Mathematics - 2021-08-05 (page 3 of 3)

Ted Shifrin

7:02 PM

Yes, negative.

Maybe light case, maybe flu. Lots of this to look forward to in the future.

leslie townes

7:19 PM

good to hear it, ted.

i had something put me out of commission last week for about 72 hours. did not appear to be covid. didn't bother to get tested.

copper.hat

my son has something. so do 3 of his friends. i am torn. it seems unlikely that all 3 fully vaxed friends would all contract covid.

leslie townes

my daughter brings every sort of thing home from day care.

copper.hat

he is masked and i wear a mask when around him just in case.

leslie townes

i am notorious for giving my office neighbor hand foot and mouth disease.

copper.hat

i'm afraid to ask

leslie townes

7:26 PM

my wife asked about that one too. it's highly contagious!

copper.hat

the joys of kids. i suspect (read hope) that is confers some broad immunity that is useful later

leslie townes

i think so. i was kept mostly at home until about age 5 and had a number of bad infections. our daughter has been around other children since age 1.5.

she gets everything but never seems to get sick. she just brings it home to share with the family.

copper.hat

we had a lot of stuff growing up, mumps, measles, chicken pox, etc.

leslie townes

i avoided two out of those three thanks to vaccination.

i had a horrific case of the chicken pox. i missed about a month of school.

it's worse when you're older. i had it at 12.

copper.hat

none were fun, but none stuck out other than the itching

my worst ever was jaundice (hep a)

lost about 15 lbs in a short period of time. discovered i had it during the fastnet disaster when i was helping our boy scout troup in a camping trip. 4 of 5 tents were flattened that night so we all ended up sharing. thankfully no one else got infected.

leslie townes

7:36 PM

i have gilberts syndrome, which means my liver doesn't process a blood pigment and i'm always a tiny bit yellow. not that you'd notice, but whenever i meet a new doctor they want to run all kinds of blood work.

it's harmless and apparently positively associated with less heart disease. i'll go a bit yellow for that.

it presents as jaundice.

my wife wants to know how to lose 15 lbs in a short period of time. i said, don't bother it's not worth it.

copper.hat

it is prevalent in persia...

i would advise against hep a as a weight loss route :-)

other advantages are that you can no longer donate blood

leslie townes

oh, that sucks. i like donating blood.

copper.hat

vampire

leslie townes

my high school has a letter from the president commending us on a blood drive. we were apparently one of the first high schools in the USA to do a blood drive.

geocalc33

hi

copper.hat

7:44 PM

excellent!

leslie townes

then conspiracist media said that the blood tested positive for HIV. it hadn't. but apparently you can't do anything nice.

i've never seen my dad yell at somebody, except when he got on the phone of the guy who wrote that.

hello geocalc.

hello leslie townes.

any math today?

no

i don't have any either.

geocalc33

7:53 PM

there hasn't been much math today

today there hasn't been much math

leslie townes

8:10 PM

hasn't there been much math today

copper.hat

i'm a monomath. if even that.

trying to persuade my daughter to take a taxi to the airport for her upcoming trip. she doesn't want to 'waste' the money and take a bus instead.

20yo girl at 3am on the streets of london/

leslie townes

yeah, take the taxi.

copper.hat

she's battling me all the way

leslie townes

tell her some guy on the internet said to do it.

copper.hat

:-) not sure she will go for it :-)

i told her i will pay all costs

she's stubborn

leslie townes

8:14 PM

i'm going to be dealing with this kind of thing soon enough. i can't wait.

copper.hat

:-)

leslie townes

my daughter thought it would be fun this morning to run around naked instead of getting dressed. then she demanded to go to the duck pond although she knows that's a weekend thing and it isn't the weekend.

copper.hat

it is good to ask.

leslie townes

she was very good about her wednesday bath though. probably the least problematic bath in weeks.

she's always looking for an angle.

copper.hat

it is good for her to negotiate. she needs to understand she has input that will be considered. she also needs to see the 'bigger picture'.

which is a little tough at 3

leslie townes

8:21 PM

my daughter is going to run the world some day. i'm sorry in advance.

copper.hat

scratch my advice above. in case she reads it.

leslie townes

she isn't even 3. i didn't mouth off the way she does at 3. it's alarming.

my wife and i both love it, it's just scary. what's she going to be at six?

