Method of constructing exactly solvable chaos

We present a new systematic method of constructing rational mappings as ergordic transformations with nonuniform invariant measures on the unit interval [0,1]. As a result, we obtain a two-parameter family of rational mappings that have a special property in that their invariant measures can be explicitly written in terms of algebraic functions of parameters and a dynamical variable. Furthermore, it is shown here that this family is the most generalized class of rational mappings possessing the property of exactly solvable chaos on the unit interval, including the Ulam=Neumann map y=4x(1-x). Based on the present method, we can produce a series of rational mappings resembling the asymmetric shape of the experimentally obtained first return maps of the Beloussof-Zhabotinski chemical reaction, and we can match some rational functions with other experimentally obtained first return maps in a systematic manner.Comment: 12 pages, 2 figures, REVTEX. Title was changed. Generalized Chebyshev maps including the precise form of two-parameter generalized cubic maps were added. Accepted for publication in Phys. Rev. E(1997

Exact solutions of the radial Schrodinger equation for some physical potentials

By using an ansatz for the eigenfunction, we have obtained the exact analytical solutions of the radial Schrodinger equation for the pseudoharmonic and Kratzer potentials in two dimensions. The energy levels of all the bound states are easily calculated from this eigenfunction ansatz. The normalized wavefunctions are also obtained.Comment: 13 page

Assessing the role of EO in biodiversity monitoring: options for integrating in-situ observations with EO within the context of the EBONE concept

The European Biodiversity Observation Network (EBONE) is a European contribution on terrestrial monitoring to GEO BON, the Group on Earth Observations Biodiversity Observation Network. EBONE’s aims are to develop a system of biodiversity observation at regional, national and European levels by assessing existing approaches in terms of their validity and applicability starting in Europe, then expanding to regions in Africa. The objective of EBONE is to deliver: 1. A sound scientific basis for the production of statistical estimates of stock and change of key indicators; 2. The development of a system for estimating past changes and forecasting and testing policy options and management strategies for threatened ecosystems and species; 3. A proposal for a cost-effective biodiversity monitoring system. There is a consensus that Earth Observation (EO) has a role to play in monitoring biodiversity. With its capacity to observe detailed spatial patterns and variability across large areas at regular intervals, our instinct suggests that EO could deliver the type of spatial and temporal coverage that is beyond reach with in-situ efforts. Furthermore, when considering the emerging networks of in-situ observations, the prospect of enhancing the quality of the information whilst reducing cost through integration is compelling. This report gives a realistic assessment of the role of EO in biodiversity monitoring and the options for integrating in-situ observations with EO within the context of the EBONE concept (cfr. EBONE-ID1.4). The assessment is mainly based on a set of targeted pilot studies. Building on this assessment, the report then presents a series of recommendations on the best options for using EO in an effective, consistent and sustainable biodiversity monitoring scheme. The issues that we faced were many: 1. Integration can be interpreted in different ways. One possible interpretation is: the combined use of independent data sets to deliver a different but improved data set; another is: the use of one data set to complement another dataset. 2. The targeted improvement will vary with stakeholder group: some will seek for more efficiency, others for more reliable estimates (accuracy and/or precision); others for more detail in space and/or time or more of everything. 3. Integration requires a link between the datasets (EO and in-situ). The strength of the link between reflected electromagnetic radiation and the habitats and their biodiversity observed in-situ is function of many variables, for example: the spatial scale of the observations; timing of the observations; the adopted nomenclature for classification; the complexity of the landscape in terms of composition, spatial structure and the physical environment; the habitat and land cover types under consideration. 4. The type of the EO data available varies (function of e.g. budget, size and location of region, cloudiness, national and/or international investment in airborne campaigns or space technology) which determines its capability to deliver the required output. EO and in-situ could be combined in different ways, depending on the type of integration we wanted to achieve and the targeted improvement. We aimed for an improvement in accuracy (i.e. the reduction in error of our indicator estimate calculated for an environmental zone). Furthermore, EO would also provide the spatial patterns for correlated in-situ data. EBONE in its initial development, focused on three main indicators covering: (i) the extent and change of habitats of European interest in the context of a general habitat assessment; (ii) abundance and distribution of selected species (birds, butterflies and plants); and (iii) fragmentation of natural and semi-natural areas. For habitat extent, we decided that it did not matter how in-situ was integrated with EO as long as we could demonstrate that acceptable accuracies could be achieved and the precision could consistently be improved. The nomenclature used to map habitats in-situ was the General Habitat Classification. We considered the following options where the EO and in-situ play different roles: using in-situ samples to re-calibrate a habitat map independently derived from EO; improving the accuracy of in-situ sampled habitat statistics, by post-stratification with correlated EO data; and using in-situ samples to train the classification of EO data into habitat types where the EO data delivers full coverage or a larger number of samples. For some of the above cases we also considered the impact that the sampling strategy employed to deliver the samples would have on the accuracy and precision achieved. Restricted access to European wide species data prevented work on the indicator ‘abundance and distribution of species’. With respect to the indicator ‘fragmentation’, we investigated ways of delivering EO derived measures of habitat patterns that are meaningful to sampled in-situ observations

