mm- Subject: Re: 2 questions on Þnitely generated solvable groups Epigone-thread: colsmymeu >YdC: >Clearly, it is NOT 2-transitive (it preserves the distance...). More >generally, an abelian group cannot act 2-transitively on an inÞnite set. > ... >YdC: >And then? This does not prove that this group is not a quotient of a >Þnitely presented solvable group (with more generators or higher >solubility length)... You are right, I goofed in 1 and was too hasty in 2. 1. There is some description of all solvable 2-transitive groups given in the book: Suprunenko D.A., Groups of Substitutions, Minsk, Navuka i Tekhnika, 1996. (Russian) First, any 2-transitive group G must be primitive. (Since any two elements from a nontrivial block can be put to different blocks by the action.) The author proves that a primitive permutaion group G acting on a set X is similar (i.e. isomorphic as a transformation group) to a group of the type: G = A.T acting on the set X = V, where V is a vector space over a prime Þeld F (GF(p) or Q) T is the full group of translations over vectors of V A is a solvable subgroup of GL(V) acting transitively on V{0}. A.T means the semidirect product of A and T So G is a subgroup of Aff(V). Since our group in question is inÞnite, the Þeld F is either Q (the rationals), or F=GF(p) and dim V = inÞnity. All that remains is to show that such groups are never Þnitely generated. If you are interested, I can provide you with the details of the proof Hope this helps, Ignat Soroko Minsk, Belarus === Subject: curvature invariants Let (M,g) be a (pseudo-)Riemannian manifold, and Lambda^2 the space of antisymmetric tensors of second rank on M. On Lambda^2 there is a metric G induced by G(X,Y,A,B) = g(X,A) g(Y,B) - g(X,B) g(Y,A) as G shares the Riemann-Christoffel symmetries G(X,Y,A,B) = -G(Y,X,A,B) = -G(X,Y,B,A) G(X,Y,A,B) = G(A,B,X,Y) The geometrical signiÞcance of G is that G(X,Y,X,Y) measures the area squared of the parallelogram deÞned by X and Y. Knowing only G, one apparently cannot reconstruct the metric g. Question: Are there curvature invariants of (M,g) that can be written in terms of only G and its Þrst and second derivatives, i.e. without using the metric g? === Subject: This week in the mathematics arXiv (11 Oct - 15 Oct) 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 modiÞcation. Titles in the mathematics arXiv (11 Oct - 15 Oct) ------------------------------------------------- AC: Commutative Algebra ----------------------- math.AC/0410304 Emanoil Theodorescu: Bivariate Hilbert Functions for the Torsion Functor math.AC/0410303 Emanoil Theodorescu: Derived Functors and Hilbert Polynomials math.AC/0410265 Marcel Morales, Apostolos Thoma: Complete Intersection Lattice Ideals math.AC/0410264 Anargyros Katsabekis: Projections of cones and the arithmetical rank of toric varieties math.AC/0410257 David Jorgensen, Liana Sega: Independence of the total reþexivity conditions for modules math.AC/0410253 Mitsuhiro Miyazaki: On the discrete counterparts of Cohen-Macaulay algebras with straightening laws math.AC/0410220 Rouchdi Bahloul: Generic and comprehensive standard bases AG: Algebraic Geometry ---------------------- math.AG/0410327 Victor Przyjalkowski: Gromov-Witten invariants of Fano threefolds of genera 6 and 8 math.AG/0410323 Brian Osserman: Mochizuki¹s crys-stable bundles: a lexicon and applications math.AG/0410313 Jean Michel: Hurwitz action on tuples of Euclidean reþections math.AG/0410309 Euisung Park: Noncomplete embeddings of rational surfaces math.AG/0410306 Tomohide Terasoma: Rational convex cones and cyclotomic multiple zeta values math.AG/0410295 Michael Lonne: On bifurcation braid monodromy of elliptic Þbrations math.AG/0410283 Alexander Polishchuk: Holomorphic bundles on 2-dimensional noncommutative toric orbifolds math.AG/0410281 Samuel Boissiere: On the McKay correspondences for the Hilbert scheme of points on the afÞne plane math.AG/0410269 Frederic Paugam: Quelques bords irrationnels de varietes de Shimura math.AG/0410268 Dominic Joyce: ConÞgurations in abelian categories. IV. Changing stability conditions math.AG/0410267 Dominic Joyce: ConÞgurations in abelian categories. III. Stability conditions and invariants math.AG/0410262 Martin Moeller: Periodic points on Veech surfaces and the Mordell-Weil group over a Teichmueller curve math.AG/0410259 Kenichiro Kimura: A rational map between some threefolds math.AG/0410258 David B. Massey: L^e Modules and Traces math.AG/0410255 Kai Behrend: On the de Rham Cohomology of Differential and Algebraic Stacks math.AG/0410254 Frederic Paugam: Three examples of noncommutative boundaries of Shimura varieties math.AG/0410252 Ivan Cheltsov: Factorial nodal threefolds in $mathbb{P}^{5}$ math.AG/0410240 Michel Brion: Lectures on the geometry of þag varieties math.AG/0410224 Carlos T. Simpson: Formalized proof, computation, and the construction problem in algebraic geometry math.AG/0410223 R. Cluckers, F. Loeser: Ax-Kochen-Er{v{s}}ov Theorems for $p$-adic integrals and motivic integration math.AG/0410221 S.Subramanian: Notes on abelian class Þeld theory AP: Analysis of PDEs -------------------- math.AP/0410330 Adrien Blanchet, Jean Dolbeault, Regis Monneau: On the one-dimensional parabolic obstacle problem with variable coefÞcients math.AP/0410287 Li Ma, DeZhong Chen: Radial Symmetry and Monotonicity Results for an Integral Equation math.AP/0410279 W. J. Golz: On the Convection-Dispersion Equation for a Finite Domain: Third-Type Boundaries as a Necessary Condition of the Conservation Law math.AP/0410229 Alexander Vasil¹ev: Evolution dynamics of conformal maps with quasiconformal extensions CA: Classical Analysis and ODEs ------------------------------- math.CA/0410320 A. Martinez-Finkelshtein, R. Orive: Riemann-Hilbert analysis