@Victor: You need to write down what you want to prove into a mathematical statement and only then one can use mathematical proofing methods to proof whether your statement is true or false.
I kinda know what you mean but I'm not at all good at writing that down in a mathematical statement.
While reading some things about analytic functions earlier tonight it came to my attention that Fourier series are not necessarily analytic. I used to think one could prove that they are analytic using induction
Let $P(n)$ be some statement parametrized by the natural number $n$ (in this case: ...
If we use the same definition, 2 would be singular - it cannot be written as union of 1 singleton. And 3 is regular. There is - in my opinion - why finite numbers should be considered in two different classes w.r.t. cofinality. I think the best possible way is not to consider them at all.
Anyway, these notions are usually important only for infinite sets.
BTW as cofinality was mentioned, I remember that Todorcevic and others had some results about cofinal types of *directed sets*. I can imagine this can be useful, I did not studied those things is detail. Is any of you familiar with this? This is mostly directed to @Asaf and @tb.
I had slight feeling that there might be kind of relation between these results and Alexei's question about two types of subnets. But I am not sure about this. (First of all because I do not know those results about cofinal types in detail.)
@AsafKaragila You've got an upvote from me. In connection with your last paragraph, wouldn't Easton's theorem be worth mentioning? At least by giving some links, perhaps Wikipedia or MO thread. (or both)
@MartinSleziak Easton's theorem has nothing to do with that. It has to do with power sets. If anything, Prikry forcing has to do with changing cofinalities without changing the cardinals. However this requires a measurable cardinal.
@tb I thought that the division is not made by the difficulty of question (from this viewpoint it's elementary), but by the topics. According to tag wiki e-s-t is for the sort of material covered in "Chapter 0" of undergraduate texts and in undergraduate set theory texts. This was the reason why I thought s-t is better.
We're like co-Norway, where there is only one week of "summer" which is really just a nice spring day here. In Beer-Sheva we have like one week of "winter" which is really just a nice autumn day in Norway.
@MartinSleziak But the first sentence for set theory reads "This tag is for set theory topics typically studied at the advanced undergraduate or graduate level"
@AsafKaragila have you looked at the OP's profile of the question you closed a few minutes ago? More than half of the questions are of this kind... Kind of frightening.
Wow... I just need to offer and give a bounty on someone else's question and then I have all the bronze badges except Precognitive, Peer Pressure and Tumbleweed.
@tb One of the 3 up votes on your amusing comment is mine! This guy isn't being told off for asking 5 questions in 2 hours he's told off for posting homework verbatim without even having had a go at it first. No, I don't see anything wrong with even asking 20 questions in 2 hours.
Now he's changed his name. : D
@tb Good job the König isn't here otherwise he might consider giving you a good spanking for using a language other than English. On top of that one that's mouth rape if spoken. : P
I'm really confused about how my university/library doesn't seem to have full access to everything on SpringerLink. Here I was thinking, for all its faults, at least it should be rich enough to afford some good subscriptions...
@MartinSleziak Because nobody uses it: and the room was not advertised in chat here (perhaps you could've used the "notice board" to the right for advertisement =)).
I went there just now seeing you and tb visited it recently.
An approach that is very effective but not quite elementary is given by nonstandard analysis (NSA). Essentially, epsilontics are delegated to some powerful set-theoretic machinery. There is a post by Terry Tao that explains how NSA does the epsilon management for you: Ultrafilters, nonstandard an...
is that true...?
a sequence is a function N -> R
he's lifting it to nonstandard numbers *N -> *R
I don't really know whether or not you can do that
yeah you can do it
so if x,y : N -> R are sequences and their lifts *x,*y : *N -> *R are real for hyperreal inputs (<=> the "limit" is defined) then *(xy) is real for hyperreal inputs so it's limit is defined?
is that correct? (it's a lot simpler than the answer given)
I guess it could be real but vary, rather than be constant..
Do you have any ideas for what other purposes could room like Jury Duty useful. I mean things that are not important enough to be starred (pinned to the noticeboard, so to say), but it would be good to have them separately somewhere, so that they do not get lost between smalltalk and other stuff and where they can be easily noticed at least for a few hours.
The original suggestion was to use it in connection with closing questions and similar things. (But we usually have enough jurors in the main chat very quickly.)
Maybe something like comments about tags, e.g. like the Srivatsan's comment? (Perhaps with link to the discussion in the main chat, if there was a discussion about it.)
Sorry to interrupt the NSA stuff, I just thought it would be useful to post something like this. Just wanted to make more people notice that room. (We should either start to really use it, or abandon it.)
Definition of Uniform Continuity
A function $f$ is said to be uniformly continuous if to each $\epsilon >0$, $\exists~\delta >0$ such that whenever $|x-y|<\delta$, $|f(x)-f(y)|<\epsilon$.
The following is just an "uninformed" intuition:
If your function will have arbitrary lon...
Can any of you go over this answer and leave comments for me?
@KannappanSampath Well, it's not entirely clear whether in the newer question asks for intuition (in that case I would definitely say duplicate), or the OP just want to see how Jacobian is computed in this specific case. To be sure, I've asked here, instead of voting to close as duplicate right away.
@KannappanSampath I strongly object to this kind of rude and non-constructive criticism. To be blunt, I find your comment obnoxious. Of course, there's always room for improvement but if you don't have any helpful thing to say please keep silent.
Yes, it is a bit hard to tell what exactly the question is, but obviously the OP invested quite a bit of effort to get his point across (and he may have failed badly). But just telling him that he should try to improve his grammar and write more clearly without pointing out what is unclear and where you have trouble following is not constructive or helpful at all.
I've been curious about them. I figure who better to ask than you all.
I have no personal experience with them, besides caffeine. I'm sure we all know a lot of people who use Adderall, but I can't vouch for it.
I'd also be interested in hearing from anyone about things you believe help your cognitive abilities, whether it's coffee or cheese, or some habit, whatever it is.
Up until a few weeks ago, I didn't even drink coffee. It would give me a rush for 20 minutes, then headaches the rest of the day, muddy thinking, and so on. My wife bought some fresh stuff, and I had some: it made me sharp for around 8 hours afterward. I had such clarity. I even began studying again. This last week, I've been studying around 8 or 10 hours a day.
So, we ran out of coffee a few days ago. We bought some, the same brand. Nothing. No effect. I'm so sleepy, my thinking is muddy again. It's terrible. I can't concentrate. I can't hold shapes and numbers in my head.
But I understand your point. However, last week was fantastic. I haven't had that clarity for years probably. If a cup of coffee is all I need to get that, then where's the harm?
But anyway, my point wasn't to talk about coffee. I wanted to find out if anyone has had a similar experience with anything other than caffeine.
I wanted to know if there was any credibility in the claims of other foods, and so on.
I think it depends on how you look at it. If you're looking for cure-all, you're not going to find it. If you're looking for a drug to make everything easy with no drawbacks, that's not going to happen.
But then there's caffeine. You may be capable of more, but most days you're running at 70%. Caffeine just stimulates you enough to allow you to perform as well as you are naturally able.
While reading some things about analytic functions earlier tonight it came to my attention that Fourier series are not necessarily analytic. I used to think one could prove that they are analytic using induction
Let $P(n)$ be some statement parametrized by the natural number $n$ (in this case: ...
Definition of Uniform Continuity
A function $f$ is said to be uniformly continuous if to each $\epsilon >0$, $\exists~\delta >0$ such that whenever $|x-y|<\delta$, $|f(x)-f(y)|<\epsilon$.
The following is just an "uninformed" intuition:
If your function will have arbitrary lon...
