Bruce R. Miller

Technical Aspects of the Digital Library of Mathematical Functions

The NIST Digital Library of Mathematical Functions (DLMF) Project, begun in 1997, is preparing a handbook and Web site intended for wide communities of users. The contents are primarily mathematical formulas, graphs, methods of computation, references, and links to software. The task of developing a Web handbook of this nature presents several technical challenges. We describe the goals of the Digital Library of Mathematical Functions Project and the realities that constrain those goals. We propose practical initial solutions, in order to ease the authoring of adaptable content: a LaTeX class which encourages a modestly semantic markup style; and a mathematical search engine that adapts a text search engine to the task.

The critical inclination in artificial satellite theory

Certain it is that the critical inclination in the main problem of artificial satellite theory is an intrinsic singularity. Its significance stems from two geometric events in the reduced phase space on the manifolds of constant polar angular momentum and constant Delaunay action. In the neighborhood of the critical inclination, along the family of circular orbits, there appear two Hopf bifurcations, to each of which there converge two families of orbits with stationary perigees. On the stretch between the bifurcations, the circular orbits in the planes at critical inclinmation are unstable. A global analysis of the double forking is made possible by the realization that the reduced phase space consists of bundles of two-dimensional spheres. Extensive numerical integrations illustrate the transitions in the phase flow on the spheres as the system passes through the bifurcations.

The Lissajous transformation III. Parametric bifurcations

We examine a parametric family of cubic perturbed 1-1 resonant harmonic oscillators with an aim to understanding the phase flows of the reduced system. Variation of the parameters leads the system through five bifurcations of different types. Some bifurcations are due to passage through cases of discrete symmetry or integrability. A conjecture correlating degenerate equilibria in reduced systems with integrability is modified and reinforced.

Simplify or Perish

Three suggestions are made for breaking the gridlock which paralyses progress in automating massive symbolic calculations in celestial mechanics: (i) simplifications by canonical transformations, (ii) object oriented programming to endow the algebra of Poisson series with various structures, (iii) massively parallel processing.

Design of a digital mathematical library for science, technology and education

The concept of a digital library is of proven worth because of its ability to provide dramatic capabilities that are impossible with traditional print media. We are interested in providing such capabilities for scientific, technical and educational users of mathematical reference data. Our attention is focused on the highly specialized field of mathematics that is concerned with the properties, application and computation of the elementary and higher mathematical functions. Calling upon domain experts worldwide for assistance, the National Institute of Standards and Technology is conducting an ambitious project to construct, ab initio, a comprehensive and authoritative Web resource on this subject. The need to make effective use of the latest developments in digital library research is a major focus, as is the development of content. We discuss our approach to such difficulties as the representation, display and manipulation of symbolic expressions, numerical data and graphical visualizations, and we describe a prototype Web site that has been constructed to test, evaluate and advance the NIST Digital Library of Mathematical Functions project.

The Quadratic Zeeman Effect in Moderately Strong Magnetic Fieldsa

Over the last several years, celestial mechanics has made a fresh and candid assessment of the discipline’s resources and of the conditions facing its research. Out of this examination, there has come agreement on strategies that we believe hold the greatest promise for improving the value of our heritage in mathematical astronomy by moving the research to the front ranks of nonlinear dynamics. Implementing these strategies will require bold strokes of creativity and acts of difficult programming that will unfold in the years ahead. Our focus is on three strategic priorities:

Normalization in the Face of Integrability

Celestial mechanics is a pleasant occupation, although at times fraught with real hazards. We are peddlers of illusions; chief among them is the illusion of integrability. That most perturbed Keplerian systems are not integrable we are not loath to admit nor are we embarrassed to confess that we do not know any other way of studying them than by making them integrable. Perturbations break symmetries; it is their nature, no matter how small they are. There are exceptions, of course, but they are rare indeed. Unavoidably a violation of symmetry results in the loss of integrability. Nonetheless, we like to convince ourselves that the disturbance is small enough that we can replace the original system with an approximation to which we restore some of the lost symmetries, enough of them to make the substitute integrable. The fakery goes nowadays by the respectable name of normalization. It came into the light of day as an averaging process. Concerned as they were with the task of fitting sparse, widely spread observations to a nominal trajectory, astronomers learned to regard the perturbations as having two kinds of effects, the short period fluctuations modulating the long trend variations, a reenacting if you will of Ptolemy’s fiction of fast epicycles rolling over roulettes in slow drift. By marching in large steps along a smoothed path, the integration of the averaged equations connects far apart observations. A high moment in the development of mechanics was reached when Delaunay equated the method of averaging to a canonical transformation that renders the mean anomaly I ignorable in the Hamiltonian representing a perturbed Keplerian system. The angle I being eliminated, its conjugate momentum L becomes an integral. But integrals beget symmetries. With enough angles eliminated from the original system, enough symmetries will have been imposed on the transformed Hamiltonian to make it integrable. Our working conditions have changed radically: telemetry data pouring down from satellites in dense showers ensure an almost uniform covering of their paths. Our interest has shifted nowadays to the task of designing trajectories to satisfy mission requirements. Yet the problem of separating short-term effects from long-trend evolutions is still with us in so far as we try to tell aerospace designers how close their satellites will adhere for how long to the anticipated track.

Three years of DLMF: web, math and search

DLMF was released to the public in May 2010 and is now completing its 3rd year online. As a somewhat early adopter of largescale MathML content online, and exposing a math-aware search engine to the public, the project encountered situations distinct from those with our previous web sites. In the hopes that our experiences may inform developers of current and future Digital Library projects, we describe some of our observations delivering MathML content and trends in both web usage and browser evolution. We will also look at the the ways our readers have used math search, attempting to assess whether they found what they sought, and ways the engine might be improved.

Gyldén systems: Rotation of pericenters

A canonical transformation in phase space and a rescaling of time are proposed to reduce a Keplerian system with a time-dependent Gaussian parameter to a perturbed Keplerian system with a constant Gaussian parameter. When the time variation is slow, the perturbation through second order in the reduced problem is conservative, and, as a result, the orbital space of the averaged system is a sphere on which the phase flow causes a differential rotation representing a circulation of the line of apsides. The flow presents two isolated singularities corresponding to circular orbits travelled respectively in the direct and in the retrograde sense, and a degenerate manifold of fixed points corresponding to the collision orbits. Normalization beyond order two does not break the degeneracy. Adiabatic invariants, which are conservative functions, may be computed from the normalized Hamiltonian evaluated here to the fourth order. Nonetheless so high an approximation gives little information because the normalizing Lie transformation depends explicitly on the time through mixed secular-periodic terms. As an application, an estimate is offered for the apsidal rotation that a second order time derivative in the mass of the sun would induce on planetary orbits. This suggests an observational method for determining the latter parameter for the solar wind, but the predicted motions are too slow for the current level of observational precision.

The LaTeXML daemon: editable math on the collaborative web

The language of the \({T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}/{\rm L\kern-.36em\raise.3ex\hbox{\sc a}\kern-.15em T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}\) typesetting system has become all-pervasive in scientific publications and has proven its stability, convenience and expressivity in its three-decade history. With the advent of the Web 2.0 paradigm, it has also become the primary choice of various technical and scientific social platforms, most prominently online encyclopedias (e.g. Planet- Math [Pla]) and question-answer forums (e.g. MathOverflow [Mat]). On the other hand, the standardization of MathML and OpenMath and the adoption of the former in HTML5, have opened the floodgates for scientific content native to the browser.