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Srivatsan
Srivatsan
11:01 AM
@FreakEnum One second.
 
Jonas Teuwen
Jonas Teuwen
@Matt What is $X$? More likely is that it is the space of Radon measures on $X$.
 
Matt
Matt
@JonasTeuwen It's some locally compact sigma compact metric space. But forget what I wrote.
 
Jonas Teuwen
Jonas Teuwen
Right.
 
Matt
Matt
I actually know what I have to read to be able to do this question (tb told me yesterday) but I did another question first and now I have 1h left before I have to go.
 
QED
QED
Hello
 
Matt
Matt
11:12 AM
Hi.
@JonasTeuwen Regular countably additive complex Borel measure, according to Riesz-Markov.
 
FreakEnum
FreakEnum
@Srivatsan waiting :(
 
Srivatsan
Srivatsan
@FreakEnum Alright, sorry.
So, let's say 1 man in 1 day does 1 unit of work. We need 45 men working for 16 days -- amounts to 45 x 16 = 720 units of work.
 
FreakEnum
FreakEnum
11:27 AM
my browser doesn't interprets \ cdot etc
 
QED
QED
 
FreakEnum
FreakEnum
@Srivatsan ok
@QED Thanks for link :)
 
Srivatsan
Srivatsan
@FreakEnum So, in 6 days, 45 men do = 6 x 45 = 270 units of work.
 
FreakEnum
FreakEnum
@Srivatsan ok
 
Srivatsan
Srivatsan
@FreakEnum How much work remaining?
 
FreakEnum
FreakEnum
11:33 AM
@Srivatsan 720 - 270 =450
 
Srivatsan
Srivatsan
Ok. Now, after 6 days, how many men are there?
 
FreakEnum
FreakEnum
@Srivatsan 75
 
Matt
Matt
Which of these is the most comfortable? link Aha, sorting with respect to votes yields six-to-8.
 
Srivatsan
Srivatsan
@FreakEnum Ok. We have 450 units of work. 75 men. So how many days does that take?
 
FreakEnum
FreakEnum
@Srivatsan 450/75 =6 days
 
Srivatsan
Srivatsan
11:37 AM
That's it. That's the answer.
 
Jonas Teuwen
Jonas Teuwen
@Matt Well those are the Radon measures no?
 
FreakEnum
FreakEnum
@Srivatsan how do you think that way of solving problem ? I spent 25 mins , but couldn't get close to answer
How do I get logical thinking of solving problems like you?
 
Srivatsan
Srivatsan
@FreakEnum Not sure how to answer that question =)
 
Matt
Matt
@FreakEnum Practice.
 
Srivatsan
Srivatsan
I second that.
 
FreakEnum
FreakEnum
11:42 AM
@Srivatsan didn't get it, what do you mean?
 
Srivatsan
Srivatsan
@FreakEnum I second Matt's suggestion that you need to practise more problems.
 
FreakEnum
FreakEnum
@Matt seriously or just joking with me ? practice will get me good logical skills also?
 
Matt
Matt
@FreakEnum Seriously.
 
FreakEnum
FreakEnum
@Matt I'll sue you in the court if that doesn't helps :D
 
Jonas Teuwen
Jonas Teuwen
Sue someone for that? Are you American?
 
FreakEnum
FreakEnum
11:46 AM
@JonasTeuwen lol , was just kidding
 
Jonas Teuwen
Jonas Teuwen
So was I 8-).
 
FreakEnum
FreakEnum
@JonasTeuwen you're from Netherlands?
 
Jonas Teuwen
Jonas Teuwen
I live there.
 
FreakEnum
FreakEnum
@JonasTeuwen den haag?
 
Jonas Teuwen
Jonas Teuwen
I'm very close to Den Haag.
 
Matt
Matt
11:48 AM
@JonasTeuwen The question is: are Radon measures (inner regular measures on the σ-algebra of Borel sets of a Hausdorff topological space X that is locally finite) the same as regular countably additive complex Borel measures on a locally compact Hausdorff space X. (just thinking aloud)
 
Jonas Teuwen
Jonas Teuwen
@Matt I can be mistaken with the terminology, in any case the dual of your space is some space of measures :-).
 
FreakEnum
FreakEnum
@JonasTeuwen I have my Big brother living there :)
 
Matt
Matt
@JonasTeuwen : )
Anyway. The deadline is too close. My brain has switched off already and I'm in panic mode.
 
