Mathematics - 2024-08-22

zeta space

00:55

Sophie Swett

I'm trying to find the "nicest" area-preserving mapping between a Euclidean equilateral triangle and a spherical equilateral triangle with 72° interior angle. Where might I look for such a thing?

Jakobian

@uhoh

zeta space

oddly enough I am looking for the same thing

Jakobian

@SophieSwett If $A, B, C$ are vertices of this spherical triangle, and also of the flat normal triangle, then wouldn't a good map between them by given by projection?

area-preserving? Do those triangles have equal area to begin with

zeta space

01:10

Sorry about that Elmo decision

@Jakobian that is a keen observation

projections seem like a good direction

Jakobian

Do I seem like I'm extroverted

I do like talking but I'm actually an introvert

I was watching a video about people who have both autism and ADHD and it kind of resonated with me. I do suspect I might be on the spectrum, but I don't really feel like an autistic person, but the experiences of people with both did feel like something close to home

Just to be clear I'm not set in stone that I do have both, or any at all

zeta space

i am an introvert and might have some sliver of ADHD

Jakobian

Do you also have that you often find yourself replaying songs in your head

like one song constantly on repeat, or a part of it

zeta space

I don't get songs "stuck in my head" as we say

although i do listen to songs on repeat a lot

Jakobian

01:28

I see, so it's different for you

Earworm

An earworm or brainworm, also described as sticky music or stuck song syndrome, is a catchy or memorable piece of music or saying that continuously occupies a person's mind even after it is no longer being played or spoken about. Involuntary Musical Imagery (INMI) is most common after earworms, but INMI as a label is not solely restricted to earworms; musical hallucinations also fall into this category, although they are not the same thing. Earworms are considered to be a common type of involuntary cognition. Some of the phrases often used to describe earworms include "musical imagery repetition...

yeah this actually happens to me a lot

they say here earworms correlate with neuroticism so maybe that's that

I also read this correlates with higher mortality rates. I'm doomed

^ this is also a joke

Sophie Swett

@Jakobian That's a pretty good map, but not an area-preserving one. If we imagine that the flat triangle is in 3D space near the sphere, then the local scale factor depends on the local distance from the center of the sphere.

@Jakobian I sure hope so!

zeta space

01:51

can someone help understand something

zeta space

02:39

?

peep

How do you cook boilabase?

^_^

Can you do bordisms with two Banach manifolds?

if you say no you were right

actually maybe I am wrong about that

Banach spaces are generally infinite dimensional and noncompact. Banach manifolds are manifolds in which each point neigborhood is homeomorphic to a banach space

Jakobian

02:59

@zetaspace you'd have to define what it means to be a Banach manifold with boundary

and then explain if we can uniquely recover that boundary

zeta space

Well the Banach manifolds are the boundary

in other words the Banach manifolds in this set up are cobordant

they should at least be compact manifolds definitely

and of the same dimension

the question is what exactly are they bounding..

Jakobian

You can't talk about bordism without pants

zeta space

I was thinking about pants recently

Jakobian

You must have obsession about pants

冥王 Hades

@Jakobian I thought the same....until I was diagnosed with high functioning Autism.

zeta space

03:15

Often I have ideas that I don't have the dictionary to express to others but the dictionary exists somewhere

Jakobian

@zetaspace yeah. Boots, pants and a shirt

But don't forget the belt. And its cousin the Mobius belt

zeta space

that just gave me a cool idea

Jakobian

You're going to open a clothes shop

@冥王Hades I don't think that excludes ADHD

Unless you're saying that this explained everything for you

zeta space

03:29

I can't understand if this is a trivial or nontrivial symmetry. Or more precisely where does it show up? $S^2$ chopped in half, rotate top half by pi/4 radians, sew everything back up

it should show up in combinatorics or as an action on the circle group

nickbros123

04:55

there is a subtle difference between the statements "$(\forall x \in X)(\forall G: G \text{ open set in }X: x \in G)....$" and $(\forall G: G \text{ open in }X)(\forall x: x \in G)....$ right? It might be the case that I have confused myself with this notation

wait, there is no difference

Soumik Mukherjee

05:14

Why you wrote for all x in X in the first if you want to restrict things to G?

nickbros123

this is how rudin defines basis for a metric space: "for every x in X, and for every open set G such that x in G...."

I think there should be no difference between the two statemtns

Soumik Mukherjee

Okay then the first statement is correct

The second statement is also correct

But if he is defining local basis then you should use the first statement

nickbros123

08:44

@SoumikMukherjee I am not sure. I think he is speaking generally about it

(more accurately, he is speaking barely about it)

I will look into shirali and vasudeva's book

uhoh

09:11

21

Q: What do all branches of Mathematics have in common to be considered "Mathematics", or parts of the same field?

