Mathematics - 2021-05-17 (page 1 of 3)

leslie townes

1:06 AM

looked like it had been raining at the duck pond.

shintuku

(nvm, solved)

copper.hat

1:37 AM

apparently there were sprinkles around the bay area. none in albany as far as i can tell. my wife is camping in ft bragg, hopefully she is not freezing

Ted Shifrin

Wow, I haven't been to Ft. Bragg in ... years.

copper.hat

i love the north coast, but not a fan of driving

my son has his aps so i stayed home to feed him

not sure he even needs his aps, i am impressed by his discipline

Ted Shifrin

He probably can feed himself better than you, too :P

So I read today that the whole UC system will no longer be considering SATs or ACTs. Of course, APs stay. I get the rationale. I'm just so not a fan of the whole AP industry.

copper.hat

no question about that

i am with you there

Ted Shifrin

I got an email, with no politeness or even a name, saying basically "Here are my questions" about things in your diff geo text. ... "Regards."

I was too nice. I should have called in @copper.

copper.hat

1:44 AM

i really do not understand people.

people do things electronically that they would never dare to do in person

Ted Shifrin

Is this a consequence of the internet? Certainly we can't blame this on COVID. I have no idea even what the status or age of this person is.

copper.hat

i attribute it to the internet.

Ted Shifrin

My experience over the years at UGA was that students were generally polite, whereas helicopter parents tended to the abusive.

(And that's in person, too.)

copper.hat

wow, parent involvement at the 3rd level?

Ted Shifrin

3rd level = college? YES!

leslie townes

1:47 AM

i had that a few times while teaching. and i didn't teach for very long.

copper.hat

wow. i thought one could invoke some over 18 sort of thing, as in i cannot discuss.

leslie townes

one time someone emailed me asking for solutions to my advisor's textbook. "asking" is even kind of an overstatement.

Ted Shifrin

I was quite abused for correcting mistaken advising when a young woman came to me. Her mother berated me on the phone for half an hour. I had parents come in livid when their children didn't exempt whatever course and expected me to say, "Of course. I realize your child is brilliant and I should do whatever you say."

That's at a whole different level of mathematics/society, @leslie.

Yes, @copper, FERPA is a thing.

But since I wasn't their actual instructor in these instances, I'm not sure it applied to that situation.

copper.hat

incomprehensible (parental interference)

leslie townes

FERPA did come up. even for under-18s who happen to be in college.

Ted Shifrin

1:49 AM

Yes, I invoked that if a parent of one of my own students interfered. But when it came to correcting misadvising, I don't see how it's germane.

Or just saying, I can't exempt person X from this course based on what the evidence is. I don't think FERPA is germane.

leslie townes

yeah, advising seems like a different kettle of fish.

copper.hat

you were helping them and they are mad at you???

Ted Shifrin

Well, it was not perceived as help. The misadvice told a CS major that (at that time) she didn't not need to take linear algebra. Many years later, that would have been correct. It was a generic Honors adviser who screwed up. Mommy was furious when I had to countermand it after the student had dropped the linear algebra course.

copper.hat

ah, i see, she was mad at you the university representative as such.

Ted Shifrin

I was the Honors adviser (who should have been advising earlier, obviously). I think in all my 33 years of advising, I screwed up one time (and I admitted to it in the appeal I helped draft).

copper.hat

1:53 AM

Everyone makes mistakes.

Ted Shifrin

Yes, but often advising mistakes are very costly to students. I don't blame the parents for being frustrated, but insisting for 30 minutes that I had ruined her child's life was unacceptable.

And mommy called during office hours when I had 10 students there.

I kept saying I can't talk now. Should have hung up.

Ah, such memories make me relish retirement. But, overall, I loved advising and teaching.

copper.hat

College is much more complicated here, I think.

Ted Shifrin

OK, time to go cook my last non-liquid food for two days.

copper.hat

Enjoy! Too late, but I think you can spend more $$$ and then there is only a one day no solids wait.

and good luck.

