Mathematics - 2018-02-21 (page 1 of 4)

Xander Henderson

00:00

someone said my name?

@MatheinBoulomenos What's the question? What are we convolving?

is it a positive summability kernel?

MatheinBoulomenos

@XanderHenderson we're convolving to approximate by continuous functions. @quallenjäger wants to extend some integral bound from continuous functions to a larger class of functions. I pinged you because you seemed to be there

Xander Henderson

I see; I was not there

what is the inequality? there are a couple of ways that one can try to make it work, depending on what kind of functions you are working with

MatheinBoulomenos

@quallenjäger are you there?

Xander Henderson

as you seem to have indicated, the typical trick is to mollify

@quallenjäger Where you at?

I WANT TO MOLLIFY!

HYPERFUNCTIONS!

quallenjäger

00:18

Yes I am here.

I have a Lipschitz continuous path $\gamma$ and I want to find a lower bound on the increment $\Delta \gamma =\gamma(t_1)-\gamma(t_2)$.

For a $C^1$ path $\gamma*$, I have the lower bound $\Delta \gamma* > C$

So what I have so far is $\Delta \gamma = \int_{t_2}^{t_1} \dot{\gamma}(t) dt$, where $\dot{\gamma}(t)$ is almost everywhere bounded.

Xander Henderson

hrm...

quallenjäger

Do you see what I want to do?

Or shall I continue?

Xander Henderson

no, please continue

$\dot{\gamma}$ is the derivative with respect to $t$, yes?

quallenjäger

yes

Xander Henderson

(I am not a physician---the physics notation is unfamiliar to me)

you can bound the integral explicitly in terms of the Lipschitz constant, no?

quallenjäger

00:25

But not from below

I need something from below

Xander Henderson

oh, yeah...

quallenjäger

I need it to be greater than 0.

Xander Henderson

oi...

hrm...

quallenjäger

So my thought is, if I can mollify my $\dot{\gamma}(t)$

Maneesh Narayanan

Documentary on Amal Kumar Raychaudhuri, the renowned theoretical physicist from Kolkata

►

Inspirational.

quallenjäger

00:28

then I know that the bound hold for the mollifier, namely $\Delta \gamma*>C$ for $\gamma*$ being mollifier and thus continuous.

I hoped that the lower bound would still hold true if I take the limit.

Maneesh Narayanan

@BalarkaSen

Xander Henderson

it seems like you should be able to use standard arguments to push the bound into $L^1$

but lower bounds are weird in this case...

that said, it is probably a standard $\frac{\varepsilon}{3}$-style argument

quallenjäger

which 3 parts do I need?

Xander Henderson

plus some reverse triangle inequality?

hrm...

no... I don't see it...

quallenjäger

What I have thought is

$|\int_{t_2}^{t_1}\dot{\gamma}(t)-\int_{t_2}^{t_1}\dot{\gamma}^*(t)|<|\dot{\gamm‌a}(t)-\dot{\gamma}^*(t)|_{L_1}$

lol what is this....

Anyway, we know that the latter term converges

because the mollifer converges against $\dot{gamma}(t)$ under $L^1$ norm.

Xander Henderson

00:34

So...$$ \left| \int_{t_1}^{t_2} \dot{\gamma}(t) - \gamma^{\ast}(t) \right| $$ converges to zero, no?

by general abstract nonsense...

quallenjäger

And I can somehow control the term $\int_{t_2}^{t_1}\dot{\gamma}(t)$ from not being to far away from the lower bound of $\int_{t_2}^{t_1}\dot{\gamma}^*(t)$

Yes, sorry for my notation.

Something is not working right.

Would this work?

Xander Henderson

so $\|\dot{\gamma} - \dot{\gamma}^{\ast}\|_{1} \to 0$.

quallenjäger

Yes

Xander Henderson

so fix $\varepsilon < \frac{C}{2}$

and pick a mollifier $\gamma^{\ast}$ such that $\|\dot{\gamma}(t) - \dot{\gamma}^{\ast}\|_1 < \varepsilon$.

By (reverse?) triangle inequality, you should get the bound you want on $[t_1, t_2]$

quallenjäger

I see, so this would work?

