Misplaced Pages

Revision as of 22:16, 7 January 2012

Archive of old talk.

Convex function

I'm glad that you showed understanding that a convex function may be defined on a higher dimensional set. However, you wrongly removed the epigraphic definition, and I hope that you would restore it soon.

Also, imho, it would be better to remove the primitive phrasings "concave up", etc. to the end of the introduction, perhaps in their own paragraph. The reader should meet the definition immediately, and not have to jump over many deprecated alternative definitions.

Thanks,  Kiefer.Wolfowitz  (Discussion) 22:18, 24 April 2011 (UTC)

Number of involutions in a group: continued...

Hi Sławomir,

I was wondering if you'd had any luck thinking of an infinite group with a positive-even number of involutions. The question on the reference desk got a bit off track (as I'm sure you noticed). Any infinite group I construct, in which I try to produce a finite number of involutions, involves taking some product of a torsion-free infinite group with a finite group (and the finite group is where the involutions come from). But, as we know, such a finite group cannot have a positive-even number of involutions. I'm not too keen to go and mess about with presentations, and given the level of the OP's first question (assuming it's from a homework sheet), it seems as if we shouldn't have to. Was just wondering if you'd thought about it any more! Thanks, Icthyos (talk) 09:54, 26 April 2011 (UTC)

Sorry, I haven't been thinking much about it. What limited thoughts I have had are not encouraging. Sławomir Biały (talk) 13:06, 26 April 2011 (UTC)

Undoing and reverting

Please try and be reasonable.

Moving just your comment is not following the continuity and doing it twice is provocative and causing ill-will. I realise that you have a history of that, but I expect more of someone who wants to be taken seriously. Chaosdruid (talk) 17:31, 1 May 2011 (UTC)

Someone removed my comment in flagrant violation of the talk page guidelines. I was putting it back where is was originally. Unfortunately, the first edit put it in the wrong place, so I had to edit a second time to get it in the right place. Sorry for the misunderstanding. There's no need make this personal, by the way. Sławomir Biały (talk) 17:43, 1 May 2011 (UTC)

This is not an RfC and I have not edited your post. I simply moved it to the correct section, this is not an RfC it is a move request. I am not edit warring as it was you that moved it back twice. It certainly was not personal until you made that ridiculous comment about me editing your post, something which I would never do.

The point is that there is a discussion section at the bottom, that is where the discussion should go and not at the top. A discussion is a continuing series of responses which is what it was; you responded to my vote, and in continuing the discussion with my response (see, that is a discussion) I moved it to the discussion section. Why you would consider that there was anything wrong with that I cannot imagine. Provoking me with comments on my talk page, saying that I had edited someone elses post on an article talk page, was both unecessary and in very bad taste. I have deleted it as I cannot imagine that you wish to continue saying that I edited your post when in fact I did not :¬) Chaosdruid (talk) 18:04, 1 May 2011 (UTC)

In your first post on my talk page directly above, you say that I have a history of provocation and causing ill-will. I do consider this a personal slight. It's hard to imagine in what way that is a civil thing to write on someone's talk page. At any rate, moving a post out of its original context is editing the post. Sławomir Biały (talk) 18:15, 1 May 2011 (UTC)

Deletion of Fractal Space Map

Hi Slawomir, Just back from holiday and trying to respond to a contact's request to view the Misplaced Pages article on Fractal Space Maps that I authored last month. Unfortunately you seem to have deleted it. Could you restore it please?

To expand: your stated reason for the deletion was that "The only source is a patent, and that's also the only thing that shows up in a Google Scholar search. Without reliable sources that are independent of the subject, the subject of the article is not presumed to be sufficiently notable".

To respond to that reason, first, the references to the 2 (not 1) patents included in the article can be used by someone familiar with the way patents work to trace the fact that Fractal Space Maps are the subject of 18 granted patents in several countries including USA, Japan, Germany, France and UK (ie the largest economies in the world other than China). I did not, however, want to bore readers by listing every single reference.

