12:19 AM
the toddler is coming home momentarily. it's going to be yelling about what she wants and fighting with the cat until about 8:30 or so.
remind me why we thought propagation of the human species was a good idea

12:30 AM
That's concupiscence for ya.
I finally decided on optimizing the Newton-Raphson reciprocal approximation that PM 2Ring gave me. I looked into it more with 3-2x as the initial guess. It turns out it really likes to be parallelized, and doesn't actually require that many operations. I'll solve this and other NP-hard problems later when I've made the tools to properly analyze these things myself.

Is there some standardised shorthand notation to represent something like $n_{1}\mathbf{a}_{1}+n_{2}\mathbf{a}_{2}+n_{3}\mathbf{a}_{3}$? I considered dot product notation first ($\mathbf{n}\cdot\mathbf{a}$), then realised that this would result in a scalar, not the vector I wanted.

You can use $\sum$

questions per day are down to 330 :)
I wonder what the goal is.
Would $0$ be ideal?

@o.9 Isn't that a bad thing?

I don't know

12:35 AM
i don't know of a 'standardised' notation for linear combinations other than what you'd do with sum notation.

I think some people read 10 words per minute so they don't like a lot of questions

i read less than that. i have slowed down as i age.

@user400188 $n_1 a_1 + n_2 a_2 ... = \sum_{k=1}^{m} n_k a_k$ could work.

@AMDG Thanks :)

12:36 AM
@user400188 My pleasure

sometimes people up against a word limit send me stuff to edit and i remove about half of the words.
one time the limit was 14,000 words and they were at the limit. i got it down to 6.
thousand. i'm not that good.
we won, too.

but at what cost?

@leslietownes sometimes people up against word limit send stuff edit I remove about half words.

8000 words

Bruh, if I had a word limit, I'd probably go over it by ten orders of magnitudes.

12:38 AM
a few hours of my time.

do you ever think about those words?

Someone asks me "How does this work?" And I begin by explaining the fabric of reality all the way up to the actual process they asked me to describe. That's me.

I sometimes help people with word limits by removing about half the words.

it's very easy to do about half.

$\sim \frac{1}{2} = \text{ezclap}$

12:41 AM
Remove half the words in a 14k word essay and tell them they should delete the original and make sure they keep this one safe. What they don't know is you removed half the words at random.
EEEEEZZZZ

sometimes in legal documents you need to keep a certain amount of stuff in to 'preserve an argument' so you don't lose the ability to appeal the decision. but you can still do close to half.
one time a guy sent me something from a case i wasn't even on, subject line, "halve," no message body.
AI is getting pretty good at this, too.

folds the document in half

how good?
The only job that's gonna be left is investor
and landlord

i've seen a product that can do a factor of 80-90% and preserve the meaning.
it's not sold yet.

I mean we now have fully autonomous cars that we can be quite confident in so...

12:45 AM
Are you supposed to read it first
or just fullsend like a chad

Pssshhh, who needs to proof-read a computer's work?
I mean computers never make mistakes so...
:P

I've never caught a computer making a mistake :/
although I did lower the voltage too low once and it would freeze

Every computer is an incarnation of Bob Ross. That's the secret.

but I found the sweetspot later and its stable af now

Computers don't make mistakes, just happy accidents.
Overclocking and AMD don't seem to mix very well.

12:51 AM
I was just undervolting

Well, at least for me, I couldn't get my RAM to be stable at the expected overclock speed of 2666 MHz.
The base is 2133
I have 3200MHz RAM now, though, and it works fine. All AMD system.

my ram is 2133 right now as well

when i was in grad school we had a postdoc who was constantly frying his hardware with aggressive overclocking. my advisor ended up buying him a better computer.

I haven't tried to overclock it though

But the thing I've noticed is that the automatic GPU overclocking, likely still because AMD driver devs are still figuring out how to write drivers, or... something. Hard to find a reasonable explanation as to why AMD drivers don't Just Work:tm:.

12:53 AM
we said, here you go, please don't overclock this. overclock your other s--t.

No, my username stands for the Latin phrase Ad maiorem Dei gloriam.

oh that's a good one

My BIOS is up-to-date enough. I can't really update it much more (and this is the secondary BIOS since the other one I updated and it wouldn't POST because I didn't realize the old CPU wasn't compatible with the new microcode).
I figured as much, but I can't quite tell what it is.

me neither
but its a dude with two eyes and a nose

12:57 AM
What do you mean? I don't see any Is, just an o and a 9.
How much precision is needed for scientific work? Anyone here know?

jesuit i presume

@copper.hat Yeah, I "took it back".

