3484:{{a,b}}" that maps each point to the set of all its open neighborhoods, and so on. I'd say, this is similar to describing a vector in this or that coordinate system. But for a vector, our level is much higher! We have the general notion of a coordinate system, a general transformation formula for vector coordinates, and (if we are physicists) we can define a vector as something that transforms this way. For topologies, even this "physical" level is still in the sky! Do we imagine the class of all (rather than these 7) equivalent definitions of a topology? Can we define a topology as something that transforms as required from one definition to another? (Yes, we can do so for the 7 definitions, but I really mean all potentially possible definitions.)
2341:
a geometric object which satisfies axioms (about parallel lines and parallelograms). An affine space (2) can be modelled by an affine space (1), the para-associative law is the existence of parallelograms. A linear space is an affine space (2) with a distinguished point called the origin. A vector space is an affine space (1) with a distinguished element called zero. The questions are, how to present this on
Knowledge, and in particular whether the slogan in question is helpful or unhelpful for the reader who wants to understand and remember the material.
40:
3005:
of gamma functions, is different from both. I am now thinking there should be separate articles. For instance, the "semifactorial" n(n-2)... is used in combinatorics, while the gamma-function-defined "double factorial" is not. On the other hand, it's possible the "double factorial" mentioned in some of the analytical articles is not the semifactorial (but it is, in articles on volumes of spheres). I am abandoning the edit of
3551:. I believe that the disciples of homotopy type theory would tell you that if two definitions give you equivalent ∞-categories, then the two definitions are indistinguishable, and therefore you shouldn't worry about the difference. I think, however, that that will not satisfy you. Your question seems to ask for a Bourbaki-style answer in terms of sets with additional structure.
3131:"Semifactorial" would be unambiguous. It "clearly" means alternating factors and nonfactors down to 1. (Cf. Michael Hardy's following comment.) "Odd factorial" would also be unambiguous; it "clearly" means only odd factors down to 1. (I personally never heard of it and have never seen any use of it.) "Double factorial" is the problem. This should be continued on the
3835:'Mathematicians are of course used to identifying isomorphic structures in practice, but they generally do so by "abuse of notation", or some other informal device, knowing that the objects involved are not "really" identical. But in this new foundational scheme, such structures can be formally identified...' (Subsection "Univalent foundations" of Introduction in book:
1976:
2447:"Take an affine space (A,V). Choose a point a as origin. Then you have a particular representation of A, together with a vector space V. You do not have V=(A,V), which is what you are saying, and which is patent nonsense. To get the vector space from an affine space you have to forget A, not choose an origin. The origin is a point in A. It is not in V."
2783:
field is auxiliary, not principal. Accordingly, no one introduces "the forgetful functor from the category of vector spaces to the category of fields". Likewise, for an affine space, its difference space is auxiliary, not principal. Indeed, "a continuously differentiable map from one finite-dimensional affine space to another" means a map from
3869:. While there are surely foundational and metamathematical sources that discuss structural equivalence in these broad terms, the idea of isomorphism occurs in all three systems of set, category and type theory and could be a good launching point for readers. There is already some material on the notions of isomorphism and equality there. --
3555:
appropriate. (For example, the category "topological spaces in terms of open sets" is isomorphic (not merely equivalent) to the category "topological spaces in terms of closed sets".) So long as you are clear on exactly what kinds of properties you are hoping the definitions to express, this has always seemed sufficient to me.
3962:"Chi-squared divergence" gets 58 hits in GScholar, not a lot, and none of the most cited papers are about this divergence in particular. It is a minor variant in a family of distribution metrics and divergences. I think notability is marginal and may not survive an AfD. The divergence is verifiable, however, and a redirect to
2307:. The section that you continue to try to delete, without consensus, is clearly not meant to be a formal description anyway (even the title of the section is "Informal descriptions".). Regarding the use of the word "forget", the assignment of the affine space underlying a vector space is indeed an example of a
3602:
Categories are very interesting, and I use this occasion for learning more on them. However, I could not like such a definition of topology that cannot answer questions like "is this set open in this topology?" I want to use the Baire category theorem (when applicable). I want to consider probability
3019:
Note 2: Possibly
Meserve was the first to use the !! notation but the semifactorial was certainly known long before that. What it was called, if anything, is not known to me. My teachers in the 1960s said the name was "semifactorial" (and I believed them implicitly), so I'm sure there is something
2856:
Such functors may superficially seem to be forgetful in one definition (of the "from" category), but do not sustain transition to an equivalent definition. A topological space may be defined, equivalently, via neighborhoods. An affine space may be defined equivalently... see above... not mentioning a
2640:
This is demonstrably equivalent to the "(A,V)" characterization of affine spaces. I leave the proof of equivalence to RQG as an exercise. And any undergraduate reading this may also find it useful to go through this exercise. By this second characterization of the concept of affine space, a vector
2460:
This says "Take an affine space (A,V)". I take that to mean A is a set and V is a vector space that acts transitively on that set in a way that satisfies certain desiderata. RQG seems to say that if you then delete A from this structure, you're left with V, so that an affine space is something more
2335:
RQG has a point, although he presents it in a way which is not entirely helpful (appeals to personal authority tend not to go down well). The question is about language and the abuse thereof, creative an otherwise. One could unpick the various concepts as follows. A vector space over a field is an
3783:
A notion of a mathematical structure arises from our intuition; and these bijections between different "implementations" are dictated by our intuition (rather than a formal requirement). Therefore the number of "implementations" must be finite (since our intuition cannot do more) (but of course some
2340:
or the origin. The axioms of a vector space are sufficiently abstract that the concept is capable of modelling a linear space. An affine space (1) is an algebraic object which is a principal homogeneous space over a vector space, or a commutative heap on which a field acts. An affine space (2) is
4244:
Insert obligatory quote from Knuth here: "The material that follows comes mostly from a larger area of mathematics known as the theory of graphs. Unfortunately, there will probably never be a standard terminology in this field, and so the author has followed the usual practice of contemporary books
3462:
Surely, in each case these definitions are equivalent. But what exactly does it mean? From the article "Topological space": "there are many other equivalent ways to define a topological space: in other words, the concepts of neighbourhood or of open respectively closed set can be reconstructed from
3224:
It's actually written in a rather impenetrable engineering jargon/style as far as I'm concerned. It's basically just a bunch of examples, and it's kinda missing all its math content/background, which is a bit non-trivial. A search found that the recommend text (up to 1980s or so) for the mathy part
3004:
Excellent idea, David
Eppstein. I'm afraid I jumped into a confused area of terminology. For instance, the "double factorial" article gave three definitions of an even "double factorial", one of which is "semifactorial", another of which is called "odd factorial", and the third of which, in terms
2782:
Yes. But, ridiculously, RQG thanks John for clarifying "that an affine space has more structure than a vector space"! I guess, the problem is, not understanding the difference between "principal base sets" and "auxiliary base sets" (in the
Bourbaki terminology). For a vector space over a field, the
1907:
It requires some thought. Left invariant vector fields constitute the Lie algebra of a Lie group, so that should probably go into one of the major articles (Lie group?). I think exponential coordinates and group topology perhaps could go in there as well, since they are natural constructs for a Lie
3475:
the equivalence means that "there is a bijection between the field orderings of F and the positive cones of F". I dislike this formulation. As for me, "there is" means "exists", and "exists a bijection" means, equal cardinalities. No, surely this is not meant! Rather, it is meant that the specific
3198:
I agree with both of you (thanks, Michael Hardy). I will still favor "semifactorial" in my own writing. However, as an encyclopedia, WP ought to follow convention. I'm now convinced that "double factorial" is overwhelmingly the most used. And thanks again to Arthur Rubin for reverting for me.
2461:
than a vector space: If you start with an affine space and discard part of the structure, you're left with a vector space. That is consistent with at least this much of the way I originally learned it: An affine space has an underlying set A and some vector space that acts on A in a certain way.