Xander Henderson

00:38

right?

maybe?

quallenjäger

Exactly what I have on my mind

Xander Henderson

it seems like a reasonable approach

quallenjäger

Alright, I will work this out, thanks

By the way

How do you make this double absolute value sign?#

I always use | on my keyboard :D

Xander Henderson

\|

quallenjäger

$\ |$

Xander Henderson

00:39

not ell, pipe

it is a slash, then the usual absolute value symbol

\|

hit the same key twice, just with the shift key down the first time

$\|$

quallenjäger

$\ ||$

shift key? I am using mac

same combination?

Xander Henderson

yes, so am I

slash pipe

\|

it is two characters

(right click on the symbol and view the math as TeX; maybe that will help

\| x - y \| is rendered as $\| x-y \|$

okay, I need to go make food; my people will be home soon

quallenjäger

Thanks, I will try this later:D

Thank you for your help.

Secret

01:13

(If the next era is mollification, what will that even mean?)

(Cause I have been seeing that word popping up quite frequently in recent weeks)

(Not that I don't know what mollification means, but rather, how broad it can be)

Sometimes, strange things happen when a meta analysis of the chat is performed

Xander Henderson

02:23

@Secret I gotta ask, man, 'cause I'm worried about you: how many hallucinogenic mushrooms do you ingest in a typical day?

Antonios-Alexandros Robotis

lol

0celo7

@BalarkaSen heeeelp

SoumyoB

02:48

@XanderHenderson on a related note, mushrooms, especially shiitake mushrooms improve gut health and have several testosterone boosting micronutrients

Gasparo

I no longer really believe what biological scientists tell me about the body and food, because it seems they keep changing their mind.

SoumyoB

the one thing that almost always remains consistent though is whatever they say about testosterone

whenever in doubt, always follow the diet/habits/lifestyle that increase testosterone

Xander Henderson

uh... sure...

I'm, like, super worried about my testosterone :eyeroll:

SoumyoB

I've found so much overlap between natural testosterone boosting foods and habits and foods and habits that boost cognitive performance and cardiovascular health

Xander Henderson

good for you

SoumyoB

02:55

Cold showers, intermittent fasting, magnesium supplementing are among the several things in that overlap.

@AlessandroCodenotti yes

Meow Mix

what about magnesium showers

Secret

@XanderHenderson well. zero. I am on an immunesupressant, which interact with many other food chemicals and drugs meaning I won't take anything else not prescribed

Balarka, Blue and many others have been wondering why on Earth I have so many weird dreams and even more weird behaviour, and has asked me whether I have take any hallucination agents, and I have took none

So I have no idea...

@MeowMix why are you showering yourself with magnesium?

Meow Mix

for my testosterone duh

Secret

Actuallly googles magnesium shower

SoumyoB

search results are pleasantly surprising

Secret

03:05

Ah I see, magnesium products, not magnesium flakes

SoumyoB

there are magnesium gels but sadly most of them would likely have BPA's that would introduce xeno-estrogens into the body that would reduce testosterone

Faust

1

Q: Finite dimensional vectors spaces external direct products and linear maps.

Suppose that V is a finite dimensional vector space over a field $\Bbb F$, and that $T \in L(V )$. Suppose also that $V = U_1 \oplus U_2 $, where $U_1 $ and $U_2$ are T-invariant. Let $\beta_1 $ and $\beta_2$ be bases for $U_1 $ and $U_2$, respectively, and let $\beta = \beta_1 \cup \beta_2$. ...

linear-algebra

can someone help me with the hint given for part 1? i cant seem to tell what $[T]_{\beta}^{\beta}$ is

i hate the fact that its been 7 years since i learned linear algerbra

have to look so much stuff

Xander Henderson

it's been nearly 20 for me ;P

Faust

yeah but i was in the tech industry for those 7 years

linear algebra doesnt come up...

the hardest question of the day, what to eat for lunch?

Xander Henderson

that is moderately surprising

food

Faust

03:10

good start

Xander Henderson

though, typically, I don't bother with lunch

Faust

where

what kind

Xander Henderson

coffee for breakfast

Faust

thats a hard question

Xander Henderson

more coffee for bunch

then a light meal of coffee for lunch

followed by a late afternoon snack consisting of coffee

and a sensible dinner!

