mm-1757 === Subject: Re: Converging product? Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > Hello > I would like your help on this problem: > I need to find out upper and lower bounds for the product (p): > p=(1-x*y^n)*(1-x*y^(n+1))*(1-x*y^(n+2))*...*(1-x*y^(m-1))*(1-x*y^m) > Where 0 n is an integer, typically >1000 > And especially I would like to know the limit of p when m->infinity. > If the solution is difficult to get, I at least would liketo know if p->0 > when m-> infinity > - as a simplification: > q=(1-x*y^m)^(m-n+1)>p > might be used since if q->0 when m->infinity the p->0 as well. In addition to the facts already presented by others there is a convergence criterion for products: If sum_k=0^infty |a_k| is finite, then prod_k=0^infty (1 - a_k) is finite. Moreover, we can have a_k = 1 in only finitely many factors, and the infinite product is nonzero unless in some factor a_n = 1. One normally proves this by esimating the log of the product. The expression q converges to 1. when m tends to infinity. You can simplify your product by setting a=xy^n and then it is (1-a)*(1-ay)*(1 - ay^2)*...*(1-ay^{m-n}). That might help when looking for information. Dan === Subject: Re: Minimum Sets - Voronoi Originator: bergv@math.uiuc.edu (Maarten Bergvelt) >Here's the basic problem. I'm looking for an algorithm that can best >be described in terms of McDonalds. >Is there an algorithm - this may fall under voronoi - that given a set >of McDonalds locations, we want to place the minimum number of >distribution centers such that the distribution center is within some >maximum distance to those centers it serves. In addition, for each >collection of McDonalds handled, there will be one with higher sales >and will need replenishing the most often, so the distribution center >should be closest to it. > Suppose your McDonalds locations are m_i, i=1...N, and you need a > distribution centre within distance r_i of m_i for each i. Draw the > N circles of radius r_i centred at m_i for i=1...N, dividing the plane > into some collection of regions. Let e_{ij} = 1 if region #j is within > the circle centred at m_i, 0 otherwise. Then your problem can be > expressed as follows, where x_j = 1 means put a distribution centre in > region #j: > minimize sum_j x_j > subject to > sum_j e_{ij} x_j >= 1 for each i=1..N > x_j in {0,1} > This is known as a Set Cover Problem. It is NP-complete. > Small instances can be solved using integer linear programming. > A naive greedy algorithm provides a surprisingly good > approximate solution: see J. Hromkovic, Algorithmics for > Hard Problems, Springer 2001, sec. 4.3.2. > Robert Israel israel@math.ubc.ca > Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia Vancouver, BC, Canada === Subject: Posynomial - Logisitc function - neural network? Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Suppose I take a basic feed forward neural network(1, 2 or 3 layer) with sigmoidal functions with no negative biases. With each neuron in every layer of the form Output = f(x1*w1 + x2*w2 .... xn*wn) where f(x) is sigmoidal in the range of 0 to 1 And we restrict all weights to be positive and all inputs to be in the range of 0 to 1.0 Would a least square cost function of its out to the desired output be considered posynomial???? Is there any books or tutorials people recommend for me to get an understanding of how build posynomial functions or to check if an neural output is posynomial. I am, as you might have guessed from the above, trying to develop a neural network and matching learning algorithm that is based on Geometric programming. Or if you are aware of current work in thsi direction please point me in the right direction. Sarvi === Subject: This week in the mathematics arXiv (18 Apr - 22 Apr) Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Here are this week's 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 modification. Titles in the mathematics arXiv (18 Apr - 22 Apr) ------------------------------------------------- AC: Commutative Algebra ----------------------- math.AC/0504340 Lars Winther Christensen, Srikanth Iyengar: Gorenstein dimension of modules over homomorphisms AG: Algebraic Geometry ---------------------- math.AG/0504434 Kieran G. O'Grady: Irreducible symplectic 4-folds numerically equivalent to Hilb^2(K3) math.AG/0504431 Alexey Zaytsev: The Galois closure of the Garcia-Stichtenoth tower math.AG/0504418 C. Consani, C. Faber: On the cusp form motives in genus 1 and level 1 math.AG/0504392 Andreas Gathmann, Hannah Markwig: The Caporaso-Harris formula and plane relative Gromov-Witten invariants in tropical geometry math.AG/0504390 Andreas Gathmann, Hannah Markwig: The numbers of tropical plane curves through points in general position math.AG/0504380 L^e D~ung Tr'ang, David B. Massey: Hypersurface Singularities and Milnor Equisingularity math.AG/0504376 Atsushi Moriwaki: Rigidity of morphisms for log schemes math.AG/0504369 Jianzhong Pan, Yongbin Ruan, Xiaoqin Yin: Gerbes and twisted orbifold quantum cohomology