I was wondering if we have an article on Jacobi iterative method ? I noticed a new variant of it scheduled relaxation Jacobi method was recently published doi: 10.1016/j.jcp.2014.06.010 -- 65.94.171.126 ( talk) 06:01, 2 July 2014 (UTC)
Someone more knowledgeable about probability theory may want to check out the new Geometric Poisson distribution article. As I explained on the talk page, 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. Huon ( talk) 17:21, 3 July 2014 (UTC)
Draft:Deformation tensor. FoCuSandLeArN ( talk) 13:46, 4 July 2014 (UTC)
I have changed Quotient Space (with capital initials) from a redirect to Equivalence class to a redirect to Quotient space (lower-case "s") and changed the latter to a disambiguation page, so far with only two main links and a "see also" link. So:
Michael Hardy ( talk) 18:38, 12 June 2014 (UTC)
Just wanted to draw attention to this user, 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. -- JBL ( talk) 13:19, 7 July 2014 (UTC)
Please comment at Wikipedia:Articles for deletion/Jacob Barnett (2nd nomination). Sławomir Biały ( talk) 12:15, 25 June 2014 (UTC)
Draft:1/ ∞. FoCuSandLeArN ( talk) 14:55, 11 July 2014 (UTC)
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 groups of Lie type that apparently fancied the authors more than classical Lie groups. I have retained most of that stuff, Classical groups over general fields or algebras, but I think it should go somewhere else. Suggestions? I opened a thread at the article talk page. YohanN7 ( talk) 18:30, 11 July 2014 (UTC)
I have proposed a reorganization of List of mathematical symbols at Talk:List of mathematical symbols#Reorganize. As it is a major change, I'd like some consensus and, if possible some help, before proceeding.-- agr ( talk) 15:26, 14 July 2014 (UTC)
Do we have Lie correspondence and Closed subgroup theorem under different names? If not, I might write stubs for them. YohanN7 ( talk) 20:20, 14 July 2014 (UTC)
Talk:Affine space#"Forgotten which point is the origin": gibberish or functor? 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. Boris Tsirelson ( talk) 10:46, 8 July 2014 (UTC)
Quote from "RQG":
End of quote from RQG
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 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. But this notion that an affine space is (A,V) where A and V satisfy certain conditions and are related in certain ways is only one way of encoding the concept of affine space. There are others. One of those other goes like this:
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 space is an affine space with this bit of additional structure: One chooses some point which we will call 0 to serve as the origin or zero or whatever you want to call it, and and then one can define a linear combination s1p1 + ... + snpn in which s1 + ... + sn need not add up to 1 by saying that it is
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.
I am pleased to see that user:John Baez has joined the discussion on the article's talk page. Michael Hardy ( talk) 03:54, 17 July 2014 (UTC)
Zaslav ( talk · contribs) has moved double factorial to semifactorial, 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. — David Eppstein ( talk) 04:40, 17 July 2014 (UTC)
"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 n!! is obnoxious because it looks like the factorial of the factorial, and that is not at all what is meant. Michael Hardy ( talk) 23:54, 17 July 2014 (UTC)
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 is Wai-Kai Chen (1971). Applied Graph Theory. Elsevier. ISBN 978-0-444-60193-3.. Chapters 3-4 in particular, but most of the book is basically just about this topic. There actually two signal-flow graphs, the Mason graph and the Coates graph and they can be converted to each other easily, but no such info can be gleaned from wikipeidia etc. 188.27.81.64 ( talk) 04:45, 21 July 2014 (UTC)
There seem to be 3 categories that more or less overlap in their actual contents:
I'm guessing the first one is intended for "core" graph theory concepts. But some concepts like interval (graph theory) 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 with various other info. Most of the stuff in the 3rd category aren't actually ways to implement graphs as data structures, e.g. adjacency list seem okay in that category, but and-inverter graph seems to belong to 2 instead—the article doesn't even say how these might be implemented. 188.27.81.64 ( talk) 13:14, 21 July 2014 (UTC)
Bothered with the recent trouble with affine spaces, I'd like to have an article about "Equivalent definitions of mathematical structures". To this end I ask myself two questions:
Item (a).
First, some case study.
Topological space has at least 7 definitions. "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".)
Uniform space has at least 3 definitions.
Differentiable manifold has at least 4 definitions.
Algebraic space has at least 2 definitions.
Ordered field has at least 2 definitions.
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 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 up to isomorphism 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.
According to the article Ordered field 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 correspondence described there in the next lines is a bijection.
Now, some thoughts.
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->{{a},{a,b}},b->{{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.)
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)->S(β), (a set)->(the set of all topologies on this set). Two equivalent definitions lead to two naturally equivalent (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.
Is it true that for every pair of naturally equivalent functors (of this kind) there exists only one natural equivalence between them?
