Mathematics - 2021-03-04 (page 1 of 3)

leslie townes

00:10

Riemann Rock is a subgenre of indie rock that had a lot of underground popularity in the late 1990s

Jakobian

Ah yes, wheels

I don't think those are even in common algebraic structures at wiki or related

en.m.wikipedia.org/wiki/Outline_of_algebraic_structures

not here, nor on any of the sites wiki directs to on that page

leslie townes

"Unneeded" is an ugly word

Jakobian

but I did hear about them before. Just as a curiosity

and I do dabble a bit sometimes with more exotic structures as well as universal algebra

so I can see someone not hearing about it, definitely not common knowledge

user2103480

00:33

@EdwardEvans compsci is easy credits

And double the careerwise benefit

leslie townes

00:57

george bergman has a good book on universal algebra. i recommend it if you haven't seen

math.berkeley.edu/~gbergman/245

love_sodam

01:12

Is there an example that for integral domains R, S, for each prime ideal $I\subset R$, there are infinitely many prime ideals $J\subset S$ such that $I = R\cap J$??

Thorgott

I assume you want $R\subset S$?

consider $k\subset k[x]$

love_sodam

yes yes sorry

k is a field?

Thorgott

ye

love_sodam

Ah, k has only zero ideal so any prime ideal of k[x] satisfy that condition

zero ideal as prime

Thorgott

indeed

love_sodam

01:23

Thanks!

Thorgott

np

Jakobian

@leslietownes I do know about it, but I read Sankappanavars book before I found it, and I wasn't that interested in reading it

but I heard Bergmans book is really good

leslie townes

01:39

i used some ideas from it in my thesis. i was a teaching assistant for bergman a few times in the late 1990s. his students tended not to like him personally very much, but they loved his materials. the book is a good book.

anakhro

Any ideas on a simple function $h$ such that $h(z+1) = z^{-1}h(z)$?

Kind of like $h(z) = e^{az}$ has $h(z+1) = e^a h(z)$

Ted Shifrin

Sounds like $1/\Gamma$.

anakhro

Does it? I figured out the previous question by the way, Ted. It was simple and I was overcomplicating it. All you needed to do was set $f(z) = e^{-az}g(z)$ and it works.

But now I am on a slight extension of this but I can't figure it out. :P

namely given a polynomial p(z), I want to find f such that $f(z+1) = p(z)f(z)$.

(entire f)

Constants are easy to do given what I wrote above.

Ted Shifrin

Factor $p$ and then use a product of shifts of $\Gamma$?

leslie townes

01:57

that's a clever idea

anakhro

But I can't use the gamma function.

But I am guessing if it works for the gamma function then we can do it with my solution for f for my previous question.

In what way would I shift the Gamma function for it? I don't think shifting the argument works.

Mike Miller

But why does one want this?

anakhro

I wish I could tell you, Mike. I really do wish I could.

But exercises are exercises, and sometimes they just seem weird.

Mike Miller

Fair enough, but not enough to hook me on the problem!

anakhro

places Mike's favourite snack on top of the exercise.

By the way, what is everyone's favourite food? I need something new to cook.

Ted Shifrin

02:13

Roast duck :)

anakhro

Have you always enjoyed roast duck, or is this a taste you acquired when you were older?

Ted Shifrin

Always, and thai spicy duck curry, yum.

Mike Miller

With a long family history of heart issues I think I'll pass on the duck most days.

Ted Shifrin

Yeah, I have heart disease, so it's a rare treat. But, properly cooked, you cook off all the fat .

leslie townes

i like sashimi. not sure that it is something to cook. i am fantasizing about ordering it tomorrow

Ted Shifrin

02:19

Simple dish: poulet au diable.

You have to buy super fresh sushi-grade fish.