math.AG/0504360 Samuel Boissiere, Alessandra Sarti: Contraction of excess fibres between the McKay correspondences in dimensions two and three math.AG/0504359 C. Diem, N. Naumann: On the Structure of the Weil Restriction of Abelian Varieties math.AG/0504330 Christopher D Hacon, James McKernan: Shokurov's Rational Connectedness Conjecture math.AG/0504327 Christopher D Hacon, James McKernan: Boundedness of pluricanonical maps of varieties of general type math.AG/0504314 Jungkai Alfred Chen, Meng Chen, De-Qi Zhang: A Nonvanishing Theorem for Q-divisors math.AG/0504309 K. Behrend, B. Noohi: Uniformization of Deligne-Mumford curves AP: Analysis of PDEs -------------------- math.AP/0504384 Jiayu Li, Yuxiang Li: Solutions for Toda systems on Riemann surfaces math.AP/0504370 Piero D'Ancona, Luca Fanelli: Decay estimates for the wave and Dirac equations with a magnetic potential math.AP/0504344 Yehuda Pinchover: On Davies' conjecture and strong ratio limit properties for the heat kernel math.AP/0504339 Hans Lindblad: Well posedness for the motion of a compressible liquid with free surface boundary math.AP/0504333 Andrej Zlatos: Sharp Transition Between Extinction and Propagation of Reaction math.AP/0504320 Yaroslav Kurylev, Matti Lassas, Erkki Somersalo: Maxwell's Equations with Scalar Impedance: Inverse Problems with data given on a part of the boundary math.AP/0504317 Yuxiang Li: Remarks on the Extremal Functions for the Moser-Trudinger Inequalities AT: Algebraic Topology ---------------------- math.AT/0504437 Tornike Kadeishvili: On the homology theory of fibre spaces math.AT/0504432 J.P.C.Greenlees: Rational S^1-equivariant elliptic cohomology math.AT/0504396 Marisa Fernandez, Vicente Mu~noz: Non-formal compact manifolds with small Betti numbers math.AT/0504334 Julia E. Bergner: Three models for the homotopy theory of homotopy theories math.AT/0504329 Luis Casian, Yuji Kodama: Toda lattice, cohomology of compact Lie groups and finite Chevalley groups math.AT/0504322 Birgit Richter: A lower bound for coherences on the Brown-Peterson spectrum math.AT/0504321 Eva Maria Feichtner, Sergey Yuzvinsky: Formality of the complements of subspace arrangements with geometric lattices CA: Classical Analysis and ODEs ------------------------------- math.CA/0504416 Szilard Gy. Revesz: Right order Turan-type converse Markov inequalities for convex domains on the plane math.CA/0504407 K. Betina: Indices d'un operateur differentiel matriciel et applications math.CA/0504394 Lawrence Baggett, Palle Jorgensen, Kathy Merrill, Judith Packer: A non-MRA $C^r$ frame wavelet with rapid decay CO: Combinatorics ----------------- math.CO/0504436 Michael Anshelevich, Edward G. Effros, Mihai Popa: Zimmermann Type Cancellation in the Free Faa di Bruno Algebra math.CO/0504430 Patrick Baier: NP-completeness of Partial Chirotope Extendibility math.CO/0504429 Satoshi Murai: A combinatorial proof of Gotzmann's persistence theorem for monomial ideals math.CO/0504425 Guoce Xin: A Generalization of Stanley's Monster Reciprocity Theorem math.CO/0504400 Brad Jackson, Frank Ruskey: Meta-Fibonacci Sequences, Binary Trees, and Extremal Compact Codes math.CO/0504397 Leonid Gurvits: A proof of hyperbolic van der Waerden conjecture : the right generalization is the ultimate simplification math.CO/0504367 Timothy Y. Chow: Reduction of Rota's basis conjecture to a problem on three bases math.CO/0504342 William Y. C. Chen, Toufik Mansour, Sherry H. F. Yan: Matchings Avoiding Partial Patterns math.CO/0504326 Raul Cordovil: Kalai orientations on matroid polytopes math.CO/0504310 Carla D. Savage, Herbert S. Wilf: Pattern avoidance in compositions and multiset permutations CV: Complex Variables --------------------- math.CV/0504387 Francesco Costantino: Stein domains and branched shadows of 4-manifolds math.CV/0504374 Jim Agler, JOhn E. McCarthy: Parametrizing distinguished varieties math.CV/0504353 A. V. Isaev: Analogues of Rossi's map and E. Cartan's classification of homogeneous strongly pseudoconvex 3-dimensional hypersurfaces nlin.SI/0504026 R. G. Halburd, R. J. Korhonen: Finite-order meromorphic solutions and the discrete Painleve equations DG: Differential Geometry ------------------------- math.DG/0504439 Manuel Ritor'e, C'esar Rosales: Rotationally invariant hypersurfaces with constant mean curvature in the Heisenberg group H^n math.DG/0504422 Albert Chau, Luen-Fai Tam: On the complex structure of Kahler manifolds with nonnegative curvature math.DG/0504421 John Lott: Remark about scalar curvature and Riemannian submersions math.DG/0504412 Laurent Mazet: A height estimate for constant mean curvature graphs and uniqueness math.DG/0504405 Daniel John: Symmetrization procedures for the isoperimetric problem in symmetric spaces of noncompact type math.DG/0504398 Mauricio Angel, Rafael D'{i}az: $N$-differential graded algebras math.DG/0504393 Andre Diatta, Peter Giblin, Brendan Guilfoyle, Wilhelm Klingenberg: Level sets of functions and symmetry sets of smooth surface sections math.DG/0504389 Andreas Cap: Two constructions with parabolic geometries math.DG/0504381 Franc{c}ois Gay-Balmaz, Tudor S. Ratiu: The Lie-Poisson structure of the LAE-$alpha$ equation math.DG/0504366 Marco Godina, Paolo Matteucci: The Lie derivative of spinor fields: theory and applications math.DG/0504358 Alexander I. Bobenko, Yuri B. Suris: Discrete differential geometry. Consistency as integrability math.DG/0504338 J.