Jonas Teuwen
Jonas Teuwen
@Matt But you have a particular $X$. Compact and Hausdorff?
In panic? Why? Failure is no shame.
 
Matt
Matt
@JonasTeuwen No it's not about failure. Time pressure disturbs me.
Can't explain it now because I have to get ready and leave.
 
Jonas Teuwen
Jonas Teuwen
11:51 AM
Bye. Good luck.
 
Matt
Matt
No luck needed, I just have to hand it in. As for the space: in the homework it's a locally compact sigma compact metric space.
Bbl.
(Panic mode is a slight exaggeration : ) )
 
QED
QED
@Matt, you're not an undergraduate are you?
 
Jonas Teuwen
Jonas Teuwen
Yes, he is an undergraduate if I recall correctly.
 
QED
QED
But how come he is doing such advanced things
 
Jonas Teuwen
Jonas Teuwen
It is not that advanced is it?
 
QED
QED
11:59 AM
I completed an undergrad course and I don't even know the things he's doing
 
Jonas Teuwen
Jonas Teuwen
It is just basic functional analysis, didn't you have that?
 
QED
QED
yes
 
Jonas Teuwen
Jonas Teuwen
Well then. This is the dual of $C_0(X)$ with $X$ compact and Hausdorff.
 
t.b.
t.b.
@JonasTeuwen drop the naught or add locally :)
 
Jonas Teuwen
Jonas Teuwen
Oops. 8-).
Well, to be honest, here you could also pick between algebra and other subjects instead of analysis. If you did analysis you would get that.
But I suppose you would get something like homological algebra or algebraic topology then.
 
QED
QED
12:05 PM
I don't think I really got anything out of functional analysis
I think it was mostly intended as practice proving things are metric spaces
 
Jonas Teuwen
Jonas Teuwen
No TVS?
 
QED
QED
Well. I got a very nice fixed point theorem which I've forgotten and I got to study proofs of those three big convergence theorems
so I did get something out of it. It was a long time ago.
No TVS
 
Jonas Teuwen
Jonas Teuwen
What convergence theorems? Fatou, monotone and Lebesgue? Isn't that measure theory?
Was it the Banach fixed point theorem?
 
QED
QED
yes
if they all measure theory then I probably don't know any actual functional analysis
I think I was something simpler and weaker than Banach
 
Jonas Teuwen
Jonas Teuwen
Well, if it was functional analysis, then it should contain stuff like: Hahn-Banach, open mapping theorem, closed graph theorem, Banach-Steinhaus.
To my taste at least.
 
QED
QED
12:12 PM
yeah I don't know any of that
 
Jonas Teuwen
Jonas Teuwen
Hmm. That should be fixed 8-).
 
QED
QED
At the moment I'm (very very slowly) getting a bit more complex analysis so I can learn the Reimann mapping theorem
 
Jonas Teuwen
Jonas Teuwen
Read this.
 
QED
QED
I can't read German sadly
 
t.b.
t.b.
@JonasTeuwen can you take care of this?
 
Jonas Teuwen
Jonas Teuwen
12:18 PM
@tb Done!
 
FreakEnum
FreakEnum
12:43 PM
@Srivatsan I am again stuck in a Q :(
 
Srivatsan
Srivatsan
@FreakEnum Well, not right now. I have to leave now.
 
FreakEnum
FreakEnum
@Srivatsan give me hint before going?
 
Srivatsan
Srivatsan
@FreakEnum Ok. Shoot the question.
 
FreakEnum
FreakEnum
Q : 2 men and 3 boy does a work in 10 days. 3 men and 2 boys does same work in 8 days. how much days will 2 men and 1 boy to do a work?
After answering my previous Q's A , do you think anyone would have solved this problem?
 
Srivatsan
Srivatsan
I am not sure about whether anyone could've solved it.
Let's say a man does x units of work in a day, and a boy does y units of work in a day. So, how much work will 2 men+3 boys do in a day?
 
FreakEnum
FreakEnum
12:50 PM
@Srivatsan 2x+3y :)
 
Srivatsan
Srivatsan
Sure. That is (1/10)th of the total work, because they (2 men+3 boys) need 10 days to complete.
So, can you write it as an equation? 2x + 3y = (1/10).
 
FreakEnum
FreakEnum
@Srivatsan 10(2x+3y)=1
 
Srivatsan
Srivatsan
What does the second condition give?
 
