Martingales and Financial Mathematics

In this expository paper, we will discuss the role played by martingales in Financial Mathematics. More precisely, we will restrict ourselves to a mathematical formulation of the economical concept of an arbitrage-free, complete market and the pricing of derivatives in such models. For a clear exposition, we only consider the discrete case. We also discuss the Cox-Ross-Rubinstein model which is still one of the most used models in Finance

Design and proof of concept for silicon-based quantum dot quantum bits

Spins based in silicon provide one of the most promising architectures for quantum computing. Quantum dots are an inherently scalable technology. Here, we combine these two concepts into a workable design for a silicon-germanium quantum bit. The novel structure incorporates vertical and lateral tunneling, provides controlled coupling between dots, and enables single electron occupation of each dot. Precise modeling of the design elucidates its potential for scalable quantum computing. For the first time it is possible to translate the requirements of fault-tolerant error correction into specific requirements for gate voltage control electronics in quantum dots. We demonstrate that these requirements are met by existing pulse generators in the kHz-MHz range, but GHz operation is not yet achievable. Our calculations further pinpoint device features that enhance operation speed and robustness against leakage errors. We find that the component technologies for silicon quantum dot quantum computers are already in hand.Comment: References adde

Probability distribution of the seismic damage cost over the life cycle of structures

In the life-cycle analysis, the total cost of damage caused by earthquakes is a significant but highly uncertain component. In the current literature, the seismic risk analysis is largely limited to the evaluation of the average cost of damage, which is not informative about the full extent of variability in the cost. The paper presents a systematic development of the stochastic modeling of seismic risk analysis problem and reformulates the damage cost analysis as a superposition of compound Poisson processes. An explicit analytical solution for the distribution of damage cost is derived in form of a recursive equation. The proposed approach extends the capability of the existing framework of seismic risk analysis, which can be used to optimize initial design and retrofitting of structures.Natural Science and Engineering Research CouncilUniversity Network of Excellence in Nuclear Engineerin

Controllable valley splitting in silicon quantum devices

Silicon has many attractive properties for quantum computing, and the quantum dot architecture is appealing because of its controllability and scalability. However, the multiple valleys in the silicon conduction band are potentially a serious source of decoherence for spin-based quantum dot qubits. Only when these valleys are split by a large energy does one obtain well-defined and long-lived spin states appropriate for quantum computing. Here we show that the small valley splittings observed in previous experiments on Si/SiGe heterostructures result from atomic steps at the quantum well interface. Lateral confinement in a quantum point contact limits the electron wavefunctions to several steps, and enhances the valley splitting substantially, up to 1.5 meV. The combination of electronic and magnetic confinement produces a valley splitting larger than the spin splitting, which is controllable over a wide range. These results improve the outlook for realizing spin qubits with long coherence times in silicon-based devices.Comment: Published version, including supplementary material

Estimating the frequency of trains approaching red signals: A case study for improving the understanding of SPAD risk

This paper describes a novel technique for estimating the frequency with which trains approach signals showing a red aspect. This knowledge is potentially important for understanding the likelihood of a signal being passed at danger (SPAD) at individual signals and also for normalisation of SPAD data, both locally and nationally, for trending and benchmarking. The industry currently uses estimates for the number of red aspect approaches based on driver surveys which are considered to have significant shortcomings. Data for this analysis is sourced from publicly available live feeds provided by Network Rail which give information on train movements and signal states. The development of the analysis model and supporting software are described and some sample results from case studies are presented. An initial study of seven signalling areas showed that approximately 3.3% of all signal approaches are to red signals. However, it also highlighted that there is a large variation in the red approach rates between signalling areas and between individual signals. SPAD risk assessment at individual signals may be significantly enhanced by the ability to estimate red approach rates for individual signals using the techniques described

Broadband Optical Serrodyne Frequency Shifting

We demonstrate serrodyne frequency shifting of light from 200 MHz to 1.2 GHz with an efficiency of better than 60 percent. The frequency shift is imparted by an electro-optic phase modulator driven by a high-frequency, high-fidelity sawtooth waveform that is passively generated by a commercially available Non-Linear Transmission Line (NLTL). We also implement a push-pull configuration using two serrodyne-driven phase modulators allowing for continuous tuning between -1.6 GHz and +1.6 GHz. Compared to competing technologies, this technique is simple and robust, and offers the largest available tuning range in this frequency band.Comment: 3 pages, 4 figure

Imposing a Lagrangian Particle Framework on an Eulerian Hydrodynamics Infrastructure in Flash

In many astrophysical simulations, both Eulerian and Lagrangian quantities are of interest. For example, in a galaxy cluster merger simulation, the intracluster gas can have Eulerian discretization, while dark matter can be modeled using particles. FLASH, a component-based scientific simulation code, superimposes a Lagrangian framework atop an adaptive mesh refinement Eulerian framework to enable such simulations. The discretization of the field variables is Eulerian, while the Lagrangian entities occur in many different forms including tracer particles, massive particles, charged particles in particle-in-cell mode, and Lagrangian markers to model fluid structure interactions. These widely varying roles for Lagrangian entities are possible because of the highly modular, flexible, and extensible architecture of the Lagrangian framework. In this paper, we describe the Lagrangian framework in FLASH in the context of two very different applications, Type Ia supernovae and galaxy cluster mergers, which use the Lagrangian entities in fundamentally different ways