1,404 research outputs found
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
Using the distribution of cells by dimension in a cylindrical algebraic decomposition
We investigate the distribution of cells by dimension in cylindrical
algebraic decompositions (CADs). We find that they follow a standard
distribution which seems largely independent of the underlying problem or CAD
algorithm used. Rather, the distribution is inherent to the cylindrical
structure and determined mostly by the number of variables.
This insight is then combined with an algorithm that produces only
full-dimensional cells to give an accurate method of predicting the number of
cells in a complete CAD. Since constructing only full-dimensional cells is
relatively inexpensive (involving no costly algebraic number calculations) this
leads to heuristics for helping with various questions of problem formulation
for CAD, such as choosing an optimal variable ordering. Our experiments
demonstrate that this approach can be highly effective.Comment: 8 page
Program Verification in the presence of complex numbers, functions with branch cuts etc
In considering the reliability of numerical programs, it is normal to "limit
our study to the semantics dealing with numerical precision" (Martel, 2005). On
the other hand, there is a great deal of work on the reliability of programs
that essentially ignores the numerics. The thesis of this paper is that there
is a class of problems that fall between these two, which could be described as
"does the low-level arithmetic implement the high-level mathematics". Many of
these problems arise because mathematics, particularly the mathematics of the
complex numbers, is more difficult than expected: for example the complex
function log is not continuous, writing down a program to compute an inverse
function is more complicated than just solving an equation, and many algebraic
simplification rules are not universally valid.
The good news is that these problems are theoretically capable of being
solved, and are practically close to being solved, but not yet solved, in
several real-world examples. However, there is still a long way to go before
implementations match the theoretical possibilities
Choosing a variable ordering for truth-table invariant cylindrical algebraic decomposition by incremental triangular decomposition
Cylindrical algebraic decomposition (CAD) is a key tool for solving problems
in real algebraic geometry and beyond. In recent years a new approach has been
developed, where regular chains technology is used to first build a
decomposition in complex space. We consider the latest variant of this which
builds the complex decomposition incrementally by polynomial and produces CADs
on whose cells a sequence of formulae are truth-invariant. Like all CAD
algorithms the user must provide a variable ordering which can have a profound
impact on the tractability of a problem. We evaluate existing heuristics to
help with the choice for this algorithm, suggest improvements and then derive a
new heuristic more closely aligned with the mechanics of the new algorithm
A "Piano Movers" Problem Reformulated
It has long been known that cylindrical algebraic decompositions (CADs) can
in theory be used for robot motion planning. However, in practice even the
simplest examples can be too complicated to tackle. We consider in detail a
"Piano Mover's Problem" which considers moving an infinitesimally thin piano
(or ladder) through a right-angled corridor.
Producing a CAD for the original formulation of this problem is still
infeasible after 25 years of improvements in both CAD theory and computer
hardware. We review some alternative formulations in the literature which use
differing levels of geometric analysis before input to a CAD algorithm. Simpler
formulations allow CAD to easily address the question of the existence of a
path. We provide a new formulation for which both a CAD can be constructed and
from which an actual path could be determined if one exists, and analyse the
CADs produced using this approach for variations of the problem.
This emphasises the importance of the precise formulation of such problems
for CAD. We analyse the formulations and their CADs considering a variety of
heuristics and general criteria, leading to conclusions about tackling other
problems of this form.Comment: 8 pages. Copyright IEEE 201
Seeding Treatments to Enhance Seedling Performance of the Bulrushes Bolboschoenus Maritimus, Schoenoplectus Acutus and S. Americanus in Wetland Restorations
A major goal in restoration is to reestablish native plant communities. There are several ways to reestablish species, but for large areas the most logistically feasible approach is to sow seed of desirable species. However, most wetland seeds are buoyant and are extremely difficult to establish in designated areas before floating away. In upland areas, tackifiers have been used to stabilize hill slopes from erosion and to keep seeds in place. The tackifier works as an adhesive that binds the seeds to the soil. However, the use of a tackifier has not been widely employed in wetland restorations, and prior to its broad implementation into wetland restoration practice, it is important to determine if tackifiers will hold up in wetland conditions. In greenhouse studies, we tested the effectiveness of different tackifier types and concentrations on Bolboschoenus maritimus seedling emergence, the influence of soil moisture and flooding on the duration of tackifier effectiveness, the effect of a mulch addition on tackifier effectiveness (Bolboschoenus maritimus, Schoenoplectus acutus and S. americanus), the effectiveness of pre-germination in enhancing Bolboschoenus maritimus seedling emergence using a tackifier, and the effectiveness of tackifier over time. We concluded that the use of a tackifier was effective at keeping seeds from washing away for at least 15 days, a mulch addition did not enhance tackifier effectiveness, and pre-germination did not benefit B. maritimus seedling emergence. The results from this study provide strong evidence that the use of a tackifier could be an effective solution to establish bulrush species in designated areas in wetland restorations
A comparison of three heuristics to choose the variable ordering for CAD
Cylindrical algebraic decomposition (CAD) is a key tool for problems in real
algebraic geometry and beyond. When using CAD there is often a choice over the
variable ordering to use, with some problems infeasible in one ordering but
simple in another. Here we discuss a recent experiment comparing three
heuristics for making this choice on thousands of examples
CMB anisotropy from spatial correlations of clusters of galaxies
The SZ effect from clusters of galaxies is a dominant source of secondary CMB
anisotropy in the low-redshift universe. We present analytic predictions for
the CMB power spectrum from massive halos arising from the SZ effect. Since
halos are discrete, the power spectrum consists of a Poisson and a correlation
term. The latter is always smaller than the former, which is dominated by
nearby bright rich clusters. In practice however, those bright clusters are
easy to indentify and can thus be subtracted from the map. After this
subtraction, the correlation term dominates degree-scale fluctuations over the
Poisson term, as the main contribution to the correlation term comes from
distant clusters. We find that the correlation term is detectable by Planck
experiment. Since the degree scale spectrum is quite insensitive to the highly
uncertain core structures of halos, our predictions are robust on these scales.
Measuring the correlation term on degree scales thus cleanly probes the
clustering of distant halos. This has not been measured yet, mainly because
optical and X-ray surveys are not sufficiently sensitive to include such
distant clusters and groups. Our analytic predictions are also compared to
adiabatic hydrodynamic simulations. The agreement is remarkably good, down to
ten arcminutes scales, indicating that our predictions are robust for the
Planck experiment. Below ten arcminute scales, where the details of the core
structure dominates the power spectrum, our analytic and simulated predictions
might fail. In the near future, interferometer and bolometer array experiments
will measure the SZ power spectrum down to arcminutes scales, and yield new
insight into the physics of the intrahalo medium.Comment: 9 pages, 4 figures. submitted to Proceedings of the 9th Marcel
Grossmann meetin
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