What is a jesuit?

An order that used to be Catholic.

an order of catholics
used to be?

1:01 AM
They follow Vatican II, so they cannot be considered Catholic.
Not anymore

just a tiny bit extreme

They are catholic in Mexico as far as I know

and contrary to the work catholic

But I didn't know normal folk could be jesuits

they are not

1:01 AM
There's a Jesuit school close to me

one needs to be ordained

most of my neighbours went there

excellent schooling

No, nothing extreme. It's a matter of what the Church teaches. Vatican II is contrary to Church teaching, therefore, Vatican II did not come from the Church. You are either Catholic or not.

I think my neighbours are catholic

1:02 AM
i lived through vatican II

Actually I've went to church to a Jesuit chapel

look up the meaning of the wwork

I press X to Doubt everytime I hear someone say they are or know someone who claims to be Catholic.
Yeah well as a Catholic, I have no idea what "the work" is.

i am not religious, but attended a franciscan seminary

Was it before the death of His Holiness, Pope Pius the XII (Oct. 9th 1958) or after?

1:03 AM
yeah, i'm sitting this one out

lol alright

usually people think f it as 1962

well there's a lot of catholics in Mexico I think

my first elementary school was named after pp 10

but maybe they don't pass your test of catholicism

1:04 AM
ireland has a few

i think that is true. there are a lot of catholics in mexico.

if one is strict, few are

I thought I was catholic but maybe I'm not ;(

What you likely get in more conservative areas are what we refer to as "conservative Novus Ordo" or "Indult".

there are extremes everywhere
must go collect dinner

1:05 AM
Although the term "indult" doesn't apply anymore to those who adhere to the Novus Ordo hierarchy.

Is the pope catholic?
the argentinian dude

A true pope is Catholic, but as an adherent to the Thesis of Cassiciacum, "Habemus papam materialiter, sed non formaliter."

is there a true pope?

Which is to say that all the popes after HH Pope Pius XII are only material popes, or simply popes in name. They enjoy the title, but not the jurisdiction, of the papacy owing to a serious impediment to that jurisdiction.

oh ok

1:08 AM
Namely, that they do what is objectively contrary to the purpose and objective intent of the papacy and the Church by teaching and attempting to impose what is objectively not Catholic, viz. Vatican II.

did someone raise flags -.-

Hm? Why?

there's a cop here

There isn't anything disruptive here, so there's no reason to flag anything here.
No one has violated any civil peace and order.
@o.9 Also, I didn't see this comment of yours. If you'd like, I'd be glad to talk to you more concerning the faith in detail elsewhere.

I don't think anything has been deleted

1:17 AM
ctrl+f would confirm that
Aight, well I'm going to go do other things now. Nice, chatting!

take it ez

ted, how do you do?

he's not in here
so I guess good
oh he is :D

Who?

nobody knows.

1:27 AM

No word yet on my test.
Is Olivia watching M Hulot?

that's what i was asking about. i got an inconclusive result on mine although i probably had it.
livvy is indeed watching that.

I’m coming right over!
Oh wait.

2:03 AM
my son has a sore throat & mild fever, apparently contracted from a friend. i am supposed to fly to ireland on monday, so a bit of panic. unusually, i have had even less in person interaction over the last few days, so i am hoping that wearing a mask inside the house will reduce the chances of contracting it (& failing the test which would be a bit of a disaster).

@copper.hat. My whole family had it. Don't worry. Drink a lot of lemon + ginger + hot water.
Stay safe folk
@copper.hat. In case you have breathing issue, this need a device that bump oxygen for 1 week before you come back normal

@Avra i am not concerned about getting covid (other than spreading) but am concerned about canceling flights, car rentals, and my daughter's flights.
i realise this is no guarantee, but i am in pretty good shape for a 60 yo. good lung & vomax capacity.

@copper.hat. Stay save and take as much precaution as possible and always keep hand sanitizer next to you as you travel
@copper.hat. Keep your self as far as possible and avoid touching your eyes, nose and face. If necessary, use sanitizer before touching your face

what a fossil. nothing like us juvenile 40 year olds.

thanks! my hygeine in that regard is reasonable.
i spend a year traveling on a budget and learned a lot of good lessons :-)

2:11 AM
@copper @leslie My test came back negative. Presumably flu.

excellent!!!

ted i had something similar last week, really awful stuff but not covid.
good to hear.

it is hard to get real numbers about the $\Delta$ variant.

