mm- Subject: Equivariant Derived Categories Received-SPF: Received-SPF: none (mailbox6.ucsd.edu: domain of news@news.stanford.edu does not designate permitted sender hosts) Originator: israel@math.ubc.ca (Robert Israel) Can anyone point me to a reference (in English) for deŽning the equivariant derived category for a discrete Abelian group. In particular, I think there ought to be an action of the group on the category given by tensoring with a representation. I would like to understand this action in the context of the correpsondence of Bridgeland, King and Reid (math.AG/9908027) which does contain some basic deŽnitions. Thanx, Aaron === Subject: Re: Comparing Rotated Point Sets Originator: israel@math.ubc.ca (Robert Israel) Wow. This gives me something to chew on. I¹ll try to get my head Tom [Lengthy quote deleted by moderator. -RI] === Subject: On unitary matrix of a complex Gaussian random matrix Originator: israel@math.ubc.ca (Robert Israel) With the singular value decomposition, complex Gaussian random matrix H1 and H2 can be decomposed to H1=U1*D1*V1 and H2=U2*D2*V2, respectively. As you might know, V1 and V2 are isotropically random unitary matrices. For example, if H1 and H2 are all 3x2 sized matrices and they have 2 same rows, given as H1=[h1,h2,h3] and H2=[h1,h2,h4], where hi, iin[1,4], are 2-sized vector. Do you guys have any idea what the relationship is between V1 and V2? It will be very appreciated if you can give me some hint or references. Yang === Subject: This week in the mathematics arXiv (27 Sep - 1 Oct) 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 (27 Sep - 1 Oct) ------------------------------------------------ AC: Commutative Algebra ----------------------- math.AC/0409608 Hans Schoutens: The asymptotic behavior of solid closure in mixed characteristic math.AC/0409591 Alexandre Tchernev: Representations of matroids and free resolutions for multigraded modules math.AC/0409514 Marco Fontana, Giampaolo Picozza: Semistar invertibility on integral domains math.AC/0409496 Uwe Nagel: Liaison classes of modules math.AC/0409462 David Cox, Alicia Dickenstein, Hal Schenck: A case study in bigraded commutative algebra AG: Algebraic Geometry ---------------------- math.AG/0409592 Amin Gholampour: On the equivariant Gromov-Witten Theory of $mathbb{P}^{2}$-bundles over curves hep-th/0409290 Falk Rohsiepe: Lattice polarized toric K3 surfaces hep-th/0409227 Gaetano Bertoldi, Stefano Bolognesi, Gaston Giribet, Marco Matone, Yu Nakayama: Zamolodchikov relations and Liouville hierarchy in SL(2,R)_k WZNW model math.AG/0409584 Ofer Gabber, Lorenzo Ramero: Foundations of $p$-adic Hodge theory -- Release zero math.AG/0409572 Michael A. van Opstall: Stable degenerations of symmetric squares of curves math.AG/0409569 Andrei Mustata, Magdalena Anca Mustata: Intermediate Moduli Spaces of Stable Maps math.AG/0409564 Bernard Le Stum, Adolfo Quir¹os: The Žltered Poincar¹e lemma in higher level (with applications to algebraic groups) hep-th/0409288 L. Griguolo, D. Seminara, R.J. Szabo: Double Scaling String Theory of QCD in Two Dimensions math.AG/0409526 Cindy De Volder, Antonio Laface: A note on the very ampleness of complete linear systems on blowings-up of P^3 math.AG/0409525 Olga V. Chuvashova: The separation properties for closures of toric orbits math.AG/0409524 Cindy De Volder, Antonio Laface: Recent results on linear systems on generic K3 surfaces math.AG/0409504 Evgenia Soprunova, Frank Sottile: Lower Bounds for Real Solutions to Sparse Polynomial Systems math.AG/0409500 Tommaso de Fernex: Length, multiplicity, and multiplier ideals math.AG/0409499 Tommaso de Fernex, Antonio Lanteri: Bad loci of free linear systems math.AG/0409495 Tom Braden, Valery A. Lunts: Equivariant-constructible Koszul duality for dual toric varieties math.AG/0409493 Claus Lehr: An analog to Deuring¹s criterion for good reduction of elliptic curves math.AG/0409490 Alexander Woo, Alexander Yong: When is a Schubert variety Gorenstein? math.AG/0409482 Thomas J. Haines: Introduction to Shimura varieties with bad reduction of parahoric type math.AG/0409469 S. Subramanian: Forms of AfŽne Space AP: Analysis of PDEs -------------------- math.AP/0409585 Jim Colliander, Sarah Raynor, Catherine Sulem, J. Douglas Wright: Ground state mass concentration in the L^2-critical nonlinear Schrodinger equation below H^1 math.AP/0409579 Jeremy Marzuola: Nonconcentration of eigenfunctions for partially rectangular billiards math.AP/0409486 D. Schertzer, M. Larchev, J. Duan, V.V. Yanovsky, S. Lovejoy: Fractional Fokker--Planck Equation for Nonlinear Stochastic Differential Equations Driven by Non-Gaussian Levy Stable Noises AT: Algebraic Topology ---------------------- math.AT/0409597 Jean-Francois: Chas and Sullivan algebra of Žber bundles math.AT/0409574 G¹abor Braun: Cobordism class of multiple points of immersions hep-th/0404013 Igor Kriz, Hisham Sati: M Theory, Type IIA Superstrings, and Elliptic Cohomology CA: Classical Analysis and ODEs ------------------------------- math.CA/0409594 Ali Taghavi: On Periodic Solutions Of Lienard Equations math.CA/0409580 Stephen Semmes: Potpourri, 4 math.CA/0409508 S. A. Abramov, M. A. Barkatou, M. van Hoeij: Apparent Singularities of Linear Difference Equations with Polynomial CoefŽcients CO: Combinatorics ----------------- math.CO/0409588 Shawn Elldge, Glenn H. Hurlbert: An application of graph pebbling to zero-sum sequences in abelian groups math.CO/0409562 Matthias Beck, Mike Develin, Sinai Robins: On Stanley¹s reciprocity theorem for rational cones math.CO/0409538 J. Haglund, M. Haiman, N. Loehr: A Combinatorial Formula for Macdonald Polynomials quant-ph/0409152 A. Horzela, P. Blasiak, G.H.E. Duchamp, K.A. Penson, A.I. Solomon: A product formula and combinatorial Želd theory math.CO/0409515 A.H. Zemanian: The Galaxies of Nonstandard Enlargements of TransŽnite Graphs of Higher Rsnks math.CO/0409509 Ralf Stephan: Prove or Disprove. 100 Conjectures from the OEIS math.CO/0409480 Alexander Berkovich, Frank G. Garvan: Dissecting the Stanley Partition Function math.CO/0409478 A.H. Zemanian: The Galaxies of Nonstandard Enlargements of InŽnite and TransŽnite Graphs math.CO/0409474 J. Richard Lundgren, K.B. Reid, Simone Severini, Dustin J. Stewart: Quadrangularity and Strong Quadrangularity in Tournaments math.CO/0409468 Guoce Xin: Constructing All Magic Squares of Order Three math.CO/0409466 Chunhui Lai: An extremal problem on potentially $K_{m}-P_{k}$-graphic sequences math.CO/0409463 Thomas Lam: Ribbon Schur Operators CT: Category Theory ------------------- math.CT/0409598 B. Toen: Towards an axiomatization of the theory of higher categories math.CT/0409477 Isar Stubbe: Categorical structures enriched in a quantaloid: orders and ideals over a base quantaloid math.CT/0409475 Isar Stubbe: Categorical structures enriched in a quantaloid: regular presheaves, regular semicategories math.CT/0409473 Isar Stubbe: Categorical structures enriched in a quantaloid: categories, distributors and functors CV: Complex Variables --------------------- math.CV/0409582 James W. Anderson, Kurt Falk, Pekka Tukia: Conformal measures associated to ends of hyperbolic n-manifolds math.CV/0409560 