Re: [math-fun] Lie-sphere lurker
Fred wrote: << . . . the rings involved are isomorphic. I'll denote this \H ~= Cl(0,2) . . . The quaternions have a natural order-3 automorphism i -> j -> k -> i, which Cl(0,2) lacks. This causes both computational inconvenience and conceptual confusion, when neophytes attmept to impose a symmetry which doesn't exist. . . .
OK, so do I have it right: You're saying the quaternions and Cl(0,2) are isomorphic as rings, but unlike the quaternions, Cl(0,2) has no natural order-3 automorphism ? (But Cl(0,2) must have *some* order-3 isomorphism, even if not "natural", by virtue of being isomorphic to H.) I'm not sure I'd call i -> j -> k -> i a "natural" isomorphism of H, because its naturalness depends on a chosen basis for H. (More precisely, a choice of oriented 2-subspace of pure quaternions.) The automorphism group of H is SO(3), so without such a choice, any 120-degree rotation in SO(3) would be equally natural. I think. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
On 12/16/09, Dan Asimov <dasimov@earthlink.net> wrote:
Fred wrote:
<< . . . the rings involved are isomorphic.
I'll denote this
\H ~= Cl(0,2)
. . .
The quaternions have a natural order-3 automorphism i -> j -> k -> i, which Cl(0,2) lacks. This causes both computational inconvenience and conceptual confusion, when neophytes attmept to impose a symmetry which doesn't exist.
. . .
OK, so do I have it right: You're saying the quaternions and Cl(0,2) are isomorphic as rings, but unlike the quaternions, Cl(0,2) has no natural order-3 automorphism ?
(But Cl(0,2) must have *some* order-3 isomorphism, even if not "natural", by virtue of being isomorphic to H.)
Yes --- as a ring (skew field, even) --- but not as a graded algebra, which turns out in practice of central importance to Clifford algebra techniques. [Not that every practitioner seems fully to appreciate this: there's a substantial community promoting an abomination called "paravectors" --- sums of scalars and vectors --- I mean, dash it all, these bounders are not even versors! Harrumph --- I digress ...]
I'm not sure I'd call i -> j -> k -> i a "natural" isomorphism of H, because its naturalness depends on a chosen basis for H. (More precisely, a choice of oriented 2-subspace of pure quaternions.)
The automorphism group of H is SO(3), so without such a choice, any 120-degree rotation in SO(3) would be equally natural. I think.
Agreed --- the point I'm trying to make here is just that the notational visibility of this particular symmetry encourages people to assume a cod non-homomorphism \H -> Cl(0,3): i -> \u, j -> \v, k -> \w ?? WFL
On 12/16/09, Hans Havermann <pxp@rogers.com> wrote:
"Alice's adventures in algebra: Wonderland solved" in New Scientist, 16 December 2009, by Melanie Bayley:
http://www.newscientist.com/article/mg20427391.600-alices-adventures-in-alge...
So of course, when the discussion on quaternions has been written up, the title will have to be "The Mad Hatter's Tea-Party". In the meantime, perhaps I may move on to what prompted raising these matters in the first place: 2. Factorisation --- but not as we know it __________________________________________ In pursuit of my protracted investigations into the murkier recesses of Geometric Algebra, I had occasion recently to study Cl(2,2), with generators \x,\y,\v,\w (say) satisfying \x \x = \y \y = +1; \v \v = \w \w = -1; \x \y + \y \x = \x \v + \v \x = \x \w + \w \x = \y \v + \v \y = \y \w + \w \y = \v \w + \w \v = 0. One element of particular interest happens to be A == (a^2 + b^2) + b \x\y + a \x\v - a \y\w + b \v\w + \x\y\v\w, A^+ = (a^2 + b^2) - b \x\y - a \x\v + a \y\w - b \v\w + \x\y\v\w, where a,b represent real coefficients. We might verify that A is a versor, painstakingly checking that A^+ (c \x + d \y + e \v + f \w) A = vector; in fact for versors of grade < 6, it suffices that the terms are all even-grade or all odd-grade, and the magnitude scalar: so here we find ||A|| = (a^2 + 2a + 1+b^2) (a^2 - 2a + 1+b^2). Another theorem (Chasles) states that any versor is a product of vectors (with components in the scalar field). There are many ways in which a given versor may be factorised: for example when a = 0, A = \x \v (b\v - \w) (b\x + \y) = -\w \y (b\y - \x) (b\w + \v). Exercise: factorise the case a = 1, b = 0, A = 1 + \x\v - \y\w + \x\y\v\w. [Why should your suspicions be aroused by this apparently innocent case? Is it really a proper versor? Are there more cases like it?] It would be reassuring to have an explicit vector factorisation of A for general a,b. And given the simple, symmetric nature of A, we might hope for reasonably simple, symmetric factors. So far the best I have is (ready? deep breath!) A = ( a\x + b\y ) ( b\v + a\w ) ( -2ab(-a-b^2+a^2+a^3+ab^2)\x + a(a^4-a^2-4ab^2-b^4+b^2)\y - b(a^2+b^2+1)(-a-b^2+a^2+a^3+ab^2)\v - (a^6-2a^4b^2+a^4+2a^3b^2-a^2b^4+2ab^2+2b^4a-b^4-b^2)\w ) ( (a^4-2ab^2+b^2+a^2b^2)\x + b(-a-b^2+a^2+a^3+ab^2)\y + a(a^2+b^2)\v ), with RHS scaled up by scalar factor (a^6+2a^4b^2-a^4-2a^3b^2+a^2b^4-2ab^2-2b^4a+b^4+b^2)(a^2+b^2)^2 Can anybody improve on this? WFL
