5,749 research outputs found
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
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
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
An Urgent Appeal to the Government of Canada to Proclaim our Northern Identity
... Looming new developments of unprecedented scale now present for Northerners both significant opportunities and challenges, especially if long-term cultural vitality and a sustainable environment are to be maintained. Northern Canada is also a key archive and bellwether for global climate change, and understandably it has become increasingly the focus of foreign researchers, whose resources vastly exceed those of Canadians. In light of this northern transformation, this article revisits Canada's ongoing unpreparedness in northern science and technology ..., contrasts Canada's inactivity with proactive steps being taken by other countries with well-structured polar agendas, and recommends recently proposed solutions as a long-awaited remedy. ..
Design, development and test of shuttle/Centaur G-prime cryogenic tankage thermal protection systems
The thermal protection systems for the shuttle/Centaur would have had to provide fail-safe thermal protection during prelaunch, launch ascent, and on-orbit operations as well as during potential abort. The thermal protection systems selected used a helium-purged polyimide foam beneath three rediation shields for the liquid-hydrogen tank and radiation shields only for the liquid-oxygen tank (three shields on the tank sidewall and four on the aft bulkhead). A double-walled vacuum bulkhead separated the two tanks. The liquid-hydrogen tank had one 0.75-in-thick layer of foam on the forward bulkhead and two layers on the larger area sidewall. Full scale tests of the flight vehicle in a simulated shuttle cargo bay that was purged with gaseous nitrogen gave total prelaunch heating rates of 88,500 Btu/hr and 44,000 Btu/hr for the liquid-hydrogen and -oxygen tanks, respectively. Calorimeter tests on a representative sample of the liquid-hydrogen tank sidewall thermal protection system indicated that the measured unit heating rate would rapidly decrease from the prelaunch rate of approx 100 Btu/hr/sq ft to a desired rate of less than 1.3 Btu/hr/sq ft once on orbit
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
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
Statistical Physics of Self-Replication
Self-replication is a capacity common to every species of living thing, and
simple physical intuition dictates that such a process must invariably be
fueled by the production of entropy. Here, we undertake to make this intuition
rigorous and quantitative by deriving a lower bound for the amount of heat that
is produced during a process of self-replication in a system coupled to a
thermal bath. We find that the minimum value for the physically allowed rate of
heat production is determined by the growth rate, internal entropy, and
durability of the replicator, and we discuss the implications of this finding
for bacterial cell division, as well as for the pre-biotic emergence of
self-replicating nucleic acids.Comment: 4+ pages, 1 figur
Generalised Elliptic Functions
We consider multiply periodic functions, sometimes called Abelian functions,
defined with respect to the period matrices associated with classes of
algebraic curves. We realise them as generalisations of the Weierstras
P-function using two different approaches. These functions arise naturally as
solutions to some of the important equations of mathematical physics and their
differential equations, addition formulae, and applications have all been
recent topics of study.
The first approach discussed sees the functions defined as logarithmic
derivatives of the sigma-function, a modified Riemann theta-function. We can
make use of known properties of the sigma function to derive power series
expansions and in turn the properties mentioned above. This approach has been
extended to a wide range of non hyperelliptic and higher genus curves and an
overview of recent results is given.
The second approach defines the functions algebraically, after first
modifying the curve into its equivariant form. This approach allows the use of
representation theory to derive a range of results at lower computational cost.
We discuss the development of this theory for hyperelliptic curves and how it
may be extended in the future.Comment: 16 page
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