12:00 AM
i'm never quite sure about the h bar

Walking the Planck?

I think that if $(E, F)$ is a dual pair and $F$ is separable then $E$ is second countable.

3 hours later…
3:06 AM
Can someone explain why in game theory it is never correct to choose a strategy that isn't a best response to some belief about what the other player will do?

3 hours later…
6:25 AM
Assume that $n$ is a multiple of 3, the other two cases should be similar.For example, $n=6$
\tag{1}\label{1}
A = \begin{pmatrix}
a_1 & 1 & 0 & 0 & 0 & 0 \\
a_2 & 0 & 1 & 0 & 0 & 0 \\
a_3 & 0 & 0 & 1 & 0 & 0 \\
a_4 & 0 & 0 & 0 & 1 & 0 \\
a_5 & 0 & 0 & 0 & 0 & 1 \\
a_n & 0 & 0 & 0 & 0 & 0
A^{-1} = \begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 1/a_n \\
1 & 0 & 0 & 0 & 0 & -a_1/a_n \\
0 & 1 & 0 & 0 & 0 & -a_2/a_n \\
0 & 0 & 1 & 0 & 0 & -a_3/a_n \\
0 & 0 & 0 & 1 & 0 & -a_4/a_n \\
0 & 0 & 0 & 0 & 1 & -a_5/a_n
Any help in getting eqn (2), (3) ? much appreciated!
3

1 hour later…
7:45 AM

will the questions per day continue to drop friends?
also why did traffic drop so much in may

First we should consider the simpler question: how many sides does a circle have?

isn't it none?

190

My son is in 2nd grade. His math teacher gave the class a quiz, and one question was this: If a triangle has 3 sides, and a rectangle has 4 sides, how many sides does a circle have? My first reaction was "0" or "undefined". But my son wrote "$\infty$" which I think is a reasonable answ...

the top answer agrees with me
right?

7:55 AM
:: nods ::
✔️

> She didn't know that a sideways "8" means infinity.
2
Eek!

Wouldn't it have infinite amount of vertices and no sides?
Because if you take a tangent line at any point, then the intersection with the object is 0-dimensional

2nd graders are only around 7 years-old

@XanderHenderson It nicely shows Alexander's dark band on the left. en.wikipedia.org/wiki/Alexander%27s_band
@user726941 Two. An inside and an outside. :)

8:04 AM
@PM2Ring Topologists answer is definitely bonkers here. There's different kinds of coffee cups, some would have one, some would no holes, some could have two. The philosopher's answers is a close second, but chemist killed me, lol.

@PM2Ring On a language acquisition level, yes.

@Jakobian Hmmm. That depends on how you define vertices. It doesn't work if you define a vertex as the intersection of two sides.

@PM2Ring by the same logic any polygon has two sides

Expecting them to grasp "infinitely many" at that age seems overly optimistic, in my humble opinion.
Or even, the concept of zero.

Sure. But that OP's son was aware of it. I'm not claiming he fully understood it.

8:11 AM
True, true,...

I've been comfortable with infinite sets, limits, etc, for decades. But I wouldn't claim that I really understand any of the various infinities that we use in mathematics.

How about we define a vertex as a non-differentiable point of the boundary. Then it'd have no vertices. Then we could define a side as a smooth curve between two vertices.

OTOH, I think a "simple" infinite entity like the set of counting numbers is easier to grasp mentally than some huge finite number. I can write 1000!, but can I honestly say that I really appreciate its magnitude?

non differentiable point of the boundary?
differentiable how?

It does agree with our understanding of triangles on the sphere

8:17 AM
@Jakobian Ok, I guess that works. Sort of. ;) But it means that a circle has the same number of sides as the empty universe, which is a bit unsatisfactory.

Then a circle should have 0 edges

@Asinomás You don't have a well-defined derivative at a vertex or cusp point on a curve.

