User talk:D.Lazard/Archive 1

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Hello, D.Lazard, 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:

I hope you enjoy editing here and being a Wikipedian! Please sign your messages on discussion pages using four tildes (~~~~); this will automatically insert your username and the date. If you need help, check out Wikipedia:Questions, ask me on my talk page, or ask your question on this page and then place {{helpme}} before the question. Again, welcome! Wikipelli Talk 11:57, 26 May 2010 (UTC)

Cubic function

Hi, I reverted your edit to Cubic function. Rationale, see edit summary. If you'd like to discuss, please do so on article talk page. Cheers - DVdm (talk) 16:31, 28 October 2010 (UTC)

I have edited (in line with wp:MOS) and moved/shortened your latest contribution to the lead of the article. Please aquaint yourself with our manual of style? Thanks. DVdm (talk) 18:18, 28 October 2010 (UTC)

Gröbner basis

Gröbner writes in [1], §2, p72: "Das Mittel aber, die Eliminationstheorie frei von diesem Mangel aufzubauen, liefert die Idealtheorie"; afterward he defines "das erste Eliminationsideal" (p.73) and performs his study completely in terms of "Idealtheorie" (p.73). Gröbner also suggests a geometrical interpretation of a constructed ideal as an algebraic manifold (p.74), and this is as close as he comes to any vector space. The generator systems of ideals are described on the p.75-76.

Obviously, the words "Gröbner basis" do not appear in the text: Gröbner introduced them but his student Buchberger named them "Gröbner basis" in his PhD thesis where he implemented existing algorithm as an effective computer program. This fact is widely known, e.g. Buchberger has got The Paris Kanellakis Theory and Practice Award for "developing the theory of Gröbner bases into a highly effective tool in computer algebra".

Heisuke Hironaka who independently developed analogous conception for local rings has got Fields Medal.

If you do not know German - you can find the English translation of the original Gröbner's paper here: —Preceding unsigned comment added by (talk) 12:50, 27 December 2010 (UTC)

Trigonometric tables

Hipparchus of Rhodes compiled trigonometric tables in the first century AD, so the answer to your question "Trigonometric tables in 11th century?" is "Yes, certainly." JamesBWatson (talk) 16:47, 6 November 2011 (UTC)

But do you have any source showing that 1/ Omar Khayyám did use them 2/ use them in connection with his geometric solution? I think that more credible that he intended to use his solution to build abaci, that is to compute graphically the solutions. As I am unable to source my opinion and I have not read the texts of Omar Khayyám, the best is to simply remove the assertion on trigonometric tables as unsourced. D.Lazard (talk) 17:14, 6 November 2011 (UTC)

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Algebraic geometry

Thank you for the heads-up! Tous les meilleurs voeux! Garald (talk) 10:41, 4 January 2012 (UTC)

PS. We are neighbours, aren't we? Garald (talk) 10:41, 4 January 2012 (UTC)

Possible. Je n'ai pas décrypté votre pseudo. Moi je suis à Jussieu en info (LIP6). D.Lazard (talk) 11:35, 4 January 2012 (UTC)
Je suis au DMA (ENS). Garald (talk) 14:36, 16 January 2012 (UTC)

Hello! (and a question close to flat modules)

Hi, I had not realized you were that Lazard whose flat module results I have read about. Nice to meet you! I hope you find the chance to visit the eastern US sometime. Do you happen to have the time to address some basic questions on flat modules (and their candidate dual notions)? Rschwieb (talk) 14:18, 21 January 2012 (UTC)

I am afraid that I will be of few help on flat modules: I have not worked on this subject since more than 35 years. Since then I am working on Computer algebra and specifically on Polynomial system solving and computational algebraic geometry. Sincerely. D.Lazard (talk) 15:07, 21 January 2012 (UTC)
Doh, that's too bad :( No matter, say hi anytime! Rschwieb (talk) 15:38, 21 January 2012 (UTC)

Algebraic structure

Hi, I've recently been starting changes to Algebraic structure. Some sort of distinction has arisen that I want to be clear on. The problem is that order seems both algebraic and topological to me. Here is the question I've been asking myself and the answer I've been giving myself:

  • Q: A lattice can be defined algebraically, and the order structure creates a topology, which is non-algebraic. Is the nature of a lattice algebraic or non algebraic?
  • A: The lattice structure is algebraic. This structure can be used to create a topology, however that will involve quantifying over subsets of the set, so the topology itself is not algebraic.

Does this seem consistent with the definition of "algebraic object" you offered earlier, where only first order logic was involved? Thanks for any feedback. Rschwieb (talk) 13:56, 2 March 2012 (UTC)

Hi, Here is my opinion, but it is essentially OR. I do not know any mathematician who has tried to define properly what is algebraic and what is not. This is probably as impossible as to define precisely the limits of a scientific area. And this changes with scientific progresses. Here are some hints for answering to your question.
  • The general definition of a topology is certainly non algebraic, as it involves quantifiers on infinite subsets.
  • Nevertheless, some topologies may be defined algebraically. This is the case of the topology of the rational numbers Q: The open intervals are a basis of the topology and an open interval is defined by, at most, two rationals. Similarly the notion of limit may be defined by a first order formula. This seems also true for the topology of the real numbers, but is not because the set of real numbers itself may not be defined algebraically.
  • Some questions about a given topology, which is algebraically defined may not be asked by a first order formula, and are thus not algebraic. This is the cases of the completeness. But the non completeness may be proved algebraically, as "∃ an infinite sequence" may be replaced by giving explicitly an infinite sequence.
Conclusion: A topology is not an algebraic structure, but some specific topologies may be defined algebraically (I think that it is the case of that of the lattices, but I do not remember exactly this definition). When it is the case, as long as one uses only first order formulas, the introduction of the topology is only for the comfort of thinking and language, as every reasoning could be done without using topology.

By the way, I remark that the page Algebraic structure does not consider ordered sets and graphs as algebraic structures. In my opinion, they are as both are are sets equipped by a binary operation with values in {true, false] (the ordering relation or the property to be joined by an edge).
D.Lazard (talk) 16:22, 2 March 2012 (UTC)
It's strange that order would be nonalgebraic and lattices would be algebraic. Rschwieb (talk) 19:47, 2 March 2012 (UTC)
I can explain: Lattices may be defined as a structure involving only one set, while orders involve two sets (one set and the booleans). Probably the authors of algebraic structure did not have in mind the definition of "binary relation" which is standard in computer science and is not even cited in relation (mathematics): A binary relation is a bivariate boolean valued function. In fact this is the only definition which is compatible with a constructivist point of view. D.Lazard (talk) 11:17, 3 March 2012 (UTC)
Yes that seems to be the case. I just found it a little surprising! Thanks for the thoughts Rschwieb (talk) 15:08, 3 March 2012 (UTC)


Hi, I noticed you recently converted many instances of "noncommutative" to "non-commutative" in the article ring theory. I'm changing all of these back for two reasons.

