Mathematics - 2021-07-23 (page 3 of 3)

copper.hat

5:10 PM

@XanderHenderson i'm tenderising the potatoes so they are not so lumpy when i sleep on them...

leslie townes

ahh, potato talk. i am home.

AMDG

Potatoes everywhere quiver at the sound of copper's name.

copper.hat

i like my mash rough

leslie townes

one time in law school i was criticized for my use of the term "sack out" meaning to rest. somehow this had not reached the hallowed halls of my alma mater.

Ted Shifrin

Yukon Gold for me.

AMDG

5:12 PM

I prefer smooth and uniform mash.

It's like colloidal cream

leslie townes

ted and i agree somehow again.

copper.hat

vomit

the golds are my favourite from what one can obtain on california shelves

leslie townes

the advantage of AMDG's preference is you are more likely to get it in most places.

rough mash you tend to have to make yourself

copper.hat

kerrs pink would be my store bought fave in ireland

i like my mash like my ...

AMDG

Well what is meant by rough mash exactly? I'm not a fan of eating "lumpy soup".

copper.hat

5:14 PM

what's wrong with you

AMDG

Ah, wait, I think I see.

Yeah, not a fan of inconsistency, so can't say I would be too much of a fan of rough mashed.

copper.hat

the unequivocal and absolute best spuds are irish new potatoes. however, i would not waste them on mash

i like variety in many ways

AMDG

The question is: what's the best way to have a potato?

copper.hat

btw, wuite by surprise

leslie townes

not mashed, in my view.

copper.hat

5:16 PM

my daughter get pfizer#2 this morning

AMDG

Assuming that it isn't sweet potato of course.

copper.hat

uurgh

leslie townes

wonderful news.

i like roasting them and eating in small chunks.

AMDG

~~Raw. Straight out of the ground. Dirt and all. Best way to eat potatoes hands down.~~

copper.hat

it is actually. most of her covid related issues are facing in the right direction at least

AMDG

5:17 PM

Only Real Men eat potatoes straight from the ground.

leslie townes

pfizer no 1 knocked me out. i felt really sick but at least that meant it was working.

copper.hat

@AMDG you may be joking, but growing up we would often head out, root up a few and cook them up

leslie townes

pfizer no 2 was just my arm being sore for an uncommonly long time. it was not a flu shot.

copper.hat

she would not take the black cab like i ordered her to

its the environment dad

i replied its the daughter daughter

AMDG

@copper.hat Well sure, but I'm talking about grabbing it out of the ground and just chomping on it as-is.

leslie townes

5:18 PM

we used to bury vegetable debris in our garden and sometimes potatoes would grow there. i've done that.

copper.hat

i have eaten raw shaved potatoes.

why i can't remember

leslie townes

raw potato is basically an apple. same experience.

AMDG

Ah, I was curious what that was like.

copper.hat

i do like spanish omelettes

nice way to consume some left over boiled spuds

leslie townes

i'd also like spanish omelettes, if we're ordering.

AMDG

5:20 PM

But the real question is ~~will it run crysis~~ can I divide it into thirds using only half cuts?

copper.hat

punishment in boarding school sometimes involved peeling masses of spuds

AMDG

That's considered a punishment? Sounds honestly like a nice, relaxing activity.

leslie townes

classic feature of KP duty before it was outsourced to contractors.

copper.hat

you need a banach tarski knife for that sort of thing

AMDG

Nahhh

You just need to solve the knapsack problem. ez

Or do a binary search

Yorch

5:22 PM

that's a good one !

copper.hat

i like potatoes anna

if we are going fancy

Yorch

who doesn't

AMDG

What is potatoes anna?

copper.hat

or proper roasted (not baked, ughhh) potatoes

Yorch

I like anything that will clog up my arteries and is extremely caloric

copper.hat

5:23 PM

cheesy potatoes

AMDG

Ah

Ted Shifrin

A gratin — cream and cheese.

leslie townes

my wife can't eat potatoes. they cause her to have dark circles under her eyes and get indigestion.

copper.hat

ah yes

Ted Shifrin

Thinly sliced potatoes.