FreakEnum
FreakEnum
wow
 
Srivatsan
Srivatsan
@FreakEnum Yes, it's the same thing. Whatever you feel comfortable with =)
 
FreakEnum
FreakEnum
12:52 PM
Thanks a lot:)
 
Srivatsan
Srivatsan
I hope you think that I am going *systematically* with all these problems. =)
 
FreakEnum
FreakEnum
@Srivatsan very , You teach very good I must say :)
 
Srivatsan
Srivatsan
@FreakEnum Thanks.
Well, there are a few patterns with these problems. Once you realise these patterns, you can tackle most problems systematically. It's not like each new problem requires something totally different from what you know.
For all these "work" problems, I think a reliable starting point is the following: Assume that a man does x units of work per day, a woman does y units of work per day, and a child does z units of work per day. From this, it is a matter of writing down the given condition as an equation. I encourage you to try it on a new problem and see if it works for you.
What say you?
 
Clash
Clash
1:23 PM
hey guys, can I interrupt shortly? I have a quick question about a limit exercise
 
QED
QED
go ahead
 
Clash
Clash
I have to calculate the following limit
$\lim_{x \to -2}\frac{x^2+4}{x+2}$. I decomposed it as $x-2 + \frac{8}{x+2}$. So when $x \rightarrow (-2)^-$, I get $-\infty$ and $x \rightarrow (-2)^-$ gets me $\infty$. Is it ok if I stop there or do I have show something more?
Or should I say this does not have a limit?
 
QED
QED
In general if you prove the one sided limits are different, that proves that the two sided limit does not exist: I recommend stating that formally and proving it in general
 
Srivatsan
Srivatsan
@Clash It does not have a limit. What you showed just now are that the function approaches $+\infty$ as $x$ approaches on one side, and $-\infty$ from the other side. Since the one-sided limits differ, the limit does not exist.
 
Clash
Clash
great, thanks guys
 
Srivatsan
Srivatsan
1:31 PM
You're welcome, Clash.
 
Clash
Clash
if both "converged" to $\infty$ would the limit exist? Or it doesn't exist if it's to $\infty$?
 
QED
QED
it's wrong to say converge to infinity
it diverges to infinity (or -infinity)
 
Clash
Clash
can I say a limit exists if it diverges to infinity?
 
QED
QED
no
 
Clash
Clash
ok, thanks again guys! srivatsan, i still want to marry you
 
QED
QED
1:36 PM
Did you manage to state it as a theorem?
 
Clash
Clash
hm? sorry, what do you mean?
 
QED
QED
this idea you're using "if the one sided limits are different the two sided limit doesn't exist"
that's actually a theorem which demands proof
 
Clash
Clash
oh, doesn't it suffice if I just show the one sided limits are different $\Rightarrow$ the limit doesn't exist? do I have to write the name of the theorem? or show something more?
 
QED
QED
yes you need to prove it. It doesn't matter if you give it a name or not
 
t.b.
t.b.
@QED theorem? it's the contrapositive of the statement "If the two-sided limit exists then it coincides with both the one-sided limits".
 
Clash
Clash
1:40 PM
@QED how can I prove that?
 
QED
QED
should I instead call it a "true mathematical statement"
theorem seems easier.
 
t.b.
t.b.
"Theorem" sounds as if something serious is going on. What I stated should be known to Clash.
 
QED
QED
When I say theorem I mean a pair consisting of a well formed formula and a proof of it
 
Clash
Clash
do I have to prove t.b.'s statement is true? how can I prove that?
 
t.b.
t.b.
That should be clear: assumption "whenever $x_n \to x$ then $f(x_n) \to f(x)$". In particular this ($f(x_n) \to f(x)$) holds whenever $x_n \searrow x$ or $x_n \nearrow x$.
(oh, you wrote $x_n \to x^+$ and $x_n \to x^-$ instead)
 
Clash
Clash
1:50 PM
I see, many thanks t.b.!
 
Matt
Matt
2:33 PM
@QED i'm an undergrad. If you read my questions here and on SE it becomes quite obvious.
 
QED
QED
not obvious
you're doin stuff much more advanced than what I saw in my undergrad
 
Matt
Matt
I should be in first year really but i'm not.
 
QED
QED
thats absurd
 
Matt
Matt
Indeed. It's like giving a toddler a rocket launcher to play with.
 