2164:
Just a small comment: it's not a good practice to confuse the zero vector with the origin, or is it? A polynomial ring (over a field) is a vector space but you don't really call the zero polynomial the origin. The term "origin" suggests positional concepts; i.e., a choice of basis/coordinates. The
3554:
Personally, I have long thought that for formal purposes, one should treat different definitions as defining different kinds of structure (whether you're working in terms of structured sets, categories, ∞-categories or whatever) and then prove that those structures are equivalent or isomorphic as
3293:
are application specific, but aren't exactly graphs themselves so don't neatly fit in the 2nd category. The third category seems to be the most problematic, as it seems to contain mostly items that should be in the 2nd one (app-specific graphs) or some variation thereof, i.e. are graphs augmented
2226:
This seems very standard as an intuitive description of an affine space. The zero in a vector space is often called an origin, especially when concerned with the geometry of affine spaces. The article includes a quotation by Marcel Berger including the term "origin" in this context. It should be
2068:
In that case, "Cartan's theorem" is decidedly inappropriate for the "Closed subgroup theorem", much more so than "Lie correspondence" for "Lie group–Lie algebra correspondence", which is actually used in the literature. "Lie correspondence" can hardly be confused with "Algebraic group–Lie algebra
1420:
should be an article in its own right, not only a disambiguation page. The concept clearly applies (under the name "quotient") throughout abstract algebra, and seems amenable to a general definition. It would naturally link to the articles for the concept as applied in subdisciplines, for example
3779:
After more thinking I see how naive is my original idea that functoriality itself can dictate a single bijection between, say, topologies as sets of open sets, and topologies as families of neighborhood filters. (Initially I wrote I do not know whether this fails... now I see it surely fails.)
3744:
Sure. But here I do not really discuss definitions of a vector space. Rather I use this as a metaphor; see my first message, the paragraph "Now, some thoughts". And I would be happy to treat topologies as an abstract data type, and the 7 definitions as 7 implementations (equally "official", or
2358:
Is not the idea of a principal homogeneous space as a space with a free and transitive group action already a geometrical one? One often describes a principal homogeneous space as a group without a fixed identity element. That is a group in which we forget the identity. So I don't think the
2261:
May I say that I have a first in mathematics from
Cambridge University, and a Ph.D. in mathematics from University of London, so I think I am by training closer to mathematics than to physics. As has previously been remarked by other, the mathematical definition of vector space, which does not
2045:
About that. "Lie correspondence" didn't seem descriptive enough for me. Considering
Knowledge is a general encyclopedia, when there is no official or standard term, it's better to use a descriptive name. This is for example why Knowledge includes the year in the name of an event article: e.g.,
1764:
Thanks for the info. I'll definitely do the closed subgroup theorem because it seems manageable enough. The Lie correspondence is quite an undertaking, and I'll let that be for a little while to come. It really surprises me that there is no such article because the Lie correspondence is truly
2187:
Tastes differ. In the theory of Banach spaces it is very usual to call 0 the origin. If I think of the vector space of polynomials geometrically, then indeed I think of 0 as the origin, of constant polynomials as a straight line through the origin, and so on. But maybe this is typical for a
3023:
Note 3: There should be a distinction between notation and concept. The function "semifactorial" ("double factorial" in the parity product meaning) is not the same as the "double factorial" notation. Is the article to be about the function(s) or the notation? (Is the answer obvious?!)
3697:
I think it's helpful here to distinguish the defining properties of a mathematical object (e.g. a vector space is a group with a scalar multiplication operation over a field (etc), its dimension if finite is the maximum number of independent vectors, etc) from the models of those objects
3317:
Many of the things in the 3rd list are data structure that take the form of graphs. I agree that they should be separate from data structures for representing graphs. The category for data structure in the form of graphs should probably be a subcategory of application-specific graphs.
3463:
other starting points and satisfy the correct axioms". But it refers to the main article "Characterizations of the category of topological spaces"; there the equivalence means isomorphism of categories. This does not make me happy. First, having the category of topological spaces
2095:? The term would work perfectly in the books with the titles containing "Lie groups" but not quite in the places like here. The same problem with "closed subgroup theorem". Without context, it's not clear it's a result in Lie theory, while "Cartan" conjures the feel of Lie :) --
3887:
Maybe. But, as noted by David (above), it is rather a cryptomorphism. Indeed, it is unusual to consider isomorphisms between different (differently described) structures. Moreover, in order to introduce such notion, a specific transition between such structures must be chosen.
2336:
algebraic object with algebraic axioms. Being a group it has a zero element. A linear space (a term which is not much in use, and here redirects to vector space) is a geometric object satisfying certain geometric axioms. As a geometric object it has a special point called
3493:(in other words, naturally isomorphic) functors. It is tempting to consider the whole equivalence class of functors (similarly to the class of all coordinate systems). Pretty elegant, and general (applies to all structures, not just topologies). However, there is a problem.
2113:(one form of it actually refers to a form of the Lie correspondence)) in the literature, but have always seen "Lie correspondence". But let us not argue about this. The descriptive names (as they are now) are, in view of this discussion, clearly better in both cases.
1338:, I suspect the current version of the article covers a non-notable distribution which shares a name with a vastly more notable one, the latter of which is also called the Pólya-Aeppli distribution. We should rewrite the article so it covers the notable topic.
3569:
I think that seeing the equivalence between defining topological spaces in terms of open sets or in terms of closed sets as categorical is somewhat weaker than what's going on the the mind of the mathematician: they are attempts to define the "same thing".
4150:(ec) Current convention is not to delete articles if the underlying topic is notable. Is there a single thing that is represented by "Grafting (ordered tree)" that could be used to cut an unsatisfactory article back to a stub and then grown out again?
3860:
I agree that type theory presents an intriguing alternative to set theory and category theory as a foundational system. In
Shulman's piece, there was a good discussion of equality and isomorphism. Getting back to your original proposal, perhaps the
2311:. If one regards vector spaces geometrically, rather than algebraically, it is precisely the functor that "forgets" the origin. This perspective is well-supported by high quality references (and standard use in the mathematical community).
2148:
Please look. Recent edits are generally constructive, but made by a person closer to physics than math (I feel so), with somewhat different philosophy. As for me, the views of physicists are welcome, but our views should not be exterminated.
3502:
For topologies in general, I do not know. (Do you?) But for some mathematical structures the answer is discouraging: yes, there exists a nontrivial natural equivalence from one such functor to itself. For groups, it happens because of
3119:, all of which coincide if the argument is an odd integer (positive or negative), and differ if the argument is a nonnegative even integer. As we (Knowledge) seem to use it only for integer arguments, it may make a difference. —
2262:
mention an origin. It has an identity element, which certainly cannot be forgotten, and which cannot be identified with an origin. It is not appropriate for mathematical articles to be written by people abusing mathematical words.
3510:
Thus, it seems, we are able to list a finite number of definitions (for a given mathematical structure) and write down a consistent (that is, commutative) system of natural equivalences between them. But we are unable to do more.
4356:
Your sarcasm is unwarranted. Deltahedron is a well-known contributor here, and his comments are calm and in line with
Knowledge policy. We all share your frustration with material that shouldn't be on Knowledge but is.
3419:. "The utility of the notion of a topology is shown by the fact that there are several equivalent definitions of this structure. Thus one chooses the axiomatisation suited for the application." (From "Topological space".)
3945:
Maybe not. Google gives me 5270 results to "Chi-squared divergence" (with the quotation marks). Surely, the author did very little. I am not enthusiastic to collaborate in that. But the topic could have some notability.
3467:
I still do not know what is a topology on a given set. Second, yes, continuous maps are most natural as morphisms, but other possibilities exist (and are sometimes used), such as open maps or even Borel measurable maps.
2860:
The true meaning of a mathematical notion, on one hand, and its specific encoding (via sets), on the other hand, should not be confused. An equivalent definition leads to the same notion, but (often) another encoding.
3726:-bit binary numbers, with bitwise exclusive or, but all three give naturally-isomorphic finite dimensional vector spaces.) You could define a finite dimensional real vector space as being a vector space isomorphic to
3358:
Some of that is just articles not yet having been sorted into appropriate subcategories of the main graph theory category. If you think they should be in a specific subcategory (as seems likely in this case) please
4098:
Well, it actually might last 8 more years. Someone "saved" it by pasting a bunch of references to papers about graphs being used for something else, papers that happen to have a guy named Muller among the authors.
2958:, claiming that this name is both more traditional and more correct, and has edited many other articles to implement the same change. Some of us disagree. Please join the discussion at Talk:whichever name it is. —
2280:" has an identity element ... which cannot be identified with an origin." This statement is just false, both as a statement about formalisms and as a statement about common usage by mathematicians. --
1570:
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4164:
In the time it took you write that you could have done the trivial work of clicking the article's history and see for yourself that the answer is no. If you really think that the editor who made
3054:
isn't willing to do it, then he needs to stop editing until this can be resolved. "Google scholar" searches are even more impressive (238 v 1880), but few of either of them are this topic. —
3924:
1509:, who has recently created two very cranky looking articles (with main citations in MDPI, a predatory low-value journal) and started adding links to them from articles on CS and fractals. --
3507:. For topologies on two-element sets, it happens because of possible swapping of "{{},{a},{a,b}}" and "{{},{b},{a,b}}" (exercise: check that continuous maps are insensitive to this swap).