As a contrast, some motivational background ... 3. A Modicum of Isometry ________________________ Why, might one ask, should this horrible object be interesting? Clifford algebras provide homogeneous coordinates for the isometries and subspaces of a number of intuitively appealing geometric spaces, composition corresponding to product: spherical, Euclidean, Moebius (conformal, inversive), Laguerre (equilong), and Lie-sphere geometry in particular. [So of course do matrices --- but that's another story.] For most purposes we can restrict consideration to isometries which are "kinematic" (aka "Spin-up"), connected to the identity and so --- by simultaneously reducing their extent and increasing iteration --- capable of extension in the limit into continuous motion. A kinematic versor B is characterised algebraically by constraints ||B|| > 0 --- which might as well be rescaled to ||B|| = 1; and B is even (comprises only even-grade terms). The isometry structure of the more familiar geometries above is reasonably well-documented, at least in low dimension. Briefly, a kinematic isometry B of (maximum) grade 2l can uniquely be factorised into orthogonal grade-2 "rotations" B = B_1 B_2 ... B_l, where B_i = S_i + L_i; here S_i is a scalar representing the extent of the rotation, and L_i a versor purely of grade 2 representing its (coline) axis. The orthogonality constraint implies that the Clifford product of any k-subset of the axes L_i is "exterior", purely of grade 2k. Each axis can now in turn be factorised uniquely into a pair of eigenvectors L = F G with an explicit geometric interpretation --- as S varies through the limiting range, under the associated motion any (object represented by a) vector proceeds continuously from the source F to the sink G. [In Euclidean 3-space there is insufficient room for a pair of finite axes: the most general kinematic isometry is a "heliform" bi-rotation, composing a rotation about a finite axis line with a (parabolic) translation along it and about an orthogonal axis lying at infinity.] What a nice theory --- but there are devils in the detail, for both decompositions rely on solving quadratic equations. One devil concerns the sign of ||L||. When ||L|| = -1, L is "hyperbolic" and F,G are real and visible: for example, dilation (in the Moebius group) has eigenvectors at the origin and infinity. When ||L|| = +1, L is "elliptic" and F,G are conjugate complex and invisible: for example, in Euclidean rotation there is plainly no actual source or sink. When ||L|| = 0, L is "parabolic" and F,G are coincident --- while we'd prefer not to have to go there too soon, these are difficult to avoid --- one example is Euclidean translation, with double eigenvector at infinity. WFL
And at long last, the crucial connection. 4. Something in the Woodshed ____________________________ And there's another devil lurking: the extents S_i are (reciprocal) roots of an elegant polynomial equation ||B(1/S)|| = 0, with B(T) = <B>_0 + <B>_2 T + <B>_4 T^2 + ... representing a kind of Taylor series in the variable scalar T. The associated axis L_i is given by the bivector (grade-2) part of the singular versor scalar + L_i = B(T) (d/dT)B(T) evaluated at T = 1/S_i --- at least, provided L_i is not parabolic (S_i = 0), and B is not "isoclinic" (S_i multiple root). [This mysterious expression parrots the formula for the tangent to the space curve defined by a vector function of a single parameter.] The equation in S has only even powers; when a root (S_i)^2 > 0, the associated rotation S + L is hyperbolic, and setting S = coth(t/2) converts it into a continuous motion of time t; when (S_i)^2 < 0, S + L is elliptic, and S = cot(t/2) for angle t. [Yes, I know there's an \I = sqrt(-1) gone missing above, and honestly, I can explain, only for the time being I'm not going to, so there!] Suppose the equation in S^2 had a pair of conjugate complex roots? Since this never seemed to happen in most of these geometries --- though I should very much like to know, I actually