Subject: International Conference on Computational Science (2nd CFP) Content-Length: 2487 Originator: rusin@vesuvius Please excuse us if you receive this announcement more than once. ************************************************************* ******** * ICCS 2005: 5th International Conference on Computational Science * * Atlanta, May 22-25, 2005 http://www.iccs-meeting.org/ * ************************************************************* ******** You are invited to submit a paper and/or a proposal to organize a workshop at ICCS 2005, Altanta, USA, May 22-25, 2005. Please see http://www.iccs-meeting.org/ for more information. The theme for ICCS 2005 in Atlanta, USA, is Advancing Science through Computation, to mark several decades of progress in computational science theory and practice, leading to greatly improved applications science. Original contributions not exceeding 8 pages are invited for publication and oral presentation. All accepted papers will be printed in the conference proceedings published by Springer-Verlag in the Lecture Notes in Computer Science series. Selected papers will also be published as special issues of appropriate journals. Topics of Interest ------------------ ICCS 2005 invites original contributions on all topics related to computational science, including, but not limited to: * ScientiÞc Computing * Problem Solving Environments * Advanced Numerical Algorithms * Complex Systems: Modeling and Simulation * Hybrid Computational Methods * Computational Science in Data Mining/Information Retrieval * Web- and Grid-based Simulation and Computing * Parallel and Distributed Computing * Visualization in Computational Science * Applications of Computation as a ScientiÞc Paradigm * New Algorithms for Computational Kernels and Applications * Education in Computational Science Important deadlines: -------------------- NotiÞcation of acceptance of papers: .............. January 31, 2005 Camera ready papers: .............................. February 14, 2005 Early registration: .................................. March 30, 2005 ------------------------- iccs2005@mathcs.emory.edu ScientiÞc Chair ..................................... Vaidy Sunderam Workshops Chair ..................................... Dick van Albada Overall Co-chair ...................................... Jack Dongarra Overall Chair ...................................... Peter M.A. Sloot === Subject: This week in the mathematics arXiv (6 Sep - 10 Sep) Content-Length: 27129 Originator: rusin@vesuvius 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 (6 Sep - 10 Sep) ------------------------------------------------ AC: Commutative Algebra ----------------------- math.AC/0409156 Clare D¹Cruz: A formula for the multiplicity of the multi-graded extended Rees algebra math.AC/0409143 Anurag K. Singh: The F-signature of an afÞne semigroup ring math.AC/0409114 Jeaman Ahn, Juan C. Migliore: Some Geometric Results arising from the Borel Fixed Property math.AC/0409107 MuÞt Sezer, R. James Shank: Coinvariants for modular representations of cyclic groups of prime order math.AC/0409097 Juergen Herzog, Takayuki Hibi: Cohen-Macaulay Polymatroidal Ideals math.AC/0409096 Clare D¹Cruz: Multigraded extended Rees algebras of $m$-primary ideals math.AC/0409090 Elena Guardo, Adam Van Tuyl: Powers of Complete Intersections: graded Betti numbers and applications math.AC/0409059 Tony J. Puthenpurakal: A short note on the non-negativity of partial Euler characteristics math.AC/0409058 Tony J. Puthenpurakal: Invariance of a length associated to a reduction math.AC/0409056 Adam Van Tuyl: On the deÞning ideal of a set of points in multi-projective space math.AC/0409051 Tony J. Puthenpurakal: The Hilbert Function of a Maximal Cohen-Macaulay Module AG: Algebraic Geometry ---------------------- math.AG/0409151 Akira Ishii, Hokuto Uehara: Autoequivalences of derived categories on the minimal resolutions of $A_n$-singularities on surfaces math.AG/0409145 Brian Osserman: Logarithmic connections with vanishing p-curvature hep-th/0408055 L. Griguolo, D. Seminara, R.J. Szabo: Two-Dimensional Yang-Mills Theory and Moduli Spaces of Holomorphic Differentials math.AG/0409129 Antonio Laface, Luca Ugaglia: On standard birational transformations of P^n and special linear systems math.AG/0409128 Antonio Laface, Luca Ugaglia: A conjecture on special linear systems of P^3 math.AG/0409127 Antonio Laface, Luca Ugaglia: A counterexample to a conjecture on linear systems on P^3 math.AG/0409126 L. F. Tabera: Tropical Cramer¹s rule and tropical Pappus¹ Theorem math.AG/0409125 Nicolas Perrin: Courbes elliptiques sur la variete spinorielle de dimension 10 math.AG/0409123 Nero Budur: On the V-Þltration of D-modules math.AG/0409118 Julianna S. Tymoczko: Paving Hessenberg varietes by afÞnes math.AG/0409116 Matt Kerr, James Lewis, Stefan Muller-Stach: The Abel-Jacobi map for higher Chow groups math.AG/0409078 Shigetaka Fukuda: On the projective fourfolds with almost numerically positive canonical divisors math.AG/0409077 Ilya Tyomkin: On Severi varieties and Moduli spaces of curves in arbitrary characteristic math.AG/0409071 Claus Mokler: Integrating inÞnite-dimensional Lie algebras by a Tannaka reconstruction (Part II) math.AG/0409066 Thomas Scanlon: Local Andr¹{e}-Oort conjecture for the universal abelian variety math.AG/0409060 Takeo Nishinou, Bernd Siebert: Toric degenerations of toric varieties and tropical curves math.AG/0409055 Dmitri I. Panyushev: An extension of Rais¹ theorem and seaweed subalgebras of simple Lie algebras math.AG/0409053 Claus Mokler: Integrating inÞnite-dimensional Lie algebras by a Tannaka reconstruction (Part I) math.AG/0409038 Ai-Ko Liu: Cosmic String, Harvey-Moore Conjecture and Family Seiberg-Witten Theory math.AG/0409037 Ai-Ko Liu: The Residual Intersection Formula of Type II Exceptional Curves AP: Analysis of PDEs -------------------- math.AP/0409153 Yuxin Ge, Ruihua Jing, Frank Pacard: Bubble