Ted Shifrin

Wow, someone just downvoted a (not particularly detailed) answer from April, 2013 ... pretty much when I got here.

@copper: Yes, that's correct (without money). Procedure is Tues morning.

So I exaggerated by half a day.

copper.hat

1:58 AM

:-). i got two downvotes on a closed question today. the answer is fine, i suspect at least one is punitive, and since it is closed, probably both.

Ted Shifrin

I remember a vendetta against me some time ago, but I don't remember if anything precipitated it. See y'all later.

copper.hat

enjoy your dinner!

love_sodam

I'm currently computing the degree of a map $f:S^1\to S^1$ by $z\mapsto z^n$ for $n>0$. To do this, I compute the local degree for each $x_i\in f^{-1}(1)$. Since $f$ is a local homeomorphism, $\deg f|x_i = \pm 1$ for each $i$. But how can I show $\deg f|x_i = 1$?

Thorgott

take orientation into account, or just write down generators and see what the map does to them

love_sodam

2:16 AM

Is there a way of finding generator of H_1(U_i,x_i) ?

U_i is a small open nbd of x_i

robjohn

@TedShifrin: I seem to have answered a question that you answered a few hours ago. I did not see your answer until I answered because I had left it open on my computer through dinner. I don't think our answers overlap.

On second viewing, maybe they are along the same lines (Mean Value Theorem, but then aren't most estimates of the Remainder?)

leslie townes

yeah, and the form of the desired result screams MVT.

robjohn

indeed

leslie townes

although not loudly enough for me to heed its call

robjohn

This uses the fact that the MVT assures of a $c$ between $a$ and $b$.

I almost gave up at one point until I remembered that

since $(x-t)^{p-1}$ vanishes at the end of the interval

Ted Shifrin

2:39 AM

Very interesting! Can you think of any applications? @robjohn

Andrew Micallef

Hallo

Im really close

The i's negate every second odd integer!

robjohn

@TedShifrin haven't yet

Ted Shifrin

Seems one of those “who cares” results, @robjohn.

robjohn

yep

it is not really surprising either

so all in all yawn inducing

glad I didn't delay dinner for it

Andrew Micallef

When i is taken to the power $4n-1$, i get -i, and taken to the power of $4n+1$ i get -i

robjohn

2:43 AM

the second one is wrong

Andrew Micallef

Huh, but thats like : 5,9,13...etc right?

robjohn

yes

$i^5\ne-i$

Andrew Micallef

And $i^5 = 2i^2 \cdot i$

robjohn

what?

Andrew Micallef

Oh i meant +i

robjohn

2:45 AM

yes

Andrew Micallef

And then all even powers i get -1?

robjohn

no

Andrew Micallef

No?

Huh

robjohn

$i^4\ne-1$

Andrew Micallef

Oh every 2nd even power

robjohn

2:48 AM

which ones?

Andrew Micallef

2,6,etc

robjohn

yes

Andrew Micallef

So 4n-2

Vs 4n

robjohn

how do you write them?

ah yes

Andrew Micallef

Sorry, got interupted

But tgat is equiv to 2n and 2n-1

And they are the basis for sin and cos series

Gtg, Thanks robjohn!

robjohn

3:05 AM

@Ted is tomorrow the day you have to drink unpleasant amounts of fluids?

or was that today?

copper.hat

i took bart to walnut creek when i had mine, and the timing would have been fine but for a 20 min delay at rockridge. i then had a mile walk on the other end. when i arrived, the staff just pointed towards the bathrooms without asking.

it was on a st partricks day too :-)

Ted Shifrin

3:21 AM

@robjohn Tomorrow ;).

@copper.hat Didn't you need a chauffeur? At least for pick-up?

copper.hat

@TedShifrin my wife collected me afterwards, i did not want to drag her for two visits (kids were 7 & 10 then). i am a total wimp with anesthetic and was completely wiped out afterwards.

