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12:00 AM
It's a new day (UTC)
12:22 AM
@BalarkaSen I love the normal families proof :)
Howdy @Krijn. Long time no see!
12:33 AM
@TedShifrin: Ready for Halloween?
Oops... just noticed it's time to take the dogs to the park. BBL
Hi, bye @robjohn. Everything is dead this year :(
@TedShifrin Hello Ted. Finished my first measure theory exam yesterday. I am glad that he put up some facts from topology in the hints, otherwise I wouldn't have gotten the questions right. I think I got 100%. But wow, has this class been tough.

I've seen several posts on Reddit this semester of students talking about how hard it is to take measure theory remotely. I can relate.
Measure Theory is technical and usually hard for people, regardless. But I always told my advisees to take topology first.
I've been seeing why since the beginning of the semester.

I think one person once told me that when you're in math grad school, topology is more or less assumed. I guess I didn't take that to heart since I thought this is more a probability course than a measure theory course, but wow, was I wrong.
12:50 AM
I already explained my rationale to you months ago, so I won't repeat it. :)
Hi @TedShifrin, I jump in and out yeah
Breaking my head atm over some stupid counting problem
1:07 AM
Oh god
I did it
After 6 pages of blood and tears
1:50 AM
When doing integration in polar coordinates, is one allowed to change the order of integration?
@user193319 If the integral is set up properly, it is just a double integral.
@TedShifrin sorry to hear that.
2:08 AM
Let $\{a_k\}_{k=0}^{\infty}$ be a sequence of complex numbers. Is it true that $\sum_{k=0}^{\infty} \frac{2k^2}{4k^2 + 8k +3} |a_k|^2 < \infty$ if and only if $\sum_{k=0}^{\infty} |a_k|^2 < \infty$.
Should I ask this on the main?
2:23 AM
Note that $\frac{2k^2}{4k^2+8k+3}<\frac{1}{2}$, so if the latter converges, so does the former. On the other hand, $3\frac{2k^2}{4k^2+8k+3}>1$ for large enough $k$, so if the former converges, so does the latter.
2:45 AM
This is actually standard in calculus books.
3:04 AM
ah, right, I knew there was a name
2 hours later…
4:41 AM
Let $f$ be an entire function such that $f(z_0) = 0$. If $$\int_{\Bbb{C}} |f(z)|^p e^{- \frac{p \alpha}{2} |z|^2} dA(z) < \infty$$, how do I argue that $$\int_{\Bbb{C}} |\frac{f(z)}{z-z_0}|^p e^{- \frac{p \alpha}{2} |z|^2} dA(z) < \infty$$?
@user193319 , I need a help : math.stackexchange.com/questions/3887105/…
@Spectre I would love to,but it's 1am where I am and I am exhausted...I have to get up in a couple hours, maybe I'll take a look then...no guarantees.
@user193319 Oh.. ok then.. take your time :)
2 hours later…
6:38 AM
How could I handle this problem?
Sanity check: If I have a partially ordered set $\mathcal{I}$ and $I$ has an initial object $e$ in it. Suppose $\mathcal{C}$ is any category that is indexed by $\mathcal{I}$, then $\underset{i \in \mathcal{I}}{\varprojlim}A_i$ always exists right because it is precisely the indexed object in $C$ corresponding to $e$?
6:59 AM
@love_sodam what do you get when you take $\frac{\partial^2}{\partial a\,\partial b}$?
f(a,b) ?
@love_sodam indeed and you also get $0$
what do you mean 0?
1 hour later…
8:05 AM
y0, anyone here ever had problem classes overlapping?
I might this afternoon
Hmm, both possibilities for my ANT2 problem class occur at the same time as problem classes for other courses :(
Yeah @Edward. I had to take classes at the same time slot this semester because I had committed to both of them beforehand without paying attention to the timings. Another reason because one of the classes was Symplectic Geometry and Classical Mechanics and I badly wanted to take that.
What does one do in that situation? Just visit one on alternating weeks or whut? The problem is that one of the classes is a Seminar and the other is a problem class for my most difficult course, so I'm not sure I wanna skip either of those on any week lol
Lol coz it was zoom I had opened it on two different devices.
