1,039 research outputs found
Sparse implicitization by interpolation: Characterizing non-exactness and an application to computing discriminants
We revisit implicitization by interpolation in order to examine its properties in the context of sparse elimination theory. Based on the computation of a superset of the implicit support, implicitization is reduced to computing the nullspace of a numeric matrix. The approach is applicable to polynomial and rational parameterizations of curves and (hyper)surfaces of any dimension, including the case of parameterizations with base points.
Our support prediction is based on sparse (or toric) resultant theory, in order to exploit the sparsity of the input and the output. Our method may yield a multiple of the implicit equation: we characterize and quantify this situation by relating the nullspace dimension to the predicted support and its geometry. In this case, we obtain more than one multiples of the implicit equation; the latter can be obtained via multivariate polynomial gcd (or factoring).
All of the above techniques extend to the case of approximate computation, thus yielding a method of sparse approximate implicitization, which is important in tackling larger problems. We discuss our publicly available Maple implementation through several examples, including the benchmark of bicubic surface.
For a novel application, we focus on computing the discriminant of a multivariate polynomial, which characterizes the existence of multiple roots and generalizes the resultant of a polynomial system.
This yields an efficient, output-sensitive algorithm for
computing the discriminant polynomial
Computing Dynamic Output Feedback Laws
The pole placement problem asks to find laws to feed the output of a plant
governed by a linear system of differential equations back to the input of the
plant so that the resulting closed-loop system has a desired set of
eigenvalues. Converting this problem into a question of enumerative geometry,
efficient numerical homotopy algorithms to solve this problem for general
Multi-Input-Multi-Output (MIMO) systems have been proposed recently. While
dynamic feedback laws offer a wider range of use, the realization of the output
of the numerical homotopies as a machine to control the plant in the time
domain has not been addressed before. In this paper we present symbolic-numeric
algorithms to turn the solution to the question of enumerative geometry into a
useful control feedback machine. We report on numerical experiments with our
publicly available software and illustrate its application on various control
problems from the literature.Comment: 20 pages, 3 figures; the software described in this paper is publicly
available via http://www.math.uic.edu/~jan/download.htm
Statistical description of the black hole degeneracy spectrum
We use mathematical methods based on generating functions to study the
statistical properties of the black hole degeneracy spectrum in loop quantum
gravity. In particular we will study the persistence of the observed effective
quantization of the entropy as a function of the horizon area. We will show
that this quantization disappears as the area increases despite the existence
of black hole configurations with a large degeneracy. The methods that we
describe here can be adapted to the study of the statistical properties of the
black hole degeneracy spectrum for all the existing proposals to define black
hole entropy in loop quantum gravity.Comment: 41 pages, 12 figure
Symbolic-numeric algorithms for univariate polynomials
Thesis (Ph. D. in Science)--University of Tsukuba, (B), no. 2485, 2010.3.25 Includes bibliographical referencesNote to the re-typeset version: This is re-typeset version of the original dissertation. While I have maintained the original contents without changing any words and/or formulas in the main body, I have added the following information: 1. Copyright notice of corresponding articles in each chapter; 2. Digital Object Identifiers (DOI) or URLs of references as many as possible.Please note that the number of pages is slightly increased in the present edition from that of the original edition, possibly by changes of page style parameters.200
Consistent Energy-based Atomistic/Continuum Coupling for Two-body Potentials in Three Dimensions
Very few works exist to date on development of a consistent energy-based
coupling of atomistic and continuum models of materials in more than one
dimension. The difficulty in constructing such a coupling consists in defining
a coupled energy whose minimizers are free from uncontrollable errors on the
atomistic/continuum interface. In this paper a consistent coupling in three
dimensions is proposed. The main achievement of this work is to identify and
efficiently treat a modified Cauchy-Born continuum model which can be coupled
to the exact atomistic model. The convergence and stability of the method is
confirmed with numerical tests.Comment: 29 pages, 1 Matlab code. Typos corrected, exposition improve
The computational complexity of the Chow form
We present a bounded probability algorithm for the computation of the Chow
forms of the equidimensional components of an algebraic variety. Its complexity
is polynomial in the length and in the geometric degree of the input equation
system defining the variety. In particular, it provides an alternative
algorithm for the equidimensional decomposition of a variety.
As an application we obtain an algorithm for the computation of a subclass of
sparse resultants, whose complexity is polynomial in the dimension and the
volume of the input set of exponents. As a further application, we derive an
algorithm for the computation of the (unique) solution of a generic
over-determined equation system.Comment: 60 pages, Latex2
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