Ted Shifrin

I remember never remembering a word the doctor says.

copper.hat

i started chatting with the theater staff and they quickly turned up the taps to shut me up...

Ted Shifrin

I imagine shutting you up is a tough task.

robjohn

Evidently I have a long colon. Immediately after the main event, I had to go over to the hospital for a lower GI. Unanesthetized, it was quite painful.

I guess because it was right after the anesthetic from the first procedure, they didn't want to give more.

copper.hat

3:39 AM

@TedShifrin I suspect many would concur. @robjohn I am squirming at the thought.

robjohn

I have put off my second one a bit for the memory.

Ted Shifrin

This us my fifth, if I count correctly.

Hawk

In Da Carmo (p128), he proves a certain self-adjoint operator S = -(\bar{\del}_xN)^T. If $X,Y$ are local extensions of $x,y$ that are tangent to $M$ and $N$ is a local extension of $n$ normal to $M$, at the last step he shows $-(Y,\bar{\del}_XN)(p) = (-\bar{\del}_xN,y)$, am I correct he evaluating at $p$ and therefore we are back to the vectors ? If so, why is it capital N on the RHS? Also can a single vector be considered a "vector field"?

robjohn

@TedShifrin more than one every 10 years I assume.

copper.hat

4:00 AM

@robjohn is it possible (for me) to search the link text in a chatroom? (as in the urls linked in the chat itself.)

robjohn

I don't think so, but i've never tried.

@copper.hat you mean like searching for "ucla.edu" in all urls used here?

copper.hat

@robjohn i am curious (provoked is probably a better word) how questions get selected to be closed. it seems that they get nominated in cured and shortly afterwards they are closed. i wanted to search for specific questions to see the timeline.

i understand much of the closing, but reasonable questions that do not have (easily found) equivalents/dups do not seem prime targets for closing.

robjohn

@copper.hat try https://chat.stackexchange.com/search?q=4141136&room=36

two from Ted and one from me?

copper.hat

@robjohn thanks!! i must have goofed, i tried something similar (i thought) in cured a few moments ago.

robjohn

For CURED: https://chat.stackexchange.com/search?q=4140266&room=2165

copper.hat

4:12 AM

yep, i am guessing the search for word matches, not partials

found it. thanks!

yep, it seems that moments after a close nomination the cured vultures all dive in together. doesn't seem quite right.

so one person basically can dictate the closing.

time for a drink :-)

epsilon-emperor

https://imgur.com/a/qrD4DDD
So it seems I need to prove that a ball of positive n-dimensional volume has Hausdorff dimension "n" - any hints, please?

Ted Shifrin

@robjohn What have I done this time?

robjohn

Those were the comments where we each mentioned that question using the MVT

@epsilon-emperor Didn't we do that just not long ago?

epsilon-emperor

@robjohn Nope, that was a cube

Ted Shifrin

But. A ball contains a … what?

epsilon-emperor

4:24 AM

and I don't remember, but if we did it for a ball, it must have been the box-counting dimension - not the Hausdorff dimension

@TedShifrin I'm inclined to say many small cubes

copper.hat

a cube contains a ball contains a cube contains a....

its a load of

epsilon-emperor

??

Martin Sleziak

@copper.hat Certainly posting a link in chat might increase the question's visibility (and thus likelihood of closure/reopening, deletion/undeletion, etc.) The same goes for posting on meta. But I don't think the word dictate is accureate here.

After all, every close voter has mind of their own - and they can decide whether they agree with the suggestion or not.

copper.hat

@MartinSleziak thanks! i'll wait. i am bit irked at the moment :-)

just seems a pity to lose content.

epsilon-emperor

@robjohn Did we?

Martin Sleziak

4:33 AM

Moreover, even without mentioning the close/reopen vote anywhere, the question automatically go into the review queue. (Of course, now that it is so big, many of questions in the clove vote review queue go unnoticed.)