Secret

03:11

@Faust My guess is the coordinate vector of the map $T$ under the basis $\beta,\beta$

Faust

now, a hint to get you started: using the $T$-invariance of the subspaces, which entries of $A$ can you guarantee will be zero?

i have nfi what hes talking about

@Secret i think so as well but have no idea how the hint follows from that

@XanderHenderson i normally when i worked would only eat one meal and it was lunch

Secret

If my 2nd memory serves: T invariance is that when you have some subspace W, then T(W)=W, meaning that any vector w in W will stay stuck inside W when mapped by T

Faust

i think the answer is that its upper triangular but i have no idea what that has to do with anything

@Secret also correct ^^

Secret

Suppose the subspace is a plane, then if there is a T such that the subspace is T invariant, then you expect the image T(w) will also be vectors on the plane, meaning that there will be a set of vectors in the whole space missing from its image, thus this suggest some of the coefficients of T has to go away otherwise the image can fall outside of the subspace

One example is a T that does sheering in the xy plane, you will not expect any vectors at the xy plane to leave it after mapping by T

Faust

well i think they are saying that A is upper triangular

question makes me feel like an idiot been working on it on and off for a week

Secret

03:24

U1, U2 are T invariant meaning that T(U1)=U1 and T(U2)=U2. The only way that can happen is if T is a direct sum (a block diagonal matrix under some basis, so that off block diagonal elements has to vanish otherwise the vectors in U1 can map to U2 and vise versa, contradicting T invariance of the subspaces)

I don't remember if block diagonal matrices depend on basis though, need to revise

Faust

hmm this actually sort of makes some sense

do we know that dim$T|_{U_1} = dim U_1 $

or more simply do we know that $dim [T]_{\beta}^{\beta}$ = dim V ?

ANyway thank for your time @Secret you defiantly helped

0celo7

@Daminark If I have $f$ holomorphic in the unit disk and $\exists r\in(0,1)$ such that $|f(1/n)|\le r^n$ for each $n\in\Bbb N$, then $f\equiv 0$ by a power series argument, right?

Also hi

$f(1/n)=a_1/n+a_2/n^2+\cdots$ causes issues if the $a_i$'s are not all zero

Secret

In the part 2 of his anawer, he proved $\beta_1\cup \beta_2=\beta$ is a basis for $V$, thus T has to be of the same dimension as V otherwise it cannot map any vectors in V to V

Faust

03:42

ok thanks =)

WDUK

hi whats the correct way to write this:
ln(cos(x) - c) where c is an arbritrary constant.

I was thinking i should add | | to state it cannot be negative (x is defined between -pi/2 and pi/2) but since c is arbitrary and not important should i still add them since c can technically be a positive number anyway?

Semiclassical

Do you have an initial condition?

WDUK

only that -pi/2 < x < pi/2

Semiclassical

well, that at least means cos(x)>0

WDUK

yup

but can we just assume c < 0

or should i apply absolute

Semiclassical

03:51

if you're not told to, I doubt you can assume it.

WDUK

okay, ill play it safe then

0celo7

@Semiclassical reading BBS string theory for my hyperbolic PDE class

I'm in the upside down

help

Xander Henderson

@WDUK what is the context in which that expression pops up?

Semiclassical

god help you

(BBS?)

0celo7

Becker, Becker, Schwarz

Xander Henderson

03:52

that should dictate (to some extent) what $c$ can be

0celo7

it has a decent section on conformal groups

Semiclassical

ah

WDUK

@XanderHenderson it was from differential equation giving me e^(-y) =cos(x) - c
needed to get it to be y = . . .

so i applied log to get -y

Semiclassical

Is that like Virasoro algebra stuff?

0celo7

no, because I want it for 3+ dimensions

2D conformal stuff is crazy

Semiclassical

03:54

@0celo7 sounds right

0celo7

It's $\mathrm{SO}(D+1,1)$

I'm just not sure what the prof really wants me to do

Semiclassical

so $D>1$, hmm

yeah

conformal mappings are fun, but conformal groups...yikes

and CFT is scary

0celo7

I think I'll talk about how the conformal group acts on light cones.

