General solution for Hamiltonians with extended cubic and quartic potentials

We integrate with hyperelliptic functions a two-particle Hamiltonian with quartic potential and additionnal linear and nonpolynomial terms in the Liouville integrable cases 1:6:1 and 1:6:8.Comment: LaTex 2e. To appear, Theoretical and Mathematical Physics 200

Modelling life trajectories and mode choice using Bayesian belief networks

Traditionally, transport mode choice was primarily examined as a stand alone problem. Given a purpose and destination, the choice of transport mode was modelled as a function of the various attributes of the transport mode alternatives. Later, transport mode choice decisions were modelled as part of more comprehensive models (activity-based approach). There is a need in the transport research community to explore and model dynamics in activity-travel patterns along various time horizons. This will lead to dynamic models of behavioural change. In this thesis, it is argued that a life course perspective offers some potential advantages in understanding and modelling activity-travel decisions, including transport mode choice. Central concepts in the life course approach are life trajectories, transitions and events. An individual life course is composed of multiple, interdependent careers (i.e. housing, household, education, occupational career) which develop over time in parallel. Earlier life transitions may have a cumulative effect on later life. The concepts of timing, sequencing, duration and spacing are used to describe life events, transitions and trajectories. The assumed effect of events on activity-travel decisions is captured in terms of a theory of learning and adaptation. Individuals develop and continuously adapt choice rules while interacting with their environment. The context is nonstationary, uncertain and highly dynamic and therefore it is assumed that individuals adapt their behaviour. Under stationary conditions, individuals will show habitual behaviour after some period of time. A life course event is seen as a trigger that may induce individuals and households to reorganise their activities in time and space. A particular event may also lead to other life course events. Thus, life course events may have direct and indirect effects on activity-travel patterns. An event does not necessarily lead to immediate changes in particular facets of activity-travel patterns. Behavioural change may also occur in anticipation of life course events. Bayesian Belief Networks is the approach adopted in this thesis to model the direct and indirect effects of life course effects on transport mode choice. More complex causation patterns can be included and results can be directly interpreted in terms of the classified events. Such networks need as input empirical data to learn the structure of the network and the conditional probability tables of the variables that are identified to be relevant. Data was collected using a retrospective Internet-based survey. Retrospective surveys, especially when administered through the Internet, are a good alternative for (quasi-)longitudinal data collection methods, like panel surveys, repeated cross sectional surveys, and cohort pseudo-panel surveys. One would expect that the quality of data coming from a retrospective survey depends on the nature of the event about which information is collected and on the time elapsed between the occurring of the events and the time of the retrospective survey. The quality of the data was tested and the results were positive. In case of life course events memory lapses are less of a problem. Life course events can be better recollected than other events. In this study, the results of the reliability and validity tests of the collected data showed that item nonresponse in general was relatively low, especially for those life course events that serve as markers unfolding one’s life. A statistical analysis suggested that memory / cohort effects were not found for the more salient life course events, such as housing, work and study related events. Memory may have an effect in reporting of events in case of income and transport mode related events (car availability and PT pass). The study illustrated that certain details of events, such as housing type and housing state are more difficult to recall. The time effect of an influence of life course events on mode choice was tested with a simple multinomial logit model. The results support the conclusion that a certain time influence exists in the response to events. The data of the retrospective Internet-based survey was used as input for two Bayesian Belief Networks, a life trajectory and a mode choice network. A year is chosen as the unit of analysis for these networks. Both networks were successfully learned from the data. The first network can be used to simulate a person’s life trajectory and the second network can be used to predict mode choice for an individual at a certain time given the individual life trajectory. The goodness-of-fit of the learned Bayesian Belief Networks was assessed on the basis of the log likelihood statistic. The values indicated that both networks perform relatively well. It was also investigated whether the life trajectory network was capable of reproducing observed characteristics of complete life trajectories. The observed and predicted life trajectories were compared in terms of the following criteria: the number of occurrences, interval times between occurrences of events, simultaneous occurrences of events and sequence of occurrences of events. The life trajectory network reproduced the number of occurrences in the life trajectories quite well. In general, the network predicted more or less the same means of interval times for the events, except for the PT pass event. The network was less successful in predicting correctly the observed incidence of synchronic events. The results of the sequence alignment analysis indicate that the network predicts the sequence of the occurrences in the life trajectories relatively good. The modal split (car, public transport and slow transport) of the predicted mode choice was compared with the observed mode choice. Results indicated a relatively small over prediction of public transport and under prediction of car and slow transport. This suggests that the mode choice network is able to simulate more or less the same mode choice as registered in the data. The learned networks were used to study direct and indirect effects of one variable on other variables in the network. The described effects seem logical. A simulation illustrated the dynamics of the lives of ten inhabitants of a newly build neighbourhood. It showed that, insight in dynamics of life trajectory events and mode choice can lead to a better understanding which can support the development of better or different policy measures

