Hello, PMajer, and welcome to Wikipedia! Thank you for your contributions. I hope you like the place and decide to stay. Here are some pages that you might find helpful:
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Thanks for providing a detailed answer to the problem involving number of progressions. Happy new year. Cheers--Shahab (talk) 06:50, 30 December 2008 (UTC)[reply]
Thank you! I don't completely understand but it sounds very nice :)--PMajer (talk) 00:50, 4 January 2009 (UTC)[reply]
Not of great importance but there is a list of folks who regularly contribute at one or more of the reference desks. I happened to notice your contributions at the Mathematics desk and so I added you to the list of people who are regular and positive contributors. I hope you don't mind :) hydnjotalk 20:14, 4 January 2009 (UTC)[reply]
Thank you very much, I'm very glad to be both regular and positive!! ;) --PMajer (talk) 21:00, 4 January 2009 (UTC)[reply]
I've reformatted this section in order to help you understand some of the wiki syntax. You seem to be catching on just fine so I'll throw in a couple of links to illustrate disambiguation (one word has several meanings). Please let me know if you would like any further help. Also, you can click "[edit]" next to this section to see all of the wiki-syntax used :)
Regular as you/I intended = Protagonist so you can show "Regular" and have it link to "Protagonist" like this: [[Protagonist|Regular]].
Positive as you/I intended = Positive (linguistics) so you can show "Positive" and have it link to "Positive (linguistics)" like this: [[Positive (linguistics)|Positive]].
That vertical line "|" is called a pipe and using it is called "piping" a piped link. It lets you separate the name of the page you want to link to any name that you want : [[real pagename | any name]]. Another usage is called the pipe trick. My apologies if I've confused you, just let me know if there is something that I can try to explain better :)
I've taken this opportunity to start your "try-out" or personal experimental page. This may be of help to you if you want to try something out without committing to an actual page just or see if it looks OK. Or, just play around - its your sandbox
so do whatever you want there! ;) hydnjotalk 02:13, 5 January 2009 (UTC)[reply]
Thank you again! I knew the pipe trick, anyway I will refer to you for help! --PMajer (talk) 14:30, 5 January 2009 (UTC)[reply]
The box to the right is the newly created userbox for all RefDesk regulars. Since you are an RD regular, you are receiving this notice to remind you to put this box on your userpage! (but when you do, don't include the |no. Just say {{WP:RD regulars/box}} ) This adds you to Category:RD regulars, which is a must. So please, add it. Don't worry, no more spam after this - just check WP:RDREG for updates, news, etc. flaminglawyerc 03:08, 6 January 2009 (UTC)[reply]
Steve, it was me?? ..Seems it is so; of course, it was completely unwillingly, for I am grateful to your response. Sorry. (By the way I noticed that you used the singular form "a dice" as a kind of support; it's a particular that reveals a very kind person). Thank you again and excuse me for the unwilled vandalism! --pma (talk) 11:45, 29 January 2009 (UTC)[reply]
Molto grazie. Ecphora (talk) 02:28, 4 February 2009 (UTC)[reply]
You are welcome! Do not esitate to ask me directly here, if you have any unsolved question, for I usually check the RD/Maths, and seldom go around for the other RD (and remember I am definitely not an expert of your topics). --pma (talk) 09:50, 4 February 2009 (UTC)[reply]
"Parte chiamavasi a'tempi della Repubblica Veneta un Decreto o Legge o Risoluzione, ch' era presa a partito da un Consiglio tanto sovrano che suddito legalmente convocato."
In an Italian dictionary I have, "partito" means not only "party", but also "resolution" or "decision". I wonder therefor if "parte presa" should best be translated as "resolution taken" or "decision taken" (or "made"). This fits very well with the various Parte presa documents I have (which prompted by original question); they all involve some problem (thefts, tax avoidance, frauds, blasphemy, etc.) and do not involve any "parties" as such. They all contain the language "l'anderà parte ...", which as you recognized, could mean here "the resolution will be that ..." What do you think? Thanks again for all the help. Ecphora (talk) 00:44, 7 February 2009 (UTC)[reply]
Yes, I think your translation is correct, and notice that it's not in contradiction with the original meaning of "parte" as "piece". As far as I see, the typical situation, in an assembly where a decision has to be taken, by discussion and votation, is that the assembly divides in two or more groups according to their opinions, plus maybe a number of people still undecided and floating; they also gather in different parts of the room (otherwise everybody quarrels with each other). So the original phisical meaning of part as "piece" then also cover "group of persons with the same opinion" (party) and then also: the opinion itself shared by a given group; when one opinion wins (after votation or other) it became the decision, and we can well translate "parte presa" as "decision taken". Note that already the Latin word "pars" has a political acceptance of "party". --pma (talk) 13:07, 7 February 2009 (UTC)[reply]
Problem solved! Thanks once again. Ecphora (talk) 15:22, 7 February 2009 (UTC)[reply]
OT for the refdesk, but you did ask, so here goes: I find it vanishingly unlikely that a person using Cambridge university computers and asking three questions from a Cambridge problem sheet on a course being taught at the moment is anything other than a Cambridge maths undergraduate. All such persons have paid supervisors. Algebraist 21:36, 5 February 2009 (UTC)[reply]
Again, since you asked: I first saw this problem two years ago, and I liked it immediately. The key point is that the sum of the jumps needs to diverge (otherwise you can't possibly hit every rational) while the sum of their squares must converge. The obvious choice for such a sequence is 1/n, so I decided the nth jump would be (approximately) 1/n. The rest is easy. Start with your favourite enumeration of the rationals, starting with 0, 100, 1, -10 say. Then let 0 be the first term of your sequence, and jump towards 100, making the nth jump of length 1/n. Since the sum of 1/n diverges, you'll get to 100 in finitely many steps (the last jump will have to be a bit undersize, of course). As you go, cross off every number you hit from the list of rationals you started with. Now choose the first rational on your list you haven't crossed off (-10 in this case) and jump towards that, again making sure the nth jump overall is about 1/n. Of course, you might not be able to jump 1/n exactly, but if you stay close (between 1/n and 1/2n say), then nothing will go wrong (since the sum still diverges and the sum of squares converges), and since the rationals are dense, you'll always have somewhere to jump to. Continue in this fashion, and you're there.
