Mathematics - 2018-10-29 (page 1 of 3)

Leaky Nun

00:52

@user451552 a problem is that, maths as a compulsory course isn't supposed to train pure mathematicians, but applied mathematicians, so there might be less emphasis on rigour and proof, but rather getting the formulas right

Lucas Henrique

Hi chat.

@Ted, I do not understand your proof of the fundamental theorem of algebra

(I'm reading your "Abstract Algebra: a geometrical approach")

Specifically when you claim things like $\lim_{|z| \to \infty }( a_n + \frac{a_{n-1}}{z} + \dots + \frac{a_0}{z^n}) = 0$

TBH I'm not too acquainted with limits regarding moduli, just usual complex limits (limit of modulus of the difference $|z - z_0| < \epsilon$ for all $\epsilon$)

CaptainAmerica16

01:23

@TedShifrin Why is the Intermediate Value Theorem important? It seems so intuitive that I don't know why it's even defined.

Ted Shifrin

02:26

@Lucas: This is just the limit of the sum is the sum of the limits plus the fact that $\lim\limits_{z\to\infty} 1/z = 0$.

@CaptainAmerica: It's a very deep fact, actually, about the real numbers. (In Spivak it's all based on the least upper bound property.) Note that if we lived in the world of rational numbers, the function $f(x)=x^2-2$ would be positive at $x=2$ and negative at $x=1$, but it would never equal $0$ on $[1,2]$.

CaptainAmerica16

02:43

@TedShifrin Wow! I would have never thought of that.

Ted Shifrin

That's why you pay me the big bucks.

CaptainAmerica16

Lol, sure.

Every time I come across something weird in math, it has some basis in the simplest of concepts. When you know about the completeness of real numbers, of course, the IVT applies to continuous functions! @TedShifrin

Ted Shifrin

Yes, but it still takes a proof to appreciate how it works. Of course it's intuitive. Most results probably are.

CaptainAmerica16

Do you know where I can find a proof?

Leaky Nun

proofwiki

CaptainAmerica16

02:47

kk

Ted Shifrin

a proof of what?

CaptainAmerica16

A proof of the IVT thing we were just talking about. Your comment implied there was a proof.

Ted Shifrin

Of course. It's in the Three Hard Theorems chapter of Spivak.

You'll get there.

CaptainAmerica16

It all points back to Spivak...

Ted Shifrin

Well, either that or my multivariable math book :P

Leaky Nun

02:50

learning some topology would be quite useful...

Ted Shifrin

Not yet, Leaky. Down.

CaptainAmerica16

@LeakyNun That's end goal.

"end"

user2236

slow down...

CaptainAmerica16

@user2236 OMG, I know. Walk before you run and stuff.

user2236

...concentrate on basics first.

CaptainAmerica16

02:53

More basic than Spivak?

user2236

sure, why not?

CaptainAmerica16

-.-

user2236

if it helps :-)

Ted Shifrin

you're doing fine, @CaptainAmerica. Well, you haven't turned in homework to me, so I'm not sure.

CaptainAmerica16

@TedShifrin You give homework to people on the interwebs? (Besides AoPS.)

Ted Shifrin

02:56

Well, you have problems in Spivak you could be writing up :P

But, yeah, I've given people in this chat homework before.

CaptainAmerica16

It's not my fault I got distracted by probability :P

Ted Shifrin

Of course it's not ... :P

CaptainAmerica16

My first goal is to finish Ch. 1 of Spivak and then do like 5 of the problems. There's a ton.

Assuming they'll take a while. Idk.

Ted Shifrin

There are tons of problems. Some are more interesting than others. I can give guidance if you ask.

CaptainAmerica16

Ok, sure. I want to do the non-trivial ones. I'm guessing some of the problems are important stuff that I don't know enough to recognize.

Ted Shifrin

03:08

When you get to that, I'll give you some recommendations in various chapters.

CaptainAmerica16

Ok :D

Talk to you later @Ted. I'm going to do some homework before bed.

