Mathematics - 2022-05-09

Ted Shifrin

12:19 AM

Now that @leslie has sullied it, I retract the rhetorical tool.

Obliv

What allows math to exist?

Akiva Weinberger

Ted Shifrin

Whatever allows you to exist.

Akiva Weinberger

Finally figured out how to do it with intervals

(The thingy from before)

On the left I have three intervals and on the right I have two, and the sum of their indicator functions is the same

(Make them shrink as they head to the right so that they are compact)

(and so that the summands on the right become only one line)

Ted Shifrin

I have no idea what you’re doing, DogAteMy.

Akiva Weinberger

12:33 AM

That's OK

Obliv

12:43 AM

@TedShifrin Can math/philosophy explain everything we experience though?

Akiva Weinberger

Obliv

Yeah this is more in the realm of metaphysics/philosophy I'm just gonna scurry away.

anak

1:03 AM

@Obliv The answer is no.

leslie townes

no it isn't

anak

GET OUT OF HERE LESLIE

leslie townes

ok, but no is not the answer

anak

Tell me, leslie, what is the answer?

leslie townes

yes is the answer. no is the question

anak

1:06 AM

yes

no

yes

$$no^{e^{2}}$$

puts everyone on ignore

leslie townes

ted: yes

Akiva Weinberger

1:12 AM

This is somewhat difficult to draw

Imagine the diagram shrinks to a point as it goes to the right, so that the two red curves in the top join up to a single curve, and the same for the two white curves

Then on the top we have $Y_1$ and $Y_2$ (the red and white curves)

and on the bottom we have $X_1$, $X_2$, and $X_3$ (the disconnected red segment, the rest of the red curve, and the white curve)

and the sum of the indicator functions of the $Y$ equal that of the $X$

and they're all homeomorphic to an interval

Akiva Weinberger

1:28 AM

Alternate version^

anak

1:44 AM

@TedShifrin I am trying to understand why the dual of Lie algebra of vector fields on S^1 is isomorphic to the quadratic differentials of the form $u\,d\theta\otimes d\theta$. This might not be unique to the circle, but do you have any ideas?

Maybe the quadratic differential part isn't supposed to be special...

Ted Shifrin

What’s $u$? How is that a Lie algebra?

anak

C^\infty. It's from an old Kirillov paper that I am reading.

Ted Shifrin

No clue.

Ajay

English exam in an hour.

leslie townes

ajay: huh? what did you say?

:)

anak

1:53 AM

Ajay

I have my english final exam in 1 hour.

Ted Shifrin

Why the dual of vector fields isn’t just $1$-forms? Both bundles are trivial. Shrug.

leslie townes

ajay: what style of exam is it? what are they testing?

anak

Well that's why I am wondering if it's not just 1-forms and quadratic forms are isomorphic, too.

Ajay

This kind of stuff drive.google.com/drive/folders/…

anak

1:55 AM

Like that it's just an isomorphism that's being brought up because it suits the needs later, rather than being something super insightful alone.

Ted Shifrin

This is too much structure for me at this hour.

anak

Can you read Russian, by the way, Ted?

Ted Shifrin

Yes, but long forgotten.

leslie townes

ajay: huh, seems like a weird mix of basic comprehension and persuasive writing.

Ajay

Its descriptive writing and persuasive writing.

Describe what you see, feel and hear in a very crowded street.

As I wandered leisurely down the street, swarms of bodies swirled around me. Stepping out of their reserved shells, exuberant adults and teenagers dressed in exquisitely crafted and snug kimonos humoured each other with high-pitched stories from work and school. Loitering young men rapidly attacked their phone screen as if determined to slay the ultimate villain at the final level in a video game. Next to the pier, young and old couples shared sphere-like lollipops while huddled together. They laughed vibrantly and unashamedly …

Ted Shifrin

1:56 AM

I am never persuasive.

leslie townes

reminds me a little bit of portions of the LSAT [exam taken in the USA to qualify for law school].

Ajay

An example I wrote

leslie townes

you will ace it.

Ajay

i'm so stressed as this whole paper is impressionistic, it depends on the marker if they want to give you the marks.