for Jacobi polynomials orthogonal on a single contour math.CA/0410284 Vivina Barutello, Susanna Terracini: A bisection algorithm for the numerical Mountain Pass math.CA/0410250 George Gasper, Mizan Rahman: Some Systems of Multivariable Orthogonal q-Racah polynomials math.CA/0410249 George Gasper, Mizan Rahman: Some Systems of Multivariable Orthogonal Askey-Wilson Polynomials math.CA/0410248 George Gasper, Mizan Rahman: q-Analogues of Some Multivariable Biorthogonal Polynomials math.CA/0410228 Stephen Semmes: Potpourri, 5 CO: Combinatorics ----------------- math.CO/0410335 Sonja Lj. Cukic, Dmitry N. Kozlov: Higher connectivity of graph coloring complexes math.CO/0410334 Raymond Hemmecke: Exploiting Symmetries in the Computation of Graver Bases math.CO/0410319 L. Friess: Die Anzahl der Faerbungen ebener Graphen math.CO/0410308 Elena Fuchs: Longest Induced Cycles on Cayley Graphs math.CO/0410301 Peter McNamara: Cylindric skew Schur functions math.CO/0410289 Raymond Hemmecke: Computation of Atomic Fibers of Z-Linear Maps math.CO/0410222 William Y.C. Chen, Qing-Hu Hou, Yan-Ping Mu: Applicability of the $q$-Analogue of Zeilberger¹s Algorithm math.CO/0410218 B. Bollobas, V. Nikiforov: The sum of degrees in cliques math.CO/0410217 B. Bollobas, V. Nikiforov: Joints in graphs math.CO/0410216 V. Nikiforov: The smallest eigenvalue of K_p-free graphs cs.DM/0410013 Alex Vinokur: Fibonacci connection between Huffman codes and Wythoff array CT: Category Theory ------------------- math.CT/0410328 Toby Bartels: CategoriÞed gauge theory: Two-bundles math.CT/0410230 Mark Weber: Operads within monoidal pseudo algebras CV: Complex Variables --------------------- math.CV/0410296 Pengfei Guan: Extremal function of intrinsic norms math.CV/0410294 Andrew McIntyre, Leon A. Takhtajan: Holomorphic factorization of determinants of laplacians on Riemann surfaces and a higher genus generalization of Kronecker¹s Þrst limit formula math.CV/0410274 Y.Peterzil, S.Starchenko: Subanalytic sets and complex analytic geometry DG: Differential Geometry ------------------------- math.DG/0410332 D. Auroux, S. K. Donaldson, L. Katzarkov: Singular Lefschetz pencils math.DG/0410314 Wayne Rossman: InÞnite Periodic Discrete Minimal Surfaces Without Self-Intersections math.DG/0410312 Mikhail G. Katz, Stephane Sabourau: Entropy of systolically extremal surfaces and asymptotic bounds math.DG/0410260 Yat-Ming Chan: Desingularizations of Calabi-Yau 3-folds with a conical singularity math.DG/0410239 Toshiki Mabuchi: An energy-theoretic approach to the Hitchin-Kobayashi correspondence for manifolds, II math.DG/0410232 Gabriela P. Ovando: Invariant pseudo Kaehler metrics in dimension four math.DG/0410215 D. Kotschick: Entropies, volumes, and Einstein metrics DS: Dynamical Systems --------------------- math.DS/0410316 John Franks: Erratum to Generalizations of the Poincar¹e-Birkhoff Theorem math.DS/0410310 A. J. Roberts, I. G. Kevrekidis: Higher order accuracy in the gap-tooth scheme for large-scale solutions using microscopic simulators math.DS/0410299 Minoru Ogawa: A sufÞcient condition for pseudointegrable systems with weak mixing property math.DS/0410237 M.F. Kondratieva, S.Yu. Sadov: Two-system of a Hamiltonian system math.DS/0410231 Stefano Luzzatto, Ian Melbourne, Frederic Paccaut: The Lorenz attractor is mixing FA: Functional Analysis ----------------------- math.FA/0410286 D. Q. Cao, Dongsheng Liu, Charles H.-T. Wang: Three Dimensional Nonlinear Dynamics of Slender Structures: Cosserat Rod Element Approach math.FA/0410256 Jesus M. F. Castillo, Yolanda Moreno: Extensions by spaces of continuous functions GM: General Mathematics ----------------------- cs.GT/0410018 Petra Berenbrink, Leslie Ann Goldberg, Paul Goldberg, Russell Martin: Utilitarian resource assignment math.GM/0410241 Sebastian Martin Ruiz: A congruence with the Euler totient function math.GM/0410234 Sergey Sadov: On a necessary and sufÞcient cyclicity condition for a quadrilateral GN: General Topology -------------------- math.GT/0410329 Louis H. Kauffman: Knot Diagrammatics GT: Geometric Topology ---------------------- math.GT/0410326 Clifford Henry Taubes: Minimal surfaces in germs of hyperbolic 3--manifolds math.GT/0410321 J. O. Button: Fibred and Virtually Fibred hyperbolic 3-manifolds in the censuses math.GT/0410300 Peter Ozsvath, Zoltan Szabo: Knot Floer homology and integer surgeries math.GT/0410288 F. Deloup, D. Garber, S. Kaplan, M. Teicher: Palindromic Braids math.GT/0410278 Martin Scharlemann: Proximity in the curve complex: boundary reduction and bicompressible surfaces math.GT/0410275 Florian Deloup: Involutive braids math.GT/0410272 Julien Marche: Surgery on a single clasper and the 2-loop part of the Kontsevich integral math.GT/0410233 Jessica S. Purcell: Cusp Shapes of Hyperbolic Link Complements and Dehn Filling KT: K-Theory and Homology ------------------------- math.KT/0410315 Denis Perrot: The equivariant index theorem in entire cyclic cohomology math.KT/0410261 Moritz C. Kerz: The complex of words and Nakaoka stability MG: Metric Geometry ------------------- math.MG/0410324 Oleg R. Musin: The kissing problem in three dimensions math.MG/0410251 D.Siersma, M. van Manen: The nine Morse generic tetrahedra MP: Mathematical Physics ------------------------ math-ph/0410038 Alexander Elgart, Laszlo Erdos, Benjamin Schlein, Horng-Tzer Yau: Gross-Pitaevskii Equation as the Mean Field Limit of Weakly Coupled Bosons math-ph/0410037 Joseph V. Pule, Andre F. Verbeure, Valentin A. Zagrebnov: A Dicke Type Model for Equilibrium BEC Superradiance math-ph/0410027 Naqing Xie: A Generalized Positive Energy Theorem for Spaces with Asymptotic SUSY CompactiÞcation cond-mat/0410320 Luis Morales-Molina, Franz G. Mertens, Angel Sanchez: Ratchet behavior in nonlinear Klein-Gordon