Even simpler: what if there exists a nontrivial natural equivalence from one such functor to itself?
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 opposite group. 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).
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.
Do you agree? Boris Tsirelson ( talk) 18:29, 20 July 2014 (UTC)
Really, now I feel that the "abstract data type" (thanks to David) is the most apt word.
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.)
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 parameters running over infinite sets could appear).
Now the question is, to which extent is it (not) Original Research? Boris Tsirelson ( talk) 08:58, 21 July 2014 (UTC)
Something that would be indirectly relevant: "From Set Theory to Type Theory" by Mike Shulman; "Why do categorical foundationalists want to escape set theory?" (Mathoverflow); Homotopy type theory; Sear (redlink to "SEAR (mathematics)"); Univalence axiom. Boris Tsirelson ( talk) 13:22, 24 July 2014 (UTC)
'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: Homotopy Type Theory: Univalent Foundations of Mathematics. The Univalent Foundations Program. Institute for Advanced Study). Boris Tsirelson ( talk) 16:59, 24 July 2014 (UTC)
Well, I did: Equivalent definitions of mathematical structures; please look. Improvements are welcome, of course. Boris Tsirelson ( talk) 19:14, 28 July 2014 (UTC)
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. — Anne Delong ( talk) 19:24, 27 July 2014 (UTC)
The article Constant curvature appears to be in a rather poor state for an old and important article. I only noticed it because someone changed a link at Hyperbolic space to point to it. JRSpriggs ( talk) 02:37, 30 July 2014 (UTC)
For some amusement, before it's gone, check out Is theta a scalar quantity or vector quantity. — David Eppstein ( talk) 07:02, 23 July 2014 (UTC)
Before reading that, I had not suspected that no scalars are negative. Michael Hardy ( talk) 19:12, 25 July 2014 (UTC)
Grafting (ordered tree) 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. JMP EAX ( talk) 19:25, 25 July 2014 (UTC)
Oh, and rooted binary tree is actually citing its source correctly, except the source doesn't make (much) sense. JMP EAX ( talk) 19:27, 25 July 2014 (UTC)
n.b. I have amused myself by completely rewriting Grafting (algorithm). The article is no more and no less than Left-child right-sibling binary tree. A redirect there might be marginally helpful, but I'm not going to argue too strenuously for it. Lesser Cartographies ( talk) 03:26, 30 July 2014 (UTC)
I was wondering if we have an article on Jacobi iterative method ? I noticed a new variant of it scheduled relaxation Jacobi method was recently published doi: 10.1016/j.jcp.2014.06.010 -- 65.94.171.126 ( talk) 06:01, 2 July 2014 (UTC)
Someone more knowledgeable about probability theory may want to check out the new Geometric Poisson distribution article. As I explained on the talk page, 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. Huon ( talk) 17:21, 3 July 2014 (UTC)
Draft:Deformation tensor. FoCuSandLeArN ( talk) 13:46, 4 July 2014 (UTC)
I have changed Quotient Space (with capital initials) from a redirect to Equivalence class to a redirect to Quotient space (lower-case "s") and changed the latter to a disambiguation page, so far with only two main links and a "see also" link. So:
Michael Hardy ( talk) 18:38, 12 June 2014 (UTC)
Just wanted to draw attention to this user, 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. -- JBL ( talk) 13:19, 7 July 2014 (UTC)
Please comment at Wikipedia:Articles for deletion/Jacob Barnett (2nd nomination). Sławomir Biały ( talk) 12:15, 25 June 2014 (UTC)
Draft:1/ ∞. FoCuSandLeArN ( talk) 14:55, 11 July 2014 (UTC)
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 groups of Lie type that apparently fancied the authors more than classical Lie groups. I have retained most of that stuff, Classical groups over general fields or algebras, but I think it should go somewhere else. Suggestions? I opened a thread at the article talk page. YohanN7 ( talk) 18:30, 11 July 2014 (UTC)
I have proposed a reorganization of List of mathematical symbols at Talk:List of mathematical symbols#Reorganize. As it is a major change, I'd like some consensus and, if possible some help, before proceeding.-- agr ( talk) 15:26, 14 July 2014 (UTC)
Do we have Lie correspondence and Closed subgroup theorem under different names? If not, I might write stubs for them. YohanN7 ( talk) 20:20, 14 July 2014 (UTC)
Talk:Affine space#"Forgotten which point is the origin": gibberish or functor? 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. Boris Tsirelson ( talk) 10:46, 8 July 2014 (UTC)
Quote from "RQG":
End of quote from RQG
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 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. But this notion that an affine space is (A,V) where A and V satisfy certain conditions and are related in certain ways is only one way of encoding the concept of affine space. There are others. One of those other goes like this:
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 space is an affine space with this bit of additional structure: One chooses some point which we will call 0 to serve as the origin or zero or whatever you want to call it, and and then one can define a linear combination s1p1 + ... + snpn in which s1 + ... + sn need not add up to 1 by saying that it is
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.
