead of cross-correlation. It looks different but it's hard to match all the different notation from these Wiki links.

> GPT4 cannot answer this.

Duh.

Sad

People without access to quality feedback turn to chat gpt and dig themselves into a hole by believing it

rather than helping the most vulnerable it feels like chat gpt just helps them lose more

the only thing chatGPT has over a parrot is the internet and a better retention

but at its core it's just a stochastic parrot

@Semiclassical bro woke up and choose to speak the language of facts