@TedShifrin. Great to hear that. Usually, Doctors magnify the signal, so usually flue get mixed with COVID cases.

Good luck, copper. Times will be scary for the indeterminate future.

2:12 AM
@TedShifrin Much appreciated. I am a bit more tense than my usual.

No doctor involved, Avra.

As my brother says, I am "wound tighter than a clam's ass at 1,000 fathoms".

i spent three days with a high fever and did not eat or drink a thing. i eventually needed about three gallons of fluid and food. so far, so good.

@TedShifrin. I mean test itself involves magnifying the signal to be safe, so some flue cases get mixed with COVID cases.

there are different sorts of tests.
pcr is pretty specific & sensitive

2:14 AM
@copper.hat. I can not speak after your folks, you are the math expert now... :/

i have some binmaxnow tests.
i am certainly no maths expert :-)
doesn't stop me from commenting though...
@TedShifrin Still, flu is a risk, take care of yourself.

I’m 10 days through now, more or less done.

@TedShifrin. Usually, flue takes from 1 week to 1 month

i never had a flu last longer than 3 days.
at least the elevated temperature.

I ger a flu shot every year … so nothing too

2:18 AM
@TedShifrin. Have you read the book about sugar and the 100 disease it cause
?
Among which is flue
When I was in China, I rarely see people get flue! I asked about that!

lots of people get flu in china!!!

They told me they eat garlic in their breakfast, dinner and launch

these are old wives tales, unsupported by evidence.

However, I wonder what classmates ,workmates would do if they smell garlic 24 hours coming from me!

some things, like turmeric, have some support.
if garlic was magic, then cultures that eat more would likely have some observable effect
unless the entire collection of allia were responsible for said effect
i am not discounting the incredible power of the placebo effect.
and spurious inferred correlations...

2:25 AM
garlic has some antibiotic, so it's not bad
better than nothing
In China, most medicine is natural, so it takes 360 days to cure flu

not all bacteria are bad, so antibiotics can affect health in different ways.

i am not sure where you are getting your numbers from.

@copper.hat. I just repeat why my brother said. He is a Doctor. So I am not sure

i have many doctors in my family.

2:28 AM
I really don't know much about biology

i know little about biology, but do know something about extracting conclusions from data.
covid resurrected many quack treatments. people have great difficulty separating causation from correlation.
they get sick, they take magic potion #1 with some rhinoceros horn and then 3 days later they are well.
however, they have no way of knowing if they would have been well anyway and rarely think to ask.
even well educated folks, well versed in reasoning, drop their guard when out of their subject matter area.
linus pauling and vitamin c is a much touted example.
if something seems to work for you by all means take it.
everyone likes to cling to 'natural' cures.

@copper.hat. You mean even vitamin C might not be true about what it promotes for?

it has little effect on colds.
except in extreme athletes where is shortens the cold duration a small amount.

@copper.hat. Maybe that is why it take long time to cure flue only by relying on natural ingredients like vitamin C

this is substantiated.
flu is viral.

2:36 AM
you mean sicne it's viral, so you don't think vitamin C has any effect on it?

anti-virals can shorten the time, but there are no (well) know natural anti-virals
none, apparently

Probably, vitamin C energize immunity system?

your mind is powerful. but it needs a hook, apparentlly
my sister is an elementary school teacher and swears by vitamin c.
but that is placebo.

@copper.hat. My mind is powerful?
@copper.hat. I understand what you said, probably just myths

well, i can't speak specifically about yours :-), but one's mind can have a huge effect on the course of disease.

2:38 AM
I don't know even Doctor promotes for this vitamin C!

if the myth works, use it

I really don't understand what is going on in this field

many doctors prescribe just because people want something
there is a lot of shoddy data analysis in the biological arena

@copper.hat. Do you think data analysis will develop this field?

partly lack of sufficient education in dealing with data & experimental techniques, but also because of funding.