systems with point-like inhomogeneities nlin.SI/0410016 Fabio Musso, Matteo Petrera, Orlando Ragnisco: Algebraic extensions of Gaudin models math-ph/0410036 Hellmut Baumgaertel: On Lax-Phillips semigroups math-ph/0410035 Richard L. Hall, Qutaibeh D. Katatbeh, Nasser Saad: A basis for variational calculations in d dimensions cond-mat/0410192 Kazumitsu Sakai, Andreas Klumper: Non-dissipative Thermal Transport and Magnetothermal Effect for the Spin-1/2 Heisenberg Chain quant-ph/0410079 D. B. Cervantes, S. L. Quiroga, L. J. Perissinotti, M. Socolovsky: Improper transformations of non relativistic spin 1/2 spinors math-ph/0410034 R. Aldrovandi, A. L. Barbosa: Wu-Yang ambiguity in connection space math-ph/0410033 Marcel Griesemer: Non-relativistic Matter and Quantized Radiation hep-th/0408079 Davide Fioravanti: Geometrical Loci and CFTs via the Virasoro Symmetry of the mKdV-SG hierarchy: an excursus quant-ph/0410050 Gerardo Adesso, Fabrizio Illuminati: Multipartite Entanglement and its Polygamy in Continuous Variable Systems nlin.SI/0409050 P.M. Santini, M. Nieszporski, A. Doliwa: An integrable generalization of the Toda law to the square lattice math-ph/0410032 T. Spencer, M.R. Zirnbauer: Spontaneous symmetry breaking of a hyperbolic sigma model in three dimensions math-ph/0410031 A. Wereszczynski: Nested Multi-Soliton Solutions with Arbitrary Hopf Index math-ph/0410030 Guido Gentile, Daniel A. Cortez, Joao C. A. Barata: Stability for quasi-periodically perturbed Hill¹s equations math-ph/0410029 Roland Friedrich: On Connections of Conformal Field Theory and Stochastic L{oe}wner Evolution math-ph/0410028 Mark Naber: Time fractional Schrodinger equation math-ph/0410026 Jining Gao: The Maurer-Cartan structure of BRST differential math-ph/0410025 Hayriye Tutunculer, Ramazan Koc: Differential Realizations of the Two-Mode Bosonic and Fermionic Hamiltonians: A uniÞed Approach hep-th/0410083 Antonino Flachi, Alan Knapman, Wade Naylor, Misao Sasaki: Zeta Functions in Brane World Cosmology cond-mat/0410159 L. Diago-Cisneros, H. Rodriguez-Coppola, R. Perez-Alvarez, P. Pereyra: Symmetries and General Principies in the Multiband Effective Mass Theory: A Transfer Matrix Study math-ph/0410024 Xu-Dong Luo, Han-Ying Guo, Yu-Qi Li, Ke Wu: Difference Discrete Variational Principle in Discrete Mechanics and Symplectic Algorithm NT: Number Theory ----------------- math.NT/0410333 Lev A. Borisov: Holomorphic Eisenstein series with Jacobian twists math.NT/0410297 Bernd C. Kellner: A conjecture about numerators of Bernoulli numbers related to Integer Sequence A092291 math.NT/0410292 Alexander Schmidt: Tame class Þeld theory for arithmetic schemes math.NT/0410270 T. W. Hilberdink, M. L. Lapidus: Beurling Zeta Functions, Generalised Primes, and Fractal Membranes math.NT/0410266 John Voight: Binary quadratic forms that represent almost the same primes math.NT/0410246 Yuri F. Bilu, Florian Luca: Divisibility of class numbers: enumerative approach math.NT/0410245 A.C. de la Maza: Classes of forms Witt equivalent to a second trace form over Þelds of characteristic two math.NT/0410244 A.C. de la Maza: Generalization of the second trace form of central simple algebras in characteristic two OA: Operator Algebras --------------------- math.OA/0410337 Matthew Neal, Bernard Russo: Representation of contractively complemented Hilbertian operator spaces and the Fock space math.OA/0410305 Marcelo Laca, Machiel van Frankenhuijsen: Phase transitions on Hecke C*-algebras and class-Þeld theory over Q math.OA/0410290 Benton L. Duncan: Explicit construction and uniqueness for universal operator algebras of directed graphs math.OA/0410235 Marius Junge: Embedding of OH and logarithmic `little Grothendieck inequality¹ math.OA/0410219 Ilwoo Cho: Random Variables in Graph W*-Probability Spaces OC: Optimization and Control ---------------------------- math.OC/0410277 Michael Malisoff, Mikhail Krichman, Eduardo Sontag: Global Stabilization for Systems Evolving on Manifolds math.OC/0410225 Raymond Hemmecke, Robert Weismantel: Integral Function Bases PR: Probability --------------- math.PR/0410336 Bela Bollobas, Oliver Riordan: The critical probability for random Voronoi percolation in the plane is 1/2 math.PR/0410331 Martin Dyer, Leslie Ann Goldberg, Mark Jerrum, Russell Martin: Markov chain comparison math.PR/0410318 B. Chauvin, A. Rouault: Connecting Yule Process, Bisection and Binary Search Tree via Martingales math.PR/0410311 Geoffrey Grimmett, Svante Janson: Branching Processes, and Random-Cluster Measures on Trees math.PR/0410298 Jianjun Tian, Xiao-Song Lin: Continuous Time Markov Processes on Graphs math.PR/0410285 Huyen Pham: On the smooth-Þt property for one-dimensional optimal switching problem math.PR/0410282 Itai Benjamini, Oded Schramm, David B. Wilson: Balanced Boolean functions that can be evaluated so that every input bit is unlikely to be read math.PR/0410276 A. Ruzmaikina, M. Aizenman: Characterization of invariant measures at the math.PR/0410236 Davar Khoshnevisan, David A. Levin, Pedro J. Mendez-Hernandez: Capacities in Wiener Space, Quasi-Sure Lower Functions, and Kolmogorov¹s Epsilon-Entropy math.PR/0410227 R. Ferriere, A. Guionnet, I. Kurkova: Timescales of population rarity and commonness in random environments QA: Quantum Algebra ------------------- math.QA/0410322 Gaetano Fiore: New approach to Hermitian q-differential operators on R_q^N math.QA/0410291 Hiroshige Kajiura, Jim Stasheff: Homotopy algebras inspired by classical open-closed string Þeld theory hep-th/0410084 Yas-Hiro Quano: Form factors, correlation functions and vertex operators in the eight-vertex model at reþectionless points math.QA/0410263 Julien Bichon, Giovanna Carnovale: Lazy cohomology: an analogue of the Schur multiplier for arbitrary Hopf