Semiclassical

nice

0celo7

Or I could just talk about GR...

idk

Semiclassical

03:56

lol

talk about twistors :P

0celo7

Duuuuuuuude

I wish I knew that

Semiclassical

same tbh

it seems like crazy stuff

0celo7

@Semiclassical do you want to read something this summer on it

Semiclassical

I might be up for that if I've got the time

0celo7

amazon.com/Introduction-Twistor-Mathematical-Society-Student/dp/… your library probably has this

Secret

03:58

[Some random rambles] Actually understanding the significance of the off block diagonal entries of a linear map is quite useful:

Semiclassical

sounds right

0celo7

I think it's fairly elementary

Semiclassical

my main exposure to twistors is in Penrose's big book

which has nice pictures

Secret

While nobody have found a way to visualise the linear map itself because it is too high dimensional, what the linear map does to the space in terms of blocks can be easily understood by its block diagonal and off block diagonal elements

Semiclassical

but probably not great for getting a grounding in it

Secret

04:00

and I really learn of this view from chemistry, when computing overlap integrals

Xander Henderson

@WDUK In that case, it is probably safe to assume that the argument of the logarithm is positive

unless you have an initial condition that tells you otherwise

0celo7

@Semiclassical I'm also looking at some CFT books for this

Xander Henderson

but in that context, the constant of integration relates to an initial value

Semiclassical

good luck tbh

0celo7

@Semiclassical her english is terrible so I'm really just guessing what she wants me to do

Secret

04:02

Given a linear map $\bigoplus_{i=1}^n U_i$ the off block diagonal components mixes the subspaces i=\=j

Semiclassical

there's a big yellow book which is standard for CFT but it's tough going from what I've heard

0celo7

she's like "you're a geometry expert, so you can tell us about these conformal transformations"

Semiclassical

:/

0celo7

@Semiclassical Francescoooo

Semiclassical

ah, you've heard of it

0celo7

04:03

I've read part of it

I was into string theory back in the day

Secret

I think we can go even further, that the 1st upper and lower diagonal quantify nearest neighbour mapping and then in general the nth upper/lower diagonal quantify the nth nearest neighbour mapping

Semiclassical

my knowledge of CFT is pretty much limited to "according to these papers, the leading term of the corrections to a certain computation one can do is dictated by the central charge"

0celo7

@Semiclassical CFT/ST/QFT is full of mystery calculations

Semiclassical

it's black magic

0celo7

@Semiclassical the mathematical equivalent is probably algebraic topology

there's just these strange formulas for things

Semiclassical

04:08

the math/physics I'd like to know better is known as resurgence theory

we had a guy here just today/yesterday for that (who actually recognized my name from some previous stuff, which was neat)

he was here as a candidate for a faculty position. no idea if he'll get it

0celo7

what on earth is resurgence theory

Semiclassical

bloody weird

not going to try to explain it in detail, but a lot of QM/QFT physics stuff is basically all to do with asymptotic series

0celo7

ah, asymptotic series

Semiclassical

usually, you just do perturbation theory calculations and accept that you're not going to do better than that

and that's not a bad attitude to have, since if it's an asymptotic series you don't expect to be able to resum the series in order to get an infinitely precise answer; asymptotic series don't converge

0celo7

asymptotic series are definitely strange

Secret

04:13

(For those who are interested, head to Rambles to hear about my ramblings about linear maps. The current conversation is too interesting to risk flooding it)

0celo7

ugh there are too many things I want to learn

@Semiclassical so I've got 1 hour to go from 0 to linear stability of perfect stars in full GR

presentation in fluids 2

it's gonna be a disaster

Semiclassical

resurgence theory is basically a whole bunch of stuff geared at overcoming that barrier in asymptotic series and getting stuff which works despite that

0celo7

@Semiclassical I definitely want to learn some more physics this summer

Semiclassical

which basically amounts to analytic continuation out the wazoo

0celo7

twistors or asymptotic series both seem worthy

Secret

04:16

taught me twistors and resurgence theory when you guys are ready, my wavefunction is already too delocalised to squeeze time from my chemistry to learn more stuff :P

Semiclassical

lol

the person who was speaking was (as I'd understand it) trying to point towards the possibility of using resurgence theory to understand the mass gap in QCD

@0celo7 a lot of this is linked with semiclassical approximation stuff, so you can see why I'd be interested

Corellian

are transversals useful in rudimentary group theory

0celo7

@Semiclassical Do you have any pubs? I don't actually know what you do

Semiclassical

a few, actually

but they're a bit hodgepodge

i like to joke that my thesis is going to be more like an anthology of short stories than a novel for that reason

0celo7

what's the physics version of mathscinet

inspire?

god, searching you there just gives me ATLAS papers

Semiclassical

04:22

i guess? i mostly rely on arxiv

0celo7

@Semiclassical you have a CFT paper :P

central charges

Semiclassical

yeah

though from my pov that one came down to finding the spectrum of an n-by-n matrix for large n

0celo7

@Semiclassicalq this Riemann surfaces paper has an insane title.

