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Award ID contains: 2212461

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  1. ; ; ; (, Numerical Algorithms)
    Abstract A standard question in real algebraic geometry is to compute the number of connected components of a real algebraic variety in affine space. This manuscript provides algorithms for computing the number of connected components, the Euler characteristic, and deciding the connectivity between two points for a smooth manifold arising as the complement of a real hypersurface of a real algebraic variety. When considering the complement of the set of singular points of a real algebraic variety, this yields an approach for determining smooth connectivity in a real algebraic variety. The method is based upon gradient ascent/descent paths on the real algebraic variety inspired by a method proposed by Hong, Rohal, Safey El Din, and Schost for complements of real hypersurfaces. Several examples are included to demonstrate the approach. 
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  2. ; (, Journal of Symbolic Computation)
    Free, publicly-accessible full text available July 1, 2027
  3. ; ; (, Journal of Symbolic Computation)
    Free, publicly-accessible full text available May 1, 2027
  4. ; ; ; (, Maple Transactions)
    Newton's method is a classical iterative approach for computing solutions to nonlinear equations. To overcome some of its drawbacks, one often considers a continuous adjoint form of Newton's method. This paper investigates the geometric structure of the trajectories produced by the continuous adjoint Newton's method for bivariate quadratics, a system of two quadratic polynomials in two variables, via eigenanalysis at its equilibrium points. The main ideas are illustrated using plots generated by a Maple program. 
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    Free, publicly-accessible full text available August 1, 2026
  5. ; ; (, Annals of Mathematics and Artificial Intelligence)
    Free, publicly-accessible full text available March 27, 2026
  6. ; (, Science China Mathematics)
    Full Text Available