Lukas Geyer, Sergei Merenkov: A Hyperbolic Surface With A Square Grid Net math.CV/0409534 Wenhua Zhao: Hessian Nilpotent Formal Power Series and Their Deformed Inversion Pairs math.CV/0409494 Sergei Treil, Brett D Wick: The Matrix-Valued $H^{p}$ Corona Problem in the Disk and Polydisk DG: Differential Geometry ------------------------- math.DG/0409605 Stefan Haller, Josef Teichmann: Smooth perfectness for the group of diffeomorphisms math.DG/0409603 TaouŽk Bouziane: ŒRegularity¹ of Energy Minimizer maps between Riemannian polyhedra math.DG/0409587 Sebastien Racaniere: Quasi-Poisson actions and massive non-rotating BTZ black holes math.DG/0409583 Sun-Yung A. Chang, Matthew J. Gursky, Paul C. Yang: An equation of Monge-Ampere type in conformal geometry, and four-manifolds of positive Ricci curvature math.DG/0409577 Gianmarco Capitanio: Legendrian graphs generated by Tangential Families math.DG/0409559 Benjamin McKay: Complete Cartan connections on complex manifolds math.DG/0409553 M. A. Aprodu, T. Bouziane: Pseudo Harmonic Morphisms on Riemannian Polyhedra math.DG/0409537 Nathan Brown, Rachel Finck, Matthew Spencer, Kristopher Tapp, Zhongtao Wu: Invariant metrics with nonnegative curvature on compact Lie groups hep-th/0409200 Paolo Aschieri, Branislav Jurco: Gerbes, M5-Brane Anomalies and E_8 Gauge Theory math.DG/0409476 Dietrich Burde, Karel Dekimpe, Sandra Deschamps: The Auslander conjecture for NIL-afŽne crystallographic groups math.DG/0409467 Claus Gerhardt: Closed hypersurfaces of prescribed mean curvature in locally conformally žat Riemannian manifolds math.DG/0409465 Claus Gerhardt: Hypersurfaces of prescribed mean curvature in Lorentzian manifolds DS: Dynamical Systems --------------------- math.DS/0409557 Romain Dujardin: Laminar currents and birational dynamics math.DS/0409556 Alexey Glutsyuk: Instability of nondiscrete free subgroups in Lie groups math.DS/0409544 Vitor Araujo: Integrability versus frequency of hyperbolic times and the existence of a.c.i.m math.DS/0409528 Gabriel P. Paternain: Magnetic Rigidity of Horocycle žows math.DS/0409492 A.Vershik: Polymorphisms, Markov processes, random perturbations of K-automorphisms math.DS/0409485 Jinqiao Duan, Kening Lu, Bjoern Schmalfuss: Invariant manifolds for stochastic partial differential equations math.DS/0409483 Jinqiao Duan, Kening Lu, Bjorn Schmalfuss: Smooth stable and unstable manifolds for stochastic partial differential equations math.DS/0409481 Igor Chueshov, Jinqiao Duan, Bj{o}rn Schmalfu{ss}: Determining functionals for random partial differential equations FA: Functional Analysis ----------------------- math.FA/0409516 Simon Eveson: Asymptotic behaviour of iterates of Volterra operators on L^p(0,1) GN: General Topology -------------------- math.GN/0409609 Francis Jordan, Frederic Mynard: Compatible relations on Žlters and stability of local topological properties under supremum and product math.GN/0409590 R. Kozhan, M. Zarichnyi: Open-multicommutativity of the probability measure functor math.GN/0409566 Roman Kozhan: Open-multicommutativity of some functors related to the functor of probability measures GR: Group Theory ---------------- math.GR/0409586 J.-F. Lafont: Rigidity results for certain 3-dimensional singular spaces and their fundamental groups math.GR/0409472 Tetsuya Hosaka: Parabolic subgroups of Coxeter groups acting by režections on CAT(0) spaces math.GR/0409461 D. Jeremy Copeland: A Special Subgroup of the Surface Braid Group GT: Geometric Topology ---------------------- math.GT/0409611 U. Hamenstaedt: Train tracks and the Gromov boundary of the complex of curves math.GT/0409606 Carlo Petronio: Spherical splitting of 3-orbifolds math.GT/0409581 Thomas W. Mattman, Owen Sizemore: Bounds on the Crosscap Number of Torus Knots math.GT/0409536 Yi-Jen Lee: Heegaard Splittings and Seiberg-Witten monopoles math.GT/0409529 J¹er^ome Dubois: A volume form on the $mathrm{SU}(2)$-representation space of knot groups math.GT/0409498 Jiming Ma, Ruifeng Qiu: Heegaard Splittings of Boundary Reducible 3-Manifolds math.GT/0409497 Ruifeng Qiu: Stabilizations of Reducible Heegaard Splittings math.GT/0409460 Gabriel Murillo, Sam Nelson: Alexander quandles of order 16 math.GT/0409459 Luis G Valdez-Sanchez: Seifert Klein bottles for knots with common boundary slopes HO: History and Overview ------------------------ math.HO/0409596 Erhard Scholz: Philosophy as a cultural resource and medium of režection for Hermann Weyl math.HO/0409578 Erhard Scholz: C.F. Gauss¹ Prazisionsmessungen terrestrischer Dreiecke und seine Uberlegungen zur empirischen Fundierung der Geometrie in den 1820er Jahren (C.F. Gauss¹ high precion measurements of terrestrial triangles and his thoughts on the empirical foundations of geometry in the 1820s) math.HO/0409576 Erhard Scholz: The changing concept of matter in H. Weyl¹s thought, 1918 -1930 math.HO/0409571 Erhard Scholz: The Introduction of Groups into Quantum Theory KT: K-Theory and Homology ------------------------- math.KT/0409573 Paul Baum, Piotr M. Hajac, Rainer Matthes, Wojciech Szymanski: The K-Theory of Heegaard-Type Quantum 3-Spheres math.KT/0409541 Do Ngoc Diep: Noncommutative Spherical Tight Frames in Žnitely generated Hilbert C*-modules LO: Logic --------- math.LO/0409567 Alexander S. Kechris, Christian Rosendal: Turbulence, amalgamation and generic automorphisms of homogeneous structures MP: Mathematical Physics ------------------------ physics/0409154 Veniamin A. Abalmassov, Dmitri A. Maljutin: Where will a pen fall to? math-ph/0409082 Pavel Bleher, Alexander Its: Asymptotics of the partition function of a random matrix model math-ph/0409081 B. Grammaticos, A. Ramani, C.-M. Viallet: Solvable Chaos math-ph/0409080 Dinghua Shi, Qinghua Chen, Liming Liu: An Markov Chain-Based Numerical Method for Calculating Network Degree Distributions math-ph/0409079 Anatoli Babin, Alexander Figotin: Nonlinear Photonic Crystals: IV. Nonlinear Schrodinger Equation Regime cond-mat/0403268 P. Di Francesco, J.-B. Zuber: On FPL conŽgurations with four sets of nested arches math-ph/0409078 D. Arnaudon, J. Avan, N. Crampe¹, A. Doikou, L. Frappat, E.Ragoucy: On sl(N) and sl(M|N) integrable open spin chains math-ph/0409077 Luis J. Boya: Spinors and Octonions math-ph/0409076 Daniel Arnaudon, Jean Avan, Nicolas Crampe, Anastasia Doikou, Luc Frappat, Eric Ragoucy: On osp(M|2n) integrable open spin chains math-ph/0409075 Alexei Borodin, Grigori Olshanski: Markov processes on partitions math-ph/0409074 David Damanik, Rowan Killip, Barry Simon: Schroedinger Operators With Few Bound States cond-mat/0409710 Andreas Lochmann, Manfred Requardt: A Proof of a Broad Phase-Transition Zone in the Small-World Model cond-mat/0409581 V. Fleurov, A. Soffer: Nonlinear effects in tunnelling escape in N-body quantum systems quant-ph/0409187 Shiro Kawabata: Information Theoretical Approach to Control of Quantum Mechanical Systems math-ph/0409073 Peter Henselder, Allen C. Hirshfeld, Thomas Spernat: Star Products and Geometric Algebra cond-mat/0409646 Jean Farago: Energy ProŽle Fluctuations in Dissipative Nonequilibrium Stationary States math-ph/0409072 Yu. G. Stroganov: Izergin-Korepin determinant reloaded