I just got some yellowtail tuna and cooked it rare.

leslie townes

that sounds delicious. i love yellowtail

Ted Shifrin

Great fish in San Diego.

leslie townes

yes, san diego is great

bear flag seafood in newport beach is a good place. i don't know what it's like now. they made another one south of newport and another one in huntington. very high quality

i wish restaurants would come back. it's coming up on a year since i've been in an office

BigSocks

blanking on regular ole logic here @anakhro math.stackexchange.com/questions/4048001/…

anakhro

I don't know anything about that question, sorry BigSocks. :(

BigSocks

02:33

no stress- kind of a tricky one

Jeff

Can anyone offer help on how to show that $a_n < 1$ where
$$a_n = \sum_{k=n+1}^{2n} \frac 1k.$$

Ted Shifrin

It's easy. Do the most basic estimate.

Jeff

02:47

What is 'the most basic estimate'?

Ted Shifrin

What is the largest term?

Jeff

1/(n+1)

Ted Shifrin

So what's the largest the sum could be?

Jeff

1/(n+1) * (2n-n)?

Ted Shifrin

Which is?

Jeff

02:49

@TedShifrin That works. You're right

@TedShifrin ...less than 1

Ted Shifrin

Yippee.

Sayan Chattopadhyay

Hello @Ted

Jeff

For the same
$$a_n = \sum_{n+1}^{2n} \frac 1k$$
is it sufficient to say that $a_1=1/2$ and $a_n = a_1 + \text{positive terms}$ to prove that $a_n \geq \frac 12$?

leslie townes

03:12

i agree that a_1 is 1/2. i don't see where you're finding a_n = a_1 + positive terms. i'd be tempted to estimate the thing with an integral. there must be something simpler

ok, a_n is a sum of n terms. the smallest is 1/2n. not a lot going on there

i'd still be tempted to use integrals

Jeff

@leslietownes, well, the goal is to estimate so that we can use integrals to show that $a_n \to \ln(2)$ as $n \to \infty$.

So we can't really use integrals

Fargle

@Jeff That's sufficient, yes.

Wait, hang on, I think I misread the thing.

Jeff

@Fargle OK, waiting....

Fargle

It's not true that $a_n = a_1 + \textrm{positive terms}$, but as Leslie noted, $a_n$ is a sum of $n$ things, each of which are at least as big as $1/2n$. Therefore their sum is at least as big as $\frac{n}{2n} = \frac{1}{2}$.

Like, $a_n$ is indeed bigger than $a_1$, but it doesn't break down into a sum of $a_1$ and other stuff in an obvious way. Leslie's proposed approximation is the right one here.

Jeff

Oh wait, that's right. I was confusing $a_n$ with $\sum a_n$ (which is mentioned in the problem, but not relevant to this part).

So one thing I'm noticing is that $a_n = a_{n-1} - \frac 1n + \frac{1}{2n-1} + \frac{1}{2n} = a_{n-1} + \frac{1}{2n(2n-1)}$ (previous term, less the first fraction, plus the last two). So then $a_n is always increasing.

Martin Ødegaard

03:33

Suppose $f(z) = ℘(z) − ℘(u)$, where $℘$ is the Weierstraß $℘$-function. Then $f$ has double zero at $z=u \bmod{L}$ if $2u \in L$, where $L$ is the lattice generated by $\left\{\lambda_1, \lambda_2\right\}$.

How do I construct an elliptic function with a double pole $z=0$ and simple zeros at $\frac{\lambda_2}{3}$ and $\frac{2\lambda_2}{3}?$

I think $f(z) = ℘(z) − ℘(\frac{\lambda_2}3)$ has a double pole at $z=0$ and a double zero at $z=\frac{\lambda_2}{3}$ and $g(z) = ℘(z) − ℘(\frac{2\lambda_2}3)$ has a double pole at $z=0$ and a double zero at $z=\frac{2\lambda_2}{3}$. Then maybe $\frac{f(z)g(z)}{(z-\lambda/2)(z-2\lambda/3)}$ is the required function?

