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Full rank[edit]

My colleagues and I agree that the property of being "full rank" makes perfect sense and has a conventional definition for rectangular matrices. However, none of the books I have on hand give a definition. Can anyone produce a RS? (Refer: [1].) --JBL (talk) 00:50, 13 May 2015 (UTC)

The book by Gentle on Matrix Algebra, section 3.36, discusses the notion of full rank for non-rectangular matrices. --Mark viking (talk) 02:58, 13 May 2015 (UTC)
That book refers to a "full [hyphen omitted] rank matrix" rather than to a full-rank matrix. Punctuation was taught in elementary school when I was there; the reason why one should write about a "full-rank matrix", with a hyphen, and also say that a matrix "is of full rank", without a hyphen, is quite simple, and the presence or absence of the hyphen can effectively convey a lot of information in some cases (e.g. the difference between a "man-eating shark", which scares people away from beaches, and a "man eating shark", who is a customer in an exotic restaurant). I think people are still accustomed to seeing the traditional standard use of hyphens in magazines, newspapers, and books on subjects in which the non-technically-trained copy-editor is not afraid to say too much, although writers of advertising copy and package labeling do not use it. I think maybe it could still be saved, if an effort were made. Michael Hardy (talk) 17:52, 14 May 2015 (UTC)

@Mark viking :

  • What do you mean by a "non-rectangular matrix"?
  • Does that book anywhere give an explicit definition of "full rank" for matrices that are not square?

BTW, there's a pretty bad typo in equation (3.122) in that book. It says

 (xA)^T C (Ax) \,

where it should say this:

 (Ax)^T C(Ax) \,

Michael Hardy (talk) 18:11, 14 May 2015 (UTC)

  • Ah sorry, that was a typo (or perhaps a thinko). I meant to say non-square.
  • On page 77, last paragraph, the books says if the rank of a matrix is the same as its smaller dimension, then the matrix is of full rank. Then it goes on to note that "full row rank" and "full column rank" are typically used when discussing non-square matrices. No hyphens in any of these definitions in the book as far as I can tell. --Mark viking (talk) 20:02, 14 May 2015 (UTC)
Our article Rank (linear algebra) seems a little confused on whether a matrix having full rank is (a) both of full column rank and full row rank (and hence a square matrix), or (b) either of full column rank or full row rank (and hence can be non-square). I find the second repugnant – it seems like what would be used when someone is too lazy to use the term full column rank or full row rank as appropriate. What is the dominant use? (I already put in a note, but no-one of knowledge has chipped in.) —Quondum 00:41, 15 May 2015 (UTC)

Over a field, a square matrix is invertible if and only if it is full-rank (right?) So, I don't think "full-rank" is particularly useful for a square marrix. For a non-square matrix, a "full-rank", I think, has the usual meaning, meaning the rank (row rank) is the maximal possible; i.e., the matrix defines a surjection when it is viewed as a linear transformation. For example, to check the submersion theorem applies one checks if the Jacobian matrix has full-rank, meaning it is surjective; see for instance [2]. At least, this (full-rank = surjective) is how I use the term in my day life. -- Taku (talk) 19:41, 15 May 2015 (UTC)