math-ph/0409071 Yeontaek Choi, Yuri V. Lvov, Sergey Nazarenko: Probability densities and preservation of randomness in wave turbulence math-ph/0409070 Henning Bostelmann: Phase space properties and the short distance structure in quantum Želd theory math-ph/0409067 Omar Foda, Ian Preston: On the correlation functions of the domain wall six vertex model nlin.SI/0406032 A. Sergyeyev: A strange recursion operator demystiŽed math-ph/0409069 J. M. Bouclet, S. De Bievre: Long time propagation and control on scarring for perturbed quantized hyperbolic toral automorphisms math-ph/0409068 Pierre Ca Grange, Ernst Werner: QED Fermi-Fields as Operator Valued Distributions and Anomalies math-ph/0409066 Ioana Dumitriu, Alan Edelman, Gene Shuman: MOPS: Multivariate Orthogonal Polynomials (symbolically) math-ph/0409065 Barry Simon, Andrej Zlatos: Higher-Order Szego Theorems With Two Singular Points cond-mat/0409612 Pavel Exner, Valentin A. Zagrebnov: Bose-Einstein Condensation in Geometrically Deformed Tubes NA: Numerical Analysis ---------------------- math.NA/0409464 Ed Bueler: Chebyshev collocation for linear, periodic ordinary and delay differential equations: a posteriori estimates NT: Number Theory ----------------- math.NT/0409607 Henry H. Kim, Freydoon Shahidi: Functorial products for $GL_2times GL_3$ and the symmetric cube for $GL_2$ math.NT/0409604 J.-H. Evertse, H. P. Schlickewei, W. M. Schmidt: Linear equations in variables which lie in a multiplicative group math.NT/0409540 Graham Everest, Gerard Mclaren, Tom Ward: Primitive divisors of elliptic divisibility sequences math.NT/0409535 Kiran S. Kedlaya, James G. Propp: In search of Robbins Stability math.NT/0409533 Filippo Viviani: RamiŽcation groups and Artin conductors of radical extensions of the rationals math.NT/0409532 Jan Minac, Andrew Schultz, John Swallow: Galois module structure of pth-power classes of cyclic extensions of degree p^n math.NT/0409531 Tsz Ho Chan: $lambda$-th moments of primes in short intervals math.NT/0409530 Tsz Ho Chan: Equivalence of higher moments of primes in short intervals math.NT/0409523 Michael Filaseta, Dmitrii V. Pasechnik: On the irreducibility of a truncated binomial expansion math.NT/0409521 Hao Pan, Zhi-Wei Sun: A characterization of covering equivalence math.NT/0409510 K. Belabas, M. van Hoeij, J. Klueners, A. Steel: Factoring polynomials over global Želds math.NT/0409507 Alex Eskin, Hee Oh: Ergodic theoretic proof of equidistribution of Hecke points math.NT/0409506 Alex Eskin, Hee Oh: Representations of integers by an invariant polynomial and unipotent žows math.NT/0409501 Denis Charles: Complex Multiplication Tests for Elliptic Curves math.NT/0409489 John C. Harris, David L. wehlau: Non-Negative Integer Linear Congruences math.NT/0409484 Nicole Lemire, Jan Minac, John Swallow: Galois module structure of Galois cohomology OA: Operator Algebras --------------------- math.OA/0409601 N. Akiho, F. Hiai, D. Petz: Equilibrium states and their entropy densities in gauge-invariant C*-systems quant-ph/0409181 Christopher King, Michael Nathanson, Mary Beth Ruskai: Multiplicativity properties of entrywise positive maps on matrix algebras math.OA/0409527 Jesse Peterson: A 1-cohomology characterization of property (T) in von Neumann algebras math.OA/0409511 Alberto E. Marrero, Paul S. Muhly: Cuntz-Pimsner Algebras, Completely Positive Maps and Morita Equivalence math.OA/0409505 Cynthia Farthing, Paul S. Muhly, Trent Yeend: Higher-rank graph C*-algebras: an inverse semigroup and groupoid approach math.OA/0409488 David Sherman: A new proof of the noncommutative Banach-Stone theorem PR: Probability --------------- math.PR/0409610 Noureddine El Karoui: An asymptotic Berry-Esseen result for the largest eigenvalue of complex white Wishart matrices math.PR/0409595 Ken Dykema: Symmetric random walks on certain amalgamated free product groups math.PR/0409554 Mark Adler, Alexei Borodin, Pierre van Moerbeke: Expectations of hook products on large partitions math.PR/0409552 Denes Petz, Julia Reffy: Large deviation for the empirical eigenvalue density of truncated Haar unitary matrices math.PR/0409548 Moshe Zakai: On mutual information, likelihood-ratios and estimation error for the additive Gaussian channel math.PR/0409547 Jean Bertoin, Alain Rouault: Asymptotical behaviour of the presence probability in branching random walks and fragmentations math.PR/0409545 Jean Bertoin, Alain Rouault: Discretization methods for homogeneous fragmentations math.PR/0409539 Roberto Fernandez, Gregory Maillard: Construction of a speciŽcation from its singleton part cond-mat/0409532 Anton Bovier, Irina Kurkova: Poisson convergence in the restricted $k$-partioning problem math.PR/0409502 P¹{a}l R¹{e}v¹{e}sz, Jay Rosen, Zhan Shi: Large time asymptotics for the density of a branching Wiener process math.PR/0409491 Davar Khoshnevisan, Yimin Xiao: Images of the Brownian Sheet math.PR/0409479 Davar Khoshnevisan, David A. Levin, Pedro J. Mendez-Hernandez: Exceptional Times and Invariance for Dynamical Random Walks QA: Quantum Algebra ------------------- math.QA/0409602 Alissa S. Crans: Lie 2-Algebras math.QA/0409600 S. Caenepeel, M. De Lombaerde: A categorical approach to Turaev¹s Hopf group-coalgebras math.QA/0409599 S. Caenepeel, Dingguo Wang, Yanmin Yin: Yetter-Drinfeld modules over weak Hopf algebras and the center construction math.QA/0409589 Lars Kadison: An action-free characterization of weak Hopf-Galois extensions math.QA/0409563 Nathan Geer: Etingof-Kazhdan quantization of Lie superbialgebras math.QA/0409520 Matilde Marcolli: Lectures on Arithmetic Noncommutative Geometry math.QA/0409503 Michael Freedman, Vyacheslav Krushkal: On the asymptotics of quantum SU(2) representations of mapping class groups math.QA/0409470 Giuseppe Dito, Remi Leandre: Produit de Moyal stochastique sur l¹espace de Wiener RA: Rings and Algebras ---------------------- math.RA/0409565 Frederick Leitner, Robert Pawloski: Unital Grobner Bases over Arbitrary Ground Rings math.RA/0409551 Francois Couchot: RD-žatness and RD-injectivity math.RA/0409550 Francois Couchot: Finitely presented modules over semihereditary rings math.RA/0409549 Francois Couchot: Local rings of bounded module type are almost maximal valuation rings math.RA/0409546 Francois Couchot: Commutative Local Rings of bounded module type math.RA/0409522 L. Grunenfelder, M. Mastnak: On Bimeasurings math.RA/0409519 Francois Couchot: Injective modules and fp-injective modules over valuation rings math.RA/0409518 Francois Couchot: Modules with RD-composition series over a commutative ring math.RA/0409517 Francois Couchot: The lambda-dimension of commutative arithmetic rings math.RA/0409513 Gabriella Bohm: Galois theory for Hopf algebroids math.RA/0409512 Dietrich Burde: On the matrix equation XA-AX=X^p RT: Representation Theory ------------------------- math.RT/0409593 Raphael Rouquier: CategoriŽcation of the braid groups math.RT/0409570 Bernt Tore Jensen, Xiuping Su, Alexander Zimmermann: Degenerations for derived categories math.RT/0409561 Ian M. Musson, Jeb F. Willenbring: Invariant Differential Operators and FCR factors of Enveloping algebras math.RT/0409543 Dennis Gaitsgory, David