mm-1046 === Subject: Re: probability problem . Let the original distributions be x_0, ..., x_m and y_0, ..., y_m. Let c be the minimal probability in the distribution of the sum. In the distribution of the sum, the probability of 0 is x_0 y_0, the probability of 2m is x_m y_m. The probability of m is x_0 y_m + ... +x_m y_0 . We keep only the first and last term. The sum of these is by the arithmetic-geometric mean inequality x_0 y_m + x_m y_0 >= 2 sqrt (x_0 y_0 x_m y_m) >= 2c. So there is at least a factor of 2 between the minimal and maximal probability. Equality occurs when x_0 = x_1 = ... = 1/(m+1) and y_0 = y_m =1/2, the rest is 0. Imre Ruzsa === Subject: Re: Smooth retraction Content-Length: 1491 Originator: rusin@vesuvius > Im a bit confused about the notion of a smooth retraction which I > encountered in several places, but unfortunately without a definition. > By a retraction I understand a continuous map $f:M to N$ from a topological > space $M$ to a subspace $N$ such that $f(x)=x$ for all $x in N$. (just as, > for instance, Bredon defines it) > Now what is a smooth retraction? > I would naively assume that this is a retraction in the above sense with the > additional properties that $M$ and $N$ are smooth manifolds, and $f$ being a > smooth map. Ive never seen the term smooth retraction, but thats a reasonable definition. > Is it already clear that $N$ is then an embedded submanifold of $M$? Its clear that, at any point n in N, f has rank dim(N). Its also clear that the rank of f is at most dim(N) everywhere in M. Since rank cannot jump down, f has rank dim(N) on a neighborhood of n in M. The implicit function theorem therefore implies that f looks locally like projection onto a coordiate subspace, so N is an embedded submanifold of M. > If $N$ were an embedded submanifold of $M$, then $f$ is clearly a submersion > in any $x in N$. > But is $f$ also a submersion in any $x in M$? No. With M = R^2 (= xy plane), N = R^1 (= x-axis), f(x, y) = x(y + 1), you can easily check that f is a smooth retraction, but it is not a submersion at (0, -1). John Mitchell === Subject: This week in the mathematics arXiv (29 Sep - 3 Oct) Content-Length: 20557 Originator: rusin@vesuvius Here are this weeks 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 (29 Sep - 3 Oct) ------------------------------------------------ AG: Algebraic Geometry ---------------------- math.AG/0310026 Giuseppe Pareschi: Generic vanishing, gaussian maps, and Fourier-Mukai transform math.AG/0310009 Alberto Calabri, Ciro Ciliberto, Flaminio Flamini, Rick Miranda: On degenerations of surfaces math.AG/0310008 Alexander Kuznetsov: Derived category of V_{12} Fano threefolds math.AG/0310003 Alicia Dickenstein, Laura Matusevich, Timur Sadykov: Bivariate hypergeometric D-modules math.AG/0309473 Marco Andreatta, Elena Chierici, Gianluca Occhetta: Generalized Mukai conjecture for special Fano varieties math.AG/0309460 Laurent Bonavero: Pseudo-index of Fano manifolds and smooth blow-ups math.AG/0309456 Herbert Lange, Christian Pauly: On Frobenius-destabilized rank-2 vector bundles over curves math.AG/0309451 Ichiro Shimada: Supersingular K3 surfaces in odd characteristic and sextic double planes math.AG/0309444 differential operators with values in the determinant bundle math.AG/0309440 Ian Goulden, David Jackson, Ravi Vakil: Towards the geometry of double Hurwitz numbers math.AG/0309439 Gregory Pearlstein: SL_2-orbits and degenerations of mixed Hodge structure math.AG/0309436 William Fulton: On the quantum cohomology of homogeneous varieties math.AG/0309435 Dan Abramovich, Alexander Polishchuk: Sheaves of t-structures and valuative criteria for stable complexes AP: Analysis of PDEs -------------------- math.AP/0310018 N. Burq, P. Gerard, N. Tzvetkov: Multilinear eigenfunction estimates for the Laplace spectral projectors on compact manifolds math.AP/0309463 Sergiu Klainerman, Igor Rodnianski: A geometric approach to the Littlewood-Paley theory math.AP/0309459 Sergiu Klainerman, Igor Rodnianski: Sharp trace theorems for null hypersurfaces on Einstein metrics with finite curvature ßux math.AP/0309428 Terence Tao: On the asymptotic behavior of large radial data for a focusing non-linear Schrodinger equation gr-qc/0309115 Mihalis Dafermos, Igor Rodnianski: A proof of Prices law for the collapse of a self-gravitating scalar field gr-qc/0307013 Mihalis Dafermos: The interior of charged black holes and the problem of uniqueness in general relativity AT: Algebraic Topology ---------------------- math.AT/0309467 Joao P Santos: Dirac operator coupled to instantons on positive definite 4 manifolds math.AT/0309455 Thomas Tradler, Mahmoud Zeinalian: Poincare Duality at the Chain Level, and a BV Structure on the Homology of the Free Loops Space of a Simply Connected Poincare Duality Space math.AT/0309434 Barry Jessup, Gregory Lupton: Free Torus Actions and Two-Stage Spaces math.AT/0309432 Gregory Lupton, Samuel Bruce Smith: Rationalized Evaluation Subgroups of a Map and the Rationalized G-Sequence CA: Classical Analysis and ODEs ------------------------------- math.CA/0310017 Isidore Fleischer: Change of Variable for Multi-dimensional Integral math.CA/0310010 L. I. Danilov: On Equi-Weyl Almost Periodic Selections of Multivalued Maps nlin.SI/0309078 A. V. Kitaev: Dessins dEnfants, Their Deformations and Algebraic the Sixth Painleve and Gauss Hypergeometric Functions math.CA/0309445 Neretin Yu. A: Some continuous analogs of expansion in Jacobi polynomials and vector valued hypergeometric orthogonal bases math.CA/0309443 A.B.J. Kuijlaars, A. Martinez-Finkelshtein: Strong asymptotics for Jacobi polynomials with varying nonstandard parameters math.CA/0309442 Sever Silvestru Dragomir: Bounding the Chebychev Functional for a Pair of Sequences in Inner Product Spaces CO: Combinatorics ----------------- math.CO/0310020 B. Fiedler: Generators of algebraic covariant derivative curvature tensors and Young symmetrizers math.CO/0310016 Marcelo Aguiar, Nantel Bergeron, Frank Sottile: Combinatorial Hopf algebras and generalized Dehn-Sommerville relations math.CO/0310015 Marc Zucker: Motions on n-Simplex Graphs with m-value memory math.CO/0309458 CT: Category Theory ------------------- math.CT/0310027 Ettore Aldrovandi: Hermitian-holomorphic (2)-Gerbes and tame symbols math.CT/0309465 Jurg Frohlich, Jurgen Fuchs, Ingo Runkel, Christoph Schweigert: Correspondences of ribbon categories CV: Complex Variables --------------------- math.CV/0310004 Michael Miller: A quadratic approximation to the Sendov radius near the unit circle math.CV/0310002 Eric Bedford, Jeffrey Diller: Energy and Invariant Measures for Birational Surface Maps math.CV/0309449 Mikhail Sodin, Boris Tsirelson: Random complex zeroes, II. Perturbed lattice DG: Differential Geometry ------------------------- math.DG/0310024 P. Gilkey, S. Nikcevic: Curvature homogeneous spacelike Jordan Osserman pseudo-Riemannian manifolds math.DG/0310007 Hao Fang, Zhiqin Lu: Generalized Hodge Metrics and BCOV torsion on Calabi-Yau Moduli math.DG/0309450 Wei-Dong Ruan: Generalized special Lagrangian fibration for Calabi-Yau hypersurfaces in toric varieties III: The smooth fibres DS: Dynamical Systems --------------------- math.DS/0309477 Sylvain Crovisier, Francois Beguin, Frederic Le Roux, Alice Patou: Pseudo-rotations of the closed annulus : variation on a theorem of J. Kwapisz math.DS/0309462 Claudio Bonanno: The Algorithmic Information Content for randomly perturbed systems FA: Functional Analysis ----------------------- math.FA/0310023 W. B. Johnson, E. Odell: The Diameter of the Isomorphism Class of a Banach Space GM: General Mathematics ----------------------- math.GM/0309474 Joseph Amal Nathan: An Elementary Number Theory Proof of Fermats Last Theorem for Exponent 3 math.GM/0309431 Jean Dezert, Florentin Smarandache: On the Generation of Hyper-powersets for the DSmT GR: Group Theory ---------------- math.GR/0310022 math.GR/0310001 Michael Kapovich: Representations of polygons of finite groups math.GR/0309472 Colette Moeglin: Stabilite en niveau 0 pour les groupes orthogonaux impairs p-adiques math.GR/0309471 Christopher Voll: Zeta functions of groups - singular Pfaffians math.GR/0309447 W. Ethan Duckworth: Describing unipotent classes in algebraic groups using subgroups math.GR/0309446 W. Ethan Duckworth: A Classification of Certain Finite Double Coset Collections in the Classical Groups GT: Geometric Topology ---------------------- math.GT/0310025 Tahl Nowik: Immersions of Non-orientable Surfaces math.GT/0310019 Ursula Ludwig: Stratified Morse Theory with Tangential Conditions math.GT/0309466 Colin Adams: Hyperbolic Knots hep-th/0306165 Sergei Gukov: Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial math.GT/0309437 David Bachman: 2-Normal Surfaces math.GT/0309427 Ryan Budney: Little cubes and long knots HO: History and Overview ------------------------ math.HO/0310014 Ferdinand Minding: On the determination of the degree of an equation obtained by elimination (translation with commentary) math.HO/0310013 Ferdinand Minding: Ueber die Bestimmung des Grades einer durch Elimination hervorgehenden Gleichung (On the determination of the degree of an equation obtained by elimination) MG: Metric Geometry ------------------- math.MG/0309430 Oleg R. Musin: The kissing number in four dimensions MP: Mathematical Physics ------------------------ quant-ph/0310001 Hans Halvorson: Generalization of the Hughston-Jozsa-Wootters theorem to hyperfinite von Neumann algebras nlin.SI/0309058 Pavel Winternitz: Symmetries of Discrete Systems math-ph/0310002 M. Stenmark: A note on regularization and renormalization hep-th/0207200 A.Agarwal, L.Akant, G.S.Krishnaswami, S.G.Rajeev: Collective potential for large N hamiltonian matrix models and free Fisher information hep-th/0111263 L. Akant, G. S. Krishnaswami, S. G. Rajeev: Entropy of Operator-valued Random Variables: A Variational Principle for Large N Matrix Models math-ph/0310001 Marek Biskup, Lincoln Chayes, Steven A. Kivelson: Order by disorder, without order, in a two-dimensional spin system with O(2) symmetry hep-th/0309265 Masashi Hamanaka, Kouichi Toda: Towards Noncommutative Integrable Equations hep-th/0301213 Masashi Hamanaka, Kouichi Toda: Noncommutative Burgers Equation cond-mat/0309680 Luigi Amico, Kazuhiro Hikami: Integrable generalization of the