towers for supercritical semilinear elliptic equations math.AP/0409098 D. Del Santo, M. Prizzi: On the absence of rapidly decaying solutions for parabolic operators whose coefÞcients are non-Lipschitz continuous in time math.AP/0409052 M. Berti, P. Bolle: Bifurcation of free vibrations for completely resonant wave equations AT: Algebraic Topology ---------------------- math.AT/0409133 Kevin P. Knudson: Invariant chains and the homology of quotient spaces CA: Classical Analysis and ODEs ------------------------------- math.CA/0409063 Stephen Semmes: Potpourri CO: Combinatorics ----------------- math.CO/0409147 Lorenz Friess: Proof of a conjecture of Hadwiger math.CO/0409134 Nurit Gazit, Michael Krivelevich: On the asymptotic value of the choice number of complete multi-partite graphs math.CO/0409105 Bridget Eileen Tenner: Tiling Parity Results and the Holey Square Solution math.CO/0409099 Henry Crapo, William Schmitt: A unique factorization theorem for matroids math.CO/0409081 Torsten Schoneborn, Gunter M. Ziegler: The Topological Tverberg Problem and winding numbers math.CO/0409080 Henry Crapo, William Schmitt: The Free product of Matroids math.CO/0409062 William M. Y. Goh, Robert Boyer: On the Zero Attractor of the Euler Polynomials math.CO/0409054 John Shareshian, Michelle L. Wachs: Torsion in the Matching Complex and Chessboard Complex math.CO/0409050 Roland Bacher: On generating series of complementary planar trees math.CO/0409041 Chunhui Lai: An extremal problem on potentially $K_{m}-C_{4}$-graphic sequences CT: Category Theory ------------------- math.CT/0409158 Benno van den Berg, Federico de Marchi: Non-well-founded trees in categories CV: Complex Variables --------------------- math.CV/0409170 Dan Popovici: $L^2$ Extension for Jets of Holomorphic Sections of a Hermitian Line Bundle math.CV/0409120 Albert Boggess, Daniel Jupiter: Global approximation of CR functions on Bloom-Graham model graphs in $C^n$ math.CV/0409049 Joerg Winkelmann: Surface Foliations with Compact complex leaves are holomorphic math.CV/0409048 Joerg Winkelmann: On a special class of complex tori DG: Differential Geometry ------------------------- math.DG/0409167 Francisco Martin Cabrera: Special Almost Hermitian Geometry math.DG/0409157 Frank Klinker: Polynomial poly-vector Þelds math.DG/0409152 Andrej A. Agrachev, Natalia N. Chtcherbakova, Igor Zelenko: On curvatures and focal points of dynamical Lagrangian distributions and their reductions by Þrst integrals math.DG/0409137 Simon G. Chiossi, Anna Fino: Conformally parallel G_2 structures on a class of solvmanifolds math.DG/0409136 Florin Belgun, Nicolas Ginoux, Hans-Bert Rademacher: Twistor spinors with zeros on compact orbifolds math.DG/0409121 Pralay Chatterjee, Dave Witte Morris: Divergent torus orbits in homogeneous spaces of Q-rank two math.DG/0409104 Florin Belgun, Andrei Moroianu, Uwe Semmelmann: Killing Forms on Symmetric Spaces math.DG/0409093 Marco Gualtieri: Generalized geometry and the Hodge decomposition math.DG/0409089 Gianmarco Capitanio: Simple tangential families and perestroikas of their envelopes math.DG/0409073 Jun-ichi Inoguchi, Magdalena Toda: Timelike Minimal Surfaces via Loop Groups math.DG/0409065 Simon P. Morgan: Mixed Dimensional Compactness with Dimension Collapsing from Sn-1 Bundle Measures math.DG/0409064 Simon P. Morgan: Harmonic maps of surfaces approaching the boundary of moduli space and the elimation of bubbling DS: Dynamical Systems --------------------- math.DS/0409144 Ivan Tyukin, Danil Prokhorov, Cees van Leeuwen: Parameter estimation and control for a class of systems with nonlinear parametrization math.DS/0409131 Jaume Llibre, Michael Todd: Periods for holomorphic maps via Lefschetz numbers math.DS/0409085 Stefano Luzzatto: Stochastic-like behaviour in nonuniformly expanding maps math.DS/0409084 Henk Bruin, Stefano Luzzatto: Topological invariance of the sign of the Lyapunov exponents in one-dimensional maps math.DS/0409061 Artur Avila, David Damanik: Generic singular continuous spectrum for ergodic Schrodinger operators FA: Functional Analysis ----------------------- math.FA/0409139 Narcisse Randrianantoanina: A weak-type inequality for non-commutative martingales and applications math.FA/0409138 Ronald G. Douglas & Gadadhar Misra: On quasi-free Hilbert modules math.FA/0409119 Daniel Jupiter, David Redett: Invariant subspaces of $RL^1$ math.FA/0409100 G. Olafsson, E. Ournycheva, B. Rubin: Multiscaled wavelet transforms, ridgelet transforms, and Radon transforms on the space of matrices GM: General Mathematics ----------------------- math.GM/0409113 Haibin Wang, Praveen Madiraju, Yanqing Zhang, Rajshekhar Sunderraman: Interval Neutrosophic Sets math.GM/0409070 Alexander Sakharov: Median Logic GN: General Topology -------------------- math.GN/0409092 Gwen Spencer, Francis Edward Su: The LSB theorem implies the KKM lemma math.GN/0409072 Boaz Tsaban: SPM Bulletin 10 math.GN/0409069 Boaz Tsaban: Some new directions in inÞnite-combinatorial topology math.GN/0409068 Heike Mildenberger, Saharon Shelah, Boaz Tsaban: The combinatorics of tau-covers GR: Group Theory ---------------- math.GR/0409149 Yacine Ait Amrane: Cohomology of Drinfeld symmetric spaces and Harmonic cochains math.GR/0409146 Anton A. Klyachko: The Kervaire-Laudenbach conjecture and presentations of simple groups math.GR/0409094 Benson Farb, G. Christopher Hruska: Commensurability invariants for nonuniform tree lattices math.GR/0409087 Sucharit Sarkar: Commutators and squares in free groups math.GR/0409074 Michael K. Kinyon, J.D. Phillips: Rectangular loops and rectangular quasigroups math.GR/0409043 Helge Glockner: Contraction groups for tidy automorphisms of totally disconnected groups GT: Geometric Topology ---------------------- math.GT/0409166 Dan Burghelea, Stefan Haller: The geometric complex of a Morse-Bott-Smale pair and an extension of a theorem by Bismut-Zhang math.GT/0409086 Tomomi Kawamura: Links associated