Mathematics - 2024-01-22

Jakobian

00:03

Let $X$ be a non-empty set and $\mathcal{F}$ be an upward closed family of non-empty subsets of $X$. I think that there always is a topology on $X$ for which $\mathcal{F}$ is the set of dense subsets

Ted Shifrin

@SineoftheTime rate of change of volume under flow?

Ted Shifrin

00:29

This is also related to the Divergence Theorem.

Jakobian

I never read a proper proof of Stokes theorem

Juliamisto

02:23

Imagine that you meet a question with two parts, where part 2 is harder than part 1 and is a slight generalization of part 1. Suppose further that you have solved part 1 but you have difficulty in solving the other. What should you do?

The question was inspired by math.stackexchange.com/questions/4848470 .

Jakobian

02:50

vague

Utkarsh

04:33

Hi I'm confused how to solve $\sum_{i=1}^5 \frac{1}{x^i} = 3.56$. Can anyone suggest any method?

Ted Shifrin

05:04

@Utkarsh Use the formula for summing a geometric series.

Although that gives a tough equation. Where did this come from?

Utkarsh

@TedShifrin Thanks. This is used to find Internal Rate of Return (IRR) often used in Financial Analysis.

Ted Shifrin

You just have to use numerical methods, then. Let your calculator or computer solve it.

Utkarsh

Actually there is no scientific calculator allowed in exam. So I just have to compute it manually.

Ted Shifrin

There's no way.

Utkarsh

Just a basic calculator with +,-, x, /, and sqrt() functions is allowed.

Ted Shifrin

05:13

Then they must know some tricky math that I don't know.

Presumably you were taught this, if you're supposed to take an exam.

Utkarsh

I'm a self learner and there is no calculator trick mentioned in my book. So I'm just looking at the internet for some help.

Ted Shifrin

Oh, no, I take it back. You will still need a calculator with more functions.

The formula for $\sum_{i=1}^n r^i = \frac{r^{n+1}-1}{r-1}$ is something you need to know cold.

That's what I said in my first response.

It still turns into something you can't solve, though.

The book must show you how they expect you to solve such things.

Utkarsh

Yeah.

Ted Shifrin

You end up with the equation $2.56x^6-3.56x^5+1=0$. You need Newton's method or a fancier calculator.

Soumik Mukherjee

The solution is 9/8, but I don't see any direct way to get there

Utkarsh

05:20

@TedShifrin I think I can atleast find an approximate value by hit and trial method.

Ted Shifrin

No way in hell.

My equation is wrong.

Utkarsh

Then I think I should consult my friends who are also preparing for this exam.

Ted Shifrin

Yes, precisely.

The equation is right. $x=1$ is an obvious root. We need to get rid of that one and then numerically find a root near $1$.

Utkarsh

05:38

@TedShifrin if it is given that the value of $x$ lies between $1.1$ & $1.13$ then is it possible to solve it? Using interpolation method?

leslie townes

there are all sorts of numerical methods, but in a classroom context you would want to use exactly what they expect you to use, and not some internet thing.

finding IRR is one of the best examples of a problem that more or less instantly requires numerical methods. there are probably a handful of back of the envelope ones that business students use, but no universal standard.

which always raises the interesting question, what happens if you answer numerically with a result that is closer to the exact answer than the method you were expected to use. sometimes, nothing good comes from that

Utkarsh

I saw there are annuity factors given on the back of the book. And they used them to approximate the value for $x$. Then probably I can use interpolation (I'm not aware about any other numerical methods). This is not a classroom test or a university exam BTW. They have given values 3.791 at 10% rate and 3.433 at 14% rate. And I want to calculate rate at which it will be 3.561.

Ted Shifrin

So then do a graphing calculator or learn Newton’s method.

copper.hat

05:59

Or bisection.

Ted Shifrin

06:16

Numerics needed one way or another …

Dannyu NDos

07:11

Is $\text{cis} : \mathbb{Q} \to S^1$ an imbedding?