Kazhdan: Algebraic groups over a 2-dimsnional local Želd: irreducibility of certain induced representations math.RT/0409487 Shantala Mukherjee: On some representations of nilpotent Lie algebras and superalgebras SG: Symplectic Geometry ----------------------- math.SG/0409568 Felix Schlenk: Packing symplectic manifolds by hand math.SG/0409555 William D. Kirwin, Siye Wu: Geometric Quantization, Parallel Transport and the Fourier Transform math.SG/0409542 Mei-Lin Yau: Cylindrical contact homology of subcritical Stein-Žllable contact manifolds SP: Spectral Theory ------------------- math.SP/0409575 E R Johnson, Michael Levitin, Leonid Parnovski: Existence of eigenvalues of a linear operator pencil in a curved waveguide -- localized shelf waves on a curved coast math.SP/0409558 Alexander K. Motovilov, Alexei V. Selin: Some sharp norm estimates in the subspace perturbation problem ST: Statistics -------------- math.ST/0409471 Cristina Butucea, Alexandre B. Tsybakov: Sharp optimality for density deconvolution with dominating bias -- / 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: Multiplication and Division in Triplets; division by zero-sized vectors Originator: israel@math.ubc.ca (Robert Israel) I can only add and subtract [triplets]. (W.R.Hamilton, speaking to his son in the 1840¹s.) Triplet Multiplication. DeŽne A={a1,a2,a3}, B={b1,b2,b3} and their triplet product AB={a1 b1 +a3 b2 +a2 b3, a2 b1 +a1 b2 +a3 b3, a3 b1 +a2 b2 +a1 b3}, where a1 etc. are real or complex numbers (this should also work with quaternion and octonion numbers, but I have not tested it). Triplet Multiplication uses the C3 group as a multiplication table. Conserved properties. For any triplet X={x1,x2,x3}, the functions Xs and Xp, Xs={x1+x2+x3}, Xp={((x1-x2)^2 +(x2-x3)^2 +(x3-x1)^2)/2}, are conserved properties (I call them sizes) for triplet algebra, i.e. ABs =As Bs, ABp =Ap Bp. Xs & Xp are the factors of the determinant of the C3 multiplication table with x1 etc. mapped onto the indices. They are divisors in the partial-fraction multiplicative inverse Xinv, which provides division by X:- (Triplet division. First version, B/A=B.Ainv) DeŽne Xinv={1/(3Xs) +(2x1-x2-x3)/(3Xp), 1/(3Xs) +(2x3-x1-x2)/(3Xp), 1/(3Xs) +(2x2-x1-x3)/(3Xp)} Ex. 1. Random values are used for A & B to show that the sizes are conserved and AB.Ainv=B. Note that A.Ainv={1,0,0}, the unit in this algebra:- A={3,1,4}, As=8, Ap=7, Ainv={5/56, 13/56, -11/56} B={1,5,2}, Bs=8, Bp=15, AB={25,24,15}, ABs=64=8x8, ABp=91=7x13 AB.Ainv={1,5,2}, i.e. =B. A.Ainv={1,0,0}. If one of As and Ap is zero, A is a zero-sized triplet, and Ainv appears to involve division by zero. (As=0, Ap=0 is the trivial vector, {0,0,0}.) Recall that division by natural numbers, or by integers, creates a dividend and a remainder. This idea is used to avoid division by zero in a second version of triplet division. It creates a dividend BAd and a remainder BAr, B/A =BAd +BAr, by projecting the dividend onto a constrained sub-space where either BAds=0 (if As=0) or Badp=0 (if Ap=0), and ejecting a remainder with the other constraint. If neither As nor Ap is zero, the remainder is zero; if both=0, the dividend is zero and the remainder is all of B:- (* Triplet division, second version, B/A=BAd + BAr *) Which[ As==0 And Not(Ap==0), BAd=B.{(2a1-a2-a3)/(3Ap), (2a3-a1-a2)/(3Ap), (2a2-a1-a3)/(3Ap)}, Ap==0 And Not(Ap==0), BAd=B.{1/(3As) , 1/(3As), 1/(3As)}, As==0 And Ap==0, Print[Division by {0,0,0}];BAd={0,0,0}, Not(As==0)And Not(Ap==0)(*i.e. Else*), BAd=B.{1/(3As) +(2a1-a2-a3)/(3Ap), 1/(3As) +(2a3-a1-a2)/(3Ap), 1/(3As) +(2a2-a1-a3)/(3Ap)} ]; BAr=B-A.BAd (This is done more elegantly via genInverse, a Mathematica function for any conservative algebra, in [1].) Ex. 2. Divide by C={3,3,3}, with Cs=9, Cp=0, Cinv={1/27, 1/27, 1/27}:- AC={24,24,24}, ACs=72=8x9, ACp=0; AC.Cinv gives AC/C={8/3,8/3,8/3}, (ACd)s=8, (ACd)p=0, with remainder ACr={1/3,-5/3,4/3}, (ACr)s=0, (ACr)p=7. Ex. 3. Divide by D={1,-3,2}, with Ds=0, Dp=21, Dinv={1/21,2/21,-1/7}:- AD={-7,0,7}, ADs=0, ADp=147=7x21; AD.Dinv gives {1/3,-5/3,4/3}, with remainder {8/3,8/3,8/3}, exchanging the dividend and remainder of Example 3. Generalization to Hoop Algebras Triplet algebra is the simplest conservative partial-fraction-division algebra. Every multiplication table with the Moufang division property deŽnes a conservative division algebra (I call them Hoops because they are rings or algebraic loops that conserve a shape); a few (including R, C, H, O, Clifford2 & Pauli-sigma algebras) have only one (repeated) factor and so do not have a partial-fraction inverse. (Most algebras have several conserved sizes. The appendix demonstrates that the Klein-group algebra conserves four.) Every Moufang loop (all groups, plus octonions) generates one or more division algebra; the majority of these have partial-fraction-division. Examples - Davenport Algebra, most Clifford algebras, over 70 others in [1]. Consequently, almost all conservative algebras are renormalizing, for division by any non-trivial vector gives a Žnite product, ejecting a remainder if the denominator-zeroes do not match the numerator-zeroes. Division by zero-sized vectors is always free from inŽnities in conservative algebras with more than one size. Roger Beresford. [1] http://library.wolfram.com/infocenter/MathSource/4894/ Appendix. The Klein-group algebra conserves a shape of four sizes. This edited transcript of a Mathematica session (using the Hoops.m package from [1]) demonstrates Klein-group multiplication and division (AB.Ainv=B), and conservation of the shape consisting of four sizes. genTimes[{a1,a2,a3,a4},{b1,b2,b3,b4}, K] = {a1 b1+a2 b2+a3 b3+a4 b4, a2 b1+a1 b2+a4 b3+a3 b4, a3 b1+a4 b2+a1 b3+a2 b4, a4 b1+a3 b2+a2 b3+a1 b4} A={3,1,2,5}; B={5,0,3,6}; AB=genTimes[A,B,K]={51,32,25,46} Ainv=genInverse[A]={-43/165, 23/165, -32/165, 67/165} genTimes[AB,Ainv] ={5,0,3,6}(* i.e.=B*) shape[{x1,x2,x3,x4}]= {x1+x2-x3-x4, x1-x2+x3-x4, x1-x2-x3+x4, x1+x2+x3+x4} shape[A], shape[B], shape[AB]} {-3,-1,5,11},{-4,2,8,14},{12,-2,40,154}} One really wants to have a division algebra (Stephen Adler, writing about physics.) === Subject: Re: Multiplication and Division in Triplets; division by zero-sized vectors Originator: israel@math.ubc.ca (Robert Israel) > I can only add and subtract [triplets]. (W.R.Hamilton, speaking to > his son in the 1840¹s.) > Triplet Multiplication. > DeŽne A={a1,a2,a3}, B={b1,b2,b3} and their triplet product AB={a1 b1 > +a3 b2 +a2 b3, a2 b1 +a1 b2 +a3 b3, a3 b1 +a2 b2 +a1 b3}, where a1 > etc. are real or complex numbers (this should also work with > quaternion and octonion numbers, but I have not tested it). Triplet > Multiplication uses the C3 group as a multiplication table. So this is the group algebra of C_3 over R or C. > Conserved properties. > For any triplet X={x1,x2,x3}, the functions Xs and Xp, > Xs={x1+x2+x3}, Xp={((x1-x2)^2 +(x2-x3)^2 +(x3-x1)^2)/2}, are conserved > properties (I call them sizes) for triplet algebra, i.e. ABs =As Bs, > ABp =Ap Bp. If K is a Želd of characteristic zero, then the group algebra K C_3 is isomorphic to K x K(omega) or K x K x K according to whether K contains a root of x^2 + x + 1 = 0 or not. Here omega is a