-F. Lafont, B. Schmidt: Simplicial volume of closed locally symmetric spaces of non-compact type math.DG/0504337 Andriy Panasyuk: Algebraic Nijenhuis operators and Kronecker Poisson pencils math.DG/0504324 Francisco Martin; William H. Meeks, III; Nicolai Nadirashvili: Bounded domains which are universal for minimal surfaces math.DG/0504319 Boris Doubrov, Igor Zelenko: A Canonical Frame for Nonholonomic Rank Two Distributions of Maximal Class DS: Dynamical Systems --------------------- q-bio.NC/0501021 Michael Stiber: Spike timing precision and neural error correction: local behavior nlin.CD/0504028 Ricardo Lopez-Ruiz, Daniele Fournier-Prunaret: Periodic and Chaotic Events in a Discrete Model of Logistic Type for the Competitive Interaction of Two Species FA: Functional Analysis ----------------------- quant-ph/0504151 Dimitri Gioev, Israel Klich: Entanglement entropy of fermions in any dimension and the Widom conjecture math.FA/0504406 Massimiliano Berti, Michela Procesi: Quasi-periodic solutions of completely resonant forced wave equations GM: General Mathematics ----------------------- math.GM/0504426 Gerard Maze: Existence of a Limiting Distribution for the Binary GCD Algorithm math.GM/0504355 Dhananjay P. Mehendale: Some Observations on the 3x+1 Problem GN: General Topology -------------------- math.GN/0504325 Gady Kozma: On removing one point from a compact space GR: Group Theory ---------------- math.GR/0504438 Alexey Muranov: On torsion-free groups with finite regular file bases math.GR/0504410 Mark Pankov: On geometry of symplectic involutions math.GR/0504409 Mark Pankov: On geometry of linear involutions math.GR/0504401 Adam Piggott: Algorithmic constructions and primitive elements in the free group of rank 2 math.GR/0504354 Helge Glockner: Locally compact groups built up from p-adic Lie groups, for p in a given set of primes math.GR/0504350 A.Yu. Olshanskii, M. V. Sapir: A finitely presented group with two non-homeomorphic asymptotic cones math.GR/0504349 A.Yu. Olshanskii: Groups with quadratic-non-quadratic Dehn functions math.GR/0504312 Miklos Abert: On the probability of satisfying a word in a group GT: Geometric Topology ---------------------- math.GT/0504415 Sergei Matveev: Roots of knotted graphs and orbifolds math.GT/0504404 Peter Ozsvath, Zoltan Szabo: Knot Floer homology and rational surgeries math.GT/0504385 Moulay Benameur, James Heitsch: Index theorey and Non-Commutative Geometry. II. Dirac operators and index bundles math.GT/0504356 Jose Ignacio Cogolludo, Vincent Florens: Twisted Alexander polynomials of Plane Algebraic Curves math.GT/0504346 Rui Pedro Carpentier: Representations of non-singular planar tangles by operators math.GT/0504345 Scott Baldridge, Paul Kirk: On symplectic 4-manifolds with prescribed fundamental group math.GT/0504328 Jason Behrstock, Dan Margalit: Curve complexes and finite index subgroups of mapping class groups HO: History and Overview ------------------------ math.HO/0504335 Roberto Volpe: Notes on Theory of Quadratic Residues KT: K-Theory and Homology ------------------------- math.KT/0504372 Damien Calaque, Vasiliy Dolgushev, Gilles Halbout: Formality theorems for Hochschild chains in the Lie algebroid setting LO: Logic --------- math.LO/0504375 Dmytro Taranovsky: Extending the Language of Set Theory math.LO/0504351 Joel David Hamkins, Alexei Miasnikov: The halting problem is decidable on a set of asymptotic probability one quant-ph/0504115 Radhakrishnan Srinivasan: Logical analysis of the Bohr Complementarity Principle in Afshar's experiment under the NAFL interpretation MG: Metric Geometry ------------------- math.MG/0504357 Wieslaw Kubi's, Matatyahu Rubin: Extension theorems and reconstruction theorems for the Urysohn Universal Space math.MG/0504341 Iwan Praton: The Erdos and Campbell-Staton conjectures about square packing MP: Mathematical Physics ------------------------ math-ph/0504066 Vladimir Entov, Pavel Etingof: On a Generalized Two-Fluid Hele-Shaw Flow math-ph/0504065 G. Marmo, G. Scolarici, A. Simoni, F. Ventriglia: Quantum Bi-Hamiltonian systems, alternative Hermitian structures and Bi-Unitary transformations math-ph/0504064 G. Marmo, G. Scolarici, A. Simoni, F. Ventriglia: Classical and Quantum Systems: Alternative Hamiltonian Descriptions math-ph/0504063 HR Dullin, JM Robbins, H Waalkens, SC Creagh, G Tanner: Maslov Indices and Monodromy math-ph/0504062 Hajime Moriya: Necessity of Fermion grading symmetry for automorphisms that commute with an arbitrary asymptotically abelian group of automorphisms hep-th/0504146 N.M. Nikolov, K.