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

Stability of the magnetic Schr\"odinger operator in a waveguide

The spectrum of the Schr\"odinger operator in a quantum waveguide is known to be unstable in two and three dimensions. Any enlargement of the waveguide produces eigenvalues beneath the continuous spectrum. Also if the waveguide is bent eigenvalues will arise below the continuous spectrum. In this paper a magnetic field is added into the system. The spectrum of the magnetic Schr\"odinger operator is proved to be stable under small local deformations and also under small bending of the waveguide. The proof includes a magnetic Hardy-type inequality in the waveguide, which is interesting in its own

Schr\"odinger operator on homogeneous metric trees: spectrum in gaps

The paper studies the spectral properties of the Schr\"odinger operator $A_{gV} = A_0 + gV$ on a homogeneous rooted metric tree, with a decaying real-valued potential $V$ and a coupling constant $g\ge 0$. The spectrum of the free Laplacian $A_0 = -\Delta$ has a band-gap structure with a single eigenvalue of infinite multiplicity in the middle of each finite gap. The perturbation $gV$ gives rise to extra eigenvalues in the gaps. These eigenvalues are monotone functions of $g$ if the potential $V$ has a fixed sign. Assuming that the latter condition is satisfied and that $V$ is symmetric, i.e. depends on the distance to the root of the tree, we carry out a detailed asymptotic analysis of the counting function of the discrete eigenvalues in the limit $g\to\infty$. Depending on the sign and decay of $V$, this asymptotics is either of the Weyl type or is completely determined by the behaviour of $V$ at infinity.Comment: AMS LaTex file, 47 page

Low cost management of replicated data in fault-tolerant distributed systems

Many distributed systems replicate data for fault tolerance or availability. In such systems, a logical update on a data item results in a physical update on a number of copies. The synchronization and communication required to keep the copies of replicated data consistent introduce a delay when operations are performed. A technique is described that relaxes the usual degree of synchronization, permitting replicated data items to be updated concurrently with other operations, while at the same time ensuring that correctness is not violated. The additional concurrency thus obtained results in better response time when performing operations on replicated data. How this technique performs in conjunction with a roll-back and a roll-forward failure recovery mechanism is also discussed

Exploiting replication in distributed systems

Techniques are examined for replicating data and execution in directly distributed systems: systems in which multiple processes interact directly with one another while continuously respecting constraints on their joint behavior. Directly distributed systems are often required to solve difficult problems, ranging from management of replicated data to dynamic reconfiguration in response to failures. It is shown that these problems reduce to more primitive, order-based consistency problems, which can be solved using primitives such as the reliable broadcast protocols. Moreover, given a system that implements reliable broadcast primitives, a flexible set of high-level tools can be provided for building a wide variety of directly distributed application programs

Reliable broadcast protocols

A number of broadcast protocols that are reliable subject to a variety of ordering and delivery guarantees are considered. Developing applications that are distributed over a number of sites and/or must tolerate the failures of some of them becomes a considerably simpler task when such protocols are available for communication. Without such protocols the kinds of distributed applications that can reasonably be built will have a very limited scope. As the trend towards distribution and decentralization continues, it will not be surprising if reliable broadcast protocols have the same role in distributed operating systems of the future that message passing mechanisms have in the operating systems of today. On the other hand, the problems of engineering such a system remain large. For example, deciding which protocol is the most appropriate to use in a certain situation or how to balance the latency-communication-storage costs is not an easy question