Tavis-Cummings model with counter-rotating terms cond-mat/0309692 Zohar Nussinov, Marek Biskup, Lincoln Chayes, Jeroen van den Brink: Orbital order in classical models of transition-metal compounds cond-mat/0309691 Marek Biskup, Lincoln Chayes, Zohar Nussinov: Orbital ordering in transition-metal compounds: I. The 120-degree model math-ph/0309067 Nasser Saad, Richard L. Hall, Attila B. von Keviczky: Perturbation expansions for a class of singular potentials math-ph/0309066 Hakan Ciftci, Richard L. Hall, Nasser Saad: Asymptotic iteration method for eigenvalue problems math-ph/0309065 J. Dolbeault, M.J. Esteban, M.Loss, L. Vega: An analytical proof of Hardy-like inequalities related to the Dirac operator math-ph/0309064 Filippo Colomo, Andrei Pronko: On the partition function of the six-vertex model with domain wall boundary conditions math-ph/0309063 P.Contucci, C. Giardina, C. Giberti, F.Unguendoli, C. Vernia: Interpolating Greedy and Reluctant Algorithms math-ph/0309062 Sergei M. Grudsky, Kira V. Khmelnytskaya, Vladislav V. Kravchenko: On a quaternionic Maxwell equation for the time-dependent electromagnetic field in a chiral medium math-ph/0309061 Martin Greiter, Dirk Schuricht: Imaginary in all directions: an elegant formulation of special relativity and classical electrodynamics math-ph/0309060 James Lindesay: Group Structure of an Extended Lorentz Group math-ph/0309059 Yuri A. Kubyshin: Geometrical formalism in gauge theories quant-ph/0309194 Robert Alicki, Artur Lozinski, Prot Pakonski, Karol Zyczkowski: Quantum dynamical entropy and decoherence rate quant-ph/0305163 Sabine Kreidl, Gebhard Gruebl, Hans G. Embacher: Bohmian arrival time without trajectories nlin.CD/0309068 U.Frisch, K.Khanin, T.Matsumoto: Multifractality of the Feigenbaum attractor and fractional derivatives math-ph/0309058 Peter J. Forrester: Growth models, random matrices and Painleve transcendents math-ph/0309057 Jean-Bernard Zuber: On the Counting of Fully Packed Loop Configurations. Some new conjectures hep-th/0309243 Jasbir Nagi: Superconformal Primary Fields on a Graded Riemann Sphere hep-th/0309208 Andrei Okounkov, Nikolai Reshetikhin, Cumrun Vafa: Quantum Calabi-Yau and Classical Crystals hep-th/0309196 Armen Nersessian, Armen Yeranyan: 3D Oscillator and Coulomb Systems reduced from Kahler spaces NT: Number Theory ----------------- math.NT/0310005 Alexander Borisov, Yang Wang, Melvyn B. Nathanson: Quantum integers and cyclotomy math.NT/0309478 Stephen S. Gelbart, Stephen D. Miller: Riemanns Zeta Function and Beyond math.NT/0309475 Tom Weston: Power residues of Fourier coefficients of modular forms math.NT/0309433 J. Arias-de-Reyna: X-Ray of Riemann zeta-function OA: Operator Algebras --------------------- math.OA/0309464 M. I. Merklen: Resultados motivados por uma caracterizac{c}~ao de operadores pseudo-diferenciais conjecturada por Rieffel (Ph. D. thesis, in Portuguese) math.OA/0309438 Ronald G. Douglas: Ideals in Toeplitz Algebras math.OA/0309429 Nathan Brownlowe, Nadia S. Larsen, Ian F. Putnam, Iain Raeburn: Subquotients of Hecke C^*-algebras OC: Optimization and Control ---------------------------- math.OC/0310028 Stephane Gaubert, Ricardo Katz: Reachability problems for products of matrices in semirings PR: Probability Theory ---------------------- math.PR/0310012 Lancelot F. James: Functionals of Dirichlet processes, the Markov Krein Identity and Beta-Gamma processes math.PR/0310006 Timothy C. Wallstrom: The marginalization paradox does not imply inconsistency for improper priors math.PR/0309457 D.E. Yakovlev, D.N. Zhabin: Exact Solution of Discrete Hedging Equation for European Option math.PR/0309441 David Gamarnik, Tomasz Nowicki, Grzegorz Swirscsz: Maximum Weight Independent Sets and Matchings in Sparse Random Graphs. Exact Results using the Local Weak Convergence Method quant-ph/0307225 A. S. Holevo: Asymptotic estimation of shift parameter of a quantum state QA: Quantum Algebra ------------------- math.QA/0309468 A. I. Molev, V. N. Tolstoy, R. B. Zhang: On irreducibility of tensor products of evaluation modules for the quantum affine algebra math.QA/0309461 Alexander Molev, Vladimir Retakh: Quasideterminants and Casimir elements for the general Lie superalgebra math.QA/0309453 V. Hinich: Erratum to Homological algebra of homotopy algebras math.QA/0309452 Giovanni Felder, Alexander Varchenko: Hypergeometric theta functions and elliptic Macdonald polynomials math.QA/0309425 Michael E. Hoffman: Algebraic Aspects of Multiple Zeta Values RA: Rings and Algebras ---------------------- math.RA/0310029 M. Domokos, P. E. Frenkel: Mod 2 indecomposable orthogonal invariants math.RA/0310011 A. M. Cohen, D. A. H. Gijsbers, D. B. Wales: BMW algebras of simply laced type math.RA/0309448 Shouchuan Zhang, Yao-Zhong Zhang: Hopf Galois Extension in Braided Tensor Categories RT: Representation Theory ------------------------- math.RT/0309476 Andrew Hubery: The composition monoid and the composition algebra at q=0 of the Kronecker quiver math.RT/0309469 Toshihiko Matsuki: Equivalence of domains arising from duality of orbits on ßag manifolds II math.RT/0309454 Hisayosi Matumoto: The homomorphisms between scalar generalized Verma modules associated to maximal parabolic subalgebras math.RT/0309426 Matthias Kunzer, Andrew Mathas: Elementary divisors of Specht modules SG: Symplectic Geometry ----------------------- math.SG/0309470 M.L. Bialy, R.S. MacKay: Symplectic twist maps without conjugate points SP: Spectral Theory ------------------- math.SP/0310021 Evgeni Korotyaev, Alexander Pushnitski: A trace formula and high energy spectral asymptotics for the perturbed Landau Hamiltonian -- / Greg Kuperberg (UC Davis) / / Visit the Math ArXiv Front at http://front.math.ucdavis.edu/ / * All the math thats fit to e-print * === Subject: maximal geodesics (Was Re: radial directed subgraphs) Epigone-thread: ßermgronfrand Content-Length: 254 Originator: rusin@vesuvius maximal geodesics, as I realized after my posting. Ild still be interested in anything else anybody has to say about this. I am mostly interested in random unit disc graphs though. === Subject: Kontsevich (?) cubic chains Content-Length: 1141 Originator: rusin@vesuvius This is elementary, but I couldnt find a reference. Operads and Motives in Deformation Quantization Letters in Mathematical Physics 48: 35-72, 1999. Maxim Kontsevich uses a particular kind of cubic chain complex of a topological space in which it takes the quotient of all cubic chains -i.e. linear combinations of continuous maps f: k-cube ---> X - by the submodule of degenerate ones -those not depending on all variables- _plus_, and this is the point, the relation f circ (permutation of variables) = = (sign of permutation) x f I can see the reason for such a relation in order to obtain an operad of chain complexes from a topological one and also that this does no harm in order to compute the homology of the topological space. But the cubical chains I knew before reading Kontsevich were those of Masseys book Singular Homology Theory, Springer GTM 70, which dont include this relation. So my question is: are Kontsevich cubical chains also well-known? Is Agust.92 Roig === Subject: A problem involving binomial random variables Content-Length: 529 Originator: rusin@vesuvius Let X_i be a binomail random variable with parameters (n_1,p), ie n_1 trials and the probability of success is p Let Y_j be a binomail random variable with parameters (n_2,p) X = max{X_i} where 1 <= i <= m_1 Y = max{Y_j} where 1 <= j <= m_2 What is P(X>Y+c)? Can anyone give some tips on how to find a closed form expression for this? If it is not possible, at least with the current state of mathematics, then can we have a nice upper bound on this probability. Any help will be greatly appreciated. Manan === Subject: Re: NY Times profile of I. M. Gelfand Content-Length: 645 Originator: rusin@vesuvius >> The northern NJ edition of the NY Times for Sunday October 5 has a profile of >> Israel M. Gelfand on page 4 of section 14, the local New Jersey section. I >> I dont know whether it will appear in non-NJ editions. Lexis-Nexis by searching for Gelfand and restricting to the new york times. David (remove no sp am to reply) === Subject: `A question about acyclic space Content-Length: 382 Originator: rusin@vesuvius Hi all, I know that an acyclic space means that a nonempty topological space with all of its reduced Cech homology groups over rationals vanish. Is it true that the product space of any number (finite or infinite) of acyclic spaces is still acyclic ? Any comments are appreciated. -- çÁ Origin: îM[Micro]ø.b9¨.b2.82.b9[Register edTrademark]ój©.97[CapitalEth].87 (bbs.math.nthu.edu.tw) === Subject: This week in the mathematics arXiv (6 Oct - 10 Oct) Content-Length: 22146 Originator: rusin@vesuvius Here are this weeks 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 (6 Oct - 10 Oct) ------------------------------------------------ AC: Commutative Algebra ----------------------- math.AC/0310130 M. Caboara, M. Kreuzer, L. Robbiano: Efficiently Computing Minimal Sets of Critical Pairs math.AC/0310129 Romain Bondil: Geometry of superficial elements math.AC/0310122 Giulio Caviglia, Enrico Sbarra: Characteristic-free bounds for the Castelnuovo-Mumford regularity math.AC/0310051 Susumu Oda: Endomorphisms of Polynomial Rings with Invertible Jacobians AG: Algebraic Geometry ---------------------- math.AG/0310139 Michael Dettweiler, Stefan Wewers: Variation of local systems and parabolic cohomology math.AG/0310137 Sylvain Maugeais: Deformations Ôequivariantes des courbes stables I : ÔEtude cohomologique math.AG/0310136 Sylvain Maugeais: Theorie des deformations Ôequivariantes des morphismes localement dintersection compl`ete hep-th/0310057 Anton Kapustin: Topological strings on noncommutative manifolds math.AG/0310116 Vladimir Hinich: Deformations of sheaves of algebras math.AG/0310108 David Cox, Alicia Dickenstein: Vanishing and Codimension Theorems for Complete Toric Varieites math.AG/0310107 Michel Brion, Roy Joshua: Intersection Cohomology of reductive varieties math.AG/0310085 Samuel Grushevsky, Riccardo Salvati Manni: Gradients of odd theta functions math.AG/0310084 Andras Nemethi: Line bundles associated with normal surface singularities math.AG/0310076 Al Vitter: Restricting Semistable Bundles on the Projective Plane to Conics