Stable Computation of the Complex Roots of Unity

In this correspondence, we show that the problem of computing the complex roots of unity is not as simple as it seems at first. In particular, the formulas given in a standard programmer's reference book (Knuth, Seminumerical Algorithms, 1981) are shown to be numerically unstable, giving unacceptably large error for moderate sized sequences. We give alternative formulas, which we show to be superior both by analysis and experiment

A note on Keen's model: The limits of Schumpeter's "Creative Destruction"

This paper presents a general solution for a recent model by Keen for endogenous money creation. The solution provides an analytic framework that explains all significant dynamical features of Keen's model and their parametric dependence, including an exact result for both the period and subsidence rate of the Great Moderation. It emerges that Keen's model has just two possible long term solutions: stable growth or terminal collapse. While collapse can come about immediately from economies that are nonviable by virtue of unsuitable parameters or initial conditions, in general the collapse is preceded by an interval of exponential growth. In first approximation, the duration of that exponential growth is half a period of a sinusoidal oscillation. The period is determined by reciprocal of the imaginary part of one root of a certain quintic polynomial. The real part of the same root determines the rate of growth of the economy. The coefficients of that polynomial depend in a complicated way upon the numerous parameters in the problem and so, therefore, the pattern of roots. For a favorable choice of parameters, the salient root is purely real. This is the circumstance that admits the second possible long term solution, that of indefinite stable growth, i.e. an infinite period.Comment: 25 pages, 12 figures, JEL classification: B50, C62, C63, E12, E4

Singularities of the moduli space of level curves

We describe the singular locus of the compactification of the moduli space $R_{g,l}$ of curves of genus $g$ paired with an $l$-torsion point in their Jacobian. Generalising previous work for $l\le 2$, we also describe the sublocus of noncanonical singularities for any positive integer $l$. For $g\ge 4$ and $l=3,4, 6$, this allows us to provide a lifting result on pluricanonical forms playing an essential role in the computation of the Kodaira dimension of $R_{g,l}$: for those values of $l$, every pluricanonical form on the smooth locus of the moduli space extends to a desingularisation of the compactified moduli space.Comment: 37 pages, 9 figures, to appear in J Eur Math So