Like, $\text{cis}(x) = \exp (ix)$

Nvm, it's easy to see it's a continuous injection.

It's counterintuitive enough, tho.

Now I have a standard way of topologizing algebraic closure of prime fields.

oscarmetal break

07:40

Hi, after computing a faithful representation of a group, how can I show this is unique up to isomorphism?

For example, the group of quaternion

user unkown

08:02

Show that there is an absolute constant $c$ such that for each $k>1$, every
tournament that satisfies $S_k$ has at least $c \,k2^k$ vertices.
Hint: Show that for every set $W$ of at most $k-1$ vertices, there are at least $k+1$
vertices that all point to each vertex of $W$.

Any hints on proving the hint, and any ideas on how the hint relates to the main proof.

MXO120

Let $T \in End(\mathbb{R^3})$ be the linear operator such that $T(e_1) = e_2, T(e_2) = e_3, T(e_3) = e_1$.
Can you find a vector $v$ such that the cyclic subspace for $T$ generated by $v$ is not the whole space $\mathbb{R^3}$ and $v \neq e_1+e_2+e_3, 0$?

leslie townes

08:17

MX: any nonzero multiple of e_1 + e_2 + e_3 would also work, as would e.g. any nonzero multiple of v := e_1 - 2 e_2 + e_3 (note that T^2 v = -v - Tv)

MXO120

08:43

@leslietownes So basically any vector with coordinates x, y, z that satisfies x + y + z = 0. Is that correct?

leslie townes

that is a way of describing my additional set of examples that maybe makes it clearer that they are examples, yes

Lucky Chouhan

09:18

Hello @robjohn , How are you doing?

Summerday

10:15

can someone help me?

0

Q: How can I show that $y^2+y=x^3-x^2$ is elliptic over $\Bbb{Q}$?

I have given the curve $E: y^2+y=x^3-x^2$ over $\Bbb{Q}$ and I want to check this is elliptic. I first define $f(x,y)=y^2+y-x^3+x^2$ then the curve is given by $f(x,y)=0$. Now I need to show that all points on $E$ are smooth. But for this we first need to homogenize $f$, so $f(x,y,z)=y^2z+yz^2-...

PNDas

10:41

What is $E_0^1(\Omega)$?

Any ideas?

Jakobian

11:11

@DannyuNDos continuous injections are not embeddings

You should be able to obtain a sequence $x_n\to 1$ in the image such that preimages diverge to infinity

user unkown

Can anyone explain why $$\mathbb{P}(e\text{ is not rainbow})=4\,\left(\frac{\color{red}3}{4}\right)^k=\frac{{\color{red}{3^k}}}{4^{k-1}}\,.$$ in this answer math.stackexchange.com/questions/3758906/…

Jakobian

Just choose $x_n$ to be in $\mathbb{Q}\cap [2\pi n,2\pi(n+1)]$ with $|1-\exp(ix_n)|\leq 1/n$

robjohn

11:32

@TedShifrin the sum should start at $i=0$ or the formula should be $\frac{r^{n+1}-r}{r-1}$

user unkown

11:45

@robjohn Could you please explain this?
Why is $$\mathbb{P}(e\text{ is not rainbow})=4\,\left(\frac{\color{red}3}{4}\right)^k=\frac{{\color{red}{3^k}}}{4^{k-1}}\,.$$ in this answer math.stackexchange.com/…

Summerday

13:59

@Thorgott can you help me with my elliptic curve question?

Ted Shifrin

15:02

@robjohn Oops. Yup.

robjohn

@Ted Raining there?

I’m delaying my dogs’ walk this morning, hoping it stops for a bit.

Xander Henderson

15:22

Raining here, too!

Yay!

Oh, no... I lied. The sun has come out in the last five minutes. Drat.

robjohn

@XanderHenderson Wow! Is there a lot forecast there? Oh, sorry.

