The dot product formula allows one to measure an angle determined by two vectors, and a formula known to Euler and Lagrange outputs the measure of a solid angle in R 3 given its three spanning vectors. However, there appears to be no closed form expression for the measure of an n-dimensional solid angle for n > 3. We derive a multivariable (infinite) Taylor series expansion to measure a simplicial solid angle in terms of the inner products of its spanning vectors. We then analyze the domain of convergence of this hypergeometric series and show that it converges within the natural boundary for solid angles.
In Tangencies Apollonius of Perga showed how to construct a circle that is tangent to three given circles. More generally, Apollonius' problem asks to construct the circle which is tangent to any three objects that may be any combination of points, lines, and circles. The case when all three objects are circles is the most complicated case since up to eight solution circles are possible depending on the arrangement of the given circles. Within the last two centuries, solutions have been given by J. D. Gergonne in 1816, by Frederick Soddy in 1936, and most recently by David Eppstein in 2001. In this report, we illustrate the solution using the geometry software Cinderella™, survey some connections among the three solutions, and provide a framework for further study.
While benchmarks are important tools for portfolio managers and investors alike, we challenge the conventional wisdom that they are paradigms of investing excellence. Despite the effective marketing campaigns that have brought benchmarks into the public consciousness and attracted significant capital to passive investment strategies, few investors fully understand how these benchmarks are calculated or what they represent. We dissect the benchmark construction process and reveal how decisions made by index providers can lead to unintended factor exposures in various benchmarks. Using the Russell Midcap ® Value Index as our primary example, we find evidence of size, momentum and sector tilts, as well as outsized exposure to interest rates, clientele effects and low-quality businesses. We demonstrate that without full understanding of benchmark construction, evaluations of active-manager performance are unreliable. Moreover, factors that have led to outperformance by index funds in recent years could easily reverse.
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