Ted Shifrin

Night, @CaptainAmerica.

user2236

03:24

YOUR BOSTON #REDSOX ARE THE 2018 WORLD SERIES CHAMPIONS! #DAMAGEDONE https://t.co/ax05nkT8k7

Tweeted by RedSox on October 29, 2018 at 3:17 AM

Ted Shifrin

Yea for my once home team. I think I only went to one of their games in the 10 years I lived in the Boston area.

user2236

lol

Ted Shifrin

I went to more SF Giants and ATL Braves games.

But now I don't care about baseball.

user2236

what happened to your interest?

Ted Shifrin

I was never that much into baseball. It's largely boring.

user2236

03:27

I see.

Daminark

Hey Ted!

Fargle

Good evening chat

Ted Shifrin

heya Demonark and @Fargle.

I'm sorry to hear you were sickly, Demonark.

Daminark

Hey Fargle!

And yeah, but I think/hope I'm starting to recover now, thanks!

Fargle

Just got back from playing Werewolf with a group of friends. Always a good time, that game.

Daminark

03:36

How've things been going for you guys?

Oh Werewolf is great

Ted Shifrin

You are indubitably a werewolf, @Fargle.

Fargle

I asked if I could run one game as the narrator. And the roles I picked were 7 villagers, 1 bodyguard. No werewolves.

Those games are always the best.

Ted Shifrin

I know not whereof you speaketh.

Daminark

Replace the bodyguard with a Tanner :P

Fargle

@TedShifrin Essentially, it's a big social intrigue game. Some people in the "town" are villagers, some are werewolves.

Ted Shifrin

03:38

And I thought you were antisocial :P

Fargle

Every "day", the villagers get together and try to figure out who the werewolves are and hang them (but the werewolves look like villagers during the day, so they play a part in this process too and can throw a wrench in).

Daminark

Oh you do the longer version

Ted Shifrin

Wish we could make this more politically relevant.

Daminark

I usually play One Night Werewolf

Fargle

Every "night", the werewolves devour a villager. The game ends either when all the werewolves are dead, or when the number of vills equals the number of werewolves, in which case the wolves win.

@Daminark I'm not familiar with that ruleset.

@TedShifrin There's a game in a similar vein called Secret Hitler.

Ted Shifrin

03:40

Oh great. Just what I need.

Fargle

Basically, America's playing it and losing.

(My condolences to any Brazilians in the chat, on that same note.)

Ted Shifrin

I am sure Eric will be permanently depressed. The guy more right-wing than Trompolini was just elected president of Brazil.

Right, sniped.

Fargle

Honestly frightening. I fear for the notion of democracy itself.

But I digress.

Ted Shifrin

Veritably.

Daminark

@Fargle so, you have a bunch of roles, each given by a card. Every player has a card, and there are 3 in the middle. One person gets to either see another person's card, or look at two in the middle. One person switches himself with someone, one person switches two other people, etc

Fargle

03:43

Interesting.

Daminark

So then when daytime comes around, everyone tries to find a werewolf and execute him. If the village ends up executing a villager, the werewolves win. If a werewolf is executed, village wins. There's a character named Tanner that is only sometimes played, but if he's playing, he wins if he dies

Ted Shifrin

This sounds very Salem/witches.

Secret

When what once mere fantasy becomes a real nightmare, you knew our days are numbered

2

climate change, far right rise, genocides, impeding economic collapse, mass survellience etc.

Fargle

@TedShifrin There is actually a version of Werewolf for the computer called Town of Salem.

Ted Shifrin

How appropriate.

Fargle

03:52

@Daminark Huh. Tanner is also a role in Ultimate Werewolf (the edition we were playing) but I think the rules for it are slightly different? I'd have to check

Secret

so is tanner a villager or werewolf?

Daminark

Tanner is his own team, his win is at the expense of both of the other teams

Fargle

In the non-one-night game, he counts as a villager for purposes of determining whether werewolves win, but can only win with his win condition (which, as I just checked, is the same, but his winning doesn't end the game).

