5,645 research outputs found
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
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
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
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
Improving the use of equational constraints in cylindrical algebraic decomposition
When building a cylindrical algebraic decomposition (CAD) savings can be made
in the presence of an equational constraint (EC): an equation logically implied
by a formula.
The present paper is concerned with how to use multiple ECs, propagating
those in the input throughout the projection set. We improve on the approach of
McCallum in ISSAC 2001 by using the reduced projection theory to make savings
in the lifting phase (both to the polynomials we lift with and the cells lifted
over). We demonstrate the benefits with worked examples and a complexity
analysis
High-altitude gravity waves in the Martian thermosphere observed by MAVEN/NGIMS and modeled by a gravity wave scheme
First high-altitude observations of gravity wave (GW)-induced CO density
perturbations in the Martian thermosphere retrieved from NASA's NGIMS
instrument on board the MAVEN satellite are presented and interpreted using the
extended GW parameterization of Yi\u{g}it et al. [2008] and the Mars Climate
Database as an input. Observed relative density perturbations between 180-220
km of 20-40 % demonstrate appreciable local time, latitude, and altitude
variations. Modeling for the spatiotemporal conditions of the MAVEN
observations suggests that GWs can directly propagate from the lower atmosphere
to the thermosphere, produce appreciable dynamical effects, and likely
contribute to the observed fluctuations. Modeled effects are somewhat smaller
than the observed but their highly variable nature is in qualitative agreement
with observations. Possible reasons for discrepancies between modeling and
measurements are discussed.Comment: Accepted for publication in Geophysical Research Letters (GRL).
Special section: First Results from the MAVEN Mission to Mar
Improved cross-validation for classifiers that make algorithmic choices to minimise runtime without compromising output correctness
Our topic is the use of machine learning to improve software by making
choices which do not compromise the correctness of the output, but do affect
the time taken to produce such output. We are particularly concerned with
computer algebra systems (CASs), and in particular, our experiments are for
selecting the variable ordering to use when performing a cylindrical algebraic
decomposition of -dimensional real space with respect to the signs of a set
of polynomials.
In our prior work we explored the different ML models that could be used, and
how to identify suitable features of the input polynomials. In the present
paper we both repeat our prior experiments on problems which have more
variables (and thus exponentially more possible orderings), and examine the
metric which our ML classifiers targets. The natural metric is computational
runtime, with classifiers trained to pick the ordering which minimises this.
However, this leads to the situation were models do not distinguish between any
of the non-optimal orderings, whose runtimes may still vary dramatically. In
this paper we investigate a modification to the cross-validation algorithms of
the classifiers so that they do distinguish these cases, leading to improved
results.Comment: 16 pages. Accepted into the Proceedings of MACIS 2019. arXiv admin
note: text overlap with arXiv:1906.0145
The Paradox of Compacts: final report to the Home Office on monitoring the impact of Compacts
The Compact is an important building block in achieving a better relationship between
Government and the voluntary and community sector. We are fully committed to partnership
working with the sector and increasing their role in civil society and in the delivery of public
s e rvices. The Compact helps us to work better together, so that we can better meet the
needs of communities
Recommended from our members
On the lifetime and one-year views of reserve risk, with application to IFRS 17 and Solvency II risk margins
This paper brings together analytic and simulation-based approaches to reserve risk in general (P&C) insurance, applied to the traditional actuarial view of risk over the lifetime of the liabilities and to the one-year view of Solvency II. It also connects the lifetime and one-year views of risk. The framework of the model in Mack (1993) is used throughout, although the results have wider applicability.
The advantages of a simulation-based approach are highlighted, giving a full predictive distribution, which is used to estimate risk margins under Solvency II and risk adjustments under IFRS 17. We discuss methods for obtaining capital requirements in a cost-of-capital risk margin, and methods for estimating risk adjustments using risk measures applied to a simulated distribution of the outstanding liabilities over their lifetime
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