Mathematics - 2024-02-23 (page 1 of 2)

Obliv

00:21

does $-\Delta x + \ln(1+\Delta x)$ imply the total is less than 0

i.e. is $\Delta x > \ln(1+\Delta x)$ for all $\Delta x > 0$?

Jakobian

@Obliv yes

equality holds only for $\Delta x = 0$

Ted Shifrin

Hint: First degree Taylor + remainder

Jakobian

00:37

or use that $\ln$ is strictly concave

Ted Shifrin

or Hint: Beginning of $1/(1+x) = $ …

Jakobian

01:06

-2

Q: A property of morphisms about Euclidean spaces

In 2019 I gave definition of "space" that encompasses all kinds of spaces in general topology (metric spaces and locales/frames and everything in between: topological spaces, uniform spaces, etc.) (This paragraph is just to give a background, this is not used in the question.) The definition of a...

Hmmm

its an interesting question. A bit unclear on what author means but looks creative

does anyone have an idea what is he talking about?

Jakobian

01:59

thanks to me he corrected his post like 5 times already

seems like it wasn't just me being confused

Thorgott

02:17

the defn does not make sense to me

Jakobian

yea

Daniel Donnelly

07:34

@Shaun hey mn

Tears For Fears - Everybody Wants To Rule The World (Official Archive Video)

►

BAYMAX

08:19

Hi chat!

I am trying to solve the simultaneous equations for $x,y,z,u$. The simultaneous equations are \begin{aligned*} \begin{equation*} \frac{a-x}{b} - z - u &= 0,\\ x - min(\beta u, c y) &=0,\\ x -y &= 0,\\ x-1 &= 0\\ \end{equation*}\end{aligned*}

$ \frac{a-x}{b} - z - u = 0,\\ x - min(\beta u, c y) =0,\\ x -y = 0,\\ x-1 = 0$

From last two equations, we have $x = 1, y =1$, then what is $u,z$ ?

For the second equation we will have $1 = min(\beta u, c.1)$

$a,b,\beta,c$ are constants

Sahaj

08:40

hi guys

Daniel Donnelly

09:14

Hiiigh

🌿🍀🍁

Kripke Platek

@EE18 I don't really know about Halmos, but if you want to study set theory formally from the beginning, you can check out Enderton's textbook "Elements of Set Theory". I have personally read it and so far I had no problems with it.

Shaun

10:24

@DanielDonnelly Hi :)

Daniel Donnelly

Hey mn

:D

Not so sure about my point now. Working on TPC at elementary level is going in circles!

Shaun

How are you? I saw your email this morning.

Daniel Donnelly

Doing good

Hw are you?

Made $1500 last month, but this mnth no work

Shaun

I could be better. I had to cancel plans for the day as I'm struggling with avolition :/

@DanielDonnelly That's cool :)

Daniel Donnelly

Oh not wanting to do stuff?

Same here

Maybe do new things for novelty feelings

Shaun

10:27

Yeah, but more clinical; it's a symptom of schizophrenia.

Daniel Donnelly

I have that too

They had me with BP, but then changed it

I'm not sure what I am

Shaun

I want to do things. I just can’t bring myself to do them.

I'm sorry to hear that :(

Daniel Donnelly

Try serotonin - how much do you exercise?

A swift jog / walk will boost your sertonin then you'll want to do stuff even quit smoking e.g.

Shaun

I don't exercise at all. I used to hit the gym twice a week before the pandemic though l

Daniel Donnelly

There you go that's always the reason, fr me too !

Shaun

10:29

My OT and I are working towards getting back into that habit.

I always feel bad cancelling plans over avolition. It feels like I'm just being lazy.

Daniel Donnelly

Don't worry about it too much. I'm probably worse than you

Shaun

I was in great shape, too. I had a skinny guy six pack going on.

