Mathematics - 2023-03-08

Silly Goose

02:21

I was wondering if I could get some advice on how to study mathematics well. I am currently taking real analysis, and study by thoroughly reading the text and taking notes and making sure i understand the proofs of each theorem. and then i practice via our weekly problem sets, which I trust are representative problems enough. and, i do not have time to do any more problems (since they take a few to several hours each :P). but this way of attempting to learn math doesn't really seem to work well

i'm thinking i have some like fundamental thought processes relevant to mathematics that have just never been developed. since it takes me several hours to read a few pages in Rudin sometimes (when each page is littered with 2-4 theorems)

leslie townes

you won't like to hear this, but that is very normal with rudin.

Silly Goose

02:36

I suppose it is somewhat consoling :P

Kamal Saleh

I have a question, why are there so many users with "Monica" in their username?

Silly Goose

but idk it feels a lot of the time especially at the end of this chapter 2 that the proofs are so much like "i would never think of this"

leslie townes

this is especially true of chapter 2 of rudin. it's probably the densest material in the book.

Silly Goose

like in dummit and foote maybe just for the huge theorems like the sylow theorems and such the proofs might be long and convoluted and such but all else was quite reasonable and like "oh okay i guess it is reasonable to try and go down this route"

:P I am having trouble with the perfect sets and cantor sets bit and I happen to have to miss class on the day we're covering such material hah

leslie townes

it's hard stuff. you're giving as examples some of the most likely to be unfamiliar material in the book.

the chapters on sequences and continuity and stuff will at least be somewhat recognizable from calculus.

when some people teach rudin they skip chapter 2 almost entirely (as they skip the construction of real numbers in chapter 1). it doesn't come up that often later, particularly in the 'calc 1' part of the book.

Silly Goose

02:41

ah interesting

i wouldn't have minded skipping 1 :P

user4539917

@KamalSaleh there was a big controversial removal of a mod with that name and a lot of users disapproved of it and either quit the site or put her name up as protest.

Silly Goose

but the idea of connected and compact sets seems important for some physics stuff i am interested in eventually studying

though perhaps its sufficient to know the defs i suppose no need for these theorems probably

leslie townes

yeah, all of the proofs involving dedekind cuts just suck. the definitions are not easy to work with and you put in 100x effort to verify something that is going to work out anyway.

Silly Goose

what is up with all the theorems on non-empty intersections of a collection of sets in chapter 2? what is that all for :P

leslie townes

the concepts in chapter 2 are super important and useful outside of that book. but there are a few chapters - maybe from 3-6 - where he is not going to be using that material a lot, because you're just in R, and not a random metric space.

there's something called "topology" which is slightly more general than what he's covering. you'll notice the word "topology" is in the title of chapter 2 but not defined anywhere.

Silly Goose

02:44

oh yeah i recall you mentioning that hah

leslie townes

when you're in another class and you learn topology, it will be familiar because of what he is doing there.

Silly Goose

i found a PDF of some supplementary notes to rudin which made that same remark

and i thought of when you said that :P

maybe things will get better as we move on to other material. this topology stuff is quite interesting but yeah quite condensed. i just feel like im not learning as much as i did in say abstract algebra--or at least am not improving my mathematical framework very much

ehh i guess that's not so true. it is nice to know about metric spaces and so on now :)

leslie townes

you'll find that the triangle inequality in R and C gets used quite a lot in the later chapters, and a big part of chapter 2 is, there's a world where you can just "do the triangle inequality" type stuff even if you aren't working with numbers.

and then general topology is one more level of abstraction beyond that.

it's cool, but not easy reading. i hated that chapter.

Silly Goose

huh i did not see that triangle inequality bit in chapter 2

i have accepted that im not going to understand the perfect set theorem myself XD so i am waiting to return to campus to ask the prof

D.C. the III

Advice on how to create partitions for integrting?.....this isn't clicking for me...

leslie townes

02:52

profs do skip a lot of stuff in chapter 2, even if they cover it. our prof didn't assign any of the difficult exercises but chose to write her own.

user4539917

What would you do if you had to teach out of it :^)

Silly Goose

or what book would u teach out of if you could choose :D

leslie townes

depends on the audience, at a selective school or in an 'honors' course, i would cover the basics but not the more obscure stuff and skip most of the exercises too.

at a less selective or non honors course i would just skip it or not use rudin if i could choose not to.

D.C. the III

surely there must be better books even for honurs courses?

