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12:46 AM
you wrote "the field with one element F1 does not exist.", however just now @MartinBrandenburg on this question[1] wrote to me "Of course F1 exists, in many ways." After looking into things myself, it appears to me that some authors dont make the rule $1 \neq 0$ in the field axioms, but then every theorem/proof needs to make a remark about the zero ring being excluded. Therefore I am leaning towards F1 does exist as a logical concept
[1] https://math.stackexchange.com/questions/4983752/what-is-the-reason-for-needing-to-have-an-element-1-neq-0-in-the-commutative?noredirect=1#comment1067678
amusingly, before i left to go out, i had posted a question and written "F1 does not exist as a logical concept as a field for this reason" and when I came back someone had edited it to "F1 does exist as a logical concept as a field for this reason" and I am still trying to process this and can't find who made that edit in the edit history
@MattCalhoun no one edited your post in that way. You are likely misremembering what you typed.
 
2 hours later…
2:53 AM
a statement like "F1 does exist as a logical concept" is absolutely void of meaning
in mathematics, things are only what we define them to be
it is not possible for a field to have only one element, when people refer to "the field with one element", it is a philosophical/moral notion
if you take a look at the paper Martin Brandenburg linked, you will see that the point of this philosophy is not to define a field with one element, but rather to define a notion of a "variety/scheme over F1" in name only
the point of that being that those are suppose to behave to varieties/schemes over Fp in the usual sense as 1 behaves to p
that's all there is to it
pie
pie
3:22 AM
I’m really stuck and could use some guidance. I’ve been studying math on my own, but I’m not sure how to do it effectively. I keep forgetting what I’ve learned, especially topics like real analysis, and it feels like I’m stuck in a loop of relearning instead of making progress.

I also don’t know if I should memorize long proofs, like the implicit function theorem, or just get a general understanding. I don’t have anyone to guide me, so I’m hoping for advice from more experienced people here. Any help would be really appreciated!
Bob
Bob
4:12 AM
@pie what is the goal of you learning math?
pie: bob's question is the key question. without knowing the answer i'd only add, there is almost no point in memorizing specific proofs. even in a classroom context (which is where most people who do that end up doing that) it is almost useless. totally useless for self study.
Bob
Bob
I was once asked in a job interview if I could prove the central limit theorem
and I had to answer no
I had seen the proof but I could not recall it at all
OK, but that is rare enough use case, and also it is totally different from memorizing a specific proof of a theorem. i didn't say that it was totally useless to know how to prove theorems.
Bob
Bob
if I had memorized the proof, I could have said yes
as it turns out, they offered me the job but I turned it down
OK, if you find that you can't prove theorems without memorizing specific proofs of them, then yes, if you want to use math you should memorize everything you can see.
pie: also without knowing the answer, i'd say there is no general reason why anybody would need to be able to state, let alone prove, any version of the implicit function theorem, unless they expected to use that theorem in some context. there isn't any "general mathematical culture" where that is going to come up in a way that is divorced from some motivation for needing to know it.
there would be people in almost any university math department who don't know or care about the implicit function theorem.
Bob
Bob
4:22 AM
who would care about the implicit funciton theorem?
any context where you might need to be able to understand the foundations of "multivariable calculus" or things that use that kind of toolkit. differential geometry being one example.
even some applications oriented folks where there are large numbers of potentially related "variables" (e.g. physics, e.g. at least some statistics) might find it helpful for purposes of understanding where the formulas they are using are coming from, or for deriving formulas of their own.
Bob
Bob
will knowing the implicit function theorem and things like it get you a good job?
I am wondering if @pie should start applying for jobs
and again, learning why you might care about a theorem, or understanding what theorems like that are generally used, is very separate from memorizing proofs of them. memorizing proofs of things is basically only useful for contexts where "recite a memorized proof of X" is going to be helpful.
Bob
Bob
understood
the level at which anybody "needs" to dive into any one thing is going to be a function of what you hope to do with it. if it's self study just for self study's sake, i'd say, do whatever feels good. skip every proof, memorize all the proofs, whatever you like. there are no constraints.