Fargle

@Jeff That would be sufficient for the reasoning as well.

Jeff

@fargle Thought so. TY

Martin Ødegaard

In trying to remove the extra zeros I increased the poles I think.

I would appreciate if someone could give me a source that gives examples of how to construct elliptic functions from poles and zeros

robjohn

03:54

@TedShifrin: you're following me...

Ted Shifrin

@robjohn Where are we going?

robjohn

@TedShifrin I don't know. I floated into the room, and then you floated in after.

Ted Shifrin

Ah ... Well, I don't expect to float anywhere.

robjohn

Faculty and staff at UCLA have been green-lighted for vaccination.

one of my friends at the library got theirs today

Ted Shifrin

Great!

robjohn

04:00

he had a bad reaction, but I bet he'll be better tomorrow

he had bad flu symptoms

Karim Mansour

Hi @TedShifrin

Hi @robjohn

robjohn

hey there

leslie townes

my wife, also an educator, had some symptoms from the first round of the two-shot vaccine. mostly a very sore arm. apparently the second shot can be worse

Ted Shifrin

Yes, second is typically worse. Sore arm is totally standard.

Hi Karim..

dc3rd

Any more interesting stories from the good ol' days at Cal between the three of you? Been fun reading the small anecdotes you guys have been discussing as I snoop around and eavesdrop...................

Jeff

04:10

@leslietownes My mom had the same reaction to the first shot, but not to the second.

Ted Shifrin

Different Cals. Two Berkeley, two UCLA ....

Jeff

@TedShifrin Really? That's opposite of my mom's experience.

Ted Shifrin

What I said seems to be typical.

leslie townes

i've got one. grigory barenblatt, a student of kolmogorov and a gifted lecturer, has now passed on. he always reminded me of a james bond villain. one time i was in the elevator and he got on, and pointed at me, and said "your system is incomplete." i didn't know what he was talking about. i thought he was being very figurative. then my officemate told me that the top button on my shirt was undone.

Ted Shifrin

I had chills, fever, headache after second.

Karim Mansour

04:11

@TedShifrin I want to study something associated with smooth projective surface.

Ted Shifrin

Lol

Karim Mansour

What is the best way to consider a family ? I didn't learn about this stuff yet. So far I have in mind hilbert schemes and spreading the variety

i.e varrying the coefficients and constructing $f : \mathcal{X} \rightarrow S$ where S is the original variety.

Ted Shifrin

I don't know what you're talking about, karim. I've said that before. So you need other sources, not me.

Remember that in general deformations may be non-algebraic, even non-Kähler.

Karim Mansour

alright

dc3rd

did he pass on recently @leslietownes? or was that just stating where he is now?

Ted Shifrin

04:15

So just perturbing coefficients is not a good generic family.

Karim Mansour

right

yeah exactly this is something that I thought about. I want the best way to generate a family

leslie townes

he passed a few years ago. i just think of him whenever i fail to button my shirt all the way. someday i hope to tell somebody else that their system is incomplete

Ted Shifrin

You can compute dimensions and see that.

Karim Mansour

right

Ted Shifrin

This is 50+ years old stuff.

Karim Mansour

04:18

I see

leslie townes

holy shit vaughan jones died. i somehow missed that. what an awful year

Karim Mansour

ok I have some idea

love_sodam

0

Q: Criterion of map $M_f\to Z$ to be continuous

I was about to prove the following proposition A map $M_f\to Z$ is continuous if and only if the induced maps $X\times I\to Z$ and $Y\to Z$ are both continuous. Here, $M_f$ is a mapping cylinder of $f:X\to Y$ i.e. $M_f = X\times I\sqcup Y/f(x)\sim (x,0)$ for $x\in X$. I first let $p:X\to X\time...

Anyone can check if my proof is valid?

Bredon says 'the proof is an easy application of the definition of the quotient topology on a mapping cylinder' but my proof seems not very easy or trivial.