Silly Goose

our psets are 3-4 non-rudin problems and then like 3-5 rudin problems :P but i usually just don't have time to even attempt the rudin problems and end up just skimming what they're asking

the rudin problems are mandatory but are not turned in :P

leslie townes

02:56

i dunno, rudin is maybe a good experience for an honors course. and i don't actually know of better.

some books deliberately adopt the goal of, "we are just going to justify everything in a calculus 1-2 type class, with proofs," so a lot of the more rudiny rudin stuff goes right out the window. but in an honors course i think you need to do more than that.

Silly Goose

do you think such students really retain all the theorems and so on :P

leslie townes

what's funny to me, and i think we've discussed this before, sometimes people assume that because rudin is so often taught or regarded as 'a tough but rewarding book,' then rudin's 'real and complex analysis' must be like, an even MORE tough and rewarding book, like "the next level" of analysis. which is kinda funny because it's not.

and he makes some weird pedagogical choices in R&C.

Silly Goose

i think i understand now at least the choice to introduce compactness with respect to an embedded space. as you need to start from that perspective and then prove that actually the embedded space is unnecessary. but yeah perhaps he could be a little less economical with words on explaining that :P

D.C. the III

interesting....I thought that, then read some reviews about it

Mr. Feynman

05:01

I don't understand how using the chain rule they say $(5.11)$ can be derived using $(5.10)$. The composition of $\exp$ and $X(t)$ is $t\mapsto\exp[X(t)]$, how would I use $(5.10)$ here? Also, $(5.10)$ is evaluated at $t=0$.

Franklin

Can anyone please help me with this ( now edited question ) : math.stackexchange.com/questions/4653956/…

PNDas

05:48

What does convergence in $L^p_{\text{loc}}(\Omega)$ mean?

Semiclassical

05:59

is it legit to write $f,g,h:\mathbb{R}\to \mathbb{R}$ to mean "f,g,h each map reals to reals"? rather than writing "$f:\mathbb{R}\to \mathbb{R}$ and $g:\mathbb{R}\to \mathbb{R}$ and $h:\mathbb{R}\to \mathbb{R}$" separately?

leslie townes

semi: you get to decide. it is common usage and the alternative really sucks imvho

pndas: probably equivalent to convergence in L^p(K) for all compact subsets K of Omega, but maybe not defined that way

Semiclassical

thx

I guess a less formal way is "Let $f,g,h$ be functions $\mathbb{R}\to \mathbb{R}$..."

leslie townes

yeah, even with just one function i slightly prefer using words instead of the colon.

Franklin

Add on: math.stackexchange.com/questions/4653956/… in this question I am not getting how to solve the recurrence relation $ (n+1)(n) c_{n+1}- (n+2)(n+1) c_{n+2}+(n) c_{n}+ c_{n+1}=0$ , specifically ? I need some help solving this mentioned recurrence relation for $c_n$ ....

Silly Goose

@Mr.Feynman fancy seeing you here

Yai0Phah

06:56

@ILikeMathematics Tablets become more common during and after pandemic, since there are more online talks. It is not really something essential, as far as I understand. Do whatever that fits you. However, it seems useful to typeset mathematical notes, say, in graphical tools like TeXmacs and LyX, or LaTeX.

@TedShifrin Not even for fields of char 2.

one potato two potato

07:12

It seems Stein's functional analysis textbook is not a very popular choice for self-study or course textbook?

Mr. Feynman

08:40

@SillyGoose I'm delocalized

Silly Goose

09:00

heh

boy i started drawing pictures for these analysis problems and it helps quite a bit :P i had resistance to doing so because it seemed a bit unrigorous but it helps get ideas started in a good way

user2236

09:25

@SillyGoose Arranging the given information in a chart may also be helpful.

It helps keep those ideas organized...

one potato two potato

11:12

0

Q: $E$ and $E^*$ are orthogonal subspaces

The question is a sentence in Farkas/Kra Riemann surfaces. Note that from Proposition II.3.2 we deduce that $E$ and $E^*$ are orthogonal subspaces. Proposition II.3.2 states that Let $\alpha\in L^2(M)$ be of class $C^1$. Then $\alpha\in E^{*\perp}$ (respectively, $E^\perp$) if and only if $\alp...

one potato two potato

11:24

I think I get it. The post will be deleted soon.

ZaWarudo

Premise: English is not my first language. How can someone write in English the statements $\exists a \in A[P(a) \wedge (\neg Q(a) \lor \neg R(a))]$ and $\exists a \in A[(P(a) \wedge \neg Q(a)) \lor \neg R(a))]$ so that they are distinguishable?

Using an example, to me they both translate to: "There exists a person in the world that eat meat and not eat vegetables or fish".

Is the sentence "There exists a person in the world that eat meat and not eat vegetables or fish" actually ambiguous in English? Or could be write in a way that is unambiguous?

