105 research outputs found
Polynomial Optimization with Applications to Stability Analysis and Control - Alternatives to Sum of Squares
In this paper, we explore the merits of various algorithms for polynomial
optimization problems, focusing on alternatives to sum of squares programming.
While we refer to advantages and disadvantages of Quantifier Elimination,
Reformulation Linear Techniques, Blossoming and Groebner basis methods, our
main focus is on algorithms defined by Polya's theorem, Bernstein's theorem and
Handelman's theorem. We first formulate polynomial optimization problems as
verifying the feasibility of semi-algebraic sets. Then, we discuss how Polya's
algorithm, Bernstein's algorithm and Handelman's algorithm reduce the
intractable problem of feasibility of semi-algebraic sets to linear and/or
semi-definite programming. We apply these algorithms to different problems in
robust stability analysis and stability of nonlinear dynamical systems. As one
contribution of this paper, we apply Polya's algorithm to the problem of
H_infinity control of systems with parametric uncertainty. Numerical examples
are provided to compare the accuracy of these algorithms with other polynomial
optimization algorithms in the literature.Comment: AIMS Journal of Discrete and Continuous Dynamical Systems - Series
Note on Integer Factoring Methods IV
This note continues the theoretical development of deterministic integer
factorization algorithms based on systems of polynomials equations. The main
result establishes a new deterministic time complexity bench mark in integer
factorization.Comment: 20 Pages, New Versio
Causal inference via algebraic geometry: feasibility tests for functional causal structures with two binary observed variables
We provide a scheme for inferring causal relations from uncontrolled
statistical data based on tools from computational algebraic geometry, in
particular, the computation of Groebner bases. We focus on causal structures
containing just two observed variables, each of which is binary. We consider
the consequences of imposing different restrictions on the number and
cardinality of latent variables and of assuming different functional
dependences of the observed variables on the latent ones (in particular, the
noise need not be additive). We provide an inductive scheme for classifying
functional causal structures into distinct observational equivalence classes.
For each observational equivalence class, we provide a procedure for deriving
constraints on the joint distribution that are necessary and sufficient
conditions for it to arise from a model in that class. We also demonstrate how
this sort of approach provides a means of determining which causal parameters
are identifiable and how to solve for these. Prospects for expanding the scope
of our scheme, in particular to the problem of quantum causal inference, are
also discussed.Comment: Accepted for publication in Journal of Causal Inference. Revised and
updated in response to referee feedback. 16+5 pages, 26+2 figures. Comments
welcom
Integrable quadratic Hamiltonians on so(4) and so(3,1)
We investigate a special class of quadratic Hamiltonians on so(4) and so(3,1)
and describe Hamiltonians that have additional polynomial integrals. One of the
main results is a new integrable case with an integral of sixth degree.Comment: 16 page
Analogue of Newton-Puiseux series for non-holonomic D-modules and factoring
We introduce a concept of a fractional-derivatives series and prove that any
linear partial differential equation in two independent variables has a
fractional-derivatives series solution with coefficients from a differentially
closed field of zero characteristic. The obtained results are extended from a
single equation to -modules having infinite-dimensional space of solutions
(i. e. non-holonomic -modules). As applications we design algorithms for
treating first-order factors of a linear partial differential operator, in
particular for finding all (right or left) first-order factors
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
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
Sparse implicitization by interpolation: Geometric computations using matrix representations
Based on the computation of a superset of the implicit support,
implicitization of a parametrically given hyper-surface is reduced to computing
the nullspace of a numeric matrix. Our approach exploits the sparseness of the
given parametric equations and of the implicit polynomial. In this work, we
study how this interpolation matrix can be used to reduce some key geometric
predicates on the hyper-surface to simple numerical operations on the matrix,
namely membership and sidedness for given query points. We illustrate our
results with examples based on our Maple implementation
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