Ted Shifrin

A while ago, @leslie.

love_sodam

04:33

Is there any simper proof?

leslie townes

yeah i've been distracted by work stuff, it's like a whole year of news is still news to me. and none of it is good

Ted Shifrin

05:25

Better not to catch up!

leslie townes

my daughter has no idea how weird all of this has been. she puts on a mask and goes to day care and has never been in the same room as her grandparents

it truly is better not to catch up

Leonhard Euler

06:33

if infinite monkeys are kept at a place with a typewriter

then atleast one of them will start by writing the proof of the riemann hypothesis

in fact infinitely many will start by writing the proof of RH

but not all

Leonhard Euler

07:37

I am having trouble simplifying algebraic expressions in c++

my code is 150 lines long and it is still not completed :(

Rover

08:08

Any hints guys!

Mary Star

09:25

Hello !! Which of the transformations rotation, translation, line reflection, glide reflection has no invariant points?

love_sodam

09:57

Proving $M_f$ is T_2 when two given spaces are $T_2$ is harder than I first thought...

Jack Ohara

10:20

@LeakyNun Hi I have a question about logic

Leaky Nun

just ask it

Jack Ohara

10:32

can you give example of a theory

that is consistent , complete but not deductivly closed?

@LeakyNun I saw you on the logic forum haha that is why I'm asking you

am proving this statement

a theory is maximally consistent if and only if it is consistent , complete and deductively closed,

but in the reserve direction, there is no need for deductive closure

Leaky Nun

@JackOhara suppose T is a consistent complete theory. suppose T proves phi. By completeness, either phi in T or (not phi) in T. but if (not phi) in T then T isn't consistent. so phi must be in T. so T is deductively closed

love_sodam

11:05

1

Q: Mapping cylinder is Hausdorff

If $f:X\to Y$ is a map between Hausdorff spaces, then $M_f$ and $C_f$ are Hausdorff. Here, $M_f$ and $C_f$ are mapping cyliner and mapping cone i.e. $M_f = X\times I\sqcup Y/(x,0)\sim f(x)$ for $x\in X$ and $C_f = M_f/(X\times\{1\})$ I first tried to prove $M_f$ is $T_2$. Before, as $X$ and $Y$...

general-topology

Anyone give me some hint of this problem?

Sarabsri mt

11:28

Guys . I am going to start making notes for physics and chemistry on IPAD using Apple pen. It saves a lot of time. But it will hurt eyes more. Do you think I should do it ?

robjohn

@Sarabsrimt isn't there software that can take your handwritten notes into some better format?

Sarabsri mt

@robjohn I didn’t get you exactly ?

@robjohn There is an app on iPad that is really good for handwriting notes

Also , I can arrange things better and write things that I may have left on the same using iPad

robjohn

@Sarabsrimt yeah, it allows you to write your notes with the pen and then convert it to text so that it is easier to read

Sarabsri mt

My only fear is that people say kids who wrote on notebooks are better in remembrance than kids on digital media.

Better in learning , thinking also

Sioud Abdou

12:37

are we learning the same programm of math all over the world of the same years

Jakobian

No

People learn at different pace in different countries

Mike Miller

Even in one country the curriculum can vary quite widely between institutions.

geocalc33

12:55

Am I correct in saying this series converges only for real $s>0?$ $$ \rho(s)=\sum_{\rho\in\Bbb P} e^{-\rho^s} $$

user2103480

man how should we know you're using two different rhos in that series and dont even specify what $\Bbb P$ is. Are those the primes?

geocalc33

I used one rho

user2103480

yes, I mean semantically different rhos

geocalc33

$\Bbb P$ is the set of primes

Astyx

is s real?

