Collapsing sequences of solutions to the Ricci flow on 3-manifolds with almost nonnegative curvature

We study sequences of 3-dimensional solutions to the Ricci flow with almost nonnegative sectional curvatures and diameters tending to infinity. Such sequences may arise from the limits of dilations about singularities of Type IIb. In particular, we study the case when the sequence collapses, which may occur when dilating about infinite time singularities. In this case we classify the possible Gromov-Hausdorff limits and construct 2-dimensional virtual limits. The virtual limits are constructed using Fukaya theory of the limits of local covers. We then show that the virtual limit arising from appropriate dilations of a Type IIb singularity is always Hamilton's cigar soliton solution. Partially supported by NSF grant DMS-0203926.     

The Superconvergence of the Composite Trapezoidal Rule for Hadamard Finite Part Integrals

The composite trapezoidal rule has been well studied and widely applied for numerical integrations and numerical solution of integral equations with smooth or weakly singular kernels. However, this quadrature rule has been less employed for Hadamard finite part integrals due to the fact that its global convergence rate for Hadamard finite part integrals with (p+1)-order singularity is p-order lower than that for the Riemann integrals in general. In this paper, we study the superconvergence of the composite trapezoidal rule for Hadamard finite part integrals with the second-order and the third-order singularity, respectively. We obtain superconvergence estimates at some special points and prove the uniqueness of the superconvergence points. Numerical experiments confirm our theoretical analysis and show that the composite trapezoidal rule is efficient for Hadamard finite part integrals by noting the superconvergence phenomenon. The work of this author was partially supported by the National Natural Science Foundation of China(No.10271019), a grant from the Research Grants Council of the Hong Kong Special Administractive Region, China (Project No. City 102204) and a grant from the Laboratory of Computational Physics The work of this author was supported in part by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 102204).     

Bi-Lipschitz geometry of weighted homogeneous surface singularities

We show that a weighted homogeneous complex surface singularity is metrically conical (i.e., bi-Lipschitz equivalent to a metric cone) only if its two lowest weights are equal. We also give an example of a pair of weighted homogeneous complex surface singularities that are topologically equivalent but not bi-Lipschitz equivalent. L. Birbrair was supported under CNPq grant no. 300985/93-2. A. Fernandes was supported under CNPq grant no. 300393/2005-9. W. D. Neumann was supported under NSA grant H98230-06-1-011 and NSF grant no. DMS-0206464.     

Local and parallel finite element algorithms for the stokes problem

Based on two-grid discretizations, some local and parallel finite element algorithms for the Stokes problem are proposed and analyzed in this paper. These algorithms are motivated by the observation that for a solution to the Stokes problem, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fine grid by some local and parallel procedure. One technical tool for the analysis is some local a priori estimates that are also obtained in this paper for the finite element solutions on general shape-regular grids. Y. He was partially subsidized by the NSF of China 10671154 and the National Basic Research Program under the grant 2005CB321703; A. Zhou was partially supported by the National Science Foundation of China under the grant 10425105 and the National Basic Research Program under the grant 2005CB321704; J. Li was partially supported by the NSF of China under the grant 10701001. J. Xu was partially supported by Alexander von Humboldt Research Award for Senior US Scientists, NSF DMS-0609727 and NSFC-10528102.     

A bifurcation analysis of the nonlinear parametric programming problem

The structure of solutions to the nonlinear parametric programming problem with a one dimensional parameter is analyzed in terms of the bifurcation behavior of the curves of critical points and the persistence of minima along these curves. Changes in the structure of the solution occur at singularities of a nonlinear system of equations motivated by the Fritz John first-order necessary conditions. It has been shown that these singularities may be completely partitioned into seven distinct classes based upon the violation of one or more of the following: a complementarity condition, a constraint qualification, and the nonsingularity of the Hessian of the Lagrangian on a tangent space. To apply classical bifurcation techniques to these singularities, a further subdivision of each case is necessary. The structure of curves of critical points near singularities of lowest (zero) codimension within each case is analyzed, as well as the persistence of minima along curves emanating from these singularities. Bifurcation behavior is also investigated or discussed for many of the subcases giving rise to a codimension one singularity.This work was supported by the National Science Foundation through NSF Grants DMS-85-10201 and DMS-87-04679 and by the Air Force Office of Scientific Research through grant number AFOSR-88-0059.     

Anti de Sitter horospherical flat timelike surfaces

We investigate a special timelike surfaces in Anti de Sitter 3-space.We call such a timelike surface an Anti de Sitter horospherical flat surface which belongs to a class of surfaces given by one parameter families of Anti de Sitter horocycle.We give a generic classification of singularities and study the geometric properties of such surfaces from the viewpoint of Legendrian singularity theory.     