Matt
Matt
3:15 PM
Here is a question: Find a sequence which converges to 0 but is not in any space $\mathcal{l}^p$ where 1<=p< infinity
I was thinking maybe $ \frac{1}{\log n}$
But now i'm unsure how to see whether $\int_2^\infty \frac{1}{\log^n x} dx $ is also infinite.
 
Daniil
Daniil
3:32 PM
Hi, guys!
What is a good mode for LaTeX which allows you to preview and stuff? That dynamic-dvi viewer I was using does not work under Winblows :(
 
robjohn
robjohn
3:45 PM
I wonder what is the maximum number of stars a chat comment has ever gotten.
3
 
t.b.
t.b.
19?
 
robjohn
robjohn
$$
\int_2^\infty\frac{1}{\log^n(x)}\mathrm{d}x=\int_{\log(2)}^\infty\frac{1}{u^n}e^u\mathrm{d}u
$$
@Matt Does that help at all?
@tb That's why I was asking. :-)
@Matt at about $u=n$, the integrand starts to increase monotonically (and exponentially).
 
t.b.
t.b.
@robjohn I quickly glanced over this list and it seems to me that this 9* post is the highest starred one, followed by a few having 7 stars.
 
robjohn
robjohn
4:00 PM
@tb I should look to see if I can do an SQL query to find this out.
 
Daniil
Daniil
How can I use monospace font in math mode in LaTeX/
 
t.b.
t.b.
@Daniil \texttt{...}
 
Daniil
Daniil
Thanks
 
Clash
Clash
Hey guys I have the following formula $x=\frac {-3q-2}{1-q^2}$. I would like to express now in terms of $m/n$, where $m,n \in \mathbb N$. How should I proceed as clearly not every value $\in \mathbb N$ is valid for $m$ or $n$? Thanks!
 
QED
QED
whats q?
 
Clash
Clash
4:07 PM
sorry, $q \in \mathbb N (q > 4)$
urgh
 
QED
QED
I still don't understand the question
 
Clash
Clash
jesus this was right on my face wasnt it
$m=-3q-2$ and $n=1-q^2$ right?
 
QED
QED
no
 
Clash
Clash
why not?
oh m and n need to be written with q as parameter
 
robjohn
robjohn
@tb \mathtt{uaiii}comes out as $\mathtt{uaiii}$
@tb I don't think that works here.
$texttt{uaiii}$
 
t.b.
t.b.
4:12 PM
@robjohn \texttt is not implemented in MathJax, right, but \mathtt is. (I overlooked mathmode) in Daniil's question.
@Clash the way you wrote it $m$ nad $n$ are negative...
 
Clash
Clash
wow, nice find!!!
thanks!
 
robjohn
robjohn
@tb Yes, \texttt{...} works in text mode in a $\LaTeX$ doc, but not here where $\LaTeX$ is only used in math mode.
 
t.b.
t.b.
I somewhat like this title :)
martingales --- huh!? --- What is it good for? ... aaaabsolutely nothing
 
robjohn
robjohn
According to the User’s Guide for the amsmath Package, the basic set of math font commands include \mathbf, \mathrm, \mathcal, \mathsf, \mathtt, and \mathit.
However, \mathtt is not included on the LaTeXRefCard v2.0
and that list from the User's Guide doesn't include \mathbb
@tb: you're #4 on this list
Except I think that the results are for m.SE instead of SO as the title indicates.
 
t.b.
t.b.
4:32 PM
@robjohn woow :)
 
robjohn
robjohn
@tb = t ype b onanza
 
t.b.
t.b.
@robjohn I like this better than TheoBromine
 
robjohn
robjohn
@tb and that's bad for dogs...
As seen here
@Daniil Did you follow the discussion above? The final answer is to use \mathtt{...}
 
t.b.
t.b.
@robjohn Oh, I knew that chocolate was bad for dogs, but I didn't know why. This makes me object all the more to this abuse of my initials...
 
robjohn
robjohn
Yes, I was mainly pointing out that theobromine is the substance that makes chocolate bad for dogs, not that chocolate is bad for dogs.
@tb I guess they want to know how to get a metric, but it is not clear.
 
t.b.
t.b.
4:52 PM
"One of the thing I hate about Analysis, is that the proofs aren't visual. Heavy definition type proofs with weird tricks. Hard to see what going on." oh, well.
 
robjohn
robjohn
@tb That's not really so. I usually visualize most proofs and the symbol pushing is simply to make it rigorous.
@tb or is this a quote from somewhere?
 