Bob
Bob
4:28 AM
sounds good
I am going to sleep now
it is late here
nice chatting
bye
pie if you have a specific goal in mind it will shape everything. if the self study is just for purposes of educating yourself, unfortunately the answer is going to be very subjective. there isn't really any established canon of things that every "math person" should know. or if you find yourself in a social environment where there is such a thing, whatever the fashionable thing is will be determined by subjective social rules.
i've never heard of such an environment in math but who knows. it strikes me as something that might have existed, if at all, in fairly aristocratic circles in centuries past and not now
like maybe there was some club in victorian england you couldn't get into, or wouldn't fit into, if you couldn't discuss euclid's elements in latin, or something. then you should study euclid's elements in latin.
5:02 AM
@MattCalhoun I see no contradiction with my message, he links to the paper which explains in what way F1 exists. People define categories of schemes over F1, but not F1 as a field. And it has nothing to do with the trivial ring
pie
pie
5:35 AM
@Bob:" what is the goal of you learning math?". I like math I want to be able to publish paper and be a mathematician however math major in my country is very poor so I am not a math major I just study math in my "free time"(although I deidcate a large portion of my time for it) .
Since I study all alone I don't know what should I really do.
6:02 AM
@Bob just remember tricks. chebyshev inequality
if u forgot chebyshev, try to recall markov inequality
actually i forgot markov inequality
lol
@pie i struggle with this too. Id say spend maybe 20 mins a day relearning old stuff, rest of the time on new stuff
i think markov was $P(\mu(x)\geq c)\leq E(\mu(x))/c$ for positive valued function $\mu$
How to solve the Diophantine equation, $ab(a+b)=T$, for positive integers, where $a$ is any given Triangular number & $T$ are also certain Triangular numbers? As of, $(a,b,T)=(3,7,210), (3,82,20910)...$
6:32 AM
Apr 16, 2022 at 20:33, by copper.hat
Mathematics is just a depressing subject. All you learn is how much there is that you do not know and how much one can forget.
 
2 hours later…
8:16 AM
Hi
Hi
9:08 AM
@SineoftheTime But once I get here, then i have to replace $u = \arctan(y/x) \quad 2I=\sec u \tan u +\log(\sec u +\tan u)+k$
Then another question, the integral $\int \frac{1}{\sin(x)^3}$, I saw that with the Weiestrass substitution it is easier than $\int \frac{1}{\cos(x)^3}$
But I was thinking if for this integrale with $\sin$ too, for example, we can use that $\frac{1}{\sin(x)^3} = \csc(x)^3$
Maybe now I'll try to see for myself
Oh no sorry, but before with the backwards substitution of $u$ I was referring to the fact that I started from $\int \sqrt{x^2+y^2} dx$ , sorry
Oh yes by the way also for $\int \csc(x)^3 dx$, I followed the same procedure and it works
 
1 hour later…
10:36 AM
@Pizza $\tan u= \frac yx $ and $\sec u =\sqrt{1+\tan^2 u}$
@Pizza ok
 
1 hour later…
12:06 PM
what's the standard example here? I know $V=\{(1+1/n)e_n\}\subseteq \ell^2$
I don't think there really is a standard example
@SineoftheTime your example reminds me of this answer of mine math.stackexchange.com/a/4799056/476484
which is about two disjoint closed subsets of a complete metric space which have zero distance between them
12:32 PM
@VladimirLysikov gotcha, thx. I'm not trying to argue with anyone, I am simply agreeing with whatever the last person who would know better than me told me lol I absolutely was 100% convinced you were correct, and clearly I need to carefully read this stuff so I can understand it for myself. Before asking about this any further I have a lot of reading to do
12:43 PM
What is the maximum possible monetary sum of coins value 1 2 3 4 6 9 12 36, such that no subset sums to 72. How can I prove that finding the optimum can be done using a greedy algorithm so using a smany coins of vakue 36 as possible(1) as many of vakue (12) etc. ?