algebras math.QA/0410247 Jining Gao: $L_{infty}$ algebra structures of Lie algebra deformations math.QA/0410238 Marta M. Asaeda, Jozef H. Przytycki, Adam S. Sikora: A categoriÞcation of the skein module of tangles hep-th/0410086 N.A. Gromov, V.V. Kuratov: Quantum kinematics RA: Rings and Algebras ---------------------- math.RA/0410317 Heide Gluesing-Luerssen, Wiland Schmale: On doubly-cyclic convolutional codes math.RA/0410226 Bilbo Baggins, Laurent Bartholdi: Branch Rings, Thinned Rings, Tree Enveloping Rings RT: Representation Theory ------------------------- math.RT/0410339 Volodymyr Mazorchuk, Catharina Stroppel: On functors associated to a simple root math.RT/0410325 K. Baur: A normal form for admissible characters in the sense of Lynch math.RT/0410302 Toshihiko Matsuki: Equivalence of domains arising from duality of orbits on þag manifolds III math.RT/0410293 I. Gordon, J.T.Stafford: Rational Cherednik algebras and Hilbert schemes II: representations and sheaves math.RT/0410273 Franc{c}ois Rodier: Errata `a ``Sur les repr¹esentations non ramiÞ¹ees des groupes r¹eductifs $p$-adiques; l¹exemple de ${rm GSp}(4)$¹¹ math.RT/0410242 Yuri A. Neretin: On compression of Bruhat-Tits buildings SG: Symplectic Geometry ----------------------- math.SG/0410338 Michael Entov, Leonid Polterovich: Quasi-states and symplectic intersections math.SG/0410243 D. Kotschick: Free circle actions with contractible orbits on symplectic manifolds SP: Spectral Theory ------------------- math.SP/0410307 R.F. Efendiev: The characterization problem for one class high order ordinary differential operators with periodic coefÞcients math.ST/0410280 Olivier Catoni: Improved Vapnik Cervonenkis bounds ST: Statistics -------------- q-bio.GN/0410012 Satya N. Majumdar, Sergei Nechaev: Exact Asymptotic Results for a Model of Sequence Alignment math.ST/0410271 J.J. Lok: Statistical modelling of causal effects in continuous time -- / 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 Þt to e-print * === Subject: new products in geometric algebra I¹ve been studying some of the additional operations deÞned in the literature on geometric algebra, AKA Clifford algebra. There are some very interesting constructions, but I am having a hard time mapping them to more common operations on the exterior algebra. One question I have: has anyone come up with a clean deÞnition in terms of Clifford (geometric) multiplication for the inner product as deÞned on the exterior algebra to deÞne the Hodge star? For A and B k-blades (i.e. the exterior products of k vectors A_i and B_j) this is the construction := det() with the result deÞned to be zero for blades of different grade (composed of different numbers of vectors). Even an operation that coincided with this for two k-blades would be of interest. I¹ve looked into various operations including: - inner product A.B := _|j-k| where A and B are j- and k-vectors - commutator product A x B := (AB - BA)/2 - contraction A J B := (A / (Bw))w^-1 where w is the volume element (pseudoscalar) and J is supposed to be a backwards L but none seem to do the job. A related item: I¹ve been putting the deÞnition of the Hodge star in terms of geometric algebra operations *A = ((w^-1)A)^+ where w^-1 is the inverse of the volume element (pseudovector), and ^+ indicates reversion into some more usable (for what I¹m doing) forms. Can anyone verify these results? Here A is a k-blade, w is the volume element (pseudoscalar), and s is the number of negative signs in the signature of the inner product deÞning the Clifford algebra. *A = (-1)^(k(k-1)/2 + s) Aw *A = (w^+w)(A^+)w === Subject: Re: new products in geometric algebra Mark Adams schrieb im Newsbeitrag > One question I have: has anyone come up with a clean deÞnition in terms > of Clifford (geometric) multiplication for the inner product as > deÞned on the exterior algebra to deÞne the Hodge star? For A and B > k-blades (i.e. the exterior products of k vectors A_i and B_j) this is > the construction > := det() > with the result deÞned to be zero for blades of different grade > (composed of different numbers of vectors). Even an operation that > coincided with this for two k-blades would be of interest. I¹ve looked > into various operations including: > - inner product A.B := _|j-k| where A and B are j- and k-vectors > - commutator product A x B := (AB - BA)/2 > - contraction A J B := (A / (Bw))w^-1 where w is the volume element > (pseudoscalar) and J is supposed to be a backwards L > but none seem to do the job. I think what you are looking for is _0 , (where ^+ is reversion). You can check that this gives the desired result by looking at a convenient orthonormal basis. You can explicitly derive this expression by noting that the exterior bundle supports two mutually supercommuting copies of the Clifford algebra that you have in mind, constructed from sums and differences of operators of exterior and interior multiplication, respectively. Noting that the latter are mutual adjoints with respect to the Hodge inner product and using the symbol map you can then derive the above (up to a global sign, depending on some conventions) from the Hodge inner product. If you want to see the details have a look at pp.279 of http://www-stud.uni-essen.de/~sb0264/sqm.html . > A related item: I¹ve been putting the deÞnition of the Hodge star in > terms of geometric algebra operations > *A = ((w^-1)A)^+ where w^-1 is the inverse of the volume element > (pseudovector), and ^+ indicates reversion > into some more usable (for what I¹m doing) forms. Can anyone verify > these results? Here A is a k-blade, w is the volume element > (pseudoscalar), and s is the number of negative signs in the signature > of the inner product deÞning the Clifford algebra. > *A = (-1)^(k(k-1)/2 + s) Aw > *A = (w^+w)(A^+)w This is also discussed at the above reference. I have found it very helpful to pass between Clifford calculus and exterior calculus this way when dealing with supersymmetric quantum systems. For instance hep-th/0311064 and math-ph/0407005 makes use of this. === Subject: Library for Subgraph matching and graph dissimilarity Hi all... I have two graphs G1 and G2 labeled along nodes and edges. I have to get a pattern/ subgraph from these two graphs such that this subgraph captures those nodes and edges in G1 which are not included in G2. i.e there is a subgraph of G1 which is changed in G2. Rest of G1 in G2 is not changed and I need to capture this subgraph. These are NP complete problems as far as I know. I request you to suggest me an optimum way to do this. Is there any library or tool which does this. Vipindeep === Subject: Re: p-adic transcendentals David Madore a ecrit: > So, is there a known way of proving that log(3)/log(2) is > transcendental in Q_p? I don¹t know the answer, but your question reminds me the following, [roughly translated from Exercice 4 of Chapter IV 5 of Borevitch-Chafarevitch Thorie des Nombres] : Find a mistake in the following proof of the irrationality of pi. The number pi is the smallest number >0 such that sin(pi)=0. Suppose that pi is rational. Then since pi>3, the numerator of pi has to be divisible by some odd prime p, or by 2^2. In that case set p=2. It follows that the series sin(x) and cos(x) are convergent in Q_p at x=pi. but since sin(x+y)=sin(x)cos(y)+sin(y)cos(x) it follows that sin(n pi)=0 for any integer n. Hence the function sin(x) has inÞnitely many zeros inside its convergence area, hence it is identically zero (from Exercice 1 at the same page). This is a contradiction. Serge. === Subject: Re: p-adic transcendentals > David Madore a ecrit: > > So, is there a known way of proving that log(3)/log(2) is > transcendental in Q_p? > Is there a sense in which this p-adic number x satisÞes 2^x = 3 ? > I don¹t know the answer, but your question reminds me the following, > [roughly translated from Exercice 4 of Chapter IV 5 of > Borevitch-Chafarevitch Theorie des Nombres] : > Find a mistake in the following proof of the irrationality of pi. > The number pi is the smallest number >0 such that sin(pi)=0. > Suppose that pi is rational. Then since pi>3, the numerator of pi > has to be divisible by some odd prime p, or by 2^2. In that case > set p=2. It follows that the series sin(x) and cos(x) are convergent > in Q_p at x=pi. but since > sin(x+y)=sin(x)cos(y)+sin(y)cos(x) > it follows that sin(n pi)=0 for any integer n. Hence the function > sin(x) has inÞnitely many zeros inside its convergence area, > hence it is identically zero (from Exercice 1 at the same page). > This is a contradiction. I guess the error is: A series of rationals that convervges both in the usual absolute value and in a p-adic absolute value need not converge to the same thing in the two cases, even if both limits are rationals. === Subject: Re: p-adic transcendentals Epigone-thread: kanþarcho Originator: bergv@math.uiuc.edu (Maarten Bergvelt) >> David Madore a ecrit: >> >> So, is there a known way of proving that log(3)/log(2) is >> transcendental in Q_p? >> >Is there a sense in which this p-adic number x satisÞes 2^x = 3 ? I believe the OP covered this, if we deÞne 2^x to be exp(x*log(2)). (We suppose p is neither 2 nor 3.) The point is that the usual MacLaurin expansions for exp(x), log(1+x) give well-deÞned mutually inverse homomorphisms exp: pZ^_p --> 1 + pZ^_p log: 1 + pZ^_p --> pZ^_p Now of course 2 does not belong to 1 + pZ^_p, but there is a unique root of unity w such that 2w belongs to 1 + pZ^_p, and log(2) is to be deÞned as log(2w). In other words, there is an identiÞcation 1 + pZ^_p ~ (Z^_p)*/((p-1)th roots of unity) so we may say 2^{(log 3)/(log 2)} and 3 differ by a factor which is a root of unity. Todd Trimble === Subject: Re: p-adic transcendentals Originator: bergv@math.uiuc.edu (Maarten Bergvelt) G. A. Edgar a .8ecrit: > I guess the error is: A series of rationals that convervges both in > the usual absolute value and in a p-adic absolute value need not > converge to the same thing in the two cases, even if both limits > are rationals. That was also my guess, concerning this exercise. But I also guess I would have accepted this proof as correct if the mistake had not been explicitly announced in the exercice : this proof really looks correct at Þrst sight !:-) Serge. === Subject: Re: p-adic transcendentals Originator: bergv@math.uiuc.edu (Maarten Bergvelt) G. A. Edgar in litteris scripsit: >> David Madore a ecrit: >> So, is there a known way of proving that log(3)/log(2) is >> transcendental in Q_p? > Is there a sense in which this p-adic number x satisÞes 2^x = 3 ? Unless I¹m mistaken: Write x as the limit, for the p-adic topology, of a sequence of integers x_i all congruent mod p-1 (I mean all congruent to some Þxed quantity mod p-1). Then (2^{x_i}) converges (again, in the p-adics) to some number which is equal to 3 up to multiplication by a (p-1)-th root of unity, the root in question being determined by the class mod p-1 of the x_i (and for one particular class we do get exactly 3). In particular, x cannot be rational (I believe I¹m not making the error which Serge points out, here), otherwise some power of 2 would be equal to some power of 3 (up to multiplication by a root of unity, but that cannot be) *in the rationals*. >> Find a mistake in the following proof of the irrationality of pi. >> The number pi is the smallest number >0 such that sin(pi)=0. >> Suppose that pi is rational. Then since pi>3, the numerator of pi >> has to be divisible by some odd prime p, or by 2^2. In that case >> set p=2. It follows that the series sin(x) and