Daniel Donnelly

What I mean is, everybody can't fit this normal mold perfectly

Shaun

Indeed :)

Daniel Donnelly

Oh you studied $\textbf{Ab}$s aka abelian grups?

lol

Shaun

10:33

Lol :)

Daniel Donnelly

Let's study together bro

we could make it a rule to do pushups or something for 5 minutes every 30 minutes

Shaun

I'd like that!

Daniel Donnelly

You know I ran out of paper I write so much

Shaun

I can hardly do one pushup lately though

Daniel Donnelly

here I am scribbling notes in an EMFT shaums outline book

I have carpel tunnel from both keyboard and majorly writing now

have to do yoga or something

@Shaun if you are interested in elementary number theory stuff like sums over divisors, I could teach that

It's strange, each integer is like a simplex

nvm ╰（‵□′）╯╰（‵□′）╯

Shaun

10:37

Why? What's up?

Daniel Donnelly

Oh, if you take $q_1^{k_1} \lt q_2^{k_2} \lt \dots q_n^{k_n}$ where $k_i \geq 1$ and our input number $X = \prod_{i = 1}^n q_i^{k_i}$, then the above inequalities are strict and give the "natural ordering" (of the simplex) such that the following map is a chain complex on certain $\Bbb{Z}$-modules of functions. Anyway, I could show you that some time. I don't know how to apply it though to ANYTHING!!!

Shaun

I could use books as dumbbells:)

Daniel Donnelly

$q_i$ primes

You need a 40lb kettlebell. I got one

So you're studying matrix applications in group theory?

That goes deep

So I hear, I've onlly seen glimpses from Lang

Shaun

Yes :) Mostly $SL(n,q)$.

Daniel Donnelly

Special linear group of $n\times n$ mats over $\Bbb{Z}/q$ the field?

Shaun

10:43

The project started out as studying linear algebraic groups. And yes :)

Daniel Donnelly

I know absolutely nothing about that :D

Shaun

But with det 1.

Daniel Donnelly

But you get all sorts of interesting theorems when you do that

matrixify things

namely rings lol

Shaun

Every finite group is a linear algebraic group.

Daniel Donnelly

what's that?

Shaun

10:45

But SL(n,q) and other groups are more direct in how that's so.

Daniel Donnelly

They nearly solved the graph isomorphism problem using permutation groups

What's weird is first they encode the graphs as strings and then solve the string isomorphism problem using groups

Shaun

A l.a.g. is a group whose elements are solutions to polynomials, essentially.

Daniel Donnelly

A string is just "abababaccacaba"

Shaun

There's a bunch of ways do define them.

They're app complicated.

*all

Daniel Donnelly

So the elements together solve a single polynomial?

Shaun

10:47

Yes :)

Daniel Donnelly

So strange!

Shaun

The Zariski topology plays a role.

Daniel Donnelly

I know that

Shaun

Topology isn't my forte :(

Daniel Donnelly

Closed sets are algebraic zeros

to familys of polys

Shaun

10:49

I could use an HPC soon.

Daniel Donnelly

O__O only made it through the first chapter of AG lol

What's HPC?

Shaun

I did calculations on my laptop and they took ages, so that's why.

Daniel Donnelly

Oh, IC

You need to learn C++ or D

or C#

Shaun

"High Performance Computer", almost a supercomputer.

Daniel Donnelly

something faster than Python / Sage

But I believe GAP is already implemented in C, so that's good to go

Shaun

10:51

Yeah, I use GAP.

Daniel Donnelly

I got a Getac rugged laptop with tiny touch screen and convertibility into tablet for $400

Shaun

The amount of issues we've had getting the HPC to work in stupid.

Daniel Donnelly

At school?

Shaun

Anyway, I'm computing a function Delta of SL(n,q).

Yeah.

They have two here.

Daniel Donnelly

Is it linux env or something?

Shaun

10:52

I'm using the less powerful one for now.

I don't know.

Daniel Donnelly

Write her in assembler using SSE instructions

Shaun

I'll leave it to the IT team!

Daniel Donnelly

Maybe the HPC has virii

Someone made a supercomputer virus at school lol

Shaun

Maybe!

