Mathematics - 2023-04-26 (page 1 of 2)

AMDG

00:08

Ted Shifrin

Not to mention both USCs!

AMDG

There's a new one now?

vine boom sound effect

AMDG

00:57

General sum to product formuler not involving circular functions wen

I seem to have narrowed analysis of $e^{-\lfloor\ln b\rfloor} b$ down to finding the relationship of the fractional part of this expression to the fractional part of the logarithm.

What we get is necessarily some $1 + x = e^{-\lfloor\ln b\rfloor} b$ which is interesting considering that the infinite sums available for the logarithm are explicitly of this form.

And it also meets the domain requirements iirc

AMDG

01:11

I wonder... we have $\ln (1 + x) = \ln b - \lfloor\ln b\rfloor$. If we compute the terms $R^n$ as given above for $1 + x$ but with $R = e$, here, $n \leq 0$ for all $n$ which suggests a direct relationship to the fractional part of the RHS as $x = e^{-\lfloor\ln b\rfloor}b - \lfloor e^{-\lfloor\ln b\rfloor}b\rfloor = e^{-\lfloor\ln b\rfloor}b - 1$ to the logarithm itself. There's probably something worth investigating about the logarithm $\ln(1 + x)$, its infinite sum, and its $R^n$ sum.

Wolgwang

What does it mean for a function to be symmetrical about a point (say $a$ ), $f(a-k)=f(a+k)$ or $f(a-k)=-f(a+k)$ where $k \in \mathbb R$ ?

Ted Shifrin

Who knows? With no context, we might assume even symmetry, but it’s way vague.

AMDG

01:32

Has anyone tried computing the limit for certain presumably "nice" respective parts of the logarithm if we take the infinite sum for $\ln(1 + a)$ and modify it to be a sum $\ln(a + b)$ and found anything interesting via the binomial theorem applied to its terms?

Ted Shifrin

No clue what you’re talking about.

AMDG

Well you have for each term of this function in such case as $\sum_{k=1}^{\infty} \frac{(-1)^{k-1} (a + b - 1)^k}{k}$, and then applying the binomial theorem to each term, we can expand the term for each $k$. I suppose it's equivalent to asking whether or not anything interesting or useful comes out of analyzing $\sum_{k=1} (a + b)^k$, or perhaps the multinomial theorem for our trinomial $a + b - 1$.

In terms of the ways in which it might be simplified or otherwise represented equivalently through the binomial and multinomial theorems when the expressions are fully expanded.

I mean idk... maybe there's a representation here that can be found which we can split into multiple sums with at least one of them having a known closed form behind it that would permit splitting the logarithm into parts of some kind.

The fact that these kinds of hyperubermega transcendental functions that sound like final bosses doesn't make them any less scary you know...

*The Lerch Transcendent: Master of the 8th Dimension Between Worlds*
Level: don't bother
Health: 😂
Stamina: NaN

Wolgwang

02:00

One gets the answer by odd symmetry. I had tried even symmetry before.

D Left Adjoint to U

02:20

@AMDG How is this? I switched from Card group to Flexbox container

uhjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjj

My cat typed that

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

AMDG

the best kind of neural net

@DLeftAdjointtoU Looks great!

D Left Adjoint to U

@AMDG I took your comment seriously and fixed all the extra space

I would though make the diagrams bigger

AMDG

Good on ya.

D Left Adjoint to U

That uses @varkor 's Quiver, lol. I hacked it

I'm liking web dev. It's just incredibly difficult

AMDG

Yeah about making them bigger, maybe you might try adding an "expand diagram" button that can enlarge the diagram in-place that makes the card enlarge but makes the other cards force wrap. Make sure it has a quick transition at minimum, though. Instant changes are jarring to the user in general.

Or you might make it bring up a closeable diagram viewer overlay.

Or even simpler: open it in a new page with the diagram enlarged

Which choice of implementation however depends on the objective purpose of this UI and therefore your website.

If it's more of a gallery sort of thing, then opening in a new page wouldn't be appropriate for example.

If it could be for any purpose, then you account for and provide the means for the user to implement whichever choice of possibility they desire since the objective purpose of all UIs is to be a means for a user to bring about some arbitrary end, so everything has to be oriented around that.

