arXiv subjects

P. Robert Kotiuga

Publications and source records attributed to P. Robert Kotiuga.

5 recordsLinked to original sources

arXiv2024

An electrical engineering perspective on naturality in computational physics

We look at computational physics from an electrical engineering perspective and suggest that several concepts of mathematics, not so well-established in computational physics literature, present themselves as opportunities in the field. We discuss elliptic complexes and highlight the category theoretical background and its role as a unifying language between algebraic topology, differential geometry, and modelling software design. In particular, the ubiquitous concept of naturality is central. Natural differential operators have functorial analogues on the cochains of triangulated manifolds. In order to establish this correspondence, we derive formulas involving simplices and barycentric coordinates, defining discrete vector fields and a discrete Lie derivative as a result of a discrete analogue of Cartan's magic formula. This theorem is the main mathematical result of the paper.

math.NA↗

arXiv2019

Cuts for 3-D magnetic scalar potentials: visualizing unintuitive surfaces arising from trivial knots

A wealth of literature exists on computing and visualizing cuts for the magnetic scalar potential of a current carrying conductor via Finite Element Methods (FEM) and harmonic maps to the circle. By a cut we refer to an orientable surface bounded by a given current carrying path (such that the flux through it may be computed) that restricts contour integrals on a curl-zero vector field to those that do not link the current-carrying path, analogous to branch cuts of complex analysis. This work is concerned with a study of a peculiar contour that illustrates topologically unintuitive aspects of cuts obtained from a trivial loop and raises questions about the notion of an optimal cut. Specifically, an unknotted curve that bounds only high genus surfaces in its convex hull is analyzed. The current work considers the geometric realization as a current-carrying wire in order to construct a magnetic scalar potential. Moreover, we consider the problem of choosing an energy functional on the space of maps, suggesting an algorithm for computing cuts via minimizing a conformally invariant functional utilizing Newton iteration.

physics.comp-ph↗

arXiv2015

On the Topological Characterization of Near Force-Free Magnetic Fields, and the work of late-onset visually-impaired Topologists

The Giroux correspondence and the notion of a near force-free magnetic field are used to topologically characterize near force-free magnetic fields which describe a variety of physical processes, including plasma equilibrium. As a byproduct, the topological characterization of force-free magnetic fields associated with current-carrying links, as conjectured by Crager and Kotiuga, is shown to be necessary and conditions for sufficiency are given. Along the way a paradox is exposed: The seemingly unintuitive mathematical tools, often associated to higher dimensional topology, have their origins in three dimensional contexts but in the hands of late-onset visually impaired topologists. This paradox was previously exposed in the context of algorithms for the visualization of three-dimensional magnetic fields. For this reason, the paper concludes by developing connections between mathematics and cognitive science in this specific context.

math.DG↗

arXiv2015

Inter-winding Distributed Capacitance and Guitar Pickup Transient Response

Simple RLC circuit models of guitar pickups do not account for audible features that characterize the pickup. Psycho-acoustic experiments reveal that any acoustically accurate model has to reproduce the first 30 milli-seconds of the transient response with extreme precision. The proposed model is impractical for simple-minded model reduction or brute force numerical simulations yet, by focusing on modeling electromagnetic details and exposing a connection to spectral graph theory, a framework for finding the transient response to sufficient accuracy is exposed.

math.OC↗

arXiv2010

Iron Rings, Doctor Honoris Causa Raoul Bott, Carl Herz, and a Hidden Hand

The degree of Doctor of Sciences, honoris causa, was conferred on Raoul Bott by McGill University in 1987. Much of the work to make this happen was done by Carl Herz. Some of the author's personal recollections of both professors are included, along with some context for the awarding of this degree and ample historical tangents. Some cultural aspects occurring in the addresses are elaborated on, primarily, the Canadian engineer's iron ring. This paper also reprints both the convocation address of Raoul Bott and the presentation of Carl Herz on that occasion.

math.HO↗