I don't feel that just because in some instances another definition happens to be equivalent that one should consider one of them "not particularly useful". Your argument of being equivalent to being surjective is far more persuasive. (Your first argument would argue against using a new term for 'surjective', though!) In the context of matrices, one cannot call a matrix 'surjective' though: it is left-multiplication by that matrix which would be surjective, or right-multiplication my that matrix. So again, one is looking at calling it 'full row rank' or 'full column rank', with 'full rank' being only sensible where the two are equivalent (e.g. square matrices over a field). —Quondum 20:24, 15 May 2015 (UTC)
Full-rank ⇔ surjective is just not right. But I have never seen 'full row rank' or 'full column rank'. 'Full rank' is unambiguous, but it may be more common to call it 'maximal rank', at least in differential geometry (it is still referring to matrices). YohanN7 (talk) 23:00, 15 May 2015 (UTC)
The term 'maximal rank' makes much more sense than 'full rank' when this meaning is intended, and if it is more common, the article could be updated accordingly, subject to sourcing/dominance. —Quondum 01:04, 16 May 2015 (UTC)
Right or not, my point was using "full-rank" for "surjectivity" seems fairly common at lease in differential geometry. The reason I think is that it doesn't make sense to say whether a matrix is surjective or not; thus, "full-rank" becomes a shorthand for the linear transformation given as the left multiplication by the matrix being surjective (doesn't roll well on the tongue, does it?). For a square matrix, there is no need for the term "full-rank" (except in the pedagogical context) since "invertible matrix" is simpler and more precise. -- Taku (talk) 02:32, 16 May 2015 (UTC)
You need to be careful about the precise meaning: words used in mathematics are often used imprecisely, with a lot implied by context. For example, 'Jacobian matrix' may be used to refer to the matrix, but implied may be the mapping between tangent spaces that it represents, which implies 'left multiplication by'. This does not apply to matrices in general, where properties often are not referenced to the properties of the operators they might represent, but typically more directly in terms of the components of the matrix. So in matrices, my perception is that the rank seems to be mostly defined in terms of the dimension of the space spanned by the rows or columns, respectively. This is not the same thing as the dimension of the image of its left and right multiplication, or whether it is surjective, because these are determined by the dimension of the domain and the dimension of the codomain, which can be less or more respectively than the number of columns and rows. So the fit is really quite poor. —Quondum 03:19, 16 May 2015 (UTC)
There seems some confusion. By definition, I agree, the row/column rank is the dimension of the space spanned by rows/columns (they turn out to be the same number). But the rank of the matrix can be equally characterized by the dimension of the image of the linear transformation determined by a matrix. Via the use of a transpose, we only need to consider the case by the left multiplication. Then the rank of the matrix is the dimension of the image of the matrix viewed as a linear transformation. In other words, there might be some "a priori" distinction that can be made from matrix point of view and operator point of view, but the distinction is not too important to be concerned in practice. A case in point: one speaks of finite-rank operator (by the way, as Michael Hardy noted, the universe collapses if you forget hyphen here) even though it is not really a matrix. Taku (talk) 14:10, 17 May 2015 (UTC)
You seem to be missing what I'm saying. The rank of a vector space can be less than the dimension of its representation (it is equal to the size of the basis, and hence the rank of a map can be less than the rank of the matrix chosen to represent it). When defining the rank of a matrix, Bourbaki explicitly uses vector spaces of dimension equal to the dimensions of the matrix. Without this, your second statement in the above paragraph does not hold.
Bourbaki covers the rank of both a linear map and of a matrix in detail, but does not mention the concept of maximal rank in any form. I'd posit that the concept of a maximal-rank matrix has little utility – little enough that I would trim the definition to an observation that when the terms maximal rank or full rank are used, these terms typically mean "[description here]". We have sufficient sources to make this diminished statement. —Quondum 00:25, 18 May 2015 (UTC)
It seems to me like you're extrapolating too heavily from your own experience. "Full rank" was immediately recognized and understood by my office mates; I wouldn't bat an eyelash seeing (or using) it in a research paper. And we have at least two RSs in this thread with definitions (thanks very much to those who found them!), while I doubt very strongly that anyone will produce a RS that deprecates the term in the way you suggest. --JBL (talk) 01:10, 18 May 2015 (UTC)
I think a suitable definition of maximal rank has become clear (and I suppose we can assume that 'full rank' is a synonym). But what do you make of four out of six RSs mentioned in this thread that cover matrix rank simply failing to mention it? Should we present it as though it is mainstream and significant as if every RS had mentioned it? I was hoping some consensus would emerge, but it seems to be slow in coming; the proposals I make are merely strawmen for consideration. —Quondum 03:32, 18 May 2015 (UTC)
Okay, never mind. I've tweaked the article in the direction of maximal or full rank being defined for nonsquare matrices. At least that gets rid of the internal inconsistency, and does not change the article much. I'm not going to belabour this any further. —Quondum 04:26, 18 May 2015 (UTC)
Thank you, I am happy with the final result. --JBL (talk) 20:46, 18 May 2015 (UTC)

In case you're still looking, I found a reliable source: David C. Lay, Linear Algebra and Its Applications, 1994, Addison-Wesley, p. 242, exercise 26: "In statistical theory, a common requirement is that a matrix be of full rank. That is, the rank should be as large as possible." [Italics are in the original.] Mgnbar (talk) 00:01, 16 May 2015 (UTC)

And by the way books by Kolman, Hoffman and Kunze, and Halmos don't seem to mention full rank. Mgnbar (talk) 00:04, 16 May 2015 (UTC)

