4,907 research outputs found
Bounding the number of stable homotopy types of a parametrized family of semi-algebraic sets defined by quadratic inequalities
We prove a nearly optimal bound on the number of stable homotopy types
occurring in a k-parameter semi-algebraic family of sets in , each
defined in terms of m quadratic inequalities. Our bound is exponential in k and
m, but polynomial in . More precisely, we prove the following. Let
be a real closed field and let with . Let be a
semi-algebraic set, defined by a Boolean formula without negations, whose atoms
are of the form, . Let be the projection on the last k co-ordinates. Then, the number of
stable homotopy types amongst the fibers S_{\x} = \pi^{-1}(\x) \cap S is
bounded by Comment: 27 pages, 1 figur
Approximation of definable sets by compact families, and upper bounds on homotopy and homology
We prove new upper bounds on homotopy and homology groups of o-minimal sets
in terms of their approximations by compact o-minimal sets. In particular, we
improve the known upper bounds on Betti numbers of semialgebraic sets defined
by quantifier-free formulae, and obtain for the first time a singly exponential
bound on Betti numbers of sub-Pfaffian sets.Comment: 20 pages, 2 figure
Algorithmic Semi-algebraic Geometry and Topology -- Recent Progress and Open Problems
We give a survey of algorithms for computing topological invariants of
semi-algebraic sets with special emphasis on the more recent developments in
designing algorithms for computing the Betti numbers of semi-algebraic sets.
Aside from describing these results, we discuss briefly the background as well
as the importance of these problems, and also describe the main tools from
algorithmic semi-algebraic geometry, as well as algebraic topology, which make
these advances possible. We end with a list of open problems.Comment: Survey article, 74 pages, 15 figures. Final revision. This version
will appear in the AMS Contemporary Math. Series: Proceedings of the Summer
Research Conference on Discrete and Computational Geometry, Snowbird, Utah
(June, 2006). J.E. Goodman, J. Pach, R. Pollack Ed
Cylindrical Algebraic Decomposition Using Local Projections
We present an algorithm which computes a cylindrical algebraic decomposition
of a semialgebraic set using projection sets computed for each cell separately.
Such local projection sets can be significantly smaller than the global
projection set used by the Cylindrical Algebraic Decomposition (CAD) algorithm.
This leads to reduction in the number of cells the algorithm needs to
construct. We give an empirical comparison of our algorithm and the classical
CAD algorithm
Certification of Real Inequalities -- Templates and Sums of Squares
We consider the problem of certifying lower bounds for real-valued
multivariate transcendental functions. The functions we are dealing with are
nonlinear and involve semialgebraic operations as well as some transcendental
functions like , , , etc. Our general framework is to use
different approximation methods to relax the original problem into polynomial
optimization problems, which we solve by sparse sums of squares relaxations. In
particular, we combine the ideas of the maxplus estimators (originally
introduced in optimal control) and of the linear templates (originally
introduced in static analysis by abstract interpretation). The nonlinear
templates control the complexity of the semialgebraic relaxations at the price
of coarsening the maxplus approximations. In that way, we arrive at a new -
template based - certified global optimization method, which exploits both the
precision of sums of squares relaxations and the scalability of abstraction
methods. We analyze the performance of the method on problems from the global
optimization literature, as well as medium-size inequalities issued from the
Flyspeck project.Comment: 27 pages, 3 figures, 4 table
Algorithmic and topological aspects of semi-algebraic sets defined by quadratic polynomial
In this thesis, we consider semi-algebraic sets over a real closed field
defined by quadratic polynomials. Semi-algebraic sets of are defined as
the smallest family of sets in that contains the algebraic sets as well
as the sets defined by polynomial inequalities, and which is also closed under
the boolean operations (complementation, finite unions and finite
intersections). We prove new bounds on the Betti numbers as well as on the
number of different stable homotopy types of certain fibers of semi-algebraic
sets over a real closed field defined by quadratic polynomials, in terms of
the parameters of the system of polynomials defining them, which improve the
known results. We conclude the thesis with presenting two new algorithms along
with their implementations. The first algorithm computes the number of
connected components and the first Betti number of a semi-algebraic set defined
by compact objects in which are simply connected. This algorithm
improves the well-know method using a triangulation of the semi-algebraic set.
Moreover, the algorithm has been efficiently implemented which was not possible
before. The second algorithm computes efficiently the real intersection of
three quadratic surfaces in using a semi-numerical approach.Comment: PhD thesis, final version, 109 pages, 9 figure
Matrix Convex Hulls of Free Semialgebraic Sets
This article resides in the realm of the noncommutative (free) analog of real
algebraic geometry - the study of polynomial inequalities and equations over
the real numbers - with a focus on matrix convex sets and their projections
. A free semialgebraic set which is convex as well as bounded and open
can be represented as the solution set of a Linear Matrix Inequality (LMI), a
result which suggests that convex free semialgebraic sets are rare. Further,
Tarski's transfer principle fails in the free setting: The projection of a free
convex semialgebraic set need not be free semialgebraic. Both of these results,
and the importance of convex approximations in the optimization community,
provide impetus and motivation for the study of the free (matrix) convex hull
of free semialgebraic sets.
This article presents the construction of a sequence of LMI domains
in increasingly many variables whose projections are
successively finer outer approximations of the matrix convex hull of a free
semialgebraic set . It is based on free analogs of
moments and Hankel matrices. Such an approximation scheme is possibly the best
that can be done in general. Indeed, natural noncommutative transcriptions of
formulas for certain well known classical (commutative) convex hulls does not
produce the convex hulls in the free case. This failure is illustrated on one
of the simplest free nonconvex .
A basic question is which free sets are the projection of a free
semialgebraic set ? Techniques and results of this paper bear upon this
question which is open even for convex sets.Comment: 41 pages; includes table of contents; supplementary material (a
Mathematica notebook) can be found at
http://www.math.auckland.ac.nz/~igorklep/publ.htm
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