We revisit the static coverage control problem for placement of vehicles with simple motion on the real line, under the assumption that the cost is a polynomial function of the locations of the vehicles. The main contribution of this paper is to demonstrate the use of tools from numerical algebraic geometry, in particular, a numerical polynomial homotopy continuation method that guarantees to f...

Periodic travelling waves (wavetrains) are an important solution type for many partial differential equations. In this paper I review the use of numerical continuation for studying these solutions. I discuss the calculation of the form and stability of a given periodic travelling wave, and the calculation of boundaries in a two-dimensional parameter plane for wave existence and stability. I als...

Polyhedral homotopy continuation is a promising approach to solving large-scale polynomial systems. However, the method faces a computational bottleneck in constructing the homotopy functions because we need to find all mixed cells for polynomial systems. The search for mixed cells can be formulated as a vertex enumeration problem for polyhedra. Among the search methods for solving the problem ...

Our Maple package PHCmaple came to existence in 2004 when it provided a convenient interface to the basic functionality of phc, a program which is a part of PHCpack and implements numeric algorithms for solving polynomial systems using polynomial homotopy continuation. Following the recent development of PHCpack the package has been extended with functions that deal with singular polynomial sys...

To decompose solution sets of polynomial systems into irreducible components, homotopy continuation methods generate the action of a natural monodromy group which partially classi es generic points onto their respective irreducible components. As illustrated by the performance on several test examples, this new method achieves a great increase in speed and accuracy, as well as improved numerica...

We begin with formulating the stationary equations of the Kuramoto model as a system of polynomial equations in a novel way. Then, based on an algebraic geometric root count, we give a prescription of computing an upper bound on the number of equilibria of the Kuramoto model for the most general case, i.e., defined on an arbitrary graph and having generic values of natural frequencies and inhom...

In this paper, we investigate a class of nonlinear eigenvalue problems resulting from quantum physics. We first prove that for any open set G , there exists an eigenfunction cannot be polynomial on which may reviewed as refinement the classic unique continuation property. Then apply non-polynomial behavior to show adaptive finite element approximations are convergent even if initial mesh is not...

Multi-homogeneous homotopy continuation method is one of the most efficient approaches in solving all isolated solutions of a polynomial system of equations. Finding the optimal partition of variables with the minimal multihomogeneous Bézout number is clearly an optimization problem. Multi-homogeneous continuation using the optimal partition of variables reduces the computational cost in path t...

The reduced Gassner representation is a multi-parameter representation of Pn, the pure braid group on n strings. Specializing the parameters t1, t2,...,tn to nonzero complex numbers x1,x2,...,xn gives a representation Gn(x1,...,xn): Pn GL( n 1 ) which is irreducible if and only if x1...xn 1.We find a sufficient condition that guarantees that the tensor product of an irreducible Gn(x1,...,xn)wit...

This is a continuation of our previous investigations on polynomials orthogonal with respect to the linear functional L : P → C, where L = ∫ 1 −1 p(x) dμ(x), dμ(x) = (1 − x2)λ−1/2 exp(iζx) dx, and P is a linear space of all algebraic polynomials. Here, we prove an extension of our previous existence theorem for rational λ ∈ (−1/2, 0], give some hypothesis on three-term recurrence coefficients, ...