An implementation of CAD in Maple utilising problem formulation, equational constraints and truth-table invariance

Cylindrical algebraic decomposition (CAD) is an important tool for the investigation of semi-algebraic sets, with applications within algebraic geometry and beyond. We recently reported on a new implementation of CAD in Maple which implemented the original algorithm of Collins and the subsequent improvement to projection by McCallum. Our implementation was in contrast to Maple's in-built CAD command, based on a quite separate theory. Although initially developed as an investigative tool to compare the algorithms, we found and reported that our code offered functionality not currently available in any other existing implementations. One particularly important piece of functionality is the ability to produce order-invariant CADs. This has allowed us to extend the implementation to produce CADs invariant with respect to either equational constraints (ECCADs) or the truth-tables of sequences of formulae (TTICADs). This new functionality is contained in the second release of our code, along with commands to consider problem formulation which can be a major factor in the tractability of a CAD. In the report we describe the new functionality and some theoretical discoveries it prompted. We describe how the CADs produced using equational constraints are able to take advantage of not just improved projection but also improvements in the lifting phase. We also present an extension to the original TTICAD algorithm which increases both the applicability of TTICAD and its relative benefit over other algorithms. The code and an introductory Maple worksheet / pdf demonstrating the full functionality of the package are freely available online.Comment: 12 pages; University of Bath, Dept. Computer Science Technical Report Series, 2013-02, 201

Higher Genus Abelian Functions Associated with Cyclic Trigonal Curves

We develop the theory of Abelian functions associated with cyclic trigonal curves by considering two new cases. We investigate curves of genus six and seven and consider whether it is the trigonal nature or the genus which dictates certain areas of the theory. We present solutions to the Jacobi inversion problem, sets of relations between the Abelian function, links to the Boussinesq equation and a new addition formula

Formulating problems for real algebraic geometry

We discuss issues of problem formulation for algorithms in real algebraic geometry, focussing on quantifier elimination by cylindrical algebraic decomposition. We recall how the variable ordering used can have a profound effect on both performance and output and summarise what may be done to assist with this choice. We then survey other questions of problem formulation and algorithm optimisation that have become pertinent following advances in CAD theory, including both work that is already published and work that is currently underway. With implementations now in reach of real world applications and new theory meaning algorithms are far more sensitive to the input, our thesis is that intelligently formulating problems for algorithms, and indeed choosing the correct algorithm variant for a problem, is key to improving the practical use of both quantifier elimination and symbolic real algebraic geometry in general.Comment: To be presented at The "Encuentros de \'Algebra Computacional y Aplicaciones, EACA 2014" (Meetings on Computer Algebra and Applications) in Barcelon

An implementation of Sub-CAD in Maple

Cylindrical algebraic decomposition (CAD) is an important tool for the investigation of semi-algebraic sets, with applications in algebraic geometry and beyond. We have previously reported on an implementation of CAD in Maple which offers the original projection and lifting algorithm of Collins along with subsequent improvements. Here we report on new functionality: specifically the ability to build cylindrical algebraic sub-decompositions (sub-CADs) where only certain cells are returned. We have implemented algorithms to return cells of a prescribed dimensions or higher (layered {\scad}s), and an algorithm to return only those cells on which given polynomials are zero (variety {\scad}s). These offer substantial savings in output size and computation time. The code described and an introductory Maple worksheet / pdf demonstrating the full functionality of the package are freely available online at http://opus.bath.ac.uk/43911/.Comment: 9 page

Computing with CodeRunner at Coventry University:Automated summative assessment of Python and C++ code.