And then the rest is just, as a matter of GUI principles as just the same objective purpose as UIs but with graphics, here the practicals to apply consist of 1) directing the focus or awareness of the user to the elements he's looking for, and with regard to what he's looking for, 2) minimize the effort required by the user to find what he's looking for.

D Left Adjoint to U

02:40

I like your idea of the popup with the background area around the popup dimmed so that hte diagram is highlighted (and larger)

Right now I've got this going on:

AMDG

Still looks great

D Left Adjoint to U

Remember, it's a flexbox, so I made it wrap, but I'll try to make at least 3 fit on the first row on widescreen

So it wraps when the screen shrinks down to tablet or phone because I'm using bootstrap everything

@Thanks !

@AMDG

AMDG

RIP Thanks' inbox

Well here principle 2 is applicable. It all reduces to finding information. You want as much information on screen as possible without overwhelming the user.

D Left Adjoint to U

Pagination is going to be a b*ch

That's next on that page, right now they're just static buttons

AMDG

What I was suggesting about force wrap only applies to the idea I had about expanding the diagram in-place, though softwrap might work.

Flexbox is pretty nice for customizing layout via CSS I must say, and probably quite a good choice in terms of compatibility.

D Left Adjoint to U

02:44

It's weird, when you work on something you can't really see it, but others can

AMDG

:)

"Every way of a man seemeth right to himself: but the Lord weigheth the hearts" as it's written. drbo.org/cgi-bin/d?b=drb&bk=22&ch=21&l=2#x

We have a certain blindness to our own evil, and a tendency to exaggerate our good.

This extends, too, to what we make :)

D Left Adjoint to U

i know mon, commutative diagrams are evil, because I'm using server and it's heating the planet

AMDG

KEKW

D Left Adjoint to U

I want one day to integrate with MSE posts

so people can craft CD-heavy posts on my site and it will post it nicely to MSE for them

AMDG

I mean you can technically integrate anything immediately using what was it... iframes?

D Left Adjoint to U

02:49

Either that or MSE needs a special feature

I don't think they let you embed stuff

with iframe

AMDG

try it and see if it works

D Left Adjoint to U

I know it doesn't

AMDG

How do you know?

D Left Adjoint to U

If it did I would have been simply embedding a long time ago

Okay, I'll try it

AMDG

There is no surer source of truth than reality itself! It either works or it doesn't. :)

D Left Adjoint to U

02:51

@AMDG Can't: math.stackexchange.com/questions/4686686/…

0

Q: Test post - testing whether I can embed Quiver diagrams here

Testing whether I can embed Quiver here

diagram-chasing

AMDG

Oh you mean embed in a post?

You want the diagram itself embedded in a post?

D Left Adjoint to U

Yes of course, so that MSE is like a textbook

You can do it by uploading an image

AMDG

Hm, well if you can make something like an HTML renderer, you can do whatever you want serverside, then provide an absolute URL to that image representation.

D Left Adjoint to U

or copy paste

I could put something in their copy buffer

@AMDG yes, that's brilliant

Maybe a Django view that does that

I use iframe's like crazy in my site

AMDG

Why?

D Left Adjoint to U

02:56

one for embedding Quiver CD editor and one for each embedded preview (no editor)

Only where I need it lol

AMDG

Those can easily contribute to poor UX due to loading times if it's cross-site is all I'm thinking of.

D Left Adjoint to U

It's all inhouse

AMDG

I remember using edmodo. It was pretty cringe. Implements its own page as an iframe for the whole site ecks dee

D Left Adjoint to U

though I've plugged in varkor's host for quiver and that works too

AMDG

Hm, that all is fair and well then it seems

D Left Adjoint to U

02:57

Yep! It's going to be a neat site. I want to have proofs there as well

Like you start with some CD's and glue them together where possible and produce a proof

A proof is the gluing instructions

or deleting, or functoring, etc

AMDG

CDs?

D Left Adjoint to U

commutative diagrams

AMDG

I'm sure that means something in math, but I'll be content with not knowing. Got it.