Of course, there's nothing special about 1/n: I could have used any sequence whose sum diverges but whose sum of squares converges. Also, the result still holds if the 2 in the question is replaced by any real strictly greater than 1. Algebraist 00:34, 6 February 2009 (UTC)[reply]
Thanks, yes, that's also what I thought, more or less. Clearly the point is to follow a first list of the rationals, to be sure not to miss any of them. --pma (talk) 13:45, 7 February 2009 (UTC)[reply]
Thanks for your helpful reply on the ref desk... The Bombing of Helsinki in World War II makes interesting reading on your point about a spread-out population and where the bombing went on. Cheers, Julia Rossi (talk) 22:11, 5 February 2009 (UTC)
Aaah! I just went to your article tip Aki Kaurismaki and discovered that I enjoyed the Total Balalaika Show and the Leningrad Cowboys without knowing about the Finnish connection. How coincidental and how clever, thanks again, Julia Rossi (talk) 22:24, 5 February 2009 (UTC)[reply]
No, pma, not at all. You brought another view to the events and the feelings behind them. I'm moved too by your experience in the alps. Conflict is always a tragedy, events getting out of hand and wiping out incandescent lives and people suffering without protection or choice in the matter. Then there's the humour, the joke against everything. The Finns were very clever to move the fires around to deceive the bombers that the city was elsewhere. The odds were amazing. The Leningrad Cowboys documentary made a big impression on me to see all those people enjoying themselves, regardless of history. You might be interested in this guy: Simo Häyhä. Julia Rossi (talk) 23:21, 5 February 2009 (UTC)[reply]
I can't believe it! That was from Teorema, one of my favorite movies, and all I associated were goats and cows :-)! Well spotted and thanks for your input. I hope you keep visiting the language desk. There has been a lack of native Italian speakers there, as far as I'm aware. ---Sluzzelintalk 08:52, 10 April 2009 (UTC)[reply]
Thank you, my pleasure! I'm very glad to meet people who know Pasolini. --pma (talk) 09:03, 10 April 2009 (UTC)[reply]
Excellent contributions to Wikipedia:Reference desk/Mathematics, pma! I've not noticed you away from the reference desk for even a day, and a question on topology that you haven't answered. Good work! PST 10:52, 16 April 2009 (UTC)[reply]
Thank you, I'm honored to get a barnstar of topology from a topologist. I must confess that I answer questions at the RD/M also to be allowed to put sometimes silly jokes ;-) --pma (talk) 12:17, 16 April 2009 (UTC)[reply]
"Personally I'm a bit sad about the fact that one day I must die, but the thought that everybody must die as well, greatly cheers me up."
Did you make it up yourself, or did you borrow it from somewhere. Not to accuse you or anything, it is just one of those perfect timeless quotes that sounds as if it should have being uttered by Dr. Johnson (or maybe Frankie Boyle!). I Google it, but got nothing. Frank Bruno's Laugh (talk) 16:19, 14 June 2009 (UTC)[reply]
Hi Frank, I'm glad you appreciated. In fact, I meant to write something in style of philosophic consolatory, you know, appropriate for the circumstance, and the funny effect was not scheduled ;-) --pma (talk) 18:30, 14 June 2009 (UTC)[reply]
Dear User, in reference to your talk[2], can you tell me how to obtain the 'First' WP page referring to the Boubaker Polynom?
thanks Rirunmot (talk) 17:30, 16 June 2009 (UTC)[reply]
Hi, actually, I don't know. Apparently, there is no more page on "Boubaker's polynomials"; I do not why, it has been removed. It was worth keeping at least the definition, I think. However there should be a way to read the removed pages, but I fear I don't know more than you; if you can't find it, try at the helpdesk... --pma (talk) 18:30, 16 June 2009 (UTC)[reply]
Thanks, please, if you find this way, inform me in my talk page (or transmit to me the user name of he who can help), many thanks Rirunmot (talk) 21:20, 16 June 2009 (UTC)[reply]
Hello. Thanks for all your help recently. I am very thankful. I will probably stop asking questions soon and spend time reviewing things I already know since my test is Thursday. But, before I do that, I have come to ask for help once more on a question from the past about a double integral. I never truly understood what you were saying here and I sort of forgot about it. But, this is a very important topic on the test and I am not good with this sort of thing. One thing you said was I could generalize some of the methods you mentioned on another question. So, I just used one of your methods to do the 1D case, I believe, which is at the bottom there. But, I'm not sure how that would generalize to the 2D case. So, if you are willing, will you please help me further in trying to understand this double integral problem? Thanks for all your help either way! I have also left a few additional comments at the double integral question to explain more. StatisticsMan (talk) 20:34, 15 August 2009 (UTC)[reply]
Ok, I will resume and paste here the main facts, numbered, so you can ask me which point you are interested in and I'll expand it. The two dimensional statement was " an L1 function on whose integral over all rectangles [a,b]x[c,d] vanishes, is zero (a.e.)". You can prove it in several ways:
Via the L1 density of step functions. The set of all "step functions" (that is, linear combinations of characteristic functions of rectangles) is a subspace of , dense in the L1 norm. This result implies in particular that there exists a sequence in converging almost everywhere to the function sgn(f) (note that it's only locally L1). Moreover, since for all x, we can take for all x (because if needed we an replace it with the truncate sequence , that still converges a.e. to sgn(f(x)) and of course is between -1 and 1). Thus the sequence converges a.e. to , and it is dominated by , so by the Lebesgue convergence theorem converges to . But is zero for all k because it is a linear combination of integrals of f over rectangles, and we conclude . RMK: the L1 density of step functions is a not difficult consequence of the L1 density of simple functions (that is, linear combinations of characteristic functions of measurable sets of finite measure), but still requires a little work.
Via convolution and approximation of the identity. Consider the characteristic function of the square [-1,1]x[-1,1], normalized by 4. Define as usual . Then converges to f in L1 as : this is the most simple result about the approximation of the identity (warning: at the moment the wiki article about mollifiers is wrong in many points). By definition, for all x in , is the integral mean of f on the square , so it is identically zero, and we conclude as before. You may use as well the characteristic function of any other bounded set of positive measure in place of the square [-1,1]x[-1,1], and the same conclusion follows assuming "the integral of f vanishes over every domain obtained from by means of homoteties and translations", since this ensures that the corresponding convolution vanishes identically.
Of course both proofs work in any dimension; actually "squares" or "cubes" in place of "rectangles" [a1,b1]x[a2,b2]..x[an,bn] are sufficient, and f in L1loc instead of L1 is also sufficient. In the second proof, you can use the segment [-1,0] for the one dimensional case: so define as the characteristic function of [-1,0], then . You can recognize that where , and the proof can be rephrased in terms of properties of absolutely continuous functions, as you did.