  • One is that the unhyphenated version seems to be far more common in the literature. Using mathscinet for example, "noncommutative" appears in 4367 titles and "non-commutative" in 1509. A similar 3:1 ratio arises if you use the two adjectives in the "anywhere" field. Using googlebooks in title search function, there are ~6000 hits with the hyphen, and ~16600 hits without the hyphen.
  • Secondly, it seems like most article titles in wikipedia use the unhyphenated version, so it'd be nice to match them.

I did note the presence of numerous subsections using a hyphenated version, but I'm not really sure if there is some merit I am overlooking. Rschwieb (talk) 13:12, 30 March 2012 (UTC)

This is not a problem for me. The main object of my edit was to clarify the relationship between commutative algebra, algebraic geometry and number theory. Indeed the previous version presented wrongly algebraic geometry as a subfield of commutative algebra. I have inserted the hyphens because it makes reading easier and I have learned recently that, normally, non is hyphenated. But as non native English speaker, I am not sure of myself for such a question. D.Lazard (talk) 13:41, 30 March 2012 (UTC)
You definitely clarified what was there. Writing a lead for such an article must be challenging... it seems like a very broad experience is required! Rschwieb (talk) 15:04, 30 March 2012 (UTC)

Degree of a monomial

Hello! Thanks for assuming my edit on the "Monomial" page (regarding multivariate degree) was in good faith. I agree that my "this definition is seldom used" was overly strong.

That said, I suppose I meant "This definition is seldom used outside of a graduate and post-graduate context." I've done a better job of revising the page now, so as to be much more objective. Let me know what you think.

As far as original research, my justification for my claim consists of spending an hour perusing the online literature, and finding no other context for why one would define the degree of a multivariate monomial thusly. Hence my conclusion that it was most useful in the circumstances I stated. How would I cite this, given that I'm largely raising the fact that there's an absence of evidence otherwise? (talk) 23:13, 21 April 2012 (UTC)

I think that your online research gave a biased answer, because the degree of a multivariate monomial can not be dissociate to the degree of a multivariate polynomial. Thus the usage of "degree of a monomial" is most of the time implicit. I have edited the article for explaining this. I have removed the reference to monomial ordering, because of wp:DUE, keeping it would either need to add a lot of other examples or make an artificial distinction between "degree of a monomial" and "degree of a polynomial". D.Lazard (talk) 12:23, 22 April 2012 (UTC)
Ha, I agree with you that my latest edit did place undue weight on using monomial degree for monomials only -- because I was treating its use in the polynomial case as so implicit as to be basic! Your edits, pointing out how the degree definition is used implicitly, are well-taken. I've added back in a link to the Gröbner basis page as an example of a more explicit use. I don't think it's too minority of a viewpoint so as not to be featured. Cheers. (talk) 15:50, 22 April 2012 (UTC)

Historical revision on symbolic integration is false

The symbolic integration scheme based on the exploitation of special functions was pioneered by Maple and certainly not by Axiom, a system whose implementastion of this approach of integration came AFTER Maple. The method was shown at the MIT conference and emulated by the systems whose representatives attended the conference. TonyMath (talk) 23:04, 3 June 2012 (UTC)

To reiterate, definite integrals from special functions did exist in a limited way in systems older than Maple, like Macsyma, but the method pioneered by developers of Maple involved, for example, derivatives of special functions with respect to its parameters and providing a much more varied class of integrals. Axiom emulated the method because a couple of developers for Maple had become developers of Axiom. If you insist that the method as stated in the references was really invented earlier by developers of Macysma or Reduce, then you should provide proof with an older reference to a peer-reviewed paper. TonyMath (talk) 23:26, 3 June 2012 (UTC)

A suggestion for Diophantine approximations

It looks to me like the section Measure of the accuracy of approximations should come immediately after the introduction -- it sort of sets up the analysis in the rest of the article, and explains why in the "best approximations" we focus on the denominator q. However, there's a small amount of rewriting that would have to take place as a result (the talk of convergents is not explained until later, for example), and I'm not comfortable doing it myself. What do you think? --JBL (talk) 20:02, 1 July 2012 (UTC)

The ordering of the sections is a matter of taste. I have left "Best approximation" at the beginning for several reasons:
  • It was there and I did not see why changing this.
  • It is the section for which the proofs are the easiest, and WP:MOS recommend to order the sections by increasing difficulty
  • Before measuring something, it is better to know it
  • Section "Measure" is clearly an introduction to the sections on bounds. It would be strange if an introducing section would be followed by a section that is not introduced by irt.
  • This section is used by for upper bounds (at least implicitly), but not for lower bounds. Thus, to put it just after Section "Measure" would imply to exchange the places of "upper bounds" and "lower bounds. I am not willing to proceed to such an exchange, because lower bound results are IMHO more notable and more useful for the applications to transcendence theory and Diophantine equations.
Again, this is somewhat a matter of taste. If the question is asked on the task page, and if there is a consensus for another organization of the sections, I would accept it. D.Lazard (talk) 09:27, 2 July 2012 (UTC)
(Also, another minor comment: currently three separate things are denoted by φ in the article; are there other standard or acceptable choices for any of them?) — Preceding unsigned comment added by Joel B. Lewis (talkcontribs) 20:07, 1 July 2012 (UTC)
In fact, there are only two meanings for φ: either the golden ration, for which it is a standard notation, or a unspecified function of the denominator. This is why I used \varphi for the latter (note that \varphi and φ have the same shape, which is different of that of \phi, used for the golden ratio). It is probably better to use f for an unspecified function. D.Lazard (talk) 09:27, 2 July 2012 (UTC)
Thanks for your reply! On the first point, I am happy to defer to you. On the second, I had changed \varphi to \phi yesterday because the symbol {{math|''φ''}} appeared the same as \phi on the computer I was using at the time; however, at my present computer it appears the same as (or at least similar enough to) \varphi. Any idea what could be causing this/how it can be avoided? --JBL (talk) 12:42, 2 July 2012 (UTC)

Elementary algebra GA Review

Thank you for you input to the situation, I have given you a barnstar for it. Contrary to that what you said and what I said and believe is that this article is "quickfail" since it is sourced with books and no websites and nothing can really be verified, and furthermore it sounds like a Wikiversity page and is confusing to beyond repair. If so, I will quickfail this article. ObtundTalk 04:14, 5 August 2012 (UTC)

Barnstar of Diligence Hires.png The Barnstar of Diligence
Thank you for your input to the situation, I really appreciate it. ObtundTalk 04:14, 5 August 2012 (UTC)

Credo Reference

I'm sorry to report that there were not enough accounts available for you to have one. I have you on our list though and if more become available we will notify you promptly.