CodeRunner is a free open-source Moodle plugin for automatically marking student code. We describe our experience using CodeRunner for summative assessment in our first year undergraduate programming curriculum at Coventry University. We use it to assess both Python3 and C++14 code (CodeRunner supports other languages also). We give examples of our questions and report on how key metrics have changed following its use at Coventry.Comment: 4 pages. Accepted for presentation at CEP2

Building Abelian Functions with Generalised Baker-Hirota Operators

We present a new systematic method to construct Abelian functions on Jacobian varieties of plane, algebraic curves. The main tool used is a symmetric generalisation of the bilinear operator defined in the work of Baker and Hirota. We give explicit formulae for the multiple applications of the operators, use them to define infinite sequences of Abelian functions of a prescribed pole structure and deduce the key properties of these functions. We apply the theory on the two canonical curves of genus three, presenting new explicit examples of vector space bases of Abelian functions. These reveal previously unseen similarities between the theories of functions associated to curves of the same genus

Times of change? Insights into the Government of India's water policy and management response to climate change

This thesis examines how climate change is being integrated within India's national and state government water policy and management practices. Climate change poses significant challenges to the management of non-stationary hydro-meteorological conditions, whilst meeting rising water demand. The nature and orientation of the Indian government's water institutional approach compounds this challenge, due to the1r focus on large-scale infrastructure-based supply-side water management. This research takes an interdisciplinary political ecology approach to examine the Indian hydrocracy's response, namely, the Ministry of Water Resources' (MWR) policy response to climate change, and the state level response by the Andhra Pradesh (AP) Irrigation Department. The analysis is based on policy documents and other government reports, interviews with policy makers and water managers, and non-government water experts 1n India, conducted between 2008 and 2011. The research draws on theoretical groundings of the linear and interactive models to understand public policy processes, water management paradigms including the hydraulic mission, river basin trajectory and institutional reform theory to understand the process and pace of government change. The Indian water policy experience will generate insights into the use of water policy to respond to climate change. The results indicate that climate change is being integrated within policy and water management practices as a continuation of infrastructure-based supply approaches to water management. This approach is facilitated by the uncertainty of climate change projections and impacts, which provide plasticity for it to be used to strengthen a sanctioned 'water for food' government discourse and hence continue India's hydraulic mission. The MWR and AP Irrigation Department appear resistant to change their strategic approach to water management. However, certain reformist actors within the margins of government are endeavouring to operationalise demand management strategies and institutional reform measures, broadly representing a reflexive modernity stage of water management. Insights into the Indian water policy process highlight numerous challenges to implementation, consistent with an interactive theoretical model of public policy. Implementation challenges of paramount importance include the politically contested nature of water management which serves vested political and financial interests, and the inertia of government, characterised by centralised and hierarchical structures and procedures. The government appears to be operating within the limits of a linear theoretical model of public policy, recommending demand management and institutional reform 'statements of policy intent', but without offering a suitable institutional approach to address implementation challenges. The hydrocracy is largely permitted to continue its approach within the wider political context in India, with other actors implicitly supporting and benefiting from large-scale water infrastructure. In conclusion, this research finds that both continuity and change co-exist within government water management in India. Resistance to change endures, whilst at the same time, certain reformist actors are intent to navigate the complex and uncertain nature of institutional reform

Higher genus Abelian functions associated with algebraic curves

We investigate the theory of Abelian functions with periodicity properties defined from an associated algebraic curve. A thorough summary of the background material is given, including a synopsis of elliptic function theory, generalisations of the Weierstrass σ and 0functions and a literature review. The theory of Abelian functions associated with a tetragonal curve of genus six is considered in detail. Differential equations and addition formula satisfied by the functions are derived and a solution to the Jacobi Inversion Problem is presented. New methods which centre on a series expansion of the σ function are used and discussions on the large computations involved are included. We construct a solution to the KP equation using these functions and outline how a general class of solutions can be generated from a wider class of curves. We proceed to present new approaches used to complete results for the lower genus trigonal curves. We also give some details on the the theory of higher genus trigonal curves before finishing with an application of the theory to the Benney moment equations. A reduction is constructed corresponding to Schwartz-Christoffel maps associated with the tetragonal curve. The mapping function is evaluated explicitly using derivatives of the σ function