3487:
Given a cardinality α, introduce the category S(α) of all sets of this cardinality, with bijections (not all maps!) as morphisms. Each definition of topology leads to a functor S(α)-: -->
3603:
measures in topological spaces, and see that (under appropriate conditions on the topology) each measure sits on some sigma-compact set. Etc, etc. Can I do it in the category language?
4133:
is another pearl. Lesson learned: I'm not mentioning it on Jimbo's page. This article is already in "saved" format, meaning it consists of a bunch of non-sequiturs with some ref tags.
3089:
There is still the issue of the three contradictory definitions of "double factorial" as well as the ambiguously related term "odd factorial". According to the WP article, the use of
3416:
2087:. Of course, the perception depends on references one uses. But let me try this way too: "Lie correspondence" isn't precise enough; why "Lie" here has to refer to Lie groups? Why not
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4168:
could possibly write something intelligent in a math/CS article, then you probably drank too much wikicoolaid about the encyclopedia that anyone can (but should?) edit and so forth.
3156:"Semifactorial" seems like a good name for the concept, because you're only multiplying half the integers. I don't recall having heard it before. I've always thought the notation
2379:
terminology to make the discussion easier -- I'm not claiming these are universally accepted terms or that
Knowledge should use them: I'm asking how they should be presented.
3730:, but you'd need to have already defined vector spaces and isomorphisms already for that to even make sense, and it would be an ugly definition to choose as the primary one. —
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That sound good to me. Still, the smaller problem of categorizing app/domain specific concepts (rather than whole graphs) remains. They seem to be all over the place. E.g.
4371:
Thanks for that. As far as unwanted content is concerned, I simply want us to distinguish between useful/useless topics for articles and useful/useless article content.
2754:
Viewed in that way, a vector space is an affine space with some additional structure. And this way of viewing it is demonstrably equivalent to the "(A,V)" point of view.
2083:
I don't think so since Cartan's theorem, in my impression, is fairly standard usage, while the term "Lie correspondence" doesn't appear in any of sources that are used in
1445:
that is a reference to the more-inclusive concept (but not to all the links in the current disambiguation page), when none of the individual articles will do as a target.
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3930:
Dear mathematicians: Some time ago I asked about this draft article, but received no reply. I am assuming that it is not a notable topic and should be deleted. —
4516:
21:
3097:
is not well defined. There needs to be some work on that. Some of my edits were to put the "correct" definition in some articles. That still should be done.
49:
3482:
If I ask you "give me an example of a topology on the 2-element set {a,b}", you may give me the set "{{},{a},{a,b}}" of all open sets, or the function "a-: -->
2109:
I have exactly the opposite impression from the literature. I have never seen "Cartan's theorem" (much more ambiguous than "Lie correspondence" according to
1716:
3641:. One definition is taken as the official definition and the equivalent definitions are rolled into a theorem stating that they are all the same thing.
2413:
I think this seems like a much more productive direction than continued quibbling over the informal description. There is good stuff in that article.
1638:
I have rewritten the article completely. As it was before, it managed to miss the majority of the classical groups, but instead had bits and pieces on
3046:
references to "semifactorial" outside of
Knowledge. "Semi-factorial" and "Semi factorial" are sometimes used. I think this needs to be reverted.
2084:
2028:
1804:
2853:, from the category of topological spaces to the category of lattices. However, would anyone call this functor "forgetful"? Definitely, I would not.
2303:
has a long and distinguished tradition in mathematical discourse, being frequently employed even by such luminaries of the formalist perspective as
1742:
mentions the Lie correspondence for nilpotent and solvable cases but doesn't expound upon it. There are existing red links in various articles for
17:
2165:
definition of "affine space" that I like is that it is an element if a quotient space V/W. The problem with this definition is that it is not
3235:
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3533:. The article is very specific to matroid theory, though, and I don't know if that specific word has been used in a lot of other contexts. —
4490:
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3086:
Thank you, Arthur Rubin, for your help. All in all, I made a stupid mistake. I will check to see if there are any further reverts needed.
2027:
I can live with naming the article I wrote either "Closed subgroup theorem" or "Cartan's theorem", either way is fine. But I really think
2811:). Accordingly, I think, "the forgetful functor from the category of affine spaces to the category of vector spaces" is not a good idea.
2550:
involves an underlying set A whose members let us call "points", and and a field F whose members are called scalars, and an operation of
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Is it true that for every pair of naturally equivalent functors (of this kind) there exists only one natural equivalence between them?
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3702:). (Coming from a CS point of view, I think of these as analogous to abstract data types vs their implementations.) I wouldn't use
3679:
This is exactly what I'd like to avoid. In the vector metaphor, it means: "officially" define a finite-dimensional vector space as
2477:
involves an underlying set V whose members are called vectors, and a field F whose members are called scalars, and an operation of
3242:. Chapters 3-4 in particular, but most of the book is basically just about this topic. There actually two signal-flow graphs, the
2210:
Presumably in an encyclopedic article it would be good to have at least a brief discussion of all these different perspectives. --
3294:
with various other info. Most of the stuff in the 3rd category aren't actually ways to implement graphs as data structures, e.g.
2879:
Isn't "The location of the gap in somebody's understanding?" quite a nasty subject title? There may be holes in your brain too.
2051:
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1391:(lower-case "s") and changed the latter to a disambiguation page, so far with only two main links and a "see also" link. So:
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3718:-tuples of members of GF(2), with pointwise addition, as having a different data type than subsets of the numbers 0, 1, ...,
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appears to be in a rather poor state for an old and important article. I only noticed it because someone changed a link at
1827:(not submitted yet, you are welcome to comment on its talk page), I bumped into the following missing articles candidates;
3273:
2031:
should be renamed to "Lie correspondence". It sounds better and seems to be a little more common, including in Knowledge.
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The Estonian and Farsi Wikipedias use "double", while Swedish uses "semi". The more I look, the more confused I get. —
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vector space. The functors are still well-defined, but do not seem to be "forgetful" anymore. Since really they are not.
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this notion that an affine space is (A,V) where A and V satisfy certain conditions and are related in certain ways is
1297:
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1483:, trying to cover enough for a reader to understand both generalized and particular meanings and uses of the term. ᛭
2988:
3802:
2841:
is in fact a partially ordered set (by inclusion), moreover, a lattice. It is legitimate to consider the functor (
4494:
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to serve as the origin or zero or whatever you want to call it, and and then one can define a linear combination
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In being rooted, I would imagine. Propose a merge and redirect if you think the difference is insufficient.
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3160:!! is obnoxious because it looks like the factorial of the factorial, and that is not at all what is meant.
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space is an affine space with this bit of additional structure: One chooses some point which we will call
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2100:
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2050:, even the term is not commonly used in media. Also, "Lie correspondence" is a little ambiguous; there is
2007:
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1963:
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that apparently fancied the authors more than classical Lie groups. I have retained most of that stuff,
1430:
4425:
4010:
now, but the text is available in the history if anyone wants to add some of it to the main article. —
2166:
4493:. A redirect there might be marginally helpful, but I'm not going to argue too strenuously for it.
3810:
3499:
Even simpler: what if there exists a nontrivial natural equivalence from one such functor to itself?
3250:
and they can be converted to each other easily, but no such info can be gleaned from wikipeidia etc.
2912:
1832:
1673:. As it is a major change, I'd like some consensus and, if possible some help, before proceeding.--
1646:, but I think it should go somewhere else. Suggestions? I opened a thread at the article talk page.
1594:. I will never be able to figure out why people think that content like this is worth having in WP.
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appear to be substantially identical, despite having been created by apparently different users,
4271:
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on graph theory, namely to use words that are similar but not identical to the terms used in any
4208:
4186:. If there is no prospect of writing even a short stub, then that is an argument for deletion.
4117:
4081:
4035:
3299:
3161:
3016:
Note 1: I made a bad mistake by assuming this was a simple matter. I acknowledge it was my bad.
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I'm guessing the first one is intended for "core" graph theory concepts. But some concepts like
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Lots of pages link to the new disambiguation page. Those need to get appropriatedly directed.
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of a finite dimensional vector space; it (together with the right tuple of operations) is an
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I agree it seems a good name. I've just never seen it used, and we need a related name for
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1462:, but somehow with reference to the structure-preserving concept, is that the right name? —
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seems to belong to 2 instead—the article doesn't even say how these might be implemented.
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Really, now I feel that the "abstract data type" (thanks to David) is the most apt word.
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is actually citing its source correctly, except the source doesn't make (much) sense.
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Another similar case (for example). A topological space is usually defined as a pair (
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For those following the story, an admin decided to keep. The decision has since been
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3683:, and then discuss arbitrary coordinate systems in addition to the "official" one.
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1981:
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There seem to be 3 categories that more or less overlap in their actual contents:
1330:
Someone more knowledgeable about probability theory may want to check out the new
1843:(coordinates associated with the group topology, rarely mentioned in Knowledge).