2:39 AM
Do you think this is the begining for new data analysis world, i.e., this centuray?

it will take time because there is no money in it

@copper.hat. I heard that it shrinks!

i don't know

Companies are shrinking monetary support for DS

A sine curve with a height (amplitude) of 13 and period of 16 lines remarkably well onto a grid

The heights are 0, 5, 9, 12, 13, 12, etc

2:41 AM
I don't know how they can do such a thing while statistics and statistical inference is ever lasting field!

i think that is not the issue, the issue is that it costs a lot of money to do large experiments
i should get off my soap box now :-)

I suppose this means it should be feasible to do 16 evenly spaced points around a circle of radius 13 on a grid
(0,13), (5,12), (9,9), (12,5), (13,0), etc

Trick or Treatment by ernst & singh is a lightweight quick read that discusses some alternative medicine
how many lattice points will a circle of radius $r$ at some centre $(x,y)$ intersect?

2:56 AM
In mathematics, the Gauss circle problem is the problem of determining how many integer lattice points there are in a circle centered at the origin and with radius r {\displaystyle r} . This number is approximated by the area of the circle, so the real problem is to accurately bound the error term describing how the number of points differs from the area. The first progress on a solution was made by Carl Friedrich Gauss, hence its name. == The problem == Consider a circle in R...

@AkivaWeinberger thanks :-) i want 'on' the circle :-)

Ohh I thought you wanted in it

i know that one, my question was triggered by your comments above

Related question: what are the possible circumcenters of three lattice points?
Because if $(x,y)$ is not one of those points, then surely there's a maximum of 2
In fact: Consider the union of all perpendicular bisectors of pairs of lattice points. There's countably many of them, and they're all measure zero, so the union has measure 0 too (despite being, I assume, dense)
Therefore, a random point $(x,y)$ will almost surely have a maximum of 1 for its possible answers!
I'm honestly not sure what points are circumcenters of lattice points though
Separately from that: Suppose a triangle has lattice points as vertices but does not intersect any other lattice points (neither on the boundary nor in the interior). What are the possible areas of such a triangle? @copper.hat
Surely 1/2 is possible because the triangle with vertices (0,0), (0,1), and (1,0) doesn't intersect any other lattice points
Anything else? (I know the answer)
Continuing the circumcenter thing: I calculate that the circumcenter of $(-1,0)$, $(1,0)$, and $(a,b)$ is$$\left(0,\frac{a^2+b^2-1}{2b}\right)$$
I don't know a quick way to check if a rational number can be written in that form...

3:24 AM
0

What points in the plane can be written as the circumcenter of three lattice points? The points that cannot be written this way have an interesting property: for any $r$, at most two lattice points are a distance $r$ away from them. I know at least that all circumcenters are rational points, beca...

Decided to ask the question on main

@Avra No, but I eat no junk food and very few sweets other than fresh fruit.

3:42 AM
Suppose a sequence $(a_n)$ is be bounded and satisfies $a_{n+1}\ge a_n-\frac 1{2^n}$, then it is to be proven that the sequence $(a_n)$ converges.
The proof follows if we consider an auxiliary sequence $x_n =a_{n+1}+\frac 1{2^n}$ and then show that $(x_n)$ is monotonic.

i used that result as a hack in a penalty function algorithm decades ago.

No problem in that so far.
Hi copper.

hi @Koro

Then I tried to prove the result alternatively- by defining $L=\{n\in \mathbb N: a_n\gt a_{n+1}\}$ and $R=\{n\in \mathbb N: a_n\le a_{n+1}\}$ and then I noted that neither $L$ nor $R$ can be finite (else the implication is trivial by monotone convergence theorem) so both $L$ and $R$ are infinite and infact define a complimentary pair of subsequences of $(a_n)$ and hence the problem is now reduced to showing that these two subsequences converge to the same limit. And I got stuck there.
And I think that the accepted answer is not correct because in order to prove Cauchy one must show that $|s_m-s_n|$ can be made arbitrarily small by choosing $m,n$ large enough.

Hm, it's almost there, but bungles it at the end because $(m-n-1)/2^n$ goes to infinity as $m$ goes to infinity
but the immediately previous expression doesn't

3:50 AM
However, the answer only shows that $s_m-s_n$ can be bounded from above by arbitrary small positive number. And, it is to be noted here that this does not say anything about lower bound for $s_m-s_n$

Replacing that last expression of the answer with $\le1/(2^{n-1})$ should fix it

let $a= \limsup_n a_n$ and show that $\liminf_n a_n = a$.