with generic immersions of graphs KT: K-Theory and Homology ------------------------- math.KT/0409164 J. Brodzki, R.J. Plymen: Entire cyclic homology of Schatten ideals math.KT/0409039 M. A. Farinati: Hochschild duality, localization and smash products LO: Logic --------- math.LO/0409142 Mark Burgin: Axiomatic Theory of Algorithms: Computability and Decidability in Algorithmic Classes math.LO/0409110 Arnold W. Miller, Juris Steprans: The number of translates of a closed nowhere dense set required to cover a Polish group math.LO/0409109 Y. Firat Celikler: Dimension theory and parameterized normalization for D-semianalytic sets over non-Archimedean Þelds math.LO/0409103 Laurent Moret-Bailly: Elliptic curves and Hilbert¹s tenth problem for algebraic function Þelds over real and p-adic Þelds math.LO/0409095 Y. Firat Celikler: Parameterized stratiÞcation and piece number of D-semianalytic sets MG: Metric Geometry ------------------- math.MG/0409067 Xiangdong Xie: Foldable cubical complexes of nonpositive curvature MP: Mathematical Physics ------------------------ quant-ph/0408151 Maurice Robert Kibler, Tidjani Negadi: On the q-analogue of the hydrogen atom math-ph/0409020 Plamen Stefanov: Approximating resonances with the Complex Absorbing Potential Method math-ph/0409019 Juerg Froehlich, Enno Lenzmann: Mean-Field Limit of Quantum Bose Gases and Nonlinear Hartree Equation math-ph/0409018 Khosrow Chadan, Reido Kobayashi: The Absence of Positive Energy Bound States for a Class of Nonlocal Potentials math-ph/0409017 A. Elgart, G. M. Graf, J. H. Schenker: Equality of the bulk and edge Hall conductances in a mobility gap math-ph/0409016 Kazuhiro Hikami: Quantum Invariant, Modular Form, and Lattice Points math-ph/0409015 H. S. Bhat: Semidirect product reduction theory: a user¹s guide hep-lat/0409013 David H. Adams: Testing universality and the fractional power prescription for the staggered fermion determinant gr-qc/0409030 Christian G. Boehmer: Static perfect þuid balls with given equation of state and cosmological constant cond-mat/0409127 Pierluigi Contucci, Francesco Unguendoli: Long-Range Order in Finite-Dimensional Spin-Glass Models quant-ph/0409012 O. Chavoya-Aceves: Generalization of Hamilton-Jacobi method and its consequences in classical, relativistic, and quantum mechanics math-ph/0409014 Yan V. Fyodorov: On Hubbard-Stratonovich Transformations over Hyperbolic Domains math-ph/0409013 Doug Pickrell: An invariant measure for the loop space of a simply connected compact symmetric space math-ph/0409012 James P. Kelliher: Navier-Stokes equations with Navier boundary conditions for a bounded domain in the plane math-ph/0409011 James P. Kelliher: The inviscid limit for two-dimensional incompressible þuids with unbounded vorticity math-ph/0409010 Lukasz Bratek: Linear stability analysis of hedgehogs in the Skyrme model on a three-sphere. Critical phenomena and spontaneously broken reþection symmetry astro-ph/0409081 Marc Lachieze-Rey: Harmonic projection and multipole Vectors nlin.SI/0406059 Ken-ichi Maruno, Gino Biondini: Resonance and web structure in discrete soliton systems: the two-dimensional Toda lattice and its fully discrete and ultra-discrete versions math-ph/0409009 Andrei Gabrielov, Dmitry Novikov, Boris Shapiro: Mystery of point charges math-ph/0409008 Yong Moon Park: Improvement of Uncertainty Relations for Mixed States math-ph/0409007 Peter D. Hislop, Frederic Klopp, Jeffrey H. Schenker: Continuity with respect to Disorder of the Integrated Density of States hep-th/0409026 Jorgen Rasmussen: On SU(2) Wess-Zumino-Witten models and stochastic evolutions hep-th/0409005 A. Chervov, L. Rybnikov: Deformation quantization of submanifolds and reductions via Duþo-Kirillov-Kontsevich map hep-th/0207048 Yong Zhang: Hopf Algebraic Structures in Proving Perturbative Unitarity NT: Number Theory ----------------- math.NT/0409141 Gaetan Chenevier: On number Þelds with given ramiÞcation math.NT/0409122 Thomas Geisser, Lars Hesselholt: Bi-relative algebraic K-theory and topological cyclic homology math.NT/0409115 Luis Dieulefait: Elliptic mod ell Galois representations which are not minimally elliptic math.NT/0409102 representations and rigid Calabi-Yau threefolds math.NT/0409083 Damian Roessler: A note on the Manin-Mumford conjecture OA: Operator Algebras --------------------- math.OA/0409169 Hiroyuki Osaka, N. Christopher Phillips: Furstenberg transformations on irrational rotation algebras math.OA/0409168 Hiroyuki Osaka, N. Christopher Phillips: Stable and real rank for crossed products by automorphisms with the tracial Rokhlin property math.OA/0409124 Marius Dadarlat, Cornel Pasnicu: Continuous Fields of Kirchberg C*-algebras math.OA/0409091 Gilles Pisier: A similarity degree characterization of nuclear $C^*$-algebras math.OA/0409075 Alan Hopenwasser, Jurtin R. Peters, Stephen C. Power: Subalgebras of Graph C*-Algebras math.OA/0409044 V. Toledano-Laredo: Fusion of Positive Energy Representations of LSpin(2n) math.OA/0409040 Slawomir Klimek: A note on noncommutative holomorphic and harmonic functions on the unit disk OC: Optimization and Control ---------------------------- math.OC/0409111 Igor Borovikov: Order Reduction of Optimal Control Systems PR: Probability --------------- math.PR/0409155 O. G. Somlyanov, H. v Weizsaecker, O. Wittich: Chernoff¹s Theorem and Discrete Time Approximations of Brownian Motion on Manifolds math.PR/0409135 Carles Rovira, amy Tindel: On the Brownian directed polymer in a Gaussian random environment math.PR/0409112 Galin L. Jones: On the Markov chain central limit theorem math.PR/0409088 Mathew D. Penrose, J. E. Yukich: Normal Approximation in Geometric Probability math.PR/0409079 T. Bodineau: Translation invariant Gibbs states for the Ising model math.PR/0409076 Ted Theodosopoulos: Uncertainty relations in models of market microstructure math.PR/0409057 Martin Hairer, Jonathan C. Mattingly, Etienne Pardoux: Malliavin calculus