12:09 AM
Just one last thing

You sound like Lieutenant Columbo.

I've written up 80% of the method, with an invitation for him to work out the last 20%

Yes, he needs practice writing stuff up, too.

who is columbo?

Famous American TV program from the 70s.

12:10 AM
oh the detective

Police lieutenant who investigated murders ... and always bumbling along pestered the criminal with "excuse me, sir, just one more question."

hahah
what would you call the equation we are trying to show here? is it a constraint for a?
or is it just an equation

It is the equation $a$ must satisfy in order to solve the problem, so it's a constraint in that sense.

righto, I'll just say that
maybe leave a not about the approx value of a so he can sanity check, and also mention the teacher is wrong
now to work out how to put labels on shaded areas in tikz and I'm good to go

You've already wasted a LOT of time on this.
(Not to mention our time.) :D

12:14 AM
jee he sure is getting his money's worth (first sesssion is free)

After this, I demand royalties for asking "Self ..."

lol. I'll consider it. how does a few cents a shout out sound
actually I'm insulted on your behalf by that suggestion

12:35 AM
Ok. I re-did it. :) Ted's right. The proper quartic is $2a^4-8a^3+27$. And $a\approx 1.84281729560283$
If you want more digits: 1.842817295602831355173491746527940

1 hour later…
1:45 AM
@JoeShmo Ah, I get your point now.
Yeah, I don't know much about physics, so I wouldn't know myself

2 hours later…
4:11 AM
just one more question

4:30 AM
i always miss the fun.

i was just at a family gathering where someone said grace before a meal that went on, to my mind, far too long. it was a catholic crowd. had not experienced that before.
going to ask my priest friend what the church's position is on this.

@TedShifrin apparently bumbling, I'd say.

beats saying the angelus twice a day and the rosary on a friday night

1 hour later…
6:07 AM
HI
anyone online ?

2 hours later…
7:40 AM
hi

8:28 AM
In the definition of $\mathrm{limsup}$ as given in the link, is it always the case that one wants to look at the interval [x,a] or [a,x] getting smaller and smaller, i.e. $0<|x-a|< \delta\to 0^+$?
Would one not be interested in intervals of a greater lengths as well?
I guess the intervals should be, assuming $x>0$, $[a,a+x]$ and $[a-x,a]$.

8:59 AM
@XanderHenderson Sorry for the ping, but on a slightly related note about the linked pdf from yesterday (see below), why are $\delta$ and $x_0$ as introduced in the second table needed? Why must there exist a constant $\delta >0$ such that $|s-s_0| \leq \delta$ as well as $x_0$ such that $x\geq x_0$? What is the difference between these two $O$ notations?

9:29 AM
mathematics crack exchange :D
@robjohn hey
@LeakyNun hi

9:49 AM
@OlympicComputerChairSitter yes?

It's a good brain teaser
Basically, if $F(n) =$ a union of cosets and the number of distinct cosets in the union equals $G(n)$ and $G(n) = \prod_{i=3}^n (p_{i} -2)$ is proved. Then can you take the limit of $F(n)$ as a union of sets and say that $|F(n)| = \infty$?
The proof that $G(n)$ is equal to that is just to show that indeed the number of cosets making up $F(n)$ is indeed growing (quite rapidly as well).
@robjohn thx for the comment :)

@OlympicComputerChairSitter Sorry, I missed that it had been stated at the beginning.

Lol, you're too funny
^_^
Yes, the $p_n$, the dread of number theorists. I am a victim of twin primes and prime numbers themselves
@robjohn left you a comment on the post

10:04 AM
@OlympicComputerChairSitter yes, I missed that at the beginning of the post, it was stated that $p_n$ is the $n^\text{th}$ prime.