copper.hat

if there is one thing i learned from management (which i truly hate) it is that you must think of the long term effect of decisions.

not so easy when there is a lot of screaming going on

i am emotionally weak. only saved by some irish inadequacy of expressing feelings.

leslie townes

if my daughter learns how to express her feelings in an emotionally healthy way, it won't be from me.

the last funeral i was at was a masterpiece of dysfunction. the priest said his things and nobody else said anything. i shook hands with someone and left.

copper.hat

8:39 PM

generally irish are unstoppable in the talking department, even at funerals

but just being there and expressing care helps.

leslie townes

i couldn't think of anything so i just shook hands and left. no hugging. it was implicit. i took off work to go to the thing.

copper.hat

my daughter seems to be an improvement on me in that regard.

my son, i'm not so sure. a seething pot of emotional tension.

leslie townes

i was an emotional tire fire from 15-25. i cooled down after that.

copper.hat

i can't speak for others, but i think just being there is good.

leslie townes

i met my wife when i was 21. she had a few years of annoyance.

it's really hard to learn how to become a man. if you're like most of us you have no good influences.

copper.hat

8:44 PM

my dad combined many attributes. i think as kids we learned from the good and worked around the worst.

to be fair, we had no really bad family influences.

which is a bit surprising.

leslie townes

my dad did his best. he had a horrible upbringing. i didn't appreciate that until later.

copper.hat

my dad & myself never got along. but i appreciate him more as i get older.

he threatened to not come to our wedding two weeks prior.

leslie townes

his dad was fun when he was around, and very charming (i met him once), but he was never around because he was constantly gambling and fooling around with other women.

copper.hat

that was, to put it mildly, awkward

leslie townes

and he spent most of his evenings at the neighbor's house because they could afford to feed him and his mom couldn't. awful stuff.

my daughter's only antagonist is the cat. that's me doing a good job.

copper.hat

8:47 PM

my da could be a charmer when he wanted. he was (by irish standard of the time) tall, dark & handsome and was (relatively) successful.

but he had a dark side.

not his hair colour

leslie townes

my grandfather was a master at BSing people. just nonstop BS. sentimental BS if he thought it would work.

copper.hat

i guess we learned that the two come together :-)

leslie townes

he mostly worked in K-12 education and would do speeches on the holidays and really wring it out of people. he could put you in tears. it was manipulative and mostly false but he could do it.

copper.hat

unfortunately none of us had the real gift of the gab...

my brother the doctor is pretty good by family standards.

leslie townes

if anyone had it, my grandfather did. it was largely phony, but he had it.

his brother was a sociopath and also a psychiatrist. weird combination.

copper.hat

8:50 PM

nothing like the healy-raes en.wikipedia.org/wiki/Healy-Rae family

just put a target on myself

leslie townes

uh oh

one of my dad's cousins is an anglican priest and maybe one of the worst people i have ever met in my life. he is malignant.

the first time i met him, he spent 20 minutes telling me how his daughter was a failure. his daughter was in the room.

i wanted to throw up

copper.hat

i have no patience at the best of time.

geocalc33

how do you make an aperiodic pattern?

leslie townes

look for the period, and then don't do it.

they awarded me a phd for that level of insight. in all seriousness i think you just look at previously discovered patterns and do them. i don't know if there is an algorithmic machine that cranks them out.

geocalc33

are aperiodic 2D patterns always slices of some higher dimensional space?

leslie townes

8:59 PM

there's a lot of geometry floating around tilings but i am not aware of the details.

geocalc33

I am not certain but I think I can construct an aperiodic point pattern using my own method

i don't think it's mathematical though

i basically take thickened lines and overlay these lines using rotations about a central point

and where no lines cross there's a gap and then I take that to be a point

and then the set of these discrete points form an aperiodic pattern

Dubias

hey ive got a rather silly question. if we know that $0 \leq x_1,x_2 < \infty$ and $y = \frac{x_1}{x_2}$, then what would $y$ be bounded by?

geocalc33

but the discrete points are finite

DIRAC1930

Could someone help me with some questions about representation and group theory?

leslie townes

dirac, you won't know until you ask. i have some knowledge in this area but would not put myself forth as an expert.

geocalc33

9:09 PM

@leslietownes what do you think?