have no idea why not --- as far as I am aware, nobody else bothered about this possibility much. But I bothered about it. For a start --- if such an exotic beast existed --- what physical meaning might be assigned to its single complex extent S, an apparently inextricable entanglement of angle and length? And behold: it actually does exist, in n-space Lie-sphere geometry, whose isometries are represented by the algebra Cl(n+1,2). The associated pair of rotations is also conjugate complex, their eigenvectors E + \I F, G + \I H combining in conjugate pairs across rotations, rather than (as it were) along a single elliptic rotation. The canonical grade-4 example has eigenvectors \x +/- \I\w, \y +/- \I\v, and associated conjugate complex rotation axes (\x\y + \v\w) -/+ \I(\x\v + \y\w), with product essentially \x\y\v\w. This can now be transmogrified into a continuous motion with two real parameters, by attaching complex extents S = 2(a + b\I), S^- = 2(a - b\I) to the axes, yielding A = (S + \x\y + \v\w - \I\x\v - \I\y\w)(S^- + \x\y + \v\w + \I\x\v + \I\y\w) = |S|^2 + 2 Re(S)(\x\v - \y\w) + 2 Im(S)(\x\y + \v\w) + 4\x\y\v\w = (a^2+b^2) + b\x\y + a\x\v - a\y\w + b\v\w + \x\y\v\w (dropping scalar factor 4), as previously presented. Isometries of this type turn out to be surprisingly common: an early attempt at implementation of factorisation into orthogonal rotations encountered a specimen as its very first random test datum, causing the hapless programmer (me) to suspect the early onset of senile dementia. Conjugates of A, along with compositions of hyperbolic-hyperbolic, hyperbolic-elliptic, and elliptic-elliptic of two types, are the only types of bi-rotation in Lie-sphere geometry having positive density [Bertrand's paradox preventing any more specific assertion concerning the relative values involved]. WFL
By now I'm probably talking to myself --- so just 5. Postscript _____________ The case a = 1, b = 0 of A surreptitiously introduced earlier can of course be cracked using the general factorisation given subsequently; however, if tackled in isolation there's no obvious algorithm available to unearth say A = 1 + \x\v - \y\w + \x\y\v\w = \x \y (\y - \w) (\x + \v) . This pathological horror is non-invertible, with magnitude zero; a = -1, b = 0 is the only other such case. It conjugates every vector remorselessly into zero (representing no object). More strangely still, it has complex conjugate rotation factors, even though the curve tangent formula would surely yield a (single) real factor for real S --- the explanation being that S = S^- is a double root, illustrating inter alia how this formula is inherently inapplicable to isoclinic cases! Of course, there's much more I could ramble on about --- proofs and algorithms, interactive Java movies of these isometries, plausible canards (I stopped counting at 12); the mapping from the algebra to the geometry; Bott periodicity; Dupin cyclides; a neat connection with linked spheres ... And then there are parabolic and isoclinic isometries, leading to an exhaustive classification of the conjugacy types of isometry. The last topic seems curiously neglected --- part of the reason may simply be that nineteenth century sources gave the impression that it had all been sorted out, at least in low dimension. Close inspection reveals that --- often without explicit mention --- they work over \C rather than \R, severely restricting any application to concrete, intuitive geometry; and no means of verifying correctness or completeness of any claimed classification is likely to be apparent. A more currently active area which has many obvious parallels with, and ought surely to bear some relevance to, these investigations is Lie algebra. Alas --- despite having embarked upon the consumption of numerous tracts upon this subject --- I have yet to acquire substantive understanding of what it might actually be useful for, and in particular of how it might be applied to the classification problem. Can anybody out there enlighten me? WFL
Have you tried math overflow? http://mathoverflow.net/ On Sat, Dec 19, 2009 at 10:10 AM, Fred lunnon <fred.lunnon@gmail.com> wrote:
A more currently active area which has many obvious parallels with, and ought surely to bear some relevance to, these investigations is Lie algebra. Alas --- despite having embarked upon the consumption of numerous tracts upon this subject --- I have yet to acquire substantive understanding of what it might actually be useful for, and in particular of how it might be applied to the classification problem.