Maybe differentiable under some parametrization of the circle into our geometric figure

For those interested in going down another rabbit hole

8:35 AM
I thought I posted this a few months ago, but Search can't find it. Rational parameterisation of the full unit circle using t = tan(theta/4)

1 hour later…
9:54 AM
what do you mean derivative

@Asinomás It's not given locally by a graph of a differentiable function

10:45 AM

2 hours later…
12:54 PM
Hello. I have a dummy question: is the n-dimensional sphere without the north-pole $S^n\setminus\{N\}$ an open set in $\mathbb{R}^{n+1}$?
Let $S$ be the south pole. $S^n\setminus\{N\}\cup S^n\setminus\{S\}=S^n$ why does the union of two open sets give a compact (thus closed) set? What is the problem? Maybe the punctured sphere is open only with respect to the topology induced on the sphere a regarding the sphere as a topological space, it would be an open set as well

@Feynman_00 What do you think?

1:28 PM
@Thorgott I think what I wrote should be the answer
So is that correct?

I don't see where you answered the question of $S^n\setminus\{N\}$ being open in $\mathbb{R}^{n+1}$. I see that you have postulated that $S^n\setminus\{N\}$ is open in $S^n$ with the subspace topology, which is a correct and good observation, but not an answer to the first question.

Oh, okay. I asked the first question as a preamble for the second one that was my actual question. Regarding the question about $S^n\setminus\{N\}$ being open in $\mathbb{R}^{n+1}$, I would say no. If $S^n\setminus\{N\}$ and $S^n\setminus\{S\}$ were open sets in $\mathbb{R}^{n+1}$ their union being a closed set (i.e. the sphere) would be absurd

1:50 PM
What about the open sets $\mathbb{R}^n\times(-\infty,1)$ and $\mathbb{R}^n\times(-1,\infty)$ in $\mathbb{R}^{n+1}$? Is their union not closed in $\mathbb{R}^{n+1}$?

2:00 PM
Is it? It looks like an open set to me
Oh wait, I've mistaken $1$ and $-1$

Is $S^n \setminus \{N\}$ open in $\mathbb{R}^{n+1}$?
So you aren't taking the union of two open sets. That seems to answer the original question...

@XanderHenderson I think it does
@Thorgott in that case, though, the union is the entire topological space that by definition is closed and open at the same time (together with the empty set)

@Feynman_00 In $\mathbb{R}^n$, the only clopen sets are the empty set and the entire space. Hence the union of open sets will not be closed unless it is the entire space. Other spaces permit more interesting examples of unions of open sets which are closed (or compact, even).
The example of $S^n \setminus \{N\} \cup S^n \setminus \{S\}$ is a good example, in the subspace topology on $S^n$.
Or give the integers the discrete topology. Every set is clopen, and every finite set is compact.

The first point you raise is clear to me. Regarding the second point (the one about other spaces), I don't see how the union of a countable number of open sets can be closed (except they cover the entire space or they all are the empty set)

@Feynman_00 Consider the space $\mathbb{R}\setminus \{0\}$. Cover $(0,\infty)$ with a countable collection of open intervals.
The union of those intervals is closed in $\mathbb{R}\setminus\{0\}$, and is not the whole space.
However, if $A = \bigcup_\alpha U_\alpha$ is closed, where each $U_\alpha$ is open and the $\alpha$ come from some (possibly infinite, possibly uncountably infinite index set), then $A$ is a connected component (or the union of several connected components).
To see this, suppose that $X$ is the ambient space containing $A$. Then $B = X \setminus A$ is open (since $A$ is assumed to be closed). Then $A \cup B = X$, and $A \cap B = \varnothing$. Thus the space is disconnected, and, in particular, no point of $B$ lives in any of the connected components of $A$.

2:17 PM
I'm sorry but I'm weak in topology and I need more time to think about this. How does this not contradict the definition of a topology as a collection of open sets such that, among other things, any arbitrary (finite or infinite) union of members of the topology belongs to the topology?

"To belong to the topology" is to say that the set is open.
So any union of open sets is open.
In my example, $A$ is open.
It is also closed.
(i.e. "clopen").
"Open" and "closed" are not opposite to each other, nor contradictory.