We're continually working to bring resources like Credo to Wikipedia editors, and this will very hopefully not be your last opportunity to sign up for one. If you haven't already, please check out WP:HighBeam and WP:Questia, where accounts are still available. Cheers, Ocaasi 19:11, 22 August 2012 (UTC)

Reversion of "Klein 4 ring" comment

The original poster appears to have been talking about this, which is nothing like the endomorphism ring of the Klein 4 group. At any rate, the main thing I objected to was the terminology and the use of a ring without identity, so your fix seems ok. Rschwieb (talk) 14:35, 7 November 2012 (UTC)

I did not notice that. In any case the "Klein 4 ring" is a subring of the endomorphisms of the Klein 4 group: the matrices with zero second column, when the endomorphisms are represented by matrices over the field with 2 elements. D.Lazard (talk) 15:20, 7 November 2012 (UTC)
Since our convention is to favor rings with identity, and the matrix ring you describe is far easier and nicer to describe, I'm glad you used it there. At any rate, the term "Klein 4-ring" appears to have been coined by an author on Planet Math, and nowhere else: I couldn't let it stand :) Rschwieb (talk) 16:26, 7 November 2012 (UTC)

"In Galois theory,....."

Please see this edit. I don't think the phrase "In Galois theory,..." succeeds in informing the lay reader that mathematics is what the article is about. "Algebra", on the other hand is a word that everybody knows. Michael Hardy (talk) 18:25, 8 November 2012 (UTC)

I agree. In fact, I have created this stub by copying three lines I had written sometimes ago in the dab page resolvent, because this well established topic was not even mentioned in this page. Recently an aficionado of MOS:DAB has removed these lines and replaced them by links redirecting to their source. The creation of the stub was the best way to solve the problem while respecting MOS:DAB (see talk: Resolvent). As I do not like not understandable texts, I have added some details and, now, a complete definition. D.Lazard (talk) 22:15, 8 November 2012 (UTC)

(Non) speedy deletion of Mathomatic

I've replied to your message on my talk page. I hope my comments will be helpful to you. Feel free to keep in touch on this, if you like. JamesBWatson (talk) 15:18, 12 November 2012 (UTC)

Comparison of computer algebra systems

After this [1] I am inclined to think that discussion at the talk page is needed before either of you make further reverts. Deltahedron (talk) 17:29, 17 November 2012 (UTC)

OK D.Lazard (talk) 17:52, 17 November 2012 (UTC)


I have answered your question [2] at Talk:Diophantine approximation#Terminology. In general, please don't try to conduct conversations through edit summaries: take a moment to post your question on the relevant talk page where others are more likely to see it and will find it easier to respond. Deltahedron (talk) 07:43, 18 November 2012 (UTC)

Algebraic curve

I have reverted your edit because the left superscript for the h of the homogenization was intended: A right superscript could be confusing with exponentiation, and, mainly, is not compatible with the prime of the derivative. The \mbox{} is for a correct spacing after "=" (otherwise "h" appears as an exponent of the "=". D.Lazard (talk) 11:05, 19 November 2012 (UTC)

May be correct, but if Wikipedia won't render it, it is of no use. Incompatible subscripts are better than broken but compatible tex. But I see that you've found a new markup. Great, thanks. -lethe talk + 13:13, 19 November 2012 (UTC)

Rank conditions on augmented and unaugmented matrices

For my future reference, is the following statement true regardless of the numbers of equations, independent equations, and unknowns?

A system of linear equations Ax = b is consistent if and only if the rank of the augmented matrix [A b] equals the rank of the unaugmented matrix A, and is inconsistent if and only if the rank of the augmented matrix is greater than the rank of A.

I want to put this or a corrected version into overdetermined system, underdetermined system, augmented matrix, and System of linear equations#Consistency. Duoduoduo (talk) 01:13, 22 November 2012 (UTC)

Never mind, I found it in Rank (linear algebra)#Applications. Duoduoduo (talk) 01:44, 22 November 2012 (UTC)

Möbius group is isomorphic to PGL(2,C)

I believe your reversion of my correction at Möbius transformation is wrong because your have misunderstood the notation. PGL(2,C) does indeed act on the complex projective line which has dimension 1, but it is called PGL(2,C) because it is a projection of GL(2,C). PGL(1,C) (if that notation is ever used) would be the projection of GL(1,C), and so would be the trivial group. To confirm that the Mobius group is indeed isomorphic to PGL(2,C), see the section Projective matrix representations further down in the article, or projective linear group which says "the projective linear groups therefore generalise the case PGL(2,C) of Möbius transformations (sometimes called the Möbius group), which acts on the projective line" or Google "mobius pgl" for numerous sources. Gandalf61 (talk) 16:34, 15 December 2012 (UTC)

You may be right. I have not checked on the literature, but Projective group and Fano plane use the same convention as you. I'll revert my edit. D.Lazard (talk) 18:54, 15 December 2012 (UTC)
Thank you. Gandalf61 (talk) 19:06, 15 December 2012 (UTC)

Jean-François Monteil wanted to contact D.Lazard

( (talk) 14:33, 20 December 2012 (UTC))

More spam?

Re this: did you have a look at recent Special:Contributions/Mrjohncummings? Not sure, but more of the same, it somewhat seems. Cheers - DVdm (talk) 22:34, 2 January 2013 (UTC)

Speedy deletion declined: Differential equationsof mathematical physics

Hello D.Lazard. I am just letting you know that I declined the speedy deletion of Differential equationsof mathematical physics, a page you tagged for speedy deletion, because of the following concern: Not a recently created redirect - consider WP:RfD. Thank you. — Malik Shabazz Talk/Stalk 04:31, 31 January 2013 (UTC)

Projective module

Thanks for working with the recent edit I made about the diagram there involving torsion-free modules. I had asserted that the right-to-left implication was true for domains only because I was having trouble tracking down a citation for the general statement.