Can anybody out there enlighten me?
WFL
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On 12/20/09, Mike Stay <metaweta@gmail.com> wrote:
Have you tried math overflow? http://mathoverflow.net/
At first impression --- this looks a very interesting site, well worth a visit. At second impression --- the technology they use seems a tad elaborate ("reputation points", anyone?), and perhaps over reliant on latest-version user software: I've used jsMath without trouble in the past, yet their scripts are only half-decoded by my browser (Firefox 2). If I can work out how to fins my way round it more comfortably, it should prove most useful ... Thanks for the tip! WFL
6. A Few References ___________________ There is a flourishing community publishing and discussing "Geometric Algebra" --- the application of Clifford algebra to analytic geometry --- which rarely seems to make contact with the outside world, and expends much energy on recycling stale material. Below I've disinterred a selection of the worthwhile minority hidden among the dross [there goes my invited address down the tube!] Most of my material has surely been folk-lore for a century of more; but if I knew of a satisfactory account of such matters, I shouldn't have felt obliged to write these notes. Once again, any suggestions? Rafal Ablamowicz Structure of Spin Groups Associated with Degenerate Clifford Algebras Journal of Mathematical Physics vol.27, pp.1--6 (1986); "CLIFFORD" (Algebra package for Maple V) http://math.gannon.edu/rafal/cliff3/ [One of very few to tackle degenerate Cl(p,q,r); concerned with implementation for theoretical physics.] S. L. Altmann "Rotations and Quaternions and Double Groups" Oxford (1986) [Does what it says on the tin, after a traditional fashion; but somehow, the wood tends to get lost among the trees.] T. E. Cecil "Lie Sphere Geometry" Springer ed.1 (1992), ed.2 (2008) [Introductory chapters discuss Lie-sphere group.] J. L. Coolidge "A Treatise on the Circle and the Sphere" Clarendon Press (1916); Chelsea (1971) [Extensive, idiosyncratic account of XIX-century analytic Lie-sphere geometry in later chapters.] C. J. L. Doran, A. N. Lasenby et al www.mrao.cam.ac.uk/~Clifford/ [Cavendish geometric algebra site: papers, Cl(4,1) software for 3-space Moebius group.] Jean Gallier "Clifford Algebras, Clifford Groups, and a Generalization of the Quaternions: The Pin and Spin Groups" www.cis.upenn.edu/~jean/home.html (online May 2008) [Good compact online account of (Bott) algebraic structure of Cl(p,q).] David Hestenes et al http://ModelingNTS.la.asu.edu/GC_R&D.html [Early modern proselyte for geometric algebra, though uninterested in nuts-and-bolts; papers passim, some online.] Pertti Lounesto "Clifford Algebras and Spinors" Cambridge University Press (2001) http://www.helsinki.fi/~lounesto/CLICAL.htm [Much detail about spinors, ideal structure, representation theory; unclear to me how this relates to practicality.] H. Pottmann & S. Leopoldseder "Geometries for CAGD" in "Handbook of 3D Modeling" pp.43--73 Elsevier (2002); online at http://www.geometrie.tuwien.ac.at/geom/leopoldseder/geom4cagd.pdf [Pottman's group at Vienna on Laguerre geometry.] Ian R. Porteous "Clifford Algebras and the Classical Groups" Cambridge University Press (1995, 2000, 2009) [Impressively complete account of classical theory of Cl(p,q), with relation to classical Lie groups.] Jon M. Selig "Geometrical Methods for Robotics" Springer ed.1 (1996), ed.2 (2005) [Sole published account of a properly thought-out, working metrical algebra, Cl(0,3,1) for Euclidean solid geometry; better is Cl(3,0,1)?]