Martin Sleziak
Martin Sleziak
@robjohn From the question he linked: math.stackexchange.com/questions/91479/…
You did not seriously think that this was supposed to be t.b's opinion...?
 
robjohn
robjohn
@MartinSleziak I didn't think so, but I did not see it in the question, since I had not refreshed recently enough.
 
t.b.
t.b.
Thanks Martin. In fact, I can't recall a more visual proof in basic point set topology than the onion peeling proof of Urysohn's lemma.
 
robjohn
robjohn
That quote was added a few minutes after I loaded the question :-)
 
t.b.
t.b.
5:01 PM
I guess simplicity thought the post was inordinately civil and rant-less.
 
robjohn
robjohn
@tb If the question didn't get an answer, at least it might start some flame war :-)
 
user18991
Good evening.
 
robjohn
robjohn
@WTP How goes?
 
user18991
Very good, thank you.
 
QED
QED
hi
 
robjohn
robjohn
5:06 PM
I should shorten my nick to 3 capitals.
TMS
 
user18991
Like I did?
 
robjohn
robjohn
and QED...
 
user18991
:p
 
robjohn
robjohn
@WTP what are your mathematical interests?
 
user18991
@robjohn everything that's related to mathematics.
 
robjohn
robjohn
5:11 PM
@WTP you like algebra as well as analysis, etc?
 
user18991
@robjohn yes. But especially algebra and geometry.
 
robjohn
robjohn
I barely scratch the surface of algebra, I am more into the analytical side of things.
I also like combinatorics and number theory, but I am still learning those.
I should have added geometry to that last list :-)
I did teach a course in discrete math at UCLA for a couple of years.
but I think I was teaching while learning :-)
@WTP: If you don't mind my asking, what does WTP stand for, since those don't appear to be your initials?
@WTP: I see you have a bit of ajax experience. We've just added MathJax support to the chat via a bookmark.
 
user18991
WTP means "What's the point?" :p
 
robjohn
robjohn
5:27 PM
I see :-)
 
t.b.
t.b.
Hey, robjohn, I gotta step out for a while. See you later!
 
robjohn
robjohn
actually the bookmark adds MathJax support to any webpage.
@tb have a good one!
afk for a bit, too.
 
The Chaz
The Chaz
What's the intuition behind why there are so many questions of the form "what's the intuition behind...?"?
 
Matt
Matt
@robjohn Thanks! Very nice : ) So I got the answer right. Can you think of any simpler sequence that is also in none of the $\mathcal{l}^p$?
 
Asaf Karagila
Asaf Karagila
@Matt You mean $\ell^p$. \ell > \mathcal{l}
 
Matt
Matt
5:38 PM
@AsafKaragila Probably I do. Does it make any difference though?
 
Asaf Karagila
Asaf Karagila
Yes. $\ell$ is prettier.
 
N3buchadnezzar
N3buchadnezzar
I know N Z and Q are name for sets of numbers
do the even and odd numbers also have similar names?
 
Asaf Karagila
Asaf Karagila
No.
Often we just use "Your mother." and "Your father" instead.
 
N3buchadnezzar
N3buchadnezzar
I saw someone use E for the set of even numbers, It kinda confused me
learning about cardinality
 
Martin Sleziak
Martin Sleziak
The notation 2Z and 2Z+1 is sometimes used for the set of even and odd numbers.
 
Matt
Matt
5:47 PM
Sorry for the rude formulation of my earlier question. I was on the iPhone and was in a lecture. It ended up sounding quite wrong. At least to my ears.
 
Jonas Teuwen
Jonas Teuwen
6:08 PM
I'm correcting homeworks functional analysis. I'm starting to doubt myself.
Say if you define an element in $\ell^2$ by some series, then the bloody series itself is a sequence right?
Is it common for students to misunderstand the concept "sequences of sequences"? Since everybody up to now did it wrong I'm doubting myself 8-).
 
asd
asd
Hi
My question is here
is there anyone can help
 
Jonas Teuwen
Jonas Teuwen
No.
Be patient.
 
Dylan Moreland
Dylan Moreland
It's definitely out of place on MathOverflow.
You could ask it on the main site here. I think that would be best.
 