I suggest we define either of 'full rank' or 'maximal rank' and provide the other one in an or-clause. While 'full column rank' and 'full row rank' both make sense, they don't seem to appear in the literature. Both imply 'full rank' and 'full rank' implies one or the other (or both) of them (if they were defined) supported by the theorem that says row-rank = column-rank. Lee defines 'maximal rank' for an m × n or n × m-matrix, n > m as one having a m × m minor of rank m. YohanN7 (talk) 17:05, 16 May 2015 (UTC)
I would hope we make no definition without it being a fairly standard term. That was why I initially requested some input: some sources clearly define full rank as the maximal rank for the matrix size, but if that is a minority definition, I would prefer to see to noted as such. If most sources use a different term, or define it some other way, we should document it accordingly. So far, no-one seems to have done more than consider an isolated source or so. I we have a few highly notable secondary sources defining it (how widely accepted is Lee?), we could do as you say. —Quondum 23:12, 16 May 2015 (UTC)
Well, Lee is no Bourbaki, he defines things on the fly (and is therefore readable as opposed to Bourbaki). My point is that it doesn't matter very much, just take one "reputable" reference at random, use it for the def and provide the alternative term. It can't get very much wrong. Unfortunately, my own supply of books in linear algebra is limited. YohanN7 (talk) 14:58, 17 May 2015 (UTC)

With a risk of complicating the discussion further, I would like to mention yet another point of view: "full-rank" = "maximal rank" = "generic-rank". Here, I'm using "generic" in the following way. Let X be the (vector) space of all matrices of some fixed size (possibly non-square). We view X as an (affine) algebraic variety (X is simply a vector space.) Then the matrices of maximal possible rank form an open subset with respect to Zariski topology (it is the complement of the vanishing locus of minors.) So, a matrix in a general position has maximal possible rank and that's the generic rank of a matrix. (Do I make sense?) By definition, a matrix is full-rank if it has generic rank or equivalently maximal possible rank. -- Taku (talk) 14:38, 17 May 2015 (UTC)

This vaguely resembles an example in Lee's book. The set of m × n matrices with full rank is open in M(m, n) in the subspace topology, hence a submanifold. YohanN7 (talk) 14:58, 17 May 2015 (UTC)
It is correct that "full rank" = "maximal rank", when considering the set of all m × n matrices. But, in the case of the a differentiable mapping, it may occur that the Jacobian matrix is never full rank, that is "maximal rank" < "full rank". On the other hand, when "generic rank" is defined (that is in the context of algebraic geometry), it is true that "maximal rank" = "generic rank". As an example of a situation where the term "full rank" is useful, and probably widely used, is the following result: Given a set of polynomials that generate a prime ideal, the algebraic variety of their common zeros is a complete intersection if the Jacobian matrix is "full-rank" at, at least, one point of the variety. In this case, the singular points are exactly the points where the Jacobian matrix is not full-rank. D.Lazard (talk) 09:56, 18 May 2015 (UTC)
In essence then, "maximal rank" is generally used for the supremum of ranks of a set (e.g. Jacobian matrices of a map) of matrices, while "full rank" applies to one matrix? YohanN7 (talk) 11:53, 18 May 2015 (UTC)
There is another case to consider: infinite-dimensional matrices. Because we can zero a column of a full-rank matrix without changing the rank, the rank remains maximal, but because the column span is no longer that of the resulting vector space (the map is no longer surjective), it is no longer of full rank. But this argument depends on whether the rank of a matrix is defined for infinite-dimensional matrices. —Quondum 14:19, 18 May 2015 (UTC)
I don't think "rank" is helpful for infinite-dimensional matrices unless finite; but my "natural" inclination is to consider "injective" rather than "surjective" as the definition of "full rank" for infinite-dimensional matricies. — Arthur Rubin (talk) 20:34, 18 May 2015 (UTC)

Isn't the matter settled? Don't we have two reliable sources for full rank (Gentle and Lay), ignoring bad spelling in the former? More sources don't explicitly define the term because they don't need it or its meaning is obvious? (I don't want to dictate conclusion of discussion. I'm just trying to figure out whether I need to keep paying attention.) Mgnbar (talk) 13:45, 18 May 2015 (UTC)

I'm treating the original question as settled (despite my personal reservations), though any further discussion may lead to tweaks, e.g. distinctions between "maximal" and "full". —Quondum 14:19, 18 May 2015 (UTC)
I agree. --JBL (talk) 20:46, 18 May 2015 (UTC)

I realize I'm late to the party, but thought I would chime in. The term "maximal rank" can be ambiguous. Consider the following statement:

Suppose that f:\mathbb R^n\to\mathbb R^m is a continuously differentiable function, and let U be the set of points where the derivative of f has maximal rank.

A rather trivial example of the ambiguity is the constant function f(x) = 0 for all x. The rank of the derivative is zero everywhere, so the maximum value of the rank of f is equal to zero! Thus (under this interpretation) U=\mathbb R^n. Now, clearly for "most" applications, this is not the interpretation that would be intended by the statement. Rather, we would mean

Suppose that f:\mathbb R^n\to\mathbb R^m is a continuously differentiable function, and let U be the set of points where the derivative of f has full rank.