Secondly, the criteria in all these countries for the granting of a patent is that the idea expressed is: novel, inventive and industrial applicable. Novel is straight-forward: nobody has thought of it before. Inventive is more subtle: the novelty is not merely the result of a trivial tweak to a pre-existing idea, but is instead the result of a genuine leap in thinking. Industrially applicable is also fairly straight-forward, but often overlooked: essentially the idea must be genuinely useful. In short, therefore, highly trained and experienced patent examiners in the leading developed economies all agree that Fractal Space Maps are new, interesting and useful.

If this is not sufficient to make the subject "notable" from Misplaced Pages users' point of view, I could look out a bunch of coverage the concept has received from the Financial Times, Gartner, Waters and other highly respected publications, though I would prefer the article to remain focused on the maths and human/computer interaction aspects rather than the potential commercial applications which these articles tend to focus. —Preceding unsigned comment added by Gervasecb (talk • contribs) 10:53, 4 May 2011 (UTC)

Hi Slawomir, Thank you for the quick response. Misplaced Pages:DRV says that the deletion review should only be used in cases "where someone is unable to resolve the issue in discussion with the administrator (or other editor) in question". Are you the editor in question? If not could you tell me how to find out who is? If possible, I would like to have the article undeleted, so I can add in some references to (dedicated in depth) articles and awards from leading industry organizations concerning the technique. Ironically I left these out in the first draft because I did not want the article to seem biased or commercially focused! By the way, the process for having a patent granted is much more rigorous than you seem to imagine:) —Preceding unsigned comment added by Gervasecb (talk • contribs) 15:01, 4 May 2011 (UTC)

Again thank you for the quick response Slawomir, indeed the article is a little difficult to pigeon-hole. Certainly Fractal Maps have significant potential commercial applications, and I am happy to cite plenty of references to back up this statement. However, the comparison of their characteristics to other known deterministic fractals such as the Sierpinski Triangle which somewhat counter-intuitively has a higher Hausdorff dimension than a Fractal Map modulo 3, and Pascal Triangle series whose Hausdorff dimension is, unlike the Fractal Map series, unbounded as the modulo tends to infinity, throws some interesting light on the validity of using Hausdorff dimensions in the fundamental definition of a fractal. In fact I have an open invitation to take up a post-graduate position at my old Oxford University college to expand on this line of thinking, but don't really think a Misplaced Pages article is the right place to go into too much detail - it's an encyclopaedia not a mathematical journal! Regarding patents, they have to go through an extremely rigorous and objective process in order to be granted, far more rigorous than most, probably all, academic journals' publication process, so I would be interested in encouraging Misplaced Pages to think through the appropriate status to accord them as citations. Finally, however, and most importantly, can you please arrange for the article to be undeleted? Recreating it from scratch will be a significant waste of my time. Gervasecb (talk) 10:55, 5 May 2011 (UTC)

MaxTutor

FYI -- On the discussion at WikiProject Mathematics I edited your remark by adding "" after "it". I had added a reply to a comment just above yours which caused ambiguity in your pronoun.

CRGreathouse (t | c) 19:53, 4 May 2011 (UTC)

Misplaced Pages talk:WikiProject Mathematics/Straw poll regarding lists of mathematics articles

In light of your participation in the discussion(s) regarding the treatment of disambiguation pages on the "Lists of mathematics articles" pages, please indicate your preference in the straw poll at Misplaced Pages talk:WikiProject Mathematics/Straw poll regarding lists of mathematics articles. Cheers! bd2412 T 18:42, 23 May 2011 (UTC)

Just a reminder, the straw poll is set to close in about three hours. Cheers again! bd2412 T 02:58, 2 June 2011 (UTC)

A light-hearted limerick.

