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8:05 AM
Hah, coding done.
Hey @rob...
@JM Yes?
Can you help Srivatsan? ;)
@Srivatsan so do you see that limit b_n exists?
Oh, =) // Yes, lim b_n exists.
What should I see about c_{n+1}/c_n?
next, 0<=c_{n+1}/c_n = a_{n+1}/a_n b_n/b_{n+1} <= 1
because 0<=a_{n+1}/a_n<=1+1/n^2
and b_{n+1}/b_n=1+1/n^2
8:13 AM
Brilliant. So c_{n+1} is monotonic. So, limit exists.... =)
Thanks, @robjohn.
no problem... :-)
Clearest solution =). [No votes though, I'll upvote later...]
Thanks :-)
8:16 AM
But you might want to at least add the magic word "monotonic" for those of us who don't see it... =) Just a suggestion.
It looks to me that @Asaf will appreciate today's XKCD...
user image
(I uploaded the comic since it is of interest to all =))
I like the x\in\phi on the board
If you hover over the image, you get: Proof of Zermelo's well-ordering theorem given the Axiom of Choice: 1: Take S to be any set. 2: When I reach step three, if S hasn't managed to find a well-ordering relation for itself, I'll feed it into this wood chipper. 3: Hey, look, S is well-ordered.
8:33 AM
Nice comic!
@robjohn I don't really see it, but what does it suppose to mean?
@Daniil You have to hover over the image at xkcd.com/982
It is more about Proof by Intimidation
I think he meant "see" in the sense of understand. =)
@robjohn no, I mean the x \in \phi part
Well ordering S by intimidation...
@Daniil Oh, that's the second line on the board. The first line says x \in S.
8:37 AM
Oh, I guess I didn't know what wasn't being seen :-)
@Srivatsan I thought it says x \in \O
@Daniil That, too...
\O is correct, I just didn't remember the LaTeX symbol right off
(Somebody should star the comic for posterity =))
8:41 AM
@Srivatsan done
Thanks =). I starred the hovertext...
It's \varnothing or \emptyset, by the way.
How do you write formulas here again?
@Srivatsan \varnothing looks better
upload that URL from the web
Oh, cool
8:46 AM
@Daniil Spaces should be percent-encoded, though (%20).
@Srivatsan I think that is what caused Daniil to ask
Not sure about that. His question appeared earlier. Only we two have been talking since then.
Hello, can anyone tell if Sivaram's Edit part math.stackexchange.com/q/85450/1281 is correct?
I feel his is right.
I'm out for a while. Later.
@Tim I am not so sure. What kind of integral is it and how are we allowed to do integration by parts?
8:52 AM
Just suppose now P denotes cdf of X instead of measure of X. Then the integrals are Riemann-Stieltjes integrals
That looks like Didier's answer, no? // But I guess you want to apply integration by parts to the integral in Didier's answer.
Didier actually used some tail formula for expectation. I think it is a different approach.
In Didier's part, P is still measure of X, not cdf of X.
@robjohn: Thanks for bringing up the tail formula of expectation.
@Tim I was going to use that in an answer yesterday, but I found out that Markov's Inequality included that.
9:01 AM
@robjohn: Does the formula you wrote require X to be a continuous random variable? Can it be discrete?
@Matt cya
bbl = bye bye l?
@Tim I believe it can be discrete...
bbl = be back later
@robjohn: I didn't see that tail formula is used in proving Markov's inequality. Maybe I am missing something?
9:06 AM
@Tim P(|x|>t) will have a discontinuity, but E(|x|) is computed okay.
@Tim Note that for a decreasing f, \int_x^\infty f dt <(\int_0^\infty f dt)/x
That and the integral should prove Markov
Um, sorry that should be...
f(x)<=(\int_0^\infty f dt)/x
@Srivatsan I like the "Proof by intimidation" idea
Intimidate the students to memorize the proofs
That is how math is usually taught!
@robjohn: Then proving Markov's inequality does not need tail formula. en.wikipedia.org/wiki/…
@Skullpatrol: Math is not supposed to be taught that way. t.b. said chat.stackexchange.com/transcript/message/2527983#2527983
9:25 AM
@Tim I agree that it is not "supposed to" be taught that way, but the reality is that memorization does play an important role in this subject.