D Left Adjoint to U

Bassically take a 3x3 grid of objects with grid-like morphism arrangement in an abelian category. The number of equalities is huge and there's no easy way to remember them all. but to rememmber a 3x3 grid is easy

AMDG

Too many new words for me to be able to understand.

D Left Adjoint to U

02:59

CD's are just an advanced notation that is not textual

AMDG

Two in particular

Cool

D Left Adjoint to U

There's text in them, but the language is mainly visual

Well, category theorists will love the idea one day. Once the database has enough entries in it to attract math studiers

of arrow math

AMDG

Maybe it's just me, but it seems a lot of times, the pioneer or discoverer of a thing puts in enough effort to get something working, and then no one bothers to improve the thing afterwards... it just gets cemented into society regardless of how well it works (or doesn't).

Phone numbers are still a thing in 2023. Stroads are still built like they are the epitome of road design here in America.

D Left Adjoint to U

@TedShifrin look at my screenshot above

AMDG

and as perceived from a recent Veritasium video, they look to quantum computers to give them the answer to factoring large prime... residues? I can't quite remember. They give all possibilities, but... have they looked into just applying Fourier arithmetic to the usual stuff and exploiting things like $\frac 1 2 n (n + 1)$?

D Left Adjoint to U

03:07

@AMDG yes, they try all avenues

Cotton Headed Ninnymuggins

That was a good video, as with every Veritasium video

AMDG

I hope "all" doesn't equate to trying the infinite set of integers.

D Left Adjoint to U

The integers are quite mysterious

AMDG

Empirical means are vastly inferior to the powers of deduction and induction unique to man.

Ted Shifrin

@Wolgwang Symmetry about a point in the plane means odd (180° rotation).

AMDG

03:11

@DLeftAdjointtoU I mean again, I'd just like to point out that, for example, the 2-adic valuation of the even integers $n$ is $n\land (\neg n + 1)$.

That extends to any number evaluated similarly in higher bases.

The definition of negation for higher bases would have to be modified accordingly.

So then for $b$-adic valuation of $n\equiv 0 \mod b$, $n\land_b (\neg_b n + 1)$

Because the number of trailing zeros of $n$ represented in base $b$ are directly proportional to the number of factors of $b$ in $n$.

See now if I just knew how to algebraically compute the number of digits a number requires for its base $b$ representation, including zero terms interspersed between non-zero terms, ...

But that's why I'm looking into the relationship of $\ln x - \lfloor\ln x\rfloor$ to $x$ itself.

Wolgwang

For a regular tetrahedron of $l$ edge:
$$Volume=\frac13(Base\ area)(height)=\frac13(\frac{\sqrt3}{4}l^2)(l\frac{\sqrt3}{2})=\dfrac{l^3}{6}$$

What is wrong?

D Left Adjoint to U

@AMDG There is the final widescreen page look. As you can see it's flexbox and wrapping and will fillout the first row before wrapping

one potato two potato

If $D, D'$ are special divisors over a compact Riemann surface of genus $g$ with $\deg DD' = 2g-2$, then $DD'$ is a canonical divisor?

D Left Adjoint to U

@TedShifrin there's the screenshot

I'm thinking I should go with blue / yellow to remind people of springer textbooks subconsciously

AMDG

@DLeftAdjointtoU Great

one potato two potato

03:33

Oh it's special divisor of degree $2g-2$ so $DD'$ is equivalent to canonical divisor

D Left Adjoint to U

Sounds like AG

I changed the background to yellow. Now the page really pops

Any pythonistas here

one potato two potato

03:49

0

A: Understanding theorem III.8.7 in Farkas & Kra Riemann surfaces

$1$. If $D$ is a special divisor over a compact Riemann surface with degree $2g-2$, then $D\sim Z$ (equivalent). So $D$ is contained in a canonical class. In particular, $D$ is a canonical divisor. $2$. If $Z = (\omega)$ for some abelian differential $\omega$, then $\Omega(Z) = \{\eta:\eta\text{...

