454 research outputs found
Cylindrical Algebraic Sub-Decompositions
Cylindrical algebraic decompositions (CADs) are a key tool in real algebraic
geometry, used primarily for eliminating quantifiers over the reals and
studying semi-algebraic sets. In this paper we introduce cylindrical algebraic
sub-decompositions (sub-CADs), which are subsets of CADs containing all the
information needed to specify a solution for a given problem.
We define two new types of sub-CAD: variety sub-CADs which are those cells in
a CAD lying on a designated variety; and layered sub-CADs which have only those
cells of dimension higher than a specified value. We present algorithms to
produce these and describe how the two approaches may be combined with each
other and the recent theory of truth-table invariant CAD.
We give a complexity analysis showing that these techniques can offer
substantial theoretical savings, which is supported by experimentation using an
implementation in Maple.Comment: 26 page
Using the distribution of cells by dimension in a cylindrical algebraic decomposition
We investigate the distribution of cells by dimension in cylindrical
algebraic decompositions (CADs). We find that they follow a standard
distribution which seems largely independent of the underlying problem or CAD
algorithm used. Rather, the distribution is inherent to the cylindrical
structure and determined mostly by the number of variables.
This insight is then combined with an algorithm that produces only
full-dimensional cells to give an accurate method of predicting the number of
cells in a complete CAD. Since constructing only full-dimensional cells is
relatively inexpensive (involving no costly algebraic number calculations) this
leads to heuristics for helping with various questions of problem formulation
for CAD, such as choosing an optimal variable ordering. Our experiments
demonstrate that this approach can be highly effective.Comment: 8 page
Computing the Rank and a Small Nullspace Basis of a Polynomial Matrix
We reduce the problem of computing the rank and a nullspace basis of a
univariate polynomial matrix to polynomial matrix multiplication. For an input
n x n matrix of degree d over a field K we give a rank and nullspace algorithm
using about the same number of operations as for multiplying two matrices of
dimension n and degree d. If the latter multiplication is done in
MM(n,d)=softO(n^omega d) operations, with omega the exponent of matrix
multiplication over K, then the algorithm uses softO(MM(n,d)) operations in K.
The softO notation indicates some missing logarithmic factors. The method is
randomized with Las Vegas certification. We achieve our results in part through
a combination of matrix Hensel high-order lifting and matrix minimal fraction
reconstruction, and through the computation of minimal or small degree vectors
in the nullspace seen as a K[x]-moduleComment: Research Report LIP RR2005-03, January 200
Numerical Algebraic Geometry: A New Perspective on String and Gauge Theories
The interplay rich between algebraic geometry and string and gauge theories
has recently been immensely aided by advances in computational algebra.
However, these symbolic (Gr\"{o}bner) methods are severely limited by
algorithmic issues such as exponential space complexity and being highly
sequential. In this paper, we introduce a novel paradigm of numerical algebraic
geometry which in a plethora of situations overcomes these short-comings. Its
so-called 'embarrassing parallelizability' allows us to solve many problems and
extract physical information which elude the symbolic methods. We describe the
method and then use it to solve various problems arising from physics which
could not be otherwise solved.Comment: 36 page
Structural synthesis of a stiffened cylinder
Structural synthesis of cylindrical shell reinforced with rectangular cross section stiffener
Algorithmic Contributions to the Theory of Regular Chains
Regular chains, introduced about twenty years ago, have emerged as one of the major
tools for solving polynomial systems symbolically. In this thesis, we focus on different
algorithmic aspects of the theory of regular chains, from theoretical questions to high-
performance implementation issues.
The inclusion test for saturated ideals is a fundamental problem in this theory.
By studying the primitivity of regular chains, we show that a regular chain generates
its saturated ideal if and only if it is primitive. As a result, a family of inclusion tests
can be detected very efficiently.
The algorithm to compute the regular GCDs of two polynomials modulo a regular
chain is one of the key routines in the various triangular decomposition algorithms. By
revisiting relations between subresultants and GCDs, we proposed a novel bottom-up
algorithm for this task, which improves the previous algorithm in a significant manner
and creates opportunities for parallel execution.
This thesis also discusses the accelerations towards fast Fourier transform (FFT)
over finite fields and FFT based subresultant chain constructions in the context of
massively parallel GPU architectures, which speedup our algorithms by several orders
of magnitude
Time-Domain Minimization of Voltage and Current Total Harmonic Distortion for a Single-Phase Multilevel Inverter with a Staircase Modulation
This paper presents the optimization technique for minimizing the voltage and current total harmonic distortion (THD) in a single-phase multilevel inverter controlled by staircase modulation. The previously reported research generally considered the optimal THD problem in the frequency domain, taking into account a limited harmonic number. The novelty of the suggested approach is that voltage and current minimal THD problems are being formulated in the time domain as constrained optimization ones, making it possible to determine the optimal switching angles. In this way, all switching harmonics can be considered. The target function expression becomes very compact and existing efficient solvers for this kind of optimization problems can find a solution
in negligible processor time. Current THD is understood as voltage frequency weighted THD that assumes pure inductive load—this approximation is practically accurate for inductively dominant RL-loads. In this study, the optimal switching angles and respective minimal THD values were obtained for different inverter level counts and overall fundamental voltage magnitude (modulation index) dynamic range. Developments are easily modified to cover multilevel inverter grid-connected applications. The results have been verified by experimental tests
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