Semiclassical

lol

0celo7

@Semiclassical I think that's what K theory is?

Semiclassical

04:27

it's definitely related

0celo7

better hope Mathein doesn't show up

algebraic K theory is def just linear algebra 1

@Semiclassical what the heck is nonhermitian QM

Semiclassical

non-self adjoint operators

0celo7

I figured that, but what does that mean physically

Daminark

@MatheinBoulomenos

Semiclassical

it means you're not actually doing quantum mechanics :P

in the context of our problem, it showed up because $e^{i n x}$ had an interpretation of a creation operator for a +n ion. so if you had a gas with +1 and -1 charges then it'd be just e^ix+e^-ix

Gasparo

04:30

Hmm, I think I am gonna delete my account soon, this time just after a few days, lol.

Secret

again?

0celo7

@Gasparo do you have some condition that forces you to do that

Gasparo

@0celo7 It's not related to my OCD. I'm just weird.

Semiclassical

but when you want to have +2 and -1 charges then you're forced to have something genuinely complex

Gasparo

@Secret I hope you will find peace in your heart soon, since you seem very troubled.

Semiclassical

04:31

so it's reflecting a certain charge balance condition in the system

Gasparo

@Semiclassical What does many-body theory in the context of advanced quantum mechanics study? I heard about it only recently...

Semiclassical

ultimately it ends up not being a big deal, since the potential is symmetric enough that all eigenvalues end up being either real or coming in complex conjugate pairs

but you can also play around for fun with examples that don't even have that

0celo7

@Gasparo lmao

Semiclassical

and what's weird is that the semiclassical approximation still works pretty well

@Gasparo at a certain level, it's just what it sounds like: a system composed of many particles and their interactions

Gasparo

@Semiclassical Do you know about WKB approximation methods?

Semiclassical

04:33

yeah. that's what the semiclassical approximation more or less is

though I was always much more happy to work with the formulas derived from WKB vs. deriving stuff myself from it

WKB is a pain to work with

Gasparo

Currently, I am deciding between buying Feynman's Lectures On Physics 1,2,3 and Nolting's Theoretical Physics 1,2,3,4,5,6,7,8,9.

Secret

uh, feymann's lectures are online, so you don't need to buy them

Gasparo

@Secret I use only physical books for serious reading.

Secret

That's true, but isn't the online version of the feymann lecture webpage always the most up to date?

Semiclassical

i don't know Nolting, but Feynman is a good one

Gasparo

04:37

I really want to mention the Nolting books. Like I said, the last two volumes are still being translated from German into English. They are really excellent and come with exercises and full solutions for every volume. It is a masterpiece. Nolting is a good man for writing them.

@Secret The latest print version is as up to date as it can get. Hundreds of errata have been fixed. Currently the latest is the so called New Millennium Edition published by Basic Books.

Secret

Hmm I see

Gasparo

However, note that the nine volumes of Nolting obviously cost a bomb.

Semiclassical

tradeoffs, yeah

0celo7

I don't think I just justify reading Feynman when I could learn about nonconvolution singular integral operators instead

i.e. I think basic physics might kill me

Semiclassical

lol

@0celo7 Figure out Riemann-Hilbert problems and then explain it to me

0celo7

04:41

@Semiclassical you sound like Balarka

Semiclassical

(they're related to singular integral equations iirc)

Gasparo

There are many good calculus texts, that is, not those two hundred dollar colourful ones, on the market. But for basic physics, hmm, hard to find the equivalent.

0celo7

I don't need a calculus book either

Semiclassical

@ÍgjøgnumMeg ah, here we go: people.kth.se/~duits/RHPcourse/Lecture2.pdf

Gasparo

Of course, my favourite calculus books are Calculus And Linear Algebra 1,2 by Kaplan and Lewis. The best for me.