I hope I'm correct

If $f:M\to\Bbb C_{\infty}$ is a meromorphic function then $f(z)dz$ in local coordinate $z$ on $M$ defines an abelian differential. If $\omega$ is an abelian differential and $1$ is an abelian differential with $dz$ for each local coordinate $z$, then $\omega/1$ is a meromorphic function so the coefficient function of $\omega$ defines a meromorphic function. Does it make sense?

Wait does $1$ make sense?

gewbDog5

04:56

Can someone help me understand this: math.stackexchange.com/questions/93463/… (the second answer down)

The proof shows $Q(\sqrt 2, \sqrt 3)$ is isomorphic to $Q(\sqrt 2 + \sqrt 3)$ as vector spaces over $Q$. But how does that show the fields are equal? It doesn't make any sense to me.

First of all, why do so many algebra proofs not draw a distinction between stuff being equal and stuff being ismorphic? This proof actually shows that these two fields are isomorphic and not equal, right?

leslie townes

uh, it isn't that clearly written to begin with, but i don't think the basic logic of the argument intends to show isomorphism, it intends to show equality

Alessandro Codenotti

@gewbDog5 You already know that both are subfield of $\overline{\Bbb Q}$, so you only know to show that they contain the same elements to show that they are the same field. Obviously $\sqrt2+\sqrt3\in\Bbb{Q}(\sqrt2,\sqrt3)$, so only the other inclusion is not needed

@gewbDog5 Usually this doesn't matter, but in this case the fields are actually equal

gewbDog5

OK so the fields are equal but people go around showing they're ismorphic as vector spaces and every says bravo great proof

what am i not getting?!

leslie townes

the argument explicitly or implicitly uses that Q(sqrt(2) + sqrt(3)) is a subfield of Q(sqrt(2), sqrt(3)), and also that Q(sqrt(2) + sqrt(3)) is degree-4 extension of Q, and concludes from the apparently known fact that Q(sqrt(2),sqrt(3)) is a degree-4 extension of Q (indeed, with a particular basis) that Q(sqrt(2) + sqrt(3)) = Q(sqrt(2), sqrt(3))

it isn't clear to me how much of that is being proved vs. assumed by the writer of that post

but if F is a finite extension of Q, and E is a subfield of F that contains Q, one way of showing that E = F is to show that [F:E] = 1 or equivalently that [E:Q] = [F:Q]

we're definitely using the fact that E is a subfield of F in this style of argument, it's not just some random other extension of Q that is out there flapping in the breeze

gewbDog5

OK so this is helpful. but will take me a second to process.

leslie townes

05:12

or to be clear, in general, if E and F are just random finite extensions of Q, the fact that [E:Q] = [F:Q] would not be enough to conclude E = F, or even that there is a field isomorphism between E and F

for example even within this example we have [Q(sqrt(2)):Q] and [Q(sqrt(3)):Q] both being 2, while the fields Q(sqrt(2)) and Q(sqrt(3)) themselves are neither equal nor isomorphic

Ted Shifrin

@gewbDog5 One is a subset of the other, so they are equal if they have the same dimension.

This is a powerful technique to master.

I guess leslie already told you that.

gewbDog5

still appreciated ^

leslie townes

ted: we swam with a duck today.

Ted Shifrin

The usual duck or a neophyte?

gewbDog5

I'm def a neophyte but am I this neophyte ^^

leslie townes

05:27

we think it's the usual one.

Ted Shifrin

I meant the duck, gewb :)

gewbDog5

07:26

I have to know is saying "we swam with a duck today" a way of secretly putting down a person who doesn't know as much math as you?

I want to know what that little exchange means ^^

leslie townes

07:41

uh, i swim with my daughter in a nearby pool every day, and on most days, we are joined by a mallard duck

Apr 22 at 17:44, by leslie townes

we think that it's one particular duck, who for whatever reason isn't shy about sharing the water with us.

there is nothing figurative about this

gewbDog5

Right OK- that sounds great. The "neophyte" did make it appear figurative

geocalc33

09:09

How do you obtain a parametrization of a surface from the collection of its level sets?

one potato two potato

09:58

So if we allow multi-valued functions, Mittag-Leffler theorem is true for compact Riemann surfaces

geocalc33

Anyone good with finding transformations that leave a measure invariant?