Yorch

5:23 PM

imagine if our mody was trained to enjoy celery.

copper.hat

i like proper french fries

leslie townes

maybe also my daughter. we're still figuring out the eye circle thing.

AMDG

Potatoes au gratin sounds great.

copper.hat

omg, poor things

none of mine really came over to the green side

they are all on the asian side of the genes

Yorch

I always have dark circles under my eyes

no indigestion though

copper.hat

5:25 PM

my daughter does have a passage west accent

which my sister in law vowed she would rid her of

even though she herself is of said origin

AMDG

@leslietownes I could be wrong, but I believe those are called shiners and are related to allergy because I have an allergy to dust mites. This is what I've been told from my mother who heard it from a doctor when I was examined when I was very young. No idea how credible that info is, however.

Yorch

their mother is asian?

copper.hat

yup

leslie townes

i have traces of my parents' boston accents. which for some reason i was only ridiculed for when i lived near boston.

Yorch

there was like a 6/10 probability

copper.hat

5:26 PM

a shiner is just a black eye, usually from a punch

leslie townes

AMDG yes it is an allergy thing. my wife had a ton of blood work done at a very expensive doctor and they determined potatoes as the source.

AMDG

Hm, well it was probably called something else then.

Yorch

If you take a person and their two parents there is a (6/10)^3 probability at least one of them is asian

AMDG

Ah, hm, wow.

copper.hat

i am very sceptical of food allergies

Yorch

5:27 PM

assuming they are all independent events

which they should be

copper.hat

we went though a whole bs wheat allergy thing with my daughter

leslie townes

i do think a lot of the time a preference masquerades as an allergy.

AMDG

Is there a difference if she eats GMO vs. non-GMO potatoes? Sometimes GMO vegetables can cause allergic reactions but not non-GMO vegetables.

leslie townes

my wife said she had an allergy to bell peppers. i said, OK, let's use serrano peppers. and half the time i was using bell peppers. no result.

copper.hat

even the allergist agreed with my line of questioning, along the lines of "you really don't know, do you?"

Yorch

5:28 PM

yes GMO vegetables cause the clap

copper.hat

i can attest to that

there is no other way

toilet seat maybe

AMDG

I knew someone at an SSPX chapel that I used to attend who was of some sort of hispanic ethnicity. She's allergic to GMO corn, but not non-GMO corn.

copper.hat

that does not make sense

vitamin d

hello

copper.hat

hi, we are discussing maths in a generalised sense

Yorch

5:29 PM

Hello

copper.hat

very inclusive of spuds and clap

AMDG

I mean... if you have DNA that is modified, then it kind of makes sense.

Yorch

except it doesn't

copper.hat

it is knee jerk pseudo science

Yorch

it's not like your digestive system processes chunks of dna

AMDG

5:30 PM

On what basis? I understand this is anecdotal, but I have no reason not to believe her.

leslie townes

it is not outside the realm of possibility. as one example most forms of sugar are completely indistinguishable to most people, but if you have diabetes, the subtleties of the form of sugar suddenly start to matter.

copper.hat

it makes a good story, and i am certainly guilty of spinning a yarn

and i am no fan of monsanto

and their wonderful bunch of ip lawyers

Yorch

sure but carbohydrates get processed

dna doesn't

copper.hat

it is possible, but i would say very unlikely

Yorch

perhaps GMO potatoes contain different macro nutrients for some reason

or vitamins or w/e

copper.hat

5:31 PM

as in got it from the toilet seat

people love to attribute stuff to food.

leslie townes

i've used that excuse. the toilet seats out there are something else.

copper.hat

not so much drugs

AMDG

DNA gets processed with everything else we consume. It happens every time you eat anything that was once living or alive at the time of consumption. What happens beyond the chemical decomposition of DNA among other things? I don't know.

copper.hat

swimming pool was another one

possibility and probability are two different things :-)

AMDG

But what seems quite clear to me is that the digestive tract is a selectively permeable membrane.

copper.hat

5:32 PM

fuzzy logic science

time for breakfast

Yorch

that's like saying books that contain heresy burn hotter than normie books because the fire consumes the ink

copper.hat

no potatoes unfortunately

i like that saying

leslie townes

i just ate a horrible bagel from starbucks. they need a different bagel chef.

