Mark McCartney

Behaviour of a whole-link travel time model used in dynamic traffic assignment

Whole-link models of traffic flows have been widely used in mathematical programming models for dynamic traffic assignment (DTA). In this paper, we consider a well-known whole-link model in which the link travel time, for traffic entering at time t, is a function of the number of vehicles on the link, and may also be a function of the inflow rate or outflow rate at time t. Instead of considering this in a network context, we examine its behaviour for a single link, for given inflow profiles, so as to distinguish behaviour within a link from network behaviour. We consider steady state solutions, for constant inflows and outflows, note that various model forms can yield the same solution, and that under certain conditions the model may admit multiple values for the link travel time. We derive the complete analytic solution for a model where the travel time depends linearly only on the number of vehicles on the link, and show that the solution exhibits pseudo-periodicity, and converges to a steady state solution. The results indicate that the analytic solution is quite complex even for very simple cases, and that care has to be exercised in the choice of parameters. We illustrate the solutions numerically.

Mathematics of Zernike Polynomials: A Review

Monochromatic aberrations of the eye principally originate from the cornea and the crystalline lens. Aberrometers operate via differing principles but function by either analysing the reflected wavefront from the retina or by analysing an image on the retina. Aberrations may be described as lower order or higher order aberrations with Zernike polynomials being the most commonly employed fitting method. The complex mathematical aspects with regards the Zernike polynomial expansion series are detailed in this review. Refractive surgery has been a key clinical application of aberrometers; however, more recently aberrometers have been used in a range of other areas ophthalmology including corneal diseases, cataract and retinal imaging.

Pseudo periodicity in a travel-time model used in dynamic traffic assignment

In the past several years, in network models for dynamic traffic assignment, link travel times have frequently been treated as a function of the number of vehicles on the link. In an earlier paper, the present authors considered the linear form of this link travel-time function and showed that if there is a step increase in the inflow pattern this causes an infinite sequence of steps or jumps in the outflow profile, gradually damping out over time. This paper extends the analysis of this phenomenon to nonlinear travel-time functions and to more general inflow patterns. We show that the phenomenon occurs with general travel-time functions, and occurs whether the flow changes in discontinuous steps or more smoothly, and whether flows increase or decrease. We illustrate the results with numerical examples. We find, and prove, some surprising results, in particular that, in the travel-time model, outflows can take a much longer time to adjust to small falls in inflows than to large falls in inflows.

Lyapunov exponents for multi-parameter tent and logistic maps

The behaviour of logistic and tent maps is studied in cases where the control parameter is dependent on iteration number. Analytic results for global Lyapunov exponent are presented in the case of the tent map and numerical results are presented in the case of the logistic map. In the case of a tent map with N control parameters, the fraction of parameter space for which the global Lyapunov exponent is positive is calculated. The case of bi-parameter maps of period N are investigated.

A model of activation of the protein tyrosine phosphatase SHP-2 by the human leptin receptor

Signalling through the leptin receptor has been shown to activate the SH2 domain-containing tyrosine phosphatase SHP-2 through tyrosine phosphorylation. The human leptin receptor contains five tyrosine residues in the cytoplasmic domain that may become phosphorylated. We show here using BIAcore studies, wherein binding of peptides to SHP-2 was detected, that peptides corresponding to sequences containing phosphotyrosines 974 and 986 (LR974P and LR986P, respectively) from the leptin receptor cytoplasmic domain were the only two peptides that bound to the enzyme. Binding of LR974P to SHP-2 was inhibited in a dose-dependent fashion by orthovanadate, whereas binding of LY986P was not, indicating that the enzyme binds to these peptides through different sites. Only the leptin receptor-derived peptide corresponding to tyrosine 974 was dephosphorylated by recombinant purified SHP-2. Time courses of the reaction were complex, and fitted a two exponent rate equation. Preincubation of SHP-2 with LR986P markedly activated the enzyme at early time points and time courses of the activated enzyme fitted a single exponential first order rate equation. We propose that LR974P binds to the active site of SHP-2, whereas LR986P may bind to the N- and C-terminal SH2 domains of SHP-2, thus activating the phosphatase activity. These data support a model in which SHP-2 binds to phosphotyrosine 986 in the activated leptin receptor and is activated to dephosphorylate phosphotyrosine 974, downregulating signalling events emanating from SH2 domain-containing proteins that bind here.

Inattentive Drivers: Making the Solution Method the Model.

A simple car following model based on the solution of coupled ordinary differential equations is considered. The model is solved using Eulers method and this method of solution is itself interpreted as a mathematical model for car following. Examples of possible classroom use are given.

The routes of unity

A model for car following on a closed loop is defined. The stability of the solutions of the model is investigated by considering the evolution of the roots of the corresponding characteristic equation in the complex plane. The solution provides a motivation for investigating the behaviour of the roots of a simple class of algebraic equation.

Cannibalism and chaos in the classroom

ABSTRACT Two simple discrete-time models of mutation-induced cannibalism are introduced and investigated, one linear and one nonlinear. Both form the basis for possible classroom activities and independent investigative study. A range of classroom exercises are provided, along with suggestions for further investigations.

Analysis of a Class of Low-Dimensional Models of Mutation and Predation

We consider a class of simple two-dimensional discrete models representative of a system incorporating both mutation and predation. A selection of analytic and numerical results are presented, classifying the dynamic behavior of the system by means of Lyapunov exponents over a biologically-reasonable region of parameter space, and illustrating the occurrence of hyperchaos and a Neimark–Sacker bifurcation producing regions of quasiperiodicity.

Dynamics of Conflicting Beliefs in Social Networks

This paper analyzes two proposed models for simulating opinion dynamics in social networks where beliefs might be considered to be competing. In both models agents have a degree of tolerance, which represents the extent to which the agent takes into account the differing beliefs of other agents, and a degree of conflict, which represents the extent to which two beliefs are considered to be competing. In this paper, we apply different tolerance and conflict degrees to different groups in a network, and see how these groups affect each other. Simulations show that the groups having different tolerance degrees do not have significant effect upon each other in both Models I and II. On the other hand, the group perceiving a conflict causes more diversity in the agents based on Model I, but introduces a higher consensus level among agents when the fraction becomes larger in Model II.