math.AG/0310073 Al Vitter: On Stable Bundles of Ranks 2 and 3 on P^3 math.AG/0310066 Xiaotao Sun: Minimal rational curves in moduli spaces of stable bundles math.AG/0310060 Leonid Makar-Limanov, Peter van Rossum, Vladimir Shpilrain, Jie-Tai Yu: The Stable Equivalence and Cancellation Problems math.AG/0310054 Ziv Ran: Normal bundles of rational curves in projective spaces math.AG/0310053 Sadok Kallel, Denis Sjerve: On the Group of Automorphisms of Cyclic Covers of the Riemann Sphere math.AG/0310050 Eric Edward Katz: Toplogical Recursion Relations by Localization math.AG/0310049 Stefano De Leo, Gisele Ducati: Real linear quaternionic operators math.AG/0310045 T. A. Nevins, J. T. Stafford: Sklyanin algebras and Hilbert schemes of points math.AG/0310043 Antonio Laface, Luca Ugaglia: Special linear Systems on Toric Varieties math.AG/0310040 U. Bruzzo, D. Hernandez Ruiperez: Semistability vs. nefness for (Higgs) vector bundles math.AG/0310036 James Pommersheim, Hugh Thomas: Cycles representing the Todd class of a toric variety math.AG/0310035 Indranil Biswas an A. J. Parameswaran: On the equivariant reduction of structure group of a principal bundle to a Levi subgroup math.AG/0310032 Joseph Lipman, Suresh Nayak, Pramathanath Sastry: Pseudofunctorial behavior of Cousin complexes on formal schemes AP: Analysis of PDEs -------------------- math.AP/0310110 Alessio Pomponio: Singularly perturbed Neumann problems with potentials math.AP/0310071 Francesco Chiacchio, Tonia Ricciardi: Multiple vortices for a self-dual CP(1) Maxwell-Chern-Simons model math.AP/0310052 Tristan Riviere, Gang Tian: The Singular Set of 1-1 Integral Currents math.AP/0310039 Vlasov equations with singular potential math.AP/0310030 Helge Kristian Jenssen, Gregory Lyng: Low Frequency Stability of Multi-dimensional Inviscid Planar Detonation Waves CA: Classical Analysis and ODEs ------------------------------- math.CA/0310077 David M. Bradley: A Pair of Difference Differential Equations of Euler-Cauchy Type math.CA/0310063 S. Sadov: Coupling of the Legendre polynomials with kernels $|x-y|^alpha$ and $ln|x-y|$ math.CA/0310062 Douglas Bowman, David M. Bradley: Multiple Polylogarithms: A Brief Survey math.CA/0310061 Douglas Bowman, David M. Bradley: Resolution of Some Open Problems Concerning Multiple Zeta Evaluations of Arbitrary Depth CO: Combinatorics ----------------- math.CO/0310132 Frederic Chyzak, Marni Mishna, Bruno Salvy: Effective Scalar Products for D-finite Symmetric Functions math.CO/0310121 Nathan Reading: The cd-index of Bruhat intervals math.CO/0310120 R.M. Green, J. Losonczy: Freely braided elements in Coxeter groups, II math.CO/0310109 Dan Romik: Shortest paths in the Tower of Hanoi graph and finite automata math.CO/0310103 Kevin Purbhoo: Root games on Grassmannians math.CO/0310082 Douglas Bowman, David M. Bradley: The Algebra and Combinatorics of Shufßes and Multiple Zeta Values math.CO/0310079 J.-F. Fortin, P. Jacob, P.Mathieu: Jagged partitions math.CO/0310078 J. Orestes Cerdeira, Raul Cordovil: A note on mixed graphs and matroids math.CO/0310068 Anatol N. Kirillov, Toshiaki Maeno: Noncommutative algebras related with Schubert calculus on Coxeter groups math.CO/0310057 S. V. Shadrin: Some relations for one-part double Hurwitz numbers math.CO/0310056 Eric Babson, Dmitry N. Kozlov: Complexes of graph homomorphisms CT: Category Theory ------------------- math.CT/0310134 Raphael Rouquier: Dimensions of triangulated categories math.CT/0310055 Eric Babson, Dmitry N. Kozlov: Group Actions on Posets CV: Complex Variables --------------------- math.CV/0310119 Su-Jen kan: On rigidity of Grauert tubes math.CV/0310069 Su-Jen Kan: On rigidity of Grauert tubes over homogeneous Riemannian manifolds math.CV/0310033 Vladimir Ezhov, Alexander Isaev: On the dimension of the stability group for a Levi non-degenerate hypersurface DG: Differential Geometry ------------------------- math.DG/0310126 Tedi Draghici: Symplectic obstructions to the existence of $omega$-compatible Einstein metrics math.DG/0310124 N. A. Daurtseva: $U(n+1)times U(p+1)$ - invariant Hermitian metrics with Hermitian tensor Ricci on the manifold $S^{2n+1}times S^{2p+1}$ hep-th/0310084 Sergey A. Cherkis, Nigel J. Hitchin: Gravitational Instantons of Type D_k math.DG/0310118 P. Gilkey, S. Nikcevic, V. Videv: Manifolds which are Ivanov-Petrova or k-Stanilov math.DG/0310102 Raphael Ponge: Spectral Asymmetry, Zeta Functions and the Noncommutative Residue math.DG/0310097 Thomas Bouetou Bouetou: On an algebraical computaion of the tensor and the curvature for 3-Webs math.DG/0310096 Thomas Bouetou Bouetou: On the structure of Bol algebras math.DG/0310095 Frederic Helein, Pascal Romon: Hamiltonian stationary tori in the complex projective plane math.DG/0310092 Jose A. Galvez, Pablo Mira: The Cauchy problem for Liouville equation and Bryant surfaces math.DG/0310075 Sergiu Moroianu: Weyl laws on open manifolds math.DG/0310072 Janusz Grabowski, Giuseppe Marmo, Peter W. Michor: Homology and modular classes of Lie algebroids math.DG/0310047 Kim A. Froyshov: Monopoles over 4-manifolds containing long necks, I math.DG/0310046 Kai Cieliebak, Edward Goldstein: A note on mean curvature, Maslov class and symplectic area of Lagrangian immersions math.DG/0310041 Michael T. Anderson: Einstein metrics on some exotic negatively curved manifolds DS: Dynamical Systems --------------------- math.DS/0310135 Jean Dezert, Florentin Smarandache: Combining Uncertain and Paradoxical Evidences for DSm Hybrid Models math.DS/0310089 Marco Abate: Discrete local holomorphic dynamics math.DS/0310058 Philip Boyland: Dynamics of two-dimensional time-periodic Euler ßuid ßows FA: Functional Analysis ----------------------- math.FA/0310094 Ralf Meyer: Smooth group representations on bornological vector spaces math.FA/0310086 Yury Grabovsky, Omar Hijab, Igor Rivin: Differentiability of functions of matrices math.FA/0310081 Gelu Popescu: Commutator lifting inequalities and interpolation GR: Group Theory ---------------- math.GR/0310127 Patrick Bahls: Automorphisms of Coxeter groups math.GR/0310125 Anatoliy V. Tushev: On controllers of prime ideals in group algebras of torsion-free abelian groups of finite rank math.GR/0310065 Jason Fox Manning: Quasi-actions on trees and Property (QFA) math.GR/0310038 Cornelius J Griffin: The Fesenko groups have finite width GT: Geometric Topology ---------------------- math.GT/0310112 Hossein Abbaspour: On String Topology of Three Manifolds math.GT/0310111 Julien Marche: On Kontsevich integral of torus knots math.GT/0310100 Se-Goo Kim, Charles Livingston: Knot mutation: 4-genus of knots and algebraic concordance math.GT/0310083 Andras Nemethi: On the Ozsvath-Szabo invariant of negative definite plumbed 3-manifolds math.GT/0310074 Ka Yi Ng: Addendum to Groups of Ribbon Knots math.GT/0310067 Sergey Maksymenko: Homotopy groups of orbits of Morse functions on surfaces math.GT/0310034 Hao Wu: Legendrian Vertical Circles in Small Seifert Spaces KT: K-Theory and Homology ------------------------- math.KT/0310088 M. Khalkhali, B. Rangipour: A Note on Cyclic Duality and Hopf Algebras LO: Logic --------- math.LO/0310131 Dietmar Schweigert: Polynomial Functions on a Central Relation MG: Metric Geometry ------------------- math.MG/0310101 Corran Webster, Adam Winchester: Boundaries of Hyperbolic Metric Spaces MP: Mathematical Physics ------------------------ math-ph/0310015 A.N.F. Aleixo, A.B. Balantekin, M.A. Candido Ribeiro: An Algebraic q-Deformed Form for Shape-Invariant Systems math-ph/0310014 Adel Mohammadpour, Ali Mohammad-Djafari: An alternative inference tool to total probability formula and its applications math-ph/0310013 Paul Federbush: A Criterion for Ferromagnetism in the Isotropic Quantum Heisenberg Model math-ph/0310012 Vladimir Gerdt, Denis Yanovich, Miloslav Znojil: On Exact Solvability of Anharmonic Oscillators in Large Dimensions math-ph/0310011 George Pogosyan, Alexey Sissakian, Pavel Winternitz: Separation of variables and Lie algebra contractions. Applications to special functions math-ph/0310010 Luis J. Boya: Obstructions, Extensions and Reductions. Some applications of Cohomology math-ph/0310009 W.D. van Suijlekom: The noncommutative Lorentzian cylinder as an isospectral deformation gr-qc/0310039 M. Carfora, C. Dappiaggi, A. Marzuoli: The conformal geometry of Random Regge Triangulations cond-mat/0310148 Saul Ares, Mario Castro: Hidden structure in the randomness of the prime number sequence math-ph/0310008 A.Kokotov, D.Korotkin: Bergmann tau-function on Hurwitz spaces and its applications math-ph/0310007 S.P. Gavrilov, D.M. Gitman, A.A. Smirnov: Green functions of the Dirac equation with magnetic-solenoid field hep-th/0310012 Sergey N. Solodukhin: Horizon State, Hawking Radiation and Boundary Liouville Model gr-qc/0310024 Felix Finster, Niky Kamran, Joel Smoller, Shing-Tung Yau: An Integral Spectral Representation of the Propagator for the Wave Equation in the Kerr Geometry math-ph/0310006 D. Levi, J. Negro, M.A. del Olmo: Discrete q-derivatives and symmetries of q-difference equations math-ph/0310005 Wung-Hong Huang: On Tachyon Condensation of Intersecting Noncommutative Branes in M(atrix) Theory hep-th/0308020 Roland Friedrich, Jussi Kalkkinen: On Conformal Field Theory and Stochastic Loewner Evolution math-ph/0310004 Simon Gravel: Superintegrability, isochronicity, and quantum harmonic behavior math-ph/0310003 Julia Breiderhoff, Fabio Musso, Orlando Ragnisco: Exact solution of a supersymmetric Gaudin model hep-th/0310016 R.C. King, T.D. Palev, N.I. Stoilova, J. Van der Jeugt: A non-commutative NT: Number Theory ----------------- math.NT/0310123 Amilcar Pacheco: On the variation of the rank of Jacobian varieties on unramified abelian towers over number fields math.NT/0310106 Samuel Grushevsky, Riccardo Salvati Manni: Two generalizations of Jacobis triple product identity math.NT/0310105 Yoichi Motohashi: A Functional Equation for the Spectral Fourth Moment of Modular Hecke L-Functions math.NT/0310104 Roelof W. Bruggeman, Yoichi Motohashi: A New Approach to the Spectral Theory of the Fourth Moment of the Riemann Zeta-Function math.NT/0310064 Yoichi Motohashi: A multiple sum involving the Moebius function OA: Operator Algebras --------------------- math.OA/0310138 Alan L. T. Paterson: The Fourier-Stieltjes and Fourier algebras for locally compact