AMDG

@Yorch What is?

copper.hat

i like cold pizza for breakfast

leslie townes

5:34 PM

cold pizza is one of my favorites. also apple pie.

Yorch

The statement where you say DNA gets processed with everything else we consume

AMDG

Then I don't follow your reasoning, Yorch. It's a chemical like any other, and whatever enters your digestive tract must be processed properly. To do so, that requires a means of identifying what is eaten in order to decide what to do about it.

copper.hat

what i have found strange is that i know some super smart people whose rational & logic goes out the window with certain things such as food.

AMDG

This is plausible especially if we consider the great bundle of neurons right around your intestines.

copper.hat

hic sunt dracones

AMDG

5:36 PM

Why would you need neurons here except that you need something to be dynamic and adapt to change in an ordered way?

leslie townes

copper about 2/3 of my wife's food sensitivities are preferences. some of them are real but not a lot of them.

copper.hat

i am being extreme for my morning entertainment

Yorch

@AMDG I don't know, but I'll give you my car if it's to detect if a vegetable is GMO

copper.hat

have 'saved' some kids from real life threatening reactins

before they were known

sorta obvious, bright red, choking, etc

but at a party nobody pays attention to little kids

leslie townes

there's also sensitivity vs. anything life threatening. a lot of stuff is in the former category where the worst consequence is discomfort and a long time in the bathroom.

copper.hat

5:38 PM

except helicopter parents

leslie townes

where's grandad?

Yorch

hero

copper.hat

exactly

name of my current project

AMDG

@Yorch Bruh, of course I'm not suggesting there's a built-in mechanism that literally detects if something is GMO or not. What I'm stating is simply that there is the ability to distinguish between different substances and react to each one uniquely.

copper.hat

my sister has some reaction to shell fish of some sort, but she forgets from time to time...

its the gmo gene

Yorch

5:39 PM

yeah but dna isn't a "substance"

AMDG

._.

Yorch

your digestive system doesn't parse it

copper.hat

it was planted by the gates vaccine in conjunction with windows 11.0001 and monsanto

there is a zero sum game with windows functionality and release number

Yorch

People in my family are allergic to shrimp

some people

not me

copper.hat

i like shrimp but only when i have a clue where they came from

bottom feeders bother me a little

leslie townes

5:40 PM

mostly vietnam, in the united states.

copper.hat

part of my irrationality

leslie townes

i can't stand shellfish. it's not a sensitivity i just don't eat sea bugs

if people press me on it i say it's for an unspecified religious reason.

copper.hat

dublin bay used to be a good supply, now i would not bathe there in the irridescent sellafield radiated murk

Yorch

sea bugs are delicious

AMDG

k, maybe I'm just not being clear, or you're failing to see something... but I'm about ready to exit this conversation. I have no clue what you could possibly mean by claiming "dna isn't a substance" which has no reasonable basis. Everything has substance, and here, I mean substance in the sense of a unique chemical whose being or is-ness differs from that of another chemical.

leslie townes

5:42 PM

enough religions discourage the eating of sea bugs that i don't have to specify

copper.hat

i do like clams, mussels and especially scallops.

leslie townes

AMDG one thing that is true about DNA is that it is expressed. it isn't just the sequence in there, but what the sequence tells the organism to do. vary the sequence, vary the expression.

copper.hat

but crabs are not a substance

leslie townes

clams mussels and scallops are sea bugs to me

copper.hat

ill take yours grandad

AMDG

5:43 PM

> That which has mass and occupies space; matter.