primitive cube root of unity. Your maps X |--> X_s and X |--> X_p are related to the projections of K C_3 onto the factors of this decomposition. In all cases X_s is the projection of X onto the Žrst factor. In the case where K = R, then X_p = pi(X) sigma(pi(X)) where pi is the projection from R C_3 to R(omega) = C and sigma is complex conjugation: more explicitly pi(x_1,x_2,x_3) = x_1 + x_1 omega + x_2 omega^2. In the case where K = C, then X_p = pi_2(X) pi_3(X) where pi_2 and p_3 are the projections from R C_3 to R(omega) = C deŽned by pi_2(x_1,x_2,x_3) = x_1 + x_1 omega + x_2 omega^2 and pi_3(x_1,x_2,x_3) = x_1 + x_1 omega^2 + x_2 omega. > Xs & Xp are the factors of the determinant of the C3 multiplication > table with x1 etc. mapped onto the indices. Frobenius created the theory of group characters in his study of the factorization of the determinants of multiplication tables of arbitrary Žnite groups. > (Triplet division. First version, B/A=B.Ainv) > DeŽne Xinv={1/(3Xs) +(2x1-x2-x3)/(3Xp), 1/(3Xs) +(2x3-x1-x2)/(3Xp), > 1/(3Xs) +(2x2-x1-x3)/(3Xp)} As this shows, an element of K C_3 is invertible iff its projections to K and or K(omega) are all nonzero. > If one of As and Ap is zero, A is a zero-sized triplet, and Ainv > appears to involve division by zero. appears? Such elements of the group ring are not invertible. > (As=0, Ap=0 is the trivial > vector, {0,0,0}.) Recall that division by natural numbers, or by > integers, creates a dividend and a remainder. This idea is used to > avoid division by zero in a second version of triplet division. It > creates a dividend BAd and a remainder BAr, B/A =BAd +BAr, by > projecting the dividend onto a constrained sub-space where either > BAds=0 (if As=0) or Badp=0 (if Ap=0), and ejecting a remainder with > the other constraint. If neither As nor Ap is zero, the remainder is > zero; if both=0, the dividend is zero and the remainder is all of B:- > (* Triplet division, second version, B/A=BAd + BAr *) > Which[ > As==0 And Not(Ap==0), > BAd=B.{(2a1-a2-a3)/(3Ap), (2a3-a1-a2)/(3Ap), (2a2-a1-a3)/(3Ap)}, > Ap==0 And Not(Ap==0), > BAd=B.{1/(3As) , 1/(3As), 1/(3As)}, > As==0 And Ap==0, > Print[Division by {0,0,0}];BAd={0,0,0}, > Not(As==0)And Not(Ap==0)(*i.e. Else*), > BAd=B.{1/(3As) +(2a1-a2-a3)/(3Ap), 1/(3As) +(2a3-a1-a2)/(3Ap), > 1/(3As) +(2a2-a1-a3)/(3Ap)} > ]; > BAr=B-A.BAd This seems to be analogous to the Moore-Penrose inverse of matrices. There the M-P inverse A^+ of A satisŽes A A^+ A = A. In your division the inverse 1/A_d has the property that A(1/A_d) is an idempotent in K C_3. But there is still something a bit aribtrary and unsaisfying about this: A(1/A_d) is not necessarily the generator of the ideal in K C_3 generated by A. For example let A = (1, omega, omega^2) in C C_3. Then A_s = A_p = 0 so your algorithm gives an inverse of zero. But this can be obviated. We can deŽne A^+ as follows for K = C (I¹ll omit the case K = R). Let A = (a_1, a_2, a_3), b_0 = a_1 + a_2 + a_3, b_1 = a_1 + a_2 omega + a_3 omega^2 and b_2 = a_1 + a_2 omega^2 + a_3 omega. Let c_j = 1/b_j if c_j =/= 0 and c_j = 0 if b_j = 0. Let A^+ = (1/3) (c_0 + c_1 + c_2, c_0 + c_1 omega^2 + c_2 omega, c_0 + c_1 omega + c_2 omega^2). Then A A^+ is the idempotent generator of the ideal generated by A, and BA^+ is a solution of AX = B provided such a solution exists. (Your inverse BA_d lacks this property in general). More generally one can consider a general group algebra C G over a Žnite group, and deŽne a Moore-Penrose type inverse by decomposing CG as a product of matrix algebras and applying Moore-Penrose inversion in each. I was going to suggest this with C replaced by an arbitrary characteristic zero Želd K, but that would require Moore-Penrose inverses in matrix algebras over skew Želds. I don¹t know if there are such things .... anyone? Of course there are no problems if, as here G is abelian, -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: On Hodge and Betti numbers Epigone-thread: sherdphendspoy Originator: israel@math.ubc.ca (Robert Israel) While giving a seminar today, I got the following question to which I don¹t know the answer. On a Kaehler manifold, one can quite easily prove that the p-th Betti number b^p equals the sum of the Hodge numbers b^{r,s}, where the sum is over r and s such that r+s=p. The proof uses Hodge¹s theorem (saying that in each cohomology class there is a unique harmonic form) and the fact that the De Rham Laplacian equals (two times) the Dolbeault Laplacian, so the harmonic forms with respect to the two exterior derivatives are the same. For a non-Kaehler complex manifold, this proof does not work, since the two Laplacians may differ. (a) Does this mean the statement is also not true in this case? (b) If so, does anyone know a counter-example? Marcel Vonk === Subject: Re: On Hodge and Betti numbers Originator: israel@math.ubc.ca (Robert Israel) >On a Kaehler manifold, one can quite easily prove that the p-th Betti >number b^p equals the sum of the Hodge numbers b^{r,s}, where the sum >is over r and s such that r+s=p. The proof uses Hodge¹s theorem > ... >For a non-Kaehler complex manifold, this proof does not work, since >the two Laplacians may differ. >(a) Does this mean the statement is also not true in this case? >(b) If so, does anyone know a counter-example? Your question is equivalent to asking whether the Hodge to De Rham (aka Frolicher) spectral sequence degenerates. While it does for certain non Kahler manifolds, it will fail in general. Perhaps the simplest counterexample is the Iwasawa manifold (see the book of GrifŽths & Harris for details). - Donu Arapura === Subject: Re: On Hodge and Betti numbers Originator: israel@math.ubc.ca (Robert Israel) > While giving a seminar today, I got the following question to which I > don¹t know the answer. > On a Kaehler manifold, one can quite easily prove that the p-th Betti > number b^p equals the sum of the Hodge numbers b^{r,s}, where the sum > is over r and s such that r+s=p. The proof uses Hodge¹s theorem > (saying that in each cohomology class there is a unique harmonic form) > and the fact that the De Rham Laplacian equals (two times) the > Dolbeault Laplacian, so the harmonic forms with respect to the two > exterior derivatives are the same. > For a non-Kaehler complex manifold, this proof does not work, since > the two Laplacians may differ. > (a) Does this mean the statement is also not true in this case? > (b) If so, does anyone know a counter-example? > Marcel Vonk There is a spectral sequence whose E_1-term is the Hodge cohomology with the map induced by the exterior differential as E_1-differential and converging to de Rham cohomology. Hence the sum of the b^{r,s} equals b^p for all p precisely when this spectral sequence degenerates. In particular if there are non-closed holomorphic forms then the sum of the b^{r,s} is strictly larger than the b^p. An example of a compact complex manifold with non-closed is obtained as follows: Let H(C) be the group of upper triangular complex 3x3-matrices with 1¹s along the diagonal and let H(Z[i]) be the subgroup of matrices with entries in the Gaussian integers, Z+Zi. H(Z[i]) is a discrete subgroup and as