Given a real valued function f (X, Y ), a box region B 0 ⊆ R 2 and ε > 0, we want to compute an ε-isotopic polygonal approximation to the restriction of the curve S = f −1 (0) = {p ∈ R 2 : f (p) = 0} to B 0 . We focus on subdivision algorithms because of their adaptive complexity and ease of implementation. Plantinga & Vegter gave a numerical subdivision algorithm that is exact when the curve S is bounded and non-singular. They used a computational model that relied only on function evaluation and interval arithmetic. We generalize their algorithm to any bounded (but possibly non-simply connected) region that does not contain singularities of S. With this generalization as a subroutine, we provide a method to detect isolated algebraic singularities and their branching degree. This appears to be the first complete purely numerical method to compute isotopic approximations of algebraic curves with isolated singularities.

A natural way to measure protein surface curvature is to generate the least squares fitted (LSF) sphere to a surface patch and use the radius as the curvature measure. While the concept is simple, the sphere-fitting problem is not trivial and known means of protein surface curvature measurement use alternative schemes that are arguably less straightforward to interpret. We have developed an approach to solve the LSF sphere problem by turning the sphere-fitting problem into a solvable plane-fitting problem using a transformation known as geometric inversion. The approach works on any arbitrary surface patch, and returns a radius of curvature that has direct physical interpretation. Additionally, it is flexible in its ability to find the curvature of an arbitrary surface patch, and the "resolution" can be adjusted to highlight atomic features or larger features such as peptide binding sites. We include examples of applying the method to visualization of peptide recognition pockets and protein conformational change, as well as a comparison with a commonly used solid-angle curvature method showing that the LSF method produces more pronounced curvature results. Proteins 2005; 61:1068 -1074.

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Subdivision-based algorithms recursively subdivide an input region until the smaller subregions can be processed. It is a challenge to analyze the complexity of such algorithms because the work they perform is not uniform across the input region. Continuous amortization was introduced in Burr et al. (2009) as a way to bound the complexity of subdivision-based algorithms. The main features of this new technique are that (1) the technique can be applied, uniformly, to a variety of subdivision-based algorithms, (2) the technique considers a function directly related to the subdivision-based algorithm under consideration, and (3) the output of the technique is often explicitly expressed in terms of the intrinsic complexity of the problem instance.In this paper, the theory of continuous amortization is generalized and applied in several directions. The theory is generalized (1) to allow the domain to be higher dimensional or an abstract measure space, (2) to allow more general subdivisions than bisections, and (3) to bound the value of general functions on the regions of the final partition. The theory is applied to seven examples of subdivision-based algorithms. These applications include (1) bounding the number of subdivisions performed by algorithms for isolating real and complex roots of polynomials, (2) bounding the bit-complexity of subdivision-based algorithms for isolating the real roots of polynomials, and (3) bounding the expected run-time of an algorithm for approximating a biased coin. In each of these applications, by using continuous amortization, we achieve or improve the best-known complexity bounds.Here, w(J) denotes the length of the interval J. 2 In previous work (Burr et al. (2009), Sharma and Yap (2012), and Burr and Krahmer (2012)), a local size bound was called a stopping function. That name, however, proved to be confusing.3 A local size bound is not logically equivalent to define a local size bound asbecause, by choosing intervals containing x in different positions, e.g., intervals extending to the left or right of x, it is possible to have a large interval where the stopping criterion is True and a smaller interval where the stopping criterion is False. However, one could use the sup-based formula to derive a lower bound on the complexity of a bisection algorithm. 5The local size bound can be seen visually in Figure 1. In this figure, the local size bound at x must be smaller than the widths of intervals J 1 , J 4 and J 5 because each of these intervals contains x, and the stopping criterion is False for these intervals. Notice that it is not important for x to be centered in the interval, see the interval J 1 . Also, note that J 2 does not impose a condition on the local size bound because C(J 2 ) = True. Additionally, J 3 does not put a restriction on the local size bound at x because it does not contain the point x. Finally, even though the interval J 2 both contains x and C(J 2 ) = True, the local size bound at x will be smaller than w(J 2 ) because J 4 is a smaller interval with C(J 4 ...

Let f be a univariate polynomial with real coefficients, f ∈ R [X]. Subdivision algorithms based on algebraic techniques (e.g., Sturm or Descartes methods) are widely used for isolating the real roots of f in a given interval. In this paper, we consider a simple subdivision algorithm whose primitives are purely numerical (e.g., function evaluation). The complexity of this algorithm is adaptive because the algorithm makes decisions based on local data. The complexity analysis of adaptive algorithms (and this algorithm in particular) is a new challenge for computer science. In this paper, we compute the size of the subdivision tree for the SqFreeEVAL algorithm.The SqFreeEVAL algorithm is an evaluation-based numerical algorithm which is well-known in several communities. The algorithm itself is simple, but prior attempts to compute its complexity have proven to be quite technical and have yielded sub-optimal results. Our main result is a simple O(d(L + ln d)) bound on the size of the subdivision tree for the SqFreeEVAL algorithm on the benchmark problem of isolating all real roots of an integer polynomial f of degree d and whose coefficients can be written with at most L bits.Our proof uses two amortization-based techniques: First, we use the algebraic amortization technique of the standard Mahler-Davenport root bounds to interpret the integral in terms of d and L. Second, we use a continuous amortization technique based on an integral to bound the size of the subdivision tree. This paper is the first to use the novel analysis technique of continuous amortization to derive state of the art complexity bounds.