Adaptive local grid refinement algorithms for finite-element collocation

An adaptive grid refinement procedure allows accurate solutions to advection-dominated, time-dependent flows using finite-element collocation. The technique relies on a data structure that is readily amenable to parallel computing. The paper discusses computational aspects of the method.     

Difference approximations for wave equations via finite elements. I. Construction and performance

Many practical applications of wave equations involve media in which there are interfaces, or discontinuities in material properties. The accurate numerical representation of these interfaces is important in mathematical models. One can develop generalizations of standard finite-difference methods that accommodate sharp interfaces by modifying a straightforward finite-element approach. In two space dimensions, these methods yield explicit, 5-point or 9-point difference schemes that accurately capture reflection, transmission, and refraction at interfaces. The approach also extends readily to the simulation of waves in elastic media. A companion article presents an error analysis for the approach. © 1995 John Wiley & Sons, Inc.     

A modified method of characteristics incorporating streamline diffusion

A hybrid finite-element method, combining ideas from a modified method of characteristics and the streamline diffusion method, delivers accurate solutions to the advection–diffusion equation. An error analysis for the case of tensorial diffusion shows that the lowest-order version of the scheme, which allows one to use a symmetric linear solvers at each time step, possesses first-order accuracy in time and space. Numerical experiments demonstrate the scheme's ability to model advection-dominated transport of solute plumes without distorting sharp fronts. © 1995 John Wiley & Sons, Inc.     

A Generally Covariant Wave Equation for Grand Unified Field Theory

A generally covariant wave equation is derived geometrically for grand unified field theory. The equation states most generally that the covariant d'Alembertian acting on the vielbein vanishes for the four fields which are thought to exist in nature: gravitation, electromagnetism, weak field and strong field. The various known field equations are derived from the wave equation when the vielbein is the eigenfunction. When the wave equation is applied to gravitation the wave equation is the eigenequation of wave mechanics corresponding to Einstein's field equation in classical mechanics, the vielbein eigenfunction playing the role of the quantized gravitational field. The three Newton laws, Newton's law of universal gravitation, and the Poisson equation are recovered in the classical and nonrelativistic, weak-field limits of the quantized gravitational field. The single particle wave-equation and Klein-Gordon equations are recovered in the relativistic, weak-field limit of the wave equation when scalar components are considered of the vielbein eigenfunction of the quantized gravitational field. The Schrödinger equation is recovered in the non-relativistec, weak-field limit of the Klein-Gordon equation). The Dirac equation is recovered in this weak-field limit of the quantized gravitational field (the nonrelativistic limit of the relativistic, quantezed gravitational field when the vielbein plays the role of the spinor. The wave and field equations of O(3) electrodynamics are recovered when the vielbein becomes the relativistic dreibein (triad) eigenfunction whose three orthonormal space indices become identified with the three complex circular indices (1), (2), (3), and whose four spacetime indices are the indices of non-Euclidean spacetime (the base manifold). This dreibein is the potential dreibein of the O(3) electromagnetic field (an electromagnetic potential four-vector for each index (1), (2), (3)). The wave equation of the parity violating weak field is recovered when the orthonormal space indices of the relativistic dreibein eigenfunction are identified with the indices of the three massive weak field bosons. The wave equation of the strong field is recovered when the orthonormal space indices of the relativistic vielbein eigenfunction become the eight indices defined by the group generators of the SU (3) group.     

Photooxygenation of Furylalkylamines: Easy Access to Pyrrolizidine and Indolizidine Scaffolds

A highly adaptable method targeting the ubiquitous and very important pyrrolizidine and indolizidine scaffolds is presented. The general synthetic utility of the method is underscored by its application to the rapid and easy synthesis of five natural products starting from readily accessible alkylfuran precursors. These unprotected primary furylalkylamines are subjected to photooxygenation conditions, which initiate a complex cascade reaction sequence concluding with the production of high value motifs. This sequence can be tailored to need by varying the choice of both photosensitizer and base additive.     

Precise orders of strong unicity constants for a class of rational functions

Let denote a certain class of rational functions. For each f ε , consider the polynomial of degree at most n that best approximates f in the uniform norm. The corresponding strong unicity constant is denoted by M_n(f). Then there exist positive constants α and β, not depending on n, such that an M_n(f) βn, N = 1,2,….     

The Equations of Grand Unified Field Theory in Terms of the Maurer-Cartan Structure Relations of Differential Geometry

The first and second Maurer-Cartan structure relations are combined with the Evans field equation [1] for differential forms to build a grand unified field theory based on differential geometry. The tetrad or vielbein plays a central role in this theory, and all four fields currently thought to exist in nature can be described by the same equations, the tangent space index of the tetrad in general relativity being identified with the tetrad's internal (gauge group) index guage theory.