Xander Henderson

But it rained most of yesterday, too, so we did get some water.

Which is needed.

And the forecast is for scattered showers over the next couple of days, too. And snow in the mountains.

Thorgott

@Summerday no

robjohn

15:39

@XanderHenderson we had no rain yesterday, but ½” Saturday. We’re expecting about 1” today.

Xander Henderson

@robjohn That is positively BIBLICAL. :D

robjohn

We’ve had ⅚” overnight, and maybe ¾” more, so actually perhaps 1½”

Xander Henderson

(In the summer, we sometimes get an inch or two in half an hour, but we've apparently only managed about a tenth of an inch in the last two days).

It might be better to say that we have had "heavy fog" for the last two days. :D

robjohn

The forecast keeps saying it will stop in 30 minutes or 1 hour, but they lie; it keeps going. My dogs are getting restive.

We may have to walk while it’s raining and then we have to dry them off after.

Xander Henderson

The dogs'll love it! Wetness! Water! Cold! Yay!

Ted Shifrin

16:22

@robjohn indeed. Yesterday when I drove north to the farmers msrket there were 5 cars either off the road in the mud or backwards on the freeway. Idiots.

@robjohn They needed a bath anyway?

Soumik Mukherjee

16:34

300+ AQI here. The smog is terrible.

Xander Henderson

16:53

The AQI here is... 12.

Oh, it's 2 in Heber (45 minutes south of here), and 8 in Winslow (30 minutes to the west).

leslie townes

xander you should hawk real estate down there. you know what they say, only three things matter in real estate, AQI, AQI, and AQI.

robjohn

17:15

@XanderHenderson Holly doesn't mind water. Rosie prefers dryness.

Xander Henderson

@leslietownes What about location?

leslie townes

only matters to the extent it effects AQI

robjohn

@TedShifrin Driving on the 138 at night (2 lane desert highway with no lights), I was changing lanes and did not realize that the road was flooded there. I ended up doing a 180 and ending up off the side of the road. Kind of scary as there was a big truck behind me in the road. Luckily, my car maintained speed until I was off the road.

I guess it was a 4 lane highway; 2 lanes in each direction

Hasini

Hi. A set is a collection of well defined objects. Then when we have, A= even integers and B=beautiful singers, A is a set and B is not a set because B has no well defined definition. Is this right?

leslie townes

hasini: at the level of 'naive set theory' (without formal definitions and rules about what counts as a set), that could probably be either right or wrong. i could certainly see a textbook using that as example of the need for care in defining a set.

sometimes textbooks will use things like "the set of all people in the room" to illustrate aspects of sets in contexts where the elements have no mathematical interpretation that might distract from the point they are trying to make. that

's also an example of something that could go either way, but in that context, it would be understood that there is a particular room under discussion, and nobody is entering or exiting it for purposes of the discussion

similarly in your example i could imagine a situation where you did have, for purposes of some fixed discussion, some objective test of what a 'beautiful singer' was, and in that instance it would be a fine example of a set.

robjohn

17:27

@leslietownes like a room with no doors or windows?

Hasini

Hmm okay. @leslietownes if there is an objective test/ criteria by which we can determine the elements of the set, we can call it as a set?

Xander Henderson

@leslietownes Or even a subjective test---"singers the constructor of the set considers beautiful". At the level of English, what is a set and what is not a set can be very fuzzy.

(Which should not be confused with fuzzy sets---those are different).

leslie townes

yeah. the broader point is that to define a set means (one way or another) to be able to decide the question of whether something is or is not in the set, and merely intending to do that with some verbal formula is sometimes not enough.

robjohn

the set of fuzzy things...

Hasini

Hmm okayy

Xander Henderson

17:29

@robjohn Oh, that, again, is something else.

But it certainly includes kittens and tribbles.