@leslietownes I Hope

leslie townes

the trend in the US is to abolish reliance on big exams but i'm personally a fan of them. it's how i got into college and beyond. if it comes down to something more holistic, they'll find out who i am.

Ted Shifrin

2:00 AM

Yup, crap.

Ajay

I like some exams, just because I can "celebrate my learning".

And because they have set mark schemes

leslie townes

ted: we went to the duck pond and saw not one but two forms of baby geese, and baby ducks. dinner (on me) is sushi. my daughter will yell about sashimi as usual before eating more than her fair share.

i have no idea how i would have changed careers without the LSAT and people relying on it.

Ted Shifrin

so munchkin knows the difference between sushi and sashimi? She wants to cut down on carbs?

leslie townes

yes, she's very particular.

anak

2:17 AM

Oh, Kirillov is just using the coadjoint representation to construct the literal smooth dual.

linear_combinatori_probabi

hola, chat

anak

hola amigo

Akiva Weinberger

3:02 AM

Is it Day of the Mother or Day of the Mothers

Apparently in Spanish both Día de la Madre and Día de las Madres are used

and in English we have arguments over where to put the apostrophe

leslie townes

3:18 AM

the person who founded it wanted it in the singular.

i dunno. my name ends with s and i've been struggling with apostrophes all my life.

put 'em where you want 'em.

Ted Shifrin

Theyr’e where they do’nt bel’ong.

Akiva Weinberger

3:53 AM

@leslietownes No commen't

leslie townes

pbhhbt

Xander Henderson

@TedShifrin Ow... that hurts my head.

Ted Shifrin

4:21 AM

Go’o’d!

Rithaniel

4:34 AM

Ah, yes, my favorite contraction: Go over there -> Go o'er dere -> Go'o de -> Go'o'd

Ted Shifrin

4:50 AM

LOL

Koro

5:21 AM

I had skipped this question also in my yesterday's exam math.stackexchange.com/questions/4446340/…

:(

William John

Hi how can I access trash?

porridgemathematics

5:40 AM

hi, im trying to understand this paragraph from Jost: Riemannian geometry and geometric analysis , imgur.com/a/Pzqz83H , page 134 of the latest edition, im familiar with the definition of sobolev spaces of vector valued functions on open subsets of $\mathbb{R}^d$, but not sure what he means by $H^{k,r}(U)$ when $U$ is an open subset of a manifold , im also unsure why he claims that $\phi_{2} \circ \phi_{1}^{-1}$ has bounded derivative on the support of $s$ in $U_1 \cap U_2$

he seems to be claiming that this support is compact in $U_1 \cap U_2$, but i dont see why this has to be the case

in his appendix he has a bunch of results for sobolev spaces on open subsets of euclidean space, but nowhere in the book does he actually define these spaces on open subsets of manifolds, although these spaces are referred to

Ajay

My exam has finished!

It was much easier than my first paper!

user400188

6:03 AM

Personally I have only ever found it to be a joy.

Any of the not so joyful stuff that can come with it tends to not be mathematics.

Koro

Can anyone please take a look at this? math.stackexchange.com/questions/4446340/…

One thought: Let $\ker \pi:=K$.

$G/K\cong H$

So $H\times G/H\cong G/K\times G/H$

Define $f: G\to G/K\times G/H $ as $f(g)=(gK, gH)$.

I aim to show this $f$ to be isomorphic but the problem at hand is that $f$ might not be onto (?) as I think that if $a\ne b$ are in G then $(aK, bH)$ is not in range f.

Even if f is onto, then ker f=$K\cap H$ ,which may not be identity so this seems wrong.

William John

I am confused about theorem 1.11.

In Rudin.

Why is $\alpha\in L$?

What concludes $\alpha\in L$?

porridgemathematics

6:21 AM

hmm, I suspect jost was speaking modulo partitions of unity and some other nonsense

dc3rd

6:31 AM

@WilliamJohn draw a picture and I guarantee you will see why that is the case.

Akiva Weinberger

6:57 AM

I'll put my apostrophes where I wan't

*whe're

Wave

Is it true that if I have a uniform variable of the unit square [0,1]^2 that then the characteristic function is the product of the characteristic function in each coordinate? So I mean that $\Phi_(X,Y)(u,v)=\Phi_X(u)\cdot \Phi_Y(v)$?