Here full rank means that the rank of Df is as large as it can possibly be for an m\times n matrix. Thus, the restriction of f to U is a submersion, if n\ge m, and an immersion, if n\le m. Sławomir Biały (talk) 15:47, 18 May 2015 (UTC)

Considering the reservations expressed here (by D.Lazard and Sławomir Biały), I have removed my unsourced addition of 'maximal rank 'to Rank (linear algebra), leaving the definition of 'full rank'. —Quondum 03:57, 21 May 2015 (UTC)

Face configuration[edit]

The article title Face configuration appears to be a neologism: neither Cundy & Rollett nor Williams, both cited, use the term. Rather, they use the symbol to identify the related polyhedron (typically a Catalan solid). I do not have Grünbaum and Shephard to hand, but I have never heard the term in this connection. The article makes much of Cundy & Rollett's (non-existent) usage. Do we accept this kind of apparently fabricated usage, or should this kind of article be nominated for deletion? — Cheers, Steelpillow (Talk) 11:47, 19 May 2015 (UTC)

Vertex configuration is another article in the same genre it would seem. Although possibly slightly less WP:ORish, the term "vertex configuration" (or the other synonyms listed there) do not appear to be in wide use. I think this article should be redirected to vertex figure. The notation can be mentioned there, without creating a neologism. I don't know if there is a suitable merge target for face configuration, though. Sławomir Biały (talk) 13:34, 19 May 2015 (UTC)
Face configuration is a fairly well known concept in a facial recognition system, possibly a notable topic in the geometric/parametric approach to facial recognition. I could not find sources describing the polyhedral version. --Mark viking (talk) 16:39, 19 May 2015 (UTC)
As stated, the two sources are Cundy-Rollett for Archimedean dual polyhedra and Grünbaum and Shephard for Monohedral/Lave tilings. Williams repeats the Cundy-Rollett symbol usage. Mathworld calls it a Cundy-Rollett symbol. I'm open to merging Vertex configuration and Face configuration as one article and calling both Cundy-Rollett symbol, like at [3]. Vertex figure is functionally different than vertex configuration, not a symbol but a n-dimensional polytope existing at a vertex. Tom Ruen (talk) 16:56, 19 May 2015 (UTC)
I cannot find reference to "Cundy-Rollett symbol" on Mathworld, only a passing reference to the "Cundy and Rollett symbol" for the Archimedean solids. Either way, use for the odd table column heading does not establish notability. The "functional difference" alluded to above is trivial: a "vertex configuration" is just a symbol denoting a certain kind of vertex figure, so I agree with Sławomir Biały that merging is a good way to go there. But Face configuration is not a recognised term and the thing it denotes is not notable either - witness the fact that the purported sources use it in a very minor way and don't even bother to give it a name. If anything in the article worth keeping can be found, it can be merged across into Face (geometry), but AFAICT there is nothing even to justify a redirect and the article itself should be deleted. — Cheers, Steelpillow (Talk) 19:22, 19 May 2015 (UTC)
It is a symbol, not a figure. And I'd rather keep it separate from Face (geometry) and collect it with vertex configuration, whatever might be called. Here's another newer reference for Archimedeans Cundy-Rollett symbol. Tom Ruen (talk) 19:44, 19 May 2015 (UTC)
"Cundy-Rollett symbol" gets just ten hits on Google, most of which are scrapings from here - and Popko's book, which mentions it only briefly. That is not enough to establish notability. — Cheers, Steelpillow (Talk) 21:13, 19 May 2015 (UTC)
Whatever you'd like to call it, it's used EVERYWHERE! Would you prefer something like A universal symbol that represents the sequence of vertex valances around a face of a polyhedron or tiling?! Tom Ruen (talk) 21:40, 19 May 2015 (UTC)
When you say "it's used everywhere", you really mean that you, personally, are responsible for adding it to a large number of our articles, right? Did you have sources when you did so? —David Eppstein (talk) 22:53, 19 May 2015 (UTC)
No, by "everywhere", I means every book or paper that talks about about regular, semiregular, uniform polyhedra and their duals. Tom Ruen (talk) 23:57, 19 May 2015 (UTC)
Surely such a prominent concept has an established name. Ozob (talk) 02:06, 20 May 2015 (UTC)
For the more popular vertex configurations, I've found 9 descriptive names for the symbol: vertex configuration, vertex figure, vertex type, vertex symbol, vertex arrangement, vertex pattern, face-vector, vertex description, and Cundy-Rollett symbol. Tom Ruen (talk) 05:58, 20 May 2015 (UTC)
That is because the symbol is part of the theory of uniform vertex figures and has no significance outside of that topic. Moreover its mathematical significance even within that topic is so trivial that nobody has ever bothered to agree an accepted name. All those other terms are mere ad hoc descriptions, because that is all it deserves. The "face configuration" is even less justifiable. — Cheers, Steelpillow (Talk) 07:24, 20 May 2015 (UTC)