cos(x) are convergent >> in Q_p at x=pi. but since >> sin(x+y)=sin(x)cos(y)+sin(y)cos(x) >> it follows that sin(n pi)=0 for any integer n. Hence the function >> sin(x) has inÞnitely many zeros inside its convergence area, >> hence it is identically zero (from Exercice 1 at the same page). >> This is a contradiction. > I guess the error is: A series of rationals that convervges both in > the usual absolute value and in a p-adic absolute value need not > converge to the same thing in the two cases, even if both limits > are rationals. Yes, there¹s no reason that because sin(a/b)=0 in the reals we should also have sin(a/b)=0 in the p-adics. But perhaps there¹s a way to prove this anyway, if the coefÞcients of the sin power series expansion are tame enough in some sense. -- David A. Madore (david.madore@ens.fr, http://www.dma.ens.fr/~madore/ ) === Subject: Re: p-adic transcendentals Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Just for the record: David Madore in litteris scripsit: > Write x as the limit, for the p-adic topology, of a sequence of > integers x_i all congruent mod p-1 (I mean all congruent to some Þxed > quantity mod p-1). Then (2^{x_i}) converges (again, in the p-adics) > to some number which is equal to 3 up to multiplication by a (p-1)-th > root of unity, the root in question being determined by the class mod > p-1 of the x_i I believe the above is correct. However, this > (and for one particular class we do get exactly 3). might not be true unless we assume that 2 is primitive mod p (or, more generally, that 3 belongs to the multiplicative subgroup generated by 2 in the Þnite Þeld with p elements). For example, there is no way to approach 3 with powers of 2 in the 7-adics (the powers of 2 in the Þnite Þeld with 7 elements are 1, 2 and 4), but log(3)/log(2) still makes sense in the way I suggested (and it is precisely 3 + 3*7 + 49 + 0 + 5*7^4 + 6*7^5 + O(7^6)). -- David A. Madore (david.madore@ens.fr, http://www.dma.ens.fr/~madore/ ) === Subject: Integral equations of Þrst kind Good Morning, I am working with the integral equations of Þrst kind on an annular in R^3. I have to prove the unicity of the solution, Numerically, i have a well conditionned matrix, but theorically i don¹t know how to prove it. In fact, i don¹t know which methods use for the equations of Þrst kind. i¹ve tried with the alternative of Fredholm, it helps me to prove the unicity in the quotient space but not in the space itself (what i want to prove) ! So, i would like to ask if someone can help me , maybe with some theorems of Þxed point ..... Zimar. === Subject: Re: Free software for Homotopy Continuation Methods >Moderator¹s Note. The post in question is dated 1997, from David >Harney, Chemical Engineering Dept., University of Missouri - Rolla. >Is anything current known about David Harney or the HOMES software? >I am a engineering student at India . I was searching the fortran >program to solve Non linear systems of equations by homotopy. I got >your mail ID from the link >http://mathforum.org/epigone/sci.math.research/praxkhingglou >I was trying to resolve the turning point issue but i could not do it. >I will be very grateful if you can send me the fortran source code for >it along with all the details of Subroutines. >I will be very grateful to you for this. >Sonal jain I know nothing about HOMES 2.0 but http://plato.la.asu.edu/topics/problems/zero.html lists several good sources for continuation methods (in fortran) capable of continuation beyond turning points hth peter >>HOMES 2.0, a fortran program that solves systems of algebraic >>nonlinear systems using homotopy continuation methods, has been >>developed here at the University of Missouri - Rolla. >>A choice of homotopy functions are available as is a selection >>of path tracking algorithms. Bifurcations into the complex domain >>at turning points and real domain bounding algorithms are among >>other selections available. >>If you would like a copy of the code, please email me at >>harney@umr.edu >>It will eventually be available on a website, as soon as I muster up >>the motivation to learn how to create one! >>David Harney >>Chemical Engineering Dept. >>University of Missouri - Rolla === Subject: Re: Category of categories axiomatically >> How to deÞne the category of (locally small) categories axiomatically? > Maybe my question is on a highly sensitive issue in Category Theory, which > explains there are no replies ^_^. Indeed it is not clear that the notion > of category of categories makes sense. The question does make sense. But you have to be careful. There is a lots of things known about the category Cat of small categories: it is locally small, complete, cocomplete, cartesian closed, etc... I believe that considering three Grothendieck universes U subset V subset W is sufÞcient to solve most of the problems of size. The unique rule is then: instead of talking about sets, collections, and I dont know what next, use the term U-small set, V-small set, W-small set. A U-small set has a U-small cardinal. etc... In other terms, at each step, specify cleary to which universe the set under consideration belongs. Instead of talking about small category, use the term U-small category, i.e. a U-small set of U-small objects with U-small homsets : denote by Cat_U the corresponding V-small and locally U-small category. Instead of talking about locally small category, use the term locally U-small category, i.e. a V-small set of U-small objects with U-small homsets. Denote by CAT_U the corresponding category: if I dont make any mistake, CAT_U is certainly not V-small, but is W-small because CAT_U subset Cat_V and the latter is W-small. Since Cat_V is V-complete, it is U-complete. So I believe (but I did not check that) that CAT_U is