Collecting together the earlier discussion on quaternions --- chronologically, I suppose this should be section 1.5. 7. The Mad Hatter's Tea Party _____________________________ After the reals \R ~= Cl(0,0), and complex numbers \C ~= Cl(0,1) generated by \I of dimension 2, come the quaternions \H ~= Cl(0,2) of dimension 4, equipped with their traditional basis \i,\j,\k, where \i \i = \j \j = \k \k = -1, \i \j = -\j \i = \k, \j \k = -\k \j = \i, \k \i = -\i \k = \j; [aka the Mad Hatter, the March Hare, the Dormouse; but killing Time, who thereupon took scalar offence.] A potential source of confusion arises from the resemblance between these definitions and those for Clifford generators \u,\v,\w (say) of Cl(0,3). There is a fundamental distinction between the concepts, the basis {1,\i,\j,\k} being linearly independent, but algebraically dependent: therefore only two members --- say \i,\j --- are available as generators of Cl(0,2), \k = \i\j mapping to a grade-2 versor. As a consequence, the threefold symmetry of the basis is lost under homomorphism to this graded algebra. A more satisfactory isomorphism of quaternions for our purposes embeds it as the even subalgebra within Cl(3,0) [or Cl(0,3) would do instead]: \H ~= Vl(0,3)^0 ~= Vl(3,0)^0 = Cl(3,0)^0, defined by \i -> \z\y, \j -> \x\z, \k -> \y\x. This map both preserves symmetry, and maps quaternions to versors; they are furthermore bi-vectors (grade-2), which proves to be the natural representation for rotations. In contrast are the complex bi-quaternions \H (x) \C of dimension 8, quaternions with components in \C --- or vice-versa, or indeed 2x2 complex matrices). These are isomorphic to the entire Clifford algebra \H (x) \C ~= \C (x) \H ~= \C(2) ~= Cl(3,0) under \i -> \z\y, \j -> \x\z, \k -> \y\x, \I -> \x\y\z. Not being restricted to versors, the embedding is only of restricted computational utility; on the other hand, the versor subgroup Vl(3,0) extends quaternions by one coset of (odd) reflections to the full rotation group of the sphere, the point group for Euclidean 3-space. WFL
An aide-memoire concocted for personal use, which might also be useful to anyone else with a memory as treacherous as mine. 7.5 Atiyah-Bott-Shapiro and all that ___________________________________ The algebraic structure of the general Clifford algebra was surveyed by these fellows in 1964. Nobody actually reads that, of course --- see Gallier, Porteous, or the following viciously compressed crib-sheet.
From the point of view of practical geometric computation, more to the point would be tabulation of the structure of the versor semigroups Vl(p,q) --- not to mention Vl(p,q,r) --- rather than the full algebras.
Notation: with \P denoting any of \R,\C,\H, etc, \P (x) \P' == tensor product of \P,\P'; \P(n) == n x n matrices over \P; 2 \P == \P (+) \P == double ring, defined by (a,b)(c,d) = (ac,bd) for a,b,c,d in \P. Bott structure of Cl(p,q-1) as offset symmetric matrix (below): for p = 0,...,7, Cl(0,p-1) ~= Cl(p,0-1) ~= 2 \R(1/2), \R, \C, \H, 2 \H, \H(2), \C(4), \R(8); for min(p,q) >= 0, Cl(p+8,q-1) ~= Cl(p,q-1+8) ~= Cl(p,q-1)(16); ================================================================================ p\q = -1 0 1 2 3 4 5 6 7 8 0 0 R C H 2H H(2) C(4) R(8) 2R(8) R(16) 1 R 2R R(2) C(2) H(2) 2H(2) H(4) C(8) R(16) 2 C R(2) 2R(2) R(4) C(4) H(4) 2H(4) H(8) 3 H C(2) R(4) 2R(4) R(8) C(8) H(8) 4 2H H(2) C(4) R(8) 2R(8) R(16) 5 H(2) 2H(2) H(4) C(8) R(16) 6 C(4) H(4) 2H(4) H(8) 7 R(8) C(8) H(8) 8 2R(8) R(16) 9 R(16) [view without proportional spacing] ================================================================================ Vahlen: Cl(p+1,q-1+1) ~= Cl(q+1,p-1+1) ~= Cl(p,q-1)(2). Even subalgebra: Cl^0(p,q) ~= Cl(p,q-1) ~= Cl(q,p-1) ~= Cl^0(q,p) [versors & grades not conserved]. Splitting: == Cl(p,q) simple iff p-q <> 1 (mod 4); else Cl(p,q) ~= Cl(p,q)(1+J)/2 (+) Cl(p,q)(1-J)/2, where J denotes unit pseudar (product of