Such that at no step there is a subset that sums to 72
I'm looking to generalise this kind of problem
this is an absolutely wild guess, but this reminds me of the partition function. Would it be possible to construct some kind of generating function that creates a partition function just for the integers you named in your list? I am super rusty on partition function and in the past when I played around with this thing I seem to recall there was a way to do it where you could use any integers not just the complete set of all integers. not sure tho
https://en.wikipedia.org/wiki/Partition_function_(number_theory)
I think I understand what you mean. Are you referring to the restricted partition function section of the page?
something like $\sum{q(n)x^n}=\prod_{k \in K}{1+x^k}$ where $K=[1,2,3,4,5,9,12,36]$? would that work?
@ArjunRaghavan lol i should have scrolled down!!! I was going from memory but yes I think thats exactly what I meant
I feel like there's a simpler way since we're only looking for the maximal sum and not all subsets. Because if I understand the method you suggested correctly we would find all the possible subsets that sum to 72 and the avoid them?
And with just some messing around with different numbers I wasn't able to achieve a sum greater than the one given by a greedy algorithm.
Sorry I'm very rusty myself
Bob
Bob
1:24 PM
@pie have you tried to publish anything? is it time to try?
do re mi fa so la si do
do si la so fa mi re do
your LaTeX is broken
$\begin{bmatrix} r \\\theta \end{bmatrix} + \begin{bmatrix} R \\ \phi \end{bmatrix} = \begin{bmatrix} \sqrt{2rR\cos(\theta-\phi)} \\ \cot^{-1}\left(\frac{r\cos(\theta)+R\cos(\phi)}{r\text{ sen}(\theta)+R\text{ sen}(\phi)} \right ) +\begin{cases} \pi & r\text{ sen}(\theta)+R\text{ sen}(\phi)\geq 0 \\0 & \text{si no}\end{cases}\end{bmatrix}$
what's that?
wait, are you Italian too. The amount of Italians in this chat is unreal
re mi fa so la si do re
do si la so fa mi re do
1:29 PM
is mathematics education in Italy that good, or is it that bad
\textbf{Theorem (Existence and Uniqueness of a Linear Transformation)}

Let \( V \) and \( W \) be vector spaces. If \( \{ \mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n \} \) is a basis for \( V \) and \( \mathbf{w}_1, \mathbf{w}_2, \ldots, \mathbf{w}_n \) are vectors in \( W \) (not necessarily distinct), then there exists a unique linear transformation \( f : V \to W \) such that:
\[
\begin{cases}
f(\mathbf{v}_1) = \mathbf{w}_1, \\
f(\mathbf{v}_2) = \mathbf{w}_2, \\
\vdots \\
f(\mathbf{v}_n) = \mathbf{w}_n.
I'm not sure that Gian is Italian
@Gian'sPizzeria What are you doing ?
But can LaTeX be tested here?
are you referring to the fact that he writes sen instead of sin?
1:30 PM
well only math-mode really
@SineoftheTime also the "si no"
@Jakobian In high school my teacher didn't explain, he sent the students to the blackboard...
If I have a long thing to write in LaTeX, I test in it on main in an answer box 💀
And he told him what to write
$\textbf{Large Pizza 10.5"}$$\quad$£16.99$
well that makes sense I guess... education is kinda fucked in that a teacher has to make everyone pass or they risk their job
1:33 PM
I think Italian universities are good overall
$\text{What is}\\{1+1}\\
\text{ equal to? }\\3$
@Jakobian For oral verify you could bring an exercise you had done at home... Tell me if it makes sense
even if you're encouraged to graduate fast instead of taking you time to understand deeper
And you could also look a bit from the notebook, but there were also those who had the teacher assign them an exercise.
1:35 PM
@SineoftheTime in Poland the issue persisted even in universities. You are encouraged to let your students pass because that makes money
not sure about how universities are right now, though, but that's from the time I began studying
moreover, if the students in your uni have high marks you receive as an institution more money
so you can understand what happens
because high marks "=" good student
@Gian'sPizzeria But are you doing tests in LaTeX?