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3489:(the set of all topologies on this set). Two equivalent definitions lead to two
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4388:? Same content, "different" user. Allow me to be frustrated just a little bit.
3807:"Why do categorical foundationalists want to escape set theory?" (Mathoverflow)
1881:
mentions, but does not define, left and right invariants of various objects. --
1591:
3638:
2146:
Talk:Affine space#"Forgotten which point is the origin": gibberish or functor?
1266:
4358:
3589:
3556:
3176:
2979:
2915:
issues after RQG's insistent misunderstanding of Baez's post on the matter.
2632:, and this operation of affine combination satisfies certain algebraic laws.
2543:, and this operation of linear combination satisfies certain algebraic laws.
1972:
Ty, blushing. You have probably not yet spotted all errors bound to be there
1746:; I say go for the stub. "Closed subgroup theorem" is in a list of todos at
1339:
3722:− 1, with symmetric difference of sets, and that in turn is different from
2911:
Problems at that article appear to continue. I begin to suspect definite
4402:
Just to spell this out for the benefit of passing readers: the articles
2467:
only one way of encoding the concept of affine space. There are others.
4116:
Before reading that, I had not suspected that no scalars are negative.
3836:
1958:
Wow, nice work on short notice. It looks ready for mainspace to me. --
4324:
May I suggest to you to spend the next 500 years of your life saving
1936:. I want it to go the official way to get some sort of sanity check.
1771:
fundamental result (or rather class of results) of Lie group theory.
3787:
Now the question is, to which extent is it (not) Original Research?
1437:, quotient set ... and the like. This is exemplified by the link to
2002:? (there is no question the theorem deserves an article on it.) --
1395:
The disambiguation page would probably benefit from more work; and
3547:
The question about nontrivial natural equivalences reminds me of
3476:
correspondence described there in the next lines is a bijection.
1877:, perhaps a a defn of left and right invariants could go there?
1750:, so seems like a reasonable candidate for an article as well. --
1717:
Representation_theory_of_the_Lorentz_group#The_Lie_correspondence
1449:
would then become a disambiguation page rather than a redirect. —
3966:, where it is mentioned, seems warranted if this is not kept. --
3814:
3714:
of a finite dimensional vector space. (For instance, I think of
3391:, I'd like to have an article about "Equivalent definitions of
1723:. I found both of those articles using Knowledge's search box.
1571:
Knowledge:Articles for deletion/Jacob Barnett (2nd nomination)
1224:
33:
3639:
finite set#Necessary and sufficient conditions for finiteness
3042:
I know we cannot depend on Google searches, but Google finds
1700:
under different names? If not, I might write stubs for them.
3013:, until this is cleared up. I can't clear it all up myself.
2375:
I'm trying to unpack the various possible meanings using an
3925:
Knowledge talk:Articles for creation/Chi-squared divergence
3402:(b) Which part of the said can we find in reliable sources?
3838:
Homotopy Type Theory: Univalent Foundations of Mathematics
2188:
relatively small group of mathematicians? I do not know.
1803:
I don't think it's a good redirect. I simply started the
4386:
4325:
4227:
4165:
3111:
I see there are (possibly more than 2) definitions of
2765:
has joined the discussion on the article's talk page.
2141:
Affine space, forgot the origin: gibberish or functor?
3784:
parameters running over infinite sets could appear).
2024:
is Cartan's theorem too, at least in its global form.
3909:; please look. Improvements are welcome, of course.
1839:
mentioned anywhere, is it unusual terminology?) and
1292:
on the recent development has already been created.
4418:. The articles have been proposed for deletion by
3344:is in category #1 ("objects") from the above list.
4030:"Constant curvature" needs some tender loving care
2359:distinction you're making reflects actual usage.
3907:Equivalent definitions of mathematical structures
3867:equivalent definitions of mathematical structures
3399:(a) What can we reasonably say in this direction?
3383:Equivalent definitions of mathematical structures
1544:Knowledge:Articles for deletion/Frequency fractal
1526:Knowledge:Articles for deletion/AjoChhand Machine
4061:For some amusement, before it's gone, check out
3865:article would be a good location to expand upon
3803:"From Set Theory to Type Theory" by Mike Shulman
1644:Classical groups over general fields or algebras
3265:Graph categories: some clean-up might be needed
4184:comment on the content, not on the contributor
3637:You might want to follow the example I set in
4485:I have amused myself by completely rewriting
4063:Is theta a scalar quantity or vector quantity
3801:Something that would be indirectly relevant:
1232:This page has archives. Sections older than
57:
8:
1740:Lie algebra#Abelian, nilpotent, and solvable
1671:Talk:List of mathematical symbols#Reorganize
4489:. The article is no more and no less than
3395:". To this end I ask myself two questions:
2691:need not add up to 1 by saying that it is
1383:(with capital initials) from a redirect to
3020:more to the history than the article says.
2052:Algebraic group–Lie algebra correspondence
64:
50:
2977:but this seems like it was a bad move. --
1253:I was wondering if we have an article on
3529:There is something like this already at
2343:Obvious but important mistake corrected.
3964:f-divergence#Instances of f-divergences
2243:Gibberish, but good gibberish. Let in.
1661:Talk:List of mathematical symbols reorg
88:
4385:What can I say. Do I see double today
18:Knowledge talk:WikiProject Mathematics
4517:WikiProject Mathematics archives/2014
3841:. The Univalent Foundations Program.
2837:being the set of "open" subsets. The
2169:, but works perfectly otherwise. --
7:
4491:Left-child right-sibling binary tree
4342:Why? Do you think it's important?
3279:Category:Application-specific graphs
2893:I felt so too, and have retitled. --
2085:Lie group–Lie algebra correspondence
2029:Lie group–Lie algebra correspondence
1973:
1807:, though it's far from complete. --
1805:Lie group–Lie algebra correspondence
1665:I have proposed a reorganization of
3817:(redlink to "SEAR (mathematics)");
2469:One of those other goes like this:
2437:The nature of the misunderstanding?
1458:On review, perhaps that article is
1259:scheduled relaxation Jacobi method
45:WikiProject Mathematics archives (
32:
3389:recent trouble with affine spaces
2933:Double factorial vs semifactorial
1785:As a placeholder, I have created
1505:Just wanted to draw attention to
1236:may be automatically archived by
3298:seem okay in that category, but
1974:
1831:(often mentioned in Knowledge),
1257:? I noticed a new variant of it
38:
4274:is clearly a notable concept.
1447:Quotient space (disambiguation)
1439:Quotient space (disambiguation)
1334:article. As I explained on the
3745:equally "unofficial"). Can I?
3284:Category:Graph data structures
3186:sounds particularly weird. —
1332:Geometric Poisson distribution
1325:Geometric Poisson distribution
1:
4230:. Vertices ≠ nodes for some.
3361:go ahead and make that change
3274:Category:Graph theory objects
1284:We do have an article on the
3843:Institute for Advanced Study
3093:!! for even natural numbers
2048:2014 Northern Iraq offensive
1823:While writing the draft for
1667:List of mathematical symbols
2554:by which one takes scalars
2481:by which one takes scalars
4533:
4057:Soon-to-be-deleted article
3984:Ah, yes, a good redirect.
2395:One more source of ideas:
2227:allowed to stay as it is.
1748:Talk:List of theorems#TODO
1688:Do we have these articles?