generators). Complex scalars: with m = p+q, Cl(m;\C) == Cl(p,q) (x) \C; for m even, Cl(m;\C) ~= Cl(m/2, m/2+1) ~= Cl(m+1,0), Cl(0,m+1) as m/2 odd, even [versors & grades not conserved]; Cl(m;\C) ~= \C(2^{m/2}) for m even, ~= 2\C(2^[m/2]) for m odd. Degenerate algebra: Cl(p,q,r) has p squares +1, q squares -1, r squares 0; Cl(p,q,r) ~= Cl(p,q) (x) Cl(0,0,r) [Ablamovicz] Reduction isomorphisms: \R(l) (x) \R(m) ~= \R(lm); \C (x) \C ~= 2\C; \H (x) \H ~= \R(4); \H (x) \C ~= \C(2), via a + b \i + c \j + d \k \in \H <-> [[a - d \I, b \I + c], [b \I - c, a + d \I]] \in \C(2). Cl(1,0) ~= 2\R via \x <-> [[1,0],[0,-1]] (diagonal matrix)! WFL
Quoting Fred lunnon <fred.lunnon@gmail.com>:
On 12/20/09, Mike Stay <metaweta@gmail.com> wrote:
Have you tried math overflow? http://mathoverflow.net/
At first impression --- this looks a very interesting site, well worth a visit.
well ..... Someone wanted to know when there are pairs of complex roots for a real polynomial. Someone else offered that it can't depend on the coefficients because they can be made arbitrtarily small and so are useless !!!!! I don't have a reference in mind, but that question was solved 500 years ago, give or take a century or two. Sturm sequences, alternation of signs, whatever ... . They want problems. Maybe someone could offer them the zeta function. That reputation bit --- you get points for answering questions, it seems. Karma, in other words. -hvm ------------------------------------------------- www.correo.unam.mx UNAMonos Comunicándonos
That site has a strong /algebraic geometry / category language/ type flavor and if your question doesn't fit the mold they are likely to put you down, close the question and send you elsewhere. Maybe just my experience. Some people here might thrive over there, not me. --Jim On Sat, Dec 19, 2009 at 9:43 PM, Mike Stay <metaweta@gmail.com> wrote:
Have you tried math overflow? http://mathoverflow.net/
On Sat, Dec 19, 2009 at 10:10 AM, Fred lunnon <fred.lunnon@gmail.com> wrote:
A more currently active area which has many obvious parallels with, and ought surely to bear some relevance to, these investigations is Lie algebra. Alas --- despite having embarked upon the consumption of numerous tracts upon this subject --- I have yet to acquire substantive understanding of what it might actually be useful for, and in particular of how it might be applied to the classification problem.
Can anybody out there enlighten me?
WFL
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Well, I swallowed my reservations and posted a question --- and it looks like James was right on the ball there. What a strange system! Mind you, they do seem to have an impressive amount of traffic, even if much of it is pretty tedious ... WFL On 12/20/09, James Buddenhagen <jbuddenh@gmail.com> wrote:
That site has a strong /algebraic geometry / category language/ type flavor and if your question doesn't fit the mold they are likely to put you down, close the question and send you elsewhere. Maybe just my experience. Some people here might thrive over there, not me.
--Jim
On Sat, Dec 19, 2009 at 9:43 PM, Mike Stay <metaweta@gmail.com> wrote:
Have you tried math overflow? http://mathoverflow.net/ -- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com
Link? On Tue, Dec 22, 2009 at 2:40 PM, Fred lunnon <fred.lunnon@gmail.com> wrote:
Well, I swallowed my reservations and posted a question --- and it looks like James was right on the ball there.
What a strange system! Mind you, they do seem to have an impressive amount of traffic, even if much of it is pretty tedious ...
WFL
On 12/20/09, James Buddenhagen <jbuddenh@gmail.com> wrote:
That site has a strong /algebraic geometry / category language/ type flavor and if your question doesn't fit the mold they are likely to put you down, close the question and send you elsewhere. Maybe just my experience. Some people here might thrive over there, not me.