Ooooooh that makes things more clear

I though a set could be clopen iff it were either the topological space or the empty set. Now I understand how that makes sense
Thanks for your help and patience :)
5

@Feynman_00 This is true for some spaces, e.g. $\mathbb{R}^n$. But there are all kinds of spaces with lots of clopen sets. Every open set in the $p$-adic numbers is also closed. Every subset of a space with the discrete topology is clopen. Every component of a disconnected set is clopen. And so on.

2:21 PM
For the record, while Xander is finishing the point about connectedness, I want to point that we can also observe that $S^n\setminus\{N\}$ is not open in $\mathbb{R}^{n+1}$ directly by definition. To be open would require containing a small ball around each of its point, but actually if you take any ball around any point in$S^n\setminus\{N\}$, it will contain points not in $S^n\setminus\{N\}$, e.g. by moving slightly in the direction perpendicular to the sphere.

@XanderHenderson This is gold.
Answering to Thorgott, now that I think about it, I used the definition to answer my question about the punctured sphere a few days ago, then the problem arised with the second question but now everything is clear. Thanks.

3:24 PM
17

Here is a different proof from the ones in the comments. Let $O=(0,\dotsc,0)$. If $\mathbb{R^n}$ and $H^n$ were homeomorphic, then let $P$ be the image of $O$ in this homeomorphsim. Then $\mathbb{R^n}\setminus\{P\}$ and $H^n\setminus\{O\}$ would be homeomorphic, too. In particular, they would be...

1 hour later…
4:45 PM
1

Are the following definitions of relative countable compactness equivalent? $\overline{A}$ is countably compact Every sequence in $A$ has a convergent subnet Note that for regular spaces, the analoguous definitions are equivalent for relative compactness. Also, the equivalence always holds if...

My question was closed for no reason
Wtf

@Jakobian It was not closed for no reason. It was closed for the reason
> Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc.

@XanderHenderson this reason doesn't make any sense to me

"The reason doesn't make sense to me" is different from "It was closed for no reason."
I will also admit to some confusion about the statement "Every sequence in 𝐴 has a convergent subnet," though it could just be that my topology is rusty: why do you consider a subnet of a sequence? Nets generalize sequences. If you are already working with sequences, why do you ask for the generality of nets?

What do you mean. I justified it, it's all in the question

That being said, I likely wouldn't have voted to close the question, and a polite discussion in CURED might get you enough upvotes to have the question reopened.
(Charging in and saying "My question was closed for no reason!" is not likely to win you support.)

4:59 PM
This is equivalent to countable compactness for closed sets
You can't use sequences alone for countable compactness
And if you used only nets then that's compactness not countable compactness

Why did you accept the answer?

Because it answered my question and I wanted it to not appear in the unanswered questions
The definitions aren't equivalent even for relatively nice spaces

### CURED

For feedback/discussion/requests of Close/Undelete/Reopen/Edit...

5:15 PM
i think a somewhat generous reading of the last two sentences of the post would count as "possible strategies" and/or "your current progress" although this might not be apparent to the average SE user.

Who knew that Eric Wofsey was an esoteric point-set wizard ...

5:35 PM
Morning @TedShifrin

they should have warning labels attached to profiles like that. he answered that in 15 minutes. people should have advance warning that they're dealing with someone who's walking around thinking about the deleted tychonoff plank.

Morning Leslie
Hello Prithu

hello

Looks like you came back @PrithuBiswas
What have you been up to lately, Leslie?

howdy, Ajay
@leslie Perhaps when I studied for my point set final exam in 1972 I could have written that ... but I've never in my life thought about nets.

5:46 PM
ajay: just the usual, working in the day, goofing off at night. it's been hot lately and my daughter makes me take her swimming for at least an hour a day.
ted: i probably had a half-semester window in which i could have written that too. and i've thought about nets. so what's my excuse.

You lucky duck Leslie, down in Asia, it's been raining all the time...

We are in desperate need of rain. Lots.
Speaking of lucky ducks, how are Munchkin's ducks?

If I could somehow control the weather, I would gladly give you all the rain.