C/Z[i] is compact so is X=H(C)/H(Z[i]), which is a homogeneous variety under left multiplication by H(C). The cotangent bundle is trivial spanned by translation invariant forms and hence, by the compactness of X, the holomorphic k-forms are exactly the translation invariant forms so that the space of k-forms may be identiŽed with Lambda^k g^*, the space of k-forms on the Lie algebra g of H(C). The exterior differential d from 1-forms to 2-forms, d: g^* ---> Lambda^2 g^* is dual to the the commutator map Lambda^2 g ---> g on the Lie algebra g (by the well-known formula expressing the exterior differential in terms of commutators of vector Želds). As g is a non-commutative Lie algebra there are non-closed 1-forms. === Subject: Two papers published by AGT Originator: israel@math.ubc.ca (Robert Israel) The following two papers have been published: (1) Eta invariants as sliceness obstructions and their relation to Casson-Gordon invariants by Stefan Friedl URL: http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-39.abs.html (2) Whitehead doubling persists by Stavros Garoufalidis URL: http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-40.abs.html Full details follow: (1) Algebraic and Geometric Topology URL: http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-39.abs.html Title: Eta invariants as sliceness obstructions and their relation to Casson-Gordon invariants Author(s): Stefan Friedl Abstract: We give a useful classiŽcation of the metabelian unitary representations of pi_1(M_K), where M_K is the result of zero-surgery along a knot K in S^3. We show that certain eta invariants associated to metabelian representations pi_1(M_K) --> U(k) vanish for slice knots and that even more eta invariants vanish for ribbon knots and doubly slice knots. We show that our vanishing results contain the Casson-Gordon sliceness obstruction. In many cases eta invariants can be easily computed for satellite knots. We use this to study the relation between the eta invariant sliceness obstruction, the eta-invariant ribbonness obstruction, and the L^2-eta invariant sliceness obstruction recently introduced by Cochran, Orr and Teichner. In particular we give an example of a knot which has zero eta invariant and zero metabelian L^2-eta invariant sliceness obstruction but which is not ribbon. Keywords: Knot concordance, Casson-Gordon invariants, Eta invariant Author(s) address(es): Department of Mathematics, Rice University, Houston, TX 77005, USA Email: friedl@rice.edu URL: http://math.rice.edu/~friedl/ (2) Algebraic and Geometric Topology URL: http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-40.abs.html Title: Whitehead doubling persists Author(s): Stavros Garoufalidis Abstract: The operation of (untwisted) Whitehead doubling trivializes the Alexander module of a knot (and consequently, all known abelian invariants), and converts knots to topologically slice ones. In this note we show that Whitehead doubling does not trivialize the rational function that equals to the 2-loop part of the Kontsevich integral. Secondary: 57M25 Keywords: Whitehead double, loop Žltration, Goussarov-Habiro, clovers, claspers, Kontsevich integral Received: 27 March 2001 Author(s) address(es): School of Mathemtaics, Georgia Institute of Technology Atlanta, GA 30332-0160, USA. Email: stavros@maths.gatech.edu URL: http://www.math.gatech.edu/~stavros === Subject: Re: Ricci and Weyl tensors in GR Originator: israel@math.ubc.ca (Robert Israel) I¹d also like to recomend learning the Jacobi equation and how a Jacoby Želds along geodesics behave. It involves the Ricci tensor or the like and then, to get things on a ball , you integrate over the vectors in the circle (or sphere) and you get the Ricci tensor ughh... or something like that I guess it¹s been a while since I worked this out, though I remember writing it in my notes. I do remember That I used Dr Baez¹s website almost as my sole inspiration, to study the stuff---and I had to go to Bishop Goldberg to understand the concept of sectional curvature. === Subject: Re: Algorithmic complexity of a graph Originator: israel@math.ubc.ca (Robert Israel) > Hi! > > I don¹t know if this is the right place to ask this question, but I > have no other ideas... > > Does by chance anyone know of a deŽnition of the ALGORITHMIC (or > Kolmogorov) complexity of a graph? And, in case, could you suggest > where I could look for it? > > I have only found deŽnitions of computational complexity... > > > cat > Well, I reckon you could Žx some system of notation and count the > bits necessary to describe a graph in such a system - for instance you > could count the 1¹s in the adjacency matrix. > Apropos, the number of spanning trees is often called the complexity > of a graph. Depending on your needs, you might Žnd this a useful > messure too. Obviously it¹s the Kolmogorov complexity of the adjacency matrix, which can be easily encoded as a bitstring. (Any bitstring representation should do) Have a look at the lecture notes of Alexander Shen on the web. (section Œincompressible graphs¹ IIRC) http://user.it.uu.se/~vorobyov/Courses/KC/2000/ -- Eray === Subject: Number Theory F in Z[x]... Originator: israel@math.ubc.ca (Robert Israel) Let F(x)=x^n+ n-1 and consider the statement: ,,There are an inŽnite number of positive integres m such that F(m) has at most k, k=k(n)= k(F,n), prime factors. Question: What about k= k(n) ? Upper bounds,... ? Some known results : a) k(2)= 5 - B.V.Levin -[1960] b) k(3)= 4 - Y. Wang - [1957] Other information: In case when G(x)=x(x+2), A.Selberg-[1947] has proved that k(G,n)=5. === Subject: Diff as a closed monoidal category? Received-SPF: none (mailbox4.ucsd.edu: domain of mod-submit@uni-berlin.de does not designate permitted sender hosts) Originator: israel@math.ubc.ca (Robert Israel) Sorry: this is a very na.95f question. Is there any hope that the category of differentiable manifolds and differentiable maps could be a _closed_ monoidal category? The monoidal structure should be given by the cartesian product: M x N , which is obviously a differentiable manifold, if M and N are such manifolds. The question is: which should be the internal hom functor ( Hom ) if there has to be an adjonction hom ( M x N , X ) = hom ( M , Hom (N, X) ) ? (Here hom means the set of differentiable maps and Hom the hipothetical internal hom functor.) For instance: is there any reasonable way to give a differentiable manifold structure (of inŽnite dimension?) to the set of differentiable maps between two differentiable manifolds Hom (M , N) ? This is possible in the category (or a suitable subcategory) of topological spaces, according to Borceux, and McLane books. Is absolutely impossible with differentiable manifolds? Agust.92 Roig === Subject: Re: I have a riddle I need help solving..!! > A man is walking down a road and comes to a fork in the road. There > are two paths he can take. He knows that at the end of one path is the > town of truths where everybody always tells the truth. At the end of > the other path is the town of lies where everyone always lies. the > problem is he doesn¹t know which path leads to which town. There is > another man standing where the road breaks into two different ones. > The Žrst man can ask the second man ONE question. What is the > question? He should ask for the telephone number of a taxi Žrm. The taxi driver can then be _told_ where to go, no further questions are needed. === Subject: Re: I have a riddle I need help solving..!! !