Sourcing?[edit]

Now I find myself being reverted. The article at Vertex configuration gives "Cundy-Rollett symbol" as a synonym and cites mathworld. As I pointed out above, mathworld does not in fact use this term and the cite is therefore incorrect. As I also pointed out, one lone author does not establish notability. However when I removed the reference Tom Ruen chose to revert my edit without comment. Is this behaviour acceptable to this Project? — Cheers, Steelpillow (Talk) 07:04, 20 May 2015 (UTC)

I restored the removal, and added a second source, and commented on the second source in the revert. Tom Ruen (talk) 08:48, 20 May 2015 (UTC)
No you have not restored my removal. You have moved it to the article lead and wrapped a whole load more trivial sources around it. I have to ask again, is this behaviour acceptable to this Project? — Cheers, Steelpillow (Talk) 19:05, 20 May 2015 (UTC)
I said I restored what you removed and I added a second reference along with my revert, and I named the second reference in the comment. I did not revert "without comment". After that I continued to improve the content as a good editor should. You can see "trivia" as you like, but my intention was to demonstate varied ways the symbol was named. If someone reads the article, they can check the usages and decide for themselves which name has the best legitimacy or whatever. Tom Ruen (talk) 03:40, 21 May 2015 (UTC)
Nevertheless, you reverted me. The tenor of this discussion is that thse sources are inadequate and you should not have done so. Yet you persist in embellishing them while the discussion continues. — Cheers, Steelpillow (Talk) 08:08, 21 May 2015 (UTC)
I'll let smarter people than me validate your confusing charges against me. I've done NOTHING but try to please you, and all you see is insult. Tom Ruen (talk) 09:59, 21 May 2015 (UTC)
So, reverting me is trying to please me. I'll remember that next time I want to try and please you. Face-wink.svg — Cheers, Steelpillow (Talk) 12:24, 24 May 2015 (UTC)
Indeed. removing-without-improving and reverting-with-improving might weigh as equal acts of kindness. Tom Ruen (talk) 19:28, 24 May 2015 (UTC)

Another sourcing[edit]

Tom Ruen (talk) 09:50, 20 May 2015 (UTC)

Physical Metallurgy: 3-Volume Set, Volume 1, edited by David E. Laughlin,
Page 16

The Cundy and Rollett symbol of a vertex configuration nm means that m n-gons meet at a vertex. The vertex configuration can also be written in the form of the Schlafli symbol {n,m} or (n,m). The eight semiregular Archimedean tilings are uniform. This means they have only one type of vertex configuration, i.e. they are vertex transitive; they consist of two or more regular polygons as unit tiles. In the case of layer structures, where one layer type corresponds to one of the Archimedean tilings, the layer next to it will preferentially be the respective dual tiling (Catalan or Laves tiling). The dual to a tiling can be obtained by putting vertices into the center of the unit tiles and connecting them by lines. If the tiling is regular, then the dual tiling will be regular as well. The dual of the regular square tiling is a regular square tiling again, so this tiling is self-dual. The dual to the hexagon tiling is the triangle tiling. While the uniform semiregular tilings are described by their vertex configuration, their duals consistent of just one type of polygon (are isohedral), but have more than one vertex configuration. Therefore, they are described by their face configuration, i.e. the sequential numbers of polygons meeting at each vertex of a face. For instance, the dual to the Archimedean snub square tiling 32.4.3.4 is the Cairo pentagon tiling, V32.4.3.4. Its face configuration V32.4.3.4 means pentagonal unit tile with corners, where 3,3,4,3,4 squashed pentagons meet.

page 20

The Archimedean solids can all be inscribed in a sphere and in one of the Platonic solids. Their duals are the Catalan solids, with faces that are congruent but not regular (face-transitive); instead of circumspheres like the Archimedean solids, they have inspheres. The midsphere, touching the edges are common to both of them. The face configuration is used for the description of these face-transitive polyhedra. It is given by the sequential listing of the number of faces that meet around each vertex around a face. For instance, V(3.4)2 describes the rhombic dodecahedron, where at the vertices around one rhombic face 3,4,3,4 rhombs, respectively, meet.