In my humble opinion.
@Skullpatrol: I agree with you somehow. When we are driven by the deadline of homework and exams, and don't have time to digest the ideas, we usually try to memorize it in a hurry. That is sad...
The courses are usually arranged in such a way that they contradict their original purpose.
@Skullpatrol Not sure what your experience was, but I had quite a lot of fun learning math.
One thing I discovered since I joined this site: deep understanding is difficult.
I guess most (school, even college) teachers themselves have only a shallow understanding of the material they teach.
To me, if being driven by other things than interest, anything can go from fun to pain.
I don't blame the teachers. It is difficult to be perfect. I only hope they are more open-minded, so that they can accept students to learn in their own ways.
The reality is that memorization does play an important role in this subject. The earliest example of this is the times tables.
I'm not saying that these should not be memorized, but that they must be memorized
But you could look back and think whether what you memorized before makes sense or not.
9:35 AM
Is there a difference between and ? Post asking for help with computing integrals seem to be in both these tags.
Both tag wikis mention computing indefinite integrals.
@MartinSleziak No idea what the difference is. But tagging them makes more sense to me...
Looking in meta.
Man, it is 10.37 and nobody's here!
(at the group)
@JonasTeuwen Has tb left already?
Huh? Tb is a few 1000 km's away :-).
Ilya is in the next building, that's the closest one.
9:40 AM
Oh, you were talking about real life when you said no one's here? =)
Ok. [I was wondering whether tb has left for his vacation or whatever to Vienna or whereever.]
@Martin To make myself clear. My guess is that covers "all aspects of integration" including the computation of usual integrals, questions pertaining to integrals but not to do with computation, Lebesgue integrals and so on. seems to be a subset of the first tag having to do with the computation part.
But it seems that this is not the convention followed OR that no particular convention is followed. A meta post is in order, perhaps?
@Srivatsan Based on the names, I would think the same. But tag wiki for integration says All aspects of integration, including the definition of the integral and computing indefinite integrals (antiderivatives).
Maybe we can bring it up on meta.
Or wait to see whether some more experienced user appears here and tells us his opinion.
Disclaimer: I do not suggest to go retagging old questions, but if we had some convention/agreement about this, I would know what to do with new and recently edited questions.
@MartinSleziak I would take the second route. (Further, there are already a few big posts on tagging in meta. I am not sure whether we should create new posts or ask in the existing threads themselves.)
I'm off for lunch. @Srivatsan, please ping me if something relevant will be said here about these tags.
9:53 AM
Ok, final point: One idea is to make a synonym of . That way, we need not touch the old posts.
@JM, If you can give your opinion on the discussion between me and Martin, it'll be great. Thanks!
I'm not too active in calculus questions - so I' definitely not the right person to make suggestions about this.
@Tim I don't quite follow. All of those methods of proving Markov are based on E(|x|) = \int_0^\infty P(|x|>t) dt. The probability argument is simply disguising things using the expectation of the indicator function, but they say the same thing.
But maybe dividing post between "low level" - like calculate integral - and "higher level" - like problems with Kurzweil-Henstock integral or integration of Banach space valude functions - might be reasonable.
@MartinSleziak Well, I think you raised a fair point. (For that matter, I don't have experience either. =)) Let's see what the experienced have to say...
We will see what other people will say.
@Srivatsan BTW JM is not in the room right now: chat.stackexchange.com/rooms/info/36/mathematics I am not sure how this works in chat - will he see your ping?
9:59 AM
@MartinSleziak I am kind of positive he will see the ping. Let's see. =)
As far as the experience with calculus and integration questions is concerned, you have calculus badge as I see.
By the way, @robjohn, if you have something to add, feel free to. =)
So I think you have something to say about this.
@MartinSleziak That's from answering countless homework questions. ;)
@MartinSleziak When JM is next on MSE, the message will show on his profile page.
@Srivatsan I have been replying to comments on some answers. What are we discussing? Let me read back.