ILikeMathematics

13:34

@Yai0Phah Thanks. When taking notes from math books, would you write in your own words or copy some things too?

Franklin

14:19

Can anyone please help me with this :math.stackexchange.com/questions/4654825/… ?

Koro

are you writing JEE?

Franklin

@Koro Who?

What do you mean ?😕

I don't get it, 🥲

Koro

14:38

@Franklin: it's the name of an exam.

Thanks a lot. It's much clearer to me now. Yes, I am familiar with the pasting lemma. In the last para, don't you mean $f$ restricted to $F_\sigma$ is $\sigma$? I think so because for $(x_1,..., x_n)\in F_\sigma, f(x_1,..., x_n)= (x_{\sigma 1},..., x_{\sigma n})= \sigma (x_1,..., x_n)$. — Koro 21 hours ago

Is my understanding correct?

This is with reference to the following: math.stackexchange.com/questions/4653924/…

can we prove inverse function theorem in complex analysis using Rouche theorem?

Franklin

@Koro Ohh...But is my solution correct? 🥲

Koro

14:59

@Franklin I don't see why Mx+Ny=5x^2y^2.

Franklin

@Koro Thanks for pointing out the typo! It would be -5x^2y^2 . I have fixed it. Is it alright now?

Koro

no, please recheck.

Franklin

@Koro I am so so sorry 😞! A big mistake on my part. I have fixed it.

I hope this resolves it?

Franklin

16:31

Can anyone please help me with this : math.stackexchange.com/questions/4654909/… ?

Koro

Suppose that we have a finite dimensional subspace F of a Hilbert space H. Let x be in H but not in F.

Do we have an explicit formula to find the distance of x from F?

leslie townes

maybe, what do you allow an 'explicit' formula to involve?

terms like that make me nervous.

Koro

like some determinant with entries as inner products of some vectors.

leslie townes

well if you have a basis $(f_1, \dots, f_k)$ of $F$, the orthogonal projection of $x$ onto $F$ is $\sum \langle x, f_j \rangle f_j$, and the norm^2 of that is $\sum |\langle x, f_j\rangle|^2$. and you can write the distance from $x$ to $F$ in terms of $\|x\|$ and that sum using the pythagorean theorem.

Koro

We could say H= F+ F', + is direct sum. Then if P is projection, we have P give us the minimizing vector.

@leslietownes oh right.

leslie townes

16:42

but that formula would involve $\|x\|$, which while you can write it in terms of inner products, you'd generally need more ingredients than just the finite basis for $F$ to do it.

Koro

I think you need ONBs above, not just basis?

leslie townes

yes when i say basis in a hilbert space context i mean ONB.

Koro

ohh okay.

leslie townes

e.g. if you're willing to extend a basis for $F$ to a basis for all of $H$ by adding a potentially infinite sequence of vectors $(g_1, \dots)$, the distance^2 from $x$ to $F$ is $\sum |\langle x, g_j \rangle|^2$. but that's in terms of a basis for F-perp, which might be infinite, and i dunno if you have your hands on such a thing.

there's lots of hilbert space stuff that can look very "explicit" if you're willing to just conjure up, "okay, choose a basis for that" and ignore whether that might be computationally intense or involve infinitely many things that you don't plan on computing "explicitly."

formulas involving bases can 'feel' more concrete even if outside of the finite dimensional setting they maybe aren't, really. physics people like them. this came up a while ago, not quite this but something very close, recently.

oh yeah, dc3 had a question about partial isometries and i think we ended up dealing with it without ever working in terms of a bases, although we could have. whether that would have made it more or less confusing is a "coke or pepsi?" kind of question.

Koro

17:06

the formula gets too complicated even in 3 dimensions.

:(

I wonder then how to calculate the determinant of a Hilbert matrix with entries as inner product then.

It apparently has a recursive formula.

leslie townes

17:27

this is a slight overstatement, but almost anything involving determinants is going to be a nonstarter in the infinite dimensional setting, or it's some spacey physics thing that they will write down but never compute.

is there some reason for wanting determinants in this at all? are you familiar with the "gram determinant" (at least an example of a well-studied something made out of inner products)?

Koro

1

A: Formula for the Ratio of Gram Determinants

HINT: You can check by induction that for each $m$, the span of the vectors $(v_1, v_2, \ldots, v_m)$ equals to the span of the vectors $(e_1, e_2, \ldots, e_m)$. Since the vector $v'_{n+1} =v_{n+1}-\sum_{i=1}^{n}\langle v_{n+1},e_i\rangle e_i$ is perpendicular to all of the vectors $e_1, \ldot...