one potato two potato

10:24

I guess only translations if we assume some regularity

D Left Adjoint to U

11:18

Got rudimentary pagination working together with order by and asc/desc

D Left Adjoint to U

11:29

Got "My diagrams" list page working with order-by, direction, and pagination

►

geocalc33

0

Q: Obtaining parametrization of surface through its level sets

Consider a smooth, closed convex surface $f(x,y)\subset [0,1]^3 $ where $\mathrm{max}(f)=f(0,1)=1$ and $\mathrm{min}(f)=f(1,0)=0.$ Define the collection of level sets of $f$ as: $$L_c(f)= \big\lbrace (x,y) ~| ~f(x,y)=c\big\rbrace $$ where $c\in [0,1].$ Define the strict sublevel set: $$L^{-}_c(f)...

multivariable-calculus circles conic-sections surfaces parametrization

@DLeftAdjointtoU cool!

didn't know you had a youtube channel

D Left Adjoint to U

Yeah, I have YT mon

Look at how cool that webpage is designed (by me)

geocalc33

It is very cool

2

D Left Adjoint to U

11:44

I designed dat chit

I can teach you how it's made

if interested

mainly it's just a lot of code but luckily I can do the frontend in a webflow-like tool called BSS

Otherwise you'd spend all day just trying to position something in the web world

one potato two potato

What is an importance of Poincare inequality?

Sha Vuklia

What exactly is the data here? We have a fundamental system $\mathcal V$ such that the group operation is defined for each $x\in V$ for any $V\in\mathcal V$? If that is the case, how do we define the group structure on the total space $G$? It's clear to me how to define a topology (translating the fundamental system to any other point)

Hm, never mind, I think they mean that given a (discrete) group $G$, we have a topological group iff we have these data

So the (discrete) group operation is in that sense not determined

geocalc33

what do you call smooth maps that aren't yours?

NACHO MAPS

Xander Henderson

12:02

@geocalc33 That makes no sense. It works if the cheese isn't yours...

geocalc33

Looks like I need to look myself in the mirror and understand what went wrong

I need to go back to the 4 P's: Prior Preparation Prevents Poor Performance

Xander Henderson

There are three types of people in the world: those who can count, and those who can't.

geocalc33

I seriously cannot get a grip on the most elusive problem in mathematics it's deeply disturbing

not the Riemann hypothesis so don't even think that

PlaceReporter99

12:39

@AMDG that is equal to $\ln x \mod 1$

Soumik Mukherjee

Anyone following WCC game $12$?

PlaceReporter99

@geocalc33 isn’t that 5 P’s?

@XanderHenderson seems like the person who is cracking the joke is the second type mentioned.

This chat room can be expressed as the fraction $\frac9{10}$

Xander Henderson

@PlaceReporter99 Jokes are always funnier when you explain them.

robjohn

@TedShifrin Now there is something we could do without ;-)

geocalc33

13:07

I was only off by a constant

Ajay

Can someone explain the Banach fixed point theorem to me?

actually, don't ask to ask

So, please explain the Banach fixed point theorem to me.

Anyone...?

geocalc33

Put a map of your country on the floor, there's a point on the map that is touching the actual point
it refers to

please study this meme.

Consider the space of smooth maps:

$$ f: [0,1]^3 \to [0,1]^3 $$

such that $f$ preserves the convexity of the class of all closed surfaces satisfying $f(0,1)=1$ and $f(1,0)=0.$

What can we say about $f?$

Ajay

13:25

@geocalc33 Are you talking to me?

geocalc33

no

Ajay

I wish you were :(

anak

@Ajay what troubles you with the Banach FPT?

Is the intuition at least clear? e.g. "a contraction mapping $f$ moves all points closer together, so if you pick any point $p$, then $\lim_{n\to\infty} f^n(p)$ seems like it converges to something"?

Ajay

13:40

Yes, I understand that much. However, I need a more detailed explanation

or investigation...

I'm writing a mini thesis

And I need to be able to explain the Banach fixed point theorem to high school students

Xander Henderson

@Ajay If you don't grok it yourself, you probably shouldn't be trying to explain it to others...