We are given a basis in E And a new basis we want to be, The inverse transpose Is what we propose Is a basis of it's dual space, we see! -Travis Maron — Preceding unsigned comment added by NOrbeck (talk • contribs) 06:00, 1 June 2011 (UTC)

That's true, if by a "basis" you mean an invertible linear mapping ${\displaystyle B:\mathbf {R} ^{n}\to V}$. The transpose of B, ${\displaystyle B^{T}:V^{*}\to (\mathbb {R} ^{n})^{*}}$ is then defined by duality by

${\displaystyle B^{T}\alpha =\alpha \circ B}$

for all α in ${\displaystyle V^{*}}$. And then the inverse transpose ${\displaystyle (B^{T})^{-1}:V^{*}\to (\mathbb {R} ^{n})^{*}}$ gives the dual basis (once ${\displaystyle (\mathbf {R} ^{n})^{*}}$ is naturally identified with its dual). But it isn't the naive matrix transpose that I sense you had meant here (the sense of the word "tranpose" would need to be clarified.) At any rate, in the section under consideration, it's not the transpose but the adjoint that is relevant (since both the basis and reciprocal basis are elements of the same space ${\displaystyle V}$). This is defined by

${\displaystyle (B^{*}v)\cdot w=\langle v,Bw\rangle }$

for all ${\displaystyle v\in V,w\in \mathbb {R} ^{n}}$, so that ${\displaystyle (B^{*})^{-1}:\mathbb {R} ^{n}\to V}$ is the dual basis.

This approach seems to be largely of theoretical importance, since it is ultimately just a restatement of the definition of dual basis or reciprocal basis. To calculate the reciprocal basis directly, usually you would use the inverse of the Gram matrix as the change of basis matrix. Sławomir Biały (talk) 11:03, 1 June 2011 (UTC)

Semi-retired.

I hadn't been active in a while and noticed some activity on your talk page. When I stopped by to see what you were up to I noticed you had semi-retired. What a loss to WP. From our brief interactions you earned quite a bit of respect, for your editorial skills/mathematical breadth/mathematical depth. I sincerely hope the decision to is due to the wonderful mathematical papers I'm sure your writing and not because of any wiki-drama. If our paths don't cross again, I thought I would take this moment to wish you all the best. Thenub314 (talk) 04:25, 2 June 2011 (UTC)

Thanks for the sentiments. It's a little bit of being busy in real life, but the bigger issue is that I don't find editing Misplaced Pages as personally rewarding as I once did. Best, Sławomir Biały (talk) 23:37, 2 June 2011 (UTC)

Alternative expression for the Taylor series

Hi Sławomir,

I believe that an alternative expression for the Taylor series is

${\displaystyle \sum _{m=0}^{\infty }{\frac {1}{m!}}\ {\biggl (}\sum _{n=0}^{\infty }a^{n}{\frac {f^{(n+m)}(a)}{n!}}\ (-1)^{n}{\biggr )}x^{m}}$

which provides the polynomial in x in ascending powers of x. Might this be incorporated in the Taylor series page? Mhallwiki (talk) 20:31, 19 July 2011 (UTC)

Issues of convergence of the inner series aside, I would no longer call this expression the Taylor series. If f were analytic, the inner series converges to${\displaystyle f^{(m)}(0)}$, so your formula simplifies to give the Maclaurin series, which is already covered. Anyway, without sources this really shouldn't be added to the article. Sławomir Biały (talk) 10:24, 20 July 2011 (UTC)

Thank you for your reply. Perhaps I'm missing something but I'm unconvinced that the inner series converges to ${\displaystyle f^{(n)}(0)}$ if f is analytic. Mhallwiki (talk) 00:25, 21 July 2011 (UTC)

Take the Taylor series for ${\displaystyle f^{(m)}(x+a)}$ about ${\displaystyle x=0}$ and evaluate it at ${\displaystyle x=-a}$. Sławomir Biały (talk) 11:26, 21 July 2011 (UTC)

I agree that in the specific case of evaluation at ${\displaystyle x=-a}$ the inner series I indicated simplifies to the Maclaurin series for ${\displaystyle f^{(m)}}$. For ${\displaystyle x}$ ≠${\displaystyle -a}$, however, it does not. Would the complete formula then be a representation of the Taylor series for ƒ(x)? Mhallwiki (talk) 15:11, 21 July 2011 (UTC)