The most general definition of a torsion-free module that I'm aware of is this: "M is torsion-free if the following holds: for any element x of R which is not a right zero divisor, right multiplication by x is injective on M." I don't have access to Matlis' book atm, but the recent contributions make it look like Matlis uses a version of this with just regular elements.

If memory serves, I have seen definitions of "Dedekind ring" that were not necessarily domains, and I am wondering if the "torsion-free implies flat" conclusion might hold for them. Rschwieb (talk) 15:24, 3 February 2013 (UTC)

For my edit I have used WP definitions: Dedekind rings and Dedekind domains are defined as synonymous. For torsion-free modules, I have the feeling that, for the editor who has written the previous version, "torsion-free" is defined only over a domain. I believe that the same is true for many people. Therefore my "however". For Dedekind rings that are not domains, I think that the best thing is to exclude this case. This could be done by editing the figure to replace "Dedekind ring" by "Dedekind domain". But I do not know how to do that. D.Lazard (talk) 17:30, 3 February 2013 (UTC)
I think you're right. As an aside, I don't know if you enjoy any of Carl Faith's works, but in the fun book "Rings and Things" he gives such a definition (for commutative rings). I'm not saying this should be included, I just wanted to point you to where I saw it. Thanks! Rschwieb (talk) 22:41, 3 February 2013 (UTC)

N-ary associativity

Hello, D.Lazard, and thank you for your contributions!

An article you worked on N-ary associativity, appears to be directly copied from Please take a minute to make sure that the text is freely licensed and properly attributed as a reference, otherwise the article may be deleted.

It's entirely possible that this bot made a mistake, so please feel free to remove this notice and the tag it placed on N-ary associativity if necessary.Template:Z120 MadmanBot (talk) 21:54, 17 February 2013 (UTC)


I fear that the lay reader, seeing the sentence below, will not know whether the article is about psychology, jurisprudence, chemistry, medicine, diplomacy, comparative religion, theory of banking, computer science, or tennis.

Associativity can be generalized to n-ary operations.

I've changed it so that it tells the reader at the outset that algebra is what it's about. Michael Hardy (talk) 23:35, 22 February 2013 (UTC)

I have not written this sentence. I have simply moved this paragraph from associative property (where it was too technical) to create this stub. IMO the notability of the topic does not deserve a WP article, but I am not sure of that. I have created this article, because it was the easiest way to remove this paragraph from associative property without fighting.D.Lazard (talk) 23:45, 22 February 2013 (UTC)

Additional Revision to Ring (Mathematics) Example

I see the concept regarding the additive inverse in "Detailed example I: the ring Z4" thanks to you, but believe it may still be confusing to readers. Would a revision more along the lines of

The additive inverse of is if . In Z4, the sum of if the remainder of (as integers) when divided by 4 is 0. Therefore the additive inverse of

be appropriate? I leave this to your discretion... :) — Preceding unsigned comment added by Critical Reason (talkcontribs) 21:07, 25 February 2013 (UTC)


Hi, Thanks for your effort to engage the new editor in dialog at adequality. I wanted to make a comment in response to what you wrote there, but the discussion got so voluminous that the comment will be lost in the verbiage. My comment is that the term "adaequalitat" should not be analyzed in terms of its Latin meaning, and in particular the meaning of the Latin prefix "ada". This is because the original mathematical term is Diophantus' "parisotes" in Greek, and "adaequalitat" is merely Bachet's calque from "parisotes". Therefore what is more relevant is the meaning of the Greek prefix "para" which combined with "isotes" (equality) gives "parisotes". However, this meaning seems to be ambiguous, and one has to go to the mathematics to understand Diophantus' intention. Tkuvho (talk) 18:30, 20 February 2013 (UTC)

Thanks again for putting in the effort. By the way, you should sign your recent post at User:Klaus Barner's page. I wanted to make a quick comment concerning algorithms. Fermat's applications of the technique often go beyond the algorithm that's described in the page. He adds subtle physical and geometric considerations that allow him to derive powerful applications of the method. One such example is the application to transcendental curves. The technique as described in the page only applies to polynomials, therefore clearly additional input is needed to treat the cycloid, for example, as Fermat does. After looking at the variety of applications it is clear that they go far beyond the simple algorithm. Tkuvho (talk) 20:49, 2 March 2013 (UTC)
I wanted also to respond to your comment that In my opinion, the only thing whose analysis is relevant is the nature and the significance of e. In fact, Fermat computed with it as it were a number, but it is not a number because, at some point it was supposed to be zero, and just after, it is put to zero. I am convinced that Fermat considered it as some kind of "imaginary number" (the modern meaning was not yet invented). The question "what is e?" is double. The first question is "what intuition of e had Fermat?". The second is "what modern meaning may be given to e to make his algorithm provable?" One relevant article here is that by Stromholm in 1968. Stromholm specifically argues against the idea that Fermat sets e "equal to zero". Stromholm illustrates how Fermat bends over backwards by carefully choosing a variety of terms to emphasize that e is being discarded rather than set to zero. In fact, as the term parisotes/adequality suggests, Fermat is working with a binary relation R which is not an equality but rather an approximate equality, so that one would have 2x+e R 2x (of course Fermat never calculated derivatives and did not have the notion of local slope). Tkuvho (talk) 21:06, 2 March 2013 (UTC)
What is from a computational point of view (and from a formal point of view also) the difference between putting e to 0 and removing e? The difference lies only in the informal (intuitive) description for which the accuracy of the terminology is not very important. D.Lazard (talk) 22:49, 2 March 2013 (UTC)
First, Fermat's contribution is not limited to a computational algorithm. For example, his treatment of the cycloid involves clever geometric arguments involving a mutual replacement of a point on the curve and a nearby point on the tangent line (both points being close to the point of tangency), as an instance of "adequality". This is not part of the formal kernel as outlined in the article but rather a nontrivial additional piece of geometric data. From the conceptual point of view, it is important that Fermat does not commit the logical fallacy of first assuming that e is nonzero and then setting it equal to zero at the end of the proof. For the simplest examples involving polynomials this does not matter because everything can be done purely algebraically using the idea of the "double root" as in Pappus and Viete. However, Fermat's nontrivial applications involve additional nontrivial ideas beyond the algorithmic kernel. Tkuvho (talk) 13:32, 3 March 2013 (UTC)
I also wanted to comment on your remark that It may be infinitesimal in non-standard analysis sense. I think it is clear that Fermat's "E" cannot be viewed as an "infinitesimal in non-standard analysis sense". The construction of infinitesimals in non-standard analysis requires means and techniques that were utterly beyond the reach of 17th century mathematicians. Thus, such a construction requires algebraic results such as the existence of maximal ideals that were not only inaccessible but also inexpressible, since the relevant concepts have not been introduced. Furthermore, 17th century concepts were not precise enough to be identifiable with modern notions. Therefore any formalisation in terms of modern mathematics is also an interpretation. Modulo these remarks, one can interpret and formalize Fermat's E as such infinitesimals, much as one would interpret and formalize the definitions of the derivative and the integral by Leibniz, Newton, Euler, and Lagrange. Tkuvho (talk) 14:04, 3 March 2013 (UTC)

Integrals of absolutes


How are you? Hope fine... I wanted to discuss with you the disputed formula in the section Lists of integrals#Absolute value functions.