DIRAC1930

Okay thanks, I'm just a bit confused on how the representation theory of the Lorentz group

Specifically, why do we complexify the algebra

I'm struggling with some basic definitions such as what is the double cover of a group and what is a universal covering group.

For example, if I have a group G, is the double cover G' a group that has 2 elements that map to 1 element of G?

Thorgott

9:34 PM

covers are first and foremost a topological concept, not a group-theoretic one. in the case of Lie groups, a cover can be again made into a Lie group, but I'd advise you to understand the concept on a purely topological basis first.

pritchard

10:01 PM

Hi, can anyone help me with this understanding the answer to this question? I was wondering why we can't just take the closure of the finite cover which then is compact, map it by p inverse to $\tilde {X}$, which will then give us the finite union of compact sets showing $\tilde {X}$ is compact? Here is the question: math.stackexchange.com/a/1072276/810585

Thorgott

can you a bit more precise about what you're trying to do?

pritchard

I am thinking that instead of finding the closed (and thus compact) sets by shrinking the open cover $\{U_i\}_{[1,N]}$, why can't we just take the closure of each $U_i$ which is also compact. And it should map back the same way I think?

Thorgott

the closure of $U_i$ is not necessarily evenly covered anymore

so it is not clear that its preimage is compact

(it will in fact be compact, but this needs an argument)

pritchard

Sorry, what does evenly covered mean?

Oh I see

@Thorgott Could you give me a hint as to why the closure under preimage would still be compact?

Thorgott

10:21 PM

I do not have a good argument that is non-circular (from where you stand)

pritchard

What would be the circular argument?

SAJW

so if a mask protects against 99% of viruses, 1% spreads exponential?

Balarka Sen

@TedShifrin Fantastic news!!

Take care, drink lots of fluid.

Thorgott

well, if $K\subseteq X$ is compact, $p$ restricts to a finite covering space $p^{-1}(K)\rightarrow K$

by the way, are compact spaces Hausdorff by definition to you?

cause the answer assumes this (but the result is true with either convention)

pritchard

For this question, I am assuming $X$ is compact Hausdorff

Thorgott

10:28 PM

ok, then the argument in the answer works fine

Avra

11:03 PM

Have you ever been into this logical question please: Consider the following induction “proof” that all sheep in a flock are the same
color:
Base case: One sheep. It is clearly the same color as itself.
Induction step: A flock of n sheep. Take a sheep, a, out of the flock. The
remaining n − 1 are all the same color by induction. Now put sheep a back in
the flock, and take out a different sheep, b. By induction, the n − 1 sheep (now
with a in their group) are all the same color. Therefore, a is the same color as all

Why induction does not work here? I understand that it's supposed to hold for base case and then we assume IH (induction hypothesis) in order to prove the induction step.

So, if we follow these rules, why we can not conclude that all sheepes have the same color?

Thorgott

ah, this is a famous fallacy

examine your induction step again when n=2, this is the critical part

Avra

@Thorgott. Thanks. Is this sort of strong induction where you have to show that some cases hold before the Induction Step please?

Thorgott

I'm not sure I understand what you're asking

Avra

2

copper.hat

what is $P(n)$?

Avra

11:11 PM

@Thorgott. Do you know strong induction please?

@copper.hat. $P(n) = assume~ all ~n~ sheeps ~have~the ~same ~color$.

Thorgott

I know what strong induction is, yes

Avra

@Thorgott. In strong induction, we add more than basic and induction step, so probably it's easier to prove it with strong induction

copper.hat

why would $P(n)$ imply $P(n+1)$?

Thorgott

I'm not sure what you're trying to get at

you can't prove something that is not true

Avra

@copper.hat. If we assume that $p(n)$ holds, then we want to prove that $p(n+1)$ hold as well.

@Thorgott. I am not saying that I will prove that it's correct, but probably with strong induction it's easier to show that it's false/correct

copper.hat

11:14 PM

i understand that. if you have $n+1$ sheep and you pick one, why must that be the same colour as the rest?

Thorgott

it is not correct. the induction fails at the step n=1 => n=2 which is the same for both strong and weak induction. you don't show this is false by induction, you simply do it by example.

Avra

@copper.hat. The assumption was that all up to $n$ is true, so we have to show that for one extra step, which is $n+1$. It could be either true, or false.

