240,185 research outputs found
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
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