William John

@dc3rd Doesn't work.

user400188

How do you specify that a compliment of a set, is taken from another set?

e.g. I have a set $A$ which is a subset of $B$, which in turn is a subset of $C$. I want to talk about $B-A$ as the compliment of $A$, but I am unsure how to refer to $B$ as the set the remaining members are taken from, (as opposed to $C$).

Wave

7:44 AM

my question from above is now clear I could prove it!

Akiva Weinberger

8:05 AM

Desmos comes to the rescue for my graphic design needs

The same pattern (with multiplicity!) from two lines and four lines, respectively

Pretty

Both together

Akiva Weinberger

8:52 AM

Y'know what?

As much as I like that top pattern, this one does the same job more simply:

William John

9:29 AM

@dc3rd Well I guess I was bit sleepy. Now I see I asked a dumb question.

$\alpha$ is lower bound of B thus it is element of L

linear_combinatori_probabi

10:41 AM

hooooooooola

Question eligible for bounty in 22 hours

bilibili blablabla bububu

anak

12:00 PM

@user400188 Just use the notation you use, B-A. This is the complement of A in B.

(it's known as the "relative complement")

geocalc33

1:31 PM

This is a parametrization of a space curve: $S=\big(e^{-e^s},e^{-e^s},e^{-{e^{-s}}} \big)$ What minimum number of parameters do I need to introduce for the set of all curves in the parameter space to partition $M=(0,1)^3$ without intersections?

My attempt is to add real positive parameters $r_1,r_2,r_3$ so that $S=\big(e^{-r_1e^s},e^{-r_2e^s},e^{-r_3{e^{-s}}} \big)$

but I don't know how to show whether the curves in the parameter space partitions $M$ and without intersections

anak

what are these? exponents for ants?

Xander Henderson

@anak It's a model, sir.

I am giving final exams today. On both of the exams, about 1/3 of the questions are word-for-word copies of questions from previous exams. I am not optimistic. :/

anak

Yeah, I wouldn't be optimistic either.

Students so rarely check their mistakes on previous exams, it seems.

Xander Henderson

1:56 PM

On the other hand, I went to Gallup on Saturday for a rug auction. I got a fractal rug!

It's like FOUR "Two Grey Hills" in one!

anak

2:31 PM

Very cool looking!

Lucas Henrique

hi chat!

Xander Henderson

The best part is that I didn't catch it in the preview, and it came up after I didn't get the rug I really wanted (I dropped out when the bidding exceeded four times the opening bid). Because I didn't get what I wanted, I still had money in my budget for that gem.

Jack Ohara

Hello

I have a question about category theory

Lucas Henrique

2:52 PM

go ahead and ask

Jack Ohara

it is about the objects

the set up is

Fix a functor from two categories and fix an object from the source

when we consider ( x , h : A---> Gx )

G is the name of the functor here

geocalc33

Does anyone want a preview of my question before I release it on all platforms?

Jack Ohara

my question is, if we are going to define a map from two objects of the comma category

what to do with h?

Ie taking two objects of this form, ( x , h : A---> Gx ) and ( ( y , f : A---> Gy )

let me put G : D-->C is our functor

so f and h lives in C

how will f and h map to each other?

x to y will be a map in D

@LucasHenrique I hope my notation and what i am trying to say is clear, because I am a bit confused about this notion

Akiva Weinberger

@geocalc33 Try adding $1$ to $s$

Jack Ohara

3:08 PM

@AkivaWeinberger hello Akiva

Akiva Weinberger

Hello

I really like this diagram I discovered

2 lines = 4 lines

Jack Ohara

does anyone know a book of category theory with problems and solutions?

Akiva Weinberger

The figures are the same "counting multiplicity", meaning if a point is in two lines in one diagram, it's in two lines in the other

This is impossible in a finite diagram (with finitely many points of intersection)

Jack Ohara

the topic is very new to me and seeing something being solved after trying it on my own would be helpful

Akiva Weinberger

See how the way the frame crops the image breaks the top black line into three pieces

I do not @JackOhara

I know "Seven Sketches in Compositionality" is good but I don't remember if it has solutions

I tried reading "Category Theory in Context". I started getting lost towards the end… I do not know if there are solutions online, though

Jack Ohara

3:11 PM

I see thanks !

