Mathematics - 2024-08-13

user 20458579510081670432

2:56 AM

Meaning of ineffably in English
in a way that causes so much emotion, especially pleasure, that it cannot be described

DannyNiu

4:30 AM

Hi all. I'm just wondering, if there're computer programs that transform formulas that researchers (professional cryptographers, Nobel prize winners, and alike) use to simplify formulas. Also, I wonder if these program can take specialized axioms (e.g. non-associativity) and understand special functions (e.g. mean value).

psie

8:58 AM

I am studying the discrete metric, in particular its symmetry property. Silly question maybe, but what is it that makes it symmetric? Is the equality sign an equivalence relation? Then obviously $x=y$ is the same as $y=x$ by symmetry. My book kind of treats the equality sign as an equivalence relation, but how would one go about proving symmetry for example?

Alessandro Codenotti

Well there are a few possibilities here depending on how deep you want to go. Usually symmetry of equality is just built-in into first-order logic

But if you're talking about equality of elements of a fixed set $X$, then you can think about it as an equalivence relation on $X$, that is a subset of $X^2$, namely the diagonal $\Delta_X=\{(x,x)\mid x\in X\}$

Then a relation $R\subseteq X^2$ on a set $X$ is symmetric if $(x,y)\in R\implies (y,x)\in R$, which is immediate when $R=\Delta_X$

psie

ah, nice!

@AlessandroCodenotti ok, I am doubting what you wrote here; didn't you use some "inference rule", some intrinsic property of the equality sign, to conclude that $(y,x)$ is also on the diagonal?

You concluded that since $x=y$, $y=x$ which I don't see how you did.

psie

10:05 AM

Ok, I've thought some more about this. Maybe I went too far in showing the discrete metric is symmetric;

> Claim: $d(x,y)=d(y,x)$ for all $(x,y)\in X^2$.

> Proof: If $x=y$, then surely $d(x,y)=d(x,x)=d(y,x)$. If $x\neq y$, then $d(x,y)=1=d(y,x)$.

Ryder Rude

hi

Sahaj

12:06 PM

Hi Ryder Rude.

Hi Peter and Alessandro

Sine of the Time

12:47 PM

@Jakobian but it's a public room. I think they are the owner not because its their room

No wonder in the past some users were removed from being owners after moving to trash messages which criticised their actions

this is funny tho

nickbros123

@psie this is in rudin, it's called separability

Every separable metric space has a countable basis

Alessandro Codenotti

separability is a different (but related, in metric spaces) thing, it's really about second countability there

nickbros123

Yh I was speaking from a metric spaces perspective. Separability implies countable basis

Alessandro Codenotti

@Jakobian @SineoftheTime for what it's worth neither room owners nor high rep users can delete messages in this room, that's reserved for admins

Sine of the Time

but they can move messages to trash right?

Alessandro Codenotti

1:02 PM

Looks like it (I just discovered this despite having been an owner in this room for years now lol)

user 20458579510081670432

@SineoftheTime correct

@AlessandroCodenotti and mods

@SineoftheTime just laugh it off and move on

full disclosure: they have done it to me several times

Sine of the Time

I don't care but this behaviour made a lot of user leave this community

user 20458579510081670432

sure, some people are more sensitive than others

2 days ago, by Jakobian

best to leave children to their toys

Soumik Mukherjee

Who are the admins?

user 20458579510081670432

community managers

Soumik Mukherjee

1:14 PM

Okay

I have a topology question. If I am open minded then does that make the rest of mankind closed minded?

nickbros123

@SoumikMukherjee not in discrete topology, everyone's open :)

Sine of the Time

I see what you did there

zeta space

If $f_1$ and $f_2$ are functions satisfying a functional equation, or differential equation, and we consider a finite bounded region in the plane, $R$ s.t. $\partial R_1=f_1$ and $\partial R_2=f_2$, what is the effect of gluing $f_1$ to $f_2$ to obtain a quotient space? What happens to the differential/functional equations under this quotient relation?

I meant that $f_1$ and $f_2$ are boundaries of $R$.