-H. Rehren, I.T. Todorov: Partial wave expansion and Wightman positivity in conformal field theory nlin.SI/0504037 Sergei Sakovich: Enlarged spectral problems and nonintegrability math-ph/0504061 Hasan R. Karadayi, M.Gungormez: On Poincare Polinomials of Hyperbolic Lie Algebras math-ph/0504060 George Chavchanidze: Involutive orbits of non-Noether symmetry groups math-ph/0504059 S. R. Czapor, R. G. McLenaghan, F. D. Sasse: Complete Solution of Hadamard's Problem for the Scalar Wave Equation on Petrov type III Space-Times math-ph/0504058 Bertrand Eynard, Nicolas Orantin: Topological expansion of the 2-matrix model correlation functions: diagrammatic rules for a residue formula math-ph/0504057 Kalle Kytola: On conformal field theory of SLE(kappa; rho) math-ph/0504056 Hakan Ciftci, Richard L. Hall, Nasser Saad: Perturbation theory in a framework of iteration methods hep-th/0504009 Jasbir Nagi: Logarithmic Primary Fields in Conformal and Superconformal Field Theory cond-mat/0504417 T. Sasamoto: Spatial correlations of the 1D KPZ surface on a flat substrate quant-ph/0504102 A.J. Bracken: Complex numbers and symmetries in quantum mechanics, and a nonlinear superposition principle for Wigner functions quant-ph/0504098 Piotr Garbaczewski: A Subtlety of the Schr'{o}dinger Picture Dynamics math-ph/0504055 O. Cornejo-Perez, H.C. Rosu: Nonlinear second order ODE's: Factorizations and particular solutions math-ph/0504054 G. A. Pavliotis, A.M. Stuart: Analysis of White Noise Limits for Stochastic Systems with Two Fast Relaxation Times math-ph/0504053 T. M. Garoni, P. J. Forrester, N. E. Frankel: Asymptotic corrections to the eigenvalue density of the GUE and LUE math-ph/0504052 Johannes Kellendonk, Serge Richard: A topological version of Levinson's theorem hep-th/0504077 F. Cannata, M.V. Ioffe, D.N. Nishnianidze: Double Shape Invariance of Two-Dimensional Singular Morse Model math-ph/0504051 Alexander Elgart, Benjamin Schlein: Mean Field Dynamics of Boson Stars math-ph/0504050 W. G. Anderson, R. G. McLenaghan, F. D. Sasse: Huygens' Principle for the Non-Self-Adjoint Scalar Wave Equation on Petrov type III Space-Times math-ph/0504049 C. Jarlskog: A recursive parameterisation of unitary matrices math-ph/0504048 Vladimir V. Kornyak: On Compatibility of Discrete Relations math-ph/0504047 Josef Janyv{s}ka, Marco Modugno: Graded Lie algebra of Hermitian tangent valued forms math-ph/0504046 Plamen Iliev: Finite heat kernel expansions on the real line math-ph/0504045 Plamen Iliev: On the heat kernel and the Korteweg-de Vries hierarchy NA: Numerical Analysis ---------------------- math.NA/0504428 Mar'ia L'opez-Fern'andez, Cesar Palencia, Achim Schadle: A spectral order method for inverting sectorial Laplace transforms NT: Number Theory ----------------- math.NT/0504413 Hao Pan, Zhi-Wei Sun: A sharp result on m-covers math.NT/0504402 Luis Baez-Duarte: Moebius-convolutions and the Riemann hypothesis math.NT/0504388 Laurent Berger, Christophe Breuil: Sur la r'eduction des repr'esentations cristallines de dimension 2 en poids moyens math.NT/0504336 D. A. Goldston, C. Y. Yildirim: Small Gaps Between Primes I math.NT/0504332 Claus Mazanti Sorensen: A Generalization of Level-Raising Congruences for Algebraic Modular Forms math.NT/0504316 Emmanuel Kowalski: Exponential sums over definable subsets of finite fields OA: Operator Algebras --------------------- math.OA/0504435 F. Hiai, D. Petz: Large deviations for functions of two random projection matrices math.OA/0504423 Takeshi Katsura: Non-separable AF-algebras math.OA/0504362 Guyan Robertson: Abelian subalgebras of von Neumann algebras from flat tori in locally symmetric spaces math.OA/0504361 Ken Dykema: Multilinear function series and transforms in free probability theory math.OA/0504331 Alan Hopenwasser: The Spectral Theorem for Bimodules in Higher Rank Graph C*-algebras math.OA/0504313 Daniel Beltic{t}u{a}, Bebe Prunaru: Amenability, completely bounded projections, dynamical systems and smooth orbits OC: Optimization and Control ---------------------------- math.OC/0504419 Ali Jadbabaie, Nader Motee, Mauricio Barahona: On the stability of the Kuramoto model of coupled nonlinear oscillators math.OC/0504365 Stewart D. Johnson: Stasis Points and Approximating Two-Cycles math.OC/0504323 S'ergio Rodrigues: Navier-Stokes Equation on the Rectangle math.OC/0504308 Dionisis Stefanatos, Navin Khaneja: Semidefinite Programming and Reachable Sets of Dissipative Bilinear Control Systems PR: Probability --------------- math.PR/0504414 Mireille Capitaine, Catherine Donati-Martin: Strong asymptotic freeness for Wigner and Wishart matrices math.PR/0504408 Roger Mansuy: On processes which are infinitely divisible with respect to time math.PR/0504391 Janos Englander, Ross G. Pinsky: The compact support property for measure-valued processes math.PR/0504378 S. Roch: A Short Proof that Phylogenetic Tree Reconstruction by Maximum Likelihood is Hard math.PR/0504377 J. Englander, A. Winter: Law of Large Numbers for a Class of Superdiffusions cond-mat/0504268 N. Read: Minimum spanning trees and random resistor networks in d dimensions math.PR/0504318 