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.
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. $0<|x-a|< \delta\to 0^+$?
@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?
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).
@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
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
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?
@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 $|f(s)| < C g(x)$.
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?
@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 $f$..."
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.
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
@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.
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.
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.
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.
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)
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'
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.
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.
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.
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.
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.
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
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.
@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.
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
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 $0$.
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.
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.
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.
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.
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.
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
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.
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.
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.