4

anak

@Ajay Well I think it's important that you do most of the work in figuring it out, then!

one potato two potato

@Astyx Rotations by $2\pi/3$, now I see

Ajay

@XanderHenderson I do understand it to a certain extent. The difficulty i'm having is finding a good way to explain it.

geocalc33

Explain it right now to us

anak

13:46

@Ajay So how do you explain it for yourself? That's usually a good way to start.

Xander Henderson

@Ajay Okay, so explain it to us. I am happy to ask annoying questions about your explanation. This kind of interrogation is often a good way to solidify your own understanding.

one potato two potato

And realize $\langle\sigma_1,\sigma_2,\sigma_3\rangle\simeq\langle\sigma_1,\sigma_2,\sigma_3\mid\sigma_i^2 = (\sigma_i\sigma_j)^3 = 1\rangle$ which is a Coxeter group

geocalc33

@anak I asked that question as per your suggestion on mathoverflow

Ajay

Say we have a nonempty complete metric space and a function defined in the metric space. say we have two points that exist in this metric space. If we plug each point into the function and find the distance between them, we see that the distance is less than or equal to the distance between the points themselves times some constant.

Since this is true, this means that there is some unique point that exists within the metric space such that when you plug in the value of the limiting point into the function it produces that limiting point.

shintuku

sounds like a fixed point theorem

oh you're doing that

Ajay

13:53

I further used an example using successive approximations. I used the function $f(x) = \frac{1}{5}x$ to show that the limiting value must be 0

and that 1/5 is the constant value les than 1 in the theorem

I also explained a bit about why the constraint(less than or equal to one) is needed.

anak

@geocalc33 Did they remove it? I don't see it.

@Ajay how doo you justify the need for completeness?

geocalc33

1

Q: Constructing spacetimes

Consider a spacetime $(\zeta^{3,1},g)$ where $$g=\frac{du\,dv}{uv}-\frac{dr^2}{r^2}-\frac{dw^2}{w^2} \quad \forall u,v,r,w \in (0,1)$$ Now this is just Minkowski space in different coordinates (related to Dirac/Light cone coordinates/null coordinates). I'm looking to take a Cauchy foliation of $\...

dg.differential-geometry lorentzian-geometry semi-riemannian-geometry information-geometry

@anak it's here

anak

Oh you used a different account.

At least there is one upvote, which is a good sign it won't be obliterated in a day.

Ajay

@anak is it because of cauchy sequences?

Xander Henderson

@Ajay That seems to be the fundamental idea. What are you confused about?

Ajay

13:58

I talked with my advisor and he wants me to expand even more

And I don't know how

Xander Henderson

@Ajay I would pick a more interesting map. Something where it is harder to immediately see what the fixed point is.

Ajay

That was the motivating example

I have 3 examples

Well, I intend to...

Xander Henderson

Like, for example, $\sin(x)$ on $[0,\pi/2]$ (or where ever $\sin$ is contractive). This is my goto example, because you can hit the $\sin$ button on the calculator over and over again.

geocalc33

@anak yeah, I mean I provided the metric itself but there are nuances I don't fully understand which hopefully someone will clarify for me

anak

@Ajay you tell me! If you are unsure about why a certain ingredient is important, you should dig deeper into it and try to figure out why it is important.

Ajay

14:08

@XanderHenderson isn't it just 0?

@anak let me think for a while.

Ajay

14:20

@anak is it because in the in some situations the limit point might actually be the removed point? So we need the space to be complete.

shintuku

have you tried doing the proof first?

Ajay

@shintuku Are you talking to me?

anak

The point is not for me to tell you, or to confirm for you. This is your thesis, Ajay!

shintuku

yeah, sometimes the explanation comes more easily when you complete the proof

Ajay

I'm putting the proof in the appendix for now. I'm not gonna put it in the main body

Xander Henderson

14:23

@Ajay Oh, shoot. That's not the example I'm thinking of...

Ajay

@anak I went away and thought about it, I just need your help to steer me in the right direction.

I'm not asking for the answer.

Xander Henderson

Hang on, I know I've written an answer on Math SE about the example I like...