geocalc33

13:00

$s \in \Bbb C$

If $\Im s\ne0$ then I don't think $\rho(s)$ converges

Astyx

$s=a+ib$, $p^s = p^a e^{ib\log p}$, $||e^{-p^s}|| = e^{-p^a\cos({b\log p)}}$

I'm willing to bet that $\cos(b\log p)$ is dense in [-1,1] when $b\ne0$, which means the general term is not bounded

geocalc33

so when you want to see if you can extend a series such as $\rho(s)$ do you let $s$ be a complex number

and if $\rho(s)$ doesn't converge for $\Im s \ne 0$ then it can't be analytically extended?

$$P(s)=\sum_{p\in\Bbb P} p^{-s}$$

this function has a "natural boundary" at $\Re s=0$

and $P(s)$ converges for $s>1$

schn

13:22

Consider the Gaussian and Cauchy-Lorentzian distribution as defined in the screenshot:

They're defined in terms of the full width half maximum.

When only changing the full width half maximum, that is $\Gamma_{L,G}$, will these distributions still be normalized (so that the integral from negative infinity to positive infinity equals 1)?

Jack Ohara

@LeakyNun thanks but that is not what am trying to prove

in my prove no need for deductive closure is needed

i mean assuming complete and consistent we can get to max consistent , but not sure if there is something I missed or not

love_sodam

13:39

Why do we call this Klein bottle even if it's not Klein bottle?

Mike Miller

This is an immersion of the Klein bottle into R^3.

From this one can picture the construction (paste ends of a cylinder together in an orientation-reversing way).

Or, if you are willing to allow yourself color as an extra dimension, from this picture one immediately can see an embedded Klein bottle in R^4.

In addition usually Klein bottles are discussed for popular audiences. I am not certain you can convince a popular audience of an object which they cannot "see".

Jaakko Seppälä

13:52

Is the sum notation defined for $\sum_{k=1}^{-1}a$?

Astyx

usually it's defined to be zero

Jaakko Seppälä

okay. Thanks

Mike Miller

I would say: $\sum_{k=a}^b f(k)$ is the sum "For all $a \leq k \leq b$, evaluate $f(k)$, then add up all the results."

When $a > b$ there is no such $k$. Therefore you are adding a sum over no numbers. Traditionally the "empty sum" is defined to be zero.

After all, if I have already added a bunch of numbers, and then I add no additional numbers, that's the same as adding 0.

Thorgott

@MikeMiller So say I have an isotopy MxI->N. Switch to the track map MxI->NxI, now this becomes an embdding, so, by keeping the first component fixed and varying the time t, we obtain a bunch of "flowlines" on the embedded copy of MxI in NxI. I wanted to extend these to a vector field on NxI, then flow along that and use that to construct a diffeotopy extending the isotopy, but I haven't managed to do so. Is this at least a sensible idea or am I completely off?

Mike Miller

That's the right idea

Your field on F(M x I) subset N x I has a special property you need to demand on your extended vector field

But you've now managed to do the clever part

The rest is small technicalities

Thorgott

14:01

hmm, I probably want to demand it is constant 1 in the TI factor of T(NxI)

that should give me control on the second component of the flow, so that I can forget about it again

Mike Miller

You got it, the point of course being that $\Phi^t(m, s) = (\Phi^t(m), s+t)$ where your flow is already defined

Leaky Nun

@JackOhara what do you want to prove then

Matthew Christopher Bartsh

I answered a question in English Language & Usage and then I saw almost the same question in Mathematics. Am I allowed to answer the question in Mathematics in the same way as I answered the question in ELU?

Astyx

14:34

I don't see why not

Jack Ohara

@LeakyNun I think assuming the theory is consistent and complete is enough to show maximally consistent

sorry for late replies, somehow I don't get notifications when people text me

user2103480

@JackOhara yeah deductively closed is redundant here. Suppose a theory is complete and consistent. Let $\phi$ be any sentence. Suppose $T \vdash \phi$. Since $T$ is complete, either $\phi \in T$ or $\neg \phi \in T$, so $T \vdash \phi$ or $T \vdash \neg \phi$. The second one is impossible by consistency so $\phi \in T$ must hold

Bob Pen

Hey, I was just wondering. In Linear Algebra, why does one want minimum norm solution ?