Ah, that we don't need and ?
10:05 AM
Yes. But a pragmatic proposal is to make the former a synonym of the latter. // That was my proposal, at least =).
I think they could be combined and a synonym created.
@Srivatsan yes
In my opinion, integration is the process of taking the integral.
just like addition is the process of adding
please correct me if I'm mistaken
@Skullpatrol Good point, but a better analogy would be integration:integral::addition:sum, I feel.
@Srivatsan how about addition, adding, and add or multiplication, multiplying, and multiply; I feel.
Of course this is all context dependent
as is always the case with English
10:23 AM
@robjohn I tend to same opinion as what you said - but I'll ask anyway. This argument is not strong enough for keeping them both, is it? chat.stackexchange.com/transcript/message/2530098#2530098
Of course, I am not sure whether that was the original intention when creating the two tags.
For example questions like this, that are more about definition and properties of (some type of) integral and not about calculating integral of a given function: math.stackexchange.com/questions/84092/…
@MartinSleziak sort of like and ?
@robjohn Um, I like the tag...
Well, obviously it's empty.
But integration-theory is a better description of what I mentioned above.
@MartinSleziak yes, but it is a bit more descriptive than
(I'm not really sure if there are many questions like that.)
10:28 AM
@MartinSleziak but that would be the only reason for two tags.
that I can see.
Well, at least the only reason I am able to come up with.
But again, it seems that now we have question which would be suitable for under both tags.
This one is tagged integral and asks about the definition of gauge integral: math.stackexchange.com/questions/54953/…
@MartinSleziak Is there any example? Most questions seem to sway in one direction or another.
^ That would be a wrong application of the integral tag then. =)
And this is not about calculation, but about proving properties of Riemann integral and it is tagged integration. math.stackexchange.com/questions/72844/…
@MartinSleziak That fits our (i.e., we three) view of the world, right?
@Srivatsan Yes. I wanted to find at least one example in both tags.
So it seems that even if we decided to have two tags - e.g. for question concerning calculations and for "theoretical" questions, this cannot be obtained by simply renaming integration to integral.
And the argument for keeping two tags, which I suggested, is falling apart...
10:35 AM
@MartinSleziak No. Because the current tagging scheme permits computation questions to go to either tags, we will need to retag those at least.
@Srivatsan Yes, that's what I meant. If we want to have some tag like "integration-theory", we would have introduce a new tag and retag some questions. (Or at least tag the new questions correctly.)
And you pointed out one spuriously tagged (integral instead of integration-theory or integration) post. =)
Well, yes. (I had to look up spuriously in the dictionary.)
But I guess that if I was able to find such post quite easily, there will be a few more similar posts.
@Srivatsan I asked this same question to "Ask the Prof." english-online.org.uk/askprof/…
To sum up my point of few: I agree that making the two tags synonyms seems to be a good idea. If a new tag (or something similar) will be created, I'll do my best to retag new questions accordingly if I spot an incorrect use.
10:43 AM
Synonyms would mean less work, but might not be the "optimal" choice (like we decided to separate abstract-algebra from algebra-precalc). If we take the other option, we will need to decide what to do with the old posts. The new ones can be taken care of, let's hope.
I am relatively new to this "tag creation" business, so I am not sure to which extend there is a need to have consensus in the community, but recently for some tag-changing t.b. pointed it out in meta thread and after his answer got several upvotes, it was considered as reaching a consensus: meta.math.stackexchange.com/q/3187/8297
Wow, that was a nice typo: my point of few.
There you go, I even starred it for you... =)
@Srivatsan How did you like the link Srivatsan?
@MartinSleziak Ok. I think that since we three of us are in some agreement, we will wait for @JM and @Henning's comments tomorrow.
In fact, we are yet to decide which of the two options (keep 2 tags, or synonym+merge) we will propose in meta.
@Skullpatrol It was a good question. But I think it is addressing a slightly different poin from the one I tried to convey. If integration is the process of taking the integral, then addition is the process of taking the «what»? I think "sum" seems to be a good fit there...
@Srivatsan I agree in that context you are correct.