$g_f=\frac{\sum_{n=1}^p r_n^2\left(\frac{1}{2}f(n)\sin\left(\frac{2\pi}{f(n)}\right)-\pi\cos^2\left(\frac{\pi}{f(n)}\right)\right)}{1-\prod_{n=1}^p \cos^2\left(\frac{\pi}{f(n)}\right)}$
@SineoftheTime But do you frequent university
But is there a LaTeX exam in mathematics university?
I did two LaTeX courses
But this chat does not support all commands
1:41 PM
I've read a couple of days ago that they're planning to update LaTeX in chat
@SineoftheTime $\Huge \text{Sine of the time}$
@SineoftheTime No, they are working on upgrading MathJax on the main site. That is distinct from ChatJax, which is used in chat.
ah ok
@Gian'sPizzeria try ctrl w to enlarge the font
Ok
$\Huge 😡$
$\Huge \text{I'm angry}$
pie
pie
@Bob No, I don't have enough knowledge or know how to publish.
1:52 PM
@SineoftheTime But did you also take the algebra exams?
what they consist of
Group, ring and field theory
In the first
It was one exam
1:55 PM
I thought algebra was like decompositions of expressions, things like that
that's high school stuff
How did you find your high school exam?
there are no exams in high school
? you don't have the final exam
1:59 PM
Yes, that's what I mean
I did it during covid so it was easy
Was it online?
I didn't understand why during covid it was easy
If you did it in person
Shouldn't the difficulty be the same every year?
it was only oral
2:05 PM
I wish my exams were oral
With all the AI chatbots I expect the exams will start shifting to oral now
hi
physics describes the real world. And math is part of the real world. so physics>math
but physics uses math. so math>physics
what's the definition of real world?
Galileo says "mathematics is the language with which God has written the universe"
this quote seems to imply that math is more fundamental
@SineoftheTime But with Gauss Green formulas and consequences, after I write $$\iint\limits_D {\left( {{{\partial Q} \over {\partial x}} - {{\partial P} \over {\partial y}}} \right)da} = \int\limits_{\partial D} {Pdx + Qdy}$$
2:18 PM
e.g. C is more fundamental than a program written in C
What should I write?
@SineoftheTime the universe
@Pizza there are plenty of consequences of GG, but I don't know what did you do in class
Did you divergence and Stokes in the plane?
@RyderRude the issue is physics doesn't describe the real world in the exact sense of the word
physics describes the real world, sure, biology does too
@Jakobian this is correct, but if we accept reductionism, then all other descriptions of the real world derive from physics
e.g. chemistry is a low detail description of physics
2:27 PM
whatever, my point stands
but reductionism cannot be confirmed, so there is that
there is only evidence for reductionism. it could still be wrong
Wigner suspected that it is wrong
in the "unreasonable effectiveness" paper
what you said is irrelevant to my objection
@SineoftheTime I found that Gauss Green is used to prove the divergence theorem in the plane
you are doing gross oversimplification of what physics, as a piece of science, is
and from that oversimplification you are drawing false conclusions
@Jakobian yes. "physics" in the sense of how it is practiced, is not applied to everything. i am making a distinction between "physics the subject" and "physics the knowledge"
e.g. i wouldn't apply physics to model chemical reactions, but i do have the knowledge that physics laws apply to it
in other words, a human won't apply physics laws to everything, but we know that physics laws apply to everything
2:31 PM
@Pizza did you see that in class?
@RyderRude in here, you must have meant physics in both senses of the word, so you either purposefully are trying to deceive the reader, or you actually made a mistake when it comes to distinguishing the two
since you are saying now, that you are distinguishing the two, I guess you was trying to deceive the reader
@Jakobian sorry..I'm not trying to deceive the reader. I do see ur point. When I say "math>physics", it implies that im talking about how humans do physics, i.e. physics the subject
@SineoftheTime Yes, now I'll try to demonstrate it on a piece of paper, I'll try to understand
but i would say that physics the knowledge is also built upon math
in some abstract sense
as in, free objects move in a straight line regardless of whether or not a human choses to study it using math
i would say moving in a straight line and other deterministic rules are mathematical in nature, regardless of whether or not someone is studying these rules
philosophers try to talk about "possible worlds" which are the ways the rules of our universe "could have been"
@Pizza you usually rewrite GG differently to prove the div theorem
2:46 PM
How do you prove that a greedy algorithm gives the best value, when the length of all possible solutions isn't the same. For example maximum possible monetary sum of coins value 1 2 3 4 6 9 12 36, such that no subset sums to 72.