> A material of a particular kind or constitution.

copper.hat

a kick in the atter is better than a kick in the ass

Yorch

It doesn't matter

I'm not a professional in whatever field it is

leslie townes

one time i went to a sort of family reunion in maine, where 'reunion' is not the right term because we had mostly not met each other. the only food was lobster. which might have been great if you're into it but i was almost vomiting.

copper.hat

srt to hear it

don't get the lobster fascination

leslie townes

i haven't done the research but lobster are also sea bugs.

copper.hat

5:45 PM

its like a giant earwig

AMDG

@Yorch Neither am I. Perhaps we should move on to a different topic, then.

leslie townes

it was really cheap when i was in law school and was served at almost every event i went to.

anakhro

Wow I am impressed by the length of time since something of meaningful mathematical content was exchanged in this chat.

leslie townes

we are doing our best.

our best might not be that good. this might come up at our next performance review.

anakhro

Leslie, you need to try harder. More food, less math.

AMDG

5:46 PM

I have no idea what you mean. Obviously everything here stated is related to mathematics.

Yorch

I'm impressed you decided to type that

copper.hat

we were discussing the pde satisfied by a lumpy mash potatoe

thats potato with an e

Yorch

peetato

AMDG

Yes, and the topology of DNA.

@copper.hat I guess you could say it's "all natural".

copper.hat

what is not natural?

AMDG

5:48 PM

Things without an $e$.

I tried.

leslie townes

the supreme court has an opinion about patentability of DNA sequences saying you can patent them if they don't occur naturally. which makes abstract sense as a rule but how anyone is to figure out whether something does occur naturally is another thing.

AMDG

Yes, that's kind of my question about that sort of thing myself. How do you know such a sequence is truly "invented" and is not already present in some crevice of the earth?

leslie townes

in US examination practice the burden is usually on the examiner to show how something isn't patentable, than for the applicant to show that something is.

and examiners spend something like an average of 15 minutes on every application.

AMDG

And if we didn't listen to reason, then should some human end up with a mutation that matches a patented DNA sequence... the patent owner would now "own" part of a person and that person's off-spring?

It's obviously not a reasonable thing to claim, but y'know... legal stuff today is absurd because apparently everyone is always trying to cover every hole in their codes.

anakhro

Reminds me of those music people who made a point out of procedurally generating all the sufficiently basic melodies for purpose of demonstrating how stupid it is to pursue infringement for basic melodies.

Every Melody Has Been Copyrighted (and they're all on this hard drive)

►

leslie townes

5:56 PM

that was a funny project, although it is kind of fruitless under US law. to prove copyright infringement you need to prove copying. this can be done indirectly and the fact that the copied thing might previously exist somewhere else from some other source is not a defense to infringement if there is copying.

A and B can write the same poem and if C copies B, the fact that A wrote the same poem isn't a defense.

AMDG

And everyone ends up giving copyright a bad reputation because it is abused.

leslie townes

this never comes up in practice but it is clear as a matter of doctrine.

for a legal regime intended to incentivize creation and the protection of creative endeavors, it doesn't function very well.

AMDG

I agree. It has a number of flaws. Fair use is also needlessly uncertain.

copper.hat

@AMDG some sad recent history in regards to ip rights to human genetics

anakhro

@leslietownes are you sure that is the case? It's not uncommon that people pursue legal battles over naming and trademark conflicts, despite the chances of "copying" there being minimal or seemingly irrelevant.

leslie townes

6:01 PM

well if it's trademark, that's a different regime where prior use can be a defense.

at least in the US, the standard for copyright infringement liability does not require novelty of the copied work. it may work differently elsewhere.

AMDG

@copper.hat Tell me

leslie townes

the US patent system is more similar to the trademark system and what many would think of as reasonable: if someone did it before you, you can't sue on it.

although it is goofy in other ways.