I'm not sure this is the best source for establishing standard usage in geometry. Apart from the obvious issue that this is a metallurgy textbook (and the terminology does not seem to be standard in metallurgy either), the neologism appears only in the fifth 2014 edition, long after we had an encyclopedia article on the subject. Also, the description that appears on page 20 is a paraphrase of our article on the subject, including the same choice of example. So it seems that one source has merely picked up our neologism. An encyclopedia should merely reflect what is already standard; it should not be in the business of inventing new standards. Sławomir Biały (talk) 12:04, 20 May 2015 (UTC)
It looks like a large fraction of the content above was referenced and taken from Crystallography of Quasicrystals: Concepts, Methods and Structures by Walter Steurer, Sofia Deloudi, (2009) p.18-20, p.51-53. Tom Ruen (talk) 10:23, 21 May 2015 (UTC)
Yes, this seems like a better source. Sławomir Biały (talk) 13:15, 22 May 2015 (UTC)
The oldest source I can find using the phrase vertex configuration is this 1993 paper on the uniform star polyhedra: [4]. Pretty much ALL the online constructions of the uniform stars trace back to this paper, of course the original source is the Coxeter 1954 paper on uniform polyhedra which draws the diagrams as polygons rather than listing the n-gons. Tom Ruen (talk) 10:33, 21 May 2015 (UTC)
  • Uniform Solution for Uniform Polyhedra*
  • Zvi Har’El
  • Department of Mathematics, Technion − Israel Institute of Technology, Haifa 32000, Israel, E-Mail: rl@math.technion.ac.il
  • ABSTRACT: An arbitrary precision solution of uniform polyhedra and their duals is presented. The solution is uniform for all polyhedra given by their kaleidoscopic construction, with no need to ‘examine’ each polyhedron separately.
I'd summarize and conclude Cundy and Rollett symbol is a good name for this notation for giving credit for their 1952 book Mathematical models, first published book expressing the convenient notation for the (convex) regular, semiregular, and semiregular dual polyhedra, as integers a.b.c. ... Then Zvi Har’El's 1993 paper extend that usage as vertex configuration and may get first credit for the first consistent notation for the uniform stars and duals, which include full orientation information (prograde/retrograde) and wrap information, which makes the stars explicitly constructable: like (a/b . c/(d-c) . ... x/y)/z. Roman E. Maeder then ported it into Mathematica around 1995, using the same terminology, while Robert Webb used it in Stella around 2001, but instead calling it a vertex description. Tom Ruen (talk) 11:22, 21 May 2015 (UTC)
In parallel, and a bit earlier Tiling and Patterns (1987) uses a similar more general notation for Euclidean tilings, calling the notation tile symbols or of type. It also has hollow tilings with star polygons and retrograde orientation, using negatives for retrograde, so somewhat similar to Zvi Har’El's star polyhedra usage. Tom Ruen (talk) 11:44, 21 May 2015 (UTC)

AfC submission[edit]

Care to offer insight into Draft:Topological Functioning Model? Thanks! FoCuSandLeArN (talk) 20:01, 19 May 2015 (UTC)

Category ICM 2014 Plenary and Invited Speakers[edit]

I've just created the category for the ICM 2014 Plenary or Invited Speakers. A list with the names of the speakers can be found on http://www.mathunion.org/db/ICM/Speakers/SortedByCongress.php . If someone wants to help to add the category to more articles, be more than welcome! Lolaszvodikech (talk) 00:44, 22 May 2015 (UTC)

Off-topic: A mathematician named László Erdős is listed there! I wonder if he is a relative of Paul Erdős! Anyway, with such a name he must attract a lot of attention :). Lolaszvodikech (talk) 01:28, 22 May 2015 (UTC)

Lecture notes as a reference?[edit]

See recent edits at Brouwer fixed-point theorem, and also my talk page. I think lecture notes are sometimes okay and sometimes not depending on what is available. In this case, a simple Google search will give millions of hits, and I don't see the need to have lecture notes linked. YohanN7 (talk) 11:38, 24 May 2015 (UTC)