@Thorgott. Got it. Thanks. So, why we don't follow similar thinking to all other proof by induction examples. We used before to prove basic step, jump to IH before we prove it for induction step!

This example is one and only among proof by induction that I saw

copper.hat

if you assume that any group of $n$ sheep have the same colour, then for $n>1$ you are assuming that they are all the same colour.

Avra

Even I did not see this in graph theory

copper.hat

there is no fallacy really. it is just that the assumption is must stronger than you think.

this occurs in formal verification. people make innocuous appearing assumptions which turn out to be rather stronger than they expect.

a bit like proving that the system behaves correctly as long as the on button is not pressed.

Avra

11:21 PM

@copper.hat. Wow! Thanks, but can you please rephrase if you can, "assumptions which turn out to be rather stronger than they expect."?

what is meant by "assumption is stronger than you think" please?

copper.hat

the assumption that any group of 2 sheep have the same colour is the same as assuming that all sheep have the same colour.

(assuming of course that i can group any two sheep :-)

Avra

@copper.hat. I have one last question please. Do you agree this is odd in sample of question that we used to prove by induction?

:/

copper.hat

the key element is "any 2". that is why @Thorgott was focusing on 2

i don't understand your last question.

Avra

@copper.hat. I did not see this kind of reasoning applied to other prove by induction questions

copper.hat

which sort of reasoning?

Avra

11:25 PM

That we check for $n=2$ or "any $2$"

copper.hat

the idea is just (i) assert $P(0)$ and then prove $P(n) \implies P(n+1)$.

Or in this case, replace (i) by $P(1)$.

$P(n)$ true means any group of $n$ sheep have the same colour.

$P(1)$ is clearly true.

Avra

Okay! Thanks.

copper.hat

However, assuming $P(2)$ is the same as assuming $P(n)$ for $n \ge 2$.

for exactly the reasoning you gave in the original statement.

Avra

Strong induction is better then for this sort of problems?

copper.hat

However, it is not true that $P(1)$ implies $P(2)$.

i don't really distinguish weak from strong induction, it seems like an artificial distinction to me.

Ted Shifrin

11:31 PM

No, strong induction is crucial when you prove prime factorization.

How could factoring $n$ help you factor $n+1$?

copper.hat

in the sense that you can reword your proposition.

Avra

@Thorgott. @copper.hat. @TedShifrin. Thanks.

Ted Shifrin

@copper.hat Yes, but for most undergraduates this is too complex.

Thorgott

you prove prime factorization by Noetherian induction :P

copper.hat

i do find nets, proofs by transfinite induction a little detached from my reality

Ted Shifrin

11:36 PM

Your reals are finite? :D

copper.hat

no logical issue. just lacking any real intuition.

i live in an irrational world.

Ted Shifrin

I have never learned or taught nets. And I’m almost dead.

Thorgott

well-ordered induction is a very straightforward generalization of strong induction

transfinite induction is just well-ordered induction applied to ordinals

Balarka Sen

I liked nets

I have forgotten them

Thorgott

nets are cool

the fact we were discussing earlier - a finite covering space of a compact space is compact - is imo much easier to prove using nets

copper.hat

11:38 PM

i liked them but never really used them other than homework.

Thorgott

in fact, I do not know a proof in the non-Hausdorff case other than the one using nets

copper.hat

i am not a fan of sequence proofs either, if a reasonable alternative can be found

Ted Shifrin

I like net profits.

copper.hat

i second that

Balarka Sen

I like sequences

I use them almost always

copper.hat

11:39 PM

they are faster & more convenient, but often obscure the underlying ideas

Balarka Sen

I think in terms of sequences. Open sets do not make sense to me

A neighborhood basis is what describes a topology for me, which is basically a sequence. Or rather, a net.

In that sense, I think in terms of nets.

But I have also forgotten the language.

copper.hat

assuming some countability :-)

Balarka Sen

Yeah, given assumptions we get to go back and forth, formally. But the basic idea is all the same. There has to be some good notion of closeness. Open sets can be huge

Neighborhood bases/sequences/nets pin down closeness. Stuff converges.

Ted Shifrin

When stuff is congested, maybe.

Balarka Sen

Nice terminology.

Thorgott

11:43 PM

yeah, nets are just a way of making this tautological