N3buchadnezzar
N3buchadnezzar
"This, therefore, is mathematics: she gives life to her own discoveries; she awakens the mind and purifies the intellect; she brings light to our intrinsic ideas; she abolishes the oblivion and ignorance that are ours by birth" - Proclus ( 5th C. )
 
asd
asd
I am new here. Now how can I get the answer?
 
N3buchadnezzar
N3buchadnezzar
6:17 PM
Ask question here math.stackexchange.com
Chat is for chat, site is for homework
 
Jonas Teuwen
Jonas Teuwen
You don't "get" an answer. You ask a properly posed question and then you hope that someone wants to give you an answer. You get those by asking them on the main site and not on this chat.
 
QED
QED
no
 
Dylan Moreland
Dylan Moreland
@asd I would go to the Ask a Question page and enter your question as you did at MathOverflow.
 
QED
QED
it's not for homework
 
N3buchadnezzar
N3buchadnezzar
I see no problem with getting help with homework, I often ask for help after I have tried my best,
 
Dylan Moreland
Dylan Moreland
6:19 PM
@asd I mean, you've already typed it up. Just a simple copy-paste, really.
 
Jonas Teuwen
Jonas Teuwen
Well, I'm not really against people asking questions here, but some people just come here to get an answer and then go away, that's not the purpose of this chat.
 
N3buchadnezzar
N3buchadnezzar
Q and A ?
Oh, you said chat. I thought you said site
 
QED
QED
it's fine to post homework questions
 
asd
asd
ok, thanks for guiding
 
N3buchadnezzar
N3buchadnezzar
I really hate the quadratic formula
I think it often kills creativity
 
robjohn
robjohn
6:26 PM
@AsafKaragila Ahem
 
Jonas Teuwen
Jonas Teuwen
@robjohn Are you aware of students misunderstanding that if you say something like $\|x_n - x\|_{\ell^2} \to 0$ then $(x_n)$ is actually a sequence of sequences? Most of them seem to... mess it up.
So I'm wondering: Maybe I don't really understand.
 
robjohn
robjohn
@JonasTeuwen what don't you understand?
 
Jonas Teuwen
Jonas Teuwen
I'm not sure. They make an element in $\ell^2$ like $x_n = \sum_{i = 1}^n x_i e_i$ where $e_i$ are the unit vectors.
 
robjohn
robjohn
@JonasTeuwen @asd: Yes, if the question has some real meat to it, it is better to ask on the main site and get a broader range of answers.
 
Jonas Teuwen
Jonas Teuwen
Okay fine. Now they say $x_n \to x$ without any comment. So I think that they think they just take the limit of partial sums?
 
robjohn
robjohn
6:30 PM
@JonasTeuwen why would you look at partial sums when the question is about a sequence?
 
Jonas Teuwen
Jonas Teuwen
Yes! I don't know!
They all do stuff like that!
While $(x_n)$ is a sequence of sequences. But they think that if they truncate the series in its basis expansion that they can just take the limit without any further comment.
 
robjohn
robjohn
are they looking at $\{x_n\}$ and thinking that a question about the convergence, is a question about the convergence of $\sum x_n$?
 
Jonas Teuwen
Jonas Teuwen
Yes.
 
robjohn
robjohn
Well, when mentioning $\ell^2$, they may think of the sum of the squares of the elements of $x_n$
 
Jonas Teuwen
Jonas Teuwen
So you have $x = \sum_n x_n e_n$ for an element in $\ell^2$. Then they set $x_n = \sum_{i = 1}^n x_i e_i$ right? Then they say $\lim_{n \to \infty} x_n = x$.
Without saying why. That makes me think that they just see it as a sequence of normal partial sums and the $e_i$ as numbers and not as sequences.
 
robjohn
robjohn
6:35 PM
the $e_n$ are sequences? what does $x_ne_n$ mean then?
how do you multiply two sequences?
 
Jonas Teuwen
Jonas Teuwen
$(e_n)$ is the basis for $\ell^2$.
$(x_n)$ are just scalars.
 
robjohn
robjohn
so the $x_n$ are numbers... okay
I thought that earlier it seemed the $x_n\in\ell^2$
 
Jonas Teuwen
Jonas Teuwen
Well, $x_n$ is just $x$ with every element after the $n$-th $0$ right?
Then they need to give an argument (since it is an introductory course) why $x_n \to x$.
 
robjohn
robjohn
Are the $e_n$ orthonormal?
 
