Numerical Bifurcation Analysis of Conformal Formulations of the Einstein Constraints

The Einstein constraint equations have been the subject of study for more than fifty years. The introduction of the conformal method in the 1970's as a parameterization of initial data for the Einstein equations led to increased interest in the development of a complete solution theory for the constraints, with the theory for constant mean curvature (CMC) spatial slices and closed manifolds completely developed by 1995. The first general non-CMC existence result was establish by Holst et al. in 2008, with extensions to rough data by Holst et al. in 2009, and to vacuum spacetimes by Maxwell in 2009. The non-CMC theory remains mostly open; moreover, recent work of Maxwell on specific symmetry models sheds light on fundamental non-uniqueness problems with the conformal method as a parameterization in non-CMC settings. In parallel with these mathematical developments, computational physicists have uncovered surprising behavior in numerical solutions to the extended conformal thin sandwich formulation of the Einstein constraints. In particular, numerical evidence suggests the existence of multiple solutions with a quadratic fold, and a recent analysis of a simplified model supports this conclusion. In this article, we examine this apparent bifurcation phenomena in a methodical way, using modern techniques in bifurcation theory and in numerical homotopy methods. We first review the evidence for the presence of bifurcation in the Hamiltonian constraint in the time-symmetric case. We give a brief introduction to the mathematical framework for analyzing bifurcation phenomena, and then develop the main ideas behind the construction of numerical homotopy, or path-following, methods in the analysis of bifurcation phenomena. We then apply the continuation software package AUTO to this problem, and verify the presence of the fold with homotopy-based numerical methods.Comment: 13 pages, 4 figures. Final revision for publication, added material on physical implication

Virtual Evidence: A Constructive Semantics for Classical Logics

This article presents a computational semantics for classical logic using constructive type theory. Such semantics seems impossible because classical logic allows the Law of Excluded Middle (LEM), not accepted in constructive logic since it does not have computational meaning. However, the apparently oracular powers expressed in the LEM, that for any proposition P either it or its negation, not P, is true can also be explained in terms of constructive evidence that does not refer to "oracles for truth." Types with virtual evidence and the constructive impossibility of negative evidence provide sufficient semantic grounds for classical truth and have a simple computational meaning. This idea is formalized using refinement types, a concept of constructive type theory used since 1984 and explained here. A new axiom creating virtual evidence fully retains the constructive meaning of the logical operators in classical contexts. Key Words: classical logic, constructive logic, intuitionistic logic, propositions-as-types, constructive type theory, refinement types, double negation translation, computational content, virtual evidenc

Evaluation of IoT-Based Computational Intelligence Tools for DNA Sequence Analysis in Bioinformatics

In contemporary age, Computational Intelligence (CI) performs an essential role in the interpretation of big biological data considering that it could provide all of the molecular biology and DNA sequencing computations. For this purpose, many researchers have attempted to implement different tools in this field and have competed aggressively. Hence, determining the best of them among the enormous number of available tools is not an easy task, selecting the one which accomplishes big data in the concise time and with no error can significantly improve the scientist's contribution in the bioinformatics field. This study uses different analysis and methods such as Fuzzy, Dempster-Shafer, Murphy and Entropy Shannon to provide the most significant and reliable evaluation of IoT-based computational intelligence tools for DNA sequence analysis. The outcomes of this study can be advantageous to the bioinformatics community, researchers and experts in big biological data

Mathematical practice, crowdsourcing, and social machines

The highest level of mathematics has traditionally been seen as a solitary endeavour, to produce a proof for review and acceptance by research peers. Mathematics is now at a remarkable inflexion point, with new technology radically extending the power and limits of individuals. Crowdsourcing pulls together diverse experts to solve problems; symbolic computation tackles huge routine calculations; and computers check proofs too long and complicated for humans to comprehend. Mathematical practice is an emerging interdisciplinary field which draws on philosophy and social science to understand how mathematics is produced. Online mathematical activity provides a novel and rich source of data for empirical investigation of mathematical practice - for example the community question answering system {\it mathoverflow} contains around 40,000 mathematical conversations, and {\it polymath} collaborations provide transcripts of the process of discovering proofs. Our preliminary investigations have demonstrated the importance of "soft" aspects such as analogy and creativity, alongside deduction and proof, in the production of mathematics, and have given us new ways to think about the roles of people and machines in creating new mathematical knowledge. We discuss further investigation of these resources and what it might reveal. Crowdsourced mathematical activity is an example of a "social machine", a new paradigm, identified by Berners-Lee, for viewing a combination of people and computers as a single problem-solving entity, and the subject of major international research endeavours. We outline a future research agenda for mathematics social machines, a combination of people, computers, and mathematical archives to create and apply mathematics, with the potential to change the way people do mathematics, and to transform the reach, pace, and impact of mathematics research.Comment: To appear, Springer LNCS, Proceedings of Conferences on Intelligent Computer Mathematics, CICM 2013, July 2013 Bath, U

Computability and analysis: the legacy of Alan Turing

We discuss the legacy of Alan Turing and his impact on computability and analysis.Comment: 49 page

How could a rational analysis model explain?

Rational analysis is an influential but contested account of how probabilistic modeling can be used to construct non-mechanistic but self-standing explanatory models of the mind. In this paper, I disentangle and assess several possible explanatory contributions which could be attributed to rational analysis. Although existing models suffer from evidential problems that question their explanatory power, I argue that rational analysis modeling can complement mechanistic theorizing by providing models of environmental affordances

ScratchMaths: evaluation report and executive summary

Since 2014, computing has been part of the primary curriculum. ‘Scratch’ is frequently used by schools, and the EEF funded this trial to test whether the platform could be used to improve pupils’ computational thinking skills, and whether this in turn could have a positive impact on Key Stage 2 maths attainment. Good computational thinking skills mean pupils can use problem solving methods that involve expressing problems and their solutions in ways that a computer could execute – for example, recognising patterns. Previous research has shown that pupils with better computational thinking skills do better in maths. The study found a positive impact on computational thinking skills at the end of Year 5 – particularly for pupils who have ever been eligible for free school meals. However, there was no evidence of an impact on Key Stage 2 maths attainment when pupils were tested at the end of Year 6. Many of the schools in the trial did not fully implement ScratchMaths, particularly in Year 6, where teachers expressed concerns about the pressure of Key Stage 2 SATs. But there was no evidence that schools which did implement the programme had better maths results. Schools may be interested in ScratchMaths as an affordable way to cover aspects of the primary computing curriculum in maths lessons without any adverse effect on core maths outcomes. This trial, however, did not provide evidence that ScratchMaths is an effective way to improve maths outcomes