Cosine. Use the cosine function.

anak

@Ajay Going away and thinking about it for 12 minutes is not nearly long enough. Sometimes these sorts of things take days to grok.

Xander Henderson

41

A: Why do I get a converging result when pressing cosine multiple times on a calculator?

Let $\cos^n$ denote the $n$-fold composition of the cosine function with itself, e.g. $$ \cos^3(\theta) = \cos(\cos(\cos(\theta))). $$ Note that this is not usually what this notation means in, for example, introductory calculus texts. However, it is convenient in the current context. What you ...

@Ajay Think of an example of a very common, useful space which is not metrically complete.

Ajay

@anak Ok :) you could have just said that

And I would have thought for longer

anak

14:26

ping me in a week if you still need help.

Ajay

I don't have a week 🥲

I wish I did

Rithaniel

15:08

Hola math chat

CroCo

15:39

Hi everyone, Is there an online linear algebra handbook that lists theorems and proofs by sections, as in basic linear algebra textbooks?

Is it possible for you to remember all theorems without using a reference?

user977780

@CroCo Basic linear algebra : solving of system of linear equations.

CroCo

@user977780 yes.

Xander Henderson

@CroCo Generally speaking, you don't. You remember the results you use a lot, and tend to forget the rest (and look them up if you need them).

user977780

@CroCo Yes( at least what I have studied).You can also remember. Don't try to remember the whole proof just outlined the key steps.

CroCo

@XanderHenderson As a result, I assume we need handbooks that provide information. Having read a number of elementary textbooks on linear algebra, I am getting bored with any new book because of the details.

Xander Henderson

15:46

@CroCo I don't understand why you "need handouts".

As I said above, you learn the results you have to use. While you are learning a new topic, you should be assigned a fair amount of work which requires you to use the theorems you are studying, hence those theorems should become familiar to you.

You'll likely forget them eventually (once you stop using them), but while you are learning them, you should learn them.

CroCo

Since I don't do math for a living, I tend to forget a lot of important theorems.

user977780

For an example: Theorem : For any square matrix A, there exists \epsilon >0 such that A-\epsilon I is invertible. Proof: To show det(A-\epsilon I)\not 0 .Observe that det(A-\epsilon I) =0 implies \epsilon >0 is an eigen value. What happen if we choose 0<\epsilon<|\lambda| where \lambda is the non zero eigen value ,smallest in absolute value in spec(A).

Xander Henderson

@CroCo Has nothing to do with doing math for a living. I am sure that I have forgotten more theorems than you will ever learn. :P

There is a moment in my career when I had a lot of results memorized and ready to go. That was the quarter I took all of my qualifying exams. I've lost a lot of that fluency over time.

user977780

@CroCo Try to prove the theorem at least once and figure out key steps.If you can remember key steps then whole proof will be automatically recovered.

CroCo

My usual method of learning is to rewrite the theorems of the textbook and then save them. It's surprising to me that I can read the book again with so few pages.

@user977780 I totally agree.

As a mathematician, I assume this isn't the case since you teach math and have a great memory.

user977780

15:59

@CroCo I am not qualified to add any comment.

Xander Henderson

@CroCo You don't remember the things you don't use every day. I haven't thought about the Sylow theorems in ages, and can't remember a bit about them (something something a group of order something something subgroups dividing something something...).

CroCo

As a tutor at my university, I have tried to teach basic math at a lower price to avoid forgetting something vital. In spite of this, tutoring can be exhausting, so I'm looking for alternatives like handbooks that provide immediate results.

It surprises me how few books are available on this subject.

user4539917

A good math book should have a chapter summary that lists important ideas from the chapter.

Xander Henderson

@CroCo There is no royal road.

CroCo

I may create my own then. :)

Xander Henderson

16:09

@CroCo That is what most good instructors do.

I am fairly certain that there are actually more analysis texts out there than there are actual analysts. Because every analyst has written at least one book of their own.

(Since every other analyst DOES IT WRONG!)

user4539917

Create royal roads for their students?