Yes, you've been foiled by the primes as was I
j/k
What are your immediate thoughts on the post
Gonna stand up and do some jumping jacks
Hi, @MatsGranvik

@OlympicComputerChairSitter Hi. All you need to do to prove the Riemann hypothesis is to show the convergence of the following sum: math.stackexchange.com/a/4171621/8530

@MatsGranvik that is one monsterous summation :)
@MatsGranvik I upvoted

Thanks

You're welcome. Here's some twin primality attempt: math.stackexchange.com/questions/4171618/…
Basically $(P(\Bbb{Z}), \Delta, \cap)$ forms a boolean ring, I switch over notation to $+, \cdot$ throughout since there is no summation for $\Delta$ etc.
You can also take the ring of all finite symmetric differences of integer cosets (so not including most of $\mathcal{P}(\Bbb{Z})$) and you also get a ring that way

10:54 AM
Is there any example when set A is dense in metric space X and set of all limit points of A $\ne X$?
By definition of A dense in X, we should have closure (A)=X
So I wonder if an example as I asked above exists or not.

11:08 AM
try A=X=a point

If for two chain complexes $(C_{},d_{}), (D_{},d_{}')$ we have $C_n\cong D_n$ for each $n$, does that mean the homology groups are isomorphic?

11:26 AM
@Thorgott Nice. Thank you!

@monoidaltransform are the maps also the same?

@monoidaltransform the answer is a resounding no, try some examples

11:43 AM
Would anyone mind clarifying why the two constants $\delta$ and $x_0$ in the second table in the linked pdf (see below) are needed? Why $|s-s_0| \leq \delta$ as well as $x \geq x_0$? What is the difference between the two $O$ notations?

2 hours later…
1:15 PM
If an algorithm for solving a problem requires an infinite number of variables, does it mean that the problem is undecidable in a Gödel sense?

1:34 PM
@schn Because you are taking a limit of some kind.
Try to build some basic intuition: what are those notations meant to represent?
When we say that $f \in O(g)$, what we mean is that $g$ "eventually" dominates $f$.
"Eventually" can mean a number of different things. Perhaps what we mean is that if $|s|$ is "large enough", then $|f(s)| < C |g(s)|$. Or maybe we mean if $s$ is "close enough" to $s_0$, then $|f(s)| < C |g(s)|$.
"Large enough" means that there exists some $M$ such that $s > M$ implies that $|f(s)| < C |g(s)|$.
While "close enough to $s_0$" means that there exists some $\delta$ such that if the distance from $s$ to $s_0$ is smaller than $\delta$ (that is, $|s-s_0| < \delta$), then $|f(s)| < C g(x)$.

2:09 PM
If an algorithm for solving the Riemann hypothesis requires an infinite number of variables, does it mean that the Riemann hypothesis is undecidable in a Gödel sense?

2:21 PM
@XanderHenderson Right, those notations allude to taking a limit of some kind. Thank you for clarifying this very nicely...+star
@XanderHenderson What would you say the difference between the two is? The difference seems to be very subtle.
Well, I guess the difference is what you said, namely that $f \in O(g)$ holds if either $s$ is greater than $M$ or if $s$ is in the neighborhood of $s_0$. Do any of those consider the case if $s$ is smaller than $M'$ then $f \in O(g)$ holds?
Maybe that case is irrelevant because of "...$g$ "eventually" dominates $f$..."

3:11 PM
mats, i'm not a logician, but that question strikes me as in need of further formalization. i could imagine a question with a true or false answer, where some algorithms that could return the answer conceivably require checking infinitely many cases, and other algorithms for the same question don't.

3:32 PM
Nearly finished with Undergrad :). I may honestly just delete my SE profile and start over
Honestly when reading back on my posts I sort of feel a bit embarrassed
Sorry some of you had to put up with my bad posts and lack of clarity in my arguments.

4:03 PM
congrats on being nearly finished. i recommend against deleting, if that removes the questions/answers. they could be helpful to others.

oof you have a fair point i'm just a bit embrassed by my posts for context i've now taken Calc (1-3), Linear Algebra, ODE, Abstract Algebra and i'll be taking Real Analysis this semester
looking back at my posts I really lacked a lot maturity feel like I know less that what I ever did

i'm skimming them at random, most of them look completely fine to me. higher quality than most questions.
you could always just remove anything personally identifiable from your profile, if you were worried about that. some people explicitly market their participation in mathoverflow, math.se and the like, and it is fine not to do that.
i think people would have to do a lot of connecting the dots on my questions/answers to figure out who i am. if they want to put in the work i guess good for them.