Jakobian

for statistics estimators? for example

Bob Pen

Yeah

Given a least squares regression problem

Why does one want the solution to the minimum norm solution?

Astyx

14:47

to have the best regression possible

Bob Pen

Let us say $Ax = b$ , where $A$ is rank-deficient

But what if you minimize $||Ax-b||^2$ and you obtain an $x$ solution with a higher norm

Is it then a worse regression ?

I found this question where math.stackexchange.com/questions/3931144/…

Where there is an answer in the comments, but the answer says that there is a nice general explanation. So that got me wondering, does the minimum norm imply a better fit for a regression problem?

Jakobian

is there an s-dimensional ball in R^n

something that would be a natural set of Hausdorff dimension s

Thorgott

the unit ball from R^s?

Jakobian

15:03

no

for any positive s<n

Mike Miller

I dunno about "natural" and it certainly wouldn't deserve the name unit ball

But I agree it would be nice to see an explicit construction for all s

Especially if one can make it continuous in s

Jack Ohara

@user2103480 I see thanks ! I argued in a similar way

Mike Miller

@Jakobian This would be a nice MSE question. Write C(R^n) for the closed subsets of R^n, considered as metric space under Hausdorff distance. (Hausdorff dimension is not continuous in this space but maybe there is a variant where it is.) Can one find a continuous map F: [0,n] -> C(R^n) so that F(s) has Hausdorff dimension s?

Another natural question: is there a good topology on [some kind of] subsets of R^n so that Hausdorff dimension is continuous? This is a soft question since someone dull might answer "the discrete topology" and the only thing this violates is that it's obviously not what I mean.

I guess smoothly varying submanifolds ought to define continuous variations in whatever this space is.

Thorgott

easiest thing coming to mind would probably be some Cantor-type constructions

Mike Miller

15:20

I'm not terribly familiar with Hausdorff dimension so I don't have a sense for how plausible this ought to be

Notice how easy it is to construct an example of dimension-dropping: take [0,1]^{n-1} x [0,t]. This is certainly continuous in the Hausdorff topology and in fact it's hard for me to imagine a non-stupid topology where it's not

But that seems more related to my lack of imagination than anything else

Jakobian

honestly all I was looking for is some natural example of set of dimension s, so that I can use it to show that the approximate Hausdorff measure doesn't satisfy the Caratheodory condition for Borel measures

the one which Hausdorff measure is limit of

Mike Miller

Yeah but what does natural mean

I'm trying to encode that in terms of some continuous variation

Jakobian

just something I could work with

Mike Miller

So more like "explicit" I guess

Anyway I was outlining a question above, not something I have any answers to :)

Jakobian

that was interesting though

leslie townes

15:27

there are some topologies out there crying because they were just called stupid. i hope you're happy.

Mike Miller

They probably needed a wake up call.

leslie townes

hausdorff and i share a birthday. with frege

geocalc33

I wonder who was the last person to not share a birthday with someone

leslie townes

15:42

one time i was teaching probability and began with the birthday paradox. "for example, i'm born on november 8." by some strange coincidence, two people in the room were also born on november 8. i think this may have conveyed the wrong message re the birthday paradox. "in a group of 30 people, for any date you choose, there will be three people born on that date."

geocalc33

I remember one of my profs explaining the birthday paradox

Matthew Christopher Bartsh

@Astyx was "I don't see why not" to me?

Astyx

Yes

Matthew Christopher Bartsh

Do you know where I can read a rule about that? I couldn't find any information about it.

Astyx

You could have a look around math.stackexchange.com/help

As long as what you post is in good faith there's no issue AFAIK

If your posts are against the rules, someone will point it out and you'll know why

Don't stress about it

You could link to the original answer

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