Another role that's fun is Hunter: if he's killed, whether lynched or mauled, he gets to kill any player of his choice.

Daminark

So then the werewolves win => the tanner wins?

Fargle

Not necessarily. Werewolves win with equal numbers, tanner can still be alive at that point.

Daminark

03:57

Wait I thought Werewolves win when the village is dead

Oh

Hmm

Fargle

Nah. If there are 2 vills and 2 wolves, wolves win.

Ted Shifrin

ponders transmogrifying into a few werewolves

Fargle

Flavor is that now the werewolves can overpower the remaining population one-on-one.

I'll have to try One Night though. That sounds really neat.

Daminark

Yeah it's a good time

Daminark

04:10

So, have you guys been thinking of any interesting math?

Fargle

Related to Werewolf, I've heard people say off-hand that, if there are an even number of people left, the optimal strategy is to not hang anyone. And I'm trying to figure out why that is.

As for actual math I'm just getting further in baby Rudin

Daminark

Baby Rudin is not too bad for sure

Fargle

Yeah, just a lot of the proofs involve ass-pull quantities where it's not clear how they were arrived at.

So I have to proceed a bit more slowly when that happens.

Daminark

Ah yeah he does that

Like, I remember in chapter 1 when he was trying to show the set of rational numbers whose square is less than 2 is bounded above he pulled some really weird number out

Fargle

I remember figuring out how he did that actually.

So, given a number $p$, notice that $p - \sqrt{2} = \frac{p^2 - 2}{p + \sqrt{2}}$. If $0 < p < \sqrt{2}$, then this will be less than $\frac{p^2 - 2}{p + 2}$, as the denominator will be bigger so the other number is more negative, and if $p > \sqrt{2}$, then the other inequality will be true.

Daminark

04:20

Ah

Fargle

But that's the only one I've really figured out. There's one later in chapter 1 (I think where he proves that there's a unique positive nth root of any nonnegative real) that's just, like, uh.

Daminark

I'll see

This?

Fargle

ye

Like, I see that he's basically abusing the binomial theorem, among other things, but I just can't intuit that number at all.

Secret

04:35

I want a machine such that when I input a crazy number full of factorials, the answer it splits out is what do all those factorials mean

i cannot intuit in anything that is a rational function of powers of (x+stuff)

Geometric abstraction

Geometric abstraction is a form of abstract art based on the use of geometric forms sometimes, though not always, placed in non-illusionistic space and combined into non-objective (non-representational) compositions. Although the genre was popularized by avant-garde artists in the early twentieth century, similar motifs have been used in art since ancient times. == History == Geometric abstraction is present among many cultures throughout history both as decorative motifs and as art pieces themselves. Islamic art, in its prohibition of depicting religious figures, is a prime example of th...

if it exists in $\mathbb{R}^2$ it is not abstract enough

Secret

05:38

[Random]

A rambling about geometry

Consider the following:

This is a cube

We all knew how we can formalise a cube by the set of coordinates $\{(\pm 1,\pm 1,\pm 1 )\}$ along with the regions it defines under some coordinate system

We can also easily rotate and scale the cube by applying some linear transformation

In the case of rotation, while all the values of the coordinates changes, the "cubicess" is retained. How should we quantify this "cubicness"? Well, the first thing to note is the angles between two adjecent edges, which is given by the eucledian inner product, remains 0, indicating they remain orthogonal

We also note the volume remains the same

The above two quantities are thus invariant under rotation

Now for scaling, we can likewise found that the angles remain 90 degrees, but the volume changes

As for shearing, we found that while some angles changed, and so is the volume, there is some interrelation between the faces and the edges remains the same

It now might seemed that the essence of geometry is symmetry. Before making that conclusion. Let's have a look at the following shape:

17

Q: Difference between "space" and "mathematical structure"?