and ergodic properties of highly degenerate 2D stochastic Navier--Stokes equation cond-mat/0409060 E. Ben-Naim, S. Redner: Winning quick and dirty: the greedy random walk math.PR/0409047 U.A. Rozikov; Yu.M.Suhov: Gibbs Measures For SOS Models On a Cayley Tree math.PR/0409046 U.A.Rozikov: An Analysis of Ising Type Models On Cayley Tree by a Contour Argument math.PR/0409042 S. Satheesh: Why There Are No Gaps In The Support Of Non-Negative Integer-Valued InÞnitely Divisible Laws? QA: Quantum Algebra ------------------- math.QA/0409159 Xiao-Wu Chen, Pu Zhang: Comodules of $U_q(sl_2)$ and modules of $SL_q(2)$ via quiver math.QA/0409140 Haisheng Li: Abelianizing vertex algebras math.QA/0409130 Stefan Jansen, Stefan Waldmann: The H-Covariant Strong Picard Groupoid math.QA/0409117 Igor B. Frenkel, Konstantin Styrkas: ModiÞed regular representations of afÞne and Virasoro algebras, VOA structure and semi-inÞnite cohomology math.QA/0409106 Lars Kadison: Hopf algebroids and Galois extensions hep-th/0409007 A. Chervov, D. Talalaev: Universal G-oper and Gaudin eigenproblem RA: Rings and Algebras ---------------------- math.RA/0409163 Zhaoyong Huang: On the grade of modules over noetherian rings math.RA/0409162 E. L. Green, G. Hartman, E. N. Marcos, O. Solberg: Resolutions over Koszul algebras math.RA/0409161 Zhaoyong Huang, Hourong Qin: Duality in Auslander¹s $k$-Gorenstein rings math.RA/0409150 Zhaoyong Huang: Wakamatsu Tilting Modules, $U$-Dominant Dimension and $k$-Gorenstein Modules math.RA/0409108 Fernando Guzman: The Hyperradical and The Hopkins-Levitzki Theorem for Modular Lattices RT: Representation Theory ------------------------- math.RT/0409101 Yasufumi Hashimoto: Arithmetic expressions of Selberg¹s zeta functions for congruence subgroups SG: Symplectic Geometry ----------------------- math.SG/0409160 C. Caubel, A. Nemethi, P. Popescu-Pampu: Milnor open books and Milnor Þllable contact 3-manifolds math.SG/0409148 Tanya Schmah: A cotangent bundle slice theorem math.SG/0409082 Klaus Niederkruger: 5-dimensional contact SO(3)-manifolds and Dehn twists SP: Spectral Theory ------------------- math.SP/0409154 Dmitry Jakobson, Michael Levitin, Nikolai Nadirashvili, Iosif Polterovich: Spectral problems with mixed Dirichlet-Neumann boundary conditions: isospectrality and beyond ST: Statistics -------------- math.ST/0409165 J.J. Lok, R.D. Gill, A.W. van der Vaart, J.M. Robins: Estimating the causal effect of a time-varying treatment on time-to-event using structural nested failure time models math.ST/0409132 Lior Pachter, Bernd Sturmfels: The Mathematics of Phylogenomics math.ST/0409045 J.J. Lok: Mimicking counterfactual outcomes for the estimation of causal effects -- / 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: Re: a non-linear second-order PDE Content-Length: 2139 Originator: rusin@vesuvius > >I consider the following non-linear second-order partial differential > >equation with two non-negative variables x and y: > >(a+ b x) [ df/dx d^2f/(dx dy) - df/dy d^2f/(dx^2) ] + b y [ df/dx d^2 > >f/d(y^2)- df/dy d^2f/(dx dy)] =0 > >where a >0 and b is a real number. Additional conditions are > >f(x,0) = (a + b x)^(1-1/b) /(1/b (1-1/b)) > >and > >df(x,0)/dy =0. > I assume you mean df/dy = 0 when y = 0. > >for all x. > >(For b=1, f(x,0) converges to logarithmic function ln(a+x), but that > >should not really matter). > >One solution to the PDE is: > >f(x,y) = 1/(1/b (1-1/b)) [ (a + b x)^(1- 1/b) + c2 b y^(1-1/b) ] > This doesn¹t satisfy your second additional condition in general, however: > df/dy has a singularity at y=0 if b > 0. It does work if c2 = 0 or if > b < 0. Of course, in general there¹s also a singularity when a+bx=0. > >with constants c1, c2. > You didn¹t have a c1. > >My question: Are there other solutions? If so, which? If no, how can > >one prove uniqueness? > Solutions of your PDE include > f(x,y) = c_0 + sum_{j=1}^n c_j (a+bx)^(k_j) y^(p-k_j) > for any constants c_j, k_j and p. For example, you could add to > your solution a term in (a+bx)^(-1-1/b) y^2 and get another solution > with the same values of f(x,0) and df/dy (x,0). So, no uniqueness > there. > BTW, an interesting feature of the PDE is that if f(x,y) is a solution, so > is f(x,y)^k for any k, or ln(f(x,y)), or exp(f(x,y)). I believe I¹ve found the general solution to the PDE: f(x,y) = G((a+bx) H(y/(a+bx))) where G and H are arbitrary (twice differentiable) functions. Wlog we can take H(0) = 1, and then f(x,0) = G(a+bx) determines G, while df/dy (x,0) = 0 as long as H¹(0) = 0. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: a non-linear second-order PDE Epigone-thread: gromstunpand Content-Length: 2366 Originator: rusin@vesuvius >>>I consider the following non-linear second-order partial >differential >>>equation with two non-negative variables x and y: >>>(a+ b x) [ df/dx d^2f/(dx dy) - df/dy d^2f/(dx^2) ] + b y [ df/dx >d^2 >>>f/d(y^2)- df/dy d^2f/(dx dy)] =0 >>>where a >0 and b is a real number. Additional conditions are >>>f(x,0) = (a + b x)^(1-1/b) /(1/b (1-1/b)) >>>and >>>df(x,0)/dy =0. >>I assume you mean df/dy = 0 when y = 0. >>>for all x. >>>(For b=1, f(x,0) converges to logarithmic function ln(a+x), but that >>>should not really matter). >>>One solution to the PDE is: >>>f(x,y) = 1/(1/b (1-1/b)) [ (a + b x)^(1- 1/b) + c2 b y^(1-1/b) ] >>This doesn¹t satisfy your second additional condition in general, >however: >>df/dy has a singularity at y=0 if b > 0. It does work if c2 = 0 or >if >>b < 0. Of course, in general there¹s also a singularity when a+bx=0. >>>with constants c1, c2. >>You didn¹t have a c1. >>>My question: Are there other solutions? If so, which? If no, how can >>>one prove uniqueness? >>Solutions of your PDE include >>f(x,y) = c_0 + sum_{j=1}^n c_j (a+bx)^(k_j) y^(p-k_j) >>for any constants c_j, k_j and p. For example, you could add to >>your solution a term in (a+bx)^(-1-1/b) y^2 and get another solution >>with the same values of f(x,0) and df/dy (x,0). So, no