Symmetries of Integrable Difference Equations on the Quad-Graph

This paper describes symmetries of all integrable difference equations that belong to the famous Adler–Bobenko–Suris classification. For each equation, the characteristics of symmetries satisfy a functional equation, which we solve by reducing it to a system of partial differential equations. In this way, all five-point symmetries of integrable equations on the quad-graph are found. These include mastersymmetries, which allow one to construct infinite hierarchies of local symmetries. We also demonstrate a connection between the symmetries of quad-graph equations and those of the corresponding Toda type difference equations.     

Trace norm estimates for products of integral operators and diffusion semigroups

We give trace norm estimates for products of integral operators and for diffusion semigroups. These are applied to differences of heat semigroups. A natural example of an integral operator with finite trace which is not trace class is given.     

Long wavelength quantum well lasers: Synopsis of the RACE “AQUA” project: MOVPE/MBE/GSMBE for InGaAsP/InP and InGaAlAs/InP high speed MQW DFB lasers

The technological limits for ultra high speed devices are now rapidly expanding due to the use of quantum well (QW) materials. This new class of materials gives the opportunity of tailoring materials parameters by controlling geometries on an atomic scale. They look very promising as materials for lasers, detectors and transistors suitable even above 10 Gb/s. It will be demonstrated that state of the art MQW structures can be realized in both material systems, InGaAsP/InP and InGaAlAs/InP. Parallel lateral laser structures (e.g. SIBH, BRS and TBH) have been designed to take full benefit of QW technology. Ultimate reduction of parasitics, whilst using potential low cost fabrication technologies is the basis for achieving high bitrate (10 Gb/s) MQW lasers, even with the stronger damping in QW material. Using the DFB-SIBH laser structure 10 Gb/s large signal experiments are successfully performed with bulk, MQW and SLMQW lasers. Extremely low fall times of 44 ps are achieved. Additional MQW based improvements are observed such as: −3 times higher differential gain, increased output power (>110 mW), 2.5 times lower chirp (Δλ_−20dB = 0.40 nm at 10 Gb/s modulation), and 2 dB gain in power budget at 10 Gb/s digital transmission.     

Improved modeling and solution methods for the multi-resource routing problem

This paper presents modeling and solution method improvements for the Multi-Resource Routing Problem (MRRP) with flexible tasks. The MRRP with flexible tasks is used to model routing and scheduling problems for intermodal drayage operations in which two resources (tractors and trailers) perform tasks to transport loaded and empty equipment. Tasks may be either well defined, in which both the origin and the destination of a movement are given, or flexible, in which the origin or the destination is chosen by the model. This paper proposes methods to effectively manage the number of options considered for flexible tasks (either feasible origins for a known destination or feasible destinations for a known origin). This modeling change generates sufficient options to allow for low-cost solutions while maintaining reasonable computational effort. We also propose a new solution method that uses randomized route generation. Computational results from test cases show that these changes improve the quality of solutions by at least 5% in the test cases as compared to methods from previous studies.     

Instability analysis of semiconductor injection lasers with saturable absorption and self-sustained pulsations on the centre manifold

The small-signal analysis of a simple model of three rate equations is performed rigorously by means of both the Routh-Hurwitz theorem and Descartes' sign rule, and the relationship between bistable behaviour and self-sustained pulsations (SSPs) is investigated. Because one eigenvalue of the Jacobian of the model system is always negative, small-amplitude, sinusoidal SSPs can be described in terms of Haken's dressed-mode variables on a two-dimensional centre manifold. Possible simplifications for the switching modelling are also mentioned.     

Empirical minimization

We investigate the behavior of the empirical minimization algorithm using various methods. We first analyze it by comparing the empirical, random, structure and the original one on the class, either in an additive sense, via the uniform law of large numbers, or in a multiplicative sense, using isomorphic coordinate projections. We then show that a direct analysis of the empirical minimization algorithm yields a significantly better bound, and that the estimates we obtain are essentially sharp. The method of proof we use is based on Talagrand's concentration inequality for empirical processes. Research partially supported by NSF under award DMS-0434393. Research partially supported by the Australian Research Council Discovery Porject DP0343616.