Franc{c}ois Coquet, Sandrine Toldo: Convergence of values in optimal stopping and convergence of optimal stopping times math.PR/0504315 Sandrine Toldo: Stability of solutions of BSDEs with random terminal time physics/0504094 M.V. Simkin, V.P. Roychowdhury: A mathematical theory of citing QA: Quantum Algebra ------------------- math.QA/0504433 Takeo Kojima, Hitoshi Konno, Robert Weston: The Vertex-Face Correspondence and Correlation Functions of the Fusion Eight-Vertex Model I: The General Formalism math.QA/0504420 Vasiliy A. Dolgushev: A Proof of Tsygan's Formality Conjecture for an Arbitrary Smooth Manifold math.QA/0504386 Florin Panaite, Freddy Van Oystaeyen: L-R-smash product for (quasi) Hopf algebras math.QA/0504373 K.A. Dancer, M.D. Gould, J. Links: Lax Operator for the Quantised Orthosymplectic Superalgebra U_q[osp(m|n)] math.QA/0504368 Saeid Azam: Derivations of tensor product of algebras hep-th/0501097 Igor Batalin, Maxim Grigoriev, Simon Lyakhovich: Non-Abelian Conversion and Quantization of Non-scalar Second-Class Constraints hep-th/0101089 I.A. Batalin, M.A. Grigoriev, S.L. Lyakhovich: Star Product for Second Class Constraint Systems from a BRST Theory math.QA/0504311 to modular tensor categories in conformal field theory RA: Rings and Algebras ---------------------- math.RA/0504427 S. Caenepeel, D. Quinn, S. Raianu: Duality for finite Hopf algebras explained by corings math.RA/0504352 Yunge Xu, Yang Han, Wenfeng Jiang: Hochschild (co)homology of exterior algebras RT: Representation Theory ------------------------- math.RT/0504417 Amritanshu Prasad: On Bernstein's presentation of Iwahori-Hecke algebras and representations of split reductive groups over non-Archimedean local fields math.RT/0504411 Joseph Bernstein, Andre Reznikov: Periods, subconvexity of L-functions and representation theory math.RT/0504399 Paul-Olivier Dehaye: Averages over classical compact Lie groups and Weyl characters math.RT/0504395 A. Oblomkov: Deformed Harish-Chandra homomorphism for the cyclic quiver math.RT/0504371 Apoorva Khare: Category O over skew group rings math.RT/0504364 Eddy Ardonne, Rinat Kedem, Michael Stone: Fermionic characters of arbitrary highest-weight integrable sl_{r+1}-modules math.RT/0504363 Hadi Salmasian: A new notion of rank for unitary representations based on Kirillov's orbit method math.RT/0504343 Alexander Premet: Enveloping algebras of Slodowy slices and the Joseph ideal SG: Symplectic Geometry ----------------------- math.SG/0504403 Peter Ozsvath, Andras I. Stipsicz, Zoltan Szabo: Planar open books and Floer homology math.SG/0504379 Nicolas Roy: Regular deformations of completely integrable systems math.SG/0504348 Nicholas M. Ercolani, Guadalupe I. Lozano: A Bi-Hamiltonian Structure for the Integrable, Discrete Non-Linear Schrodinger System math.SG/0504347 Benoit Dherin: The Universal Generating Function of Analytical Poisson Structures ST: Statistics -------------- math.ST/0504424 Richard Samworth, Oliver Johnson: The empirical process in Mallows distance, with application to goodness-of-fit tests math.ST/0504383 Clementine Dalelane: Exact minimax risk for density estimators in non-integer Sobolev classes math.ST/0504382 Clementine Dalelane: Exact oracle inequality for a sharp adaptive kernel density estimator -- / 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 that's fit to e-print * === Subject: Center of Max Rings Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Is it true that the center of a left max ring (rings for which every non-zero left module has a maximal submodule) is a max ring!? === Subject: Re: Schur result on linear dimension Originator: bergv@math.uiuc.edu (Maarten Bergvelt) thank you very much for this beautiful and elementary proof of Schur result. >> Can someone give here the proof of a Schur result : >> for n>=1, the maximal linear dimension of a commutative >> subalgebra Aof the matrix algebra M_n(K) (K a commutative field) is > >>[n^2/4]+1. > There's a 1998 note in the Monthly about this: > A Simple Proof of a Theorem of Schur > M. Mirzakhani > The American Mathematical Monthly, Vol. 105, No. 3. (Mar., 1998), > pp. 260-262. > Available on JSTOR at >http://links.jstor.org/sici?sici=0002-9890%28199803%29105%3A3%3C260%3AASPOA T%3E2.0.CO%3B2-R Jacobson, N. Schur's theorems on commutative matrices. Bull. Amer. Math. Soc. 50, (1944). 