Surely this should end like$$\le\frac1{2^n}+\frac1{2^{n+1}}+\ldots+\frac1{2^{m-1}}\le\frac1{2^{n-1}}$$ — Akiva Weinberger 16 secs ago
Just made that comment on the thing

@AkivaWeinberger it shouldn't, I think. The problem is with the LHS wherein it should have been $|s_m-s_n|$ (as we don't know (apriori) whether $s_n$ is monotonic)

Ohh, fair point

3:52 AM
for any $\epsilon>0$ there is some $N$ such that $n,m \ge N$ gives $a_{n+1}-a_m \ge -\epsilon$.

Yeah, they seem to be assuming $s$ is decreasing

@copper.hat true

now choose some subsequence $a_{n_k} \to a$.

@AkivaWeinberger hmm

Incidentally: If we $a$ is bounded above and below and $a_{n+1}\ge a_n-\frac1n$, must $a$ converge?

3:55 AM
and show for any $\epsilon>0$ we have $\liminf_n a_n -\limsup_n \ge -\epsilon$.
no, you need the extra bit to be summable

I think you can oscillate, so I guess no (though it'll take really long to oscillate back down 'cause you have to wait for a section of the harmonic series to get big)

@copper.hat let me process this more in my head as something looks fishy here(I may very well be wrong though)

i am going to sleep shortly, so if you have questions about my approach ask soon or else tomorrow :-)

I think $x_n:=a_n+\sum_{n=0}^{n-1}1/{2^n}$ is the easiest way to go here

copper, no problem. please go to sleep. take care. I'll let you know if I have complaints with the approach you suggested above.
@AkivaWeinberger that's as good as taking $x_n=a_n+\frac 1{2^{n-1}}$. Isn't it?

4:00 AM
Hm, let's see
So $a_{n+1}\ge a_n-1/2^n$
so $x_{n+1}-1/2^n\ge x_n-1/2^{n-1}-1/2^{n-1}$
so $x_{n+1}-1/2^n\ge x_n-1/2^{n-2}$
so $x_{n+1}\ge x_n+1/2^n-1/2^{n-2}=x_n-3/2^n$
I don't think that guarantees $x_n$ is increasing

@Koro just for giggles, i found the old erl memo www2.eecs.berkeley.edu/Pubs/TechRpts/1988/ERL-88-64.pdf. Look at lemma 3.1 on p12.
i, as usual, did the work :-)

it's very funny to me that we were both at ERL at different times.

@AkivaWeinberger right, that doesn't :-( however, $x_n=a_n-\frac 1{2^{n-1}}$ surely does :)

Uh let's try that
You're almost certainly right but it's late so I can't do this in my head

:-).

4:09 AM
So $a_{n+1}\ge a_n-1/2^n$

i don't like my proof back then. i seem to have improved marginally with age.

ERL?

electronics research lab. some billing unit in eece

the electronics research laboratory. i researched electronics.

so $x_{n+1}+1/2^n\ge x_n+1/2^{n-1}-1/2^n$

4:10 AM
my first job out of college.

Ah and yeah now everything cancels out perfectly

does that count as a job?
i'm, not being facetious

$x_{n+1}-x_n=a_{n+1}-\frac 1{2^n}-a_n+\frac 1{2^{n-1}}\ge 0$

probably not, but i was paid more than anyone else in my family, so for me, yes.

4:11 AM
it just about covered my non resident tuition
the rest i borrowed
once i got a waiver and that was a life saver
i used to show up in rags. not to make a statement, but because that was what i had
my research advisor at some point brought in some of his old clothes for me.
i still have one of his jackets. he didn't want it, but it is rather nice.

there was a prof in the department who complained about the union trying to get us a dental plan. he said our parents should pay for our dental expenses and that we were spoiled children. i think he was the spoiled child.

i think that was when he realised i was not monied like his other grad students
i was much more vicious in response back then.
i took out a few folks verbally who had the temerity to suggest something similar
i really was on the breadline.

i told him once, i don't need anyone who was the beneficiary of world record levels of investment in education telling me what i can or can't afford, and you can [expletive deleted].
may have burned a bridge there.
he really thought people's parents would just chip in for whatever. i don't know who his parents were. ugh.

yeah, i had no patience for such crap
i do realise that it is a first world problem and that many have survival issues

i probably shouldn't have cussed him out. so many grad students do come from wealthy backgrounds.

