On the sums of two cubes

We solve the equation $f(x,y)^3 + g(x,y)^3 = x^3 + y^3$ for homogeneous $f, g \in \mathbb C(x,y)$, completing an investigation begun by Vi\`ete in 1591. The usual addition law for elliptic curves and composition give rise to two binary operations on the set of solutions. We show that a particular subset of the set of solutions is ring-isomorphic to $\mathbb Z[e^{2 \pi i / 3}]$.Comment: Revised version, to appear in the International Journal of Number Theor

Interpretation of three-dimensional structure from two-dimensional endovascular images: implications for educators in vascular surgery

AbstractPurposeEndovascular therapy has had a major effect on vascular surgery; surgeons perform tasks in three dimensions (3D) while viewing two-dimensional (2D) displays. This fundamental change in how surgeons perform operations has educational implications related to learning curves and patient safety. We studied the effects of experience, training, and visual-spatial ability on 3D perception of 2D angiographic images of abdominal aortic aneurysms (AAA).MethodsA novel computer-based method was developed to produce 3D depth maps based on subjects' interpretations of 2D images. Seven experts (certified vascular surgeons) and 20 novices (medical or surgical trainees) were presented with a 2D AAA angiographic image. With software specifically designed for this study, a depth map representing each subject's 3D interpretation of the 2D angiogram was produced. The novices were then randomized into a control group and a treatment group, who received a 5-minute AAA anatomy educational session. All subjects repeated the exercise on a second AAA image. Finally, all novices were given tests of visual-spatial ability, including the Surface Development Test and the Mental Rotations Test. Comparisons between experts and novices were made with depth map comparison, a subject's perception of overall object contour.ResultsThe depth maps were significantly different (depth map comparison, P < .001) between the expert and both novice groups for the first image. After the educational intervention, the control group and the treatment group exhibited significantly different depth maps (depth map comparison, P < .001), with treatment group depth maps more similar to those of the expert group. There were no significant correlations between the visual-spatial tests and the novice depth map comparison with the expert group.ConclusionsThis is the first study to examine perception of endovascular images in an educational context. Perception of overall surface contour of 3D structures from 2D angiographic images is affected by experience and training. With application of methods of vision science to an important problem in surgery, this research represents a first step in understanding the nature of visual perceptual processes involved in execution of an increasingly common clinical task. These results have implications for understanding and studying the endovascular learning curve.Clinical relevanceThis research represents a unique collaboration in an effort to understand and solve one of the greatest problems facing surgical educators and surgeons. This research uses applied tools in vision science to understand the perceptual constraints involved in minimally invasive surgery. Specifically, we examined the mental three-dimensional maps experts use when viewing two-dimensional displays. Furthermore, we compared experts with novices in an effort to assist surgical trainees