At some point in my life I think I've read what all branches of Mathematics had in common were numbers. But then I remembered a branch of the many Mathematics I had when I was an university student, and I didnt remember numbers in it. Then, I made a Google search to see about this, and I found th...

asked in Philosophy SE, slightly related to:

10

Q: What does "most of mathematics" mean?

After reading the question Is most of mathematics not dealing with sets? I noticed that most posters of answer or comments seemed to be comfortable with the concept of "most of mathematics". I'm not trying to ask a stickler here, I'm just curious if there is some kind of consensus how the quant...

soft-question philosophy

nickbros123

@uhoh anything that looks at a set and a structure of that set is mathematics i guess

2

uhoh

@nickbros123 I use and love math but I'm not a mathematician. Does the phrase "structure of a set" have a formal definition? I perhaps naively think of numbers and functions as "doing things" for me, but when I think I'm using them, am I simply finding structural aspects of the sets to which they belong?

@nickbros123 mathematics lives outside of time? It doesn't give me an answer when I use it, the answer was already there in the structure of the set all along? (Sorry, I've been biking in the hot Sun this afternoon and probably a bit dehydrated right now)

Or did I overlook "looks at".

Sine of the Time

sup @Binky

Soumik Mukherjee

09:29

@nickbros123 anyway, both definitions say that given any point and any open set containing it, there exists a basic open set that contains the point and is contained in the given open set

nickbros123

@uhoh I mean, you have an arbitrary set, could be anything, with a binary operation (or more than one) that obeys certain properties. ASCII- associativity, closed, identity, inverse- the structure that follows this is a "group". You derive many properties, and then u look at everyday things and see which abstract thing youve constructed explain them. R^3 is many things for example- a vector space, a metric space. R is a metric space, a field, a group, a vector space.

@uhoh I believe this, but others may, or do not. This is the age old question of "inventing" and "discovering" that many people, even experts, disagree with each other

I personally couldnt give a damn

Binky McSquigglebottom

09:47

@SineoftheTime hi

uhoh

09:58

@nickbros123 yes, I got all of that, thanks!

Binky McSquigglebottom

10:10

@SineoftheTime are you free ?

psie

Consider a statement $$\forall x(P(x)\land Q(x)).$$ I think sometimes it can be challenging to know when this statement is true for all $x$. So I'm wondering, can we instead look at when the statement is false, i.e. when the negated statement $\exists x(P(x)\lor Q(x))$ is true, and once we've determined that, negate those occurrences to arrive at when $\forall x(P(x)\land Q(x))$ is true?

Sine of the Time

@BinkyMcSquigglebottom yes

Binky McSquigglebottom

10:39

@SineoftheTime but to make integrals by substitution, do we need to learn the various cases to see which is the best substitution to make?

psie

@psie I believe what I am trying to ask is that if $$S(x)\text{ is true for }x\in A\iff \neg S(x) \text{ is true for }x\in A^c?$$Here $A$ is the universe.

Sine of the Time

11:04

@BinkyMcSquigglebottom certain integrals are evaluated by standard substitutions

you just need practice

Xander Henderson

@SineoftheTime Evaluated.

Sine of the Time

That's what I said

Xander Henderson

One does not "solve" an integral.

Sine of the Time

noted

Xander Henderson

@SineoftheTime Oh, yeah. Another question: do these gas lamps look dim to you?

:P

Sine of the Time

11:08

😦

Pizza

@SineoftheTime but for yesterday's exercise, should the parameterization describe a helical curve that wraps around the cylinder?

Sine of the Time

the curve was not an helix

the equation represents a cylinder

can you send me a picture of the whole exercise if you can?

Pizza

yes, which extends along z

I had thought of writing: x(t) = cos(t), y(t) = 1/√2 sin(t), z(t) = t

@SineoftheTime ok

Sine of the Time

source of the problem?

Pizza

It's from an exam

Sine of the Time

11:26

I can't make sense of what is asked

Binky McSquigglebottom

@XanderHenderson Can it be said "calculate the integral"?

Pizza

@SineoftheTime Do you mean in the second part of the exercise?

Xander Henderson

@BinkyMcSquigglebottom I don't like that phrasing.

Sine of the Time

yes

@xander In the problem sent by Pizza, there's a vector field in $\Bbb R^3$ and it's asked to find the work along the "curve" $x^2+2y^2=1$

this does not make sense to me

Soumik Mukherjee

@SineoftheTime I can't either, but due to different reasons

Sine of the Time

11:30

I translated it @Soumik

Soumik Mukherjee

non so l'italiano

Sine of the Time

me too

Binky McSquigglebottom

Same

Soumik Mukherjee

@SineoftheTime really?

