Using ordinary multiplication to do relativistic velocity addition

Relativistic addition of velocities in one dimension, though a mainstay of introductory physics, contributes much less physical insight than it could. For such calculations, we propose the use of velocity factors (two-way doppler factors). Velocities can easily, often by inspection, be turned into velocity factors, and vice versa. Velocity factors compose by ordinary multiplication. This simple device considerably extends the kinds of questions that can be asked and answered in an introductory course.Comment: 6 page

Proof Theory, Transformations, and Logic Programming for Debugging Security Protocols

We define a sequent calculus to formally specify, simulate, debug and verify security protocols. In our sequents we distinguish between the current knowledge of principals and the current global state of the session. Hereby, we can describe the operational semantics of principals and of an intruder in a simple and modular way. Furthermore, using proof theoretic tools like the analysis of permutability of rules, we are able to find efficient proof strategies that we prove complete for special classes of security protocols including Needham-Schroeder. Based on the results of this preliminary analysis, we have implemented a Prolog meta-interpreter which allows for rapid prototyping and for checking safety properties of security protocols, and we have applied it for finding error traces and proving correctness of practical examples

The evolution problem for the 1D nonlocal Fisher-KPP equation with a top hat kernel. Part 1. The Cauchy problem on the real line

We study the Cauchy problem on the real line for the nonlocal Fisher-KPP equation in one spatial dimension, $$u_t = D u_{xx} + u(1-\phi*u),$$ where $\phi*u$ is a spatial convolution with the top hat kernel, $\phi(y) \equiv H\left(\frac{1}{4}-y^2\right)$. After showing that the problem is globally well-posed, we demonstrate that positive, spatially-periodic solutions bifurcate from the spatially-uniform steady state solution $u=1$ as the diffusivity, $D$, decreases through $\Delta_1 \approx 0.00297$. We explicitly construct these spatially-periodic solutions as uniformly-valid asymptotic approximations for $D \ll 1$, over one wavelength, via the method of matched asymptotic expansions. These consist, at leading order, of regularly-spaced, compactly-supported regions with width of $O(1)$ where $u=O(1)$, separated by regions where $u$ is exponentially small at leading order as $D \to 0^+$. From numerical solutions, we find that for $D \geq \Delta_1$, permanent form travelling waves, with minimum wavespeed, $2 \sqrt{D}$, are generated, whilst for $0 < D < \Delta_1$, the wavefronts generated separate the regions where $u=0$ from a region where a steady periodic solution is created. The structure of these transitional travelling waves is examined in some detail

Attacking Group Protocols by Refuting Incorrect Inductive Conjectures

Automated tools for finding attacks on flawed security protocols often fail to deal adequately with group protocols. This is because the abstractions made to improve performance on fixed 2 or 3 party protocols either preclude the modelling of group protocols all together, or permit modelling only in a fixed scenario, which can prevent attacks from being discovered. This paper describes Coral, a tool for finding counterexamples to incorrect inductive conjectures, which we have used to model protocols for both group key agreement and group key management, without any restrictions on the scenario. We will show how we used Coral to discover 6 previously unknown attacks on 3 group protocols

Well-posedness and qualitative behaviour of a semi-linear parabolic Cauchy problem arising from a generic model for fractional-order autocatalysis

In this paper, we examine a semi-linear parabolic Cauchy problem with non-Lipschitz nonlinearity which arises as a generic form in a significant number of applications. Specifically, we obtain a well-posedness result and examine the qualitative structure of the solution in detail. The standard classical approach to establishing well-posedness is precluded owing to the lack of Lipschitz continuity for the nonlinearity. Here, existence and uniqueness of solutions is established via the recently developed generic approach to this class of problem (Meyer & Needham 2015 The Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations. London Mathematical Society Lecture Note Series, vol. 419) which examines the difference of the maximal and minimal solutions to the problem. From this uniqueness result, the approach of Meyer & Needham allows for development of a comparison result which is then used to exhibit global continuous dependence of solutions to the problem on a suitable initial dataset. The comparison and continuous dependence results obtained here are novel to this class of problem. This class of problem arises specifically in the study of a one-step autocatalytic reaction, which is schematically given by A→B at rate a(p)b(q) (where a and b are the concentrations of A and B, respectively, with 0<p,q<1) and well-posedness for this problem has been lacking up to the present

Aspects of Hadamard well-posedness for classes of non-Lipschitz semilinear parabolic partial differential equations

We study classical solutions of the Cauchy problem for a class of non-Lipschitz semilinear parabolic partial differential equations in one spatial dimension with sufficiently smooth initial data. When the nonlinearity is Lipschitz continuous, results concerning existence, uniqueness and continuous dependence on initial data are well established (see, for example, the texts of Friedman and Smoller and, in the context of the present paper, see also Meyer), as are the associated results concerning Hadamard well-posedness. We consider the situations when the nonlinearity is Hölder continuous and when the nonlinearity is upper Lipschitz continuous. Finally, we consider the situation when the nonlinearity is both Hölder continuous and upper Lipschitz continuous. In each case we focus upon the question of existence, uniqueness and continuous dependence on initial data, and thus upon aspects of Hadamard well-posedness.</jats:p

Micro-enterprises: small enough to care?

This report presents findings of an evaluation of micro-enterprises in social care in England, which ran from 2013 to 2015. Organisations are here classed as micro if they employ five or fewer full-time equivalent staff. The aim of the project was to test the extent to which micro-enterprises deliver services that are personalised, valued, innovative and cost-effective, and how they compare with small, medium and large providers. Working in three parts of the country, researchers compared 27 organisations providing care and support, of which 17 were micro-enterprises, 2 were small, 4 were medium and 4 were large. The project team interviewed and surveyed 143 people (staff, older people, people with disabilities and carers) who received support from the 27 providers. The findings presented are relevant to people who use services and their families; social care commissioners; regulators and policy makers at a local and national level; people who provide care services; and social entrepreneurs who are considering setting up micro forms of support. The research was based at the University of Birmingham. It was funded by the Economic and Social Research Council (ESRC), as part of a project entitled Does Smaller mean Better? Evaluating Micro-enterprises in Adult Social Care (ESRC Standard Grant ES/K002317/1)

Inductive learning spatial attention

This paper investigates the automatic induction of spatial attention from the visual observation of objects manipulated on a table top. In this work, space is represented in terms of a novel observer-object relative reference system, named Local Cardinal System, defined upon the local neighbourhood of objects on the table. We present results of applying the proposed methodology on five distinct scenarios involving the construction of spatial patterns of coloured blocks

Overcharging a Black Hole and Cosmic Censorship

We show that, contrary to a widespread belief, one can overcharge a near extremal Reissner-Nordstrom black hole by throwing in a charged particle, as long as the backreaction effects may be considered negligible. Furthermore, we find that we can make the particle's classical radius, mass, and charge, as well as the relative size of the backreaction terms arbitrarily small, by adjusting the parameters corresponding to the particle appropriately. This suggests that the question of cosmic censorship is still not wholly resolved even in this simple scenario. We contrast this with attempting to overcharge a black hole with a charged imploding shell, where we find that cosmic censorship is upheld. We also briefly comment on a number of possible extensions.Comment: 26 pages, 3 figures, LaTe