4503:03:26, 30 July 2014 (UTC)
4459:16:33, 26 July 2014 (UTC)
4398:14:39, 26 July 2014 (UTC)
4381:08:30, 26 July 2014 (UTC)
4367:23:44, 25 July 2014 (UTC)
4352:20:42, 25 July 2014 (UTC)
4338:20:35, 25 July 2014 (UTC)
4320:20:29, 25 July 2014 (UTC)
4306:20:24, 25 July 2014 (UTC)
4288:19:36, 25 July 2014 (UTC)
4259:19:33, 25 July 2014 (UTC)
4240:19:29, 25 July 2014 (UTC)
4221:19:27, 25 July 2014 (UTC)
4196:20:46, 25 July 2014 (UTC)
4178:20:23, 25 July 2014 (UTC)
4160:19:31, 25 July 2014 (UTC)
4143:19:25, 25 July 2014 (UTC)
4126:19:12, 25 July 2014 (UTC)
4109:19:25, 25 July 2014 (UTC)
4094:10:01, 23 July 2014 (UTC)
4075:07:02, 23 July 2014 (UTC)
4052:02:37, 30 July 2014 (UTC)
4020:01:55, 30 July 2014 (UTC)
3994:21:10, 27 July 2014 (UTC)
3976:20:49, 27 July 2014 (UTC)
3956:20:14, 27 July 2014 (UTC)
3940:19:24, 27 July 2014 (UTC)
3919:19:14, 28 July 2014 (UTC)
3898:19:25, 24 July 2014 (UTC)
3879:18:19, 24 July 2014 (UTC)
3855:16:59, 24 July 2014 (UTC)
3831:13:22, 24 July 2014 (UTC)
3797:08:58, 21 July 2014 (UTC)
3755:08:28, 21 July 2014 (UTC)
3740:07:44, 21 July 2014 (UTC)
3693:07:31, 21 July 2014 (UTC)
3651:07:07, 21 July 2014 (UTC)
3613:05:35, 21 July 2014 (UTC)
3598:03:24, 21 July 2014 (UTC)
3580:21:41, 20 July 2014 (UTC)
3565:21:06, 20 July 2014 (UTC)
3543:18:51, 20 July 2014 (UTC)
3524:18:29, 20 July 2014 (UTC)
3471:According to the article
3373:21:03, 22 July 2014 (UTC)
3354:19:09, 22 July 2014 (UTC)
3336:is in the main category (
3328:16:12, 21 July 2014 (UTC)
3312:13:14, 21 July 2014 (UTC)
3260:04:45, 21 July 2014 (UTC)
3220:could use a bit more math
3209:16:36, 18 July 2014 (UTC)
3194:15:38, 18 July 2014 (UTC)
3170:23:54, 17 July 2014 (UTC)
3145:16:40, 18 July 2014 (UTC)
3127:18:27, 17 July 2014 (UTC)
3107:15:14, 17 July 2014 (UTC)
3074:07:49, 17 July 2014 (UTC)
3062:06:11, 17 July 2014 (UTC)
3034:04:57, 17 July 2014 (UTC)
2998:04:50, 17 July 2014 (UTC)
2968:04:40, 17 July 2014 (UTC)
2927:18:21, 17 July 2014 (UTC)
2903:16:28, 17 July 2014 (UTC)
2889:15:50, 17 July 2014 (UTC)
2871:08:02, 17 July 2014 (UTC)
2821:05:50, 17 July 2014 (UTC)
2775:03:54, 17 July 2014 (UTC)
2761:I am pleased to see that
2425:19:38, 16 July 2014 (UTC)
2409:19:12, 16 July 2014 (UTC)
2389:18:44, 16 July 2014 (UTC)
2371:17:58, 16 July 2014 (UTC)
2354:17:25, 16 July 2014 (UTC)
2123:16:46, 16 July 2014 (UTC)
2105:16:02, 16 July 2014 (UTC)
2079:14:42, 16 July 2014 (UTC)
2064:11:31, 16 July 2014 (UTC)
2041:02:14, 16 July 2014 (UTC)
2012:01:41, 16 July 2014 (UTC)
1990:00:22, 16 July 2014 (UTC)
1968:00:16, 16 July 2014 (UTC)
1946:23:50, 15 July 2014 (UTC)
1918:23:50, 15 July 2014 (UTC)
1891:23:34, 15 July 2014 (UTC)
1853:21:11, 15 July 2014 (UTC)
1817:01:00, 15 July 2014 (UTC)
1799:23:28, 14 July 2014 (UTC)
1781:22:46, 14 July 2014 (UTC)
1760:21:11, 14 July 2014 (UTC)
1733:20:56, 14 July 2014 (UTC)
1710:20:20, 14 July 2014 (UTC)
1683:15:26, 14 July 2014 (UTC)
1656:18:30, 11 July 2014 (UTC)
1627:14:55, 11 July 2014 (UTC)
1585:12:15, 25 June 2014 (UTC)
1467:19:42, 22 June 2014 (UTC)
1454:19:24, 22 June 2014 (UTC)
1411:18:38, 12 June 2014 (UTC)
1267:10.1016/j.jcp.2014.06.010
4278:would be easy to find.
4249:books on graph theory."
4080:Well, at least that one
3409:First, some case study.
3334:dominator (graph theory)
2323:18:41, 8 July 2014 (UTC)
2290:17:52, 8 July 2014 (UTC)
2272:17:39, 8 July 2014 (UTC)
2253:16:23, 8 July 2014 (UTC)
2239:15:36, 8 July 2014 (UTC)
2220:15:03, 8 July 2014 (UTC)
2198:13:42, 8 July 2014 (UTC)
2179:12:52, 8 July 2014 (UTC)
2159:10:46, 8 July 2014 (UTC)
1604:13:28, 9 July 2014 (UTC)
1556:02:21, 8 July 2014 (UTC)
1538:21:20, 7 July 2014 (UTC)
1519:13:19, 7 July 2014 (UTC)
1493:05:56, 7 July 2014 (UTC)
1370:13:46, 4 July 2014 (UTC)
1358:Draft:Deformation tensor
1348:17:21, 3 July 2014 (UTC)
1316:07:28, 2 July 2014 (UTC)
1302:06:44, 2 July 2014 (UTC)
1279:06:01, 2 July 2014 (UTC)
1255:Jacobi iterative method
4408:Grafting (ordered tree)
4131:Grafting (ordered tree)
3433:Differentiable manifold
3393:mathematical structures
3342:interval (graph theory)
3291:interval (graph theory)
2016:How funny. It looks in
1998:Shouldn't it be called
1934:Closed subgroup theorem
1841:Exponential coordinates
1825:Closed subgroup theorem
1698:Closed subgroup theorem
1261:was recently published
4182:It is more helpful to
4008:Chi-squared divergence
1610:AfC submission - 11/07
1353:AfC submission - 04/07
1239:Lowercase sigmabot III
3338:Category:Graph theory
3227:Wai-Kai Chen (1971).
3133:Talk:Double factorial
2452:End of quote from RQG
1932:Draft now submitted:
1431:Quotient vector space
4495:Lesser Cartographies
4487:Grafting (algorithm)
4404:Grafting (algorithm)
4228:figured that one out
3811:Homotopy type theory
3491:naturally equivalent
3479:Now, some thoughts.
3229:Applied Graph Theory
2513:, one gets a vector
1416:It seems to me that
4292:And different from
4082:didn't last 8 years
4006:It's a redirect at
3488:S(β), (a set)-: -->
3483:{{a},{a,b}},b-: -->
2602:, and gets a point
4272:Rooted binary tree
4209:rooted binary tree
4036:Constant curvature
3387:Bothered with the
3300:and-inverter graph
3182:as an instance of
2983:
2552:affine combination
2479:linear combination
2022:Lie correspondence
1835:(strangely enough
1787:Lie correspondence
1744:Lie correspondence
1694:Lie correspondence
1640:groups of Lie type
1569:Please comment at
1565:is up for deletion
1435:Quotient semigroup
1375:Quotient space (2)
3465:up to isomorphism
3413:Topological space
3237:978-0-444-60193-3
3218:signal-flow graph
2981:
2973:Wow. I'm all for
2797:(surely not from
2442:Quote from "RQG":
2397:nLab:affine space
2344:
2309:forgetful functor
2301:Abuse of language
2069:correspondence".
1871:Topological group
1481:equivalence class
1460:Equivalence class
1387:to a redirect to
1385:Equivalence class
1294:FireflySixtySeven
1246:
1245:
95:Nov 2002–Dec 2003
4524:
4448:
4447:
4428:
4276:reliable sources
4042:to point to it.
4040:Hyperbolic space
3819:Univalence axiom
3241:
3113:double factorial
3011:Double factorial
2996:
2993:
2984:
2952:double factorial
2919:
2586:= 1, and points
2417:
2363:
2342:
2315:
2305:Nicolas Bourbaki
2231:
2111:Cartan's theorem
2018:Cartan's theorem
2000:Cartan's theorem
1979:
1978:
1977:
1721:Cartan's theorem
1577:
1241:
1225:
66:
59:
52:
42:
34:
4532:
4531:
4527:
4526:
4525:
4523:
4522:
4521:
4507:
4506:
4429:
4424:
4423:
4059:
4032:
3986:Boris Tsirelson
3948:Boris Tsirelson
3928:
3911:Boris Tsirelson
3890:Boris Tsirelson
3847:Boris Tsirelson
3823:Boris Tsirelson
3789:Boris Tsirelson
3747:Boris Tsirelson
3685:Boris Tsirelson
3605:Boris Tsirelson
3516:Boris Tsirelson
3443:Algebraic space
3385:
3267:
3238:
3226:
3222:
2995:
2989:
2980:
2978:
2935:
2917:
2863:Boris Tsirelson
2813:Boris Tsirelson
2810:
2803:
2796:
2789:
2741:
2732:
2725:
2717:
2708:
2702:
2690:
2681:
2674:
2666:
2657:
2651:
2631:
2623:
2614:
2608:
2601:
2592:
2585:
2576:
2569:
2560:
2542:
2534:
2525:
2519:
2512:
2503:
2496:
2487:
2439:
2415:
2401:Boris Tsirelson
2361:
2313:
2229:
2190:Boris Tsirelson
2151:Boris Tsirelson
2143:
1975:
1879:Exponential map
1867:Exponential map
1789:as a redirect.
