@BalarkaSen Ahh interesting. Why would you want to do that? I mean what extra information does topology carry?

Err, I don't understand the question. It gives you a way to solve partial differential equations without doing any analysis whatsoever, just doing topology/homotopy theory. That's something.

Because doing the analysis is the hardest part :P

sounds intriguing

Okay I see(you still dont like analysis :p). If you dont mind telling me , sorry for being to vague, what do you mean by differential equations over manifolds?

I do like analysis I just haven't learnt enough hah

Ok, let me explain the idea

Suppose you have a PDE $\Phi(f, \partial f/\partial x_i, \partial^2 f/\partial x_i \partial x_j, \cdots, \partial^r f/\partial x_{i_1} \cdots \partial x_{i_r}) = 0$ of order $r$ and in $n$ variables. Here $\Phi$ is an algebraic equation in the higher order partial derivatives of $f$, and you want a solution to it - so an $f$ which satisfies this. That's a PDE.

The partial differential relation (PDR) corresponding to is simply $\Phi$ considered as an algebraic equation. So, eg, if you had the PDE $f + \partial^2 f/\partial^2 x + \partial^2 f/\partial^2 y = 0$, the PDR corresponding to it is $F + G^2 + H^2 = 0$.

I note that your example has a linear PDE.

Yeah I am not spending too much time on finding a complicated PDE. Just demonstrating an example.

okay. I guess that means that one could have cross-terms in the PDR e.g. $FG$ not just $F^n+G^m$

Solving a PDR is just finding random functions $f$, $g_i$, $h_{ij}$, ... which, plugged in $\Phi(f, g_i, h_{ij}, \cdots) = 0$ gives a solution. It's an "analytically easy problem"

@Semiclassical Yeah that could happen.

Hi, everybody.

Hello Daniel San

(and you generically would have cross-terms if you homogenize your PDR. not sure how relevant that is tho)

what if you had a term like $\sin(FG)$

then it's not algebraic

sad beans

I wonder why latex in this chat doesn't work for me

but latex on the site works for me

have you enabled mathjax?

I highly doubt I can enable anything on here

everything is banned

maybe not. it's the link in the room desc at any rate

Context: Now say $\Phi$ is a PDE but over a manifold (that makes sense to say if you have a Riemannian metric, say, then you could define the gradient, Hessian, etc). Then the corresponding PDR has a natural place it lives - in the "jet bundle". Gromov's theorem (watered down) says if the PDR $\Phi = 0$ has a solution (this is usually a homotopy theoretic problem), then the PDE $\Phi(f, \partial_i f, \partial^2_{ij} f, \cdots) = 0$ also has a solution

at the cost of deforming the domain of the solution region a bit.

I quote

"As a workaround while this request is pending..."

question was asked in 2011

loooool

That deformation can be irregular, so a smooth solution domain may have to be made to only a $C^k$ domain for $k < \infty$. Usually $k= 0$.

Regularity of the domain will be lost

But you'll get solutions of the PDE from doing homotopy theory by figuring out if the PDR has a solution

Which is hella surprising

So a h-principle solution isn't necessarily going to be a nice one

Yeah

Correct

Hmm fascinating

That's why eg Nash embedding theorem loses regularity. Any Riemannian manifold can be isometrically embedded in a Euclidean space, but usually that embedding is $C^1$.

I vaguely remember a remark made by Gromow in an article you linked to the effect that h-principle solutions are typically not relevant for the kinds of PDEs you see in physics?

(It's a corollary of some sort of h principle)

Hmm, not sure if I have seen that. I have seen an MO post to that effect

maybe it was Mike

Nash embedding theorem is about isometric embeddings right?