I am trying to understand the difference between a "space" and a "mathematical structure". I have found the following definition for mathematical structure: A mathematical structure is a set (or sometimes several sets) with various associated mathematical objects such as subsets, sets of sub...

terminology

and... I lost track. Will try again later...

user21820

06:48

@Secret It's just simply "rigid motion". Google that for precise definition. Rotations and reflections generate all rigid motions.

In real world of course reflections usually cannot be physically done, that's why we have chirality of some molecules.

Secret

It is often said geometry is topology + mathematical structure, but rarely is mentioned on what kind of mathematical structure constitute geometry. Is there a minimalstic definition of geometry?

Meanwhile, it seemed to be a bit more clear on what abstract algebra is, as universal algebra states that any algebraic system is consists of identities (equations that holds for the elements in question), constants, operators and their signatures

Deductive system is then a generalisation on this in that iff is replaced by implications, making the rewritting of expressions directional

and then logic pops out as a subobject of deductive system where the rewritting rules carry inference

It is also clear on what topology is, since it concerns about a collection of sets obeying certain closure axioms

But it seems we don't have a clear cut definition on what geometry is

Put it in another away and abusing notations a bit, what is:

$$\bigcap_{x \in \text{Geometry}} x = ?$$

Alternately, what is:

$$\text{Mathematical Structure} \cap \text{Geometry} = ?$$

These are the questions we need to answer in order to define the most abstract and general possible notion of geometry

Interestingly, we do knew what kind of mathematical structure is insufficient to form a geometry. For example, we don't call any arbitrary linear order or partial order a geometry, nor we will necessarily call a set with such ordering a mathematical space

Inner products, metrics, ordering, cardinality, measure, topological invariants etc. are all mathematical structures, yet only a subclass of them are considered part of geometry. What determines this criteria

Lucas Henrique

09:35

Back again. Hi chat!

@TedShifrin I think that I’m almost understanding (yay!) the proof. So your approach relies strongly on the generalized triangle inequality, right?

Also, I do understand what you did by taking the minimum values etc but I wouldn’t have neither the background nor the ingenious thought to use the continuity properties. Usually proofs concerning the same objects and relations between them (specially elementary subjects like number theory) use all the same techniques. What’s the idea behind the technique you used there?

AFAIK the FTA does not have a pure algebraic solution so I’d have to rely on topology/analysis, but I don’t even know where I should start from.

(Don’t forget I’m a high schooler, LOL)

Tobias Kildetoft

@LucasHenrique It does have something fairly close to a purely algebraic one, but it does use a continuity argument to show that any odd-degree polynomial has a root.

And clearly some sort of completeness is needed, since the corresponding statement fails for other subfields of the reals.

Lucas Henrique

@Ted: also, why “by convenience”? What’s exactly $z_0$ and why is that supposition valid?

@TobiasKildetoft I think that’s ok tho. Not really difficult to prove.

And, even if not geometrically obvious, it’s algebraically obvious (IMO)

Tobias Kildetoft

09:50

What is algebraically obvious?

Lucas Henrique

That any odd degree real polynomial has a root.

(Although you did not say that the polynomial was real)

Tobias Kildetoft

but it is only true over the reals, so the reason cannot be purely algebraic.

Lucas Henrique

Yeah.

Leaky Nun

10:17

@TobiasKildetoft :P

philmcole

11:15

Hi

ACuriousMind

11:35

16 messages deleted

@Secret You've been repeatedly asked to not post swathes of unrelated material in rooms. Stop it.

4

Tobias Kildetoft

@ACuriousMind Thanks.

Prashin Jeevaganth

14:11

Can anyone verify with me that the answer for 2a is -217 and 2b is 65/3? Want to check my understanding on expectation.

Semiclassical

if you show the work you've done, we're more likely to offer helpful feedback

Mancala

Can anyone tell me the general solution of an ode of the type $Z^{\prime }\left( s\right) -2\sqrt{-1}H\left( s\right) Z\left( s\right) -1=0 $?