Phenomenology of the Lense-Thirring effect in the Solar System

Recent years have seen increasing efforts to directly measure some aspects of the general relativistic gravitomagnetic interaction in several astronomical scenarios in the solar system. After briefly overviewing the concept of gravitomagnetism from a theoretical point of view, we review the performed or proposed attempts to detect the Lense-Thirring effect affecting the orbital motions of natural and artificial bodies in the gravitational fields of the Sun, Earth, Mars and Jupiter. In particular, we will focus on the evaluation of the impact of several sources of systematic uncertainties of dynamical origin to realistically elucidate the present and future perspectives in directly measuring such an elusive relativistic effect.Comment: LaTex, 51 pages, 14 figures, 22 tables. Invited review, to appear in Astrophysics and Space Science (ApSS). Some uncited references in the text now correctly quoted. One reference added. A footnote adde

Environment and Rural Affairs Monitoring & Modelling Programme - ERAMMP Report-60: ERAMMP Integrated Modelling Platform (IMP) Land Use Scenarios

Six scenarios consisting of changes in farm-gate prices (T1 to T6) have been applied to the ERAMMP Integrated Modelling Platform (IMP) to simulate impacts on land use change, biodiversity and ecosystem services (carbon, water quality and air quality). The scenarios were based on discussions held between stakeholders in the Evidence and Scenario subgroup (Roundtable Wales and Brexit1) and Welsh Government (WG) policy officials. These discussions took place in late 2020 before the arrangements for the UK leaving the EU were agreed, therefore are based on broad assumptions around the detail of the trade agreement with the EU as well as other third countries including Australia, New Zeland and USA. It is important to note that the outputs of these discussions which were used as inputs into the ERAMMP IMP may therefore not accurately reflect the outcomes achieved within the finalised trade agreements. The T1 scenario assumes no EU trade deal and trade liberalisation, with no tariffs applied to imported products and T2 an EU trade deal with no change to the trade arrangements with third countries. These two scenarios used the changes to farm-gate prices modelled by FAPRI2. The assumptions used in the T3 to T6 scenarios were based on expert opinion from the stakeholder group, and include impacts on farm-gate prices which potentially could have resulted from different combinations of trade deals with New Zealand, Australia and USA. Scenarios which include “no EU deal” options (T1 and T4) are no longer relevant. In no way whatsoever do T1, T3, T4, T5 and T6 represent a WG position; our understanding of the nature and impact of new and emerging trade deals has evolved significantly and the WG Trade Policy Team lead in this area. The objective of this work was to gain an early understanding of how changes in farm-gate prices potentially resulting from trading relationships may influence land use and subsequently effect entry into the Sustainable Farming Scheme. We note that many other factors are also likely to influence Welsh farmgate prices, such as (but not limited to), currency exchange rates, energy prices and extreme weather events in other parts of the world. This report provides an overview of the land use implications of all these scenarios, but focuses on the T2 scenario, which represents an EU Trade Deal. This T2 scenario is being used as the counterfactual scenario against which the costs and benefits of the land use implications of the proposed Sustainable Farming Scheme will be assessed in the Regulatory Impact Assessment for the proposed Agricultural Bill. This includes the estimated environmental outcomes of the EU Trade Deal scenario and, where the ERAMMP IMP has attached monetary valuations to these, the value of these outcomes to society. In the Cost Benefit Analysis, these monetary values will inform the overall estimated Net Present Value (NPV) of this business-as-usual counterfactual. The IMP involves many assumptions and these need to be borne in mind when interpreting and using its outcomes. By necessity, all models are a simplification of the real situation, but can still provide very useful insights if applied for a specific purpose and with caution. The collaborative and iterative consortium-based approach to co-designing the IMP has meant that Welsh Government and IMP teams have clear, open channels of communication for asking questions. This ensures that the modelling represents government aspirations as well as possible and the limits of the approach are well understood. IMP outputs for the T2 scenario show that some simulated full-time farms (>1 FTE labour) come under economic pressure (7%) and are simulated to be unable to produce a sufficient Farm Business Income to be economically viable. For these farm types, no options to transition to a more alternative profitable farm type are available and they are assumed to leave full-time agriculture. A greater number of farms transition to dairying resulting in a 75% increase in the number of dairy farms. This is associated with large increases in the number of dairy cattle (73%) and reductions in sheep (-34%). A general intensification of grassland systems is simulated resulting from the farm type transitions, with a 66% increase in temporary grasslands and a 21% decrease in permanent grasslands. Overall, these changes in agriculture and land use are simulated to lead to mixed, but predominantly negative, effects on biodiversity, increases in GHG emissions and deterioration in air and water quality. The T2 scenario predicts the least change in agriculture out of the six scenarios. T1 simulates the greatest impacts on agriculture due to significant farm-gate price reductions across dairy, beef and sheep systems, with a large number of full-time farms leaving agriculture. This leads to large increases in woodland area and generally positive effects on biodiversity and ecosystem services. T3 and T4 also simulate large impacts on agriculture. These are associated with significant farm transitions to dairy (due to increases in milk prices and significant decreases in beef and lamb prices) resulting in larger increases in GHG emissions and greater declines in air and water quality, compared to the T2 scenario. The T5 and T6 scenarios fall between these extremes, with T6 projecting the second greatest impacts on agriculture (after T1) in terms of farms under pressure. These simulated changes in agriculture are associated with net benefits for air and water quality, but net costs for GHG emissions; although these costs are lower than for scenarios T3-T5