uniqueness >>there. >>BTW, an interesting feature of the PDE is that if f(x,y) is a >solution, so >>is f(x,y)^k for any k, or ln(f(x,y)), or exp(f(x,y)). >>Robert Israel israel@math.ubc.ca >>Department of Mathematics href=http://www.math.ubc.ca/~israel>http://www.math.ubc.ca/~ israel >>University of British Columbia Vancouver, BC, Canada >There must be a simpler solution. >Let us observe that d/dx (a(c+y)+bxy) =by and d/dy (a(c+y)+bxy)=a+bx >So instead od f(x,y) consider g(x,y)=(a(c+y)+bxy)*f(x,y) >The equation becomes very simple . > Be pleased hearing of you... > Alain. I cannot see how it gets simpler then. I Þnd it gets terribly messy. Can you be more precise on your proposal? Best, Andreas === Subject: Re: classiÞcation of Lie groups Content-Length: 1692 Originator: rusin@vesuvius > Since the complex simple Lie algebras are completely classiÞed, I am > trying to work backwards to construct a statement like every compact > Lie group is... Here¹s what I come up with: > The compact real forms of complex simple Lie algebras are su(n), > so(n), sp(2n), and the 5 exceptional Lie algebras. You are implicitly limiting yourself to Þnite-dimensional Lie algebras over a Þeld of characteristic 0. I know nothing about characteristic p, but there are simple inÞnite-dimensional Lie algebras already over the Þeld of complex numbers. Indeed, the simple inÞnite-dimensional Lie algebras of polynomial vector Þelds in n indeterminates, n Þnite, are W_n arbitrary vector Þelds S_n divergence-free vector Þelds H_n Hamiltonian vector Þelds K_n contact vector Þelds The corresponding Lie groups are the groups of all diffeomorphisms, volume-preserving diffeomorphisms, symplectomorphisms, and contact transformations, respectively. This list was conjectured by Sophus Lie himself and proven by Elie Cartan in 1909. Alternatively, we say that these algebras are of Þnite growth (more precisely, growth n). There are also simple Lie algebras of inÞnite growth, but I¹m pretty sure that no classÞcation is available. In the theory of Kac-Moody Lie algebras one uses a modiÞed of the word simple. A KM Lie algebra is simple if it contains no *graded* ideal. The reason is that all KM algebras contain a non- graded ideal, namely the central extension in the case of afÞne algebras, so the original deÞnition is not fruitful. === Subject: Fall PaciÞc NW Geometry Seminar Content-Length: 3113 Originator: rusin@vesuvius First announcement: PACIFIC NORTHWEST GEOMETRY SEMINAR University of Oregon Eugene, OR SPEAKERS: David Auckly (Kansas State University) The structure of maps into homogenous spaces and the Faddeev and Skyrme models Charles Doran (University of Washington) Mirror Symmetry, K-Theory, and Toric Geometry John Lott (University of Michigan) Ricci curvature for metric-measure spaces Lei Ni (UCSD) Ancient Solutions of the Kaehler-Ricci Flow Mutao Wang (Columbia) Mean curvature þows of Lagrangian submanifolds ----------------------------------------------------------- TRAVEL SUPPORT: PNGS meetings are supported by funds from the National Science Foundation and the PaciÞc Institute for the Mathematical Sciences. Limited travel support is available for participants in this meeting. First priority goes to graduate students and faculty from the participating universities (Oregon State, Portland State, Stanford, U. of Oregon, UBC, U. of Utah, U. of Washington). If you are afÞliated with one of these universities and are interested in travel support, please contact one of the PNGS organizers at your university NO LATER THAN OCTOBER 16: OSU Christine Escher (tine@math.orst.edu) Hal Parks (parks@math.orst.edu) Juha Pohjanpelto (juha@math.orst.edu) PSU Serge Preston (serge@mth.pdx.edu) Stanford Rafe Mazzeo (mazzeo@math.stanford.edu) UBC Jim Carrell (carrell@math.ubc.ca) Jingyi Chen (jychen@math.ubc.ca) Ailana Fraser (afraser@math.ubc.ca) Utah Mladen Bestvina (bestvina@math.utah.edu) Nick Korevaar (korevaar@math.utah.edu) Jesse Ratzkin (ratzkin@math.utah.edu) Nat Smale (smale@math.utah.edu) Andrejs Treibergs (treiberg@math.utah.edu) UW Dan Pollack (pollack@math.washington.edu) Jack Lee (lee@math.washington.edu) Participants from other universities may be supported if funds are available. Travel support is limited to graduate students or faculty members who do not have NSF grants that provide travel funds. We are particularly interested in providing support to young researchers, female mathematicians, and members of underrepresented groups. If you are not afÞliated with a participating university and would like to be considered for travel support, contact Jim Isenberg as soon as possible. ----------------------------------------------------------- For general information about the PNGS, visit the PNGS web site: http://www.math.washington.edu/~lee/PNGS It contains up-to-date information about this meeting, travel and lodging information, general information about the PNGS, and a historical record of all PNGS meetings and speakers. ----------------------------------------------------------- For more information about this meeting, contact the organizers: Boris Botvinnik (botvinn@math.uoregon.edu) Jim Isenberg (jim@newton.uoregon.edu) === Subject: Spectral theorem in QM Content-Length: 1202 Originator: rusin@vesuvius Could someone please explain how the following version of the spectral theorem for unbounded operators If A is self-adjoint on H then there exists a spectral measure E on sigma(A) such that A = int z dE implies the statement below If A is self-adjoint on H then there exists an orthonormal basis of H consisting of eigenvectors for A which is assumed in Sakurai. Even though the case of the quantum-mechanical position operator is an explicit counterexample to the latter statement Sakurai considers distributions as eigenvectors which I Þnd somewhat unsatisfactory. My questions are as below: 1) How does the Þrst statement of the spectral theorem imply the second? Can one, e.g., prove that under certain assumptions on the spectrum of A (discreteness?), A will satisfy the second version of the spectral theorem? 