4:16 AM
but in context, i was dealing with entitled whiners. often union supported.

it is easy to forget about those who are not as lucky

i don't think the folks like that really understood my background

i also told our union guy to f off.
haha

i really did do construction and shit surveys

i don't play well with others. ted has observed this.

4:17 AM
i am tolerant but have a hard limit

@copper.hat topographic surveys?

lack of caring is one area of low tolerance for me

i did topographic surveys too as part of my lab at college :)

@Koro i used to do sewer surveys
measure & sample the flow of effluent from factories (pig factories my specialty)
i did surveying as well, not the laser & gps variety

our union rep tried to lecture me on what students were going through. i said, my man, you have no idea what students are going through. typical burgeois crap is what you are going through, i don't give a s blank blank blank.

4:19 AM
i was the pole guy :-)
i did draughting.
my advisor needed some drawings but would not pay the draughtsman $600 back then to draw them. so he had me do it. he did not realise i was a trained draughtman$600 would have been most welcome

i hope you billed him for it

i still have billing problems.
i had a 2 hr mtg and 1hr work call today and and trying to decide if i bill the 3hrs
i am a good deal sadly

@copper.hat wow

i work very efficiently and this has come up often at performance reviews. someone always says, i don't think we should penalize leslie for working quickly.
i'm leaning on those folks but eventually i will have to leave.

i have adapted slightly, but still am pretty straight up with billing.
i am going to bill the 3 hrs.

4:23 AM
i've had people say, are you sure it only took you 90 minutes to write that?
yes it did. not inflating a damn thing

i have permanent writer's block, so no one will ever say that to me

i realize what we're up to here but i didn't spend 91 minutes on that so i'm not biling for that

i can talk up a storm. but write it???
whoa, i was going to bed...

Hours ago .

cultural hazard

2 hours later…
6:12 AM
Let $\displaystyle ( a_{n})$ be a bounded sequence that satisfies $\displaystyle a_{n+1} =a_{n} -\frac{1}{2^{n}}$.

Let $\displaystyle \lim \sup \ ( a_{n}) =a^{*} \land \ \liminf \ ( a_{n}) =a_{*}$

Let $\displaystyle m\in \mathbb{N}$ be arbitrary and let $\displaystyle n >m$, then we have

$\displaystyle a_{m+1} \geq a_{m} -\frac{1}{2^{m}}$

$\displaystyle a_{m+2} \geq a_{m+1} -\frac{1}{2^{m+1}}$

...

...

...

$\displaystyle a_{n+1} \geq a_{n} -\frac{1}{2^{n}}$

Adding all these to get: $\displaystyle a_{n+1} -a_{m} \geq \frac{1}{2^{n}} -\frac{1}{2^{m-1}} \Longrightarrow \forall \epsi I think it's correct except in the fifth last line and sentence before that wherein I should have said "now we can choose large enough$m,n$($n\ge m$) such that ..." and "$\exists M\in \mathbb N$such that$n\ge m\ge M$" respectively. Thanks a lot @copper.hat and @AkivaWeinberger 3 hours later… 9:28 AM Hello, I'm sorry if it's forbidden, but I want to promote the Online World Math Contest taking place this month. This contest is opened to everyone, any age, from anywhere. The problems will be translated in 9 languages (English, French, German, Italian, Spanish, Portuguese, Polish, Russian and Arabic) and participants will be separated in 7 categories according to their age. You can register (deadline for registration : 21.08.2021) or look for more details here : fsjm.ch/en/competitions/owmc If you want to train for this contest, you can look on the "Archives" tab of the website (fsjm.ch/en/archives), all the previous contests are available there ! At the end of the contest, there will be individual rankings for each category and a ranking between different countries. Winners will be rewarded with a diploma ! Best regards ! 9:48 AM if given a (lattice) - discrete subgroup of the multiplicative group$\Bbb R^2_+,$does anyone know how to find the volume/co-volume? 9:59 AM does the volume form of$\Bbb R^2_+$as a smooth manifold let you get to the volume of the lattice? I think that the volume/co-volume is not a constant in this case? could it depend on where you are on the lattice? 10:27 AM If$a_j>0$, how does$\lim_{j\to\infty}=L$mean$a_j$is approximately equal to$L^j\$?

10:39 AM
Just ignore me. Dumb question. Pretend I was drunk. I know answer.

1 hour later…
11:58 AM
@EvilJohnRennie What is the story behind your name ?