I'm trying to understand this answer.

I don't understand why $G(v_1,v_2,...,v_n, v_{n+1}')=G_n. \|v_{n+1}'\|^2$ is true.

Obliv

if $E[X]=\sum_{i=1}^{\infty}x_ip_i$ then how would I expand $Var(X)=E[(X-\mu^2)]$ as a sum, like $(x_1-\mu)^2p_1 + (x_2-\mu)^2p_2 + ...$ ?

leslie townes

koro: if v'_{n+1} is orthogonal to the v_1, ..., v_n (as is claimed there), it would fall right out of the definition of the gram matrix. the last row or column would have a long series of 0s corresponding to <v'_{n+1}, v_j> = 0 being 0 for each j, and then a single <v'_{n+1}, v'_{n+1}> = ||v'_{n+1}||^2 at the end.

Obliv

how do you write something as text in the middle of a latex expression again

Koro

Ah, I got it.

The last column and last row is zero except the entry at (n+1, n+1)

@leslietownes yes yes, indeed.

I got carried away in the symbols.

leslie townes

17:35

for what it's worth, while the gram determinant is a certainly known/studied object and has conceptual uses, i don't understand the motivation for OP's question or any use for the result.

Obliv

the mean is just the weighted average of a set of values with weight = 1?

leslie townes

haha, wikipedia.

> An important application is to compute linear independence: a set of vectors are linearly independent if and only if the Gram determinant (the determinant of the Gram matrix) is non-zero.

that might be the most useless characterization of linear independence ever, but do go on

Koro

:(

Obliv

why is that, @leslietownes?

leslie townes

because it is generally hard to compute determinants. because it is often easier to determine linear independence/dependence in other ways (often without using/choosing an inner product at all). and when it's difficult, the gram determinant is probably going to be even more difficult.

it's really just an issue of phrasing. there are all sorts of relationships expressible in the form "X if and only if Y" and this is one of them. "an important application of X is to compute Y," to me, suggests far more than just an if and only if kind of relationship.

if i edited wikipedia i'd litter that sentence with "citation needed"

robjohn

17:48

@冥王Hades Using the Law of Sines and the Law of Cosines, this can be done with labels on the vertices and computing the angles opposite $x$. The diagram I used.

Obliv

18:03

@LeslieTownes aren't determinants just computation? like expanding or gauss jordan stuff. How do they get more difficult to compute?

like can't you just have computers solve them

Silent

18:27

How to solve $xy''-y'-4x^3y=8x^3\sin(x^2)$ ? I mean it seems close to Cauchy-Euler equation, but multiplying wholde DE by $x$ also does not lead me anywhere!

Obliv

@Silent did you try variation of parameters

(that's all I know up to so far lol)

Silent

@Obliv Variation of parameter would need a guess for solution of complementary function, right?

Ted Shifrin

@Yai0Phah Yes, true enough.

Obliv

nope, complementary function is solution of homogeneous which you get from characteristic @Silent

you also don't guess for the particular solution using variation of parameters

you have to set up a linear system

you did that before or no?

Silent

No! I guess I need to read about it.

jay

18:36

(I am a noob in differential manifolds). At a point $x$ in $d$-dimensional euclidean space, is the cotangent space just the space of real valued $d\times d$ matrices ? since the tangent space is just $d$-dimensional euclidean, and linear operators on $d$-dimensional euclidean are just matrices

Silent

@Obliv, actually I have done problems using variation of parameters for DE with constant coefficient, from here , and they mentioned that one of the disadvantages of variation of parameter method is that complementary solution is must to use this method.

Can you please guide me how to go in this particular problem?

Ted Shifrin

@jay No, you're misunderstanding. It's the space of linear functionals, not linear operators.

Have you never worked with dual spaces and dual bases?

ILikeMathematics

@TedShifrin When you are doing math research, you use paper, right? Or do you completely work with LaTeX?

Obliv

Wait @Silent yeah you're right you'd need the complementary solution to do this

Silent

oh!

Ted Shifrin

18:40

Geez, @ILike, give it a rest.

jay

wait

ILikeMathematics

Sorry

jay

yes I did

Ted Shifrin

@jay Yes, linear maps to $\Bbb R$ :)

jay

yh ignore that of course I do

Obliv

18:41

@TedShifrin $xy'' - y' - 4x^3y = 8x^3\sin(x^2)$ without being given $y_c$, you can't do variation of parameters right

Ted Shifrin

So, if $x^1,\dots,x^n$ are local coordinates on your manifold and $\partial/\partial x^1|_p,\dots,\partial/\partial x^n|_p$ are the obvious basis for the tangent space, then the dual basis $dx^1|_p,\dots,dx^n|_p$ is the basis for the cotangent space.