I need something mathematical not applied

also if you know of some video lectures would be of great help :D

@AkivaWeinberger what did you feel about Category Theory in Context

Is it something for selflearning ?

Akiva Weinberger

I found it useful but difficult

It has a lot of prereqs though

It assumes you know topology and abstract algebra etc because that's the "context"

Jack Ohara

oh

haha thanks , that should have been added as prereq then

from what it is written about it, the prerep are none - -

Akiva Weinberger

I mean, the definitions and stuff don't require the knowledge, but then it's like "Here's an example of this concept in ten different fields of math"

(My brother, upon seeing the title of the book, remarked that "the context for math can't be more math")

Jack Ohara

haha :D

Xander Henderson

4:02 PM

@AkivaWeinberger Your brother is wrong. :D

Akiva Weinberger

Emily Riehl agrees, obviously :P

Xander Henderson

Also, no one has shown up yet for this morning's final exam. Which means that they will need to show up either tomorrow, or on Thursday. But, I get the rest of the morning off. Yay!

Akiva Weinberger

(The bold in the image is from using Twitter search to find it)

I don't think I've had an exam where I get to choose the day

Xander Henderson

@AkivaWeinberger Two of the classes I am teaching this semester are asynchronous online classes, but the "guidance" from the state is that we must have at least 50% of the final grade in a class be determined by proctored assessments. Thus I am proctoring final exams. But people work, have other classes, have lives, etc, so I have given them three choices.

I am, frankly, not surprised that no one showed up at 8:30 on a Monday morning. But today is the morning I had time.

I am guessing that there will be a few people tomorrow afternoon, and a lot Thursday evening.

Jack Ohara

@XanderHenderson what are you teaching?

Xander Henderson

4:08 PM

(Where "a lot" is still a small number, given that enrollment is so very, very low this semester.)

Jack Ohara

and do you have room for one more student? :D

Xander Henderson

@JackOhara Calc and precalc. And the semester is nearly over.

Jack Ohara

oh :(

I know those topics anyway :D

Category theory course would be much appriciated

Xander Henderson

I had a guy take calculus from me once, after having tested out of it via AP exams. He was looking for an easy A.

Jack Ohara

as online course

Xander Henderson

4:09 PM

He barely managed a C+.

Jack Ohara

well any area of math needs to be taken seriously or you cant ace it

Xander Henderson

@JackOhara His problem was that he didn't believe that college calculus was going to be different from high school calculus.

Jack Ohara

I did make that mistake before as well !

Xander Henderson

"Okay, guys, the derivative of the function $x \mapsto x^3$ is given by $$\lim_{h\to 0} \frac{(x+h)^3 - x^3}{h} = \lim_{h\to 0} \frac{x^3 + 3hx^2 + 3h^2 x + h^3 - x^3}{h} = \lim_{h\to 0} 3x^2 + 3hx + h^2 = 3x^2,$$ cool?"

"BUT THERE IS A SHORTCUT! YOU JUST BRING THE 3 DOWN AND DECREASE THE EXPONENT BY 1!!!!"

Akiva Weinberger

…Is that not covered by the AP exam?

leslie townes

4:12 PM

the AP curriculum is particularly bad in that it conveys to many the idea that it contains everything one needs to know.

Xander Henderson

"Okay, can you prove it?"

"WHY DO I HAVE TO PROVE IT?! MY HIGH SCHOOL TEACHER TOLD IT TO ME!"

Akiva Weinberger

There's something a bit odd about the unruly student arguing for not questioning authority

Xander Henderson

"Okay, well, until you can prove it (or until I have proved it for you), you don't get to use the Power Rule, because you need to have a deeper understanding of what a derivative actually is..."

@AkivaWeinberger Oh, that happens a lot. "That isn't the way my high school teacher taught it, so you must be wrong."

