Subject: Two papers published by AGT Originator: israel@math.ubc.ca (Robert Israel) GTP announces the publication of two papers in Algebraic and Geometric Topology (1) Peripheral separability and cusps of arithmetic hyperbolic orbifolds by D.B. McReynolds URL: http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-32.abs.html and (2) The braid groups of the projective plane by Daciberg Lima Goncalves and John Guaschi URL: http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-33.abs.html Details follow: (1) Algebraic and Geometric Topology URL: http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-32.abs.html Title: Peripheral separability and cusps of arithmetic hyperbolic orbifolds Author(s): D.B. McReynolds Abstract: For X = R, C, or H it is well known that cusp cross-sections of Žnite volume X-hyperbolic (n+1)-orbifolds are žat n-orbifolds or almost žat orbifolds modelled on the (2n+1)-dimensional Heisenberg group N_{2n+1} or the (4n+3)-dimensional quaternionic Heisenberg group N_{4n+3}(H). We give a necessary and sufŽcient condition for such manifolds to be diffeomorphic to a cusp cross-section of an arithmetic X-hyperbolic (n+1)-orbifold. A principal tool in the proof of this classiŽcation theorem is a subgroup separability result which may be of independent interest. Secondary: 20G20 Keywords: Borel subgroup, cusp cross-section, hyperbolic space, nil manifold, subgroup separability. Revised: 24 August 04 Author(s) address(es): University of Texas, Austin, TX 78712, USA Email: dmcreyn@math.utexas.edu (2) Algebraic and Geometric Topology URL: http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-33.abs.html Title: The braid groups of the projective plane Author(s): Daciberg Lima Goncalves and John Guaschi Abstract: Let B_n(RP^2)$ (respectively P_n(RP^2)) denote the braid group (respectively pure braid group) on n strings of the real projective plane RP^2. In this paper we study these braid groups, in particular the associated pure braid group short exact sequence of Fadell and Neuwirth, their torsion elements and the roots of the `full twist¹ braid. Our main results may be summarised as follows: Žrst, the pure braid group short exact sequence 1 --> P_{m-n}(RP^2 - {x_1,...,x_n}) --> P_m(RP^2) --> P_n(RP^2) --> 1 does not split if m > 3 and n=2,3. Now let n > 1. Then in B_n(RP^2), there is a k-torsion element if and only if k divides either 4n or 4(n-1). Finally, the full twist braid has a k-th root if and only if k divides either 2n or 2(n-1). Secondary: Secondary: 55Q52, 20F05 Keywords: Braid group, conŽguration space, torsion Author(s) address(es): Departamento de Matematica - IME-USP Caixa Postal 66281 - Ag. Cidade de Sao Paulo CEP: 05311-970 - Sao Paulo - SP - Brasil and Laboratoire de Mathematiques Emile Picard, UMR CNRS 5580 UFR-MIG Universite Toulouse III, 118, route de Narbonne 31062 Toulouse Cedex 4, France Email: dlgoncal@ime.usp.br, guaschi@picard.ups-tlse.fr === Subject: Re: Žnitely generated ideals Originator: israel@math.ubc.ca (Robert Israel) >hi, >i¹m wondering where to Žnd such results in literature, or even better, >how to prove them: >if A is a Žnitely generated algebra, and I is a Žnite-codimension >ideal, then I is Žnitely generated. A is Žnitely generated over what? An arbitrary ring is always Žnitely generated over itself. If you¹re thinking that A is Žnitely generated over Z or over a Želd, then the result you mention is true due to the theory of Noetherian rings (but in that case all ideals are of Žnite codimension, so I¹m not sure why you mention it). --Paul Vojta, vojta@math.berkeley.edu === Originator: israel@math.ubc.ca (Robert Israel) Third Announcement Centre de Recherches Math.8ematiques Montr.8eal, (Qu.8ebec) Canada The topics to be covered at this meeting include the most recent developments in algebraic K-theory and the closely allied areas of motivic homotopy theory, algebraic cycles, and motivic cohomology theory, along with applications in other areas of Mathematics. The following mathematicians have agreed to speak: Paul Balmer (ETH, Zurich) Gunnar Carlsson (Stanford) Jean-Louis Colliot-Th.8el.8fne (Paris Sud) Thomas Geisser (USC) Alexander Goncharov (Brown) Jens Hornbostel (Regensburg) Max Karoubi (Paris VII) Marc Levine (Northeastern) Ib Madsen (Aarhus) Fabien Morel (Paris VII) Oliver Roendigs (Bielefeld) Markus Rost (Bielefeld) Marco Schlichting R. Sujatha (Tata Institute) Andrei Suslin (Northwestern) Burt Totaro (Cambridge) Mark Walker (Nebraska) Organizers: Eric Friedlander, Northwestern University Dan Grayson, University of Illinois Rick Jardine, University of Western Ontario Manfred Kolster, McMaster University Housing and travel information and an online registration form are available at the CRM conference web site: http://www.crm.umontreal.ca/K/ The most up to date web page for the conference is Abstracts and a schedule for the meeting are linked to that page. Some Žnancial support for students and researchers otherwise unsupported to attend the meeting may still be available. Those wishing to apply for support should contact either Dan Grayson (if based in the U.S.) or Rick Jardine The conference is generously funded by the Centre de Recherches Math.8ematiques (CRM) of the Universit.8e de Montr.8eal and by the National Science Foundation (NSF) via a grant (NSF DMS 03-03519). === Subject: Re: Can one foliate, or rather spaghettize, 3-space with skew lines? Originator: israel@math.ubc.ca (Robert Israel) Would spaghettization be a still better name than foliation for this partitioning of 3-space into straight lines? This spaghettization is closely related to a beautiful classical 3D Žgure, a so-called linear complex. A linear complex consists of all lines that are tangent to a point on a screw line with a speciŽed axis and pitch. (I do not know whether this is the ofŽcial deŽnition of linear complex. I did not retrieve an ofŽcial deŽnition while writing this message.) In the correspondence (x, y, 0) => [ t => (x - ty, y + tx, t) ] as given previously in this newsgroup thread, look at all screw lines with the Z axis as their axis and 2pi as their pitch, and draw all tangents to these screw lines at the points where they intersect the XY plane. This yields exactly the abovementioned spaghettization. It is a plane section of the entire line complex. Think of a bundle of lemonade straws loosely held together in a circular vase with a narrow neck. Observe that the common perpendiculars of each line of the spaghettization and the central axis all lie in the same plane. BTW, another classical item is the so-called null system. This is a projective reciprocal relation between points and planes in 3D space that is, contrary to polar point - polar plane relations in 3D space, not based on a quadric. A null system is obtained by means of a plane section S of a linear complex. Each point P corresponds to the plane p perpendicular to the line in S through P; each plane p corresponds to the point P of intersection with that single line in S that is perpendicular to p. Line geometry was foremost developed by the German mathematicians Julius Pluecker and Felix Klein. >> >>>Can one foliate space with straight lines that are pairwise skew? >>> >>It looks to me offhand like this works, but I can¹t Žnd any >>more scrap paper to check. Begin by exhausting 3-space by >>the pairwise disjoint sets H(A) = {(x,y,z): x^2+y^2=A^2(1+z^2)}, >>for A greater than or equal to 0. H(0) is the z-axis, and >>for A greater than 0, H(A) is a hyperboloid of one sheet, >>and thus a ruled surface, foliated (in two ways) by straight >>lines that are pairwise skew. I *think* that it¹s both >>clear and true that for A not equal to B, each generator >>of the ruling on H(A) is skew to each generator of the >>ruling on H(B). Then choosing the rulings consistently, >>I think you get the desired foliation of 3-space. >> >It works! -- Here¹s an explicit parametrization: >To any point (x,y,0) in the plane >associate the line > t |--> (x-t*y, y+ t*x, t) >in three-space. >These lines are pairwise skew and reach every point in three-space. === Subject: Re: Can one foliate, or rather spaghettize, 3-space with skew lines? Originator: israel@math.ubc.ca (Robert Israel) > Would spaghettization be a still better name than foliation for this > partitioning of 3-space into straight lines? I don¹t have an opinion on that, but here¹s a followup to the original question: Is it possible to foliate space with skew lines so that there¹s one line in every possible direction? Note that in the foliation by hyperboloids, none of the lines are parallel to the xy-plane. Dean Hickerson dean@math.ucdavis.edu === Subject: Re: Can one foliate, or rather spaghettize, 3-space with skew lines? Epigone-thread: trangghoanerl << Is it possible to foliate space with skew lines so that there¹s one line in every possible direction? I think this has been completely settled in the negative by Thomas Mautsch¹s post. But it suggests another question: QUESTION: Is R^3 the disjoint union of straight lines, exactly one in each possible direction? (This is of course the same as Dean Hickerson¹s question except that it ignores continuity.) I hereby go on record as conjecturing that there is an example of this, probably provable by following a (1964 ?) proof by Conway & Croft showing that R^3 is the disjoint union of congruent geometric unit circles. --Dan Asimov === Subject: Re: Can one foliate, or rather spaghettize, 3-space with skew lines? > << > Is it possible to foliate space with skew lines so that there¹s one > line in every possible direction? > >> > I think this has been completely settled in the negative by Thomas > Mautsch¹s post. > But it suggests another question: > QUESTION: Is R^3 the disjoint union of straight lines, exactly one in > each possible direction? > (This is of course the same as Dean Hickerson¹s question except that > it ignores continuity.) Actually, that¹s the question that I had in mind. I wasn¹t aware that the word foliate implied some sort of continuity. William Thurston has shown that the existence of such a partition is implied by the axiom of choice. I wonder if there¹s a constructive proof. Dean Hickerson dean@math.ucdavis.edu === Subject: Re: Can one foliate, or rather spaghettize, 3-space with skew lines? > Is it possible to foliate space with skew lines so that there¹s one line > in every possible direction? I don¹t think so. - Assume that there is a map from R^3 to the space RP^2 of directions in R^3 so that the preimage of every point in RP^2 is a line in R^3. Consider the preimage of a non-contractible, closed curve in RP^2. Topologically it must be a (ruled) cylinder in R^3. Take a section of this cylinder, i.e. a closed curve on the cylinder that projects 1-to-1 down to the original curve in RP^2. This curve can be contracted to a point in R^3; its projection to RP^2 would yield a null-homotopy of the original, non-contractible curve, hence a contradiction. > Note that in the foliation by hyperboloids, none of the lines are parallel > to the xy-plane. Still, from this naive topological point of view it might be possible to construct a foliation by skew lines that misses only one direction in R^3... === Subject: Re: Can one foliate, or rather spaghettize, 3-space with skew lines? > > > > Is it possible to foliate space with skew lines so that there¹s one line > > in every possible direction? > I don¹t think so. - Assume that there is a map from R^3 > to the space RP^2 of directions in R^3 > so that the preimage of every point in RP^2 is a line in R^3. > Consider the preimage of a non-contractible, closed curve in RP^2. > Topologically it must be a (ruled) cylinder in R^3. > Take a section of this cylinder, i.e. a closed curve on the cylinder > that projects 1-to-1 down to the original curve in RP^2. > This curve can be contracted to a point in R^3; > its projection to RP^2 would yield a null-homotopy of the original, > non-contractible curve, hence a contradiction. > > Note that in the foliation by hyperboloids, none of the lines are parallel > > to the xy-plane. > Still, from this naive topological point of view > it might be possible to construct a foliation by skew lines > that misses only one direction in R^3... Following your arguments, it is not possible: the preimage of a non-contractible curve in RP^2 would be a Moebius strip of straight lines, which seems not exist in R^3. Simeon === Subject: Re: Can one foliate, or rather spaghettize, 3-space with skew lines? >>Is it possible to foliate space with skew lines so that there¹s one line >>in every possible direction? >> Precisely one line or at least one line? The following discussion assumes precisely one, but it can be adapted to the more general case. I assume that we¹re discussing a foliation in the usual technical sense. We have to check for the following arguments that the projection R^3 --> RP^2 corresponding to the foliation deŽnes a (locally trivial) Žbre bundle with typical Žbre R. This seems to work, but I¹m too lazy to verify the technical details. >I don¹t think so. - Assume that there is a map from R^3 >to the space RP^2 of directions in R^3 >so that the preimage of every point in RP^2 is a line in R^3. >Consider the preimage of a non-contractible, closed curve in RP^2. >Topologically it must be a (ruled) cylinder in R^3. Nitpickingly, it could a priori be a Moebius strip. But in any case, it is a Žbre bundle (over the circle) with typical Žbre R and thus admits a section. >Take a section of this cylinder, i.e. a closed curve on the cylinder >that projects 1-to-1 down to the original curve in RP^2. >This curve can be contracted to a point in R^3; >its projection to RP^2 would yield a null-homotopy of the original, >non-contractible curve, hence a contradiction. >>Note that in the foliation by hyperboloids, none of the lines are parallel >>to the xy-plane. >> >Still, from this naive topological point of view >it might be possible to construct a foliation by skew lines >that misses only one direction in R^3... We can ask this question for any open subset X of RP^2; you suggest here X = RP^2 {point}. The foliation deŽnes a Žbre bundle R^3 --> X with typical Žbre R. The corresponding exact homotopy sequence shows that X is contractible (since R and R^3 are contractible). But X = RP^2 {point} is not contractible (consider e.g. the Mayer/Vietoris sequence for RP^2 = X union disc: if X were contractible, then H_2(RP^2) and H_1(S^1) would coincide, but they don¹t). However, RP^2 minus the circle RP^1 is contractible; this observation corresponds to the hyperboloid example. -- Marc Nardmann === Subject: Re: Can one foliate, or rather spaghettize, 3-space with skew lines? >> >>>Is it possible to foliate space with skew lines so that there¹s one line >>>in every possible direction? >>> >Precisely one line or at least one line? The following discussion >assumes precisely one, but it can be adapted to the more general case. >I assume that we¹re discussing a foliation in the usual technical sense. >We have to check for the following arguments that the projection R^3 --> >RP^2 corresponding to the foliation deŽnes a (locally trivial) Žbre >bundle with typical Žbre R. That is far from clear; what about this hypothetical foliation would force its leaf space to be RP^2? The usual technical sense of a foliation involves a triviality condition which is local *in the foliated space*, not (necessarily) uniformly along the leaves. >This seems to work, but I¹m too lazy to >verify the technical details. >>I don¹t think so. - Assume that there is a map from R^3 >>to the space RP^2 of directions in R^3 >>so that the preimage of every point in RP^2 is a line in R^3. >>Consider the preimage of a non-contractible, closed curve in RP^2. >>Topologically it must be a (ruled) cylinder in R^3. >Nitpickingly, it could a priori be a Moebius strip. Not for sufŽciently large values of a priori. Since the hypothetical leaves are straight lines, the ruled surface (of all lines mapping to the given non-contractible, closed curve in RP^2) will have a well-deŽned locus at inŽnity that is, in fact, when interpreted as a locus in the RP^2 at inŽnity which compactiŽes R^3 to RP^3, is the very non-contractible, closed curve in RP^2 you started with. But now, interpret it instead as a locus in the S^2 at inŽnity, which is the double cover of that curve--if (as I think Thomas Mautsch meant to us to think; anyway, as he could have demanded) that curve is simple, then so is its double cover. Now interpret R^3 with its S^2 at inŽnity as D^3: we have the purported Moebius strip embedded in the interior of D^3, and bounded by a single simple closed curve on S^2 that itself, by the Jordan curve theorem, bounds a disk on S^2. The union of the Moebius strip and the disk is a topological RP^2 embedded in D^3; contradiction. >But in any case, it >is a Žbre bundle (over the circle) with typical Žbre R and thus admits >a section. >>Take a section of this cylinder, i.e. a closed curve on the cylinder >>that projects 1-to-1 down to the original curve in RP^2. >>This curve can be contracted to a point in R^3; >>its projection to RP^2 would yield a null-homotopy of the original, >>non-contractible curve, hence a contradiction. >> >>>Note that in the foliation by hyperboloids, none of the lines are parallel >>>to the xy-plane. >>> >>Still, from this naive topological point of view >>it might be possible to construct a foliation by skew lines >>that misses only one direction in R^3... >> >We can ask this question for any open subset X of RP^2; you suggest here >X = RP^2 {point}. The foliation deŽnes a Žbre bundle R^3 --> X with >typical Žbre R. The corresponding exact homotopy sequence shows that X >is contractible (since R and R^3 are contractible). But X = RP^2 >{point} is not contractible (consider e.g. the Mayer/Vietoris sequence >for RP^2 = X union disc: if X were contractible, then H_2(RP^2) and >H_1(S^1) would coincide, but they don¹t). However, RP^2 minus the circle >RP^1 is contractible; this observation corresponds to the hyperboloid >example. I¹m still worried about your cavalier assumption that the leaf space is nice. (Maybe that follows from the geometrical assumption that the leaves are straight lines, or even the weaker topological assumption that they are properly embedded copies of R. But it isn¹t clear to me, yet.) Lee Rudolph === Subject: Re: Trying to solve 2 homogenous quadratics Originator: israel@math.ubc.ca (Robert Israel) >> How would one solve the following system of equations? >> (1) 17*u^2 - 144*u*v + 1904*v^2 = 43*U^2 - 60*U*V + 258*V^2 >> (2) 162*u^2 - 8568*u*v + 18144*v^2 = 10*U^2 - 172*U*V + 60*V^2 >> and u,v,U,V are in Z. >> We seek solutions so that (u,v) and (U,V) have the same value in either (1) or >> (2) (coincidental solutions?) >> This is from trying to factor an homogenous quartic space from the elliptic >> curve [0,6800,0,11559996,0] >> Randall >Provided the discriminants of each equation satisfy suitable mutual >conditions (equal, or have the same square-free factor?) I think it¹s >possible to Žnd a linear transform of u, v, U, V, or maybe a birational >transform of the dehomogenized pair, that reduces the original pair to >the more canonical form: > ax^2 + bxy + cy^2, dx^2 + exy + fy^2 = z^2, t^2 resp >However, I¹m moving house at the moment and all my notes are in boxes. >[sci.math.research added - hope you don¹t mind, but I¹d be interested >in an expert answer to this myself!] >John R Ramsden (jramsden@glasshouse.cam) com not cam For those who don¹t want to run the 400-line Maple script, you can try to Žnd a nonzero integer solution to the homogeneous quartic 4 3 0 = denvar + (-18 a3 + 54 a6) denvar 2 2 2 + (744 a6 + 372 a3 - 580 a3 a6) denvar 2 2 - 18 (-2 a6 + 3 a3) (26 a6 - 15 a3 a6 + 13 a3 ) denvar 2 2 2 + 2 (26 a6 - 15 a3 a6 + 13 a3 ) or 4 3 0 = denvar + (18 a3 - 54 a6) denvar 2 2 2 + (-852 a3 a6 + 425 a3 + 850 a6 ) denvar 2 2 + 18 (-2 a6 + 3 a3) (84 a6 - 25 a3 a6 + 42 a3 ) denvar 2 2 2 + 2 (84 a6 - 25 a3 a6 + 42 a3 ) 1 The values of a3, a6, denvar are predicted to have about half as many digits as u, v, U, V. ----------------- u := 4*unew; v := vnew; given1 := (17*u^2 - 144*u*v + 1904*v^2) - ( 43*U^2 - 60*U*V + 258*V^2); given2 := ( 162*u^2 - 8568*u*v + 18144*v^2) - ( 10*U^2 - 172*U*V + 60*V^2); # Both equations are linear in u*v, u^2 + 112*v^2, # U*V, U^2 + 6*V^2. Denote s = u^2 + 112*v^2 and t = u*v. # The change of variables makes s = 16*(unew^2 + 7*vnew^2) # and t = 4*unew*vnew. We Žnd # U^2 + 6*V^2 = 72*t - s # U*V = 54*t - s # (U^2 - 6*V^2)^2 = (U^2 + 6*V^2)^2 - 24*(U*V)^2 # = -23*s^2 + 2448*s*t - 64800*t^2 # If we guess u and v, we can compute s and t. # Check whether # -23*s^2 + 2448*s*t - 64800*t^2 is a square, say 256*w^2. # Once we get an integer w, we can compute # U^2 - 6*V^2. We know U^2 + 6*V^2 and # can test whether the computed U^2 is a square. # The requirement w^2 > 0 lets us trim the search to # 2.035 < u/v < 2.383 or 47.00 < u/v < 55.02. # In particular, u and v have the same sign. wsq := simplify((U^2 - 6*V^2)^2/256, {given1, given2}, {U, V}); # Express wsq in terms of u and v # Observe wsq splits into four linear # factors over Q(sqrt(2), sqrt(3)). alias(sq2 = RootOf(z^2 = 2, z)); alias(sq3 = RootOf(z^2 = 3, z)); sq6 := sq2*sq3; factor(wsq, {sq2, sq3}); # Maple¹s factorization of wsq has denominator > 1. # Scale the coefŽcients of one linear factor to get fac0. # fac0 is an algebraic integer if unew, vnew are rational integers. fac0 := unew*(1 + 5*sq2/2 + sq3 + 3*sq6/2) + vnew*(4 - sq2 - 3*sq3 - sq6): evala(Rem(wsq, fac0, unew)); # Zero, conŽrming claim that fac0 divides wsq. # The product of the four conjugates of fac0 is wsq. fac0_poly := collect(evala(Norm(X - fac0)), X, factor); normal(coeff(fac0_poly, X, 0)/wsq); # 1 # We claim that if GCD(unew, vnew) = 1, then # the four conjugates of fac0 are # relatively prime as algebraic integers, # except possibly for the pairs of factors # # Powers of 2 # 3 +- sqrt(2) (norm 7) # 43 +- 5*sqrt(6) (norm 1699) # If an algebraic integer divides two # of the four conjugates of fac0, then it # divides both the X^1 and X^0 coefŽcients of fac0_poly. # We take their resultant. coef1 := coeff(fac0_poly, X, 1); ru := resultant(coef1, wsq, unew); rv := resultant(coef1, wsq, vnew); ifactor(content(ru)); # Powers of 2, 3, 7, 1699 ifactor(content(rv)); Factor(wsq) mod 2; # Vanishes only if unew + vnew is even Factor(wsq) mod 3; # No linear factors Factor(wsq) mod 7; # Vanishes only if unew == 0 (mod 7) Factor(wsq) mod 1699; # Vanishes only if unew == 101*vnew (mod 1699) # These lead to the exceptions listed above. # A corollary is that if two conjugates of # an algebraic integer divide fac0 for some integer u, v, # then GCD(u, v) > 1, except possibly for # a pair like 1 +- sqrt(3) (where the norm is -2). # For example, if both 3 + sqrt(2) and 3 - sqrt(2) # divide fac0, then 7 divides fac0, and 7 divides GCD(u, v). # Using the factor cof0 over Q(sqrt(2), sqrt(3)), # we can get a factor over Q(sqrt(2)). fac0_conj := subs(sq3 = -sq3, fac0); fac_sq2 := collect(simplify(fac0*fac0_conj), sq2, factor); # fac_sq2 has norm wsq, which we are assuming to be a square. # Q(sqrt(2)) is a unique factorization domain. # # Try to write # fac_sq2 = un2 * alpha2^2 * rat2 # where alpha2 = alpha2(sq2) in Q(sqrt(2)), rat2 is rational, # and un2 = un2(sq2) is a unit in Q(sq2). # [sq2 can designate sqrt(2) or -sqrt(2).] simplify(fac_sq2 * subs(sq2 = -sq2, fac_sq2)/wsq); # 1 # The permissible units (up to square factors) # are un2 = +-1 and +-(1 + sqrt(2)). # wsq = norm(fac_sq2) = norm(un2 * rat2 * alpha2)^2 # = norm(un2) * rat2^2 * norm(alpha2)^2, # # we Žnd norm(un2) = wsq/(rat2^2 * norm(alpha2)^2) # is a rational square, hence positive. # This restricts us to un2 = +- 1, which we can lump with rat2. # I am unable to determine the sign of un2 * rat2. # If u/v is near 50, then both embeddings # of fac_sq2 are negative, and we will need un2 * rat2 < 0. # If u/v is near 2.2, then both embeddings of fac_sq2 # are positive, and we will need un2 * rat2 > 0. # Likewise we can get a factor fac_sq3 of wsq over Q(sqrt(3)). # We try to write fac_sq3 as un3 * rat3 * alpha3^2 where # un3, alpha3 in Q(sqrt(3)) and un3 is a unit and rat3 rational. # The choices for un3 are +- 1 and +- (2 + sqrt(3)). # This time Norm(2 + sqrt(3)) = 4 - 3 = +1 # unlike Norm(1 + sqrt(2)) = 1 - 2 = -1 # last time, so we cannot decide the # existence of a 2 + sqrt(3) factor merely by # looking at signs. Perhaps we can use congruences. fac_sq3 := collect(simplify(fac0*subs(sq2 = -sq2, fac0)), sq3, factor); simplify(fac_sq3 * subs(sq3 = -sq3, fac_sq3)/wsq); # 1 # Thirdly, we can factor over Q(sqrt(6)). # The coefŽcients look familiar, involving the # u^2 + 112*v^2 and u*v from our original equations. # We can switch variables back to {U, V}. fac_sq6 := collect(simplify(fac0*subs(sq3 = -sq3, sq2 = -sq2, fac0)), sq2, factor); simplify(fac_sq6 * subs(sq3 = -sq3, fac_sq6)/wsq); # 1 fac_sq6UV := simplify(fac_sq6, {given1, given2}, {unew, vnew}); factor(fac_sq6UV); # fac_sq6UV is a square in Q(sqrt(6)), so fac_sq6 must be one too. # This time the unit and rational factors must be +1. # Now we return to the factorization over Q(sqrt(2), sqrt(3)). # The Dirichlet units group of the Želd Q(sqrt(2), sqrt(3)) has rank 3 # (the Želd has 4 real embeddings, no non-real complex embeddings). # Here are three units. There is also the torsion unit -1. unit1 := sq2 + 1; unit2 := sq3 + sq2; unit3 := (sq6 + sq2)/2; # I have not proved that these generate the full units group. # But I claim that no product # (+-) unit1^e1 * unit2^e2 * unit3^e3 # with e1, e2, e3 in {0, 1} # is a square in Q(sqrt(2), sqrt(3)), # except the obvious e1 = e2 = e3 = 0. # If a product is a square, it must be a square under all embeddings. # First look at the signs of the units under the real embeddings. # # Embedding sign Signs of units # sq2 sq3 sq6 -1 unit1 unit2 unit3 # + + + - + + + # + - - - + - - # - + - - - + - # - - + - - - + # # The only nontrivial unit product which is everywhere positive # is unit4 = unit1*unit2*unit3. We conŽrm that # all real embeddings of unit4 are positive. # However the embeddings of unit4 modulo 23 are not squares. # Q(sqrt(2), sqrt(3)) splits modulo 23 since # 2 == 5^2 and 3 == 7^2 are quadratic residues. unit4 := simplify(unit1*unit2*unit3); # Potential square of unit unit4poly := evala(Norm(X - unit4)); # Minimal polynomial of unit4 fsolve(unit4poly, X); # Four positive real roots Factor(unit4poly) mod 23; # Embeddings of unit4 modulo 23 # are in GF(23) but are not squares. # We aim to Žnd a unit un = un(sq2, sq3), a rational rat, # and alpha = alpha(sq2, sq3) such that # fac0(sq2, sq3) = un(sq2, sq3) * rat * alpha(sq2, sq3)^2 # un(sq2, sq3) must have the form (modulo a square) # un(sq2, sq3) = +- unit1(sq2)^e1 # * unit2(sq2, sq3)^e2 # * unit3(sq2, sq3)^e3 # where e1, e2, e3 are 0 or 1 and # unit1(sq2) = sq2 + 1 # unit2(sq2, sq3) = sq3 + sq2 # unit3(sq2, sq3) = (sq6 + sq2)/2 # Earlier we found that fac_sq6 is a square in Q(sqrt(6)). # Its value is # fac_sq6 = fac0(sq2, sq3) * fac0(-sq2, -sq3) # = un(sq2, sq3) * un(-sq2, -sq3) * rat^2 # * (alpha(sq2, sq3) * alpha(-sq2, -sq3))^2 # = (unit1(sq2) * unit1(-sq2))^e1 # * (unit2(sq2, sq3) * unit2(-sq2, -sq3))^e2 # * (unit3(sq2, sq3) * unit3(-sq2, -sq3))^e3 # * (element of Q(sqrt(6))^2 # = (-1)^e1 * (-(sq3 + sq2)^2)^e2 * 1^e3 * (square in Q(sqrt(6)) # = (-1)^e1 + (-5 - 2*sq6)^e2 * (square in Q(sqrt(6)) # # This is possible only if e1 = e2 = 0. # We also found that fac_sq2 is rat2 * (a square in Q(sqrt(2)), where # fac_sq2 = fac0(sq2, sq3) * fac0(sq2, -sq3) # = un(sq2, sq3) * un(sq2, -sq3) * rat^2 # * (alpha(sq2, sq3) * alpha(sq2, -sq3))^2 # = (unit3(sq2, sq3) * unit3(sq2, -sq3))^e3 # * (element of Q(sqrt(2))^2 # = (-1)^e3 * (square in Q(sqrt(2)) # We learn that rat2 = (-1)^e3 * (square in Q(sqrt(2)). # Unfortunately this tells us nothing about e3. # With this background, we are ready to start a search. # Guess the unit factor unitfac = +- unit3^e3 . # The choice of a leading factor +- does not matter -- # we¹ll get the same linear combinations if # u, v are replaced by their negatives. # Trying e3 = -1 rather than e3 = 1 # is computationally convenient because the # polynomial in denvar (given later) turns out to be monic. for unitfac in [1, 1/unit3] do print(************ Case unitfac = , unitfac); # Look for a square in Q(sqrt(2), sqrt(3)) # which has the form fac0/unitfac for some rational u and v. potential_square := collect(fac0/unitfac, {unew, vnew}, factor); test1 := simplify((a1 + a2*sq2 + a3*sq3 + a6*sq6)^2 - potential_square); # We have four linear equations in (rational) u and v, # by equating coefŽcients of 1, sq2, sq3, sq2*sq3. # Choose two of the equations. # Substitute into the two others. uvs := solve({coeff(test1 + subs(sq2 = -sq2, test1), sq3, 1), coeff(test1 + subs(sq3 = -sq3, test1), sq2, 1)}, {unew, vnew}); test1 := collect(subs(uvs, test1), sq2, factor); # # We get two homogeneous linear equations in # # a1^2 + 2*a2^2 + 3*a3^2 + 6*a6^2 # a1*a2 + 3*a3*a6 # a1*a3 + 2*a2*a6 # a1*a6 + a2*a3 # We started with two homogeneous quadratic equations # in four variables. Now we have two different # homogeneous quadratic equations in four variables. # Actually we have eight pairs of homogeneous quadratic # equations, due to the choice of unitfac. # The original quadratics separate nicely so one side # involved only (U, V) and the other side only (u, v). # The new ones seem not to separate so easily. # What have we gained? The magnitudes of a1, a2, a3, a6 # are smaller than those of u, v, U, V. # If u and v are around 10^10 (say), then # the real embeddings of unitfac*fac0 are near 5*10^11. # The embeddings of alpha and its conjugates will be around 7e5, # so the coefŽcients a1, a2, a3, a6 will be around 10^6, # slightly over the square roots of u and v. # # Instead of looping over u and v, we might loop # over a3 and a6. Even if it costs 50 times # as long to test for an integer root of # a quartic as to test for a perfect square, # and we eliminate neither potential value of unitfac, # our work factor will drop from (10^10)^2 = 10^20 # to 2 * 50 * (10^6)^2 ~= 10^14. a1sub := factor(solve(coeff(test1, sq2, 1), a1)); # Linear noa1 := collect(primpart(numer(subs(a1 = a1sub, test1))), a2, factor); # Quartic in a2, a3, a6 which we want to vanish # Suppose we are given a3 and a6, and want to check # whether noa1 has a rational root a2. # The last equation is not monic in a2, and we # don¹t know the factorization of its constant term. # This may complicate checking for rational roots, # If we make denvar = denom(a1) an unknown, in place of a2, # then we get a monic quartic with partially factored constant term. # (The monicity happens for unitfac = 1 or 1/unit3, # but not necessarily other values.) # In both cases the quartic has the form (with X = denvar) # X^4 + c3*X^3 + c2*X^2 + 2*c1*C*X + 2*C^2 = 0 # where c3, c2, c1, C are integers dependent on a3, a6. # Any rational root must be an integer dividing 2*C^2. # Actually, it must divide C. If we substitute X = (2*C)/Y, then # then we get (after multiplying by Y^4/2C^2) # # 8*C^2 + 4*C*c3*Y + 2*c2*Y^2 + 2*c1*Y^3 + Y^4 # Since Y is rational, it must be an even integer. # This proves any rational root X is an integer dividing C. # This observation may simplify the search for rational roots, # given a3 and a6. # # The check for integer roots might view the # real roots as continuous functions of a3 and a6. # As a3 and a6 vary, a Newton-Raphson update # may be adequate to revise the real roots. # C is a quadratic polynomial in a3 and a6, # so min(abs(X), abs(C/X)) will be O(abs(a3) + abs(a6)), # and we should be able to check whether this is close to an integer. # Or we might try a p-adic method of searching for roots. # Find a prime q0 which never divides the discriminant # of the quartic, perhaps between 100 and 1000. # A table indexed by (a3 mod q0) and (a6 mod q0) can # give all potential (denvar mod q0) (zero to four values). # Use a Hensel lifting to get denvar modulo a power of q2. # The table can be compressed (or its generation speeded) # by noting that the noa1 polynomial is homogeneous in denvar, a3, a6, # and by using a3/a6 (mod q0) as the index. After # computing a potential root modulo a power of q0, # check it with computations modulo 2^32 -- # if both tests pass, save (denvar, a3, a6) for # more processing. a2sub := solve(denom(a1sub) = denvar, a2); # Change variable to denvar noa1 := collect(primpart(expand(subs(a2 = a2sub, noa1))), denvar, factor); print(******** This is the denvar polynomial when unitfac = , unitfac); od; # for unitfac ;quit; -- During a wedding ceremony, members of the audience cried. Scientists examined the tears. They determined the liquid was eye dew. pmontgom@cwi.nl Microsoft Research and CWI Home: Bellevue, WA === Subject: Proof for no integer solution of y^2 = x^3+23 Originator: israel@math.ubc.ca (Robert Israel) This or similar questions may have been posted before, but I¹d greatly appreciate if anyone can give me an outline proof of the following: y^2 = x^3 + 23 has no integer solutions. Xiao-an Wang === Subject: Re: Proof for no integer solution of y^2 = x^3+23 >> I¹d greatly appreciate if anyone can give me >> an outline proof of the following: >> y^2 = x^3 + 23 >> has no integer solutions. >It proceeds much along the same lines as the usual proof that >the equation y^2 = x^3 + 7 has no integer solutions. Here¹s brief notes I took outlining that proof. y^2 = x^3 + 7 has no integral solutions. Otherwise: x odd. Otherwise: y^2 = 3 (mod 4) which cannot be x^2 - 2x + 4 = 3 (mod 4); y^2 + 1 = x^3 + 8 = (x + 2)(x^2 - 2x + 4) some prime p = 3 (mod 4) with p | y^2 + 1; y^2 = -1 (mod p) -1 quadradic residual; (-1/p) = (-1)^(p-1)/2 = -1 not so! (-1/p) Lagrange symbol. >First, deduce some congruence conditions on x and y mod a suitable >power of 2. Then, rewrite the equation in such a way as to introduce >a convenient factorisation of the RHS, and use the congruence >conditions to deduce that some prime p = -1 mod 4 must divide one of >the factors. Looking at the LHS, realise that this cannot happen, >and thus there are no solutions. y^2 + 2^2 = x^3 + 3^3 = (x + 1)(x^2 - 3x + 9) x odd; y even. Otherwise y^2 = 3 (mod 4) which cannot be. x^2 - 3x + 9 = 3 (mod 4) some prime p = 3 (mod 4) with p | x^2 - 3x + 9 p | y^2 + 2^2; y^2 = -2^2 (mod p) Ah ha, that¹s what I was missing before, to use y is even. some n with y = 2n; p /= 2; n^2 = -1 (mod p) Then as before -1 not quadradic residual, thus no solution. Email: privacy.net = agora.rdrop.com ---- === Subject: Re: Proof for no integer solution of y^2 = x^3+23 > This or similar questions may have been posted before, but I¹d greatly > appreciate if anyone can give me an outline proof of the following: > y^2 = x^3 + 23 > has no integer solutions. It proceeds much along the same lines as the usual proof that the equation y^2 = x^3 + 7 has no integer solutions. First, deduce some congruence conditions on x and y mod a suitable power of 2. Then, rewrite the equation in such a way as to introduce a convenient factorisation of the RHS, and use the congruence conditions to deduce that some prime p = -1 mod 4 must divide one of the factors. Looking at the LHS, realise that this cannot happen, and thus there are no solutions. Geoff. ------------------------------------------------------------- --------------- - Geoff Bailey (Fred the Wonder Worm) | Programmer by trade -- ftww@maths.usyd.edu.au | Gameplayer by vocation. ------------------------------------------------------------- --------------- - === Subject: Re: Proof for no integer solution of y^2 = x^3+23 Xiao-an === Subject: This week in the mathematics arXiv (23 Aug - 27 Aug) Originator: israel@math.ubc.ca (Robert Israel) Here are this week¹s titles in the mathematics arXiv, available at: http://front.math.ucdavis.edu/ http://front.math.ucdavis.edu/submissions This week in the mathematics arXiv may be freely redistributed with attribution and without modiŽcation. Titles in the mathematics arXiv (23 Aug - 27 Aug) ------------------------------------------------- AC: Commutative Algebra ----------------------- math.AC/0408368 K. Divaani-Aazar, R. Sazeedeh, M. Tousi: On vanishing of generalized local cohomology modules math.AC/0408351 Ana L. Branco Correia, Santiago Zarzuela: Some asymptotic properties of the Rees powers of a module math.AC/0408346 A. V. Jayanthan, Tony J. Puthenpurakal, J. K. Verma: On Žber cones of ${mathfrak m}$-primary ideals AG: Algebraic Geometry ---------------------- math.AG/0408350 Robin de Jong: Faltings¹ delta-invariant of a hyperelliptic Riemann surface math.AG/0408338 Brendan Hassett: Classical and minimal models of the moduli space of curves of genus two math.AG/0408337 B. Toen: The homotopy theory of dg-categories and derived Morita theory math.AG/0408336 J¹anos Koll¹ar, Karen E. Smith, Alessio Corti: A correction to: Rational and nearly rational varieties math.AG/0408335 S. Kaplan, A. Shapiro, M. Teicher: Braid Monodromy Type and Rational Transformations of Plane Algebraic Curves math.AG/0408318 Quang Minh Nguyen: The moduli space of rank-3 vector bundles with trivial determinant over a curve of genus 2 and duality math.AG/0408315 Jakob Stix: A monodromy criterion for extending curves hep-th/0408167 Brian Forbes: Computations on B-model geometric transitions math.AG/0408311 Manfred Einsiedler, Mikhail Kapranov, Douglas Lind: Non-archimedean amoebas and tropical varieties math.AG/0408301 Qi Zhang: Rational connectedness of log $Q$-Fano varieties math.AG/0408295 J. Wildeshaus: The boundary motive: deŽnition and basic properties math.AG/0408294 Claude Sabbah: Fourier-Laplace transform of irreducible regular differential systems on the Riemann sphere math.AG/0408288 Vicentiu Pasol, Alexander Polishchuk: Universal triple Massey products on elliptic curves and Hecke¹s indeŽnite theta series math.AG/0408283 Igor V. Dolgachev: Luigi Cremona and cubic surfaces math.AG/0408274 Yoshinori Namikawa: Birational geometry of symplectic resolutions of nilpotent orbits II math.AG/0408272 Katsuhisa Mimachi, Masaaki Yoshida: Regularizable cycles associated with a Selberg type integral under some resonance condition hep-th/0408142 Yang-Hui He: Lectures on D-branes, Gauge Theories and Calabi-Yau Singularities AP: Analysis of PDEs -------------------- math.AP/0408352 Khalil El Mehdi, Mokhless Hammami: Blowing up Solutions for a Biharmonic Equation with Critical Nonlinearity math.AP/0408332 Ross Pinsky: Reaction diffusion equations with super-linear absorption: universal bounds, uniqueness for the Cauchy problem, boundedness of stationary solutions math.AP/0408273 Andrew Hassell, Terence Tao, Jared Wunsch: Sharp Strichartz estimates on non-trapping asymptotically conic manifolds AT: Algebraic Topology ---------------------- math.AT/0408333 Mark Brightwell, Paul Turner: Relative differential characters CA: Classical Analysis and ODEs ------------------------------- math.CA/0408369 V.P. Spiridonov: Short proofs of the elliptic beta integrals math.CA/0408366 V.P. Spiridonov: A multiparameter summation formula for theta functions math.CA/0408317 Robert S. Maier: The 192 Solutions of the Heun Equation math.CA/0408269 Raimundas Vidunas: Algebraic transformations of Gauss hypergeometric functions math.CA/0408268 Stephen Semmes: Notes on groups and representations CO: Combinatorics ----------------- math.CO/0408365 Vadim E. Levit, Yulia Kempner: Quasi-concave functions on antimatroids math.CO/0408364 Masao Ishikawa, Hiroyuki Kawamuko, Soichi Okada: A PfafŽan-Hafnian analogue of the Borchardt¹s identity math.CO/0408363 I. Cahit: Three Colorability of an Arrangement Graph of Great Circles math.CO/0408354 Federico Ardila: The number of halving circles math.CO/0408349 Leroux Philippe: A LL-lattice reformulation of arithmetree over planar rooted trees. Part II math.CO/0408348 Leroux Philippe: Free dendriform dialgebras: reformulation and application in free probability. Part I math.CO/0408343 Edward Swartz: Lower bounds for h-vectors of k-CM, independence and broken circuit complexes math.CO/0408331 Michael Joswig, Marc E. Pfetsch: Computing Optimal Morse Matchings math.CO/0408326 Chunhui Lai: An extremal problem on potentially $K_{p_{1},p_{2},...,p_{t}}$-graphic sequences math.CO/0408312 Petter Br, Œen: A counter-example to the Neggers-Stanley conjecture math.CO/0408292 Chunhui Lai: An extremal problem on potentially $K_{p,1,1}$-graphic sequences math.CO/0408289 Alex Iosevich, Mischa Rudnev: Distinct distances on a sphere hep-th/0408145 Hector Figueroa, Jose M. Gracia-Bondia: Combinatorial Hopf algebras in quantum Želd theory I CT: Category Theory ------------------- math.CT/0408306 Philip J. Higgins: Thin Elements and Commutative Shells in Cubical omega-categories math.CT/0408298 Thomas M. Fiore: Pseudo Limits, Bi-Adjoints, and Pseudo Algebras: Categorical Foundations of Conformal Field Theory CV: Complex Variables --------------------- math.CV/0408284 Karl Oeljeklaus, Dan Zaffran: Steinness of bundles with Žber a Reinhardt bounded domain DG: Differential Geometry ------------------------- hep-th/0408188 Martin Rocek, Neal Wadhwa: On Calabi-Yau supermanifolds math.DG/0408347 