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~( 5eZ41to5f%E@¹ELIi $t^ VcLWP@J5p^rst0+(Œ>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw >>A man is walking down a road and comes to a fork in the road. There >> are two paths he can take. He knows that at the end of one path is the >> town of truths where everybody always tells the truth. At the end of >> the other path is the town of lies where everyone always lies. the >> problem is he doesn¹t know which path leads to which town. There is >> another man standing where the road breaks into two different ones. >> The Žrst man can ask the second man ONE question. What is the >> question? > The question is: If I were to ask you which path leads to the town of > truths, which one would you indicate. That¹s just where you came from. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: numerically determine area and circumference by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iA71kvi26532; Hey guys I¹m currently making a small game (simple space game) and have a problem with numerically determine area and circumference of a ball that is being thrown. The problem is presented here: http://trasigkondensator.tripod.com/ball.htm === Subject: numerically determine area and circumference by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iA71kwm26566; Hey guys I¹m currently making a small game (simple space game) and have a problem with numerically determine area and circumference of a ball that is being thrown. The problem is presented here: http://trasigkondensator.tripod.com/ball.htm === Subject: close harmonian/bla by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iA71l0326644; how do i solve: does u=exp(2xy)cos(x^2-y^2) have close harmonian? === Subject: Re: Question about powers by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iA71kuR26482; >> ... it is not true >> that if p and q are rationals and p=q, then a^p = a^q. It depends on >> the representation of p and of q. >and later >> Usually, you deŽne a^x as e^{x*log(a)}. >hmmm... if p=q, and yet a^p is not identical to a^q, then I do not see >how the ^ operator can have any meaningful deŽnition at all, whether >it be the one given or any other? Yes, that¹s the point -- that the function f(z) = z^p is not well-deŽned (as an complex analytic function of z) if p is not an integer. In the second deŽnition, log(z) is not a well-deŽned analytic function of non-zero z: it has inŽnitely many branches where the values differ by multiples of 2pi*i. Todd Trimble === Subject: Re: Question about powers days. My association with the Department is that of an alumnus. >> ... it is not true >> that if p and q are rationals and p=q, then a^p = a^q. It depends on >> the representation of p and of q. >and later >> Usually, you deŽne a^x as e^{x*log(a)}. ->IN THE COMPLEX NUMBERS<-, and the exponentiation function already takes into account the normal form restriction given above. >hmmm... if p=q, and yet a^p is not identical to a^q, then I do not see >how the ^ operator can have any meaningful deŽnition at all, whether >it be the one given or any other? Then you aren¹t paying attention to the deŽnition. It is ->PERFECTLY LEGITIMATE<- to deŽne an operation based on a normal form for the elements of the sets. The rational numbers have multiple representations, but it is legitimate to deŽne an operation in terms of a single such representation. -- 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: riddle >I have hands but no thumbs. I have arms but can not reach. I run all >day but have no legs. what am I. > This is NOT a riddle group . It IS a math group. Perhaps the answer (which you have not guessed, and I do not know) has something to do with math. === Subject: Re: Uniqueness of physical objects in the physical universe. > It¹s a matter of scale. It¹s really very, very simple. > > As the universe isn¹t repeating itself, it¹s taking inŽnitely long. > > All I¹m saying is that it has an outer boudary - relative to an > observer. > Beyond this approximate distance, nothing exists relative to an > observer. > > Read up about inžation during the early stages of the Big Bangup Job. > However, if you traveled to the edge, you would Žnd yourself at the > center > of a similar looking place. And if two observers became separated by a > distance which was great enough, they would cease to exist relative to > one > another. It is very straightforward and I would say even stupidly > simple, > but kind of wierd. > > Get to the edge? You can¹t even get to Mars. However, such real > tangibles aside have you noticed how the horizon edge of the observable > universe keeps moving ahead of you while you¹re moving toward it and were > you to look behind in your soggy saga, you would Žnd the horizon edge > creeping up on you. Eventually when you run out of time, it will catch > you and for all of your travels, you will have gone from the center of the > observable to the center of the observable universe, ie nowheres. > Exactly. > Existence is relative. > The justiŽcation for the boundary itself is not really a bad one, could be > wrong, but seems pretty credible. Can be derived with no math whatsoever. Just to recap, lets say you wanted to build a clock so that you could observe the passage of time. You want to make observations of time passing. You could use our solar system, as has been done for thousands of years. The Earth goes around the sun marking off the years as if it were one of the hands of a clock. You could also use a galaxy, but you would have to wait a long, long time for it to make even a single rotation. You could, in principle, base your clock on the rotation of a galaxy. It should actually work quite well. However, if your clock becomes too big, like the outer limits of the universe, it is simply so vast that motion is very near zero relative to man, and so it simply cannot be used as a clock because it¹s motion is so close to zero. It is motionless (relative to man). And so your clock wont work if you base it on rotations of outer boundaries of universe. Therefore, time becomes unobservable, and spacetime no longer exists relative to an observer here on Earth, because spacetime really has no meaning if time is unobservable. This is why the universe is Žnite, but open, and existence is relative. Now, On a scale of 0 to 10, 0 being not a crackpot at all, and 10 being completely wacky crackpot material, what do you think ? 