Soumik Mukherjee

@nickbros123 ^_^

zeta space

Naïvely one might expect there is some new diff eq./func. eq that is a combination of the individual eq's

the gluing of f_1 to f_2 makes sense, but not implications for the resultant diff eq/func. eq on the qotient space

Jakobian

2:03 PM

@zetaspace what do you mean by $\partial R_i = f_i$

zeta space

@Jakobian I meant to say $F=f_1 \cup f_2$ encloses $R$. $R$ is just some connected region in the plane

I realized that the subscript on R was wrong

Jakobian

2:58 PM

@psie metric being symmetric has nothing to do with symmetry of a relation

sure, it represents that "$x$ and $y$ are distance $r$ apart from each other" is a symmetric relation etc., but it doesn't directly speak of any relation

@psie although I don't think there is such thing as "the" discrete metric, I think most people think of $d(x, y) = \begin{cases} 0 & x = y \\ 1 & x\neq y\end{cases}$

If $x = y$ then $d(x, y) = 0 = d(y, x)$ as $y = x$ as well

and if $x\neq y$ then $d(x, y) = 1 = d(y, x)$ as $y\neq x$ as well

@psie without diving into logic too much, equality is symmetric i.e. $x = y$ then $y = x$. Its also transitive and reflexive.

if you're wondering about this, this has nothing to do with metric spaces or those other things

First-order logic

First-order logic—also called predicate logic, predicate calculus, quantificational logic—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables, so that rather than propositions such as "Socrates is a man", one can have expressions in the form "there exists x such that x is Socrates and x is a man", where "there exists" is a quantifier, while x is a variable. This distinguishes it from propositional logic, which does not use...

zeta space

What do we know about complex 1dim. manifolds?

Jakobian

the substitution for formulas property implies that if $x = y$ then $y = x$

hmm, well, from what I hear on the internet authors do refer to this standard discrete metric as "the" discrete metric

Jakobian

4:46 PM

what's a good/your favourite German dish?

and I'm not talking about desserts

zeta space

stroganoff

Jakobian

that's not German though, I think

I also like stroganoff but I think its actually Russian, according to wiki

zeta space

oh yeah

Jakobian

I mean there is some intersection of cuisines, but neither does the stroganoff article lists Germany, nor does German cuisine article lists stroganoff anywhere

so I doubt they consider that a German dish in any sense of the word

zeta space

yeah I was incorrect

Jakobian

4:51 PM

I'm wondering because I heard opinion that German cuisine is bland, and I don't think I necessarily agree with the statement

zeta space

I don't know any German dishes actually

Jakobian

I mean sure, e.g. schnitzel with potatos, we eat that too so it seems bland to me

nickbros123

Its really nice the interplay between group theory and number theory. Makes me appreciate number theory a lot more

Jakobian

5:10 PM

@nickbros123 can you give an example

I am more of an impression that ring and field theory are important to number theory

perhaps you mean Galois correspondence

The Empty String Photographer

100 divided by 7 is approximately 14.

interesting rule of thumb

100 / 7 ≈ 2 * 7

Why is this site so much better than Quora?

Xander Henderson

$100 / 7 = 2\cdot 50 / 7 \approx 2 \cdot 49 / 7 = 2\cdot 7 = 14$. Sure.

@TheEmptyStringPhotographer Because we have standards?

The Empty String Photographer

I suppose

I mean, you’d probably close a question that asked for the answer to 8+8… However that’s on Quora…

Jakobian

$8^T+8^T = 8^T$

Xander Henderson

Cute.

psie

5:33 PM

@Jakobian here it seems to me you made use of the fact that equality is symmetric and thus you needed this fact in order to prove that "a" discrete metric is symmetric (though I am not sure what other discrete metrics are out there except for the one you gave)

Xander Henderson

@psie The metric depends on the space. So the discrete metric on $\mathbb{N}$ is not the same as the discrete metric on $[0,1]^2$ (if one wants to be really anal).

psie

@XanderHenderson ok, but isn't it still going to be $1$ if $x\neq y$ and $0$ if $x=y$? How will it be different?

Xander Henderson

@psie It is on a different space.

psie

ok, I see, so different domain, different function

Xander Henderson

A function (of which a metric is an example) is a triple of domain, codmain, and "rule".

The domain and codomain matter.

Jakobian

5:40 PM

@psie natural numbers are discrete yet have different metric

And something discrete can even have different uniform structure

Xander Henderson

Oh, that too. I was thinking of "the" discrete metric.