Daniel Donnelly

But it was just a randomly buggy group theory program somene wrote that spawned the viral code into the machine

Shaun

10:55

They're pretty strict on security for it though.

The HPC version of GAP is in its alpha stage still. So maybe!

Daniel Donnelly

Self replicating & mutating programs wuld be cool, but it's very non-trivial

Shaun

I might just use GAP instead, the ordinary version, but that's got the same issues on HPC so far anyway.

Daniel Donnelly

Buy online supercomputer HPC time

perhaps

then you can scale with what you require

Shaun

I didn't know you could do that!

Daniel Donnelly

I don't know where, you'd have to look into it

Shaun

10:58

But with two at my disposal already, I should be fine!

Daniel Donnelly

But just general purpose "cloud" things such as Amazon AWS have pay per usage and you can upgrade hardware for dough

Shaun

I'm going to get something to eat now. It's nearly 11:00 and I haven't had breakfast.

Daniel Donnelly

I've gt it

Shaun

See you later :)

Daniel Donnelly

build your own photonic computer (analog) that just solves the one Delta Function cmputation

psie

11:33

I have some basic questions regarding a sentence in my measure notes;

> For example, if $N\subset \mathbb R$ is not a Borel measurable subset of $\mathbb R$, then $\{0\}\times N$ is not a Borel-measurable subset of $\mathbb R^2$ even though it is contained in the subset $\{0\}\times \mathbb R$ of zero Lebesgue measure on $\mathbb R^2$.

Why is $\{0\}\times N$ not a Borel-measurable subset of $\mathbb R^2$? Why does $\{0\}\times \mathbb R$ have zero Lebesgue measure on $\mathbb R^2$?

Xander Henderson

Read the FIRST word of the example.

"If".

The product of a non Borel measurable set and a Borel measurable set is not Borel measurable.

psie

ok, thanks, that clarifies the first part

Xander Henderson

But the product of a Lebesgue measurable set and a Lebesgue null set has Lebesgue measure zero---in this case, the set is contained in a line segment, which had 2d Lebesgue measure zero.

psie

11:49

ok, I need to look this statement up further, i.e. that a product of a Lebesgue measurable set and a Lebesgue null set has Lebesgue measure zero

Jakobian

This follows from lower dimensional linear subspaces having Lebesgue measure zero

Xander Henderson

How do you define the two dimensional Lebesgue measure?

As I recall, the usual way to do this is to define a premeasure on $\mathbb{R}^2$ by assigning the measure $\lambda^2(A\times B) = \lambda(A)\lambda(B)$ for any measurable subsets $A,B \subseteq \mathbb{R}$.

Then construct the complete measure in the "usual" manner.

psie

@XanderHenderson in my notes, a subset $N\subset \mathbb R^d$ has Lebesgue measure zero if and only if for every $\epsilon>0$ there is a countable collection $\{I_n : n\in\mathbb N\}$ of, not necessarily disjoint, $d$-dimensional rectangles such that $$N\subset \bigcup_{n=1}^\infty I_n, \quad \sum_{n=1}^\infty m(I_n)<\epsilon.$$

Jakobian

But here $N$ is not necessarily Lebesgue measurable

psie

I'm using these notes, example 6.

Above, $m$ is the Lebesgue measure, sorry.

Xander Henderson

11:57

@psie Each natural number $n$ is contained in the rectangle $[n-1/2, n+1/2] \times [-\varepsilon/2^{n+1}, \varepsilon/2^{n+1}]$.

(That is, each point $(n,0)$ in $\mathbb{N}\times\{0\}$).

Jakobian

Why $N\times \{0\}$ is not Borel measurable in $\mathbb{R}^2$, is that otherwise it would be Borel measurable in $\mathbb{R}\times \{0\}$. So $N$ would be Borel in $\mathbb{R}$.