431--436. And also to Manuel Ojanguren for the reference : R.C.Cowsik A short note on the Scur-Jacobson Theorem Proc.Amer.Math.Soc.118 (1993)p.675-676. Georges === Subject: Two papers published by Geometry and Topology Originator: bergv@math.uiuc.edu (Maarten Bergvelt) The following two related papers have been published: Geometry and Topology, Volume 9 (2005) Paper no. 17, pages 699--755 Geometry and Topology, Volume 9 (2005) Paper no. 18, pages 757--811 URL's: http://www.maths.warwick.ac.uk/gt/GTVol9/paper17.abs.html http://www.maths.warwick.ac.uk/gt/GTVol9/paper18.abs.html Titles: Complete intersection singularities of splice type as universal abelian covers and Complex surface singularities with integral homology sphere links Author(s): Walter D Neumann, Jonathan Wahl Abstracts: (1) It has long been known that every quasi-homogeneous normal complex surface singularity with Q-homology sphere link has universal abelian cover a Brieskorn complete intersection singularity. We describe a broad generalization: First, one has a class of complete intersection normal complex surface singularities called splice type singularities, which generalize Brieskorn complete intersections. Second, these arise as universal abelian covers of a class of normal surface singularities with Q-homology sphere links, called splice-quotient singularities. According to the Main Theorem, splice-quotients realize a large portion of the possible topologies of singularities with Q-homology sphere links. As quotients of complete intersections, they are necessarily Q-Gorenstein, and many Q-Gorenstein singularities with Q-homology sphere links are of this type. We conjecture that rational singularities and minimally elliptic singularities with Q-homology sphere links are splice-quotients. A recent preprint of T Okuma presents confirmation of this conjecture. (2) While the topological types of normal surface singularities with homology sphere link have been classified, forming a rich class, until recently little was known about the possible analytic structures. We prove in the first paper of this pair, that many of them can be realized as complete intersection singularities of splice type, generalizing Brieskorn type. We show that a normal singularity with homology sphere link is of splice type if and only if some naturally occurring knots in the singularity link are themselves links of hypersurface sections of the singular point. The Casson Invariant Conjecture (CIC) asserts that for a complete intersection surface singularity whose link is an integral homology sphere, the Casson invariant of that link is one-eighth the signature of the Milnor fiber. In this paper we prove CIC for a large class of splice type singularities. The CIC suggests (and is motivated by the idea) that the Milnor fiber of a complete intersection singularity with homology sphere link Sigma should be a 4-manifold canonically associated to Sigma. We propose, and verify in a non-trivial case, a stronger conjecture than the CIC for splice type complete intersections: a precise topological description of the Milnor fiber. We also point out recent counterexamples to some overly optimistic earlier conjectures in [Trends in Singularities, Birkhauser (2002) 181--190 and Secondary: 57M25, 57N10 Keywords: Surface singularity, Gorenstein singularity, rational homology sphere, complete intersection singularity, abelian cover, Casson invariant, integral homology sphere, surface singularity, monomial curve, plane curve singularity Revised: 18 April 2005 Accepted: 6 March 2005 Published: 28 April 2005 Proposed: Robion Kirby Seconded: Ronald Fintushel, Ronald Stern Author(s) address(es): Department of Mathematics, Barnard College, Columbia University New York, NY 10027, USA and Department of Mathematics, The University of North Carolina Chapel Hill, NC 27599-3250, USA Email: neumann@math.columbia.edu, jmwahl@email.unc.edu === Subject: Re: finite divisibility in probability Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > The problem of finite divisibility > is complicated; one can also talk about what powers of a > distribution are distributions, and for infinitely divisible > distributions it is all powers. But for the distribution > with density (sin(pi*x)/(pi*x))^2, it is all real numbers > greater than or equal to 1. A multivariate distribution > which has similar properties is the Wishart distribution; > it exists for all n greater than or equal to the dimension > of the space, and only for integer n smaller. The set of such powers needs to be closed under addition. One could wonder whether EVERY set closed under addition occurs in this way. For example, consider the set { (0,0) } union { (a,b) : a is real, b is a positive integer }. Is there some family of probability distributions that, under the operation of convolution of probability distributions, is isomorphic to this set with the operation of addition, and that cannot be extended to a larger set? -- Mike Hardy === Subject: Re: McDowell-Mansouri gravity Originator: bergv@math.uiuc.edu (Maarten Bergvelt) With so much stuff happening, the probability of something very improbable occuring is very high. It turns out that in my previous posts to this thread a trailing . got put on a line by itself... which killed the post, since it's routed through unix mail. Here's the meat of the post on the Clifford algebra version of McDowell-Monsouri gravity that I had tried to send earlier, and a question for help at the end: Lets just deal in Cl_4 (or CL_(1,3) to be more precise) the whole time. In differential geometry terms, a Clifford fiber bundle. Standard GR can be formulated as follows. The dynamic variables are the frame, a Clifford vector valued 