Koji Fujiwara, Koichi Nagano, Takashi Shioya: Fixed point sets of parabolic isometries of CAT(0)-spaces math.DG/0408316 C. Dunn, P. Gilkey, S. Nikcevic: Curvature homogeneous signature (2,2) manifolds math.DG/0408285 R.J. Miatello, R.A. Podesta, J.P. Rossetti: (Z_2^k)-manifolds are isospectral on forms nlin.SI/0301042 Sergei A. Igonin: Coverings and the fundamental group for partial differential equations gr-qc/0408072 Michael T. Anderson: Existence and stability of even dimensional asymptotically de Sitter spaces gr-qc/0408071 S. Brian Edgar, Jos¹{e} M. M. Senovilla: A local potential for the Weyl tensor in all dimensions DS: Dynamical Systems --------------------- math.DS/0408344 Jairo Bochi, Bassam Fayad, Enrique Pujals: A remark on conservative diffeomorphisms math.DS/0408340 Erik Boczko, Todd Young: Basins of attraction for cascading maps math.DS/0408328 E. Glasner, B. Weiss: On the interplay between measurable and topological dynamics math.DS/0408323 Christoph Gugg, Jinqiao Duan: A Markov jump process approximation of the stochastic Burgers equation math.DS/0408322 Wei Wang, Jianhua Sun, Jinqiao Duan: Ergodic Dynamics of the Stochastic Swift-Hohenberg System math.DS/0408307 Wenxiang Sun, Todd Young: Liao Standard Systems and Nonzero Lyapunov Exponents for Differential Flows math.DS/0408297 E. Akalin, M.U. Akhmet: On the principles of B-smooth discontinuous žows math.DS/0408290 Artur Avila, Mikhail Lyubich: Hausdorff dimension and conformal measures of Feigenbaum Julia sets FA: Functional Analysis ----------------------- math.FA/0408329 Tomonari Suzuki: The set of common Žxed points of a one-parameter continuous semigroup of mappings is F(T(1)) cap F(T(sqrt 2)) math.FA/0408300 V.A. Babalola: Endomorphic Elements in Banach Algebras math.FA/0408287 Joseph M. Renes: Equiangular Tight Frames from Paley Tournaments math.FA/0408278 Claudia Garetto: Topological structures in Colombeau algebras II: investigation into the duals of $Gc(Om)$, $G(Om)$ and $GS(R^n)$ GM: General Mathematics ----------------------- math.GM/0408313 Ashwin Vaidya, Bong-Jae Chung: An Axiomatization of Realities GN: General Topology -------------------- math.GN/0408339 Ramiro de la Vega: Homogeneity properties in large compact S-spaces math.GN/0408305 Bernardo Lafuerza--Guillen, Jose Antonio Rodriguez--Lallena, Carlo Sempi: Normability of Probabilistic Normed Spaces GR: Group Theory ---------------- math.GR/0408355 Chris Connell, Roman Muchnik: Harmonicity of quasiconformal measures and Poisson boundaries of hyperbolic spaces math.GR/0408330 Valerij G. Bardakov: Extending representations of braid groups to the automorphism groups of free groups math.GR/0408299 Kai-Uwe Bux, Kevin Wortman: Finiteness properties of arithmetic groups over function Želds math.GR/0408277 D. N. Azarov, D. Tieudjo: On root-class residuality of generalized free products GT: Geometric Topology ---------------------- math.GT/0408361 Xiaobo Liu: The quantum Teichmuller space as a noncommutative algebraic object math.GT/0408353 Leonardo N Carvalho: Tightness and efŽciency of irreducible automorphisms of handlebodies math.GT/0408325 Greg Friedman: There exist non-trivial PL knots whose complements are homotopy circles math.GT/0408310 Jerome Levine: The Lagrangian Žltration of the mapping class group and Žnite-type invariants of homology spheres math.GT/0408286 Blake Mellor: Tree Diagrams for String Links II: Determining Chord Diagrams LO: Logic --------- math.LO/0408370 Giovanni Panti: Bernoulli automorphisms in many-valued logic math.LO/0408282 Dominic Hughes: Proofs Without Syntax math.LO/0408279 Peter Cholak, Leo Harrington: Extensions Theorems, Orbits, and Automorphisms of the Computably Enumerable Sets MP: Mathematical Physics ------------------------ math-ph/0408052 T. Christiansen: Schrodinger operators with complex-valued potentials and no resonances math-ph/0408051 R. Jackiw: Chern-Simons Integral as a Surface math-ph/0408050 Želds fulŽlling Yang-Feldman equations math-ph/0408049 H. Gottschalk: A Characterisation of Locality in Momentum Space math-ph/0408048 H. Gottschalk: Complex velocity transformations and the Bisognano--Wichmann theorem for quantum Želds acting on Krein spaces math-ph/0408047 H. Gottschalk, H. Thaler: An indeŽnite metric model for interacting quantum Želds with non-stationary background gravitation math-ph/0408046 M. R. Dennis: Canonical representation of spherical functions: Sylvester¹s theorem, Maxwell¹s multipoles and Majorana¹s sphere math-ph/0408045 J. Mark Heinzle, Alan D. Rendall, Claes Uggla: Theory of Newtonian self-gravitating stationary spherically symmetric systems math-ph/0408044 Gerald Kaiser: Eigenwavelets of the Wave equation math-ph/0408043 Martin Hallnas, Edwin Langmann, Cornelius Paužer: Generalized local interactions in 1D: solutions of quantum many-body systems describing math-ph/0408042 Jan de Gier, Vladimir Rittenberg: ReŽned Razumov-Stroganov conjectures for open boundaries math-ph/0408041 Alexander I. Aptekarev, Pavel M. Bleher, Arno B.J. Kuijlaars: Large $n$ limit of Gaussian random matrices with external source, part II math-ph/0408040 Adonai S. Sant¹Anna: Entropy is complexity math-ph/0408039 Alex Kasman: On the Quantization of a Self-Dual Integrable System math-ph/0408038 Michael Gekhtman, Alex Kasman: Integrable Systems and Rank One Conditions for Rectangular Matrices cond-mat/0408497 Paul Fendley, Kareljan Schoutens, Hendrik van Eerten: Hard squares with negative activity math-ph/0408037 Yuri N. Fedorov, Bozidar Jovanovic: Integrable nonholonomic geodesic žows on compact Lie groups math-ph/0408036 Paolo Amore: Convergence acceleration of series through a variational approach math-ph/0408035 V P Belavkin: Contravariant Densities, Operational Distances and Quantum Channel Fidelities math-ph/0408025 Pascal Baseilhac: An integrable structure related with tridiagonal algebras gr-qc/0408060 O.I. Streltsova, E.E. Donets, E.A. Hayryan, D.A. Georgieva, T.L. Boyadjiev: Unstable even-parity eigenmodes of the regular static SU(2) Yang-Mills-dilaton solutions math-ph/0408034 Han-Ying Guo, Jianzhong Pan, Bin Zhou: The Generalized Liouville¹s Theorems via Euler-Lagrange Cohomology Groups on Symplectic Manifold math-ph/0408033 Thomas Guhr, Heiner Kohler: Supersymmetry and Models for Two Kinds of hep-th/0408161 Urs Schreiber: Super-Pohlmeyer invariants and boundary states for non-abelian gauge Želds quant-ph/0408124 Roger Balian, Bertrand Duplantier: Geometry of the Casimir Effect math-ph/0408032 Margarida de Faria, Maria Joao Oliveira, Ludwig Streit: Feynman integrals for non-smooth and rapidly growing potentials math-ph/0408031 S. H. Djah, H. Gottschalk, H. Ouerdiane: Feynman graph representation of the perturbation series for general functional measures NT: Number Theory ----------------- math.NT/0408371 Andrew Bremner, Nikos Tzanakis: Lucas sequences whose 8th term is a square math.NT/0408359 Matthew Young: Lower-Order Terms of the 1-Level Density of Families of Elliptic Curves math.NT/0408341 Benjamin Lundell, Jason McCullough: A Generalized Floor Bound for the Mnimum Distance of Goemetric Goppa codes and its Application to Two-Point Codes math.NT/0408319 Andrew Granville, Greg Martin: Prime Number Races math.NT/0408309 Tobias Muhlenbruch: Hecke operators on period functions for $Gnull{n}$ math.NT/0408304 Jan-Hendrik Evertse: Distances between the conjugates of an algebraic number math.NT/0408293 Tetsuya Takahashi: Epsilon factor for GL_l times GL_{l¹}; lneq l¹ primes math.NT/0408271 Bjorn Poonen, Alexandra Shlapentokh: Diophantine deŽnability of inŽnite discrete non-archimedean sets and Diophantine models over large subrings of number Želds OA: Operator Algebras --------------------- math.OA/0408324 Nathan Brownlowe, Iain Raeburn: Exel¹s Crossed Product and Relative Cuntz-Pimsner Algebras math.OA/0408314 Ilwoo Cho: Graph W*-probability Theory math.OA/0408296 N. Christopher Phillips: Examples of different minimal diffeomorphisms giving the same C*-algebras math.OA/0408291 Huaxin Lin, N. Christopher Phillips: Crossed products by minimal homeomorphisms math.OA/0408281 David P. Blecher, Bojan Magajna: Duality and Operator Algebras II: Operator Algebras as Banach Algebras math.OA/0408275 Sang Hyun Kim, Gabriel Nagy: Spectral Symmetry in II_1 Factors PR: Probability --------------- math.PR/0408372 mean-Želd interaction math.PR/0408367 A.N. Borodin, Paavo Salminen: On some exponential integral functionals of BM($mu$) and BES(3) math.PR/0408360 Greg Kuperberg: Special moments math.PR/0408327 Nina Gantert, Wolfgang Konig, Zhan Shi: Annealed deviations of random walk in random scenery math.PR/0408276 Daniel Egloff: Monte Carlo Algorithms for Optimal Stopping and Statistical Learning QA: Quantum Algebra ------------------- math.QA/0408362 Hiroyuki Yamane: A Žnite number of deŽning relations and a UCE theorem of the elliptic Lie algebras and superalgebras with rank $geq 2$ math.QA/0408358 Qi Chen, Thang Le: Quantum Invariants of Periodic Links and Periodic 3-Manifolds math.QA/0408357 Qi Chen, Thang Le: Almost integral TQFTs from simple Lie algebras math.QA/0408356 Qi Chen: On certain integral tensor categories and integral TQFTs math.QA/0408308 Alexander Varchenko: Selberg Integrals math.QA/0408303 A. I. Molev: A quantum Sylvester theorem and skew representations of twisted Yangians RA: Rings and Algebras ---------------------- math.RA/0408334 S. Caenepeel, F. Van Oystaeyen, Y.H. Zhang: The equivariant Brauer group of a group math.RA/0408320 S. Amghibech: A note on recurrence sequences RT: Representation Theory ------------------------- math.RT/0408302 Jeb F. Willenbring, Gregg Zuckerman: Small semisimple subalgebras of semisimple Lie algebras SG: Symplectic Geometry ----------------------- math.SG/0408345 Yi Lin: Examples that the strong Lefschetz property does not survive symplectic reduction math.SG/0408342 Bertram Kostant, Nolan Wallach: Gelfand-Zeitlin theory from the perspective of classical mechanics. I math.SG/0408280 Alberto Abbondandolo, Matthias Schwarz: On the Floer homology of cotangent bundles ST: Statistics -------------- math.ST/0408321 Muneya Matsui, Akimichi Takemura: Some improvements in numerical evaluation of symmetric stable density and ts derivatives math.ST/0408270 Serkan Hosten, Amit Khetan, Bernd Sturmfels: Solving the Likelihood Equations -- / Greg Kuperberg (UC Davis) / Home page: http://www.math.ucdavis.edu/~greg/ / Visit the Math ArXiv Front at http://front.math.ucdavis.edu/ / * All the math that¹s Žt to e-print * === Subject: A question about classical invariant theory Originator: israel@math.ubc.ca (Robert Israel) In the paper J.A.,Todd: Ternary quadratic types Philos. Trans. Roy. Soc. London. Ser. A, 241 (1948) (p. 399 - 456) J.A. Todd obtains, [using previous results of H.W. Turnbull appeared in Proc. London Math. Soc (2) (1910)], a complete system of concomitants for 5 ternary quadratics. This result leads - via a theorem of Peano- to a complete knowledge of the concomitant types of any number of ternary quadratics. Todd gives the concomitants in symbolic notation. Does anybody know if this result had been proved again later by other methods? Blind Bruno === Subject: Synergetics Section 260.42 root Žnding Received-SPF: none (mailbox8.ucsd.edu: domain of news@gnilink.net does not designate permitted sender hosts) Originator: israel@math.ubc.ca (Robert Israel) The last graphic in the section Finding Roots Of Equations Numerically With Bucky Numbers at: http://users.adelphia.net/~cnelson9/Links/index_lnk_11.html which is one of the sections of Synergetics Coordinates Applications at: http://users.adelphia.net/~cnelson9/ is followed by a question about the speed of the convergence of a general root Žnding method. The guesses for the third iteration on, could be found with the vector equation method shown in the section Solving Matrix Problems Using Bucky Numbers at: http://users.adelphia.net/~cnelson9/Links/index_lnk_10.html I don¹t think Newton¹s method can be beat, I¹ve tried and tried, but if anyone could beat Newton it would be Bucky Fuller. How does the speed of convergence compare for polynomials in general? What is the traditional name for this kind of root Žnding method? Has it been done with the Cartesian coordinate system? What is the closest to Synergetics Section 260.42? at: http://mathworld.wolfram.com/topics/Root-Finding.html Cliff Nelson Dry your tears, there¹s more fun for your ears, Forward Into The Past 2 PM to 5 PM, Sundays, California time, at: http://www.kspc.org/ Don¹t be a square or a blockhead; see: http://users.adelphia.net/~cnelson9/ === Subject: nonnegative integer linear combinations Originator: israel@math.ubc.ca (Robert Israel) I have noticed that if a, b, c, a¹, b¹, c¹ are integers such that 0 <= a¹ <= a 0 <= b¹ <= b 0 <= c¹ <= c then (a-a¹)(b-b¹)(c-c¹) + a¹b¹c¹ is (seemingly) always a nonnegative integer combination of a, b and c. Does anyone know how to prove this? (I¹ve checked it by computer for all values of a, b, c, a¹, b¹, c¹ such that max(a,b,c) <= 100.) John NB: I originally posted this to sci.math, but got no answers, so I repost it here. === Subject: Re: nonnegative integer linear combinations Epigone-thread: komskorfoi Originator: israel@math.ubc.ca (Robert Israel) Answered in sci.math === Subject: Re: nonnegative integer linear combinations Originator: israel@math.ubc.ca (Robert Israel) === Subject: nonnegative integer linear combinations >I have noticed that if a, b, c, a¹, b¹, c¹ are integers such that >0 <= a¹ <= a >0 <= b¹ <= b >0 <= c¹ <= c >then >(a-a¹)(b-b¹)(c-c¹) + a¹b¹c¹ >is (seemingly) a nonnegative integer combination of a, b and c. >Does anyone know how to prove this? Consider 1 <= x,y,z and (x - 1)(y - 1)(z - 1) + 1 = xyz - xy - xz + x - yz + y + z - 1 + 1 = x + y + x - xy - yz - zx + xyz How are there a,b,c in Z+ with ax + by + cy = x + y + x - xy - yz - zx + xyz = x(1 - y) + y(1 - z) + z(1 - x) + xyz = x(1 - y + uyz) + y(1 - z + vzx) + z(1 - x + wxy) where u + v + w = 1. (Wouldn¹t it be nice if 0 <= x,y,z <= 1?) Ok , Žnd u,v,w with u + v + w = 1 y <= 1 + uyz, z <= 1 + vzx, x <= 1 + wxy As 1 <= x,y,z, it may sufŽce if y - 1 <= uy, z - 1 <= vz, x - 1 <= wx What next I know not. Now if this onerous chore can be Žnished, then tackle the general case as guided by the experience of this pilot project. (Email: privacy.net = agora.rdrop.com) ---- === Subject: Exposed Venn Diagrams Originator: israel@math.ubc.ca (Robert Israel) I have been asked a question about Venn diagrams, as deŽned here: http://www.combinatorics.org/Surveys/ds5/VennWhatEJC.html We follow Gr.9fnbaum [Gr75] in Žrst deŽning a more general concept, an independent family. Let C = { C1, C2, ..., Cn } be a collection of simple closed curves drawn in the plane. The collection C is said to be an independent family if the intersection of X1, X2, ..., Xn is nonempty, where each Xi is either int(Ci ) (the interior of Ci ) or is ext(Ci ) (the exterior of Ci ). If, in addition, each such intersection is connected and there are only Žnitely many intersections, then C is a Venn diagram, or an n-Venn diagram if we wish to emphasize the number of curves in the diagram. The condition that there are only a Žnite number of intersection points is usually assumed in the literature, but often not stated explicitly. It rules out segments of curves from intersecting. *** The questioner wants to know what the *maximum* number of sets that can be incorporated into an *exposed* venn diagram, using only ellipses and rectangles, is. An exposed Venn diagram is deŽned as follows: http://www.combinatorics.org/Surveys/ds5/VennOtherEJC.html A Venn diagram is exposed if each of its curves touches the outer face at some point of non-intersection. [] Stated in terms of n-Venn graphs, being exposed is the same as the condition that the vertex corresponding to the outer face has degree n. Every symmetric Venn diagram must be exposed (obvious) and every convex Venn diagram must be exposed ([Gr92a]). A Venn diagram has a hidden curve if it has a curve that does not touch the outer region. *** Can anyone assist? I am neither an algebraist or a geometer, so this falls outside my area of expertise. Mark H Wilkinson === Subject: Why are there more Mersenne primes than Fermat primes??? Epigone-thread: khonjeiglai Originator: israel@math.ubc.ca (Robert Israel) I¹ve been looking around, and everyone says that there are so MANY Mersenne primes (compared to a Žnite set), but almost NO Fermat primes! I just don¹t understand this! Just because Fermat numbers grow faster, that means that there should be less primes in the sequence? But why? These heuristic arguments don¹t really have much sense, no offense intended... I mean, the sequence of primes is inŽnite, so what are chances that there are no sub-sequences that grow faster than e^(n^2) ??? Not to mention the existance of Mill¹s constant(s). I know that from actual computational expirience, it seems that Elliptic Divisibility Sequences have only Žnitely many primes, but if I had to actually guess, I would guess that there are inŽnitely many sequences who have inŽnitely many primes. What do you people think? === Subject: This week in the mathematics arXiv (30 Aug - 3 Sep) Originator: israel@math.ubc.ca (Robert Israel) Here are this week¹s titles in the mathematics arXiv, available at: http://front.math.ucdavis.edu/ http://front.math.ucdavis.edu/submissions This week in the mathematics arXiv may be freely redistributed with attribution and without modiŽcation. Titles in the mathematics arXiv (30 Aug - 3 Sep) ------------------------------------------------ AC: Commutative Algebra ----------------------- math.AC/0409007 Mitsuyasu Hashimoto: Another proof of global $F$-regularity of Schubert varieties math.AC/0408437 Jan Snellman: A conjecture on the Poincar{Œe}-Betti series of modules of differential operators on a generic hyperplane arrangement math.AC/0408429 Mitsuyasu Hashimoto: Another proof of theorems of De Concini and Procesi math.AC/0408399 Sean Sather-Wagstaff: Semidualizing modules and the divisor class group AG: Algebraic Geometry ---------------------- math.AG/0409031 A. Gerasimov, S. Kharchev, D. Lebedev, S. Oblezin: On a class of representations of the Yangian and Moduli Space of Monopoles math.AG/0409030 Daniel Huybrechts, Paolo Stellari: Equivalences of twisted K3 surfaces math.AG/0409029 L.Chiantini, C.Madonna: ACM bundles on general hypersurfaces in ${bf P}^5$ of low degree math.AG/0409024 Vladimir L. Popov: Generically multiple transitive algebraic group actions math.AG/0409004 N. Lemire, V. L. Popov, Z. Reichstein: Cayley groups math.AG/0409002 Thierry Zell: Topological complexity of the relative closure of a semi-PfafŽan couple math.AG/0408439 Ruxandra Moraru: Stable bundles on Hopf manifolds math.AG/0408428 Helge Maakestad: The Chern character for Lie algebroids math.AG/0408426 Jun Li, Chiu-Chu Melissa Liu, Kefeng Liu, Jian Zhou: A Mathematical Theory of the Topological Vertex math.AG/0408408 Nero Budur, Mircea Mustata, Morihiko Saito: Bernstein-Sato Polynomials of Arbitrary Varieties math.AG/0408394 Hung-Jen Chiang-Hsieh, Joseph Lipman: A numerical criterion for simultaneous normalization math.AG/0408391 Misha Verbitsky: Holomorphic bundles on diagonal Hopf manifolds math.AG/0408382 Robin de Jong: Jacobian Nullwerte associated to hyperelliptic Riemann surfaces AP: Analysis of PDEs -------------------- math.AP/0409015 N. Burq, P. Gerard, N. Tzvetkov: Multilinear Eigenfunction Estimates And Global Existence For The Three Dimensional Nonlinear SchrOdinger Equations math.AP/0408438 Marius Mitrea, Michael Taylor, Andras Vasy: Lipschitz domains, domains with corners and the Hodge Laplacian math.AP/0408376 S. A. Denisov: Absolutely continuous spectrum of multidimensional Schrodinger operator AT: Algebraic Topology ---------------------- math.AT/0409036 Emanuele Delucchi: Diagram models for the covers of the Salvetti complex math.AT/0408436 Daniel Dugger: Notes on the Milnor conjectures math.AT/0408417 Pavle Blagojevic, Vladimir Grujic, Rade Zivaljevic: Symmetric products of surfaces; a unifying theme for topology and physics math.AT/0408403 Daniel Henry Gottlieb: Transfers and Periodic Orbits CA: Classical Analysis and ODEs ------------------------------- math.CA/0409023 Wadim Zudilin: Approximations to -, di- and tri- logarithms math.CA/0409018 Sorina Barza, Anna Kaminska, Lars-Erik Persson, Javier Soria: Mixed norm and multidimensional Lorentz spaces math.CA/0409017 Joaquim Martin, Javier Soria: Characterization of rearrangement invariant spaces with Žxed points for the Hardy-Littlewood maximal operator math.CA/0408424 