On 4-fold covering moves

We prove the existence of a finite set of moves sufficient to relate any two representations of the same 3-manifold as a 4-fold simple branched covering of S^3. We also prove a stabilization result: after adding a fifth trivial sheet two local moves suffice. These results are analogous to results of Piergallini in degree 3 and can be viewed as a second step in a program to establish similar results for arbitrary degree coverings of S^3.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-5.abs.html Version 2: correction added on page 13

Eigenvalue bounds in the gaps of Schrodinger operators and Jacobi matrices

We consider $C=A+B$ where $A$ is selfadjoint with a gap $(a,b)$ in its spectrum and $B$ is (relatively) compact. We prove a general result allowing $B$ of indefinite sign and apply it to obtain a $(\delta V)^{d/2}$ bound for perturbations of suitable periodic Schrodinger operators and a (not quite)Lieb-Thirring bound for perturbations of algebro-geometric almost periodic Jacobi matrices

An elementary approach to the mapping class group of a surface

We consider an oriented surface S and a cellular complex X of curves on S, defined by Hatcher and Thurston in 1980. We prove by elementary means, without Cerf theory, that the complex X is connected and simply connected. From this we derive an explicit simple presentation of the mapping class group of S, following the ideas of Hatcher-Thurston and Harer.Comment: 62 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol3/paper17.abs.htm

Spectral approach to homogenization of an elliptic operator periodic in some directions

The operator $$A_{\varepsilon}= D_{1} g_{1}(x_{1}/\varepsilon, x_{2}) D_{1} + D_{2} g_{2}(x_{1}/\varepsilon, x_{2}) D_{2}$$ is considered in $L_{2}({\mathbb{R}}^{2})$, where $g_{j}(x_{1},x_{2})$, $j=1,2,$ are periodic in $x_{1}$ with period 1, bounded and positive definite. Let function $Q(x_{1},x_{2})$ be bounded, positive definite and periodic in $x_{1}$ with period 1. Let $Q^{\varepsilon}(x_{1},x_{2})= Q(x_{1}/\varepsilon, x_{2})$. The behavior of the operator $(A_{\varepsilon}+ Q^{\varepsilon}$ as $\varepsilon\to0$ is studied. It is proved that the operator $(A_{\varepsilon}+ Q^{\varepsilon})^{-1}$ tends to $(A^{0} + Q^{0})^{-1}$ in the operator norm in $L_{2}(\mathbb{R}^{2})$. Here $A^{0}$ is the effective operator whose coefficients depend only on $x_{2}$, $Q^{0}$ is the mean value of $Q$ in $x_{1}$. A sharp order estimate for the norm of the difference $(A_{\varepsilon}+ Q^{\varepsilon})^{-1}- (A^{0} + Q^{0})^{-1}$ is obtained. The result is applied to homogenization of the Schr\"odinger operator with a singular potential periodic in one direction.Comment: 3

Lagrangian mapping class groups from a group homological point of view

We focus on two kinds of infinite index subgroups of the mapping class group of a surface associated with a Lagrangian submodule of the first homology of a surface. These subgroups, called Lagrangian mapping class groups, are known to play important roles in the interaction between the mapping class group and finite-type invariants of 3-manifolds. In this paper, we discuss these groups from a group (co)homological point of view. The results include the determination of their abelianizations, lower bounds of the second homology and remarks on the (co)homology of higher degrees. As a by-product of this investigation, we determine the second homology of the mapping class group of a surface of genus 3.Comment: 20 pages. The proof of Lemma 4.2 is corrected. To appear in Algebraic & Geometric Topolog

Homogenization of the elliptic Dirichlet problem: operator error estimates in L2L_2

Let $\mathcal{O} \subset \mathbb{R}^d$ be a bounded domain of class $C^2$. In the Hilbert space $L_2(\mathcal{O};\mathbb{C}^n)$, we consider a matrix elliptic second order differential operator $\mathcal{A}_{D,\varepsilon}$ with the Dirichlet boundary condition. Here $\varepsilon>0$ is the small parameter. The coefficients of the operator are periodic and depend on $\mathbf{x}/\varepsilon$. A sharp order operator error estimate $\|\mathcal{A}_{D,\varepsilon}^{-1} - (\mathcal{A}_D^0)^{-1} \|_{L_2 \to L_2} \leq C \varepsilon$ is obtained. Here $\mathcal{A}^0_D$ is the effective operator with constant coefficients and with the Dirichlet boundary condition.Comment: 13 page

How robust are distributed systems

A distributed system is made up of large numbers of components operating asynchronously from one another and hence with imcomplete and inaccurate views of one another's state. Load fluctuations are common as new tasks arrive and active tasks terminate. Jointly, these aspects make it nearly impossible to arrive at detailed predictions for a system's behavior. It is important to the successful use of distributed systems in situations in which humans cannot provide the sorts of predictable realtime responsiveness of a computer, that the system be robust. The technology of today can too easily be affected by worn programs or by seemingly trivial mechanisms that, for example, can trigger stock market disasters. Inventors of a technology have an obligation to overcome flaws that can exact a human cost. A set of principles for guiding solutions to distributed computing problems is presented