Well, that's your first homework assignment.

we haven't been to the duck pond in weeks due to the heat. we did see the coyote while checking the mailbox on thursday. and last night i saw two coyotes outside our porch.

5:50 PM
They're circling their wagons.

Do any of you have any advice on dealing with stress?

If necessary, talk with a counselor!
All students have to deal with stress to a certain extent. I know my classes stressed out some of my students, but others relished them. I am a worrier by nature, so I do stress, but not usually out of control; there have been a few exceptional times — and I did go talk to a professional for a few weeks during one of those.

Same, in prior months, due to stress from exams and other work, i've become impulsive and am quick to anger. I have been seeing a therapist for a while, and it has been helping.
Well, I guess as we all grow older we find better ways of dealing with stress...
Anyways, enough with the boring talk... How have you been these few months?

I think the best way not to get stressed by exams is to prepare well ahead and not procrastinate. In my 40+ years of teaching experience, I found that the procrastinators put themselves in binds repeatedly.
7

6:07 PM
Procrastination is one of my worst issues, I have been in one too many situations where I race to finish my HW before my teacher comes around.
Well, that is my fault, I was mostly watching youtube vids in class and not paying attention.

So ... my advice is. Change. Discipline yourself to get your work done early. That will serve you well in college and the rest of your life.
13

01010100 01101000 01100001 01101110 01101011 00100000 01111001 01101111 01110101 00101100 00100000 01010100 01100101 01100100 00101110 00001010
That means "Thank you, Ted" in Binary code.

LOL ... or perhaps something far more sarcastic in reverse binary.

I'm gonna be off again for the next few months. I will definitely come back changed!

6:22 PM
All best to you!! :)

Hello friends!
I've a doubt related to applications of derivative.
___
I was learning about maxima and minima using second order derivative test in which first we differentiate the given function $f(x)$ and then solve for $f'(x) = 0$. Putting the critical points so obtained in $f''(x)$ tells us about maxima and minima...
If $f''(x) > 0$ implies x is local minima.
While $f''(x) < 0$ implies that x is local maxima.
___
My question is that -- how is second order derivative able to tell whether the point is local maxima or minima. Just like first order derivative can be defined as the rate of

minima and maxima are PLURAL words. Just so you know. $x$ is a local minimum or maximum.

putting aside where f''(x) = 0 (where the test is inconclusive, but is worth mentioning - this 'test' fails says nothing about even a simple function like f(x) = x^4) one way of looking at it is that the sign of f'' tells you concavity. f'' being positive means the graph is smiling near the point (so if it has an extremum its a minimum). f'' being negative means the graph is frowny so if it has an extremum it will be a max.

Draw pictures. Draw a parabola upward and a parabola downward.
5
At a minimum the derivative is increasing (from negative to positive). At a maximum, the reverse.

you can also frame stuff about f'' simply in terms of the first derivative (of a function that happens to be f'). if f''(c) is positive, this suggests that f' is increasing near c. so if f' is 0 at c, then f' should be negative before c and positive after c. and step back and ask what that tells you about what f is doing. (decreasing and then increasing, i.e. hitting a local min.)

6:32 PM
Right. To emphasize. $f'' = (f')'$.
2

@leslietownes How is second order derivative able to determine concavity of the graph? -- Smiling or Frowny? apologies if it's a senseless question.

it's definitely not a senseless question. it basically does that for reasons relating to my second comment.
the smiley/frowny stuff is more of a mnemonic than an explanation.

Ohh thanks. I think I got it now. :)

Did you read my sentences above?

a graph is "smiley" when as you move from left to right the slopes of the tangent lines increase. that means f' should be an increasing function, or as ted would have it (f')' should be positive.

6:36 PM
"sign of f'' tells you concavity" and "minima and maxima are PLURAL words" were really new things for me. Just got to know about them.