Yeah but I feel like the writing could be a lot better ._., during undergrad I really learned about the process of doing Mathematics espcially the importance of examples
A lot of those posts were made when I was in High School

these days i think everyone is fairly used to being able to google up stuff from when people were in high school. unless you plan on running for congress and said something politically hot, who cares? and maybe even then.
i removed most of the stuff i put about how the CIA runs hollywood and the lizard people but some of it might still be up.

Oh lol ehh I just kinda feel embarrassed you know. 4+ years later my writing skills went up in terms of technical communication and I still feel like theres a lot more I could be doing

4:20 PM
In step 1, why is it true that g$[x/g]\in G$ where [.] is floor function?
I disagree with this.

why do you disagree?

if g is in G (which it seems to be), any integer multiple of g will also be in G

@leslietownes Wow. I feel like an idiot now.
@Thorgott Thor, I somehow forgot the fact that G is group when I saw that smart multiplication.

you too. or winter, depending on hemisphere. :)

4:27 PM
Itâ€™s raining here. heavily!

@koro i would not be too hard on yourself. that proof seems to have been written as to minimize the use of explanatory language. i would have made different expository choices but maybe that is the 'house style' of proofwiki, or something.

Thx looking to make some new friends in this community :)

Leslie, I feel that thatâ€™s why I am elaborating every step replacing the portions by alternatives wherever I can fit them

i do actually like aspects of that style. when a proof involves manipulation of an equation, i like LHS = RHS1 [reason], = RHS2 [reason], = RHS3 [reason], = RHS4 [reason], therefore done. but when you don't have a reason on every line i begin to like it less.
if it's not important enough for you to state the reason maybe it's not important enough to be included. i dunno.
i think it's also helpful to preface a chain of equation manipulations by some description of what will be going on in the equations. if there is some way of describing the idea of what is about to happen in a sentence, even if it's technically not 100% formal, it is helpful to add in.

proofwiki is entirely formal, it's not expository

4:33 PM
For example: step 1: I argue like this: 0 is clearly a limit point of G. Then I take any open interval (a,b) and choose a g such that $0<g<b-a$ and then by well ordering principle there must exist natural number m such that $mg>a$ and $(m-1)g\le a$

that makes sense. it could be more formal, if it's going to be formal. :) you could imagine having everything sufficiently formal that people could choose differing level of display styles and have further stuff automatically generated to match that.
it doesn't seem like it's quite there.

Then I show that mg>b leads to contradiction. Thatâ€™s how my first pet is complete.
@leslietownes hmm

i'm the kind of dude where, if your proof isn't entirely symbols, i don't want to see anything like $\forall$ and $\exists$ in it. make it machine readable and formally verifiable by software, or speak to me in human language. no hybrids.
i am definitely in the minority on this. :)

Dear @leslietownes did my proof prototype above use these symbols at all ? I tried to avoid those
Ah I didnâ€™t use the symbols:)

oh, i don't know, i don't think so. i was riffing on what i saw on proofwiki, where you had the doubt.

5:25 PM
is there a way I can make a room I created read only ?
er i mean , is there a way i can prohibit others from being able to read whats written in this room
besides another person i created the room for to discuss some maths he finds sensitive about?

no

alright then, no problem
so its not a rep thing?

like if you have high enough rep you can do it
ok got it

if you want to go to extreme lengths, you could create a new room for you and that person, make both of you moderators and delete every single message you post instantaneously. they wont be visible publicly anymore, but moderators can still see the deleted message (though that also applies to the network-wide moderators)

5:30 PM
noted, this is actually about a mathbi.n link I pasted in the room but that cant be deleted now since it was a few hours ago
i highly doubt the contents of the link are sensitive, the other person is being a little paranoid about it, but now i cant really acquiesce him with more than 'its extremely unlikely anyone is going to be bothered enough to see if what is in that link has publishable value'

i've already duplicated his proof of the twin prime conjecture and taken public credit for it. it's over.
i'll see both of you in court

haha
i think mathb.in links get cleared eventually, so its not the end of the world

a lot of those paste sites have a fairly short time horizon, at least if people don't keep accessing the paste.

cool, ill tell the other person to copy paste it and not access it again

but as i said, it doesn't matter. i'm getting the nobel prize now.
the best way of disclosing something to another person without creating a record of it is a phone call. unless people are monitoring your phone calls, and for that, like, you're probably just going to jail for something at that point.
in the US anyway. some regimes might run a tighter ship.
a friend of mine once had to work in a country with a fairly autocratic regime, and i'm absolutely certain that someone was listening to all of our phone calls.
and probably recording.