This shows that the inner series converges to ${\displaystyle f^{(m)}(0)}$. Sławomir Biały (talk) 22:25, 21 July 2011 (UTC)

I agree when ${\displaystyle x=-a}$. Mhallwiki (talk) 23:15, 21 July 2011 (UTC)

But the inner series doesn't involve x! Here's another explanation. Consider the Taylor series expansion of ${\displaystyle f^{(m)}(t)}$ about ${\displaystyle t=a}$:

${\displaystyle f^{(m)}(t)=\sum _{n=0}^{\infty }{\frac {f^{m+n}(a)}{n!}}(t-a)^{n}.}$

Evaluating both sides of this identity at ${\displaystyle t=0}$ gives

${\displaystyle f^{(m)}(0)=\sum _{n=0}^{\infty }{\frac {f^{m+n}(a)}{n!}}a^{n}(-1)^{n}.}$

This says that, independently of x, the inner series in your formula is just equal to ${\displaystyle f^{(m)}(0)}$. Sławomir Biały (talk) 23:27, 21 July 2011 (UTC)

The independance of x has really helped me to understand and agree with your arguments. Thank you very much for taking the time to expalin the matter. Mhallwiki (talk) 14:31, 22 July 2011 (UTC)

A barnstar for you!

	The Citation Barnstar
	hi Qoscsynth (talk) 03:44, 22 July 2011 (UTC)

Why did you revert this?

Why this? I just want to make a note of how to tell when the parallelogram's area is positive and when it is negative, and I thought I had the rule, from a cited source. Wnt (talk) 21:51, 26 July 2011 (UTC)

It's wrong. The wedge product is neither positive nor negative (it's a "vector" quantity), and counterclockwise only makes sense in two dimensions (and only then when there is an established orientation). Sławomir Biały (talk) 00:01, 27 July 2011 (UTC)

It's a vector? If so... which way does it point? I thought this source suggested it was actually a scalar. Though looking at the other page from the same site it looks backward about which sign is which - somewhere there must be a clear source about it... anyway, if the wedge product is a vector, then this article has a long, long way to go before being even slightly useful to anyone, because I don't see anything in it that is going to get me a vector value. Wnt (talk) 06:07, 27 July 2011 (UTC)

It lives in an abstract vector space. This is discussed in the article. Sławomir Biały (talk) 10:13, 27 July 2011 (UTC)

The source you cited cites Marsden and Tromba as its source. For the record, I have taught undergraduate courses in vector analysis using this textbook. It does not define the wedge product as a scalar (and indeed only confines attention to the wedge product of differential forms). Their approach is strictly formal: no attempt is made to give a geometrical context for the wedge product. Sławomir Biały (talk) 11:23, 27 July 2011 (UTC)

Bourbaki

So I picked up a copy of Bourbaki ch. 1-3 a little while ago. Nowhere near the tensor stuff, but I'm enjoying it a lot, so I wanted to thank you for recommending it. The symbol-agnostic language they use for algebraic structures reminded me of how polynomials' properties (zeros, inflection points, whatever you want to look at) determine them in full. Is it the same for algebraic structures? Do the properties of a law of composition comprehensively classify structures? Does the underlying set matter? ᛭ LokiClock (talk) 03:18, 27 July 2011 (UTC)

Cool. I found their formalist approach to be most gratifying when I was first learning the subject. Bourbaki is not afraid to explain things carefully and clearly from first principles. Many algebraic structures can be completely classified by known invariants (like polynomials are known by their degree, intercepts, maxima, inflection points, and so on). This is one of the most important general problems in algebra. Sławomir Biały (talk) 11:05, 27 July 2011 (UTC)

Do you know what concepts would address how/why invariants induce shapes? ᛭ LokiClock (talk) 05:44, 29 July 2011 (UTC)

I don't really have a complete answer to this. The overall philosophy is that of representation theory: how do we represent an algebraic structure as something geometrical. This is a really huge area of contemporary research. Somewhat mysteriously, many seemingly different algebraic structures in representation theory can ultimately be reduced to the study of Coxeter groups, which are groups generated by reflections in Euclidean space. The most important invariants can be classified by graphs called Dynkin diagrams. Each Dynkin diagram represents a lattice in Euclidean space, and the Coxeter group is generated by the reflections in the chambers of the lattice.