8 ? === Subject: Re: Uniqueness of physical objects in the physical universe. <8Ahjd.55759$HA.16732@attbi_s01> === Subject: Uniqueness of physical objects in the physical universe. If you push this button time goes backwards goes time button this push you if. > Just to recap, lets say you wanted to build a clock so that you > could observe the passage of time. You want to make observations of > time passing. Objective or surjective time? > You could use our solar system, as has been done for thousands of > years. The Earth goes around the sun marking off the years as if it > were one of the hands of a clock. Wouldn¹t work, too slow, I¹d be late for work. > You could also use a galaxy, but you would have to wait a long, long > time for it to make even a single rotation. You could, in principle, > base your clock on the rotation of a galaxy. It should actually work > quite well. Naw totally inadequate and too clumbersome. More practical is the vibration of a cesium atom. > However, if your clock becomes too big, like the outer limits of the > universe, it is simply so vast that motion is very near zero > relative to man, and so it simply cannot be used as a clock because > it¹s motion is so close to zero. It is motionless (relative to man). > And so your clock wont work if you base it on rotations of outer > boundaries of universe. You¹re getting too big for your britches, think small. What¹s the smallest interval of time that can be measured and if the interals of time got shorter and shorter, would time vanish? > Therefore, time becomes unobservable, and spacetime no longer exists > relative to an observer here on Earth, because spacetime really > has no meaning if time is unobservable. Nope, last I looked it¹s still there. > This is why the universe is Žnite, but open, and existence is > relative. Ya, I inherited existence from my relatives. > On a scale of 0 to 10, 0 being not a crackpot at all, and 10 > being completely wacky crackpot material, what do you think ? Hey, if you¹re going to be a crackpot, let¹s at least get a laugh out of it. ;-) In no time at all, time had begun, for it¹s time had come. ---- === Subject: Re: Uniqueness of physical objects in the physical universe. === > Subject: Uniqueness of physical objects in the physical universe. > If you push this button time goes > backwards > goes time button this push you if. > Just to recap, lets say you wanted to build a clock so that you > could observe the passage of time. You want to make observations of > time passing. > Objective or surjective time? > You could use our solar system, as has been done for thousands of > years. The Earth goes around the sun marking off the years as if it > were one of the hands of a clock. > Wouldn¹t work, too slow, I¹d be late for work. > You could also use a galaxy, but you would have to wait a long, long > time for it to make even a single rotation. You could, in principle, > base your clock on the rotation of a galaxy. It should actually work > quite well. > Naw totally inadequate and too clumbersome. > More practical is the vibration of a cesium atom. > However, if your clock becomes too big, like the outer limits of the > universe, it is simply so vast that motion is very near zero > relative to man, and so it simply cannot be used as a clock because > it¹s motion is so close to zero. It is motionless (relative to man). > And so your clock wont work if you base it on rotations of outer > boundaries of universe. > You¹re getting too big for your britches, think small. What¹s the > smallest interval of time that can be measured and if the interals of time > got shorter and shorter, would time vanish? > Therefore, time becomes unobservable, and spacetime no longer exists > relative to an observer here on Earth, because spacetime really > has no meaning if time is unobservable. > Nope, last I looked it¹s still there. > This is why the universe is Žnite, but open, and existence is > relative. > Ya, I inherited existence from my relatives. > On a scale of 0 to 10, 0 being not a crackpot at all, and 10 > being completely wacky crackpot material, what do you think ? > Hey, if you¹re going to be a crackpot, > let¹s at least get a laugh out of it. ;-) > In no time at all, time had begun, for it¹s time had come. working on the James Harris equation of squaring the circle via revolutionary new cubic equation prime number generator such that disingenuous algebraic violinists have been harrassing him with tinfoil hats and torturing his mind intentionally with circular logic mind maze. If you dont believe me, just look at this http://www.timecube.com/ My tin foil hat is shaped like that of Paul Revere. This is because my ideas are so revolutionary. === Subject: Re: Uniqueness of physical objects in the physical universe. <8Ahjd.55759$HA.16732@attbi_s01> > Hey, if you¹re going to be a crackpot, > let¹s at least get a laugh out of it. ;-) > In no time at all, time had begun, for it¹s time had come. > If you dont believe me, just look at this > http://www.timecube.com Oh, that¹s your problem, having dirty thoughts. -- === Subject: algebraic geometry by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iA76FAu15071; If F is a Želd of characteristic p, then how could one show that every line passing through 0 is a tangent line to the curve y=x^(p+1)? === Subject: Re: Please help me solve this riddle by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iA76F7x15014; >what runs but doesn¹t walk, >what has a month but doesn¹t talk, >what has a bed but doesn¹t sleep, >what has a head but doesn¹t weep. A RIVER === Subject: Re: Precalculus Help! by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iA76F8J15033; help! solve for x: 0.5(a^x-a^-x)3, a>0 solve for x: log(base 5)(x+1)<2 solve for x:(base b)=(b) log(b)(3x+2)+log(b)64+log(b)2-log(b)(3x-2) what characterizes exponential functions? === Subject: Re: Precalculus Help! >help! >solve for x: >0.5(a^x-a^-x)3, a>0 [snip more problems] >what characterizes exponential functions? We¹re not averse to helping with homework, but it is not help simply to give you answers that you could get by reading your textbook. Please show us what you¹ve done to solve these problems, and then we can give you speciŽc, focused help where you got stuck. -- Stan Brown, Oak Road Systems, Tompkins County, New York, USA http://OakRoadSystems.com/ And if you¹re afraid of butter, which many people are nowa- days, (long pause) you just put in cream. --Julia Child === Subject: Re: Precalculus Help! memogoez@yahoo.com wants help: >help! >solve for x: >0.5(a^x-a^-x)3, a>0 That is an expression. What is the relation? What is the expression supposed to equal? G C === Subject: Re: Numbers with Non-Decreasing Digits There are countably inŽnitely many such numbers. Consider the number 19 Add a digit to get 119 Again, you get 1119 11119 111119 1111119, and so on. How many times can you do this ? Aleph null. Filling in the gaps with the rest of the possible numbers under consideration still gives aleph null, because aleph null + aleph null = aleph null. === Subject: Re: Numbers with Non-Decreasing Digits alt.math.undergrad: > There are countably inŽnitely many such numbers. > Consider the number 19 > Add a digit to get 119 > Again, you get 1119 > 11119 > 111119 > 1111119, None of these meets the requirements of Bernd¹s problem: one of the requirements is that the number in position k be at most k. And even if they did satisfy the conditions, your answer shows that you¹ve completely missed the point of the question, which is to count how many solutions of length n there are _for_each_n_. [...]