2,899 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
Validity proof of Lazard's method for CAD construction
In 1994 Lazard proposed an improved method for cylindrical algebraic
decomposition (CAD). The method comprised a simplified projection operation
together with a generalized cell lifting (that is, stack construction)
technique. For the proof of the method's validity Lazard introduced a new
notion of valuation of a multivariate polynomial at a point. However a gap in
one of the key supporting results for his proof was subsequently noticed. In
the present paper we provide a complete validity proof of Lazard's method. Our
proof is based on the classical parametrized version of Puiseux's theorem and
basic properties of Lazard's valuation. This result is significant because
Lazard's method can be applied to any finite family of polynomials, without any
assumption on the system of coordinates. It therefore has wider applicability
and may be more efficient than other projection and lifting schemes for CAD.Comment: 21 page
Ga-actions on affine cones
We give a criterion of existence of a unipotent group action on the affine
cone over a projective variety or, more generally, on the affine quasicone over
a variety which is projective over another affine variety.Comment: 16p. In a formula in the proof of Lemma 2.3, the direct sum was
replaced by the usual sum . The authors are grateful to Kevin Langlois for
mentioning this inaccurac
CAD Adjacency Computation Using Validated Numerics
We present an algorithm for computation of cell adjacencies for well-based
cylindrical algebraic decomposition. Cell adjacency information can be used to
compute topological operations e.g. closure, boundary, connected components,
and topological properties e.g. homology groups. Other applications include
visualization and path planning. Our algorithm determines cell adjacency
information using validated numerical methods similar to those used in CAD
construction, thus computing CAD with adjacency information in time comparable
to that of computing CAD without adjacency information. We report on
implementation of the algorithm and present empirical data.Comment: 20 page
Gluing Localized Mirror Functors
We develop a method of gluing the local mirrors and functors constructed from
immersed Lagrangians in the same deformation class. As a result, we obtain a
global mirror geometry and a canonical mirror functor. We apply the method to
construct the mirrors of punctured Riemann surfaces and show that our functor
derives homological mirror symmetry.Comment: 69 pages, 39 figures, comments are welcom
Hard hexagon partition function for complex fugacity
We study the analyticity of the partition function of the hard hexagon model
in the complex fugacity plane by computing zeros and transfer matrix
eigenvalues for large finite size systems. We find that the partition function
per site computed by Baxter in the thermodynamic limit for positive real values
of the fugacity is not sufficient to describe the analyticity in the full
complex fugacity plane. We also obtain a new algebraic equation for the low
density partition function per site.Comment: 49 pages, IoP styles files, lots of figures (png mostly) so using
PDFLaTeX. Some minor changes added to version 2 in response to referee
report
An Algorithm for Computing the Limit Points of the Quasi-component of a Regular Chain
For a regular chain , we propose an algorithm which computes the
(non-trivial) limit points of the quasi-component of , that is, the set
. Our procedure relies on Puiseux series expansions
and does not require to compute a system of generators of the saturated ideal
of . We focus on the case where this saturated ideal has dimension one and
we discuss extensions of this work in higher dimensions. We provide
experimental results illustrating the benefits of our algorithms
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