8:15 AM
hahaha I thought about doing that but how tf does one pay attention to both
I was mostly paying attention to the symplectic geometry course and reading the material for the other course before hand sometimes so that I can answer questions if something came up lol
hmm fair, I might also ask if the problem class for ANT2 will be recorded
But yeah you cannot pay attention on both of them, that's for sure
just means I can't ask questions of my own
Don't you guys have some specific tutorial days? Maybe you can ask your questions on those days.
8:17 AM
lol tutorial is the word I was looking for
problem class
No, the tutorials are just randomly distributed throughout the week
I see, then you might be in some trouble, if the course is too hard.
aye, the ANT2 course will (strangely) be the easiest of the three courses, so maybe I'll be alright if the tutorials are recorded
or if many others are in a similar situation then the lecturer might move it
Yeah, that is a possibility
meh I need to wait and see what the program is for Hodge theory anyway, it might be way too hard, in which case I'll drop it for AG1
Damn I wish we had a course on Hodge theory, the only interesting course I will get next sem is maybe Homological Algebra or some Global Analysis.
8:27 AM
(p-adic Hodge theory tho)
Sure, why not. Though it seems too hard.
shrugs in self-overestimation
Is sheaf cohomology a prerequisite ?
it depends what the program is. It might be that the punchline is a correspondence between p-adic Galois representations and a weird category of modules
in which case it should be okay
if the punchline is some correspondence between étale cohomology and algebraic de Rham cohomology then I'ma ditch it hahaha
Hmm I see. The second punchline seems more "Hodge theoretic" but what do I know lol
8:32 AM
The tutor says "in principle you should be fine with ANT1 and Alg2"
yeah you're right, but I looked at the master's theses of the lecturer's PhD students that focus on aspects of the first punchline and then CTRL-F'd some algebraic geometry buzzwords and got 0 results so
Man I hate courses. Because of them my reading list just keeps on increasing and the lockdown isn't helping in decreasing it at all. Currently I have to read Arnold, read some Vakil, get to Lurie and field theories someday, more symplectic geometry, and decide a thesis topic
I'm gonna skim Lang's chapter on homological algebra during the ~first month of the ANT course, he's got a big chunk of exercises on group cohomology
well "skim" is optimistic, I'll probably read it in the evenings lol
Leisure reading lol
3 hours later…
11:13 AM
@Balarka can you convince me again that the Homeo group of a top. manifold acts $n$-transitively?
I guess this boils down to "given finitely many points in $M$ are they contained in a single chart?"
I want to do something like join them with a non self intersecting path and fatten it up a little
My notes say that if $X$ is irreducible then $\mathcal{O}_X(U)$ is the set of all rational functions $k(X)$, which are defined at every point in $U$. I do not see why do you need $X$ to be irreducible here
Let $f$ be an entire function such that $f(z_0) = 0$. If $$\int_{\Bbb{C}} |f(z)|^p e^{- \frac{p \alpha}{2} |z|^2} dA(z) < \infty$$, how do I argue that $$\int_{\Bbb{C}} |\frac{f(z)}{z-z_0}|^p e^{- \frac{p \alpha}{2} |z|^2} dA(z) < \infty$$?
11:41 AM
@SayanChattopadhyay probably just convention that every variety is assumed to be irreducible
math.stackexchange.com/questions/2534369 For this question, I want to show that the set of points of discontinuity is exactly equal to $[0,1]\times [0,1]\cap \mathbb{Q}\times \mathbb{Q}$. But when I use the sequential continuity definition, I get a larger set. For eg: $p_n$ be a sequence of rational points converging to $\frac{1}{\sqrt{2}}$ but the sequence $f(\frac{1}{2},p_n)$ doesn't converge to zero. What is wrong with this method?
@LeakyNun I was wondering the same thing, thanks!
@Kumar why doesnt it converge to zero?
@LeakyNun because $f(\frac{1}{2},p_n)=\frac{1}{2}$ for all $n\in\mathbb{N}$ but $f(\frac{1}{2},\frac{1}{\sqrt{2}})=0$
it says y=p/q not x
11:55 AM
@LeakyNun sorry, I messed up. It is the other way round i.e. $f(p_n,\frac{1}{2})=\frac{1}{2}$ for all $n\in\mathbb{N}$ but $f(\frac{1}{\sqrt{2}},\frac{1}{2})=0$
@LeakyNun Maybe here's why you maybe need $X$ to be irreducible. To define $k(X)$ you need to have your coordinate ring $\Gamma(X)$ be an integral domain. That only happens when $I(X)$ is a prime ideal which only happens when $X$ is irreducible.