3:09 PM
My paper is 8 pages so far. In the acknowledgements I am acknowledging a few people. Is it customary to wait until the paper is complete and then email the professors to be acknowledged asking their permission, or just email them what I currently have?
@ModularMindset You don't generally need to ask permission to acknowledge others. It is fine to do so, but you don't need to (as acknowledgements have no real meaning---they are not co-authorships, nor are they citations).
Okay thanks :)
If you are going to ask (which is fine to do), I would do so only have having a completed paper, and I would include a copy of the paper when you ask.
How to solve the Diophantine equation, $ab(a+b)=T$, for positive integers, where $a$ is any given Triangular number & $T$ are also certain Triangular numbers? As of, $(a,b,T)=(3,7,210), (3,82,20910)...$
3:28 PM
@RajeshBhowmick you've asked this a bunch of times, if you don't get answers apparently users in chat are not interested. If you want to receive an answer, you'll have more chances if you ask on main
@SineoftheTime Why you are so rude? I did not ask any dumb question. May be you are not interested, but you can not represent the whole chat room.
I don't think I'm being rude. I did not say "stop asking here" or something similar. Since you've asked the question here at least five times and did not find a suitable answer, I'm saying that if you ask on main you'll have more chances of receiving an answer
3:46 PM
@RajeshBhowmick Sine of the Time was not being rude. I have seen you ask that same question at least twice, and you have gotten no answer. As Sine of the Time said, this is an indication that no one active in this room (a) has the knowledge to answer your question, or (b) has the time and inclination to answer your question. When you do the same thing multiple times and get the same result, it is time to do something else.
If anything, I think that it is somewhat rude to repeatedly ask the same question when it is apparent that no one is going to answer it.
(Not hugely rude, but at least mildly impolite.)
@XanderHenderson I'm sorry for my impoliteness.
4:45 PM
@SineoftheTime $\int_{\partial{D}} -F_2 dx + F_1 dy$
what are $F_i$ ?
It is the vector field $F = (F_1 , F_2)$
yes? What are you trying to say?
$\iint_D \frac{\partial{F_1}}{\partial{x}} = \int_{\partial{D}} F_1 dy$
$\iint_D \frac{\partial{F_2}}{\partial{y}} = - \int_{\partial{D}} F_2 dx$
Adding member to member I get the formula above
are you sure you did not confuse $F_1$ with $F_2$?
5:00 PM
Why
Mm no?
It is to demonstrate the divergence theorem in the plane
Wait
I'm following from here
ok
I always got confused when I look at GG
Don't know why
I have a question
But this: $\int_{S} (\nabla \times \mathbf F) \,\cdot d\mathbf s = \oint_{\partial S} \mathbf F \cdot d \mathbf r$
It's the same thing, right?
I'm confused with the notations, but it should be the same
$\nabla \times F$ is the rotor of $F$
5:15 PM
Which notation do you prefer?
I use $\nabla \times F$ to remember the components of the rotor
but I prefer rot
I saw the proof of Schwarz's theorem, it's quite long...
yeah, but it's not hard
5:32 PM
@SineoftheTime What would you write about differentiability in two variables?
@SineoftheTime Yes, I'm taking a look at it now.
@Pizza that's up to you
you can write the definition and write the statement of the important theorems
5:47 PM
today I'm going to get drunk as hell
I already feel departing from my consciousness to be honest
6:34 PM
and that was only an hour ago!