Any lecture notes, particularly the terabytes of pdf notes floating around, should just be external links not references, they are not published mainstream secondary sources. M∧Ŝc2ħεИτlk 10:37, 26 May 2015 (UTC)
These are external links (didn't start out that way though), so I guess it is fine. Not a huge fan of the practice though. Why should we "promote" or "endorse" them? YohanN7 (talk) 10:45, 26 May 2015 (UTC)
It isn't "promotion" or "endorsement", I thought the purpose of external links is to point to other sources of info which are not considered reliable and are not secondary but have at least some usefulness. M∧Ŝc2ħεИτlk 10:49, 26 May 2015 (UTC)
There is a difference between neutral links to MathWorld and the like and lecture notes. But I have myself committed the crime (actual references, not external links) on occasion when nothing else has been available to me. See this more like me asking a question than pushing a POV (thought my posts look and sound like POV pushingFace-smile.svg) YohanN7 (talk) 11:18, 26 May 2015 (UTC)
I would say it depends on the circumstances. Even lecture notes by a well-known authority can be used as references (with care, and assuming there are no better sources available), per WP:SPS. I would say that this source is a good one. The chapter discusses the degree of a mapping from the perspective of multivariable calculus. The Brouwer theorem is a consequence of the homotopy invariance of the degree of a function. This is something that could, in principle, be added to the article, since the connection with degree is not really made clear. Sławomir Biały (talk) 12:19, 26 May 2015 (UTC)
I can buy that in full. As a side note, both Steven Willard (General Topology) and John M. Lee (Introduction to Topological Manifolds) handle it using homotopy theory (unless my memory fails). Especially Willard's proof was kind of neat, I recall, (and super-short). Will check this out. YohanN7 (talk) 13:38, 26 May 2015 (UTC)

First article about Indonesian mathematician on WP-En[edit]

Hello. I've just created the first page about any mathematician from Indonesia at the English Wikipedia. The name of the page is Moedomo Soedigdomarto. My English is not very good, and the sources are all in Indonesian (my 3rd language, which I don't know very well too...) Anyway, if someone wants to help. Please feel free to join the effort. Chinese-Indonesian (talk) 08:46, 26 May 2015 (UTC)

Looks like a nice start, especially for someone who is not a native English-speaker, good job! But links (like the first two references 1 and 2) in Latin-alphabetized Indonesian are not too helpful for English speakers in English Wikipedia. Also, strong claims like "He was one of the first Indonesians" need to be cited. If possible please add more English sources. M∧Ŝc2ħεИτlk 10:34, 26 May 2015 (UTC)

Mathematical logic and Goedel sentences[edit]

Please take a look at mathematical logic. --Trovatore (talk) 18:30, 26 May 2015 (UTC)

Also axiom, please. --Trovatore (talk) 04:19, 27 May 2015 (UTC)
What is the problem with those articles? — Arthur Rubin (talk) 23:53, 28 May 2015 (UTC)
If I read the talk page correctly, it involves a philosophical debate between a Platonic position in which one can say that a first-order sentence about the integers is true of "the" integers (even if it might be false for some models of the Peano axioms), and a radically relativist position in which sentences may be stated to be true of models but not true in any absolute sense and in which no model of Peano's axioms (not even the model given by the finite ordinals) is privileged as being "the" integers. This affects the article in terms of whether it is more accurate to say that the Gödel sentence is "true, but unprovable" or "neither it nor its negation are provable". —David Eppstein (talk) 01:13, 29 May 2015 (UTC)
Not quite: it's about whether a first-order sentence should be said to be true without qualification, or true of the integers. But anyway it's resolved. --Sammy1339 (talk) 04:13, 29 May 2015 (UTC)

Fundamental lemma of calculus of variations[edit]

Much weakened version of that lemma is formulated in our article. The reason (articulated on the talk page) is that the simple proof given in the article does not give more. As for me, irrespective of the proof, a stronger formulation should be given. But for now I have only lecture notes as sources. I wonder, do you know better sources? Mine are:

Boris Tsirelson (talk) 11:53, 29 May 2015 (UTC)