AMDG

Y'know, it brings up an interesting discussion concerning things such as modding. Consider popular ROM hacks of old Nintendo games.

While it is perfectly legal in the US, Nintendo is, for whatever reason, seemingly paranoid about their IP being abused.

It would be justified to say that Nintendo has a phobia, however, I noticed that Nintendo's legal page no longer exists that mentions things about emulators and ripping games... and I couldn't find mention of this stuff in the new legal section on Nintendo's website, either, so perhaps things will move in a better direction.

leslie townes

this is a disadvantage of the fair use doctrine. there are no clear lines so it is hard to say that any specific thing, uncategorically, is OK.

sometimes you just see companies getting smarter about not bothering people.

AMDG

I know. It would be quite easy to solve the fair use clause today, however, at least on a per-service basis.

leslie townes

6:07 PM

nintendo seems to have calmed down, they used to be really bad about that.

copper.hat

to reduce the lack of knowledge in this subject i would suggest reading up on lacks.

AMDG

Concerning YouTube, a simple solution is to require videos uploaded with a claim of fair use to be under a pre-determined CC license.

copper.hat

don't really want to make light of the matter, but i cannot resist

leslie townes

levi's jeans and mattel also used to be really aggressive but have chilled out, mostly after a number of expensive losses.

AMDG

Yeah, I'd like to know more about Nintendo's stance, but all I can find are old articles online unfortunately.

leslie townes

6:08 PM

youtube is so funny. i don't know how you can manage anything at that scale.

copper.hat

try the eda world if you want to see caustic ip protection

AMDG

@leslietownes I don't think they do manage it, really. They seem to leave it to their own devices, an only intervene when actively require to.

leslie townes

music licensing used to be premised on having goons from performance rights organizations being able to go to bars and stores and snitch on people playing music without a license. that was at scale, but still very discrete and premised upon humans monitoring a fairly small number of locations.

the youtube copyright claim procedure is a disaster.

but if they hired me to fix it i don't know what i would do.

copper.hat

laws are for the law abiding

AMDG

That statement is sadly far too true.

copper.hat

6:11 PM

you don't need to enforce maths, but you need to enforce law

gosh i am full of it this morning

AMDG

Without law, there is chaos an disorder.

copper.hat

must be the vaccination

leslie townes

copper what is the issue in the eda world? crappy patents?

copper.hat

yes or overly broad and the being obvious to one skilled with farts

lemme see if i can dig up some

leslie townes

the patent system is fairly good in fields where people write about what they're doing all the time and also patent it. examiners mostly consult written works, and mostly patent publications.

copper.hat

6:13 PM

my memory fails me for specific names. mostly to do with a pair of brothers and a spice simulator if i recall

leslie townes

if people are just out there making products and not trying to patent everything, the thing breaks down

because the examiners don't have anything to point to. they effectively aren't allowed to do non-patent research.

copper.hat

that described much of eda in its infancy

to be fair, that is why so much developed so fast

AMDG

What is EDA?

copper.hat

electronic design automation

leslie townes

my wife's uncle had a company that created high-sensitivity motion sensors. any serious player in that game was just doing products and not writing about what they were doing. so you had a thicket of crap patents in that area.

copper.hat

6:15 PM

i have designed software that was used to design something you own :-)

AMDG

Ah

copper.hat

the #1 analog/mixed signal design environment still generates many m of revenue for the company that i worked for.

tooting my horn briefly

anakhro

The geodesic equation in Diff(R^n) with the L^2 metric is simply $\ddot\varphi = 0$ (i.e. flat). This paper says that GL(n) is totally geodesic in Diff(R^n) because if $\varphi(0)\in GL(n)$ and $\dot\varphi(0)$ is linear, then the solution to the geodesic equation $\ddot\varphi = 0$ remains linear. I don't quite understand what they mean.

leslie townes

what does 'linear' even mean in that context? i am ignorant if this is standard terminology.

anakhro

linear map R^n \to R^n I think.