U-complete as well: the only thing you have to check is that a U-small limit in CAT_U stays inside CAT_U. CAT_U is probably U-cocomplete as well. CAT_U is probably not cartesian closed however because the obvious candidat for HOM(C,D) is the category of functors from C to D. And it is well-known that HOM(C,D) is in CAT_U iff C is essentially U-small (i.e. equivalent to a U-small category). etc... pg. === Subject: Re: Category of categories axiomatically Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > CAT_U. CAT_U is probably U-cocomplete as well. CAT_U is probably not > cartesian closed however because the obvious candidat for HOM(C,D) is > the category of functors from C to D. And it is well-known that > HOM(C,D) is in CAT_U iff C is essentially U-small (i.e. equivalent to > a U-small category). etc... A small mistake : D=Set is understood. For over category D, the statement above may be false. The very-known statement I had in mind is the: a category of presheaves is locally small iff the base category (C here) is essentially small. pg. === Subject: Does this deÞnition of dim 2 n-free poset exterior ring any bells with anyone? Let H be the Hasse diagram of any dim 2 n-free poset and let Hp be any natural plane embedding of H. Then inasmuch as Hp is a DAG whose edges are deÞned by the immediate covering relations in H, one can assign the integer d to each vertex v of Hp, where d is the depth of v in Hp interpreted as a DAG (i.e. the number of vertices in Hp which are ancestors of v, excluding v itself.) Further, since Hp is embedded in the plane relative to the geometry of the paper, a left-right or horizontal ordering of the vertices of Hp results from its embedding in the plane. That is, we can assign the numbers p and t to each vertex v of Hp, where: p is the number of vertices to the left of v in Hp t is the number of vertices to the right of v in Hp. And it is readily shown that because Hp is both dim 2 and n-free: a) the values (p0+d0),...,(pi+di),...,(pn+dn) for the n+1 vertices of Hp are 0,...,n+1 b) the values (t0+d0),...,(ti+di),...,(tn+dn) for the n+1 vertices of Hp are a permutation of 0,...,n+1. Suppose now, however, that one uses the integers pi, ti, di to assign each vertex v of Hp the triple of integers (pi,ti,di). And further, suppose that one deÞnes the exterior of Hp as just those vertices in Hp for which at least one of pi,ti,di is 0. And Þnally, suppose that one deÞnes the exterior union of Hp as the union of all the exteriors of all the subgraphs of Hp which are themselves interpretable as Hasse diagrams of dim 2 n-free posets. Does this notion of exterior union ring any bells with anyone ?? considering this matter. === Subject: TOC / Index for the 3 Notebooks of Ramanujan Epigone-thread: doichayþon [Firstly, I would like to know if there is a facility in this User Group to upload a document.] Pl read on ... published by TIFR in 2 Volumes. Then, as I wanted to read the proofs of Ramanujan¹s Theorems as published by Bruce C Berndt in his 5-Part books, I noticed that there is an obvious strong refernce made to the 3 Notebooks and the 2 Volumes of TIFR. I then thought that it is very essential at least for a beginner like me to have some form of a quick-reference TOC / index to these creations of Ramanujan. This effort was relatively easier wrt the so-called organized parts, the 21 Chapters ; however, I need some help to update (and correct) parts that I have described as Misc in my prepared document. I have categorized Misc itself into 3 main types. It would be great if someone could help me state the split of appropriate subject(s) for these Misc Pages. For eg, in Volume I, ie, NB 1, I could count a total of 121 pages - marked *under my so-called Misc category. Since it is claimed that there are 100 Unorganized pages in NB 1, I would like to know which 100 pages of these 121 pages fall in the Unorganized category. Now, how do I upload my document (SR_NBs_TOC_Index.doc) ? If there is no such facility to upload documents in this User Group, interested readers can request me by sending me a mail to my email id : sundarkrishnan@hotmail.com Sundar Krishnan [ Moderator¹s Note. sci.math.research newsgroup does not include documents. Offering to send by email is one way. Another is to post it on a web page, then tell us the URL here. ] === Subject: HyperGeometric Series Originator: bergv@math.uiuc.edu (Maarten Bergvelt) I am not an expert in HyperGeometric Series and Functions [from now on, I use the short form HG below.] Along with other basic Maths books, I have been trying to learn this subject also by reading the famous book : A Course of Modern Analysis (Cambridge Mathematical Library), by E. T. Whittaker, G. N. Watson. [from now on, I call it the W & W book below.] Also, I realised that many of Srinivasa Ramanujan¹s works relate to HG Fns and Series, as can also be seen in the 5-Part book authored by Bruce C Berndt. At many places, I have been able to follow the derivation of complex expressions of the W & W book, but often with slightly different expressions like say, an extra snippet of expression, a change of sign etc. And even the so-called (simple, but very appropriate to the subject at that point) Examples are sometimes difÞcult to answer. I would therefore like to know if there is a Solutions Manual to this book so that it is easier to learn and a little faster too - something similar to Donald Knuth¹s books with Solutions on The Art of Computer Programming. My brief background : After working for many years in a number of companies like ABB, HP, Motorola, I have now taken up some studies in Video Compression, Error Correction Codes, DSP and Maths - Elliptical Integrals, HyperGeometric Series etc. So, I am neither a student nor a Professor, but very much