Hasini

If there is no objective test/ criteria, can we call it as the empty set rather than saying it is not a set?

robjohn

watching kittens, one gets a feeling for fuzzy logic

Xander Henderson

@Hasini No---if there is no way of determining whether or not some object is an element of a set, then you have not defined a set. You should be able to point to a singer and answer the question "is this singer beautiful?" with either a "yes" or "no".

leslie townes

hasini: it would be unusual to use the terminology of set theory, even 'naive' set theory, to describe things that weren't clearly defined. you could do that, but it would be an unusual use of language, and it would maybe be a better practice to not use 'set' language

Xander Henderson

Though, funnily enough, the notion of a "fuzzy set" might be more appropriate to this kind of subjective definition, where membership can be... fuzzy.

Hasini

17:32

Hmm okay Thanks a lot @leslietownes and @XanderHenderson and @robjohn

leslie townes

the set of all true scotsmen

Xander Henderson

@leslietownes T'ain't no such thing!

Hasini

In the definition of a set, should I specifically say as "A set is a collection of well defined distinct objects" rather than "A set is a collection of well defined objects"? Or is the "distinct" part implied through the second version already?

Xander Henderson

I would say that your emphasis is in the wrong place: A set is a well-defined collection of objects.

And yes, the objects are distinct, so it might be helpful to add that in there.

Hasini

Ahh okay :)

What is to be said that if someone says the empty set doen't have a collection of elements?

leslie townes

17:36

at some point you have to either stop doing naive set theory, or accept that a naive "definition" is not going to resolve every issue. the empty set puts a lot of pressure on intuition that people have even when they formalize stuff.

this thing about elements being "distinct" is already showing that. there's like a paragraph of explanation about what that word is attempting to capture. i'm not sure that someone unfamiliar with mathematical sets more generally would understand what it is doing.

Hasini

Where can I see a paragraph of explanation?

leslie townes

which isn't to say that it isn't helpful to have it there, just... these word formulas aren't that helpful for understanding details of set theory.

sorry, i was speaking figuratively. i believe the point of the word "distinct" there is that as it is used in math, the set concept does not encode "repetition" of an element. the thing that captures the idea of "okay, i have a bag that includes 5 identical copies of the number 1" is not "the set with five 1s in it" but something like a "list" which in math is thought of as different from a set.

Hasini

Okay. Thank you very very much again!!!

Have a nice day all of you!

leslie townes

which then gets into, okay, but many people conceptualize of lists and sets as being the same thing. or a set is a list "where order and repetition don't matter." and a whole lot of people internalize order in their mental "sets" even though the math concept forgets that.

put that together for a paragraph :)

Xander Henderson

@leslietownes A list is as set...

...of ordered pairs, where the first element is a natural number.

BUT WHAT IS A NATURAL NUMBER?!

IT'S SETS, ALL THE WAY DOWN! :(

leslie townes

17:43

"X is 'really' this specific set-theoretic encoding of X" is what people do before they learn category theory. it is a gateway drug to that.

Xander Henderson

@leslietownes Look, a "set" is just an object in the category of something something something... (gawd, I hate myself...)

robjohn

I'm against gory cats

set or not

leslie townes

hasini has probably left now but the word "collection" in "a well-defined collection of [distinct] objects" is also doing a tiny amount of work in the naive set theory setting. books use it because it does not suggest any inherent order (like "group" or "bag of", but not like "list"). but again i'm not sure how clear that actually is to someone who isn't already familiar with mathy-math sets.

Peter

Why is the tool factordb totally overloaded during the last few weeks ?

Jakobian

set of ordered pairs is product of sets and not a list

Xander Henderson

17:51

@Jakobian It's a good thing that I didn't say that any set of ordered pairs is a list, then, isn't it? It's also a good thing that I was being totally serious and attempting to give a 100% pedantic definition, and not making a joke, eh?

leslie townes

jakobian didn't cotton on to what you were "really just" doing

Xander Henderson

Seems so. It happens a lot with him. He likes to tell me that I am wrong.

Ted Shifrin

@robjohn We're turning into a lake near me now.