2) Can one combine some of the theory of distributions with the Þrst version of the spectral theorem to make rigorous sense out of the notion of distributions as a kind of eigenvectors? Or are there other ways of remedying the horrible mathematical MESS in Sakurai? Any answer or reference to bibliography much appreciated! Kasper J. Larsen === Subject: Re: Spectral theorem in QM Content-Length: 1611 Originator: rusin@vesuvius >Could someone please explain how the following version of the spectral >theorem for unbounded operators >If A is self-adjoint on H then there exists a spectral measure E on >sigma(A) such that A = int z dE >implies the statement below >If A is self-adjoint on H then there exists an orthonormal basis of H >consisting of eigenvectors for A >which is assumed in Sakurai. Even though the case of the >quantum-mechanical position operator is an explicit counterexample to >the latter statement Sakurai considers distributions as eigenvectors >which I Þnd somewhat unsatisfactory. Yuck. I don¹t know Sakurai, but it sounds like a typical physicist¹s wrong. >My questions are as below: >1) How does the Þrst statement of the spectral theorem imply the >second? Can one, e.g., prove that under certain assumptions on the >spectrum of A (discreteness?), A will satisfy the second version of >the spectral theorem? Yes. The key term is pure point spectrum. >2) Can one combine some of the theory of distributions with the Þrst >version of the spectral theorem to make rigorous sense out of the >notion of distributions as a kind of eigenvectors? I think the theory of rigged Hilbert spaces does more or less exactly that. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Penalty Epigone-thread: snermskoabe Content-Length: 786 Originator: rusin@vesuvius I encountered the following statement from an optimization book of which I am quite unclear. minimize f(x) subject to h(x) = 0 Applying penalty to the objective function and by treating it with a term [v/2 * ||x-x¹||^2], we obtain minimize f(x) + k/2 * ||h(x)||^2 + v/2 * ||x-x¹||^2 subject to x in S where S = {x : ||x-x¹|| <= e, e > 0}, k is the iteration number, v > 0 is an appropriate constant value, and x¹ is a local minimum. The last term in the latter objective funcion is to ensure x¹ is ``strictly local minimum of the function f(x) + v/2 * ||x-x¹||^2 subject to h(x) = 0. The book claims that f(x_k) + k/2 * ||h(x_k)||^2 + v/2 * ||x_k - x¹||^2 <= f(x¹) .... My question is, why the last expression has its inequality being `<=¹ and not `>=¹ ? === Subject: Cube-free numbers, and the reciprocal of the Ramanujan tau/Dirichlet function charset=iso-8859-1 Content-Length: 3362 Originator: rusin@vesuvius reader/contributor who might be able to help me with this. The other day, playing around on my (incredibly slow but still trusty old) TI-92Plus, I looked at the coefÞcients for the inverse of the Ramanujan tau/Diricihlet function, and, to my surprise ... but let me be more speciÞc. Let tau(n) be the coefÞcients of the Ramanujan tau/Dirichlet function, deÞned by the identity: Sum(tau(n)*z^n,n=1..inÞnity) = z * Product(1-z^m,m=1..inÞnity)^24 and hence the Ramaunjan tau/Dirichlet function is deÞned, in some suitable half-plane such that Re(s)>sigma_0, by: tau(s) = Sum(tau(n)/n^s,n=1..inÞnity). Now consider the reciprocal of tau(s). Starting with the coefÞcients tau(n), form new ones, say theta(n), such that, for all s: Sum(tau(n)/n^s,n=1..inÞnity) * Sum(theta(n)/n^s,n=1..inÞnity) = (Def.) 1 and the latter series is, formally at least, the Dirichlet series for 1/tau(s) - in the same way that, if mu(n) be the M.9abius mu function (or lattice of divisors), then: Sum(mu(n)/n^s,n=1..inÞnity) represents the reciprocal of the Riemann zeta function: it (the former) has the property that mu(n) vanishes if n is divisible by the square of any natural number (/i.e./ is not square-free: a product of distinct primes only). Now, I assert that I have found, on my old programmable calculator that, for some range of small natural numbers n (<=1272, in fact), the coefÞcients theta(n) to which I alluded above, vanish if n is divisible by the cube of a natural number (in other words, is not cube-free), and I strongly suspect that this result holds generally, although I don¹t have a proof (as yet). Furthermore, I don¹t believe that this result has not previously been noticed. I wondered whether Srinivasa Ramanujan himself had observed this, but, even though I do not have my copies of Bruce Berndt¹s superlative compendium of the former¹s notebooks to hand, I think that I might (or even, should) have recalled such a result. Therefore, I wondered who Þrst noticed it, and whether anyone who reads the group knew of comparable results for numbers free of fourth, Þfth, /etc./ powers. My search attempts on the WWW have failed to resolve the matter. I¹ve found a number of results involving cube-free numbers, but not this one. Additionally, the possibility of similar results for fourth-free, Þfth-free (and so on ) numbers might also be of interest ... Is there anyone out there who can enlighten me - please? John johnDOTmorrisonATtescoDOTnet -- Anyone who dreams of a mathematician¹s heaven had better reconsider, if of all its angels there be more than one mathematician. - Charles Hoy Fort, in Lo!, Chapter VIII (1931) === Subject: Re: Cube-free numbers, and the reciprocal of the Ramanujan tau/Dirichlet function Received-SPF: none (mailbox7.ucsd.edu: domain of news@theplanet.net does not designate permitted sender hosts) Content-Length: 1321 Originator: rusin@vesuvius > Let tau(n) be the coefÞcients of the Ramanujan tau/Dirichlet > function, > deÞned by the identity: > Sum(tau(n)*z^n,n=1..inÞnity) = z * > Product(1-z^m,m=1..inÞnity)^24 > and hence the Ramaunjan tau/Dirichlet function is deÞned, in some > suitable half-plane