"Losing content" is a vague term. Clearly, we do not want to keep everything. A an extreme example, I don't think many people are opposed to removing spam.

copper.hat

@MartinSleziak it was really more a (now deleted) comment that got to me. i'll be over it tomorrow :-)

i understand and am copacetic with the broad goals

Martin Sleziak

Wiktionary: copacetic.

At least we are already the lucky ones - we can easily get to the list of our own deleted posts.

copper.hat

sorry, usa slang

epsilon-emperor

@TedShifrin Could you elaborate on this?

copper.hat

@epsilon-emperor if $A \subset B$ then $H^d(A) \le H^d(B)$.

robjohn

4:45 AM

@epsilon-emperor: it might be useful to prove that $A$ and $rA$ have the same dimension where $r$ is any positive real.

epsilon-emperor

@robjohn That follows from the scaling property of Hausdorff measures!

@copper.hat Hmm, I know this.

copper.hat

then you know the ball and cube have the same Hausdorff dimension

epsilon-emperor

To show \dim_H F \ge n, I'm using exactly the same argument - just that I need to show a ball's dimension must be "n"

@copper.hat How? :'(

OH

copper.hat

maybe i misunderstood, i thought you knew the dimension of one of them?

robjohn

Any cube can be contained in a sphere of radius $s\frac{\sqrt{n}}2$

epsilon-emperor

4:49 AM

If I consider a cube inside a ball, and a cube outside, I can show that the ball's dimension is the same as that of the cube

and I do know that a cube has dimension "n"

robjohn

there

epsilon-emperor

but hey, I have a question

Here (imgur.com/a/qrD4DDD) the author could have used a cube instead of a ball too, right?

robjohn

yes

epsilon-emperor

Now the confusing part is

What does the fact that F is open have to do anything here?

If I have a non-empty set, I can still find a ball or cube inside it?

robjohn

because an open set contains a ball around all its points

epsilon-emperor

4:50 AM

even if I have a closed set, it works??

robjohn

no

epsilon-emperor

@robjohn Agreed

@robjohn Why not?

robjohn

A line in $\mathbb{R}^3$ still has dimension $1$

and a line is a closed set

epsilon-emperor

Hmm, you're right

So how about claiming that: the closures of all open sets also have dimension 'n'

That seems to work.

robjohn

yes because they contain open sets

epsilon-emperor

4:52 AM

Is it true that the only closed sets that have full Hausdorff dimension in R^n are closures of open sets?

robjohn

how does the book you are reading handle sets like the union of a sphere and a line?

copper.hat

no. take any closed set union a small closed ball

epsilon-emperor

@robjohn Haven't handled it so far - but what's your point?

copper.hat

the $H^d$ of a collection of parts is roughly the $\max$ of the $H^d$ of the prts

robjohn

just trying to figure out how they handle mixed dimensions

epsilon-emperor

4:54 AM

@copper.hat Hmm this certainly has full dimension

copper.hat

i mean that very informally

and $H^d$ \le $ dimension of the ambient space, of course.

epsilon-emperor

@copper.hat Closed ball union some closed set contains the closed ball, which contains an open ball --- all these have full dimension n

copper.hat

if the set contains any open ball at all it has full dimension

robjohn

so a tootsie pop has full dimension (sphere on a stick)

epsilon-emperor

@robjohn but it is the closure of some open set right?

namely the interior?

copper.hat

4:56 AM

why but?

robjohn

@epsilon-emperor so the answer to that question is no

@epsilon-emperor no

epsilon-emperor

@robjohn Why?

Oh the stick is one dimensional

robjohn

a line union a ball is full dimension, but it is not the closure of an open set

copper.hat

a tootsie pop contains an open set but the stick does not

epsilon-emperor

It does make sense now, thanks!

