2,321 research outputs found
Solving polynomial systems via symbolic-numeric reduction to geometric involutive form
AbstractWe briefly survey several existing methods for solving polynomial systems with inexact coefficients, then introduce our new symbolic-numeric method which is based on the geometric (Jet) theory of partial differential equations. The method is stable and robust. Numerical experiments illustrate the performance of the new method
Real solution of DAE and PDAE System
General systems of differential equations don\u27t have restrictions on the number or type of equations. For example, they can be over or under-determined, and also contain algebraic constraints (e.g. algebraic equations such as in Differential-Algebraic equations (DAE) and Partial differential algebraic equations (PDAE). Increasingly such general systems arise from mathematical modeling of engineering and science problems such as in multibody mechanics, electrical circuit design, optimal control, chemical kinetics and chemical control systems. In most applications, only real solutions are of interest, rather than complex-valued solutions. Much progress has been made in exact differential elimination methods, which enable characterization of all hidden constraints of such general systems, by differentiating them until missing constraints are obtained by elimination. A major problem in these approaches is related to the exploding size of the differentiated systems. Due to the importance of these problems, we outline a Symbolic-Numeric Method to find at least one real point on each connected component of the solutions set of such systems
Calcium supplements with or without vitamin D and risk of cardiovascular events : reanalysis of the Women's Health Initiative limited access dataset and meta-analysis
Peer reviewedPublisher PD
Structural Analysis by Modified Signature Matrix for Integro-differential-algebraic Equations
Integro-differential-algebraic equations (IDAE)s are widely used in
applications of engineering and analysis. When there are hidden constraints in
an IDAE, structural analysis is necessary. But if derivatives of dependent
variables appear in their integrals, the existing definition of the signature
matrix for an IDAE cannot be satisfied. Moreover, if an IDAE has a singular
Jacobian matrix after structural analysis by the Sigma-method, improved
structural analysis methods are proposed to regularize it. However, the optimal
value of an IDAE may be negative which can not ensure the termination of the
regularization. Furthermore, overestimation of the signature matrix may also
lead to failure of its structural analysis.
In this paper, firstly, we redefine the signature matrix and introduce a
definition of the degree of freedom for IDAEs. Thus, the termination of
improved structural analysis methods can be guaranteed. Secondly, the detection
method by points is proposed to deal with the problem of overestimation of
signature matrix. Thirdly, the embedding method has proved to suitable for
structural unamenable IDAEs, including those types that arise from symbolic
cancellation and numerical degeneration. Finally, the global numerical method
is applied to an example of two-stage drive system which can help to find all
solutions for IDAEs by witness points. Hopefully, through the example of
pendulum curtain, the approach for IDAEs proposed in this paper can be applied
to integro-partial-differential-algebraic equations (IPDAE)s.Comment: 33 pages, 4 figures, conferenc
On approximate triangular decompositions in dimension zero
Abstract. Triangular decompositions for systems of polynomial equations with n variables, with exact coefficients are well-developed theoretically and in terms of implemented algorithms in computer algebra systems. However there is much less research about triangular decompositions for systems with approximate coefficients. In this paper we discuss the zero-dimensional case, of systems having finitely many roots. Our methods depend on having approximations for all the roots, and these are provided by the homotopy continuation methods of Sommese, Verschelde and Wampler. We introduce approximate equiprojectable decompositions for such systems, which represent a generalization of the recently developed analogous concept for exact systems. We demonstrate experimentally the favourable computational features of this new approach, and give a statistical analysis of its error. Keywords. Symbolic-numeric computations, Triangular decompositions, Dimension zero, Polynomial system solving
Calcium supplements and cancer risk : a meta-analysis of randomised controlled trials
Peer reviewedPublisher PD
Effect of calcium supplements on risk of myocardial infarction and cardiovascular events : meta-analysis
Peer reviewedPublisher PD
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