groupoids math.OA/0310117 Alan L. T. Paterson: The analytic index for proper, Lie groupoid actions math.OA/0310115 Alan L. T. Paterson: The Fourier algebra for locally compact groupoids math.OA/0310113 Gelu Popescu: Similarity and ergodic theory of positive linear maps math.OA/0310037 M. I. Merklen: Boundedness of Pseudodifferential Operators of C^*-Algebra-Valued Symbol PR: Probability Theory ---------------------- math.PR/0310091 Serban Nacu: Increments of Random Partitions math.PR/0310059 Mark Huber: Exact Sampling from Perfect Matchings of Dense Nearly Regular Bipartite Graphs math.PR/0310044 Timo Seppalainen: Second-order ßuctuations and current across characteristic for a one-dimensional growth model of independent random walks QA: Quantum Algebra ------------------- math.QA/0310128 M.Pevzner, Ch.Torossian: Isomorphisme de Dußo et la cohomologie tangentielle math.QA/0310087 Alexander Kirillov Jr: On the modular functor associated with a finite group math.QA/0310080 Stefano Capparelli, James Lepowsky, Antun Milas: The Rogers-Selberg recursions, the Gordon-Andrews identities and intertwining operators math.QA/0310070 Roland Berger, Nicolas Marconnet: Koszul and Gorenstein properties for homogeneous algebras nlin.CG/0310002 A. Kuniba, T. Takagi, A. Takenouchi: Factorization, reduction and embedding in integrable cellular automata hep-th/0310026 A. P. Balachandran, S. Kurkcuoglu: Topology Change for Fuzzy Physics: Fuzzy Spaces as Hopf Algebras math.QA/0310042 Tatsuro Ito, Paul Terwilliger: Tridiagonal pairs and the quantum affine algebra $U_q({hat {sl}}_2)$ RT: Representation Theory ------------------------- math.RT/0310140 Ivan Penkov, Gregg Zuckerman: Generalized Harish-Chandra Modules: A New Direction math.RT/0310031 Xuhua He: Total positivity in the De Concini-Procesi compactification SG: Symplectic Geometry ----------------------- math.SG/0310133 James Montaldi, Juan-Pablo Ortega, Tudor S. Ratiu: The relation between local and global dual pairs math.SG/0310114 Eduardo Gonzalez: Quantum cohomology and $S^1$-actions with isolated fixed points math.SG/0310099 Dave Auckly: Topologically knoted Lagrangians in simply connected four manifolds math.SG/0310098 Jiang-Hua Lu, Sam Evens: Thompsons conjecture for real semi-simple Lie groups math.SG/0310048 Yi Lin, Reyer Sjamaar: Equivariant symplectic Hodge theory and the $d_Gdelta$-lemma SP: Spectral Theory ------------------- math.SP/0310093 Yu. Netrusov, Yu. Safarov: Weil asymptotic formula for the Laplacian on domains with rough boundaries math.SP/0310090 Tomio Umeda: Generalized eigenfunctions of relativistic Schroedinger operators I -- / Greg Kuperberg (UC Davis) / / Visit the Math ArXiv Front at http://front.math.ucdavis.edu/ / * All the math thats fit to e-print * === Subject: Paper published by Geometry and Topology Content-Length: 2334 Originator: rusin@vesuvius The following paper has been published: URL: http://www.maths.warwick.ac.uk/gt/GTVol7/paper16.abs.html Title: Equivariant Euler characteristics and K-homology Euler classes for proper cocompact G-manifolds Author(s): Wolfgang Lueck and Jonathan Rosenberg Abstract: Let G be a countable discrete group and let M be a smooth proper cocompact G-manifold without boundary. The Euler operator defines via Kasparov theory an element, called the equivariant Euler class, in the equivariant K$-homology of M. The universal equivariant Euler characteristic of M, which lives in a group U^G(M), counts the equivariant cells of M, taking the component structure of the various fixed point sets into account. We construct a natural homomorphism from U^G(M) to the equivariant KO-homology of M. The main result of this paper says that this map sends the universal equivariant Euler characteristic to the equivariant Euler class. In particular this shows that there are no `higher equivariant Euler characteristics. We show that, rationally, the equivariant Euler class carries the same information as the collection of the orbifold Euler characteristics of the components of the L-fixed point sets M^L, where L runs through the finite cyclic subgroups of G. However, we give an example of an action of the symmetric group S_$ on the 3-sphere for which the equivariant Euler class has order 2, so there is also some torsion information. Secondary: 19K35, 19K56, 19L47, 58J22, 57R91, 57S30, 55P91 Keywords: Equivariant K-homology, de Rham operator, signature operator, Kasparov theory, equivariant Euler characteristic, fixed sets, cyclic subgroups, Burnside ring, Euler operator, equivariant Euler class, universal equivariant Euler characteristic Received: 2 August 2002 Proposed: Steve Ferry Seconded: Martin Bridson, Frances Kirwan Author(s) address(es): Institut fur Mathematik und Informatik, Westfalische Wilhelms-Universtitat Einsteinstr. 62, 48149 Munster, Germany and Department of Mathematics, University of Maryland College Park, MD 20742, USA Email: lueck@math.uni-muenster.de, jmr@math.umd.edu URL: wwwmath.uni-muenster.de/u/lueck, www.math.umd.edu/~jmr === Subject: Paper published by Algebraic and Geometric Topology Content-Length: 1174 Originator: rusin@vesuvius The following paper has been published: Algebraic and Geometric Topology URL: http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-34.abs.html Title: Deformation of string topology into homotopy skein modules Author(s): Uwe Kaiser Abstract: Relations between the string topology of Chas and Sullivan and the homotopy skein modules of Hoste and Przytycki are studied. This provides new insight into the structure of homotopy skein modules and their meaning in the framework of quantum topology. Our results can be considered as weak extensions to all orientable 3-manifolds of classical results by Turaev and Goldman concerning intersection and skein theory on oriented surfaces. Secondary: 57M35, 57R42 Keywords: 3-manifold, string topology, deformation, skein module, torsion, link homotopy, free loop space, Lie algebra Author(s) address(es): Department of Mathematics, Boise State University 1910 University Drive, Boise, ID 83725-1555, USA Email: kaiser@math.boisestate.edu URL: http://diamond.boisestate.edu/~kaiser/ === Subject: Paper published by Algebraic and Geometric Topology Content-Length: 918 Originator: rusin@vesuvius The following paper has been published: Algebraic and Geometric Topology URL: http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-35.abs.html Title: On the domain of the assembly map in algebraic K-theory Author(s): Arthur C. Bartels Abstract: We compare the domain of the assembly map in algebraic K-theory with respect to the family of finite subgroups with the domain of the assembly map with respect to the family of virtually cyclic subgroups and prove that the former is a direct summand of the later. Secondary: 19A31, 19B28 Keywords: K-theory, group rings, isomorphism conjecture Author(s) address(es): SFB 478, Westfalische Wilhelms-Universitat 48149 Munster, Germany Email: bartelsa@math.uni-muenster.de === Subject: Pluecker coordinates of symmetric powers Content-Length: 903 Originator: rusin@vesuvius Let V be a finite dimensional vector space with a given ordered basis e_1, ..., e_n . Let W be a p-dimensional linear subspace. If x_1, ..., x_p is a basis of W, express the x_i as linear combinations in the e_j. The coefficients give a p x n-matrix. The maximal minors of this matrix are known as the *Pluecker coordinates* of the subspace W . Up to a scalar multiple they are independent of the basis x_1, ..., x_p and, considered as homogeneous coordinates, characterize W uniquely. For any k, consider the k-th symmetric power S^kV of V. The ordered basis e_1, ..., e_n determines an ordered basis of S^kV, and the subspace W of V determines a subspace S^kW of S^kV. So the Pluecker coordinates of S^kW as a subspace of S^kV are determined by the Pluecker coordinates of W in V and, in fact, should be polynomials in the latter ones. Does someone know how these look Boudewijn Content-Length: 20931 Originator: rusin@vesuvius ------------------------------------------------------------- --------------- -- Call For Papers and Applications Coupling Approaches, Coupling Media and Coupling Languages for Information Retrieval University of Avignon (Vaucluse), France Organized by: CENTRE DE HAUTES ETUDES INTERNATIONALES DINFORMATIQUE DOCUMENTAIRE (C.I.D., France)in cooperation with the LIA (Laboratoire dInformatique dAvignon- Universit.8e dAvignon) and with technical support of IRIT (Institut de Recherche en Informatique de Toulouse) CALL FOR PAPERS Current content-based information management involves many different disciplines. Information must be retrieved from video, from sound, and from images and graphs. Question answering involves both syntax and semantics. Information classification and filtering involve machine learning and linguistics. In addition, as information technology spreads throughout the world, a wider variety of languages in increasingly complex combinations must be handled. papers covering the coupling of techniques from different domains to research and developments from all areas of multi-media and multi-language information retrieval. Submissions, demonstrating combination of techniques from disparate domains, may treat retrieval from either a single medium, or across media (indexing one medium for finding information in another), or from coupling unstructured and structured information (e.g. exploiting both text and XML structure), or from across languages. Conference Themes: Paper submissions should cover one or more of the following themes: Multimedia information: * Media-specific indexing techniques (text, speech, fixed and animated images, music) * Indexing composite documents * Querying multimedia documents * Automatically generating text from images and from video * Indexing interactive documents Multilingual Information: * Cross-lingual information retrieval, especially involving rarer languages * Automatic construction of bilingual lexicons and term banks * Production of multilingual documents Man Machine Combinations: * Coupling search and browsing * Coupling search and semantic mapping (ontologies, SOM, etc) * Multimodal interfaces * Coupling access through structure and through content * Automatic presentation of search aids (e.g. key words, phrases) * Neuroscience applied to information recognition