I was imagining a cylindrical stick because that's how we get tootsie pops in real life

Hehe

copper.hat

4:58 AM

it was an ideal toosie pop

infinitely long

How was Martin able to pop in yet his icon does not show? If you exit explicitly does the icon disappear?

robjohn

@copper.hat watch my icon...

copper.hat

@robjohn thanks!

robjohn

@copper.hat Did you mention something to Martin in another room? Just wondering why he popped in.

copper.hat

no, but i had a little diatribe in cured

Martin Sleziak

@robjohn Am I unwelcome here?

copper.hat

5:05 AM

:-) of course you are welcome here

Martin Sleziak

If that's the case, I can simply stay away from this room. Although that would make me a bit said.

robjohn

@MartinSleziak no, just haven't seen you here much

copper.hat

it is me that they are trying to get rid of

Martin Sleziak

I do try not to be seen.

robjohn

lurking in the transcripts?

Martin Sleziak

5:07 AM

I look at the chatrooms more often without visiting them. I only enter when I want to say something.

But sometimes I have to enter. For example, to look for somebody who can help on MO with ChatJax.

robjohn

@MartinSleziak nobody here can do that ;-)

copper.hat

nobody who says nobody

robjohn

I was glad to be able to figure out what had been plaguing Chrome users

shintuku

hello again, it is me, of course, as you well know. can anyone hint at me the proof technique to go from the fact that if a map $\Theta$ from an affine space to a vector space produces a vector whose span is that of a subspace $F$, the map $\Theta$ maps an affine subspace to $F$?

i got the part that proves the span is that of $F$, but now, what do I do

Russian Bot Who Knows Your IP

yo

copper.hat

5:13 AM

@shintuku i have difficulty (it may be just me) parsing your sentence.

shintuku

uh, no, it is most definitely my fault. give me a second

robjohn

@shintuku a vector spans all real multiples of itself

shintuku

$\mathcal{F}$ is an affine subspace of $\mathcal{E}$, directed by $F$ which is a vector subspace of $E$. $\Theta: \mathcal{F} \to F$. I've shown that, for $X \in \mathcal{F}$, $span[\Theta(X)] = F$. How do I show $\Theta(\mathcal{F}) = F$

robjohn

"directed by"? does that mean that all displacements in $\mathcal{E}$ are vectors in $E$?

not real familiar with affine spaces

shintuku

yep, I mean by that that $E$ is the vector space corresponding to the affine space $\mathcal{E}$

robjohn

5:19 AM

any vector in $E$ is the difference of two points in $\mathcal{E}$?

for any vector, $v$, in $E$ we can find two points in $\mathcal{E}$, $p_1,p_2$, so that $\Theta(p_2)-\Theta(p_1)=v$?

shintuku

Difference of points in $\mathcal{E}$ isn't defined yet, but for vector spaces it is. I have, $\forall A, B, C \in \mathcal{F}$, $\vec{AB} = \vec{AC} + \vec{CB}$

robjohn

@shintuku I thought that was how displacements were defined.

equivalence classes of differences of points.

copper.hat

Is $\Theta$ an affine map?

robjohn

If I understand, $\Theta$ maps from an affine space into the vector space

shintuku

hm, we're given $B = A + \vec{AB}$, but difference hasn't been specified yet and proper algebra hasn't been introduced yet

well, suppose we're given difference

I guess in that case it is a bit more obvious

robjohn

5:30 AM

$B-A=\overrightarrow{AB}$

shintuku

alright, but then we still only have a span, right?

robjohn

is a span a displacement, not sure.

shintuku

hm, Ill tinker a bit more with this. the author begins the book with affine geometry without coordinates, so I'm completely lost on proof technique

Martin Sleziak

So your $\mathcal F$ is given as $P+F$ where $P$ is some point (and $F$ is a given subspace)?

The way I've seen the definition of affine spaces, they are defined as the pair $(\mathcal P,V)$, where $\mathcal P$ is a set of points and $V$ is a vector space.

shintuku

$\Theta_B: M \mapsto \vec{BM}$, where $B, M \in \mathcal{F}$ and $\vec{BM} \in F$

Martin Sleziak

5:41 AM

Ok, so we want $\Theta_B(\mathcal F)=F$. I was wondering how $\Theta(\mathcal F)$ makes sense.

