Showing papers by "Saint Anselm College published in 1998"

Four-thirds power law for knots and links

Abstract: Physical knot theory has recently been applied to polymer dynamics, and specifically to gel electrophoresis of DNA1,2. Knot energies3,4,5,6 measure the complexity of a knot conformation; minimum energy conformations are considered canonical or ‘ideal’ conformations. The rope length of a knot is one such measure of energy6, and an approximately linear relationship between rope length and the average crossing number for minimum rope-length conformations of simple knots has been reported7. Here I show that a linear relationship cannot hold in general: the rope length required to tie an N-crossing knot or link varies at least between ˜N3/4 and ˜N.

Automata, Living and Non-Living: Descartes' Mechanical Biology and His Criteria for Life

Abstract: Despite holding to the essential distinction between mind and body, Descartes did not adopt a life-body dualism Though humans are the only creatures which can reason, as they are the only creatures whose body is in an intimate union with a soul, they are not the only finite beings who are alive In the present note, I attempt to determine Descartes' criteria for something to be 'living' Though certain passages associate such a principle with the presence of a properly functioning heart, I show that there are important reasons for also understanding life in terms of a degree of complexity of design

Most smooth closed space curves contain approximate solutions of the n -body problem

Abstract: The determination of the exact trajectories of mutually interacting masses (the n-body problem1,2) is apparently intractable for n ⩾ 3, when the generic solutions become chaotic. A few special solutions are known, which require the masses to be in certain initial positions; these are known as ‘central configurations’ (refs 1,2,3,4,5,6) (an example is the equilateral triangle formed by the Sun, Jupiter and Trojan asteroids). The configurations are usually found by symmetry arguments. Here I report a generalization of the central-configuration approach which leads to large continuous families of approximate solutions. I consider the uniform motion of equidistributed masses on closed space curves, in the limit when the number of particles tends to infinity. In this situation, the gravitational force on each particle is proportional to the local curvature, and may be calculated using an integral closely related to the Biot–Savart integral. Approximate solutions are possible for certain (constant) values of the particle speed, determined by equating this integral to the mass times the centrifugal acceleration. Most smooth, closed space curves contain such approximate solutions, because only the local curvature is involved. Moreover, the theory also holds for sets of closed curves, allowing approximate solutions for knotted and linked configurations.

The Liturgical Argument in Apollinarius: Help and Hindrance on the Way to Orthodoxy

Abstract: In the essay “Creed or Chaos?” written in the midst of the turmoil of World War II, British mystery novelist Dorothy L. Sayers defended the relevance of the creeds produced during the doctrinal debates of the fourth and fifth centuries to the lives of modern Christians. The theological dogmas contained in such documents as the Nicene Creed (325) or the Chalcedonian Definition (451) are not, she notes wittily, “a set of arbitrary regulations invented a priori by a committee of theologians enjoying a bout of all-in dialectical wrestling,” but were “hammered out under pressure of urgent practical necessity” to resolve theological controversies that had real impact on the discipleship of ordinary Christians. To put matters at their simplest, the trinitarian controversies revolved around the question of whether Christ was divine, and so capable of saving humankind from sin and death. The christological controversies, at least in their earliest stage, debated whether Christ was really human, truly the God-made-man capable of healing wounded humanity and providing aviable role model for Christians to follow in the living of a redeemed life. At stake in both controversies was a convincing explanation of the central Christian tenet that “Jesus saves” for those who profess to be his followers.

On the distribution of mass in collinear central configurations

Abstract: Moulton’s Theorem says that given an ordering of masses, m1,m2, . . . ,mn, there exists a unique collinear central configuration with center of mass at the origin and moment of inertia equal to 1. This theorem allows us to ask the questions: What is the distribution of mass in this standardized collinear central configuration? What is the behavior of the distribution as n → ∞? In this paper, we define continuous configurations, prove a continuous version of Moulton’s Theorem, and then, in the spirit of limit theorems in probability theory, prove that as n → ∞, under rather general conditions, the discrete mass distributions of the standardized collinear central configurations have distribution functions which converge uniformly to the distribution function of the unique continuous standardized collinear central configuration which we determine.