Architecture for Combined Approaches: * Architecture for coupling techniques (e.g. Machine Learning for Content Management) * Architecture for coupling media * Architecture for treating multilingual information Specific Systems Combining Diverse Approaches: * Systems for Collaborative Information Retrieval * Question answering systems * Multidocument or multilingual summarization * Automatic translation, translation memory Combining Linguistic and Statistics for Retrieving Content: * Improved linguistic analyzers in information retrieval * Exploiting linguistic knowledge in search and retrieval * Knowledge Extraction for Information Retrieval * Semantics in indexation and retrieval Composite Documents and Content: * Exploiting document structure * Semantic Web, and Ontologies for Full-Scale Information Retrieval * Exploiting new multimedia norms for content-based information management Evaluation of Combined Approaches: * User oriented retrieval metrics * New retrieval metrics * Question-Answering systems evaluation metrics Application domains combining techniques: (descriptions of systems involving the following domains): * Cultural heritage * Indexation and retrieval of medical images * Applications concerning security * Protection of intellectual property * Protection of minors * E-learning * Technology Watch Important dates: Submissions should be about 6000 words (about 20 pages, double spaced), include and abstract and be submitted in PDF or PS format. Submissions for communications will be made electronically on its web site : http://www.riao.org. The working language of the conference is English. However, in agreement with the French regulations of the Loi Toubon, submission of papers in French and presentation of papers, if selected, in French will be accepted. CALL FOR APPLICATIONS Innovative applications and products related to the conference topics are sought for demonstrations during the three days of the Conference. They will be selected by the international Application committee, on the basis of their innovation and future or present marketability. Selected applications will be given free demonstrations spaces. Application submissions should cover one or more of the following topics: * Multimedia indexing and retrieval systems (text, sound, speech, images, video) * Cross-lingual indexing and retrieval systems * Peer-to-peer text search engines * Cooperative Information Retrieval (grids) * Automatic XML structuring of documents * Automatic metadata generation for text, sound, and images, automatic annotators * Automated ontology construction and annotators * Topic detection and event detection in streaming documents, technology watch, strategy watch * Intelligent message filtering * Intelligent text agents * E-learning - response interpretation * Document summarisation -- mono or multilingal, mono or multidocument, profile driven * Topic maps * Domain-specific application of information retrieval and multimedia retrieval: medicine, e-commerce, computer-assisted teaching, video production, etc Important dates: Program Committee Co-Chairs Christian Fluhr Gregory Grefenstette Bruce Croft CEA, France Clairvoyance UMass,Amherst,USA Europe, Africa Asia, Oceania Americas Catherine Berrut IMAG, France Francine Chen PARC, USA Claude Chrisment IRIT, Toulouse, France Franciska de Jong Univ. Twente, Netherland Claude de Loupy Sinequa, France Renato De Mori Univ. Avignon, France Marc El-B.8fze Univ. Avignon, France Pascale Fung Scienc. Tech. Univ., Hong Kong Sadaoki Furui Tokyo Inst. Tech., Japan Jean-Luc Gauvain LIMSI, France Edouard Geoffrois ETCA/DGA, France Julio Gonzalo UNED, Spain Donna Harman NIST, USA Ulrich Heid Univ. Stuttgart, Germany Eduard Hovy ISI, Univ. S. California, USA Christian Jacquemin LIMSI, France Boris Katz MIT, USA Elisabeth Liddy Univ. Syracuse, USA Simone Marinai Univ. Florence, Italy Jos.8e Martinez Univ. Nantes, France Christof Monz Univ. Amsterdam, Netherland Frank Nack CWI, Netherland Chahab Nastar LTU, France Jian-Yun Nie Univ. Montr.8eal, Canada Douglas Oard Univ. Maryland, USA J.9arg Ontrup Univ. Bielefeld, Germany Gabriella Pasi Univ. Milano, Italy Marie Theresa Pazienza Univ. Roma, Italy Marc Pic Advestigo, France Jean-Marie Pierrel INALF, France Jean-Marie Pinon INSA Lyon, France Yan Qu Clairvoyance, USA Steve Renals Univ. Sheffield, Great Britain Tetsuya Sakai Toshiba, Japan Fr.8ed.8erique Segond Xerox, France Bernadette Sharp Staffordshire, Great Britain Alan Smeaton Univ. Dublin, Ireland Tokunaga Takenobu Tokyo Inst. Tech., Japan Simone Teufel Univ. Cambridge, Great Britain Evelyne Tzoukermann ACM, USA Keith Van Rijsbergen Univ. Glasgow, Great Britain Ross Wilkinson CSIRO, Australia Zhiping Zheng Univ. Saarland, Germany (Final list forthcoming) Applications Committee Chair Chantal Soul.8e-Dupuy Universit.8e de Toulouse, France (Forming Committee) Organisation and Coordination Committee Chair Agn.8fs Beriot D.8el.8egu.8ee G.8en.8erale du C.I.D., France Peter Brodnitz Ogilvy & Mather, Japan Jean Louis dArc F.8ed.8eration France-Polonge,France Jean Perri.8fre Administrator, Secretary General,C.I.D., France Saryn Rosart CASIS, USA Anne Tabutiaux Recherche et Diffusion, France (Final list forthcoming) Local Organisation Committee Aur.8elia Barri.8fre Univ. Avignon, France St.8ephane Igounet Univ. Avignon, France (Final list forthcoming) Technical Committee President Luc Boulianne C.I.D., Canada Jonathan Albert C.I.D., Canada Max Chevalier Univ. Toulouse, France Jean-Jacques Guilbart Coll.8fge de France, France C.8ecile Laffaire Univ. Toulouse, France (Final list forthcoming) Contact Information Centre de Hautes Etudes Internationales dInformatique Documentaire (C.I.D.) 36 bis rue Ballu 75009 Paris France Tel: (33 / 0) 1 42 85 04 75 Fax: (33 / 0) 1 48 78 49 61 Email: information@le-cid.org Web: http://www.le-cid.org ------------------------------------------------------------- --------------- -- === Subject: A couple of point-set topology questions Content-Length: 1497 Originator: rusin@vesuvius I have two questions. I believe that both of them are standard, with well-known answers, but I am having trouble finding the answers. References to the literature would be very acceptable. 1. Let X be a compact hausdorff space, whose product with a closed unit interval is homeomorphic to a closed square. How does one prove that X is a closed unit interval? What happens when one takes away compactness assumptions? 2. Suppose I have an uncountable number of disjoint copies of a compact space Y embedded in a nice space Z. For example Z could be the the closed unit square. What deductions can I make? It ought to be true that one can find a sequence of disjoint copies of Y converging to a copy of Y. How does one show that? Its probably easy, but I dont see it. Are there reasonable conditions on Z which would enable one to deduce that there is a sequence of homeos converging to the identity in some sense, and taking one of the given Ys to another? Can the second Y be fixed? Question 2 is of course relevant to the well-known result that you cant embed an uncountable set of disjoint compact triods in the plane. However, my question does NOT concern the triod. I am interested in other spaces, and Y stands for any space, not for a triod. I would be grateful if answers can be copied to me at dbae@maths.warwick.ac.uk -- Phone: +44-24-76522677 (work) +44-1926-853955 (home) +44-24-76524182 (fax) David Epstein === Subject: Re: A couple of point-set topology questions Originator: israel@math.ubc.ca (Robert Israel) > I have two questions. I believe that both of them are standard, with > well-known answers, but I am having trouble finding the answers. > References to the literature would be very acceptable. > 2. Suppose I have an uncountable number of disjoint copies of a > compact space Y embedded in a nice space Z. For example Z could be the > the closed unit square. What deductions can I make? It ought to be > true that one can find a sequence of disjoint copies of Y converging > to a copy of Y. How does one show that? Its probably easy, but I > dont see it. Are there reasonable conditions on Z which would enable > one to deduce that there is a sequence of homeos converging to the > identity in some sense, and taking one of the given Ys to another? > Can the second Y be fixed? See http://www.ams.org/mathscinet-getitem?mr=94f54034 for a fairly comprehensive answer to question 2 You even get the product $Ytimes C$ in $Z$ (where $C$ is the Cantor set*).* I assume separable completely metrizable is nice enough. KP -- E-MAIL: K.P.Hart@EWI.TUDelft.NL PAPER: Faculty EWI PHONE: +31-15-2784572 TU Delft FAX: +31-15-2786178 Postbus 5031 URL: http://aw.twi.tudelft.nl/~hart 2600 GA Delft the Netherlands === Subject: Hyperexponential Epigone-thread: cenzirquu Content-Length: 267 Originator: rusin@vesuvius I have noticed that if one sets f(x)=sum((n+1)^(n-1))/n!*x^n starting with n=0 then if g(x)=f(ln(x+1)) we have (1+x)^g(x)=g(x). Is this a well-known result? Can it be easily proved? Id appreciate if someone could take a look at this. Rico === Subject: Re: Hyperexponential Originator: israel@math.ubc.ca (Robert Israel) > I have noticed that if one sets f(x)=sum((n+1)^(n-1))/n!*x^n > starting with n=0 then if g(x)=f(ln(x+1)) we have (1+x)^g(x)=g(x). Is > this a well-known result? Can it be easily proved? Id appreciate > if someone could take a look at this. It is a well known result. Your notation is mixed up, however. Heres the actual result: If you let your f = LambertW(0,x), (and denote it by LW(x)) then the LW has the xpansion: LW(x) = Sum((-n)^(n-1)*x^n/n!, n=1..