Prashin Jeevaganth

@Semiclassical is the answer not sufficient?

Semiclassical

@Mancala H(s) is what?

heaviside step function or something?

Mancala

$H:I\rightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ is functions real of one variable

Prashin Jeevaganth

14:16

Pardon me for using the wrong jargons/terminologies

2a) Let X be the random variable of the monetary profit gained from a single die roll
E[X] = 1/6 (2) (2+3+5) + 1/6 (-3)(1+4+6) = -13/6

We are finding E[100X] so by the linearity of Expectation it's simply the product and -217 when rounded off to nearest int

Semiclassical

@PrashinJeevaganth looks right

hands

Prashin Jeevaganth

2b) Let X be the random variable of increments to the total test score by answering a single question.

E[X] = 1/3 (1) + 2/3 (-1/4) = 1/6

We are finding 20 + E[10X] = 65/3

Semiclassical

looks good

(ignore the hands thing, that's a random typo)

Prashin Jeevaganth

@Semiclassical Would you mind looking at my handwritten proof for Q4?

haha hands = have a nice day son

Sure u wanna take it back?

Semiclassical

not right now, no

Prashin Jeevaganth

14:21

ah ok

@Semiclassical do u know any good books for combinatorics and probability to brush up my concepts? Preferably with practice problems and solutions

Semiclassical

the weird thing about that last one is that it's not really a probability question

not off the top of my head, no

Prashin Jeevaganth

Isn't Pigeonhole principle a probability question?

Semiclassical

Pigeonhole is combinatorics, not probability, in my book

you're not asking for the likelihood of some event

you're showing that there's no way to pick 5 points from an equilateral triangle such that they're all more than X distance apart

Prashin Jeevaganth

Well these questions stem from probability or counting

Semiclassical

obv you could ask for the probability that they're closer than that

Prashin Jeevaganth

14:24

they don't only limit to probability

Semiclassical

hmm, fair enough

the connection to pigeonhole is a nice way to look at that one anyways

i think?

Prashin Jeevaganth

Well actually the problems I've been receiving have very unique ways of solving

this is one of them

Semiclassical

nice

Prashin Jeevaganth

I'm still not that good with the bijection principle actually ...

The idea of modelling a problem into lines and balls for advanced problems

Semiclassical

like, stars-and-bars?

i.e. *|**|* stuff

Prashin Jeevaganth

14:34

@Semiclassical yes

I mean I'm not really clear about the "hidden prerequisites" to allow me to use that approach

That's my thoughts about doing counting or probability problems, knowing the constraints and fitting it to an appropriate approach

Semiclassical

for better or worse, combinatorics is not so much about concepts as it is about techniques

e.g. there's no 'fundamental theorem of combinatorics.' there's just a bunch of different questions people have asked, some of which arise frequently enough for us to remember their formulas

Prashin Jeevaganth

Well I don't have a good exposure to these techniques ... every time I'm exposed to a new problem, definitely in the exam haha

Semiclassical

my pool of techniques isn't that great either tbh

Prashin Jeevaganth

do you know anyone lurking here to be an expert in these techniques?

haha

Semiclassical

14:50

not off the top of my head

Adam

15:56

what is the notation for the group of all natural numbers under elementary arithmetic operations?

Slereah

$(\mathbb{N}, +, \times)$?

Tobias Kildetoft

@Adam The natural numbers do not form a group.

Slereah

yeah not a group indeed

Adam

sorry the natural numbers union with the set ${\{0}\}$

Slereah

I always use $\mathbb{N}$ to mean that

It's the common notation in France

Adam

15:59

well that's not very helpful to anyone you are very naughty

Tobias Kildetoft

I use it to mean with or without $0$ depending on what happens to be the most useful right then.

Fargle

@Slereah Doesn't "positive" mean "greater than or equal to zero" in France as well? If so, that would at least be consistent with using $\mathbb{N}$ to mean $0$

Slereah

yeah

We use "strictly positive" for $> 0$

Fargle

I guess you could say "nonnegative" too, but we poor, silly Americans get confused by other people's conventions.