Are fourier series just one specific set of orthogonal functions (trig functions) but in general we can represent any function as a set of orthogonal functions that might have infinite number of terms and not always converge
did joseph fourier prove the latter or the case of trig functions or what was the actual statement
@Obliv we have in general a formula $x = \sum_{i=1}^\infty \langle e_i, x\rangle x$
and thats sometimes called a Fourier series but this is an equality in $L^2$
but people have a lot of ways in which the Fourier series are true!
they studied when sum of trigonometric functions converges to the original function and in what sense
so its not really the most general you sacrifice generality for something else you know
What are e_i's? and L^2
its always sacrifice one thing for the other in mathematics (well, it usually is)
@Obliv oh okay this is just any type of orthonormal basis and by $L^2$ I really should have wrote any infinite-dimensional (in this case separable) Hilbert space
i figured that was the case just wanted to make sure
6:39 PM
obliv "what did fourier actually do" is not as useful a inquiry as it might sound, because he was not working to modern standards of rigor and this was kind of debatable even at the time. his work predated modern and commonly accepted rigorous definition of things like 'continuity' for example and his vocabulary of known stuff was completely different. you have to read between a few lines to put his stuff in modern language
Its like, in a Hilbert space you have a generalization from finite-dimensional spaces where instead of algebraic bases you have orthonormal bases
but its not always the general way in which people understand Fourier series
roughly he showed that under nice enough conditions on a function f on a closed interval, the "fourier series" of that function converges pointwise to that function
in this specific topic you can get better with pointwise convergence
and a little but not much about circumstances where the convergence is not quite pointwise everywhere but very close
in Hilbert spaces you really only get $L^2$ convergence, so its okay, but maybe not for all purposes
6:41 PM
in hindsight, pointwise convergence of fourier series is one of the most hardest problems in fourier analysis and it's really a cruel joke that it's what interested people first
start with the stuff about orthogonality where the theory is a lot easier and is what everyone actually starts with
yeah, orthogonality, leslie is making a lot of sense right now
basically orthogonality is Hilbert spaces, Perseval theorem and Bessel inequality
people made criteria when pointwise convergence is possible
but even like, continuity is not enough for that
there's two main theorems from that from what I recall, and there was this book from Fourier analysis I can recommend
Fourier analysis by Javier Duoandikoetxea Zuazo
you can see different criteria for pointwise convergence, well, somtimes it not pointwise but average of limits from left and the right, not sure why, I guess Fourier series and Fourier analysis is just like that
I'll take a look. Thanks
there was a similar theorem about Fourier inverse transoforms we were talkinng about earlier
for piecewise smooth functions
maybe this is related (I don't know)
In mathematics, the Fourier inversion theorem says that for many types of functions it is possible to recover a function from its Fourier transform. Intuitively it may be viewed as the statement that if we know all frequency and phase information about a wave then we may reconstruct the original wave precisely. The theorem says that if we have a function f : R → C {\displaystyle f:\mathbb {R} \to \mathbb {C} } satisfying certain conditions, and we use the convention for the Fourier...
and when I said I'm not sure why, I really mean a "deeper" reason, and not just surface reason
but I guess Fourier things are like that, they kind of react to principal values and what not, its just their nature
I'm not that experienced in Fourier transforms so I'll let Leslie take the floor (if they want)
(or anyone else)
@Obliv yeah while I'm not actually that experienced, this is the book that one of my teachers (really good at analysis apparently) liked
so I think I can say its pretty damn good
I didn't read it in full, maybe first chapter or so
but if you're interetsed in that then I do encourage you to try and learn Fourier analysis - knowledge is really good for you, even when it comes to something as useless as mathematics
 
1 hour later…
8:12 PM
Could an average computer nowadays calculate the exact value if you consider amongst each spacing the numbers surrounding as a combination problem where you pick the lower number amongst the larger for this sequence? 220 126 5 8 165 78. Or by the time it was finished would the number be too large for computers?
