On the asymptotic magnitude of subsets of Euclidean space

Magnitude is a canonical invariant of finite metric spaces which has its origins in category theory; it is analogous to cardinality of finite sets. Here, by approximating certain compact subsets of Euclidean space with finite subsets, the magnitudes of line segments, circles and Cantor sets are defined and calculated. It is observed that asymptotically these satisfy the inclusion-exclusion principle, relating them to intrinsic volumes of polyconvex sets.Comment: 23 pages. Version 2: updated to reflect more recent work, in particular, the approximation method is now known to calculate (rather than merely define) the magnitude; also minor alterations such as references adde

The best quintic Chebyshev approximation of circular arcs of order ten

Mathematically, circles are represented by trigonometric parametric equations and implicit equations. Both forms are not proper for computer applications and CAD systems. In this paper, a quintic polynomial approximation for a circular arc is presented. This approximation is set so that the error function is  of degree $10$ rather than $6$; the Chebyshev error function equioscillates $11$ times rather than $7$; the approximation order is $10$ rather than $6$. The method approximates more than the full circle with Chebyshev   uniform error  of  $1/2^{9}$. The examples show the competence and simplicity of the proposed approximation, and that it can not be improved

High Temperature Symmetry Nonrestoration and Inverse Symmetry Breaking on Extra Dimensions

We study $D$-dimensional gauge theory with an extra dimension of a circle at finite temperature. We mainly focus on the expectation value of the gauge field for the direction of the extra dimension, which is the order parameter of the gauge symmetry breaking. We evaluate the effective potential in the one-loop approximation at finite temperature. We show that the vacuum configuration of the theory at finite temperature is determined by a $(D-1)$-dimensional gauge theory defined by removing the Euclidean time coordinate as well as all of the fermions from the original $D$-dimensional gauge theory on the circle. It is pointed out that gauge symmetry nonrestoration and/or inverse symmetry breaking can occur at high temperature in a class of gauge theories on circles and that phase transitions (if they occur) are, in general, expected to be first order.Comment: 18 pages, 4 figures, version to appear in Phys. Rev.

Maty's Biography of Abraham De Moivre, Translated, Annotated and Augmented

November 27, 2004, marked the 250th anniversary of the death of Abraham De Moivre, best known in statistical circles for his famous large-sample approximation to the binomial distribution, whose generalization is now referred to as the Central Limit Theorem. De Moivre was one of the great pioneers of classical probability theory. He also made seminal contributions in analytic geometry, complex analysis and the theory of annuities. The first biography of De Moivre, on which almost all subsequent ones have since relied, was written in French by Matthew Maty. It was published in 1755 in the Journal britannique. The authors provide here, for the first time, a complete translation into English of Maty's biography of De Moivre. New material, much of it taken from modern sources, is given in footnotes, along with numerous annotations designed to provide additional clarity to Maty's biography for contemporary readers.Comment: Published at http://dx.doi.org/10.1214/088342306000000268 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

A moving boundary problem motivated by electric breakdown: I. Spectrum of linear perturbations

An interfacial approximation of the streamer stage in the evolution of sparks and lightning can be written as a Laplacian growth model regularized by a `kinetic undercooling' boundary condition. We study the linear stability of uniformly translating circles that solve the problem in two dimensions. In a space of smooth perturbations of the circular shape, the stability operator is found to have a pure point spectrum. Except for the zero eigenvalue for infinitesimal translations, all eigenvalues are shown to have negative real part. Therefore perturbations decay exponentially in time. We calculate the spectrum through a combination of asymptotic and series evaluation. In the limit of vanishing regularization parameter, all eigenvalues are found to approach zero in a singular fashion, and this asymptotic behavior is worked out in detail. A consideration of the eigenfunctions indicates that a strong intermediate growth may occur for generic initial perturbations. Both the linear and the nonlinear initial value problem are considered in a second paper.Comment: 37 pages, 6 figures, revised for Physica