Xander Henderson

12:28

Oh, I just realized that I misread the question $N$, not $\mathbb{N}$. Doink. Still, enumerate the integers in some manner (e.g. $(0,1,-1,2,-2,3,-3,\dotsc)$). Then cover $N\times\{0\}$ by $[z_n-1/2,z_n+1/2]\times [-\varepsilon/2^{n+1},\varepsilon/2^{n+1}]$, where $z_n$ is the $n$-th integer in your enumeration.

Sha Vuklia

12:44

anyone here familiar with the term backward error from numerical analysis?

psie

13:04

thanks for the help 👍

Xander Henderson

@ShaVuklia I am familiar with it, but I haven't thought about it since my masters degree (so it's been... uh... more than a decade).

Sha Vuklia

let me give it a shot then :)

above is a definition of backward error

Xander Henderson

Sha Vuklia

I'm trying to understand why the backward error is useful (and easier to compute)

I found one explanation in another book (which works with a slight modifcation of the definition)

Xander Henderson

My recollection is that the backward error is a measure of how much you can perturb the initial data before you aren't looking at the right solution any more. This is practical, since we actually have some control over how precise our measuring instruments are.

Sha Vuklia

13:13

they say that the forward error is hard to compute, because we don't know the true value of $f(x)$. however, for the backward error we still need to solve $\hat y=f(x+\Delta x)$ using the true function $f$ (instead of the algorithm), so wouldn't that be equally hard if not harder?

Xander Henderson

What is Figure 1.1? Have you looked at that?

Sha Vuklia

@XanderHenderson that is sensible, but the definition already works with the measured $\hat y$, so it's not as if we know beforehand whether $\hat y$ is close to $f(x)$

figure 1.1 explains what's going on formally

Xander Henderson

Yeah, I figured it was going to be that picture.

Sha Vuklia

but to me it doesn't clarify why the backward error is useful (or easier to compute)

Xander Henderson

You compute $\hat{y}$, right?

That is the thing your algorithm gives you?

Sha Vuklia

13:16

yes, that's our numerical result

one interpretation I can get behind:

if the backward error is big, then we would have to deviate a lot from $x$ to obtain the numerical solution exactly

and I suppose if $f$ is continuous, and our backward error is small, then we know that the actual solution is close to the real value?

Xander Henderson

No, I don't think that is the right way of thinking about it.

Sha Vuklia

I am glad I mentioned it then, because I would like to know the right way :)

Xander Henderson

It is more about how much you can perturb the input, and still get the same output, which may or may not be close to the "true" solution.

Like you compute $\hat{y}$ using $x$. But $\hat{y}$ isn't actually $f(x)$. It is actually $f(x + \Delta x)$ for some error term $\Delta x$. The backward error is the smallest size of that error term $\Delta x$.

Sha Vuklia

I assume small $\Delta x$ is good?

which would then indicate that we can apply some perturbation? (I don't see that implication)

Xander Henderson

Yeah, in general, you want your errors to be small.

No, it isn't about "applying" a perturbation.

It is about understanding how much the data can be perturbed until things go wonky.

If $f$ is sufficiently nice (e.g. $C^1$), then small perturbations in the initial data should lead to small perturbations in the output. The backward error is, in some sense, giving a bound on this (smaller is better).

My very vague recollection is that the forward error can be estimated in terms of the backward error and the condition number...?

Sha Vuklia

13:25

yea, that works by definition

Xander Henderson

But, like I said, it's been more than a decade.

Sha Vuklia

I'm afraid still don't see the implication: if $\Delta x$ such that $f(x+\Delta x)=\hat y$ is small, then "the data can be perturbed without things going wonky"

it seems like there is a tacit assumption there that $\hat y$ is already pretty good

but if we knew that, we would be done (since backwards error is presented as a more computation friendly thing to check than forwards error)

so we don't know if $\hat y$ is good, and therefore I don't see how $\Delta x$ being small tells us anything

(if we already knew that $\hat y$ is good, then we would know that a small perturbation still yields a good result, so then it makes sense)

but maybe I shouldn't see them as conditions with the same end in mind

Xander Henderson

Look at it this way: you have a ruler which measures in cm. You obtain some measurement $x$ cm, and get a result $\hat{y}$. You know that $\hat{y}$ isn't the actual value of $f(x \text{ mm})$, but you know that the backward error is 0.5 mm. So this means that the $\hat{y} = f(x+\Delta x)$, where $\Delta x$ is smaller than 0.5 mm. Your ruler measures in cm, so you are probably close to the actual value.