LOL, I was more concerned about my telling you to look at upward- and downward-pointing parabolas. Draw tangent lines and look how their slopes behave as you move left to right.
Most textbooks explain this, and I certainly would hope every calculus teacher explains it carefully.

one of the two calc teachers at my hs only did frowny-smiley. the other one could go into it a little more than that
thankfully, i got the latter one, and here i am today

@TedShifrin Yes I'm trying to understand that slowly slowly... kind of new stuff for me. Reading that again and again 😁

For several terms, I worked with grad students their first time teaching calculus — discussed how they should present things, worked with them on exams, and watched them teach several times. This was certainly one item we covered.
Ultimately, the department hired someone to do this officially as one of their courses — both for precalculus and for calculus.
It was always amusing dealing with grad students who figured they were going to be the genius that does everything "right" ...
I still shudder when I think about some of the very experienced professors at MIT who lectured multivariable to 300+ students for the first time and just were so bad at it.

6:52 PM
@TedShifrin That's really amazing. Just understood this.

Oh, great :)
I mean, @HelpMeToUnderstandContours, that is literally what taking the derivative of the derivative means :P

@TedShifrin Ohh yes yes 😅
You are great.

@robjohn @leslie @copper @Xander This was a fun comment to write.
No need for that, @HelpMe. But, truly, drawing pictures and seeing things before your very eyes helps more than formulas. Even @leslie will grudgingly agree with me.

But sadly, graphs are not allowed in exams. :(

Well, I don't mean using a computer. I mean you should try to visualize concepts with examples/pictures.
No one can stop you from drawing a sketch on your paper to remind you of what's going on.

7:00 PM
Yes.

Some teachers will agree with me and some will disagree. But when I taught integral calculus and taught different methods of finding volumes, I would not give complete credit unless students drew a sketch and showed me how they thought through the problem. So many students struggle on this because they try to do it by rote memorization and don't understand what's going on.

@TedShifrin You are correct but I believe that I'm not tend to draw pictures as I've never drawn pictures :|
no one insisted on me for drawing pictures

@TedShifrin It is too bad that the question is several months old, and the original asker is likely long gone.

I think I need a teacher like you. 😄

But the question is now closed.

7:10 PM
@Xander Well, having seen some questions from that answerer, I was a bit taken aback by his tone of authority.
@HelpMe Yes, sadly, too many teachers are rigid and don't understand that different students have different learning styles. You can still learn these skills and apply them.

@TedShifrin Yes Exactly.

Your name would suggest you're studying multivariable calculus.

@TedShifrin I'm actually a calculus 1 student. But during the COVID-19 period, I learnt many things in maths which are not even related to my course. Recently I saw the symbol $\oint$ and was greatly attracted to that symbol LOL. And then I started learning complex analysis.

7:32 PM
notation, an underappreciated gateway drug to mathematics
5

good afternoon. another sweltering 21c day here in albany, ca
my dad was in finance and he had some schaum outline book related to financial calculations (we're talking 60s here). i think my gateway to math was all the $\sum_{n=1}^N$ symbols.
i was stymied by a simple convex question recently, it ended up with a simple answer on MO that took me hours to unwind.

7:50 PM
@copper.hat What is that in real, patriotic, American units?

personally i prefer rankine

Isn't that around 70°F? Seems quite pleasant to me.

i would have thought that Fahrenheit would have been seen as colonial
@XanderHenderson i cycle with a friend whose idea of tolerably cold is my idea of tolerably hot
we had a nice onshore breeze as we cycled along the bay, threading between the heads down peloton on sideway groups and dog walkers.
saw a pelican catch & eat a fish
really a bit nervous about my labour day trip to southern AZ
i need to hitch a ride in a 4WD to get to the ranch, it seems

@copper.hat Southern Arizona should be warm, but tolerable, in September.

for you...

8:01 PM
Particularly if you can get up into the mountains a bit.

i'm not sure what the plans are, i'll be there for 5 days

You should come up to Holbrook, then drive over to Gallup for the rug auction.
Though that is late September, so you'll be gone already.
Oh, well. No rugs for you.

we'll be down near the MX border

Oh, fun. You'll get to talk to the Border Patrol!

much alcohol will be consumed, i imagine
use to be they were suspicious of irish
coming in from the north & south were standard tricks back then

8:41 PM