It is interesting that in the one-dimensional case the same conclusion holds if you ask the condition only on the intervals of length 1. Indeed, this implies, by the sigma-additivity of integral, that the integral is 0 on all half-lines. By subtraction then the integral on intervals of any length also vanishes, and you are lead to the preceding case. This easily generalizes to more dimensions: if an integrable function on has vanishing integrals over all unit cubes with edges parallel to the axis, it is zero a.e. You just have to play with the sigma-additivity and with translations, gaining the previous hypotheses. The reduced hypotheses are not sufficient in the case of functions as shown by .
Still true, but less elementary, the following generalization to integrable functions on Rn : if the integral of over any translated of a given bounded set of positive measure vanishes, then again a.e. Indeed, the assumption is equivalent to say that , where is the characteristic function of . Applying the Fourier transform you have that the pointwise product of and is 0 a.e. But is a non-zero analytic function by the Paley-Wiener theorem, so it is a.e. different from 0, hence is 0 a.e., and since the Fourier transform is injective, is 0 a.e.
You can also prove the result in the two-dimensional case (and in the n-dimensional too) as a consequence of the one dimensional case. Indeed you can reduce the problem to 1 variable using Fubini's theorem. For fixed a and b consider the function . It has vanishing integral over all intervals [b,c]. Therefore it is identically zero, according to the 1 dimensional case. This means that for a.e. y, the function has vanishing integral over the fixed interval [a,b]. But this is true for all a and b, so we can also say that for a.e. y the function f(x,y) has vanishing integral in x over all bounded intervals. To be precise, we sould argue in the usual way when dealing with "a.e. properties": let be an enumeration of all finite intervals with rational end-points. The integral of f(x,y) in x over vanishes for almost all y, that is for all , where has measure zero. So for all , hence for almost all y, it is true that for all k. But for any such y the equality immediately extends to any bounded interval I by continuity of the integral. By the one dimensional result you conclude.
You could also translate the two-dimensional (or n dimensional) hypotheses in terms of the integral function of f(x,y): that is, proving that has vanishing distributional partial derivatives, , , or equivalently, it has a vanishing distributional gradient. Therefore F is constant, hence f is 0. But the proof of this fact would be essentially one of the above.
Okay, it would be helpful if you showed me more about #2. I guess I figured it out for the 1D case but I'm not sure about 2D.
And, for #6, I think I am actually understanding that. A previous result was if for every interval , then f = 0 a.e. Are you saying it is also true if we only have that integral is 0 for every interval with ? That makes sense I guess since if the endpoints are irrational, you can use the intervals inside or outside with rational endpoints and get arbitrarily close. Now, assuming that part is good, in the end you have for almost all y, for every . From the 1D case, this implies for all such y, f(x, y) = 0 for almost all x. So for almost all y, we have for almost all x that f(x, y) = 0. So, let E be the set of all (x, y) where the integral is not 0. Then, this says for almost all y, has measure 0. And, I already did another qual problem proving this implies mE = 0. Is that all right? Thanks for your help! StatisticsMan (talk) 18:49, 16 August 2009 (UTC)[reply]
Yes, your conclusion of #6 is perfect. As to #2, it's not clear to me what's your doubt. If is a rectangle (or any measurable set of positive and finite measure) and if we denote the characteristic function of the set (normalized! I think I forgot to say it above), that is
if and otherwise,
then we have , for any
(because the support of is the set of all points such that , that is ). Therefore is the integral mean of on the set ; analogously is the integral mean of on the set .
So in our hypotheses is identically 0 for all and if is a rectangle. As you see, everything follows from the general result:
If , then converges to in the norm.
Do you know the proof? The usual way to show it is: first, prove it for the case when is a continuous function with compact support. Indeed in this case the uniform continuity of gives immediately the uniform convergence of the sequence, within the same bounded support, hence the convergence. Then one proves that the set of the for which the thesis hold true, is a closed linear subspace of . Therefore it is all because of the density of the continuous functions with compact support. (It is true for any integrable function with unit integral, btw, and one also has the a.e. convergence: but that's not needed here ). Is it clear?
I think everything you say makes sense. I'm not sure I understand it well enough to repeat it though. But, I'm too the point where I think I need to just review everything I've done so far. I'm currently reading every qual problem I have written out, which is taking a long time, and I want to review theorems and definitions and all that. Thanks for all your help. You have definitely helped me out in a lot of ways and I am definitely more prepared to pass because of it. StatisticsMan (talk) 00:32, 19 August 2009 (UTC)[reply]
Good, I'm glad that I've been of help. It seems you are in a good shape for your qual. Do it, then I'll toast to it too, with the Ocean in between.
I passed! Thanks again. As I said on the Reference Desk, I believe I got the minimum correct to pass (I will see for sure in a couple days) and one of the ones I got was showing that if an integrable function has integral 0 over every interval, then it is 0 almost everywhere, a problem you helped me solve! StatisticsMan (talk) 14:14, 29 August 2009 (UTC)[reply]
Very good, great! And the next time you have to get the maximum! ;-)
Luckily, there is no next time. I just need to take classes and do research for the next 3 years. No more major tests! StatisticsMan (talk) 15:06, 29 August 2009 (UTC)[reply]
Hello there, I hope you're well. Your addition to the envelope article is a very nice piece of mathematics. But I think it might be a bit out of place in the article as it stands. The idea of an envelope is a simple one from differential geometry. When people talk about envelopes they are, by and large, talking about the envelopes of families of smooth submanifolds. Your addition seems very algebraic, and a little out of place. For example: Hölder's inequality? (and in turn Lp-spaces?) It seems to be a very algebraic and topological addition to what was, and ought to be, a differential geometric article. You obviously know a lot about the topic. Why don't you start another article or at least a new section? I think there's a lot of milage to be had in looking at envelopes from this point of view, but at the moment the example looks right out of place. It had me scratching my head, and I have a PhD in singularity thoery (e.g. discriminants, bifurcations sets, families of functions, etc) and differential geometry. If I was left scratching my head then the interested undergraduate (God help the laymen) would be totally lost. ~~ Dr Dec (Talk) ~~ 12:50, 10 September 2009 (UTC)[reply]
Dear Dec, of course, you are free to revert it or move it elsewhere, if you think it is out of place there; maybe also after hearing other people's opinion. I will consider in any case any solution of yours as aimed to improve the quality. However, for what concerns me, I wouldn't feel like making an article or a section out of it --but if somebody will, he's welcome of course. My idea was to provide a particular example of envelope that can be treated by elementary inequalities. In the same spirit, for instance, that it is didactically useful the elementary proof of the implicit function theorem in the case F(x,y(x))=0 with a function F:R2 →R , even if the general proof, at the level of Banach spaces, uses the local inverse mapping theorem in a definitely clarifying way, &c.