1-form, e, and the connection, a Clifford bi-vector valued 1-form, W. Using the unit Clifford four-vector element, g = gamma_0 gamma_1 gamma_2 gamma_3, (which satisfies g g = -1) the action for gravity is S = int < e e R(W) g + lambda e e e e g > Where the < > means take the Clifford scalar part, or, if you like using matrices, the trace. Lambda is the cosmological constant (times some factor) and R is the Clifford bi-vector valued curvature 2-form, R = d W + (1/2) W W I think that's pretty cool in itself -- no indices. :) One thing to note is, using Clifford algebra, the Hodge star transformation, or duality, is just multiplication by g. OK, now lets do something along the lines of McDowell-Mansouri. Taking advantage of the fact that in a Clifford algebra it's OK to add and multiply vectors, bi-vectors, and four-vectors, you can just write a Clifford connection as A = e + W The curvature of this connection, a Clifford valued 2-form, is F = d A + (1/2) A A = ( d e + (1/2) (e W + W e) ) + ( d W + (1/2) W W + (1/2) e e ) Where the first term in ( ) is a Clifford vector (the torsion) and the second term is a Clifford bi-vector, the curvature R plus a frame term. The action we cook up to start with is S = int < B (F - (1/2) B g) > with B some arbitrary 2-form with Clifford vector and bi-vector parts. The Clifford vector part of B is the Lagrange multiplier that makes the vector part of F, the torsion, vanish. Varying the rest of B then gives its bi-vector part as B = - (R + (1/2) e e) g and plugging this back into the action gives the effective action S = int < (R + (1/2) e e) (R + (1/2) e e) g > = int < R R g + e e R g + e e e e g > which we also could have gotten just by starting with S = int < F F g > Scaling e by a constant gets the cosmological constant in there, and the < R R g > is a topological term that doesn't contribute to the dynamics -- so what's left is GR. Nifty, eh? One thing I just realized is, via this formalism, diffeomorphisms and local frame rotations enter through the same infinitesimal gauge transformation of the connection: A' = A + d C + (1/2)( A C - C A ) with the Cillford vector part of C giving diffeomorphisms and the bivector part giving frame rotations. That's pretty cool. Of course, the next thing to try is going whole hog and letting the connection, A, be an arbitrary Clifford element. Then we'd get GR and a bunch of other stuff. Now, if you want to think about this in terms of group theory, the best thing to do is think of the basis elements of the Clifford algebra as Lie algebra generators. Then I think, for example, the Lie algebra corresponding the 16 generators of Cl_4 is... u(4)? And I think the Lie algebra corresponding to just the 10 Cl_4 vectors and bi-vectors is sp(2), but I'm not at all sure of that, as my group theory is lacking. Maybe someone out there knows how to correlate the Clifford algebras of various dimensions, and their multi-vector subalgebras, to the corresponding Lie algebras? I've seen how to represent Lie algebra generators as Clifford bi-vectors, but I'd like to see how the arbitrary Clifford subalgebras, and not just bivector subalgebras, correlate the other way. I'd love to see that. Best, Garrett === Subject: Re: Diffeomorphisms of Lie groups Originator: bergv@math.uiuc.edu (Maarten Bergvelt) David Roberts asks: << . . . Can we find an automorphism of a Lie group that is homotopic to a given self-diffeomorphism? Very interesting question! This is not true in general. E.g., the Lie group SO(3) -- topologically projective 3-space P^3 -- does not admit an orientation-reversing automorphism. One way to see this is that its Lie algebra so(3) is isomorphic to the pure imaginary quaternions H/R (with bracket corresponding to quaternion multiplication, i.e., cross product in R^3). But the relations ij=k, jk=i, ki=j determine an orientation on H/R, and so its Lie algebra automorphisms must preserve this orientation. So any orientation-reversing diffeomorphism of SO(3) that carries the identity element e to itself, must reverse the god-given orientation on T_e(SO(3)), and so cannot be an Lie group automorphism. === Subject: Re: Diffeomorphisms of Lie groups Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > Just a question on diffeomorphisms of Lie groups - what is known about > these being homotopic to homomorphisms. > Specifically, Can we find an automorphism of a Lie group that is > homotopic to a given self-diffeomorphism? If you want to look at the Lie algebra problem first, there are some relevant works related to Poincar.8e's famous paper on linearization. Given a semisimple Lie algebra L of real-analytic vector fields on $R^n$, vanishing at 0, it can be linearized by a real-analytic diffeomorphism p; see the back-to-back papers by Robert Hermann and by Guillemin & Sternberg in Trans. Amer. Math. Soc. vol. 130 (1968). J. L. Sedwick and I in J. Differential Equations 25 (1977) 377--390 showed that transitivity on $R^n setminus 0$ suffices; such L is of the form semisimple + center, and the transitive