1690:
1663:
1636:
1633:Classical group
1612:
1575:
1567:
1503:
1379:I have changed
1377:
1355:
1328:
1251:
1237:
1226:
1220:
1211:
1097:
87:
86:
73:
70:
30:
29:
28:
12:
11:
5:
4530:
4528:
4520:
4519:
4509:
4508:
4480:
4479:
4478:
4477:
4476:
4475:
4474:
4473:
4472:
4471:
4470:
4469:
4468:
4467:
4466:
4465:
4464:
4463:
4462:
4461:
4354:
4264:
4263:
4262:
4261:
4226:Well, someone
4205:
4204:
4203:
4202:
4201:
4200:
4199:
4198:
4114:
4113:
4112:
4111:
4067:David Eppstein
4058:
4055:
4031:
4028:
4027:
4026:
4025:
4024:
4023:
4022:
3999:
3998:
3997:
3996:
3979:
3978:
3959:
3958:
3927:
3922:
3903:
3902:
3901:
3900:
3882:
3881:
3774:
3773:
3772:
3771:
3770:
3769:
3768:
3767:
3766:
3765:
3764:
3763:
3762:
3761:
3760:
3759:
3758:
3757:
3732:David Eppstein
3664:
3663:
3662:
3661:
3660:
3659:
3658:
3657:
3656:
3655:
3654:
3653:
3624:
3623:
3622:
3621:
3620:
3619:
3618:
3617:
3616:
3615:
3552:
3535:David Eppstein
3531:cryptomorphism
3514:Do you agree?
3505:opposite group
3404:
3403:
3400:
3384:
3381:
3380:
3379:
3378:
3377:
3376:
3375:
3365:David Eppstein
3320:David Eppstein
3296:adjacency list
3287:
3286:
3281:
3276:
3266:
3263:
3236:
3221:
3215:
3214:
3213:
3212:
3211:
3184:multifactorial
3177:multifactorial
3154:
3153:
3152:
3151:
3150:
3149:
3148:
3147:
3109:
3087:
3079:
3078:
3077:
3076:
3064:
3037:
3036:
3021:
3017:
3014:
3001:
3000:
2987:
2960:David Eppstein
2934:
2931:
2930:
2929:
2918:Sławomir Biały
2908:
2907:
2906:
2905:
2876:
2875:
2874:
2873:
2858:
2854:
2823:
2808:
2801:
2794:
2787:
2763:user:John Baez
2759:
2758:
2752:
2751:
2749:
2748:
2747:
2737:
2730:
2721:
2713:
2706:
2700:
2686:
2679:
2670:
2662:
2655:
2649:
2638:
2637:
2634:
2633:
2627:
2619:
2612:
2606:
2597:
2590:
2581:
2574:
2565:
2558:
2544:
2538:
2530:
2523:
2517:
2508:
2501:
2492:
2485:
2458:
2457:
2449:
2448:
2438:
2435:
2434:
2433:
2432:
2431:
2430:
2429:
2428:
2427:
2416:Sławomir Biały
2393:
2392:
2391:
2362:Sławomir Biały
2356:
2328:
2327:
2326:
2325:
2314:Sławomir Biały
2295:
2294:
2293:
2292:
2275:
2274:
2258:
2257:
2256:
2255:
2241:
2230:Sławomir Biały
2224:
2223:
2222:
2203:
2202:
2201:
2200:
2182:
2181:
2142:
2139:
2138:
2137:
2136:
2135:
2134:
2133:
2132:
2131:
2130:
2129:
2128:
2127:
2126:
2125:
2025:
2020:like that the
1996:
1995:
1994:
1993:
1992:
1951:
1950:
1949:
1948:
1927:
1926:
1925:
1924:
1923:
1922:
1921:
1920:
1898:
1897:
1896:
1895:
1894:
1893:
1858:
1857:
1856:
1855:
1833:Group topology
1829:Left invariant
1821:
1820:
1819:
1801:
1736:
1735:
1689:
1686:
1662:
1659:
1635:
1630:
1611:
1608:
1607:
1606:
1576:Sławomir Biały
1566:
1560:
1559:
1558:
1548:David Eppstein
1540:
1530:David Eppstein
1502:
1499:
1498:
1497:
1496:
1495:
1470:
1469:
1456:
1423:Quotient group
1418:Quotient space
1400:
1399:
1396:
1389:Quotient space
1381:Quotient Space
1376:
1373:
1354:
1351:
1327:
1322:
1321:
1320:
1319:
1318:
1250:
1247:
1244:
1243:
1231:
1228:
1227:
1222:
1218:
1216:
1213:
1212:
1210:
1209:
1154:
1098:
1096:
1095:
1040:
985:
930:
875:
820:
765:
710:
655:
600:
545:
490:
435:
380:
325:
270:
215:
160:
105:
84:
83:
82:
79:
78:
75:
74:
69:
68:
61:
54:
46:
43:
37:
31:
15:
14:
13:
10:
9:
6:
4:
3:
2:
4529:
4518:
4515:
4514:
4512:
4505:
4504:
4500:
4496:
4492:
4488:
4484:
4460:
4456:
4452:
4445:
4442:
4439:
4436:
4433:
4427:
4421:
4417:
4413:
4412:User:Mmmzeta0
4409:
4405:
4401:
4400:
4399:
4395:
4391:
4387:
4384:
4383:
4382:
4378:
4374:
4370:
4369:
4368:
4364:
4360:
4355:
4353:
4349:
4345:
4341:
4340:
4339:
4335:
4331:
4327:
4323:
4322:
4321:
4317:
4313:
4309:
4308:
4307:
4303:
4299:
4295:
4291:
4290:
4289:
4285:
4281:
4277:
4273:
4270:
4269:
4268:
4267:
4266:
4265:
4260:
4256:
4252:
4248:
4243:
4242:
4241:
4237:
4233:
4229:
4225:
4224:
4223:
4222:
4218:
4214:
4210:
4197:
4193:
4189:
4185:
4181:
4180:
4179:
4175:
4171:
4167:
4163:
4162:
4161:
4157:
4153:
4149:
4148:
4147:
4146:
4145:
4144:
4140:
4136:
4132:
4128:
4127:
4123:
4119:
4118:Michael Hardy
4110:
4106:
4102:
4097:
4096:
4095:
4091:
4087:
4083:
4079:
4078:
4077:
4076:
4072:
4068:
4064:
4056:
4054:
4053:
4049:
4045:
4041:
4037:
4029:
4021:
4017:
4013:
4009:
4005:
4004:
4003:
4002:
4001:
4000:
3995:
3991:
3987:
3983:
3982:
3981:
3980:
3977:
3973:
3969:
3965:
3961:
3960:
3957:
3953:
3949:
3944:
3943:
3942:
3941:
3937:
3933:
3926:
3923:
3921:
3920:
3916:
3912:
3908:
3905:Well, I did:
3899:
3895:
3891:
3886:
3885:
3884:
3883:
3880:
3876:
3872:
3868:
3864:
3859:
3858:
3857:
3856:
3852:
3848:
3844:
3840:
3839:
3833:
3832:
3828:
3824:
3820:
3816:
3812:
3808:
3804:
3799:
3798:
3794:
3790:
3785:
3781:
3777:
3756:
3752:
3748:
3743:
3742:
3741:
3737:
3733:
3729:
3725:
3721:
3717:
3713:
3709:
3705:
3701:
3696:
3695:
3694:
3690:
3686:
3682:
3678:
3677:
3676:
3675:
3674:
3673:
3672:
3671:
3670:
3669:
3668:
3667:
3666:
3665:
3652:
3648:
3644:
3640:
3636:
3635:
3634:
3633:
3632:
3631:
3630:
3629:
3628:
3627:
3626:
3625:
3614:
3610:
3606:
3601:
3600:
3599:
3595:
3591:
3588:categories.