Frederik J. Simons, F. A. Dahlen, Mark A. Wieczorek: Spatiospectral concentration on a sphere CO: Combinatorics ----------------- math.CO/0409028 Henry Crapo, William Schmitt: A free subalgebra of the algebra of matroids math.CO/0409022 Marcelo Aguiar, Frank Sottile: Structure of the Loday-Ronco Hopf algebra of trees math.CO/0409003 Federico Ardila: Semimatroids and their Tutte polynomials math.CO/0408397 Joshua N. Cooper: Line ConŽgurations and the Erdos-Hajnal Conjecture math.CO/0408396 J. Cooper, J. Solymosi: Collinear Points in Permutations math.CO/0408384 Peter Doerre: Every planar graph is 4-colourable and 5-choosable - a joint proof math.CO/0408377 Guoce Xin: A Fast Algorithm for MacMahon¹s Partition Analysis CV: Complex Variables --------------------- math.CV/0409009 Takashi Ichikawa, Masaaki Yoshida: A family of Schottky groups arising from the hypergeometric equation math.CV/0408407 A. Rashkovskii, R. Sigurdsson: Green functions with singularities along complex spaces math.CV/0408373 Finnur Larusson: Mapping Cylinders and the Oka Principle DG: Differential Geometry ------------------------- math.DG/0408423 Qihua Ruan, Zhihua Chen: Non-ancient solution of the Ricci žow math.DG/0408422 Qihua Ruan, Zhihua Chen: Manifolds with non-negative Ricci curvature and Nash inequalities math.DG/0408413 Juan Carlos Alvarez Paiva, Gautier Berck: What is wrong with the Hausdorff measure in Finsler spaces math.DG/0408410 Francisco Martin, Valerio Ramos-Batista: The embedded singly periodic Scherk-Costa surfaces math.DG/0408409 Leonardo Biliotti: Coisotropic and polar actions on compact irreducible Hermitian symmetric spaces math.DG/0408387 Radu Slobodeanu: Biconformal changes of metric and pseudo-harmonic morphisms gr-qc/0405028 Aleks Kleyn: Metric-AfŽne Manifold DS: Dynamical Systems --------------------- math.DS/0409019 J.J. Duistermaat: Chaplygin¹s sphere math.DS/0409001 Anthony Quas, Mate Wierdl: Rates of Divergence of non-Conventional Ergodic Averages math.DS/0408431 Thierry Monteil: A counter-example to the theorem of Hiemer and Snurnikov math.DS/0408430 Viviane Baladi: Anisotropic Sobolev spaces and dynamical transfer operators: C^infty foliations math.DS/0408386 J. Duan, H. Gao, B. Schmalfuss: Stochastic Dynamics of a Coupled Atmosphere--Ocean Model math.DS/0408385 Hongjun Gao, Jinqiao Duan: Dynamics of a Coupled Atmosphere-Ocean Model FA: Functional Analysis ----------------------- math.FA/0408388 Volker Runde: Fourier and Fig`a-Talamanca-Herz algebras on amenable, locally compact groups GM: General Mathematics ----------------------- math.GM/0409014 Gerard Maze, Lorenz Minder: A New Family of Almost Identities GR: Group Theory ---------------- math.GR/0408412 Ruth Charney, John Crisp: Automorphism groups of some afŽne and Žnite type Artin groups math.GR/0408400 G¹abor Elek, Endre Szab¹o: Hyperlinearity, essentially free actions and $L^2$-invariants. The soŽc property math.GR/0408393 E. A. Ivanova: On the conjugacy separability in the class of Žnite $p$-groups of Žnitely generated nilpotent groups GT: Geometric Topology ---------------------- math.GT/0409033 Elmas Irmak, Mustafa Korkmaz: Automorphisms of the Hatcher-Thurston complex math.GT/0409005 Tam¹as K¹alm¹an: One parameter families of Legendrian torus knots math.GT/0408398 V.Kurlin: Explicit description of compressed logarithms of all Drinfeld associators math.GT/0408379 Siddhartha Gadgil: Open manifolds, Ozsvath-Szabo invariants and Exotic R^4¹s math.GT/0408375 Sam Nelson: Signed ordered knotlike quandle presentations math.GT/0408374 Constance Leidy: Higher-order linking forms for knots KT: K-Theory and Homology ------------------------- math.KT/0408416 Masoud Khalkhali: Very Basic Noncommutative Geometry math.KT/0408389 Jorge A. Guccione, Juan J. Guccione: Relative cyclic homology of square zero extensions LO: Logic --------- math.LO/0409034 Dan Seabold, Stefan Waner, Steve Warner: A Calculus of Inconsistency I: Sentential Logic MG: Metric Geometry ------------------- math.MG/0408414 Juan Carlos Alvarez Paiva: Dual spheres have the same girth MP: Mathematical Physics ------------------------ math-ph/0409006 Bruno Nachtergaele: Quantum Spin Systems math-ph/0409005 Takashi Aoki, Takahiro Kawai, Shunsuke Sasaki, Akira Shudo, Yosugu Takei: Virtual turning points and bifurcation of Stokes curves for higher order ordinary differential equations math-ph/0409004 G. Cicogna. G. Gaeta, P. Morando: On the relation between standard and $mu$-symmetries for PDEs hep-th/0408093 A.Albeed, M.S.Shikakhwa: SO(3) Algebra and the Conservation of Helicity in the First Order Aharonov-Bohm Scattering Amplitude hep-th/0405207 M.S.Shikakhwa: Algebraic Properties of the Interaction Hamiltonian in the quant-ph/0408181 D. Han, Y. S. Kim, Maryln E. Noz: Wigner Rotations and Iwasawa Decompositions in Polarization Optics math-ph/0409003 Avinash Khare: Supersymmetry in Quantum Mechanics math-ph/0409002 N.I. Stoilova, J. Van der Jeugt: A classiŽcation of generalized quantum statistics associated with classical Lie algebras math-ph/0409001 Daniel Henry Gottlieb: Maxwell¹s Equations hep-th/0407150 David Kutasov, Jens Marklof, Gregory W. Moore: Melvin Models and Diophantine Approximation math-ph/0408061 Boris Noyvert: Ramond sector of superconformal algebras via quantum reduction math-ph/0408060 Jonathan C. Mattingly, TouŽc M. Suidan: The Small Scales of the Stochastic Navier Stokes Equations under Rough Forcing math-ph/0408053 Igor Mencattini, Dirk Kreimer: The Structure of the Ladder Insertion-Elimination Lie algebra math-ph/0408044 Gerald Kaiser: Eigenwavelets of the Wave equation cond-mat/0408657 Maxime Clusel, Jean-Yves Fortin: 1D action and partition function for the 2D Ising model with boundary magnetic Želd quant-ph/0407222 Sibel Baskal, Y. S. Kim: The language of Einstein spoken by optical instruments math-ph/0408059 Ioan Sturzu: Taylor expansion for an operator function math-ph/0408058 Monique Combescure: About Quantum Revivals, Quantum Fidelity, A semiclassical Approach math-ph/0408057 Jan Derezi{Œn}ski, Krzysztof A. Meissner: Quantum massless Želd in 1+1 dimensions math-ph/0408056 Thomas {O}stergaard S{o}rensen: The large-Z behaviour of pseudo-relativistic atoms math-ph/0408055 Gerald Kaiser: Making electromagnetic wavelets II: Spheroidal shell antennas math-ph/0408054 J. D. Bondurant, S. A. Fulling: The Dirichlet-to-Robin Transform math-ph/0407018 I.Krichever: Analytic theory of difference equations with rational and elliptic coefŽcients and the Riemann-Hilbert problem quant-ph/0408132 Ali Mostafazadeh, Ahmet Batal: Physical Aspects of Pseudo-Hermitian and $PT$-Symmetric Quantum Mechanics NA: Numerical Analysis ---------------------- math.NA/0408419 Anton Leykin, Jan Verschelde, Ailing Zhao: Newton¹s method with dežation for isolated singularities of polynomial systems math.NA/0408418 Alexander Kayumov: Least-squares approximation by second-order L-splines NT: Number Theory ----------------- math.NT/0409035 Zhi-Wei Sun, Hao Pan: Identities concerning Bernoulli and Euler polynomials math.NT/0409020 Noam D. Elkies: Shimura curves for level-3 subgroups of the (2,3,7) triangle group, and some other examples hep-th/0409011 math.NT/0409008 Roland Bacher: On Minkowksi¹s bound for lattice-packings math.NT/0408421 Adrian Vasiu: Projective integral models of Shimura varieties of Hodge type with compact factors math.NT/0408406 Jan Hendrik Bruinier, Jens Funke: Traces of CM values of modular functions math.NT/0408404 Laurent Berger, Christophe Breuil: Towards a $p$-adic Langlands programme math.NT/0408383 Daniel Maisner, Enric Nart: Zeta functions of supersingular curves of genus 2 math.NT/0408381 Jan-Hendrik Evertse, Roberto G. Ferretti: A generalization of the Subspace Theorem with polynomials of higher degree math.NT/0408380 U.K. Anandavardhanan, C.S. Rajan: Distinguished representations, base change, and reducibility for unitary groups OA: Operator Algebras --------------------- math.OA/0409025 Franz Lehner: Cumulants in Noncommutative Probability Theory IV. De Finetti¹s Theorem, $L^p$-Inequalities and Brillinger¹s Formula math.OA/0409011 Louis Labuschagne: Pure state transformations induced by linear operators math.OA/0408435 Gabriel Nagy: Abelian self-commutators in Žnite factors math.OA/0408434 Stefan Teodor Bildea: Generalized amalgamated free products of operator algebras math.OA/0408433 Marius Ionescu: Mauldin-Williams graphs, Morita Equivalence and Isomorphisms OC: Optimization and Control ---------------------------- math.OC/0409010 Luc Moreau: Stability of continuous-time distributed consensus algorithms math.OC/0408378 S. A. Belbas: Dynamic programming in mixed continuous-discrete time scales PR: Probability --------------- math.PR/0409026 Anna Karczewska, Piotr Rozmej: On numerical solutions to stochastic Volterra equations math.PR/0409021 Noam Berger: A lower bound for the chemical distance in sparse long-range percolation models math.PR/0409013 Kurt Johansson: Non-intersecting, simple, symmetric random walks and the extended Hahn kernel math.PR/0408395 Alan Hammond, Fraydoun Rezakhanlou: The kinetic limit of a system of QA: Quantum Algebra ------------------- math.QA/0409006 Robert L Grossman, Richard G Larson: Differential Algebra Structures on Familes of Trees hep-th/0408196 Angel Ballesteros, N. Rossano Bruno, Francisco J. Herranz: On 3+1 anti-de Sitter and de Sitter Lie bialgebras with dimensionful deformation parameters math.QA/0408405 Dominique Manchon: Hopf algebras, from basics to applications to renormalization math.QA/0408401 Olivier Schiffmann: Canonical bases of loop algebras via quot schemes II math.QA/0408390 Paul Terwilliger: Two linear transformations each tridiagonal with respect to an eigenbasis of the other; an algebraic approach to the Askey scheme of orthogonal polynomials RA: Rings and Algebras ---------------------- math.RA/0408425 Graham J. Leuschke, Ragnar-Olaf Buchweitz: The adjoint of an even size matrix factors math.RA/0408420 Zinovy Reichstein, Nikolaus Vonessen: Group actions on central simple algebras: a geometric approach math.RA/0408402 Yang Han: Hochschild (co)homology dimension math.RA/0408392 Masha Vlasenko: Description of the center of the afŽne Temperley-Lieb algebra of type $widetilde{A_N}$ RT: Representation Theory ------------------------- math.RT/0409016 Shun-Jen Cheng, Weiqiang Wang, R.B. Zhang: A super duality and Kazhdan-Lusztig polynomials math.RT/0408432 Jonathan Korman: On the local constancy of characters math.RT/0408427 David Kazhdan, Yakov Varshavsky: On endoscopic decomposition of certain depth zero representations SG: Symplectic Geometry ----------------------- math.SG/0409032 Joshua M. Sabloff: Augmentations and Rulings of Legendrian Knots math.SG/0409027 O.V. Kulikova: On the fundamental groups of the complements of Hurwitz curves math.SG/0408415 Juan Carlos Alvarez Paiva: Dual mixed volumes and isosystolic inequalities math.SG/0408411 Tobias Ekholm, John Etnyre, Michael G. Sullivan: Orientations in Legendrian Contact Homology and Exact Lagrangian Immersions SP: Spectral Theory ------------------- math.SP/0409012 Maksim Sokolov: Introduction to the spectral theory of self-adjoint differential vector-operators cond-mat/0408507 B. Kaulakys, J. Ruseckas: Stochastic nonlinear differential equation generating 1/f noise -- / Greg Kuperberg (UC Davis) / Home page: http://www.math.ucdavis.edu/~greg/ / Visit the Math ArXiv Front at http://front.math.ucdavis.edu/ / * All the math that¹s Žt to e-print * === Subject: classiŽcation of Lie groups Originator: israel@math.ubc.ca (Robert Israel) Hello everybody, I¹ve been reviewing the classiŽcation of Lie algebras, with a view towards forming corresponding classiŽcation statements regarding Lie groups. For some reason this is rarely done in textbooks, maybe it¹s considered obvious? Or maybe it¹s just very complicated and I¹m trying to get too simple of an answer. Anyway, I was hoping someone would point out any errors or conŽrm the statements I¹ve come up with, along with answering the accompanying questions. DeŽnitions: a simple Lie algebra is a nonabelian one that has no nonzero proper ideal; a simple Lie group is one with a simple Lie algebra. This entire discussion only concerns Žnite-dimensional Lie groups and algebras. Here are the related facts I can conŽrm in books: (1) Every non-connected Lie group consists of some number of diffeomorphic copies of the connected component containing the identity. Thus, every Lie group is a set of diffeomorphic copies of a connected Lie group. (2) Every connected Lie group G is obtained via a homomorphism from its universal covering group G* with discrete abelian kernel isomorphic to the fundamental group of G. Thus, every connected Lie group is completely determined by its fundamental group and its Lie algebra. (3) A connected Lie group G has a connected normal subgroup N iff the subgroup¹s Lie algebra is an ideal in the Lie algebra of G. Thus, every connected simple Lie group has no connected normal Lie subgroup. (4) Every complex simple Lie algebra has a unique compact real form, which is the only one with a corresponding compact Lie group. Thus, every compact connected simple Lie group has a Lie algebra that is the compact real form of a complex simple Lie algebra. (5) Every compact connected Lie group has a Lie algebra that is the direct sum of simple Lie algebras and abelian Lie algebras. Thus, every compact connected Lie group is the direct product of compact connected simple Lie groups and some number of copies of U(1). Since the complex simple Lie algebras are completely classiŽed, I am trying to work backwards to construct a statement like every compact Lie group is... Here¹s what I come up with: The compact real forms of complex simple Lie algebras are su(n), so(n), sp(2n), and the 5 exceptional Lie algebras. Q: Is it correct that the 5 exceptional Lie algebras correspond to compact Lie groups? Clearly these algebras correspond to the compact connected simple Lie groups SU(n), U(n), SO(n), Spin(n), Sp(2n), and the exceptional Lie groups. Q: Is it correct that these are the only compact connected simple Lie groups? If so, we can ignore the complication of division of discrete subgroups, and we have the very simple statement: Every compact connected Lie group is a direct product of SU(n), U(n), SO(n), Spin(n), Sp(2n), and the exceptional Lie groups. Q: Is this statement correct? I found it in an astrophysics paper on the arxiv and I really hope it is correct, but if so I¹m at a loss as to why no one ever puts it this way... If not, we would like to say something like every connected Lie group is the semidirect product of a discrete abelian Lie group and a Lie group with no discrete subgroups, but we can¹t since a Lie group is not generally a Lie subgroup of a covering Lie group. So the best I can come up with is the up to division by a discrete subgroup statement or every connected Lie group is completely determined by its fundamental group and its Lie algebra. Q: Are there any more direct statements possible? Is there any kind of product that can be deŽned so that we can write G x pi_1(G) = G*? Moving on to non-compact Lie groups, the only statement I¹ve seen is that any connected Lie group is topologically the product of a compact Lie group and a Euclidean space. As far as actually classifying Lie groups, including the group structure, here is what I come up with: A listing of the complex simple Lie algebras and some of their real forms are: sl(n, C), su(n), su(r, s), sl(n, R) so(n, C), so(n), so(r, s) sp(2n, C), sp(2n), sp(2n, R) The 5 exceptional Lie algebras Q: Are there any other real simple Lie algebras? Are any of the above isomorphic? For each of the above simple Lie algebras, we can list the connected component of the identity of the corresponding Lie group, and its universal cover. Q: Beyond these, are there any other simple Lie groups? If not, is there any way to express *any* Lie group as a direct product of these and perhaps another Žnite list? Finally, there is the question of when a Lie group is a matrix group, i.e. when it has a faithful linear representation. The only example I¹ve seen of a Lie group that doesn¹t is the universal cover SL(n, R), since it is inŽnite-sheeted (and, I guess, SL(n, C) is as well?) Q: Is there an easy way to list the Lie groups that do not have faithful linear representations? (e.g., the Lie groups that are a direct product of SL(n, R) or SL(n, C)). === Subject: Re: classiŽcation of Lie groups Epigone-thread: thumsloisterl more things: >(3) A connected Lie group G has a connected normal subgroup N iff the >subgroup¹s Lie algebra is an ideal in the Lie algebra of G. Thus, >every connected simple Lie group has no connected normal Lie subgroup. This is in fact not true. The necessary requirement to be a Lie subgroup is that a subgroup N is arcwise connected, not just connected (Yamabe¹s theorem). A counterexample in the plane is given here: http://www.austms.org.au/Publ/Gazette/1997/Apr97/ subgroups.html (see Example 2 in the end there) >A listing of the complex simple Lie algebras and some of their real >forms are: >sl(n, C), su(n), su(r, s), sl(n, R) >so(n, C), so(n), so(r, s) >sp(2n, C), sp(2n), sp(2n, R) >The 5 exceptional Lie algebras >Q: Are there any other real simple Lie algebras? Are any of the above >isomorphic? Yes, there are other real simple Lie algebras. At Žrst, every complex Lie algebra g of dimension n can be viewed as the algebra g^R over R of dimension 2n, which is not isomorphic to any of the above. These are sometimes called realliŽcations of complex Lie algebras, or real Lie algebras that allow complex structure. Such algebra is unique for any complex simple Lie algebra. Besides, there are some other algebras that allow no complex structure: sl(n/2,H) if n is even (H is the skew-Želd of quaternions) for sl(n,C), sp(p,q) (p+q=n) for sp(2n,C), u^*(n,H) for so(2n,C). E6 has 5 real forms, E7 has 4, E8 and F4 have 3, and G2 has 2. Lest some algebras be isomorphic, one has to impose conditions on parameters: su(p,n-p), so(p,n-p), sp(p,n-p): 1 <= p <= (n-1)/2. There are some other isomorphisms: so(3)~su(2)=sp(1) so(1,2)~su(1,1)~sl(2,R)=sp(2,R) so(4)~su(2)+su(2) so(6)~su(4) so(1,5)~sl(2,H) so(2,4)~su(2,2) so(1,3)~sl(2,C)^R so(2,2)~sl(2,R)+sl(2,R) so(5)~sp(2) so(1,4)~sp(1,1) so(2,3)~sp(4,R) so(3,3)~sl(4,R) u^*(2,H)~su(2)+sl(2,R) u^*(3,H)~su(1,3) u^*(4,H)~so(2,6) Apart from the compact real form, every complex simple Lie algebra has one more distinguished real form -- the so-called *split* real form. It is a real simple Lie algebra which has the same root system as the complex one. It is an antipode of compact Lie algebra, and all other real forms are in a certain sense intermediate between the compact form (which has empty root system) and the split one. For example, the split forms of sl(n,C), so(2k+1,C), so(2k,C), sp(2n,C) are sl(n,R), so(k,k+1), so(k,k) and sp(2n,R) correspondingly. >Q: Beyond these, are there any other simple Lie groups? If not, is >there any way to express *any* Lie group as a direct product of these >and perhaps another Žnite list? Such idea cannot be realized due to an innumerable quantity of nilpotent Lie algebras that exist. In the dimensions of 7 and above, there are continual families of non-isomorphic nilpotent Lie algebras (and hence Lie groups). See the classiŽcation of 7-dimensional nilpotent Lie algebras here: http://etd.uwaterloo.ca/etd/mpgong1998.pdf >Finally, there is the question of when a Lie group is a matrix group, >i.e. when it has a faithful linear representation. The only example >I¹ve seen of a Lie group that doesn¹t is the universal cover SL(n, R), >since it is inŽnite-sheeted (and, I guess, SL(n, C) is as well?) First, SL(n,C) is simply-connected so it is its own universal cover. (By the way, the cover of SL(2,R) cannot be complexiŽed.) The question if a simple real Lie group has an exact representation is solved. The answer is the following: every simply connected semisimple real Lie group G has a maximal linear quotient group G_{lin}, which is obtained by factoring G by a certain discrete central subgroup L(G), which is called the linearizer of G. For all simple real Lie algebras the linearizer is known and isomorphic to either an inŽnite cyclic group, or Z/2Z, or either is trivial. Examples of simply connected simple linear Lie groups are: SL(p,H), Sp(p,q), and any realliŽcation of a simply connected complex simple Lie group. A readable account of these topics is given in the book: A.Onishchik and E.Vinberg, Seminar on Lie groups and algebraic groups, Springer, 1990. -- Ignat Soroko