--Jim
On Sat, Dec 19, 2009 at 9:43 PM, Mike Stay <metaweta@gmail.com> wrote: > Have you tried math overflow? > http://mathoverflow.net/ > -- > Mike Stay - metaweta@gmail.com > http://math.ucr.edu/~mike > http://reperiendi.wordpress.com >
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On 12/22/09, Mike Stay <metaweta@gmail.com> wrote:
Link?
http://mathoverflow.net/questions/9555/ga-factor-problem-closed
On Tue, Dec 22, 2009 at 2:40 PM, Fred lunnon <fred.lunnon@gmail.com> wrote:
Well, I swallowed my reservations and posted a question --- and it looks like James was right on the ball there.
What a strange system! Mind you, they do seem to have an impressive amount of traffic, even if much of it is pretty tedious ...
WFL
On 12/20/09, James Buddenhagen <jbuddenh@gmail.com> wrote:
That site has a strong /algebraic geometry / category language/ type flavor and if your question doesn't fit the mold they are likely to put you down, close the question and send you elsewhere. Maybe just my experience. Some people here might thrive over there, not me.
--Jim
On Sat, Dec 19, 2009 at 9:43 PM, Mike Stay <metaweta@gmail.com> wrote:
Have you tried math overflow? http://mathoverflow.net/ -- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com
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--
Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com
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Yeah, that is annoying. I know Tom Leinster; he's not the one who closed it, just one who suggested some formatting changes that would make people more likely to read far enough to answer it (use latex markup instead of mathematica markup). The other guy seems like one of those editors on wikipedia that go around deleting articles just because they can. On Tue, Dec 22, 2009 at 4:38 PM, Fred lunnon <fred.lunnon@gmail.com> wrote:
On 12/22/09, Mike Stay <metaweta@gmail.com> wrote:
Link?
http://mathoverflow.net/questions/9555/ga-factor-problem-closed
On Tue, Dec 22, 2009 at 2:40 PM, Fred lunnon <fred.lunnon@gmail.com> wrote: > Well, I swallowed my reservations and posted a question --- > and it looks like James was right on the ball there. > > What a strange system! Mind you, they do seem to have an > impressive amount of traffic, even if much of it is pretty tedious ... > > WFL > > On 12/20/09, James Buddenhagen <jbuddenh@gmail.com> wrote: >> That site has a strong /algebraic geometry / category language/ type >> flavor and if your question doesn't fit the mold they are likely to >> put you down, close the question and send you elsewhere. Maybe just >> my experience. Some people here might thrive over there, not me. >> >> --Jim >> >> >> On Sat, Dec 19, 2009 at 9:43 PM, Mike Stay <metaweta@gmail.com> wrote: >> > Have you tried math overflow? >> > http://mathoverflow.net/ >> > -- >> > Mike Stay - metaweta@gmail.com >> > http://math.ucr.edu/~mike >> > http://reperiendi.wordpress.com >> > >
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--
Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com
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-- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com
On 12/23/09, Mike Stay <metaweta@gmail.com> wrote:
Yeah, that is annoying. I know Tom Leinster; he's not the one who closed it, just one who suggested some formatting changes that would make people more likely to read far enough to answer it (use latex markup instead of mathematica markup).
Reasonably enough ...
The other guy seems like one of those editors on wikipedia that go around deleting articles just because they can.
Leading one to conclude that spending more time tuning the formatting would be unlikely to prove a worthwhile investment. Shame about that! WFL
On Tue, Dec 22, 2009 at 4:38 PM, Fred lunnon <fred.lunnon@gmail.com> wrote:
On 12/22/09, Mike Stay <metaweta@gmail.com> wrote:
Link?
http://mathoverflow.net/questions/9555/ga-factor-problem-closed
On Tue, Dec 22, 2009 at 2:40 PM, Fred lunnon <fred.lunnon@gmail.com> wrote:
Well, I swallowed my reservations and posted a question --- and it looks like James was right on the ball there.
What a strange system! Mind you, they do seem to have an impressive amount of traffic, even if much of it is pretty tedious ...
WFL
On 12/20/09, James Buddenhagen <jbuddenh@gmail.com> wrote:
That site has a strong /algebraic geometry / category language/ type flavor and if your question doesn't fit the mold they are likely to put you down, close the question and send you elsewhere. Maybe just my experience. Some people here might thrive over there, not me.
