8,517 research outputs found
Polar Varieties, Real Equation Solving and Data-Structures: The hypersurface case
In this paper we apply for the first time a new method for multivariate
equation solving which was developed in \cite{gh1}, \cite{gh2}, \cite{gh3} for
complex root determination to the {\em real} case. Our main result concerns the
problem of finding at least one representative point for each connected
component of a real compact and smooth hypersurface. The basic algorithm of
\cite{gh1}, \cite{gh2}, \cite{gh3} yields a new method for symbolically solving
zero-dimensional polynomial equation systems over the complex numbers. One
feature of central importance of this algorithm is the use of a
problem--adapted data type represented by the data structures arithmetic
network and straight-line program (arithmetic circuit). The algorithm finds the
complex solutions of any affine zero-dimensional equation system in non-uniform
sequential time that is {\em polynomial} in the length of the input (given in
straight--line program representation) and an adequately defined {\em geometric
degree of the equation system}. Replacing the notion of geometric degree of the
given polynomial equation system by a suitably defined {\em real (or complex)
degree} of certain polar varieties associated to the input equation of the real
hypersurface under consideration, we are able to find for each connected
component of the hypersurface a representative point (this point will be given
in a suitable encoding). The input equation is supposed to be given by a
straight-line program and the (sequential time) complexity of the algorithm is
polynomial in the input length and the degree of the polar varieties mentioned
above.Comment: Late
Polar Varieties and Efficient Real Elimination
Let be a smooth and compact real variety given by a reduced regular
sequence of polynomials . This paper is devoted to the
algorithmic problem of finding {\em efficiently} a representative point for
each connected component of . For this purpose we exhibit explicit
polynomial equations that describe the generic polar varieties of . This
leads to a procedure which solves our algorithmic problem in time that is
polynomial in the (extrinsic) description length of the input equations and in a suitably introduced, intrinsic geometric parameter, called
the {\em degree} of the real interpretation of the given equation system .Comment: 32 page
Polar Varieties and Efficient Real Equation Solving: The Hypersurface Case
The objective of this paper is to show how the recently proposed method by
Giusti, Heintz, Morais, Morgenstern, Pardo \cite{gihemorpar} can be applied to
a case of real polynomial equation solving. Our main result concerns the
problem of finding one representative point for each connected component of a
real bounded smooth hypersurface. The algorithm in \cite{gihemorpar} yields a
method for symbolically solving a zero-dimensional polynomial equation system
in the affine (and toric) case. Its main feature is the use of adapted data
structure: Arithmetical networks and straight-line programs. The algorithm
solves any affine zero-dimensional equation system in non-uniform sequential
time that is polynomial in the length of the input description and an
adequately defined {\em affine degree} of the equation system. Replacing the
affine degree of the equation system by a suitably defined {\em real degree} of
certain polar varieties associated to the input equation, which describes the
hypersurface under consideration, and using straight-line program codification
of the input and intermediate results, we obtain a method for the problem
introduced above that is polynomial in the input length and the real degree.Comment: Late
Pretreatment tumour-antigen Ta-4 in serum of patients with squamous cell carcinoma of the uterine cervix.
Using Elimination Theory to construct Rigid Matrices
The rigidity of a matrix A for target rank r is the minimum number of entries
of A that must be changed to ensure that the rank of the altered matrix is at
most r. Since its introduction by Valiant (1977), rigidity and similar
rank-robustness functions of matrices have found numerous applications in
circuit complexity, communication complexity, and learning complexity. Almost
all nxn matrices over an infinite field have a rigidity of (n-r)^2. It is a
long-standing open question to construct infinite families of explicit matrices
even with superlinear rigidity when r = Omega(n).
In this paper, we construct an infinite family of complex matrices with the
largest possible, i.e., (n-r)^2, rigidity. The entries of an n x n matrix in
this family are distinct primitive roots of unity of orders roughly exp(n^2 log
n). To the best of our knowledge, this is the first family of concrete (but not
entirely explicit) matrices having maximal rigidity and a succinct algebraic
description.
Our construction is based on elimination theory of polynomial ideals. In
particular, we use results on the existence of polynomials in elimination
ideals with effective degree upper bounds (effective Nullstellensatz). Using
elementary algebraic geometry, we prove that the dimension of the affine
variety of matrices of rigidity at most k is exactly n^2-(n-r)^2+k. Finally, we
use elimination theory to examine whether the rigidity function is
semi-continuous.Comment: 25 Pages, minor typos correcte
Serendipitous discovery of a projected pair of QSOs separated by 4.5 arcsec on the sky
We present the serendipitous discovery of a projected pair of quasi-stellar
objects (QSOs) with an angular separation of arcsec. The
redshifts of the two QSOs are widely different: one, our programme target, is a
QSO with a spectrum consistent with being a narrow line Seyfert 1 AGN at
. For this target we detect Lyman-, \ion{C}{4}, and
\ion{C}{3]}. The other QSO, which by chance was included on the spectroscopic
slit, is a Type 1 QSO at a redshift of , for which we detect
\ion{C}{4}, \ion{C}{3]} and \ion{Mg}{2}. We compare this system to previously
detected projected QSO pairs and find that only about a dozen previously known
pairs have smaller angular separation.Comment: 4 pages, 3 figures. Accepted for publication in A
Determining the fraction of reddened quasars in COSMOS with multiple selection techniques from X-ray to radio wavelengths
The sub-population of quasars reddened by intrinsic or intervening clouds of
dust are known to be underrepresented in optical quasar surveys. By defining a
complete parent sample of the brightest and spatially unresolved quasars in the
COSMOS field, we quantify to which extent this sub-population is fundamental to
our understanding of the true population of quasars. By using the available
multiwavelength data of various surveys in the COSMOS field, we built a parent
sample of 33 quasars brighter than mag, identified by reliable X-ray to
radio wavelength selection techniques. Spectroscopic follow-up with the
NOT/ALFOSC was carried out for four candidate quasars that had not been
targeted previously to obtain a 100\% redshift completeness of the sample. The
population of high quasars (HAQs), a specific sub-population of quasars
selected from optical/near-infrared photometry, is found to contribute
of the parent sample. The full population of bright spatially
unresolved quasars represented by our parent sample consists of
reddened quasars defined by having , and
of the sample having assuming the extinction
curve of the Small Magellanic Cloud. We show that the HAQ selection works well
for selecting reddened quasars, but some are missed because their optical
spectra are too blue to pass the color cut in the HAQ selection. This is
either due to a low degree of dust reddening or anomalous spectra. We find that
the fraction of quasars with contributing light from the host galaxy is most
dominant at . At higher redshifts the population of spatially
unresolved quasars selected by our parent sample is found to be representative
of the full population at mag. This work quantifies the bias against
reddened quasars in studies that are based solely on optical surveys.Comment: 22 pages, 10 figures, accepted for publication in A&A. The ArXiv
abstract has been shortened for it to be printabl
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