659 research outputs found
Isolating the real roots of the piecewise algebraic variety
AbstractThe piecewise algebraic variety, as a set of the common zeros of multivariate splines, is a kind of generalization of the classical algebraic variety. In this paper, we present an algorithm for isolating the zeros of the zero-dimensional piecewise algebraic variety which is primarily based on the interval zeros of univariate interval polynomials. Numerical example illustrates that the proposed algorithm is flexible
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
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
A 1-parameter family of spherical CR uniformizations of the figure eight knot complement
We describe a simple fundamental domain for the holonomy group of the
boundary unipotent spherical CR uniformization of the figure eight knot
complement, and deduce that small deformations of that holonomy group (such
that the boundary holonomy remains parabolic) also give a uniformization of the
figure eight knot complement. Finally, we construct an explicit 1-parameter
family of deformations of the boundary unipotent holonomy group such that the
boundary holonomy is twist-parabolic. For small values of the twist of these
parabolic elements, this produces a 1-parameter family of pairwise
non-conjugate spherical CR uniformizations of the figure eight knot complement
Maximally inflected real rational curves
We introduce and begin the topological study of real rational plane curves,
all of whose inflection points are real. The existence of such curves is a
corollary of results in the real Schubert calculus, and their study has
consequences for the important Shapiro and Shapiro conjecture in the real
Schubert calculus. We establish restrictions on the number of real nodes of
such curves and construct curves realizing the extreme numbers of real nodes.
These constructions imply the existence of real solutions to some problems in
the Schubert calculus. We conclude with a discussion of maximally inflected
curves of low degree.Comment: Revised with minor corrections. 37 pages with 106 .eps figures. Over
250 additional pictures on accompanying web page (See
http://www.math.umass.edu/~sottile/pages/inflected/index.html
Algorithm for connectivity queries on real algebraic curves
We consider the problem of answering connectivity queries on a real algebraic
curve. The curve is given as the real trace of an algebraic curve, assumed to
be in generic position, and being defined by some rational parametrizations.
The query points are given by a zero-dimensional parametrization. We design an
algorithm which counts the number of connected components of the real curve
under study, and decides which query point lie in which connected component, in
time log-linear in , where is the maximum of the degrees and
coefficient bit-sizes of the polynomials given as input. This matches the
currently best-known bound for computing the topology of real plane curves. The
main novelty of this algorithm is the avoidance of the computation of the
complete topology of the curve.Comment: 10 pages, 2 figure
Existence of Solutions to the Bethe Ansatz Equations for the 1D Hubbard Model: Finite Lattice and Thermodynamic Limit
In this work, we present a proof of the existence of real and ordered
solutions to the generalized Bethe Ansatz equations for the one dimensional
Hubbard model on a finite lattice, with periodic boundary conditions. The
existence of a continuous set of solutions extending from any positive U to the
limit of large interaction is also shown. This continuity property, when
combined with the proof that the wavefunction obtained with the generalized
Bethe Ansatz is normalizable, is relevant to the question of whether or not the
solution gives us the ground state of the finite system, as suggested by Lieb
and Wu. Lastly, for the absolute ground state at half-filling, we show that the
solution converges to a distribution in the thermodynamic limit. This limit
distribution satisfies the integral equations that led to the well known
solution of the 1D Hubbard model.Comment: 18 page
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