Wikipedia isn't in the business of giving proofs we should be pointing to textbooks for that, though of course trying to explain it is an important part of Wikipedia's business. I must admit I can't see why it has been split or forked out of the article on the calculus of variations, it isn't as though that article is too big and this is a very important part of it. Dmcq (talk) 12:36, 29 May 2015 (UTC)
Of course if you can expand that article with another formulation then of course it deserves its own article, it just seemed rather short and a duplication of bits of the main article to me at the moment. Dmcq (talk) 12:53, 29 May 2015 (UTC)
Well, I am neutral about possible merge. I bother that the optimal function should be proved to be smooth, rather than assumed. That is, in the fundamental lemma, the given function "orthogonal" to all smooth functions must vanish even if not assumed smooth. I could expand; the problem is, whether "my" sources are reliable enough. Boris Tsirelson (talk) 13:44, 29 May 2015 (UTC)
I'm not entirely clear on your objection. Do you want to strengthen the lemma by saying that h \in C^{\infty} instead of just C^{k}? That would be the grownup way to write this; it would involve bump functions instead of the current proof (which is kind of cute nonetheless). --Sammy1339 (talk) 15:06, 29 May 2015 (UTC)
Oh I see. Yes it should be strengthened. --Sammy1339 (talk) 15:06, 29 May 2015 (UTC)
Smoothness of h matters, too. But I mostly bother about smoothness of f. Boris Tsirelson (talk) 15:18, 29 May 2015 (UTC)
(Edit conflict) Really, that lemma should be treated as a special case of the fact that every weak solution of a linear ODE is also a strong solution. Regretfully, I did not find this fact in our article "weak solution" (nor in "distribution"). I also did not find this connection in texts on calculus of variations. Is it my Original Research?! Boris Tsirelson (talk) 15:36, 29 May 2015 (UTC)
I just looked in Gelfand & Fomin and interestingly they follow the exact same steps as your sources above, but this formulation is neither stronger nor weaker than what appears in the article. You require a stronger hypothesis (\forall h \in C^{2} instead of just \forall h \in C^{k}.) Then there is Lemma 4 in your sources above (which is also Lemma 4 in G&F on p. 11) which is just integration by parts basically showing that if f is continuous and weakly differentiable then it is differentiable. --Sammy1339 (talk) 15:28, 29 May 2015 (UTC)
Ah, yes, it is nice to know that Gelfand & Fomin is a good source! Given that you have this book while I do not (for now), maybe you'll improve the article accordingly? Boris Tsirelson (talk) 15:39, 29 May 2015 (UTC)
And yes, "if f is ... weakly differentiable then it is differentiable" is just what I want to see. Boris Tsirelson (talk) 15:42, 29 May 2015 (UTC)
I'm looking for a source for that which states it full generality. --Sammy1339 (talk) 15:58, 29 May 2015 (UTC)

See theorems 1.2.4 and 1.2.5 in volume 1 of Hörmander. If f is a continuous (resp. locally integrable) function st for all compactly supported smooth φ

\int f \phi=0

then f=0 (resp f=0 a.e.) Sławomir Biały (talk) 16:47, 29 May 2015 (UTC)

Formal linear combination[edit]

This notion is used, for example, in "Chain (algebraic topology)", with a link to "Free abelian group"; there, "formal sums" are defined (but "formal linear combinations" are not). In some books I see "formal linear combinations" used with no definition (and often, with no explanation). Once I used it on an undergraduate lecture and was asked by students: "what's it?"

Should we redirect "Formal linear combination" to "Free abelian group"? Should we enlarge the latter, including non-integer coefficients? Boris Tsirelson (talk) 17:27, 31 May 2015 (UTC)

Free module seems like a better target, although that article could use improvement. Sławomir Biały (talk) 17:56, 31 May 2015 (UTC)
For now it does not mention "formal linear combinations". Boris Tsirelson (talk) 19:02, 31 May 2015 (UTC)
The book Introduction to Topological Manifolds by John Lee introduces "formal linear combinations" in the context of free Abelian groups. --Mark viking (talk) 18:39, 31 May 2015 (UTC)
I guess, it uses only integer coefficients; but (at least) real coefficients are really needed in the context of Stokes theorem.
I also bother that "free module unique up to isomorphism" is demanding for some readers that could be satisfied with "finitely supported functions on the set of (...)". Boris Tsirelson (talk) 19:02, 31 May 2015 (UTC)

I have rewritten the corresponding section of Free module for defining explicitly formal linear combination (and also for being less technical). I have also created the redirect Formal linear combination. D.Lazard (talk) 09:17, 1 June 2015 (UTC)

Very nice; now I can recommend it to my undergraduates. Boris Tsirelson (talk) 15:32, 1 June 2015 (UTC)
Now if only someone could do something about that lead paragraph ;). --JBL (talk) 19:44, 1 June 2015 (UTC)

No new articles[edit]

At Wikipedia:WikiProject Mathematics/Current activity, we find no new articles since May 25. Jitse's bot, run by Jitse Niesen, which edits that page, has done so more recently, and mathbot, which, among other things, lists new articles, whose lists are used by Jitse's bot, has been active more recently. Are there no new articles? Michael Hardy (talk) 18:44, 31 May 2015 (UTC)

Are you looking at the same current activity that I am? Because I see 14 new articles on May 26 (starting with Alicia Dickenstein), 6 new articles on May 28, and a large number of new articles (possibly caused by moving some category involving tilework into the ones the bot lists) on May 29. —David Eppstein (talk) 19:17, 31 May 2015 (UTC)
Yes, there a many entries dated after May 25. A diff shows them. I see "This page was last modified on 31 May 2015" at the bottom of Wikipedia:WikiProject Mathematics/Current activity. If you se an older date then try to bypass your cache. PrimeHunter (talk) 19:26, 31 May 2015 (UTC)