Jonas Teuwen
Jonas Teuwen
Just the canonical basis.
$e_n$ is the sequence with everything $0$ except the $n$ element a $1$.
 
robjohn
robjohn
6:40 PM
so they are orthonormal with the standard dot product
 
Jonas Teuwen
Jonas Teuwen
Yes.
 
robjohn
robjohn
So they could write $X_n=\sum_{k=1}^nx_ke_k$
 
Jonas Teuwen
Jonas Teuwen
Oh well, I'll just ask the professor tomorrow what to do with that. I seem to understand what is going on 8-). Phew.
 
robjohn
robjohn
cool.
If I heard from them what they were thinking, it might be easier to deduce where their confusion lies.
 
Jonas Teuwen
Jonas Teuwen
I was just wondering if it is common for many students to have trouble understanding what a sequence of sequence is or how to calculate with that.
 
robjohn
robjohn
6:43 PM
second hand confusion is difficult :-)
 
Jonas Teuwen
Jonas Teuwen
Yes :-).
And you've got the third hand confusion :-).
 
robjohn
robjohn
It has been a long time since I've taught sequences, much less sequences of sequences, that I don't remember what the students had trouble with.
or why
 
QED
QED
I think the biggest difficulty for all these things is not knowing how to prove quantifiers, or what to do when you do have proofs of quantified statements
 
robjohn
robjohn
@JonasTeuwen This statement is what confused me. I thought from this that $x_n$ was a sequence since $x\in\ell^2$
 
QED
QED
I'm guessing most people were taught the same amount of logic as me: none
 
robjohn
robjohn
6:47 PM
@QED The only time I taught any logic was in my concrete math course.
in all the others, it was assumed that they knew logic.
 
Jonas Teuwen
Jonas Teuwen
I don't like to teach, if people don't understand it I have failed :(. And they often don't understand.
 
robjohn
robjohn
@JonasTeuwen It is a combination, sometimes the combination can be all theirs and sometimes all yours.
 
QED
QED
I don't understand why.. but I seems like students are expected to "aquire" mathematical language in the same fuzzy way as you do natural language, rather than actually teaching it to anyone.
 
robjohn
robjohn
Sometimes even the most eloquent teacher cannot get through to some students.
sometimes a bone-headed professor can confuse even the best students.
 
Jonas Teuwen
Jonas Teuwen
They use very very very ugly notation. Stuff like $\|f(s)\|_1$.
That's not how I did it.
 
robjohn
robjohn
6:51 PM
That's the norm where 1 has weight 1 and all other integers have weight 0 :-)
 
Jonas Teuwen
Jonas Teuwen
They mean $\|f\|_{L^1(\mathbf R^d)}$.
 
robjohn
robjohn
I know.
 
Jonas Teuwen
Jonas Teuwen
The subscript is fine, it is the $(s)$.
 
robjohn
robjohn
ah.
 
QED
QED
If you assume that what the teacher is saying is correct then you're bound to get confused.
 
Jonas Teuwen
Jonas Teuwen
6:52 PM
Well, better assume what the teacher says is correct then not think at all.
 
robjohn
robjohn
I can understand a bit; you want to know what variable to integrate in to get the norm... but yes, it seems that you want the norm at a given $s$
what is $\|f(1)\|_1$?
 
Jonas Teuwen
Jonas Teuwen
Well, even then. $L^1$ "functions" are equivalence classes.
So they also fail to understand that then.
I'm not sure how that is even possible.
 
robjohn
robjohn
well, I'd say that an $L^1$ function is a function that belongs to an equivalence class...
just as 3 is a number mod 5, but we shorthand the equivalence class of 3 mod 5 by just saying 3 mod 5
That leads to confusion, too.
 
Jonas Teuwen
Jonas Teuwen
Hmm. Maybe I wrote it wrong then I see it as the space of integrable functions modulo the null set stuff.
 
robjohn
robjohn
there are a bunch of assumed conventions that students don't pick up on, either because they are not explained clearly, or because the students are trying to learn so much at once that they ignore conventions sometimes.
teachers take these conventions as given and often gloss over them.
 
Jonas Teuwen
Jonas Teuwen
7:05 PM
I'm sure that the professor didn't do that.
(gloss over them).
 
robjohn
robjohn
7:47 PM
I wasn't accusing your particular professor. However, I've known some who, because they deal with this stuff all the time, forget that others don't know it, and don't give it as much attention as the students may need.
playing around with the SQL stuff, it seems that this answer of mine has more than twice the length of any of the other of my posts.
 