Sine of the Time

nah I was just trolling

Soumik Mukherjee

11:32

And here I was doubting my memory as I saw you to speak Italian in chat

Sine of the Time

@Pizza Anyways, I don't think it's important to focus only one particular problem. Since $\omega$ it's conservative, if the curve is closed the the work is $0$, otherwise you compute at the endpoints

Xander Henderson

@SineoftheTime I haven't really even gotten out of bed yet. I'm not ready to think math.

Sine of the Time

K

Xander Henderson

NPR seems to think that the Moonlight Sonata is good wake up music...

But it is go to bed music!

Peer Gynt is wake up music!

Stupid local(ish) NPR station!

(also, now that my alarm has gone off, I should probably start moving...)

Sine of the Time

put the alarm far from the bed so that when it rings you have to get up

it told this to one of my friends who struggles to get up and he said it's cruel

Binky McSquigglebottom

11:45

@SineoftheTime why?

Sine of the Time

does it make sense to you?

Binky McSquigglebottom

The problem can be solved as a two-dimensional case, setting $z=z_0$

Xander Henderson

@SineoftheTime the alarm is on a desk on the other side of the room. But it doesn't ring, it just plays music, and then the news.

But I rarely sleep until it goes off.

Pizza

@SineoftheTime Maybe when they did this exercise the teacher will have made some clarifications

Xander Henderson

So waking up is not really the issue.

And because they STILL haven't fixed the fire alarms in my office, I can't go to work for another two hours. So I have no real motivation to get moving quickly.

Sine of the Time

11:51

@BinkyMcSquigglebottom if you set $z=z_0$ for some $z_0>0$, the curve is still in the space

but the problem makes sense

but that's not what is written

Pizza

I'm going to eat bye :)

Sine of the Time

@Pizza move on and do other problem in my opinion

Pizza

@SineoftheTime Yes, in fact I'll leave this one like this

Binky McSquigglebottom

@SineoftheTime Oh...

Pizza

Maybe I'll try asking someone if they did it

Sine of the Time

11:53

let me know

Pizza

👍

Binky McSquigglebottom

Why Is $4\frac{1}{2}=\frac{9}{2}$?

Sine of the Time

it's a notation for $4+1/2$

Binky McSquigglebottom

Why isn't it an *?

Sine of the Time

it's a matter of notation

instead of writing $4.5$, some authors write $4\frac12$

Binky McSquigglebottom

12:01

👍 thanks

but there is another way to do it, what is this stuff?

Sine of the Time

12:37

@BinkyMcSquigglebottom Stirling formula

Sine of the Time

12:49

I've seen this limit solve with different methods, but it's the first time I see it solved using Riemann sums

Xander Henderson

@BinkyMcSquigglebottom It is notation which predates most of modern mathematics.

It is bad notation to teach students in a mathematical setting, but it shows up enough in other places that it does still need to be taught.

(eg it is the notation used in most cookbooks)

Binky McSquigglebottom

13:10

thank you all

Sine of the Time

13:46

@Pizza you should answer this :)

冥王 Hades

16:20

@XanderHenderson I bought a dedicated alarm, placed it close to my bed, and destroyed it the very first morning....for daring to disturb my slumber

Soumik Mukherjee

@BinkyMcSquigglebottom is that blackpenredpen?

Sine of the Time

@BinkyMcSquigglebottom if you're interested, see this

ZaWarudo

16:36

In the photo above, $F(S)$ is the set of functions with domain $S$ and codomain a field $K$. I am a little confused by the notation $f=\sum_{s \in S} f(s)\delta_s$ when evaluating $f$ in a point $s \in S$, because it leads to $f(s)=\left(\sum_{s \in S} f(s) \delta_s\right)(s)$ and, since sum of functions is defined pointwise in a previous instance of the book (that is, $(f_1+f_2)(x)=f_1(x)+f_2(x)$), we have $\left(\sum_{s \in S} f(s) \delta_s\right)(s)=\sum_{s \in S}(f(s) \delta_s)(s)$.