I think you can still define $k(X)$ in general, it just won't be a field
but ok what you said makes sense
How would you? If $\Gamma(X)$ is just a ring, then won't you need some specific multiplicative subset (localisation) if you want to "invert" elements?
@Kumar ok, and why do you know that the points of discontinuity would be exactly equal to $([0,1]\times[0,1]) \cap (\Bbb Q \times \Bbb Q)$?
@SayanChattopadhyay I just define it as "functions defined on some nonempty open set"
I think it will be the product of $k(X)$ over all the irreducible components
@LeakyNun This is claimed in the textbook. Also, I believe it must be true because it is just $\mathbb{R}^2$ version of Thomae's function.
12:30 PM
math.stackexchange.com/questions/2534369 For this question, I want to show that the set of points of discontinuity is exactly equal to $[0,1]\times [0,1]\cap \mathbb{Q}\times \mathbb{Q}$. But when I use the sequential continuity definition, I get a larger set. For eg: $p_n$ be a sequence of rational points converging to $\frac{1}{\sqrt{2}}$ but the sequence $f(p_n,\frac{1}{2})$ doesn't converge to zero i.e. $f(\frac{1}{\sqrt{2}},\frac{1}{2})$ What is wrong with this method?
1:12 PM
What's up with this number theory chat room that I somehow have been added to?
1:46 PM
@AlessandroCodenotti That's it.
@AlessandroCodenotti Your recipe is too hard, just do it inductively. It suffices to prove that the compactly supported homeomorphism group of a connected topological manifold acts transitively. And sure, you can try some geometric argument like that, but I don't know how you'll get a nice embedded path; it sounds very fiddly.
Just use the standard argument: there is an equivalence relation, where two points are equivalent if there is a compactly supported homeomorphism taking one to the other. Equivalence classes are open since you know how to do this in open balls via explicit formula. Connectedness implies there is only one equivalence class.
Hey @MikeMiller!
It seems easier to me to use this statement to prove that any n points lie in a single chart.
How's life?
@user193319 What is $A(z)$? That could make or break the answer.
1:57 PM
no more mask ?
It's ok, I'm grading which is never the most fun time
1 hour later…
3:12 PM
@Lukas hälst du einen Vortrag für's Langlands Seminar?
3:23 PM
@LeakyNun Why use alternating column sums in the definition of Specht module (or rather, polytabloids, which span the Specht module), instead of (non-alternating) row sums?
They're the same theory right?
3:38 PM
Just use the cell structure coming from positivity of the Kazhdan-Lusztig basis. That is much clearer :)
@MikeMiller Ah ok that's a nice argument! thanks
I swear this is the most common way to use connectedness, too
A space is connected iff an equivalence relation with open equivalence classes is, how shall I say, the constant relation?
TBH I always forget how to do this in balls
The hard part is to take a point away from the origin and move it to the origin
Oh, just work with the unit cube instead
Professor had a question for us today: $\bar{f}$ is pronounced "$f$-bar." How do you pronounce the same thing but the bar under the $f$?
Who uses that notation
@MikeMiller Used in describing upper and lower integrals (just missed how) as far as I understand
3:48 PM
Agree with Tobias
I would simply not use that notation
I know statisticians use that notation to emphasize that the quantity is a vector. Ugh. I'm going to have to figure out how to type that.
@EdwardEvans probably not. Want to focus on finishing my bachelor
4:10 PM
@Lukas Ah I see, someone called "Lukas" was listed as a "Gastvortragende" because not many people signed up, thought it might've been you
there are many Lukases doing math in Heidelberg
yeah, but I happen to know one Lukas in Heidelberg who does a load of number theory
Sometimes you meet your destiny in the path you chose to avoid it.
@Edward also I already did a seminar on local Langlands for GL(2), so I fear that it will be mostly the same stuff
Nice :)
4:15 PM
I suppose you're following Bushnell-Henniart
the book is really nice, they only use rep theory that is developed mostly from scratch to prove the correspondence
Yeah and I'm enjoying the fact that it's basically all algebraic rofl
but I'll be speaking about L-functions so the fun will end eventually
one thing that is a bit annoying is that they use other normalization than some other source such as Jacquet-Langlands or Bump
they aren't doing anything analytic with the L-functions
then I'm even happier
4:20 PM
I'm actually including some GL(2) stuff in my thesis as well
since it's closely related to automorphic forms
Did someone say automorphic forms?