So if all points $\mathcal F$ are of the form $B+\vec v$, where $\vec v\in F$, then you get $\Theta_B(B+\vec v)= \vec v$. So you get every vector in $F$ as an image of some point from $\mathcal F$.

Ted Shifrin

@robjohn A vector space with no distinguished origin.

Subtraction of elements is well-defined but addition is not.

Martin Sleziak

@shintuku We're looking at page 10 of Audin's geometry, right?

shintuku

yes! exactly

Russian Bot Who Knows Your IP

bitcoin price

shintuku

right, but wouldn't that still only give us the span, and not the identity $\Theta_B(\mathcal{F}) = F$?

if, as you say, we get every vector in $F$ as an image of some point in $\mathcal{F}$, then it means, for any $M \in \mathcal{F}$, $span[\Theta_B(M)] = F$

then, I'm missing the extra step

Martin Sleziak

5:46 AM

I see, the point of this exercises is to show that $\Theta_B(\mathcal F)$ is the same for every point $B\in\mathcal F$.

@shintuku I do not really follow what is the problem here.

By definition $\Theta_B(\mathcal F)=\{\Theta_B(M); M\in\mathcal F\}$.

I mean, this is the definition of the image of a set.

In any case, in Aubin's book $\Theta_A(\mathcal F)=F$ is part of the definition of an affine subspace. The point of this exercise is to show that we get the same vector subspace as $\Theta_B(\mathcal F)$ for every point $B$.

shintuku

right, using Audin's definitions, I wrote down this:

so we can get any $\Theta_B(M)$ as a linear combination of vectors that come from $\Theta_A(\mathcal{F})$

which means, any $\Theta_B(M)$ can be written using a basis of $F$

but I feel like I haven't proven yet that the map maps the entire affine subspace $\mathcal{F}$ to $F$, only for an arbitrarily given $M$

I think I must be missing some linear algebra definition somewhere here

Martin Sleziak

You have expressed every vector in $\Theta_B(\mathcal F)$ in the form $\vec u+\vec v$ where $\vec v$ is fixed and $\vec u\in\Theta_A(\mathcal F)$.

Moreover, $\vec v=\Theta_A(B)\in F$.

So basically this boils down to verifying that if $F$ is a vector subspace and $\vec v\in F$ is a fixed vector then $$F=\{\vec u+\vec v; \vec u\in F\}.$$

This also corresponds nicely to the geometrical intuition behind this. If we fix some point $A$, we'll get vectors of the form $\Theta_A(M)=\overrightarrow{AM}$. If we fix some other point $B$ and take vectors with the endpoints in $\mathcal F$, then the difference is exactly the vector $\overrightarrow{AB}$. (Or $\overrightarrow{BA}$, depending on the direction we're looking at.)

shintuku

6:02 AM

don't we need to prove $F$ is a vector subspace, or do we immediately get that from showing it is spanned by $\vec u + \vec v$?

Martin Sleziak

It depends on what you denote by $F$.

I am using $\Theta_A(\mathcal F)=F$, so I know (directly from Aubin's definition) that $F$ is a vector subspace.

shintuku

right of course

Martin Sleziak

Since I see that affine space are being discussed here, I will mention that there is also a room called Linear & Abstract Algebra. Although I have to admit that it is usually quite silent.

shintuku

one last question: we've expressed every vector in $\Theta_B(\mathcal{F})$ in the form $\vec u + \vec v$. But why does this imply that the vector subspace $F$ is the set $\vec u + \vec v$

does the fixed vector here play an important role?

I'm unable to bridge this gap, going from the $\vec u + \vec v$ representation, which to me seems to only denote a span, to the vector subspace

both $\vec u$ and $\vec v$ are vectors from $F$, of course. but how does the fact that we can express every vector in $\Theta_B(\mathcal{F})$ as a combination of vectors in $F$ prove to us that, actually, $\Theta_B(\mathcal{F})$ is $F$ itself

Martin Sleziak

6:18 AM

I would suggest to forget about the affine structure for a bit and look only at the problem in the language of vector space.