@jay Just do linear algebra and forget manifolds. If you represents vectors in $\Bbb R^n$ by column vectors, then the dual covectors are naturally row vectors.

@Obliv I haven't thought about this in 50 years. What is $y_c$, anyhow?

jay

@TedShifrin ahh. I am reading something atm where they are indeed using that fact, which confused me, hence why I asked the question in the first place

Obliv

LOL, it's the complementary solution. $y'' - \frac{1}{x}y' - 4x^2y = 0$

Oh wait yes you can

no wait no you can't

Ted Shifrin

@jay They are probably talking about $(1,1)$-tensors, which are the linear operators on the tangent space.

Obliv

18:59

@Silent did you figure it out

Ted Shifrin

@Obliv I've never heard of a that terminology. You're talking about solutions of the homogeneous differential equation. Don't you need two linearly independent solutions of the homogeneous? That's how you do variation of parameters.

Obliv

Right, if you're not explicitly given those solutions, you can't do the variation of parameters. I'd need $y_1,y_2$ so I could find the particular solution

Out of curiosity, how come you haven't thought of this in so long @TedShifrin are there just better ways to solve D.E. or is this just automatic to you :P

Ted Shifrin

I last taught ODE as a course when I was a beginning graduate student. I never taught it again, other than as bits and pieces of a year-long applied mathematics course I loved teaching. Although some ODE definitely show up in differential geometry, nothing like this.

Obliv

19:17

That's interesting, I wonder why it doesn't come up as often in higher up math stuffs.

I guess I'll find out.

leslie townes

@Obliv it's a question of scale. the even the best algorithms are slow at scale (without a lot assumed about the structure of the matrix). and without exact arithmetic, telling whether something is nonzero or not on a computer is its own problem, even not at scale.

Ted Shifrin

Well, I cheated and asked Mathematica to solve the homogeneous equation. The trick is to look for functions of $x^2$.

Obliv

Oh very true, stuff like irrationals?

@Silent If you still need the solution I found that the trick is to look for functions of $x^2$ (I know, how am I that good)

I cheated by asking you to ask mathematica, since I don't have a license @TedShifrin

Ted Shifrin

You still can do Wolfram Alpha.

Obliv

That solution looks complicated, I doubt I'd have to do something like this in my ODE class

(thankfully)

Mr. Feynman

19:32

Any hint for this?

15 hours ago, by Mr. Feynman

I don't understand how using the chain rule they say $(5.11)$ can be derived using $(5.10)$. The composition of $\exp$ and $X(t)$ is $t\mapsto\exp[X(t)]$, how would I use $(5.10)$ here? Also, $(5.10)$ is evaluated at $t=0$.

Akiva Weinberger

Conjecture

Let $\frac n{2?}:=\begin{cases}n/2,&n\equiv0\pmod2\\
n,&n\equiv1\pmod2\end{cases}$ and let $T_n:=\frac{n(n+1)}2$ be the $n$th triangular number. Then$$\boxed{\text{$T_k$ is a square iff $\frac k{2?}$ and $\frac{k+1}{2?}$ are both squares.}}$$

Ted Shifrin

@Mr.Feynman Start by writing $X(t)=X(t_0)+(t-t_0)X'(t_0)+o(t-t_0)$?

leslie townes

akiva: i love that notation.

Ted Shifrin

19:47

I would endorse $\dfrac n{¿2?}$.

Akiva Weinberger

Is there an n¡

or ¡n!

Mr. Feynman

@TedShifrin Mh, even then I would obtain $(5.10)$ with the substitutions $X\rightarrow X(t_0)$ and $Y\rightarrow X'(t_0)$

Ted Shifrin

Which is what you want, as long as you get rid of the higher-order stuff.

Mr. Feynman

Oh, yes that's actually it. Thank you!

I had this idea but I didn't think of using $t_0$ as the center of expansion, and then rely on the fact it is arbitrary

Ted Shifrin

I think it'll work :)

Mr. Feynman

20:00

It did, thank you again!

Ted Shifrin

Sure thing :)

robjohn

20:16

@Mr.Feynman I have an answer that bears on that.

Ted Shifrin

@robjohn Mr. Feynman was trying to deduce (5.11) from (5.10), but presumably your proof might replace that of (5.10). :)

robjohn

I just mentioned it because it relates. Some of the things proven there might be useful. Don't know for sure.

Ted Shifrin

You starting to feel stronger? :)

robjohn

I'm pretty much back physically, but I do get tired. Probably healing inside still. Still working on the continence.