Semiclassical

tbh my sense of what's natural depends on context

Slereah

16:02

Also $]a,b[$ for $(a,b)$

Semiclassical

if i'm doing generating functions, then stuff starting at n=0 makes sense

but if i'm doing anything involving primes and multiplication then excluding 0 makes sense

Fargle

@Adam I'm not sure what terminology should be used with two operations, but, for example, if I were taking the natural numbers with 0 under addition, that would be called a monoid rather than a group. Closed, associative, identity, but not inverses.

Semiclassical

(which I'd argue is reflected in the fact that Dirichlet series, unlike other generating functions, start at n=1 not n=0)

Tobias Kildetoft

And of course, I don't mean $0$ to be positive, except when using the term "positively graded" since there disallowing $0$ would be absurd.

Fargle

@Semiclassical That jives with my intuition. My number theory class excludes $0$, and every combinatorics class I've taken has included $0$.

Semiclassical

16:06

It's not a perfect distinction, mind: Number theory naturally leads to talking about modular arithmetic, and you definitely want to allow zero there

that's what divisibility means: d divides a if a=0 mod d

Adam

all of you are incredibly naughty and I will be forming a secret society for those with atypical autism to guard them against this kind of torture, write software to translate into understandable notation when they are presented with these kinds of shenanigans

Semiclassical

lol

can't avoid dealing with context

Slereah

@Semiclassical You could say $\nexists n. n \times d = a$

Just as good!

Tobias Kildetoft

@Slereah Except you negated it.

Semiclassical

sure. why not.

Adam

16:12

@Semiclassical false currently on my breadboard with my Arduino microcontroller prototyping a context sensor

measures the magnitude of the shenanigan's number of layers of abstraction precisely

Semiclassical

"you being on a breadboard with an arduino microcontroller" = context tho

Tobias Kildetoft

@Adam What level does a Frobenius kernel fall under?

Semiclassical

can only build a context sensor while within a context

Kaustabha Ray

16:31

$U = memused / memtotal $

$\frac{dU}{dt} = L$

where memused can range between 0 and 4 and memtotal is always 4, would these equations classify as differential equations?

Adam

16:54

@TobiasKildetoft I'm not sure haha but now that I've actually looked it up it has me interested particularly en.wikipedia.org/wiki/Feit–Thompson_theorem#An_outline_of_the_proof

Tobias Kildetoft

@Adam Ahh, that was not actually the notion of Frobenius kernel I had in mind :)

Adam

ah right well it was the part that interested me the most, a lot of it is too advanced for me at this point, group theory concepts and abstract algebra concepts have been a very slow yet rewarding learning process for me

but I am looking for the terminology for a non commutative group, but for which the elements can be made to commute via a finite number of algebraic steps using the same method, an example is a finite group under Concatenation, in other words, an alphabet for which words of a language are formed by performing Concatenation

Abcd

17:16

In how many ways can $15$ identical blankets be distributed among $6$
beggars such that everyone gets at least one blanket and two
particular beggars get equal blankets and another three particular
beggars get equal blankets.

Adam

ie you can take any selection of the group, concate them in the same fashion we do for the digits of a number (ie from left to right), and this will of course not be the same

if we the select the same elements in a different order, but all arrangements of those elements selected can be easily changed to any other element selected via a finite number of algebraic steps to adjust their ordering in the initial concatenation of one ordering in a manner that makes the end result the same as the other's orderings initial concatenation

Abcd

First, $6$ blankets are given and then $9$ blankets are left.
After that, according to me there are only two possibilities:

- Possibility $1$:

First set of *particular* beggars receives: $(3,3)$

Second set of *particular* beggars receives: $(1,1,1)$

- Possibility $2$:

First set of *particular* beggars receives: $(0,0)$

Second set of *particular* beggars receives: $(3,3,3)$

However answer given is $12$.

Please tell me my mistake.