such that Re(s)>sigma_0, by: > tau(s) = Sum(tau(n)/n^s,n=1..inÞnity). It is well-known that tau is multiplicative: tau(mn)= tau(m) tau(n) when m and n are coprime, and that tau(p^{r+1}) = tau(p) tau(p^r) - p^11 tau(p^{r-1}) for primes p and integers r >= 1. See for instance Serre¹s Course in Arithmetic. It follows that this function (you surely can¹t also call it tau) has an Euler product product_p [1 - tau(p)/p^s + p^11/p^{2s}]^{-1} > Now consider the reciprocal of tau(s). Starting with the coefÞcients > tau(n), form new ones, say theta(n), such that, for all s: > Sum(tau(n)/n^s,n=1..inÞnity) * > Sum(theta(n)/n^s,n=1..inÞnity) = (Def.) 1 Hence this new series equals product_p [1 - tau(p)/p^s + p^11/p^{2s}]. Expanding this out gives theta(n) = 0 for cube-free n. -- 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: contructing permutation groups Content-Length: 907 Originator: rusin@vesuvius I just bumped into the following composition mechanism for Þnite groups. I could not seem to Þnd anything similar in the basic literature, so any help (pointers to advanced literature, etc...) would be really appreciated here. The construction is as follows: given two subgroups G, H of the symmetric group S_n , consider the sets K_g = G H g for each g of G. Then take the intersection K of all such sets. The set K is a group (the proof is relatively straightforward, or so it seems to me), it always contains at least G, and sometimes it contains G strictly (e.g. when H = {e} , then K = G, but when H = S_n then K = S_n). I¹m looking for a better characterization of this trick (e.g. what pieces of H does K inherit? how? is there a fast way to build K from G and H? etc...) Again, any help will be greatly appreciated, m.damiani. symbolics, inc. san francisco, ca 94114 === Subject: Re: contructing permutation groups Content-Length: 1375 Originator: rusin@vesuvius > I just bumped into the following composition mechanism for Þnite groups. > I could not seem to Þnd anything similar in the basic literature, > so any help (pointers to advanced literature, etc...) > would be really appreciated here. > The construction is as follows: > given two subgroups G, H of the symmetric group S_n , consider the > sets > K_g = G H g > for each g of G. > Then take the intersection K of all such sets. > The set K is a group (the proof is relatively straightforward, or so it > seems to me), it always contains at least G, > and sometimes it contains G strictly > (e.g. when H = {e} , then K = G, but when H = S_n then K = S_n). > I¹m looking for a better characterization of this trick (e.g. what pieces of > H does K inherit? how? is there a fast way to build K from G and H? etc...) That looks interesting. Of course, the construction does not depend on the overgroup being S_n in particular. I think it could be any group - certainly any Þnite group, and probably also any group at all. I believe that your group K is the (well deÞned) largest group K that satisÞes G <= K <= GH. Equivalently, K = GL, where L is the (again well deÞned) largest subgroup of H such that GL is a group. Derek Holt. === Subject: Re: contructing permutation groups Received-SPF: none (mailbox10.ucsd.edu: domain of root@news.free.fr does not designate permitted sender hosts) Content-Length: 665 Originator: rusin@vesuvius > I¹m looking for a better characterization of this trick (e.g. what pieces of > H does K inherit? how? is there a fast way to build K from G and H? etc...) This construction works for any ambient Þnite group A (not just A=S_n): the group K is the stabilizer of the subset GH of A under left multiplication, i.e. K={xin A| xGH=GH}. In particular, it contains G (as you observe), but also the intersection of H with the normalizer of G in A. So K contains G(Hcap N_A(G)), but this inclusion may be proper : for an example, consider the case where A is simple and A=GH, for two proper subgroups G and H of A. I hope this helps. Serge. === Subject: chain of regular tetrahedra Received-SPF: pass (mailbox6.ucsd.edu: domain of mail2news-moderator-return@nym.alias.net designates 18.26.0.252 as permitted sender) receiver=mailbox6.ucsd.edu; client_ip=18.26.0.252; envelope-from=mail2news-moderator-return@nym.alias.net; Mail-To-News-Contact: postmaster@nym.alias.net Content-Length: 241 Originator: rusin@vesuvius Hello All! Is it possible to make a chain of regular tetrahedra of the same size in R^3 like a circle (so that each tetrahedron has the same face with next one and the last has the same face with the Þrst one?) Sergei Markelov . === Subject: Re: chain of regular tetrahedra Content-Length: 1360 Originator: rusin@vesuvius > Is it possible to make a chain of regular tetrahedra of the same size in > R^3 like a circle (so that each tetrahedron has the same face with next > one and the last has the same face with the Þrst one?) I don¹t know the answer, (I would suppose no), but it more or less reduces to a problem in group theory. (It exactly reduces to this problem if you allow your tetrahedra to overlap.) Let T_0, T_1, ... be the tetrahedra. Then T_{j+1} is produced from T_j by a reþection in one of its faces. It¹s apparent then that each T_j is obtained from T_0 by a sequence of reþections in the faces of T_0. So we get a chain if there is a sequence of reþections in the faces of T_0, reduced in the sense that no reþection occurs twice in a row, taking T_0 to itself. Repeating this sequence would eventually result in the identity. We are asking if there are any relations among the generators R_1,...,R_4 apart from R_j^2 = I where these are the reþections in the faces of T_0. Using barycentric coordinates and extending to R^4 we can get matrices R_1 = (-1 2/3 2/3 2/3) ( 0 1 0 0 ) ( 0 0 1 0 ) ( 0 0 0 1 ) etc. for the R_j. I don¹t know if there are any nontrivial relations between these. Maybe some combinatorial group theorist out there does? -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html === Subject: Re: chain of regular tetrahedra Content-Length: 673 Originator: rusin@vesuvius >> Is it possible to make a chain of regular tetrahedra of the same size in >> R^3 like a circle (so that each tetrahedron has the same face with next >> one and the last has the same face with the Þrst one?) >I don¹t know the answer, (I would suppose no), but it more or less >reduces to a problem in group theory. (It exactly reduces to this problem >if you allow your tetrahedra to overlap.) Essentially the same question was asked about dodecahedra in 1998; see http://www.math-atlas.org/98/dodec_prf for some rough notes. dave === Subject: Satellites of simple groups Content-Length: 301 Originator: rusin@vesuvius Let G be a Þnite simple group. Is there a description of all possible two-sided extensions M.G.A for all pairs (M,A) of groups, where M is a factor group of the Schur multiplier of G and A is a subgroup of Out(G)? (I once heard such extensions called the satellites of G). Anvita === Subject: Re: Importance sampling at high dimension Content-Length: 1344 Originator: rusin@vesuvius >I am reading some material which suggests that importance sampling is >a powerful variation reduction technique even for high dimension >problem. When I read the textbooks, they often provide evaluates a >1-D integral and show how to reduce the variance from say, (s^2)/n to >c(s^2)/n, where s is the std deviation and c is some number between 0 >to 1. >I tried to extend the analysis to m-dim problem, e.g. multivariate >normal distribution. Clearly, the simple Monte Carlo sampling gives >(s^2m)/n. This is deÞnitely news to me. If it is the case, and one can make s^2 < 1, just increase the dimension and the accuracy goes up very fast. The variance of a product is generally not the product of the variances unless the means are 0 and the random variables are independent. If one is estimating a product integral, deÞnitely do the one-dimensional integrals. Otherwise, the dimension, for integration using random variables, is completely irrelevant. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: A simple(?) hyperbolic PDE in 2 vars Epigone-thread: choxbimsten Content-Length: 948 Originator: rusin@vesuvius The PDE is u_t + u_x * u_xt = 0. Here u is the unknown function of two vars u = u(x,t). The partial wrt t is u_t, similarly wrt x is u_x, and the mixed partial is denoted u_xt. I have one solution u(x,t)=x*t - x^2/2 - t^2/2. Also if u(x,t) is a solution so is w(x,t)=u(x,h(t)) where h(t) is differentiable nonzero. So the above solution gives a family of solutions based on the arbitrary function h(t), namely w(x,t) = x*h(t)-x^2/2 - [h(t)]^2/2. My meagre knowledge of this type of second order PDE is that it should have a general solution dependent on two arbitrary functions, not just one. I am interested in Þnding the general solution of the PDE, or at least in what are its properties. The PDE came up while I was investigating surface areas of volumes of revolution. === Subject: Re: A simple(?) hyperbolic PDE in 2 vars Content-Length: 1468 Originator: rusin@vesuvius >The PDE is u_t + u_x * u_xt = 0. >Here u is the unknown function of two vars u = u(x,t). >The partial wrt t is u_t, similarly wrt x is u_x, >and the mixed partial is denoted u_xt. >I have one solution u(x,t)=x*t - x^2/2 - t^2/2. >Also if u(x,t) is a solution so is w(x,t)=u(x,h(t)) where >h(t) is differentiable nonzero. >So the above solution gives a family of solutions based on the >arbitrary function h(t), namely > w(x,t) = x*h(t)-x^2/2 - [h(t)]^2/2. >My meagre knowledge of this type of second order PDE is that it should >have a general solution dependent on two arbitrary functions, >not just one. >I am interested in Þnding the general solution of the PDE, or >at least in what are its properties. The PDE came up while I was >investigating surface areas of volumes of revolution. Integrate with respect to to obtain u + (u_x)^2/2 = C(x) where C(x) is arbitrary (let¹s say continous). If C is a constant, this becomes u_x = sqrt(2C - 2u) (on x-intervals where u is increasing in x) and u_x = -sqrt(2C - 2u) on intervals where u is decreasing. This is a separable equation with the general solution u(x) = C - (x - h)^2/2 = C - x^2/2 - h^2/2 + x*h, where h does not depend on x. Now h may depend on t again, and the result is your family of solutions. If C is not constant, a solution in closed form does not appear possible in general. Hope this helps! Hans Engler === Subject: One problem on linear algebra Content-Length: 394 Originator: rusin@vesuvius I am unable to solve the problem.Any help is welcome. This is not an assignment problem. PROBLEM: Let V be an n-dimensional vector space over R and let w be a NON ZERO exterior k-form, 1<= k <=n-1 on V. Let us consider the following subspace of V, S = {x is in V | x#w = 0 }, where # denotes the exterior product. Then, what is the maximum possible dimension of S? === Subject: Re: One problem on linear algebra Content-Length: 1064 Originator: rusin@vesuvius > Let V be an n-dimensional vector space over R and let w be a NON ZERO > exterior k-form, 1<= k <=n-1 on V. Let us consider the following subspace > of V, > S = {x is in V | x#w = 0 }, where # denotes the exterior > product. > Then, what is the maximum possible dimension of S? Let me Þrst clarify what I think you mean. w is an element of the k-th exterior product of V (a k-form would be in the dual), right? And # is just the usual wedge product. Under those assumptions, I believe the maximum dimension of ker( x --> x#w ) occurs when w is decomposable, w=(v_1)#...#(v_k). In that case, then the maximum dimension would of course be k. The proof would be a messy case analysis, writing w in general as a sum of decomposable elements and dividing up the possibilities for x#(each term), with possible cancellations. -- David L. Johnson __o | You will say Christ saith this and the apostles say this; but _`(,_ | what canst thou say? -- George Fox. (_)/ (_) |