Jakobian

wait I'm wrong

a list is a set of ordered pairs

leslie townes

xander: just remember that you're not wrong, you're "just" right in the opposite category

robjohn

17:52

@TedShifrin The rain has lightened here, but not stopped. On streets with no storm drains, the gutters are rivers.

Ted Shifrin

My neighborhood in SD notoriously floods the worst. I had to go out to go to the chiropractor. I did make it back :)

@leslie You mean he is contraindicated.

Xander Henderson

@leslietownes Oh, golly, now I hate myself even more. :(

Jakobian

If people are abolishing things like law of excluded middle then they should abolish opposite categories as well. Opposite category? Which one?

leslie townes

munchkin got to see a few cool splashes as i hit deep puddles on the way to school this morning, but thankfully no flooding along that route. i did see someone almost plant themselves in a ditch

jakobian: reverse some of the arrows

Jakobian

there should be something called fuzzy category theory where there can be multiple different opposites

Ted Shifrin

17:55

As I mentioned earlier, yesterday when I drove north to the farmers market, there were at least 5 cars off in the mud, some turned backwards on I5.

The Prius's low construction is not good in flooding situtations. My Honda is better, but not like a Ford pickup.

Xander Henderson

@TedShifrin Yeah, that is the biggest problem I have with Prii---they are too close to the ground. My next biggest complaint is that visibility is poor (the back window is a joke, and the A bar is in an annoying playce.

robjohn

@TedShifrin I have a Lexus RX 450h, much better clearance than my son's Prius.

Ted Shifrin

Yes, robjohn ... You drive a behemoth.

robjohn

@TedShifrin Actually, it is small for an SUV. Much smaller than some of the behemoths

It fits in compact spots.

Ted Shifrin

Hell, my Civic fits in compact spots, but then I can't get out of it.

Some of the parking lots have gotten ridiculous and the cars are too big.

leslie townes

18:00

the honda civic is the perfect car

robjohn

My wife has a Lexus ES 300. I have a hard time getting in and out of it.

Xander Henderson

@leslietownes I like my little orange Prius-C.

Second favorite car I've ever owned.

Ted Shifrin

@leslie My 2015 is the size of the Accords from 15 years ago.

leslie townes

xander: your favorite being the lifted f-350 with punisher and "thin blue line" decals all over it?

Xander Henderson

@leslietownes Oh, gosh, no. I've never had one of those. I had an older Tacoma for a while (it was my father's; I got it when he died).

leslie townes

18:03

ted: yeah, that's my one gripe with the civic (even toyota got rid of the prius c)

Xander Henderson

My favorite car was an early Nissan Leaf.

leslie townes

xander i have a prius c too (2015, in red). the other day i was taking an uber to pick it up from the mechanic, and the uber was a prius c (2014, in white).

Xander Henderson

@leslietownes Mine is also a 2015!

Whee!

But orange.

Balarka Sen

Hi, all

leslie townes

wow, balarka!

Ted Shifrin

18:06

My 2002 Civic 5-speed was probably my favorite car amongst the ones I've owned.

Wow, it's a Balarka!!! I mentioned you a few days ago.

Xander Henderson

@TedShifrin You summoned him, it seems.

Balarka Sen

Hi, Leslie, Ted! Hope you're doing alright

@TedShifrin Huh, let me check.

Ted Shifrin

It was all good, Balarka. Don't worry.

Xander Henderson

Jan 17 at 22:32, by Ted Shifrin

After I gave him enough comments, Balarka — whom we miss seeing — turned into a great teacher here!

Balarka Sen

Thanks, Dr Ted! That's a kind compliment.

(Unlike a kind complement)

leslie townes

18:07

balarka is a very slow-arriving demon. say his name once, and a few days later, look out

Ted Shifrin

That was I, silly goose.

Balarka Sen

I thanked Xander for saving me the effort to namesearch :)

Ted Shifrin

Oh.