There's a beautiful book "Indra's Pearls" by David Mumford (and some other folks) that discusses some other aspects of the "geometry of algebra", notably connections to hyperbolic geometry. It's aimed at an audience of non-mathematicians, but unlike so many books like that it manages to be very informative (even to a mathematical audience). Sławomir Biały (talk) 14:05, 29 July 2011 (UTC)

I'll chew on that a while. I'm used to interpreting Coxeter-Dynkin diagrams as a sort of encoding for ND polytopes. Since I just now started learning about group representations, this will be a big shift. This lattice & reflection perception of geometry is unlike any obvious empirical description of the same objects. ᛭ LokiClock (talk) 08:24, 31 July 2011 (UTC)

Quadratic approximation image for e^x

It looks like you've done a lot of really great work on the Taylor's theorem article. It seems like there's an error in one of the figures ( though). The red line doesn't appear to actually be the graph of 1+x+x^2/2. The figure shows 1+x+x^2/2 as about 0.8 at x=-.5, when the value should be 0.625. Similarly, the value at x=1 should be 2.5, but it's displayed as 3.

Am I misunderstanding something about the diagram? If not, do you still have the Mathematica source files?

-- Creidieki 17:30, 29 July 2011 (UTC)

Looks like the red and blue are switched. I'll fix it when I get the chance. Sławomir Biały (talk) 01:06, 30 July 2011 (UTC)

Recent prods related to User:Marshallsumter

If you coming here to tell me that you have removed a recent WP:PROD of mine relating to User:Marshallsumter, I would like to ask that you please familiarize yourself with the consensus involving this editor's articles, and the many issues involved: see ANI thread. That said, the last thing I want to be is overzealous when it comes to deletion of content. If you feel strongly that one of these articles should not be deleted, then by all means remove the prod notice. I would appreciate it if you could try to clean up the article in question (bearing in mind the many potential issues, most especially copyright violations). Failing that, please notify me and an appropriate Wikiproject. There are simply too many effected articles for Misplaced Pages volunteers to vet all of them, but if a few are worth salvaging, then probably someone can be found who is willing to work on them. Sławomir Biały (talk) 14:37, 11 September 2011 (UTC)

Time for a ban of Marshallsumter?

I noticed from the ANI thread that Marshallsumter could presumably return once his "research project" is finished. However, looking at the articles you prodded for copyvio reasons, I'm thinking it's time to seal the vault--unless he can already be considered banned. What do you think?

It's hard to gauge that from the ANI thread at the moment. There's a poor signal-to-noise ratio. The discussion about a community ban there got immediately sidetracked by whether or not an absolute deletion policy is preferable to dealing with the articles on a per case basis. I emphatically support a ban, subject to the version of the Misplaced Pages:Standard offer that I tried to articulate at ANI. Hopefully more community members will continue to comment on this question. Sławomir Biały (talk) 22:45, 11 September 2011 (UTC)

Well, it looks like all of his articles got spiked. And after reading that Wikiversity page of his, it's more than warranted. Good grief, there's all kinds of ethical problems here--not to mention the fact that what he's doing is a quantum leap away from what Misplaced Pages is. Blueboy96 12:40, 12 September 2011 (UTC)

In Response To The AfD "irrelevant comment"

"Sigh. Do some content work instead of harassing serious editors."

This is the irrelevant comment.Curb Chain (talk) 16:01, 8 October 2011 (UTC)

To put this comment into context... I (User:Nageh) am the author of the quoted comment, which I directed towards User:Curb Chain in response to a frustrating and fruitless deletion discussion initiated by him. Nageh (talk) 19:02, 9 October 2011 (UTC)

A barnstar for you!

