Gregory Buck Selected Publications	gbuck@anselm.edu Department of Mathematics Saint Anselm College Vita
The self-induced motion of filaments Filaments are ubiquitous -- they appear on every scale studied by science. We present here a new system of equations for the self induced motion for an attracting or repelling filament or filamentary distribution of matter. The equations allow us to solve several natural problems involving filaments of varying shape and/or varying mass or charge density. Applications include celestial distributions such as spiral arms of galaxies, charged strings, and biofilaments such as DNA.
Galactic spiral arms can be sustained by self-gravity Using an equation of motion for a self-gravitating filament, we show how galactic spiral arms might be created and sustained. We find that the combination of differential rotation of the galactic disk and the self-gravity of the arm (as given by the equation) leads to a rotating spiral structure. Moreover, using this analysis, we find a differential equation that explicitly relates this spiral structure to the rotation curve of the galaxy. In particular, the equation connects spiral shape to pattern speed. We also describe a simple many-body numerical experiment that supports our approach. The findings are with consistent with observational evidence concerning arm structure and rotation curves, including leading arm structures.
The Wondrous Mathematics of Winter …when winter comes I think of Plato’s theory of ideal forms. If I say circle or square, you know what I mean, though in some sense you have never seen a circle—the shape in the plane where all the points are exactly equidistant from the center—because in reality everything is always at least a little off. Plato thought truth ought to work this way, that what we understand as truth is always an approximation to an ideal form. Winter is the Platonist’s season…
The spectrum of filament entanglement complexity and an entanglement phase transition G. Buck and J. Simon DNA, hair, shoelaces, vortex lines, rope, proteins, integral curves, thread, magnetic flux tubes, cosmic strings and extension cords; filaments come in all sizes and with diverse qualities. Filaments tangle, with profound results: DNA replication is halted, field energy is stored, polymer materials acquire their remarkable properties, textiles are created and shoes stay on feet. We classify entanglement patterns by the rate with which entanglement complexity grows with the length of the filament. We show which rates are possible and which are expected in arbitrary circumstances. We identify a fundamental phase transition between linear and nonlinear entanglement rates. We also find (perhaps surprising) relationships between total curvature, bending energy and entanglement.
A Mathematician Goes to the Beach Like everyone else I know, when I go to the beach I think mathematics. Archimedes, my favorite mathematician, did too -- his perhaps best-known work is the 'Sand Reckoner,' wherein he counts the grains of sand it would take to fill the universe. He must have been at the beach when he thought how to do it. I'm not sure Archimedes picked the most interesting thing about the beach for analysis, but he might be forgiven -- because the ancient Greeks didn't wear swimsuits. Swimsuits often remind me of Carl Friedrich Gauss, a superb nineteenth-century mathematician who really understood a curved surface...
Accessibility and occlusion of biopolymers, ray tracing of radiating tubes, and the temperature of a tangle G. Buck, R. Scharein, J. Schnick, J. Simon We introduce a measure of complexity, an energy, for any conformation of filaments. It is the occlusion, the portion hidden when viewed from an arbitrary exterior point. By inverting we get the exposure, a first approximation of the accessibility of the filaments. Assuming the filament is a source, we get the self-irradiation, which leads to both an interpretation as the temperature and a visualization technique: ray tracing as a virtual laboratory. There is a wide variety of applications, from enzyme action on and radiation damage of biopolymers, to the geometry of light bulb filaments. Energy minimization provides automatic detangling, resulting in symmetric and pleasing conformations.
DNA disentangling by type-2 topoisomerases G. Buck and E. Zechiedrich A type-2 topoisomerase cleaves a DNA strand, passes another through the break, and then rejoins the severed ends. Because it appears that this action is as likely to increase as to decrease entanglements, the question is: how are entanglements removed? We argue that type-2 topoisomerases have evolved to act at “hooked” juxtapositions of strands (where the strands are curved toward each other). This type of juxtaposition is a natural consequence of entangled long strands. Our model accounts for the observed preference for unlinking and unknotting of short DNA plasmids by type-2 topoisomerases and well explains experimental observations.
Most smooth closed space curves contain an approximate solution to the N-body problem The determination of the exact trajectories of mutually interacting masses (the n-body problem) is apparently intractable for n greater than or equal to 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' (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.
Four-thirds power law for knots and links Physical knot theory has recently been applied to polymer dynamics, and specifically to gel electrophoresis of DNA^.Knot energiesmeasure 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 energy and an approximately linear relationship between rope length and the average crossing number for minimum rope-length conformations of simple knots has been reported. 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 ~N ^3/4 and ~N. Since it is known that there are only a finite number of knots for any given finite N number of crossings, this result gives a bound on the number of different knots that can be tied in a finite length of rope.