The absolute cosine formula there is partly incorrect. The correct anti-derivative which is continuous as well, is:

The problem with this is that I have derived this equation myself and have verified it, both graphically and numerically, but have no external reference or citations. If this equation is eligible to enter that article, do you think it suitable to replace the old, incorrect one with this?

Sγεd Шαмɪq Aнмεd Hαsнмɪ (тαʟк)

01:55, 10 April 2013 (UTC)

Normally this kind of questions is better placed in the talk page of the relevant article. The rule of Wikipedia is that citation must exist for every assertion. But WP:CALC says "Routine calculations do not count as original research". Here we are (in my opinion) in a limit case, because the derivation of such formulas is routine computation only for rather experimented mathematicians. This is why I have added the tag "citation needed" to the formula for |sin| and I'll add it to your formula (or to any similar one) if it is inserted.
Nevertheless your formula is not fully correct if the usual definition of the sign function (sgn(0)=0). To be correct, one have to choose sgn(0)=1. On the other hand, the floor function is more usual than the ceil function, and I would prefer the similar formula with the floor function (which would imply sgn(0)=-1) This is a personal opinion, that does not really matter.
All these considerations may be useful to many readers. Therefore, I'll add comments to the section to emphasize on this question of continuity. D.Lazard (talk) 08:26, 10 April 2013 (UTC)
Thanks for pointing that problem with the sgn function... A solution to this could be replacing by , whose output is 1 when x≥0 and is -1 otherwise.

This solves the problem with the usual definition of the sgn function. How about considering this instead of a redifinition of sgn?

Sγεd Шαмɪq Aнмεd Hαsнмɪ (тαʟк) 11:20, 10 April 2013 (UTC)


If you are interested, you could help us improve the article on quartic functions, as you did with the cubic ones. The general formula for roots obviously fails to cover the special cases when either t or Delta or u + v are 0. — (talk) 00:50, 7 May 2013 (UTC)

IMO, the priority for this article is to rewrite all the material after the general formula to have something coherent, in particular for a description of how to derive the formula, which is compatible with both Ferrari solution and the correct formula you have given. I have done this some time ago for the cubic case, and partly for the quintic case (the structure of this article is not yet clear). I'll not have the time to do this for the quartic in the near future, being busy with other articles that are more important (for me) to improve (like Gröbner basis) and other articles related to computer algebra, a field badly covered by WP). D.Lazard (talk) 08:57, 7 May 2013 (UTC)


What was wrong with this suggestion? (talk) 14:09, 7 May 2013 (UTC)

  1. The author of this suggestion is suspected of WP:Sockpuppetry, and this is the main reason to delete this comment. If it is wrong, then you are invited to post a comment at Wikipedia:Sockpuppet investigations/Echigo mole#Comments by other users to give evidence that it is wrong. If it will be considered to be true, then the account will be blocked indefinitely.
  2. There are already many recommendations about writing math articles at MOS:MATH. Proposing further recommendations has not any sense without reference to this manual of style.

D.Lazard (talk) 15:48, 7 May 2013 (UTC)

Your comments to WP:AE

Hello. In my capacity as an administrator responding to requests at WP:AE, I have removed your recent comment, because, as you said, it was entirely unrelated to the appeal being discussed. An appeal discussion is not the place to bring forward unrelated accusations against others. If you have problems with another user, please use the normal dispute resolution process (WP:DR). Thanks,  Sandstein  13:24, 14 May 2013 (UTC)

Hi, I noticed your comment at AE that Sandstein removed. While Sandstein is right that The Devil's Advocate's appeal is not the right time or place to raise that issue, I or TDA might make an arbitration request about Mathsci sometime in the future, and it would be appropriate to raise the Deltahedron issue there. If you want, whenever the request happens you can be included as a party so you can present evidence about it. Akuri (talk) 20:41, 14 May 2013 (UTC)

Echigo mole socks

Hi there; you seem to be taking a stand against the blocking of socks of Echigo Mole. While it of course true that you can edit exactly as you wish, could I ask that ib the case of this multiple sockpuppeteer you do so with extreme caution? He is quite evasive, and can be difficult to detect. Supporting him can damage the encyclopedia. Before you ask, no, I am not a skilled mathematician. But I am very experienced in Wikipedia.--Anthony Bradbury"talk" 20:51, 19 May 2013 (UTC)

I am not against the blocking of socks of Echigo Mole. But I am definitively against blocking a good faith editor who is not a sockpuppet. In the case of Hyperbaric oxygen, nothing in his edits suggests that he is a sockpuppet. On the contrary, all his edits are constructive. In the SPI of this case, I understand the checkusers comments as no evidence of sockpupperty. Thus the accusation of sockpuppetry relies only on the conviction of a single editor who gives a description of Hyperbaric oxygen's edits that do not correspond to their real content. IMHO, the assertions of this editor are not credible, not only for the reasons that I gave in my posts at user talk:Hyperbaric oxygen, but also for his recent behavior here (section Hurwitz's theorem and related articles) and here (section Conjugation and real inner product on the quaternions), where his systematic WP:personal attacks has pushed an excellent editor (Deltahedron) to retire from wp. This has already damaged Wikipedia. D.Lazard (talk) 23:01, 19 May 2013 (UTC)
+1! Rschwieb (talk) 13:49, 20 May 2013 (UTC)
Given Anthony Bradbury's explanation, this response is not helpful. First of all I have an extremely long record of creating mathematical content. The incident being discussed above is stale now. It was in the end about minor and trivial notational issues within wikipedia not in mathematics. Secondly, and far more significantly, the disruptive editing by Echigo mole with multiple simultaneous socks is not something that is open to debate. The disruption falls within the category long term abuse, is not something new (it started in 2009) and does not have to be explained three or four times to ech person previously unaware if it. Just today an ipsock of Echigo mole was blocked by Future Perfect at Sunrise after posting here on this user talk page and on User talk:AGK. It was exactly the same user as Hyperbaric ozygen from one of the usual IP range he uses in Britain: Template:Ipuser Because of the disruption caused by this particular banned user, a motion was passed by the arbitration committee about their edits and the enabling of their edits by others.[3] After coordinated editing from other socks plus a clear explanation from an admnistrator, any further attempts to argue for an unblock of this particular sock would probably lead to an enforcement request at WP:AE. That has never happened before after over 250 socks and ipsocks. Mathsci (talk) 15:07, 20 May 2013 (UTC)
"Would probably lead to an enforcement request at WP:AE": Is that a threat against me? D.Lazard (talk) 16:41, 20 May 2013 (UTC)