Ah yes never mind

Yeah it is

@DanielSank Oh, I had a question for you in your career capacity

This idea of thinking about a PDR, isn't this a lot like how we deal with simple second order ordinary differential equations by associating a polynomial to the differential equation

Seems right

@Albas That assumes a linear PDE tho

Ah yes

It'd be interesting to do this to different pde's arising in physics

Obviously, you work out in California on quantum electronics physics. What I was curious is to what extent there's jobs in that area in the midwest

(or, at least, whether you were aware of such companies.)

yeah, see the Gromov quote at the bottom of the first page of this article: math.uzh.ch/fileadmin/math/preprints/15_11.pdf

Aha

Interesting quote

Though the thrust of that article seems to be that, despite initial expectations, there are variations of the h-principle that are relevant for fluid mechanics

Hmm interesting

I'm in no position to assess it either way, so I dunno

The kind of examples of partial differential relations I gave (arising from a PDE) are usually not which arise in practice, actually.

Because these are zero sets of algebraic equations, so are closed submanifolds (of the jet space)

There is not much space to "move around"

Jiggling is an essential technique to solve PDR's, so one can expect that these are generally not solvable

Hi @Eric

Sup

interesting

@EricSilva Did you meet this chat's martyr

Do u mean 0celo7?

Indeed

Yes

We are sitting in a lecture on minkowski geo rn

The worlds are colliding big time

@EricSilva Wow. Nice

It's p good

The gr summer school is aight, a lil slow

a whole lecture on minkowski geometry?

I hope to reach that level in relativity

Well it's an intro to lorentzian geo so RN he's just drawing a bunch of cones and I'm bored and that's why I'm here

is he drawing 4-D cones?

That'd be sick

indeed

@EricSilva Don't tell him you are on Physics.SE chat. :P

hmmmm

@Blue interesting, wondering is that a summer program or ...? tifr.res.in

Anyone able to find the answer?? ;) physics.stackexchange.com/questions/411392/…

@Blue am aware you were talking about BaSe

@EricSilva you are a colleague of ...?

@Blue (sometimes others in here know/ say more than he is willing to share personally...)

@Blue lol sometimes have to be creative around here. :| :( (ps sorry to hear of your early mod issues)

Afaik, TIFR doesn't have a normal summer internship program

they do

they have the VSRP

which is happening this month

Oh didn't know that

@vzn being an undergrad IDT I would call anyone a colleague

Too stuffy

@EricSilva lol, sigh what would you call 0celo7 then

A peer I guess? We're attending the same summer school

the king of hbar

@Blue there are misc problems in here. am not the source of all of em.

@EricSilva nice/ cool, are you in TN?

No the summer school is at mit

I'm a uchicago student

what is up with vzn trying to coerce information out of people's business

@BalarkaSen lol, sigh

lol

R u trying to assassinate me @vzn

thats worse than dying

lol

It's fine I embrace the void tbh

you can take the red pill or the blue pill™! :P

← would congratulate BaSe on attending but he might reject the compliment o_O

thank you but i do not think visiting a university is much of a compliment-worthy act

@EricSilva so is it mostly on physics or math? it would be interesting to read more

Mathy

If it were too physicsy I'd be lost

@BalarkaSen "beg to differ"

@BalarkaSen he preempted ur humility

tifr.res.in/~vsrp

@EricSilva I did not reject the compliment tho

ur argoment is invalid

you accepted that )( much

@BalarkaSen a rejection by any other name

@EricSilva btw congrats 2u2! :) so whats the coolest part for you

tsk tsk, radicalites do not think of subtleties

i accepted it, but rejected the basis of it's acceptance

@BalarkaSen thx for clarifying that :)

you're welcome but i do not think my clarification was worthy of a thanks

airhorns

@vzn it's only been a day for me so idk, so far it hasnt gotten to exciting stuff yet

hopefully soon

← outsnarked, clearly a mere amateur among pros o_O

also i reject ur congrats on principle

@EricSilva and what "principle" would that be... (afraid to ask...)

@BalarkaSen being wRoNg isnt a subtlety

stackoverflow.com/questions/50816364/…

i was on like 20 layers of irony so dont take it seriously @vzn

@JohnRennie can you help me in this ?