Akiva Weinberger

It's been ages since I took that test but I thought the limit definition of the derivative was on it. I guess not?

Xander Henderson

I think that this accounts for a lot of the pushback against Common Core in the US, too---"That isn't they way I learned it, so why are you teaching it to my kids all different?"

@AkivaWeinberger It probably is, but my guess is that it gets rushed through pretty quickly.

And that students aren't made to do very many basic examples.

leslie townes

4:14 PM

the limit definition of the derivative was definitely part of the AP curriculum when i took it. it did not play a central role.

Akiva Weinberger

Of course, we all know the real definition of the derivative is $f'(x)=f(x)y-yf(x)$ in the noncommutative ring where $xy-yx=1$.

(It's a neat exercise to show that that actually works, at least for polynomials. The main clue is to check that $fg'+f'g=(fg)'$.)

Xander Henderson

Yeah, yeah, yeah. Lie brackets and other abstract nonsense.

:P

northerner

4:34 PM

Hello

Given a numeric value that changes once each day, what are some ways that a person can get a sense of how that number changes between two specific dates? I was thinking the average of the difference between adjacent days.

Simplicity would be preferred as it would be used by an average person, not a specialist.

leslie townes

it might depend on what the number is. just the difference itself might be meaningful and most helpful. people often like to report percentage changes, which is kind of interesting, because percentages can be confusing, particularly time series of percentages.

Derivative

Can someone give me an example of a complete algebraically closed field other than the complex numbers?

1

A: Do the properties of metric completeness and algebraic closure characterize the complex numbers up to isomorphism?

I cannot find the topic where I thought I saw this a couple days ago, so here it is: Short answer: No. There are infinitely many fields which are complete and algebraically closed and not isomorphic to $\Bbb C$. You want a minimal algebraically closed, complete, archimedean field containing $\...

maths researcher

4:59 PM

Could one do that $H_k(\mathbb{S}^1\vee \mathbb{S}^1)=H_k(\mathbb{S}^1)\oplus H_k(\mathbb{S}^1)$ without appealing to good pairs

because, $\mathbb{S}^1\vee \mathbb{S}^1 = \mathbb{S}^1\cup \mathbb{S}^1$ where $\mathbb{S}_1 \cap \mathbb{S}^1$ is a point

I am identifying $\mathbb{S}^1$ with its image under canonical map $\mathbb{S}^1\rightarrow \mathbb{S}^1\vee \mathbb{S}^1$

Ted Shifrin

Every Calc I university course makes the students do a derivative from the definition (even on the final). But not on the AP. With only 5 free response questions, it’s just not worth it.

@mathsresearcher I don’t understand the question. So you want to use the axioms but not Mayer-Vietoris? And you mean $k>0$?

Prithu biswas leftmse

5:22 PM

Does anyone know about this theorem? [in green]

Akiva Weinberger

@mathsresearcher Let $X_1$ be $S^1\vee S^1$ minus a point in one of the circles, and $X_2$ be it minus a point in another

and then they're open sets and Mayer-Vietoris applies

and also those things are homotopic to circles

@Prithubiswasleftmse What's the definition of convex?

Surely $\begin{cases}-x+1, & x\le-1 \\ 0, & -1\le x\le1 \\ x-1, & 1\le x\end{cases}$ is a counterexample, no? The tangent at $0$ is the $x$-axis

Ted Shifrin

What does “no good pairs” mean? Whence my question.

Akiva Weinberger

@Prithubiswasleftmse

Prithu biswas leftmse

@AkivaWeinberger A function f is convex on an interval, if for all a and b in the interval, the line segment joining (a,f(a)) and (b,f(b)) lies above the graph of f.

Ted Shifrin

Convex for Spivak is strictly convex.

Akiva Weinberger

5:29 PM

Ah. So my example is only weakly convex and you want strictly convex

I suppose the key is to show that the (infinite) line joining (a,f(a)) and (b,f(b)) lies below the graph when x is not between a and b

Ted Shifrin

He is also talking about everywhere diff here, I think.

Akiva Weinberger

and then take the limit as a and b kiss

It shouldn't matter I think @TedShifrin

Last night, it finally happened - my water bottle cap wasn't tightly screwed before I put it into my backpack

Ted Shifrin

Marinated textbooks?