Minsk, Belarus === Subject: Re: classiŽcation of Lie groups > more things: >> (3) A connected Lie group G has a connected normal subgroup N iff the >> subgroup¹s Lie algebra is an ideal in the Lie algebra of G. Thus, >> every connected simple Lie group has no connected normal Lie > subgroup. > This is in fact not true. The necessary requirement to be a Lie > subgroup is that a subgroup N is arcwise connected, not just connected > (Yamabe¹s theorem). A counterexample in the plane is given here: > http://www.austms.org.au/Publ/Gazette/1997/Apr97/ subgroups.html > (see Example 2 in the end there) I¹m not sure I understand which part of (3) this shows is false. Your counterexample shows that a connected subgroup of a Lie group may not be a Lie group itself, while Yamabe¹s theorem says that an arcwise connected subgroup is always a Lie subgroup. The above statements only concern Lie subgroups, so even if there are connected but not arcwise connected subgroups that are not Lie subgroups, I don¹t see how this contradicts the statements. Perhaps it¹s because I didn¹t explicitly say the normal subgroup with a Lie algebra was a Lie group...? algebra isomorphisms. I assume = denotes an isomorphism with isomorphic associated Lie groups, while ~ denotes an isomorphism with only homomorphic associated Lie groups. Does u^*(3,H) denote the anti-hermitian quatrionic matrices, i.e. the Lie algebra of Sp(3), the compact symplectic group? === Subject: Re: classiŽcation of Lie groups Received-SPF: none (mailbox6.ucsd.edu: domain of mod-submit@uni-berlin.de does not designate permitted sender hosts) Originator: israel@math.ubc.ca (Robert Israel) > (2) Every connected Lie group G is obtained via a homomorphism from > its universal covering group G* with discrete abelian kernel > isomorphic to the fundamental group of G. Thus, every connected Lie > group is completely determined by its fundamental group and its Lie > algebra. I can¹t see how do you deduce from the Žrst assertion that every connected Lie group is completely determined by its fundamental group and its Lie algebra. Let G and H be two Lie groups with isomorphic Lie algebras and isomorphic fundamental groups pi(G) and pi(H). Then G* and H* are isomorphic and what you know is that G is isomorphic to the quotient of G* by a subgroup of its center isomorphic to pi(G) and that H is isomorphic to the quotient of H* by a subgroup of its center isomorphic to pi(H). Even knowing that G* and H* are isomorphic and that pi(G) and pi(H) are isomorphic, it¹s not obvious (at least, not to me) that G*/pi(G) and H*/pi(H) are isomorphic. > (3) A connected Lie group G has a connected normal subgroup N iff the > subgroup¹s Lie algebra is an ideal in the Lie algebra of G. Thus, > every connected simple Lie group has no connected normal Lie subgroup. Yes, assuming that the dimension of your subgroup is at least 1. > (4) Every complex simple Lie algebra has a unique compact real form, > which is the only one with a corresponding compact Lie group. Thus, > every compact connected simple Lie group has a Lie algebra that is the > compact real form of a complex simple Lie algebra. Wrong. Take the Lie group SL(2,R), for instance. Its Lie algebra is sl(2,R), which is not the compact real form of a complex simple Lie algebra. > Every compact connected Lie group is a direct product of SU(n), U(n), > SO(n), Spin(n), Sp(2n), and the exceptional Lie groups. No. Where is the quotient of SO(4) by {Id,-Id}? Jose Carlos Santos === Subject: Re: classiŽcation of Lie groups > I can¹t see how do you deduce from the Žrst assertion that every > connected Lie group is completely determined by its fundamental group > and its Lie algebra. Let G and H be two Lie groups with isomorphic > Lie algebras and isomorphic fundamental groups pi(G) and pi(H). Then > G* and H* are isomorphic and what you know is that G is isomorphic > to the quotient of G* by a subgroup of its center isomorphic to pi(G) > and that H is isomorphic to the quotient of H* by a subgroup of its > center isomorphic to pi(H). Even knowing that G* and H* are isomorphic > and that pi(G) and pi(H) are isomorphic, it¹s not obvious (at least, not > to me) that G*/pi(G) and H*/pi(H) are isomorphic. Hmm, good point, this is not obvious at all. Maybe this is true for compact connected Lie groups? My new, and hopefully better, understanding is: The connected simple compact Lie groups are obtained by taking the quotient of each simply connected simple compact Lie group G* by discrete normal groups, which all lie in the center of G*. For example Spin(2n) has center Z_4 for 2n odd, Z_2 + Z_2 for 2n even (n divisible by 4). Both Z_4 and Z_2 + Z_2 have 2 nontrival proper subgroups, so the quotient of G* by either the entire center or one of these subgroups gives SO(n). My question then is: what does the quotient of G* by the other discrete subgroups give? If the subgroups are isomorphic (e.g. the two copies of Z_2) does this necessarily give isomorphic quotients? Perhaps this can help answer the (new improved) question above: is a compact connected Lie group completely determined by its Lie algebra and fundamental group? === Subject: Re: classiŽcation of Lie groups >>I can¹t see how do you deduce from the Žrst assertion that every >>connected Lie group is completely determined by its fundamental group >>and its Lie algebra. Let G and H be two Lie groups with isomorphic >>Lie algebras and isomorphic fundamental groups pi(G) and pi(H). Then >>G* and H* are isomorphic and what you know is that G is isomorphic >>to the quotient of G* by a subgroup of its center isomorphic to pi(G) >>and that H is isomorphic to the quotient of H* by a subgroup of its >>center isomorphic to pi(H). Even knowing that G* and H* are isomorphic >>and that pi(G) and pi(H) are isomorphic, it¹s not obvious (at least, not >>to me) that G*/pi(G) and H*/pi(H) are isomorphic. > Hmm, good point, this is not obvious at all. Maybe this is true for > compact connected Lie groups? Not good enough. Take SO(4,R) and SU(2) x SO(3,R). They have isomorphic Lie algebras (so(4,R)) and isomorphic fundamental groups (Z_2), but they are not isomorphic, in spite of the fact that they are both compact and connected. Jose Carlos Santos === Subject: Re: classiŽcation of Lie groups >> (4) Every complex simple Lie algebra has a unique compact real form, >> which is the only one with a corresponding compact Lie group. Thus, >> every compact connected simple Lie group has a Lie algebra that is the >> compact real form of a complex simple Lie algebra. > Wrong. Take the Lie group SL(2,R), for instance. Its Lie algebra is > sl(2,R), which is not the compact real form of a complex simple Lie > algebra. But SL(2,R) is also not a compact Lie group, as can be seen by considering the sequence |n 0 | |0 1/n| which has no convergent subsequence. Further, Adams¹ statement is correct, except that there may be several compact real forms (as subalgebras of the complex algebra; all are isomorphic under an automorphism of the complex algebra). This is all in Helgason (Chapter III, sections 6 and 7, as well as Chapter X), and is due to Cartan. In particular, the complexiŽcation of sl(2,R) is of course sl(2,C), with compact real form su(2). As others have suggested, the original poster has to worry more about discrete normal subgroups, coverings, and which Lie algebras are actually simple. I suggest that he look at Helgason¹s book. It is the best reference I know of for this material. -- David L. Johnson __o | Accept risk. Accept responsibility. Put a lawyer out of _`(,_ | business. (_)/ (_) | === Subject: Re: classiŽcation of Lie groups >>> (4) Every complex simple Lie algebra has a unique compact real form, >>> which is the only one with a corresponding compact Lie group. Thus, >>> every compact connected simple Lie group has a Lie algebra that is the >>> compact real form of a complex simple Lie algebra. >> Wrong. Take the Lie group SL(2,R), for instance. Its Lie algebra is >> sl(2,R), which is not the compact real form of a complex simple Lie >> algebra. > But SL(2,R) is also not a compact Lie group, as can be seen by considering > the sequence > |n 0 | > |0 1/n| > which has no convergent subsequence. Further, Adams¹ statement is > correct, except that there may be several compact real forms (as > subalgebras of the complex algebra; all are isomorphic under an > automorphism of the complex algebra). This is all in Helgason (Chapter > III, sections 6 and 7, as well as Chapter X), and is due to Cartan. Quite right. The OP posted the same questions at the sci.math newsgroup (that¹s multiposting and it should be avoided, for the reasons stated at http://www.blakjak.demon.co.uk/mul_crss.htm ) and there Robin Chapman detected my mistake right away; as I told him, I missed the word compact from the second sentence. I should have posted that correction here too. Jose Carlos Santos === Subject: Re: classiŽcation of Lie groups >> Every compact connected Lie group is a direct product of SU(n), U(n), >> SO(n), Spin(n), Sp(2n), and the exceptional Lie groups. > No. Where is the quotient of SO(4) by {Id,-Id}? Take SO(6)/{Id,-Id} instead. Jose Carlos Santos === Subject: Re: classiŽcation of Lie groups Originator: baez@math-cl-n03.math.ucr.edu (John Baez) Originator: israel@math.ubc.ca (Robert Israel) >The compact real forms of complex simple Lie algebras are su(n), >so(n), sp(2n), and the 5 exceptional Lie algebras. >Q: Is it correct that the 5 exceptional Lie algebras correspond to >compact Lie groups? Yes. To get compact Lie groups with these Lie algebras, you can just take the groups of automorphisms of these Lie algebras. They may be disconnected (I guess this is only true for E_6, which is the only exceptional Lie algebra whose Dynkin diagram has symmetries), so take the connected component if you like. Alternatively you can Žnd explicit descriptions of these exceptional compact Lie groups here: http://math.ucr.edu/home/baez/Octonions/node13.html with more details in the references, especially: John F. Adams, Lectures on Exceptional Lie Groups, eds. Zafer Mahmud and Mamoru Mimira, University of Chicago Press, Chicago, 1996 >Clearly these algebras correspond to the compact connected simple Lie >groups SU(n), U(n), SO(n), Spin(n), Sp(2n), and the exceptional Lie >groups. Okay, except leave out U(n): it isn¹t simple! U(1) is abelian and all the rest contain this as a normal subgroup. In other words, all the Lie algebras u(n) have u(1) as an abelian ideal. >Q: Is it correct that these are the only compact connected simple Lie >groups? Given your deŽnition of a simple Lie group, which is the usual one, a simple Lie group can have normal subgroups if they are discrete. This indeed happens. So, if you hand me a compact connected simple Lie group, I can mod out by such a normal subgroup and get another one. For this reason it¹s best to start by listing the *simply connected* compact connected simple Lie groups: all the rest can be gotten from these by modding out by discrete normal subgroups. Such subgroups always lie in the *center*! So if you know the centers of the simply connected compact connected simple Lie groups, it¹s easy to work out the complete list of compact connected simple Lie groups. You can Žnd these centers tabulated in, for example, Helgason¹s book Differential Geometry, Lie Gruops and Symmetric Spaces. For example: Spin(n) is simply connected, and the center of Spin(n) has either 2 or 4 elements depending on whether n is odd or even. If n is even, the center is Z/2 x Z/2 or Z/4 depending on whether or not n is divisible by 4. In every case, the center contains a normal subgroup N of Spin(n) such that Spin(n)/N = SO(n). But, when n is even, there is at least one other compact connected simple Lie group whose Lie algebra is so(n)! Naturally, every good mathematician spends some time Žguring out all the possibilities. :-) Some of the more unusual ones actually show up in string theory. Or: SU(n) is simply connected, and the center of SU(n) is Z/n. So, there are as many compact connected Lie groups with Lie algebra su(n) as there are subgroups of Z/n. That can be a lot! >Every compact connected Lie group is a direct product of SU(n), U(n), >SO(n), Spin(n), Sp(2n), and the exceptional Lie groups. >Q: Is this statement correct? No. But every compact connected Lie group is a quotient of such a direct product by a discrete normal subgroup. I would prefer to say it this way: Every compact connected Lie group is of the form G/N where G is a direct product of copies of U(1), SU(n), Spin(n), Sp(2n), and simply connected compact exceptional Lie groups, and N is a discrete normal subgroup. >Q: Are there any more direct statements possible? One can work out all possible discrete normal subgroups that show up in the statement above, and also work out when different quotients give isomorphic groups. But, it¹s tiresome! Better to tackle individual cases when the need arises. >Moving on to non-compact Lie groups, the only statement I¹ve seen is >that any connected Lie group is topologically the product of a compact >Lie group and a Euclidean space. Right; that comes in handy. >As far as actually classifying Lie groups, including the >group structure, here is what I come up with [...] You¹re not going to Žnd a complete classiŽcation of all noncompact Lie groups, since this is a wild sort of classiŽcation problem, sort of like classifying all Žnite groups. But, to see some of the progress that¹s been made, try: A. L. Onishchik and E. B. Vinberg, eds., Lie Groups and Lie Algebras III, Springer, Berlin, 1991. Among lots of other more useful things, they describe the classiŽcation of all Lie algebras of dimensions 1, 2, 3, 4, ... until it gets too hard. You¹ll see why it¹s sort of hopeless. === Subject: Re: classiŽcation of Lie groups > For example: Spin(n) is simply connected, and the center of Spin(n) > has either 2 or 4 elements depending on whether n is odd or even. > If n is even, the center is Z/2 x Z/2 or Z/4 depending on whether > or not n is divisible by 4. In every case, the center contains a > normal subgroup N of Spin(n) such that Spin(n)/N = SO(n). But, when > n is even, there is at least one other compact connected simple > Lie group whose Lie algebra is so(n)! Naturally, every good > mathematician spends some time Žguring out all the possibilities. :-) > Some of the more unusual ones actually show up in string theory. OK, just to make sure I¹m clear: the connected simple compact Lie groups are obtained by taking the quotient of each simply connected simple compact Lie group G* by discrete normal groups, which all lie in the center of G*. For example Spin(n) has center Z_4 for 2n odd, Z_2 + Z_2 for 2n even (n divisible by 4). But I¹m missing something on your comments concerning elements and subgroups in the center. Both Z_4 and Z_2 + Z_2 have 4 elements and 2 nontrival proper subgroups, right? So the quotient of G* by at least one of these gives SO(n); what does the other give? Or do they both give isomorphic copies of SO(n) in different ways...? === Subject: Re: classiŽcation of Lie groups > But I¹m missing something on your comments concerning elements and > subgroups in the center. Both Z_4 and Z_2 Z_2 have 4 elements and 2 > nontrival proper subgroups, right? So the quotient of G* by at least > one of these gives SO(n); what does the other give? Or do they both > give isomorphic copies of SO(n) in different ways...? Whoops! Of course they don¹t both have 2 subgroups...my error and an answer to my question was provided by anonymous email reply; the exchange went as follows: No, Z_4 has 1 subgroup of order 2, and Z_2 + Z_2 has 3 subgroups of order 2. > So the quotient of G* by at least > one of these gives SO(n); what does the other give? Or do they both In the Z_4 case, there is only one quotient. In the other case, I do not know, except in the case of Spin(4), which is isomorphic to SU(2)xSU(2). There are there quotients by central subgroups of order 2, giving: SO(3)xSU(2), SO(4), SU(2)xSO(3). It is clear that the Žrst and the third are isomorphic, but are not isomorphic to the second. I think that a similar picture works for all SO(4n), but it needs to be checked. It means that we must solve the question: how does the outer automorphism of Spin(4n) acts on its center. If the action is trivial, the three possible quotients by subgroups of order 2 are pairwise non-isomorphic. Otherwise, this would give two isomorphic quotients and one other, and I think this is the case. > Perhaps this can help answer the (new improved) question above: is a > compact connected Lie group completely determined by its Lie algebra > and fundamental group? I already gave you an counterexample: SO(3)xSU(2) and SO(4) have so(3)xso(3) as Lie algebra, Z_2 as pi_1, but they are not isomorphic. I also gave you the arguments to show that there exists, for all n, a connected, compact Lie group with Lie algebra so(4n), with fundamental group Z_2, which is NOT isomorphic to SO(4n). > clariŽcation: one can take the quotient by the entire center as well as the > subgroups, correct? So in the Z_4 case, there are two quotients. Yes. If g is a Lie algebra and G the corresponding simply connected Lie group, the connected Lie groups with Lie algebra g are the G/Z, where Z is any discrete subgroup of the center Z(G) of G. Moreover, if Z and Z¹ are discrete subgroups of Z(G), then G/Z and G/Z¹ are isomorphic if and only if there exists an automorphism U of G [note that Aut(G) and Aut(g) are canonically isomorphic] such that U(Z)=Z¹. === Subject: New list-server (and BB) for discussion re DAGs, posets, and permutations Originator: israel@math.ubc.ca (Robert Israel) Cumulative Inquiry (CI) has started a new list-server to facilitate discussion of certain theoretical questions arising in the study of DAGs on n vertices whose transitive closures are dim 2 posets, in relation to permutations of 0,...,n-1. These questions arise when: i) considering the set of permutations which: a) can be built in a natural way from strings over a 4-letter alphabet whose symbols are {++,-+,--,+-}, e.g strings over the four DNA bases {t,c,a,g} or the mRNA bases {u,c,a,g}; ii) considering which DAGs corresponding to these permutations are invariant under a half-turn when considered as embedded in a torus rather than the plane. iii) considering the skip-Fib sequence which arises by enumerating the number of dim 2 posets whose Hasse diagrams which are the isomorphism classes of the DAGs in (ii). To subscribe to the list, Žrst send a plain-text message with NO SUBJECT LINE to: Majordomo@www.cumulativeinquiry.com and in the body of the message, place the single command: subscribe ci_discuss_001 You will receive a reply message containing an auth command with a key. Then send a second message back to the CI major domo with NO SUBJECT line and the entire auth command as the Žrst and only line in the message body itself. Any questions or problems with subscription should be reported to dhalitsky@cumulativeinquiry.com This list-server will be augmented by a full Bulletin Board in the near future. For background information on the class of problems germane to the ci_discuss_001 list, please see URL http://www.CumulativeInquiry.com/Problems or go to URL http://www/CumulativeInquiry.com and then navigate to Theoretical Foundations. David Halitsky Cumulative Inquiry === Subject: Shintani¹s Zeta function Epigone-thread: swansneuqual Originator: israel@math.ubc.ca (Robert Israel) For an r by n matrix A = (a_jk) of positive reals, an r-tuple x = (x_j) of positive reals, and r-tuple z = (z_j) of complex numbers whose moduli are bounded above by one, and complex number s whose real part exceeds r/n, Shintani¹s Zeta function zeta(s,A,x,z) is deŽned by the following series: SUM_{n_1,...,n_r >= 0} [ PRODUCT_{k=1...r} z_k^n_k ] [ PRODUCT_{j=1...n} ( SUM_{l=1...r} a_lj (n_l+x_l) )^{-s} ) ] . zeta(s,A,x,z) can be analytically continued in its variables. I need a variation on this function, namely, the I need to sum over ALL integer values of n_1,...,n_r, with |z_k| = 1, s = 1, x_1=...