So, there are, if I understand, two objections: 1. it's not based on classical methods of differential geometry 2. it is difficult. I tend to have a unitary view of mathematics, so I wouldn't consider your first objection as an insurmountable difficulty, that is, we can work a little on it, showing the connections between different methods and languages of maths. In maths, connections are important: this is etymologically a truth. However, after all, what are the methods of a discipline is not a completely definite matter, for they vary with different schools, times, places.
As to the other point, maybe you can just show me the obscure points. I tried to make it short, because it was just intended to be an example. Also, you can certainly improve the language.
In any case, maybe we can stuff it in the astroid article, as a quick proof --generalized with no additional price, of the characterization of an astroid as envelope of unit segments with endpoints on the axes. What do you think?
PS: you are right on that Hölder inequality: I didn't check the link, I meant this Hölder inequality, in R2 , that has nothing or little to do with general Lp spaces (no, indeed the link it was ok, it goes to Rn)
Well, I've posted the same message (more or less) on the talk page. I doubt that we'll get much feed back: the last new thread on that page, before mine, was more than a year and five months ago. Reading your last post, I get the feeling that you're over-complicating the issues I raised. The point is that your example doesn't fit into the page as it stands. If you add something to a page then you should modify the page so that it is capable of accepting the new information. E.g. you need to add sufficient new background and motivation so that whatever you add doesn't look out of place and doesn't just seem to appear out of the blue. It isn't just that your example doesn't use classical methods of differential geometry; it's that the example doesn't have any connection to anything else written on that page, before or after. For example, the article begins "In mathematics, an envelope of a family of manifolds (especially a family of curves) is a manifold that is tangent to each member of the family at some point." As far as I can tell, your Ts,t aren't even manifolds: they're manifolds with boundary. Anyone coming to look at the article is going to be looking for the envelope of families of smooth submanifolds, probably smooth curves in the plane. Adding your example will confuse people, and isn't really what anyone would be coming to the article to see. If the example is to stay then I think you need to add enough detail and motivation for it not to look so out of place. If you don't want to do that then I might suggest that you should think about removing it until a time that the article is at a stage so that it may be re-introduced. ~~ Dr Dec (Talk) ~~ 15:06, 10 September 2009 (UTC)[reply]
OK, good idea; and if we still don't get much feedback, we can decide something by ourselves. In the meantime, I'd be glad to have your impression of mathematical reader, about the readability of that example, regardless to its collocation. In particular, if in your opinion there is something to be explained in more detail. As to the laymen, I wouldn't worry for them more than enough -just because with no mathematical background not every mathematical article is accessible. I am more interested in the general mathematical readers (everybody: undergraduates, graduates, phd's, postdocs &c), that are the people that really look for this more technical information, as the RD/M experience shows.
So, as I mentioned in the article, the idea was making a classical example of an envelope of curves that in particular are boundaries of some sets. More precisely, an example of a case in which
boundary of a union of subdomains = envelope of the boundaries of the subdomains.
You can either see it as (i) an application of the notion of envelope to understanding a union of sets, or vice versa, (ii) the (classical) technique of treating an envelope of curves (more generally, of codimension one submanifolds) when they can be seen as boundary of some smooth subdomains.
How would you prefer to have the example explained, as (i) or (ii) ? I think (ii) is more natural here.
Is it clear to you what is the family of curves (the boundaries) in the example?
I understand that you are not familiar with the Hölder inequality in R2. Is the Cauchy-Schwarz inequality in R2 better known to you? In this case, do you think that doing the example in the particular case of α=1 would make it any easier?
Would you like to find a further explanation about why is the envelope of the family of segments?
Other remarks?
Thank you very much for letting me know your impression! pma.
I've just got back to the house. I want to add three alternative definitions to the Enevelope article. I think your point about "boundary of a union of subdomains = envelope of the boundaries of the subdomains" will come under the third definition. I'm sorry, I don't have much time now. I'll address each of your points tomorrow evening. Sorry! ~~ Dr Dec (Talk) ~~ 22:24, 10 September 2009 (UTC)[reply]
Done! Let me know what you think. I'm off to bed now. ~~ Dr Dec (Talk) ~~ 23:03, 10 September 2009 (UTC)[reply]
Ok, good. I have few time till the next week, but I'll go and check it. The picture you added certainly improves the comprehension of the example 2. As I told you, one can keep it there, or move in the astroid page, or even in the Hoelder inequality page, as an example of a geometric application. My aim was, mainly, to have it as a quick reference.
As to the case of "boundary of a union=envelope of boundaries", I think that, of course in a rigorous form, it may provide a natural class of envelopes, but note that it is not as general as to cover all cases of envelopes even for curves in R2, so it couldn't be used as a definition.
Would you mind giving me a reference which discusses the convergence of the Binomial expansion for (1+x)^r where x and r are complex numbers? thanks a lot! HowiAuckland (talk) 08:22, 3 October 2009 (UTC)[reply]
Hi, well I provided a self-contained proof in that section on binomial series because I did not have a better reference. I tried to make it as plain as possible. If you want a book reference, maybe a classic one on complex analysis is ok, like Titschmarsh's book. PS: Yes, sorry, it was mispelled; I answered in your TP.
Thanks for replying so fast. I am not familar with that book. Would you mind giving me the exact reference? i.e. title, full author name, year. Are you sure the spelling of his name is correct? I have searched in MathSciNet and found no articles by Titschmarch last name. HowiAuckland (talk) 08:28, 3 October 2009 (UTC)[reply]
Given that you claim to be an Italian, I am surprised by how you mix up the spellings Davis and Davies. This is a common problem in the UK since one is an Irish surname and one is Welsh. Quite why an "Italian" might be subject to the problem is beyond me. ~~Dr Dec (Talk)~~ 23:45, 5 October 2009 (UTC)[reply]
And then? --pma (talk) 00:13, 6 October 2009 (UTC)[reply]
I haven't really followed this fiasco but I do read the maths desk regularly. Just wanted to point out that the RD will quickly devolve into something like Yahoo! Answers if its good contributors retire every time they let their tempers get ahead of them. Zain Ebrahim (talk) 13:22, 7 October 2009 (UTC)[reply]
Hello PMajer, Shahab has smiled at you! Smiles promote WikiLove and hopefully this one has made your day better. Spread the WikiLove by smiling at someone else, whether it be someone you have had disagreements with in the past or a good friend. Go on, smile! Cheers, and happy editing! Smile at others by adding {{subst:Smile}} to their talk page with a friendly message.