@Lukas how long is your thesis? hahaha
it's still growing
But yeah what's up Lukas and Edward?
Do you have length requirements/constraints in Heidelberg?
4:21 PM
Hey @Amin
And Alessandro!
not really
Hey @Amin
depends on the prof I think
my prof doesn't have any
Hi @Amin how are you doing?
Ah nice, in Bonn there are hard limits of at least 10 and at most 100 pages
Doing alright, how's it going with you?
Oh interesting Alessandro
Is that just for masters theses or also PhD?
@Lukas I'm still yet to find a satisfactory topic for a master's thesis, I know I wanna do something Iwasawa theory-y but classical Iwasawa theory won't cut it for a master's thesis lol
4:22 PM
Masters, I don't know if they have limits for PhDs
ha, I think I'll go over 100 pages in the end
@AminIdelhaj pretty well, I started my PhD and I'm doing some interesting maths
@LukasHeger i wrote 45 pages, counting bibliography, index, cover, white pages and everything
So maybe 35 pages of content
Ah that's a fun time
Everyone I know in Bonn wrote twice that amount but I'm lazy af optimized
Wait what do white pages do? I guess a long time ago when people read books or smth they were so that chapters could start on the right side? I think?
4:26 PM
Yeah they want printed copies of the thesis in Bonn
And that's exactly what the white pages were doing
@AlessandroCodenotti My Master's thesis and my PhD dissertation ended up with about the same number of pages of actual content. My master's thesis just had a bunch of extra pages with some code and stuff
Hey Tobias!
@EdwardEvans have you talked to Venjakob about it?
4:29 PM
And Balarka too what's up? We've got the party going
I am here to chew bubblegum and do math
I am out of math
@Lukas Not yet, when I first spoke to him he wanted me to take at least one of his Veranstaltungen first, so I'm taking both of them this semester lol
and then I'll speak to him after
sounds good
there's also an elliptic curves seminar this semester
I'm actually chewing gum as we speak and procrastinating on math lol
Also according to Rustam, one should be fine with AZT1 and Alg2 in p-adic Hodge theory, so I can't imagine we'll be doing étale cohomology
@Lukas I'm also taking that lol
4:31 PM
Doing all that in one semester would be etale order
@AminIdelhaj haha nice
I'm considering doing the Siegel modular forms course lol
Don't celebrate Halloween tomorrow. Celebrate Weierstrass's birthday.
Do it hahaha
@epic_math Why not both?
4:33 PM
Math requires full focus. Halloween distracts it.
full focus lol
Dress up as Weierstrass' function that's nowhere differentiable for Halloween
Fully focussing on math is for nerds, I like to scroll up and down facebook for 16 hours a day while I feel guilty for not doing math
spooky indeed
@EdwardEvans mood
But replace Facebook with Among Us
4:34 PM
I got For the King the other day too
Some 18 year old just moved into the flat opposite me and I don't think he knows that you can see through net curtains in the evening when his light is on
Unless he's literally an exhibitionist
Halloween is a differentiable function
No one understands my nerdy jokes.
I am a math nerd.
Unlucky man
Math jokes aren't allowed in this room
smh my head
4:38 PM
Why was 6 afraid of 7
Because 7 ate 9.
7 murdered 6's family in cold blood the other night
that took a cold turn
I am gonna differentiate 7.
He will disappear.
4:40 PM
7 ate 9 by teaching 9 infinity categories
So Edward thinking about any fun math?
7 tried to kill 8. He kicked eight but rather than dying, eight became infinitely bigger.
Errr I'm basically just preparing for the semester that starts on monday; smooth representations, recapping some stuff on local fields and homological algebra
How about you`?
Number theory beats algebra.
what about algebraic number theory tho
4:42 PM
In number theory rn we're doing Minkowski theorem stuff
Oh nice
So tomorrow we are supposed to resurrect Weierstrass, is that what you guys are saying?
Also trying to learn about automorphic representations
Number theory is the queen of math, and and algebra is the fool of math.
Tobias: Okay that's a good idea
4:43 PM
@Amin coooool
So algebraic number theory is a fool queen.
fair one
guess I'll just kms then
Where are my algebra backup people?
you're a fool, @Tobias
Imma nerd
4:45 PM
More like a studier of fools
I'm the nerd of fools
And fool of nerds.