A hybrid iterative method for symmetric positive definite linear systems

A hybrid iterative scheme that combines the Conjugate Gradient (CG) method with Richardson iteration is presented. This scheme is designed for the solution of linear systems of equations with a large sparse symmetric positive definite matrix. The purpose of the CG iterations is to improve an available approximate solution, as well as to determine an interval that contains all, or at least most, of the eigenvalues of the matrix. This interval is used to compute iteration parameters for Richardson iteration. The attraction of the hybrid scheme is that most of the iterations are carried out by the Richardson method, the simplicity of which makes efficient implementation on modern computers possible. Moreover, the hybrid scheme yields, at no additional computational cost, accurate estimates of the extreme eigenvalues of the matrix. Knowledge of these eigenvalues is essential in some applications.Research supported in part by NSF grant DMS-9409422.Research supported in part by NSF grant DMS-9205531.     

On computing the center of a convex quadratically constrained set

We propose a method for finding analytic center of a convex feasible region whose boundaries are defined by quadratic functions. The algorithm starts from an arbitrary initial point and approaches to the desired center by simultaneously reducing infeasibility or slackness of all constraints. A partial Newton step is taken at each iteration.Research supported in part by the ONR under grant N00014-87-K-0214 and by the NSF under grant CCR-8810107.Research supported in part by the NSF under grant ECS-8721709.     

Universal abelian covers of quotient-cusps

 The quotient-cusp singularities are isolated complex surface singularities that are double-covered by cusp singularities. We show that the universal abelian cover of such a singularity, branched only at the singular point, is a complete intersection cusp singularity of embedding dimension 4. This supports a general conjecture that we make about the universal abelian cover of a ℚ-Gorenstein singularity. Received: 3 February 2001 / Revised version: 8 March 2002 / Published online: 10 February 2003 Mathematics Subject Classification (2000): 14B05, 14J17, 32S25 This research was supported by grants from the Australian Research Council and the NSF (first author) and the the NSA (second author).     

The scalar curvature equation on 2- and 3-spheres

In this article we obtain a priori estimates for solutions to the prescribed scalar curvature equation on 2- and 3-spheres under a nondegeneracy assumption on the curvature function. Using this estimate, we use the continuity method to demonstrate the existence of solutions to this equation when a map associated to the given curvature function has non-zero degree.Research of first author supported in part by NSF grant 91-03949Research of second author supported by a NSF Postdoctoral Fellowship.Research of third author supported in part by NSF grant 91-02872 and the Ellentuck Fund.     

Initial segments of the degrees of constructibility

We show that any constructible, constructibly countable, (dual) algebraic lattice is isomorphic to the degrees of constructibility of reals in some generic extension ofL. Research partially supported by NSF grant DMS-8601777. Research partially supported by NSF grant DMS-8601048 and grant 84-00067 from the U.S.-Israel Binational Science Foundation. Thanks are also due to Uri Abraham and Mati Rubin for helpful discussions about initial segments of the degrees of constructibility and to Bill Lampe for information on algebraic lattices.     

Weierstrass weight of Gorenstein singularities with one or two branches

We obtain formulas for the Weierstrass weight of singularities with either one or two branches on a Gorenstein curve in characteristic zero. These formulas generalize results of C. Widland in the cases of simple cusps and ordinary nodes. The formulas arise from a study of the semigroup of values of such a singularity and the relation between this semigroup and a basis for the dualizing differentials on the curve adapted to the singular point. Partially supported by a CNPq grant Partially supported by NSF grant INT-9101337     

An Adaptive Finite Element Method for the H- ψ Formulation of Time-dependent Eddy Current Problems

In this paper, we develop an adaptive finite element method based on reliable and efficient a posteriori error estimates for the H − ψ formulation of eddy current problems with multiply connected conductors. Multiply connected domains are considered by making “cuts”. The competitive performance of the method is demonstrated by an engineering benchmark problem, Team Workshop Problem 7, and a singular problem with analytic solution.W. Zheng was supported in part by China NSF under the grant 10401040.Z. Chen was supported in part by China NSF under the grant 10025102 and 10428105, and by the National Basic Research Project under the grant 2005CB321701.     