And, again, my vague recollection is that you can bound the error in $\hat{y}$ using the condition number and the backwards error.

Sha Vuklia

@XanderHenderson hm, to generalise this: if we know that $f$ is not sensitive to small errors, then a small $\Delta x$ indicates that we found a good approximation

I think the first point mentioned here is what you just illustrated

we generally have "noise"

so if our backward error has the same magnitude as the noise

then we shouldn't expect our algorithm to distinguish $f(x)$ and $f(x+\Delta x)$ anyways

copper.hat

13:48

I wonder sometimes where folks get their energy to track 4, year old questions to give little lectures math.stackexchange.com/a/3690071/27978

Sha Vuklia

The first point in the last screenshot I shared is now clear to me (and I can get behind it), and I suppose the second point about perturbation is what you were talking about Xander. I'll have a look in the book where they elaborate the point on perturbation further

Jakobian

@copper.hat not sure what Anne is trying to do here

copper.hat

,@Jakobian I have no idea, seems a bit pointless:-)

Xander Henderson

@copper.hat Yeah, that's not really okay.

Sha Vuklia

gotta admit, she got our attention

@XanderHenderson this seems to be the perturbation argument (in the book from the last screenshot)

Xander Henderson

14:03

The term on the left looks like the relative error in the output, being bound by the product of the condition number and the backward error.

Sha Vuklia

yes

Xander Henderson

Yeah, the second quote.

Isn't that exactly what I said before? :P

Sha Vuklia

the condition number is some cases (like evaluating $f$ for $f$ $C^2$) is easy to compute

@XanderHenderson yes, but I wasn't aware that the condition number was sometimes easy to compute

to me the condition number was just by definition "relative forward error"/"relative backward error"

@XanderHenderson hence my reaction to this with "yea, that works by definition"

I'm starting to see now why it is useful

Xander Henderson

It is a theoretical tool, which lets you prove bounds. Super useful.

Xander Henderson

14:18

In other news, the finance people at the college can't do basic arithmetic, and it scares me.

In a recent meeting, they said something to the effect of "Faculty have gotten 5% pay raises in each of the last five years, for a total of a 25% pay raise" (note: these numbers are completely made up.

$1.05^5 \approx 1.28$, so that is a 28% pay raise (the magnitude of the actual mistake made by these folk is closer to a 5% error).

I'm totally okay with them making this mistake, as it benefits me, but the same error has lead them to under report CPI over the same period, and that has compounded much more to the detriment of faculty.

Sha Vuklia

that's wild

Xander Henderson

Of course, the administration is also on a kick to pay everyone at the college a minimum wage of \$20/hr (which is going to cost us about half a million dollars, out of an annual budget of about \$40 million, in a year when we are looking at a $2-3 million shortfall due to the legislature being a bunch of idiots). And they want to make summer classes tuition-free.

I am fairly well convinced that our current president is looking to go somewhere else, and is trying to do things which look good on a resume ("Hey! I increased the minimum wage at the college to \$20/hr! I gave students free classes! Aren't I the best?!"), while making no real concessions to the long-term viability of the college.

It is very frustrating.

Jakobian

14:35

@XanderHenderson fail them

Xander Henderson

@Jakobian They aren't in any of my classes. :(

Jakobian

Oh its not the students?

Xander Henderson

@Jakobian No. It is the finance people at the college. Like, the Vice President of Finance.

I guess her actual title is "Vice President of Administrative Services / CFO".

Jakobian

She's not made for the job

冥王 Hades

15:11

Well, one of our professors rick-rolled the entire class, including me.

Xander Henderson

@冥王Hades That doesn't seem comparable. Your professor seems like a class act.

John Zimmerman