Balarka Sen

Too many people to thank

Xander Henderson

@TedShifrin I fixed it for you. :P

Ted Shifrin

18:08

Have you finished your thesis yet? :D

Balarka Sen

Which one? It's only been a couple months since I officially signed up for a PhD! I finished my Masters' thesis, if that's what you meant.

Ted Shifrin

Oh, no, I meant Ph.D.

Balarka Sen

I have some published results, trying to prove more things for my PhD.

Taking my time

Ted Shifrin

So are you ending up doing topology/geometry?

Balarka Sen

Yep, I am thinking about symplectic/contact topology now.

Ted Shifrin

18:10

Very cool. Mike might even be proud.

Balarka Sen

During my MSc days I thought about h-principles (which is what my published work is about), which is not too unrelated but I got bored with that.

Ted Shifrin

Well, you were obsessed for a few years.

Balarka Sen

@TedShifrin I talk to Mike regularly! He's doing well, watching a lot of movies of late.

Hah, true!

Ted Shifrin

Mike used to keep in touch a bit, but no longer.

Balarka Sen

I think he may finally have some time to get around to that.

But hard to tell

Ted Shifrin

18:13

He used to bounce teaching ideas off me, but I think he realized my style doesn't fit his that well :D

I hope he's doing well.

Balarka Sen

Ah, well. He gets very involved with teaching and occasionally becomes gloomy when some (from what I can tell, only a handful) of his students don't do well. I tell him he can't help everyone, especially those not receptive to learning, but...

Ted Shifrin

As I've said before, one of the reasons I retired early was that I was no longer motivating all my students to work hard — some, yes, but some not.

Balarka Sen

Right, it's stressful.

Speaking of teaching, today I gave a student seminar talk where I proved every closed oriented 3-manifold admits a contact structure.

Ted Shifrin

Have you done any official teaching yet ... or just all your seminar talks? :)

Balarka Sen

Just talks, I am afraid.

Ted Shifrin

18:17

Hmm, that's something I once knew how to prove.

No need to be afraid ... if you're doing an exemplary job :D

Balarka Sen

I did some origami to explain the "Thurston jiggling lemma": given a distribution on a 3-manifold and a triangulation, one can subdivide the simplices and C^0-perturb the new simplices a little bit so that the 1- and 2-simplices are transverse to the distribution.

It was fun, people seemed to like it.

Ted Shifrin

a 2-d distribution

Balarka Sen

Right, I should have said that. Plane distribution.

In a month I have a 30 minute research talk in the department where I explain some stuff I have been working on to a general math audience. Slightly daunting.

Have to make slides and whatnot

Ted Shifrin

The 30 minutes makes it dauntinger. An hour would be better.

Balarka Sen

Yeah.

Ted Shifrin

18:25

@robjohn Just saw this. Lucky you weren't hurt!

robjohn

@TedShifrin Yes, it was a dark and stormy night. My car needed realignment, but that was all that happened.

Ted Shifrin

I'm too old for this much excitement.

robjohn

this was 6 or 7 years ago

Ted Shifrin

When I was in my 20s, I enjoyed driving in blizzards and through feet of snow. No longer.

robjohn

what kind of shoes does one put on feet of snow?

snowshoes?

Ted Shifrin

18:30

Feet of clay ...

Xander Henderson

@TedShifrin Nope, I've never enjoyed that. My first car (which I started driving when I was 14) was a terrible, terrible Mustang. One of my earliest memories of driving was coming to a stop sign, stopping, and then sliding backwards down a hill.

*shudder*

Ted Shifrin

I learned on a 66 Dodge Coronet. But drove a Saab 96 from senior year of college until I finally gave up on it (repeated fuel pump issues, some on cross-country trek) in GA in 1983.

robjohn

In Princeton, a friend was driving a few of us to a movie. He turned to go into the driveway, but we kept moving in the same direction, sliding on the black ice, completely out of control. Luckily, we came to a stop before hitting anything.