Jakobian

When it comes to equality, that should be a problem which on the proper timeline should be the first thing you think about, before even set theory

I mean I get the sudden realization but it's a totally different problem

Accept what equality is for now, and focus on what's important. You can come back to reviewing what equality is later

psie

ok :) I will not bother about it anymore, but I still feel like you made use of the fact that equality is symmetric in showing that a discrete metric is symmetric...I have a hard time letting go of that thought

Xander Henderson

@Jakobian This may be a matter of language use. In my experience, a "discrete metric" is specifically the metric $$ d(x,y) = \begin{cases} 0 & \text{if $x=y$, and} \\ 1 & \text{otherwise}. \end{cases} $$ The usual metric on $\mathbb{N}$ induces the discrete topology (as does the discrete metric), but I would not normally expect the usual metric to be called "a" discrete metric.

Jakobian

@psie so I understand it that you didn't understand me

Xander Henderson

5:48 PM

Rather, I would call it a metric which induces the discrete topology.

(Which is a long way of saying that there is some potential ambiguity in your use of "a discrete metric".)

psie

@Jakobian sorry, I probably didn't

Jakobian

@XanderHenderson I don't think its a problem that one normally encounters in education, there's simply no need for discrete metric to mean anything else than that particular metric on a set

But I find the concept of "a discrete metric" to still be nice to have

in particular when discussing different discrete uniform structures one can have

Xander Henderson

@Jakobian I'm not sure what it is that you are telling me isn't a problem. In the courses that I have taken and the books that I have read, "the discrete metric" is a specific metric, and a metric which is not that metric but which induces the discrete topology would not be called a discrete metric.

This is not a criticism of anything you've written, but an FYI for you, so that you might understand the confusion which might be caused by calling the usual metric on $\mathbb{N}$ a discrete metric.

Jakobian

well it is somewhat unfortunate name

for a metric to be complete we mean that the induced metric space is complete etc.

so "discrete metric" should probably be interpreted as "metric which induces a topological space which is discrete" according to this rule

but I need more examples to get myself convinced of this

I don't believe in what I'm saying actually. I don't find it unfortunate

Xander Henderson

6:04 PM

@Jakobian In my experience, that is not the way it works. And expecting naming conventions to be consistent is a lost cause.

Jakobian

yes. It was so early that I've learned what the discrete metric is, that I never had a need to think about this. But it seems like a non-problem

bringing it up made no sense

Sahaj

6:26 PM

hi

Jakobian

@Sahaj hi Sahaj

Sine of the Time

hi Sahaj

Jakobian

9:47 PM

@psie The point is that equality is symmetric, and that you should use that fact without anything holding you back, on an intuitive subconscious level. It doesn't need to be justified

And if you are wondering, why is there such thing as equality, why can we consider all those things, why can we consider sets and why is there such thing as $\in$, then you are dividing into fundamentals of math.

And it wouldn't be wrong to get confused and ask about it in excessive, if you were to be learning those subjects. But for what you're currently learning, you just need to accept that this what equality is, and we can really make it all work, and it does work this way.

And even if logic were to later tell you otherwise, you need to assume that you can make it all work

But, if you want to learn fundamentals, then you need to start learning fundamentals to alleviate your confusions. But who has time for that? If you just want to do math, then you accept that those things might be slightly complicated and you don't want to burden yourself with them.

Equality is some symbol that has properties what we think it should have. Two things are equal, when they are literally interchangable in any setting possible that we can come up with

Joe

I tend to view mathematical logic as formalising and codifying what we already know, rather than being the "ultimate" justification for the mathematics that mathematicians do on a day-to-day basis.

Allie

hi

leslie townes

there should be a better and more widely used term than "fundamentals" for this kind of thing (i don't know of one in english and maybe there isn't one). if someone was setting out to learn to read, and immediately began going down rabbit holes like why there are 26 letters in the alphabet, the history of how the capital letter A has been written, how lowercase g and a came to be commonly typeset in two entirely different ways, etc., you wouldn't say they were exploring the "fundamentals"

Allie

RIP teed

leslie townes

although letters certainly are fundamental to reading

Allie

9:57 PM

hes left us

i was gonna ask for a rec for a linear algebra book..

leslie townes

and a lot of people who zoom straight to "foundational" questions in math, whether out of personal interest, or a feeling that they ought to, or need to, strike me as doing something very similar to that. just, diving down any rabbit hole they can find relating to some building block instead of 'doing anything'