0celo7

04:42

@Semiclassical Do you mean nonlocal PDE?

Gasparo

Again, that pair of books is unheard of these days. Michigan students might know about it though.

Semiclassical

actually, that link isn't great. i shouldn't have jumped to it

0celo7

why did you send that to @ÍgjøgnumMeg

Semiclassical

...i have no idea

0celo7

@ÍgjøgnumMeg get busy!

Semiclassical

04:45

if i do @ on its own, without typing anything further, then his is the only one that shows up momentarily

so probably what's happening is that i'm trying to do "@0" and tab to finish it, but I do it too fast

it's weird that I do that, though. i don't quite get it

0celo7

I send the messages to the wrong people all the time

Semiclassical

@0celo7 well, what I have in mind is stuff like this (which is from a random abstract)

Gasparo

@0celo7 It's more important not to send the wrong message to the right person.

Semiclassical

"The Cauchy singular integral operator, S, is one of the main actors in the theory of Fourier convolutions, Toeplitz operators, Riemann-Hilbert problems, Wiener-Hopf and singular integral equations."

0celo7

the PV operator?

Semiclassical

04:49

it's related, I think. not sure it's identical

@0celo7 it's the article which starts here (Google books link) books.google.com/…

schrodinger_16

Hello. I've a suggestion to put out there, to the users of Math SE. I think it's a great idea to create a WhatsApp group for Math SE discussion, working together on problems, sharing ideas and contributing to the growth of the community. It'll be a much more convenient and efficient way for the best minds to stay connected. What say?

skullpatrol

That's why we create other rooms :-)

Semiclassical

I think you should start by getting a list of people who would be interested and see if there's a critical mass there. I'm not especially interested myself, but others may be.

0celo7

@Semiclassical that's actually in the book I would read if I had so much time I could :P

Semiclassical

lol

that's not an article I plan on reading myself, tbh

way too much stuff that's over my head

Corellian

04:55

the ending sound in Sylow is like fan or van

0celo7

@Semiclassical The amount of Fourier analysis in hyperbolic PDE is shocking.

singular integral operators are fourier analysis

skullpatrol

Sy-low

schrodinger_16

@skullpatrol The fact that chat rooms can't be accessed via the app is a bit inconvenient. Any suggestions on how I may spread the idea?

Corellian

soo-lof? syuh-lov?

skullpatrol

Ask on meta?

schrodinger_16

04:58

Also, I request you all to have a look at this question-

0

Q: Binomial coefficient: Search for multiples of 13

Question: The coefficients of how many terms in the expansion of (1+x)2018 are multiples of 13? So, we've to investigate the powers of 13 in ${2018 \choose r}$, where 0 ≤ r ≤ 2018 I tried using the following: $$s_p(N!) = \left \lfloor \frac{N}{p} \right \rfloor + \left \lfloor \frac{N}{p^2} \ri...

number-theory elementary-number-theory prime-numbers binomial-coefficients binomial-theorem

Corellian

cannot norsk

schrodinger_16

Any ideas?

skullpatrol

You're right @schrodinger_16 I don't use the mobile app for that reason, instead I use the full site option.

Semiclassical

Basically, $(1+x)^{2018}$ with coefficients mod 13

0celo7

@Semiclassical my advisor wants me to give a talk in the junior colloquium (undergrad seminar)

why am I giving so many talks

this is absurd

skullpatrol

05:04

because he thinks you like to talk?

like an old woman :P

Semiclassical

@schrodinger_16 fwiw, mathematica agrees with 1395

schrodinger_16

Yes, 1395 is right. I tried a Python code that gave the same result. @Semiclassical How do I calculate it with hand? Without using advanced computational tools like Wolfram, Mathematica and stuff?

Semiclassical

Count[CoefficientList[(1+x)^2018,x],u_/;Divisible[u,13]]

schrodinger_16

LOL

Did you understand my question?

I wish to calculate it without advanced computational tools.

Semiclassical

I was including it for reference.

schrodinger_16

05:09

Ooh, thanks for that

WDUK

05:36

anyone here use maxima

Adeek

06:07

@0celo7 hi

want to discuss something related to frames ?

Faust

@Secret finally got it.

@TedShifrin cowardice lies you know tons of interesting things.

Transcript for

$Mathematics$ Mathematics