I am pleased to see that user:John Baez has joined the discussion on the article's talk page. Michael Hardy ( talk) 03:54, 17 July 2014 (UTC)
Zaslav ( talk · contribs) has moved double factorial to semifactorial, 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. — David Eppstein ( talk) 04:40, 17 July 2014 (UTC)
"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 n!! is obnoxious because it looks like the factorial of the factorial, and that is not at all what is meant. Michael Hardy ( talk) 23:54, 17 July 2014 (UTC)
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 is Wai-Kai Chen (1971). Applied Graph Theory. Elsevier. ISBN 978-0-444-60193-3.. Chapters 3-4 in particular, but most of the book is basically just about this topic. There actually two signal-flow graphs, the Mason graph and the Coates graph and they can be converted to each other easily, but no such info can be gleaned from wikipeidia etc. 188.27.81.64 ( talk) 04:45, 21 July 2014 (UTC)
There seem to be 3 categories that more or less overlap in their actual contents:
I'm guessing the first one is intended for "core" graph theory concepts. But some concepts like interval (graph theory) 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 with various other info. Most of the stuff in the 3rd category aren't actually ways to implement graphs as data structures, e.g. adjacency list seem okay in that category, but and-inverter graph seems to belong to 2 instead—the article doesn't even say how these might be implemented. 188.27.81.64 ( talk) 13:14, 21 July 2014 (UTC)
Bothered with the recent trouble with affine spaces, I'd like to have an article about "Equivalent definitions of mathematical structures". To this end I ask myself two questions:
Item (a).
First, some case study.
Topological space has at least 7 definitions. "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".)
Uniform space has at least 3 definitions.
Differentiable manifold has at least 4 definitions.
Algebraic space has at least 2 definitions.
Ordered field has at least 2 definitions.
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 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 up to isomorphism 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.
According to the article Ordered field 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 correspondence described there in the next lines is a bijection.
Now, some thoughts.
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->{{a},{a,b}},b->{{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.)
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)->S(β), (a set)->(the set of all topologies on this set). Two equivalent definitions lead to two naturally equivalent (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.
Is it true that for every pair of naturally equivalent functors (of this kind) there exists only one natural equivalence between them?
Even simpler: what if there exists a nontrivial natural equivalence from one such functor to itself?
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 opposite group. 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).
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.
Do you agree? Boris Tsirelson ( talk) 18:29, 20 July 2014 (UTC)
Really, now I feel that the "abstract data type" (thanks to David) is the most apt word.
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.)
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 parameters running over infinite sets could appear).
Now the question is, to which extent is it (not) Original Research? Boris Tsirelson ( talk) 08:58, 21 July 2014 (UTC)
Something that would be indirectly relevant: "From Set Theory to Type Theory" by Mike Shulman; "Why do categorical foundationalists want to escape set theory?" (Mathoverflow); Homotopy type theory; Sear (redlink to "SEAR (mathematics)"); Univalence axiom. Boris Tsirelson ( talk) 13:22, 24 July 2014 (UTC)
'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: Homotopy Type Theory: Univalent Foundations of Mathematics. The Univalent Foundations Program. Institute for Advanced Study). Boris Tsirelson ( talk) 16:59, 24 July 2014 (UTC)
Well, I did: Equivalent definitions of mathematical structures; please look. Improvements are welcome, of course. Boris Tsirelson ( talk) 19:14, 28 July 2014 (UTC)
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. — Anne Delong ( talk) 19:24, 27 July 2014 (UTC)
The article Constant curvature appears to be in a rather poor state for an old and important article. I only noticed it because someone changed a link at Hyperbolic space to point to it. JRSpriggs ( talk) 02:37, 30 July 2014 (UTC)
For some amusement, before it's gone, check out Is theta a scalar quantity or vector quantity. — David Eppstein ( talk) 07:02, 23 July 2014 (UTC)
Before reading that, I had not suspected that no scalars are negative. Michael Hardy ( talk) 19:12, 25 July 2014 (UTC)
Grafting (ordered tree) 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. JMP EAX ( talk) 19:25, 25 July 2014 (UTC)
Oh, and rooted binary tree is actually citing its source correctly, except the source doesn't make (much) sense. JMP EAX ( talk) 19:27, 25 July 2014 (UTC)
n.b. I have amused myself by completely rewriting Grafting (algorithm). The article is no more and no less than Left-child right-sibling binary tree. A redirect there might be marginally helpful, but I'm not going to argue too strenuously for it. Lesser Cartographies ( talk) 03:26, 30 July 2014 (UTC)