=x_r=0, and A = (a_jk) a rank-r matrix of integers, not necessarily positive. My basic question is this: Is it known that Lerch¹s Transformation Formula for Lerch¹s Phi function (a.k.a. Lerch Transcendent) can(or cannot) be adapted to handle Shintani¹s Zeta function? === Subject: Re: Shintani¹s Zeta function Epigone-thread: swansneuqual I¹ve pretty much Žgured out what I wanted to know, but here is the key to it, in case somebody wants to discuss this. When the entries in the matrix A are allowed to be any real numbers (not necessarily positive any longer), then Shintani¹s Zeta function can be expressed via n nested contour integrals, each of which follows half of the imaginary axis (which half depends on A and x) from inŽnity to the origin, wraps around the origin once counter clockwise, and then goes back to inŽnity along the same half of the imaginary axis used previously. The details of this multiple integral are straightforward to discover by adapting the derivation in Shintani¹s original paper (on his Zeta function) and by using formula 1.6(5) in volume one of Erdelyi et al¹s Higher Trans. Functions, the Bateman Man. Project. === Subject: binary vector packings Originator: israel@math.ubc.ca (Robert Israel) Could someone please help me identify the following problem? Consider binary arrays {u_i} of length n with k 1¹s (and n-k 0¹s). How many can we choose such that all pairwise inner products sum_i u_i v_i < t ? Equivalently, what is the maximum number of k-subsets of the n-set with pairwise intersections less than t elements? Does this problem, or some equivalent, have a name? Any references? e.g. for (n,k,t)=(9,3,2) the following 12 binary arrays seem to be maximal. 111000000 000111000 000000111 100100100 010010010 001001001 100010001 010001100 001100010 100001010 010100001 001010100 Andrew. === Subject: Re: binary vector packings > Could someone please help me identify the following problem? > Consider binary arrays {u_i} of length n with k 1¹s (and n-k 0¹s). How > many can we choose such that all pairwise inner products > sum_i u_i v_i < t ? > Equivalently, what is the maximum number of k-subsets of the n-set with > pairwise intersections less than t elements? > Does this problem, or some equivalent, have a name? Any references? See http://www.research.att.com/projects/OEIS?Anum=A001839 for (k,t) = (3,2). Rob Pratt === Subject: Re: binary vector packings > Could someone please help me identify the following problem? > Consider binary arrays {u_i} of length n with k 1¹s (and n-k 0¹s). How > many can we choose such that all pairwise inner products > sum_i u_i v_i < t ? > Equivalently, what is the maximum number of k-subsets of the n-set with > pairwise intersections less than t elements? > Does this problem, or some equivalent, have a name? Any references? > e.g. for (n,k,t)=(9,3,2) the following 12 binary arrays seem to be > maximal. You should Žnd some information about this problem in the book Linear Algebra Methods in Combinatorics With Applications to Geometry and Computer Science Preliminary Version 2 (September 1992) (216 pages) by Laszlo Babai and Peter Frankl which may be purchased from http://www.cs.uchicago.edu/research/publications/combinatorics I have an earlier version of these notes which discusses the problem of restricted intersections. The uniform case is: Given a set L of non-negative integers L and k >=0. What is the maximum size of a collection A of k-subsets of an n-set satisfying |X intersect Y| in L for all X,Y in A. In your case L = {0,1,...,t-1}. At least in 1988 no general answer to the problem was known or even conjectured, but various partial results are given in these notes. A MathSciNet search on restricted partition* brings up several papers. Most probably treat the problem with more generality than you are interested in. --Edwin Clark === Subject: Re: Relation between period length and fundamental unit Received-SPF: pass (mailbox7.ucsd.edu: domain of jpr2718@aol.com designates 64.12.137.9 as permitted sender) receiver=mailbox7.ucsd.edu; client_ip=64.12.137.9; envelope-from=jpr2718@aol.com; Originator: israel@math.ubc.ca (Robert Israel) > [In Length of the period of a quadratic irrational, > E.V. Podsypanin claims] > ******************************************* > Let $xi$ be a reduced real quadratic irrational of discriminant > $D > 0$. > Denote by $p_k / q_k$ the k-th convergent of the expansion of $xi$ > in a (simple) continued fraction. We will start counting by $k = 0$, < so $p_0 / q_0 = a_0 / 1$. > Then for the fundamental unit $epsilon$ of $Q(sqrt{D})$: > epsilon = q_{l-1}xi + q_{l-2} > where $l$ denotes the period length of the (purely periodic) > continued fraction expansion of $xi$. > ******************************************* > I think that there is introduced an error during translation from the > russian original (to which I do not have access) or an original error > because > 1) the structure of the formula does not seem right > 2) the norm of $epsilon$ is not 1 or -1 in each case. Here¹s a proof that the norm of epsilon is 1 or -1. A quadratic irrational xi = (P_0 + sqrt{D})/Q_0 satisŽes: D > 0 is not a square, P_0^2 == D (mod Q_0), Q_0 is not zero. The quadratic irrational xi is reduced if xi > 1 and -1 < xi¹ < 0, where xi¹ is the conjugate of xi. It is a theorem that xi is reduced if and only if xi has a purely periodic (simple) continued fraction expansion [1, Theorem 5.3.2, p. 241] [2, p. 1] [3, p. 45] ([3] attributes this theorem to Galois!). Let L be the length of the continued fraction expansion of xi. Then, because xi is reduced, Tr(xi) = xi + xi¹ = 2P_0/Q_0 = (p_{L-1} - q_{L-2})/q_{L-1} [1, Exercise 5.3.9, p. 251]. Cross multiplying and rearranging gives Q_0 p_{L-1} - P_0 q_{L-1} = Q_0 q_{L-2} + P_0 q_{L-1}. Again because xi is reduced, Q_L = Q_0 where (P_L + sqrt(D))/Q_L is the L-th complete quotient of xi. Taking G_{L-1} = Q_0 p_{L-1} - P_0 q_{L-1}, we have [G_{L-1} + q_{L-1} sqrt(D)]/Q_L = q_{L-2} + q_{L-1} xi. G_i^2 - D q_i^2 = (-1)^{i+1} Q_0 Q_{i+1} [1, p. 246] we have that the norm of [G_{L-1} + q_{L-1} sqrt(D)]/Q_L is 1 or -1, so q_{L-2} + q_{L-1} xi is a unit. This does not prove that epsilon = q_{l-1}xi + q_{l-2} is a fundamental unit. I have not found any cases where xi is not the fundamental unit of the appropriate order of a quadratic number Želd. John Robertson References [1] Richard A. Mollin, Fundamental Number Theory with Applications, CRC Press, 1998. [2] E. P. Golubeva, Spectrum Of The Levy Constants For Quadratic Irrationalities, Journal of Mathematical Sciences, Vol. 110, No. 6, 2002. [3] Andrew M. Rockett and Peter Szusz, Continued Fractions, World ScientiŽc, 1994. === Subject: Re: Relation between period length and fundamental unit Originator: israel@math.ubc.ca (Robert Israel) > I am just reading Length of the period of a quadratic irrational by > E.V. Podsypanin. > ******************************************* > Let $xi$ be a reduced real quadratic irrational of discriminant > $D > 0$. > Denote by $p_k / q_k$ the k-th convergent of the expansion of $xi$ > in a (simple) continued fraction. We will start counting by $k = 0$, < so $p_0 / q_0 = a_0 / 1$. > Then for the fundamental unit $epsilon$ of $Q(sqrt{D})$: > epsilon = q_{l-1}xi + q_{l-2} > where $l$ denotes the period length of the (purely periodic) > continued fraction expansion of $xi$. > ******************************************* > I think that there is introduced an error during translation from the > russian original (to which I do not have access) or an original error > because > 1) the structure of the formula does not seem right > 2) the norm of $epsilon$ is not 1 or -1 in each case. > Moreover I was not able to Žnd this formula in the reference he gave > (Borevich/Shafarevich: Number Theory). > So I have the question: > Does somebody know the right version of this formula (maybe even with > a reference for the proof)? Do you have an example where the norm of $epsilon$ is not 1 or -1? In a search on small cases, I did not Žnd one. My understanding is that a ``reduced quadratic irrational¹¹ is a fairly speciŽc item, meeting the following criteria: xi = (P_0 + sqrt{D})/Q_0, where D > 0 is not a square, P_0^2 == D (mod Q_0), Q_0 is not zero, xi > 1, the conjugate of xi is strictly between -1 and 0. Generally, G_i^2 - D q_i^2 = (-1)^{i+1} Q_0 Q_{i+1} (see Mollin) where G_i = Q_0 p_i - P_0 q_i and (P_i + sqrt{D})/Q_i is the i-th complete quotient for (P_0 + sqrt(D))/Q_0. As such, the q_{L-1} xi (for L the length of the period of the continued fraction) term seems appropriate. If Q_0 = 1 or 2, then p_{L-1} = q_{L}, where p_i/q_i is a convergent to xi, as above. Combined with the Mollin formula just above, I think this gives what is needed to prove Podsypanin¹s formula for these cases. John Robertson Reference [1] Richard A. Mollin, Fundamental Number Theory with Applications, CRC Press, 1998, Theorem 5.3.4, page 246. === Subject: Re: Relation between period length and fundamental unit > This does not prove that epsilon = q_{l-1}xi + q_{l-2} is a fundamental > unit. > I have not found any cases where xi is not the fundamental unit of the > appropriate order of a quadratic number Želd. Blame on me, I Žrst made a systematic calculation error. Now I have found a proof in some leture notes by W. Ruppert that this is always a fundamental unit. Anke Pohl === Originator: israel@math.ubc.ca (Robert Israel) International Journal of Wavelets, Multiresolution and Information Processing View table-of-contents and abstracts at http://www.worldscinet.com/ijwmip.html Contents: A Neural Network-Wavelet Model For Generating ArtiŽcial Accelerograms Gene F. Sirca, Jr. And Hojjat Adeli Time-Shifts Generalized Multiresolution Analysis Over Dyadic-Scaling Reducing Subspaces Nhan Levan And Carlos S. Kubrusly JPEG2000 And JPEG: A Statistical Approach For Lossily Compressed Medical Images Quality Evaluation Tick Hui Oh And Rosli Besar Compression Of Segmented 3d Seismic Data Valery A. Zheludev, Dan D. Kosloff And Eugene Y. Ragoza Wavelet Directional Histograms Of The Spatio-Temporal Templates Of Human Gestures Arun Sharma, Dinesh K. Kumar, Sanjay Kumar And Neil Mclachlan Monitoring Machining Processes Based On Discrete Wavelet Transform And Statistical Process Control Xiaoli Li And R. Du Experimental Study Of Translation-Invariant DWT Face Feature Estimation Kun Ma And Xiaoou Tang For more information, go to http://www.worldscinet.com/ijwmip.html === Subject: SD Special Issue:Foundations of Nonautonomous Dynamical Systems Originator: israel@math.ubc.ca (Robert Israel) Stochastics and Dynamics Special Issue: Foundations of Nonautonomous Dynamical Systems View table-of-contents and abstracts at http://www.worldscinet.com/sd.html Contents: Preface Towards A Morse Theory For Random Dynamical Systems Hans Crauel, Luu Hoang Duc And Stefan Siegmund Robust Asymptotic Controllability Under Time-Varying Perturbations Lars Gr.9fne A Contribution To Nonautonomous Inertial Manifolds Norbert Koksch A Nonautonomous Saddle-Node Bifurcation Pattern Roberta Fabbri, Russell Johnson And Francesca Mantellini Almost All Nonautonomous Linear Stochastic Differential Equations Are Regular Nguyen Dinh Cong Equi-Attraction And The Continuous Dependence Of Pullback Attractors On Parameters Desheng Li And P. E. Kloeden Finite Dimensionality Of Attractors For Non-Autonomous Dynamical Systems Given By Partial Differential Equations J. A. Langa And B. Schmalfuss Pullback And Forward Attractors For A Damped Wave Equation With Delays T. Caraballo, P. E. Kloeden And J. Real Attracting Sets In Cocycle Delay Differential Equations And In Their RungeKutta Discretizations Johannes Schropp Invariant Measures And Their Projections In Nonautonomous Dynamical Systems Zvi Artstein On Lipschitz Continuity Of The Top Lyapunov Exponent Of Linear Parameter Varying And Linear Switching Systems Fabian Wirth Frequency Domain Conditions For The Existence Of Almost Periodic Solutions In Evolutionary Variational Inequalities Volker Reitmann And Holger Kantz For more information, go to http://www.worldscinet.com/sd.html === Subject: Eigenvalues of a complex matrix, Perron-Frobenius, Gersgorin disk Originator: israel@math.ubc.ca (Robert Israel) If I have a matrix [0 1 0 0 .... 0 0] [0 0 1 0 .... 0 0] [0 0 0 1 .... 0 0] ......... [0 0 0 0 .... 0 1] [c_1 c_2 c_3...c_n] and the c_n terms sum to exactly 1, then obviously [1 1 1 ... 1]¹ is an eigenvector with an eigenvalue 1. My intuition suggests that if the sum of c_n were greater than 1, there would be an eigenvalue with absolute value > 1. Does anyone know if this is true or not? Or any conditions under which its true? I thought of Perron-Frobenius theory, but The c_n may be any complex numbers, not just non-negative reals. I also know that by Gersgorin¹s disk th¹m that if c_n > 1 + [sum of the others], then this is true, but I want something more general than that, if possible! === Subject: Re: Eigenvalues of a complex matrix, Perron-Frobenius, Gersgorin disk X-UBC-Relayed: relayed through mail-relay2.ubc.ca >If I have a matrix >[0 1 0 0 .... 0 0] >[0 0 1 0 .... 0 0] >[0 0 0 1 .... 0 0] > ......... >[0 0 0 0 .... 0 1] >[c_1 c_2 c_3...c_n] >and the c_n terms sum to exactly 1, then obviously [1 1 1 ... 1]¹ is >an eigenvector with an eigenvalue 1. >My intuition suggests that if the sum of c_n were greater than 1, >there would be an eigenvalue with absolute value > 1. >Does anyone know if this is true or not? >Or any conditions under which its true? Your matrix is known as a companion matrix. Its characteristic polynomial is P(z) = z^n - sum_{j=0}^{n-1} c_{j+1} z^j. In particular P(1) = 1 - sum_{k=1}^{n} c_k. If the c_j are real with sum_{k=1}^n c_k > 1, since P(1) < 0 and lim_{z -> +inŽnity} P(z) = +inŽnity the Intermediate Value Theorem shows there is some real eigenvalue > 1. If the c_j are complex, what we can say is: if sum_{k=1}^n |c_k| <= 1, then every eigenvalue has absolute value >= 1. However, it is not true that if sum_{k=1}^n c_k > 1 there is an eigenvalue of absolute value > 1. Consider for example P(z) = (z - (1+i)/2)^4. The only root has absolute value sqrt(1/2) < 1, but P(1) = -1/4. This corresponds to c_1 = 1/4 c_2 = -1+i c_3 = -3 i c_4 = 2 + 2 i with c_1+c_2+c_3+c_4 = 5/4. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Reference correction Epigone-thread: skiwhelimp Originator: israel@math.ubc.ca (Robert Israel) Please help me with the following offtopic question. There is an excellent survey of literature on functional equations (aggregated by year) at the end of the famous book: Aczel J., Dhombres J. Functional Equations in Several Variables (Encyclopedia of Mathematics and its Applications). Cambridge University Press, 1989. In my Russian version (translation) of the book I have, among others, the following reference for 1982 year: Dhombres J. Moyennes. Nantes: Math.8ematiques de lUniversit.8e de Nantes, 1982. But I can¹t Žnd the book or even it¹s mentioning in any other sources. I am afraid that it has been distorted during translation. If it possible, please, give me full reference (included amounts of pages) on it from original version of Aczel/Dhombres book. === Subject: Re: Reference correction On 15 Sep 04 05:13:30 -0400 (EDT), Mikhail V. Sokolov >Please help me with the following offtopic question. >There is an excellent survey of literature on functional equations >(aggregated by year) at the end of the famous book: >Aczel J., Dhombres J. Functional Equations in Several Variables >(Encyclopedia of Mathematics and its Applications). Cambridge >University Press, 1989. >In my Russian version (translation) of the book I have, among others, >the following reference for 1982 year: >Dhombres J. Moyennes. Nantes: Mathmatiques de l.89Universit de Nantes, >1982. >But I can¹t Žnd the book or even it¹s mentioning in any other >sources. I am afraid that it has been distorted during translation. >If it possible, please, give me full reference (included amounts of >pages) on it from original version of Aczel/Dhombres book. [2] J. Aczl and J. Dhombres, Functional Equations in Several Variables, Encyclopedia of Mathematics and Its Applications, vol. 31, Cambridge University Press, Cambridge, 1989. xiv+462 pp. ISBN 0-521-35276-2 Found at the following URL: http://www.ams.org/mathscinet-getitem?mr=90h:39001 It may require subscribing to MathSciNet to view this. Dan -- Dan Luecking Department of Mathematical Sciences University of Arkansas Fayetteville, Arkansas 72701 To reply by email, change Look-In-Sig to luecking === Subject: Re: Reference correction >On 15 Sep 04 05:13:30 -0400 (EDT), Mikhail V. Sokolov >>Please help me with the following offtopic question. >>There is an excellent survey of literature on functional equations >>(aggregated by year) at the end of the famous book: >>Aczel J., Dhombres J. Functional Equations in Several Variables >>(Encyclopedia of Mathematics and its Applications). Cambridge >>University Press, 1989. >>In my Russian version (translation) of the book I have, among others, >>the following reference for 1982 year: >>Dhombres J. Moyennes. Nantes: Mathmatiques de lUniversit de Nantes, >>1982. >>But I can¹t Žnd the book or even it¹s mentioning in any other >>sources. I am afraid that it has been distorted during translation. >>If it possible, please, give me full reference (included amounts of >>pages) on it from original version of Aczel/Dhombres book. >[2] J. Aczl and J. Dhombres, Functional Equations in Several Variables, >Encyclopedia of Mathematics and Its Applications, vol. 31, Cambridge >University Press, Cambridge, 1989. xiv+462 pp. ISBN 0-521-35276-2 Oops, never mind, I misinterpreted pretty much everything. MathSciNet doesn¹t list any papers of J. Dhombres from 1982, so I suspect this is a local technical report. The papers from 1983 and 1980 (those nearest 1981) are: MR0726668 (84k:16053) Dhombres, J. Sur les fonctions simultanment suradditives et surmultiplicatives. (French) [Simultaneously superadditive and supermultiplicative functions] C. R. Math. Rep. Acad. Sci. Canada 5 (1983), no. 5, 207--210. 