5:39 PM
whew
how were you certain?

he was one of a small number of US citizens who was there. the regime was notorious for doing it, mostly i think to monitor journalists. there was sometimes a weird electronic clicking on the calls. it seems like you could do better with that.
no.
i wouldn't answer a call from there. :)

aw
you probably cant make outgoing calls from there

we talked mostly about the food we were eating. sadly i do not live a life of intrigue.
and some boring stuff about the timing of writing some checks to vendors in his business in the USA.
i bet most intelligence agencies are bored stiff with what they're told to monitor.
if someone's monitoring my email i hope they enjoy how much i talk about how i just ate sushi, or soon want to eat sushi. there are also cat pictures.

6:13 PM
They pay vendors by checks?
@Leslie

this guy did, yeah.
we do that too at my company.
unless someone demands a wire transfer, which does happen.
i hate getting a check. what do you want me to do with paper, i ask.

at my company we transfer online. We don't do cash payments anymore.
Or check payments anymore.

i have one now on my desk, my mortgage lender reimbursed some kind of overpayment or overcharge, but only in paper check form.
the US is so behind the rest of the world in terms of using paper for payment.

Due to corona, checks have been further discouraged.

i hate having the paper. my daughter gleefully shredded a check i hadn't cashed. i had to ask them to reissue it.

6:16 PM
Hi! I'm happy to be here again. I'd like to know what is useful of discovering a new largest known prime number? Is it only for fun? Anyone has any opinion about this!

i think at the extremes, identifying the largest known prime number is more of a hobby than something worthwhile. what you need for encryption does not reach those heights.

since I'm a theoretical CS, I don't see any great things at looking at discovering a new prime number.

but all frontiers will be pushed forward eventually and the mechanisms used to identify large primes may inform development of other useful things.
i kinda agree with you, i don't care what the largest identified prime number is.

@leslietownes Your daughter really is a chip off the old block!

We have many and many primes with length of 1000 digits that is enough for secure anything that is important. Thank you Leslie for your sharing words!

6:21 PM
we went to a party yesterday, our first in 1.5+ years. she couldn't stop pestering people. nonstop stories about her cat, her toys, ducks. reminded me of somebody.

i have a feeling child me had much higher 'intuition' than older me

i think i have more intuition than my daughter. knowing more gives you more intuition.

Is that an intuitive feeling, @porridge?

yeah more or less an intuitive feeling :)

6:36 PM
my daughter's intuition at the moment is, if you see the cat's pupils widen unexpectedly and hear a meow, back away. i have a lot more intuition than that.

My kitten screeches instead of meowing.

i guess what I mean to say is im not really talking about 'learned' intuition or even math intuition, more like 'how fast can you learn totally new things' intuition, and the reason i think so is because a lot more things are totally new at that points, so you are probably forming really novel neural pathways (novel for you) really fast
and its sorta harder to do that now, but i agree i have more learned math intuition, and i can learn math a lot faster than i could when i was younger, because of whatever 'mathematical maturity' is
i think its just like speaking a language for longer

Definitely acquire new experiences/knowledge faster when a child.

my daughter understands spanish because she gets it at school. she doesn't speak it but she absorbs it like a sponge.

right, i feel way less spongey now :(
that being said im still pretty spongey for learinng certain new math

6:39 PM
we sometimes watch something in spanish and i ask her to explain what she saw, and she gives me english that more than covers the spanish. and she isn't cheating by reading the subtitles like i am.
although i do remember some spanish.
it's frightening, the rate that kids absorb things.