Mixed Boundary Value Problems of Thermopiezoelectricity for Solids with Interior Cracks

We investigate asymptotic properties of solutions to mixed boundary value problems of thermopiezoelectricity (thermoelectroelasticity) for homogeneous anisotropic solids with interior cracks. Using the potential methods and theory of pseudodifferential equations on manifolds with boundary we prove the existence and uniqueness of solutions. The singularities and asymptotic behaviour of the mechanical, thermal and electric fields are analysed near the crack edges and near the curves, where the types of boundary conditions change. In particular, for some important classes of anisotropic media we derive explicit expressions for the corresponding stress singularity exponents and demonstrate their dependence on the material parameters. The questions related to the so called oscillating singularities are treated in detail as well. This research was supported by the Georgian National Science Foundation grant GNSF/ST07/3-170 and by the German Research Foundation grant DFG 436 GEO113/8/0-1.     

A fourth-order numerical method for the planetary geostrophic equations with inviscid geostrophic balance

The planetary geostrophic equations with inviscid balance equation are reformulated in an alternate form, and a fourth-order finite difference numerical method of solution is proposed and analyzed in this article. In the reformulation, there is only one prognostic equation for the temperature field and the velocity field is statically determined by the planetary geostrophic balance combined with the incompressibility condition. The key observation is that all the velocity profiles can be explicitly determined by the temperature gradient, by utilizing the special form of the Coriolis parameter. This brings convenience and efficiency in the numerical study. In the fourth-order scheme, the temperature is dynamically updated at the regular numerical grid by long-stencil approximation, along with a one-sided extrapolation near the boundary. The velocity variables are recovered by special solvers on the 3-D staggered grid. Furthermore, it is shown that the numerical velocity field is divergence-free at the discrete level in a suitable sense. Fourth order convergence is proven under mild regularity requirements. R. Samelson was supported by NSF grant OCE04-24516 and Navy ONR grant N00014-05-1-0891. R. Temam was supported by NSF grant DMS-0604235 and the research fund of Indiana University. S. Wang was supported by NSF grant DMS-0605067 and Navy ONR grant N00014-05-1-0218.     

Singularities of admissible normal functions

In a recent paper, M. Green and P. Griffiths used R. Thomas’ work on nodal hypersurfaces to sketch a proof of the equivalence of the Hodge conjecture and the existence of certain singular admissible normal functions. Inspired by their work, we study normal functions using Morihiko Saito’s mixed Hodge modules and prove that the existence of singularities of the type considered by Griffiths and Green is equivalent to the Hodge conjecture. Several of the intermediate results, including a relative version of the weak Lefschetz theorem for perverse sheaves, are of independent interest. P. Brosnan’s research was supported in part by an NSERC discovery grant. H. Fang’s research was supported in part by NSF grant number DMS 0606721. G. Pearlstein’s research was supported in part by NSF grant number DMS 0703956. N. Fakhruddin School of Mathematics, Tata Institue of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India e-mail: naf@math.tifr.res.in     

A minimal cutset of the boolean lattice with almost all members

Two almost explicit constructions are given satisfying the title.This research was done while the authors visited the Institute for Mathematics and its Applications at the University of Minnesota, Minneapolis, MN 55455.Research supported in part by Hungarian National Foundation for Scientific Research, grant no. 1812.Research supported in part by NSF Grant MDS 87-01475.Research supported in part by NSF grant DMS 86-06225 and Airforce Grant OSR-86-0076.     

Multi-valued solutions and branch point singularities for nonlinear hyperbolic or elliptic systems

Multi-valued solutions are constructed for 2 × 2 first-order systems using a generalization of the hodograph transformation. The solution is found as a complex analytic function on a complex Riemann surface for which the branch points move as part of the solution. The branch point singularities are envelopes for the characteristics and thus move at the characteristic speeds. We perform an analysis of stability of these singularities with respect to perturbations of the initial data. The generic singularity types are folds, cusps, and nondegenerate umbilic points with non-zero 3-jet. An isolated singularity is generically a square root branch point corresponding to a fold. Two types of collisions between singularities are generic: At a “tangential” collision between two singularities moving at the same characteristic speed, a cube root branch point is formed, corresponding to a cusp. A “non-tangential” collision, between two square root branch points moving at different characteristic speeds, remains a square root branch point at the collision and corresponds to a nondegenerate umbilic point. These results are also valid for a diagonalizable n-th order system for which there are exactly two speeds. © 1993 John Wiley & Sons, Inc.     

Singularities on complete algebraic varieties

We prove that any finite set of n-dimensional isolated algebraic singularities can be afforded on a simply connected projective variety. The first author was partially supported by NSF grant DMS-01591 and the third author was partially supported by NSF grant DMS-03693