The worst thing to deal with was the catheter for a month.

Ted Shifrin

Yes, I can only imagine. It was bad enough having drains in my arm after they removed the tumor in my upper arm. But I was thrilled still to have the arm. Different sort of drain, I realize. But you're on the mend and will soon be leaping mountains in one bound.

robjohn

20:30

I did go to UCLA on Monday to proctor an exam and have lunch with a friend. I made many pit stops, but it was a pretty successful trip.

Ted Shifrin

Step by step. Great!

I haven't been back to downtown LA since a year before Covid.

Obliv

Anyone know of some type of compendium of types of math in general that kinda just lays out everything from scratch?

Ted Shifrin

What in the world are you thinking of?

Obliv

Idk, i kind of wish I could get something that starts with formal axioms, then moving on to something like equations, all of the different types of equations and stuff

robjohn

@TedShifrin I made my first trip to UCLA after Covid in November last year

Koro

20:45

does Rudin have a book on complex analysis?

Obliv

like for diff eq, all of the different kinds of diff eqs with examples

leslie townes

koro, stop trolling. :)

Koro

I was wondering as I didn't find it in RCA.

Sorry for saying so. I probably didn't look carefully.

shintuku

@Obliv you can sort of get something like this by just doing standard set theory and then real analysis

leslie townes

koro: i mean, maybe it doesn't, if you define that term narrowly. anything specific you were looking for?

Obliv

20:49

@Shintuku like if you wanted to list out all types of diff eq what would be your way of writing them?

Koro

like extending maps analytically, something about poles, something about essential singularities etc.

my macbook is behaving nicely today and is not interrupting Airdrop as it does habitually.

leslie townes

that stuff is definitely in there. knowing rudin, some of it may be blended in with generalizations to function spaces or harmonic analysis on R^n, but it's in there.

Obliv

also is this considered a diff eq $dy=1-dx$

Koro

I also want insights on how to prove part of inverse function theorem using Rouche theorem.

shintuku

@Obliv there's no generalized theory of differential equations atm, but there are a couple of reference books that list the known sorts

give me a sec I have one somewhere

Koro

20:53

i.e., suppose f is holomorphic on open connected $\Omega$, and $f'(z_0)\ne 0$ for some $z_0\in \Omega$, then there exists a ball around $z_0$ in which f is 1-1.

So there exists a ball around $z_0$ in which f' is non zero.

shintuku

@Obliv the book I was thinking of was Handbook of First Order Partial Differential Equations

it's a reference work

Koro

Set $w_0=f(z_0)$. By open mapping theorem, there exists a small ball around $z_0$, which gets mapped to a ball around $w_0$.

Obliv

ok ill check it out thanks

shintuku

but in any case, like i said, there's no current general theory of differential equations @Obliv, real analysis, functional analysis and more differential equations are what you'll meet when studying them seriously

leslie townes

koro: i mean, we've gone from "does he have a book on complex analysis" to something very specific, it wouldn't surprise me if that isn't stated in that specific form, or maybe left for an exercise. but that's true of a lot of choices made in complex analysis books. theorem in one, exercise in another

Koro

20:58

😅

leslie townes

the book maybe isn't organized like a "normal" complex analysis book, so it's harder to navigate for specific things.

Ted Shifrin

@Obliv This makes absolutely no sense, by the way.

leslie townes

but even e.g. conway and ahlfors have versions of this problem (in my view)

AMDG

So... I like how Wolfram doesn't like solving things that have floor in it, yet when you convert it to a function of arctan(tan(x)), it just works. Presumably, this gives the number of powers of two in an integer that has powers of two: wolframalpha.com/…

This should be solving, ultimately $2^{-y} x \equiv 1 \pmod 2$

And that obviously generalizes to other factors.

Anyways, just thought that was worth sharing.

Mr. Feynman

@robjohn I was working on that theorem to finally prove the BCH, so we can say that the content of the answer was my endgame :P

Sine of the Time

21:48

Can you have a look at this answer? It seems wrong to me

@leslietownes @TedShifrin

leslie townes

a negative-voted answer to a closed question? sure, why not?

in integers, pq = pr and p nonzero allows you to deduce q = r. whether you regard that as "dividing by p" is kinda up to you

Sine of the Time

@leslietownes but is it formally correct to say "dividing"?

Ted Shifrin

In the rationals, yes

Sine of the Time

but in $\Bbb Z$ I think it's imprecise

leslie townes

i don't know or care? it's up to you? it's not formally wrong, in the sense of, nowhere is it written that a problem about integers must only stay in the integers

depending on where this occurred in a class, and what we had seen so far, and what the point of the class was, it's the kind of thing i might take off a single point for, but that's about it

Ted Shifrin

21:54

I would use no zero-divisors (which takes a proof) and write $p(q-r)=0$.