220 choose 126 already has like 63 digits
63 digits is nothing
Well there would be like what 13 more iterations, at least 4 or so stacked
And for combinations there are smart ways to estimate them, so you can first check the approximations, and then compute with more precision if needed
Interesting. The reason I bring up this instance is because I've got a sequence that emerges from what I feel is an interesting algorithmic process involving binary and a third placeholding symbol.
Ah, I didn't understand. You want to compute combinations for pairs and then do it again?
8:20 PM
And perhaps there is something to where this interpretation would 'end up' whether that be a numeric answer to a problem perhaps mathematicians have asked previously or maybe something universally relevant
Yeah until there's only 1 number remaining
Ah, this is complicated. But for your example I think doable
I like the optimistic thought. Now I'm thinking perhaps not an average computer but supercomputer ?
Sadly if the number is that large it might be difficult to find out if it's useful for anything. Would be hilarious if it dealt with Riemann Zeta
How can super large n th zeta function zeroes be checked?
Or can they?
8:47 PM
Odlyzhko has papers about this
I don't know the specifics
Thanks Vladimir
would anyone here agree with me that we can have a rapidly decreasing surface?
clearly and might i add, obviously, we can have a rapidly decreasing function
these are usually called Schwartz functions
A foliation $\mathcal{F}$ on a smooth manifold $ X $ is a smooth decomposition of $ X $ into disjoint connected submanifolds $ \{L_\alpha\}_{\alpha \in A} $, called leaves, where each leaf $ L_\alpha $ is locally diffeomorphic to
$ \mathbb{R}^k $ for some $ k \leq n $. The local trivialization of the foliation is given by a smooth submersion
$$
\phi: U \to \mathbb{R}^k \times \mathbb{R}^{n-k},
$$
where $ U \subset X $ is a local chart.

Assume $X=(0,1)^3$ and $\partial X=[0,1]^3-(0,1)^3$ and our foliation $\mathcal F$ satisfies the condition:
There is my purported rigorous formulation of my proposal.
The last sentence is not terribly rigorous.
9:13 PM
@ModularMindset Yes you can have a rapidly decreasing function, just multiply the fast growing hierarchy by $-1$.
I feel silly for asking, but if $\phi_n\to f$ pointwise, does then $(\phi_n)_x\to f_x$? Here the subscript is the $x$-section of a function, i.e. $f_x(y)=f(x,y)$. I feel like it all boils down to a simple observation, which I don't see.
9:27 PM
The simple observation is that for a convergent sequence converging to $y$ you get convergent sequence converging to $(x,y)$ simply by taking always $x$ in the first variable.
@MartinBrandenburg ok, I will meditate on this, thank you :)
There seems to be a lot more questions on the main site lately where the OP doesn't even bother with basic checking.
@copper.hat it's the start of the term :)
I am wondering about a special subring of $Z[T]$, the polynomials in one variable over the integers
@BenSteffan ah yes, make sense.
9:33 PM
Namely the subring generated by all $p^2$ for a polynomial $p$. It contains $T^2$, sure, but it also contains $(T+1)^2 - 1 - T^2$ which is $2T$
The ring is also closed under all endomorphisms, i.e. under substitutions. Then it also contains $2p$ for every polynomial $p$
But I am wondering about a full description. Could it be that this subring is just the whole ring?
(I am a bit embarrassed to ask this in the forum)
i am probably missing your point, but $T$ would not be in the $p^2$ subring. $2T$ would, but not $T$.
why not $T$?
Ok. But how do you prove that?
Consider this subring modulo 2, so every element is a sum of squares. But in char 2 we have that a sum of squares is a square
Right, so we know the image of the subring in $F_2[T]$, it must be $F_2[T^2]$
In particular the subring cannot contain $T$. Hmm
What about polynomials then whose coefficients of odd monomials are even... Is this the result?
9:50 PM
Yes. As with 2T, for every polynomial p the subring contains 2p
Yes Leslie Townes is here. Finally!
Ok so for example $5T^2$ is contained because this is $T^2 + 2 \cdot 2T^2$
but where is grandda?
Ok also these polynomials form a subring, containing every square via direct calculation. Cool
Thanks Vladimir

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