::runs::

user977780

🏃💨🏃💨🏃💨

@CroCo 🤞

Soumik Mukherjee

Forgetting vital things sometimes causes so much trouble, some days ago, in the final exam of this semester, i forgot which classroom i was sitting at when I left to use the washroom :/

user977780

Some theorem requires special techniques( tricks) and this may be difficult to remember.

PNDas

Example of a sequence which is weak* convergent but not weakly convergent?

user977780

16:21

Road map for analytic number theory?

Ted Shifrin

@user4539917 I disagree. This is not high school.

Xander Henderson

@TedShifrin No, I agree. Every book should have a chapter summary.

It's called "the title of the chapter."

:P

user977780

16:40

Funny :)

Hi @XanderHenderson , how are you?

Xander Henderson

@user977780 Still breathing.

user977780

16:57

@XanderHenderson :)

Ted Shifrin

17:22

In the times of Covid, that is a nontrivial accomplishment!

Rithaniel

17:52

@XanderHenderson The briefest of summaries

PlaceReporter99

The devil is hiding in the golden ratio: $ϕ = -2 \sin(666 °)$

from WA

wolframalpha.com/input?i=golden+ratio

Obliv

I want to find the volume between two elliptic paraboloids, $z = x^2 + y^2, z = 8 - x^2 - y^2$, using cylindrical coordinates.

I know the z limits are 0 to 8

I think the "top" function is $z=8-x^2-y^2$ since it opens down

So I guess I do top - bottom from 0 to 8 but in cylindrical

May I have a hint

Ted Shifrin

18:11

You know wrong.

leslie townes

@PNDas a standard example would be the sequence e_n in in ell^1(positive integers) given by letting e_n's nth entry be 1 and have its other entries be 0. converges weak-star to 0, but not weakly. here you think of ell^1(positive integers) as the dual of the sequences indexed by positive integers that tend to 0 (itself normed by the sup norm).

3

Obliv

@TedShifrin So I set them equal to each other and found $4 = x^2 + y^2$ is the circle of intersection between the two surfaces

Ted Shifrin

So what does that tell you?

Obliv

The radius runs from 0 to 2

Ted Shifrin

OK. And theta?

PlaceReporter99

18:20

@Obliv that made me think about multivariable calculus

Obliv

$[0,2\pi]$

Ted Shifrin

OK, Obliv. Now, fix a point inside the circle of radius $2$. For that point (fixed $r,\theta$), what does $z$ do? When do you enter the region? When do you exit the region?

PlaceReporter99

@TedShifrin I just stopped caring about covid. We are in the endemic phase now.

Ted Shifrin

Well, convenient for you, but not for me. I just got it for the first time two weeks ago.

No. Remember, $r,\theta$ are some fixed value in the acceptable range.

Obliv

Ah i see

z is the circle that goes from 0 = x^2 + y^2 to 4 = x^2 + y^2

:D

Ted Shifrin

18:29

Huh? Draw a picture. You are drawing a vertical line ... Where does it enter and exit?

Obliv

shintuku

ah, yes, the alien chrysalid incubator

Obliv

I'm unsure what you mean by "what does z do" because I think the solid itself ranges from 0 to 8

but in cylindrical coordinates I guess it changes

I'm guessing it ranges from 0 to 2

because $4 = x^2 + y^2$ implies that

Ted Shifrin

Go back to basics and how you set up limits of integration in the 2D case. You're not thinking. How do you set up the integral between $y=x^2$ and $y=4-x^2$ in $xy$-coordinates?

Obliv

set them equal $x^2 = 2$ so you have $-\sqrt{2} \leq x \leq \sqrt{2}$

with the top function $4-x^2$

yes that makes sense

Ted Shifrin

18:41

And where does the vertical line enter the region?

Obliv

z=0

Ted Shifrin

I'm still on the 2D example.

You have no logical approach to these problems.

You should go watch a few of my videos on setting up double and triple integrals.

Obliv

at $(-\sqrt{2},2)$

Ted Shifrin

I give up.

Obliv

Wait how is $(-\sqrt{2},2)$ not the point of entry for a vertically simple integration $\int_{-\sqrt{2}}^{\sqrt{2}}4-2x^2 dx$

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$Mathematics$ Mathematics