Problems and Techniques

Abstract

With major elections in many countries during 2004, what could be more timely than the two papers in this SIAM Review issue about preferential voting---the Survey and Review paper by Roberto Serrano and our first Problems and Techniques article, "Single Transferable Votes with Tax Cuts," by Eivind Stensholt? In the Problems and Techniques paper, the author begins by explaining the philosophy and strategy of different ways to tally preferential votes, giving sometimes surprising illustrations of their strengths and weaknesses. Even using the "single transferable vote" (STV) systems described in the paper, which are designed to avoid certain defects of other systems, there will always be collections of individual preference relations that allow election manipulation---for example, under some conditions a voter can actually hurt candidate x by giving x the top rank. A further wrinkle is learning how a "tax cut" (not income tax...) can be included to address the "free ride" problem, in which some voters end up with more influence than others. The paper includes data from a real case of preference voting, analyzing (among several interesting insights) the situation of an "anti-establishment" voter. Moving from voting to fundamental numerical algorithms in scientific and engineering applications, our second paper, by Leslie Greengard and June-Yub Lee, concerns acceleration of the nonuniform fast Fourier transform (FFT). For applications with an irregular sampling of data in the frequency domain---for example, image reconstruction or radio astronomy---the standard FFT is inappropriate because it requires uniform sampling. Greengard and Lee stress that three separate issues arise in reconstructing functions from nonuniform data in the Fourier domain: acquisition of data, choice of a quadrature scheme, and existence of a fast algorithm for computing the discrete transform. Given that a quadrature approach has been chosen, the authors describe techniques for rapid and accurate computation of the needed discrete sums---in effect, combining an interpolation scheme with the standard FFT by replacing a Dirac delta function source with a sharply peaked Gaussian function, thereby "smearing" the source strength onto a regular grid. An especially interesting part of the paper is its characterization of fast Gaussian gridding, which relies on the fact that the values to be evaluated are negligible except at nearby grid points. Numerical examples are included that verify the speed and accuracy of the proposed method, illustrate the benefits of fast gridding, and display results on the ubiquitous Shepp--Logan phantom. The third paper, "Symplectic Rotational Geometry in Human Biomechanics," by V. Ivancevic, begins with informative background on the study of human motion, characterized by complex, multidimensional, highly nonlinear, and hierarchical dynamics---i.e., a very difficult problem! Classical biomechanics relies on vector geometry for the more than six hundred skeletal muscles, each generating forces and torques. With the widely used technique of inverse dynamics, segmental movements are measured using motion capture, often combined with other sensor data. For the more unusual forward dynamics approach, in which translational and rotational motion are simulated based on joint forces and torques, the author proposes the formalism of rotational symplectic geometry. A review of symplectic geometry in manifolds leads to an analysis of how different features of human motion (such as humanoid joint angles, "hinge" joints, and "ball-and-socket" joints) can be modeled in terms of constrained Lie groups, followed by a mathematical tour of (among other topics) segment dynamics, full body dynamics, joint dynamics, and muscular dynamics. We thank the authors of all three papers for proving by example the rich variety of the Problems and Techniques section.