Adam

@Abcd educate the beggars that the lack of morality of shop lifting blankets is trivial, much like all of morality, it depends on context, and in their context they may steal as many as they require

parvin

hi ! Sorry but I've got some off-topic question from English natives ! I'm writing an SOP and it's urgent;
is it correct to say " it built up my self-confidence" ?

Eric Silva

17:31

@Semi do u know what physicists do w the legendre transform

Alessandro Codenotti

Physics

Eric Silva

shit bro i didn’t even think about that

@parvin this isn’t incorrect but it does sound kind of facile on the face of it but it might sound good in context idk ur story

Ted Shifrin

Hi demonic @Alessandro and @Eric (w/heartfelt regrets).

Eric Silva

ya fuck me everythings fucked

Semiclassical

@EricSilva there’s a few big uses

One is relating the Lagrangian to the Hamiltonian in classical mechanics

Eric Silva

17:40

oh word are they transforms of each other

Ted Shifrin

Well, not everything, @Eric. Mathematically you're doing fine. With regard to the world, we're all pretty fucked right now.

Eric Silva

@TedShifrin ya i’m tryna bury my head in math so i don’t think about all my minority and lgbt brazilian friends and fam who are about to get fucked by this racist homophobic troglodyte pos

Semiclassical

Right. And it’s sorta intuitive: Lagrangian mechanics cares about positions + velocities, so the Lagrangian is a function on the tangent bundle of your configuration space

Ted Shifrin

@Eric: I heard an interview on NPR of someone in the Brazilian know who was slightly optimistic re this particular issue.

Semiclassical

Hamiltonian mech by contrast cares about positions and momenta, with the Hamiltonian as a function on the cotangent bundle

Eric Silva

17:44

idt optimism would’ve been a wise position to take if he won or lost but hey i hope i’m wrong

Ted Shifrin

Anyhow, I seem to be getting embroiled in all sorts of questions about line bundles on main these days, so get to it :)

Alessandro Codenotti

Hi @Ted

Eric Silva

all i can hope is that he turns out to be an impotent monster

@Semiclassical ty for phrasing this in my native language lol

Semiclassical

Plus, the Lagrangian formulation is based on a variational principle, so it shouldn’t be so surprising that the Legendre transform matters

That said, physicists tend to be rather lazy and just write stuff like H= pv -L

Ted Shifrin

Legendre transform even appears in one or two of my papers ... :)

Eric Silva

17:48

@Semiclassical is this lazy? i thought this worked in a v restricted context just fine

and physicists seem to live in restricted context

Semiclassical

Whereas a proper Legendre transform involves taking the min/max of an expression like that. That’s still happening in the background of the physics but not explicitly

Eric Silva

oh sure

Semiclassical

Lol, that we do

Ted Shifrin

Get lots learned this weekend, demonic @Alessandro?

Semiclassical

You similarly are stuff like that in thermodynamics

Alessandro Codenotti

17:48

My weekend was spent on the algebraic geometry problem set mostly

Ted Shifrin

well, you should have liked that :)

Semiclassical

With Legendre transforms relating the various thermodynamic potentials

Eric Silva

if ur diff and convex then i think that formula might just be honest

so i think that might be the context

MatheinBoulomenos

my alg geo problem sets so far were too easy

Semiclassical

Yeah

Ted Shifrin

17:49

LOL, typical @Mathein complaint.

Eric Silva

@Semiclassical this is probably the context this came up in

i got asked by someone in a statmech class

Alessandro Codenotti

Turns out algebraic geometry is 99.9% algebra and 0.1% geometry, this is not what I signed up for!