Jakobian

@leslietownes but when you learn a language, you also wouldn't say that all those rules which people teach are really helpful to you when learning, but they are fundamentals

leslie townes

which i realize is a little unfair to logicians or whoever but there really aren't that many of them

Jakobian

like, you don't need to understand language on the deep fundamental level, but the things people teach are still fundamentals I'd think

leslie townes

allie one dividing point for linear algebra recommendations is whether you want the subject to give you facility with how people use linear algebra to calculate things, or whether you do not

Jakobian

10:00 PM

and I feel like what we do in mathematics is similar to what people do when learning a language

leslie townes

allie another dividing point would be, do you plan on learning other kinds of pure mathy things in the "abstract algebra" realm or not (which may not be the same thing as the calculational goal, but probably at least correlates somewhat with it)

Allie

well im entering my phd in computational chemistry next year which from what i understand is entirely linear algebra based, so i would think so?

Jakobian

and natives don't learn fundamentals at all (they might but they don't need it/don't need a lot of it), because they know on intuitive level how to speak

Allie

maybe not entirely but largely

leslie townes

allie so yes you probably want something that is very calculational and matrix focused

Allie

10:02 PM

i definitiely plan on learning other pure math stuff though. i love math and theres so much i want to explore

Jakobian

well, natives do need to learn some fundamentals of their language, you go to school and you do need to learn some of it

I did in elementary school at least, some of it

Allie

especially because my interest in comp chem has to do with the more physics side of things

leslie townes

allie you might ask people in your program for their recommendations as i have no idea what would be specifically relevant to computational chemistry, but friedberg/insel/spence linear algebra or horn/johnson matrix algebra are both solid linear algebra books that talk about a lot of stuff in terms of matrices

what they don't have is a lot of calculational problems or examples of the sort that might actually appear in an application. i don't know of any good books that do - they tend to be tied to specific applications of interest, or specific software packages, and so get pedagogically 'obsolete' pretty quickly

Allie

hmm

leslie townes

some statistics books sometimes cover quite a bit of linear algebra (including e.g. the singular value decomposition), but i'm not sure that the examples of interest in some statistical context would necessarily carry over to being of interest in your field

Allie

10:10 PM

well im not in a program for comp chem. i go to a small liberal arts college and nobody here speciailizes in comp chem. ive just been doing research with other colleges

leslie townes

and there's a ton of really mathematically incompetent applied stat books, even at a 'high level'

Allie

ill ask the PI of the lab im probably gonna work at this year

leslie townes

allie yeah i meant ask whoever you'll be going to work with (or people like the people you'd be going to work with) later

there's a real tension in applied linear algebra especially of, if you discuss any application that really appeals to some subset of students in detail, some other subset will be left out. engineers and physicists and chemists and statistics people and machine learning people don't all find the exact same stuff interesting

so you can give like really silly toy examples from a ton of different applications (kind of like what calculus books do in their 'applications' or 'word problems'), in utterly no depth at all, or you can just teach the subject in a way that doesn't specialize to any one of them

Allie

i mean honestly i dont necessarily know if i want somethign geared towards a specific application

leslie townes

the books i recommended above are agnostic as to application

Allie

10:14 PM

okay sorry i think i missed that message

thank you!!!

Jakobian

@leslietownes agnostic? Unacceptable. I only accept books that have no applications at all

leslie townes

linear algebra for the nihilist

monoidaltransform

@Jakobian Hi Jakobian, do you know if every complete metric ANR embed isometrically into some hilbert space?

Xander Henderson

10:34 PM

@Jakobian No applications is still too many.

Jakobian

@monoidaltransform I think that not every Banach space embedds isometrically, as a metric space, as a subset of a Hilbert space

I don't think there's any $f:c_0\to \ell^2(X)$ such that $\|f(x)-f(y)\| = x-y$ for example

If $1 < p <\infty$ then $L^p$ is uniformly convex so by Mazur-Ulam theorem, a metric space isometry $f:L^p\to \ell^2(X)$ can be assumed to be linear

and I don't think that $\ell^2(X)$ contains a copy of $L^p$ for $1 < p < \infty, p\neq 2$

@monoidaltransform

and this is true because then $L^p$ would be a Hilbert space which it's not

so $L^p$, as a metric space, is not an isometrically embedded in any Hilbert space

note that any Banach space is a complete metric space which is also an AR (which is stronger than being an ANR)

Transcript for

$Mathematics$ Mathematics