Claim 1. If $F$ is a vector subspace and $\vec v\in F$ is a fixed vector then $$F=\{\vec u+\vec v; \vec u\in F\}.$$

Is this claim (about linear subspaces) clear, or should we go through the proof of this?

shintuku

It makes sense: it is a vector $\vec v$ to which you add some vector in $F$

Martin Sleziak

Every vector from the set $\{\vec u+\vec v; \vec u\in F\}$ clearly belongs to $F$. (A subspace is closed under addition.)

We also need to show the opposite inclusion $F\subseteq\{\vec u+\vec v; \vec u\in F\}$.

shintuku

right, that makes sense

Martin Sleziak

I.e. we want to show that if $\vec x\in F$, it can be expressed as $\vec u+\vec v$ for some $\vec u\in F$.
To do this we can simply take $\vec u=\vec x-\vec v$.

@shintuku So is it now clear that Claim 1 holds?

shintuku

yes

Martin Sleziak

6:24 AM

If claim 1 (which is the "vector" part of our situation) is clear, then we can look at the "affine" part in our problem.

Claim 2. Let $\mathcal F$ be an affine subspace of $\mathcal E$ and $A,B\in\mathcal E$. Then $$\Theta_B(\mathcal F)=\{\vec u+\overrightarrow{BA}; u\in\Theta_A(\mathcal F)\}.$$

shintuku

ahhhhhhhh

Martin Sleziak

Any vector from $\Theta_B(\mathcal F)$ is of the form $\vec x=\overrightarrow{BM}$.
So we get $\vec x=\overrightarrow{BM}=\overrightarrow{BA}+\overrightarrow{AM}=\overrightarrow{BA}+\Theta_A(M)$.

This shows $\Theta_B(\mathcal F)\subseteq\{\vec u+\overrightarrow{BA}; u\in\Theta_A(\mathcal F)\}.$

We need to check also that $\{\vec u+\overrightarrow{BA}; u\in\Theta_A(\mathcal F)\}\subseteq\Theta_B(\mathcal F)$ to finish the argument in this way.

shintuku

riiiiiiiiiiight

Martin Sleziak

But the manipulation in which we can get this inclusion should be somewhat similar to the first one.

shintuku

it finally clicked!

thank you so much!

I've spent like three hours on this problem

honestly, thank you for the time spent spelling it out

Martin Sleziak

6:30 AM

I guess this is much easier if one has possibility to draw pictures. (Some plane in 3D representing $\mathcal F$, then drawing points and vectors between them.)

But that would be difficult to do in Stack Exchange chat.

shintuku

well, I had the intuition down and images and everything

I just couldn't make the logic work out

Martin Sleziak

That's good.

copper.hat

it is surprising to me that we still cannot easily draw & share little pictures on our computers.

key word being 'easily'

Martin Sleziak

I think that what we did here was basically just writing down a formal proof of the same thing one can imagine in the picture for some specific example (such as a plane).

shintuku

I'll be going back to Audin now, again thank you so much!

copper.hat

6:33 AM

restores my faith in mse humanity

Martin Sleziak

@copper.hat During the last two academic year I have been using a graphics tablet (Wacom) a lot.

Most of my teaching was done in the way that I was writing by hand and talking to students.

But using something like that here in Stack Exchange chat would not work well. I would have to draw the picture and then paste it here.

copper.hat

@MartinSleziak about 2 decades ago i tried to go paperless, but still struggle. i have a pen, used to have a motion computer for a while, but nothing really replaces my pencil & paper :-(

Martin Sleziak

Sequence of chat messages is completely different from sharing a screen, where the other participant(s) can see the things in the way the are added.

copper.hat

that is true, but even static pics can be of big value

Martin Sleziak

That's right I use that only very seldom to write something that is just for me.

copper.hat

6:35 AM

even with work, slack/teams/whatever, there really is nothing that functions well as a whiteboard

Martin Sleziak

But since I had to teach remotely, I had to use something and this seemed as a reasonable way. (With an advantage that I do not need physical blackboard. Some colleagues actually went to an empty classroom and broadcasted what they were writing an a board.)