I see too many $s$': maybe the author should have used a different letter for the value we calculate $f$…

Binky McSquigglebottom

16:54

@SoumikMukherjee yes

@SineoftheTime thanks

ZaWarudo

17:09

Moreover, the sentence "any function from $F(S)$ can be uniquely represented by a linear combination of delta functions: $f=\sum_{s \in S} f(s)\delta_s$" means that if $\varphi \in F(S)$ then $\varphi = \sum_{s \in S} \varphi(s) \delta_s$ and if $\varphi = \sum_{s \in S} \varphi(s)\delta_s$ then $\varphi \in F(S)$ and this representation is unique in the sense that if $\varphi = \sum_{s \in S} \varphi(s) \delta_s$ and $\varphi = \sum_{t \in S} \varphi(s) \delta_s'$ then $\delta_s=\delta_s'$?

Pizza

@SineoftheTime 😂

Binky McSquigglebottom

$\iint_D 1-2x-3y \ dxdy \ \ D=x^2+y^2\leq x$

is it right? pls

This is how I did it

Sine of the Time

you should get a number, not a function of $x$

Pizza

It should be $(x-\frac{1}{2})^2 + y^2 \leq \frac{1}{4}$

C = (1/2,0) r = 1/2

@SineoftheTime in this case can we use a translated polar coordinate system?

Sine of the Time

yes

by simmetry, $\iint_D y\,dxdy=0$

Binky McSquigglebottom

17:27

I'm not understanding what I should do now

Sine of the Time

the domain is wrong

read what Pizza said

Pizza

It should be as I wrote above

Binky McSquigglebottom

But if y≤√(x-x^2) why do I also have to integrate below the x axis

@SineoftheTime But I solved for y and I get this

Sine of the Time

it does not make sense to solve for $y$

plus you solved wrong

Binky McSquigglebottom

I saw that you don't have to take the root of y

Why?

Sine of the Time

17:36

why would you do it? It's a circle

Binky McSquigglebottom

@SineoftheTime To solve the inequality for y

Sine of the Time

anyways, your mistake is $\sqrt{y^2}=y$, in fact it's $\sqrt{y^2}=|y|$

Binky McSquigglebottom

So a circle would come out

And therefore it is better to use polar coordinates

@SineoftheTime Why is there only y left in the integral?

Sine of the Time

or you can use the properties of the center of mass

@BinkyMcSquigglebottom does it matter if it is multiplied by $-3$? $0\cdot(-3)=0$

Binky McSquigglebottom

17:53

I'm trying to do this without polar coordinates, but I keep getting it wrong...

By symmetry doesn't that mean making •2 and integrating over half the graph?

Sine of the Time

no because it's not symmetric with respect to the $y-$axis

plus the order of integration is wrong

Binky McSquigglebottom

@SineoftheTime why ?

Sine of the Time

18:09

because you integrate first $y$

when you fix $x$, $|y|\le \sqrt{x-x^2}$ and $x$ varies in $[0,1]$

Binky McSquigglebottom

@SineoftheTime yes

Sine of the Time

the fastest way to solve it is: $I=\iint_D(1-2x-3y)dxdy=\mu(D)-2\mu(D)x_{\text{cm}}-3y_{\text{cm}}=\mu(D)-2\mu(D)\frac12-3\cdot0=0$

Binky McSquigglebottom

Thanks very much

Binky McSquigglebottom

18:54

Joe

19:15

In many textbooks on algebraic geometry, it is proven using the Nullstellensatz that if $K$ is an algebraically closed field, then the category of affine algebraic sets is equivalent to the category of finite-type reduced $K$-algebras. Are these categories isomorphic? For my question to make sense, I suppose I have to specify that by an algebraic set, I mean a subset of $K^n$ for some $n\ge0$ which is the common zero locus of a collection of polynomials in $K[z_1,\dots,z_n]$.

I suppose my question is quite unnatural from a categorical perspective, but I am still curious to know the answer.

zeta space

19:39

What do you call $4,$ $\lbrace 4,1 \rbrace $-gons intersecting

correction: mutually intersecting

You can call it a stacked abstract polytope

stacked because it's a composition of (identical in this case) abstract polytopes. Abstract because the faces can be curved. You could even say, "stacked abstract non-convex polytope"

Soumik Mukherjee

19:54

I have not seen Thorgott online for quite a while

zeta space

20:29

Xander Henderson

20:43

@BinkyMcSquigglebottom I don't know who the guy on the bottom left is...

zeta space

European football theory studies $S^2$ and American football theory studies $S^2$ quotiented by a discrete finite linear group $G$. Speficifically in the American theory we usually take $G$ s.t. the orbifold $\mathcal O$ has two fixed points.

In this way American football generalizes soccer

Tsundoku

20:59

@BinkyMcSquigglebottom Is this from a CAPTCHA on a maths forum?

Minsky

22:13

plz help math.stackexchange.com/questions/4961811/…

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$Mathematics$ Mathematics