Hi. Thank you for the detailed response.-Shahab (talk) 14:37, 30 November 2009 (UTC)[reply]
Hi pma. I noticed that you reverted some vandalism recently[3]. Thanks for that. When you come across a user that has vandalised a page then it's a good idea to leave a warning on the user's talk page. If the user continues to vandalise they will build up a warning history and then can be reported to WP:AIAV and they will be blocked. For an example of a warning history leading to a block please see here. A proper warning history is necessary for a casual vandal to be blocked. (Of course there are many exceptions to this rule, e.g. WP:3RR.) You can cut-and-paste warning templates onto the user's talk page. A list can be found here. Alternatively, you could activate Twinkle in the gadget tab of your preferences section. This is an automated way of dealing with vandalism. It will revert the vandalism and, with some input from you, will leave a warning on the user's talk page. I've left a warning on the user's talk page, and I'll keep an eye on their future edits. ~~ Dr Dec(Talk) ~~ 15:27, 15 December 2009 (UTC)[reply]
Hi, thanks for the info. It will be useful (in the present case I guess it was unintentional, but you are right, it's a good idea to leave a warning)
Thanks again for correcting my silly mistake at the reference desk. I knew that you had to be right, but I just could not see why (a bad day at the reference desk...). --PST 12:20, 19 December 2009 (UTC)[reply]
Don't worry, it happens to all of us... --pma (talk) 17:57, 19 December 2009 (UTC)[reply]
For no particular reason, except to say hello, and to say that your contributions to the project are appreciated:
Hello PMajer, Sławomir Biały has smiled at you! Smiles promote WikiLove and hopefully this one has made your day better. Spread the WikiLove by smiling at someone else, whether it be someone you have had disagreements with in the past or a good friend. Go on, smile! Cheers, and happy editing! Smile at others by adding {{subst:Smile}} to their talk page with a friendly message.
Hi Sławomir thank you I appreciate. Happy new year! --pma 19:28, 6 January 2010 (UTC)[reply]
I was surprised at your comment on rckrone's comments on the circle problem. "A pretty good description"? That one person would be that confused is not surprising; that a mathematician would then endorse the comment is surprising. Michael Hardy (talk) 04:59, 15 January 2010 (UTC)[reply]
Well, we all agree that the original question was unclear and ill-posed, though not silly. As I understood it, Rckrone's approach was to start from the simplest and more direct interpretation of the question (which is a good rule for the RD, I think): precisely, the set of points is given, the radius R is given: how should we describe the admissible positions of the center of the surrounding circle of radius R and possibly locate it? I found his description clear, and satisfactory as a first answer. In fact it is the best answer one can give to the original question ("where is the center?"): the center is somewhere in a certain region, and that's it. I also suggested that simple variation of the question in order to save the wanted conclusion (namely, the unicity). One may further investigate on a distribution of probability for the center as a random variable; however this means changing the question, and possibly adding assumptions. Anyway, I'm glad you left a message here, and I hope you didn't take my comment on that comment as an indirect criticism, or underestimation of your mathematical comments, which I have the highest opinion of. --pma 14:28, 15 January 2010 (UTC)[reply]
Also, since you entered this matter, as to your non-mathematical comments, specifically, comments on other people's posts at the RD/M, my opinion its that they often reflect an excessive rigidity from your side. It is possible that somebody uses a term in a less common acceptance, or even improperly, even if the idea is correct. Also, it is possible that somebody has his own iterpretation of a previous unclear statement of somebody else, that differs from yours. I think maybe you should not necessarily decide that he's wrong, totally ignorant, &c and start biting. pma 12:34, 17 January 2010 (UTC)[reply]
pma, would you be willing to help me a bit more on a reference desk question I asked a couple weeks ago? If you are willing, just go to the link as I have asked my question there. Thanks. StatisticsMan (talk) 15:47, 18 January 2010 (UTC)[reply]
Hi, sure, but would you precise better the question? --pma 18:02, 18 January 2010 (UTC)[reply]
Hi, I think I was precise. If you click on the link I put a few lines up, it goes to the ref desk archives of a question I asked a couple weeks ago. At the bottom of that, I added my understanding of what has been said so far and asked a question or two that I was not 100% on. Those are the questions I am talking about. Does that make sense? Thanks StatisticsMan (talk) 17:32, 19 January 2010 (UTC)[reply]
Let's define, for and for
For any we have, by the mean value theorem (see Rckrone bound)
We wish to show that the family is locally normally summable in the uniform norm, that is, for any there exists a neighborhood of ) such that
This implies that the double sum
converges uniformly to a continuous function on
Consider an open covering of by open sets of the form
for real numbers and Let be one of these.
It is convenient to bipartite the set of indices into the subsets:
Therefore, for any there are at most values of such that and in any case are among them.
Since for and we have
so
we can bound the sum on as follows:
On the other hand, for all either or In both cases, for any and
Note that for any one has so and the last inequality holds.