@EricSilva lost you all around the 1st. maybe having trouble keeping up at my age.

@Koolman float and double are needed because of the varied use of them by programmers

@BalarkaSen w the wisdom of the sages

@Semiclassical yes?

For someone to save size, they would prefer float. For someone requiring higher precision, they would use double

@Semiclassical Ah.

@DanielSank the question is a little bit further in the transcript

yeah

Ummmm, no I'm not sure I am aware of such a company.

@Sid so it all depend on the size

Fair enough.

tbh I'm just trying to figure out what I can do without leaving MN

what language is that

The quantum computing companies I'm aware of are in California, Massachusetts(sp?), and .... California...

Java?

@EricSilva hm, i cannot find a refutation

C++?

@DanielSank Yeah, I figured it might all be on the coasts

but wanted to make sure I wasn't missing anything obvious. thanks

@Koolman How can you use "char" for integers?

@user929304 I'm around now, but if you had just left me your actual question, I could even try to answer it right away! :P

@Semiclassical intel is getting into it. MS has some. china is big on it. theres some EU. etc. but its not very widespread in US. this could chg that. gizmodo.com.au/2018/06/…

@enumaris For Java

@Sid I think I have used it in C

JavaScript?

Java is different than Javascript

sounds right.

I think he means Java

@Koolman You really need to specify the language there - some languages don't even have all of the types you list there!

i'm not wedded to QC stuff, mind.

And some convert so implicitly between them that the distinction becomes academic

i really am just trying to figure out what's around in MN

That too ^^(what ACM said)

@BalarkaSen i was jok boi

@ACuriousMind I am talking about JAVA

i cannot find a refutation of that joke

@Koolman Then that should be mentioned clearly in the question and it should be tagged as such.

@ACuriousMind I will edit it .

@Koolman the different data types are structures which can hold different types of data. The int type is specifically built to hold integers only, it can not deal with floats (decimals). The char type is built to hold single characters. The String type is not a primitive type like the others, but actually a class. It was built in Java to help facilitate the manipulation of strings of characters - i.e. words/sentences.

Lastly, the float type holds 32 bit floating point numbers while the double type holds 64 bit floating point numbers.

@enumaris Then what is the use of int we can directly use float type

@Koolman Float operations are much costlier on the CPU than int operations.

@Koolman if you have items which should not be decimals - for example iterators, then they should be int type.

@Blue So it the role of size .

(I thought Java had become obsolete. Apparently not.)

I don't know what Java does under the hood (it might cleverly use int arithmetic if it detects only integer values in your floats), but in days gone by the int vs. float performance hit was very real.

@BalarkaSen idk man if i ever stop to think abt the things i say i realize the crushing weight of the void within

so best not to think too hard my dude

escapism is the only way

Floats also have rounding issues. With an int, you can be sure you'll never get 2 + 2 = 3.9999999999998 :P

@Koolman also with floats, you have to worry about floating point operations. I.e. rounding of some sort. Since a float is stored in memory in a specific way, and there is a finite size, it doesn't have the nice properties as integer operations.

@ACuriousMind yeah nice point

Double ninja. Nice B)

FGITW

stabs ACM

@Blue Is it not the case with int and double ?

doubles have all the problems that floats have

they are just bigger

@Koolman double has higher precision.

and higher precision

Ninja! Nice.

@Koolman A double is a float(ing point number). It is an abbreviation of Double-precision floating-point

It may be worth to point out that "float" is not some mystery name: It literally refers to the decimal dot "floating around" when you do operations, i.e. the number of decimals after the dot potentially varying after each step.

@enumaris Why there are rounding issues ?

@Koolman because no computer has infinite memory so no computer can represent a decimal to infinite precision.

So using float we can get 2 + 2 = 3.9999999999998

@enumaris Wrong. Computers can represent 0.25 to infinite precision :P

You meant "arbitrary decimal" ;)

@ACuriousMind well, I think in a computer ".25" gets represented as ".2500000000..." (but in binary) or some such. The specific memory size is allocated and you don't get to truncate that.