3587:
3583:
3582:
3581:
3577:
3573:
3568:
3567:
3566:
3562:
3558:
3553:
3550:
3546:
3545:
3544:
3540:
3536:
3532:
3528:
3527:
3526:
3525:
3521:
3517:
3512:
3508:
3506:
3500:
3497:
3494:
3492:
3485:
3480:
3477:
3474:
3473:Ordered field
3469:
3466:
3460:
3458:
3457:2 definitions
3455:has at least
3454:
3453:Ordered field
3450:
3448:
3447:2 definitions
3445:has at least
3444:
3440:
3438:
3437:4 definitions
3435:has at least
3434:
3430:
3428:
3427:3 definitions
3425:has at least
3424:
3423:Uniform space
3420:
3418:
3417:7 definitions
3415:has at least
3414:
3410:
3407:
3401:
3398:
3397:
3396:
3394:
3390:
3382:
3374:
3370:
3366:
3362:
3357:
3356:
3355:
3351:
3347:
3343:
3339:
3335:
3331:
3330:
3329:
3325:
3321:
3316:
3315:
3314:
3313:
3309:
3305:
3301:
3297:
3292:
3285:
3282:
3280:
3277:
3275:
3272:
3271:
3270:
3264:
3262:
3261:
3257:
3253:
3249:
3245:
3239:
3234:
3230:
3219:
3216:
3210:
3206:
3202:
3197:
3196:
3195:
3192:
3189:
3185:
3181:
3180:Semifactorial
3178:
3174:
3173:
3172:
3171:
3167:
3163:
3162:Michael Hardy
3159:
3146:
3142:
3138:
3134:
3130:
3129:
3128:
3125:
3122:
3118:
3117:semifactorial
3114:
3110:
3108:
3104:
3100:
3096:
3092:
3088:
3085:
3084:
3083:
3082:
3081:
3080:
3075:
3072:
3069:
3065:
3063:
3060:
3057:
3053:
3049:
3045:
3041:
3040:
3039:
3038:
3035:
3031:
3027:
3022:
3018:
3015:
3012:
3008:
3007:Semifactorial
3003:
3002:
2999:
2994:
2992:
2985:
2976:
2972:
2971:
2970:
2969:
2965:
2961:
2957:
2956:semifactorial
2953:
2949:
2946:
2943:
2939:
2932:
2928:
2924:
2920:
2914:
2913:WP:COMPETENCE
2910:
2909:
2904:
2900:
2896:
2892:
2891:
2890:
2886:
2882:
2878:
2877:
2872:
2868:
2864:
2859:
2855:
2852:
2848:
2844:
2840:
2836:
2832:
2828:
2824:
2822:
2818:
2814:
2807:
2800:
2793:
2786:
2781:
2780:
2779:
2778:
2777:
2776:
2772:
2768:
2767:Michael Hardy
2764:
2757:
2756:
2755:
2750:
2745:
2740:
2736:
2729:
2724:
2720:
2716:
2712:
2705:
2699:
2696:
2695:
2694:
2693:
2692:
2689:
2685:
2678:
2673:
2669:
2665:
2661:
2654:
2648:
2644:
2636:
2635:
2630:
2626:
2622:
2618:
2611:
2605:
2600:
2596:
2589:
2584:
2580:
2573:
2568:
2564:
2557:
2553:
2549:
2545:
2541:
2537:
2533:
2529:
2522:
2516:
2511:
2507:
2500:
2495:
2491:
2484:
2480:
2476:
2472:
2471:
2470:
2468:
2464:
2456:
2455:
2454:
2453:
2446:
2445:
2444:
2443:
2436:
2426:
2422:
2418:
2412:
2411:
2410:
2406:
2402:
2398:
2394:
2390:
2386:
2382:
2378:
2374:
2373:
2372:
2368:
2364:
2357:
2355:
2351:
2347:
2339:
2334:
2333:
2332:
2331:
2330:
2329:
2324:
2320:
2316:
2310:
2306:
2302:
2299:
2298:
2297:
2296:
2291:
2287:
2283:
2279:
2278:
2277:
2276:
2273:
2269:
2265:
2260:
2259:
2254:
2250:
2246:
2242:
2240:
2236:
2232:
2225:
2221:
2217:
2213:
2209:
2208:
2207:
2206:
2205:
2204:
2199:
2195:
2191:
2186:
2185:
2184:
2183:
2180:
2176:
2172:
2168:
2163:
2162:
2161:
2160:
2156:
2152:
2147:
2140:
2124:
2120:
2116:
2112:
2108:
2107:
2106:
2102:
2098:
2094:
2090:
2086:
2082:
2081:
2080:
2076:
2072:
2067:
2066:
2065:
2061:
2057:
2053:
2049:
2044:
2043:
2042:
2038:
2034:
2030:
2026:
2023:
2019:
2015:
2014:
2013:
2009:
2005:
2001:
1997:
1991:
1987:
1983:
1971:
1970:
1969:
1965:
1961:
1957:
1956:
1955:
1954:
1953:
1952:
1947:
1943:
1939:
1935:
1931:
1930:
1929:
1928:
1919:
1915:
1911:
1906:
1905:
1904:
1903:
1902:
1901:
1900:
1899:
1892:
1888:
1884:
1880:
1876:
1872:
1868:
1864:
1863:
1862:
1861:
1860:
1859:
1854:
1850:
1846:
1842:
1838:
1834:
1830:
1826:
1822:
1818:
1814:
1810:
1806:
1802:
1800:
1796:
1792:
1791:RockMagnetist
1788:
1784:
1783:
1782:
1778:
1774:
1770:
1769:
1763:
1762:
1761:
1757:
1753:
1749:
1745:
1741:
1738:
1737:
1734:
1730:
1726:
1725:RockMagnetist
1722:
1718:
1714:
1713:
1712:
1711:
1707:
1703:
1699:
1695:
1687:
1685:
1684:
1680:
1676:
1672:
1668:
1660:
1658:
1657:
1653:
1649:
1645:
1641:
1634:
1631:
1629:
1628:
1624:
1620:
1619:FoCuSandLeArN
1616:
1609:
1605:
1601:
1597:
1593:
1589:
1588:
1587:
1586:
1582:
1578:
1572:
1564:
1563:Jacob Barnett
1561:
1557:
1553:
1549:
1545:
1541:
1539:
1535:
1531:
1527:
1523:
1522:
1521:
1520:
1516:
1512:
1508:
1500:
1494:
1490:
1486:
1482:
1478:
1474:
1473:
1472:
1471:
1468:
1465:
1461:
1457:
1455:
1452:
1448:
1444:
1440:
1436:
1432:
1428:
1427:Quotient ring
1424:
1419:
1415:
1414:
1413:
1412:
1408:
1404:
1403:Michael Hardy
1397:
1394:
1393:
1392:
1390:
1386:
1382:
1374:
1372:
1371:
1367:
1363:
1362:FoCuSandLeArN
1359:
1352:
1350:
1349:
1345:
1341:
1337:
1333:
1326:
1323:
1317:
1313:
1309:
1308:65.94.171.126
1305:
1304:
1303:
1299:
1295:
1291:
1287:
1286:Jacobi method
1283:
1282:
1281:
1280:
1276:
1272:
1271:65.94.171.126
1268:
1264:
1260:
1256:
1249:Jacobi method
1248:
1240:
1235:
1230:
1229:
1215:
1214:
1208:
1204:
1200:
1196:
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85:Earlier years
81:
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36:
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27:
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4434:
4426:188.27.81.64
4420:User:JMP EAX
4390:188.27.81.64
4246:
4206:
4129:
4115:
4086:188.27.81.64
4060:
4034:The article
4033:
3929:
3904:
3866:
3837:
3834:
3800:
3786:
3782:
3778:
3775:
3727:
3723:
3719:
3715:
3711:
3707:
3703:
3699:
3680:
3585:
3549:∞-categories
3513:
3509:
3501:
3498:
3495:
3486:
3481:
3478:
3470:
3464:
3461:
3451:
3441:
3431:
3421:
3411:
3408:
3405:
3386:
3346:188.27.81.64
3304:188.27.81.64
3288:
3268:
3252:188.27.81.64
3248:Coates graph
3231:. Elsevier.
3228:
3223:
3188:Arthur Rubin
3157:
3155:
3121:Arthur Rubin
3094:
3090:
3068:Arthur Rubin
3056:Arthur Rubin
3047:
3043:
2990:
2950:) has moved
2944:
2936:
2850:
2846:
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2555:
2551:
2548:affine space
2547:
2539:
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2531:
2527:
2520:
2514:
2509:
2505:
2498:
2497:and vectors
2493:
2489:
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2475:vector space
2474:
2466:
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2459:
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2441:
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2144:
2093:Lie groupoid
1836:
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822:
767:
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632:
602:
547:
492:
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382:
327:
272:
217:
162:
107:
103:Sep–Dec 2004
99:Jan–Aug 2004
44:
4451:Deltahedron
4373:Deltahedron
4344:Deltahedron
4312:Deltahedron
4294:binary tree
4280:Deltahedron
4188:Deltahedron
4152:Deltahedron
4012:Anne Delong
3968:Mark viking
3932:Anne Delong
3871:Mark viking
3863:isomorphism
3572:Deltahedron
3244:Mason graph
3009:, formerly
2570:satisfying
2381:Deltahedron
2346:Deltahedron
2089:Lie algebra
1960:Mark viking
1883:Mark viking
1752:Mark viking
1692:Do we have
1443:Isomorphism
1290:new section
4416:User:Exe89
3708:definition
3406:Item (a).
