2:56 AM
Meaning of ineffably in English
in a way that causes so much emotion, especially pleasure, that it cannot be described

2 hours later…
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).

4 hours later…
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?
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 $R=\Delta_X$
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.
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 $d(x,y)=1=d(y,x)$.

2 hours later…
12:06 PM
Hi Ryder Rude.
Hi Peter and Alessandro
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
@psie this is in rudin, it's called separability
Every separable metric space has a countable basis
separability is a different (but related, in metric spaces) thing, it's really about second countability there
Yh I was speaking from a metric spaces perspective. Separability implies countable basis
@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
but they can move messages to trash right?
1:02 PM
Looks like it (I just discovered this despite having been an owner in this room for years now lol)
@SineoftheTime correct
@AlessandroCodenotti and mods
@SineoftheTime just laugh it off and move on
full disclosure: they have done it to me several times
I don't care but this behaviour made a lot of user leave this community
sure, some people are more sensitive than others
2 days ago, by Jakobian
best to leave children to their toys
community managers
1:14 PM
Okay
I have a topology question. If I am open minded then does that make the rest of mankind closed minded?
@SoumikMukherjee not in discrete topology, everyone's open :)
I see what you did there
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 $R$.
@nickbros123 ^_^
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
2:03 PM
@zetaspace what do you mean by $\partial R_i = f_i$
@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
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 logicâ€”also called predicate logic, predicate calculus, quantificational logicâ€”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...
What do we know about complex 1dim. manifolds?
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

2 hours later…
4:46 PM
what's a good/your favourite German dish?
and I'm not talking about desserts
stroganoff
that's not German though, I think
I also like stroganoff but I think its actually Russian, according to wiki
oh yeah
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
yeah I was incorrect
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
I don't know any German dishes actually
I mean sure, e.g. schnitzel with potatos, we eat that too so it seems bland to me
Its really nice the interplay between group theory and number theory. Makes me appreciate number theory a lot more
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
100 divided by 7 is approximately 14.
interesting rule of thumb
100 / 7 â‰ˆ 2 * 7
Why is this site so much better than Quora?
$100 / 7 = 2\cdot 50 / 7 \approx 2 \cdot 49 / 7 = 2\cdot 7 = 14$. Sure.
@TheEmptyStringPhotographer Because we have standards?
I suppose
I mean, youâ€™d probably close a question that asked for the answer to 8+8â€¦ However thatâ€™s on Quoraâ€¦
$8^T+8^T = 8^T$
Cute.
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)
@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).
@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?
@psie It is on a different space.
ok, I see, so different domain, different function
A function (of which a metric is an example) is a triple of domain, codmain, and "rule".
The domain and codomain matter.
5:40 PM
@psie natural numbers are discrete yet have different metric
And something discrete can even have different uniform structure
Oh, that too. I was thinking of "the" discrete metric.
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
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
@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.
@psie so I understand it that you didn't understand me
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".)
@Jakobian sorry, I probably didn't
@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
@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.
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
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.
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
6:26 PM
hi
@Sahaj hi Sahaj
hi Sahaj

3 hours later…
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
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.
hi
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"
RIP teed
although letters certainly are fundamental to reading
9:57 PM
hes left us
i was gonna ask for a rec for a linear algebra book..
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'
@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
which i realize is a little unfair to logicians or whoever but there really aren't that many of them
like, you don't need to understand language on the deep fundamental level, but the things people teach are still fundamentals I'd think
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
10:00 PM
and I feel like what we do in mathematics is similar to what people do when learning a language
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)
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?
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
maybe not entirely but largely
allie so yes you probably want something that is very calculational and matrix focused
10:02 PM
i definitiely plan on learning other pure math stuff though. i love math and theres so much i want to explore
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
especially because my interest in comp chem has to do with the more physics side of things
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
hmm
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
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
and there's a ton of really mathematically incompetent applied stat books, even at a 'high level'
ill ask the PI of the lab im probably gonna work at this year
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
i mean honestly i dont necessarily know if i want somethign geared towards a specific application
the books i recommended above are agnostic as to application
10:14 PM
okay sorry i think i missed that message
thank you!!!
@leslietownes agnostic? Unacceptable. I only accept books that have no applications at all
linear algebra for the nihilist
@Jakobian Hi Jakobian, do you know if every complete metric ANR embed isometrically into some hilbert space?
10:34 PM
@Jakobian No applications is still too many.
@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)