16A86 MR0647614 (83e:01087) Dhombres, Jean L¹enseignement des mathmatiques par la mthode rvolutionnaire. Les leons de Laplace l¹cole Normale de l¹an III. (French) [Mathematical instruction by the ``revolutionary method¹¹. The lectures of Laplace at the Ecole Normale during Year III] Rev. Histoire Sci. Appl. 33 (1980), no. 4, 315--348. Dan -- Dan Luecking Department of Mathematical Sciences University of Arkansas Fayetteville, Arkansas 72701 To reply by email, change Look-In-Sig to luecking === Subject: Eigenvalue problem Epigone-thread: hiclusmeu Originator: israel@math.ubc.ca (Robert Israel) I have a problem to Žnd a diagonal matrix A (n*n) with diagonal terms a_i that can make the matrix (A-B) semideŽnite under the constraint sum a_i=c where c is a constant and B is a hermitian matrix. This problem can be written as A^1/2(I-A^{-1/2}BA^{-1/2})A^1/2 The problem become Žnd diagonal matrix A that the eigenvalues of A^{-1/2}BA^{-1/2} are less than 1 under the constranit sum a_i=c. So if the eigenvalues of B is known, how to obtain the eigenvalue of Yue === Subject: Re: Eigenvalue problem >I have a problem to Žnd a diagonal matrix A (n*n) with diagonal terms >a_i that can make the matrix (A-B) semideŽnite under the constraint >sum a_i=c where c is a constant and B is a hermitian matrix. >This problem can be written as >A^1/2(I-A^{-1/2}BA^{-1/2})A^1/2 >The problem become Žnd diagonal matrix A that the eigenvalues of >A^{-1/2}BA^{-1/2} are less than 1 under the constranit sum a_i=c. So >if the eigenvalues of B is known, how to obtain the eigenvalue of >Yue the eigenvalues of A^{-1/2}BA^{-1/2} are those of the generalized eigenvalue problem Bx=lambda*A*x and these cannot be easily obtained from those of B alone. you can solve it as a semideŽnite program but I guess this is just what you wanted to avoid. hth peter === Subject: IJGMMP Special Issue: Advanced Geometric Techniques in Gauge Theory International Journal of Geometric Methods in Modern Physics Special Issue: Advanced Geometric Techniques in Gauge Theory Abstracts and Table-of-contents area available at http://www.worldscinet.com/ijgmmp.html Contents: Preface Short Communications Symplectic Reduction For Yang.9aMills On A Cylinder Ambar N. Sengupta Surface Holonomy And Gauge 2-Group Amitabha Lahiri Papers On The Structure Of The Space Of Generalized Connections J. M. Velhinho Brst Cohomological Results On The Massless Tensor Field With The Mixed Symmetry Of The Riemann Tensor C. Bizdadea, C. C. Ciobirca, E. M. Cioroianu, S. O. Saliu And S. C. Sararu A Report On Gauge Invariant Forms And Variational Problems On The Bundle Of Connections Of A Principal U(1)-Bundle And On Associated Vector Bundles M. Castrilln Lpez And J. Muoz Masqu A Boundary Value Problem For Monopoles Over A 3-Dimensional Disk Antonella Marini Gribov Problem For Gauge Theories: A Pedagogical Introduction Giampiero Esposito, Diego N. Pelliccia And Francesco Zaccaria On The Gauge Natural Structure Of Modern Physics L. Fatibene, M. Ferraris And M. Francaviglia Geometrical Aspects Of Brst Cohomology In Augmented SuperŽeld Formalism R. P. Malik Gauge Theory Deformations And Novel Yang.9aMills Chern.9aSimons Field Theories With Torsion Stephen C. Anco Topical Review Topics On D-Brane Charges With B-Fields Juan Jos Manjarn For more information, go to http://www.worldscinet.com/ijgmmp.html === Subject: existence and uniqueness Epigone-thread: streiskaifran I am facing a problem of an almost Painleve form, where u is a matrix. L(u¹¹)=F(u¹,u,z) u is formed by three functions of z: r,s,u; L is linear; F is a function that allows permutation between the lines of the 3x1 matrix, which is u. I do not have access to databases at the moment and seek for help regarding Žnding out theorems to help proving existence and uniqueness theorems for this system. === Subject: Help me solving: (f(x)+3*x )^[5/3]=f(x /(x+1))+3*x /(x+1) . Epigone-thread: rerdbreldyul f is a continuous real function,g(x)^[r] means r iterate of g, r is real. Is it a known method to solve such equations, Alain. === Subject: Re: Help me solving: (f(x)+3*x )^[5/3]=f(x /(x+1))+3*x /(x+1) . > f is a continuous real function,g(x)^[r] means r iterate of g, > r is real. > Is it a known method to solve such equations, I would proceed like this. I ignore the fact that non-integer iterates need not be unique (and this may be a real problem...). The proposed problem (f(x)+3*x )^[5/3]=f(x /(x+1))+3*x /(x+1) I take to mean: let g(x) = f(x)+3*x, then solve g^[5/3] = g h, where h(x) = x/(x+1). { g h means composition, * means multiplication. } So this is g^[2/3] = h , thus g = h^[3/2]. Now I happen to know this one-parameter group: h_t(x) = x/(t*x+1) where h_1 = h and h_(s+t) = h_s h_t, so (ignoring as stated above) h^[3/2](x) = x/((3/2)*x+1) so g(x) = x/((3/2)*x+1) so f(x) = x/((3/2)*x+1) - 3*x = (-4*x-9*x^2)/(3*x+2) . We can check that g g g g g = r r r, where r(x) = f(x/(x+1))+3*x/(x+1) and g(x) = f(x)+3*x { they are both x -> 2*x/(15*x+2) }, and that is presumably the meaning of the assertion g^[5/3] = r. -- Gerald A. Edgar edgar at math.ohio-state.edu Department of Mathematics telephone: 614-292-0395 (OfŽce) The Ohio State University 614-292-4975 (Math. Dept.) Columbus, OH 43210 614-292-1479 (Dept. Fax) === Subject: the intersection can be empty if... Epigone-thread: yirshermskay Can anyone Žgure out an example with the following properties: If {E_n} is a sequence of closed and bounded sets in a complete metric space X, if E_n+1 is subset of E_n. The intersection should be empty and lim(n-->oo)diam not equal to 0. please help me Mild === Subject: Re: the intersection can be empty if... > Can anyone Žgure out an example with the following properties: > If {E_n} is a sequence of closed and bounded sets in a complete metric > space X, if E_n+1 is subset of E_n. The intersection should be empty > and lim(n-->oo)diam not equal to 0. Take X to be the positive integers, with a new metric deŽned by d(m,n) = 1 when m and n are different. This is a complete metric space. All subsets are closed, and all containing more than one point have diameter 1. Let E_n be the integers greater than n. William C. Waterhouse Penn State === Subject: Re: Preprint archives considered harmful Epigone-thread: whendtwarsan I do not consider the whole archive system harmful at all. In fact, it has perhaps helped bring differing research and research Institutes from across the globe together in ways never possible before. I would agree that perhpas the system ought to be used more for polished up for prime time, so to speak. The whole point of the archive systems is to allow the žow of information. In many ways it provides the best version of real unbiased peer review where the opinion comes from all researchers in general instead of some paid subscription or otherwise reviewers where information counts and time also counts, such a system is very vital to the health of research in general. [ Moderator¹s Note: This is a reply to a discussion from 1999. See Is there really anything new to say about it? ] === Subject: Re: Preprint archives considered harmful One aspct of the math arXiv system that I Žnd very irritating is that they do not accept submissions from people not at educational institutions. I tried to submit some work of mine as an individual, and they refused to accept it. Martin Cohen === Subject: Re: Preprint archives considered harmful ^OX9W/.#XpUmm`>TD2zNE-t}emfPkFR.Z5`žY:3QYT$>dUwN^sm;MBV: F7aL9x*q!` ln!l}>Y6_45$%R|P7DSrBkEph@1-;P*s~F_28vO@e4p/¹>}Pc?@rl8cz] d9RXOt One aspct of the math arXiv system that I Žnd very irritating is that > they do not accept submissions from people not at educational > institutions. I tried to submit some work of mine as an individual, and > they refused to accept it. I believe you are talking about the old system. There is a new system of endorsement: http://arxiv.org/help/endorsement If you can get your work endorsed, then you should be able to submit your work. === Subject: Re: Preprint archives considered harmful > > One aspct of the math arXiv system that I Žnd very irritating is that > > they do not accept submissions from people not at educational > > institutions. I tried to submit some work of mine as an individual, and > > they refused to accept it. > I believe you are talking about the old system. There is a new system > of endorsement: > http://arxiv.org/help/endorsement > If you can get your work endorsed, then you should be able to submit > your work. The link above says If you¹re looking for an endorsement, you can Žnd somebody qualiŽed to endorse by clicking on the link titled ŒWhich of these authors are endorsers?¹ at the bottom of every abstract. But I looked at several abstracts, and yet never found any link titled Which of these authors are endorsers? Did I somehow overlook something? David Cantrell === Subject: Re: Preprint archives considered harmful ^OX9W/.#XpUmm`>TD2zNE-t}emfPkFR.Z5`žY:3QYT$>dUwN^sm;MBV: F7aL9x*q!` ln!l}>Y6_45$%R|P7DSrBkEph@1-;P*s~F_28vO@e4p/¹>}Pc?@rl8cz] d9RXOt > > One aspct of the math arXiv system that I Žnd very irritating is that > > > they do not accept submissions from people not at educational > > > institutions. I tried to submit some work of mine as an individual, and > > > they refused to accept it. > > I believe you are talking about the old system. There is a new system > > of endorsement: > > http://arxiv.org/help/endorsement > > If you can get your work endorsed, then you should be able to submit > > your work. > The link above says If you¹re looking for an endorsement, you can Žnd > somebody qualiŽed to endorse by clicking on the link titled ŒWhich of > these authors are endorsers?¹ at the bottom of every abstract. > But I looked at several abstracts, and yet never found any link titled > Which of these authors are endorsers? Did I somehow overlook something? I think it depends on how you¹re accessing the arXiv. If you go straight to arxiv.org (at Cornell), you should be able to Žnd these links. Not all the arXiv servers and such have been updated to show the endorsement link. For example, the one in France does, but the one in Australia does not. Also, front.math.ucdavis.edu has not been updated for that information. === Subject: Publication of the Casson Proceedings Publication of the following monograph has started: Edited by Cameron Gordon and Yoav Rieck Papers are being published in reverse alphabetical order of author name. Publication starts today with the publication of the Žrst 100 pages comprising three papers: 1. Pages 1--26 Poincare duality in dimension 3 by CTC Wall 2. Pages 27--68 Seifert Klein bottles for knots with common boundary slopes by Luis G Valdez-Sanchez 3. Pages 69--100 Minimal surfaces in germs of hyperbolic 3-manifolds by Clifford Henry Taubes The following URL is the main page for the monograph and publication can be monitored from there: http://www.maths.warwick.ac.uk/gt/gtmcontents7.html Announcements of details of published papers will be made periodically. --- Details of the three papers published today follow. (1) Proceedings of the Casson Fest Paper no. 1, pages 1--26 URL: http://www.maths.warwick.ac.uk/gt/GTMon7/paper1.abs.html Title: Poincare duality in dimension 3 Author(s): CTC Wall Abstract: The paper gives a review of progress towards extending the Thurston programme to the Poincare duality case. In the Žrst section, we Žx notation and terminology for Poincare complexes X (with fundamental group G) and pairs, and discuss Žniteness conditions. For the case where there is no boundary, pi_2 is non-zero if and only if G has at least 2 ends: here one would expect X to split as a connected sum. In fact, Crisp has shown that either G is a free product, in which case Turaev has shown that X indeed splits, or G is virtually free. However very recently Hillman has constructed a Poincare complex with fundamental group the free product of two dihedral groups of order 6, amalgamated along a subgroup of order 2. In general it is convenient to separate the problem of making the boundary incompressible from that of splitting boundary-incompressible complexes. In the case of manifolds, cutting along a properly embedded disc is equivalent to attaching a handle along its boundary and then splitting along a 2-sphere. Thus if an analogue of the Loop Theorem is known (which at present seems to be the case only if either G is torsion-free or the boundary is already incompressible) we can attach handles to make the boundary incompressible. A very recent result of Bleile extends Turaev¹s arguments to the boundary-incompressible case, and leads to the result that if also G is a free product, X splits as a connected sum. The case of irreducible objects with incompressible boundary can be formulated in purely group theoretic terms; here we can use the recently established JSJ type decompositions. In the case of empty boundary the conclusion in the Poincare duality case is closely analogous to that for manifolds; there seems no reason to expect that the general case will be signiŽcantly different. Finally we discuss geometrising the pieces. Satisfactory results follow from the JSJ theorems except in the atoroidal, acylindrical case, where there are a number of interesting papers but the results are still far from conclusive. The latter two sections are adapted from the Žnal chapter Keywords: Poincare complex, splitting, loop theorem, incompressible, JSJ theorem, geometrisation Author(s) address(es): Department of Mathematical Sciences, University of Liverpool Liverpool, L69 3BX, UK Email: ctcw@liv.ac.uk (2) Proceedings of the Casson Fest Paper no. 2, pages 27--68 URL: http://www.maths.warwick.ac.uk/gt/GTMon7/paper2.abs.html Title: Seifert Klein bottles for knots with common boundary slopes Author(s): Luis G Valdez-Sanchez Abstract: We consider the question of how many essential Seifert Klein bottles with common boundary slope a knot in S^3 can bound, up to ambient isotopy. We prove that any hyperbolic knot in S^3 bounds at most six Seifert Klein bottles with a given boundary slope. The Seifert Klein bottles in a minimal projection of hyperbolic pretzel knots of length 3 are shown to be unique and pi_1-injective, with surgery along their boundary slope producing irreducible toroidal manifolds. The cable knots which bound essential Seifert Klein bottles are classiŽed; their Seifert Klein bottles are shown to be non-pi_1-injective, and unique in the case of torus knots. For satellite knots we show that, in general, there is no upper bound for the number of distinct Seifert Klein bottles a knot can bound. Secondary: 57N10 Keywords: Seifert Klein bottles, knot complements, boundary slope Author(s) address(es): Department of Mathematical Sciences, University of Texas at El Paso El Paso, TX 79968, USA Email: valdez@math.utep.edu (3) Proceedings of the Casson Fest Paper no. 3, pages 69--100 URL: http://www.maths.warwick.ac.uk/gt/GTMon7/paper3.abs.html Title: Minimal surfaces in germs of hyperbolic 3-manifolds Author(s): Clifford Henry Taubes Abstract: archetypal element is a pair that consists of a metric and second fundamental form from a compact, oriented, positive genus minimal surface in some hyperbolic 3-manifold. This moduli space is a smooth, Žnite dimensional manifold with canonical maps to both the cotangent bundle of the Teichmueller space and the space of SO(3,C) representations for the given genus surface. These two maps embed the universal moduli space as a Lagrangian submanifold in the product of the latter two spaces. Secondary: 53D30 Keywords: Hyperbolic 3-manifold, minimal surface Author(s) address(es): Department of Mathematics, Harvard University Cambridge, MA 02138, USA Email: chtaubes@math.harvard.edu === Subject: Proofs of theorems and Symmetries I¹m not a specialist in Logic, formal languages (etc), but I was wondering if the following conjecture makes sense and/or is true. Comments appreciated. Eric Chopin http://perso.wanadoo.fr/eric.chopin Let¹s decompose a proovable theorem T in the following components: 1: hypothesis 2: proof 3: conclusion (result) My conjecture is that, if T is a proovable theorem such that there exists one proof which does not require the axiom of choice, and S a symmetry property such that both the hypothesis and the conclusion that I have added the requirement that the axiom of choise must not be necessary to proove the theorem because I have the feeling that some counter-examples can be found otherwise, though I¹ve not found such an example so far. I will also illustrate this on some examples. Let¹s consider Žrst the Engel¹s Theorem. Engel¹s theorem states that, if V is a Žnite dimensional vector space and E is a Lie algebra made exclusively of nilpotent endomorphisms of V, then there exists a non-vanishing vector in V such that u(x)=0 for any u in E. The proof that is commonly given in almost all textbooks consists in Žrst building a speciŽc basis of V in which all the elements of E are upper triangular. However, both the hypothesis and the conclusion do not depend on a particular basis of V. I was therefore pesuaded that there is a way to proove Engel¹s theorem without specifying any particular basis. Eventually, I found such a proof on the web recently. Other example: in a euclidean vector space, one can Žnd an orthogonal basis. The common proof consists in using an arbitrary basis on which we use the Gram-Schmidt algorithm to build the orthogonal basis. However, a permutation of the vectors of the Žrst basis does not give the same orthogonal basis at the end (up to the same permutation). But there¹s still a way to respect this symmetry when building the orthogonal basis. The method consists in taking the unique square root of the Gram matrix of the initial basis, the columns of which give the orthogonal basis, expressed in the basis of the initial one. Here is an example for which I¹ve not yet the solution: Bezout. If p and q are relatively prime, how to Žnd u and v such that pu+qv=1. The standard method uses the Euclide algorithm, but it explicitely breaks the symmetry between p and q.... === Subject: Re: Proofs of theorems and Symmetries Originator: tchow@maclaurin.mit.edu.mit.edu (Timothy Chow) >My conjecture is that, if T is a proovable theorem such that there >exists one proof which does not require the axiom of choice, and S a >symmetry property such that both the hypothesis and the conclusion are This is an interesting idea, and is probably believed at some level by many mathematicians. But no such principle is known at the level of generality that you¹re seeking. For example, depending on how broadly we interpret your conjecture, it might predict that Frobenius¹s theorem should have a purely group-theoretic proof that doesn¹t use character theory. But no such proof is known. In his book On Numbers and Games, John Conway remarks that he originally deŽned the multiplication of numbers in terms of normal forms (what you might call a basis-dependent deŽnition), but later became convinced that there must exist a genetic (in your terms, symmetric) deŽnition. In this case he found one, but it required ad hoc original thinking. Certain grand generalizations of your conjecture are known to be false. For example, one might hope that theorems statable in terms of elementary number theory should be provable using only elementary means. But if you Google for Paris-Harrington or Goodstein¹s theorem or Harvey Friedman large cardinals then you will learn that this hope is in vain. I think your conjecture is a useful heuristic that may help you create beautiful mathematics. However, it will be very difŽcult to formulate and prove a precise version that covers a signiŽcant swath of mathematics. -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences === Subject: Requirement of primality for size of Žnite Želd- where does it come from? Epigone-thread: thermudwun Requirement of primality for size of Žnite Želd- where does it come from? I can follow the arguments that Žnite Želds which have p^n elements exist, where p prime, 0 <= n in integers, but I can¹t Žnd the other end of the statement- that only Želds with p^n elements exist. I¹m suspecting it has to do with irreducibility of polynomials. Could someone elucidate this for me or point me to a book on this topic? Paul === Subject: Re: Requirement of primality for size of Žnite Želd- where does it come from? > Requirement of primality for size of Žnite Želd- > where does it come from? > I can follow the arguments that Žnite Želds which have p^n elements > exist, where p prime, 0 <= n in integers, but I can¹t Žnd the other > end of the statement- that only Želds with p^n elements exist. > I¹m suspecting it has to do with irreducibility of polynomials. > Could someone elucidate this for me or point me to a book on this > topic? see the Žrst few sentences in Serre¹s Course d¹Arithmetique. Look at the image of the integers in your Želd: n mapsto 1 + 1+...+1 (n times) in your Želd. We must have some p mapping to zero, for Žniteness, and p must be prime, otherwise you would get zero divisors in your Želd. -- Maarten Bergvelt [ Moderator¹s Note: This shows 1 has prime (additive) order p. And in a Želd, multiplication by any nonzero element is a group isomorphism, so all nonzero elements have order p. If the number of elements of the Želd were not a power of p, then it would have another prime q as divisor, and there would then be an element with order q. ] === Subject: Re: [Q] Ronald M. Foster (of Foster Census) Epigone-thread: khoitweldstun I hope you will still get this very belated response to your 10 year old message which I just saw by accident. Prof Ronald M. Foster died in Feb 1998. In was once Prof Foster¹s student, in the Žfties. I went to a small family rememberance. Bede Liu >I had a deafening silence to the following posting on sci.math. >Maybe I will fare better on sci.math.research ... >Having come across the book The Foster Census >containing a listing of connected symmetric trivalent graphs >of orders up to 512, I have the following question: >Is R. M. Foster still alive? >(The biographical sketch states that he was in his 92nd year >in October 1987.) >Paul Hafner >+----------------------------------------------------------- ------+ >| Paul Hafner Œphone +64 9 373-7599 x 5748 | >| Department of Mathematics fax +64 9 373-7457 | >| University of Auckland e-mail hafner@mat.auckland.ac.nz | >| Private Bag 92019 | >| Auckland, New Zealand. time gmt +12 (13) (summer time)| >| latitude S 36^ 51¹ 15.3¹¹ longitude E 174^ 45¹ 46.6¹¹ | === Subject: Name for a polynomial a root of which generates its splitting Želd? Hello - how do you call a polynomial f a root x of which generates the splitting Želd of f (over the Želd of deŽnition of f)? I vaguely remember there is a name for such a thing, but I have been unable to Žnd it. Harald helfgott AT dms DOT umontreal DOT ca === Subject: Lemma 5.23 of Jech¹s AC book I spent some time this summer trying to Žnally work through some details of Jech¹s book, The Axiom of Choice. It was quite educational to try, but I didn¹t succeed and now I think that there might be something wrong with Lemma 5.23. I am a little worried that this might have disastrous implications for the rest of the book, whereas I¹ve always considered it a very nice book, one of my favorites, so I really hope it can be Žxed. More on the sky falling at the end of this posting. Before I say anything else, I should mention that I¹ve already exchanged some email with Jech about these difŽculties and they seem, at least for the moment, to be genuine. It is best if the reader has access to a copy of Jech¹s book, even if it means leaving the computer terminal. There is too much to quote to do it justice. However, I will try. I apologize in advance for any errors of transmission on my part. Let M be a transitive model of ZF+AC, N the symmetric extension of M deŽned in Section 5.3, which I¹ll describe briežy. The set P of Žnite partial functions from w x w (where w denotes omega, the set 0,1,2,... of natural numbers) to {0,1} is partially ordered by containment, i.e. an extension of a partial function to a larger Žnite set is considered to be smaller than the function it extends. Give P the order topology and and B, one constructs the class of B-valued sets M^B; B-valued sets are also names. The group of permutations of w acts on P by acting on the Žrst coordinate of w x w. It preserves order and therefore acts on B and ultimately on M^B. A name x is called symmetric if there is a Žnite subset e of w such that every permutation of w Žxing e pointwise also Žxes x. In that case, we say e is a support for the name x. A name x is said to be hereditarily symmetric if either x is the empty function or if x is symmetric and every element of x is hereditarily symmetric. One denotes by HS the class of hereditarily symmetric names. One gets the actual forcing extension M[G] associated to M^B by pretending there is an M-generic Žlter G and modding out by it in an inductive manner. I¹ll refer to elements of M[G] as sets. The image of HS in M[G] is what was meant by N above. N is a model of the Zermelo-Frankel axioms ZF but not of AC. A Žnite subset e of w is called a support of a set x of N if it supports some name for x. The set x of N actually has a lot of names in HS and Lemma 5.22 of Jech¹s book asserts that the set of supports for these various names has a unique minimal element, i.e. there is a Žnite set e of w such that e is a support of every name for x. Lemma 5.23 is used to prove Lemma 5.22. To explain Lemma 5.23, it is necessary to mention that not only do sets in M[G] have names, but also the formulas phi expressing relations among sets in M[G] can be lifted to formulas Phi involving names in M^B, and that each such lift can be assigned a truth value [Phi] which is an element of the Boolean algebra B. The original formula phi involving sets of M[G] holds in M[G] iff [Phi] belongs to the Žlter G. We identify P with a subset of B and we say that a condition p of P forces the formula Phi if p.