3340:) whereas
2091:? Why not
1873:. We have
1615:Draft:1/ ∞
1306:Thanks --
91:Motivation
4326:this edit
4166:this edit
4044:JRSpriggs
3643:JRSpriggs
2675:in which
2167:intrinsic
1875:Invariant
1715:There is
1507:this user
1485:LokiClock
1477:a section
1336:talk page
4511:Category
4438:contribs
4207:Oh, and
3246:and the
2948:contribs
2851:topology
2847:topology
2839:topology
2835:topology
2831:topology
2733:− ... −
2709:+ ... +
2682:+ ... +
2658:+ ... +
2615:+ ... +
2577:+ ... +
2526:+ ... +
2054:too. --
1865:We have
1596:Rschwieb
1592:appealed
1501:CS crank
1475:I added
1288:, and a
24: |
20: |
4330:JMP EAX
4298:JMP EAX
4251:JMP EAX
4232:JMP EAX
4213:JMP EAX
4170:JMP EAX
4135:JMP EAX
4101:JMP EAX
3712:example
3706:as the
3135:page.
2975:WP:BOLD
2881:YohanN7
2849:)-: -->
2833:), the
2726:+ (1 −
2245:YohanN7
2115:YohanN7
2071:YohanN7
2033:YohanN7
1982:YohanN7
1938:YohanN7
1910:YohanN7
1908:group.
1845:YohanN7
1773:YohanN7
1702:YohanN7
1648:YohanN7
1542:...and
1524:Now at
1464:Quondum
1451:Quondum
1234:15 days
22:Archive
3584:But I
3201:Zaslav
3191:(talk)
3137:Zaslav
3124:(talk)
3099:Zaslav
3071:(talk)
3059:(talk)
3052:Zaslav
3050:. If
3026:Zaslav
2938:Zaslav
2377:ad hoc
4444:WHOIS
4296:how?
4247:other
2593:,...,
2561:,...,
2504:,...,
2488:,...,
16:<
4499:talk
4483:n.b.
4455:talk
4432:talk
4422:and
4414:and
4406:and
4394:talk
4377:talk
4363:talk
4359:Ozob
4348:talk
4334:talk
4316:talk
4302:talk
4284:talk
4255:talk
4236:talk
4217:talk
4192:talk
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4105:talk
4090:talk
4071:talk
4048:talk
4016:talk
3990:talk
3972:talk
3952:talk
3936:talk
3915:talk
3894:talk
3875:talk
3851:talk
3827:talk
3815:Sear
3793:talk
3751:talk
3736:talk
3689:talk
3647:talk
3609:talk
3594:talk
3590:Ozob
3586:like
3576:talk
3561:talk
3557:Ozob
3539:talk
3520:talk
3369:talk
3350:talk
3324:talk
3308:talk
3256:talk
3233:ISBN
3205:talk
3166:talk
3141:talk
3103:talk
3030:talk
2982:Kinu
2964:talk
2942:talk
2923:talk
2899:talk
2885:talk
2867:talk
2817:talk
2771:talk
2421:talk
2405:talk
2385:talk
2367:talk
2350:talk
2319:talk
2286:talk
2268:talk
2249:talk
2235:talk
2216:talk
2194:talk
2175:talk
2171:Taku
2155:talk
2119:talk
2101:talk
2097:Taku
2075:talk
2060:talk
2056:Taku
2037:talk
2008:talk
2004:Taku
1986:talk
1964:talk
1942:talk
1914:talk
1887:talk
1869:and
1849:talk
1813:talk
1809:Taku
1795:talk
1777:talk
1756:talk
1729:talk
1719:and
1706:talk
1696:and
1679:talk
1652:talk
1623:talk
1600:talk
1581:talk
1552:talk
1534:talk
1515:talk
1489:talk
1407:talk
1366:talk
1344:talk
1340:Huon
1312:talk
1298:talk
1275:talk
1158:2024
1103:2023
1044:2022
989:2021
934:2020
879:2019
824:2018
769:2017
714:2016
659:2015
604:2014
549:2013
494:2012
439:2011
384:2010
329:2009
274:2008
219:2007
164:2006
109:2005
26:2014
4065:. —
3845:).
3363:. —
3225:is
3048:Now
2954:to
2895:JBL
2843:set
2827:set
2804:to
2790:to
2546:An
2463:But
2282:JBL
2264:RQG
2212:JBL
1837:not
1768:the
1675:agr
1669:at
1573:.
1546:. —
1528:. —
1511:JBL
1479:to
1441:in
1269:--
1263:doi
1207:Dec
1203:Nov
1199:Oct
1195:Sep
1191:Aug
1187:Jul
1183:Jun
1179:May
1175:Apr
1171:Mar
1167:Feb
1163:Jan
1152:Dec
1148:Nov
1144:Oct
1140:Sep
1136:Aug
1132:Jul
1128:Jun
1124:May
1120:Apr
1116:Mar
1112:Feb
1108:Jan
1093:Dec
1089:Nov
1085:Oct
1081:Sep
1077:Aug
1073:Jul
1069:Jun
1065:May
1061:Apr
1057:Mar
1053:Feb
1049:Jan
1038:Dec
1034:Nov
1030:Oct
1026:Sep
1022:Aug
1018:Jul
1014:Jun
1010:May
1006:Apr
1002:Mar
998:Feb
994:Jan
983:Dec
979:Nov
975:Oct
971:Sep
967:Aug
963:Jul
959:Jun
955:May
951:Apr
947:Mar
943:Feb
939:Jan
928:Dec
924:Nov
920:Oct
916:Sep
912:Aug
908:Jul
904:Jun
900:May
896:Apr
892:Mar
888:Feb
884:Jan
873:Dec
869:Nov
865:Oct
861:Sep
857:Aug
853:Jul
849:Jun
845:May
841:Apr
837:Mar
833:Feb
829:Jan
818:Dec
814:Nov
810:Oct
806:Sep
802:Aug
798:Jul
794:Jun
790:May
786:Apr
782:Mar
778:Feb
774:Jan
763:Dec
759:Nov
755:Oct
751:Sep
747:Aug
743:Jul
739:Jun
735:May
731:Apr
727:Mar
723:Feb
719:Jan
708:Dec
704:Nov
700:Oct
696:Sep
692:Aug
688:Jul
684:Jun
680:May
676:Apr
672:Mar
668:Feb
664:Jan
653:Dec
649:Nov
645:Oct
641:Sep
637:Aug
633:Jul
629:Jun
625:May
621:Apr
617:Mar
613:Feb
609:Jan
598:Dec
594:Nov
590:Oct
586:Sep
582:Aug
578:Jul
574:Jun
570:May
566:Apr
562:Mar
558:Feb
554:Jan
543:Dec
539:Nov
535:Oct
531:Sep
527:Aug
523:Jul
519:Jun
515:May
511:Apr
507:Mar
503:Feb
499:Jan
488:Dec
484:Nov
480:Oct
476:Sep
472:Aug
468:Jul
464:Jun
460:May
456:Apr
452:Mar
448:Feb
444:Jan
433:Dec
429:Nov
425:Oct
421:Sep
417:Aug
413:Jul
409:Jun
405:May
401:Apr
397:Mar
393:Feb
389:Jan
378:Dec
374:Nov
370:Oct
366:Sep
362:Aug
358:Jul
354:Jun
350:May
346:Apr
342:Mar
338:Feb
334:Jan
323:Dec
319:Nov
315:Oct
311:Sep
307:Aug
303:Jul
299:Jun
295:May
291:Apr
287:Mar
283:Feb
279:Jan
268:Dec
264:Nov
260:Oct
256:Sep
252:Aug
248:Jul
244:Jun
240:May
236:Apr
232:Mar
228:Feb
224:Jan
213:Dec
209:Nov
205:Oct
201:Sep
197:Aug
193:Jul
189:Jun
185:May
181:Apr
177:Mar
173:Feb
169:Jan
158:Dec
154:Nov
150:Oct
146:Sep
142:Aug
138:Jul
134:Jun
130:May
126:Apr
122:Mar
118:Feb
114:Jan
4513::
4501:)
4457:)
4449:.
4396:)
4379:)
4365:)
4350:)
4336:)
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4318:)
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4286:)
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1277:)
1205:·
1201:·
1197:·
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65:e
58:t
51:v
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