Or at least, that's what they mean by $\dot\varphi(0)$ being linear. They might mean a linear interpolation or something for the final mention.

leslie townes

6:23 PM

oh, yes, i should have clarified. "remains linear" is not clear to me, at all.

but i am not a g--m-t-r and miss this kind of thing.

anakhro

I was thinking it meant $(1-t)A_0 + tA_1$ is the geodesic between $A_0$ and $A_1$.

Which is...flat.

Not sure I really know what they are calling $\varphi$, if this is the geodesic or the diffeomorphism, or both with an implicit $t$

I figured it was like $\varphi_t\colon\mathbb R^n\to\mathbb R^n$ is a geodesic in Diff(R^n).

copper.hat

what, maths?

vitamin d

Can we follow from $f(n):=M(n)/sqrt(n)$ and $\liminf f(n)<-1.009, \limsup f(n)>1.06$ that $M(n)=-2$ for infinitely many $n$, where $M$ denotes the Mertens function?

anakhro

And so if this is a geodesic in $GL(n)$ (so each \varphi_t is linear) with $\left.\frac{d\varphi_t}{dt}\right|_{t=0}$ being a linear map (i.e. part of the tangent space), then... I am not sure why they are solving the geodesic equation, when they should just be testing that a geodesic in GL(n) satisfies the equation...

schn

7:16 PM

Dear @robjohn,
Could you explain your reasoning to equation 4 (https://math.stackexchange.com/questions/4188804/a-little-oh-dilemma-or-the-expectation-of-kde/4200496?noredirect=1#comment8718105_4200496) some more? We have already conversed somewhat about it and I'm not questioning it. However, I claimed that it should be $f(hu)=u^{m+1}g(h)$ for $f(x),g(x)\in o(x^m)$, since the first function that comes to mind is $f(x)=x^{m+1}\in o(x^m)$ and for this function it seems more natural to just write it as $f(hu)=u^{m+1}g(h)$ where $g(h)=h^{m+1}$. It also seemed like one could pull out a factor $…

vitamin d

7:29 PM

btw I think the answer to my question above is that liminf M = -infty and limsup M =+infty. Since the mertens function only moves up +1 or -1 (or 0), it takes all values infinitely many times.

Ted Shifrin

8:08 PM

I guess I don't understand how the $L^2$ metric gives a Riemannian metric.

Maybe I'm being stooopid (and overheated from walking in the hot, humid day).

leslie townes

8:34 PM

the weather today really sucks.

my cat is normally in a windowsill right now soaking up some sun but today, nope. she's in the closet.

Sayan Chattopadhyay

Hello chat

copper.hat

i have not wandered outside yet, but it is a beautiful albany day, replete with fully vaccinated offspring

hello!

Sayan Chattopadhyay

I never noticed this before, but Japanese jazz artists derive a lot from bebop greats like Charlie Parker, Oscar Peterson, whereas jazz coming from South America seems to be more along the lines of smooth jazz in the spirit of Chet Baker, Paul Desmond etc.

copper.hat

8:57 PM

my jazz is dated, armstrong, fitzgerald, holiday, joplin, flack, etc.

leslie townes

tony williams.

there's a line of japanese artists who have taken smooth jazz to the next dimension. i have a youtube playlist of it somewhere.

but i think you're right, most composers from japan go in a charlie parker direction.

japanese culture is amazing. moving it back to food, i think if you lived in tokyo you could probably eat the best cuisine of the universe within a few train stops because they perfect absolutely everything.

there are probably downsides to a culture of perfection but food is not one of them.

Sayan Chattopadhyay

9:13 PM

Yeah fast paced licks, using the harmonic minor scale to improvise. I do not know why maybe but maybe this has something to do with the traditional roots of Japanese music. Just like how Bossa nova comes from a very peculiar groove deeply rooted in south american culture, something similar must be happening here.