6:56 PM
Does anyone know why isogenies (between elliptic curves) over $\mathbb Q$ are always separable?

7:06 PM
Anyone here good at asymptotic density in the integers?

two questions in a row where i have no idea. anybody have a third?

We agree again, @Leslie.

not again.

@leslietownes $\dfrac{(p_2 - 2) (p_3 - 2) \cdots (p_n -2)}{p_2 p_3 \cdots p_n} \to 1$ as $n \to \infty$, right?
I'm wondering how this is reflected in an intersection that equals the numerator number of cosets unioned together modulo the denominator.

7:12 PM
Yea, I got helped in the forum
I had forgotten that any field extension of a field of characteristic zero is separable

Oh, even Ted knew that.

i must confess i knew that too. but who knows what an isogeny is.

I'm in the same boat as the two above me

x''D

is $p_k$ the $k$th prime?

7:16 PM
@porridgemathematics yes
that fraction equals precisely the number of cosets that are present in the formula fo $A(n)$.

this is way beyond me and my knowledge, but it sounds like whether that tends to one or not depends on the distribution of prime gaps
and it seems like the answer is yes (based on that distribution)
but i dont have a proof :(

@OlympicComputerChairSitter If I write that as $$\lim_{n\to \infty} \prod_{j=1}^{n} \frac{p_j-2}{p_j},$$ I feel like we end up multiplying a large number of terms which are all smaller than $1$, hence the product should tend to something smaller than $1$ (zero, possibly). Indeed, if $m < n$, then $$\prod_{j=1}^{m} \frac{p_j-2}{p_j} > \prod_{j=1}^{n} \frac{p_j - 2}{p_j},$$ no? So the sequence is monotonically decreasing.

@XanderHenderson yes, Erik Wong has confirmed that in the post

Ah. Okay. I should read the post.

But that was just an approach to the first question, so that approach probably won't work on the first question

7:31 PM
My next step was to expand that series as a sum to get a sense of how quickly it decreases. :D

whoops. made a fool of myself :)
multiplying a bunch of $<1$ numbers.. doh

8:23 PM
@leslietownes i never told you stories about my ducks...

8:46 PM
@copper.hat I'll be sure to duck out when you do.

@TedShifrin part of an intricate web of narratives...

Thereby hangs a spider.
Isn't this amusing?

9:16 PM
Does $1+\cdots$ converge or diverge?

yes
@ShaVuklia What is you definition of an isogeny ? And of separable?

9:42 PM
I think this is true for any abelian varieties over a field $k$ of char zero, because it boils down to a finite algebraic extension of $k(Y)$ being separable, which is always the case in char 0
elliptic curves being dim 1 abelian varieties

9:53 PM
@copper.hat No.

Salut @Astyx

Semi salute @Semiclassical o/

I haven't seen Semiclassic in a month.

10:13 PM
the law of the included middle...

@copper I take that personally. My middle includes far too much.

@TedShifrin I'm working on my included middle...

I'm sure you're back to your usual arduous bicycling regime.

10:53 PM
quack quack quack quack
one of my friends has stories about being attacked by almost every kind of bird. birds just hate her. i thought she was making it up and then one day i was out with her and she got repeatedly dive bombed by a scrub jay. no nest or young in sight. i was there too.

quick question Leslie.....I have a function, I've shown it has directional derivatives at $0$ in all directions, which means the partials are continuous, but I have to show it is not differentiable at $0$. The only thing I feel I can resort to is showing that it is not continuous at $0$.

does it mean the partials are continuous?

well I used the directional derivative definition and got partials which themselves are both $0$ along their respective axes.

That doesn't make them continuous, does it?

putting that aside, if the only conceivable derivative is the zero map, which maybe it is, this would impose a growth condition at the origin. maybe work with that.
and maybe i shouldn't put that aside, there might be other things going on here.

11:06 PM
no it wouldn't because that is only one specific direction.
or path may be more appropriate to say
Ok got it now then.

that's a good distinction to make. when a function is differentiable it's more natural to think of the value of the directional derivative as associated with a 'direction' than it is when a function is not and you can approach a point in two different ways, arguably from the same direction, and not get the same result.
daughter creeped us out this afternoon by suddenly asking "what's [my hometown]?" over and over. we don't know where she heard about it. had to explain it was where i grew up.
genetic memory, maybe.