Thus
Remark. In the definition of there is no need of all the parameters a,b,c,d : you may better take just , and , so the open sets are a sequence and still cover the domain. But as I wrote it, it should be easier to check the inequalities. In fact you do not need precise bounds, and they are a bit annoying: it should have better made a more qualitative argument, but maybe this way it makes everything more concrete. --pma 10:04, 20 January 2010 (UTC)[reply]
Dear Sir, I found your reply to my question most unhelpful. I stated clearly that I wanted an explicit, worked example, and I gave a sample function to be worked with. Reading your reply makes me think that you have not read my original question. I wanted to see some εδ-proofs involving my sample function and its truncated Taylor polynomials. As I said: I know that my sample function is holomorphic and so equal to its power series, but I wanted it to be used as an example. Your reply is not dissimilar to other replies I have seen on that page. Many replies seem to attack the validity and rigour of the question instead of attacking the stated problem. This is most saddening. •• Fly by Night (talk) 21:55, 12 March 2010 (UTC)[reply]
Dear Fly by Night, I'm sorry that you were not satisfied with the answer. Note that I didn't, and don't have anything to object about the rigour of your question. I was just not sure about what you really wanted, because you presented two different situations. First, a generic function from C to C, etc. As you stated it, it was very natural to wonder whether you meant real differentiability. Second, the function exp(sin(z)). I tried to provide you with an answer covering all possible interpretations and stating the most relevant facts; this way you could then put a furthur question focusing better on what you really had in mind. Of course I'm not giving εδ-proofs before I'm sure of what a questioner want, otherwise I'm wasting my time and his time at the same time. The general idea is that after the first reply you may put a more precise question etc. After your reply, I understand that your interest was whether the Taylor polynomials of exp(sin(z)) converge uniformly on the complex plane, and whether they converge in some other senses. Is it this what you want? --pma 23:24, 12 March 2010 (UTC)[reply]
Dear Sir, I was clearly speaking about complex differentiability. My definition of used the notation . This, I believe, is standard notation for ordinary differentiation, i.e. d^kf/dz^k (instead of partial differentiation). Since I used z as my variable, and used ordinary differentiation notation I would have thought that it were clear that I was differentiating with respect to z, with respect to the complex variable. •• Fly by Night (talk) 16:09, 13 March 2010 (UTC)[reply]
As I told you your question sounded a bit strange to me, since you were speaking of a function defined on C, but only differentiable on a neighborhood of 0, without any continuity assumptions outside the neighborhood. The point was not complex vs real differentiability, but since you required it to be infinitely differentiable on the neighborhood, incidentally, it could be really possible to me that you even meant real differentiability (for in the complex sense one usually just speak of "differentiability", since it implies C∞). But what is still not clear to you, is that mine is definitely not a criticism about your questions. My concern is only understanding what people want, and try to give an answer. I think I answerred your question under all possible interpretation of it and I hope it will of help. --pma 00:00, 14 March 2010 (UTC)[reply]
...at the reference desk. Since we are both Italian mathematicians may I ask you where do you work? I'm in Tor Vergata university.--Pokipsy76 (talk) 16:23, 13 March 2010 (UTC)[reply]
facile: pma=Pietro Majer a Pisa!
Molto onorato :) I'm just a PhD student instead... --Pokipsy76 (talk) 19:27, 17 March 2010 (UTC)[reply]
Hi Pmajer. I hope you are well. I am glad you read my question. I'll try to describe the whole problem to you, in fact I will be glad if you can give me any general advice regarding the problem.
Consider the equation v+x+y-z = b (b is an even natural number). It is known that there exists a least natural number r(b) such that if {1,2...r(b)} is partitioned into 2 classes arbitrarily, at least one contains a solution to the given equation. My final goal is to find out a formula for r(b). To this end, I have written a computer program (using brute force) to estimate various values of r(b) for different b, and I use the bound r(b)<= b/2 as v=x=y=z=b/2 is a trivial solution. The problem is that the program starts taking an awful lot of time to estimate r(b) as b increases because the number of possible partitions increases as a power of 2.
Is there any general advice on how to approach this problem, any techniques etc you can suggest. I will really appreciate it.--Shahab (talk) 02:33, 29 June 2010 (UTC)[reply]
Thank you Henning, nice construction and very clear explanation. It also leaves interesting questions (e.g. what about not using GCH...). It seems worth add some information to the articles on vector spaces and duals. --pma 10:17, 18 October 2010 (UTC)[reply]
I managed to find a brute-force approach that works without GCH. More elementary, but less elegant. –Henning Makholm (talk) 05:38, 19 October 2010 (UTC)[reply]
Very good. So summarizing, if is infinite, and at least in the case the right inequality is an equality; all without using CH.--pma 08:47, 20 October 2010 (UTC)[reply]
Well, never happens. I'm not sure about the right inequality at all. A slight modification of my latter argument will prove , with equality when the scalar field is small enough. But I don't think we've excluded the possibility of for huge scalar fields. –Henning Makholm (talk) 09:59, 20 October 2010 (UTC)[reply]
In fact, for any set and any field there is an extension field such that . Namely, construct by adjoining independent (formal) transcendentals to . Then the set
Thanks, very interesting! I made a typo above, of course I meant and, yes, the right inequality should not be true for large fields. I will think about it as soon as I have moer free time. --pma 17:16, 21 October 2010 (UTC)[reply]
You are allusive, but elusive, in criticism of the article on Archimedes Palimpsest, particularly where it touches on Stomachion. It would surely help other users were you to present criticism in greater detail, preferably with references. —Preceding unsigned comment added by 133.31.18.68 (talk) 05:45, 13 March 2011 (UTC)[reply]
I think you mean this. You are right that I gave no details and references; my post is exactly an invitation/request to people in the field in order to fill the gap, and present the criticism in detail and with complete references, as you are saying. --pma 15:08, 14 March 2011 (UTC)[reply]
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Thank you for the notification (I didn't realize it was linking to a disambiguation page). Fixed now. --pma 00:13, 17 February 2012 (UTC)[reply]
You're right, my additional observations did follow from what was already there, but they provided a perspective from the point of view of α as opposed to x. They also provided a link to the subject of conditional convergence. I believe they served a useful purpose. Would you reconsider including them? Rickhev1 (talk) 14:27, 21 February 2012 (UTC)[reply]
Yes, your observation is indeed meaningful, so I have re-included. My concern was about not expanding the article too much, as happened to some other ones. Cheers! --pma 19:34, 26 February 2012 (UTC)[reply]
Thanks very much. I think you have done an excellent job on this article. Rickhev1 (talk) 15:09, 28 February 2012 (UTC)[reply]
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I think you oversimplified Polya's proof. The inequality
for all
is wrong: the best you can get in the right-hand side is e/n. This does not yield the sharp constant. To get the sharp constant, you need to apply the MA-MG inequality to akck, where ck are defined by
Best regards,
Sasha (talk) 15:37, 22 September 2012 (UTC)[reply]
Dear Sasha, actually I don't think that inequality is wrong (although it is a bit less obvious than the inequality with e/n). And I provided a simple derivation of it from a well-known inequality on the Stirling approximation. Therefore, can you please now
provide a counterexample to that inequality (i.e. a particular value of n for which it does not hold); or
show where is the flaw in the argument I mentioned; or
just explain what is unclear to you, or what makes you suspicious about the validity of that inequality (note that I was not referring to any Polya's proof).
Thank you. I will in any case appreciate any contribution to improve that article. --pma 19:43, 22 September 2012 (UTC)[reply]
Dear P.,
I am sorry, it seems that the inequality is correct (your explanation works for n=2,3,..., and for n=1 it is directly verified). I was in a hurry, my apologies.
sorry for bothering you again. Could you please clarify the phrase "...optimal -otherwise it would be attained by some non-negative and non-identically zero convergent series."? I agree that the constant is optimal, and that it follows from the proof, but I do not understand the argument that you propose.