Akiva Weinberger

The real worry was my laptop, but it seems fine miraculously

Ted Shifrin

@Prithu Look in the later editions. He proves it there.

Akiva Weinberger

5:35 PM

I'm still unreasonably proud of this picture

2 lines = 4 lines

This is the simplest way I could figure out how to make that happen

Ted Shifrin

Still makes no sense to I.

Akiva Weinberger

In the top, I have two lines (black and red); in the bottom, I have four lines (two black and two red)

The figures are the same "counting multiplicity" (if a point is in two lines in one figure, then it's in two lines in the other)

This (breaking the same figure into different numbers of lines) is impossible if you have finitely many points of intersection

(see how the cropping of the image breaks the top black line into three pieces)

(with some light manipulation - cutting it off at the left and compactifying it at the right - you can turn it into compact arcs (three at the top and two at the bottom))

Ted Shifrin

Aha …

Akiva Weinberger

Also, "thickening up" the lines gives you an answer to my open disks question that I asked on main

^Trivial variation

Akiva Weinberger

6:06 PM

Alternate version (doesn't turn into the compact version as nicely though)

(also, even though the decomposition is technically simpler, I'm not a fan of the colors… Desmos doesn't have very good options)

Final form maybe?

Akiva Weinberger

6:46 PM

Joe Shmo

7:36 PM

what does this answer?

Xander Henderson

@JoeShmo If I understand correctly: divide the plane into some number of subsets. Given any point in the plane, that point will be contained in some number of those sets. Now divide the plane into a different collection of subsets so that if a point is contained in $n$ of the original sets, it is also contained in $n$ of the new sets. Is this possible? How?

The sets @AkivaWeinberger is considering are all "nice" curves.

And the pictures are purty.

Joe Shmo

they are.. looks like a coloring problem?

quite literally :-)

it resembles his question from the other night, which come to think of it might itself be a coloring problem

Xander Henderson

@JoeShmo Coloring problems, as I understand them, are typically about coloring graphs. This seems to be somewhat distinct.

Joe Shmo

yeah yeah but you might be able to construct a graph from all this, although it doesn't look obvious

Xander Henderson

It depends on how @AkivaWeinberger has phrased the problem.

Joe Shmo

7:45 PM

although, since clearly the pattern repeats, each node would be a period, etc.

Xander Henderson

As I see it, it is about subsets of the plane---the edges matter, as well as the points of intersection.

Joe Shmo

yes, but although I don't know what I'm talking about, I suspect you might be able to do something with (planar) graph theory

his question from the other night was $X_i, Y_j \sim \mathbb{R}^2$ are homeomorphic, $\sum_{i=1}^m \mathbf{1}_{X_i} = \sum_{j=1}^n \mathbf{1}_{Y_j} \implies m = n$, where each $X_i, Y_j$ are open intervals essentially, so for example you might correspond a node to each interval, and then maybe edges go between overlapping intervals, and then by graph theory magic (the graph is bipartite?) blah blah.. $m = n$. Or not.

Akiva Weinberger

For this one in particular I have $X_i,Y_i$ homeomorphic to $\Bbb R^2$

Joe Shmo

sorry, not intervals

maybe suffices to consider $\mathbb{R}$?

open balls

including balls with infinite radius

Akiva Weinberger

I mean, $\Bbb R^2$ and the open unit disc ($\{x\in\Bbb R^2:\|x\|<1\}$) are homeomorphic

I also asked a version where they're all homeomorphic to $[0,1]$. That's a bit trickier, but I think you can do this sort of thing:

Joe Shmo

7:52 PM

yeah, so what?

Akiva Weinberger

Oh, sorry, I misread what you said

Yeah, you're all good

^That image has two compact arcs

2

Q: Can you tell how many open disks are added together?

As a disclaimer, I know the answer to this question; I'm sharing it here because I think others may enjoy tackling it. (Notation: for $X\subseteq\Bbb R^2$, $~\mathbf1_X$ is the indicator function of $X$, defined by $p\mapsto\begin{cases}1,&p\in X\\0,&p\notin X\end{cases}$.) Suppose $X_1,\dots,X_j...