Ted Shifrin

But with radial tires it was great in snow ... I even drove through serious snow in the Sierras with no chains (although I was afraid the cops would stop me).

Black ice is treacherous.

robjohn

We actually had some here a few weeks ago, and I almost slipped on it. So did my dog.

Ted Shifrin

18:35

Wow. It gets cold enough where you are?

I guess you forgot to put studded snow tires on your dog.

robjohn

There was some on the walkway under the math building (Fine Hall). I remember saying hi to the department secretary, then having her staring down at me as I came to on the ground.

Ted Shifrin

Slips like that (I had some in Athens wearing flip-flops after rain) lead to bad damage to back disks.

robjohn

@TedShifrin It got to 32°, but horizontal surfaces can get lower than ambient due to heat loss to the sky. It was cloudless and water from the sprinklers had frozen on the sidewalk.

Ted Shifrin

@robjohn Ah.

robjohn

@TedShifrin I wear Tevas, they are really good flip-flops. I have hiked in them and have never slipped on wet sidewalks.

Ted Shifrin

18:39

Never heard of those. Where do you find them?

Just googled. They be expensive.

robjohn

Here are the ones I get.

@TedShifrin They are good and they last well

Ted Shifrin

There are different ones on that page. You mean Mush II?

robjohn

and there is a discound if you buy 5 or more.

yep

Ted Shifrin

I won't live that long. :)

robjohn

I hike in Mammoth with them.

Xander Henderson

18:42

@TedShifrin I had a Forrester for a while that did fine in the Sierras in snow. Huzzah for Subaru and all-wheel drive by default.

leslie townes

18:55

free admission to any indigo girls concert is also included in the base package

robjohn

19:14

@TedShifrin Actually, now, it looks as if you get a discount for getting 2. Even better.

Ted Shifrin

19:29

@robjohn Yes, I saw that. I’m hesitant to get 2 without being 100% sure of size/fit.

Jakobian

@BalarkaSen hii

I'm going to be doing this exercise because its used in the proof that $\text{SO}(3)$ and $\mathbb{R}\text{P}^3$ are homeomorphic

its like my worst nightmare

Ted Shifrin

19:47

That exercise obscures the geometry.

Jakobian

you mean this? Yes its here too, its just not formal enough description for the proof

Ted Shifrin

It’s easy to make it as formal as you need.

The starting point is that every non-trivial element is rotation about a unique axis.

Thorgott

yeah, I like this picture

this map is in fact induced by the Lie group exponential of $SO(3)$

and it's also linked to the description of RP^3 as Thom space of RP^2

I remember discussing this with Mike a couple years ago

Thorgott

20:18

also, even in linear algebra style, i think you can cut down the effort through realizing the universal cover $S^3\rightarrow SO(3)$ by thinking of $S^3$ as unit quaternions

Sha Vuklia

20:41

@Jakobian I guess the relation with Alaoglu is as follows:

Thorgott

it agrees with the WOT?

Sha Vuklia

on bounded sets it says

robjohn

20:57

@TedShifrin My shoe size is 10.5-11. I get size 11. They feel very steady and the piece between the toes is comfortable.

Ted Shifrin

21:07

@robjohn Thanks. That seems consistent with what the website recommends.

gf.c

are these two definitions for continuity at a point equivalent?

1. $f : X \to Y$ is continuous at $x \in X$ iff for every open neighborhood $N$ of $f(x)$ in $Y$ there exists an open neighborhood $M$ of $x$ in $X$ such that $f(M) \subseteq N$

2. $f: X \to Y$ is continuous at $x \in X$ iff for every open neighborhood $N$ of $f(x)$ in $Y$, $f^{-1}(N)$ is an open neighborhood of $x$ in $X$

I understand they are both equivalent to the usual "global" notion of continuity when we consider continuity at *all* points in $X$, so I'm interested in the case of a fixed point $x \in X$.