I really like Japan as a country, one of the faculties in my uni is from Nagoya, and I was asked to look there for a doctoral position, but the whole cultural seismic shift scares me.

leslie townes

one of my friend lives there and he says, you can speak the language, you can have your friends, but you always feel like a guest.

you will never actually live there, is what he said to me.

i wish i knew more about indian music. i heard someone play the sarangi once and it tore my heart out. that is an amazing instrument.

Sayan Chattopadhyay

Yeah I wish I knew too, my family has a whole line of classical musicians, but I took the jazz route when I was being taught stuff.

leslie townes

there's an irish instrument similar to the sarangi. i forget what it is but it also sounds good. i think because it gets close to the sound of a human voice.

Sayan Chattopadhyay

Hurdy-gurdy?

leslie townes

i'm looking for it now.

that might be it.

Sayan Chattopadhyay

9:22 PM

Yeah I have seen people play a hurdy-gurdy, fascinating beasts

shintuku

the solution curves to a differential equation looks like level curves to some function. isn't there some sort of link between the graph of 3d functions and the graph of the solutions to differential equations?

i've been trying to parametrize a 3d function which sort of used this idea but failed miserably hehe

Lucas Henrique

hey chat

quick question: in general (i.e. infinite dimension), $(V\otimes W)^* \not\cong V^*\otimes W^*$

is this correct?

leslie townes

this stuff is really hard. really hard. what are V and W.

Lucas Henrique

vector spaces

I thought about an inclusion morphism $f \otimes g \mapsto ( u \otimes v \mapsto f(u)g(v) )$

but I can only argue about isomorphisms in the finite case, where dimensions are all well behaved

I found this while googling, but I'm always skeptical about infinite dimensional vector spaces

32

Q: Why is the inclusion of the tensor product of the duals into the dual of the tensor product not an isomorphism?

Let $V$ and $W$ be vector spaces (say over the reals). There is a linear injection $V^* \otimes W^* \to (V \otimes W)^*$ which sends $\sum_i f_i \otimes g_i \in V^* \otimes W^*$ to the unique functional in $(V \otimes W)^*$ sending $v \otimes w \mapsto \sum_i f_i(v) \cdot g_i(w)$ for all $(v,w) \...

vector-spaces

robjohn

9:39 PM

@schn Don't just take one example and think that that covers all cases. what happens if $f(x)=x^{m+1/2}$ or $f(x)=\frac{x^m}{\log(x)}$. Those are both in $o\!\left(x^m\right)$, but neither will decay like $x^{m+1}$ as $x\to0$.

copper.hat

Uilleann pipes

shintuku

rephrasing: given the graph of the solution curves of a differential equation does there always exist a 3D function s.t. these curves are its level curves? If so is there always a way to determine its graph?

Ted Shifrin

10:08 PM

@shin: You're talking about level curves and yet a 3D function, so this is confused. What does "3D function" mean?

shintuku

oh! you're right, it doesn't need to be a function, only some graph

sorry i'm trying to work out the terms and I'm not to sure how to express it hehe

Ted Shifrin

The function is a function of two variables. There's nothing 3D here unless you graph it. That just confuses things.

So, the point is this. If you have an ODE $\omega = 0$ with an integrating factor, then there is a nonvanishing function $\lambda$ so that $\lambda\omega = df$. The integral curves of the differential equation are then level curves of $f$.

vitamin d

Can someone give me a "1 sentence definition" of sympletic topology?

shintuku

I'm thinking through your comment. Is it $\lambda$ of $\omega$ or multiplied by $\omega$?

copper.hat

depressed to discover that ireland has a white nationalist movement

would think we would learn

anakhro

10:22 PM

@TedShifrin metric tensor is $g_\varphi(\dot\varphi,\dot\varphi) = \int_{R^n}\|\dot\varphi(x)\|\mu$.

shintuku

ok, I'll work on this and think it through, but is there a theorem about this? or somewhere I can read up on examples?

anakhro

@vitamind Symplectic topology is the study of the global phenomenon of symplectic geometry (i.e. the study of manifolds endowed with a closed, non-degenerate 2-form).