	The Barnstar of Diplomacy
	This is to thank you for saving the little old rhyme on Sieve of Eratosthenes from deletion, and for resolving a long and tiring even if seemingly petty conflict through a nice and inventive solution! WillNess (talk) 17:20, 11 October 2011 (UTC)

I'd like to add to this. You wrote to me once with kind words, seeking advice. Now I am impressed by your contributions, both at the Sieve and the ongoing RfC/U. It seems to me that you have "got it", and that Misplaced Pages is lucky indeed to have you as a contributor. All the best, Geometry guy 20:50, 11 October 2011 (UTC)

Thanks guys! I'm touched. Sławomir Biały (talk) 10:43, 12 October 2011 (UTC)

If you have at all tolerance for verbosely formulated vague ideas, I tried to express some of mine in this post as pertaining to the WP edit process (in parts :) ). Perhaps you could take a look. Your opinion will be much appreciated. :) WillNess (talk) 11:57, 12 October 2011 (UTC)

Infinitesimal differential

Hey Sławomir Biały,
I wanted to discuss about this. You said my version is confusing and potentially misleading, but I think the same about the current version. I come from the non-standard world and I'm really disturbed by the word "of course" in the sentence. It is of course not a real number (I guess we agree on that), but if you come from a (mathematical rigorous!) "standard" world, the word "infinitely small" is not defined and there it can't be infinitely small. But the current version indicates that it is "of course not infinitely small", which doesn't make sense in a standard framework as it's not defined there. Imho this prohibits to see it as a non-standard infinitesimal (which on the contrary is defined). I tried to formulate it in different ways, but I couldn't come up with anything better than what I posted. I would really appreciate if you have a better way of expressing what I just tried to say here (and I hope it made sense). --EoD (talk) 17:59, 10 November 2011 (UTC)

Thanks for the explanation. That makes sense. I'm reluctant to include wording that suggests that dy/dx could be infinitely small, or that Leibniz would have regarded it as such, because that introduces a needless uncertainty at this stage. This article will be read mostly by students of the "standard" calculus, who are at least somewhat familiar with infinitesimals as a pseudo-mathematical metaphor, but are on solid mathematical footing where real numbers are concerned. I have removed the term "of course". Although this probably doesn't completely satisfy your objection, I couldn't think of any other way that we could both be completely happy with it sort of a more extensive rewrite. Sławomir Biały (talk) 11:35, 11 November 2011 (UTC)

a simple question

Hey Sławomir !!!

I see you are talking about (recalling the infamous history of Boubaker polynomials) !

What do you mean by infamous history?

Thank you for answer.--Nourja (talk) 12:19, 26 November 2011 (UTC)

delta function (distribution)

Hi, just wanted to let you know that I agree with you. I just wanted to make it clear that it is NOT a real definition. It said it was a heuristic definition and I wanted to stress the heuristic bit. It is still under the definition section though. I agree that it is a characterization instead. I am not the one that wrote definition to begin with. Any suggestions to make it clearer? Thanks. --kupirijo (talk) 00:33, 27 November 2011 (UTC)

Hmmm... Things are reiterated a bit between the Overview and the heuristic part of the Definition section. This can probably be streamlined a bit, but I think it's going to take more than just moving text around to achieve that. Ideally there should be a natural transition between the two sections (that currently is somewhat lacking). Sławomir Biały (talk) 13:45, 27 November 2011 (UTC)

Eigenvalues and eigenvectors/Matrix theory

Hi there, I see you have reverted my edit in which I removed Category:Linear algebra from Eigenvalues and eigenvectors with the claim it was idiotic. I removed it because I knew eigenvalues and eigenvectors are specifically apart of matrix theory. Upon looking at Category:Matrix theory I noticed that the only parent category of it was Category:Linear algebra, hence I removed what I thought was a unnecessary parent category, but you have made me aware of the error, so I thank you. Just wondering though, what other categories should be parents of Category:Matrix theory? Thanks Brad7777 (talk) 15:11, 27 November 2011 (UTC)

Eigenvectors and eigenvalues are part of linear algebra per se. Yes, they are also a big part of matrix theory, but they make sense for arbitrary linear transformations (even in infinite dimensions: spectral theory). Arguably most eigenvalues in physics have nothing to do with matrices. Sławomir Biały (talk) 15:19, 27 November 2011 (UTC)

A simple question (2)

Hey Sławomir !!!
you responded to the two messages above, but not to mine, any problem??