@Koolman you can get issues that are in the same vein as that one yeah.

@enumaris Yes, but the binary rep of 0.25 is also terminating, roughly as $0.01_2$ - it is stored as its exact value, unless you demand of the computer to store an infinite number of zeroes to be "precise" :P

The trouble is with numbers whose binary representation is not finite, or larger than what the allocated memory can fit.

but the binary representation of .25 is the same as .2500000000000000000000000000000000000000000000000001 in a computer so, I'm not sure you can truly say that .25 is represented to infinite precision since there's degeneracy....might be semantics though...

Either way, I think the point is clear :P

What's not clear to me is what you think precision means :P

in a computer are they not the same?

tack on some more 0's if they aren't the same

@enumaris It entirely depends on in which format you are trying to store them.

at some point, there will be enough 0's that they are the same -.-

unless somehow computer representations of number is not a monotonic function

There are ways to store numbers such as yours - even with a arbitrarily many zeroes - e.g. by encoding them as a ratio of two integers. If you use a format with no MaxInt, the computer will run out of RAM before the two numbers ever become equal if you keep adding zeroes :P

well you can construct some arbitrary method to disprove my 2 examples, but at that point, what's the marginal benefit? -.-

The benefit is to point out that "computers" don't have a single native way to treat numbers. The way numbers are stored and processed is very much a feature of the program/programming language

the OP was asking about Java

I think we left that question far behind when we started quibbling about precision :P

I will concede that I'm wrong.

I don't know where we got with this conversation though lol

@enumaris ¯\_(ツ)_/¯

Which is the best book for all this ?

I guess at some point we would need a computer hardware engineer to tell us about all the different ways numbers are represented on different chips :P

when Java creates a double, how does that code get translated to assembly language, and then how is that ultimately stored on the chip.

Java is also special because it's machine-agnostic

which is why you need the JVM to interpret java code on different machines

@enumaris that's probably the only reason why it is special

so I guess the answer might be different if you're on a linux vs windows machine

and then probably also different depending on the chipset you use...

Not so special, many interpreted languages can do that

Anyone wanna do a crazy integral?

not really :P

OH SNAP, MATHEMATICA DID IT

:-D

(This MATLAB thing is fun. So far.)

Java is not an interpreted language tho

so I guess it might be special in the compiled languages

who knows

@enumaris Well...it is compiled to an interpreted language...

is it?

@enumaris Sure, the bytecode that the JVM executes is an interpreted language

I feel like you are a stickler for technicalities

but sure

"The World's 8 richest men now have the same wealth as the poorest 3.6bn people in the world"

You will have to explain the exact and precise definition of interpreted language.

Lenin needs to rise from the grave now to save this world. :P

This is not just technicality.

@enumaris I never said I wasn't :D

The word "compile" means, generically, conversion from one set of code to another.

In the old days, "compile" usually meant "compile to machine code".

@Sid I am become Lenin, destroyer of the bourgeois

These days, the word is far less specific. For example, we have compilers that go from {all kinds of things} to Javascript.

We also have things like the Java compiler that convert text code to byte code.

Java is interesting in that it also has a Just-In-Time compiler (JIT).

@Sid the ghosts of strangled revolutions wont save us now

The JIT compiles the byte code to machine code at runtime!

That is different from C-Python, where the interpreter executes each bytecode command.

There's some issues with interpreted languages that don't appear in compiled languages...for example see the GIL

@Blue Did you read my previous messages?

So the take away is...meh...I don't really care too much about these technicalities...lol

@BalarkaSen All Hail BaSe!!

@Blue JIT does often not compile all code - at least a clever JIT will only compile code where it's worth it, e.g. short routines that are called thousands of times, and still interpret the rest, so JIT and interpreting are not mutually exclusive

@enumaris I'm not sure the GIL is inexorably linked to the fact that CPython uses an interpreter. Would have to do some research on that.