I would allow “cancelling” rather than “dividing.”

Sine of the Time

Sure, cancelling is more appropriate and it's used also in groups

leslie townes

you do run into exactly this kind of construction when trying to formalize what Q is and what it means to divide, and that's the kind of setting where i'd say, don't be this imprecise. but it isn't clear from OP's context that that's where they are

it could have been a random exercise in a section about equivalence relations where the focus was something else

now that the question is closed, we'll never know :~(

Ted Shifrin

I think we need to know $\Bbb Z$ is an integral domain in order to define its field of fractions. This is sorta a big point.

leslie townes

the existence of sqrt(2) is sorta a big point, unless it's not

robjohn

I thought all points were very small, infinitesimal, actually.

Ted Shifrin

22:06

In the meantime, someone edited to remove the "dividing" language and make it the argument I suggested.

@robjohn What about surreal points?

Sine of the Time

that's why I prefer calc over abstract algebra. Doing too much set theory and pure math make you going crazy :(

Ted Shifrin

Well, if you do calc as a mathematician and not as an engineer, it's not all that different.

In other words, once you get past formulas and plug-and-chug, analysis is far harder than algebra.

shrug On that note, I'm going to the store.

Sine of the Time

@TedShifrin I hope it's harder than algebra :)

Xander Henderson

22:23

@TedShifrin Oh, man... I so disagree. :P

Algebra is too rigid. I find it far harder than analysis---there is never any real weakening of bounds or relaxing of conditions which can allow you to force some estimate through.

Either a thing Just Works™, or you have to find a completely different approach.

Which, admittedly, is probably a failing on my part, rather than anything about either algebra or analysis being "harder". But... still... algebra is hard, man.

robjohn

@TedShifrin I'll have to check with Georges Seurat.

Xander Henderson

@SineoftheTime It can be. It depends on what, precisely, you mean by "dividing".

robjohn

@TedShifrin I'd have to agree with Xander on that.

@XanderHenderson for me, I agree.

Xander Henderson

Even in the context of the integers, it is possible to define a division operation which acts on some subset of $\mathbb{Z}\times\mathbb{Z}$, and then show that "dividing both sides by $d$" is a reasonable thing to do, so long as the division operation is defined.

I wouldn't want to do things this way, and would prefer to describe this as "multiplicative cancelation", but there is no reason that you couldn't, in (5-odd) principal, describe this as "division".

Also, I get to teach the Mean Value Theorem in 20 minutes! Yay! I love the Mean Value Theorem.

It is, far and away, the best theorem in first semester calculus. It might be the best theorem in the first year of calculus. I like Taylor better, but I'm willing to entertain counterarguments.

Presumably, he is teaching that it is opposite over hypotenuse.

Sine of the Time

22:42

@XanderHenderson maybe algebra is harder because you're good at calculus :)

robjohn

@XanderHenderson It is between $-1$ and $1$

Xander Henderson

@robjohn Could be. Or perhaps that it is the $y$-coordinate of the point where the angle in question intersects the unit circle?

@SineoftheTime I don't believe in the concept of a person being "good" or "bad" in a mathematical field. I very strongly believe that anyone can get good, if they apply themselves.

robjohn

Or it is the imaginary part of $e^{ix}$ when $x$ is real.

Xander Henderson

But I never found algebra very compelling or intuitive, whereas I find much of analysis to be really interesting. Hence I put more effort into learning analysis, and less into learning algebra.

@robjohn Ah! That must be it.

Ted Shifrin

@XanderHenderson I'm talking about introductory coursework for undergraduates. One quantifier (typically) in algebra. Three (typically) in analysis.

Sine of the Time

22:46

@XanderHenderson Don't know, but when I attend the lessons, I understand the concepts of analysis when the teacher is explaining, Obiuously, I've to study at home, but I can grasp the concepts. But I can't do this in general when attending algebra or geometry lessons

Xander Henderson

@TedShifrin Ah, okay.

That's fair.

I still found analysis much more intuitive and easy as an undergrad, but I can definitely see how a typical undergrad might find algebra easier. Of course, maybe the problem was the text---Hungerford is not really an undergraduate text.

Ted Shifrin

Note, also, that both you and @robjohn were/are professional analysts. I have much more natural intuition for analysis than for algebra. But if you teach enough undergraduates both algebra and analysis (as I did for 36 years), you find that what I said is correct. Starting with (Rudin) analysis, as I did in 1970, is not going to work with most math majors. Algebra is far more straightforward to learn to do proofs in.