Request now at WP:AE

An enforcement request has been made concerning you at WP:AE. Mathsci (talk) 07:02, 21 May 2013 (UTC)

Premature closing of MathSci's RfE against D.Lazard by Future Perfect at Sunrise?

I wish to notify you of a discussion that you were involved in.[4] Thanks. A Quest For Knowledge (talk) 15:42, 22 May 2013 (UTC)

In "Quartic function" please restore "Ferrari's solution in the special case of real coefficients"

In the article on the Quartic function you made this edit which removed the section: "Ferrari's solution in the special case of real coefficients" with this justification: "Remove a misplaced section, which is WP:OR".

The section you removed is _not_ Original Research since the published, verifiable source that the section quotes provides all the information found in the section.

Please restore this section to the article at your earliest convenience.

Thank you.

Lklundin (talk) 18:57, 26 May 2013 (UTC)

Could you help me ?

I know that you're very busy, and have very little time on your hands, but I have absolutely NO idea how to prove this without retorting to Euler's beta function, or an equivalent thereof:

I've asked this question on the Math Reference Desk here at Wiki, but all the answers that I've got so far are more-or-less unsatisfactory (beta functions, hypergeometric functions, Cauchy matrixes, Möbius inversion formula, Taylor series, finite differences, etc). I personally would want to find a simple proof based probably on induction, or perhaps on something even more basic than that. — (talk) 19:40, 11 June 2013 (UTC)

I do not know much of combinatorics (indeed, this is a question on combinatorics). I believe that the state of the art for these question is the book "A=B" ( Its goal is more automatic proofs than simple proofs. But automatic proofs requires a deep understanding of the problems, and this should provide simple proofs if any. D.Lazard (talk) 08:32, 12 June 2013 (UTC)

Thanks! Somebody else recommended to me the exact same book a few days ago. It generally deals with hypergeometric functions, and indeed, if the sum would not alternate, i.e. if the (-1)k term were missing, the sum does become an expression in 2F1. It also contains a theorem and an algorithm detailing how to transform such sums into recursive equations by dividing two of its consecutive terms. On a different note, I've arrived at the problem in question by noticing the following:

The first integral above has already been studied by John Wallis three or four centuries ago (see here, on page 49), and its expression becomes that of Euler's beta function by making the simple substitution The second one becomes the expression of Euler's famous gamma function when making the same substitution. Furthermore, the first integral above shows the connection between factorials, combinations, and geometric shapes described by equations of the form xm + yp = 1 (if we make both k and n - k smaller than 1), as well as elliptic integrals and arithmetic-geometric means (for algebraic arguments of the form as shown by Borwein and Zucker in the early `90's, who also mention that some of those expressions were already known to Gauss two centuries ago: see Gauss's constant). — (talk) 17:51, 12 June 2013 (UTC)

Diophantine approximation

I thought that the additional condition and the '<' would be right because it is written in this way in the book of Khinchin. Also the current definition for best approximation of the second kind does not exactly fit the ones given in the cited books of Lang and Cassels. I think that the difference in the definitions do not matter for irrational alpha anyway, but for rational alpha some care is needed. Compare also

A best rational approximation to a real number Template:Mvar is a rational number {{ safesubst:#invoke:Unsubst||$B=n/d}}, d > 0, that is closer to Template:Mvar than any approximation with a smaller or equal denominator.

in Continued_fraction#Best_rational_approximations and §15 in Perron's book. -- KurtSchwitters (talk) 19:54, 13 June 2013 (UTC)

I agree that both definitions are equivalent for irrational numbers, and that the one with strict inequalities may be better sourced. However, the main reason of my revert was the additional condition , which may be confusing, and has confused me, suggesting that non-irreducible fractions are considered and inserting a single latex formula inside a sentence in which the other formulas are in HTML. Therefore, I'll not be opposed that you revert my revert, if you replace this condition by simply saying "for every rational number p'/q' different of p/q and such that 0< q' ≤ q ". D.Lazard (talk) 14:23, 14 June 2013 (UTC)
Thanks. I am not used to the way {{math| is used here. That was the reason I used the latex formula. As it is not urgent to change it, I will think about it for a while and update the paragraph on Monday. -- KurtSchwitters (talk) 16:50, 14 June 2013 (UTC)

Not Solvable Through Radicals

Uhm... Silly question, but I have to ask: What does it mean that certain equations (like the quintic) are "not solvable through radicals" ? I thought it meant that although such radical expressions do theoretically exist, their formulas cannot be guessed or deduced merely by looking at their coefficients (as in the case of linear, quadratic, cubic, or quartic ones). I mean, if a number would not even be expressible through radicals, then it would have to be transcendent, right ? And, as such, it couldn't obviously be the solution to an algebraic equation, no ? Or ? — (talk) 17:34, 17 June 2013 (UTC)