+oo) It is known that the LW solves the equation x^y = y, and the solution has the form: y = -LW(-ln(x))/ln(x) = exp(-LW(-ln(x)). In other words: x^[-LW(-ln(x))/ln(x)] = -LW((-ln(x))/ln(x), or x^g(x) = g(x). Its just a matter of rearranging your notation to match the LW. Rico, do yourself a favor. Read my pages on hyperexponentials FIRST, instead of trying to re-invent the wheel: Hi. > Ive become interested in polyhedron approximations of the 3D sphere. > Now, Im looking for a list of polyhedrons, for which the > volume/surface ratio is maximalized. It is obvious that the more > polygons/faces/sides a polyhedron has, it is possible to approximate > the maximum volume/surface ratio scored by the sphere shape. However, > for a fixed number of faces, there must be a Ôwinning polyhedron. > Has this been researched? Is there a table that lists for each value > of Ôthe number of faces the polyhedron that maximalizes the > volume/surface ratio? > Robert Lukassen > The Netherlands I dont have an answer to your question, but it might be worthwhile to have a look at the polyhedra in http://www.research.att.com/projects/OEIS?Anum=A081314 and the links given there. Hugo Pfoertner === Subject: Re: Volume/surface ratio in (3D) polyhedrons Originator: israel@math.ubc.ca (Robert Israel) > Im looking for a list of polyhedrons, for which the > volume/surface ratio is maximalized. That reminds me of related threads in January 2002, e.g. I did get some valuable information when I asked this question then. Not just the sci.math.research-level but rather alt.math.recreational (where I created a certain Mumpf :-) but nontheless maybe interesting for you. Google for Tetrakaidecahedron and Kelvin. http://www.susqu.edu/brakke/kelvin/kelvin.html I forgot about the mumpf-discussion, but here it is Rainer Rosenthal r.rosenthal@web.de === Subject: Is this a Bipartite Matching or TSP ? Originator: israel@math.ubc.ca (Robert Israel) I have 2n vertices, n for group A, n for group B. The vertices are fully connected by edges, each edge has a different weight. What I want to solve is this: Go trough all vertices, without repetition, and minimize the sum of weights, under this rule: ... -A-A-B-B-A-A-B-B-A-A-B-B-... where A represents some vertex of A. This is different than Bipartite TSP, which is ...-A-B-A-B-A-... Can anyone tell me if there are algorithms to solve my problem? === Subject: Re: Is this a Bipartite Matching or TSP ? Originator: israel@math.ubc.ca (Robert Israel) > I have 2n vertices, n for group A, n for group B. > The vertices are fully connected by edges, each edge has a different > weight. > What I want to solve is this: > Go trough all vertices, without repetition, and minimize the sum of > weights, under this rule: > ... -A-A-B-B-A-A-B-B-A-A-B-B-... > where A represents some vertex of A. > This is different than Bipartite TSP, which is ...-A-B-A-B-A-... The classical TSP is a special case, so your problem is NP-hard. Given a graph G with n vertices, create the graph on A and B as follows: Weight between A_i and A_j, and B_i and B_j, the same as between G_i and G_j. Weight 0 between A_i and B_i. Very high weight between A_i and B_j if i is not equal to j. A circuit on the new graph which goes only from A_i to B_i has the same weight as the corresponding cycle on G. A circuit which goes from A_i to B_j where j is not equal to i has one edge of very high weight and thus cannot be the shortest circuit. -- David Grabiner, grabiner@alumni.princeton.edu, http://remarque.org/~grabiner Baseball labor negotiations FAQ: http://remarque.org/~grabiner/laborfaq.html Shop at the Mobius Strip Mall: Always on the same side of the street! Klein Glassworks, Torus Coffee and Donuts, Projective Airlines, etc. === Subject: Two-planes in Four-space Originator: israel@math.ubc.ca (Robert Israel) Let G(2,4) be the Grassmanian of 2-dimensional subspaces in R^4. Map G(2,4) -> RP^2 as follows. Given a plane g, choose an orthogonal basis e1, e2. Identifying R^4 with the quaternions in the obvious way, form the quaternion u = e1 * e2^(-1). Then u is a square root of -1, well defined up to a sign. The square roots of -1 are naturally identified with the unit ball S^2 sitting in R^3 sitting in R^4 via (a,b,c) |-> (0,a,b,c), so u gives a well-defined element of S^2/(plus or minus 1) = RP^2. Question 1: Is there a standard name for this map G(2,4) -> RP^2 ? Now consider the fiber F_u over the point in RP^2 represented by the quaternion u. We can (unless Ive made an error) say the following things about this fiber: a) F_u has a natural group structure: Given two planes g1 and g2 in F_u, it follows that the set g1 g2 = {p q |p in g1 and q in g2} is also a plane in F_u. This makes the fiber naturally isomorphic to H^*/(R+Ru)^* . b) F_u consists of all planes that are fixed by the 90 degree rotation represented by u. Thus: c) Two planes sit in the same fiber if and only if there exists a 90 degree rotation of R^4 that fixes both of them. d) Two planes g1 and g2 sit in the same fiber if and only if there is a quaternion p such that p g1 = g2. Question 2: Is there a standard name for the equivalence relation described in c) (or equivalently d))? Steven E. Landsburg www.landsburg.com/about2.html -- === Subject: Re: Two-planes in Four-space Originator: israel@math.ubc.ca (Robert Israel) > Let G(2,4) be the Grassmanian of 2-dimensional subspaces in R^4. > Map G(2,4) -> RP^2 as follows. Given a plane g, choose an orthogonal > basis e1, e2. Identifying R^4 with the quaternions in the obvious way, > form the quaternion u = e1 * e2^(-1). Then u is a square root of -1, well > defined up to a sign. The square roots of -1 are naturally identified > with the unit ball S^2 sitting in R^3 sitting in R^4 via (a,b,c) |- (0,a,b,c), so u gives a well-defined element of S^2/(plus or minus 1) = > RP^2. > Question 1: Is there a standard name for this map G(2,4) -> RP^2 ? I think that this will be essentially the same map, modulo the obvious covers, as the map from the oriented Grassmannian to S^2 defined by the decomposition of G_o(2,4) as S^2 x S^2. Since the Hodge star * is a symmetric operator on Lambda^2(R^4), and satisfies **=1, choose an orthonormal basis of Lambda^2(R^4) consisting of +1 (dim 3) and -1 (3 also) eigenvectors of *. Then any element of G_o(2,4) inside Lambda^2(R^4) is a sum of elements in the two eigenspaces, each of length 1/sqrt(2). My guess is that your map, for appropriate choices everywhere, is a projection onto one of these factors. The construction of this map is in an old paper by Singer and Thorpe, dealing with the curvature of Einstein 4-manifolds, in a collection called Global Analysis -- I believe its Although your construction is quite different, I dont see how it could not be the same map. > Now consider the fiber F_u over the point in RP^2 represented by the > quaternion u. We can (unless Ive made an error) say the following things > about this fiber: > a) F_u has a natural group structure: Given two > planes g1 and g2 in F_u, it follows that the set g1 g2 = {p q |p > in g1 and q in g2} is also a plane in F_u. This makes the fiber > naturally isomorphic to H^*/(R+Ru)^* . Well, if my identification is right, this would have to be an S^2. I dont see why this group structure is well-defined. > b) F_u consists of all planes that are fixed by the 90 > degree rotation represented by u. Hmm. u is an imaginary quaternion, so it is also a complex structure on a decomposition of H into C^2. Planes fixed by that are complex lines for this complex structure, so are an S^2. This is not consistent with a), but is consistent with my construction, since the +1 eigenvectors of * of length 1/sqrt(2) plus a single (-1) eigenvector of the same length, the fiber under my construction, is a CP^1 inside G_o(2,4) for some complex structure. -- David L. Johnson __o | Deserves death! I daresay he does. Many that live deserve _`(,_ | death. And some that die deserve life. Can you give it to (_)/ (_) | them? Then do not be too eager to deal out death in judgement. -- J. R. R. Tolkien === Subject: Re: Two-planes in Four-space Originator: israel@math.ubc.ca (Robert Israel) educate me. I was of course wrong about the group structure on the fiber. The correct statement about products (I think) is this: Given two planes g1 and g2, define their product to be {p q | p in g1 and q in g2}. We have two fibrations G(2,4)->RP^2 corresponding to the two projections of G+(2,4) onto S^2. Call the fibers overs u F_u and G_u. Suppose g1 is in F_u and g2 is in G_v. Then g1 g2 is a plane if and only if u=v. Thus there is a product F_u x G_u -> G(2,4). -- === Subject: Re: Two-planes in Four-space Originator: israel@math.ubc.ca (Robert Israel) >Let G(2,4) be the Grassmanian of 2-dimensional subspaces in R^4. And while were at it, let G+(2,4) be the Grassmannian of oriented 2-dimensional subspaces in R^4 (already identified with the quaternions, if we like). >Map G(2,4) -> RP^2 as follows. Given a plane g, choose an >orthogonal basis e1, e2. Identifying R^4 with the quaternions in >the obvious way, form the quaternion u = e1 * e2^(-1). Then >u is a square root of -1, well defined up to a sign. The square >roots of -1 are naturally identified with the unit ball S^2 sitting >in R^3 sitting in R^4 via (a,b,c) |-> (0,a,b,c), so u gives a >well-defined element of S^2/(plus or minus 1) = RP^2. >Question 1: Is there a standard name for this map G(2,4) -> RP^2 ? Well, if you do the same construction starting with G+(2,4), the map goes to S^2, and its standard name could be projection on the factor, for indeed G+(2,4) is (pretty damned naturally) the product S^2times S^2 in such a way that thats your map (and the ambiguity in pretty damned matches the ambiguity earlier). This makes the rest of your construction, below, worrisome to me: for, in my construction (which I learned ages ago from Bob MacPherson, though clearly it wasnt original with him, either), the fiber over (the unit pure quaternion) u is (of course) a 2-sphere (namely, the complex projective line of all complex 1-subspaces of the complex 2-space which is the real vectorspace H with [left] multiplication by u as its J-operator); so it seems to me that the fiber F_u in *your* construction is still S^2, and thus cannot have a (very) natural group structure, as claimed in a). (Your b) is fine.) >Now consider the fiber F_u over the point in RP^2 represented by >the quaternion u. We can (unless Ive made an error) say the >following things about this fiber: > a) F_u has a natural group structure: Given two > planes g1 and g2 in F_u, it follows that the set > g1 g2 = {p q |p in g1 and q in g2} is also a plane in F_u. > This makes the fiber naturally isomorphic to > H^*/(R+Ru)^* . > b) F_u consists of all planes that are fixed by the 90 > degree rotation represented by u. >Thus: > c) Two planes sit in the same fiber if and only if there > exists a 90 degree rotation of R^4 that fixes both of > them. In my language (transfered to your context), two unoriented planes sit in the same fiber if and only if they can be oriented