Hungerford has an undergrad text (rings first) and a grad text.

Xander Henderson

@TedShifrin Yes, I know. We were taught undergraduate "Groups, Rings, and Fields" (a one semester class) out of the yellow Springer text.

Sine of the Time

I found analysis intuitive, but it's difficult to compare because the programm is different

Ted Shifrin

@Xander a bizarre choice.

Sine of the Time

22:49

take for example what's called calculus 1

Xander Henderson

@TedShifrin Indeed.

Sine of the Time

here it's a unique exam, namely calc 1 + real analysis.

Ted Shifrin

In Europe it's totally different from the US, @Sine. In Europe calculus is much more analysis, not just formulas and problem-solving.

Xander Henderson

In any event, I also think that Rudin is a bad choice of text. It is not a very pedagogical text.

Sine of the Time

@TedShifrin formulas and problem solving are a small fraction

Ted Shifrin

22:50

Well, it was the standard text in the 70s, Xander. MIT ultimately made their analysis course into a three-pronged option.

Xander Henderson

@TedShifrin Sure. What was the comparable algebra text in the 70s?

Sine of the Time

for example this year I've calc 2 and we're studying now mesure theory

Ted Shifrin

Yes, @Sine. I know. In the US college calculus is "calculus for science and engineering," most definitely not for mathematicians.

Herstein.

But people were already using Fraleigh in more accessible courses.

Artin wrote his algebra text in the 70s (we were early guinea pigs for a few sections of it), but officially was using Herstein at the time.

Sine of the Time

@TedShifrin I've it but the notation is unconventional sometimes

I'm referring to Herstein

Ted Shifrin

Sometimes? It's horrible. He writes functions on the right.

Xander Henderson

22:51

@TedShifrin I am unfamiliar with that text.

@TedShifrin Oh, yuck.

Sine of the Time

@TedShifrin yes, functions and composition

Ted Shifrin

I dislike the book for many reasons. One is that he makes algebra too much symbol-pushing and doesn't make the concepts sing the way Artin does, for example.

Xander Henderson

Artin is a nice text. Comparing Rudin to Artin is apples and oranges. :/

Ted Shifrin

He also separates algebra from linear algebra, whereas Artin unifies them throughout the book.

Artin is still graduate level for most colleges in the US, Xander. It's a tough book.

Xander Henderson

@TedShifrin I don't disagree.

I just think that Rudin, also, really should be thought of as either a graduate review text, or a simple reference text.

Sine of the Time

22:53

Is "real and complex analysis" good to study mesure theory? I'm referring to the first chapters

Xander Henderson

I don't think that Rudin makes any effort to actually teach or instruct---he simply presents the relevant theorems, with the shortest, most elegant possible proofs.

Ted Shifrin

I did teach out of it in 1982-3 at UGA (at that point they were xeroxed notes, but pretty much getting to final form). I had some great students in that class. The next time I taught algebra at UGA I was writing my own book ...

@Xander The problem is that there is no standard excellent option as a Rudin alternative. Abbott is too watered-down. Pugh is too hard. I sorta like Wade's analysis book, although I have never taught out of it.

Xander Henderson

@TedShifrin Yeah, I agree.

There are not really any good alternatives.

Which is really sad. :/

Ted Shifrin

@Sine For measure theory, per se, I would suggest Royden or Halmos.

Xander Henderson

And probably why nearly every analyst I know has their own unpublished undergraduate analysis text.

@TedShifrin I'll second the recommendation of Royden.

Sine of the Time

22:56

@TedShifrin ty

Ted Shifrin

I haven't read it, but why hasn't Tao published his?

Xander Henderson

@TedShifrin An Epsilon of Room.

Yeah, I think he's published it.

Ted Shifrin

So why isn't the world using Tao? He seems to be a much better expositor than Rudin.

Xander Henderson

I haven't read most of it. And what I have read is from the blog, so, presumably, it has been edited significantly.

@TedShifrin Dunno. I like his style quite a lot. Anywho... time for class.

Ted Shifrin

Happy class.

Sine of the Time

22:58

@ted have you met Tao ?

Ted Shifrin

No.

Sine of the Time

sad

Imagine if he's an MSE account

Ted Shifrin

Most of the well-known research mathematicians I know are on MO, not here.

Appropriately so.

Sine of the Time

I've never understood the difference between MSE and MO

shintuku

mse: not research level, mo: research level

Sine of the Time

23:04

ty shin

shintuku

you too

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$Mathematics$ Mathematics