The old message:

I see you are talking about (recalling the infamous history of Boubaker polynomials) !

What do you mean by infamous history?

Thank you for answer.--Nourja (talk) 15:26, 27 November 2011 (UTC)

A barnstar for you!

	The Writer's Barnstar
	Thank you! Alansfault (talk) 12:30, 1 December 2011 (UTC)

Diff. identities

Congratulations × c on your awards - unlike me your'e not a stupid failure so you definitley deserve them.

Just to let you know - I left a few final comments here for you. Only answer when you can. Yours--Maschen (talk) 19:52, 2 December 2011 (UTC)

Re: Upright "d"

Ok, where is the style manual? I've been searching for it in the past... Nevertheless, the upright "d" issue is somewhat similar to how "naïve" should be written. People who don't care, write it as "naive", while people with a sense for nuance (or should I say nuancé?) write it upright. Imho, it should be written in roman all the time, not to be confused with yet another variable and multiplication. Furthermore, this is the prescribed style in Physics journals and publications, and at least in Physics, italics "d" looks more and more outdated and gives an impression of sloppiness of the author. Do you think the style should be decided by people that do not care for beauty of mathematical expressions set in (la)tex? — Preceding unsigned comment added by 193.2.120.35 (talk) 14:32, 5 December 2011 (UTC)

Misplaced Pages:Reference desk/Mathematics‎

Could you disambiguate lambda expression, i.e. which link there? Thanks. --Bob K31416 (talk) 00:38, 13 December 2011 (UTC)

Non-implemented integral signs

Hello - sorry to intrude, but I notice you are an experianced editor on wikipedia maths. If you don't mind me gaining your attention: I have copied and pasted a section (I posted) from the wikiproject help desk here on something you may be interested in. Cheers.

________________________________________

Can everyone please see this link when they have time? It is a proposed workaround for including the closed double and triple integral symbols not possible with LaTeX. They are rendered JPEG images:

\oiint and \oiiint:

• ],
• ].

and as templates: {{oiint}}, {{oiiint}}, and {{Oiint+Oiiint}}. I had high hopes for templates, unfortuatley they didn't work out as well the the pure images.

Thanks - feel free to criticize as heavy as you will, all comments very welcome...

________________________________________

--F=q(E+v^B) (talk) 00:49, 13 December 2011 (UTC)

"Sort of irrelevant"?

Hi, Slawekb. You described as "sort of irrelevant" my introduction of the solution of basic linear differential equations in terms of e in the article on e. But how can we possibly omit from an article on e one of the most important uses of e in mathematics? If this is not relevant, what is the relevance of the bit that was already there, about the first-order case? And if the latter is relevant, it is also insufficient because it gives the appearance that e appears only trivially in differential equations.

Maybe they should both go in the section "Applications" instead of the section "Properties"? Duoduoduo (talk) 20:08, 24 December 2011 (UTC)

But it has nothing at all to do with the number e. The same thing that you wrote holds in any base. We already have the article exponential function for discussing general applications of those. We should keep this article dedicated (as much as reasonably possible) to the actual mathematical constant itself. The first order example is included because it characterizes this constant. Sławomir Biały (talk) 23:09, 24 December 2011 (UTC)

Not true. The thing that distinguishes continuous time and differential equations from discrete time and difference equations is that in continuous time, you cannot avoid base e in the solutions. If you try to write the solution in any other base, the ith term becomes ${\displaystyle b_{i}^{x}}$; but ${\displaystyle b_{i}}$ equals ${\displaystyle e^{\lambda _{i}}}$, so you haven't avoided e. Duoduoduo (talk) 01:07, 25 December 2011 (UTC)