groups are known: have all but one, spin(9,1) which Linus Kramer has given in a paper on another subject in For the final word on this see Ellen Livingston & DLE in the same journal vo. 55 (1984) 289-299. To get homotopies, you can require that p be an element of a pseudogroup of diffeomorphisms arising from a differential equation dp/dt = g(p), p(0) = I. Lifting this to your question is natural, but I don't know of any papers. I'd be glad to hear of any such work, since once again I'm working on this linearization problem. D. L. Elliott, delliott at umd.edu === Subject: Diffeomorphisms of Lie groups Originator: bergv@math.uiuc.edu (Maarten Bergvelt) >> Just a question on diffeomorphisms of Lie groups - what is known >> about >> these being homotopic to homomorphisms. >> Specifically, Can we find an automorphism of a Lie group that is >> homotopic to a given self-diffeomorphism? >If you want to look at the Lie algebra problem first, there are >some relevant works related to Poincar.8e's famous paper on >linearization. Actually I'm looking at it from a category theoretic point of view, so arrows, 2-arrows and similar. I'll still look at the papers but I'm afraid my application (partial categorification of bundle gerbes) may not be helpful. David Roberts Pure Maths Adelaide Univ. === Subject: Q: Classification of Hermitian Lattices Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Hello. Recently, while studying Hermitian lattices, I need the classification of positive definite Hermitian lattices. But the known results and informations are dispersed and, even worse, I cannot find all related journals in my university. What I want to know is the class numbers and the representatives according to the ranks and discriminants of lattices over the imaginary quadratic fields. I have found some data through MathSciNet. Some of them are listed in the last lines. But these are not enough, especially, for non-unimodular cases. Could you help me? If you have any related materials, could you send me them? My email is puzzlist at gmail dot com. Some papers about classification of Hermitian lattices: imaginary quadratic fields Q(sqrt{-m}), r = rank, d = discriminant Zhu, Classification of positive definite unimodular Hermitian forms, Chinese Ann. Math. Ser. A 12 (1991), no.6, 720-730. (But I CANNOT find this journal in my university. If you have this paper, could you send me? This paper is written in Chinese.) m=1, r=2,3,4,5, d=1 m=2, r=2,3,4,5, d=1 m=3, r=2,3,4,5, d=1 m=7, r=2,3,4, d=1 m=11, r=2,3, d=1 Elstrodt et al., Zeta-functions of binary Hermitian forms and special values of Eisenstein series on three-dimensional hyperbolic space, Math. Ann. 277 (1987), no.4, 655-708. m=1, r=2, d=1,2,...,9 m=2, r=2, d=1,2,...,5 m=3, r=2, d=1,2,...,6 Otremba, Zur Theorie der hermiteschen Formen in imaginaer-quadratischen Zahlkoerpern, J. Reine Angew. math. 249 (1971), 1-19. diagonal forms of 1-class genus Feit, Some lattices over Q(sqrt{-3}), J. Algebra 52 (1978), no.1, 248-263. m=3, r=2,...,12, d=1 === Subject: Paper published by Algebraic and Geometric Topology Originator: bergv@math.uiuc.edu (Maarten Bergvelt) The following paper has been published: Algebraic and Geometric Topology Volume 5 (2005), paper no. 16, pages 369--378 URL: http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-16.abs.html Title: All integral slopes can be Seifert fibered slopes for hyperbolic knots Author(s): Kimihiko Motegi, Hyun-Jong Song Abstract: Which slopes can or cannot appear as Seifert fibered slopes for hyperbolic knots in the 3-sphere S^3? It is conjectured that if r-surgery on a hyperbolic knot in S^3 yields a Seifert fiber space, then r is an integer. We show that for each integer n, there exists a tunnel number one, hyperbolic knot K_n in S^3 such that n-surgery on K_n produces a small Seifert fiber space. Keywords: Dehn surgery, hyperbolic knot, Seifert fiber space, surgery slopes Received: 10 March 2005 Revised: 25 March 2005 Accepted: 12 April 2005 Published: 30 April 2005 Author(s) address(es): Department of Mathematics, Nihon University Tokyo 156-8550, Japan and Division of Mathematical Sciences, Pukyong National University 599-1 Daeyondong, Namgu, Pusan 608-737, Korea Email: motegi@math.chs.nihon-u.ac.jp, hjsong@pknu.ac.kr === Subject: hypergeometric function inequality Originator: bergv@math.uiuc.edu (Maarten Bergvelt) hello, I've come across the inequality below (*) in the course of some work on hypergeometric function. However, I have been unable to prove it or find a proof in the literature. Does anyone know any references to such a result or have any ideas for how to prove it? Any help at all would be greatly appreciated, Matthew Baxter Suppose a, b are real numbers with 0 < alpha, beta < 1. For any integer r>1, put G_{r}(x) = _{2}F_{1} (-r,-r-a; 1-a; x), where _{2}F_{1} is the standard hypergeometric function (here a polynomial, since -r is negative). (*) For 1/2 < x < 1, G_{r}'(x) (x^{b}+x) - rG_{r}(x) > 0, where G_{r}'(x) is the derivative of G_{r}(x) with respect to x.