10:53 AM
@Henning, We are discussing why we have both the and tags. For the complete discussion, go here: chat.stackexchange.com/transcript/message/2529959#2529959.
Comments welcome!
What link is that? It seems that you created a room or something, but I cannot find it anywhere.
@Srivatsan All I'm saying is that it depends on the content of the question and in what context it is being asked. This is what determines what form of the "tag" that should be used i.e. integral or integration.
The link I provided gives the grammatical guide lines to follow, but concludes with "There is no rule as to which form the noun will take."
Just my humble opinion.
11:11 AM
Hi people! May I ask a simple question here that I think it's not worth becoming a real question on the site?
@PantelisSopasakis I can't answer for "the people," but I would like to see it.
Ok. So, here is the question: Assume that $\{C_n\}_{n\in\mathbb{N}}$ is a sequence of sets in $\mathcal{X}$. Assume that $\{C_n\}_{n\in\mathbb{N}}$ is increasing ($C_1 \subseteq C_2 \subseteq \ldots$. Then I need to prove that $\overline{\bigcup_{k\in\mathbb{N}C_{n_k}}}=\overline{\bigcup_{knin\mathbb{N}C_{‌​n}}}$. Is this true if we remove the closure?
(How do I write math in here?)
BTW, to me what I need to prove look obvious? Is it?
11:47 AM
@Srivatsan It is a bookmarked conversation - see here: chat.stackexchange.com/rooms/info/36/…
I might be mistaken, but I thought that they are visible to everyone.
1 hour later…
1:11 PM
Good afternoon.
@Srivatsan I don't think I have any intelligent opinion about the integrals tags.
Ok, thanks.
It's not my core area of competence anyway.
Good morning.
Good morning? I'm already packing to go home :-).
@JonasTeuwen How was your day, Jonas? Looking forward for the weekend?
1:22 PM
@Srivatsan I sure do! It was fine. Yours has just started? :-).
Full disclosure: I wasn't pinged by any of your comments (odd), but since I back-read the transcript anyway... ;)
As for - that was one of the first tags I made. I'm not sure how came up, but I'd call seniority here... :D
i.e., I call for folding in into .
@JonasTeuwen Yes, my day just started. Will soon know «who» amassed «what» in their Thanksgiving day (should I say night?) sale. =)
Ah... Black Friday...
@JM Oh ok. Thanks. The folding seems to be in the wrong direction though: [integral] looks like a subset of [integration] (henceforth, integration-theory]?).
I'd go with changing the tag-wiki description myself, actually. :)
So, merge the newbie into the grandfather, then modify the description so that everything's nicely covered.
1:29 PM
@JM You know of it? People braving the cold at 2 and 3 in the morning to get items?
@JM We don't need the community to approve?
@Srivatsan Stampedes happen there. Brr.
@Srivatsan That's just my proposal. There should still be a community consensus.
On the other hand, Mike Spivey already brought up the topic in that big thread on tags...
(Maybe I should bump...)
Oh, and a tip for people reading this beastie: set the answer view to "active" instead of "votes".
hi also
1:46 PM
@Martin, if I may ask: aren't there English translations of the jokes posted on your website?
Sorry, no I do not have an English version.
Some mathematical texts and the papers are in English - but they are not that funny. (I hope...)
No, I'm not complaining about your papers... :)
Those jokes I collected as a student.
And when I started PhD-study and get my new website on the department, I copied them there.
@JM We should have noticed that post, when we were discussing integral-related tags here....
2:03 PM
@MartinSleziak It was buried deep down. We could (should) have searched for integral or integration, but I guess that did not strike us...
What can we do about it? Just remove the tag slowly from posts and then delete the tag?
@Srivatsan Hence the tip I gave. :)
@Srivatsan What do you guys think? Only five questions...
@JM Yes, now it appears in the active order = awesome. Thanks for the tip.
@JM I detagged one of the posts.
@Srivatsan Maybe not only "edit day" but also "retagging day" should be organized from time to time :-) chat.stackexchange.com/transcript/36?m=2443654#2443654
One untag from me.