This means that for some (in fact almost all) polynomials of degree 5 or higher, there can not exist any expression for the roots (formula for the roots) that may be constructed from the integers and the coefficients by using only the following operations: addition, subtraction, multiplication, division and nth root extraction for an integer n. This is Abel-Ruffini theorem. The simplest equation that cannot be solved in terms of radicals is D.Lazard (talk) 08:02, 18 June 2013 (UTC)
So what you're telling me is that there really are algebraic (non-transcendental) numbers which simply cannot be written as combinations of integers (or even rationals), +, -, ×, /, and n√ ? (If so, then does this weird sub-class of algebraics bear a name ?) — (talk) 16:22, 18 June 2013 (UTC)
Yes, this is exactly what the Abel-Ruffini theorem says. As far as I know, the only specific usual name is radical extension for a field extension generated by algebraic numberrps that may be expressed in terms of radicals. Note that if one restricts oneself to square roots instead of nth roots one gets the algebraic numbers that may be constructed with compass and straightedge. D.Lazard (talk) 16:39, 18 June 2013 (UTC)
Wow! This all seems so shockingly surreal... :-| It's definitely "news" to me... Thanks! — (talk) 18:20, 18 June 2013 (UTC)
similar to (talk) 23:29, 18 June 2013 (UTC)
This is true, but Abel-Ruffini theorem concerns finite formulas. In fact such infinite formulas express the solutions as a limit and are not algebraic. Moreover, every transcendental numbers may be represented by this kind of infinite formulas. That is the purpose of continued fractions theory. D.Lazard (talk) 09:09, 19 June 2013 (UTC)
Yes, I know, I wasn't implying anything, I was just happy to come up with something prettier-looking than a Bring radical... :-) On a somehow related note, are there some interesting or relevant conclusions to be drawn from the fact that the graphic of the zeta function so closely resembles that of a hyperbole of equation (x-1)(y-1) = 1 ? I'm asking this because also approximates the same hyperbolic function. — (talk) 15:26, 19 June 2013 (UTC)

Real multivariable function

Considering your post at Wikipedia talk:WikiProject Mathematics‎#A wild idea: multi-tiered maths articles to match the target audience?, I had a try starting this article, currently written starting from pure mathematics and ending in physics and engineering. It seems you were looking for an article on this and I couldn't find one either. By all means edit if inclined. Thanks and regards, M∧Ŝc2ħεИτlk 14:21, 26 June 2013 (UTC)

A Suggestion

I'm trying to come up with a single-letter name for the set of algebraic numbers that can be expressed as combinations of radicals, fractions, subtractions, sums, and products of integers. I thought of V for the combinations containing irreducible radicals (since the letter's shape reminds me of the symbol for the radical sign), but I don't really know what to call the reunion of V and Q. Any suggestions ? — (talk) 21:01, 5 August 2013 (UTC)

Big oops

Thank you very much for having detected the confusion between radical and root. The fr:Duplication du cube involves the cubic root of two, and this is a radical while it's not a constructible number. I should have paid attention to that. I rephrased the whole section to focus on square roots instead of generic radicals (which can even be complex, as you pointed out). --MathsPoetry (talk) 12:07, 14 August 2013 (UTC)

Union? What union? Where is the union?

"The angle is not the intersection of the two rays, but their union.."

Tell me, where in "In geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle." is there a union? I was trying to add that thought which you took away. I might not have the best way of stating it, but please don't just revert it without editing it yourself to where it reads better. John W. Nicholson (talk) 13:02, 10 September 2013 (UTC)

The sentence that I have restored asserts correctly that an angle consists in two rays and their intersection at their common endpoint (the vertex), all together. Thus, when drawing an angle, you are representing the set theoretic union of the two rays. The edit that I have reverted asserted wrongly that an angle consists in the intersection of the two rays. This is wrong because the intersection of the two rays is not the angle; it is its vertex. If you think that the given definition is incorrect or unclear, you have to explain why in the talk page before inserting your WP:OR thought. D.Lazard (talk) 16:27, 10 September 2013 (UTC)

Theorem in commutative algebra

Hi there: been a while since we talked :) Might you be able to help out with this question?

Thank you for quoting this link. However, I cannot help here: I have never read this part of Eisenbud's book and I have not it at hand. More, I do not even know if Eisenbud's proof is mine, Govorov's one or a new proof. D.Lazard (talk) 17:33, 3 September 2013 (UTC)
OK, thanks for looking anyway! I notice now that someone pointed out that the poster is correct in his concerns, and that Eisenbud corrected it in some errata. See you around! Rschwieb (talk) 13:38, 23 September 2013 (UTC)

Books and Bytes: The Wikipedia Library Newsletter

Books and Bytes

Volume 1, Issue 1, October 2013

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by Template:User-multi, Template:User-multi

Greetings Wikipedia Library members! Welcome to the inaugural edition of Books and Bytes, TWL’s monthly newsletter. We're sending you the first edition of this opt-in newsletter, because you signed up, or applied for a free research account: HighBeam, Credo, Questia, JSTOR, or Cochrane. To receive future updates of Books and Bytes, please add your name to the subscriber's list. There's lots of news this month for the Wikipedia Library, including new accounts, upcoming events, and new ways to get involved...

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Read the full newsletter

Thanks for reading! All future newsletters will be opt-in only. Have an item for the next issue? Leave a note for the editor on the Suggestions page. --The Interior 20:44, 27 October 2013 (UTC)

SMath Studio

I would like to ask you a few questions, D.Lazard, regarding your Oct 28 2013 edit:

1. Why did you delete link to the official site?

2. Why do you insist that SMath Studio doesn't meet noteability guidelines and the artcile should be deleted?

3. What is it you find wrong about the feature remark (since you deleted it as well)?

Regards, Andy Monakov (talk) 09:04, 29 October 2013 (UTC)

  1. The deletion of the link to the official site is a side effect of the revert. I'll not oppose if you reinsert it again.
  2. Notability guideline says "if no reliable third-party sources can be found on a topic, then it should not have a separate article". For the moment, no such reliable third-party source is provided, although asked for since about one year.
  3. The feature remark is an editor's opinion, not supported by any reliable source. As such it is WP:original research, and is forbidden by Wikipedia policy.
D.Lazard (talk) 10:50, 29 October 2013 (UTC)

2. Well, are you aware that over the years SMath Studio was covered in multiple reviews? Or do you perceive it as insufficient?

3. I'm afraid you didn't look well at the edit at all. By definition, opinion is a person's subjective judgment, and the remark doesn't contain anything of the sort. Andy Monakov (talk) 14:19, 29 October 2013 (UTC)

Free graph theory software

I want to discuss adding a link in the article "graph theory" to the website of "Free graph theory software". It was recently deleted with the following argument: "There are any number of free graph software libraries out there; what makes this one special?".

I want to correct a misunderstanding. "Free graph theory software" is no library. It is a free graph theory software tool to construct, analyse, and visualise graphs for science and teaching. It has a graphical user interface and works online without installation.

Why it is less special than other external links, that are not deleted? E.g., In "Free graph theory software" you can import an arbitrary list of graphs in graph6 format and choose graphs from it, by entering their graph parameters, which i consider a really useful feature. I do not know any other software that has this function. And it has a lot of other features listed on the website.