11:23 PM
i want to believe thats a thing (past lives included) but that cant be a thing right?
genetic memory

the best we can do so far is that yesterday we were outdoors at a garden party with two people from my hometown. it may have come up in some conversation. i didn't hear it.

interesting, bar that its a mystery

it's very, very weird. we've had this before and eventually work out what it was.
one time she kept saying there was a ghost in her room. she was blending together a scene in one of her books where someone has a blanket over her head, and the fact that we'd absentmindedly put a blanket over one of her dolls' heads.
but it was super weird to have her insisting that there was a ghost in her room.
because she's young it's easy to think she's kind of a blank slate when she wakes up and doesn't remember something she read in a book from two months ago. the reality is she can still remember quite a lot from her early years.
i have a memory of going to preschool on the first day, and someone asking me my age, and i held up two fingers instead of saying. my daughter would yell two AND A HALF.

she's surprisingly unshy with people she's just met. she will run the world some day.
her latest thing is asking for a fruit, for example a blueberry, and then disagreeing with the format when it is presented to her. "i wanted a frozen blueberry," if it's fresh, and "i wanted a fresh blueberry," if it's frozen. she would not have been out of place as a roman emperor.

11:31 PM
natural born contrarian

i do know where she gets that from.
i had sort of a howard hughes angle to it. i would accuse people of having put "cat food" in my food, as if to poison me. if you asked me to eat a plate of vegetables, instead of saying no, i'd say, someone has put cat food in this, so i can't eat it.
it's a very effective tactic. once you've changed the topic to whether someone has or has not put cat food into your food, you've won. there's no outcome where you're like, OK, i will eat these brussels sprouts.

itd be inhumane to put pressure on you to , at that point!

framing, narratives, the anchor effect. i appreciated this very early on.

that reminds me, I noticed some dead ants in my food when I was a child, and my grandmother told me not to let them distract me from the meal, it did get me to eat it

oh, that's funny. my dad said something similar once, "it's extra protein, you need that."

11:36 PM
I didn't taste the dead ants, I don't know what that says about my grandmother, besides that odd event she was a lovely lady

i don't think ants are a protein rich food, but it convinced me at the time.
he grew up very poor. he would eat a lot at the houses of relatives and neighbors and sometimes they didn't have a lot either. he used to have 'salads' from leaves from a bush in his aunt's yard. turns out the bush was slightly poisonous and not exactly food. probably better to eat ants.

yikes, yeah I think ants are (pretty much?) safe to eat

some of them are probably even rich in nutrients. just not the ones we grew up around.
my daughter is roaring at the cat. i asked, "why are you roaring at the cat?" she said, "i'm roaring because she's running away." maybe it's the other way around.

thats funny, I get similar sentences logically mixed up in the same way occasionally, usually when Im tired, not saying she meant it the other way around
it reminds me of how (apparently) babies see things upside down before a certain age

have you heard about those studies where they give people glasses or a mask that flips things upside down? apparently it takes you only a very short time to switch over.

11:47 PM
i haven't, sounds interesting

i forget what the time scale is. i think it's more than hours but less than days.
and then you take the mask off and, you have some recuperation time.

i wonder what peoples vision is like during the recuperation time, does it discretely flip the image after a certain point, or is it some weird continuous hodge podge until normal vision is restored

i don't know. it's fascinating to me that people can just flip over. i guess it's probably something like learning the controls on a video game. the first x minutes, you're sort of in the dark, and then it becomes second nature.
there's lots of weird research with people who are partially lobotomized or had the halves of the brain surgically separated in an attempt to reduce seizures. the bottom line is that nobody understands anything about the brain and we are probably still very primitively getting at that stuff.
some technologies can probe individual neurons in real time, but they are surgically invasive and i don't know that they have been multiplexed to allow for resolution of data across large numbers of cells.

TIL Pulsations of cerebrospinal fluid occur 60 times a minute in our brain.