What you talking about man
I am talking about... being a fool.
What should be Euler doing right now?
Probably resting in peace
He's resting on the zeta function.
4:47 PM
Wait zeta function I think came way later
That was Riemann
I mean Euler factors sure but the real analysis of it
@Amin are you doing additive or multiplicative Minkowski theory, or are you doing one first and then the other afterwards? lol
The Riemann zeta function is aka the Euler zeta function.
Euler would be talking with gauss.
Don't call him youler.
Based on my impression of what those words mean, additive. We basically showed that convex symmetric sets of size blah have lattice points
And used it to prove that every number is a sum of squares
Ye nice, I think additive probably always comes first idk
that's how it is in Neukirch anyway lol
And I think we're about to use it to show that representatives of every ideal class can be found with norm bounded by blah blah blah
4:50 PM
yeah right, and then class group is finite
Let's talk about number theory or ighta imma head out
Then you have a multiplicative theory which you use for Dirichlet's unit theorem
Ah we may do that later I think
We're pretty much following these notes very closely: jmilne.org/math/CourseNotes/ANT.pdf
Once you have those two proven, I'm fairly sure Minkowski theory becomes more or less irrelevant
ah yeah I used those notes for my bachelor thesis
4:52 PM
@epic_math okay I'm gonna have to be honest with you here it was amusing for the first 4 messages but now this spiel is just getting obnoxious
(Also what's up Ted?)
The proper definition of the Riemann zeta function is $\zeta(s)=\int_{\widehat{\Bbb Z}^\times} |x|^s d^\times x$ where $d^\times x$ is a Haar measure on $\widehat{\Bbb Z}^\times$ and $|x|$ is the restriction of the adele norm
@amin sorry...
Hey I got an article on primes.
@Lukas rofl
4:54 PM
Ah that's a fun one yeah
What are your favorite fields of mathematics?
local fields
commutative algebra, noncommutative algebra, representation theory and algebraic number theory
Mine is (many would have guessed) analytic number theory.
4:56 PM
My tentative research area at this point is kinda analytic NT
But like Sarnak style not Tao style
@loch eyy what's up
Gauss is getting angry. Everyone should say that analytic NT is his/her favorite field.
Gauß is a dead guy
@EdwardEvans I'm gonna cry
behold the beauty in this abstract
4:58 PM
@AminIdelhaj long time no see!
Ugh Lukas that's painful
How've you been Loch?
$\frac{\log X \log \log X \log \log \log \log X}{(\log \log \log X)^2}$
simply beautiful
I mean idk if I can diss on it too hard, I don't have too many logs in my stuff but it is at the moment looking somewhat boundsy
@epic so I'm guessing you're thinking of the "number theory is the queen of math" quote but why analytic in particular?
One of his things was quadratic reciprocity which builds more in an algebraic direction anyway
5:01 PM
@Amin it's because I also like analysis.
He also built the guns on the back of Warthogs in Halo
@loch are you done yet
one more?
I really don't like that kind of analytic number theory anyway
I find it remarkably ugly
@AminIdelhaj Oh no
5:03 PM
Alessandro don't worry as I said it's not like prime gaps and all that shit
@EdwardEvans please don't call it ugly...
It's more like automorphic forms and spectral theory
Its the number theory gang, scary.
modular forms et al are cool, but all this prime gaps and bounds on pi(x) and shit doesn't turn me on at all
Number theory gang has the most fierce members.
5:05 PM
One thing that's kinda cool I've been looking at (by which I mean I wrote about for a fellowship research statement and will table it for now) is arithmetic quantum chaos
Change my mind
@Amin is that like
the weird Riemann hypothesis is eigenvalues of a Hamiltonian or some shit
Riemann hypothesis called weird! I'm triggered.
Pretty much the setup is, let's say you're on a Riemannian manifold. Classical dynamics on it is given by geodesic flow, quantum mechanics (up to some simplifications) by the Laplacian
@EdwardEvans What the hell is this?