Who said anything at all about discrete time? What you wrote in that article is word-for-word exactly true if you replace each instance of e by any other positive real number ≠1. For instance, with 2 instead of e, we have

More broadly, the differential equation

${\displaystyle y^{(n)}(x)+a_{n-1}y^{(n-1)}(x)+\cdots +a_{1}y^{(1)}(x)+a_{0}y(x)+b=0,}$

where ${\displaystyle y^{(k)}}$ denotes the kth derivative of y, has a solution of the form

${\displaystyle y(x)={\frac {-b}{a_{0}}}+c_{1}2^{\lambda _{1}x}+c_{2}2^{\lambda _{2}x}+\cdots +c_{n}2^{\lambda _{n}x},}$

where the ${\displaystyle c_{i}}$ and ${\displaystyle \lambda _{i}}$ depend nonlinearly on the coefficients ${\displaystyle a_{i}}$.

I hope you can see that this sort of thing clearly belongs at exponential function rather than at e (mathematical constant).

If you want to argue that all exponential functions must necessarily use the base e, then this is a separate issue having very little to do with differential equations. It isn't strictly true as you have said it. One can define b (at least for real x) without any reference to base e. However, it's clear that e is the natural base to work with for exponential functions of a real (or complex) variable. Presumably someone has written something on this that the article could reference, and a subsection could be added to the applications section. I would not, however, agree that the emphasis should be on differential equations. Sławomir Biały (talk) 03:36, 25 December 2011 (UTC)

You say that one could write

${\displaystyle y(x)={\frac {-b}{a_{0}}}+c_{1}2^{\lambda _{1}x}+c_{2}2^{\lambda _{2}x}+\cdots +c_{n}2^{\lambda _{n}x},}$

where the ${\displaystyle c_{i}}$ and ${\displaystyle \lambda _{i}}$ depend nonlinearly on the coefficients ${\displaystyle a_{i}}$.

No, not without your ${\displaystyle \lambda _{i}}$ being defined in terms of e. ${\displaystyle \lambda _{i}}$ as I used it in the conventional way refers to an algebraic number, a root of the characteristic equation, which is a polynomial equation. This is utterly standard elementary treatment of differential equations. In your equivalent equation, what you've unconventionally denoted as ${\displaystyle \lambda _{i}}$ depends on e: specifically, ${\displaystyle 2^{your\lambda _{i}x}=e^{\lambda _{i}x}}$ so ${\displaystyle 2^{your\lambda _{i}}=e^{\lambda _{i}}}$ so ${\displaystyle your\lambda _{i}=\lambda _{i}\cdot \log _{2}e}$; hence your lambda is a transcendental number that depends on e. It's impossible to get from the characteristic equation, which has roots ${\displaystyle \lambda _{i}}$ that of course are algebraic numbers, to the transcendental number ${\displaystyle e^{\lambda _{1}}}$ without using specifically e.

In any event, I'm not planning to put it back into the article, because you feel so strongly about it. Duoduoduo (talk) 18:06, 27 December 2011 (UTC)

Actually, it is your lambdas that depend on 2, not mine that depend on e! Sławomir Biały (talk) 19:31, 27 December 2011 (UTC)

Irony aside, what you wrote mentions nothing about the characteristic equation, only that the lambdas depend on the coefficients. That is true in any base at all. Whether on has a useful formula for them is an entirely different issue. Indeed, even in base e, there are no useful general formulas for the eigenvalues. But as I've said already, this is beside the point. The point that you should be making is that there are good reasons for using e as the base of the exponential function. So, go and find sources! Sławomir Biały (talk) 19:48, 27 December 2011 (UTC)

Differentiation rules

You may not know that it was decided to merge differentiation rules with list of differentiation identities. You removed much of content that was merged from list of differentiation identities into differentiation rules. If you oppose the merge I suggest you to explain it and restore the list of differentiation identities article, not just remove content.--Anuclanus (talk) 11:40, 30 December 2011 (UTC)