14 mins ago, by Xander Henderson

@JoeShmo If I understand correctly: divide the plane into some number of subsets. Given any point in the plane, that point will be contained in some number of those sets. Now divide the plane into a different collection of subsets so that if a point is contained in $n$ of the original sets, it is also contained in $n$ of the new sets. Is this possible? How?

@XanderHenderson Your description of the problem is so much clearer than mine was, lol

I should have phrased it like that

Joe Shmo

wait so whats the answer to the question from the other night?

Xander Henderson

@JoeShmo Language, please.

@JoeShmo What question?

Joe Shmo

that was to Akiva....

maths researcher

@TedShifrin No. I want to use Mayer vietoris. My decomposition of $\mathbb{S}^1\vee \mathbb{S}^1$ is $\mathbb{S}^1\cup \mathbb{S}^1$ such that $\mathbb{S}^1\cap \mathbb{S}^1$ is the wedged point.

Akiva Weinberger

8:09 PM

@JoeShmo Which one

Joe Shmo

homeo subsets

Akiva Weinberger

Homeo to R^2? It's what I drew

In one figure we have 4; in the other we have 2; their sums are the same, therefore $j$ and $k$ don't need to be equal

Joe Shmo

ok, where did you get that from

I see

Ted Shifrin

@mathsresearcher So you do the usual thing of “thickening” up to get open subsets. One circle union an open arc of the other, etc.

maths researcher

yupp

@TedShifrin

Ted Shifrin

8:20 PM

So what’s your question?

ioch

9:43 PM

is there no variant of cauchy schwartz in a riemannian manifold

ignore that

geocalc33

10:03 PM

can one equip the space $\scr S^1$ of smooth convex curves s.t. f(0)=0 and f(1)=1, with a natural algebraic structure?

Akiva Weinberger

10:14 PM

What do you mean by algebraic structure?

Subtracting $x$ from everything gives you convex curves with f(0)=f(1)=0, which might be easier to think about

I suppose it's an like an affine subspace

Well…if you do curves that are pure convex or pure concave maybe

geocalc33

I think it could be a monoid

$y=x^n$ is convex on $(0,1).$ and $x^n x^k=x^{n+k}$

so that means you can associate the monoid structure $(\Bbb R_+,\times)$ to the class $y=x^n.$

and then I'm guessing that you have all these monoids attached to every class. then prove that they are all homeomorphic?

I don't know

Akiva Weinberger

10:36 PM

It's also a convex set

in the sense that a convex combination of convex functions is convex

If $f$ is convex, is $2f-x$ convex?

I suppose what I want to say is that it's a "translate of a convex cone"

Alek Murt

11:14 PM

Hello guys, guys I'm trying to study convex conex and I would like to see some good examples, by any chance someone has a good reference for this ?

Ted Shifrin

So you can draw all sorts of pictures, @Alek. What specifically are you having trouble with?

Alek Murt

I'm struggling a lot with this cone $K_{d}^{\mathrm{lc}}=\left\{p \in \mathbb{R}_{d}^{\mathrm{lc}}[x]: p(x) \geq 0\right.$ for every $\left.x \in\{0,1\}^{n}\right\}$ where $\mathbb{R}_{d}^{\mathrm{lc}}[x]$ are the free square polynomials. I would like to prove that it has a compact base and talk about the extreme rays. But i feel kinda lost with this, so I would like to go in more detail with some more theory

Ted Shifrin

Polynomials of fixed degree?

I don’t know a source for general theory.

Alek Murt

11:31 PM

degree is bounded by $d$. Finding some reference for general theory seems quite hard, I have read Boyd's book, and Barbinok's book in this topic but I feel like is not enough

Akiva Weinberger

What does $\rm lc$ stand for?

(Free square… do you mean square free?)

Ted Shifrin

So the unit sphere in finite dimensions (which this now is) is compact. The cone is closed.

Alek Murt

Yes I mean square free, srry for that

The "lc" thing mean square free for me, I labeled like that XD

user400188

@anak Ah, thank you, that was the term I was looking for :)