Best regards, Sasha (talk) 20:35, 22 September 2012 (UTC)[reply]
Dear Sasha,
Yes, I think you are right, that sentence is obscure. The fact that the constant e is optimal should be just a consequence of the fact that it can be approached arbitrarily by a sequence of the form (1,1/2,1/3,...,1/N,0,0,0,..) taking N large enough, right? Maybe I had something in mind, but at the moment I think it is better to remove it. --pma 22:36, 22 September 2012 (UTC)[reply]
Dear P.,
thanks again! I agree with both things you wrote.
I have looked again at Polya's paper (which is a didactic exposition of his paper from the 20's), it seems that he chose the coefficients in a more complicated way in order not to use very precise asymptotics for n!. Asymptotically his c_k behave like k, so the difference between the two choices is not very dramatic. Which argument is better is of course a matter of taste.
Hi,
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Thanks for your update on "Function" , benifit a lot :) Xfcjscn (talk) 08:00, 12 January 2018 (UTC)[reply]
Thank you Xfcjscn! Receiving a wiki barnstar makes me feel proud but also a bit guilty, since I've not been here for quite a long time.
Still I don't get what update you mean about Function :) --pma 11:40, 12 January 2018 (UTC)[reply]
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You have made a bold edit (introducing, for unclear reasons, yet another proof of the same result). It has been reverted. You can now seek to convince some other editors to agree with you, by using the talk page. --JBL (talk) 13:09, 25 July 2019 (UTC)[reply]
P.S. The condescending offer to explain the math is a nice touch (hint: I have a PhD, and correctness is not the problem with your poor addition). Maybe instead you should worry about understanding Wikipedia's policies. --JBL (talk) 13:11, 25 July 2019 (UTC)[reply]
Hi, I was writing on the meanwhile on your page. I did not mean to offend you offering to explain the maths, my apologies. Explaining to each other is what we usually do here. Just saying the point is not about rearranging as you were saying: is it clear? pma 13:20, 25 July 2019 (UTC)[reply]
Let me add that your tone is not quite appropriate. As to policies: before deleting another user's edit, it should be good to explain one's doubts about it to whom made it, and in case ask for explanations. Anyway, I'm still waiting for a more clear and constructive explanation of your reasons. Bye pma 13:35, 25 July 2019 (UTC)[reply]
I agree that the tone was hostile, I apologize. (I must point out that I explained my reasons for reverting initially, and your first response was completely non-substantive, leaving nothing to discuss. So I think a certain amount of frustration was understandable.) I appreciate that you have begun the discussion and I will add my comments there later. --JBL (talk) 13:08, 26 July 2019 (UTC)[reply]
I see that you changed back to an older version on 2019-07-12, thereby undoing a number of changes I had made. I want to mention some possible problems in the current version.
1. In proving (iii) you state that if |x| = 1 and Re(alpha) <= -1, then \left|{\alpha \choose k}\; x^k \right|
\geq 1. While this is true, I don't see that it is obvious. Instead of proving this, it would be simpler to observe that the series diverges since its terms are bounded away from zero by formula (5).
2. In your proof of (iv) it is true that formula (5) implies divergence if Re(alpha) < 0, but I don't see that it does if Re(alpha) = 0 but alpha neq 0.
3. In your proof of "Elementary bounds on the coefficients" you show that \left|{\alpha \choose k} \right|\leq\frac {M}{k^{1+\mathrm{Re}\,\alpha}},\qquad\forall k\geq1. I don't believe this establishes formula (5), since you also need that \left|{\alpha \choose k} \right| {k^{1+\mathrm{Re}\,\alpha}} is bounded away from zero; i.e. is >= m.
If you agree that these are indeed gaps in the proof, please let me know whether you want me to make the necessary changes or would prefer to do them yourself.Rickhev1 (talk) 22:01, 14 September 2019 (UTC)[reply]
Dear Rickhev1, thank you for your comments, which I will consider carefully.
3. Of course you are correct! When I wrote it (Revision as of 12:27, 7 September 2009, wow, 10 years ago) the statement to be proved was just
Later on, it seemed it was better to refer to a more precise inequality, that is current formula (5). So for the moment I'd suggest to make a more suitable introductory sentence of the section "Elementary bounds" (like "here we just show the upper bound in (5)") and see if it is worth completing the proof with the other inequality (I'd say: yes, unless the computation is not simpler than deriving the above Gauss formula!). pma 08:34, 15 September 2019 (UTC)[reply]
1. Let me first recall the proof of the claim. We write the binomial coefficient by its formula, as a product of terms :
If , then all factors in the RHS have also a real part which is , so they all have modulus greater than or equal to 1, and their product as well. So, the divergence in this case (Reα≤-1 and |x|≥1) has an explanation which is really simpler than the lower bound of (5), and also more apparent (an infinite sum of terms of modulus not less than 1). So I would keep this explanation; we may then explain further the reason of in more or less detail, and also mention that the divergence also follows by (5). (I would be careful not to expand too much the proofs, however, otherwise a reader may miss the key points) pma 09:13, 15 September 2019 (UTC)[reply]
2. Correct! Indeed there is a typo: the last sentence should read "whence (iv) follows using again." So the summation formula for the n-th partial sum Sn for x=-1 together with the asymptotics (4) gives and everything reduces to the elementary conditions of convergence of the sequence n-α for n→+∞. pma 10:47, 15 September 2019 (UTC). The possible objection is that this is not covered by both "Elementary Bounds".[reply]
I'll do the minor changes to mend incorrect/incomplete/ambiguous sentences, as per your remarks; then we may decide to expand further. Thank you for now! pma 11:18, 15 September 2019 (UTC)[reply]
I've also just added a few more lines of explanation about the case you mentioned in point 2.pma 21:17, 15 September 2019 (UTC)[reply]
I'll defer to you on how to proceed. My earlier version did provide a self-contained proof of the whole shebang.
1. To keep things simple, I suggest in proving the part of (iii) for |x| = 1 & Re(alpha) <= -1 that you just use form. (5) to show that the terms |(alpha choose k) x^k| >= m. Thus, they can't converge to 0, whence the series must diverge.
2. Your new proof of (iv) is valid, but it depends on the gamma function. Your statement following form. (5) that it is sufficient for our needs is no longer true.