Integration of a generalized H\'enon-Heiles Hamiltonian

The generalized H\'enon-Heiles Hamiltonian $H=1/2(P_X^2+P_Y^2+c_1X^2+c_2Y^2)+aXY^2-bX^3/3$ with an additional nonpolynomial term $\mu Y^{-2}$ is known to be Liouville integrable for three sets of values of $(b/a,c_1,c_2)$. It has been previously integrated by genus two theta functions only in one of these cases. Defining the separating variables of the Hamilton-Jacobi equations, we succeed here, in the two other cases, to integrate the equations of motion with hyperelliptic functions.Comment: LaTex 2e. To appear, Journal of Mathematical Physic

Completeness of the cubic and quartic H\'enon-Heiles Hamiltonians

The quartic H\'enon-Heiles Hamiltonian $H = (P_1^2+P_2^2)/2+(\Omega_1 Q_1^2+\Omega_2 Q_2^2)/2 +C Q_1^4+ B Q_1^2 Q_2^2 + A Q_2^4 +(1/2)(\alpha/Q_1^2+\beta/Q_2^2) - \gamma Q_1$ passes the Painlev\'e test for only four sets of values of the constants. Only one of these, identical to the traveling wave reduction of the Manakov system, has been explicitly integrated (Wojciechowski, 1985), while the three others are not yet integrated in the generic case $(\alpha,\beta,\gamma)\not=(0,0,0)$. We integrate them by building a birational transformation to two fourth order first degree equations in the classification (Cosgrove, 2000) of such polynomial equations which possess the Painlev\'e property. This transformation involves the stationary reduction of various partial differential equations (PDEs). The result is the same as for the three cubic H\'enon-Heiles Hamiltonians, namely, in all four quartic cases, a general solution which is meromorphic and hyperelliptic with genus two. As a consequence, no additional autonomous term can be added to either the cubic or the quartic Hamiltonians without destroying the Painlev\'e integrability (completeness property).Comment: 10 pages, To appear, Theor.Math.Phys. Gallipoli, 34 June--3 July 200

Resonant tunneling in Y(Dy)Ba2Cu3O7−δ/PrBa2Cu3−xGaxO7−δ/Y(Dy)Ba2Cu3O7−δ ramp-type Josephson junctions

We have investigated both experimentally and theoretically the normal state resistance and Josephson critical current of ramp-type Josephson junctions having YBCO (DyBCO) electrodes and 8–30 nm thick Ga-doped barriers PrBa2Cu3−xGaxO7−δ with Ga content x = 0, 0.05 and 0.1. Analysis of the data shows that the behavior of the junctions can be well described by the model assuming transport through a finite number of localized states in the barrier

On reductions of some KdV-type systems and their link to the quartic He'non-Heiles Hamiltonian

A few 2+1-dimensional equations belonging to the KP and modified KP hierarchies are shown to be sufficient to provide a unified picture of all the integrable cases of the cubic and quartic H\'enon-Heiles Hamiltonians.Comment: 12 pages, 3 figures, NATO ARW, 15-19 september 2002, Elb