If you didn't mean "as related to symplectic geometry", then you can view it as the study of global phenomena of phase spaces that arise in Hamiltonian/Lagrangian mechanics.

Ted Shifrin

@shintuku Product.

For example, $y^2\,dx + xy\,dy$ is not exact. What is the integrating factor?

@anakhro: Interesting. I assume that should have a square exponent?

anakhro

@TedShifrin Ah yes, sorry, typo!

shintuku

am working on your question, I'm new to diff eq so it'll take a few minutes

Ted Shifrin

10:36 PM

It helps to be comfortable with differential forms (and you can see lots of my posts on these things on main), but it's not necessary for the classical stuff in the plane.

anakhro

differential equations needs to be taught differently. :(

Hirsch and Smale's book is pretty good start in the right direction, but it's kind of assuming basics in DEs, I guess.

Ted Shifrin

Things have moved more towards systems and away from the horrid Boyce/dePrima courses of the past 50 years.

Hirsch-Smale is for advanced math students, not general audience.

shintuku

took a bit: $\mu(x) = x\mu'(x)$

s.t. $\frac{\partial}{\partial y}(\mu(x)y^2) = \frac{\partial}{\partial x}(\mu(x)xy)$

Ted Shifrin

For mine? You missed the exponent?

shintuku

hm.. i'll redo it my bad

I get $\mu(x) = x\mu'(x)$ again

but it works, doesn't it? I verified $\frac{\partial}{\partial y}(\mu(x)y^2) = \frac{\partial}{\partial x}(\mu(x)xy)$

copper.hat

10:46 PM

the $y$ is just noise there

sequoia13

If $F:X\times I\to Y$ is a homotopy s.t. $x_1\sim x_2\implies f(x_1,t)=f(x_2,t)\forall t\in I$, how to show that $G:(X/\sim)\times I\to Y$ defined so $([x],t)\mapsto f(x,t)$ is continuous?

shintuku

oh wait, $\mu(x) = x$

I improvised this, hadn't seen in class that you could get rid of the function characterization of the integrating factor but it makes sense

Ted Shifrin

I guess $x$ might work. I intended $1/y$.

vitamin d

can you help me to reopen this question and close the other one as its dupe?

actually doesn't matter

Ted Shifrin

Reopen a 6-yr old closed question? Surely you jest!

anakhro

10:57 PM

I think they desire the closed one to be the open one and the other one the duplicate because the closed one is 2013 and is closed because it is a duplicate of the 2014 one.

But I don't see what difference it makes.

Since they are both linked anyway.

shintuku

oh, so if the differential equation is an exact equation, the solution curves of the differential equation are the level curves of $f$

vitamin d

8.5 haha. sorry guys forget it

shintuku

hm.. but what is the characteristic of exact differential equations that other diff eqs don't have that makes its solution curves the level curves of $f$?

Ted Shifrin

11:37 PM

In the plane, every de with no singular points has an integrating factor.

Not so in higher dimensions. See my posts on main.

shintuku

been perusing your posts hehe

I see it is the fact of having an integrating factor rather than being exact that makes the solution be some level curve

Ted Shifrin

Right.

shintuku

my real analysis is weak to understand this, would you have suggestions on what I could work on to get a better grasp of why this is a fact?

Ted Shifrin

No, this is more advanced differential forms/manifolds stuff.

shintuku

back to the shifrin lectures then sunglasses

Ted Shifrin

11:42 PM

In 3D you can give the exactness criterion in terms of curl of the vector field… another post.

shintuku

thanks for the search terms, I'll be spending some time on this so it's super appreciated!

Ted Shifrin

For basic differential equations courses you don’t need fancy stuff.

shintuku

yeah the diff eq course I had was pure calculation, but I felt there was too much stuff being taken for granted so I'm working towards the theory/geometry behind it

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