Ron Perline

Poisson geometry of the filament equation

SummaryThe Hasimoto transformation (relating vortex filament flow to the nonlinear Schrödinger equation) is interpreted in the context of Poisson geometry with the aid of a compact formula for its differential. A useful relationship is derived between Killing fields for soliton solutions of the filament flow and the sequence of commuting Hamiltonian flows.

CURVE MOTION INDUCING MODIFIED KORTEWEG-DE VRIES SYSTEMS

Abstract A straightforward vector generalization of the modified Korteweg-deVries equation is shown to be intimately related to the geometry of curves in n-dimensional Euclidean spaces and spheres. This mKdV system, which is coupled in a particularly a simple way, describes the dynamics of the natural curvature vector of a unit speed curve subject to an elementary geometric evolution equation. The underlying structure of these equations is related to generalizations of the nonlinear Schrodinger and localized induction equations in the context of Hermitian symmetric Lie algebras.

Local geometric invariants of integrable evolution equations

The integrable hierarchy of commuting vector fields for the localized induction equation of 3D hydrodynamics, and its associated recursion operator, are used to generate families of integrable evolution equations which preserve local geometric invariants of the evolving curve or swept‐out surface.

Localized induction equation and pseudospherical surfaces

We describe a close connection between the localized induction equation hierarchy of integrable evolution equations on space curves and surfaces of constant negative Gauss curvature.

Double-mirror catadioptric sensors with ultrawide field of view and no distortion

We compute a family of double-mirror catadioptric sensors with ultrawide field of view and no distortion. The two concentric mirrors are rotationally symmetric, and the inside mirror is a revolved conic section. The mapping between the object and the image planes was linear, hence the lack of distortion. The equations describing the outer mirror were determined by the projection induced by the inside mirror and the rectifying property of the sensor. Solving the resulting nonlinear ordinary differential equations yielded the cross section of the secondary mirror. The sensors we present require no further digital processing.

A class of vortex filament solitons in fluids, plasmas, and superconductors

We consider the Uby–Isichenko–Yankov (UIY) equation for vortex filament dynamics in plasmas and superconductors. This is a perturbation of the localized induction equation (LIE), which is itself an integrable model of filament motion in an ideal fluid. We present a novel class of solutions for a time-modulated modification of UIY; namely, vortex configurations which are evolving spherical curves of varying radius. These solutions can be considered as generalizations of a well-known class of soliton solutions for LIE.

Localized induction hierarchy and Weingarten systems

Abstract We describe a method of constructing Weingarten systems of triply orthogonal coordinates, related to the localized induction equation hierarchy of integrable geometric evolution equations.

Padé Approximants for Some q-Hypergeometric Functions

We show that a large number of explicit formulas for Pade approximants for the ratios of basic hypergeometric functions result from an explicit expression given by Ismail and Rahman for the associated Askey-Wilson polynomials. By specializing this result and using a new transformation for basic hypergeometric series, we are able to recover a result due to Andrews, Goulden and Jackson. We also show how Pade approximants off the main diagonal can be constructed in this latter case.

Developing the Soliton Explorer: A problem solving environment for soliton surface investigation

With the CtCoq experiment, we have developed a strong competence around ways to use the mouse to express logical operations. A feature provided in CtCoq makes it possible to perform proofs of predicate calculus without equality by simply pointing at the sub-expressions of logical formulas that are subject of the next step. We intend to show a sequence of windows, indicating the operations performed with the mouse and the consequence. A second feature provided in CtCoq makes it possible to perform proofs based on rewriting by dragging data around algebraic expressions. This tool is very much inspired ]by the direct manipulation provided in tools such as Theorist (a product distributed in Europe under the name MathPlus). We will also show a little scenario of this kind of interaction. Work in this progress concerns ways to merge the two interaction techniques into one, using the drag-and-drop concept in new ways, etc. We have also worked around the rendering of proof attempts as text in a form close to natural language. This makes it possible to present the formal proofs checked by the computer in a manner that als0 seems natural to readers that are not computer specialists. This rendering mechanism can actually be used to display proofs-in-progress. In this case, incomplete proofs appear like proof texts in which unfinished parts appear as conjectures. Using mouse interaction facilities, users can then proceed by filling in the gaps, with the illusion of simply transforming some proof text. If our implementation progresses enough, we hope to include an example of incomplete proof text in the poster, and maybe steps that show how the text construction progresses. Mutatis mutandi, this work can also be used to write functional programs in a programming style close to ML. In this case, the program and the proof of its correctness are developed in parallel. In its most basic used the proof engine can be used to force users to spell out the proofs they performed. In this respect, it can be used effectively to teach the basic techniques of mathematical reasoning. Also, because proofs can be performed in advance by teachers and collected in some form of interactive books, with high-quality display of formulas, rich comments, and illustrations, this tool can also be used to provide a reference to mathematical knowledge. As an interactive book, the tool can be used to enliven mathematical results, as students may try …