Tim
Tim
8:04 PM
@robjohn: Out of curiosity, how did you play around with the SQL stuff?
 
robjohn
robjohn
8:19 PM
@Tim try here
 
Tim
Tim
@robjohn Thanks! If I just want to learn and experiment, when I compose a new query, can it not be published?
 
robjohn
robjohn
@Tim sure, you have to ask for it to be published.
 
Tim
Tim
or if it will be published, does it matter if it is a bad one.
 
FreakEnum
FreakEnum
I've a problem with a Q, May I ask?
 
robjohn
robjohn
@Tim Probably not. I think if you save it, it becomes public, but I'm not sure.
@FreakEnum sure
 
FreakEnum
FreakEnum
8:28 PM
Q: Given x=123456 and z=x-y then for how many values of y , z is divide by 48,98,105 ?
 
robjohn
robjohn
@FreakEnum do you mean for y and z positive?
 
FreakEnum
FreakEnum
@robjohn yes
 
N3buchadnezzar
N3buchadnezzar
123456 = y mod 48 98 105 ?
 
robjohn
robjohn
that is, you want to know how many numbers less than or equal to 123456 are divisible by 48, 98, and 105?
 
FreakEnum
FreakEnum
@robjohn yes :)
 
robjohn
robjohn
8:32 PM
$48=2^4\cdot3$, $98=2\cdot7^2$, and $105=3\cdot5\cdot7$
what is the lcm of the three numbers?
$2^4\cdot3\cdot5\cdot7^2=11760$
$10<\frac{123456}{11760}<11$
 
FreakEnum
FreakEnum
@robjohn can you give me some 5 mins? I think I need to revise some prerequisites before starting with Q
 
robjohn
robjohn
okay
 
N3buchadnezzar
N3buchadnezzar
So there are 11 numbers that are overcounted?
Oh, I tought he said 48, 98 or 105
 
robjohn
robjohn
If it were "or" we would perhaps need to use some inclusion-exclusion.
 
N3buchadnezzar
N3buchadnezzar
yeah
Let p_1 , p_2 and p_3 be three distinct prime numbers
Define n to be ( p_1 - 1 )( p_2 - 1)( p_3 - 1 )
 
robjohn
robjohn
8:39 PM
three primes? where are they?
 
N3buchadnezzar
N3buchadnezzar
How many positive numbers below or equal to n are divible by at least one of p_1 , p_2 and p_3
 
QED
QED
the most possible would be when the primes are of the form 2^r+1 wouldn't it? If so you could get an upper bound that way
 
N3buchadnezzar
N3buchadnezzar
@robjohn For a speficic case I think 7 11 13 would suffice
 
robjohn
robjohn
$\lceil\frac{123456}{48}\rceil+ \lceil\frac{123456}{98}\rceil+ \lceil\frac{123456}{105}\rceil- \lceil\frac{123456}{2352}\rceil- \lceil\frac{123456}{1680}\rceil- \lceil\frac{123456}{1470}+ \lceil\frac{123456}{11760}\rceil$
 
N3buchadnezzar
N3buchadnezzar
Haha 1+
 
robjohn
robjohn
8:45 PM
where 2352 = LCM(48,98), 1680=LCM(48,105), 1470=LCM(98,105), and 11760=LCM(48,98,105)
actually instead of ceil(x), those should be floor(x+1)
I compute 4809 non-negative integers not greater than 123456 that are divisible by 48,98, or 105.
 
N3buchadnezzar
N3buchadnezzar
I could write a matlab script to verify it
 
robjohn
robjohn
feel frree.
include 0 and 123456
 
N3buchadnezzar
N3buchadnezzar
although I have a physics exam in two days, not practiced for it at all.
 
FreakEnum
FreakEnum
@robjohn How did you find the lcm of three numbers? ( I am used to 2 only :(
 
robjohn
robjohn
LCM(a,b,c)=LCM(LCM(a,b),c)
or you just take the maximum of the exponent of each prime in their factorization.
$48=2^4\cdot3$, $98=2\cdot7^2$, and $105=3\cdot5\cdot7$, so the LCM is $2^4\cdot3\cdot5\cdot7^2=11760$
@FreakEnum have you installed the MathJax bookmark?
 
FreakEnum
FreakEnum
8:58 PM
@robjohn yes I already did :)
 
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