so as to both be complex lines for the same orthogonal complex structure on R^4. > d) Two planes g1 and g2 sit in the same fiber if and only > if there is a quaternion p such that p g1 = g2. >Question 2: Is there a standard name for the equivalence relation >described in c) (or equivalently d))? I dont think theres any name less long-winded than whats written above. >Steven E. Landsburg >www.landsburg.com/about2.html Lee Rudolph === Subject: Distribution of Garch Process Originator: israel@math.ubc.ca (Robert Israel) Hi All, Can anybody tell me about what is the distribution for GARCH process? I tried at my known resources but no luck.Also is it possible to approximate it with Normal distribution in a single time period? Please excuse me if I am asking a very siple question considering my background is OR. Further I am also looking for any web resource where I can find details of GARCH process. Sharad === Subject: Re: Distribution of Garch Process Originator: israel@math.ubc.ca (Robert Israel) > Hi All, > Can anybody tell me about what is the distribution for GARCH process? > I tried at my known resources but no luck.Also is it possible to > approximate it with Normal distribution in a single time period? > Please excuse me if I am asking a very siple question considering my > background is OR. > Further I am also looking for any web resource where I can find > details of GARCH process. > Sharad Not knowing whether the Q for the distribution makes sense, in OR you may have continous variants. You will find a lot of lectures through Google for Econometrics (+Garch). One is Christoffersen, Chap 2 for free at http://www.intranet. management.mcgill.ca/homepage/profs/christop/research.htm Another is Carol Alexander at ISMA http://www.isma.rdg.ac.uk /noframes/static/faculty/carolalexander.htm, may be you have to dig for her papers on that site. Quite readable (i do nor say i got them) and very carefull are books by W Haerdle and some are for download (but not printable) at http://www.xplore-stat.de/ebooks/ebooks.html === Subject: Re: Distribution of Garch Process Originator: israel@math.ubc.ca (Robert Israel) Sharad Hi All, > Can anybody tell me about what is the distribution for GARCH process? > I tried at my known resources but no luck.Also is it possible to > approximate it with Normal distribution in a single time period? > Please excuse me if I am asking a very siple question considering my > background is OR. > Further I am also looking for any web resource where I can find > details of GARCH process. > Sharad > Not knowing whether the Q for the distribution makes sense, > in OR you may have continous variants. You will find a lot > of lectures through Google for Econometrics (+Garch). One > is Christoffersen, Chap 2 for free at http://www.intranet. > management.mcgill.ca/homepage/profs/christop/research.htm > Another is Carol Alexander at ISMA http://www.isma.rdg.ac.uk > /noframes/static/faculty/carolalexander.htm, may be you > have to dig for her papers on that site. > Quite readable (i do nor say i got them) and very carefull > are books by W Haerdle and some are for download (but not > printable) at http://www.xplore-stat.de/ebooks/ebooks.html === Subject: Re: A conjecture harder than Goldbach !? Epigone-thread: yirchamcin Originator: israel@math.ubc.ca (Robert Israel) |I encountered a wonderful conjecture. |Take every composite number other than 1. Let a_i denotes the number |of single divisors (i.e., if 2*2*3, then only 2, 2, and 3, not |including 2*3) of the ith composite other than 1 and itself. Then, the |sequence of partial sums {a_1, a_1 + a_2, a_1 + a_2 + a_3,...} |contains EVERY square other than 1^2. |We take composites from 4 to 55: |4 -> 2 (the number of divisors) ; 2 (partial sum) |6 -> 2 ; 4 :) |8 -> 3 ; 7 |9 -> 2 ; 9 :) |10-> 2 ; 11 |12 2,2,3-> 3 ; 14 |14 2,7, -> 2 ; 16 :) |15 3,5, -> 2 ; 18 |16 2,2,2,2 -> 4 ; 22 |18 3,3,2 -> 3 ; 25 :) |20 2,2,5 -> 3 ; 28 |21 3,7 -> 2 ; 30 |22 2,11 -> 2 ; 32 |24 2,2,2,3 -> 4 ; 36 :) |25 5,5 -> 2 ; 38 |26 2,13, -> 2 ; 40 |27 3,3,3, -> 3 ; 43 |28 2,2,7 -> 3 ; 46 |30 2,3,5 -> 3 ; 49 :) |32 2,2,2,2,2 -> 5 ; 54 |33 3,11, -> 2 ; 56 |34 2,17, -> 2 ; 58 |35 5,7, -> 2 ; 60 |36, 2,3,2,3, -> 4 ; 64 :) |38, 2,19, -> 2 ; 66 |39, 3,13, -> 2 ; 68 |40, 2,2,2,5, -> 4 ; 72 |42, 2,3,7, -> 3 ; 75 |44, 2,2,11 -> 3 ; 78 |45, 3,3,5 -> 3 ; 81 :) |46, 2,23 -> 2 ; 83 |48, 2,2,2,2,3-> 5 ; 88 |49, 7,7 -> 2 ; 90 |50, 5,5,2 -> 3 ; 93 [ moderators note: as Kevin Buzzard pointed out, 51 is composite so 100 is missed. -ri ] |52, 2,2,13 -> 3 ; 96 |54, 2,27 -> 2 ; 98 |55, 5,11 -> 2 ; 100 :) Rodriguez points out: >I have this refutation for your conjecture: >Calling S = Sum of the number of separate factors of composites from 4 >to N. >Suppose that N = 2^n-1 (2^n-1 being composite) and S = K^2 - b for >certain K and b. If b is smaller than n, then the next S will be K^2 - >b + n , and the K^2 will be missed. The table will appear as: > N S > 2^n - 1 K^2 - b > 2^n K^2 - b + n > n being as large as we want. [ moderators note: This is hardly a refutation, unless theres some reason that b < n when N = 2^n-1. In fact, the only cases for n <= 14 where b < n are when b = 0. -ri ] So, could we state formally our conjecture? Take every composite number other than 1. Let a_i denotes the number of prime divisors (i.e., if 2*2*3, then only 2, 2, and 3, not including 2*3) of the ith composite other than 1 and itself. Then the sequence of partial sums {a_1, a_1 + a_2, a_1 + a_2 + a_3,...} contains infinitely many squares. H. Shinya === Subject: Re: A conjecture harder than Goldbach !? Originator: israel@math.ubc.ca (Robert Israel) > Take every composite number other than 1. Let a_i denotes the number > of prime divisors (i.e., if 2*2*3, then only 2, 2, and 3, not > including 2*3) of the ith composite other than 1 and itself. Then the > sequence of partial sums {a_1, a_1 + a_2, a_1 + a_2 + a_3,...} > contains infinitely many squares. Most numbers have about log log n prime factors. So if the a_i behave like random numbers of about that size, then the probability of hitting t^2 will be on the order of 1/(t log log t). The sum of these diverges. Indeed, the biggest a_i can be is something like log i, and if the a_i were like random numbers *that* size then the probability of hitting t^2 would be like 1/(t log t), and that sum diverges too. So I expect that (1) there are infinitely many squares in your sequence, but (2) it may be difficult to prove. -- Gareth McCaughan .sig under construc === Subject: Number Theory Originator: israel@math.ubc.ca (Robert Israel) I would like for capable and interested people to critique the papers located at http://home.bellsouth.net/p/s/community.dll?ep=16&ext=1& groupid=55687&ck= or Jere Housworths Spot Homepage.url. A good start would be a one page paper Up-lim~. I have no knowledge of how these groups work just as I did not know the name of my hompage automatically ended with the word. === Subject: Cutsets in planar graphs Originator: israel@math.ubc.ca (Robert Israel) Let G be a planar graph with minimum degree five. I wish to know: 1) Under what conditions will there be a cutset of size three? 2) What is the maximum size of a minimal cutset of G? 3) If C={v_1,v_2,v_3} is a cutset of G, where v_1 - v_2 - v_3 is a path, is it always possible, by adding an edge, to change C into K_3 while maintaining planarity? What if the only restriction is that C be of size three? (Intuition says yes to both, but..) An answer to question one especially would be a great help. === Subject: Re: Cutsets in planar graphs Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > Let G be a planar graph with minimum degree five. I wish to know: > 1) Under what conditions will there be a cutset of size three? > 2) What is the maximum size of a minimal cutset of G? > 3) If C={v_1,v_2,v_3} is a cutset of G, where v_1 - v_2 - v_3 is a path, > is it always possible, by adding an edge, to change C into K_3 while > maintaining planarity? What if the only restriction is that C be of > size three? (Intuition says yes to both, but..) > An answer to question one especially would be a great help. For question (1), triangle-free would do. For question (2), the answer is 5 without any assumptions and could be made smaller by making assumptions on the girth. Both these facts follow from a theorem of Cook [1]. if you have trouble obtaining it, I can send you a copy. HTH, Felix. [1] R.J. Cook, Heawoods theorem and connectivity, Mathematika 20 (1973), 201-207. === Subject: Re: Cutsets in planar graphs Originator: bergv@math.uiuc.edu (Maarten Bergvelt) >>Let G be a planar graph with minimum degree five. I wish to know: >>1) Under what conditions will there be a cutset of size three? >>2) What is the maximum size of a minimal cutset of G? >>3) If C={v_1,v_2,v_3} is a cutset of G, where v_1 - v_2 - v_3 is a path, >>is it always possible, by adding an edge, to change C into K_3 while >>maintaining planarity? What if the only restriction is that C be of >>size three? (Intuition says yes to both, but..) >>An answer to question one especially would be a great help. > For question (1), triangle-free would do. Unfortunately, the theoretical graph Im interested in has plenty of triangles...any others? > Both these facts follow from a theorem of Cook [1]. if you have > trouble obtaining it, I can send you a copy. === Subject: A conjecture on norming Banach subspaces Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Consider only vector spaces over the field or reals. Below S(X) denotes the unit sphere of a normed space X. Let X be a normed space, let Z be a subspace of X, and let Y be a subspace of X (the dual of X). Say that Z norms Y if sup{y(z) : z in S(Z)}=1 for all y in S(Y). Say that Y norms Z if sup{y(z) : y in S(Y)}=1 for all z in S(Z). Say that Z strongly norms Y if for every y in S(Y) there is z in S(Z) such that y(z)=1. Say that Y strongly norms Z if for every z in S(Z) there is y in S(Y) such that y(z)=1. Conjecture ---------- Let a Banach space X and a Banach subspace Y of X satisfy the following conditions: (1) Y strongly norms X; (2) X strongly norms Y. Then no proper Banach subspace of X norms Y. Condition (2) is probably superßuous. As for (1), it cannot be weakened by assuming that Y just norms X (there is an immediate counterexample). It is easy to see that the conjecture