Explicit concave fillings of contact three-manifolds

In this paper we give explicit, handle-by-handle constructions of concave symplectic fillings of all closed, oriented contact 3-manifolds. These constructions combine recent results of Giroux relating contact structures and open book decompositions of 3-manifolds, earlier results of the author on attaching 4-dimensional symplectic 2-handles along transverse links, and some tricks with mapping class groups of compact surfaces with non-empty boundary.Comment: 15 pages. Accepted for publication in the Mathematical Proceedings of the Cambridge Philosophical Society. Current version is identical to final version submitted to the journal, differs from original version only in some notation and minor editorial change

Representing Symmetric Rank Two Updates

Various quasi-Newton methods periodically add a symmetric "correction" matrix of rank at most 2 to a matrix approximating some quantity A of interest (such as the Hessian of an objective function). In this paper we examine several ways to express a symmetric rank 2 matrix [delta] as the sum of rank 1 matrices. We show that it is easy to compute rank 1 matrices [delta1] and [delta2] such that [delta] = [delta1] + [delta2] and [the norm of delta1]+ [the norm of delta2] is minimized, where ||.|| is any inner product norm. Such a representation recommends itself for use in those computer programs that maintain A explicitly, since it should reduce cancellation errors and/or improve efficiency over other representations. In the common case where [delta] is indefinite, a choice of the form [delta1] = [delta2 to the power of T] = [xy to the power of T] appears best. This case occurs for rank 2 quasi- Newton updates [delta] exactly when [delta] may be obtained by symmetrizing some rank 1 update; such popular updates as the DFP, BFGS, PSB, and Davidon's new optimally conditioned update fall into this category.

On Modifying Singular Values to Solve Possible Singular Systems of Non-Linear Equations

We show that if a certain nondegeneracy assumption holds, it is possible to guarantee the existence of a solution to a system of nonlinear equations f(x) = 0 whose Jacobian matrix J(x) exists but maybe singular. The main idea is to modify small singular values of J(x) in such away that the modified Jacobian matrix J^(x) has a continuous pseudoinverse J^+(x)and that a solution x* of f(x) = 0 may be found by determining an asymptote of the solution to the initial value problem x(0) = x[sub0}, x’(t) = -J^+(x)f(x). We briefly discuss practical (algorithmic) implications of this result. Although the nondegeneracy assumption may fail for many systems of interest (indeed, if the assumption holds and J(x*) is non-singular, then x is unique), algorithms using(x) may enjoy a larger region of convergence than those that require(an approximation to) J[to the -1 power[(x).

Some Convergence Properties of Broyden's Method

In 1965 Broyden introduced a family of algorithms called(rank-one) quasi—New-ton methods for iteratively solving systems of nonlinear equations. We show that when any member of this family is applied to an n x n nonsingular system of linear equations and direct-prediction steps are taken every second iteration, then the solution is found in at most 2n steps. Specializing to the particular family member known as Broyden’s (good) method, we use this result to show that Broyden's method enjoys local 2n-step Q-quadratic convergence on nonlinear problems.

Reconstructing 4-manifolds from Morse 2-functions

Given a Morse 2-function $f: X^4 \to S^2$, we give minimal conditions on the fold curves and fibers so that $X^4$ and $f$ can be reconstructed from a certain combinatorial diagram attached to $S^2$. Additional remarks are made in other dimensions.Comment: 13 pages, 10 figures. Replaced because the main theorem in the original is false. The theorem has been corrected and counterexamples to the original statement are give

Constructing symplectic forms on 4-manifolds which vanish on circles

Given a smooth, closed, oriented 4-manifold X and alpha in H_2(X,Z) such that alpha.alpha > 0, a closed 2-form w is constructed, Poincare dual to alpha, which is symplectic on the complement of a finite set of unknotted circles. The number of circles, counted with sign, is given by d = (c_1(s)^2 -3sigma(X) -2chi(X))/4, where s is a certain spin^C structure naturally associated to w.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol8/paper20.abs.htm