3. I don't see the point of establishing just the upper bound for |(alpha choose k)|. The way things are now, you need the Gamma function, not just form. (5). And even if you use my method or some other to prove (iv) using only form. (5), it's very unsatisfying to establish the upper bound but not the lower one for form. (5). So I would favor either using Gamma and eliminating the section on "Elementary bounds on the coefficients" or using a proof depending only on form. (5) and either giving a full proof of form. (5) in "Elementary bounds on the coefficients" or eliminating that section. Rickhev1 (talk) 23:35, 15 September 2019 (UTC)[reply]
Thank you for these comments, which I think go to the point. So I think here we have a typical dilemma concerning proofs in maths wiki articles, the choice being between:
(A) The technological approach, that leans on some more advanced facts --here, the asymptotics (4) via the Gamma function. Pro: it's simpler, shorter, easier to remember, as everything comes from formula (4), which is of independent interest. It provides a nice application of the theory of Gamma function, and it's especially suitable for whom already knows it, but, also, a good motivation to learn it, for who doesn't. Contra: a less prepared reader may be forced to believe (4), and miss a self-contained proof, and maybe be led to believe that things are more advanced than they are.
(B) An elementary and self-contained treatment. Pro: really elementary, self-contained, available even to a (well motivated) freshman. It may be of interest also for the advanced reader, who usually also likes to know if a result really requires more or less deep facts, or is a triviality (for example, the mentioned "only if" part of (iii) is really elementary and does not require (4)). Contra: longer to follow, difficult to memorize. Besides, to make it complete, we need more than the double inequality (5), but in fact some equivalent form of (4). This will make the proof a bit too long for an encyclopedia, and, although elementary, I fear too long for the freshman.
In general, the choice depends on the target of the article, which however is usually heterogeneous, and should be evaluated case by case. One could also keep both approaches (that was my initial idea, when I wrote the section on "Elementary bounds", but, as said, completing it would make it a bit too long). In this case, I think it is important to keep (A) and (B) separated, otherwise a less careful reader may remain with the impression that the proof is both non-elementary like (A) and complicated like (B).
Conclusion: considering Pro-and-Con, having to choose in this case I'd go for (A), so I'll cut the section on "Elementary bounds" as you say. In any case we can resume it, together with your earlier version (btw, my apologies for the bold edit, which I hope I have by now explained). But then I think, it would be nice to put it in form of a self-contained proof of formula (4), maybe with a C in place of , if we don't want to mention the Gamma function). Let me know your opinion. pma 16:28, 16 September 2019 (UTC)[reply]
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Thank you, my fault, fixedpma 13:30, 24 August 2020 (UTC)[reply]
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Please see MOS:REFPUNCT: All ref tags should immediately follow the text to which the footnote applies, with no intervening space. Refs are placed after adjacent punctuation, not before. --JBL (talk) 12:48, 27 September 2021 (UTC)[reply]
Thank you, fixed
Also, concerning this edit: can you clarify whether you are correcting something by reference to the source or based on your own personal interpretation / research? --JBL (talk) 12:50, 27 September 2021 (UTC)[reply]
Hi JBL. Yes, from the source: the one quoted in the article. (And, of course, because the result is obvious to me). Bye, pma 15:57, 27 September 2021 (UTC)[reply]
Also, I tried to give a clearer statement for the other quoted result (Wilczyński's), that seemed a bit obscure as it was written. pma 16:08, 27 September 2021 (UTC)[reply]
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Hello, can you add a reference to your paragraph or describe a bit more the proof of the inequality you use please (Forgive me if i do not use well the talk page).
--FRCNico (talk) 15:21, 16 January 2022 (UTC)[reply]
Hello FRCNico, I've added there a couple of lines to justify that inequality. Let me know if it looks sufficiently clear now. Since these bounds are quite standard, and can be directly checked by anybody, I didn't bother to look for a reference.pma 18:59, 16 January 2022 (UTC)[reply]
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Hey, I noticed you added the power series of [W(x)]^r (the Lambert W function), with r an integer. It says there that it is obtained using the Lagrange inversion formula. I don't think i fully understand how, and i also didn't find any reference for that statement.
The coefficients of the series seems to be just right, and I would love to understand how to obtain them myself. Thanks alot! BarakBud (talk) 14:02, 30 November 2022 (UTC)[reply]
Dear BarakBud, the expansion of with real or complex exponent is just an extension of the Lagrange inversion formula to series with real or complex exponents, in the simplest case, namely series whose exponents are an arithmetic sequence of real or complex numbers (an instance of Hahn_series). At the time I also put a remark on this generalisation on the section on the Lagrange formula here Formal_power_series#The_Lagrange_inversion_formula: just follow the formal proof given there for the standard case, and adapt it to this case.
You may also like to see a similar application of this Lagrange inversion formula for inverting a "polynomial" with non-integer exponents , which is a just a bit more challenging exercise (In case I'll add the details there; for the moment I won't spoil the pleasure of trying the computation :) but do not hesitate asking for details, because clarifying things would be useful for all users --me included).
This formulation enabled me to calculate the power series of some inverse function, but not the series of it raised to an arbitrary power (at least not in a straightforward way).
Following you reply, I understood that other formulations of the theorem exist, so I found this article, which puts everything together in a very clear and didactic manner:
Anyway, you helped me and my colleagues alot with a paper we are currently working on, and we would like to give you a credit for that. Please contact me if you are intersted: barak.budnick@mail.huji.ac.il
I'm glad I happened to help to clarify things. Yes, I know that Ira Gessel's paper, it's very informative and very well written. Also, the book by Flajolet and Sedgewick (Analytic Combinatorics, freely available on the authors' page) has a beautiful part on formal series, and so has Stanley's book (Enumerative combinatorics). The formal series approach is indeed more elementary, simpler, and more clear than the complex analysis formulation. In fact, there is a kind of general abstraction procedure that I find extremely intriguing. When we have settled a mathematical theory on some mathematical objects (e.g. holomorphic functions) we make a calculus out of it, and the objects become just symbols. So for instance the "formal residue" is just the coefficient of order -1 of a formal Laurent series; we retain the rules and forget the origin as path integral. The abstraction also works in the opposite direction: when we have a calculus, we start treating symbols as live mathematical objects! For instance, people used to make computations with expressions like or (that they used to call "forms", i.e. formal expressions of some kind) then somebody had the idea of dropping the summation symbols and introducing a double sum convention. Then somebody also dropped the indices, and wrote or , and symbols became "real" objects, vector spaces and linear maps. Math is fun! pma 19:58, 2 December 2022 (UTC)[reply]
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