is valid if X or Y is reßexive (in which case Y=X). Any ideas, similar facts, or known similar open problems will be very much appreciated. -- Alexander E. Gutman gutman@math.nsc.ru === Subject: Re: A conjecture on norming Banach subspaces Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > Consider only vector spaces over the field or reals. Oops, a misprint. Please replace or with of: I mean the field of reals of course. -- Alexander E. Gutman === Subject: Paper published by Algebraic and Geometric Topology Originator: bergv@math.uiuc.edu (Maarten Bergvelt) The following paper has been published: Algebraic and Geometric Topology URL: http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-36.abs.html Title: Resolutions of p-stratifolds with isolated singularities Author(s): Anna Grinberg Abstract: Recently M. Kreck introduced a class of stratified spaces called p-stratifolds [M. Kreck, Stratifolds, Preprint]. He defined and investigated resolutions of p-stratifolds analogously to resolutions of algebraic varieties. In this note we study a very special case of resolutions, so called optimal resolutions, for p-stratifolds with isolated singularities. We give necessary and sufficient conditions for existence and analyze their classification. Secondary: 58K60 Keywords: Stratifold, stratified space, resolution, isolated singularity Author(s) address(es): Department of Mathematics, UC San Diego 9500 Gilman Drive, La Jolla, CA, 92093-0112 Email: agrinber@math.ucsd.edu === Subject: fixed-length alternate paths in random graphs Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Suppose I have a random graph. I am interested in random graphs that have large clustering coefficients, yet are sparse, for example: (1) Nodes are placed randomly on a plane. Two nodes are connected by an edge if they are separated by less than a distance D. (2) Power-law graphs. But answers for other random graphs would be helpful too. Suppose I randomly choose an edge (v1,v2) in that graph. I want to find the probability vertices v1 and v2 are connected by an alternate path of length less than some upper bound K. For example if K=4, I want the probability some random pair (v1,v2) connected by an edge also have an alternate path between them, of length 1, 2, 3, or 4 hops. Is this a solved problem? I think I see how to do this for Erdos-Renyi style graphs, but Im not sure how to do it for graphs with more clustering. Any thoughts, pointers to papers etc. would be very helpful. Matt === Subject: Maximization problem. Epigone-thread: frenddrehun Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Please help me with the following maximization problem. It is required to solve (desirable analitically): L(X) = [sum_{i=1,...,n} x(i)*a(i)] / sqrt(sum_{i,j=1,...,n} x(i)*x(j)*c(i,j)) -> max, by x(i), i=1,...,n. Or in matrix form: L(X) = A*X / sqrt(X*C*X) -> max, by X. Where: A - row of parameters (n); X - column of variables (n), X - transposed to X; C - square symmetric and positively defined matrix (non-singular covariation matrix of some random vector) (n*n); sqrt() - square root. Of course, the solution (X) of the problem is with precision to an arbitrary multiplier. One approach to the problem is to reduce the task to a Generalized Eigenvalue Problem: If at least one a(i) > 0, then maximum achieved at positive value of L. So it belongs to one of the stationary points of the same maximization problem, but for L(X)^2. L(X)^2 is ratio of two symmetric quadratic forms (with some matrics B=( b(i,j)=a(i)*a(j) ) and C=( c(i,j) ), which is known as Rayleigh quotient. The problem of seeking stationary points of Rayleigh quotient is equivalent to so called Generalized Symmetric Eigenvalue Problem for a regular (as C is positively defined) sheaf of quadratic forms: lambda*B*X = C*X, where lambda - eigenvalues and X - corresponding eigenvectors. If matrix B has inverse B^(-1) (i.e. a(i) <> 0 for all i=1,...,n) then it can be reduced to a standard eigenvalue problem for matrix B^(-1)*C: lambda*X=B^(-1)*C*X. (But due to a simple structure of A, B) Is there another (more direct) way to solve the problem? May be some reference... === Subject: Re: Maximization problem. Originator: bergv@math.uiuc.edu (Maarten Bergvelt) >Please help me with the following maximization problem. >It is required to solve (desirable analitically): >L(X) = [sum_{i=1,...,n} x(i)*a(i)] / sqrt(sum_{i,j=1,...,n} >x(i)*x(j)*c(i,j)) -> max, >by x(i), i=1,...,n. >Or in matrix form: >L(X) = A*X / sqrt(X*C*X) -> max, by X. > Where: >A - row of parameters (n); >X - column of variables (n), X - transposed to X; >C - square symmetric and positively defined matrix (non-singular The term is positive definite. >covariation matrix of some random vector) (n*n); >sqrt() - square root. >Of course, the solution (X) of the problem is with precision to an >arbitrary multiplier. The optimal solution (up to scalar multiples) is X = C^(-1) A, for which A X/sqrt(X C X) = sqrt(A C^(-1) A). Note that for any X we have A X <= sqrt(A C^(-1) A) sqrt(X C X) by Cauchy-Schwarz. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: Maximization problem. (Wertung=-0.5, benoetigt 5, REFERENCES -0.50) Originator: bergv@math.uiuc.edu (Maarten Bergvelt) >Please help me with the following maximization problem. >It is required to solve (desirable analitically): >L(X) = [sum_{i=1,...,n} x(i)*a(i)] / sqrt(sum_{i,j=1,...,n} >x(i)*x(j)*c(i,j)) -> max, >by x(i), i=1,...,n. >Or in matrix form: >L(X) = A*X / sqrt(X*C*X) -> max, by X. > Where: >A - row of parameters (n); >X - column of variables (n), X - transposed to X; >C - square symmetric and positively defined matrix (non-singular >covariation matrix of some random vector) (n*n); >sqrt() - square root. >Of course, the solution (X) of the problem is with precision to an >arbitrary multiplier. >One approach to the problem is to reduce the task to a Generalized >Eigenvalue Problem: >If at least one a(i) > 0, then maximum achieved at positive value of >L. >So it belongs to one of the stationary points of the same maximization >problem, but for L(X)^2. >L(X)^2 is ratio of two symmetric quadratic forms (with some matrics >B=( b(i,j)=a(i)*a(j) ) >and C=( c(i,j) ), which is known as Rayleigh quotient. >The problem of seeking stationary points of Rayleigh quotient is >equivalent to so called Generalized >Symmetric Eigenvalue Problem for a regular (as C is positively >defined) sheaf of quadratic forms: >lambda*B*X = C*X, >where lambda - eigenvalues and X - corresponding eigenvectors. >If matrix B has inverse B^(-1) (i.e. a(i) <> 0 for all i=1,...,n) then >it can be reduced to a standard eigenvalue problem for matrix >B^(-1)*C: >lambda*X=B^(-1)*C*X. >(But due to a simple structure of A, B) Is there another (more direct) >way to solve the problem? >May be some reference... from your assumptions, C has a Cholesky factorization C=L*L with a lower triangular L with stictly positive diagonal entries. set y=L*X then A*X/sqrt(X*C*X) = A*inverse(L)*y/sqrt(Y*Y) hence you maximize the linear form A*inverse(L)*y on the unit ball in y-space. this is of course y=L*A/norm(L*A) hope I got it right. peter === Subject: Re: Maximization problem. Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > Please help me with the following maximization problem. > It is required to solve (desirable analitically): > L(X) = [sum_{i=1,...,n} x(i)*a(i)] / sqrt(sum_{i,j=1,...,n} > x(i)*x(j)*c(i,j)) -> max, > by x(i), i=1,...,n. > Or in matrix form: > L(X) = A*X / sqrt(X*C*X) -> max, by X. If you normalize the denominator to 1, you get the equivalent problem min a^Tx s.t. x^TCx=1 This can be reduced with a lagrange multiplier to solving a linear system. Arnold Neumaier === Subject: Sphere packings, kissing numbers and lattices Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Consider a sphere in n-dimensional Euclidean space. We are given a set S of points on the surface of the sphere; our task is to bound the cardinality of S from above. We are told that every two distinct points in S are separated by at least 60 degrees. Then the most we know is that #S << 2^{0.401... * n} (Kabtjanskii and Levenshtein, 1978, MR 0514023 (58 #24018)) Now suppose we are told that all points on S lie on a lattice; that is, they can all be expressed as linear combinations over Z of at most n fixed elements of R^n. Can we improve on the bound on #S given above at all? It would seem that lying on a lattice is a rather tight restriction, but I do not know how take advantage of it in this context at all. Any help will be appreciated. Harald harald.helfgott@yale.edu === Subject: Re: Sphere packings, kissing numbers and lattices Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > Consider a sphere in n-dimensional Euclidean space. We are given a set S of > points on the surface of the sphere; our task is to bound the cardinality of > S from above. We are told that every two distinct points in S are separated > by at least 60 degrees. > Now suppose we are told that all points on S lie on a lattice; that is, > they can all be expressed as linear combinations over Z of at most n fixed > elements of R^n. Can we improve on the bound on #S given above at all? > It would seem that lying on a lattice is a rather tight restriction, but I do > not know how take advantage of it in this context at all. If the lattice contains the center of the sphere, this implies that the configuration on the sphere is antipodal (at least if you assume that the configuration is exactly the intersection of the lattice with the sphere - if not, there is no hope to get a better answer, I think). Antipodality can be exploited, see recent work by Anstreicher. It gives much better bounds for small n. Whether it can be used to improve the asymptotics, I dont know. Arnold Neumaier === Subject: Question for Lie group or representation theory experts Originator: bergv@math.uiuc.edu (Maarten Bergvelt) How to evaluate the following integral int dU exp( sum_{i,j=1}^N U_{ij} U_{ij}^* a_i b_j ) ? here U runs through all U(N) matrices, dU is the Haar measure normalized so that the volume of U(N) is 1. a_i, b_js are given numbers. For example, one can expand out the exponential and the leading terms of the integral are 1+{1over N} sum a_i b_j + ... If the full answer for general N is not available, is there an approximate answer for large N? Xi