I have been using Google Meet during the previous academic year, MS Teams this academic year. (The choice of software was basically decided by the university.)

copper.hat

amazing to me that in 2021 we still struggle with basic drawings :-)

i used meet on friday for the first time

it was a meeting with google no surprise :-)

Martin Sleziak

As far as Whiteboard goes, I have experimented a bit with various things, the one I used the most was MS WhiteBoard.

copper.hat

my work is in eda, mostly unix base folks

Martin Sleziak

But most of the time I simply write the thing locally - some shared Whiteboard where both me and the students can write things is needed only somethimes.

Anyway, it was nice chatting here, I guess it's time for me to get some breakfast.

copper.hat

6:39 AM

where are yo???

im about to hit the sack

ok, bye!

Martin Sleziak

Bratislava, Slovakia: timeanddate.com/worldclock/slovakia/bratislava

robjohn

@TedShifrin it was the "directed by" terminology I was asking about.

copper.hat

closest i've been to there was wien. over 4 decades ago :-(

robjohn

@copper.hat That's best if said with a British accent... Been to Wien...

copper.hat

such was my naivete that i did not realise that wien was vienna

robjohn

6:46 AM

In the US, it's like Ben to Wien

just no magic to it

copper.hat

:-)

good night! time for my ugly sleep

Martin Sleziak

Good night, see you later!

robjohn

@MartinSleziak Later. Thanks for dropping by!

Though I know you'll see us while we don't see you

epsilon-emperor

7:21 AM

imgur.com/YbCL7Mn

I feel this proof in Falconer is incomplete, they only consider one case - i.e. s > \dim_H F

If s < \dim_H F I do realise that H^s(F) = \infty, but we still don't know anything about H^s(f(F))

Oh in that case it seems sufficient to say that the inequality holds trivially. Got it.

One thing still seems off: they say "implying that \dim_H f(F) \le s/\alpha...". It should be without equality, right?

s > \dim_H F and s < \dim_H F cases are handled, but what if s = \dim_H F?

Mad Spaces

9:05 AM

when we have an element say $x \in \mathbb{R} $ and the author writes $ x > 0 $ what does this mean? the scalar product of x with itself bigger than 0 ? each element in the tupel x is bigger than zero? the distance between the element x and the (0,0,0....) is bigger than 0 (in relation to the defined metric)

i am asking in regard to this answer
@MISC {1871279,
TITLE = {Prove that $\mathbb{R}^n\setminus \{0\} $ is connected for $n > 1$},
AUTHOR = {Aloizio Macedo (https://math.stackexchange.com/users/59234/aloizio-macedo)},
HOWPUBLISHED = {Mathematics Stack Exchange},
NOTE = {URL:https://math.stackexchange.com/q/1871279 (version: 2016-07-26)},
EPRINT = {https://math.stackexchange.com/q/1871279},
URL = {https://math.stackexchange.com/q/1871279}
}

Nevermind i think he means each element of the tupel x is bigger than 0

robjohn

10:37 AM

@MadSpaces when you want to refer to a post, do not copy and paste the citation from the Cite link. Instead, copy the Share link and use it in a [link to the post](https://math.stackexchange.com/a/1871279) link to the post.

That is a lot less spammy and just as easy to use.

Thorgott

11:12 AM

@epsilon-emperor there is no need for cases

$x\le s/a$ for all $s>y$ implies $x\le y/a$

robjohn

@AndrewMicallef: did the suggestion for Chrome on Android work for Firefox on Android?

shintuku

11:29 AM

ah

i just learned about the extensionality axiom

i guess that explains why I couldn't finish most proofs