Some New Addition Formulae for Weierstrass Elliptic Functions

We present new addition formulae for the Weierstrass functions associated with a general elliptic curve. We prove the structure of the formulae in n-variables and give the explicit addition formulae for the 2- and 3-variable cases. These new results were inspired by new addition formulae found in the case of an equianharmonic curve, which we can now observe as a specialisation of the results here. The new formulae, and the techniques used to find them, also follow the recent work for the generalisation of Weierstrass' functions to curves of higher genus.Comment: 20 page

Do people with intellectual disabilities understand their prescription medication? A scoping review

© 2019 The Authors. Journal of Applied Research in Intellectual Disabilities Published by John Wiley & Sons Ltd.Background: People with intellectual disabilities are more likely to experience poor health than the general population and are frequently prescribed multiple medications. Therefore, it is important that people with intellectual disabilities understand their medication and potential adverse effects. Method: A scoping review explored people with intellectual disabilities' knowledge of prescription medications, their risks and how medication understanding can be improved. Results: Ten journal articles were included. People with intellectual disabilities often lacked understanding of their medication, including its name, purpose and when and how to take it. Participants were often confused or unaware of adverse effects associated with their medication. Information was sometimes explained to carers rather than people with intellectual disabilities. Some interventions and accessible information helped to improve knowledge in people with intellectual disabilities. Conclusion: There is a need for accessible and tailored information about medication to be discussed with people with intellectual disabilities in order to meet legal and best practice standards.Peer reviewe

Using the Regular Chains Library to build cylindrical algebraic decompositions by projecting and lifting

Cylindrical algebraic decomposition (CAD) is an important tool, both for quantifier elimination over the reals and a range of other applications. Traditionally, a CAD is built through a process of projection and lifting to move the problem within Euclidean spaces of changing dimension. Recently, an alternative approach which first decomposes complex space using triangular decomposition before refining to real space has been introduced and implemented within the RegularChains Library of Maple. We here describe a freely available package ProjectionCAD which utilises the routines within the RegularChains Library to build CADs by projection and lifting. We detail how the projection and lifting algorithms were modified to allow this, discuss the motivation and survey the functionality of the package

Cylindrical Algebraic Sub-Decompositions

Cylindrical algebraic decompositions (CADs) are a key tool in real algebraic geometry, used primarily for eliminating quantifiers over the reals and studying semi-algebraic sets. In this paper we introduce cylindrical algebraic sub-decompositions (sub-CADs), which are subsets of CADs containing all the information needed to specify a solution for a given problem. We define two new types of sub-CAD: variety sub-CADs which are those cells in a CAD lying on a designated variety; and layered sub-CADs which have only those cells of dimension higher than a specified value. We present algorithms to produce these and describe how the two approaches may be combined with each other and the recent theory of truth-table invariant CAD. We give a complexity analysis showing that these techniques can offer substantial theoretical savings, which is supported by experimentation using an implementation in Maple.Comment: 26 page

Truth Table Invariant Cylindrical Algebraic Decomposition by Regular Chains

A new algorithm to compute cylindrical algebraic decompositions (CADs) is presented, building on two recent advances. Firstly, the output is truth table invariant (a TTICAD) meaning given formulae have constant truth value on each cell of the decomposition. Secondly, the computation uses regular chains theory to first build a cylindrical decomposition of complex space (CCD) incrementally by polynomial. Significant modification of the regular chains technology was used to achieve the more sophisticated invariance criteria. Experimental results on an implementation in the RegularChains Library for Maple verify that combining these advances gives an algorithm superior to its individual components and competitive with the state of the art

Thermochemical generation of hydrogen and carbon dioxide

Mixing of carbon in the form of high sulfur coal with sulfuric acid reduces the temperature of sulfuric acid decomposition from 830.degree. C. to between 300.degree. C. and 400.degree. C. The low temperature sulfuric acid decomposition is particularly useful in thermal chemical cycles for splitting water to produce hydrogen. Carbon dioxide is produced as a commercially desirable byproduct. Lowering of the temperature for the sulfuric acid decomposition or oxygen release step simplifies equipment requirements, lowers thermal energy input and reduces corrosion problems presented by sulfuric acid at conventional cracking temperatures. Use of high sulfur coal as the source of carbon for the sulfuric acid decomposition provides an environmentally safe and energy efficient utilization of this normally polluting fuel

Oxidation processes in magneto-optic and related materials

The surface oxidation processes of thin films of magneto-optic materials, such as the rare-earth transition metal alloys have been studied, starting in ultrahigh vacuum environments, using surface analysis techniques, as a way of modeling the oxidation processes which occur at the base of a defect in an overcoated material, at the instant of exposure to ambient environments. Materials examined have included FeTbCo alloys, as well as those same materials with low percentages of added elements, such a Ta, and their reactivities to both O2 and H2O compared with materials such as thin Fe films coated with ultrathin adlayers of Ti. The surface oxidation pathways for these materials is reviewed, and XPS data presented which indicates the type of oxides formed, and a critical region of Ta concentration which provides optimum protection

Atlantic Ocean Heat Transport Enabled by Indo-Pacific Heat Uptake and Mixing

The ocean transports vast amounts of heat around the planet, helping to regulate regional climate. One important component of this heat transport is the movement of warm water from equatorial regions toward the poles, with colder water flowing in return. Here, we introduce a framework relating meridional heat transport to the diabatic processes of surface forcing and turbulent mixing that move heat across temperature classes. Applied to a (1/4)° global ocean model the framework highlights the role of the tropical Indo‐Pacific in the global ocean heat transport. A large fraction of the northward heat transport in the Atlantic is ultimately sourced from heat uptake in the eastern tropical Pacific. Turbulent mixing moves heat from the warm, shallow Indo‐Pacific circulation to the cold deeper‐reaching Atlantic circulation. Our results underscore a renewed focus on the tropical oceans and their role in global circulation pathways