@MartinSleziak May be retagging is more important than minor edit.
@JM I edited the post to say we are untagging.
2:25 PM
Okay, maybe only those two for now? Let's do the three others in a few hours...
That sounds good. You don't have to worry about it; I'll take care. =)
Some body decided to gift me some votes to get me to 9k rep. =)
Can somebody please help me to think up some small model and its' property? I am doing a presentation on model checking and temporal logic and I need some small yet complex property
@Srivatsan OVER 9000!!!1! :D
@Daniil No, don't know much about it. Sorry..
Thanks, @JM.
3:00 PM
10k+ people: can anyone make sense of this deletion? The question seems legit and even shows some work.
Huh. That's a surprise...
Hey @Willie!
3:26 PM
@robjohn: Let $f$ be a measurable function from $(\Omega, \mathcal{F}, \mu)$ to $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$. The Markov inequality is

$$\int_\Omega |f| \, d\mu
\geq \epsilon \mu( |f| \geq \epsilon ) $$

for any $\epsilon >0$.

My understanding of the proof without the tail formula of expectation is

$$\int_\Omega |f| \, d\mu
\geq \int_{\{ |f| \geq \epsilon \}} \epsilon \, d \mu
= \epsilon \mu( |f| \geq \epsilon ) $$

I guess your understanding of the proof based on the tail formula of expectation is:
@Tim Yes, they are...
@robjohn: Even if $\mu(|f|=t)>0$?
Just a]
Just a moment...
This is discussed a little in this question math.stackexchange.com/questions/63756/tail-sum-for-expectation
3:42 PM
The functions $\mu(\{|f| \geq t\})$ and $\mu(\{|f| > t\})$ differ only at points where $\mu(\{|f| = t\})>0$ and those are the points of discontinuty of $\mu(\{|f| > t\})$ and since it is a monotonic function, it can only have countably many dscontinuities. The integrals of two functions that differ only at countably many points are equal.
am I making some gross error, or is this making sense?
I think it make sense. I was not saying you are not correct. I am just not sure.
I have not dealt with this stuff in a long time, so I could be overlooking something, but I think this is right.
I have to go out for a bit. bbl
So the proof of Markov inequality can be either based ontail formula or without it. c u, @robjohn
I think the proofs are both essentially the same, just naming things differently.
Not really. My understanding of Wiki proof is without the tail formula, and more straightforward. later
3:57 PM
Once again, I find your solution the clearest, @robjohn.
Though I don't see the very first step of math.stackexchange.com/q/85550/13425. I know about Lebesgue's characterization of Riemann integrable functions as continuous a.e.. The statement you are quoting -- does that follow from this?
Presumably it's much simpler than the hammer than I am thinking about...
Never mind, I think I figured it out. Thanks
4:22 PM
I am sure tb would love this comment: math.stackexchange.com/questions/85548/… =)
4:32 PM
Not another one. Gah.
4:43 PM
What subjects are you guys studying?
5:00 PM
@Srivatsan I'm not sure that Dimitrije is aware of that can of worms that he's presenting...
@JM I found the suggestion of mailing Wiles a little hilarious =)
Not that farfetched I think, but I would however make sure that if I were asking Wiles about a result of his, I'd already gone through his stuff at least thrice.
(Have you seen that paper for proving Taniyama-Shimura? It's looong!)
@JM Hundreds of pages?
I have some kind of sympathy for iyengar. I can see some of myself on him. As more I read about him, it becomes a mixture of feelings.
Let me see if there's a nice link...
5:08 PM
I think most of us have sympathy.
Yes, and I just heard of him.
But just how do you get through that he's going about it wrong?
@Srivatsan Have a look at this.
But it converts into frustration pretty quickly.
@JM You don't, evidently.
@JM Thanks. Little over a hundred pages long.
5:11 PM
It's as if he's trying to build a skyscraper when his foundation's made of plaster of Paris...
I agree with you, although I don't have the knowledge to understand his discussion with others. I am not sure if he has realized it now.
From his questions on SE, he seems to have realized it more than when he was on MO in August.