[Phi]=p, where . denotes the product in B. Lemma 5.23 says that if x,y are names in HS and if p is a condition that forces x=y and if e1 is a support of x and e2 is a support of y, then there exists a hereditarily symmetric name z such that the intersection of e1 and e2 is a support of z and p forces x=z. To prove Lemma 5.23, Jech Žrst observes (Lemma 5.24) that if Phi has n free variables and e is a support for names x1,...,xn in HS and if p forces Phi(x1,...,xn), then p:e also forces Phi(x1,...,xn), where p:e is the restriction of p to e x w. Next, he observes that every name u in HS has a least support, which he denotes s(u). If p is a forcing condition, he denotes by s(p) the set of all n in w such that there is an m in w for which (n,m) is in the domain of p. Returning to the notation of the statement of Lemma 5.23, we may as well take e1=s(x), e2=s(y). Jech then makes the simplifying assumption that e1 and e2 are disjoint, promising to sketch at the end of the proof the modiŽcations needed to handle the more general case, which he does. With this assumption, he needs to produce a name z in HS with empty support such that p forces x=z. At this point, he states formula (5.26), p.74, which says that if Phi denotes the formula w is an element of x, then p.[Phi] equals p.([Phi]:s(w)), where the operation q |-> q:s(w) for conditions q is extended to all of B by taking sups. To prove (5.26), Jech lets e=s(w) and claims that it sufŽces to prove, for every condition q less than or equal to p, the following two assertions (displayed as (5.27)): (i) If q forces Phi then the union of p and q:e also forces Phi (ii) Ditto for the negation of Phi where Phi is still as above. I was able to verify Jech¹s proof of (5.27), but I am unable to verify the assertion that this sufŽces to prove equation (5.26). Here is the basic difŽculty: (1) The left side of (5.26) is clearly bounded above by the right side, so we have to prove the reverse inequality. (2) The right side is the supremum of all p.(r:e) where r runs over all conditions which force Phi. (3) Basically, we want to prove that such a p.(r:e) is bounded above by the Boolean truth value of Phi. (4) The Žrst line of (5.27) does tell us that p.(r:e) is bounded above by the Boolean truth value of Phi in case r forces Phi and is bounded by p. (5) Unfortunately, it is at least conceivable that there could be an r forcing Phi such that r is incompatible with p but r:e is compatible with p. In that case, the term p.(r:e) in the supremum deŽned by the right side of (5.26) would not obviously be bounded by the Boolean truth value of Phi. (6) In that case, I don¹t see how to complete the proof of (5.26). (7) There are admittedly two lines in (5.27), and I tried to put the second one to work also, using the deŽnition of forcing a negation of a formula, but I didn¹t see how to get that to work either. So, that is the problem. Now for my concerns about the sky falling in. Lemma 5.23 is used to prove Lemma 5.22, which asserts that the basic Cohen model is a support model, and that in turn is used to prove that the Axiom of Choice (AC) is independent of the Ordering Principle (Thm. 5.21). That result isn¹t in doubt, just the method adopted for proving it in Jech¹s book. However, that method is of independent interest. In Ch.6, the proof of the Support Theorem (Thm.6.6), which greatly generalizes Lemma 5.22, is modeled on the proof of Lemma 5.23 and is also in jeopardy; I don¹t know how much of Ch.6 survives if Lemma 5.23 can¹t be salvaged. Chapters 7 and 8 seem to be intended to illustrate the utility of the transfer principles derived from the results of Ch.6. I¹ve always found the approach to be quite beautiful and it would be a real shame if it turned out to be wrong. If someone knows how to Žx the proof of Lemma 5.23 and of Theorem 6.6 and all that depend on them, or knows how to prove that they can¹t be Žxed, please let me know. -- Ignorantly, Allan Adler * Disclaimer: I am a guest and *not* a member of the MIT CSAIL. My actions and * comments do not režect in any way on MIT. Also, I am nowhere near Boston. === Subject: direct sums and products from category theory Hello again, I¹ve been trying to organize the various direct sums and products by relating them to products and coproducts in category theory. Any help would be much appreciated. Note: I write the direct sum symbol as O+ and the tensor product notation as Ox; rings are all assumed commutative with unity Products are simpler, since in the categories I¹m considering they are just Cartesian products with operations applied component-wise. In terms of terminology and notation, we have: Category Product Sets Cartesian product, x Groups External direct product, x (for subgroups) Internal direct product, O+ Abelian groups Direct sum, O+ Rings Direct sum, O+ Vector spaces Direct sum, O+ In terms of coproducts, here¹s what I have: Category Coproduct Sets Disjoint union, U_d Groups Free product, * (for subgroups) Not used (?) Abelian groups Direct sum, O+ Rings Tensor product, Ox Vector spaces Direct sum, O+ The notation for groups seems especially confusing. Here is my understanding: - For a Žnite collection of abelian groups, the product and coproduct are identical, which explains why the product of abelian groups is called the direct sum - For an inŽnite collection of abelian groups, the coproduct differs from the product in that elements only contain a Žnite number of non-identity elements - This concept seems to have been grafted over to non-abelian groups, where the direct sum of an inŽnite collection of groups similarly signiŽes that elements only contain a Žnite number of non-identity elements - Adding to the confusion, the symbol for this direct sum of groups is O+, same as for the internal direct product Q: Is this last item related to any relationship between the two types of products? Q: For rings, I understand the generalized tensor product to be deŽned as the normal tensor product of abelian groups, with ring multiplication deŽned component-wise, i.e. (r Ox s).(r¹ Ox s¹) = (r.r¹) Ox (s.s¹). Is this correct? Does this generalized tensor product have a special name of any kind? Q: I¹ve seen the categorical coproduct called the direct sum; is this standard, given the generally inconsistent use of this term in practice? === Subject: Re: direct sums and products from category theory Something went wrong with the encoding, I¹m going to try again so it is readable. Also, another question: what is the coproduct in the category of algebras? Hello again, I¹ve been trying to organize the various direct sums and products by relating them to products and coproducts in category theory. Any help would be much appreciated. Note: I write the cartesian product symbol as x, the direct sum symbol as 0+, and the tensor product symbol as Ox; all rings are assumed commutative with unity. Products are simpler, since in the categories I¹m considering they are just Cartesian products with operations applied component-wise. In terms of terminology and notation, we have: Category Product Sets Cartesian product, x Groups External direct product, x (for subgroups) Internal direct product, O+ Abelian groups Direct sum, O+ Rings Direct sum, O+ Vector spaces Direct sum, O+ In terms of coproducts, here¹s what I have: Category Coproduct Sets Disjoint union, U_d Groups Free product, * (for subgroups) Not used (?) Abelian groups Direct sum, O+ Rings Tensor product, Ox Vector spaces Direct sum, O+ The notation for groups seems especially confusing. Here is my understanding: - For a Žnite collection of abelian groups, the product and coproduct are identical, which explains why the product of abelian groups is called the direct sum - For an inŽnite collection of abelian groups, the coproduct differs from the product in that elements only contain a Žnite number of non-identity elements - This concept seems to have been grafted over to non-abelian groups, where the direct sum of an inŽnite collection of groups similarly signiŽes that elements only contain a Žnite number of non-identity elements - Adding to the confusion, the symbol for this direct sum of groups is O+, same as for the internal direct product Q: Is this last item related to any relationship between the two types of products? Q: For rings, I understand the generalized tensor product to be deŽned as the normal tensor product of abelian groups, with ring multiplication deŽned component-wise, i.e. (r Ox s).(r¹ Ox s¹) = (r.r¹) Ox (s.s¹). Is this correct? Does this generalized tensor product have a special name of any kind? Q: I¹ve seen the categorical coproduct called the direct sum; is this standard, given the generally inconsistent use of this term in practice? === Subject: Re: direct sums and products from category theory days. My association with the Department is that of an alumnus. >Something went wrong with the encoding, I¹m going to try again so it is >readable. >Also, another question: what is the coproduct in the category of >algebras? If you mean k-algebras (rings containing k as a subring of their center), then it is the tensor product over k, unless I am much mistaken. If you mean algebras in the General Algebra sense, then of course the answer is the coproduct, which is highly uninformative, so I suspect you do not mean that... >Products are simpler, since in the categories I¹m considering they are >just Cartesian products with operations applied component-wise. In general, if your category has a left adjoint to the underlying set functor, then the underlying sets of the products will be the category-of-sets-products of the underlying sets. Since in the category of sets the categorical product is the cartesian product, you get that. This is the case for abelian groups (the free abelian group), groups (the free group), rings (the polynomial ring over the integers), and many other categories. In fact, any category of algebras (in the universal algebra sense) has the notion of free object, which is none other than the image of the left adjoint to the underlying set functor. If your category has a right adjoint to the underlying set functor, then the underlying set of a coproduct is the coproduct of the underlying sets. For example, in topological spaces, the underlying set functor has both a left adjoint (the discrete topology) and a right adjoing (the indiscrete topology). So, in topological spaces, the coproduct is the disjoint union, whose underlying set is the categorical coproduct of the underlying sets of the spaces. More generally, left adjoints respect limits, and right adjoints respect colimits; products are a kind of limits and coproducts a kind of colimits (perversely, though, inverse limits are limits and direct limits are colimits... which explains why inverse limits tend to be easier to understand in categories of algebras). > In terms >of terminology and notation, we have: >Category Product > Sets Cartesian product, x >Groups External direct product, x > (for subgroups) Internal direct product, O+ >Abelian groups Direct sum, O+ This is incorrect. The categorical product for abelian groups is the direct product, not the direct sum. Remember that given a family of objects {O_i} i in I, the product of the O_i is an object P, together with projection maps p_i: P->O_i, such that for every object Q and maps f_i:Q-> O_i, there exists a unique map F:Q->P such that p_i f = f_i. The coproduct of the family is an object C together with inclusion maps i_j: O_j -> C, such that for every object D and maps g_j: O_j->D, there exists a unique map g:C->D such that g i_j = g_j. So the direct sum is the coproduct of abelian groups, but not the product. It just happens that when the family is Žnite, they coincide, so some abuse of notation exists. >Rings Direct sum, O+ Again, the categorical product of rings is the direct product, not the direct sum; they coincide when the family is Žnite. >Vector spaces Direct sum, O+ Categorically, you still need the direct product, not the direct sum. >In terms of coproducts, here¹s what I have: >Category Coproduct >Sets Disjoint union, U_d >Groups Free product, * > (for subgroups) Not used (?) The distinction between internal and external is not normally used, no. >Abelian groups Direct sum, O+ >Rings Tensor product, Ox >Vector spaces Direct sum, O+ Correct. >The notation for groups seems especially confusing. It is a consequence of abuse of notation, since the Žnite direct product coincides with the direct sum. > Here is my understanding: > - For a Žnite collection of abelian groups, the product and coproduct >are identical, Only for Žnite families. If you are talking about binary product and coproduct, you are correct. > which explains why the product of abelian groups is >called the direct sum > - For an inŽnite collection of abelian groups, the coproduct differs >from the product in that elements only contain a Žnite number of >non-identity elements > - This concept seems to have been grafted over to non-abelian groups, >where the direct sum of an inŽnite collection of groups similarly >signiŽes that elements only contain a Žnite number of non-identity >elements Not so much grafted. Under some circumstances, there are natural maps from the objects {O_i} into their product. This is the case, for example, in groups, where there is a natural injection mapping O_i to the elements which have trivial j-coordinates, when j is different from i (no such Œnatural map¹ exists in the category of sets, for example). In that case, since you have a family of maps from the O_i to P, the universal property of the coproduct gives you a unique map C->P. The direct sum is the image of this map. > - Adding to the confusion, the symbol for this direct sum of groups is >O+, same as for the internal direct product (Aside: I¹ve personally never had much use for the internal direct product as a distinct entity). >Q: Is this last item related to any relationship between the two types >of products? I¹m not sure what this means. >Q: For rings, I understand the generalized tensor product to be >deŽned >as the normal tensor product of abelian groups, with ring multiplication >deŽned component-wise, i.e. > (r Ox s).(r¹ Ox s¹) = (r.r¹) Ox (s.s¹). >Is this correct? Does this generalized tensor product have a special >name >of any kind? I¹ve always known it as the tensor product. >Q: I¹ve seen the categorical coproduct called the direct sum; is this >standard, given the generally inconsistent use of this term in practice? According to Mac Lane¹s _Categories for the Working Mathematician_, direct sum is reserved for special categories that have a biproduct. These are Ab-categories (a category in which each hom-set is an additive abelian group, and composition of arrows is bilinear relative to this addition); a biproduct of the objects A and B is an object C together with arrows p_1:C->A, p_2:C->B, i_i:A->C, i_2:B->C, wuch that p_1i_1 = Id_A, p_2i_2 = Id_B, i_1p_1 + i_2p_2 = Id_C. In these circumstances, there is no inconsistency, since the term is only used in such categories (note that the direct sum of abelian groups, rings, and vector spaces has this property; it is a bit of a misuse for groups, since Hom(G,H) is not an abelian group for arbitrary groups G and H, but the properties of the product do satisfy the given identities, since the two components commute). -- It¹s not denial. I¹m just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: direct sums and products from category theory days. My association with the Department is that of an alumnus. Sigh... I always get these two confused. My most sincere apologies! >More generally, left adjoints respect limits, and right adjoints >respect colimits; Exactly the other way around: left adjoints respect colimits; right adjoints respect limits. Since underlying set is the right adjoint of the underlying set-free object pair, that means that the underlying set of the limit is the limit of the underlying sets. E.g., the underlying set of the product is the product of the underlying sets. Free object is the left adjoint, which is why the free object on a disjoint union of sets is the coproduct of the free objects on the sets; e.g., the free group on X (disjoint union) Y is the free product of F(X) and F(Y). -- It¹s not denial. I¹m just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Hopf ideals in Hopf algebras I¹m wondering if anybody has a clue to the following: Let G be a group, and kG is its group ring, which is a Hopf algebra. Let I be a *Hopf* ideal in kG. Then there exists N normal in G with I = < n-1 : n in N > I can prove it if G is Žnite; indeed then (kG)^* is a Hopf algebra; it is an algebra of functions on G. set B = { tau in (kG)^* : tau(I) = 0 }. Then, since B is a subalgebra of (kG)^*, it is the algebra of functions on an equivalence relation over G. Since B is a sub-coalgebra, this equivalence relation is the quotient by a normal subgroup. Does anybody know what happens for inŽnite groups? Is there a good reference for such things? TIA, laurent -- Laurent Bartholdi laurent.bartholdiepžch EPFL, IGAT, B.89timent BCH T.8el.8ephone: +41 21-6930380 CH-1015 Lausanne, Switzerland Fax: +41 21-6930385 === Subject: expand the right-hand side on the basis of the eigenvectors of the matrix Hello! I have a problem: Ax=b, where A = [ 1001 1000; 1000 1001] x = [x1 ; x2] b = [ 2001; 2001] a small change of b db = [1; -1] Could somebody explain why the small variation in the right hand side produces large variations in the solution? Here is a hint: expand the right-hand side on the basis of the eigenvectors of the matrix. SO. I found eigenvectos: x01 = [1; x02 = [1; 1] -1] It forms an orthogonal basis, because the matrix A is symmetric. Then, when we expand the right hand side in the basis of the eigenvectors, then we have: b = c1 * x01 + c2 * x02, Hence c1 = 2001 and c2 = 0 But, when we use (b + db) = c1 * x01 + c2 * x02, then c1 = 2001.5 and c2 = 0.5 So, in the Žrst case c2 = 0 and in the second c2 = 0.5 . I believe that the answer to the question: why the small variation in the right hand side produces large variations in the solution? should be somewhere here... But what is a name of a theorem or a method which will help me to anser such question. Sinserely, Taglit