The intention of "Free graph theory software" is to help scientists and students all over the world. It is a graph theory software that may be used for free by everyone. Why should it not deserve an external link in the article "graph theory" of wikipedia? — Preceding unsigned comment added by Yloreander (talkcontribs) 15:24, 8 November 2013 (UTC)

Please read WP:What Wikipedia is not and more specifically WP:ADVOCATE. Your arguments supporting "Free graph theory software" look like advocacy. For the Wikipedia policy about external links, see WP:ELNO. There are many graph theory software, free and not free. To be linked to, a software needs to be notable, and its notability must be attested by reliable secondary sources. This is apparently not the case for "Free graph theory software". D.Lazard (talk) 15:43, 8 November 2013 (UTC)

Reducible polynomials

Is the definition of a polynomial be reducible over the integers the only one. For instance in Stewart's Galois Theory, Third Edition (Definition 3.10) he writes

"A polynomial over a subring R of C is reducible if it is a product of two polynomials over R of smaller degree. Otherwise it is irreducible."

He then goes on to give an example of a polynomial that is irreducible over Z(t) of 6t+3=3(2t+1). This was the basis for the change that I made on the irreducible polynomial page that you reverted back. If different books use different definitions then shouldn't this be included in the article. — Preceding unsigned comment added by Uwhoff (talkcontribs) 20:18, 17 November 2013 (UTC)

I have answered in Talk:Irreducible polynomial#Irreducibility aver the integers. However, I did not answered about this particular example. Are you sure that he wrote "the polynomial 6t+3=3(2t+1) is irreducible over Z[t]" (which is incorrect) and not "the polynomial 6t+3=3(2t+1) (defined) over Z[t] is irreducible over Q[t]" or "the polynomial 6t+3=3(2t+1) (defined) over Z[t] is irreducible" (which are both correct, if one considers only the irreducibility over a field). The place of "over" is essential. D.Lazard (talk) 14:30, 18 November 2013 (UTC)

Modular arithmetic

"(Reverted good faith edits by Jtle515 (talk): Translating the latin ablatif as "by" is not sourced and WP:OR. (TW))"

Knowing another language counts as original research now? Huh? --Jtle515 (talk) 06:25, 23 November 2013 (UTC)

It is not the knowledge of Latin which is original research, but the choice of translating ablatif, in this particular case, by "by". This choice is controversial. D.Lazard (talk) 10:18, 23 November 2013 (UTC)
What else could it be, in context? --Jtle515 (talk) 11:02, 23 November 2013 (UTC)
For a mathematician, "modulo" means "up to modulus". However I do not know if this translation is linguistically correct. D.Lazard (talk) 11:12, 23 November 2013 (UTC)

The Wikipedia Library Survey

As a subscriber to one of The Wikipedia Library's programs, we'd like to hear your thoughts about future donations and project activities in this brief survey. Thanks and cheers, Ocaasi t | c 15:14, 9 December 2013 (UTC)

Javascript Exetended Euclidean Algorithm


I think the pseudocode is not so clear on the page Extended Euclidean algorithm, so I put a implementation in Javascript.

But you revert my edits.

I see that on the talk page there are more people talking it, so I think that it could bee good put some "real" code there. I am not saying that we need put the Javascript code, but, some code.

What do you think about?


Lp.vitor (talk) 12:53, 1 December 2013 (UTC)

The pseudo-code is be very close to Pascal or Maple code. Therefore Javascript of other codes seem not useful for the article. In fact, the difference with Pascal code lies only in the absence of "end" keywords and in multiple assignations. Maybe, it would be clearer if they would be split in pairs of simple assignations, which is complicated by the need an auxiliary temporary variable. D.Lazard (talk) 16:41, 1 December 2013 (UTC)
Or maybe someone could explain how the multiple assignations works... Lp.vitor (talk) 21:51, 1 December 2013 (UTC)
Done in the article D.Lazard (talk) 15:21, 9 December 2013 (UTC)

All right

Penrose pentagram.svg Diligent work
Ok, no links, I'm sorry Ignacitum (talk) 20:26, 18 December 2013 (UTC)

Reverted my edit

Hi, you reverted my edit on "Algebra" today. I made an svg version of an equation that was a png, in line with the guidance here. Was there a reason to undo the edit? Could the svg be improved? I'm fairly new to editing Wikipedia so any help would be... well... helpful. Thanks Jamietwells (talk) 20:03, 14 January 2014 (UTC)

Template:Ping The only reason of my revert was that the copyright status of the image is not fixed. The file is tagged with "This file does not have information on its copyright and licensing status. Unless the copyright and licensing status is provided, the image will be deleted after Tuesday, 21 January 2014. Please remove this template if a correct copyright license tag has been added." When the copyright status will be clarified, there will be no problem to reinsert the svg version. D.Lazard (talk) 21:18, 14 January 2014 (UTC)

Ok, yeah, I made the image so I had already fixed it, but I don't know how to get rid of the copyright notice thing. I don't think you can copyright a formula anyway, It think they're all public domain (at least, the quadratic equation is). Can I put the image back then? Jamietwells (talk) 22:28, 14 January 2014 (UTC)

January 2014

Hello, I'm BracketBot. I have automatically detected that your edit to Computer algebra system may have broken the syntax by modifying 1 "[]"s. If you have, don't worry: just edit the page again to fix it. If I misunderstood what happened, or if you have any questions, you can leave a message on my operator's talk page.

List of unpaired brackets remaining on the page:
  • common divisor]]s is systemically used for the simplification of expressions involving fractions]].

Thanks, BracketBot (talk) 16:40, 16 January 2014 (UTC)

Hello, I'm BracketBot. I have automatically detected that your edit to Complex dimension may have broken the syntax by modifying 1 "()"s. If you have, don't worry: just edit the page again to fix it. If I misunderstood what happened, or if you have any questions, you can leave a message on my operator's talk page.

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  • x^2+y^2+z^2=0</math> is a variety of (complex) dimension 2 (a surface), but of real dimension 0 — (it has only one real point, (0, 0, 0).

Thanks, BracketBot (talk) 14:47, 28 January 2014 (UTC)

From user Arpan Mathur

Hi, I am Arpan Mathur. You have reverted my edit on article Derivative today. I agree with you that the example put by me

was too technical. But, having more than one example will clarify the process of differentiation by first principles as is used in the first example. So, i have posted another example which is simpler to understand than the previous one. I hope you will agree with me. Arpan Mathur (talk) 15:27, 4 February 2014 (UTC)

  1. Wolfgang Gröbner (1950). Über die Eliminationstheorie, Monatshefte für Mathematik, Zeitschriftenband 54, pp.71-78