Luckily, there were nice people on MO trying to reach to him
"I have been mailing tons of emails asking for help, nobody responded" - problematic part highlighted...
5:27 PM
Yes, it is problematic.
I'm bushed. Later, y'all.
5:44 PM
That's true, I'll never understand taniyama-shimura if I sit here watching cartoons all day
I think that should explain what might be bothering you.
@robjohn Ok, I will look at it in a but. Thanks.
@Srivatsan: The space of continuous and compactly supported functions is dense in L^p(R^d) from en.wikipedia.org/wiki/Lp_space#Dense_subspaces
I think that works for Lebesgue and Riemann integrals.
5:59 PM
Why would someone star that? :-/
Probably the typo.
It's not a mistake when typing - it's a mistake when clicking.
I don't know how it should be called.
@MartinSleziak I am seeing the in a but, what are you seeing?
Well, I did not expect this: urbandictionary.com/define.php?term=clicko
There is such a word.
@robjohn I did not see that. I thought that you are saying that someone clicked on the star by mistake...
@MartinSleziak It's hard to come up with anything original these days.
6:11 PM
I'm feeling a little dyslectic now.
That's probably correct.
Pun was not intended.
@robjohn I was hoping to star that typo in retaliation against the world. :-/
@Srivatsan :-)
6:54 PM
In this, do the limits not commute? That would make it very easy.
No clue. (By the way, is this question of yours answered: math.stackexchange.com/q/78129?)
@Srivatsan No I was going to answer it myself. Just been too busy : (
@Matt No problem. I was just rummaging through some old unanswered posts when this one came up. (No hurry...)
Any opinions?
@Matt Er, not sure.
But what's your justification for switching the limits?
7:08 PM
At the moment none. If I knew that the space had finite measure I could apply the dominated convergence theorem.
The function is the constant 1 function, I just need to know that that's integrable.
@Srivatsan: What were you referring to when you asked "Why would someone star that?"?
@Matt The previous comment of mine that was starred -- thanks to a typo.
Oh! Of course! I thought it was referring to question on SE : )
@Matt Um, that makes sense for finite measure.
Have to leave for lunch. // See you later.
Bye, see you!
7:48 PM
@Matt is each \mu_k a positive measure?
@Matt This question seems to be related: math.stackexchange.com/questions/15240/…
@QED what's up?
Still a bit fed up not doing any maths
@QED looking for a topic?
7:53 PM
@Matt is \mu_k(E) an increasing function of k?
8:08 PM
Good afternoon
How goes it?
I feel as if I am badgering Andre Nicolas.
@robjohn No, it's not positive but increasing. It's this question here
@robjohn There is an answer now.
@MartinSleziak Ah, thanks!
Good night folks!
8:53 PM
Does anyone know how Riemann surfaces come into play in physics? VI Arnold likes to talk about them when he's ranting about pure mathematics.
Hi everybody, may i ask you: This is the question ask recently, but it is not so self contain that makes me understand the question, is there a interpretion of this question where it use simple intuition?
i am not able to guess what it mean...
@AsafKaragila How so?
9:09 PM
Consider a complex analytic function f(z), a point z_0, and a parametarized circle re^(i/theta) around z_0. Why does f(z+re^(i/theta)) tend uniformly to f(z) as r tends to 0?
I now have a smartphone! Now I want an alarm that requires me to compute integrals before it will stfu.
@JonasTeuwen If you find such a thing, let me know. I have been looking for one for a while.
There are ones with simple challenges like 5 - 4.
@Srivatsan On the Banach-Tarski on [0,1] question, he answered and I kept correcting him.
9:29 PM
This whisky smells like old folks.
This arak smells like arak.
You're not very refined right? Smell better. Maybe it smells like strawberries.
Nope. It has a clear smell of arak. Nothing more and nothing less.
I wish I had some cola to cut it with.
9:49 PM
@Potato This one asked me for (1023 - 327) x 2
@JonasTeuwen We should pay a programmer to make one that presents trigonometric integrals.
@Potato I find the captchas hard enough. No need for trig integrals... =)
@Potato Then the damn phone is ringing for 15 minutes!

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