--Jim
On Sat, Dec 19, 2009 at 9:43 PM, Mike Stay <metaweta@gmail.com> wrote:
Have you tried math overflow? http://mathoverflow.net/ -- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com
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--
Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com
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-- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com
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I wrote Tom Leinster about this, and he said I could forward his reply: ---------------------- Hi Mike, I'm sorry your friend/acquaintance had this bad experience. It may be that the moderator was overeager - and that particular moderator can be on the trigger-happy side - but having said that, I'm not enormously surprised it was closed. I don't think there was any problem with the fact that it wasn't a category theory question. (After all, number theory is even more popular a subject for Math Overflow questions.) What I imagine went through the moderator's mind was that the question looked like a whole string of formulas with very little context or explanation. It might have helped for the author to explain why he wanted to know, or whether he had any idea what the answer might be - any kind of background, really. As for formatting, I don't know Mathematica notation, but I'm guessing that \x just means x, and of course in plain text it becomes much more readable if you take out the backslash. As I said, Latex is also an option. (In case my meaning wasn't clear, the site does actually compile Latex, so readers wouldn't see the dollar signs etc.) In case you want to forward this mail to him (feel free), I have a couple of other thoughts: 1. Many of the really low-quality questions come from "unknown (google)", i.e. people without logins. The converse isn't true - i.e. not every question from "unknown (google)" is crappy - but the moderators are only human, and I'm sure they can't help forming an association in their minds. In general, everyone's encouraged to use their real names. 2. When the moderator suggested editing and then flagging for reopening, I think that was perfectly sincere. Questions do get reopened, and since there's clearly a genuine mathematical question here, I can't see a reason not to follow this suggestion. There are a lot of smart people reading the site, and someone might well provide a good answer. All the best, Tom PS - Incidentally, the question was closed and the moderator's comment added before I made my comment about formatting, not the other way round. PPS - I have no idea why the word "opinionated" was used. Maybe there was no reason at all. ---------------------- On Tue, Dec 22, 2009 at 6:02 PM, Fred lunnon <fred.lunnon@gmail.com> wrote:
On 12/23/09, Mike Stay <metaweta@gmail.com> wrote:
Yeah, that is annoying. I know Tom Leinster; he's not the one who closed it, just one who suggested some formatting changes that would make people more likely to read far enough to answer it (use latex markup instead of mathematica markup).
Reasonably enough ...
The other guy seems like one of those editors on wikipedia that go around deleting articles just because they can.
Leading one to conclude that spending more time tuning the formatting would be unlikely to prove a worthwhile investment. Shame about that!
WFL
On Tue, Dec 22, 2009 at 4:38 PM, Fred lunnon <fred.lunnon@gmail.com> wrote: > On 12/22/09, Mike Stay <metaweta@gmail.com> wrote: >> Link? > > http://mathoverflow.net/questions/9555/ga-factor-problem-closed > >> >> On Tue, Dec 22, 2009 at 2:40 PM, Fred lunnon <fred.lunnon@gmail.com> wrote: >> > Well, I swallowed my reservations and posted a question --- >> > and it looks like James was right on the ball there. >> > >> > What a strange system! Mind you, they do seem to have an >> > impressive amount of traffic, even if much of it is pretty tedious ... >> > >> > WFL >> > >> > On 12/20/09, James Buddenhagen <jbuddenh@gmail.com> wrote: >> >> That site has a strong /algebraic geometry / category language/ type >> >> flavor and if your question doesn't fit the mold they are likely to >> >> put you down, close the question and send you elsewhere. Maybe just >> >> my experience. Some people here might thrive over there, not me. >> >> >> >> --Jim >> >> >> >> >> >> On Sat, Dec 19, 2009 at 9:43 PM, Mike Stay <metaweta@gmail.com> wrote: >> >> > Have you tried math overflow? >> >> > http://mathoverflow.net/ >> >> > -- >> >> > Mike Stay - metaweta@gmail.com >> >> > http://math.ucr.edu/~mike >> >> > http://reperiendi.wordpress.com >> >> > >> > >> >> > _______________________________________________ >> > math-fun mailing list >> > math-fun@mailman.xmission.com >> > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> > >> >> >> >> -- >> >> Mike Stay - metaweta@gmail.com >> http://math.ucr.edu/~mike >> http://reperiendi.wordpress.com >> >> >> _______________________________________________ >> math-fun mailing list >> math-fun@mailman.xmission.com >> http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> > > _______________________________________________ > math-fun mailing list > math-fun@mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >
-- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com
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-- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com
participants (5)
-
Dan Asimov -
Fred lunnon -
James Buddenhagen -
mcintosh＠servidor.unam.mx -
Mike Stay