5:07 PM
It's trash ignore it
@Sayan idk there was some thing released years ago about it and I read it for like 10 seconds and was like "whats a Hamiltonian" and then never looked at it again
So you sorta expect from physics that in the high energy limit, quantum mechanics starts to resemble classical. In particular, if you're on a hyperbolic manifold, then classical dynamics is chaotic
You can obscure the ideas in a statement to make people think it's really about their field
I put that on the same level as the "topological" proof of infinitude of primes
So this sorta gives conjectures on the behavior of high energy Laplace eigenfunctions
@Amin the fuhh
5:08 PM
But often these are only known in the setting of arithmetic manifolds
The sort of pioneering paper here is Iwaniec-Sarnak
@MikeMiller nope 2022
@AminIdelhaj ive been slacking off to figure out industry options so i have to catch up on math now lol
Let's say $X$ is a compact Riemannian surface, and let's say $\Delta \phi + \lambda \phi = 0$. We know in general that $\|\phi\|_{\infty} \le C\lambda^{1/4}$
($\phi$ here is $L^2$-normalized)
This bound is sharp for the sphere but we expect better in the hyperbolic case. Some mix of numerical evidence and that this might be what one would consider "chaotic", that function kinda evens out or smth
There is a hidden Terence tao among us.
It seems interesting that prime gaps have maximums at multiples of 6.
But for general hyperbolic surfaces all you can really get (at least at the time I-S was written) is $\|\phi\|_{\infty} \le C\frac{\lambda^{1/4}}{\log(\lambda)}$
But if you're dealing with arithmetic surfaces you're able to get a power saving bound by playing with Hecke operators
Can anyone give some intuition for this?
5:14 PM
@Amin Hecke operators! You said a word that I know!
In particular $\|\phi\|_{\infty} = O(\lambda^{5/24 + \varepsilon})$
@loch That's definitely not unproductive! I need to figure out that stuff as well, in particular (prob not this summer but in a future one) I may apply for finance-type internships just to be safe lol
What are the maps $f$ such that $f\circ g$ continuous implies $g$ continuous?
I start by naming open injections
Hi. If the arithmetic mean $\sigma_n = \frac{1}{n} (s_1 + s_2 + \cdots + s_N)$ converges to some finite value as $n \to \infty$, can we also say that $(s_n)$ and $(\sum_n s_n)$ converge?
I want to show that if S is a regular surface and p\in S and if P\subset R^3 is a plane such that P contains p and S lies on one side of P then T_pS = P.
My definition of T_p S is {f'(0): f is a regular curve in S with f(0)=p}
5:31 PM
@S.D. what do you think?
@love_sodam both are two-dimensional subspaces, so it's enough to show containment in one direction, I believe $T_pS\subseteq P$ is the easier one
@Thorgott I can't think of any strong reason why those two implications should hold, but it seems like they've used that fact here in the 2nd page, to prove that Cesaro convergence => Abel convergence
Notice the statement: "If σn converges to a limit and r < 1, then both (N + 1)σN+1r N and sN r N+1 will vanish"
Do you have any idea?
that does not require the implications you suggest
@Thorgott Oh really? :O
Could you explain?
they explicitly deny that the implication "convergence => cesaro convergence" can be reversed
@love_sodam One suggestion would be to rotate $\Bbb R^3$ so that $P$ is the $xy$-plane.
5:37 PM
note that $r<1$, so $r^n\rightarrow0$
@Thorgott Interesting! I see your point. But then why does $(N+1)\sigma_{N+1} r^N$ and $s_N r^{N+1}$ vanish as $N \to \infty$?
Given only that $\sigma_N \to \sigma$ as $N \to \infty$ where $\sigma \in \mathbb R$
Election Day is coming up!
@Thorgott Agree. Maybe up to rigid motion, we may assume p is the origin and P is the xy-plane(@TedShifrin). If f'(0)\in T_pS then f'(0) should contained in xy-plane why..?
Write $f(tj=(x(t),y(t),z(t))$. Write down what you know.
@TedShifrin Maybe we can think of extreme point
By assumption, as S is contained in R^2\times R^{+} and f is contained in S, f has a local minimum at the origin.
wait that's strange
5:51 PM
Hey everyone!
Oh, z(t) has local minimum as 0 so that z'(0)=0
so that f'(0) should contained in xy-plane what do you think? @TedShifrin
Max or min, one of them, yes.
Oh I assumed nonnegative z-axis so, minimum
@S.D. you also need to note that $(N+1)r^N\rightarrow0$ for the first one, the other term is bounded
the second term can be expressed as the difference of two terms looking like the first one
00:00 - 18:0018:00 - 00:00

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