keen on the above subjects. Sundar Krishnan ... PS : Interested readers may also refer to my other query : TOC / Index for the 3 Notebooks of Ramanujan. ... === Subject: Solving Quintics Using One Fifth Root Extraction X-mailer: epigone Epigone-thread: cymingþu Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Hello all, For those who like solvable quintics, here¹s another paper: Solving Solvable Quintics Using One Fifth Root Extraction ABSTRACT: We prove that all irreducible but solvable equations of degree n can be transformed in radicals into the binomial form y^n+c=0 using a Tschirnhaussen transformation of degree n-1. The resulting equation is then solvable by a single nth root extraction. In particular, we illustrate the method using the solvable quintic. Mathematics Subject ClassiÞcation. Primary: 12E12; Secondary: 11D09 http://www.geocities.com/titus_piezas/morequintics.html Just click at the link above. It¹s the 2nd pdf Þle. -Titus (tpiezasIII@uap.edu.ph -> remove III for email) === Subject: Re: Open questions related to periodic continued fractions Received-SPF: Received-SPF: none (mailbox4.ucsd.edu: domain of news@newsread1.news.pas.earthlink.net does not designate permitted sender hosts) Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > I am starting research for a thesis on continued fractions, and want > to look at open questions related to periodic continued fractions. Is > anyone aware of current open questions of interest? > As you may know, the outstanding problem in this area is to improve on known > conditions for the length of the period of the continued fraction expansion of > sqrt{D} to be odd (D > 0, D not a square, equivalent to x^2 - Dy^2 = -1 having > solutions). The following might be a current summary of known conditions: > B. D. Beach and H. C. Williams, A Numerical Investigation of the > Diophantine Equation x^2 - dy^2 = -1, Congressus Numerantium VI, > Proceedings of the Third Southeastern Conference on Combinatorics, > Graph Theory and Computing, Utilitas Mathematica Publishing Inc., > Winnipeg, Canada, 1972, pages 37 to 52. > A less well known problem is as follows. As far as I know, this is an open > problem, but I cannot be sure. For D > 0 not a square, D congruent to 1 modulo > 4, let L1(D) and L4(D) denote the lengths of the periods of the continued > fraction expansions of sqrt{D} and (1+sqrt{D})/2 respectively. The question > is whether it is possible to have L1(D) = L4(D) + 4 when L4(D) == 3 (mod 6). > As far as I can tell, this is not prohibited by results in the literature. I > have empirical evidence that it is not possible based on testing those D up to > 30 billion that have L4(D) <= 255. It is not hard to show this for one > particular case, namely if L4(D) = 3 then L1(D) cannot be 7. > See discussion under the heading ``Periods of Continued Fractions¹¹ in April > and May of 2000 in the archives of the Number Theory Listserver at > http://listserv.nodak.edu/archives/nmbrthry.html > for related comments and some references that might be of interest. > John Robertson === Subject: Paper published by Algebraic and Geometric Topology Originator: bergv@math.uiuc.edu (Maarten Bergvelt) The following paper has been published: Algebraic and Geometric Topology URL: http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-41.abs.html Title: Partition complexes, duality and integral tree representations Author(s): Alan Robinson Abstract: We show that the poset of non-trivial partitions of 1,2,...,n has a fundamental homology class with coefÞcients in a Lie superalgebra. Homological duality then rapidly yields a range of known results concerning the integral representations of the symmetric groups S_n and S_{n+1} on the homology and cohomology of this partially-ordered set. Secondary: 17B60 Keywords: Partition complex, Lie superalgebra Author(s) address(es): Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK Email: car@maths.warwick.ac.uk === Subject: The extent of a plane curve X-RFC2646: Format=Flowed; Original Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Call an arc in the plane convex if it is simple (i.e., 1-1 except that its endpoints may coincide) and it lies on the boundary of its convex hull. DeÞne the extent of a curve c of length L in the plane to be the length of the shortest convex arc whose convex hull contains the (range of the) curve c. So, for example, the extent of a triangle is the sum of its two longer sides, and the extent of a circle of radius r is (2 + pi)r. This notion has a number of interesting properties, but it seems awkward to investigate. My inquiry here is whether anyone knows of any mention of the notion in the literature. --J. Wetzel === Subject: algebraic sets Originator: bergv@math.uiuc.edu (Maarten Bergvelt) topology and that the following question comes from optimization: Let p_i be a Þnite set of k relatively prime polynomials, each of which maps R^n into R. I am interested in describing the boundary of the set A = {x in R^n : p_i(x) >= 0 for all i, 1 <= i <= k} as a union of manifolds. Can I do this? SpeciÞcally, I would like to say the following: Let I be an index set and let A_I = {x in R^N : p_i(x) = 0 for i in I and p_i(x) > 0 for i notin I}. Then A_I is the Þnite union of manifolds of dim <= cardinality(I). I have been told that a semialgebraic subset of R^n (like A) can be written as a union of manifolds of decreasing dimension, but I have not seen this result stated anywhere. I have seen a result that says a semialgebraic set is composed of a disjoint union of connected semialgebraic sets, but these are not quite what I want. Could someone point out some results for me in this direction and where I might Þnd them written down? Even results requiring additional hypotheses would be welcome.