MatheinBoulomenos

@AlessandroCodenotti sounds good to me

Semiclassical

Yeah

Eric Silva

my alg geo class is 50% geometry 49% analysis and 1% algebra

Semiclassical

17:51

with stat mech there’s some pretty neat stuff with Legendre transforms that I’ve forgotten

Alessandro Codenotti

@EricSilva doesn't sound too bad

Semiclassical

Though there’s at least one use of it in a physics paper that I remain convinced makes no sense as written

Eric Silva

in fact it's all ive ever dreamed of my dude

Alessandro Codenotti

@TedShifrin I'm not liking algebraic geometry as much as I thought I would so far

Ted Shifrin

@Alessandro: That's the typical modern algebraic style, but not the approach that I followed ...

Semiclassical

17:53

Which I’d find easier to dismiss as nonsense were it not for the fact that it seems to corroborate other calculations I’ve done

Ted Shifrin

@Eric: There actually is some algebra that comes in at various places, but not like Mathein and Alessandro are doing.

Eric Silva

sure

Ted Shifrin

@Alessandro: I can recommend some more geometric/analytic reading if you want. But I'm not surprised. And even your topology class sounds horrid (or did).

Tobias Kildetoft

@AlessandroCodenotti What sort of objects are you starting with? Varieties or schemes?

Eric Silva

so far it's never been mega hard algebra to me but we're still kind of floating around ch 0 and ch 1

Alessandro Codenotti

17:54

So far all we're doing is sheaves and prevarieties, we started talking about separetedness of prevarieties today

Eric Silva

he told me last class 2 think about chern class of complex bundly bois

Ted Shifrin

@Alessandro: It would help to have studied Riemann surfaces = algebraic curves first ... to have concrete examples.

MatheinBoulomenos

we're doing classical first, but only the basic definitions so far. We defined "spaces with functions" which basically means a subsheaf of the sheaf of all mappings $U \to k$, but it's less abstract because sections are actually functions

Ted Shifrin

Good, @Eric.

Let me know which of my exercises you're doing/enjoying :P

Semiclassical

@EricSilva as an example in thermo: the internal energy of a gas is written as U(S,V) where S is the entropy and V is the volume of the gas

Alessandro Codenotti

17:55

I actually like the topology course @Ted. We started doing singular homology today but it was cool to see how many things you can prove from the axioms only

Eric Silva

so rn im tryna understand this curvature form boi and do some calcs

Ted Shifrin

@Alessandro: As long as you do some nontrivial examples and computations, and not just formal crap.

parvin

@EricSilva thank you, so is that ok if I write that in a letter to an american professor !? or does it sound like I'm making up expressions of my own!?

Eric Silva

@parvin it's a thing real people actually say

Semiclassical

With the temperature T and pressure P being expressed in terms of the first partials of U

Eric Silva

17:56

we talk about building confidence all the time

so it doesnt sound like something u made up

Alessandro Codenotti

We computed homology of spheres, wedge sums, the torus and that kind of space which can be done from the axioms directly actually

parvin

@EricSilva thank you

Ted Shifrin

@Alessandro: Yeah, that's totally standard. I want more interesting "bois," as Eric would say.

Eventually, at least.

Alessandro Codenotti

I suppose we'll do more concrete examples with singular homology

Semiclassical

However, experimentally you often don’t have direct control over the entropy and the volume. Rather, you’d control the pressure and temperature

MatheinBoulomenos

17:57

for computations cellular is the most useful homology theory I know

Eric Silva

word

Ted Shifrin

@Alessandro: You really need cellular to have any power.

Ah, Mathein sniped me.

<-- shuts up now

Alessandro Codenotti

@MatheinBoulomenos I thought true algebraists only did homology with values in modules over Nagata's infinite dimensional Noetherian ring

Semiclassical

To trade the entropy for the tenperature, you introduce the Helmholtz free energy F = U-TS

Alessandro Codenotti

We did simplicial breafly

Eric Silva

17:59

@TedShifrin hopefully my lingo catches on

Ted Shifrin

@Semiclassic: You're reminding me of my freshman days in (chemistry) thermo. I loved that course.

@Eric: It is pretty sexist, so it should.

Semiclassical

And since there’s convexity/differentiability in the background, this is a Legendre transform

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$Mathematics$ Mathematics