I see how (2) implies (1) but not the other way around

I could only deduce that $M \subseteq f^{-1}(N)$, but not sure how this would imply that $f^{-1}(N)$ is itself open

Ted Shifrin

The other way won’t work if continuity at other preimages of $f(x)$ fails.

You win only if $f$ is one-to-one.

gf.c

ok this is what I suspected, so effectively they are the same notion only if we consider a function that is continuous at all points?

Ted Shifrin

Or at least at all preimages of $f(x)$.

Good question, however!

gf.c

right, thanks!

Sine of the Time

21:25

consider the operator $T_n:L^2\to L^p$, where $T_n f(x)=n^{3/4}\int_x^{x+1/n} f(t) dt$. For which $p\in [1,+\infty]$ $T_n$ is continuous ? Does anyone have an idea on how to procede?

Ted Shifrin

Why is $T_n(f)$ in $L^p$?

Sine of the Time

It's a function of $x$

in the bounds of integration

Ted Shifrin

I understand that.

Sine of the Time

$\|T_n f\|_p=n^{3/4} (\int_{\Bbb R} |\int_x^{x+1/n} f(t) dt|^p dx)^{1/p}$ right?

Ted Shifrin

Yes.

Start with $p=\infty$, maybe.

Sine of the Time

21:35

I need $\| f \|_2$ so I tried with CS but did not conclude anything

Ted Shifrin

In general, surely Hölder and not CS will be relevant.

Sine of the Time

with $p=\infty$, $\|T_n f \|_{\infty} \le \|f \|_2 n^{-1/4}$

the exponent of $n$ is wrong

But I don't think it's an issue

it should be $1/4$

Ted Shifrin

OK, so you’ve answered the question for $p=\infty$.

Sine of the Time

yes

Ted Shifrin

Try $p=2$ and $p=1$?

Sine of the Time

21:44

let me try

Sine of the Time

21:54

I'm not making progress :(

monoidaltransform

is hyperbolic distance always greater than or equal to euclidean distance?

Sine of the Time

maybe an idea is to use MInkowsky's integral inequality

monoidaltransform

I mean on poincare disk

psie

22:34

I'm currently working a problem where I want to apply the following result from measure theory;

> Suppose that $f: X\to \overline{\mathbb R}$ is an extended real-valued measurable function, then $\int |f|d\mu=0$ if and only if $f=0$ $\mu$-a.e.

In my problem, I have a non-negative integrand which is the product of a continuous function and a measurable function. I'm integrating over a compact set in $\mathbb R$, and I know the integral vanishes, but I'm a little unsure if I can apply the above theorem or not, since not all continuous functions are measurable. If we use the Borel $\sigma$-algebra, I guess the continuous function is measurable. On the other hand, the Borel $\sigma$-algebra is not complete under Lebesgue measure.

I do not really understand the consequences of incompleteness yet, and therefor I'm unsure if the theorem applies. If anyone understands this more concretely, I'd be grateful for a comment or two.

Incompleteness when talking about metric spaces seems pretty "bad", so I'm wondering if the word has the same status in measure theory...

Xander Henderson

22:49

Wait.. what?

What continuous function is not measurable?!

psie

Well, I found this.

As I said, it depends on the $\sigma$-algebra I believe.

Jakobian

thats not what measurable means

we always equip the image with Borel sigma-algebra

when unspecified

and yes - all continuous functions are measurable

Xander Henderson

Time for a hot toddy. The weather is right.

Jakobian

@psie its significant but the relationship is a bit different

as you seen with Borel sigma-field, we want to allow incomplete measure spaces

not like for metric spaces where we fight incompleteness like fire

psie

23:04

yeah

Jakobian

@psie here in this result we equip extended real line with Borel sigma-field anyway

psie

true

Jakobian

the standard meaning of what it means to be measurable, measurability of $\{x : f(x) > a\}$ or $\{x : f(x) < a\}$ is the measurability into the Borel sigma-field

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