Exploring the impact of different cost heuristics in the allocation of safety integrity levels

Contemporary safety standards prescribe processes in which system safety requirements, captured early and expressed in the form of Safety Integrity Levels (SILs), are iteratively allocated to architectural elements. Different SILs reflect different requirements stringencies and consequently different development costs. Therefore, the allocation of safety requirements is not a simple problem of applying an allocation "algebra" as treated by most standards; it is a complex optimisation problem, one of finding a strategy that minimises cost whilst meeting safety requirements. One difficulty is the lack of a commonly agreed heuristic for how costs increase between SILs. In this paper, we define this important problem; then we take the example of an automotive system and using an automated approach show that different cost heuristics lead to different optimal SIL allocations. Without automation it would have been impossible to explore the vast space of allocations and to discuss the subtleties involved in this problem

Octonionic Mobius Transformations

A vexing problem involving nonassociativity is resolved, allowing a generalization of the usual complex Mobius transformations to the octonions. This is accomplished by relating the octonionic Mobius transformations to the Lorentz group in 10 spacetime dimensions. The result will be of particular interest to physicists working with lightlike objects in 10 dimensions.Comment: Plain TeX, 12 pages, 1 PostScript figure included using eps

Hazard Rates and Probability Distributions: Representation of Random Intensities

Recent attempts to apply the results of martingale theory in probability theory have shown that it is first necessary to interpret this abstract mathematical theory in more conventional terms. One example of this is the need to obtain a representation of the dual predictable projections (compensators) used in martingale theory in terms of probability distributions. However, up to now a representation of this type has been derived only for one special case. In this paper, the author gives probabilistic representations of the dual predictable projection of integer-valued random measures that correspond to jumps in a semimartingale with respect to the sigma-algebras generated by this process. The results are of practical importance because such dual predictable projections are usually interpreted as random intensities or hazard rates related to jumps in trajectories: applications are found in such fields as mathematical demography and risk analysis

Understanding and Calculating the Odds: Probability Theory Basics and Calculus Guide for Beginners, with Applications in Games of Chance and Everyday Life

This book presents not only the mathematical concept of probability, but also its philosophical aspects, the relativity of probability and its applications and even the psychology of probability. All explanations are made in a comprehensible manner and are supported with suggestive examples from nature and daily life, and even with challenging math paradoxes

Deformations of M-theory Killing superalgebras

We classify the Lie superalgebra deformations of the Killing superalgebras of some M-theory backgrounds. We show that the Killing superalgebras of the Minkowski, Freund--Rubin and M5-brane backgrounds are rigid, whereas the ones for the M-wave, the Kaluza--Klein monopole and the M2-brane admit deformations, which we give explicitly.Comment: 20 pages (v3: a number of signs and a couple of factors have changed without affecting the result. v4: yet more sign changes, but results remain unchanged. v5: this is becoming absurd... but the signs ought to be correct now! v6: no more sign changes, but section 5.2 on the MKK monopole has been partially rewritten and some relevant references have been added.